The stiffness properties of a short narrow rectangular beam as modified by a primary bending moment and shear stress distribution in the major plane are presented. The beam is a segment taken from a longer member the "structure." A distribution of bending stress is assumed over the beam segment length and its effect on the stiffness properties in lateral bending and torsion obtained. The stiffness matrix is used to obtain the critical value of load for a number of well known examples of narrow rectangular beams and the results are shown to be in good agreement. The results of an energy solution, which produces a symmetrical matrix, are presented. Comparison with classical examples shows accurate results with the added benefit that the symmetrical matrix lends itself much more readily to more complicated problems. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Plane stress stiffness matrices are derived explicitly for square Isotropic elements under different assumptions on the stress distribution. An explicit (8 x 8) matrix is obtained under the assumption of uniform σx, σy, linear τxy and thus it is shown that the Gallagher matrix belongs to the class of parametric matrices. Two (10 x 10) matrices are obtained under the assumption of linear σx, σy, τxy using interior nodal translations and corner edge rotations respectively as additional generalized displacements. These two matrices do not appear suitable for general usage but will perform as well as the Turner matrix under the same nodal loads. A (12 x 12) matrix is derived under the assumption of hyperbolic σx, σy, and parabolic τxy, again exemplifying the use of corner edge rotations as additional generalized displacements. This matrix behaves unexpectedly with varying Boisson's ratio. A method of evaluating stiffness matrices, which reduces the necessity of comparing finite element solutions with analytical ones, is formulated. In this method a comparison is made of the strain energy of deformation produced within a finite element by the different matrices under the same nodal loads. It is shown that such comparisons require the study of special matrices i.e. the stiffness difference matrix and the inverse difference matrix which are obtained from the matrices under comparison. It is proved that the results of the element matrix comparisons apply to the structure. It is shown that the strain energy of a finite element under normalised loads is bounded between the maximum and minimum eigenvalues of the inverse matrix. The strain energy comparison criterion is used in the study of parametric matrices. An explicit parametric inverse is obtained. Explicit parametric eigenvalues are obtained for the inverse difference matrix and the stiffness difference matrix, and it is verified that they give identical results for the matrix comparisons. It is proved that the parametric matrices produce the exact strain energy under uniform nodal loads. It is shown that the stiffness matrix parameter and the inverse matrix parameter represent a measure of the strain energy under non-uniform nodal loads so that the strain energy can always be bounded by varying the parameter. It is proved that if strain energy curves are drawn with respect to structure sub-division then no two curves will intersect. It is proved that all parametric strain energy curves will converge towards the true solution with progressive structure subdivision. A strain energy ordering is obtained for the parametric matrices and the following conclusions are drawn. The Pian matrix is the best displacement matrix. The Gallagher matrix is inferior to the Turner, Pian, and Argyris-Melosh matrices. Constant stress tri-nodal triangles are generally inferior to the use of square elements. Matrices satisfying microscopic equilibrium or capable of representing uniform stresses will not necessarily yield good results. A method is proposed for obtaining upper bounds on the strain energy of a region under plane stress by replacing the continuum with a psuedo-truss system, the bar forces of which provide the equilibrium and self-straining solutions. Two examples of its application are presented, and an indication is obtained that upper bounding by varying the matrix parameter will give better results for the same structure subdivision. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Les travaux présentés dans cette thèse s'inscrivent dans le cadre de l"algorithmique et de la combinatoire du texte et s'appliquent à la bio-informatique. Plus particulièrement, ils concernent la localisation de motifs pondérés modélisés par des matrices score-position dans un texte non pondéré. Ces travaux sont appliqués au problème biologique de la recherche de sites de fixation de facteurs de transcription dans un génome. Cette application contribue à la compréhension de la régulation des gènes. Nous nous sommes attaqués à deux problèmes complémentaires, la recherche d'une seule matrice dans un texte puis la recherche simultanée d'un ensemble de matrices. Pour accélérer les algorithmes existant, nous nous sommes inspiré des algorithmes de recherche de motifs exacts connus pour leur efficacité. La différence est que les matrices score-position sont des motifs probabilistes, utilisant des fonctions de score. Nous devons donc intégrer la distribution de ces fonctions dans les algorithmes de recherche. Concernant le premier problème nous proposons une extension de l'algorithme de Knuth, Morris et Pratt qui repose sur un pré-traitement du motif pour optimiser le parcours le long du texte. Concernant le second problème nous avons utilisé une structure d'indexation afin de factoriser l'ensemble des matrices. Cette structure tire partie des distributions de scores associées à chaque matrice. Dans les deux cas, nous traitons en amont une partie des données de départ. Nous avons choisi de pré-traiter les matrices par rapport à l'application bio-informatique car les sites de fixation de facteurs de transcription sont des données relativement stables dans le temps. Ces algorithmes ont été mis en oeuvre dans un logiciel disponible en ligne appelé TFMscan. Ils ont fait l'objet d'une validation à grande échelle sur les bases de données de facteurs de transcription Jaspar et Transfac. / The work presented in this thesis is a part of the algorithmic and the combination of text and is apply to bio-informatic. More precisely, they relate to the location of grounds weighted modeled by position weIght matrices scoring in an unweighted text. This work is applied to the problem of biological research binding sites of transcription factors in a genome. This applrcation contributes to understanding the regulation of genes. We tackled two problems, find a single. matnx in a text and then the simultaneous search of a whole matrices. To accelerate existing algonthms, we have been inspired by exact research algorithms known for their effectiveness. The difference is that the dice are scoring position probabilistic grounds usmg. functions score. We must integrate the distribution of these functions in the search algonthms. Regardmg the first problem we proposed an extension of the Knuth. Morris and Pratt algorithm witch is based on a pre-treatment ta optimize route of ground along the text .Regarding the second problem we used an indexing structure in order to factor all matnces. ThIs structure uses distributions of scores associated to each matrix. ln both cases we are dea!mg with upstream part of baselines. We chose to pre-treat matrices in relation to the bio-mformatic application because the binding sites of transcription factors data are relatively stable over time. These algorithms have been implemented in a software available online called TFMscan. They have been validated on a large-scale bases data transcription factors Jaspar and Transfac .

This paper is the result of a study of triangular matrices with particular emphasis on those which are convergence preserving transformations.

[EN] The essential topic of this memory is the study of H-matrices as they were introduced by Ostrowski and hereinafter extended and developed by different authors. In this study three slopes are outlined: 1) the iterative or automatic determination of H-matrices, 2) the properties inherent in the H-matrices and 3) the matrices related to H-matrices. H-matrices acquire every time major relevancy due to the fact that they arise in numerous applications so much in Mathematics, since in the Industry between. Between these applications we can mention the following ones: 1) in the discretization of certain parabolic non-linear equations, 2) in the system resolution of linear equations, assuring his presence the convergence of iterative classic methods and 3) in the resolution of problems of free contour in Analysis of Fluids. It is very important to observe that some H-matrices transform in H- matrices for the action of some matrix operation on them. Such it is the case of the matrix operation known as Hadamard's Product, that is to say, the product element to element of two matrices. If this product realizes between the elements of a matrix and the elements of its inverse transpose then this matrix product is called combined matrix. The combined matrix is an H- matrix under certain conditions of the original matrix and, in addition, the combined matrix is linked to applications very important as the Relative Gain in chemical processes or the relation between the eigenvalues of the original matrix and the elements of a diagonalizable matrix. In addition, provided that the sum of every row and of every column is equal to one, in those cases in which the combined matrix is not negative, C(A) is a doubly stochastic matrix and therefore it is of great usefulness in the Statistical Theory. The present memory is structured of the following way. In the first chapter, after the introduction, we present the notation, the basic concepts and previous results developed by other authors and that are going to be used largely in the memory. xiii xiv In the Chapter 2 we present and analyze different algorithms that have been proposed by the aim to determine when a given matrix is or is not an H-matrix. It is emphasized in the study of those algorithms that have turned out to be the most efficient and in the most relevant part of this chapter we present a new algorithm that turns out to be a contribution to the literature of the algorithms for the determination or identification of H-matrices, as well as of his character. In the Chapter 3 we widely studied the combined matrix of a nonsingular H-matrices and we obtain new and important properties of the combined matrix of H-matrices. In the Chapter 4 we calculate the combined matrix of diagonally dominant and equipotent matrices and also we obtain new and important results that relate the combined matrix of these diagonally dominant and equipotent matrices to H-matrices. In Chapter 5, like summary, we outline the principal achievements reached during the development of this memory and, in addition, enumerate the works on which already we are working and also we present some of the principal lines of investigation for the near future. Finally, in the appendices we present, in format MATLAB, different algorithms studied in Chapter 2 that make the automatic determination of H-matrices as a purpose. Especially, is outlined the codification of the new algorithm proposed with each of its parts in the correct order to be run in the computer. / [ES] El tema esencial de esta memoria es el estudio de las H-matrices tal y como fueron introducidas por Ostrowski y más adelante ampliadas y desarrolladas por diferentes autores. En ese estudio se destacan tres vertientes: 1) la determinación iterativa o automática de las H-matrices, 2) las propiedades inherentes a las H- matrices y 3) las matrices relacionadas con las H-matrices. Las H-matrices adquieren cada vez mayor relevancia debido a que surgen en numerosas aplicaciones tanto en la ciencia Matemática como en la Industria. Entre esas aplicaciones podemos citar las siguientes: 1) en la discretización de ciertas ecuaciones parabólicas no lineales, 2) en la resolución de sistemas de ecuaciones lineales, asegurando su presencia la convergencia de métodos iterativos clásicos y 3) en la resolucion de problemas de contorno libre en Análisis de Fluidos. Es de suma importancia observar que algunas matrices devienen en H- matrices por la acción de alguna operación matricial sobre ellas. Tal es el caso de la operación matricial conocida como Producto de Hadamard, es decir, el producto elemento a elemento de dos matrices. Si este producto se realiza entre los elementos de una matriz y los elementos de su matriz inversa traspuesta, entonces la matriz producto, denominada Matriz Combinada, puede ser una H-matriz bajo determinadas condiciones de la matriz original y, además, la matriz combinada está vinculada a aplicaciones muy importantes como la Ganancia Relativa en procesos químicos o la relación entre los valores propios de la matriz original y los elementos de una matriz diagonalizable. Además, dado que la suma de cada fila y de cada columna de una matriz combinada es exactamente igual a 1, en aquellos casos en que la matriz combinada sea no negativa, C(A) es una matriz doblemente estocástica y por tanto puede ser de gran utilidad en Estadística y Probabilidad. La memoria está estructurada por capítulos de la siguiente manera. En cada uno de ellos se presentan las aportaciones de la misma. ix x En el Capítulo 1, luego de la introducción, se da la notación y se definen los conceptos básicos y, además, se enuncian los resultados previos de ámbito general desarrollados por otros autores y que van a ser utilizados en gran parte de la memoria. En el Capítulo 2 se presentan y analizan diferentes algoritmos que han sido propuestos con el objetivo de determinar cuándo una matriz dada es o no es una H-matriz. Se hace hincapié en el estudio de aquellos algoritmos que han resultado ser los más eficientes y en la parte más relevante de este capítulo se presenta un nuevo algoritmo de menor coste computacional que los anteriores y más sencillo de programar, que resulta ser un aporte a la literatura de los algoritmos para la determinación o identificación de las H-matrices, así como de su carácter y también determina los bloques diagonales irreducibles. En el Capítulo 3 se estudia ampliamente la matriz combinada de H- matrices no singulares y se obtienen también nuevos e importantes resultados sobre las propiedades de la matriz combinada de H-matrices. Se demuestra que la matriz combinada de una H-matriz de la clase invertible es también H-matriz de la misma clase. Además, se prueba que la matriz combinada de una H-matriz de la clase mixta no singular es también H-matriz. En el Capítulo 4 se calcula la matriz combinada de matrices diagonalmente dominantes equipotentes. En particular, se demuestra que la matriz combinada de una H-matriz, denominada DmP es siempre una H-matriz de la clase mixta pero singular. Para otras H-matrices que no son DmP se prueba que su matriz combinada es H-matriz de la clase invertible. Se conjetura que todas las H-matrices de la clase mixta que no son DmP tienen esta última propiedad. En el Capítulo 5 se recogen, a modo de resumen, los principales logros alcanzados durante el desarrollo de esta memoria y, además, se enumeran los trabajos sobre los cuales ya se está trabajand / [CAT] El tema essencial d'aquesta memòria és l'estudi de les H-matrius tal com van ser introduïdes per Ostrowski i més endavant ampliades i desenvolupades per diferents autors. En aqueix estudi es destaquen tres vessants: 1) la determinació iterativa o automàtica de les H-matrius, 2) les propietats inherents a les H-matrius i 3) les matrius relacionades amb les H-matrius. Les H-matrius adquireixen cada vegada major rellevància a causa que sorgeixen en nombroses aplicacions tant en la ciència Matemàtica com en la Indústria. Entre aqueixes aplicacions podem citar les següents: 1) en la discretització de certes equacions parabòliques no-lineals, 2) en la resolució de sistemes d'equacions lineals, assegurant la seua presència la convergència de mètodes iteratius clàssics i 3) en la resolució de problemes de contorn lliure en Anàlisi de Fluids. És de summa importància observar que algunes matrius esdevenen en H-matrius per l'acció d'alguna operació matricial sobre elles. Tal és el cas de l'operació matricial coneguda com a Producte de Hadamard, és a dir, el producte element a element de dues matrius. Si aquest producte es realitza entre els elements d'una matriu i els elements de la seua matriu inversa trasposada, llavors la matriu producte, denominada Matriu Combinada, pot ser una H-matriu sota determinades condicions de la matriu original i, a més, la matriu combinada està vinculada a aplicacions molt importants com el Guany Relatiu en processos químics o la relació entre els valors propis de la matriu original i els elements d'una matriu diagonalitzable. A més, atès que la suma de cada fila i de cada columna d'una matriu combinada és exactament igual a 1, en aquells casos en què la matriu combinada siga no negativa, C(A) és una matriu doblement estocàstica i per tant pot ser de gran utilitat en Estadística i Probabilitat. La memòria està estructurada per capítols de la següent manera. En cadascun d'ells es presenten les aportacions de la mateixa. En el Capítol 1, després de la introducció, es dóna la notació i es defixi xii neixen els conceptes bàsics i , a més, s'enuncien els resultats previs d'àmbit general desenvolupats per altres autors i que van a ser utilitzats en gran part de la memòria. En el Capítol 2 es presenten i analitzen diferents algorismes que han sigut proposats amb l'objectiu de determinar quan una matriu donada és o no és una H-matriu. Es posa l'accent en l'estudi d'aquells algorismes que han resultat ser els més eficients i en la part més rellevant d'aquest capítol es presenta un nou algorisme de menor cost computacional que els anteriors i mes senzill de programar, que resulta ser una aportació a la literatura dels algorismes per a la determinació o identificació de les H-matrius, així com del seu caràcter i també determina els blocs diagonals irreductibles. En el Capítol 3 s'estudia àmpliament la matriu combinada d'H-matrius no singulars i s'obtenen també nous i importants resultats sobre les propietats de la matriu combinada d'H-matrius. Es demostra que la matriu combinada d'una H-matriu de la classe invertible és també H-matriu de la mateixa classe. A més, es prova que la matriu combinada d'una H-matriu de la classe mixta no singular és també H-matriu. En el Capítol 4 es calcula la matriu combinada de matrius diagonalment dominants equipotents. En particular, es demostra que la matriu combinada d'una H-matriu, denominada DmP és sempre una H-matriu de la classe mixta però singular. Per a altres H-matrius que no són DmP es prova que la seua matriu combinada és H-matriu de la classe invertible. Es conjectura que totes les H-matrius de la classe mixta que no són DmP tenen aquesta última propietat. En el Capítol 5 s'arrepleguen, a manera de resum, els principals assoliments aconseguits durant el desenvolupament d'aquesta memòria i, a més, s'enumeren els treballs sobre els quals ja s'està treballant i s'esbossen algunes de les principal / Scott Guilleard, JA. (2015). Determinación y propiedades de H-matrices [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/58766 / TESIS

Nous étudions dans cette thèse une méthode hybride de résolution des systèmes linéaires GMRES/LS-Arnoldi qui accélère la convergence grâce à la connaissance des valeurs propres calculées parallèlement par la méthode d’Arnoldi dans les cas réels. Le caractère asynchrone de cette méthode présente l’avantage de fonctionner avec une architecture hétérogène. Une étude de cas complexe est également faite en effectuant la transformation de la matrice complexe en une matrice réelle de dimension double. Nous avons mis en oeuvre la méthode GMRES hybride ainsi que la méthode GMRES générale sur trois différents types de plates-formes matérielles. Il s’agit respectivement de supercalculateurs IBM série SP, plates-formes matérielles typiquement centralisées; de Grid5000, une plate-forme matérielle typiquement distribuée, et de Tsubame (Tokyo-tech Supercomputer and Ubiquitously Accessible Massstorage Environment) supercalculateur, dont certains noeuds sont munis d’une carte accélératrice. Nous avons testé les performances de GMRES général et de GMRES hybride sur ces trois plates-formes, en observant l’influence des nombreux paramètres sur les performances. Des résultats significatifs ont ainsi été obtenus, nous permettant non seulement d'améliorer les performances du calcul parallèle, mais aussi de préciser le sens de nos efforts futurs. / In this thesis, we have studied an effective parallel hybrid method of solving linear systems, GMRES / LS-Arnoldi, which accelerates the convergence through knowledge of some eigenvalues calculated in paralled by the Arnoldi method in real cases. The asynchronous nature of this method has the advantage of working with a heterogeneous architecture. A study in complex cases is also done by transforming the complex matrix into a real matrix of double dimension. We have implemented our hybrid GMRES method and the general GMRES method on three different types of hardware platforms. They are respectively the IBM SP series supercomputer, a typically centralized hardware platform; Grid5000, a fully distributed hardware platform, and the Tsubame (Tokyo-tech Supercomputer and Ubiquitously Accessible Massstorage Environment) supercomputer, where some nodes are equipped with an accelerator card. We have tested the performance of general GMRES and hybrid GMRES on these three platforms, observing the influence of various parameters for the performance. A number of meaningful results have been obtained; we can not only improve the performance of parallel computing but also specify the direction of our future efforts.

It is the purpose of this paper to discuss some important properties of unitary matrices and particularly to discuss the real unitary, or real orthogonal, matrices. The main results stated in this paper may be found in the works listed in the bibliography, but it is believed that the organization of the paper is such that the material will be more readily understandable and useful to the reader in the form in which it is here given. / Typescript. / "June, 1953." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: T. L. Wade, Professor Directing Paper. / Includes bibliographical references (leaf 65).

No description available.

Le célèbre théorème de Birkhoff affirme que l'espace Dₙ des matrices doublement stochastiques d'ordre n est un polytope convexe dont les matrices de permutation constituent les points extrémaux. De cette structure particulière émerge une structure géométrique intéressante que nous explorons en détail dans ce mémoire. Plus précisément, nous explorons quelques propriétés géométriques de Dₙ, vu comme un espace métrique muni de deux différents types de normes, à savoir les p-normes de Schatten et les normes d'opérateurs induites par les normes vectorielles ℓᵖ. En particulier, nous étudions la norme des matrices doublement stochastiques ainsi que le rayon de Tchebychev, les centres de Tchebychev et le diamètre de Dₙ. Ce faisant, de nouvelles connexions avec le célèbre problème d'affectation sont établies. Nous utilisons également les propriétés géométriques de Dₙ établies dans ce mémoire pour améliorer un résultat de Štefan Schwarz sur la convergence de produits infinis de matrices doublement stochastiques. / The celebrated Birkhoff theorem states that the space of n × n doubly stochastic matrices Dₙ is a convex polytope whose extreme points are the permutation matrices. From this particular structure emerges an interesting geometric structure that we explore in detail in this dissertation. Specifically, we explore some geometric properties of Dₙ, seen as a metric space equipped with two different type of norms, which are the Schatten p-norms and the operator norms induced by the ℓᵖ vector norms. In particular, we study the norm of the doubly stochastic matrices along with the Chebyshev radius, the Chebyshev centers and the diameter of Dₙ. In doing so, new connections with the well-known assignment problem are made. We also use the geometric properties of Dₙ established in this dissertation to improve a result of Štefan Schwarz about the convergence of infinite product of doubly stochastic matrices.

Le célèbre théorème de Birkhoff affirme que l'espace Dₙ des matrices doublement stochastiques d'ordre n est un polytope convexe dont les matrices de permutation constituent les points extrémaux. De cette structure particulière émerge une structure géométrique intéressante que nous explorons en détail dans ce mémoire. Plus précisément, nous explorons quelques propriétés géométriques de Dₙ, vu comme un espace métrique muni de deux différents types de normes, à savoir les p-normes de Schatten et les normes d'opérateurs induites par les normes vectorielles ℓᵖ. En particulier, nous étudions la norme des matrices doublement stochastiques ainsi que le rayon de Tchebychev, les centres de Tchebychev et le diamètre de Dₙ. Ce faisant, de nouvelles connexions avec le célèbre problème d'affectation sont établies. Nous utilisons également les propriétés géométriques de Dₙ établies dans ce mémoire pour améliorer un résultat de Štefan Schwarz sur la convergence de produits infinis de matrices doublement stochastiques. / The celebrated Birkhoff theorem states that the space of n × n doubly stochastic matrices Dₙ is a convex polytope whose extreme points are the permutation matrices. From this particular structure emerges an interesting geometric structure that we explore in detail in this dissertation. Specifically, we explore some geometric properties of Dₙ, seen as a metric space equipped with two different type of norms, which are the Schatten p-norms and the operator norms induced by the ℓᵖ vector norms. In particular, we study the norm of the doubly stochastic matrices along with the Chebyshev radius, the Chebyshev centers and the diameter of Dₙ. In doing so, new connections with the well-known assignment problem are made. We also use the geometric properties of Dₙ established in this dissertation to improve a result of Štefan Schwarz about the convergence of infinite product of doubly stochastic matrices.