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The logarithmic distribution is a continuous distribution for a variate X in [a,b] with probability function P(x)=(lnx)/(b(lnb-1)-a(lna-1)) (1) and distribution function ...

The MacBeath circle, a term coined here for the first time, is the circumcircle of the MacBeath triangle. It has a fairly complicated radius, center function, and circle ...

Polynomials s_n(x) which form the Sheffer sequence for f^(-1)(t)=1+t-e^t, (1) where f^(-1)(t) is the inverse function of f(t), and have generating function ...

A generalization of Student's t-distribution known as the noncentral Student's t-distribution is given by (1) where Gamma(z) is the gamma function and _1F_1(a;b;z) is a ...

The number of "arrangements" in an ordering of n items is given by either a combination (order is ignored) or a permutation (order is significant). An ordering (or order) is ...

A result in control theory. Define H(psi,x,u)=(psi,f(x,u))=sum_(a=0)^npsi_af^a(x,u). Then in order for a control u(t) and a trajectory x(t) to be optimal, it is necessary ...

The power polynomials x^n are an associated Sheffer sequence with f(t)=t, (1) giving generating function sum_(k=0)^inftyx^kt^k=1/(1-tx) (2) and exponential generating ...

The ratio X_1/X_2 of uniform variates X_1 and X_2 on the interval [0,1] can be found directly as P_(X_1/X_2)(u) = int_0^1int_0^1delta((x_1)/(x_2)-u)dx_1dx_2 (1) = ...

Given a succession of nonsingular points which are on a nonhyperelliptic curve of curve genus p, but are not a group of the canonical series, the number of groups of the ...

The constant polynomial P(x)=0 whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. ...