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For every topological T1-space X, the following conditions are equivalent. 1. X is regular and second countable, 2. X is separable and metrizable. 3. X is homeomorphic to a ...

A function, continuous in a finite closed interval, can be approximated with a preassigned accuracy by polynomials. A function of a real variable which is continuous and has ...

Let X_1,X_2,...,X_N be a set of N independent random variates and each X_i have an arbitrary probability distribution P(x_1,...,x_N) with mean mu_i and a finite variance ...

For every positive integer n, there exists a circle which contains exactly n lattice points in its interior. H. Steinhaus proved that for every positive integer n, there ...

If two perpendicular lines are drawn through the orthocenter H of any triangle, these lines intercept each side (or its extension) in two points (labeled P_(12), P_(12)^', ...

Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes ...

The identity sum_(y=0)^m(m; y)(w+m-y)^(m-y-1)(z+y)^y=w^(-1)(z+w+m)^m (Bhatnagar 1995, p. 51). There are a host of other such binomial identities.

If one root of the equation f(x)=0, which is irreducible over a field K, is also a root of the equation F(x)=0 in K, then all the roots of the irreducible equation f(x)=0 are ...

If a plane cuts the sides AB, BC, CD, and DA of a skew quadrilateral ABCD in points P, Q, R, and S, then (AP)/(PB)·(BQ)/(QC)·(CR)/(RD)·(DS)/(SA)=1 both in magnitude and sign ...

If theta is a given irrational number, then the sequence of numbers {ntheta}, where {x}=x-|_x_|, is dense in the unit interval. Explicitly, given any alpha, 0<=alpha<=1, and ...