DOCSLIB.ORG

List of Symbols

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
List of Symbols List of Symbols SYMBOL MEANING PAGE REFERENCE Q field of rational numbers 1 R field of real numbers 1 C field of complex numbers 1 ==} implies <=} if and only if E is an element of 2 is not an element of 2 •Ix I P(x)} the class of all x such that P(x) is true 2 C is a subclass (or subset) of 2 525 empty set 3 peA) or 2A power set of A 3 UAi union of the sets Ai 3 i.I nAi intersection of the sets Ai 3 iel B-A relative complement of A in B 3 A' complement of A 3 I: A---+B fis a function from A to B 3 a~ I(a) the function Imaps a to I(a) 3 liS restriction of the function Ito S 4 { identity function on the set A 4 lA identity element of the ring A 115 {composite function of land g 4 golor gl composite morphism of I and g 52 Iml image of the function I 4,31 l-l(T) inverse image of the set T 4 485 486 LIST OF SYMBOLS SYMBOL MEANING PAGE REFERENCE {Cartesian product of sets A and B 6 A XB direct product of groups A and B 26 { is equivalent to 6 is equipollent with 15 a equivalence class of a 6 (Cartesian) product of the sets Ai; 7 product of the fomUy of obi""" I A, I i ,II 53 IT Ai ! iol direct product of the family of groups [or rings or modules] {Ai liE II 59,130,173 Z set of integers 9 N set of nonnegative integers (natural numbers) 9 N* set of positive integers 9 alb a divides b 11, 135 a{'b a does not divide b 11, 135 (a"a2, . , an) { greatest common divisor of a" . , an 11 ideal generated by a" . , an 123 a== b (mod m) a is congruent to b modulo m 12 16 r'dina' number of the "t A IAI order of the group A 24 determinant of the matrix A 351 No aleph-naught 16 D4* group of symmetries of the square 26 Sn symmetric group on n letters 26 G(£JH direct sum of additive groups G and H 26 Zm integers modulo m 27 Q/Z group of rationals modulo one 27 Z(p"") Sylow p-subgroup of Q/Z 30, 37 '" is isomorphic to 30 Ker f kernel of the homomorphism f 31, 119, 170 H<G H is a subgroup of G 31 <X> subgroup generated by the set X 32 <a> cyclic (sub)group generated by a 32 H V K,H+K the join of subgroups Hand K 33 Qs quaternion group 33 lal order of the element a 35 a=Tb(modH) ab-1 EH 37 a =Ib (mod H) a-1b EH 37 Ha,aH right and left cosets of a 38 [G:H] index of a subgroup H in a group G 38 HK {ab la EH, b EK} 39 N<JG N is a normal subgroup of G 41 GIN factor group of G by N 42 LIST OF SYMBOLS 487 SYMBOL MEANING PAGE REFERENCE sgn T sign of the permutation T 48 An alternating group on n letters 49 Dn dihedral group of degree n 50 () Ai disjoint union of the sets Ai 58 i.l rrGi weak direct product of the groups Gi 60 i.I L: Gi direct sum of the groups (or modules) Gi 60,173 i.I II* Gi free product of the groups Gi 68 i.1 G[m] {u E G I mu = 01 77, 224 G(p) {u EG I u has order a power of p 1 77,222 G, torsion subgroup [submodule] of G 78,220 G;r; stabilizer of x 89 CH(x) centralizer of x in H 89 NH(K) normalizer of Kin H 89 C(G) center of G 91 C.. (G) n-th term of ascending central series 100 G' commutator subgroup of G 102 G(n) n-th derived subgroup of G 102 End A endomorphism ring of A 116 G) binomial coefficient 118 char R characteristic of the ring R 119 Rop opposite ring of R 122, 330 (X) ideal generated by the set X 123 (a) principal ideal generated by a 123 S-lR ring of quotients of R by S 143 Rp localization of R at P 147 R[x] ring of polynomials over R 149 R[xJ, ... , x n] ring of polynomials in n indeterminates over R 151 R[[x)] ring of formal power series over R 154 degf degree of the polynomial f 157, 158 C(f) content of the polynomial f 162 HomR(A,B) set of all R-module homomorphisms A -+ B 174 dimDV dimension of the D-vector space V 185 RAs R-S bimodule A 202 RA,[AR] left [resp. right] R-module A 202 A* dual module of A 203 <a,f> f(a) 204 ~ii Kronecker delta 204 mt(A,B) category of middle linear maps on A X B 207 488 LIST OF SYMBOLS SYMBOL MEANING PAGE REFERENCE A@RB tensor product of modules A and B 208 /@g induced map on the tensor product 209 0. order ideal of a 220 [F:Kl dimension of field F as a K-vector space 231 K[ul, ... , Un], subring generated by K and u), ... , Un [resp. Xl 232 [resp. K[Xll K(ul, ... , Un) subfield generated by K and Ul, ••• , Un [resp. K(X)l [resp. Xl 232 K(xl, ... , xn) field of rational functions in n indeterminates 233 AutKF Galois group of F over K 243 ~ discriminant of a polynomial 270 Fpn {u pn I u EF; char F = pI 285 [F:Kl, separable degree of F over K 285 [F:Kli inseparable degree of F over K 285 NKF(U) norm of u 289 TKF(U) trace of u 289 gnCx) n-th cyclotomic polynomial 298 tr.d. F/K transcendence degree of F over K 316 KI/pn {U Eel upn EKl 320 K1I pco {u Eel upn EK for some n :2: 0 I 320 In n X n identity matrix 328 MatnR ring of n X n matrices over R 328 At transpose of the matrix A 328 A-I inverse of the invertible matrix A 331 En,m r a certain matrix 337 Aa classical adjoint of the matrix A 353 q.p(x), qA(X) minimal polynomial of 4J [resp. Al 356 TrA trace of the matrix A 369 Rad I radical of the ideal I 379 V(S) affine variety determined by S 409 a(B) left annihilator of B 417 roa r + a + ra 426 J(R) Jacobson radical of R 426 peR) prime radical of R 444 hom(A,B) or set of morphisms A ---> B in a category e 52,465 home(A,B) hA covariant hom functor 465 hB contravariant hom functor 466 eT category formed from e and T 470 °C,D zero morphism from C to D 482 Bibliography All books and articles actually referred to in the text are listed below. The list also includes a number of other books that may prove to be useful references. No attempt has been made to make the list complete. It contains only a reasonable selection of English language books in algebra and related areas. Almost all of these books are accessible to anyone able to read this text, though in some cases certain parts of this text are prerequisites. For the reader's convenience the books are classified by topic. However, this classification is not a rigid one. For instance, several books classified as "general" contain a fairly complete treatment of group theory as well as fields and Galois theory. Other books, such as [26] and [39], readily fit into more than one classifica­ tion. BOOKS GENERAL 1. Chevalley, c., Fundamental Concepts of Algebra. New York: Academic Press, Inc., 1956. 2. Faith, c., Algebra: Rings, Modules and Categories I. Berlin: Springer-Verlag, 1973. 3. Goldhaber, J. and G. Ehrlich, Algebra. New York: The Macmillan Compa­ ny, 1970. 4. Herstein, I., Topics in Algebra. Waltham, Mass.: Blaisdell Publishing Company, 1964. 5. Lang, S., Algebra. Reading, Mass.: Addison-Wesley, Publishing Company, Inc., 1965. 6. MacLane, S. and G. Birkhoff, Algebra. New York: The Macmillan Company, 1967. 7. Van der Waerden, B. L., Algebra. (7th ed., 2 vols.), New York: Frederick Ungar Publishing Co., 1970. SET THEORY 8. Eisenberg, M., Axiomatic Theory of Sets and Classes. New York: Holt, Rinehart and Winston, Inc., 1971. 9. Halmos, P., Naive Set Theory, Princeton, N. J.: D. Van Nostrand Company Inc., 1960. 10. Suppes, P., Axiomatic Set Theory. Princeton, N. J.: D. Van Nostrand Com­ pany, Inc., 1960. 489 -------------------------490 BIBLIOGRAPHY GROUPS 11. Curtis, C. W. and I. Reiner, Representation Theory of Finite Groups and Asso­ ciative Algebras. New York: Interscience Publishers, 1962. 12. Dixon, J., Problems in Group Theory. Waltham, Mass.: Blaisdell Publishing Company, 1967. 13. Fuchs, L., Infinite Abelian Groups. New York: Academic Press, Inc., 1970. 14. Gorenstein, D., Finite Groups. New York: Harper and Row, Publishers, 1968. 15. Hall, M., The Theory of Groups. New York: The Macmillan Company, 1959. 16. Hall, M. and 1. K. Senior, The Groups of Order 2n(n ~ 6). New York: The Macmillan Company, 1964. 17. Kaplansky, I., Infinite Abelian Groups (2d ed.), Ann Arbor, Mich.: University of Michigan Press, 1969. 18. Kurosh, A. G., The Theory of Groups (2 vols.), New York: Chelsea Publishing Company, 1960. 19. Rotman, J., The Theory of Groups (2d ed.).
Load more

Recommended publications
  • 500 Natural Sciences and Mathematics
    500 500 Natural sciences and mathematics Natural sciences: sciences that deal with matter and energy, or with objects and processes observable in nature Class here interdisciplinary works on natural and applied sciences Class natural history in 508. Class scientific principles of a subject with the subject, plus notation 01 from Table 1, e.g., scientific principles of photography 770.1 For government policy on science, see 338.9; for applied sciences, see 600 See Manual at 231.7 vs. 213, 500, 576.8; also at 338.9 vs. 352.7, 500; also at 500 vs. 001 SUMMARY 500.2–.8 [Physical sciences, space sciences, groups of people] 501–509 Standard subdivisions and natural history 510 Mathematics 520 Astronomy and allied sciences 530 Physics 540 Chemistry and allied sciences 550 Earth sciences 560 Paleontology 570 Biology 580 Plants 590 Animals .2 Physical sciences For astronomy and allied sciences, see 520; for physics, see 530; for chemistry and allied sciences, see 540; for earth sciences, see 550 .5 Space sciences For astronomy, see 520; for earth sciences in other worlds, see 550. For space sciences aspects of a specific subject, see the subject, plus notation 091 from Table 1, e.g., chemical reactions in space 541.390919 See Manual at 520 vs. 500.5, 523.1, 530.1, 919.9 .8 Groups of people Add to base number 500.8 the numbers following —08 in notation 081–089 from Table 1, e.g., women in science 500.82 501 Philosophy and theory Class scientific method as a general research technique in 001.4; class scientific method applied in the natural sciences in 507.2 502 Miscellany 577 502 Dewey Decimal Classification 502 .8 Auxiliary techniques and procedures; apparatus, equipment, materials Including microscopy; microscopes; interdisciplinary works on microscopy Class stereology with compound microscopes, stereology with electron microscopes in 502; class interdisciplinary works on photomicrography in 778.3 For manufacture of microscopes, see 681.
    [Show full text]
  • Field Theory Pete L. Clark
    Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9.
    [Show full text]
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • A New Rule-Based Method of Automatic Phonetic Notation On
    A New Rule-based Method of Automatic Phonetic Notation on Polyphones ZHENG Min， CAI Lianhong (Department of Computer Science and Technology of Tsinghua University, Beijing, 100084) Email: [email protected], [email protected] Abstract: In this paper, a new rule-based method of automatic 2. A rule-based method on polyphones phonetic notation on the 220 polyphones whose appearing frequency exceeds 99% is proposed. Firstly, all the polyphones 2.1 Classify the polyphones beforehand in a sentence are classified beforehand. Secondly, rules are Although the number of polyphones is large, the extracted based on eight features. Lastly, automatic phonetic appearing frequency is widely discrepant. The statistic notation is applied according to the rules. The paper puts forward results in our experiment show: the accumulative a new concept of prosodic functional part of speech in order to frequency of the former 100 polyphones exceeds 93%, improve the numerous and complicated grammatical information. and the accumulative frequency of the former 180 The examples and results of this method are shown at the end of polyphones exceeds 97%. The distributing chart of this paper. Compared with other approaches dealt with polyphones’ accumulative frequency is shown in the polyphones, the results show that this new method improves the chart 3.1. In order to improve the accuracy of the accuracy of phonetic notation on polyphones. grapheme-phoneme conversion more, we classify the Keyword: grapheme-phoneme conversion; polyphone; prosodic 700 polyphones in the GB_2312 Chinese-character word; prosodic functional part of speech; feature extracting; system into three kinds in our text-to-speech system 1.
    [Show full text]
  • Multilinear Algebra and Applications July 15, 2014
    Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3.
    [Show full text]
  • Normal Subgroups of the General Linear Groups Over Von Neumann Regular Rings L
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 96, Number 2, February 1986 NORMAL SUBGROUPS OF THE GENERAL LINEAR GROUPS OVER VON NEUMANN REGULAR RINGS L. N. VASERSTEIN1 ABSTRACT. Let A be a von Neumann regular ring or, more generally, let A be an associative ring with 1 whose reduction modulo its Jacobson radical is von Neumann regular. We obtain a complete description of all subgroups of GLn A, n > 3, which are normalized by elementary matrices. 1. Introduction. For any associative ring A with 1 and any natural number n, let GLn A be the group of invertible n by n matrices over A and EnA the subgroup generated by all elementary matrices x1'3, where 1 < i / j < n and x E A. In this paper we describe all subgroups of GLn A normalized by EnA for any von Neumann regular A, provided n > 3. Our description is standard (see Bass [1] and Vaserstein [14, 16]): a subgroup H of GL„ A is normalized by EnA if and only if H is of level B for an ideal B of A, i.e. E„(A, B) C H C Gn(A, B). Here Gn(A, B) is the inverse image of the center of GL„(,4/S) (when n > 2, this center consists of scalar invertible matrices over the center of the ring A/B) under the canonical homomorphism GL„ A —►GLn(A/B) and En(A, B) is the normal subgroup of EnA generated by all elementary matrices in Gn(A, B) (when n > 3, the group En(A, B) is generated by matrices of the form (—y)J'lx1'Jy:i''1 with x € B,y £ A,l < i ^ j < n, see [14]).
    [Show full text]
  • Irreducibility of Ideals in a One-Dimensional Analytically Irreducible Ring Vol
    Troisième Rencontre Internationale sur les Polynômes à Valeurs Entières Rencontre organisée par : Sabine Evrard 29 novembre-3 décembre 2010 Valentina Barucci and Faten Khouja Irreducibility of ideals in a one-dimensional analytically irreducible ring Vol. 2, no 2 (2010), p. 91-93. <http://acirm.cedram.org/item?id=ACIRM_2010__2_2_91_0> Centre international de rencontres mathématiques U.M.S. 822 C.N.R.S./S.M.F. Luminy (Marseille) France cedram Texte mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Irreducibility of ideals in a one-dimensional analytically irreducible ring Valentina Barucci and Faten Khouja Abstract Let R be a one-dimensional analytically irreducible ring and let I be an integral ideal of R. We study the relation between the irreducibility of the ideal I in R and the irreducibility of the corresponding semigroup ideal v(I). It turns out that if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition of a nonzero ideal. A numerical semigroup is a subsemigroup of N, with zero and with finite complement in N. Pν The numerical semigroup generated by d1, . , dν ∈ N is S =< d1, . , dν >= { i=1 nidi, ni ∈ N}. M = S\{0} will denote the maximal ideal of S, e the multiplicity of S, that is the smallest positive integer of S, g the Frobenius number of S, that is the greatest integer which does not belong to S.
    [Show full text]
  • Växjö University
    School of Mathematics and System Engineering Reports from MSI - Rapporter från MSI Växjö University Geometrical Constructions Tanveer Sabir Aamir Muneer June MSI Report 09021 2009 Växjö University ISSN 1650-2647 SE-351 95 VÄXJÖ ISRN VXU/MSI/MA/E/--09021/--SE Tanveer Sabir Aamir Muneer Trisecting the Angle, Doubling the Cube, Squaring the Circle and Construction of n-gons Master thesis Mathematics 2009 Växjö University Abstract In this thesis, we are dealing with following four problems (i) Trisecting the angle; (ii) Doubling the cube; (iii) Squaring the circle; (iv) Construction of all regular polygons; With the help of field extensions, a part of the theory of abstract algebra, these problems seems to be impossible by using unmarked ruler and compass. First two problems, trisecting the angle and doubling the cube are solved by using marked ruler and compass, because when we use marked ruler more points are possible to con- struct and with the help of these points more figures are possible to construct. The problems, squaring the circle and Construction of all regular polygons are still im- possible to solve. iii Key-words: iv Acknowledgments We are obliged to our supervisor Per-Anders Svensson for accepting and giving us chance to do our thesis under his kind supervision. We are also thankful to our Programme Man- ager Marcus Nilsson for his work that set up a road map for us. We wish to thank Astrid Hilbert for being in Växjö and teaching us, She is really a cool, calm and knowledgeable, as an educator should. We also want to thank of our head of department and teachers who time to time supported in different subjects.
    [Show full text]
  • Right Ideals of a Ring and Sublanguages of Science
    RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance.
    [Show full text]
  • Problems in Abstract Algebra
    STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • Multilinear Algebra
    Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R.
    [Show full text]