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Bertini L.,University of Rome La Sapienza | De Sole A.,University of Rome La Sapienza | Gabrielli D.,University of L'Aquila | Jona-Lasinio G.,University of Rome La Sapienza | And 2 more authors.
Reviews of Modern Physics | Year: 2015

Stationary nonequilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. A detailed review of this theory including its main predictions and most relevant applications is given. © 2015 American Physical Society.

Landim C.,IMPA | Landim C.,University of Rouen
Communications in Mathematical Physics | Year: 2014

It has been observed (Evans in Braz J Phys 30:42-57, 2000 ; Jeon et al. in Ann Probab 28:1162-1194, 2000) that some zero-range processes exhibit condensation, a macroscopic fraction of particles concentrates on one single site. We examined in (Beltrán and Landim in Probab Theory Relat Fields 152:781-807, 2012) the asymptotic evolution of the condensate in the case where the dynamics is reversible, the number of sites is fixed, and the total number of particles diverges. We proved in that paper that in an appropriate time-scale the condensate evolves according to a symmetric random walk whose transition rates are proportional to the capacities of the underlying random walk. In this article, we extend this result to the condensing totally asymmetric zero-range process, a non-reversible dynamics. © 2014 Springer-Verlag Berlin Heidelberg.

Sagastizabal C.,IMPA
Mathematical Programming | Year: 2012

Modern electricity systems provide a plethora of challenging issues in optimization. The increasing penetration of low carbon renewable sources of energy introduces uncertainty in problems traditionally modeled in a deterministic setting. The liberalization of the electricity sector brought the need of designing sound markets, ensuring capacity investments while properly reflecting strategic interactions. In all these problems, hedging risk, possibly in a dynamic manner, is also a concern. The fact of representing uncertainty and/or competition of different companies in a multi-settlement power market considerably increases the number of variables and constraints. For this reason, usually a trade-off needs to be found between modeling and numerical tractability: The more details are brought into the model, the harder becomes the optimization problem. For structured optimization and generalized equilibrium problems, we explore some variants of solution methods based on Lagrangian relaxation and on Benders decomposition. Throughout we keep as a leading thread the actual practical value of such techniques in terms of their efficiency to solve energy related problems. © Springer and Mathematical Optimization Society 2012.

Sagastizabal C.,IMPA
Mathematical Programming | Year: 2013

We consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping. This is a sufficiently rich class that includes max-functions, largest eigenvalue functions, and norm-1 regularized functions. The bundle method uses an oracle that is able to compute separately the function and subgradient information for the convex function, and the function and derivatives for the smooth mapping. With this information, it is possible to solve approximately certain proximal linearized subproblems in which the smooth mapping is replaced by its Taylor-series linearization around the current serious step. Our numerical results show the good performance of the Composite Bundle method for a large class of problems. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Pontiveros G.F.,IMPA
Combinatorics Probability and Computing | Year: 2013

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in ℤp behave like integer sets. Thus, given A ⊂ ℤp of sufficiently small density, it is straightforward to show that |λ1A + λ2A + ⋯ + λκ A| ≥ (Σi |λi|) |A| - o(|A|). On the other hand, the behaviour for sets of large density turns out to be rather surprising. Indeed, for any ε > 0, we construct subsets of density 1/2-ε such that |A + λ A| ≤ (1-δ)p, showing that there is a very different behaviour for subsets of large density. © 2012 Cambridge University Press.

Hammerlindl A.,IMPA
Nonlinearity | Year: 2012

If the stable, centre and unstable foliations of a partially hyperbolic system are quasi-isometric, the system has Global Product Structure. This result also applies to Anosov systems and to other invariant splittings. If a partially hyperbolic system on a manifold with an abelian fundamental group has quasi-isometric stable and unstable foliations, the centre foliation is without holonomy. If, further, the system has Global Product Structure, then all centre leaves are homeomorphic. © 2012 IOP Publishing Ltd & London Mathematical Society.

Monteiro R.D.C.,Georgia Institute of Technology | Svaiter B.F.,IMPA
SIAM Journal on Optimization | Year: 2013

In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point-to-set maximal monotone operator with a separable two-block structure and introduce a framework of block-decomposition prox-type algorithms for solving it which allows for each one of the single-block proximal subproblems to be solved in an approximate sense. Moreover, by showing that any method in this framework is also a special instance of the hybrid proximal extragradient (HPE) method introduced by Solodov and Svaiter, we derive corresponding convergence rate results. We also describe some instances of the framework based on specific and inexpensive schemes for solving the single-block proximal subproblems. Finally, we consider some applications of our methodology to establish for the first time (i) the iteration-complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and, as a consequence, the ergodic iteration-complexity of the Douglas-Rachford splitting method and (ii) the ergodic iteration-complexity of the classical alternating direction method of multipliers for a class of linearly constrained convex programming problems with proper closed convex objective functions. © 2013 Society for Industrial and Applied Mathematics.

Bounemoura A.,IMPA
Communications in Mathematical Physics | Year: 2011

For perturbations of integrable Hamiltonian systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is analytic. This fundamental result has been extended in two distinct directions. The first one is due to Niederman, who showed that under the analyticity assumption, the result holds true for a prevalent class of integrable systems which is much wider than the steep systems. The second one is due to Marco-Sauzin but it is limited to quasi-convex integrable systems, for which they showed exponential stability if the system is assumed to be only Gevrey regular. If the system is finitely differentiable, we showed polynomial stability, still in the quasi-convex case. The goal of this work is to generalize all these results in a unified way, by proving exponential or polynomial stability for Gevrey or finitely differentiable perturbations of prevalent integrable Hamiltonian systems. © 2011 Springer-Verlag.

Bounemoura A.,IMPA
Annales Henri Poincare | Year: 2012

In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result proves this conjecture in the linear case. © 2011 Springer Basel AG.

Grossman P.,IMPA | Snyder N.,Columbia University
Communications in Mathematical Physics | Year: 2012

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the "quantum subgroups" in the sense of Ocneanu), we find all irreducible subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors. © 2012 Springer-Verlag.

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