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In homogeneous coordinates, the first positive quadrant joins (0,1) with (1,0) by "points" (f_1,f_2), and is mapped onto the hyperbolic line -infty<u<+infty by the ...

A divisor d of a positive integer n is biunitary if the greatest common unitary divisor of d and n/d is 1. For a prime power p^y, the biunitary divisors are the powers 1, p, ...

Any bounded planar region with positive area >A placed in any position of the unit square lattice can be translated so that the number of lattice points inside the region ...

The blow-up lemma essentially says that regular pairs in Szemerédi's regularity lemma behave like complete bipartite graphs from the point of view of embedding bounded degree ...

The finite zeros of the derivative r^'(z) of a nonconstant rational function r(z) that are not multiple zeros of r(z) are the positions of equilibrium in the field of force ...

If a continuous function defined on an interval is sometimes positive and sometimes negative, it must be 0 at some point. Bolzano (1817) proved the theorem (which effectively ...

An ordinary differential equation of the form x^my^'=f(x,y), where m is a positive integer, f is analytic at x=y=0, f(0,0)=0, and f_y^'(0,0)!=0. Zwillinger (1997, p. 120), ...

A hypothetic building design problem in optimization with constraints proposed by Bhatti (2000, pp. 3-5). To save energy costs for heating and cooling, an architect wishes to ...

Calabi's triangle is the unique triangle other that the equilateral triangle for which the largest inscribed square can be inscribed in three different ways (Calabi 1997). ...

A set of positive integers S is called sum-free if the equation x+y=z has no solutions x, y, z in S. The probability that a random sum-free set S consists entirely of odd ...