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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 ...

Any entire analytic function whose range omits two points must be a constant function. Of course, an entire function that omits a single point from its range need not be a ...

The meeting point of lines that connect corresponding points from homothetic figures. In the above figure, O is the homothetic center of the homothetic figures ABCDE and ...

Two planes always intersect in a line as long as they are not parallel. Let the planes be specified in Hessian normal form, then the line of intersection must be ...

The 21 assumptions which underlie the geometry published in Hilbert's classic text Grundlagen der Geometrie. The eight incidence axioms concern collinearity and intersection ...

Voronin (1975) proved the remarkable analytical property of the Riemann zeta function zeta(s) that, roughly speaking, any nonvanishing analytic function can be approximated ...

Let T be an ergodic endomorphism of the probability space X and let f:X->R be a real-valued measurable function. Then for almost every x in X, we have 1/nsum_(j=1)^nf ...

Given A = |a_(11)-x a_(12) ... a_(1m); a_(21) a_(22)-x ... a_(2m); | | ... |; a_(m1) a_(m2) ... a_(mm)-x| (1) = x^m+c_(m-1)x^(m-1)+...+c_0, (2) then ...

Let p and q be partitions of a positive integer, then there exists a (0,1)-matrix A such that c(A)=p, r(A)=q iff q is dominated by p^*.