For a graph G, let     max : is an edge cut of b G D D G  . For graphs G and H, a map     :V G V H   is a graph homomorphism if for each   e uv E G   ,       u v E H    . In 1979, Erdös proved by probabilistic methods that for p ≥ 2 with

A graph homomorphism is a vertex map which carries edges from a source graph to edgesin a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem(MAX H -COL) are edge-weighted graphs G and the objective is to find a subgraph of G that hasmaximal total edge weight, under the condition that the subgraph has a homomorphism to H ;note that for H = Kk th...

In this note we study a certain class of groups w for which the homotopy classification of (rr, m)-complexes is independent of the fc-invariant for small m. 1. Homotopy trees. A (tt, w)-complex is a finite, connected w-dimensional CW-complex such that ttx(X) = m and ttj(X) = 0 for 1 < i < m. The homotopy tree HT(tt, m) is a directed tree whose vertices are homotopv classes of (tt, w)-complexes....

Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing via of endomorphism an abelian group and set all homomorphisms modules form [1] along with Ch. 2 set. 1 [2]. The formalized items are shown below list notations: M ab for Abelian suffix “ ” without is used ring. 1. denoted by End ( ). 2. A pair homomorphism <m:math xmlns:m="ht...

Recall that if R is a commutative ring, then the set R× ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to abelian groups. In general, there is no obvious relation between the additive group of a ring R and the multiplicative group of units R×. However, under certain circumstances one can define a homomorp...

This paper is concerned with ideals in a commutative Noetherian ring R of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of R generated by regular sequences exhibit a desirable type of ‘uniform’ behaviour. The principal technical tool used is a result, proved by R. Hartshorne and R. Speiser in the case where R is local and contains its residue fi...

Abstract We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory corresponding monoid T ( ). Results Levy–Wiegand and Levy–Odenthal together with local case yield an explicit description The is typically neither factorial nor cancellative. Nevertheless, we construct transfer homomorphism to graph agglomerat...

We show that the spherical subalgebra Uk,c of the rational Cherednik algebra associated to Sn o C`, the wreath product of the symmetric group and the cyclic group of order `, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size `. This confirms a version of [EG, Conjecture 11.22] in the case of cyclic groups. The ...

We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a quasi-Frobenius bimodule and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of qua...

The following definition is an example of defining things by mapping properties, that is, by the way the object relates to other objects, rather than by internal structure. The first proposition, which says that there is at most one such thing, is typical, as is its proof. Let R be a commutative ring with 1. Let S be a set. A free R-moduleM on generators S is an R-module M and a set map i : S →...