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Let omega(n) be the number of distinct prime factors of n. If Psi(x) tends steadily to infinity with x, then lnlnx-Psi(x)sqrt(lnlnx)<omega(n)<lnlnx+Psi(x)sqrt(lnlnx) for ...

Let X be a continuum (i.e., a compact connected metric space). Then X is hereditarily unicoherent provided that every subcontinuum of X is unicoherent. Any hereditarily ...

On the class of topological spaces, a homeomorphism class is an equivalence class under the relation of being homeomorphic. For example, the open interval (-pi/2,pi/2) and ...

If f:D->Y is a map (a.k.a. function, transformation, etc.) over a domain D, then the image of f, also called the range of D under f, is defined as the set of all values that ...

The finite Fourier cosine transform of an apodization function, also known as an apparatus function. The instrument function I(k) corresponding to a given apodization ...

Let alpha(x) be a monotone increasing function and define an interval I=(x_1,x_2). Then define the nonnegative function U(I)=alpha(x_2)-alpha(x_1). The Lebesgue integral with ...

A function f(x) is logarithmically concave on the interval [a,b] if f>0 and lnf(x) is concave on [a,b]. The definition can also be extended to R^k->(0,infty) functions ...

For a given function f(x) over a partition of a given interval, the lower sum is the sum of box areas m^*Deltax_k using the infimum m of the function f(x) in each subinterval ...

The winding number W(theta) of a map f(theta) with initial value theta is defined by W(theta)=lim_(n->infty)(f^n(theta)-theta)/n, which represents the average increase in the ...

A measure space is a measurable space possessing a nonnegative measure. Examples of measure spaces include n-dimensional Euclidean space with Lebesgue measure and the unit ...