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In machine learning theory, the Vapnik-Chervonenkis dimension or VC-dimension of a concept class C is the cardinality of the largest set S which can be shattered by C. If ...

Let X=(X,tau) be a topological vector space whose continuous dual X^* separates points (i.e., is T2). The weak topology tau_w on X is defined to be the coarsest/weakest ...

There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems with commutative multiplication and addition are ...

An apex graph is a graph possessing at least one vertex whose removal results in a planar graph. The set of vertices whose removal results in a planar graph is known as the ...

Consider a Boolean algebra of subsets b(A) generated by a set A, which is the set of subsets of A that can be obtained by means of a finite number of the set operations ...

A function built up of a finite combination of constant functions, field operations (addition, multiplication, division, and root extractions--the elementary operations)--and ...

Faà di Bruno's formula gives an explicit equation for the nth derivative of the composition f(g(t)). If f(t) and g(t) are functions for which all necessary derivatives are ...

The falling factorial (x)_n, sometimes also denoted x^(n__) (Graham et al. 1994, p. 48), is defined by (x)_n=x(x-1)...(x-(n-1)) (1) for n>=0. Is also known as the binomial ...

In the fields of functional and harmonic analysis, the Littlewood-Paley decomposition is a particular way of decomposing the phase plane which takes a single function and ...

The terms "measure," "measurable," etc. have very precise technical definitions (usually involving sigma-algebras) that can make them appear difficult to understand. However, ...