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Abstract Let R be a noncommutative ring with unity. The commuting graph of R , denoted by Γ (R) , is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if x y = y x. Let F be a finite field and n ≥ 2. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if Γ (R) ≅ Γ (M n (F)) , then R ≅ M n (F). In this paper, we prove the conjecture whenever n is of the form 2 k 3 l with k ≠ 0. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the functional code introduced by G. Lachaud (1996) in the special case where X is a non-singular Hermitian variety in and . In , F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) on the minimum distance of this code for a Hermitian variety X in . In this paper, we will answer the question about the minimum distance in general dimension N, with . We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes. [Copyright &y& Elsevier]

Abstract: In this paper we treat cyclotomic binary duadic codes. The conjecture of Ding and Pless is that there are infinitely many cyclotomic duadic codes of prime lengths that are not quadratic residue codes. We shall prove this conjecture by using the special case of Tschebotareff''s density theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. [Copyright &y& Elsevier]

Abstract: In 1988 Manickam and Singhi conjectured that for every positive integer and every , every set of real numbers whose sum is non-negative contains at least subsets of size whose sums are non-negative. In this paper we make use of Hall’s matching theorem in order to study some numbers related to this conjecture. [Copyright &y& Elsevier]

Abstract: The main result of this paper is the following: Any minimal counterexample to Hadwiger''s Conjecture for the k-chromatic case is -connected. This improves the previous known bound due to Mader [W. Mader, Über trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243–252], which says that any minimal counterexample to Hadwiger''s Conjecture for the k-chromatic case is 7-connected for . This is the first result on the vertex connectivity of minimal counterexamples to Hadwiger''s Conjecture for general k. Consider the following problem: There exists a constant c such that any ck-chromatic graph has a -minor. This problem is still open, but together with the recent result in [T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces large complete bipartite graph minors, preprint], our main result implies that there are only finitely many minimal counterexamples to the above problem when . This would be the first step to attach the above problem. We also prove that the vertex connectivity of minimum counterexamples to Hadwiger''s Conjecture is at least -connected. This is also the first result on the vertex connectivity of minimum counterexamples to Hadwiger''s Conjecture for general k. [Copyright &y& Elsevier]

Gut and Spa˘taru (J. Math. Anal. Appl. 248 (2000) 233–246) proved a precise asymptotic theorem for random variables in the normal domain of attraction of some stable distribution with exponent α, 1<α&les;2. They also conjectured that the theorem should hold for the general case of the domain of attraction of a stable distribution. In this paper, we shall provide an example to deny their conjecture and study some further properties. [Copyright &y& Elsevier]

Let p be a prime, u be a linear recurring sequence of integers of order d and let S=3d2+9d/2+1. The main result of the present paper: if u is uniformly distributed mod pS, then it is uniformly distributed mod ps for all s&ges;1. This solves a longstanding folklore conjecture. [Copyright &y& Elsevier]

Let A2(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c∈(0,1), such that whenever &z.sfnc;f(z)&z.sfnc;⩽&z.sfnc;g(z)&z.sfnc; (f,g∈A2(D)) in the annulus c<&z.sfnc;z&z.sfnc;<1, then &z.dfnc;f&z.dfnc;⩽&z.dfnc;g&z.dfnc;. In 1999 Hayman proved Korenblum''s conjecture. But the sharp value of c (we use γ to denote this sharp value) is still unknown. In this paper we give an upper bound on γ, that is, γ<0.67795, which improves an earlier result of the author. [Copyright &y& Elsevier]