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Two topological spaces X and Y are homotopy equivalent if there exist continuous maps f:X->Y and g:Y->X, such that the composition f degreesg is homotopic to the identity ...

A function f(x) increases on an interval I if f(b)>=f(a) for all b>a, where a,b in I. If f(b)>f(a) for all b>a, the function is said to be strictly increasing. Conversely, a ...

A real-valued univariate function f=f(x) is said to have an infinite discontinuity at a point x_0 in its domain provided that either (or both) of the lower or upper limits of ...

Let sumu_k be a series with positive terms and let f(x) be the function that results when k is replaced by x in the formula for u_k. If f is decreasing and continuous for ...

An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable. For example, a linear first-order ordinary ...

If p_1, ..., p_n are positive numbers which sum to 1 and f is a real continuous function that is convex, then f(sum_(i=1)^np_ix_i)<=sum_(i=1)^np_if(x_i). (1) If f is concave, ...

If mu is a real measure (i.e., a measure that takes on real values), then one can decompose it according to where it is positive and negative. The positive variation is ...

A real-valued univariate function f=f(x) has a jump discontinuity at a point x_0 in its domain provided that lim_(x->x_0-)f(x)=L_1<infty (1) and lim_(x->x_0+)f(x)=L_2<infty ...

An algorithm for partitioning (or clustering) N data points into K disjoint subsets S_j containing N_j data points so as to minimize the sum-of-squares criterion ...

Kakutani's fixed point theorem is a result in functional analysis which establishes the existence of a common fixed point among a collection of maps defined on certain ...