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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 ...

If a matrix group is reducible, then it is completely reducible, i.e., if the matrix group is equivalent to the matrix group in which every matrix has the reduced form ...

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 a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...

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 ...