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Encyclopedia > Support (mathematics)

In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. The most common situation occurs when X is a topological space (such as the real line) and f is a continuous function. In this case, the support of f is defined as the smallest closed subset of X outside of which f is zero. The topological support is the closure of the set-theoretic support. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a function returns a unique output for a given input. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the real line is simply the set of real numbers. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...


In particular, in probability theory, the support of a probability distribution is the closure of the set of possible values of a random variable having that distribution. Probability theory is the mathematical study of probability. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...


Compact support

Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are examples of functions that vanish at infinity. In good cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. Note that every function on a compact space has compact support since every closed subset of a compact space is compact. In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the function defined on the real line vanishes at infinity. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... In topology and related areas of mathematics a subset A of a topological space X is called dense (in X) if the only closed subset of X containing A is X itself. ...


It is possible also to talk about the support of a distribution, such as the Dirac delta function δ(x) on the real line. In that example, we can consider test functions F, which are smooth functions with support not including the point 0. Since δ(F) (the distribution δ applied as linear functional to F) is 0 for such functions, we can say that the support of δ is {0} only. Since measures (including probability measures) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way. Look up distribution in Wiktionary, the free dictionary. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In linear algebra, a branch of mathematics, a linear functional is a linear function from a vector space to its field of scalars. ... Measure can mean: To perform a measurement. ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...


Singular support

In Fourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a function. Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...


For example, the Fourier transform of the Heaviside step function can up to constant factors be considered to be 1/x (a function) except at x = 0. While this is clearly a special point, it is more precise to say that the transform qua distribution has singular support {0}: it cannot accurately be expressed as a function in relation to test functions with support including 0. It can be expressed as an application of a Cauchy principal value improper integral. The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of control theory and signal processing to represent a signal... In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite number where b is a point at which the behavior of the function f is such that for any a < b and for any c > b (one sign is + and the other is −). or...


For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails - essentially because the singular supports of the distributions to be multiplied should be disjoint). In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but more precisely also with respect to its Fourier transform at each point. ... Huygens principle (named for Dutch physicist Christiaan Huygens) is a method of analysis applied to problems of wave propagation. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...


Family of supports

An abstract notion of family of supports on a topological space X, suitable for sheaf theory, was defined by Henri Cartan. In extending Poincaré duality to manifolds that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example Alexander-Spanier cohomology. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential mathematician. ... In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ... A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... In mathematics, particularly in algebraic topology Alexander_Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. ...


Bredon, Sheaf Theory (2nd edition, 1997) gives these definitions. A family Φ of closed subsets of X is a family of supports, if it is down-closed and closed under finite union. Its extent is the union over Φ. A paracompactifying family of supports satisfies further than any Y in Φ is, with the subspace topology, a paracompact space; and has some Z in Φ which is a neighbourhood. If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...


  Results from FactBites:
 
Math Forum - Key Questions: Public Support of Mathematics (1616 words)
Mathematics provides us with powerful theoretical and computational techniques to advance our understanding of the modern world and societal problems, and to develop and manage the technology industries that are the backbone of our economy.
When mathematics contributes to the understanding of particular social issues (e.g., statistical modeling to develop more accurate census estimates, formulation and interpretation of economic models (inflation, Social Security scenarios, budget models), epidemiology models, ATM security), we need to encourage the media to include this fact in their reporting.
Ironically, this scenario means that the task of representing the value of mathematics to the general public falls disproportionately to the applications community, regarded by many within the profession as the proverbial fl sheep of the family.
Panel Reports On State Of U.S. Mathematics (453 words)
Federal support for mathematics -- except for that provided by NSF -- is falling rapidly, according to the panel, so NSF bears a special responsibility for the future of mathematics in the United States.
It is urgent to give more support to mathematics and to utilize that support more effectively, he added.
Mathematics is also crucial to managing and "mining" the massive amounts of data now typically collected in many scientific experiments.
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