Neste trabalho consideramos casos especiais de uma versão modificada do Modelo de Isard-Liossatos para crescimento econômico espacial, levando em consideração a interação entre as distribuições de capital e mão-de-obra. Por um lado, consideramos que o capital se move de regiões com alta densidade de capital para regiões com baixa densidade de capital de uma forma difusiva, e, por outro lado, que ele se move para regiões com uma maior densidade de mão-de-obra (o que chamamos de movimento por taxia do capital). De forma similar, consideramos um movimento difusivo e por taxia para a mão-de-obra. No primeiro caso, consideramos um sistema de reação-difusão-taxia governando a evolução espaço-temporal das densidades de capital e trabalho, e encontramos quatro tipos de comportamentos para o sistema: (i) a economia converge para um estado estacionário homogêneo; (ii) a economia converge para um estado estacionário não-homogêneo; (iii) a economia desenvolve ciclos periódicos; e (iv) a economia desenvolve ciclos irregulares e aperiódicos. No segundo caso, consideramos um modelo dinâmico e hiperbólico, derivado a partir da Lei de Fourier Modificada proposta por Cattaneo (1948), a qual implica que a informação se propaga com velocidade finita através da economia. Finalmente, introduzimos um estoque de recursos naturais não-renovável no modelo de reação-difusãotaxia, e mostramos que, para o caso (ii) acima, a economia como um todo se beneficia. / In this work we consider special cases of a modi ed version of the Isard-Liossatos Model for spatial economic growth, taking into account the interplay between the distributions of capital and labor. On one hand we consider that capital moves from regions with high density of capital to regions with low density of capital in a di use way, and, on the other hand, we consider that it moves into regions with a higher density of labor available (what we call the capital taxis motion). In the same fashion, we consider a di usive and a taxis motion for the labor force. In the rst case we consider a taxis-reaction-di usion system governing the spatio-temporal evolution of capital and labor densities, and nd four kinds of spatio-temporal behaviors for the system: (i) the economic converges to a homogeneous steady-state; (ii) the economy converges to a non-homogeneous steadystate; (iii) the economy develops periodic cycles; and (iv) the economy develops irregular and aperiodic cycles. In the second case we consider a dynamic and hiperbolic model, derived from the modi ed Fourier Law proposed by Cattaneo (1948), which implies that information propagates throughout the economy in a nite speed. Finally, we introduce a non-renewable natural resource in the taxis-reaction-di usion model, and show that, in the case (ii) above, the economy as a whole bene ts from it.

Orientador: Euclydes Custodio de Lima Filho / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Ciência da Computação / Made available in DSpace on 2018-07-21T04:57:34Z (GMT). No. of bitstreams: 1 Pinho_AndreLuisSantosde_M.pdf: 2639435 bytes, checksum: caa21ed212b2c79fb9d970e328a8e006 (MD5) Previous issue date: 1995 / Resumo: Não informado. / Abstract: Not informed. / Mestrado / Mestre em Estatística

Orientador: Jose Mario Martinez / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-07-26T03:15:42Z (GMT). No. of bitstreams: 1 Pilotta_ElvioAngel_D.pdf: 3939443 bytes, checksum: 1c19786cda4ad43e405851398d5d1a07 (MD5) Previous issue date: 2000 / Resumo: Nesta tese apresentamos métodos numéricos para problemas de minimização com restrições. O Capítulo 1 está baseado no artigo "Validation of an Augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of Hard-Spheres Problems", de Krejié, Martínez, Mello e Pilotta. O Capítulo 2 está baseado no artigo "Inexact-Restoration algorithm for constrained optimization" , de Martínez e Pilotta, onde é considerado um novo método de tipo restauração inexata para um problema de minimização com restrições gerais. O Capítulo 3 estábaseado no artigo "Spectral Gradient method for linearly constrained optimization" , de Martínez, Pilotta e Raydan, onde é considerado um novo método para um problema de minimização com restrições lineares e canalizações usando gradiente espectral precondicionado e penalização exponencial. O Capítulo 4 está baseado no artigo "A limited-memory multipoint secant method for bound constrained optimization", de Burdakov, Martínez e Pilotta, onde é considerado um novo método para um problema de minimização com canalizações usando uma estratégia de restrições ativas e um método secante simétrico multipoint com memória limitada para resolver um subproblema quadrático em cada face / Abstract: We present numerical methods for constrained minimization problems. Chapter 1 is based on the paper "Validation of an Augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of Hard-Spheres Problems", by Krejié, Martínez, Mello and Pilotta. Chapter 2 is based on the paper "InexactRestoration algorithm for constrained optimization" , by Martínez and Pilotta, where we introduce an inexact-restoration method for solving a general constrained minimization problem. Chapter 3 is based on the paper "Spectral Gradient method for linearly constrained optimization", by Martínez, Pilotta, and Raydan, where we introduce a new method for this problem which uses exponential penalization. Chapter 4 is based on the paper " A limited-memory multipoint secant method for bound constrained optimization", by Burdakov, Martínez and Pilotta, where we introduce a new method for bound constrained optimization that uses active set methods for solving a quadratic subproblem in each face / Doutorado / Doutor em Matemática Aplicada

Orientador: Ubiratan D'Ambrosio / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Ciencia da Computação / Made available in DSpace on 2018-07-15T16:14:12Z (GMT). No. of bitstreams: 1 Vila_MariadoCarmo_M.pdf: 4698880 bytes, checksum: 1768a31856b12325ec27153c3446e39e (MD5) Previous issue date: 1982 / Resumo: A Matemática é atualmente tida, para uma grande maioria de pessoas, como uma ciência difícil, complexa e accessível apenas a umas poucas mentes privilegiadas. Por isso mesmo, essas pessoas sentem uma surpreendente admiração pelos profissionais da área e confessam, publicamente, o desgosto pelo pouco que aprenderam sobre essa disciplina durante a vida escolar. Por que se aprende tão pouco sobre a Matemática? Por que tantas pessoas odeiam essa disciplina? Procurando uma resposta para tais perguntas, verificamos que as dificuldades de aprendizagem dos conteúdos matemáticos surgem desde os primeiros contatos com o estudo da disciplina. A busca de uma solução para esse problema nos levou à construção de um modelo operatório de metodologia de Matemática para as séries iniciais do 1º Grau e à construção de materiais didáticos que possibilitassem aos professores a sua aplicação em sala de aula. O modelo e os materiais elaborados foram aplicados durante 4 anos no Centro Pedagógico da Universidade Federal de Minas Gerais e, nesse período, foram oferecidos cursos de treinamento para professores do Estado. O presente estudo descreve e analisa as opiniões de alunos e professores sobre o modelo de metodologia vivenciado e sobre os materiais didáticos utilizados. / Abstract: Not informed. / Mestrado / Ensino de Ciencias e Matematica / Mestre em Matemática

Orientador: Carlos Alberto Dutra Fraga Filho / Banca: Messias Meneguette Júnior / Banca: Vanessa Avansini Botta Pirani / Banca: Andriana Susana Lopes de Oliveira Campanharo / Resumo: Neste trabalho, apresenta-se o método Lagrangiano de partículas livre de malhas, Smoothed Particle Hydrodynamics (SPH). Neste método, o domínio do problema é discretizado empregando partículas que possuem propriedades físicas calculadas a partir das propriedades das partículas vizinhas, por meio de uma interpolação, utilizando uma função de suavização. O método foi aplicado para discretizar e adaptar para SPH um modelo de ruptura de barragem sobre um leito úmido conhecido como Dam-Break in Wet-Bed. Tal modelo, considerado ideal, é empregado nos testes de desempenho, precisão e confiabilidade de modelos matemáticos para validação de métodos numéricos aplicados a problemas de vertedouros em barragens hidrelétricas e ondas em praias. / Abstract: In this work, we present the Lagrangian mesh-free method of particles, Smoothed Particle Hydrodynamics (SPH). In this method, the problem domain is discretized using particles having physical properties calculated from neighboring particles by means of interpolation using a smoothing function. The method was applied to discretize and to adapt the SPH for a model of dam rupture on a wet-bed known as Wet-Dam-Break. This model, considered as ideal, is used in tests of performance, accuracy and reliability of mathematical models for validation of numerical methods applied to spillways problems in hydroelectric dams and waves at beaches. / Mestre

This work aims at presenting different approaches to epidemics modelling. It consists of two main topics, which cover both theoretical and computational approaches to the development and analysis of mathematical models of infectious diseases. The first half regards the formulation and the analysis of SAIRS (Susceptible - Asyptomatics infected - Infected symptomatic - Recovered - Susceptible) epidemic models, including the possibility of vaccination. The model is formulated as a system of ordinary differential equations (ODEs), for which we provided a complete global stability analysis, combining two different approaches: the classical Lyapunov stability theorem, and a geometric approach, which generalises the Poincaré-Bendixon theorem. Afterwards, the model has been generalised using heterogeneous networks, which may describe different groups of individuals or different cities. For this model, the global stability analysis has been developed using the graph-theoretic approach to find Lyapunov functions. The second part of the thesis covers simulations based approaches to modelling heterogeneous humans interactions in epidemics. The first example we provide is an application with synthetic data. We investigate a stochastic SIR (Susceptible - Infected symptomatic - Recovered) dynamics on a network, by using a specialised version of the Gillespie algorithm. The other two examples we show consist of real data applications. Both regard the cost-benefit analysis of the introduction of new influenza vaccines. Both analyses have been performed using a multi-group SEIR (Susceptible - Exposed - Infected - Recovered) epidemiological model divided by age classes.

Mathematical models represent a powerful tool for investigating the dynamics of human infection diseases, providing useful predictions about the spread of a disease and the effectiveness of possible control measures. One of the central aspects to understand the dynamics of human infection is the heterogeneity in behavioral patters adopted by the host population. Beyond control measures imposed by public authorities, human behavioral changes can be triggered by uncoordinated responses driven by the diffusion of fear in the general population or by the risk perception. In order to assess how and when behavioral changes can affect the spread of an epidemic, spontaneous social distancing - e.g. produced by avoiding crowded environments, using face masks or limiting travels - is investigated. Moreover, in order to assess whether vaccine preventable diseases can be eliminated through not compulsory vaccination programs, vaccination choices are investigated as well. The proposed models are based on an evolutionary game theory framework. Considering dynamical games allows explicitly modeling the coupled dynamics of disease transmission and human behavioral changes. Specifically, the information diffusion is modeled through an imitation process in which the convenience of different behaviors depends on the perceived risk of infection and vaccine side effects. The proposed models allow the investigation of the effects of misperception of risks induced by partial, delayed or incorrect information (either concerning the state of the epidemic or vaccine side effects) as well. The performed investigation highlights that a small reduction in the number of potentially infectious contacts in response to an epidemic and an initial misperception of the risk of infection can remarkably affect the spread of infection. On the other hand, the analysis of vaccination choices showed that concerns about proclaimed risks of vaccine side effects can result in widespread refusal of vaccination which in turn leads to drops in vaccine uptake and suboptimal vaccination coverage.

The main object of our research is the notion of "intrinsic regular surfaces" introduced and studied by Franchi, Serapioni, Serra Cassano in a Carnot group G. More precisely, an intrinsic regular hypersurface (i.e. a topological codimension 1 surface) S is a subset of G which is locally defined as a non critical level set of a C^1 intrinsic function. In a similar way, a k-codimensional intrinsic regular surface is locally defined as a non critical level set of a C^1 intrinsic vector function. Through Implicit Function Theorem, S can be locally represented as an intrinsic graph by a function phi. Here the intrinsic graph is defined as follows: let V and W be complementary subgroups of G, then the intrinsic graph of phi defined from W to V is the set { A \cdot phi(A) | A belongs to W}, where \cdot indicates the group operation in G. A fine characterization of intrinsic regular surfaces in Heisenberg groups (examples of Carnot groups) as suitable 1-codimensional intrinsic graphs has been established in [1]. We extend this result in a general Carnot group introducing an appropriate notion of differentiability, denoted uniformly intrinsic differentiability, for maps acting between complementary subgroups of G. Finally we provide a characterization of intrinsic regular surfaces in terms of existence and continuity of suitable "derivatives" of phi introduced by Serra Cassano et al. in the context of Heisenberg groups. All the results have been obtained in collaboration with Serapioni. [1] L.Ambrosio, F. Serra Cassano, D. Vittone, \emph{Intrinsic regular hypersurfaces in Heisenberg groups}, J. Geom. Anal. 16, (2006), 187-232.

In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. In conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while there exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is completely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators with quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc Hilbert spaces. Clearly, this property turns out to be crucial.

In this thesis we study intrinsic Lipschitz functions. In particular we provide a regular approximation result and a Poincarè type inequality for this class of functions. Moreover we study the obstacle problem in the Heisenberg group and we prove a geometric Poincarè inequality for a class of semilinear equations in the Engel group.