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Probability and Statistics

The following integral transform relationship, known as the Abel transform, exists between two functions f(x) and g(t) for 0<alpha<1, f(x) = int_0^x(g(t)dt)/((x-t)^alpha) (1) ...

Given a subset A of a larger set, the characteristic function chi_A, sometimes also called the indicator function, is the function defined to be identically one on A, and is ...

A second-order partial differential equation, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called elliptic if the matrix Z=[A B; B C] (2) is positive ...

A general integral transform is defined by g(alpha)=int_a^bf(t)K(alpha,t)dt, where K(alpha,t) is called the integral kernel of the transform.

Roughly speaking, isospectral manifolds are drums that sound the same, i.e., have the same eigenfrequency spectrum. Two drums with differing area, perimeter, or genus can ...

A Maclaurin series is a Taylor series expansion of a function about 0, (1) Maclaurin series are named after the Scottish mathematician Colin Maclaurin. The Maclaurin series ...

The Mellin transform is the integral transform defined by phi(z) = int_0^inftyt^(z-1)f(t)dt (1) f(t) = 1/(2pii)int_(c-iinfty)^(c+iinfty)t^(-z)phi(z)dz. (2) It is implemented ...

Define q=e^(2piitau) (cf. the usual nome), where tau is in the upper half-plane. Then the modular discriminant is defined by ...

A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical coordinates about the x-axis, ...