Dissertations / Theses on the topic 'Density eigenvalue'

The new developments in coding theory research have revolutionized the application of coding to practical systems. Low-Density Parity Check (LDPC) codes form a class of Shannon limit approaching codes opted for digital communication systems that require high reliability. This thesis investigates the underlying relationship between the spectral properties of the parity check matrix and LDPC decoding convergence. The bit error rate of an LDPC code is plotted for the parity check matrix that has different Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. It is found that for a given (n,k) LDPC code, large SSEM has better error floor performance than low SSEM. The value of SSEM decreases as the sparseness in a parity-check matrix is increased. It was also found from the simulation that long LDPC codes have better error floor performance than short codes. This thesis outlines an approach to analyze LDPC decoding based on the eigenvalue analysis of the corresponding parity check matrix.

Low-density parity check (LDPC) codes are very popular among error correction codes because of their high-performance capacity. Numerous investigations have been carried out to analyze the performance and simplify the implementation of LDPC codes. Relatively slow convergence of iterative decoding algorithm affects the performance of LDPC codes. Faster convergence can be achieved by reducing the number of iterations during the decoding process. In this thesis, a new approach for faster convergence is suggested by choosing a systematic parity check matrix that yields lowest Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. MATLAB simulations are used to study the impact of eigenvalues on the number of iterations of the LDPC decoder. It is found that for a given (n, k) LDPC code, a parity check matrix with lowest SSEM converges quickly as compared to the parity check matrix with high SSEM. In other words, a densely connected graph that represents the parity check matrix takes more iterations to converge than a sparsely connected graph.

In this thesis we present results about some eigenvalue statistics of the beta-Hermite ensembles, both in the classical cases corresponding to beta = 1, 2, 4, that is the Gaussian orthogonal ensemble (consisting of real symmetric matrices), the Gaussian unitary ensemble (consisting of complex Hermitian matrices) and the Gaussian symplectic ensembles (consisting of quaternionic self-dual matrices) respectively. We also look at the less explored general beta-Hermite ensembles (consisting of real tridiagonal symmetric matrices). Specifically we look at the empirical distribution function and two different scalings of the largest eigenvalue. The results we present relating to these statistics are the convergence of the empirical distribution function to the semicircle law, the convergence of the scaled largest eigenvalue to the Tracy-Widom distributions, and with a different scaling, the convergence of the largest eigenvalue to 1. We also use simulations to illustrate these results. For the Gaussian unitary ensemble, we present an expression for its level density. To aid in understanding the Gaussian symplectic ensemble we present properties of the eigenvalues of quaternionic matrices. Finally, we prove a theorem about the symmetry of the order statistic of the eigenvalues of the beta-Hermite ensembles.
I denna kandidatuppsats presenterar vi resultat om några olika egenvärdens-statistikor från beta-Hermite ensemblerna, först i de klassiska fallen då beta = 1, 2, 4, det vill säga den gaussiska ortogonala ensemblen (bestående av reella symmetriska matriser), den gaussiska unitära ensemblen (bestående av komplexa hermitiska matriser) och den gaussiska symplektiska ensemblen (bestående av kvaternioniska själv-duala matriser). Vi tittar även på de mindre undersökta generella beta-Hermite ensemblerna (bestående av reella symmetriska tridiagonala matriser). Specifikt tittar vi på den empiriska fördelningsfunktionen och två olika normeringar av det största egenvärdet. De resultat vi presenterar för dessa statistikor är den empiriska fördelningsfunktionens konvergens mot halvcirkel-fördelningen, det normerade största egenvärdets konvergens mot Tracy-Widom fördelningen, och, med en annan normering, största egenvärdets konvergens mot 1. Vi illustrerar även dessa resultat med hjälp av simuleringar. För den gaussiska unitära ensemblen presenterar vi ett uttryck för dess nivåtäthet. För att underlätta förståelsen av den gaussiska symplektiska ensemblen presenterar vi egenskaper hos egenvärdena av kvaternioniska matriser. Slutligen bevisar vi en sats om symmetrin hos ordningsstatistikan av egenvärdena av beta-Hermite ensemblerna.

La présente thèse est consacrée à l’étude de l’effet d’une perturbation sur le spectre d’une matrice hermitienne perturbée par une matrice aléatoire de petite norme opérateur et dont les entrées dans la base propre de la première matrice sont indépendantes, centrées et possèdent un profil de variance. Ceci est réalisé au travers de développements perturbatifs de divers types des lois spectrales des grandes matrices perturbées considérées. Dans un premier temps, nous démontrons différents développements perturbatifs de la mesure spectrale empirique dans les cas du régime perturbatif et du régime semi-perturbatif et mettons en évidence des modèles heuristiques bien connus en physique, comme la transition entre les régimes semi-perturbatifs et perturbatifs. Dans un deuxième temps, nous proposons une étude approfondie du régime semi-perturbatif et prouvons le fait nouveau que ce régime peut être décomposé en un nombre infini de sous-régimes. Enfin, nous démontrons, au travers d’un développement perturbatif des mesures spectrales associées à un vecteur donné, un développement perturbatif des coordonnées des vecteurs propres des matrices perturbées que nous considérons
The present thesis is devoted to the study of the effect of a perturbation on the spectrum of a Hermitian matrix by a random matrix with small operator norm and whose entries in the eigenvector basis of the first one were independent, centered and with a variance profile. This is carried out through perturbative expansions of various types of spectral laws of the considered perturbed large matrices. First, we demonstrate different perturbative expansions of the empirical spectral measure in the cases of the perturbative regime and the semi-perturbative regime and highlight well known heuristic patterns in Physics, as the transition between semi-perturbative and perturbative regimes. Secondly, we provide a thorough study of the semi-perturbative regime and prove the new fact that this regime could be decomposed into infinitely many sub-regimes. Finally, we prove, through a perturbative expansion of spectral measures associated to the state defined by a given vector, a perturbative expansion of the coordinates of the eigenvectors of the perturbed matrices

In this thesis we study the dependence of the eigenvalues of elliptic partial differential operators upon mass density perturbations on open subsets of the N-dimensional euclidean space. We prove continuity and analyticity results for the eigenvalues of poly-harmonic operators and apply them to certain optimization problems. In order to prove analyticity, we use a general technique of P.D. Lamberti and M. Lanza de Cristoforis, and we obtain formulas for the Frechet differentials of the eigenvalues which are used to characterize critical mass densities under the constraint that the total mass is preserved. Then we state a sort of `maximum principle' in spectral optimization problems for elliptic operators subject to mass density perturbations. Moreover, we consider a special class of densities, namely densities which concentrate near the boundary of open subsets of the N-dimensional euclidean space. We study the asymptotic behavior of the eigenvalues of Neumann-type problems for the Laplace and the biharmonic operator. By adapting a general technique of J.M. Arrieta, we prove that the Neumann eigenvalues converge to the appropriate limiting Steklov eigenvalues. In this way, we formulate a genuine Steklov eigenvalue problem for the biharmonic operator. In the case of the Laplace operator we prove the validity of an asymptotic expansion of the Neumann eigenvalues and eigenfunctions and provide formulas for the first terms in the expansions. We adapt to our case asymptotic analysis techniques used by M.E. Perez and S.A. Nazarov to describe vibrating systems with masses concentrated at points or along curves. Moreover, we consider the problem of domain perturbations for the biharmonic Steklov problem obtained with this mass concentration procedure and prove that balls are critical domains for all the eigenvalues. Then we adapt the arguments of F. Brock and R. Weinstock to prove that the ball is actually a maximizer for the rst positive eigenvalue among bounded domains of given measure. Moreover, we provide a quantitative version of such an isoperimetric inequality, showing also that it is sharp.
In questa tesi studiamo la dipendenza degli autovalori di operatori differenziali alle derivate parziali di tipo ellittico da perturbazioni della densità di massa su aperti dello spazio euclideo N-dimensionale. In particolare, proviamo risultati di dipendenza continua e analitica degli autovalori di operatori poliarmonici e li applichiamo ad alcuni problemi di ottimizzazione. Per provare i risultati di analiticità, adoperiamo una tecnica generale sviluppata da P.D. Lamberti e M. Lanza de Cristoforis, ottenendo formule per i differenziali di Frechet degli autovalori che ci permettono di caratterizzare le densità critiche sotto il vincolo di massa fissata. Inoltre, enunciamo un `principio di massimo' per la classe di problemi di ottimizzazione considerata. In seguito, prendiamo in esame una famiglia particolare di densità di massa, ovvero densità che si concentrano al bordo degli aperti dove i problemi differenziali sono definiti. In questo caso, studiamo il comportamento asintotico degli autovalori e delle autofunzioni dei problemi di Neumann per l'operatore di Laplace e l'operatore biarmonico quando la massa si concentra al bordo. Proviamo in entrambi i casi, adattando una tecnica generale sviluppata da J.M. Arrieta, che gli autovalori e le autofunzioni del problema di Neumann convergono agli autovalori e alle autofunzioni di appropriati problemi limite di tipo Steklov. In particolare, il problema di tipo Steklov per l'operatore biarmonico così formulato viene introdotto per la prima volta in questa tesi, dove ne vengono poi studiate alcune proprietà. Nel caso dell'operatore di Laplace, proviamo la validità di un'espansione asintotica degli autovalori e delle autofunzioni del problema di Neumann fino al primo ordine ed otteniamo formule esplicite per i primi termini delle espansioni. Per ottenere questi risultati adattiamo al nostro problema delle tecniche di analisi asintotica utilizzate da M.E. Perez e S.A. Nazarov per lo studio di sistemi vibranti con masse concentrate in punti o lungo certe curve. Per quanto riguarda il problema di Steklov per l'operatore biarmonico, consideriamo anche il problema della dipendenza degli autovalori dal dominio. Utilizzando sempre la tecnica generale sviluppata da P.D. Lamberti e M. Lanza de Cristoforis, proviamo che le palle sono domini critici per tutti gli autovalori. Inoltre, adattando l'argomento di F. Brock e R.Weinstock per il problema di Steklov per l'operatore di Laplace, riusciamo a mostrare che la palla massimizza il primo autovalore positivo del problema di Steklov per l'operatore biarmonico tra tutti gli aperti limitati di misura fissata. Proviamo infine una versione quantitativa di questa disuguaglianza isoperimetrica, mostrando poi che l'esponente che compare nella disuguaglianza è ottimale.

Cette thèse traite de certaines propriétés spectrales de deux classes spécifiques des opérateurs périodiques. Nous nous intéressons tout d’abord à un modèle périodique perturbée par un opérateur dépendant d’un petit paramètre semi-classique. Nous obtenons alors le comportement asymptotique de la fonction du comptage des valeurs propres dans les gaps spectrales avec une estimation optimale du reste. Le second modèle étudié dans cette thèse est un modèle elliptique périodique d’ordre deux perturbée par un opérateur dépendant d’une grande constante de couplage. Nous donnons également la description de la fonction de compactage des valeurs propres lorsque la constante de couplage tend vers l’infini. La dernière partie de cette thèse discute l’étude du spectre discret de l’opérateur de Schrödinger avec un potentiel très oscillent dépendant d’un petit paramètre semi-classique
This Ph.D thesis deals with some spectral properties of two specific classes of two periodic operators. We are firstly interested in the model periodic perturbed by operator depending on a small semi-classical constant. We obtain an asymptotic behavior of the eigenvalue counting function in the spectral gaps with scharp remainder estimate. The second model studied in this thesis is a two-dimensional periodic elliptic second order opera-tor perturbed by operator depending on a large coupling constant. We also give the description of the counting function of eigenvalues when the coupling constant tends to infinity. The last part of this thesis highlights the study the spectrum of a Schrödinger operator perturbed by a fast oscillatingdecaying potential depending on a small parameter

Le but de cette thèse est triple : INÉGALITÉS DE SOBOLEV AVEC DES CONSTANTES EXPLICITES SUR DES VARIÉTÉS RIEMANNIENNES À DENSITÉ ET À BORD CONVEXE : On obtient des inégalités de Sobolev à densité, avec des constantes géométriques explicites pour des variétés à courbure de m-Bakry-Émery Ricci minorée par une constante positive et à bord convexe. Ceci permet de généraliser de nombreux résultats connus dans le cas riemannien aux variétés avec densité. Nous montrons aussi comment déduire des inégalités de Sobolev obtenues, un résultat d’isolement pour les applications f -harmoniques. Nous présenterons également une nouvelle et très simple méthode pour la preuve de l’inégalité de Moser-Trudinger-Onofri [Onofri, 1982] dans le cas du disque euclidien
The purpose of this thesis is threefold: SOBOLEV INEQUALITIES WITH EXPLICIT CONSTANTS ON A WEIGHTED RIEMANNIAN MANIFOLD OF CONVEX BOUNDARY: We obtain weighted Sobolev inequalities with explicit geometric constants for weighted Riemannian manifolds of positive m-Bakry-Emery Ricci curvature and convex boundary. As a first application, we generalize several results of Riemannian manifolds to the weighted setting. Another application is a new isolation result for the f -harmonic maps. We also give a new and elemantry proof of the well-known Moser-Trudinger-Onofri [Onofri, 1982] inequality for the Euclidean disk

Traditionally, the dynamic analysis of Multi degree Systems are based on the study of modal Shapes or energy balances given an established oscillation figure. The expert user of those methods could determine how resonance affects the dynamic behaviour of the analysed structure under time dependent variables as wind, moving loads, human steps, etc. or response spectrums (seismic). This Doctoral thesis propases a calculus methodology whose purpose is to obtain the characteristic modal shapes and energy balances of a tensile structure. Thus, an idealized tensile structure which is modeled using beams with only axial forces could be analyzed in order to get its natural vibrations, modal shapes and tridimensional dynamic behaviour. The expert use of the obtained results conforms the base for the analysis of the most common dynamic situations that can be considered when a tensile structure is designed. Proposed methodology is based on: . Force density method. . Classic analysis of dynamics of structures. . Modal Spectrum Analysis . . Energy balances: Rayleigh Ritz method. Starting from these fundamentals, a sequence of physical and mathematical principies is used to obtain and verify the natural vibration modes for a given tensile structure. The highlights of . the procedure are: . Deduction of the stiffness matrix using the force density method: an adaption of the general method is used to substantiate a stiffness relation among applied forces and displacements on movable nodes. This is possible due to a "static condensation" of part of the system and the deduction of a Dynamic Force Density for each beam that links load and length for a given prestressing level, taking in consideration section, materials and other corrections. Finally, it is concluded that geometry, loads on movable nodes, supports, prestressing loads and beam properties are linked to each other through a "dynamic stiffness matrix" . Classic Dynamic Analysis and Modal Spectrum Methods: taking as the starting point the previous "dynamic stiffness matrix", a relation to the mass of the system (having into consideration geometry, loads and materials) is formulated. Furthermore, it is possible to outline the classic eigenvalue's problem whose resolution is necessary to obtain the modal shapes of a system with multiple degrees of freedom . lt is important to consider the singularities of the problem because X, Y, and Z are independent variables. Synchronization among directions is required because it affects to the tridimensional analysis of the movement when the natural frequencies are studied and the total movement is composed. . Energy balances: They are mostly used to verify results previously obtained. The proposal contains a software as part of the Thesis, whose development is based in the exposed method. The software allows the simulation of the most common tensile structures, showing its modal shapes and natural frequencies
El análisis dinámico de los sistemas de múltiples grados de libertad se plantea tradicionalmente estudiando sus Modos Propios de oscilación o mediante los balances de energia que se establecen sobre formas de oscilación supuestas. A partir de dichas técnicas el analista determinará la sensibilidad a la resonancia del sistema ante acciónes externas dinámicas (viento, circulación de vehículos, paso humano, etc ... ), o acotará el potencial comportamiento del sistema en base a los espectros de respuesta (sismo). Esta Tesis Doctoral propone una metodología de cálculo que permitirá obtener las diferentes formas modales y balances energéticos asociados a la dinámica de una malla tesa, de manera que para una tenso estructura idealizada en barras se determinarán sus frecuencias fundamentales, las formas características de los modos propios asociados a las diferentes frecuencias, así como la composición de la oscilación en "X, Y, Z". Con dicha información el analista tendrá la base con la cual interpretar el comportamiento dinámico potencialmente resultante en una malla tesa en las situaciones ingenierilmente más comunes. La metodologia propuesta se basa en los siguientes fundamentos del cálculo de estructuras. . Método de la Densidad de Fuerza. . Dinámica clásica de sistemas estructurales. . Análisis Modal Espectral. . Planteamiento Energético . Rayleigh Ritz. Partiendo de los fundamentos anteriormente expuestos realizaremos una secuenciada asociación de principios físicos y matemáticos que nos permitirá deducir y verificar las formas modales de vibración y las frecuencias asociadas a una tenso estructura dada. El proceso sigue la siguiente pauta: . Deducción de la rigidez dinámica a partir del método de la Densidad de Fuerza: Se realizará una adaptación del método general de cálculo que nos permitirá plantear para los nudos móviles del sistema una relación de rigidez entre los desplazamientos y las fuerzas aplicadas. Ello se logra gracias a la "condensación en X,Y,Z" de la parte del sistema asociada a los apoyos, y a la deducción de unas "densidades dinámicas de fuerza" que vinculan -para un pretensado determinado- la tensión y la longitud de la barra en equilibrio con la sección y el material correspondiente a la estructura real, incluyéndose además las pertinente correcciones asociadas al módulos de elasticidad Tangente. Se relacionan pues las coacciones externas, la geometría, el pretensado de equilibrio, y las propiedades de las barras que vinculan entre si a los nudos en la estructura real. Entendemos por nudo móvil del sistema aquellos que no son apoyos y que por tanto son susceptible de oscilación. . Dinámica clásica de sistemas estructurales y Análisis Modal: Planteada la citada "rigidez dinámica" estableceremos una asociación de masas (basada en la geometría, materiales y cargas actuantes) que nos permitirá desarrollar el problema clásico de análisis dinámico para múltiples grados de libertad, pero atendiendo a las singularidades del caso por ser un movimiento cuyo análisis se realiza de forma disociada en "X, Y, Z". Se deducirán de ello las formas y frecuencias modales de oscilación (idénticas para los distintos ejes por ser el movimiento forzosamente sincronizado en los ejes) y se analizará la influencia de la composición del movimiento "X, Y, Z" en la frecuencia. Principios energéticos: Realizaremos la verificación energética para los resultados obtenidos de la dinámica clásica. La propuesta se complementa con un software informático que materializa la aplicación del método.

This thesis presents a means of enhancing the iterative calculation techniques used in electronic structure calculations, particularly Kohn-Sham DFT. Based on the subspace iteration method of the FEAST eigenvalue solving algorithm, this nonlinear FEAST algorithm (NLFEAST) improves the convergence rate of traditional iterative methods and dramatically improves their robustness. A description of the algorithm is given, along with the results of numerical experiments that demonstrate its effectiveness and offer insight into the factors that determine how well it performs.

碩士
國立中山大學
應用數學系研究所
93
In this thesis, we prove 2 theorems. First let ρ0 be a minimizing (or maximizing) density function for the first antiperiodic eigenvalue λ1'' in E[h,H,M], then ρ0=hχ(a,b)+Hχ[0,π]/(a,b) (or ρ0=Hχ(a,b)+hχ[0,π]/(a,b)) a.e. Finally, we prove minλ1''=minμ1=minν1 where μ1 and ν1 are the first Dirichlet and second Neumann eigenvalues, respectively. Furthermore, we determine the jump point X0 of ρ0 and the corresponding eigenvalue λ1'', assuming that ρ0 is symmetric about π/2 We derive the nonlinear equations for this jump point X0 and λ1'',then use Mathematica to solve the equations numerically.

In this thesis we analysed the problem of a single structural change occurring at some unknown data in multivariate time series. Our results rely on the assumption that the multivariate time series is generated by a dynamic factor model. Dynamic factor analysis is a very rich methodology which can be extended in many way to get a closer approximation to complex economic reality. They attempt to capture the correlation structure of a large number of original variables with a small set of common factors, in order to reduce the dimensionality of the vector space of the original variables. Three di_erent type of breaks has been analysed: the break in the mean level, the break in the factor loadings and the break in the factor moments. For each of them we suggest a model and therefore we focus the attention on the population and sample moments. When a break occurs, the data-generating process is not stationary anymore. The break in level affects the first moment of the process but the variance is still stationary whereas the other break types affect the second order moments. Furthermore we showed that the estimates are always affected by the break. Given these preliminary results we are interested in the Fourier transform of the estimated variance covariance matrices. For Geweke (1977) we know that all variation in the observed data may be decomposed into variance across frequencies using spectral techniques and, under some restrictions, much of the variation of the observable variables at low frequencies can be attributed to the common factors. Then, in order to investigate on the common factors number, we analyse the eigenvalues of estimated Fourier transform of the variance covariance matrices evaluated at frequencies close to zero. The target has been to understand if and in which way the break affects these matrices and their eigenvalues.