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 series – Stone-Weierstrass Theorem series    ARTICLE   A series is an infinite sum, where the nth summand is the nth term of a sequence. A series is usually denoted using “sigma notation,” i.e.,The index n may begin with 0, 1, or k for any natural number k, as a matter of convenience. The nth partial sum Sn of a series is the (finite) sum of the first n terms of the series. A series is said to converge if and only if its sequence of partial sums {S 1, S 2, . . . , Sn, . . . } is a convergent sequence. There are several important types of series and several tests for the convergence of a series. Additionally, most useful functions have Taylor series representations, which makes them very important in the study of differential equations. See the article for a complete description. set   Naively, any well-defined collection considered as a single, abstract object. By “well-defined” is meant that it is always possible to determine for a given set when something is an element of the set and when not. In formal set theory, the term “set” is not defined, but is a primitive term whose meaning is informed purely by the axioms in which it appears.Cf. ZF, ZFC. set algebra   See algebra of sets. set difference   The set difference of two sets A and B, denoted A - B or A \ B, is the set of elements that is contained in A but not in B. This is equivalent to the intersection between A and the complement of B. set function   A function whose domain of definition is a collection of sets. set ring   See ring of sets. set theory   Naive set theory: The study of sets (i.e., well-defined collections of objects) which have a binary extensional relation (set membership) defined on them.Abstract set theory: As naive set theory, but with all sets built using only elements which are themselves sets (beginning with the empty set, which has no members).Formal set theory: Any of several axiom systems of abstract set theory in the language of first-order logic, such as Zermelo Fraenkel set theory, Gödel-Bernays set theory, Quine’s New Foundations, etc. Sid’s Paradox    ARTICLE   A paradox of knowledge: is the distinction between matters of opinion and matters of fact a matter of opinion or a matter of fact? See the article for discussion. signed measure   Given a set X together with a s-algebra of sets M defined on it, a signed measure on (X, M) is an extended real-valued function m with domain M satisfying: The signed measure of the empty set is zero.The signed measure m assumes at most one of the values +/- infinity.(Countable additivity) Given a countable sequence of disjoint sets in M, the signed measure of the union of the sequence is equal to the sum of the signed measures of the sets in the sequence, where this sum converges absolutely if the signed measure of the union is finite. Technically speaking, every measure is a signed measure; ordinary (i.e., nowhere negative) measures are sometimes called positive measures. similar   Graph Theory: Two vertices or edges of a graph are called similar if there is an automorphism of the graph that takes one to the other. sin   See sine. sine    ARTICLE   A periodic trigonometric function defined on angles. Usually abbreviated “sin.”See the article for a complete exposition.Cf. tangent, cosine. singleton set   A set with exactly one element. singular cardinal   A cardinal that is not regular. slope   A line in the Cartesian plane which passes through two points (x 1, y 1) and (x 2, y 2) has a slope m given byThe slope may easily be remembered as “rise over run.” It is evident that the slope of a horizontal line is 0, and the slope of a vertical line is undefined.Cf. linear function. space   Any abstract set with a structure defined on it, such as an order relation, metric, etc.Cf. Euclidean space, Hilbert space, metric space, topological space. sphere   A closed surface, all points of which are equidistant from a given point, called the center. In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) isThe term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1. square   A regular polygon having four equal sides and four right angles. square matrix   A matrix that has the same number of rows as columns. stationary set   If a is an ordinal, a set S in a is called stationary if S has non-empty intersection with every closed unbounded subset of a. Stone-Weierstrass Theorem   If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X). series – Stone-Weierstrass Theorem
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