Towards Optimal Degree-distributions for Left-perfect Matchings in Random Bipartite Graphs

Consider a random bipartite multigraph $G$ with $n$ left nodes and $m \geq n \geq 2$ right nodes. Each left node $x$ has $d_x \geq 1$ random right neighbors. The average left degree $\Delta$ is fixed, $\Delta \geq 2$. We ask whether for the probability that $G$ has a left-perfect matching it is advantageous not to fix $d_x$ for each left node $x$ but rather choose it at random according to some (cleverly chosen) distribution. We show the following, provided that the degrees of the left nodes are independent: If $\Delta$ is an integer then it is optimal to use a fixed degree of $\Delta$ for all left nodes. If $\Delta$ is non-integral then an optimal degree-distribution has the property that each left node $x$ has two possible degrees, \floor{\Delta} and \ceil{\Delta}, with probability $p_x$ and $1-p_x$, respectively, where $p_x$ is from the closed interval $[0,1]$ and the average over all $p_x$ equals \ceil{\Delta}-\Delta. Furthermore, if $n=c\cdot m$ and $\Delta>2$ is constant, then each distribution of the left degrees that meets the conditions above determines the same threshold $c^*(\Delta)$ that has the following property as $n$ goes to infinity: If $c<c^*(\Delta)$ then there exists a left-perfect matching with high probability. If $c>c^*(\Delta)$ then there exists no left-perfect matching with high probability. The threshold $c^*(\Delta)$ is the same as the known threshold for offline $k$-ary cuckoo hashing for integral or non-integral $k=\Delta$

«Zusammenarbeit über Grenzen hinweg war mir ein Anliegen»

Zuerst unterrichtete er an der Ingenieurschule Biel deutsche Sprache, dann wurde er Dozent für Kommunikation: Mit der Gründung der Berner Fachhochschule BFH veränderte sich auch der Aufgabenbereich von Diego Jannuzzo, der im Februar pensioniert wurde. Ein Rückblick

«Beruflich weiterkommen und sich weiterentwickeln»

Weiterbildung ist ein wichtiger Pfeiler der Berner Fachhochschule BFH. Und – Zufall oder nicht – die Zahlen des 25-Jahr-Jubiläums spiegeln sich auch im Bereich Weiterbildung des Departements Architektur, Holz und Bau BFH-AHB wider: Das Angebot umfasst 5 MAS sowie 25 CAS, und der Umsatz betrug im letzten Jahr just 2,5 Millionen Franken. Ein Gespräch über Trends, Misserfolge und Dauerbrenner im Angebot

The Hall algebra and the composition monoid

Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid, as introduced by M. Reineke, and the generic composition algebra, as introduced by C. M. Ringel, specialised at q=0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then the composition algebra at q=0 is isomorphic to the monoid algebra of the composition monoid. Moreover, if Q is an acyclic, extended Dynkin quiver, we show that there exists an epimorphism from the composition algebra at q=0 to the monoid algebra of the composition monoid, and we describe its non-trivial kernel. Our main tool is a geometric version of BGP reflection functors on quiver Grassmannians and quiver flags, that is varieties consisting of filtrations of a fixed representation by subrepresentations of fixed dimension vectors. These functors enable us to calculate various structure constants of the composition algebra. Moreover, we investigate geometric properties of quiver flags and quiver Grassmannians, and show that under certain conditions, quiver flags are irreducible and smooth. If, in addition, we have a counting polynomial, these properties imply the positivity of the Euler characteristic of the quiver flag.Comment: 111 pages, doctoral thesis University of Paderborn (2009