Motion of discrete interfaces through mushy layers

We study the geometric motion of sets in the plane derived from the homogenization of discrete ferromagnetic energies with weak inclusions. We show that the discrete sets are composed by a `bulky' part and an external `mushy region' composed only of weak inclusions. The relevant motion is that of the bulky part, which asymptotically obeys to a motion by crystalline mean curvature with a forcing term, due to the energetic contribution of the mushy layers, and pinning effects, due to discreteness. From an analytical standpoint it is interesting to note that the presence of the mushy layers imply only a weak and not strong convergence of the discrete motions, so that the convergence of the energies does not commute with the evolution. From a mechanical standpoint it is interesting to note the geometrical similarity of some phenomena in the cooling of binary melts.Comment: 20 pages, 3 figure

Discrete double-porosity models for spin systems

We consider spin systems between a finite number $N$ of "species" or "phases" partitioning a cubic lattice $\mathbb{Z}^d$. We suppose that interactions between points of the same phase are coercive, while between point of different phases (or, possibly, between points of an additional "weak phase") are of lower order. Following a discrete-to-continuum approach we characterize the limit as a continuum energy defined on $N$-tuples of sets (corresponding to the $N$ strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part which describes the combined effect of lower-order terms, weak interactions between phases, and possible oscillations in the weak phase.Comment: arXiv admin note: text overlap with arXiv:1406.175

A Relaxation result for energies defined on pairs set-function and applications

We consider, in an open subset Ω of RN, energies depending on the perimeter of a subset E С Ω (or some equivalent surface integral) and on a function u which is defined only on E. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” Ω \ E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application, a new proof of convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah functional

Nonlocal-interaction vortices

We consider sequences of quadratic non-local functionals, depending on a small parameter \e, that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis and Mironescu. Similarly to what is done for hard-core approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by |\log\e|^{-1} and restrict them to $S^1$-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equi-coercive, and show the converge to a vortex energy, similarly to the limit behaviour of Ginzburg-Landau energies at the vortex scaling

Asymptotic behaviour of ground states for mixtures of ferromagnetic and antiferromagnetic interactions in a dilute regime

We consider randomly distributed mixtures of bonds of ferromagnetic and antiferromagnetic type in a two-dimensional square lattice with probability $1-p$ and $p$, respectively, according to an i.i.d. random variable. We study minimizers of the corresponding nearest-neighbour spin energy on large domains in ${\mathbb Z}^2$. We prove that there exists $p_0$ such that for $p\le p_0$ such minimizers are characterized by a majority phase; i.e., they take identically the value $1$ or $-1$ except for small disconnected sets. A deterministic analogue is also proved

Compactness by coarse-graining in long-range lattice systems

We consider energies on a periodic set ${\mathcal L}$ of ${\mathbb R}^d$ of the form $\sum_{i,j\in{\mathcal L}} a^\varepsilon_{ij}|u_i-u_j|$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the interactions is $R_\varepsilon$ with $R_\varepsilon\to +\infty$, i.e., if $\|i-j\|\le R_\varepsilon$ then $a^\varepsilon_{ij}\ge c>0$. In a discrete-to-continuum analysis, we prove that the overall behaviour as $\varepsilon\to 0$ of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on $\varepsilon{\mathcal L}$ with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded $R_\varepsilon$ and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ${\mathcal L}={\mathbb Z}^d$

Discrete double-porosity models for spin systems

We consider spin systems between a finite number N of “species” or “phases” partitioning a cubic lattice Zd . We suppose that interactions between points of the same phase are coercive while those between points of different phases (or possibly between points of an additional “weak phase”) are of lower order. Following a discrete-to-continuum approach, we characterize the limit as a continuum energy defined on N-tuples of sets (corresponding to the N strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part that describes the combined effect of lowerorder terms, weak interactions between phases, and possible oscillations in the weak phase

Estudo Empírico de Métodos de Estimação Robusta de Modelos PAR e Aplicação a Dados de Poluição do Ar

Os estudos de poluição atmosférica geralmente envolvem medições e análises de dados de concentrações de poluentes, como é o caso do MP10 (material particulado de diâmetro inferior a 10 µm), do SO2 (dióxido de enxofre) e de outros poluentes. Esses dados normalmente possuem características estatísticas importantes como autocorrelação, sazonalidade, periodicidade e a presença de picos na série que apesar de não serem observações atípicas (outliers) pela alta frequência com a qual ocorrem, podem ser modelados como tais pelo efeito que tem na série. Todas essas características exigem atenção especial durante a análise dos dados. Com esse motivação, esse estudo comparou o desempenho dos estimadores robustos para modelos periódicos autorregressivos (PAR) propostos por Sarnaglia, Reisen & Levy-Leduc (2010) (Yule-Walker Robusto) e Shao (2008) (Mínimos Quadrados Robusto), por meio de um estudo de Monte Carlo em diferentes cenários, incluindo: contaminação por observações atípicas aditivas e desvios da normalidade. Para efeito de comparação, também foi considerada a metodologia clássica de Yule-Walker que pode ser vista em McLeod (1994), por exemplo. O interesse prático em poluição do ar é avaliar se ambas as metodologias robustas captam melhor a estrutura de correlação da série do que a metodologia clássica. Isso pode ser verificado, por exemplo, pela ordem dos modelos autorregressivos obtidas por cada procedimento de estima¸c ao. Esses três métodos foram aplicados para o ajuste do modelo PAR a` dados de MP10 da esta¸c ao da Enseada do Suá da rede de monitoramento da qualidade do ar da Grande Vitória-ES