Let k = (k1, k2, · · · , kt) be a t-tuple of integers, and m be a positive integer. For a subset A ⊂ Zm and any n ∈ Zm, let rk A(n) denote the number of solutions of the equation k1a1 + · · ·+ ktat = n with a1, · · · , at ∈ A. In this paper, we give a necessary and sufficient condition on (k,m) such that there exists a subset A ⊂ Zm satisifying rk A = r k Zm\A. This settles a problem of Yang an...

‎Let $S$ be an inverse semigroup with the set of idempotents $E$‎. We prove that the semigroup algebra $ell^{1}(S)$ is always‎ ‎$2n$-weakly module amenable as an $ell^{1}(E)$-module‎, ‎for any‎ ‎$nin mathbb{N}$‎, ‎where $E$ acts on $S$ trivially from the left‎ ‎and by multiplication from the right‎. ‎Our proof is based on a common fixed point property for semigroups‎.  

Background and objectives: Cisplatin (CP), a highly effective antineoplastic drug, causes testicular damage. Zataria multiflora Boiss. (ZM), a medicinal plant, has antioxidant and anti-inflammatory properties. The aim of this study was to investigate the effects of ZM against cisplatin-induced testicular toxicity. Methods: Thirty-two adult male mice wer...

The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories XAss ' Cat-Ass we consider this problem in Cat-Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of catego...

The mitochondrial transcription termination factor (mTERF) proteins are nucleic acid binding proteins characterized by degenerate helical repeats of ∼30 amino acids. Metazoan genomes encode a small family of mTERF proteins whose members influence mitochondrial gene expression and DNA replication. The mTERF family in higher plants consists of roughly 30 members, which localize to mitochondria or...

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, the...

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 3 and standard module V . Recently Ito and Terwilliger introduced four direct sum decompositions of V ; we call these the (μ, ν)–split decompositions of V , where μ, ν ∈ {↓, ↑}. In this paper we show that the (↓, ↓)–split decomposition and the (↑, ↑)–split decomposition are dual with respect to the standard Hermitian form on V...