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In this dissertation, we solve multivariable Nevanlinna-Pick type interpolation problems. Particularly, we consider the left tangential interpolation problems on the commutative or noncommutative unit ball. For the commutative setting, we discuss left-tangential operator-argument interpolation problems for Schur-class multipliers on the Drury-Arveson space and for the noncommutative setting, we discuss interpolation problems for Schur-class multipliers on Fock space. We apply the Krein-space geometry approach (also known as the Grassmannian Approach). To implement this approach J-versions of Beurling-Lax representers for shift-invariant subspaces are required. Here we obtain these J-Beurling-Lax theorems by the state-space method for both settings. We see that the Krein-space geometry method is particularly simple in solving the interpolation problems when the Beurling-Lax representer is bounded. The Potapov approach applies equally well whether the representer is bounded or not. / Ph. D.

A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\mathscr{H}$ of functions with the property that the values $f(x)$ for $f \in \mathscr{H}$ are reproduced from the inner product in $\mathscr{H}$. Recent applications are found in stochastic processes (Ito Calculus), harmonic analysis, complex analysis, learning theory, and machine learning algorithms. This research began with the study of RKHSs to areas such as learning theory, sampling theory, and harmonic analysis. From the Moore-Aronszajn theorem, we have an explicit correspondence between reproducing kernel Hilbert spaces (RKHS) and reproducing kernel functions—also called positive definite kernels or positive definite functions. The focus here is on the duality between positive definite functions and their boundary spaces; these boundary spaces often lead to the study of Gaussian processes or Brownian motion. It is known that every reproducing kernel Hilbert space has an associated generalized boundary probability space. The Arveson (reproducing) kernel is $K(z,w) = \frac{1}{1-_{\C^d}}, z,w \in \B_d$, and Arveson showed, \cite{Arveson}, that the Arveson kernel does not follow the boundary analysis we were finding in other RKHS. Thus, we were led to define a new reproducing kernel on the unit ball in complex $n$-space, and naturally this lead to the study of a new reproducing kernel Hilbert space. This reproducing kernel Hilbert space stems from boundary analysis of the Arveson kernel. The construction of the new RKHS resolves the problem we faced while researching “natural” boundary spaces (for the Drury-Arveson RKHS) that yield boundary factorizations: \[K(z,w) = \int_{\mathcal{B}} K^{\mathcal{B}}_z(b)\overline{K^{\mathcal{B}}_w(b)}d\mu(b), \;\;\; z,w \in \B_d \text{ and } b \in \mathcal{B} \tag*{\it{(Factorization of} $K$).}\] Results from classical harmonic analysis on the disk (the Hardy space) are generalized and extended to the new RKHS. Particularly, our main theorem proves that, relaxing the criteria to the contractive property, we can do the generalization that Arveson's paper showed (criteria being an isometry) is not possible.

Resumo: O método de segunda quantificação é utilizado para construir o operador espalhamento S no espaço de Fock, no contexto de Thermofield Dynamics (TFD), para o campo de Dirac sujeito a um potencial eletromagnético externo dependente do tempo. Esta descrição é baseada na abordagem construtiva do espaço de Fock, a qual é aplicada ao sistema original e a seu dual. Seguindo a prescrição de TFD, o operador S é utilizado para avaliar o processo de produção de pares elétron-pósitron à temperatura finita, e uma análise do limiar de produção é feita a partir do cálculo da probabilidade total de transição. / Abstract: The second quantization methods is used to build the scattering operator S in Fock space, in the contex of Thermofield Dynamics (TFD), for the Dirac field subject to an external time-dependent electromagnetic potential. This description is based on the constructive approach to the Fock space, wich is applied to the original system and to its dual. Following TFD prescription, the operator S is used to estimate the process of electron-positron pair production at finite temperature, and an analysis of the production threshold is done based on the calculation of the total transition probability. / Orientador: Jeferson de Lima Tomazelli / Coorientador: Bruto Max Pimentel Escobar / Banca: José David Mangueira Vianna / Banca: Ademir Eugênio de Santana / Banca: Fernando Luiz de Campos Carvalho / Banca: Maria Cristina Batoni Abdalla / Doutor

Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2002-08Bitstream added on 2014-06-13T19:21:27Z : No. of bitstreams: 1 placido_hq_dr_ift.pdf: 696431 bytes, checksum: 6ab3e4a3aecfcc9e8f376ba3a9ba8dde (MD5) / O método de segunda quantificação é utilizado para construir o operador espalhamento S no espaço de Fock, no contexto de Thermofield Dynamics (TFD), para o campo de Dirac sujeito a um potencial eletromagnético externo dependente do tempo. Esta descrição é baseada na abordagem construtiva do espaço de Fock, a qual é aplicada ao sistema original e a seu dual. Seguindo a prescrição de TFD, o operador S é utilizado para avaliar o processo de produção de pares elétron-pósitron à temperatura finita, e uma análise do limiar de produção é feita a partir do cálculo da probabilidade total de transição. / The second quantization methods is used to build the scattering operator S in Fock space, in the contex of Thermofield Dynamics (TFD), for the Dirac field subject to an external time-dependent electromagnetic potential. This description is based on the constructive approach to the Fock space, wich is applied to the original system and to its dual. Following TFD prescription, the operator S is used to estimate the process of electron-positron pair production at finite temperature, and an analysis of the production threshold is done based on the calculation of the total transition probability.

Mankind has come a long way since the invention of wheel to accessing information in the quintillionth of a second. At the heart of every invention ever made, there has been only one objective, to ease the way of living. The progeny of this philosophy automatically came to be known as technology. It was technology that led to the design of the wheel for fast human transportation and the same motivation let him design more sophisticated machines. In mankind’s journey to improve technology, it began to learn efficient or correct ways to utilize and understand resources around it, creating a whole new philosophy called science. Ingeniously, it was science that let humans understand what they were made of: matter, to discovering what matter itself was composed of: atoms and what puts these together: forces. Science and technology has been of tremendous comfort for mankind and has helped it evolve throughout history. However, it is not always that science and technology go hand in hand. Technology has always helped man design devices and instruments which often bring physical comfort. Science on the other hand has made sure that loss in manual labor is compensated by increased inquisitiveness. There were times when technology was more developed than science. This was the time when machines were taking mankind by fire, resulting in the first and second industrial revolutions. During that same time, science was develop-ing slowly by increasing human curiosity to learn the way nature functioned at finer details. This led to the discovery of the electron by Joseph John Thomson, who proved the electron to be a negatively charged particle. Consequently, he was awarded the 1906 Nobel Prize in Physics for his work on electricity conduction in gases. Later, his son, George Paget Thomson, counter-proved that electrons are actually waves. He was also awarded the 1937 Nobel Prize in Physics, along with Clinton Joseph Davisson for their discovery of electron diffraction caused by crystals. Despite the ambiguity, mankind today accepts electrons to have dual properties. It is both a wave and a particle. This duality is not limited to electrons but is applicable to all matter, as proposed by Louis de Broglie and is one of the fundamental principles in science. With the help of well-developed technology, mankind can now design machines that allow controlled flow of electrons establishing the world of electronics, allowing faster human communication. The study of electronic properties and its usage in designing efficient devices is what electronics is all about. Electrons are the protagonist of mankind today. The presence of electrons is unanimously accepted by everyone. All physical and chemical processes are a result of electrons getting transported. Electron transfer processes are ubiquitous in nature, be it in photosynthesis or energy production in mitochondria . It is the fundamental process in all chemical reactions and all physical processes related to electricity. Every piece of hi-tech gadget practically uses the electron, and the whole of humanity is being serviced by it. In fact, a life without utilizing the electrons is abysmally mundane. Electronics has evolved from designing the first millimeter sized point contact transistor to silicon chip processors that contain billions of nanosized transistors. Studying electron transport has also led to the discovery of light emission during conduction popularly known as LED, an abbreviation for light emitting diode. Heating up of devices during electron transport forced mankind to study heat transport and design materials that have highly efficient electron transfer processes. Electron transfer is also the basic principle behind the Scanning Tunneling Microscope (STM), Scanning Electron Microscope (SEM) and the Transmission Electron Microscope (TEM) which replaced the conventional idea of using light (photons) as a source to observe matter at the nanolevel. However, mankind is still in the process of developing a technology which exploits both properties of the electron simultaneously. Today, science and technology work together to overcome this barrier. Indeed, science and technology today have come as far as controlling electron transport up to a single atomic level where quantum effects (discretization and interference of states that make up the system) are very pronounced. This branch can be referred to as quantum electronics or quantronics. It is one of the possible alternatives to conventional silicon based electronics, and is made of three separate fields. The first one that exploits the quantum nature of electron transport in nanoscopic systems, is usually called molecular electronics or moletronics. The second involves ex-ploiting the spin of the electron and is termed as spintronics. The third is the most challenging where neither science nor technology has been able to fully grasp the characteristics, i.e utilizing the heat quanta in designing thermal de-vices at the single atomic level. In general, for ultimate exploitation of both the wave and particle characteristics of the electron, a proper comprehension of the quantum effects during electron transport is necessary to design a quantronic device. Also, in any quantronic device, apart from quantum effects, fluctuations in temperature cause changes in the flow of electrons. Since electron flow is a random process, fluctuations need to be analyzed from a statistical point of view. Moreover, to address issues related to efficiency and power of these quantronic devices, a proper understanding of the thermodynamic aspects is required. The aim of the work in the thesis is to theoretically analyze the fluctuations, quantum effects and thermodynamics, that in principle, affect the basic physics and chemistry during electron and heat transport in a specific class of out of equilibrium quantum systems. This class of quantum systems are prototypes for designing quantronic devices, where both wave and particle nature of the electrons are pronounced. These are called molecular junctions or quantum junctions. It will in turn help the field of quantronics in the long run. However, in this thesis, it is the science that I address and not the technological aspects.

Cette thèse est consacrée à l’étude des représentations modulaires des algèbres d’Ariki-Koike, et des liens avec la théorie des cristaux et des bases canoniques de Kashiwara via le théorème de catégorification d’Ariki. Dans un premier temps, on étudie, grâce à des outils combinatoires, les matrices de décomposition de ces algèbres en généralisant les travaux de Geck et Jacon. On classifie entièrement les cas d’existence et de non-existence d’ensembles basiques, en construisant explicitement ces ensembles lorsqu’ils existent. On explicite ensuite les isomorphismes de cristaux pour les représentations de Fock de l’algèbre affine quantique Uq(sle). On construit alors un isomorphisme particulier, dit canonique, qui permet entre autres une caractérisation non-récursive de n’importe quelle composante connexe du cristal. On souligne également les liens avec la combinatoire des mots sous-jacente à la structure cristalline des espaces de Fock, en décrivant notamment un analogue de la correspondance de Robinson-Schensted-Knuth pour le type A affine. / This thesis is devoted to the study of modular representations of Ariki-Koike algebras, and of the connections with Kashiwara’s crystal and canonical bases theory via Ariki’s categorification theorem. First, we study, using combinatorial tools, the decomposition matrices associated to these algebras, generalising the works of Geck and Jacon. We fully classify the cases of existence and non-existence of canonical basic sets, and we explicitely construct these sets when they exist. Next, we make explicit the crystal isomorphisms for Fock spaces representations of the quantum affine algebra Uq(sle). We then construct of a particular isomorphism, so-called canonical, which gives, inter alia, a non-recursive description of any connected component of the crystal. We also stress the links with the combinatorics of words underlying the crystal structure of Fock spaces, by describing notably an analogue of the Robinson-Schensted-Knuth correspondence for affine type A.