Theorem. Let R be a Cohen-Macaulay ring (locally, always) 1 c R an ideal o f height at least 2, S the Rees ring of R with respect to I, and G = S /S I the associated graded ring. Assume that S and G are Cohen-Macaulay rings, and that S has a canonical module cos. Then G has a canonical module r and: (i) I f co s can be embedded into S such that cos (considered as an ideal now) is not contained ...

Given a graph G, a subset M of V (G) is a module of G if for each v ∈ V (G) ∖M , v is adjacent to all the elements of M or adjacent to none of them. For instance, V (G), ∅ and {v} (v ∈ V (G)) are modules of G called trivial. Given a graph G, ωM(G) (respectively αM(G)) denotes the largest integer m such that there is a module M of G which is a clique (respectively a stable) set in G with ∣M ∣ = ...

One of the challenging problems regarding learning decision trees is whether decision trees over the domain Z~, in which each internal node is labeled by a module function axi(mod N) for some i E (1 .... ,n), are efficiently learnable with equivalence and membership queries [5]. Given any decision tree T, let s be the number of its leaf nodes. Let N = p~l...p~ be its prime decomposition, and le...

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module S has non-trivial cyclic vertex if and only if λ has p-weight 1. 2. If p does not divide n and S(n−r,1 ) is indecomposable then its vertex is a p-Sylow subgroup of Sn−r−1 × Sr. Mathematics Subject Classification...

One of the major concerns in the power sector is the day-today increasing power demand but the unavailability of enough resources to meet the power demand using the conventional energy sources. Energy is the prime mover of economic growth and is vital to the sustenance of a modern economy .Future economic growth crucially depends on the long-term availability of energy from sources that are aff...

Let H be a bialgebra. An H-module-algebra is an associative algebra A that is also an H-module such that the multiplication map on A becomes an H-module morphism. This algebraic structure arises often in algebraic topology, quantum groups [11, Chapter V.6], Lie and Hopf algebras theory [3, 16, 19], and group representations [1, Chapter 3]. For example, in algebraic topology, the complex cobordi...

The set of all endomorphisms over -module is a non-empty denoted by . From we can construct the ring addition and composition function. prime ideal an which satisfies properties like numbers. In this paper, take integer number module such that ring. Furthermore, show existences on We also applied property to

OBJECTIVE Individuals visiting a primary care practice were screened to determine the prevalence of depressive disorders. The DSM-IV-TR research criteria for minor depressive disorder were used to standardize a definition for subthreshold symptoms. METHOD Outpatients waiting to see their physicians at 3 community family medicine sites were invited to complete a demographic survey and the Prim...