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.

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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.

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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

8 area of surface antibody A protein (immunoglobulin) pro- approximate solution Consider the differ- duced by the immune system of an organism in ential equation (*) x = f(x,t) , x ∈ ; ⊂ response to exposure to a foreign molecule (anti- Rn,t ∈ [a,b]. The vector valued function y(t) gen) and characterized by its speciﬁc binding to is an "-approximate solution of (*) if y(t) − a site of that molecule (antigenic determinant or f(t,y(t))Rn <", for all t ∈ [a,b]. epitope). arc (1) A segment, or piece, of a curve. anticommutator If A, B are two linear (2) The image of a closed interval [a,b] under operators, their anticommutator is {A, B}= a one-to-one, continuous map. AB + BA. arc length Let σ :[a,b] → Rn be a C1 antiderivation A linear operator T on a curve. The arc length l(σ) of σ is deﬁned as graded algebra (A, ·) satisfying T(a· b) = Ta· b b + (−1)(degree of b)a · Tbfor all a,b ∈ A. l(σ) = σ (t)dt. a F(x) antiderivative A function is called an arccosecant The inverse trigonometric f(x) F (x) = antiderivative of a function if function of cosecant. The arccosecant of a f(x) . number x is a number y whose cosecant is x, y = −1(x) = (x) antigen A substance that stimulates the written as csc arc csc , i.e., x = (y). immune system to produce a set of speciﬁc csc antibodies and that combines with the antibody arccosine The inverse trigonometric func- through a speciﬁc binding site or epitope. tion of cosine. The arccosine of a number x is a number y whose cosine is x, written as antimatter Matter composed of antipar- y = cos−1(x) = arc cos(x), i.e., x = cos(y). ticles. arccotangent The inverse trigonometric antiparticle A subatomic particle identical function of cotangent. The arccotangent of a to another subatomic particle in mass but oppos- number x is a number y whose cotangent is ite to it in the electric and magnetic properties. x, written as y = cot−1(x) = ctn−1(x) = antiselfdual A gauge ﬁeld F such that F = arc cot(x), i.e., x = cot(y). −∗F , where ∗ is the Hodge-star operator. arcsecant The inverse trigonometric func- x aphelion The point in the path of a celestial tion of secant. The arccosecant of a number y x body (as a planet) that is farthest from the sun. is a number whose secant is , written as y = sec−1(x) = arc sec(x), i.e., x = sec(y). apogee The point in the orbit of an object (as a satellite) orbiting the earth that is the greatest arcsine The inverse trigonometric function x distance from the center of the earth. of sine. The arcsine of a number is a number y whose sine is x, written as y = sin−1(x) = applied potential The difference of poten- arc sin(x), i.e., x = sin(y). tial measured between identical metallic leads arctangent to two electrodes of a cell. The applied poten- The inverse trigonometric func- tangent arctangent x tial is divided into two electrode potentials, each tion of . The of a number y tangent x of which is the difference of potential existing is a number whose is , written as y = −1(x) = (x) x = (y). between the bulk of the solution and the interior tan arc tan , i.e., tan of the conducting material of the electrode, an area of surface Consider the surface S given iR or ohmic potential drop through the solution, by z = f(x,y) that projects onto the bounded and another ohmic potential drop through each region D in the xy-plane. The area A(S) of the electrode. surface S is given by In the electroanalytical literature this quan- voltage 2 2 tity has often been denoted by the term , A(S) = fx + fy + 1 dD. whose continued use is not recommended. D

9 area under curve area under curve Let a curve be given by arithmetic sum The sum of an arithmetic y = f(x) a ≤ x ≤ b N (a + nd) , . The area A under this sequence n=0 . curve from a to b is given by the integral

b arity The number of arguments of a relation. A = f(x)dx. a array A display of objects in some regular arrangements, as a rectangular array or matrix Argand diagram The basic idea of complex in which numbers are displayed in rows and numbers is credited to Jean Robert Argand, a columns, or an arrangement of statistical data in Swiss mathematician (1768–1822). An Argand order of increasing (or decreasing) magnitude. diagram is a rectangular coordinate system in which the complex number x + iy is represented array index In a rectangular array such as a by the point whose coordinates are x and y. The matrix the element in the ith row and jth column x y -axis is called real axis and the -axis is called is indexed as aij . imaginary axis. artiﬁcial intelligence A branch of computer argument The collection of elements satis- science dealing with the simulation of intelligent r fying some relation is called the set of argu- behavior of computers. ments of r.

ascending sequence A sequence {an} is argument of complex number See ampli- called ascending (increasing) if each term is tude of a complex number. a ≥ a greater than the previous term, i.e., n n−1. If an >an− then it is called monotone ascend- arithmetic 1 The study of the positive integers ing/increasing. 1, 2, 3, 4, 5, ... under the operations of addition, subtraction, multiplication, and division. ASCII American Standard Code for Information Interchange. A code for represent- arithmetic difference The arithmetic dif- ing alphanumeric information. ference of two numbers a and b is |a − b|. Ascoli Giulio Ascoli (1843–1896), Italian arithmetic division To determine the arith- a a analyst. metic quotient b of two nonnegative integers and b, where [x] is the greatest integer, which is Ascoli’s theorem Let {f } be a family of not bigger than x. n uniformly bounded equicontinuous functions on , {f } n [0 1]. Then some subsequence n(i) converges arithmetic mean The arithmetic mean of , a ,a , ..., a uniformly on [0 1]. numbers 1 2 n is a + a +···+a assembler A computer program that auto- x = 1 2 n . n matically converts instructions written in assem- bly language into computer language. arithmetic progression A sequence of a ,a , ..., a , ... numbers 1 2 n in which each follow- assembly Computation of a ﬁnite element ing number is obtained from the preceding num- stiffness matrix A from the element matrices AK r a = a + ber by adding a given number , i.e., n 1 belonging to the cells K of the underlying mesh (n − )r 1 . ;h. The general formula is arithmetic quotient arithmetic divi- T See A = IK AK IK , sion. K∈;h arithmetic sequence A sequence a,(a + where the IK are rectangular matrices reﬂect- d),(a + 2d),··· ,(a + nd), ···, in which each ing the association of local and global degrees term is the arithmetic mean of its neighbors. of freedom.

10 Atiyah assembly language A programming lan- asymptotic freedom Property of quantum guage that consists of instructions that are ﬁeld theories which says that interactions are mnemonic codes for corresponding machine lan- weak at high energies (momenta). Mathemat- guage instructions. ically this means that suitably normalized correl- ation functions tend to the correlation function of associated bundle If (P,M,p : G) is a a free theory when the momenta go to inﬁnity. principal bundle and λ : G × F → F is a left action of the Lie group G and a manifold F , then asymptotic series A function f(x) on n one can deﬁne a right action of G on P ×F using (0,a), a>0, is said to have anx as asymp- the canonical right action R of G on P as follows: totic series (expansion) as x ↓ 0, written f(x)∼ n anx ,x↓ 0, if, for each N −1 (p, f ) · g = (Rgp, λ(g , f )). N n N The quotent space P ×λ F ≡ (P × F)\G lim f(x)− anx x = 0. x↓0 has a canonical structure of a bundle (P ×λ n=0 F,M,π; F)and it is called an associated bundle {V } H to P . asymptotically dense Let h h∈H, some Trivializations of P induce trivializations of index set, be a family of ﬁnite dimesional sub- spaces of the Banach space V . This family is P ×λ F , principal connections on P induce con- called asymptotically dense,if nections on P ×λ F , right invariant vector ﬁelds on P induce vector ﬁelds on P ×λ F . The transi- Vh = V, tion functions of P ×λ F are the same transition functions of P represented on F by means of the k∈H action λ. where the closure is with respect to the norm of V . association The assembling of separate molecular entities into any aggregate, especially asymptotically equal Two functions f and of oppositely charged free ions into ion pairs or g are said to be asymptotically equal (at inﬁnity) larger and not necessarily well-deﬁned clusters if, for every N>0, we have of ions held together by electrostatic attraction. lim xN (f (x) − g(x)) = 0. The term signiﬁes the reverse of dissociation,but x→∞ is not commonly used for the formation of def- inite adducts by colligation or coordination. asymptotically optimal A ﬁnite-element solution of a variational problem is regarded as associative Describing an operation among asymptotically optimal,ifitisquasi optimal, pro- objects x,y,z,..., denoted by • , such that (x • vided that the meshwidths of the underlying tri- y) • z = x • (y • z). For example, addition angulations stay below a certain threshold. and multiplication of numbers associative (x + y) + z = x + (y + z), (x · y) · z = x · (y · z), asymptotically stable Let X be a vector for all numbers x,y,z. ﬁeld on the manifold M and Ft its ﬂow. A crit- m X ical point 0 of is called asymptotically stable V m asymptote A straight line associated with a if there is a neighborhood of 0 such that for plane curve such that as a point moves along an each m ∈ V there exists an integral curve cm(t) inﬁnite branch of the curve the distance from the of X starting at m for all t>0, Ft (V ) ⊂ Fs (V ) t>s F (V ) ={m } m point to the line approaches zero and the slope of if and limt→+∞ t 0 . 0 the tangent to the curve at the point approaches is asymptotically unstable if it is asymptotically the slope of the line. stable as t →−∞. asymptote to the hyperbola The standard Atiyah Sir Michael Atiyah (1929–), Dif- form of the equation of the hyperbola in the plane ferential Geometer/Mathematical Physicist, Pro- is x2/a2 − y2/b2 = 1. The lines y = bx/a and fessor emeritus University of Edinburgh. Fields y =−bx/a are its asymptotes. Medal 1966, Knighted 1983.

11 Atiyah-Singer index theorem

Atiyah-Singer index theorem The Atiyah- scattering and luminescence. It was formerly Singer index theorem gives the equality between called extinction. the analytic index and the topological index of an elliptic complex over a compact manifold. The attractor Consider a dynamical system u(t) analytical index of an elliptic complex {D ,E } p p in a metric space (M, d) which is described by is deﬁned as a semigroup S(t), i.e., u(t) = S(t)u(0), S(t + s) = S(t) · S(s) S( ) = I {D ,E }= (− )p ker . and 0 .Anattractor for index p p 1 dim p u(t) A ⊂ M p is a set with the following proper- ties: The topological index of an elliptic complex (i.) A is an invariant set, i.e., S(t)A = A, {D ,E } is deﬁned as p p for all t ≥ 0. A U topindex{Dp,Ep} (ii.) possesses an open neighborhood u ∈ U S(t)u such that, for every 0 , 0 converges to ∗ = ch(K(D))Lρ todM, A as t →∞: ψ(M) dist(S(t)u ,A)→ 0 as t →∞, where ch(K(D)) is the Chern character of the 0 symbol bundle K(D), ρ the projection of the where the distance d(x,A) = inf d(x,y). compactiﬁed cotangent bundle ψ(M) onto M, y∈A and tod(M) the Todd class. Theorem (Atiyah-Singer) index{Dp,Ep}= augmented matrix If a system of linear topindex{Dp,Ep}. equations is written in matrix form Ax = b, then the matrix [A|b] is called the augmented matrix. atlas An atlas on a manifold M is a collec- tion of charts whose domains cover M. autocatalysis A reaction in which the prod- uct also serves as a catalyst. Hence this reaction atmosphere (of the earth) The entire mass is nonlinear with a positive feedback. Autocata- of air surrounding the earth which is composed lysis is an important ingredient for an oscillatory largely of nitrogen, oxygen, water vapor, clouds chemical reaction. (liquid or solid water), carbon dioxide, together with trace gases and aerosols. automorphism An isomorphism of a set with itself. Also an isomorphism of an object f(t ,...,t ) of a category into itself. atomic formula A term 1 n , where f is a relation. Comment: Note this is “atomic” in the com- autonomous system A system of differen- puter science and linguistic, not the chemistry, dx = F(x) tial equations dt is called autonomous senses. if the independent variable t does not appear explicitly in the function F . atomic units System of units based on four base quantities: length, mass, charge, and action autoparallel A vector ﬁeld X on a Riemann- (angular momentum) and the corresponding base ian manifold M is called autoparallel along a a units the Bohr radius, 0, rest mass of the elec- curve c(t) if the covariant derivative of X along tron, me, elementary charge, e, and the Planck c vanishes, i.e., ∇c(t)˙ X = 0. In local coordinates constant divided by 2π, . i i j k c¨ (t) + Njk(c(t))c˙ (t)c˙ (t) = 0 . attenuance, D Analogous to absorbance, but taking into account also the effects due to Hence c˙ is autoparallel along c if c is a geodesic.

12 azimuth average The average value of a function axis of rotation A straight line about which f(x)over an interval [a,b] is given by the num- a body or geometric ﬁgure is rotated. ber 1 b f(x)dx axis of symmetry A straight line with b − a a respect to which a body or geometric ﬁgure is symmetrical. axial gauge For a ﬁxed vector n, there exists a gauge transformation A → A such that n · A(x) = 0 . This is called the axial gauge. azimuth Horizontal direction expressed as the angular distance between the direction of a axiom A statement that is accepted without ﬁxed point and the direction of an object. proof. The axioms of a mathematical theory In polar coordinates (r, θ) in the plane the are the basic propositions from which all other polar angle θ is called the azimuth of the propositions can be derived. point P .

13

barycentric coordinates

Banach algebra A Banach space X together with an internal operation, usually called multi- plication, satisfying the following: for all x, y,z ∈ X, α ∈ C B (i.) x(yz) = (xy)z (ii.) (x +y)z = xz+yz, x(y+z) = xy +xz α(xy) = (αx)y = x(αy) B¨acklund transformations Transforma- (iii.) tions between solutions of differential equations, (iv.) xy≤xy in particular soliton equations. They can be used (v.) X contains a unit element e ∈ X, such to construct nontrivial solutions from the trivial that xe = ex = x solution. (vi.) e=1. Formally: Two evolution equations ut = K(x, u, u , ..., u ) v = G(y,v,v, ..., v ) 1 m and t 1 n Banach ﬁxed point theorem Let (X, d) be are said to be equivalent under a B¨acklund a complete metric space and T : X → X a transformation if there exists a transformation contraction map. Then T has a unique ﬁxed point y = ψ(x, u, u , ..., u ), v = φ(x, x ∈ X T(x ) = x of the form 1 n 0 , i.e., 0 0. u, u , ..., u ) 1 n . Banach manifold A manifold modeled on a bag An unordered collection of elements, Banach space. including duplicates, each of which satisﬁes Banach space A normed vector space which some property. An enumerated bag is delimited is complete in the metric deﬁned by its norm. by braces ({x}). Comment: Unfortunately, the same delimiters barrel A subset of a topological vector space are used for sets and bags. See also list, sequence, which is absorbing, balanced, convex, and set, and tuple. closed.

Baire space A space which is not a count- barreled space A topological vector space able union of nowhere dense subsets. Example: E is called barreled if each barrel in E is a neigh- A complete metric space is a Baire space. borhood of 0 ∈ E, i.e., the barrels form a neigh- borhood base at 0.

Baker-Campell-Hausdorff formula For barycenter The barycenter (center of mass) any n × n matrices A, B we have σ = (a , ..., a ) of the simplex 0 p is the point s2 e−sABesA = B +s A, B + A, A, B +··· 1 [ ] [ [ ]] bσ = (a +···+ap). 2 p + 1 0 p , ..., p balanced set A subset M of a vector space V barycentric coordinates Let 0 n be over R or C such that αx ∈ M, whenever x ∈ M n + 1 points in n-dimensional Euclidean space n and |α|≤1. E that are not in the same hyperplane. Then for each point x ∈ En there is exactly one set of real numbers (λ , ..., λ ) such that ball Let (X, d) be a metric space.Anopen 0 n B (x ) a x ball a 0 of radius about 0 is the set of all x = λ p + λ p +···+λnpn x ∈ X d(x,x )

17 bilinear map bilinear map Let X, Y, Z be vector spaces. biochemical motif A motif describing a bio- A map B : X × Y → Z is called bilinear if it is chemical relationship between two compounds linear in each factor, i.e., in the donor-acceptor formalism. Comment: The constraint for bimolecular B(αx + βy, z) = αB(x, z) + βB(y, z), relationship permits use of the common donor- acceptor language. A reaction may have more B(z,αx + βy) = αB(z, x) + βB(z, y). than one such relationship. Note that the bio- chemical donor-acceptor relationship is often binary A binary number system is based on opposite to that of the chemical one: thus a the number 2 instead of 10. Only the digits 0 and phosphoryl donor is a nucleophile acceptor. See 1 are needed. For example, the binary number also chemical, dynamical, functional, kinetic, 101110 = 1 · 25 + 0 · 24 + 1 · 23 + 1 · 22 + mechanistic, phylogenetic, regulatory, thermo- 0 · 20 = 46 in decimal notation. dynamic, and topological motifs. A binary operation is an operation that depends on two objects. Addition and, multi- biochemical network A mathematical net- (V, E, P, L) plication are binary operations. work N representing a system R of biochemical reactions, their participating molecular species; descriptive, transformational, binomial The formal sum of two terms, thermodynamic, kinetic, and dynamic param- e.g., x + y. eters describing the reactions singly and com- posed together; and labels giving the names of binomial coefﬁcients The coefﬁcients in the reactions, molecules, and subnetworks. V is the (x + y)n (k + ) expansion of . The 1 st binomial bipartite set of vertices: Vm representing molecu- n xn−kyk coefﬁcient of order is the coefﬁcient of , lar species; Vr representing reactive conjunctions and it is given by of molecules, V = Vm ∪ Vr . E = Es ∪ Ed ∪ Ec is the set of relations between molecule and reactive n n! = . conjunction vertices, e(λ, vm,i ,vr,j ) ∈ E, where k k!(n − k)! for each pair (vm,i ,vr,j ), λ is one and only one {s,d,c}=L This is also the number of combinations of n of : a molecule is a member of things k at a time. the set of coreacting species that appear sinis- tralaterally, dextralaterally, or catalytically in the reaction equation. Members of the parameter set bioassay A test to determine whether a P apply to vertices, edges, and connected graphs chemical has any biological function (sometimes of vertices and edges as biochemically appro- also called activity). This is usually accom- priate and as such information is available. If plished by a set of chemical reactions leading there are no parameters (P =∅), the network to an observable change in biological systems or N(V, E, P, L) reduces to its graph N(V, E, L). in test tubes. Labels apply to vertices, edges, and subnetworks and take the form of one of the elements of biochemical graph A set of biochemical {l ,l ,l ,l } m,i r,j ((m,i),(r,j)) {Vm,Vr ,E} . reactions, their participating molecules, and Comment: The network is a biochemical labels for reactions, molecules, and subgraphs, graph whose nodes, edges, and subgraphs have represented as a graph. qualitative and quantitative parameters. Thus Comment: The considered biochemical concentration is a property of a compound node; graphs are sometimes hypergraphs, mathemat- G0 is a property of a set of compound and reac- ically. However, the key results and algorithms tive conjunction nodes, and their incident edges; of the two objects are equally applicable; the kcat is a property of the edge joining an enzyme to common usage in computer science is to use its reaction; molecular structure is a property of a the word “graph.” Notice this is simply the compound node; etc. Not all nodes or edges need biochemical network with an empty parameter be so marked; and in fact much known informa- set. See also biochemical network. tion is at present unavailable electronically.

18 Bohr radius biochemical reaction A biochemical reac- molecule always has more than one function; at tion is any spontaneous or catalyzed transfor- a minimum it must be made and degraded. mation of covalent or noncovalent molecular bonds which occur in biological systems, written biometrics The ﬁeld of study that uses as a balanced, formal reaction equation, includ- mathematical and statistical tools to solve bio- ing all participating molecular species, whose logical problems and solving mathematical and kinetic order equals the sum of the partial orders statistical problems arising from biology. In of the reactants, including the active form of any recent years, it has a much narrower meaning catalyst(s). Equally, a composition of a set of in practice: it mainly deals with statistical analy- bichemical reactions. sis and methodology applicable to biology and Comment: The deﬁnition places no restric- medicine. tions on the level of resolution of the description, or size and complexity of reacting species; thus, it bipartite Describing a graph G(V, E) permits the recursive speciﬁcation of processes. whose chromatic number χ(G) = 2. “Distinct origin” means molecular species aris- Comment: Informally, a graph will be bipar- ing from different precursors. Thus, two pro- tite if it has two distinct sets of nodes and if nodes tons, if one came from water and the other from of each type are always adjacent to the other. a protein, would be individually recorded in the Thus the biochemical graph is bipartite because equation. By “kinetic signiﬁcance” is meant it has a set of compound nodes and a set of reac- any molecular species which at any concentra- tive conjunction nodes, and each is connected to tion contributes a term to the empirical rate law the other. of the overall reaction. From the empirical rate law, the reaction’s apparent kinetic order is the black hole (general relativity) A hypothet- sum of the partial orders of the reactants (includ- ical object in space with so intense a gravitational ing catalysts). The restriction to active forms ﬁeld that light and matter cannot escape. of the catalyst includes those instances where the catalyst must be activated, by either covalent blob See denser subgraph. modiﬁcation or ligand binding, or is inhibited by block A portion of a macromolecule, com- those means, so that not all molecules present prising many constitutional units that has at least are equally capable of catalysis. The deﬁnition one feature which is not present in the adjacent places no restrictions on the level of resolution of portions. Where appropriate, deﬁnitions relating the description, or size and complexity of reac- to macromolecule may also be applied to block. ting species, thus permitting the recursive speci- ﬁcation of processes. The recursion scales over block matrix If a matrix is partitioned in any size or complexity of process. submatrices it is called a block matrix. bioinformatics See computational biology. Bogomolny equations The self-dual Yang- Mills-Higgs equations are called Bogomolny biological functions The roles a molecule equations. They are plays in an organism. Comment: By function (called here biological 1 Fµν = "µνρσ Fρσ function to distinguish it from the mathematical 2 sense of function), biologists mean both how F = 1 " B ,F = D φ, a,b,c= a molecule interacts with its milieu and what where ab 2 abc c a4 a , , results from those interactions. The results 1 2 3. The solutions of the Bogomolny equa- are often decomposed into biochemical, physio- tions are called magnetic monopoles. logical, or genetic functions, but it is equally h2 Bohr radius rB = 2 2 = 0.529 × plausible to consider, for example, the ultrastruc- 4π mee −8 tural function of a molecule (what part of the 10 cm, where h is the Planck constant, me is cell’s microanatomy does it build, how strong the rest mass of the electron, and e is the electron is it, etc.). What is critical is to realize that a charge.

19 Boltzmann constant

Boltzmann constant The fundamental Bose-Einstein statistics In quantum statis- physical constant k = R/L = 1.380 = 658 × tics of the distribution of particles among vari- 10−23JK−1, where R is the gas constant and ous possible energy values there are two types of L the Avogadro constant. In the ideal gas particles, fermions and bosons, which obey the law PV = NkT, where P is the pressure, V Fermi-Dirac statistics and Bose-Einstein statis- the volume, T the absolute temperature, N tics, respectively. In the Fermi-Dirac statis- the number of moles, and k is the Boltzmann tics, no more than one set of identical particles constant. may occupy a particular quantum state (i.e., the Pauli exclusion principle applies), whereas in the Bose-Einstein statistics the occupation number is Boltzmann equation Boltzmann’s equation not limited in any way. for a density function f(x,v,t)is the equation of continuity (mass conservation) Boson A particle described by Bose- ∂f ∂f ∂f Einstein statistics. (x,v,t)+˙x (x,v,t)+˙v (x,v,t)= . ∂t ∂x ∂v 0 boundary Let A ⊂ S be topological spaces. The boundary of A is the set ∂A = A¯−A◦, where Bolzano-Weierstrass theorem (for the real line) A¯ is the closure and A◦ is the interior of A in S. If A ⊂ R is inﬁnite and bounded, then there exists at least one point x ∈ R that is an accu- boundary layer The motion of a ﬂuid of low mulation point of A; equivalently every bounded viscosity (e.g., air, water) around (or through) sequence in R has a convergent subsequence. a stationary body possesses the free velocity of In metric spaces: compactness and sequential an ideal ﬂuid everywhere except in an extremely compactness are equivalent. thin layer immediately next to the body, called the boundary layer. bond There is a chemical bond between two atoms or groups of atoms in the case that boundary value problem The problem of the forces acting between them are such as to ﬁnding a solution to a given differential equation lead to the formation of an aggregate with suf- in a given set A with the solution required to meet ﬁcient stability to make it convenient for the certain speciﬁed requirements on the boundary chemist to consider it as an independent “molecu- ∂A of that set. lar species.” See also coordination. bounded linear operator A bounded linear (X , . ) operator from a normed linear space 1 1 to another normed linear space (X , . ) is a bond order, prs The theoretical index of the 2 2 rs T X → X degree of bonding between two atoms relative to map : 1 2 which satisﬁes that of a single bond, i.e., the bond provided by (i.) T(αx+ βy) = αT (x) + βT (y) for all x,y ∈ X ,α,β ∈ R one localized electron pair. In molecular orbital 1 ; linearity theory it is the sum of the products of the cor- Tx ≤ Cx (ii.) 2 1, for some constant responding atomic orbital coefﬁcients (weights) C ≥ x ∈ X 0, all 1 ; boundedness. over all the occupied molecular spin-orbitals. Boundedness is equivalent to continuity. Borel sets The sigma-algebra of Borel sets Bourbaki, N. A pseudonym of a changing of Rn is generated by the open sets of Rn. group of leading French mathematicians. The An element of this algebra is called Borel Association of Collaborators of Nicolas Bour- measurable. baki was created in 1935. With the series of monographs Ele´ments de Mathe´matique they Bose-Einstein gas A gas composed of par- tried to write a foundation of mathematics based ticles with integral spin. on simple structures.

20 BRST

Boussinesq equation bridge An edge spanning two connected components of a graph. u − u + (u2) − u = . tt xx 3 xx xxxx 0 Comment: The mathematical word accurately Bramble-Hilbert lemma A crucial tool for conveys the structure: two subgraphs joined by proving local interpolation estimates for para- a single edge. metric equivalent ﬁnite elements. Given a Brouwer’s ﬁxed point theorem Every con- ; ⊂ Rn bounded domain with Lipschitz- tinuous map from the closed ball in Rn into itself k ∈ N continuous boundary, it states that, for 0, has at least one ﬁxed point. 0 ~~ P (;) the simulation. This naturally leads to motion where 0 and k stands being stochastic and hence the dynamic of the for the space of multivariate polynomials of total ≤ k ; molecule is Brownian motion–like. One essen- degree on . The proof of this lemma Brownian dynamics H s (;) tial difference between and relies on the compact embedding of in MD is that the water molecules are explicit in the L2(;) , a fact that is known as Rellich’s lemma. latter, whereas they contribute as a random force The Bramble-Hilbert lemma can be extended to Brownian ≤ k and a viscous medium in the former. spaces of polynomials with separate degree dynamics have the advantage of large time steps. in each independent variable and to non-standard Its diffculty lies in the uncertainty about the inter- anisotropic Sobolev spaces. atomic interaction on the large time step and in branch point (in polymers) A point on a implicit water, known as coarse-graining. chain at which a branch is attached. Brownian motion A stochastic process f Notes: (1)A branch point from which linear {X(t) : t ≥ 0} is a Brownian motion process f chains emanate may be termed an -functional (or Wiener process)if branch point, e.g., ﬁve-functional branch point. X( ) = Alternatively, the terms trifunctional, tetrafunc- (i.) 0 0; tional, pentafunctional, etc. may be used, e.g., (ii.) for each t, X(t)is a normal random vari- pentafunctional branch point. able with zero mean; (2)A branch point in a network may be termed (iii.) if a~~

21 bulk sample bulk sample The sample resulting from the the bundle morphism is usually required to planned aggregation or combination or sample preserve that structure (e.g., to be linear on units. ﬁbers). A bundle morphism (), φ) is called a strong morphism if φ is a diffeomorphism. It is called bundle A triple (E,B,π), consisting of two vertical if M = M and φ = id .If) topological spaces E and B and a continous M is surjective, (), φ) is called a bundle epimor- surjective map π : E → B. E is called the phism.If) is injective, (), φ) is called a bundle total space, B the base space, and π −1(x) the monomorphism.If) is a diffeomorphism, then φ ﬁber at x ∈ B.Atrivial bundle is of the form is also a diffeomorphism and ()−1,φ−1) is a bun- π : B × E → B with π = pr the projec- 1 dle morphism, called the inverse morphism.In tion onto the ﬁrst factor. If all ﬁbers π −1(x) are that case (), φ) is called a bundle isomorphism. homeomorphic to a space F , and the bundle is If (xµ,yi ), (xµ,yi ) are ﬁbered coordinates locally trivial; i.e., there are homeomorphisms on B and B, the local expression of a ﬁbered ψ : U × F → π −1(U ) ( U an open cover of i i i i morphism is: B) with transition maps which are homeomorph- isms, then the bundle is called a ﬁber bundle and xµ = f µ(x) F is called the typical ﬁber. If the typical ﬁber yi = Y i (x, y) F is a vector space, the bundle is called a vector bundle. If the spaces E,B are smooth mani- u = u + u2 folds and all the maps above are smooth, then the Burger’s equation t xx x . bundles are called smooth ﬁber bundles. Exam- ples are the tangent bundle and cotangent bundle bursting Some biological cells exhibit brief of a smooth manifold B.Aprincipal ﬁber bun- bursts of oscillations in their membrane electric dle consists of a smooth ﬁber bundle (E,B,π) potential interspersed with quiescent periods and a Lie group G acting freely on E on the during which the membrane potential changes right (u, g) ∈ E × G → ug ∈ E satisfying only slowly. The ﬁrst mathematical model for this phenomenon was proposed by T.R. Chay ψi (x, gh) = ψi (x, g)h , x ∈ Ui ,g,h∈ G. and J. Keizer in terms of ﬁve coupled nonlin- ear ordinary differential equations in which a bundle morphisms If B = (B,M,π,F) slow oscillator modulates a high frequency oscil- and B = (B ,M ,π ,F ) are two ﬁber bun- lation. When the slow oscillating variable (S) dles,abundle morphism is a pair of maps (), φ) passes some numerical value µ, the high fre- such that ) : B → B , φ : M → M and quency oscillation occurs; while when the slow π ◦ ) = φ ◦ π; i.e., ) sends ﬁbers into ﬁbers. oscillating variable is below the critical value, the One usually summarizes this property by saying fast oscillation disappears and the corresponding that the following diagram: variable F changes slowly. Hence the overall F ) dynamics of shows bursts of high frequency B −→ B oscillations when S>µ, interspered with qui- ↓↓ escent periods when S<µ.(cf. J. Rinzel, Burst φ M −→ M oscillations in an excitable membrane model. In: Ordinary and Partial Differential Equations, is commutative. If the bundles are endowed with Eds. B.D. Sleeman and R.J. Jarvis, Springer- some additional structure (e.g., vector bundles) Verlag, New York, 1985).

22 canonical transformation

Campbell-Hausdorff formula Also known as Baker-Campbell-Hausdorff formula, is the formula for the product of exponentials in a Lie algebra exp A exp B = exp{A + B+ 1 [A, B]+ 1 [A, [A, B]]+ 1 [B,[B,A]]+···+ C 2 12 12 cn[A[A...[A, B] ...]]}.

candidate device A subgraph of the bio- Calabi-Yau spaces Complex spaces with a chemical graph with particular topological vanishing ﬁrst Chern class or, equivalently, with properties, without necessarily having distinct trivial canonical bundle (canonical class). They biochemical or dynamical properties. are used to construct possibly realistic (super) string models. canonical bracket On a symplectic manifold (M, ω) with canonical coordinates (q1,...qn,p ,...p ) calcium-induced calcium release A posi- 1 n we have the canonical tive feedback component in the calcium dynam- (Poisson) bracket between any two functions F ics of biological cells (see autocatalysis). It and G on M as is known from experiments that some calcium n ∂F ∂G ∂F ∂G {F,G}= − . channels (see ion channel) which are respon- i i ∂q ∂qi ∂pi ∂q sible for calcium inﬂux into cytosol are positively i=1 modulated by the calcium in the cytosol. This canonical coordinates (Darboux’s the- mechanism has been suggested as being respon- orem) On any symplectic manifold (M, ω) there sible for the widely observed calcium oscillation (q1,...qn,p,...p ) exist local coordinates 1 n , in cells. called canonical coordinates, such that n calibration curve See calibration function. i ω = dq ∧ dpi . i=1 calibration function (in analysis) The canonical one-form On any cotangent functional (not statistical) relationship for the ∗ manifold T M there is a unique one-form T chemical measurement process, relating the ∗ such that α T = α for any one form α on M. expected value of the observed (gross) signal or In canonical coordinates (q1,...qn,p ,...p ), E(y) 1 n response variable to the analyte amount T is given by x. The corresponding graphical display for a n single analyte is referred to as the calibration T = p dqi . curve. When extended to additional variables or i i= analytes which occur in multicomponent analy- 1 sis, the “curve” becomes a calibration surface or canonical symplectic form On any cotan- hypersurface. gent manifold T ∗M there is a unique symplectic form ; deﬁned as ; =−dT where T is the Callan-Symanzik equation A type of canonical one-form.Incanonical coordinates (q1,...qn,p ,...p ) ; renormatization group equation in quantum ﬁeld 1 n , is given by theory which studies the variation of the Green’s n i function with respect to the physical mass. ; = dq ∧ dpi . i=1 Camassa-Holm equation A shallow water canonical transformation A smooth map equation which is a completely integrable system f (M ,ω ) between two symplectic manifolds 1 1 having peakon solutions (M ,ω ) and 2 2 is called canonical or symplec- tic if it preserves the symplectic forms, i.e., 2 2 3 ∗ ∂t u − ∂t ∂x u + 3u∂x u − 2∂x u∂x u − u∂x u = 0. f (M ,ω ) → (M ,ω ) f ω = ω : 1 1 2 2 , 2 1.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 23 cardinality cardinality The number of elements in a set. products of a reaction is called autocatalysis. Comment: Often denoted by S or |S|, where Catalysis brought about by a group on a reactant S is any set. See also countable set, denumerably molecule itself is called intramolecular catalysis. inﬁnite set, ﬁnite set, inﬁnite set, and uncountably The term catalysis is also often used when inﬁnite set. the substance is consumed in the reaction (for example: base-catalyzed hydrolysis of esters). carotenes Hydrocarbon carotenoids (a sub- Strictly, such a substance should be called an class of tetraterpenes). activator. Comment: Reactions are written at many C carotenoids Tetraterpenoids ( 40), formally levels of resolution, and those reactions which derived from the acylic parent ψ, ψ-carotene I detail how the catalyst functions will not meet by hydrogenation, dehydrogenation, cyclization, this definition. But that is appropriate, since oxidation, or combination of these processes. the molecule which functions catalytically in one This class includes carotenes, xanthophylls, and reaction will be a substrate in the reactions which certain compounds that arise from rearrangement describe the catalysis. of the skeleton of I or by loss of part of this structure. Retinoids are excluded. Cauchy-Goursat theorem Suppose that f : D → Cisanalytic on a disk D ⊂ C; then f Cartan matrix For a semisimple Lie alge- has an antiderivative F on D, i.e., F (z) = f(z), l A bra of rank the Cartan matrix [ ij ] is deﬁned which is analytic in D, and, if γ is any closed (α ,α ) 2 i j D f = by Aij = where α , ··· ,αl is the system curve in , then γ 0. (αi ,αi ) 1 of simple roots. Cauchy integral formula Let f be analytic γ z Casimir Elements in the universal envelop- within and on a simple closed curve and 0 a ing algebra of a Lie algebra that commute with point in the interior of γ . Then all other elements of the Lie algebra are called 1 f(z) Casimirs. f(z ) = dz. 0 πi z − z 2 γ 0 Casimir function On a Poisson manifold (P, { , }) C Cauchy-Riemann equations (Cauchy- , a function that Poisson commutes ; ⊂ F Casimir func- Riemann theorem) Let C be an open set with every function is called a f ; → f(z) = u(x, y) tion, i.e., {C, F}=0 for all F . and : C be a function +iv(x, y) f (z ) f . Then 0 exists if and only if is Casorati-Weierstrass theorem Let f have differentiable in the sense of real variables and, z = (x ,y ) u, v z w ∈ at 0 0 0 , satisfy the Cauchy- an essential singularity at 0 and let C. Then z ,z ,z ,... z → z Riemann equations there exists 1 2 3 such that n 0 and f(zn) → w. ∂u ∂v ∂u ∂v = , =− . ∂x ∂y and ∂y ∂x catalyst A substance that increases the rate of a reaction without modifying the overall stand- Cauchy-Riemann theorem See Cauchy- ard Gibbs energy change in the reaction; the pro- Riemann equations. cess is called catalysis. The catalyst is both a reactant and product of the reaction. The words Cauchy-Schwarz inequality In any inner catalyst and catalysis should not be used when product space (V, < , >) for any x,y ∈ V the added substance reduces the rate of reaction (see inhibitor). |

24 characteristic function

Cauchy theorem If a function f is analytic an end-group,abranch point or an otherwise- in a region ; and γ is a closed curve in ; which designated characteristic feature of the macro- is homotopic to a point in ;, then γ f = 0. See molecule. homotopy. Notes: (1) Except in linear single-strand macromolecules, the deﬁnition of a chain may Cayley transform Let A be a closed sym- be somewhat arbitrary. metric operator on a Hilbert space. Then (A − (2)A cyclic macromolecule has no end groups iI)(A + iI)−1 is called the Cayley transform but may nevertheless be regarded as a chain. of A . (3) Any number of branch points may be present between the boundary units. (4) Where appropriate, deﬁnitions relating to Cea’s lemma Let a be a bilinear/sesqui- macromolecule may also be applied to chain. linear form on a real/complex Banach space V , which satisﬁes: chain rule Let X, Y, Z be Banach spaces Continuity: and f : X → Y , g : Y → Z be differen- |a(u, v)|≤Cu v , ∀u, v ∈ V, k V V tiable of class C . Then g ◦ f : X → Z is of V-ellipticity: class Ck and 2 |R{a(u, v)}| ≥ αuV , ∀u ∈ V. D(g ◦ f )(x) = Dg(f (x)) ◦ Df (x). Suppose that Vh ⊂ V is a closed subspace of V .Forf ∈ V , let u ∈ V,u ∈ V stand for h h See derivative. solutions of the variational problems characteristic classes Chern classes a(u, v) = f(v) ∀v ∈ V, c ,...,c 1 k are deﬁned for a complex vector bundle of dimension k (or equivalently for a a(uh,vh) = f(vh) ∀vh ∈ Vh. 2i GL(k, C) principal bundle) ci ∈ H (M). p ,...,p These are unique according to the Bramble- Pontrjagin classes 1 j are deﬁned for k Hilbert lemma. Then Cea’s lemma asserts that a real vector bundle of dimension (or equiv- alently for a GL(k, R) principal bundle) pi ∈ C H 4i (M). u − uh≤ infv ∈V u − vhV . α h h w ,...,w Stiefel-Whitney classes 1 k are deﬁned for a real vector bundle of dimension a C If is symmetric/Hermitian the constant α can k (or equivalently for a GL(k, R) principal be replaced with 1. A generalization of Cea’s bundle). They are Z2 characteristic classes lemma to sesqui-linear forms that satisfy an inf- w ∈ H i (M; ) i Z2 . sup condition is possible. Assume that instead of being V-elliptic the bilinear form a satisﬁes, characteristic cone The principal sym- α with α>0, bol Pm(x, ξ) = |α|=m cα(x)ξ of a (lin- ear partial) differential operator P(x,D) = |a(uh,vh)| α ≥ αu ∀u ∈ V . |α|≤m cα(x)D is homogeneous of degree m supvh∈Vh h V h h vhV in ξ . The set C (x) ={ξ ∈ Rn | P (x, ξ) = } Then, provided that a unique continuous solution Pm m 0 u ∈ V of the above variational problem exists, P x there holds is called the characteristic cone of at . −1 characteristic equation For an n×n matrix u − uhV ≤ 1 + Cα infv ∈V u − vhV . h h A the equation det(A − λI) = 0. chain (in polymers) The whole or part of characteristic function The characteristic a macromolecule,anoligomer molecule or a function of a set A is deﬁned by block, comprising a linear or branched sequence x ∈ A of constitutional units between two boundary χ (x) = 1if A x ∈ A constitutional units, each of which may be either 0if

25 characteristic polynomial characteristic polynomial See characteris- chemical equation A balanced elementary tic equation. reaction equation, bimolecular or less on one side, which describes the organic chemistry of characteristic set The characteristic set of the noncatalytic species of an overall biochem- P a pseudodifferential operator is the subset ical reaction by one of ﬁve fundamental organic Char(P ) ⊂ T ∗M − p(x, ξ) = 0 deﬁned by 0, mechanisms (substitution, addition, elimina- p(x, ξ) T ∗M − → R where : 0 is the principal tion, rearrangement, and oxidation-reduction). P symbol of . Equally, a set of such reaction equations which characteristic x-ray emission X-ray emis- sums to the overall biochemical reaction. sion originates from the radiative decay of Comment: These mechanisms distinguish electronically highly excited states of matter. all the various subtypes (nucleophilic, elec- Excitation by electrons is called primary excita- trophilic, free-radical; heterolytic, homolytic, tion and by photons, secondary or ﬂuorescence pericyclic; direct electron, hydride, hydrogen excitation. Particle induced x-ray emission atom transfer, ester intermediates, displacement, (PIXE) is produced by the excitation of heav- and addition-elimination reactions). ier particles such as protons, deuterons, or heavy atoms in varying degrees of ionization. Emis- chemical measurement process (CMP) An sion of photons in the x-ray wavelength region analytical method of deﬁned structure that has occurs from ionized gases or plasmas at high been brought into a state of statistical control, temperatures, nuclear processes (low-energy end such that its imprecision and bias are ﬁxed, of the gamma-ray spectrum) and from radiative given the measurement conditions. This is transitions between muonic states. prerequisite for the evaluation of the performing Characteristic x-ray emission consists of a characteristics of the method, or the development series of x-ray spectral lines with discrete fre- of meaningful uncertainty statements concerning quencies, characteristic of the emitting atom. analytical results. Other features are emission bands from transi- tions to valence levels. In a spectrum obtained chemical motif A motif describing a chem- with electron or photon excitation the most ical relationship between two compounds in the intense lines are called diagram lines or normal donor-acceptor formalism. x-ray lines. They are dipole allowed transitions Comment: The constraint for bimolecular between normal x-ray diagram levels. relationship permits use of the common donor- acceptor language. A reaction may have more ψ = charge conjugation For a Dirac spinor than one such relationship. See also biochem- ψ L ψc = ical, dynamical, functional, kinetic, mechanistic, ψ the charge conjugate spinor is R phylogenetic, regulatory, thermodynamic, and σ 2ψ∗ R ∗ topological motives. −σ 2ψ∗ , where means the complex con- L 0 −i jugate and σ 2 = is the secomd Pauli chemical potential In thermodynamics, the i 0 partial molar Gibbs free energy. In a thermal ∗ ∗ spin matrix. Note that (ψ ) = ψ. equilibrium with many chemical reactions, all components have to have the same chemical chart A chart on a manifold M is a pair potential. (U, φ) where U is an open subset of M and φ is bijection form U onto an open subset of Rn. − chemical reaction A process that results in U is called a coordinate patch, φ φ 1 are coor- i j the interconversion of chemical species. Chem- dinate transition functions. ical reactions may be elementary reactions or Chebyshev’s inequality If f ∈ (Lp,µ) stepwise reactions. (It should be noted that (0

0 this deﬁnition includes experimentally observ- p able interconversions of conformers.) f p µ{x : |f(x)| >α}≤ . Detectable chemical reactions normally α involve sets of molecular entities, as indicated

26 chromophore by this deﬁnition, but it is often concep- chirality The components ψL,ψR of a iβγ tually convenient to use the term also for Dirac spinor ψ → e 5 ψ are called chiral com- changes involving single molecular entities (i.e., ponents or Weyl components. “microscopic chemical events”). chemical species An ensemble of chem- cholesteric phase See liquid-crystal transi- ically identical molecular entities that can tions. explore the same set of molecular energy lev- els on the time scale of the experiment. The term Christoffel symbols on a (pseudo)- is applied equally to a set of chemically identi- Riemannian manifold (M, g), the coefﬁcients of cal atomic or molecular structural units in a solid µ the Levi-Civita connection.Ifg = gµν (x) dx array. ν ⊗ dx is the local expression of the metric For example, two conformational isomers tensor, its Christoffel symbols are given by: may be interconverted sufﬁciently slowly to be detectable by separate NMR spectra and hence {α } = 1 gα" −∂ g + ∂ g + ∂ g to be considered to be separate chemical species βµ g 2 " βµ β µ" µ "β on a time scale governed by the radiofrequency gα" covariant inverse metric of the spectrometer used. On the other hand, in a where denotes the , ∂ slow chemical reaction the same mixture of con- and µ denotes the partial derivative with respect xµ formers may behave as a single chemical species, to . xµ = i.e., there is virtually complete equilibrium popu- Under changes of local coordinates xµ(x) lation of the total set of molecular energy levels , Christoffel symbols transform as: belonging to the two conformers. Except where {α } = J α {γ } J¯δJ¯ν + J¯γ the context requires otherwise, the term is taken βµ g γ δν g β µ βµ to refer to a set of molecular entities containing J α = ∂xα isotopes in their natural abundance. The word- where γ ∂xγ is the jacobian of the coordinate J¯γ = ∂xγ ing of the deﬁnition given in the ﬁrst paragraph is transformation, α ∂xα is the inverse Jaco- intended to embrace both cases such as graphite, J¯γ = ∂2xγ ∇ bian, and we set βµ ∂xβ ∂xµ .If denotes the sodium chloride, or a surface oxide, where the covariant derivative operator associated to the basic structural units may not be capable of iso- Levi-Civita connection, then one has: lated existence, as well as those cases where they are. λ ∇∂ ∂ν ={ µν }g ∂λ. In common chemical usage generic and spe- µ ciﬁc chemical names (such as radical or hydrox- ide ion) or chemical formulae refer either to a chromatic number The minimum number chemical species or to a molecular entity. of colors needed to color the nodes of a graph, such that no two adjacent nodes have the same Chern classes See characteristic classes. color. Denoted χ(G). chiral group In QCD (quantum chromo SU × SU dynamics) the group 3 3. See chromo- chromodynamics, quantum (QCD) The dynamics, quantum. quantum ﬁeld theory describing the strong inter- actions of quarks and gluons. chiral transformation The transformation iβγ of a Dirac spinor ψ → e 5 ψ is called chiral transformation,orchiral symmetry, where β is chromophore The part (atom or group of γ = iγ 0γ 1γ 2γ 3 × a constant and 5 with the 4 4 atoms) of a molecular entity in which the elec- gamma matrices deﬁned by the Pauli matrices tronic transition responsible for a given spectral −i σ 1 = 01 σ 2 = 0 σ 3 = 10 band is approximately localized. The term arose , i , − 10 0 0 1 in the dyestuff industry, referring originally to the 0 −σ i groupings in the moleculaer that are responsible by γ i = . σ i 0 for the dye’s color.

27 chromosome chromosome A self-replicating structure classical symbol A symbol a(x,ξ) of a consisting of DNA complexes with various pro- pseudodifferential operator (or Fourier integral teins and involved in the storage and transmission operator) of order m is called classical if there ∞ of genetic information; the physical structure that exist C functions aj (x, ξ), positively homoge- contains genes (cf. plasmid). Eukaryotic cells neous of degree m − j in ξ (i.e., a(x,τξ) = m−j have a characteristic number of chromosomes τ aj (x, ξ), τ > 0) such that asymptotically per cell (cf. ploidy) and contain DNA as linear ∞ duplexes. The chromosomes of bacteria consist a(x,ξ) ∼ aj (x, ξ). of double-standed, circular DNA molecules. j=0 circulation In the theory of Markov pro- Clifford algebra Clifford algebra is a for- cesses, the probability ﬂux in a stationary mulation of algebra which uniﬁes and extends state which can be achieved by two types of complex numbers and vector algebra. It is based on the Clifford product of two vectors a and b balance: detailed balance and circular balance. which is written ab. The product has two parts, For most Markov processes, the sufﬁcient and a scalar part and a bivector part. The scalar part necessary condition for zero circulation is time- is symmetric and corresponds with the usual dot reversibility. a · b = 1 (ab + ba) product 2 . The bivector part is antisymmetric and can be thought of as a directed Ck f U ⊂ Rn → Rm class A function : area, deﬁning a plane a ∧ b = 1 (ab − ba). Then is differentiable of class Ck,0≤ k ≤∞, if all 2 α the Clifford product can be written as ab = a · ∂ f ≤|α|≤k f partial derivatives ∂xα ,0 ,of up to b + a ∧ b. order k exist and are continuous. closed curve A closed curve (or closed path) M γ a,b → M classical Fourier integral operator A in a space is a curve :[ ] such that Fourier integral operator Au = eφ(x,ξ) γ(a)= γ(b). a(x, ξ)u(y)dy is called classical if it has a clas- closed form An exterior form ω on a mani- sical symbol a(x,ξ). fold having vanishing exterior differential dω = 0. Exact forms are closed. The converse is not GL(n), classical groups The matrix groups true in general. It is true on contractible mani- SL(n), U(n), SU(n), O(n), SO(n), Sp( n) 2 are folds (e.g., on star-shaped open sets of Rm by called classical Lie groups. Poincar´e’s lemma). De Rham cohomology stud- ies the topological properties of manifolds by classical limit In quantum mechanics when classifying closed forms that are not exact. the Planck constant h→¯ 0 classical mechanics is Example: The form ω = (x2 + y2)−1 recovered. (xdy − ydx) is a closed (but not exact) form deﬁned on R2 −{0}; in fact, it locally reduces to classical mechanics Classical mechanics dϕ on the unit circle x2 + y2 = 1,butitisnot vis-a-vis` quantum mechanics and relativistic an exact form since ϕ cannot extend to a single- mechanics. The equations of motion in classical valued coordinate function on the whole of the mechanics are Hamilton’s equations or equiva- circle. lently the Euler-Lagrange equations, formulated closed graph theorem X Y on a ﬁnite dimensional symplectic manifold. If and are Banach spaces and T : X → Y is a closed linear operator deﬁned on all of X then T is bounded classical path A distinction between clas- (i.e., continuous). sical and quantum mechanics. In classical mechanics a particle takes only one path to go closed operator A linear operator T : X → from a point q to a point q, while all paths Y , between two Banach spaces X and Y and contribute in quantum mechanics. The three having a dense domain D(T ) ⊂ X is closed if its colors R, G, and B belong to the representation graph N(T ) ={(x, T x) : x ∈ D(T )} is closed of SU(3). in X × Y .

28 coisotropic (sub)manifold closed orbit An orbit γ(t) of a vector ﬁeld coboundary operator See cochain com- X is called closed if there is a τ>0 such that plex. γ(t + τ) = γ(t)for all t. cochain complex A cochain complex con- closed set The complement of an open set in sists of a sequence of modules and homomor- a topological space (X, τ(X)). The class χ(X) phisms of closed sets has the following properties: ···→Cq−1 → Cq → Cq+1 →··· (i.) The empty set ∅ and the whole space X are elements in χ(X); such that at each stage the image of a given (ii.) The union of a ﬁnite number of elements homomorphism is contained in the kernel of the q q q+ in χ(X) is still in χ(X); and next. The homomorphism d : C → C 1 is d2 = (iii.) The intersection of a (possibly inﬁnite) called coboundary operator.Wehave 0. Zq = ker dq q family of elements in χ(X) is still in χ(X). The kernel is the module of th degree cocycles and the image Bq = im dq−1 q closure For a subset A of a topological is the module of th degree coboundaries. The qth H q H q = space, the smallest closed set containing A. cohomology module is deﬁned as Zq /Bq Denoted A¯. . coacervation The separation into two liquid cocycle See cochain complex phases in colloidal systems. The phase that is more concentrated in colloid component is the codifferential On a Riemannian manifold (M, g) δ ;k(M) → coacervate, and the other phase is the equilibrium the codifferential : ;k−1(M) solution. is deﬁned by n(n+k)+1+Ind(g) k ∗ δα = (−1) ∗ d ∗ α,α∈ ; (M), coadjoint action The coadjoint action Ad of a Lie group G on the dual of its Lie algebra ∗ d ∗ where is the Hodge operator and the exterior g is deﬁned as the dual of the adjoint action of derivative.Wehaveδ2 = 0. G on g.Itisgivenby

29 col col (saddle point) A mountain-pass in a commutative Referring to a set A with a potential-energy surface is known as a col or sad- binary operation * satisfying dle point. It is a point at which the gradient is zero along all coordinates, and the curvature is a ∗ b = b ∗ a positive along all but one coordinate, which is the a,b ∈ A reaction coordinate, along which the curvature is for all . See also Abelian group. negative. commutator (in an associative algebra A) colligation The formation of a covalent The binary operation given by bond by the combination or recombination of two radicals (the reverse of unimolecular homolysis). [A, B] = A · B − B · A. ˙OH + H C˙ → CH OH For example, 3 3 . For example, this is the deﬁnition for the com- colloid A short synonym for colloidal sys- mutator of the Lie algebra GL(n, R) of n × n tem. matrices. Notice that it satisﬁes the Jacobi iden- tities colloidal The term refers to a state of sub- division, implying that the molecules or poly- [[A, B],C] + [[B,C],A] + [[C, A],B] = 0. molecular particles dispersed in a medium have at least in one direction a dimension roughly The commutator of two vector ﬁelds X = µ µ µ between 1 nm and 1 m, or that in a system dis- X ∂µ, Y = Y ∂µ is deﬁned by continuities are found at distances of that order. µ ν µ ν [X, Y ] = (X ∂µY − Y ∂µX )∂ν . color (1) (in mathematics) A nonlabel token for an edge or node of a graph. A coloring of More generally any binary operation deﬁning the graph G is the set of colors assigned to the a Lie algebra, namely, the Lie-product obeying nodes, such that no two adjacent nodes have the Jacobi identities. See also Lie algebra. same color. An edge coloring is the set of colors assigned to edges; a proper edge coloring is the compact (1)Atopological space (X, τ(X)) set of colors such that no two adjacent edges have is compact if from any open covering of X one the same color. can always extract a ﬁnite subcovering.IfX is a Comment: These need not be physical colors, topological subspace of a metric space, “com- though it is perhaps easiest to think of them that pact” is equivalent to “closed and bounded.” way. Note that colorings are by deﬁnition proper, Thence closed intervals are compact in R; closed whereas edge colorings need not be. The differ- balls are compact subsets of Rm (as well as in any ence between deﬁnitions of node and edge color- metric space). ings may have been motivated in the beginning (2) (locally) A topological space X is locally by the four color map problem. compact if every point p ∈ X has a compact (2) (in quantum QCD) Gluons and quarks neighborhood. have an additional type of polarization (degree of freedom) not related to geometry. “The idiots compact operator Let X and Y be Banach physicists, unable to come up with any wonder- spaces and T : X → Y a bounded linear oper- ful Greek words anymore, called this type of ator. T is called compact or completely continu- polarization by the unfortunate name of color.” ous if T maps bounded sets in X into precompact (R.P. Feynman in QED: The Strange Theory of sets (the closure is compact) in Y . Equivalently, Light and Matter, Princeton University Press, {x }⊂X {Tx } Princeton, NJ 1985). for every bounded sequence n , n has a subsequence convergent in Y . commutation relations If a classical Hamiltonian H(qi ,pi ) is quantized, one con- compact set A subset M of a topological siders qi and pi as operators on a Hilbert space X is called compact if every system of space satisfying the commutation relations open sets of X which covers M contains a ﬁnite [qi ,pj ] = δij and [qi ,qj ] = [pi ,pj ] = 0. subsystem also covering M.

30 computational biology compact support A continuous function f complex conjugate The complex conjugate has compact support if its support supp(f ) = of the complex number z = x + iy is the complex {x : f(x)= 0} is a compact set. number z¯ = x − iy. competition One class of models for popu- complex number A number of the form x + lation dynamics in which species are competing iy, where x and y are real numbers and i2 =−1. for the same, limited resource. Hence, the growth rate of one species decreases with increasing complex structure A complex structure on population of the other and vice versa. Such a real vector space V is a linear map J : V → V a system is likely to exhibit extinction of the such that J2 = Id. Setting iz = J(z) gives V weaker species. Stable co-existence is possible, the structure of a complex vector space. however, under a delicate balance. complex vector space A complex vector complement Given a set A, the set of all c space V is a vector space over the ﬁeld of com- elements not in A, denoted A¯ or A . plex numbers C; i.e., scalar multiplication λz is complete space A metric space (M, d) is deﬁned for complex number λ ∈ C,z∈ V . complete if every Cauchy sequence {Xn} in M converges in M. That is, there is x ∈ M such composition Consider two functions, f A → B g C → D that d(xn,x)→ 0, as n →∞ : and : , over four sets A, B, C, D.Iff (A) ⊆ C, then the result of complete vector ﬁeld A vector ﬁeld X on a applying f and g in succession is equivalent to manifold M is complete if its ﬂow φt is deﬁned the application of a single composite function, t ∈ R d φ (x) = X(φ (x)) for all ,(dt t t for all denoted g ◦ f : A → D. g ◦ f is deﬁned for t ∈ R). any x ∈ A as (g ◦ f )(x) = g(f (x)). completely continuous See compact oper- compound Any molecule or assembly of ator. molecules. Comment: These range in size from single completely integrable system A Hamilto- + nian system deﬁned over a symplectic manifold nuclei (H ) to DNA to transport complexes embedded in cell membranes on up. One often (P, ω) of dimension 2n with n ﬁrst integrals Fi distinguishes between enzymes and metabolites, in involution, i.e., such that {Fi ,Fj }=0 for all pairs (i, j). The equations of motion of a com- smaller molecules. The boundary between pletely integrable system can be formally inte- “macromolecular” and “smaller” is not ﬁxed pre- grated. Explicitly, a Hamiltonian system (vector cisely, but is perhaps about 1000 daltons. 2n ﬁeld) XH on R is called completely integrable n f ,...f compound’s reactions A compound’s reac- if there exist constants of the motion 1 n which are linearly independent and in involution, tions is the set of reactions in which that com- pound participates. i.e., {fi ,fj }=0 for all i, j = 1, ..., n. Comment: See reaction’s compounds. complex A molecular entity formed by loose association involving two or more component computational biology The development molecular entities (ionic or uncharged), or the and application of theory, algorithms, heuristics, corresponding chemical species. The bonding and computational systems, including electronic between the components is normally weaker than databases, to biological problems; and equally, in a covalent bond. the application of biological concepts, materials, The term has also been used with a variety or processes to problems not originating in bio- of shades of meaning in different contexts; it logy; and activities at the interface of these two. is therefore best avoided when a more explicit Comment: This deﬁnition is broad, but still alternative is applicable. In inorganic chemistry useful; it includes areas such as the representa- the term “coordination entity” is recommended tion of information in electronic databases, the instead of “complex.” construction and maintenance of those databases

31 computational complexity

(including interfaces), the development of bio- (ii.) recognizing Ki,O (the set of outputs logically realistic automated reasoning methods, actually produced by the computational step) and any underlying mathematical techniques, from KO (the set of possible outputs for the total the management of computations over widely computation); distributed, very heterogeneous systems like (iii.) mapping ci : Ki,I → Ki,O (its compu- the World Wide Web; molecular computing; tation for that step), ci ∈ C, the set of all compu- nanotechnology (where computational issues are tations. raised or where biological artifacts are used in computation); and the problems that biology Comment: This deﬁnition is a bit unusual in poses for the foundations of computer science. that it explicitly includes the recognition of the In a nutshell, computational biology is “com- actual inputs and outputs from among the sets puting about biology and computing with biol- of possible ones, rather than just the mapping ogy.” among them. The reason is that one can then Occasionally, one sees a distinction between place discrete, Turing computations ﬁrmly in a computational biology and bioinformatics, with framework which accommodates both them and the notion that computational biology is theory the analog computations of molecular machines, and algorithms and bioinformatics are databases for example, for DNA computers. See determin- and retrieval. This is a distinction without a dif- istic, nondeterministic computations. ference, since most biological problems require both. To the extent that biological problems are concerted process Two or more primitive molecular, computational biology overlaps with changes are said to be concerted (or to consti- computational chemistry. Often the distinction tute a concerted process) if they occur within is more one of investigator origin (chemist vs. the same elementary reaction. Such changes biologist), degree of resolution of the question will normally (though perhaps not inevitably) be (quantum-mechanical vs. tissue), and platform “energetically coupled.” (In the present context choice (SGI vs. Sun). the term “energetically coupled” means that the simultaneous progress of the primitive changes computational complexity The time and involves a transition state of lower energy than space an algorithm requires to solve a problem. that for their successive occurrence.) In a con- Comment: Time and space have a mathemat- certed process the primitive changes may be syn- ical relationship to the size N and complexity chronous or asynchronous. of the problem. Thus algorithms are described as being linear, polynomial, etc., depending on conﬁguration space The conﬁguration the function of N in the relation between it space of a mechanical system is an n dimen- and time or space. Functions are designated sional manifold Q with local (position) coord- O(f(N)), meaning the algorithm requires on the inates qi ,...,qn. Then T ∗Q is the momentum order of f(N)resources. Algorithms which are phase space. polynomial represent the practical upper bound of feasibility for large N, with lower values conformal mapping A map which pre- for exponents and determinism of the algorithm serves angles. A conformal diffeomorphism on obviously preferable. Many problems requir- a Riemannian manifold (M, g) is a diffeomor- ing supra-polynomial resources can be addressed phism φ : (M, g) → (M, g) such that φ∗g = in favorable cases. Occasionally other metrics, f 2g for a nowhere vanishing function f . such as the number of statements in the program, are deﬁned for particular purposes. conformation The spatial arrangement of the atoms affording distinction between computational step A computational step stereoisomers which can be interconverted by ∈ si S consists of three components: rotations about formally single bonds. Some (i.) recognizing Ki,I (the set of inputs authorities extend the term to include inversion actually presented for that computational step) at trigonal pyramidal centers and other polytopal from KI (the set of possible inputs for the total rearrangements. computation); See also tub conformation.

32 contact form conjugate momenta For a Lagrangian conservation law A conservation law or L(qi , q˙i ) the conjugate momenta pi are conserved quantity of a vector ﬁeld X is any func- tion F such that F ◦ φ = F where φ is the ﬂow ∂L t t p = of X. See constant of motion. i ∂q˙i conservative vector ﬁeld A vector ﬁeld F conjugated system (conjugation) In the on a region in Rn is called conservative if it is conjugated system original meaning a is a the gradient of a function f , i.e., F =∇f . molecular entity whose structure may be represented as a system of alternating single and conserved current For a time-independent multiple bonds. For example, CH = CH − 2 Lagrangian system a conserved current is a ﬁrst CH = CH , CH = CH − C ≡ N.In 2 2 integral. such systems, conjugation is the interaction of In ﬁeld theory, it is a (m − 1)-form E over one p-orbital with another across an intervening some jet-prolongation J hB such that it is closed σ -bond in such structures. (In appropriate molec- once evaluated on a critical section σ , i.e., such ular entities d-orbitals may be involved.) The that term is also extended to the analogous interac- d[(j hσ)∗E] = 0 tion involving a p-orbital containing an unshared Cl − CH = CH electron pair, e.g., 2. See Lagrangian system. connected Describing a topological space constant of motion A function f : M → R (X, τ(X)) in which the empty set, ∅, and the on a manifold M is called a constant of motion X whole space, , are the only subsets which of a vector ﬁeld X on M if f ◦ Ft = f , where Ft are both closed and open. The open interval is the ﬂow of X.IfXH is a Hamiltonian vector I = ( , ) ⊂ R 0 1 is connected in the standard ﬁeld, then f is a constant of motion of XH if the topology. The union of I and the open interval Poisson bracket {f, H}=0. See conservation (−1, 0) is not connected. In fact, both I and law. (−1, 0) are at the same time open and closed, since they complement each other. constitutional unit An atom or group of See pathwise connected. atoms (with pendant atoms or groups, if any) comprising a part of the essential structure of a (V, E) connected graph A graph G is con- macromolecule,anoligomer molecule,ablock, nected if there is a path between any two vertices, or a chain. {vi ,vj }∈V, for all pairs of vertices. Comment: Visually, connected graphs are contact form A 1-form over J kB which those which are in “one piece.” Note that if a con- identically vanishes on holonomic sections j kσ . nected graph contains a cycle, breaking the cycle They are locally expressed as once is guaranteed to preserve the connectedness of the graph. The effect of subsequent breaks will ωi = yi − yi xν d ν d depend on the topology of the graph. i i i ν ωµ = dy − yµν dx ... connection of a bundle A distribution (of i i i ν m = (M) ⊂ T B ωµ ...µ = dy − yµ ...µ ν dx . rank dim ) of planes b b over 1 k−1 1 k−1 the total space B of a bundle (B,M,π; F). The spaces b ⊂ TbB are required to be nonvertical One deﬁnes three different ideals of contact so that b ⊕ Vb - TbB, where Vb denotes the forms: k subspace of vertical vectors. ;c(J B) is the bilateral ideal gen- erated algebraically by contact 1-forms i i i (ω ,ωµ,...,ωµ ...µ ). connectivity In a chemical context, the 1 k−1 ∗ k information content of a line formula, but omit- ;c (J B) is the differential bilateral ideal ting any indication of bond multiplicity. generated by contact 1-forms (i.e., it includes

33 continuity equation products and exterior differentials of gener- the contraction. However, no other nonmathe- ators). Accordingly, this includes also the 2- matical information is required for contrac- i i ν forms dωµ ...µ =−dyµ ...µ ν ∧ dx . tions of models of biological systems, such as 1 k−1 1 k−1 ¯ ∗ k ;c (J B) is the ideal of all forms vanishing biochemical networks (unlike expansion). Such on all holonomic sections, including all (m+h)- information may be beneﬁcial, for example, in forms with h>0. suggesting appropriate approximations. An expansion is the reverse operation. continuity equation The law of conserva- tion of mass of a ﬂuid with density ρ(x,t) and contraction mapping theorem Let T : velocity v is called continuity equation (M, ρ) → (M, ρ) be a contraction mapping of (M, ρ) ∂ρ a complete metric space into itself. Then + div(ρv) = 0. T has a unique ﬁxed point; i.e., there exists a ∂t x ∈ M Tx = x unique 0 such that 0 0. continuous function A function between T topological spaces such that the inverse image contravariant/covariant tensor A tensor E r of every open set is open. on a vector space , contravariant of order and covariant of order s is a multilinear map continuous spectrum The continuous spec- T : E∗ ×···×E∗ × E ×···×E → R trum σc(T ) of a linear operator T on a Hilbert space H consists of all λ ∈ C such that T − λI is (r copies of E∗ and s copies of E). a one-to-one mapping of H onto a dense proper subspace of H. convergent sequence A sequence {xn} L continuously differentiable A function f : having a limit . That is, for every neighbor- ∂f U ⊂ Rn → Rm i hood U of L,wehavexn ∈ U for all except whose partial derivatives ∂x j n exist and are continuous is called continuously ﬁnitely many . differentiable or of class C1. convex combinations See convex hull. contractible A topological space X is con- tractible when the identity map id : M → M convex hull The convex hull of a set A in a (deﬁned by id(x) = x)) and the constant map vector space X is the set of all convex combina- A cx : M → M (deﬁned by cx (y) = x)) are homo- tions of elements of , i.e., the set of all sums t x +···t x x ∈ A, t ≥ , t = ,n topic. See homotopy. 1 1 n n, with i i 0 i 1 arbitrary. contraction (1) (in metric topology) A map T : (M, ρ) → (M, ρ) of a metric space (M, ρ) convex set A set C ⊂ X is convex if for any into itself such that there exists a constant λ, 0 < x,y ∈ C, tx+(1−t)y ∈ C for all t (0

34 covariant derivative coordination The formation of a covalent (x) = eix+e−ix cosine The function cos 2 . Geo- bond, the two shared electrons of which have metrically, it is the ratio of the lengths of adjacent come from only one of the two parts of the side to hypotenuse for a right triangle with angle molecular entity linked by [iut,] as in the reaction x

35 covariant tensor

The covariant derivative of a section σ of a kinds of symmetry operations, screw axes and bundle (B,M,π; F) with respect to a connec- glide planes. (cf. G.H. Stout and L.H. Jensen, tion N is deﬁned by X-Ray Structure Determination: A Practical Guide, 2nd ed., John Wiley & Sons, New York, ∇Xσ = T σ (X) − N(X) 1989). µ i i = X (∂µσ − Nµ(x, σ ))∂i F = (F , curl The curl of a vector ﬁeld 1 µ F ,F ) R3 where X = X ∂µ is a vector ﬁeld over M, T 2 3 on is is the tangent map and N(X) is the horizontal curl F =∇×F lift of X with respect to the connection N.Itis σ ∂F ∂F ∂F ∂F a vertical vector ﬁeld deﬁned over the section , = 3 − 2 i + 1 − 3 j TB → M ∂y ∂z ∂z ∂x i.e., a section of the bundle which projects over the section σ : M → B. ∂F ∂F + 2 − 1 k ∂x ∂y covariant tensor See contravariant tensor. curve (in a topological space X) A continu- γ R → X covering space of M A space C and a pro- ous map : . Sometimes the domain I ⊂ R jection π : C → M which is a local diffeomor- is restricted to an interval ; if the origin ∈ R I phism. A covering space can be regarded as a 0 is a point of , then the curve is said to x = γ( ) X bundle with a discrete standard ﬁber. Covering be based at 0 .If is a differentiable γ spaces are characterized by the property of the manifold, the curve is usually required to be lifting of curves: if γ is a curve in M based at the differentiable. point x ∈ M and b ∈ B is a point projecting on Not to be confused with the trajectory or path x = π(b), then there exists a unique curve γˆ in B which is the image of the curve, i.e., the subset .(γ ) ⊂ X X based at b which projects over γ(t) = π(γ(t))ˆ . of the space . Furthermore, the lift of two homotopic curves One can suitably deﬁne the composition of γ λ , → X produces two curves of B which are still homo- two curves , :[01] provided that γ( ) = λ( ) topic. 1 0 . Notice that even if the two curves γ and λ are differentiable, the composition λ ∗ γ can be nondifferentiable. critical conﬁguration See Hamilton prin- ciple. cycle A path through the graph G(V, E) V ={v ,v ,v ,...,v } consisting of nodes 1 2 3 n critical point A critical point of a vector E ={(v ,v ), (v ,v ),...,(v ,v )} and edges 1 2 2 3 n 1 , X x X(x ) = ﬁeld is a point 0 such that 0 0. V ⊆ V, E ⊆ E. The length of the cycle is n. Comment: It is the property of “pathness,” crystallographic group Crystals are that there is a sequence of edges connecting the formed with symmetries. In three-dimensional nodes in the cycle, which distinguishes a cycle space, these symmetries are represented by from a disconnected subgraph. This is the only the crystallographic groups. These include part of the deﬁnition which might otherwise not rotation, reﬂection (point group), and translation be intuitive. (lattice symmetry) as basic elements, and their composition (e.g., screw axes and glide planes) cytoskeleton A dynamic polymer network lead to a ﬁnite group. More speciﬁcally, 32 underneath the cell membrane. It is mostly point groups combined with 14 Bravais lattices responsible for the mechanics, i.e., shape and (7 primitive and 7 nonprimitive) result in 230 motility, of cells. The main types of ﬁlaments unique space groups which describe the only in the network are actin ﬁlaments, microtubules, ways in which identical objects may be arranged and intermediate ﬁlaments. There are many in an inﬁnite 3D lattice. A simpler, practical other different proteins involved in the network. approach leading to the same result studies They regulate the state and dynamics of the translations in a lattice, introducing two new network.

36 denser subgraph

degree The number of edges incident to a node; here notated as d(v). Comment: The number of reactions in which a compound participates is its degree. Simi- D larly, the number of compounds participating in a reaction is the degree of the reaction. Since the best-known biochemical network described has dalton The unit of molecular weight, in an enzyme for every reaction, one usually means g/mol. nonenzymatic degree, when applying the word Comment: A mole (abbreviated mol) is an to reactions. Note this would not be the case for Avogadro’s number (N ) of molecules, approxi- other versions of the biochemical network which mately equal to 6.02252 × 1023 molecules/mol. include spontaneous reactions, or detailed kinetic Darboux’s theorem On any symplectic and mechanistic views of reactions in which the manifold (M, ω) there exist local coordi- enzyme does not necessarily appear in the reac- nates (canonical coordinates) (x1,...,xn, tion equations as a formal catalyst. y1,...,yn) in which the symplectic form ω degree of polymerization The number of takes the form monomeric units in a macromolecule or oligomer n ω = dxi ∧ dyi . molecule,ablock,orachain. i=1 degrees of freedom A system of N particles, dark current See responsivity. free from constraints, has 3N independent coor- dinates (positions and momenta) called degrees dark output See responsivity. of freedom. dark resistance See responsivity. delocalization A quantum mechanical con- data model A computational method for cept most usually applied in organic chemistry implementing a domain model. to describe the π-bonding in a conjugated sys- Comment: This is the database management tem. This bounding is not localized between two system, object-oriented model, set of MatLab atoms: instead, each link has a “fractional double functions, etc., which [reify] the ideas in the bond character” or bond order. There is a cor- domain model. Thus, it is the computational responding “delocalization energy,” identiﬁable machinery, not (per se) the intellectual basis of with the stabilization of the system compared the abstract model of the domain. with a hypothetical alternative in which formal (localized) single and double bonds are present. f(x) decreasing function A function such Some degree of delocalization is always present f(y)≤ f(x) x ≤ y that whenever . and can be estimated by quantum mechanical cal- deﬁnite integral Let f be a function deﬁned culations. a,b P a = on a closed interval [ ] and a partition delta function See Dirac delta function. x

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 37 denumerably inﬁnite set denumerably inﬁnite set A set which is provided that this limit exists, f is called dif- equivalent to the set of natural numbers, N. ferentiable at the point x with derivative f (x). (There is a bijection between the set and N.) More generally, if X, Y are Banach spaces U ⊂ See also cardinality, ﬁnite set, inﬁnite set, and X open, and f : U ⊂ X → Y , then the Frechet uncountably inﬁnite set. derivative of f is map Df : U → L(X, Y ), where L(X, Y ) is the vector space of bounded deoxyribonucleic acids (DNA) High- linear operators for X to Y deﬁned by molecular-weight linear polymers, composed of f(x+ th) − f(x) nucleotides containing deoxyribose and linked Df (x)h = lim . by phosphodiester bonds; DNA contain the t→0 t genetic information of organisms. The double- If this limit exists f is called differentiable at x stranded form consists of a double helix of with (total or Frechet) derivative Df (x). two complementary chains that run in opposite directions and are held together by hydrogen bonds between pairs of the complementary deterministic computation A deterministic nucleotides and Hoogsteen (stacking) forces. computation Cd speciﬁes a computation C : KI → KO such that the cardinalities (denoted S, deRham cohomology group The quotient where S is any set) of the sets of presented inputs groups of closed forms by exact forms on a and produced outputs (KI and KO , respec- manifold M are called the deRham cohomol- tively) are 1 (KI = KO = 1); each compu- ogy groups of M. The kth deRham cohomology tational step is a one-to-one mapping between group of M is the presented input and the produced output (∀ci , ci ∈ Cd is one-to-one); and the probability that H k(M) = ker dk/range dk−1, each symbol in the sets of presented inputs and produced outputs exists is 1 (∀σi,I , σi,I ∈ KI , k k where d is the exterior derivative on forms. ∀σi,O , σi,O ∈ KO , Pe(σi,I ) = Pe(σi,O ) = 1). Comment: As with the deﬁnitions of compu- derivation (of an R-algebra A)GivenR tation, nondeterministic, and stochastic compu- a ring,anR-linear map D : A → A (i.e., tations, the goal here is to place the usual theory D(λx + µy) = λD(x) + µD(y)) such that the of computing within a framework that accom- Leibniz rule holds, i.e., ∀x, y ∈ AD(xy)= modates molecular computers. Like any other D(x)y + xD(y). A derivation on a manifold system of chemical reactions, a molecular com- M is a derivation of the function algebra A = puter is surrounded by many copies of its possible ∞ C (M). inputs and outputs. These copies of symbols Example: let F(R) be the R-algebra of real are populations of symbols, each type of sym- valued (differentiable) functions f : R → R bol occuring at some frequency in its respective R D F(R) → over . The ordinary derivative : population (Pe). Pe(σi ) for any particular σi can F(R) is a derivation. vary depending on the constitution of KI ,KO, A Notice that is not required to be an associ- and the properties of a particular ci ; so there ative algebra. In particular the deﬁnition applies exists a probability density function over K.For to Lie algebras where the Leibniz rule reads as computation over a number of steps, a vector P D( x,y ) = D(x),y + x,D(y) e [ ] [ ] [ ]. of probabilities for each step to each σi would L x ∈ Example: let be a Lie algebra. For all be assigned. Notice that a unit cardinality does L L → L y → , the map adx : deﬁned by adx : not imply that the length of the input and output x,y [ ]isaderivation. tokens is one. See also nondeterministic and stochastic computations. derivative The derivative of a function f : R → R at a point x is the function f deﬁned by device A subnetwork of the biochemical net- f(x+ h) − f(x) work with distinct dynamical (and perhaps bio- f (x) = lim h→0 h chemical) properties.

38 diffusion-driven instability dextralateral The set of obligatorily co- diameter For any bounded domain K ⊂ Rn reacting species arbitrarily written on the right- we deﬁne its diameter by hand side of a formal reaction equation. (K) = {| − |, , ∈ K}. Comment: Formal reaction equations repre- diam : sup x y x y sent two sets of molecular species, one written on diffeomorphism A diffeomorphism of class the left and the other on the right-hand side of the Ck is a Ck differentiable map f which is invert- equation. The placement of a set on a side of the − k ible such that f 1 is also of class C . equation is completely arbitrary. The equation is understood to be symmetric in that the reaction’s differentiable See derivative. chemistry proceeds in both the “forward” (left to right) and “backward” (right to left) simultan- differentiable manifold A topological space eously, until dynamic equilibrium is achieved. M is called differentiable manifold of class Cp if For the (bio)chemistry to proceed, each member each point x ∈ M has a neighborhood U(x) ⊂ of a set of coreacting species must be present. For M (called local chart or coordinate patch) which each, the higher its concentration, the easier it is is homeomorphic to an open set in a vector to observe the reaction and the faster the reaction space (Rn), and such that the change of coor- will occur. dinates (coordinate transition functions) are dif- Since the reactions are reversible, the equa- ferentiable maps of class Cp. See chart. tions are symmetric, and the placement of sets of coreacting species in the equation is arbitrary, differential equation An equation involv- the sets of species are designated sinistralateral ing a function and some of its derivatives, e.g., and dextralateral.These designations distinguish f (x) + f(x)= 0. them from the sets of substrates and products: these terms indicate the role a molecule plays in differential form A differential form of r r U the reaction. See also direction, dynamic equi- degree (or an -form) on an open set in a X ω U → Lr X∗ librium, formal reaction equation, microscopic vector space is a smooth map : U r X∗ reversibility, product, rate constant, reversibil- from into the th alternating product of . ity, sinistralateral, and substrate. differential operator Let E and F be vector bundles over a manifold M and C∞(E), C∞(F ) T diagonalizable A linear transformation : the spaces of smooth sections. A differential V → V V ∞ ∞ on a vector space is called diagonal- operator is a linear map L : C (E) → C (F ) izable (or semi-simple) if there exists a basis of such that L(fg) = (Lf)g +(Lg)f for all f, g ∈ V T ∞ consisting entirely of eigenvectors of . See C (E). linear. diffusion equation The diffusion equation diagram level (in x-ray spectroscopy) A or heat equation is of the following form, for n level described by the removal of one electron u = u(x, t), x ∈ R from the conﬁguration of the neutral ground (∂ − )u = . state. These levels form a spectrum similar to t 0 that of a one-electron or hydrogen-like atom but, ∂2u(x,t) ∂u(x,t) e.g., in two dimensions = . being single-valency levels, have the energy scale ∂x2 ∂t reversed relative to that of single-electron levels. diffusion-driven instability This is a bifur- Diagram levels may be divided into valence cation phenomenon in nonlinear diffusion- levels and core levels according to the nature of reaction equations. If when all the diffusion the electron vacancy. Diagram levels with orbital terms are absent, the remaining nonlinear ODE angular momentum different from zero occur in has a stable ﬁxed point, but when a diffusion term pairs and form spin doublets. is present, the spatially homogeneous solution corresponding to the stable ﬁxed point is unsta- diagram line (in x-ray spectroscopy) See ble, we say the system has a diffusion-driven characteristic x-ray emission and x-ray satellite. instability.

39 Dini’s theorem

Dini’s theorem Let {fn} be a sequence of Dirac operator A linear ﬁrst-order partial continuous functions converging pointwise to differential operator with nonconstant coefﬁ- a continuous function f .If {fn} is monotone cients, deﬁned between sections of the spin bun- increasing sequence, then the convergence is uni- dle. It is deﬁned as covariant derivative followed form. by Clifford product. Locally the Dirac operator is given by dipole–dipole interaction Intermolecular δ or intramolecular interaction between molecules Dψ = γ µ ψ δxmu or groups having a permanent electric dipole µ moment. The strength of the interaction depends γ µ on the distance and relative orientation of the where are the Dirac gamma matrices. dipoles. The term applies also to intramolecu- Dirac spinors Elements in the full complex lar interactions between bonds having permanent spin representation as opposed to elements in the dipole moments. half spin representation which are called Weyl spinors Dirac Paul Adrien Maurice Dirac (1902– and elements in the real spin representa- tion which are called Majorana spinors. 1984), Swiss/English theoretical physicist, founder of relativistic quantum mechanics. directed edge A sequence of two nodes Nobel prize for physics 1933 (shared with (ordered pair) in a graph; often denoted Schr¨odinger). /v ,v 0 i i+1 . Comment: These are the edges that are drawn Dirac delta function The generalized func- with arrows, indicating a direction of travel or tion δ(x − x ) deﬁned by 0 ﬂow. See edge. ∞ f(x)δ(x− x )dx = f(x ). 0 0 direction In a formal reaction equation, the −∞ consumption of sinistralateral coreactants (the

Dirac delta measure The measure δy “forward” direction); equally, the consumption located at some arbitrary, but ﬁxed y ∈ Rn is of dextralateral coreactants (the “reverse” direc- deﬁned for A ⊂ Rn as tion). Comment: Chemical and biochemical reac- y ∈ A δ (A) = 1if tions can be thought of as a composite of two y y ∈ A. 0if reactions. One that consumes sinistralateral core- actants, producing dextralateral ones or proceed- Dirac equation The equation ing left to right and the opposite. These two (i ∂ − m)ψ = 0 directions are often represented as half reactions which are formal equations that specify only where ψ is a wave function describing a relativis- one of the two directions. The intrinsic rates of µ µ tic particle of mass m and ∂ := γ δµ with γ the two directions, as measured by their respec- the Dirac gamma matrices. tive rate constants can differ very signiﬁcantly. See also dextralateral, dynamic equilibrium, for- × Dirac gamma matrices The 4 4 matrices mal reaction equation, microscopic reversibility, I σ j γ 0 = 0 γ j = 0 , product, rate constant, reversibility, sinistralat- −I , −σ j 0 0 eral, and substrate. j = 1, 2, 3, where I is the 2 × 2 identity matrix 01 and σ j are the Pauli matrices σ 1 = , Dirichlet boundary condition Speciﬁca- 10 tion of the value of the solution to a partial dif- 0 −i 10 ferential equation along a bounding surface. σ 2 = , σ 1 = , i2 =−1. i 0 0 −1 Dirichlet problem The problem of ﬁnding Dirac Laplacian The square root of the solutions of the Laplace equation u = 0(uxx + 2 Dirac operator, i.e., we have D ψ = ψ, the uyy = 0 in two dimensions) that take on given d’Alembert operator. boundary values.

40 domain model discrete spectrum The discrete spectrum or divergence theorem Also called Gauss’s 3 point spectrum σp(A) of a linear operator A is theorem. Let ; be a region in R and ∂; the the set of all λ for which (λI − A) is not one- oriented surface that bounds ;, and denote by n to-one, i.e., σp(A) is the set of all eigenvalues the unit outward normal vector to ∂;. Let X be A ; of . a vector ﬁeld deﬁned on . Then discrete topology In the discrete topology (divX)dV = (X · n)dS. on a set S every subset of S is an open set. ; ∂; divergent sequence A sequence that is not disjoint Two collections, especially sets convergent. such as A and B, are said to be disjoint if A ∩ B =∅. dividing surface A surface, usually taken to be a hyperplane, constructed at right angles dissipative A linear operator T : H → H to the minimum-energy path on a potential- on a Hilbert space (H, < , >) is called dissi- energy surface. In conventional transition-state pative if Re

41 dominated convergence theorem dominated convergence theorem Let {fn} If V is ﬁnite dimensional, then dim(V ) = 1 ∗ be a sequence in L such that fn → f almost dim(V ). See also dual basis. everywhere and there exists a nonnegative g ∈ 1 L such that |fn|≤galmost everywhere for all n f ∈ L1 f = f . Then and limn→∞ n. duality techniques (Aubin-Nitsche trick) These are used to establish a priori estimates donor A compound which breaks a chem- for the discretization error of ﬁnite element ical bond, yielding a substituent group which schemes in norms weaker than the natural norms forms a new bond in a bimolecular chemical or associated with the continuous variational biochemical reaction. See acceptor. problem. For example, let H,V be Hilbert spaces, V continuously embedded in H , and a : V × V → C a continuous and V-elliptic dot product See angle between vectors. sesqui-linear form. Write u ∈ V,uh ∈ Vh for the solutions of double helix The structure of DNA in all biological species is in the form of a double helix a(u, v) = f(v) ∀v ∈ V, made of two chain molecules. Each chain is a polymer made of four types of nucleotide: A

(adenine), G (guanine), T (thymine), and C (cyto- a(uh,vh) = f(vh) ∀vh ∈ Vh, sine). The structure that was ﬁrst proposed by J.D. Watson and F.H.C. Crick immediately leads where Vh is some closed subspace of V (a con- to a possible mechanism of biological heredity. forming ﬁnite element space) and f ∈ V .For This was later conﬁrmed by experiments and φ ∈ H denote by g(φ) ∈ V the solution of hence provides a molecular basis for genetics.

a(v, g(φ)) = (φ, v)H ∀v ∈ V. drawing See rendering.

Then, choosing φ := u − uh we obtain via dual See dual vector space, complement. Galerkin orthogonality

u − u 2 = a(u − u ,g(u− u )) ≤ dual basis Let E be a ﬁnite dimensional vec- h H h h (e ,...,e ) tor space with basis 1 n . The dual basis (α1,...,αn) of the dual space E∗ is deﬁned by j j j au − uhV infv ∈V g(u − uh) − vhV α (ei ) = δi , where δi = 1ifj = i and 0 other- h h wise. If g(φ) possesses extra regularity beyond merely belonging to V , the second term on the right- V dual vector space Let be a vector space hand side will become very small compared to (over the ﬁeld K); the set V ∗ of all linear func- u − uhH , if the resolution of the ﬁnite ele- tionals α : V → K can be endowed with a struc- ment space is increased. Thus, when considering ture of vector space by deﬁning (λα + µβ)(v) = families of ﬁnite element spaces, the H -norm of λ α(v) + µ β(v) (where λ, µ ∈ K; v ∈ V ). The the discretization error may converge asymptot- vector space V ∗ so obtained is called the (alge- ically faster than the V -norm. braic) dual vector space of V . If V is a topological vector space the set of all linear and continuous functionals α : V → K with the standard linear structure introduced Dufﬁng equation The Dufﬁng equation above is called the topological dual space of V , and it is still denoted by V ∗. x¨ + x + "x3 = 0

42 dynamical system is an example of weakly nonlinear oscillators, dynamical motif A conserved pattern of i.e., small perturbations of the linear oscillator dynamical regimes for a reaction or group of x¨ + x = 0. reactions. Comment: Notice this is distinct from a device Dym’s equation The nonlinear evolution in that the latter is not required to exhibit equation conservation. See also biochemical, chemical, −1/2 functional, kinetic, mechanistic, phylogenetic, ut = 2(u )xxx. regulatory, thermodynamic, and topological motives dynamic equilibrium In a chemical or bio- . chemical reaction, the continuous reaction of dynamical system 1 F vec- sinistralateral and dextralateral sets of coreac- ( ) The ﬂow t of a tor ﬁeld X manifold M F M → M tants, such that no net change in the concentra- on a ; i.e., t : tions of each members of both sets occurs. is a one-parameter group of diffeomorphisms, F = F ◦F differential equa- Comment: What determines which direc- t+s t s , and satisﬁes the tion tion of a reaction forward or backward will d F (x) = X(F (x)) . predominate is the relative concentration of dt t t reactants forming the two sets of obligatorily co- (2)(autonomous dynamical system) a pair reacting species. High concentrations of one set (M, X) where M is a manifold and X is a vector will drive the chemistry in the direction which ﬁeld over M.Anintegral curve γ : R → X is consumes the reactants of that set, until the two such that sets are in equilibrium. See also dextralateral, direction, formal reaction equation, microscopic γ˙ = X ◦ γ reversibility, product, rate constant, reversibility, where γ˙ is the tangent vector to γ . sinistralateral, and substrate. (3)(non-autonomous dynamical system over M ) a dynamical system (M,ˆ X)ˆ over Mˆ = R × ˆ dynamic viscosity, η For a laminar ﬂow of M such that X = ∂t + X(t,x), where X(t,x) is a ﬂuid, the ratio of the shear stress to the velocity a time-dependent vector ﬁeld over M. gradient perpendicular to the plane of shear. See equilibrium point, ﬁrst integral.

43

electromagnetism

Einstein tensor On a pseudo-Riemannian (M, g) G = R − 1 Rg manifold , the tensor µν µν 2 µν where Rµν is the Ricci tensor and R is the Ricci scalar of the Levi-Civita connection of the met- E ric g. By extension, on a manifold M with a con- α nection Nβµ,itisthetensor with the same local edge An unordered pair of nodes in a graph, expression where Rµν is now the Ricci tensor usually denoted as a tuple of arity two with the and R is the Ricci scalar of the connection ﬁxed (v ,v ) = (v ,v ) two nodes as arguments. Thus i j j i on M. (the edge is symmetric). Because of Bianchi identities we have Comment: Edges are usually represented as µ ∇µG ·µ = 0. lines or arcs in renderings of a graph or network. Incidence relation and undirected edge are syn- onyms. See directed edge. electric charge, Q Integral of the electric current over time. The smallest electric charge effective Lagrangian An effective Lagran- found on its own is the elementary charge, e, the gian describes the behavior of a quantum ﬁeld charge of a proton. theory at large distances (low energy). electrode potential, E Electromotive force eigenspace See eigenvalue. of a cell in which the electrode on the left is a standard hydrogen electrode and the electrode eigenvalue For a linear operator A : X → on the right is the electrode in question. X a scalar λ is called eigenvalue of A if there is a nontrivial solution x to the equation Ax = λx. Such an x is called eigenvector corresponding to electrodynamics Classical electrodynamics λ. The subspace of all solutions of the equation is described by Maxwell’s equations of an elec- (A − λI)x = 0 is called the eigenspace of A tromagnetic ﬁeld. Quantum electrodynamics corresponding to λ. (QED) is the theory of interaction of light with matter and is described by the Dirac equations eigenvector See eigenvalue. γµ(i∇µ − eAµ)ψ = mψ. eikonal equation In geometrical optics the Hamilton-Jacobi equation H (x, dS(x)) = 0. electromagnetic ﬁeld In modern geometric terms a 2-form on space-time M given as fol- A Einstein Albert Einstein (1879–1955) Ger- lows. Let be a connection 1-form (vector F man-American physicist. The inventor of special potential). Its curvature 2-form A, given by F = dA + 1 A ∧ A and general relativity theory. Nobel prize in A 2 [ ], is the electromagnetic F = 1 F dxµdxν physics 1922. Popularly regarded as a genius ﬁeld. In local coordinates A 2 µν . L =−1 (F ∧∗F ) ∗ among geniuses, the greatest scientist in history. From the Lagrangian 2 A A ( Everything should be formulated as simple as the Hodge star operator) we obtain that classical possible, but not simpler. equations of motion d ∗ FA = 0. These together with the Bianchi identity dFA = 0 are Maxwell’s Einstein equations Einstein’s ﬁeld equa- equations (in empty space). tion of general relativity (for the vacuum) is the system of second-order partial differential equa- electromagnetism One of the four elemen- tions tary forces in nature. Classical electrodynamics R = 0 µν is governed by Maxwell’s equations for the elec- where Rµν is the Ricci curvature tensor. tromagnetic ﬁeld.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 45 element matrix element matrix Given a linear variational empty collection For sets, bags, lists, and problem, based on the sesqui-linear form a : V × sequences, the empty collection is the corres- V → C, and V-conforming ﬁnite element (K, ponding collection which has no elements: thus V ,X ) b , ··· ,b ,M = K K with shape functions 1 M the empty set, empty list, etc. It is denoted by the dimVK , the corresponding element matrix is corresponding delimiters with nothing between given by them; thus {}, [ ], and /0for empty set, empty list, and empty sequence, respectively. A := (a(b ,b ))M . K i j i,j=1 Comment: If a bag is empty, it reduces to the The bilinear form a could be replaced by a dis- empty set, also denoted ∅. See also bag, list, crete approximation ah (see variational crime). sequence, set, and tuple. elementary charge Electromagnetic funda- end-group A constitutional unit that is an mental physical constant equal to the charge extremity of a macromolecule or oligomer of a proton and used as atomic unit of charge molecule. e = 1.602 177 33(49) × 10−19C. An end-group is attached to only one con- See electric charge. stitutional unit of a macromolecule or oligomer molecule. elementary forces The four elementary forces in nature are gravitation, electromag- endomorphism (1) A map from a set to netism, weak nuclear force, and strong nuclear itself, satisfying certain conditions depending on force. the nature of the set. For example, f(x ∗ y) = f(x)∗ f(y), if the set is a group. elementary reaction A reaction in which (2)Amorphism (not necessarily invertible) of no reaction intermediates have been detected, or an object of a category into itself. need to be postulated in order to describe the reaction on a molecular scale. Until evidence to energy function For a Hamiltonian system, the contrary is discovered, an elementary reac- the Hamiltonian is also called the energy function tion is assumed to occur in a single step and to of the system. pass through a single transition state. energy, kinetic In classical mechanics, that elementary symbol See semiote. part of the energy of a body which the body pos- sesses as a result of its motion. A particle of mass elimination The reverse of an addition reac- m and speed v has kinetic energy E = 1 mv2. tion or transformation. In an elimination two 2 groups (called eliminands) are lost most often energy-momentum tensor In classical ﬁeld from two different centers (1/2/elimination or theory invariance of the Lagrangian under trans- 1/3/elimination, etc.) with concomitant forma- lations implies, via the Noether theorem, conser- tion of an unsaturation in the molecule (double vation of energy-momentum. Let the Lagranian bond, triple bond) or formation of a new ring. on space-time be given by L(φ, ∂µφ); then the tensor T elliptic equation A linear partial differen- conserved energy-momentum is given n by tial equation (PDE)onR of order m with con- j stant coefﬁcients is of the form |j|≤m aj D u = ∂L Tµν = ∂ φ − g L ,∂Tµν = 0. f . It is called elliptic if the equation in p, ∂(∂ φ ) ν i µν j µ i |j|=m aj p = 0 has no real solution p = 0. In general relativity the energy-momentum ellipticity A sesqui-linear form a : V × tensor is deﬁned as the conserved current V → C on a Banach space V is said to be (via Noether’s theorem) given by varying the V-elliptic, if metric. And in Yang-Mills theory (including 2 Maxwell’s equation) the energy-momentum ten- |Ra(u, v)|≥αuV , ∀u ∈ V sor is deﬁned as conserved current obtained by with a constant α>0. An inf-sup condition for varying the gauge invariant Lagrangian with a is an immediate consequence. respect to the Yang-Mills gauge invariant frame.

46 equivalence relation entropy The quantitative measure of disor- the sinistralateral (s)ordextralateral (d) side der, which in turn relates to the thermodynamic of the reaction equation for reaction ri , Xs|d,i functions, temperature, and heat. is the number of reactants on the sinistralat- eral and dextralateral sides, respectively (| used enzyme A catalyst occurring naturally in a here as logical “or”), and ki ,k−i are the forward biochemical system, or derived from or modeled and reverse rate constants, respectively. For the K upon a naturally occurring enzyme. Enzymes are entire system of reactions, eq,R is a class of proteins made of polypeptide. Comment: Until about 15 years ago all known N K = K . enzymes were proteins, but since then a number eq,R eq,i of naturally occurring, catalytic RNAs have been i=1 discovered. This deﬁnition includes the products Comment: This deﬁnition sets up conditions of laboratory manipulation as well as molecules that allow one reasonably to approximate found in nature. thermodynamic activities by concentrations, and reactant order by stoichiometries. It also epimorphism A surjective morphism be- assumes that experiments measuring the appar- tween objects of a category. For example, an ent equilibrium constant have been done by epimorphism of vector spaces is a linear surjec- measuring the constant at several different con- tive map; an epimorphism of groups is a surjec- centrations of reactants and extrapolating the tive group homomorphism; an epimorphism of results to zero concentration. Under these con- manifolds is a surjective differentiable map. See ditions, the equilibrium constant as used by also bundle morphism. biochemists comes reasonably close to the equi- epitope Any part of a molecule that acts as librium constant as deﬁned in thermodynamics. an antigenic determinant. A macromolecule can See also formal reaction equation. contain many different epitopes, each capable x of stimulating production of a different speciﬁc equilibrium point (1) A point 0 is called antibody. an equilibrium point, critical point,orsingular X X(x ) = F point of a vector ﬁeld if 0 0. If t is X F (x ) = x t equations of motion The equations which the ﬂow of then t 0 0 for all . select the evolution of a mechanical system, or (2) (of a dynamical system (M, X ) A point more generally any system with only one inde- x ∈ M such that X(x) = 0. The constant curve c (t) = x pendent variable. x in an equilibrium point is an integral See Hamilton principle and Euler-Lagrange curve. equations. equilibrium solution See coacervation. equicontinuous Referring to a family F of functions with the property that for every ">0 equivalence classes See equivalence rela- there exists a δ>0 such that |f(x)− f(y)| <" tion. whenever |x − y| <δfor all f ∈ F . equivalence relation A relation R ={(a, b) equilibrium constant For any reaction ri |a,b ∈ A} that is, a ∼ b ⇔ (a, b) ∈ R from A that is at equilibrium and whose solutes are at to itself such that: ∈ inﬁnite dilution, ri R (where R is a system of ∀a ∈ A a ∼ a K (i.) , ; reactions), the equilibrium constant, eq,i is (ii.) ∀a, b ∈ A, a ∼ b ⇒ b ∼ a; X n d,i x i,d,j (iii.) ∀a, b, c ∈ A, a ∼ b, b ∼ c ⇒ a ∼ c. K = j=1 j = k /k , eq,i X n i −i s,i x i,s,j If ∼ is an equivalence relation, the equiva- j=1 j lence class of a ∈ A is the subset [a] ={b ∈ where xj is the concentration of reactant xj , A : b ∼ a}.Ifc ∈ A is not in the equivalence ni,s|d,j is the stoichiometry of that reactant on class of a (i.e., c ∈ [a] or equivalently a ∼ c)

47 equivalent norms then [c] and [a] are disjoint. The set A/ ∼ of Euler Leonhard Euler (1707–1783) Swiss all equivalence classes is a partition of the set A, mathematician. Some say the most proliﬁc math- and it is called the quotient of A with respect to ematician in history and the ﬁrst modern math- the relation ∼. ematician universalist. He worked at the St. Example: Let ≡n be the relation in Z deﬁned Petersburg Academy and the Berlin Academy of by a ≡n b if and only if n divides a − b, Science. i.e., if there exists an integer k ∈ Z such that a − b = nk. It is an equivalence relation. If Euler equations The motion of a perfect a ≡n b we say that a is congruent to b modulo n. ﬂuid in a domain M ( a smooth Riemannian mani- The equivalence classes are [a]n ={a + kn : k ∈ Z}. The quotient space is denoted by fold with boundary) is governed by the Euler equations Zn ={[0]n, [1]n,...,[n − 1]n} and it is ﬁnite. Z The structure of additive group of induces a ∂u +∇ u =−∇p ∂t u structure of additive group on Zn with respect to divu = 0 the operation [a]n +[b]n = [a+b]n which is well deﬁned; i.e., it does not depend on the representa- where u is the velocity ﬁeld of the ﬂuid and p is tives chosen for the equivalence classes. In fact, the pressure. if a ∈ [a]n and b ∈ [b]n, then [a ]n + [b ]n = The motion of a rigid body with angular [a + b ]n = [a + b]n = [a]n + [b]n. momentum X = (X ,X ,X ) Specializing to the case n = 2, the quotient 1 2 3 and principal = (I ,I ,I ) space Z is made of two elements [0] and [1] . moments of inertia I 1 2 3 is governed 2 2 2 by the Euler equations The class [0]2 is the neutral element with respect + to the additive structure, while one has [1]2 I −I X˙ = 2 3 X X I I [1] = [0] . By using the function exp(iπn), 1 2 3 2 3 2 2 I −I Z X˙ = 3 1 X X 2 with the additive structure is mapped into the 2 I I 1 3 I 1−I3 X˙ = 1 2 X X . group of signs with the obvious multiplicative 3 I I 1 2 group structure. 1 2

Euler-Lagrange equations The Lagrang- equivalent norms Two norms 1 and 2 normed space X ian formulation of a classical mechanical sys- on a vector such that there is a 1 n c c−1x ≤x ≤ tem described by a Lagrangian L(q ,...q , positive number such that 1 2 n cx x ∈ X q˙1,...q˙ ) is given by the principle of least action, 1 for all .Onﬁnite dimensional vector spaces all norms are equivalent. which leads to the Euler-Lagrange equations d ∂L ∂L equivariant See principal bundle. − = 0 ,i = 1,...,n. dt ∂q˙i ∂qi Erlanger program A plan initiated by Felix These are equivalent to Hamilton’s equations. Klein in 1872 to describe geometric structures in In ﬁeld theory where the Lagrangian L terms of their groups of automorphisms. depends on ﬁelds ϕi (x) and their derivatives ∂ ϕ (x) essential singularity A singularity of an µ i , the Euler-Lagrange equations become analytic function that is not a pole. ∂L(x) ∂L(x) essential spectrum For a linear operator − ∂µ = 0 ,i= 1,...,n. ∂ϕ (x) ∂[∂ ϕ (x)] A : X → X on a normed vector space X the i µ i set of scalars λ such that either ker(A − λI) is inﬁnite dimensional or the image of A − λI fails to be a closed subspace of ﬁnite codimension. For a mechanical Lagrangian system (Q, L) over the conﬁguration space Q with local essentially self-adjoint operator See self- coordinates qi the Lagrangian is a function adjoint operator. L(t, qi ,ui )with (qi ,ui ) local coordinates over

48 expansion the tangent bundle TQ. Euler-Lagrange equa- evolution equation The equations of motion tions are given by of a dynamical system. They describe the time evolution of a system and are given by differential ui =˙qi equations of the form ∂L − d ∂L = ∂qi dt ∂ui 0 dx(t) = F (x(t)). where dot denotes time derivative. dt For a Lagrangian system (B,L) of a ﬁeld theory over the conﬁguration bundle B = evolution operator (B,M,π,F) let us choose local ﬁbered coord- As time passes, the state µ i i i ψ of a physical system evolves. If the state is ψ inates (x ,y ,yµ,...,yµ ...µ ) over the k-jet 0 1 k t = ψ J kB at time 0 0 and it changes to at a later prolongation and the Lagrangian of the t F (ψ ) = ψ F µ i i i time , one sets t 0 . The operator t form L = L(x ,y ,yµ,...,yµ ...µ ) ds, where 1 k ds is the standard local volume element of the is called the evolution operator. Determinism F ◦ F = base manifold M. Euler-Lagrange equations are is expressed by the group property t s F ,F = given by t+s 0 identity. i i i i yµ = dµy ,..., yµ ...µ = dµ ...µ y k 1 k 1 k exact form An exterior k-form ω ∈ ; (M) ∂L − ∂L +···+(− )k ∂yi dµ ∂yi 1 dµ ...µk (k − ) µ 1 which is the exterior differential of a 1 - ∂L ω ω = θ i = 0 form , i.e., d . Of course, exact forms are ∂yµ ...µ 1 k closed, i.e., dω = 0. See also closed form. 2 where dµ denotes the total derivative with respect Example: The form ω =xdx + ydyin R is µ i i ν 1 2 2 to x (i.e., dµ = ∂µ + y ∂i + y ∂ +···). ω = (x + y ) µ µν i an exact form, since d 2 . See Hamilton principle and Lagrangian sys- tem. excitability The concept of excitability ﬁrst Euler’s formula eiθ = cos θ + i sin θ appeared in the literature on neural cells reveals a profound relationship between complex (neurons). It was shown experimentally that a numbers and the trigonometric functions. small trigger in the membrane current can lead to large, transient response in membrane elec- Euler’s integral The representation for the trical potential. This is known as action potential, gamma function and it is responsible for the rapid communica- ∞ tions between neurons. The mathematical model z− −t N(z) = t 1e dt. for this phenomenon was Huxley and it exhibits, 0 among many other features, the threshold phe- Euler’s method The simplest numerical nomenon (see also threshold phenomenon). integration scheme for a differential equation x˙ = f(x). The update rule is xn+ = xn + 1 excluded volume A polymer is made of f(xn) + t. a chain of molecules. These molecules cannot evaluation map Let F(M) denote a set of occupy the same position in three-dimensional functions on a manifold M. The evaluation map space. In the simple theory for polymers (see is given by ev : F(M) × M → R : ev(f, x) = Gaussian chain), one neglects this effect. A more f(x), f ∈ F(M), x ∈ M. realistic models for a polymer must to consider this effect. even function A function f(x) such that f(−x) = f(x), for all x.Ifg(−x) =−g(x), for all x, then g is an odd function. expansion In a network (or graph), the replacement of a smaller subnetwork by a larger event In probability theory, a measurable one using a sequence of operations on nodes, set. In relativity, a point in space-time. edges, parameters, and labels.

49 expectation

Comment: In applying it to models of biol- The exterior algebra L(V ) =⊕Lk(V ) is the ogy such as biochemical networks, expansion quotient of this algebra by the bilateral ideal requires additional information that speciﬁes generated by the elements of the form ei ⊗ ej − which expansion operation, with its components, ej ⊗ ei .IfV is of dimension dim(V ) = m, is appropriate. This is quite distinct from con- L(V ) is a ﬁnite dimensional Z-graded algebra traction, which needs only its mathematical oper- of dimension 2m. The induced multiplication ations. See contraction. ∧ : L(V ) × L(V ) → L(V ) is called the exter- ior product. An element in Lk(V ) is called an expectation In probability theory the exterior k-form on V . f expectation of an random variable (measur- Example: Let ;k(M) denote the space of able function) is the integral fdµ. all exterior k-forms on a manifold M, k = 1,...,n= dimM. The exterior algebra ;(M) exponential (1) The unique solution to the or Grassman algebra of M is the direct sum of differential equation f (x) = f(x), with initial the spaces ;k(M), i.e., ;(M) =⊕n ;k(M). condition f(0) = 1. It can also be deﬁned by its k=1 power series

2 3 exterior derivative A family of operators, x x x e = 1 + x + + +··· on a manifold M, d : ;k(M) → ;k+1(M) 2! 3! (k = 1,2,. . . n = dim M), ;k(M) the space of exte- (2) The exponential map of a Lie group G rior k − forms on M, such that is deﬁned as follows: Let g be the Lie algebra of G.Anyξ ∈ g corresponds to a left invariant (i.) d is a ∧-antiderivative, i.e., d(α ∧ β) = k k vector ﬁeld Xξ on G. Let cξ be the integral curve dα ∧ β + (−1) α ∧ dβ,α∈ ; (M); X e ∈ G, c( ) = e of ξ passing through 0 . The (ii.) d2 = 0; exponential map is now deﬁned as exp : g → f ∈ ;0(M) = C∞(M) G :exp(ξ) = c (1). (iii.) for we have ξ ( f) = ∂f Example: For G = GL(n),exp(A) = 1 + A + d i ∂xi . A2 + A3 +··· 2! 3! (3) The exponential map on a Riemannian manifold (M, g) is deﬁned as exp : T M → M : exterior differential The derivation d of the x M exp(v) = γ(1) where γ is the (unique) geodesic exterior algebra of a manifold deﬁned as fol- 1 µ µ lows: if ω = ωµ ...µ dx 1 ∧ ... ∧dx k is a with initial conditions (x, v), i.e., γ(0) = x and k! 1 k k ω (k + γ(˙ 0) = v ∈ Tx M. -form then its exterior differential d is a 1)-form locally expressed as extension In this context only, the use of 1 µ µ dω = (k + 1)∂ µωµ ...µ dx ∧ dx 1 ∧ an explicit, fully instantiated, representation of (k+1)! [ 1 k ] µ a datum in a database or model. ···∧dx k Comment: The restriction is meant to avoid philosophical wrangling and conﬁne the dis- The exterior algebra with the exterior differential course to that of databases. See also intension. deﬁnes a cohomology (i.e., one has d ◦ d = 0), and it is called deRahm cohomology. exterior algebra (of a vector space V ) The A form ω in the kernel ker(d) (i.e., dω = 0) is Z-graded algebra of skew-symmetric covariant called a closed form; a form ω in the image .(d) tensors on V .IfV is ﬁnite dimensional, {e } is i ω = θ θ a basis of V and {ei } the dual basis in V ∗ then (i.e., d for some form ) is called exact. An exact form is always closed. The converse is we denote by T(V) =⊕Tk(V ) the Z-graded algebra of covariant tensors on V . It is inﬁnite true only provided some topological conditions M dimensional, and it is spanned by are met on . For example, the Poincar´e lemma proves that all closed forms in M are exact when i i j i i I,e,e ⊗ e , ..., e 1 ⊗···⊗e k , ... the manifold M is contractible.

50 external photoelectric effect exterior differential algebra (of a manifold M) exterior product The exterior product or k The Z-graded algebra L(M) =⊕kLk(M), wedge product of an exterior k-form α ∈ ; (M) l where Lk(M) is the set of k-forms over M.An and an l-form β ∈ ; (M) is the k + l-form k+l element ω ∈ Lk(M) is a skew-symmetric map α ∧ β ∈ ; (M) deﬁned by k (X ,...,X ) that associates to vector ﬁelds 1 k a ω(X ,...,X ) F(M) α ∧ β(X ,...X ) = (sign σ )α function 1 k . Such a map is - 1 k+l linear in all its arguments, e.g., ∀f , g ∈ F(M) ×(X ,...,X )β(X ,...X ), σ(1) σ(k) σ(k+1) σ(k+l) ω(fX + gY ,...,X ) 1 1 k where the sum is taken over all permutations σ = fω(X ,...,X ) + gω(Y ,...,X ) 1 k 1 k of {1, 2,...,k+ l}. k ω In other words, a -form is a skew-symmetric, Basic properties of the ∧-product are F(M)-linear form on the F(M)-module X(M) of vector ﬁelds over M. (i.) ∧ is bilinear, i.e., α ∧ (tβ + sγ) = t(α∧ β) + s(α ∧ γ), t,s ∈ R; exterior form Let X (M) denote the space α ∧ (β ∧ γ)= (α ∧ β) ∧ γ ; of smooth vector ﬁelds on a manifold M.An (ii.) kl k exterior k-form (or exterior differential form of (iii.) α ∧ β = (−1) β ∧ α, α ∈ ; (M), β ∈ l order k)isak-multilinear map α : X (M)×···× ; (M). X (M) → C∞(M), k factors, which is skew symmetric, i.e., changes sign if two arguments external photoelectric effect See photo- are interchanged. emissive detector.

51

Feynman diagram

Fahrenheit scale A temperature scale devised by Fahrenheit (1686–1736) on which the freezing point of water is 32 degrees and the boiling point of water is 212 degrees, both at F standard pressure. (This scale is still used in the U.S.) f -functional branch point See branch faithful action An action )g : M → M, point. g ∈ G, of a group G on a space M is called ) = id g = e faithful or effective if 9 M implies . facilitated diffusion The Fickian diffusion of a solute, J = D(cb − ca)/(b − a), can be faradaic current A current corresponding enhanced by the presence of a protein which to the reduction or oxidation of some chemical combines reversibly with the solute. The dif- substance. The net faradaic current is the alge- fusion coefﬁcient of the protein is, of course, braic sum of all the faradaic currents ﬂowing signiﬁcantly lower than that of the solute. Never- through an indicator or working electrode. theless, it enhances the overall diffusion of the solute across a distance b − a with concentration fast Fourier transform (FFT) A discrete difference cb − ca. The resolute to this counter- Fourier transform algorithm which converts a intuitive phenomenon is that the free solute sampled complex-valued function of time into concentration ca at a is different from the total a complex-valued function of frequency. It typ- solute concentration at a due to the reversible ically reduces the number of computations from binding. There is an amount of solute bound to order N 2 to N log N. the protein. [PS]a = Kca[P ]a where [P ]a is the concentration for unbound protein at a, K is Fermi-Dirac statistics In the Fermi-Dirac the binding constant. Hence the total solute at statistics or quantum statistics, no more than one a is (1 + K[P ]a)ca which can be signiﬁcantly of a set of identical particles may occupy a par- greater than ca if K[P ]a 3 1. (cf. J.D. Murray ticular quantum state (i.e., the Pauli exclusion and J. Wyman, J. Biol. Chem., 246, 5903, 1971). principle applies), whereas in the Bose-Einstein statistics, the occupation number is not limited. Faddeev-Popov ghost (not the ghost of Fad- Particles described by these statistics are called deev or Popov) In quantum ﬁeld theory fermions and bosons, respectively. with gauge invariant Lagrangian one can inte- grate over the gauge group by ﬁxing a gauge. fermion A particle described by Fermi- This leads to the effective Lagrangian, which Dirac statistics. is no longer gauge invariant, by introducing new anticommuting auxiliary scalar ﬁelds, so- Feynman diagram A sketch in momentum called Faddeev-Popov ghosts. They have sim- space of the Green’s functions describing the ilar properties as the Maurer-Cartan form and interactions of particles. External lines in a dia- can be interpreted as Lie algebra cohomologies. gram represent incoming particles and vertices Integration over such a slice introduces a represent interactions. Straight lines represent determinant term change of coordinates which electrons and wavy lines photons. Feynman’s is lifted to the exponent of the action, i.e., idea was that expectation values of quantities in added to the Lagrangian as a Gaussian integral. quantum theory should be obtained by summing This determinant is called the Faddeev-Popov the values of these quantities over all histories, determinant. This leads to BRST quantization i.e., over all diagrams and for each diagram over theory. This method of quantization of a system all positions in space-time for the particle inter- with symmetry is known as the Faddeev-Popov actions. These summations over all histories are procedure. known as Feynman path integrals.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 53 Feynman rules

Feynman rules The rules that describe the manifold M and the standard ﬁber F are pro- manipulations with Feynman diagrams. These vided, then there exists a bundle (B,M,π; F) allow physicists to compute Green’s functions in (unique up to isomorphisms) (see bundle mor- quantum ﬁeld theory. phisms) having g(αβ) as transition functions. The total space of# this bundle is identiﬁed with the ﬁber bundle A quadruple B = (B, M, disjoint union α∈I Uα ×F [modulo] the equiva- π; F) where B, M, and F are differentiable lence relation manifolds and π : B → M is a surjective map of maximal rank. B is called the total space, M (α, x, f ) ∼ (β,y,g) F π the base manifold, the standard ﬁber, and if x = y,g = g(βα)(f ) the projection. Furthermore, a trivialization is required, i.e., a family {(Uα,tα)}α∈I such that: Fiber bundles are tightly related to variational calculus. In this framework coordinates xµ on (i.) Uα!are open sets which cover the whole the base manifold are regarded as independent of M, i.e., α∈I Uα = M; variables while yi are identiﬁed with dependent (ii.) t : π −1(U ) → U × F are α α α variables (i.e., dynamical ﬁelds). A conﬁgura- diffeomorphisms. tion is a (global) section of the bundle, i.e., a Each (Uα,tα) is called a local trivialization. map σ : M → B such that π ◦ σ = idM . If a global trivialization tα : B → M × F exists I ={} U ={M} (i.e., if one can take 1 and 1 ) ﬁeld (1)Inalgebra, a commutative ring K the bundle is said to be trivial. Trivializations with at least two elements such that for each B amount to requiring that the total space of a,b ∈ K,a = 0 there exists a unique x ∈ K a bundle is locally diffeomorphic to the triv- such that a · x = b. Examples are the rational, M × F ial model . Examples: The cylinder real, and complex numbers. (S1 × R,S1,p , R) 1 is a trivial bundle, while the (2) In physics, functions (real or complex val- Moebius strip S1 is ﬁbered over but it is not triv- ued) on conﬁguration space are called scalar ial, not being globally isomorphic to a Cartesian ﬁelds and in general, a ﬁeld can be a section of product. any vector bundle over space-time. A preferred class of coordinates (xµ; yi ) is then selected on B such that xµ are coordinates on M and yi are coordinates on F . These coor- ﬁeld equations The equations which select dinates are called ﬁbered coordinates; the bundle the evolution of a ﬁeld theory. structure actually amounts to being allowed to See Hamilton principle and Euler-Lagrange separate coordinates in B into two subsets (xµ equations. and yi ) in a coherent way. One can restrict trivializations to the ﬁber ﬁnite element Let K ⊂ Rn be a domain with π −1(x) x ∈ U over a point α, obtaining diffeo- piecewise smooth boundary, and write VK for −1 morphisms tα(x) : π (x) → F . When two a ﬁnite dimensional space of piecewise smooth x ∈ U = local" trivializations are involved (i.e., αβ functions or vector ﬁelds on K.IfXK is a basis of U U g F → α β ), we can deﬁne the maps (αβ) : the dual space VK , then the triple (K, VK ,XK ) is −1 F given by g(αβ) : tα(x) ◦ tβ (x). The maps a valid ﬁnite element. The functionals in the set g(αβ) : Uαβ → Diff(F ) are called transition XK are referred to as local degrees of freedom, functions. They satisfy the cocycle property, i.e., VK as local space, and K is called a (geomet- ric) element. For a meaningful ﬁnite element g = αα idF the functionals in XK have to be localized in −1 gαβ = [gβα] the sense that they are associated with vertices, gαβ ◦ gβγ ◦ gγα = idF edges, faces, etc. of the geometric element. This means that, for φ ∈ XK , the value φ(v), v ∈ VK , If a family of maps g(αβ) : Uαβ → Diff(F ), only depends on the restriction of v onto a par- which satisﬁes the cocycle property, the base ticular vertex, edge, face, etc.

54 ﬂux

ﬁnite element space Given a triangulation ﬁrst integral For a dynamical system ;h of ;, consider a ﬁnite element (K, VK ,XK ) (M, X) a function F : M → R such that X(F) for each cell K ∈ ;h. A related (global) ﬁnite = 0. The deﬁnition is equivalent to requiring element space Vh has to satisfy (for suitable that F ◦ γ is constant for any integral curve γ m ∈ N) of X. ∗ More generally, a function F : R × M → R Vh ⊂ Vh := which is constant along the curves of motion. M ˆ {v : ; → C deﬁned a.e. in ;, Equivalently, let us deﬁne X = ∂t + X; F is a ﬁrst integral if X(F)ˆ = 0. v|K ∈ VK ∀K ∈ ;h}. x ∈ M ﬁxed point A point 0 is called a ﬁxed T M → M T(x ) = x The global ﬁnite element space is said to be point of a map : if 0 0. conforming with respect to a space V of func- tions/vector ﬁelds deﬁned a.e. on ;,ifVh ⊂ V . ﬂow The ﬂow of a vector ﬁeld X on a Often, this leads to the deﬁnition manifold M is the one-parameter group of diffeomorphisms Ft : M → M,(Ft+s = Ft ◦Fs ) ∗ Vh = Vh ∩ V. such that d In many cases, the requirement of conformity F (x) = X(F (x)), x ∈ M. dt t t for all amounts to conditions on the continuity of global ﬁnite element functions at boundaries between In other words, t → Ft (x) is an integral curve geometric elements. This is due to the fact (trajectory) of X with initial condition x ∈ M. ∗ that functions in V are piecewise smooth.For Flows are also called dynamical systems. instance, conformity in H 1(;) entails global continuity, and ﬁnite element functions in H 2(;) ﬂow rate (of a quantity) Quantity X (e.g., even have to be globally continuously differen- heat, amount, mass, volume) transferred in a time tiable. interval divided by that time interval. General ˙ Actually, the glue between the (local) ﬁnite symbols: qX, X. elements is provided by the degrees of free- F,Z,H dom, because in practice Vh is introduced as ﬂuence, 0 At a given point in space, ∗ Vh :={v ∈ Vh , ∀K,T ∈ ;h : φ(v|k) = the radiant energy incident on a small sphere κ(v|T ) if φ ∈ XK ,κ ∈ XT are of the same type divided by the cross-sectional area of that sphere. and are associated with the same vertex, edge, It is used in photochemistry to specify the energy face, etc. of the global mesh}. It goes without delivered in a given time interval (for instance, saying that this construction imposes tight con- by a laser pulse). straints on the choice of the local ﬁnite elements. ∂x /∂t Local degrees of freedom of the ﬁnite elements ﬂux The ﬂux of a compound xj is j x belonging to adjacent cells have to match. or jt. This makes it possible to convert the local Comment: Biochemically, ﬂux is frequently degrees of freedom into nodal degrees of freedom. used to describe the consumption of indi- vidual molecules by a series of reactions in These are obtained by collecting all local degrees dynamic equilibrium. Experimentally, ﬂux can of freedom into one set X and weeding out func- h be determined by giving a ﬁxed amount of a par- tionals that agree on V . By deﬁnition of V , the h h ticular compound with a radioactive or heavy remaining functionals in X :={φ , ··· ,φ }, h 1 N isotope to a population of cells, then measur- N = dimV , form a basis of the dual space. h ing the amount and rate of movement of the isotope into compound(s) derived from the ﬁrst. ﬁnite set A set whose cardinality is either 0 Since the cells are usually in a metabolic steady or a natural number. See also cardinality, count- state, the assumption is that all the reactions able set, denumerably inﬁnite set, inﬁnite set, and are in dynamic equilibrium. Compound con- uncountably inﬁnite set. centrations do not change over time (x˙i = 0).

55 Fock space

Mathematically, the deﬁnition of ﬂux varies. The where the sets of reactants written sinistralat- alternative taken here is simply to write the erally (s) and dextralaterally (d) in the formal ordinary differential velocity equations as partial reaction equation are Xs,i and Xd,i, respectively; differential equations, so that x˙i becomes ∂xi /∂t Xs,i and Xd,i are the number of reactants in each or xit. An alternative is to deﬁne ﬂux relative set (see cardinality); ni,{s|d},j is the stoichiometry to some other compound, which allows one to of reactant xj in Xs,i or Xd,i ( | is logical or); and incorporate the dynamic equilibrium condition ki and k−i are the forward and reverse reaction directly; the form will vary, much as it does in rate constants, respectively. A species appearing a physical context. Still another is to use ﬂux in catalytically in a reaction equation is included other functions without deﬁning what it is math- on both sides. R is then described by the set of ematically. In general, the word is best used only formal reaction equations, one for each ri ∈ R. when it is mathematically explicit. See also dextralateral, direction, dynamic equi- librium, microscopic reversibility, product, rate Fock space In quantum mechanics, the full constant, reversibility, sinistralateral, stoichiom- Hilbert space of states. etry, and substrate.

Focker-Plank equation Let U ⊂ Rn be Fourier Jean Babtiste Joseph, Baron de open and u : U × R → R. The Focker-Plank Fourier (1768–1830). French analyst and mathe- equation for u is matical physicist. n n ij i Fourier coefﬁcients Let {xi }i∈I be an ut − (a u)x x − (b u)x = 0. i j i orthonormal basis in a Hilbert space (H,<,>). i,j=1 i=1 Then every x ∈ H can be written as a Fourier form See differential form. series x =

56 functional motif frame A set of n linearly independent vec- free action An action )g : M → M,g ∈ tors in the tangent space to an n dimensional G, of a group G on a space M which has no ﬁxed manifold at a point. points, i.e., )g(x) = x implies g = e. Frechet derivative Let V,W be Banach spaces, U ⊂ V open. A map F : U → W is free Lagrangian In ﬁeld theory, a called Frechet differentiable at the point x ∈ U Lagrangian without interaction of the involved if it can be approximated by a linear map in the ﬁelds. The corresponding theory is called a free form theory. F(x + h) = F(x)+ DF (x)h + RF (x, h) function An association of exactly one where DF (x), called the Frechet derivative of F object from a set (the range) with each object at x, is a bounded (continuous) linear map from V from another set (the domain). This is equivalent to W, i.e., DF (x) ∈ L(V, W) and the remainder to deﬁning the function f as a set, f ⊆ A × B. RF (x, h) satisﬁes For f to be a function, it must be the case that if RF (x, h) (x, y) ∈ f (x, z) ∈ f y = z lim = 0. and , then . h→0 h Comment: This last condition is equivalent to If the map DF : U → L(V, W) is continuous in saying if a = b, f(a) = f(b). Functions are x, then F is called continuously differentiable also called mappings and transformations. See or of class C1.IfF is Frechet differentiable also bijection, biological functions, and relation. at x then F is Gateaux-Levi differentiable at x DF (x)h = DF (x, h) F and .If is Gateaux- functional The term functional is usually Levi differentiable at each point x ∈ U and the used for functions whose ranges are R or C. map DF : U → L(V, W) is continuous, then A functional f on a vector space V is a lin- F is Frechet differentiable at each x. In ﬁnite ear functional if f(x+ y) = f(x)+ f(y), and dimensions, i.e., V = Rn,W = Rm, this means n m f(αx)= αf (x), for all x,y ∈ V and scalar α. F : R → R is Frechet differentiable if all partial derivatives of F exist and are continuous. functional analysis The theory of inﬁnite Frechet space vector space A topolocical dimensional linear algebra, i.e., the theory of which is metrizable and complete. Examples are topological vector spaces (e.g., Hilbert, Banach, Banach and Hilbert spaces. or Frechet spaces) and linear operators between Fredholm alternative If T is a compact them (bounded or unbounded). operator on a Banach space, then either (I−T)−1 Tx = x exists or has a nonzero solution. functional derivative Let V be a Banach ∗ Fredholm integral equation The equation space, V its dual space, and F : V → R differ- x ∈ V F b entiable at . The functional derivative of x δF ∈ V ∗ f(x)= u(x) + k(x,t)f(t)dt. with respect to is the unique element δx a (if it exists), such that Fredholm operator A bounded linear oper- δF ator T : V → W between Banach spaces V DF (x) · y =< ,y >, y ∈ V δx for all and W such that − (i.) Ker(T) = T 1(0) is ﬁnite dimensional; where <, >: V ∗ ×V → R is the pairing between ∗ (ii.) Rang(T ) = T(V)is closed; V and V and DF is the Frechet derivative of F , DF (x) · y = 1 F(x + ty) − F(x) (iii.) W/T(V) is ﬁnite dimensional. i.e., limt→0 t [ ]. Example: If V = W and K is a compact oper- ator, then I − K is Fredholm. functional motif A pattern present more The integer ind T = dim T −1(0) − dim W/ than once in a set of biochemical reactions, T(V)is called the index of T. described from any point of view.

57 functor

Table 1 Some Examples of Functional Motives Type An Example Biochemical Methyl transfer reaction G = G Thermodynamic Reaction ri has i , reaction ri+1 has G = G G > G < |G | > |G | i+1, i 0, i+1 0, i+1 i Chemical Aldol condensation Mechanistic Phosphoenzyme intermediate Kinetic Non-allosteric sequential enzyme Dynamical Connected reactions exhibiting birythmicity Topological Reactions and compounds forming a cycle of length n,4≤ n ≤ 7, with at least one reaction requiring an additional compound not a member of the cycle Regulatory Rate increased upon binding of ligand Phylogenetic Mammalian phosphoglycerate mutases

Comment: Types of motif and some examples futile cycle A cycle of alternating com- can be found in Table 1 below. See also bio- pound and reactive conjunction nodes which, chemical, chemical, dynamical, kinetic, mecha- stoichiometrically, regenerates all compounds nistic, phylogenetic, regulatory, thermodynamic, in the cycle and consumes more nucleotide and topological motives. or coenzyme molecules than it produces. Comment: The biochemical connotation of functor (1) A function between categories. the word is strongly dependent on the notion (2) An operator denoting the relation satisﬁed of futility. A disproportionately large energy by a tuple’s arguments. or substituent consumption for no apparent syn- Comment: Where the functor, also called an thetic or catabolic change. It also depends on operator in some contexts, is written is largely a stoichiometry. Clearly for futility to occur, all matter of convention. Some operators are written “nonenergetic” molecules which enter the cycle as preﬁxes (e.g., derivatives, logical predicates); must remain in it. Thus if some proportion are others are inﬁx operators, such as the common diverted out of the cycle to other fates, so that arithmetic ones; and still others are postﬁx oper- the stoichiometry condition is broken, the cycle ators, such as exponentiation. Consider the equa- will decay and energy or substituent consump- tion x = y + a. This equation uses two binary tion will decline. operators, = and +, seen more easily by writing The quotation marks of “nonenergetic” are the operations as relations = (+(y, a), x). meant to warn of elastic biological language. Every molecule has the intrinsic energy of its fundamental theorem of algebra Every chemical bonds, so strictly speaking no molecule n ≥ polynomial of degree 1 with complex coef- is nonenergetic. But in a biochemical context, ﬁcients has at least one root in the complex num- certain bonds of certain molecules such as ATP C bers . and NADH are broken in many reactions to yield particularly convenient amounts of energy for the fundamental theorem of calculus Let f be reaction or a substituent group for transfer to continuous (hence integrable) on [a,b] and let F another molecule. Compounds used as energy be an antiderivative of f (i.e., F (x) = f(x)), or substituent sources are regenerated by many then b other reactions. The net result is that energy f(x)dx = F(b)− F(a). or substituent groups (or both) are transferred a among molecules by these “energetic” or “cur- fusion (in biotechnology) The amalgama- rency” metabolites. tion of two distinct cells or macromolecules into a single integrated unit.

58 gauge group

Galilei group The closed subgroup of GL(5, R) consisting of Galilei transformations, i.e., of matrices of the following block structure Rva G t 01 001 G-graded algebra An algebra A together where R ∈ SO(3), v, a ∈ R3,r ∈ R. with a map # : g → Ag called the degree map which associates a vector subspace Ag to any element g ∈ G. The subspaces Ag are not neces- sarily subalgebras. However, the product of A gamma function The complex function is compatible with the degree map, in the sense given by that if a ∈ Ag and b ∈ Ah then a · b ∈ Agh. ∞ N(z) = tz−1e−t dt The elements a ∈ Ag are called homogeneous of degree g. 0 for complex z with positive real part. Galerkin method An approach to the dis- cretization of a variational problem Garding˙ inequality The inequality satisﬁed by a coercive sesquilinear form on some Sobolev u ∈ V :< A(u), u >= 0 ∀v ∈ V, space.

V a Banach space, A : V → V continu- Gateaux derivative Let V,W be Banach ous. It relies on two ﬁnite dimensional subspaces spaces, U ⊂ V open. The Gateaux derivative or Vh,Wh ⊂ V to obtain the discrete variational directional derivative of a map F : U ⊂ V → problem W at the point x ∈ U in direction h ∈ V is deﬁned by uh ∈ Vh : < A(uh), vh >= 0 ∀vh ∈ Wh. (4) 1 DF (x, h) = lim [F(x + th) − F(x)]. The space Vh is called the trial space, the space t→0 t W is known as the test space.IfV = W we h h h F x ∈ U confront a Ritz-Galerkin method, the more gen- is called Gateaux differentiable at if DF (x, h) h ∈ V eral case is often referred to as a Petrov-Galerkin exists for all . method. Necessary and sufﬁcient conditions for V,W existence and uniqueness of solutions of (4) are Gateaux-Levi derivative Let be U ⊂ V F U → supplied inf-sup conditions. Banach spaces, open. A map : W is called Gateaux-Levi differentiable at the point x ∈ U if it is Gateaux differentiable at x Galerkin orthogonality Let a be a sesqui- and the map h ∈ V → DF (x, h) from V to W is linear form on a Banach space V and V be a h linear and bounded. If F is Gateaux-Levi differ- subspace of V that may represent a ﬁnite element entiable, we can set DF (x)h = DF (x, h) and space.Forf ∈ V the functions u ∈ V and get uh ∈ Vh are to satisfy F(x + h) = F(x)+ DF (x)h + RF (x, h) a(u, v) = f(v) ∀v ∈ V, where a(uh,vh) = f(vh) ∀vh ∈ Vh. RF (x, th) lim = 0, f or each h ∈ V. The Galerkin orthogonality refers to the straight- t→0 t forward relationship gauge group The group of gauge transfor- a(u − uh,vh) = 0 ∀vh ∈ Vh. mations.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 59 gauge invariant gauge invariant In ﬁeld theory, a Lagran- geodesic In Riemannian geometry a curve γ gian which admits a gauge transformation. whose velocity γ˙ is autoparallel to γ , i.e., satis- ﬁes the geodesic equations ∇γ(t)˙ γ(t)˙ = 0, or, in gauge theory A quantum ﬁeld theory with a local coordinates gauge invariant Lagrangian. i i j k γ¨ + Njkγ˙ γ˙ = 0, gauge transformation In ﬁeld theory a Ni transformation of the ﬁelds which leaves the where jk are the Christoffel symbols. For exam- Rn equations of motion invariant. For example, in ple, in geodesics are straight lines. M electrodynamics if we add to the electromagnetic More generally, given a manifold with a N γ˙ potential A the gradient of a function ∇φ, then connection , a curve whose tangent vector is parallel transported along the curve itself, i.e., the Maxwell equations are invariant under this ∇ γ˙ = gauge transformation A → A +∇φ. γ˙ 0. In terms of principal ﬁber bundles a gauge geodesic equations See geodesic. transformation is an automorphism of the total space that covers the identity of the base space, geodesic motion For the kinetic energy i.e.: If π : P → M is a principal G bundle, Lagrangian, with metric tensor gij then a diffeomorphism φ : P → P is called a gauge transformation if φ(p · g) = φ(p) · g n i i 1 i j for all p ∈ P,g ∈ G and π(f(p)) = π(p). L(q , q˙ ) = gij (q)q˙ q˙ 2 Gauge transformations form an inﬁnite dimen- i,j=1 Lie group sional (under composition), called the the Euler-Lagrange equations are called group of gauge transformations gauge group or . geodesic ﬂow or geodesic motion; they are Elements of its Lie algebra are called inﬁnitesi- equivalent to the geodesic equations. mal gauge transformations. Gibbs energy diagram A diagram showing Gaussian chain The simplest mathematical the relative standard Gibbs energies of reactants, model for a polymer chain molecule. The model transition states, reaction intermediates and treats the molecule as a random walk and neglects products, in the same sequence as they occur in the excluded volume effect. Each step in the walk a chemical reaction. These points are often con- is assumed to be a random variable. In the limit of nected by a smooth curve (a “Gibbs energy pro- a large number of steps, the distribution for the ﬁle,” commonly still referred to as a “free energy end-to-end distance is approximately Gaussian proﬁle”) but experimental observation can pro- distributed. (cf. P.J. Flory, Statistical Mechanics vide information on relative standard Gibbs ener- of Chain Molecules, John Wiley & Sons, New gies only at the maxima and minima and not at York, 1969.) the conﬁgurations between them. The abscissa expresses the sequence of reactants, products, n2 general linear group The dimensional reaction intermediates, and transition states and Rn Rn Lie group of linear isomorphisms form to , is usually undeﬁned or only vaguely deﬁned by GL(n, R) denoted by . The group of linear iso- the reaction coordinate (extent of bond break- Cn Cn morphisms form to is called the (complex) ing or bond making). In some adaptations the GL(n, C) A ∈ general linear group, denoted by . abscissas are, however, explicitly deﬁned as bond GL(n) A n × n if is an invertible real (complex) orders, Bronsted exponents, etc. A = matrix, i.e., det 0. Contrary to statements in many textbooks, the highest point on a Gibbs energy diagram does not ; ⊂ Rn generalized function Let be open necessarily correspond to the transition state of C∞(;) and 0 the Frechet space of inﬁnitely dif- the rate-limiting step. For example, in a stepwise ferentiable functions with compact support in reaction consisting of two reaction steps: ;.Ageneralized function or distribution is A + B ←→ C a continuous linear functional on the space (i.) ; C∞(;) C + D → E 0 . (ii.) .

60 graph

One of the transition states of the two reaction Green’s theorem A special case of Stokes’ steps must (in general) have a higher standard theorem for the plane. Let P,Qbe differentiable Gibbs energy than the other, whatever the con- functions in a region ; ⊂ R2, then centration of D in the system. However, the value ∂Q ∂P of that concentration will determine which of the − = Pdx + Qdy. reaction steps is rate-limiting. If the particular ; ∂x ∂y ∂; concentrations of interest, which may vary, are chosen as the standard rate, then the rate-limiting grammar For any language L, the grammar step is the one of highest Gibbs energy. X is the set of rules specifying the syntax of well- formed constructs in L. Gibbs energy of activation (standard free Comment: A synonym for the rules (and con- energy of activation), ‡G0 The standard fusingly, sentences formed by applying the rules) Gibbs energy difference between the transition is “productions.” Grammars have three func- state of a reaction (either an elementary reaction tions: to generate and to recognize constructs or a stepwise reaction) and the ground state of the in a language, and to transform one language reactants. It is calculated from the experimental to another. The most familar example of a rate constant k via the conventional form of the grammar comes from string grammars built from absolute rate equation: natural languages, which specify the syntactic properties a sentence must fulﬁll for it to be ‡ “legal.” Many grammars, and the languages G = RT [ln(kB /h)− ln(k/T )] they describe, fall into a hierarchy of increasing where kB is the Boltzmann constant and h mathematical complexity ﬁrst devised by Noam the Planck constant (kB /h = 2.08358 × Chomsky. A context-sensitive grammar, one of 1010 K−1s−1). The values of the rate constants, the more complex types, speciﬁes that a token’s and hence Gibbs energies of activation, depend output depends on its context. Examples in upon the choice of concentration units (or of the English are a little contrived: perhaps the best thermodynamic standard state). is “Dick and Jane went north and south, respec- tively.” Here “respectively” signals a mapping function, so that Dick went north and Jane south. Gram-Schmidt orthogonalization A pro- Grammars are commonly applied to recognize cess to construct an orthonormal basis in a features of DNA and protein sequence. In that Hilbert space out of an arbitrary Hilbert basis. context they are usually called string grammars. They are also used to recognize and generate pat- Green’s functions Auxiliary functions used terns of chemical and biochemical structure and to solve nonhomogeneous boundary value prob- function. See graph grammar. lems. Example: The general solution of the boundary value problem graph A graph G(V, E) consists of a set V V = ∅ −y = f(x), y(0) = 0,y(1) = 0 of vertices , and a set of edges e(λ, vi ,vj ) ∈ E, E ≥∅, where λ ∈ L, L = ∅ can be written in the form is the type of relation the edge expresses, and {vi ,vj }∈V,i = j are the (possibly empty) ver- 1 y = φ(x) = G(x, s)f (s)ds tices associated with that edge. 0 Comment: This deﬁnition has edges express- ing relationships but allows them to be G(x, s) where is the Green’s function deﬁned by unbounded by vertices on either or both sides (“free” edges). This latter feature is particularly s( − x), ≤ s ≤ x, G(x, s) = 1 0 useful in specifying certain types of graphs and x( − s), x ≤ s ≤ . 1 1 operators upon them. All the graphs considered here are ﬁnite; have one and only one edge join- ing any pair of nodes (are not multigraphs); and

61 graph grammar have no edges with only one node (loops; not a Comment: Graph theory is a rich source for pseudograph). Derived from G. Rozenberg (per- pattern recognition algorithms. sonal communication). Grassman algebra See exterior algebra. graph grammar A grammar over a set G of graphs G(V, E). ground In logic and logic programming, a Comment: Like a natural language or string term which is not, or does not include, a variable. grammar, a graph grammar provides rules to Comment: The restriction is meant to avoid generate, recognize, and transform graphs; for wiring confusion. example, expanding a token for a substituent group into the atoms and bonds of that group, group (1) A deﬁned linked collection of or contracting the group to an abbreviation. Mir- atoms or a single atom within a molecular entity. roring the chemistry, one could form bonds and The use of the term in physical organic and gen- specify orientation by examining an atom’s con- eral chemistry is less restrictive than the deﬁni- text. This suggests the grammar is context- tion adopted for the purpose of nomenclature of sensitive. However, to date the only formal organic compounds. proofs for molecular languages are that linear (2) A set G with a binary operation which is polymers such as DNA are greater than context- associative. Each element is assumed to have an free (noncontext-free), a superset which includes inverse and G contains an identity element. the context-sensitive grammars among its mem- bers. So while it is reasonable to conjecture that group homomorphism A map φ : G → H molecules form a context-sensitive language, between two groups such that there is no formal proof yet. φ(e ) = e φ(g · g ) = φ(g ) · φ(g ) G H 1 2 1 2 graph theory The branch of mathematics dealing with the topological relationships among where eG and eH denote the identity elements of abstract graphs. the groups G and H , respectively.

62 Hammett equation

Hamilton principle The principle accord- ing to which the action functional determines conﬁgurations of motion (also called critical con- ﬁgurations). Critical conﬁgurations are those H for which the action functional is stationary with respect to all compactly supported deformations. For this reason, the Hamilton principle is also h-version of ﬁnite elements A strategy that called the principle of stationary action. seeks to achieve a sufﬁciently small discretiza- Lagrangian systems tion error of a ﬁnite element scheme for a bound- In the Hamilton principle ary value problem by using ﬁne meshes. This can is equivalent to the Euler-Lagrange equations, mean a small global meshwidth or local reﬁne- which, depending on the number of independent n = n> ment. The latter is employed in the context of variables ( 1or 1), are called equations adaptive reﬁnement. of motion or ﬁeld equations, respectively. See also Lagrangian system and action func- t half-life The time 1/2 required for one half tional. of a population to change randomly from some ∫ ∫ state to another state . At least one of these Hamiltonian system Let (M, ω) be a sym- two states must be observable. plectic manifold and H : M → R a smooth Comment: The classic example of half-life is function. The corresponding Hamiltonian vec- radioactivity. In any sample of matter, some of tor ﬁeld XH is determined by the condition the nuclei will be radioactive isotopes and the ω(XH ,Y) = dH · Y. The ﬂow of XH in canon- remainder not. The decline in the radioactivity ical coordinates satisﬁes Hamilton’s equations of the sample is governed by a ﬁrst-order Poisson and is called a Hamiltonian system. process N = N e−λt More generally, an evolution equation is 0 called a Hamiltonian system if it can be written in 1 N N ˙ where and 0 are the current and original the form F ={F,H} with respect to some Pois- number of radioactive atoms in the sample and son bracket { , }. H is called the Hamiltonian of λ the characteristic decay constant for an iso- the system. t = /λ tope. Thus 1/2 ln 2 . Judged by the metric of radioactivity, nonradioactivity is a nonobserv- Hamiltonian vector ﬁeld Let (P, { , }) be able state (though it can of course be observed ∞ Poisson manifold and H ∈ C (P ). The Hamil- by other assays). tonian vector ﬁeld XH of H is deﬁned by Hamilton equations Let (q1,...,qn,p , 1 X (G) ={H,G}, G ∈ C∞(P ). ...,pn) be canonical coordinates and H a H for all smooth function. Hamilton’s equations for H are Hammett equation (Hammett relation) The ∂H ∂H equation in the form: q˙i = , p˙ =− ,i= 1,...,n. ∂p˙ i ∂q˙i i (k/k ) = ρσ lg 0 Hamilton-Jacobi equation Let U ⊂ Rn be open and u : U ×R → R. The Hamilton-Jacobi or u (K/K ) = ρσ equation for is lg 0

ut + H(Du,x) = 0, applied to the inﬂuence of meta-orpara- substituents X on the reactivity of the func- where Du = Dx u = (ux ,...,ux ) denotes the 1 n Y m gradient of u with respect to the spatial variable tional group in the benzene derivative -or x = (x ,...,x ) p-XC H Y . k or K is the rate or equilibrium 1 n . 6 4 constant, respectively, for the given reaction of 1This notation is restricted to this discussion and char- m p XC H 4Y k K -or - 6 ; 0 or 0 refers to the reac- acteristic of that seen in other texts on this subject; do not C H Y X = H σ confuse it with notation on reactions elsewhere, especially tion of 6 5 , i.e., ; is the substituent the use of N and λ. constant characteristic of m-orp-X: ρ is the

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 63 harmonic form reaction constant characteristic of the given reac- Heisenberg uncertainty principle In quan- tion of Y . The equation is often encountered in tum mechanics, a principle set forth by Heisen- k K a form with lg 0 or lg 0 written as a separate berg which asserts that the simultaneous exact term on the right-hand side, e.g., measurements of the values of position and momentum is impossible. If q is the range lg k = ρσ + lg k 0 of values found for the coordinate q of a particle, or and p is the range in the simultaneous meas- K = ρσ + K lg lg 0 urement of the corresponding momentum, then q · p ≥ h h It then signiﬁes the intercept corresponding to , where is the Planck constant. X = H in a regression of lg k or lg K on σ. helicity A quantitative measure of the See also Yukawa-Tsuno equation. amount of helix in a polymer molecule with heli- harmonic form A k-form α on a manifold cal structure. M is called harmonic if α = 0, where = dδ + δd is the Laplace-deRham operator on M. helix In biochemistry, a molecular structure having a given skew symmetry. For proteins, harmonic frequency generation Produc- there is α-helix which was ﬁrst proposed by tion of coherent radiation of frequency kν(k = L. Pauling. For DNAs, there is a double helix. 2, 3,...) from coherent radiation of frequency ν. In general, this effect is obtained through the helix-coil transition A mathematical model interaction of laser light with a suitable optical originally develped for the probability distribu- medium with nonlinear polarizability. The case tion of α-helix formation in a polypeptide. The k = 2 is referred to as frequency doubling, k = 3 model is formulated based on the transfer matrix is frequency tripling, and k = 4 is frequency method. It is a generalization of one-dimensional quadrupling. Even higher integer values of k are Ising model (cf. D. Poland and H.A. Scher- possible. aga, Theory of Helix-Coil Transitions, Academic Press, New York, 1970). harmonic function A function u satisfying u = 0. See Laplace equation. Helmholtz equation Let U ⊂ Rn be open u U ⊂ Rn → R harmonic oscillator The function H = and : . The Helmholtz equation 1 (−d2/dx2 + x2) or eigenvalue equation for u is 2 is the Hamiltonian of the har- m = monic oscillator (assuming 1) . The integral −u = λu. curves (trajectories) are circles. heat, q,Q Energy transferred from a hotter Hermitian operator See self-adjoint oper- to a cooler body due to a temperature gradient. ator. heat equation Let U ⊂ Rn open and u : heterolysis (heterolytic) The cleavage of a U × R → R. The heat equation or diffusion covalent bond so that both bonding electrons equation for u is remain with one of the two fragments between which the bond is broken. ut − u = 0. heat kernel On Rn × Rn the function Higgs mechanism In quantum ﬁeld theory, the spontaneous breaking of gauge symmetry, in which a massless gauge boson (Goldstone boson) |x − y|2 et(x, y) = (4πt)−n/2exp − and a massless scalar ﬁeld combine to form a 4t massive gauge boson, called a Higgs boson. The action of the heat kernel on a function f is deﬁned by Hilbert-Schmidt operator A bounded lin- ear operator T on a Hilbert space H is called ∗ (etf )(x) = et(x, y)f (y)dy. Hilbert-Schmidt if traceT T<∞, that is, n ∗ R λj < ∞ for the eigenvalues of T T .

64 hyperbola

Hilbert space A vector space H which has (i.) F(0,x)= ϕ(x); <, > an inner product (scalar product) and (ii.) F(1,x)= φ(x). is complete with respect to the induced norm x2 =

homeomorphism (between two topological horizontal lift (induced by a connection b) spaces X and Y ) A map ϕ : X → Y which The unique vector N(X) ∈ b projecting onto is continuous with a continuous inverse map X. Local generators of the space b are of the − ∂ − Ni (x, y)∂ X = Xµ∂ ϕ 1 : Y → X. form µ µ i .If µ is a vec- tor ﬁeld over M, then the horizontal lift of X is locally given by homolysis (homolytic) The cleavage of a bond (“homolytic cleavage” or “homolytic ﬁs- µ i N(X) = X (∂µ − Nµ(x, y))∂i ∈ b sion”) so that each of the molecular fragments between which the bond is broken retains one of the bonding electrons. A unimolecular reac- hydrocarbons Compounds consisting of tion involving homolysis of a bond (not forming carbon and hydrogen only. part of a cyclic structure) in a molecular entity containing an even number of (paired) electrons H + results in the formation of two radicals. hydron The general name for the cation ; H − H It is the reverse of colligation. Homolysis is the species is the hydride anion and is also commonly a feature of bimolecular substitu- the hydro group. These are general names to tion reactions (and of other reactions) involving be used without regard to the nuclear mass of radicals and molecules. the hydrogen entity, either for hydrogen in its natural abundance or where it is not desired to distinguish between the isotopes. homotopy Let X and Y be topological spaces and ϕ : X → Y and φ : X → Y be two continuous maps from X to Y . They hyperbola The conic section with equation are homotopic if there exists a continuous map x2/a2 − y2/b2 = 1. See asymptote to the hyper- F :[0, 1] × X → Y such that bola.

65 hyperbolic critical point hyperbolic critical point A critical point hyperbolic equation A second-order par- m X X(m ) = 0 of a vector ﬁeld (i.e., 0 0) is called tial differential equation of the form hyperbolic or elementary if none of the eigenval- X(m ) Au + Bu + Cu + Du + Eu + Fu = G ues of the linearization 0 (called character- xx xy yy x y istic exponents) has zero real part. Liapunov’s theorem shows that near a hyperbolic critical such that B2 − 4AC > 0. It is called parabolic point the ﬂow looks like that of its linearization. if B2 − 4AC = 0 and elliptic B2 − 4AC < 0.

66 inf-sup condition

implicit function theorem Let V,W,Z be U ⊂ V,U ⊂ W Banach spaces, 1 2 open and f U × U → Z x ∈ : 1 2 differentiable. For some 0 U ,y ∈ U D f(x ,y ) W → Z 1 0 2 assume 2 0 0 : is I an isomorphism. Then there are neighborhoods U x Z f(x ,y ) 0 of 0 and 0 of 0 0 and a unique differ- g U × Z → U entiable map : 0 0 2 such that for all (x, z) ∈ U × Z ideal (of an algebra A)Aleft ideal is a vector 0 0 subspace I ⊂ A such that ∀a ∈ A, i ∈ I, a· i ∈ f(x,g(x,z)) = z. I. Analogously, a right ideal is a vector subspace I ⊂ A such that ∀a ∈ A, i ∈ I, i · a ∈ I.A incidence relation See edge. bilateral ideal is left ideal which is also a right ideal. incident A node vi is incident to an edge The quotient of an algebra by a bilateral ideal (vi ,vj ), since it is an end point of the edge. is an algebra. In the category of algebras with unity, proper ideals are not subalgebras. In fact, index See Fredholm operator, Atiyah- ∈ I I = A if the unit 1 belongs to the ideal, then . Singer index theorem. On the contrary, ideals of Lie algebras are always subalgebras. inf-sup condition A sesqui-linear form a : V × V → C on a Hilbert space V satisﬁes an identity (1) An element e of a set X with a inf-sup condition if, for some α>0, binary operation * satisfying |a(u, v)| supv∈V ≥ αuV ∀u ∈ V, a ∗ e = e ∗ a = a vV |a(u, v)| for all a ∈ X. > ∀u ∈ V. supv∈V v 0 (2) A true mathematical equation. V This is necessary and sufﬁcient for the variational image Let r be a relation with domain A and problem codomain B, and let a ∈ A, r(a) ∈ B. Then the u ∈ V a(u, v) = f(v) ∀v ∈ V image of a under r, r(a) ∈ B, is produced by : r a applying to . to possess a unique solution for any f ∈ V . The Comment: See the comment on relation for −1 solution satisﬁes uV ≤ α f V . more details. See also codomain, domain, range, For a linear symmetric variational saddle and relation. point problem with sesqui-linear forms a : V × V → C and b : V × W → C the suitable inf- immersional wetting A process in which a sup conditions claim the existence of constants solid or liquid, β, is covered with a liquid, α, α, β > 0 such that both of which were initially in contact with a gas 2 or liquid, δ, without changing the area of the αδ- R{a(u, u)}≥αuV ∀u ∈ Ker(B), interface. |b(v, p)| ≥ βp ∀p ∈ V supv∈V v V immunoglobulin (Ig) A protein of the V globulin-type found in serum or other body ﬂuids where that possesses antibody activity. An individual Ig Ker(B) := {v ∈ V : b(v, q) = 0 ∀q ∈ W} . molecule is built up from two light (L) and two heavy (H) polypeptide chains linked together by Then the variational saddle point problem, which disulﬁde bonds. Igs are divided into ﬁve classes seeks u ∈ V,p ∈ W such that based on antigenic and structural differences in the H chains. a(u, v) + b(v, p) = f(v) ∀v ∈ V

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 67 inﬁnite set

b(u, q) = g(q) ∀q ∈ W, injection A map φ : A → B which is injective, so that whenever φ(a ) = φ(a ) then has a unique solution. It satisﬁes 1 2 a = a . 1 2 1 a 1 u ≤ f + + g , A B A V α V α 1 β W injective Let and be two sets, with the domain and B the codomain of a function f . Then the function f is injective if, for any pW ≤ x and x ∈ A, x = x, their images f(x) = 1 a a a f(x). Injective functions are also called one-to- + 1 f V + + 1 gW . β α α β2 one transformations of A into B. See also into, Similar results are known for nonsymmetric sad- onto, bijection, and surjection. dle point problems. inner product An inner product or scalar product complex vector space V inﬁnite set A set whose cardinality is not on a is a map <, >: V × V → C such that, for all x,y,z ∈ ﬁnite. See also cardinality, countable set, denu- V,α ∈ C merably inﬁnite set, ﬁnite set, and uncountably inﬁnite set. (i.)

68 inverse function theorem integral (sub)manifold (of a distribution of sub- into In mathematics, sometimes the word is spaces of rank n) A submanifold N ⊂ M used to identify a one-to-one mapping, some- such that Tx N ⊂ x for all x ∈ N. Usually N times simply a mapping of a set X which trans- is required to be of dimension n = rank() forms points of X into points of another set Y , e.g., y = x2 is a mapping of the real numbers integrable system A Hamiltonian system into the real numbers, or onto the nonnegative XH ona2n-dimensional symplectic manifold real numbers. See also onto, bijection, injection, (Mω) such that there exist n linearly indepen- and surjection. H = F ,...,F dent functions 1 n in involution, i.e., {Fi ,Fj }=0,i,j= 1,...,nand dFi (m) intramolecular (1) Descriptive of any pro- are linearly independent at each point m ∈ M. cess that involves a transfer (of atoms, groups, electrons, etc.) or interactions between different X integral curve Let be a vector ﬁeld on a parts of the same molecular entity. M X manifold .Anintegral curve of with initial (2) Relating to a comparison between atoms x t = c condition 0 at 0, is a smooth curve : or groups within the same molecular entity. a,b → M c( ) = x [ ] such that 0 0 and intramolecular isotope effect A kinetic iso- c (t) = X(c(t)), for all t ∈ [a,b]. tope effect observed when a single substrate, See also dynamical system. in which the isotopic atoms occupy equivalent reactive positions, reacts to produce a nonstatis- intension In this context only, the use of an tical distribution of isotopomeric products. In implicit representation of a datum in a database such a case the isotope effect will favor the path- or model, which is not fully instantiated within way with lower force constants for displacement the database and whose value must be computed of the isotopic nuclei in the transition state. according to the representation. Comment: The restriction is meant to avoid intrinsic rate constant See rate constant. philosophical wrangling and conﬁne the dis- course to that of databases. A common inten- intrinsic viscosity (of a polymer) The lim- η /c sional representation is a declarative rule, but the iting value of the reduced viscosity, i , or the η output of a statistical algorithm, federated to a inherent viscosity, inh, at inﬁinite dilution of the database and performed on data derived from it polymer, i.e., or other computations, is another example. See [η] = lim(ηi /c) = lim ηinh also extension. c→0 c→0 interior (of a set) For a subset A of a topo- Notes: (1) This term is also known in the lit- logical space, the set of points p ∈ A such that erature as the Staudinger index. 3 −1 A also contains a neighborhood of p. (2) The unit must be speciﬁed; cm g is rec- ommended. intermediate A molecular entity with a life- (3) This quantity is neither a viscosity nor a time appreciably longer than a molecular vibra- pure number. The term is to be looked on as a tion (corresponding to a local potential energy traditional name. Any replacement by consistent minimum of depth greater than RT ) that is terminology would produce unnecessary confu- formed (directly or indirectly) from the reactants sion in the polymer literature. Synonymous with and reacts further to give (either directly or indi- limiting viscosity number. rectly) the products of a chemical reaction; also the corresponding chemical species. inverse function theorem Let V,W be Banach spaces, U ⊂ V open and f : U ⊂ U V → W Df (x ) internal energy, Quantity of change differentiable. If 0 is a lin- q Df (x ) V → W f which is equal to the sum of heat, , brought ear isomorphism 0 : , then to the system and work, w, performed on it, is a local diffeomorphism, i.e., a diffeomorphism U = q + w x . Also called thermodynamic from a neighborhood of 0 onto a neighborhood f(x ) energy. of 0 .

69 inverse inequality inverse inequality Inequalities bounding ion pair A pair of oppositely charged ions stronger norms (in the sense that they contain held together by Coulomb attraction without for- higher derivatives) of functions in ﬁnite dimen- mation of a covalent bond. Experimentally, an sional spaces by weaker norms. An impor- ion pair behaves as one unit in determining con- tant example are bounds for the H 1(;)-norm of ductivity, kinetic behavior, osmotic properties, ﬁnite element functions in terms of their L2(;)- etc. norm. The constants in these inequalities are Following Bjerrum, oppositely charged ions invariably for families of approximating spaces with their centers closer together than a distance that are asymptotically dense. For instance, for q = . × 6z+z−/(" T) the above pair of norms and h-version families 8 36 10 r pm H 1(;) ﬁnite element spaces of -conforming on are considered to constitute an ion pair (“Bjerrum quasi-uniform and shape regular meshes the con- ion pair”). [z+ and z− are the charge numbers of stant will grow like O(h−1) where h is the mesh the ions, and "r is the relative permittivity (or width. dielectric constant) of the medium.] An ion pair, the constituent ions of which are inverse relation If R ⊆ A × B is a rela- in direct contact (and not separated by an inter- tion from A to B, then the set R−1 ={(b, a) | vening solvent or other neutral molecule) is des- (a, b) ∈ R} is a subset of B × A. R−1 is called ignated as a “tight ion pair” (or “intimate” or X+ the inverse of R. “contact ion pair”). A tight ion pair of and Y − is symbolically represented as X+Y −. By contrast, an ion pair whose constituent invertible A map φ : A → B is invertible if ions are separated by one or several solvents or − there exists a map φ 1 : B → A such that other neutral molecules is described as a “loose ion pair,” symbolically represented as X+||Y −. −1 −1 φ ◦ φ = idA φ ◦ φ = idB The members of a loose ion pair can readily inter- change with other free or loosely paired ions in The map φ−1 is called the inverse of φ. the solution. This interchange may be detectable (e.g., by isotopic labeling) and thus affords an experimental distinction between tight and loose involution of functions Two functions f ion pairs. and g on a symplectic manifold are in involu- A further conceptual distinction has some- tion Poisson bracket when their vanishes, i.e., times been made between two types of loose {f, g}= 0. ion pairs. In “solvent-shared ion pairs” the ionic constituents of the pair are separated by involutive distribution A distribution of only a single solvent molecule, whereas in subspaces on M with generators XA such that “solvent-separated ion pairs” more than one sol- [XA,XB ] ∈ where [ , ]isthecommutator.A vent’s molecule intervenes. However, the term distribution is involutive if and only if it is inte- “solvent-separated ion pair” must be used and grable (Frobenius theorem). interpreted with care since it has also widely been used as a less speciﬁc term for “loose” ion pair. ion channel In biochemistry, a protein, ionizing radiation Any radiation consisting embedded in cell membrane, which conducts of directly or indirectly ionizing particles or a ionic current of speciﬁc ions. An ion channel mixture of both, or photons with energy higher can be either passive or modulated by the electri- than the energy of photons of ultraviolet light or cal voltage across the cell membrane. The latter a mixture of both such particles and photons. is often called voltage-gated ion channel. (cf. B. Hille, Ionic Channels of Excitable Membranes, irradiance, E Radiant power received by 2nd ed., Sinauer Associates, Sunderland, MA, a surface divided by the area of that surface. 1992). For collimated beams this quantity is sometimes

70 isotropy group called intensity and given the symbol I. See also isotopically substituted Describing a com- photon irradiance. pound that has a composition such that essen- tially all the molecules of the compound have irradiation Exposure to ionizing radiation. only the indicated nuclide(s) at each designated position. For all other positions, the absence of isomer One of several species (or molecular nuclide indication means that the nuclide com- entities) that have the same atomic composition position is the natural one. (molecular formula) but different line formulae or different stereochemical formulae and hence isotopically unmodiﬁed Describing a com- different physical and/or chemical properties. pound that has a macroscopic composition such that its constituent nuclides are present in the pro- isomorphic Two sets are isomorphic if there portions occurring in nature. is an isomorphism between them. Two graphs, G and G, are isomorphic if their nodes can be isotopologue A molecular entity that differs , ,...,p labeled with the numbers 1 2 such that only in isotopic composition (number of isotopic v v v˙ v˙ whenever i is adjacent to j in G, i and j are substitutions), e.g., CH CH D, CH D . adjacent in G,1≤ i, j ≤ p, i = j. 4 3 2 2 isotopomer Isomers having the same num- isomorphism (1)Abijection between two ber of each isotopic atom but differing in their sets which preserves all structure shared by the positions. The term is a contraction of “iso- sets (e.g., group structure). topic isomer.” Isotopomers can be either con- (2)A morphism between objects of a category CH DCH = stitutional isomers (e.g., 2 0 and which is both surjective and injective. CH CD = 3 0) or isotopic stereoisomers (e.g., (R)- and (S) − CH CHDOH or (Z)- and isotope effect The effect on the rate or equi- 3 (E) − CH CH = CHD). librium constant of two reactions that differ only 3 in the isotopic composition of one or more of their otherwise chemically identical components isotropic (sub)manifold (in a symplectic mani- P,ω M ⊂ P is referred to as a kinetic isotope effect or a fold [ ]) A submanifold of a (P, ω) thermodynamic (or equilibrium) isotope effect, symplectic manifold such that at any point p ∈ M respectively. the tangent space contains its symplectic polar, i.e., (T M)o ⊂ T M. isotopically labeled Describing a mixture p p of an isotopically unmodiﬁed compound with one The dimension of an isotropic manifold M is at or more analogous isotopically substituted com- most half of the dimension of P . pound(s). isotropy group Let ) : G × M → M be a isotopically modiﬁed Describing a com- smooth action of the Lie group G on the manifold pound that has a macroscopic composition such M. For each x ∈ M the group that the isotopic ratio of nuclides for at least one element deviates measurably from that occurring Gx ={g ∈ G | )(g, x) = x} in nature. It is either an isotopically substituted compound or an isotopically labeled compound. is called the isotropy group of ) at x.

71

junction point

meant to be symmetric with respect to the lower µ ...µ

J k µ i i i j σ(x) = (x ,y (x), ∂µy (x),...,∂µ ...µ y (x)) 1 k jet bundle Let (B,M,π; F)be a bundle and which is called the k-jet prolongation of σ . N (π) x be the set of local sections deﬁned around Any bundle morphism ) : B → B (pro- x ∈ M x J kB . The jet space at is the space x of jecting over a diffeomorphism φ : M → M ) j kσ N (π) k k k equivalence classes, denoted by x ,in x , induces a bundle morphism J ) : J B → J B k x of sections having contact at (i.e., two local deﬁned by sections are equivalent if they have the same k- order Taylor polynomial). The union of all such k k k −1 j )(jx σ) = jf(x)() ◦ σ ◦ f ) jet spaces k k J B = Jx B which is called the k-jet prolongation of ). x∈M Analogously, any projectable vector ﬁeld X is a bundle over M (as well as over B and over all over B induces a projectable vector ﬁeld j kX J hB for h ≤ k), and it is called the k-jet prolon- over J kB which is called the k-jet prolongation gation of B.If(xµ; yi ) are ﬁbered coordinates of X. µ i i i of B, then (x ,y ,yµ,...,yµ ...µ ) are ﬁbered 1 k k i coordinates of JB . The coordinates yµ ...µ are junction point See branch point. 1 h

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 73

kinetic equivalence

kinetic energy, Ek Energy of motion. For 2 a body of mass m, Ek = mv /2, where v is the speed. K kinetic equation A balanced reaction equa- tion which describes an elementary step in the kinetic sequence of an overall biochemical Karle-Hauptman method In x-ray crystal- reaction among unambiguously identiﬁed reac- lography, the diffraction pattern is essentially tants. There is no constraint on the number of the Fourier transform of the periodic crystal participating species. Equally, a set of such equa- structure which has both an amplitude and a tions which can be combined by any allowed phase. In practice, the phase information cannot operation or combination of operations to pro- be obtained; hence, one has to invert the Fourier duce kinetics identical to those measured for the transform without the phase and, hence, the algo- overall biochemical reaction. rithm is not unique. J. Karle and H. Haupt- Comment: The requirement that the equa- man developed a system of inequalities which the tion be elementary ensures that its order will be amplitudes have to satisfy. Using these inequal- the sum of the molecularities of the reactants ities as constraints, one can uniquely determine (otherwise, it will be the reactants’ activities). the inverse Fourier transform if the number of Kinetic sequences here considered correspond atoms in a unit cell is sufﬁciently small. The to the common word descriptors (sequential, inequalities are based on the fact that the elec- ping-pong, etc.). The requirement for identiﬁed tron density is a positive function of which the species distinguishes among different proteins diffraction is the Fourier transform (cf. J. Karle, which catalyze the “same” reactions but at differ- J. Chem. Inf. Comput. Sci., 34, 381, 1994). ent rates. Since most kinetic sequences consist of more than one elementary step, the equations killing ﬁeld On a Riemannian manifold will usually occur in groups. Since not all reac- (M, g) X a vector ﬁeld such that the Lie deriva- tions are sequential, the deﬁnition provides that L g = tive X 0. they can be combined by operators other than summation, such as Boolean operators or coefﬁ- killing vector Over a pseudo-Riemannian µ cients representing the frequency of a particular manifold (M, g),avector ﬁeld ξ = ξ (x) ∂ ∈ µ type of event in a population of simultaneously X(M) such that the Lie derivative $ξ g vanishes. α occurring events. If { βµ}g are the Christoffel symbols of the metric g, ξ is a killing vector if and only if it satisﬁes kinetic equivalence Referring to two reac- the killing equation: tion schemes which imply the same rate law.For

∇µξν +∇ν ξµ = 0 example, consider the two schemes (i.) and (ii.) for the formation of C from A: λ λ ξ = g ξ ∇ ξ = ∂ ξ −{ } ξ − k where ν λν and µ ν µ ν νµ g λ k ,OH 2 1 −→ (i.) A ←− B −→ C Accordingly, killing vectors are inﬁnitesimal k ,OH − −1 generators of isometries of g. A killing vec- providing that B does not accumulate as a reac- tor is uniquely determined once one speciﬁes its tion intermediate. ξ µ(x ) x ∈ M value 0 at a point 0 together with − µ d[C] k k [A][OH ] its derivatives ∂ν ξ (x ). (In fact, by deriving of = 1 2 0 t k + k OH− (1) the killing equation one obtains a Cauchy prob- d 2 −1[ ] lem which uniquely determines the component k µ k −→2 ξ M m −→1 functions .) On a manifold of dimension (ii.) A ←− B OH− C k m(m + )/ −1 one can have at most 1 2 killing vec- Providing that B does not accumulate as a reac- m(m + )/ tors. The manifolds with exactly 1 2 tion intermediate: killing vectors are called maximally symmetric; d C k k A OH− e.g., the plane, the sphere, and the hypersphere [ ] = 1 2[ ][ ] dt k + k OH− (2) are maximally symmetric. −1 2[ ]

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 75 kinetic motif

Both equations for d[C]/dt are of the form Comment:As deﬁned here, the biochemical reac- tion corresponds to that for an elementary chem- d C r A OH− [ ] [ ][ ] ical reaction. The nice correspondence among = − (3) dt 1 + s[OH ] molecularity, stoichiometry, and order is a direct consequence of the restriction to elementary where r and s are constants (sometimes called reactions. In practice, the (partial) order is sim- “coefﬁcients in the rate equation”). The equa- ply that exponent for a molecular species in a tions are identical in their dependence on con- kinetic equation that fully accounts for the pro- centrations and do not distinguish whether OH− duction or consumption of that species, the over- catalyzes the formation of B, and necessarily also all order being the sum of the partial orders for its reversion to A, or is involved in its further all species. Thus until a reaction is established to transformation to C. The two schemes are there- be elementary, one cannot assume that the expo- fore kinetically equivalent under conditions to nents will be the stoichiometries. If the concen- which the stated provisos apply. tration of a reactant is so large compared to the others as to be “effectively inﬁnite,” then the reac- kinetic motif A kinetic sequence and asso- tion becomes zeroth order for that reactant. ciated constants, represented as a set of balanced reactions and their parameters, conserved over kinetic proofreading A mathematical different molecules, reactions, or combinations model based on the kinetics of transcription thereof. (from DNA to RNA) or translation (from RNA Comment: One of several different appli- to protein) which explains how an extremely cations of graph theory to biochemistry is to high accuracy is achieved in these biological represent such kinetic motifs. Notice there is processes in the presence of thermal noise. The no constraint on the number of species in the key idea is that these biological processes utilize reaction equation. The deﬁnition may later be free energy in order to obtain high ﬁdelity (cf. expanded to include noninitial studies. The J.J. Hopﬁeld, Proc. Natl. Acad. Sci. U.S.A., 71, words used to characterize qualitatively enzyme 4135, 1974). kinetics (ping-pong, ordered) are useful but incomplete. In practice, recognition of the values Kolmogorov equation Let U ⊂ Rn open of the constants will clearly be within tolerances and u : U × R → R. The Kolmogorov equation to minimize experimental variation. See also for u is chemical, dynamical, functional, mechanistic, n n phylogenetic, regulatory, thermodynamic, and ij i ut − a ux x − b ux = 0. topological motives. i j i i,j=1 i=1 kinetic order The kinetic order of a bio- Korteweg-deVries (KdV) equation Let chemical reaction is the sum of the molecular- U ⊂ Rn open and u : U × R → R. The ities of the reaction. A reaction is said to be of Korteweg-deVries (KdV) equation is the shallow nth order in a reactant (partial order) if the stoi- water wave equation chiometry of that reactant in the reaction equation is n. ut + uux + uxxx = 0 .

76 Lagrangian

First, consider the unit simplex K in Rn. The local space VK agrees with the space of multi- variate polynomials of total degree ≤ k,k ∈ N,

VK L α α = → a x 1 ···x n ,a ∈ C . : x α 1 a α α∈Nn,|α|≤k label A unique identiﬁer for a node or edge 0 of a graph, network, or a subnetwork (subgraph). The local degrees of freedom are based on point Comment: Thus, compound names are labels evaluations for those nodes, and an edge is labeled by giving the tuple of nodes which it joins, so (vi ,vj ). X k = φ C(K) → C, φ(u) = u( ), ∈ P , laboratory sample The sample or subsam- : p : p p ple(s) sent to or received by the laboratory. When the laboratory sample is further prepared where (reduced) by subdividing, mixing, grinding, or by combinations of these operations, the result P is the test sample. When no preparation of the n = 1(l , ··· ,l )T ,l ∈ N , l ≤ k ⊂ K. laboratory sample is required, the laboratory : k 1 n i 0 i sample is the test sample. A test portion is i=1 removed from the test sample for the perfor- Obviously, we have mance of the test or for analysis. The laboratory sample is the ﬁnal sample from the point of view n + k V = . of sample collection but it is the initial sample dim K k from the point of view of the laboratory. Several laboratory samples may be prepared and sent to On the unit hyper-cube K ⊂ RN , the geomet- different laboratories or to the same laboratory ric reference element for quadrilateral and hexa- for different purposes. When sent to the same hedral meshes, the local spaces VK are given by laboratory, the set is generally considered as a polynomials with degree ≤ k in each indepen- single laboratory sample and is documented as a dent variable single sample. V K Ladyshenskaja-Babuˇska-Brezzi (LBB) condi- α α 1 n := x → α∈Nn,α ≤k aαx ···xa ,aα ∈ C . tion A sufﬁcient condition for the stability 0 i 1 of ﬁnite element schemes for saddle point prob- lems. It amounts to requiring uniform inf-sup Local degrees of freedom are given by point conditions for the pair of ﬁnite element spaces evaluation in the points used in a Galerkin discretization of the saddle point problem. P = 1(l , ··· ,l )T ,l ∈ N ,l ≤ k ⊂ K, : k 1 n i 0 i Lagrange ﬁnite elements A parametric equivalent family of ﬁnite elements giving rise to which means H 1(;)-conforming ﬁnite element spaces. Para- n metric equivalence is based on the following pull- dimVK = (k + 1) . back mapping for functions Lagrangian The Lagrangian formulation F)(u)(x) := u()(x)) x ∈ K. of mechanics is based on the variational principle Thus, it is sufﬁcient to specify the ﬁnite elements of Hamilton. To describe a mechanical system for reference elements. Lagrangian ﬁnite ele- one chooses a conﬁguration space Q with coor- ments can be distinguished by their polynomial dinates qi ,i = 1,...,n. Then one introduces degree k ∈ N. the Lagrangian function L = K − V , where K

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 77 Lagrangian (sub) manifold is the kinetic and V the potential energy. The called the conﬁguration bundle, and L is a hori- variational principle states zontal m-form over the k-jet prolongation J kB [m being the dimension of the base manifold b i i m = (M) dynamical system δ L(q , q˙ , t)dt = 0 dim ]. The is induced a by the action functional on the sections σ of the conﬁguration bundle: which leads to the Euler-Lagrange equations of motion k ∗ AD(σ ) = (j σ) L d ∂L ∂L D − = ,i= ,...,n. i i 0 1 dt ∂q˙ ∂q where (j kσ)∗L denotes evaluation of the Lagrangian L along the (jet prolongation) of the See also Lagrangian system. conﬁguration σ . See also action functional.

Lagrangian (sub)manifold (in a symplectic lamp A source of incoherent radiation. manifold [P,ω]) A submanifold M ⊂ P of a symplectic manifold (P, ω) which is both language A language L = (K, N ,c,X) isotropic and coisotropic, i.e., such that can be described either as a tuple of the lan- guage’s alphabet (K), the set of nonterminals o TpM = (TpM) . (N ), the symbol denoting a construct in the lan- guage (c), and the set of grammar rules (X); or The dimension of a Lagrangian manifold M is equally, the nonempty countable set of allowed exactly half of the dimension of P . constructs C, C ∈ K∗, where K∗ is the set of all possible constructs formed over the alphabet K. Lagrangian symmetry A transformation Comment: Programs (for example, databases leaving a Lagrangian system invariant. and queries upon them) are necessarily expressed For example, let (M, L) be a in some arbitrary language. time-independent Lagrangian system;a transformation ) : M → M, locally given Laplace equation Let U ⊂ Rn open and by )(x) = x , can be lifted to the tangent u : U → R. The Laplace equation for u is space T) : TM → TM, locally given by n T )(x, v) = (x ,v). ItisaLagrangian u = ux x = 0. symmetry if the following identity is satisﬁed i i i=1 L(x,v) = L(x, v). Lax-Milgram lemma This lemma asserts the existence and uniqueness of solutions of vari- In ﬁeld theory, a bundle morphism ) : B → k ational problems with elliptic sesquilinear forms. B, which lifts to the jet prolongation j ) : k k It is a particular case of more general results for J B → J B, such that sesquilinear forms satisfying inf-sup conditions. (j k))∗L = L. left action (of a group on a space X ) A map λ G × X → X Lagrangian system A dynamical system : such that: which is deﬁned by the action functional induced (i.) λ(e, x) = x; by a Lagrangian. (ii.) λ(g · g ,x)= λ(g , λ(g ,x)); A time-independent Lagrangian system is a 1 2 1 2 pair (Q, L) where Q is a manifold, called the where G is a group, e its neutral element, · the conﬁguration space, and L : TQ→ R is a func- product operation in G, and X a topological tion on the tangent space of Q. The function L space. The maps λg : X → X deﬁned by is called the Lagrangian of the system. λg(x) = λ(g, x) are required to be homeomor- A Lagrangian ﬁeld theory is a pair (B,L) phisms. A map λˆ : G → Hom(X) deﬁned ˆ where B = (B,M,p; F) is a ﬁber bundle, by λ(g) = λg is thence associated to a left

78 Lie derivative action λ. The orbit of a point x ∈ X is the subset Levi-Civita connection The symmetric α ox ={x ∈ X : ∃g ∈ G, λ(g, x) = x }. The (linear) connection Nβµ on a (pseudo)- action is transitive if ox = X (for any x ∈ X). Riemannian manifold (M, g) uniquely deﬁned The action is free if it has no ﬁxed points, by the requirement that the metric g is parallel, i.e., if there exists an element x ∈ X such that i.e., ∇g = 0. Equivalently, it is the only α λ(g, x) = x, then one has g = e. Example: connection Nβµ such that R3 Rotations in are not free and not transitive; (i.) it is symmetric, i.e., torsionless, i.e., translations α α are free and transitive. Nβµ = Nµβ ; If X has a further structure, one usually (ii.) it is compatible with the (pseudo)-metric requires λ to preserve the structure. For exam- structure, i.e., ple, if X is a topological space, λg are required " " to be continuous for any g ∈ G. ∂λ gµν − Nµλ g"ν − Nνλ gµ" = 0. If X = V is a vector space, λg are required to The coefﬁcients of the Levi-Civita connection be linear for any g ∈ G and in this case the left are usually denoted by Nα = α and can be action is also called a representation of G on V . βµ βµ g expressed as a function of the (pseudo)-metric If X = M is a manifold, λg are required to be diffeomorphisms for any g ∈ G. tensor and its ﬁrst derivatives; they are called If λ is a left action, then one can deﬁne a right Christoffel symbols. ρ(x,g) = λ(g−1,x) action by setting . Lewis acid A molecular entity (and the cor- left invariance The property that an object responding chemical species) that is an electron- on a manifold M is invariant with respect to a left pair acceptor and, therefore, able to react with a action of a group on M. For example, a vector Lewis base to form a Lewis adduct, by sharing ﬁeld X ∈ X(M) is left invariant with respect to the electron pair furnished by the Lewis base. λ M → M the left action g : if and only if: Lewis adduct The adduct formed between a Lewis acid and a Lewis base. Tx λg X(x) = X(g · x). left translations (on a group G) The left Lewis base A molecular entity (and the cor- action of G onto itself deﬁned by λ(g, h) = responding chemical species) able to provide a pair of electrons and thus capable of coordina- λg(h) = g · h. Notice that, if G is a Lie group, tion to a Lewis acid, thereby producing a Lewis λg ∈ Diff(G) is a diffeomorphism but not a homomorphism of the group structure. See also adduct. right translations and adjoint representations. Lie algebra A vector space L endowed with an operation [ , ]:L × L → L having the Legendre transformation Given a Lagran- following properties gian L : TQ→ R, the Legendre transformation FL : TQ→ T ∗Q is deﬁned in local coordinates (i.) [ , ] is bilinear; (qi , q˙i ) of TQby (ii.) [ , ] is antisymmetric (or skew- ∂L symmetric), i.e., [A, B] =−[B,A]; FL(qi , q˙i ) = (qi ,p ), where p = . i i ∂q˙i (iii.) [ , ] satisﬁes Jacoby identity (i.e., [[A, B],C] + [[B,C],A] + [[C, A],B] = 0). This gives an equivalence between the Euler- The operation [ , ] is called the Lie bracket or Lagrange equations and the Hamilton equations commutator. Notice that usually Lie algebras are of motion. not associative. See Lie group for the deﬁnition of a Lie algebra of the Lie group G. length In the context of graph theory, the Lie derivative Let α be an exterior k-form length of a path or cycle is the number of edges and X a vector ﬁeld with ﬂow ϕt . The Lie deriva- in the particular subgraph. tive of α along X is given by Comment: Lengths are an instance of the more general class of measures (such as string length, 1 ∗ LXα = lim [(ϕt α) − α]. absolute value of a number, etc.). t→0 t

79 Lie group

Lie group A group G which has a compati- value. Statistically, it represents the mean life ble manifold structure such that both the product · expectancy of an excited species. In a reacting : G × G → G and the inversion i : G → G : system in which the decrease in concentration g → g−1 are differentiable maps. of a particular chemical species is governed by Left translations Lg : G → G deﬁned by a ﬁrst-order rate law, it is equal to the recipro- Lg(h) = g · h are diffeomorphisms.Ifv ∈ TeG cal of the sum of the (pseudo)unimolecular rate is a tangent vector in the unit e ∈ G we can constants of all processes which cause the decay. deﬁne a vector ﬁeld λv(g) = TeLg(v) which is When the term is used for processes which are left-invariant, i.e., TLgλv(h) = λv(g · h).Any not ﬁrst order, the lifetime depends on the initial left-invariant vector ﬁeld on G is obtained in this concentration of the species, or of a quencher, way. The set XL(G) of left-invariant vector ﬁelds and should be called apparent lifetime instead. form a Lie subalgebra of the Lie algebra of all limiting current The limiting value of a vector ﬁelds X(G). The Lie algebra X (G) is L faradaic current that is approached as the rate of isomorphic to TeG as a vector space, so that it is (X (G)) = T G = the charge-transfer process is increased by vary- ﬁnite dimensional (dim L dim e applied G G ing the potential. It is independent of the dim ). It is called the Lie algebra of . potential over a ﬁnite range, and is usually evalu- R G → G Analogously, right-translations g : ated by subtracting the appropriate residual cur- R (h) = h · g deﬁned by g are diffeomorphisms. rent from the measured total current. A limiting v ∈ T G If e is a tangent vector we can deﬁne current may have the character of an adsorption, ρ (g) = T R (v) a vector ﬁeld v e g which is right- catalytic, diffusion,orkinetic current, and may TR ρ (h) = ρ (h · g) invariant, i.e., g v v .Any include a migration current. right-invariant vector ﬁeld on G is obtained in this way. The set XR(G) of right-invariant vector line formula A two-dimensional represen- ﬁelds form a Lie subalgebra of the Lie algebra tation of molecular entities in which atoms are of vector ﬁelds X(G). The Lie algebra XR(G) shown joined by lines representing single or is canonically isomorphic to XL(G) as a vec- multiple bonds, without any indication or impli- tor space, so that it is also ﬁnite dimensional cation concerning the spatial direction of bonds. (dim(X (G)) = dim G). Clearly, if the group R linear A vector space is often called a linear G is Abelian, then X (G) and X (G) coincide. L R space. A map T : V → W from a linear space V into a linear space W is called linear or a linear Lie-Poisson bracket Let g be a Lie algebra, transformation or a linear operator if T(x+ y) g∗ its dual space and <, >: g∗ × g → R = T(x) + T(y) and T(αx) = αT (x) for all the natural pairing, <µ,ξ>= µ(ξ). The Lie- x,y ∈ V , α ∈ C. Poisson bracket of any F,G : g∗ → R is deﬁned by linear chain A chain with no branch points intermediate between the boundary units. δF δG {F,G} (µ) =±/µ, , 0 ± δµ δµ linear functional A linear scalar valued function f : V → C on a vector space V . Some- δF ∈ F f where δµ g is the functional derivative of at times continuity of is also assumed. µ deﬁned by linear group (of a vector space V ) The 1 δF group of automorphisms of V (which is a group lim [F(µ+ tν) − F (µ)] =/ν, 0. t→0 t δµ with respect to composition). It is denoted by GL(V ). With the Lie-Poisson bracket, g∗ is a Poisson Also the matrix group of all invertible (ﬁnite) manifold. matrices. It can be noncanonically identiﬁed with GL(Rm), and it is denoted by GL(m, R). lifetime (mean lifetime), τ The lifetime of If dim(V ) = m, then there exists a group iso- a chemical species which decays in a ﬁrst-order morphism between GL(V ) and GL(m, R) which process is the time needed for a concentration is induced by the choice of a basis of V ; both of this species to decrease to 1/e of its original GL(V ) and GL(m, R) are Lie groups.

80 loop linear macromolecule A macromolecule, other (for example, they satisfy a generating rela- the structure of which essentially comprises tion like y = x2 or are all employees in a com- the multiple repetition in linear sequence of pany), but either may or may not have repeated units derived, actually or conceptually, from elements. The elements of a list need not have molecules of low relative molecular mass. any relationship to each other. Notice further that none of these deﬁnitions requires that the mem- linear operator See linear. bers of the collection be ordered in some way. If they were, one would speak of an ordered set, linear transformation See linear. bag,orlist. For example, one might sort the elements of a set alphabetically or in UNIX sort U ⊂ Rn Liouville’s equation Let open and order. See also bag, sequence, set, and tuple. u : U × R → R. The Liouville equation for u is n load vector The right-hand side of a vari- u − (bi u) = . t xi 0 ational problem posed over the Banach space i=1 V is an element of the dual space V . When discretization by means of a conform- liquid-crystal transitions A liquid crystal ing ﬁnite element space Vh is performed, is a molecular crystal with properties that are f has to be evaluated for the nodal basis both solid- and liquid-like. Liquid crystals are functions bi ,i = 1, ··· ,N := dim Vh, composed predominantly of rod-like or disk-like N of Vh. The resulting vector (f (bi )) has molecules, that can exhibit one or more differ- i=1 been dubbed load vector in calculations of ent, ordered ﬂuid phases as well as the isotropic linear elasticity. ﬂuid; the translational order is wholly or partially destroyed but a considerable degree of orien- tational order is retained on passing from the logistic equation A nonlinear equation with x( − x) crystalline to the liquid phase in a mesomorphic a 1 term. It was ﬁrst motivated from transition. modeling biological population growth. In the (1) Transition to a nematic phase. A meso- context of ordinary differential equations, the dx/dt = ax( − x) morphic transition that occurs when a molecular equation, 1 can be solved. x = crystal is heated to form a nematic phase in which In the context of difference equations, n+1 ax ( −x ) the mean direction of the molecules is parallel or n 1 n . This equation exhibits a wide range antiparallel to an axis known as the director. of interesting nonlinear phenomena: bifurcation (2) Transition to a cholesteric phase. A meso- and periodic doubling to chaos. morphic transition that occurs when a molecu- lar crystal is heated to form a cholesteric phase London forces Attractive forces between in which there is simply a spiraling of the local apolar molecules, due to their mutual polariz- orientational order perpendicular to the long axes ability. They are also components of the forces of the molecules. between polar molecules. Also called “disper- (3) Transition to a smectic state. A mesomor- sion forces.” phic transition that occurs when a molecular crys- tal is heated to yield a smectic state in which x ∈ X γ I → X loop (based at 0 ) A curve : there is a one-dimensional density wave which γ( ) = γ( ) = x I = , such that 0 1 0 and [0 1]. produces very soft/disordered layers. X x ∈ X The set of all loops of based at 0 can be endowed with a group product. Let λ, γ : I → list An unordered collection of elements, X be two loops; the product λ ∗ γ : I → X is that may include duplicates. An enumerated list the following loop x is delimited by brackets ([ ]). Comment: Notice the deﬁnitions of set, bag, γ(2t) 0 ≤ t ≤ 1/2 and list progressively release constraints on the λ ∗ γ(t)= λ( t − ) / ≤ t ≤ . elements in the various collections. The elements 2 1 1 2 1 of sets and bags have some relationship to each

81 Lorentz group

η = η =− i> The group structure is not commutative. It deﬁned by 00 1, ii 1 when 0 and is compatible with the homotopy equivalence ηij = 0 when i = j. An endomorphism α : relation so that it induces the group structure of Rm → Rm is an element of the Lorentz group if π (X, x ) X x the homotopy group 0 0 of (based at 0). and only if Let us now consider the homotopy groups a b j π (X, x ) π (X, x ) A ηab A = ηij α(Ei ) = A Ej . 0 0 and 0 1 at two different points i j i of X and a path γ : I → X connecting the two γ( ) = x γ( ) = x Lorentzian manifold A pair (M, g) formed points, i.e., 0 0 and 1 1. We can i π (X, x ) by a manifold M and a Lorentzian metric g on deﬁne a group isomorphism between 0 0 and π (X, x ) given by M, i.e., a pseudo-Riemannian metric of signature 0 1 ( ,m− ) (m − , ) i(λ) = γ −1∗λ∗γ ∈ π (X, x )λ∈ π (X, x ) either 1 1 or 1 1 . 0 0 0 1 where γ −1 is the path γ −1(t) = γ(1 − t). luminescence Spontaneous emission of radiation from an electronically or vibrationally Lorentz group The special orthogonal excited species not in thermal equilibrium with group SO(1,m − 1) of isometries of Rm with its environment. the standard indeﬁnite metric η of signature (1,m−1). Let us choose an orthonormal basis Ei lyate ion The anion produced by hydron m i j in R , the metric is expressed by η = ηij E ⊗E removal from a solvent molecule. For example, i (where E is the dual basis). The matrix ηij is the hydoxide ion is the lyate ion of water.

82 mass matrix

For example, the sphere S2 ⊂ R3 (x2 + y2 + z2 = 1) is both a real and a complex manifold. The real structure is given by the real charts x y ϕ (x,y,z)= , ,z= M N 1 and − z − z 1 1 x y ϕS (x,y,z)= , ,z= −1 macromolecule (polymer molecule) A 1 + z 1 + z molecule of high relative molecular mass, the structure of which essentially comprises the while the complex atlas is given by multiple repetition of units derived, actually or x + iy conceptually, from molecules of low relative ϕN (x,y,z)= ,z= 1 and molecular mass. 1 − z Notes: (1) In many cases, especially for syn- x − iy ϕS (x,y,z)= ,z= −1. thetic polymers, a molecule can be regarded as 1 + z having a high relative molecular mass if the addi- tion or removal of one or a few of the units has mapping See function. a negligible effect on the molecular properties. This statement fails in the case of certain macro- mass, m Base quantity in the system of molecules for which the properties may be criti- quantities upon which SI is based. cally dependent on ﬁne details of the molecular structure. mass lumping Approximation of the mass (2) If a part of the whole of the molecule matrix by a diagonal matrix. This is usually has a high relative molecular mass and essen- achieved by replacing the L2(;)-inner product tially comprises the multiple repetition of with a mesh-dependent inner product (., .)h on a units derived, actually or conceptually, from ﬁnite element space Vh that is based on numer- molecules of low relative molecular mass, it may ical quadrature. A prominent example is sup- be described as either macromolecular or poly- plied by linear Lagrangian ﬁnite elements and meric, or by polymer used adjectivally. vertex based quadrature: on a triangular mesh ;h it boils down to manifold A (usually connected) topologi- cal space M such that there exists an atlas 3 |T | T T {(Uα,ϕα)}α∈I , i.e., a collection such that: (uh,vh)L2(;) ≈ uh(ai )v¯h(ai ) 3 T ∈Th i=1 (i.) {Uα}α∈I is an open covering of M;

(ii.) ϕα : Uα → Wα is a local homeomor- × uh,vh ∈ Vh, m phism onto Wα = ϕα(Uα) ⊂ R ; where aT , aT , aT stand for the vertices of the ϕ ◦ ϕ−1 W → W 1 2 3 (iii.) β α : α β are local homeo- triangle T . As the degrees of freedom for V are Ck h morphisms of class . associated with vertices of the mesh, the diagonal −1 The maps ϕβ ◦ ϕα : Wα → Wβ are called structure of the resulting “lumped” mass matrix transition functions. The integer m is the dimen- is evident. sion of M.Amanifold M is usually assumed to be paracompact. mass matrix The matrix arising from a ﬁnite If k = 0, M is called a topological (real) element discretization of the L2(;)-inner prod- ; ⊂ Rn ψ ··· ,ψ manifold;ifk =∞, it is called a differentiable uct ( a bounded domain): if 1 N (real) manifold. If the transition functions are stands for the nodal basis of the ﬁnite element 2 analytical, then M is called an analytical (real) space Vh ⊂ L (;), then the corresponding mass (U ,ϕ ) Cm ((ψ ,ψ ) )N manifold.Ifcharts α α are valued in matrix is given by i j L2(;) i,j=1. The term and transition functions are, holomorphic, then mass matrix has its origin in the ﬁnite element the manifold is called a complex manifold. analysis of problems of linear elasticity.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 83 mass spectrometer mass spectrometer An instrument in which Maxwell-Vlasov equations Let f(x,v,t) beams of ions are separated (analyzed) according denote the plasma density and E(x, t), B(x, t) to the quotient mass/charge, and in which the ions the electric and magnetic ﬁelds, respectively. are measured electrically. This term should also The equations for a collisionless plasma of a sin- be used when a scintillation detector is employed. gle species of charged particles with charge e and mass m are described by the Maxwell-Vlasov matrix (1) (in analysis) A rectangular array equations of scalars. A matrix with n rows and m columns represents a linear operator from an m dimen- ∂f + v · ∂f + e ( + 1 v × ) · ∂f = , ∂t ∂x m E c B ∂v 0 sional vector space to an n dimensional vector space. ∂ 1 B =−curl E, (2) (in analytical chemistry) The components c ∂t of the sample other than the analyte. 1 ∂E =−curl − 1 , c ∂t B c jf matrix effect (1) (in analytical chemistry) The combined effect of all components of the div E = ρf and divB = 0 sample other than the analyte on the measure- f = ment of the quantity. If a speciﬁc component where the current of is given by jf can be identiﬁed as causing an effect then this is e vf (x, v, t)dv, and the charge density by referred to as interference. See matrix. ρf = e f (x, v, t)dv. (2) (in surface analysis) Effects which cause changes in Auger-electron, photoelectron, sec- measurable set A measure on a set S is fre- ondary ion yield, or scattered ion intensity, the quently deﬁned only on certain subsets of S, energy or shape of the signal of an element in forming a σ algebra (closed under countable any environment as compared to these quantities unions and differences of its members). Sets in in a pure element. this σ algebra are called measurable sets. (a) Chemical matrix effects: changes in the chemical composition of the solid which affect measure (1) A numerical determination of the signals as described above. size. For example, the cardinality, if x is a set; (b) Physical matrix effects: topographical and/or or the length, if x is a sequence; or the absolute crystalline properties which affect the signal as value, if x is a number. described above. (2) A nonnegative, real valued function µ deﬁned on certain subsets of a set S and satis- maximal common subgraph Two graphs, fying G1 and G2,haveamaximal common subgraph if there exists a graph, G , which is the largest (i.) µ()) = 0 subgraph common to G and G . 1 2 (ii.) µ(∪Un) = µ(Un) when {Un} is a maximal element (in a partially ordered set pairwise disjoint, countable sequence of sets. [A, ]) Referring to an element m ∈ A such that there is no element a ∈ A except a = m such mechanism (of a reaction) A detailed that am. See ordering and minimum. description of the process leading from the reac- tants to the products of a reaction, including a A, maximum (in a partially ordered set [ ]) characterization as complete as possible of the m ∈ A ∀a ∈ A Referring to an element such that , composition, structure, energy, and other prop- ma . See ordering. erties of reaction intermediates, products, and Maxwell’s equations In electrodynamics, transition states. An acceptable mechanism of a for an electric ﬁeld E(x, t) and magnetic ﬁeld speciﬁed reaction (and there may be a number of B(x, t) Maxwell’s equations in the vacuum are such alternative mechanisms not excluded by the evidence) must be consistent with the reaction = curl Et B stoichiometry, the rate law, and with all other =−curl Bt E available experimental data, such as the stereo- div = div = . B E 0 chemical course of the reaction. Inferences

84 mesopause concerning the electronic motions which dynam- meiosis The reductive cell division which ically interconvert successive species along the results in daughter cells containing one copy of reaction path (as represented by curved arrows, each of the chromosomes of the parent. The for example) are often included in the description entire meiotic process involves two separate of a mechanism. divisions (meiosis I and meiosis II). The ﬁrst divi- It should be noted that for many reactions sion is a true reductive division with the chromo- all this information is not available and the sug- some number being halved, whereas the second gested mechanism is based on incomplete exper- division resembles mitosis in many ways. Thus, imental data. It is not appropriate to use the term a diploid parental cell will give rise to haploid mechanism to describe a statement of the proba- daughter cells (cf. ploidy). ble sequence in a set of stepwise reactions. That should be referred to as a reaction sequence, and memory A memory is a storage device of not a mechanism. ﬁxed half-life for data and programs that per- mit any combination of reading from and writing mechanistic equation A balanced elemen- to it. tary reaction equation, bimolecular or less on mesh width The mesh width of a triangula- one side, which describes the ligand binding, tion ;h is the maximal diameter of its cells. organic chemical, and conformational rearrange- ment reactions which occur on an unambigu- mesomeric effect The effect (on reaction ously identiﬁed entity, whose net effect in the rates, ionization, equilibria, etc.) attributed to a biochemical reaction is catalytic, to produce an substituent due to overlap of its p-orπ-orbitals overall biochemical reaction. Equally, a set of with the p-orπ-orbits of the rest of the molecu- such reaction equations which can be combined lar entity. Delocalization is thereby introduced by summation or logical operators to produce the or extended, and electronic charge may ﬂow to overall biochemical reaction. or from the substituent. The effect is symbolized Comment: The requirement that the equation by M. be elementary ensures that its order will be the Strictly understood, the mesomeric effect sum of the molecularities of the reactants. All operates in the ground electronic state of the types of interactions are included. Since the aim molecule. When the molecule undergoes elec- of the equations is to describe interactions at the tronic excitation or its energy is increased on the catalytic species, it follows that that molecule waytothetransition state of a chemical reac- will not appear in these equations as a catalyst. tion, the mesomeric effect may be enhanced by The requirement for clearly identiﬁed species the electromeric effect, but this term is not much distinguishes among different catalysts carrying used, and the mesomeric and electronic effects out the “same” reactions at different rates. The tend to be subsumed in the term resonance effect deﬁnition provides that they can be combined by of a substituent. summation or logical operators to represent alter- mesomorphic transition A transition that native paths through the space of possible reac- occurs between a fully ordered crystalline solid tions, depending on the consequences of prior and an isotropic liquid. Mesomorphic transitions reactions in the set. can occur mechanistic motif A repeated pattern over (i.) from a crystal to a liquid crystal, mechanistic equations. (ii.) from a liquid crystal to another liquid Comment: See also biochemical, chemical, crystal, and dynamical, functional, kinetic, phylogenetic, (iii.) from a liquid crystal to an isotropic liq- regulatory, thermodynamic, and topological uid. motives. mesopause (in atmospheric chemistry) That medium The phase (and composition of the region of the atmosphere between the meso- phase) in which chemical species and their reac- sphere and the thermosphere at which the tem- tions are studied in a particular investigation. perature is a minimum.

85 mesosphere mesosphere That region of the atmosphere Comment: Once equilibrium is attained, any which lies above the stratopause (about 4752 reaction will continue in both directions, shifting km) and below the mesopause (about 8090 km) the reaction away from and back toward equilib- and in which temperature decreases with increas- rium from instant to instant, but by inﬁnitesi- ing height; this is the region in which the lowest mally small changes. See also dextralateral, temperatures of the atmosphere occur. direction, dynamic equilibrium, formal reaction equation, product, rate constant, reversibility, metabolite A molecule which participates sinistralateral, and substrate. noncatalytically in metabolism and is not an inor- ganic ion or element. migration (1) The (usually intramolecular) Comment: This seems a bit circular, because transfer of an atom or group during the course of it depends on being able to recognize what a molecular rearrangement. metabolism is. Metabolites tend to be relatively (2) The movement of a bond to a new position, small (usually less than 1000 daltons molecu- within the same molecular entity, is known as lar weight) and structurally simple. They can be “bond migration.” monomers or polymers (for example, fatty acids), Allylic rearrangements, e.g., but are not usually considered to be macro- RCH = CHCH X −→ RCCHCH = CH molecules or their smaller polymers (oligonu- 2 | 2 X cleotides, peptides). As should be obvious, the notion of a metabolite is rather elastic. exemplify both types of migration. metric A metric on a set M is a map d : minimal biochemical network The min- M × M → R satisfying for all x,y,z ∈ M, imal biochemical network is the network

(i.) d(x,y) ≥ 0, equal 0 iff x = y, positive NB ({vm, ,vm, ,vr, }, {e(s, vm, ,vr, ), 0 1 2 1 1 1 deﬁniteness, e(d, v ,v )}, ∅, {l ,l ,l }). (ii.) d(x,y) = d(y,x), symmetry, m,2 r,1 m,1 m,2 r,1 (iii.) d(x,z) ≤ d(x,y) + d(y,z), triangle Comment: This is a network of a single spon- inequality. taneous reaction, without any known parameters but for which the identities of the reactants (but A metric space is a pair (M, d). not their stoichiometry, a parameter of the edges) are known. metric space See metric. minimal element (of a partially ordered set Michaelis-Menten kinetics The mathemat- [A, ]) An element m ∈ A such that there is ical model for enzyme kinetics based on the law no element a ∈ A except a = m such that ma. of mass action. The resulting equations are non- See ordering and minimum. linear. However, due to the peculiar nature of n the enzymatic reaction in which the concentra- minimal surface equation Let U ⊂ R tion of the enzyme is usually signiﬁcantly greater open and u : U × R → R. The minimal surface than that of the substrate, the system of nonlinear equation for u is ordinary differential equations can be treated by Du singular perturbation. This treatment gives the div = 0, ( +|Du|2)1/2 steady-state approximation well known in bio- 1 chemical literature. where Du = Dx u = (ux ,...,ux ) denotes the 1 n gradient of u with respect to the spatial variable microscopic reversibility x = (x ,...,x ) The continuous 1 n . reaction of sinistralateral and dextralateral sets of coreactants, such that no net change in the minimum (of a partially ordered set [A, ]) concentrations of coreactants occurs over meas- An element m ∈ A such that ∀a ∈ A, am. urable time. See ordering.

86 molecular entity minimum-energy reaction path The path This is a symmetric saddle point problem, corresponding to the steepest descent from the which can be discretized by means of v ∈ col of a potential-energy surface into the two H(div;;)-conforming mixed ﬁnite elements for valleys. The reaction coordinate corresponds the ﬂux unknown. Solving the second-order to this minimal path. Some workers refer to elliptic boundary value problem is equivalent the minimum-energy reaction path as simply the to minimizing a convex energy functional on reaction path but this is not recommended as it H 1(;) 0 . Convex analysis teaches that this gives leads to confusion. rise to primal and dual variational problems. The latter leads to the mixed variational formulation. Minkowski space The metric vector space (R4,η)with the signature (1, 3). module (over a ring R) A set M endowed mixed ﬁnite elements A rather general term with an inner binary operation, called the sum for ﬁnite element spaces meant to approximate + : M × M → M and a binary operation called vector ﬁelds. Prominent representatives are the product (by a scalar) · : R × M × M. The (i.) H(div; ;)-conforming Raviart-Thomas structure (M, +) is a commutative group and the elements required for the discretization of mixed product by scalars satisﬁes the following axioms: variational formulations of second-order elliptic ∀λ µ ∈ R ∀v ∈ M λ(µ·v) = (λµ)·v boundary value problems. (i.) , , : λ · (v + w) = λ · v + λ · w (ii.) H(curl; ;)-conforming N´ed´elec ele- (ii.) ments (edge elements) used to approximate elec- (iii.) (λ + µ) · v = λ · v + µ · v tromagnetic ﬁelds. (iv.) IR · v = v (iii.) a variety of schemes for the approxi- mation of velocity ﬁelds in computational ﬂuid If R is a ﬁeld then an R-module is nothing but mechanics. a vector space. mixed variational formulation Consider a second-order elliptic boundary value problem mole An Avogadro’s number of molecules (N ≈ 6.023 × 1023). n −div(Agradu) = f on ; ⊂ R ,u|∂; = gD.

Formally its mixed formulation is obtained by molecular dynamics Based on Newton’s introducing the ﬂux j := Agradu as new law of motion and treating atoms in a molecule unknown, which results in a system of ﬁrst-order as classical particles with interaction, molecu- differential equations lar dynamics simulate the motion of the entire molecule on a computer. This approach has been − A u = , − = f. j grad 0 divj successful in studying the dynamics of small molecules but has not yielded a complete pic- The ﬁrst equation is tested with v ∈ H(div;;) and integration by parts is carried out. The sec- ture of the dynamics of biologically important ond equation is tested with q ∈ L2(;).We molecules, e.g., proteins. The difﬁculty is mainly end up with the variational problem: seek j ∈ the stiff nature of the many-body system and the H(div;;), u ∈ L2(;) such that limited computational power. A−1j · vdx + divvudx = gv · ndS ; ; N molecular entity Any constitutionally or isotopically distinct atom, molecule, ion, ion ∀v ∈ H(div;;), pair, radical, radical ion, complex, conformer, etc., identiﬁable as a separately distinguishable divjqdx =− fqdx ∀q ∈ L2(;). ; ; entity.

87 molecularity

Molecular entity is used here as a general term Monge-Ampere equation Let U ⊂ Rn be for singular entities, irrespective of their nature, open and u : U × R → R. The Monge-Ampere while chemical species stands for sets or ensem- equation for u is bles of molecular entities. Note that the name of a compound may refer to the respective molecular det(D2u) = f, entity or to the chemical species; e.g., methane, CH where Du = Dx u = (ux ,...,ux ) denotes the may mean a single molecule of 4 (molecu- 1 n lar entity) or a molar amount, speciﬁed or not gradient of u with respect to the spatial variable x = (x ,...,x ) (chemical species), participating in a reaction. 1 n . The degree of precision necessary to describe a molecular entity depends on the context. For monomer A substance composed of mono- example “hydrogen molecule” is an adequate mer molecules. deﬁnition of a certain molecular entity for some purposes, whereas for others it is necessary to monomer molecule A molecule which can distinguish the electronic state and/or vibrational undergo polymerization thereby contributing state and/or nuclear spin, etc. of the hydrogen constitutional units to the essential structure of a molecule. macromolecule. molecularity The number of reactant monomeric unit (nomoner unit, mer) The molecular entities that are involved in the largest constitutional unit contributed by a single microscopic chemical event constituting an monomer molecule to the structure of a macro- elementary reaction. (For reactions in solution molecule or oligomer molecule. this number is always taken to exclude molecular Note: entities that form part of the medium and which The largest constitutional unit con- are involved solely by virtue of their solvation tributed by a single monomer molecule to of solutes.) A reaction with a molecularity the structure of a macromolecule or oligomer of one is called “unimolecular,” one with a molecule may be described either as monomeric molecularity of two “bimolecular,” and of three or by monomer used adjectivally. “termolecular.” monomorphism An injective morphism moment of a force, M, about a point The between objects of a category. For example, a vector product of the radius vector from this point monomorphism of vector spaces is a linear injec- to a point on the line of action of the force and tive map; a monomorphism of groups is an injec- the force, M = r × r × F . tive group homomorphism;amonomorphism of manifolds is an injective differentiable map. See momentum, p Vector quantity equal to the also bundle morphisms. product of mass and velocity. motif For any collection of objects X = momentum map Let a Lie group G act on a {x ,x ,...,xn}, xi is a motif if there exists an Poisson manifold P and let g be the Lie algebra 1 2 x ∈ X, i = j such that f(x ) = x .Ifx of G.Anyξ ∈ g generates a vector ﬁeld ξ (the j i j i P occurs in X at least twice, that is, if x = x for inﬁnitesimal generator of the action) on P by i j 1 ≤ i, j ≤ n, i = j,itisanexact motif or an d exact match. ξ (x) = (tξ) · x | . P dt [exp ] t=0 Comment: The relationship f is a transfor- mation between x and x . For nucleic acid and J → C∞(P ) i j Suppose there is a map : g such protein sequences, the most common motives are X = ξ ξ ∈ J that J(ξ) P , for all g. The map : the percent identity and percent similarity rela- P → g∗ deﬁned by tionships, which are usually set to some thresh- /J(x),ξ0=J (ξ)(x) , ξ ∈ g,x ∈ P old value such that the identity (similarity) of the resulting string is at least that threshold value. is called the momentum map of the action. When X is a set of graphs, then f is either the

88 motif relationship of subgraph (for motif ) or isomor- biological pattern recognition, most commonly phism (for exact motif ). in DNA and protein sequence motives. For our purposes the term “pattern” is synony- Ofﬁcially, motif may be pluralized either mous with motif. It is not obvious that the notion motives (with the accent on the last syllable) or of motif as used in biology is isomorphic to that motifs. One will ﬁnd both in the biological liter- of pattern. Deﬁnitions of motif and pattern have ature. been developed to cover problems arising from See also biochemical, chemical, dynamical, tessellation of surfaces (such as tiling a plane functional, kinetic, mechanistic, phylogenetic, like your bathroom ﬂoor). Instead, our intent regulatory, thermodynamic, and topological is to capture the notion of motif as it is used in motives.

89

nondeterministic computation

nodal interpolation operator Given a ﬁnite element space Vh with set Xh of nodal degrees of freedom, we can deﬁne a projection I S ∪ V → V ,φ(Iu) = φ(u) ∀φ ∈ X , N h : h h h h where S is a space of sufﬁciently smooth func- tions/vector ﬁelds for which the evaluation of Navier-Stokes equations Let U ⊂ Rn be degrees of freedom is well deﬁned. Note that for open and u : U × R → Rn. The Navier-Stokes many ﬁnite element spaces that are intended as equations for incompressible, viscous ﬂows are conforming approximating spaces for a function V S V space , will be strictly contained in . Nodal ut + u · Du =−Dp interpolation operators are local in the sense that divu = 0 for each cell K of the underlying triangulation the restriction of Ihu onto K depends on u|K¯ . where Du = Dx u = (ux ,...,ux ) denotes the 1 n v i gradient of u with respect to the spatial variable node A node, i ( indexing the nodes of the graph), is a member of the set of nodes (V)ofa x = (x ,...,xn). 1 mathematical graph or network. Comment: Each node is commonly rendered neighborhood (of a point p in a topological by a geometric point, but in fact it is simply an space) A set containing an open set contain- element of any set; so in a set of integers, each ing p. integer would be a node of a graph. A com- mon synonym is vertex. When reactions and compounds are both nodes in the representation nematic phase See liquid-crystal transi- of a biochemical network, because there are two tions. types of nodes, the network is said to be bipartite.

nonconforming ﬁnite elements A ﬁnite network A mathematical graph N(V, E, element space Vh is called nonconforming with P, L) consisting of a set of nodes (or vertices) respect to a function space V if V ⊂ V . Using V E h , a set of edges between the nodes , a set of V to discretize a variational problem over V P h parameters describing properties of subgraphs amounts to a variational crime. However, it L of the network, and a set of labels . might be feasible, if V satisﬁes certain consis- P =∅ h Comment: If the network reduces to its tency conditions expressed in Strang’s second (V, E, L) corresponding graph [denoted either N lemma. or G(V, E, L) according to one’s taste]. In gen- eral, all graph algorithms apply to networks, but nondeterministic computation Let the sets there are some specialized algorithms, relying on of presented inputs and produced outputs be KI the parameters, which do not apply to graphs. and KO , respectively; the cardinalities of sets be denoted by the double overset bars; and σi,I and σ be the particular symbols presented to or {b , ··· ,b } i,O nodal basis The basis 1 N of a produced by the mapping c . V X i ﬁnite element space h dual to the set h of Then a nondeterministic computation C φ (b ) = n global degrees of freedom, that is, k i speciﬁes a computation C : KI → KO , such δ ,i,k = , ··· ,N ik 1 , is called the nodal basis that KI ≥ 1; KO ≥ 1; at least one mapping ci ∈ V of h. Thanks to the localization of the degrees Cn transforming presented inputs to presented of freedom, the nodal basis functions are locally outputs is many-to-many; and ∀σi,I ,σi,I ∈ b supported. More precisely, supp i is contained KI , ∀σi,O ,σi,O ∈ KO , the probabilities that in the closure of the union of all those cells of ;h each exists, Pe(σi,I ) and Pe(σi,O ), are unity. that share the vertex, face, edge, etc. to which the Comment: This computation is not stochastic, related global degree of freedom is associated. but it is nondeterministic. There is at least one

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 91 nonenzymatic degree step in the computation where the output pro- symmetries; the associated conserved current is duced is not a function of the input presented, given by ∂L even though all inputs and outputs exist with E = Xλ probability one. See deterministic and stochastic ∂uλ computations. where (xλ,uλ) are local coordinates on the tan- gent bundle TM. nonenzymatic degree The degree of a react- ive conjunction node excluding the macro- nuclear decay A spontaneous nuclear trans- molecular catalyst. formation. nonlinear optical effect An effect brought nuclear ﬁssion The division of a nucleus about by electromagnetic radiation the magni- into two or more parts with masses of equal order tude of which is not proportional to the irradi- of magnitude, usually accompanied by the emis- ance. Nonlinear optical effects of importance sion of neutrons, gamma radiation, and, rarely, to photochemistry are harmonic frequency small charged nuclear fragments. generation, lasers, Raman shifting, upconver- sion, and others. nucleic acids Macromolecules, the major organic matter of the nuclei of biological cells, norm (of a vector) See angle between vec- made up of nucleotide units, and hydrolyzable tors, normed space. into certain pyrimidine or purine bases (usually adenine, cytosine, guanine, thymine, or uracil), normal Perpendicular, as a vector to a sur- d-ribose or 2-deoxy-d-ribose, and phosphoric face or to another vector. acid. See deoxyribonucleic acids. normed space A vector space X with a norm ||x|| x ∈ X deﬁned, for , satisfying nucleosides Ribosyl or deoxyribosyl de- (i.) ||x|| ≥ 0 and equality only when x = 0, rivatives (rarely, other glycosyl derivatives) of (ii.) ||x + y||≤||x||+||y||, and certain pyrimidine or purine bases. They are thus glycosylamines or N-glycosides related to ||λx||=|λ|||x|| (iii.) . nucleotides by the lack of phosphorylation. It has also become customary to include among nucleo- N¨other’s theorem (1)IftheLie algebra sides analogous substances in which the glycosyl P g acts canonically on the Poisson manifold , group is attached to carbon rather than nitrogen admits a momentum map J : P → g∗, and ∞ (“C-nucleosides”). See also nucleic acids. H ∈ C (P ) is g-invariant, i.e., ξP (H ) = 0 for all ξ ∈ g, then J is a constant of the motion for nucleotides Compounds formally obtained H , i.e., J ◦ ϕ = J by esteriﬁcation of the 3 or 5 hydroxy group of t nucleosides with phosphoric acid. They are the where ϕt is the ﬂow of the Hamiltonian vector monomers of nucleic acids and are formed from ﬁeld XH . them by hydrolytic cleavage. (2) (of a Lagrangian system) A map between 1-parameter families of Lagrangian symmetries nuclide A species of atom, characterized by and conserved currents. For example, let (M, its mass number, atomic number, and nuclear L) be a time-independent Lagrangian system energy state, provided that the mean life in that λ and X = X ∂λ an inﬁnitesimal generator of state is long enough to be observable.

92 order of reaction

covering {Uα}α∈I such that for any x ∈ M there is an open neighborhood Ux which intersects just a ﬁnite number of elements Uα of the covering. On a paracompact manifold M every open covering O contains a good covering, i.e., an open covering {Uα}α∈I such that for any ﬁnite set of indices α ,α ,...,αk ∈ I the intersection Uα α ...α = "1 2 1 2 k oligomer A substance composed of k U i=0 αi (which is an open set by deﬁnition) is oligomer molecules.Anoligomer obtained by topologically trivial, i.e., it is contractible. telomerization is often termed a telomer. open neighborhood (of a point x of a topo- oligomer molecule A molecule of inter- logical space [X, τ(X)]) An open subset mediate relative molecular mass, the structure U ⊂ X containing x. of which essentially comprises a small plurality of units derived, actually or conceptually, from open set A subset U ⊂ X of a topological molecules of lower relative molecular mass. space X which is an element of its topology τ(X). Notes: (1) A molecule is regarded as having In Rn, with the standard metric topology, a an intermediate relative molecular mass if it has subset U ⊂ Rn is open if any point x ∈ U is con- properties which do vary signiﬁcantly with the tained in U together with an open n-ball centered r n removal of one or a few of the units. at x of radius r, i.e., Bx ={y ∈ R : |y−x|

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 93 ordering hence the rate of reaction) are not themselves order is then the same as the molecularity.For measurable, provided it is possible to measure a stepwise reactions there is no general connec- chemical ﬂux. For example, if there is a dynamic tion between stoichiometric numbers and partial equilibrium according to the equation: orders. Such reactions may have more complex rate laws, so that an apparent order of reaction aA pP may vary with the concentrations of the chemi- cal species involved and with the progress of the and if a chemical ﬂux is experimentally found, reaction. In such cases, it is not useful to speak line-shape analysis (e.g., by NMR ) to be related of orders of reaction, although apparent orders to concentrations by the equation of reaction may be deducible from initial rates. φ−A/α = k[A]α[L]λ In a stepwise reaction, orders of reaction may in principle always be assigned to the elementary then the corresponding reaction is of order α with steps. respect to A,... and of total (or overall) order n(= α + λ + ...). The proportionality factor ordering (of a set) (1) Preordering: a rela- k above is called the (nth order) “rate coefﬁ- tion on a set A such that cient.” Rate coefﬁcients referring to (or believed (i.) ∀a ∈ Aaa; to refer to) elementary reactions are called “rate (ii.) ∀a, b, c ∈ Aab, bc ⇒ ac. constants” or, more appropriately, “microscopic” (hypothetical, mechanistic) rate constants. (2) Partial ordering: a preordering such that The (overall) order of a reaction cannot be (i.) ∀a, b ∈ Aab and ba ⇒ a = b. deduced from measurements of a “rate of appear- ance” or “rate of disappearance” at a single value (3) Total ordering: a partial ordering such that of the concentration of a species whose concen- (i.) ∀a, b ∈ Aab or ba. tration is constant (or effectively constant) during the course of the reaction. If the overall rate of Examples: The inclusion is a partial ordering reaction is, for example, given by in the power set P(X) of a set X. The relation ≥ is a total ordering in the real line R. The relation α β ν = k[A] [B] zw if and only if |z|≥|w| deﬁned on the com- plex plane C is a preordering but not a partial but [B] stays constant, then the order of the reac- ordering. tion (with respect to time), as observed from the concentration change of A with time, will oregonator R.M. Noyes and R.J. Field at be α, and the rate of disappearance of A can be the University of Oregon developed a mathemat- expressed in the form ical model, consisting of three coupled nonlinear ordinary differential equations, for the BZ reac- ν = k [A]α. A obs tion (see Belousov-Zhabotinskii reaction). The model was shown to have a limit cycle, hence The proportionality factor kobs deduced from such an experiment is called the “observed rate ﬁrmly established the theoretical basis of chemi- coefﬁcient,” and it is related to the (α+β)th order cal oscillation (cf. R.M. Noyse, J. Chem. Educ., rate coefﬁcient k by the equation: 66, 190, 1989). β m M kobs = k[B] . orientation (of an -dimensional manifold ) A nondegenerate m-form over M.If(M, g) is a 1 For the common case when α = 1, kobs is often (pseudo)-Riemannian manifold and ds = dx ∧ x2 ∧···∧ xm m referred to as a “pseudo-ﬁrst-order rate coefﬁ- d d is the canonical√ local basis of - cient” (kψ ). forms over M, then η = gds is an orientation. For a simple (elementary) reaction a partial The manifolds that allow orientations are order of reaction is the same as the stoichio- called orientable, and they allow oriented metric number of the reactant concerned and atlases, i.e., atlases with transition functions with must therefore be a positive integer. The overall deﬁnite positive Jacobians.

94 orthonormal set

Ornstein-Uhlenbeck process A Gaussian, Thence the inverse of an orthogonal matrix coin- Markov stochastic processes deﬁned by a linear cides with its transpose. Example: Rotations in stochastic differential equation R3 (as well as reﬂections) are elements of O(3). Analogously, the orthogonal group O(r,s) dX =−bXdt + adW(t) (with r + s = m) is the group of isometries of the indeﬁnite metric of signature (r, s) on Rm. a,b > W(t) in which 0, and is the Wiener pro- Example: The Lorentz group is the orthogonal dW/dt cess, i.e., is the white noise. group O(1,m− 1). The subgroup of isometries α : Rm → Rm V V m orthogonal Describing two elements 1, 2 which preserve the orientation of (R ,η) is /v v 0= of a Hilbert space such that 1, 2 0. denoted by SO(r, s), and it is called the special orthogonal group. orthogonal group The group O(m) is the Rm group of isometries of with the standard posi- orthonormal basis An orthonormal set tive deﬁnite metric δ. Let us choose an ortho- m which is also a basis. Synonym: complete normal basis Ei in R , the metric δ is expressed i j i orthonormal set. by δ = δij E ⊗ E (where E is the dual α Rm → Rm basis). An endomorphism : is an orthonormal set A set {uα} of a Hilbert O(m) element of the orthogonal group if and space H satisfying only if a b j 0ifα = β Ai δab Aj = δij α(Ei ) = Ai Ej . /uα,uβ 0= . 1ifα = β

95

partition of unity

to pull-back. An important consequence is that nodal interpolation operators and pull-backs commute: F (I u ) = I (F (u)) , P ) h |T h ) |K for all sufﬁcient smooth functions/vector ﬁelds u T p-Laplacian equation Let U ⊂ Rn be open on . {; } and u : U × R → R. The p-Laplacian equation Given an inﬁnite family of meshes h h∈H, H for u is an index set, a related family of ﬁnite ele- div(|Du|p−2Du) = 0, ment spaces is considered parametric equivalent, if there is a small number of ﬁnite elements to where Du = Dx u = (ux ,...,ux ) denotes the 1 n which all elements of the family are parametric gradient of u with respect to the spatial variable equivalent. These elements are called reference x = (x ,...,x ) 1 n . elements. Parametric equivalence is a key tool in most proofs of local a priori interpolation estimates for p-version of ﬁnite elements The ﬁnite families of ﬁnite element spaces and their associ- element discretization of a boundary value prob- ated nodal interpolation operators. If the meshes lem is built upon a ﬁxed mesh. A family of are shape regular, it is often possible to gauge ﬁnite elements is employed that supplies ﬁnite the change of norms under pull-back. First the element spaces for a wide range of local polyno- interpolation error is estimated on the reference mial degrees. The idea of the p-version of ﬁnite element(s). Then, the relationship of commuta- elements is to get an acceptable discrete solu- tivity given above is used to conclude an estimate tion by using a sufﬁciently high degree polyno- for an arbitrary element. mial. A local variant of the p-version raises the polynomial degree only on cells where a better partial differential equation A differential approximation is really needed to reduce the dis- equation involving a function of more than one cretization error; cf., adaptive reﬁnement. variable, and hence partial derivatives. parallel transport A vector ﬁeld X deﬁned partition A set of subgraphs, {G(V, E), along a curve γ in a manifold M with a connec- G(V, E),...} of a graph G(V, E), such that tion N such that the subgraphs are disjoint.

∇γ˙ X = 0 partition of unity Let M be a paracompact manifold and {U } a locally ﬁnite covering γ˙ α α∈I where denotes the tangent vector to the (see open covering). A partition of unity relative γ curve . to {Uα}α∈I is a family of (smooth) local functions fα : M → R such that: parametric equivalence Two ﬁnite ele- ∀x ∈ M f (x) ≥ ments (K, VK ,XK ) and (T , VT ,XT ) are para- (i.) , α 0; metric equivalent, if there is a piecewise smooth (ii.) suppfα ⊂ Uα; ) K → T bijective mapping : and a bijective (iii.) ∀x ∈ M, α∈I fα(x) = 1; linear mapping F : V → V , the pull-back, ) T k f such that where supp α denotes the support of the function fα, i.e., the closure of the set {x ∈ M : fα(x) = } XT ={κ : VT → C, κ(u) = φ(F)u), φ ∈ Xk}. 0 . Notice that the sum in P3 is ﬁnite since the Sloppily speaking, this means that both the geo- covering {Uα}α∈I is locally ﬁnite. The functions metric elements and the local spaces can be fα of a partition of unity can be required to be mapped onto each other, and that the local real-analytic only in trivial situations. In fact, degrees of freedom are invariant with respect any analytical function which is identically zero

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 97 parts of collections outside Uα is zero everywhere. On the contrary, valid subsequences, but /d,c0 is not a subse- the functions fα can be required to satisfy less quence of S. See also bag, empty collection, list, stringent regularity conditions such as C∞. sequence, set, and superset. Partitions of unity are often used to prove existence of a global object once local objects are known to exist in M. For example, let M be path An alternating sequence of nodes and ∞ (V, E) a (paracompact) C manifold and {(Uα,xα)}α∈I edges drawn from a graph G , such that they (V, E) an atlas of M. This means that Uα are diffeo- form a connected subgraph, G , where m V ⊆ V E ⊆ E morphic (via the maps xα : Uα → R )toan and . m open set xα(Uα) ⊂ R (m = dim(M)). Con- Comment: A path through a graph is simply sequently, there exist local (strictly) Riemann- a connected subgraph whose nodes and edges ian metrics gα induced on Uα (via the maps appear in sequence: just begin at the beginning m xα : Uα → R ) by the standard metric and walk along the path to the end. If one permits m δαβ on R . Notice that metrics are second oneself to let the nodes of the path be implicitly rank, nondegenerate, positive deﬁnite tensors. represented by the edges, the path becomes a If {(Uα,fα)} is a partition of unity relative to sequence of edges. See also pathway, sequence, the open covering {Uα}, we can then deﬁne the and terminal nodes. second-order tensors fα gα, one for each α ∈ I. They are global tensors but they identically van- pathway A sequence of biochemical reac- ish outside Uα so that they are not suitable to tions and their compounds whose nodes and deﬁne a Riemannian metric over M. Let us, how- edges form a path and have historically been con- ever, consider the combination g = α∈I fα gα (which exists since {Uα} is locally ﬁnite). Then sidered by biochemists to be a pathway. g is a global Riemannian metric over M, since Comment: This deﬁnition seems a little linear combinations with positive coefﬁcients of circular, but in fact what we deﬁne as biochem- positive deﬁnite tensors are still positive deﬁnite. ical pathways is largely determined by the his- The same argument cannot be applied in gen- tory and results of the experiments involved in eral to arbitrary signature and in particular to the their discovery. Deﬁnitions of particular path- Lorentzian case. In those cases, in fact, further ways vary slightly among different sources. For topological conditions have to be satisﬁed for a example, some authors include phosphorylation global metric to exist. of d-glucopyranose as part of glycolysis; others do not. Still others refer to the next step as the parts of collections For any bag, list, ﬁrst “committed step” in glycolysis. See also sequence, or set, a subpart (subbag, sublist, sub- path, sequence, and terminal nodes. sequence, or subset) is a portion of the original collection. If the part is less than the original pathwise connected The property of a topo- collection, it is a proper subbag, sublist, subse- logical space (X, τ(X)) that any two points can quence, or subset, and we denote the relationship be joined by a curve. See connected. Pathwise between the part and the collection by part ⊂ col- connectedness implies connectedness. The con- lection. Otherwise, the relationship is denoted by verse is false. ⊆ to indicate the part may be equal to the whole. Example: The subset in R2 given by the union Comment: A set, bag, list, or sequence may L ={( ,x) x ∈ , } contain another of its type. For example, {a,b}⊂ of the set 0 : [0 1] and the set S ={(x, ( 1 )) ≤ x} {a,b,c} and /a,b0⊂/a,b,c0. sin x :0 is connected but not In forming parts of collections, bear in pathwise connected. mind that the part must satisfy the same properties as the whole. For example, if we pattern See motif. have S =/a, b, c, d, e0, then any derived subsequence must have the same precedence relations: {/a,b0, /b, c, d0, /d,e0} is a set of pendant node See singleton node.

98 photon irradiance pericyclic reaction A chemical reaction in photon ﬂow, )p The number of photons which concerted reorganization of bonding takes (quanta, N) per unit time. (dN/dt, simpliﬁed place throughout a cyclic array of continu- expression: )p = N/t when the number of pho- ously bonded atoms. It may be viewed as a tons is constant over the time considered.) The − reaction proceeding through a fully conjugated SI unit is s 1. Alternatively, the term can be used cyclic transition state. The number of atoms in with the amount of photons (mol or its equivalent −1 the cyclic array is usually six, but other num- Einstein), the SI unit then being mol s . bers are also possible. The term embraces a 0 photon ﬂuence, Hp The integral of the cycloadditions, p variety of processes, including amount of all photons (quanta) which traverse cheletropic reactions, electrocyclic reactions, a small, transparent, imaginary spherical target, and sigmatropic rearrangements, etc. (provided divided by the cross-sectional area of this tar- they are concerted). E0 get. The photon ﬂuence rate, p, integrated over 0 the duration of the irradiation ( Epdt, simpli- 0 0 0 photoelectrical effect The ejection of an ﬁed expression: Hp = Ept when Ep is constant electron from a solid or a liquid by a photon. over the time considered). Photons per unit area (quanta m−2). The SI unit is m−2. Alternatively, the term can be used with the amount of photons photoelectron spectroscopy (PES) A spec- (mol or its equivalent Einstein), the SI unit then troscopic technique that measures the kinetic being mol m−2. energy of electrons emitted upon the ionization E0 of a substance by high energy monochromic photon ﬂuence rate, p The rate of photon photons. A photoelectron spectrum is a plot ﬂuence. Four times the ratio of the photon ﬂow, ) of the number of electrons emitted vs. their p, incident on a small, transparent, imaginary kinetic energy. The spectrum consists of bands spherical volume element containing the point due to transitions from the ground state of an under consideration divided by the surface of that sphere, S .( L dw, simpliﬁed expression: atom or molecular entity to the ground and K 4π p E0 = 4) /S when the photon ﬂow is constant excited states of the corresponding radical cation. p p K over the solid angle considered). The SI unit Approximate interpretations are usually based on is m−2s−1. Alternatively, the term can be used “Koopmans theorem” and yield orbital energies. with the amount of photons (mol or its equivalent PES and UPS (UV photoelectron spectroscopy) Einstein), the SI unit then being mol m−2 s−1.It refer to the spectroscopy using vacuum ultra- reduces to photon irradiance for a parallel and violet sources, while ESCA (electron spec- normally incident beam not scattered or reﬂected troscopy for chemical analysis) and XPS use by the target or its surroundings. X-ray sources. photon ﬂux Synonymous with photon irra- diance. photoemissive detector A detector in which a photon interacts with a solid surface, photon irradiance, Ep The photon ﬂow, which is called the photocathode, or a gas, )p, incident on an inﬁnitesimal element of sur- releasing a photoelectron. This process is called face containing the point under consideration the external photoelectric effect. The photoelec- divided by the area (S) of that element (d)p/ds, trons are collected by an electrode at positive simpliﬁed expression: Ep = )p/S when the electric potential, i.e., the anode. photon ﬂow is constant over the surface con- sidered). The SI unit is m−2s−1. Alternatively, the term can be used with the amount of photons photon Particle of zero charge, zero rest (mol or its equivalent Einstein), the SI unit then mass, spin quantum number 1, energy hν, and being mol−2s−1. For a parallel and perpendicu- momentum hν/c (h is the Planck constant, ν the larly incident beam not scattered or reﬂected by frequency of radiation, and c the speed of light); the target or its surroundings photon ﬂuence rate 0 carrier of electromagnetic force. (Ep) is an equivalent term.

99 phylogenetic motif phylogenetic motif Any motif conserved Functions on P with Poisson bracket form a over biological species. See also biochemical, Lie algebra. There exists a Lie algebra homo- chemical, dynamical, functional, kinetic, mech- morphism between the Lie algebra of functions anistic, regulatory, thermodynamic, and topo- on P and vector ﬁelds, given by logical motives. H → XH (f ) ={H,f}. physical quantity (measurable quantity) An attribute of a phenomenon, body, or substance It is an isomorphism when constants are quo- that may be distinguished qualitatively and deter- tiented out of functions. mined quantitatively. More generally, let P be a smooth manifold and C∞(P ) the space of smooth functions. A P planar graph A graph that can be drawn in Poisson bracket or Poisson structure on is an { , } C∞(P ) × C∞(P ) → C∞(P ) the plane with no edges crossing. operation : satisfying the following plasmid An extrachromosomal genetic ele- (i.) {F,G} is real, bilinear in F and G, ment consisting generally of a circular duplex (ii.) {F,G,}=−{G, F }, skew symmetric, of DNA which can replicate independently of {{F,G},H}+{{H,F},G}+{{G, H }, chromosomal DNA. R-plasmids are responsible (iii.) F }= for the mutual transfer of antibiotic resistance 0, Jacobi identity, among microbes. Plasmids are used as vectors (iv.) {FG,H}=F {G, H }+{F,H}G, Leib- for cloning DNA in bacteria or yeast host cells. niz rule. ploidy A term indicating the number of sets Poisson equation Let U ⊂ Rn open and u : of chromosomes present in an organism, e.g., U × R → R. The (nonlinear) Poisson equation haploid (one) or diploid (two). for u is −u = f(x). Poisson-Boltzmann equation A mathemat- ical model for the ionic gas (plasma) or ionic solution. The nonlinear equation is established Poisson manifold A smooth manifold P based on two physical laws: the Poisson equation with a Poisson bracket { , }. Examples: Sym- relating the electric potential to charge distribu- plectic manifolds are Poisson manifolds, where tion, and the Boltzmann law relating the charge the Poisson bracket is deﬁned by distribution to electric potential. It has been shown that this equation is a good model for {F,G}(x) = ω(x)(XF (x), XG(x)), x ∈ M many applications even though it suffers from some thermodynamic inconsistency. The lin- where XF is the Hamiltonian vector ﬁeld of F . earized equation and its solution is known as Debye-H¨uckel theory. The one-dimensional ver- polar coordinates The parameterization of sion of the nonlinear equation can be applied the plane R2 as X = (r, θ), where r>0 is the to modeling the charge distribution near a ﬂat, absolute value |X| and θ is the angle (in radians) charged membrane; this is known as Guy- between the horizontal axis and the segment from Chapman equation. O to X.

Poisson bracket Let (P, ω) be a symplectic A z manifold, ξ a system of canonical coordinates pole A complex number 0 is a pole of a func- f(z) f(z) < |z − z | <" and f , g two real functions on P . Then the Pois- tion if is analytic in 0 0 , "> z son bracket of f and g is deﬁned as the function for some 0 and not analytic at 0,but (z − z )nf(z) z 0 is analytic at 0, for some positive −1 AB {f, g}=ω (df, dg) = ω (∂Af )(∂B g). integer n.

100 potential-energy proﬁle pollution effect If some feature of a problem Post production system (PPS) A ﬁnite has an indirect and nonobvious adverse impact grammar X over a ﬁnite set of symbols K which on the accuracy that can be achieved by a dis- produces a set of constructs, C, from an initial c ∈ C C cretization scheme, a pollution effect can be construct 0 , such that each construct in observed. A prominent example is the deteriora- can be obtained from one or more symbols in tion of certain asymptotic rates of convergence of K by a ﬁnite derivation, where each step in the ﬁnite element schemes for second-order elliptic derivation is a rule πi ∈ X. boundary value problems in the presence of reen- Comment: The constructs were strings repre- trant corner. Another case is the loss of accuracy senting logical predicates, and the grammar the suffered by standard ﬁnite element schemes for rules of logical derivation and proof, in Post’s the Helmholtz equation −u − κu = f when κ original formulation (E.L. Post, “Finite combin- becomes large. atory processes. Formulation I”, J. Symbolic Logic, 1, 103–105 (1936)). The class of Post- generated construct sets is a member of the class polymer A substance composed of macro- of recursively enumerable sets. The grammar molecules. may be deterministic or nondeterministic, but its execution is not stochastic. See universal Turing polymerization The process of converting machine and von Neumann machine. a monomer or a mixture of monomers into a postﬁx An operator written after its polymer. operands. Thus for two operands x and y and operator f , the syntax is xyf . polynomial A linear combination of non- Comment: Remember calculators with negative, integer powers of a variable (or reverse Polish notation? In many instances this unknown) x, is the most natural order of all; for example, exponentiation. See also functor, inﬁx, preﬁx, p(x ) = a + a x +···+a xn. 0 1 n and relation.

posynomial A positive sum of polynomials population genetics One can study genetics a(i,j) i cj Xj x(j) , where cj > 0. Each mono- from either a molecular point of view (see double mial is the product helix) or from a population point of view. Popula- a(i,j) a(i,1) a(i,2) a(i,n) tion genetics studies where the gene is located in Xj x(j) = x(1) ·x(2) ·...·x(n) a chromosome, and how it is passed from gener- a(i,j) ation to generation from the pedigree of heredi- and [ ] is called the exponent matrix. tary markers. This is the fundamental function in geometric programming. population model A system of mathemat- potential energy, Ep,V Energy of position ical equations representing the proposed relation or orientation in a ﬁeld of force. between the population sizes and their growth rates. The often used equations in textbooks, potential-energy proﬁle A curve describ- such as exponential growth, logistic equation, ing the variation of the potential energy of the predator-prey, and competition are simpliﬁed system of atoms that make up the reactants and population models. products of a reaction as a function of one geo- metric coordinate, and corrresponding to the “energetically easiest passage” from reactants to U ⊂ Rn porous medium equation Let products (i.e., along the line produced by join- u U × R → R open and : . The porous medium ing the paths of steepest descent from the transi- equation for u is tion state to the reactants and to the products).

γ For an elementary reaction, the relevant geo- ut − (u ) = 0. metric coordinate is the reaction coordinate.For

101 potential-energy (reaction) surface a stepwise reaction it is the succession of reaction power set (of a set A) The set of all subsets coordinates for the successive individual reac- of A (the empty subset included). It is denoted by tion steps. (The reaction coordinate is some- P(A) or 2A. For example, the empty set ∅ has no times approximated by a quasi-chemical index of element. Its power set P(∅) = {∅} has exactly reaction progress, such as degree of atom transfer one element. The power set of A ={a,b} is or bond order of some speciﬁed bond.) P(A) ={∅, {a}, {b},A}. In general, if A has n elements, the power set has 2n elements. potential-energy (reaction) surface A geo- metric hypersurface on which the potential predator-prey population model A math- energy of a set of reactants is plotted as a func- ematical model, in terms of a pair of ordinary tion of the coordinates representing the molecu- differential equations, representing the popula- lar geometries of the system. tions of two species with prey-predator relation. For simple systems two such coordinates Such systems are likely to exhibit oscillation or (characterizing two variables that change during even limit cycles. Generalization to more than the progress from reactants to products) can be two species is possible and, in fact, is widely selected, and the potential energy plotted as a used in applications. contour map. For simple elementary reactions, e.g., A − preﬁx An operator written before its B + C → A + B − C, the surface can show operands. Thus for two operands x and y and the potential energy for all values of the A,B,C operator f , the syntax is fxy or f(x,y). See geometry, providing that the ABC angle is ﬁxed. also functor, inﬁx, postﬁx, and relation. For more complicated reactions a different choice of two coordinates is sometimes pre- prey-predator relation Two biological spe- ferred, e.g., the bond orders of two different cies are in the following relation: the growth bonds. Such a diagram is often arranged so that rate of the predator increases with increasing reactants are located at the bottom left corner and prey population, and the growth rate of the prey products at the top right. If the trace of the rep- decreases with increasing predator population. resentative point characterizing the route from The popular examples are ﬁsh and sharks, and reactants to products follows two adjacent edges deer and wolves. of the diagram, the changes represented by the two coordinates take place in distinct succes- primary sample The collection of one of sion. If the trace leaves the edges and crosses more increments or units initially taken from a the interior of the diagram, the two changes are population. concerted. In many qualitative applications it is The potions may be either combined (com- convenient (although not strictly equivalent) for posited or bulked sample) or kept separate (gross the third coordinate to represent the standard sample). If combined and mixed to homogeneity, Gibbs energy rather than potential energy. it is a blended bulk sample. The term “bulk sam- Using bond orders is, however, an oversim- ple” is commonly used in the sampling literature pliﬁcation, since these are not well deﬁned, even as the sample formed by combining increments. for the transition state. (Some reservations con- The term is ambiguous since it could also mean cerning the diagrammatic use of Gibbs energies a sample from a bulk lot, and it does not indicate are noted under Gibbs energy diagram.) whether the increments or units are kept separ- The energetically easiest route from reactants ate or combined. Such use should be discour- to products on the potential-energy contour map aged because less ambiguous alternative terms deﬁnes the potential-energy proﬁle. (composite sample, aggregate sample) are avail- able. “Lot sample” and “batch sample” have also power series A formal summation of the been used for this concept, but they are self- ∞ c (x − a)n c ,c ,... limiting terms. The use of “primary” in this sense form n , where a and 0 1 are n=0 is not meant to imply the necessity for multistage complex numbers. sampling.

102 pseudodifferential operator primitive change One of the conceptually reversible reaction. See also dextralateral, direc- simpler molecular changes into which an ele- tion, dynamic equilibrium, formal reaction equa- mentary reaction can be notionally dissected. tion, microscopic reversibility, rate constant, Such changes include bond rupture, bond for- reversibility, sinistralateral, and substrate. mation, internal rotation, change of bond length or bond angle, bond migration, redistribution of projectable vector ﬁeld A projectable vec- (B,M,π; F) charge, etc. tor ﬁeld is a vector ﬁeld on a bundle The concept of primitive change is helpful inducing by projection a vector ﬁeld on the base M in the detailed verbal description of elementary manifold . Locally, it is expressed as µ i reactions, but a primitive change does not repre- \ = ξ (x)∂µ + ξ (x, y)∂i sent a process that is by itself necessarily observ- µ i able as a component of an elementary reaction. where (x ; y ) are ﬁbered coordinates. The ﬂow of a projectable vector ﬁeld is formed by ﬁbered principal bundle A bundle (P,M,p; G) automorphisms. Vector ﬁelds that are not pro- with a Lie group G as standard ﬁber and transi- jectable do not preserve the bundle structure. tion functions valued in the same G and acting protein folding The dynamical process of on G by means of left translations L : G → g obtaining its structure (see protein structure), G : h → g · h.Onprincipal bundles a global under appropriate conditions such as tempera- and canonical right action R of G is deﬁned by g ture, pH, etc., from an arbitrary initial structure. In test tubes, it is known experimentally that Rg : P → P : (x, h) → (x, h · g). many proteins can reach their respective, almost This action is free, transitive on ﬁbers and verti- unique three-dimensional structures. How this cal (i.e., it is formed by vertical automorphisms), process works is still not completely known, and it completely characterizes the principal even though it is generally accepted that the pro- bundle. An object is called equivariant if it pre- cess is governed by the intramolecular interac- serves the canonical right action Rg. tions between atoms and interaction between the On a principal bundle there is a one-to-one molecule and water in which the process occurs. correspondence between local trivializations and protein structure A protein molecule has local sections. Hence, a principal bundle is triv- a well-deﬁned three-dimensional structure in ial if and only if it allows a global section. terms of the relative spatial coordinates of all its atoms. Most of our current knowledge about principal connection A connection over a the protein structures are derived from labora- principal bundle (P,M,p; G) which preserves tory determinations of the structures of many the canonical right action, i.e., proteins using the methods of x-ray crystallogra- phy or nuclear magnetic resonance spectroscopy. TRg(Hp) = Hp·g. The former requires a protein to form a crystal, In physics it is also called a gauge ﬁeld. while the latter can obtain the structure of a pro- tein in aqueous solution. No reliable computa- product In chemistry and biochemistry, the tional method exists yet for obtaining the spatial molecular species yielded by a reaction running structure of a protein from its chemical struc- in a particular direction. ture (known as its amino acid sequence in bio- Comment: The distinction is between the chemical literature). molecular result of a reaction and where a com- pseudodifferential operator A pseudo- pound is represented in the arbitrarily written for- P k mal reaction equation. When a reaction occurs differential operator of order on a com- pact manifold M is locally of the form: for any in a particular direction (either forward or back- u ∈ C∞(M) ward), the substrates and products are known, c and this identiﬁcation of molecules with chem- P u(x) = ical roles is independent of where the molecules ( π)−n ei(x−y)·ξ a(x,y,ξ) × u(y)dydξ, appear in the formal equation representing the 2

103 psychometry where a(x,y,ξ) is a symbol of order k. This purine bases Purine and its substitu- deﬁnes a bounded linear operator between the tion derivatives, especially naturally occurring s s−k Sobolev spaces P : Hc (M) → Hc (M). examples. psychometry The use of a wet-and-dry bulb pyrimidine bases Pyrimidine and its substi- thermometer for measurement of atmospheric tution derivatives, especially naturally occurring humidity. examples.

104 quotient space

property that the discrete solutions uh satisfy u − u ≤ C u − v . h V infvh∈Vh h V

Here, u ∈ V denotes the solution of the continu- Q ous variational problem. In a strict sense, the constant C>0 must not depend on the choice of Vh. In a loose sense, it may depend on the family QCD See chromodynamics, quantum. of ﬁnite elements, but not on the triangulation, on which Vh is built. In a very loose sense, if fam- quantum yield, ) The number of deﬁned ilies of triangulations are involved, one demands events which occur per photon absorbed by the that the constant C be independent of the mesh system. The integral quantum yield is: width, but it may depend on shape regularity. ) = (number of events)/(number of photons absorbed) {; } , H For a photochemical reaction: quasi-uniform A family h h∈H an ; ⊂ Rn ) = (amount of reactant consumed or product index set, of triangulations of a domain C> formed)/(amount of photons absorbed) is called quasi-uniform, if there exists a 0 such that d[x]/dt ) = max{diam(K), K ∈ ;h} n sup ,h∈ H ≤ C. min{diam(K), K ∈ ;h} where d[x]/dt is the rate of change of a mea- surable quantity, and n the amount of photons Sloppily speaking, the cells of all meshes of (mol or its equivalent einstein) absorbed per unit a quasi-uniform family have about the same time. ) can be used for photophysical processes diameter. This permits us to provide asymptotic or photochemical reactions. a priori estimates for interpolation and approxi- mation errors of ﬁnite element spaces in terms of quasi-optimal Describing a V -conforming the mesh widths. ﬁnite element scheme used to discretize a vari- ational problem posed over the space V with the quotient space See equivalence relation.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 105

radius of gyration

restricted to those radicals which do not form parts of radical pairs. Depending upon the core atom that pos- sesses the unpaired electron, the radicals can be R described as carbon-, oxygen-, nitrogen-, metal- centered radicals. If the unpaired electron occu- pies an orbital having considerable s or more or radiant (energy) ﬂux, P,) Although ﬂux less pure p character, the respective radicals are is generally used in the sense of the “rate of trans- termed σ -orπ-radicals. fer of ﬂuid, particles or energy across a given sur- In the past, the term “radical” was used to des- face,” the radiant energy ﬂux has been adopted ignate a substituent group bound to a molecular by IUPAC as equivalent to radiant power, P . entity, as opposed to “free radical,” which now- (P = ) = dQ/dt, simpliﬁed expression: P = adays is simply called radical. The bound entities ) = Q/t when the radiant energy, Q, is constant may be called groups or substituents, but should ) over the time considered). In photochemistry, no longer be called radicals. is reserved for quantum yield. radioactive The property of a nuclide of radiant exposure, H The irradiance, E undergoing spontaneous nuclear transformations integrated over the time of irradiation ( Edt, with the emission of radiation. simpliﬁed expression H = Et when the irradi- ance is constant over the time considered). For a parallel and perpendicularity, incident beam not radioactive decay Nuclear decay in which scattered or reﬂected by the target or its surround- particles or electromagnetic radiation are emit- (H ) ted or the nucleus undergoes spontaneous ﬁssion ings, ﬂuence 0 is an equivalent term. or electron capture. radiant power, P,) Same as radiant (energy) ﬂux, ). Power emitted, transferred, or radioactivity The property of certain received as radiation. nuclides of showing radioactive decay. radiation A term embracing electromag- radiochemistry That part of chemistry netic waves as well as fast moving particles. In which deals with radioactive materials. It radioanalytical chemistry the term usually refers includes the production of radionuclides and to radiation emitted during a nuclear process their compounds by processing irradiated mate- (radioactive decay, nuclear reaction, nuclear ﬁs- rials or naturally occurring radioactive materials, sion, accelerators). the application of chemical techniques to nuclear studies, and the application of radioactivity to radical (free radical) A molecular entity the investigation of chemical, biochemical, or CH ,˙SnH ,Cl· such as 3 3 possessing an unpaired biomedical problems. electron. (In these formulae the dot, symboliz- ing the unpaired electron, should be placed so radioluminescence Luminiscence arising as to indicate the atom of highest spin density, if from excitation by high energy particles or this is possible.) Paramagnetic metal ions are not radiation. normally regarded as radicals. However, in the “isolobal analogy” the similarity between certain paramagnetic metal ions and radicals becomes radionuclide A nuclide that is radioactive. apparent. At least in the context of physical organic radius of gyration, s A parameter charac- chemistry, it seems desirable to cease using the terizing the size of a particle of any shape. adjective “free” in the general name of this type For a rigid particle consisting of mass ele- of chemical species and molecular entity, so that ments of mass mi , each located at a distance ri the term “free radical” may in the future be from the center of mass, the radius of gyration s,

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 107 range is deﬁned as the square root of the mass-average which exerts a strong effect, stronger than that 2 of the ri for all the mass elements, i.e., of any other rate constant, on the overall rate. It is recommended that the expressions rate- m r2 1/2 s = i i i controlling, rate-determining, and rate-limiting i mi be regarded as synonymous, but some special meanings sometimes given to the last two expres- For a nonrigid particle, an average overall con- sions are considered under a separate heading. formation is considered, i.e., A rate-controlling step can be formally 2 1/2 deﬁned on the basis of a control function (or / mi r 0 /s201/2 = i i control factor) CF, identiﬁed for an elementary m 1/2 i i reaction having a rate constant ki by

The subscript zero is used to indicate unperturbed CF = (∂ ln v∂ ln k )K ,k /s201/2 i j j dimensions, as in 0 . where v is the overall rate of reaction. In per- r A range Let be a relation with domain and forming the partial differentiation all equilibrium B a ∈ A r(a) ∈ B codomain , and let , . Then the constants K and all rate constants except k are r r(a ), ∀a ∈ A j i range of is the set of all images, i i . held constant. The elementary reaction having r(A) I The range is denoted by either or . the largest control factor exerts the strong inﬂu- Comment: See the comment on relation for ence on the rate v and a step having a CF much more details. In particular, note that the range larger than any other step may be said to be rate- codomain and of a relation are not necessarily controlling. equivalent. See also domain, image, and rela- A rate-controlling step deﬁned in the way tion. recommended here has the advantage that it is directly related to the interpretion of kinetic iso- rate coefﬁcient See order of reaction and tope effects. kinetic equivalence. As formulated, this implies that all rate con- stants are of the same dimensionality. Consider, rate constant The parameter expressing the however, the reaction of A and B to give an inter- intrinsic rate at which a reaction can proceed in mediate C, which then reacts further with D to a particular direction; denoted k, usually with a give products subscript indicating which direction of the reac- tion is meant. k −→1 A + B ←− C(1) Comment: This parameter is also called the k −1 “intrinsic rate constant.” For any reaction the k quotient of the rate constants (forward over C + D −→2 Products.(2) reverse) is the equilibrium constant for the reac- tion as written in a particular formal reaction Assuming that C reaches a steady state, then the equation. The actual rate achieved in a particu- observed rate is given by lar direction is the product of the rate constant k k A B D and the concentrations of each of the obliga- v = 1 2[ ][ ][ ]. k + k D torily coreacting species for that direction. See −1 2[ ] also dextralateral, direction, dynamic equilib- k D rium, formal reaction equation, microscopic Considering 2[ ] a pseudo-ﬁrst-order rate con- k D >> k reversibility, product, reversibility, sinistra- stant, then 2[ ] −1, and the observed rate v = k A B k = k lateral, and substrate. 1[ ][ ] and obs 1. Step (1) is said to be the rate-controlling step. k D << k rate-controlling step The rate-controlling If 2[ ] −1, then the observed rate (rate-determining or rate-limiting) step in a reac- k k tion occurring by a composite reaction sequence v = 1 2 A B D . k [ ][ ][ ] is an elementary reaction, the rate constant for −1

108 reduced viscosity rate law (empirical differential rate equation) reaction coordinate is typically chosen to follow An expression for the rate of reaction of a particu- the path along the gradient (path of shallowest lar reaction in terms of concentrations of chem- ascent/deepest descent) of potential energy from ical species and constant parameters (normally reactants to products. rate coefﬁcients and partial orders of reaction) The term has also been used interchangeably only. For examples of rate laws see equations with the term transition coordinate, applicable (1)-(3) under kinetic equivalence. to the coordinate in the immediate vicinity of the potential energy maximum. Being more speciﬁc, rate of change (of a function f(t), with respect the name transition coordinate is to be preferred t t t to )(1) Average rate of change over [ 1, 2] in that context. is f(t ) − f(t ) U ⊂ Rn 2 1 . reaction-diffusion equation Let t − t u U × R → Rn 2 1 be open and : . The reaction- u t = t diffusion equation for is (2) Instantaneous rate of change at 1 is df ut − u = f (u). (t ). dt 1 reaction path (1) A synonym for mech- reactant A substance that is consumed in anism. the course of a chemical reaction. It is some- (2) A trajectory on the potential-energy times known, especially in the older literature, as surface. a reagent, but this term is better used in a more (3) A sequence of synthetic steps. specialized sense as a test substance that is added reaction’s compounds The set of molecular to a system in order to bring about a reaction or to species, including catalysts, which participate in see whether a reaction occurs (e.g., an analytical that reaction. See compound’s reactions. reagent). reactive (reactivity) As applied to a chem- reaction Any chemical or biochemical ical species, the term expresses a kinetic prop- transformation of matter and energy. erty. A species is said to be more reactive or Comment: This deﬁnition includes both spon- to have a higher reactivity in some given con- taneous and catalyzed reactions. The word text than some other (reference) species if it has “transformation” is used in the general English a larger rate constant for a speciﬁed elemen- sense, not the mathematical or linguistic sense. tary reaction. The term has meaning only by reference to some explicitly stated or implicitly reaction coordinate A geometric param- assumed set of conditions. It is not to be used eter that changes during the conversion of one for reactions or reaction patterns of compounds (or more) reactant molecular entities into one in general. (or more) product molecular entities and whose The term is also more loosely used as a phe- value can be taken for a measure of the progress nomenological description not restricted to ele- of an elementary reaction (for example, a bond mentary reactions. When applied in this sense length or bond angle or a combination of bond the property under consideration may reﬂect not lengths and/or bond angles; it is sometimes only rate, but also equilibrium, constants. approximated by a nongeometric parameter, such as the bond order of some speciﬁed bond). reduced viscosity (of a polymer) The ratio In the formalism of “transition-state theory,” the of the relative viscosity increment to the mass reaction coordinate is that coordinate in a set of concentration of the polymer, c, i.e., ηi /c, where curvilinear coordinates obtained from the con- ηi is the relative viscosity increment. ventional ones for the reactants which, for each Notes: (1) The unit must be speciﬁed; cm3g−1 reaction step, leads smoothly from the conﬁgur- is recommended. ation of the reactants through that of the transition (2) This quantity is neither a viscosity nor a state to the conﬁguration of the products. The pure number. The term is to be looked on as

109 reference material a traditional name. Any replacement by con- and f : A → B, with f : a → b. Notice that sistent terminology would produce unnecessary according to this deﬁnition, which is standard in f x → 1 confusion in the polymer literature. Synony- algebra, the map : x is not a function mous with viscosity number. f : R → R since it is not deﬁned in 0 ∈ R. Of course, it is a function on the smaller space reference material A substance or mixture R −{0}. of substances, the composition of which is known within speciﬁed limits, and one or more of the relative molar mass Molar mass divided by properties of which is sufﬁciently well estab- 1 g mol−1 (the latter is sometimes called the lished to be used for the calibration of an appar- standard molar mass). atus, the assessment of a measuring method or for assigning values to materials. Reference materials are available from national laborator- relative molecular mass, Mr Ratio of the ies in many countries [e.g., National Institute for mass of a molecule to the uniﬁed atomic mass Standards and Technology (NIST), U.S., Com- unit. Sometimes called the molecular weight or munity Bureau of Reference, U.K.]. relative molar mass. regulatory motif A conserved topological motif which reduces or increases the activity of relative responsivity See responsivity. a gene, enzyme, or macromolecular complex. Comment: See also biochemical, chemical, relative viscosity The ratio of the viscosity dynamical, functional, kinetic, mechanistic, η of the solution to the viscosity ηs of the sol- phylogenetic, thermodynamic, and topological vent, i.e., η = η/η . Synonymous with viscos- motives. r s ity ratio. relation (between two sets) A relation between A and B is any subset R ⊂ A×B of the relative viscosity increment The ratio of Cartesian product of the two sets. If (a, b) ∈ R the difference between the viscosities of solution we say that a is related to b (in that order), and and solvent to the viscosity of the solvent, i.e., we write aRb.IfR can be produced by some ηi = (η − ηs )/ηs , where η is the viscosity of the operation or procedure r, then we often call r the solution and ηs is the viscosity of the solvent. relation. The use of the term “speciﬁc viscosity” for For example, consider A ={a, b,c} and B = this quantity is discouraged, since the relative {1, 2, 3}, and let r(ai ,bi ), ai ∈ A, bi ∈ B,1 ≤ viscosity increment does not have the attributes i ≤ 3. Then R ={(a, 1), (b, 2), (c, 3)}. of a speciﬁc quantity. Consider a relation r between two sets A and B. A is called the domain of r and B its codomain. The elements of B related, via a rela- relaxation oscillator For some oscillatory tion r to an element a ∈ A is called the image of a dynamics with a limit cycle, parts of the cycle are under r, and denoted by r(a). The union r(A) of traversed quickly in comparison with other parts. all r(a),a ∈ A, is called the range of r, and can This is often referred to as a relaxation oscilla- also be denoted I. Thus, r(A) may be a proper tor. This phenomenon suggests that the under- subset of B. So, the range is not necessarily the lying ordinary differential equations for such a same as the codomain. system have a small parameter which is present See also domain, image, and range, and rela- in a crucial place to cause this rapid variation in tion. the solution. A widely used class of models for A relation f is called a function if for each such phenomena is a system du/dt = f(u,α), a ∈ A there exists a unique b ∈ B such that afb. dα/dt = "g(u,α) which can be analyzed by If f is a function and afbwe writef(a) = b singular perturbation for small ".

110 representative sample rendering A physical model or drawing color, or reaction rate. In general expressive rep- of an object intended for direct perception by resentations will mirror the object closely and humans. in ways that reﬂect how humans conventionally Comment: The key notion is that the model represent that object in their heads, often via is directly perceived by humans, usually visu- language. However, it must be remembered that ally. For example, renderings can be images strictly speaking, there is no requirement for any drawn on a raster screen, a physical molecular relationship between a term and a representa- model, or a projection in a virtual reality environ- tion for an object, apart from the properties of ment. A rendering is distinct from the informa- the transformations between them. Note that tion it contains or its abstract speciﬁcation, and a database can contain multiple representations is strongly determined by the device used for its which upon (internal) transformation are infor- achievement and the intended method of percep- mationally equivalent, provided that each pre- tion. Thus renderings of the same landscape in serves a different property. Thus a molecule can oils and water colors can have distinctly different characters, even if the landscapes are completely be represented by a conﬁguration rule, a key- isomorphic. Synonymous with drawing. pair list, or the terminal form. Each preserves a different property of the molecule (conﬁguration of substituent groupings, edges in the structural representation (1)Arepresentation of an graph, atoms, and bonds); all are interconvert- object a ∈ A is an image ψq (a) ∈ Z which preserves a property q such that ible by the grammar; and each is optimized for a speciﬁc class of computation. ψ (i.) uniqueness: the mapping, q , between The second is that if the representation is to A Z q and for is one-to-one and onto (notated be useful for computations which do something ψ a −→ ψ (a) a ∈ A ψ (a) ∈ Z q : q , , q , and more sophisticated than simply looking up data, (ii.) equivalent transformations: for some it must permit one to deﬁne a computational transformation ζ over the set of objects A, transformation which mimics a “real-world oper- ζ : a −→ a, ation” sufﬁciently so that for the preserved prop- erty of interest, the result of the computational a,a ∈ A, there exists a transformation ζ over transformation is the image of the result of the the set of representations, real-world operation. The word is used as both a singular and a collective noun. See also seman- ζ : ψ (a) −→ ψ (a), q q tics and semiote. 2 G left action ψ (a), ψ (a ) ∈ Z, such that the result of ψ ( ) (of a group ) A linear of a q q q V applied to ζ(a) is identical to the result of ζ group on a vector space . To any group ele- ment g ∈ G an automorphism of V is associ- applied to ψq (a),or ated. Example, the group of rotations SO(3) is R3 ψq ◦ ζ(a) = ζ ◦ ψq (a) = ψq (a ) represented on by matrix multiplication.

where ◦ is the composition operator. ψq (ζ ) = ζ , so we say that ζ and ζ are equivalent transfor- representative sample A sample resulting mations. from a sampling plan that can be expected to Equally, a set of such representations Z. reﬂect adequately the properties of interest of the Comment: The intent here is to capture two parent population. of the most salient features of a representation A representative sample may be a random as the term is commonly used in artiﬁcial intel- sample or, for example, a stratiﬁed sample, ligence and databases. The ﬁrst is that the rep- depending upon the objective of sampling and resentation is simply an image of something in the characteristics of the population. The degree the “real world” which preserves some property of representativeness of the sample may be lim- important to the user, geometric isomerism, hair ited by cost or convenience.

111 resolvent resolvent See spectrum. Ricci scalar On a manifold M with a con- nection N and a metric g the contraction of the resonance effect See mesomeric effect. Ricci tensor of N with the inverse metric µν R = g Rµν . responsivity (in detection of radiation), R Detector input can be, e.g., radiant power, irradi- ation, or radiant energy. It produces a measurable Ricci tensor The contraction of the Riemann detector output which may be, e.g., an electri- tensor deﬁned by α α cal charge, an electrical current or potential or Rβν = R · βαν =−R · βνα. a change in pressure. The ratio of the detector output to the detector input is deﬁned as the It is symmetric in the indices (β, ν). responsivity. It is given in, e.g., ampere/watt, volt/watt. The responsivity is a special case of Riemannian manifold A pair (M, g) the general term sensitivity. formed by a manifold M of dimension m and Dark current is the term for the electrical out- a Riemannian metric g on it. If the metric is put of a detector in the absence of input. This is a positive deﬁnite (i.e., of signature (m, 0)) then special case of the general term dark output.For it is called strictly Riemanniann; if the metric is photoconductive detectors the term dark resist- indeﬁnite then it is called pseudo-Riemannian. ance is used. If the responsivity is normalized with regard Riemannian metric A positive deﬁnite to that obtained from a reference radiation, the inner product on the tangent space to a manifold resulting ratio is called relative responsivity.For at x, for each point x of the manifold, varying measurements with monochromatic radiation at continuously with x. a given wavelength λ the term spectral responsiv- ity R(λ) is used. In some cases the relative spec- X tral responsivity, where the spectral responsivity right action (of a group on a space )A ρ X × G → X is normalized with respect to the responsivity at map : such that: some given wavelength, is used. The dependence (i.) ρ(x,e) = x; of the spectral responsivity on the wavelength is ρ(x,g · g ) = ρ(ρ(x,g ), g ) described by the spectral responsivity function. (ii.) 1 2 1 2 ; The useful spectral range of the detector should where G is a group, e its neutral element, · be given as the wavelength range where the rela- the product operation in G, and X a topologi- tive responsivity does not fall below a speciﬁed cal space. The maps ρg : X → X deﬁned by value. ρg(x) = ρ(x,g) are required to be homeomor- phisms.IfX has a further structure one usually rest point (of a balance) The position of the requires ρ to preserve the structure. If ρ is a right pointer with respect to the pointer scale when the action, then one can deﬁne a left action by setting motion of the beam has ceased. λ(g, x) = ρ(x,g−1). See also left action. retinoids Oxygenated derivatives of 3,7- right invariance The property that an object dimethyl-1-(2,6,6-trimethylcyclohex-1-enyl) on a manifold M is invariant with respect to a nona-1,3,5,7-tetraene and derivatives thereof. right action of a group on M. For example, a vector ﬁeld X ∈ X(M) is right invariant with reversibility In chemistry and biochemistry, respect to the right action ρg : M → M if and the notion that any reaction can proceed in only if: both directions, at least to some extent. See T ρ X(x) = X(x · g). also dextralateral, direction, dynamic equilib- x g rium, formal reaction equation, microscopic reversibility, product, rate constant, sinistra- right translations (on a group G) The right lateral, and substrate. action of G onto itself deﬁned byρ(h, g) =

112 rotation

ρg(h) = h · g. Notice that, if G is a Lie group, (ii.) · is associative: a · (b · c) = (a · b)· c for ρg ∈ Diff(G) is a diffeomorphism but not a all a,b,c ∈ R; homomorphism of the group structure. See also (iii.) + and · satisfy the distributive laws: left translations and adjoint representations. a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a, for a,b,c ∈ R. R ring A nonempty set R, with two binary is called a commutative ring if it is commu- · operations + and · , which satisfy the following tative with respect to . axioms: rotation An orthogonal tranformation of a (i.) with respect to +, R is an Abelian group; Euclidean space (V, g). See orthogonal group.

113

scaling

sample (in analytical chemistry) A portion of material selected from a larger quantity of material. The term needs to be qualiﬁed, e.g., bulk sample, representative sample, primary S sample, bulked sample, or test sample. The term “sample” implies the existence of a sampling error, i.e., the results obtained on the portions taken are only estimates of the saddle point Let f : X × Y −→ R. Then, concentration of a constituent or the quantity of (x∗,y∗) saddle point f x∗ is a of if minimizes a property present in the parent material. If there f(x,y∗) X y∗ f(x∗,y) Y on , and maximizes on . is no or negligible sampling error, the portion Equivalently, removed is a test portion, aliquot, or specimen. The term “specimen” is used to denote a portion f(x∗,y)≤ f(x∗,y∗) ≤ f(x,y∗) taken under conditions such that the sampling variability cannot be assessed (usually because for all x in X, y in Y. the population is changing), and is assumed for convenience, to be zero. The manner of selection von Neumann (1928) proved this equivalent to: of the sample should be prescribed in a sampling plan. Inf [Sup{f(x,y) : y ∈ Y } : x ∈ X] sample unit The discrete identiﬁable por- = Sup[Inf{f(x,y) : x in X} : y ∈ Y ] tion suitable for taking as a sample or as a portion of a sample. These units may be different at dif- = f(x∗,y∗). ferent stages of sampling.

satisﬁability problem Find a truth assign- saddle point problem On Banach spaces ment to logical propositions such that a (given) V,W consider the functional J : V ×W → R.A collection of clauses is true (or ascertain that saddle point problem seeks a pair (u, p) ∈ V ×W at least one clause must be false in every truth such that assignment). This fundamental problem in com- putational logic forms the foundation for NP- completeness J (u, p) = infv∈V supq∈W J(v,q). V Necessary conditions for this kind of stationary scalar Given a vector space , a member point often lead to symmetric linear variational of the ﬁeld from which scalar multiplication of vectors in V is deﬁned. problems with saddle point structure: seek u ∈ V,p ∈ W such that scalar product See inner product. a(u, v) + b(v, p) = f(v) ∀v ∈ V, scaling Changing the units of measurement, usually for the numerical stability of an algo- b(u, q) = g(q) ∀q ∈ W, rithm. The variables are transformed as x = Sx, where S = diag(sj ). The diagonal elements are where a : V × V → C,b : V × W → C s ,...,s > the scale values, which are positive: 1 n are sesqui-linear forms, and f, g stand for lin- 0. Constraint function values can also be scaled. ear forms on V and W, respectively. Specimens For example, in an LP, the constraints Ax = b, of variational saddle point problems are encoun- can be scaled by RAx = Rb, where R = diag(ri ) tered in the case of mixed variational formula- such that r>0. (This affects the dual values.) tions, the Stokes problem of ﬂuid mechanics, and Some LP scaling methods simply scale each col- whenever a linear constraint is taken into account umn of A by dividing by its greatest magnitude by a Lagrangian multiplier. (null columns are identiﬁed and removed).

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 115 scaling argument

Example A column scaling A row scaling 10x+100y=500 .333x+y=500 x+10y =50 30x +.3y =.2 x+.003y =.2 300x +3y =2

Another method is logarithmic scaling, which Schr¨odinger equation Let U ⊂ Rn be open scales by the logarithm of the greatest magnitude. and u : U × R → R. The Schr¨odinger equation More sophisticated methods are algorithmic, tak- for u is ing both row and column extremes into account. iut + u = 0.

scintillators Materials used for the mea- scaling argument (homogeneity argument) surement of radioactivity, by recording the The idea is to exploit the behavior of norms radioluminescence. They contain compounds of functions or vector ﬁelds under pull-backs (chromophores) which combine a high ﬂuores- related to scaling transformations x → δx, cence quantum efﬁciency, a short ﬂuorescence δ>0. Very closely related to techniques relying lifetime, and a high solubility. These compounds on parametric equivalence. are employed as solutes in aromatic liquids and polymers to form organic liquid and plastic scin- scatter search A population-based meta- tillators, respectively. heuristic, starting with a collection of reference points, usually obtained by the application of search tree The tree formed by a branch and some heuristic. A new point is created by tak- bound algorithm strategy. It is a tree because ing combinations of points in the population, and at each (forward) branching step the problem is rounding elements that must be integer valued. It partitioned into a disjunction. A common one is bears some relation to a genetic algorithm, except to dichotomize the value of some variable, x ≤ v that scatter search uses linear combinations of or x ≥ v + 1. This creates two nodes from the the population, while the GA crossover operation parent: can be nonlinear. [parent node] x ≤ vx≥ v + 1 scheduling (e.g., jobs) A schedule for a 89 j ,...,j sequence of jobs, say 1 n, is a speciﬁca- [left child] [right child] t ,...,t tion of start times, say 1 n, such that cer- tain constraints are met. A schedule is sought secant The function that minimizes cost and/or some measure of time, like the overall project completion time (when the (x) = 1 sec x last job is ﬁnished) or the tardy time (amount by cos which the completion time exceeds a given dead- See cosine. line). There are precedence constraints, such as in the construction industry, where a wall cannot secant method A method to ﬁnd a root of a be erected until the foundation is laid. univariate function, say F . The iterate is There is a variety of scheduling heuristics. F(xk)[xk − x(k−1)] Two of these for scheduling jobs on machines x(k+1) = xk − . are list heuristics: the Shortest Processing Time F(xk) − F(x(k−1)) (SPT) and the Longest Processing Time ( LPT). F C2 F (x) = These rules put jobs on the list in non-decreasing If is in and 0, the order of con- g = and non-increasing order of processing time, vergence is the golden mean, say, (approx. respectively. 1.618), and the limiting ratio is: Other scheduling problems, which might not (g− ) 2F (x) 1 involve sequencing jobs, arise in production . F (x) planning.

116 semantics second-order conditions A descendant self-concordance Properties of a function from classical optimization, using the that yield nice performance of Newton’s method second-order term in Taylor’s expansion. used for line search when optimizing a barrier For unconstrained optimization, the second- function. Speciﬁcally, let B be a barrier function order necessary condition (for f in C2)is for S ={x ∈ X : g(x) ≤ 0} with strict interior that the Hessian is negative semideﬁnite (for a S0. Let x be in S and let d be a direction vector max). Second-order sufﬁcient conditions are the in Rn such that the line segment [x − td,x+ td] ﬁrst-order conditions plus the requirement that is in S for t in [0,t∗], where t∗ > 0. Then, deﬁne the Hessian be negative deﬁnite. F :[0,t∗] −→ R by: For constrained optimization, the second- order conditions are similar, using projection F(t) = B(x + td) for a regular mathematical program and the F x d Lagrange multiplier rule. They are as follows (while noting that depends on and ). The F (all functions are in C2, and the mathematical function is self-concordant if it is convex in ∗ C3 x d program is in standard form, for x a local maxi- and satisﬁes the following for all and : mum): |F (0)|≤2F (0)(3/2). (i.) Second-order necessary conditions. There exist Lagrange multipliers, (u, v), such One calls Fk-self-concordant in an open convex that u ≥ 0 and ug(x∗) = 0 for which: (1) domain if ∗ ∗ gradx [L(x ,u,v)] = 0, and (2) Hx [L(x ,u,v)] |F ( )|≤ kF ( )(3/2). is negative semi-deﬁnite on the tangent plane. 0 2 0 (ii.) Second-order sufﬁcient conditions. The The logarithmic barrier function, associated with above necessary conditions hold but with (2) linear programming, is self-concordant with k = ( )H L(x∗,u,v) replaced by 2 x [ ] is negative def- 1. This further extends naturally to functions inite on the tangent plane. in Rn. selectively labeled An isotopically labeled semantic mapping A bijective, partial func- compound is designated as selectively labeled tion between each member of the set of symbols, when a mixture of isotopically substituted com- σj ∈ K, and its semantics: ω : σj −→ ω(σj ), pounds is formally added to the analogous ω the mapping operator. The set of semantic isotopically unmodiﬁed compound in such a way mappings, ;, is deﬁned for each element of the that the position(s) but not necessarily the num- domain and codomain to which they apply. ber of each labeling nuclide is deﬁned. A selec- Comment: Notice that ω is a partial func- tively labeled compound may be considered as a tion. There will be elements of the domain (the mixture of speciﬁcally labeled compounds. symbol set of the language) for which a given mapping will not be deﬁned. See also semantics self-adjoint operator A linear operator T and semiote. on a Hilbert space (H,<,>) such that T ∗ = T T ∗ , where the (Hilbert)-adjoint is deﬁned by semantics For each symbol σj in the alpha- ∗

117 semideﬁnite program

K = K −{σ } K a universe of discourse, has a unique and pre- 0 . A member of the set is cise meaning. This deﬁnition, which is appro- denoted σj , j a positive integer indexing K . σ priate for computation but not for linguistics, Then 0 is considered to be an elementary sym- eliminates multiple connotations for a single bol,orsemiote, if three conditions are fulﬁlled. term except as it can be recursively fragmented to (i.) well-formed. There is one and only one other terms having unique semantics. In that case, ω ω σ −→ well-formed mapping such that : 0 however, the ultimate terms would necessarily be ω(σ ) 0 . used in preference to the more connotation-rich σ (ii.) unique. The symbol 0 and its seman- term. The language can be formal or not, though ω(σ ) it is more likely that any arbitrary mapping will tics, 0 , are unique, or be unique if it is derived from a formal language. σ = σ ,ω(σ) = ω(σ ), 0 j 0 j Of course, natural language is used to describe the phenomenal universe and the objects within ∀ σj ∈ K , ω(σj ) ∈ ;(K ). it (see representation); the meanings of terms in (iii.) elementary. Denote the set of all possi- the natural language are mental constructs. Thus ble constructs of symbols in K by C, and a par- in deﬁning the natural language term “reaction” ticular construct, ck, k a positive integer index. corresponding to a biochemical transformation σ Then 0 is elementary if, for a well-formed map- (the object), one speciﬁes the type of informa- ping ω, the semantics of every construct, ω(ck), tion understood — chemical mechanism, kinetic σ ω(σ ) is not equal to the semantics of 0, 0 ,or regime, formal equation, etc., fragmenting the ω(σ ) = ω(c ), ∀ c ∈ C. term into a set of terms, then mapping each term 0 k k to a meaning. The representation of the object The semiote σ is denoted ς. Every semiote has in the database is arbitrary and independent of 0 four properties: the term referencing that object. Semantics are distinct from database schemata, which describe (i.) its formally deﬁned, computable seman- the relationships among objects internal to the tics; database but depend on the observer to recognize (ii.) its formally deﬁned, computable syntax; the mappings. This notion of semantics echoes (iii.) its informally deﬁned, natural language that of conceptual graphs, but without depending semantics; and upon its graphical apparatus. See also represen- tation, semiote, and term semantics. (iv.) its informally deﬁned, natural language syntax. semideﬁnite program Min {cx : S(x) ∈ The set of all semiotes is denoted Kς , and is P }, where P is the class of positive semideﬁ- also called the semantic basis set. A construct S(x) = S + {x(j)S } nite matrices, and 0 Sumj j , of semiotes is called a bundle of semiotes or a where each Sj , for j = 0,...,nis a (given) sym- semiotic bundle. metric matrix. This includes the linear program Comment: For the natural world there are as a special case. as many mental models and domain models as there are scientists and databases. Some of semi-inﬁnite program A mathematical pro- their terms’ semantics and representations will gram with a ﬁnite number of variables or con- be isomorphic among people and databases (and straints, but an inﬁnite number of constraints between people and databases), but many will or variables, respectively. The randomized pro- not. Semiotes, singly or more usually com- gram is a semi-inﬁnite program because it has an bined, are intended to map between the multiple inﬁnite number of variables when X is not ﬁnite. meanings humans assign to terms of a lan- guage describing the phenomenal world and semiote A semiote is a symbol denoting the the multiple ways in which objects and oper- semantics of a useful elementary part of an idea, ations on them from that world can be repre- datum, or computation. sented in databases. The proposition is that an σ More formally, let a symbol be denoted 0 and abstract layer of semiotes, incrementally and dis- σ the set of all symbols other than 0 be denoted tributively formulated and maintained, clearly

118 separable program speciﬁes the semantics of these models for the sensitivity analysis The concern with how purposes of exchanging computations. Each site the solution changes if some changes are made offering data or computations to the network in either the data or in some of the solution values would locally maintain a publicly readable list (by ﬁxing their value). Marginal analysis, which of mappings between its internal representation is concerned with the effects of small perturba- and the semiotes and a parser implementing those tions, may be measurable by derivatives. Para- mappings as needed by the local engine. Simi- metric analysis is concerned with larger changes larly, sites posing computations would maintain in parameter values that affect the data in the mappings and parser between their internal rep- mathematical program, such as a cost coefﬁ- resentation and the semiotes. All sites are free ciency or resource limit. to change their schema, engines, and proffered Under suitable assumptions, the multipliers in services at any time in any manner. Flexibil- the Lagrange multiplier rule provide derivatives ity of use and ease of deﬁnition is promoted by of the optimal response function, i.e., under cer- ∗ making semiotes representing the smallest use- tain conditions, (u, v) = grad f (b, c). ful part, and combining these together. See also representation and semantics. separable program The functions are sep- f(x) = f (x ), g(x) = g (x ) arable: j j j j j j , and h(x) = h (x ). The classical (LP) sensitivity (1) (in mass spectrometry) Two j j j approaches to separable programming are called different measures of sensitivity are recom- lambda-form and delta-form, both using piece- mended. The ﬁrst, which is suitable for relatively wise linear approximations. involatile materials as well as gases, depends Let {xk} be a speciﬁed set of points, where upon the observed change in ion current for a k x = [x(k,1), x(k, 2),...,x(k,n)], and let y = particular amount or change of ﬂow rate of sam- {y(k,j)} be decision variables that are the coef- ple through the ion source. A second method of ﬁcients of convex combinations, giving the fol- stating sensitivity, that is most suitable for gases, lowing linear program: depends upon the change of ion current related to the change of partial pressure of the sample in Max Sumkj{y(k,j)fj (x(k, j))} : y ≥ 0, the ion source. It is important that the relevant experimental y(k,j) = 1 for each j, conditions corresponding to sensitivity measure- k ment should always be stated. These include in a y(k,j)gj (x(k, j)) ≤ 0, typical case details of the instrument type, bom- kj barding electron current, slit dimensions, angular y(k,j)hj (x(k, j)) = 0. collimation, gain of the multiplier detector, scan kj speed, and whether the measured signal corre- sponds to a single mass peak or to the ion beam A restricted basis entry rule is invoked during integrated over all masses. Some indication of the simplex method to yield an approximate solu- the time involved in the determination should be tion. (However, this is dominated by the general- given, e.g., counting time or bandwidth. The sen- ized Lagrange multiplier method, which can be sitivity should be differentiated from the detec- viewed as generating the approximating break- tion limit. points a posteriori, getting successively ﬁner (2) (in metrology and analytical chemistry) near the solution.) u(k, j) = The slope of the calibration curve. If the curve The delta form uses the differences: x(k,j)− x(k − ,j) is in fact a “curve,” rather than a straight line, 1 . The associated functional then of course sensitivity will be a function of differences are: analytic concentration or amount. If sensitivity Df (k, j) = f (x(k, j)) − f (x(k − 1, j)), is to be a unique performance characteristic, it j j Dg(k, j) = g (x(k, j)) − g (x(k − , j)), must depend only on the chemical measurement j j 1 process, not upon scale factors. Dh(k, j) = hj (x(k, j)) − hj (x(k − 1, j)).

119 separating hyperplane

Then, the approximating LP is: sequential quadratic programming (SQP) Solving a nonlinear program by a sequence of Max Df (k, j)u(k, j) :0≤ u ≤ 1, quadratic approximations and using quadratic kj programming to solve each one. The approxima- tions are usually done by using the second-order Dg(k, j)u(k, j) ≤ b, kj Taylor expansion. Dh(k, j)u(k, j) = c, kj sequential unconstrained minimization tech- nique (SUMT) This is the penalty function b =− g (x( ,j)) c =− h where j j 0 and j j approach. (x(0,j)) (a similar constant was dropped from the objective). Another restricted basis rule is invoked: u(k, j) > 0 implies u(k, q) = 1 for sesqui-linear Describing a complex-valued all q

120 simplex method shape regular Let ;h,h ∈ H, H an index (note its dimension is n − 1). The open simplex n set, be a family of meshes of a domain ; ⊂ R . excludes the axes: {x ∈ Sn : x>0}. For each cell K ∈ ;h we deﬁne More generally, some authors deﬁne a sim- plex to be the convex hull of any n + 1 afﬁnely rK = independent vectors, and refer to the special case n sup[r>0,∃x∈K :{y∈R :|y−x|

121 simplical subdivision simplicial subdivision Given a simplex, S, Slater’s (interiority) condition Originally its simplicial subdivision is a collection of sim- for the purely inequality system with g convex, plices, say {Ti } such that ∨i {Ti }=S and for any it means there exists x for which g(x) < 0. More i and j, either the intersection Ti ∧Tj is empty or generally, for a mathematical program in stand- equals the closure of a common face. The mesh ard form, it means there exists x in X for which of the subdivision is the diameter of the largest g(x) < 0 and h(x) = 0. subsimplex. This arises in a ﬁxed point approach to compute an economic equilibrium. slope Of a nonvertical straight line with equation y = mx + b is by deﬁnition the num- simulated annealing An algorithm for solv- ber m. ing hard problems, notably combinatorial opti- mization, based on the metaphor of how smetic state See liquid-crystal transitions annealing works: reach a minimum energy state and mesomorphic phase. upon cooling a substance, but not too quickly in order to avoid reaching an undesirable ﬁnal smooth Referring to a continuously differ- state. As a heuristic search, it allows a non- entiable function. improving move to a neighbor with a probabil- ity that decreases over time. The rate of this solitons There are types of wave equations decrease is determined by the cooling sched- ule, often just a parameter used in an exponen- called soliton type equations that admit special tial decay (in keeping with the thermodynamic solitary wave solutions called solitons. These metaphor). With some (mild) assumptions about solitons have the following property: suppose the cooling schedule, this will converge in prob- two solitons are moving left to right, well separ- ability to a global optimum. ated with the smaller one to the right. After some time the bigger one catches up, the waves overlap sine The function and interact. Still later the bigger wave separates eix − e−ix from the smaller one, and eventually regains its (x) = initial shape and velocity. The only effect of the sin i 2 interaction is a phase shift. Geometrically, it is the ratio of the lengths of Example: The Korteweg-deVries equation has opposite side to hypotenuse of a right triangle solitons of the form x

122 stable sparsity The fraction of zeros in a matrix.If spectrum of a matrix The set of eigenval- A is m by n, and A(i, j) = 0 for k of its elements, ues of A. its sparsity is k/mn. Large linear programs tend to be very sparse, increasing as the dimensions stability region The set of parameter values get large. For example, consider the standard for which an optimal solution remains optimal. transportation problem with s sources and d des- This arises naturally in combinatorial optimiza- tinations. This has m = (s + d) constraints and tion, where a solution is often a subgraph, such n = sd variables. Each column, however, has as a tree, and the question is for what range exactly two nonzeros since A is the incidence of arc weights is this subgraph optimal (such matrix of the network, so its sparsity is 2n/mn, as a spanning tree that is minimum for given or simply 2/m, which decreases as the number weights). More generally, x could be a solu- of sources and/or destinations grows large. The A sparsity of a simple graph (or network) is the tion generated by some algorithm, , from an x0 sparsity of its adjacency matrix. More generally, initial value . Then, suppose the feasibility F(p) p the sparsity of a multigraph refers to the average region depends on the parameter and f(x p) p degree of its nodes. the objective j also depends on . Let X(p,A,x0) denote the generated solution from specially ordered set (SOS) Certain sets of algorithm A, starting at x0, with parameter value nonnegative variables that are required to sum to p. Let the parameter set be P (which includes 1. For computational efﬁciency, it is sometimes p∗). The stability region of x∗ = X(p∗,A,x0) better to deﬁne these sets by some marking data is {p ∈ P : x∗ = X(p,A,x0)}. The algorithm structure, rather than include them along with may be a heuristic,sox∗ need not be optimal. other equality constraints. There are two types For example, one could use an n-opt heuristic of SOSs, distinguished by what they represent. for the traveling salesman problem, so x repre- A Type 1 SOS is one in which each variable is sents a tour. The parameters could be the costs, binary, so the constraint is one of multiple choice. or they could be the location of each point in a A Type 2 SOS is one in which a restricted basis euclidean TSP. The stability region is the set of entry rule is used, as in the lambda-form of sep- costs, or coordinates in the plane, for which the arable programming. tour generated by n-opt is the same. speciﬁcally labeled An isotopically labeled compound is designated as speciﬁcally labeled stable As applied to chemical species, the when a unique isotopically substituted com- term expresses a thermodynamic property, which pound is formally added to the analogous isotopi- is quantitatively measured by relative molar cally unmodiﬁed compound. In such a case, both standard Gibbs energies. A chemical species A 0 position(s) and number of each labeling nuclide is more stable than its isomer B if r G > 0 for are deﬁned. the (real or hypothetical) reaction A → B, under A standard conditions. If for the two reactions: spectral radius (of a matrix, ) The radius P → X + Y(G0) r 1 of the following disk that contains the spectrum: Q → X + Z(G0) r(A) = Max{|y| : y is an eigenvalue of A}. r 2 0 0 r G >r G , P is more stable relative to spectral responsivity function See respon- 1 2 the product Y than is Q relative to Z. Both in sivity. qualitative and quantitative usage the term stable spectrum Let T be a linear operator on a is therefore always used in reference to some Banach space V . A complex number λ is said explicitly stated or implicitly assumed standard. to be in the resolvent set ρ(T) of T if λI − T The term should not be used as a synonym is a bijection with bounded inverse. Rλ(T ) = for unreactive or “less reactive” since this con- (λI − T)−1 is called the resolvent of T at λ.If fuses thermodynamics and kinetics. A relatively λ ∈ ρ(T), then λ is said to be in the spectrum more stable chemical species may be more reac- σ(T) of T . The set of all eigenvalues of T is tive than some reference species toward a given called the point spectrum of T . reaction partner.

123 stable mathematical program stable mathematical program Roughly, steady state (stationary state) In a kinetic one whose solution does not change much under analysis of a complex reaction involving unstable perturbation. For the inequality case, we have intermediates in low concentration, the rate of the following stability conditions: change of each such intermediate is set equal to (i.) {x ∈ X : g(x) ≤ b} is bounded for some zero, so that the rate equation can be expressed b>0. as a function of the concentrations of chemical cl{x ∈ X g(x) < }={x ∈ X g(x) species present in macroscopic amounts. For (ii.) : 0 : X ≤ 0}. example, assume that is an unstable interme- diate in the reaction sequence:

The ﬁrst stability condition pertains to upper k A + BA −→1 X semicontinuity and the second, called the closure ←− k− condition, pertains to lower semicontinuity. 1 k 2 The conditions are not only sufﬁcient to X + C −→ D. ensure the respective semicontinuity, but they are Conservation of mass requires that: necessary when: A + X + D = A (i.) {x ∈ X : g(x) ≤ 0} is bounded, [ ] [ ] [ ] [ ]0 {x ∈ X g(x) < } (ii.) : 0 is not empty. A which, since [ ]0 is constant, implies: standard deviation, s The positive square −d X /dt = d A /dt + d D /dt. root of the sum of the squares of the deviations [ ] [ ] [ ] between the observations and the mean of the X series, divided by one less than the total num- Since [ ] is negligibly small, the rate of forma- D ber in the series. The standard deviation is the tion of is essentially equal to the rate of dis- A X positive square root of the variance, a more fun- appearance of , and the rate of change of [ ] damental statistical quantity. can be set equal to zero. Applying the steady state approximation (d[X]/dt = 0) allows the state A state of a system, sj ∈ S, where elimination of [X] from the kinetic equations, S is the set of states of the system, is a distinct whereupon the rate of reaction is expressed: observable or derivable variable. k k A C Comment: The deﬁnition is meant to cover d D /dt =−d A /dt = 1 2[ ][ ] [ ] [ ] k + k C both the intensive states of thermodynamics and −1 2[ ] the states of computations and computational devices. By these deﬁnitions Post production Notes: (1) The steady state approximation X systems have neither memory nor states; instead, does not imply that [ ] is even approximately the set of constructs C formed during a derivation constant, only that its absolute rate of change A is discussed. is very much smaller than that of [ ] and [D]. Since according to the reaction scheme d D /dt = k X C X stationary point Usually this is used to [ ] 2[ ][ ], the asusmption that [ ] mean a Kuhn-Tucker point, which specializes to is constant would lead, for the case in which C one for which gradf(x)= 0 if the mathematical is in large excess, to the absurd conclusion that program is unconstrained. In the context of an formation of the product D will continue at a algorithm, it is a ﬁxed point. constant rate even after the reactant A has been consumed. stationary policy In a dynamic program, a ∗ (2) In a stirred ﬂow reactor a steady state policy that is independent of time, i.e., x (s, t) = implies a regime in which all concentrations are T(s)(some function of state, but not of time, t). independent of time. statistical genetics Genetics is a stochastic process. Statistical genetics studies the genetics steel beam assortment problem A steel by using the concepts and methods from the the- corporation manufactures structured beams of a ory of probability and statistics. See population standard length, but a variety of strengths. There genetics. is a known demand of each type of strength, but a

124 stiffness matrix stronger one may fulﬁll demand (or part thereof ) (No direction vector is sought if gradf(x) = 0; for another beam (but not conversely). The manu- such algorithms stop when reaching a stationary facture of each type of steel beam involves a point.) ﬁxed charge for its setup. In addition, there is Other norms, such as d2 = d Qd, where Q a shipping cost proportional to the difference in is symmetric and positive deﬁnite, lead to other the demanded strength and the actual strength, directions that are steepest relative to that norm. and proportional to the quantity shipped. In particular, if Q = Hf (x), this yields the modi- ﬁed Newton method. Let Steiner problem Find a subgraph of a N = number of varieties of strengths graph, say G = [V ,E], such that V contains D(t) = demand for beam of strength st V ∗ (a speciﬁed subset of nodes), and Sum{c(e) : ( where s ≥ s ≥ ... ≥ s ) 1 2 N e ∈ E } x(t) = amount of beams of strength s is minimized. It is generally assumed t c ≥ |V ∗|= manufactured 0. When 2, this is the shortest path problem. When |V ∗|=|V |, this is the (min- pt (x(t)) = manufacturing cost of x(t) units imum) spanning tree problem. of beam of strength st (including) ﬁxed charge step size A scalar (s) in an algorithm of the y(t) = total excess of beams of strength x = x + sd d s , ..., s before fulﬁlling demand form: , where is the direction 1 t d D(t + 1), ...,D(N) vector. After is chosen (nonzero), the step size is speciﬁed. One step size selection rule ht (y(t)) = shipping cost( = c[s(t + 1) − st ] Min{y(t), D(t)}). is line search; another is a ﬁxed sequence, like sk = 1/k. t Although does not index time, the mathemat- stepwise reaction A chemical reaction with ical program for this problem is of the same form at least one reaction intermediate and involving as the production scheduling problem, using the at least two consecutive elementary reactions. inventory balance equations to relate y and x. s ≥ s ≥ ...≥ s This is valid because 1 2 N implies stereochemical formula (stereoformula) A y(t) D(t + ) + can be used to fulﬁll demand 1 three-dimensional view of a molecule either as D(t + ) + ...+ D(N) h 2 . (Also, note that here t such or in a projection. is not a holding cost.) sticking coefﬁcient (in surface chemistry) steepest ascent (descent, if minimizing) A The ratio of the rate of adsorption to the rate at class of algorithms, where x = x + sd, such which the adsorptive strikes the total surface, i.e., that the direction vector d is chosen by maximiz- covered and uncovered. It is usually a function ing the initial velocity of change, and the step of surface coverage, of temperature and of the size (s) is chosen by line search. Generally used details of the surface structure of the adsorbent. in the context of unconstrained optimization, the n mathematical program is Max{f(x) : x ∈ R }, stiffness matrix The Galerkin discretiza- 1 where f is in C .(Fordescent, change Max to tion of a linear variational problem by means of a d Min.) Then, is chosen to maximize the ﬁrst- ﬁnite element space Vh leads to a linear system of order Taylor approximation, subject to a normal- equations. If a stands for the sesqui-linear form ization constraint: Max{gradf(x)d : d=1}, of the variational problem, the system matrix is where d denotes the norm of the direction vec- given by tor, d. When the Euclidean norm is used, this A = (a(b ,b ))N , yields the original steepest ascent algorithm by : i j i,j=1 Cauchy, which moves in the direction of the gra- {b , ··· ,b },N = V dient: where 1 N dim h, is the nodal basis of Vh. If the sesqui-linear form a arises d = gradf(x)/gradf(x). from the weak formulation of a boundary value

125 stochastic computation problem for a partial differential equation, the hold for all values of p, except on a set of mea- interaction of the basis functions will be local in sure zero). Some models separate constraints the sense that with several levels: P g (x; p) ≤ i I ≥ a meas(supp bi ∩ supp bj ) = 0 ⇒ a(bi ,bj ) = 0. [ i 0 for all in k] k k = , ..., K. This leads to considerable sparsity of the stiffness for 1 matrix, which is key to the efﬁciency of ﬁnite The case of one chance constraint with the element schemes. only random variable being the right-hand side is particularly simple. Suppose F is the cumu- stochastic computation Let the set of sym- lative distribution function of b for the chance bols of the computation be K, and the probability constraint P [g(x) ≤ b] ≥ a.Ifb is a continuous F that a particular symbol σi exists be Pe(σi ). Then random variable and is continuous and strictly a computation C is executed stochastically if for increasing, the chance constraint is equivalent to g(x) ≤ F −1( − a) F −1 at least one σi ∈ K,Pe(σi ) ≤ 1; or if at least one 1 (where is the inverse F g(x) = Ax ci ∈ C is a stochastic function. function of ). In particular, if , the Comment: This deﬁnition explicitly posits program remains linear. that determinism is a property of a computation (3) Recourse model. This assumes decisions and stochasticity that of its execution. An exam- are made over time where the effect of an early decision can be compensated by later decisions. ple of the latter is when at least one σi has a half- life which is not orders of magnitude greater than The objective is to optimize the expected value. The two-stage model has the form that of tC, the time needed to execute the com- σ putation, so that one cannot rely upon i to per- Maxf (x ; p ) + f (x ; p ) : sist throughout the lifetime of the computation 1 1 1 2 2 2 x ∈ X ,x ∈ X ,g(x ; p ) + g(x ; p ) ≤ . (including the step of actually detecting it). See 1 1 2 2 1 1 2 2 0 deterministic and nondeterministic computation. (Sums could be replaced by other operators.) x p Once 1 is implemented, 1 becomes known and stochastic matrix A nonnegative matrix x is chosen. The certainty equivalent is whose row sums are each 1. (A column stochas- 2 E f (x ; p ) + F (x |p ) x ∈ X tic matrix is one whose column sums are 1.) This Max [ 1 1 1 2 1 1 ]: 1 1, arises in dynamic programs whose state transi- where tion is described by a stochastic matrix contain- F (x |p ) = Sup{E[f (x ; p )]: ing the probabilities of each transition. 2 1 1 2 2 2 x ∈ X (p ), g(x ; p ) ≤−g(x ; p )} 2 2 2 2 2 1 1 stochastic program A program written in for all p (except on set of measure zero). the form of a mathematical program extended 2 (E[ ] denotes the expected value.) The “Sup” is to a parameter space whose values are random F used to deﬁne 2, the second stage value for a variables (generally with a known distribution x x particular value of 1, because the choice of 1 function). This is converted to a standard form might be infeasible. The nature of the recourse by forming a certainty equivalent. Here are some model is pessimistic: x must be chosen such that certainty equivalents: the original constraints hold no matter what the (1) Average value. Replace all random vari- values of the random variables. With a ﬁnite ables with their means. number of possibilities, this means a system of (2) Chance constraint. Given a stochastic constraints for each possible realization of p = p program with a random variable, , in its con- (p ,p ). This sextends recursively to a k-stage g(x; p) ≤ 1 2 straint: 0, a certainty equivalent is model. P g(x; p) ≤ to replace this with the constraint, [ The linear two-stage recourse model has the ≥ a P 0] , where [ ] is a (known) probability form: function on the range of g, and a is some accep- tance level (a = 1 means the constraint must max E[c]x + E[F(x; p)]:

126 strictly complementary

Ax = b, x ≥ 0, then the ﬁrst of Strang’s lemmas tell us where u − uhV F(x; p) = max d(p)y : W(p)y ≤ C( (u − v infvh∈Vh h V = w(p) − T(p)x,y ≥ 0. |a(v ,w ) − a (v ,w )| + h h h h h ) supwh∈Vh Here the second stage variable is denoted y.Itis whV x determined after has been set and the random |f(wh) − fh(wh)| + sup ), variable p has been realized. The LP data depend wh∈Vh whV on p as functions, d(p), W(p), w(p), and T(p). C = C(α, a, a )> The ﬁxed recourse model has W(p) = W. The with h 0. The second complete recourse model assumes it is ﬁxed and Strang’s lemma targets a non-conforming choice V V ⊂ V . {Wy : y ≥ 0} is all of Rm (where m = number of h, that is h .As V is not necessarily V of rows of W). This means that no matter what well deﬁned for functions of , we have to intro- . V + V value of x is chosen for the ﬁrst stage, there is duce a mesh-dependent norm h on h for a feasible recourse (y). This is simple recourse if which h is elliptic W = [I − I], so we can think of y as having |R{ah(vh,uh)}| ≥ αvhh ∀vh ∈ Vh. two parts: ypos and yneg. The second stage LP simpliﬁes to the following: Moreover, we have to require continuity dpos(p)ypos + dneg(p)yneg max : |ah(u, vh)|≤cuhvhh ypos,yneg ≥ 0 ∀u ∈ Vh + V, vh ∈ Vh. pos neg y − y = w(p) − T(p)x. Then with C = C(α, c) > 0 The certainty equivalent depends upon the u − uhV ≤ C(infv ∈V u − vhV underlying decision process. If it is adaptive, the h h recourse model applies (but RO might be more |a(v ,w ) − a (v ,w )| + h h h h h ). practical). The chance constraint model repre- supwh∈Vh whV sents a notion of an allowed frequency of viola- tions, as in environmental control models. stratopause That region of the atmosphere which lies between the stratosphere and the stoichiometry The number of moles of a mesosphere and in which a maximum in the tem- reactant used in a reaction, normalized to the perature occurs. number of moles of all the other reactants. Comment: Stoichiometries are always inte- stratosphere The atmospheric shell lying gers, because they are determined by chemical just above the troposphere which is characterized indivisibility of atoms. See mole. by an increasing temperature with altitude. The stratosphere begins at the tropopause (about 10– Strang’s lemmas Consider a linear vari- 15 km height) and extends to a height of about a(u, v) = f (v), v ∈ V ational problem posed 50 km, where the lapse rate changes sign at the V over a Hilbert space . Commiting a vari- stratopause and the beginning of the mesosphere. ational crime the corresponding discrete vari- ational problem reads strict interior Let {x ∈ X : g(x) ≤ b} be the level set of g. Then, its strict interior is {x ∈ u ∈ V : a (u ,v ) = f (v ) ∀v ∈ V , h h h h h h h h h X : g(x)

127 strictly concave function strictly concave function Negative of a subbag See parts of collections. strictly convex function. subdifferential (of f at x) ∂f (x) ={y : x strictly convex function A convex function is in argmax {vy − f(v) : v ∈ X}}.Iff that also satisﬁes the deﬁning inequality strictly is convex and differentiable with gradient, for distinct points, say x and y: gradf, ∂f (x) ={gradf(x)}. Example: f(x)= |x|. Then,∂f (0) = [−1, 1]. f(ax+ (1 − a)y)

128 surface tension subset See parts of collections. reaction and where a compound is represented in the arbitrarily written formal reaction equation. subspace A subset of a vector space that is, See also dextralateral, direction, dynamic equi- itself, a vector space. An example is the null librium, formal reaction equation, microscopic space of a matrix, as well as its orthogonal com- reversibility, product, rate constant, reversibil- plement. ity, and sinistralateral. substituent atom (group) An atom (group) successive approximation The iterative that replaces one or more hydrogen atoms scheme by which an approximation is used for attached to a parent structure or characteristic the basic design of an algorithm. The sequence group except for hydrogen atoms attached to a generated is of the form x(k+1) = xk + A(xk), chalcogen atom. where A is an algorithm map speciﬁed by its approximation to some underlying goal. Typ- substitution In logic and logic program- ically, this is used to ﬁnd a ﬁxed point, where θ ming, a substitution is a ﬁnite set of pairs of A(x) = 0 (e.g., seeking f(x) = x, let A(x) = X = t X ∈ X (k+ ) k the form i i , where i are unique vari- f(x)− x, so the iterations are x 1 = f(x ), X = X i = j X ∈ t ∗ ∗ ables ( i j for all and i j for any converging to x = f(x ) if f satisﬁes certain i j t ∈ t and ), and i are ground terms. Thus, the conditions, such as being a contraction map). result, A, of applying a substitution θ to a com- plex term A, denoted Aθ, is the term obtained by Here are some special types: replacing each Xi in A by ti in the new term A , (i.) Inner approximation X = t ∈ θ for every pair i i . (ii.) Outer approximation Comment: It is important to realize that θ is a set of mappings or transformations (substi- (iii.) Successive linear approximation tutions) between the variables in the variable- (iv.) Successive quadratic approximation containing term in the original language, and a term from the transformed language with all sufﬁcient matrix Let A be an n × n matrix. variables fully substituted. This mapping pro- Then, A is column sufﬁcient if t duces the ground terms . Thus, one can as well x (A ) ≤ i ⇒ θ = X = t ,X = t ,...,X = t [ i x i 0 for all ] write 1 1 2 2 n n.Do not confuse this usage with the chemical mean- [xi (Ax )i = 0 for all i]. ing of substitution. Aisrow sufﬁcient if its transpose is column sufﬁ- substitution reaction A reaction, elemen- cient. A is sufﬁcient if it is both column and row tary or stepwise, in which one atom or group in a sufﬁcient. One example is when A is symmetric molecular entity is replaced by another atom or and positive semideﬁnite. This arises in linear group. For example, complementarity problems. CH Cl + OH− → CH OH|Cl− 3 3 superset A set which contains another set. If the superset and the contained set can be equal, substrate (1) (in biochemistry) The speciﬁc the relation between them is denoted superset ⊇ molecules which are recognized by an enzyme. contained set; otherwise it is denoted ⊃. (2) (in chemistry) A chemical species, the Comment: The same idea can be applied to reaction of which with some other chemical other types of collections, just considering parts reagent is under observation (e.g., a compound of collections in the opposite sense. See also that is transformed under the inﬂuence of a cat- bag, empty collection, list, parts of collections, alyst). sequence, and set. Comment: The term should be used with care. Either the context or a speciﬁc statement should surface tension, γ,σ Work required to always make it clear which chemical species in increase a surface area divided by that area. a reaction is regarded as the substrate. The dis- When two phases are studied it is often called tinction is between the molecular material of a interfacial tension.

129 surjection surjection A map φ : A → B which is sur- operation. So what is left over are the elements jective. which are present in only one of the two sets, for each of the two sets. See also set difference. surjective Let A and B be two sets, with A being the domain and B the codomain of a symmetric operator An operator T , function f . Then the function f is surjective if, densely deﬁned on a Hilbert space, satisfying for any y ∈ B, there is at least one element < T u, v> = __for all u, v in the x ∈ A with f(x) = y. Thus, the range of domain of T . a surjective functions is equal to its codomain. Surjective functions are also said to be onto B, symplectic manifold A pair (P, ω) where and may be many-to-one. See also bijection and P is a manifold and ω is a closed non-degenerate injection. 2-form on P . As a consequence P is of even dimension 2n. symmetric difference Given two sets, A Canonical coordinates are coordinates (qλ, and B, their symmetric difference is deﬁned as pλ) such that the local expression of ω is of the λ A ⊗ B = (A − B) ∪ (B − A). form ω = dpλ ∧ dq . Canonical coordinates Comment: To visualize this more clearly, always exist on a symplectic manifold (Darboux think about the elements of the two difference theorem). A − B B − A d ∈ A − B sets, and . An element 1 d ∈ A d ∈ B σ if and only if 1 and 1 . This must syntax For a symbol , a unique and precise be true for all of the elements of A − B, by the deﬁnition of the form of the term and all terms deﬁnition for set difference. Similarly, for an composing it. d ∈ B − A element 2 . So the intersection of the Comment: Syntax deﬁnes such things as two difference sets, A − B and B − A must whether a term is a constant or a variable, how be empty: any elements shared between A and many arguments it has, and the syntax of those B would already be removed by the set difference arguments.__

130 tangent plane

tangent cone Let S be a subset of Rn and let x∗ be in S. The tangent cone, T(S,x∗),is the set of points y such that there exist sequences n n ∗ {an},an ≥ 0 and {x } in S such that {x }→x {a (xn −x∗)−y} → T and n 0. This arises in con- nection with the Lagrange multiplier rule much like the tangent plane, though it allows for more tableau (pl. tableaux) A detached coef- general constraints, e.g., set constraints. In par- ﬁcient form of a system of equations, which can ticular, when there are only equality constraints, h(x) = ,T(S,x∗) = h(x∗) change from x + Ay = b to x + Ay = b. 0 null space of grad h(x∗) The primes denote changes caused by multiply- if grad has full row rank. (There are some ing the ﬁrst equation system by the basis inverse subtleties that render the tangent cone more gen- tangent plane (a sequence of pivots in the simplex method). eral, in some sense, than the or null space. It is used in establishing a necessary Other information could be appended, such as constraint qualiﬁcation.) the original bound values. tangent lift A general procedure to associate tabu search This is a metaheuristic to solve canonically an object on the tangent bundle TM global optimization problems, notably combina- of a manifold M once an object is given on M. torial optimization, based on multilevel mem- In particular: ory management and response exploration. It (i.) Tangent lift of a parametrized curve γ : requires the concept of a neighborhood for a trial dγ I ⊂ R → M: let us deﬁne τγ (t ) = |t=t solution (perhaps partial). In its simplest form, a 0 dt 0 γ t = t ∈ I tabu search appears as follows: the tangent vector to the curve at 0 . We can deﬁne the tangent lift of γ as the curve µ (i.) Initialize. Select x and set Tabu List T = γˆ : I → TM : t → (γ (t), τγ (t)).Ifγ (t) is null. If x is feasible, set x∗ = x and f ∗ = f(x∗); the local expression of γ , then (γ µ(t), γ(t))˙ is otherwise, set f ∗ =−inf (for minimization set the local expression of γˆ. ∗ f = inf). (ii.) Tangent lift of a map φ : M → N:if (ii.) Select move. Let S(x) = set of neigh- tangent vectors are identiﬁed with derivations of bors of x.IfS(x)\T is empty, go to update. the algebra of local functions, the tangent map Otherwise, select y in argmax {E(v) : v in Tφ : TM → TN : v → w is deﬁned by µ µ S(x)\T }, where E is an evaluator function that w(f ) = v(f ◦ φ).Ifx = φ (x) is the local measures the merit of a point (need not be the expression of φ, then the local expression of the original objective function, f ). If y is feasible tangent map Tφis given by: f(y)>f∗ x∗ = y f ∗ = f(x∗) and , set and . xµ = φµ(x) Set x = y (i.e., move to the new point). wµ = vν ∂ φµ(x)K (iii.) Update. If some stopping rule holds, ν T stop. Otherwise, update (by some tabu update (iii.) Tangent lift of a vector ﬁeld ξ = µ ˆ rule) and return to select move. ξ (x) ∂µ ∈ X(M): a vector ﬁeld ξ = ξ µ(x) ∂ + ξˆ µ(x) ∂ TM There are many variations, such as aspiration ∂xµ ∂vµ over locally given ˆ µ ν µ levels, that can be included in more complex by ξ = ξ ∂ν ξ . Notice that the tangent lift of speciﬁcations. a commutator coincides with the commutator of lifts, i.e., [ξ,ζ]ˆ = [ξ,ˆ ζˆ]. tangent (function) The function tangent plane Consider the surface, S = n 1 sin x {x ∈ R : h(x) = 0}, where h is in C .A tan(x) = . x∗ S cos x differentiable curve passing through in is {x(t) : x(0) = x∗ and h(x(t)) = 0 for all t in See cosine, sine. (−e, e)}, for which the derivative, x(t), exists,

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 131 target analysis where e>0. The tangent plane at x∗ is the where f (k) is the kth derivative of f . Truncating set of all initial derivatives: {x(0)}. (This is the series at the nth term, the error is given by: a misnomer, except in the special case of one n (k) function and two variables at a nonstationary f (h) k |En(h)|=f(x)− (x − h) . point.) An important fact that underlies the clas- k! k=0 sical Lagrange multiplier theorem when the rank of grad h(x∗) is full row (x∗ is then called a reg- This is a Taylor expansion, and for the Taylor ular point): the tangent plane is {d : grad h(x∗) series to equal the functional value, it is necessary d = 0}. that the error term approaches zero for each n: Extending this to allow inequalities, the lim En(h) = 0. equivalent of the tangent plane for a regular point h→0 (x∗) is the set of directions that satisfy ﬁrst-order y conditions to be feasible: In any case, there exists in the line segment [x,x + h] such that {d : grad h(x∗)d = 0 and f (n+1)(y) E (h) = (y − h)n+1. ∗ ∗ n (n + ) grad gi (x )d ≤ 0 for all i : gi (x ) = 0}. 1 ! Taylor theorem Let f : (a−h, a+h) → R target analysis This is a metaheuristic to be in Cn+1. Then, for x in (a, a + h), solve global optimization problems, notably combinatorial optimization, using a learning f(x)= f(a)+ [f (1)(a)][x − a] + mechanism. In particular, consider a branch and (n) n bound strategy with multiple criteria for branch ... + [f (a)][(x − a) ]/n! + Rn(x, a), selection. After solving training problems, hind- where Rn(x, a), called the remainder, is given by sight is used to eliminate dead paths on the search the integral: tree by changing the weights on the criteria: set w> wV ≤ i x (x − t)n 0 such that i 0 at node with value (n+1) V wV > f (t) dt. i , that begins a dead path, and i 0 at each a n! node, i, on the path to the solution. If such weights exist, they deﬁne a separating hyper- This extends to multivariate functions and plane for the test problems. If such weights do is a cornerstone theorem in nonlinear program- not exist, problems are partitioned into classes, ming. Unfortunately, it is often misapplied as using a form of feature analysis, such that each an approximation by dropping the remainder, x → a class has such weights for those test problems in assuming that it goes to zero as . the class. After training is complete, and a new U ⊂ Rn problem arrives, it is ﬁrst classiﬁed, then those telegraph equation Let be open u U × R → R weights are used in the branch selection. and : . The telegraph equation for u is u + du − u = 0. Taylor expansion For f in Cn, Taylor’s the- tt t xx orem is used by dropping the remainder term. temperature inversion (in atmospheric chem- The ﬁrst-order expansion is f(x) = f(y) + istry) A departure from the normal decrease gradf (x)(x − y), and the second-order expan- of temperature with increasing altitude. A sion is f(x) = f(y) + gradf (x)(x − y)+ (x − y)t H (x)(x − y)/ temperature inversion may be produced, for f 2. example, by the movement of a warm air mass over a cool one. Intense surface inversions may f Taylor series For a function, , having all form over the land during nights with clear skies order derivatives, the series and low winds due to the radiative loss of heat from the surface of the earth. The temperature ∞ f (k)(h) (x − h)k, increases as a function of height in this case. k! k=0 Poor mixing of the pollutants generally occurs

132 thermodilatometry below the inversion, since the normal convective the set of all semantically well-formed mappings process which drives the warmer and lighter air operating on K and ;(K) is the set of deﬁned at ground-level to higher altitudes is interrupted semantics for the members of K. as the rising air parcels encounter the warmer air Comment: This notion of term semantics above. Temperature inversions near the surface extends in the domain direction the notions of are particularly effective in trapping ground-level semantics of programs, and is consistent with it. emissions. See semantics and semiote. temperature jump A relaxation technique term, T Energy divided by the product of in which the temperature of a chemical system the Planck constant and the speed of light, when is suddenly raised. The system then relaxes to a new state of equilibrium, and analysis of the of wave number dimension, or energy divided by relaxation processes provides rate constants. the Planck constant, when of frequency dimen- sion. temperature lapse rate (in atmospheric chem- istry) The rate of change of temperature terminal nodes The ﬁrst and last nodes in with altitude (dT /dz). The rate of tempera- the sequence of nodes and edges forming a path. ture decrease with increase in altitude which is If the path is a connected tree, the last nodes are expected to occur in an unperturbed dry air mass the leaves of the tree (letting the root of the tree ◦ − is 9.8 × 103 C min 1. This is called the dry be the ﬁrst node of the sequence). See also path adiabatic lapse rate. The lapse rate is taken and sequence. as positive when temperature decreases with increasing height. For air saturated with H O 2 test sample the lapse rate is less because of the release of The sample, prepared from the the latent heat of water as it condenses. The laboratory sample, from which test portions are average tropospheric lapse rate is about 6.5 × removed for testing or for analysis. 103◦C min−1. The lapse rate has a negative value within an inversion layer. theorem of the alternative Any of several theorems that establish that two systems are alter- tensor See contravariant tensor. natives. See, for example, Fredholm alternative. term (1) A variable, constant, or complex thermal conductance, G Heat ﬂow rate expression of the form f(σ ,σ ,...,σ ) where 1 2 n divided by the temperature difference. there exists at least one σi (that is, the arity of f is always positive). If f is a relation (or predicate) f(σ ,σ ,...,σ ) λ symbol, then 1 2 n is an atomic for- thermal conductivity, Tensor quantity mula. relating the heat ﬂux, J q to the temperature gra- Comment: Note that this deﬁnition is recur- dient, J q =−λ grad T . sive (a term is either a term or a function of terms), and that it includes both variable and constant (or thermal resistance, R Reciprocal of the ground) terms. It is taken to be synonymous with thermal conductance. token. (2) (in x-ray spectroscopy) A set of levels which have the same electron conﬁguration and thermodilatometry A technique in which a the same value of the quantum numbers for total dimension of a substance under negligible load is spin S and total orbital angular momentum, L. measured as a function of temperature while the substance is subjected to a controlled tempera- term semantics If for each σj ∈ K there is ture program. at least one semantic mapping ω that is semantic- Linear thermodilatometry and volume ally well formed, then we say the term semantics thermodilatometry are distinguished on the basis of ;: K −→ ;(K)are well-formed, where ; is of the dimensions measured.

133 thermodynamic isotope effect thermodynamic isotope effect The effect thermosphere Atmospheric shell extending of isotopic or substitution on an equilibrium con- from the top sof the mesosphere to outer space. stant is referred to as a thermodynamic (or equi- It is a region of more or less steadily increasing librium) isotope effect. For example, the effect temperature with height, starting at 70 or 80 km. of isotopic substitution in reactant A that partici- It includes the exosphere and most or all of the pates in the equilibrium: ionosphere (not the D region). A + B −→←− C E threshold energy, 0 Potential energy gap is the ratio K1/Kh of the equilibrium constant for between reactants and the transition state, some- the reaction in which A contains the light isotope times involving the zero point energies, but to that in which it contains the heavy isotope. The usually not. ratio can be expressed as the equilibrium constant for the isotopic exchange reaction threshold phenomenon For a linearly sta- A1 + Ch −→←− Ah + C1 ble ﬁxed point in a system of ordinary differ- ential equations, returning to the ﬁxed point is in which reactants such as B that are not isotopi- monotonic for small perturbations. But for per- cally substituted do not appear. turbations greater than a threshold, the dynamic The potential energy surfaces of isotopic variables can undergo large excursion before molecules are identical to a high degree returning to the ﬁxed point (see excitability). of approximation, so thermodynamic isotope effects can only arise from the effect of isotopic tight constraint Same as active constraint, mass on the nuclear motions of the reactants and but some authors exclude the redundant case, products, and can be expressed quantitatively in where an inequality constraint happens to hold terms of partition function ratios and for nuclear with equality, but it is not binding. motion: K1 (Q1 /Qh ) = nuc nuc C . time constant (of a detector), τc If the h 1 h K (Qnuc/Qnuc)A output of a detector changes exponentially with Although the nuclear partition function is a prod- time, the time required for it to change from its − (−t/τ ) uct of the translational, rotational, and vibra- initial value by the fraction [1 exp c ] t = τ tional partition functions, the isotope effect is (for c) of the ﬁnal value, is called the time determined almost entirely by the last named, constant. speciﬁcally by vibrational modes involving motion of isotopically different atoms. In the time-staged A model with a discrete time case of light atoms (i.e., protium vs. deuterium parameter, t = 1, ..., T , as in dynamic program- or tritium) at moderate temperatures, the isotope ming, but the solution technique need not use the effect is dominated by zero-point energy differ- DP recursion. The number of time periods (T) is ences. called the planning horizon. thermodynamic motif A conserved pattern tint The edge coloring corresponding to the of changes in the thermodynamic quantities type of biochemical relationship between two G, H, or S for a set of reactions. See also biochemical, chemical, dynamical, functional, nodes of the biochemical network. kinetic, mechanistic, phylogenetic, regulatory, Comment: The three fundamental relation- and topological motives. ships are sinistralateral, dextralateral, and catalyst. The sum of the sinistralateral and dex- thermolysis The uncatalyzed cleavage of tralateral relationships is the reactant relation- one or more covalent bonds resulting from expo- ship. These relationships are directly speciﬁed in sure of a compound to a raised temperature, or the database. Note that tint is not a proper edge a process in which such cleavage is an essential coloring as two adjacent edges can have identical part. colors.

134 topological transformation titre (titer) The reacting strength of a stand- In this case, the residual is r = Ax − b, and the ard solution, usually expressed as the weight test has the form: (mass) of the substance equivalent to 1 cm3 of r≤t + t b, the solution. a r where is some norm. token See term. topological invariant A quantity enjoyed tolerance approach An approach to sen- by a topological space which is invariant with sitivity analysis in linear programming that respect to homeomorphisms. expresses the common range that parameters can occupy while preserving the character of the topological motif A subnetwork of the bio- solution. In particular, suppose B is an optimal chemical network invariant under topological basis and rim data changes by (Db, Dc). The transformation. tolerance for this is the maximum value of t for Comment: See also biochemical, chemical, which B remains optimal as long as |Dbi |≤t dynamical, functional, kinetic, mechanistic, for all i and |Dcj |≤t for all j. The tolerance for phylogenetic, regulatory, and thermodynamic the basis, B, can be computed by simple linear motives. algebra, using tableau information. topological sort This sorts the nodes in a network such that each arc, say kth, has Tail(k) < tolerances Small positive values to control Head(k) in the renumbered node indexes. This elements of a computer implementation of an arises in a variety of combinatorial optimization algorithm. When determining whether a value, problems, such as those with precedence con- v v>−t , is nonnegative, the actual test is , where straints. If the nodes cannot be topologically t is an absolute tolerance. When comparing two sorted, the network does not represent a partially u ≥ v values to determine if , the actual test is ordered set. This means, for example, there is u − v ≤ t + t |v|, an inconsistency in the constraints, such as jobs a r that cannot be sequenced to satisfy the asserted precedence relations. where ta is the absolute tolerance (as above), and t r is the relative tolerance (some make the rela- topological space A pair (X, τ(X)) where u v tive deviation depend on as well as on , such X is a set and τ(X) its topology. Different |u|+|v| as the sum of magnitudes, ). Almost choices of the topology τ(X)of a space X corre- every MPS has a tolerance for every action it spond to different topological structures on X. takes during its progression. In particular, one Examples: If (X, d) is a metric space, then zero tolerance is not enough. One way to test we can deﬁne U ∈ τ(X) if and only if for all feasibility is usually one that is used to deter- r x ∈ U there exists an open ball Bx ={y ∈ X : mine an acceptable pivot element. In fact, the r d(x,y) < r} such that x ∈ Bx ⊂ U ⊂ X. This use of tolerances is a crucial part of an MPS, is called the metric topology of (X, d). including any presolve that would ﬁx a variable On any set X we can deﬁne the trivial top- when its upper and lower bounds are sufﬁciently ology τ(X) ={∅,X} and the discrete topology close (i.e., within some tolerance). A tolerance in which τ(X)is the set of all subsets of X (so that is dynamic if it can change during the algorithm. any subset of X is open in the discrete topology). An example is that a high tolerance might be used for line search early in an algorithm, reducing it topological transformation A one-to-one as the sequence gets close to an optimal solu- correspondence between the points of two geo- tion. The Nelder-Mead simplex method illus- metric ﬁgures A and B which is continuous in trates how tolerances might change up and down both directions. If one ﬁgure can be transformed during the algorithm. into another by a topological transformation, Another typical tolerance test applies to re- the two ﬁgures are said to be topologically siduals to determine if x is a solution to Ax = b. equivalent.

135 topology

Comment: This is one of two standard total ion current (in mass spectrometry) deﬁnitions. Common examples are smooth (1) (after mass analysis) The sum of the separate deformation of a triangle into a circle; a sphere ion currents carried by the different ions con- into a beaker (a cup without a handle); and a tributing to the spectrum. This is sometimes trefoil knot into a circle. In the last case, the called the reconstructed ion current. transformation is allowed to cut the perimeter of (2) (before mass analysis) The sum of all the the ﬁgure so long as the cut ends are rejoined in separate ion currents for ions of the same sign their original manner. Note that expansions and before mass analysis. contractions of the network are not topological transformations. totally unimodular matrix See unimodular matrix. topology (1) (of a network) The topology of toxicity (1) Capacity to cause injury to a a network G(V, E) is the set of nodes V and their living organism deﬁned with reference to the incidence relations in the network, E. quantity of substance administered or absorbed, Comment: Speciﬁed here are two particular the way in which the substance is administered topological properties which are to remain invari- (inhalation, ingestion, topical application, injec- ant under a topological transformation, such as a tion) and distributed in time (single or repeated continuous deformation. The standard deﬁnition doses), the type and severity of injury, the time requires only that the edges remain invariant to needed to produce the injury, the nature of the transformation. This is perfectly reasonable for organism(s) affected and other relevant condi- networks derived from mathematics, but does tions. not ﬁt the biological case as well. Many net- (2) Adverse effects of a substance on a living works will include singleton nodes which are organism deﬁned with reference to the quantity important, but whose edges are not yet known. of substance administered or absorbed, the way Hence the standard deﬁnition is here augmented in which the substance is administered (inhala- to cover this case as well. There are a number of tion, ingestion, topical application, injection) and other important senses of the word which are not distributed in time (single or repeated doses), the directly relevant here. type and severity of injury, the time needed to (2) (on a set X ) A set τ(X) of subsets of produce the injury, the nature of the organism(s) X, called open sets, which satisfy the following affected, and other relevant conditions. axioms: (3) Measure of incompatibility of a substance with life. This quantity may be expressed as the (i.) the empty set ∅ and the whole space X reciprocal of the absolute value of median lethal / / are elements in τ(X); dose (1 LD50) or concentration (1 LC50). (ii.) the intersection of a ﬁnite number of ele- trace element Any element having an aver- ments in τ(X) is still in τ(X); and age concentration of less than about 100 parts per (iii.) the union of a (possibly inﬁnite) family million atoms (ppma) or less than 100 µg per g. of elements in τ(X) is still in τ(X). trajectory (in reaction dynamics) A path torque, T Sum of moments of forces not taken by a reaction system over a potential- acting along the same line. energy surface, or a diagram or mathematical description that represents that path. A trajec-

α tory can also be called a reaction path. torsion tensor Let Nβµ be a (linear) connection on a manifold M. The torsion of transfer Movement of a component within α α α the connection is the tensor Tβµ = Nβµ − Nµβ . a system or across its boundary. It may be Despite the fact that the connection is not a ten- expressed using different kinds of quantities, sor, the torsion is a tensor since nonhomogeneous e.g., rates of change dQ/dt or Q/t. terms in the transformation rules of connections Examples are mass rate, dmB /dt or mB /t; cancel out. substance rate, dnB /dt or nB /t.

136 transition state theory transformation (1) (in chemistry) The con- frequency. The assembly of atoms at the transi- version of a substrate into a particular prod- tion state has been called an activated complex. uct, irrespective of reagents or mechanisms (It is not a complex according to the deﬁnition in involved. For example, the transformation of this compendium.) (C H NH ) N aniline 6 5 2 into -phenylacetamide It may be noted that the calculations of reac- (C H NHCOCH ) 6 5 3 may be effected by use of tion rates by the transition state method and acetyl chloride or acetic anhydride or ketene. A based on calculated potential-energy surfaces transformation is distinct from a reaction, the full refer to the potential energy maximum at the description of which would state or imply all the saddle point, as this is the only point for which reactants and all the products. the requisite separability of transition state coor- (2) (in mathematics) See function. dinates may be assumed. The ratio of the num- ber of assemblies of atoms that pass through to transient (chemical) species Relating to the products to the number of those that reach a short-lived reaction intermediate. It can be the saddle point from the reactants can be less deﬁned only in relation to a timescale ﬁxed by than unity, and this fraction is the transmission κ the experimental conditions and the limitations coefﬁcient . (There are reactions, such as the of the technique employed in the detection of the gas-phase colligation of simple radicals, that do intermediate. The term is a relative one. not require activation and which therefore do not Transient species are sometimes also said to involve a transition state.) be metastable. However, this latter term should be avoided, because it relates a thermodynamic transition state theory A theory of the rates term to a kinetic property, although most of elementary reactions which assumes a special transients are also thermodynamically unstable type of equilibrium, having an equilibrium con- ‡ with respect to reactants and products. stant K , existing between reactants and acti- vated complexes. According to this theory the transient phase (induction period) The rate constant is given by period that elapses prior to the establishment of k = (k T /h)K‡ a steady state. Initially the concentration of a B reactive intermediate is zero, and it rises to the where kA is the Boltzmann constant and h is the steady-state concentration during the transient Planck constant. The rate constant can also be phase. expressed as transition function manifold ‡ 0 ‡ 0 See . k = (kB T/h)exp( S /R) exp(− H /RT ) transition state In theories describing ele- where ‡S0, the entropy of activation, is the mentary reactions it is usually assumed that standard molar change of entropy when the acti- there is a transition state of more positive molar vated complex is formed from reactants and Gibbs energy between the reactants and the prod- ‡H 0, the enthalpy of activation, is the cor- ucts through which an assembly of atoms (ini- responding standard molar change of enthalpy. tially composing the molecular entities of the The quantities Ea (the energy of activation) and reactants) must pass on going from reactants to ‡H 0 are not quite the same, the relationship products in either direction. In the formalism between them depending on the type of reaction. of transition state theory the transition state of Also an elementary reaction is that set of states (each ‡ 0 characterized by its own geometry and energy) k = (kB T/h)− exp(− G /RT ) in which an assembly of atoms, when randomly placed there, would have an equal probability of where ‡G0, known as the Gibbs energy of forming the reactants or of forming the products activation, is the standard molar Gibbs energy of that elementary reaction. The transition state change for the conversion of reactants into is characterized by one and only one imaginary activated complex. A plot of standard molar

137 transition structure

Gibbs energy against a reaction coordinate is where x ≥ 0, known as a Gibbs-energy proﬁle; such plots, {x(i,j) : j in T }≤s(i) for all i in S, unlike potential-energy proﬁles, are temperature- j dependent. In principle, the expressions for k above must {x(i,j) : i in S}≥d(j) for allj in T. i be multiplied by a transmission coefﬁcient, κ, which is the probability that an activated com- The decision variables (x) are called ﬂows, and plex forms a particular set of products rather than the two classes of constraints are called supply reverting to reactants or forming alternative prod- limits and demand requirements, respectively. ucts. (Some authors use equality constraints, rather It is to be emphasized that ‡S0,‡H 0, and than the inequalities shown.) An extension is ‡G0 occurring in the former three equations the capacitated transportation problem, where x ≤ U are not ordinary thermodynamic quantities, since the ﬂows have bounds . one degree of freedom in the activated complex is ignored. transpose (of a matrix [aij ]) The matrix Transition-state theory has also been known [aji]. as the absolute rate theory, and as activated- complex theory, but these terms are no longer transposition theorem Same as a theorem recommended. of the alternative. transition structure A saddle point on a transshipment problem This is an exten- potential-energy surface. It has one negative sion of the transportation problem whereby the force constant in the harmonic force constant network is not bipartite. Additional nodes serve matrix. See also transition state. as transshipment points, rather than providing translation The action of a group (V, +) supply or ﬁnal consumption. The network is N = V,A V regarded as a group of transformations on V itself [ ], where is an arbitrary set of nodes, through the action T : V × V → V given by except that it contains a nonempty subset of sup- ply nodes (where there is external supply) and a T : (T , v) → v + T. nonempty subset of demand nodes (where there is external demand). A is an arbitrary set of arcs, transport equation Let U ⊂ Rn be open and there could also be capacity constraints. and u : U × R → R. The (linear) transport u equation for is traveling salesman problem (TSP) Given n n points and a cost matrix, [c(i, j)], a tour is u + bi u = . a permutation of the n points. The points can t xi 0 i=1 be cities, and the permutation the visitation of each city exactly once, then returning to the transportation problem Find a ﬂow of ﬁrst city (called home). The cost of a tour, least cost that ships from supply sources to con- /i ,i , ..., i ,i ,i 0 1 2 n−1 n 1 , is the sum of its costs: sumer destinations. This is a bipartite network, N = [S∗T,A], where S is the set of sources, T is c(i ,i ) + c(i ,i ) + ... + c(i ,i ) + c(i ,i ), 1 2 2 3 n−1 n n 1 the set of destinations, and A is the set of arcs. In N A (i ,i , ..., i ) { , ..., n} the standard form, is bi-complete ( contains where 1 2 n is a permutation of 1 . all arcs from S to T ), but in practice networks The TSP is to ﬁnd a tour of minimum total cost. tend to be sparsely linked. Let c(i, j) be the unit The two common integer programming formula- i S j T s(i) = cost of ﬂow from in to in , supply tions are: i d(j) = j c x x ∈ P, x ∈{ , } at th source, and demand at th destina- ILP: min ij ij ij : ij 01 tion. Then, the problem is the linear program Subtour elimination constraints: i,j∈V xij ≤|V |−1 for ∅=V ⊂{1,...,n} (V = Minimize {c(i, j)x(i, j) : i in S,j inT } ij {1,...,n})

138 tropopause

where (iii.) The intersection of the closures of any two cells is either empty or a vertex, edge, face, 1 j follows city i in tour etc., of both. xij = 0 otherwise This means that a triangulation constitutes c n−1 x x + x x ; QAP: min ij ij k=1 ik jk+1 in j1 : a nondegenerate cellular decomposition of . Special types of meshes are simplicial meshes, x ∈ P, xij ∈{0, 1}, for which all the cells are n-simplices. In two dimensions the cells of quadrilateral meshes where have four, possibly curved, edges each. Their three-dimensional counterparts are hexaedral i j 1 if tour has city in position meshes, whose cells are bricks (with curved faces xij = 0 otherwise and edges). In a straightforward fashion the con- cept can be generalized to the notion of a triangu- In each formulation, P is the assignment poly- lation of a compact piecewise smooth manifold tope. The subtour elimination constraints in ILP (with or without boundary). In the case of adap- eliminate assignments that create cycles. tive reﬁnement it is often desirable to relax the ( → → → ) For example, in subtours 1 2 3 1 above requirements by admitting hanging nodes. ( → → ) (length 3) and 4 5 4 (length 2), the ﬁrst These are vertices of some cells that lie in the V ={, , } subtour is eliminated by 1 2 3 , which interior of edges of other cells. x + x + x ≤ requires 12 23 31 2. The second sub- tour is eliminated by V ={4, 5}, which requires x + x ≤ triple point 45 54 1. The point in a one-component system at which the temperature and pressure of tree A connected graph containing no three phases are in equilibrium. If there are p cycles. possible phases, there are p!/(p − 3)!3! triple points. Example: In the sulfur system four pos- triangle inequality A property of a distance sible triple points (one metastable) exist for the S function: f(x,y) ≤ f(x,z) + f(z,y) for all four phases comprising rhombic (solid), mono- S S S x,y,z. clinic (solid), (liquid), and (vapor). triangular matrix A square matrix A,is triplet state A state having a total electron called upper triangular if all elements are zero spin quantum number of 1. below the main diagonal, i.e., A(i, j) = 0 for i>j. It is called lower triangular if its transpose is upper triangular. We sometimes call triprismo- An afﬁx used in names to denote a matrix triangular if it is either lower or upper six atoms bound into a triangular prism. triangular. trivial bundle A bundle (B,M,π; F) triangulation/mesh Given a bounded which has a global trivialization so that the total n domain ; ⊂ R with piecewise smooth bound- space is diffeomorphic to the Cartesian product ary a triangulation/mesh ;h of ; is a ﬁnite set B - M × F . Trivial bundles always allow {K }M ,M ∈ N i i=1 , of piecewise smooth open global sections, and they are the local model of subsets of ;, called cells, such that all bundles. (i.) the interior of the closure of each cell is the cell itself. ! tropopause The region of the atmosphere K¯ ;¯ which joins the troposphere and stratosphere, (ii.) the union K∈;h coincides with and Ki ∩ Kj =∅,ifi = j (open partition prop- and where the decreasing temperature with alti- erty). tude,characteristic of the troposphere ceases, and

139 troposphere the temperature increase with height which is Using the Euclidean norm and applying the characteristic of the stratosphere begins. Lagrange multiplier rule to the subproblem yields p from the equation troposphere The lowest layer of the atmos- [H (x) − uI]p =−grad (x) for some u ≥ 0. phere, ranging from the ground to the base of f f the stratosphere (tropopause) at 10–15 km of Note that for u = 0, the iteration is Newton’s altitude depending on the latitude and meteoro- method, and for very large u, the iteration is logical conditions. About 70% of the mass of the nearly Cauchy’s steepest ascent. atmosphere is in the troposphere. This is where most of the weather features occur and where the tub conformation A conformation (of sym- D chemistry of the reactive anthropogenic species metry group 2d ) of an eight-membered ring in released into the atmosphere takes place. which the four atoms forming one pair of diamet- rically opposite bonds in the ring lie in one plane Trotter product formula If A and B are and all other ring atoms lie to one side of that self-adjoint operators and A + B is essentially plane. It is analogous to the boat conformation self-adjoint, then of cyclohexane.

lim (eitA/neitB/n)n = ei(A+B)t. Tung distribution (of a macromolecular assem- n→∞ bly) A continuous distribution with the dif- ferential mass-distribution function of the form: true value (in analysis), τ The value that b−1 b characterizes a quantity perfectly in the condi- fw(x)dx = abx exp(−ax )dx tions that exist when that quantity is considered. x It is an ideal value which could be arrived at only where is a parameter characterizing the chain if all causes of measurement error were elim- length, such as relative molecular mass or degree a b inated, and the entire population was sampled. of polymerization and and are positive adjustable parameters. truncated gradient Projection of the gradi- tunneling The process by which a par- f C1 a,b ent of a function, (in ) on a box, [ ], to ticle or a set of particles crosses a barrier on put zeros in coordinates if the sign of the par- its potential-energy surface without having the tial derivative is negative at the lower bound or energy required to surmount this barrier. Since it is positive at the upper bound. This yields a the rate of tunneling decreases with increasing feasible direction. reduced mass, it is signiﬁcant in the context of isotope effects of hydrogen isotopes. trust region method The iteration is deﬁned as x = x + p, where p is the (complete) change tuple A collection of elements which satisfy (no separate step size), determined by a subprob- some relation r. A tuple is delimited by paren- lem of the form theses ((x, y)), and is usually written with the relation as the functor of the tuple; thus r(x,y). Max F(p) : p≤D, Comment: Note that a sequence is not equiv- alent to a tuple. The functor is omitted when it where F depends on the iterate and is an approxi- is absolutely clear what relation it satisﬁes. See mation of the change in objective function value. also bag, list, relation, sequence, and set. The particular norm and the magnitude of D determine the set of admissible change values Turing pattern A.M. Turing was the ﬁrst (p), and this is called the trust region. A com- person who proposed a mechanism for pattern mon choice of F is the quadratic form using the formation in a spatially homogeneous system Taylor expansion about the current iterate, x,as to involve chemical reactions and diffusion of two species. The pattern formation is due to F(p) = gradf (x)p + p [Hf (x)]p/2. diffusion-driven instability.

140 unitary group

unimodal function A function which has one mode (usually a maximum, but could mean a minimum, depending on context). If f is deﬁned on the interval [a,b], let x∗ be its mode. Then, f strictly increases from a to x∗ and strictly U ∗ decreases from x to b (reverse the monotonicity on each side of x∗ if the mode is a minimum). unbounded mathematical program The (For line search methods, like Fibonacci, the objective is not bounded on the feasible region mode could occur in an interval, [a∗,b∗], where (from above, if maximizing; from below, if min- f strictly increases from a to a∗, is constant (at imizing). Equivalently, there exists a sequence its global max value) on [a∗,b∗], then strictly k k of feasible points, say {x } for which {f(x )} decreases on [b∗,b].) diverges to inﬁnity (minus inﬁnity, if minimiz- ing). unimodular matrix A nonsingular matrix whose determinant has magnitude 1. A square unconstrained mathematical program One matrix is totally unimodular if every nonsingular with no constraints (can still have X be a proper submatrix from it is unimodular. This arises in subset of Rn, such as requiring x to be integer- (linear) integer programming because it implies valued.) a basic solution to the LP relaxation is integer- valued (given integer-valued right-hand sides), unconstrained optimization Taken liter- thus obtaining a solution simply by a simplex ally, this is an unconstrained mathematical method. An example of a totally unimodu- program. However, this phrase is also used in lar matrix is the node-arc incidence matrix of a context that X could contain the strict inte- a network, so basic solutions of network ﬂows rior, with constraints of the form g(x) < 0, are integer-valued (given integer-valued supplies but the mathematical program behaves as uncon- and demands). strained. This arises in the context of some algo- rithm design, as the solution is known to lie in the interior of X, such as with the barrier function. unimolecular See molecularity. uncountably inﬁnite set An inﬁnite set unisolvence A set of functionals on a ﬁnite which is not denumerably so. See also cardinal- dimensional vector space Vh is called unisolvent, ity, countable set, denumerably inﬁnite set, ﬁnite if it provides a basis of the dual space of Vh. set, and inﬁnite set. Unisolvence is an essential property of degrees of freedom in the ﬁnite element method. undirected edge See edge. unit An identity element. uniﬁed atomic mass unit Non-SI unit of mass (equal to the atomic mass constant), deﬁned as one twelfth of the mass of a carbon-12 atom unit circle A circle of radius 1. Usually, the in its ground state and used to express masses term refers to the circle of radius 1 and center 0 of atomic particles, u = 1.660 5402(10) × in the complex plane ({z : |z|=1}). 10−27 kg. unitary group An automorphism α : F m m uniformly bounded Referring to a family C → C is called unitary if, once a basis Ei has m j of functions such that the same bound holds for been chosen in C , the matrix Ui representing all functions in F . For example, j the automorphism by α(Ej ) = Ui Ej satisﬁes − f(x)≤ µ U 1 = U †, i.e., if the inverse of U coincides with the transpose of the complex conjugated with the same µ, for all f ∈ F . matrix.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 141 unitary matrix

The group of all such matrices is called the each step. One of the symbols input to C refers unitary group, and it is denoted by U(m). The to an emulation of a particular Turing machine subgroup of unitary matrices with det U = 1is stored in the UTM memory. This emulation is (m) (s ,σ ,s ,σ ,a ) denoted by SU . the set of all tuples i i,I i+1 i,O i for that machine and forms the algorithm. The algo- unitary matrix A nonsingular matrix whose rithm may be deterministic or nondeterministic, Hermitian adjoint equals its inverse (same as but the machine executes it nonstochastically. orthogonal for real-valued matrices). See self- Comment: This deﬁnition differs slightly adjoint operator. from Minsky’s (cf. M.L. Minsky, Computation: Finite and Inﬁnite Machines, Prentice-Hall, Englewood Cliffs, NJ, 1967). See Post produc- univariate optimization A mathematical tion system and von Neumann machine. program with a single variable.

universe of discourse The nonempty set, universal set The set containing all sets, or Ud , of all possible constant terms of a program. sets of interest; denoted U. See also universe of More generally, the area of nature, thought, discourse. or existence described by a program or set of Comment: In databases, one frequently programs. speaks of a universe of discourse, the set of Comment: For many purposes, this is equiva- all terms, facts, relations, and functions used in lent to the universal set. See universal set. reifying the database’s model of its world (its domain model). For many purposes the two are equivalent. unreactive Failing to react with a speciﬁed chemical species under speciﬁed conditions. The universal Turing machine A universal term should not be used in place of stable, since Turing machine (UTM) is a discrete automaton a relatively more stable species may neverthe- that executes a computation C. It has a read-write less be more reactive than some reference species head that reads symbols from and writes them toward a given reaction partner. to an unbounded but ﬁnite, immutable memory. The memory stores symbols from a ﬁnite symbol unstable The opposite of stable, i.e., the set chemical species concerned has a higher molar K = KI ∪ KO ∪ KC, Gibbs energy than some assumed standard. The term should not be used in place of reactive K = , K = K where I 2 O 1, and C are the sym- or transient, although more reactive or transient bols internal to the computation. At each step species are frequently also more unstable. the automaton assumes one of a ﬁnite number Very unstable chemical species tend to s ,s ∈ S of discrete states, i i . A computation undergo exothermic unimolecular decomposi- C is deﬁned as a set of tuples each of the form tions. Variations in the structure of the related (s ,σ ,s ,σ ,a ) s i i,I i+1 i,O i , where i is the automa- chemical species of this kind generally affect the i σ ton’s state at the th step, i,I the symbol read energy of the transition states for these decom- K = K = into the automaton at that step ( i,I i,O positions less than they affect the stability of the s 1), i+1 is the new state the automaton assumes decomposing chemical species. Low stability i upon completion of step (which will be its state may therefore parallel a relatively high rate of as it commences step i + 1), σ is the sym- i,O unimolecular decomposition. bol output at step i, and ai is the action altering the position of the tape in the head that is per- formed at the end of that step (move it left, right, upper semicontinuity (or upper semicontinuous or nowhere). The values for each σ and ∫ persist [USC]) Suppose {xk}→x. in the automaton long enough for it to execute Of a function, lim sup f(xk) = f(x).

142 utility function √ √ √ N S Of a point-to-set map, let e[ ]beaneigh- {√x : g(x) ≤ b}=[− 2 − 1 + b, − 2 + borhood of the set S. For each e>0, there exists 1 + b]/[− log b, ∞), which is not bounded. k K such that for all k > K, A(x ) is a subset of The map fails to be (USC) at 0 due to the lack of Ne[A(x)]. Here is an example of what can go stability of its feasibility region when perturbing wrong. Consider the feasibility map with its right-hand side (from above). √ (x + 2)2 − 1ifx<0 upper triangular matrix A square matrix, g(x) = A A(i, j) = i ≥ j , such that 0 for . exp{−x} if x ≥ 0 Note√ g is continuous√ and its level set is utility function A measure of beneﬁt, used [− 2 − 1, − 2 + 1]. However, for any b>0, as a maximand in economic models.

143

variational inequality

have Cauchy’s steepest ascent. If f is concave and one chooses H equal to the negative of the inverse Hessian, we have the modiﬁed Newton’s V method. variable upper bound (VUB) A constraint of the form: xi ≤ xj . valid inequality An inequality constraint variational calculus An approach to solv- added to a relaxation that is redundant in the ori- ing a class of optimization problems that seek a ginal mathematical program. An example is a functional (y) to make some integral function (J ) ax ≤ b linear form, , used as a cutting plane in an extreme. Given F in C1, the classical uncon- the LP relaxation of an integer program. strained problem is to ﬁnd y in C1 to minimize A linear form that is a facet of the integer poly- (or maximize) the following function: hedron. x 1 J(y) = F(x,y,y)dx. x value iteration This is an algorithm for 0 inﬁnite horizon [stochastic] dynamic programs An example is to minimize arc length, where that proceeds by successive approximation to sat- 2 F = (1 + y ). Using the Euler-Lagrange isfy the fundamental equation equation, the solution is y(x) = ax + b, where a and b are determined by boundary conditions: F(s) = Opt {r(x,s)+ a P(x,s,s )F (s )}, s y(x ) = y and y(x ) = y . 0 0 1 1 If constraints take the form G(x,y,y) = 0, where a is a discount rate. The successive this is called the problem of Lagrange; other approximation becomes the DP forward equa- forms are possible. tion. If 0

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 145 vector vector (1) A directed line segment in R2 or vertex cover Given a graph, G = [V,E], a R3, determined by its magnitude and direction. vertex cover is a subset of V , say C, such that for (2) An element of a vector space. each edge (u, v) in E, at least one of u and v is in C. Given weights, {w(v)} for v in V , the weight vector bundle A bundle (B,M,π; V)with of a vertex cover is the sum of weights of the a vector space V as a standard ﬁber and such nodes in C. The minimum weight vertex cover that the transition functions act on V by means problem is to ﬁnd a vertex cover whose weight is of linear transformations. Vectorbundles always minimum. allow global sections. vertical automorphism A bundle mor- ) B → B (B,M,π; F) vector ﬁeld A vector-valued function V , phism : on a bundle which projects over the identity, i.e., such that deﬁned in a region D (usually in R3). The vector π ≡ id ◦ π = π ◦ ). Locally a vertical mor- V(p), assigned to a point p ∈ D, is required to M phism is of the following form have its initial point at p. x = x x = (x , vector product For two vectors 1 y = Y(x,y). x ,x ) y = (y ,y ,y ) R3 2 3 and 1 2 3 in , the vector vertical vector ﬁeld A vector ﬁeld \ over x ×y = (x y −x y ,x y −x y ,x y −x y ). 2 3 3 2 3 1 1 3 1 2 2 1 a bundle (B,M,π; F) which projects over the zero vector of M. Locally, it is expressed as vector space A set closed under addition and \ = ξ i (x, y)∂ scalar multiplication (by elements from a given i ﬁeld). One example is Rn, where addition is where (xµ; yi ) are ﬁbered coordinates. The ﬂow the usual coordinate-wise addition, and scalar of a vertical vector ﬁeld is formed by vertical t(x , ..., x ) = (tx , ..., tx ) multiplication is 1 n 1 n . automorphisms. Another vector space is the set of all m × n matrices. If A and B are two matrices (of the viscosity See dynamic viscosity. same size), so is A + B. Also, tA is a matrix for voltage (in electroanalysis) The use of this any scalar, t in R. Another vector space is the set X Rn term is discouraged, and the term applied poten- of all functions with domain and range in . tial f g f + g should be used instead, for nonperiodic sig- If and are two such functions, so are nals. However, it is retained here for sinusoidal tf t R and for all in . Note that a vector space and other periodic signals because no suitable t = must have a zero since we can set 0. See substitute for it has been proposed. also module. von Neumann machine (vNM) A device vehicle routing problem (VRP) Find opti- that stores a program specifying an algo- mal delivery routes from one or more depots to a rithm or heuristic in memory; includes separ- set of geographically scattered points (e.g., popu- able arithmetic and logic processing units and lation centers). A simple case is ﬁnding a route input/output facilities and executes a discrete for snow removal, garbage collection, or street computation on the input data using the stored sweeping (without complications, this is akin to program. It assumes a ﬁnite number of distinct a shortest path problem). In its most complex internal states and operates on a ﬁnite number of form, the VRP is a generalization of the TSP, as symbols K. It may or may not store the input it can include additional time and capacity con- and output, as desired. For convenience allow straints, precedence constraints, plus more. execution to be either sequential on a single pro- cessor or sequential within each of a set of pro- velocity, v,c Vector quantity equal to the cessors joined together by various schemes for derivative of the position vector with respect to message passing and storage. The machine is time (symbols, u, v, w for components of c). a nonstochastic device executing deterministic or nondeterministic algorithms. See universal vertex See node. Turing machine and Post production system.

146 wavelength converter

wash out (in atmospheric chemistry) The removal from the atmosphere of gases and some- times particles by their solution in or attachment to raindrops as they fall.

W n wave equation Let U ⊂ R open and u : U ×R → R. The (linear) wave equation for u is wall-coated open-tubular (WCOT) column (in chromatography) A column in which the liq- utt − u = 0. uid stationary phase is coated on the essentially unmodiﬁed smooth inner wall of the tube. The (nonlinear) wave equation for u is

utt − u = f (u). warehouse problem The manager of a warehouse buys and sells the stock of a certain wave function (state function), Z, ψ, φ The commodity, seeking to maximize proﬁt over a solution of the Schr¨odinger equation, eigen- period of time, called the horizon. The ware- function of the Hamiltonian operator. house has a ﬁxed capacity, and there is a hold- ing cost that increases with increasing levels of inventory held in the warehouse (this could vary wave height (electrochemical) The limit- period to period). The sales price and purchase ing current of an individual wave, frequently cost of the commodity ﬂuctuate. The warehouse expressed in arbitrary units for convenience. is initially empty and is required to be empty at the end of the horizon. This is a variation of the wave number, σν The reciprocal of the production scheduling problem, except demand wavelength λ, or the number of waves per unit is not ﬁxed. (Level of sales is a decision variable, length along the direction of propagation. Sym- which depends on whether cost is less than price.) bols ν in a vacuum, σ in a medium.

Let wavelength, λ Distance in the direction of x(t) = level of production in periodt propagation of a periodic wave between two suc- (before sales) cessive points where at a given time the phase is y(t) = level of inventory at the end of the same. period t z(t) = level of sales in period t wavelength converter Converts radiation W = warehouse capacity at one wavelength to radiation at another h(t, u) = holding cost of inventory u detectable wavelength or at a wavelength of from period t to t + 1 improved responsivity of the detector. The clas- p(t) = production cost (per unit of produc- sical wavelength converter consists of a screen of tion) luminescent material that absorbs radiation and s(t) = sales price (per unit) radiates at a longer wavelength. Such materials T = horizon are often used to convert ultraviolet to visible radiation for detection by conventional photo- Then, the mathematical program is: tubes. In X-ray spectroscopy, a converter that emits optical radiation is called a scintillator.In Minimize h(t, y(t)) + px − sz : most cases wavelength conversion is from short t to long wavelength, but in the case of conver- sion of long to short wavelength the process is x,y,z ≥ ; y( ) = y(T ) = ; 0 0 0 sometimes called upconversion. Wavelengths of y(t) = y(t − 1) + x(t) − z(t) ⇐ W coherent sources can be converted using non- linear optical techniques. A typical example is for t = 1, ..., T . frequency doubling.

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 147 wavelength dispersion wavelength dispersion (in x-ray emission form of a boundary value problem is through inte- spectroscopy) Spatial separation of characteristic gration by parts, using Green’s formulas. x-rays according to their wavelengths. wedge projection A stereochemical projec- wavelength-dispersive x-ray ﬂuorescence tion, roughly in the mean plane of the molecule, analysis A kind of x-ray ﬂuorescence in which bonds are represented by open wedges, analysis involving the measurement of the tapering off from the nearer atom to the farther wavelength spectrum of the emitted radiation atom. e.g., by using a diffraction grating or crystal. wavelength error (in spectrochemical analysis) Weierstrass theorem See Bolzano-Weier- The error in absorbance which may occur if there strass theorem. is a difference between the (mean) wavelength of the radiation entering the sample cell and the weight, G Force of gravity acting on a body, indicated wavelength on the spectrometer scale. G = mg, where m is its mass and g the acceler- ation of free fall. wavelet An orthonormal basis of L2(Rn). They are used as phase space localization meth- weighted mean If in a series of observations ods. A typical example of a family of wavelets a statistical weight (wi ) is assigned to each value, ψj,k(x) in one dimension is given by a weighted mean xw can be calculated by the

−j/2 −j formula ψj,k(x) = 2 ψ(2 x − k) wi xi xw = j wi −j x − 2 k = 2 2 ψ( ), j, k ∈ Z, 2j Comment: Unless the weights can be assigned where ψ is a smooth function with reasonable objectively, the use of the weighted mean is not −(1+") normally recommended. decay (say, |ψ(x)|

148 wormlike chain

wW αβδ = γ βδ − γ αβ where γ αβ , γ αδ, and γ βδ are the surface ten- sions between two bulk phases α, β; α, δ and γ αβ γ βδ where and are the surface ten- β,δ, respectively. α β β,δ sions between two bulk phases , and , The work of adhesion as deﬁned above, and respectively. traditionally used, may be called the work of separation. White’s formula A closed DNA molecule modeled as a ribbon has a twist (Tw) with respect work of cohesion per unit area Of a single to its central line (the line of centroids) and α pure liquid or solid phase α, wC is the work done a writhe (Wr) which represents the contortion on the system when a column α of unit area is of the central line in space. It was shown by split, reversibly, normal to the axis of the column J.H. White, that Tw + Wr = linking number, a to form two new surfaces each of unit area in topological invariant. White’s formula relates contact with the equilibrium gas phase. DNA structure to knot theory (see also DNA supercoil; cf. J.H. White, Am. J. Math., 91, 693 α α wC = 2γ (1969)). where γ α is the surface tension between phase α wind rose A diagram designed to show the and its equilibrium vapor or a dilute gas phase. distribution of wind direction experienced at a given location over a considerable period of time. work softening The application of a ﬁnite Usually shown in polar coordinates (distance shear to a system after a long rest may result from the origin being proportional to the prob- in a decrease of viscosity or consistency. If the ability of the wind direction being at the given decrease persists when the shear is discontinued, angle usually measured from the north). Similar this behavior is called work softening (or shear diagrams are sometimes used to summarize the breakdown), whereas if the original viscosity or average concentrations of a given pollutant seen consistency is recovered this behavior is called over a considerable period of time as a function thixotropy. of direction from a given site (sometimes called a pollution rose). working electrode An electrode that serves as a transducer responding to the excitation signal wood horn A mechanical device that acts by and the concentration of the substance of interest absorption as a perfect photon trap. in the solution being investigated, and that per- mits the ﬂow of current sufﬁciently large to effect wood lamp A term used to describe a low- appreciable changes of bulk composition within pressure mercury arc. See lamp. the ordinary duration of a measurement. w W F work, , Scalar produce of force, , and working set Constraints believed to be d r,w= F × d r. position change, active at a solution. work hardening Opposite of work soft- wormlike chain (in polymers) A hypo- ening, in which shear results in a permanent thetical linear macromolecule consisting of an increase of viscosity or consistency with time. inﬁnitely thin chain of continuous curvature; the direction of a curvature at any point is ran- work of adhesion The work of adhesion per αβδ dom. The model describes the whole spectrum unit area, w , is the work done on the system A of chains with different degrees of chain stiffness when two condensed phases α and β, forming from rigid rods to random coils, and is particu- an interface of unit area, are separated reversibly larly useful for representing stiff chains. In the to form unit areas of each of the αδ- and βδ- literature this chain is sometimes referred to as a interfaces. Porod-Kratky chain. Synonymous with continu- αβδ αδ βδ αβ wA = γ + γ − γ ously curved chain.

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x-ray satellite

x-ray intensity Essentially all x-ray meas- urements are made by photon counting tech- niques but the results are seldom converted to radiant ﬂux or irradiance or radiant exposure. X The term photon ﬂux would be appropriate if the measurements were corrected for detector efﬁciency but this is seldom done for x-ray chem- ξ - (xi-) A symbol used to denote unknown ical analysis. Therefore, the term x-ray inten- conﬁguration at a chiral center. sity, I, is commonly used and expressed as photons/unit time detected. Likewise the term xanthophylls carotenoids A subclass of relative x-ray intensity, Ir , is used to mean the consisting of the oxygenated carotenes. intensity for the analyte in an unknown specimen divided by the intensity for a known concentra- xenobiotics Manmade compounds with tion of the analyte element. chemical structures foreign to a given organism. xenon lamp An intense source of ultravio- x-ray level An electronic state occurring as let, visible, and near-infrared light produced by the initial or ﬁnal state of a process involving electrical discharge in xenon under high pressure. the absorption or emission of x-ray radiation. It represents a many electron state which, in the xerogel A term used for the dried out open purely atomic case, has total angular momentum structures which have passed a gel stage during (J = L + S) as a well-deﬁned quantum number. preparation (e.g., silica gel); and also for fried out compact macromolecular gels such as gelatin or rubber. x-ray photoelectron spectroscopy (XPS) Any technique in which the sample is bombarded with XPRESS-MP A mathematical program- x-rays and photoelectrons produced by the sam- ming modeling system and solver. ple are detected as a function of energy. ESCA (Electron Spectroscopy for Chemical Analysis) XPS See photoelectron spectroscopy. refers to the use of this technique to identify ele- ments, their concentrations, and their chemical x-radiation Radiation resulting from the state within the sample. interaction of high-energy particles or photons with matter. x-ray satellite A weak line in the same x-ray escape peak In a gamma or x-ray energy region as a normal x-ray line. Another spectrum, the peak due to the photoelectric effect name used for weak features is non-diagram in the detector and escape, from the sensitive part line. Recommendations as to the use of these of the detector, of the x-ray photon emitted as a two terms have conﬂicted. Using the term dia- result of the photoelectric effect. gram line as deﬁned here, non-diagram line may well be used for all lines with a different x-ray ﬂuorescence The emission of charac- origin. The majority of these lines originate from teristic x-radiation by an atom as a result of the the dipole-allowed de-excitation of multiply ion- interaction of electromagnetic radiation with its ized or excited states, and are called multiple- orbital electrons. ionization satellites. A line where the initial state has two vacancies in the same shell, notably x-ray ﬂuorescence analysis A kind of the K-shell, is called a hypersatellite. Other analysis based on the measurement of the ener- mechanisms leading to weak spectral features gies and intensities of characteristic x-radiation in x-ray emission are, e.g., resonance emission, emitted by a test portion during irradiation with the radiative auger effect, magnetic dipole, and electromagnetic radiation. electric quadrupole transitions and, in metals,

1-58488-050-3/01/$0.00+$.50 c 2003 by CRC Press LLC 151 x-ray spectroscopy plasmon excitation. Atoms with open electron x-ray spectroscopy Consists of three steps: shells, i.e., transition metals, lanthanides, and (i.) excitation to produce emission lines actinides, show a splitting of certain x-ray lines characteristic of the elements in the material, due to the electron interaction involving this open shell. Structures originating in all these (ii.) measurement of their intensity, and ways as well as structures in the valence band of (iii.) conversion of x-ray intensity to concen- molecules and solid chemical compounds have tration by a calibration procedure which may in the past been given satellite designations. include correction for matrix effects.

152 Yukawa-Tsuno equation

a constant. The value given corresponds to the Gregorian calendar year (a = 365.2425 d).

yield, Y (in biotechnology) Ratio express- Y ing the efﬁciency of a mass conversion process. The yield coefﬁcient is deﬁned as the amount of cell mass (kg) or product formed (kg, mol) Yang-Mills theory A nonabelian gauge the- related to the consumed substrate (carbon or ory. For a ﬁxed compact Lie group G with Lie nitrogen source or oxygen in kg or moles) or to algebra g, the ﬁeld in this theory is vector poten- the intracellular ATP production (moles). tial A, i.e., a connection 1-form on some principal G bundle. Let FA be the curvature 2-form of the yield stress The shear stress σ or τ at connection A. The fundamental Lagrangian in 0 0 which yielding starts abruptly. Its value depends pure gauge theory is on the criterion used to determine when yielding 1 n occurs. L(A) = |FA||d x|, 2 p, q, r ≥ where |FA| denotes the norm in the Lie algebra. Young’s inequality Let 1 and /p + /q + /r = f ∈ Lp(Rn), g ∈ From the variational principle we get the equa- 1 1 1 2. Let Lq (Rn), h ∈ Lr (Rn) tions of motion, the Yang-Mills equations . Then d ∗ F = , A A 0 f (x)(g ∗ h)(x)dx d Rn where A is the covariant derivative with respect ≤ C f g h . to A and ∗ is the Hodge star operator. For any A p,q,r,n p q r we also have the Bianchi identity d F = 0. In A A Yukawa-Tsunoequation A multiparameter local coordinates we have: extension of the Hammett equation to quan- tify the role of enhanced resonance effects on Fµν = ∂µAν − ∂ν Aν + i[Aµ,Aν ], and L = Tr(F F µν ). the reactivity of meta- and para-substituted µν benezene derivatives, e.g., Then the Yang-Mills equations become k = k + ρ σ + r(σ+ − σ) . lg lg 0 [ ] ∂µF + i[Aµ,F ] = 0. µν µν The parameter r gives the enhanced resonance year Unit of time, a = 31 556 952 s. The effect on the scale (σ + − σ)or (σ − σ), respec- year is not commensurable with the day and not tively.

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Zorn’s lemma

Zimm-Rouse model A stochastic math- ematical model for the dynamics of an ideal- ized polymer chain. The linear chain is modeled as a collection of N beads connected by N − 1 Z springs. A system of stochastic differential equa- tions for the N particles deﬁnes a multivariate Gaussian process. The equations can be solved Zeeman effect The splitting or shift of spec- using normal mode method. This model is a gen- tral lines due to the presence of an external mag- eralization of the Ornstein-Uhlenbeck process. netic ﬁeld. Zorn’s lemma If S is a partially ordered zigzag phenomenon Successive directions set such that each totally ordered subset has an are orthogonal, causing Cauchy’s steepest ascent upper bound in S, then S has a maximal ele- to converge slowly. The successive orthogonal- ment. Partially ordered means that for some pairs ity comes from optimizing the (scalar) step size x,y ∈ S there is an ordering relation x ≤ y; with (exact) line search, but the issue runs deeper. totally ordered means for each pair x,y ∈ S one In general, the zigzag phenomenon causes small either has x ≤ y or y ≤ x. The Zorn lemma is steps around a ridge. equivalent to the axiom of choice.

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