Narrow Results

A nonempty circular string C(x) of length n is said to be covered by a set U_k of strings each of fixed length k≤n iff every position in C(x) lies within an occurrence of some string u∈U_k. In this paper we consider the problem of determining the minimum cardinality of a set U_k which guarantees that every circular string C(x) of length n≥k can be covered. In particular, we show how, for any positive integer m, to choose the elements of U_k so that, for sufficiently large k, u_k≈σ^k–m, where u_k=|U_k| and σ is the size of the alphabet on which the strings are defined. The problem has application to DNA sequencing by hybridization using oligonucleotide probes.

We present an algorithm that for a given simple non-convex polygon P finds an approximate inner-cover by large convex polygons. The algorithm is based on an initial partitioning of P into a set C of disjoint convex polygons which are an exact tessellation of P. The algorithm then builds a set of large convex polygons contained in P by constructing the convex hulls of subsets of C. We discuss different strategies for selecting the subsets and we claim that in most cases our algorithm produces an effective approximation of P.

A complex C of R-modules is called #-injective if all terms Cⁱ are injective R-modules for i ∈ ℤ. In this paper, we first give some characterizations and properties of #-injective complexes. Some relations between #-injective (pre)envelopes of a complex X and injective (pre)envelopes of the R-modules Xⁱ are also given. Finally, we study the -dimensions of complexes, where is the class of #-injective complexes.

A complex C is called Gorenstein cotorsion if Ext¹(G, C) = 0 for any Gorenstein flat complex G. It is shown that a complex C of left R-modules is Gorenstein cotorsion if and only if Cⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ℤ and is exact for any Gorenstein flat complex G; and is a hereditary cotorsion theory over a right coherent ring R, where and denote the classes of all Gorenstein flat and Gorenstein cotorsion complexes respectively. Also Gorenstein cotorsion envelopes and Gorenstein flat covers of complexes are considered.

Let be the variety of polynilpotent Lie algebras of class row (c₁, …, c_n). The paper is devoted to present the concepts of polynilpotent multiplier and cover of a Lie algebra L with respect to the variety , and compute for some known Lie algebras. Also, we determine the structure of all covers of Lie algebras whose polynilpotent multipliers are finite-dimensional, and investigate some common properties between covers.

Given an $R$-module $C$ and a class of $R$-modules ${𝒟}$ over a commutative ring $R$, we investigate the relationship between the existence of ${𝒟}$-envelopes (respectively, ${𝒟}$-covers) and the existence of $\mathrm{\text{Hom}}(C,{𝒟})$-envelopes or $C\otimes {𝒟}$-envelopes (respectively, $\mathrm{\text{Hom}}(C,{𝒟})$-covers or $C\otimes {𝒟}$-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by $C$-projective, $C$-flat, $C$-injective and $C$-$FP$-injective modules, where $C$ is a semidualizing $R$-module.

In this paper, by taking the class of all $C3$ (or $D3$) right $R$-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via $D3$-covers (or $D3$-envelopes) and a right $V$-ring (or a right noetherian $V$-ring) via $C3$-covers (or $C3$-envelopes). By using isosimple-projective preenvelope, we obtained that if $R$ is a semiperfect right FGF ring (or left coherent ring), then every isosimple right $R$-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

Partially ordered monoids (or pomonoids) $S$ acting on a partially ordered set (or poset), briefly $S$-posets, appear naturally in the study of mappings between posets, and play an essential role in pomonoid theory. The study of flatness properties of $S$-posets was initiated by Fakhruddin in the 1980s, and an extensive theory of flatness properties has been developed in the past several decades. The obtained results have prompted a new progress in the research area of $S$-posets. To date, a large number of familiar properties have been generalized from acts to $S$-posets (involving free and projective $S$-posets, flat $S$-posets of various sorts, $S$-posets satisfying Conditions $(P)$, $(WP)$ and $(PWP)$, and torsion free $S$-posets). Some new properties in $S$-posets, such as Conditions $(P_w)$, ${(WP)}_w$ and ${(PWP)}_w$, have also been discovered. This paper continues the study of flatness properties of $S$-posets. We first introduce Condition $(P^{\prime })$ in the context of $S$-posets, and characterize pomonoids by this new property of $S$-posets. Unlike the case for acts, pomonoids over which all right $S$-posets satisfy Condition $(P^{\prime })$ are stronger than pogroups. Thereby, we introduce Condition $(P_w^{\prime })$ similar to Condition $(P_w)$. Furthermore, we describe Conditions $(P^{\prime })$ and $(P_w^{\prime })$ covers of cyclic $S$-posets. Finally, we investigate direct products of $S$-posets satisfying Conditions $(P^{\prime })$ and $(P_w^{\prime })$.

In this paper we study the existence of ℱℐ_n-envelopes, -envelopes and ℱℐ_n-covers, where ℱℐ_n denotes the class of all n-absolute pure modules for an integer n > 0 or n=∞. We prove that -envelopes and ℱℐ_n-covers exist over an n-coherent ring R, and ℱℐ₁-covers and special ℱℐ_n-preenvelopes exist over any ring R.

In this paper, we prove that the Schur multiplier of the direct sum of two arbitrary Lie algebras is isomorphic to the direct sum of the Schur multipliers of the direct factors and the usual tensor product of the Lie algebras, which is similar to the work of Miller (1952) in the group case. Also, a cover for the direct sum of two Lie algebras in terms of given covers of them will be constructed.

Let (L, N) be a pair of finite-dimensional nilpotent Lie algebras, in which N is an ideal in L. In this paper we derive some inequalities for the dimension of the Schur multiplier of the pair (L, N) in terms of the dimension of the commutator subalgebra [L, N].