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Given a set X, let F be a nonempty set of subsets of X. Then F is a ring if, for every pair of sets in F, the intersection, union, and set difference is also in F. F is ...

Given any assignment of n-element sets to the n^2 locations of a square n×n array, is it always possible to find a partial Latin square? The fact that such a partial Latin ...

kappa(d)={(2lneta(d))/(sqrt(d)) for d>0; (2pi)/(w(d)sqrt(|d|)) for d<0, (1) where eta(d) is the fundamental unit and w(d) is the number of substitutions which leave the ...

Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial p(x) is irreducible in the polynomial ring Q[x]. The polynomial ...

Elliptic alpha functions relate the complete elliptic integrals of the first K(k_r) and second kinds E(k_r) at elliptic integral singular values k_r according to alpha(r) = ...

An existential sentence is a statement claiming the existence of an object with given properties. In the language of set theory it can be formulated as follows, exists x in U ...

A extension ring (or ring extension) of a ring R is any ring S of which R is a subring. For example, the field of rational numbers Q and the ring of Gaussian integers Z[i] ...

The extouch triangle DeltaT_1T_2T_3 is the triangle formed by the points of tangency of a triangle DeltaA_1A_2A_3 with its excircles J_1, J_2, and J_3. The points T_1, T_2, ...

A fallacy is an incorrect result arrived at by apparently correct, though actually specious reasoning. The great Greek geometer Euclid wrote an entire book on geometric ...

The Fibonacci Q-matrix is the matrix defined by Q=[F_2 F_1; F_1 F_0]=[1 1; 1 0], (1) where F_n is a Fibonacci number. Then Q^n=[F_(n+1) F_n; F_n F_(n-1)] (2) (Honsberger ...