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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. Source


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. Source


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. Source


Bertini L.,University of Rome La Sapienza | De Sole A.,University of Rome La Sapienza | Gabrielli D.,University of LAquila | 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. Source


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. Source

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