Theses & Dissertations


Ph.D Students

Samuel Pacitti Gentil

Advisor: Marcos Craizer
Thesis Title: Discretization of "four-vertex type" theorems for spatial and spherical polygons
Defense: 22/02/2024 | Abstract | Thesis
Departamento de Matemática PUC-Rio Sala L856

The aim of this work is to study a certain class of spatial polygons and prove theorems on the minimal number of flattenings that such polygons must have. In order to do this, we investigate spherical polygons which are not contained in any closed hemisphere and deduce, among many results, that under certain hypotheses such spherical polygons have a nontrivial lower bound on the number of spherical inflections.

Master’s Students

Márcio dos Guimarães Peixoto

Advisor: Marcos Craizer
Thesis Title: não divulgado
Defense: 30/09/2024 | Abstract | Thesis
Sala 856 - Depto de Matemática - PUC Rio

não divulgado

Vinicius Oliveira da Costa

Advisor: Sinésio Pesco
Thesis Title: Multistep Forecast Amazon Deforestation using Regression and Recurrent Neural Network approaches
Defense: 27/09/2024 | Abstract | Thesis
Sala L856 - Depto de Matemática PUC-Rio

The Amazon rainforest, the largest tropical biome in the world, plays an essential role in both society and global environmental balance. Through its vast biodiversity and carbon storage capacity, it also supports local cultures and provides resources for sustainable development.

Deforestation prediction occupies a significant role mainly in monitoring, control, and conservation planning. The ability to predict where and when deforestation will occur allows authorities and organizations to take more effective preventive measures, allocate resources more strategically and develop policies that can mitigate negative impacts.

Therefore, the study of methods to predict deforestation has been increasingly developed in recent years. This work aims to apply supervised machine learning methods and statistical methods, such as autoregression, LightGBM, and Long Short Term Memory (LSTM) neural network to predict multi-step deforestation in the Brazilian Legal Amazon, using past observations of deforestation and climatic variables from the region.

By enhancing strategies for monitoring and controlling deforestation, this study has the potential to positively impact public policies, promoting a balance between economic development and environmental preservation and climate regulation.

Paulo Ricardo de França Silva

Advisor: Boyan Sirakov
Thesis Title: A equação de Laplace e o método de Perron
Defense: 25/09/2024 | Abstract | Thesis
Defesa realizada por meios de comunicação remota

Esta dissertação versa sobre a equação de Laplace e dos resultados que obtemos tentando resolvê-la. Nosso principal objetivo é resolver o problema clássico de Dirichlet utilizando o método de Perron.

Fabricio Barbosa dos Santos

Advisor: Marcos Craizer
Thesis Title: Singularidades genéricas de superfícies pseudoesféricas
Defense: 13/09/2024 | Abstract | Thesis
Sala 856 - Depto de Matemática - PUC Rio

A presente dissertação foi desenvolvida com o objetivo principal de estudar e classificar as singularidades genéricas existentes nas superfícies com curvatura Gaussiana K = −1 em R^3. Mais especificamente, começamos o nosso estudo definindo as parametrizações assintóticas e as redes de Tchebyshev, obtendo a chamada equação de sine-Gordon. Em seguida, apresentamos características a respeito da função geradora e fizemos uma correlação entre ela e as singularidades da superfície associada.
Por fim, enunciamos e demonstramos um teorema que permite a localização e classificação das singularidades sobre superfícies pseudoesféricas genéricas.
 
Antônio Carlos Bruno de Andrade

Advisor: Sinésio Pesco (Co-orientador: Abelardo Barreto)
Thesis Title: Reservatórios Limitados com e sem Manutenção de Pressão no Limite Externo: Comportamento da Pressão durante Testes de Injeção
Defense: 05/07/2024 | Abstract | Thesis
Defesa realizada por meios de comunicação remota

O teste de injetividade é um procedimento que tem como objetivo otimizar a produção e maximizar a recuperação de óleo dos reservatórios de petróleo. Esse teste consiste em injetar um fluido (comumente água ou gás), em um poço de produção para coletar informações sobre ele. Com base no comportamento da pressão medida durante o período de injeção, é possível identificar uma série de características relacionadas ao reservatório, incluindo volume de óleo recuperável, parâmetros do reservatório, tais como permeabilidade, zonas de baixa permeabilidade e condição de fronteira de reservatório. Atualmente, são encontrados poucos estudos relativos a testes de injetividade em reservatórios homogêneos em regimes de fluxo permanente e pseudopermanente. Por esse motivo, este trabalho tem como objetivo propor uma solução analítica aproximada para o comportamento de pressão em reservatórios homogêneos em regimes de fluxo permanente e pseudopermanente. Os resultados calculados apresentaram uma aproximação satisfatória em todos os casos.

Anselmo de Souza Pontes Junior

Advisor: Silvius Klein
Thesis Title: Limit Laws for Dynamical Systems with Some Hyperbolicity
Defense: 17/04/2024 | Abstract | Thesis
Departamento de Matemática PUC Rio - Sala L856

The study of statistical properties of dynamical systems has been an active research area in recent decades.  Its main goal is to investigate when certain deterministic chaotic systems exhibit stochastic behavior when examined through the lens of a relevant invariant measure. Some of the key tools employed in deriving such results  are the spectral properties of the transfer operator. However, certain skew product systems, including random and mixed random-quasiperiodic linear cocycles, do not fit this approach. Very recent works have obtained limit laws  for these systems by studying the Markov Operator. The purpose of this dissertation  is to explain how these operators can be used to derive limit laws, such as Large Deviations Estimates and Central Limit Theorem, for certain skew-product dynamical systems.

Raul Steven Rodriguez Chavez

Advisor: Lorenzo Justiniano Díaz Casado
Thesis Title: Statistical behavior of skew products: Schwarzian derivative and arc-sinelaws
Defense: 05/04/2024 | Abstract | Thesis
Departamento de Matemática PUC-Rio - Sala L856

We consider skew products over Bernoulli shifts, whose fibred dynamics is given by diffeomorphisms of the interval. We study the statistical and/or non-statistical behavior, referring to convergence and/or non-convergence, almost everywhere, of the Birkhoff average, respectively. We employ the Schwarzian derivative of the fiber maps and the arc-sine law to identify conditions under which these skew products exhibit these types of behavior. We identify distinct types of behavior according to the Schwarzian derivative. When the Schwarzian derivative is negative, the skew product has intermingled basins. Conversely, when the Schwarzian derivative is positive, the skew product has a physical measure. Finally, when the Schwarzian derivative is zero, the skew product has non-statistical behavior. In the latter scenario, we establish a connection between non-statistical behavior and the arc-sine law that allows us to obtain results in other settings independent of the sign of the Schwarzian derivative.

Antônio Maria Vasconcellos M. D. da Costa

Advisor: Carlos Tomei
Thesis Title: Recovery of tridiagonal matrices from spectral data
Defense: 29/02/2024 | Abstract | Thesis
Departamento de Matemática PUC Rio - Sala L856

Algorithms relating Jacobi matrices and spectral variables are standard objects in numerical analysis. The recent discovery of bidiagonal coordinates led to the search of an appropriate algorithm for these new variables. The new algorithm is presented and compared with previous techniques.

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