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Let K be a number field and let O be an order in K. Then the set of equivalence classes of invertible fractional ideals of O forms a multiplicative Abelian group called the ...

Ramsey's theorem is a generalization of Dilworth's lemma which states for each pair of positive integers k and l there exists an integer R(k,l) (known as the Ramsey number) ...

A fundamental structural result in extremal graph theory due to Szemerédi (1978). The regularity lemma essentially says that every graph can be well-approximated by the union ...

Every sufficiently large odd number is a sum of three primes (Vinogradov 1937). Ramachandra and Sankaranarayanan (1997) have shown that for sufficiently large n, the error ...

Let a random n×n (0,1)-matrix have entries which are 1 (with probability p) or 0 (with probability q=1-p) and numbers are assigned to the edges of a grid. A b-cluster is an ...

A class of knots containing the class of alternating knots. Let c(K) be the link crossing number. Then for knot sum K_1#K_2 which is an adequate knot, ...

The set E of edges of a loopless graph (V,E), being a set of unordered pairs of elements of V, constitutes an adjacency relation on V. Formally, an adjacency relation is any ...

In a graph G, two graph vertices are adjacent if they are joined by a graph edge.

Every finite-dimensional Lie algebra of characteristic p=0 has a faithful finite-dimensional representation.

The algebraics, sometimes denoted A (Derbyshire 2004, p. 173), are the set of algebraic numbers. The set of algebraic numbers is implemented in the Wolfram Language as ...