DICTIONARY OF Applied math for engineers and scientists Comprehensive Dictionary of Mathematics
Douglas N. Clark Editor-in-Chief Stan Gibilisco Editorial Advisor
PUBLISHED VOLUMES
Analysis, Calculus, and Differential Equations Douglas N. Clark Algebra, Arithmetic, and Trigonometry Steven G. Krantz Classical and Theoretical Mathematics Catherine Cavagnaro and William T. Haight, II Applied Mathematics for Engineers and Scientists Emma Previato
FORTHCOMING VOLUMES
The Comprehensive Dictionary of Mathematics Douglas N. Clark a Volume in the Comprehensive Dictionary of Mathematics
DICTIONARY OF applied math for engineers and scientists
Edited by Emma Previato
CRC PRESS Boca Raton London New York Washington, D.C.
3122 disclaimer Page 1 Friday, September 27, 2002 9:47 AM
Library of Congress Cataloging-in-Publication Data
Dictionary of applied math for engineers and scientists/ edited by Emma Previato. p. cm. ISBN 1-58488-053-8 1. Mathematics—Dictionaries. I. Previato, Emma.
QA5 .D49835 2002 510¢.3—dc21 2002074025
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA The fee code for users of the Transactional Reporting Service is ISBN 1-58488-053-8/03/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com
© 2003 by CRC Press LLC
No claim to original U.S. Government works International Standard Book Number 1-58488-053-8 Library of Congress Card Number 2002074025 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper PREFACE
To describe the scope of this work, I must go back to when Stan Gibilisco, editorial advisor of the dictionary series, asked me to be in charge of this volume. I appreciated the idea of a compendium of mathematical terms used in the sciences and engineering for two reasons. Firstly, mathematical definitions are not easily located; when I need insight on a technical term, I turn to the analytic index of a monograph that seems related; recently I was at a loss when trying to find “Vi`ete’s formulas ∗,” a term used by an Eastern-European student in his homework. I finally located it in the Encyclopaedic Dictionary of Mathematics, and that brought home the value of a collection of esoteric terms, put together by many people acquainted with different sectors of the literature. Secondly, at this time we do not yet have a tradition of cross-disciplinary terms; in fact, much interaction between mathematics and other scientific areas is in the making, and times (and timing) could not be more exciting. The EPSRC∗∗ newsletter Newsline (available on the web at www.epsrc.ac.uk), devoted to mathematics, in July 2001 rightly states “Even amongst fellow scientists, mathematicians are often viewed with suspicion as being interested in problems far removed from the real world. But ...things are changing.” Rapidly, though, my enthusiasm turned to dismay upon realizing that any strategy I could devise was doomed to fail the test of “completeness.” What is a dictionary? At best, a rapidly superseded record of word/symbol usage by some groups of people; the only really complete achievement in that respect is, in my view, the OED. Not only was such an undertaking beyond me, the very attempt at bridging disciplines and importing words from one to another is still an ill-defined endeavor — scientists themselves are unsure how to translate a term into other disciplines. As a consequence what service I can hope this book to provide, at best, is that of a pocket manual with which a voyager can at least get by in a basic fashion in a foreign-speaking country. I also hope that it will have the small virtue to be a first of its kind, a path-breaker that will prompt others to follow. Not being an applied mathematician myself, I relied on the generosity of the following team of authors: Lorenzo Fatibene, Mauro Francaviglia, and Rudolf Schmid, experts of mathematical physics; Toni Kazic, a biologist with broad and daring interdisciplinary experience; Hong Qian, a mathematical biologist; and Ralf Hiptmair, who works on numerical solution of differential equations. For oper- ations research, Giovanni Andreatta (University of Padua, Italy), directed me to H.J. Greenberg’s web glossary, and Toni Kazic referred me to the most extensive web glossary in chemistry, authored by A.D. McNaught and A. Wilkinson. To all these people I owe much more than thanks for their work. I know the reward that would most please them is for this book to have served its readers well: please write me any comments or suggestions, and I will gratefully try to put them to future use. Emma Previato, Department of Mathematics and Statistics Boston University, Boston, MA 02215-2411 – USA e-mail: [email protected]
∗They are just the elementary symmetric polynomials, in case anyone beside me didn’t know ∗∗Engineering and Physical Sciences Research Council, UK.
v
CONTRIBUTORS
Lorenzo Fatibene To Professor Greenberg and Dr. McNaught, a Istituto di Fisica Matematica great many thanks are due for a most courteous, Universit`a di Torino prompt and generous permission to use of their Torino, Italy glossaries.
Harvey J. Greenberg Mauro Francaviglia Mathematics Department Istituto di Fisica Matematica University of Colorado at Denver Universit`a di Torino Denver, Colorado, U.S. Torino, Italy A.D. McNaught Ralf Hiptmair General Manager, Production Division Mathematisches Institut RSC Publishing, Royal Society of Chemistry Universit¨at T¨ubingen Cambridge, U.K. Tubingen,¨ Germany
Toni Kazic Department of Computer Engineering and Computer Science University of Missouri — Columbia Columbia, Missouri, U.S.
Hong Qian Department of Applied Mathematics University of Washington Seattle, Washington, U.S.
Rudolf Schmid Department of Mathematics and Computer Science Emory University Atlanta, Georgia, U.S.
In addition, the two following databases have been used with permission: IUPAC Compendium of Chemical Terminology, 2nd ed. (1997), compiled by Alan D. McNaught and Andrew Wilkinson, Royal Society of Chem- istry, Cambridge, U.K. http://www.iupac.org/ publications/compendium/index.html H. J. Greenberg. Mathematical Programming Glossary http://carbon.cudenver.edu/˜hgreenbe/ glossary/ glossary.html ,1996-2000.
vii
absorbance
aberration The deviation of a spherical mir- ror from perfect focusing. abscissa In a rectangular coordinate system (Cartesian coordinates) (x, y) of the plane R2, x A is called the abscissa, y the ordinate. absolute convergence A series xn is said a posteriori error estimator An algorithm to be absolute convergent if the series of absolute for obtaining information about a discretization values |xn| converges. error for a concrete discrete approximation uh of the continuous solution u. Two principal features absolute convergence test If |xn| con- are expected from such device: verges, then xn converges. (i.) It should be reliable: the estimated error absolute error The difference between the (norm) must be proportional to an upper bound exact value of a number x and an approximate for the true error (norm). Thus, discrete solutions value a is called the absolute error a of the that do not meet a prescribed accuracy can be approximate value, i.e., a =|x − a|. The quo- δ = a detected. tient a a is called the relative error. (ii.) It should be efficient: the error estimator absolute ratio test Let xn be a series of |x | should provide some lower bound for the true n+1 = nonnegative terms and suppose limn→∞ |x | error (norm). This helps avoid rejecting a discrete n ρ. solution needlessly. (i.) If ρ<1, the series converges absolutely In the case of a finite element discretization an (hence converges); additional requirement is the locality of the a (ii.) If ρ>1, the series diverges; posteriori error estimator. It must be possible to ρ = extract information about the contributions from (iii.) If 1, the test is inconclusive. individual cells of the mesh to the total error. This absolute temperature −273.15◦C. is essential for the use of an a posteriori error esti- mator in the framework of adaptive refinement. absolute value The absolute value of a real number x, denoted by |x|, is defined by |x|=x abacus Oldest known “computer” circa if x ≥ 0 and |x|=−x if x<0. 1100 BC from China, a frame with sliding beads absolute value of an operator Let A be a for doing arithmetic. bounded linear operator on a Hilbert space, H. A |A|= Abbe’s sine condition (Ernst Abbe 1840– Then√ the absolute value of is given by A∗A A∗ A 1905) n l sin β = nl sin β where n, n ,β,β , where is the adjoint of . are the refraction indices and refraction angles, absolutely continuous A function x(t) respectively. defined on [a,b] is called absolutely continuous on [a,b] if there exists a function y ∈ L1[a,b] Abelian group (Niels Henrik Abel 1802– t such that x(t) = y(s)ds + C, where C is a group (G, ·) a 1829 ) A is called Abelian or com- constant. mutative if a · b = b · a for all a,b ∈ G. absorbance A logarithm of the ratio of inci- ∞ a xn Abelian theorems (1) Suppose n=0 n dent to transmitted radiant power through a converges for |x|
1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 1 absorbing set absorbing set A convex set A ⊂ X in a vec- adduct formation) for the members of a series tor space X is called absorbing if every x ∈ X of structurally similar indicator bases (or acids) lies in tA for some t = t(x) > 0. of different strengths. The best known of these H functions is the Hammett acidity function 0 (for acceleration The rate of change of velocity uncharged indicator bases that are primary aro- with time. matic amines).
v acceleration vector If is the velocity vec- action (1) The action of a conservative a = dv tor, then the acceleration vector is dt ;orif dynamical system is the space integral of the total s is the vector specifying position relative to an momentum of the system, i.e., v = ds a = d2s origin, we have dt and hence dt2 . P 2 dri mi · dri acceptor A compound which forms a chem- P dt ical bond with a substituent group in a bimolecu- 1 i lar chemical or biochemical reaction. where mi is the mass and ri the position of the Comment: The donor-acceptor formalism is ith particle, t is time, and the system is assumed necessarily binary, but reflects the reality that few P P to pass from configuration 1 to 2. if any truly thermolecular reactions exist. The (2) Action of a group: A (left) action of a bonds are not limited to covalent. See also donor. group G on a set M is a map ) : G × M −→ M such that: accumulation point Let {zn} be a sequence of complex numbers.Anaccumulation point of (i.) )(e, x) = x, for all x ∈ M, e is the {zn} is a complex number a such that, given any identity of G; "> n 0, there exist infinitely many integers such (ii.) )(g, )(h, x)) = )(g · h, x), for all |z − a| <" that n . x ∈ M and g,h ∈ G.(g · h denotes the group operation (multiplication) in G. accumulator In a computing machine, an adder or counter that augments its stored number If G is a Lie group and M is a smooth mani- by each successive number it receives. fold, the action is called smooth if the map ) is smooth. accuracy Correctness, usually referring to An action is said to be: numerical computations. (i.) free (without fixed points)if)(g, x) = x, for some x ∈ M implies g = e; acidity function Any function that meas- ures the thermodynamic hydron-donating or (ii.) effective (faithful) if )(g, x) = x for all -accepting ability of a solvent system, or a closely x ∈ M implies g = e; related thermodynamic property, such as the ten- (iii.) transitive if for every x,y ∈ M there dency of the lyate ion of the solvent system exists a g ∈ G such that )(g, x) = y. to form Lewis adducts. (The term “basicity function” is not in common use in connection See also left action, right action. with basic solutions.) Acidity functions are not unique properties of the solvent system alone, action angle coordinates A system of gen- but depend on the solute (or family of closely eralized coordinates (Qi ,Pi ) is called action related solutes) with respect to which the thermo- angle coordinates for a Hamiltonian system dynamic tendency is measured. defined by a Hamiltonian function H if H Commonly used acidity functions refer to depends only on the generalized momenta Pi but concentrated acidic or basic solutions. Acidity not on the generalized positions Qi . In these functions are usually established over a range coordinates Hamilton’s equations take the form of composition of such a system by UV/VIS ∂Pi ∂Qi ∂H spectrophotometric or NMR measurements of = 0 , = the degree of hydronation (protonation or Lewis ∂t ∂t ∂Pi
2 adaptive refinement action, law of action and reaction (New- action principle (Newton’s second law) ton’s third law) The basic law of mechanics Any force F acting on a body of mass m induces asserting that two particles interact so that the an acceleration a of that body, which is pro- forces exerted by one upon another are equal in portional to the force and in the same direction magnitude, act along the straight line joining the F = ma. particles, and are opposite in direction. action, principle of least The principle action functional In variational calculus (Maupertius 1698–1759) which states that the (and, in particular, in mechanics and in field actual motion of a conservative dynamical sys- P P theory) is a functional defined on some suitable tem from 1 to 2 takes place in such a way that space F of functions from a space of independent the action has a stationary value with respect to P P variables X to some target space Y ; for any all possible paths between 1 and 2 correspond- regular domain D and any configuration ψ of ing to the same energy (Hamilton principle). the system it associates a (real) number AD[ψ]. activation energy (Arrhenius activation A regular domain D is a subset of the space X energy) An empirical parameter characterizing (the time t ∈ R in mechanics and the space-time the exponential temperature dependence of the point x ∈ M in field theory) such that the action rate coefficient k,E = RT 2(d ln k/dT ), where functional is well-defined and finite; e.g., if X is a R is the gas constant and T the thermodynamic a manifold, D can be any compact submanifold temperature. The term is also used for threshold of X with a boundary ∂Dwhich is also a compact energies in electronic potential surfaces, in submanifold. which case the term requires careful definition. By the Hamilton principle, the configurations ψ which are critical points of the action func- activity In biochemistry, the catalytic power tional are called critical configurations (motion of an enzyme. Usually this is the number of sub- curves in mechanics and field solutions in field strate turnovers per unit time. theory). In mechanics one has X = R and the relevant adaptive refinement A strategy that aims to space is the tangent bundle TQto the configura- reduce some discretization error of a finite ele- tion manifold Q of the system. Let γˆ = (γ, γ)˙ ment scheme by repeated local refinement of the be a holonomic curve in TQwhich projects onto underlaying mesh. The goal is to achieve an the curve γ in Q and L : TQ → R be the equidistribution of the contribution of individ- Lagrangian of the system, i.e., a (real) func- ual cells to the total error. To that end one relies tion on the space TQ. The action is given by on a local a posteriori error estimator that, for K ; AD[γ ] = L(γ (t), γ(t))dt˙ . D can be any each cell of the current mesh h, provides an D η closed interval. If suitable boundary conditions estimate K of how much of the total error is due are required on γ one can allow also infinite inter- to K. ; vals in the parameter space R. Starting with an initial mesh h, the refine- In field theory X is usually a space-time mani- ment loop comprises the following stages: fold M and the relevant space is the k-order (i.) Solve the problem discretized by means k jet extension J B of the configuration bundle of a finite element space built on ;h; (B,M,π,F) σˆ of the system. Let be a holo- (ii.) Determine guesses for the total error of J kB nomic section in which projects onto the the discrete solution and for the local error contri- σ B L J kB → R section in and : be the butions ηh. If the total error is below a prescribed Lagrangian of the system, i.e., a (real) func- threshold, then terminate the loop; tion on the space J kB. The action is given (iii.) Mark those cells of ; for refinement by A [σ ] = L(σ(x))ˆ ds, where L(σ(x))ˆ h D D whose local error contributions are above the denotes the value which the Lagrangian takes average error contribution; over the section; D ⊂ M can be any regular domain and ds is a volume element. If suitable (iv.) Create a new mesh by refining marked ; boundary conditions are required on the sections cells of h and go to (i.). σ one can allow also infinite regions up to the Algorithms for the local refinement of simplicial whole parameter space M. and hexaedral meshes are available.
3 addition reaction addition reaction A chemical reaction of Comment: Note that in this version any node two or more reacting molecular entities, resulting is present at least twice: as the key to each sublist in a single reaction product containing all atoms (X−[...] and as a member of some other sublist of all components, with formation of two chem- (−[X]). This representation is a more compact ical bonds and a net reduction in bond multipli- version of the connection tables often used to city in at least one of the reactants. The reverse represent compound structures. process is called an elimination reaction. If the reagent or the source of the addends of an add- adjacent For any graph G(V, E), two nodes ition are not specified, then it is called an addition v , v are adjacent if they are both incident to transformation. i i See also [addition, α-addition, cheletropic the same edge (share an edge); that is, if the edge (v ,v ) ∈ E (v ,v ) reaction, cycloadition.] i i . Similarly, two edges i i , (vi ,vi ) are adjacent if they are both incident to {v ,v }∩{v ,v }=∅ adduct Anewchemical species AB, each the same vertex; that is if i i i i . molecular entity of which is formed by direct Comment: Two atoms are said to be adjacent combination of two separate molecular entities if they share a bond; two reactions (compounds) A and B in such a way that there is change in are said to be adjacent if they share a compound connectivity, but no loss, of atoms within the (reaction). moieties A and B. Stoichiometries other than 1:1 are also possible, e.g., a bis-adduct (2:1). An adjoint representations (on a [Lie] group G) intramolecular adduct can be formed whenA and (1) The action of any group G onto itself defined B are groups contained within the same molecu- by ad : G → Hom(G) : g → adg. The lar entity. group automorphism adg : G → G is defined This is a general term which, whenever appro- −1 by adg(h) = g · h · g . priate, should be used in preference to the less (2) On a Lie algebra. If G is a Lie group explicit term complex. It is also used specifically the adjoint representation above induces by deri- for products of an addition reaction. vation the adjoint representation of G on its Lie g T g → g adiabatic lapse rate (in atmospheric chemistry) algebra . It is defined by eadg : where T G The rate of decrease in temperature with increase e denotes the tangent map (see tangent lift). If in altitude of an air parcel which is expanding is a matrix group, then the adjoint representation T (ξ) = g · ξ · g−1 slowly to a lower atmospheric pressure without is given by eadg . exchange of heat; for a descending parcel it is the (3) Also defined is the adjoint representa- g → (g) rate of increase in temperature with decrease in tion Ad : Hom of the Lie algebra altitude. Theory predicts that for dry air it is equal g onto itself. For ξ, ζ ∈ g, the Lie algebra to the acceleration of gravity divided by the spe- homomorphism Adξ : g → g is defined by cific heat of dry air at constant pressure (approx- commutators Adξ (ζ ) = [ξ,ζ]. imately 9.8◦Ckm−1). The moist adiabatic lapse rate is less than the dry adiabatic lapse rate and adsorbent A condensed phase at the surface depends on the moisture content of the air mass. of which adsorption may occur. adjacency list A list of edges of a graph G of the form adsorption An increase in the concentration of a dissolved substance at the interface of a con- [vi − [vj ,vk,...,vn],vj − [vi ,vl,...,vm]], densed and a liquid phase due to the operation of ...,vn − [vi ,vp,...vq ], surface forces. Adsorption can also occur at the where interface of a condensed and a gaseous phase.
E ={(vi ,vj ), (vi ,vk),...,(vi ,vn), (vj ,vl), adsorptive The material that is present in ...,(v ,v ),...,(v ,v ),...,(v ,v )}, j m n p n q one or other (or both) of the bulk phases and and i, j, k, l, m, n, p, and q are indices. capable of being adsorbed.
4 affine space affine bundle A bundle (A,M,π; A) which affine geometry The geometry of affine has an affine space A as a standard fiber and tran- spaces. sition functions acting on A by means of affine affine map Let X and Y be vector spaces transformations. and C a convex subset of X. A map T : C → If the base manifold is paracompact then any Y is called an affine map if T((1 − t)x + ty) affine bundle allows global sections. Examples = (1 − t)Tx + tTy for all x,y ∈ C, and all of affine bundles are the bundles of connections 0 ≤ t ≤ 1. (transformation laws of connections are affine) k+1 k+1 k and the jet bundles πk : J C → J C. affine mapping (1) Let A be an affine space with difference space E. Let P → P be a map- affine connection A connection on the ping from A into itself subject to the following frame bundle F(M)of a manifold M. conditions: (i.) P Q = P Q implies P Q = P Q ; affine coordinates An affine coordinate sys- 1 1 2 2 1 1 2 2 φ E → E tem (0; x ,x , ..., xn) in an affine space A con- (ii.) The mapping : defined by 1 2 φ(PQ) = P Q sists of a fixed pont 0 ∈ A, and a basis {xi } (i = is linear. 1, ..., n) of the difference space E. Then every Then P → P is called an affine mapping. pont P ∈ A determines a system of n numbers If a fixed origin 0 is used in A, every affine {ξ i } (i = , ..., n) P = n ξ i x x → x 1 by 0 i=1 i . The mapping can be written in the form numbers {ξ i } (i = 1, ..., n) are called the affine x = φx + b, where φ is the induced linear map- coordinates of P relative to the given system. ping and b = 00 . The origin 0 has coordinates ξi = 0. (2) Let M,M be Riemannian manifolds.A map f : M → M is called an affine map if the affine equivalence A special case of para- tangent map Tf : TM → TM maps every hori- ) metric equivalence, where the mapping is an zontal curve into a horizontal curve. An affine )( ) = + affine linear mapping, that is, x : Ax t map f maps every geodesic of M into a geodesic A ∈ Rn,n ∈ Rn n with a regular matrix and t ( the of M . dimension of the ambient space). For affine equivalent families of finite ele- affine representation A representation of a G V ment spaces on simplicial meshes in dimension Lie group which operates on a vector space φ V → V n the usual reference element is the unit simplex such that all g : are affine maps. Rn spanned by the canonical basis vectors of . affine space Let E be a real n-dimensional {; } In the case of a shape regular family h h∈H vector space and A a set of elements P, Q, ... of meshes and affine equivalence, there exist which will be called points. Assume that a rela- C > i = , ..., constants i 0, 1 4, such that tion between points and vectors is defined in the C (K)n ≤| A |≤C (K)n, following way: 1diam det K 2diam (i.) To every ordered pair (P,Q) of A there AK ≤C diam(K), is an assigned vector of E, called the difference 3 A−1≤C (K)−1 ∀K ∈ ; ,h∈ H. vector and denoted by PQ; K 4diam h (ii.) To every point P ∈ A and every vector Here . denotes the Euclidean matrix norm, and x ∈ E there exists exactly one point Q ∈ A such the matrix A belongs to that unique affine map- that PQ = x; ping taking a suitable reference element on K. P, Q, R These relationships pave the way for assessing (iii.) If are arbitrary points in A, then PQ + QR = PR the behavior of norms under pullback. . Then A is called an n-dimensional affine space affine frame An affine frame on a manifold with difference space E. n M at x ∈ M consists of a point p ∈ Ax (M) The affine n-dimensional space A is distin- n (where Ax (M) is the affine space with differ- guished from R in that there is no fixed origin; n ence space E = Tx M) and an ordered basis thus the sum of two points of A is not defined, (X , ..., X ) T M 1 n of x (called a linear frame at but their difference is defined and is a vector x (p; X , ..., X ) Rn ). It is denoted by 1 n . in .
5 Airy equation
Airy equation The equation y − xy = 0. almost K¨ahler An almost K¨ahler manifold is an almost Hermitian manifold (M,J,g)such AKNS method A procedure developed by that the fundamental two-form ; defined by Ablowitz, Kaup, Newell, and Segur (1973) that ;(u, v) = g(u, J v) is closed. allows one, given a suitable scattering problem, to derive the nonlinear evolution equations solv- almost periodic A function f(t) is called able by the inverse scattering transform. almost periodic if there exists T(")such that for any " and every interval I" = (x, x + T(")), algebra An algebra over a field F is a ring there is x ∈ I" such that, |f(t+ x) − f(t)| <". R which is also a finite dimensional vector space over F , satisfying (ax)(by) = (ab)(xy) for all α-limit set Consider a dynamical system a,b ∈ F and all x,y ∈ R. u(t) in a metric space (M, d) which is described by a semigroup S(t), i.e., u(t) = S(t)u(0), n algebraic equation Let f(x) = anx + S(t + s) = S(t) · S(s) and S(0) = I. The α- a xn−1 + ...+ a x + a u ∈ M A ⊂ M n−1 1 0 be a polynomial in limit set, when it exists, of 0 ,or , R[x], where R is a commutative ring with unity. is defined as a xn + a xn−1 + ... + a x + The equation n n−1 1 a = α(u ) = S(−t)−1u , 0 0 is called an algebraic equation. 0 0 s≤0 t≤s ALGOL A programming language. or α(A) = S(−t)−1A. algorithm A process consisting of a specific s≤ t≤s sequence of operations to solve certain types of 0 φ ∈ α(A) problems. Notice, if and only if there exists a sequence ψn converging to φ in M and a alignment In dealing with sequence data sequence t →+∞, such that φn = S(tn)ψn ∈ such as DNAs and proteins, one compares two A, for all n. such molecules by matching the sequences. Sequence alignment means finding optimal alphabet A set of letters or other characters matching, defined by some criteria usually called with which one or more languages are written. “scores.” Between two binary sequences, for alternating series A series that alternates example, the Hamming distance is a widely used signs, i.e., of the form (−1)na , a ≥ 0. score function. n n n alternation For any covariant tensor field almost complex manifold A manifold with K on a manifold M the alternation A is defined an almost complex structure. as almost complex structure A manifold M is 1 (AK)(X , ..., X ) = (sign π) said to possess an almost complex structure if it 1 r r ! π carries a real differentiable tensor field J of type 2 K(X , ..., X ) (1, 1) satisfying J =−I. π(1) π(r) where the summation is taken over all r! permu- almost everywhere A property holds almost tations π of (1, 2, ..., r). everywhere (a.e.) if it holds everywhere except on a set of measure zero. amplitude of a complex number The angle θ is called the amplitude of the complex number almost Hermitian manifold M iθ A with a z = re = r(cos θ + i sin θ). Riemannian metric g invariant by the almost complex structure J , i.e., g and J satisfy amplitude of oscillation The simplest equa- m d2x =−kx g(Ju,Jv) = g(u, v) tion of a linear oscillator is √dt2 . It has the solution x(t) = A cos(t k/m − c). A is for any tangent vectors u and v. called the amplitude.
6 angle between lines analog In contrast to digital, analog means a analytical function A function which dynamic variable taking a continuum of values, relates the measure value Ca to the instrument e.g., a timepiece having hour and minute hands. reading, X, with the value of all interferants, Ci , remaining constant. This function is expressed analog computation Instead of using by the following regression of the calibration binary computation as in a digital computer, one results: C = f(X) uses a device which has continuous dynamic a variables such as current (or voltage) in an elec- The analytical function is taken as equal to the trical circuit, or displacement in a mechanical inverse of the calibration function. device. analytical index The analytical index of an analog computer A device that computes elliptic complex {Dp,Ep} is defined as using analog computation. A computer that p operates with numbers represented by directly index{Dp,Ep}= (−1) dim ker p, measurable quantities (e.g., voltage). p
where p = dδ + δd is the Laplacian on p- analog multiplier Using analog computa- forms. tion to obtain the product, as output, of two (input) quantities. analytical unit (analyser) An assembly of subunits comprising: suitable apparatus permit- analog variable See analog and analog ting the introduction and removal of the gas, computation. liquid, or solid to be analyzed and/or calibration materials; a measuring cell or other apparatus which, from the physical or chemical properties analytic dynamics A dynamical system is of the components of the material to be analyzed, called analytic if the coefficients of the vector gives signals allowing their identification and/or field are analytic functions. measurement; signal processing devices (ampli- fication, recording) or, if need be, data processing analytic function A function f(x)is called devices. analytic at x = a if it can be represented by a power series angle A system of two rays extending from ∞ the same point. The numerical measure of an f(x)= c (x − a)n , angle is the degree (measured as a fraction of n ◦ n=0 360 , the entire angle from a line to itself ) or the radian (= 180/π degrees). convergent for |x|
7 angle between planes angle between planes The angle between angular variables Let M be a manifold and two planes is given by the angle between the two S1 the unit circle. A smooth map ω : M → S1 normal vectors to these planes. is called an angular variable on M.
n angle between vectors Let u ∈ R and v ∈ angular velocity If a particle is moving in n n R be two vectors in R . The angle θ between u a plane, its angular velocity about a point in the and v is given by plane is the rate of change per unit time of the u ·v angle between a fixed line and the line joining θ = , the moving particle to the fixed point. cos uv u ·v = n u v anion A monoatomic or polyatomic species where the dot product i=1 i i and the u2 = n u2 u = (u , ..., u ) having one or more elementary charges of the norm is i=1 i ,if 1 n and v = (v , ..., v ) electron. 1 n . angle of depression The angle between the annihilation operator For the harmonic H = 1 (p2 + ω2q2) horizontal plane and the line from the observer’s oscillator with Hamiltonian 2 eye to some object lower than the line of her eyes. the annihilation operator a is given by a = √1 (p −iωq). The creation operator a∗ is given 2ω angle of elevation The angle between the by a∗ = √1 (p + miωq). Then H = ω (aa∗ + horizontal plane and the line from the observer’s 2ω √ 2 a∗a) and we have aψ(N) = Nψ(N − 1) and eye to some object above her eyes. √ a∗ψ(N) = N + 1ψ(N + 1). angle of incidence The angle that a line (as X X∗ a ray of light) falling on a surface or interface annihilator Let be a vector space, its Y X makes with the normal vector drawn at the point dual vector space, and a subspace of . The M⊥ M M⊥ ={f ∈ of incidence to that surface. annihilator of is defined as X∗ | f(x)= 0,forallx ∈ M}. angle of reflection The angle between a reflected ray and the normal vector drawn at the annulus The region of a plane bounded by point of incidence to a reflecting surface. two concentric circles in the plane. Let R>r, the annulus A determined by the two circles of angle of refraction The angle between a radius R and r, respectively, (centered at 0) is refracted ray and the normal vector drawn at the given by point of incidence to the interface at which the 2 refraction occurs. A ={x = (x, y) ∈ R | r<x