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If the average number of envelopes per pigeonhole is a, then some pigeonhole will have at least a envelopes. Similarly, there must be a pigeonhole with at most a envelopes.

The canonical generator of the nonvanishing homology group on a topological manifold.

If M is continuous and int_a^bM(x)h(x)dx=0 for all infinitely differentiable h(x), then M(x)=0 on the open interval (a,b).

The series sum_(k=1)^inftykr^k=sum_(k=1)^inftysum_(i=k)^inftyr^i=r/((1-r)^2), valid for 0<r<1.

A mathematical object invented to solve irreducible congruences of the form F(x)=0 (mod p), where p is prime.

For n players, n-1 games are needed to fairly determine first place, and n-1+lg(n-1) are needed to fairly determine first and second place.

gamma_r=(kappa_r)/(sigma^(r+2)), where kappa_r are cumulants and sigma is the standard deviation.

year winner 2010 Yves Meyer

If a distribution has a single mode at mu_0, then P(|x-mu_0|>=lambdatau)<=4/(9lambda^2), where tau^2=sigma^2+(mu-mu_0)^2.

If (1+xsin^2alpha)sinbeta=(1+x)sinalpha, then (1+x)int_0^alpha(dphi)/(sqrt(1-x^2sin^2phi))=int_0^beta(dphi)/(sqrt(1-(4x)/((1+x)^2)sin^2phi)).