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If a_1>=a_2>=...>=a_n (1) b_1>=b_2>=...>=b_n, (2) then nsum_(k=1)^na_kb_k>=(sum_(k=1)^na_k)(sum_(k=1)^nb_k). (3) This is true for any distribution.

A complete metric space is a metric space in which every Cauchy sequence is convergent. Examples include the real numbers with the usual metric, the complex numbers, ...

A requirement necessary for a given statement or theorem to hold. Also called a criterion.

Any pair of equations giving the real part of a function as an integral of its imaginary part and the imaginary part as an integral of its real part. Dispersion relationships ...

F_x[1/pi(1/2Gamma)/((x-x_0)^2+(1/2Gamma)^2)](k)=e^(-2piikx_0-Gammapi|k|). This transform arises in the computation of the characteristic function of the Cauchy distribution.

A pseudoanalytic function is a function defined using generalized Cauchy-Riemann equations. Pseudoanalytic functions come as close as possible to having complex derivatives ...

The method for solving the Goursat problem and Cauchy problem for linear hyperbolic partial differential equations using a Riemann function.

Any motion of a rigid body in space at every instant is a screw motion. This theorem was proved by Mozzi and Cauchy.

The topological completion C of a field F with respect to the absolute value |·| is the smallest field containing F for which all Cauchy sequences or rationals converge.

Let psi_1(x) and psi_2(x) be any two real integrable functions in [a,b], then Schwarz's inequality is given by |<psi_1|psi_2>|^2<=<psi_1|psi_1><psi_2|psi_2>. (1) Written out ...