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Given a pick-7 lottery with 23 numbers that pays a prize to anyone matching at least 4 of the 7 numbers, there is a set of 253 tickets that guarantees a win. This set ...

The word quantile has no fewer than two distinct meanings in probability. Specific elements x in the range of a variate X are called quantiles, and denoted x (Evans et al. ...

Given a homogeneous linear second-order ordinary differential equation, y^('')+P(x)y^'+Q(x)y=0, (1) call the two linearly independent solutions y_1(x) and y_2(x). Then ...

An affine variety V is an algebraic variety contained in affine space. For example, {(x,y,z):x^2+y^2-z^2=0} (1) is the cone, and {(x,y,z):x^2+y^2-z^2=0,ax+by+cz=0} (2) is a ...

The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers, exists x(emptyset in x ^ forall y in x(y^' in x)), where ...

A Belphegor prime (also known as a Beelphegor prime) is a prime Belphegor number, i.e., a palindromic prime of the form 1(0...)666(0...)1. The first few Belphegor primes are ...

Polynomials b_n(x) which form a Sheffer sequence with g(t) = t/(e^t-1) (1) f(t) = e^t-1, (2) giving generating function sum_(k=0)^infty(b_k(x))/(k!)t^k=(t(t+1)^x)/(ln(1+t)). ...

In order to find a root of a polynomial equation a_0x^n+a_1x^(n-1)+...+a_n=0, (1) consider the difference equation a_0y(t+n)+a_1y(t+n-1)+...+a_ny(t)=0, (2) which is known to ...

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) = ...

The covariant derivative of the Riemann tensor is given by (1) Permuting nu, kappa, and eta (Weinberg 1972, pp. 146-147) gives the Bianchi identities ...