Finding a maximum <Emphasis Type="Italic">k</Emphasis>-club using the <Emphasis Type="Italic">k</Emphasis>-clique formulation and canonical hypercube cuts

Detecting low-diameter clusters is an important graph-based data mining technique used in social network analysis, bioinformatics and text-mining. Low pairwise distances within a cluster can facilitate fast communication or good reachability between vertices in the cluster. Formally, a subset of vertices that induce a subgraph of diameter at most k is called a k-club. For low values of the parameter k, this model offers a graph-theoretic relaxation of the clique model that formalizes the notion of a low-diameter cluster. Using a combination of graph decomposition and model decomposition techniques, we demonstrate how the fundamental optimization problem of finding a maximum size k-club can be solved optimally on large-scale benchmark instances that are available in the public domain. Our approach circumvents the use of complicated formulations of the maximum k-club problem in favor of a simple relaxation based on necessary conditions, combined with canonical hypercube cuts introduced by Balas and Jeroslow.     

Parallel tensors and Ricci solitons in <Emphasis Type=Italic>N</Emphasis> (<Emphasis Type=Italic>k</Emphasis>)-quasi Einstein manifolds

The Eisenhart problem of finding parallel and symmetric tensors is considered in the framework ofN (k)-quasi Einstein manifolds and the result is connected with Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is: i) shrinking on a class of conformally flat perfect fluid space-times and on quasi-umbilical hypersurfaces, in particular unit spheres; ii) expanding if the generator is of torse-forming type.     

The <Emphasis Type=Italic>k</Emphasis>-Fibonacci matrix and the Pascal matrix

We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.     

Total <Emphasis Type=Italic>k</Emphasis>-domination in Cartesian product graphs

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more detail, namely a problem that origins from a conjecture of Rényi and Erd?s (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gave at the Joint Austrian-Hungarian Mathematical Conference 2015, August 25–27, 2015 in Gy?r (Hungary).     

<Emphasis Type=Italic>k</Emphasis>-Phase Oscillator Synchronization for Graph Coloring

This paper investigates ways of applying oscillator synchronization to graph coloring. A previous method based on the generalization of the Aihara model is sensitive to the varying degree of the vertices in the graph and there is a strong tendency for the network to form suboptimal limit cycles on regular graphs. Other models such as those by Wu and Nakaguchi, Jin’no and Tanaka do not generalize well into greater than 2-coloring. In this paper, we present ways to overcome these problems and describe the results of our experiments on graphs requiring more than two colors. Our k-phase model enhances the coloring performance over the previous similar models. We further attempt to formalize and analyze the categorical behavior of these systems and discuss connections to other optimization methods.     

About the computation of the signature of surface singularities <Emphasis Type=Italic>z</Emphasis><Superscript><Emphasis Type=Italic>N</Emphasis></Superscript> + <Emphasis Type=Italic>g</Emphasis>(<Emphasis Type=Italic>x,y</Emphasis>) = 0

In this article we describe our experiences with a parallel Singular implementation of the signature of a surface singularity defined by z ^N + g(x; y) = 0.     

Twins of <Emphasis Type=Italic>k</Emphasis>-free numbers in arithmetic progressions

We give a new upper bound of Barban–Davenport–Halberstam type for twins of k-free numbers in arithmetic progressions.     

Radicals of <Emphasis Type=Italic>l</Emphasis>-rings and one-sided <Emphasis Type=Italic>l</Emphasis>-ideals

We consider the plane deformation of an infinite body with a cylindrical inclusion whose mechanical and thermophysical characteristics differ from those of the material of the body for a heat flow and tensile forces given far from the inclusion. Stress intensity factors in the vicinity of a crack located on the continuation of the diameter of the inclusion are determined by the solution of a singular integral equation using the method of mechanical quadratures. We investigate the ratio of the length of the crack to the distance between it and the inclusion at which the regular kernel of the integral equation can be neglected in calculations. In this case, analytic formulas for the determination of the stress intensity factors are obtained.     

About <Emphasis Type=Italic>I</Emphasis>(<Emphasis Type=Italic>J</Emphasis>)-approximately continuous functions

A generalization of density point with respect to category and I(J)-density topology has been presented in [5]. This papers deals with some basic properties of the corresponding I(J)-approximately continuous functions and gives answer when these functions are of the first Baire class and when they are Darboux functions.     

The Perron integral of order <Emphasis Type=Italic>k</Emphasis> in Riesz spaces

A Perron-type integral of order k for Riesz-space-valued functions is defined in terms of the Peano derivatives. Some fundamental properties of this integral, including an integration by parts formula, are presented.     

<Emphasis Type=Italic>N</Emphasis>-homogeneous <Emphasis Type=Italic>C</Emphasis>*-algebras generated by idempotents

In this paper we consider n-homogeneous C*-algebras generated by idempotents. We prove that a finitely generated unital n-homogeneous (when n is greater than or equals 2) C*-algebra A can be generated by a finite set of idempotents if and only if the algebra A contains at least one nontrivial idempotent.