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If, in the above commutative diagram of modules and module homomorphisms the columns and two upper rows are exact, then so is the bottom row.

Let f be analytic on the unit disk, and assume that 1. |f(z)|<=1 for all z and 2. f(0)=0. Then |f(z)|<=|z| and |f^'(0)|<=1. If either |f(z)|=|z| for some z!=0 or if ...

If, in a plane or spherical convex polygon ABCDEFG, all of whose sides AB, BC, CD, ..., FG (with the exception of AG) have fixed lengths, one simultaneously increases ...

Let f be a family of meromorphic functions on the unit disk D which are not normal at 0. Then there exist sequences f_n in F, z_n, rho_n, and a nonconstant function f ...

When ac is divisible by a number b that is relatively prime to a, then c must be divisible by b.

int_0^pi(sin[(n+1/2)x])/(2sin(1/2x))dx=1/2pi, where the integral kernel is the Dirichlet kernel.

If G is a perfect group, then the group center of the quotient group G/Z(G), where Z(G) is the group center of G, is the trivial group.

Let R be a number ring of degree n with 2s imaginary embeddings. Then every ideal class of R contains an ideal J such that ||J||<=(n!)/(n^n)(4/pi)^ssqrt(|disc(R)|), where ...

The image on the Riemann sphere of any circle under a complex rational mapping with numerator and denominator having degrees no more than n has length no longer than 2npi.

If, for n>=0, beta_n=sum_(r=0)^n(alpha_r)/((q;q)_(n-r)(aq;q)_(n+r)), (1) then beta_n^'=sum_(r=0)^n(alpha_r^')/((q;q)_(n-r)(aq;q)_(n+r)), (2) where alpha_r^' = ...