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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 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 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…

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 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 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…

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), 1…

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…

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…

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 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…

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 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 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…