We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and nonlinear parts, and studied how their interact ion affects the local and global compactness of the problem. Our approach is based on the Caffarelli-Kohn-Nirenberg inequalities.

The idea of automatic programming using the genetic programming paradigm is a concept that has been explored in the work of Koza and several works since. Most problems attempted using genetic programming are finite in size, meaning that the problem involved evolving a function that operates over a finite domain, or evolving a routine that will only run for a finite amount of time. For problems with a finite domain, the internal representation of each individual is typically a finite automaton that is unable to store an unbounded amount of data. This thesis will address the problem of applying genetic programming to problems that have a ``discrete unbounded domain", meaning the problem involves evolving a function that operates over an unbounded domain with discrete quantities. For problems with an discrete unbounded domain, the range of possible behaviors achievable by the evolved functions increases with more versatile internal memory schemes for each of the individuals. The specific problem that I will address in this thesis is the problem of evolving a real-time deciding program for a fixed language of strings. I will discuss two paradigms that I will use to attempt this problem. Each of the paradigms will allow each individual to store an unbounded amount of data, using an internal memory scheme with at least the capabilities of a Turing tape. As each character of an input string is being processed in real time, the individual will be able to imitate a single step of a Turing machine. While the real-time restriction will certainly limit the languages for which a decider may be evolved, the fact that the evolved deciding programs run in real-time yields possible applications for these paradigms in the discovery of new algorithms. The first paradigm that I will explore will take a naive approach that will ultimately prove to be unsuccessful. The second paradigm that I will explore will take a more careful approach that will have a much greater success, and will provide insight into the design of genetic programming paradigms for problems over a discrete unbounded domain.

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A new automatic algorithm for sampling from monotone, unbounded densities is presented. The user has to provide a program to evaluate the density and its derivative and the location of the pole. Then the setup of the new algorithm constructs different hat functions for the pole region and for the tail region, respectively. For the pole region a new method is developed that uses a transformed density rejection hat function of the inverse density. As the order of the pole is calculated in the setup, conditions that guarantee the correctness of the constructed hat functions are provided. Numerical experiments indicate that the new algorithm works correctly and moderately fast for many different unbounded densities. The proposed algorithm is the first black-box method that works for unbounded densities suggested in the literature. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

A new algorithm for sampling from largely abitrary monotone, unbounded densities is presented. The user has to provide a program to evaluate the density and its derivative and the location of the pole. Then the setup of the new algorithm constructs different hat functions for the pole region and for the tail region, respectively. For the pole region a new method is developed that uses a transformed density rejection hat function of the inverse density. As the order of the pole is calculated in the setup, conditions that guarantee the correctness of the constructed hat functions are provided. Numerical experiments indicate that the new algorithm works correctly and moderately fast for many different unbounded densities. (c) ACM, (2007). This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

The idea of automatic programming using the genetic programming paradigm is a concept that has been explored in the work of Koza and several works since. Most problems attempted using genetic programming are finite in size, meaning that the problem involved evolving a function that operates over a finite domain, or evolving a routine that will only run for a finite amount of time. For problems with a finite domain, the internal representation of each individual is typically a finite automaton that is unable to store an unbounded amount of data. This thesis will address the problem of applying genetic programming to problems that have a ``discrete unbounded domain", meaning the problem involves evolving a function that operates over an unbounded domain with discrete quantities. For problems with an discrete unbounded domain, the range of possible behaviors achievable by the evolved functions increases with more versatile internal memory schemes for each of the individuals. The specific problem that I will address in this thesis is the problem of evolving a real-time deciding program for a fixed language of strings. I will discuss two paradigms that I will use to attempt this problem. Each of the paradigms will allow each individual to store an unbounded amount of data, using an internal memory scheme with at least the capabilities of a Turing tape. As each character of an input string is being processed in real time, the individual will be able to imitate a single step of a Turing machine. While the real-time restriction will certainly limit the languages for which a decider may be evolved, the fact that the evolved deciding programs run in real-time yields possible applications for these paradigms in the discovery of new algorithms. The first paradigm that I will explore will take a naive approach that will ultimately prove to be unsuccessful. The second paradigm that I will explore will take a more careful approach that will have a much greater success, and will provide insight into the design of genetic programming paradigms for problems over a discrete unbounded domain.

Neste trabalho, apresenta-se o resultado da existência de soluções fracas em domínios não-limitados para as equações de Navier-Stokes, desde que a fronteira satisfaça uma certa condição de regularidade que é necessária para a obtenção de estimativas em domínios não-limitados semelhantes à desigualdade de Poincaré em domínios limitados. Apresenta-se o desenvolvimento detalhado do método de Galerkin para as equações de Navier-Stokes em domínios não-limitados com cálculo explícito de várias constantes e com forças externas não nulas. Apresenta-se dois teoremas fundamentais: um fornecendo condições para existência de soluções do problema estacionário e o outro fornecendo condições para existência de soluções do problema não-estacionário. / In the work it is presented results of existence of weak solutions in unbounded doroains for the Navier-Stokes equations. The roain condition to obtain similar results as those for bounded doroains; for e."'Carople the Poincaré inequality; is a certain condition of regularity at the boundary of the doroain. It is presented the detailed developroent of the Galerkin roethod for the t.he Navier-Stokes equations in unbounded doroains ~vith the explicit calculat ions of many constants and ''rith non null externai forces. It is presented two basic theorern: one presenting condition for the existence of solutions for the stationary problem and the other presenting conditions for existence of solution for the non stationary problem.

Uma revisão dos algoritmos e conjuntos de instâncias presentes na literatura do Problema da Mochila com Repetições (PMR) é apresentada nessa dissertação de mestrado. Os algoritmos e conjuntos de instâncias usados são brevemente descritos nesse trabalho, afim de que o leitor tenha base para entender as discussões. Algumas propriedades bem conhecidas e específicas do PMR, como a dominância e a periodicidade, são explicadas com detalhes. O PMR é também superficialmente estudado no contexto de problemas de avaliação gerados pela abordagem de geração de colunas aplicada na relaxação contínua do Bin Packing Problem (BPP) e o Cutting Stock Problem (CSP). Múltiplos experimentos computacionais e comparações são realizadas. Para os conjuntos de instâncias artificiais mais recentes da literatura, um simples algoritmo de programação dinâmica, e uma variante do mesmo, parecem superar o desempenho do resto dos algoritmos, incluindo aquele que era estado-da-arte. O modo que relações de dominância é aplicado por esses algoritmos de programação dinâmica têm algumas implicações para as relações de dominância previamente estudadas na literatura. O autor dessa dissertação defende a tese de que a escolha dos conjuntos de instâncias artificiais definiu o que foi considerado o melhor algoritmo nos trabalhos anteriores. O autor dessa dissertação disponibilizou publicamente todos os códigos e conjuntos de instâncias referenciados nesse trabalho. / A review of the algorithms and datasets in the literature of the Unbounded Knapsack Problem (UKP) is presented in this master's thesis. The algorithms and datasets used are brie y described in this work to provide the reader with basis for understanding the discussions. Some well-known UKP-speci c properties, such as dominance and periodicity, are described. The UKP is also super cially studied in the context of pricing problems generated by the column generation approach applied to the continuous relaxation of the Bin Packing Problem (BPP) and Cutting Stock Problem (CSP). Multiple computational experiments and comparisons are performed. For the most recent arti cial datasets in the literature, a simple dynamic programming algorithm, and its variant, seems to outperform the remaining algorithms, including the previous state-of-the-art algorithm. The way dominance is applied by these dynamic programming algorithms has some implications for the dominance relations previously studied in the literature. In this master's thesis we defend that choosing sets of arti cial instances has de ned what was considered the best algorithm in previous works. We made available all codes and datasets referenced in this master's thesis.

Les travaux de recherche présentés dans cette thèse sont consacrés à l’étude de la stabilité dans divers problèmes inverses associés à l’équation de Schrödinger magnétique. Dans la première partie, on s’intéresse à un problème inverse concernant l’équation de Schrödinger autonome posée dans un domaine cylindrique non borné, avec potentiel magnétique périodique. On démontre à l’aide d’une construction de solutions particulières, dites solutions de type "optique géométrique", que le champ magnétique induit par le potentiel périodique est déterminé de façon stable à partir une infinité d’observations latérales de la solution, contenues dans l’opérateur de Dirichlet-Neumann. La deuxième partie de la thèse porte sur le même type de problèmes inverses mais associés à l’équation de Schrödinger magnétique non autonome. Dans un premier temps, on montre l’existence d’une unique solution régulière de cette équation dans un domaine borné ou non. Ensuite, on s’intéresse au problème inverse de la détermination simultanée des potentiels magnétique et électrique dans un domaine borné, à partir d’un nombre fini d’observations latérales de la solution. Enfin, on prouve que dans un domaine cylindrique infini, le potentiel magnétique peut être reconstruit de façon Lipschitz stable à partir d’un nombre fini d’observations de type Neumann. / This thesis, is devoted to the study of inverse problems related to the Schrödinger equation. The first partof the thesis is devoted to study the boundary inverse problem of determining the alignedmagnetic field appearing in the magnetic Schrödinger equation in a periodic quantum cylindricalwaveguide. From the Dirichlet-to-Neumann map of the magnetic Schrödinger equation,we prove a Hölder stability estimate with respect to the Dirichlet-to-Neumann map, by meansof the geometrical optics solutions of the magnetic Schrödinger equation.The second part of this thesis deals with the inverse problem of determining the magnetic field and the electricpotential appearing in the magnetic Schrödinger equation, from the knowledge of a finitenumber of lateral observations of the solution.

Made available in DSpace on 2015-05-15T11:46:17Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1816441 bytes, checksum: 97d911fa4449e54f12d9f203b8b50c90 (MD5) Previous issue date: 2013-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study qualitative properties of solutions of the semilinear elliptic equation class 8<: u + f(u) = 0, em , u = 0, em @ , defined in different kinds of unbounded domains of Rn, among them, infinite cylinders, half spaces and Lipschitz domains. We analyze properties like convergence, monotonocity and symmetry of solutions of the problem (1), when f satisfy certain conditions suitable. For this purpose, we will use various kinds of maximum principles, the moving planes method,elliptic estimates and compacity theorems. We also studied some results about Schrödinger operators and we prove the De Giorge conjecture in dimension n = 2. / Neste trabalho, estudamos propriedades qualitativas de soluções da seguinte classe de equações elípticas semilineares 8<:u + f(u) = 0, em u = 0, em @ , (1) definidas em vários tipos de domínios não limitados do Rn, dentre eles, cilíndros infinitos, semi espaços e domínios Lipschitzianos. Analisamos propriedades de convergência, monotonicidade e simetria de soluções de (1), quando f satisfaz certas condições adequadas. Para tanto, utilizaremos várias versões do princípio do máximo, o método dos planos móveis (moving planes), estimativas elípticas e teoremas de compacidade. Estudamos ainda resultados sobre operadores de Schrödinger e, como consequência, provamos a conjectura de De Giorgi em dimensão n = 2.