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A sphere is rigid.

Let rho(x) be an mth degree polynomial which is nonnegative in [-1,1]. Then rho(x) can be represented in the form {[A(x)]^2+(1-x^2)[B(x)]^2 for m even; ...

Let f(x) be a finite and measurable function in (-infty,infty), and let epsilon be freely chosen. Then there is a function g(x) such that 1. g(x) is continuous in ...

A necessary and sufficient condition for all the eigenvalues of a real n×n matrix A to have negative real parts is that the equation A^(T)V+VA=-I has as a solution where V is ...

Simple majority vote is the only procedure which is anonymous, dual, and monotonic.

Somewhere on the Earth, there is a pair of antipodal points having simultaneously the same temperature and pressure.

If the period of a repeating decimal for a/p, where p is prime and a/p is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves ...

If each of two nonparallel transversals with nonminimal directions meets a given curve in finite points only, then the ratio of products of the distances from the two sets of ...

In any tree, the number of dissimilar points minus the number of dissimilar lines plus the number of symmetry lines equals 1.

The eigenvalues lambda satisfying P(lambda)=0, where P(lambda) is the characteristic polynomial, lie in the unions of the disks |z|<=1 |z+b_1|<=sum_(j=1)^n|b_j|.