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If a minimal surface is given by the equation z=f(x,y) and f has continuous first and second partial derivatives for all real x and y, then f is a plane.

If g(theta) is a trigonometric polynomial of degree m satisfying the condition |g(theta)|<=1 where theta is arbitrary and real, then g^'(theta)<=m.

Two curves which, at any point, have a common principal normal vector are called Bertrand curves. The product of the torsions of Bertrand curves is a constant.

A convergence test also called "de Morgan's and Bertrand's test." If the ratio of terms of a series {a_n}_(n=1)^infty can be written in the form ...

J_n(x)=1/piint_0^picos(ntheta-xsintheta)dtheta, where J_n(x) is a Bessel function of the first kind.

Let x^__1 and s_1^2 be the observed mean and variance of a sample of N_1 drawn from a normal universe with unknown mean mu_((1)) and let x^__2 and s_2^2 be the observed mean ...

Another "beta function" defined in terms of an integral is the "exponential" beta function, given by beta_n(z) = int_(-1)^1t^ne^(-zt)dt (1) = ...

A distribution with probability function P(x)=(x^(alpha-1)(1+x)^(-alpha-beta))/(B(alpha,beta)), where B is a beta function. The mode of a variate distributed as ...

The free part of the homology group with a domain of coefficients in the group of integers (if this homology group is finitely generated).

Let the nth composition of a function f(x) be denoted f^((n))(x), such that f^((0))(x)=x and f^((1))(x)=f(x). Denote the composition of f and g by f degreesg(x)=f(g(x)), and ...