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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…

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 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…

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…

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.…

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…

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 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…

A topological group with only countably many connected components whose identity component is open and is an analytic group. An analytic group or connected Lie group is a topological group with the additional structure of a smooth manifold such that…

An area of mathematics concerned with the minimization (or maximization) of a linear function of several variables subject to linear equations and inequalities. Linear programming developed from three main areas: transportation problems, game…

The subject that investigates, formulates, and establishes principles of valid reasoning.The first attempt to investigate acceptable modes of reasoning systematically was made by Aristotle, in whose Organon reasoning was recognized as the subject of…

The study of matrices and their properties, and of linear transformations on vector spaces, which can be represented by matrices. A matrix is a rectangular array of numbers, with the numbers that appear in the matrix being called entries. For…

A branch of mathematical analysis connected with the theory of integration. In order to discuss this subject, a formal definition of the term measure must be given. Sigma algebras Let X be an arbitrary set. Let m be a fixed collection of subsets of…

A technique for estimating the solution, x, of a numerical mathematical problem by means of an artificial sampling experiment. The estimate is usually given as the average value, in a sample, of some statistic whose mathematical expectation is equal…

A graphical relationship between a set of variables that are related by a mathematical equation or law. The fundamental principle involved in the construction of a nomographic or alignment chart consists of representing an equation containing three…

The area of applied mathematics and operations research concerned with finding the largest or smallest value of a function subject to constraints or restrictions on the variables of the function. Nonlinear programming is sometimes referred to as…

The study of the properties and relationships between integers and other special types of numbers. There are many sets of positive integers of particular interest, such as the primes and the perfect numbers. Number theory, of ancient and continuing…

The mathematical theory of the formation and behavior of queues or waiting lines. The name is also applied loosely to the mathematical study of a wide variety of problems connected with traffic congestion and storage systems. Uneven flow through a…

One of the four fundamental operations of arithmetic and algebra. The symbol + of addition is thought to be a ligature for “et,” used in a German manuscript of 1456 to denote addition. Its first printed appearance is in Johann Widman's Behennede und…

The study of zero sets of polynomial equations. Examples 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 defined by Eqs. (1), 1 in the coordinate space with coordinates w,…

Total variation in experimental data is partitioned into components assignable to specific sources by the analysis of variance. This statistical technique is applicable to data for which (1) effects of sources are additive, (2) uncontrolled or…

A branch of mathematics in which algebra is applied to the study of geometry. Because algebraic methods were first systematically applied to geometry in 1637 by the French philosopher-mathematician René Descartes, the subject is also called…

A framework for solving a problem. The analytic hierarchy process is a systematic procedure for representing the elements of any problem. It organizes the basic rationality by breaking down a problem into its smaller constituents and then calls for…

The superficial contents of a geometrical figure of two dimensions. The area of any rectangle or square is the product of two adjacent sides, one of which may be called the base and the other the altitude. In general, any line segment that partially…

A branch of mathematics dealing with numbers, operations on numbers, and computation. Arithmetic is useful in solving many practical problems, such as buying, selling, budgets, sports statistics, and measurement. The usual numbers of arithmetic are…

If the graph of a function y = f (x) goes out to infinity, each “end” of the graph may get closer and closer to a straight line, which is then called an asymptote. In other words, some straight line may well approximate the graph as the function…

The application of the properties of the centroid of a set of weighted points to geometry. For certain new operations such as the addition of point and point or of point and vector, it is possible to give their vector equivalents and work within the…

An approach to statistics in which estimates are based on a synthesis of a prior distribution and current sample data. Bayesian statistics is not a branch of statistics in the way that, say, nonparametric statistics is. It is, in fact, a…

By definition the solutions of Bessel's differential equation, Eq. (1). 1Bessel functions, also called cylinder functions, are examples of special functions which are introduced by a differential equation. Bessel functions are of great interest in…

One of the most important algebraic identities, with many applications in a variety of fields. The binomial theorem, discovered by Isaac Newton, expresses the result of multiplying a binomial by itself any number of times, as given in Eq. (1),…

The application of mathematical and statistical methods to describe and analyze data concerning the variation of biological characteristics obtained from either observation or experiment. The concept of a population and the samples derived from it…

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

The curve formed by an ideal heavy uniform string hanging freely from two points of support. The lowest point A (Fig. 1) is the vertex. The portion AP is an equilibrium under the horizontal tension H at A, the tension F directed along the tangent at…

The curve that is the locus of points in a plane with equal distance from a fixed point. One of the most basic geometric figures, a circle is perfectly round. In elementary mathematics, "circle" often refers to the finite portion of the plane…

A procedure in which the basic problem is to pass a curve through a set of points, representing experimental data, in such a way that the curve shows as well as possible the relationship between the two quantities plotted. It is always possible to…

A member of the class of curves that are intersections of a plane with a cone of revolution. It is obtained (see illustration) when the cutting plane is parallel to an element of the cone. Conic section Parabola as (a) conic section and (b) locus…

An auxiliary variable, functions of which give the coordinates of a curve or surface. The coordinates of a curve are functions of one parameter. A curve in 3-space has parametric equations (1). 1 The coordinates of a surface are functions of two…

A type of mathematical equation used, typically, to represent curves in a plane or in space of three dimensions. In principle, however, there is no limitation to any particular number of dimensions. A parameter is actually an independent variable.…

A mathematical operation performed on functions of more than one variable. In this article only two or three variables are considered; however, the principles apply to functions of n variables, for any positive integer n > 1. If z = f(x,y), the…

A scenario for the transition of a natural process from regular motion to chaos. Various natural processes develop in time in a way that depends upon prevailing environmental details. A quantity that specifies the particular state of the environment…

The field of knowledge concerned with collecting, analyzing, and presenting data. Statistical methods are regularly used by researchers in the physical, biological, and social sciences, as well as engineers, business managers, government officials…

How will rapid urbanization affect how cities develop sustainably in the future? What are the best interventions for reducing crime rates in an area? Can we balance human resource requirements with environmental preservation? These are not trivial…

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…

The study of systems consisting of arbitrary sets of elements of unspecified type, together with certain operations satisfying prescribed lists of axioms. Abstract algebra has been developed since the mid-1920s and has become a basic idiom of…

A branch of mathematics that was first developed systematically, because of its applications to logic, by the English mathematician George Boole, around 1850. Closely related are its applications to sets and probability.Boolean algebra also…

An extension of the part of differential calculus which deals with maxima and minima of functions of a single variable. The functions of the calculus of variations depend in an essential way upon infinitely many independent variables. Classically…

In its simplest form, a vector is a directed line segment. Physical quantities, such as velocity, acceleration, force, and displacement, are vector quantities, or simply vectors, because they can be represented by directed line segments. The algebra…

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…

Complex dynamics is a field of mathematics in which one studies dynamical systems on mathematical spaces, where these spaces are defined over the complex numbers C. More generally, a dynamical system is a pair of objects (X, f), where X is a space…

A natural and extremely useful extension of the familiar real numbers. They can be introduced formally as follows. Consider the two-dimensional real vector space consisting of all ordered pairs (a1, a2) of real numbers. Geometrically this space can…

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…

One of the class of curves in which a plane may cut a cone (surface) of revolution. They were extensively studied by the ancient Greeks. The section is a parabola if the plane is parallel to an element of the cone, an ellipse or circle if the plane…

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…

A parallelepiped whose six faces are all squares. The cube (see illustration) is one of the five regular solids known to the ancient Greeks, who proposed the famous problem (now proved to be unsolvable) of constructing, with the use of the compass…

A curve traced in the plane by a point on a circle that rolls, without slipping, on a line. If the line is the x axis of a rectangular coordinate system, at whose origin O the moving point P touches the axis, parametric equations of the cycloid are…

The solid of revolution obtained by revolving a rectangle about a side is called a cylinder, or more precisely a right circular cylinder. More generally the word cylinder is used in solid geometry to describe any solid bounded by two parallel planes…

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…

The concept of a determinant can best be explained with reference to a matrix A. Let be an n by n square array of numbers. Such an array is called an n by n square matrix, and the numbers aij in the array are called elements of the matrix.…

A relationship between one or more independent variables, one or more dependent variables, and differences of those variables. Difference equations arise in the analysis of discrete systems (for example, a string loaded along its length with small…

A relationship between a function and its derivatives.If there is one independent variable, the differential equation is called an ordinary differential equation. The general form of such an equation is shown in Eq. (1), 1where t is the independent…

A branch of mathematics that deals with intrinsic properties of curves and surfaces in three-dimensional euclidean space. The intrinsic properties are those which are independent of the geometrical object's orientation or location in space. The…

In arithmetic and algebra, the process of finding one of two factors of a number (or polynomial) when their product and one of the factors are given. The symbol ÷ now used mostly in elementary English and American arithmetics to denote division…

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 method of quantitative multivariate analysis with the goal of representing the interrelationships among a set of continuously measured variables (usually represented by their intercorrelations) by a number of underlying, linearly independent…

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…

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…

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…

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…

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' theoremStokes'…

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…

An abstract notion of great power and beauty, which has been central to the development of mathematical analysis and forms the backdrop for many applications of analysis to science and engineering. Its essence lies in the fact that the objects of…

A curve cut from a cone of revolution by a plane that intersects both nappes of the cone and does not contain the apex (Fig. 1). In analytic geometry it is shown, as shown in Fig. 2, that a hyperbola is the locus of points P in a plane, such that PF…

The hyperbolic sine and cosine of a real or complex variable z are defined by Eqs. (1). 1 Both sinh z and cosh z have a period 2πi of ez. From Dez = dezdz = ez, Eqs. (2) 2are obtained. Since by definition, Eqs. (3) hold and the series converge for…

A geometry that obeys a consistent set of axioms differing from Euclid's in implying that the angle sum of a triangle is less than 180°. In the early nineteenth century, K. F. Gauss proved the existence of such a geometry. He questioned which…