L'inégalité de Hausdorff-Young a été généralisée aux groupes localement compacts par R. Kunze dans le cas unimodulaire puis par M. Terp dans le cas général. Une version de cette inégalité a été donnée par B. Russo pour les opérateurs intégraux. Dans cette thèse, on établit une inégalité de Hausdorff-Young pour les groupoïdes mesurés qui recouvre ces résultats. Comme dans les cas des groupes non commutatifs, on utilise la théorie non commutative de l'intégration. La majeure partie de ce travail est l'identification des espaces Lp de l'algèbre de von Neumann du groupoïde dans les cas p=1, 2 comme espaces de fonctions et aussi comme espaces d'opérateurs aléatoires.

Ultra-fast laser heating of nano-films is investigated using 3-D Dual Phase Lag heat transport equation with laser heating at different locations on the metal film. The energy absorption rate, which is used to model femtosecond laser heating, is modified to accommodate for three-dimensional laser heating. A numerical solution based on an explicit finite-difference method is employed to solve the DPL equation. The stability criterion for selecting a time step size is obtained using von Neumann eigenmode analysis, and grid function convergence tests are performed. DPL results are compared with classical diffusion and hyperbolic heat conduction models and significant differences among these three approaches are demonstrated. We also develop an implicit finite-difference scheme of Crank-Nicolson type for solving 1-D and 3-D DPL equations. The proposed numerical technique solves one equation unlike other techniques available in the literature, which split the DPL equation into a system of two equations and then apply discretization. Stability analysis is performed using a von Neumann stability analysis. In 3-D, the discretized equation is solved using delta-form Douglas and Gunn time splitting. The performance of the proposed numerical technique is compared with the numerical techniques available in the literature.

Quantum entanglement is a phenomenon which has shown great potential use in modern technical implementations, but there is still much development needed in the field. One major problem is how to measure the amount of entanglement present in a given entangled state. There are numerous different entanglement measures suggested, all satisfying some conditions being of either operational, or more abstract, mathematical nature. However, in contradiction to what one might expect, the measures show discrepancies in the ordering of entangled states. Concretely this means that with respect to one measure, a state can be more entangled than another state, but the ordering may be opposite for the same states using another measure. In this thesis we take a closer look at some of the most commonly occurring entanglement measures, and find examples of states showing inequivalent entanglement ordering for the different measures. / Kvantmekanisk sammanflätning är ett fenomen som visat stor potential för framtida tekniska tillämpningar, men för att kunna använda oss av detta krävs att vi hittar lämpliga modeller att mäta omfattningen av sammanflätningen hos ett givet tillstånd. Detta har visat sig vara en svår uppgift, då de modeller som finns idag är otillräckliga när det gäller att konsekvent avgöra till vilken grad olika tillstånd är sammanflätade. Exempelvis kan en modell visa att ett tillstånd är mer sammanflätat än ett annat, medan en annan modell kan visa på motsatsen - att det första tillståndet är mindre sammanflätat än det andra. En möljig orsak kan ligga i de olika modellernas deifnition, då vissa utgår från operativa definitioner, medan andra grundas på matematiska, abstrakta villkor. I denna uppsats tittar vi lite närmre på några av de mätmodeller som finns, och hittar exempel på tillstånd som uppvisar olika ordning av sammanflätningsgrad beroende på vilken modell som används.

Die vorliegende Arbeit beschäftigt sich mit der Bures-Geometrie auf Zustandsräumen über von-Neumann-Algebren. Diese basiert auf jenem Abstandsbegriff für normale Zustände, der von Bures im Jahre 1969 eingeführt wurde. Eng damit verbunden ist der Begriff der algebraischen Übergangswahrscheinlichkeit, der von Uhlmann 1976 vorgeschlagen wurde. An einem Beispiel wird gezeigt, dass man den Bures-Abstand unter Umständen nicht implementieren kann, wenn man einen der implementierenden Vektoren vorgeben will. Im weiteren wird der vom Bures-Abstand induzierte Paralleltransport von Vektoren entlang Loops von normalen Zuständen untersucht. Um die Holonomiegruppe im unendlichdimensionalen Fall zu untersuchen, werden Sätze über Produkte positiver Operatoren hergeleitet. Diese Sätze, die durchaus auch von eigenständigem Interesse sein könnten, werden mit Ergebnissen aus der Literatur verglichen. Schließlich wird der Bures-Abstand unter infinitesimalem Blickwinkel betrachtet. Die so entstehenden Bures-geodätischen Bögen werden untersucht. Speziell wird gefragt, ob gewisse Strata stets geodätisch konvex sind, also als Beispiel für Umgebungen dienen können. Um diese Frage am Ende negativ zu beantworten, werden mehrere Sätze über Sakaische Radon-Nikodym-Operatoren hergeleitet, die auch ohne Bezug zur Bures-Geometrie interessant sein könnten. Das entscheidende Gegenbeispiel nutzt Gohbergs Ergebnis zum Spektrum bestimmter Toeplitzoperatoren aus. Ein Nebeneffekt des beschriebenen Verfahrens ist, dass es auch zur Konstruktion von Operatoren mit hinreichend nichttrivialem Spektrum benutzt werden kann. / The present paper deals with Bures' geometry in the state space over von-Neumann algebras. This geometry is based on the distance introduced by Bures in 1969. Closely related with it is the concept of algebraic transition probability as proposed by Uhlmann in 1976. It is shown by an example that there are cases where one can not implement Bures' distance if one of the implementing vectors is given. In the following, the parallel transport of vectors along loops of normal states, which is induced by Bures' distance, is examined. In order to investigate the holonomy group in the infinite-dimensional case, theorems on products of positive operators are derived. These theorems, which could be of interest on their own, are compared with the literature. Finally, Bures' distance is examined infinitesimally. The thus arising Bures-geodesic arcs are investigated. Especially, it is asked whether certain strata are geodesically convex and can therefore serve as examples of neighbourhoods. In order to finally give a negative answer, several theorems on Sakai's Radon-Nikodym operators, which could also be of interest without a connection to Bures' geometry, are derived. The critical counterexample exploits Gohberg's result on the spectrum of certain Toeplitz operators. A by-product of the described procedure is that it can be used to construct operators which have a sufficiently non-trivial spectrum.

Power distribution networks (PDNs) are conducting structures employed in semiconductor systems with the aim of providing circuits with reliable and constant operating voltage. This network has non-neglible electrical parasitics. Consequently, when digital circuits inside the chip switch, the supply voltage delivered to them does not remain ideal and exhibits spatial and temporal voltage fluctuations. These fluctuations in the supply voltage, known as the power-supply noise (PSN), can affect the functionality and the performance of modern microprocessors. The design of this PDN in the chip is an important part in ensuring power integrity. Modeling and simulation of the PSN in on-chip PDNs is important to reduce the cost of processors. These PDNs have irregular geometries, which affect the PSN. As a result, they have to be modeled. The problem sizes encountered in this simulation are usually large (on the order of millions), necessitating computationally efficient simulation approaches. Existing approaches for this simulation do not guarantee at least one of the following three required properties: computationally efficiency, accuracy, and numerically robustness. Therefore, there is a need to develop accurate, numerically robust, and efficient algorithms for this simulation. For many interconnects (e.g., transmission lines, board connectors, package PDNs), only their frequency responses and SPICE circuits (e.g., nonlinear switching drivers, equivalent circuits of interconnects) terminating them are known. These frequency responses are usually available only up to a certain maximum frequency. Simulating the electrical behavior of these systems is important for the reliable design of microprocessors and for their faster time-to-market. Because terminations can be nonlinear, a transient simulation is required. There is a need for a transient simulation of band-limited frequency-domain data characterizing a multiport passive system with SPICE circuits. The number of ports can be large (greater than or equal to 100 ports). In this simulation, unlike in traditional circuit simulators, normal properties like stability and causality of transient results are not automatically met and have to be ensured. Existing techniques for this simulation do not guarantee at least one of the following three required properties: computationally efficiency for a large number of ports, causality, and accuracy. Therefore, there is a need to develop accurate and efficient time-domain techniques for this simulation that also ensure causality. The objectives of this Ph.D. research are twofold: 1) To develop accurate, numerically robust, and computationally efficient time-domain algorithms to compute PSN in on-chip PDNs with irregular geometries. 2) To develop accurate and computationally efficient time-domain algorithms for the causal cosimulation of band-limited frequency-domain data with SPICE circuits.

Made available in DSpace on 2014-06-11T19:24:01Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-04-30Bitstream added on 2014-06-13T18:51:25Z : No. of bitstreams: 1 brandao_c_me_sjrp.pdf: 2464888 bytes, checksum: 22ba55e346e2ad76af2e1695d3998ff4 (MD5) / Nesta dissertação, além da apresentação de um ensaio teórico sobre a fundamentação da Mecânica Quântica, Computação, Informação Quântica, Criptografia e Entropias Quânticas, serão mostradas, de forma inédita, algumas implementações sobre o efeito da Entropia no Emaranhamento Quântico, importante para processos de transmissão da Informação Quântica, com o uso dos programas Mathematica e Matlab. Primeiramente e apresentado um breve histórico sobre a Computação Quântica e a Informação Quântica, junto com uma perspectiva do futuro. Logo em seguida uma breve introdu cão sobre a Mecânica Quântica, com o estudo de autovetores e autovalores e seus postulados, produtos tensoriais e o micro-universo. Na sequência um texto sucinto com os conceitos fundamentais da Computação Quântica como os bits quânticos, e portas lógicas. Além dos principais algoritmos quânticos. Depois passa-se a estudar a Informa ção Quântica, as operações quânticas, canais de inversão e polarização, para então chegar-se a Entropia, quando e feito um estudo comparativo entre as entropias de Von Neumann e Tsallis. E por fim um pouco de Criptografia Quântica. / In this dissertation, beyond the presentation of a theoretical essay on the basis of the Quantum Mechanics, Computation, Quantum information, Quantum Criptografy and Entropies, it will also be shown, for rst time, some implementations on the e ect of the Entropy tests on Quantum Entanglement for processes of transmission of Quantum Information, through the uses Mathematica and Matlab Programs. First I present a historical brie ng on the Quantum Computation and Quantum Information, together with a perspective of the future. Afterwards it will shown on introduction on the Quantum Mechanics, and its postulates, and the micro-universe. In sequence, a brief text with the fundamental concepts of the Quantum Computation, as the quantum bits, logic gates, and the main quantum algorithms. Later we will start to study Quantum Information, the quantum operations, channels of inversion and polarization. Furthermore we will go to discuss Entropy, where it is made a comparative study of Entropies of Von Neumann and Tsallis. And nally a little of Quantum Criptografy will be worked out.

<p>This work explores how randomization can be exploited to deliver sophisticated</p><p>algorithms with provable bounds for: (i) The approximation of matrix functions, such</p><p>as the log-determinant and the Von-Neumann entropy; and (ii) The low-rank approximation</p><p>of matrices. Our algorithms are inspired by recent advances in Randomized</p><p>Numerical Linear Algebra (RandNLA), an interdisciplinary research area that exploits</p><p>randomization as a computational resource to develop improved algorithms for</p><p>large-scale linear algebra problems. The main goal of this work is to encourage the</p><p>practical use of RandNLA approaches to solve Big Data bottlenecks at industrial</p><p>level. Our extensive evaluation tests are complemented by a thorough theoretical</p><p>analysis that proves the accuracy of the proposed algorithms and highlights their</p><p>scalability as the volume of data increases. Finally, the low computational time and</p><p>memory consumption, combined with simple implementation schemes that can easily</p><p>be extended in parallel and distributed environments, render our algorithms suitable</p><p>for use in the development of highly efficient real-world software.</p>

In this paper we derive the eigenfunctions to the Hamiltonian operator associated with the Harmonic Oscillator, and show that they are given by the Hermite functions. Then we prove that the Hermite functions form an orthonormal basis in the underlying Hilbert space. We also classify the inverse to the Hamiltonian operator as a Schatten-von Neumann operator. Finally, we derive the fundamental solution to the Schrödinger Equation corresponding to the Harmonic Oscillator using Mehler’s formula.

This doctoral thesis provides several new results in which we leverage the inner structure of non-singular rings, in particular of self-injective von Neumann regular rings. First, we describe categorical and set-theoretical conditions under which all products of compact objects remain compact, where the notion of compactness is relativized with respect to a fixed subclass of objects. A special instance when such closure property holds are the classic module categories over rings of our interest. Moreover, we show that a potential counterexample for Köthe's Conjecture might be in the form of a countable local subring of a suitable simple self-injective von Neumann regular ring. 1

The paper is an analysis of a special class of the master equations such that the Dissipation superoperator is L(ρ) = [M, [M, ρ]], where M is an hermitian andunitary operator and ρ a density matrix. It mainly investigates the dynamics ofρ and its properties such as boundness of the operators of the master equation,the eigenvalues of these operators, the purity of the states, the steady states. In the study of the temporal evolution of ρ it has been done an analysis of Decoherence free subspaces(DFS). A special attention is given to von Neumannentropy. For what it regards this last topic there are also specific referencesto the camel-like behaviour, a phenomenon regarding the entropy that happenswhen certain conditions of the dissipation superaoperator are not satisfied.There are Python simulations of the expectation values of some operators, andof the von Neumann entropy, and Linear Entropy.