Um estudo sobre polinômios matriciais

This research work aims to make a study of the algebraic theory of matrix monic polynomials, as well as the definitions, concepts and properties with respect to block eigenvalues, block eigenvectors and solvents of P(X). We investigte the main relations between the matrix polynomial and the Companion and Vandermonde matrices. We study the construction of matrix polynomials with certain solvents and the extention of the Power Method, to calculate block eigenvalues and solvents of P(X). Through the relationship between the dominant block eigenvalue of the Companion matrix and the dominant solvent of P(X) it is possible to obtain the convergence of the algorithm for the dominant solvent of the matrix polynomial. We illustrate with numerical examples for diferent cases of convergence.

Advanced thermal analysis of microelectronics using spreading resistance models

Thermal analysis of electronic devices is one of the most important steps for designing of modern devices. Precise thermal analysis is essential for designing an effective thermal management system of modern electronic devices such as batteries, LEDs, microelectronics, ICs, circuit boards, semiconductors and heat spreaders. For having a precise thermal analysis, the temperature profile and thermal spreading resistance of the device should be calculated by considering the geometry, property and boundary conditions. Thermal spreading resistance occurs when heat enters through a portion of a surface and flows by conduction. It is the primary source of thermal resistance when heat flows from a tiny heat source to a thin and wide heat spreader. In this thesis, analytical models for modeling the temperature behavior and thermal resistance in some common geometries of microelectronic devices such as heat channels and heat tubes are investigated. Different boundary conditions for the system are considered. Along the source plane, a combination of discretely specified heat flux, specified temperatures and adiabatic condition are studied. Along the walls of the system, adiabatic or convective cooling boundary conditions are assumed. Along the sink plane, convective cooling with constant or variable heat transfer coefficient are considered. Also, the effect of orthotropic properties is discussed. This thesis contains nine chapters. Chapter one is the introduction and shows the concepts of thermal spreading resistance besides the originality and importance of the work. Chapter two reviews the literatures on the thermal spreading resistance in the past fifty years with a focus on the recent advances. In chapters three and four, thermal resistance of a twodimensional flux channel with non-uniform convection coefficient in the heat sink plane is studied. The non-uniform convection is modeled by using two functions than can simulate a wide variety of different heat sink configurations. In chapter five, a non-symmetrical flux channel with different heat transfer coefficient along the right and left edges and sink plane is analytically modeled. Due to the edge cooling and non-symmetry, the eigenvalues of the system are defined using the heat transfer coefficient on both edges and for satisfying the orthogonality condition, a normalized function is calculated. In chapter six, thermal behavior of two-dimensional rectangular flux channel with arbitrary boundary conditions on the source plane is presented. The boundary condition along the source plane can be a combination of the first kind boundary condition (Dirichlet or prescribed temperature) and the second kind boundary condition (Neumann or prescribed heat flux). The proposed solution can be used for modeling the flux channels with numerous different source plane boundary conditions without any limitations in the number and position of heat sources. In chapter seven, temperature profile of a circular flux tube with discretely specified boundary conditions along the source plane is presented. Also, the effect of orthotropic properties are discussed. In chapter 8, a three-dimensional rectangular flux channel with a non-uniform heat convection along the heat sink plane is analytically modeled. In chapter nine, a summary of the achievements is presented and some systems are proposed for the future studies. It is worth mentioning that all the models and case studies in the thesis are compared with the Finite Element Method (FEM).

Periodic nonlinear Fourier transform for fiber-optic communications, Part II:eigenvalue communication

In this paper we propose the design of communication systems based on using periodic nonlinear Fourier transform (PNFT), following the introduction of the method in the Part I. We show that the famous "eigenvalue communication" idea [A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993)] can also be generalized for the PNFT application: In this case, the main spectrum attributed to the PNFT signal decomposition remains constant with the propagation down the optical fiber link. Therefore, the main PNFT spectrum can be encoded with data in the same way as soliton eigenvalues in the original proposal. The results are presented in terms of the bit-error rate (BER) values for different modulation techniques and different constellation sizes vs. the propagation distance, showing a good potential of the technique.

Using phase-space localized basis functions to obtain vibrational energies of molecules

For decades scientists have attempted to use ideas of classical mechanics to choose basis functions for calculating spectra. The hope is that a classically-motivated basis set will be small because it covers only the dynamically important part of phase space. One popular idea is to use phase space localized (PSL) basis functions. This thesis improves on previous efforts to use PSL functions and examines the usefulness of these improvements. Because the overlap matrix, in the matrix eigenvalue problem obtained by using PSL functions with the variational method, is not an identity, it is costly to use iterative methods to solve the matrix eigenvalue problem. We show that it is possible to circumvent the orthogonality (overlap) problem and use iterative eigensolvers. We also present an altered method of calculating the matrix elements that improves the performance of the PSL basis functions, and also a new method which more efficiently chooses which PSL functions to include. These improvements are applied to a variety of single well molecules. We conclude that for single minimum molecules, the PSL functions are inferior to other basis functions. However, the ideas developed here can be applied to other types of basis functions, and PSL functions may be useful for multi-well systems.

La metrica di Agmon ed il decadimento esponenziale delle soluzioni di equazioni alle derivate parziali

The aim of this master thesis is to study the exponential decay of solutions of elliptic partial equations. This work is based on the results obtained by Agmon. To this purpose, first, we define the Agmon metric, that plays an important role in the study of exponential decay, because it is related to the rate of decay. Under some assumptions on the growth of the function and on the positivity of the quadratic form associated to the operator, a first result of exponential decay is presented. This result is then applied to show the exponential decay of eigenfunctions with eigenvalues whose real part lies below the bottom of the essential spectrum. Finally, three examples are given: the harmonic oscillator, the hydrogen atom and a Schrödinger operator with purely discrete spectrum.

Generating a state t-design by diagonal quantum circuits

We investigate protocols for generating a state t-design by using a fixed separable initial state and a diagonal-unitary t-design in the computational basis, which is a t-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary t-design generates a O (1/2(N))-approximate state t-design, where N is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary t-design. We also show that it is necessary and sufficient to use O (log(2)(t)) -qubit gates with random phases to generate a diagonal-unitary t-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary t-design by non-diagonal two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.

Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.

Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

Spectral graph theory with applications to quantum adiabatic optimization

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.

Knowledge about dietary fibres (KADF): development and validation of an evaluation instrument through structural equation modelling (SEM)

Objectives: Because there is scientific evidence that an appropriate intake of dietary fibre should be part of a healthy diet, given its importance in promoting health, the present study aimed to develop and validate an instrument to evaluate the knowledge of the general population about dietary fibres. Study design: The present study was a cross sectional study. Methods: The methodological study of psychometric validation was conducted with 6010 participants, residing in ten countries from 3 continents. The instrument is a questionnaire of self-response, aimed at collecting information on knowledge about food fibres. For exploratory factor analysis (EFA) was chosen the analysis of the main components using varimax orthogonal rotation and eigenvalues greater than 1. In confirmatory factor analysis by structural equation modelling (SEM) was considered the covariance matrix and adopted the Maximum Likelihood Estimation algorithm for parameter estimation. Results: Exploratory factor analysis retained two factors. The first was called Dietary Fibre and Promotion of Health (DFPH) and included 7 questions that explained 33.94 % of total variance ( = 0.852). The second was named Sources of Dietary Fibre (SDF) and included 4 questions that explained 22.46% of total variance ( = 0.786). The model was tested by SEM giving a final solution with four questions in each factor. This model showed a very good fit in practically all the indexes considered, except for the ratio 2/df. The values of average variance extracted (0.458 and 0.483) demonstrate the existence of convergent validity; the results also prove the existence of discriminant validity of the factors (r2 = 0.028) and finally good internal consistency was confirmed by the values of composite reliability (0.854 and 0.787). Conclusions: This study allowed validating the KADF scale, increasing the degree of confidence in the information obtained through this instrument in this and in future studies.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let H^k[G] be the lexicographic product of H^k and G, where H^k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H^k[G]H and H^k when G and H are regular and the Laplacian spectrum of H^k[G] and H^k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10^100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

Bethe graphs attached to the vertices of a connected graph: a spectral approach

A weighted Bethe graph $B$ is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family $F$ of graphs. The operation of identifying the root vertex of each of $r$ weighted Bethe graphs to the vertices of a connected graph $\mathcal{R}$ of order $r$ is introduced as the $\mathcal{R}$-concatenation of a family of $r$ weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when $F$ has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in $F$ are all regular) of the $\mathcal{R}$-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph.

On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface

We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds