Talks 2015

Organizers: Pablo Rodriguez and Renato Gava (Mar to Jul 2015), Sandro Gallo and Paulo da Veiga (Aug - Dec 2015).

Abstracts

Um formalismo termodinâmico generalizado

Speaker: Samuel Senti (UFRJ)
Date: 27/11/2015 • Time: 16h00 • Room: 4-111 Auditório Fávaro ICMC-USP

Abstract: Nos anos 1970, Sinai, Ruelle e Bowen desenvolveram o formalismo termodinâmico que permite estabelecer a existência e unicidade de medidas de equilíbrio para sistemas hiperbólicos e potenciais do tipo Hölder. Apresentaremos técnicas que permitem generalizar esse formalismo para classes de potenciais mais gerais. Trata-se de encontrar uma codificação do sistema que permita estudar a mesma questão para o deslocamento enumerável, em vez do sistema original. Outros resultados, tais como decaimento de correlações e teorema central do limite, também seguem por esse método.

A dynamical proof to a Poisson limit theorem for continued fraction expansion

Speaker: Xuan Zhang (UFRJ)
Date: 27/11/2015 • Time: 17h00 • Room: 4-111 Auditório Fávaro ICMC-USP

Abstract: Doeblin, in 1940, discovered a Poisson limit theorem for continued fraction expansions. His result was later clarified and extended by Iosifescu in the 1970?s. In this talk, we re-prove Doeblin?s result using perturbation theory for transfer operators of Keller and Liverani and discuss related results.

Phase transitions for layered systems

Speaker: Maria Eulália Vares (IM-UFRJ)
Date: 13/11/2015 • Time: 16h00 • Room: 4-111 Auditório Fávaro ICMC-USP

Abstract: I will discuss some of the results of a research project in collaboration with L.R. Fontes, D. Marchetti, I. Merola, and E. Presutti. We consider a system of Ising spins on Z × Z, where on each horizontal line {(x, i), x ∈ Z} the interaction is given by a ferromagnetic Kac potential with coupling strength Jγ (x, y) ∼ γ J(γ(x-y)). We then add a nearest neighbor ferromagnetic vertical interaction of small strength and investigate the occurrence of phase transition provided γ > 0 is small enough. Open questions and difficulties to treat more general systems will be discussed at the end.

On the KyFan Inequality

Speaker: Salimeh Yasaei Sekeh (UFSCar)
Date: 23/10/2015 • Time: 16h00 • Room: 4-111 Auditório Fávaro ICMC-USP

Abstract: T.B.A.

Infinite systems of interacting chains with memory of variable length - a stochastic model for biological neural nets

Speaker: Antonio Galves (IME-USP and NeuroMat)
Date: 02/10/2015 • Time: 16h00 • Room: 4-111 Auditório Fávaro ICMC-USP

Abstract: In this talk a new class of non Markovian processes with a countable number of interacting components will be presented. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian, and the stochastic chains with memory of variable length which have finite state space. These features make it suitable to describe the time evolution of biological neural systems. In this talk I will briefly review what is known for this class of processes and present some interesting open questions.

Synchronization in Complex Networks: Structure and Dynamics

Speaker: Tiago Pereira (ICMC-USP)
Date: 25/09/2015 • Time: 16h00 • 43 DEs-UFSCar

Abstract: Our everyday life depends on network synchronization at various levels. In power grids, power stations must keep a proper synchronization to avoid energy supply disturbances and blackouts. Sensor networks rely on synchronization among sensors to transmit information. In the brain, epileptic seizures and Parkinson′s diseases are a strong manifestation of synchronization. In this talk, I will discuss how the network microscopic details can profoundly affect the overall synchronization. In particular, I will show that improvements in the linking structure of a directed network can lead to synchronization loss. These results have an impact on planning the optimization of power grids and real-world networks.

Mean field reduction for coupled maps in evolving networks

Speaker: Francesco Ricci (Imperial College London)
Date: 18/08/2015 • Time: 16h00 • 4-001 ICMC-USP

Abstract: I study the dynamics of expanding circle maps interacting in a heterogeneous random network which changes in time under a process that leaves invariant the probability measure. The heterogeneity of the graphs is related to the considerable disparity between the degrees of its nodes: few nodes are very well connected, while the remaining nodes are poorly connected. The approach to the problem is probabilistic, and under suitable conditions the dynamics of each massively connected nodes can be reduced to a macroscopic equation depending only on the state of the node itself. This reduction allows one to explore the coherent properties of the network.

O teorema de Curtis-Hedlund-Lyndon para espaços shift sobre alfabetos enumeráveis

Speaker: Marcelo Sobottka (UFSC)
Date: 03/07/2015 • Time: 16h00 • Room: 4-111 Auditório Fávaro ICMC-USP

Abstract: Seja A um conjunto enumerável (chamado alfabeto) sobre o qual consideramos a topologia das partes, e seja AΝ com a topologia produto. Seja σ:AΝ → AΝ a aplicação dada para todo (xi)i ∈ Ν ∈ AΝ por σ((xi)i ∈ Ν)=(xi+1)i ∈ Ν (denominada shift map). Um espaço shift é um conjunto fechado Λ ⊂ AΝ tal que σ(Λ) ⊂ Λ. Seja Φ: Λ → Γ uma aplicação entre dois espaços shifts.

No caso em que A é finito o teorema de Curtis-Hedlund-Lyndon garante que Φ é contínuo e comuta com σ se, e só se, Φ é um sliding block code (i.e., existe K ≥ 0 tal que para todo n ∈ Ν, (Φ)n é função de (xn … xn+K)). No entanto, quando A é infinito o espaço AΝ não é compacto e o teorema de Curtis-Hedlund-Lyndon não é válido.

Nesse trabalho nós utilizamos a definição de Ott-Tomforde-Willis para espaços shifts, a qual coincide com a definição usual quando A é finito, e entrega um espaço compacto quando A é infinito. Definimos, então uma generalização do conceito de sliding block code (que coincide com o conceito usual no caso A finito) e recuperamos o teorema de Curtis-Hedlund-Lyndon para aplicações entre espaços shifts sobre alfabetos enumeráveis.

Este é um trabalho conjunto com o Dr. Daniel Gonçalves (UFSC) e Dr. Charles Starling (uOttawa).

Palavras chaves: Dinâmica Simbólica, Sistemas Dinâmicos, Teorema de Curtis-Hedlund-Lyndon.

Nonparametric conditional density estimation in high dimensions: some theoretical aspects

Speaker: Rafael Izbicki (UFSCar)
Date: 12/06/2015 • Time: 16h00 • Room: Sala de Seminários DEs-UFSCar

Abstract: In some applications (e.g., in cosmology and economics), the regression E[Z|x] is not adequate to represent the association between a predictor x and a response Z because of multi-modality and asymmetry of f(z|x); using the full density instead of a single-point estimate can then lead to less bias in subsequent analysis. However, there are currently no effective ways of estimating f(z|x) when x represents high-dimensional, complex data. Here we propose a fully nonparametric approach to conditional density estimation (CDE) that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with, for example, a large number of irrelevant components or nonlinear manifold structure in the data. We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on real-world as well as simulated data.

Convergência Fraca de grafos aleatórios a uma transformação da Teia Browniana

Speaker: Leon Alexander Valencia (Universidad de Antioquia)
Date: 03/06/2015 • Time: 14h00 • Room: Sala de Seminários DEs-UFSCar

Abstract: A Teia Browniana (BW) é uma variável aleatória que formalmente consiste de movimentos Brownianos coalescentes unidimensionais começando de qualquer ponto do espaço tempo R × R. Neste seminário apresentaremos alguns exemplos de sequências de grafos aleatórios que convergem fracamente a transformações (homeomorfismos) da BW.

On Critical Phenomena and Power Laws

Speaker: Thiago Mosqueiro (IFSC-USP)
Date: 29/05/2015 • Time: 16h00 • Room: 4-111

Abstract: After heated over a certain temperature, a common refrigerator magnet looses its magnetization. This can be explained quite well investigating a set magnetic dipoles with a short-distance interaction, often studied in the form of Ising Model (and its variations). What is counter-intuitive, however, is that this behavior happens abruptly: there is a critical temperature at which the magnet's magnetization simply vanishes. Ising model shows that, at this precise temperature, the correlation between one dipole and another at some distance decays very slowly, namely with a Power Law. Thus, with an interaction involving only nearby dipoles, at the critial temperature all dipoles become somehow linked. Moreover, several important variables present unexpected behaviors, such as divergences or non-analyticity. Over the years, this has become a new area in Physics known today as Critical Phenomena, for which Kenneth Wilson, for instance, was awarded the Nobel prize in '85. Since then, this notion of criticality gained momentum, especially in systems biology. In fact, a whole new sub-area was created that seems to suit well biology and several natural phenomena: Self-Organized Criticality, where the system spontaneously evolves towards its own critical point. A decade ago, appealing observations appeared in neuroscience with evidences of a self-organized behavior working underneath several basic functions of the brain. For instance, poising the brain at the edge of two very different states may be one reason why our eyes can operate in an enormous range of light intensities. In this talk I review some recent papers on this subject and present timely gaps in the current literature from a point of view of Dynamical Systems and Stochastic Processes. In special, I will focus on the actual role of Power Laws in these observations and a possible biases in recent investigations that do not only apply to neuroscience, but any experimental research on criticality. My main objective, then, is to present problems where contributions from modeling Stochastic Processes would have a significantly impact and may contribute to the understanding of the brain.

Ferromagnetic Ising model with periodical external field

Speaker: Manuel González Navarrete (IME-USP)
Date: 22/05/2015 • Time: 16h00 • Room: 4-111

Abstract: We study the low-temperature phase diagram for a ferromagnetic Ising model on Z², with a periodical external magnetic field. The external field takes two values: h and -h, where h>0. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides L1 × L2 sites, such that the total value of the external field is zero. As a main result, we show the presence of a first-order phase transition. The phase transition holds if h< (2J ⁄ L1)+ (2J ⁄ L2), where J is an interaction constant. We use the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate. Furthermore, we prove uniqueness for Gibbs measure in h>4J, using a uniqueness condition obtained in terms of disagreement percolation. Joint work with Eugene Pechersky and Anatoly Yambartsev.

Regularity theory for mean field games

Speaker: Edgard Pimentel (UFC)
Date: 15/05/2015 • Time: 16h00 • Room: 4-111

Abstract: In this talk, we put forward a brief introduction to the mean field games theory and present a series of results on the regularity of solutions. We prove the existence of smooth solutions, under general assumptions on the datum of the problem. The mean field games theory, as introduced in the trailblazing works of J.-M. Lasry and P.-L. Lions, is among the most dynamic areas in analysis of partial differential equations. With many applications ranging from optimal transport to regularity theory of Hamilton-Jacobi equations, this class of problems has attracted the attention of several authors in the recent years. We conclude the talk by discussing applications of this framework to problems arising in economic theory and finance. This is based on joint works with D. Gomes, L. Nurbekyan, H, Sánchez-Morgado and V. Voskanyan.

Invariance under quasi-isometries of subcritical and supercritical behaviour in Boolean Percolation Model

Speaker: Cristian Coletti (CMCC-UFABC)
Date: 08/05/2015 • Time: 16h00 • Room: 4-111

Abstract: In this work we study the Boolean model of percolation in locally compact Polish metric spaces , both discrete and continuous, and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exists for a Boolean model of percolation in N. Then we apply these results to understand the phenomena of phase transition in a large families of metric spaces. Indeed, we study the Poisson Boolean model of percolation in some Riemannian Manifolds, a large family of nilpotent Lie groups and Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.

Métodos estocásticos em sistemas complexos

Speaker: Alexandre Ferreira Ramos (EACH-USP)
Date: 24/04/2015 • Time: 16h00 • Room: 4-111

Abstract: Geralmente chamamos de complexos sistemas compostos por múltiplos elementos cujos acoplamentos são diversos. Essa combinação resulta em comportamentos coletivos que não são trivialmente previstos a partir da dinâmica das partes. Exemplos desse tipo de sistema abundam em Biologia, como é o caso da regulação da expressão gênica ou dos processos de tumorigênese. Nesse seminário pretendemos abordar esses dois fenômenos via modelamento estocástico. No primeiro, consideramos um modelo binário para um gene baseado num processo de Markov a tempo contínuo e apresentamos o mecanismo estatístico subjacente à maior precisão do controle por auto-regulação negativa. Também mostraremos a ocorrência do limite em que ocorrem os chamados “bursts” de expressão gênica. Em seguida, apresentaremos alguns resultados preliminares de nossa pesquisa utilizando o modelo de Widom-Rowlinson para o estudo da proliferação celular.

Caminho de retorno mais curto: o que ele nos diz do processo?

Speaker: Miguel Abadi (IME-USP)
Date: 17/04/2015 • Time: 16h00 • Room: 4-111

Abstract: No espaço das realizações de um processo estocastico, definimos uma seqüência de partições refinadas. Para cada elemento desta partição definimos o caminho de retorno mais curto a ele mesmo, que define uma propriedade topologica. Mostramos o comprimento típico dele, suas fluctuações e grandes desvios.

Beyond Anderson localization: Anomalous transmission of waves through media with Levy disorder

Speaker: Jose Antonio Mendez-Bermudez (Instituto de Fisica, Benemerita Universidad Autonoma de Puebla)
Date: 10/04/2015 • Time: 16h00 • Room: 4-111

Abstract: It is widely known that the presence of disorder leads to an exponential localization of waves in one-dimensional random media, as predicted by Anderson. In this work, however, we provide experimental evidence that waves can be anomalously localized, in relation to the standard Anderson localization, by introducing random configurations of the disorder that follow a distribution with a power-law tail, i.e., a Levy-type distribution. Using a microwave waveguide with dielectric slabs randomly placed, we show that if the spacing between slabs follow a Levy-type distribution, unconventional properties in the microwave-transmission fluctuations appear, revealing the presence of anomalous localization. Our analytical model describes the experimental results for the statics of the transmission fluctuations. Effects of anomalous localization on the transmission are compared with those from the standard Anderson localization as well.

Imitative Learning as a Connector of Collective Brains

Speaker: Jose Fernando Fontanari (IFSC-USP)
Date: 20/03/2015 • Time: 16h00 • Room: 4-111

Abstract: The notion that cooperation can aid a group of agents to solve problems more efficiently than if those agents worked in isolation is prevalent in computer science and business circles. Here we consider a primordial form of cooperation - imitative learning - that allows an effective exchange of information between agents, which are viewed as the processing units of a social intelligence system or collective brain. In particular, we use agent-based simulations to study the performance of a group of agents in solving a cryptarithmetic problem. An agent can either perform local random moves to explore the solution space of the problem or imitate a model agent - the best performing agent in its influence network. There is a trade-off between the number of agents N and the imitation probability p, and for the optimal balance between these parameters we observe a thirtyfold diminution in the computational cost to find the solution of the cryptarithmetic problem as compared with the independent search. If those parameters are chosen far from the optimal setting, however, then imitative learning can impair greatly the performance of the group.

A large deviations principle for the Maki-Thompson rumour model

Speaker: Elcio Lebensztayn (IMECC-UNICAMP)
Date: 13/03/2015 • Time: 16h00 • Room: 4-005

Abstract: We consider the stochastic model for the propagation of a rumour within a population which was formulated by Maki and Thompson. Sudbury has established that, as the population size tends to infinity, the proportion of the population never hearing the rumour converges in probability to 0.2032. We prove a corresponding large deviations principle, with an explicit formula for the rate function.