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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.

The matrix direct sum of n matrices constructs a block diagonal matrix from a set of square matrices, i.e., direct sum _(i=1)^nA_i = diag(A_1,A_2,...,A_n) (1) = [A_1 ; A_2 ; ...

The Poisson sum formula is a special case of the general result sum_(-infty)^inftyf(x+n)=sum_(k=-infty)^inftye^(2piikx)int_(-infty)^inftyf(x^')e^(-2piikx^')dx^' (1) with x=0, ...

A sum which includes both the Jacobi triple product and the q-binomial theorem as special cases. Ramanujan's sum is ...

A sum-free set S is a set for which the intersection of S and the sumset S+S is empty. For example, the sum-free sets of {1,2,3} are emptyset, {1}, {2}, {3}, {1,3}, and ...

The exponential sum function e_n(x), sometimes also denoted exp_n(x), is defined by e_n(x) = sum_(k=0)^(n)(x^k)/(k!) (1) = (e^xGamma(n+1,x))/(Gamma(n+1)), (2) where ...

Given a real number x, find the powers of a base b that will shift the digits of x a number of places n to the left. This is equivalent to solving b^x=b^nx (1) or x=n+log_bx. ...

The axiom of Zermelo-Fraenkel set theory which asserts the existence for any set a of the sum (union) x of all sets that are elements of a. The axiom may be stated ...

A zero-sum game is a game in which players make payments only to each other. In such a game, one player's loss is the other player's gain, so the total amount of "money" ...

The group direct sum of a sequence {G_n}_(n=0)^infty of groups G_n is the set of all sequences {g_n}_(n=0)^infty, where each g_n is an element of G_n, and g_n is equal to the ...