In this dissertation, we establish existence and multiplicity of positive solutions for semilinear elliptic equations with subcritical and critical nonlinearities. We treat problems invariant under subgroups of the orthogonal group. Roughly speaking, we prove that if enough "mass " is concentrated around special orbits, then among the functions with prescribed symmetry, there is a solution for the original problem. Our results can be regarded as a further development of the work of Z.-Q. Wang, where existence of local minima in the space of symmetric functions was studied for the Schrödinger equation. We illustrate the general theory with three examples, all of which produce new results. Our method allows the construction of solutions with prescribed symmetry, and it represents a step further in the classification of positive solutions for certain nonlinear elliptic problems.

Cette thèse est consacrée à l’étude et à la discussion de deux questions importantes qui se posent dans le domaine de la Programmation Mathématique: les symétries et les distances. En arrière-plan, nous examinons la Programmation Semidéfinie (PSD) et sa pertinence comme l’un des principaux outils employés aujourd’hui pour résoudre les Programmes Mathématiques (PM) durs. Après le chapitre introductif, nous discutons des symétries au Chapitre 2 et des distances au Chapitre 5. Entre ces deux chapitres, nous présentons deux courts chapitres que nous préférons en fait appeler entr’actes: leur contenu ne mérite pas d’être publié pour le moment, mais ils fournissent un lien entre les deux Chapitres 2 et 5 apparemment distincts, qui contiennent les principales contributions de cette thèse. Il est bien connu que les PMs symétriques sont plus difficiles à résoudre pour l’optimalité globale en utilisant des algorithmes du type Branch-and-Bound (B&B). Il est également bien connu que certaines des symétries de solution sont évidentes dans la formulation, ce qui permet d’essayer de traiter les symétries en tant qu’étape de prétraitement. L’une des approches les plus simples consiste à rompre les symétries en associant les Contraintes de Rupture de Symétrie (CRS) à la formulation, en supprimant ainsi des optima globaux symétriques, puis à résoudre la reformulation avec un solveur générique. Ces contraintes peuvent être générés à partir de chaque orbite de l’action des symétries sur l’ensemble des indices des variables. Cependant, il est difficile de savoir si et comment il est possible de choisir deux ou plus orbites distinctes pour générer des CRSs qui sont compatibles les unes avec les autres (elles ne rendent pas tous les optima globaux infaisables). Dans le Chapitre 2, nous discutons et testons un nouveau concept d’Indépendance Orbitale (IO) qui clarifie cette question. Les expériences numériques réalisées à l’aide de PLMEs et de PNLMEs soulignent l’exactitude et l’utilité de la théorie de l’IO. Programmation Quadratique Binaire (PQB) est utilisée pour étudier les symétries et SDP dans Entr'acte 3. Programmes quadratiques binaires symétriques ayant une certaine structure de symétrie sont générés et utilisés pour illustrer les conditions dans lesquelles l'utilisation de CRSs est avantageuse. Une discussion préliminaire sur l'impact des symétries et des CRSs dans la performance des solveurs PSD est également réalisée. Le Problème Euclidien de l'Arbre de Steiner est étudié dans Entr'acte 4. Deux modèles sont dérivés, ainsi que des relaxations SDP. Un algorithme heuristique basé à la fois sur les modèles mathématiques et sur les principes d'IO présentés au Chapitre 2 est également proposé. Concernant ces méthodes, des résultats préliminaires sur un petit ensemble d'exemples bien connus sont fournis. Finalement, dans le Chapitre 5, nous abordons le problème fondamental qui se pose dans le domaine de la Géométrie de Distance: il s’agit de trouver une réalisation d’un graphe pondéré non orienté dans RK pour un certain K donné, de sorte que les positions pour les sommets adjacents respectent la distance donnée par le poids de l’arête correspondante. Le Problème de la Géométrie de Distance Euclidienne (PGDE) est d’une grande importance car il a de nombreuses applications en science et en ingénierie. Il est difficile de calculer numériquement des solutions, et la plupart des méthodes proposées jusqu’à présent ne sont pas adaptées à des tailles utiles ou sont peu susceptibles d’identifier de bonnes solutions. La nécessité de contraindre le rang de la matrice représentant des solutions réalisables du PGDE rend le problème si difficile. Nous proposons un algorithme heuristique en deux étapes basé sur la PSD (en fait basé sur le paradigme de la PDD) et la modélisation explicite de Contraintes de Rang. Nous fournissons tests informatiques comprenant des instances générées de façon aléatoire ainsi que des exemples réalistes de conformation de protéines. / This thesis is mostly dedicated to study and discuss two important challenges existing not only in the field of Mathematical Programming: symmetries and distances. In the background we take a look into Semidefinite Programming (SDP) and its pertinency as one of the major tools employed nowadays to solve hard Mathematical Programs (MP). After the introductory Chapter 1, we discuss about symmetries in Chapter 2 and about distances in Chapter 5. In between them we present two short chapters that we actually prefer to call as entr’actes: their content is not necessarily worthy of publication yet, but they do provide a connection between the two seemingly separate Chapters 2 and 5, which are the ones containing the main contributions of this thesis. It is widely known that symmetric MPs are harder to solve to global optimality using Branch-and-Bound (B&B) type algorithms, given that the solution symmetry is reflected in the size of the B&B tree. It is also well-known that some of the solution symmetries are usually evident in the formulation, which makes it possible to attempt to deal with symmetries as a preprocessing step. Implementation-wise, one of the simplest approaches is to break symmetries by adjoining Symmetry-Breaking Constraints (SBC) to the formulation, thereby removing some symmetric global optima, then solve the reformulation with a generic solver. Sets of such constraints can be generated from each orbit of the action of the symmetries on the variable index set. It is unclear, however, whether and how it is possible to choose two or more separate orbits to generate SBCs which are compatible with each other (in the sense that they do not make all global optima infeasible). In Chapter 2 we discuss and test a new concept of Orbital Independence (OI) that clarifies this issue. The numerical experiences conducted using public MILPs and MINLPs emphasize the correctness and usefulness of the OI theory. Binary Quadratic Programming (BQP) is used to investigate symmetries and SDP in Entr'acte 3. Symmetric Binary Quadratic Programs having a certain symmetry structure are generated and used to exemplify the conditions under which the usage of SBCs is majoritarily advantageous. A preliminary discussion about the impact of symmetries and SBCs in the performance of SDP solvers is also carried out. The Euclidean Steiner Tree Problem is studied in Entr'acte 4. Two models (which are exact reformulations of an existing formulation) are derived, as well as SDP relaxations. A heuristic algorithm based on both the mathematical models and the OI principles presented in Chapter 2 is also proposed. As concerns these methods, preliminary results on a small set of well-known instances are provided. Finally and following up the Distance Geometry subject, in Chapter 5 we cope with the most fundamental problem arising in the field of Distance Geometry, the one of realizing graphs in Euclidean spaces: it asks to find a realization of an edge-weighted undirected graph in RK for some given K such that the positions for adjacent vertices respect the distance given by the corresponding edge weight. The Euclidean Distance Geometry Problem (EDGP) is of great importance since it has many applications to science and engineering. It is notoriously difficult to solve computationally, and most of the methods proposed so far either do not scale up to useful sizes, or unlikely identify good solutions. In fact, the need to constrain the rank of the matrix representing feasible solutions of the EDGP is what makes the problem so hard. Intending to overcome these issues, we propose a two-steps heuristic algorithm based on SDP (or more precisely based on the very recent Diagonally Dominant Programming paradigm) and the explicitly modeling of Rank Constraints. We provide extensive computational testing against randomly generated instances as well as against feasible realistic protein conformation instances taken from the Protein Data Bank to analyze our method.

Student Number : 9202983Y - PhD thesis - School of Computational and Applied Mathematics - Faculty of Science / A derivation is given of the constitutive equation for an incompressible transversely isotropic hyperelastic material in which the direction of the anisotropic director is unspecified. The field equations for a transversely isotropic incompressible hyperelastic material are obtained. Nonlinear radial oscillations in transversely isotropic incompressible cylindrical tubes are investigated. A second order nonlinear ordinary differential equation, expressed in terms of the strain-energy function, is derived. It has the same form as for radial oscillations in an isotropic tube. A generalised Mooney-Rivlin strainenergy function is used. Radial oscillations with a time dependent net applied surface pressure are first considered. For a radial transversely isotropic thin-walled tube the differential equation has a Lie point symmetry for a special form of the strain-energy function and a special time dependent applied surface pressure. The Lie point symmetry is used to transform the equation to an autonomous differential equation which is reduced to an Abel equation of the second kind. A similar analysis is done for radial oscillations in a tangential transversely isotropic tube but computer graphs show that the solution is unstable. Radial oscillations in a longitudinal transversely isotropic tube and an isotropic tube are the same. The Ermakov-Pinney equation is derived. Radial oscillations in thick-walled and thin-walled cylindrical tubes with the Heaviside step loading boundary condition are next investigated. For radial, tangential and longitudinal transversely isotropic tubes a first integral is derived and effective potentials are defined. Using the effective potentials, conditions for bounded oscillations and the end points of the oscillations are obtained. Upper and lower bounds on the period are derived. Anisotropy reduces the amplitude of the oscillation making the tube stiffer and reduces the period. Thirdly, free radial oscillations in a thin-walled cylindrical tube are investigated. Knowles(1960) has shown that for free radial oscillations in an isotropic tube, ab = 1 where a and b are the minimum and maximum values of the radial coordinate. It is shown that if the initial velocity v0 vanishes or if v0 6= 1 but second order terms in the anisotropy are neglected then for free radial oscillations, ab > 1 in a radial transversely isotropic tube and ab < 1 in a tangential transversely isotropic tube. Radial oscillations in transversely isotropic incompressible spherical shells are investigated. Only radial transversely isotropic shells are considered because it is found that the Cauchy stress tensor is not bounded everywhere in tangential and longitudinal transversely isotropic shells. For a thin-walled radial transversely isotropic spherical shell with generalised Mooney-Rivlin strain-energy function the differential equation for radial oscillations has no Lie point symmetries if the net applied surface pressure is time dependent. The inflation of a thin-walled radial transversely isotropic spherical shell of generalised Mooney-Rivlin material is considered. It is assumed that the inflation proceeds sufficiently slowly that the inertia term in the equation for radial oscillations can be neglected. The conditions for snap buckling to occur, in which the pressure decreases before steadily increasing again, are investigated. The maximum value of the parameter for snap buckling to occur is increased by the anisotropy.

Na análise global e local de sistemas dinâmicos assumimos, em geral, que as equações estão numa forma normal. Em presença de simetrias, as equações e o domínio do problema são invariantes pelo grupo formado por estas simetrias; neste caso, o campo de vetores é equivariante pela ação deste grupo. Quando, além das simetrias, temos também ocorrência de anti-simetrias - ou reversibilidades - as equações e o domínio do problema são ainda invariantes pelo grupo formado pelo conjunto de todas as simetrias e anti-simetrias; neste caso, o campo de vetores é reversível-equivariante. Existem muitos modelos físicos onde simetrias e anti-simetrias aparecem naturalmente e cujo efeito pode ser estudado de uma forma sistemática através de teoria de representação de grupos de Lie. O primeiro passo deste processo é colocar a aplicação que modela tal sistema numa forma normal e isto é feito com a dedução a priori da forma geral dos campos de vetores. Esta forma geral depende de dois componentes: da base de Hilbert do anel das funções invariantes e dos geradores do módulo das aplicações reversíveis-equivariantes. Neste projeto, nos concentramos principalmente na aplicação de resultados recentes da literatura para a construção de uma lista de formas gerais de aplicações reversíveisequivariantes sob a ação de diferentes grupos. Além disso, adaptamos ferramentas algébricas da literatura existentes no contexto equivariante para o estudo sistemático de acoplamento de células idênticas no contexto reversível-equivariante / In the global and local analysis of dynamical systems, we assume, in general, that the equations are in a normal form. In presence of symmetries, the equations and the problem domain are invariant under the group formed by these symmetries; in that case, the vector field is equivariant by the action of this group. When, in addition to the symmetries, we have the occurrence of anti-symmetries - or reversibility - the equations and the problem domain are still invariant by the group formed by the set of all symmetries and anti-symmetries; in this case, the vector field is reversible-equivariant. There are many physical models where both symmetries and anti-symmetries occur naturally and whose effect can be studied in a systematic way through group representation theory. The first step of this process is to put the mapping that model the system in a normal form, and this is done with the deduction of the general form of the vector field. This general form depends on two components: the Hilbert basis of the invariant function ring and also the generators of the module of the revesible-equivariants. In this work, we mainly focus on the applications of recent results of the literature to build a list of general forms of reversible-equivariant mappings under the action of different groups. We also adapt algebraic tools of the existing literature in the equivariant context to the systematic study of coupling of identical cells in the reversible-equivariant context

A proposta deste trabalho é apresentar resultados para o estudo sistemático de sistemas dinâmicos reversíveis-equivariantes, ou seja, em presença simultânea de simetrias e antisimetrias. Este é o caso em que o domínio e as equações que regem o sistema são invariantes pela ação de um grupo de Lie compacto &#915; formado pelas simetrias e anti-simetrias do problema. Apresentamos métodos de teoria de Singularidades e teoria de invariantes para classificar bifurcações a um parâmetro de pontos de equilíbrio destes sistemas. Para isso, separamos o estudo de aplicações &#915;-reversíveis-equivariantes em dois casos: auto-dual e não auto-dual. No primeiro caso, a existência de um isomorfismo linear &#915;-reversível-equivariante estabelece uma correspondência entre a classificação de problemas &#915;-reversíveis-equivariantes e a classificação de problemas &#915;-equivariantes associados, para os quais todos os elementos de &#915; agem como simetria. Os resultados obtidos para o caso não auto-dual se baseiam em teoria de invariantes e envolvem técnicas algébricas que reduzem a análise ao caso polinomial invariante. Dois algoritmos simbólicos são estabelecidos para o cálculo de geradores para o módulo das funções anti-invariantes e para o módulo das aplicações reversíveis-equivariantes. / The purpose of this work is to present results for the sistematic study of reversible-equivariant dynamical systems, namely in simultaneous presence of symmetries and reversing simmetries. This is the case when the domain and the equations modeling the system are invariant under the action of a compact Lie group &#915; formed by the symmetries and reversing symmetries of the problem. We present methods in Singularities and Invariant theory to classify oneparameter steady-state bifurcations of these systems. For that, we split the study of the ¡¡reversible-equivariant mapping into two cases: self-dual and non self-dual. In the first case, the existence of a &#915;-reversible-equivariant linear isomorphism establishes a one-toone correspondence between the classification of &#915;-reversible-equivariant problems and the classification of the associated &#915;-equivariant problems, for which all elements in &#915; act as symmetries. The results obtained for the non self-dual case are based on Invariant theory and involve algebraic techniques that reduce the analysis to the invariant polynomial case. Two symbolic algorithms are established for the computation of generators for the module of anti-invariant functions and for the module of reversible-equivariant mappings.

Na análise global e local de sistemas dinâmicos assumimos, em geral, que as equações estão numa forma normal. Em presença de simetrias, as equações e o domínio do problema são invariantes pelo grupo formado por estas simetrias; neste caso, o campo de vetores é equivariante pela ação deste grupo. Quando, além das simetrias, temos também ocorrência de anti-simetrias - ou reversibilidades - as equações e o domínio do problema são ainda invariantes pelo grupo formado pelo conjunto de todas as simetrias e anti-simetrias; neste caso, o campo de vetores é reversível-equivariante. Existem muitos modelos físicos onde simetrias e anti-simetrias aparecem naturalmente e cujo efeito pode ser estudado de uma forma sistemática através de teoria de representação de grupos de Lie. O primeiro passo deste processo é colocar a aplicação que modela tal sistema numa forma normal e isto é feito com a dedução a priori da forma geral dos campos de vetores. Esta forma geral depende de dois componentes: da base de Hilbert do anel das funções invariantes e dos geradores do módulo das aplicações reversíveis-equivariantes. Neste projeto, nos concentramos principalmente na aplicação de resultados recentes da literatura para a construção de uma lista de formas gerais de aplicações reversíveisequivariantes sob a ação de diferentes grupos. Além disso, adaptamos ferramentas algébricas da literatura existentes no contexto equivariante para o estudo sistemático de acoplamento de células idênticas no contexto reversível-equivariante / In the global and local analysis of dynamical systems, we assume, in general, that the equations are in a normal form. In presence of symmetries, the equations and the problem domain are invariant under the group formed by these symmetries; in that case, the vector field is equivariant by the action of this group. When, in addition to the symmetries, we have the occurrence of anti-symmetries - or reversibility - the equations and the problem domain are still invariant by the group formed by the set of all symmetries and anti-symmetries; in this case, the vector field is reversible-equivariant. There are many physical models where both symmetries and anti-symmetries occur naturally and whose effect can be studied in a systematic way through group representation theory. The first step of this process is to put the mapping that model the system in a normal form, and this is done with the deduction of the general form of the vector field. This general form depends on two components: the Hilbert basis of the invariant function ring and also the generators of the module of the revesible-equivariants. In this work, we mainly focus on the applications of recent results of the literature to build a list of general forms of reversible-equivariant mappings under the action of different groups. We also adapt algebraic tools of the existing literature in the equivariant context to the systematic study of coupling of identical cells in the reversible-equivariant context

A proposta deste trabalho é apresentar resultados para o estudo sistemático de sistemas dinâmicos reversíveis-equivariantes, ou seja, em presença simultânea de simetrias e antisimetrias. Este é o caso em que o domínio e as equações que regem o sistema são invariantes pela ação de um grupo de Lie compacto &#915; formado pelas simetrias e anti-simetrias do problema. Apresentamos métodos de teoria de Singularidades e teoria de invariantes para classificar bifurcações a um parâmetro de pontos de equilíbrio destes sistemas. Para isso, separamos o estudo de aplicações &#915;-reversíveis-equivariantes em dois casos: auto-dual e não auto-dual. No primeiro caso, a existência de um isomorfismo linear &#915;-reversível-equivariante estabelece uma correspondência entre a classificação de problemas &#915;-reversíveis-equivariantes e a classificação de problemas &#915;-equivariantes associados, para os quais todos os elementos de &#915; agem como simetria. Os resultados obtidos para o caso não auto-dual se baseiam em teoria de invariantes e envolvem técnicas algébricas que reduzem a análise ao caso polinomial invariante. Dois algoritmos simbólicos são estabelecidos para o cálculo de geradores para o módulo das funções anti-invariantes e para o módulo das aplicações reversíveis-equivariantes. / The purpose of this work is to present results for the sistematic study of reversible-equivariant dynamical systems, namely in simultaneous presence of symmetries and reversing simmetries. This is the case when the domain and the equations modeling the system are invariant under the action of a compact Lie group &#915; formed by the symmetries and reversing symmetries of the problem. We present methods in Singularities and Invariant theory to classify oneparameter steady-state bifurcations of these systems. For that, we split the study of the ¡¡reversible-equivariant mapping into two cases: self-dual and non self-dual. In the first case, the existence of a &#915;-reversible-equivariant linear isomorphism establishes a one-toone correspondence between the classification of &#915;-reversible-equivariant problems and the classification of the associated &#915;-equivariant problems, for which all elements in &#915; act as symmetries. The results obtained for the non self-dual case are based on Invariant theory and involve algebraic techniques that reduce the analysis to the invariant polynomial case. Two symbolic algorithms are established for the computation of generators for the module of anti-invariant functions and for the module of reversible-equivariant mappings.

In this thesis, the symmetry structure of gravitational theories at null infinity is studied further, in the case of pure gravity in four dimensions and also in the case of Einstein-Yang-Mills theory in d dimensions with and without a cosmological constant.<p><p>The first part of this thesis is devoted to the presentation of asymptotic methods (symmetries, solution space and surface charges) applied to gravity in the case of the BMS gauge in three and four spacetime dimensions.<p><p>The second part of this thesis contains the original contributions.<p>Firstly, it is shown that the enhancement from Lorentz to Virasoro algebra also occurs for asymptotically flat spacetimes defined in the sense of Newman-Unti. As a first application, the transformation laws of the Newman-Penrose coefficients characterizing solution space of the Newman-Unti approach are worked out, focusing on the inhomogeneous terms that contain the information about central extensions of the theory. These transformations laws make the conformal structure particularly transparent, and constitute the main original result of the thesis.<p>Secondly, asymptotic symmetries of the Einstein-Yang-Mills system with or without cosmological constant are explicitly worked out in a unified manner in $d$ dimensions. In agreement with a recent conjecture, a Virasoro-Kac-Moody type algebra is found not only in three dimensions but also in the four dimensional asymptotically flat case.<p><p>These two parts of the thesis are supplemented by appendices. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The term spatiotemporal chaos refers to physical phenomena that exhibit irregular oscillations in both space and time. Examples of such phenomena range from cardiac dynamics to fluid turbulence, where the motion is described by nonlinear partial differential equations. It is well known from the studies of low dimensional chaotic systems that the state space, the space of solutions to the governing dynamical equations, is shaped by the invariant sets such as equilibria, periodic orbits, and invariant tori. State space of partial differential equations is infinite dimensional, nevertheless, recent computational advancements allow us to find their invariant solutions (exact coherent structures) numerically. In this thesis, we try to elucidate the chaotic dynamics of nonlinear partial differential equations by studying their exact coherent structures and invariant manifolds. Specifically, we investigate the Kuramoto-Sivashinsky equation, which describes the velocity of a flame front, and the Navier-Stokes equation for an incompressible fluid in a circular pipe. We argue with examples that this approach can lead to a theory of turbulence with predictive power.

This thesis primarily concerns kinematic dynamo action by the 1:1:1 ABC flow, in the highly conducting limit of large magnetic Reynolds number Rm. The flow possesses 24 symmetries, with a symmetry group isomorphic to the group O24 of orientation-preserving transformations of a cube. These symmetries are exploited to break up the linear eigenvalue problem into five distinct symmetry classes, which we label I-V. The thesis discusses how to reduce the scale of the numerical problem to a subset of Fourier modes for a magnetic field in each class, which then may be solved independently to obtain distinct branches of eigenvalues and magnetic field eigenfunctions. Two numerical methods are employed: the first is to time step a magnetic field in a given symmetry class and obtain the growth rate and frequency by measuring the magnetic energy as a function of time. The second method involves a more direct determination of the eigenvalue using the eigenvalue solver ARPACK for sparse matrix systems, which employs an implicitly restarted Arnoldi method. The two methods are checked against each other, and compared for efficiency and reliability. Eigenvalue branches for each symmetry class are obtained for magnetic Reynolds numbers Rm up to 10^4 together with spectra and magnetic field visualisations. A sequence of branches emerges as Rm increases and the magnetic field structures in the different branches are discussed and compared. All symmetry classes are found to contain a dynamo, though dynamo effectiveness varies greatly between classes, suggesting that the symmetries play an important role in the field amplification mechanisms. A closely related problem, that of linear hydrodynamic stability, is also explored in the limit of large Reynolds number Re. As the same symmetry considerations apply, the five symmetry classes of the linear instability can be resolved independently, reducing the size of the problem and allowing exploration of the effects of the symmetries on instability growth rate. Results and visualisations are obtained for all five classes for Re up to 10^3, with comparisons drawn between the structures seen in each class and with those found in the analogous magnetic problem. For increasing Re, multiple mode crossings are observed within each class, with remarkably similar growth rates seen in all classes at Re=10^3, highlighting a lack of dependence on the symmetries of the instability, in contrast with the magnetic problem. This thesis also investigates the problem of large-scale magnetic fields in the 1:1:1 ABC flow through the introduction of Bloch waves that modify the periodicity of the magnetic field relative to the flow. Results are found for a field with increased periodicity in a single direction for Rm up to 10^3; it is established that the optimal scale for dynamo action varies as Rm increases, settling on a consistent scale for large Rm. The emerging field structures are studied and linked with those of the original dynamo problem. On contrasting this method with a previous study in which the flow is instead rescaled, it is shown that the use of Bloch waves drastically increases the range of possible scales, whilst cutting required computing time. Through a multiple-scale analysis, the contribution from the alpha-effect is calculated for the 1:1:1 ABC flow and is seen in growth rates for Rm &lt;&lt; 1.