ABSTRACT In this contribution, we present a predictive tool developed to help in the management of the evolution of the COVID-19 pandemic situation in Rio Grande do Sul (RS)-Brazil. This tool is the result of georeferenced data analysis, mathematical modeling, and parameter calibration for the dynamics of a SIR-type model defined on a spatial structure that allows distinct subpopulations to interact, similar to the controlled distancing (A l for l=1, ···, 21)) groupings proposed by the RS government and public health authorities. The predictive analysis, updated biweekly, provides three distinct scenarios per month (milder, average, and severe) and is made available as WebSIGs (Geographic Information System-GIS). The forecast of the average scenario for each A l group is the result of a simulation of the proposed SIR-type dynamics with calibrated parameters derived from an augmented Lagrangian maximum a posteriori estimation and data on the number of infected cases made available by the RS Health Secretariat. The milder and severe scenarios are obtained from the average scenario, with changes in the contagion rates of each A l group. When compared to the number of infections reported in each A l group, the modeling predictions for a biweekly time window (the first two weeks) were quite satisfactory, with errors ranging from 0% to 5.13%, gradually increasing over time. Therefore, we suggest a biweekly re-calibration of the parameters and corresponding forecasts as a wise strategy.

Resumo Este trabalho interdisciplinar propõe utilizar objetos astronômicos como ferramenta lúdica para despertar a curiosidade dos alunos, oportunizar e potencializar o processo de ensino-aprendizagem nas aulas de Física II e Mecânica Estatística. Seu objetivo consiste em discutir se a lei de distribuição de velocidades de Maxwell-Boltzmann para gases ideais também descreve o comportamento das velocidades das estrelas em aglomerados estelares. Para isso, os pensamentos investigativos dos alunos serão provocados com dados de velocidades de estrelas, obtidos através de observações do telescópio Blanco e KPNO (Observatório Nacional Kitt Peak), fornecidos na literatura para o aglomerado globular Ômega Centauri e para o aglomerado aberto Plêiades. Foi possível concluir que, em aglomerados globulares relaxados dinamicamente, embora possuam naturezas físicas muito diferentes da de gases ideais, as estrelas seguem a lei de distribuição de velocidades de Maxwell-Boltzmann. Isto ocorre porque, de fato, qualquer sistema clássico de muitos corpos que atenda as duas hipóteses usadas na dedução da distribuição de velocidades vai satisfazer a distribuição de Maxwell-Boltzmann, independente da interação entre os corpos serem de curto ou longo alcance e de serem objetos microscópios ou macroscópicos. Então, ao demonstrar que aglomerados de estrelas seguem leis semelhantes à dos gases ideais tornou-se possível utilizar o lúdico da Astronomia para incentivar os alunos no processo de aprendizagem do conteúdo de teoria cinética dos gases.

Abstract This interdisciplinary work proposes to use astronomical objects as a ludic tool to arouse student's curiosity, provide opportunities and enhance the teaching-learning process in Physics II and Statistical Mechanics classes. Its purpose is to discuss whether the Maxwell-Boltzmann velocity distribution law for ideal gases also describes the behavior of star velocities in star clusters. For this, the student's investigative thoughts it will be triggered by star velocity data, obtained through observations from the Blanco telescope and KPNO (Kitt Peak National Observatory), provided in the literature for the Omega Centauri globular cluster and the Pleiades open star cluster. It can be concluded that in dynamically relaxed globular clusters, although physically very different from ideal gases, stars follow Maxwell-Boltzmann's law of velocity distribution. This is because, in fact, any classical many-body system that satisfies the two assumptions used in the computations of velocity distribution will satisfy the Maxwell-Boltzmann distribution, regardless of whether the interaction between the bodies is of short or long-range and whether they are microscopic or macroscopic objects. Thus, by demonstrating that star clusters follow ideal gas-like laws, it became possible to use the ludic of astronomy to encourage students in the process of learning the content of the kinetic theory of gases.

We obtain through a Matrix Product Ansatz the exact solution of the most general inhomogeneous spin chain with nearest neighbor interaction and with U(1)² and U(1)³ symmetries. These models are related to the one loop mixing matrix of the Leigh-Strassler deformed N = 4 SYM theory, dual to type IIB string theory in the generalized Lunin-Maldacena backgrounds, in the sectors of two and three kinds of fields, respectively. The solutions presented here generalizes the results obtained by the author in a previous work for homogeneous spins chains with U(1)N symmetries in the sectors of N = 2 and N = 3.

We obtain through a Matrix Product Ansatz (MPA) the exact solution of the most general N-state spin chain with U(1)N symmetry and nearest neighbour interaction. In the case N = 6 this model contain as a special case the integrable SO(6) spin chain related to the one loop mixing matrix for anomalous dimensions in N = 4 SYM, dual to type IIB string theory in the generalised Lunin-Maldacena backgrounds. This MPA is construct by a map between scalar fields and abstract operators that satisfy an appropriate associative algebra. We analyses the Yang-Baxter equation in the N = 3 sector and the consistence of the algebraic relations among the matrices defining the MPA and find a new class of exactly integrable model unknown up to now.

The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product ansatz. Due to the similarity of the master equation and the Schrodinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. Initially, we present the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size (s = 0; 1; 2; ...) in units of lattice spacing, diffuses asymmetrically on the lattice. The solution of the more general problem where we have the diffusion of particles belonging to N distinct classes of particles (c = 1; ... ; N), with hierarchical order and arbitrary sizes, is also presented. Our matrix product ansatz asserts that the amplitudes of an arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed by a product of matrices. The algebraic properties of the matrices defining the ansatz depend on the particular associated Hamiltonian. The absence of contradictions in the algebraic relations defining the algebra ensures the exact integrability of the model. In the case of particles distributed in N > 2 classes, the associativity of this algebra implies the Yang-Baxter relations of the exact integrable model.