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Planets beyond our solar system. For millennia, humankind has known about most of the major planets in our solar system because these worlds are visible to the naked eye. Exoplanets, however, were not indisputably known until 1995, when astronomers…

A space telescope mission dedicated to detecting exoplanets. The tenth mission launched under the National Aeronautics and Space Administration's (NASA's) Discovery Program, Kepler was the agency’s first mission dedicated to the search for planets…

The planet fourth outward from the Sun and the planet most like Earth in terms of geology, climate, and suitability for life. Mars is nicknamed the "Red Planet" for its reddish hue (Fig. 1). Besides Earth, Mars is the most-studied world in the solar…

The apparent change in direction of a source of light caused by an observer's component of motion perpendicular to the impinging rays.To visualize the effect, first imagine a stationary telescope (illus.a) aimed at a luminous source such as a star,…

A discipline that combines the means to know the location and heading of an aircraft with respect to some map or coordinate system, with guidance which means steering the aircraft along some desired path toward a destination. The construction of the…

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 closest spiral galaxy to the Milky Way Galaxy. The name of the Andromeda Galaxy (Fig. 1) derives from its appearance in the constellation of Andromeda. The galaxy is also known as M31 and NGC 224, based on its entries in two widely used…

A gravitationally bound, condensed body whose mass is too small to sustain nuclear fusion reactions in its core. Brown dwarfs are often likened to "failed stars." The approximate mass range for brown dwarfs is between 13 and 80 times the mass of…

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…

An algorithm that gives a name to each day. These day names are generally distinct, but not always. For example, the day of the week is a calendar, in a trivial sense, with infinitely many days having the same day name. The design of calendars has…

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…

A force whose line of action is always directed toward a fixed point. The central force may attract or repel. The point toward or from which the force acts is called the center of force. If the central force attracts a material particle, the path of…

Light emitted by a high-speed charged particle when the particle passes through a transparent, nonconducting, solid material at a speed greater than the speed of light in the material. The blue glow observed in the water of a nuclear reactor, close…

A quantitative measure of a star's color. Even a casual look at the night sky reveals the stars to be colored. From spectral analysis and the laws of radiation, it has been known since early in the twentieth century that such color is closely…

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…

The Southern Cross, a small southern constellation. Because of the absence of a star at the center of the cross, this constellation really looks like a kite (see illustration). To its side, the Coalsack is a giant dark silhouette of dust against the…

A unit of time equal to the period of rotation of Earth. Different sorts of day are distinguished, according to how the period of rotation is reckoned with respect to one or another direction in space. A day is normally defined as 86,400 SI (Système…

A form of navigation that determines position of a craft by advancing a previous position to a new one on the basis of assumed distance and direction moved. The name probably stems from the early practice of determining speed by throwing overboard a…

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…

Equipment used for determining the direction from which a received signal is being radiated. Direction finders serve a variety of applications. Direction finders can be used for navigation purposes to determine the heading of a vehicle relative to a…

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…

An internationally standardized navigation system which allows an aircraft to measure its distance from a selected ground-based beacon. Such beacons are used throughout the world by all airliners, most of the military aircraft of the West, and a…

A star whose state of evolution resembles that of the Sun. The term “dwarf star” derives from the early work of Ejnar Hertzsprung and Henry Norris Russell, who distinguished two kinds of stars: large ones, called giants (or supergiants), and smaller…

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…

Systems that locate suitably equipped users by means of measurements made by electronic sensors. Typical measurements include distance or bearing from a vehicle to a known point, or direct determination of the present position of the vehicle in a…

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…

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.…

A nearby neutron star that emits pulsed x-rays and gamma rays, steady optical radiation, and possibly radio and optical pulsations. Since the 1970s, it has been studied at a level of detail very unusual for a neutron star. As a result, not only are…

A small, opaque nebula seen in silhouette against a rich star field or a bright nebula. Globules were first cataloged in the 1920s. In 1947, B. J. Bok called attention to their potential significance for star formation, and since then they have been…

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 system carried on many aircraft to warn the pilot that the aircraft may be in danger of inadvertent contact with the ground. It is intended to reduce the occurrence of controlled-flight-into-terrain (CFIT) accidents, in which aircraft with no…

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…

A process of navigation by which a destination is approached by keeping some navigation parameter constant. In its early uses, the most commonly chosen parameter was the relative bearing from the vehicle to destination as determined from a signal…

Measurement of the time dimension. In practice, horology is the search for a steady or repetitive action, and the design of an instrument to perform that action and to indicate (read out) a measure of the action. Until early in the twentieth…

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 =…

The three laws of planetary motion discovered by Johannes Kepler during the early years of the seventeenth century. Kepler, a German astronomer (Fig. 1), published the first two laws in 1609 in a book entitled Astronomia nova, and the third law in…

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…