The k-Zero-Divisor Hypergraph of a Commutative Ring

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and the k-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. Let R be a commutative ring and k an integer strictly larger than 2. A k-uniform hypergraph Hk(R) with the vertex set Z(R,k), the set of all k-zero-divisors in R, is associated to R, where each k-subset of Z(R,k) that satisfies the k-zero-divisor condition is an edge in Hk(R). It is shown that if R has two prime ideals P1 and P2 with zero their only common point, then Hk(R) is a bipartite (2-colorable) hypergraph with partition sets P1−Z′ and P2−Z′, where Z′ is the set of all zero divisors of R which are not k-zero-divisors in R . If R has a nonzero nilpotent element, then a lower bound for the clique number of H3(R) is found. Also, we have shown that H3(R) is connected with diameter at most 4 whenever x2≠0 for all 3-zero-divisors x of R. Finally, it is shown that for any finite nonlocal ring R, the hypergraph H3(R) is complete if and only if R is isomorphic to Z2×Z2×Z2

A mixed finite element method for solving coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term

This paper is concerned with the numerical approximation of the solution of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term using a mixed finite element method. The Raviart-Thomas mixed finite element method is one of the most prominent techniques to discretize the second-order wave equations; therefore, we apply this scheme for space discretization. Furthermore, an L2-in-space error estimate is presented for this mixed finite element approximation. Finally, the efficiency of the method is verified by a numerical example. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd

Protein complex prediction based on k-connected subgraphs in protein interaction network

<p>Abstract</p> <p>Background</p> <p>Protein complexes play an important role in cellular mechanisms. Recently, several methods have been presented to predict protein complexes in a protein interaction network. In these methods, a protein complex is predicted as a dense subgraph of protein interactions. However, interactions data are incomplete and a protein complex does not have to be a complete or dense subgraph.</p> <p>Results</p> <p>We propose a more appropriate protein complex prediction method, CFA, that is based on connectivity number on subgraphs. We evaluate CFA using several protein interaction networks on reference protein complexes in two benchmark data sets (MIPS and Aloy), containing 1142 and 61 known complexes respectively. We compare CFA to some existing protein complex prediction methods (CMC, MCL, PCP and RNSC) in terms of recall and precision. We show that CFA predicts more complexes correctly at a competitive level of precision.</p> <p>Conclusions</p> <p>Many real complexes with different connectivity level in protein interaction network can be predicted based on connectivity number. Our CFA program and results are freely available from <url>http://www.bioinf.cs.ipm.ir/softwares/cfa/CFA.rar</url>.</p