Chip-firing via open covers

We investigate chip-firing with respect to open covers of discrete graphs and metric graphs. For the case of metric graphs we show that given an open cover and a sink q, stabilization of a divisor D is unique and that there is a distinguished configuration equivalent to D, which we call the critical configuration. Also, we show that given a double cover of the metric graph by stars, which is the continuous analogue of the sandpile model, the critical configurations are in bijection with reduced divisors. Passing to the discrete case, we interpret open covers of a graph as simplicial complexes on the vertex and observe that chip-firing with respect to a simplicial complex is equivalent to the model introduced by Paoletti [G. Paoletti. July 11 2007: Master in Physics at University of Milan, defending thesis “Abelian sandpile models and sampling of trees and forests”; supervisor: Prof. S. Caracciolo. http://pcteserver.mi.infn.it/caraccio/index.html]. We generalize this setup for directed graphs using weighted simplicial complexes on the vertex set and show that the fundamental results extend. In the undirected case we present a generalization of the Cori-Le Borgne algorithm for chip-firing models via open covers, giving an explicit bijection between the critical configurations and the spanning trees of a graph.(http://www.elsevier.com/locate/endm)     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Three-dimensional cellular automaton simulation of tumour growth in inhomogeneous oxygen environment

Cellular automaton theory has previously been used to study cell growth. In this study, we present a three-dimensional cellular automaton model performing the growth simulation of normal and cancerous cells. The necessary nutrient supply is provided by an artificial arterial tree which is generated by constrained constructive optimization. Spatial oxygen diffusion is approximated again by a cellular automaton model. All results could be illustrated dynamically by three-dimensional volume visualization. Because of the chosen modelling approach, an extension of the model to simulate angiogenic processes is possible.     

Hierarchical Model-Based Clustering for Large Datasets

In recent years, hierarchical model-based clustering has provided promising results in a variety of applications. However, its use with large datasets has been hindered by a time and memory complexity that are at least quadratic in the number of observations. To overcome this difficulty, this article proposes to start the hierarchical agglomeration from an efficient classification of the data in many classes rather than from the usual set of singleton clusters. This initial partition is derived from a subgraph of the minimum spanning tree associated with the data. To this end, we develop graphical tools that assess the presence of clusters in the data and uncover observations difficult to classify. We use this approach to analyze two large, real datasets: a multiband MRI image of the human brain and data on global precipitation climatology. We use the real datasets to discuss ways of integrating the spatial information in the clustering analysis. We focus on two-stage methods, in which a second stage of processing using established methods is applied to the output from the algorithm presented in this article, viewed as a first stage.     

Homogeneous Deformations of Brauer Tree Algebras

Abstract

Based on tilting theory, we demonstrate the existence of homogeneous deformations for the Brauer tree algebras, which are derived naturally from a tilting complex.     

Edge-forwarding index of star graphs and other Cayley graphs

We present a technique for building, in some Cayley graphs, a routing for which the load of every edge is almost the same. This technique enables us to find the edge-forwarding index of star graphs and complete-transposition graphs.     

Partitioned Tensor Products and Their Spectra

A pleasant family of graphs defined by Godsil and McKay is shown to have easily computed eigenvalues in many cases.     

Mode Trees for Multivariate Data

We consider visualizing scales of multivariate density estimates with the help of mode trees. Mode trees visualize the locations of the modes of density estimates, when the smoothing parameter of the estimates ranges over an interval. We define multiframe mode graphs which generalize classical mode trees to the multivariate setting. We give examples of the application of multiframe mode graphs with kernel estimates and with multivariate adaptive histograms.     

A Localized Implementation of the Iterative Proportional Scaling Procedure for Gaussian Graphical Models

In this article, we propose localized implementations of the iterative proportional scaling (IPS) procedure by the strategy of partitioning cliques for computing maximum likelihood estimations in large Gaussian graphical models. We first divide the set of cliques into several nonoverlapping and nonempty blocks, and then adjust clique marginals in each block locally. Thus, high-order matrix operations can be avoided and the IPS procedure is accelerated. We modify the Swendsen–Wang Algorithm and apply the simulated annealing algorithm to find an approximation to the optimal partition which leads to the least complexity. This strategy of partitioning cliques can also speed up the existing IIPS and IHT procedures. Numerical experiments are presented to demonstrate the competitive performance of our new implementations and strategies.     

矩阵乘法的推广及运用

利用矩阵的Kronecker积定义了一种矩阵乘积"*积",并且对这种乘积的性质进行了研究,发现它对于任意两个矩阵都有意义而且具有通常矩阵乘积的所有性质,并且在一些特殊情况下它比通常的矩阵乘积更和谐对称,而且当在"合适维数"下它就是通常的矩阵乘积,所以可以把这种"*积"看作是对通常矩阵乘积的推广.