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As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

If the random variates X_1, X_2, ... satisfy the Lindeberg condition, then for all a<b, lim_(n->infty)P(a<(S_n)/(s_n)<b)=Phi(b)-Phi(a), where Phi is the normal distribution ...

A branch of mathematics that is a sort of generalization of calculus. Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a ...

Let M be a compact n-dimensional manifold with injectivity radius inj(M). Then Vol(M)>=(c_ninj(M))/pi, with equality iff M is isometric to the standard round sphere S^n with ...

A generalization of Schröter's formula.

Taking the locus of midpoints from a fixed point to a circle of radius r results in a circle of radius r/2. This follows trivially from r(theta) = [-x; 0]+1/2([rcostheta; ...

Any collineation from P(V) to P(V), where V is a three-dimensional vector space, is associated with a semilinear map from V to V.

If a sequence of double points is passed as a closed curve is traversed, each double point appears once in an even place and once in an odd place.

A primitive subgroup of the symmetric group S_n is equal to either the alternating group A_n or S_n whenever it contains at least one permutation which is a q-cycle for some ...

Suppose that {f_n} is a sequence of measurable functions, that f_n->f pointwise almost everywhere as n->infty, and that |f_n|<=g for all n, where g is integrable. Then f is ...