BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry circle – concave function circle   In a plane, the locus of all points equidistant from a given point, called the center. The general equation for a circle in the Cartesian plane is given by (x - h) 2 + (y - k) 2 = r 2, where r is the radius of the circle (distance from the center to the locus of points), and (h, k) are the coordinates of the center.The interior of a circle is referred to as an open disk.A circle is also a conic section; a special case of an ellipse in which the foci coincide. Related article: Conics circumference   Geometry: The distance around a circle in the plane, or around a great circle of a sphere.Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles. closed   General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.See the following listings for other uses of the word “closed” in mathematics.Topology: A set is topologically closed if it is not open. closed interval   An interval of the real number line (or any other totally ordered set) which includes its endpoints. An interval containing only one of its endpoints is called half-open. Cf. open interval. closed set   Topology: A subset E of a topological space X is closed if X - E (set difference) is open. In a metric space, E is closed if every convergent sequence in E converges in E; equivalently, if every accumulation point of E is in E.Set Theory: If a is a limit ordinal, then a set C contained in a is called closed if and only if for every limit ordinal b less than a, if C b is unbounded in b, then b C. C is called c.u.b. (“cub set” or “club set”) if and only if C is closed and unbounded in a.Cf. stationary set. closed set system   If X is a set (or proper class) and F is a family of subsets of X, then F is called a closed set system providedX is a member of F, andF is closed under arbitrary intersections.Cf. filter. closure   Topology: The closure of a subset E of a topological space is the smallest closed set containing E. It may also be expressed as the union of E with its accumulation points. If E is closed, then it is equal to its closure.Algebra: An algebraic closure of a field F is a field G containing F such that every polynomial with coefficients from F has a root in G. closure operator   If X is a set, then a function C from P(X) into P(X) (i.e., a function on the power set of X) is called a closure operator providedY is contained in C(Y) for every subset Y of X, C(C(Y)) = C(Y) for every subset Y of X, andIf Y and Z are both subsets of X, with Y a subset of Z, then C(Y) is a subset of C(Z). Closure operators induce closed set systems. coefficient   See polynomial. combination   A subselection of a set of r elements from a set of n elements. The number of such combinations, i.e., the number of ways in which r elements may be chosen from a set of n elements, is given by the formulaThis operation is also sometimes denoted by nC r, and is read “n choose r.”Cf. permutation, factorial. common logarithm   A logarithm with base 10. commutative   An operation “ · ” on elements of a set A is commutative if for all elements a, b in A, a · b = b · a.Cf. associative, distributive property. commutative property   A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative. compact   Topology: In a topological space, a set E is compact if every open covering of E has a finite subcover, i.e., a finite subcollection which also covers E. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. E is called s-compact if there exists a sequence of compact sets {Ci} such that E is contained in their union.Cf. locally compact, Bolzano-Weierstrass property, Heine-Borel property.Set Theory: A cardinal k is called weakly compact if it is uncountable and Equivalently, k is weakly compact if it is strongly inaccessible and there are no k-Aronszajn trees.Lattices: an element a of a lattice L is called compact if whenever a is dominated by the join of a subset X of L then a is dominated by the join of a finite subset of X. Symbolically: complement   The complement of a set A is the set of all elements that are not elements of A.Graph Theory: The complement of a simple graph G with vertex set V is the simple graph Gc, which also has vertex set V, and in which two vertices are adjacent if and only if they are not adjacent in G. complementary angles   Two angles are complementary if they add up to a right angle. complete   Analysis: A metric space X is complete if every Cauchy sequence in X converges in X.Logic: a system of axioms for a mathematical theory is complete if every theorem in the theory is deducible from the axioms. Gödel's incompleteness theorem states that any axiom system which includes or allows the operations of arithmetic is necessarily incomplete.Set Theory: If F is a filter on a set X and k is a regular, uncountable cardinal, then we say that F is k-complete (k-closed) if AF for every AF with |A| < k. Every filter is w-complete. If k is the first uncountable cardinal (1), then F is called countably complete. Related article: Gödel's Theorems complete measure   See measure. complex number   An element of the set C of numbers of the form a + bi, where a and b are real numbers and i 2 = -1. The number i is called the imaginary number. concave   A region of space is concave if there are two points of the region such that a line joining the two points is not entirely contained within the region. In particular, a polygon is concave if any of its interior angles is greater than 180°.Cf. convex. concave function   A function is concave if the chord connecting any two points of its graph lies entirely below the graph.Cf. convex function. circle – concave function
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