c 2001 by CRC Press LLC Comprehensive Dictionary of

Douglas N. Clark Editor-in-Chief Stan Gibilisco Editorial Advisor


Analysis, , and Differential Equations Douglas N. Clark Algebra, Arithmetic and Trigonometry Steven G. Krantz


Classical & Theoretical Mathematics Catherine Cavagnaro and Will Haight for Engineers and Scientists Emma Previato The Comprehensive Dictionary of Mathematics Douglas N. Clark

c 2001 by CRC Press LLC a Volume in the Comprehensive Dictionary of Mathematics


Edited by Steven G. Krantz

CRC Press Boca Raton London New York Washington, D.C.

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The second volume of the CRC Press Comprehensive Dictionary of Mathematics covers algebra, arithmetic and trigonometry broadly, with an overlap into differential , , and other related fields. The authorship is by well over 30 mathematicians, active in teaching and research, including the editor. Because it is a dictionary and not an encyclopedia, definitions are only occasionally accompanied by a discussion or example. In a dictionary of mathematics, the primary goal is to define each term rigorously. The derivation of a term is almost never attempted. The dictionary is written to be a useful reference for a readership that includes students, scientists, and engineers with a wide range of backgrounds, as well as specialists in of analysis and differential equations and mathematicians in related fields. Therefore, the definitions are intended to be accessible, as well as rigorous. To be sure, the degree of accessibility may depend upon the individual term, in a dictionary with terms ranging from Abelian cohomology to z intercept. Occasionally a term must be omitted because it is archaic. Care was taken when such circum- stances arose to ensure that the term was obsolete. An example of an archaic term deemed to be obsolete, and hence not included, is “right line”. This term was used throughout a turn-of-the-century textbook we needed to consult, but it was not defined there. Finally, reference to a contemporary English language dictionary yielded “straight line” as a synonym for “right line”. The authors are grateful to the editor, Stanley Gibilisco, for dealing with our seemingly endless procedural questions and to Nora Konopka, for always acting efficiently and cheerfully with CRC Press liaison matters.

Douglas N. Clark Editor-in-Chief


Edward Aboufadel Neil K. Dickson Grand Valley State University of Glasgow Allendale, Michigan Glasgow, United Kingdom

Gerardo Aladro David E. Dobbs Florida International University University of Tennessee Miami, Florida Knoxville, Tennessee

Mohammad Azarian Marcus Feldman University of Evansville Washington University Evansville. Indiana St. Louis, Missouri

Susan Barton Stephen Humphries West Virginia Institute of Technology Brigham Young University Montgomery, West Virginia Provo, Utah

Albert Boggess Shanyu Ji Texas A&M University University of Houston College Station, Texas Houston, Texas

Robert Borrelli Kenneth D. Johnson Harvey Mudd College University of Georgia Claremont, California Athens, Georgia

Stephen W. Brady Bao Qin Li Wichita State University Florida International University Wichita, Kansas Miami, Florida

Der Chen Chang Robert E. MacRae Georgetown University University of Colorado Washington, D.C. Boulder, Colorado

Stephen A. Chiappari Charles N. Moore Santa Clara University Kansas State University Santa Clara. California Manhattan, Kansas

Joseph A. Cima Hossein Movahedi-Lankarani The University of North Carolina at Chapel Hill Pennsylvania State University Chapel Hill, North Carolina Altoona, Pennsylvania

Courtney S. Coleman Shashikant B. Mulay Harvey Mudd College University of Tennessee Claremont, California Knoxville, Tennessee

John B. Conway Judy Kenney Munshower University of Tennessee Avila College Knoxville, Tennessee Kansas City, Missouri

c 2001 by CRC Press LLC Charles W. Neville Anthony D. Thomas CWN Research University of Wisconsin Berlin, Connecticut Platteville. Wisconsin

Daniel E. Otero Michael Tsatsomeros Xavier University University of Regina Cincinnati, Ohio Regina, Saskatchewan,Canada

Josef Paldus James S. Walker University of Waterloo University of Wisconsin at Eau Claire Waterloo, Ontario, Canada Eau Claire, Wisconsin

Harold R. Parks C. Eugene Wayne Oregon State University Boston University Corvallis, Oregon Boston, Massachusetts

Gunnar Stefansson Kehe Zhu Pennsylvania State University State University of New York at Albany Altoona, Pennsylvania Albany, New York

c 2001 by CRC Press LLC its Galois is an . See Galois group. See also Abelian group.

Abelian extension A Galois extension of a A field is called an Abelian extension if its Galois group is Abelian. See Galois extension. See also Abelian group. A-balanced mapping Let M be a right mod- ule over the A, and let N be a left module Abelian A function f(z1,z2,z3, n over the same ring A. A mapping φ from M ×N ...,zn) meromorphic on C for which there ex- n to an Abelian group G is said to be A-balanced ist 2n vectors ωk ∈ C , k = 1, 2, 3,...,2n, if φ(x,·) is a from N to called period vectors, that are linearly indepen- G for each x ∈ M,ifφ(·,y)is a group homo- dent over R and are such that morphism from M to G for each y ∈ N, and + = if f (z ωk) f(z) φ(xa,y) = φ(x,ay) holds for k = 1, 2, 3,...,2n and z ∈ Cn. holds for all x ∈ M, y ∈ N, and a ∈ A. Abelian function field The set of Abelian A-B-bimodule An Abelian group G that is a functions on Cn corresponding to a given set of left module over the ring A and a right module period vectors forms a field called an Abelian over the ring B and satisfies the associative law function field. (ax)b = a(xb) for all a ∈ A, b ∈ B, and all x ∈ G. Abelian group Briefly, a commutative group. More completely, a set G, together with a binary Abelian cohomology The usual cohomology , usually denoted “+,” a unary opera- with coefficients in an Abelian group; used if tion usually denoted “−,” and a distinguished the context requires one to distinguish between element usually denoted “0” satisfying the fol- the usual cohomology and the more exotic non- lowing : Abelian cohomology. See cohomology. (i.) a + (b + c) = (a + b) + c for all a,b,c ∈ G, Abelian differential of the first kind A holo- (ii.) a + 0 = a for all a ∈ G, morphic differential on a closed Riemann sur- (iii.) a + (−a) = 0 for all a ∈ G, face; that is, a differential of the form ω = (iv.) a + b = b + a for all a,b ∈ G. a(z)dz, where a(z) is a holomorphic function. The element 0 is called the identity, −a is called the inverse of a, (i.) is called the Abelian differential of the second kind A associative axiom, and axiom (iv.) is called the meromorphic differential on a closed Riemann commutative axiom. surface, the singularities of which are all of greater than or equal to 2; that is, a differential Abelian ideal An ideal in a Lie algebra which of the form ω = a(z)dz where a(z) is a mero- forms a commutative subalgebra. morphic function with only 0 residues. Abelian integral of the first kind An indef- Abelian differential of the third kind A inite integral W(p) = p a(z)dz on a closed p0 differential on a closed Riemann surface that is Riemann surface in which the function a(z) is not an Abelian differential of the first or sec- holomorphic (the differential ω(z) = a(z)dz ond kind; that is, a differential of the form ω = is said to be an Abelian differential of the first a(z)dz where a(z) is meromorphic and has at kind). least one non-zero residue. Abelian integral of the second kind An in- Abelian equation A polynomial equation definite integral W(p) = p a(z)dzon a closed p0 f(X) = 0 is said to be an Abelian equation if Riemann surface in which the function a(z) is

c 2001 by CRC Press LLC meromorphic with all its singularities of order as the product of prime ideals of K of absolute at least 2 (the differential a(z)dz is said to be an degree 1 if and only if p is a principal ideal. Abelian differential of the second kind). The term “absolute class field” is used to dis- tinguish the Galois extensions described above, Abelian integral of the third kind An in- which were introduced by Hilbert, from a more definite integral W(p) = p a(z)dzon a closed p0 general concept of “class field” defined by Riemann surface in which the function a(z) is Tagaki. See also class field. meromorphic and has at least one non-zero resi- due (the differential a(z)dzis said to be an Abel- absolute covariant A covariant of weight 0. ian differential of the third kind). See also covariant.

Abelian A Lie group for which absolute inequality An inequality involving the associated Lie algebra is Abelian. See also variables that is valid for all possible substitu- Lie algebra. tions of real for the variables. Abelian projection operator A non-zero absolute invariant Any quantity or property projection operator E in a von Neumann algebra of an algebraic variety that is preserved under M such that the reduced von Neumann algebra birational transformations. ME = EME is Abelian.

Abelian subvariety A subvariety of an absolutely irreducible character The char- that is also a . See also acter of an absolutely irreducible representation. Abelian variety. A representation is absolutely irreducible if it is irreducible and if the representation obtained by making an extension of the ground field remains Abelian surface A two-dimensional Abelian irreducible. variety. See also Abelian variety.

Abelian variety A complete algebraic vari- absolutely irreducible representation A ety G that also forms a commutative algebraic representation is absolutely irreducible if it is group. That is, G is a group under group oper- irreducible and if the representation obtained by ations that are regular functions. The fact that making an extension of the ground field remains an is complete as an algebraic irreducible. variety implies that the group is commutative. See also regular function. absolutely A group that con- tains no serial subgroup. The notion of an ab- Abel’s Theorem Niels Henrik Abel (1802- solutely simple group is a strengthening of the 1829) proved several results now known as concept of a simple group that is appropriate for “Abel’s Theorem,” but perhaps preeminent infinite groups. See serial subgroup. among these is Abel’s proof that the general quintic equation cannot be solved algebraically. absolutely uniserial algebra Let A be an al- Other theorems that may be found under the gebra over the field K, and let L be an extension heading “Abel’s Theorem” concern power se- field of K. Then L ⊗K A can be regarded as ries, Dirichlet series, and on Riemann an algebra over L. If, for every choice of L, surfaces. L ⊗K A can be decomposed into a direct sum of ideals which are primary rings, then A is an absolute class field Let k be an algebraic absolutely uniserial algebra. number field. A Galois extension K of k is an absolute class field if it satisfies the following absolute multiple covariant A multiple co- property regarding prime ideals of k: A prime variant of weight 0. See also multiple covari- ideal p of k of absolute degree 1 decomposes ants.

c 2001 by CRC Press LLC absolute number A specific number repre- acceleration parameter A parameter chosen 3 sented by numerals such as 2, 4 , or 5.67 in con- in applying successive over-relaxation (which trast with a literal number which is a number is an accelerated version of the Gauss-Seidel represented by a letter. method) to solve a system of linear equations nu- merically. More specifically, one solves Ax = b absolute value of a complex number More iteratively by setting commonly called the modulus, the absolute val- = + ue of the complex number z a ib, where a xn+1 = xn + R (b − Axn) , and b are real, is denoted√ by |z| and equals the non-negative real number a2 + b2. where −1 −1 absolute value of a vector More commonly R = L + ω D called the magnitude, the absolute value of the vector with L the lower triangular submatrix of A, D −→ the diagonal of A, and 0 <ω<2. Here, ω v = (v1,v2,...,vn) is the acceleration parameter, also called the |−→| is denoted by v and equals the non-negative relaxation parameter. Analysis is required to 2 + 2 +···+ 2 real number v1 v2 vn. choose an appropriate value of ω. absolute value of real number For a real acyclic chain complex An augmented, pos- number r, the nonnegative real number |r|,given itive chain complex by r if r ≥ 0 ···−→∂n+1 −→∂n −→∂n−1 |r|= Xn Xn−1 ... −r if r<0 . ∂2 ∂1  ···−→ X1 −→ X0 → A → 0 abstract algebraic variety A set that is anal- forming an exact sequence. This in turn means ogous to an ordinary algebraic variety, but de- that the of ∂ equals the of ∂ + fined only locally and without an imbedding. n n 1 for n ≥ 1, the kernel of  equals the image of ∂  X A abstract function (1) In the of gen- 1, and is surjective. Here the i and are eralized almost-periodic functions, a function modules over a commutative unitary ring. mapping R to a Banach space other than the complex numbers. addend In arithmetic, a number that is to be (2) A function from one Banach space to an- added to another number. In general, one of the other Banach space that is everywhere differen- of an operation of . See also tiable in the sense of Fréchet. addition. abstract variety A generalization of the no- addition (1) A basic arithmetic operation tion of an algebraic variety introduced by Weil, that expresses the relationship between the in analogy with the definition of a differentiable number of elements in each of two disjoint sets manifold. An abstract variety (also called an and the number of elements in the union of those abstract algebraic variety) consists of (i.) a two sets. family {Vα}α∈A of affine algebraic sets over a (2) The name of the in an given field k, (ii.) for each α ∈ A a family of Abelian group, when the notation “+” is used open subsets {Wαβ }β∈A of Vα, and (iii.) for each for that operation. See also Abelian group. pair α and β in A a birational transformation be- (3) The name of the binary operation in a tween Wαβ and Wαβ such that the composition ring, under which the elements form an Abelian of the birational transformations between sub- group. See also Abelian group. sets of Vα and Vβ and between subsets of Vβ (4) Sometimes, the name of one of the opera- and Vγ are consistent with those between sub- tions in a multi-operator group, even though the sets of Vα and Vγ . operation is not commutative.

c 2001 by CRC Press LLC addition formulas in trigonometry The for- for all s ∈ S. Such an is nec- mulas essarily unique and usually is denoted by “0.” cos(φ + θ) = cos φ cos θ − sin φ sin θ, In ordinary arithmetic, the number 0 is the additive identity because 0 + n = n + 0 = n sin(φ + θ) = cos φ sin θ + sin φ cos θ, holds for all numbers n. tan φ + tan θ (φ + θ) = . tan − 1 tan φ tan θ In any with a commutative operation referred to as ad- addition of algebraic expressions One of dition and denoted by “+,” for which there is the fundamental ways of forming new algebraic an additive identity 0, the additive inverse of an expressions from existing algebraic expressions; element a is the element b for which a + b = the other methods of forming new expressions b + a = 0. The additive inverse of a is usu- from old being , , divi- ally denoted by −a. In arithmetic, the additive sion, and root extraction. inverse of a number is also called its opposite. See additive identity. addition of In elementary geometry or trigonometry, the resulting from the additive set function Let X be a set and let A process of following rotation through one an- be a collection of subsets of X that is closed un- gle about a center by rotation through another der the union operation. Let φ : A → F , where angle about the same center. F is a field of scalars. We say that φ is finitely ∈ A additive if, whenever S1,...,Sk are pair- addition of complex numbers One of the ∪k = k wise disjoint then φ( j=1Sj ) j=1 φ(Sj ). fundamental operations under which the com- We say that φ is countably additive if, when- plex numbers C form a field. If w = a + ib, ··· ∈ A ever S1,S2, are pairwise disjoint then z = c + id ∈ C, with a, b, c, and d real, then ∞ ∞ φ(∪ = Sj ) = = φ(Sj ). w + z = (a + c)+ i(b+ d) is the result of addi- j 1 j 1 tion, or the sum, of those two complex numbers. additive valuation Let F be a field and G be a totally ordered additive group. An addi- addition of vectors One of the fundamental tive valuation is a function v : F → G ∪ {∞} operations in a vector space, under which the set satisfying of vectors form an Abelian group. For vectors n n (i.) v(a) =∞if and only if a = 0, in R or C ,ifx = (x1,x2,...,xn) and y = = + (y1,y2,...,yn), then x + y = (x1 + y1,x2 + (ii.) v(ab) v(a) v(b), y2,...,xn + yn). (iii.) v(a + b) ≥ min{v(a), v(b)}. additive group (1) Any group, usually adele Following Weil, let k be either a finite Abelian, where the operation is denoted +. See algebraic extension of Q or a finitely generated group, Abelian group. extension of a finite prime field of transcendency (2) In discussing a ring R, the commutative degree 1 over that field. By a place of k is meant group formed by the elements of R under the the completion of the image of an isomorphic addition operation. embedding of k into a local field (actually the equivalence class of such completions under the additive identity In an Abelian group G, the equivalence induced by isomorphisms unique element (usually denoted 0) such that of the local fields). A place is infinite if the local g + 0 = g for all g ∈ G. field is R or C, otherwise the place is finite. For a place v, kv will denote the completion, and if additive identity a binary operation that is v is a finite place, rv will denote the maximal called addition and is denoted by “+.” In this compact subring of k .Anadele is an element ∈ v situation, an additive identity is an element i S of that satisfies the equation k × r , + = + = v v i s s i s v∈P v/∈P

c 2001 by CRC Press LLC where P is a finite set of places containing the the adjoint representation into the space of linear infinite places. endomorphisms of g. See also adjoint represen- tation. adele group Let V be the set of valuations on the global field k.Forv ∈ V , let kv be adjoint matrix For a matrix M with complex the completion of k with respect to v, and let entries, the adjoint of M is denoted by M∗ and Ov be the ring of elements in kv. The is the complex conjugate of the transpose of M; ∗ adele group of the G is so if M = mij , then M has m¯ ji as the entry the restricted direct product in its ith row and jth column. Gkv GOv adjoint representation (1) In the context of v∈V Lie , the adjoint representation is the mapping sending X to [X, ·]. which, as a set, consists of all sequences of el- (2) In the context of Lie groups, the adjoint ements of G , indexed by v ∈ V , with all but kv representation is the mapping sending σ to the finitely many terms in each sequence being ele- differential of the automorphism ασ : G → G ments of GOv . −1 defined by ασ (τ) = στσ . adele ring Following Weil, let k be either a (3) In the context of representations of an al- finite algebraic extension of Q or a finitely gen- gebra over a field, the term adjoint representa- erated extension of a finite prime field of tran- tion is a synonym for dual representation. See scendency degree 1 over that field. Set dual representation. adjoint system Let D be a curve on a non- kA(P ) = kv × rv , singular surface S. The adjoint system of D is v∈P v/∈P |D + K|, where K is a canonical on S. where P is a finite set of places of k contain- ing the infinite places. A ring structure is put adjunction formula The formula on kA(P ) defining addition and multiplication componentwise. The adele ring is 2g − 2 = C.(C + K) kA = kA(P ) . relating the genus g of a non-singular curve C P on a surface S with the intersection pairing of C and C + K, where K is a canonical divisor on A locally compact topology is defined on kA by S. requiring each kA(P ) to be an open subring and using the product topology on kA(P ). admissible homomorphism For a group G with a set of operators , a group homomor- adjoining (1) Assuming K is a field exten- phism from G to a group G on which the same ⊂ sion of k and S K, the field obtained by ad- operators act, such that joining S to k is the smallest field F satisfying k ⊂ F ⊂ K and containing S. ω(ab) = (ωa)(ωb) (2)IfR is a commutative ring, then the ring of polynomials R[X] is said to be obtained by holds for all a,b ∈ G and all ω ∈ . Also called adjoining X to R. an -homomorphism or an operator homomor- phism. adjoint group The image of a Lie group G, under the adjoint representation into the space admissible isomorphism For a group G with of linear endomorphisms of the associated Lie a set of operators , a group isomorphism from algebra g. See also adjoint representation. G onto a group G , on which the same operators act, such that adjoint Lie algebra Let g be a Lie algebra. The adjoint Lie algebra is the image of g under ω(ab) = (ωa)(ωb)

c 2001 by CRC Press LLC holds for all a,b ∈ G and all ω ∈ . Also called affectless equation A polynomial equation an -isomorphism or an operator isomorphism. for which the Galois group consists of all per- mutations. See also affect. admissible Let G be a group. It is easily seen that a subset N of G is affine algebraic group See linear algebraic a normal subgroup if and only if there is some group. equivalence relation ∼ on G such that ∼ is com- patible with the multiplication on G, meaning affine morphism of schemes Let X and Y be schemes and f : X → Y be a morphism. If a ∼ b, c ∼ d ⇒ (ac) ∼ (bd) , there is an open affine cover {Vi} of the scheme −1 Y for which f (Vi) is affine for each i, then f and N is the equivalence class of the identity. is an affine morphism of schemes. In case G also has an operator domain ,an admissible normal subgroup is defined to be the affine scheme Let A be a commutative ring, equivalence class of the identity for an equiva- and let Spec(A) = X be the set of all prime lence relation ∼ that is compatible with the mul- ideals of A, equipped with the spectral or Zariski O tiplication as above and that also satisfies topology. Let X be a sheaf of local rings on X. The ringed space (X, OX) is called the affine a ∼ b ⇒ (ωa) ∼ (ωb) for all ω ∈ . scheme of the ring A.

affine space Let V be a real, linear n-dimen- admissible representation Let π be a uni- sional space. Let A be a set of points, which are tary representation of the group G in a Hilbert denoted P,Q. Define a relation between points space, and let M be the von Neumann algebra in A and vectors in V as follows: (i.) To every ordered pair (P, Q) ∈ A×A, there generated by π(G). The representation π is said −→ to be an admissible representation or a trace ad- is associated a “difference vector” PQ∈ V. missible representation if there exists a trace on (ii.) To every point P ∈ A and every vector + v ∈ V there is associated precisely one point M which is a character for π. −→ Q ∈ A such that PQ= v. Ado-Iwasawa Theorem The theorem that (iii.) If P, Q, R ∈ A then every finite dimensional Lie algebra (over a field −→ −→ −→ of characteristic p) has a faithful finite dimen- PQ+ QR = PR . sional representation. The characteristic p = 0 case of this is Ado’s Theorem and the charac- In this circumstance, we call A an n-dimensional teristic p = 0 case is Iwasawa’s Theorem. See affine space. also Lie algebra. affine variety A variety (common zero set Ado’s Theorem A finite dimensional Lie al- of a finite collection of functions) defined in an gebra g has a representation of finite degree ρ affine space. such that g =∼ ρ(g). A-homomorphism For A-modules M and While originally proved for Lie algebras N, a group homomorphism f : M → N is over fields of characteristic 0, the result was called an A-homomorphism if extended to characteristic p by Iwasawa. See Ado-Iwasawa Theorem. f(am)= af (m) for all a ∈ A, m ∈ M. affect For a polynomial equation P(X)= 0, the Galois group of the equation can be consid- Albanese variety For V a variety, the Al- ered as a group of permutations of the roots of banese variety of V is an Abelian variety A = the equation. The affect of the equation is the Alb(V ) such that there exists a rational f : index of the Galois group in the group of all V → A which generates A and has the uni- permutations of the roots of the equation. versal mapping property that for any rational

c 2001 by CRC Press LLC g : V → B, where B is an Abelian variety, In this case, M is called the kernel of the ex- there exist a homomorphism h : A → B and a tension because it is the kernel of the canonical constant c ∈ B such that g = hf + c. homomorphism.

Alexander If A is a compact subset algebra homomorphism Suppose A and B of Rn, then for all indices q and all R-modules are algebras of the same type, meaning that for G, each n-ary operation fA on A there is a corre- sponding n-ary operation fB on B. A mapping n n n−q−1 H q (R , R \ A; G) = H (A; G) . φ : A → B is called a homomorphism from A to B if, for each pair of corresponding operations fA and fB , algebra (1) The system of symbolic ma- nipulation formalized by François Viéte (1540Ð φ (fA (a1,a2,...,an)) 1603), which today is known as elementary al- = f (φ (a ) ,φ(a ) ,...,φ(a )) gebra. B 1 2 n (2) The entire of mathematics in which holds for all a ,a ,...,a ∈ A. one studies groups, rings, fields, etc. 1 2 n (3) A vector space (over a field) on which is Typically, an algebra A is a ring that also has also defined an operation of multiplication. the structure of a module over another ring R,so (4) A synonym for , which that an algebra homomorphism φ must satisfy + = + ∈ includes structures such as Boolean algebras. (i.) φ(a1 a2) φ(a1) φ(a2) for a1,a2 A, (ii.) φ(a1a2) = φ(a1)φ(a2) for a1,a2 ∈ A, algebra class An equivalence class of central (iii.) φ(ra) = rφ(a), for r ∈ R and a ∈ A. simple algebras under the relation that relates a pair of algebras if they are both isomorphic to algebraic (1) An adjective referring to an full matrix rings over the same algebra. object, structure, or theory that occurs in algebra Algebras in the same algebra class are said to be or arises through application of the processes “similar.” See also central simple algebra. used in algebra. (2) An adverb meaning a process that in- algebra class group Let K be a field. Two volves only the operations of algebra, which are central simple algebras over K are said to be addition, subtraction, multiplication, division, similar if they are isomorphic to full matrix rings and root extraction. over the same division algebra. Similarity is an equivalence relation, and the equivalence algebraic addition In , classes are called algebra classes. The product the addition of algebraic expressions which ex- of a pair of algebra classes is defined by choos- tends the operation of addition of numbers in ing an algebra from each class, say A and B, and arithmetic. letting the product of the classes be the algebra class containing A ⊗K B. This product is well algebraic addition formula For an Abelian defined, and the algebra classes form a group un- function f , an equation that expresses f(a+ b) der this multiplication, called the algebra class rationally, in terms of the values of a certain group or Brauer group. (p + 1)-tuple of Abelian functions, evaluated at the points a,b ∈ C. See also Abelian function. algebra extension Let A be an algebra over the commutative ring R. Then by an algebra algebraic algebra An algebra A over a field extension of A is meant either K such that every a ∈ A is algebraic over K. (i.) an algebra over R that contains A;or See algebra. (ii.) an algebra A containing a two-sided R- module M which is a two-sided ideal in A and algebraically closed field A field k, in which is such that every polynomial in one variable, with coeffi-

A /M = A. cients in k, has a root.

c 2001 by CRC Press LLC algebraic The smallest algebraically element of K if it satisfies a non-trivial polyno- closed extension field of a given field F . The mial equation with coefficients in k. algebraic closure exists and is unique up to iso- morphism. algebraic equation An equation of the form P = 0 where P is a polynomial in one or more algebraic correspondence Let C be a non- variables. singular algebraic curve. By an algebraic cor- respondence is meant a divisor in the product algebraic equivalence Two cycles X1 and variety C × C. More generally, an algebraic X2 in a non-singular algebraic variety V are al- correspondence means a Zariski closed subset gebraically equivalent if there is a family of cy- T of the product V1 × V2 of two irreducible va- cles {X(t) : t ∈ T } on V , parameterized by rieties. Points P1 ∈ V1 and P2 ∈ V2 are said t ∈ T , where T is another non-singular alge- to correspond if (P1,P2) ∈ T . See also corre- braic variety, such that there is a cycle Z in V ×T spondence ring. for which each X(t) is the projection to V of the intersection of Z and V ×{t}, and X1 = X(t1), algebraic curve An algebraic variety of di- X2 = X(t2), for some t1,t2 ∈ T . Such a family mension one. See also algebraic variety. of cycles X(t) is called an algebraic family. algebraic cycle By an algebraic cycle of di- algebraic equivalence of divisors Two di- mension m on an algebraic variety V is meant a visors f and g on an irreducible variety X are finite formal sum algebraically equivalent if there exists an alge- ∈ c V braic family of divisors, ft , t T , and points i i ∈ = = t1 and t2 T , such that f ft1 , and g ft2 . where the ci are and the Vi are irre- Thus, algebraic equivalence is an algebraic ana- ducible m-dimensional subvarieties of V . The log of homotopy, though the analogy is not par- cycle is said to be effective or positive if all the ticularly fruitful. coefficients ci are non-negative. The support of Algebraic equivalence has the important the cycle is the union of the subvarieties hav- property of preserving the degree of divisors; ing non-zero coefficients. The set of cycles of that is, two algebraically equivalent divisors have dimension m forms an Abelian group under ad- the same degree. It also preserves principal dition, which is denoted Zm(V ). divisors; that is, if one divisor of an algebrai- cally equivalent pair is principal, then so is the algebraic dependence The property shared other one. (A divisor is principal if it is the di- by a set of elements in a field, when they - visor of a rational function.) Thus, the group isfy a non-trivial polynomial equation. Such an D0/P is a subgroup of the divisor class group 0 equation demonstrates that the set of elements Cl (X) = D/P. Here, D0 is the group of divi- is not algebraically independent. sors algebraically equivalent to 0, P is the group of principal divisors, and D is the group of di- algebraic (1) An equa- visors of degree 0. The group D0/p is exactly tion of the form the subgroup of the divisor class group realized

F x,y,y ,y ,...,y(n) = 0 by the group of points of the Picard variety of X. See algebraic family of divisors, divisor. See in which F is a polynomial with coefficients that also integral divisor, irreducible variety, Picard are complex analytic functions of x. variety. (2) An equation obtained by equating to zero a differential polynomial in a set of differential algebraic An expression formed variables in a differential extension field of a from the elements of a field and one or more differential field. See also differential field. variables (variables are also often called inde- terminants) using the algebraic operations of ad- algebraic element If K is an extension field dition, subtraction, multiplication, division, and of the field k, an element x ∈ K is an algebraic root extraction.

c 2001 by CRC Press LLC algebraic extension An extension field K of algebraic group An algebraic variety, to- a field k such that every α in K, but not in k, gether with group operations that are regular is algebraic over k, i.e., satisfies a polynomial functions. See regular function. equation with coefficients in k. algebraic homotopy group A generalization algebraic family A family of cycles {X(t) : of the concept of homotopy group, defined for t ∈ T } on a non-singular algebraic variety V , an algebraic variety over a field of characteris- parameterized by t ∈ T , where T is another tic p>0, formed in the context of finite étale non-singular algebraic variety, such that there coverings. is a cycle Z in V × T for which each X(t) is the projection to V of the intersection of Z and algebraic identity An algebraic equation in- ×{ } V t . volving a variable or variables that reduces to an arithmetical identity for all substitutions of algebraic family of divisors A family of di- numerical values for the variable or variables. visors ft , t ∈ T , on an irreducible variety X, where the index set T is also an irreducible va- algebraic independence Let k be a subfield = ∗ riety, and where ft φt (D) for some fixed of the field K. The elements a ,a ,...,a of K × ∈ 1 2 n divisor D on X T and all t T . Here, for are said to be algebraically independent over k ∈ ∗ each t T , φt is the map from divisors on if, for any polynomial p(X ,X ,...,X ) with × 1 2 n X T to divisors on X induced by the embed- coefficients in k, p(a ,a ,...,a ) = 0 implies : → × = 1 2 n ding φt X X T , where φ(t) (x, t), p ≡ 0. When a set of complex numbers is said × and X T is the Cartesian product of X and T . to be algebraically independent, the field k is The variety T is called the for the algebraic understood to be the rational numbers. family ft , t ∈ T . See also Cartesian product, irreducible variety. algebraic integer A complex number that satisfies some monic polynomial equation with algebraic function A function Y = f(X , 1 integer coefficients. X2,...,XN ) satisfying an equation R(X1,X2, ...,XN ,Y)= 0 where R is a rational function over a field F . See also rational function. algebraic Lie algebra Let k be a field. An algebraic group G, realized as a closed subgroup algebraic function field Let F be a field. of the GL(n, k), is called a Any finite extension of the field of rational func- linear algebraic group, and its tangent space at tions in the identity, when given the natural Lie algebra structure, is called an algebraic Lie algebra. X1,X2,...,Xn over the field F is called an algebraic function algebraic multiplication In elementary al- field over F . gebra, the multiplication of algebraic expres- sions, which extends the operation of multipli- algebraic fundamental group A generaliza- cation of numbers in arithmetic. tion of the concept of fundamental group defined for an algebraic variety over a field of character- algebraic multiplicity The multiplicity of an istic p>0, formed in the context of finite étale eigenvalue λ of a matrix A as a root of the char- coverings. acteristic polynomial of A. See also geometric multiplicity, index. algebraic geometry Classically, algebraic geometry has meant the study of geometric prop- A complex number z is erties of solutions of algebraic equations. In an algebraic number if it satisfies a non-trivial modern times, algebraic geometry has become polynomial equation P(z) = 0, for which the synonymous with the study of geometric objects coefficients of the polynomial are rational num- associated with commutative rings. bers.

c 2001 by CRC Press LLC algebraic number field A field F ⊂ C, algebraic torus An algebraic group, isomor- which is a finite degree extension of the field phic to a direct product of the multiplicative of rational numbers. group of a universal domain. A universal do- main is an algebraically closed field of infinite In elementary algebra, transcendence degree over the prime field it con- the operations of addition, subtraction, multipli- tains. cation, division, and root extraction. In a gen- eral algebraic system A, an algebraic operation algebraic variety Classically, the term “al- may be any function from the n-fold cartesian gebraic variety” has meant either an affine al- product An to A, where n ∈{1, 2,...} (the case gebraic set or a projective algebraic set, but in n = 0 is sometimes also allowed). See also the second half of the twentieth century, various algebraic system. more general definitions have been introduced. One such more general definition, in terms of algebraic pencil A linear system of divi- sheaf theory, considers an algebraic variety V O sors in a projective variety such that one divisor to be a pair (T , ), in which T is a topological O passes through any point in general position. space and is a sheaf of germs of mappings from V into a given field k, for which the topo- { }N algebraic scheme An algebraic scheme is a logical space has a finite open cover Ui i=1 O| scheme of finite type over a field. Schemes are such that each (Ui, Ui) is isomorphic to an generalizations of varieties, and the algebraic affine variety and for which the image of V un- schemes most closely resemble the algebraic va- der the diagonal map is Zariski closed. See also rieties. See scheme. abstract algebraic variety. algebra isomorphism An algebra homomor- algebraic space A generalization of scheme phism that is also a one-to-one and onto mapping and of algebraic variety due to Artin and in- between the algebras. See algebra homomor- troduced to create a category which would be phism. closed under various constructions. Specifically, an algebraic space of finite type is an affine algebra of matrices The n×n matrices with scheme U and a closed subscheme R ⊂ U × U entries taken from a given field together with the that is an equivalence relation and for which both operations of matrix addition and matrix multi- the coordinate projections of R onto U are étale. plication. Also any nonempty set of such ma- See also étale morphism. trices, closed under those operations and con- taining additive inverses, and thus forming an algebraic subgroup A Zariski closed sub- algebra. group of an affine algebraic group. algebra of vectors The vectors in algebraic surface A two-dimensional alge- three-dimensional space, together with the oper- braic variety. See also algebraic variety. ations of vector addition, scalar multiplication, the scalar product (also called the inner prod- algebraic system A set A, together with var- uct or the dot product), the vector product (also ious operations and relations, where by an oper- called the cross product), and the vector triple ation we mean a function from the n-fold carte- product. sian product An to A, for some n ∈{0, 1, 2,...}. algebroidal function An analytic function f algebraic system in the wider sense While satisfying a non-trivial algebraic equation an algebraic system is a set A, together with n n−1 various operations and relations on A, an alge- a0(z)f + a1(z)f +···+an(z) = 0 , braic system in the wider sense may also include higher level structures constructed by the power in which the coefficients aj (z) are meromorphic set operation. functions in a domain in the complex z-plane.

c 2001 by CRC Press LLC all-integer An algorithm for two equations which the entire will be carried out = = in integers, provided the given data is all given ax1 b and x2a b in integers. Such are of interest for are uniquely solvable for x1 and x2. Also called linear programming problems that involve addi- alternative skew-field. tional integrality conditions. A notable example of such an algorithm was given in the early 1960s amalgamated product Given a family of by Gomory. groups {Gα}α∈A and embeddings {hα}α∈A of a fixed group H into the Gα, the amalgamated allowed submodule In a module M with op- product is the group G, unique up to isomor- erator domain A,anallowed submodule is a sub- phism, having the universal properties that (i.) module N ⊂ M such that a ∈ A and x ∈ N there exist homomorphisms {gα}α∈A such that implies ax ∈ N. Also called an A-submodule. gα ◦ hα = gβ ◦ hβ for all α, β ∈ A and (ii.) for any family {α}α∈A of homomorphisms of almost integral Let R be a subring of the the groups G to a fixed group L satisfying ∈ α ring R . An element a R is said to be almost  ◦ h =  ◦ h for all α, β ∈ A, there exists ∈ α α β β integral over R if there exists an element b R a unique homomorphism  : G → L such that which is not a zero divisor and for which anb ∈ α =  ◦ gα. R holds for every positive integer n. For the case of two groups G1 and G2 with isomorphic H ⊂ G and H ⊂ G , For fixed n, the subgroup 1 1 2 2 the amalgamated product of the groups can be of the group of permutations of {1, 2, ...,n}, identified with the set of finite sequences of el- consisting of the even permutations. More spe- ements of the union of the two groups with the cifically, the set of permutations σ :{1, 2,..., equivalence relation generated by identifying a n}→{1, 2,...,n} such that sequence with the sequence formed when adja- cent elements are replaced by their product if (σ (j) − σ(i))> 0 . they are in the same Gi or with the sequence 1≤i

c 2001 by CRC Press LLC (n) S ∈ CR and n a nonnegative integer, let S a function has a convergent power series expan- denote the n-fold tensor product of S over R. sion about each point of its domain. (n+1) For n a nonnegative integer, let Ei : S → (n+2) S (i = 0, 1,...,n) be the CR-morphisms analytic homomorphism A homomorphism defined by between two Lie groups which is also an ana- lytic function (i.e., expandable in a power series E ⊗···⊗ = i (x0 xn) at each point in the Lie group, using a local co- x0 ⊗···⊗xi−1 ⊗ 1 ⊗ xi ⊗···⊗xn . ordinate system). + + Define dn : F(S(n 1)) → F(S(n 2)) by setting analytic isomorphism An analytic n homomorphism between two Lie groups which n i is one-to-one, onto and has an inverse that is d = (−1) F (Ei) . i=0 also an analytic homomorphism. See analytic homomorphism. Then {F(S(n+1)), dn} defines a cochain com- plex called the Amitsur complex and the coho- analytic structure A structure on a differen- mology groups are called the Amitsur cohomol- tiable manifold M which occurs when there is ogy groups. an atlas of charts {(Ui,ϕi) : i ∈ I} on M, where the transition functions Amitsur cohomology groups See Amitsur − ϕ ◦ ϕ 1 : ϕ U ∩ U → ϕ U ∩ U cohomology. j i i i j j i j are analytic. Amitsur complex See Amitsur cohomology. analytic variety A set that is the simulta- ample See ample vector bundle, ample divi- neous zero set of a finite collection of analytic sor. functions. ample divisor A divisor D such that nD is analytic vector A vector v in a Hilbert space very ample for some positive integer n. A divi- H is called an analytic vector for a finite set { }m H sor is very ample if it possesses a certain type of Tj j=1 of (unbounded) operators on if there canonical projective immersion. exist positive constants C and N such that T ···T v ≤ CNkk! ample vector bundle A vector bundle E j1 jk H ∨ where the line bundle OE∨ (1) on P(E ) is am- for all ji ∈{1,...,m} and every positive integer ple. That is, there is a morphism f from P(E∨) k. n ⊗m ∗ to a projective space P with OE∨ (1) = f O V P N (1). anisotropic A vector space with an inner product (·, ·) and containing no non-zero iso- amplification The process of increasing the tropic vector. A vector x ∈ V is isotropic if magnitude of a quantity. (x, x) = 0. analytically normal ring An analytically un- antiautomorphism An isomorphism of an ramified ring that is also integrally closed. See algebra A onto its opposite algebra A◦. See analytically unramified ring. opposite. analytically unramified ring A local ring antiendomorphism A mapping τ from a ring such that its completion contains no non-zero R to itself, which satisfies nilpotent elements. (An element x of a ring is τ(x + y) = τ(x)+ τ(y), τ(xy) = τ(y)τ(x) nilpotent if x · x = 0.) for all x,y ∈ R. The mapping τ can also be analytic function Same as a holomorphic viewed as an endomorphism (linear mapping) function, but with emphasis on the fact that such from R to its opposite ring R◦. See opposite.

c 2001 by CRC Press LLC antihomomorphism A mapping σ from a (iv.) if two apartments  and  contain two group G into a group H that satisfies σ(xy) = elements A, A ∈ , then there exists an iso- σ(y)σ(x) for all x,y ∈ G. An antihomor- morphism of  onto  which leaves invariant phism can also be viewed as a homomorphism A, A and all their faces. σ : G → H ◦ where H ◦ is the opposite group to H . See opposite. approximate functional equations Equa- tions of the form f(x) = g(x) + Ev(x) where anti-isomorphism A one-to-one, surjective f(x) and g(x) are known functions and the map f : X → Y that reverses some intrinsic growth of Ev(x) is known. property common to X and Y .IfX and Y are ∗ groups or rings, then f reverses multiplication, approximately finite algebra A C -algebra = that is the uniform closure of a finite dimensional f(ab) f(b)f(a).IfX and Y are lattices, then ∗ f reverses the operations, f(a ∩ b) = C -algebra. f(a)∪ f(b)and f(a∪ b) = f(a)∩ f(b). approximately finite dimensional von Neu- mann algebra A von Neumann algebra, M, antilogarithm For a number y and a base b, = which contains an increasing sequence of finite the number x such that logb x y. dimensional subalgebras, An ⊆ An+1, such that ∪∞ A is dense in M. (Density is defined in antipode Let S be a sphere in Euclidean n=1 n terms of any of a number of equivalent topolo- space and s a point of S. The line through s and ∗ gies on M, e.g., the weak topology, or the the center of the sphere will intersect the sphere strong operator topology in any normal repre- in a uniquely determined second point s that is sentation.) called the antipode of s. The celebrated Borsuk- Ulam Theorem of consid- approximate number A numerical approx- →− ers the antipodal map P P . The theory imation to the actual value. of Hopf algebras contains a notion of antipode which is analogous to the geometric one just de- approximation theorem A theorem which scribed. states that one class of objects can be approxi- mated by elements from another (usually antisymmetric decomposition The decom- smaller) class of objects. A famous example position of a compact Hausdorff space X con- is the following. sists of disjoint, closed, maximal sets of anti- symmetry with respect to A, where A is a closed Weierstrass A. T. Every con- subalgebra of C(X), the algebra of all complex- tinuous function on a closed inter- valued continuous functions on X. A is called val can be uniformly approximated antisymmetric if, from the condition that f, f¯ ∈ by a polynomial. That is, if f(x) A, it follows that f is a constant function. A is continuous on the closed inter- subset S§X is called a set of antisymmetry with val [a,b] and >0, then there ex- respect to A if any function f ∈ A that is real ists a polynomial p(x) such that on S is constant on this set. |f(x) − p(x)| <for all x ∈ [a,b]. apartment An element of A, a set of sub- complexes of a complex  such that the pair (, A) is a building. That is, if the following The numbers 0, 1, 2, 3, 4, hold: 5, 6, 7, 8, and 9. These numbers can be used to represent all numbers in the system. (i.)  is thick; (ii.) the elements of A are thin chamber com- arbitrary constant A constant that can be set plexes; to any desired value. For example, in the calcu- (iii.) any two elements of  belong to an lus expression 2xdx = x2 + C, the symbol apartment; C is an arbitrary constant.

c 2001 by CRC Press LLC arc cosecant The multiple-valued inverse of is satisfied. the trigonometric function csc θ, e.g., arccsc(2) (iii.) For each x ∈ F there exists a positive = π/6 + 2kπ where k is an arbitrary integer integer n such that n · 1 >x. (k = 0 specifies the principal value of arc cose- The rational numbers are an Archimedian or- cant). The principal value yields the length of dered field, and so are the real numbers. The the arc on the , subtending an angle, p-adic numbers are a non-Archimedian ordered whose cosecant equals a given value. field. The arc cosecant function is also denoted − csc 1x. Archimedian valuation A valuation on a ring R, for which v(x − y) ≤ max(v(x), v(y)) arc cosine The multiple-valued inverse of the is false, for some x,y ∈ R. See valuation. trigonometric function cos θ, e.g., arccos(−1) = π +2kπ where k is an arbitrary integer (k = 0 arcsecant The multiple-valued inverse of specifies the principal value of arc cosine). The the trigonometric function sec x, sometimes de- principal value yields the length of the arc on the noted sec−1x. unit circle, subtending an angle, whose cosine equals a given value. arc The multiple-valued inverse of the The arc cosine function is also denoted trigonometric function sin θ, e.g., arcsin(1) = − cos 1x. π/2+2kπ where k is an arbitrary integer (k = 0 specifies the principal value of arc sine). The arc cotangent The multiple-valued inverse principal value yields the length of the arc on of the√ trigonometric function cotan θ, e.g., arc- the unit circle, subtending an angle, whose sine cot ( 3) = π/6 + 2kπ where k is an arbitrary equals a given value. integer (k = 0 specifies the principal value of The arc sine function is also denoted sin−1x. arc cotangent). The principal value yields the length of the arc on the unit circle, subtending arc tangent The multiple-valued inverse of an angle, whose cotangent equals a given value. the√ trigonometric function tan θ, e.g., arctan The arc cotangent function is also denoted ( 3) = π/3 + 2kπ where k is an arbitrary in- − cot 1x. teger (k = 0 specifies the principal value of arc tangent). The principal value yields the length Archimedian ordered field If K is an or- of the arc on the unit circle, subtending an angle, dered field and F a subfield with the property whose tangent equals a given value. that no element of K is infinitely large over F , The arc tangent function is also denoted then we say that K is Archimedian. tan−1x.

Archimedian ordered field A set which, in ArensÐRoyden Theorem Let C(MA) denote addition to satisfying the axioms for a field, also the continuous functions on the maximal ideal possesses an Archimedian ordering. That is, the space MA of the Banach algebra A. Suppose field F is ordered in that it contains a subset P that f ∈ C(MA) and f does not vanish. Then and the following properties hold: there exists a g ∈ A, for which g−1 ∈ A, and for { } − (i.) F is the disjoint union of P , 0 , and P . which f/gˆ has a continuous on M . ∈ A In other words, each x F belongs either to P , (Here gˆ denotes the Gelfand transform of g.) or equals 0, or −x belongs to P , and these three possibilities are mutually exclusive. arithmetic The operations of addition, sub- (ii.) If x,y ∈ P , then x +y ∈ P and xy ∈ P . traction, multiplication, and division and their The ordered field is also Archimedian in that properties for the integers. the absolute value function  arithmetical equivalence An equivalence  ∈ x, if x P relation on the integers which is consistent with |x|= 0, if x = 0 the four operations of arithmetic. (a ∼ b and  −x, if x ∈−P c ∼ d imply a ± c ∼ b ± d, etc.) An example

c 2001 by CRC Press LLC would be congruence mod n where n is a positive arithmetic subgroup For a real algebraic integer. Here, two integers j and k are equiva- group G ⊂ GL(n, R), a subgroup  of G, com- lent if j − k is divisible by n. See equivalence mensurable with GZ = G ∩ GL(n, R). That relation. is, : ∩ ∞ : ∩ ∞ arithmetically effective Referring to a divi- [  GZ] < and [GZ  GZ] < . sor on a nonsingular algebraic surface, which is numerically semipositive, or numerically effec- Arrow-Hurewicz-Uzawa gradient method tive (nef). A technique used in solving convex or concave programming problems. Suppose ψ(x,u) is arithmetic crystal class For an n-dimen- concave or convex in x ∈ A ⊂ Rn and convex in sional Euclidean space V , an equivalence class u ∈ 0 ⊂ Rm. Usually ϕ(x,u) = ψ(x)+u·g(x) of pairs (, G) where  is a lattice in V and where ϕ is the function we wish to minimize or G is a finite subgroup of O(V). Two pairs maximize and our constraints are given by the (1,G1) and (2,G2) are equivalent if there functions g (x) ≤ 01≤ j ≤ m. The method ∈ = j is a g GL(V ) such that g1 2, and devised by Arrow-Hurewicz and Uzawa consists −1 = gG1g G2. of solving the system of equations    =  arithmetic genus An integer, defined in terms  0ifxi 0   ∂ψ  of the characteristic polynomial of a homoge- dxi and ∂x < 0, U = i neous ideal in the ring of polynomials, dt  i = 1,...,n  [ ]   k x1,...,xn , in the variables x1,...,xn over ∂ψ otherwise a commutative ring k.Ifχ(¯ U; q) denotes this ∂xi characteristic polynomial, then    0ifuj = 0   ∂ψ  ¯ U; = q + q +···+ q + duj and ∂u > 0, χ( q) a0 r a1 r−1 ar−1 1 ar = j dt  j = 1,...,m   −  ∈ { q } ∂ψ otherwise where a0,...,ar k and (j ) are the binomial ∂uj − r − coefficients. The integer ( 1) (ar 1) is the If (x(t), u(t)) is a solution of this system, un- U arithmetic genus of . der certain conditions, lim x(t) = x solves the t→∞ arithmetic mean For a positive integer n, the programming problem. arithmetic mean of the n real numbers a ,..., 1 artificial variable A variable that is intro- a is (a +···+a )/n. n 1 n duced into a linear programming problem, in order to transform a constraint that is an inequal- arithmetic of associative algebras An area ity into an . For example, the problem of mathematics devoted to the study of simple of minimizing algebras over local fields, number fields, or func- tion fields. C = 3x1 + 2x2 A sequence {sn} of subject to the constraints real numbers such that 4x1 − 5x2 ≤ 7 x1 + x2 = 9 sn = sn−1 + r, for n>1 . with x1 ≥ 0, x2 ≥ 0, is transformed into The number s1 is the initial term, the number = + + r is the difference term. The general term sn C 3x1 2x2 0A1 satisfies s = s + (n − 1)r. n 1 subject to the constraints

arithmetic series A series of the form 4x1 − 5x2 + A1 = 7 ∞ ≥ = + n=1 an where for all n 1,an+1 an d. x1 + x2 + 0A1 = 9

c 2001 by CRC Press LLC p with x1 ≥ 0, x2 ≥ 0, A1 ≥ 0, by introducing K,Pai is a root of x − x − ai = 0, L/K is the artificial variable A1. This latter version is Galois, and the Galois group is an Abelian group in the standard form for a linear programming of exponent p. problem. Artin’s conjecture A conjecture of E. Artin Artin-Hasse function For k a p-adic num- that the Artin L-function L(s, ϕ) is entire in s, ber field with k0 a maximal subfield of k unram- whenever ϕ is irreducible and s = 1. See Artin ified over Qp, a an arbitrary integer in k0 and L-function. ∈ = − x k, the function E(a,x) exp L(a, x) ∞ σ i i pi where L(a, x) = = ((a ) /p )x and σ is Artin’s general law of reciprocity If K/k i 0 F the Frobenius automorphism of ko/Qp. is an Abelian field extension with conductor and AF is the group of ideals prime to the con- K/k Artinian module A (left) module for which ductor, then the Artin map A → A is a every descending sequence of (left) submodules homomorphism AF → Gal(K/k). The reci- procity law states that this homomorphism is ⊃ ⊃···⊃ ⊃ ⊃ M1 M2 Mn Mn+1 ... an isomorphism precisely when A lies in the subgroup HF of AF consisting of those ideals is finite, i.e., there exists an N such that M = n whose prime divisors split completely. That is, Mn+1 for all n ≥ N. ∼ AF /HF = Gal(K/k). Artinian ring A ring for which every de- Artin’s symbol The symbol K/k defined scending sequence of left ideals ℘ as follows. Let K be a finite Abelian Galois extension of a number field k with σ the princi- I1 ⊃ I2 ⊃···⊃In ⊃ In+1 ⊃ ... pal order of k and D the principal order of K. = K/k is finite. That is, there exists an N such that For each prime ℘ of K there is a σ ℘ In = In+1 for all n ≥ N. ∈ G =Gal(K/k) such that Aσ ≡ AN(℘) (mod ℘), A ∈ D ; Artin L-function The function L(s, ϕ), de- fined as follows. Let K be a finite Galois exten- K/k is called the Artin symbol of ℘ for the = ℘ sion of a number field k with G Gal(K/k). Abelian extension K/k. For an ideal a = ℘e Let ϕ : G → GL(V ) be a finite dimensional rep- of k relatively prime to the relative discriminant resentation (characteristic 0). For each prime ℘ e K/k K/k =  K/k = − −s −1 of , define a ℘ . of k, set L℘(s, ϕ) det(I ϕ℘N(℘) ) , = 1 where ϕ℘ e τ∈T ϕ(σ τ), T is the inertia ascending central series A sequence of sub- | |= group of ℘, T e and σ is the Frobenius groups automorphism of ℘. Then {1}=H0

1 . of a group G with identity 1, where H + is the ℘ n 1 unique normal subgroup of Hn for which the group Hn+1/Hn is the center of G/Hn. Artin-Rees Lemma Let R be a Noetherian ring, I an ideal of R, F a finitely generated sub- ascending chain of subgroups A sequence module over R, and E a submodule of F . Then, of subgroups there exists an integer m ≥ 1 such that, for all H < ···

Artin-Schreier extension For K a field of associate A relation between two elements a characteristic p = 0, an extension of the form and b of a ring R with identity. It occurs when L = K(Pa1,...,PaN ) where a1,...,aN ∈ a = bu for a unit u.

c 2001 by CRC Press LLC associated factor sets Related by a certain augmentation An augmentation (over the equivalence relation between factor sets belong- integers Z) of a chain complex C is a surjective ing to a group. Suppose N and F are groups and α ∂1  homomorphism C0→Z such that C1→C0→Z G is a group containing a normal subgroup N →0 isomorphic to N with G/N =∼ F .Ifs : F → G equals the trivial homomorphism C1 Z (the is a splitting map of the sequence 1 → N → trivial homomorphism maps every element of G → F → 1 and c : F × F → N is the C1 to 0). map, c(σ, τ) = s(σ)s(τ)s(στ)−1 (s,c) is called augmented algebra See supplemented alge- a factor set. More generally, a pair of maps (s, c) bra. where s : F → AutN and c : F × F → N is called a factor set if augmented chain complex A non-negative (i.) s(σ)s(τ)(a) = c(σ, τ)s(σ τ)(a)c(σ,  chain complex C with augmentation C→Z.A τ)−1(a ∈ N), chain complex C is non-negative if each Cn ∈ C (ii.) c(σ, τ)c(στ, ρ) = s(σ )(c(τ, ρ))c(σ, with n<0 satisfies Cn = 0. See augmentation. τρ). Two factor sets (s, c) and (t, d) are said to automorphic form Let D be an open con- be associated if there is a map ϕ : F → N nected domain in Cn with  a discontinuous sub- − such that t(σ )(a) = s(σ )(ϕ(σ )(a)ϕ(σ ) 1) and group of Hol(D).Forg ∈ Hol(D) and z ∈ D − d(σ,τ) = ϕ(σ )(s(σ )(ϕ(τ)))c(σ, τ)ϕ(σ τ) 1. let j(g,z) be the determinant of the Jacobian transformation of g evaluated at z. A mero- associated form Of a projective variety X in morphic function f on D is an automorphic Pn, the form whose zero set defines a particular form of weight  (an integer) for  if f(γz) = projective hypersurface associated to X in the f (z)j (γ, z)−,γ ∈ , z ∈ D. Chow construction of the parameter space for X. The construction begins with the irreducible automorphism An isomorphism of a group, algebraic correspondence (x, H0,...,Hd ) ∈ or algebra, onto itself. See isomorphism. n n X × P ×···×P : x ∈ X ∩ (H0 ∩···∩Hd ) between points x ∈ X and projective hyper- automorphism group The set of all auto- n planes Hi in P , d = dim X. The projection morphisms of a group (vector space, algebra, of this correspondence onto Pn × ··· × Pn is etc.) onto itself. This set forms a group with a hypersurface which is the zero set of a single binary operation consisting of composition of multidimensional form, the associated form. mappings (the automorphisms). See automor- phism. associative algebra An algebra A whose multiplication satisfies the associative law; i.e., average Often synonymous with arithmetic for all x,y,z ∈ A, x(yz) = (xy)z. mean. Can also mean integral average, i.e., 1 b f(x)dx, associative law The requirement that a bi- b − a nary operation (x, y) → xy on a set S satisfy a x(yz) = (xy)z for all x, y, z ∈ S. the integral average of a function f(x) over a closed interval [a,b],or asymmetric relation A relation ∼, on a set 1 ∼ ⇒ ∼ fdµ, S, which does not satisfy x y y x for µ(X) X some x,y ∈ S. the integral average of an integrable function f over a measure space X having finite measure asymptotic ratio set In a von Neumann al- µ(A). gebra M, the set axiom A statement that is assumed as true, ={∈] [: ⊗ r∞(M) λ 0, 1 M Rλ without proof, and which is used as a basis for is isomorphic to M}. proving other statements (theorems).

c 2001 by CRC Press LLC axiom system A collection (usually finite) of Azumaya algebra A central separable alge- axioms which are used to prove all other state- bra A over a commutative ring R. That is, an ments (theorems) in a given field of study. For algebra A with the center of A equal to R and ◦ example, the axiom system of Euclidean geom- with A a projective left-module over A ⊗R A etry, or the Zermelo-Frankel axioms for set the- (where A◦ is the opposite algebra of A). See ory. opposite.

c 2001 by CRC Press LLC base See base of logarithm. See also base of number system, basis.

base of logarithm The number that forms the B base of the exponential to which the logarithm is inverse. That is, a logarithm, base b, is the in- verse of the exponential, base b. The logarithm back substitution A technique connected is usually denoted by logb (unless the base is with the method for solv- Euler’s constant e, when ln or log is used, log is ing simultaneous linear equations. After the sys- also used for base 10 logarithm). A conversion tem formula, from one base to another, is

a x + a x +···+a nxn = b 11 1 12 2 1 1 loga x = logb x loga b. a21x1 + a22x2 +···+a2nxn = b2 ...... an1x1 + an2x2 +···+annxn = bn base of number system The number which is used as a base for successive powers, com- is converted to triangular form binations of which are used to express all posi- tive integers and rational numbers. For example, t x + t x +···+t nxn = c 11 1 12 2 1 1 2543 in the base 7 system stands for the number t22x2 +···+t2nxn = c2 ......       2 73 + 5 72 + 4 71 + 3 . tnnxn = cn . − . One then solves for xn and then back substitutes Or, 524 37 in the base 8 system stands for the number this value for xn into the equation          − − − 5 82 + 2 81 + 4 + 3 8 1 + 7 8 2 . tn−1 n−1xn−1 + tnnxn = cn−1 The base 10 number system is called the deci- and solves for xn− . Continuing in this way, all 1 mal system. For base n, the term n-ary is used; of the variables x ,x ,...,xn can be solved for. 1 2 for example, ternary, in base 3. backward error analysis A technique for base point The point in a set to which a bun- estimating the error in evaluating f(x ,...,xn), 1 dle of (algebraic) objects is attached. For exam- assuming one knows f(a ,...,an) = b and has 1 ple, a vector bundle V defined over a manifold control of |xi − ai| for 1 ≤ i ≤ n. M will have to each point b ∈ M an associated vector space Vb. The point b is the base point Baer’s sum For given R-modules A and C, for the vectors in Vb. the sum of two elements of the Abelian group ExtR(C, A). base term For a spectral sequence E = {Er ,dr } where dr : Er → Er ,r = Bairstow method of solving algebraic equa- p,q p−r,q+r−1 2, 3,... and Er = 0 whenever p<0or tions An iterative method for finding qua- p,q q<0, a term of the form Er . dratic factors of a polynomial. The goal being p,o to obtain complex roots that are conjugate pairs. basic feasible solution A type of solution of the linear equation Ax = b.IfA is an m × n Banach algebra An algebra over the com- matrix with m ≤ n and b ∈ Rm, suppose A = plex numbers with a norm ·, under which it A + A when A is an m × m nonsingular is a Banach space and such that 1 2 1 matrix. It is a solution of the form A(x1, 0) = m xy≤xy A1x1 + A20 = b where x1 ∈ R and x1 ≥ 0. for all x,y ∈ B.IfB is an algebra over the real basic form of linear programming problem numbers, then B is called a real Banach algebra. The following form of a linear programming

c 2001 by CRC Press LLC problem: Find a vector (x1,x2,...,xn) which infinite (and convergent). If only finite sums are minimizes the linear function permitted, a basis is referred to as a Hamel basis. C = c x + c x +···+c x , 1 1 2 2 n n Bernoulli method for finding roots An it- erative method for finding a root of a polyno- subject to the constraints n n−1 mial equation. If p(x) = a0x + an−1x + ···+an is a polynomial, then this method, ap- a11x1 + a12x2 +···+a1nxn = b1 plied to p(x) = 0, consists of the following a21x1 + a22x2 +···+a2nxn = b2 ... steps. First, choose some set of initial-values x ,x− ,...,x−n+ . Second, define subsequent am1x1 + am2x2 +···+amnxn = bm 0 1 1 values xm by the recurrence relation and x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0. a b c a1xm−1 + a2xm−2 +···+anxm−n Here ij , i, and j are all real constants and xm =− m

c 2001 by CRC Press LLC Bezout’s Theorem If p1(x) and p2(x) are for all x,y,z ∈ V and α, β ∈ F . See also two polynomials of degrees n1 and n2, respec- quadratic form. tively, having no common zeros, then there are two unique polynomials q1(x) and q2(x) of de- bilinear function See multilinear function. grees n1 − 1 and n2 − 1, respectively, such that bilinear mapping A mapping b : V × V → p1(x)q1(x) + p2(x)q2(x) = 1 . W, where V and W are vector spaces over the field F , which satisfies biadditive mapping For A-modules M,N b(αx + βy, z) = αb(x, z) + βb(y, z) and L, the mapping f : M × N → L such that     and   f x + x ,y = f(x,y) + f x ,y , and     b(x, αy + βz) = αb(x, y) + βb(x, z) f x,y + y = f(x,y) + f x,y , for all x,y,z ∈ V and α, β ∈ F . x,x ∈ M, y,y ∈ L. bilinear programming The area dealing with A k bialgebra A vector space over a field that finding the extrema of functions is both an algebra and a coalgebra over k. That (A,µ,η,*,ε) (A,µ,η) f (x ,x ) = Ct x + Ct x + xt Qx is, is a bialgebra if is 1 2 1 1 2 2 1 2 an algebra over k and (A,*,ε) is a coalgebra over k, µ : A ⊗k A → A (multiplication). over η : k → A * : A → A ⊗ A  (unit), k (co- n n X = (x ,x ) ∈ 1 × 2 : A x ≤ b , multiplication). ε : A → k (counit) and these 1 2 R R 1 1 1 maps satisfy A2x2 ≤ b2,x1 ≥ 0,x2 ≥ 0} µ ◦ (µ ⊗ I ) = µ ◦ (I ⊗ µ) , A A where Q is an n1 × n2 real matrix, A1 is an n × n , real matrix and A is an n × n real µ ◦ (η ⊗ I ) = µ ◦ (I ⊗ η) , 1 1 2 2 2 A A matrix. (* ⊗ IA) ◦ * = (IA ⊗ *) ◦ *, binary A Diophan- (ε ⊗ I ) ◦ * = (I ⊗ ε) ◦ *. A A tine equation in two unknowns. See Diophan- tine equation. bialgebra-homomorphism For (A, µ, η, *, ε) and (A,µ,η,*,ε) bialgebras over binary operation A mapping from the Carte- a field k, a linear mapping f : A → A where sian product of a set with itself into the set. f ◦η = η,f◦µ = µ ◦(f ⊗f),(f⊗f)◦* = That is, if the set is denoted by S, a mapping   b : S × S → S 3 * ◦ f, ε = ε ◦ f . See bialgebra. . A notation, such as , is usually adopted for the operation, so that b(x,y) = x3y. biideal A linear subspace I of A, where (A, µ, η, *, ε) is a bialgebra over k, such that µ(A binomial A sum of two monomials. For ex- x y α β ⊗k I) = I and *(I) ⊂ A ⊗k I + I ⊗k A. ample, if and are variables and and are constants, then αxpyq +βxr ys, where p, q, r, s bilinear form A mapping b : V × V → F , are integers, is a binomial expression. where V is a vector space over the field F , which binomial coefficients The numbers, often satisfies n denoted by (k ), where n and k are nonnegative b(αx + βy, z) = αb(x, z) + β(y,z) integers, with n ≥ k, given by

n and = n! k b(x, αy + βz) = αb(x, y) + βb(x, z) k!(n − k)!

c 2001 by CRC Press LLC where m!=m(m−1) ···(2)(1) and 0!=1 and blowing up A process in algebraic geometry 1!=1. The binomial coefficients appear in the whereby a point in a variety is replaced by the set Binomial Theorem expansion of (x +y)n where of lines through that point. This idea of Zariski n is a positive integer. See Binomial Theorem. turns a singular point of a given manifold into a smooth point. It is used decisively in Hironaka’s binomial equation An equation of the form celebrated “resolution of singularities” theorem. xn − a = 0. blowing up Let N be an n-dimensional com- binomial series The series (1 + x)α = pact, complex manifold (n ≥ 2), and p ∈ N. ∞ (α)xn |x| < Let {z = (zi)} be a local coordinate system, in n=0 n . It converges for all 1. a neighborhood U, centered at p and define  Binomial Theorem For any nonnegative in- n− n n n j n−j U˜ = (z, l) ∈ U × P 1 : z ∈ l , tegers b and n, (a + b) = j=o(j )a b . P n−1 l birational isomorphism A k-morphism ϕ : where is regarded as a set of lines in   n π : U˜ → U G → G , where G and G are algebraic groups C . Let denote the projection π(z,l) = z π −1(p) P n−1 defined over k, that is a group isomorphism, . Identify with and U˜ \π −1(p) U\{p} π whose inverse is a k-morphism. with , via the map and set

N˜ = (N\{p}) ∪ U,B˜ p(N) = N/˜ ∼ , birational mapping For V and W irreducible algebraic varieties defined over k, a closed irre- where z ∼ w if z ∈ N\{p} and w = (z, l) ∈ U˜ . ducible subset T of V × W where the closure of The blowing up of N at p is π : Bp(N) → N. the projection T → V is V , the closure of the projection T → W is W, and k(V ) = k(T ) = BN-pair A pair of subgroups (B, N) of a k(W). Also called birational transformation. group G such that: (i.) B and N generate G; birational transformation See birational (ii.) B ∩ N = H*N; and mapping. (iii.) the group W = N/H has a set of genera- tors R such that for any r ∈ R and any w ∈ W Birch-Swinnerton-Dyer conjecture The (a) rBw ⊂ BwB ∩ BrwB, rank of the group of rational points of an el- (b) rBr = B. liptic curve E is equal to the order of the 0 of L(s, E) at s = 1. Consider the Bochner’s Theorem A function, defined on E : y2 = x3 − ax − b where a and b are inte- R, is a Fourier-Stieltjes transform if and only if it gers. If E(Q) = E ∩ (Q × Q), by Mordell’s is continuous and positive definite. [A function Theorem E(Q) is a finitely generated Abelian f , defined on R, is defined to be positive definite group. Let N be the conductor of E, and if if  p | N, let ap + p be the number of solutions of f(y)f(x− y)dy > 0 y2 = (x3 − ax − b) (mod p). The L-function R of E, for all x-values.]   −s L(s, E) = 1 − εpp Borel subalgebra A maximal solvable sub- p|N algebra of a reductive Lie algebra defined over   an algebraically closed field of characteristic 0. −s 1−2s 1 − app + p p|N Borel subgroup A maximal solvable sub- group of a complex, connected, reductive Lie where εp = 0or ± 1. group. block A term used in reference to vector bun- Borel-Weil Theorem If Gc is the complex- dles, permutation groups, and representations. ification of a compact connected group G,any

c 2001 by CRC Press LLC irreducible holomorphic representation of Gc is bounded variation Let I =[a,b]⊆R be holomorphically induced from a one-dimension- a closed interval and f : I → R a function. al holomorphic representation of a Borel sub- Suppose there is a constant C>0 such that, for group of Gc. any partition a = a0 ≤ a1 ≤ ··· ≤ ak = b it holds that boundary (1) (Topology.) The intersection k of the complements of the interior and exterior |f(a ) − f(a )|≤C. of a set is called the boundary of the set. Or, j j−1 j= equivalently, a set’s boundary is the intersection 1 of its closure and the closure of its complement. Then f is said to be of bounded variation on the (2) (Algebraic Topology.) A boundary in interval I. a differential group C (an Abelian group with ∂ : C → C ∂∂ = homomorphism satisfying 0) bracket product If a and b are elements of ∂ is an element in the range of . a ring R, then the bracket product is defined as [a,b]=ab − ba. The bracket product satisfies ∂ boundary group The group Im , which is a the distributive law. subgroup of a differential group C consisting of the image of the boundary operator ∂ : C → C. branch and bound integers programming At step j of branch and bound integers pro- boundary operator A homomorphism gramming for a problem list P a subproblem Pj ∂ : C → C of an Abelian group C that satisfies is selected and a lower bound is estimated for its ∂∂ = 0. Used in the field of algebraic topology. optimal objective function. If the lower bound See also boundary, boundary group. is worse than that calculated at the previous step, then Pj is discarded; otherwise Pj is separated bounded homogeneous domain A bounded into two subproblems (the branch step) and the domain with a transitive group of auto- process is repeated until P is empty. morphisms. In more detail, a domain is a con- N N nected open subset of complex space C . branch divisor The divisor iXX, where A domain is homogeneous if it has a transi- iX is the differential index at a point X on a tive group of analytic (holomorphic) automor- nonsingular curve. phisms. This means that any pair of points z and w can be interchanged, i.e., φ(z) = w,by Brauer group The Abelian group formed by an invertible analytic map φ carrying the do- the tensor multiplication of algebras on the set of main onto itself. For example, the unit ball in equivalence classes of finite dimensional central 2 complex N space, {z = (z1,...,zN ) :|z1| + simple algebras. ···+|zN |2 < 1}, is homogeneous. A domain is bounded if it is contained in a ball of finite Brauer’s Theorem Let G be a finite group radius. A bounded homogeneous domain is a and let χ be any character of G. Then χ can be bounded domain which is also homogeneous. written as nkχψk , where nk is an integer and N Thus, the unit ball in C is a bounded homo- each χψk is an induced character from a certain geneous domain. There are many others. See linear character ψk of an elementary subgroup also Siegel domain, Siegel domain of the sec- of G. ond kind. Bravais class An arithmetic crystal class de- bounded matrix A continuous linear map termined by (L, B(L)), where L is a lattice and K : ?2(N) ⊗ ?2(N) → ?1(N) where N is the set B(L) is the Bravis group of L. See Bravais of natural numbers. group. bounded torsion group A torsion group T Bravais group The group of all orthogonal where there is an integer n ≥ 0 such that tn = 1 transformations that leave invariant a given lat- for all t ∈ T . tice L.

c 2001 by CRC Press LLC Bravais lattice A representative of a Bravais apartments of C) such that every two simplices type. See Bravais type. of C belong to an apartment and if A, B are in S, then there exists an isomorphism of A onto Bravais type An equivalence class of arith- B that fixes A ∩ B elementwise. metically equivalent lattices. See arithmetical equivalence. building of Euclidian type A building is of Euclidean type if it could be used like a sim- Brill-Noether number The quantity g − plical decomposition of a Euclidean space. See (k + 1)(g − k + m), where g is the genus of building. a nonsingular curve C and k and m are posi- tive integers with k ≤ g. This quantity acts as a building of spherical type A building that lower bound for the dimension of the subscheme has finitely many chambers. See building. ϕ(D) : l(D)>m,deg D = k of the Jacobian variety of C, where ϕ is the canonical function Burnside Conjecture A finite group of odd from C to this variety. order is solvable.

Bruhat decomposition A decomposition of Burnside problem (1) The original Burn- a connected semisimple algebraic group G,asa side problem can be stated as follows: If every union of double cosets of a Borel subgroup B, element of a group G is of finite order and G with respect to representatives chosen from the is finitely generated, then is G a finite group? classes that comprise the Weyl group W of G. Golod has shown that the answer for p-groups For each w ∈ W, let gw be a representative in is negative. the normalizer N(B∩ B−) in G of the maximal (2) Another form of the Burnside problem is: torus B ∩ B− formed from B and its opposite If a group G is finitely generated and the orders subgroup B−. Then G is the disjoint union of of the elements of G divide an integer n, then is the double cosets BgwB as w ranges over W. G finite? building A thick chamber complex C with a system S of Coxeter subcomplexes (called the

c 2001 by CRC Press LLC canonical coordinates of the first kind For each basis B1,...,Bn of a Lie algebra L of the Lie group G, there exists a positive real number r with the property that {exp( biBi) : C |bi|

c 2001 by CRC Press LLC (2) Let the ring R = i∈I Ri be the direct cap product (1) In a lattice or Boolean alge- product of rings Ri. The mapping φi : R → Ri bra, the fundamental operation a ∧b, also called that assigns to each element r of R its ith com- the meet or product, of elements a and b. ponent ri is called a canonical homomorphism (2) In cohomology theory, where Hr (X, Y ; (of direct product of rings). G) and H s(X, Y ; G) are the homology and co- homology groups of the pair (X, Y ) with coef- G canonical injection For a subgroup H of a ficients in the group , the operation that as- G θ H → sociates to the pair (f, g), f ∈ Hr+s(X, Y ∪ group , the injective homomorphism : Z; G ), g ∈ H r (X, Y ; G ) f ∪g ∈ G θ(h) = h h ∈ H θ 1 2 the element , defined by for all .( is also H (X, Y ∪ Z; G ) called the natural injection.) s 3 determined by the composi- tion

0 canonically bounded complex Let F (C) Hr+s (X, Y ∪ Z; G1) and F m+1(C) (m an integer) be subcomplexes −→ Hr+s ((X, Y ) × (X, Z); G1) of a complex C such that F 0(C) = C, and −→ H r (X, Y ; G ) ,H (X, Z; G ) F m+1(Cm) = 0, then the complex C is called a Hom 2 s 3 canonically bounded complex. where the first map is induced by the diagonal map 0 : (X, Y ∪ Z) → (X, Y ) × (X, Z). canonically polarized Jacobian variety A pair, (J, P ), where J is a Jacobian variety whose Cardano’s formula A formula for the roots polarization P is determined by a theta divisor. of the general over the complex numbers. Given the cubic equation ax3 +bx2 + cx + d = 0, let A = 9abc − 2b3 − 27a2d canonical projection Let S/ ∼ denote the and B = b2 − 3ac. Also, let y and y be set of equivalence classes of a set S, with re- 1 2 solutions of the quadratic equation Y 2 − AY + spect to an equivalence relation ∼. The mapping B3 = 0. If ω is any root of 1, then (−b+ω µ:S → S/ ∼ that carries s ∈ S to the equiva- √ √ 3 y +ω2 3 y )/3a is a root of the original cubic lence class of s is called the canonical projection 1 2 equation. (or quotient map). cardinality A measure of the size of, or num- canonical surjection (1) Let H be a normal ber of elements in, a set. Two sets S and T are subgroup of a group G. For the factor group said to have the same cardinality if there is a G/H , the surjective homomorphism θ:G → function f : S → T that is one-to-one and G/H such that g ∈ θ(g), for all g ∈ G, is called onto. See also countable, uncountable. the canonical surjection (or natural surjection) to the factor group. Cartan integer Let R be the root system of a Lie algebra L and let F ={x1,x2,...,xn} be (2) Let G = G × G × ...× Gn be the di- 1 2 a fundamental root system of R. Each of the rect product of the groups G1,G2,... ,Gn. The n2 integers xij =−2(xi,xj )/(xj ,xj )(1 ≤ i, mapping (g ,g ,... ,gn) → gi (i = 1, 2,... , 1 2 j ≤ n) is called a Cartan integer of L, relative n) from G to Gi is a surjective homomorphism, to the fundamental root system F . called the canonical surjection on the direct prod- uct of groups. Cartan invariants Let G be a finite group and let n be the number of p-regular classes capacity of prime ideal Let A be a separable of G. Then, there are exactly n nonsimilar, algebra of finite degree over the field of absolutely irreducible, modular representations, of a Dedekind domain. Let P be a prime ideal M1,M2,...,Mn,ofG. Also, there are n non- of A and let M be a fixed maximal order of similar, indecomposable components, denoted A. Then, M/P is the matrix algebra of degree by R1,R2,..., Rn, of the regular representation d over a division algebra. This d is called the R of G. These can be numbered in a such a way capacity of the prime ideal P . that Mn appears in Rn as both its top and bottom

c 2001 by CRC Press LLC component. If the degree of Mn is mn and the Cartan-Weyl Theorem A theorem that as- degree of Rn is rn, then Rn appears mn times sists in the characterization of irreducible repre- in R and Mn appears rn times in R. The multi- sentations of complex semisimple Lie algebras. plicities µnt of Mt in Rn are called the Cartan Let G be a complex semisimple Lie algebra, H a invariants of G. Cartan subalgebra, 8 the root system of G rela- tive to H, α = σ∈8 rσ σ , rσ ∈ R, a complex- H ρ : G → Cartan Let G be a connected valued linear functional on , and ( ) G α semisimple Lie group with finite center and let GLn C a representation of . The functional M be a maximal compact subgroup of G. Then is a weight of the representation if the space of v ∈ n ρ(h)v = α(h)v there exists a unique involutive automorphism of vectors C that satisfy for h ∈ H n G whose fixed point set coincides with M. This all is nontrivial; C decomposes as a di- automorphism is called a Cartan involution of rect sum of such spaces associated with weights α ,...,α the Lie group G. 1 k. If we place a lexicographic linear order ≤ on the set of functionals α, the Cartan- Weyl Theorem asserts that there exists an irre- Cartan-Mal’tsev-Iwasawa Theorem Let M ducible representation ρ of G having α as its be a maximal compact subgroup of a connected highest weight (with respect to the order ≤)if Lie group G. Then M is also connected and G is [α,σ] and only if 2 is an integer for every σ ∈ 8, homeomorphic to the direct product of M with [σ,σ] n and w(α) ≤ α for every permutation w in the a Euclidean space R . Weyl group of G relative to H.

Cartan’s criterion of semisimplicity A Lie Carter subgroup Any finite algebra L is semisimple if and only if the Killing contains a self-normalizing, nilpotent subgroup, form K of L is nondegenerate. called a Carter subgroup.

Cartan’s criterion of solvability Let gl(n, Cartesian product If X and Y are sets, then K) be the general linear Lie algebra of degree the Cartesian product of X and Y , denoted X × n over a field K and let L be a subalgebra of Y , is the set of all ordered pairs (x, y) with x ∈ X gl(n, K). Then L is solvable if and only if and y ∈ Y . tr(AB) = 0(tr(AB) = trace of AB), for every A ∈ L and B ∈[L, L]. Cartier divisor A divisor which is linearly equivalent to the divisor 0 on a neighborhood of each point of an irreducible variety V . Cartan’s Theorem (1) E. Cartan’s Theo- rem. Let W1 and W2 be the highest weights of Casimir element Let β ,...,βn be a basis of irreducible representations w ,w of the Lie al- 1 1 2 the semisimple Lie algebra L. Using the Killing gebra L, respectively. Then w is equivalent to 1 form K of L, let mij = K(βi,βj ). Also, let w if and only if W1 = W2. ij 2 m represent the inverse of the matrix (mij ) and (2) H. Cartan’s Theorem. The sheaf of let c be an element of the quotient associative ij ideals defined by an analytic subset of a com- algebra Q(L), defined by c = m βiβj . This plex manifold is coherent. element c is called the Casimir element of the semisimple Lie algebra L. Cartan subalgebra A subalgebra A of a Lie algebra L over a field K, such that A is nilpotent Casorati’s determinant The n × n determi- and the normalizer of A in L is A itself. nant D(c (x),...,c (x)) = 1 n Cartan subgroup A subgroup H of a group c (x) c (x) ··· cn(x) 1 2 G such that H is a maximal nilpotent subgroup c (x + )c(x + ) ··· cn(x + ) 1 1 2 1 1 of G and, for every subgroup K of H of finite , ··· index in H, the normalizer of K in G is also of K finite index in . c (x + n − )c(x + n − ) ··· cn(x + n − ) 1 1 2 1 1

c 2001 by CRC Press LLC If A = an and B = bn, then cn = AB c (x),...,c (x) n where 1 n are solutions of the (if all three series converge). The Cauchy prod- homogeneous linear difference equation uct series converges if an and bn converge n and at least one of them converges absolutely pk(x)y(x + k) = 0 . (Merten’s Theorem). k=0 Cauchy sequence (1) A sequence of real {r } casting out nines A method of checking numbers, n , satisfying the following condi- B> base-ten and divisions. See ex- tion. For any 0 there exists a positive inte- N |r −r | N cess of nines. ger such that m n , for all . (2) A sequence {pn} of points in a metric casus irreducibilis If the cubic equation space (X, ρ), satisfying the following condition: ax3 + bx2 + cx + d = 0 is irreducible over ρ (p ,p ) → n, m →∞. the extension Q(a,b,c,d)of the rational num- n m 0as ber field Q, and if all the roots are real, then it is still impossible to find the roots of this cubic Cauchy sequences are also called fundamen- equation, by only rational operations with real tal sequences. radicals, even if the roots of the cubic equation are real. Cauchy transform The Cauchy transform, µˆ , of a measure µ, is defined by µ(ζˆ ) = (z − − category A graph equipped with a notion ζ) 1dµ(z). If K is a compact planar set with of identity and of composition satisfying certain connected complement and A(K) is the algebra standard domain and range properties. of complex functions analytic on the interior of K, then the Cauchy transform is used to show Cauchy inequality The inequality that every element of A(K) can be uniformly approximated on K by polynomials. n 2 n n 2 2 aibi ≤ a b , i i Cayley algebra Let F be a field of charac- i=1 i=1 i=1 teristic zero and let Q be a quaternion algebra a ,...,a ,b ,...,b for real numbers 1 n 1 n. Equal- over F .Ageneral Cayley algebra isatwo- a = cb c ity holds if and only if i i, where is a dimensional Q-module Q + Qe with the mul- constant. tiplication (x + ye)(z + ue) = (xz + vuy) + (xu + yz)e, where x,y,z,u ∈ Q, v ∈ F and n Cauchy problem Given an th order partial z, u are the conjugate quarternions of z and z differential equation (PDE) in with two inde- u, respectively. A Cayley algebra is the special x y @ pendent variables, and , and a curve in the case of a general Cayley algebra where Q is the xy -plane, a Cauchy problem for the PDE con- quaternian field, F is the real number field, and z = φ(x,y) sists of finding a solution which v =−1. meets prescribed conditions ∂j+kz Cayley-Hamilton Theorem See Hamilton- = f ∂xj ∂yk jk Cayley Theorem. j + k ≤ n − j,k = , ,...,n− @ 1, 0 1 1on . Cayley number The elements of a general Cauchy problems can be defined for systems Cayley algebra. See Cayley algebra. of partial differential equations and for ordinary differential equations (then they are called ini- Cayley projective plane Let H be the set of tial value problems). all 3 × 3 Hermitian matrices M over the Cayley algebra such that M2 = M and tr M = 1. The Cauchy product The Cauchy product of two ∞ ∞ ∞ set H , with the structure of a projective plane, is series an and bn is cn where n=1 n=1 n=1 called the Cayley projective plane. See Cayley cn = a1bn + a2bn−1 +···+anb1 . algebra.

c 2001 by CRC Press LLC Cayley’s Theorem Every group is isomor- chain equivalent Let C1 and C2 be chain phic to a group of permutations. complexes. If there are chain mappings α: C1 → C β C → C α ◦ β = 2 and : 2 1 such that 1C2 and β ◦ α = C Cayley transformation The mapping be- 1C1 , then we say that 1 is chain equiv- tween n × n matrices N and M, given by M = alent to C2. See chain complex, chain mapping. (I − N)(I + N)−1, which acts as its own in- verse. The Cayley transformation demonstrates chain homotopy Let C1 and C2 be chain a one-to-one correspondence between the real complexes. Let α, β: C1 → C2 be two chain alternating matrices N and proper orthogonal mappings, and let R be a ring with identity. If matrices M with eigenvalues different from −1. there is an R-homomorphism γ : C1 → C2 of degree 1, such that α−β = γ ◦α +β ◦γ , where   CCR algebra A C*-algebra A, which is (C1,α) and (C2,β ) are chain complexes over mapped to the algebra of compact operators un- R. Then γ is called a chain homotopy of α to der any irreducible ∗-representation. Also called β. See chain complex, chain mapping. liminal C*-algebra. chain mapping Let (C1,α) and (C2,β) be center (1) Center of symmetry in Euclidean chain complexes over a ring R with identity. An geometry. The midpoint of a line, center of a R-homomorphism γ : C1 → C2 of degree 0 triangle, circle, ellipse, regular polygon, sphere, that satisfies β ◦ γ = γ ◦ α is called a chain ellipsoid, etc. mapping of C1 to C2. See chain complex. (2) Center of a group, ring, or Lie algebra X. The set of all elements of X that commute with chain subcomplex Let R be a ring with iden- every element of X. tity and let (C, α) be a chain complex over R.If (3) Center of a lattice L. The set of all central H = n Hn is a homogeneous R-submodule elements of L. of C such that α(H) ⊂ H , then H is called a chain subcomplex of C. See chain complex. central extension Let G, H, and K be groups such that G is an extension of K by H .IfH is Chain Theorem Let A, B, and C be alge- contained in the center of G, then G is called a braic number fields such that C ⊂ B ⊂ A and central extension of H. let 0A/C, 0A/B, and 0B/C denote the relative difference of A over C, A over B, and B over C, centralizer Let X be a group (or a ring) and respectively. Then 0A/C = 0A/B0B/C. See let S ⊂ X. The set of all elements of X that different. commute with every element of S is called the centralizer of S. chamber In a finite dimensional real affine space A, any connected component of the com- central separable algebra An R-algebra plement of a locally finite union of hyperplanes. which is central and separable. Here a central See locally finite. R-algebra A which is projective as a two-sided A-module, where R is a commutative ring. chamber complex A complex with the prop- erty that every element is contained in a chamber central simple algebra A simple algebra A and, for two given chambers C, C, there exists over a field F , such that the center of A coincides a finite sequence of chambers C = C0,C1,..., F C = C (C ∩ with . (Also called normal simple algebra.) r in such a way that codimCk−1 k Ck−1) = codimCk (Ck ∩ Ck−1) ≤ 1, for k = chain complex Let R be a ring with iden- 1, 2,... ,r. See chamber. tity and let C be a unitary R-module. By a chain complex (C, α) over R we mean a graded character A character X of an Abelian group R-module C = n Cn together with an R- G is a function that assigns to each element x of homomorphism α: C → C of degree −1, where G a complex number X (x) of absolute value 1 α ◦ α = 0. such that X (xy) = X (x)X (y) for all x and y in

c 2001 by CRC Press LLC G.IfG is a topological Abelian group, then X with constant coefficients ai (i = n−1,...,n− must be continuous. k) and then divide by λn−k, we obtain λk + a λk−1 +···+a λ + a = , character group The set of all characters n−1 n−k+1 n−k 0 of a group G, with addition defined by (X + 1 which is again called the characteristic equation X )(x) = X (x)·X (x). The character group is 2 1 2 associated with this given difference equation. Abelian and is sometimes called the dual group (3) The above two definitions can be extended of G. See character. for a system of linear differential (difference) characteristic Let F be a field with identity equations. M = (m ) 1. If there is a c such that c1 = (4) Moreover, if ij is a square ma- n F 1 +···+1 (c 1s) = 0, then the smallest such c trix of degree over a field , then the algebraic |λI − M|= is a p, called the characteristic of equation 0 is also called the char- M the field F . If there is no natural number c such acteristic equation of . that c1 = 0, then we say that the characteristic characteristic linear system Let S be a non- of the field F is 0. singular surface and let A be an irreducible al- characteristic class (1)OfanR-module gebraic family of positive divisors of dimension d S A extension 0 → N → X → M → 0, on such that a generic member M of is an irreducible non-singular curve. Then, the char- the element 00(idN ) in the extension module 1 acteristic set forms a (d − 1)-dimensional linear ExtR(M, N), where idN is the identity map on ∼ system and contains TrM |M| (the trace of |M| on N in HomR(N, N) = Ext0 (N, N) and 00 is R M) as a subfamily. This linear system is called the connecting homomorphism Ext0 (N, N) → R the characteristic linear system of A. 1 ExtR(M, N) obtained from the extension se- quence. See connecting homomorphism. characteristic multiplier Let Y(t)be a fun- X (2) Of a vector bundle over base space , damental matrix for the differential equation any of a number of constructions of a particu-  lar cohomology class of X, chosen so that the y = A(t)y . (∗) bundle induced by a map f : Y → X is the ω A(t) image of the characteristic class of the bundle Let be a period for the matrix . Suppose H over X under the associated cohomological map that is a constant matrix that satisfies f ∗ : H ∗(X) → H ∗(Y ) . See Chern class, Euler Y(t + ω) = Y(t)H, t ∈ (−∞, ∞). class, Pontrjagin class, Stiefel-Whitney class, Thom class. Then an eigenvalue µ for H of index k and mul- tiplicity m is called a characteristic multiplier characteristic equation (1) If we substitute (∗) A(t) λx for , or for the periodic matrix , of index y = e in the general nth order linear differen- k and multiplicity m. tial equation (n) (n−1) characteristic multiplier Let Y(t)be a fun- y (x) + an−1y (x) + ...  damental matrix for the differential equation +a1y (x) + a0y(x) = 0 y = A(t)y . (∗) with constant coefficients ai (i = n − 1,...,0) λx and then divide by e , we obtain Let ω be a period for the matrix A(t). Suppose n n−1 that H is a constant matrix that satisfies λ + an−1λ +···+a1λ + a0 = 0 , which is called the characteristic equation asso- Y(t + ω) = Y(t)H, t ∈ (−∞, ∞). ciated with the given differential equation. n Then an eigenvalue µ for H of index k and mul- (2) If we substitute yn = λ in the general tiplicity m is called a characteristic multiplier, kth order difference equation of index k and multiplicity m, for (∗), or for the yn + an−1yn−1 +···+an−kyn−k = 0 periodic matrix A(t).

c 2001 by CRC Press LLC characteristic of logarithm The integral part Chebotarev Density Theorem Let F be an of the common logarithm. algebraic number field with a subfield f , F/f be a Galois extension, C be a conjugate class of characteristic polynomial The polynomial the Galois group G of F/f, and I(C)be the set on the left side of a characteristic equation. See of all prime ideals P of k such that the Frobenius characteristic equation. automorphism of each prime factor Fi of P in F is in C. Then the density of I(C)is |C|/|G|. characteristic series Let G be a group. If we take the group Aut(G) (the group of automor- Chern class The ith Chern class is an ele- phisms of G) as an operator domain of G, then ment of H 2i(M; R), where M is a complex man- a composition series is called a characteristic ifold. The Chern class measures certain proper- series. See composition series. ties of vector bundles over M. It is used in the Riemann-Roch Theorem. characteristic set A one-dimensional set of positive divisors D of a nonsingular curve of Chevalley complexification Let G be a com- dimension n so that, with respect to one such pact Lie group, r(G) the representative ring of generic divisor D of the curve, the degree of 0 G, A the group of all automorphisms of r(G), the specialization of the intersection D ·D over  0 and G the centralizer of a subgroup of A in A. the specialization of D over D is a divisor of  0 If G is the closure of G relative to the Zariski degree equal to that of D · D .   0 topology of G , then G is called the Chevalley complexification of G. character module Let G be an algebraic group, with the sum of two characters X1 and X2 of G defined as (X + X )(x) = X (x) · X (x), Chevalley decomposition Let G be an alge- 1 2 1 2 F R for all x ∈ G. The set of all characters of G braic group, defined over a field and u the G F forms an additive group, called the character unipotent radical of .If is of characteristic module of G. See character of group, algebraic zero, then there exists a reductive, closed sub- C G G group. group of such that can be written as a of C and Ru. See algebraic character of a linear representation For the group. representation ρ : A → GLn(k) of the algebra A over a field k, the function χρ on A given by Chevalley group Let F be a field, f an ele- χρ(a) = tr(ρ(a)). ment of F , LF a Lie algebra over F , B a basis of LF over F and tθ (f ) the linear transformation character of group A rational homomor- of LF with respect to B, where θ ranges over the phism α of an algebraic group G into GL(1), root system of LF . Then, the group generated where GL(1) is a one-dimensional connected by the tθ (f ), for each root θ and each element algebraic group over the prime field. See alge- f , is called the Chevalley group of type over F . braic group. Chevalley’s canonical basis Of a complex, character system For the quadratic field k, semisimple Lie algebra G with Cartan subalge- d I ∼ with discriminant and ideal class group = bra H and root system 8, a basis for G consisting F/H F H ( the group of fractional ideals and the of a basis H1, ...,H s of H and, for each root subgroup of principal ideals generated by posi- σ ∈ 8, a basis Xσ of its root subspace Gσ χ (N(A)) tive elements), a collection p (p|d) of that satisfy: (i.) σ(Hi) is an integer for every 2 numbers, indexed by the prime factors of d,in σ ∈ 8 and each Hi; (ii.) β(Xσ ,X−σ ) = (σ,σ ) which χp is the Legendre symbol mod p and for every σ ∈ 8, where ( , ) represents the inner A is any representative ideal in its ideal class product on the roots induced by the Killing form mod H. The character system is independent of β on G; (iii.) if σ , τ, and σ + τ are all roots and the choice of representative and uniquely deter- [Xσ ,Xτ ]=nσ,τ Xσ+τ , then the numbers nσ,τ mines each class in I. are integers that satisfy n−σ,−τ =−nσ,τ .

c 2001 by CRC Press LLC Chevalley’s Theorem Let G be a connected difficulties as Russell’s paradox, concerning the algebraic group, defined over a field F , and let set of all sets that do not belong to themselves. N bea(F -closed) largest, linear, connected, closed, normal subgroup of G.IfC is a closed, class field Let F be an algebraic number field normal subgroup of G, then the factor group and E be a Galois extension of F . Then, E G/C is complete if and only if N ⊂ C. is said to be a class field over F , for the ideal group I (G), if the following condition is met: a Cholesky method of factorization A method prime ideal P of F of absolute degree 1 which is of factoring a positive definite matrix A as a relatively prime to G is decomposed in E as the product A = LLT where L is a lower triangular product of prime ideals of E of absolute degree matrix. Then the solution x of Ax = b is found 1 if and only if P is in I (G). by solving Ly = b, LT x = y. class field theory A theory created by E. Choquet boundary Let X be a compact Artin and others to determine whether certain Hausdorff space and let A be a function algebra primes are represented by the principal form. on X. The Choquet boundary is c(A) ={x ∈ F X : the evaluation at x has a unique representing class field tower problem Let beagiven F = F ⊂ F ⊂ measure}. algebraic number field, and let 0 1 F2 ⊂ ··· be a sequence of fields such that Fn F F Chow coordinates Of a projective variety is the absolute class field over n−1, and ∞ is X the union of all Fn. Now we ask, is F∞ a finite , the coefficients of the associated form of the F variety, viewed as homogeneous coordinates of extension of ? The answer is positive if and Fk k See points on X. See associated form. only if is of class number 1 for some . absolute class field. Chow ring Of a nonsingular, irreducible, class formation An axiomatic structure for projective variety X, the graded ring whose ob- class field theory, developed by Artin and Tate. jects are rational equivalence classes of cycles A class formation consists of on X, with addition given by addition of cycles (1) a group G, the Galois group of the for- and multiplication induced by the diagonal map mation, together with a family GK : K ∈ Z 0 : X → X × X. The ring is graded by codi- of subgroups of G indexed by a collection 8 of mension of cycles. fields K so that V Chow variety Let be a projective variety. (i.) each GK has finite index in G; The set of Chow coordinates of positive cycles that are contained in V is a projective variety (ii.) if H is a subgroup of G containing some  called a Chow variety. GK , then H = GK for some K ; {G } circulant See cyclic determinant. (iii.) the family K is closed under intersec- tion and conjugation; circular units The collection of units of the s G G 1−ζ ζ pn p (iv.) 8 K is the trivial subgroup of ; form 1−ζ t , where is a th root of unity, is a prime, and s ≡ t(modp) (and p |s,t). (2)aG-module A, the formation module, such that A is the union of its submodules A(GK ) class (1)(Algebra.) A synonym of set that that are fixed by GK ; is used when the members are closely related, (3) cohomology groups H r (L/K), defined r (G ) like an equivalence class or the class of residues as H (GK /GL,A K ), for which H 1(L/K) modulo m. = 0 whenever GL is normal in GK ; (2)(.) A generalization of set, includ- (4) for each field K, there is an isomorphism ing objects that are “too big” to be sets. Con- A → invK A of the Brauer group H 2(∗ /K) sideration of classes allows one to avoid such into Q/Z such that

c 2001 by CRC Press LLC (i.) if GL is normal in GK of index n, clearing of An equation is cleared of fractions if both sides are multiplied by a com- 2 1 invK H (L/K) = Z /Z mon denominator of all fractions appearing in n the equation.

and Clebsch-Gordon coefficient One of the co- efficients, denoted (ii.) even when GL is not normal in GK , j1m1j2m2 j1j2jm invL ◦ resK,L = n invK in the formula where resK,L is the natural restriction map ψ(jm) = (j m j m j j jm) H 2(∗/K) → H 2(∗/E). 1 1 2 2 1 2 −j≤m1,m2≤j

× ψ(j1m1)ψ(j2m2) classical compact real simple Lie algebra A compact real simple Lie algebra of the type which relates the basis elements of the represen- An, Bn, Cn,orDn, where An, Bn, Cn, and Dn tation space C2 ⊗···⊗C2 of 2j copies of C2 are the Lie algebras of the compact Lie groups for a representation of SO(3) =∼ SU(2)/ {±I}. SU(n + 1),SO(2n + 1),Sp(n), and SO(2n), The coefficients are determined by the formula respectively. j m j m j j jm = δ 1 1 2 2 1 2 m1+m2,m classical compact simple Lie group Any of (n + ) ( j + )(j + j − j)!(j + j − j )!(j + j − j )! the connected compact Lie groups SU 1 , × 2 1 1 2 1 2 2 1 (j + j + j + )! SO(nl +1),Sp(n),orSO(2n), with correspond- 1 2 1 ing compact real simple Lie algebra An (n ≥ 1), ν (j1 + m1)!(j1 − m1)!(j2 + m2)! Bn (n ≥ 2), Cn (n ≥ 3),orDn (n ≥ 4) as its × (−1) ν ν!(j1 + j2 − j − ν)!(j1 − m1 − ν)! Lie algebra. (j − m )!(j + m)!(j − m)! × 2 2 . (j + m − ν)!(j − j + m + ν)!(j − j − m + ν)! classical complex simple Lie algebra Let 2 2 2 1 1 2 An, Bn, Cn, and Dn be the Lie algebras of the complex Lie groups SL(n + 1,C),SO(2n + L n 1,C),Sp(n, C), and SO(2n, C). Then An (n ≥ Clifford algebra Let be an -dimensional F Q 1), Bn (n ≥ 2), Cn (n ≥ 3), and Dn (n ≥ 4) are linear space over a field , a quadratic form on called classical complex simple Lie algebras. L, A(L) the tensor algebra over L, I(Q) the two- sided ideal on A(L) generated by the elements classical group Groups such as the general l ⊗ l − Q(l) · 1 (l ∈ L), where ⊗ denotes tensor linear groups, orthogonal groups, symplectic multiplication. The quotient associative algebra groups, and unitary groups. A(L)/I (Q) is called the Clifford algebra over Q. classification Let R be an equivalence rela- tion on a set S. The partition of S into disjoint Clifford group Let L be an n-dimensional union of equivalence classes is called the clas- linear space over a field F , Q a quadratic form sification of S with respect to R. on L, C(Q) the Clifford algebra over Q, G the set of all invertible elements g in C(Q) such that class number The order of the ideal class gLg−1 = L. Then, G is a group with respect group of an algebraic number field F . Similarly, to the multiplication of C(Q) and is called the the order of the ideal class group of a Dedekind Clifford group of the Quadratic form Q. See domain D is called the class number of D. Clifford algebra. class of curve The degree of the tangential Clifford numbers The elements of the Clif- equation of a curve. ford algebra. See Clifford algebra.

c 2001 by CRC Press LLC closed boundary If X is a compact Haus- (k), M) is bijective for any algebraically closed dorff space, then a closed boundary is a bound- field k, and ary closed in X. (ii.) for any scheme N and any natural transfor- mation ψ : M → Hom(−,N)there is a unique closed image Let µ:V → V  be a morphism natural transformation λ : Hom(−,M) → of varieties. If V is not complete, then µ(V ) Hom(−,N)such that ψ = λ ◦ ϕ. may not be closed. The closure of µ(V ) which is in V , is called the closed image of V . coarser classification For R,S ⊂ X×X two equivalence relations on X, S is coarser than R closed subalgebra A subalgebra B1 of a Ba- if R ⊂ S. nach algebra B that is closed in the norm topol- ogy. B1 is then a Banach algebra, with respect coboundary See cochain complex. to the original algebraic operations and norm of B. coboundary operator See cochain complex. closed subgroup A subgroup H of a group cochain See cochain complex. G such that x ∈ H, whenever some nontrivial power of x lies in H. cochain complex A cochain complex C = {Ci,δi : i ∈ Z} is a sequence of modules {Ci : closed subsystem Let M be a character mod- i ∈ Z}, together with, for each i, a module s r homomorphism δi : Ci → Ci+1 such that δi−1◦ ule. Let be a subset of a root system , and let i s be the submodule of M generated by s.If δ = 0. Diagramatically, we have s ∩ r = s, then s is called a closed subsystem. δi−2 δi−1 δi δi+1 See character module. ...−→ Ci−1 −→ Ci −→ Ci+1 −→ ... , closed under operation A set S, with a bi- where the composition of any two successive nary operation ∗ such that a ∗ b ∈ S for all δj is zero. In this context, the elements of Ci a,b ∈ S. are called i-cochains, the elements of the ker- nel of δi, i-cocycles, the elements of the im- closure property The property of a set of be- age of δi−1, i-coboundaries, the mapping δi, ing closed under a binary operation. See closed the ith coboundary operator, the factor module under operation. H i(C) = ker δi/im δi−1, the ith cohomology module and the set H(C) ={H i(C) : i ∈ Z},  coalgebra Let ρ and ρ be linear mappings the cohomology module of C.Ifc is an i-cocycle  i defined as ρ : V → V ⊗F V , and ρ : V → (i.e., an element of ker δ ), the corresponding el- F , where F is a field, V is vector space over ement c + im δi−1 of H i(C) is called the coho- F , and ⊗ denotes tensor product. Then, the mology class of c.Twoi-cocycles belong to the  triple (V,ρ,ρ ) is said to be a coalgebra over same cohomology class if and only if they dif- F , provided (1V ⊗ ρ) ◦ ρ = (ρ ⊗ 1V ) ◦ ρ and fer by an i-coboundary; such cocycles are called   (1V ⊗ ρ ) ◦ ρ = (ρ ⊗ 1V ) ◦ ρ = 1V . cohomologous. coalgebra homomorphism A k-linear map cochain equivalence Two cochain com-     f : C → C , where (C,0,ε)and (C ,0,ε) plexes C and C˜ are said to be equivalent if are coalgebras over a field k, such that (f ⊗f)◦ there exist cochain mappings φ : C → C˜ and   0 = 0 ◦ f and ε = ε ◦ f . ψ : C˜ → C such that φψ and ψφ are homo- topic to the identity mappings on C and C˜ , re- coarse moduli scheme A scheme M and a spectively. natural transformation ϕ : M → Hom(−,M) where M is a contravariant functor from cochain homotopy Let C ={Ci,δi : i ∈ Z} schemes to sets such that and C˜ ={C˜ i,Bi : i ∈ Z} be two cochain com- (i.) ϕ(Spec(k)) : M(Spec(k)) → Hom (Spec plexes and let φ,ψ : C → C˜ be two cochain

c 2001 by CRC Press LLC mappings. A homotopy ζ : φ → ψ is a se- coefficient of equation A number or constant quence of mappings {ζ i : i ∈ Z} such that, for appearing in an equation. (For example, in the each i, ζ i is a homomorphism from Ci to C˜ i and equation 2 tan x = 3x + 4, the coefficients are 2, 3, and 4.) φi − ψi = δiζ i+1 + ζ iBi−1 . coefficient of linear representation When a When such a homotopy exists, φ and ψ are said representation χ (of a group or ring) is isomor- to be homotopic and H(φ) = H(ψ), where phic to a direct sum of linear representations and H(φ) and H(ψ) are the morphisms from the other irreducible representations, the number of cohomology module H(C) to the cohomology ˜ times that a particular linear or irreducible rep- module H(C) induced by φ and ψ. resentation φ occurs in the direct sum is called the coefficient of φ in χ. cochain mapping Let C ={Ci,δi : i ∈ } ˜ ={C˜ i,Bi : i ∈ } Z and C Z be two cochain coefficient of polynomial term In a polyno- φ : → ˜ complexes. A cochain mapping C C is mial a sequence of mappings {φi : i ∈ Z} such that, i i i for each i, φ is a homomorphism from C to C˜ 2 n a0 + a1x + a2x + ...+ anx , and φiBi = δiφi+1 . the constants a0,a1,... ,an are called the coef- a The mapping φi induces a homomorphism from ficients of the polynomial. More specifically, 0 a the ith cohomology module H i(C) to the ith is called the constant term, 1 the coefficient of x a x2 ... a cohomology module H i(C˜ ). Hence φ can be , 2 the coefficient of , , n the coefficient xn a regarded as inducing a morphism φ∗ from the of ; n is also called the leading coefficient. cohomology module H(C) to the cohomology module H(C˜ ). The morphism φ∗ is also de- coefficient ring Consider the set of numbers noted H(φ). This enables H(·) to be regarded as or constants that are being used as coefficients in a functor from the category of chain complexes some algebraic expressions. If that set happens to a category of graded modules; it is called the to be a ring (such as the set of integers), it is cohomological functor. called the coefficient ring. Likewise, if that set happens to be a field (such as the set of real cocommutative algebra A coalgebra C,in numbers) or a module, it is called the coefficient which the comultiplication 0 is cocommutative, field or coefficient module, respectively. i.e., has the property that δτ = δ, where τ is the A n×n (k, P) flip mapping, i.e., the mapping from C ⊗ C to cofactor Let be an matrix. The A A (− )k+P itself that interchanges the two C factors. cofactor of written kP is 1 times the determinant of the n−1×n−1 matrix obtained cocycle See cochain complex. by deleting the kth row and jth column of A. codimension Complementary to the dimen- The trigonmetric function that is sion. For example, if X is a subspace of a vector the function of the complementary angle. For space V and V is the direct sum of subspaces X example, cotan is the cofunction of tan. and X, then the dimension of X is the codi- mension of X. cogenerator An element A of category C such that the functor Hom(−,A) : C → A is coefficient A number or constant appearing faithful where A is the category of sets. Also in an . (For example, in called a coseparator. 3 + 4x + 5x2, the coefficients are 3, 4, and 5.) Cohen’s Theorem A ring is Noetherian if coefficient field See coefficient ring. and only if every prime ideal has a finite basis. There are several other theorems which may be coefficient module See coefficient ring. called Cohen’s Theorem.

c 2001 by CRC Press LLC coherent algebraic sheaf A sheaf F of OV - coideal Let C be a coalgebra with comulti- modules on an algebraic variety V , such that if, plication 0 and counit B. A subspace I of C is for every x ∈ V , there is an open neighborhood called a coideal if 0(I) is contained in U of x and positive integers p and q such that the OV |U -sequence I ⊗ C + C ⊗ I

p q and B(I) is zero. OV |U → OV |U → F|U → 0 φ : M → N is exact. coimage Let be a homomor- phism between two modules M and N. Then M/ φ φ coherent sheaf of rings A sheaf of rings A the factor module ker (where ker de- φ on a topological space X that is coherent as a notes the kernel of ) is called the coimage of φ φ sheaf of A-modules. , denoted coim . By the First Isomorphism Theorem, the coimage of φ is isomorphic to the φ cohomological dimension For a group G, image of , as a consequence of which the term the number m, such that the cohomology group coimage is not often used. H i(G, A) is zero for every i>mand every G- m cokernel Let φ : M → N be a homomor- module A, and H (G, A) is non-zero for some phism between two modules M and N. Then the G-module A. factor module N/ im φ (where im φ denotes the image of φ) is called the cokernel of φ, denoted cohomology The name given to the subject coker φ. area that comprises cohomology modules, co- homology groups, and related topics. collecting terms The name given to the proc- ess of rearranging an expression so as to com- cohomology class See cochain complex. bine or group together terms of a similar nature. For example, if we rewrite cohomology functor See cochain mapping. x2 + 2x + 1 + 3x2 + 5x as 4x2 + 7x + 1 cohomology group (1) Because any Z- module is an Abelian group, a cohomology we have collected together the x2 terms and the module H(C), where C is a cochain complex, is x terms; if we rewrite called a cohomology group in the case where all the modules under consideration are Z-modules. x2+y2+2x−2y as x2 + 2x + y2 − 2y (2) Let G be a group and A a G-module. Then i H (G, A), the ith cohomology group of G with we have collected together all the terms involv- coefficients in A, is defined by ing x, and all the terms involving y. H i(G, A) = i ( ,A). ExtG Z color point group A pair of groups (K, K1) such that K = G/T , G is a space group, T is (When interpreting the Ext functor in this con- the group of translations, and for some positive G text, Z should be interpreted as a trivial - integers r, K1 is the group of conjugacy classes module.) of all subgroups G1 of G with T ⊂ G1 and [G : G1]=r. cohomology module See cochain complex. color symmetry group A pair of groups (G, cohomology set The set of cohomology G) where G is a space group and [G : G] < ∞. classes, when these classes do not possess a group structure. column finite matrix An infinite matrix with an infinite number of rows and columns, such cohomology spectral sequence See spectral that no column has any non-zero entry beyond sequence. the nth entry, for some finite n.

c 2001 by CRC Press LLC column in matrix See matrix. Here (x−1)(x+1) is the common denominator. column nullity For an m × n matrix A, the common divisor A number that divides all number m − r(A), where r(A) is the rank of A. the numbers in a list. For example, 3 is a com- mon divisor of 6, 9, and 12. See also greatest column vector A matrix with only one col- common divisor. umn, i.e., a matrix of the form   a/b a1 common A quotient of the form   a,b b =  a2  where are integers and 0.  .  .  .  a common logarithm Logarithm to the base m 10 (the logarithm that was most often used in Also called column matrix. arithmetic before electronic calcu- lators were invented). See also logarithm. combination of things When r objects are selected from a collection of n objects, the r common multiple A number that is a multi- selected objects are called a combination of r ple of all the numbers in a list. For example, 12 objects from the collection of n objects, pro- is a common multiple of 3, 4, and 6. See also vided that the selected objects are all regarded . as having equal status and not as being in any particular order. (If the r objects are put into a commutant If S is a subset of a ring R, the particular order, they are then called a permu- commutant of S is the set S ={a ∈ R : ax = tation, not a combination, of r objects from n.) xa for all x ∈ S}. Also called: commutor. The number of different combinations ofr ob- n n nC jects from is denoted by r or r and (1) The name given to n! the subject area that considers rings and modules in which multiplication obeys the commutative equals r!(n − r)!. law, i.e., xy = yx for all elements x,y. commensurable Two non-zero numbers a (2) An algebra in which multiplication obeys and b such that a = nb,√ for some rational√ num- the commutative law. See algebra. ber n. For example, 2√ 2 and 3√2 are com- = 2 ( ) commutative field A field in which multi- mensurable because 2 2 3 3 2 . All ra- tional numbers are commensurable with each plication is commutative, i.e., xy = yx, for all other. No is commensurable elements x,y. (The axioms for a field require with any rational number. that addition is always commutative. Some ver- sions of the axioms insist that multiplication has common denominator When two or more to be commutative too. When these versions of fractions are about to be added, it is helpful to re- the axioms are in use, a commutative field be- express them first, so that they have the same de- comes simply a field, and the term division ring nominator, called a common denominator. This is used for structures that obey all the field ax- process is also described as putting the fractions ioms except commutativity of multiplication.) over a common denominator. For example, to See also field. 3 − 2 simplify x−1 x+1 we might write commutative group See Abelian group. 3 − 2 x − 1 x + 1 commutative law The requirement that a bi- (x + ) (x − ) = 3 1 − 2 1 nary operation ∗, on a set X, satisfy x∗y = y∗x, (x − 1)(x + 1) (x − 1)(x + 1) for all x,y ∈ X. Addition and multiplication of x + real numbers both obey the commutative law; = 5 . (x − 1)(x + 1) matrix multiplication does not.

c 2001 by CRC Press LLC commutative ring A ring in which multi- compact simple Lie group A Lie group that plication is commutative, i.e., xy = yx, for all is compact as a topological space and whose Lie elements x,y. (The axioms for a ring ensure algebra is simple (i.e., not Abelian and having that addition is always commutative.) The real no proper invariant subalgebra). numbers are a commutative ring. Rings of ma- trices are generally not commutative. compact topological space A topological space X with the following property: whenever −1 commutator (1) An element of the form x O ={Oα}α∈A is an open covering of X, then −1 −1 −1 y xy or xyx y in a group. Such an el- there exists a finite subcovering O1,O2,...,Ok. ement is usually denoted [x,y] and it has the property that it equals the identity element if companion matrix Given the monic poly- and only if xy = yx, i.e., if and only if x and y nomial over the complex field commute. n n−1 (2) An element of the form xy −yx in a ring. p(t) = t + an−1t +···+a1t + a0 , Such an element is denoted [x.y] and is also n × n called the Lie product of x and y. It equals zero the matrix if and only if xy = yx.   ∗ 00... 0 −a0 In a ring with an involution , the element  ... −a  ∗ ∗ 10 0 1  x x − xx is often called the self-commutator  . .   .. . .  of x. A = 01 . . .  . . .  . .. .. −a  commutator group See 0 n−2 commutator sub- ... −a group. 0 01 n−1 is called the companion matrix of p(t). The commutator subgroup The subgroup G of characteristic polynomial and the minimal poly- a given group G, generated by all the commu- nomial of A are known to coincide with p(t).In tators of G, i.e., by all the elements of the form fact, a matrix is similar to the companion ma- x−1y−1xy (where x,y ∈ G). G consists pre- trix of its characteristic polynomial if and only if cisely of those elements of G that are expressible the minimal and the characteristic polynomials as a product of a finite number of commutators. coincide. So G may contain elements that are not them- selves commutators. G is a characteristic sub- complementary degree Let F be a filtra- group of G. G/G is Abelian and in fact G is tion of a differential Z-graded module A. Then the unique smallest normal subgroup of G with {FpAn} is a Z-bigraded module and the module the property that the factor group of G by it is FpAn has complementary degree q = n − p. Abelian. G is also called the commutator group G or the derived group of . complementary law of reciprocity A reci- procity law due to Hasse and superseded by commutor See commutator. Artin’s general law of reciprocity. Let p be a prime number. If α ∈ k is such that α ≡ compact group A that is 1 mod p(1 − τp) where τp = exp 2πi/p∈ k, compact, as a topological space. A topologi- G then cal group is a group , with the structure of a p = τ c , topological space, such that the map d p p (x, y) → xy−1 G × G G c = α ; from to Trk/Q p(1−τp) is continuous. 1 − τp d = τp , α p compact real Lie algebra A real Lie alge- bra whose Lie group is a compact group. See α − 1 d =−Trk/Q . compact group. p(1 − τp)

c 2001 by CRC Press LLC complementary series The irreducible, uni- complete intersection A variety V in P n(k) tarizable, non-unitary, principal series represen- of dimension r, where I(V)is generated by n−r tations of a G. homogeneous polynomials. See variety. complementary slackness This refers to complete linear system The set of all effec- Tucker’s Theorem on Complementary Slackness, tive divisors linearly equivalent to a divisor. which asserts that, for any real matrix A, the in- equalities Ax = 0,x ≥ 0 and t uA ≥ 0have complete local ring See local ring. solutions x,u satisfying At u + x>0. completely positive mapping For von Neu- mann algebras A and B, a linear mapping T : N complementary submodule Let be a sub- A → B such that for all n ≥ 1 the induced M module of a module . Then a complementary mappings submodule to N in M is a submodule N  of M M = N ⊕ N  such that . In ⊗ T : Mn(A) = Mn(C) ⊗ A → Mn(B) complementary are positive. The functions cosine, cosecant, and cotangent (cos θ, cscθ and cot θ), so called because the completely reduced module An R module cosine, cosecant, and cotangent of an acute an- (R a ring) which is the direct sum of irreducible gle θ equal, respectively, the sine, secant, and R modules. An R module is irreducible if it has tangent of the complementary angle to θ, i.e., no sub R-modules. the acute angle φ such that θ and φ form two of π k the angles in a right-angled triangle. (φ = −θ completely reducible Let be a commuta- 2 E k R in radians or 90 − θ in degrees.) tive ring and a module over . Let be a k-algebra and let ϕ : R → Endk(E) be a repre- sentation of R in E. We say that ϕ is completely complete cohomology theory A cohomol- reducible (or semi-simple)ifE is an R-direct ogy theory of the following form. Let π be a sum of R-submodules Ei, finite and B = B(Z(π)) the bar resolution. If A is a π-module define E = E1 ⊕···⊕Em , H ∗(π, A) = H ∗(B, A). See cohomology. with each Ei irreducible. complete field A field F with the following k property: whenever p(x) = a0 +a1x +···akx completely reducible representation A rep- is a polynomial with coefficients in F then p has resentation σ such that the relevant R-module E a root in F . is an R-direct sum of R-submodules Ei, E = E ⊕ E ⊕···⊕E , complete group A group G whose center 1 2 m is trivial and all its automorphisms are inner. such that each Ei is irreducible. See irreducible Thus, G and the automorphism group of G are representation. canonically isomorphic. Also called semi-simple. complete integral closure Let O be an in- completely solvable group A group that is tegral domain in a field K and M an O-module the direct product of simple groups. contained in K. Let S be the set of all val- uations on K that are nonnegative on O.If complete pivoting A process of solving an υ ∈ S, let Rυ be the valuation ring of υ. Then n × n linear system of equations Ax = b.By  M = v∈S RnM is the completion of M.IfO a succession of row and column operations, one is the integral closure of O, M is the complete may solve this equation once A has been trans- integral closure of the O-module M = OM. formed to an upper triangular matrix. Suppose

c 2001 by CRC Press LLC A(0) = A, A(1), ··· ,A(k−1) have been deter- completing the square When a quadratic (k−1) (k−1) (k−1) 2 mined so that A = (aij ). Let apq be ax + bx + c is rewritten as the entry so that b ax2 + bx + c = a x2 + x + c a 2 2 (k− ) (k− ) b b a 1 = max a 1 : i, j ≥ k . = a x + − + c pq ij 2a 4a2 b 2 b2 = a x + − − c a a Interchange the kth row and pth row and the 2 4 kth column and qth column to obtain a matrix = aX2 − d B(k−1).NowA(k) is obtained from B(k−1) by (k− ) (k− ) b b2 − ac subtracting b 1 /b 1 times the kth row of X = x + d = 4 ik kk where a and a B(k−1) from the ith row of B(k−1). 2 4 the process is called completing the square in x. It is often used when solving equations or de- complete resolution Let π be a finite mul- termining the of an expression because the tiplicative group. The bar resolution BZ((π)) absence of an X term in the final form aX2 − d of B is also called the complete free resolution. makes it easy to determine whether the expres- Here Bn is the free V-module with generators sion is positive, negative, or zero. [x1|···|xn] for all x1 = 1,...,xn = 1inπ. So, Bn is the free Abelian group generated by completion The act of enlarging a set (mini- all x[x1| ...|xn] with x ∈ π and no xi = 1. mally) to a complete space. This occurs in ring Define ∂ : Bn → Bn−1 for n>0 by setting theory, measure theory, and metric space theory.

complex (1) Involving complex numbers. See complex number. ∂ [x | ...|xn] = x [x | ...|xn] 1 1 2 (2) A set of elements from a group (not nec- n− 1 essarily forming a group in their own right). + (− )j x | ...|x x | ...|x 1 1 j j−1 n (3) A sequence of modules {Ci : i ∈ Z}, j= 1 together with, for each i, a module homomor- n i i i+1 i−1 i + (−1) x1| ...|xn−1 phism δ : C → C such that δ ◦ δ = 0. Diagramatically, we have

i−2 i−1 i i+1 ...−→δ Ci−1 −→δ Ci −→δ Ci+1 −→δ ... where [y1| ...|yn]=0 if some yi = 1. where the composition of any two successive δj complete scheme A scheme X over a field is zero. K together with a morphism from X to Spec K complex algebraic variety See algebraic va- that is proper and of finite type. See scheme. riety. complete valuation ring Let k be a field. A complex analytic geometry (1) Analytic subring R of k is called a valuation ring if, for geometry, i.e., the study of geometric shapes any x ∈ k, we have either x ∈ R or x−1 ∈ R.A through the use of coordinate systems, but valuation ring gives rise to a valuation, or norm, within a complex vector space rather than the on K. The valuation ring is complete if every more usual real vector space so that the coor- Cauchy sequence in this valuation converges. dinates are complex numbers rather than real numbers. (2) The study of analytic varieties (the sets of complete Zariski ring See Zariski ring. common zeros of systems of analytic functions),

c 2001 by CRC Press LLC as opposed to algebraic geometry, the study of complex orthogonal matrix A square ma- algebraic varieties. trix A such that its entries are complex num- bers and it is orthogonal, i.e., has the property complex conjugate representation Let G that AAT = I, or equivalently that AT = A−1 be a group and let φ be a complex representa- (where AT denotes the transpose of A). Such a tion of G (so that φ is a homomorphism from matrix has determinant ±1. The set of all n × n G to the group GL(n, C) of all n × n invert- complex orthogonal matrices forms a group un- ible matrices with complex entries, under matrix der matrix multiplication, called the complex or- multiplication). If we define a new mapping thogonal group O(n, C). The set of all n × n ψ : G → GL(n, C) by setting ψ(g) = φ(g) complex orthogonal matrices of determinant 1 for all g ∈ G, where φ(g) is the matrix ob- is a normal subgroup of O(n, C), called the com- tained from the matrix φ(g) by replacing all its plex special SO(n, C). entries by their complex conjugates, then ψ is also a complex representation of G, called the The set C of complex num- complex conjugate of φ. bers can be represented geometrically as the points of a plane by identifying each complex complex fraction An expression of the form number a + ib (where a and b are real) with z z w w where and are expressions involving com- the point with coordinates (a, b). This geomet- plex numbers or complex variables. A complex z rical representation of C is called the complex fraction w is frequently simplified by observ- plane or Argand diagram. In this representation, zw¯ ing that it equals ww¯ (where w¯ is the complex points on the x-axis correspond to real numbers conjugate of w), which is simpler because the and points on the y-axis correspond to numbers denominator ww¯ is real. of the form ib (where b is real). So the x- and y-axes are often called the real and imaginary complex Lie algebra See Lie algebra. axes, respectively. Note also that the points rep- resenting a complex number z and its complex complex Lie group See Lie group. conjugate z¯ are reflections of each other in the real axis. complex multiplication (1) The multiplica- tion of two complex numbers a + ib and c + id complex quadratic field A field of the form (where a, b, c, d are real) using the rule √ Q[ m] (the smallest√ field containing the ratio- (a + ib)(c + id) = (ac − bd) + i(ad + bc) , nal numbers and m) where m is a negative integer. which is simply the usual rule for multiplying binomials, coupled with the property that i2 = complex quadratic form An expression of −1. the form (2) The multiplication of two complexes of n a group, which is defined as follows. Let G be a z2 + a z z a group and let A, B be complexes, i.e., subsets i i ij i j of G. Then AB ={ab : a ∈ A, b ∈ B}. i=1 i=j This definition obeys the associative law, i.e., i j z A(BC) = (AB)C for all complexes A, B, C of where, for all and , i is a complex variable, a a G. and i and ij are complex constants. complex number A number of the form z = complex representation A homomorphism x + iy where x and y are real and i2 =−1. The φ from a group G to the general linear group set of all complex numbers is usually denoted C GL(n, C), for some n. (GL(n, C) is the group or C. of all n × n invertible matrices with complex entries, the group operation being matrix multi- complex orthogonal group See complex or- plication.) n is called the degree of φ. Complex thogonal matrix. representations of degree 1 are called linear.

c 2001 by CRC Press LLC I n × n complex root A root of an equation that is a and denotes the 2 2 identity matrix) forms complex number but not a real number. For ex- a group under matrix multiplication, called the x3 = ,e2πi/3, ( n , ) ample, the equation 1 has roots 1 complex Sp 2 C . e4πi/3. Of these, 1 is called the real cube root of unity, while e2πi/3 and e4πi/3 are called the complex torus A torus of the form Cn/@ complex cube roots of unity. where @ is a lattice in Cn. complex semisimple Lie algebra A complex complex variable A variable whose values Lie algebra that is semisimple, i.e., does not have are complex numbers. an Abelian invariant subalgebra. component (1) When a vector or force is (a ,a ) complex semisimple Lie group A complex expressed as an ordered pair 1 2 in two di- (a ,a ,a ) Lie group whose Lie algebra is semisimple, i.e., mensions, as a triple 1 2 3 in three dimen- n (a ,a ,... ,a ) n does not have an Abelian invariant subalgebra. sions, or as an -tuple 1 2 n in di- mensions, the numbers a1,a2,... ,an are called complex simple Lie algebra A complex Lie its components. V algebra that is simple, i.e., not Abelian and hav- (2) When a vector v in a vector space is a + a +···+a ing no proper invariant subalgebra. expressed in the form 1e1 2e2 nen where a1,a2,...,an are scalars and e1, e2,..., en is a basis of V , the scalars a ,a ,... ,an are complex simple Lie group A complex Lie 1 2 called the components of v with respect to the group whose Lie algebra is simple, i.e., not basis e , e ,...,en. Abelian and having no proper invariant subal- 1 2 gebra. (3) When a vector or force v is expressed as w+x, where w is parallel to a given direction and x is perpendicular to that direction, w is called complex special orthogonal group See com- the component of v in the given direction. plex orthogonal matrix. (4) The word component is also used loosely to mean simply a part of a mathematical expres- complex sphere A complex n-sphere with sion. center z0 and radius r is the set of all points at distance r from z in the n-dimensional com- 0 composite field The smallest subfield of a plex metric space Cn.(Cn is the set of all n- given field K containing a given collection {kα : tuples (a ,a ,...,an), where each ai is a com- 1 2 α ∈ A} of subfields of K. plex number.) composite number An integer that is not complex spinor group The universal cover zero, not 1, not −1, and not prime. of SO(n, C) ={A ∈ GL(n, C) : At A = I and A = } (n, ) det 1 it is denoted Spin C . composition algebra An alternative algebra A over a field F (characteristic = 2), with iden- complex structure A complex analytic struc- tity 1 and a quadratic norm n : A → F such ture on a differentiable manifold. Complex that n(x, y) = n(x)n(y). structures may also be put on real vector spaces and on pseudo groups. See analytic structure. composition factor See composition series. complex symplectic group The set of all n× composition factor series For a group G n A matrices with complex numbers as entries with composition series G = G0 ⊃ G1 ⊃ AT JA = J and having the property that (where ··· ⊃ Gr ={e}, the sequence G0/G1,..., AT A J is the transpose of , is the matrix Gr−1/Gr . 0 I composition series A series of subgroups −I 0 G0,G1,...,Gn of a group G such that G0 = 1,

c 2001 by CRC Press LLC Gn = G and, for each i, Gi is a proper normal computation by This is the name subgroup of Gi+1 such that Gi+1/Gi is simple. given to a method of solving an equation A = B The Jordan-Hölder Theorem states that if H0, where A and B are complicated expressions in- H1,...,Hm is another composition series for volving products and powers. The method is to G, then m = n and there is a one-one corre- take logarithms of both sides, i.e., to say that spondence between the two sets of factor groups log A = log B, and then to use the laws of log- {Gi+1/Gi : i = 0,...,n− 1} and {Hi+1/Hi : arithms to simplify and rearrange that equation i = 0,...,n− 1} such that corresponding fac- so as to obtain the logarithm of the unknown tor groups are isomorphic. The factor groups variable and hence obtain that variable itself. Gi+1/Gi : i = 0,... ,n−1 are therefore called the composition factors of G. There are similar comultiplication See coalgebra. definitions for composition series and compo- sition factors of other algebraic structures such concave programming The subject dealing as rings. These are obtained by making obvi- with problems of the following type. Suppose X n ous changes to the definitions for groups. For is a closed convex subset of R and g1,...,gm example, in the case of rings, take the above are convex functions on X. Let f be a concave X x ∈ X definitions, replace G0 = 1byG0 = 0, and function on . Determine such that replace the words group, subgroup, and normal f(x) = min {f(x) : x ∈ X and g1(x) ≤ subgroup by ring, subring, and ideal throughout. 0 for i = 1,...,m}.

Composition Theorem (class field theory) conditional equation An equation involving Let K1 and K2 be class fields over k for the variable quantities which fails to hold for some respective ideal groups H1 and H2. Then the values of the variables. composite field K1K2 is the class field over k conditional inequality An inequality involv- for H1 ∩ H2. ing variable quantities which fails to hold for compound matrix Given positive integers some values of the variables. n, P (P ≤ n), denote by QP,n the P-tuples of (A(·(A−1( {1, 2,...,n} with elements in increasing order. condition number The quantity , n where (A( is the norm of the matrix A. QP,n has (P ) members ordered lexicographically. For any m×n matrix A and ∅=α ⊆{1, 2,..., K/k m}, ∅=β ⊆{1, 2,...,n}, let A[α | β] denote conductor (1) Of an Abelian extension , F = F the submatrix of A containing the rows and col- the product ℘ ℘ (over all prime umns indexed by α and β, respectively. ideals) of the conductors of the local fields K /k F = ℘n n Given an integer P,0

c 2001 by CRC Press LLC v(x) ≥ 0}. The ideal ℘f(χ), where ℘ is the conjugate complex number For z = x + iy maximal ideal of the ring of integers in K,is a complex number, the complex conjugate of z is also called the (Artin) conductor of χ. written as z¯ or z∗ and is given by z¯ = x−iy. This operation preserves multiplication and addition Conductor Ramification Theorem (class field in the sense that z + w =¯z +¯w and zw =¯zw¯ . theory) If F is the conductor of the class field A consequence is that, if P is a polynomial with P(z)= P(z)¯ K/k, then it is prime to all unramified prime real coefficients, then , so that roots P divisors for K/k, and F factors as F = ℘ F℘, of occur in complex conjugate pairs. where each F℘ is the ℘-conductor of the local field K℘/k℘ at some ramified prime ℘. See conjugate field To the field extension F over ¯ conductor, class field. a base field k (within some algebraic closure k), any subfield F  of k¯ isomorphic to F . In one conformal transformation A mapping of of the fundamental theorems of , Riemannian manifolds that preserves angles in it is found that if F is a subfield of a normal the respective tangent spaces. In classical com- extension E of k, then the conjugate fields of F inside E are precisely those fields F  for which plex analysis, the same as a holomorphic or an-  alytic function with nonzero derivative. the Galois groups Gal(E : F) and Gal(E : F ) are conjugate subgroups of Gal(E : k). congruence A form of equivalence relation A of two sets or collections of objects. The term conjugate ideal To a fractional ideal of a K/k ϕ(A) will have different specific meanings in different number field , the image ideal of the ϕ(K) k contexts. conjugate field under some -isomorphism ϕ : k¯ → k¯. congruence zeta function The (complex val- 1 conjugate√ radicals √Expressions of the form ued) function ζK (s) = s where the sum √ √ N(A) a + b and a − b. More generally, the is over all integral divisors A of the algebraic √ √n expressions n a − ζ i b, i = 0, 1,...,n− 1, function field K over k(x) where k is a finite where ζ is an of unity, are conjugate field. radicals. congruent integers With respect to a positive  conjugate subgroup For a subgroup G of integer (modulus) m, two integers a and b are  − a group G, any of the subgroups gG g 1 = congruent modulo m, written a ≡ b (mod m), −  {ghg 1 : h ∈ G }. when a − b is divisible by m. conjugation mapping An automorphism of S conjugacy class Assume is a set. A binary a group G of the form a → gag−1 for some R S S × S relation on is a subset of . The con- g ∈ G. jugacy class determined by an element a ∈ S is b ∈ S (a, b) ∈ R the set of elements so that .In conjugation operator Given a uniform al- R (a, a) ∈ R a ∈ S case is reflexive ( for all ), gebra (function algebra) A on some compact (a, b) ∈ R (b, a) ∈ R symmetric (if then ) and Hausdorff space X with ϕ in the maximal ideal (a, b), (b, c) ∈ R (a, c) ∈ R transitive (if then ), space of A, and µ a representing measure for then distinct conjugacy classes are disjoint and ϕ, the operator that assigns to each continuous S the union of all the conjugacy classes is . real-valued function u ∈)A, the continuous real-valued function ∗u so that u + i ∗ u ∈ A conjugate (1) Of a complex number z = and ∗udµ = 0. See function algebra, maxi- a + bi (a, b real), the related complex number mal ideal space, representing measure. z¯ = a − bi. (2) Of a group element h, the group element connected graded module A graded module ghg−1 g M = ∞ M k M , where is another element of the group. n=0 n, over a field , for which 0 is See also conjugate radicals. isomorphic to k. See graded module.

c 2001 by CRC Press LLC connected group An algebraic group which connecting the corresponding cohomology is irreducible as a variety. See algebraic group, groups. variety. connecting morphism See connected se- connected Lie subgroup A Lie subgroup quence of functors. which is connected, as a differentiable manifold. connected sequence of functors A sequence consistent equations A set of equations which F i : C → C of functors between Abelian cat- have some simultaneous solution. egories for which there exist connecting mor- phisms constant A function F : A → B such that there is a c ∈ B with F(x) = c, for all x ∈ A. i i−1 ∂∗ : F (C) → F (A) constant of proportionality The constant (or k ∂∗ : F i(C) → F i+1(A)) , relating one quantity to others in a relation of direct, inverse, or joint proportionality. For for every exact sequence 0 → A → B → C → example, quantity x is directly proportional to 0 of objects in C that turns quantity y if there is a constant k = 0 such that x = ky. In this case, the constant k is ∂ ···→F i+1(C) −→∗ F i(A) −→ F i(B) the constant of proportionality. See also direct proportion, inverse proportion, joint proportion. ∂ −→ F i(C) −→∗ F i−1(A) −→··· constant term Given an equation in a vari- (respectively, able x, any part of the equation that is inde- ∗ x g(x) = ···→F i−1(C) −→∂ F i(A) −→ F i(B) pendent of is a constant term. If sin(x) + x2 + 3π, then the constant term is 3π. ∗ −→ F i(C) −→∂ F i+1(A) −→···) constituent A Z-representation of a finite into a chain complex, and, whenever group G, Z the rational integers.

0 −→ A −→ B −→ C −→ 0 ↓ f ↓ g ↓ h    constituent Z 0 −→ A −→ B −→ C −→ 0 Let denote the rational inte- gers and Q the rational field. Let T be a Z- is a morphism of exact sequences, then representation of a finite group G. Then T is called a constituent of the group G. i i−1 ∂∗ ◦ F (h) = F (f ) ◦ ∂∗ F (respectively, constructible sheaf A sheaf on a scheme X, decomposable into locally closed sub- ∂∗ ◦ F i(h) = F i+1(f ) ◦ ∂∗). schemes so that the restriction of F to each sub- scheme is locally constant. See chain complex. continuation method of finding roots A connecting homomorphism The boundary method for approximating roots of the equation homomorphism f(x)= 0 on the closed interval [a,b] by intro- ducing a parameter t, so that f(x)≡ g(x,t)|t=b ∂∗ : Hr (K, L; G) → Hr− (L; G) , 1 and so that g(x,a) = 0 is easily solved to ob- x = x connecting the homology groups of the simpli- tain 0. We partition the interval to give a = t

c 2001 by CRC Press LLC continued fraction A number of the form tion on a convex set obtained as the solution set to a family of inequalities. 1 a0 + 1 coordinate ring Of an affine algebraic set X a1 + a + 1 (over a field k), the quotient ring 2 . .. a3 + k[X]=k[x1,x2,...,xn]/I (X) where the ai are real numbers and a ,a ,... 1 2 k are all positive. A continued fraction is simple of the ring of polynomials over by the ideal of X if all the ai are integers and finite, as opposed to polynomials that vanish on . infinite, if the sequence of ai is finite. For typo- A i ∈ I graphical convenience, the continued fraction is coproduct Of objects i ( ) in some category, the universal object Ai, with mor- often written as [a0; a1,a2, ...]. phisms ϕi : Ai → Ai satisfying the condition X ψ : A → X continuous analytic capacity Of a subset A that if is any other object and i i of C, the measure supf (∞) over all contin- are morphisms, there will exist a unique mor- ψ : A → X ψ = ψ ◦ ϕ uous functions on the Riemann sphere S2 that phism i so that i i.If vanish at ∞, are analytic outside some compact the category is Abelian, the coproduct coincides A subset of A, and have sup norm (f (≤1onS2. with the direct sum i. continuous cocycle A matrix that arises as coradical Of a coalgebra, the sum of its sim- the derivative of a group action on a manifold. ple subcoalgebras. See coalgebra, simple co- algebra. contragredient representation Of a repre- sentation ρ of a group G on some vector space coregular representation The contragredi- V , the representation ρ∗ of G on the dual space ent representation of the right regular represen- V ∗ defined by ρ ∗ (g) = ρ(g−1)∗. See repre- tation reciprocal to a given left regular represen- sentation, dual space. tation. See contragredient representation. f ,...,f contravariant See functor. Corona Theorem If 1 n belong to H ∞, the set of bounded analytic functions in convergent matrix An n × n matrix A such the unit disk D in the complex plane, and if m |f (z)|+···+|f (z)|≥δ> D that every entry of A converges to 0 as m ap- 1 n 0in , then there ∞ proaches ∞. Convergent matrices arise in nu- exist g1,...,gn ∈ H with fj gj = 1inD. merical analysis; for example, in the study of In functional analytic terms, the theorem says ∞ iterative methods for the solution of linear sys- this. H is a vector space under pointwise ad- tems of equations. It is known that A is conver- dition and scalar multiplication and a Banach gent if and only if its spectral radius is strictly space with the norm (f (=sup |f(z)|, for z ∈ less than 1; namely, all eigenvalues of A have D. Further, since the pointwise product of two modulus strictly less than 1. bounded functions is bounded it is an algebra with pointwise multiplication. As a Banach ∞ ∞ convex hull The smallest convex set contain- space, H has a dual space H , which is ing a given set. (A set S is convex if, whenever the set of all continuous linear mappings from ∞ x,y ∈ S, then the straight line joining x and H into the complex numbers C. A functional ∞ y also lies in S.) If E is a vector space and λ ∈ H may have the added property of being e ,...,e E λ(fg) = λ(f )λ(g) 1 k are elements of , then the convex multiplicative, . We denote {e ,...,e } { k c e : hull of the set 1 k is the set 1 j j the set of all such multiplicative linear function- cj ≥ 0 and cj = 1}. als by M. For example, if a ∈ D, then the linear functional λ(f ) = f(a)is in M and is called a  convex programming The theory that deals point evaluation. The set H ∞ has a topology ∞ with the problem of minimizing a convex func- whereby λa converges to λ if, for each f ∈ H ,

c 2001 by CRC Press LLC the numbers λa(f ) converge to λ(f ). M inher- in radians) so that the angle is one of the angles  its this method of convergence from H ∞ and in a right triangle with adjacent side a, opposite a the Corona Theorem states that the “point eval- side b, and hypotenuse c, then cos α = c . uations” are dense in M in this topology. This was proved by Carleson. cospecialization Let A and B be sets and ! : A → B a function. Cospecialization is correspondence ring Of a nonsingular curve a process of selecting a subset of B with refer- X, the ring C(X) whose objects, called corre- ence to subsets of A, using the mapping ! as a spondences, are linear equivalence classes of referencing operator. divisors of the product variety X × X, mod- ulo the relation that identifies divisors if they cotangent function The quotient of the co- are linearly equivalent to a degenerate divisor. sine and the sine functions. Also the reciprocal θ = cos θ = The addition in this ring is addition of divisors, of the tangent function. Hence, cot sin θ and the multiplication is defined by composi- 1 tan θ . See sine function, cosine function. tion, that is, if C1, C2 are correspondences of X and x ∈ X, then C1 ◦ C2 is the correspondence cotangent of angle Written cot α, the x-coor- C1(C2(x)) where C(x) is the projection on to dinate of the point where the terminal ray of the the second component of C(x,X). angle α whose initial ray lies along the positive x-axis intersects the line P with equation y = 1. cosecant function The reciprocal of the sine If α measures more than π radians (= 180◦), the θ θ = 1 function, denoted csc . Hence, csc sin θ . terminal is taken to extend back to intersect the See sine function. P <α< π α line .If0 2 ( in radians), so that the angle is one of the angles in a right triangle with cosecant of angle The reciprocal of the sine a b 1 adjacent side , opposite side , and hypotenuse of the angle. Hence, csc θ = θ . See sine of a sin c, then cot α = b . angle. coterminal angles Directed angles whose cosemisimple coalgebra A coalgebra which terminal sides agree. (Their initial sides need is equal to its coradical. See coalgebra, coradi- not agree.) cal. cotriple A functor T : C → C on a category cosine function One of the fundamental C for which there exist natural transformations trigonometric functions, denoted cos x.Itis(1) ε : IdC → T , δ : T → T 2, for which the periodic, satisfying cos(x + 2π) = cos x; (2) following diagrams commute: bounded, satisfying −1 ≤ cos x ≤ 1 for all x real ; and (3) intimately related with the sine IT(X) IT (X) function, sin x, satisfying the important identi- T(X) ←− T(X) −→ T(X) x = ( π − x) 2 x + 2 x = BT(X) , δX ↓-T(BX) ties cos sin 2 , sin cos 1, and many others. It is related to the exponen- T 2(X) eix+e−ix tial function via the identity cos x = √ 2 δ (i = −1), and has series expansion T(X) −→X T 2(X) δ ↓↓δ 2 4 6 X T(X) x x x T(δ ) cos x = 1 − + − +··· T 2(X) −→X T 3(X) 2! 4! 6! valid for all real values of x. See also cosine of angle. countable set A set S such that there is a one-to-one mapping f : S → N from S onto cosine of angle Written cos α, the x-coor- the set of natural numbers. See also cardinality. dinate of the point where the terminal ray of the angle α whose initial ray lies along the positive numbers The positive integers 1, x <α< π α ... -axis intersects the unit circle. If 0 2 ( 2, 3,

c 2001 by CRC Press LLC Courant-Fischer (min-max) Theorem Let of C lying in the star complex St(C) of objects A be an n×n Hermitian matrix with eigenvalues containing C.Aroot is the image of the complex under some idempotent endomorphism, called a λ1 ≥ λ2 ≥···≥λn . folding.

Then Coxeter diagram A labeled graph whose ∗ nodes are indexed by the generators of a Cox- λi = max min x Ax dim X=i x∈X, x∗x=1 eter group which has (Pi,Pj ) as an edge la- beled by Mij whenever Mij > 2. Here Mij and are elements of the Coxeter matrix. Also called ∗ Coxeter-Dynkin diagrams. These are used to λi = min max x Ax . dim X=n−i+1 x∈X, x∗x=1 visualize Coxeter groups.

This theorem was first proved by Fischer for Coxeter group A group with generators ri, a matrices (1905) and later (1920) it was extended i ∈ I, and relations of the form (rirj ) ij = 1, by Courant to differential operators. where all aii = 1 and whenever i = j, aij > 1 (aij may be infinite, implying no relation be- covariant A term describing a type of func- tween ri and rj in such a case). tor, in constrast with a contravariant functor. A F : C → D covariant functor assigns to every Cramer’s rule Assume we are given n linear c C F(c) object of the domain category an object equations in n unknowns. That is, we are given D n of the codomain category and to every arrow the linear equations Li(X) = Cij Xj , α : c → c C F(α) : F(c) → F(c) j=1 of an arrow where the Cij are given numbers, the Xj are D F( ) = of in such a way that idC idD, where idC unknowns and i = 1,...,n. We are asked and idD are the identity arrows of the respective to solve the n equations Li(X) = bj , where T(β◦ α) = T(β)◦ T(α) ◦ categories, and that ( j = 1,...,n. One can write these equations in represents composition of arrows in both cate- matricial form AX = B, where A is a square gories) whenever β ◦ α is defined in C. See also n × n matrix and X = (X1,...,Xn) and B = contravariant. n (b1,...,bn) are vectors in R . Cramer gave a formula for solving these equations provid- covering Of a (nonsingular) algebraic curve ing the matrix A has an inverse. Let |·|de- X k Y over the field , a curve for which there note the determinant of a matrix, which maps a k Y → X exists a -rational map which induces square matrix to a number. (See determinant.) k(x) X→ k(Y) an inclusion of function fields The matrix A will have an inverse provided the k(Y) k(X) making separably algebraic over . number |A| is not zero. So, assuming A has an inverse (this is also referred to by saying that A covering family A family of morphisms in a is non-singular), the solution values of Xj are category C which define the Grothendieck topol- given as follows. In the matrix A, replace the ogy on C. ith column of A (that is, the column made up of Ci1,...,Cin)byB. We again obtain a square coversine function The function |Ai | matrix Ai and the solution numbers Xi = |A| . X n covers α = 1 − sin α. If is the vector made up of these numbers, it will solve the system and this is the only so- lution. Coxeter complex A thin chamber complex in which, to every pair of adjacent chambers, there Cremona transformation A birational map exists a root containing exactly one of the cham- from a projective space over some field to itself. bers. When C ⊂ C we say that C is a face of C. See birational mapping. Two chambers are adjacent if their intersection has codimension 1 in each: the codimension of cremona transformation A birational map C in C is the number of minimal nonzero faces of the projective plane to itself.

c 2001 by CRC Press LLC criterion of ruled surfaces A theorem of graphic group @ is cyclic of order q, where the Nakai, characterizing ample divisors D on a only possible values of q are 1, 2, 3, 4, or 6. X ruled surface as those for which the inter- The terms used are defined as follows. The D section numbers of with itself and with every group @ is a discrete subgroup of the Euclidean X irreducible curve in are all positive. See am- group E(2) = R2 × O(2). Let j : E(2) → ple, ruled surface, intersection number. O(2) be the homomorphism (x, A) → A. Then H = j(@) is a subgroup of O(2), called the R crossed product Of a commutative ring point group of @. Let L be the kernel of j,a G with a commutative monoid (usually a group) discrete subgroup of R2. If the rank of L is 2, a ∈ R : g,h ∈ with respect to a factor set g,h then @ is called a crystallographic group. Then G .IfG acts on R in such a way that the map- H0 is H ∩ SO(2). ping r → rg is an automorphism of R, then the crossed product of R by G with respect to the crystallographic space group A crystallo- factor set is the R-algebra with canonical basis g graphic group of motions in Euclidean 3-space. elements b for g ∈ G and multiplication law See crystallographic group.    rgbg shbh crystal system A classification of 3-dimen- g∈G h∈G sional crystallographic lattice groups. Where a b c the lattice constants , , represent the lengths g hg gh = ag,hr (s b ). of the three linearly independent generators and g,h∈G α, β, γ the angles between them, the seven crys- tal systems are (where x, y are distinct and not To ensure that this multiplication is associative, 1, and θ = 90◦): the elements of the factor set must satisfy the g relations ag,hagh,k = a ag,hk for all g, h, k ∈ h,k a : b : c (α, β, γ ) G. Further, if e ∈ G is the identity element, Name the unit element of the algebra is be, which also cubic 1 : 1 : 1 (90◦, 90◦, 90◦) requires that, in the factor set, ae,g = ag,e equals the unit element in R for all g ∈ G. tetragonal 1 : 1 : x(90◦, 90◦, 90◦) Crout method of factorization A type of rhombic : x : y(◦, ◦, ◦) LU-decomposition of a matrix in which L is 1 90 90 90 lower triangular, U is upper triangular, and U monoclinic : x : y(◦,θ, ◦) has 1s on the diagonal. 1 90 90 ◦ ◦ ◦ crystal family A collection of crystallo- hexagonal 1 : 1 : x(90 , 90 , 120 ) graphic groups whose point groups, @, are all rhombohedral : : (θ,θ,θ) conjugate in GL3(R) and whose lattice groups 1 1 1 are minimal in their crystal class with respect to the ordering (Y, @) ≤ (Y,@), defined as: triclinic other than above  there exists a g ∈ GL3(R) so that (Y ) = g(Y), @ = g@g−1 B(Y) ⊆ B(Y) B(Y) and where See crystallographic group. is the Bravais group g ∈ O(3) : (Y) = (Y) . See crystallographic group, Bravais group. cube (1) In geometry, a three-dimensional crystallographic group A of solid bounded by six square faces which meet motions in Rn containing n linearly independent orthogonally in a total of 12 edges and, three translations. See also crystal system. faces at a time, at a total of eight vertices. One of the five platonic solids. crystallographic restriction The proper sub- (2) In arithmetic, referring to the third power 3 group H0 of the point group H of a crystallo- x = x · x · x of the number x.

c 2001 by CRC Press LLC cube root Of a number x, a number t whose G which is the product of (i.) a reductive Lie x t3 = x x G cube is : .If is real, it has exactly√ one subgroup 1, stable under the Cartan involution, t = 3 x G real cube root, which is denoted . (ii.) a vector subgroup 2 whose centralizer in G is G2G1, and (iii.) a group G3 = exp( Gα) p(X) = cubic (1) A polynomial of degree 3: where G is the Lie algebra of G1Gα = X ∈ 2 3 a0 + a1X + a2X + a3X , where a3 = 0. G :[H,X]=α(H)X for all H ∈ H , where (2) A curve whose analytic representation in H is the Lie algebra of G2, α is some functional some coordinate system is a polynomial of de- on H, and the sum that defines G3 is over all gree 3 in the coordinate variable. positive α for which Gα is nonzero.

3 cubic equation An equation of the form ax + cycle (1) In , a graph Cn on ver- bx2 +cx +d = a b c d 0, where , , , are constants tices v1,v2,... ,vn whose only edges are be- a = (and 0). tween v1 and v2, v2 and v3,...,vn and v1. (2) In a permutation group Sn, a permutation cup product (1) In a lattice or Boolean alge- with at most one orbit containing more than one bra, the fundamental operation a ∨b, also called object. the join or sum of the elements a and b. 3 r ( ) In homology theory, any element in the (2) In cohomology theory, where H (X, Y ; kernel of a homomorphism in a homology com- G) is the cohomology of the pair (X, Y ) with plex. coefficients in the group G, the operation that r sends the pair (f, g), f ∈ H (X, Y ; G1), g ∈ cyclic algebra A crossed product algebra of s H (X, Z; G2), to the image a cyclic extension field F (over a base field k) G =g f ∪ g ∈ H r+s(X, Y ∪ Z; G ) with its (cyclic) Galois group with 3 respect to a factor set that is determined by a of the element f ⊗ g under the map single nonzero element a of k. The elements α + r s of this algebra are uniquely of the form 0 H (X, Y ; G ) ⊗ H (X, Z; G ) n−1 1 2 α1b+···+αn−1b , where the αi come from n− r+s F , 1,b,b2,... ,b 1 are the formal basis el- → H (X, Y ∪ Z; G3), ements of the algebra, and n is the order of G. induced by the diagonal map The multiplication is determined by the relations g i j i+j n 0 : (X, Y ∪ Z) → (X, Y ) × (X, Z) . bα = α b, b b = b and b = a.

cyclic determinant A determinant of the Cup Product Theorem A theo- form rem of Eilenberg-MacLane in the theory of co- a0 a1 ··· an−1 homology of groups. Suppose the group G can an−1 a0 ··· an−2 , be presented as the quotient of the F ··· with relation subgroup H, so that 1 → H → a1 a2 ··· a0 G → F → A G 1 is exact. If is a -module, in which the entries of successive rows are G → then the induced extension of givenby1 shifted to the right one position (modulo n). H/[H,H]→F/[H,H]→G → 1isa2- Also called a circulant. cocycle ζ in H 2(G, A). Let K = H/[H,H]. ϕ ∈ H r (G, (K, A)) If Hom , the cup product A group, all of whose elements ϕ ∪ ζ ∈ H r+2(G, A) n (which makes use of the are powers a (n = 0, 1,...) of a single gen- (K, A)⊗K → A natural map Hom ) determines erating element a. Such a group is finite if, for an isomorphism some d, ad equals the identity element. The r r+ a H (G, Hom(K, A)) =∼ H 2(G, A) . group is often denoted . cyclic representation A unitary representa- cuspidal parabolic subgroup A closed sub- tion ρ of a topological group G, having a cyclic group G1G2G3 of a connected algebraic group vector x, i.e., an element x of the representation

c 2001 by CRC Press LLC space for which the span of vectors ρ(g)x,asg cyclotomic polynomial Any of the polyno- runs through G, is dense. See representation. mials !n(x) whose roots are precisely those roots of unity of degree exactly n. That is, H cyclic subgroup Any subgroup of a group !1(x) = x − 1 and, for n>1, G H an in which all elements of are powers n n = , ,... a ∈ H x − 1 ( 0 1 ) of a single element , the !n(x) = , generator of the subgroup. d|n (!d (x)) cyclotomic field Any extension of a field ob- where the product is over the proper divisors d tained by adjoining roots of unity. of n.

c 2001 by CRC Press LLC di, then another representation of x can be ob- tained by replacing di with di − 1 and defining di−1 = di−2 =···=9. For example, 238.75 = D 238.74999···. decomposable operator A bounded linear operator T , on the separable Hilbert space L2 Danilevski method of matrix transformation (, µ; H) of square-integrable, measurable, H- A method for computing eigenvalues of a matrix valued functions on some measure space (, µ) M, involving application of row and column op- where H is also a Hilbert space, so that for each erations that produce the companion matrix of measurable ξ(γ), the function γ → T(γ)ξ(γ) M. is measurable, and so that, for each ξ ∈ L2(, µ; H), T can be represented as the direct inte- decimal number system The base 10 po- gral sitional system for representing real numbers. Every real number x has a representation of the Tξ = ⊕T(γ)ξ(γ)dµ(γ). form 

dn−1dn−2 ···d1d0 · d−1d−2 ··· , decomposition field Let the ring A be closed in its quotient field K. Suppose that B is its in which the di are the digits of x; d0 and d−1 integral closure in a finite Galois extension L, are separated by the decimal point (.). The digits with group G. Then B is preserved by elements of x are determined recursively by the formulas of G. Let ℘ be a maximal ideal of A and B x dn− = and for k>1, B ℘ 1 10n−1 a maximal ideal of that lies above .Now GB is the subgroup of G consisting of those k− x − 1(d n−j ) elements that preserve B. Observe that GB acts j=1 n−j 10 dn−k = , in a natural way on the residue class field B/B 10n−k and it leaves A/℘ fixed. To any σ ∈ GB,we can associate an element σ ∈ B/B over A/℘; where n is the unique integer that satisfies 10n−1 n the map ≤ x<10 and t is the floor function (the σ −→ σ 3 greatest integer ≤ t). For example, if x = 238 , G 4 thereby induces a homomorphism of B into then n = 3,d2 = 2,d1 = 3,d0 = 8,d−1 = the group of automorphisms of B/B over A/℘. 7,d−2 = 5. The only possible digits are 0, 1, 2, The fixed field in GB is called the decompo- dec 3, 4, 5, 6, 7, 8, 9. The digit d0 is called the ones sition field of B, and is denoted L . digit, d1 the tens digit, d2 the hundreds digit, d3 A the thousands digit, etc.; also, d−1 is the tenths decomposition field Let the ring be closed K B digit, d−2 the digit, etc. It can be in its quotient field . Suppose that is its inte- shown that, precisely when x is rational, the se- gral closure in a finite Galois extension L, with quence of digits of x eventually repeats. That is, group G. Then B is preserved by elements of there is a smallest integer p for which di−p = di G. Let ℘ be a maximal ideal of A and B a maxi- for every i less than some fixed index. Here we mal ideal of B that lies above ℘.NowGB is the G call the string of digits di−1di−2 ···di−p a re- subgroup of consisting of those elements that peating block and p the period of the represen- preserve B. Observe that GB acts in a natural tation of x. In the special case that the repeating way on the residue class field B/B and it leaves block is the single digit 0, the convention is to A/℘ fixed. To any σ ∈ GB we can associate an drop all the trailing zeros from the representation element σ ∈ B/B over A/℘; the map x and say that has a finite or terminating decimal σ −→ σ expansion. Further, it is possible in this case to give a second, distinct expansion of x:ifx has thereby induces a homomorphism of GB into the finite decimal expansion with final nonzero digit group of automorphisms of B/B over A/℘.

c 2001 by CRC Press LLC The fixed field in GB is called the decompo- defect If k is a field which is complete under sition field of B, and is denoted Ldec an arbitrary valuation, and if E is a finite exten- sion of degree n, with ramification e and residue decomposition group Of a prime ideal ℘ for class degree f , then ef divides n: n = ef δ , and a Galois extension K/k, the stabilizer of ℘ in δ is called the defect of the extension. Gal(K/k). See stabilizer, Galois group. defective equation An equation, derived decomposition number The multiplicity of from another equation, which has fewer roots an absolutely irreducible modular representa- than the original equation. For example, if x2 + tion of a group G, in the splitting field K (for x = 0 is divided by x, the resulting equation which it is the Galois group) as it appears in a x + 1 = 0 is defective because the root 0 was decomposition of one of the irreducible repre- lost in the process of division by x. sentations of G in some number field for which K is the residue class field. See absolutely irre- defective number A positive integer which ducible representation, modular representation. is greater than the sum of all its factors (except itself). For example, the number 10 is defective, Dedekind cut One of the original ways of since the sum of its factors (except itself) is 1 + defining irrational numbers from rational ones. 2 + 5 = 8. (L, R) A Dedekind cut is a decomposition of If a positive integer is equal to the sum of L R the rational numbers into two sets and such all its factors (except itself), then it is called a L R that (i.) and are nonempty and disjoint; (ii.) perfect number. If the number is less than the x ∈ L y ∈ R x1 and the sum extends over all non- plex) Hilbert space. A Hermitian form is a func- zero ideals of Ok, while the product runs over tion f : H × H → C, such that f(x,y) is lin- all nonzero prime ideals of Ok. ear in x, and conjugate linear in y, and f(x,x)

c 2001 by CRC Press LLC is real. A Hermitian form f is called posi- any way without tearing. Additional conditions, tive definite [resp., nonnegative definite, nega- such as continuity, are usually attached to a de- tive definite, nonpositive definite] if, for x = 0, formation. Thus, one can talk about a continu- f(x,x) > 0 [resp., ≥ 0,<0, ≤ 0]. ous deformation, or smooth deformation, etc. definite quadratic form Let E be a finite di- degenerate A term found in numerous sub- mensional vector space over the complex num- jects in mathematics. For example, in algebraic bers C . Let L be a symmetric bilinear form on geometry, when considering the homogeneous E.(See bilinear form, symmetric form.) The bar resolutions of Abelian groups, one uses sub- quadratic form associated with L is the func- groups generated by (n + 1)-tuples (y0,y1, ..., tion Q(x) = L(x, x).IfQ(x) > 0, [resp., yn) with yi = yi+1 for at least one value of i; Q(x) ≥ 0,< 0, ≤ 0] the form is called posi- such an (n + 1)-tuple is called degenerate. tive definite [resp., nonnegative definite, nega- tive definite, nonpositive definite]. As an im- degree of divisor If a polynomial p(x) is portant application, assume F(x,y) is a smooth factored as follows: function of two real variables x and y. Let f be n1 nk the quadratic part of the Taylor expansion of F p(x) = a0 (x − x1) ···(x − xk) ,

2 2 f(x,y) = ax + 2bxy + cy , where x1,...,xk are distinct. Then each x − xi is called a divisor and the corresponding ni is a,b c where , and are determined as the appro- called the degree of the divisor. priate second partial derivatives of F , evaluated at a given point. Questions involving the mini- degree of equation In a polynomial equa- F mum points of can be solved by considering tion, the highest power is called the degree of the two by two matrix equation. In a differential equation, the highest   ab order of differentiation is called the degree of A =   , equation. cd degree of polynomial A polynomial is an with its determinant and the upper left one by expression of the form one determinant a both positive. In this case we p(x) = a + a x +···+a xn,a= . are considering 0 1 n n 0   x The integer n is called the degree of polynomial (x, y)A y = f(x,y) . p(x). deflation A process of finding other eigenval- degree of polynomial term See degree of ues of a matrix when one eigenvalue and eigen- polynomial. vector are known. More specifically, if A is an De Moivre’s formula See De Moivre’s The- n × n matrix with eigenvalues λ1,...,λn, and orem. Av = λ1v, with v a nonzero (column) vector, then, for any other vector u, the eigenvalues of n the matrix B = A − vuT are De Moivre’s Theorem For any integer and any angle θ the complex equation (De Moivre’s T λ1 − u v,λ2,...,λn . formula) T n By choosing u to be a multiple of the first row (cos θ + i sin θ) = cos(nθ) + i sin(nθ) T of A and scaled so that u v = λ1, the first row of B becomes identically zero. holds. deformation A deformation is a transfor- denominator The quantity B, in the fraction A mation which shrinks, twists, expands, etc. in B (A is called the numerator).

c 2001 by CRC Press LLC density Weightper unit (volume, area, length, Lie subalgebra generated by all [s,t] for s ∈ S etc.). and t ∈ T . dependent variable In a function y = f(x), Descartes’s Rule of Signs A rule setting an x is called the independent variable and y the upper bound to the number of positive or nega- dependent variable. tive zeros of a function. For example, the pos- itive zeros of the function f(x) cannot exceed derivation A map D : A → M, from a the number of changes of sign in f(x). commutative ring A to an A-module M such that descending central series The series of nor- D(a + b) = D(a) + D(b) mal subgroups and G = N ⊇ N ⊇ N ⊇··· , D(ab) = aD(b) + bD(a) 0 1 2 G N = G for all a and b in A. of a group , defined recursively by: 0 , Ni+1 =[G, Ni], where derivation of equation A proof of an equa-   −1 −1 tion, by modifying a known identity, using cer- [G, Ni] = x y xy : x ∈ G, y ∈ Ni tain rules. is a commutator subgroup. See also commutator derivative If a function y = f(x)is defined subgroup. on a real interval (a, b), containing the point x0, then the limit descending chain of subgroups A (finite or infinite) sequence {Gi} of subgroups of a group f(x)− f(x0) G G G lim , , such that each i+1 is a subgroup of i. See x→x 0 x − x0 also subgroup. if it exists, is called the derivative of f at x and 0 determinant A number, defined for every may be denoted by square matrix, which encapsulates information df about the matrix. Common notation for the de- f  (x ) , (x ) ,D f (x ) ,f (x ) , 0 dx 0 x 0 x 0 etc . terminant of A is det A and |A|.Fora1× 1 matrix, the determinant is the unique entry in If the function y = f(x)has a derivative at the matrix. For a 2 × 2 matrix, every point of (a, b), then the derivative function    f (x) is sometimes simply called the derivative a11 a12 of f . a21 a22 a a − a a derived equation See derivation of equation. the determinant is 11 22 12 21. For a larger n×n matrix A, the determinant is defined recur- derived functor If T is a functor then its sively, as follows: Let Ai,j be the (n−1)×(n−1) left-derived functors are defined inductively as matrix created from A by removing the ith row j follows: Let T0 = T .IfSn is any connected and the th column. Then sequence of (additive) functors, then each natu- . n ral transformation S → T extends to a unique i+j 0 0 det A = (−1) aij det Ai,j morphism {Sn : n ≥ 0}→{Tn : n ≥ 0} of con- i=1 nected sequences of functors. The right derived functors are defined similarly. This sum can be computed, and is the same, for any choice of j between 1 and n. derived series Given a Lie algebra G,we define its derived series G0,G1,..., inductively determinant factor If A is a matrix with ele- by G0 = G, Gn+1 =[Gn,Gn], n ≥ 0, where, ments in a principal ideal ring (for example, the for any subsets S and T of G, [S,T ] denotes the integers or a polynomial ring over a field), then

c 2001 by CRC Press LLC the determinant factors of A are the numbers for each i ∈{1, 2,...,n}. When the inequality d1,...,dr where di is the greatest common di- above holds strictly for every i ∈{1, 2,...,n}, visor of all minors of A of degree i and r is the we say that A is strictly row diagonally domi- rank of A. nant. Similarly, we can define diagonal domi- nance with respect to the sums of the moduli of determinant of coefficients The determi- the off-diagonal entries in each column. nant of coefficients of a set of n linear equations diagonal matrix An n×n matrix (aij ) where a x +···+a nxn = b 11 1 1 1 aij = 0ifi = j. . . . . diagonal sum The sum of the diagonal en- a x +···+a x = b n1 1 nn n n tries of a square matrix A, which also equals the A in n unknowns over a commutative ring R is sum of the eigenvalues of . If the entries in A denoted by are complex and the diagonal sum is positive   (negative), then at least one of the eigenvalues a11 ··· a1n of A has a positive (negative) real part. Also  . . .  det  . .. .  , called spur or trace. an ··· ann 1 difference The (set theoretical) difference of or by two sets A and B is defined by: a11 ··· a1n . . A\B ={x : x ∈ A and x/∈ B} . . .. . . . . . A B an1 ··· ann The (algebraic) difference of subsets and G ( P)a ···a of a group is defined by: Its value is P sgn 1p1 npn , where the sum is over all permutations P = (p1,...,pn) A − B ={x ∈ G : x = a − b, a ∈ A, b ∈ B} . of 1, 2,...,n. Usually, R is the real or the com- plex numbers. The set of equations is uniquely solvable if and only if the determinant of the difference equation An equation of the form coefficients is nonzero. xn+1 = F (xn,xn−1,...,x0) , diagonalizable linear transformation A lin- which defines a sequence of numbers, provided V ear transformation from a vector space into that initial values (e.g., x ) are given. Difference W 0 another vector space which can be repre- equations are the discrete analog of differential sented by a diagonal matrix (with respect to equations. The difference equation known as some choice of bases for V and W). See diago- the logistic equation, xn+1 = axn(1 − xn), a nal matrix. See also diagonalizable operator. constant, is one of the original examples of a system that exhibits chaotic behavior. diagonalizable operator A linear transfor- V mation of a vector space into itself which can difference group The quotient group of an be represented by a diagonal matrix with respect additive group G by a subgroup H (written as V to some basis of . An operator is diagonaliz- G − H ). For example, the additive group of V able if and only if there is a basis for made integers has the even integers as a subgroup, and up entirely of eigenvectors of the operator. See the difference group is the mod2 group {0, 1}. also Jordan normal form. See quotient group. n × n diagonally dominant matrix An ma- difference of like powers The factorization: trix A = (aij ) with entries from the complex field is called row diagonally dominant if an − bn = (a − b) an−1 + an−2b |aii| ≥ |aik| + an−3b2 +···+ abn−2 + bn−1 . k=i

c 2001 by CRC Press LLC difference of the nth order If y(x)is a func- differential form of the first kind Suppose tion of a real variable x and 9x is a fixed number, that  is a nonsingular curve and that ω is a then the first order difference 9y(x) is defined differential form on .If(ω) is a positive divisor by 9y(x) = y(x + 9x) − y(x). Scaling 9x to of the free Abelian group generated by points of 1, the difference of the nth order is defined by , then ω is a differential form of the first kind (or a regular 1-form). If, for any point P of , 9ny(x) = 9 9n−1y(x) there is a rational function fP such that ω −dfP n   is a regular 1-form, then ω is a differential form n = (− )n−k y(x + k) . of the second kind. If ω has nonzero residues, 1 k k=0 then it is a differential form of the third kind.

differential form of the second kind See difference of two squares The factorization: differential form of the first kind. a2 − b2 = (a − b)(a + b). See also difference of like powers. differential form of the third kind See dif- difference product The polynomial defined ferential form of the first kind. p(x ,...,x ) = over an integral domain by 1 n differential ideal Suppose that R is a differ- i

c 2001 by CRC Press LLC and D(xy) = Dx·y+x·Dy, for x,y ∈ R.) The if each Hi is a normal subgroup of G, G = ring of all real-valued differentiable functions of H1H2 ...Hn, and H1 ...Hi−1 ∩ Hi ={e}, i = a real variable is a differential ring. 2,... ,n. differential subring If R is a differential ring directed set A set X equipped with a partial with derivations D1,...,Dk, then a subring of ordering and such that if x,y ∈ X then there S of R is a differential subring if DiS ⊂ S for exists z ∈ X such that x ≤ z and y ≤ z. all i. See differential ring. direct factor Either H or K, in the direct differential variable See differential poly- product H × K or H ⊗ K. See direct product. nomial. direct integral See integral direct sum. differentiation (1) In a chain complex, a map, usually denoted dn or δn, from one module direct limit Suppose {Gµ}µ∈I is an indexed to the next. An example of a differentiation is family of Abelian groups, where I is a directed the boundary operator encountered in the study set. Suppose that there is also a family of ho- of simplicial complexes. See also chain com- momorphisms ϕµν : Gµ → Gν, defined for all plex, boundary operator. µ<ν, such that: ϕµµ : Gµ → Gµ is the iden- (2) Of a function f(x)of a real variable, at a tity, and if µ<ν<κthen ϕνκ ◦ ϕµν = ϕµκ . real number x = a, the limit Consider the disjoint union of the groups Gµ and form an equivalence relation by xµ ∼ xν,  f(a+ h) − f(a) x ∈ G x ∈ G f (a) = lim . µ µ and ν ν, if for some upper bound h→0 h κ of µ and ν we have ϕµκ (xµ) = ϕνκ(xν). Then the direct limit is defined to be the set of equiv- Many generalizations to other topological spaces alence classes and is denoted by exist. lim Gµ . → µ∈I digit A symbol in a number system. For example, in the system, the only See also directed set. digits that are used are 0 and 1. See also duodec- imal number system. direct product (1) A group G is called the internal direct product of subgroups H and K An algebraic group Dn, gen- if the following three conditions hold: H and erated by two elements: a, which is a rotation K are normal subgroups of G, H ∩ K contains of the Euclidean plane about the origin through only the identity element, and G = HK. This 2π angle of n , and b, which is reflection through internal direct product is denoted H × K. the y-axis. Dn is the group of symmetries of the (2)IfH and K are any two groups, then the regular n-gon, and has order 2n. external direct product of H and K, denoted H ⊗ K, is the Cartesian product: {(h, k) : h ∈ dimension The number of vectors in the ba- H, k ∈ K}. H ⊗K is a group, defining multipli- sis of a vector space V . If the basis is finite, cation componentwise, i.e., (h1,k1)·(h2,k2) = then V is called finite dimensional. In this case, (h1 · h2,k1 · k2). See also internal product, ex- if V is a vector space of dimension n over the ternal product. real numbers R, then it is isomorphic to the Eu- clidean space Rn. If the basis is infinite, then V direct proportion Quantity x is directly pro- is called infinite dimensional. See also basis. portional to quantity y,orvaries directly as y, if there is a constant k = 0 such that x = ky. Diophantine equation An equation in which (k is called the constant of proportionality.) See solutions are restricted to the integers. also inverse proportion, joint proportion. direct decomposition A group G has a di- direct sum (1) In the case where V1, V2, ... , rect decomposition G = H1 × H2 ×···×Hn Vn are all vector spaces over the same field F ,

c 2001 by CRC Press LLC one can define the direct sum V to be a vector where χ is a character of the group of classes space made up of n-tuples of the form (v1,v2, coprime to some positive integer m and ... ,vn), where vi ∈ Vi. The common notation  χ((n)) if (n,m)=1 for this direct sum is: χ(n) = 0if(n,m)=1 , V = V ⊕ V ⊕···⊕V . 1 2 n where (n) is the residue class of n(mod m). The function L(s) converges absolutely for V V ... V (2) In the case where 1, 2, , n are all (s) > 1, and is used widely in the study of vector subspaces of the same vector space W, rational number fields and of quadratic and cy- and Vi⊥Vj if i = j, we can define the direct clotomic number fields. See also L-function. sum V to be a vector space made up of sums of the form: v1 + v2 +···+vn where vi ∈ Vi. Dirichlet Unit Theorem Suppose that k is The same notation is used as above. See also an extension field (of first degree) of the rational orthogonal subset. subfield Q of the complex number field. Then (3) In the case where H1, H2, ... , Hn are all the group of units of k is the direct product of subgroups of the same Abelian group G, we can a cyclic group and a free Abelian multiplicative define the direct sum H to be a subgroup of G group. made up of sums of the form: h1 +h2 +···+hn where hi ∈ Hi, provided that each x ∈ H has discrete filtration A finite collection n a unique representation as the sum of elements F 1,...,F of submodules of a module A i i+ n from the subgroups {Hi}. The same notation is such that F ⊃ F 1 and F = 0. used as above. (4)IfA is a k × l matrix and B is an m × n discrete series Suppose that G is a con- matrix, then the direct sum of A and B is the nected, semisimple Lie group, with a square in- (k + m) × (l + n) partitioned matrix tegrable representation. The set of all square G   integrable representations of is the discrete A series of the irreducible unitary representations 0 . 0 B of G.

discrete valuation A non-Archimedean val- v v direct trigonometric functions The usual uation is discrete if the valuation ideal of is trigonometric functions (sine, cosine, tangent, a nonzero principal ideal. In this case the valu- v etc.) as opposed to the inverse trigonometric ation ring for is also said to be discrete. functions. discrete valuation ring See discrete valua- direct variation See direct proportion. tion. discriminant (1) For the quadratic equation Dirichlet algebra A closed subalgebra A of ax2 +bx +c = 0, the number 9 = b2 −4ac.If the continuous complex-valued functions C(X) 9>0, then the equation has two real-valued on a compact Hausdorff space X such that (i.) solutions. If 9<0, then the equation has A contains the constant functions, (ii.) A sep- two complex-valued solutions which are com- arates points of X, and (iii.) { (f ), f ∈ A} is plex conjugates. If 9 = 0, then the equation dense in C (X), the space of all real-valued and R has a double root which is real valued. continuous functions on X. (2) For the conic section Ax2 +Bxy+Cy2 + Dx+Ey +F = 0, the number 9 = B2 −4AC. Dirichlet L-function The function L(s), de- If 9>0, then the conic section is a hyperbola. fined by If 9<0, then the conic section is an ellipse. 9 = ∞ If 0, then the conic section is a parabola. L(s) = χ(n)/ns , The discriminant is invariant under rotation of n=1 the axes.

c 2001 by CRC Press LLC discriminant of equation See discriminant. division (1) Finding the quotient q and the r in the division algorithm. See divi- disjoint unitary representations A pair, U1 sion algorithm. and U2, of unitary representations of a group (2) A binary operation which is the inverse of such that no subrepresentation of one is equiva- the multiplication operation. See also quotient. lent to a subrepresentation of the other. division algebra An algebra such that every disjunctive programming In mathematical nonzero element has a . programming, the task is to find an extreme value a of a given function f , which maps a set A into division algorithm Given two integers and b an ordered set R. Usually A is a closed subset , not both equal to zero, there exist unique q r a = qb + r of a Euclidean space and (usually) A is defined integers and such that and ≤ r<|q| q r by a collection of inequalities or equalities. In 0 . is called the quotient and disjunctive programming, the set A is not con- is called the remainder. Also called Euclidian nected. Algorithm. The division algorithm is used in the develop- ment of a number of ideas in elementary num- distributive algebra A linear space A, over ber theory, including greatest common divisor a field K, such that there is a bilinear mapping and congruence. There are other situations in (or multiplication) A × A → A. If the multipli- which a division algorithm holds. See greatest cation does not satisfy the associative law, the common divisor, congruence, division of poly- algebra is nonassociative. nomials, Gaussian integer. distributive law A law from algebra that x division by logarithms To compute y , first states that if a, b, and c belong to a set with x compute c = log x − log y. Then = bc. two binary operations + and ·, then it is true that b b y This can be an aid to computation, when a table a·(b+c) = a·b+a·c and (b+c)·a = b·a+c·a. of logarithms to the base b is available. See also logarithm. dividend The quantity a in the division al- x gorithm. It is a quantity which is to be divided For any real number x, is a a 0 by another quantity; that is, the number in b . not defined. See division algorithm. division of complex numbers For real num- divisibility relation Suppose that a, b, and bers a, b, c, and d c lie in a ring R and that a = bc. Then we a + bi ac + bd bc − ad say that b divides a (written b|a), and call this a = + i divisibility relation where b is a divisor or factor c + di c2 + d2 c2 + d2 of a. This formula comes from multiplying the nu- merator and denominator of the original expres- divisible An integer a is divisible by an inte- sion by c − di. ger b if there exists another integer k such that a = bk . division of polynomials If polynomials f(x) and g(x) belong to the polynomial ring F [x], G divisible group An Abelian group (under and the degree of g(x) is at least 1, then there the operation of addition) such that, for every exist unique polynomials q(x) and r(x) in F [x] g ∈ G n ∈ x ∈ G and every N, there exists such that such that g = nx. In other words, each element in G is divisible by every natural number. An f(x)= q(x)g(x) + r(x) example of a divisible group is the factor group G/T , where T is the torsion subgroup of G. See where r(x) ≡ 0 or the degree of r(x)is less than torsion group. the degree of q(x). The process is sometimes

c 2001 by CRC Press LLC called synthetic division. See also division al- Doolittle method of factorization An “LU- gorithm, degree of polynomial. factorization” method for a square matrix A. The method concerns factoring A into the prod- division of whole numbers See division al- uct of two square matrices: L is a lower triangu- gorithm. See also divisible. lar matrix and U is an upper triangular matrix. All of the diagonal elements of L are required divisor (1) The quantity b in the division to be 1. Explicit formulas can then be created L U algorithm. See division algorithm. for the rest of the entries in and . See also Gaussian elimination. (2) For an integer a, the integer d is called a a b divisor of if there is another integer so that double chain complex A double complex a = bd d a d × . Colloquially, is a divisor of if of chains B over  is an object in MZ Z, to- a  “evenly divides” . gether with two endomorphisms ∂ : B → B and ∂ : B → B of degree (−1, 0) and (0, −1), divisor class An element of the factor group respectively, called the differentials, such that of the group of divisors on a Riemann surface         by the subgroup of meromorphic functions. The ∂ ∂ = 0,∂∂ = 0,∂∂ + ∂ ∂ = 0 . factor group is called the divisor class group. In other words, we are given a bigraded family divisor class group If X is a smooth alge- of -modules {Bpq}, p, q ∈ Z, and two families  braic variety, then the divisor class group is the of -module homomorphisms free Abelian group on the irreducible codimen-     ∂ : B → B , ∂ : B → B , sion one subvarieties of X (these are called “di- pq pq p−1q pq pq pq−1 visors”), modulo divisors of the form (f ) = (f )0 − (f )∞, for all rational (or meromorphic) such that the earlier three equations involving   functions f on X; here (f )0 is the divisor of the operators ∂ and ∂ hold. zeros and (f )∞ is the divisor of poles. If X is not smooth, “divisor” in “divisor class double invariance Let  be a dense subgroup group” means Cartier divisor, i.e., a divisor which of the additive group R of real numbers, with the is locally given by one equation; that is to say, discrete topology. Let G be the character group one which is locally of the form (f ) for a rational of . Any element a ∈ , as a character of G, function f on X. defines a continuous function χa on G. Let σ be the Haar measure of G. A closed subspace M of Divisor class groups also exist in commuta- L2(σ ) is called invariant if χaM ⊆ M, for all tive algebra, and in geometry for singular germs. a ∈  with a ≥ 0. M is called doubly invariant See divisor class. if χaM ⊆ M for all a ∈ . Such invariance is called double invariance. divisor of set Let X be a non-singular vari- ety defined over an algebraically closed field k. Douglas algebra Let L∞ be the space of A closed irreducible subvariety Y ⊆ X having bounded functions on the unit circle and H ∞ codimension 1 is called a prime divisor. An ele- be the subspace of L∞ consisting of functions ment of the free Abelian group generated by the whose harmonic extension to the unit disk is set of prime divisors is called a divisor on X. bounded and analytic. An inner function is a function in H ∞ whose modulus is 1 almost ev- domain The domain of a function is the set erywhere. A Douglas algebra is a subalgebra ∞ ∞ on which the function is defined. For example, of L generated by H and the conjugates the function f(x) = sin x has domain R since of finitely many inner functions (in the uniform the sine of every real number is defined,√ while topology). the domain of the function f(x) = x is the A theorem of Chang and Marshall asserts that non-negative real numbers. See also integral every uniform algebra A, with H ∞ ⊂ A ⊂ L∞ domain, unique factorization domain, range. is a Douglas algebra.

c 2001 by CRC Press LLC downhill method of finding roots To find dual Hopf algebra Suppose that (A,φ,ψ) the real roots of is a graded Hopf algebra. Then (A∗,φ∗,ψ∗) is also a graded Hopf algebra which is called the f(x,y) = , g(x, y) = , 0 0 dual Hopf algebra. See graded Hopf algebra. find points (a, b) where φ = f + g has its ex- duality If X is a normed complex vector treme values. In the downhill method the co- space, then the set of all bounded linear func- ordinates of (a, b) are approximated by choos- tionals on X is called the dual of X and is usu- ing an estimate (x ,y ), selecting a value for h, ∗ 0 0 ally denoted X . The dual space can be de- evaluating φ at the nine points xj = x + M h, 0 1 fined for many other classes of spaces, includ- yk = y + M h where Mi =−1, 0, 1, and con- 0 2 ing topological vector spaces, Banach spaces, structing the quadratic surface, φ = a +a x + 1 0 1 and Hilbert spaces. An identification of the dual a y +a (3x2 −2)+a (3y2 −2)+a xy, where 2 3 4 5 space is usually referred to as duality. For ex- the values of the coefficients aj are calculated p p∗ ample, the duality of L spaces, where L = by applying the method of least squares and us- q − − L ,p 1 + q 1 = 1, 1 ≤ p<∞. ing the values of φ1 at the nine points (xj ,yk). (x ,y ) Then replace the 0th approximation 0 0 to duality principle in projective geometry (a, b) (x ,y ) by the center 1 1 of the quadratic sur- Suppose that P n is a finite dimensional projec- φ face defined by 1. The process is repeated with tive geometry of dimension n. Suppose that T h smaller and smaller values of . is a proposition in P n and P n−r−1 (0 ≤ r ≤ n) and that “contains” and “contained in” are re- Drazin inverse See generalized inverse. versed, in the statement of T , obtaining in this way a new statement Tˆ , called the dual of T . dual algebra Suppose that (A,µ,η) is an Then T is true if and only if Tˆ is true. algebra over a field k (with multiplication µ and unit mapping η), and (C,9,M)is a coalgebra of Duality Theorem A theorem in the study (A,µ,η). Then (C∗,µ,η)is the dual algebra of linear programming. The Duality Theorem of (C,9,M) if C∗ is the dual space of C and states that the minimum value of c x + c x + µ, η and 9,M are correspondingly dual. 1 1 2 2 ···+cnxn in the original problem is equal to the maximum value of b y + b y +···+bmym dual coalgebra Suppose that (A,µ,η)is an 1 1 2 2 ◦ in the dual problem, provided an optimal solu- algebra over a field k and that A is the collection ∗ tion exists. If an optimal solution does not exist, of elements of the dual space A whose kernels then there are two possibilities: either both fea- each contain an ideal I where A/I is finite di- ◦ sible sets are empty, or else one is empty and mensional. Then (A ,9,M), where 9 and M are the other is unbounded. See dual linear pro- induced dually by µ and η, respectively, is the gramming problem. dual coalgebra. dual curve Suppose that  is an irreducible dual linear programming problem (1) An- plane curve of degree m ≥ 1 in a projective other linear programming problem which is in- plane. The dual curve ˆ in the dual projective timately related to a given one. Consider the c x + plane is the closure of the set of tangent lines to linear programming problem: minimize 1 1 c x +···+c x x ≥  at its nonsingular points. The dual of ˆ is . 2 2 n n, where all i 0 and subject to the system of constraints: A = A dual graded module Suppose n≤0 n ∗ a11x1 + a12x2 +···+a1nxn ≥ b1 is a graded module over a field k.IfAn is the ∗ ∗ a21x1 + a22x2 +···+a2nxn ≥ b2 dual of the module An, then A = An is the . dual graded module of A. . am1x1 + am2x2 +···+amnxn ≥ bm . dual homomorphism A mapping φ : L1 → L2, of one lattice to another, such that φ(x∩y) = The following linear programming problem is φ(x) ∪ φ(y) and φ(x ∪ y) = φ(x) ∩ φ(y). known as the dual problem: maximize b1y1 +

c 2001 by CRC Press LLC b2y2 +···+bmym where all yi ≥ 0 and subject dual space (1) For a vector space V over a to the system of constraints: field F , the vector space V ∗, equal to the set of all linear functions from V to F . V ∗ is called a y + a y +···+a y ≤ c 11 1 21 2 m1 m 1 the algebraic dual space. a y + a y +···+a y ≤ c 12 1 22 2 m2 m 2 (2) In the case where H is a normed vector . . space over a field F , the continuous dual space H ∗ is the set of all bounded linear functions from a1ny1 + a2ny2 +···+amnym ≤ cn . H to F . By bounded, we mean that each linear The Duality Theorem describes the relation be- function f satisfies tween the optimal solution of the two problems. sup |f(x)| See Duality Theorem. < ∞ . x=0 x (2) The definition above may also be stated x∈H in matrix notation. If c( is an 1 × n vector, b( is a m × 1 vector, and A is an m × n matrix, then This type of dual space is the focus of theorems the original linear programming problem above in such as the Riesz Repre- can be stated as follows: Find the n × 1 vector sentation Theorem. x( which minimizes c(x(, subject to x( ≥ 0 and duodecimal number system A number sys- Ax( ≥ b(. The dual problem is to maximize y(b(, tem using a base of 12 rather than 10. The Ara- subject to y( ≥ 0 and yA( ≤ c. bic numerals 0 through 9 are utilized, along with (3) The dual of a dual problem is the original two other symbols X and , which are used to problem. represent 10 (base 10) and 11 (base 10). The dual module For a module M overanar- number 12 (base 10) is then represented in the bitrary ring R, the module denoted M∗, equal duodecimal system as 10. to the set of all module homomorphisms (also Durand-Kerner method of solving algebraic called linear functionals) from M to R. This equations A method of solving an algebraic dual module is also denoted HomR(M, R) and n n− equation f(z) = z + a z 1 +···+an = 0 is, itself, a module. See also homomorphism, 1 (with complex coefficients and an = 0) using an linear function. iteration formula for approximating the n roots z ,...,z f dual quadratic programming problem 1 n of i: P Suppose that is the quadratic programming n problem: zi,k+1 = zi,k − f zi,k / zi,k − zj,k , T T j=1 maximize z = c x − (1/2)x Dx , j = i, i = 1,... ,n, k = 0, 1, 2,.... The subject to the constraints Ax ≤ b, x ≥ 0, x in n f(z) n method approximates all roots of simul- R . taneously. Speed of convergence is second or- Then the dual quadratic programming prob- der. lem PD is: dynamic programming An approach to a w = bT y + ( / )xT Dx , minimize 1 2 multistep decision process, in which an outcome subject to the constraints AT y + Dx ≥ c, all is calculated for each stage. In Richard Bell- man’s approach to dynamic programming, an components xi ≥ 0, yi ≥ 0. It can be shown that if x∗ solves P, then PD optimal policy has the property that, for each ini- has solution x∗,y∗, and max z = max w. tial state and decision, the subsequent decisions must generate an optimal policy with respect to dual representation Let π : G → GL(V ) the outcome of the initial state and initial deci- be a representation of the group G in the linear sion. This is now the most widely used approach space V . Then the dual representation π ∨ : to dynamic programming. ∗ ∨ −1 ∗ G → GL(Vπ ) is given by π (g) = π(g ) .

c 2001 by CRC Press LLC See also eigenvalue, eigenvector.

eigenspace in the weaker sense Suppose that L is a linear space, T is a linear transfor- E mation on L, and λ is an eigenvalue of T . Then the collection of all elements v in L such that (T − λI)kv = 0, for some integer k>0, is the e One of the most important constants in eigenspace in the weaker sense, corresponding mathematics. The following are two common to the eigenvalue λ. Such an element v is some- ways of approximating this number: times called a root vector of T . n 1 e = lim 1 + eigenvalue For an n × n matrix (or a linear n→∞ n operator) A, a number λ, such that there exists a non-zero n × 1 vector v satisfying the equation and ∞ Av = λv. The product of the eigenvalues of a e = 1 , n! matrix equals its determinant. See also eigen- n=0 vector. where n!=1 · 2 · 3 ···n is the factorial. The number e is transcendental and its value eigenvector A non-zero n × 1 vector v that is approximately 2.7182818 ... . The letter e satisfies the equation Av = λv for some num- stands for Euler. ber λ, for a given n × n matrix A. See also eigenvalue. effective divisor See integral divisor. Eisenstein series One of the simplest exam- effective genus Suppose that  is an irre- ples of a modular form, defined as a sum over ˜ ducible algebraic curve and  is a nonsingular a lattice. In detail: Let  be a discontinuous curve that is birationally equivalent to . Then group of finite type operating on the upper half ˜  the genus of is the effective genus of . An al- plane H , and let κ1,...,κh be a maximal set gebraic curve  is rational if its effective genus of cusps of  which are not equivalent with re- is 0 and elliptic if its effective genus is 1. spect to . Let i be the stabilizer in  of κi, and fix an element σi ∈ G = SL(2, R) such that f −1 eigenfunction A function which satisfies σi∞=κi and such that σi iσi is equal to the T(f) = λf λ the equation , for some number , group 0 of all matrices of the form where T is a linear transformation from a space of functions into itself. For example, if T is 1 b the linear transformation on the space of twice 1 differentiable functions of one variable: with b ∈ Z. Denote by y(z) the imaginary part d2f T(f)= of z ∈ H . The Eisenstein series Ei(z, s) for the dx2 cusp κi is then defined by then cos(x) is an eigenfunction of T , with λ = E (z, s) = y(σ−1σz)s ,σ∈  \ , −1. Eigenfunctions arise in the analysis of par- i i 1 tial differential equations such as the heat equa- s tion. See also eigenvalue. where is a complex variable. f(x)= a +a x + eigenspace For an eigenvalue λ of a matrix Eisenstein’s Theorem If 0 1 a x2 +···+a xn (or linear operator) A, the set of all eigenvectors 2 n is a polynomial of positive de- associated with λ, along with the zero vector. gree with integral coefficients, and if there exists p p The eigenspace is the space of all possible solu- a prime number such that divides all of the f(x) a p2 tions of the vector equation coefficients of except n, and if does not divide a0, then f(x)is irreducible (prime) over (λI − A)x = 0 . the field of rational numbers; that is, it cannot

c 2001 by CRC Press LLC be factored into the product of two polynomials sional Hilbert and Banach spaces. The connec- with rational coefficients and positive degrees. tion between shift operators and elementary Jor- dan matrices is that the nilpotent operator T −λI elementary divisor (of square matrix) For may be thought of as the finite dimensional ana- a matrix A with entries in a field F , one of the log of the shift S. finitely many monic polynomials hi,1≤ i ≤ s, over F , such that h1|h2|···|hs, and A is simi- elementary symmetric polynomial A poly- lar to the block diagonal sum of the companion nomial of several variables that is invariant un- matrices of the hi. der permutation of its variables, and which can- not be expressed in terms of similar such polyno- elementary divisor of a finitely generated mials of lower degree. For polynomials of two module For a module M over a principal variables, x + y and xy are all the elementary ideal domain R, a generator of one of the finitely symmetric polynomials. many ideals Ii of the polynomial ring R[X], 1 ≤ i ≤ s, such that I1 ⊆ I2 ⊆ ··· ⊆ Is elementary symmetric polynomials For n M and is isomorphic to the direct sum of the variables x1,...,xn, the elementary symmetric R[X]/I modules i. polynomials are σ1, ··· ,σn, where σk is the sum of all products of k of the variables x1,...,xn. n × n elementary Jordan matrix An square For example, if n = 3, then σ1 = x1 + x2 + x3, matrix of the form σ = x x + x x + x x and σ = x x x .   2 1 2 1 3 2 3 3 1 2 3 λ 00... 0   elimination of variable A method used to  1 λ 0 ... 0    solve a system of equations in more than one  ..   01 .  . variable. First, in one of the equations, we solve  . .   . ..  for one of the variables. We then substitute that 00... 1 λ solution into the rest of the equations. For ex- ample, when solving the system of equations: In the special case where n = 1, every matrix (λ) is an elementary Jordan matrix. (In other 2x + 4y = 10 words, one can forget about the 1s below the 8x + 9y = 47 diagonal if there is no room for them.) It is easier to understand this if we phrase it in we can solve the first equation for x, yielding: the language of linear operators rather than ma- x = 5−2y. Then, substituting this solution into trices. An elementary Jordan matrix is a square the second equation, we have 8(5 − 2y)+ 9y = matrix representing a linear operator T with re- 47, which we can then solve for y. Once we spect to a basis e1,...,en, such that Te1 = have a value for y, we can then determine the λe1+e2, Te2 = λe2+e3, ..., Ten−1 = λen−1+ value for x. See also Gaussian elimination. n en, and Ten = λen. Thus (T − λI) = 0, that is T −λI must be nilpotent, but n is the smallest elliptic curve A curve given by the equation positive integer for which this is true. Here, I is λ 2 3 2 the identity operator, and the scalar is called y + a1xy + a2y = x + a3x + a4x + a5 the generalized eigenvalue for T . It is a theorem of that every where each ai is an integer. Elliptic curves were matrix with entries in an algebraically complete important in the recent proof of Fermat’s Last field, such as the complex numbers, is similar Theorem. See Fermat’s Last Theorem. to a direct sum of elementary Jordan matrices. See Jordan canonical form. The infinite dimen- elliptic function field The field of functions sional analog of this theorem is false. How- of an√ elliptic curve; a field of the form ever, shift operators, that is operators S such that k(x, f(x)), where k is a field, x is an inde- Sei = ei+1, for i = 1, 2, 3,..., play a promi- terminant over k, and f(x)is a separable, cubic nent role in operator theory on infinite dimen- polynomial, with coefficients in k.

c 2001 by CRC Press LLC elliptic integral One of the following three entire linear transformation Consider a lin- types of integral. ear fractional transformation (also known as a Type I: Möbius transformation) in the complex plane; that is, a function given by x dt φ dt = . az + b 0 (1 − t2)(1 − k2t2) 0 1 − k2 sin2 t S(z) = , cz + d Type II: where a, b, c, and d are complex numbers. If x 1 − k2t2 φ c = 0 and d = 0, S(z) becomes a typical linear dt = − k2 2 tdt. 2 1 sin transformation, which is also an entire function. 0 1 − t 0 Type III: enveloping algebra Let B be a subset of an A x dt associative algebra with multiplicative iden- tity 1. The subalgebra B† of A containing 1 and 0 (t2 − a) (1 − t2)(1 − k2t2) generated by B is called the enveloping algebra B A φ dt of in . See associative algebra. = . 0 (sin2 t − a) 1 − k2 sin2 t enveloping von Neumann algebra Let A de- note a C∗-algebra and S denote the state space Here 0

c 2001 by CRC Press LLC the category. A function e from set X to set of the variables is the equality true? The task of Y is surjective or onto if e(X) = Y , that is if answering this question is referred to as solving every element y ∈ Y is the image e(x) of an el- the equation. ement x ∈ X. See also morphism in a category, monomorphism, surjection. equivalence class Given an equivalence re- lation R on a set S and an x ∈ S, the equivalence Epstein zeta function A function, ζQ(s, M), class of x, usually denoted by [x], consists of all defined by Epstein in 1903. Let V be a vector y ∈ S such that (x, y) ∈ R. Clearly, x ∈[x], space over R having dimension n. Let M be a and (x, y) ∈ R if and only if [x]=[y].As lattice in V and Q(X) a positive definite qua- a consequence, the equivalence classes of R in- dratic form defined on V . Then, ζQ(s, M) is duce a partition of the set S into non-overlapping defined for complex numbers s by subsets. 1 equivalence properties The defining prop- ζQ(s, M) = . Q(x)s erties of an equivalence relation R ⊆ S × S; x∈M\{0} namely, that R is reflexive ((x, x) ∈ R for all x ∈ The Epstein zeta function is absolutely conver- S), symmetric ((x, y) ∈ R whenever (y, x) ∈ s>m Q(x) R), and transitive ((x, z) ∈ R whenever (x, y) ∈ gent when 2 .If is a positive integer for all x ∈ M \{0}, then R and (y, z) ∈ R). See also relation, equiva- lence class. ∞ a(k) ζ (s, M) = , Q ks equivalence relation A relation R on a set k=1 S (that is, a subset of S × S), which is reflex- ive, symmetric, and transitive. (See equivalence where a(k) denotes the number of distinct x ∈ properties.) For example, let S be the set of all M with Q(x) = k. The Epstein zeta function, integers and R the subset of S × S defined by in general, has no Euler product expansion. See (x, y) ∈ R if x − y is a multiple of 2. Then R is also Riemann zeta function. an equivalence relation because for all x,y,z ∈ S (x, x) ∈ R (x, y) ∈ R equal fractions Two fractions (of positive , (reflexive), whenever m k (y, x) ∈ R (x, z) ∈ R integers), and , are equal if m7 = nk.If (symmetric), and when- n 7 (x, y) ∈ R (y, z) ∈ R the greatest common divisor of m and n is 1 and ever and (transitive). the greatest common divisor of k and 7 is also equivalent divisors See algebraic equiva- 1, then the two fractions are equal if and only if lence of divisors. m = k and n = 7. equivalent equations Equations that are sat- equality (1) The property that two mathemat- isfied by the same set of values of their respec- ical objects are identical. For example, two sets tive variables. For example, the equation x2 = A, B are equal if they have the same elements; 3y−1 is equivalent to the equation 6w−2z2 = 2 we write A = B. Two vectors in a finite dimen- because their solutions coincide. sional vector space are equal if their coefficients with respect to a fixed basis of the vector space + equivalent valuations Let φ : F → R be are the same. + a valuation on a field F (here R denotes the set (2) An equation. See equation. of all nonnegative real numbers). The valuation φ gives rise to a metric on F , where the open equation An assertion of equality, usually neighborhoods are the open spheres centered at between two mathematical expressions f , g in- a ∈ F defined by volving numbers, parameters, and variables. We f = g + write . When the equation involves one {b ∈ F | φ(b − a)<9},9∈ R . or more variables, the equality asserted may be true for some or all values of the variables. A Two valuations are called equivalent if they in- natural question then arises: For which values duce the same topology on F .

c 2001 by CRC Press LLC error In the context of , an that Y is G2inX. Let X be the normaliza- error occurs when a real number x is being ap- tion of X in K(X) , and let f : X → X be  proximated by another real number xˆ. A typical the natural map. Then there is a subvariety Y example arises in the implementation of numeri-   of X which is G3inX and such that f is cal operations on computing machines, where an Y  error occurs whenever a real number x is made an isomorphism of Y  onto Y , and such that f machine representable either by rounding off or is étale at points of X in a suitable neighbor- by truncating at a certain digit. For example, if hood of Y . Then (X,Y) is an étale neighbor- x = 4.567, truncation at the third digit yields hood of (X, Y ). The étale neighborhoods form xˆ = 4.56 while rounding off at the third digit a subbasis for the étale topology. See also étale yields xˆ = 4.57. morphism. There are two common ways to measure the error in approximating x by xˆ: The quantity Euclidean domain Let R be a ring. Then R is an integral domain if x,y ∈ R and xy = 0 |x −ˆx| implies either x = 0ory = 0. An integral domain is Euclidean if there is a function d from is referred to as the absolute error, and the quan- the non-zero elements of R to the non-negative tity integers such that |x −ˆx| (x = 0) (i.) For x = 0, y = 0, both elements of R,we |x| have d(x) ≤ d(xy); is referred to as the relative error. The concept (ii.) Given non-zero elements x,y ∈ R, there of error is also used in the approximation of a exists s,t ∈ R such that y = sx + t, where vector x ∈ Rn by xˆ ∈ Rn; the absolute value in either t = 0ord(t)n. solute value entry of xˆ has approximately k cor- It can be described as follows: rect significant digits. 1. Divide n into m, i.e., find a positive p r étale Let X and Y be schemes of finite type integer and a real number so that m = pn + r ≤ r

c 2001 by CRC Press LLC iterations. For example, the modified algorithm which converges for all real numbers s>1. for the above numbers yields Euler observed that 954 = 30 × 32 − 6, ζ(s) = 1 , 32 = 5 × 6 + 2, 1 − p−s 6 = 3 × 2 + 0. where p runs over all prime numbers. This in- Hence g.c.d. (954, 32) = 2 in three iterations. finite product is called the Euler product. One of the main questions regarding zeta functions ring An integral domain R such that is whether they have similar infinite product ex- there exists a function d(·) from the nonzero el- pansions, usually referred to as Euler product ements of R into the nonnegative integers which expansions. satisfies the following properties: (i.) For all nonzero a,b∈ R, d(a) ≤ d(ab). Euler’s formula The expression a,b ∈ R (ii.) For any nonzero , there exist eiz = z + i z, p, r ∈ R a = pb + r cos sin such that , where either √ r = 0ord(r)

The elements of C+(Q) and C−(Q) are called Euler-Poincaré characteristic Let K be a the even and odd elements, respectively, of the simplicial complex of dimension n, and let αr Clifford algebra. denote the number of r-simplexes of K. The Euler-Poincaré characteristic is even number An integer that is divisible by n 2. An even number is typically represented by χ(K) = (− )r α . 1 r 2n, where n is an integer. r=0 χ(K) is a generalization of the Euler charac- evolution Another term for the process of teristic, V − E + F , where V,E,F are, re- extracting a root. See extraction of root. spectively, the numbers of vertices, edges, and faces of a simple closed polyhedron (namely, a exact sequence Consider a sequence of mod- {M } polyhedron that is topologically equivalent to ules i and a sequence of homomorphisms {h } a sphere). Euler’s Theorem in combinatorial i with h : M −→ M topology states that V − E + F = 2. See also i i−1 i Lefschetz number. for all i = 1, 2,... The sequence of homomor- phisms is called exact at Mi if Imhi = Kerhi+1. Euler product Consider the Riemann zeta The sequence is called exact if it is exact at every function Mi. For a sequence

1 1 h ζ(s) = 1 + +···+ +··· , −→ M −→2 M , 2s ks 0 1 2

c 2001 by CRC Press LLC it is understood that the first arrow (from 0 to then the associated covariant exact sequence of M1) represents the 0 map, so the sequence is ext is a certain long exact sequence of the form exact if and only if h2 is injective. 0 → Hom(M, A) → Hom(M, B) exact sequence of cohomology (1) Suppose → Hom(M, C) G is an Abelian group and A is a subspace of → Ext1 (M, A) → Ext1 (M, B) X. Then there is a long exact sequence of coho- R R 1 mology: → ExtR(M, C) n n ···→H n(X,A,G)→ H n(X, G) →···→ExtR(M, A) → ExtR(M, B) n → ExtR(M, C) →··· . → H n(A, G) → H n+1(X,A,G)→··· .

n A Here H are the cohomology groups. exact sequence of homology (1) Suppose X (2)If0→ A → B → C → 0 is a short ex- is a subspace of . Then there exists a long act sequence of complexes, then there are natural exact sequence maps δi : H i(C) → H i+1(A), giving rise to an ···→Hn(A) → Hn(X)s exact sequence → Hn(X, A) → Hn− (A) →··· , ···→H i(A) → H i(B) → H i(C) 1 called the exact sequence of homology, where → H i+1(A) →··· Hn(X) denotes the nth homology group of X and Hn(X, A) is the nth relative homology group. of cohomology. (2)If0→ A → B → C → 0 is a short ex- act sequence of complexes, then there are natural exact sequence of ext (contravariant) If maps δi : Hi(C) → Hi−1(A), giving rise to an exact sequence 0 −→ A −→ B −→ C −→ 0 ···→Hi(A) → Hi(B) → Hi(C) is a short exact sequence of modules over a ring → H (A) →··· R and M is an R-module (on the same side as i−1 A), then the associated contravariant exact se- of homology. quence of ext is a certain long exact sequence of the form exact sequence of Tor Let A be a right - module and let B → B → B be an exact 0 → Hom(C, M) → Hom(B, M) sequence of left -modules. Then there exists → Hom(A, M) an exact sequence → 1 (C, M) → 1 (B, M)      ExtR ExtR Torn (A, B ) → Torn (A, B) → Torn (A, B ) 1    → ExtR(A, M) →···→ (A, B ) → (A, B) Tor1 Tor1 → (A, B) → A ⊗ B → A ⊗ B →···→ n (C, M) → n (B, M) Tor1   ExtR ExtR  → A ⊗ B → 0 . n → ExtR(A, M) →··· . exceptional compact real simple Lie algebra exact sequence of ext (covariant) If Since compact real semisimple Lie algebras are in one-to-one correspondence with complex 0 −→ A −→ B −→ C −→ 0 semisimple Lie algebras (via complexification), the classification of compact real simple Lie al- is a short exact sequence of modules over a ring gebras reduces to the classification of complex R and M is an R-module (on the same side as A), simple Lie algebras. Thus, the compact real

c 2001 by CRC Press LLC simple Lie algebras corresponding to the ex- exhaustive filtration A filtration {Mk : k ∈ ceptional complex simple Lie algebras El (l = Z} of a module M is called exhaustive (or con- 6, 7, 8), F4 and G2 are called exceptional com- vergent from above)if pact real simple Lie algebras. ∪kMk = M. exceptional complex simple Lie algebra In the classification of complex simple Lie algebras See filtration. See also discrete filtration. α, there are seven categories: A,B,C,D,E,F,G, resulting from all possible Dynkin diagrams. Existence Theorem (class field theory) For The notation in each category includes a sub- any ideal group there exists a unique class field. script l (e.g., El), which denotes the rank of α. See ideal group, class field. The algebras in categories El (l = 6, 7, 8), F4 n × n and G2 are called exceptional complex simple expansion of determinant Given an Lie algebras (in contrast to the classical com- matrix A = (aij ), the determinant of A is for- plex simple Lie algebras). mally defined by exceptional complex simple Lie group A detA = sgn(σ )a1σ(1)a2σ(2) ...anσ (n) , complex connected Lie group associated with σ∈Sn one of the exceptional complex simple Lie alge- where Sn denotes the of de- bras El (l = 6, 7, 8), F4 or G2. gree n (the group of all n! permutations of the { , ,...,n} (σ ) exceptional curve of the first kind Given set 1 2 ), and where sgn denotes the σ (σ ) = σ two mutually nonsingular surfaces F,F and a sign of the permutation . (sgn 1if is (σ ) =− σ birational transformation T : F → F , the total an even permutation and sgn 1if is transform E of a simple point in F  by T −1 an odd permutation.) The formula in the equa- is called an exceptional curve. It is called an tion above is referred to as the expansion of the A exceptional curve of the first kind if, in addition, determinant of . T is regular along E. Otherwise, it is called an a exceptional curve of the second kind. See also exponent Given an element of a multiplica- a·a·...·a algebraic surface. tive algebraic structure, the product of , in which a appears k times, is written as ak and k is referred to as the exponent of ak. The sim- exceptional curve of the second kind See plest example is when a is a real number. In exceptional curve of the first kind. this case, we can also give meaning to negative exponents by a−k = 1/ak (assuming that k is exceptional Jordan algebra A Jordan alge- a positive integer and that a = 0). We define bra that is not special. See special Jordan alge- a0 = 1. The basic laws of exponents of real bra. numbers are the following: (1) akam = ak+m, excess of nines A method for verifying the k m k−m (2) a /a = a (a = 0), accuracy of operations among integers, also k m km (3) (a ) = a , known as the method of casting out nines. It where k and m are nonnegative integers. We uses the sums of the digits of the integers in- can extend the definition to rational exponents volved, in modulo 9 arithmetic. We illustrate m/n, where m is any integer and n is a positive the method for addition of integers. Consider √ integer, by defining am/n = n am. Synonyms the sum for the exponent are the words index and power. 683 + 256 = 939 . See also exponential mapping.

The sum of digits of 683, 256, and 939 are 17 ≡ of a matrix Let A be 8 mod 9, 13 ≡ 4 mod 9 and 21 ≡ 3 mod 9, an n × n matrix over the complex numbers C. respectively. Indeed, 8 + 4 = 12 ≡ 3 mod 9. The exponential function f (A) = eA is defined

c 2001 by CRC Press LLC by the infinite series expression A mathematical statement, using mathematical quantities such as scalars, vari- eA = I + A + 1 A2 + 1 A3 +··· ables, parameters, functions, and sets, as well as 2! 3! relational and logical operators such as equality, ∞ conjunction, existence, union, etc. = 1 Ak . k! k=0 exsecant function A trigonometric function defined via the secant of an angle as exsecθ = !·! Letting denote the Euclidean vector norm, secθ − 1. Similarly, we define the excosecant as well as the induced matrix norm, we can show function as excosecθ = cscθ − 1. that the exponential function of a matrix is well defined (i.e., the series is convergent) by show- extension Given a subfield E of a field F , ing that it is absolutely convergent. Indeed, in- namely, a subset of F that is a field with respect voking well-known inequalities, and the defini- to the operations defined in F , we call F an ex tion of the real exponential function ,wehave extension field of E. The field F can be regarded !eA! that equals as a vector space over E. The dimension of F E ∞ ∞ over is called the degree of the extension 1 1 F E f ,...,f ∈ F Ak ≤ !A!k = e!A! < ∞ . field over .If 1 p , then by k! k! E(f ,...,fp) we denote the smallest subfield k=0 k=0 1 of F containing E and f1,...,fp. E(f1) is A called a simple extension of E. One basic property of e is that its eigenval- E F ues are of the form eλ, where λ is an eigenvalue If every element of is algebraic over ,we E of A. It follows that eA is always a nonsin- call an algebraic extension. Otherwise, we E gular matrix. Also, the exponential property call a transcendental extension. eA+B = eAeB holds if and only if A and B The notion of extension also applies to rings. commute, that is, AB = BA. See also number field. The exponential function of a matrix arises extension of coefficient ring Let R[t] be the in the solution of systems of linear differential ring of polynomials over the (coefficient) ring equations in the vector form dx(t)/dt = Ax(t), n R in the indeterminate t. As the notion of ex- t ≥ 0, where x(t) ∈ C . If the initial condition tension can also concern a ring, an extension of x(0) = x is specified, then the unique solution 0 R is usually referred to as the extension of the to this differential problem is given by x(t) = tA coefficient ring of R[t]. See extension, ring of e x . 0 polynomials. exponential mapping The mapping (func- x extension of valuation If v is a valuation on tion) f(x)= e , where x ∈ R and e is the base a field F and if K is an extension of F , then an of the . (See e.) The expo- extension of v to K is a valuation w on K such nential mapping is the inverse mapping of the x that w(x) = v(x), for x ∈ F . See valuation. natural logarithm function: y = e if and only if x = ln y. Mclaurin’s Theorem yields exterior algebra Let V be a vector space F T (V ) ∞ k over a field . Let also 0 denote the direct x x e = . sum of the tensor products V ⊗ ...⊗ V . T0(V ) k! k=0 is called the contravariant tensor algebra over V and is equipped with the product ⊗ as well More generally, the exponential mapping to base as addition and scalar multiplication. Let S be x a (a = 1) is defined by f(x)= a . formed by all elements of T0(V ) of the type v ⊗ v, as well as their sums, scalar multiples and The process of evaluating their products with arbitrary elements in T0(V ). ak, that is, evaluating the product a · a · ...· a, Then S is a subgroup of (the Abelian group) in which a appears k times. See also exponent. T0(V ) and the quotient group T0(V )/S can be

c 2001 by CRC Press LLC considered. This quotient group can be made Ext group The Ext group is defined in several into an algebra by defining the operations · and subjects. For example, if A is an Abelian group as follows: and if 0 → R → F → A → 0 c · (t + S) = ct + S(c∈ F), A is any free resolution of , then for every Abel- ian group B, there exists a group Ext(A, B)such (t1 + S) (t2 + S) = (t1 ⊗ t2) + S. that The operation is called the exterior product T (V )/S V → (A, B) → (F, B) of the exterior algebra 0 of , which is 0 Hom Hom denoted by V . V is also known as the → Hom(R, B) → Ext(A, B) → 0 Grassmann algebra of V . The image of v1 ⊗ ...⊗ v T (V ) → p under the natural mapping 0 is exact; moreover, the group is independent of V v ∧...∧v is denoted by 1 p and is called the the choice of the free resolution of A. The el- v ,...,v ∈ V exterior product of 1 p . In general, ements of Ext(A, B) are equivalence classes of the image of V ⊗ ...⊗ V with p factors under short exact sequences 0 → B → M → A → 0 the above natural mapping is called the p-fold p and addition is induced by Baer sum. exterior power of V and is denoted by V . The Ext group is also defined in homological The exterior product satisfies some important algebra, topology, and operator algebras. The rules and properties. For example, it is multilin- group Ext(A, B) is called the group of exten- v ∧ ...∧ v = v ,...,v ear, 1 p 0 if and only if 1 p sions of B by A. are linearly dependent, and u∧v = (−1)pq v∧u p q whenever u ∈ V and v ∈ V . extraction of root The process of finding a See also algebra, tensor product. root of a number (e.g., that the fifth root of 32 is 2) or the process of finding the roots of an G ,G external product Given two groups 1 2, equation. the external (direct) product of G1,G2 is the group G = G1 × G2 formed by the set of all extraneous root A root, obtained by solving pairs (g1,g2) with g1 ∈ G1 and g2 ∈ G2; the an equation, which does not satisfy the origi- operation in G is defined to be nal equation. Such roots are usually introduced when exponentiation or clearing of fractions is (g ,g ) g ,g = g g ,g g . 1 2 1 2 1 1 2 2 performed.

This definition can be extended in the obvious extreme terms of proportion Given a pro- way to any collection of groups G1,G2,.... a portion, namely, an equality of two ratios b = The operation in each component is carried out c d , the numbers a and d are called the extreme (or in the corresponding group. The external prod- outer) terms of the proportion. See proportion. uct coincides with the external sum of groups if the number of groups is finite. See also internal product.

c 2001 by CRC Press LLC factorial series The infinite series, involving factorials, ∞ 1 1 1 = 1 + + +··· k! 1! 2! F k=0 This series converges to the number e. See fac- factor (1) An integer n is a factor (or divisor) torial. of an integer m if m = nk for some integer k. Thus, ±1, ±2, and ±4 are all the factors of 4. factoring The process of finding factors; of More generally, given a commutative monoid an integer or of a polynomial, for example. Given M that satisfies the cancellation law, b ∈ M is a an integer n>1, the Fundamental Theorem factor of a ∈ M if a = bc for some c ∈ M.We of Arithmetic states that n can be expressed as then usually write b|a. See also prime, factor of a product of positive prime numbers, uniquely, polynomial. apart from the order of the factors. The process (2) A von Neumann algebra whose center is of finding these prime factors is referred to as the set of scalar multiples of the identity opera- the prime factoring or the prime factorization tor. The study of von Neumann algebras is car- of n. See also division algorithm, factoring of ried out by studying the factors which are type I, polynomials. II, or III with subtypes for each. See also type-I factor, type-II factor, type-III factor, Krieger’s factoring of polynomials The process of factor. finding factors of a polynomial f(x) ∈ F [x] over a field F .Iff(x)has positive degree, then factorable polynomial A polynomial that f(x)can be expressed as a product has factors other than itself or a constant poly- f(x)= cg (x)g (x)...g (x) , nomial. See also factor of polynomial. 1 2 r where c ∈ F and g ,g ,...,gr are irreducible G H 1 2 factor group Let be a group and let and monic polynomials in F [x]. This is referred G denote a normal subgroup of . The set of left to as the prime factoring or the prime factoriza- G (or right) cosets of , denoted by tion of f(x)and is unique apart from the order G/H ={aH : a ∈ G} , of the factors. If f(x) ∈ F [x] is monic and has positive forms a group under the operation (aH )(bH ) = degree, then there is an extension field E of F , (ab)H and is called the factor group or the quo- so that f(x)can be factored into tient group of G relative to H. f(x)= (x − r )(x − r ) ...(x − r ) The factor groups of a group G can be useful 1 2 k G in revealing important information about .For in E[x]. The field E is the splitting field of f(x) example, letting C(G) represent the center of and it satisfies E = F(r1,r2,...,rk), where G,ifG/C(G) is cyclic it follows that G is an r1,r2, ...,rk are the roots of f(x) in E.If Abelian group. the field F is algebraically closed (for example the complex numbers), then E = F . See also factorial The factorial of a positive integer n Factor Theorem, division algorithm. (read n factorial) is denoted by n! and is defined by factor of polynomial A polynomial g(x) ∈ F [x] F f(x) ∈ F [x] n!=n(n− 1)(n− 2)...321. over a field is a factor of if f(x)= g(x)h(x) for some h(x) ∈ F [x].For For example, 4!=4321= 24. By definition, example, g(x) = x − 1 is a factor of f(x) = 0!=1. An approximation of n! for large values x2 − 1. See also factoring of polynomials. of n is given by Stirling’s formula: n √ factor representation Consider a nontriv- n!≈ n πn . e 2 ial Hilbert space H and a topological group G

c 2001 by CRC Press LLC (i.e., a group with the structure of a topological of bounded linear operators acting on a Hilbert space so that the mappings (x, y) → xy and space H . See partially ordered space. x → x−1 are continuous). Let U be a unitary representation of G, namely, a homomorphism faithful R-module A module M over a ring of G into the group of unitary operators on H R such that, whenever r ∈ R satisfies rM = 0, that is strongly continuous. If the von Neumann then r = 0. Also called faithfully flat. algebra M generated by {Ug : g ∈ G} and its M commutant satisfy false position The method of false position  (or regular falsi method) is a numerical method M ∩ M ={z1 : z ∈ C} , for approximating a root r of a function f(x), r r then U is called a factor representation of G. given initial approximations 0 and 1 that sat- isfy f(r0)f (r1)<0. The next approximation r x factor set Suppose A is an Abelian group 2 is chosen to be the -intercept of the line (r ,f(r )) (r ,f(r )) and G is an operator group acting on A. Then through the points 0 0 and 1 1 . r f(r )f (r )< every element σ ∈ G defines an automorphism Then 3 is chosen as follows: If 1 2 0 σ t σ στ r x a → a of A such that (a au) = a .Afactor we choose 3 to be the -intercept of the line (r ,f(r )) (r ,f(r )) set is a collection of elements {aσ,τ : σ, τ ∈ G} through the points 1 1 and 2 2 . r x in A such that Otherwise, we choose 3 as the -intercept of the line through the points (r0,f(r0)) and σ aσ,τ aστ,ρ = aτ,ρaσ,τρ . (r2,f(r2)), and swap the indices of p0 and p1 to continue. This process ensures that succes- sive approximations enclose the root r. See also (x − a) Factor Theorem The linear term is secant method. a factor of the polynomial f(x) ∈ F [x] over a F f(a)= field if and only if 0. feasible region The set of all feasible solu- It is an immediate corollary of the Remainder tions of a linear programming problem. See also See Fac- Theorem. ( Remainder Theorem.) The feasible solution. tor Theorem can be useful in finding factors of polynomials: If f(x)= x4 + x3 + x2 + 3x − 6, feasible solution In linear programming, the then one knows that if a is an integer, then (x−a) objective is to minimize or maximize a linear is a factor of f(x)only if a divides 6. Hence it function of several variables, subject to one or makes sense to search for integer roots of f(x) more constraints that are expressed as linear among the factors of 6. equations or inequalities. A solution (choice) faithful function Let V and W be partially of the variables that satisfies these constraints ordered vector spaces with positive proper cones is called a feasible solution. See also feasible V + and W +, respectively. A function f : V → region. W is said to be faithful if f(x) = 0, with x ∈ V +, occurs only in the case where x = 0. Feit-Thompson Theorem Every non- As an example, let V be the complex vec- Abelian simple group must have even order. This tor space of n × n matrices with positive cone result was conjectured by Burnside and proved V + consisting of all positive semidefinite ma- by Feit and Thompson in 1963. It was an impor- trices. Let W be the complex field and W + = tant step and the driving force behind the effort [0, ∞), and let f : V → W be the trace func- to classify the finite simple groups. n tion: f(x) = trace(x) = xii, for all i=1 n x = (xij ) ∈ V . Then f is a faithful function. Fermat numbers Integers of the form 22 + Faithful functions arise in the theory of C∗- 1, where n is a nonnegative integer. For n = algebras, as follows: Let A be a C∗-algebra. 1, 2, 3, 4, the Fermat numbers are prime inte- By a theorem of Gelfand, Naimark, and Segal, gers. Euler proved, contrary to a conjecture by there is a faithful C∗-algebra homomorphism Fermat, that the Fermat number for n = 5isnot ρ : A → B(H ), where B(H ) is the C∗-algebra a prime.

c 2001 by CRC Press LLC Fermat’s Last Theorem There are no posi- of multiplication and addition. Also the residue tive integers x, y, z, and n, with n>2, satisfying ring modulo p, Zp, is a field when p is a prime xn + yn = zn. integer. See ring. Pierre Fermat wrote a version of this theo- rem in the margin of his copy of ’ field of quotients Let D = 0 be a com- ∗ Arithmetica. He commented that he knew of a mutative integral domain and let D denote its marvelous proof but that there was not enough nonzero elements. Consider the relation ≡ in ∗ space in the margin to present it. D × D defined by (a, b) ≡ (c, d) if ad = bc. This assertion is known as Fermat’s Last The- It can be shown that ≡ is an equivalence relation. orem, because it was the last unresolved piece (See equivalence relation.) Denote the equiva- of Fermat’s work. A proof eluded mathemati- lence class determined by (a, b) as a/b (called cians for over 300 years. In 1993, A. J. Wiles a quotient or a fraction) and let F ={a/b} be announced a proof of the conjecture. Some gaps the quotient set determined by ≡. We can now and errors that were found in the original proof equip F with addition, multiplication, an ele- were corrected and published in the Annals of ment 0, and an element 1 so that it becomes a Mathematics in 1995. field as follows:

fiber (1) Preimage, as of an element or a set; a/b + c/d = (ad + bc)/bd , inverse image. (a/b) · (c/d) = ac/bd , f : X → Y (2) In homological algebra, if is a 0 = 0/1, and 1 = 1/1 . morphism of schemes, y ∈ Y , k(y)is the residue y k(y) → Y field of , and Spec is the natural It can be shown that the above operations +, · f morphism, then the fiber of the morphism over define single-valued compositions in F and that y the point is the scheme F with the above 0 and 1 is a commutative ring. Moreover, if a/b = 0, then a = 0 and b/a is Xy = X ×Y Spec k(y) . the inverse of b/a. This shows that F is a field; it is called the field of quotients or the field of Also spelled fibre. fractions (or rational expressions)ofD. numbers The sequence of num- field of rational expressions See field of quo- bers fn given by the recursive formula fn = tients. fn−1 + fn−2 for n = 2, 3,..., where f1 = f = 1 (or sometimes f = 0,f = 1). It can 2 1 2 field of values See numerical range. be shown that every positive integer is the sum of distinct Fibonacci numbers. Any two con- field theory In algebra, the theory and re- secutive Fibonacci numbers are relatively prime. f search area associated with fields. See field. The ratios n+1 form a convergent sequence fn √ n →∞ 5+1 figure (1) A symbol used to denote an integer. whose limit as is the golden ratio, 2 . (2) In topology, a set of points in the space fibre See fiber. under consideration.

field A commutative ring F with multiplica- filtration A filtration of a module M is a tion identity 1, all of whose nonzero elements collection {Mk : k ∈ Z}, of submodules of M, are invertible with respect to multiplication: for such that Mk+1 ⊂ Mk for all k ∈ Z. See also any nonzero a ∈ F there exists c ∈ F such that exhaustive filtration, filtration degree. ac = 1. We usually write c = a−1. It fol- lows that if a,b ∈ F are nonzero elements of a filtration degree Suppose that M is a graded field, then so is ab, namely, every field is an inte- module with differentiation d of degree 1, i.e., k gral domain. Well-known examples of fields are M is a complex, and that the filtration {F M}k∈Z the rational numbers, the real numbers, and the of M is homogeneous. Define the graded mod- E (M) Ek(M) complex numbers with the familiar operations ule 0 to be the direct sum k∈Z 0

c 2001 by CRC Press LLC Ek(M) = F kM/Fk+1M { , ,...,n} where 0 . Define elements of the set 1 2 . We may then |S|=n S k,l k+l k k k+l write . The fact that a set is finite is de- F M = M ∩ F M = F M noted by |S| < ∞ and we say that S consists of a k,l k,l k+ ,l− finite number of elements. See also cardinality, and E (M) = F M/F 1 1M, where 0 finite function. k,l ∈ Z. Then E0(M) is doubly graded as the k,l direct sum E (M). In the same way, k,l∈Z 0 finite Abelian group A group G that is finite, the module E (H (M)) is doubly graded by the 0 as a set, and commutative; namely for any a,b ∈ Ek,l(H (M)) = F kH k+l(M)/F k+1 modules 0 G, ab = ba. See also finite, Abelian group. H k+l(M), where H(M) is the homology mod- M F kH k+l(M) = F k,lH(M) ule of and . finite basis A basis of a vector space V over ≤ r ≤∞ For 1 define a field F , comprising a finite number of vectors. k,l k+l k k+r More precisely, a finite basis of V is a finite set Zr (M) = Im H F M/F M of vectors B ={v1,v2,...,vn}⊂V with two properties: → H k+l F kM/Fk+1M , (i.) B is a spanning set of V , that is, every vector of V is a linear combination of the vectors Bk,l(M) = H k+l−1 F k−r+1M/FkM r Im in B. (ii.) B consists of linearly independent vec- → H k+l F kM/Fk+1M , tors (over the specified field F ). k,l k,l k,l A (finite) basis of a vector space is not, in Er (M) = Zr (M)/Br (M) . general, unique. However, all the bases of V E (M) Then r is doubly graded by identifying have the same number of vectors. This number Ek(M) r with the direct sum is called the dimension of V . Here the dimen- V n V Ek,l(M) . sion of is and thus is referred to as a finite r dimensional vector space. l∈Z n The set of vectors {ei}i= , consisting of the n 1 Finally, the differentiation operator dr : Er → vectors ei ∈ R whose ith entry is one and all k,l k,l Er is composed of homomorphisms dr : Er other entries equal zero, is a finite basis for the k+r,l−r+1 n → Er , in other words, dr has bi-degree vector space R . It is usually called the standard n (r, 1 − r). basis of R . In all of these doubly graded modules the first degree k is called the filtration degree, the sec- finite continued fraction Let q be a contin- ond degree l is called the complementary degree, ued fraction defined by q = p1 + 1/q1, where and k + l is called the total degree. q1 = p2 + 1/q2, q2 = p3 + 1/q3,..., and where pi,qi are numbers or functions of a vari- fine moduli scheme A moduli scheme Mg able. See continued fraction. If the expression of curves of genus g such that there exists a flat q terminates after a finite number of terms, then family F → Mg of curves of genus g such that, q is a finite continued fraction. For example, for any other flat family X → Y of curves of g Y → M 1 genus , there is a unique map g via q = 1 + X F + 1 which is the pullback of . 2 3

finer If T and U are two on a space is a finite continued fraction usually denoted by X T ⊆ U U + 1 1 , and if , then is said to be finer than 1 2+ 3 . T . finite field A field F which is a finite set. In finite A term associated with the number of such a case, the prime field of F can be identified elements in a set. A set S is finite if there ex- with Zp, the field of residue classes, modulo p, ists a natural number n and a one-to-one corre- for some prime integer p.(See prime field.) It spondence between the elements of S and the follows that |F |=pn for some integer n.We

c 2001 by CRC Press LLC thus have the fundamental fact that the num- group generated by b1,b2,...,bn. Then there ber of elements (cardinality) of a finite field is is a homomorphism h of F onto G. If the ker- a power of a prime integer. Moreover, all finite nel of h is the minimal normal subgroup of F fields of the same cardinality are isomorphic. containing the classes of words

finite function (1) A function f : X → Y w1 (a1 ...,an) ,...,wm (a1 ...,an) , which is finite, when thought of as a subset of the Cartesian product X × Y . It follows that f then is finite if and only if its domain X is a finite set. w (b ...,bn) = 1,...,wm (b ...,bn) = 1 (2) A function from a set X to the extended 1 1 1 ∪ {±∞} ∪ real or complex numbers (R or C are the defining relations of G.Ifm and n are {±∞} ±∞ ), never taking the values . See also both finite, we call G a finitely presented group. semifinite function. Finiteness Theorem (1)(Finiteness theo- finite graded module See graded module. rem of Hilbert concerning first syzygies) There are only finitely many first syzygies. See first finite group A group, which is finite as a set. syzygy. An example of a finite group is the symmetric (2)(Completeness of predicate calculus)Ifa S n n! group, n, of degree , having elements. In proposition H is provable (derivable) from a set n fact, any finite group with elements is isomor- of statements X, then there exists a finite subset S phic to some subgroup of n. See symmetric X∗ ⊂ X from which H can also be derived. group. finite A group G, which G finitely generated group A group that is both finite and nilpotent. See nilpotent group. consists of all possible finite products of the el- The following conditions are equivalent to being S ements of a finite set . We usually write nilpotent for a finite group: G G =S={s1s2 ...sk : si ∈ S} (i.) has at least one central series. (ii.) The upper central series of subgroups Zi S S and we call a set of generators of . The of G, simplest example is a cyclic group, namely, a G a group generated by one element . We then {e}≡Z0 ⊂ Z1 ⊂ Z2 ⊂ ... write G ={a}={ak : k ∈ Z}. ends with Zn ≡ G for some finite n ∈ N. finitely generated ideal An ideal I in a ring R (iii.) The lower central series of G such that I contains elements i1,...,ik which, G ≡ H˜ ⊃ H˜ ⊃ ... under sums and products by elements of R, gen- 0 1 erate all of I. ˜ ends with Hm ={e} for some finite m ∈ N. finitely generated module Let A be a ring (iv.) Every maximal (proper) subgroup is and M an A-module. We say that M is finitely normal. (v.) Every subgroup differs from its normal- generated if there is a finite set {x1,x2,...,xk} of elements of M such that, for each element izer. G p x ∈ M, there exist scalars ai ∈ A, so that (vi.) is a direct product of Sylow -sub- groups of G. k x = a x . i i finite prime divisor An equivalence class i=1 of non-Archimedean valuations of an algebraic We refer to {x1,x2,...,xk} as a system of gen- number field K. erators of M. finite simple group A group G of finite order finitely presented group Let F be the free |G| (|G| > 1) that contains no proper normal group generated by a1, a2,...,an and G be a subgroup.

c 2001 by CRC Press LLC finite solvable group A group G that has a them. Requiring that deg (gj ) = deg (fj ), j = composition series 1,...,m, we supply F with the structure of a graded R-module. The kernel of a graded R- G ≡ G ⊃ G ⊃···⊃G ≡{e} , 0 1 n homomorphism φ : F → M, M = φ(F), de- fined by φ(gj ) = fj is referred to as the first whose factor groups Gi/Gi+1,i= 0,...,n− 1, are of prime order. Equivalently, the composi- syzygy. It is uniquely determined by M up to a R tion factors Gi/Gi+ are simple Abelian groups graded -module isomorphism. 1 R (i.e., cyclic groups of prime order). (More generally: Let be a Noetherian ring, M a finitely generated R-module. Then one can first factor of class number The class num- find a finitely generated free R-module F and an ber h of the p-th cyclotomic field Kp, where p R-homomorphism φ : F → M (onto), whose M is a prime, is a product h = h1h2 of two factors kernel defines the first syzygy of .) h1 and h2, called, respectively, the first and the second factor of the class number h. fixed component The maximal positive divi- 2πi/m sor < that is contained in all divisors of a linear If Km = Q(ξm), ξm = e , then h2 is the 0  < < class number of the real subfield Km = Q(ξ + system is called the fixed component of . − V f , ξ 1). Explicitly Let be a complete irreducible variety, 0   f1,...,fn the elements of the function field p− (− )r+1 r+1 1 k(V ) of V , and D a divisor ring on V such that h = 1  jχ (j) , 1 r i (fi) + D ≥ 0, for all i. Then the set < of the (2p) i=1 j=1 divisors of the form (

c 2001 by CRC Press LLC flat morphism of schemes A morphism of X . A section of O over Q is a continuous map schemes f : X → Y such that, for each point σ : A ⊂ X → O such that π ◦ σ = 1A, where x ∈ X, a stalk OX,x at that point is a flat OY,f (x)- π : O → X is such that π(Ox) = x,Ox being module. If f is surjective, f is faithfully flat. a stalk over x.

flat resolution Let B be a left R module, form ring Let (R, P ) be a local ring and Q where R is a ring with unit. A flat resolution of its P -primary ideal. Set B is an exact sequence, i i+1 0 Fi = Q /Q ,i= 0, 1,...; Q = R, φ φ φ ···−→2 E −→1 E −→0 B −→ 0 , 1 0  i+  and for A = A (mod Q 1) ∈ Fi B = B (mod E R j+ where every i is a flat left module. (We Q 1) ∈ Fj , require shall define exact sequence shortly.) There is a companion notion for right R modules. Flat   i+j+1 AB = A B modQ ∈ Fi+j . resolutions are important in homological alge- bra and enter into the dimension theory of rings Then the form ring of R with respect to Q is and modules. See also flat dimension, flat mod- defined as a graded ring F generated by F over ule, injective resolution, projective resolution. 1 F and equal to the direct sum of modules An exact sequence is a sequence of left R 0 modules, such as the one above, where every φi ∞ is a left R module homomorphism (the φi are F = Fi , called connecting homomorphisms), such that i=0 Im(φi+1) = Ker(φi). Here Im(φi+1) is the im- F age of φi+1, and Ker(φi) is the kernel of φi.In where i is a module of homogeneous elements the particular case above, because the sequence of degree i. ends with 0, it is understood that the image of φ0 is B, that is φ0 is onto. There is a companion formula (1) A formal expression of a propo- notion for right R modules. sition in terms of local symbols. (2) A formal expression of some rule or other formal group Formal groups are analogs of results (e.g., Frenet formula, Stirling formulas, the local Lie groups, for the case of algebraic etc.). k-groups with a nonzero prime characteristic. (3) Any sequence of symbols of a formal cal- culus. formal scheme A topological local ringed space that is locally isomorphic to a formal spec- forward elimination A step in the Gauss trum Spf(A) of a Noetharian ring A. elimination method of solving a system of linear equations formal spectrum Let R be a Noetherian ring which is complete with respect to its ideal I n (in the I-adic topology), so that its completion aij xj = bi,(i= 1,...,n) along I can be identified with R. The formal j=1 spectrum Spf(R),ofR, is a pair (X , OX ) con- sisting of a formal scheme X = V(I) ⊂ Spec consisting of the following steps (R) and a sheaf of topological rings OX defined as follows: (m+1) (m) (m) (m) (m) aij = aij − aim amj /amm , >(D(f ) ∩ X , O ) = R /I nR ,f∈ R. X lim← n>0 f f (m+1) (m) (m) (m) (m) bi = bi − aim bm /amm , Here V(I)is a set of primitive ideals of R con- taining I, D(f ) = Spec(R) − V(f),f ∈ R are elementary open sets forming a base of the i, j = m + 1,...,n, with Zariski topology, and >(Q, O) designates the (1) (1) set of sections over Q of a sheaf space O over aij ≡ aij ,bi = bi ,

c 2001 by CRC Press LLC for all i, j. After the forward elimination, we so that get a system with a triangular coefficient matrix 1 ck = (ak − ibk) =¯c−k . n 2 (m) (m) amj xj = bm ,m= 1,...,n j=m Fourier’s Theorem Consider an nth degree and apply backward elimination to solve the sys- polynomial in x, tem. f(x)= a xn + a xn−1 +···+a , Also called forward step. 0 1 0

with ai ∈ R and a0 = 0, and the algebraic four group The simplest non-cyclic group equation (of order 4). It may be realized by matrices f(x)= 0 .  ±I 0 Designate by V(c1,c2,...,cp) the number of ±I 0 sign changes in the sequence c1,c2,...,cp of real numbers, defined by (any two elements different from the identity generate V ), or as a non-cyclic subgroup of the q−1 1 alternating group A4, involving the permuta- V c ,c ,...,cp = 1 − sgn cν cν , 1 2 2 j j+1 tions j=1 ( ), ( )( ), ( )( ), ( )( ). c ,c ,...,c c ,c 1 12 34 13 24 14 23 where ν1 ν2 νq is obtained from 1 2, ...,cp by deleting the vanishing terms ci = 0. It is Abelian and, as a transitive permutation Defining, finally group, it is imprimitive with the imprimitivity system {12}, {34}. W(x) = V f(x),f(x),...,f(n)(x) Also called Klein’s four group or Vierergruppe V. and

Fourier series An infinite trigonometric se- N ≡ N(a,b) = W(a)− W(b), ries of the form the number m ≡ m(a, b) of real roots in the ∞ 1 interval (a, b) equals a + [an cos nx + bn sin nx] 2 0 n=1 m = N(mod2), m ≤ N, a ,a,...,b , ··· ∈ with 0 1 1 R, referred to as i.e., m = N or m = N − 2orm = N − 4,..., (real) Fourier coefficients. Here or m = 0or1. π The precise value of m can be obtained us- a = 1 f(t) nt dt , ing the theorem of Sturm that exploits the se- n cos  π quence f(x),f (x), R (x),...,Rm− (x), Rm, −π 1 1 where −Ri is the remainder when dividing Ri−2 π   by Ri−1, with R0 = f and R−1 = f . Then b = 1 f(t) nt dt , m = W(a)− W(b). n π sin −π Fourier’s Theorem (on algebraic equations) is also called the Budom-Fourier Theorem. n = 0, 1, 2,..., with f(t) a periodic function with period 2π. The relationship between f(x) fraction A ratio of two integers m/n, where and the series is the subject of the theory of m is not a multiple of n and n = 0, 1, or any Fourier series. The complex form is number that can be so expressed. In general, ∞ any ratio of one quantity or expression (the nu- ikx cke , merator) to another nonvanishing quantity or ex- k=−∞ pression (the denominator).

c 2001 by CRC Press LLC fractional equation An algebraic equation An infinite cyclic group is one generated by a with rational integral coefficients. simple element x such that all integral powers of x are distinct. See cyclic group, direct product. fractional exponent The number a,inan a expression x , where a is a rational number. free additive group The direct sum of ad- ditive groups Ai,i ∈ C, such that each Ai is fractional expression See fraction. isomorphic to Z. See additive group. fractional ideal (1)(Of an algebraic num- free group A free product of infinite cyclic ber field k) Let k be an algebraic number field groups. The number of free factors is called the of finite degree (i.e., an extension field of Q of rank of the group F . Alternatively, (i.) a free I finite degree) and an integral domain consist- group Fn on n generators is a group generated ing of all algebraic integers. Further, let p be by a set of free generators (i.e., by the generators the principle order of k (i.e., an integral domain that satisfy no relations other than those implied k ∩ I whose field of quotients is k) and a an by the group axioms); (ii.) a free group is a group integral ideal of k (i.e., an ideal of the principle with an empty set of defining relations. k order p). Then a fractional ideal of is a p- To form a free group we can start with a free k ⊂ module that is contained in (i.e., pa a) such semigroup, defined on a set of symbols S = that αa ⊂ p for some α(α ∈ k,α = 0). See {a1,a2,...}, that consists of all words (i.e., fi- algebraic number, algebraic integer. nite strings of symbols from S, repetitions being R R (2)(of a ring )An -submodule a of the allowed), including an empty word representing Q R  ring of total quotients of such that there the unity. Next, we extend S to S that contains R ⊂ exists a non-zero divisor q of such that qa the inverses and the identity e R . See ring of total quotients.   S = e, a ,a−1,a ,a−1,... . fractional programming Designating: x an 1 1 2 2 n-dimensional vector of decisive variables, m an n-dimensional constant vector, Q a positive Then a set of equivalence classes of words S definite, symmetric, constant n × n matrix and formed from with the law of composition de- αβ (, )the scalar product, the problem of fractional fined by juxtaposition (to obtain a product of α β β programming is to maximize the expression two words and , we attach to the end of α; to obtain the inverse of α reverse the order of −1 (x, m) ai ai a √ , symbols while replacing by i and vice (x, Qx) versa) is the free group F on the set S. The equivalence relation (designated by “∼”) subject to nonnegativity of x (x ≥ 0) and linear used to define the equivalence classes is defined ≤ −1 constraints Ax b. via the elementary equivalences ee ∼ e, aiai −1 −1 −1 ∼ e, ai ai ∼ e, aie ∼ ai,ai e ∼ ai ,eai = fractional root A root of a polynomial p(x) −1 −1 ai,ea = a , so that α ∼ β if α is obtain- with integral coefficients i i able from β through a sequence of elementary n n−1 equivalences. p(x) = a0x + a1x +···+an that has the form r/s. One can show that r and s free module For a ring R,anR-module that are such that an is divisible by r and a0 is divisi- has a basis. ble by s. Any rational root of monic polynomial Remarks: (i.) If R is a field, then every R- having integral coefficients is thus an integer. module is free (i.e., a linear space over R). Also called rational root. (ii.) A finitely generated module V is free if there is an isomorphism φ : Rn → V , where R free Abelian group The direct product (fi- is a commutator ring with unity. nite or infinite) of infinite cyclic groups. Equiv- (iii.) A free Z-module is also called a free alently, a free Abelian group is a free Z-module. Abelian ring.

c 2001 by CRC Press LLC free product Consider a family of groups gebra over k). Defining a new product by {Gi}i∈C and their disjoint union as sets S.A word is either empty (or void) or a finite se- x ∗ y = (xy + yx)/2 , quence a a ...an, of the elements of S. Des- 1 2 we obtain a Jordan algebra A(J ). The subalgebra ignate by W the set of all words and define the (J ) of A generated by 1 and the xi is called the following binary relations on W:  free special Jordan algebra (of n generators). (i.) The product of two words w and w is (J )  Remark: A special Jordan algebra A arises obtained by juxtaposition of w and w . A  from an associative algebra by defining a new (ii.) w ' w if either w contains successive product x ∗ y,asabove. elements akak+1 belonging to the same Gi and  w results from w by replacing akak+1 by their Frobenius algebra An algebra A over a field product a = akak+1,orifw contains an identity k such that its regular and coregular representa-  element and w results by deleting it. tions are similar. (iii.) w ∼ w if there exists a finite sequence Remarks: (i.) Any finite dimensional semi-  of words w = w0,w1,...,wn = w such that simple algebra is a Frobenius algebra. for each j(j= 0,...,n−1) either wj ' wj+1 (ii.) An algebra A is a Frobenius algebra, ∗ or wj+1 ' wj . if the left A-module A and a dual module A Clearly, the definition of the product imme- of the right A-module A are isomorphic as left diately implies the associativity and “∼” rep- A-modules. resents an equivalence relation that is compati- ble with multiplication. We can thus define the Frobenius automorphism An element of a product for the quotient set G of W by the equiv- Galois group of a special kind that plays an im- alence relation ∼, and take as the identity the portant role in algebraic number field theory. K/F equivalence class containing the empty word. Designate by a relative algebraic num- K F The resulting group G is called the free product ber field, with a Galois extension of of [K : F ]=n G = G(K/F ) of the system of groups {Gi}i∈C. degree , the cor- F Remarks: (i.) The free product is the dual con- responding Galois group, and the principal K cept to that of the direct product (and is called order of . In Hilbert’s decomposition theory F K/F the coproduct in the theory of categories and of prime ideals of , for a Galois extension G functors). in terms of the Galois group , the subgroup   (ii.) If each Gi is an infinite cyclic group H = σ ∈ G : Pσ = P , generated by ai, then the free product of {Gi}i∈C is the free group generated by {ai}i∈C. called the decomposition group of a prime ideal P of F over F , plays an important role. The nor- free resolution (Of Z) A certain cohomo- mal subgroup H of H , called the inertia group logical functor, defined by Artin and Tate, that of P over F , is defined by   can be described as a set of cohomology groups σ concerning a certain complex in a non-Abelian H = σ ∈ H : a ≡ a(modP), a ∈ F . theory of homological algebras. The quotient group H/H is a cyclic group of order k, where k is the relative degree of P. free semigroup All words (i.e., finite strings This cyclic group is generated by an element of symbols from a set of symbols S = σo ∈ H that is uniquely determined (modH) by {a ,a ,...}) with the product defined by jux- 1 2 the requirement taposition of words [and the identity being the σ N(π) empty (void) word when a semigroup with iden- a o ≡ a (modP), a ∈ F tity is considered]. See also free group. where π is a prime ideal of F that is being de- free special Jordan algebra Let A = k[x1, composed. This element σo is referred to as the ...,xn] be a noncommutative free ring in the in- Frobenius automorphism or the Frobenius sub- determinates x1,...xn (i.e., the associative al- stitution of P over F .

c 2001 by CRC Press LLC See also ramification group, ramification where F(α)(x) = J(x)s(α)(x) is the Frobenius numbers, ramification field, Artin’s symbol. generating function (q.v.) and J[λ](x) is a gen- eralized Vandenmonde determinant Frobenius endomorphism For a commuta- [λ] (λ λ ...λ ) J (x) ≡ J 1 2 n (x ,x ,...,x ) tive ring R with identity and prime characteristic 1 2 n   p, the ring homomorphisn F : R → R, defined  λ +n−1 λ +n−1 λ +n−1   x 1 x 1 ... xn1  by  1λ +n− 2λ +n− λ +n−   x 2 2 x 2 2 ... x 2 2   1 2 n  F(a) = ap,( a ∈ R) . =  . . .  . for all  . . .     xλn xλn ... xλn  Clearly, for any a,b ∈ R, we have that 1 2 n (a + b)p = ap + bp (a · b)p = ap · bp . An alternative form is and [x] s(α)(x) = χ(α)S[λ](x) , For a Galois extension K/Fp over a prime field m λ Fp of degree m := [K : Fp] (with p = q distinct elements), referred to as Galois field Fq where S[λ](x) is Schur polynomial and s(α)(x) = n (s (x))αi α of order q, the Frobenius endomorphism r=1 r is the product of ith powers of p Newton polynomials F : Fq → Fq ,F: x (→ x n s (x) = xr . is injective and thus an automorphism of Fq . r i (See also Frobenius automorphism.) In fact, F i=1 G( / ) generates the cyclic Galois group Fq Fp . See also Frobenius generating function. Generally, for a scheme X over a finite field n Fq of q(= p ) elements, the Frobenius endo- Frobenius generating function For an arbi- morphism is an endomorphism ψ : X → X trary partition (α) of n, written in the form such that ψ = Id (the identity mapping) on the n set of Fq –points of X (i.e., on the set of points of α α α (α) ≡ 1 1 2 2 ...n n , iαi = n, X defined over Fq ) and the mapping of the struc- ∗ i=1 ture sheaf ψ : OX → OX raises the elements of OX to the qth power. the Frobenius generating function F(α)(x) ≡ X ⊂ n Thus, for an affine variety A defined F(α)(x1,...,xn) is given by the product over Fq ,wehave F(α)(x) = J(x)s(α)(x) , ψ (x ,...,x ) = xq ,...,xq . 1 n 1 n where J(x) is a polynomial, expressible as a Vandermonde determinant [λ] Frobenius formula The characters χ(α) of J(x) ≡ J (x1,x2,...,xn) = xi − xj the irreducible representation [λ]≡(λ1,λ2, α i

c 2001 by CRC Press LLC See also Frobenius formula. set V admits a partition into disjoint subsets of vertices so that in each such subset the vertices A (nonregular) transitive have access to each other via a directed path (se- permutation group on a set M, each element of quence of edges). The subsets of this partition which has at most one fixed point, i.e., the iden- are called the strongly connected components of tity of the group is its only element that leaves G(A). more than one element of M invariant. See reg- If there is only one strongly connected com- ular transitive permutation group. ponent, we call A irreducible. Otherwise, A is reducible. If A is reducible, there exists a Frobenius homomorphism (For commuta- permutation matrix P such that PAPT ,isin tive rings.) Let A be a commutative ring of Frobenius normal form. For instance, if G(A) pn prime characteristic p, so that (a + b) = has t strongly connected components, then its pn pn a + b for any a,b ∈ A and any n ∈ N. Frobenius normal form is Thus, the map φ : a (→ ap is an endomorphism   p of the additive group of A. Since further 1 = 1 A11 A12 ...... A1t p p p   and (ab) = a b for a,b ∈ A, this is also an  0 A22 A23 ... A2t   .  endomorphism of A, which is referred to as the  .. .. .   00 . . .  , Frobenius homomorphism of A. It is injective  . . .   . .. ..  when 0 is the only nilpotent element of A, which . At−1,t−1 At−1,t is the case when A is an integral domain. See 0 ...... 0 Att also Frobenius endomorphism for schemes. where each Aii for i = 1, 2,...,t is square Frobenius inequality Let f : X → Y, g : and irreducible. The Frobenius normal form is Y → Z, and h : Z → V be linear maps of finite not, in general, unique. As for any permutation −1 T dimensional vector spaces over a division ring matrix P , P = P , spectral properties of A (a skew-field). Then can be studied by studying spectral properties of the irreducible blocks Aii in its Frobenius rank (hg)+rank (gf ) ≤ rank (g)+rank (hgf ) . normal form.

Frobenius Reciprocity Theorem Let G and Frobenius morphism See Frobenius auto- H be finite groups, > and γ some irreducible morphism, Frobenius homomorphism, Frobe- representations of G and H , respectively, and nius endomorphism. H a subgroup of G, H ⊂ G. Designate by (γ ↑ G) the representation of G induced by γ and Frobenius norm For an n × n matrix A, by (> ↓ H) the representation of H subduced the length, denoted )A), of the n2-dimensional (restricted) from >. Then the multiplicity of > vector in (γ↑G) equals the multiplicity of γ in (>↓H). (a11,a12,...,a1n,a21,a22,...,a2n,...,ann), Equivalently, i.e.,   1/2 χ (>),χ(γ ↑G) =χ (>↓H),χ(γ ) , n   G H  2 )A)= aij  . χ (C) i,j=1 where designates the character of the rep- resentation C of X and Frobenius normal form An n × n matrix χ (C),X(C) = 1 χ (C)(x)χ (C)(x) , A = (a ) X |X| ij that is block (upper) triangular, with x∈X the diagonal blocks being square irreducible ma- trices that correspond to the strongly connected is a normalized Hermitian inner product on the components of G(A). Here, G(A) is the di- class function space of X, X = H or G. rected graph (digraph), G(A) = (V, E), con- sisting of vertices V ={1, 2,...,n} and di- Frobenius substitution See Frobenius auto- rected edges E ={(i, j) : aij = 0}. The morphism.

c 2001 by CRC Press LLC Frobenius Theorem There are a number of e, form a normal subgroup N of G, theorems associated with the name of Frobenius. (See also Frobenius Theorem on Non-Negative N = (G\H)∪{e} , Matrices, Frobenius Reciprocity Theorem.) G = NH, H ∩ N ={e} G/N =∼ (1)(Frobenius Theorem on Division Alge- such that and H bras.) The fields R (the field of real numbers) . See also Frobenius group. and C (the field of complex numbers) are the (6)(Frobenius Theorem on Abelian Varieties.) G only finite dimensional real associative and com- Let be an additive group of divisors on an A, X A Aˆ mutative algebras without zero divisors (i.e., di- Abelian variety a divisor on , and the A φ vision algebras), while H (the skew-field of Picard variety of . Denote by X a rational A Aˆ a ∈ A quaternions or Hamilton’s quaternion algebra) homomorphism of into that maps is the only finite dimensional real associative, into the linear equivalence class of the divisor X − X X X but noncommutative, division algebra. a , where a is the image of under the A → A b (→ a+b For nonassociative algebras, the only alterna- translation defined by . There a ,...,a tive algebra without zero divisors is the Cayley are elements 1 n such that the product Xa •···•Xa n = A algebra. See Cayley algebra, alternative algebra. (intersection) 1 n , dim is de- fined. Designating the degree of the zero cycle (2)(Frobenius Theorem for Finite Groups.) (n) Xa •···•Xa (X ) φx For a finite group G of order |G|=g, the num- 1 n by , the degree of is (X(n))/n ber of solutions of the equation xn = c, where given by ! c belongs to a class C having h conjugated ele- (7)(Frobenius Theorem on Subduced Repre- G ments, is given by g.c.d. (hn, g). sentations.) Let be a (finite) group having r C (N = ,...,r) r The original, simpler version of this theorem classes N, 1 with N elements |C |=r {> } states that the number m of solutions of xn = 1, each, N N. Further, let i be the set of G χ where n|g, is divisible by n, i.e., n|m. irreducible representations (irreps) of , and i > H (3)(Frobenius Theorem for Transitive Per- the character of i. Similarly, let be a sub- G s D s mutation Groups.) For a transitive permutation group of having classes k with k elements {J } group G of degree n, whose elements, other than each, and having the irreps j and characters φ H the identity, leave at most one of the permuted j . Designate, further, the representation of > > = > ↓ H symbols invariant, the elements of G displacing subduced by i by i i and its char- χ rs all the symbols form, together with the identity, acter by i. Then there exist nonnegative c a normal subgroup of order n. integers ij such that (4)(Also called Zolotarev-Frobenius Theo- ∼ s >i = cij Jj ,(i= 1,...,r) rem.) Let a be an arbitrary integer and b any odd j=1 integer such that a and b are relatively prime, and i.e., g.c.d.(a, b) = 1. Further, let πa desig- nate multiplication by a in the additive group s χ = c φ ,(i= ,...,r). H := Z/bZ. Then πa (regarded as an automor- i ij j 1 phism of H) represents a permutation of the set j=1 H and as such possesses the parity (or sign) δ Clearly, given by a δ (π ) = , s a b 1 cij = skχi(k)φ (k) . a |H | j where b is the Jacobi (generalized Legendre) k=1 symbol. (5)(Frobenius Theorem in The- Finally, ory.) Let H be a selfnormalizing subgroup of a r |G| G NG(H ) = H  finite group , , and cij χi(N) = sN φj (N [N]), − rN|H | H ∩ x 1Hx ={e} , i=1 N  for all x ∈ G. Then the elements of G that do where N [N] labels the classes DN of H that are not lie in H, together with the identity element contained in (CN ∩ H)and have sN elements.

c 2001 by CRC Press LLC Frobenius Theorem on Non-Negative Matri- respect to a rotation by 2π/h when represented ces (Also called Peron-Frobenius Theorem.) by points in the complex plane. Let A be an indecomposable (or irreducible), (iv.) Finally, if h>1, the matrix A can be non-negative n×n matrix over R, A ≡)aij )n×n brought to the following cyclic form (i.e., all entries aij of A are non-negative and there are no invariant coordinate subspaces 0A12 0 ... 0 n when we regard A as an operator on R ; in other 00A23 ... 0 . . . . words, aij ≥ 0 and there are no permutations of = . . . . , A . . . . rows and columns that would reduce the matrix 000... Ah−1h to the following block form Ah 00... 0  1 A12 Q ). A A with square blocks along the diagonal, by a suit- 21 22 able permutation of rows and columns.

Further, let λ0,...,λn−1 be eigenvalues of A, labeled in such a way that Fuchsian group A special case of a Kleinian     group, i.e., a finitely generated discontinuous ρ ≡ |λ | = |λ | =···=λ  > |λ | ≥ λ  0 1 h−1 h h+1 group of linear fractional transformations acting on some domain in the complex plane. (See Kleinian group, linear fractional function.) ≥···≥ |λn− | ,(1 0} or of the unit disc Xd ={z ∈ C :|z| < 1} onto the and complex plane. In the former case (X = Xu), ρB := max |µi| , 0≤i 0) or all negative (ci < 0). a Fuchsian group can be regarded as a discrete group of motions in this plane that preserve the (iii.) If A has h characteristic values λ0 = orientation. A Fuchsian group is referred to as ρ,λ1,...,λh−1 of modulus r, as in (i), then these eigenvalues are all distinct and are given elementary if it preserves a straight line in the by the roots of the equation λh − ρh = 0, i.e., Lobachevski plane (or, equivalently, some point in the closure X of X). For a nonelementary i > λj = ω ρ, (j = 0, 1,...,h− 1) Fuchsian group , one then defines the limit set of >, designated as L(>), as the set of limit where ω is the hth root of unity, ω = e2πi/h. points of the orbit of a point x ∈ X located on Moreover, any eigenvalue of A, multiplied by the circle ∂X and independent of x. We then ω, is again an eigenvalue of A. Thus, the entire distinguish Fuchsian groups of the first and sec- spectrum {λi, 0 ≤ i

c 2001 by CRC Press LLC L(>) = ∂X, while in the second case L(>) is a Yf = f(X), and the image of the element x ∈ nowhere dense subset of ∂X. Xf under f is y = f(x). Often one simply sets For any z ∈ C ∪ {∞} (the extended com- X = Xf . The set of ordered pairs f ={(x, y)}, plex plane) and any sequence {γi} of distinct regarded as a subset of X × Y , is referred to as elements of >, we define a limit point of > as a the graph of f .(See graph.) cluster point of {γiz}. If there are at most two The mapping property of a function f is usu- limit points, > is conjugate to a group of mo- ally expressed by writing f : X → Y , and tions of a plane. Otherwise, the set L of all limit for (x, y) ∈ f one often writes y = f(x) or points of > is infinite, and > is called a Fuchsoid f : x (→ y,oreveny = fx or y = xf . In lieu f(x ) f(x)| group. Such a group is Fuchsian if it is finitely of the symbol 0 one also writes x=x0 , generated. and often the function itself is denoted by the symbol f(x)rather than f : x (→ y, since this Fuchsoid group See Fuchsian group. Also notation is more convenient for actual computa- called Fuchsoidal group. tions. The set of elements of X that are mapped into function One of the most fundamental con- agiveny0 ∈ Y is called the pre-image of y0 and −1 cepts in mathematics. Also referred to as a is designated by f (y0), so that mapping, correspondence, transformation, or f −1 (y ) = {x ∈ X : f(x)= y } . morphism, particularly when dealing with ab- 0 0 −1 stract objects. This concept gradually crystal- For y0 ∈ Y \Yf we have clearly f (y0) =∅ ized from its early implicit use into the present (the empty set). day abstract form. The term “function” was first The notation f : X → Y indicates that the used by Leibniz, and it gradually developed into set X is mapped into the set Y . When X = Y ,we a general concept through the work of Bernoulli, say that X is mapped into itself. When Y = Yf , Euler, Dirichlet, Bolzano, Cauchy, and others. we say that f maps X onto Y or that f is sur- The modern-day definition as a correspondence jective (or a surjection). Thus, f : X → Y is a between two abstract sets is due to Dedekind. surjection (or onto) if for each y ∈ Y there ex- Generally, a function f is a relation between ists at least one x ∈ X such that f : x (→ y.If two sets, say X and Y , that associates a unique the images of distinct elements of X are distinct, element f(x) ∈ Y to an element x ∈ X.(See i.e., if x = x implies that f(x) = f(x) for relation.) The sets X and Y need not be distinct. any x,x ∈ X, we say that f is one-to-one, or Formally, this many-to-one relation is a set of or- univalent, or injective (or an injection). Thus, f dered pairs f ={(x, y)}, x ∈ X, y ∈ Y , that is, is injective if the preimage of any y ∈ Yf con- a subset of the Cartesian product X×Y , with the tains precisely one element from X, i.e., card     − property that for any (x ,y ) and (x ,y ) from f 1(y) = 1,y ∈ Yf . The mapping f : X → Y     f , the inequality y = y implies that x = x . that is simultaneously injective and surjective The first element x ∈ X of each pair (x, y) ∈ f (or one-to-one and onto) is referred to as bi- is called the argument or the independent vari- jective (or a bijection). For a bijective func- able of f and the second element y ∈ Y is re- tion one defines the by f −1 = − ferred to as the abscissa, dependent variable or {(y, f 1(y)), y ∈ Yf . See inverse function. the value of f for the argument x. For two functions f : X → Y and g : Y → The sets Xf and Yf of the first and second Z, with Yf ⊂ Yg, the function h : X → Z that elements of ordered pairs (x, y) ∈ f are called is defined as the domain (or the set) of definition of f and the h(x) = g(f (x)), for all x ∈ Xf , range (or the set) of values of f , respectively, while the entire set X is simply called the domain is called the composite function of f and g (also and Y the codomain of f . For any subset A ⊂ the superposition or composition of f and g), X, the set of values of f , {y = f(x) ∈ Y : and is designated by h = g ◦ f . x ∈ A} is called the image of A under f and is designated by f (A). In particular, the image of function algebra For a compact Hausdorff the domain of f is f(Xf ), i.e., Yf = f(Xf ) or space X, let C(X) [or CR(X)] be the algebra

c 2001 by CRC Press LLC of all complex- [or real-] valued functions on X. associating with each object X of C, X ∈ Ob C Then a closed subalgebra A of C(X)[or CR(X)] an object F(X) of D, F (X) ∈ ObD, and with is referred to as a function algebra on X if it each morphism α : X → Y in C, α ∈ Mor C,a contains the constant functions and separates the morphism F(α) : F(X) → F(Y)in D, F (α) ∈ points of X [i.e., for any x,y ∈ X, x = y, there MorD, in such a way that the following hold: f ∈ A f(x) = f(y) exists an such that ]. A (i.) F(1X) = 1F(X), for all X ∈ Ob C and typical example is the disk algebra, that is, the (ii.) F(α ◦ β) = F(α)◦ F(β) for all mor- complex-valued functions, analytic in the unit phisms α ∈ HomC(X, Y ) and β ∈ HomC(Y, Z). disk D ={|z| < 1}, which extend continuously Note that a functor F : C → D defines a to the closure of D, with the supremum norm. mapping of each set of morphisms HomC(X, Y ) Alternatively, a function algebra is a semi- into HomD(F (X), F (Y )), associating the mor- simple, commutative Banach algebra, realized phism F(α) : F(X) → F(Y)to each morphism as an algebra of continuous functions on the α : X → Y . It is called faithful if all these maps space of its maximal ideals (recall that a com- are injective, and full if they are surjective. The mutative Banach algebra is semi-simple if its identity functor IdC or 1C of a category C is the { } radical reduces to 0 , the radical being the set identity mapping of C into itself. of generalized nilpotent elements). A contravariant functor F : C → D asso- ciates with a morphism α : X → Y in C a mor- function field A type of extension of the field phism F(α) : F(Y) → F(X)in D (or, equiva- of rational functions C(x) that plays an impor- lently, acts as a covariant functor from the dual tant role in algebraic geometry and the theory of ∗ category C to D), with the second condition analytic functions.  (ii.) replaced by (ii. ) F(α◦ β) = F(β)◦ F(α) V(P) For an (irreducible) affine variety , for all morphisms α ∈ HomC(X, Y ), β ∈ P [x]≡ [x ,... where is a prime ideal in C C 1 , HomC(Y, Z). xn], the function field of V(P) is the field of A generalization involving a finite number quotients of the affine coordinate ring C[x]/P. of categories is an n-place functor from n cate- Similarly, for a projective variety V(P), where gories C ,...Cn into D that is covariant for the P is a homogeneous prime ideal in a projective 1 indices i ,i ,...,ik and contravariant in the re- n-space Pn, the function field of V(P) is the sub- 1 2 maining ones. This is a functor from the Carte- field of the quotient field of the homogeneous ⊗n C˜ D C˜ = C [x]/P sian product i=1 i into where i i for coordinate ring C of zero degree [i.e., the ˜ ∗ i = i ,...,ik and Ci = C otherwise. A two- ring of rational functions f(x ,x ,...xn)/g(x , 1 i 0 1 0 place functor that is covariant in both arguments x1,...,xn) (f, g being homogeneous polyno- mials of the same degree and g/∈ P) modulo is called a bifunctor. the ideal of functions (f/g), with f ∈ P]. See also Abelian function field, algebraic function fundamental curve A concept in the theory field, rational function field. of birational mappings (or correspondences) of algebraic varieties. See birational mapping. function group A Kleinian group > whose Consider complete, irreducible varieties V and W and a birational mapping F between region of discontinuity has a nonempty, con-  nected component, invariant under >. them, F : V → W. A subvariety V of V is called fundamental if dim F [V ] > dim V .  functor A mapping from one category into When V is a point, it is called a fundamental  another that is compatible with their structure. point with respect to F and, likewise, when V is Specifically, a covariant functor (or simply a a curve, it is referred to as a fundamental curve functor) F : C → D, from a category C into a with respect to F . category D, represents a pair of mappings (usu- See also Cremona transformation for a bira- ally designated by the same letter, i.e., F ). tional mapping between projective planes. Remarks: A variety V is irreducible if it is F : Ob C → ObD,F: Mor C → MorD , not the union of two proper subvarieties and any

c 2001 by CRC Press LLC algebraic variety can be embedded in a complete be a subset of the root system J of a semi- variety. simple Lie algebra g, relative to a chosen Cartan A projective variety is always complete, while subalgebra h,={α1,α2,...,αr }. Then an affine variety over a field F is complete when is called a fundamental root system of J if it is of zero dimension. (i.) any root α ∈ J is a linear combination of the αi with integral coefficients, fundamental exact sequence A concept in r the theory of homological algebras. i α = miαi,mi ∈ Z , Let H (G, A) be the ith cohomology group i= of G with coefficients in A, where A is a left G- 1 module (that can be identified with a left Z[G]- and module). (See cohomology group.) Designate, (ii.) the mi are either all non-negative (when further, the submodule of G-invariant elements α ∈ J+ is a positive root) or all non-positive. G i in A by A and assume that H (H, A) = 0,i= The roots belonging to are usually referred 1,...,n, for some normal subgroup H of G. to as simple roots. They can be defined as pos- Then the sequence itive roots that are not expressible as a sum of two positive roots. They are linearly indepen- n H n O → H G/H, A → H (G, A) dent and constitute a basis of the Euclidean vec- J n G tor space spanned by whose (real) dimension → H H,A equals the rank of g. → H n+1 G/H, AH fundamental subvariety See fundamental → H n+1(G, A) curve.

(composed of inflation, restriction, and trans- fundamental system (Of solutions of a sys- gression mappings) is exact and is referred to as tem of linear homogeneous equations) the fundamental exact sequence. n aij xj = 0,i= 1,...,m). (1) fundamental operations of arithmetic The j=1 operations of addition, subtraction, multiplica- tion, and division. See operation. Often, extrac- n tion of square roots is added to this list. Let fi = aij Xj ,(i= 1,...,m) be linear Starting with the set N of natural numbers, j=1 F, f : F n → F which is closed with respect to the first three forms over a field i . Clearly, fundamental operations, one carries out an ex- the solutions of (1) form a (right) linear space i (k) (k) tension to the field of rational numbers Q rep- V over F , since if xk ≡ (x ,...,xn ) ∈ n 1 resenting the smallest domain in which the four F ,(k = 1,...,r) are solutions of (1), so r c fundamental operations can be carried out in- is their (right) linear combination j=1 xj j , discriminately, excepting division by zero. Ad- cj ∈ F . In fact, V is the kernel of the (left) lin- n m joining the operation of extractions, ear mapping G : F → F given by G : x (→ we arrive at the field of complex numbers C. (f1(x),...,fm(x)). Designating the dimension These operations can be defined for various of V by d, d = dim V , we can distinguish the algebraic systems, in which case the commuta- following cases: tive and associative laws may not hold. (i.) d = 0: In this case the system (1) has only the trivial solution x = 0. fundamental point See fundamental curve. (ii.) d>0: Choosing a basis {x1,...,xd } for V , we see that any solution of (1) is a (right) fundamental root system (of a (complex) linear combination of the xk, (k = 1,...,d). semi-simple Lie algebra g) A similar concept One then says that x1,...,xd form a fundamen- arises in Kac-Moody algebras, in algebraic tal system of solutions of (1). Clearly, a nontriv- groups, algebraic geometry, and other fields. Let ial solution is found if and only if r

c 2001 by CRC Press LLC the rank of the matrix A =)aij ) [or, equiva- Fundamental Theorem of Galois Theory lently, the number of linearly independent lin- Let K be a Galois extension of a field M with a ear forms fi,(i= 1,...,m)], and the number Galois group G = G(K/M). Then there exists of linearly independent fundamental solutions is a bijection H ↔ L between the set of subgroups d = n − r. Since m ≥ r, we see that a non- {H } of G and the set of intermediate fields {L}, trivial solution always exists if the number of K ⊃ L ⊃ M. ForagivenH , the corresponding equations is less than the number of unknowns. subfield L = L(H ) is given by a fixed field KH of H (consisting of all the elements of K fundamental system of irreducible represen- that are fixed by all the automorphisms of H ). tations (Of a complex semisimple Lie alge- Conversely, to a given L corresponds a subgroup bra g.) Consider a complex semisimple Lie al- H = H (L) of G that leaves each element of gebra g and fix its Cartan subalgebra h. Let L fixed, i.e., H (L) = G(K/L), so that [K : ∗ k = dim h = rank g, h the dual of h, consist- L]=|H |. This bijection has the property that ∗ ing of complex-valued linear forms on h, and hR [L : M]=[G : H ], where [L : M] is the ∗ the real linear subspace of h , spanned by the degree of the extension L/M and [G : H ] is the root system J. Let, further, ={α1,...,αk} index of H in G. be the system of simple roots (see fundamental Also called Main Theorem of Galois Theory. root system) and define Fundamental Theorem of Proper Mapping 2αi α∗ = , A basic theorem in the theory of formal schemes, i (α ,α ) i i also called formal geometry, in algebraic geom- where (α, β) is a symmetric bilinear form on h∗ etry. See formal scheme. K(·, ·) Let f : S → T be a proper morphism of defined by the Killing form as follows:  locally Noetherian schemes S and T , T a closed   (α, β) = K tα,tβ , subscheme of T , S the inverse image of T  (given by the fiber product S ×T T ) and, finally, −1 tα being a star vector associated to α, tα = ν Sˆ and Tˆ the completions of S and T along S and (α), where ν designates the bijection ν : h → T , respectively. Then fˆ : Sˆ → Tˆ (the induced ∗ ,..., ∗ h . Let, further, 1 k be a basis of hR proper morphism of formal schemes) defines the α∗,...,α∗ ( ,α∗) = δ that is dual to 1 N , i.e., i j ij . canonical isomorphism Then the set of irreducible representations {> , 1 n n R f (X) =∼ F fˆ (X  ), n ≥ ...,>k} that have 1,...,k as their highest ∗ |T  ∗ |S 0 weights is called the fundamental system of irre- O X S ducible representations (irreps) associated with for every coherent S-module on , i.e., a O . sheaf of S-modules. For a coherent sheaf X on S, X|S denotes X S Rnf Fundamental Theorem of Algebra Every the completion of along . ∗ is the right f (X) X nonconstant polynomial (i.e., a polynomial with derived functor of the direct image ∗ of . a positive degree) with complex coefficients has a complex root. Fundamental Theorem of Symmetric Poly- n Also called Euler-Gauss Theorem. nomials Any symmetric polynomial in vari- ables, p(x) ≡ p(x1,x2,...xn), from a poly- R[ ]≡R[x ,x ,...x ] Fundamental Theorem of Arithmetic Ev- nomial ring x 1 2 n , can be ery nonzero integer n ∈ Z can be expressed as uniquely expressed as a polynomial in the ele- a product of a finite number of positive primes mentary symmetric functions (or polynomials) s ,s ,...,s x times a unit (±1), i.e., 1 2 n in the variables i. See elementary symmetric polynomial. n = Cp1p2 ...pk , In other words, for each p(x) ∈ R[x] there exists a unique polynomial π(z) ∈ R[z], z ≡ where C =±1,k≥ 0 and pi,i = 1,...,k (z1,z2,...,zn) such that are positive primes. This expression is unique except for the order of the prime factors. p (x1,x2,...,xn) = π (s1,s2,...,sn) ,

c 2001 by CRC Press LLC where si are the elementary symmetric polyno- sin nα = mials (functions) [(n− )/ ] s = n x 1 2 1 i=1 i , n (− )j 2j+1 α n−(2j+1) α, s = x x 2j+1 1 sin cos 2 i

c 2001 by CRC Press LLC tion f(X) = 0anAbelian equation or a cyclic equation, respectively. (See Abelian equation.) When the extension field K can be obtained by adjoining a root α of f(X) to k, K = k[α], G then the equation f(X) = 0 is called a Galois equation.

Galois cohomology Let K/k be a finite Ga- Galois extension A finite field extension K/k lois extension with the Galois group G(K/k). such that the order of the Galois group G(K/k) Suppose, further, that G(K/k) acts on some is equal to the degree [K : k]=dimk K of the Abelian group A. The Galois cohomology field extension K, i.e., groups H n(G(K/k), A) ≡ H n(K/k, A), n ≥ |G(K/k)|=[K : k]= K. 0 are then the cohomology groups defined by the dimk (F n,d) F n (cochain) complex , with consisting Remarks: G(K/k)n → A d of all mappings and designat- (i.) The degree [K : k] of the field exten- ing the coboundary operator (see cohomology sion K/k, k ⊂ K, equals the dimension of K K/k groups). When the extension is of an in- as a k-vector space, [K : k]=dimk K. One finite degree, one also requires that the Galois distinguishes quadratic ([K : k) = 2), cubic topological group acts continuously on the dis- ([K : k]=3), biquadratic (K : k]=4), finite A crete group and the mappings for the cochains ([K : k] < ∞), etc., extensions. F n in are also continuous. (ii.) All quadratic and biquadratic extensions One also defines Galois cohomology for a are Galois extensions. non-Abelian group A, in which case one usually (iii.) For any finite field extension K/k, the restricts oneself to zero- and one-dimensional order of the Galois group g ≡|G(K/k)| divides 0 1 cohomology groups, H and H , respectively. the degree of the extension [K : k], i.e., g|[K : 0 G(K/k) In the first case, H (K/k, A) = A repre- k]. sents a set of fixed points in A under the action (iv.) For a Galois extension K/k with the Ga- of the Galois group G(K/k), while in the sec- lois group G(K/k), the fixed field KG is given 1 ond case H (K/k, A) is the quotient set of the by k, i.e., KG = k. See also Galois theory. set of 1-dimensional cocycles. n The concept of Galois cohomology enables Galois field Finite fields Fq ,q = p , are one to define the cohomological dimension of also called Galois fields. See finite field. the Galois group Gk of a field k. See coho- mological dimension. Non-Abelian Galois co- Galois group For an extension K of a field k, homology enables the classification of principal the group of all k-automorphisms of K is called homogeneous spaces of group schemes and, in the Galois group of the field extension K/k and particular, to classify types of algebraic varieties is denoted by G(K/k). (using Galois cohomology groups of algebraic Remarks: groups). (i.) A k-automorphism of an extension field Also called cohomology of a Galois group. K is an automorphism that acts as the identity See also Tamagawa number, Tate-Shafarevich on the subfield k. It is also referred to as an group. automorphism of a field extension K. (ii.) Since every Galois extension is a split- Galois equation Let K/k be a finite Galois ting field of some polynomial f(x) ∈ k[x], extension with the Galois group G(K/k). Then and any two splitting fields K of a polynomial K is a minimal splitting field of a separable poly- f(x) ∈ k[x] are isomorphic, the Galois group nomial f(X) ∈ k[X], and we call G(K/k) the G(K/k) depends only on f (up to isomorphism). Galois group of f(X) or the Galois group of See Galois extension. Thus, if K is the splitting the algebraic equation f(X) = 0. (See min- field of f(x)∈ k[x], the Galois group G(K/k) imal splitting field, separable polynomial.) If is also referred to as the Galois group of the G(K/k) is Abelian or cyclic, we call the equa- polynomial over k.

c 2001 by CRC Press LLC Galois theory In a broad sense, a theory of an irreducible polynomial p(x) ∈ k[x] with- studying various mathematical objects on the out multiple roots, the Galois group G(K/k) is basis of their automorphism groups (e.g., Galois referred to as the Galois group of the equation of rings, topological spaces, etc.). In a (or polynomial) p(x) = 0. This group can be narrower sense, it is the Galois theory of fields computed without actually solving the equation that originated in the problem of finding of roots and can be regarded as a subgroup of the group of algebraic equations of higher degrees (e.g., of permutations of the roots of p(x). In the gen- quintic and higher). This problem was solved eral case, it is simply the permutation group of by Galois in his famous letter that he wrote on all the roots, i.e., the symmetric group of de- the eve of his execution (1832) and laid unread gree n = deg[p(x)]. Since such a group is not for more than a decade. In today’s language, solvable for n ≥ 5, there are no solutions in rad- this theory may be summarized as follows. icals for the quintic and higher degree algebraic Consider an arbitrary field k.Anextension equations. This theory can also be employed to (field) K of k is any field containing k as a sub- decide which problems of geometry are solvable field, k ⊂ K, and may be regarded as a lin- by ruler and compass (by reducing them to an ear space over k (finite or infinite dimensional; equivalent problem of solving some algebraic dimk K ≡[K : k] is called the degree of the ex- equation over the field of rational numbers in tension K/k). One says that α ∈ K is algebraic terms of quadratic radicals). over k if it is a root of a non-zero (irreducible) Galois theory had an enormous impact on the polynomial p(x) ∈ k[x] from a polynomial ring development of algebra during the nineteenth k[x] (i.e., with coefficients from k). The small- century. It was extended and generalized in var- est extension of k containing α is usually de- ious directions (Galois topological groups, class noted by k(α), and the smallest extension of k field theory, inverse problem of Galois theory, that contains all the roots of an irreducible poly- etc.), even though many important problems of nomial p(x) ∈ k[x] is called the splitting field the classical Galois theory remain unsolved. of p(x). The degree of such an extension is di- visible by the degree of p(x) and is equal to this Galois theory of differential fields Let K be degree if all the roots of p(x) can be expressed a differential field and N a field extension. The as polynomials in one of these roots. A finite ex- corresponding Galois group, G(N/K), is the set tension K/k is separable if K = k(α) and the of all differential isomorphisms of N over K. irreducible polynomial p(x) with α as a root has no multiple roots, and normal if it is the split- gap value A concept from the theory of al- ting field of some polynomial in k[x]. When gebraic functions. the extension is both separable and normal it is Consider a closed Riemann surface R of called a Galois extension. See Galois extension. genus g. When R carries no meromorphic func- If char k = 0 (e.g., k is a number field), any tion whose only pole of multiplicity m is at a finite extension is separable. point p ∈ R, then m is called a gap value of The group of all automorphisms of a Galois p ∈ R. extension K, leaving all elements of k invari- The Riemann-Roch Theorem implies that if ant, is the Galois group G(K/k). The relation- g = 0, then no point has gap values, while if ship between its subgroups H, H ⊂ G, and the g ≥ 1, then every p ∈ R has exactly g gap corresponding intermediate extension fields L, values. A point p ∈ R is an ordinary point k ⊂ L ⊂ K, is described by the Fundamental if m at p equals 1, 2,...,g and a Weierstrass (or Main) Theorem of Galois theory. See Funda- point otherwise. See ordinary point, Weierstrass mental Theorem of Galois Theory. In this way, point. See also Riemann-Roch Theorem. the difficult problem of finding all subfields of K is reduced to a much simpler problem of deter- gauge transformation A concept which mining the subgroups of G(K/k). Moreover, if arose in Maxwell’s formulation of the electro- H is normal, L is a Galois extension. These re- magnetic field theory and was later extended to sults are then exploited in studying the solutions more general field theories. In mathematics, it of algebraic equations. If K is the splitting field is used to designate bundle automorphisms of a

c 2001 by CRC Press LLC principle fiber bundle over a (space-time) man- For example, for the complex valued fields (x) ifold that is endowed with a group structure. interacting via the electromagnet field Ai(x) that In general, gauge transformations (in both clas- is generated by the electric charges, the theory sical and quantum field theories) change non- (i.e., the field equations and the Lagrangians) observable field properties (potentials) without should be invariant with respect to gauge trans- affecting the physically observable quantities formations of the type (observables). if (x) In electromagnetic field theory, the gauge (x) → (x) = e (x) , ∗ ∗ −if (x) ∗ transformation (also called gradient transfor- (x) → (x) = e (x) , A (x) → A (x) = A (x) + ∂f (x) . mation or gauge transformation of the second j j j ∂xj kind) has the form ∂f Such gauge transformations form an Abelian φ → φ = φ + , A → A = A − f, group of transformations [with the binary op- ∂t grad eration f(x)= g(x) + h(x) for two successive where φ and A designate, respectively, the scalar gauge transformations g and h]. and vector potentials of the field and f is an ar- The concept of gauge transformations has bitrary (twice differentiable) scalar function of been generalized to various field theories. In space and time. Equivalently, using the formal- mathematics, one employs this concept in ex- ism of special relativity, the 4-component elec- ploring principle fiber bundles over a manifold tromagnetic vector potential Aj (x), j = 0, 1, 2, endowed with the group structure. A (gauge) 3 and x = (x0,x1,x2,x3), transforms as fol- potential is then a connection on this bundle lows: and a gauge transformation is a bundle auto- ∂f morphism that leaves the underlying manifold A (x) → A (x) = A (x) + . j j j ∂xj pointwise invariant. These automorphisms form a group of gauge transformations. This transformation does not change the fields involved implying the gauge invariance of the Gaussian elimination A method for succes- underlying field theory. It may be used to sim- sive elimination of unknowns when solving a plify the relevant field equations through a suit- system of linear algebraic equations ("0), able choice of gauge, i.e., of a function f(x) (Coulomb gauge, Lorentz gauge, etc.). n In Weyl’s unified field theory (which origi- aij xj − ai0 = 0,i= 1,...,m, ("0) nated from the theory of Cartan’s connections), j=1 one employs a space (time) whose structure is where aij are elements of some field F . As- defined by the fundamental tensor gij , the co- suming that a = 0 (otherwise, renumber the variant derivative of which is defined in terms 11 equations), the key algorithmic step can be de- of the electromagnetic potential Ai as follows: scribed as follows: ∇igjk = 2Aigjk . Multiply the first equation by (a21/a11) and subtract it (term by term) from the second equa- This equation is invariant with respect to the tion. Next, multiply the first equation by (a / g → g = ρ2g 31 scale transformation ij ij ij and a ) and subtract it from the third equation, etc., the gauge transformation 11 until the first equation is multiplied by (am1/a11) ∂ ρ (i = m) A → A = A − log . and subtracted from the last equation i i i i ∂x of the system ("0). Designate the resulting system of equations For complex valued fields (required in quan- with the first equation deleted by (" ), and carry tum field theories), the theory must also be in- 1 out the same set of operations on (" ) obtain- variant with respect to gauge transformations (of 1 ing (" ), etc. Assuming that the rank of the the first kind) of the wave functions (x) in- 2 coefficient matrix of (" ) [which is also called volved, which have the general form 0 the rank of the system of equations ("0)], r = ig(x) (x) → (x) = e (x) . rank("0), is smaller than m, r

c 2001 by CRC Press LLC after the rth step, a system ("r ) in which all the alent system of linear algebraic equations, and coefficients of the unknowns vanish. The sys- the process of Gauss elimination can thus be tem ("0) [or ("r )] is called compatible, if in represented by a product of corresponding ele- ("r ) all the absolute terms vanish as well (i.e., mentary matrices. when the rank of the coefficient matrix equals In practical applications, when numerical ac- the rank of the augmented matrix); otherwise it curacy is at stake, one can also require that the is incompatible and has no solution. diagonal coefficients (the so-called pivots) are To obtain a solution of a compatible system, not only different from zero, but the largest ones (0) (0) we choose some solution (xr , ··· ,xn ) of the possible: in partial pivoting one chooses the ab- a i system ("r−1) and proceed by the back sub- solutely largest ii (from the th column), and in stitution to ("r−2), ("r−3), etc., until ("0) is complete pivoting the absolutely largest element reached. See back substitution. In general, we of the entire coefficient matrix (by appropriately can choose a solution for ("r−1) by assigning ar- renumbering the unknowns). bitrary values to the n−r variables xr+1,...,xn, Also called Gauss method, Gauss elimina- say xj = cj−r (j = r + 1,...,n), so that tion method or Gauss algorithm for solving lin- ear systems of algebraic equations. See also n (0) (r−1) (r−1) (r−1) forward elimination. x = (a − a cj−r )/a r r0 rj rr j=r+ 1 Gaussian integer A complex number (a + (0) bi) with integer a and b. The Gaussian inte- and x = cj−r for j = r + 1,...,n. The j gers are thus the points of a square lattice in the (general) solution will then depend on r − n complex plane, forming the ring arbitrary parameters cj ∈ F , j = 1,...,n− r. (" ) Once we have a solution of r−1 , the back Z[i]={a + bi : a,b ∈ Z} . substitution proceeds by assigning the values (0) (0) [i] xr ,...,xn to the unknowns xr ,..., xn in In fact, Z is an integral domain (with four ± , ±i the first equation of ("r−2), obtaining xr−1 = units 1 ) and, using the absolute value (0) (0) (0) (0) squared as a size function, it is also a Euclidean xr− , and thus a solution (xr− ,xr , ...,xn ) 1 1 domain (and, hence, principal ideal domain and of ("r−2). These values for xr−1, ...,xn are thus a unique factorization domain). The prime then substituted into the first equation of ("r−3), (0) elements (called Gauss primes) are either ra- obtaining xr− = x , etc., until a solution 2 r−2 tional primes that are congruent to 3 modulo 4 (x(0),x(0),...,x(0)) (" ) 1 2 n of 0 is obtained. The (i.e., 3, 7, 11, 19, etc.), or the complex numbers c (j = ,..., general solution results when the j 1 (a + ib) whose norm squared N = a2 + b2 is n − r) are regarded as free parameters. either a rational prime congruent to 1 modulo 4 This method can be generalized in various or 2 (i.e., 1 + i, 1 + 2i, 3 + 4i, etc.). ways. (See Gauss-Jordan elimination.) It can (Also called Gauss integer or Gauss number.) also be formulated in terms of a general m × n (or m×n+1) matrix A over F [representing the Gaussian ring A unique factorization do- coefficient (or augmented) matrix of a system main. See unique factorization domain. ("0)], in which case it is normally referred to as the row reduction of A. The algorithm can then Gaussian sum Let χ(n, k) be a numerical be conveniently expressed through the so-called character modulo k. Then a trigonometric sum elementary row operations, which in turn can be of the form represented by the elementary matrices of three k−1 basic types [(I + aeij ), i = j, replacing the ith 2πi(an) G(a, χ) = χ(n, k)e k row Xi by Xi+aXj , I+eij +eji−eii−ejj , inter- n=0 changing rows i and j, and I +(c−1)eii,c = 0, multiplying the ith row by c] acting from the left is referred to as the Gauss(ian) sum modulo k.It on A. Clearly, the action of the elementary ma- is thus fully defined by specifying the character trices and of their inverses on the augmented χ(n, k) and the number a. Note that when a ≡ matrix A of a system ("0) produces an equiv- b(mod k), then G(a, χ) = G(b, χ).

c 2001 by CRC Press LLC The Gauss sum is exploited in number the- called the single step method, for approximat- ory where it enables one to establish a relation ing the solution to a system of linear equations. between the multiplicative and additive charac- In more detail, suppose we wish to approximate ters. the solution to the equation Ax = b, where A is an n × n square matrix, and x and b are n Gauss-Jordan elimination A variant of the dimensional column vectors. Write A = L + Gauss elimination method, in which one zeros D + U, where L is lower triangular, D is di- out the elements in the entire column, rather than agonal, and U is upper triangular. The matrix only below the diagonal. See Gaussian elimina- L + D is easy to invert, so replace the exact tion. equation (L + D)x =−Ux + b by the relation The initial step is identical with that of Gauss (L + D)xk =−Uxk−1 + b, and solve for xk in elimination, while in the subsequent steps one terms of xk−1: subtracts the ith equation multiplied by (aji/aii) x =−(L + D)−1Ux + (L + D)−1b. from all the other equations. Consequently, the k k−1 upper left submatrix of the coefficient matrix This gives us the core of the Gauss-Seidel iter- after the ith iteration is diagonal or, by dividing ation method. We choose a convenient starting each equation by the diagonal coefficient, it is a vector x0 and use the above formula to compute unit matrix. In this way, one obtains the solution successive approximations x1,x2,x3,... to the of the system directly, without performing the actual solution x. Under suitable conditions, the substitution. The coding of this algorithm is sequence of successive approximations does in- simpler than for Gauss elimination, although the deed converge to x. See also iteration matrix. required computational effort is larger. Also called sweeping-out method. Gauss’s Theorem (1) See Fundamental The- orem of Algebra. Gauss-Manin connection A way to differ- (2) Let R be a unique factorization domain. entiate cohomology classes with respect to pa- Every polynomial from R[X] or R[X1,..., rameters. See cohomology class. Xn] (which are also unique factorization do- The first de Rham cohomology group mains) can be uniquely expressed as a product of certain primitive polynomials and an element H 1 (X/K) dR of R. See primitive polynomial. Then a product of primitive polynomials is primitive. of a smooth projective curve X over a field K A number of other theorems in analysis are can be identified with the space of differentials associated with Gauss’s name, e.g., the so-called (of the second kind) on X modulo exact differ- “Theorema Egregium” or Gauss curvature for entials. Each derivative θ of K can be lifted in a regular surfaces in E3, Mean Value Theorem for canonical way to a mapping ∇θ of H 1 (X/K) dR Harmonic Functions, Gauss-Bonnet Theorem, into itself such that etc.

∇θ (f ω) = f ∇θ (ω) + θ(f)ω , GCR algebra A generalization of CCR [com- 1 pletely continuous (= compact) representation] where f ∈ K and ω ∈ HdR(X/K). This im- plies an integrable connection algebras that are also referred to as liminal or liminary algebras. See CCR algebra. 1 1 1 C∗ ∇:HdR(X/K) → -K ⊗ HdR(X/K) A GCR algebra is a -algebra having a (possibly transfinite) composition series whose called the Gauss-Manin connection. This can be factor algebras are CCR (i.e., if Iλ is a compo- ∗ generalized to higher dimensions as well as to sition series of our C -algebra, then Iλ+1/Iλ is other algebraic or analytic structures. See also CCR). See composition series. This is equiva- Hodge theory. lent to requiring that the trace quotients be con- tinuous. Gauss-Seidel method for solving linear equa- Equivalently, a C∗-algebra is GCR if the im- tions An iterative numerical method, also age of every nontrivial representation contains

c 2001 by CRC Press LLC some nonzero compact operator. Thus, starting [i.e., the coset containing x ∈ R] can be re- ∗ with a largest CCR ideal I0 ofagivenC -algebra garded as a complex number and the functional A that consists of all elements a ∈ A whose im- xˆ : x → x(M) is multiplicative and linear, age π(a)is compact for all irreducible represen- i.e., xy(M) = x(M)y(M). Conversely, with tations π, we construct the factor algebra A/I0. each multiplicative linear functional one can as- Continuing this process, we eventually obtain sociate a regular maximal ideal. Designating by the largest GCR ideal of A. This will turn out to X , the set of all multiplicative functionals on be the C∗-algebra A itself if it is GCR. Clearly, R, endowed with Gel’fand topology, we obtain any CCR algebra is GCR while the converse is the Gel’fand representation, associating with an false. element of R a continuous function on X vanish- Also called postliminary algebra. ing at infinity. See Gel’fand topology. In fact, X is a locally compact Hausdorff space and, Gel’fand-Mazur Theorem A complex Ba- when R has a unit element, a compact Haus- nach algebra is a field if and only if it coincides dorff space. with the field of complex numbers C. Also called Gel’fand transform. See Banach algebra. Gel’fand tableau A triangular pattern Gel’fand-Naimark Theorem Any C∗-alge-   bra admits a faithful (i.e., injective) representa- (mn)  (m )  tion on some Hilbert space.  n−1   .  C∗ A [m]=: .  := More precisely, any -algebra is isomet-  .  rically ∗-isomorphic to a C∗-subalgebra of some   (m2) algebra B(H ) of bounded linear operators on a (m1) Hilbert space H. Thus, a C∗-algebra is a Ba-  nach algebra of (bounded linear) operators on m1n m2n a Hilbert space H (with the usual operations of   m1n−1 m2n−1  addition, multiplication by scalars, and product  ··· of operators) which is closed under the taking  m12 of adjoints. m11 Moreover, if A is separable, then H can be  assumed to be separable as well. ··· mnn  ··· mn−1n−1   Gel’fand-Pyatetski-Shapiro Reciprocity Law   Let G be a connected semisimple Lie group, 3 a m22 discrete subgroup of G and T the regular repre- sentation of G on 3\G [defined by (Tgf )(x) = that is employed to label the basis vectors of the f (xg), f ∈ L2(3\G)]. Then the multiplicity carrier spaces for the irreducible representations of a unitary, irreducible representation γ in the 3(mn) of the U(n) [or SU(n) set- regular representation T on 3\G equals the di- ting mnn = 0] relying on the Gel’fand-Tsetlin mension of the vector space formed by all auto- group chain. See Gel’fand-Tsetlin basis. The morphic forms of 3 of type γ . See automorphic irreducible representations 3(mn) of U(n) are form. uniquely labeled by their highest weight (mn) ≡ (m nm n ...mnn), where Gel’fand representation A correspondence 1 2 between the elements of a commutative Banach m1n ≥ m2n ≥···≥mnn ≥ 0 . algebra R and the continuous functions on the space of regular maximal ideals M of R. The entries of the lexicographical Gel’fand Recall that a maximal ideal M of R is called tableaux satisfy the so-called “betweenness con- regular if the quotient algebra R/M is a field, ditions” in which case R/M is isomorphic to C, the field of complex numbers. Thus, each coset x(M) mij ≥ mi,j−1 ≥ mi+1,j ,

c 2001 by CRC Press LLC i = 1,...,n−1; j = 2,...,n, reflecting Weyl’s a3 = a1(a2a3) = a1a2a3 (representing the bi- branching law for the subduction of 3 nary operation involved by a juxtaposition). The (mn) from U(n) to U(n − 1). The ith row of general associative law requires that any finite the Gel’fand tableau thus represents the highest ordered subset of n elements, say a1,a2,...,an, weight of those U(i) irreducible representations ai ∈ G, (i = 1,...,n),n>2, uniquely deter- that result by a successive subduction of 3(mn), mines their “product” a1a2 ···an, irrespective implied by the Gel’fand-Tsetlin chain. of the sequence of binary operations employed. Arranging the basis vectors in a lexicograph- ical order, we have, for example, for the (210) general equation See also Galois equation, irreducible representation of U(3) or SU(3), Galois group, Galois theory. Recall that the Ga-         G(K/k) 210 210 210 210 lois group of the finite field extension  21  ,  21  ,  20  ,  20  , K/k is also called the Galois group of the alge- 2 1 2 1 braic equation f(X)= 0, when K is a minimal         f(X) 210 210 210 210 splitting field of a separable polynomial  20  ,  11  ,  10  ,  10  . ∈ k[X]. The algebraic equation f(X) = 0is 0 1 1 0 also called a Galois equation when the exten- Gel’fand topology The weak topology on sion field K can be obtained by adjoining some the space of multiplicative linear functions on a root of f(X)to k, i.e., when K = k(α) for some commutative Banach algebra R. See Gel’fand root α of f(x). We also note that G(K/k) has a representation. faithful permutation representation based on the permutation group of the roots of f(X). When Gel’fand transform See Gel’fand represen- this representation is primitive, f(X) = 0is tation. called a primitive equation. See primitive equa- tion. See also affect (of f(X)). Gel’fand-Tsetlin basis A basis for the carrier Now, when α ,α ,...,αn are algebraically space of U(n) or SU(n) irreducible representa- 1 2 independent elements over k, then the equation tions exploiting the group chain F (n)(X) = 0 for the nth degree polynomial (n) ⊃ (n − ) ⊃···⊃ ( ) ⊃ ( ), ( ) U U 1 U 2 U 1 1 (n) n n−1 n−2 F (X) = X − α1X + α2X where U(i) ⊃ U(i−1) represents schematically n−3 n −α X ±···+(−1) αn the imbedding U(n − 1) ⊕ (1) [i.e., U(n − 1) 3 represents a subgroup of blocked n × n uni- from k(α1,α2,...,αn)[X] is called a general tary matrices (linear operators) consisting of a equation of degree n. See algebraic indepen- (n − 1) × (n − 1) unitary matrix and the 1 × 1 dence. The Galois group of this equation is then matrix (1); clearly this subgroup of U(n) is iso- isomorphic with Sn, the symmetric (or permu- morphic with U(n − 1)]. According to Weyl’s tation) group of degree n. Moreover, if char k = branching law, each irreducible representation 2, then the quadratic subfield L corresponding of U(i) subduced to U(i−1) is simply reducible. to the alternating group An of the same degree Since, moreover, U(1) is Abelian, the canoni- is obtained by adjoining the square root√ of the cal chain (1) provides an unambiguous labeling discriminant D of F (n)(X), i.e., L = k( D). for mutually orthogonal one-dimensional sub- spaces spanning the carrier space of a given ir- generalization Motto: Be wise! Generalize! reducible representation of U(n) through a set of “Picayune Sentinel” and M. Artin’s Algebra. highest weights characterizing the subduction at (1) An extension of a statement (or a con- each level. Collecting these highest weights we cept, a principle, a theorem, etc.) that applies obtain a Gel’fand tableau. See Gel’fand tableau. or is valid for some system or structure A to Also called Gel’fand-Zetlin basis or canonical all members of a larger class of systems B con- basis. taining A as one of its elements (or a proper subsystem). general associative law The associative law (2) The process of inferring such a statement for a given binary operation implies that (a1a2) (or concept, etc.).

c 2001 by CRC Press LLC (3) In logic, generalization implies the for- lar characters of the centralizer CG(y) of y in mal derivation of a general statement from a G, then particular one by replacing the subject of the (y) (y) statement with a bounded variable and prefixing χi(yz) = cij φj (z) , a quantifier (in predicate calculus). For exam- j ple, the statement “the hypothesis H(i) holds (y) for some i ∈ Z” is the (existential) generaliza- and the coefficients cij are referred to as the tion of “the hypothesis H(i) holds for i = 1 generalized decomposition numbers of G. and 3.” A universal generalization applies to When the order of y is a power pr of p, then (y) all members of a given class while an existential the coefficients cij are algebraic integers of the generalization applies only to some unspecified field of the pr th roots of unity. See also de- members of such a class. composition number, modular representation of finite groups. generalized Borel embedding A general- ization of a Borel embedding of a symmetric generalized eigenvalue problem The prob- bounded domain to an embedding of an arbi- lem of finding scalars λ ∈ F and vectors x (from trary homogeneous bounded domain. Let D be a linear space V over F ) for linear operators (or a homogeneous bounded domain and let Gh be matrices) A and B on V satisfying the equation the identity component of the full automorphism group Gh(D) of D. Let gh be the Lie algebra of Ax = λBx .(1) Gh, then gh is a j-algebra with a collection of c A B endomorphisms (j). Let gh be the complexifi- Usually, and are required to be Hermitian − c B cation of gh and let gh ={x + ij x ∈ gh : x ∈ and, moreover, to be positive definite. When c B = I V gh,j∈ (j)}.IfGh is the analytic subgroup of , the identity (matrix) on , the problem c the full linear group corresponding to gh, and reduces to the standard or classical eigenvalue − c problem. Gh is the complex closed subgroup of Gh gen- − A further generalization examines the non- erated by gh , then D can be embedded as the Gh orbit of the origin in the complex homogeneous linear problem − space Gc /G .  h h n n−1 Anλ + An−1λ +···+A1λ + A0 x = 0. generalized decomposition number Let G be a finite group of order |G|=g and K a split- Clearly, for n = 1 we obtain (1). ting field of G of characteristic char K = p = 0 (i.e., K is a splitting field for the group ring generalized Eisenstein series Let 3 ⊂ SL K[G]). If p is a divisor of g, p|g, we can gener- (2, R) be a Fuchsian group, acting on the up- ate modular representations of G. See modular per half-plane H of the complex plane C, and representation. designate by 3\H the quotient space of H by Let, further, L be an algebraic number field 3. See Fuchsian group. The Selberg zeta func- that is a splitting field of G having a prime ideal tion Zp(s, :), defined for s ∈ H and a matrix π dividing p, and :i,i= 1,...,n, be the non- representation : of 3, has a number of inter- similar irreducible representations of G in L and esting properties when 3\H is compact and 3 χi,i= 1,...,n, their standard characters. is torsion-free. See Selberg zeta function. For Now, any x ∈ G can be uniquely decom- a more general case, when 3\G is noncompact posed as follows: (though has a finite volume), the decomposition of L2(3\G) into the irreducible representation x = yz = zy , spaces has a continuous spectrum. Generalized Eisenstein series (defined by Selberg) give the where y is the so-called p-factor of x (whose or- explicit representation for the eigenfunctions of der is a power of p) and z is a p-regular element this continuous spectrum. In the special case in of G (whose order is prime to p). If, further, which 3 is the elliptic , SL(2, Z) φ(y),...,φ(y) 1 ky are absolutely irreducible modu- (or the corresponding group of linear functional

c 2001 by CRC Press LLC transformations) 3 = SL(2, Z), this series is and, similarly, H ∞(µ) as the weak-star closure of A in L∞(µ). Note that L∞(µ) is a com- y2 mutative Banach ∗-algebra (with the pointwise |cτ + d|2s multiplication and the involution given by com- (c,d)=1 plex conjugation), which is isometrically iso- where y = z, z ∈ H. morphic to the algebra of complex-valued con- Similar generalized Eisenstein series can be tinuous functions on the maximal ideal space of ∞ defined for semisimple algebraic groups G and L (µ). their arithmetic subgroups. See also Eisenstein series. generalized inverse For an m × n matrix A, with entries from the complex field, the unique + generalized Hardy class A concept arising n × m matrix A satisfying + in the theory of function algebras (or uniform (i.) AA A = A, + + + algebras) generalizing that of the Hardy class (ii.) A AA = A , + ∗ + from the theory of analytic functions. (iii.) (AA ) = AA , + ∗ + [Recall that the Hardy class H p (p > 0) con- (iv.) (A A) = A A. sists of analytic functions on the open unit disc This definition (by Penrose) is equivalent to + D ={z :|z| < 1} having the property the definition (by Moore) that A is the unique matrix so that p /p AA+ sup { |f(rζ)| dµ(ζ )}1 < ∞ , (1) is the (orthogonal) projection matrix 0

c 2001 by CRC Press LLC where GK is a closed subgroup of the profinite is the maximal Kummer extension of exponent group G that is associated with the field K and n. G A /NGA is the norm residue group relative to Also called generalized reciprocity law. the norm group NGA ={NGa = σ∈G σa : a ∈ A}. One further defines the Galois group of generalized nilpotent element An element L/K as from the kernel of the Gel’fand representation of a commutative Banach algebra R, i.e., an el- G(L/K) = GK /GL , ement x ∈ R such that    n1/n assuming that GL is a normal subgroup of GK . lim x = 0 . n→∞ In such a case, the extension L/K is called a normal or Galois extension. See Galois exten- The kernel of R is referred to as the radical of sion. Note that for a finite extension L/K, when R. When this radical reduces to {0}, R is said to AK ⊆ AL, there is a normal map be semisimple.  NL/K : AL → AK ,NL/K (a) = σa , generalized nilpotent group An extension σ of the concept of nilpotency to infinite groups, in a nonstandard manner. Thus, a group G is a with σ ranging over the right representatives G /G L/K A generalized nilpotent group if any homomorphic of K L. When is Galois, L is a G {e} G(L/K) A = AG(L/K) image of that is different from has a center -module and K L . that is different from {e}. This definition reduces Recall also that a prime element πK of AK , G π ∈ A v (π ) = to the standard one when is finite, but not K K , is an element with K K 1, when G is an infinite group. where vK designates a homomorphism generalized peak point A point x in a com- 1  vK = v ◦ NK/k : AK → Z , pact Hausdorff space such that {x} is a gener- fK alized peak set. See peak set, generalized peak with v a henselian valuation v : Ak → Z and k set. the ground field for which Gk = G. Further, σ˜ is a Frobenius lift of σ if σ =˜σ|L, generalized peak set An intersection of peak sets of a compact Hausdorff space. See peak set. σ˜ ∈φ(L/K)˜   generalized A general- = σ˜ ∈ G(L/K)˜ : (σ)˜ ∈ , degK N ization of the quaternion group G, which is the group of order 8 with two generators σ and τ where degK is a surjective homomorphism satisfying the relations σ 4 = e,τ2 = σ 2 and : G(L/K)˜ →   − − degK Z, Z the Prüfer ring τστ 1 = σ 1, which is isomorphic to the mul- tiplicative group of quaternion units {±1, ±i, Z = lim Z/nZ , ← n∈N ±j,±k}. The generalized quaternion group is a group of order 2n,(n≥ 3), again with two ←lim  = where denotes projective limit. In fact, Z generators σ and τ, which, however, satisfy the p p Zp, where is a prime. following relations: Finally, for every finite extension K/k of the ˜ ˜ n−1 n−2 − − ground field, one defines K = K · k, σ 2 = e,τ2 = σ 2 and τστ 1 = σ 1 . ˜ fK =[K ∩ k : k] The standard quaternion group corresponds to n = 3. and 1  : : GK → Z , D degK f deg generalized Siegel domain A domain in K Cn × Cm (n, m ≥ 0) with the following prop- where √ erties. (i.) D is holomorphically equivalent to ˜ n ∗ K = K K a bounded domain. (ii.) D contains a point of

c 2001 by CRC Press LLC n (s) the form (z, 0) where z ∈ C . (iii.) D is in- orthogonal idempotents by {ei }, so that variant under the following holomorphic trans- formations of Cn × Cm for some c ∈ R, and n di n (s) for all s ∈ R and t ∈ R : (z, u) → (z + s,u), ei = 1 (z, u) → (z, eitu), and (z, u) → (et z, ect u).In i=1 s=1 the above situation, D is said to be a generalized and Siegel domain with exponent c. n di n di (s) (s) generalized solvable group An extension of A = Aei = ei A, the concept of solvability to infinite groups in a i=1 s=1 i=1 s=1 nonstandard way. For example, a group G is a generalized solvable group if any homomorphic the latter relationship providing a decomposi- image of G that is different from {e} contains a tion of A into a direct sum of indecomposable nontrivial (i.e., different from {e}) Abelian nor- left (right) ideals. Here n denotes the number of mal subgroup. As with generalized nilpotency, simple algebra components Ai in the decompo- this definition reduces to a standard one for fi- sition of the semisimple quotient algebra A/N, nite groups but not in the case of infinite groups. N being the radical of A, See generalized nilpotent group. n A/N ≡ A = Ai , Generalized Tauberian Theorem A theo- i=1 rem, due to Wiener, originating in the theory of the Fourier transform. Here we give a version each Ai being a full matrix ring of degree di, that is pertinent for the exploitation of Banach which thus decomposes into the direct sum of algebras in the theory of topological groups. di minimal left (right) mutually A-isomorphic The theorem pertains to the following prob- (s) (s) ideals Aei (ei Ai), i = 1,...,n; s = 1,..., lem of spectral synthesis. See spectral synthesis. (s) di, where e designate the orthogonal idempo- Consider an Abelian topological group G and i A e(s) its L1-algebra (or group algebra) R. Let, further, tents of i. The idempotents i in the above  (s) G designate the character group of G. Then any are representatives from each residue class ei , closed ideal I in R determines a set Z(I) in which are chosen in such a way that the relations G (s) consisting of common zeros of the Fourier hold. In the following, we designate ei simply transforms of the elements of I. [Recall that by ei. the Fourier transform of x ∈ R is given by its We call A a generalized uniserial algebra if Gel’fand transform, in the present case by each indecomposable left (right) ideal Aei (eiA) of A has a unique composition series. See com- x(γ)ˆ = x(g)γ(g)dµ(g), position series. An algebra A is a generalized uniserial alge- where γ is a character of G and µ is a (left- bra if its radical N is a principal left and right invariant) measure on G (with G regarded as a ideal, and a uniserial algebra if and only if every locally compact Hausdorff group).] The above- two-sided ideal of A is a principal left and right mentioned problem then asks whether the con- ideal, i.e., if and only if every quotient algebra of verse also holds, namely, whether I can be A is a Frobenius algebra. See uniserial algebra, uniquely characterized by Z(I). Frobenius algebra. A special case when Z(I) is empty is ad- An algebra A is an absolutely uniserial al- K dressed by the Generalized Tauberian Theorem gebra if the algebra A over K, where K is an which states that I must coincide with R when extension field of a field F , is uniserial for any Z(I) =∅. such extension K/F. This is the case if and only if the radical N of A is a principal ideal generalized uniserial algebra Consider a generated by an element from the center Z of A finite dimensional, unitary (associative) algebra and Z decomposes into a direct sum of simple A over a field F and designate its system of extensions F [α] of F .

c 2001 by CRC Press LLC generalized valuation A valuation of a gen- general solution As a rule, the general solu- eral rank. See valuation. tion of a problem is a solution involving a cer- The rank of an additive valuation (or simply tain number of parameters from which any other of a valuation) v : F → G ∪ {∞} of the field F , solution (except for singular solutions) can be with G a (totally) ordered additive group and the obtained, through a suitable choice of these pa- element ∞ is defined to be greater than any ele- rameters. ment of G, is defined to be the Krull dimension Specifically, in analogy to differential equa- of the valuation ring Rv ={a ∈ F : v(a) ≥ 0}. tions, one understands, by a general solution of See Krull dimension. a nonhomogeneous linear difference equation n general linear group On a finite dimensional pi(x) y(x + i) = q(x) (1) V V = n K linear space , with dim , over a field , i=0 the group of (nonsingular) linear mappings of V onto V , the group operation being defined a solution of the form as composition of mappings, denoted GL(V ). n More generally, the automorphism group y(x) = ci(x)φi(x) + ψ(x) , AutK (V ) of the free right K-module V with n i=1 K generators, for an associative ring with unit. where φi(x), i = 1,...,n are linearly inde- Equivalently, a general linear group of de- pendent solutions of the corresponding homo- gree n over K, usually designated GL(n, K) geneous equation, ψ(x) is an arbitrary solution (K) n × n or GLn , is a group of invertible ma- of the nonhomogeneous equation (1), and ci(x) trices with entries in K. Clearly, GL(V ) and are arbitrary periodic functions of period 1. GL(n, K) are isomorphic. When viewed as an affinite variety, GL(n, K) can also be regarded general term (Of a series, of a polynomial, as an algebraic group. See algebraic group. In of an equation, of a language, etc.) An expres- most applications, K is a field. The structure of sion or an object that forms a separable part of GL(n, K) over a ring K is studied in algebraic some other expression or object, in particular K-theory. expressions separated by the plus sign (in a se- Also called full linear group. See also special ries or a polynomial), comma (in a sequence), linear group, projective general linear group. inequality or identity sign (in an inequality or a chain of inequalities or equation(s)), etc. Also general linear Lie algebra (Of degree n over called generic term. a commutative ring with unity [or over a field] K), a Lie algebra, denoted gl(n, K), which re- generating representation Let G be a com- sults from the total matrix algebra M(n, K) of pact Lie group. Designate, further, the com- (all) n × n matrices with entries from K, when mutative, associative algebra of complex-valued endowed with a Lie (or bracket) product defined continuous functions on G by C(0)(G, C) ≡ by the commutator [X, Y ]:=XY−YX. See Lie C(0)(G), and its subalgebra of representative algebra. A general linear Lie algebra gl(n, K) functions referred to as the representative ring is the Lie algebra of GL(n, K), a general lin- of G by R(G,C) ≡ R(G). See representa- ear group, and may thus also be regarded as the tive function, representative ring. Note that, tangent space to GL(n, K) at the identity (repre- with the supremum norm )f )=supg∈G |f(g)|, sented by the identity matrix). See also general C(0)(G) is a Banach space and the actions of linear group. G on this space (given by left and right transla- tions) are continuous. At the same time, C(0)(G) x ,...,x general position Let 0 k be points in may be completed with respect to the norm aris- Euclidean space. These points are said to be ing from the inner product (u, v) = uv, yield- in general position if they are not contained in G any plane of dimension less than k. This con- ing the Hilbert space L2(G) of square integrable cept may be generalized to geometric objects of functions on G. The actions of G on L2(G) higher dimension. (again by left and right translations) are unitary.

c 2001 by CRC Press LLC In view of the Peter-Weyl theorem, R(G) (iv.) Cyclic groups are generated by a single is dense in both C(0)(G) and L2(G) so that generator (which, clearly, is not unique), e.g., there exists a faithful (i.e., injective) represen- Z8 =*1+=*3+=*5+=*7+. tation ρ : G → GL(n, C) of G and R(G) may (v.) See also finitely generated group. be regarded as a finitely generated algebra over (1a) In the analytic theory of semigroups, C. Thus, there exists a faithful representation one defines the infinitesimal generator F of an g → {rij (g)} such that the functions rij and rij equicontinuous semigroup T of class (C0), T ≡ generate R(G). Such a representation is referred {Tt | t ≥ 0}, as a limit to as a generating representation. −1 Fx = lim t (Tt − e) x. t→0+ generator(s) (1)Ingroup theory, the ele- ments of a nonempty subset S of a group G such (2)Inring theory, we say that an ideal I of that G is generated by S, i.e., G =*S+, where a (commutative) ring R is generated by a finite *S+ consists of all possible products of the ele- subset S ={s1,...,sn} of R, when ments of S and of their inverses that are possibly   n subject to a certain number of conditions (called I = r s : r ∈ R , relations) of the type i i i i=1 n n n s 1 s 2 ···s m = e,si ∈ S, ni ∈ Z . 1 2 m in which case we also write When S is finite, G is said to be finitely gener- n ated or finitely presented. See relation, finitely I = (s1,...,sn) or I = Rsi , presented group. For a cyclic group, S consists i=1 of a single element. s ,i= ,...,n More precisely, let G be a group generated and refer to the ring elements i 1 , I by some set S ≡{s ,s ,...,sn} and let us des- as generators of . An ideal generated by a 1 2 I = (s) ignate by F the free group on S. Further, let T single generator, , is called a principal ideal. An extension to noncommutative rings is be a subset of F , T ≡{t1,t2,..., tm}⊂F , and NT the smallest normal subgroup of F contain- obvious. ing T (given by the intersection of all normal (3) An analogous definition also applies to subgroups of F containing T ). Then, G is said modules over a ring (or a field). A M to be given by the generators si,(i= 1,...,n) Consider an -module (i.e., a module over A X ={x } subject to the relations t = t =···=tm = e, a ring ) and a family i i∈I of elements 1 2 M A N M if there is an isomorphism G ≈ F/NT that as- of . The smallest sub- -module of con- X sociates si with siNT . This is usually expressed taining all the elements of consists of all linear by writing combinations of these elements and we write   G =*s ,...,s : t = t =···=t = e+ , 1 n 1 2 m N = Ax = a x : a ∈ A, i ∈ I . ( ) i i i i 1 i∈I i∈I and one also says that G has the presentation given by the right-hand side of (1). The family X is then referred to as a system of Remarks: (i.) Although we have used a finite generators for N. We also say that N is gener- number of generators and relations in the above ated by X. When X is finite, i.e., Card(X) < definition, this is not necessary. ∞, N is said to be finitely generated, and an − − (ii.) The relation ri = e, e.g., a 1b 2ab = A-module Ax, generated by a single element e, is often more convenient to write in the form x ∈ M, is called a monomial. avoiding inverses, i.e., ab = b2a in our exam- When A is a field, the A-module M becomes ple. a linear space over A, and the same terminol- (iii.) Recall that a free group is “free" of ogy is sometimes employed even in this case, al- relations, so that we can regard any element in though one often defines M to be spanned rather NT as the identity, as in fact the notation of (1) than generated by X, and X is referred to as a suggests. spanning set (or a basis, if linearly independent).

c 2001 by CRC Press LLC (4) The method of defining groups by gen- (6)Incoding theory, with a linear code de- erators and relations can be also applied to Lie fined as a subspace W of Vn ={0, 1,...,q− algebras. Let L be a Lie algebra over a field F 1}n, with q being a prime power q = pα, one generated by a set X ={Xi}i∈I .IfL is free on associates with each element a ∈ Vn the poly- X (in which case the vector space V = Span(x) nomial can be given an L-module structure by assign- a(X) = a + a X +···+a Xn−1 ing to each x ∈ X an element of the general 1 2 n (V ) linear Lie algebra gl and extending canon- GF (q) V ically to L) and M is the ideal of L generated over a Galois field [note that n can be regarded as an n-dimensional vector space over by a family of elements A ={aj }j∈J , J being GF (q)]. Then the cyclic code W modulo g(X) some index set, then the quotient algebra L/M is defined by is referred to as the Lie algebra with generators Xi,(i∈ I)and relations aj = 0,(j ∈ J), with W = {a ∈ Vn : a(X) ≡ 0(modg(X))} Xi the image of the element xi ∈ X in L/M. This is referred to as a presentation of L. and g(X) is referred to as the generator of N. Remarks: (i.) For semisimple Lie algebras L over an algebraically closed field F , charF = generic point Consider an irreducible vari- 0, one can give a presentation of L in terms of ety V in Kn, where K is the universal domain generators and relations that depends only on and designate by k a field of definition for V the root system of L.(See Serre’s Theorem.) (which is a subfield of K, k ⊂ K), having a finite (ii.) In the physics literature, one often refers transcendence degree over the underlying prime to the basis elements of the defining (or stan- field. See irreducible variety, universal domain, dard) representation of matrix Lie groups or al- transcendence degree, prime field. Then a point gebras as generators. Thus, for example, the (x) ≡ (x1,...,xn) of V , (x) ∈ V , with the matrix units eij =)δikδjJ) of gl(n, F ) are re- property that all points of V are specializations ferred to as raising (i < j), lowering (i > j), of (x) over k, is called a generic point of V over and weight (i = j)generators according as they k. See specialization. Note that in general a raise, lower, or preserve the weight of a repre- generic point of V is not unique. sentation involved. Remarks: (5)Inhomological algebra, an object G of (i.) In some texts, generic point refers to an an Abelian category A (with all functions be- arbitrary point of a nonempty Zariski open set ing additive) is called a generator if the natural of a variety V . See Zariski open set. mapping (ii.) A generic point x in a topological (sub)- space A is a generic point of A when A = {x} (A, B) → (h (A), h (B)) Hom Hom G G (where the bar designates the closure in the hull- is one to one. Here Hom designates the functor kernel topology). defining the category A, genus (1) For an algebraic variety, a discrete, Hom : A × A → (Ab), numerical invariant representing an important parameter then enables a classification of such with (Ab) designating the category of Abelian varieties. These invariants may be defined in groups, and with various ways, the most useful ones being based on the concept of differential forms on a vari- h (·) := (G, ·). G Hom ety and on the cohomology of coherent sheaves. p (X) Similarly, one defines a cogenerator G when One distinguishes the algebraic genus a p (X) X  and the geometric genus g of a variety . G G Hom(A, B) → Hom h (B), h (A) See geometric genus. For a one-dimensional algebraic variety C k is one to one, where now over a field , referred to as an algebraic curve, the genus g of C is defined as the dimension G h (·) := Hom(·,G). of the space of regular differential 1-forms on

c 2001 by CRC Press LLC C, assuming that C is smooth and complete. lent to a given divisor a is referred to as a com- See algebraic curve. We recall here that an al- plete linear system determined by a and is des- gebraic curve can be transformed into a plane ignated by |a|. Further, one defines a finite di- curve with only ordinary multiple points by a fi- mensional vector space V(a) over K as follows: nite number of plane Cremona transformations (quadratic transformations of a projective plane V(a) := {f ∈ K(C) : (f ) + a - 0} , into itself). See Cremona transformation. A plane algebraic curve C of degree n is a point whose one-dimensional subspaces are in a bi- | | set in an affine 2-space defined by the zero set jective correspondence with the elements of a . | | | | f(X,Y)= 0, of an nth degree polynomial f(X, We then define the dimension of a , dim a ,by Y) X Y F(X ,X ,X ) = Xn in and . Setting 0 1 2 0 dim|a|:=dimKV(a) − 1 , f(X1/X0,X2/X0), the homogeneous polyno- mial F defines an algebraic curve C of degree and the genus of C, g ≡ g(C), as the supremum n in a projective plane P2. We say that C is ir- of non-negative and bounded integers deg(a) − reducible if f(X,Y) is irreducible. Clearly, a dim |a|, curve of degree 1 is a line. A point P = (a, b) C f(X+ a,Y + b) on is an r-ple point if has no g := sup (deg(a) − dim |a|). terms of degree less than r in X and Y . There a∈G(C) are r tangent straight lines (counting multiplic- ity) at such a point: if these tangents are all dis- One can show that for any g>0, g ∈ Z, tinct, P is referred to as an ordinary point and there exists an algebraic curve of genus g. The an ordinary double point is called a node. An curves with g = 1 are the so-called elliptic r-ple point with r>1 is called a multiple or a curves and those with g>1 are subdivided into singular point. the classes of hyperelliptic and non-hyperelliptic For a nonsingular irreducible curve C, we de- curves. See elliptic curve, hyperelliptic curve. fine a divisor a as an element of the free Abelian See also specialty index, Riemann-Roch Theo- group G(C) generated by the points of C that is rem. of the form For an r-dimensional projective variety Y in Pn (a projective n-space over an algebraically a = niPi,(ni ∈ Z)(1) closed field k), the arithmetic genus of Y can i be defined in terms of the constant term of the Hilbert polynomial PY (X), X = (X0,X1,..., and has a degree deg(a) = ni. We say that i Xn),ofY , as follows: the representation (1) for a is reduced if Pi = Pj i = j r for .Apositive or an integral divisor, writ- pa(Y ) = (−1) (PY (0) − 1) . ten as a - 0, involves only positive coefficients ni,(ni > 0). We further designate by w a dis- It can be shown that pa(Y ) is independent of crete valuation whose value group is the additive projective embeddings of Y . When Y is a plane group of integers, thus representing a normal (or curve of degree d (see above), then normalized) valuation wP of K(C) defined by a valuation ring RP given by the subset of the 1 pa(Y ) = (d − 1)(d − 2), function field K(C) of C (K designating the uni- 2 versal domain) consisting of those functions f and when it is a hypersurface of degree d, then that are regular at P . The integer wP (f ) is re- f P P  ferred to as the order of at , and is said d−1 pa(Y ) = . to be a zero of f when wP (f ) > 0 and a pole n when wP (f ) < 0. The linear combination Equivalently, for a projective scheme Y over a "wP (f )P =: (f ) (2) field k, the arithmetic genus pa(Y ) can be de- fined by is called the divisor of the function f , and the set r of all positive divisors that are linearly equiva- pa(Y ) = (−1) [χ (OY ) − 1] ,

c 2001 by CRC Press LLC where OY designates a sheaf of rings of regular (3)Forquadratic forms Q over an algebraic functions from Y to k and χ(F) is the so-called number field k of finite degree, one defines equiv- Euler characteristic of a coherent sheaf F on alence classes of forms having the same genus Y that is defined in terms of the cohomology by requiring that Q and Q be equivalent over the i groups H (Y, F) of F as follows: principle order op in kp for all non-Achimedean prime divisors p of k and, at the same time, that χ(F) := "(− )i H i(Y, F). 1 dimk they are equivalent over kp for all the Archime- dian prime divisors p of k. (Recall that prime Y See projective scheme. Thus, when is a curve, divisors of k are equivalence classes of nontriv- we have that ial multiplicative valuations over k and are re- 1 ferred to as Archimedean or non-Archimedian pa(Y ) = dimk H (Y, OY ) , accordingly if these valuations are Archimedean while for a complete smooth algebraic surface or not.) See valuation. Y ,wehave (4) For a complex matrix representation of a finite group G. Consider a finite group G and pa(Y ) = χ (OY ) − 1 an algebraic number field K. Recall that there exists a bijective correspondence between linear = H 2 (Y, O ) − H 1 (Y, O ) . dimk Y dimk Y and matrix representations of G and that every On the other hand, the geometric genus pg(Y ) linear representation of G over K is equivalent of a nonsingular projective variety Y over k is to some linear representation of the group ring defined as the dimension of the (global) section K[G]. When K is the ring of rational integers, a of the canonical sheaf ωY of Y (defined as the linear representation over K is called an integral nth exterior power of the sheaf of differentials representation. where n = dim Y ), 3(Y,ωY ), i.e., Further, for a given algebraic number field K, every K[G]-module V contains G-invariant p (Y ) := 3 (Y, ω ) . g dimk Y R-lattices (called G-lattices for short), where R is the ring of integers in K, that may be viewed For a projective nonsingular curve C, the arith- as finitely generated R[G]-modules, providing metic and the geometric genuses coincide, i.e., an integral representation of G. pa(C) = pg(C) = g, while for varieties of di- P R mension greater than or equal to 2, they are not Designating by a prime ideal of and by R necessarily equal, and their difference is referred P the ring of quotients (or fractions) of the ring R P to as the irregularity of Y . See irregularity. with respect to the prime ideal , also called P R (2) For an algebraic function field K over the local ring at , one can also explore the P - k of dimension 1 (or of transcendence degree representations (this approach is closely related 1) (i.e., a finite separable extension of a purely with modular representation theory). Thus, with R[G] M transcendental extension k(x) of k such that k any -module we can associate a family R [G] M = R ⊗ M P is maximally algebraic in K), the genus of K/k of P -modules P P , with R is defined similarly as for a curve C. Thus, it ranging over all prime ideals of . Then, when, G R[G] M is achieved by replacing C with K/k, K(C) by for two -lattices (or -modules) and N K[G] V K, and K by k, while points on C now become , belonging to a -module ,wehave M ∼ N P M N prime divisors of K/k. that P = P for all , we say that and We can also say that the genus of a function have (or are of) the same genus. (See also class field is given by the genus of its Riemann surface number.) (recall that the Riemann surface S of a function (5) Similarly, for a connected algebraic group field is homeomorphic to the complement of a G over an algebraic number field k of finite de- finite set of points in a compact oriented two- gree, we consider a ring R of integers in k and dimensional manifold S, while the genus of the designate by L a general R-lattice (also called a latter is defined, loosely speaking, as the number transformation space) in the vector space V on of “holes” in S, i.e., g = 0ifS is a sphere, g = 1 which G acts. Defining then an action of GA, if S is a torus, etc.). the adele group of G, on the set {L} of R-lattices

c 2001 by CRC Press LLC in a natural way, one defines the genus of L as geometric fiber The fiber X ×Y Spec(K) of the orbit GAL of L with respect to GA. a geometric point Spec(K) → Y , where X and (6) For an imaginary quadratic field k.(See Y are schemes and K is an algebraically closed also principal genus of k.) Each coset of the field. ideal class group G of k modulo the subgroup Specifically, for a morphism of schemes φ : H of all ideal classes of k satisfying the condi- X → Y and a point y ∈ Y , the fiber of the tion (P1,...,Pt ) = (1,...,t) is referred to as morphism φ over the point y is the scheme a genus of k. See ideal class, ideal class group. Here, P1 = χi(N(a)), i = 1,...,t, where a Xy = X ×Y Spec k(y) , is an integral ideal with (a,(d)) = 1, N desig- nates the norm, the χi are the Kronecker sym- where k(y) is the residue field of y. See fiber. A bols, and d is the discriminant of k. See Kro- point of Y with values in a field K is a morphism necker symbol. For each genus, the values of Spec(K) → Y . Also, when K is algebraically Pi, (i = 1,...,t)are unique and (P1,...,Pt ) is closed, such a point is referred to as a geometric called the character system of this genus. See point. See geometric point. character system. A suitable embedding of k(y) in K is as- sumed, in the above definition. genus of function field See genus (2). Also called geometric fiber. geometrical equivalence Let V be an n- geometric fibre See geometric fiber. dimensional Euclidean space and let O(V ) be the orthogonal group of V .IfT is an n- geometric genus (Of an algebraic surface dimensional lattice in V and K is a finite sub- S) a numerical invariant characterizing this sur- group of O(V ), let (T , K) denote a faithful face given by the number of linearly independent linear representation of K on T . Every pair holomorphic 2-forms on S. (T , K) corresponds to a set of crystallographic More generally, for an n-dimensional irre- (T ,K ) (T ,K ) groups. Two pairs 1 1 and 2 2 are ducible variety V , the geometric genus is given said to be geometrically equivalent if there ex- by the number of linearly independent differen- g ∈ (V ) K = gK g−1 ists a GL such that 2 1 . This tial forms of the first kind of degree n. (T ,K ) ∼ equivalence relation is denoted by 1 1 See also genus. (T2,K2), and the equivalence classes are called geometric crystal classes. geometric mean Given any n positive num- n bers a1,...,an, the positive number G = geometric crystal class See geometrical √ a1a2 ···an. equivalence. geometric multiplicity Given an eigenvalue geometric difference equation A difference λ of a matrix A, the geometric multiplicity of λ equation of the form is the dimension of its eigenspace. It coincides y(px) = f(x,y(x)) , with the size of the diagonal block with diagonal element λ in the Jordan normal form of A. See where p ∈ C is an arbitrary complex number. also algebraic multiplicity, index. For example, the standard form of a linear dif- ference equation, geometric point A morphism Spec(K) → n X, where X is a scheme and K is an algebraically ak(x) y(x + k) = b(x) , closed field. k=0 can be transformed to this form via the change geometric programming A special case of of variables z = px, yielding nonlinear programming, in which the objective n  function and the constraint functions are linear k Ak(z) Y zp = B(z) . combinations of (not necessarily integral) pow- k=0 ers of the variables.

c 2001 by CRC Press LLC A sequence of non- Givens transformation See Givens method zero numbers, having the form ar,ar2,..., of matrix transformation. arn(,...). See also geometric series. Gleason cover The projective cover in the geometric quotient Let Z and Y be alge- category of compact Hausdorff spaces and con- braic schemes over a field. Suppose that G is tinuous maps. In this category, the projective a reductive algebraic group that operates on Z, cover of a space X always exists, and is called and f : Z → Y is a G-invariant morphism of the Gleason cover of the space. It may be con- schemes. Denote by f∗ the homological map- structed as the Stone space (space of maximal ping induced by f , and by OZ and OY the lattice ideals) of the of regu- sheaves of germs of regular functions on Z and lar open subsets of X. A subset is regular open Y , respectively. The morphism f is called a if it is equal to the interior of its closure. Al- geometric quotient if ternatively, it may be constructed as the inverse (i.) f is a surjective affine morphism and limit of a family of spaces mapping epimorphi- G f∗(OZ) = OY , cally onto X. See also epimorphism, projective (ii.) f(X)is a closed subset of Y whenever cover, Stone space. X is a G-stable closed subset of Z, and (iii.) for each z1 and z2 in Z, the equality global dimension For an analytic set A, the f(z1) = f(z2) holds if and only if the G-orbits number dim(A) = supz∈A dimz(A), where of z1 and z2 are equal. dimz(A) is the local dimension of A at z. geometric series A series of the form global Hecke algebra Let F be either an al- gebraic number field of finite degree or an alge- ∞ arj = ar + ar2 +···+arn +··· . braic function field of one variable over a finite field. Consider the general linear group GL2(F ) j=0 of degree 2 over F , and denote by GA the group (F ) Sometimes also referred to as a geometric pro- of rational points of GL2 over the adele ring F v F H gression. of . For each place of , let v be the Hecke If |r| < 1, the above series converges to the algebra of the standard maximal compact sub- sum ar/(1 − r). group of the general linear group of degree 2 over the completion of F at v. Denote by εv H Geršgorin’s Theorem The eigenvalues of an the normalized Haar measure of v. The re- n × n matrix A = (aij ), with entries from the stricted tensor product of the local Hecke alge- H {ε } complex field, lie in the union of n closed discs bras v with respect to the family v is called G (known as the Geršgorin discs), the global Hecke algebra of A.   n   G-mapping Suppose that G is a group, and X Y G G z ∈ C : |z − aii| ≤ |aik| . and are -sets. A -mapping is a mapping   f : X → Y f(gx) = gf (x) i=1 k=i such that for each g in G and each x in X. As a consequence of Geršgorin’s Theorem, every strictly diagonally dominant matrix is in- GNS construction A means of construct- ∗ vertible. The latter fact is also known as the ing a cyclic representation of a C -algebra on ∗ Lévy–Desplanques Theorem. See also ovals of a Hilbert space from a state of the C -algebra. Cassini. The letters G, N, and S refer to Gel’fand, Na˘ımark (Neumark), and Siegel (Segal), respec- Givens method of matrix transformation tively. A method for transforming a symmetric ma- trix M into a tridiagonal matrix N = PMP−1. good reduction For a discrete valuation ring Here P is a product of two-dimensional rotation R, with quotient field K, and an Abelian variety matrices. A over K, we denote by A the Neron minimal

c 2001 by CRC Press LLC model of A. Thus, A is a smooth group scheme graded object An objectO which can be of finite type over Spec(R) such that, for every written as a direct sum O = a∈A Oa, where scheme B which is smooth over Spec(R), there A is a monoid. is a canonical isomorphism graded ring A ring R with a direct sum de- composition R = R ⊕ R ⊕ ... (so that the Ri HomSpec(R)(B ,A) → HomK (B K, A) . 0 1 are groups under addition) satisfying RmRn ⊆ R m, n > If A is proper over Spec(R), then we say that A m+n for all 0. The elements of each R has a good reduction at R. See Neron minimal i are called homogeneous elements of degree i model. . A primary example is the ring R = k[x1, ...,xn] of homogeneous polynomials in n Gorenstein ring An algebraic ring which ap- hom variables over a coefficient ring k. In this ring, pears in treatments of duality in algebraic geom- Rn = 0, for n<0 and, for n ≥ 0,Rn consists of etry. Let R be a local Artinian ring with m its the polynomials that are homogeneous of degree maximal ideal. Then R is a Gorenstein ring if n. the annihilator of m has dimension 1 as a vector space over K = R/m. gradient method for solving non-linear pro- gramming problem An iterative method for R graded algebra An algebra with a direct solving non-linear programming problems. Sup- R = R ⊕ R ⊕ ... sum decomposition 0 1 (so pose that the problem is to maximize the func- R that the i are groups under addition) satisfying tion f subject to some constraints. At each stage RmRn ⊆ Rm+n, for all m, n > 0. The elements of the iteration, one uses the current iterate xk, of each Ri are called homogeneous elements of the gradient of f at xk, and a positive number degree i. λk to compute the point xk+1 according to the formula xk+ = xk − λknk, where nk is the unit (C,R,P) 1 graded coalgebra A coalgebra vector in the direction of the gradient of f at the C ,n≥ such that there exist subspaces n 0, such point xk. that C = C0 ⊕ C1 ⊕ ... and R(Cn) ⊂ i+j=n C ⊗C n> P(C ) ={ } i j , for all 0 and such that n 0 , Gramian For complex valued functions fi : for all n>0. [a,b]→C, i = 1,...,n, the determinant of  b the n × n matrix whose i, j entry is a fifj dx. graded Hopf algebra Suppose that R is a commutative ring with a unit, (A, ε) is a supple- Gram-Schmidt process A way of converting mented graded R-algebra, and ψ : A → A ⊗ A a basis x1,...,xn for an inner product space is a map such that V,(·, ·) into an orthonormal basis v1,...,vn,so that we have (vi,vj ) = 0, if i = j and (vi,vi) = i, j = ,...,n α1 ◦ (1A ⊗ ε) ◦ ψ = 1A = α2 ◦ (ε ⊗ 1A) ◦ ψ, 1, for 1 . The relationship between {x1, ...,xn} and {v1,...,vn} is given by A α x with 1A denoting the identity mapping on , 1 v = 1 , A ⊗ R 1 denoting an algebra isomorphism from )x1) A α to , and 2 denoting an algebra isomorphism x − (x ,v )v from R ⊗ A to A. Then A is called a graded v = 2 2 1 1 , 2 )x − (x ,v )v ) Hopf algebra. 2 2 1 1 x3 − (x3,v1)v1 − (x3,v2)v2 v3 = . graded module Let R = R0 ⊕ R1 ⊕ ... )x3 − (x3,v1)v1 − (x3,v2)v2) R ... be a graded ring. A graded -module is an √ R-module M which has a direct decomposition Here )x)= (x, x), for any x ∈ V . ∞ M = j=−∞ Mj , such that RiMj ⊂ Ri+j , for i ≥ 0 and j ∈ Z. The elements of each Mi are Gram’s Theorem (1) Suppose that P is a called homogeneous elements of degree i. convex polytope in Euclidean 3-space. For k =

c 2001 by CRC Press LLC 0, 1, 2, denote the kth angle sum of P by αk(P ). x1,...,xn, the set G(E) ={(a1,...,an) ∈ n Then α0(P ) − α1(P ) + α2(P ) = 1. More gen- K : E(a1,...,an) = 0}. erally, if P is a d-dimensional convex polytope in Euclidean d-space and αk(P ) is the kth angle graph of function For a function f : V → sum of P for k = 0, 1, 2, ... , d − 1, then W, where V and W are sets, the set G(F ) = {(x, f (x)) : x ∈ V }. Thus the graph G(f ) is d−1 i d−1 a subset of the Cartesian product V × W of the (−1) αi(P ) = (−1) . sets. i=0 (2) Suppose that F is a field of characteristic graph of inequality For an equation E = G zero, and denote by the general linear group E(x1,...,xn), over the ordered field K in the F of some degree over . Consider matrix rep- variables x1,...,xn, the graph of the inequality ρ ... ρ G F n n resentations 1, , q of over , with i E>0 is the set G(E) ={(a1,...,an) ∈ K : ρ ρ the degree of i, such that each i is either the E(a1,...,an)>0}. rational representation of G induced by some element of G or is the contragredient map κ. greater than If R is an Abelian group and R+ + Suppose that ρi = κ for i ≤ s and ρi = κ for is a subset of R not containing 0, such that R is (i) + + i>s. Let {xj : 1 ≤ i ≤ q,1 ≤ j ≤ ni} be closed under addition, R = R ∪ (−R ) ∪{0}, algebraically independent elements over F .For and R+∩(−R+) is empty, then, for all a,b ∈ R, + + each b = 1, 2, ... , s, let Hb be a polynomial in we have either a − b ∈ R or a − b ∈ (−R ) x(i) i>s x(i) ... or a − b = 0. In the first case, we write a>b j for that is homogeneous in 1 , , (i) and say that a is greater than b (or b is less than xn for each i, and suppose that the set V of com- i a); in the second case, we write assuch that (...,am ,... ,aj ,...)is (b) a a zero point of C for each am with b ≤ s. greatest common divisor For integers and b, not both zero, the unique positive integer, de- graph A simple graph is a pair (V, E), where noted gcd(a, b), which divides both a and b and V is a set and E is a set of distinct, unordered is divisible by any other divisor of both a and b. pairs of elements of V .Inadirected graph (or Also called greatest common factor or highest digraph), the elements of E are ordered pairs common factor. of distinct elements of V . Allowing multiplici- ties for the elements of E or allowing loops (an greatest common factor See greatest com- element of E of the form (v, v)) gives a gen- mon divisor. eral graph, also called a graph. The elements of V are called vertices and the elements of E greatest lower bound See infimum. are called edges. Thus a pair (u, v) ∈ E may be visualized as a line segment joining points Grössencharakter See Hecke character. u, v ∈ V . Graphs are used widely in and Grothendieck category A category C satis- algebra since they model relationships of vari- fying: (i.) C is Abelian; (ii.) C has a generator; ous kinds among sets. (iii.) direct sums always exist in C; and (iv.) for any object P of C, any sub-object Q of P , and for graphing See graph of equation, graph of any totally ordered family {Ri} of sub-objects, function. we have (∪Ri) ∩ Q =∪(Ri ∩ Q). graph of equation For an equation E = Grothendieck topology Suppose that C is E(x1,...,xn) over the field K, in the variables a category. A Grothendieck topology on C is a

c 2001 by CRC Press LLC collection of families of morphisms, indexed by algebra freely generated by the gα, so that an el- C r g +r g + the objects of (such a family that corresponds ement is a (formal) finite sum α1 α1 α2 α2 to an object S of C is called a covering family ···+rαn gαn . When multiplying such elements of S), such that one simplifies by writing gαgβ as gγ , for some (i.) {ϕ : T → S} is a covering family of S unique γ , and then collecting terms. whenever ϕ : T → S is an isomorphism, (ii.) if I is an index set and {ϕi : Ri → group character A representation of a group S}i∈I is a covering family of S, then, for each G is a homomorphism α : G → H , where H morphism ϕ : S → S, the fiber product Ri is a linear group over a field F . The charac- defined by Ri = Ri ×S S exists and the induced ter of α is the function Xα : G → F , de- family {ϕi : Ri → S } is a covering family of fined by Xα(g) = trace(α(g)). In the case of

S , and finite groups G, characters characterize repre- (iii.) if I is an index set, Ai is an index set sentations up to equivalence. for each i in I, the family {ϕi : Ri → S}i∈I S {s : is a covering family of , and the family i,a group extension The group G is an extension S → R } R i,a i a∈Ai is a covering family of i for of the group H by the group F if there is a short i I {ϕ ◦s : S → S} each in , then the family i i,a i,a exact sequence 1 → H → G → F → 1. See S is a covering family of . exact sequence. See also Ext group. ground field (1)IfE is an extension field of grouping of terms The rearranging of terms a field F , then F is called the ground field (or in an expression into a form more convenient for the base field). some purpose. For example, one may group the (2)IfV is a vector space over a field F , then terms involving x in the expression 5x2+ey +3x F is called the ground field of V (or field of to produce the expression (3x + 5x2) + ey. scalars of V ). ground form Consider the general linear group inverse See generalized inverse. group of some degree over a field F of charac- teristic zero. A finite number of homogeneous group minimization problem Suppose that forms with coefficients in F that are algebrai- A is an m × n matrix with real entries, b a vec- cally independent is called a set of ground forms. tor in Euclidean m-space Rm, and c a vector in n See general linear group. R . Consider the vector xB of basic variables associated with a dual feasible basis of the linear group One of the basic structures in algebra, programming problem of minimizing cT x sub- consisting of a set G and a (composition) map ject to the conditions that Ax = b and xj ≥ 0 m : G × G → G, usually written as m(g, g ) = for j = 1, ... , n. Denote by S the set of vectors g ◦ g ,g+ g or g · g , satisfying the following x such that Ax = b, xj ≥ 0 for j = 1, ... , n, axioms: and some of the coordinates of x are restricted (i.) (associativity) g ◦(g ◦g ) = (g ◦g )◦g to be integers. The problem of minimizing cT x for all g,g ,g ∈ G; subject to the condition that x is an element of (ii.) (identity) there is an element e ∈ G such the group generated from S by ignoring the non- that e ◦ g = g = g ◦ e, for all g ∈ G. negativity constraints on the coordinates of the

(iii.) (inverses) for each g ∈ G, there is g ∈ vector xB is called a group minimization prob- G (called the inverse of g) such that g ◦ g = lem. One may solve this problem by finding e = g ◦ g. the shortest path on a directed graph that has a One can think of a group as describing sym- special structure. If each coordinate of the vec- metries of certain objects. tor of basic variables for the optimal solution of the group minimization problem is nonnegative, group algebra Let R be a commutative ring then that solution is optimal for the original lin- with identity and let G be a group with ele- ear programming problem; otherwise one may ments {gα}α. The R-group algebra is the R- use a branch-and-bound algorithm to investigate

c 2001 by CRC Press LLC lattice points near the optimal solution for the all inner automorphisms if G. See group of inner group minimization problem. automorphisms. group of automorphisms A one-to-one map- group of quotients Let S be a commutative ping α : A → A, where A is an algebraic ob- semigroup with cancellation (so that ab = ac ject, for example an algebra or a group, such implies b = c), then there is an embedding of S that α preserves whatever algebraic structure A in a group G such that any g ∈ G can be written −1 has. Any such α has an inverse which also pre- as g = st , where s,t ∈ G. Here G is called serves the structure of A. The composition of the group of quotients of S. two such automorphisms does this as well. The L set of all such α forms the automorphism group group of symmetries If is a geometric L of A (where the composition map in this group object, then a symmetry of is a one-to-one L is a composition of functions). See also auto- mapping of to itself which preserves the ge- morphism group. ometry of the object (e.g., preserves the metric, if L is a subset of a metric space). The set of all such symmetries forms a group called the group group of classes of algebraic correspondences of symmetries. Suppose that C is a nonsingular curve. Denote by G the group of divisors of the product variety group of twisted type Suppose that F is a C × C, and denote by D the subgroup of divi- field. Denote by L a simple Lie algebra over sors that are linearly equivalent to degenerate the complex numbers that corresponds to the divisors. The quotient group G/D is called the Dynkin diagram of some type X, and denote by group of classes of algebraic correspondences. C the Chevalley group of type X over F . Con- sider the Lie algebra L spanned by Chevalley’s m> Z group of congruence classes Let 0be canonical basis of L over the ring Z of integers. a b an integer. Two integers and are said to be Then C is generated by linear transformations m a ≡ b m congruent mod (written mod )if xa(t) of the Lie algebra F ⊗ L , with a a root a − b m Z Z is divisible by . This is an equivalence of L, and t in F . relation. (See equivalence relation.) Denote the (1)IfX equals An, Dn,orE , then the Dynkin a [a] 6 equivalence class of the integer by ; thus diagram of type X has a nontrivial symmetry τ. [a]={b : b ≡ a m} mod . Then the set of If τ(a) = b, and if F has an automorphism equivalence classes for this equivalence relation σ such that the order of σ equals the order of [a]+ is a group under “addition” defined by τ, then denote by θ the automorphism of C [b]=[a + b]. This group is called the group of σ that sends xa(t) to xb(t ). Denote by U the congruence classes or group of congruences of θ-invariant elements of the subgroup of C gen- the integers modulo m. erated by {xa(t) : a>0,t ∈ F }, and denote by V the θ-invariant elements of the subgroup group of inner automorphisms For each of C generated by {xb(t) : b<0,t ∈ F }. The element g of a group G, let i(g) : G → G be group generated by U and V is called a group − the map defined by i(g)(g ) = gg g 1. Then of twisted type. i(gh) = i(g)i(h) g,h ∈ G we have , for all and (2) Suppose that X equals either B2 or F4 and −1 −1 also i(g ) = i(g) . It follows that i(g) is that the field F has characteristic 2, or suppose G an automorphism of and the set of all such that X = G2 and the field F has characteris- i(g) (for all g ∈ G) is called the group of inner tic 3. If F has an automorphism σ such that automorphisms of G. It is a normal subgroup of (tσ )σ = tp for each t in F , then one may ap- the group of automorphisms of G. ply a procedure similar to that described in (1) above to obtain in each case another group of group of outer automorphisms The quo- twisted type. tient group Aut(G)/Inn(G), where Aut(G) is the group of all automorphisms of G and Inn(G) groupoid A small category in which all mor- is the normal subgroup of Aut(G) consisting of phisms are invertible. See category.

c 2001 by CRC Press LLC group scheme A group scheme over a scheme groups, finite , linear groups, per- S is a scheme X, together with a morphism to the mutation groups, group actions, Galois theory, scheme S such that there is a section e : S → X and more. (thought of as the identity map), a morphism r : X → X over S (thought of as the inverse group variety A variety V , together with a map), and a morphism m : X×X → X (thought morphism m : V × V → V such that the set of as the composition map) such that (i.) the of points given by V is a group and such that − composition m ◦ (Id × r) is equal to the pro- the inverse map (v → v 1) is also a morphism jection X → S followed by e and (ii.) the two of V . Here a morphism of varieties X, Y (over morphisms m ◦ (Id × r) and m ◦ (r × Id) from a field k) is a continuous map f : X → Y , X × X × X to X are the same. such that, for every open set V in Y , and for every regular function g : V → k, the function Group Theorem Suppose that R is a ring. g ◦ f : f −1(V ) → k is regular. The set of equivalence classes of fractional ide- als of R that contain elements that are not zero G-set A set S for which there is a map f : G× divisors forms a group. S → S such that f(gg,s) = f(g,f(g,s)) and f(Id,s) = s for all g,g ∈ G and s ∈ S. group-theoretic integer programming A Here, G is a group. method that involves transforming an integer linear programming problem into a group mini- Guignard’s constraint qualification Sup- mization problem. If each coordinate of the vec- pose that X is a closed connected subset of real tor of basic variables for the optimal solution of Euclidean n-space, g is a vector-valued function the corresponding group minimization problem defined on X, and C is the set of all points in X is nonnegative, then that solution is optimal for such that gi(x) ≤ 0 for each i = 1, ... , n. Sup- the original programming problem; otherwise pose that x is a point on the boundary of X that one may use a branch-and-bound algorithm to is not an extreme point of X, denote by ∇gi(x) investigate lattice points near the optimal solu- the gradient of gi at x, and denote by Y the set tion for the group minimization problem. of vectors y for which ∇gi(x) · y ≤ 0 for each i such that gi(x) = 0. If Y is a subset of the con- group theory The study of groups. Group vex hull spanned by the vectors tangent to C at theory includes representation theory, combina- x, then g is said to satisfy Guignard’s constraint torial group theory, geometric group theory, Lie qualification at the point x.

c 2001 by CRC Press LLC the half-spin representations of the complex or- thogonal Lie algebra so2nC.

Hall subgroup Let P be a set of prime num- H bers. A P -number is a number all of whose prime divisors are in P .AP -number is a num- ber none of whose prime divisors are in P .A m × n Hadamard product Given two ma- finite group G is a P -group if |G| is a P -number. A = (a ) B = (b ) trices ij and ij , the Hadamard A subgroup H of a finite group G is a Hall P - A B product (or Schur product)of and is their subgroup if H is a P -group and [G : H ] is a P  entrywise product, usually denoted by number. Further, H is a if H is P P A ◦ B = aij bij . a -subgroup for some . This is equivalent to the condition gcd(|H |, [G : H ]) = 1. half-angle formulas The trigonometric iden- Hamilton-Cayley Theorem A matrix sat- tities cos2 θ = (1 + cos(2θ))/2 and sin2 θ = isfies its characteristic polynomial. That is, if (1 + cos(2θ))/2. p(x) is the characteristic polynomial of a matrix M, then p(M) = 0. Also called the Cayley- half-side formulas Suppose that α, β, and γ Hamilton Theorem. are the angles of a spherical triangle. Denote by a α the side of the triangle that is opposite , and Hamilton group A non-Abelian group in S = (α + β + γ)/ put 2 and which each subgroup is normal. See normal subgroup. − S R = cos . (S − α) (S − β) (S − γ) cos cos cos Hamilton’s quaternion algebra The non- commutative ring generated over the real num- The formulas 2 bers by 1, i, j, k where 1 is the identity, i = − , 2 =−, 2 =− =− 1 − cos S cos(S − α) 1 j 1 k 1 and ijk 1. Thus, sin a = , an arbitrary element has the form q = a + 2 sin β sin γ bi + cj + dk, where a, b, c, d are real num- bers. The conjugate of q is the quaternion q¯ = 1 cos(S − β)cos(S − γ) a − b − c − d cos a = , i j k, which has the property that 2 sin β sin γ qq¯ = a2 + b2 + c2 + d2. This allows one to show that each non-zero q has an inverse and − 1 q 1 =¯q/(qq)¯ . Thus, we have an example of tan a = R cos(S − α) 2 a non-commutative division algebra. The set of quaternions of norm 1 (qq¯ = 1) forms a are called half-side formulas. group under multiplication which is isomorphic to SU(2). half-spinor An element of the representa- tion space of either half-spin representation of n the complex spinor group. See half-spin repre- harmonic mean Given non-zero real num- x ,i= ,...,n sentation. bers i 1 , their harmonic mean is H , where 1/H = (1/x1 +···+1/xn)/n. half-spin representation Suppose that V is a vector space of dimension 2n and Q is a qua- harmonic motion Simple harmonic motion dratic form on V . Write V as a direct sum of is the motion of a particle subject to the differ- two n-dimensional isotropic spaces W and W  ential equation for Q. The representations corresponding to the sum of all even exterior powers of W and to the d2x W =−n2x, sum of all odd exterior powers of are called dt2

c 2001 by CRC Press LLC where n is a constant. Its solution may be writ- is the Hasse-Minkowski character of A. Here p ten as x = R cos(nt + b), where R and b are is a prime and (a, b)p is the Hilbert norm residue arbitrary constants. Damped harmonic motion symbol. is the motion of a particle subject to the differ- ential equation Hasse-Minkowski character Also called Hasse-Minkowski symbol, Hasse symbol, and d2x dx + 2p + n2x = 0 , Minkowski-Hasse character. An invariant of a dt2 dt quadratic form which, when considered together dx with the discriminant of the form and the number where 2p dt is a resistance proportional to the velocity and p is a constant. of variables, determines the class of a quadratic form over a local field. Let F be either a com- harmonic progression See harmonic series. plete archimedean field or a local field of charac- teristic not equal to 2. Let (a, b) = 1ifax2+by2 harmonic series The series represents 1, otherwise let (a, b) =−1. If f is F 1 1 1 a nondegenerate quadratic form over equiv- 1 + + +··· +··· . alent to a x2 + a x2 + ··· + anx2, then the 2 3 n 1 1 2 2 n Hasse-Minkowski character is usually defined It is well known that this series diverges; a proof as either is given by Bernoulli. One can also see this by noting that 1/2 ≥ 1/2, 1/3+1/4 > 1/2, 1/5+ χp(f ) = ai,aj or 1/6 + 1/7 + 1/8 > 1/2, 1/9 + 1/10 +···+ i 1/2, etc., showing that the series is at ∗ χ (f ) = ai,aj . least as big as n(1/2) for any n. p i≤j

Hartshorne conjecture If X1 and X2 are ∗ Note that χp = χp · (d(f ), d(f )) where d(f) smooth submanifolds with ample normal bun- is the discriminant of f . The Hasse-Minkowski dles of a connected, projective manifold Z such character depends only on the equivalence class that of the form f and not on the diagonalization dimX + dimX > dimZ, 1 2 used. It may also be defined in terms of the then is X1 ∩ X2 nonempty? Hasse algebra of f by setting χp(f ) = 1ifthe Hasse algebra associated with f splits and -1 if Hasse invariant Let E be an elliptic curve it does not. The Hasse-Minkowski character is over a perfect field k of characteristic p>0 sometimes called the Hasse invariant, since the and let F : E → E be the Frobenius mor- Hasse invariant is either 1 or the unique element ∗ phism (induced by the p-power map). Let F : of order 2 in the Brauer group of F . See also H 1(E, OE) → H 1(E, OE) be the induced map Hasse invariant, Hasse-Minkowski Theorem. on cohomology. If F ∗ = 0, then E has Hasse invariant 1; otherwise E has Hasse invariant 1. Hasse-Minkowski symbol See Hasse- Here H 1(E, OE) is a one-dimensional vector Minkowski character. space over k, since E is elliptic, and OE is the sheaf of regular functions on the variety E. Hasse-Minkowski Theorem If q is a qua- dratic form over a global field F , with charac- Hasse-Minkowski character Let A be an teristic not equal to 2, then q is isotropic over F n × n non-singular symmetric matrix with ra- if and only if q is isotropic over all F℘, where tional elements and let Di (i = 1, 2,...,n)be F℘ is the completion of F at the place ℘. the leading principal minor determinant of order If F and F℘ are defined as above, this the- i in the matrix A. Suppose further that none of orem leads to the following statement: Two the Di is zero. Then the integer quadratic forms are equivalent over F if and F n−1 only if they are equivalent over all ℘, if and only if they have the same dimension, discrim- cp = cp(A) = (−1, −Dn)p (Di, −Di+1)p i=1 inant, the same Hasse-Minkowski character for

c 2001 by CRC Press LLC m non-archimedean F℘, and the same signature of FP -rational points of VP . Denote by ZP the over the real completions of F . See Hasse- formal power series defined by Minkowski character. ∞ m Nmu ZP (u) = exp . X m Hasse principle Let be a smooth projec- m=1 tive variety over an algebraic number field k.A Denote by P the set of all prime ideals P of F fundamental Diophantine problem for X is to such that VP is defined, and denote by N(P)the decide whether there are any k-rational points absolute norm of P . Then on X and, if so, to describe them. When X is a −s geometrically integral quadric, the Hasse prin- ζ(s) = ZP N(P) ,VP . ciple affirms that if X has rational points in every P ∈P completion of k, then it has rational points in k. Hausdorff space A topological space X such Hasse’s conjecture Let m1, m2, and m3 be that, whenever p, q are points of X, then there coprime integers greater than 2. If A(m1,m2, is an open neighborhood U of p and an open m3) denotes the number of lattice points (x1,x2, neighborhood V of q such that U ∩ V =∅. x ) (x ,m ) = 3 with i i 1 in the tetrahedron Also called T2-space.

3 haversine The complex valued function h(z) (x /m ) < x /m < , 2 max i i i i 1 = (1 − cos z)/2. i=1 H then A(m ,m ,m ) is even. In 1943, Hasse Hecke algebra (1) Let be a subgroup of 1 2 3 G g G made the conjecture, which arose in his inves- a group , and suppose that for each in , H ∩ gHg−1 H tigations of class numbers of Abelian number the index of in is finite. Denote H \G/H G fields. by the set of double cosets of by H .IfR is a commutative ring, then the module RH\G/H R Hasse symbol See Hasse-Minkowski char- has an -algebra structure and is called (G, H ) R acter. the Hecke algebra of over . (2) Let H be a subgroup of a finite group G. F e Hasse-Witt map See Hasse-Witt matrix. Suppose that is a field, and is an idempotent in FH such that the left ideal FHe affords an Hasse-Witt matrix Given a complete smooth F -character. Then the subalgebra e · FG· e is algebraic variety X of dimension n defined over called a Hecke algebra. a perfect field k of positive characteristic p, there is a natural Frobenius morphism F : X → X, Hecke character Consider an algebraic num- whose action on the structure sheaf OX of X is ber field of finite degree, denote its idele group J C just as the pth power map. This action induces by and its idele class group by . A char- J C another action on the nth cohomology group acter of that is a character of is called a n H (X, OX), called the Hasse-Witt map of X Hecke character (or Grössencharakter). See n and denoted also by F . The group H (X, OX) idele class, idele group. is a finite-dimensional k-vector space and the L L(s) matrix corresponding to F is called the Hasse- Hecke -function A function , of a com- s Witt matrix of X. plex variable , defined as follows. Suppose that F is an algebraic number field of finite degree, m F χ Hasse zeta function The function ζ(s),ofa and is an integral divisor of . Denote by F complex variable s, defined as follows. Suppose the character of the ideal class group of mod- m a F that V is a nonsingular complete algebraic vari- ulo . For each integral ideal of , denote its N(a) I ety defined over a finite algebraic number field absolute norm by . Denote by the set of F P F V all integral ideals of F . Then . For each prime ideal of , denote by P the reduction of V modulo P , denote by FP the L(s) = χ(a)/N(a)s . residue field of P , and denote by Nm the number a∈I

c 2001 by CRC Press LLC height (1) Let R be a non-trivial commutative also exists and h(M) = h(M). See Herbrand ring with an identity. A prime chain of length quotient. n is a sequence I0 ⊃ I1 ⊃ I2 ⊃ ··· ⊃ In (proper inclusions) of prime ideals (i.e., Ii ∈ Hermitian form A form (·, ·) : V × V → Spec(R)). If I ∈ Spec(R), then the supremum C, where V is a real or complex vector space, of the lengths of such prime chains is called the such that (u, v) = (v, u), for all u, v ∈ V . The n height of the prime ideal I.IfI is a (proper) standard Hermitian form on C is defined by R (I) (u, v) = n u v not necessarily prime ideal of , by height i=1 i i . we will mean the minimum of the heights of the prime ideals which contain I. Hermitian matrix A matrix M, such that its n (2)If; ⊂ R is a root system and< is a base transpose and its complex conjugate (the matrix for ;, then, for β ∈ ;,wehaveβ = α∈; kαα, with entries equal to the complex conjugates of M where the kα are integers. Then the height of β the entries of ) are equal. See transpose. is α∈; kα. Hermitian operator An operator M, on a ∗ Heisenberg group The group of all upper- real or complex vector space, such that M = ∗ triangular integral (or sometimes real) 3×3 ma- M, where M is the adjoint operator. Thus, if the trices with 1s on the diagonal. The Heisenberg Hermitian form is the standard Hermitian form, ∗ T group is nilpotent. then M = MT , where stands for transpose.

Held group A finite simple group of order Hessenberg method of matrix transformation 4,030,387,200. A method for transforming a matrix into a matrix (bij ) by means of a triangular similarity trans- Hensel’s Lemma Let A be a complete lo- formation so that bij = 0 whenever i − j ≥ 2. cal ring with m its maximal ideal. Assume that A is m-adically complete. Let f(x) ∈ A[x] Hessian The Jacobian of the first order deriva- be a monic polynomial of degree n. Let π : tives of a differentiable function f = f(x1,..., A → A/m be the canonical map and extend xn) of n real variables; i.e., H(f) = J (∂f/∂x1, 2 π to π : A[x]→(A/m)[x].Ifg1(x) and ..., ∂f/∂xn) =|∂ f/∂xi∂xj |. g2(x) are relatively prime monic polynomials over A/m of degree r and n − r (respectively), Hey zeta function The function ζ(s),ofa such that π(f(x)) = g1(x)g2(x), then there ex- complex variable s, defined as follows. Sup- ist h1(x), h2(x) ∈ A[x] having degree r and n− pose that A is a simple algebra over an algebraic r such that f(x)= h1(x)h2(x) and π(hi(x)) = number field of finite degree, o is a maximal or- gi(x), for i = 1, 2. der of A, and a is an integral left o-ideal. Denote by N(a) the number of elements in o/a, and by Herbrand quotient Let G ={g1,...,gn} I the set of all integral left o-ideals. Then be a finite group and let M be a G-module. Let N : M → M be the norm defined by N(m) = ζ(s) = N(a)−s . g1m +···+gnm. Then N(M) is contained in a∈I G G M . Let h0 = M /N(M) and let h1 be the first 1 cohomology group H (G, M).Ifh0 and h1 are finite, then the quotient of their orders is h(M), higher algebra (1) Algebra at the undergrad- the Herbrand quotient. The Herbrand quotient uate level. is multiplicative for short exact sequences of G- (2) Modern, or abstract, algebra. modules: if we have 1 → M → M → M → 0, then h(M) = H(M)h(M). higher-degree Diophantine equation An al- gebraic equation of degree three or higher whose Herbrand’s Lemma Let G be a finite group. coefficients are integers, such that the solutions If M is a sub-G-module of a G-module M and sought are to be integers. See also Diophantine the Herbrand quotient h(M) exists, then h(M) equation.

c 2001 by CRC Press LLC higher differentiation For a commutative PSL(2,Od ) to an irreducible lattice in G (where ring R with a unit and N the nonnegative inte- G = PSL(2, R) × PSL(2, R)). This last is gers, a sequence called a Hilbert modular group. It is known that any irreducible lattice in G is commensurable to {δi : R → R}i∈N one of the Hilbert modular groups. of maps such that, for every x and y in R and Hilbert modular group If M ∈ PSL(2,Od ) every i and j in N, is a matrix and M stands for the matrix ob- (i.) δi(x + y) =δix + δiy; tained by replacing each entry of M by its Galois (ii.) δi(xy) = δpx · δq y; M → (M, M) p,q∈N:p+q=i conjugate, then the map sends i+j PSL(2,Od ) to an irreducible lattice in G (where (iii.) δi(δj x) = i δi+j x; and G = PSL(2, R)×PSL(2, R)). This last is called (iv.) δ0x = x. a Hilbert modular group. It is known that any G highest common factor See greatest com- irreducible lattice in is commensurable to one mon divisor. of the Hilbert modular groups. G highest weight (1) Suppose that g is a com- Hilbert modular surface Denote by a H 2 plex semisimple Lie algebra, h is a Cartan subal- Hilbert modular group, and denote by the gebra of g, and O is a lexicographic ordering on product of the upper half-space with itself. After H 2/G the real linear subspace of the complex-valued adding a finite number of points to and forms on h, spanned by the root system of g rel- obtaining the minimal resolution of the space, ative to h.Ifρ is a representation of g, then the one obtains a nonsingular surface over the com- maximal element of the set of weights of ρ with plex numbers. This surface is called the Hilbert respect to O is called the highest weight of ρ. modular surface. (2)IfV is a vector space over the field C of complex numbers, g is a semisimple Lie algebra Hilbert norm-residue symbol Suppose that F over C, π is an irreducible finite-dimensional is an algebraic number field that contains a n a b representation of g in V , and h is a Cartan sub- primitive th root of unity, that and are F P algebra of g, then g has a Cartan decomposition nonzero elements of , and that is a prime divisor of F . Put g = h ⊕ (⊕αgα), with each α being an eigen-  √   value of the action of h on π. Such an eigen- a,F n b /F value is called a weight of π. A nonzero vector σ =   , v ∈ V is called a highest weight vector of π if P v is an eigenvector for the action of h on π and if gα(v) = 0 for each positive root α of g. The where the symbol on the right is the norm-residue π a,b highest weight of is the weight of the highest symbol. The symbol P defined by π n weight vector of . √ σ a,b n Higman-Sims group A finite simple group = b P n of order 44,352,000. is called the Hilbert norm-residue symbol. See Hilbert-Hasse norm residue symbol For a also norm-residue symbol. n fixed prime p, let ζn be a primitive p th root of 1 in some algebraic closure of Qp. Let Ln = Hilbert polynomial Let R be an Artinian n Qp(ζn), and let (·, ·)n be the p th Hilbert norm ring and let R[x1,..., xn] be the (graded) poly- ∗ residue symbol of Ln. nomial ring (graded by degree). Let M = M0 ⊕ M1 ⊕ ... be a finitely generated graded R[x1, Hilbert modular group If M ∈ PSL(2,Od ) ...,xn]-module. Let LM (n) = L(Mn) denote is a matrix and M stands for the matrix ob- the length of the R-module. Then there is a tained by replacing each entry of M by its Galois polynomial f(x)with rational entries such that conjugate, then the map M → (M, M) sends LM (n) = f (n), for sufficiently large n. This

c 2001 by CRC Press LLC is the Hilbert polynomial of the R[x1,...,xn]- R-modules F0, F1, ... , Fd and there are R- module M. See length of module. homomorphisms f0 : F0 → G and fi : Fi → Fi+1 for i = 1, 2, ... , d such that the sequence Hilbert’s Basis Theorem If R is a Noethe- f f f rian ring, then so is the polynomial ring R[x]. d 1 0 0 → Fd →···→ F0 → G → 0 See Noetherian ring. is exact. Hilbert scheme Let k be an algebraically closed field. The Hilbert scheme parametrizes Hilbert’s Zero-point Theorem For any field P n R all closed subschemes of k . Here, if is a F , any finitely generated F -algebra A and any P n n ring, then k is the projective space over the ideal I of A, the radical of I is the intersection ring R. It satisfies the following condition: to A I n of all the maximal ideals of which contain . give a closed subscheme S in PT which is flat over T (for any scheme T ) is the same as giving a Hirzebruch surface The CP1-bundle over morphism f : T → H. Here the map f acts on CP1 with cross-section C where C2 =−n. t ∈ T by f(t)= the point of H corresponding S P n to the fiber t in k(T ). Hochschild’s cohomology group Suppose that A is an algebra over a commutative ring. Hilbert’s Irreducibility Theorem Suppose If M is a two-sided A-module, then consider that P is an irreducible polynomial in the n vari- the complex obtained from the module of all x ... x ables 1, , n over an algebraic number field n-cochains and the coboundary operator. The F , and suppose that 0

c 2001 by CRC Press LLC of complex vector spaces Cp,q such that the In each case, equality holds if and only if there complex conjugate of Cp,q is isomorphic to exist two real numbers a and b such that ab = 0 Cq,p. and af p = bgq almost everywhere.

Hodge theory A body of results concern- Hölder’s Theorem If n = 2, 6, then the sym- ing cohomology groups of manifolds. Three of metric group Sn is complete. See symmetric these results are listed here. group, complete group. (1) Suppose that X is a closed oriented Rie- mannian manifold. Then each element of a co- homology group of X with complex coefficients holomorphic function See analytic function. has a unique harmonic representative. (2) Suppose that V is a compact complex holomorphic functional calculus If X is a manifold of dimension n that is the image of complex Banach space (a complete normed vec- a holomorphic mapping from a compact Kähler tor space), T : X → X is a bounded linear manifold of dimension n.Ifq = max(n − p + operator and f(z)is a holomorphic function on 1, 0), then H n(V, C) carries the Hodge structure (a neighborhood of) the spectrum of T (the set induced by the type (p, q)-decomposition of the of eigenvalues of T ,ifX is finite dimensional), space of differential forms on the manifold. then one can define f(T)via a Cauchy type inte- (3)IfX is a smooth noncomplete irreducible gral. The map f → f(T)is a homomorphism variety, then H n(X, C) carries a mixed Hodge from the algebra of holomorphic functions in a structure that is independent of the choice of neighborhood of the spectrum to T into the Ba- complete algebraic variety X such that X − X nach algebra of bounded linear operators on T . is a subvariety. See Hodge structure. holosymmetric class Suppose that T is an n- dimensional lattice in n-dimensional Euclidean p = , /p + /q = Hölder inequality If 0 1, 1 1 space, and that K is a finite subgroup of the or- a ,b > 1 and i i 0, then we have thogonal group. Denote by A, B, and G the sets 1/p 1/q of arithmetic crystal classes, of Bravais types, p q and of geometric crystal classes of (T , K), re- aibi ≤ ai bi , i i i spectively. There are surjective mappings s1 : A → B and s2 : A → G, and there is an in- if p>1 and jective mapping i : B → A such that s1 ◦ i is B 1/p 1/q the identity mapping on .Aholosymmetric p q aibi ≥ a b , class is a class that belongs to the image of the i i s ◦ i i i i mapping 2 . See arithmetic crystal class, Bravais type, ge- if 0

Hölder integral inequality Suppose that f homogeneous Of the same kind. For ex- and g are measurable positive functions defined ample, a homogeneous polynomial is a polyno- on a measurable set E, and suppose that p and q mial each of whose monomials has the same de- are positive real numbers such that 1/p+1/q = gree. See homogeneous polynomial. See also 1. If p>1, then homogeneous bounded domain, homogeneous coordinate ring, homogeneous equation, homo- 1/p 1/q fg ≤ f p gq . geneous ideal, homogeneous n-chain. E E E If 0

c 2001 by CRC Press LLC homogeneous coordinate ring Let K be a homogeneous ideal An ideal I in a graded field, V a projective variety over K and I(V)the ring R such that I is generated by homogeneous ideal of K[x0,...,xn] generated by the homo- elements. See graded ring. geneous polynomials in K[x0,...,xn] that van- ish on V . Then K[x0, ...,xn]/I (V ) is called homogeneous linear equation A linear equa- the homogeneous coordinate ring of V . tion in the variables x1,...,xn, having the form c x +···+c x = , homogeneous difference equation A linear 1 1 n n 0 difference equation of the form where the ci are constants. x + c x +···+c x = , n 1 n−1 n 0 0 homogeneous n-chain Let G be a group and n+1 let G act on G diagonally: g(g0,...,gn) = where c1,...,cn are given real or complex num- (gg0,...,ggn). Define a boundary oper- bers and {xj } is an unknown infinite sequence. ator by d(g0,...,gn) = (g1,...,gn) − The notion is similar to that of homogeneous n (g0,g2,...,gn) +···+(−1) (g0,..., gn−1). linear equation except that difference equations Then the free Abelian group with basis the ele- x are often written in terms of n and powers of the ments of Gn+1, with this action of G on Gn+1

c 2001 by CRC Press LLC R-homomorphisms f : A → B for a pair of R- B can be regarded as both a left and right R- modules A and B.ByanR-homomorphism we module. mean a group homomorphism of A into B with Further development of these ideas leads to n the property that f (λa) = λf (a), for all λ ∈ R certain derived functors called ExtR(A, B) in n and a ∈ A, when A and B are left R-modules the case of the Hom functor and TorR(A, B) in and, when A and B are right R-modules, we the case of the tensor product functor. There is assume that f(aλ) = f(a)λ. These categories one such functor for each n ≥ 0 and, in case have two fundamental functors, both of which n = 0, these coincide with Hom and the ten- take their values in the category of all Abelian sor product. This development is too lengthy groups (which is, in the terminology of this def- and technical to describe further here. See Ext inition, the category of left Z-modules). We group, Tor. describe them separately. Given two left R-modules A, B, HomR(A, homological dimension Let C be the cate- B) denotes the Abelian group of all R-homo- gory of left R-modules where R is an associa- morphisms of A into B. When R is commuta- tive ring with identity. We construct a projective tive, HomR(A, B) is also both a left and right resolution of an R-module A by first taking a R-module. This functor is covariant in the sec- projective module P0, together with an epimor- ond variable and contravariant in the first. By phism P0 → A → 0 and kernel K0. There covariant in the second variable we mean that if always exists such a projective module since a B → C is a morphism then there is an induced free module on any set of generators for A will morphism work. If K0 is not projective, repeat this proc- ess with an epimorphism P1 → K0 → 0 and R(A, B) → R(A, C) . Hom Hom kernel K1. Continue until a kernel Kn which is projective is obtained. The index n is called By contravariant in the first variable we mean the projective dimension or homological dimen- that there is an induced morphism sion of the module A. Two facts must be proved A HomR(B, A) ← HomR(C, A) . before this notion makes sense: (i.) has ho- mological dimension 0 if and only if A is, itself, This functor is left exact in both variables. By projective and (ii.) the index n is independent this we mean that, given any exact sequence of the particular sequence of projective modules B → C → 0, the induced group homomor- used. See projective module. phisms homological functor A sequence of additive 0 → HomR(C, A) → HomR(B, A) covariant functors of Abelian categories H =  {Hi : A → A } defined for −∞

(a,b1 + b2) ≡ (a,b1) + (a,b2) is a complex which is always exact. (aλ, b) ≡ (a, λb) for all λ ∈ R. homological mapping A homomorphism of A ⊗R B is covariant in both variables and right homology modules induced by a chain map- exact in both. When R is commutative, A ⊗R ping. Let A be a ring with unit, and let (X, δ)

c 2001 by CRC Press LLC and (Y, δ) be chain complexes over A with ho- momorphisms defined on groups, rings, or R- mology modules H(X) and H(Y). The A- modules. homomorphism f∗ : H(X) → H(Y), of degree zero, induced by a chain mapping f : X → Y , Homomorphism Theorem of Groups Let is called the homological mapping induced by G be a group and let H and N be subgroups of f . G such that N is normal (aNa−1 ⊆ N for all a ∈ G.) Then (i.) HN = NH ={x ∈ G : x = homology Let {An : n ∈ Z} be a set of R- nh, for some n ∈ N} is a subgroup of G, (ii.) modules over some ring R. Assume that there N is a normal subgroup of NH, (iii.) N ∩ H is a family of R-homomorphisms fn : An → is a normal subgroup of H , and (iv.) the factor An−1 with the property that the composite ho- group NH/N is isomorphic to the factor group momorphisms fn−1fn = 0 for all n. This data H/(N ∩ H). defines a chain complex. The latter condition implies that Im(fn) ⊆Ker(fn−1), where Im(f ) homotopy Let X and Y be topological spaces is the image of f and Ker(f ) is the kernel of and let f and g be continuous functions from f . Then the factor module ker(fn−1)/Im(fn) = X to Y . We will say that f is homotopic to g Hn−1 is the (n − 1)st homology module of the (often written f ∼ g) to mean that there is a complex. The sequence of these modules is continuous function F : X × I → Y defined, called the homology of the complex. on the Cartesian product of X with the unit in- terval I =[0, 1], such that F(x,0) = f(x)and homology class The residue class of a cycle F(x,1) = g(x) for all x ∈ X. The relation modulo a boundary in the group of chains. Let of being homotopic is an equivalence relation. A be an Abelian category and let C = (Cn,dn) The function F that defines the relation is often be a chain complex in A. Let Zn =Kerdn and Bn spoken of as the homotopy. =Imdn+1. A residue class of Zn/Bn is called a When X is the unit interval (i.e., when we homology class. are talking about paths in the space Y ), it is cus- tomary to make the additional assumption that homology group When a chain complex of F(0,s) ≡ f(0) = g(0) and F(1,s) ≡ f(1) = Abelian groups is given, then its homology con- g(1). sists of a sequence of homology groups. See chain complex. homotopy associative Let G be a topologi- cal space carrying the structure of an H -space, homology module When a chain complex of defined by the continuous map µ : G×G → G. R-modules over some ring R is given, then its See H-space. Given points x,y,z ∈ G, we can homology consists of a sequence of homology define the maps f(x,y,z) = µ(x, µ(y, z)) and modules. See chain complex. g(x,y,z) = µ(µ(x, y), z) from G × G × G → G. We say that µ is homotopy associative if f homomorphism Algebraic structures such and g are homotopic as maps. as groups, rings, fields, and modules are all de- fined with one or more binary operations. Sup- homotopy commutative Let G be a topolog- pose that ◦:S × S → S is one such binary ical space carrying the structure of an H -space, operation which is, simply, a function from the defined by the continuous map µ : G×G → G. Cartesian product S × S to S. We customarily (See H-space.) Given points x,y ∈ G,we write a ◦ b instead of the more standard func- can define the maps f(x,y) = µ(x,y) and tional notation ◦(a, b) to describe the action of g(x,y) = µ(y,x) from G × G to G. We say this function. Then a homomorphism is a func- that µ is homotopy commutative if f and g are tion f : S → T from one such object to another homotopic as maps. with the property that f(a◦ b) = f(a)◦ f(b) for all a,b ∈ S. Additional qualifiers such homotopy identity Let G be a topological as group homomorphism, ring homomorphism, space carrying the structure of an H -space, de- or R-homomorphism are used to describe ho- fined by the continuous map µ : G × G → G.

c 2001 by CRC Press LLC (See H-space.) A constant map e(x) = e is Hopf coproduct The image of an element called a homotopy identity if the maps defined under a Hopf comultiplication. See Hopf co- by µ(x, e) and µ(e, x) are both homotopic to multiplication. the identity map i on G defined by i(x) = x for all x ∈ G. Horner method of solving algebraic equations An iteration method for finding the real roots of homotopy inverse Let G be a topological an algebraic equation. Locate a positive root space carrying the structure of an H-space, de- between two successive integers. If a1 is the fined by the continuous map µ : G × G → greatest integer less than the root, use the sub- G and having a homotopy identity e. See H- stitution x1 = x − a1 to transform the equation space, homotopy identity. A continuous map into one that has a root between 0 and 1. Lo- ν : G → G defines a homotopy inverse if the cate this root between successive tenths. Use maps defined by µ(x,ν(x)) and µ(ν(x),x) are the substitution x2 = x1 − a2 to transform the both homotopic to the homotopy identity e. equation to one that has a root between 0 and one-tenth. Continue this process. The desired Hopf algebra Two types of Hopf algebras root is then approximately a1 + a2 +···+an. have been defined. The first one was introduced by Hopf and is used in the study of homology Householder method of matrix transforma- and cohomology of Lie groups. The second type tion A method of transforming a symmet- was introduced by Sweedler and has applica- ric matrix A into a tridiagonal matrix B. The tions in the study of algebraic groups. method uses a similarity transformation of the − (1) A (graded) Hopf algebra (A,φ,ψ)over form A → H 1AH , where H represents an ∗ a field k is a graded algebra with multiplication orthogonal matrix of the form I − 2uu with ∗ φ which is also a graded co-algebra with comul- u u = 1. In the above situation, I represents the ∗ tiplication ψ such that φ : (A, ψ) ⊗ (A, ψ) → identity matrix and u is the conjugate transpose (A, ψ) is a homomorphism of graded co-alge- of u. bras. This type of Hopf algebra is frequently assumed to be connected, commutative, cocom- Householder transformation The matrix u = Hv mutative, and of finite type. transformation where H is a symmet- H = (2) A Hopf algebra with an antipode S is a ric and orthogonal matrix of the form I − xxT xT x = bialgebra with antipode S. See also antipode. 2 , where 1. In the above sit- uation, xT represents the transpose of x, and I Hopf algebra homomorphism A bialgebra represents the identity matrix. homomorphism f , between a Hopf algebra H n with antipode S and a Hopf algebra H  with H-series Since the th factor of a Blaschke   product antipode S such that S f = fS. See also Hopf an an − z algebra. |an| 1 − anz Hopf comultiplication A degree preserv- can be written in the form 1 + C(z,an), where 2 ing linear map defined as follows. Let X be C(z,a) = (1−|a|)/|a|−(1−|a| )/|a|(1−az), a topological space with a base point x0 and let the Blaschke product converges absolutely at a z C(z ,a ) µ be a continuous base point preserving map- point 0 if and only if 0 n converges H ping from X × X to X. Let ι1(x) = (x, x0) absolutely. Thus, in , an - ι (x) = (x ,x) µ ◦ ι series is a series of the form and 2 0 .If i is homotopic to 1X, then µ induces a degree preserving linear c /( − a z) , ∗ ∗ n 1 n map, µ , from the cohomology group H (X) to ∗ ∗ H (X) ⊗ H (X) viaaKunneth¨ isomorphism. where 0 < |an| < 1 (n = 1, 2, ···) and (1 − ∗ In this situation µ is called a Hopf comultipli- |an|)<∞. The set of points on C at which cation and (X, µ) is called an H-space. If α is the H -series converges is called its set of con- an element of H ∗(X), then µ∗(α) is called the vergence. The subject of representation theory Hopf coproduct of α. also considers H -series.

c 2001 by CRC Press LLC H -series hyperalgebra A bialgebra, whose underly- ing co-algebra is co-commutative, pointed, and H-space A topological space G, together irreducible. with a continuous map µ : G × G → G, from the Cartesian product of G with itself to G.In hyperbolic cosecant function See hyper- practice, various additional conditions are im- bolic function. posed on such maps that make them imitate the properties normally associated with a multipli- hyperbolic cosine function See hyperbolic cation. See homotopy. function. hull-kernel topology A topology on the set hyperbolic cotangent function See hyper- of primitive ideals (the structure space I)ofa bolic function. Banach algebra R. Under this topology, the clo- sure of a set U is the set of primitive ideals con- hyperbolic function The six complex func- taining the intersection of the ideals in U. tions: (z) = exp z−exp(−z) hyperbolic sine: sinh 2 (z) = exp z+exp(−z) Hurwitz’s relation A relation between the hyperbolic cosine: cosh 2 T sinh(z) Riemann matrices of two Abelian varieties 1 hyperbolic tangent: tanh(z) = T cosh(z) and 2 that implies the existence of a homo- (z) = cosh(z) hyperbolic cotangent: coth sinh(z) morphism from T1 to T2. Let T1 and T2 be hyperbolic secant: sech(z) = 1 Abelian varieties with Riemann matrices F1 = cosh(z) (ω(1),... ,ω(1)) F = (ω(2),... ,ω(2)) (z) = 1 1 2n and 2 1 2m . hyperbolic cosecant: csch sinh(z) . There is a homomorphism λ : T1 → T2 if and Due to the fact that sinh(iz) = i · sin(z) and only if there is a complex matrix W, and a ma- cosh(iz) = cos(z), where sin(z) and cos(z) are trix M with integer entries, such that WF1 = the ordinary sine and cosine of the complex an- F2M. In particular, for every homomorphism gle z, the are rarely used λ : T1 → T2 there is a representation matrix outside of certain specialized applications. W(λ) with complex coefficients, and a represen- tation matrix M(λ) with respect to the real coor- hyperbolic secant function See hyperbolic (ω(1),... ,ω(1)) (ω(2),..., function. dinate systems 1 2n and 1 (2) ω m), that has coefficients in Z, such that W(λ) 2 hyperbolic sine function See hyperbolic F1 = F2M(λ). function. Hurwitz’s Theorem If every member of a normal family (of analytic functions on a con- hyperbolic tangent function See hyperbolic nected, open, planar set F) is never zero on function. F, then the limit functions are either identically hyperbolic transformation A linear frac- zero or never zero on F. az+b tional function f(z)= where a, b, c, and A family of analytic functions on a set F is cz+d d are complex constants with ad − bc = 0 has normal on F if every sequence from the fam- a pair of (not necessarily distinct) fixed points. ily contains a subsequence that converges uni- (See linear fractional function.) This is so be- formly on compact subsets of F. az+b cause the equation z = cz+d is linear or qua- dratic in z. When the two fixed points coin- Hurwitz zeta function The function cide, the transformation is said to be parabolic. ∞ When there are distinct fixed points, say α and 1 ζ(s,a) = , 0

c 2001 by CRC Press LLC hyperbolic trigonometry One of the two mial ring k[X, Y ]. Since F(X,Y) generates classical types of non-Euclidean plane geom- a maximal ideal in k[X, Y ], the factor ring etry, which are distinguished from each other k[X, Y ]/(F (X, Y )) = K is also a field and con- and from Euclidean plane geometry by the form tains a canonical copy of k. If we denote the of the parallel axiom that holds. For Euclidean canonical images of X and Y in K by x and geometry, there is, through any point P not on y, respectively, then K = k(x,y) is the field a line L, exactly one line parallel to L. For el- extension of k generated by x and y. The field liptic (also Lobachevskian) geometry there is, K = k(x,y)is called the field of algebraic func- through any point P not on a line L, no line tions on the algebraic curve defined by F(X,Y). parallel to L. For hyperbolic geometry there is, A hyperelliptic curve is an algebraic curve of the through any point P not on a line L, more than special form Y 2 = P(X), where P(X)is a poly- one (actually infinitely many) lines parallel to nomial that has no repeated roots. L. It is also usually assumed that k is algebrai- Within the language of Riemannian geome- cally closed in K. That is to say, every element try these three types of spaces are those of, re- of K that is not in k is transcendental over k. spectively, zero, positive, and negative constant When this is so, k is said to be the exact con- curvature. Models that exist within Euclidean stant field for the curve. geometry are given by the following construc- tions. For elliptic geometry, the space is the hyperelliptic integral Let F(X,Y) = Y 2 − surface of a sphere and the lines are great cir- g(X) define a hyperelliptic curve over the field cles on the sphere. We must regard antipodal C of complex numbers, where the polynomial points on the sphere as being identified in order g(X) is square free and of degree n. Let K = that lines intersect in no more than one point. C(z, w) be the function field for this curve. Here, For hyperbolic geometry, the space is the inte- z and w are the canonical images of X and Y in rior of a disk and the lines are arcs of circles the residue class field K = C[X, Y ]/(F (X, Y )). that intersect the boundary of the disk at right The branch points of the function field K over the angles or are straight lines through the center projective line C(z) are the roots of g(X) and, of the disk. An alternative model is the upper also, the point at infinity, when n is odd. When half plane (y > 0) and the lines are arcs of cir- n is even, the point at infinity is not a branch cles centered on the real axis or straight lines point. We can, however, move one of the roots orthogonal to the x-axis. of g(X) to ∞ by a linear change of variable in An important property of these z. Thus there is no loss of generality in assum- concerns the sum of the angles of a triangle. In ing that n is odd. The Riemann surface X for g = − + 1 (n + ) , the sum of the angles of this curve has genus 1 2 2 1 , ac- any triangle must be exactly equal to π (that is cording to the standard formula for the genus in to say, a straight angle). In elliptic geometry the case [K : C(z)]=2 and there are n + 1 branch sum of the angles is always greater than π while points of index 2. According to the Riemann– in hyperbolic geometry the sum of the angles is Roch Theorem, there are g linearly independent always less than π. holomorphic differential forms defined on the If the non-Euclidean plane is embedded in a Riemann surface. A holomorphic differential Euclidean space of sufficiently high dimension, form is an expression of the form f(z)dz, such then it becomes possible to measure areas of re- that, if π denotes a local parameter at a point P dz gions on the plane using the Euclidean measure on the Riemann surface, then f(z)dπ , which is of area relative to the enveloping space. It was a member of K, has no pole at P . These are shown by Gauss that the area of a triangle, so commonly referred to as differentials of the first measured, equals the difference between π and kind. Each of them gives rise to an Abelian inte- the sum of the angles of the triangle. gral that is defined everywhere on the universal covering surface for the Riemann surface. These hyperelliptic curve An algebraic curve, are hyperelliptic integrals of the first kind. They over a ground field k, is defined by an ir- are more commonly referred to as Abelian inte- reducible polynomial F(X,Y) in the polyno- grals of the first kind. For hyperelliptic curves,

c 2001 by CRC Press LLC these integrals can be given quite explicitly as b in S, there exist elements x and y in S such that b ∈ ax and b ∈ ya. zr √ dz g(z) hypergroupoid A set S, with a multiplication that associates every two elements a and b in S ≤ r

c 2001 by CRC Press LLC aF ⊆ R, for some nonzero a ∈ R. This in- cludes all ordinary nonzero ideals. A fractional ideal is principal if it is of the form aR where a is some nonzero member of K. A fractional I ideal F is invertible if there is another fractional ideal G such that FG = R. The invertible frac- tional ideals form an Abelian group under ordi- I-adic topology A topology that endows a nary multiplication and this group contains the ring (module) with the structure of a topological group of principal fractional ideals. The fac- ring (module). Let I be an ideal in a ring R and tor group is isomorphic to the ideal class group let M be an R-module. The I-adic topology on defined earlier. M is formed by taking the cosets x + I nM, x ∈ M, n a positive integer, as a base of open sets. ideal class group The group of ideal classes The I-adic topology on R is formed by letting of invertible ideals of an integral domain R or, M = R. equivalently, the group of invertible fractional ideals of R, modulo the subgroup of principal icosahedral group The group of rotations fractional ideals. See ideal class. of a regular icosahedron. It is a simple group of order 60 and is isomorphic to the alternating ideal class in the narrow sense Let Ik be the group of degree 5. Abelian group consisting of the fractional ideals + of an algebraic number field k. Let Pk be the ideal In any ring (usually associative with subgroup of Ik consisting of all the principal identity), any subgroup J of the additive group ideals generated by totally positive elements of of R is called a left ideal if RJ ⊆ J and is called k.Anideal class of k in the narrow sense is a + a right ideal if JR ⊆ J and, if both of these coset of Ik modulo Pk . conditions hold, is called, simply, an ideal. ideal group Let K be an algebraic number ideal class Let R be an integral domain (com- field and let R be its ring of algebraic integers. mutative ring with no divisors of zero). Two (1) The group of fractional R-ideals of K ideals J and K are said to be linearly equivalent forms an Abelian group called the ideal group if there exist nonzero elements a and b in R such of K. that aJ = bK. This establishes an equivalence (2) Let m be an integral ideal and let Km = relation on the collection of all nonzero ideals {a/b : a,b ∈ R,aR and bR relatively prime of R which we can write as J ≈ K. Any equiv- to m}. Denote the group of all ideals of K that ∗ alence class under this relation is said to be an are relatively prime to m by IK (m). Let m be ideal class. a formal product of m and a finite number of These classes respect multiplication in the real infinite prime divisors of K, then m∗ is an sense that if J1 ≈ J2 and K1 ≈ K2, then J1J2 ≈ integral divisor of K.Forα ∈ Km let α1 and α2 K1K2. In most cases, this notion is only applied be elements of Km ∩ R such that α = α1/α2. to the collection of invertible ideals of R. That is Let S(m∗) be the group of all principal ideals to say, those ideals J for which there is another generated by elements α of Km such that α1 ≡ ideal K such that JK is linearly equivalent to α2(mod m) and α ≡ 1(mod p) for all the real the unit ideal R. For this case, the collection infinite prime divisors p included in the formal of classes of invertible ideals forms an Abelian product above. Any subgroup of IK (m) which group referred to as the ideal class group of the contains S(m∗) is called an ideal group modulo ring. m∗. An alternative and more modern way of defin- (3) Sometimes the definition given above is ing this construction is contained in the notion called a congruence subgroup. In this case an of fractional ideals. A fractional ideal of an in- ideal group is an equivalence class of a congru- tegral domain R is a nonzero R-submodule F ence subgroup under the following equivalence of the quotient field of R with the property that relation. Two congruence subgroups H1 and

c 2001 by CRC Press LLC H m∗ m∗ A 2, modulo 1 and 2, respectively, are said commutative Banach algebra , then there is a to be equivalent if there is a modulus n∗ such unique element f of A such that f 2 = f and that H1 ∩ IK (n) = H2 ∩ IK (n). the Gel’fand transform of f is1onE and 0 everywhere else. This theorem is sometimes idele Let K be a global field. By this we referred to as Shilov’s Idempotent Theorem. mean either an algebraic number field (a finite extension of the rational field Q) or a field of al- identity (1) An equation. For example, a true gebraic functions in one variable over a ground formula such as field k (a finite separable extension K of the field N N(N + ) k(x) of rational functions over the ground field j = 1 k, such that k is itself algebraically closed in 2 j=1 K). These are also referred to as product for- mula fields since each of these types possesses is an identity. a class of absolute values, which are real valued (2) That element of a group, usually denoted functions, defined on K with the properties that by the symbol 1, with the property that 1a = |a|≥0 for all a ∈ K, |a|=0 if and only if a1 = a for all a in the group (or denoted 0, with a = 0 and |a + b|≤|a|+|b|, for all a,b ∈ K. 0 + a = a + 0 = a, in the case of an additive See also valuation. group). Moreover this class of absolute values sat- isfies, for each nonzero a ∈ K, the relation identity character The character on a group |a| = G χ(g) ≡ g ∈ G all ℘ ℘ 1. Consider now the direct prod- with the property that 1 for all . uct of copies of the multiplicative group K∗ of See character. K, indexed by absolute values. That is to say, functions from the set of absolute values to the identity element A member e of a set S with a group K∗. We will denote such a function by a, binary operation x◦y, such that x◦e = e◦x = x. and its value at ℘ by a℘.Anidele is such a func- If the binary operation is not commutative, a left tion with the additional property that |a℘|=1, and right identity may be defined separately. A for almost all ℘. Since each element of K∗ gives left identity e satisfies e ◦ x = x and a right rise to such a function (a℘ = a for all ℘)we identity e satisfies x ◦ e = x. may regard K∗ as a subgroup of the group of all ideles. These are called principal ideles and the identity function A function i with a domain factor group is called the idele class group of the set X such that i(x) = x for all x ∈ X. See also field. identity map, identity morphism. idele class Members of the idele class group identity map See identity function. of a global field. See idele. identity matrix The identity element E in the n×n idele class group The group of all idele ring of matrices over some coefficient ring. E classes of a global field. See idele. The entries of are given by the Kronecker delta-function idele group The group of all ideles of a global 1ifi = j δij = field. See idele. 0ifi = j. idempotent element An element a of a ring, with the property that a2 = a. identity morphism Let C be a category, let A be an object in C, and let i : A → A be a mor- idempotent subset A subset S of a ring hav- phism in C.(Amorphism is a generalization of a ing the property that S2 = S. function or mapping.) i is the identity morphism for the object A if f ◦ i = f for every object B Idempotent Theorem If E is an open-closed and morphism f : A → B in C, and i ◦ g = g subset of the maximal ideal space of a unital for every object C and morphism g : C → A in

c 2001 by CRC Press LLC the category C. One of the axioms of category imaginary number Any member of the field theory is that every object in a category has a of complex numbers of the√ form iy, where y is (necessarily unique) identity morphism. a real number and i = −1 . The field of In most familiar categories, where objects complex numbers is the set of numbers of the are sets with some additional structure and mor- form x + iy, where x and y are real. phisms are particular kinds of functions, the identity morphism for an object A is simply the imaginary part of a complex number If z = identity function i; that is the function i such x+iy is a complex number, then the real number that i(a) = a for all a ∈ A. See also identity y is called the imaginary part of z, denoted y = function, morphism. z, and the real number x is called the real part of z, denoted x =z. Ihara zeta function Let R be the real num- bers, kp be a p-adic field, and let ' be a sub- imaginary prime divisor One of two types group of G = PSL (R)× PSL (kp) for which 2 2 of infinite prime divisors on an algebraic number the following properties hold: (i.) ' is discrete; field K. Let a ∈ K and let σ be any injection of (ii.) the projection, 'R,of' in PSL (R) is dense 2 K into the complex number field which does not in PSL (R); (iii.) the projection, 'p,of' in 2 map K into the real number field. The equiva- PSL (kp) is dense in PSL (kp); (iv.) the only 2 2 lence class of an archimedean valuation on K, torsion element of ' is the identity; (v.) the quo- given by v(a) =|σ(a)|2, is called an imaginary tient '\G is compact. Let H be the upper half (infinite) prime divisor. See also real prime di- plane, {x+iy : y>0}, and, for every z ∈ H , let visor, infinite prime divisor. 'z ={γ ∈ ' : γ(z) = z}, where ' acts on H ˜ ∼ via 'R. Let P(') ={z ∈ H : 'z = Z} and let K P(') = P(')/'˜ .IfQ ∈ P(')is represented imaginary quadratic field A field which by z ∈ H and γ is a generator of 'z, then γ is a quadratic (degree two) extension of the ra- is equivalent to a diagonal matrix in 'p whose tional number field Q is called√ a quadratic num- K = ( m) diagonal entries are λ and λ−1. Let deg(Q)be ber field. Since Q , for some square m |v(λ)| where v is the valuation of kp. The Ihara free integer , we may further distinguish these m< m> zeta function of ' is fields according to whether 0or 0. In the first case, the field is called an imaginary deg(Q) −1 Z'(u) = (1 − u ) . quadratic field and, in the second, a real qua- Q∈P(') dratic field.

f(z) ill-conditioned system A system of equa- imaginary root A root of a polynomial , r f(r)= tions for which small errors in the coefficients, i.e., a number such that 0, which is an or in the solving process, have a large effect on imaginary number. See imaginary number. the solution. imaginary unit Any number field K (finite image For a function f : S → T from a extension of the field Q of rational numbers) set S into a set T , the set Im(f ) of all elements contains a ring of integers, denoted R. This ring of T of the form f(s)for some s in S. Another consists of all roots in K of monic irreducible popular notation is f(S). Also called range. polynomials f(x) ∈ Z[x], where Z is the ring of rational integers. Alternatively, this ring is imaginary axis in the complex plane Com- the integral closure of Z in K. The units of R plex numbers√ x + iy, where x and y are real and are those elements u ∈ R for which there is i = −1, can be identified with the points of an element v ∈ R such that uv = 1. In an the real plane with Cartesian coordinates (x, y). isomorphic embedding of K into the field of The plane is then referred to as the complex complex numbers, if the image of a unit u is plane. The coordinate axis x = 0 is called the imaginary, then u is called an imaginary unit. imaginary axis. Similarly, the axis y = 0is Note that this depends on which embedding is referred to as the real axis. used.

c 2001 by CRC Press LLC imperfect field Any field that admits proper incommensurable Two members a,b ∈ S of inseparable algebraic extensions. This is the op- a partially ordered set S such that neither a ≤ b posite of perfect. The simplest example of such nor b ≤ a. a field is the field k(x) where k = GF (q) is a finite field. The extension given by the equation incomplete factorization A method of pre- tq − x = 0 is inseparable. Fields of character- conditioning the system of equations Ax = b, istic zero are perfect as are all finite fields. in which A is approximated by LU, where L is a lower triangular matrix and U is an upper tri- implicit enumeration method of integer pro- angular matrix. In an incomplete factorization, gramming A method of solving zero-one small elements are dropped, making the approx- linear programming problems. Values are as- imation sparse. signed to some of the variables; if the solution of the resulting linear program is either infeasi- inconsistent system of equations A set of ble or not optimal, then all solutions containing equations for which there does not exist a com- the assigned values may be ignored. mon solution (in an appropriate solution space).

increasing directed set A set S, together with Implicit Function Theorem A theorem guar- a ≤, having the properties (i.) a ≤ anteeing that an implicit equation F(x,y) = 0 a for all a ∈ S, (ii.) a ≤ b and b ≤ c implies can be solved for one of the variables. One of a ≤ c, and (iii.) for all a,b ∈ S, there is an many different forms of the theorem is the fol- element c ∈ S such that a ≤ c and b ≤ c. Also lowing assertion. Let F(x ,...,xn,y)be a con- 1 called directed set. tinuously differentiable function of n + 1 vari- ables. Let P be a point in n+1 space and assume ∂F indecomposable group A group G (G = that F(P) = c and (P ) = 0. Then, there is ∂y {e}), that is not isomorphic to the direct product a neighborhood of P and a unique continuously of two subgroups, unless one of those subgroups differentiable function f(x ,...,xn) defined on 1 is {e}. this neighborhood, such that indecomposable module A module M over F (x1,...,xn,f (x1,...,xn)) ≡ c a ring R such that M is not the direct sum of two proper R-submodules. on this neighborhood. indefinite Hermitian form A Hermitian form equivalent to imprimitive transitive permutation group A permutation group G on a set X with the fol- p q lowing two properties: (i.) for each x and y in X x¯ixi − x¯p+j xp+j there is a t ∈ G such that t(x) = y; (ii.) the sub- i=1 j=1 group of G consisting of all permutations which leave x fixed is not a maximal subgroup. See in which x¯ is the conjugate transpose and neither also permutation group, transitive permutation p nor q is zero. group. indefinite quadratic form A quadratic form F improper fraction A rational number m/n, over an ordered field , which represents both where m and n are integers and m>n>0. positive and negative elements. If every positive m number in F is a square (for example, if F = R), Such a fraction can be written in the form n = r then an indefinite quadratic form is equivalent to n + q where r

c 2001 by CRC Press LLC in which neither p nor q is zero. The rank of For example an infinite sequence {a1,...,an} is the quadratic form is p + q where p and q are a set indexed by the natural numbers. uniquely determined by the quadratic form. (2) There are many theorems in mathematics that are referred to as “index theorems" and these indefinite sum A concept analogous to the generally describe properties of certain special indefinite integral. Given a function f(x)and a types of indices. For example, the index of a lin- fixed quantity ;x (;x = 0), let F(x)be a func- ear function T : X → Y between two complex tion such that F(x+ ;x) − F(x) = f(x)· ;x. vector spaces is, by definition, dim ker(T )− dim If c(x) is an arbitrary function, periodic, with ker T ∗, where T ∗ is the adjoint, or conjugate a period ;x, then F(x)+ c(x) is an indefinite transpose, of T . See also index of specialty. sum of f(x). index of eigenvalue (1) The number Independence Theorem A corollary of the ∗ dim ker(A − λI) − dim ker A − λI¯ , Approximation Theorem. Let e1,e2,...,en be real numbers and let v1,v2,...,vn be mutually where A is a square matrix (or a Fredholm lin- nonequivalent and nontrivial multiplicative val- ear operator on a Banach space), I is the identity ei ∗ uations of a field K.If i vi(a) = 1 for all matrix of the same size as A, and A is the con- a ∈ K\{0}, then ei = 0 for i = 1, 2,...,n. jugate transpose (Banach space adjoint) of A. (2) The order of the largest (Jordan) block independent linear equations A system of corresponding to λ in the Jordan normal form of linear equations in which deleting any equation A. The index of λ is the smallest positive integer would expand the solution set. m such that rank(A−λI)m = rank(A−λI)m+1. independent variable A symbol that rep- index of specialty Let D be a divisor on resents an arbitrary element in the domain of a complete nonsingular curve over an algebrai- a function. If the domain of the function is a cally closed field k, and let K be a canonical di- Cartesian product set X1 × X2 ×···×Xn, then visor of X. The index of specialty of the divisor 0 2 the independent variable may be denoted (x1, D is dimkH (X, O(K − D)) =dimkH (X, O x2,... ,xn), where each xi represents an arbi- (D)), where O (D) is the invertible sheaf associ- trary element in Xi. In addition, each xi is some- ated with D. This definition also applies to divi- times called an independent variable. See also sors on nonsingular surfaces. If D is a divisor on dependent variable. a curve on genus g, then the Riemann-Roch The- orem may be applied to give a second formula indeterminate form An expression of the for the index of specialty. In this case, the spe- ∞ form 0/0, ∞/∞, 0·∞, ∞−∞, ∞0, 00, or 1 . cialty index of D is equal to g−degD+dim|D|, Such expressions are undefined. Indeterminate where |D| is a complete linear system of D. forms may appear when a limit is improperly evaluated as the quotient (product, etc.) of lim- Index Theorem of Hodge Let M be a com- its and not the limit of a quotient (product, etc.). pact Kahler¨ manifold of complex dimension 2n But the term may properly appear when such and let hp,q denote the dimension of the space of limits are classified, so that they can be evalu- harmonic forms of type (p, q) on M. The signa- p p,q ated. ture of M is p,q(−1) h . The sum may be restricted to the case where p + q is even since, indeterminate system of equations A sys- on a compact Kahler¨ manifold, hp,q = hq,p. tem of equations with an infinite number of so- Any complex, projective, nonsingular, algebraic lutions. variety is a Kahler¨ manifold, and the follow- ing application of the Riemann-Roch Theorem index (1) Most commonly, a subscript which is also called the Hodge Index Theorem. is used to distinguish members of a set S. Thus, Let H be an ample divisor on a nonsingu- in this sense, a function defined on a set (called lar projective surface X, and let D be a divisor the index set) and taking its values in the set S. which has a nonzero intersection number with

c 2001 by CRC Press LLC some divisor E. If the intersection number of smallest member. Call this member b. This is D and H equals zero, then the self-intersection not the smallest member of S as we have seen. number of D is less than zero. This implies that However, T contains all a such that a

c 2001 by CRC Press LLC years until it was realized just in this century that in these definitions. If L is a Galois extension they were equivalent. of K with Galois group G, it can be shown that e = e1 = ··· = er and f = f1 = ··· = fr , inequality An expression that involves mem- so, in this case, [L : K]=ef r . Now, for each bers of some partially ordered set, commonly j, there is a subgroup Gj of G consisting of taking the form a

c 2001 by CRC Press LLC where i+(A), i−(A), i0(A) are, respectively, the property that its fixed field is precisely K.(See number of positive, negative, and zero eigenval- Galois group.) The fixed field is the subfield of ues (counting multiplicities), of a given Hermi- L that is left elementwise fixed by the Galois tian matrix A. group. Sylvester’s Law of Inertia states that two Her- mitian matrices A and B satisfy i(A) = i(B) if infinite height (1) An element a in an Abel- and only if there exists a nonsingular matrix C ian p-group A is said to have infinite height if such that A = CBC∗. the equation pky = a is solvable in A for every The notion of inertia can be extended to ar- nonnegative integer k. bitrary square matrices with complex entries, (2) A prime ideal p is said to have infinite i (A), i (A), i (A) where + − 0 are, respectively, the height if there exist chains of prime ideals p0 < numbers of eigenvalues with positive, negative, p1

c 2001 by CRC Press LLC infinity (1) The symbol ∞, used in the nota- inhomogeneous polarization (1) The alge- tion for the sum of an infinite series of numbers braic equivalence class of a nondegenerate divi- ∞ sor on an Abelian variety. X an ; (2) Let be a proper scheme over an algebra- k (X) n=0 ically closed field and let Pic be the Picard group of X. Let Picτ (X) be the subgroup of an infinite interval, such as Pic(X) consisting of the set of all the invertible sheaves F on X, for which there exists a non- [a,∞) ={x : x ≥ a}; zero integer n, such that F n represents a point of the Picard scheme of X in the same component or a limit, as x tends to ∞, as the identity. A coset of Picτ (X) in Pic(X) lim f(x). consisting of ample invertible sheaves is called x→∞ an inhomogeneous polarization of X. (2) The point at ∞ as, for example, the north i : A → B pole of the Riemann sphere. injection A function such that i(a1) = i(a2) whenever a1 = a2. Injections inflation A map of cohomology groups in- are also called injective functions, injective map- duced by lifting the cocycles of a factor group pings, one-to-one functions, and univalent func- G/H to G. Let G be a group, M a G-module, tions. The notion of an injection generalizes to H a normal subgroup of G, MH ={m ∈ M : the notion of a monomorphism or monic mor- hm = m, for all h ∈ H}, π the canonical epi- phism in a category. See monomorphism. See morphism from G to G/H , and i : MH → M also epimorphism, surjection. the inclusion map. The inflation map is the map of cohomology groups injective class A class of objects in a cate- gory, each member of which is injective. The n H word “class” is used here rather than “set” be- inf =(π, i)∗ : H G/H, M cause we cannot talk about the set of all injec- n → H (G, M) , tive objects in most categories without running into the sort of logical difficulties, for example n ≥ (for 1), which is achieved by lifting the Russell’s paradox, connected with the “set of all (π, i) : (G/H, MH ) → homomorphism of pairs sets.” (G, M). injective dimension A left R module B, inflection point A point on a plane curve at where R is a ring with unit, has injective di- which the curve switches from being concave to mension n if there is an injective resolution convex, relative to a fixed line; a non-singular point P on a plane curve C such that the tan- 0 −→ B −→ E0 −→ · · · −→ En −→ 0 , gent line to C at P has intersection multiplicity but no shorter injective resolution of B. The greater than or equal to 3. See also intersection definition of the injective dimension of a right multiplicity. R module is entirely similar. See injective res- olution. See also flat dimension, projective di- inhomogeneous difference equation A lin- mension. ear difference equation of the form Injective dimensions have little to do with

a0yk+n + a1yk+n−1 +···+anyk = rk , more elementary notions of dimension, such as the dimension of a vector space, but they are where rk differs from 0 for some values of k.In related to a famous theorem called “Hilbert’s the opposite case where rk is identically equal Theorem on Syzygies.” Suppose the ring R is to 0, the linear difference equation is homoge- commutative. Define the global dimension of neous. These notions are directly analogous R to be the largest injective dimension of any to the corresponding ones for linear differential R module, or ∞ if there is no largest dimen- equations. sion. Let D(R) denote the global dimension of

c 2001 by CRC Press LLC R, and let R[x] be the ring of polynomials in x injective object An object I in a category C with coefficients in R. Then in modern termi- satisfying the following mapping property: If i : nology, Hilbert’s Theorem on Syzygies states B → C is a monomorphism in the category, and that D(R[x]) = D(R) + 1. See also syzygy. f : B → I is a morphism in the category, then there exists a (usually not unique) morphism g : injective envelope An object E in a cate- C → I in the category such that g ◦ i = f . gory C is the injective envelope of an object A This is summarized in the following “universal if it has the following three properties: (i.) E mapping diagram”: is an injective object, (ii.) there is a monomor- i phism i : A → E, and (iii.) there is no injective B −→ C object properly between A and E. In a gen- f %&∃g eral category, this means that if j : A → E! I and k : E! → E are monomorphisms and E! is injective, then k is actually an isomorphism. See also injective module, monomorphism, pro- See also injective object, monomorphism, pro- jective module, projective object. jective cover. In most familiar categories, objects are sets In most familiar categories, objects are sets with structure (for example, groups, Banach with structure (for example, groups, topological spaces, etc.), and morphisms are particular spaces, etc.), morphisms are particular kinds of kinds of functions (for example group homo- morphisms, bounded linear transformations of functions (for example, group homomorphisms, ≤ continuous functions, etc.), and monomor- norm 1, etc.), so monomorphisms are one- phisms are one-to-one functions of a particu- to-one functions (injections) of particular kinds. lar kind. In these categories, property (iii.) can Here are two examples of injective objects in be phrased in terms of ordinary set containment. specific categories: (1) In the category of Abel- For example, in the category of topological ian groups and group homomorphisms, the in- jective objects are just those Abelian groups spaces and continuous functions, property (iii.) G reduces to the assertion that there is no injective which are divisible. (An Abelian group is divisible g ∈ G topological space containing A as a topological if for each and each integer E n, there exists a group element gn ∈ G such subspace, and contained in as a topological ng = g subspace, except for E itself. that n .) Thus, for example, the rational numbers Q are an injective object in this cat- egory. (2) The Hahn-Banach Theorem asserts injective mapping See injection. that the field of complex numbers, thought of as a one-dimensional complex Banach space, is an injective module An injective object in the injective object in the category of complex Ba- R R category of left modules and left module nach spaces and bounded linear transformations R homomorphisms, where is a ring with unit. of norm ≤ 1. Symmetrically, an injective object in the cate- gory of right R modules and right R module injective resolution Let B be a left R mod- homomorphisms. See injective object. See also ule, where R is a ring with unit. An injective flat module, projective module, projective ob- resolution of B is an exact sequence, ject. φ φ φ R 0 1 2 In the important case where the ring is a 0 −→ B −→ E0 −→ E1 −→··· , principal ideal domain, the injective R modules are just the divisible ones. (An R module M is where every Ei is an injective left R module. divisible if for each element m ∈ M and each There is a companion notion for right R mod- r ∈ R, there exists mr ∈ R such that rmr = m.) ules. Injective resolutions are extremely impor- Thus, for example, the rational numbers, Q, are tant in homological algebra and enter into the an injective Z module, where Z is the ring of dimension theory of rings and modules. See integers. See also flat module, injective object, also flat resolution, injective dimension, injec- projective module, projective object. tive module, projective resolution.

c 2001 by CRC Press LLC An exact sequence is a sequence of left R over F . (This means that α satisfies a polyno- modules, such as the one above, where every mial equation P(α)= 0 with coefficients in F .) φi is a left R module homomorphism (the φi are Among all polynomials P with coefficients in called “connecting homomorphisms”), such that F such that P(α) = 0, there is one of smallest Im(φi) = Ker(φi+1). Here Im(φi) is the image positive degree, called the minimal polynomial of φi, and Ker(φi+1) is the kernel of φi+1.In of α. The algebraic element α is inseparable if the particular case above, because the sequence its minimal polynomial is inseparable. See also begins with 0, it is understood that the kernel inseparable polynomial. Antonym: separable of φ0 is 0, that is φ0 is one-to-one. There is a element. companion notion for right R modules. Inseparable elements can only occur if the field F has characteristic n = 0. In particular, inner automorphism A group automor- inseparable elements can never occur if F is the −1 phism of the form φa(g) = aga . In more field of rational numbers, or an extension of the detail, an automorphism of a group G is a one- rationals. See also inseparable extension. to-one mapping of G onto itself which preserves φ(g g ) = φ(g )φ(g ) the group operation, 1 2 1 2 for inseparable extension An algebraic exten- g g G a G all 1 and 2 in .If is a fixed element of , sion field containing an inseparable element. φ the mapping a defined above is easily seen to See inseparable element. Antonym: separable be an automorphism. Automorphisms of this extension. form are rather special and form a group under composition called the group of inner automor- inseparable polynomial An irreducible poly- phisms of G. nomial with coefficients in a field which factors over its splitting field with repeated factors or, inner derivation A derivation of the form more generally, a polynomial which has an in- Da(x) = xa − ax. In more detail, a deriva- separable polynomial among its irreducible fac- tion is a linear mapping D of a (possibly non- tors. associative) algebra A into itself, satisfying the F P familiar product rule for derivatives, D(xy) = In more detail, let be a field and let be a D(x)y + xD(y). Thus, derivations are alge- polynomial of positive degree with coefficients F P braic generalizations of the derivatives of calcu- in . may or may not factor into linear factors F x2 − lus. Now let A be associative. If a isafixed over (for example, 2 does not factor over element of A, the mapping Da defined above is the rationals), but there always exists a smallest F P easily seen to be a derivation. Derivations of extension field of over which √does factor√ x2− (x− )(x+ ) this form are rather special, and are called inner (for example, 2 factors as 2√ 2 G derivations. over the field formed by adjoining 2 to the The concept extends to Lie algebras, but in rationals). This smallest extension field is called P this case inner derivations are derivations of the the splitting field for . form Da(x) =[xa], where [xa] is the Lie prod- Let F [x] denote the ring of polynomials with uct. coefficients in F . Suppose first that P is irre- ducible in F [x], that is P does not factor into inner topology The topology of a Lie sub- two or more polynomials in F [x] of positive de- group H, as a submanifold of a Lie group G. gree. P is called separable if its factorization This topology need not be the relative topol- over its splitting field has no repeated factors. In ogy of H, viewed as a subspace of a topological the general case where P is not irreducible, P space G. is called separable if each irreducible factor is separable. Finally, P is called inseparable if it inseparable element An element of an ex- is not separable. Antonym: separable polyno- tension field with an inseparable minimal poly- mial. nomial. In more detail, let G be an extension of a field F . (This means that G is a field and integer (1) Intuitively, an integer is one of the G ⊇ F .) Let α be an algebraic element of G signed whole numbers 0, ±1, ±2, ±3,..., and

c 2001 by CRC Press LLC a natural number is one of the counting num- clid proved centuries ago that there are infinitely bers, 1, 2, 3,.... many prime numbers. But it is still unknown (2) Semi-formally, the ring of integers is the whether there are infinitely many pairs of prime set Z consisting of the signed whole numbers, numbers, pn and pn+1, which differ by 2, i.e., together with the ordinary operations of addi- pn+1 − pn = 2. The conjecture that there are tion, +, and multiplication, ·. The ring of in- infinitely many such pairs of primes is called tegers forms the motivating example for many the twin prime conjecture. Perhaps the deepest of the concepts of mathematics. For example, and most important unsolved question in math- the ring of integers, (Z, +, ·), satisfies the fol- ematics is the Riemann hypothesis. Although lowing properties for all x, y, and z ∈ Z: (1) the Riemann hypothesis is stated in terms of the x + y ∈ Z. (2) x + (y + z) = (x + y) + z. behavior of a certain analytic function called the (3) x + 0 = 0 + x, (4) given x, there exists an Riemann zeta function, it too involves the prop- element −x such that x +−x =−x + x = 0. erties of the integers at its heart. For example, if These are precisely the axioms for a group, so the Riemann hypothesis were true, then the twin the integers under addition, (Z, +), form the first prime conjecture (and most of the other great un- example of a group. Furthermore, addition is solved conjectures of ) would be commutative, (5) x + y = y + x. Properties true. (1) through (5) are precisely the axioms for an (3) Formally, an integer is an element of the Abelian group, so the integers under addition ring of integers. The ring of integers is the small- form the first example of an Abelian group. In est ring containing the semi-ring of natural num- addition, the integers satisfy the following ad- bers. The semi-ring of natural numbers is de- ditional properties for all x, y, and z ∈ Z: (6) fined in terms of the set of natural numbers. The x · y ∈ Z. (7) x · (y + z) = x · y + x · z, set of natural numbers is any set N, together with and (y + z) · x = y · x + z · z. Properties (1) a S carrying N to N, satisfy- through (7) are precisely the axioms for a ring, ing the Peano postulates: so the ring of integers, (Z, +, ·), form the first, (1) There is an element 1 ∈ N. and one of the best, examples of a ring. Further- (2) S : N → N is a function, that is the more, the integers satisfy (8) 1 · x = x · 1 = x. following two properties hold: (a) given n ∈ N, The number 1 is called a unit element because there is only one element S(n), and (b) for each it satisfies this identity, so the ring of integers n ∈ N, S(n) ∈ N. forms one of the best examples of a ring with (3) For each n ∈ N, S(n) = 1. unit element, or ring with unit for short. In ad- (4) S is a one-to-one function, that is if S(m) dition, the ring of integers satisfies the commu- = S(n) then m = n. tative law, (9) x · y = y · x, so it forms one of (5) (The axiom of induction) Suppose I is a the best examples of a commutative ring. subset of N satisfying the following two prop- The ring of integers has a far richer structure erties: (a) 1 ∈ I, and (b) if i ∈ I then S(i) ∈ I. than described above. For example, the ring of Then I = N. integers is an integral domain, or domain for Addition and multiplication, + and ·, are de- short, because it satisfies property (10) x ·y = 0 fined inductively in terms of the successor func- implies x = 0ory = 0 (or both). Thus, the tion S, so that S(n) = n + 1, and the semi-ring ring of integers satisfies the familiar cancella- of natural numbers is defined to be the set of tion law: If we know x · y = x · z for x = 0, natural numbers N, together with the operations then we know y = z. Finally, the integers form of + and ·. one of the best examples of a Euclidean domain, shows us that there are and of a principal ideal domain. See also Abel- fundamentally different models of the natural ian group, commutative ring, Euclidean domain, numbers, and thus of the ring of integers. (A principal ideal domain, ring, unit element. model of the natural numbers is simply a par- Many of the most profound open (unsolved) ticular set N and successor function S satisfying questions in mathematics revolve around the in- the Peano postulates.) For example, if we begin tegers. For example, a prime number is a (pos- by believing we understand a particular model of itive) integer divisible only by itself and 1. Eu- the natural numbers, and call this object (N,S),

c 2001 by CRC Press LLC then it is possible to construct out of our pre- constant K.) There are several efficient solu- existing (N,S) a new object (N∗,S∗) with the tion methods for particular classes of integer following remarkable properties: programming problems, but it is unlikely that (1) (N∗,S∗) also satisfies the Peano postu- there is an efficient solution method for all in- lates, and thus equally deserves to be called “the teger programming problems, since this would natural numbers.” imply NP = P , a conjecture widely believed (2) Every n ∈ N also belongs to N∗. to be false. (3) If n ∈ N, then S∗(n) = S(n). (4) There exist elements of N∗ larger than any integrable family of unitary representations element of N. (These elements of N∗ are called Let G be a topological group. A unitary rep- “infinite elements” of N∗.) resentation of G is a group homomorphism L Warning to the reader: One also has to rede- from G into the group of unitary operators on fine what one means by set and subset for this a Hilbert space H , which is continuous in the to work. Otherwise, (4) could not be true and following sense: For each fixed h1 and h2 in H , ∗ ∗ (N ,S ) would simply equal (N,S). the function g )→ (L(g)h1,h2) is continuous. This construction forms the basis of Abra- Here, (L(g)h1,h2) denotes the inner product of ham Robinson’s non-standard analysis, and re- L(g)h1 and h2 in the Hilbert space H . (In other lated constructions lie at the heart of the Gödel words, L is a continuous map from G into the undecideability theorem. set of unitary operators endowed with the weak (4) Other usage: In , operator topology.) See unitary representation. algebraic integers are frequently called integers, In the theory of unitary representations, it is and then ordinary integers are called rational frequently desirable to express a given unitary integers. An algebraic integer is an element α representation L as a direct integral or integral of an extension field of the rationals which is direct sum of simpler unitary representations, integral over the rational integers. See integral element. L = l(x) dµ(x) X integer programming The general linear where X is a set and µ is a measure on X,or, programming problem asks for the maximum more precisely, where X is a set, B is a σ -field value of a linear function L of n variables, sub- of subsets of X, and µ is a measure on B. See ject to linear constraints. In other words, the integral direct sum. Here, l(x) is a unitary rep- G x ∈ X L! problem is to maximize L(x1,x2,... ,xn) sub- resentation of for each .If is another ! ject to the conditions AX ≤ B, where X is the unitary representation of G, and if L can also column vector formed from x1,... ,xn, B is a be represented as a direct integral, column vector, and A is a matrix. The general ! ! integer programming problem is the same, ex- L = l (x) dµ(x) , X cept the solution (x1,... ,xn) is to consist of integers. and if l!(x) is unitarily equivalent to l(x) except Integer programming problems frequently possibly on a set of µ measure 0, then L! is uni- arise in applications, and many important com- tarily equivalent to L. In other words, direct binatorial problems, such as the travelling sales- integrals preserve the relation of unitary equiv- man problem, are equivalent to integer pro- alence. See unitary equivalence. gramming problems. The formal statement of This leads to the consideration of functions this equivalence is that integer programming from X into the set E of unitary equivalence is one of a class of hardest possible problems classes of unitary representations of G. (Here, solvable quickly by inspired guessing; in other two representations are equivalent if they are words, integer programming is NP complete. unitarily equivalent.) Such a function L is said (Technically, the NP complete problem is the to be an integrable family of unitary represen- one of determining whether an integer vector tations, or an integrable unitary representation (x1,...,xn) exists, subject to the constraints for short, if the following holds: There is a func- and making L(x1,...,xn)>a predetermined tion l defined on X such that (1) l(x) is a unitary

c 2001 by CRC Press LLC representation in the equivalence class L(x) for the interval [a,b] is defined to be each x ∈ X, and (2) for each pair of elements b n h1 and h2 in the Hilbert space H, and group ele- f(x)dx = lim f (ti) ment g ∈ G, the function x )→ *l(x)(g)h1,h2+ a |xi −xi− |→0 1 i=1 is µ measurable. (x − x ) ; An integrable family of unitary representa- i i−1 L l tions, , and any of its associated functions, , in other words, the Riemann integral is defined are exactly what is needed to form a new unitary as the limit of Riemann sums. It is non-trivial to representation via the direct integral, prove that the limit exists and is independent of the particular choice of intermediate points ti. L = l(x)dµ(x) . (ii.) There are several variants of this defini- X tion. In the most common one, f(ti) is replaced f Because of the aforementioned preservation of by the supremum (least upper bound) of on [x ,x ] unitary equivalence, one may write the interval i−1 i to obtain the upper sum U(f,P), and by the infimum (greatest lower bound) of f on [xi−1,xi] to obtain the lower L = L dµ(x) sum L(f, P). (Here, P refers to the partition X a = x0 ≤ x1 ≤ ... ≤ xn = b.) The upper and instead. lower integrals of f are

b integrable unitary representation See inte- f(x)dx = inf U(f,P) grable family of unitary representations. a P

and integral (1) Of or pertaining to the integers, b as in such phrases as “integral exponent,” i.e., f(x)dx = sup L(f, P). an exponent which is an integer. a P (2) In calculus, the anti-derivative of a con- (Here, inf stands for infimum and sup for supre- tinuous, real-valued function. In more detail, let mum. See infimum, supremum.) The function f be a continuous real-valued function defined f is defined to be Riemann integrable if the up- on the closed interval [a,b]. Let F be a func- per and lower integrals of f are equal, and their ! tion defined on [a,b] such that F (x) = f(x) common value is called the Riemann integral for all x in the interval [a,b]. F is called an of f over the interval [a,b]. It is a theorem anti-derivative of f , or an indefinite integral of that all continuous functions are Riemann inte- f , and is denoted by f(x)dx. The number grable. This definition has the advantage that F(b)− F(a)is called the definite integral of f it extends the class of integrable functions be- over the interval [a,b], or the definite integral yond the continuous ones. It is a theorem that b of f from a to b, and is denoted by a f(x)dx. a bounded function f is Riemann integrable if By the Fundamental Theorem of Calculus, if and only if it is continuous almost everywhere. f(x)≥ 0 for all x in [a,b], the definite integral (iii.) Definitions (i.) and (ii.) generalize of f over the interval [a,b] is equal to the area from intervals [a,b] to suitable regions in higher under the curve y = f(x),a ≤ x ≤ b. dimensions. There is a sequence of rigorous and increas- (iv.) The Stieltjes integral. Everything is as in ingly general definitions of the integral. In order (ii.), except that xi −xi−1 is replaced by α(xi)− of increasing generality, they are α(xi−1) in the definition of upper and lower α (i.) The Riemann integral. Let a = x0 ≤ sums, where is a monotone increasing (and x ≤ ··· ≤ xn = b be a partition of [a,b]. possible discontinuous) function. The Stieltjes 1 b Choose intermediate points t1,t2,...,tn so that integral of f is denoted by a f (x) dα(x). The x ≤ t ≤ x x ≤ t ≤ x ... x ≤ t ≤ 0 1 1, 1 2 2, , n−1 n Stieltjes integral is more general than the Rie- x n f(t )(x − x ) n. The sum, 1 i i i−1 is called a mann integral, not in that the class of integrable Riemann sum. The Riemann integral of f over functions is enlarged, but rather in that the class

c 2001 by CRC Press LLC of things we can integrate against (the functions a simple function is called measurable.)Now α) is enlarged. suppose f is a bounded function on X. Define (v.) The Lebesgue integral. Let µ be a count- the upper integral of f on X as ably additive set function on a σ-field of sets. µ The set function is called a measure. See f (x) dµ(x) = inf g(x)dµ(x) , sigma field, measure. Examples are: X g∈G X X A X (a) is a finite set, and if is a subset of , G µ(A) = A µ where equals the set of measurable simple then the number of elements in . is g g ≥ f X functions such that . Define the lower called counting measure on . f X X σ integral of on similarly, (b) is any set, finite or infinite. The -field of sets is the set of all subsets of X. Let a be X A X f (x) dµ(x) = sup h(x) dx , a fixed point of .If is a subset of , then X h∈H X µ(A) = 1ifa ∈ A, and µ(A) = 0 otherwise. µ is called a point mass at a, or the Dirac delta where H equals the set of measurable simple measure at a. functions h such that h ≤ f . The bounded (c) X is the interval [a,b]. The σ-field of sets function f is Lebesgue integrable, or simply in- is the set of Borel subsets of X. (The Borel sets tegrable, if the upper and lower integrals agree, include all subintervals, whether open or closed, and then their common value is called the of [a,b], and many other sets besides.) If I is Lebesgue integral of f with respect to the mea- an interval, then µ(I) is the length of I. µ is sure µ, or simply the integral of f with respect called Lebesgue measure on [a,b]. to the measure µ, and is denoted by X f(x)dx. (d) X and the σ-field are as in (c). Let α be The Lebesgue integral extends both the class a monotone increasing function on [a,b], and of integrable functions (all the way to bounded for convenience suppose it is continuous from measurable functions), and the class of things the right. If I = (x1,x2] is a right half closed we can integrate against (arbitrary measures). subinterval of X, then µ(I) = α(x2)−α(x1). µ The theory extends to unbounded functions as is called a Lebesgue-Stieltjes measure on [a,b]. well. (e) X is an open subset of n-dimensional Eu- (vi.) The Denjoy integral. Similar to the clidean space, Rn. The σ-field is the σ -field of Lebesgue integral, except that the upper inte- Borel subsets of X.IfC is a small n-dimen- gral is defined as the infimum of the integrals of sional cube contained in X, then µ(C) is the an appropriate family of lower semi-continuous n-dimensional volume of C. (In the familiar functions, and the lower integral is defined as the case n = 2, an n-dimensional cube is simply a supremum of a family of upper semi-continuous square, and the n-dimensional volume is simply functions. See also lower semi-continuous func- the area of the square. In the equally familiar tion, upper semi-continuous function. case n = 3, an n-dimensional cube is an ordi- The Denjoy integral requires extra structure nary 3-dimensional cube, and the n-dimensional on X, X must be a topological space, and the volume is the ordinary 3-dimensional volume of measure µ must be a Radon measure. However, the cube.) the Denjoy integral is particularly well suited for A simple function on X is a function which dealing with certain technical difficulties con- takes only finitely many values. If g is a sim- nected with the integration of functions taking ple function taking values c1,... ,cn on the sets values in a non-separable topological vector A1,... ,An (so c1,... ,cn exhaust the finite space. set of values of taken by g) and Ai ={x ∈ (vii.) Definition (i.) of the Riemann integral X, g(x) = ci}, then easily extends to continuous vector valued func- tions f taking values in a complete topological n vector space. The analogous definition for the g(x) dµ(x) = ciµ (Ai) . X Stieltjes integral also extends to this setting. See i=1 also topological vector space. (This assumes, of course, that the Ai belong to (viii.) The Pettis integral, also called the the σ-field, i.e., that the µ(Ai) are defined. Such Dunford Pettis integral. Let f be a vector valued

c 2001 by CRC Press LLC function, and let µ be a measure on a space X, for each gamma fine tagged partition. Of course, as in (v.) or (vi.). Suppose f takes values in a the Henstock integral of f is still denoted by V locally convex topological vector space . The b Pettis integral of f is defined by the conditions, f(x)dx. a f (x) dµ(x), λ = (f(x),λ) dµ(x) , If f is positive, then Henstock integrabil- X X ity and Lebesgue integrability coincide, and the Henstock integral of f equals the Lebesgue in- ∗ ∗ for all λ ∈ V . Here, V is the dual of V , tegral of f . But if f varies in sign, and |f | consisting of all continuous linear functionals on is not Henstock (and thus not Lebesgue) inte- V . Of course, this presupposes that the scalar grable, then the Henstock integral of f may still valued functions (f (x), λ) are integrable, and exist, even though the Lebesgue integral of f that the infinite system of equations, cannot exist under these circumstances. Thus, the Henstock integral is more general than the (e, λ) = (f (x), λ) dµ(x), λ ∈ E∗ , Lebesgue integral. The Henstock integral ob- X tains its added power because it captures cancel- lation phenomena related to improper integrals has a solution e ∈ V . See also locally convex that the Lebesgue integral cannot. topological space. The Henstock integral extends to a Henstock- (ix.) The definition of the Stieltjes integral Stieltjes integral in a rather simple way. Hen- extends to vector valued measures and scalar stock integration also extends to functions of valued functions, and even to operator valued several variables. However, Henstock integra- n measures and vector valued functions. The Spec- tion on subsets of n-dimensional space, R ,is tral Theorem is phrased in terms of such an in- still an open area of investigation, as is the exten- tegral. See also Spectral Theorem. sion of the Henstock integral to abstract settings (x.) Recently, a seemingly minor variant of similar to measure spaces. the classical Riemann integral has been discov- ered which has all the power of the Lebesgue integral character In number theory, a char- integral and more. This new integral is variously acter which takes on only integral values. named the Henstock integral, the Kurzweil- S Henstock integral, or the generalized Riemann integral closure Let be a commutative ring R S integral. The Henstock integral is defined in with unit, and let be a subring of . The R S terms of gauges. Define a gauge to be a func- integral closure of in is the set of all elements S R tion γ which assigns to each point x of the in- of which are integral over . See integral R S R terval [a,b] a neighborhood of x. (The neigh- element. is integrally closed in if equals S borhood may be an open interval containing x, its integral closure in . half open if x is one of the endpoints a or b.) integral dependence Let S be a commutative Let a = x 0, there is a on F.(See Hilbert space, integral, measure, gauge γ such that sigma field.) Let H(x), x ∈ X,beafam- X H ily of Hilbert spaces indexed by , and let L − f (ti)(xi − xi−1)

c 2001 by CRC Press LLC be a set of functions f defined on X such that integral divisor (1) In elementary algebra (a) f(x) ∈ H(x) for all x ∈ X; (b) the func- and arithmetic, a factor or divisor which is an tion x )→ ,f(x), is measurable; (c) there ex- integer, as in, for example, 3 is an integral divisor ists a countable family f1,f2,f3,... ∈ L such of 12. that the set {f1(x), f2(x), f3(x),...} is dense (2) In algebraic geometry, a divisor with pos- in H(x) for each x ∈ X. Assume also the fol- itive coefficients. In more detail, let X be an lowing closure property for L, (d) if g satisfies algebraic variety. (This simply means that X is property (a) and the function x )→ (f (x), g(x)) the solution set to a system of polynomial equa- is measurable for each f ∈ L, then g ∈ L. tions. See algebraic variety.) A divisor on X is a Here, ,f(x), denotes the Hilbert space norm of formal sum D = a1C1 +···+akCk, where the f(x) and (f (x), g(x)) the Hilbert space inner ai are integers and the Ci are distinct irreducible product in the Hilbert space H(x). subvarieties of X of codimension 1. (Codimen- Let sion 1 means the dimension of Ci is one less than X C L2(L, dµ) the dimension of . Irreducible means i is not the union of two proper [i.e., strictly smaller] X = f ∈ L : ,f(x),2 dµ(x) < ∞ . subvarieties. In the simplest case where is X an algebraic curve, the Ci are just points of X.) The divisor D is integral if all the coefficients The Hilbert space L2(L, dµ) is called an inte- ai are positive. Integral divisors are also called gral direct sum, or a direct integral, and is fre- positive divisors or effective divisors. quently denoted by The notions of divisor and integral divisor extend to various related and/or more general H(x)dµ(x). X X contexts, for example to the situation where is an analytic variety. See analytic variety. (2) The corresponding representation of a lin- Perhaps the clearest examples of integral di- T ear transformation between two integral direct visors occur in elementary algebra and elemen- sums, tary complex analysis. Consider a polynomial or analytic function f defined in the complex H1(x) dµ(x) and H2(x) dµ(x) , X X plane. Let C1,C2,C3,... be a listing of the zeros of f . The corresponding integral divisor as an integral of bounded linear transformations D = a C + a C + a C +··· describes the t(x). In more detail, let H (x) and H (x), x ∈ 1 1 2 2 3 3 1 2 zeros of f , counting multiplicity, and one con- X, be two families of Hilbert spaces indexed by siders such divisors repeatedly in these subjects, X as in (1) above, and let t(x), x ∈ X be a family for example in the statement that a polynomial of linear transformations from H (x) to H (x) 1 2 of degree n has exactly n zeroes counting multi- indexed by X.Iff lies in H (x) dµ(x), de- X 1 plicity. In elementary algebra and complex anal- fine T(f) by T (f )(x) = t(x)(f (x)), except ysis, integral divisors are often called “sets with possibly on a set of µ measure 0. (Of course, the multiplicity,” and one thinks of the coefficient domain of T will be all of H (x) dµ(x), that X 1 ak as meaning that the point Ck is to be counted is, T(f)will lie in H (x) dµ(x), only when X 2 as belonging to the set ak times. the family t(x), x ∈ X is uniformly bounded, µ except possibly on a set of measure 0, that is integral domain A commutative ring R with K x only when there is a constant independent of the property that ab = 0 implies a = 0orb = 0. ,t(x),≤K such that the norm , except possibly Here, a and b are elements of R. For example, µ on a set of measure 0. In this case, the oper- the ring of integers is an integral domain. ator T will be bounded, with norm ≤ K.) The T linear transformation is called the integral di- integral element Let S be a commutative t(x) rect sum, or a direct integral of the family , ring with unit, and let R be a subring of S.An x ∈ X, and is frequently denoted by α ∈ S R α element is integral over if is the solution to a polynomial equation P(α) = 0, t(x)dµ(x) . P R X where the polynomial has coefficients in

c 2001 by CRC Press LLC and leading coefficient 1. (Such a polynomial is integral form (1) A form, usually a bilinear, called a monic polynomial. For example, x2 +5 sesquilinear, or quadratic form, with integral co- is monic, but 2x2 + 5 is not monic.) efficients. For example, the form 2x2 −xy+3y2 The most important application of the con- is integral. cept of an integral element lies in algebraic num- (2) A form, usually a bilinear, sesquilinear, or ber theory, where an element of an extension quadratic form, expressed by means of integrals. field of the rational numbers which is integral For example, the inner product on the Hilbert over the ring of integers is called an algebraic space L2 of square integrable functions, integer, or often just an integer. In this case, ordinary integers are often called rational inte- (f, g) = f(t)g(t)dµ(t) , gers. X R S If both and are fields, an integral element is an integral (sesquilinear) form. is called an algebraic element. integral ideal A non-zero ideal of the ring R integral equivalence (1) For modules, Z- of algebraic integers in an algebraic number field equivalence. Here Z is the ring of integers. See F . See algebraic integer, algebraic extension, R-equivalence, Z-equivalence. ideal. See also integer, integral element, integral M M (2) For matrices, two matrices 1 and 2 extension. In the elementary case where F is are integrally equivalent if there is an invertible the field of rational numbers and R is the ring P P P −1 matrix , such that both and have integer of ordinary integers, an integral ideal is simply M = P tM P P t entries and 2 1 . Here, denotes the a non-zero ideal of the ring of integers. P transpose of . In algebraic number theory, a distinction is (3) For quadratic forms, two quadratic forms drawn between fractional ideals and integral Q (x) = xtM x Q (x) = xtM x 1 1 and 2 2 are in- ideals. An integral ideal is as defined above. M M tegrally equivalent if the matrices 1 and 2 By contrast, a fractional ideal is an R module Q are integrally equivalent. In other words, 1 lying in the algebraic number field F . and Q2 are integrally equivalent if each can be transformed to the other by a matrix with integer integrally closed Let A be a subring of a entries. ring C. Then the set of elements of C which are integral over A is called the integral closure of S integral extension A commutative ring A in C.IfA is equal to its integral closure, then R with unit and containing a subring is an in- A is said to be integrally closed. tegral extension of R if every element of S is integral over R. See integral element. integral quotient In , In the important special case where R and a quotient in which both the numerator and the S are fields, an integral extension is called an denominator are integers. For example, 3/4is algebraic extension. an integral quotient, whereas 3.5/4.5 is not, even The classic example of an integral extension though the latter represents (equals) the rational is the ring S of algebraic integers in some al- number 7/9. gebraic extension field of the rational numbers. Here, the ring R is the ring of ordinary (i.e., ra- integral representation (1)Arepresentation tional) integers. See algebraic integer, integral of a group G is a homomorphism φ from G into element. a group Mn of n × n matrices. The representa- If S is an integral domain, there is an im- tion is integral if each matrix φ(g) has integer portant relationship between being an integral entries. extension and possessing certain finiteness con- (2) Any representation of a quantity by means ditions. Specifically, an integral domain is an of integrals. See integral. integral extension of a subring R if and only if S is module finite over R. Module finite sim- integral ringed space A ringed space is a ply means that S is finitely generated as an R topological space X together with a sheaf of module. See also integral domain. rings OX on X. This means that to each open

c 2001 by CRC Press LLC set U of X, there is associated a ring OX(U). X, and let y1,y2,y3,... be a finite or infi- (The remaining properties of sheaves need not nite sequence of elements of Y . The set A = concern us here.) The ringed space (X, OX) {a1,a2,a3,...} is an H interpolating subset is integral if each ring OX(U) is an integral do- for the sequence y1,y2,y3,... if there exists n main. For example, if X is an open subset of C , a function h ∈ H such that h(a1) = y1 for complex n-space, or a complex analytic mani- i = 1, 2, 3,.... In this case, the sequence fold, and OX(U) is the ring of analytic functions a1,a2,a3,... is called an H interpolating se- defined on U, then (X, OX) is an integral ringed quence for y1,y2,y3,.... space. See integral domain. The classic examples of interpolating se- quences occur in the case where X is the field integral scheme A scheme is a particular of complex numbers and H is the set of all sort of ringed space. See scheme. An integral polynomials with complex coefficients. The La- scheme is a scheme which is an integral ringed grange interpolation theorem asserts that any fi- space. See integral ringed space. nite sequence of complex numbers, a1,...,an, all terms of which are different, is an H inter- intermediate field Let F , G, and H be fields, polating sequence for any sequence y1,...,yn with F ⊆ G ⊆ H. G is called an intermediate of complex numbers with the same number of field. terms. Although the notion of an interpolating sub- internal product Let G1 and G2 be Abel- set is usually reserved for discrete sets A as ian groups, and let R be a commutative ring. A above, it makes sense in greater generality. Let group homomorphism π : G1 ⊗ G2 → (R, +), A be an arbitrary subset of X, and let f be a where (R, +) is the underlying additive group function from A to Y . Then A is an H interpo- of R,isaninternal product if it satisfies π(g1 ⊗ lating subset for the function f if there exists a g2) = π(g1) · π(g2), where · is the multiplica- function h ∈ H such that h(a) = f(a) for all tion in the ring R. Here, G1 ⊗ G2 is the tensor a ∈ A. product of G1 and G2. The notion is most fre- quently used in homological algebra, in which intersection multiplicity A variety V is the case π becomes a homomorphism of chain com- set of common zeros of a set I of polynomials. V ={(x ,...,x ) : P(x ,..., plexes of groups, and the relation π(g1 ⊗ g2) = In other words, 1 n 1 x ) = P ∈ I} π(g1) · π(g2) only has to hold for cycles (or co- n 0 for all . Intuitively, the inter- cycles). See also chain complex, cocycle, cycle, section multiplicity of varieties V1,...,Vn at a tensor product. point x = (x1,...,xn) where they intersect is the degree of tangency (or order of contact) of internal symmetry A symmetry that is an the intersection at x plus 1. For example, the invertible mapping of a set onto itself. If the set parabolas has additional structure, then the mapping and x − x2 = 0 and x − x2 − x = 0 its inverse must preserve that structure. For ex- 2 1 2 1 1 ample, if the set is in addition a differentiable have intersection multiplicity 1 at (x1,x2) = manifold, then the mapping (and automatically (0, 0) because they intersect transversally (are its inverse) must be differentiable. If the set is not tangent to each other) there. However, in addition a topological space, then the map- x − x2 = 0 and x − 5x2 = 0 ping and its inverse must both be continuous. 2 1 2 1 If the set is in addition a metric space, then the have intersection multiplicity 2 at (x1,x2) = mapping (and automatically its inverse) must be (0, 0) because they intersect tangentially to first an isometry. See also inverse function, inverse order there. One must also include the possibil- mapping. ity of x being a multiple point, in which case the intersection multiplicity should be ≥ the order interpolating subset Let H be a set of func- of the multiple point. tions from a set X to a set Y . Let a1,a2,a3,... Rigorously, if D1,...,Dn are effective divi- be a finite or infinite sequence of elements of sors on a smooth n-dimensional variety X and

c 2001 by CRC Press LLC are in general position at a point x ∈ X, then the words, V ={(x1,...,xn) : P(x1,...,xn) = 0 intersection multiplicity of D1,...,Dn at x is for all P ∈ I}. Intuitively, the intersection number of varieties V1,...,Vn is the number (D ,...,Dn) = dim (Ox/ (f ,...,fn)) . 1 x 1 of points of intersection, counting multiplicity. Here is what all this means: X is smooth if it has Rigorously, if D1,...,Dn are effective divisors no singular points. A divisor on X is a formal on a smooth n-dimensional variety X (see in- sum D = a1C1+···+akCk, where the ai are in- tersection multiplicity for brief definitions of tegers and the Ci are distinct irreducible subva- these terms), then the intersection number of rieties of X of codimension n − 1. (Irreducible D1,...,Dn is means Ci is not the union of two proper [i.e., (D ,...,D ) = (D ,...,D ) , strictly smaller] subvarieties. In the simplest 1 n 1 n x case where X is an algebraic curve, the Ci are x∈S X D just points of .) The divisor is effective (also in other words, it is the sum of the intersection called integral or positive) if all the coefficients multiplicities over the finitely many points of ai are positive. Ox is the local ring of X at x. intersection of the divisors D1,...,Dn. Here, X x n The local ring of at consists of quotients of S = Si, Si = Ci,j , and Di = ai,j f/g x i=1 j j polynomial functions, , defined at and near Cj . X x (i.e., on an open subset of containing ) where It is also possible to define intersection num- g(x) = 0, and two such functions are identified bers for fewer than n divisors, say D ,...,Dk. x 1 if they agree on an open subset containing . (In However, this definition is the culmination of an O other words, x is the ring of terms of regular entire theory. functions at x.) Locally, each divisor Di is the divisor of a function fi, and that is where the fi intersection product (1) Let i(A,B; C) be (f ,...,f ) O come from. 1 n is the ideal in x gen- the intersection multiplicity of two irreducible f ,...,f erated by 1 n (by their terms actually). subvarieties A and B of an irreducible variety O /(f ,...,f ) The quotient ring x 1 n is not only a V , along a proper component C of A ∩ B. The ring but also a finite dimensional vector space. intersection product of A and B is The intersection multiplicity (D1,...,Dn) is the dimension of this vector space. See general A · B = i(A,B; Cn)Cn position, term, integral divisor, quotient ring. n The intersection multiplicity of effective di- where the sum is taken over all the proper com- visors not in general position at x is defined in ponents Cn of A ∩ B.IfX = aαAα and terms of the intersection multiplicity of equiva- α Y = bβ Bβ are two cycles on V such that lent divisors which are in general position. The β each component Aα of X intersects properly intersection multiplicity of fewer than n effec- with each component Bβ of Y , then the inter- tive divisors, say D ,... ,Dk, is defined in 1 section product is terms of module length rather than the less gen- eral concept of vector space dimension. Let C X · Y = aαbβ Aα · Bβ . be one of the irreducible components of the va- α β riety Ci,j , where Di = ai,j Ci,j . Then the intersection multiplicity of D1,...,Dk in the (2)IfM is an oriented n-dimensional mani- component C is fold and a and b are members of the homology groups Hp(M) and Hq (M), then the intersec- (D ,...,Dk)C = G (OC/ (f ,...,fk)) 1 1 tion product of Lefschetz is a · b = D−1aI −1 −1 where G(OC/(f1,...,fk)) is the module length b = D(D aJD b) ∈ Hp+q−n where D is of the OC module OC/(f1,...,fk) and OC is the Poincare-Lefschetz´ duality. See also inter- the local ring of the irreducible subvariety C. section multiplicity, cup product, cap product, See also local ring, module of finite length. Poincaré-Lefschetz duality. intersection number A variety V is the set of intransitive permutation group A permu- common zeros of a set I of polynomials. In other tation group G is a group of one-to-one and onto

c 2001 by CRC Press LLC functions from a set X to itself. The group oper- ric function is a sum of elementary symmetric ation is understood to be a composition of func- functions, p1 = 1, p2 = the sum of all pairs of τσ (x) = τ ◦ σ(x) = τ(σ(x)) p = x2 + x x +···+x x + x2 + tions, . Usually, variables, 2 1 1 2 1 n 2 X 2 but not always, the set is finite. The permu- x2x3 + ...+ x2xn +···+xn, p3 = the sum of G tation group is intransitive if for some (and all triples of variables, pn = x1x2 ···xn. x ∈ X O(x) ={σ(x) : hence for all) , the set Example (c): this is an important special case σ ∈ G} X O(x) is not equal to all of . The set is of example (a). G is a group of conformal trans- x called the orbit of , so we can say the permu- formations of the unit disk (in the complex plane) G tation group is intransitive if the orbit of any into itself. (Thus, G is a group of linear frac- x ∈ X X element fails to be all of . Synonym: tional transformations.) X is the unit disk, and intransitive transformation group. Antonyms: L is the set of analytic functions defined on the transitive permutation group, transitive trans- unit disk. A function in L which is invariant formation group. under the action of G is called an automorphic function. See conformal transformation, linear invariance As in ordinary non-technical En- fractional transformation. glish, the property of being unchanged with re- Example (d): Let G be an Abelian group. Let spect to some action or set of actions. K be a field and KG the group algebra over K. L KG invariant 1 L G Let be a left module. Define the action ( ) Let be a set. Let be an- G g · l = gl other set which acts on L. This means that there of by module multiplication, .In is a binary operation · so that g · l is an element the theory of group representations, the set of all L g ∈ G l ∈ L l ∈ L which are invariant under the action of G of for each and . Usually, but not G L G l ∈ L in- are called the -invariants of . The set of all always, is a group. An element is G L KG variant under the action of G G invariant, -invariants of forms a left submodule of ,ora L if g · l = l for each g ∈ G. which plays a key role in the theory of group representations. See also group algebra. Here are some examples: G L V Example (a): Let G be a group of functions (2) Let and be as in (1). A subset of L G g · v ∈ V from a set X to itself. Each g ∈ G is assumed to is invariant under the action of if v ∈ V be one-to-one and onto, and the group operation for all . Here is an important example: T is a composition of functions, g g (x) = g ◦ Let be a bounded linear operator on a Hilbert 1 2 1 H G ={T } g (x) = g (g (x)). Let L be a set of functions space . Let , the set consisting of 2 1 2 T L = H from X to some set Y . The action of G on L is alone. Let . Define the action of G L T · l = T(l) defined via a composition of functions: g · l = on by operator application, . L l ◦ g, i.e. g · l(x) = l(g(x)). A subspace of which is invariant under the G Example (b): Let G equal the symmetric action of is called an invariant subspace for T group on n letters. In other words, G is the . A famous unsolved problem is the invari- permutation group of all permutations (one-to- ant subspace problem, often called the invariant one and onto functions) of the set {1,... ,n}. subspace conjecture: Does every bounded lin- The group operation is a composition of func- ear operator on an infinite dimensional complex H { } tions. Let L be the ring of all polynomials in n Hilbert space have a proper (not 0 , not all of H ) closed invariant subspace? variables, x1,... ,xn.Ifg ∈ G, and l is a mono- k k k k x 1 ···x n g · l = x 1 ···x n (3) A bilinear form f on a Lie algebra L is mial 1 n , then g(1) g(n).In other words, g · l is formed from l by rearrang- called an invariant form if f([ac],b) + f(a,[bc]) = [ac] ing the variables. If l is an arbitrary polynomial, 0. Here, is the Lie algebra a c then l is a sum of monomials, l= aili. De- product of and . fine g · l by linearity, g · l = aig · li. The (4) A quantity which is left unchanged un- polynomials which are invariant under the ac- der the action of a prescribed class of functions tion of g are called symmetric functions. Thus, between sets is also called an invariant. For ex- the symmetric functions are those polynomials ample, the Euler characteristic is a topological which remain unchanged after rearranging their invariant because it is preserved under topolog- variables. It is a theorem that each symmet- ical homeomorphisms. (A one-to-one and onto

c 2001 by CRC Press LLC function from one topological space to another trix B1, obtained from J by taking the direct is called a homeomorphism if both it and its in- sum of k Jordan blocks, one for each distinct verse are continuous.) eigenvalue λi, having maximal order among all (5) Let X be a set and let E be an equivalence Jordan blocks corresponding to λi. Next con- relation on the set X. Let f : X → Y be a sider B2, obtained similarly to B1, but from the J function from X to another set Y .Iff(x1) = remaining Jordan blocks in . Continue in this J f(x2) whenever (x1,x2) ∈ E, that is whenever manner until all Jordan blocks of have been B ,B ,...,B x1 is equivalent to x2 modulo the equivalence used, thus obtaining a sequence 1 2 s relation E, then f is called an invariant of E. of matrices whose sizes are non-increasing and The function f is called a complete invariant of whose direct sum is by construction permuta- J E if (x1,x2) ∈ E if and only if f(x1) = f(x2). tionally similar to . Finally, a finite or infinite set F of functions on The characteristic polynomials of the matri- X is called a complete system of invariants for ces Bj , j = 1, 2,...,s are known as the in- A E if (x1,x2) ∈ E if and only if f(x1) = f(x2) variant factors of . It is worth noting that for for all functions f ∈ F . each j = 1, 2,...,s, by construction, the char- Here is a well-known example: Let X be the acteristic polynomial of Bj coincides with the set of all m × n matrices with coefficients in a minimal polynomial of Bj . In particular, the B field F . Define two such matrices M1 and M2 minimal polynomial of 1 is the minimal poly- to be equivalent if there exist invertible square nomial of A. It follows that two matrices are matrices P and Q such that M1 = PM2Q, and similar if and only if they have the same invari- let E be the resulting equivalence relation. It ant factors. is a theorem of linear algebra that the rank of a matrix M is a complete invariant for E, that is invariant field Let G be a group of field au- F H F m×n matrices M1 and M2 are equivalent under tomorphisms of a field . A subfield of the above definition if and only if they have the is invariant for G,orG-invariant, if g(h) = h same rank. for all g ∈ G and h ∈ H ; in other words, the subfield H is G-invariant if G fixes the elements invariant derivation Let D be a derivation of H . Synonym: fixed field. on the function field K(A) of an Abelian variety A, where K is the universal domain of A. Let Ta invariant form (1) A bilinear or quadratic be translation by an element a ∈ A.If(Df ) ◦ form which is invariant under the action of a set Ta = D(f ◦ Ta), for every f ∈ K(A), then D of transformations. See invariant. f L is called an invariant derivation on A. (2) A bilinear form on a Lie algebra such that f([ac],b)+ f(a,[bc]) = 0. Here, [ac] invariant element Let T : X → X be a is the Lie algebra product of a and c. function or mapping. If x ∈ X has the property G that T(x) = x, then x is called an invariant invariant of group Let be a finite Abelian element of the operator T . This concept arises group. By the Fundamental Theorem of Abelian in analysis, topology, algebra, and many other Groups, branches of mathematics. G = σ (m1) ⊕ σ(m2) ⊕···⊕σ(ms), invariant element Let T : X → X be a where σ(mi) is a cyclic group of order mi and x ∈ X function or mapping. If has the property mi divides mi+1. The numbers m1,m2,... ,ms that T(x) = x, then x is called an invariant are uniquely determined by G, are invariant un- element of the operator T . This concept arises der group isomorphism, and are called the in- in analysis, topology, algebra, and many other variants of the group G. branches of mathematics. invariant of weight w Let R be a ring and let invariant factor Let A be an n × n matrix L be a left R module. Let G be a set which acts with distinct eigenvalues λi,, i = 1, 2,...,k, on L. This means there is a binary operation · and Jordan normal form J . Consider the ma- so that g · l is an element of L for each g ∈ G

c 2001 by CRC Press LLC and l ∈ L. Usually, G is a group. Let w be a inverse function Let f be a function from a function from G to R. An element l ∈ L is an set X to a set Y . Diagrammatically, f : X → invariant of weight w under the action of G, or Y . The inverse function to f , if it exists, is the a G invariant of weight w,ifg · l = w(g)l for function g : Y → X such that f ◦ g = iY and each g ∈ G. g ◦ f = iX. (Here, iX and iY are the identity Here is an example: Let L be the set of all maps on X and Y , and ◦ denotes composition analytic functions in the upper half plane of the of functions.) In other words, f(g(y)) = y complex plane. Let G be the modular group. for all y ∈ Y , and g(f (x)) = x for all x ∈ The modular group is the group of all linear X. The function f has an inverse if and only if fractional transformations f is one-to-one and onto, and the inverse g is g(y) = x az + b defined by the unique element such g(z) = , that f(x) = y. The inverse function to f is cz + d frequently denoted by f −1. See also identity where a, b, c, and d are integers and the determi- map, inverse mapping, inverse morphism. ad − bc = G L nant 1. acts on by composition Example: Let f(x) = 10x, for x real. The of functions, inverse function to f is g(x) = log x, for x> 10 az + b 0. g · l(z) = l ◦ g(z) = l(g(z)) = l . cz + d inverse limit Suppose {Gµ}µ∈I is an indexed k G A modular form of weight is a invariant of family of Abelian groups, where I is a pre- w w(g) = (cz + d)k weight , where . In other ordered set. Suppose that there is also a family l k l ∈ L words, is a modular form of weight if of homomorphisms ϕµν : Gµ → Gν, defined and for all µ<ν, such that if µ<ν<κ, then az + b ϕνκ ◦ ϕµν = ϕµκ . Consider the direct product l = (cz + d)kl(z) , G π cz + d of the groups µ and define µ to be the pro- jection onto the µth factor in this direct product. whenever a, b, c, and d are integers and ad − Then the inverse limit is defined to be the sub- bc = 1. group G∞ ={x : µ<νimplies πµ(x) = There is a companion notion of weight for ϕνµ ◦ πν(x)}. See also preordered set. right R modules. Furthermore, the notion of a G w invariant of weight extends to the situation inverse mapping See inverse function. where R and L are simply sets, and R acts on L. inverse matrix The matrix B, if it exists, inverse Let G be a set with a binary operation such that AB = BA = I. In more detail, let A n × n I n × n · and an identity e. This means g1 · g2 ∈ G be a square matrix, and let be the n × n whenever g1 and g2 belong to G, and g · e = identity matrix, that is the matrix with en- e · g = g for all g ∈ G. The element h ∈ G is try 1 in each diagonal position and 0 elsewhere. an inverse of g ∈ G if g · h = e and h · g = e. An n×n matrix B is the inverse of A if AB = I Often g is an element of a group G. What is and BA = I. Either condition implies the other. very special about this case is that (a) either of The inverse of A, if it exists, is uniquely deter- − the conditions gh = e and hg = e implies the mined by A and is denoted by A 1. It is a the- other, and (b) the inverse of g always exists and orem of linear algebra that a matrix is invertible is uniquely determined by g. The inverse of g is (i.e., has an inverse) if and only if its determi- frequently denoted by g−1. See group. See also nant is non-zero. There is a determinantal (in- inverse function, inverse morphism. volving determinants) formula for the inverse of A, A−1 = det(A)−1adj(A), where det(A) is the An element of a set G which determinant of A and adj(A) is the classical ad- is the inverse of another element of G. See in- joint of A, that is the transpose of the matrix of verse. cofactors. See cofactor, determinant, transpose.

c 2001 by CRC Press LLC inverse morphism Let C be a category, let A inverse transformation See inverse func- and B be objects in C, and let f : A → B be a tion. morphism in C.(Amorphism is a generalization of a function or mapping.) A morphism g : inverse trigonometric function The inverse B → A is the inverse morphism of f if f ◦ g functions to the trigonometric functions sin, cos, is the identity morphism on object B, and g ◦ tan, cot, sec, and csc; that is, the arcsin, the arc- f is the identity morphism on object A. The cosine, the arctangent, the arccotangent, the arc- inverse morphism of f , if it exists, is uniquely secant, and the arccosecant, respectively. They determined by f . are denoted by arcsin, arccos, arctan, arccot, −1 −1 −1 There are also notions of left and right in- arcsec, and arccsc, or by sin , cos , tan , −1 −1 −1 verse morphisms. Morphism g is a left inverse cot , sec , and csc . Note that, in formulas morphism of f if g ◦ f is the identity morphism involving trigonometric functions and inverse −1(x) on object A, and is a right inverse morphism of trigonometric functions, sin is not equal / (x) f if f ◦ g is the identity morphism on B. See to the number 1 sin , but rather to the value x also identity morphism, inverse function. of the arcsin of . Similar comments apply to cos−1(x), etc. inverse operation See inverse function. Because the trigonometric functions sin, cos, etc. are not one-to-one, the inverse trigonometric functions are really inverse relations, although inverse proportion Quantity a is inversely they may be thought of as multiple valued func- proportional to quantity b,orvaries inversely tions. Thus arcsin(x) is any angle y such that with quantity b, if there is a constant k different sin(y) = x, and similarly with arccos from 0 such that a = k/b. For example, New- (x), etc. To make the inverse trigonometric ton’s law of universal gravitation, “The grav- functions into single valued functions, one must itational force between two masses is directly specify a branch or, equivalently, an interval in proportional to the product of the masses and which the trigonometric function is one-to-one. inversely proportional to the square of the dis- For example, the principal branch of the arcsin tance between them,” is given by the formula (x) 2 is the inverse of sin restricted to the inter- F = GmM/r , where G is a constant of nature π π val − ≤ y ≤ ; it is denoted by Arcsin or called the gravitational constant. 2 2 Sin−1, with a capital letter. Thus Arcsin(x) is π π the unique angle y in the interval − ≤ y ≤ inverse ratio (1) The inverse ratio to a/b is 2 2 such that sin(y) = x. The principal branches b/a. of the other inverse trigonometric functions are (2) Inverse ratio also means inverse propor- also denoted by capital letters, Arccos = Cos−1, a b − tion,asin“ varies in inverse ratio to .” See Arctan = Tan 1, etc. The principal branch inverse proportion. of the arctan takes values in the same interval as the principal branch of the arcsin, namely inverse relation The relation formed by re- − π ≤ y ≤ π 2 2 , but the principal branch of the versing the ordered pairs in a given relation. In arccos takes values in the interval 0 ≤ y ≤ π. more detail, a relation R is a subset of the Carte- sian product X × Y , where X and Y are sets. inverse variation See inverse proportion. (The Cartesian product X × Y of X and Y is simply the set of all ordered pairs (x, y), where inversion The act of computing the inverse. x ∈ X and y ∈ Y .) The inverse relation to R is See inverse. the relation {(y, x) : (x, y) ∈ R}. The inverse relation to R is usually denoted by R−1. Note inversion formula Any of a number of for- that if the relation R is a subset of the Cartesian mulas for computing the inverse of a quantity. product X×Y , then the inverse relation R−1 is a The two most celebrated probably are: subset of Y × X. Example: An inverse function (i.) the inversion formula for computing the in- is a special sort of inverse relation. See inverse verse of a matrix, A−1 = det(A)−1adj(A), function. where det(A) is the determinant of A and adj(A)

c 2001 by CRC Press LLC is the classical adjoint of A, that is the transpose sheaf of OX modules is defined similarly, except of the matrix of cofactors. See inverse matrix. that to each open subset U of X, there is asso- (ii.) The Fourier inversion formula, ciated an OX module F(U). A sheaf F of OX modules is locally free of rank 1, or invertible, f(x)= X U α ∈ A if can be covered by open sets α, , 1 +∞ +∞ i(x−λ)µ such that F(Uα) is isomorphic to OX(Uα). See π −∞ −∞ f (λ)e dλ dµ . 2 also ringed space, sheaf. The Fourier inversion formula is really a state- φ X ment that if f has Fourier transform, involution (1) A function from a set φ2 = φ φ2(x) = to itself, such that . Here, +∞ φ ◦ φ(x) = φ(φ(x)). F (µ) = √1 f (λ)e−iλµ dλ , A 2π −∞ (2) Let be an algebra over the complex numbers. A function φ from A to itself is an then the inverse Fourier transform is given by involution if it satisfies the following four prop- the inversion formula, erties for all x and y in A and all complex num- +∞ bers λ: (i.) φ(x + y) = φ(x) + φ(y), (ii.) 1 f(x)= √ F (µ)eiµx dµ . φ(λx) = λφ(x)¯ , (iii.) φ(xy) = φ(y)φ(x),(iv.) π 2 −∞ φ((φ(x)) = x. φ(x) x∗ (There have to be hypotheses on f , say f ∈ is frequently denoted by , and then L1 ∩ L2, for this to work.) the four properties take the more familiar form: (i.) (x +y)∗ = x∗ +y∗, (ii.) (λx)∗ = λx¯ ∗, (iii.) ∗ ∗ ∗ ∗∗ invertible element (1) An element with an (xy) = y x ,(iv.)x = x. inverse element. See inverse, inverse element. Examples: (i.) A is the complex numbers. ∗ (2) Let R be a ring with unit e. An element x =¯x, the complex conjugate of x. (ii.) A r of R, for which there exists another element is the algebra of bounded linear operators on a ∗ a in R such that ar = e and ra = e, is called Hilbert space. T is the adjoint of T . an invertible element of R. This is, of course, a special case of (1) above. The element a,if irrational equation An equation with irra- it exists, is uniquely determined by r, and is tional coefficients. See also irrational number. denoted by r−1. See unit. If only the condition ar = e holds, then r is irrational exponent An exponent which is π said to be left invertible. Similarly, if only the irrational. For example, the expressions 2 and π condition ra = e holds, then r is said to be right e involve irrational exponents. See also irra- invertible. tional number. invertible function A function f : S → T irrational expression An expression involv- such that there is a function g : T → S with ing irrational numbers. See also irrational num- f ◦ g = idT and g ◦ f = idS. Often the word ber. Although the word expression is often used “function” is used to specify that T is a field. loosely in elementary mathematics without a rigorous definition, it is possible to define ex- invertible map A function f : S → T such pression rigorously by specifying the rules of a that there is a function g : T → S with f ◦ g = formal grammar. idT and g ◦ f = idS. Often the word “map” is used to specify that T is not a field of scalars. irrational number A real number r which cannot be expressed in the form p/q, where p invertible sheaf A locally free sheaf of rank and q are integers. Equivalently, a real num- 1. In more detail, a ringed space is a topological ber which is not a rational number. It was the space X together with a sheaf of rings OX on X. great discovery of the mathemati-√ This means that to each open set U of X, there is cian and philosopher that 2 is irra- associated a ring OX(U). (The remaining prop- tional. The numbers e and π are also irrational. erties of sheaves need not concern us here.) A These last two are irrational in a very strong

c 2001 by CRC Press LLC sense, they are transcendental, but it took until ponents of V are the two lines x − y = 0 and the nineteenth century to prove this. See also x + y = 0. . The notion of an irreducible component ex- tends to varieties in other contexts, for example irreducible algebraic curve An algebraic to analytic varieties. curve is a variety of dimension 1 in 2- (2) In combinatorial group theory, every Cox- dimensional affine or projective space. (A vari- eter group can be written as the direct sum of ety is the solution set to a system of polynomial (possibly infinitely many) irreducible Coxeter equations.) An algebraic curve is irreducible groups, called the irreducible components of if it is not the union of two proper (strictly the Coxeter group. See also irreducible Cox- smaller) subvarieties. For example, the parabola eter group. y −x2 = 0 is an irreducible algebraic curve, but the pair of lines x2 − y2 = 0 is a reducible irreducible constituent Let Z denote the algebraic curve because it can be decomposed rational integers and Q the rational field. Let T into the union of the two lines x − y = 0 and be a Z-representation of a finite group G.IfT x + y = 0. is Q-irreducible, then T is called an irreducible constituent of the group G. irreducible R-module Let R be a ring and let M be a module over R. We say that M is irreducible constituent Let Z denote the ra- irreducible over R if R has no submodules. In tional integers and Q the rational field. Let T some contexts, we say that M is irreducible if be a Z-representation of a finite group G.IfT M cannot be written as a direct sum of proper is Q-irreducible, then T is called an irreducible sub-modules. These are also sometimes called constituent of the group G. simple modules. irreducible Coxeter complex A Coxeter irreducible character A character of a finite complex for which the associated Coxeter group group which is not a sum of characters different is an irreducible Coxeter group. In more detail, from itself. Every character of a finite group is a let (W, S) be a Coxeter group. For now, it suf- sum of irreducible characters. See also character fices that W is a (possibly infinite) group and of group. S is a set of generators for W. Define a spe- cial coset to be a coset of the form w*S!+, where irreducible co-algebra A co-algebra in w ∈ W, S! ⊆ S, and *S!+ is the group gener- which any two non-zero subco-algebras have ated by S!. The Coxeter complex Q associated non-zero intersection. A co-algebra C is irre- with (W, S) is the partially ordered set of special ducible if and only if C has a unique simple cosets, ordered by reverse inclusion: B ≤ A if subco-algebra. and only if B ⊇ A. Q is an irreducible Cox- eter complex if its Coxeter group (W, S) is an irreducible component (1) In algebraic ge- irreducible Coxeter group. See also irreducible ometry, a variety is the solution set of a system Coxeter group. of polynomial equations, usually in more than Although Coxeter complexes are abstractly one variable. A variety is irreducible if it is not defined, there is a rich geometry associated with the union of two proper (strictly smaller) subva- them, resembling the geometry of simplicial rieties. An irreducible component of a variety complexes. is a maximal irreducible subvariety. That is, a subvariety W ⊆ V is an irreducible component irreducible Coxeter group A Coxeter group of a variety V if (i.) W is irreducible, and (ii.) which cannot be written as the direct sum of two there is no irreducible variety W ! properly be- other Coxeter groups. In more detail, Coxeter tween W and V (W ⊂ W ! ⊂ V , W = W !, groups are generalizations of finite reflection W ! = V ). Every variety is the finite union groups. Let W be a (possibly infinite) group, of its irreducible components. Example: V = and let S be a set of generators for W. The pair {(x, y) : x2 − y2 = 0}. The irreducible com- (W, S) is called a Coxeter group if two things are

c 2001 by CRC Press LLC 2 true: (i.) Each element of S has order 2 (s = s if S1 and S2 are Siegel domains, S = S1 × S2 = if s ∈ S), and (ii.) W is defined by the system of {s = (s1,s2) : s1 ∈ S1,s2 ∈ S2} will also generators and relations: set of generators = S; be a Siegel domain. A Siegel domain is irre- set of relations = {(st)m(s,t) = 1}, where m(s, t) ducible if it is not the Cartesian product of two is the order of the element st in the group W, other Siegel domains. A Siegel domain is ho- and there is one relation for each pair (s, t) with mogeneous if it has a transitive group of analytic s and t in S and m(s, t) < ∞. (holomorphic) automorphisms. An irreducible A Coxeter group is irreducible if it cannot homogeneous Siegel domain is a homogeneous be written as the direct sum of two other Cox- Siegel domain which is not the Cartesian prod- eter groups. In other words, the Coxeter group uct of two other homogeneous Siegel domains. (W, S) is irreducible if it cannot be written as See homogeneous domain, Siegel domain. (W, S) = (W ! × W !!,S! ∪ S!!), where (W !,S!) and (W !!,S!!) are themselves Coxeter groups. irreducible linear system A system of linear (Equivalently, a Coxeter group is irreducible if equations where no equation is a linear com- its Coxeter diagram is connected.) Every Cox- bination of the others. It is a theorem that a eter group can be written as the direct sum of system of n linear equations in n unknowns is (possibly infinitely many) irreducible Coxeter irreducible if and only if the determinant of the groups, called the irreducible components of the matrix of coefficients is not 0. The methods of group. See also Coxeter diagram, irreducible row and column reduction provide computation- Coxeter complex. ally efficient tests for irreducibility. Synonym: linearly independent system of linear equations. irreducible decomposition Informally, a de- See also linear combination, linearly indepen- composition of an object into irreducible com- dent elements. ponents or elements. See irreducible compo- nent, irreducible element. irreducible matrix See Frobenius normal form. irreducible element (1) An element a of a ring R with no proper factors in the ring. This irreducible module The module analog of a means that there do not exist elements b and simple group. Specifically, an R module, c in R, different from 1 and a, such that a = where R is a ring, is an irreducible module if bc. Example: If R is the ring of integers, the it contains no proper R submodules. For ex- irreducible elements are the prime numbers. See ample, Zp, the integers modulo a prime number also prime number. p, is an irreducible Z module. (Here, Z is the (2) A join or meet irreducible element of a ring of integers.) In the case where the ring R lattice. See join irreducible element, meet irre- is not commutative, the notion of irreducibility ducible element. extends to left and right R modules. irreducible equation A polynomial equa- irreducible polynomial A polynomial with tion P(x) = 0, where the polynomial P is irre- no proper factors. In greater detail, if R is a ring ducible. See also irreducible polynomial. and R[x] denotes the ring of polynomials with coefficients in R, then a polynomial P in R[x] is irreducible fraction A fraction a/b, where irreducible if it is an irreducible element of the the integers a and b have no common factors ring R[x]. See irreducible element. Example: If other than 1 and −1. In other words, a fraction R is the field of real numbers and C is the field reduced to lowest terms. of complex numbers, the polynomial P(x) = x2 + 1 is irreducible in R[x] but reducible (it irreducible homogeneous Siegel domain factors as (x + i)(x − i))inC[x]. Siegel domains are special kinds of domains in complex N space, CN . An easy way to con- irreducible projective representation A struct new Siegel domains is to take the Carte- projective representation of a group G is a func- sian product of two given Siegel domains. Thus tion T from G into the group GL(V ) of invertible

c 2001 by CRC Press LLC linear transformations on a vector space V , sat- See Siegel domain. See also irreducible homo- isfying two additional axioms. (See projective geneous Siegel domain. representation.) T is irreducible if there does not exist a proper (= 0, = V ) subspace W of irreducible tensor An element of the tensor V such that T (g)(w) ∈ W for all g ∈ G and product V ⊗ W of two vector spaces, which w ∈ W. See also irreducible representation, cannot be written as v ⊗ w, for v ∈ V and w ∈ irreducible unitary representation. W. Also called irreducible tensor operators or spherical tensor operators. [a,b]=ab − ba irreducible representation A representa- Classically, let and let j ,j ,j x y z tion of a group G is a homomorphism T from G x y z be the -, -, and - components of into the group GL(V ) of invertible linear trans- the angular momentum j.Anirreducible ten- k T k formations on a vector space V . T is an irre- sor of rank is a dynamical quantity q , where q = k,k − ,... ,−k ducible representation if there does not exist a 1 , that satisfies the follow- proper (= 0, = V ) subspace W of V such that ing commutation relations: T (g)(w) ∈ W for all g ∈ G and w ∈ W. k k jz,Tq = qTq The notion extends to other contexts. For ex- ample, V may be a Hilbert space and GL(V ) k jx ± ijy = (k ∓ q)(k ± q + 1)Tq∓ . may be replaced by the topological group of in- 1 vertible bounded linear operators on V . In this case, the homomorphism T is required to be con- irreducible unitary representation A uni- G tinuous, and the subspaces W are required to be tary representation of a (topological) group T G closed. See also irreducible unitary representa- is a (continuous) homomorphism from into U(H) tion. the group of unitary operators on a Hilbert space H . T is an irreducible unitary representa- tion if there does not exist a proper (= 0, = H ) irreducible R-module Let R be a ring and closed subspace W of H such that T (g)(w) ∈ let M be a module over R. We say that M is W for all g ∈ G and w ∈ W. See also irre- irreducible over R if R has no submodules. In ducible representation. some contexts, we say that M is irreducible if M cannot be written as a direct sum of proper irreducible variety A variety is the solution sub-modules. These are also sometimes called set of a system of polynomial equations (usu- simple modules. ally in several variables). An irreducible vari- ety is a variety V which is not the union of two irreducible scheme A scheme whose under- proper (= ∅, = V ) subvarieties. For example, lying topological space is irreducible. In more the parabola y−x2 = 0 is an irreducible variety, detail, a scheme is a particular type of ringed but the variety x2 − y2 = 0 is reducible because (X, O ) X space, X . Here, is a topological space it can be decomposed into the union of the two O X and X is a sheaf of rings on . The scheme lines x − y = 0 and x + y = 0. (X, OX) is irreducible if X is not the union of two proper (= ∅, = X) closed subsets. See also irredundant In a lattice L, a representation ringed space, scheme. of an element a as a join a = a1 ∨ ··· ∨ an is irredundant if omitting any of the elements irreducible Siegel domain Siegel domains ai from the join produces an element b strictly are special kinds of domains in complex N space, smaller than a. There is a dual notion for meets: CN . An easy way to construct new Siegel do- A representation of an element a as a meet a = mains is to take the Cartesian product of two a1 ∧···∧an is irredundant if omitting any of the given Siegel domains. Thus if S1 and S2 are elements ai from the meet produces an element Siegel domains, S = S1 × S2 ={s = (s1,s2) : b strictly larger than a. See join, lattice, meet. s1 ∈ S1,s2 ∈ S2} will also be a Siegel domain. Example: R is a Noetherian ring (a commu- A Siegel domain is irreducible if it is not the tative ring satisfying the ascending chain condi- Cartesian product of two other Siegel domains. tion). L is the lattice of ideals of R, ordered

c 2001 by CRC Press LLC by inclusion. It is a theorem of , a covering space map of the underlying mani- the Lasker-Noether Theorem, that every ideal folds. The map φ is a covering space map if in R has an irredundant representation as an in- it is continuous and, for each h ∈ H , there ex- tersection of primary ideals. This theorem al- ists a neighborhood U of h such that φ−1(U) most completely describes the ideal theory of is a disjoint union of open sets in G mapping Noetherian rings, including such rings as the homeomorphically to U under φ. ring of polynomials in several variables with co- (2) A topological group homomorphism (a efficients in a field, and the ring of terms of holo- continuous group homomorphism) φ : G → H , morphic (analytic) functions in several complex where G and H are topological groups, which variables. See also Noether, Noetherian ring. is a covering space map of the underlying topo- logical spaces. irregularity In algebraic geometry, the di- (3) An epimorphism φ : G → H of group mension of the Picard variety of a non-singular schemes (over a ground scheme S) such that the projective algebraic variety. See also Picard va- kernel of φ is a flat, finite group scheme over S. riety. See also epimorphism, scheme. irregular prime A prime number p which isolated component Let I be an ideal in a divides the numerator of one or more of the commutative ring R, and let I = Q1 ∩···∩Qk Bernoulli numbers B2,B3,... ,Bp−3. A prime be a short representation of I as an intersection number which is not irregular is called a regular of primary ideals. (See short representation.) prime. Irregular primes were of interest because Let P1,... ,Pk be the prime ideals belonging they were the class of exceptional primes for to Q1,... ,Qk. (The easiest way to specify which Kummer’s proof of Fermat’s Last Theo- Pi is to note that Pi is the radical of Qi, i.e., n rem does not work. Wiles’ recent proof of Fer- Pi ={p ∈ R : p ∈ Qi for some integer n}.) mat’s Last Theorem probably makes the distinc- Renumbering the primary ideals Qi if necessary, tion between regular and irregular primes unin- an ideal J = Q1 ∩···∩Qr (with 1 ≤ r ≤ k)is teresting, but one never knows. See Bernoulli an isolated component of I if none of the prime number, Fermat’s Last Theorem. ideals P1,... ,Pr contains a prime ideal Pj not in the set {P1,... ,Pr }. irregular variety A non-singular projective Isolated components are of interest because algebraic variety with non-zero irregularity. A they introduce uniqueness into the representa- variety of zero regularity is called a regular va- tion theory of ideals in commutative Noetherian riety. See also irregularity. rings. Although there are often many different short representations of an ideal I, the isolated isogenous Abelian varieties A pair of Abel- components of I are uniquely determined. See ian varieties of equal dimension, for which there also isolated primary component, Noetherian is a rational group homomorphism from one va- ring, primary ideal, prime ideal, radical, short riety onto the other. In more detail, an Abelian representation. variety is, among other things, an algebraic va- riety which is also an Abelian group. A rational isolated primary component An isolated group homomorphism from one Abelian variety component J of an ideal I in a commutative to another is a rational map which is also a group ring, such that J is a primary ideal. Isolated homomorphism. See also rational map. primary components are of interest because they must occur among the primary ideals of every isogenous groups A pair of topological short representation of I. See also isolated com- groups (usually Lie groups) for which there is an ponent, short representation. isogeny from one to the other. See also isogeny. isomorphic Two groups G and H are iso- isogeny (1) A Lie group map (a continuous, morphic if there is an isomorphism φ : G → H differentiable group homomorphism) φ : G → between them. Isomorphic groups are regarded H , where G and H are Lie groups, which is as being “abstractly identical,” or different re-

c 2001 by CRC Press LLC alizations of the same abstract group. The no- such examples as isomorphisms of topological tion extends to other algebraic structures such as groups, where it is required that an isomorphism rings, to the completely general algebraic struc- φ be a group isomorphism and that φ and φ−1 be tures defined in universal algebra, and even to continuous. See also category, homomorphism, the theory of categories. See also isomorphism. morphism, identity morphism. isomorphism (1) In group theory, a map- Isomorphism Theorem of Class Theory φ : G → H G ping between two groups, and Let k be an algebraic number field. Let I (m) be H , which is one-to-one (injective), onto (sur- the multiplicative group of all fractional ideals jective), and which preserves the group opera- of k which are relatively prime to a given integral φ(g · g ) = φ(g ) · φ(g ) tion, that is 1 2 1 2 . The divisor m of k. The Galois group of a class field φ notion extends to rings, where is required to K/k for an ideal group H (m) is isomorphic to preserve the ring addition and multiplication, to I (m)/H (m). Therefore, every class field K/k φ vector spaces, where is required to preserve is an Abelian extension of k. vector addition and scalar multiplication (i.e., φ(λ v + λ v ) = λ φ(v ) + λ φ(v )), and to 1 1 2 2 1 1 2 2 isomorphism theorems of groups The three completely general algebraic contexts (see (2) standard theorems describing the relationship below). between homomorphisms, quotient groups, and (2) In universal algebra, a mapping φ : G → normal subgroups. Let G and H be groups, and H between two (universal) algebras G and H, let φ : G → H be a homomorphism with ker- which is one-to-one (injective), onto (surjec- nel K. (The kernel of φ is the set K ={g ∈ tive), and which preserves the operations of G G : φ(g) = e}, where e is the group identity el- and H. In more detail, G = (G, FG) and H = ement in H .) The First Isomorphism Theorem (H, FH), where G is a set and FG is a set of states that K is a normal subgroup of G (i.e., functions from finite Cartesian products of G gK = Kg for every g ∈ G), and the quotient with itself to G (FG is called the set of operations group (factor group) G/K is isomorphic to the on G), and similarly for H. Thus the functions image of φ. Let S and T be subgroups of G, with in FG are G-valued functions f(g ,... ,gn) of 1 T normal. The Second Isomorphism Theorem nG-valued variables, and the value of n may states that S ∩ T is normal in S, and S/(S ∩ T) vary with the function f . To be an isomor- is isomorphic to TS/T. Let K ⊂ H ⊂ G, phism, φ is required to be a one-to-one and onto with both K and H normal in G. The Third function from the set G to the set H , and for Isomorphism Theorem states that H/K is a nor- every function f ∈ FG, there must be a func- mal subgroup of G/K, and (G/K)/(H/K) is tion h ∈ FH, such that φ(f(g ,... ,gn)) = 1 isomorphic to G/H . h(φ(g1),...,φ((gn)), and vice-versa. (This is what is meant by “preserving the operations of There is an additional theorem which is some- G and H.”) times called the Fourth Isomorphism Theorem, In the special case where G and H are groups, but is more commonly called Zassenhaus’s A A B B FG equals the singleton set containing the group Lemma. Let 0, 1, 0, and 1 be subgroups G A A B operation of G (the group multiplication), and of . Suppose 0 is normal in 1, and 0 B similarly for FH. We thus recapture the moti- is normal in 1. Zassenhaus’s Lemma states A (A ∩ B ) A (A ∩ B ) vating case of group isomorphisms. that 0 1 0 is normal in 0 1 1 , B (A ∩ B ) B (A ∩ B ) (3) In , a morphism φ : A → 0 0 1 is normal in 0 1 1 , and A (A ∩ B )/A (A ∩ B ) B between two objects of a category with an 0 1 1 0 1 0 is isomorphic to B (A ∩ B )/B (A ∩ B ) inverse morphism. In other words, for φ to be 0 1 1 0 0 1 . See also factor an isomorphism, there must also be a morphism group, normal subgroup. ψ : B → A (the inverse of φ) such that ψ ◦φ = ιA and φ ◦ ψ = ιB . Here, ιA and ιB are the isotropic (1) In physics and other sciences, identity morphisms on A and B, respectively. a material or substance which responds the The category theoretic definition captures all same way to physical forces in all directions is of cases (1) and (2) above, and also includes isotropic. Antonym: anisotropic.

c 2001 by CRC Press LLC (2) Let V be a vector space equipped with tions x1,x2,x3,... to a vector x, according to a bilinear form (, ). A subspace W of V is the formula xk = Mxk−1 + c. (Of course, one isotropic (sometimes called totally isotropic or must have a conveniently chosen starting vec- ⊥ ⊥ an isotropy subspace)ifW ⊆ W , where W tor x0.) For example, suppose we wish to solve is defined in the usual way, W ⊥ ={v ∈ V : the equation Ax = b approximately, where A (v, w) = 0 for all w ∈ W}. For example, if V = is an n × n square matrix, and x and b are n- R2, 2-dimensional real space, and (, ) is the dimensional column vectors. Write A = L + Lorentz form, ((x1,t1), (x2,t2)) = x1x2 − t1t2, D + U, where L is lower triangular, D is diag- then each of the lines forming the edge of the onal, and U is upper triangular. If we choose light cone, {(x, t) : x2 − t2 = 0}, is an isotropic the iteration matrix M =−D−1(L + U) and subspace. c = D−1b, we obtain the Jacobi method for (3) If a differentiable manifold M has enough solving linear equations. On the other hand, if additional structure so that its tangent space we choose the iteration matrix M =−(L+D)U − comes equipped with a bilinear form, for exam- and c = (L + D) 1b, we obtain the Gauss- ple if M is a symplectic manifold, then a sub- Seidel method for solving linear equations. See manifold S of M is isotropically embedded if at also iteration function, Gauss-Seidel method for each point s ∈ S, TSs is an isotropic subspace solving linear equations, Jacobi method for solv- of TMs. Here, TSs is the tangent space of S at ing linear equations. s, and similarly for TMs. See also symplectic manifold, tangent space. iterative calculation A calculation which proceeds by means of iteration. See iteration. isotropy subgroup A group of transforma- tions leaving a given point fixed. In more detail, iterative improvement (1) Any one of the let G be a group of transformations acting on a many algorithms for the approximate numerical set X, and let x ∈ X. The subgroup of transfor- 0 solution of problems which proceed by obtain- mations T ∈ G leaving x fixed (T(x ) = x ) 0 0 0 ing a better approximation at each step. is called the isotropy subgroup of G at the point x0. The isotropy subgroup is also called the sta- (2) See iterative refinement. bilizer of x0 with respect to G. iterative method An algorithm or calcula- isotropy subspace See isotropic. tional process which uses iteration. A classic ex- ample is the Newton-Raphson method for com- iteration (1) Repetition; step-by-step repeti- puting the roots of an equation. Another classic tion of a mathematical operation or construction. example is the Gauss-Seidel iteration method for (2) The use of loops as opposed to recursion solving systems of linear equations. See iter- in algorithms or programs. ation, iteration function, Gauss-Seidel method for solving linear equations, Newton-Raphson iteration function In numerical analysis, a method of solving algebraic equations. function φ, used to compute successive approx- x ,x ,x ,... x imations 1 2 3 , to a quantity , accord- iterative process See iterative method. ing to the formula xn = φ(xn−1). For example, if we choose φ(x) = x − f(x)/f!(x) as an iter- ation function and then select a suitable start- iterative refinement (1) See iterative im- provement (1). ing point x0, we obtain the Newton-Raphson method for approximating a zero of the func- (2) In numerical analysis, a process for solv- tion f (approximating a solution to the equation ing systems of linear equations which begins f(x) = 0). See also Newton-Raphson method by obtaining a first solution using elimination of solving algebraic equations. (Gaussian elimination or row reduction) which is somewhat inaccurate due to roundoff errors, iteration matrix In numerical analysis, a ma- and then improves the accuracy of the solution trix M used to compute successive approxima- using one of many iterative methods.

c 2001 by CRC Press LLC Iwahori subgroup If G is a reductive group where e + n = λn + µp n+ν, for all sufficiently defined over a local field, then in addition to large n. Here, p is a prime, k is an algebraic the standard BN-pair structure G has a second number field; k∞ is a Zp extension field of k (an BN-pair structure whose associated building is extension field with Galois group isomorphic to Euclidean. In this case, the subgroups conjugate Zp, the integers modulo p); kn is an intermedi- n to B are called Iwahori subgroups. ate field of degree p over k,Cl(kn)p is the pth component of the ideal class group of the field Iwasawa decomposition (1) A decomposi- kn, and |Cl(kn)p| is the number of elements in tion of a semisimple Lie algebra g over the field Cl(kn)p. For cyclotomic Zp extensions, the in- of real numbers as g = k + a + n, where k variant µ = 0. is a maximal compact subalgebra of g, a is an Abelian subalgebra of g, a + n is a solvable Lie Iwasawa’s Main Conjecture (1) A conjec- algebra, and n is a nilpotent Lie algebra. ture relating the characteristic polynomials of (2) A decomposition of a connected Lie particular Galois modules to p-adic L-functions. group G as G = KAN, where K is an (essen- The conjecture is an attempt to extend a classic tially) maximal compact subgroup, A is an Abel- theorem of Weil, which states that the character- ian subgroup, and N is a nilpotent subgroup. istic polynomial of the Frobenius automorphism Here, G has Lie algebra g which is semisimple, of a particular type of curve is the numerator of g = k + a + n is the Iwasawa decomposition the zeta function of the curve. The conjecture of g as in (1) above, K, A, and N are analytic was originally written over the field Q, although subgroups of G with Lie algebras k, a, and n, it has been reformulated as a conjecture over and the mapping (x,y,z))→ xyz is an analytic any totally real field. It has been proved for real of K ×A×N onto G. Further- Abelian extensions of Q and odd primes p by more, the groups A and N are simply connected. Mazur and Wiles. Some work has also been The classic example of an Iwasawa decompo- done in the general case. sition is provided by the group G = SL(m, C), (2) A conjecture in number theory, relating the group of m × m matrices with determinant certain Galois actions to p-adic L-functions. ˜ 1 over the complex numbers. In this case, K = The conjecture asserts: fχ (T ) = gχ (T ).Iwa- SU(m), the group of m × m unitary matrices of sawa’s Theorem, which describes the behavior determinant 1, A = the group of m×m diagonal of the p-part of the class number in a Zp-exten- matrices of determinant 1 with positive entries sion, can be regarded as a local version of the on the diagonal, and N = the group of m×m up- Main Conjecture. per triangular matrices with 1 in every diagonal entry. See Lie algebra, Lie group, semisimple Iwasawa’s Theorem The characteristic p = Lie algebra, semisimple Lie group. 0 case of the Ado-Iwasawa Theorem: Every finite dimensional Lie algebra (over a field of Iwasawa invariants The integers λ, µ, and characteristic p) has a faithful finite dimensional ν defined by the relation representation. The characteristic p = 0 case of this is Ado’s Theorem. See Lie algebra. en Cl (kn)p = p

c 2001 by CRC Press LLC proximations to the eigenvalues of A down the diagonal. The name is most frequently applied to the Jacobi rotation method, where the ma- trices Uk are chosen to be particularly simple J unitary matrices called planar rotation matri- ces. See Jacobi rotation method.

Jacobian variety The Picard variety of a Jacobi method of finding key matrix A step smooth, irreducible, projective curve. See Pi- in solving a linear system Ax = b by the linear card variety. stationary iterative process. If the linear sta- tionary iterative process is written as x(k+1) = (x · y)· z + (y · (k) (k) Jacobi identity The identity x + R(b − Ax ), then the Jacobi method z) · x + (z · x) · y = 0 satisfied by any Lie alge- chooses R to be the inverse of the diagonal sub- A bra. For example, if is any associative alge- matrix of A. See also linear stationary iterative [x,y] [x,y]= bra, and denotes the commutator, process. xy−yx, then the commutator satisfies the Jacobi [x,y],z + [y,z],x + [z, x],y = identity [ ] [ ] [ ] 0. Jacobi rotation method An iterative method See Lie algebra. for approximating all of the eigenvalues (char- acteristic values) of a Hermitian matrix A. The Jacobi method for solving linear equations method begins by choosing A0 = A, and then An iterative numerical method, also called the produces a sequence of matrices A1,A2,..., total-step method, for approximating the solu- AN , culminating in a nearly diagonal (all entries tions to a system of linear equations. In more off the diagonal are small) matrix AN with good detail, suppose we wish to approximate the solu- approximations to the eigenvalues of A down Ax = b A n×n ∗ tion to the equation , where is an the diagonal. At each step, Ak = U Ak− Uk, A = L + D + U L k 1 square matrix. Write , where where Uk is the (unitary) matrix of a planar rota- D U is lower triangular, is diagonal, and is up- tion annihilating the off diagonal entry of Ak− D 1 per triangular. The matrix is easy to invert, so with largest modulus, hence the name rotation replace the exact equation Dx =−(L+U)x+b U ∗ method, and k is the conjugate transpose of by the relation Dxk =−(L + U)xk−1 + b, and (k) Uk. The matrix Uk = ui,j differs from the solve for xk in terms of xk−1: u(k) u(k) −1 −1 identity matrix only in four entries, p,p, q,q, xk =−D (L + U)xk−1 + D b. (k) (k) up,q, and uq,p. The formulas for these entries This gives us the core of the Jacobi iteration are particularly simple in the case where the method. We choose a convenient starting vec- original matrix Ais a real symmetric matrix: If x (k−1) (k) (k) tor 0 and use the above formula to compute Ak−1 = ai,j , then up,p = uq,q = cos(φ), successive approximations x1,x2,x3,... to the (k) (k) and up,q =−uq,p = sin(φ), where actual solution x. Under suitable conditions, the sequence of successive approximations does in- (k−1) x 2ap,q deed converge to . See iteration matrix. tan(2φ) = , (k−1) (k−1) ap,p − aq,q Jacobi method of computing eigenvalues − π ≤ φ ≤ π Any of the several iterative methods for approxi- and 4 4 . In the commonly oc- mating all of the eigenvalues (characteristic val- curring case where the original matrix A has ues) of a Hermitian matrix A by constructing no repeated eigenvalues, the method converges a finite sequence of matrices A0, A1,...,AN , quadratically. The method is named after its where A0 = A and, for 0

c 2001 by CRC Press LLC Let (ω1,...,ωg) be a basis of Abelian differen- system {G, H, (j), ω} is called a j-algebra if tials of the first kind on R and let P1,...,Pg be the following conditions are satisfied: (i.) j ≡ a given set of fixed points on R. For any given j mod H and jH ⊂ H for j and j in (j), p 2 vector (u1,...,ug) ∈ C , the problem is to find (ii.) j ≡−1 mod H , (iii.) [h, jx]≡j[h, x] a representation of all of the possible symmetric mod H for h ∈ H and x ∈ G,(iv.)[jx,jy]≡ rational functions of Q1 ...Qg as functions of j[jx,y]+j[x,jy]+[x,y] mod H for x and y u1,...,ug that satisfy in G,(v.)ω([h, x]) = 0 for h ∈ H , (vi.) ω([jx, jy]) = ω([x,y]), (vii.) ω([jx,x])>0ifx/∈ g Q j H . ωi = ui . j= Pj 1 Janko groups Any of the exceptional finite J J J J J In the above situation the path of integration is simple groups 1, 2, 3, and 4. 1 has order 3 · · · · · J 7 · 3 · 2 · the same in each of the g equations. If the path 2 3 5 7 11 19. 2 has order 2 3 5 7 is not assumed to be the same, then the system and is also called the HJ or Hall-Janko group. J 7 · 5 · · · J is actually a system of congruences modulo the 3 has order 2 3 5 17 19 and 4 has order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43. periods of the differentials (ω1,...,ωg).

Jacobson radical The set of all elements Janko-Ree group Any member of the family r in a ring R such that rs is quasi-regular for of all finite simple groups for which the central- all s ∈ R. In more detail, let R be an arbi- izer of every involution (element of order 2) has × (q) q trary ring, possibly non-commutative, possibly the form Z2 PSL2 , odd. These groups 2G ( n) n without a unit element. An element r ∈ R is consist of the Ree groups 2 3 , for odd, J quasi-regular if there is an element r ∈ R such and the Janko group 1. See Janko groups, Ree that r + r + rr = 0. (In the special case group. where R has a unit element 1, this is equiva- µ lent to (1 + r)(1 + r ) = 1.) For example, every Jensen measure (1) A positive measure ' n nilpotent element (rn = 0 for some n) is quasi- on the closure of an open subset of C (C the complex numbers) such that for x ∈ ', regular, but there are often other quasi-regular elements. The Jacobson radical of R is the set log |f(x)|≤ log |f(t)| dµ(t) , J of all elements r ∈ R such that rs is quasi- s ∈ R regular for all . for all f belonging to some appropriate class of The Jacobson radical is a two-sided ideal, and holomorphic functions (such as the holomorphic radical it generalizes the notion of the ordinary functions on ' with continuous extensions to the (the set of all nilpotent elements) of a commuta- closure of '). Jensen measures are named after tive ring, though even in the special case where R Jensen’s inequality, which states that normal- is commutative, the Jacobson radical often ized Lebesgue measure on the unit circle (1/2π differs from the ordinary radical. Both derive times arclength measure) is a Jensen measure. much of their importance from the following ' R R In this case, is taken to be the open unit disk theorem: Let be commutative. Then (i.) is {z ∈ C :|z| < 1} in the complex plane. isomorphic to a subring of a direct sum of fields µ R (2) More generally, a positive measure on if and only if its radical vanishes, and (ii.) the maximal ideal space M of a commutative is isomorphic to a subdirect sum of fields if and Banach algebra A is called a Jensen measure See also only if its Jacobson radical vanishes. for an element , ∈ M if radical, subdirect sum of rings, Wedderburn’s Theorem. log |,(f )|≤ log |t(f)| dµ(t) j-algebra A concept which reduces the study for all f ∈ A. See also Banach algebra, Jensen’s of homogeneous bounded domains to algebraic inequality, maximal ideal space. problems. Let G be a Lie algebra over R, H a subalgebra of G, (j) a collection of linear endo- Jensen’s inequality (1) In complex variable morphisms of G, and ω a linear form on G. The theory. Let f be holomorphic on a neighbor-

c 2001 by CRC Press LLC hood of the closed disc D(0,r)in the complex and S is the relation R S, with set of attribute plane. Assume that f(0) = 0. Then Jensen’s names A ∪ B and a set of possible attribute val- inequality is ues X ∪ Y , defined as the set of all functions f : A ∪ B → X ∪ Y such that f |A ∈ R and 1 2π |f( )|≤ |f(reit)| dt . f |B ∈ S. Here, f |A is the restriction of f to A, log 0 π log 2 0 and f |B is the restriction of f to B. Thus the (2) In measure theory. Let (X, µ) be a mea- formation of the natural join reduces to the fa- sure space of total mass 1. Let f be a non- miliar and ubiquitous mathematical problem of negative function on X. Let φ be a convex func- extending classes of functions. See restriction. tion of a real variable. Then Jensen’s inequality a is join irreducible element An element of a lattice L which cannot be represented as the φ f (x) dµ(x) ≤ φ ◦ f(x)dµ(x). join of lattice elements b properly smaller than X X a (b

c 2001 by CRC Press LLC theory of Lie algebras, which is modeled on the (3) The decomposition of a linear transfor- algebra of n×n matrices with the multiplication mation T into a product T = SU, where S is di- [A, B] = AB − BA. See also Lie algebra. agonalizable, U is unipotent, and S and U com- mute. (Unipotent means that U −I is nilpotent, Jordan canonical form A matrix of the form where I is the identity transformation.) This de-   composition is called the multiplicative Jordan T S U J1 0 ... 0 decomposition of . and , if they exist, are   T  0 J2 ... 0  uniquely determined by , and then the multi-  . . .  , plicative Jordan decomposition is related to the  . . ..  additive Jordan decomposition T = Ts + Tn by 00... Jk − S = Ts, U = I + S 1Tn. (4) The decomposition of a linear transfor- J where each i is an elementary Jordan matrix, mation T into a product T = EHU, where E is is in Jordan canonical form. It is a theorem of elliptic, H is hyperbolic, U is unipotent, and all n × n linear algebra that every matrix with en- three commute. (Elliptic means E is diagonal- tries from an algebraically complete field, such izable and all complex eigenvalues have mod- as the complex numbers, is similar to a matrix ulus = 1. Hyperbolic means H is diagonaliz- in Jordan canonical form. See also elementary able and all complex eigenvalues have modulus Jordan matrix. < 1.) This decomposition is called the com- pletely multiplicative Jordan decomposition of Jordan decomposition (1) The decomposi- T . E, H , and U, if they exist, are uniquely tion of a linear transformation T : V → V , determined by T . where V is a vector space, into a sum T = (5) In analysis, the decomposition of a Ts + Tn, where Ts is diagonalizable, Tn is nilpo- bounded additive set function µ (defined on a tent, and the two commute. (Nilpotent means field of sets 8) into the difference of two non- k Tn = 0 for some integer k, and diagonalizable negative bounded additive set functions, µ = means there is a basis for V with respect to which µ+ − µ−, via the formulas T can be represented by a diagonal matrix.) Ts + and Tn, if they exist, are uniquely determined µ (E) = sup µ(F ), by T . It is a theorem of linear algebra that if F ⊆E − V is a finite dimensional vector space over an µ (E) =− inf µ(F ) , F ⊆E algebraically closed field, such as the complex T T T numbers, then s and n always exist, that is where F is restricted to belong to 8. Here sup always has a Jordan decomposition. However, and inf refer to the supremum and infimum, re- this need not be true if the field is, for example, spectively. (See supremum, infimum.) The set the field of real numbers which is not algebrai- functions µ+ and µ− are called the positive or T cally closed. The Jordan decomposition of is upper variation of µ, and the negative or lower T + − equivalent to the representation of by a matrix variation of µ. The sum |µ|=µ + µ is in Jordan canonical form. See Jordan canonical called the total variation of µ. See also additive form. set function. (2) The decomposition of an element a of a (6) In analysis, the decomposition of a func- Lie algebra A into a sum a = s + n, where tion of bounded variation into the difference of a s is a semisimple element of A, n is a nilpo- monotonically increasing function and a mono- tent element of A, and s and n commute. This tonically decreasing function. (Sometimes decomposition is called the additive Jordan de- stated monotonically non-decreasing and mono- composition of a. The elements s and n, if they tonically non-increasing.) This is a special case exist, are uniquely determined by a. It is a theo- of (5). See also bounded variation, monotone rem that if A is a semisimple finite dimensional function. Lie algebra over an algebraically complete field, such as the complex numbers, then s and n al- Jordan-Hölder Theorem (1) The theorem ways exist. See semisimple Lie algebra. that any two composition series of a group are

c 2001 by CRC Press LLC equivalent. A composition series of a group G the projectivity relation and the second isomor- is a finite sequence of groups phism theorem for groups.) Unfortunately, the classical Jordan-Hölder theorem for composi- G = G0 ⊃ G1 ⊃···⊃Gn ={1} , tion series of groups (see (1) above) is not so easy to derive from the lattice theorem because G such that each group i+1 is a maximal normal the lattice of all subgroups may not be modu- G G subgroup of i. (Equivalently, each i+1 is lar. See also characteristic series, isomorphism a normal subgroup of Gi and the factor group theorems of groups. Gi/Gi+1 is simple and not equal to {1}.) Two composition series of G are equivalent if they Jordan homomorphism A mapping φ be- have the same length and isomorphic factor tween Jordan algebras A and B which respects groups. The theorem extends to groups with addition, scalar multiplication, and the Jordan operators, thus to R modules, for example, and multiplication. In other words, φ(a + b) = even to lattices (see (2) below). See also factor φ(a) + φ(b), φ(λa) = λφ(a), and φ(a · b) = group, normal subgroup, simple group. φ(a)·φ(b), for all scalars λ and for all a,b ∈ A. (2) The theorem that any two composition See Jordan algebra. chains connecting two elements a and b in a modular lattice are equivalent. A lattice is mod- Jordan module Let A be a Jordan algebra ular if it satisfies the weakened distributive law, over a field of scalars K.AJordan A module is a vector space V over the same field K, together x ∧ (y ∨ z) =(x ∧ y) ∨ (x ∧ z) with a multiplication operation · from A × V to whenever x ≥ y. V satisfying (i.) a · (v + w) = a · b + a · w; (ii.) a · (λv) = λ(a · v), and (iii.) (a · b) · v = y ∨z 1 a·(b·v)+ 1 b·(a·v) a,b ∈ A v,w ∈ V Here, denotes the lattice join or supremum 2 2 , for all , , (least upper bound) of y and z, and y ∨ z de- and scalars λ. notes the lattice meet or infimum (greatest lower Property (iii.) seems odd; indeed the reader bound) of y and z.(See join.) A composition familiar with R modules (R a ring) would think chain connecting a to b is a finite sequence of it should be replaced by (a · b) · v = a · (b · v). lattice elements, a = a0 ≥ a1 ≥ ···an = b, However, property (iii.) is easier to understand such that there is no lattice element x strictly if one realizes that the Jordan multiplication · between ai and ai+1. (Equivalently, each in- induces a mapping a →Ta between elements a terval [ai+1,ai] is a two element lattice.) Two of the Jordan algebra and linear transformations composition chains are equivalent if they have Ta.Givena ∈ A, Ta is defined by Ta(v) = a ·v. the same length and projective intervals. Two Property (iii.) is chosen to guarantee that Ta·b intervals [w, x] and [y,z] are projective if there 1 (T T + T T ) will be the Jordan product 2 a b b a in is a finite sequence of intervals, the Jordan algebra of linear transformations of the vector space V . [w, x]= w ,x , w ,x ,..., w ,x [ 1 1] [ 2 2] [ n n] In fact, the mapping a →Ta is a Jordan =[y,z] homomorphism of A into the Jordan algebra of all linear transformations on V . A homomor- such that each pair of intervals [wi,xi] and phism between a Jordan algebra A and a Jor- [wi+1,xi+1] are transposes. Finally, two inter- dan algebra of linear transformations is called vals are transposes if there are lattice elements a Jordan representation of A. The definition c and d such that one interval is [c, c ∨ d] and of a Jordan module has been designed so there the other is [c ∧ d,d]. is a one-to-one correspondence between Jordan If G is a group, the lattice of its normal sub- representations and Jordan modules. See also groups is a modular lattice. Thus, the Jordan- Jordan algebra, Jordan homomorphism, Jordan Hölder theorem for lattices gives us several representation. Jordan-Hölder like theorems for groups, for in- stance for chief series and characteristic series. Jordan normal form See Jordan canonical (The isomorphism of factor groups comes from form.

c 2001 by CRC Press LLC Jordan representation A Jordan homomor- the ring of integers, and let G be a Z-order in phism between a Jordan algebra A and a Jor- A. Let L∗ be a left A module, and let σ(L∗) be dan algebra of linear transformations on a vec- the set of all left G modules L, having a finite Z tor space V , equipped with the standard Jordan basis, which are contained in L∗, and such that T ·S = 1 (T S+ST ) L = L∗ product, 2 . There is a one-to- Q . The Jordan-Zassenhaus Theorem one correspondence between Jordan representa- states: The set σ(L∗) splits into a finite number tions and Jordan modules. See also Jordan al- of classes under Z-equivalence. gebra, Jordan homomorphism, Jordan module. The Jordan-Zassenhaus Theorem is a far reaching generalization of the theorem that the Jordan-Zassenhaus theorem Let A beafi- number of ideal classes in an algebraic number nite dimensional semisimple algebra with unit field is finite. See class field, ideal class, Z-basis, over the field of rational numbers, Q. Let Z be Z-equivalence, Z-order.

c 2001 by CRC Press LLC then M1 is the closure of A1 with respect to the weak operator topology. The theorem remains true if restated using the strong operator topology instead of the weak K operator topology. Sometimes, the statement of the theorem includes the following additional information. The set of self-adjoint elements of K3 surface A class of algebraic surface in A1 is strongly dense in the set of self-adjoint abstract algebraic geometry, defined in a pro- elements of M1, the set of positive elements of jective space over an algebraically closed field. A1 is strongly dense in M1, and if A contains 1, In projective 3-space they can be regarded as de- the unitary group of A is strongly dense in the formations of quartic surfaces. A K3 surface is unitary group of M. characterized as a nonsingular, nonrational sur- face, in several ways including: k-compact group A connected algebraic (i.) irregularity, Kodaira dimension, and the group defined over a perfect field k whose k- canonical divisor are zero; Borel subgroups are reduced to the identity (ii.) irregularity is zero and the arithmetic, group. The name k-anisotropic group is used geometric, and first plurigenus are all one; also. See also k-isotropic group. (iii.) as a compact complex analytic surface, the first Chern class is zero and it has Betti num- k-complete scheme Let f : X −→ Y be a bers b0 = 1,b1 = 0,b2 = 22,b3 = 0,b4 = 1. morphism of schemes X, Y . When f has a prop- The space of one-dimensional differential erty, it is customary to say that X has the prop- forms on a K3 surface is zero. An example of a erty over Y , or that X is a Y -(property) scheme. K3 surface is any smooth surface of order four The property of being complete is connected in projective three-dimensional space. with the property of being proper. A morphism K3 surfaces were early examples of surfaces f : X −→ Y is proper if it is separated, of fi- satisfying Weil’s conjecture concerning the ana- nite type, and is universally closed. Then X is log of the Riemann Hypothesis for algebraic va- called proper over Y . See separated morphism, rieties. morphism of finite type. Now, let k be an algebraically closed field. Kakeya-Eneström Theorem Let f be a Let X be a scheme of finite type over k which is polynomial with real coefficients, say reduced (i.e., for any element x the local ring at x has no nilpotent elements) and is irreducible f(x)= a xn + a xn−1 +···+a , n n−1 0 (i.e., the underlying topological space is not the union of proper closed subsets). If X is proper for each real number x. Suppose over k (actually over the spectrum of k), then X is called a k-complete scheme. an ≥ an−1 ≥···≥a0 > 0 .

Let r be any root of the polynomial. Then |r|≤ kernel (1) In algebra, where a homomor- f 1. phism is defined between two algebraic sys- tems A and B, if the group identity of B is de- e f Kaplansky’s Density Theorem A funda- noted by , then the kernel of is mental theorem from the theory of von Neu- ker(f ) ={x ∈ A : f(x) = e} . mann algebras proved by Kaplansky in 1951. The closure M with respect to the weak oper- Alternately, ker(f ) may be denoted f −1({e}). ator topology of a C∗-subalgebra A of the set of The kernel is a subset of A that usually has bounded linear operators on a separable Hilbert special properties. If A and B are groups, then space is a von Neumann algebra. Furthermore, the kernel of a homomorphism is a normal sub- if A1 is the set of elements of A with norm ≤ 1 group of A.IfA and B are R-modules over a in A (unit ball of A) and M1 is the set of ele- ring R, the kernel is a submodule of A.IfA and ments of M with norm ≤ 1 (unit ball of M), B are topological linear spaces, then the kernel

c 2001 by CRC Press LLC of a continuous linear operator is a closed linear It is named after W. Killing who studied it in subspace. The kernel of a semi-group homo- 1888. The Killing form is fundamental in the morphism is the smallest two-sided ideal in the study of Killing-Cartan classification of semi- semi-group. Similar remarks hold for kernels of simple Lie algebras over fields of characteristic homomorphisms or morphisms in category the- 0. ory, sheaf theory, and kernels of linear operators between spaces. k-isomorphism (1) Let k be a field and (2) In topology, for a nonempty set S in a K,L extension fields of k. An isomorphism topological space, the kernel of S is the largest σ : K −→ L such that σ(x) = x for all subset T of S such that every element of T is an x ∈ k is called a k-isomorphism. Alternately, accumulation or cluster point of T . a k-isomorphism from K onto L is an isomor- (3) The word kernel is used in various other phism of the k-algebra K onto the k-algebra L. to denote a function. In (2) For other algebraic structures over a field the study of integral equations, for example, the k,ak-isomorphism is essentially an isomor- function K in the integral phism (a bijective map that preserves the bi- b nary operations) and a “regular” mapping (pre- K(x,y)f(y)dy serving the particular structure on the sets). a For example, for linear algebraic groups a k- is called a kernel. isomorphism is an isomorphism that is also a bi- rational mapping. For homogeneous k-spaces, k-form (1) In linear algebraic group theory, a a k-isomorphism is an isomorphism that is an k-form of an algebraic group G defined over an everywhere defined pre-k-mapping. extension field K of a field k is another algebraic group H defined over k that is K-isomorphic k-isotropic group A connected algebraic to G. Much work has been done in classify- group, defined over a perfect field k, whose k- ing the “k-forms” of various types of algebraic Borel subgroups are nontrivial. For a reductive groups defined over K (e.g., semisimple alge- k-group G defined over an arbitrary field k, G braic groups or almost simple algebraic groups). is k-isotropic if the k-rank of G is greater than (2) More generally, if G is an algebraic group zero. See k-compact group. See also k-rank. defined over k and K/k is a finite Galois exten- G sion, an algebraic group G1 is said to be a K/k- Kleinian group A subgroup of the group form of G if there is a K-isomorphism from G of linear fractional functions defined on the ex- ˆ onto G . For example, let k be a field and  tended complex plane C such that there is an 1 ˆ a universal domain containing k. Let T be an element x in C which has a neighborhood U n-dimensional algebraic k-torus with splitting such that g(U) ∩ U =∅, for each nontrivial field K. Then since T is K-isomorphic to the g ∈ G. Such groups were first studied by Klein direct product of n copies of GL(1) (the multi- and Poincaré in the 19th century and were named plicative group of non-zero elements of ), T is by Poincaré. See linear fractional function. a K/k-form of the n-dimensional K-split torus GL(1)n. KMS condition A condition originally con- See also quadratic form. cerning finite-volume Gibbs states and later pro- posed for time evolution and the equilibrium Killing form In Lie algebra theory, a sym- states in quantum lattice systems in statistical metric bilinear form associated with the adjoint mechanics (mathematically, within the frame- representation of a Lie algebra. Specifically, if work of C∗-dynamical systems and a one- g is a Lie algebra over a commutative ring K parameter group of automorphisms that describe with 1, ρ is the adjoint (linear) representation the time evolution of the system). The condition of g and Tr denotes the trace operator, the sym- was first noted by the physicists R. Kubo in 1957 metric bilinear form B : g × g −→ K given and C. Martin and J. Schwinger in 1959. The by B(x,y) = Tr(ρ(x)ρ(y)) is called the Killing letters K, M, and S are derived from their names. form. The equilibrium states are called KMS states.

c 2001 by CRC Press LLC Let M be a von Neumann algebra. Let φ (See reductive; examples of reductive groups are be a faithful normal positive linear functional semisimple groups, any torus, and the general on M. Let {σt } be a strongly continuous one- linear group.) Let T be a k-split torus in G of parameter group of ∗-automorphisms of M. Let largest possible dimension. The dimension of S be the closed strip in the complex plane {z : T is called the k-rank of G. The k-rank is 0 if 0 ≤(z) ≤ 1}. Then the group {σt } will be said and only if G is anisotropic. to satisfy the KMS condition if for any x,y ∈ M, The nonzero weights of the adjoint of T are there is F : S −→ C such that F is bounded called k-roots. In case T is a maximal torus, and continuous on S, analytic in the interior of k-roots are the usual roots of G with respect to S, and satisfies the conditions T . Z T G F(t) = φ (σ (x)y) Let denote the centralizer of in ; i.e., t and F(t + i) = φ (yσt (x)) . Z = {x ∈ G : xy = y} . y∈T The theory of Tomita-Takesaki shows the exis- {σ } tence of such a group t and also that such Let N denote the normalizer of T in G; i.e., groups are characterized by the condition. The KMS condition is a very important concept in the N = x ∈ G : xT x−1 = T . construction of type-III von Neumann algebras. See type-III von Neumann algebra. Then the finite quotient group Z/N is called the k-Weyl group. The group is named after the Kostant’s formula A formula (named after German mathematician H. Weyl (1885–1955). B. Kostant) that gives the multiplicities of the weights of a finite dimensional irreducible rep- k-rational divisor (1) Let K be the alge- resentation constructed from a root system of a braic closure (Galois extension) of a finite field complex semisimple Lie algebra. It is a con- k (with q elements) and let X be an algebraic sequence of the Weyl character formula. See curve defined over k. Then the automorphism Weyl’s character formula. In order to under- σ : k −→ k defined by σ(x) = xq defines an stand the (very explicit) formula, some defini- automorphism σ : X −→ X defined by tions are in order. Let g be a complex semisim- V ρ σ (x ,x ,...,x ) = xq ,xq ,...,xq ple Lie algebra. Let be a C-module and 1 2 n 1 2 n a linear irreducible representation of g over V . k X Let )+ be the set of positive roots. Let δ be the that leaves all -rational points in fixed (called δ = 1 α the Frobenius automorphism). Let half sum of the positive roots ( 2 α∈)+ ). Let W be the Weyl group of the root system. Let - be the highest weight of ρ. Let P be a non- d = axx negative integer valued function (called the par- x∈X tition function) defined on the lattice of weights. be a divisor in X. (Recall that all ak ∈ Z (inte- For each weight µ, P (µ) is the number of ways gers) and all except at most a finite number are µ can be expressed as a sum of positive roots. zero.) Then d is a member of the free Abelian Let m-(λ) denote the multiplicity of a weight λ group of divisors with base X, called Div(X). of ρ. Then Kostant’s formula is The divisor d is a k-rational divisor if m-(λ) = det(w)P (w(- + δ) − (λ + δ)) . d = σ(d) = a σ(x). w∈W x x∈X This sum is very difficult to compute in practice and is thus of more theoretical than computa- The set of k-rational divisors is a subgroup of tional use. See also positive root, Weyl group. Div(X). (2) Another use of the term k-rational divi- k-rank Let K be an algebraically closed sor involves a finite extension k of the field of k K G d = n P field, an arbitrary subfield of , and a re- rational numbers Q. A divisor i=1 i ductive linear algebraic group defined over k. is a k-rational divisor if all rational symmetric

c 2001 by CRC Press LLC functions of the coordinates of the points Pi with product of a commutative von Neumann alge- coefficients in Q are elements of the field k. bra with one ∗-automorphism. A Krieger’s fac- tor can be identified with approximately finite k-rational point The term k-rational point dimensional von Neumann algebras (over sepa- occurs in several areas of algebraic geometry, rable Hilbert spaces) of type III0. Krieger’s fac- algebraic varieties, and linear algebraic groups. tor was named after W. Krieger who has studied Examples of how the term is used follow. them extensively. (1) Let K be an algebraically closed field and k a subfield of K. Let A2 be the set of pairs Kronecker delta A symbol, denoted by δi,j , (a, b) of elements a,b ∈ K (the affine plane). defined by Let f be a polynomial from A2 into K with i = j coefficients from K. Recall that a plane alge- δ = 1if i,j i = j. braic curve is {(x, y) ∈ A2 : f(x,y) = 0}. 0if P = (x, y) ∈ A2 k-rational point Then is a if It is a special type of characteristic function de- x,y ∈ k K = C k = Q .If and , then a point fined on the Cartesian product of a set with itself. (x, y) k-rational point is a if both coordinates Specifically, let S be a set. Let D ={(x, x) : are rational numbers. x ∈ S}. The value of the characteristic function K pr (2)If is the finite field consisting of ele- of D is1if(x, y) ∈ D (meaning that x = y) p K ments ( a prime number) and is the algebraic and is 0 otherwise. closure, then the set of k-rational points of a k curve with coefficients in coincides with the set Kronecker limit formula (1)Ifζ is the ana- f(x,y) = x,y ∈ k r = of solutions of 0, .If 1, lytic continuation of the Riemann ζ -function to k so that is a prime number field, then the set of the complex plane C, then k-rational points is equivalent to the set of solu- tions of the congruence f(x,y) ≡ 0 (mod p). 1 lim ζ(s)− = (3) More generally, let k be a subfield of an s→1 s − 1 n arbitrary field K. Let x = (x1,...,xn) ∈ A . n− Then x is a k-rational point if each xi ∈ k, 1 1 i = ,...,n x = (x ,...,x ) ∈ n lim + 1 − log(n) = γ 1 . Next, let 1 n P n→∞ m + 1 (projective space). Then x is a k-rational point m=1 if there is a (n + 1)-tuple of homogeneous co- where γ is called Euler’s constant. This can be ordinates (λx0,λx1,...,λxn), λ = 0 such that expressed by saying that “near s = 1” λxi ∈ k, for each i = 0, 1,...,n.Ifxi = 0, this 1 is equivalent to xj /xi ∈ k,∀ j = 0, 1,...,n. ζ(s) = + γ + O(s − ). s − 1 (4) Let G be a linear algebraic group defined 1 k p over a field . An element that has all of its This last formula is called the Kronecker limit coordinates in k is called a k-rational point. formula. (5)IfX is a scheme over k, a point p of X (2) In the theory of elliptic integrals, modular is called a k-rational point of X if the residue forms, and theta functions, the function E(z,s) class field (with respect to the inclusion map of (with z = x +iy) defined by the Eisenstein type k into X at p)isk. series s (6)IfK is an algebraically closed transcen- y k V |m(z) + n|2s dental extension of ; is an algebraic variety m,n defined over k; and k is a subfield of K, then  x ∈ y> a k -rational point of V is an element of V that (where R, 0, the is over  (m, n) = ( , ) (s) > has all of its coordinates in k . all pairs 0 0 , and 1) can be extended to a meromorphic function on C Krieger’s factor The study of von Neumann whose only pole is at s = 1. The Kronecker algebras is carried out by studying the factors limit formula for E(z,s) is which are of type I, II, or III, with subtypes for π +2π(γ−log(2))−4π log |η(z)|+O(s−1) each. (See factor.) Krieger’s factor is a crossed s − 1

c 2001 by CRC Press LLC where for each linear functional L : V −→ k and each ∞ linear functional M : W −→ k. Define the πiz/ πizn η(z) = e 12 1 − e2 . Kronecker product of V and W with respect to n=1 k as the quotient space U/N.IfV and W are rings with unity instead, and multiplication in U The formula can be generalized to arbitrary num- is defined componentwise, then N becomes an ber fields. More general Eisenstein type series ideal of U and the Kronecker product becomes lead to similar (but more complicated) limit for- a residue class ring. mulas. (4) Sometimes, the tensor product of algebras is referred to as their Kronecker product. Kronecker product (1) Let A = (aij ) de- note an m × n matrix of complex numbers and Kronecker’s Theorem Several theorems in B = (bij ) an r × s matrix of complex numbers. several fields of mathematics honor L. Kronecker Then the Kronecker product of the matrices A (1823–1891). and B is defined as the mr ×ns matrix described (1)Iff is a monic irreducible element of k[x] by the mn blocks Cij given by over a field k, there is an extension field L of k Cij = aij B, 1 ≤ i ≤ m, 1 ≤ j ≤ n. containing a root c of f such that L = k(c). Sometimes such an L is called a star field. For mnrs Sometimes other permutations of the el- example, the polynomial x2 + 1 over the real mr ns ements arranged in rows and columns is number field has the field of complex numbers called the Kronecker product. This product is as a star field. Further, if f has degree n, there is used, for example, in studying modulii spaces an extension field of k in which f factors into n of Abelian varieties with endomorphism struc- linear factors, so f has exactly n roots (counting ture. multiplicities). V W (2) Let and be finite dimensional vec- (2) A field extension of the field of rational k {x ,... x } tor spaces over a field with bases 1 , n numbers which has an Abelian Galois group is {y ,...,y } V W and 1 r for and , respectively. Let a subfield of a cyclotomic field. L V M be a linear transformation on and a lin- (3) Kronecker was among several mathemati- W A n × n ear transformation on . Let be an cians who studied the structure of subgroups and L B r × r matrix associated with and an ma- quotient groups of Rn generated by a finite num- M trix associated with , determined by using the ber of elements. The following theorem was stated ordering of the basis elements. If lexico- proved by him in 1884 and is also called Kro- graphic ordering is used for the tensor products necker’s Theorem. Let m, n be integers ≥ 1. In of the basis elements in determining a basis for the following i will denote an integer between 1 V W the tensor product of and , then the ten- and m while j will denote an integer between 1 sor product V ⊗W of the linear transformations and n. Let ai = (ai1,ai2,...,aim) be m points has the nr × nr Kronecker product matrix de- n n of R and b = (b1,b2,...,bm) ∈ R . For ev- scribed in (1) as its matrix representation. See ery B>0 there are m integers qi and n integers lexicographic linear ordering. pj such that for each j V W (3) Now let and be arbitrary vector k U m spaces over a field . Let be a vector space q a − p − b

c 2001 by CRC Press LLC Kronecker symbol (1) An alternate name Krull-Akizuki Theorem Let R be a Noethe- for the Kronecker delta. See Kronecker delta. rian integral domain, k its field of quotients, and (2) In number theory, a generalization of the K a finite algebraic extension of k. Let A be a Legendre symbol, used in solving quadratic con- subring of K containing R. Assume every non- gruences zero prime ideal of R is maximal (i.e., assume R has Krull dimension 1). Then A is a Noethe- ax2 + bx + c ≡ 0 (mod m) rian ring of Krull dimension 1. Also, for any ideal g = (0) of A, A/g is a finitely generated where a,b,c are integers (members of Z) and A-module. m is a positive integer (m ∈ Z+). Here, the integral closure of a Noetherian d ∈ d d ≡ Let Z, not a perfect square, domain of dimension one is Noetherian. This ( ) m ∈ + 0or1 mod 4 and Z . For the following, remains true for a two-dimensional Noetherian d m x2 = recall that is a quadratic residue of if domain but not for dimension three or higher. d(mod m) is solvable. Then the Kronecker sym- d bol for d with respect to m, denoted by ( m ),isa + + − Krull-Azumaya Lemma Let R be a com- mapping from {d}×Z onto {0, 1, 1} defined mutative ring with unit, M = 0 a finite R- by: N R M J d module, and an -submodule of . Let be = 1 , the Jacobson radical of R. The Krull-Azumaya 1  Lemma is a name given to any of the following d statements: d  0ifis even = +1ifd ≡ 1 (mod 8) (i.) If MJ + N = M, then N = M; 2  − 1 d ≡ 5 (mod 8) (ii.) M = MJ; if m is an odd prime and m|d, then (iii.) If M/MJ is spanned by a finite set {xi + MJ}, then M is spanned by {xi}. d = 0 , This lemma is also known by the name m Nakayama’s Lemma and by the name Azumaya- Krull-Jacobson’s Lemma. It is a basic tool in the m m |d if is an odd prime and , then study of non-semiprimitive rings.   + d  1ifis a quadratic d m Krull dimension The supremum of the = residue of m  −1ifd is not a quadratic lengths of chains of distinct prime ideals of a  R R residue of m ring . It is sometimes called the altitude of . The definition was first proposed by W. Krull in if m is the product of primes p1,p2,...,pr , 1937 and is now considered to be the “correct” then definition not only for Noetherian rings but also for arbitrary rings. d d d d = · ····· . With this definition any field κ has dimension m p p p 1 2 r zero and the polynomial ring κ[x] has dimension one. The Kronecker symbol is used to determine the Legendre symbol, which in turn is used to find One reason for the acceptance of this def- quadratic residues. For certain values of d and inition for rings comes from a comparison to m the Kronecker symbol can be used to count the situation in finite dimensional vector spaces. n the number of quadratic residues. In a vector space of dimension over a field κ (3) The Kronecker symbol has uses and gen- , the largest chain of proper vector subspaces n eralizations in more advanced areas of number has length . The corresponding polynomial κ[x ,...,x ] theory as well; e.g., quadratic field theory and ring 1 n has a decreasing sequence (of n class field theory. length or height ) of distinct prime ideals k-root See k-rank. (x1,...,xn) ,...,(x1) ,(0)

c 2001 by CRC Press LLC where the notation (x1,...,xn) denotes the ideal Combinations of the names W. Krull, R. generated by the elements x1,...,xn. This se- Remak, O. Schmidt, J.H.M. Wedderburn, and quence is of maximal length. G. Azumaya are used to refer to this theorem. Wedderburn first stated the theorem for groups, Krull Intersection Theorem Let R be a Remak gave a proof for finite groups, Schmidt Noetherian ring, I an ideal of R, and M a finitely gave a proof for groups with an arbitrary sys- generated R-module. Then tem of operators, Krull extended the theorem (i.) there is an element a ∈ I such that to rings, and Azumaya found extensions of the   theorem to other algebraic structures. ∞ ( − a) I j M = . 1 0 Krull ring A commutative integral domain A j= 1 for which there exists a family {νi}i∈I of discrete valuations on the field of fractions K of A such (ii.) If 0 is a prime ideal or if R is a local ring, that the intersection of all the valuation rings of and if I is a proper ideal of R, then the {νi}i∈I is A and νi(x) = 0 for all nonzero x ∈ ∞ K and for all except (possibly) a finite number j I = 0 . of indices i ∈ I. A Krull ring is also called a j=1 Krull domain. Every discrete valuation ring is a Krull ring as is a factorial ring and a principal In some formulations, part (i.) is given as: ideal domain. ∞ Krull rings were studied by W. Krull. They I j M = represent an attempt to get around the problem that the integral closure of a Noetherian domain j=1 is (generally) not finite. Since the valuations {x ∈ M : (1 − a)x = 0, for some x ∈ I} . described above may be identified with the set of prime ideals of height one, a Krull ring may be The theorem is important for the theory of defined alternately using prime ideal of height Noetherian rings and is an application of the one. Artin-Rees Lemma concerning stable ring fil- trations. See Artin-Rees Lemma. Krull’s Altitude Theorem Also called Krull’s Principal Ideal Theorem. This theorem Krull-Remak-Schmidt Theorem The the- has several forms and characterizations. orem arises in various branches of algebra and (1) Let R be a Noetherian ring, let x ∈ R addresses the common length and isomorphisms and let P be minimal among prime ideals of between elements of the decomposition of an al- R containing x. Then the height of P (or the gebraic structure. codimension or the altitude of P )is≤ 1. See (1) In group theory. Suppose a group G satis- Krull dimension. fies the descending or ascending chain condition R for normal subgroups. If G × G ×···×Gn (2) Let be a Noetherian ring containing 1 2 x ,...,x P and H ×H ×···×Hm are two decompositions 1 n. Let be minimal among prime 1 2 R x ,x ,...,x of G consisting of indecomposable normal sub- ideals of containing 1 2 n. Then the P ≤ n groups, then n = m and each Gi is isomorphic height of . to some Hj . (3) Let R be a Noetherian local ring with m R (2) In ring theory. Let A = A1×A2×···×An maximal ideal . Then the dimension of is n n and B = B1 × B2 ×···×Bn be Artinian and the minimal number such that there exist Noetherian modules where each Ai and Bj are elements x1,x2,...,xn not all contained in any indecomposable modules. Then m = n and prime ideal other than m. each Ai is isomorphic to some Bj . Consequences of the theorem include the fact (3) The theorem may be formulated for other that the prime ideals in a Noetherian ring sat- algebraic structures, e.g., for local endomor- isfy the descending chain condition so that the phism modules or for modular lattices. number of generators of a prime ideal P bounds

c 2001 by CRC Press LLC the length of a chain of prime ideals descending Recall that a one-dimensional k-cocycle f is a from P . k-mapping from A × A into G such that f(x,z)= f(x,y)f(y,z), Krull topology A topology that makes the (x,y,z)∈ H . Galois group G(L/K) (for the Galois extension for all A3/k L K of the field over the field ) into a topological If f is a one-dimensional k-cocycle such that group. A fundamental system of neighborhoods there exists a k-mapping h from A × A into M L of the field unity element of is obtained by such that f = δh, then f is said to k-split in M. G(L/M) taking the set of all groups of the form This definition is used in k-cohomology. M L where is both a subfield of and a finite (4) There is a similar definition used in the co- K Galois extension of . homolgy of k-algebras involving the k-splitting The Krull topology is discrete if L is a finite of singular extensions of k-algebras bi-modules. extension of K. This topology is named after (5)IfG is a connected semisimple linear al- W. Krull, who extended Galois theory to infinite gebraic group defined over a field k, then G is algebraic extensions and laid the foundations for called k-split if there is a maximal k-split torus Galois cohomology theory. in G. Such a G is also said to be of Chevalley type. k-split A term used in several parts of linear algebraic group theory and homology theory. Kummer extension Any splitting field F of k  a polynomial (1) Let be an arbitrary field. Let denote a k n n n universal domain containing , that is, an alge- x − a1 x − a2 ... x − ar , braically closed field that has infinite transcen- i = , ,...,r a dence degree over k. Let Ga denote the alge- where for each 1 2 the i are ele- k n braic group determined by the additive group of ments of a field that contains a primitive th  and let Gm denote the algebraic group deter- root of unity. A Kummer extension is character- mined by the multiplicative group of non-zero ized by the property of being a normal extension elements of . Let G be a connected, solvable having an Abelian Galois group and the fact that k-group. Then G is k-split if it has a composi- the least common multiple of the orders of ele- n tion series ments of the Galois group is a divisor of . Kummer’s criterion The German mathe- G = G ⊃ G ⊃···⊃G ={ } 0 1 n 0 matician E. Kummer’s attempts in the mid-19th century to prove Fermat’s Last Theorem gave composed of connected k-subgroups with the rise to the theory of ideals and the theory of cy- property that each Gi/Gi+1 is k-isomorphic to clotomic fields and led to many other theories Gm or Ga. which are now of fundamental importance in (2)Ifak-torus T of dimension n is k-isomor- several areas of mathematics. Among his many phic to the direct product of n copies of Gm, then contributions to the mathematics that was devel- T is said to be k-split. For a k-torus, definitions oped to prove (or disprove) Fermat’s Last The- (1) and (2) are equivalent. orem, Kummer obtained congruences (3) Let A be a k-set, G be a k-group, M dl−2n log(x + evy) Bn ≡ 0 (mod l) be a principal homogeneous k-space for G, and dvl−2n v=0 HX/k be the set of k-generic elements of the k- k X k n = , ,..., l−3 B n components of any -set . For any -mapping for 1 2 2 where n is the th h from A into G, let δh denote the k-mapping Bernoulli number. The congruences are called from A × A into G defined by Kummer’s criterion in his honor. In 1905 the mathematician D. Mirimanoff established the δh(x, y) = h(x)−1h(y), equivalence of these congruences to the condi- tions (which also are called Kummer’s criterion) (x, y) ∈ H . for all A2/k of the following theorem.

c 2001 by CRC Press LLC Let x,y,z be nonzero integers. Let l be a (1) The first Künneth Theorem exhibits the prime > 3. Suppose x,y,z are relatively prime Künneth formula for complexes. Let A be a ring to each other and to l.If{x,y,z} is a solution with identity, L a complex of left A-modules and to the Fermat problem R a complex of right A-modules. Assume that for each n the boundary modules Bn(R) and cy- xl + yl = zl , cle modules Cn(R) are flat. Then for each n there is a homomorphism β such that the se- then quence Bnfl− n(t) ≡ 0 (mod l) , 2 x y y z x z α β for all t ∈{−y , − x , − z , − y , − z , − x } and for 0 → Hm(R)⊗Hq (L) → Hn(R⊗L) → n = , ,..., l−3 m+q=n 1 2 2 where β l− 1 → Tor1(Hm(R) ⊗ Hq (L)) → 0 m−1 i fm(t) = i t , m+q=n−1 i=1 is exact. for m>1, and Bernoulli number Bn. (2) The next Künneth Theorem exhibits the isomorphism. Let A be a ring with identity. Let Kummer surface A class of K3 surface L be a complex of left A-modules and R be in abstract algebraic geometry first studied by a complex of right A-modules. Assume that E. Kummer in 1864. It is a quartic surface which for each n the homology modules Hn(R) and has the maximum number (16) of double points cycle modules Cn(R) are projective (in this case possible for a quartic surface in projective three- Tor1(Hm(R) ⊗ Hq (L)) = 0). Then for each n dimensional space. It can be described as the there is an isomorphism α such that the sequence quotient variety of a two-dimensional Abelian ∼ variety by the automorphism subgroup gener- Hm(R) ⊗ Hq (L) = Hn(R ⊗ L). ated by the sign-change automorphism s(x) = m+q=n −x. See also K3 surface. G In projective three-dimensional space, the (3) Let be an Abelian group. For simplicial L R L×R surface given by complexes and and Cartesian product , the homology group Hn(L × R; G) splits into x4 + y4 + z4 + w4 = 0 the direct sum ∼ is a Kummer surface. Hn(L × R; G) = Hm(L) ⊗ Hq (R) m+q=n Künneth’s formula See Künneth Theorem. ⊕ Tor Hm(L) ⊗ Hq (R) . Künneth Theorem The theorem is formu- m+q=n−1 lated for several different areas of mathemat- (4) Similar Künneth formulas and Künneth ics. Occurring in the statement of the theorem Theorems generalized (e.g., to spectral se- are one or more formulas concerning exact se- quences) or stated with other criteria on quences known as Künneth formulas. The theo- the groups (e.g., torsion free groups) have been rem itself is sometimes called Künneth formula. studied. All Künneth-type formulas are related to study- ing the theory of products (e.g., tensor products) k-Weyl group See k-rank. in homology and cohomology for various math- ematical objects and structures. Examples fol- low.

c 2001 by CRC Press LLC (ζ, ) = nj , since p does not divide n. So, k() is generated by (ζ, ).

Lagrangian density It is customary in the L fields of and quantum me- chanics to honor the Italian mathematician J.L. Lagrange (1736–1813) by using his name l-adic coordinate system Let A be an Abel- or the letter L to name a certain expression (in- ian variety of dimension n over a field k of char- volving functions of space and time variables acteristic p ≥ 0. Let l be a prime number, Ql the and their derivatives) used as integrands in varia- l-adic number field, Zl the group of l-adic inte- tional principles describing equations of motion gers, and Pl the direct product of the 2n quotient and numerous field equations describing phys- groups of Ql/Zl. Let Bl(A) denote the group ical phenomena. Thus, a “Lagrangian” occurs of points of A whose order is a power of l.If in the statements of the “principle of least ac- l = p, then Pl is isomorphic to Bl(A). The iso- tion” (first formulated by Euler and Lagrange for morphism yields thel-adic coordinate system of conservative fields and by Hamilton for noncon- Bl(A). servative fields), as well as in Maxwell’s equa- tions of electromagnetic fields, relativity theory, A, B l-adic representation Let be Abelian electron and meson fields, gravitation fields, etc. n m varieties of dimensions and , respectively, In such systems, field equations are regarded as k p ≥ l over a field of characteristic 0. Let be a sets of elements describing a mechanical system p λ : A → B prime number different from and with infinitely many degrees of freedom. a rational homomorphism. Let Bl(A), Bl(B) m Specifically, let  ⊂ R , [t0,t1) denote a denote the group of points of A, respectively B, 1 2 n time interval, and f = (f ,f ,...,f ) :[t0, l γ : B (A) → n whose order is a power of and l t ) −→ R denote a vector valued function. B (B) λ 1 l the homomorphism induced by . Then Let L be an algebraic expression featuring sums, m × n the 2 2 matrix representation (with respect differences, products, and quotients of the func- l γ to the -adic coordinate system) of is called tions of f and their time and space derivatives. l λ the -adic representation of . L may include some distributions. Define Lagrange multiplier To find the extrema of L(t) = L dx , a function f of several variables, subject to the  constraint g = 0, one sets ∇f = λ ·∇g. The m scalar λ is called a Lagrange multiplier. where x = (x1,x2,...,x ) and t k 1 Lagrange resolvent Let be a field of char- V(f) = L(t) dt . acteristic p containing the nth roots of unity such t0 that p does not divide n. Let k() be an exten- Field equations are derived from considering sion field of degree n over k with cyclic Galois variational problems for V(f). In such a setup, group. Let σ be a generator of the Galois group. L is called the Lagrangian and L the Lagrangian Let ζ be an nth root of unity. Then the Lagrange density. resolvent, denoted by (ζ, ) is defined by n−1 Lanczos method of finding roots A proce- (ζ, ) = 0 + ζ1 +···+ ζ n−1 dure (or attitude) used in the numerical solu- j where j = σ , for each j = 1, 2,...,n−1 tion (especially manual) of algebraic equations n and n = σ  = 0 = . whose principal purpose is to find a first ap- Since σj = j+1, for each j, the La- proximation of a root. Then methods (espe- grange resolvent has the property that σ(ζ,) cially Newton’s method) are used to improve = ζ −1(ζ, ). This implies that σ(ζ,)n = accuracy (by iteration). For equations with real n n (ζ, ) , meaning that (ζ, ) ∈ K. The j roots, the basic idea is to transform the equation −j can be determined from the equations ζ ζ into one which has a root between 0 and 1; then

c 2001 by CRC Press LLC reduce the order of the equation by using an ap- Lanczos method of matrix transformation proximating function (e.g., a Chebychev poly- In the numerical solution of eigenvalue prob- nomial); solve the reduced associated equation lems of linear differential and integral operators, exactly for a root between 0 and 1 and convert matrix transformations play a fundamental role. the root found back to a root of the original equa- Lanczos’ method, named after C. Lanczos who tion by using the transformation functions. For introduced it in 1950, was developed as an iter- complex roots the procedure is similar but finds ative method for a nonsymmetric matrix A and a root of largest modulus. involves a series of similarity transformations As an explicit use of the ideas consider a to reduce the matrix A to a tri-diagonal matrix cubic equation with real coefficients a, b, c, d T with the same (but more easily calculable) (with ad = 0) eigenvalues as A. Specifically, given an n × n matrix A, a tri- ax3 + bx2 + cx + d = 0 . diagonal matrix T is constructed so that T = S−1AS. Assume that the n columns of S are First change the polynomial to a monic polyno- denoted by x1,x2,...,xn which will be deter- mial mined so as to be linearly independent. Assume 3 2 x + b1x + c1x + d1 = 0 that the main diagonal of matrix T ={tij } is de- noted by tii = bi for i = 1, 2,...,n; the super- where b1 = b/a, c1 = c/a, d1 = d/a.If diagonal tii+1 = ci for i = 1, 2,...,n− 1; and d1 > 0, transform the equation by replacing x the subdiagonal tii− = di, for i = 2, 3,...,n. everywhere by −x obtaining 1 In this method the di will all be 1. The method x3 + b x2 + c x − d = , consists of constructing a sequence {yi} of vec- 2 2 2 0 T tors such that yi xi = 0, if i = j. Let x ,y ,c be zero vectors and let x and with d2 > 0, where b2 =−b1,c2 = c1,d2 = 0 0 0 1 y be chosen arbitrarily. Define {xk} and {yk} d1.√ Now, by dividing√ by d2 and setting√ y = 1 3 3 3 2 recursively, by x/ d2,b3 = b2/ d2,c3 = c2/( d2) , one gets the equation xk+1 = Axk − bkxk − ck−1xk−1 , 3 2 y = AT y − b y − c y , y + b3y + c3y − 1 = 0 . k+1 k k k k−1 k−1

Now f(0)<0. Also, for large positive y, with yT Ax f(y) > 0. If f(1)>0, there is a root be- b = k k k T tween 0 and 1. If f(1)<0, there is a root > 1. yk xk t = /y In this case substitute 1 into the equation and T obtaining y Axk c = k−1 . k−1 T 3 2 yk− xk−1 t − c3t − b3t − 1 = 0 . 1 R Assuming that y xj = 0 for each j, it can be 3 3 j In either case, substitute for t or y the quadrat- T shown that y xj = 0ifi = j, the {xi} are ic term of the Chebychev polynomial i linearly independent, and that if the recursions are used for k = n, that the resulting xn+ = 0. t3 = (1/32) 48t2 − 18t + 1 1 These results lead to the equations and solve the resulting quadratic equation ex- Ax1 = x2 + b1x1 actly. Discard any negative root and use the remaining root. Convert it back to the origi- Axk = xk+1 + bkxk + ck−1xk−1 , nal equation using all the conversions. This is for k = 2, 3,...,n− 1 and the Lanczos method for cubic equations. There are Lanczos methods for fourth and higher de- Axn = bnxn + cn−1xn−1 gree equations with real coefficients and ones for complex coefficients. which yield the desired tri-diagonal matrix T .

c 2001 by CRC Press LLC Problems with the method involve a judicious chosen from the n numbers 1, 2,...,n. Then T choice of x1 and y1 so that yj xj = 0 for each the Laplace Expansion Theorem says: j = 1, 2,...,n−1. Numerically, the weakness λRC of the method is due to a possible breakdown in (−1) det (ARC) det (AS C ) = det(A) this biorthogonalization of the sequences {xi} C and {yi} even when the matrix A is well condi- if R = S (and =0, if R = S), where the sum is tioned. The numerical procedure is likely to be n T taken over the r selection of columns C. Also, unstable, for example, when y xi is small. i For symmetric matrices with x1 = y1 the λRC (−1) det (ARC) det (AR S ) = det(A) {xi} {yi} sequences and are identical. In this case R the numerical stability is comparable to either Givens’ or Householder’s methods resulting in if R = S (and = 0, if R = S), where the n the same tri-diagonal matrix — but with more sum is taken over the r selection of rows R. cost in deriving it. The resulting in the sum det(A) are Theoretically, the indicated algorithm occurs called Laplace expansions of the determinant. If in the conjugate-gradient method for solving a r = 1, the “usual” expansion of a determinant linear system of equations, in least square poly- is obtained. nomial approximations to experimental data, and as one way to develop the Jordan canonical form. large semigroup algebra Let R be a com- S Lanczos method is sometimes called the mutative ring, S a semigroup, and R the set method of minimized iterations. of sequences of elements of R where in each sequence only a finite number of elements is S Laplace Expansion Theorem A theorem different from zero. Then R is the direct sum concerning the finite sum of certain products of of isomorphic copies of R using S as the index S determinants of submatrices of a given square set. A canonical basis of R consists of {bi}i ∈S matrix. The theorem delineates which products where for a given i ∈ S, bi has the component yield a sum equal to the determinant of the whole indexed by i equal to 1 and the rest of the com- matrix, so the expansions described are com- ponents equal to 0. The product monly used to evaluate determinants. b b = b i, j ∈ S Let A be an n × n matrix with elements i j ij for all from a commutative ring, where it can be as- S determines the multiplication in R and makes sumed that n>1. Let A have elements aij S R into an algebra called a semigroup algebra where i denotes the row and j the column. Let of S over R. Now suppose S has the property 1 ≤ r

c 2001 by CRC Press LLC that every nonempty set of left ideals contains a lattice group (1) The subgroup of transla- minimal element (with respect to inclusion), the tions of the n-dimensional crystallographic Jacobson radical is the largest nilpotent ideal. group C defined on n-dimensional Euclidean space En. The lattice group is a commutative lattice (1) A system consisting of a set S normal subgroup of C. Recall that C is a discrete together with a partial ordering ≤ on S and two subgroup of the group of motions on En that con- binary operations ∧ and ∨ defined on S such tains exactly n linearly independent translations. that, for all a,b,c ∈ S, The lattice group is generated by these transla- tions. In applied areas of crystallography, the a ∨ b ≤ c (a ≤ c b ≤ c) , if and only if and group of motions is called a space group. Then and the lattice group is called the lattice of the space group. c ≤ a ∧ b if and only if (c ≤ a and c ≤ b) . (2) Let L be an n-dimensional homogeneous lattice in En.(See lattice.) Then The element a ∨ b is called the supremum n of a and b and the element a ∧ b is called the v ∈ V : v = σ(P), for some P ∈ L infimum of a and b. L The set of all subsets of a nonempty set S is called the lattice group of . This lattice ordered by the subset relation together with the group is a free module generated by the basis {v ,v ,...,v } operations of set union and set intersection form 1 2 n . a lattice. The whole numbers, ordered by the lattice ordered group A lattice which is also relation divides, together with the operations of an ordered group. See lattice. greatest common divisor and least common mul- tiple, form a lattice. The real numbers, ordered law A property, statement, rule, or theorem in by the relation less than or equal to, together a mathematical theory usually considered to be with the operations of maximum and minimum, fundamental or intrinsic to the theory or govern- form a lattice. ing the objects of the theory. Elementary exam- (2) Let Rn denote the set of all n-tuples of ples include commutative, associative, distribu- real numbers. Let Rn be endowed with the tive, and trichotomy laws of arithmetic, (usual) Euclidean metric space structure. Call DeMorgan’s laws in mathematical logic or set this metric space En. Let V n denote the (usual) theory, and laws of , cosines, or tangents in n-dimensional real vector space defined over R trigonometry. by adding componentwise and multiplying by scalars componentwise. Let Z denote the set of Law of Quadrants Also called rule of spe- σ En integers. Let denote the mapping between cies. In , if ABC is a right V n En and that identifies points in with vectors spherical triangle, with right angle C and sides V n L in . Let be the set of points a,b, and c (measured in terms of the angle at    n  the center of the sphere subtended by the side), n a A P ∈ E : σ(P) = αj vj ,αj ∈ Z , then: (i.) if and are both acute or both ob-   tuse angles (said to be of like species), then so j=1 are b and B; (ii.) if c<90◦, then a and b are of n ◦ for some basis {v1,v2,...,vn} of V . Then L like species; (iii.) if c>90 , then a and b are is called an n-dimensional homogeneous lattice of unlike species. in En. This law is used whenever one has a formula which gives two possible values for the sine of lattice constant Let L denote the lattice group the side or angle, since it indicates whether the of an n-dimensional crystallographic group de- side or angle is acute or obtuse. fined on n-dimensional Euclidean space. Then L is generated by n translations. Each inner Law of Signs In arithmetic, a rule for sim- product of two of the generating translations is plifying expressions where two (or more) plus called a Lattice constant. See lattice group. (“+”) or minus (“−”) signs appear together. The

c 2001 by CRC Press LLC rule is to change two occurrences of the same lengths of sides of a triangle and sines of its sign to a “+” and to change two occurrences of angles. For a triangle ABC in plane trigonom- opposite signs to a “−.” etry, where A, B, and C represent vertices (and In using the Law of Signs, one is ignoring the angles) and a,b, and c represent, respectively, different usages of the minus sign; namely, for the sides opposite the angles, subtraction, for additive inverse, and to denote a b c “negative.” The same is true for the plus sign; = = . namely, for addition and to denote “positive.” sin(A) sin(B) sin(C) Thus, x − (−y) or x −−y becomes x + y; x + (−y) or x +−y becomes x − y;4− −3 In spherical trigonometry, using the same becomes 4+3; 4+ −3 becomes 4−3; x +(+y) designations for the angles and for the sides or x ++y becomes x + y, etc. (measured in terms of the angle at the center of the sphere subtended by the side), In plane trigonometry, a ra- sin(a) sin(b) sin(c) tio relationship between the lengths of sides of = = . (A) (B) (C) a triangle and tangents of its angles. For a trian- sin sin sin gle ABC, where A, B, and C represent vertices a,b c (and angles) and , and represent, respec- leading coefficient The nonzero coefficient tively, the sides opposite the angles, of the highest order term of a polynomial in a a − b tan 1 (A − B) polynomial ring. In the expression = 2 . a + b 1 (A + B) n n−1 tan 2 anx + an−1x +···+a1x + a0 ,

an is the leading coefficient. There are situations Laws of Cosines In plane and spherical trigo- when it is appropriate to consider an expression nometry, relationships between the lengths of when the leading coefficient is zero. In this case, sides of a triangle and cosines of its angles. For such a coefficient is called a formal leading co- a triangle ABC in plane trigonometry, where efficient. A, B, and C represent vertices (and angles) and a,b, and c represent, respectively, the sides op- least common denominator In arithmetic, posite the angles, the least common multiple of the denominators c2 = a2 + b2 − 2 ab cos(C) . of a set of rational numbers. See least common multiple. Here, the length of a side is determined by the lengths of the other two sides and the cosine of least common multiple (1) In number the- the angle included between those sides. ory, if a and b are two integers, a least common In spherical trigonometry, using the above multiple of a and b, denoted by lcm{a,b},isa designations for the angles and for the sides positive integer c such that a and b are each di- (measured in terms of the angle at the center visors of c (which makes c a common multiple of the sphere subtended by the side), there are of a and b), and such that c divides any other two relationships: common multiple of a and b. The product of the least common multiple and positive greatest (C) =− (A) (B) cos cos cos common divisor of two positive integers a and + sin(A) sin(B) cos(c) b is equal to the product of a and b. The concept can be extended to a set of more and than two nonzero integers. Also, the definition cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C) . can be extended to a ring where a least common multiple of a finite set of nonzero elements of the ring is an element of the ring which is a common Laws of Sines In plane and spherical trigo- multiple of elements in the set and which is also nometry, the ratio relationship between the a factor of any other common multiple.

c 2001 by CRC Press LLC (2) In ring theory, the ideal generated by the be a continuous map. For each nonnegative in- intersection of two ideals is known as the least teger k, f induces a homomorphism fk of the common multiple of the ideals. homology group Hk(X; Q) (with coefficients in In a polynomial ring k[x1,...,xr ] over a field the rational number field Q). If the ranks of the k, the least common multiple of two monomials homology groups (considered as vector spaces m m n n x 1 ·····x r x 1 ·····x r 1 r and 1 r is defined as over Q) are finite, then there is a matrix repre- (m ,n ) (m ,n ) xmax 1 1 ·····xmax r r sentation for each fk and a trace tk of the matrix 1 r . which is an invariant of fk. The Lefschetz num- least upper bound See supremum. ber of f , denoted by L(f ), is then given by ∞ Lefschetz Duality Theorem Let X be a com- k L(f ) = (−1) tk . pact n-dimensional manifold with boundary X˙ k= and orientation U over the ring R. For all R- 0 modules G and non-negative integers q there If everything is well defined, then L(f ) is an in- exist isomorphisms teger and, if different from zero, guarantees that n−q f has a fixed point. Everything will be well de- Hq (X; G) ← Hq (X \ X˙ ; G) → H (X, X˙ ; G) fined, for example if X is a finite complex. Alter- natives for X include a chain or cochain complex and of free Abelian groups (f an endomorphism of degree 0) or a finite polyhedron of degree n with ˙ n−q ˙ n−q Hq (X, X; G) → H (X\X; G) ← H (X; G) . integral coefficients (f continuous, sum from 0 to n). There is a similar result if X is a closed orientable manifold. Also, if f is the identity j : X \ X˙ ⊂ X [Here .] map, then L(f ) is the Euler characteristic. (2) For complex normal Abelian varieties, the Lefschetz fixed-point formula One of the l difference between the second Betti number and main properties of the -adic cohomology. It the Picard number is sometimes called a Lef- counts the number of fixed points for a mor- schetz number. phism on an algebraic variety or scheme. X Let be a smooth and proper scheme of fi- Lefschetz pencil Let X be a smooth com- n nite type of dimension defined over an alge- plete (thus projective) algebraic variety over an k p ≥ braically closed field of characteristic 0. infinite algebraically closed field k. In the the- l p Let be a prime number different from . Let ory of algebraic varieties it can be shown that Q l l be the quotient field of the ring of -adic there is a birational morphism integers. For each nonnegative integer i, let i H (X, Ql) denote the l-adic cohomology of X. π : X˜ −→ X Let f : X −→ X be a morphism that has iso- ˜ lated fixed points each of multiplicity one. Let from a smooth complete (projective) variety X f ∗ be the induced map on the cohomology of X. and a mapping Let Tr be the trace mapping. Then the number f : X˜ −→ P1 of fixed points of f is equal to k

n that is singular at most a finite number of points. 2 − i ∗ i An inverse image f 1(x) is called a fiber and (−1) Tr f ; H (X, Ql) . contains at most one singular point which, if it i=0 exists, is an ordinary double point. The family of fibers is called a Lefschetz pencil of X. Some- Lefschetz number (1) A concept applica- times the map f is called a Lefschetz pencil. In ble in several fields of mathematics, first intro- the theory establishing the existence of X˜ ,itis duced in 1923 by S. Lefschetz (1884–1972) after shown that fibers are hyperplane sections of X. whom it is named. The idea is as follows. Let Sometimes it is said that Lefschetz pencils “fiber X be a topological space. Let f : X → X a variety.”

c 2001 by CRC Press LLC Lefschetz pencils are involved in the proof of Artinian rings are named after E. Artin (1898– the Weil conjecture. They are named in honor of 1962). S. Lefschetz (1884–1972) who studied, among many other things, nonsingular projective sur- left balanced functor Let R1, R2, and R be C C C faces. rings. Let R1 , R2 , and R be categories of R1-modules, R2-modules, and R-modules, re- A A T : C × C → C left A-module Let be a ring. A left - spectively. Let R1 R2 R be an module is a commutative group G, together with additive functor that is covariant in the first vari- a mapping from A×G into G, called scalar mul- able and contravariant in the second variable. A ∈ C T : C −→ C tiplication and denoted here by juxtaposition, Let R1 . Let A R2 R be the satisfying for all x,y ∈ G and all r, s ∈ A: functor defined by

x = x, T(B)= T(A,B) B ∈ C . 1 A for all R2 B ∈ C T : C −→ C r(x + y) = rx + ry , Let R2 . Let B R1 R be the functor defined by (r + s)x = rx + sx , TB (A) = T(A,B)for all A ∈ CR . and 1 (rs)x = r(sx) . Then T is called left balanced if (i.) AT is exact for each projective module A ∈ CR and (ii.) TB Right A-modules are defined similarly. 1 is exact for each injective module B ∈ CR . Sometimes the term left or right is omitted. If 2 The definition can be extended to additive the ring is denoted by R, then the terminology functors T of several variables (some of which is left R-module. If addition and scalar multipli- are covariant and some contravariant) as fol- cation are just the ring operations, then the ring lows. T is called left balanced if: (i.) T be- itself can be regarded as a left A-module. If the comes an exact functor of the remaining vari- ring is a field, then the modules are the vector ables whenever any one of the covariant vari- spaces over the field. ables is replaced by a projective module; and (ii.) T left annihilator The left annihilator of an becomes an exact functor of the remaining ideal S of an algebra A (over a field k)isaset variables whenever any one of the contravariant variables is replaced by an injective module. {a ∈ A : aS = 0} . left coset Any set, denoted by aS, of all left The left annihilator is an ideal of A. Similarly, multiples as of the elements s of a subgroup S the left annihilator of S in R where S is a subset of a group G and a fixed element a of G. Every of an R-module M (where R is a ring) is the set left coset of S has the same cardinality as S.For each subgroup S of G, the group G is partitioned {r ∈ R : rS = 0} . into its left cosets, so that for a finite group the number of elements in the group (order) is a multiple of the order of each of its subgroups. left Artinian ring A ring having the property that (considered as a left module over itself) ev- left derived functor Let R1, R2, and R be C C C R ery nonempty set of submodules (meaning left rings. Let R1 , R2 , and R be categories of 1- ideals in this context) has a minimal element. modules, R2-modules, and R-modules, respec- T : C × C → C For such a ring, any descending chain of left tively. Let R1 R2 R be an additive ideals is finite. Compare with left Noetherian functor that is covariant in the first variable and ring. contravariant in the second variable. Let X be a For a left Artinian ring, the radical is the definite projective resolution for each module A C Y largest nilpotent ideal. See largest nilpotent of R1 and let be a definite injective resolution B C ideal. For a left Artinian ring, any left module for each module of R2 . From homology the- is Artinian if and only if it is Noetherian. ory, T(X,Y) is a well-defined complex which

c 2001 by CRC Press LLC can be regarded as being over T(A,B). Also, and because of the homotopies involved, up to natu- p(e, x) = x. ral isomorphisms, T(X,Y)is independent of the Then S is called a left G-set and G is said to act chosen resolutions X and Y and depends only on on S from the left. If juxtaposition ax is used A and B. Therefore, the nth homology modules for the value p(a, x), then the two conditions Hn(T (X, Y )) are functions of A and B.   become (ab)x = a(bx), ex = x. If f : A −→ A and g : B −→ B, there exist chain transformations F : X −→ X and left hereditary ring A ring in which every G : Y  −→ Y over f and g. Any two such are left ideal is projective. Alternately, a ring whose homotopic. The induced chain transformation left global dimension is less than or equal to one. T(f,g): T(X,Y)−→ T(X,Y) See left global dimension. If a ring A is left hereditary, every submodule is determined up to homotopy, is independent of a free A-module is a direct sum of modules of the choice of F and G, and depends only on isomorphic to left ideals of A. The converse is f and g. This yields a well-defined homomor- also true. phism left ideal A nonempty subset L of a ring R   Hn(T (X, Y )) −→ Hn(T (X ,Y )) . such that whenever x and y are in L and r is in R, x−y and rx are in L. For example, if R is the n L T : Define the th left derived functor n ring of n × n real matrices, the matrices whose CR × CR → CR 1 2 by first column is identically zero is a left ideal. L T(A,B)= H (T (X, Y )) Ideals play a role in ring theory analogous to n n the role played by normal subgroups in group theory. and let LnT(f,g) be the well-defined homo- morphism of the last paragraph left invariant Used in various fields to indi-   LnT(f,g): LnT(A,B)−→ LnT(A,B ). cate not being altered or changed by a transfor- mation, usually a left translation. For example, (in linear algebra) a subalgebra S of a linear al- left global dimension For a ring A with unity, gebra A is called left invariant if S contains, for the supremum of the set of projective (or homo- any x ∈ S, all left multiples ax for each a ∈ A. logical) dimensions (i.e., the set of minima of If the linear algebra A has a unity element, then the lengths of left projective resolutions) of el- the left invariant subalgebra is a left ideal. In Lie ements of the category of left A-modules. It group theory and transformation group theory is also equal to the infimum of the set of in- there are left invariant measures, left invariant jective dimensions of the category. A theorem integrals, left invariant densities, left invariant of Auslander (1955) shows that the left global tensor fields, etc. dimension is also the supremum of the set of projective dimensions of finitely generated left left inverse element Let S be a nonempty set A-modules. As simple examples of left global on which is defined an associative binary oper- dimension, the left global dimension of the in- ation, say ∗. Assume that there is a left identity tegers is 1, while the left global dimension of a element e in the set with respect to ∗. Let a de- field is 0. See projective dimension, homologi- note any element of the set S. Then a left inverse cal dimension, injective dimension. element of a is an element b of S such that left G-set Let G be a group with identity e b ∗ a = e. and juxtaposition denoting the group operation. Let S be a set. Let p : G × S → S satisfy, for If every element of the set described above each x ∈ S, has a left inverse element, then the set is called a group. For a group, a left inverse element is p(ab, x) = p(a, p(b, x)) unique and is also a right inverse element. For

c 2001 by CRC Press LLC commutative groups, the inverse element of a is left resolution Let A be a ring with unit and usually denoted by −a. Otherwise, the notation M an A-module. If the sequence a−1 is used. An element that has a left inverse ∂ ∂ ∂ < left invertible. n n−1 1 is sometimes called ···→Xn → Xn−1 → ···→ X0 → M → 0 left Noetherian ring A ring having the prop- is exact, where X is a positive chain complex erty that (considered as a left module over itself) of A-modules Xk, then the homomorphisms ∂k every nonempty set of submodules (meaning left have the property that the composition ∂k∂k+1 = ideals in this context) has a maximal element. 0 for all k, and < : X → M is a sequence For such a ring, any ascending chain of left ide- of homomorphisms

c 2001 by CRC Press LLC A, A ∈ C g : A → A T is exact. Let R1 . Let . If is covariant, Then there are exact sequences Sn+1T (A) = SnT (M)), n ≥ 1 . β → M →α P → A → 0 0 If T is contravariant, ↓ g α β S T (A) = S T (N)), n ≥ . 0 → M → P  → A → 0 n+1 n 1

 Left satellites may also be defined for general (with P and P projective) and a homomorphism   Abelian categories in a similar fashion. f : P → P such that gβ = β f . f defines uniquely a homomorphism f  : M → M such left semihereditary ring A ring in which that fα = αf . One then gets the commutative every finitely generated left ideal is projective. diagram Compare with left hereditary ring. T(α) Every regular ring is left semihereditary (and T(M) −→ T(P) right semihereditary). ↓ T(f) ↓ T(f) T(α) T(M) −→ T(P). left translation A left translation of a group G by an element a of G is a function fa : G → T(f) induces a homomorphism G defined by

 f (x) = ax , (g) : Ker(T (α)) → Ker(T (α )) . a x ∈ G It can be shown that (g) is independent of the for each . choice of f . Define Lehmer’s method of finding roots A meth-  od, developed in 1960 by D.H. Lehmer, for find- S1T(g): S1(A) −→ S1(A ) ing numerical approximations to the (real or com- by S1T(g) = (g). With this definition, S1T plex, simple or multiple) roots of a polynomial becomes a covariant functor called the left satel- using a digital computer. Lehmer’s method pro- lite of T . vides a single algorithm for automatic computa- (ii.)IfT is contravariant, the procedure starts tion applicable for all polynomials, in contrast to with an exact sequence other methods, the choice of which, for a partic- ular polynomial, is dependent on human judg- β α ment, experience, and intervention. 0 → A → Q → N → 0 , Lehmer’s method includes a test for deter- with Q injective. Similar reasoning to the co- mining whether or not the given polynomial has a root inside a given circle. The test is applied variant case yields S1T (A) = Ker(T (α)) where  at each iteration of the process (to be described T(α): T(N)−→ T (Q) and S1T(g): S1T(A) next) on circles of decreasing size. −→ S1T (A). Here, S1T is contravariant. (iii.) The sequence of function SnT is defined Assume that zero is not a root. Starting (in recursively from the exact sequences the complex plane) with the circle with center 0 and radius 1 apply the test to look for a root 0 → M → P → A → 0 inside the circle. If one is found, halve the radius and retest for the smaller circle. If a root is not and found, double the radius and apply the test. In a 0 → A → Q → N → 0 , finite number of steps, an annulus with P projective and Q injective as follows: {x : R<|z| < 2R}

S0T = T, will be determined (where R is a power of 2) that contains a root in the disk of radius 2R but not S1T as given above . in the disk of radius R. Next, cover the annulus

c 2001 by CRC Press LLC with 8 overlapping disks of radius 5R/6 and cen- (1) Let A be an Abelian variety of dimension ters (5R/3) exp(2πi/8), for k = 0, 1,...,7. n over a field k of characteristic p ≥ 0. Let m be Test each of the 8 circles until a root is found a positive integer that is not a multiple of p.An trapped in a new annulus centered at (say) α m-level structure with respect to A is a set of 2n with radii R1, 2R1 points on A which form a basis for the abstract group Bm(A) of points of order m on A. {x : R < |z − α| < R } , 1 2 1 (2) For polarized complex Abelian varieties, 5R the concept of a level structure can be consid- with R = s for some s. The new annulus 1 6(2 ) ered as a replacement of all or part of the concept is similarly covered by 8 disks and tested. After of a symplectic basis. If the variety is of type (say) k iterations of this process, one gets a circle k D, then the D-level structure for the variety is a of radius ≤ 2(5/12) which contains a root. One certain symplectic isomorphism. continues until a prescribed accuracy is reached. The strength of Lehmer’s method is its uni- There are generalized level m-structures, or- versality. The weakness is the rate of conver- thogonal level D structures, etc. The precise gence of the method which in practice may be definitions are quite detailed and require consid- several times slower than less general methods erable background and notational development. developed for specific types of roots. Thus, they will not be given here. The associ- ated modulii spaces for polarized Abelian vari- length of module The common number of eties with level structures are used in studying quotients Mi/Mi+1 of submodules of a module properties of geometry and arithmetic. M over a ring R in a Jordan-Hölder or compo- sition series Levi decomposition Let g be a finite dimen- sional Lie algebra over the field of real or com- M = M0 ⊃ M1 ⊃···⊃ Ms ={0} plex numbers. Let radg denote the Lie subal- gebra of g, called the radical of g.(See radi- M M of ,if has such a series. Any such chain cal.) Let l be any subalgebra of g. Then the has the same number of terms. A necessary and direct sum of vector spaces radg and l (radg ⊕l) M sufficient condition that have such a series forms a Lie subalgebra of g.Ifl exists such that is that it be both Artinian and Noetherian. See radg ⊕ l = g, then the sum radg ⊕ l is called composition series, Artinian module, Noethe- the Levi decomposition of g. The decomposi- rian module. tion is named after E.E. Levi. The subalgebra l involved in the splitting is called a Levi sub- Leopoldt’s conjecture The assertion in the algebra of g. A theorem of Levi (1905) states study of algebraic number fields that the Zp- that g has such a decomposition and a theorem p rank of the -adic closure of the group of units by Malcev (1942) establishes the uniqueness of of a number field is the same as the Z-rank of the the decomposition. group of units. This conjecture is an open ques- tion in many cases. There are Abelian analogs There is a similar Levi (product) decompo- G = of the conjecture and analogs for function fields sition in algebraic group theory given by G×H G G of characteristic p. The conjecture is named af- Rad where is an algebraic group, Rad G H ter H.W. Leopoldt, an extensive contributor to is a unipotent radical of , and is an alge- G the study of p-adic L-functions. braic subgroup of called the reductive Levi subgroup of G. It is useful in reducing many less than See greater than. problems to the study of reductive groups. less than or equal to See greater than. Levi subgroup Let G be a Lie group. Let RadG denote the radical of G. A Lie subgroup level structure A notion occurring in the L of G is called a Levi subgroup of G if (i.) the study of modulii spaces of Abelian varieties. identity imbedding of L into G is a Lie group Usually a letter intrinsic to the variety discussed homomorphism, (ii.) G = (RadG) L, (iii.) the is appended to the term. dimension of (RadG) ∩ L is zero. If the Lie

c 2001 by CRC Press LLC group is connected, there is always a (connected) finite fields, representations of Galois groups, Levi subgroup. See Levi decomposition. p-adic number fields, etc. (1) The most direct generalization of the Rie- lexicographic linear ordering Sometimes mann ζ-function is the Dirichlet L-function. Let called dictionary order, an ordering which treats Z denote the integers. Let m ∈ Z with m>0. numbers as if they were letters in a dictionary. Let χ : Z −→ C be a Dirichlet character mod- Suppose, for example, S is a set of monomials ulo m (i.e., χ = 0, χ(a) = 0ifa and m are of degree n in m variables: that is, not relatively prime, χ(ab) = χ(a)χ(b), and   χ(a + m) = χ(a)). Let s ∈ C with Re(s) > 1.  m  k k Define L(s) as S = x 1 x 2 ...xkm : k = n . 1 2 m j   ∞ j=1 χ(n) L(s) = . s k k k n x 1 x 2 ...x m ≤ n=1 then 1 2 m is less than or equal to ( ) l l l x 1 x 2 ...x m 1 2 m if the first nonzero difference of The definition exhibits the Dirichlet series ex- k − l k − corresponding exponents j j satisfies j pansion which converges absolutely and makes l ≤ j 0. For example, L a holomorphic function for #(s) > 1. It is equal to the Euler product x x x2x3 ≤ x x x3x ≤ x2x4x x . 1 2 3 4 1 2 3 4 1 2 3 4 1 A lexicographic ordering for the field of com- χ(p) p 1 − ps plex numbers C is defined as follows: if z1 = x + y i, z = x + y i ∈ C, then z ≤ z if 1 1 2 2 2 1 2 over primes p. It can be extended as a mero- and only if morphic function over C.Ifχ is defined to be identically 1, the Riemann ζ -function itself is (x1 ≤ x2) or (x1 = x2 and y1 ≤ y2) . obtained. The two functions have many sim- Since in both of these cases the order ≤ is re- ilar properties. For example, the study of the flexive, antisymmetric, transitive, and satisfies location of zeroes of the L-function leads to a the trichotomy law, the order is a linear order, so generalized Riemann hypothesis. the order is called a lexicographic linear order. (2) Hecke L-functions and Hecke L-functions O ,O ,...,O More generally, if 1 2 m are ordered with grössencharakters are generalizations of p = m O L sets, the product set j=1 j is given a the Dirichlet -functions to algebraic number lexicographic ordering < as follows: if a = fields. Other L-functions that can be considered (a1,a2,...,am) and b = (b1,b2,...,bm) ∈ generalizations of these include those of Artin p, then a

c 2001 by CRC Press LLC [a,b],x ∈ Kn,y ∈ Kn we have The fact that the three Lie algebras defined above are isomorphic allows some flexibility in $f(t,x)− f(t,y)$≤L$x − y$ . applications of the theory.

In such a circumstance, the initial value problem Lie group A group G, together with the struc- dx ture of a differentiable manifold over the real or = f (t, x), x(a) = c dt complex number field, such that the functions f : G × G → G and g : G → G defined − has a unique solution for all c ∈ Kn. by f(x,y) = xy and g(x) = x 1 are differen- tiable. Lie algebra A theory named after the Norwe- The additive groups of the fields of real or gian mathematician M.S. Lie (1842–1899) who complex numbers are Lie groups. The group of pioneered the study of what are now called Lie invertible n×n matrices over the real or complex Groups. number fields is a Lie group. Over a commutative ring with unit K,aLie In addition to providing a thriving branch of algebra is a K-module together with a K-module mathematics, Lie groups are used in many mod- homomorphism x ⊗ y →[x,y] of L ⊗ L into ern physical theories including that of gravita- L such that, for all x,y,z ∈ L, tion and quantum mechanics.

[x,x]=0 Lie-Kolchin Theorem Let V be a finite di- mensional vector space over a field k. Let G and be a connected solvable linear algebraic group. Let GL(V ) be the general linear group. Let [x,[y,z]]+[y,[z, x]]+[z, [x,y]] = 0 . f : G → GL(V ) be a linear representation. f (G) The product [x,y] is called a bracket product Then can be put into triangular form. and is an alternating bilinear function. The sec- Alternate conclusions in the literature include f (G) ond equation is called the Jacobi identity. the following: (i.) Then leaves a flag in V An example of a Lie Algebra is obtained for invariant. (ii.) Then there exists a basis in V f (G) an associative algebra L over K by defining in which elements of can be written as triangular matrices. [x,y]=xy − yx . This theorem has been generalized by Mal’tsev.

Lie algebra of Lie group There are three Lie’s Theorem There are several theorems Lie algebras that are called a Lie algebra of a which bear the name and honor the Norwe- Lie group G.(See Lie algebra.) gian mathematician M.S. Lie (1842–1899) who, (1) The Lie algebra of all left invariant deriva- among other things, founded the theory of Lie tions on G with the bracket product [D1,D2] groups. For solvable Lie algebras, the follow- defined for two derivations D!,D2 as D1D2 − ing version of Lie’s Theorem is considered an D2D1. important tool. (2) The Lie algebra of all left invariant vector The irreducible representations of a finite di- fields on G with the bracket product of two left mensional, solvable Lie algebra over an algebra- invariant vector fields X and Y defined using the ically closed field of characteristic zero all are natural isomorphism between the vector space one-dimensional. of left invariant vector fields and the vector space of left derivations of G. Lie subalgebra Let K be a commutative ring (3) The Lie algebra of all tangent vectors to with unit. A Lie subalgebra is a subalgebra of G at e with the bracket product between tangent a Lie algebra (meaning a K-submodule that is vectors defined using the natural isomorphism closed under the bracket product of the Lie al- between the space of left invariant vector fields gebra). A Lie subalgebra is a Lie algebra. See of G and the space of tangent vectors of G at e. Lie algebra.

c 2001 by CRC Press LLC Lie subgroup A subset of a Lie group G scribed as the set of all points (x, y) such that which is both a subgroup of G and a submani- fold of G (with both structures being induced by x = x1 if x2 = x1 those of G). (See Lie group.) As an example, an open subset which is also a subgroup of a Lie and group is a Lie subgroup. y2 − y1 y − y1 = (x − x1) if x2 = x1 . x2 − x1 like terms In an algebraic expression in sev- eral variables, terms with identical factors raised The equation is called the two point form of the to identical powers (excluding values of coeffi- equation of a line. Also, a line has a parametric cients). Like terms are usually collected and representation as the set of pairs (x, y) such that, combined using rules of arithmetic and associa- for each real number t, tive, commutative, and distributive laws. (x, y) = (1 − t)(x1,y1) + t (x2,y2) . 3x2y5z and − 2x2y5z (3) In affine geometry, where V is a finite k are like terms. dimensional vector space over a field ,aline is a one-dimensional affine space. If a,b are a b limit ordinal In the theory of ordinal num- distinct points, the line joining and can be bers, a non-zero ordinal which does not have a described as the set of all points of the form predecessor. (1 − t)a + tb, for t ∈ k. line (1) In classical Greek geometry, a line is (4) In projective geometry, a line is a one- an undefined term characterized by axioms and dimensional projective space. understood by intuition. A line is considered to (5) Many areas of mathematics endow a line be straight and to extend without bound. In ev- with properties, resulting in such concepts as eryday usage, the words line and straight line are parallel lines, half line, the real line, broken line, used interchangeably, and although in this con- tangent line, , line of curvature, long text the word straight is somewhat redundant, it line, extended line, normal line, perpendicular is still universally used. (An archaic synonym line, orthogonal line, line method (in numerical is right line.) Curves that can be drawn without analysis), etc. interruption are sometimes called lines. (2) In plane analytic geometry, the graph or linear (1) Like or resembling a line. One set of points described by a linear equation in of the most used terms in mathematics. Many two variables x and y, given by ax + by = c fields of mathematics are separated into linear (where a,b,c are real numbers) is a line. This and non-linear subfields; e.g., the field of dif- equation, or alternately ax+by+c = 0, is called ferential equations. The adjective is also used the general equation of a line. The graph of a to specify a subfield of investigation; e.g., linear real valued function f of a real variable, given algebra, linear topological space, linear (or vec- (for real numbers a,b)byf(x) = ax + b, for tor) space, linear algebraic group, general lin- each real number x, is a non-vertical line. The ear group, etc. Also, the term is used to specify number a is called the slope of the line. With a special subtype of object; e.g., linear combi- b = 0, there is a one-to-one correspondence nation, linear equation, linear function, linear between non-vertical lines and slopes. For a order, linear form, etc. See the nouns modified curve, the word slope is used more generally at for further explanation. a point on the curve as the slope of the tangent (2)IfV,W are vector (linear) spaces over a line to the curve at the point (if it exists). In this field k, a mapping L : V −→ W is called linear context, a line can be described as a curve with if L is additive and homogeneous. Equivalently, a constant slope. L is linear if, for each α, β ∈ k and x,y ∈ V , The line passing through the two points (x1, y1) and (x2,y2) in the Euclidean plane is de- L(αx + βy) = αL(x) + βL(y) .

c 2001 by CRC Press LLC (Note that the same symbolism for the vector a group defined by the first definition and one space operations was used for both vector spaces.) defined by the second. So, the distinction is only important when the word linear or affine linear algebra (1) A set L which is a finite is omitted from the first definition, as is some- dimensional vector (or linear) space over a field times done in textbooks. F such that for each a,b ∈ F and each α, β, γ ∈ L, linear combination Let x1,x2,...,xn be el- ements of a vector space V over a field F .Alin- α(βγ) = (αβ)γ (associative law) ear combination of the elements x1,x2,...,xn over F is an element of the form α(aβ + bγ ) = a(αβ) + b(αγ ) (bilinearity) a x + a x +···+a x (aα + bβ)γ = a(αγ) + b(βγ ) . 1 1 2 2 n n a ,a ,...,a ∈ F (2) A mathematical theory of “linear alge- where 1 2 n . bras.” University mathematics departments usu- Linear combination is a fundamental concept ally offer an undergraduate course where the lin- in the theory of vector spaces and is used, among ear algebras of matrices and linear transforma- other things, to define concepts of linear inde- tions are studied. pendence of vectors, basis, linear span, and the dimension of the vector space. linear algebraic group An abstract group linear difference equation An equation of G, together with the structure of an affine alge- the form braic variety, such that the product and inverse mappings given by a0xn + a1xn−1 +···+akxn−k = bn µ : G × G → G, (x, y) '→ xy where a0,a1,...,ak are scalars and a0ak = 0, is called a linear difference equation of order i : G → G, x '→ x−1 k. The equation is called non-homogeneous if b = b = are morphisms of affine algebraic varieties. Al- n 0 and homogeneous if n 0. Such re- ternately, such an algebraic group is called an cursive equations are important in many fields affine algebraic group. The ground field K is of numerical analysis; e.g., in finding zeros of assumed to be an arbitrary infinite field. polynomials and in determining numerical so- Probably the most important example of a lin- lutions of ordinary and partial differential equa- ear algebraic group is the general linear group tions. Writing the above difference equation in Dx = x − x GLn(K) or GLn(U), where U is a finite di- terms of the operator n n n−1 and its mensional vector space over K. Since GLn(K) powers, one can exploit the analogy with linear is an open subset (with respect to the Zariski differential equations. topology) of the algebra on n × n matrices over K, it inherits the affine variety structure. Also, linear disjoint extension fields Two fields, K and L, that are extensions of another field, k, polynomials on GLn(K) are realized as rational functions in matrix elements with denominators and are themselves contained in a common ex- being powers of determinants. It follows that the tension field are called linearly disjoint if every L k product and inverse mappings are morphisms of subset of that is linearly independent over K affine algebraic varieties. is also linearly independent over . Another definition of a linear algebraic group linear equation Let k be a commutative ring. is a subgroup of the general linear group GLn Let a,b ∈ k. Then an expression of the form (U) which is closed in the Zariski topology, U where is an algebraically closed field that ax + b = 0 has infinite transcendence degree over the prime field K. Since this subgroup is an algebraic is called a linear equation in one variable x. group, it can be called an algebraic linear group Let m ≥ 1. Let k[x1,...,xm] be a polyno- as well. Now, there is an isomorphism between mial ring over k. A linear equation is formed

c 2001 by CRC Press LLC by setting a linear form equal to zero. In other into A such that f = h ◦ φ. The function h is words, a linear equation in m variables is given called the linear extension of f and is unique. by an equation See Jacobian variety. See also canonical func- tion, Abelian variety. a0 + a1x1 +···+amxm = 0 , linear form (1) Also called linear functional. where a1,...,am ∈ k. In analytic geometry, where k is the real num- A linear function or transformation from a linear V F F ber field, a linear equation in m variables de- (vector) space over a field into the field scribes a line if m = 2, a plane if m = 3, and a (where the field is considered to be a linear space hyperplane if m>3. over itself). (2) An expression in n variables x1,...,xn linear equivalence Any one of several simi- of the type lar equivalence relations defined on schemes or a1x1 + a2x2 +···+anxn + b algebraic varieties. Generally speaking, if X is one of these structures and if the concepts of where a1,...an,bare elements of a commuta- divisor and principal divisor are defined, then tive ring and at least one coefficient is not 0. If two divisors are related if their difference is a b = 0, the form is called homogeneous. principal divisor. The relation is an equivalence relation, called a linear equivalence. The equiv- linear fractional function A function f with alence classes form a group (See linear equiva- domain and range either the complex numbers lence class.) or the extended complex numbers and defined More specifically, let X be a complete, non- for numbers a, b, c, d with ad − bc = 0by singular algebraic curve defined over an alge- az + b k X f(z)= . braically closed field . Two divisors on are cz + d said to be linearly equivalent if their difference is a principal divisor. A similar definition can Linear fractional functions are univalent, mero- be made for nonsingular algebraic surfaces. See morphic functions that map each circle and line divisor, principal divisor of functions. in the complex plane to either a circle or a line. More abstractly, let X be a Noetherian in- The class includes rigid motions, translations, tegral separated scheme which is nonsingular rotations, and dilations. It is a subset of the class in codimension one. The divisor d1 is called of rational functions, being those rational func- linearly equivalent to d2 if their difference is a tions of order 1. Using function composition as principal divisor. the binary operation, the set of linear fractional The definition can be extended to general transformations forms a group. They play an schemes by using Cartier divisors. See Cartier important role in the study of the geometric the- divisor. ory of functions. Also called linear fractional transformation, linear equivalence class An equivalence linear transformation, Möbius transformation, class determined by the equivalence relation fractional linear transformation, and conformal called a linear equivalence defined on the di- automorphism. See also linear fractional group. visors of an algebraic variety or a scheme. The equivalence classes form a group which is one of linear fractional group A group of func- the intrinsic invariants of the variety or scheme. tions formed by the set of linear fractional func- See linear equivalence. tions defined on the complex plane or extended complex plane using composition as the group linear extension Let Y be a nonsingular al- operation. The quotient group of the full linear gebraic curve over a field k. Let J denote the group modulo the subgroup of nonzero scalar Jacobian variety of Y . Let φ be the canonical matrices is isomorphic to the linear fractional function on Y (with values in J ). Let A be an group. Both groups are called linear fractional Abelian variety. Then for any function f on Y in the mathematical literature. See linear frac- into A, there exists a homomorphism h from J tional function.

c 2001 by CRC Press LLC linear fractional programming Determin- For real valued functions of a real variable, the ing the numerical solution of the problem of op- graph of a linear function (in this context) is timizing a function resembling the linear frac- a straight line. More properly, such functions tional functions of complex analysis subject to should be called affine, but the usage linear func- linear constraints. Linear programming meth- tion is universal. ods are usually employed in the solution either (3) See also linear fractional function. by a simplex-type method or by using the solu- tion of an associated linear programming prob- linear genus Let X be a nonsingular alge- lem with additional constraints to solve the lin- braic surface. If X has a minimal model M, ear fractional problem. Specifically, let A be an then the linear genus of X is an invariant equal n m × n matrix; let a, c be vectors in R ; let p be to the arithmetic genus of a canonical divisor m a vector in R ; and, let b, c ∈ R. Then a linear of M.IfX is not a ruled surface, then X has fractional programming problem is to minimize a minimal model and thus a linear genus. See minimal model. See also arithmetic genus. a·x + b c·x + d linear inequality In analytic geometry, a lin- m m subject to ear inequality in variables in Euclidean - dimensional space is a linear form set less than, Ax = p and x ≥ 0 . less than or equal to, greater than, or greater than or equal to zero. For example, the inequality linear fractional transformation Classically, a0 + a1x1 +···+ amxm > 0 , a function on the complex plane having the form H(z) =[az+ b]/[cz + d] for complex constants where a0,a1,...,am are real numbers, is a lin- a, b, c, d. There are analogs of this definition ear inequality. on any Euclidean space. If m = 2, the set of pairs of real numbers satisfying a linear inequality constitutes a half- linear function (1) A function L with domain plane bounded by the line given by the associ- a vector space U over a field F and range in a ated linear equation. vector space V over F which is additive and homogeneous. This means, for each x,y ∈ U linear least squares problem A problem and for each α ∈ F , where a linear function is to be found which best approximates a given set of data in the sense that L(x + y) = L(x) + L(y) (additivity) the sum of the squares of the errors in such an approximation is minimized. and

L(αx) = αL(x) (homogeneity) . linearly dependent elements A set of ele- ments that satisfies some linear relation. More The terms linear transformation and linear map- specifically, a set of elements {x1,x2,...,xN } ping are also used to describe a function satisfy- in a vector space is linearly dependent over a ing these conditions and L is sometimes called field F (such as the real or complex number a linear operator. If F is regarded as a one- systems) if there exists α1,...,αN ∈ F , not dimensional vector space over itself and if V = all zero, with α1x1 +···+αN xN = 0. Exam- F , then L is called a linear functional or linear ples include a set of two co-linear vectors or a form. set of three co-planar vectors. (2) If a function f has the real or complex numbers as its domain and range, it is called linearly dependent function A function linear if for some constants a,b, a = 0, and which depends linearly on its input variables. for each x, More precisely, a function f(x1,...,xN ) is a linearly dependent function if f(x1,...,xN ) = f(x)= ax + b. α1x1 +···+αN xN , where each αi belongs to

c 2001 by CRC Press LLC some given field, such as the real or complex linear representation If G is a group, a linear number system. representation of G is a group homomorphism of G into the invertible linear transformations linearly independent elements A set of el- on some vector space. ements which is not linearly dependent. More L (q) precisely, a set of elements {x ,x ,...,xN } in linear simple group Denoted n , where 1 2 q a vector space is linearly independent over a is a non-zero power of a prime number, it field F (such as the real or complex number sys- is one of the classical finite simple groups. It tems) if, whenever α x +···+αN xN = 0 for may be identified as the projective special linear 1 1 q α ,...,αN ∈ F , then necessarily all the αi must group over a finite field of order , and denoted 1 (n, q) d be zero. Examples include a set of two vectors PSL . Let be the greatest common divi- n q− which are not co-linear or a set of three vectors sor of and 1. The possible orders of a linear qn(n−1)/2( n (qi − ))/d which are not co-planar. simple group are i=2 1 ; where n ≥ 3, or n = 2 and q ≥ 4. linear mapping A function from one vec- linear stationary iterative process A meth- tor space to another L : V '→ W, which pre- od of using successive approximation to solve serves the vector space operations, i.e., L(r v + 1 1 the system of linear equations Ax = b when r v ) = r L(v )+r L(v ) for all vectors v ,v 2 2 1 1 2 2 1 2 A is a non-singular matrix. A method is called ∈ V and all scalars r and r . Examples include 1 2 iterative if the operations used to get from one multiplication of an n-dimensional vector by an n m approximation to the next are nearly the same. m × n matrix (here V and W are R and R , If successive steps are exactly the same, the respectively) or the operation of differentiation method is called stationary. The linear station- (here both V and W are spaces of functions with ary iterative process may be written as x(k+1) = appropriate differentiability). x(k) + R(b − Ax(k)), where R is an approxima- Also called linear transformation, linear op- tion to A−1 and x(0) = Rb. With this definition erator. x(r) = (I + C + C2 + ··· + Cr )Rb where C = I − RA and I is the identity. R is some- linear pencil Given two linear transforma- times referred to as the key matrix and I − RA tions, or matrices, A and B, the family A + λB, is called the iteration matrix. where λ is any scalar parameter, is called the (linear) pencil defined by A and B. linear system A finite set of simultaneous equations that depends linearly on its inputs. For linear programming The study of optimiz- example, ing a linear function over a convex, linear con- 3x + 2y = 5 straint set. Linear programming arises in eco- −x + 4y = 7 nomics from the desire to maximize revenue or minimize cost (from some economic process) is a linear system of equations involving the two subject to constrained resources. unknowns x and y. Another example is a linear system of differential equations, which is a sys- linear programming problem A problem tem of differential equations where each one is in which a linear function of one or more in- of the form dependent variables is maximized or minimized ∂f ∂f L[f ]=α (x) +···+α (x) = β(x)f over a convex polygonal region. For example, to 1 N ∂x1 ∂xN maximize the function L(x) = 3x1 + 2x2 + 5x3 over the set {2x1 +3x2 +5x3 ≤ 5,x1,x2,x3 ≥ where the αi and β are functions of the indepen- 0}. Often such a problem arises in economics, dent variables x = (x1,...,xN ) (in particular, where the linear function to be maximized or they do not depend on the function f ). The minimized represents revenue or costs and the class of linear systems of differential equations polygonal constraint region represents the limi- is the simplest class of differential equations to tation of resources. understand and solve.

c 2001 by CRC Press LLC linear transformation See linear mapping. local Gaussian sum Some constants in number theory. Hecke (1918, 1920) extended L-matrix See sign pattern. the notion of character by introducing the Grössencharackter χ and defining L-functions χ(a) local class field theory The theory of Abel- with such characters: Lk(s, χ) = a N(a). ian extensions of local fields; for example, the p- If we denote by ξk(s, χ) a certain multiple of adic number fields. Early results were obtained Lk(s, χ), then ξk(s, χ) satisfies a functional using the techniques inherited from (global) equation ξk(s, χ) = W(χ)ξk(1 − s,χ)¯ where class field theory. Other techniques in use in- W(χ)is a complex number with absolute value clude those from the theory of cohomology of 1. Such ξk(s, χ) can be represented by an in- s ∗ groups, algebraic K-theory, and the theory of ξk(s, χ) = c φ(t)χ(t)V(t) d t tegral form Jk , formal groups over local fields. Some of the where V(t)is the total volume of the ideal t, c main results of local class field theory include is a constant that depends on the Haar measure ∗ an isomorphism theorem and an existence theo- d t of the ideal group Jk of the given field k, and ∗ rem. Let K denote the multiplicative group of the φ(x) satisfies K. The set of all finite Abelian extensions L/K G −1 ∗ with Galois group is in one-to-one correspon- φ(x)χp(x)Vp(x) d x = dence with the norm subgroup N = NL/K (L∗) kp ∗ of K . This correspondence yields a canonical ∗ s isomorphism from G to K /N. The existence CpN (δf )p τp χp · µ Uf,p ,p|f. theorem states the converse. Any open subgroup Here τp(χp) is a constant called the local Gaus- of finite index in K∗ can be realized as a norm sian sum. subgroup for some Abelian extension L of K. See also class field theory. localization The process used to construct the field of fractions from an integral domain (a local coordinates A set of functions on a ring with no non-zero zero divisors). An exam- M p manifold near a point , which forms a one- ple is the construction of the rational numbers M to-one map of a neighborhood in of the point (fractions) from the integers. p onto a neighborhood in Euclidean space (i.e., N N R or C ). Local coordinates are used to per- locally constructible sheaf Let Xet´ be the form operations, such as differentiation or inte- etale´ site of a scheme X. A sheaf on Xet´ which is gration, to functions on a manifold by pulling represented by an etale´ covering of X is called a them back to Euclidean space where such oper- locally constructible (or locally constant) sheaf. ations are simpler to understand. locally convex topological space A topolog- local equation An equation of the form ical vector space with a basis for its topology, f(x)= 0 which locally (i.e., in a neighborhood whose members are convex. of some given point) describes a codimension one, irreducible variety. locally finite A term used in several different contexts. A typical example comes from the local field A field that is complete with re- theory of coverings: Let U ={Uα}α∈U be a spect to a discrete valuation and has a finite covering of a set E. If each e ∈ E is contained residue field. A local field with finite character- in only finitely many of the Uα, then the covering istic is isomorphic to the field of formal power U is said to be locally finite. series in one variable over a finite field. If the local field has characteristic 0, then it is isomor- locally finite algebra An algebra with the phic to a finite algebraic extension of the field of property that any finite number of its elements p-adic numbers (Qp). The term has also been generate a finite-dimensional subalgebra. applied to discretely valued fields with arbitrary residue fields. In particular, real and complex locally Noetherian scheme A scheme that number fields have been called local fields. can be covered by open affine subsets, each of

c 2001 by CRC Press LLC which is a Noetherian ring. See scheme. This The logarithm function is used to describe concept arises from the subject of algebraic ge- many physical phenomenon. For example, the ometry. Richter scale for earthquakes is defined in terms of the logarithm of the energy produced by earth- local Macaulay ring A local ring, which, as quakes. a module over itself, has dimension equal to the number of maximal regular sequences within its logarithm of a complex number A loga- maximal ideal. This class includes all Noethe- rithm of a complex number z is any number w rian rings. This concept arises in commutative with ew = z. The logarithm of a complex num- algebra. ber is not unique, since the exponential function is periodic with period 2πi (i.e., ew+2πi = ew), local maximum modulus principle A basic if w is a logarithm of z, then w + 2πki is also theorem in Banach algebras (due to Rossi) that a logarithm of z for any integer k. The prin- A states that the Shilov boundary of the space, , cipal value of the logarithm of z is the unique U ⊂ n of analytic functions on an open set C is logarithm whose argument is between −π and U contained in the topological boundary of . See π. Shilov boundary. More rigorously, log z may be defined as log |z|+2πi arg(z), where log |z| is defined, local parameter A mathematical quantity as the logarithm of a positive real number and that is only defined on a small open subset, such arg(z) is the argument of z. See logarithm. as a neighborhood of a particular point. Some- times a function, defined so as to depend upon such a quantity, can be described by a local pa- long division The long division of a number g f rameter near a given point, but not a global pa- by a number is the process of finding unique q r g = fq+r r rameter. numbers and with where (called the remainder) is less than g. Often long divi- r = local ring A ring = {0} which has only one sion involves repeating the process so that fq +r r

c 2001 by CRC Press LLC loop Any one-dimensional curve (defined as equation a11x1 = b1 for x1 and then forward the image of a continuous function defined on substituting in the subsequent equations to solve the unit interval) in a topological space, which for x2,x3,.... starts and ends at the same point. The space of all loops which pass through a given base point lowest terms The form of a fractional ex- is fundamental to the field of Homotopy theory, pression with no factor that is common to both a subfield of topology. numerator and denominator. For example, 3/4 is a fraction in lowest terms, but 6/8 is not, be- Another name for the group cause of the common factor of 2. O(1, 3), which is the group of all 4 × 4 matrices A with real entries that preserve the bilinear form loxodromic transformation One of several F(x,y) =−x0y0 + x1y1 + x2y2 + x3y3 (i.e., classification schemes for linear fractional func- az+b F (Ax, Ay) = F(x,y)). The Lorentz group tions w = cz+d in one complex variable. If such (as well as all the groups O(1,n)) are basic to a transformation has two distinct finite fixed the study of . The Lorentz points α and β, then the transformation can be group in particular is important in the study of put into the following normal form: relativity (mathematical physics). w − α z − α = k w − β z − β lower bound For a set S of real numbers, any number which is less than or equal to all S where a − cα elements in . For example, any number which k = . is less than or equal to zero is a lower bound for a − cβ the set {x : 0 ≤ x ≤ 1}. If arg k = 0, the transformation is called hy- perbolic; if |k|=1 the transformation is called lower central series A descending chain elliptic; otherwise the transformation is called of subalgebras D G, D G,..., associated to a 1 2 loxodromic. given Lie algebra G, defined inductively as fol- See linear fractional function. lows: D1G =[G, G] (which means the set of all Lie brackets of an element in G with any other Luroth’s Theorem Suppose L is the field element in G); DkG =[G, Dk− G]. This con- 1 obtained by adjoining an element α to a field cept is of fundamental importance in the classi- F where α is transcendental over F ;ifE is a fication of Lie algebras. field with F ⊂ E ⊂ L, then E is obtained by adjoining some element β ∈ E to F . lower semi-continuous function Let X be a topological space and f : X → R a function. Lutz-Mattuck Theorem The group of ra- Then f is said to be lower semi-continuous if tional points of an Abelian variety of dimen- f −1((β, ∞)) is open for every real β. sion n over the p-adic number field contains a subgroup of finite index that is isomorphic to n lower triangular matrix Any square matrix copies of the ring of p-adic integers. A = (aij ) with zero entries above the diagonal The theorem is named after its authors E. Lutz (aij = 0 for j>i). Any set of linear equa- (1937) and A. Mattuck (1955). tions (A · x = b) arising from a lower triangular matrix can be solved easily by solving the first

c 2001 by CRC Press LLC matrix A rectangular array of numbers or other elements:   a11 a12 ... a1n  a a ... a   21 22 2n  . M  ......  am1 am2 ... amn

Here n represents the number of columns and Macaulay local ring See local Macaulay m the number of rows. These are indicated by ring. designating the above array an m × n matrix. Matrices arise in many fields, such as linear Main Theorem (class field theory) If k is algebra. The most common use for a matrix is an algebraic number field, then any Abelian ex- to represent a linear mapping from one vector tension of fields K/k is a class field over k, for space to another. a suitable ideal group H. matrix group The set of all invertible square matrices of a given dimension, under matrix Main Theorem (Galois Theory) See Fun- multiplication. This group is one of the simplest damental Theorem of Galois Theory. and widely used groups where the multiplication operation is not commutative. Main Theorem (Symmetric Polynomials) Subgroups of the above group may also be See Fundamental Theorem of Symmetric Poly- referred to as matrix groups. nomials. matrix multiplication The process of mul- tiplying two matrices A and B: mantissa The fractional part of a number in   its decimal expansion. For example, the man- a11 a12 ... a1n  a a ... a  tissa of the number 3.478 is 0.478.  21 22 2n  ,  ......  am am ... amn mapping Another name for function. See  1 2  b b ... b function. 11 12 1r  b b ... b   21 22 2r  .  ......  mathematical programming A process of bs1 bs2 ... bsr breaking down a mathematical problem into its component steps so that it can be solved by a The number of columns of the left matrix A must computer. be equal to the number of rows of the second matrix B (n = s). The ij entry of the product, n A · B is the sum aikbkj . mathematical programming problem A k=1 The matrix product AB is the matrix of the mathematical problem that can be solved by pro- composition of the linear transformations with gramming, i.e., by breaking it down into its com- matrices B and A. ponent steps so that it can be solved by a com- puter. matrix of a quadratic form For the qua- dratic form on Rn Mathieu group Most non-Abelian finite sim- n ple groups can be classified into a list of infinite Q(x) = aij xixj , families. There are 26 exceptional “sporadic” i,j=1 groups that cannot be classified in this way. The smallest of these groups has 7920 elements and where each aij is a real number, the matrix (aij ). is named after its discoverer, E. Mathieu (1861). Usually there are further requirements, such as

c 2001 by CRC Press LLC that this matrix must be symmetric (aij = aji) fixed element g in the Lie group G, the map- and positive definite (Q(x) ≥ c|x|2 for some ping Tg : G → G defined by x → g · x is a positive constant c). differentiable map. A Mauer-Cartan form is a 1-form ν defined on G which has the property matrix of coefficients The matrix obtained that, for any g ∈ G, the pull back of ν via Tg is from the coefficients of the variables in a system again ν. of linear equations. For example, the matrix of coefficients of the linear system Mauer-Cartan system of differential equa-

2x + 3y = 5 tions The system of differential equations x + 5y = 7 satisfied by a set of left-invariant 1-forms that allows one to recover the multiplication opera- is the matrix tion of the underlying Lie group. 23 . 15 maximal deficiency A geometric invariant The solution to a system of linear equations is of an algebraic surface, defined as the first co- usually found by manipulating its matrix of co- homology group with values in the sheaf of 0- efficients (together with the right side of the forms (smooth functions). equation). matrix representation (1) A representation maximal ideal An ideal is maximal in a ring R of a group on a matrix group. See representation. if it is not properly contained in another ideal R (2) A matrix which is used to describe a math- other than itself. See ideal. ematical process or object. A common exam- ple is to represent a linear map from one vector maximal ideal space The collection X of space to another by a matrix by identifying the all multiplicative linear functionals on a Banach coefficients of the expansion of the linear map in algebra or function algebra X. X is so called terms of given bases for the domain and range. because the kernel of each such functional is a Such a representation depends strongly on the maximal ideal in X. choice of bases. For example, if V and W are vector spaces maximal independent system A linearly in- with bases {v1,...,vn} and {w1,...,wm}, the m dependent system of vectors in a vector space, map T : V → W which satisfies Tvj = aij 1 with the property that any additional vector wi has the matrix representation See   would make the system dependent. linearly independent elements. For example, a set of a11 a12 ... a1n  a a ... a  three non-coplanar vectors in R3 is a maximal  21 22 2n  .  ......  independent system because if any fourth vec- tor is added, the system becomes linearly depen- am1 am2 ... amn dent. matrix unit The i, j matrix unit is an n × maximal order The largest order of any ele- m matrix which hasa1inthei, j entry and ment in a group, where the order of an element zeros elsewhere. The collection of all matrix g is the smallest positive integer m with gm = e, units forms a basis for the vector space of n × m the identity element in the group. matrices. Despite its name, a matrix unit is not a unit in the algebraic sense (it does not have a multiplicative inverse). maximal prime divisor (Of an ideal I of a commutative ring R.) A prime ideal P of R such Mauer-Cartan differential form A differ- that P is maximal, as an ideal of R containing ential 1-form on a Lie group which is invariant I, and is disjoint from the set of elements of R under the group action. In more detail, for a which are not zero-divisors, modulo I.

c 2001 by CRC Press LLC maximal prime ideal A prime ideal of a ring function F from X to the set of bounded opera- R which is not properly contained in a prime tors on H, such that, for any vector h ∈ H, the ideal other than R itself. See prime ideal. function x → (F (x)h, h) is , measurable. maximal separable extension A separable measurable vector function The space of extension field of a field F which is not properly measurable vector functions is a set K of func- contained in any other separable extension field tions, from a measure space M to a Hilbert space of F . See separable extension. H, (·, ·) with the following properties: (i.) if 1 x ∈ K, then ||x|| = (x(ζ ), x(ζ )) 2 is a measur- maximal torus Let G be a Lie group. A able (scalar) function; (ii.) for any x and y ∈ torus H in G is a connected, compact, Abelian K, (x, y) is measurable, and (iii.) there is a H G subgroup. If, for any other torus in with countable family {x1,x2,...} from K such that H ⊇ H H = H H we have , then is said to be {x1(ζ ), x2(ζ ), ...} is dense in H . The definition maximal. also allows for the Hilbert space H to depend on ζ. mean Any of a variety of averages. (1) The mean of a set of numbers is their sum measure Let X be a space and let A be a divided by the number of entries in the set. More sigma field on X.Ameasure on X is a function precisely, the mean of x1,...,xN is (x1 +···+ µ : A → R that satisfies certain additivity or xN )/N. The mean is one of the basic statistical subadditivity properties, such as countable ad- quantities used in analyzing data. ditivity. See additive set function. A measure is (2) The mean of a function f(x), defined on a a device for measuring the length or the size of set S, is a continuous analog of the mean defined a set. above:  median A midway point in a data set. For 1 f(x)dx, |S| S a discrete data set, half the data points are less than or equal to the median and half are greater where |S| denotes the length, or volume, of S. than or equal to the median. More generally, See also mean of degree r. the median for a distribution ρ on a probability space {X,p} is a number m such that the proba- mean of degree r Of a function f , with re- bility of the event that f(x)is less than or equal spect to a weight function p, the quantity to m is greater than or equal to 1/2 and the prob- f(x)  /r ability of the event that is greater than or f r pdx 1 m  . equal to is greater than or equal to 1/2. pdx meet In a lattice, the infimum or greatest In other words, the mean of degree r of f is lower bound of a set of elements. Specifically, r the L norm of f with respect to the probability if A is a subset of a lattice L, themeet of A measure generated by the weight function p. is the unique lattice element b = {x : x ∈ A} defined by the following two conditions: (i.) mean proportional A mean proportional be- x ≥ b for all x ∈ A. (ii.) If x ≥ c for all x ∈ A, tween two numbers a and b is the number m such then b ≥ c. The meet of an infinite subset of that a/m = m/b. a lattice may not exist; that is, there may be no element b of the lattice L satisfying conditions mean terms of proportion The terms b and (i.) and (ii.) above. However, by definition, c in the proportion a/b = c/d. (In older works, one of the axioms a lattice must satisfy is that the proportion is sometimes written a : b :: c : the meet of a finite subset A must always exist. d.) The meet of two elements is usually denoted by x ∧ y. measurable operator function For a set X, There is a dual notion of thejoin of a sub- a σ -algebra , on X and a Hilbert space H,a set A of a lattice, denoted by {x : x ∈ A},

c 2001 by CRC Press LLC and defined by reversing the inequality signs in minimal ideal An ideal which does not prop- conditions (i.) and (ii.) above. The join is also erly contain any ideal except the zero ideal, {0}. called the supremum or least upper bound of the subset A. Again, the join of an infinite subset minimal model A nonsingular, projective may fail to exist, but the join of a finite subset surface which is the unique relatively minimal always exists by the definition of a lattice. model in its birational equivalence class. Except for rational and ruled surfaces, every non-empty meet irreducible element An element a of birational equivalence class has a (unique) min- a lattice L which cannot be represented as the imal model. The existence of a minimal model meet of lattice elements b properly larger than in the birational equivalence class of a higher a (a

c 2001 by CRC Press LLC minimal Weierstrass equation A Weier- linear groups. Let S and T be the matrices of strass equation for an elliptic curve E/K is one integral positive definite quadratic forms, corre- m n of the form sponding to the lattices 4S ⊂ R , 4T ⊂ R ,in the sense that qS and qT express the lengths of y2 + a xy + a y = x3 + a x2 + a x + a . 1 3 2 4 6 elements of 4S and 4T , respectively. Then an X S[X]=T We call such an equation minimal if, among all integral solution to the equation , S[X]:=Xt SX possible Weierstrass equations, it has least dis- where , determines an isometric 4 → 4 N(S,T) criminant |D|. embedding S T . Denote by the total number of such maps. The genus of qS is Minkowski-Farkas Lemma See Minkowski- defined to be the set of quadratic forms which q I Farkas Theorem. are rationally equivalent to S. Let be the set of these equivalence classes. One of Siegel’s Minkowski-Farkas Theorem For every ma- formulas gives the value of a certain weighted trix A and vector b the system Ax = b has a average of the numbers N(Sx,T)over a set of non-negative solution if and only if the system representatives Sx for classes x ∈ I of forms of uA ≥ 0, ub < 0 has no solution. a given genus: This theorem is sometimes referred to as  ˜ −1 the Minkowski-Farkas Lemma. The follow- N(S,T) = cm−ncm α∞(S, T ) αp(S, T ) , ing corollary has also been referred to as the p

Minkowski-Farkas Lemma. The following con- 1 ˜ where c = and ca = 1 for a>1, N(S,T) ditions are mutually exclusive. Either the 1 2 = 1 N(Sx ,T ) w(x) system of linear inequalities Ax ≤ b has a non- Mass(S) x∈I w(x) and is the order negative solution or the system of linear inequal- of the group of orthogonal transformations of ities uA ≥ 0, ub < 0 has a non-negative solu- the lattice 4S, and define the mass of S by tion. 1 Mass(S) = x∈I w(x). T = S N(S,T)˜ Minkowski-Hasse character See Hasse- In the special case ,wehave = 1 Minkowski character. Mass(S) , and the Siegel formula becomes the Minkowski-Siegel formula: Minkowski inequality If f and g are  −1 −1 complex-valued, measurable functions on a Mass(S) = cmα∞(S, S) αp(S, S) . measure space (X, µ) and 1 ≤ p<∞, then p

 1 p p Siegel’s formula can be deduced from an in- |f(x)+ g(x)| dµ(x) N(ϕ)˜ = vol(g/γ ) ϕ(x)dx X tegral formula: vol(G/ ;) G/g ,  1 where G = Om(A) is the locally compact group p ≤ |f(x)|p dµ(x) of orthogonal matrices with respect to S with X coefficients in the ring of adeles A, g and γ are  1 p closed subgroups of G with vol(g/γ ) finite and p + |g(x)| dµ(x) . ϕ is a continuous function with compact support X on G/γ . The quantities cm−n and cm become The left side of the inequality is the definition the Tamagawa numbers τ(Om−n) and τ(Om), of the Lp-norm of f + g. Thus, Minkowski’s respectively, and the Tamagawa measure on G inequality is the statement that the Lp-norm sat- can be defined. isfies the triangle inequality, which is one of the defining properties of a norm. Named after the Minkowski space A flat space of four di- German mathematician H. Minkowski (1864– mensions, designed to model the geometry of 1909). the physical universe as suggested by the special theory of relativity. It is also called Minkowski Minkowski-Siegel-Tamagawa Theory In space-time, the Minkowski world or the number theory, a theory on the arithmetic of Minkowski universe. There are two methods

c 2001 by CRC Press LLC of envisioning Minkowski space. Coordinates then the absolute value of the discriminant of k may be written as (x,y,z,ict)where i2 =−1 is greater than 1. and c is the speed of light. In this case (x,y,z) represents the position of a point in space, and t minor The determinant of an n − 1 × n − 1 is the time at which an event occurs at that point. submatrix of an n × n matrix obtained by delet- The distance between two points is then ing a row and a column from the larger matrix.  Minors arise as a theoretical tool for computing ds = (dx)2 + (dy)2 + (dz)2 − c2(dt)2 . a determinant of a matrix (as in expansion by minors). See expansion of determinant. It is also possible to view Minkowski space as the manifold R4, with a flat Lorentz metric. In minuend The term from which another term a a − b this case, the coordinates are written as (x1,x2, is to be subtracted. (The number in .) x3,x4) = (x,y,z,ct) and the space is associ- ated with an indefinite inner product x · y = minus sign The symbol “−” which indicates x1y1 + x2y2 + x3y3 − x4y4. Note, in some ref- subtraction in arithmetic. The minus sign also erences the time coordinate is listed first and indicates “additive inverse,” so that, for exam- − called x0. In addition, some references define ple, 3 is the additive inverse of 3. This no- the inner product as the negative of the one de- tion extends to groups (with operation +), where fined above. Using the definition given above, −x represents the element that when added to x a non-zero vector x is called time-like or space- gives the identity in the group. like, depending upon whether x · x is negative or positive. A non-zero vector x is called null, mixed decimal A number written in decimal isotropic, or lightlike if x · x = 0. form with an integer and a fractional part; for example, 34.587. Minkowski’s Theorem (1)IfK = Q, then |DK | > 1, where K is a field, k is an algebraic mixed expression A mathematical formula number field contained in k, with [K : k] < ∞, involving terms of more than one type. For ex- xy + y2x and DK is the discriminant of K. ample, 4 is a mixed expression of the x y The theorem is a consequence of the follow- variables and . ing Minkowski Lemma: let M be a lattice in M Rn, @ =vol(Rn/M), and let X ⊂ Rn be a cen- mixed group A set which can be parti- M ,M ,... trally symmetric convex body of finite volume tioned into disjoint subsets, 0 1 with a ∈ M v =vol(X).Ifv>2n@, then there exists a the following properties: (i.) for 0 and b ∈ M i = , ,... ab a \ b nonzero α ∈ M ∩ X. i, 0 1 , elements and a(a \ b) = b (2) (On convex bodies) Any convex region are defined such that ; (ii.) for b, c ∈ M b/c M in n-dimensional Euclidean space which is sym- i, an element of 0 is defined (b/c) · c = b metric about the origin and has a volume greater such that ; and (iii.) the associative (ab)c = a(bc) a,b ∈ M c ∈ M than 2n contains another point with integral co- law for 0 and ordinates. This theorem can be generalized as holds. follows. Let P be a convex region in n-dimen- I sional Euclidean space which is symmetric about mixed ideal An ideal of a Noetherian ring R the origin and let 4 be a lattice with determinant such that there exist associated prime ideals P,Q I P @. If the volume of P is greater than 2n|@|, of such that the height of is not equal Q then P contains a point of 4 other than the ori- to the height of . See height. gin. This is one of the most important theorems mixed integer programming problem A in the , and is one of the programing problem in which some of the vari- reasons that the geometry of numbers exists as a ables are required to have integer values. In ad- distinct subdivision of number theory. The fol- dition, all of the variables are usually required lowing application to algebraic number fields is to be nonnegative. Mixed integer programming also called Minkowski’s Theorem. If k is an al- once referred only to problems that were linear gebraic number field of finite degree and k = Q, in appearance. The concept has been expanded

c 2001 by CRC Press LLC to include nonlinear programs. A mixed integer modular character A group homomorphism linear programming problem may be written as from the (multiplicative) group of units of the   m n p ring of integers modulo to the multiplicative max cx + dy : Gx + Hy ≤ b, x ∈ Z+,y ∈ R+ , group of nonzero complex numbers. See also character. where the matrices c, d, G, H, and b have inte- c ger entries and the following dimensions: is modular operator A positive self-adjoint 1 × n, d is 1 × p, G is m × n, H is m × p, and b m × m operator, defined and used in Tomita-Takesaki is 1. In addition, it is assumed that theory. Let φ be a normal semifinite faithful and n + p are positive integers. weight on a von Neumann algebra M. Let Hφ be the Hilbert space associated with φ, let Nφ be mixed number A number that has both an the left ideal {A ∈ M : φ(A∗A) is finite}, and integer and a fractional part, such as 4 2 . 3 let η be the associated complex linear mapping from Nφ into a dense subset of Hφ. Let Sφ be the M-matrix A matrix A that can be written as ∗ antilinear operator defined by Sφη(A) = η(A ) ∗ ∗ A = sI − B, where A ∈ N ∩ Nφ and A is the adjoint of A. The polar decomposition of the closure of Sφ where B is an entrywise nonnegative matrix, I defines a self-adjoint operator called a modular is the identity matrix, and s is a positive number operator. greater than the spectral radius of B. From the Perron-Frobenius Theorem for entrywise non- modular representation A representation of negative matrices, it follows that all the eigen- a group which is also a finite field. See repre- values of an M-matrix have positive real parts. sentation. M-matrices arise naturally in several areas of the mathematical sciences such as optimization, module A nonempty set M is said to be an Markov chains, numerical solution of differen- R-module (or a module over a ring R)ifM is tial equations, and . an Abelian group under an operation (usually denoted by +) and if, for every r ∈ R and m ∈ Möbius transformation See linear fractional M, there exists an element rm ∈ M subject to function. the distributive laws: r(m1 + m2) = rm1 + rm2 and (r + s)m = rm + sm; as well as the model A mathematical representation of a associative law r(sm) = (rs)m for m, m1,m2 ∈ physical problem. For example, the differential M and r, s ∈ R. dy equation dt = ky is a model for any physical process that is governed by exponential growth module of A-homomorphisms The set of all or decay, such as population growth radioactive homomorphisms from an A-module M to an A- decay. module N forms a module over the ring A called the module of homomorphisms from M to N, de- Modular arithmetic with noted HomA(M, N). See homomorphism. See respect to a certain number p refers to an arith- also automorphism group. metic calculation, where at the end, only the remainder is kept after subtracting the greatest module of boundaries A concept that arises multiple of p. For example, if p = 5, then in the subject of graded modules. Regard the × × = = 3 7 (modulo 5) equals 1 (since 3 7 21 graded module X as a sequence of modules X0, × + 4 5 1). See congruent integers, group of X1,X2,... with maps ∂n : Xn → Xn−1 (called congruence classes. boundary maps) with ∂n−1◦∂n = 0. The module of boundaries is the graded module of images modular automorphism A one-parameter ∂ {X } ∂ {X } ... φ of the boundary map, i.e., 1 1 , 2 2 , ∗-automorphism σt , of a von Neumann algebra This concept arises most commonly in topology φ it −it M, where σt (A) ≡ @φ A@φ for a modular where the graded modules are chains of sim- operator @φ. See also modular operator. plicies in a topological space X. In this case,

c 2001 by CRC Press LLC Xk is the free module generated (say, over the orientation of the face. The cycles are the chains integers) by the continuous images of the stan- whose boundaries are zero. As a simple exam- 2 dard k-dimensional simplex in Euclidean space ple, a circle is a cycle in R because it is the {(x ,...,x ) ∈ k : x ≥ , k x ≤ } ≤ t ≤ 1 k R i 0 i=1 i 1 . image of the one-dimensional simplex 0 The boundary map of a k-simplex is a weighted 1 (via the complex exponential map, f(t) = sum of (the continuous images of) its k − 1 di- exp(2πit)) and because its boundary, f(1) − mensional faces where the weights are either f(0), is zero. plus one or minus one depending on the ori- entation of the face. Then the module of k − module of finite length A module M with 1-boundaries is the module generated by the the property that any chain of submodules of the boundaries of k-dimensional simplicies. form module of coboundaries A coboundary is {0}⊂M0 ⊂ M1 ⊂ M2 ···⊂Ml = M the image of a cocycle under the induced bound- ary map. (See module of cocycles.) The set of is finite in number. The above containments are coboundaries is a module over the underlying proper. ring (often the integers or the reals). M module of cocycles The dual of the module module of finite presentation A module R of cycles. More precisely, if X ,X,X,... over a ring such that there is a positive in- 0 1 2 n R is a graded chain of cycles over a ring R (often teger and an exact sequence of -modules → K → Rn → M → K the integers or the reals) with boundary maps 0 0 where is finitely generated. ∂n : Xn → Xn−1, then the module of cocycles ˆ ˆ ˆ is the graded chain X0, X1, X2,... where each ˆ Xn is the set of all ring homomorphisms from Xn module of finite type (1) A graded module ˆ ˆ Ai k Ai to R. The induced coboundary map ∂n : Xn → i≥0 over a field for which each is finite ˆ ˆ dimensional. Xn+1 is defined by ∂ncˆn(xn+1) =ˆcn(∂nxn+1) ˆ O O for cˆn ∈ Xn and xn+1 ∈ Xn+1. (2) A sheaf of -modules (called an - Module) in a ringed space (X, O), which is lo- module of cycles A concept that arises in the cally generated by a finite number of sections subject of graded modules. Regard the graded over O. module X as a sequence of modules X0,X1,X2, (3) A finitely generated module. ... with maps ∂n : Xn → Xn−1 (called bound- ary maps) with ∂n−1 ◦ ∂n = 0. A cycle cn ∈ Xn module of homomorphisms Let M and N is one that has zero boundary (∂ncn = 0). The be modules over a commutative ring R. The set set of all cycles is a module over the underlying of all homomorphisms of module M to module ring (often the integers). This concept arises N is called the module of homomorphisms of most commonly in topology where the graded M to N, with addition defined pointwise and modules are chains of simplicies in a topological multiplication given by (r · f )(x) = r · (f (x)), space X. In this case, Xk is the free module gen- for r ∈ R, f : M → N a homomorphism erated (say, over the integers) by the continuous and x ∈ M. The module of homomorphisms is k images of the standard -dimensional simplex denoted HomR(M, N). in Euclidean space   module with operator domain A module k k M, together with a set A and a map from A×M (x1,...,xk) ∈ R : xi ≥ 0, xi ≤ 1 into M satisfying the following conditions. (i.) i=1 For every a ∈ A and x ∈ M there is a unique (into X). The boundary map of a chain is a element ax ∈ M. (ii.) If a ∈ A and x,y ∈ M, weighted sum of (the continuous images of) its then a(x + y) = ax + ay. In this situation, M k − 1 dimensional faces where the weights are is called a module with operator domain A.It either plus one or minus one depending on the is also called a module over A or an A-module.

c 2001 by CRC Press LLC moduli of Abelian variety Parameters or Xn)-space by means of the equations invariants which classify the set of all Abelian   X1 = X varieties which are equivalent under some type  1   X2 = X X  of equivalence relation to a given Abelian vari-  . 1 2   .  ety. The problem of finding moduli for Abel-  .  ian varieties is approached both algebraically  X = X X   m 1 m  and geometrically, and involves coarse moduli  X = X   m+1 m+1  schemes and inhomogeneous polarizations.  .   . 

Xn = X moduli scheme See coarse moduli scheme. n or the reverse   modulus (1) The modulus of a complex num- X = X 1 1 2 2 1   ber z = a + ib is defined as |z|=(a + b ) 2 .  X = X2/X1   2 .  p  .  (2) Let be a positive integer. When two  .  a,b p p   integers are congruent modulo , then  Xm = Xm/X1  . is called the modulus of the congruence. See  X = X   m+1 m+1  congruent integers.  .   . 

X = Xn modulus of common logarithm The log- n arithm to the base 10 is called common loga- The origin in the (X1,...,Xn)-space is blown rithm. The factor by which the logarithm to a up into the linear (m−1)-dimensional subspace (X ,...,X ) L : X = given base of any number must be multiplied to of the 1 n -space given by 1 X =···=X = obtain the common logarithm of the same num- m+1 n 0. ber is called modulus of the common logarithm. x = b x Because one has log10 log10 logb , for monomial A polymonial consisting of one n any base b(b>0 and b = 1) and any x>0, single term such as anx . the modulus of the common logarithm for base b b is seen to be log10 . monomial module A module generated by a single generator. See module. Moishezon space A compact, complex, ir- monomial representation Let H be a sub- reducible space X of (complex) dimension n group of a finite group G.Ifρ is a linear rep- whose algebraic dimension (i.e., transcendence resentation of H with representation module M, degree of the field of meromorphic functions on then the linear representation ρG = K(G) X) is also equal to n. K(H) M of G is called the induced represen- tation from ρ. (Here, K is a field.) A mono- p(x) monic polynomial Any polynomial of mial representation of G is the induced repre- m R degree over a ring with leading coefficient sentation from a degree 1 representation of a R 1R , the unit of the ring . See leading coeffi- subgroup. Each monomial representation of G cient. corresponds to a matrix representation τ such that for each g ∈ G the matrix τ(g)has exactly monoidal transformation For any integer one nonzero entry in each row and each column. m with 1

c 2001 by CRC Press LLC In most familiar categories, such as the cate- homomorphisms. If C is the category of topo- gory of sets and functions, a monomorphism is logical spaces, the morphisms would be contin- simply an injective or one-to-one function in the uous maps, etc. See also functor. category. A function i : A → B is one-to-one, or injective if i(a1) = i(a2) whenever a1 = a2. morphism in a category A map by means See also morphism in a category, epimorphism, of which categorical equivalence is measured. injection. morphism of finite type For R a commuta- monotone function Let I ⊆ R be an interval tive ring with 1, and X a scheme, a morphism and f : I → R a function. If whenever a,b ∈ I f : X −→ Spec(R) is called a morphism of fi- and a

Mordell-Weil Theorem Let k be an alge- morphism of local ringed spaces Let X be braic number field of finite degree and let V be a topological space, and suppose that to each an Abelian variety of dimension n defined over x ∈ X is associated a local ring BX,x in a natural k. Then the group Vk of all k-rational points on way. Let Y be another such space. A mapping V is finitely generated. f : X → Y is called a morphism of local ringed This theorem was proved by Mordell for the spaces if, whenever, f(x)= y, there is induced special case of n = 1 in 1922 and by Weil for a homomorphism BY,y → BX,x. the general case in 1928. morphism of pointed sets A set X with a ◦ ∗ morphism Let be a binary operation on distinguished element x is called a pointed set. A ◦ ∗ ∗ a set , while is another such operation on For two pointed sets (X, x ) and (Y, y ), a map A m : (A, ◦) → (A , ◦ ) a set .Amorphism f : X −→ Y is said to be a morphism of pointed A A ∗ ∗ is a function on to which preserves the sets if f(x ) = y . operation ◦ on A onto the operation ◦ on A ,in the sense that morphism of schemes Let R be a commu- m(a ◦ b) = m(a) ◦ m(b) tative ring with 1. One obtains a sheaf of rings R˜ on Spec(R) by assigning to each point p of for all a,b ∈ A. Spec(R) the ring of quotients Rp. Then Spec(R) In a categorical approach to algebra, one de- is called an affine scheme when it is regarded fines a category C as a set (or, more generally, as a local-ringed space with R˜ as its structure a class) of objects, together with a class of spe- sheaf. A scheme is a local-ringed space X which cial maps, called morphisms between these ob- is locally isomorphic to an affine scheme. A jects. If C is the category of Abelian groups, morphism of schemes X and Y is a morphism then the morphisms would normally be group f : X −→ Y as local ringed-spaces.

c 2001 by CRC Press LLC V ,V ,...,V ,W dn : Cm,n −→ Cm,n+1 dm+1dm = multilinear function Let 1 2 k 2 such that 1 1 be vector spaces over a field F . A function dn+1dn = dmdn = dndm,m,n ∈ 2 2 0 and 1 2 2 1 Z.A multicomplex C in C is defined in an analogous f : V × V ×···×V → W n ,n ,...,n 1 2 k {C 1 2 r } way: A family n1,n2,...,nr ∈Z of ob- r that is linear with respect to each of its arguments jects and sets of differentials n is called multilinear. If k = 2, f is called bilin- i n1,...,ni ,...,nr n1,...,ni +1,...,nr di : C −→ C , ear. Thus, a bilinear function f : U × V → W n satisfies ni +1 ni ni j 1 ≤ i ≤ r, subject to di di = 0 and di dj n = d j dni , ≤ i, j ≤ r, i = j f (u, av1 + bv2) = af (u, v1) + bf (u, v2) j i 1 . and multiple covariants Let K be a field of char- acteristic 0 and let G = GL(n, K). Let F = f (au + bu ,v) = af (u ,v) + bf (u ,v) C m 1 2 1 2 i1,...,in i1,...,in be a homogeneous form of d x ,...,x a,b ∈ F u, u ,u ∈ U degree in variables 1 n with coefficients for all and all 1 2 and K i = d v,v ,v ∈ V in . (Here, r and 1 2 .  The determinant of an n×n matrix A is a mul- i1 in mi ,...,in = d!/ ir ! x ...xn .) tilinear function when the arguments are taken 1 1 to be each of the n rows (or columns) of A. g ∈ G gx (gC) For each , we define i and i1,...,in by setting, respectively, multinomial Given n real numbers a ,a ,...,a m −1 1 2 n, and a positive integer, then (gx1,...,gxn) = (x1,...,xn) g m the expression (a1 + a2 +···+an) is called  F = (gC) (gm ) multinomial. We have and i1,...,in i1,...,in . In this case, the action of G on the polynomial ring R = m (a + a +···+an) K[...,C ,...] 1 2 i1,...,in is given by either a rational  n! p p representation or the contragredient map A → = a 1 ...a m t − 1 m A 1. Then the G-invariants are called multiple p1! ...pm! covariants. where the sum is over all p1 +···+pm = m. See also monomial. multiple root A root of a polynomial, having multiplicity > 1. See multiplicity of root. multiobjective programming A mathemati- cal programming problem in which the objective multiplicand A number to be multiplied by function is a vector-valued function f : Rn −→ another number. Rk,k≥ 2, where Rk is ordered in some way (e.g., lexicographic order, etc.). multiplication (1) A binary operation on a set. In group theory, and in algebra in general, multiple Let S be a semigroup whose binary it is customary to denote by ab˙ or ab, the el- operation is multiplication. The element a ∈ S ement which is associated with (a, b) under a is called a left (right) multiple of b ∈ S if there given binary operation. The element c = ab is exists an element c ∈ S such that a = cb (a = then called the product of a and b, and the binary bc). Under this condition b is the left (right) operation itself is called multiplication. When divisor of a. the term multiplication is used for a binary op- eration, it carries with it the implication that if multiple complex Let C be an Abelian cat- a and b are in the set G, then ab is also in G. egory. A complex C in C is a family of objects (2) One of two binary operations on a field n n n n+1 {C }n∈Z with differentials d : C −→ C or ring. See field, ring. such that dn+1dn = 0,n ∈ Z. A bicomplex C m,n in C is a family of objects {C }m,n∈Z and two multiplication by logarithms The logarithm dm : Cm,n −→ Cm+1,n x b = sets of differentials 1 and function logb , especially when 10 or

c 2001 by CRC Press LLC b = e, the basis for the natural logarithm, has multiplication of vectors Any bilinear func- had extensive use in facilitating arithmetical cal- tion defined on pairs of vectors. Important ex- culations, especially before the days of comput- amples are the familiar dot product and cross ers. The rule product of calculus. See also wedge product, tensor product. logb n · m = logb n + logb m multiplicative group (1) Any group, where allows one to multiply large numbers n, m by as- the group operation is denoted by multiplica- certaining their logarithms (from a table), add- tion. Often a non-Abelian group is written as a ing the logarithms and then obtaining the prod- multiplicative group. See group, Abelian group. uct of m and n by another reference to the table. ∗ See logarithm, common logarithm, natural log- (2) The set F of all the non-zero elements arithm. of a field F . See field. multiplication of complex numbers Multi- multiplicative identity An element 1, in a plication of z1 = a + ib and z2 = c + id yields set S with a binary operation ·, regarded as mul- z1 ·z2 = (a +ib)(c +id) = (ac −bd)+i(ad+ tiplication, such that 1 · a = a · 1 = a, for all bc). Writing the complex numbers z1 and z2 a ∈ S. in terms of their absolute values r ,r and their 1 2 A multiplicative monoid is one in which the arguments φ ,φ : zj = rj (cos φj + i sin φj ), 1 2 binary operation is written as multiplication and yields z · z = r r [cos(φ + ¯φ ) + i cos(φ + 1 2 1 2 1 2 1 the multiplicative identity is 1 such that a1 = ¯φ )]. 2 1a = a, for all a in the monoid. multiplication of matrices Let F be a ring, and A =[aij ], 1 ≤ i ≤ m, 1 ≤ j ≤ n,an multiplicative inverse For an element a,in m × n matrix over F . Multiplication of A (on a set S with a binary operation ·, considered as the right) by a k × l matrix B =[bij ] (over the multiplication, and an identity element 1, an el- same field F ) is defined if n = k and the product ement a−1 ∈ S such that a−1 · a = a · a−1 = 1. m × l C =[c ] ij is the matrix ij , with -entry is (See also multiplicative identity.) obtained by cij = aihbhj , 1 ≤ i ≤ m and h F 1 ≤ j ≤ l: A field is a non-trival commutative ring     in which every non-zero element a has a multi- a b ... a b 1h h1  1h hl plicative inverse.   A·B =  ... aihbhj  ... . amhbh ... amhbhl 1 multiplicative Jordan decomposition Let M be a free right R-module. A linear transfor- mation τ on M is called semisimple if M has multiplication of polynomials Let the structure of a semisimple R[x] module de- n p termined by x(m) = τ(m); that is, if and only if k k f(x)= akx , g(x) = bkx , the minimal polynomial of τ has no square factor 0 0 different from the constants in R[x]. The multi- plicative Jordan Decomposition Theorem states D be polynomials over an integral domain . Mul- that any nonsingular linear transformation τ on tiplication of the polynomials gives M can be uniquely written as τ = τsτu, where τs M 1 is a semisimple linear transformation on and f(x)· g(x) = a0b0 + (a0b1 + a1b0) x −1 τu = 1M + τs τn for some nilpotent transfor- 2 mation τn on M. Here, τu is called the unipotent + (a0b2 + a1b1 + a2b0) x + ... component of τ. n+p k = ckx multiplicatively closed subset A subset S 0  of a ring R, which is a subsemigroup of R with c = k a b with k i=0 i k−i. respect to multiplication.

c 2001 by CRC Press LLC multiplicity of a point Let f be a function the multiplicity of w ∈ V :ifV w = {0}, w is defined in a neighborhood of a point p in RN . called a weight of V . The multiplicity of the point p is the order to which f vanishes at p; that is, f is of order m at multiplier Let G be a finite group and let ∗ p if all derivatives of f up to (but not including) C = C\{0}. Then the second cohomology ∗ order m vanish, but some mth order derivative group H 2(G, C ) is called the multiplier of G. ∗ does not. If H 2(G, C ) = 1, then G is called a closed In other contexts, if q is in the image of f group and any projective representation of G then the order of q is the number of elements in is induced by a linear representation. See also the pre-image set of q under f . Lagrange multiplier, Stokes multiplier, charac- teristic multiplier. multiplicity of a point Let f be a function ∗ N multiplier algebra For a C -algebra A, let defined in a neighborhood of a point p ∈ R . ∗∗ A denote its enveloping von Neumann alge- The multiplicity of the point p is the order to bra. The multiplier algebra of A is the set which f vanishes at p; that is, f is of order m at ∗∗ M(A) ={b ∈ A : bA + Ab ⊆ A}. p if all derivatives of f up to (but not including) m m order vanish, but some th order derivative multiply transitive permutation group does not. A permutation group G is called k-transi- q f In other contexts, if is in the image of , tive (k a positive integer) if, for any k-tuples then the order of q is the number of elements in (a ,a ,...,ak) and (b ,b ,...,bk) of distinct q f 1 2 1 2 the pre-image set of under . elements in X, there exists a permutation p ∈ G such that pai = bi, for all i = 1,...,k.If multiplicity of root Let D be an integral do- k = 2, G is called a multiply transitive permu- main and f(x)an element of D[x].Ifc belongs tation group. to D and c is a root of f(x) (f(c) = 0), then f(x)= (x−c)mg(x)where m is an integer with multistage programming A mathematical 0 ≤ m ≤ degf(x), g(x) ∈ D[x] and g(c) = 0. programming problem in which the objective The integer m is called the multiplicity of the function and the constraints have an iterative or root c of f(x). repetitive property. multiplicity of weight Let g denote a com- mutually associated diagrams An O(n) di- plex semisimple Lie algebra, h a Cartan subal- agram is a Young diagram T for which the sum gebra of g, and R the corresponding root system. of the lengths of the first column and the sec- Let V be a g-module (not necessarily finite di- ond column is ≤ n.TwoO(n) diagrams T1 ∗ mensional), and let w ∈ h a linear form on h. and T2 are called mutually associated diagrams We will let V n denote the set of all v ∈ V such if the lengths of their first columns sum up to that Hv = w(H)v, for all H ∈ h. This is a n and their corresponding columns (except the vector subspace of V . An element of V w is said first ones) have equal lengths. See Young dia- to have weight w. The dimension of V w is called gram.

c 2001 by CRC Press LLC (ii.) n + m∗ = (n + m)∗ whenever n + m is defined. The addition satisfies the following laws: A1 Closure Law: n + m ∈ N N A2 Commutative Law: n + m = m + n A3 Associative Law: m+(n+p) = (m+n)+p A4 Cancellation Law: If m + p = n + p, then Nakai-Moishezon criterion For a ringed m = n (X, O) I space , we let denote a coherent sheaf Multiplication on N is defined by O X k of ideals of . When is a -complete scheme (i.) n · 1 = n k r (where is a field), then for any -dimensional (ii.) n·m∗ = n·m+n whenever n·m is defined. W X closed subvariety of and any invertible M1 Closure Law: n · m ∈ N S (Sr · W) sheaf we let denote the intersection M2 Commutative Law: n · m = m · n Sr O/I number of with . The Nakai-Moishezon M3 Associative Law: m · (n · p) = (m · n) · p (Sr · W) > r criterion states that if 0 for any - M4 Cancellation Law: If m · p = n · p, then dimensional closed subvariety W of X, then S m = n is ample. Addition and multiplication are subject to the Naperian logarithm See natural logarithm. Distributive Law D1 For all n, m, p ∈ N,m·(n+p) = m·n+m·p natural logarithm The logarithm in the base The elements of N are called natural num- e. See e. The natural logarithm of x is denoted bers. log x or, in elementary textbooks, ln x. The adjective natural is used, due to the sim- natural positive cone For a weight w on plicity of the defining formula the positive elements of a von Neumann algebra A N ={A ∈ A : w(A∗A) < ∞}  x , we set w . 1 s x = 1 dt , Let 0 ≤ s ≤ . Then the closure W of the log 2 s 1 t set of vectors wη(A), where A ranges over all positive elements in Nw ∩ Nw∗ , has some for x real and positive. of the properties of the von Neumann algebra Also called Naperian logarithm. See also A. (Here, w is a modular operator and η is logarithm, logarithmic function. a complex linear mapping on Nw defined by ∗ setting (η(A◦), η(A)) = w(A A).) The special natural number A positive integer. The sys- W 1 tem N of natural numbers was developed by the case 4 is called the natural positive cone. It Italian mathematician Peano, using a few sim- is a self-dual convex cone and is independent of w ple properties known as Peano Postulates. Let the weight . there exist a non-empty set N such that Postulate I: 1 is an element in N. negation Given a proposition P , the propo- Postulate II: For every n ∈ N, there exists a sition (not P ) is called the negation of P and is unique n∗ ∈ N , called the successor of n. denoted by ∼ P or ¬P . Postulate III: 1 is not the successor of any ele- ment in N. negative See negative element. Postulate IV: If n, m ∈ N and n∗ = m∗ , then n = m. negative angle An angle XOP, with ver- Postulate V: Any subset K ⊂ N having the prop- tex O, initial side OP and terminal side OX, erties such that the direction of rotation is counter- (i.) 1 is an element of K clockwise. (ii.) k∗ ∈ K whenever k ∈ K satisfies K = N. negative cochain complex A cochain com- Addition on N is defined by: plex C in an Abelian category such that Cn = 0 (i.) n + 1 = n∗, for every n ∈ N for n>0. See cochain complex.

c 2001 by CRC Press LLC negative element (1) Let K be an ordered J is a subgroup of Div(S)⊗Z Q and the quotient field. (See ordered field.) If x<0, we say x is X = (Div(S) ⊗Z Q)/J is a finite dimensional negative and −x is positive. vector space over Q. This quotient group X is (2) An element g = e of a lattice ordered called the Neron-Severi group of the surface S group G such that g ≤ e. See lattice ordered and its dimension is called the Picard number of group. S. See also negative root. network programming A mathematical pro- negative exponent In a multiplicative group gramming problem related to network flow, in G with identity 1 the negative powers of an el- which the objective function and the constraints ement a are a−1, its multiplicative inverse, and are defined with reference to a graph. the elements a−k = (a−1)k, for k>0, where an element bk is bb ···b (with k factors). Newton-Raphson method of solving algebraic equations An iterative numerical method for negative number A negative element of the finding an approximate value for a zero of a func- real field. See negative element. tion f (approximating a solution to the equa- tion f(x) = 0). The method begins by choos- G negative root Let be a complex semisim- ing a suitable starting value x0. The method ple Lie algebra. For a subalgebra H of G,we proceeds by constructing a sequence of succes- H∗ ={ H} set all complex-valued forms on . sive approximations, x1,x2,x2,..., to the ex- ∗ ∗ For h ∈ H , let Hh∗ ={g ∈ G : ad(h)g = act solution x, according to the formula xk = ∗ h (h)g, h ∈ H} H ∗ =  for all .If h 0 and xk−1 − f(xk−1)/f (xk−1). Under suitable con- h∗ = h∗ 0, then we say that is a root; two roots ditions, the sequence x1,x2,x3,... converges are equivalent whenever one is a nonzero mul- to x. tiple of the other. The root system of G relative The name Newton-Raphson method is also ∗ ∗ ∗ to H is the finite set  ={h ∈ H : h = applied to a multi-dimensional generalization of ∗ 0 and Hh∗ = {0}}. Let H denote the real lin- the above. Here, the object is to approximate a ∗ R ear subspace of H spanned by . Relative to a solution to the equation f(x)= 0, but this time ∗ lexicographic linear ordering of H (associated x is a vector in an n-dimensional vector space with some basis over R), a root r is negative if V and f is a vector-valued function from some r<0. subset of V to V . The sequence of successive approximations is constructed according to the Neron minimal model Let R be a discrete formula: valuation ring with residue field k and quotient −1 field K. The Neron minimal model of an Abel- xk = xk−1 − D (f (xk−1)) f (xk−1) , ian variety A over K is a smooth group scheme Df f n×n A of finite type over Spec(R) such that for ev- where is the Jacobian of (the matrix ery scheme S over Spec(R) there is a canonical of partial derivatives). isomorphism Newton’s formulas (1) For interpolation: S ,A  (S, A). HomK ( K ) HomSpec(R) u f(x0 + ux) = f(x0) + 0 Here, SK is the pullback of S by Spec(K) → 1! Spec(R). u(u − ) u(u − )(u − ) + 1 2 + 1 2 3 2! 0 3! 0 Neron-Severi group Let Div(S) denote the u(u − )(u − )(u − ) group of all divisors of a nonsingular surface S. + 1 2 3 4 +··· , 0 By linearity and the Index Theorem of Hodge, 4!  there is a bilinear form I(D· D ) on Div(S) ⊗Z where 0 = f(x0 + x) − f(x0). Q which has exactly one positive eigenvalue. (2) For symmetric polynomials: Let p1,p2, (See Index Theorem of Hodge.) Let J ={D ∈ ...,pr be the elementary symmetric polynomi-   Div(S) ⊗Z Q : I(D· D ) = 0 for all D }. Then als in variables X1,X2,...,Xr . That is, p1 =

c 2001 by CRC Press LLC   Xi,p2 = XiXj , ..., pr = X1X2 ...Xr . called the nilpotent radical of G and we have  n For n = 1, 2,..., let sn = Xi . The Newton the inclusions I ⊂ N ⊂ S ⊂ G. See nilpotent formulas are: sn − p1sn−1 + p2sn−2 −···+ ideal. n−1 n (−1) pn−1s1 + (−1) npn = 0; and sn − p s + p s − ··· + (− )kp s = 1 n−1 2 n−2 1 k n−k 0, nilpotent subset A subset S of a ring R such for n = k + 1,k+ 2,.... that Sn = 0, for some positive integer n. nilalgebra An algebra in which every ele- nilradical The ideal intersection of all the ment is nilpotent. See algebra, nilpotent ele- prime ideals P which contain I, where I is an ment. ideal in a commutative ring R. The nilradical is denoted RadI. If the set of prime ideals con- nilpotent component Let M be a linear space taining I is empty, then RadI = R. over a perfect field. Then any linear transforma- tion τ of M can be uniquely written as τ = R τs + τn (Jordan decomposition), where τs is Noetherian domain A commutative ring semisimple and τn is nilpotent, and the two lin- with identity, which is an integral domain and ear transformations τs and τn commute; they are also a Noetherian ring. See integral domain, called the semisimple component and the nilpo- Noetherian ring. tent component of τ, respectively. See Jordan decomposition. Noetherian integral domain An integral do- main whose set of ideals satisfies the maximum nilpotent element An element a of a ring R condition: every nonempty set of ideals has a such that an = 0, for some positive integer n. maximal element. nilpotent group A group G such that Cn(G) Noetherian local ring A Noetherian ring = G, for some n, where 1 ⊂ C1(G) ⊂ C2(G) having a unique maximal ideal. ⊂ ··· is the ascending central series of G. See ascending central series. Noetherian module A left module M, over a ring R such that, for every ascending chain nilpotent ideal A (left, right, two-sided) ideal M ⊂ M ⊂ ··· of submodules of M, there I R I 1 2 of a ring is nil if every element of is nilpo- exists an integer p such that Mk = Mp, for all I I n = tent; is a nilpotent ideal if 0 for some k ≥ p. integer n. Noetherian ring A ring R is left (resp. right) nilpotent Lie algebra A Lie algebra such Noetherian if R is Noetherian as a left (resp. that there exists a number M such that any com- right) module over itself. See Noetherian mod- mutator of order M is zero. ule. R is said to be Noetherian if R is both left and right Noetherian. nilpotent Lie group A connected Lie group whose Lie algebra is a nilpotent Lie algebra. Noetherian scheme A locally Noetherian nilpotent matrix A square matrix A such scheme whose underlying topological space is that Ap = 0 where p is a positive integer. If p compact. A scheme X is a locally Noetherian is the least positive integer for which Ap = 0, scheme if it has an affine covering {Ui}, where then A is said to be nilpotent of index p. each Ui is the spectrum of some Noetherian ring Ri. nilpotent radical Let G be a Lie algebra. The radical of G is the union S of all solvable Noetherian semilocal ring A Noetherian ideals of G; it is itself a solvable ideal of G. The ring with only a finite number of maximal largest nilpotent ideal of G is the union N of all ideals. See Noetherian ring. See also Noethe- nilpotent ideals of G. The ideal I =[S, G] is rian local ring.

c 2001 by CRC Press LLC Noether’s Theorem Any ideal in a finitely non-linear algebraic equation An equation generated polynomial ring is finitely generated. in n variables having the form p(x1,x2,...,xn) = 0, where p is a polynomial of degree > 1 with non-Abelian cohomology The cohomology coefficients in a field. H 0(G, N) and H 1(G, N) of a group G using non-homogeneous cochains, where N is a non- non-linear differential equation Any differ-  (n) Abelian G-group. ential equation f(t,x,x ,...,x ) = 0, where  dx x represents an unknown function, x = dt non-Archimedian valuation A valuation and f(x0,...,xn+1) is not a linear function of which is not Archimedian. See Archimedian x0,...,xn+1. valuation. A non-linear nth degree equation, as above, can be written in the form of a system non-commutative field A non-commutative  ring whose nonzero elements form a (multiplica- xi = fi (t,x1,...,xn) ,i= 1,...,n tive) group. and uniqueness can be proved, with initial condi- x (a ) = b ,i= ,...,n, non-convex quadratic programming The tions i i i 1 under suitable f mathematical programming problem of mini- conditions on the i. mizing the objective function f(x) = u · x + (x · Ax)/2 subject to some linear equations or non-linear problem Any mathematical prob- inequalities. Here, f : Rn → R,u∈ Rn is lem which deals with non-linear mappings or fixed, and A is a symmetric (perhaps indefinite) operators and their related properties. matrix. non-linear programming A mathematical non-degenerate quadratic form A homoge- programming problem in which the objective function or (some of) the constraints are non- neous quadratic polynomial p in x1,x2,..., xn with coefficients in a field K is called a qua- linear. dratic form. When the characteristic of K is non-linear transcendental equation A non- not 2, we may write p(x) = (x, Ax), where linear analytic, but not polynomial, equation. x = (x1,x2,..., xn) and A is an n × n ma- trix with entries in K. We say that p is a non- non-primitive character A character of a degenerate quadratic form if the determinant finite group that is not a primitive character. See det(A) = 0. primitive character, character of group. non-degenerate representation L1(G) Let nonsingular matrix A matrix having an in- denote the space of all complex-valued inte- verse. See inverse matrix. grable functions on a locally compact group G L1(G) . Then is an algebra over C, where norm (1) Let V be a vector space over K = R (f ∗ multiplication is given by the convolution or C. A function ρ : V →[0, ∞) is a semi- g)(α) = f(αβ−1)g(β)dβ G . The map norm if it satisfies the following three condi- ∗ − f(α) → f (α) = D(α 1)f(α−1) is an in- tions: (i.) ρ(x) ≥ 0, for all x ∈ V , (ii.) volution of the algebra L1(G), where D is the ρ(αx) =|α|ρ(x),for all α ∈ K and x ∈ V , and modular function of G. For a unitary represen- (iii.) ρ(x+ y) ≤ ρ(x)+ ρ(y), for all x,y ∈ V . tation u of G, let Uf = f(α)uαdα. Then A semi-norm ρ for which ρ(x) = 0 ⇔ x = 0 a non-degenerate representation of the Banach is called a norm. algebra L1(G) with an involution is given by the (2) See reduced norm. map f → Uf . The map u → U is a bijection between the set of equivalence classes of unitary normal *-homomorphism A ∗-homomor- representations of G and the set of equivalence phism between two Banach algebras with in- classes of non-degenerate representations of the volutions B1 and B2 is an algebraic homomor- Banach algebra L1(G) with an involution. phism ϕ which preserves involution; that is,

c 2001 by CRC Press LLC ϕ(x∗) = ϕ(x)∗.A∗-homomorphism ϕ is called such that D is defined by means of (a part of) normal if, for every bounded increasing net Nα this system of local coordinates. A divisor D is in B1, one has supα ϕNα = ϕ(supα Nα).A∗- called a divisor with only normal crossings if it homomorphism between two operator algebras is a divisor with only normal crossings at every is continuous in the strong and weak operator x ∈ V . topologies. normal equation Let AX = b be a system of normal algebraic variety An irreducible va- m linear equations in n unknowns with m ≥ n. riety all of whose points are normal. See normal The problem of finding the X which minimizes point. the Euclidean norm b − AX is called the lin- ear least squares problem. The normal equa- normal Archimedian valuation An Archi- tion for this problem is t AAX = t Ab, where median valuation of an algebraic number field t A denotes the transpose of A. Applying the K of degree n which is one of the valuations Cholesky method to this normal equation is one vi, 1 ≤ i ≤ n, defined as follows: There are way of solving the linear least squares problem. exactly n mutually distinct injections ϕ1,...,ϕn See Cholesky method of factorization. of K into the complex number field C. We may assume that ϕi(K) ⊂ R if and only if 1 ≤ i ≤ r1 normal extension An extension field K of ϕ (a) ϕ (a) F K F and that i and n−i+1+r1 are complex a field , such that is a finite extension of conjugates for r1 ≤ i ≤ r2 = (n − r1)/2. Let and F is the fixed field of G(K, F ), the group vi(a) =|ϕi(a)| for 1 ≤ i ≤ r1 and vi(a) = of automorphism of K relative to F . 2 |ϕi(a)| for r1 ≤ i ≤ r2 = (n − r1)/2. See Archimedian valuation. normal form A canonical choice of repre- sentatives for equivalence classes with respect normal basis Let L be a Galois extension of to some group action. See also Jordan normal a field K and G(L/K) the Galois group of L/K. form. (See Galois extension, Galois group.) If L/K is a finite Galois extension, then there is an element normal function A continuous, strictly g l ∈ L such that the set {l : g ∈ G(L/K)} forms monotone function of the ordinal numbers. a basis for L over K called a normal basis. See also Normal Basis Theorem. Normalization Theorem for Finitely Gener- The existence of a normal basis implies that ated Rings Let R be a finitely generated ring the regular representation of G(L/K) is equiv- over an integral domain I. Then there exist an alent to the K-linear representation of G(L/K) element α ∈ I (with α = 0) and algebraically by means of L. independent elements Y1,...,Yn of R over I such that the ring of quotients RS is integral −1 i Normal Basis Theorem Any (finite dimen- over I[α ,Y1,...,Yn]. Here, S ={α : i = sional) Galois extension field L/K that has a 1, 2,...}. normal basis. Normalization Theorem for Polynomial normal chain of subgroups A normal chain Rings Let I be an ideal in a polynomial ring of subgroups of a group G is a finite sequence K[X1,X2,...,Xn] in n variables over a field G = G0 ⊃ G1 ⊃···⊃Gn ={e} of subgroups K. Then there exist Y1,Y2,...,Yn in K[X1, of G with the property that each Gi is a normal X2,...,Xn] such that (i.) Y1,Y2,...,Yn gen- subgroup of Gi−1 for 1 ≤ i ≤ n. erate I ∩K[Y1,Y2,...,Yn] and (ii.) K[X1,X2, ...,Xn] is integral over K[Y1,Y2,...,Yn]. normal crossings Let D ≥ 0 be a divisor (i.e., a cycle of codimension 1) on a non-singular normalized cochain A cochain f , variety V and let x ∈ V . Then D is a divisor in Hochschild cohomology, satisfying with only normal crossings at x if there is a sys- f(ł1,...,łn) = 0 whenever one of the łi tem of local coordinates (f1,...,fn) around x is 1. Here, : is an algebra over a commuta-

c 2001 by CRC Press LLC tive ring K, A is a two-sided :-module, an where M∗ denotes the adjoint (the transpose of n-cochain Cn is the module of all n-linear the complex conjugate) of M. mappings of : into A with C0 = A. This coho- mology group is defined by the n-cochains and normal point Let V be an affine variety over the coboundary operator δn(f ) : Cn → Cn+1 an algebraically closed field k and let p be a defined by point in V . Then p is called a normal point if n the local ring Rp = k[x1,...,xn]/I (p) is nor- δ (f ) : (ł ,...,łn+ ) → λ f (ł ,...,łn+ ) 1 1 1 2 1 mal. Here, I(p)is the ideal of all polynomials n   j in k[x1,...,xn] which vanish at p. + (−1) f ł1,...,łj łj+1,...,łn+1 j=1 normal representation A normal ∗- n+1 +(−1) f (ł1,...,łn) łn+1 . homomorphism of a von Neumann algebra A B The normalized cochains can be used to define into some operator algebra . the same cohomology group. normal ring An integrally closed integral normalizer (1) (In ergodic theory) Let (X, domain. A ring is integrally closed if it is equal B,µ) be a σ-finite measure space. A map f : to its integral closure in its ring of quotients. See X −→ X is called measurable if f −1(B) ∈ B integral closure. for every B ∈ B. A bimeasurable map is a bi- G jective measurable map whose inverse is also normal series Let be a finite group. We p (i = ,...,k) measurable. A non-singular bimeasurable map define i 1 to be the distinct primes G !P " f : X −→ X is said to be the normalizer for which divide the order of . Let i be the nor- another bimeasurable map g : X −→ X if for mal subgroup generated by the class of Sylow p r every ϕ ∈[g]={ϕ : ϕ is a non-singular mea- subgroups corresponding to the prime i.If is !P ", !P ",...,!P " surable map of X such that, for some n,we minimal, then a collection 1 2 r n G have ϕ(x) = g (x) for µ-almost all x} there which generates is called a minimal system G r is ψ ∈[g] such that f ◦ ϕ = ψ ◦ f . of Sylow classes for ; is called the Sylow G T (i = ,...,r) (2) (In group theory) Let S be a subset of a rank of . A chain i 0 of nor- G T = T = G group G. The normalizer of S is the subgroup mal subgroups of ( 0 identity, r ) − T T N(S) ={g ∈ G : g 1Sg = S} of G. such that i+1 is the group generated by i and !Pji ", where {ji} is a permutation of {1,...,r}, normal j-algebra A Lie algebra G over R is called a normal series for the given minimal such that there is a linear endomorphism j of system of Sylow classes. G and a linear form ω on G satisfying the fol- lowing four conditions: (i.) For every x ∈ G the normal series Let G be a finite group. We eigenvalues of ad(x) are all real; (ii.) [jx,jy]≡ define pi(i = 1, ··· ,k)to be the distinct primes j[jx,y]+j[x,jy]+[x,y], for all x,y ∈ G; which divide the order of G. Let !Pi" be the nor- (iii.) ω([jx,jy]) = ω([x,y]), for all x,y ∈ G; mal subgroup generated by the class of Sylow (iv.) ω([jx,x])>0 for x = 0. subgroups corresponding to the prime pi.Ifr is minimal, then a collection !P1", !P2", ··· , !Pr " normally flat variety Let V be a variety over which generates G is called a minimal system a field of characteristic zero and let S be a sub- of Sylow classes for G; r is called the Sylow scheme of V defined by a sheaf of ideals I. Let rank of G. A chain Ti(i = 0, ··· ,r) of nor- OS denote the sheaf of germs of regular func- mal subgroups of G (T0 = identity, Tr = G) tions on S and let OS,x be the stalk of OS over such that Ti+1 is the group generated by Ti and the point x ∈ S. Then V is said to be normally !Pji ", where {ji} is a permutation of {1, ··· ,r} flat along S if for any n and any x ∈ S, the quo- is called a normal series for the given minimal n n+1 tient module Ix /Ix is a flat OS,x-module. system of Sylow classes. normal matrix A square matrix M, with normal subgroup A subgroup H of a group complex entries, such that MM∗ = M∗M, G such that Hx = xH, for all x ∈ G.

c 2001 by CRC Press LLC normal valuation Let K be an algebraic It is known then that H defines a non-degenerate number field and let O be the principal part of pairing K.Ap-adic valuation v of K is a normal val- ∗ Kn(Xn) X ( · ) uation if, for α ∈ K, one has v(α) =|p|−r , × n → ζ tr res(φ/s) K (X ))q (X∗)q where |p|=norm of p = the cardinality of the n n n O/p r α set . Here, is the degree of with respect satisfying the norm property, i.e., to p. A normal valuation of a function field is defined analogously, except that instead of |p| H(α1,α2,...,αn+1) = 1 one uses cd , where c>0isfixedandd is the ⇐⇒ { α1,...,αn}∈Kn(Xn) degree of the residue class field of the valuation √ over the base field k. is a norm in Kn(Xn(q αn+1)). This property See also normal Archimedian valuation. gives rise to its name as the n-dimensional norm- residue symbol. norm form of Diophantine equation Let K be an algebraic number field of degree d ≥ 3 nuclear (1) Let X and Y be Banach spaces. and let c1,...,cn ∈ K. Then the norm A linear operator TX→ Y is called a nuclear operator if it can be written as a product T = S : S Norm (c1x1 +···+cnxn) 1 2 S1:S2 with X −→ l∞ −→ l1 −→ Y . Here, S S : d  1 and 2 are bounded linear operators and (j) (j) {λ ≥ } l = c x +···+c x is multiplication by a sequence n 0 in 1. 1 1 n n T T = j= Then the nuclear norm of is given by 1 1 S : S X Y inf 1 l1 2 . When and are Hilbert is a form of degree d with rational coefficients. spaces, this norm coincides with the trace norm. (j) Here, ci denotes a conjugate of ci, 1 ≤ i ≤ n. (2) A locally convex topological vector space Z is called a nuclear space if for each abso- normic form A normic form of order p ≥ 0 lutely convex neighborhood U of 0 there is an- in a field k is a homogeneous polynomial f of other absolutely convex neighborhood V ⊂ U d degree d ≥ 1inn = p variables with coeffi- of 0 such that the natural linear mapping τV,U : cients in k such that the equation f = 0 has no Z(V ) −→ Z(U) is a nuclear operator. Here, solution in k except (0,...,0). Z(U) and Z(V ) are the normed spaces obtained from seminorms corresponding to U and V , re- norm-residue symbol Let k be a finite ex- spectively. tension of Qp and null sequence Any sequence tending to 0 (in X = k{{t1}} ...{{tn−1}} a topological space with a 0 element). Especially, one has a null sequence in the I- n be an -dimensional local field. See local field. adic topology. Let I be an ideal of a ring R Assume that k contains the roots of unity µ of M R q = pν p = ζ and let be an -module. A null sequence order , 2. Let be a generator M µ in is a sequence which converges to zero of . According to Vostokov, there is a skew- I M symmetric map in the -adic topology of . A base for the neighborhood system of zero in this topology is H : X∗ ×···×X∗ → µ given by {InM : n = 1, 2,...}. tr res(φ/s) ∗ H(α1,α2,...,αn+1) = ζ ,αi ∈ X , null space A synonym for the elements of a with the property that αi +αj = 1 ⇐⇒ H(α1, matrix, when the matrix is considered as a linear α2,...,αn+1) = 1 for i = j. Here tr is the transformation. trace operator of the inertia subfield of k, s is determined in X by an expansion of ζ in X, and number (1) Any member of the real or com- φ(α1,α2,...,αn+1) is given by an expansion plex numbers. of the αsinX. Here res is the residue of φ/s, (2) We say that two sets have the same num- i.e., the coefficient of 1/t1t2 ···tn−1. ber if there is a bijection between them. This is

c 2001 by CRC Press LLC an equivalence relation on the class of sets. A numerical Referring to an approximate cal- number is an equivalence class under this rela- culation, commonly made with machines. tion. Examples are cardinal numbers or ordinal numbers. numerical radius See numerical range. number field Any subfield of the field of numerical range The numerical range (or complex numbers. field of values)ofann×n matrix A with entries from the complex field is the set number of irregularity The dimension of ∗ n ∗ the Picard variety of an irreducible algebraic va- F (A) = x Ax : x ∈ C ,xx = 1 . riety. The numerical range is a compact, convex set A number system Any one of the fields Q, R, that contains the eigenvalues of , is invari- or C (rational numbers, real numbers, or com- ant under unitary equivalence and is subadditive F(A+ B) ⊂ F (A) + F(B) plex numbers). (i.e., ). The numer- ical radius of A is defined in terms of F (A) by A numerator The quantity A, in the fraction B r(A) = {|z|:z ∈ F (A)} . (B is called the denominator). max

c 2001 by CRC Press LLC ous homomorphism (R, +) −→ G, t → ϕ(t), such that dϕ(d/dt) = X. Here, d/dt is the basis of the Lie algebra of R. Such a homomor- phism is called a one-parameter subgroup of the O Lie group G. one-to-one function See injection. objective function In a mathematical pro- gramming problem, the function which is to be onvolution Assume f and g are real or com- minimized or maximized. plex valued functions defined on Rn and such that f 2 and g2 are integrable on Rn. The con- octahedral group The symmetric group of volution of f with g, written fg, is a func- n degree 4. This is the group S4 of permutations tion h defined on R by the formula h(x) =  ∞ of 4 elements. −∞ f(x − y)g(y)dy. Important cases of this occur in the study of Sobolev spaces where one odd element (Of a Clifford algebra.) See can choose f so that it is integrable on all closed even element. and bounded (compact) subsets of Rn and choose for g a function with infinitely many derivatives odd number An integer that is not divisible (C∞) which is zero off a compact set (i.e., it has by 2. An odd number is typically represented compact support). In this special case, many by 2n − 1, where n is an integer. smoothness properties are passed on to fg and one can use convolutions to approximate f omega group Let  be a set and let G be a in various norms. group. Then G is called an -group (and  is called an operator domain of G) if there is a map operation Let A and B be sets. An operation ×G −→ G such that ω(g1g2) = ω(g1)ω(g2) of A on B is any map from (a subset of) A × B for any ω ∈ , g1,g2 ∈ G. to B.Anya ∈ A is called an operator on B and A is said to be a domain of operators on B. See omega subgroup A subgroup H of an - also binary operation. group G is an -subgroup of G if ω(h) ∈ H for each ω ∈  and h ∈ H. See omega group. operator algebra Let B(H ) denote the set of bounded linear operators on a Hilbert space H . one The smallest natural number. It is the Then B(H ) contains the identity operator and is identity of the multiplicative group of real num- an algebra with the operations of operator ad- bers. dition and operator product (composition). An operator algebra is any subalgebra of B(H ). one-cycle A collection C of Abelian groups {Gn}n∈Z and morphisms ∂n : Gn −→ Gn−1,n operator domain See omega group. ∈ Z, is called a chain complex. Any element in the kernel of ∂n is called an n-cycle of C.In operator homomorphism Let  be a set particular, a one-cycle of C is any element of the and let G1 and G2 be -groups. An operator kernel of ∂1. homomorphism from G1 to G2 is any homo- morphism ϕ : G1 −→ G2 which commutes one-parameter subgroup A one-parameter with every ω ∈ . That is, ϕ(ωg1) = ω(ϕg1) group of transformations of a smooth manifold for any ω ∈  and any g1 ∈ G1. See omega M is a family ft ,t∈ R, of group. of M such that (i.) the map R × M −→ M defined by (t, x) → ft (x) is smooth, and (ii.) opposite (1) Of a non-commutative group ∗ for s,t ∈ R, one has fs ◦ ft = fs+t . More gen- (or ring) G, the group G with the operation , erally, if G is a Lie group with Lie algebra G, defined by (a, b) → ab= b ◦ a, where ◦ is then for each element X ∈ G there is a continu- the operation of G.

c 2001 by CRC Press LLC (2) An oriented n-simplex σ of a simplicial order of radical Let A be an ideal of the ring complex K is an n-simplex of K with an equiv- R. The set alent class of total ordering of its vertices. (Two √   A = x ∈ R : xn ∈ A n ∈ N orderings are equivalent if they differ by an even for some permutation of the vertices.) Therefore, for ev- is called a radical of A. The order of√ radical of ery n-simplex, n ≥ 1, there are two oriented A is the of the set A. n-simplexes called the opposites of one another. order of vanishing Let f be a holomorphic optimal solution A mathematical program- function on an open set U in the complex plane. ming problem is the problem of finding the Suppose that P ∈ U and that f(P) = 0. Let k maximum/minimum of a given function f : be the least positive integer such that f (k)(P ) = X −→ R, where X is a Banach space. A 0 (where the superscript denotes a derivative). ∗ point x ∈ X at which f attains its maxi- Then f is said to vanish to order k at P . mum/minimum is called an optimal solution of There is an analogous notion of order of van- the problem. ishing for a real function. If F vanishes at a point x0 in Euclidean space, then the order of vanish- orbit A subset of elements that are related by ing at x0 is the order of the first non-vanishing an action. For example, let G be a Lie transfor- Taylor coefficient in the expansion about x0. mation group of a space . Then a G-invariant submanifold of  on which G acts transitively order relation A relation “≤” on a set such is called an orbit. that (i) x ≤ x, (ii) if x ≤ y and y ≤ x then x = y, and (iii) if x ≤ y and y ≤ z then x ≤ z. order The number of elements in a set. ordinary point A point in an analytical set ordered additive group A commutative ad- around which the set has a complete analytical ditive group with an ordering “<” such that structure. whenever a, b, c are elements with a0 and a

c 2001 by CRC Press LLC orthogonal transformation A linear trans-  n n n formation A : R → R (or, A : C → Cij , n C ) that preserves the inner product, i.e., for i,j∈{1,2,...,n},i=j v, w ∈ Rn Cn v ·w = A(v) · A(w) all (or ), . where      outer automorphism An automorphism of Cij = z ∈ C : |z − aii| z − ajj G a group that is not an inner automorphism,       −1   i.e., not of the form g → aga for some a ∈ ≤ |aik| ajk . G. The factor group of all automorphisms of G k=i k=j modulo the subgroup of inner automorphism is called the group of outer automorphisms. See also Geršgorin’s Theorem. ovals of Cassini The eigenvalues of an n × n overfield An overfield of a field F is a field matrix A = (aij ) with entries from the complex K containing F . This is also called an extension field lie in the union of n(n−1)/2 ovals, known field of F . as the ovals of Cassini,

c 2001 by CRC Press LLC such that

∂ ◦ ∂ = ∂ ◦ ∂ p−1,q p,q p,q−1 p,q = ∂ ◦ ∂ + ∂ ◦ ∂ = . P p,q−1 p,q p−1,q p,q 0 We define the associated chain complex (Xn,∂) p-adic numbers The completion of the ratio- by setting nal field Q with respect to the p-adic valuation   Xn = Xp,q,∂n = ∂p,q + ∂p,q . |·|p. See p-adic valuation. See also completion. p+q=n p+q=n p-adic valuation For a fixed prime integer p,  We call ∂ the total boundary operator, and ∂ , the valuation |·|p, defined on the field of rational  ∂ the partial boundary operators. numbers as follows. Write a rational number in pr m/n r m, n the form where is an integer, and partial derived functor Suppose F is a func- p are non-zero integers, not divisible by . Then tor of n variables. If S is a subset of {1,...,n}, |pr m/n| = /pr p 1 . See valuation. we consider the variables whose indices are in S as active and those whose indices are in parabolic subalgebra A subalgebra of a Lie {1,...n}\S as passive. By fixing all the passive algebra g that contains a maximal solvable sub- variables, we obtain a functor FS in the active g algebra of . variables. The partial derived functors are then k defined as the derived functors R FS. See also parabolic subgroup A subgroup of a Lie functor, derived functor. group G that contains a maximal connected G solvable Lie subgroup of . An example is the partial differential The rate of change of a subgroup of invertible upper triangular matrices function of more than one variable with respect GLn(C) n×n in the group of invertible matrices to one of the variables while holding all of the with complex entries. other variables constant. parabolic transformation A transformation partial fraction ∞ An algebraic expression of of the Riemann sphere whose fixed points are the form and another point. nj ajm paraholic subgroup A subgroup of a Lie m . z − αj group containing a Borel subgroup. j m=1 parametric equations The name given to X equations which specify a curve or surface by partially ordered space Let be a set. A X expressing the coordinates of a point in terms of relation on that satisfies the conditions: x ≤ x x ∈ X a third variable (the parameter), in contrast with (i.) for all x ≤ y y ≤ x x = y a relation connecting x, y, and z, the cartesian (ii.) and implies x ≤ y y ≤ z x ≤ z coordinates. (iii.) and implies is called a partial ordering. partial boundary operator We call (Xp,q, ∂,∂) over A a double chain complex if it is partial pivoting An iterative strategy, using Ax = b a family of left A-modules Xp,q for p, q ∈ Z pivots, for solving the equation , where A n × n b n × together with A-automorphisms is an matrix and is an 1 matrix. In the method of partial pivoting, to obtain the  Ak A0 = A ∂p,q : Xp,q → Xp−1,q matrix (where ), the pivot is chosen to be the entry in the kth column of Ak−1 at and or below the diagonal with the largest absolute  ∂p,q : Xp,q → Xp,q−1 value.

c 2001 by CRC Press LLC {α }∞ −e e partial product Let n n=1 be a given se- 1 and are orthogonal idempotent elements, quence of numbers (or functions defined on a and common domain  in Rn or Cn) with terms R = eR + (1 − e)R α = n ∈ n 0 for all N. The formal infinite prod- is the direct sum of left ideals. This is called α · α ··· ∞ α uct 1 2 is denoted by j=1 j . We call Peirce’s right decomposition. n Pell’s equation The Diophantine equations Pn = αj x2 − ay2 =±4 and ±1, where a is a positive j= 1 integer, not a perfect square, are called Pell’s its nth partial product. equations. The solutions of such equations can be found by continued fractions and are used in peak point See peak set. the√ determination of the units of rings such as Z[ a]. This equation was studied extensively peak set Let A be an algebra of functions by Gauss. It can be regarded as a starting point on a domain  ⊂ Cn. We call p ∈  a peak of modern algebraic number theory. point for A if there is a function f ∈ A such that When a<0, then Pell’s equation has only f(p) = 1 and |f(z)| < 1 for all z ∈  \{p}. finitely many solutions. If a>0, then all solu- The set P(A) of all peak points for the algebra tions xn, yn of Pell’s equation are given by A A √ n √ is called the peak set of . x + ay x + ay ± 1 1 = n n , Peirce decomposition Let A be a semisimple 2 2 F x y Jordan algebra over a field of characteristic 0 provided that the pair√ 1, 1 is a solution with and let e be an idempotent of A.Forλ ∈ F , let the smallest x1 + ay1 > 1. Using continued Ae(λ) ={a ∈ A : ea = λa}. Then fractions, we can determine x1, y1 explicitly.

A = Ae(1) ⊕ Ae(1/2) ⊕ Ae(0). penalty method of solving non-linear pro- gramming problem A method to modify a A This is called the Peirce decomposition of , constrained problem to an unconstrained prob- E e relative to . If 1 is the sum of idempotents j , lem. In order to minimize (or maximize) a func- A = A ( ) j = k A ∩ A let j,k ej 1 when and ej ek tion φ(x) on a set which has constraints (such j = k when . These are called Peirce spaces, as f (x) ≥ 0,f (x) ≥ 0,...fm(x) ≥ 0), a A =⊕ A 1 2 and j≤k j,k. See also Peirce space. penalty or penalty function, ψ(x,a), is intro- duced (where a is a number), where ψ(x,a) = 0 e Peirce’s left decomposition Let be an idem- if x ∈ X or ψ(x,a) > 0ifx/∈ X and ψ in- potent element of a ring R with identity 1. Then volves f1(x) ≥ 0,f2(x) ≥ 0,...fm(x) ≥ 0. Then, one minimizes (or maximizes) φa(x) = R = Re ⊕ R(1 − e) φ(x) + ψ(x,a) without the constraints. expresses R as a direct sum of left ideals. This percent Percent is called Peirce’s left decomposition. means hundredths. The symbol % stands for 100 . We may write a per- Peirce space Suppose that the unity element cent as a fraction with denominator 100. For ex- = 31 = 55 ,... 1 ∈ K can be represented as a sum of the mutu- ample, 31% 100 , 55% 100 etc. Simi- ally orthogonal idempotents ej . Then, putting larly, we may write a fraction with denominator 100 as a percent.

Aj,j = Aej (1), Aj,k = Aej (1/2)∩Aek (1/2), perfect field A field such that every algebraic F we have A = j≤k ⊕Aj,k. Then Aj,k are extension is separable. Equivalently, a field is called Peirce spaces. perfect if each irreducible polynomial with co- efficients in F has no multiple roots (in an alge- Peirce’s right decomposition Let e be an braic closure of F ). Every field of characteristic idempotent element of a unitary ring R, then 0 is perfect and so is every finite field.

c 2001 by CRC Press LLC perfect power An integer or polynomial corresponding to the sign of the permutation is which can be written as the nth power of another missing from each summand. integer or polynomial, where n is a positive in- teger. For example, 8 is a perfect cube, because permutation group Let A be a finite set with 8 = 23, and x2 + 4x + 4 is a perfect square, #(A) = n. The permutation group on n ele- because x2 + 4x + 4 = (x + 2)2. ments is the set Sn consisting of all one-to-one functions from A onto A under the group law: period matrix Let R be a compact Riemann surface of genus g. Let ω1,...,ωg be a ba- f · g = f ◦ g sis for the complex vector space of holomorphic differentials on R and let α1,...,α2g be a ba- for f, g ∈ Sn. Here ◦ denotes the composition sis for the 1-dimensional integral homology of of functions. R. The period matrix M is the g × 2g matrix whose (i, j)-th entry is the integral of ωj over permutation matrix An n × n matrix P , αi. The group generated by the 2g columns of obtained from the identity matrix In by permu- M is a lattice in Cg and the quotient yields a g- tations of the rows (or columns). It follows that a dimensional complex torus called the Jacobian permutation matrix has exactly one nonzero en- variety of R. try (equal to 1) in each row and column. There are n! permutation matrices of size n × n. They period of a periodic function Let f be a are orthogonal matrices, namely, P T P = PPT T − function defined on a vector space V satisfying = In (i.e., P = P 1). Multiplication from the relation the left (resp., right) by a permutation matrix permutes the rows (resp., columns) of a matrix, f(x+ ω) = f(x) corresponding to the original permutation. for all x ∈ V and for some ω ∈ V . The number permutation representation A permutation ω is called a period of f(x), and f(x) with a representation of a group G is a homomorphism period ω = 0 is call a periodic function. from G to the group SX of all permutations of a set X. The most common example is when X = period relation Conditions on an n×n matrix G and the permutation of G obtained from g ∈ which help determine when a complex torus is G is given by x → gx (or x → xg, depending n an Abelian manifold. In C , let * be generated on whether a product of permutations is read by (1, 0,...,0), (0, 1, 0,...,0), ..., (0, 0,..., right-to-left or left-to-right). 0, 1), (a11,a12,...,a1n), (a21,a22,...,a2n), n ... (an1,an2,...,ann). Then C /* is an Peron-Frobenius Theorem See Frobenius Abelian manifold if there are integers d1,d2,..., Theorem on Non-Negative Matrices. dn = 0 such that, if A = (aij ) and D = (δij di), then (i.) AD is symmetric; and (ii.) (AD) is Perron’s Theorem of Positive Matrices If positive symmetric. Conditions (i.) and (ii.) are A is a positive n × n matrix, A has a positive the period relations. real eigenvalue λ with the following properties: (i.) λ is a simple root of the characteristic equa- permanent Given an m×n matrix A = (aij ) tion. with m ≤ n, the permanent of A is defined by (ii.) λ has a positive eigenvector u. (iii.) If µ is any other eigenvalue of A, then A = a a ...a , |µ| <λ perm 1i1 2i2 mim . where the summation is taken over all m- Peter-Weyl theory Let G be a compact Lie permutations (i1,i2,...,im) of the set {1, 2, group and let C(G) be the commutative asso- ...,n}. When A is a square matrix, the per- ciative algebra of all complex valued continuous manent therefore has an expansion similar to functions defined on G. The multiplicative law that of the determinant, except that the factor defined on C(G) is just the usual composition

c 2001 by CRC Press LLC of functions. Denote (1856–1941). The first Picard theorem was   proved in 1879: An entire function which is not a   polynomial takes every value, with one possible s(G) = f ∈ C(G) : CL f<∞  dim g  exception, an infinity of times. g∈G The second Picard theorem was proved in 1880: In a neighborhood of an isolated essen- where Lgf = f(g·). The Peter-Weyl theory tells us that the subalgebra s(G) is everywhere tial singularity, a single-valued, holomorphic dense in C(G) with respect to the uniform norm function takes every value, with one possible ex- ception, an infinity of times. In other words, if f ∞ = maxg∈G |f(g)|. f(z)is holomorphic for 0 < |z − z0|

c 2001 by CRC Press LLC pivoting See Gaussian elimination. bers Pi = >(iK), (i = 2, 3,...). The plurigen- era Pi, (i = 2, 3,...)are the same for any two place A mapping φ : K →{F,∞}, where K birationally equivalent nonsingular surfaces. and F are fields, such that, if φ(a) and φ(b) are defined, then φ(a+b) = φ(a)+φ(b), φ(ab) = plus sign The symbol “+” indicating the al- φ(a)φ(b) and φ(1) = 1. gebraic operation of addition, as in a + b. place value The value given to a digit, de- Poincaré Let R be a commutative ring with pending on that digit’s position in relation to the unit. Let U be an orientation over R of a com- units place. For example, in 239.71, 9 repre- pact n-manifold X with boundary. Then for all sents 9 units, 3 represents 30 units, 2 represents indices q and R-modules G there is an isomor- 7 200 units, 7 represents 10 units and 1 represents phism 1 units. 100 n−q γU : Hq (X; G) ≈ H (X; G) . Plancherel formula Let G be a unimodular locally compact group and Gˆ be its quasidual. This is called Poincaré-Lefschetz duality. The Let U be a unitary representation of G and U ∗ analogous result for a manifold X without bound- be its adjoint. For any f , g ∈ L1(G) ∩ L2(G), ary is called Poincaré duality. the Plancherel formula   Poincaré-Birkhoff-Witt Theorem Let G be ∗ K X ,..., f(x)g(x)dx = t(Ug (ξ)Uf (ξ)) dµ(ξ) a Lie algebra over a number field . Let 1 ˆ G G Xn be a basis of G, and let R = K[Y1,...,Yn]  be a polynomial ring on K in n indeterminates holds, where Uf (ξ) = ˆ f(x)Ux(ξ)dx. The G Y ,...,Yn. Then there exists a unique alge- µ 1 measure is called the Plancherel measure. bra homomorphism ψ : R → G such that ψ(1) = 1 and ω(Yj ) = Xj , j = 1,...,n. plane trigonometry Plane trigonometry is Moreover, ψ is bijective, and the jth homoge- related to the study of triangles, which were j j neous component R is mapped by ψ onto G . studied long ago by the Babylonians and an- k k kn Thus, the set of monomials {X 1 X 2 ...Xn }, cient Greeks. The word trigonometry is derived 1 2 k ,...,kn ≥ 0, forms a basis of U(G) over from the Greek word for “the measurement of 1 K. This is the so-called Poincaré-Birkhoff-Witt triangles.” Today trigonometry and trigonomet- Theorem. Here U(G) = T(G)/J is the quotient ric functions are indispensable tools not only in associative algebra of G where J is the two-sided mathematics, but also in many practical appli- ideal of T(G) generated by all elements of the cations, especially those involving oscillations form X ⊗ Y − Y ⊗ X −[X, Y ] and T(G) is the and rotations. tensor algebra over G. Plücker formulas Let m be the class, n the w = degree, and δ,χ,i, and τ be the number of Poincaré differential invariant Let α(z − z )/( − z z) |α|= |z | < nodes, cups, inflections, tangents, and bitan- ◦ 1 ◦ with 1 and ◦ |z| < gents. Then 1, be a conformal mapping of 1 onto |w| < 1. Then the quantity |dw|/(1 −|w|2) = 2 n(n − 1) = m + 2δ + 3χ |dz|/(1 −|z| ) is called Poincaré’s differen- tial invariant. The disk {|z| < 1} becomes m(m − 1) = n + 2τ + 3i a non-Euclidean space using any metric with n(n − ) = i + δ + χ 3 2 6 8 ds =|dz|/(1 −|z|2). 3m(m − 2) = χ + 6τ + 8i 3(m − n) = i − χ. Poincaré duality Any theorem general- izing the following: Let M be a com- pact n-dimensional manifold without bound- plurigenera For an algebraic surface S with a ary. Then, for each p, there is an isomor- p ∼ canonical divisor K of S, the collection of num- phism H (M; Z2) = Hn−p(M; Z2). If, in

c 2001 by CRC Press LLC addition, M is assumed to be orientable, then be simple if W has no nonzero proper subco- p ∼ H (M) = Hn−p(M). algebra. The co-algebra V is called a pointed co-algebra if all of its simple subco-algebras are Poincaré-Lefschetz duality Let R be a com- one-dimensional. See coalgebra. mutative ring with unit. Let U be an orientation over R of a compact n-manifold X with bound- pointed set Denoted by (X, p), a set X where ary. Then for all indices q and R-modules G p is a member of X. there is an isomorphism polar decomposition Every n × n matrix A n−q γU : Hq (X; G) ≈ H (X; G) . with complex entries can be written as A = PU, where P is a positive semidefinite matrix and U This is called Poincaré-Lefschetz duality. The is a unitary matrix. This factorization of A is X analogous result for a manifold without bound- called the polar decomposition of the polar form ary is called Poincaré duality. of A.

Poincaré metric The hermitian metric polar form of a complex number Let z = x + iy be a complex number. This number has ds2 = 2 dz ∧ dz (1 −|z|2)2 the polar representation z = x + iy = r(cos θ + i sin θ) is called the Poincaré metric for the unit disc in  the complex plane. 2 2 −1 y where r = x + y and θ = tan x .

Poincaré’s Complete Reducibility Theorem polarization Let A be an Abelian variety and A theorem which says that, given an Abelian let X be a divisor on A. Let X be a divisor on A X A  variety and an Abelian subvariety of , A such that m1X ≡ m2X for some positive in- Y A there is an Abelian subvariety of such that tegers m1 and m2. Let X be the class of all such A is isogenous to X × Y . divisors X. When X contains positive nonde- generate divisors, we say that X determines a point at infinity The point in the extended polarization on A. complex plane, not in the complex plane itself. More precisely, let us consider the unit sphere polarized Abelian variety Suppose that V 3 in R : is an Abelian variety. Let X be a divisor on   V and let D(X) denote the class of all divisors S = (x ,x ,x ) ∈ R3 : x2 + x2 + x2 = 1 , 1 2 3 1 2 3 Y on V such that mX ≡ nY , for some inte- gers m, n > 0. Further, suppose that D(X) de- which we define as the extended complex num- termines a polarization of V . Then the couple bers. Let N = (0, 0, 1); that is, N is the north (V, D(X)) is called a polarized Abelian variety. pole on S. We regard C as the plane {(x1,x2, 0) 3 See also Abelian variety, divisor, polarization. ∈ R : x1,x2 ∈ R} so that C cuts S along the z ∈ equator. Now for each point C consider the pole Let z = a be an isolated singularity of 3 z N straight line in R through and . This in- a complex-valued function f . We call a a pole tersects the sphere in exactly one point Z = N. f Z ∈ S z ∈ S of if By identifying with C,wehave lim |f(z)|=∞. identified with C ∪{N}.If|z| > 1 then Z is in z→a the upper hemisphere and if |z| < 1 then z is in That is, for any M>0 there is a number ε> the lower hemisphere; also, for |z|=1, Z = z. 0, such that |f(z)|≥M whenever 0 < |z − Clearly Z approaches N when |z| approaches a| <ε. Usually, the function f is assumed to ∞. Therefore, we may identify N and the point be holomorphic, in a punctured neighborhood ∞ in the extended complex plane. 0 < |z − a|

c 2001 by CRC Press LLC variety of dimension r − 1. Given f ∈ C(X) \ the usual addition and multiplication of polyno- (0), let ordY f<0 denote the order of vanish- mials. The ring R[X] is called the polynomial ing of f on Y . Then (f ) = Y (ordY f)· Y is ring of X over R. called a pole divisor of f in Y . See also smooth affine variety, subvariety, order of vanishing. polynomial ring in m variables Let R be a ring and let X1,X2,...,Xm be indetermi- polynomial If a , a , ... , an are elements 0 1 nates. The set R[X1,X2,...,Xm] of all poly- of a ring R, and x does not belong to R, then nomials in X1,X2,...,Xm with coefficients in R is a ring with respect to the usual addition and a + a x +···+a xn 0 1 n multiplication of polynomials and is called the polynomial ring in m variables X1,X2,...,Xm is a polynomial. over R. polynomial convexity Let  ⊆ Cn be a do- Pontrjagin class F main (a connected open set). If E ⊆  is a Let be a complex PL M Pontrja- subset, then define sheaf over a PL manifold . The total gin class p([F]) ∈ H 4∗(M; R) of a coset [F] [F] E ={z ∈  :|p(z)|≤sup |p(w)| of real PL sheaves via complexification of w∈E satisfies these axioms: (i.) If [F] is a coset of real PL sheaves of rank for all p a polynomial} . m on a PL manifold M, then the total Pontrja- gin class p([F]) is an element 1 + p1([F]) + E ∗ The set is called the polynomially convex hull ···+p[m/ ]([F]) of H (M; R) with pi([F]) ∈ E  E ⊂⊂  2 of in . If the implication implies H 4i(M; R); E ⊂⊂   always holds, then is said to be (ii.) p(G![F]) = G∗p([F]) ∈ H 4∗(N; R) for polynomially convex. any PL map G : N → M; (iii.) p([F]⊕[G]) = p([G]) for any cosets [F] P = polynomial equation An equation 0 and [G] over M; P where is a polynomial function of one or more (iv.) If [F] contains a bona fide real vector bun- variables. dle ξ over M, then p([F]) is the classical total Pontrjagin class p(ξ) ∈ H 4∗(M; R). polynomial function A function which is a n finite sum of terms of the form anx , where n is Pontryagin multiplication A multiplication a nonnegative integer and an is a real or complex number. h∗ : H∗(X) ⊗ H∗(X) → H∗(X) . polynomial identity An equation P(X1,X2, ...,Xn) = 0 where P is a polynomial in n (H∗(X) are homology groups of the topological variables with coefficients in a field K such that space X.) P(a1,a2,...,an) = 0 for all ai in an algebra A over K. Pontryagin product The result of Pontrya- polynomial in m variables A function which gin multiplication. See Pontryagin multiplica- n n n ax 1 x 2 ...x m tion. is a finite sum of terms 1 2 m , where n1,n2,...,nm are nonnegative integers and a is a real or complex number. For example, 5x2y3+ positive angle Given a vector v = 0in Rn, 3x4z − 2z + 3xyz is a polynomial in three vari- then its direction is described completely by the ables. angle α between v andi = (1, 0,...,0), the unit vector in the direction of the positive x1-axis. If polynomial ring Let R be a ring. The set we measure the angle α counterclockwise, we R[X] of all polynomials in an indeterminate X say α is a positive angle. Otherwise, α isaneg- with coefficients in R is a ring with respect to ative angle.

c 2001 by CRC Press LLC positive chain complex A chain complex X positive Weyl chamber The set of λ ∈ V ∗ such that the only possible non-zero terms Xn such that (β, λ) > 0 for all positive roots β, are those Xn for which n ≥ 0. where V is a vector space over a subfield R of the real numbers. positive cycle An r-cycle A = niAi such that ni ≥ 0 for all i, where Ai is not in the power Let a1,...,an be a finite sequence of singular locus of an irreducible variety V for all elements of a monoid M. We define the “prod- a ,...,a i. uct” of 1 n by the following: we define 1 a = a j=1 j 1, and positive definite function A complex val-   ued function f on a locally compact topological k+1 k   group G such that aj = aj ak+1 .  j=1 j=1 f(s− t)φ(s)φ(t)dsdt ≥ 0 G Then k m k+m for every φ, continuous and compactly supported aj ak+> = aj . on G. j=1 >=1 j=1 n If all the aj = a, we denote a · a ...an as a n × n A 1 2 positive definite matrix An matrix , and call this the nth power of a. such that, for all u ∈ Rn,wehave power associative algebra A distributive al- (A(u), u) ≥ , 0 gebra A such that every element of A generates an associative subalgebra. with equality only when u = 0. power method of computing eigenvalues positive divisor A divisor that has only pos- An iterative method for determining the eigen- itive coefficients. value of maximum absolute value of an n × n matrix A. Let λ ,λ ,...,λn be eigenval- positive element An element g ∈ G, where 1 2 ues of A such that |λ | > |λ |≥···≥|λn| G is an ordered group, such that g ≥ e. 1 2 and let y1 be an eigenvector such that (λ1I − A)y = x(0) ab 1 0. Begin with a vector such that positive exponent For an expression , the (0) (0) (y1,x ) = 0 and for some i0, xi = 1. De- exponent b if b>0. 0 termine θ (0),θ(1),...,θ(m),... and x(1),x(2), (m+ ) (j) (j) (j+ ) positive matrix An n × n matrix A with real ...,x 1 ,... by Ax = θ x 1 . Then (j) (j) entries such that ajk > 0 for each j and k. See limj→∞ θ = λ1 and limj→∞ x is the also positive definite matrix. eigenvector corresponding to λ1. positive number A real number greater than power of a complex number Let z = x + zero. iy = r(cos θ + i sin θ) be a complex number 2 2 −1 y with r = x + y and θ = tan x . Let n be positive root Let S be a basis of a root system a positive number. The nth power of z will be rn( nθ + i nθ) φ in a vector spaceV such that each root β can be the complex number cos sin . written as β = a∈S maa, where the integers ma have the same sign. Then β is a positive root power-residue symbol Let n be a positive if all ma ≥ 0. integer and let K be an algebraic number field containing the nth roots of unity. Let α ∈ K× positive semidefinite matrix An n×n matrix and let ℘ be a prime ideal of the ring such that A such that, for all u ∈ Rn,wehave ℘ is relatively prime to n and α. The nth power is a positive integer and let K be an algebraic (A(u), u) ≥ 0 . number field containing the nth roots of unity.

c 2001 by CRC Press LLC Let α ∈ K× and let ℘ be a prime ideal of the groups (or rings, modules, etc.). There is a stan- ring of integers of K such that ℘ is relatively dard procedure for constructing a sheaf from a prime to n and α. The nth power residue symbol presheaf. α ℘ is the unique nth root of unity that is n primary Abelian group An Abelian group congruent to α(N℘−1)/n mod ℘. When n = 2 in which the order of every element is a power and K = Q, this symbol is the usual quadratic of a fixed prime number. residue symbol. primary component Let R be a commuta- predual Let X and Y be Banach spaces such tive ring with identity 1 and let J be an ideal ∗ that X is the dual of Y , X = Y . Then Y is of R. Assume J = I1 ∩···∩In with each Ii called the predual of X. primary and with n minimal among all such rep- resentations. Then each Ii is called a primary preordered set A structure space for a non- component of J . empty set R is a nonempty collection X of non- empty proper subsets of R given the hull-kernel primary ideal Let R be a ring with identity topology. If there exists a binary operation ∗ on 1. An ideal I of R is called primary if I = R R such that (R, ∗) is a commutative semigroup and all zero divisors of R/I are nilpotent. and the structure space X consists of prime semi- group ideals, then it is said that R has an X - primary linear programming problem A compatible operation. For p ∈ R, let Xp = linear programming problem in which the goal {A ∈ X : p/∈ A}. A preorder (reflexive and is to maximize the linear function z = cx with n a x = b (i = transitive relation) ≤ is defined on R by the rule the linear conditions j=1 ij j i that a ≤ b if and only if Xa ⊆ Xb. Then R is 1, 2,...,m)and x ≥ 0, where x = (x1,x2,..., called a preordered set. xn) is the unknown vector, c is an n × 1 vec- tor of real numbers, bi (i = 1, 2,...,n) and preordered set A structure space for a non- aij (i = 1, 2,...,n,j = 1, 2,...,n) are real empty set R is a nonempty collection X of non- numbers. empty proper subsets of R given the hull-kernel topology. If there exists a binary operation ∗ on primary ring Let R be a ring and let N be R such that (R, ∗) is a commutative semigroup the largest ideal of R containing only nilpotent and the structure space X consists of prime semi- elements. If R/N is nonzero and has no nonzero group ideals, then it is said that R has an X - proper ideals, R is called primary. compatible operation. For p ∈ R, let Xp = {A ∈ X : p/∈ A}. A preorder (reflexive and primary submodule Let R be a commuta- transitive relation) ≤ is defined on R by the rule tive ring with identity 1. Let M be an R-module. that a ≤ b if and only if Xa ⊆ Xb. Then R is A submodule N of M is called primary if when- called a preordered set. ever r ∈ R is such that there exists m ∈ M/N with m = 0butrm = 0, then rn(M/N) = presheaf Let X be a topological space. Sup- 0 for some integer n. pose that, for each open subset U of X, there is an Abelian group (or ring, module, etc.) F(U). prime A positive integer greater than 1 with Assume F(φ) = 0. In addition, suppose that the property that its only divisors are 1 and itself. whenever U ⊆ V there is a homomorphism The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 27 are the first ten primes. There are infinitely many ρUV : F(V ) → F(U) prime numbers. such that ρUU = identity and such that ρUW = prime divisor For an integer n,aprime di- ρUV ρVW whenever U ⊆ V ⊆ W. The col- visor is a prime that occurs in the prime factor- lection of Abelian groups along with the homo- ization of n. For an algebraic number field or morphisms ρUV is called a presheaf of Abelian for an algebraic function field of one variable (a

c 2001 by CRC Press LLC field that is finitely generated and of transcen- C-characters of G, where C denotes the field of dence degree 1 over a field K), a prime divi- complex numbers. Then χ ∈ Irr(G) is called a sor is an equivalence class of nontrivial valua- primitive character if χ = ϕG for any character tions (over K in the latter case). In the number ϕ of a proper subgroup of G. See also character field case, the prime divisors correspond to the of group, irreducible character. nonzero prime ideals of the ring of integers and the archimedean valuations of the field. primitive element Let E be an extension field of the field F (E is a field containing F prime element In a commutative ring R with as subfield). If u is an element of E and x is an identity 1, a prime element p is a nonunit such indeterminate, then we have the homomorphism that if p divides a product ab with a, b ∈ R, then g(x) → g(u) of the polynomial ring F [x] into p divides at least one of a, b. When R = Z, the E, which is the identity on F and send x → u.If ∼ prime elements are of the form ±p for prime the kernel is 0, then F [u] = F [x]. Otherwise, numbers p. we have a monic polynomial f(x) of positive degree such that the kernel is the principal ideal ∼ prime factor A prime factor of an integer n (f (x)), and then F [u] = F [x]/(f (x)). Then is a prime number p such that n is a multiple of we say E = F (u) is a simple extension of F p. and u a primitive element (= field generator of E/F). prime field The rational numbers and the fields Z/pZ for prime numbers p are called primitive equation An equation f(X) = 0 prime fields. Every field contains a unique sub- such that a permutation of roots of f(X)= 0is field isomorphic to exactly one of these prime primitive, where f(X)∈ K[X] is a polynomial, fields. and K is a field. prime ideal Let R be a commutative ring primitive hypercubic set A finite subgroup with identity 1 and let I = R be an ideal of R. K of the orthogonal group O(V) is called fully Then I is prime if whenever a,b ∈ R are such transitive if there is a set S ={e1,...,es} that that ab ∈ I, then at least one of a and b is in I. spans V on which K acts transitively and K has no invariant subspace in V . In this case, one can prime number A positive integer p is said choose S as either to be prime if (i.) the primitive hypercubic type: (i.) p>1, S = {e ,...,e } , e ,e = δ ; (ii.) p has no positive divisors except 1 and p. 1 n i j ij The first few prime numbers are 2, 3, 5, 7, 11, 13, 17. or (ii.) the primitive hyperbolic type: prime rational divisor A divisor p = S = {f ,...,fn+ } , niPi on X over k satisfying the following 1 1 three conditions: (i.) p is invariant under any au- (f ,f ) = 1,i=1,...n+1,i=j σ k/k¯ j i j 1 . tomorphism of ; (ii.) for any , there exists − n ,i,j=1,...,n+1,i=j σ k/k¯ P = P σj an automorphism j of such that j 1 ; (iii.) n1 = ··· = nt =[k(P1) : k]i, where X is primitive ideal Let R be a Banach algebra. a nonsingular irreducible complete curve, k is a A two-sided ideal I of R is primitive if there is a subfield of the universal domain K such that X regular maximal left ideal J such that I is the set is defined over k. Prime rational divisors gen- of elements r ∈ R with rR ⊆ I. The regularity erate a subgroup of the group of divisors G(X), of J means that there is an element u ∈ R such which is called a group of k-rational divisors. that r − ru ∈ J for all r ∈ R. primitive character Let G be a finite group primitive idempotent element An idempo- and let Irr(G) denote the set of all irreducible tent element that cannot be expressed as a sum

c 2001 by CRC Press LLC a + b with a and b nonzero idempotents satis- over V , and I (Q) is the two-sided ideal of T(V) fying ab = ba = 0. generated by elements x ⊗ x − Q(x) · 1 for x ∈ V . Compare with principal automorphism, primitive permutation representation Let i.e., the unique automorphism α of C(Q) such G be a group acting as a group of permutations that α(x) =−x, for all x ∈ V . of a set X. This is called a permutation represen- tation of G. This representation is called primi- principal automorphism Let A be a com- tive if the only equivalence relations R(x,y) on mutative ring and let M be a module over A. Let X such that R(x,y) implies R(gx,gy) for all a ∈ A. The homomorphism x,y ∈ X g ∈ G and all are equality and the M ) x *→ ax trivial relation R(x,y) for all x,y ∈ X. is called the principal homomorphism associ- f(x) primitive polynomial Let be a poly- ated with a, and is denoted aM . When aM is nomial with coefficients in a commutative ring one-to-one and onto, then we call aM a princi- R. When R is a unique factorization domain, pal automorphism of the module M. f(x)is called primitive if the greatest common divisor of the coefficients of f(x) is 1. For an principal divisor of functions The formal arbitrary ring, a slightly different definition is sum sometimes used: f(x) is primitive if the ideal (φ) = m p +···+m p + n q +···+n q generated by the coefficients of f(x)is R. 1 1 j j 1 1 k k

where p ,...,pj are the zeros and q ,...,qk primitive ring A ring R is called left prim- 1 1 are the poles of a meromorphic function φ, mi itive if there exists an irreducible, faithful left is the order of pi and ni is the order of qi. R-module, and R is called right primitive if R there exists an irreducible, faithful right - principal genus An ideal group of K formed R module. See also irreducible -module, faithful by the set of all ideals U of K relatively prime to R -module. m such that NK/k(U) belongs to H (m), where k is an algebraic number field, m is an integral m primitive root of unity Let be a positive divisor of k, T (m) is the multiplicative group R integer and let be a ring with identity 1. An of all fractional ideals of k which are relatively ζ ∈ R m element is called a primitive th root prime to m, S(m) is the ray modulo m, H (m) ζ m = ζ k = of unity if 1but 1 for all positive is an ideal group modulo m (i.e., a subgroup of k

c 2001 by CRC Press LLC Principal Ideal Theorem There are at least cipal submatrix of A. Its determinant is called two results having this name: a principal minor of A. For example, let (1) Let K be an algebraic number field and   a a a let H be the Hilbert class field of K. Every ideal 11 12 13 A = a a a  . of the ring of integers of K becomes principal 21 22 23 a a a when lifted to an ideal of the ring of integers of 31 32 33 H . This was proved by Furtwängler in 1930. Then, by deleting row 2 and column 2 we obtain (2) Let R be a commutative Noetherian ring the principal submatrix of A with 1. If x ∈ R and P is minimal among the prime ideals of R containing x, then the codi- a11 a13 mension of P is at most 1 (that is, there is no . a31 a33 chain of prime ideals P ⊃ P1 ⊃ P2 (strict in- clusions) in R). This was proved by Krull in Notice that the diagonal entries and A itself are 1928. principal submatrices of A. principal idele Let K be an algebraic number principal value (1) The principal values of × field. The multiplicative group K injects di- arcsin, arccos, and arctan are the inverse func- agonally into the group IK of ideles. The image tions of the functions sin x, cos x, and tan x, re- − π ≤ x ≤ π ≤ x ≤ is called the set of principal ideles. stricted to the domains 2 2 ,0 π − π ) matrix obtained from A by deleting certain k rows (k columns (>

c 2001 by CRC Press LLC reflection states that if the image of the circle the following way: under a linear fractional transformation w = (az + b)/(cz + d) is again a circle (this hap- Zp,q = Xp × Yq  pens unless the image is a line), then the images ∂p,q = ∂p × 1 w1 and w2 of z1 and z2 are symmetric with re-  p ∂ = (−1) 1 × ∂q . spect to this new circle. See also linear fractional p,q function. product formula (1) Let K be a finite exten- The Schwarz Reflection Principle of complex sion of the rational numbers Q. Then |x|v = analysis deals with the analytic continuation of v 1 for all x ∈ K, x = 0, where the product is over an analytic function defined in an appropriate all the normalized absolute values (both p-adic set S, to the set of reflections of the points of S. and archimedean) of K. (2) Let K be an algebraic number field con- product A term which includes many phe- taining the nth roots of unity, and let a and b nomena. The most common are the following: be nonzero elements of K. For a place v of K a,b (as in (1) above), let ( v )n be the nth norm- (1) The product of a set of numbers is the re- a,b residue symbol. Then ( )n = 1. See also sult obtained by multiplying them together. For v v norm-residue symbol. an infinite product, this requires considerations of convergence. profinite group Any group G can be made

(2)IfA1,...,An are sets, then the product into a topological group by defining the collec- A1 × ··· × An is the set of ordered n-tuples tion of all subgroups of finite index to be a neigh- (a1,...,an) with ai ∈ Ai for all i. This defini- borhood base of the identity. A group with this tion can easily be extended to infinite products. topology is called a profinite group.

(3) Let A1 and A2 be objects in a category C. projection matrix A square matrix M such 2 A triple (P, π1,π2) is called the product of A1 that M = M. and A2 if P is an object of C, πi : P → Ai is a morphism for i = 1, 2, and if whenever X is projective algebraic variety Let K be a ¯ n ¯ another object with morphisms fi : X → Ai, field, let K be its algebraic closure, and let P (K) for i = 1, 2, then there is a unique morphism be n-dimensional projective space over K¯ . Let S f : X → P such that πif = fi for i = 1, 2. be a set of homogeneous polynomials in X0,..., n ¯ Xn. The set of common zeros Z of S in P (K) (4) See also bracket product, cap product, is called a projective algebraic variety. Some- crossed product, cup product, direct product, times, the definition also requires the set Z to be Euler product, free product, Kronecker prod- irreducible, in the sense that it is not the union uct, matrix multiplication, partial product, ten- of two proper subvarieties. sor product, torsion product, wedge product. projective class Let A be a category. A pro- P A product complex Let C be a complex of jective class is a class of objects in such 1 A ∈ A P ∈ P right modules over a ring R and let C be a that for each there is a and a 2 P f : P → A complex of left R-modules. The tensor product -epimorphism . C ⊗R C gives a complex of Abelian groups, 1 2 projective class group Consider left mod- called the product complex. ules over a ring R with 1. Two finitely gener- ated projective modules P1 and P2 are said to product double chain complex The dou- be equivalent if there are finitely generated free   ble chain complex (Zp,q,∂ ,∂ ) obtained from modules F1 and F2 such that P1⊕F1 , P2 ⊕F2. a chain complex X of right A-modules with The set of equivalence classes, with the opera- boundary operator ∂p and a chain complex Y tion induced from direct sums, forms a group of left A-modules with boundary operator ∂q in called the projective class group of R.

c 2001 by CRC Press LLC projective cover An object C in a category there is a surjection f : A → M, there is a C is the projective cover of an object A if it sat- homomorphism g : M → A such that fg is the isfies the following three properties: (i.) C is identity map of M. a projective object. (ii.) There is an epimor- phism e : C → A. (iii.) There is no projective projective morphism A morphism f : X → object properly between A and C. In a gen- Y of algebraic varieties over an algebraically eral category, this means that if g : C → A closed field K which factors into a closed im- n and f : C → C are epimorphisms and C is mersion X → P (K) × Y , followed by the pro- projective, then f is actually an isomorphism. jection to Y . This concept can be generalized to Thus, projective covers are simply injective en- morphisms of schemes. velopes “with the arrows turned around.” See also epimorphism, injective envelope. projective object An object P in a category C In most familiar categories, objects are sets satisfying the following mapping property: If e : with structure (for example, groups, topologi- C → B is an epimorphism in the category, and cal spaces, etc.) and morphisms are particu- f : P → B is a morphism in the category, then lar kinds of functions (for example, group ho- there exists a (usually not unique) morphism g : momorphisms, continuous functions, etc.), and P → C in the category such that e ◦ g = f . epimorphisms are onto functions (surjections) This is summarized in the following “universal of a particular kind. Here is an example of a mapping diagram”: projective cover in a specific category: In the P category of compact Hausdorff spaces and con- f -.∃g tinuous maps, the projective cover of a space X e B ←− C always exists, and is called the Gleason cover of the space. It may be constructed as the Stone Projectivity is simply injectivity “with the ar- space (space of maximal lattice ideals) of the rows turned around.” See also epimorphism, Boolean algebra of regular open subsets of X. injective object, projective module. A subset is regular open if it is equal to the inte- In most familiar categories, objects are sets rior of its closure. See also Gleason cover, Stone with structure (for example, groups, topologi- space. cal spaces, etc.), and morphisms are particular kinds of functions (for example, group homo- projective dimension Let R be a ring with morphisms, continuous maps, etc.), so epimor- 1 and let M be an R-module. The projective phisms are onto functions (surjections) of partic- dimension of M is the length of the smallest ular kinds. Here are two examples of projective projective resolution of M; that is, the projective objects in specific categories: (i.) In the cate- dimension is n if there is an exact sequence 0 → gory of Abelian groups and group homomor- Pn → ··· → P0 → M → 0, where each phisms, free groups are projective. (An Abelian Pi is projective and n is minimal. If no such group G is free if it is the direct sum of copies finite resolution exists, the projective dimension of the integers Z.) (ii.) In the category of com- is infinite. pact Hausdorff spaces and continuous maps, the projective objects are exactly the extremely dis- projective general linear group The quo- connected compact Hausdorff spaces. (A com- tient group defined as the group of invertible pact Hausdorff space is extremely disconnected matrices (of a fixed size) modulo the subgroup if the closure of every open set is again open.) of scalar matrices. See also compact topological space, Hausdorff space. projective limit The inverse limit. See in- verse limit. projective representation A homomorphism from a group to a projective general linear group. projective module A module M for which there exists a module N such that M ⊕ N is projective resolution Let B be a left R mod- free. Equivalently, M is projective if, whenever ule, where R is a ring with unit. A projective

c 2001 by CRC Press LLC resolution of B is an exact sequence, projective symplectic group The quotient group defined as the group of symplectic ma- φ2 φ1 φ0 ···−→ E1 −→ E0 −→ B −→ 0 , trices (of a given size) modulo the subgroup {I,−I}, where I is the identity matrix. See sym- where every Ei is a projective left R module. plectic group. (We shall define exact sequence shortly.) There is a companion notion for right R modules. Pro- projective unitary group The quotient group jective resolutions are extremely important in defined as the group of unitary matrices modulo homological algebra and enter into the dimen- the subgroup of unitary scalar matrices. See uni- sion theory of rings and modules. See also flat tary matrix. resolution, injective resolution, projective mod- ule, projective dimension. proper component Let U and V be irre- An exact sequence is a sequence of left R ducible subvarieties of an irreducible algebraic modules, such as the one above, where every variety X. A simple irreducible component of φi is a left R module homomorphism (the φi U ∩ V is called proper if it has dimension equal are called “connecting homomorphisms”), such to dim U + dim V - dim X. that Im(φi+1) = Ker(φi). Here Im(φi+1) is the image of φi+1, and Ker(φi) is the kernel of φi.In proper equivalence An equivalence relation the particular case above, because the sequence R on a topological space X such that R[K]= {x ∈ X : xRk k ∈ K} ends with 0, it is understood that the image of φ0 for some is compact for K ⊆ X is B, that is, φ0 is onto. There is a companion all compact sets . notion for right R modules. proper factor Let a and b be elements of a projective scheme A projective scheme over commutative ring R. Then a is a proper factor a scheme S is a closed subscheme of projective of b if a divides b,buta is not a unit and there space over S. is no unit u with a = bu. projective space Let K be a field and con- proper fraction A positive rational number sider the set of (n + 1)-tuples (x0,...,xn) in such that the numerator is less than the denom- Kn+1 with at least one coordinate nonzero. Two inator. See also improper fraction.  tuples (...,xi,...)and (...,xi,...)are equiv- ×  Y Z alent if there exists λ ∈ K such that xi = proper intersection Let and be subva- n λxi for all i. The set P (K) of all equivalence rieties of an algebraic variety X. If every irre- classes is called n-dimensional projective space ducible component of Y ∩ Z has codimension over K. It can be identified with the set of equal to codimY +codimZ, then Y and Z are said lines through the origin in Kn+1. The equiv- to intersect properly. alence class of (x0,x1,...,xn) is often denoted (x0 : x1 : ··· : xn). More generally, let R be a proper Lorentz group The group formed commutative ring with 1. The scheme Pn(R) is by the Lorentz transformations whose matrices given as the set of homogeneous prime ideals of have determinants greater than zero. R[X0,...,Xn] other than (X0,...,Xn), with a structure sheaf defined in terms of homogeneous proper morphism of schemes A morphism rational functions of degree 0. It is also possible of schemes f : X → Y such that f is sepa- to define projective space Pn(S) for a scheme rated and of finite type and such that for every S by patching together the projective spaces for morphism T → Y of schemes, the induced mor- appropriate rings. phism X ×T Y → T takes closed sets to closed sets. projective The quotient group defined as the group of matrices (of a fixed proper orthogonal matrix An orthogonal size) of determinant 1 modulo the subgroup of matrix with determinant +1. See orthogonal scalar matrices of determinant 1. matrix.

c 2001 by CRC Press LLC proper product Let R be an integral domain ness condition for all prime ideals P is called with field of quotients K and let A be an algebra pseudogeometric. over K. Let M and N be two finitely generated R-submodules of A such that KM = KN = A. pseudovaluation A map v from a ring R into If {a ∈ A : Ma ⊆ M}={a ∈ A : aN ⊆ N}, the nonnegative real numbers such that (i.) v(r) and this is a maximal order of A, then the product = 0 if and only if r = 0, (ii.) v(rs) ≤ v(r)v(s), MN is called a proper product. (iii.) v(r + s) ≤ v(r)+ v(s), and (iv.) v(−r) = v(r) for all r, s ∈ R. proper transform Let T : V → W be a rational mapping between irreducible varieties p-subgroup A finite group whose order is and let V  be an irreducible subvariety of V .A a power of p is called a p-group. A p-group proper transform of V  by T is the union of all that is a subgroup of a larger group is called a irreducible subvarieties W  of W for which there p-subgroup. is an irreducible subvariety of T such that V  and W  correspond. p-subgroup For a finite group G and a prime integer p, a subgroup S of G such that the order S p proportion A statement equating two ratios, of is a power of . a c b = d , sometimes denoted by a : b = c : d. The terms a and d are the extreme terms and the pure imaginary number An imaginary num- terms b and c are the mean terms. ber. See imaginary number.

a pure integer programming problem A proportional A term in the proportion b = c problem similar to the primary linear program- d . Given numbers a, b, and c, a number x which a c ming problem in which the solution vector x = satisfies b = x is a fourth proportional to a, b, (x ,x ,...,xn) is a vector of integers: the prob- and c. Given numbers a and b, a number x 1 2 a b lem is to minimize z = cx with the conditions which satisfies b = x is a third proportional to a x Ax = b, x ≥ 0, and xj (j = 1, 2,...,n)is an a and b, and a number x which satisfies = x b integer, where c is an n × 1 vector of real num- is a mean proportional to a and b. bers, A is an m × n matrix of real numbers, and b is an m × 1 matrix of real numbers. proportionality The state of being in pro- portion. See proportion. purely infinite von Neumann algebra A von Neumann algebra A which has no semifinite R Prüfer domain An integral domain such normal traces on A. that every nonzero ideal of R is invertible. Equiv- R alently, the localization M is a valuation ring purely inseparable element Let L/K be an M for every maximal ideal . Another equivalent extension of fields of characteristic p>0. If R pn condition is that an -module is flat if and only α ∈ L satisfies α ∈ K for some n, then α is if it is torsion-free. called a purely inseparable element over K.

Prüfer ring An integral domain R such that purely inseparable extension An extension all finitely generated ideals in R are inversible L/K of fields such that every element of L is in the field of quotients of R. See also integral purely inseparable over K. See purely insepa- domain. rable element. pseudogeometric ring Let A be a Noethe- purely inseparable scheme Givenanir- rian integral domain with field of fractions K.If reducible polynomial f(X) over a field k,if the integral closure of A in every finite extension the formal derivative df/dX = 0, then f(X) of K is finitely generated over A, then A is said is inseparable; otherwise, f is separable. If to satisfy the finiteness condition. A Noethe- char(k) = 0, every irreducible polynomial rian ring R such that R/P satisfies the finite- f(X)(= 0) is separable. If char(k) = p>0,

c 2001 by CRC Press LLC an irreducible polynomial f(X) is inseparable √Pythagorean field A field F that contains if and only if f(X) = g(Xp). An algebraic a2 + b2 for all a,b ∈ F . element α over k is called separable or insep- arable over k if the minimal polynomial of α Pythagorean identities The following ba- over k is separable or inseparable. An alge- sic identities involving trigonometric functions, braic extension of k is called separable if all resulting from the : elements of K are separable over k; otherwise, 2(x) + 2(x) = K is called inseparable. If α is inseparable, then sin cos 1 k p has nozero characteristic and the minimal 2(x) + = 2(x) polynomial f(X) of α can be decomposed as tan 1 sec r r r p p p 2 2 f(X)= (X − α1) (X − α2) ...(X− αm) , 1 + cot (x) = csc (x) . r ≥ 1, where α ,...,αm are distinct roots of 1 r f(X)in its splitting field. If αp ∈ k for some Pythagorean numbers Any combination of r, we call α purely inseparable over k. An al- three positive integers a, b, and c such that a2 + gebraic extension K of k is called purely in- b2 = c2. separable if all elements of the field are purely inseparable over k. Let V ⊂ kn be a reduced V Pythagorean ordered field An ordered field irreducible affine algebraic variety. is called P such that the square root of any positive ele- purely inseparable k(V ) if the function field is ment of P is in P . purely inseparable over k. Therefore, we can de- fine the same notion for reduced irreducible al- Pythagorean Theorem Consider a right tri- gebraic varieties. Since there is a natural equiva- angle with legs of length a and b and hypotenuse lence between the category of algebraic varieties of length c. Then a2 + b2 = c2. over k and the category of reduced, separated, al- k gebraic -schemes, by identifying these two cat- Pythagorean triple A solution in positive egories, we define purely inseparable reduced, integers x,y,z to the equation x2 + y2 = z2. k irreducible, algebraic -schemes. Some examples are (3, 4, 5), (5, 12, 13), and (20, 21, 29).Ifx,y,z have no common divi- purely transcendental extension An exten- sor greater than 1, then there are integers a, b L/K sion of fields such that there exists a set such that x = a2 −b2, y = 2ab, and z = a2 +b2 {x } of elements i i∈I , algebraically independent (or the same equations with the roles of x and K L = K({x }) over , with i . See algebraic inde- y interchanged). Also called Pythagorean num- pendence. bers. pure quadratic A quadratic equation of the form ax2 + c = 0, that is, a quadratic equation with the first degree term bx missing.

c 2001 by CRC Press LLC quadratic form A polynomial of the form Q(X1,...,Xn) = i,j cij XiXj , where the co- efficients cij lie in some ring.

quadratic formula The roots√ of aX2 +bX+ Q −b± b2− ac c = X = 4 0 are given by 2a (in a field of characteristic other then 2). QR method of computing eigenvalues An iterative method of finding all of the eigenvalues quadratic function A polynomial function of degree two: ax2 + bx + c. For example, of an n × n matrix A. Let A0 = A. Determine 8x2 + 3x − 1 is a quadratic function. See also A1,A2,...,Am,... in the following way: If A is a real tridiagonal matrix or a complex upper pure quadratic. Hessenberg matrix, let sm be the eigenvalue of (m) quadratic inequality Any inequality of one the 2 × 2 matrix closer to ann in the lower right of the forms: corner of Am,orifA is a real upper Hessenberg s s 2 matrix, let m and m+1 be the eigenvalues of ax + bx + c<0 the 2 × 2 matrix in the lower right corner of Am. ax2 + bx + c>0 With this (these) value(s) of sm, write Am −smI ax2 + bx + c ≤ as QmRm, where Qm is a unitary matrix and 0 2 Rm is an upper Hessenberg matrix. Then de- ax + bx + c ≥ 0 . fine Am+1 = RmQm − smI. Then the elements on the diagonal of limm→∞ Am converge to the eigenvalues of A. quadratic polynomial A polynomial of the form aX2 + bX + c. quadratic Of degree 2, as, for example, a quadratic programming Theoretical as- aX2 + bX + c a = polynomial , with 0. pects and methods pertaining to minimizing or maximizing quadratic functions on sets X with quadratic differential On a Riemann sur- constraints determined by linear equations and S face , a rule which associates to each local pa- linear inequalities. rameter z mapping a parametric neighborhood U ⊂ S into the extended complex plane C quadratic programming problem A qua- (z : U → C), a function Qz : z(U) → C such dratic programming problem in which one wants t t that for any local parameter z1 : U1 → C and z = − 1 D to maximize or minimize c x 2 x x with z2 : U2 → C with U1 ∩ U2 = ∅, the following the constraints Ax ≤ b and x ≥ 0, where A is holds in this intersection: an m × n matrix of real numbers, c ∈ Rn, D is n×n ()t   an matrix of real numbers, and denotes Q (z (p)) dz (p) 2 z2 2 1 transposition. = , ∀p ∈ U1 ∩U2 . Qz (z1(p)) dz2(p) 1 quadratic reciprocity Let p and q be dis- tinct odd primes. If at least one of p and q is z(U) U z Here is the image of in C under . congruent to 1 mod 4, then p is congruent to a square mod q if and only if q is congruent to a quadratic equation A polynomial equation square mod p. If both p and q are congruent to 2 of the form aX +bX+c = 0. The solutions are 3 mod 4, then p is congruent to a square mod q given by the quadratic formula. See quadratic if and only if q is not congruent to a square mod formula. p. √ quadratic√ field A field of the form Q( d) = quadratic residue An integer r is a quadratic {a + b d : a,b ∈ Q}, where Q is the field of residue mod n if there exists an integer x with rational numbers and d ∈ Q is not a square. r ≡ x2 mod n. See modular arithmetic.

c 2001 by CRC Press LLC quadratic transformation The map from Neumann algebra generated by the set π1(A) the projective plane to itself given by (x0 : x1 : onto the von Neumann algebra generated by the x2) → (x1x2 : x0x2 : x0x1). See also projective set π2(A) such that space. ' (π1(a)) = π2(a), for all a ∈ A. quartic Of degree 4, as, for example, a poly- nomial aX4 +bX3 +cX2 +dX+e, with a = 0. quasi-Frobenius algebra Let A be an alge- A∗ quartic equation A polynomial equation of bra over a field and let be the dual of the A A A the form aX4 + bX3 + cX2 + dX + e = 0. right -module . Decompose into a di- rect sum of indecomposable left ideals and a direct sum of indecomposable right ideals: A = quasi-affine algebraic variety A locally   (j)   (j) (j) closed subvariety of affine space. See affine i j Aei = i j ei A, where Aei   space. (j )  Aei if and only if i = i , and similarly for the right ideals. A is called quasi-Frobenius if there quasi-algebraically closed field A field F exists a permutation π of the indices i such that such that, for every d ≥ 1, every homogeneous (1) (1) ∗ Aei  (eπ(i)A) . polynomial equation of degree d in more than d F variables has a nonzero solution in . Finite quasi-Fuchsian group A Fuchsian group is K(X) fields and rational function fields with a group of conformal homeomorphisms of the K algebraically closed are quasi-algebraically Riemann sphere which leaves invariant a round closed. circle in the sphere (equivalently a group of isometries of hyperbolic 3 space which leaves (X, R) quasi-coherent module Let be a invariant a flat plane). A quasi-Fuchsian group R ringed space. A sheaf of -modules is called is a quasiconformal deformation of a Fuchsian x ∈ X quasi-coherent if, for each , there exists group: there is a quasiconformal homeomor- U x M| an open set containing such that U can phism of the sphere conjugating the action of the A/B A B be expressed as where and are free Fuchsian group to the group in question. It fol- R| U -modules. lows that abstractly the group is a surface group. Quasi-Fuchsian groups are extremely important quasi-dual space The quotient space of the for hyperbolic 3-manifolds, for instance, play- space of quotient representations of a locally ing a central role in the geometrization of a large G compact group , considered with the Borel class of 3-manifolds. See also Kleinian group. structure subordinate to the topology of uniform convergence of matrix entries on compact sets. quasi-group A set with a not necessarily as- sociative law of composition (a, b) → ab such quasi-equivalent representation Two uni- that the equation ab = c has a unique solution π π G tary representations 1, 2 of a group (or when any two of the variables are specified. symmetric representations of a symmetric alge- bra A) in Hilbert spaces H1 and H2, respectively, quasi-inverse Let x,y be elements of a ring. satisfying one of the following four equivalent Then y is called a quasi-inverse of x if x + y − conditions: xy = x+y−yx = 0. If 1 exists, this means that (i.) there exist unitarily equivalent representa- (1−y)is the inverse of (1−x). An element that tions µ1 and µ2 such that µ1 is a multiple of π1 has a quasi-inverse is called quasi-invertible. and µ2 is a multiple of π2; (ii.) the non-zero subrepresentations of π2 are quasi-inverse element See quasi-inverse. not disjoint from π1; (iii.) π2 is unitarily equivalent to a subrepresen- quasi-invertible element See quasi-inverse. tation of some multiple representation µ2 of π2 that has unit central support; quasi-local ring A commutative ring with (iv.) there exists an isomorphism ' of the von exactly one maximal ideal. Often such rings are

c 2001 by CRC Press LLC called local rings, but some authors reserve the quaternion algebra See Hamilton’s quater- term local ring for Noetherian rings with unique nion algebra. maximal ideals. quaternion group The group of order 8 equal quasi-projective algebraic variety An open to {±1, ±i, ±j, ±k}, with relations i2 = j 2 = subset of a projective variety. k2 =−1, ij = k, ji =−k and (−1)a = a(−1) for all a. See also Hamilton’s quaternion quasi-projective morphism A morphism f : algebra. X → Y of varieties, or schemes, that factors X → X as an open immersion followed by a quintic Of degree 5, as, for example, a poly- X → Y projective morphism . See projective nomial aX5 + bX4 + cX3 + dX2 + eX + f , morphism. with a = 0. quasi-projective scheme An open sub- quintic equation A polynomial equation of See scheme of a projective scheme. projective the form aX5+bX4+cX3+dX2 +eX+f = 0. scheme. quotient (1) The result of dividing one num- quasi-semilocal ring A commutative ring ber by another. See also numerator, denomina- with only finitely many maximal ideals. Of- tor. ten such rings are called semilocal rings, but some authors reserve the term semilocal ring (2) The result of starting with a set (group, for Noetherian rings with finitely many maxi- ring, topological space, etc.) and forming a new mal ideals. set consisting of equivalence classes under some equivalence relation. See also quotient algebra, quasi-symmetric homogeneous Siegel domain quotient chain complex, quotient G-set, quotient A Siegel domain S = S(U,V,,,H)satisfying group, quotient Lie algebra, quotient Lie group, the following conditions: quotient representation, quotient ring, quotient (i.) , is homogeneous and self-dual; set. (ii.) Ru is associated with Tu for all u ∈ U; where U is a vector space over R of dimension quotient algebra The algebra of equivalence m; V is a vector space over C of dimension n; classes of an algebra modulo a two-sided ideal. , is a non-degenerate open convex cone in U with vertex at the origin; H is a Hermitian map quotient bialgebra The quotient space H/I H : V × V → U = U ⊗ u ∈ U c R C; for c, with the induced bialgebra structure, where I is  Ru ∈ End(V ) is defined by (u, H (v, v )) = bi-ideal of a bialgebra H , i.e., I is an ideal of   2h(v, Ruv ), v, v ∈ V ; for each u ∈ U, there an algebra (H,µ,η)which is also a co-ideal of exists a unique element Tu ∈ p(,) such that a co-algebra (H,4,5), and (H,µ,η,4,5)is a Tue = u; and g(,) = t(,) + p(,) is the Car- bialgebra. See also bialgebra. tan decomposition. Notice that a Siegel domain S symmetric is if and only if (i.) and (ii.) above quotient bundle Let π : X → B be a vector and the following condition (iii.) hold: π  : X → B R bundle and let be a subbundle of (iii.) u satisfies the relation π. The quotient bundle of π is the union of vec- tor spaces π −1(b)/(π )−1(b) with the quotient H(R(H(v ,v))v, v ) " " topology.  = H(v , R(H (v, v"))v"). quotient chain complex Let (X, ∂) be a chain complex, where X = Xn, and let Y = Yn quaternion An element of the Hamiltonian be a subcomplex (so ∂Y ⊆ Y ). Then X/Y with quaternions. See Hamilton’s quaternion alge- the map induced from ∂ forms the quotient chain bra. complex. See also chain complex.

c 2001 by CRC Press LLC quotient co-algebra The quotient space C/I quotient Lie group Let G be a Lie group with a co-algebra structure induced naturally and let H be a closed normal subgroup. The from a co-algebra (C,4,5), where I is a co- quotient group can be given the structure of a ideal of C. See also coalgebra. Lie group and is called a quotient Lie group. See also quotient group. quotient group Let G be a group and let H be a normal subgroup. The set of all cosets forms a quotient representation Let ρ : G → GL(V ) group, denoted G/H , where the group operation be a representation of a group G as a group of is defined by (aH ) ∗ (bH ) = (a ∗ b)H (where linear transformations of a vector space V . Let a ∗ b denotes the group operation in G). For W ⊆ V be a subspace such that ρ(G)W ⊆ W. example, if G = Z, the integers under addition, The quotient representation is the induced map and H = nZ, the multiples of a fixed integer G → GL(V /W). See induced representation. n, then G/H is the additive group of integers modulo n. quotient ring Let R be a ring and let I be a two-sided ideal. The set R/I of cosets with quotient G-set Let G be a group acting on a respect to the additive structure can be given the set X and let R be a G-compatible equivalence structure of a ring by defining (a+I)+(b+I) = relation on X, that is, xRy implies gxRgy, for a + b + I and (a + I)(b + I) = ab + I, where all x,y ∈ X, and all g ∈ G. The quotient set a,b ∈ R. This is called a quotient ring. The X/R, which has a natural action of G, is called standard example is when R = Z, the integers, a quotient G-set. See also G-set. and I = nZ, the multiples of a fixed integer n. Then R/I is the ring of integers modulo n. quotient Lie algebra Let g be a Lie algebra with bracket [·, ·], and let a be a Lie subalgebra quotient set The set of equivalence classes, such that [g, a]⊆a (so a is an ideal of g). The denoted X/R, of a set X with respect to an equiv- quotient g/a with the bracket induced from that alence relation R on X. of g is called a quotient Lie algebra. See also quotient algebra.

c 2001 by CRC Press LLC (4) The radical of an algebraic group is the largest connected closed solvable normal sub- group. (5) The radical of a Lie algebra is the largest R solvable ideal.

radical equation An equation in which the Racah algebra Let A be a dynamical quan- √variable is under a radical. For example, tity and let φ and φ be irreducible components √ 1 2 x2 + 1 − 2 3 x = 3x is a radical equation. of the state obtained by combining n angular mo- menta. Racah algebra is a systematic method radical of a group The maximal solvable of calculating the matrix element (φ ,Aφ ) in 1 2 normal subgroup of a group G. In other words, quantum mechanics. a solvable normal subgroup R of G, which is not Racah coefficient Certain constants in Racah contained in a larger solvable normal subgroup G algebra. Consider irreducible representations of of . See also solvable group, normal subgroup. the rotation groups R = SO(3). The reduction √ n a of the tensor product of three irreducible repre- radicand In an expression , the quantity a. sentations can be done in two ways, (D(j1) ⊗ D(j2)) ⊗ D(j3) and D(j1) ⊗ (D(j2) ⊗ D(j3)) and two corresponding sets of basis vectors. The The base of a system of numbers. For transformation coefficient for the two ways of example, 10 is the radix of the decimal system. reduction is written in the following form: ramification field See ramification group. j1j2 (j12) j3; j|j1,j2j3 (j23) ; j ramification group Let L/K be a Galois ex- = (2j12 + 1)(2j23 + 1)W (j1j2jj3; j12j23) . tension of algebraic number fields and let R be L P W(abcd; ef ) Racah coeffi- the ring of algebraic integers in . Let be Here are called the R Z cients. a prime ideal of and let be the decompo- sition group for P (the set of Galois elements g gP = P m radian A measure of the size of an angle. If such that ). The th ramification V (m) {g ∈ Z : gx ≡ the vertex of an angle is at the center of a is defined to be x( P m+1) x ∈ } of radius 1, the length of the arc on the circle mod for all R . The fixed field of V (m) m P determined by the angle is the radian measure is called the th ramification field of . of the angle. For example, a right angle is π/2 S/R radians. One radian equals 180/π, which is ap- ramification index (1)If is a finite ex- tension of Dedekind domains and ℘ is a prime proximately 57.3 degrees. e e R ℘S = ℘ 1 ...℘ g ideal of , then 1 g , where ℘ ,...,℘ S radical A term that has many meanings: 1 g are prime ideals of . The number (1) The nth root of a number. ei is called the ramification index of ℘i. (2)IfR is a commutative ring with 1 and (2)IfL/K is an extension of fields and v is a I is an ideal in R, the radical of I is the set of (multiplicative) valuation on L, then the group × × r ∈ R such that rn ∈ I for some n (depending on index [v(L ) : v(K )] is called the ramifica- r). The radical of the zero ideal is often called tion index of v. the radical of R, or the nilradical of R.Itis the intersection of all prime ideals of R. The ramification numbers Let N be a normal Jacobson radical of R is the intersection of all subgroup of a finite group G. Suppose ϕ ∈ IrrN maximal ideals of R. (the irreducible characters of N) is invariant in (3) In an arbitrary ring, the radical is the G.Ifχ is an irreducible constituent of ϕG, largest ideal that is contained in the set of quasi- then the ramification number e(χ) is the pos- invertible elements. itive integer such that χN = e(χ)ϕ. See also

c 2001 by CRC Press LLC normal subgroup, invariant element, irreducible K : v(x) ≥ 0}; that is, the supremum of the constituent. lengths n of chains of prime ideals ℘0 ⊃ ··· ⊃ ℘n (strict inclusion) in R. Ramification Theorem Let K be a number field (finite extension of Q). Then a prime p rational This term often refers to a quantity ramifies in K if and only if p divides the dis- equal to the ratio of two integers or polynomi- criminant of K. als. It is also used to describe a point on an algebraic variety with coordinates in a field (the ramified covering A triple (X,Y,π), where point is then said to be rational over this field). X and Y are connected normal complex spaces See rational point. and π is a finite surjective proper holomorphic map. For any such (X,Y,π), there exists a rational action The action of a matrix group proper analytic subset of X, outside which π G by means of a rational representation of G. is an unramified covering. See rational representation. ramified extension An extension of alge- rational cohomology group A cohomology braic number fields, for example, in which at group for transformation spaces of linear alge- least one prime ideal is ramified (i.e., has rami- braic groups G over a field K introduced by fication index greater than 1). See ramification Hochschild (1961). In particular, if G is a unipo- index. tent algebraic group over a field K of character- istic 0, then H(G,A)is isomorphic to H(g,A) S/R ramified prime ideal Let be a finite where g is the Lie algebra of G and A is a g- P extension of Dedekind domains and let be a module. prime ideal of S. If the ramification index of P is P ramified. greater than 1, the ideal is said to be rational curve An algebraic curve of genus See ramification index. 0. Such curve in n-dimensional affine space can be described by rational functions x = range See image. 1 R1(t),...,xn = Rn(t) in terms of a single pa- rameter t. rank of a group Let G be a finitely gener- ated group, having g1,g2,...,gn as generators. g g g rational equivalence Let X and X be cy- Let G˜ have generators 1 , 2 ,..., n . Then the 1 2 1 1 1 cles on a nonsingular irreducible algebraic vari- rank of the group G is the dimension of the finite ety V over a field K, and let A1 be the affine dimensional rational vector space G˜ . K line over K. Suppose there exists a cycle Z on V × A1 such that the intersection Z · (V × a) rank of element The number of minimal K a ∈ 1 nonzero faces of an element A of a complex ). is defined for all AK and such that there are a ,a ∈ 1 Z · (V × a ) − Z · (V × Here a complex ) is a set with an order relation 1 2 AK such that 1 a ) = X − X X X ⊂ (read “is a face of”) such that, for a given el- 2 1 2. Then 1 and 2 are rationally ement A, the ordered subset S(A) of all faces of equivalent. A is isomorphic to the set of all subsets of a set. rational expression Any algebraic expres- rank of matrix The maximal number of lin- sion which can be written as a polynomial di- early independent columns of a rectangular ma- vided by a polynomial. trix A = (aij ) with entries in a field. This also equals the maximal number of linearly indepen- rational function The quotient of two poly- dent rows of A and is the size of the largest nomials (possibly in several variables). nonzero minor of A. rational function field A field K(X1,..., rank of valuation For an additive valuation Xn), where K is a given field and X1,...,Xn v on a field K, the Krull dimension of R ={x ∈ are indeterminates.

c 2001 by CRC Press LLC rational homomorphism A homomorphism let X be irreducible. (A variety is the solution G1 → G2 of algebraic groups which is also a set of a system of polynomial equations. Irre- rational map of algebraic varieties. ducible means the variety X is not the union of two proper (= 0, = X) subvarieties.) A rational rational injectivity The notion that was used map F is an equivalence class of pairs (f, U), by Hochschild (1961) to introduce the rational where U is an open subset of X and f : U → Y cohomology group for transformation spaces of is a continuous function which preserves regular linear algebraic group G over a field k. See functions. Preserving regular functions means rational cohomology group. that if h : W → k is a regular function, where W is an open subset of Y , then h ◦ f is regular rationalization The process of transforming on the open set {x ∈ U : f(x)∈ W}. Two such a radical expression into an expression with no pairs (f, U) and (g, V ) are deemed equivalent radicals, or with all radicals in the numerators. if f = g on U ∩ V . It is a theorem that, given See also rationalizing the denominator. a rational map F , there is a largest open set UM such that a pair (f, UM ) ∈ F . Thus, one may rationalization of denominator Removing think of a rational map as an actual function de- radicals from the denominator of an algebraic fined on this maximal open set UM . The set UM expression.√ For example,√ in an expression of the is referred to as the set of points on which F (a + b n)/(c + d n) a, b, c, d, n form , where is defined. See also irreducible variety, regular are rational numbers, multiplying√ numerator function. and denominator√ by c − d n. The result is (r + s n)/t r, s, t of the form , where are ratio- rational number A number that can be ex- nal numbers. In some elementary mathematics pressed as the quotient of two integers m/n, with courses, it is taught that radicals in the denomi- n = 0. For example, 1/2, −9/4, 3, and 0 are ra- nator are undesirable. tional numbers. The set of all rational numbers forms a field, denoted Q. See field. rationalization of equation The process of transforming a radical equation into an equation rational operation Any of the four oper- with no radicals. The method is to manipulate, ations of addition, subtraction, multiplication, algebraically, the equation so only radicals are and division. on one side of the equal sign and then raise both sides of the equation to the index of one of the rational point Let V be an algebraic variety radicals. The process may need to be repeated. defined over a field k. A point P on V is rational over k if the coordinates of P (in some affine rationalizing the denominator The process open set containing P ) lie in k. See also k- of removing radicals from the denominator of rational point. a fraction. The method is to multiply the nu- merator and the denominator of the fraction by rational rank of valuation Let v be an addi- a rationalizing factor. For example, tive valuation on a field K with values in an or- √ √ 1 1 2 2 dered additive group. The supremum of the car- √ = √ · √ = . dinalities of linearly independent (over Z) sub- 2 2 2 2 sets of v(K×) is called the rational rank of v. rational map (1) Informally, a rational func- rational representation Let R be a commu- tion. See rational function. tative ring with 1 and let G be a subgroup of (2) In algebraic geometry, a function from an GLn(R) for some n. Let ρ : G → GLm(R) open subset U of a variety X to another variety for some m be a homomorphism. Suppose Y , which preserves regular functions. (Regu- there exist m2 rational functions in n2 vari- lar functions are functions which can be rep- ables pst({Xij }) ∈ R[{Xij }] with ρ((gij )) = resented locally as quotients of polynomials.) (pst(guv)) for all (gij ) ∈ G. Then ρ is called a In more detail, let X and Y be varieties, and rational representation.

c 2001 by CRC Press LLC Rational Root Theorem Let R be a unique realizable linear representation Let R and factorization domain with quotient field F and S be commutative rings with identity and let n n−1 suppose f(x) = anx + an−1x +···+a0 σ : R → S be a homomorphism. Let M be p σ is a polynomial with coefficients in R.Ifc = q a left R-module and let M denote the scalar with p and q relatively prime, and c is a root of extension of M by σ ; Mσ is a left S-module. f , then p divides a0 and q divides an. Let A be an associative R algebra and let ρ : This is most often used in the particular case A → EndR(M) be a linear representation of A σ when R is the integers and F is the rationals. in M. Let A denote the scalar extension of A and let ρσ denote the scalar extension of ρ.An rational surface An algebraic surface (an arbitrary linear representation of the S algebra σ σ algebraic variety of dimension 2) that is bira- A in M is called realizable in R if it is similar σ tionally equivalent to 2-dimensional projective to ρ for some linear representation ρ over R. space. See also rational variety. See scalar extension. rational variety A variety over a field k real Lie algebra A Lie algebra over the field which, for some n, is birationally equivalent to of real numbers. See Lie algebra. an n-dimensional projective space over k. See also birational mapping, variety. real number The real numbers, denoted R, form a field with binary operations of + (addi- R-basis Let R be a ring contained in a pos- tion) and · (multiplication). This field is also an sibly larger ring S. A left S module V has an ordered field: There exists a subset P ⊂ R such R-basis if V is freely generated as an R mod- that (i.) if a,b ∈ P then a +b ∈ P and a ·b ∈ P ule by a family of generators, vα,α ∈ V . This and (ii.) the sets P , −P ={x ∈ R :−x ∈ P } , means (i.) each vα ∈ V ; (ii.) if a finite sum {0} (0 denotes the additive identity) are pairwise r v +···+r v = r ,...r ∈ R 1 α1 n αn 0, where 1 n , disjoint and their union is R. Furthermore, the then ri = 0 for i = 1,...n; (iii.) the set of all fi- field R satisfies the completeness axiom: If S r v +···+r v r ,...r ∈ S nite sums 1 α1 n αn , where 1 n is a non-empty subset of R and if is bounded R, is equal to V . The family vα,α ∈ A, is called above, then S has a least upper bound in R. With an R-basis for V . these axioms, R is known as a complete ordered The most important case of this occurs when field. The existence of such a field is taken as R is chosen to be the ring of integers, Z, in which an assumption by many. Others prefer to postu- case an R basis is called a Z-basis. See also Z- late the existence of the natural numbers N ⊂ R basis. by means of the Peano postulates. See natu- ral number. With these postulated, the integers, real axis in complex plane A complex num- Z are constructed from N and then the rational ber p + iq, p, q real, can be identified with the numbers Q are constructed as the field of quo- point (p, q) in Cartesian coordinates. Under tients of Z. To construct the real numbers from this identification, numbers of the form p + i0, the rationals requires more work. One way is to that is, real numbers, are identified with points use the method of Dedekind cuts and another in- of the form (p, 0). The collection of all such volves viewing the reals as equivalence classes points forms one of the axes in this coordinate of Cauchy sequences. See also field. system; this axis is called the real axis. real part of complex number If z = a + bi, real closed field A real field F such that any where a and b are real, is a complex number, the algebraic extension of F which is real must be real part of z is a. This is denoted as (z) = a. equal to F . That is, F is maximal with respect to the property of reality in an algebraic closure. real prime divisor Let K be an algebraic See real field. number field of degree n; then there are n injec- tions σ1,...,σn of K into C. We may assume real field A field F such that −1isnotasum that these are ordered so that n = r1 + 2r2, and of squares in F . so that for 1 ≤ j ≤ r1, σi maps to the real num-

c 2001 by CRC Press LLC j ≥ r σ σ = u = x + 1 bers, for 1, j maps to C, and j+r1+r2 etc., and consider the variable x , so that σj+r for j = 1,...,r . For 1 ≤ j ≤ r set x2 + 1 = u2 − ,x3 + 1 = u3 + u,... 1 2 1 x2 2 x3 3 and vj (a) =|σ(a)|, a ∈ K and for r1 + 1 ≤ j ≤ r2 the equation reduces to an equation of degree m v (a) =|σ(a)|2 v ,...,v set j . The 1 r1+r2 form a in u. set of mutually nonequivalent valuations on K. Equivalence classes of the v ,...,vr are called 1 1 reciprocal linear representation Let R be real prime divisors. a commutative ring with identity and let M be an R module. Let A be an associative algebra real quadratic field A field of the form Q √ over R. A homomorphism A → EndR(M) is ( m) where m is square free and positive. called a linear representation of A in M.An antihomomorphism (ρ is an antihomomorphism real quadratic form Let A =[aij ] be an n× if ρ(ab) = ρ(b)ρ(a)) A → EndR(M) is called n symmetric matrix. This generates a quadratic Q(x) = xT Ax = n a x x a reciprocal linear representation. form: i,j=1 ij i j . The quadratic form is called real if all the aij are real. reciprocal of number The reciprocal of a 1 number a = 0 is the number b = a . Then ab = real representation A real analytic homo- 1, that is, the reciprocal of a number is its morphism from a Lie group G to GL(V ), where multiplicative inverse. Note also that the recip- b a V is a finite dimensional vector space over R. rocal of is ; in other words, the reciprocal of the reciprocal of a nonzero number is the num- real root A root that is a real number. See ber itself. root. reciprocal proportion The reciprocal of the real simple Lie algebra A Lie algebra that is proportion of two numbers. If a and b are two a real and simple. Every real simple Lie algebra numbers, the proportion of a to b is b ; the re- b of classical type is the Lie algebra of one of ciprocal proportion is a . two types of subgroups of a group of invertible matrices. reduced Abelian group An Abelian group with no nontrivial divisible subgroups. real variable A variable whose domain is un- derstood to be either the real numbers or a sub- A set of the real numbers. This is most often used reduced algebra Let be a semisimple Jor- K α ∈ K for emphasis to distinguish the domain from the dan algebra over a field . For an and e ∈ A A (α) ={x ∈ A : complex numbers. Functions whose domain of an idempotent set e ex = αx} definition is understood to be a subset of the real . Suppose that 1 is written as a sum e i numbers are called functions of a real variable. of idempotents i. If, for every , we can write Aei (1) = Kei + Ni, with Ni a nilpotent ideal of A ( ) A reciprocal equation A polynomial equation ei 1 , then is called a reduced algebra. n n−1 of the form anx +an−1x +···+a1x+a0 = 0 where an = a0, an−1 = a1, etc. This has the reduced basis Let G be a discrete group of property that the reciprocal of any solution is motions in n-dimensional Euclidean space and also a solution. Notice that if n is odd, then let T denote the subgroup of all translations in this equation has root x =−1. Division of G. Let {t1,...,tn} be a basis for T and let A this equation by x + 1 then yields a recipro- denote the Gram matrix aij = (ti,tj ). (Here we cal equation of degree n − 1. If n is even, say simply identify a translation with a vector; this n = 2m, then the equation may be rewritten we may take as the image of 0 under the transla- m m−1 as: a2mx + a2m−1x +···+am +···+ tion.) If the quadratic form q(x) = aij xixj , 1 m−1 1 m a1( x ) + a0( x ) = 0. Because of the sym- determined by A, is reduced then {t1,...,tn} metry of the coefficients, we associate the first is called a reduced basis for T .(See reduced and last terms, the second and next-to-last terms, quadratic form.)

c 2001 by CRC Press LLC reduced character Let ρ be a finite-dimen- say that a module problem is defined for > if sional linear representation of an associative al- there is a non-empty class P of conformally in- gebra A over a field K; that is, ρ : A → variant metrics ρ(z)|dz| on R such that ρ(z) is EndK (M) is an algebra homomorphism, where square integrable in the z-plane for each local M is a K-module. Fix a basis of M and let Ta uniformizing parameter z such that denote the matrix (relative to this basis) asso- 2 ciated to the endomorphism of M which is the Aρ(R) ≡ ρ dxdy image of a ∈ A.Fora ∈ A set χρ(a) = Tr Ta; R χρ is a function on A and is called the charac- ρ is defined and such that Aρ(R) and ter of . A character of an absolutely irreducible representation is called an absolutely irreducible Lρ(R) ≡ inf ρ |dz| character. (See absolutely irreducible represen- γ ∈> γ tation.) The sum of all the absolutely irreducible ∞ characters of A is called the reduced character are not simultaneously 0 or . We designate of A. See also reduced norm, reduced represen- the quantity tation. Aρ(R) m(γ ) = inf ρ∈R (Lρ(>))2 reduced Clifford group Let K be a field, E a finite dimensional vector space over K, q(x) as the module of >. a quadratic form on E, and C(q) the Clifford Now let A be a simply connected domain of z ∈ A r> algebra of the quadratic form q. Let G ={s ∈ hyperbolic type in C with 0 .If 0is C(q) : s is invertible, s−1Es = E}. G forms small, then a group relative to the multiplication of C(q); A ∩{z :|z − z0| >r} G is called the Clifford group of q. C(q) is the direct sum of two subalgebras, C+(q) and is a doubly connected domain A(r). Let A(r) C−(q), where C+(q) = K + E2 + E4 +··· have module m(r) for the class of curves sepa- and C−(q) = E + E3 + ···. The subgroup rating its boundary components. The limit G+ = G ∩ C+(q) G of is called the special 1 Clifford group. C(q) has an antiautomorphism lim m(r) + log r r→0 2π β which is the identity when restricted to E. This can be used to define a homomorphism is called the reduced module. N from G+ to the multiplicative group of K by N(s) = β(s)s, s ∈ G+. The kernel of reduced module Let > be a family of locally N G+ rectifiable curves on a Riemann surface R.We , denoted 0 , is called the reduced Clifford group of q. An element s ∈ G induces a lin- say that a module problem is defined for > if − ear transformation φs on E by φs(x) = sxs 1, there is a non-empty class P of conformally in- x ∈ E. Such a φs belongs to the orthogo- variant metrics ρ(z)|dz| on R such that ρ(z) is nal group of E relative to q and the mapping square integrable in the z-plane for each local s → φs is a homomorphism. This gives a uniformizing parameter z such that representation φ : G → O(q). Furthermore, 2 φ(G+) = SO(q) φ(G+) Aρ(R) ≡ ρ dxdy . The subgroup 0 of SO(q) is called the reduced orthogonal group. R See Clifford algebra, Clifford group. is defined and such that Aρ(R) and reduced dual Let G be a unimodular locally Lρ(R) ≡ inf ρ |dz| compact group with countable base. The sup- γ ∈> γ G port of the Plancherel measure in is called the ∞ reduced dual of G. are not simultaneously 0 or . We designate the quantity reduced module Let > be a family of locally Aρ(R) m(γ ) = inf rectifiable curves on a Riemann surface R.We ρ∈R (Lρ(>))2

c 2001 by CRC Press LLC as the module of >. A is called the reduced representation of A. Its Now let A be a simply connected domain of character is the reduced character. See abso- hyperbolic type in C with z0 ∈ A.Ifr>0is lutely irreducible representation, reduced char- small, then acter. (2) Let I be an ideal in a commutative ring A ∩{z :|z − z0| >r} with identity. The ideal I is said to have a pri- is a doubly connected domain A(r). Let A(r) mary representation (or primary decomposition) I = Q ∩ Q ∩···∩Q Q have module m(r) for the class of curves sepa- if 1 2 n, where each i is rating its boundary components. The limit a primary ideal. If no Qi contains the intersec- tion of all the others, and if the radicals of the Qi 1 (which are prime ideals) are distinct, the primary lim m(r) + log r r→0 2π representation is said to be reduced. is called the reduced module. reduced scheme A scheme (X, OX) such U ⊆ X O (U) reduced norm Let ρ : A → EndK (M) be that, for every open set , the ring X a finite dimensional linear representation of an has no nilpotent elements. See also scheme. associative algebra A over a field K in a finite di- mensional K-module M. Fix a basis for M and, reduced von Neumann algebra Let H be a for a ∈ A, let Ta denote the matrix representa- Hilbert space and let B(H ) denote the algebra tion, with respect to this basis, of the endomor- of bounded operators on H . A subalgebra M ∗ phism ρ(a).Fora ∈ A, define Nρ(a) = det Ta; of B(H ) is called a ∗-subalgebra if T ∈ M this is called the norm of a and it depends only whenever T ∈ M.A∗-subalgebra is called on the representation class of ρ. The norm is, up a von Neumann algebra if it contains the iden- r to a constant of the form (−1) (where r is the tity and is closed in the weak operator topol- degree of the representation), equal to the con- ogy. Let P ∈ B(H ) be a projection operator stant term in the characteristic polynomial of Ta. in a von Neumann algebra M. Then the set If pα(x) denotes the principal polynomial of α, MP ={PTP : T ∈ M} is a ∗-subalgebra then the reduced norm of α is the constant term of B(H ) which is closed in the weak operator r of pα(x) multiplied by (−1) . topology but does not contain the identity. Nev- ertheless, because P is a projection, the oper- reduced orthogonal group See reduced Clif- ators in this subalgebra map the Hilbert space ford group. P(H)to itself and MP is a von Neumann alge- bra on the Hilbert space P(H); MP is called a V n reduced quadratic form Let be an - reduced von Neumann algebra on P(H). dimensional vector space over R and q(x) = aij ξiξj a quadratic form on V , where the reducible component A component of a aij , i, j ∈{1,...,n} are given elements of R, topological space X that is not irreducible. Here and the ξi ∈ R are the coordinates of the vec- Y X tor x ∈ V relative to a fixed basis of V . Set a nonempty subset of is irreducible if it can- not be expressed as the union Y = Y ∪ Y of A =[aij ]. The quadratic form q is said to be 1 2 reduced if the matrix A is positive definite and two nonempty proper subsets, each of which is closed in Y . satisfies: (i.) aii+1 ≥ 0 for 1 ≤ i ≤ n − 1 and T (ii.) xk Axk ≥ akk whenever xk isa1×n vector T p(x) of the form xk =[y1,...,yn] , where the yi are reducible equation Suppose is a poly- integers such that the greatest common divisor nomial with coefficients in a field K, i.e., p(x) ∈ K[x] p(x) = of yk,yk+1,...,yn is 1. . An equation of the form 0 is said to be irreducible if p(x)is irreducible over K[x]; reduced representation (1) Suppose A is an otherwise the equation is said to be reducible. associative algebra over a field. The direct sum See also irreducible equation, reducible polyno- of all absolutely irreducible representations of mial.

c 2001 by CRC Press LLC n reducible linear representation (1) Let K S, that is, the set of all (a1,...,an) ∈ K such be a commutative ring with identity and let M be that f(a1,...,an) = 0 for every f ∈ S,is a K-module. Let A be an associative K algebra. called an affine K-variety. Likewise, a set in n- A linear representation of A in M is an algebra dimensional projective space over K consisting homomorphism, ρ : A → EndK (M) of A into of common zeros of a set of homogeneous poly- the algebra of all K endomorphisms of M. Such nomials is called a projective variety. An affine a representation can be used to make M into a or projective K-variety V is reducible if it can left A module by defining am = ρ(a)m, when- be written as V = W1 ∪ W2, where each Wi is ever a ∈ A, m ∈ M.IfM is a simple A module, a K-variety and a proper subset of V . then ρ is said to be irreducible; otherwise it is called reducible. reduction formulas of trigonometry The (2) Let G be a group and K a field. A linear formulae: sin(θ + 2πk) = sin(θ), cos(θ + G K πk) = (θ) k ( π − θ) = representation of over is a homomorphism 2 cos for an integer; sin 2 ρ G (K) (θ) ( π − θ) = (θ) (θ + π) = from to the matrix group GLm . The cos , cos 2 sin and sin representation is said to be reducible if there is − sin(θ), cos(θ + π) =−cos(θ). These allow another representation σ : G → GLm(K) such one to express a trigonometric function of any that there exists a matrix T ∈ GLm(K) with angle in terms of the sine and/or cosine of an T −1ρ(g)T = σ(g) g ∈ G [ , π ) for all and so that angle in the interval 0 2 . σ11(g) σ12(g) σ(g) has the form σ(g) = σ 0 σ22(g) reduction modulus The construction of ρ for all g ∈ G. from ρ, where K,L are commutative rings with identity, σ : K → L is an isomorphism such reducible linear system Let X be a com- that σ : K → K/m is the canonical homo- σ plete irreducible variety. Consider f1,...,fn, morphism, m is an ideal of K, M is the scalar ∗ elements of the function field of X, and let D be extension σ M = M ⊗K L of a K-module M a divisor on X with D+(fi) positive for every i. relative to σ , and ρ is the linear representation (Here (fi) denotes the divisor of fi. See princi- associated with M. The linear representation σ σ pal divisor of functions.) The set of all divisors ρ over L associated with M is the scalar ex- of the form ( kifi) + D, where the ki are el- tension of ρ relative to σ and is the conjugate ements of the underlying field, and not all zero, representation of ρ relative to σ . is called a linear system. The linear system is irreducible if all its elements are irreducible; if reduction of algebraic expression A modi- not, then it is called reducible. fication of an algebraic expression to a simpler or more desirable form. For example, (x+2)2 − reducible matrix An n × n matrix A such (x + 2)(x − 2) can be reduced to 4(x + 2). that either (i.) n = 1 and A = 0 or (ii.) n ≥ 2 and there is an n × n permutation matrix P reductive A Lie algebra G is reductive if it is and an integer m with 1 ≤ m

c 2001 by CRC Press LLC linear form h in f0,f1,...,fr such that h is the collection of all reflections of its constituent monic in f0 and h is G invariant. See rational points in the line. representation. If (x, y) is a point in the plane, the reflection of (x, y) in the origin is the point (−x,−y); reductive group Let G be an algebraic group. the reflection of a geometric object in the origin Let N denote the maximal connected solvable is the collection of reflections of its constituent closed normal subgroup of G. If the identity points. element is the only unipotent element of N, then In a similar fashion, define the reflection of G is called a reductive group. a point in a plane; for example, the reflection of (x,y,z) in the x,y plane in three dimensions reductive Lie algebra A Lie algebra A such is the point (x, y, −z). Likewise, reflection in that the radical of A is the center of A. This the origin is defined in the same way in higher is equivalent to the condition that the adjoint dimensions. representation of A is completely reducible. (2) In the study of groups of symmetries of The radical of A is the union of all solvable geometric objects, a group element correspond- ideals of A. ing to a transformation reflecting the object in a line or plane. In greater generality, a reflection redundant equation (1) An equation con- is a mapping of an n-dimensional simply con- taining extraneous roots that have been intro- nected space of constant curvative which has an n − 1 dimensional hyperplane as its set of fixed duced by an algebraic√ operation. For example, the equation x − 2 = x when squared yields points. In this case, the set of fixed points of the the equation x2 − 5x + 4 = 0. This has roots mapping is called the mirror of the mapping. 4 and 1; however, 1 is not a root of the original equation. The equation x2 − 5x + 4 = 0 is said reflexive property The property of a relation to be redundant because of this extraneous root. R on a set S: 2 ( ) An equation in a system of equations that (a, a) ∈ R a ∈ S. is a linear combination of other equations in the for every system. The solution to the system remains un- In other words, a relation has the reflexive prop- changed if this equation is excluded. erty if every element is related to itself. See relation. Ree group Any member of the family of finite simple groups of Lie type, 2G (3n) and 2 reflexive relation A relation having the re- 2F (2n), for n odd. 4 flexive property. See reflexive property. C refinement of chain A chain 1 in a partially regula falsi Literally, “rule of false position,” M C M ordered set is a refinement of a chain 2 in an iterative method for finding a real solution of if the least and greatest elements of each chain an equation f(x)= 0. Suppose that f(x ) and C 1 agree, and if every element of the chain 2 be- f(x ) have opposite sign. The method is to con- C C C 2 longs to 1. 1 is a proper refinement of 2 if sider the point of intersection of the x-axis with C there is at least one element of 1 which does the through (x ,f(x )) and (x ,f(x )). C 1 1 2 2 not belong to 2. This point has x coordinate reflection (1) The mirror image of a point or x − x x = x − f (x ) 2 1 . object. 3 1 1 f(x2) − f(x1) A reflection in a line of a point P1 is the point P2 on the opposite side of the line equidistant The procedure is repeated with the point (x3, from the line; that is, the point P2 such that the f(x3)) and either (x1,f(x1)) or (x2,f(x2)), de- line is the perpendicular bisector of the segment pending on which of f(x1) or f(x2) has sign P1P2, joining the point and its reflection. The opposite to f(x3). This converges more slowly reflection of a geometric object in a line is just than Newton’s method.

c 2001 by CRC Press LLC regular cone A subset C ⊂ V of a finite let aij denote the probability that a process cur- dimensional vector space over R such that C is rently in state j will in the next step be in state convex, C contains no lines, and λc ∈ C when- i. The matrix A =[aij ] is called the transition ever c ∈ C and λ>0. matrix of the chain. Note that the entries of A are all nonnegative and the sum of the elements regular extension A field extension K/k in each column of A is 1. The matrix A is said such that k is algebraically closed in K and K to be regular if there is a positive integer n such is separable over k. Equivalently, the extension that An has all positive entries. ¯ is regular if K is linearly disjoint from k (the (2) Sometimes, a synonym for non-singular algebraic closure of k) over k. or invertible matrix. See inverse matrix. regular function (1) In complex analysis, an regular module Let R be a commutative ring analytic or holomorphic function. See analytic with 1 and M an R-module. An element x ∈ M function, holomorphic function. is said to be regular if, for every r ∈ R, there (2) In algebraic geometry, a function which exists t ∈ R such that rx = rtrx. A submodule can be represented locally as a quotient of poly- N of M is regular if each element of N is regular. nomials. In more detail, let X be a variety. (A In particular, M is regular if each of its elements variety is the solution set of a system of poly- is regular. nomial equations.) Let V be an open subset of X. Let f : V → k be a function from V to a regular module Let R be a commutative ring field k. The function f is regular at the point with 1 and M an R-module. An element x ∈ M p ∈ V if there is an open neighborhood U of is said to be regular if for every r ∈ R, there p, contained in V , on which f = g/h, where exists t ∈ R such that rx = rtrx. A submodule g and h are polynomials and h = 0onU. f is N of M is regular if each element of N is regular. regular if it is regular at each point of its domain In particular, M is regular if each of its elements V . is regular. regular ideal A left ideal J in a ring R, for regular polyhedral group The group of mo- which there exists an e ∈ R such that r −re ∈ J tions which preserve a regular polygon in the for every r ∈ R. Similarly, a right ideal J , for plane, or the group of motions in three dimen- which there exists an e ∈ R such that r −er ∈ R sional space that preserve a regular polyhedron. for every r ∈ R. Note that if R has an identity, The group of motions preserving a regular n-gon then all ideals are regular. in the plane is called the dihedral group Dn;it has order 2n. The alternating group A4 can be regular local ring Let R be a local ring and realized as the group of motions preserving a set k = R/m. Then k is a field. A local Noethe- tetrahedron, and A5 can be realized as the group rian ring is called regular if dimk m/m2 = of motions preserving an iscosahedron; they are dim R. (Here dim is the Krull dimension. See called the tetrahedral and iscosahedral groups, Krull dimension.) See also local ring. respectively. Similarly, the symmetric group S4 is called the octahedral group because it can be regular mapping (1) A differentiable map realized as the group of motions preserving an with nonvanishing Jacobian determinant be- octahedron. tween two manifolds of the same dimension. (2) A mapping given by an n-tuple of regular regular prime A prime number which is not functions from a variety to a variety contained irregular. See irregular prime. in affine n-space. See regular function. regular representation (1) Let K be a com- regular matrix (1) A Markov chain is a prob- mutative ring with identity and let A be a K alge- ability model for describing a system that ran- bra. An element a ∈ A can then be viewed as an domly moves among different states over suc- element of the algebra of K endomorphisms on cessive periods of time. For such a process, A via x "→ axfor all x ∈ A. Such a representa-

c 2001 by CRC Press LLC tion is called a left regular representation. If A regulator Let K be an algebraic number field has an identity, this is a faithful representation. and let U(K) denote the units of K.IfK has In a similar way, define right regular represen- degree n, then there are n field embeddings of K tation. into the complex numbers C. Let r1 be the num- r (2) Let G be a group and, for a ∈ G, define ber of real embeddings, and 2 be the number a mapping g "→ ag. In this way each a ∈ G of complex embeddings (these occur in pairs) r + r = n σ ,...,σ induces a permutation on the set G. This gives so that 1 2 2 . Write 1 r1 for σ ,...,σ a representation of G as a permutation group on the real embeddings and r1+1 n for the G; this is called the left regular representation complex embeddings and order the latter so that σ = σ i = ,...,r of G. In a similar way, define right regular rep- r1+i r1+r2+i for 1 2. It is a theo- resentation. See also left regular representation, rem of Dirichlet that U(K)is a finitely generated r = r + r − right regular representation. Abelian group of rank 1 2 1 and so there exists a fundamental system of units for K u ,...,u x ∈ regular ring (1) An element a in a ring R ; these we denote by 1 r .For C #x# x x is called regular in the sense of Von Neumann let denote the absolute value of if is x x if there exists x ∈ R such that axa = a.If real, and the square of the modulus of if K every element of R is regular, then R is said to is complex. The regulator of is the abso- be a regular ring. Note that every division ring lute value of the determinant of any (all have the r×r r×(r+ ) is regular, whereas the ring of integers is not. same value) submatrix of the 1 ma- trix [ ln#σj (ui)#],1≤ i ≤ r,1≤ j ≤ r + 1. (2) A regular local ring. See regular local ring. related angle Given an angle α whose ter- minal side does not necessarily lie in the first regular system of equations (1) A nonsingu- quadrant, the related angle of α is the angle in n n lar system of linear equations in unknowns. the first quadrant whose trigonometric functions If we denote such a system as Ax = b, where π have the same absolute value. For example, 6 A is an n × n matrix, and x and b are n × 1 π radians is the related angle of 7 radians, 70◦ is matrices, then A has nonzero determinant and 6 the related angle of 290◦. for each n×1 vector b there is a unique solution x. Also called a Cramer system. relation For A and B be sets, a relation be- p (x ,...,x ),... (2) More generally, if 1 1 m , tween A and B is a subset R of A × B. Most p (x ,...,x ) n m n 1 m are polynomials in vari- often considered is the case when A = B; in this k ables over a field , the system of equations case it is said that R is a relation on A. For exam- p = ,...,p = 1 0 n 0 is called regular if it has ple, = is a relation on the set of real numbers; a finite number of solutions in an algebraically for each pair (a, b) ∈ R × R it can be deter- k closed field containing . mined whether or not (a, b) ∈ R ={(a, a)}. Similarly, <, >, ≤, and ≥ are relations on the regular transitive permutation group A real numbers. See also reflexive relation, sym- permutation group P is a subgroup of the sym- metric relation, transitive relation, equivalence metric group of permutations of a set S.IfS has relation. at least two elements and if, for every α, β ∈ S, there exists a p ∈ P such that pα = β, then P is relative algebraic number field If K is an said to be transitive. If, in addition, no element algebraic number field and k a subfield of K, of P , except for the identity, fixes any element the extension of K over k is called a relative of S, then P is called a regular transitive per- algebraic number field. See algebraic number mutation group. field. regular variety A non-singular projective relative Bruhat decomposition The decom- algebraic variety with zero irregularity. A va- position riety which is not regular is called an irregular $ variety. See irregular variety, irregularity. (G/P )k = Gk/Pk =∪w∈kW π Uw k ,

c 2001 by CRC Press LLC where G is a connected reductive group defined M are intermediate fields: K ⊆ L ⊆ M ⊆ F . over a field k, P is a minimal parabolic k-sub- The relative degree (or dimension) of L and M is group of G containing a maximal k-split torus the number [M : L]. This terminology allows a S, U = Rn(P ) is the unipotent radical of P , concise statement of the Fundamental Theorem kW = N/Z is the k-Weyl group of G relative of Galois Theory: If F is a finite dimensional to S, Z = ZG(S) is the centralizer of S in G, Galois extension of K, then the relative degree N = NG(S) is the normalizer of S in G,twok- of two intermediate fields is equal to the relative $ $$ $ subgroups Uw and Uw are such that U = Uw × index of their corresponding subgroups of the $$ Uw whenever w ∈k W, and π is the projection Galois group. G → G/P . relative derived functor A homological al- relative chain complex If R is a ring, a chain gebra concerns a pair of algebraic categories complex of R-modules is a family {Kn,∂n}, n ∈ (U, M) and a fixed functor ) : U → M.A Z,ofR-modules Kn and R-module homomor- short exact sequence of objects of U ∂ : K → K ∂ ∂ = phisms n n n−1 such that n n+1 0 0 → A → B → C → 0 for every n. A subcomplex of such a chain com- plex is a family of R-submodules Sn ⊂ Kn is called admissible if the exact sequence ∂ S ⊂ S n such that n n n−1 for every . In this 0 → )A → )B → )C → 0 case, each ∂n induces a well-defined homomor- $ splits in M. By means of the class E of admis- phism of quotient modules: ∂n : Kn/Sn → $ $ sible exact sequences, the class of E-projective Kn−1/Sn−1; furthermore ∂n∂n+ = 0 for every $ 1 E n. {Kn/Sn,∂ } is called a relative chain com- (resp. -injective) objects is defined as the class n P Q) plex. For a chain complex K ={Kn,∂n} of R- of objects (resp. for which the functor (P, −) (−, Q)) modules, the nth homology module of the com- HomU (resp. HomU is exact on U plex is defined as the quotient module Hn(K) = the admissible short exact sequences. If con- tains enough E-projective or E-injective objects, Ker ∂n/Im ∂n+1.IfS ={Sn} is a subcomplex of K, then the n-th relative homology module then the usual construction of homological alge- H (K, S) = ∂$ / ∂$ bra makes it possible to construct derived func- is defined as n Ker n Im n+1. See also relative cochain complex. tors in this category, which are called relative derived functors. See also derived functor. relative cochain complex If R is a ring, a relative different Let K be a relative alge- cochain complex of R is a family {Kn,dn}, n ∈ braic number field over k. Let D and d de- Z,ofR-modules Kn and R-module homomor- note the integers of K and k, respectively; that phisms dn : Kn → Kn+1 such that dn+1dn = 0 is, D = K ∩ I and d = k ∩ I, where I is for every n. (In other words, {K−n,d−n} is D ={α ∈ K : a chain complex.) A cochain subcomplex of the algebraic integers. Set (αD) ⊂ d} such a cochain complex is a sequence of R- TrK/k , where TrK/k denotes the trace. D K (D)−1 = D submodules Sn ⊂ Kn such that dnSn ⊂ Sn+ is a fractional ideal in and K/k 1 K for every n. In this case each dn induces a is an integral ideal of . It is called the relative K k well-defined homomorphism of quotient mod- different of over . $ ules: dn : Kn/Sn → Kn+1/Sn+1; furthermore, $ $ $ relative discriminant Suppose K is a rela- d d = 0 for every n. {Kn/Sn,d } is called n+1 n n k a relative cochain complex. Cohomology and tive algebraic number field over . The relative K k relative cohomology are then defined in a way discriminant of over is the norm, with re- K/k K/k completely analogous to the definition of homol- spect to , of the relative different of . ogy and relative homology. See relative chain See relative different. complex. relative homological algebra A homolog- ical algebra associated with a pair of Abelian relative degree Let F be a finite dimensional categories (U, M) and a fixed functor field extension of a field K, and denote the de- gree of the extension by [F : K]. Suppose L and ) : U → M .

c 2001 by CRC Press LLC The functor ) : U → M is taken to be additive, ated by the U (i) is the extension of an ideal a of exact, and faithful. k; a is called the relative norm of U over k. relative invariant (1) Let K be a field and let remainder (1)Ifa is an integer and b is a F be an extension field of K. Let G be a group positive integer, then there is a unique pair of of automorphisms of F over K, and suppose that integers q,r with 0 ≤ r

c 2001 by CRC Press LLC representation A mapping from an algebraic representation space is H . See representation. system to another system, usually to one which We give several examples; all are fairly similar. is better understood. The idea is to preserve the (i.) Let k be a commutative ring with identity structure of the original system as an image and and A an associative algebra over k. Let E be to take advantage of knowledge of the system a module over k and let ρ : A → Endk(E) (the containing this image. There are a multitude of space of k endomorphisms of E) be a represen- examples of this and here we list only the more tation. E is called the representation space. common representations. (ii.) Let A be a Banach algebra and X a Ba- (1) Let G be a group. Let ρ : G → H be nach space. For a representation of A on X, X a homomorphism of G to another group H ; ρ is the representation space. is then called a representation of G. In partic- (iii.) Let G be a topological group and let ρ ular, a homomorphism ρ : G → Sn of G into be a representation of G to the group of unitary a permutation group Sn of permutations of a set operators on some Hilbert space H . H is called of n elements is known as a permutation repre- the representation space. sentation. Cayley’s Theorem is that every finite group can be so represented using a monomor- representation without multiplicity Let U phism. A homomorphism of G into GL(n, K), be a unitary representation of a topological group the set of invertible n × n matrices over a com- G on a Hilbert space H , that is, a homomor- mutative ring K with identity, is called a matrix phism U from G to a group of unitary operators representation of G. on H . A subspace N of H is called invariant U (N) ⊂ N g ∈ G g ∈ G (2) Let R be a commutative ring with identity if g for every .(For , Ug and let M be a module over R. Let EndR(M) de- denotes the unitary operator that is the image g g ∈ G V = U | note the ring of R-linear maps of M onto itself. of .) For let g g N . This gives V G N Let A be an associative R-algebra. A represen- a unitary representation of on ; this is U N H tation of A in M is an R-algebra homomorphism called the subrepresentation of on .If H = H ⊕ H ρ : A → EndR(M). can be written as the direct sum 1 2 where the Hi are orthogonal, closed, and invari- (3) Let X be a Banach space and A a Banach ant, then U is said to be the direct sum of the sub- algebra. Let B(X) denote the space of bounded representations U and U of U on H and H , linear operators on X.Arepresentation of A in 1 2 1 2 respectively. A unitary representation is called X is an algebra homomorphism ρ : A → B(X) a representation without multiplicity if, when- which satisfies #ρ(a)#≤#a# for every a ∈ A. ever U is the direct sum of subrepresentations G H (4) Let be a topological group and let U and U , then U and U have the property U 1 2 1 2 be a Hilbert space. A homomorphism from that no subrepresentation of U is equivalent to G H 1 to the group of unitary linear operators on a subrepresentation of U . (Two unitary repre- G U 2 is called a unitary representation of if is sentations U : G → H and U : G → H are x ∈ H 1 1 2 2 strongly continuous, that is, for every , said to be equivalent if there exists an isometry g "→ U x G g is a continuous mapping from to T : H → H such that TU g = U gT for all H 1 2 1 2 . g ∈ G.) representation module Let R be a com- representative function Let G be a compact mutative ring with identity and let ρ : A → Lie group, and let C(G) denote the algebra of EndR(M) be a linear representation of an R- continuous complex valued functions on G.For algebra A in an R-module M. M can be made each g ∈ G, let Tg denote the operator of left into a left A-module, called the representation translation: Tg(x) = gx for x ∈ G. Tg acts on module of ρ, by defining am = ρ(a)m, for C(G) by Tg(f ) = f ◦Tg. A function f ∈ C(G) a ∈ A, m ∈ M. is called a representative function if the vector space over C generated by the set of all Tg(f ), representation space For a representation g ∈ G, has finite dimension. The collection from an algebraic structure G to the space of of all such functions is called the representative endomorphisms on an algebraic structure H , the ring of G.

c 2001 by CRC Press LLC representative of equivalence class An The most important case of this occurs when equivalence relation on a set S partitions the set R is chosen to be the ring of integers, Z, in which S into disjoint sets, each consisting of related el- case R-equivalence is called Z-equivalence. Z- ements, that is, the set is partitioned into sets of equivalence is related to the classical notion of the form Ax ={y ∈ S : x ∼ y}, called equiv- integral equivalence for matrices. See also in- alence classes. Any element of an equivalence tegral equivalence, Z-equivalence. class is called a representative of that equiva- lence class. See equivalence relation. residue class Suppose R is a ring and I is an ideal in R.Forx,y ∈ R, we say that x is representative ring See representative func- congruent to y modulo I if x − y ∈ I.Itis tion. straightforward to check that congruence mod- ulo I is an equivalence relation on R. A coset representing measure (1) A measure that of I in R is called a residue class modulo I.If represents a functional on a function space in the I is a principal ideal generated by an element sense that the value of the functional at an ele- a ∈ R, the residue class modulo I is sometimes ment f can be realized simply as integration of f referred to as the residue class modulo a; in par- with respect to the measure. A typical theorem ticular, this is used in the case when R = Z. See regarding the existence of such a measure is the also residue ring. Riesz representation theorem: Let X be a locally compact Hausdorff space and let T be a positive linear functional defined on the function space residue class field Suppose R is a commuta- consisting of continuous functions on X of com- tive ring with identity 1 = 0, and M is an ideal pact support. Then there exists a σ-algebra A in R. The quotient ring or residue ring, R/M, containing the Borel sets of X and a unique pos- is a field if and only if M is a maximal ideal. R/M itive measure µ on A such that Tf = X fdµ In this case is known as the residue class for every f which is continuous and of compact field or residue field. See also residue ring. support on X. In this case, µ is said to represent T the functional . residue field See residue class field. (2) On a function space with the property that the functionals f "→ f(ζ) are bounded, for ζ in some planar set O, the representing measures residue ring Let R be a ring and I an ideal for the points of O are measures representing of R. Define a relation ∼ on R by x ∼ y if the evaluation functionals in the sense of (1). x − y ∈ I; this is an equivalence relation. Since That is, representing measures are given by µ = I is a normal subgroup of R under the operation + µζ , carried on a set λ, often contained in the , the equivalence classes are cosets which we boundary of O, such that write additively: a+I. The set of all such cosets can be made into a ring with addition defined by (a+I)+(b+I) = (a+b)+I f(ζ)= f(z)dµζ (z) . and multiplication λ defined by (a + I)(b + I) = ab + I. This ring is called the residue ring, residue class ring, or R/I R-equivalence Let R be a ring contained in quotient ring and is denoted . a possibly larger ring S. Let V and W be S modules with finite R bases BV and BW . The Residue Theorem (1) Suppose f(z) is an modules V and W are R-equivalent if there is analytic function with an isolated singularity at a an S module isomorphism between V and W point a. The residue of f at a is the coefficient of − which, relative to the bases BV and BW , has a (z−a) 1 in the Laurent expansion of f at a;we matrix U with all entries in the ring R, and such denote this by res[f ; a]. The Residue Theorem that the inverse matrix U −1 also has all entries states that if f is analytic in a simply connected in the ring R. It is a theorem that R-equivalence domain A, except for isolated singularities, and is independent of the particular choice of the if γ is a simple closed rectifiable curve lying in bases, BV and BW . See also isomorphism. A and not passing through any singularities of

c 2001 by CRC Press LLC f , then normal subgroup of G; it is called the restricted direct product (or weak direct product)ofthe 1 f(z)dz= res f ; aj , groups Gi. 2πi γ restricted Lie algebra A Lie algebra L over a where the sum is taken over the singularities aj field K of characteristic p = 0 such that, for ev- of f that lie inside γ . ery a ∈ L, there exists an element a[p] ∈ L with (2) Let X be a complete nonsingular curve the following properties: (i.) (λa)[p] = λpa[p] over an algebraically closed field k. Let K be the whenever λ ∈ K and a ∈ L, (ii.) [a,b[p]]= function field of X. Let AX denote the sheaf of [[...[a,b],...],b] (a p-fold commutator), and differentials of X over k and AK be the module (iii.) (a + b)[p] = a[p] + b[p] + s(a,b) where of differentials of K over k.ForP ∈ X, let AP s(a,b) = s1(a, b) +···+sp−1(a, b) and (p − denote its stalk at P . It can be shown that for p−i− i)si(a, b) is the coefficient of λ 1 in the ex- each closed point P ∈ X, there exists a unique pansion of the p-fold commutator k-linear map resP : AK → k which has the following properties: (i.) resP (τ) = 0 for all [[...[a,λa + b],...],λa+ b] . n τ ∈ AP , (ii.) resP (f df ) = 0 for all n = −1 and all f in the multiplicative group of K, − (iii.) resP (f 1df ) = vP (f )·1, where vP is the restricted minimum condition A commuta- valuation associated to P . The Residue Theorem tive ring R, with identity, satisfies the restricted states that for any τ ∈ AK , minimum condition if R/I is Artinian for every nonzero ideal I of R. See Artinian ring. resP (τ) = 0 . P ∈X restriction Suppose A and B are sets and f : A → B is a function. Let S ⊂ A. The f S S B resolution of singularities The technique of restriction of to is the function from to a "→ f(a) a ∈ S applying birational transformations to an irre- given by for . This is often f |S f | ducible algebraic variety in hope of producing denoted by or S. an equivalent projective variety without singu- larities. resultant (1) The sum of two or more vectors. (2) A determinant formed from the coeffi- restricted Burnside problem A group with cients of two polynomial equations. Consider identity e is said to have exponent n if gn = the two equations e for every element g of the group. The re- n n−1 a x + a x +···+an = 0 stricted Burnside problem is the conjecture that 0 1 if a group of exponent n is generated by m el- m m−1 b0x + b1x +···+bm = 0 ements, then its order is bounded by a quan- tity S(n, m) which depends only on n and m. where a0 = 0 and b0 = 0. The resultant is the For n = p a prime was solved affirmatively by determinant of an (m + n) × (m + n) matrix Kostrikin. a0 ... an 0 ...... 0 I 0 a0 ... an 0 ...... 0 restricted direct product Let be an index- {G } i ∈ I ...... ing set, and let i , be a family of groups I G = G 0 ...... 0 a0 ... an indexed by . Consider the set i∈I i G b0 ...... bm 0 ...... 0 and put a group structure on by defining mul- 0 b0 ...... bm 0 ... 0 tiplication componentwise: ...... {gi}i∈I {hi}i∈I = {gihi}i∈I . 0 ...... 0 b0 ...... bm

G is called the direct product of the groups Gi. Here the first m rows are formed with the coeffi- The set of all elements {gi}i∈G of G such that cients ai and the last n rows are formed with the gi = e for all but a finite number of i forms a coefficients bi. The two polynomial equations

c 2001 by CRC Press LLC have a common root if and only if their resultant Riemann matrix An n × 2n matrix A, for is zero. which there exists a nonsingular skew symmet- ric 2n × 2n matrix P with integer entries so that Richardson method of finding key matrix AP AT = 0 and iAPA∗ (where A∗ is the con- Consider a real linear system of equations Ax = jugate transpose matrix of A) is positive definite b, where A is an n × n matrix and x and b are Hermitian. n × 1 matrices. One method for approximat- ing the solution, x, involves finding a sequence Riemann-Roch inequality (1) Let k be a of approximations xk by means of an iterative field and let X be a nonsingular projective curve k X D = formula xk = xk−1 + R(b − Axk−1), where R over . A divisor on is a formal sum −1 is chosen to approximate A . If the spectral p∈X npP where each np ∈ Z and all but a radius of I − RA is less than 1, the xk con- finite number of np = 0. For such aD, the de- verge to the solution for all choices of x0. The gree of D is defined as: deg(D) = np. Let Richardson method is the choice R = αAT , K denote the function field of X over k. For a <α< 2 D L(D) ={f ∈ K : (f ) ≥ where 0 #A#2 . divisor set ordP −np for all P in X}. (Here ordP (f ) denotes f P L(D) Riemann-Hurwitz formula Let M1 and M2 the order of at .) is a finite dimensional be compact connected Riemann surfaces. If f : vector space over k; set Q(D) = dimkL(D). M1 → M2 is holomorphic and p ∈ M1,we The Riemann inequality (sometimes called the may choose coordinate systems at p and f(p) Riemann-Roch inequality) states that there ex- so that locally f can be expressed as f(w) = ists a constant g = g(X) such that Q(D) ≥ wnh(w) where h(w) is holomorphic and h(0) = deg(D) + 1 − g for all divisors D. 0. (Here 0 is the image of both p and f(p)under (2) Let X be a surface. For any two divi- the respective coordinate maps.) Set bf (p) = sors C and D on X we let I(C.D)denote their n − 1; this is called the branch number. The intersection number. For a divisor D let L(D) total branching number of f is defined as B = denote the associated invertible sheaf and set 0 bf (p), where the sum runs over all p ∈ M. Q(D) = dimH (X, L(D)). Let pa denote the The equidistribution property states that there arithmetic genus of X. The Riemann-Roch in- exists a positive integer m such that every q ∈ equality states that if D is any divisor on X and M m W Q(D) + Q(W − 2 is assumed times counting multiplicity, any canonical divisor, then (b (p) + ) = m q ∈ M D) ≥ 1 I (D.(D − W))+ + p that is, f 1 for every 2, 2 1 a. where the sum is taken over all p ∈ f −1(q). For i = 1, 2, let gi denote the genus of Mi. The Riemann-Roch Theorem (1) Let X be a Riemann-Hurwitz formula states nonsingular projective curve. Let g be the genus of X. The Riemann-Roch Theorem states that W X 2 (g1 − 1) − 2m (g2 − 1) = B. if is a canonical divisor on , then for any divisor D, Q(D)−Q(W −D) = deg(D)+1−g. See Riemann-Roch inequality. Riemann hypothesis The Riemann zeta func- (2) Let X be a surface. For any two divisors tion has zeros at −2, −4, −6,.... It is known C and D on X let I(C.D)denote their intersec- that all other zeros of the zeta function must lie tion number. For a divisor D, let L(D) denote in the strip of complex numbers {z : 0 < (z) < the associated invertible sheaf and set Q(D) = 1}. The Riemann hypothesis is the yet unproved dimH 0(X, L(D)) and s(D) = dimH 1(X, conjecture that all of the zeros of the zeta func- L(D)). Let pa denote the arithmetic genus of (z) = 1 X tion in this strip must lie on the line 2 . . The Riemann-Roch Theorem states that if Hardy proved that an infinite number of zeros lie D is any divisor on X and if W is any canoni- on this line. There are numerous other equiva- cal divisor, then Q(D) − s(D) + Q(W − D) = 1 I (D.(D − W))+ + p lent formulations of this conjecture. Resolution 2 1 a. of this conjecture would have important impli- cations in the theory of prime numbers. See Riemann-Roch Theorem for the adjoint sys- Riemann zeta function. tem Let X be a surface. For any two divisors

c 2001 by CRC Press LLC C and D on X, let I(C.D)denote their intersec- circle. Respectively, the surface is then called tion number. For a divisor D let L(D) denote parabolic, elliptic, or hyperbolic. the associated invertible sheaf and set Q(D) = H 0(X, L(D)) and s(D) = H 1(X, L(D)). Let Riemann theta function Let R be a compact pa denote the arithmetic genus of X. The Riemann surface of genus g. Let {ω ,...,ωg} Riemann-Roch Theorem for the adjoint system 1 be a basis for the space of Abelian differentials states that if C is a curve with r components of the first kind, and let {α ,...,α g} be normal and K is a canonical divisor, then Q(C + K) = 1 2 1 sections forming a basis for the homology, with s(−C) − r + I ((K + C).C) + pa + 2. 2 coefficients in Z,ofR.(See Riemann’s period inequality.) Set A = (ωij ), where ωij = α ωi. Riemann’s period inequality Let R be a j It is possible to choose the ωi so that A has the compact Riemann surface of genus g. The set form A = (Ig,F), where Ig is a g × g identity of all differentials of the first kind, that is, those matrix and F is a g × g complex symmetric a(z)dz a(z) of the form , where is holomorphic, matrix whose imaginary part is positive definite. is a vector space of dimension g over C; let For u = (u1,...,ug) set {ω1,...,ωg} denote a basis. Take normal sec- tions α ,...,α g as a basis for the homology 1 2 group of R, with coefficients in Z, set ωij = θ(u) = πi muT + 1mF mT α ωi, and let A denote the g×2g matrix whose exp 2 j m 2 0 Ig ijth entry is ωij . Set E = , where −Ig 0 Ig denotes the g × g identity matrix. Riemann’s where the sum is taken over all m = AEAT = g period relation states that 0. Rie- (m1,...,mg) ∈ Z . This sum converges uni- mann’s period inequality states that the Hermi- formly on compact subsets. θ is the Riemann ¯ T tian matrix iAEA is positive definite. See also theta function. Riemann matrix.

Riemann zeta function The function de- Riemann’s period relation See Riemann’s ζ(z) = ∞ n−z z fined by n=1 for in the com- period inequality. (z) > plex plane with 1. Euler showed that −z −1 ζ(z) = p (1 − p ) where p runs over all Riemann surface There√ are analytic “func- primes. There are also various other represen- w = z tions” (for example ) that are naturally tations of this involving integral formulas. This considered as multiple valued. Such functions function can be continued analytically to a mero- are better understood using Riemann surfaces: morphic function on the complex plane with a the idea is to consider the image as a surface con- simple pole at z = 1. See also Riemann hy- sisting of a suitable number of sheets so that the pothesis. mapping can be considered as one-to-one from the z to w surfaces. For example, the function 1 Riesz group An ordered Abelian group G w = z 3 can be viewed as a three-sheeted sur- which satisfies: (i.) whenever n is a positive face (all elements of the complex plane, except integer and g ∈ G has gn ≥ 0 then g ≥ 0, and zero, have three cube roots) over the complex (ii.) if G and G are finite sets in G, and if plane. Locally, this mapping is one-to-one from 1 2 g ≤ g for every g ∈ G and g ∈ G then the plane to the surface. More precisely, a con- 1 2 1 1 2 2 there exists an h ∈ G such that g ≤ h ≤ g for nected Hausdorff space is called a Riemann sur- 1 2 all g ∈ G and g ∈ G . face if it is a complex manifold of one com- 1 1 2 2 plex dimension. Any simply connected Rie- mann surface can be conformally mapped to one right A-module Let A be a ring. A right A- of the following: the finite complex plane, the module is an additive Abelian group G, together complex plane together with the point at infinity with a mapping G × A → G, called scalar (the Riemann sphere), or the interior of the unit multiplication, and denoted by (x, r) "→ xr,

c 2001 by CRC Press LLC satisfying, for all r, s ∈ A and x,y ∈ G, right G-set A set M, acted on, on the right, by a group G, satisfying m(g1g2) = (mg1)g2 x1 = x, for all g1,g2 ∈ G and m ∈ M, and me = m for (x + y)r = xr + yr , every m ∈ M, where e is the identity of G. x(r + s) = (xr + xs) , right hereditary ring A ring in which every and right ideal is projective. See also left hereditary x(rs) = (xr)s . ring, left semihereditary ring, right semiheredi- tary ring. right annihilator Let S be a subset of a ring right ideal A nonempty subset I of a ring R R. The set {r ∈ R : Sr = 0} is called the right such that, whenever x and y are in I and r is in annihilator of S. Note that the right annihilator R, x − y and xr are in I. See also left ideal, of S is a right ideal of R and is an ideal of R if ideal. S is a right ideal. right injective resolution See right resolu- right Artinian ring A ring that satisfies the tion. descending chain condition on right ideals; that is, for every chain I1 ⊃ I2 ⊃ I3 ⊃ ... of right right invariant (1) Let G be a locally com- ideals of the ring, there is an integer m such that pact topological group. A Borel measure µ on Ii = Im for all i ≥ m. See also right Noethe- G is said to be right invariant if µ(Ea) = µ(E) rian ring, left Artinian ring, left Noetherian ring, for every a ∈ G and Borel set E ⊂ G. Artinian ring, Noetherian ring. (2) Let G be a Lie group. An element g ∈ G defines an operator of right translation by g on right-balanced functor Let T be a functor of G: Rg(x) = xg for x ∈ G.IfT is a tensor field several variables, some covariant and some con- on G and if RgT = T for every g ∈ G (where travariant, from the category of modules over a RgT is defined in the natural way), then T is ring R to the category of modules over a ring S. said to be right invariant. T is said to be right-balanced if (i.) when any of the covariant variables of T is replaced by an right inverse element Suppose S is a non- injective module, T becomes an exact functor in empty set with an associative binary operation the remaining variables, and (ii.) when any one ∗. Assume that there is a right identity element of the contravariant variables of T is replaced by e ∈ S, with respect to ∗.Ifa ∈ S, a right inverse a projective module, T becomes an exact functor element for a is an element b ∈ S such that of the remaining variables. a ∗ b = e. right coset Any set, denoted by Sa, of all right multiples sa of the elements s of a sub- If every element of such an S has an inverse group S of a group G and a fixed element a of element, then S is called a group. See also left G. See left coset. inverse element. right derived functor See left derived func- right Noetherian ring A ring which satisfies tor. the ascending chain condition on right ideals. That is, for every chain I1 ⊂ I2 ⊂ I3 ⊂ ... right global dimension For a ring A, the of right ideals there exists an integer m such smallest n ≥ 0 for which A has the property that Ii = Im for every i ≥ m. See also right that every right A-module has projective dimen- Artinian ring, left Noetherian ring, left Artinian sion ≤ n. This is equal to the smallest n ≥ 0 for ring, Noetherian ring, Artinian ring. which each right A-module has injective dimen- sion ≤ n. See projective dimension, injective right order Let g be an integral domain in dimension. See also left global dimension. which every ideal is uniquely decomposed into

c 2001 by CRC Press LLC a product of prime ideals (i.e., a Dedekind do- right semihereditary ring A ring in which main). Let F be the quotient field of g. Let A every finitely generated right ideal is projective. be a separable algebra of finite degree over F . See also left semihereditary ring, left hereditary Let L be a g-lattice of A. For such a g-lattice L, ring, right hereditary ring. {x ∈ A : Lx ⊂ L} is called the left order of A. g L g A A -lattice is a -submodule of that is right translation The function fa : G → G, g A = FL finitely generated over and satisfies . on a group G, defined by fa(x) = xa, for x ∈ G.Ifg is a function on G, a right translation R right regular representation (1) Let be of g by a is the function ga defined by ga(x) = a commutative ring with identity and let M be g(xa−1). an R-algebra. For r ∈ R, consider the mapping ρr : M → M given by right translation (i.e., right triangle A triangle with a right angle. ρr (x) = xr, for x ∈ M). The mapping ρ : The side opposite the right angle is called the R → Hom (M, M) given by ρ(r) = ρr is an Z hypotenuse and the sides adjacent the right an- antihomomorphism; it is called the right regular gle are often called the legs of the triangle. The representation of M.IfM has an identity, this Pythagorean Theorem states that in a right tri- is a faithful representation. angle the sum of the squares of the lengths of G a ∈ G (2) Let be a group, and for define the legs is equal to the square of the length of g → ga a ∈ G a mapping . In this way, each the hypotenuse. induces a permutation on the set G. This gives a representation of G as a permutation group R on G (again, the representation is an antihomo- ring A nonempty set , with two binary op- + · morphism): this is called the right regular rep- erations, usually denoted by and , which sat- resentation. See also regular representation, left isfy the following axioms: (i.) With respect to + R · regular representation. , is an Abelian group. (ii.) is associative: a · (b · c) = (a · b)· c for all a,b,c ∈ R. (iii.) + · a · (b + c) = right resolution Consider a cochain com- and satisfy the distributive laws: a · b + a · c (b + c) · a = b · a + c · a plex (X, δ) of A-modules Xn, n ∈ Z, and A and for all n n+ a,b,c ∈ R. The operation + is called addition module homomorphisms δn : X → X 1 with · δn+1δn = 0. Suppose ε : M → X0 is an aug- and the operation is called multiplication. If mentation of X over M, that is, an A module multiplication is commutative, the ring is called homomorphism from an A module M to X0 such a commutative ring. If there is an identity for ε δ0 multiplication, that is, an element 1 which has 0 1 that the composition M −→ X −→ X is triv- x · 1 = 1 · x = x for all x ∈ R, the ring is called ε δ0 δ0 δn ial. If 0 −→ M −→ X0 −→ · · · −→ Xn −→ a ring with identity. The identity for the additive ··· is exact, then X is called a right resolution Abelian group is denoted as 0. A division ring of M. If, in addition, each Xn is an injective A- is a ring whose nonzero elements form a group module, then the cochain complex X is called a under multiplication. A commutative division right injective resolution of M. ring is called a field. right satellite Let A be an Abelian category ringed space A topological space X, together and let R be a selective Abelian category. If with a sheaf of rings OX on X. The main import A has enough proper injectives, then a P con- of this statement is that to each open set U of X, nected pair (T , E∗,S) of covariant functors is there is associated a ring OX(U). Example: Let right universal if and only if each proper short X be an open subset of Cn, or, more generally, a exact sequence in A,0→ C → J → K → 0, complex analytic manifold. For each open sub- with J proper injective, induces a right exact set U, let OX(U) = the ring of analytic (holo- sequence T(J) → T(K) → S(C) → 0inR. morphic) functions defined on U. Then the pair Given T , the S with this property is uniquely (X, OX) is an important example of a ringed determined. It is called the right satellite of T . space. The sheaf OX is called the Oka sheaf. See also left satellite. See sheaf.

c 2001 by CRC Press LLC ring homomorphism Let R and S be rings, such that ai = 0 for all but a finite number of in- each with binary operations denoted by + and ·. dices i and defining addition and multiplication A ring homomorphism from R to S is a mapping by: (a0,a1,...)+(b0,b1,...)= (a0+b0,a1+ f : R → S f(a + b) = b ,...) (a ,a ,...)(b ,b ,...)= (c ,c , which satisfies: (i.) 1 and 0 1 0 1 0 1 f(a)+ f(b) f(a · b) = f(a)· f(b) ...) c = n a b and (ii.) where n i=0 n−i i. With these op- for all a,b ∈ R. erations, it is easy to see that R[x] is a ring. Usu- ally, however, the element (a0,..., an, 0, 0,...) n ring isomorphism A ring homomorphism is denoted by a0 + a1x +···+anx . Polyno- that is both injective and surjective. See ring mial rings in several variables can be defined homomorphism. inductively: R[x,y]=R[x][y]. M ring of differential polynomials Let K be a ring of scalars If is a module over a ring R, R is called the ring of scalars. See module. field and δ1,...,δn a set of commuting deriva- tions on K. An extension field L of K is called ring of total quotients Let R be a commu- a differential extension field of K if the δi ex- U tend to derivations (which we again denote by tative ring with identity and let be the multi- plicative subset of R consisting of all elements δi)onL such that δi(k) ∈ K for every k ∈ K. of R that are not zero divisors. The ring of total Let x1,...,xm be elements of a differential ex- quotients is the ring of fractions U −1R. See ring tension field of K with derivations δ1,...,δn r r r δ 1 δ 2 ···δ n x r of fractions. and suppose that 1 2 n i (where the i are nonnegative integers and 1 ≤ i ≤ m) are al- gebraically independent over K. The collection ring operations The two binary operations of all polynomials (over K) in these expressions that occur in the definition of ring. These are + · is called the ring of differential polynomials in usually denoted by and and are called addi- tion and multiplication. See ring. the variables x1,...,xm. ring theory The study of rings and their prop- ring of fractions Let R be a commutative erties. See ring. ring with identity, and S a multiplicative sub- set of R, that is, 1 ∈ S and S has the prop- Ritt’s Basis Theorem In the theory of dif- erty that ab ∈ S whenever a,b ∈ S. De- ferential rings, a polynomial generated by the fine an equivalence relation on the set R × S derivatives of a finite set S of elements is called by (r, s) ∼ (r$,s$) ⇔ s (rs$ − r$s) = 0 for 1 a differential polynomial of the elements of S. some s ∈ S. Note that in the case when S con- 1 Ritt’s Basis Theorem states that any set A of dif- tains no zero divisors and 0 ∈/ S, then (r, s) ∼ ferential polynomials contains a finite subset B (r$,s$) ⇔ rs$ − r$s = 0. The equivalence class r such that, if u is any element of A, then there of (r, s) is usually denoted as and the set of s exists an integer power of u that is a linear com- all equivalence classes defined in this way is de- bination of the elements of B and their deriva- noted S−1R. Note that if 0 ∈ S, then S−1R tives. This theorem on differential polynomials consists only of the element 0 . It can be shown 1 is the analog of Hilbert’s Basis Theorem in the S−1R that is a commutative ring with identity, ring of ordinary polynomials. See also Hilbert’s where addition and multiplication are defined by r r$ (rs$+r$s) r r$ rr$ Basis Theorem. s + s$ = ss$ and ( s )( s$ ) = ss$ . The ring S−1R is called the ring of fractions of R by S. root (1) (Of an equation) A number which satisfies the equation (makes the equation true ring of polynomials Let R be a ring and x when it is substituted in for the variable). an indeterminate. The ring of polynomials over (2) (Of a polynomial p(x)) A root of the R, denoted R[x], is the set of all polynomials in equation p(x) = 0. the variable x, with coefficients in R. This may (3) (Of a number) A number which satisfies n be formally defined by considering the set of x = b. This number√ is called the nth root of b, n all sequences (a0,a1,a2,...)of elements of R and is denoted by b.

c 2001 by CRC Press LLC (4) (Of a complex number) A number which is not minimal, then p decreases in the direc- n satisfies z = a + bi. This number is√ called the tion of the gradient of −p. Because p is a con- th n n root of a + bi, and is denoted by a + bi. vex differentiable function, p(x)p(u1)> root of a number. p(u2)... If the process converges, p is mini- mized at the limit of the sequence u0,u1,.... roots of unity If n is a positive integer, then any complex z such that zn = 1 is called an nth rotation Rigid motion about a fixed axis root of unity. There are n distinct nth roots of wherein every point not on the axis is rotated unity. A theorem that is useful for calculating through the same angle about the axis. A ro- roots of unity is De Moivre’s Theorem, which tated point set retains its shape. A rotation of states that for a complex number z in polar form Euclidian n-space is a linear transformation that z = r[cos θ + i sin θ], preserves distances and the orientation of the space. n n [r(cos θ + i sin θ)] = r (cos nθ + i sin nθ) . rotation group The group formed by the set of all rotations of Rn, also called the orthogonal n root subspace Let E be a subfield of a field group of R . The rotation group of 3-space K, and let p(x) be a polynomial with coeffi- appears prominently in theoretical mechanics. cients in E.Ifc is a root of p(x) (i.e., p(c) = 0), the smallest subfield of K that contains both E Roth’s Theorem A result in the theory of and c is called a root subspace. rational approximations of irrational numbers. If k>2 and a is a real algebraic number, then root system A finite subset U of a real vector there are only a finite number of integer pairs space V which satisfies three conditions: (i.) (x, y) with x>0 such that 0 ∈ U and U generates V ; (ii.) for each α ∈ U y 1 there exists an element α∗ in the dual space V ∗ a − < . x xk of V such that the reflection map rα : β → β − ∗ ∗ α (β)α stabilizes U and such that α, α =2, See algebraic number. where x,α∗=α∗(x); and (iii.) if α, β ∈ U, ∗ ∗ then ηβ,α ≡α, β ≡β (α) is an integer. Rouché’s Theorem Let C be a simple rec- tifiable curve in the complex plane and suppose R-order Let R be a ring. Let A be a finite that the functions F and f are analytic in and dimensional algebra with unit element e over a on C.If|F | > |f | at each point on C, then the field F . A subring G of A is called an R-order functions F and f + F have the same number in A if it satisfies (i.) e ∈ G, (ii.) G contains a of zeros in the finite region bounded by C. The basis of A as a vector space over the field F (an theorem is helpful for proving the existence of F -basis of A), and (iii.) G is a finitely generated and locating zeros of a complex function. R module. The most important case of this occurs when rounding of number A method of approxi- R is chosen to be the ring of integers, Z, in which mating numbers. To round a decimal to the nth case an R-order is called a Z-order. See also Z- place, one sets all of the digits after the nth place order. to zero, and changes the nth place according to the following rule: If the first digit after the nth Rosen’s gradient projection method An al- place is greater than or equal to 5, the nth place gorithm which seeks to minimize a convex dif- is increased by one. If the first digit after the nth ferentiable function p on a non-empty convex place is less than five, the nth place is unaffected set F . Starting with some u0 in F ,ifp(u0) by the rounding. For example, π rounded to the

c 2001 by CRC Press LLC hundredths place is 3.14; 35560 rounded to the r-ple point on a curve Suppose C is a plane nearest thousand is 36000. curve, P is a point on C, and F is a field. Write + C = Cr + Cr+1 +···+Cr+s, r, s ∈ Z , where row finite matrix An infinite matrix wherein each Ci is a form of degree i in F [X, Y ] and there are a finite number of non-zero entries in Cr = 0. Then P is called an r-ple point on the each row. curve C. See also multiplicity of a point. row nullity For A an m×n matrix, the dimen- ruled surface A surface S that contains, for sion of the row null space (the space of solutions each point p in S, a line that contains p. Some described by Ax = 0). If A is of rank r, the row simple examples of ruled surfaces are planes and nullity is equal to m − r. hyperboloids of one sheet. row of matrix A single horizontal sequence Rule of Three A philosophy of mathemati- of elements, extending across a matrix. For ex- cal pedagogy first stressed by the Calculus Con- ample, in the matrix sortium based at Harvard University, in which   graphical, numerical, and analytical approaches a a ··· a 11 12 1n are used to aid in conceptualizations of prob-  a a ··· a   21 22 2n  , lems and solutions. The same group has re-  ···· cently coined the term Rule of Four to imply a a ··· a m1 m2 mn the addition of a verbal approach to calculus. The term was used later by the University of Al- any of the arrays aj aj ...ajn isarow. 1 2 abama in instructing probability through math- ematics, simulation, and data analysis. row vector A horizontal array of elements; a1×n matrix.

c 2001 by CRC Press LLC scalar extension Suppose ϕ : R1 → R2 is a ring homomorphism and suppose M is an R2-module. Consider M to be an R1-module by R M = scalar restriction. Define the 1-module R2 R2 ⊗R M. Then MR may be considered to S 1 2  R r · (r ⊗ m) be an 2-module by defining 2 2 to r · r ⊗ m r ,r ∈ R be 2 2 for all 2 2 2 and for all m ∈ M R M saddle point A point at which the first two . This 2-module R2 is called a scalar partial derivatives of a function f(x,y) in both extension. See also scalar restriction. variables are zero, yet which is not a local ex- tremum. Also called a hyperbolic point. The scalar matrix A square matrix, wherein all term saddle hails from the appearance of the of the elements on the main diagonal are equal, plotted surface, wherein the saddle point appears i.e., all aii are the same, and all other elements as a mountain pass between two hills. Similar are 0, i.e., aij = 0, for i = j. saddle phenomena occur with three hills; they are sometimes called monkey saddles, with an scalar multiple (1) (In a linear space) The extra place for a monkey’s tail. product, ax, of an element, x, in a set L and an element, a, in a field K; K is the scalar field. Satake diagram Used in the theory of Lie (2) (Of a linear operator) The product of a algebras to describe the classification of a non- scalar a and a linear operator T : (aT )x = compact real simple Lie Algebra L that arises a(T x). from the comparison of the conjugation opera- (3) (In a module) The product ax of an el- tion of L with the complexification of L. ement x in a module M and an element a in a ring A (where M is a module over A); A is the satisfy Meet specified conditions, as in an ring of scalars. equation or set of equations. Any values that re- (4) (Of a vector) The product cv of a vector duce an equation to an identity satisfy that equa- v and a real number c. The vector cv is on the tion. Also, to meet a set of hypotheses, such as same line as the line containing v, and the ratio satisfying the conditions of a theorem. of cv to v is c (if v is the zero vector, then cv is also the zero vector). Sato’s conjecture A conjecture asserting

limx→∞ number of primes p ≤ x : θp ∈[α, β] scalar multiplication See vector space. number of primes

c 2001 by CRC Press LLC See also perfect field, separable extension. Schwarz inequality The inequality which states that for complex numbers x1,...,xn and Schottky group Suppose C1,C−1,...,Ck, y1,...,yn, C−k are 2k circles in the complex plane. Denote C I n 2 n n the interior of i by i and the exterior, includ- 2 2 xiy¯i ≤ |xi| |yi| . ing the point at infinity, by Ei, i =±1,...,±k. i=1 i=1 i=1 Let µ1,...,µk be Möbius transformations sat- isfying µi(Ci) = C−i,µi(Ei) = I−j ,µi(Ii) = The inequality is commonly generalized in −1 E−i. Further, µ−i is defined to be µi ,i = abstract inner product spaces to mean |(x, y)|≤ 1,...,k. Then the group generated by µ1,..., xy for vectors x and y, where (·, ·) is the in- µk is called a Schottky group. See also Möbius ner product and ·is its norm. Also known as transformation. the Cauchy-Schwarz or Bunyakovskiiˇ inequal- ity, it is a special case of the Hölder integral Schreier conjecture A conjecture which inequality. See Hölder’s Theorem. states that for an arbitrary finite simple group, G, the outer automorphisms of G form a solv- scientific notation The convention in applied able group. The conjecture is true for all known mathematics wherein real numbers are repre- finite simple groups. sented as (between one and ten) mul- tiplied by powers of ten. For example, 36000 is Schur index Let G be a finite group, F ⊂ S 3.6×104 and 0.00031 is 3.1×10−4 in scientific where S is a splitting field for G, and ϕ ∈ notation. The notation is particularly useful in IrrS(G).If, is an irreducible S-representation dealing with numbers that are very large or very which affords ϕ and - is an irreducible F - small in magnitude. representation such that , is a constituent of -S, then the multiplicity of , as a constituent secant function One of the fundamental of -S is called the Schur index of ϕ over F . See trigonometric functions, denoted sec x.By x = 1 also splitting field, irreducible representation, definition, sec cos x , and hence the se- constituent. cant function is (i.) periodic, satisfying sec(x + 2π) = sec x; (ii.) bounded below, satisfying Schur product See Hadamard product. 1 ≤|sec x|, for all real x, and (iii.) undefined x =±π , ± 3π ,... where the cosine is 0 ( 2 2 ). Schur’s Lemma (1) Let M and N be sim- Many other properties of the secant function can ple A-modules, and let f : M → N be an A- be derived from those of the cosine and sine. See homomorphism. Then f is either an isomor- cosine function. See also secant of an angle. phism or f is the zero homomorphism. (2) Let A be a ring and let M be a simple A- secant method The iterative sequence of ap- module. Let D = EA(M) be the endomorphism proximate solutions of the equation f(x) = 0, ring of M. Then D is a skew field. given by x − x x = x − f(x ) · n n+1 Schur subgroup The subgroup of the Brauer n+1 n n f(x ) − f(x ) group Br(k) consisting of those algebra classes n n+1 A k k that contain a Schur algebra over , where for f(xn) = f(xn+1), n ≥ 1. is a field of characteristic 0. secant of an angle Written secα, the recip- Schur-Zassenhaus Theorem Suppose N is rocal of the x-coordinate of the point where the a normal subgroup of a finite group G such that terminal ray of the angle α whose initial ray lies |N| and |G: N| are relatively prime. Then G along the positive x-axis intersects the unit cir- |G: N| <α< π α contains subgroups of order and any two cle. If 0 2 ( in radians) so that the of these subgroups are conjugate in G. angle is one of the angles in a right triangle with See also normal subgroup, conjugate sub- adjacent side a, opposite side b, and hypotenuse c group. c, then sec α = a .

c 2001 by CRC Press LLC second difference In difference equations, P2,... are the conjugacy classes of primitive where the difference of y at x is 2y(x) = y(x+ hyperbolic elements of 4, γj ∈ Pj are their rep- 2x) − y(x), the second difference is defined as resentatives, γ → M(g) is a matrix represen- 22y(x) = 2(2y(x)) = y(x + 2x) − y(x + 4 N(γ) = ξ 2 2 2 tation of , 2 the norm of hyperbolic 2x) + y(x). γ where ξ1,ξ2 are two eigenvalues of γ . Here, when the two eigenvalues of γ ∈ 4 are dis- second factor of class number See first fac- tinct real numbers ξ1,ξ2 (ξ1ξ2 = 1, ξ1 <ξ2), tor of class number. we call γ hyperbolic. When γ is hyperbolic, γ n(n = 1, 2,...)is also hyperbolic. When ±γ secular equation The characteristic equation is not a positive power of another hyperbolic ele- of a square matrix A. See characteristic equa- ment, γ is called a primitive hyperbolic element. tion. For a square symmetric matrix of real ele- ments, all of the solutions to the secular equation self-dual regular cone A cone which is self- (the eigenvalues of A) are real. dual and regular. Let X be a vector space with an inner product (·, ·). A subset C ⊂ X is called A A sedenion Any element of 4, where n is a cone if (i.) x,y ∈ C implies x + y ∈ C; (ii.) λ =− A the Clifford algebra over R with ii 1. 4 x ∈ C and t>0 imply tx ∈ C; (iii.) if x and −x is important in spinor theory and the theory of are in C, then x = 0. A cone is called regular if Dirac’s equation. the relations x1 ≤ x2 ≤···≤z imply that {xn} is norm-convergent. A subset Cd := {x ∈ X : G Selberg trace formula Let be a connected (x, y) ≥ 0, for all y ∈ C}⊂X is called a dual 4 semisimple Lie group and a discrete subgroup. cone. C is called self-dual if C = Cd . Let T be the regular representation of G on 4\G (T f )(x) = f (xy), f ∈ L2(4 \ defined by g self-intersection number On an algebraic G). When 4 \ G is compact, one has T = ∞ (k) surface, the intersection number (or Kronecker k= T as the irreducible decomposition of 1 index) I(d1 · d2) is defined for any divisors d1 T . Let χk be the character of the irreducible and d2. See intersection number. The self- unitary representation T (k). Then one has the intersection number is the case in which d1 = trace formula d , i.e., the number I(d · d) = (d2) is the self- 2 ∞ intersection number of the divisor d. −1 f(g)χk(g)dg = f(x γx)dx G D k=1 {γ } γ semidefinite Hermitian form A Hermitian where {γ } is the conjugate class of γ in 4 and form H with n variables such that the signature Dγ is the quotient space of the centralizer Gγ of H is either (r, 0) or (0,r), with 1 ≤ r

Selberg zeta function The zeta function semidirect product A group G which can be ∞ written as the product of a normal subgroup N of −s−n Z4(s, M) = det I − M (γi) N (γi) , G and a subgroup H of G where N ∩ H ={1}. i n=0 where 4 ⊂ SL(2, R) is a Fuchsian group, op- semifinite function A measurable function erating on the complex upper half-plane H, P1, f with the property that {x :|f(x)|=∞}is σ -

c 2001 by CRC Press LLC finite (the countable union of sets of finite mea- semilinear mapping A Z-linear mapping φ : sure). E → E, where E and E are modules, such that φ(ax) = aφ(x) for all x ∈ E. semigroup Generalization of a group in which a binary operation is defined and asso- semilocal ring A ring R with finitely many ciative on a set. The existence of an identity and maximal ideals. multiplicative inverses are not guaranteed in a semigroup as they are in a group. semiprimary ring A ring R such that, for any ideal A and any positive number n ∈ N semigroup algebra An algebra over a field with An = 0, we have A = 0. K which has a multiplicative semigroup G as its basis. semiprime differential ideal A differential ideal which is semiprime. Let R be a com- semigroup bialgebra An algebra ,(A) over mutative ring with an identity. If a mapping a field , with a basis A that is at the same time δ : R → R is such that, for any x,y ∈ R, (i.) a multiplicative semigroup. δ(x+y) = δx+δy; (ii.) δ(xy) = δx·y +x ·δy, then δ is called a differentiation. A ring R pro- semi-invariant (1) A common eigenvector vided with a finite number of mutually commu- of a family of endomorphisms of a vector space tative differentiations in R is called a differential or module. ring. Let δ1,...,δm be the differentiations of a (2) A numerical characteristic of random differential ring R. An ideal a of R with δj a ⊂ a variables related to the concept of moment of (i = 1, 2,...,m) is called a differential ideal. k higher order. If v = (v1,...,vk) ∈ R is a ran- We say that a differential ideal a is semiprime i(u,v) dom vector, φv(u) = E(e ) is its moment if a contains all those elements x that satisfy = (u ,...,u ) ∈ g generating function, where u 1 k x ∈ a for some natural number g. k ( , ) = n v u n ≥ R and u v j=1 j j . If for some 1 n the nth moment E(|vj | )<∞, j = 1,...,k, semiprime ideal See semiprime ring. then the (mixed) moments (α ,...,α ) α α semiprime ring A ring such that 0 is the only m 1 k = E v 1 ···v k v 1 k nilpotent ideal.

k exist for all multi-indices (α1,...,αk) ∈ (Z+) semiprime ring A ring R is semiprime if 0 such that |α|=α1 +···+αk ≤ n. Under these is the only nilpotent ideal. conditions, φv(u) = semiprimitive ring A ring in which the rad- ical is the set containing only the zero element. α1+···+αk i (α ,...,α ) α α n m 1 k u 1 ···u k +o |u| , See radical. α !···α ! v 1 k |α|≤n 1 k semireductive action A rational action de- where |u|=|u |+···+|uk| and for sufficiently 1 fined by ρ such that N = f1K + ···fr K is small |u|, the principal value of log φ (u) can be v a G-admissible module (f1,...,fr ∈ R) and expressed by Taylor’s expansion as that if f0 mod N(f0 ∈ R) is G-invariant, then there is a homogeneous form h in f ,...,fr φ ( ) = 0 log v u of positive degree with coefficients in K such that h is monic in f0 and is G-invariant, where α1+···+αk i (α ,...,α ) α α n a 1 k u 1 ···u k + o |u| K is a commutative ring with identity and G is α !···α ! v 1 k |α|≤n 1 k a matrix group over K (i.e., a subgroup of the general linear group GL(n, K)). A homomor- (α1,...,αk) where the coefficients av are called phism ρ of G into GL(m, K) is called a ratio- (mixed) semi-invariants, or cumulants, of order nal representation of G if there exist rational 2 α = (α1,...,αk) of the vector v = (v1,...,vk). functions φkl, 1 ≤ k,l ≤ m,inn variables xij

c 2001 by CRC Press LLC (1 ≤ i, j ≤ n) with coefficients in K such that sion n over a field F such that V contains a basis ρ((σij )) = (φkl(σij )) for all (σij ) ∈ G. As- of eigenvectors for T . sume that R is a commutative ring generated by x1,...,xn over K and that an action of the group semisimple matrix A diagonalizable matrix. t −1 G on R is defined such that (σ x1,...,σxn) = Hence a matrix A = SMS , where S is nonsin- t t σ (x1,...,xn) where means the transpose of gular and M is diagonal. See diagonal matrix, a matrix. In this case we say that G acts on R as nonsingular matrix. a matrix group. Assume that ρ is a rational rep- resentation of a matrix group G and ρ(G) acts semisimple module An R-module M of a on a ring R as a matrix group. Then we have ring R such that M is the sum of simple sub- an action of G on R defined by σf = (ρσ )f modules. Equivalent conditions are (σ ∈ G, f ∈ R), called the rational action de- (i.) M is the direct sum of simple submodules fined by ρ. and (ii.) for every submodule E there exists a sub- semisimple Banach algebra A Banach al- module F such that M = E + F . gebra A, such that its regular module A◦ is com- pletely reducible. See also Banach algebra, reg- semisimple part (1) A nonsingular ma- ular module, completely reducible. trix can be represented uniquely as a product (commutatively) of a diagonalizable matrix (the semisimple component The semisimple lin- semisimple part) and a unipotent matrix (the ear transformation φs of the Jordan decomposi- unipotent part) of identical dimensions. tion of a linear transformation φ : L → L, (2) The semisimple part of a group is the set where L is an n-dimensional linear space over a of all semisimple elements in that group. perfect field. By the Jordan decomposition, any such φ is represented as the sum of the φs and a semisimple ring A ring R such that, as a nilpotent linear transformation φn. module, R is semisimple. semisimple Jordan algebra A Jordan alge- Semistable Reduction Theorem A theorem bra A such that its regular module A◦ is com- proved by Grothendieck that asserts: Given an pletely reducible. See also Jordan algebra, reg- Abelian variety over F , there is a finite extension ular module, completely reducible. of F , over which it has a semistable reduction. Let F be a field with a discrete valuation v, semisimple Lie algebra A Lie algebra whose valuation ring ov, and maximal ideal mp. Let only Abelian ideal is the zero ideal. AF be an Abelian variety over F . Let A be a Néron model, A0 the connected Néron model, 0 semisimple Lie group A Lie group G is k = k(v)and Ak the special fiber over the residue called simple if class field k. We say that AF has semistable 0 (i.) dim(G) is greater than 1; reduction if Ak is an extension of an Abelian G 0 (ii.) has only finitely many connected com- variety by a torus (so that Ak is a semi-Abelian ponents; and variety). (iii.) any proper normal subgroup of the identity component of G is finite. sense of inequality The direction of an in- We say that G is reductive if it has finitely equality is sometimes also referred to as the many connected components, and some finite sense of an inequality. For example, x>5 cover of the identity component Ge is a product and y>zhave the same sense. of simple and Abelian Lie groups. It is semisim- ple if there are no Abelian factors in this decom- sensitivity analysis The analysis of response position. to imposed conditions, i.e., variations of solu- tions in response to variations in a problem’s semisimple linear transformation A linear parameters. A particularly important topic in transformation T on a vector space V of dimen- chaotic dynamics, wherein small changes in ini-

c 2001 by CRC Press LLC tial conditions or parameters (such as a butterfly fine ϕ(0) =∞where ∞ exceeds any ordinal. flapping its wings in China) often produce dra- Then associated to each x there is a sequence of i matically different behavior at later times (such ordinals {µi : µi = ϕ(p x),i ≥ 0} called the as the formation of a hurricane that proceeds to sequence of Ulm factors. See also complete val- ravage much of Florida). uation ring, reduced module, transfinite series, limit ordinal. separable algebra A finite dimensional alge- bra A over a field K such that the tensor product serial subgroup A subgroup H of a group G E ⊗K A is semisimple over E for every exten- is serial in G, denoted H ser G, if there exists a sion field E of K. totally ordered set H and a set {(I ,V ) : σ ∈ H} separable element An algebraic element a σ σ over a field K which is a root of a separable polynomial over K. such that (i.) H ≤ Vσ ≡ Iσ ≤ G ∀σ ∈ H; I ≤ V τ<σ separable extension An extension field F (ii.) τ σ if ; G \ H = {I \ V : σ ∈ H} of a field K whose elements are all separable (iii.) σ σ . elements over K. series (1) The summation of a finite sequence separable polynomial An irreducible poly- of terms, written with the Greek letter sigma as nomial with a nonzero formal derivative. follows: N separable scheme See purely inseparable an = a0 + a1 +···+aN scheme. n=0 separated morphism A morphism f : X → where n is the index of summation, 0 and N are S such that the image of the diagonal morphism the limits of the summation, and each an is a 2X/S : X → X ×S X is closed, where X and S term. are local-ringed spaces. (2)Aninfinite series is a formal sum of the above type for which N =∞. An infinite series separated scheme A scheme X such that is said to converge if limN→∞(a1 +···+aN ) = S S the diagonal subscheme 2X/Spec(A)(X) = X for some finite ; it is said to diverge oth- ×Spec(A) X is closed. erwise. There are many methods used to test the convergence or divergence of infinite series, separating transcendence basis A subset S such as the root test, ratio test, and comparison of a field extension F of a field K such that S test. Infinite series are also particularly useful is algebraically independent and F is algebraic in analysis. and separable over K(S). (3) Power series, or series of the form ∞ sequence of factor groups G n Let be a group anx , and let G = G G ··· Gn be a chain of nor- 0 1 n=0 mal subgroups of G. Then the sequence of fac- tor groups is given by Gi−1/Gi, i = 1, 2,...,n. for complex an, can be used to represent some functions on intervals; these functions are called sequence of Ulm factors Suppose R is a real analytic in the interval (as opposed to being complete valuation ring with prime p and re- singular at some point in the interval). Real ana- duced R-module M. Define {Mµ} to be the lytic functions can be differentiated or integrated descending transfinite series obtained by suc- by differentiating or integrating their represen- cessive multiplication by p where intersections tative power series term by term. Moreover, the are taken at the limit ordinals. Define ϕ by value of a function can be approximated at a ϕ(x) = µ if x ∈ Mµ but x/∈ Mµ+1; and de- point c by evaluating its truncated power series

c 2001 by CRC Press LLC HN a (x −c)n p = n=0 n , with increasing accuracy in the (vi.) If 2, approximation as N approaches infinity. − ,E a,πi (4) Series appear in a variety of other con- 1 texts, including differential equations (in which ∞ s s+ i s+1 solutions can sometimes be represented by = π, E i2 F 1(a), π 2 power series) and Fourier analysis, in which se- s=0 ries of trigonometric terms are used to represent for α ∈ DT , p | i, where K is an algebraic num- otherwise complicated functions such as pulse ber field, p ∈ K is a prime ideal and π isafixed functions. See also Fourier series. prime element of K, T is the inertia field of K, E(α) is the Artin-Hasse function, and E(α, χ) Serre conjecture Let F [X1,X2,...,Xn] be is the Shaferevich function with α ∈ DT and a ring of polynomials over a field F . Then ev- χ ∈ p. The explicit reciprocity law is a gen- ery finitely generated projective module over eralization of Gauss’s quadratic reciprocity law, F [X ,X ,...,Xn] is free. Serre’s conjecture 1 2 which allows one to decide whether an integral was proved by Quillen and Suslin and is also number in a field is an nth power residue with known as the Quillen-Suslin Theorem. respect to a prime ideal. Serre’s Theorem For a positively graded, sheaf A presheaf F on X which satisfies the commutative algebra C, the category of coher- following conditions for every open U ⊆ X ent sheaves on the scheme Proj C is equivalent to such that ∪iUi is an open covering of U: (i.) For a certain category which is completely defined  every i,ifs, s ∈ F(U) are such that s|Ui = in terms of C-gr.   s |Ui , then s = s ; and (ii.) for every i and j, if si ∈ F(Ui) and sj ∈ F(Uj ) are such that set of antisymmetry See antisymmetric de- s | = s | s ∈ F(U) composition. i Ui ∩Uj j Ui ∩Uj , then there is an such that for every i, s|Ui = si. See presheaf. Shaferevich’s reciprocity law The explicit sheaf of germs of regular functions reciprocity law which asserts that A sheaf  of rings OV on a variety V where the stalk OV,x (i.) (π, E(α, π )) = 1 for p | i, α ∈ DT . α (x ∈ V) is the local ring of the ring regular (ii.) (π, E(α)) = ζ Tr if α ∈ DT ,Trα := functions of open sets on V at x. TrT/Qp α. (iii.) (π, π) = (π, −1). × sheaf theory A mathematical method for (iv.) (E(α), ) = 1 for α ∈ DT ,  ∈ D . K providing links between the local and global (v.) If p = 2, properties of topological spaces. Sheaf theory E α, π i ,E β,πj is also used to study problems in mathematical areas such as algebra, analysis, and geometry. = π j ,E αβ, π i+j ; Shilov boundary Let B be a Banach algebra, M if p = 2, the set of all multiplicative linear functionals λ (λ : B → C linear and λ(fg) = λ(f )λ(g)), E α, π i ,E α, π j along with the weakest topology which makes the mapping M % m → m(f ) ∈ C continu- = −π j ,E αβ, π i+j ous for every f ∈ B, and fˆ the Gel’fand trans- form of f ∈ B. Then the Shilov boundary S of ∞ s i+jps M is the smallest closed subset of M such that −1,E αF (β), π ˆ ˆ supS |f |=supM |f |, f ∈ B. See also Banach s=1 ∞ algebra, Gel’fand transform. r ipr +j · −1,E F (α)β, π short representation A representation of an r=1 ideal I in a commutative ring R as an intersec- for α, β ∈ DT , p | i, p | j. tion of primary ideals, I = Q1 ∩ Q2 ∩···∩Qk,

c 2001 by CRC Press LLC satisfying two additional conditions: (i.) The Siegel domains are divided into three classes, representation is irredundant, that is, it is not Siegel domains of the first kind, Siegel domains possible to omit one of the primary ideals Qi of the second kind, and Siegel domains of the from the intersection. (ii.) The prime ideals third kind. There is a deep connection between P1,P2,...,Pk belonging to the Q1,Q2,..., Siegel domains of the second kind and perfectly Qk are all different. (Pi is the prime ideal be- general bounded homogeneous domains. (A do- longing to Qi if Pi contains Qi, but there is no main is bounded if it is contained in a ball of  smaller prime ideal Pi containing Qi. Equiva- finite radius. A bounded homogeneous domain lently, Pi is the radical of Qi, i.e., Pi ={p ∈ is a domain which is both bounded and homoge- R : pn ∈ I for some integer n}.) See also iso- neous. An open ball of finite radius in complex lated component, isolated primary component, N space is a bounded homogeneous domain. A irredundant, primary ideal, prime ideal, radical. generalized Siegel half plane is a homogeneous domain, but it is not bounded.) The connection Siegel domain An open subset S of com- between Siegel domains and bounded homoge- N plex N space C of the form S ={(z, w) ∈ neous domains is provided by a famous theorem m n C ×C : Im z −N(w, w) ∈ O}. Here, O is an of Vinberg, Gindikin, and Pyatetskii-Shapiro: m open convex cone in real m space R with vertex Every bounded homogeneous domain in CN is n n m at the origin, N : C × C → C is O Hermi- isomorphic to a homogeneous Siegel domain of tian, Im z denotes the imaginary part of z, and the second kind. See also bounded homoge- m+n = N. In addition, the cone O is usually as- neous domain, irreducible homogeneous Siegel sumed to be acute; that is, O is usually assumed domain, Siegel domain of the first kind, Siegel not to contain an entire line. Finally, O Hermi- domain of the second kind, Siegel domain of the tian means (i.) N(v, w) is linear with respect to third kind. v, (ii.) N(v, w) = N(w, v), (iii.) N(w, w) = 0 if and only if w = 0, and (iv.) N(w, w) ∈ O.In Siegel domain of the first kind A domain property (ii.), Phi(w,v) denotes the complex D(V ) ={x + iy ∈ Rc|y ∈ V }, where V is a conjugate of N(w, v), whereas in property (iv.) regular cone in an n-dimensional vector space O denotes the topological closure of O. R and Rc is the complexification of R. An example of a Siegel domain is the un- bounded domain H consisting of all n × n sym- Siegel domain of the second kind A do- metric matrices with complex entries, and with main D(V,F) ={(x + iy, u) ∈ Rc × W|y − positive definite imaginary parts. (A matrix M F (u, u) ∈ V }, where V is a regular cone in an is symmetric if Mt = M, where Mt is the trans- n-dimensional vector space R, Rc is the com- pose of M. A matrix M with complex entries plexification of R, W is a complex vector space, can be written uniquely as M = A + iB, where and F : W × W → Rc is a V -Hermitian form. A and B have real entries. The matrix B is the If W = 0, then a Siegel domain of the first kind imaginary part of M.Ann × n matrix B with is obtained. See also Siegel domain of the first real entries is positive definite if xt Bx ≥ 0 for kind. all n dimensional column vectors x with real entries. Here, xt is just x, made into a row vec- Siegel domain of the third kind A do- tor.) The Siegel domain H is called a general- main which is holomorphically equivalent to ized Siegel half plane. In the special case where a bounded domain and can be written as n = 1, the generalized Siegel half plane is just D(V,L,B) ={(x + iy,u,p) ∈ Rc × W × the familiar upper half plane of elementary com- X|y − Re Lp(u, u) ∈ V,p ∈ B}, where V is plex analysis. a regular cone in an n-dimensional vector space A Siegel domain is homogeneous if it is ho- R, Rc is the complexification of R, W and X are mogeneous as a domain, that is, if it has a tran- complex vector spaces, B is a bounded domain c sitive group of analytic (holomorphic) automor- in X and L, Lp : W ×W → R are nondegener- phisms. The generalized Siegel half planes are ate semi-Hermitian forms, for p ∈ B. See also homogeneous Siegel domains. See homoge- Siegel domain of the first kind, Siegel domain neous domain. of the second kind.

c 2001 by CRC Press LLC Siegel’s Theorem Let χ be a real Dirichlet sign-nonsingular matrix See sign pattern. character of modulus k such that sign pattern The sign pattern of a real m×n (n, k) = χ(n) = 1if 1 . matrix A is the (0, 1, −1)–matrix obtained from 0if(n, k) = 1 A when zero, positive, and negative entries are replaced by 0, 1, −1, respectively. Thus, A de- Siegel’s Theorem states that, for any >0, there termines a (qualitative) class of matrices, Q(A), exists c > 0 such that consisting of all matrices with the same sign pat- tern as A. ∞ χ(n) c >  . Sign patterns are one of the objects of study in n k n=1 combinatorial matrix analysis, concerned with the study of properties of matrices that are de- termined from its combinatorial structure (such ζ Siegel zeta function A -function attached to as the directed graph) and other qualitative in- an indefinite, quadratic form, meromorphic on formation. For example, A is called an L-matrix the whole complex plane and satisfying certain provided that every matrix in Q(A) has linearly functional equations. independent rows. The sign pattern   sigma field Consider an arbitrary non-empty 11 1−1 A X subclass of the power set of a set such that 11−11 P Q A P ∩ Q P ∪ bQ if and are in , then , , and 1 −11 1 cP are also in A. Such an A is called a field. If, in addition, A is closed under countable unions, is an example of an L-matrix. Such matrices first then it is called a σ-field. arose in the study of sign-solvable systems in economics models. A sign-nonsingular matrix (p, q) signature The ordered pair corre- A is a square L-matrix; that is, every matrix in Q sponding to a real quadratic form of rank Q(A) is nonsingular. p + q, where Q is equivalent to p q+p similar (1) (Similar figures) Geometric fig- 2 2 ures such that the ratio of the distance between xi − xj . pairs of points in one figure and the distance be- i=1 j=p+1 tween the corresponding pairs of points in an- See also signature of Hermitian form. other figure is constant for every pair of points in the first figure and corresponding pair of points signature of Hermitian form The ordered in the second figure. pair (p, q) corresponding to a Hermitian form (2) (Similar terms) Terms in an expression H of rank p + q, where H is equivalent to which have the same unknowns and each un- known is raised to the same power in each term. p q+p For example, in the expression 5x2y3 + 8x3 + xixi − xj xj . 3y3+7x2y3,5x2y3 and 7x2y3 are similar terms. i=1 j=p+1 (3) (Similar triangles) Triangles which have proportional corresponding sides. signed numbers Numbers that are either pos- itive or negative are said to be signed or directed similar decimals Numbers that have the same numbers. number of decimal places are said to be similar. For example, 6.003, 2.232, and 100.000 are all significant digits In a real number in decimal similar decimals. form, the digits to the left of the decimal point beginning with first non-zero digit, together with similar fractions Simple fractions in which the digits to the right of the decimal point ending the denominators are equal. See simple fraction, with the last non-zero digit. denominator.

c 2001 by CRC Press LLC similar isomorphism An isomorphism be- simple extension An extension E of a field F tween two ordered fields, under which positive such that E = F(x), for some element x ∈ E, elements are mapped to positive elements. where F(x) denotes the subfield of E over F , generated by x. See also extension, subfield. similar matrices Two square matrices A and −1 a B such that A = P BP for some nonsingular simple fraction A fraction b , in which both matrix P . See nonsingular matrix. a and b are integers, and b = 0. Also known as a common fraction. similar permutation representations Two permutation representations, such that there ex- simple group A group that has no normal ists a G-bijection between the corresponding G- subgroup other than itself and its identity {e}. sets, where G is a group. Let SM be the group of all permutations of a set M.Apermutation simple Lie algebra A Lie algebra whose only representation of a group G in M is a homo- ideals are itself and the zero ideal. morphism G → SM . In general, if the product ax ∈ M of a ∈ G and x ∈ M is defined and sat- simple Lie group A Lie group whose only isfies (ab)x = a(bx),1x = x, for all a,b ∈ G proper Lie subgroup is the trivial subgroup. and for all x ∈ M, with the identity element 1, then G is said to operate on M from the left and simple module A module M, with more than M is called a left G-set. We call a left G-set a one element, whose only submodules are itself G-set. A mapping f : M → M of G-sets is and the zero submodule. called a G-mapping if f(ax) = af (x) for any a ∈ G and x ∈ M. G-injection, G-surjection, simple point A point of an irreducible variety ∂fi G V of dimension d over a field K such that  -bijection are defined naturally. ∂xj is of rank n − d, where f1(x0,x1,...,xn) = 0, similar terms See like terms. f2(x0,x1,...,xn) = 0, ..., fr (x0,x1,...,xn) = 0 form a basis for equations of V . simple Abelian variety A commutative al- gebraic group without Abelian subvarieties (ex- simple ring A ring R with more than one cept itself and its zero element). See algebraic element whose only ideals are itself and the zero group, Abelian subvariety. ideal. simple algebra An algebra A such that the simple root A root of an equation that is ring A is both a simple ring and a semisimple not repeated, i.e., a root of multiplicity 1. See ring. multiplicity of root. simple co-algebra A co-algebra which has simplest alternating polynomial The poly- no nontrival proper subco-algebras. See coalge- nomial p(X1,...,Xn) = (X2 − X1)(X3 − X2) bra. ...(Xn − Xn−1). See alternating polynomial. simple component (1) Suppose R is a ring simplex (1) In geometry, the convex hull of written as R = R1 ⊗···⊗Rn where R1,...,Rn an affine independent set of points, the vertices are ideals satisfying the condition that if Ri = of the simplex.     Ri ⊗Ri , where Ri and Ri are ideals, then either (2) In topology, a homeomorphic image of a   Ri = 0orRi = 0. Then the ideals R1,...,Rn geometric simplex (as in (1)). are called the simple components of R. (2) Suppose P is a polynomial in n variables simplex method A method for optimizing a F P = n P αi over a field . Write i=1 i where the basic feasible solution of a primary linear pro- Pi are the distinct, irreducible (over F ) factors gramming problem in a finite number of itera- of P . Pj is called a simple component of P if tions. Let I be the set of basic variables and J be αj = 1. the set of non-basic variables. Let x be a basic

c 2001 by CRC Press LLC feasible solution of a primary linear program- is a system of simultaneous equations with no ming problem, and write x in the basic form: solution. xi + dij xj = gi (i ∈ I) simultaneous inequalities Any system of j∈J two or more inequalities that must both/all be satisfied by the same solutions. For example, z + f x = v.  j j y2 + x2 ≤ 1 j∈J y ≥ 0 If, for all j ∈ J , fj ≥ 0, then x, given by are simultaneous inequalities satisfied by the up- per half of the unit circle. gT if T ∈ I xT = , 0ifT ∈ J sine function One of the fundamental trigono- metric functions, denoted sin x. It is (i.) peri- is an optimal solution. If there is a j ∈ J such odic, satisfying sin(x+2π) = sin x; (ii.) bound- gi fj < ri∗ = − ≤ x ≤ that 0, then let min dij , where the ed, satisfying 1 sin 1, for all real minimum is taken over all i ∈ I such that dij > x; and (iii.) intimately related with the cosine i∗ I j x x = ( π − x) 0. Then, replace in with , obtaining a new function, cos , satisfying sin cos 2 , basis. Rewrite the basic feasible solution into a sin2 x + cos2 x = 1, and many others. It is re- (new) basic form, and repeat the procedure. lated to the exponential function√ via the identity x = eix−e−ix (i = − sin 2i 1), and has series simplex tableau See simplex method. expansion simplification of an expression Changing x3 x5 x7 sin x = x − + − +··· the form of an expression (but not the content) 3! 5! 7! with the purpose of either making the expression valid for all real values of x. more concise or readying the expression for the See also sine of angle. next step in a method or proof or solution. sine of angle Written sin α, the y-coordinate simply connected covering Lie group The of the point where the terminal ray of the angle G simply connected Lie group 0, among those α whose initial ray lies along the positive x-axis G g π Lie groups which have as their Lie algebra, intersects the unit circle. If 0 <α< (α in g 2 where is a finite-dimensional Lie algebra over radians) so that the angle is one of the angles G R. Such 0 is unique up to isomorphism. in a right triangle with adjacent side a, opposite b side b, and hypotenuse c, then sin α = . simply connected group A set G which has c the structure of a topological group (making the single step method for solving linear equa- group operation continuous) and which is sim- tions See Gauss-Seidel method for solving ply connected. linear equations. simultaneous equations Any system of two single-valued function A relation (a set R or more equations that must both/all be satisfied of pairs (x, y), x from a set X, and y from a by the same solutions. For example, set Y ) such that (x, y1), (x, y2) ∈ R implies  y = y . Thus, the function f : X → Y defined y = 3x + 5 1 2 by f(x) = y if (x, y) ∈ R assigns only one y y = 2x + 6 toagivenx. constitutes a system of two simultaneous equa- Indeed, every function has the above prop- tions satisfied by x = 1,y = 8; erty. However, it is occasionally necessary to  include multiple-valued “functions” (such as in- y = x − 1 verses of functions) in function theory and the y = 3x − 3 term single-valued is required for emphasis.

c 2001 by CRC Press LLC singular locus The subvariety of an irre- Slater’s constraint qualification The gen- ducible variety V of dimension d over a field eral nonlinear programming problem is the fol- K consisting of all points V which are not sim- lowing: let X0 be a connected closed set and let ple points. θ,g = (g1,...,gm) be real-valued functions defined on X0. Determine the set of x ∈ X0 singular matrix An n × n, non-invertible which minimizes (or maximizes) θ, subject to matrix. A square matrix A is singular if and only given constraints. If X0 is convex and gi are if its null space contains at least one nonzero convex, then the convex vector function g is vector. Equivalently, A is singular if and only said to satisfy Slater’s constrait qualification on if detA = 0 or if and only if at least one of its C0 provided there exists an x ∈ X0 such that eigenvalues is 0. gi(x) < 0, for all i. singular point A point at which a function slope-intercept form An equation of a line y = mx + b m ceases to be regular, even though the function in the plane of the form , where b remains regular at the points near the singular is the slope of the line and is its intercept on y y = x − point. See regular function. The term is often the -axis. Example: 3 2 is a line with y − y =− used with “regular” replaced by “differentiable,” a slope of 3 and a -intercept of 2( 2, ∞ x = “n times differentiable,” “C ,” etc. when 0). singular value decomposition Let A be an small category If the objects of a category n × n real normal matrix. A singular value form a set, then the category is called a small decomposition of A is the representation A = category or a diagram scheme. USU∗, where U is a unitary matrix, U ∗ is the adjoint of U, and S is a diagonal matrix of which small category If the objects of a category the diagonal entries are the singular values of A. form a set, then the category is called a small category or a diagram scheme. singular values of a matrix Eigenvalues of smooth affine variety Let k be a field and a matrix A; i.e., the values of λ such that λI −A X ⊆ kn. We call X a variety if it is the common is singular. See eigenvalue. zero set of a collection of polynomials. The va- riety is said to be smooth at a point x ∈ X if the F skew field A skew field is a ring in which collection of polynomials has non-degenerate the non-zero elements form a multiplicative Jacobian at x. group. smooth affine variety Let k be a field and A n skew-Hermitian matrix A matrix of com- X ⊆ k . We call X a variety if it is the common A¯T plex elements (necessarily square) such that zero set of a collection of polynomials. The va- =−A ; i.e., the conjugate of its transpose is riety is said to be smooth at a point x ∈ X if the equal to its negation. Also called anti-Hermitian. collection of polynomials has non-degenerate Jacobian at x. skew-symmetric matrix A matrix A (nec- T essarily square) such that A =−A, that is, its Snapper polynomial Let X be a k-complete transpose is equal to its negation. Also called scheme, F be a coherent OX-module, and L antisymmetric. be an invertible sheaf. The Snapper polynomial ⊗ is the polynomial in m given by χ(F ⊗ L m ), q q slack variable The linear inequality a1x1 + where χ(F) = q (−1) dim H (X, F ). a2x2 +···+anxn ≤ b may be converted to a linear equality by adding a nonnegative variable solution Anything that satisfies a given set z, called a slack variable. Namely, a1x1+a2x2+ of constraints is called a solution to that set of ···+anxn ≤ b becomes a1x1 + a2x2 +···+ constraints. A number may be a solution to an anxn + z = b. See also linear programming. equation or a problem. A function may be a

c 2001 by CRC Press LLC solution to a differential equation. A region on consists of using three given parts (for exam- a plane that satisfies a set of inequalities is a ple, the two sides and the included angle) to find solution of that set of inequalities. the remaining parts. See also solution of right spherical triangle. solution by radicals Any method of solving equations using arithmetic operations (including solution of plane triangle A plane triangle nth roots). For instance, the quadratic formula has six principal parts: three angles and three is a well-known method of solving second order sides. Given any three of these parts, which equations. See quadratic formula. Some equa- are sufficient to determine the triangle (e.g., two tions of fifth order and higher cannot be solved sides and the included angle), the solution of the by radicals. triangle consists of determining the remaining three parts. solution of an equation (1) For an equation F(x1,...,xn) = 0, with unknowns x1,...,xn solution of right spherical triangle A spher- in some set S,anyn-tuple of numbers (a1,..., ical triangle is formed by the intersection of the an), ai ∈ S satisfying F(a1,...,an) = 0. arcs of three great circles on the surface of a (2) For a functional equation F(x,y) = 0 sphere. A spherical triangle has six parts, the with unknown function y, any function f(x) three angles α, β, and γ , and three sides AC, such that F(x,f(x)) = 0. AB, and BC. The angle α is specified by the (3) For a differential equation dihedral angle between the plane containing the AB AC  (n) arc and the plane containing the arc , F t,x,x ,...,x = 0 , and similarly for β and γ . If one (or more) of the angles is a right angle, the triangle is called x(m) = dmx n where dtm ,any times differentiable a right spherical triangle. Note that in contrast function f(t)such that to a plane triangle, a spherical triangle may have  (n) one, two, or three right angles. The solution of F t,f(t),f (t),...,f (t) = 0 . a right spherical triangle consists of finding all Many other types of equations are possible, the parts of the triangle, given the right angle including differential equations with values of and any two other elements. The many possi- the function and certain derivatives specified at ble cases which can occur were all summarized certain points. in two formulas discovered by J. Napier (1550- 1617), known as Napier’s rules. solution of an inequality For an inequality A F(x ,...,xn) ≥ 0, with unknowns x ,...,xn solvable algebra Given an algebra , define 1 1 A(1) A(2) A(3) ... in some ordered set S,anyn-tuple of numbers the sequence of algebras , , , , A(1) = A A(i+1) = (A(i))2 i = , , ... (a ,...,an), ai ∈ S satisfying F(a ,...,an) by , , 1 2 3 , 1 1 (A(i))2 ≥ 0. For instance, the region beneath the line where denotes the span of squares of A(i) A A(j) = y = x + 2 contains solutions to the inequality elements of . Then is solvable if 0 j y − x<2. for some . solution of oblique spherical triangle A solvable by radicals From antiquity, it has ax2 + spherical triangle is formed by the intersection been known that the quadratic equation bx + c = of the arcs of three great circles on the surface of 0 has solutions √ a sphere. A spherical triangle has six parts, the −b ± b2 − 4ac three angles α, β, and γ , and three sides AC, x = . 2a AB, and BC. The angle α is specified by the dihedral angle between the plane containing the In the , analogous formulas for the arc AB and the plane containing the arc AC, solution of general cubic and quartic equations and similarly for β and γ . An oblique spherical involving arithmetic combinations of the coef- triangle is one in which none of the three an- ficients of the polynomial and their roots were gles is a right angle, and solving such a triangle discovered, and in the nineteenth century it was

c 2001 by CRC Press LLC proved that no such formula exists for the roots point in space, according to some convention of general polynomials of degree five or higher. to establish such location. Examples of such In general one says that a polynomial p(x), with coordinates are Cartesian coordinates, spheri- coefficients in a field F (with characteristic 0) is cal coordinates, or cylindrical coordinates. Note solvable by radicals if there exists an algorithm that while the coordinates must specify a unique for computing the roots of p(x) which consists point in space, a given point in space may be only of performing the field operations on the represented by several (or even infinitely many) coefficients of p, together with extracting roots different sets of coordinates. of such expressions. Since taking roots may take one to an extension field of F , a more precise space curve A one-dimensional variety in formulation would say that there exists a radical three-dimensional projective space. extension of F which contains a splitting field of p(x). See also Cardano’s formula. space group A discrete subgroup G of the Eu- clidean group, for which the subgroup of trans- solvable group Given a group G, a nor- lations in G forms a three-dimensional lattice {N }n group. Synonyms are crystallographic group mal series is a finite set of subgroups, j j=1, which satisfy: (i.) {e}=Nn ⊂ Nn−1 ⊂ ··· ⊂ and crystallographic space group. See also lat- N2 ⊂ N1 = G, where e is the identity element tice group. in G. (ii.) Nj+1 is a normal subgroup of Nj for j = 1 ...n− 1. G is solvable if there exists span of vectors The span of the vectors x1,x2, n ...,x V F a normal series {Nj }j= for which each of the k in a vector space over a field ,is 1 X groups Nj /Nj+1 is Abelian. the set of all possible linear combinations of x1,x2,..., xk.Ifx1,x2,...,xk are linearly in- solvable ideal An ideal in which the derived dependent vectors, then they constitute a basis series becomes zero. for their span, X. solvable ideal An ideal in which the derived spatial ∗-isomorphism Suppose that Hi is series becomes zero. a Hilbert space and that Mi is a von Neumann algebra contained in the set of bounded linear solvable Lie group A Lie group G, which, operators on Hi, for i = 1, 2, and consider a ignoring its Lie group structure and considering ∗-isomorphism π : M1 → M2. If there exists it only as an abstract group, is solvable. See a bijective isometric linear mapping U : H1 → ∗ solvable group. H2 such that UAU = πA, for each A in M1 (with U ∗ denoting the adjoint of U), then π is solving a triangle The act of using the size called spatial. of certain sides and angles of a triangle to deter- ∗ mine the remaining sides and angles. See also spatial tensor product The C -algebra re- solution of oblique spherical triangle, solution sulting when the algebraic tensor product of two C∗ algebras A and B is equipped with the spa- of plane triangle, solution of right spherical tri- ∗ angle. tial, or minimal, C cross-norm,

xmin = sup (ρ ⊗ η)(x) space (1) A set of objects, usually called ρ,η points. Often equipped with some additional structure, e.g., a topological space, if equipped where x ∈ A ⊗ B, and the supremum runs over with a topology, or a metric space, if equipped all ∗-representations of A and B. There may with a metric. be more than one norm on the algebraic tensor (2) The unbounded three-dimensional region product A ⊗ B whose completion gives a C∗- R3 in which all physical objects exist. algebra. space coordinates A set of three numbers special Clifford group If R is a ring with sufficient to specify uniquely the location of a identity, M a free R-module, and Q a quadratic

c 2001 by CRC Press LLC form, define the Clifford algebra C(Q) as the determinant 1, such that the transpose of M is quotient algebra T(M)/I(M) where T(M) = equal to M−1. The notation SO(n) is used. R ⊕ T 1M ⊕ T 2M ⊕ ..., T j M is the j-fold tensor product of M with itself, and I(M) is special representation If J is a Jordan alge- + the ideal generated by all elements of the form bra, A an associative algebra, and A the com- x ⊗ x − Q(x)1. Define C+(Q) to be the sub- mutative algebra obtained from A, by defining C(Q) x ∗ y = 1 (xy + yx) algebra of generated by an even number a new multiplication via 2 ,a of elements of M. The special Clifford group special representation of J is a homomorphism + G+(Q) consists of the set of invertible elements ρ : J → A . g ∈ C+(Q) for which gMg−1 = g. specialty index (1) Of a divisor a of a nonsin- special divisor (1) A positive divisor a of a gular complete irreducible curve C, the number nonsingular, complete, irreducible curve C such d − n + g, where d is the dimension of the com- that the specialty index of a is positive. plete linear system determined by a, n is the (2) A divisor D on an algebraic function field degree of a, and g is the genus of C. F in one variable such that the specialty index (2) Of a divisor D of an algebraic function of D is positive. field F in one variable, the number d −n+g−1, See positive divisor, specialty index. where D has dimension d and degree n, and F has genus g. specialization (1) Suppose that D and E are (3) Of a curve D on a nonsingular surface S, extension fields of a field F , and denote by Dn the dimension of H 2(S, O(D)), where O(D) is and En the n-dimensional affine spaces over D the sheaf of germs of holomorphic cross-sections and E, respectively. Suppose that x is a point of D. of Dn and that y is a point of En. If every polynomial p with coefficients in F that satisfies The multiplicative p(x) = 0 also satisfies p(y) = 0, then y is group of n × n matrices M such that the in- −1 called a specialization of x over F . verse M is equal to the adjoint of M and the M (n) (2) Suppose that S is a scheme, that s and t determinant of is 1. The notation SU is are geometric points of S, and that t is defined used. by an algebraic closure of the residue field of a point of the spectrum of the strict localization of special universal enveloping algebra Given J S at s. Then s is called a specialization of t. a Jordan algebra , its special universal envelop- ing algebra is a pair (U, ρ) where U is an as- ρ : J → U + special Jordan algebra Let A be an associa- sociative algebra, and is a special J tive algebra over a field F of characteristic not representation of with the property that for any + ρ˜ J equal to 2. Let A be the commutative algebra other special representation of , there exists h ρ˜ = hρ obtained by defining a new multiplication via a homomorphism , such that . x ∗ y = 1 (xy + yx). A special Jordan algebra 2 v J is a commutative algebra satisfying the Jordan special valuation A valuation of a field v identity such that the rank of is 1. Also called expo- nential valuation. See rank of valuation. (xy)x2 = x yx2 spectral radius n × n x,y J For a square ( ) matrix for every in , which is isomorphic to a A, the real number subalgebra of A+ for some A. ρ(A) = max{|λ|:λ an eigenvalue of A} . special linear group The multiplicative group of n×n matrices with determinant 1. The More generally, given a normed vector space X notation SL(n) is used. and a linear operator T : X → X, the spectral radius of T coincides with   /k special orthogonal group The multiplicative  k1 lim T  . group of n×n matrices M, with real entries and k→∞

c 2001 by CRC Press LLC This limit is independent of the choice of the complement of the set of complex numbers λ norm ·. for which the operator (L − λI)−1 is bounded. In the case in which L is an n × n matrix, the spectral sequence A sequence M = spectrum is equal to the set of eigenvalues of L. k k k {M ,d }, in which each M , k = 2, 3, 4,... (2) Let x be an element of a commutative Ba- k is a Z-bigraded module and each d is a dif- nach algebra A, with identity e. The spectrum k ferential of bidegree (−r, r − 1) mapping Mp,q of x is the set of complex numbers λ such that Mk x − λe fails to have an inverse in A. to p−r,q+r−1. Furthermore, one requires that H(Mk,dk) =∼ Mk+1, where H(Mk,dk) is the homology of Mk with respect to the differen- spherical excess The amount by which the k sum of the three angles in a sph