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For any real alpha and beta such that beta>alpha, let p(alpha)!=0 and p(beta)!=0 be real polynomials of degree n, and v(x) denote the number of sign changes in the sequence ...

No subspace of R^n can be homeomorphic to S^n.

Let Gamma be a representation for a group of group order h, then sum_(R)Gamma_i(R)_(mn)Gamma_j(R)_(m^'n^')^*=h/(sqrt(l_il_j))delta_(ij)delta_(mm^')delta_(nn^'). The proof is ...

If a sphere is covered by three closed sets, then one of them must contain a pair of antipodal points.

Two real algebraic manifolds are equivalent iff they are analytically homeomorphic (Nash 1952).

Any irreducible curve may be carried by a factorable Cremona transformation into one with none but ordinary singular points.

If the difference between the order and the dimension of a series is less than the curve genus, then the series is special.

The Lyapunov condition, sometimes known as Lyapunov's central limit theorem, states that if the (2+epsilon)th moment (with epsilon>0) exists for a statistical distribution of ...

Also known as the difference of squares method. It was first used by Fermat and improved by Gauss. Gauss looked for integers x and y satisfying y^2=x^2-N (mod E) for various ...

The lengths of the tangents from a point P to a conic C are proportional to the cube roots of the radii of curvature of C at the corresponding points of contact.