compound transformation

mation on a figure that can be regarded as one transformation followed by another transformation. For example, a compound transformation could consist of a rotation followed by a translation through space. Another example is a translation followed by a reflection. If the first and second trans- formations on the object O are denoted T1

and T2 respectively, then the compound

transformation is denoted by T2T1(0). In

general, performing the two transforma- tions in different order leads to two differ- ent compound transformations. This is consistent with the fact that transforma- tions can be described by matrices.

computability

/kŏm-pyoo-tă-bil-ă-tee/ Intuitively, a problem or function is com- putable if it is capable of being solved by an ideal machine (computer) in a finite time. In the 1930s it was discovered that some problems had no algorithmic solution and could not be solved by computers. This led many mathematicians to try to formulate a precise definition of the intuitive concept of computability, and Turing, Gödel, and Church independently came up with three very different abstract definitions, which all turned out to define exactly the same set of functions. The definition given by Tur- ing is that a function f is computable if for each element x of its domain, when some representation of x is placed on the tape of a TURING MACHINE, the machine stops in a finite time with a representation of f(x) on the tape.

computer

Any automatic device or ma- chine that can perform calculations and other operations on data. The data must be received in an acceptable form and is processed according to instructions. The most versatile and most widely used com- puter is the digital computer, which is usu- ally referred to simply as a computer. See also analog computer; hybrid computer.

A digital computer is an automatically controlled calculating machine in which in- formation, generally known as data, is rep- resented by combinations of discrete electrical pulses denoted by the binary dig- its 0 and 1. Various operations, both arith- metical and logical, are performed on the data according to a set of instructions (a program). Instructions and data are fed into the main store or memory of the com- puter, where they are held until required. The instructions, coded like the data in bi- nary form, are analyzed and carried out by the central processor of the computer. The result of this processing is then delivered to the user.

The technology used in digital comput- ers is so highly advanced that they can op- erate at extremely high speeds and can store a huge amount of information. The tube valves used in early computers were replaced by transistors; transistors, resis- tors, etc., were subsequently packed into

integrated circuits, which have become more and more complicated. As the elec- tronic circuits used in the various devices in a computer system have decreased in size and increased in complexity, so the com- puters themselves have grown smaller, faster, and more powerful. The microcom- puter has been developed as a somewhat simpler version of the full-size mainframe computer. Computers now have an im- mense range of uses in science, technology, industry, commerce, education, and many other fields.

computer graphics

The creation and re- production of pictures, photographs, and diagrams using a computer. There are many different formats for storing images but they fall into two main classes. In raster graphics the picture is stored as a series of dots (or pixels). The information in the computer file is a stream of data indicating the presence or absence of a dot and the color if present. Images of this type are sometimes known as bitmaps. This format is used for high-quality artwork and for photographs. Diagrams are more conve- niently stored using vector graphics, in which the information is stored as mathe- matical instructions. For example, it is pos- sible to specify a circle by its center, its radius, and the thickness of the line form- ing the circumference. More complicated curves are usually drawn using BEZIER CURVES. Vector images are easier to change and take up less storage space than raster images.

computer modeling

The development of a description or mathematical representa- tion (i.e. a model) of a complicated process or system, using a computer. This model can then be used to study the behavior or control of the process or system by varying the conditions in it, again with the aid of a computer.

concave

/kong-kayv, kong-kayv/ Curved inwards. For example, the inner surface of a hollow sphere is concave. Similarly in two dimensions, the inside edge of the cir- cumference of a circle is concave. A con- cave polygon is a polygon that has one (or

more) interior angles greater than 180°. Compare convex.

concentric

Denoting circles or spheres that have the same center. For example, a hollowed out sphere consists of two con- centric spherical surfaces. Compare eccen- tric.

conclusion

The proposition that is as- serted at the end of an argument; i.e. what the argument sets out to prove.

condition

In logic, a proposition or state- ment, P, that is required to be true in order that another proposition Q be true. If P is a necessary condition then Q could not be true without P. If P is a sufficient condi- tion, then whenever P is true Q is also true, but not vice versa. For example, for a quadrilateral to be a rectangle it must sat- isfy the necessary condition that two of its sides be parallel, but this is not a sufficient condition. A sufficient condition for a quadrilateral to be a rhombus is that all its sides have a length of 5 centimeters, but this is not a necessary condition. For a rec- tangle to be a square it is both a necessary and a sufficient condition that all its sides are of equal length.

In formal terms, if P is a necessary con- dition for Q, then Q → P. If P is a sufficient condition, then P → Q. If P is a necessary and sufficient condition for Q then P ≡Q. See also biconditional, symbolic logic.

conditional

(conditional statement; con-

ditional proposition) An if… then… state-

ment.

conditional convergence

See absolute

convergence.

conditional equation

See equation.

conditional probability

See probability.

cone

A solid defined by a closed plane curve (forming the base) and a point out- side the plane (the vertex). A line segment from the vertex to a point on the plane curve generates a curved lateral surface as the point moves around the plane curve. The line is the generator of the cone and the plane curve is its directrix. Any line seg- ment from the vertex to the directrix is an element of the cone.

If the directrix is a circle the cone is a circular cone. If the base has a center, a line from the vertex to this is an axis of the cone. If the axis is at right angles to the base the cone is a right cone; otherwise it is an oblique cone. The volume of a cone is one third of the base area multiplied by the altitude (the perpendicular distance from the vertex to the base). For a right circular cone

V = πr2_h/3

where r is the radius of the base and h the altitude. The area of the curved (lateral) surface of a right circular cone is πrs, where s is the length of an element (the slant height).

cone

convex concave

Concave and convex curvatures

hyperbola ellipse

parabola

The three sections of a cone – the ellipse, the parabola, and the hyperbola.

If an extended line is used to generate the curved surface (i.e. extending beyond the directrix and beyond the vertex), an ex- tended surface is produced with two parts (nappes) each side of the vertex. More strictly, this is called a conical surface

confidence interval

An interval that is

In document The Facts on File Dictionary of Mathematics (Page 50-53)