Counting Contours on Trees

We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.Comment: 12 pages, 2 figure

Phase Transitions in Ferromagnetic Ising Models with spatially dependent magnetic fields

In this paper we study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field which vanishes as $|x|^{-\alpha}$, $\alpha >0$, as $|x|\to \infty$. We prove that in dimensions $d\ge 2$ for all $\beta$ large enough if $\alpha>1$ there is a phase transition while if $\alpha<1$ there is a unique DLR state.Comment: to appear in Communications in Mathematical Physic

Weak KAM methods and ergodic optimal problems for countable Markov shifts

Let $\sigma:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ be the left shift acting on $\boldsymbol{\Sigma}$, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $\sigma$-invariant Borel probabilities that maximize the integral of a given locally H\"older continuous potential $A : \boldsymbol{\Sigma} \to \mathbb R$. Under certain conditions, we are able to show not only that $A$-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).Comment: 15 pages. To appear in Bulletin of the Brazilian Mathematical Society

Entropic repulsion and lack of the gg-measure property for Dyson models

We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.Comment: 22 pages, 4 figure

Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields

We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-\alpha}}$ with $\alpha \in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fr\"ohlich-Spencer contours for $\alpha \neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr\"ohlich and Spencer for $\alpha=0$ and conjectured by Cassandro et al for the region they could treat, $\alpha \in (0,\alpha_{+})$ for $\alpha_+=\log(3)/\log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any $\alpha \in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-\gamma}$ and $\gamma >\max\{1-\alpha, 1-\alpha^* \}$ where $\alpha^*\approx 0.2714$, the transition still persists.Comment: 13 page

Recuperação enzimática de P(3HB) produzido por Bacillus megaterium

Polihidroxialcanoatos (PHAs) são poliésteres biodegradáveis com potencial para competir com plásticos derivados de fonte fóssil, possuindo características térmicas e mecânicas semelhantes a estes polímeros. O poli(3-hidroxibutirato) (P(3HB)) é um dos PHAs mais estudados, sendo este sintetizado em corpos de inclusão e acumulado como reserva de carbono e energia por uma variedade de microrganismos. A separação do polímero de dentro da célula é uma etapa fundamental deste processo com grande impacto no custo do produto final, que tem como principal aplicação à área médica e biotecnológica. Existem diferentes métodos de recuperação do biopolímero, entre eles estão os métodos químicos, destacando-se a extração com uso de solventes halogenados, mecânicos, como exemplo o moinho de esferas, e os bioquímicos, em especial utilizando enzimas. Estes métodos podem ser utilizados de forma combinada. Este trabalho visou avaliar o potencial de algumas enzimas na recuperação de P(3HB) produzido pela bactéria Bacillus megaterium. Inicialmente, foram realizados cultivos da B. megaterium e a biomassa obtida foi liofilizada e homogeneizada. Após, foi realizada a etapa de recuperação enzimática do biopolímero, seguida da quantificação do P(3HB) por método analítico. Foram realizados ensaios de triagem para avaliar a eficiência destas enzimas na recuperação do biopolímero e, com isso, selecionaram-se cinco enzimas para estudos mais detalhados: Alcalase, Neutrase, Papaína, Bromelina e Pancreatina. Para cada uma das enzimas foi realizado um conjunto de experimentos seguindo planejamento do tipo Box Behnken para avaliar os efeitos dos parâmetros estudados na recuperação (pH, temperatura e concentração de enzima em relação à biomassa). A pancreatina proporcionou a maior recuperação e melhor custo-benefício (preço do kg enzima, concentração enzimática e percentual de pureza do P(3HB) recuperado), obtendo-se P(3HB) com 85 % de pureza.Polyhydroxyalkanoates (PHAs) are biodegradable polyesters with the potential to compete with plastics derived from fossil sources, having similar thermal and mechanical characteristics to these polymers. Poly(3-hydroxybutyrate) (P(3HB)) is one of the most studied PHAs, which is synthesized as inclusion bodies and accumulated as carbon and energy reserves by a variety of microorganisms. The separation of the polymer from the cell is a fundamental step of this process, possessing great impact on the final cost of the product, which has, as its main applications, the medical and biotechnology areas.There are different methods for recovering this biopolymer, among them the chemical methods, mainly by using halogenated solvents, mechanical methods such as ball mills, and biochemicals, using enzymes. These methods can be used in combination. This work aimed at evaluating the potential of some enzymes in the recovery process of P(3HB) from Gram-positive bacterium Bacillus megaterium. Initially, B. megaterium was cultivated and the biomass obtained was lyophilized and homogenized. After that, the enzymatic recovery step was performed, followed by P(3HB) quantification by analytical method. Screening tests were performed to evaluate the efficiency of enzymes in the biopolymer recovery and five enzymes were selected for more detailed studies: Alcalase, Neutrase, Papain, Bromelain, and Pancreatin. For each of the enzymes, a set of experiments was performed following Box Behnken Design to evaluate the effects of the studied parameters on the recovery (pH, temperature, and enzyme concentration per biomass). Pancreatin provided the highest recovery and the most cost-effective (price of kg, enzyme concentration and purity of recovered P(3HB)), obtaining P(3HB) with 85 % of purity