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The branch of mathematics dealing with the solution of equations. These equations involve unknowns, or variables, along with fixed numbers from a specified system (Fig. 1). The origins of algebra were based on the need to develop equations that…

The study of zero sets of polynomial equations. Examples of zero sets of polynomial equations studied in algebraic geometry are the parabola y − x2 = 0, thought of as sitting in the (x, y)-plane, and the locus of all points (t, t2, t3, t4), which is…

The branch of mathematics dealing with two fundamental operations, differentiation and integration, which are carried out on functions. The subject of calculus (Fig. 1), as traditionally developed in college textbooks, is partly an elementary…

A heart-shaped curve generated by a point of a circle that rolls (without slipping) on a fixed circle of the same diameter. In point-wise construction of the curve, let O be a fixed point of a circle C of diameter a, and Q a variable point of C. Lay…

A theory of mathematical structure in which smooth continuous inputs lead to discontinuous responses. Water suddenly boils, ice melts, a building crashes to the ground, or the earth unexpectedly buckles and quakes. The French mathematician René Thom…

The branch of mathematics which studies arrangements of elements (usually a finite number) into sets under certain prescribed constraints. Problems combinatorialists attempt to solve include the enumeration problem (how many such arrangements are…

The theory of complex systems. Complexity theory arises from the need to understand the richness in structure and behavior often seen in large systems. The property that distinguishes complex systems from systems that are large but simple is the…

The solid of revolution obtained by revolving a right triangle about one of its shorter sides is called a cone, or more precisely a right circular cone (see illustration). More generally, the term cone is used in solid geometry to describe a solid…

A special operation in mathematics in which a set of points in one coordinate system is mapped or transformed into a corresponding set in another coordinate system, preserving the angle of intersection between pairs of curves.A mapping or…

Schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference. These quantities are the coordinates of a point. To each set of coordinates there corresponds just one point in any…

An applied branch of decision theory. Decision analysis offers individuals and organizations a methodology for making decisions; it also offers techniques for modeling decision problems mathematically and finding optimal decisions numerically.…

A broad range of concepts which have been developed to both describe and prescribe the process of decision making, where a choice is made from a finite set of possible alternatives. Normative decision theory describes how decisions should be made in…

A mathematical operation performed on a function to determine the effect of a change in the value of the independent variable. Functions of one variable are considered in this article. For differentiation of functions of several variablesPartial…

For a polynomial f(x) = an xn + an−1 xn−1 + + a1 x + a0, the discriminant is given by the expression where x1, x2, …, xn are the roots of the equation f(x) = 0. There are n(n − 1)/2 terms (xi − xj)2 in the product corresponding to all possible…

A mathematical abstraction corresponding to an expression of the type βγ + δ∊ + · · · , in which the elements (dyad symbols) consist of two vector symbols in juxtaposition without the intervention of either the dot (·) or cross (×). Essentially a…

One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the…

One of the values of the parameter in an eigenvalue equation for which the equation has a solution. A linear eigenvalue equation is a parameter-dependent equation of the form (1) 1that possesses nonvanishing solutions, |v(n), only for particular…

A member of the class of curves that are intersections of a plane with a cone of revolution. The ellipse is obtained when the plane cuts all the elements of one nappe, and does not go through the apex. In the illustration, denote the distance…

The elliptic functions were developed originally by K. F. Gauss, N. H. Abel, K. G. J. Jacobi, and others in order to solve the problem of finding the arc length of an ellipse, but they proved to have remarkable applications in physics, probability…

A curve traced by a point on a circle that rolls on the convex side of a fixed circle. The term is also occasionally applied to the curve generated by a point on the prolongation of the radius of a circle as the circle rolls on a straight line. If…

A branch of probability and statistics concerned with deriving information about properties of random variables, stochastic processes, and systems based on observed samples. Some of the important applications of estimation theory are found in…

The chief subject matter of the monumental 13-volume work called The Elements, written about 300 BCE by the Greek mathematician Euclid, who taught and founded a school of geometry at Alexandria. One of the milestones in the history of scientific…

In mathematics, a symbol or number written to the right of and above another symbol or number to denote how many times the latter is to be multiplied by itself. For example, 73 = 7 · 7 · 7, and a3 = a · a · a. By use of this convenient mathematical…

A process in mathematics used to find the value of a function outside its tabulated values. This is done as in interpolation by assuming that over a small range of x the function may be closely approximated by a polynomial or some other readily…

A decomposition of a space E into a family of identical subspaces (fibers). The space of subspaces is called the base space. Fiber bundles arise naturally in many physical situations. The theory of fiber bundles has been applied to gauge theory in…

In algebra, the term "field" is used to designate an algebraic system or structure containing at least two elements and having two binary rules of composition: addition and multiplication (that is, if a and b are any two elements of the field, then…

A numerical analysis technique for obtaining approximate solutions to many types of engineering problems. The need for numerical methods arises from the fact that for most engineering problems analytical solutions do not exist. While the governing…

These parts of mathematics which deal with finite sets. More generally, finite mathematics is taken to include those fields which make no use of the concept of limit; thus linear programming, which finds maxima and minima through the four arithmetic…

Mathematical tools for the analysis of functions through decomposition into sinusoids. In 1822, J. Fourier proposed that ordinary mathematical functions could be represented by a sum of sinusoids, even though at first sight it might seem that the…

Geometrical objects that are self-similar under a change of scale, for example, magnification. The concept is helpful in many disciplines to allow order to be perceived in apparent disorder. For instance, in the case of a river and its tributaries,…

A fuzzy set is a generalized set to which objects can belong with various degrees (grades) of memberships over the interval [0,1]. In general, fuzziness describes objects or processes that are not amenable to precise definition or precise…

The theory of games of strategy can briefly be characterized as the application of mathematical analysis to abstract models of conflict situations. The first such models analyzed by the theory were parlor games such as chess, poker, and bridge.…

A particular mathematical function that can be used to express many definite integrals. There are, however, no significant applications where the gamma function by itself constitutes the essence of the solution. Instead it occurs usually in…

The assertion, under certain light restrictions, that the volume integral through a volume V of the divergence of vector function (x, y, z) is equal to the surface integral of the exterior normal component of over the boundary surface S of V, or in…

Thurston's geometrization conjecture (now, a theorem of Grigory Perelman), which aims to answer the question: How could one describe the possible shapes of our universe? Here we will be assuming (although, some scientists think otherwise) that our…

A branch of mathematics concerned with the properties of space, including points, lines, curves, planes and surfaces in space, and figures bounded by them. Euclidean geometry Geometry as a high school subject is largely based on the Elements of…

The result, proved by K. Gödel in 1931, that any sufficiently advanced mathematical system must be incomplete in that there must always be a true sentence that is not provable in the system. Roughly speaking, Gödel showed how, for each such system,…

The result of the application of the distributive vector differential operator ∇, ∇ = i∂/∂x + j∂/∂y + k∂/∂z, to a differentiable scalar function S(x, y, z); thus ∇S = i∂S/∂x + j∂S/∂y + k∂S/∂z. The letters i, j, k are symbols for the base vectors…

A branch of mathematics that belongs partly to combinatorial analysis and partly to topology. Its applications occur (sometimes under other names) in electrical network theory, operations research, organic chemistry, theoretical physics, and…

Procedures and techniques for visually representing on paper or a screen information pertaining to data analysis and decision making or relationships between variables by means of diagrams or charts.Structural information is usually depicted by a…

A solution of a partial differential equation for the case of a point source of unit strength within the region under examination. The Green's function is an important mathematical tool that has application in many areas of theoretical physics…

A term used variously in mathematical literature to denote either (1) the Gauss divergence theorem (shown below)or some one of several forms or immediate consequences of their theorem, or (2) the plane case of Stokes' theorem.Gauss' theorem Stokes'…

Any set of elements which is equipped with an operation (called multiplication) satisfying the requirements (1), (2), (3), and (4) is called a group. Group theory is the branch of mathematics devoted to the properties of groups. Group theory has…

Any nonplanar curve all of whose tangents make the same angle with a fixed line. Other characteristic properties are that all principal normals are parallel to a plane and that the ratio of torsion to curvature is constant. If a helix has constant…

The application to biological systems of the theory of encoding effects on signal transmission and communications efficiency. In everyday language, information is associated with knowledge, meaning (semantics), and the influence of information on…

A term applied to a curve C′ that cuts at right angles all tangents of a curve C (see illustration). Each curve C has infinitely many involutes and the distance between corresponding points of any two involutes is constant. If xi = xi (s), with i =…

A method, developed originally by A. M. Legendre, of obtaining the best values (the ones with least error) of unknown quantities supposed to satisfy a system of linear equations of the form shown as notation (1),1where n > m. Since there are more…

Solutions to the differential equation (1 − x2)y′ − 2xy′ + v(v + 1)y = 0. Legendre polynomials The most elementary of the Legendre functions, the Legendre polynomial Pn(x) can be defined by the generating function in Eq. (1). More explicit…

A curve shaped like the figure eight (see illustration), referred to by Jacques Bernoulli in 1694. Let F1, F2 be points of a plane π, with F1 F2 = 2a, a > 0. The locus of a point P of π which moves so that PF1 · PF2 = b2, where b is a positive…