Dissertations / Theses on the topic 'Nonsmooth critical point theory'

We consider the eigenvalue problem associated to the 1-Laplace operator. We define higher eigensolutions by means of weak slope and establish existence of a sequence of eigensolutions by using nonsmooth critical point theory. In addition, we deduce a new necessary condition for the first eigenvalue of the 1-Laplace operator by means of inner variations
Wir betrachten das zum 1-Laplace-Operator gehörige Eigenwertproblem. Wir definieren höhere Eigenlösungen mittels weak slope und weisen die Existenz einer Folge von Eigenlösungen nach, indem wir die nichtglatte Theorie kritischer Punkte anwenden. Zusätzlich leiten wir eine neue notwendige Bedingung für den ersten Eigenwert des 1-Laplace-Operators mittels innerer Variationen her

2012/2013
We investigate by different techniques, the solvability of a class of capillarity-type problems, in a bounded N-dimensional domain. Since our approach is variational, the natural context where this problem has to be settled is the space of bounded variation functions. Solutions of our equation are defined as subcritical points of the associated action functional.
We first introduce a lower and upper solution method in the space of bounded variation functions. We prove the existence of solutions in the case where the lower solution is smaller than the upper solution. A solution, bracketed by the given lower and upper solutions, is obtained as a local minimizer of the associated functional without any assumption on the boundedness of the right-hand side of the equation. In this context we also prove order stability results for the minimum and the maximum solution lying between the given lower and upper solutions. Next we develop an asymmetric version of the Poincaré inequality in the space of bounded variation functions. Several properties of the curve C are then derived and basically relying on these results, we discuss the solvability of the capillarity-type problem, assuming a suitable control on the interaction of the supremum and the infimum of the function at the right-hand side with the curve C. Non-existence and multiplicity results are investigated as well. The one-dimensional case, which sometimes presents a different behaviour, is also discussed. In particular, we provide an existence result which recovers the case of non-ordered lower and upper solutions.
XXV Ciclo
1985

As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered.

This study was motivated by the observation that most smooth bundles do not admit a smooth function that is Morse when restricted to every fibre. The complexity c of a critical point of a smooth map is measured by an appropriate codimension of its germ. The subset of smooth maps from a bundle to a manifold with complexity on fibres not exceeding c is studied. Bounds for c are established such that this subset is open and dense in the set of all smooth maps, where sets of smooth maps are always given the Whitney C^∞ topology. The bounds are calculated in terms of the dimensions of the base space, the fibre and the manifold into which the bundle is mapped and are proved using the theory of finite germs and a suitable adaptation of the Thom Transversality Theorem. Recent work of Vasil'ev is used to investigate real-valued functions on compact principal S¹-bundles. The existence is established of a function with complexity on fibres no more than roughly half of the minimum value for c for the open and dense subsets mentioned above. For certain bundles with fibre of dimension one, the set of smooth real-valued functions that are Morse when restricted to every fibre is shown to be C⁰ dense but not, in general, C¹ dense. For all n-sphere bundles over the circle the set is shown to be C⁰ dense. The homotopy type of the space of smooth Morse functions on the circle is derived. Arnold's determination of the fundamental group of the generalised Morse functions on the circle is included.

Strongly correlated metals are known to give rise to a variety of exotic states. In particular, if a system is tuned towards a quantum critical point, new ordered phases may arise. Sr₃Ru₂O₇ is a quasi-two dimensional metal in which field-tuned quantum criticality has been observed. In very pure single crystals of this material, a phase with unusual transport properties forms in the vicinity of its quantum critical point. Upon the application of a small in-plane field, electrical resistivity becomes anisotropic, a phenomenon which has led to the naming of this phase as an `electron nematic'. The subject of this thesis is a study of the electrical transport in high purity crystals of Sr₃Ru₂O₇. We modified an adiabatic demagnetisation refrigerator to create the conditions by which the entire temperature-field phase diagram can be explored. In particular, this allowed us to access the crossover between the low-temperature Fermi liquid and the quantum critical region. We also installed a triple axis `vector magnet' with which the applied magnetic field vector can be continuously rotated within the anisotropic phase. We conclude that the low- and high-field Fermi liquid properties have a complex dependence on magnetic field and temperature, but that a simple multiple band model can account for some of these effects, and reconcile the measured specific heat, dHvA quasiparticle masses and transport co-efficients. At high temperatures, we observe similarities between the apparent resistive scattering rate at critical tuning and those observed in other quantum critical systems and in elemental metals. Finally, the anisotropic phase measurements confirm previous reports and demonstrate behaviour consistent with an Ising-nematic, with the anisotropy aligned along either of the principal crystal axes. Our observations are consistent with the presence of a large number of domains within the anisotropic phase, and conclude that scattering from domain walls is likely to contribute strongly to the large measured anisotropy.

Scopo di questa tesi è mostrare i risultati ottenuti per tre equazioni differenziali alle derivate parziali ellittiche di tipo Schrödinger. Queste equazioni, sebbene condividano la particolarità di essere definite in tutto lo spazio, sono state affrontate con diversi metodi, e per ognuna è stato fornito un risultato di esistenza di soluzioni. Rimarchiamo che l'ultima equazione ha un forte legame con le equazioni di Maxwell. Problema 1) Consideriamo un'equazione di tipo Schrödinger, con potenziale di tipo convolutivo ed una nonlinearità perturbata e pesata derivante dalla fisica quantistica con gravitazione Newtoniana. Se consideriamo questa equazione definita in tutto lo spazio R2, otterremo un potenziale di tipo logaritmico, che rende l'analisi più delicata, in quanto il funzionale associato non è ben definito. Va dunque introdotto un opportuno setting variazionale per dimostrare la buona positura del problema. Successivamente, per gestire la perturbazione utilizziamo la tecnica perturbativa della teoria dei punti critici. Assumendo opportune ipotesi sulla funzione peso, si dimostra l'esistenza di soluzioni locali e globali. Problema 2) La seconda equazione è un'equazione di tipo Choquard governata da un operatore semirelativistico di Schrödinger, dove il potenziale ha una parte singolare ed è presente una nonlinearità generale, definita in tutto lo spazio RN. Grazie alla rappresentazione mediante trasformata di Fourier dell'operatore semirelativistico, si può dimostrare che la norma generata dalla forma quadratica associata al problema è equivalente alla norma standard. Grazie ad un risultato astratto, si prova prima l'esistenza di una successione di Cerami e successivamente la sua limitatezza. Adattando un argomento di decomposizione per successioni di Palais-Smale, si dimostra poi la convergenza di questa successione ad un punto critico non banale. Infine, viene fornita una quasi-caratterizzazione per l'esistenza di soluzioni di tipo ground-state (i.e. soluzioni corrispondenti al livello di energia minima del sistema). Per queste soluzioni, è fornito anche un risultato di compattezza rispetto al termine singolare. Problema 3) Viene fornito un Teorema astratto di tipo linking infinito-dimensionale che permette lo studio di problemi fortemente indefiniti (cioè il punto origine appartiene ad un gap spettrale dell'operatore) e con nonlinearità generali di segno variabile. Come applicazione, questo Teorema viene applicato ad un'equazione di tipo Schrödinger fortemente indefinita con potenziale singolare e nonlinearità a segno variabile, definita in tutto lo spazio RN. Per questa equazione viene dimostrata l'esistenza di una soluzione non banale. Sfruttando un risultato di equivalenza, viene fornita l'esistenza di una soluzione nonbanale anche per un'equazione di tipo curl-curl: questo tipo di equazioni sono strettamente legate alle equazioni di Maxwell.
The purpose of this thesis is to show the results obtained for three Schrödinger type elliptic partial differential equations. These equations, although sharing the feature of being entire, i.e. defined in the whole the space, have been approached with different methods, and for each a result of the existence of solutions has been provided. We emphasize that the last equation has a strong connection with Maxwell's equations. Problem 1) Let us consider a Schrödinger type equation, with convolutive potential and a perturbed and weighted nonlinearity copuling from quantum physics with Newtonian gravitation. If we consider this equation defined in all the space R2, we will obtain a logarithmic type potential, which makes the analysis more delicate, since the associated functional is not well defined. Therefore, a suitable variational setting must be introduced to show the well-posedness of the problem. Next, to manage the perturbation we use the perturbation technique of the critical point theory. Assuming suitable hypotheses on the weight function, the existence of local and global solutions is proved. Problem 2) The second equation is a Choquard type equation driven by a semirelativistic Schrödinger operator, defined in the whole space RN, where the potential has a singular part and a general nonlinearity is considered. Using the Fourier transform representation of the semirelativistic operator, it can be shown that the norm generated by the quadratic form associated with the problem is equivalent to the standard one. Thanks to an abstract result, we first prove the existence of a Cerami-sequence and then its boundedness. By adapting a Palais-Smale-sequence decomposition argument, the strong convergence of this sequence to a non-trivial critical point is then showed. Finally, an almost-characterization criterion is provided for the existence of ground-state solutions (i.e. solutions corresponding to the minimum energy level of the system). For these solutions, a compactness result is also given with respect to the singular term. Problem 3) An abstract infinite-dimensional linking-type Theorem is provided, which allows the study of strongly indefinite problems (i.e. the origin belongs to a spectral gap of the operator) and with general sign-changing nonlinearities. As an application, this Theorem is applied to a strongly indefinite Schrödinger equation with singular potential and sign-changing nonlinearity, defined in the whole space RN. For this equation the existence of a non-trivial solution is proved. By exploiting an equivalence result, the existence of a non-trivial solution is also provided for a curl-curl equation: this type of equations are closely related to Maxwell's equations.

Under adequate conditions on $g$, we show the density in $L^2((0,\pi),(0,2\pi))$ of the set of functions $p$ for which \begin{equation*} u_{tt}(x,t)-u_{xx}(x,t)= g(u(x,t)) + p(x,t) \end{equation*} has a weak solution subject to \begin{equation*} \begin{aligned} u(x,t)&=u(x,t+2\pi)\\ u(0,t)&=u(\pi,t)=0. \end{aligned} \end{equation*} To achieve this, we prove a Saddle Point Principle by means of a refined variant of the deformation lemma of Rabinowitz. Generally, inf-sup techniques allow the characterization of critical values by taking the minimum of the maximae on some particular class of sets. In this version of the Saddle Point Principle, we introduce sufficient conditions for the existence of a saddle-structure which is not restricted to finite-dimensional subspaces.

Abstract The dissertation considers a degree theory and the index of a critical point of demi-continuous, everywhere defined mappings of the monotone type. A topological degree is derived for mappings from a Banach space to its dual space. The mappings satisfy the condition (S+), and it is shown that the derived degree has the classical properties of a degree function. A formula for the calculation of the index of a critical point of a mapping A : X→X* satisfying the condition (S+) is derived without the separability of X and the boundedness of A. For the calculation of the index, we need an everywhere defined linear mapping A' : X→X* that approximates A in a certain set. As in the earlier results, A' is quasi-monotone, but our situation differs from the earlier results because A' does not have to be the Frechet or Gateaux derivative of A at the critical point. The theorem for the calculation of the index requires a construction of a compact operator T = (A' + Γ)-1Γ with the aid of linear mappings Γ : X→X and A'. In earlier results, Γ is compact, but here it need only be quasi-monotone. Two counter-examples show that certain assumptions are essential for the calculation of the index of a critical point.

L'obiettivo di questa tesi è di studiare alcuni aspetti di un potente strumento ampiamente utilizzato in analisi matematica, che è rappresentato dalle stime a priori. Infatti, le stime a priori hanno un ruolo chiave nella teoria delle equazioni differenziali a derivate parziali e nel calcolo delle variazioni, perché sono intimamente legate all'esistenza di soluzione per un dato problema. Nella tesi vengono presentati tre lavori scritti durante il periodo del dottorato, in ciascuno dei quali vengono utilizzate le stime a priori. Il primo lavoro, scritto in collaborazione con il Prof. S. Mosconi, riguarda l'esistenza di soluzione per la seguente equazione differenziale ordinaria del quarto ordine (equazione di Swift-Hohenberg), $ u''''+ qu''+ F'(u)= 0$, dove $q$ è un parametro reale e $F$ è una funzione $C^2$, coerciva e quasi-convessa. Il secondo lavoro, scritto in collaborazione con il prof. P. Winkert, riguarda stime a priori per un problema ellittico in cui gli operatori hanno crescita critica, sia nel dominio che sulla frontiera. Il terzo lavoro, scritto in collaborazione con i Prof. S.A. Marano e A. Moussaoui, riguarda l'esistenza di soluzione per un sistema ellittico definito in tutto lo spazio $\R^N$, in cui le nonlinearità contengono termini singolari, cioè che possono tendere a $+\infty$ quando la variabile tende a zero.

Orientador: Francisco Odair Vieira de Paiva
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Doutorado
Matematica
Doutor em Matemática

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CNPq
Capes
Neste trabalho usamos métodos variacionais para mostrar a existência de solução fraca para dois tipos de problema. O primeiro trata-se de uma Equação Diferencial Ordinária. O segundo é referente ao sistema Hamiltoniano. *Para Visualisar as equações ou formulas originalmente escritas neste resumo recomendamos o downloado do arquivo completo.
In this work we use variational methods to show the existence of weak solutions for two types problems. The first, is related with a following Ordinary Differential Equations. The second is relating at the Hamiltonian Systems. *To see the equations or formulas originally written in this summary we recommend downloading the complete file.

Dans cette thèse, nous étudions des problèmes aux limites impliquant le modèle micro-magnétique et les formes différentielles. Dans la première partie, nous considérons un modèle non-local de Ginzburg-Landau apparaissant en micromagnétisme avec une condition au bord de type Dirichlet. Le modèle typique implique une fonctionelle d'énergie définie pour des applications des valeurs dans la sphère S² et qui depend de plusieurs paramètres, qui représentent des quantités physiques. Une première question concerne la compacité des aimantations ayant les énergies de quelques parois de Néel de longueur finie et des défauts topologiques lorsque ces paramètres convergent vers 0. Notre méthode utilise des techniques développées pour les problèmes de type Ginzburg-Landau sur la concentration d'énergie autour des vortex, avec un argument d'approximation des champs de vecteurs dans S² par des champs de vecteurs dans S¹ éloignés des vortex. Nous effectuons également en détail la preuve de la régularité C^infini à l'intérieur et la régularité C(^1,alpha) au bord, pour tous les alpha appartiennent à (0, 1/2 ), des points critiques du modèle. Dans la deuxième partie, nous étudions le lemme de Poincaré qui affirme que sur un domaine simplement connexe chaque forme fermée est exacte. Nous prouvons le lemme de Poincaré sur un domaine avec une condition aux limites de Dirichlet sous une hypothèse naturelle sur la régularité du domaine : une forme fermée ƒ dans l'espace C(^r,alpha) est la différentielle d'une forme C(^r+1,alpha) à condition que le domaine lui-même soit C(^r+1,alpha). La preuve est basée sur une construction par approximation, avec un argument de dualité. Nous établissons également le résultat correspondant dans le cadre d'espaces de Sobolev d'ordre supérieur
In this thesis, we study some boundary value problems involving micromagnetic models and differential forms. In the first part, we consider a nonlocal Ginzburg-Landau model arising in micromagnetics with an imposed Dirichlet boundary condition. The model typically involves S²-valued maps with an energy functional depending on several parameters, which represent physical quantities. A first question concerns the compactness of magnetizations having the energies of several Néel walls of finite length and topo- logical defects when these parameters converge to 0. Our method uses techniques developed for Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-valued vector fields by S¹-valued vector fields away from the vortex balls. We also carry out in detail the proofs of the C^infinite regularity in the interior and C(^1,alpha) regularity up to the boundary, for all alpha belong to (0, 1/2), of critical points of the model. In the second part, we study the Poincaré lemma, which states that on a simply connected domain every closed form is exact. We prove the Poincaré lemma on a domain with a Dirichlet boundary condition under a natural assumption on the regularity of the domain: a closed form ƒ in the Hölder space C(^r,alpha) is the differential of a C(^r+1,alpha) form, provided that the domain itself is C(^r+1,alpha). The proof is based on a construction by approximation, together with a duality argument. We also establish the corresponding statement in the setting of higher order Sobolev spaces

Durante la última década y paralelamente al incremento de la velocidad de computación, las técnicas de simulación molecular se han erigido como una importante herramienta para la predicción de propiedades físicas de sistemas de interés industrial. Estas propiedades resultan esenciales en las industrias química y petroquímica a la hora de diseñar, optimizar, simular o controlar procesos. El actual coste moderado de computadoras potentes hace que la simulación molecular se convierta en una excelente opción para proporcionar predicciones de dichas propiedades. En particular, la capacidad predictiva de estas técnicas resulta muy importante cuando en los sistemas de interés toman parte compuestos tóxicos o condiciones extremas de temperatura o presión debido a la dificultad que entraña la experimentación a dichas condiciones. La simulación molecular proporciona una alternativa a los modelos termofísicos utilizados habitualmente en la industria como es el caso de las ecuaciones de estado, modelos de coeficientes de actividad o teorías de estados correspondientes, que resultan inadecuados al intentar reproducir propiedades complejas de fluidos como es el caso de las de fluidos que presentan enlaces de hidrógeno, polímeros, etc. En particular, los métodos de Monte Carlo (MC) constituyen, junto a la dinámica molecular, una de las técnicas de simulación molecular más adecuadas para el cálculo de propiedades termofísicas. Aunque, por contra del caso de la dinámica molecular, los métodos de Monte Carlo no proporcionan información acerca del proceso molecular o las trayectorias moleculares, éstos se centran en el estudio de propiedades de equilibrio y constituyen una herramienta, en general, más eficiente para el cálculo del equilibrio de fases o la consideración de sistemas que presenten elevados tiempos de relajación debido a su bajos coeficientes de difusión y altas viscosidades. Los objetivos de esta tesis se centran en el desarrollo y la mejora tanto de algoritmos de simulación como de potenciales intermoleculares, factor considerado clave para el desarrollo de las técnicas de simulación de Monte Carlo. En particular, en cuanto a los algoritmos de simulación, la localización de puntos críticos de una manera precisa ha constituido un problema para los métodos habitualmente utilizados en el cálculo de equlibrio de fases, como es el método del colectivo de GIBBS. La aparición de fuertes fluctuaciones de densidad en la región crítica hace imposible obtener datos de simulación en dicha región, debido al hecho de que las simulaciones son llevadas a cabo en una caja de simulación de longitud finita que es superada por la longitud de correlación. Con el fin de proporcionar una ruta adecuada para la localización de puntos críticos tanto de componentes puros como mezclas binarias, la primera parte de esta tesis está dedicada al desarrollo y aplicación de métodos adecuados que permitan superar las dificultades encontradas en el caso de los métodos convencionales. Con este fin se combinan estudios de escalado del tamaño de sitema con técnicas de "Histogram Reweighting" (HR). La aplicación de estos métodos se ha mostrado recientemente como mucho mejor fundamentada y precisa para el cálculo de puntos críticos de sistemas sencillos como es el caso del fluido de LennardJones (LJ). En esta tesis, estas técnicas han sido combinadas con el objetivo de extender su aplicación a mezclas reales de interés industrial. Previamente a su aplicación a dichas mezclas reales, el fluido de LennardJones, capaz de reproducir el comportamiento de fluidos sencillos como es el caso de argón o metano, ha sido tomado como referencia en un paso preliminar. A partir de simulaciones realizadas en el colectivo gran canónico y recombinadas mediante la mencionada técnica de "Histogram Reweighting" se han obtenido los diagramas de fases tanto de fluidos puros como de mezclas binarias. A su vez se han localizado con una gran precisión los puntos críticos de dichos sistemas mediante las técnicas de escalado del tamaño de sistema. Con el fin de extender la aplicación de dichas técnicas a sistemas multicomponente, se han introducido modificaciones a los métodos de HR evitando la construcción de histogramas y el consecuente uso de recursos de memoria. Además, se ha introducido una metodología alternativa, conocida como el cálculo del cumulante de cuarto orden o parámetro de Binder, con el fin de hacer más directa la localización del punto crítico. En particular, se proponen dos posibilidades, en primer lugar la intersección del parámetro de Binder para dos tamaños de sistema diferentes, o la intersección del parámetro de Binder con el valor conocido de la correspondiente clase de universalidad combinado con estudios de escalado. Por otro lado, y en un segundo frente, la segunda parte de esta tesis está dedicada al desarrollo de potenciales intermoleculares capaces de describir las energías inter e intramoleculares de las moléculas involucradas en las simulaciones. En la última década se han desarrolldo diferentes modelos de potenciales para una gran variedad de compuestos. Uno de los más comunmente utilizados para representar hidrocarburos y otras moléculas flexibles es el de átomos unidos, donde cada grupo químico es representado por un potencial del tipo de LennardJones. El uso de este tipo de potencial resulta en una significativa disminución del tiempo de cálculo cuando se compara con modelos que consideran la presencia explícita de la totalidad de los átomos. En particular, el trabajo realizado en esta tesis se centra en el desarrollo de potenciales de átomos unidos anisotrópicos (AUA), que se caracterizan por la inclusión de un desplazamiento de los centros de LennardJones en dirección a los hidrógenos de cada grupo, de manera que esta distancia se convierte en un tercer parámetro ajustable junto a los dos del potencial de LennardJones.
En la segunda parte de esta tesis se han desarrollado potenciales del tipo AUA4 para diferentes familias de compuesto que resultan de interés industrial como son los tiofenos, alcanoles y éteres. En el caso de los tiofenos este interés es debido a las cada vez más exigentes restricciones medioambientales que obligan a eliminar los compuestos con presencia de azufre. De aquí la creciente de necesidad de propiedades termodinámicas para esta familia de compuestos para la cual solo existe una cantidad de datos termodinámicos experimentales limitada. Con el fin de hacer posible la obtención de dichos datos a través de la simulación molecular hemos extendido el potencial intermolecular AUA4 a esta familia de compuestos. En segundo lugar, el uso de los compuestos oxigenados en el campo de los biocombustibles ha despertado un importante interés en la industria petroquímica por estos compuestos. En particular, los alcoholes más utilizados en la elaboración de los biocombustibles son el metanol y el etanol. Como en el caso de los tiofenos, hemos extendido el potencial AUA4 a esta familia de compuestos mediante la parametrización del grupo hidroxil y la inclusión de un grupo de cargas electrostáticas optimizadas de manera que reproduzcan de la mejor manera posible el potencial electrostático creado por una molecula de referencia en el vacío. Finalmente, y de manera análoga al caso de los alcanoles, el último capítulo de esta tesis la atención se centra en el desarrollo de un potencial AUA4 capaz de reproducir cuantitativamente las propiedades de coexistencia de la familia de los éteres, compuestos que son ampliamente utilizados como solventes.
Parallel with the increase of computer speed, in the last decade, molecular simulation techniques have emerged as important tools to predict physical properties of systems of industrial interest. These properties are essential in the chemical and petrochemical industries in order to perform process design, optimization, simulation and process control. The actual moderate cost of powerful computers converts molecular simulation into an excellent tool to provide predictions of such properties. In particular, the predictive capability of molecular simulation techniques becomes very important when dealing with extreme conditions of temperature and pressure as well as when toxic compounds are involved in the systems to be studied due to the fact that experimentation at such extreme conditions is difficult and expensive.
Consequently, alternative processes must be considered in order to obtain the required properties. Chemical and petrochemical industries have made intensive use of thermophysical models including equations of state, activity coefficients models and corresponding state theories. These predictions present the advantage of providing good approximations with minimal computational needs. However, these models are often inadequate when only a limited amount of information is available to determine the necesary parameters, or when trying to reproduce complex fluid properties such as that of molecules which exhibit hydrogen bonding, polymers, etc. In addition, there is no way for dynamical properties to be estimated in a consistent manner.
In this thesis, the HR and FSS techniques are combined with the main goal of extending the application of these methodologies to the calculation of the vaporliquid equilibrium and critical point of real mixtures. Before applying the methodologies to the real mixtures of industrial interest, the LennardJones fluid has been taken as a reference model and as a preliminary step. In this case, the predictions are affected only by the omnipresent statistical errors, but not by the accuracy of the model chosen to reproduce the behavior of the real molecules or the interatomic potential used to calculate the configurational energy of the system.
The simulations have been performed in the grand canonical ensemble (GCMC)using the GIBBS code. Liquidvapor coexistences curves have been obtained from HR techniques for pure fluids and binary mixtures, while critical parameters were obtained from FSS in order to close the phase envelope of the phase diagrams. In order to extend the calculations to multicomponent systems modifications to the conventional HR techniques have been introduced in order to avoid the construction of histograms and the consequent need for large memory resources. In addition an alternative methodology known as the fourth order cumulant calculation, also known as the Binder parameter, has been implemented to make the location of the critical point more straightforward. In particular, we propose the use of the fourth order cumulant calculation considering two different possibilities: either the intersection of the Binder parameter for two different system sizes or the intersection of the Binder parameter with the known value for the system universality class combined with a FSS study. The development of transferable potential models able to describe the inter and intramolecular energies of the molecules involved in the simulations constitutes an important field in the improvement of Monte Carlo techniques. In the last decade, potential models, also referred to as force fields, have been developed for a wide range of compounds. One of the most common approaches for modeling hydrocarbons and other flexible molecules is the use of the unitedatoms model, where each chemical group is represented by one LennardJones center. This scheme results in a significant reduction of the computational time as compared to allatoms models since the number of pair interactions goes as the square of the number of sites. Improvements on the standard unitedatoms model, where typically a 612 LennardJones center of force is placed on top of the most significant atom, have been proposed. For instance, the AUA model consists of a displacement of the LennardJones centers of force towards the hydrogen atoms, converting the distance of displacement into a third adjustable parameter. In this thesis we have developed AUA 4 intermolecular potentials for three different families of compounds. The family of ethers is of great importance due to their applications as solvents. The other two families, thiophenes and alkanols, play an important roles in the oil and gas industry. Thiophene due to current and future environmental restrictions and alkanols due ever higher importance and presence of biofuels in this industry.

As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered.

Many questions in mathematics and physics can be reduced to the problem of finding and classifying the critical points of a suitable functional on an appropriate manifold. In this thesis, we will be concerned with the problems of existence, location and structure of critical points by building upon the well known min-max methods that are presently used in non-linear differential equations. The thesis consists of two parts: In the first part, we exploit the new powerful mountain pass principle of Ghoussoub and Preiss and its higher dimensional extensions by Ghoussoub to classify the critical points generated by Min-Max methods. The functionals under study are only assumed to be C1and therefore the classical Morse theory is not available. In order to do this, we isolate various topological indices that can be associated with certain critical sets and points. If the functionals are C2 and the critical points are non-degenerated, these indices can then be used to recover the standard results on Morse indices. The study in this direction was first initiated in the case of the mountain pass theorem by Hofer and was expanded later by Pucci-Serrin and Ghoussoub-Preiss. We shall extend and simplify all the previously obtained structural results in this setting, but more importantly, we consider the case of the saddle point theorem and various other higher dimensional settings. In the last part, we construct an almost critical sequence (xn). by min- max procedures for a C2-functional co on a Hilbert space with some Morse type information. Actually we obtain some analytical (second order) properties concerning the Hessian d2co(x,i) which can be viewed as the asymptotic version of the information on the Morse index of the limit of(x„)„ whenever such a limit exists. As noted by P.L. Lions in his studies of the Hartree-Fock equations for Coulomb systems, this type of additional information about an almost critical sequence can sometimes be crucial in the proof of its convergence and in solving the corresponding variational problems. Examples are given as applications of the general theory developed in the thesis.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2015.
Recently there has been signi cant interest in new types of metals called non-Fermi liquids, which cannot be described by Landau Fermi liquid theory. Landau Fermi liquid theory is a theoretical model used to describe low energy interacting fermions or quasiparticles. There is a growing interest in constructing an e ective eld theory for these types of metals. One of the paradigms to understand these metals is by the use of Wilsonian renormalization group (RG) to study a theoretical toy model consisting of fermions coupled to a gapless order parameter eld. Here we will study fermions coupled to gapless bosons (order parameter) below the upper critical dimension (d = 3). We will treat both fermions and bosons on equal footing and construct an e ective eld theory which only integrates out high momentum modes. Then we compute the one-loop RG ows for the Yukawa coupling and four-Fermi interaction. We will discuss log2 and log3 subleties associated with the one loop RG ows for the four-Fermi interaction and how they can be circumvented.

In the first part of this thesis, we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let X1, X2,..., Xk be i.i.d Ginibre matrices of size n ×n whose entries are standard complex Gaussian random variables. We derive eigenvalue density for matrices of the form X1 ε1 X2 ε2 ... Xk εk , where εi = ±1 for i =1,2,..., k. We show that the eigenvalues form a determinantal point process. The case where k =2, ε1 +ε2 =0 was derived earlier by Krishnapur. In the case where εi =1 for i =1,2,...,n was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.

Economics is a direction in which there becomes visible to be many occasions for applications of time scales. The time scales approach will not only unify the standard discrete and continuous models in economics, but also, for example, authorizes for payments that reach unequally spaced points in time. We are going to study dynamic optimization problems from economics, construct a time scales model, and apply variational methods and critical point theory to obtain the existence of solutions. We derive several conditions ensuring existence of solutions of dynamic Sturm{Liouville boundary value problems. Variational methods are utilized in the proofs. We discuss the existence of at least one, three and in¯nitely many solutions for the problems under di®erent conditions on the data. Examples are also given to illustrate the main results.

by Chu Lap-foo.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1994.
Includes bibliographical references (leaves 68-69).
Introduction --- p.iii
Chapter 1 --- Basic Properties of Singular Solutions --- p.1
Chapter 1.1 --- An Asymptotic Radial Result --- p.2
Chapter 1.2 --- Local Uniqueness of Radial Solutions --- p.8
Chapter 2 --- Dirichlet Problem : Existence Theory I --- p.11
Chapter 2.1 --- Formulation --- p.12
Chapter 2.2 --- Explicit Solutions on Balls --- p.14
Chapter 2.3 --- The Moser Inequality --- p.19
Chapter 2.4 --- Existence of Solutions in General Domains --- p.24
Chapter 2.5 --- Spectrum of the Problem --- p.26
Chapter 3 --- Dirichlet Problem : Existence Theory II --- p.29
Chapter 3.1 --- Mountain Pass Lemma --- p.29
Chapter 3.2 --- Existence of Second Solution --- p.31
Chapter 4 --- Dirichlet Problem : Non-Existence Theory --- p.36
Chapter 4.1 --- Upper Bound of λ* in Star-Shaped Domains --- p.36
Chapter 4.2 --- Numerical Values --- p.41
Chapter 5 --- The Neumann Problem --- p.42
Chapter 5.1 --- Existence Theory I --- p.43
Chapter 5.2 --- Existence Theory II --- p.47
Chapter 6 --- The Schwarz Symmetrization --- p.49
Chapter 6.1 --- Definitions and Basic Properties --- p.49
Chapter 6.2 --- Inequalities Related to Symmetrization --- p.58
Chapter 6.3 --- An Application to P.D.E --- p.63
Bibliography --- p.68

Une nouvelle notion d'enlacement pour les paires d'ensembles $A\subset B$, $P\subset Q$ dans un espace de Hilbert de type $X=Y\oplus Y^{\perp}$ avec $Y$ séparable, appellée $\tau$-enlacement, est définie. Le modèle pour cette définition est la généralisation de l'enlacement homotopique et de l'enlacement au sens de Benci-Rabinowitz faite par Frigon. En utilisant la théorie du degré développée dans un article de Kryszewski et Szulkin, plusieurs exemples de paires $\tau$-enlacées sont donnés. Un lemme de déformation est établi et utilisé conjointement à la notion de $\tau$-enlacement pour prouver un théorème d'existence de point critique pour une certaine classe de fonctionnelles sur $X$. De plus, une caractérisation de type minimax de la valeur critique correspondante est donnée. Comme corollaire de ce théorème, des conditions sont énoncées sous lesquelles l'existence de deux points critiques distincts est garantie. Deux autres théorèmes de point critiques sont démontrés dont l'un généralise le théorème principal de l'article de Kryszewski et Szulkin mentionné ci-haut.
A new notion of linking for pairs of sets $A\subset B$, $P\subset Q$ in a Hilbert space of the form $X=Y\oplus Y^{\perp}$ with $Y$ separable, called $\tau$-linking, is defined. The model for this definition is the generalization of homotopical linking and linking in the sense of Benci-Rabinowitz made by Frigon. Using the degree theory developped in an article of Kryszewski and Szulkin, many examples of $\tau$-linking pairs are given. A deformation lemma is established and used jointly with the notion of $\tau$-linking to prove an existence theorem for critical points of a certain class of functionals defined on $X$. Moreover, a characterization of a minimax nature for the corresponding critical value is given. As a corollary of this theorem, conditions are stated under which the existence of two distinct critical points is guaranteed. Two other critical point theorems are demonstrated, one of which generalizes the main theorem of the article of A new notion of linking for pairs of sets $A\subset B$, $P\subset Q$ in a Hilbert space of the form $X=Y\oplus Y^{\perp}$ with $Y$ separable, called $\tau$-linking, is defined. The model for this definition is the generalization of homotopical linking and linking in the sense of Benci-Rabinowitz made by Frigon~\cite{frigon:1}. Using the degree theory developped in~\cite{szulkin:1}, many examples of $\tau$-linking pairs are given. A deformation lemma is established and used jointly with the notion of $\tau$-linking to prove an existence theorem for critical points of a certain class of functionals defined on $X$. Moreover, a characterization of a minimax nature for the corresponding critical value is given. As a corollary of this theorem, conditions are stated under which the existence of two distinct critical points is guaranteed. Two other critical point theorems are demonstrated, one of which generalizes the main theorem of the article by Kryszewski and Szulkin cited above.