Um estudo introdutório da Teoria de Grafos através de matrizes

Pós-graduação em Matemática em Rede Nacional - IBILCE

The evolution of behavioural systems: a study of grooming in rodents

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Inferência estatística clássica para a confiabilidade de rede de coautoria com enfoque nos vértices

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Uma proposta de oficina de coloração de mapas e grafos para o ensino fundamental e médio

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The cohomological invariant E'(G,W) and some properties

Let G be a group, W a nonempty G-set and M a Z2G-module. Consider the restriction map resG W : H1(G,M) → Pi wi∈E H1(Gwi,M), [f] → (resGG wi [f])i∈I , where E = {wi, i ∈ I} is a set of orbit representatives in W and Gwi = {g ∈ G | gwi = wi} is the G-stabilizer subgroup (or isotropy subgroup) of wi, for each wi ∈ E. In this work we analyze some results presented in Andrade et al [5] about splittings and duality of groups, using the point of view of Dicks and Dunwoody [10] and the invariant E'(G,W) := 1+dimkerresG W, defined when Gwi is a subgroup of infinite index in G for all wi in E, andM = Z2 (where dim = dimZ2). We observe that the theory of splittings of groups (amalgamated free product and HNN-groups) is inserted in the combinatory theory of groups which has many applications in graph theory (see, for example, Serre [12] and Dicks and Dunwoody [10]).

Grafos em superfícies

Pós-graduação em Matemática Universitária - IGCE

Yang-Lee zeros of the two- and three-state Potts model defined on π3 Feynman diagrams

We present both analytical and numerical results on the position of partition function zeros on the complex magnetic field plane of the q=2 state (Ising) and the q=3 state Potts model defined on phi(3) Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model, an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the q=3 state Potts model, our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.

A note on permutation regularity

The existence of a small partition of a combinatorial structure into random-like subparts, a so-called regular partition, has proven to be very useful in the study of extremal problems, and has deep algorithmic consequences. The main result in this direction is the Szemeredi Regularity Lemma in graph theory. In this note, we are concerned with regularity in permutations: we show that every permutation of a sufficiently large set has a regular partition into a small number of intervals. This refines the partition given by Cooper (2006) [10], which required an additional non-interval exceptional class. We also introduce a distance between permutations that plays an important role in the study of convergence of a permutation sequence. (C) 2011 Elsevier B.V. All rights reserved.

The Tripartite Ramsey Number for Trees

We prove that for all epsilon>0 there are alpha>0 and n(0)is an element of N such that for all n >= n(0) the following holds. For any two-coloring of the edges of Kn, n, n one color contains copies of all trees T of order t <=(3 - epsilon)n/2 and with maximum degree Delta(T)<= n(alpha). This confirms a conjecture of Schelp. (c) 2011 Wiley Periodicals, Inc. J Graph Theory 69: 264300, 2012

Texture descriptor based on partially self-avoiding deterministic walker on networks

Texture image analysis is an important field of investigation that has attracted the attention from computer vision community in the last decades. In this paper, a novel approach for texture image analysis is proposed by using a combination of graph theory and partially self-avoiding deterministic walks. From the image, we build a regular graph where each vertex represents a pixel and it is connected to neighboring pixels (pixels whose spatial distance is less than a given radius). Transformations on the regular graph are applied to emphasize different image features. To characterize the transformed graphs, partially self-avoiding deterministic walks are performed to compose the feature vector. Experimental results on three databases indicate that the proposed method significantly improves correct classification rate compared to the state-of-the-art, e.g. from 89.37% (original tourist walk) to 94.32% on the Brodatz database, from 84.86% (Gabor filter) to 85.07% on the Vistex database and from 92.60% (original tourist walk) to 98.00% on the plant leaves database. In view of these results, it is expected that this method could provide good results in other applications such as texture synthesis and texture segmentation. (C) 2012 Elsevier Ltd. All rights reserved.

Molecular determinants of caste differentiation in the highly eusocial honeybee Apis mellifera

Abstract Background In honeybees, differential feeding of female larvae promotes the occurrence of two different phenotypes, a queen and a worker, from identical genotypes, through incremental alterations, which affect general growth, and character state alterations that result in the presence or absence of specific structures. Although previous studies revealed a link between incremental alterations and differential expression of physiometabolic genes, the molecular changes accompanying character state alterations remain unknown. Results By using cDNA microarray analyses of >6,000 Apis mellifera ESTs, we found 240 differentially expressed genes (DEGs) between developing queens and workers. Many genes recorded as up-regulated in prospective workers appear to be unique to A. mellifera, suggesting that the workers' developmental pathway involves the participation of novel genes. Workers up-regulate more developmental genes than queens, whereas queens up-regulate a greater proportion of physiometabolic genes, including genes coding for metabolic enzymes and genes whose products are known to regulate the rate of mass-transforming processes and the general growth of the organism (e.g., tor). Many DEGs are likely to be involved in processes favoring the development of caste-biased structures, like brain, legs and ovaries, as well as genes that code for cytoskeleton constituents. Treatment of developing worker larvae with juvenile hormone (JH) revealed 52 JH responsive genes, specifically during the critical period of caste development. Using Gibbs sampling and Expectation Maximization algorithms, we discovered eight overrepresented cis-elements from four gene groups. Graph theory and complex networks concepts were adopted to attain powerful graphical representations of the interrelation between cis-elements and genes and objectively quantify the degree of relationship between these entities. Conclusion We suggest that clusters of functionally related DEGs are co-regulated during caste development in honeybees. This network of interactions is activated by nutrition-driven stimuli in early larval stages. Our data are consistent with the hypothesis that JH is a key component of the developmental determination of queen-like characters. Finally, we propose a conceptual model of caste differentiation in A. mellifera based on gene-regulatory networks.

Coordinated Control of Robotic Swarms in Unknown Environments

This thesis gathers the work carried out by the author in the last three years of research and it concerns the study and implementation of algorithms to coordinate and control a swarm of mobile robots moving in unknown environments. In particular, the author's attention is focused on two different approaches in order to solve two different problems. The first algorithm considered in this work deals with the possibility of decomposing a main complex task in many simple subtasks by exploiting the decentralized implementation of the so called \emph{Null Space Behavioral} paradigm. This approach to the problem of merging different subtasks with assigned priority is slightly modified in order to handle critical situations that can be detected when robots are moving through an unknown environment. In fact, issues can occur when one or more robots got stuck in local minima: a smart strategy to avoid deadlock situations is provided by the author and the algorithm is validated by simulative analysis. The second problem deals with the use of concepts borrowed from \emph{graph theory} to control a group differential wheel robots by exploiting the Laplacian solution of the consensus problem. Constraints on the swarm communication topology have been introduced by the use of a range and bearing platform developed at the Distributed Intelligent Systems and Algorithms Laboratory (DISAL), EPFL (Lausanne, CH) where part of author's work has been carried out. The control algorithm is validated by demonstration and simulation analysis and, later, is performed by a team of four robots engaged in a formation mission. To conclude, the capabilities of the algorithm based on the local solution of the consensus problem for differential wheel robots are demonstrated with an application scenario, where nine robots are engaged in a hunting task.

Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

A parallel community detection algorithm

Complex networks analysis is a very popular topic in computer science. Unfortunately this networks, extracted from different contexts, are usually very large and the analysis may be very complicated: computation of metrics on these structures could be very complex. Among all metrics we analyse the extraction of subnetworks called communities: they are groups of nodes that probably play the same role within the whole structure. Communities extraction is an interesting operation in many different fields (biology, economics,...). In this work we present a parallel community detection algorithm that can operate on networks with huge number of nodes and edges. After an introduction to graph theory and high performance computing, we will explain our design strategies and our implementation. Then, we will show some performance evaluation made on a distributed memory architectures i.e. the supercomputer IBM-BlueGene/Q "Fermi" at the CINECA supercomputing center, Italy, and we will comment our results.

Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.