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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,…

A book that contains astronomical or meteorological data arranged according to days, weeks, and months of a given year and may also include diverse information of a nonastronomical character. This article is restricted to astronomical and…

Any device which measures the height of an aircraft. The two chief types are the pressure altimeter, which measures the aircraft's distance above sea level, and the radar altimeter, which measures distance above the ground. Pressure Altimeter A…

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…

An anchored or moored floating object, other than a lightship, intended as an aid to navigation. Buoys are the most numerous of all engineered aids to navigation, with some 20,000 in United States waters alone.A great variety of buoys are in use.…

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…

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…

A service to promote the safe, orderly, and expeditious flow of air traffic. Safety is principally a matter of preventing collisions with other aircraft, obstructions, and the ground; assisting aircraft in avoiding hazardous weather; assuring that…

A device that reduces the risk of midair collision by providing advisories to the flight crew. It is known in the United States as a Traffic Alert and Collision Avoidance System (TCAS), and operates independently of the ground-based air-traffic…

An instrument that indicates the bearing, or angle of the direction in which an aircraft is pointing in the horizontal plane. A compass may indicate magnetic heading or bearing, bearing referenced to a radio signal source, or bearing with respect to…

A coordinated group of instruments that provides the flight crew with information about the aircraft and its subsystems. Together with the controls, aircraft instruments form the human–machine interfaces that enable the flight crew to operate the…

A ground mapping system that uses analog radar equipment to provide surveillance of aircraft and other surface vehicles on an airport surface. It is used by air-traffic controllers in airport control towers to monitor and control the movement of…

A device that computes and displays speed of an aircraft relative to the air mass in which the aircraft is flying. Three common types are the indicated airspeed meter, true airspeed indicator, and Machmeter.The commonest type is the indicated…

The satellite navigation systems GPS (the U.S.–managed Global Positioning System) and GLONASS (the Russian global navigation satellite system) have been operational in some form for decades. Other global and regional navigational satellite systems…

A device that provides the pilot with symbols representative of the attitude of an aircraft relative to an artificial horizon. In an automatic or artificial horizon a vertical gyro moves a horizon bar relative to a fixed aircraft index. Motion of…

The means for guiding and controlling aircraft from an initial approach altitude to a point where safe contact is made with the landing surface. Such systems differ from low-approach systems in three major respects: (1) They furnish not only…

An automatic means for steering an aircraft, ship, or other vehicle. The terminology means to automatically (autonomously) control a mode of the vehicle, usually its trajectory. Examples include altitude, speed, heading, accelerations, body rates,…

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…

Navigation with the aid of celestial bodies, primarily for determination of position when landmarks are not available. In celestial navigation, position is not determined relative to the objects observed, as in navigation by piloting, but in…

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…