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Let S_n be the sum of n random variates X_i with a Bernoulli distribution with P(X_i=1)=p_i. Then sum_(k=0)^infty|P(S_n=k)-(e^(-lambda)lambda^k)/(k!)|<2sum_(i=1)^np_i^2, ...

Let chi be a nonprincipal number theoretic character over Z/Zn. Then for any integer h, |sum_(x=1)^hchi(x)|<=2sqrt(n)lnn.

Given T an unbiased estimator of theta so that <T>=theta. Then var(T)>=1/(Nint_(-infty)^infty[(partial(lnf))/(partialtheta)]^2fdx), where var is the variance.

Let f(z) be an analytic function in |z-a|<R. Then f(z)=1/(2pi)int_0^(2pi)f(z+re^(itheta))dtheta for 0<r<R.

A non-Euclidean space with constant negative Gaussian curvature.

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.

The Machin-like formula 1/4pi=12cot^(-1)18+8cot^(-1)57-5cot^(-1)239.

If a function phi is harmonic in a sphere, then the value of phi at the center of the sphere is the arithmetic mean of its value on the surface.

For positive numbers a and b with a!=b, (a+b)/2>(b-a)/(lnb-lna)>sqrt(ab).

Relates invariants of a curve defined over the integers. If this inequality were proven true, then Fermat's last theorem would follow for sufficiently large exponents. ...