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It is meant to save time, to present the desired information in the place it is first looked up, and in a manner that allows the content to be more readily assimilated. Copyright Year: Online ISBN: Author s : Steven M. Need Help? She's riding a bike. Taking off all her clothes she screams 'Take whatever you want' If you take the bike , you're an engineer. Yea, the clothes wouldn't have fit you anyway. Someone who relates to the universe in a mathematical but socially inept way.

Optimist : "The glass is half full. A kick-ass uber-genius with godly math and science abilities, the training for wich being at the expence of those abilities for spelling and talking to members of the opposite sex. Engineer unknown. Someone who solves a problem you didn't know you had in a way you don't understand. You: "Hey, wait here, I'mma go take a dump.

I was playing guitar with your amp and I noticed a parasitic capacitance between the output and the input, causing parasitic oscillation. So I really quickly soldered them a little further from eachother, so it shouldn't have any feedback anymore. It's what I do. A talented individual responsible for the design and creation of all man-made objects in the known universe , as opposed to a scientist who attempts to make new discoveries about the universe. Engineers do not do much in the way of manual labour; those tasks are allocated to skilled tradesmen. Engineers are involved in the design of everything from oil tankers to staple removers.

For example, a scientist in a lab may discover a new metal with certain properties. Society depends on engineers with their lives just as much as they depend on medical professionals. It is the responsibilty of an engineer to make sure a bridge will stay up, a car will drive straight, and that planes will remain airborne. Engineering is broken down into many streams.

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Mechanical engineers would be involved with things in motion, such as a car, or jackhammer. A civil engineer would design bridges and buildings, and other static structures. There are many more streams, which I will not list here. Banach manifold Banach space. Banach space A normed vector space which is complete in the metric defined by its norm. Baire space A space which is not a countable union of nowhere dense subsets. Example: A complete metric space is a Baire space. Polish algebraist, analyst, and topologist. The numbers a1 , Bayes formula Suppose A and B1 , With appropriate dyes, the reaction can be monitored from the color of the solution in a test tube.

This is the first reaction known to exhibit sustained chemical oscillation. Spatial pattern has also been observed in BZ reaction when diffusion coefficients for various species are in the appropriate region. Bergman kernel Let M be an n-dimensional complex manifold and H the Hilbert space of holomorphic n-forms on M. Bergman metric Let M be an n-dimensional complex manifold, in any complex coordinate system z1 , The coefficients Bk are called Bernoulli numbers. Bessel equation z2.

Jn z is a solution of the Bessel equation. Betti number Let Hp be the pth homology group of a simplicial complex K. Hp is a finite dimensional vector space and the dimension of Hp is called the pth Betti number of K. The dimension of the finite dimensional vector space H p M is called the pth Betti number of M. See also onto, into, injective, and surjective. In both cases, the connection between two connected units is unidirectional. In a bilateral network, the connection between two connected units is bi-directional.

Only the digits 0 and 1 are needed. A binary operation is an operation that depends on two objects. Addition and, multiplication are binary operations. This is also the number of combinations of n things k at a time. This is usually accomplished by a set of chemical reactions leading to an observable change in biological systems or in test tubes.

Comment: The considered biochemical graphs are sometimes hypergraphs, mathematically. See also biochemical network. Comment: The constraint for bimolecular relationship permits use of the common donoracceptor language. A reaction may have more than one such relationship. Note that the biochemical donor-acceptor relationship is often opposite to that of the chemical one: thus a phosphoryl donor is a nucleophile acceptor.

See also chemical, dynamical, functional, kinetic, mechanistic, phylogenetic, regulatory, thermodynamic, and topological motifs. Members of the parameter set P apply to vertices, edges, and connected graphs of vertices and edges as biochemically appropriate and as such information is available. Comment: The network is a biochemical graph whose nodes, edges, and subgraphs have qualitative and quantitative parameters. Not all nodes or edges need be so marked; and in fact much known information is at present unavailable electronically. Equally, a composition of a set of bichemical reactions.

Comment: The definition places no restrictions on the level of resolution of the description, or size and complexity of reacting species; thus, it permits the recursive specification of processes. Thus, two protons, if one came from water and the other from a protein, would be individually recorded in the equation. The restriction to active forms of the catalyst includes those instances where the catalyst must be activated, by either covalent modification or ligand binding, or is inhibited by those means, so that not all molecules present are equally capable of catalysis.

The definition places no restrictions on the level of resolution of the description, or size and complexity of reacting species, thus permitting the recursive specification of processes. The recursion scales over any size or complexity of process. Comment: By function called here biological function to distinguish it from the mathematical sense of function , biologists mean both how a molecule interacts with its milieu and what results from those interactions.

What is critical is to realize that a. In recent years, it has a much narrower meaning in practice: it mainly deals with statistical analysis and methodology applicable to biology and medicine. Comment: Informally, a graph will be bipartite if it has two distinct sets of nodes and if nodes of each type are always adjacent to the other. Thus the biochemical graph is bipartite because it has a set of compound nodes and a set of reactive conjunction nodes, and each is connected to the other.

Where appropriate, definitions relating to macromolecule may also be applied to block. The solutions of the Bogomolny equations are called magnetic monopoles. In metric spaces: compactness and sequential compactness are equivalent. Bose-Einstein statistics In quantum statistics of the distribution of particles among various possible energy values there are two types of particles, fermions and bosons, which obey the Fermi-Dirac statistics and Bose-Einstein statistics, respectively. In the Fermi-Dirac statistics, no more than one set of identical particles may occupy a particular quantum state i.

Boson A particle described by BoseEinstein statistics. See also coordination. The theoretical index of the bond order, p rs degree of bonding between two atoms relative to that of a single bond, i. In molecular orbital theory it is the sum of the products of the corresponding atomic orbital coefficients weights over all the occupied molecular spin-orbitals. Boundedness is equivalent to continuity.

Borel sets The sigma-algebra of Borel sets of Rn is generated by the open sets of Rn. An element of this algebra is called Borel measurable. Bose-Einstein gas A gas composed of particles with integral spin. Bourbaki, N. A pseudonym of a changing group of leading French mathematicians.

## Dictionary of Applied Math for Engineers and Scientists - CRC Press Book

The Association of Collaborators of Nicolas Bourbaki was created in Bramble-Hilbert lemma A crucial tool for proving local interpolation estimates for parametric equivalent finite elements. Notes: 1 A branch point from which f linear chains emanate may be termed an f -functional branch point, e.

Alternatively, the terms trifunctional, tetrafunctional, pentafunctional, etc. Usually both X and n take integer values, and Xn is Markovian. Let Z be the random variable representing the number of offspring in the next generation of a single individual. Comment: The mathematical word accurately conveys the structure: two subgraphs joined by a single edge.

Brownian dynamics To circumvent the difficulties of MD see molecular dynamics , one approach is to introduce a larger time step in the simulation. One essential difference between Brownian dynamics and MD is that the water molecules are explicit in the latter, whereas they contribute as a random force and a viscous medium in the former. Brownian dynamics have the advantage of large time steps. Its diffculty lies in the uncertainty about the interatomic interaction on the large time step and in implicit water, known as coarse-graining.

After the introduction of ghost fields the effective Lagrangian is no longer gauge invariant, but has a new global symmetry, called BRST symmetry. The BRST operator s is defined on the algebra of local operators making it into a differential graded algebra. If the typical fiber F is a vector space, the bundle is called a vector bundle. If the spaces E, B are smooth manifolds and all the maps above are smooth, then the bundles are called smooth fiber bundles. Examples are the tangent bundle and cotangent bundle of a smooth manifold B.

One usually summarizes this property by saying that the following diagram:. If the bundles are endowed with some additional structure e. The first mathematical model for this phenomenon was proposed by T. Chay and J. Keizer in terms of five coupled nonlinear ordinary differential equations in which a slow oscillator modulates a high frequency oscillation.

Rinzel, Burst oscillations in an excitable membrane model. Sleeman and R. Jarvis, SpringerVerlag, New York, C Calabi-Yau spaces Complex spaces with a vanishing first Chern class or, equivalently, with trivial canonical bundle canonical class. They are used to construct possibly realistic super string models. It is known from experiments that some calcium channels see ion channel which are responsible for calcium influx into cytosol are positively modulated by the calcium in the cytosol.

This mechanism has been suggested as being responsible for the widely observed calcium oscillation in cells. The corresponding graphical display for a single analyte is referred to as the calibration curve. In canonical coordinates q 1 ,. Comment: Often denoted by S or S , where S is any set. See also countable set, denumerably infinite set, finite set, infinite set, and uncountably infinite set. This class includes carotenes, xanthophylls, and certain compounds that arise from rearrangement of the skeleton of I or by loss of part of this structure.

Retinoids are excluded. Casimir Elements in the universal enveloping algebra of a Lie algebra that commute with all other elements of the Lie algebra are called Casimirs. Then there exists z1 , z2 , z3 ,. The catalyst is both a reactant and product of the reaction. The words catalyst and catalysis should not be used when the added substance reduces the rate of reaction see inhibitor.

Catalysis can be classified as homogeneous catalysis, in which only one phase is involved, and heterogeneous catalysis, in which the reaction occurs at or near an interface between phases. Catalysis brought about by one of the. Catalysis brought about by a group on a reactant molecule itself is called intramolecular catalysis. The term catalysis is also often used when the substance is consumed in the reaction for example: base-catalyzed hydrolysis of esters. Strictly, such a substance should be called an activator.

Comment: Reactions are written at many levels of resolution, and those reactions which detail how the catalyst functions will not meet this definition. But that is appropriate, since the molecule which functions catalytically in one reaction will be a substrate in the reactions which describe the catalysis. See homotopy. Cayley transform Let A be a closed symmetric operator on a Hilbert space. These are unique according to the BrambleHilbert lemma. Notes: 1 Except in linear single-strand macromolecules, the definition of a chain may be somewhat arbitrary.

See derivative. Pontrjagin classes p1 ,. Stiefel-Whitney classes w1 ,. Excitation by electrons is called primary excitation and by photons, secondary or fluorescence excitation. Particle induced x-ray emission PIXE is produced by the excitation of heavier particles such as protons, deuterons, or heavy atoms in varying degrees of ionization. Emission of photons in the x-ray wavelength region occurs from ionized gases or plasmas at high temperatures, nuclear processes low-energy end of the gamma-ray spectrum and from radiative transitions between muonic states.

Characteristic x-ray emission consists of a series of x-ray spectral lines with discrete frequencies, characteristic of the emitting atom. Other features are emission bands from transitions to valence levels. In a spectrum obtained with electron or photon excitation the most intense lines are called diagram lines or normal x-ray lines. They are dipole allowed transitions between normal x-ray diagram levels.

Equally, a set of such reaction equations which sums to the overall biochemical reaction. Comment: These mechanisms distinguish all the various subtypes nucleophilic, electrophilic, free-radical; heterolytic, homolytic, pericyclic; direct electron, hydride, hydrogen atom transfer, ester intermediates, displacement, and addition-elimination reactions.

This is prerequisite for the evaluation of the performing characteristics of the method, or the development of meaningful uncertainty statements concerning analytical results. See also biochemical, dynamical, functional, kinetic, mechanistic, phylogenetic, regulatory, thermodynamic, and topological motives. In a thermal equilibrium with many chemical reactions, all components have to have the same chemical potential. Chemical reactions may be elementary reactions or stepwise reactions. It should be noted that this definition includes experimentally observable interconversions of conformers.

Detectable chemical reactions normally involve sets of molecular entities, as indicated. The term is applied equally to a set of chemically identical atomic or molecular structural units in a solid array. For example, two conformational isomers may be interconverted sufficiently slowly to be detectable by separate NMR spectra and hence to be considered to be separate chemical species on a time scale governed by the radiofrequency of the spectrometer used.

On the other hand, in a slow chemical reaction the same mixture of conformers may behave as a single chemical species, i. Except where the context requires otherwise, the term is taken to refer to a set of molecular entities containing isotopes in their natural abundance. The wording of the definition given in the first paragraph is intended to embrace both cases such as graphite, sodium chloride, or a surface oxide, where the basic structural units may not be capable of isolated existence, as well as those cases where they are. In common chemical usage generic and specific chemical names such as radical or hydroxide ion or chemical formulae refer either to a chemical species or to a molecular entity.

See chromodynamics, quantum. Christoffel symbols on a pseudo Riemannian manifold M, g , the coefficients of the Levi-Civita connection. Eukaryotic cells have a characteristic number of chromosomes per cell cf. The chromosomes of bacteria consist of double-standed, circular DNA molecules. For most Markov processes, the sufficient and necessary condition for zero circulation is timereversibility. Clifford algebra Clifford algebra is a formulation of algebra which unifies and extends complex numbers and vector algebra. It is based on the Clifford product of two vectors a and b which is written ab.

The product has two parts, a scalar part and a bivector part. Exact forms are closed. The converse is not true in general. It is true on contractible manifolds e. De Rham cohomology studies the topological properties of manifolds by classifying closed forms that are not exact. The phase that is more concentrated in colloid component is the coacervate, and the other phase is the equilibrium solution. It is given by. The dimension of a coisotropic manifold M is at least half of the dimension of P.

It is a point at which the gradient is zero along all coordinates, and the curvature is positive along all but one coordinate, which is the reaction coordinate, along which the curvature is negative. A coloring of the graph G is the set of colors assigned to the nodes, such that no two adjacent nodes have the same color. An edge coloring is the set of colors assigned to edges; a proper edge coloring is the set of colors such that no two adjacent edges have the same color.

Comment: These need not be physical colors, though it is perhaps easiest to think of them that way. Note that colorings are by definition proper, whereas edge colorings need not be. The difference between definitions of node and edge colorings may have been motivated in the beginning by the four color map problem. See also Abelian group. More generally any binary operation defining a Lie algebra, namely, the Lie-product obeying Jacobi identities.

See also Lie algebra. T is called compact or completely continuous if T maps bounded sets in X into precompact sets the closure is compact in Y. Hence, the growth rate of one species decreases with increasing population of the other and vice versa. Such a system is likely to exhibit extinction of the weaker species. Stable co-existence is possible, however, under a delicate balance. The equations of motion of a completely integrable system can be formally integrated. Explicitly, a Hamiltonian system vector field XH on R2n is called completely integrable if there exist n constants of the motion f1 ,.

The bonding between the components is normally weaker than in a covalent bond. The term has also been used with a variety of shades of meaning in different contexts; it is therefore best avoided when a more explicit alternative is applicable. One often distinguishes between enzymes and metabolites, smaller molecules. Comment: This definition is broad, but still useful; it includes areas such as the representation of information in electronic databases, the construction and maintenance of those databases.

This is a distinction without a difference, since most biological problems require both. To the extent that biological problems are molecular, computational biology overlaps with computational chemistry. Often the distinction is more one of investigator origin chemist vs. Comment: Time and space have a mathematical relationship to the size N and complexity of the problem.

Thus algorithms are described as being linear, polynomial, etc. Functions are designated O f N , meaning the algorithm requires on the order of f N resources. Algorithms which are polynomial represent the practical upper bound of feasibility for large N , with lower values for exponents and determinism of the algorithm obviously preferable. Many problems requiring supra-polynomial resources can be addressed in favorable cases. Occasionally other metrics, such as the number of statements in the program, are defined for particular purposes.

Comment: This definition is a bit unusual in that it explicitly includes the recognition of the actual inputs and outputs from among the sets of possible ones, rather than just the mapping among them. The reason is that one can then place discrete, Turing computations firmly in a framework which accommodates both them and the analog computations of molecular machines, for example, for DNA computers. See deterministic, nondeterministic computations. In a concerted process the primitive changes may be synchronous or asynchronous.

Some authorities extend the term to include inversion at trigonal pyramidal centers and other polytopal rearrangements. See also tub conformation. In appropriate molecular entities d-orbitals may be involved. The term is also extended to the analogous interaction involving a p-orbital containing an unshared electron pair, e. See pathwise connected. The effect of subsequent breaks will depend on the topology of the graph. See constant of motion. See conservation law.

Comment: The nature of the combination operations and their operands can vary considerably depending on the intended result of. However, no other nonmathematical information is required for contractions of models of biological systems, such as biochemical networks unlike expansion.

Such information may be beneficial, for example, in suggesting appropriate approximations. An expansion is the reverse operation. Then T has a unique fixed point; i. In the former sense, it is the reverse of unimolecular heterolysis. The origin of the bonding electrons has by itself no bearing on the character of the bond formed. The term is also used to describe the number of ligands around a central atom without necessarily implying two-electron bonds. Note: Copolymers that are obtained by copolymerization of two monomer species are sometimes termed bipolymers, those obtained from three monomers terpolymers, those obtained from four monomers quaterpolymers, etc.

See also cardinality, denumerably infinite set, finite set, infinite set, and uncountably infinite set. Often, but not necessarily, corrosion results in effects detrimental to the usage of the material considered. Exclusively physical or mechanical processes such as melting or evaporation, abrasion or mechanical fracture are not included in the term corrosion.

A covering space can be regarded as a bundle with a discrete standard fiber. Furthermore, the lift of two homotopic curves produces two curves of B which are still homotopic. In three-dimensional space, these symmetries are represented by the crystallographic groups. These include rotation, reflection point group , and translation lattice symmetry as basic elements, and their composition e. More specifically, 32 point groups combined with 14 Bravais lattices 7 primitive and 7 nonprimitive result in unique space groups which describe the only ways in which identical objects may be arranged in an infinite 3D lattice.

A simpler, practical approach leading to the same result studies translations in a lattice, introducing two new. Stout and L. Not to be confused with the trajectory or path which is the image of the curve, i. The length of the cycle is n. This is the only part of the definition which might otherwise not be intuitive. It is mostly responsible for the mechanics, i.

The main types of filaments in the network are actin filaments, microtubules, and intermediate filaments. There are many other different proteins involved in the network. They regulate the state and dynamics of the network. Comment: This is the database management system, object-oriented model, set of MatLab functions, etc. Thus, it is the computational machinery, not per se the intellectual basis of the abstract model of the domain.

Comment: The number of reactions in which a compound participates is its degree. Similarly, the number of compounds participating in a reaction is the degree of the reaction. Since the best-known biochemical network described has an enzyme for every reaction, one usually means nonenzymatic degree, when applying the word to reactions. Note this would not be the case for other versions of the biochemical network which include spontaneous reactions, or detailed kinetic and mechanistic views of reactions in which the enzyme does not necessarily appear in the reaction equations as a formal catalyst.

Some degree of delocalization is always present and can be estimated by quantum mechanical calculations. Comment: Because any two denser subgraphs formed with the same values of x and y are not necessarily isomorphic, denser subgraphs are not necessarily motives. Familarly known as a blob. There is a bijection between the set and N. See also cardinality, finite set, infinite set, and uncountably infinite set.

The doublestranded form consists of a double helix of two complementary chains that run in opposite directions and are held together by hydrogen bonds between pairs of the complementary nucleotides and Hoogsteen stacking forces. Notice that A is not required to be an associative algebra.

Example: let L be a Lie algebra. If this limit exists f is called differentiable at x with total or Frechet derivative Df x. Comment: As with the definitions of computation, nondeterministic, and stochastic computations, the goal here is to place the usual theory of computing within a framework that accommodates molecular computers. Like any other system of chemical reactions, a molecular computer is surrounded by many copies of its possible inputs and outputs. These copies of symbols are populations of symbols, each type of symbol occuring at some frequency in its respective population Pe.

Notice that a unit cardinality does not imply that the length of the input and output tokens is one. See also nondeterministic and stochastic computations. Comment: Formal reaction equations represent two sets of molecular species, one written on the left and the other on the right-hand side of the equation. The placement of a set on a side of the equation is completely arbitrary. For the bio chemistry to proceed, each member of a set of coreacting species must be present.

For each, the higher its concentration, the easier it is to observe the reaction and the faster the reaction will occur. Since the reactions are reversible, the equations are symmetric, and the placement of sets of coreacting species in the equation is arbitrary, the sets of species are designated sinistralateral and dextralateral.

These designations distinguish them from the sets of substrates and products: these terms indicate the role a molecule plays in the reaction. See also direction, dynamic equilibrium, formal reaction equation, microscopic reversibility, product, rate constant, reversibility, sinistralateral, and substrate. See linear. These levels form a spectrum similar to that of a one-electron or hydrogen-like atom but, being single-valency levels, have the energy scale reversed relative to that of single-electron levels. Diagram levels may be divided into valence levels and core levels according to the nature of the electron vacancy.

Diagram levels with orbital angular momentum different from zero occur in pairs and form spin doublets. If when all the diffusion terms are absent, the remaining nonlinear ODE has a stable fixed point, but when a diffusion term is present, the spatially homogeneous solution corresponding to the stable fixed point is unstable, we say the system has a diffusion-driven instability.

The strength of the interaction depends on the distance and relative orientation of the dipoles. The term applies also to intramolecular interactions between bonds having permanent dipole moments. Dirac equation. Dirac operator A linear first-order partial differential operator with nonconstant coefficients, defined between sections of the spin bundle. It is defined as covariant derivative followed by Clifford product. Dirac spinors Elements in the full complex spin representation as opposed to elements in the half spin representation which are called Weyl spinors and elements in the real spin representation which are called Majorana spinors.

Comment: These are the edges that are drawn with arrows, indicating a direction of travel or flow. See edge. Comment: Chemical and biochemical reactions can be thought of as a composite of two reactions. One that consumes sinistralateral coreactants, producing dextralateral ones or proceeding left to right and the opposite. These two directions are often represented as half reactions which are formal equations that specify only one of the two directions. The intrinsic rates of the two directions, as measured by their respective rate constants can differ very significantly.

See also dextralateral, dynamic equilibrium, formal reaction equation, microscopic reversibility, product, rate constant, reversibility, sinistralateral, and substrate. Dirichlet boundary condition Specification of the value of the solution to a partial differential equation along a bounding surface.

Examples include unimolecular heterolysis and homolysis, and the separation of the constituents of an ion pair into free ions. In both senses dissociation is the reverse of association. One should add some regularity condition in order not to allow too odd dependence of the subspace on the point x.

Let X be a vector field defined on ;. In conventional transition-state theory it passes through the highest point on the minimum-energy path. In generalized versions of transition-state theory the dividing surface can be at other positions; in variational transitionstate theory the position of the dividing surface is varied so as to get a better estimate of the rate constant. Consider the axis of the double helix as a space curve; it is known experimentally that the curve can have non-planar geometry.

For example, it can be solenoidal itself, thus the name supercoil. DNA from some organisms have their helical axis being a closed space curve with topological linking numbers, i. Bauer, F. Crick, and J. White, Sci. In mathematics, a domain is the set on whose members a relation operates. Comment: For further comments on the mathematical sense of domain, see relation.

See also image and range. Comment: It is this that a database or artificial intelligence system, or indeed any abstract model of a phenomenon, reifies. It is distinguished from a data model, which is an implementation method such as relational, object-oriented, or declarative database. See acceptor. Each chain is a polymer made of four types of nucleotide: A adenine , G guanine , T thymine , and C cytosine. The structure that was first proposed by J. Watson and F. Crick immediately leads to a possible mechanism of biological heredity. This was later confirmed by experiments and hence provides a molecular basis for genetics.

See also dual basis. Thus, when considering families of finite element spaces, the H -norm of the discretization error may converge asymptotically faster than the V -norm. Comment: What determines which direction of a reaction forward or backward will predominate is the relative concentration of reactants forming the two sets of obligatorily coreacting species. High concentrations of one set will drive the chemistry in the direction which consumes the reactants of that set, until the two sets are in equilibrium.

See also dextralateral, direction, formal reaction equation, microscopic reversibility, product, rate constant, reversibility, sinistralateral, and substrate. Comment: Notice this is distinct from a device in that the latter is not required to exhibit conservation. See also biochemical, chemical, functional, kinetic, mechanistic, phylogenetic, regulatory, thermodynamic, and topological motives. See equilibrium point, first integral. E edge An unordered pair of nodes in a graph, usually denoted as a tuple of arity two with the two nodes as arguments.

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Comment: Edges are usually represented as lines or arcs in renderings of a graph or network. Incidence relation and undirected edge are synonyms. See directed edge. Einstein Albert Einstein — German-American physicist. The inventor of special and general relativity theory. Nobel prize in physics Popularly regarded as a genius among geniuses, the greatest scientist in history. Everything should be formulated as simple as possible, but not simpler. The smallest electric charge found on its own is the elementary charge, e, the charge of a proton. Let A be a connection 1-form vector potential.

The bilinear form a could be replaced by a discrete approximation ah see variational crime. See electric charge. Until evidence to the contrary is discovered, an elementary reaction is assumed to occur in a single step and to pass through a single transition state. An inf-sup condition for a is an immediate consequence. See also bag, list, sequence, set, and tuple. An end-group is attached to only one constitutional unit of a macromolecule or oligomer molecule. Enzymes are a class of proteins made of polypeptide. Comment: Until about 15 years ago all known enzymes were proteins, but since then a number of naturally occurring, catalytic RNAs have been discovered.

This definition includes the products of laboratory manipulation as well as molecules found in nature. For example, an epimorphism of vector spaces is a linear surjective map; an epimorphism of groups is a surjective group homomorphism; an epimorphism of manifolds is a surjective differentiable map. See also bundle morphism. A macromolecule can contain many different epitopes, each capable of stimulating production of a different specific antibody.

## Dictionary of Applied Math for Engineers and Scientists

See Hamilton principle and Euler-Lagrange equations. Comment: This definition sets up conditions that allow one reasonably to approximate thermodynamic activities by concentrations, and reactant order by stoichiometries. It also assumes that experiments measuring the apparent equilibrium constant have been done by measuring the constant at several different concentrations of reactants and extrapolating the results to zero concentration.

Under these conditions, the equilibrium constant as used by biochemists comes reasonably close to the equilibrium constant as defined in thermodynamics. See also formal reaction equation. It is an equivalence relation. On finite dimensional vector spaces all norms are equivalent.

Erlanger program A plan initiated by Felix Klein in to describe geometric structures in terms of their groups of automorphisms. Euler-Lagrange equations The Lagrangian formulation of a classical mechanical system described by a Lagrangian L q 1 ,. Some say the most prolific mathematician in history and the first modern mathematician universalist.

He worked at the St. Petersburg Academy and the Berlin Academy of Science. For a mechanical Lagrangian system Q, L over the configuration space Q with local coordinates q i the Lagrangian is a function L t, q i , ui with q i , ui local coordinates over. See Hamilton principle and Lagrangian system. In relativity, a point in space-time. The operator Ft is called the evolution operator.