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Los sistemas de ecuaciones lineales o problemas de mínimos cuadrados aparecen en un amplio abanico de aplicaciones científico-técnicas. En ocasiones la matriz ligada al problema presenta una estructura banda o bien es una matriz dispersa que puede ser convertida en una matriz banda; en estos casos explotar la estructura banda de la matriz puede reducir considerablemente el coste computacional y de almacenamiento de resolución del problema. El objetivo principal de la presente tesis es el diseño, desarrollo y evaluación de una biblioteca de rutinas para la resolución de sistemas de ecuaciones lineales y problemas de mínimos cuadrados con estructura banda sobre arquitecturas de altas prestaciones. Tras evaluar la eficiencia y funcionalidad de las bibliotecas LAPACK y BLAS para operar con matrices banda, se han propuesto nuevas implementaciones más eficientes para las operaciones contempladas en estas bibliotecas, así como nuevas rutinas que implementan operaciones que amplian su funcionalidad, como por ejemplo una rutina para el cálculo de la factorización QR de una matriz banda. Los nuevos códigos se han aplicado a problemas de resolución de modelos, demostrando su eficiencia y escalabilidad.

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Heilbronn proved that for any epsilon > 0 there exists a number C (epsilon) such that for any real numbers theta and N(≥ 1) min ||n2 theta < C(epsilon) N-1/2+epsilon In the first part of this thesis we prove various extensionsof this result. We find values of g(r, k, s) so that the inequality [equation] is soluble, where X is an integral s-dimensional vector and the f's are either polynomials (without constant term) or forms, both of degree &le;1 k. The method used depends upon estimates for certain exponential sums. Using Weyl's estimates, we look, in Chapter 2, at monomials of different degree, and, in Chapter 3, at additive forms of degree k in s variables and quadratic polynomials. Using Hua's improvement of Vinogradov's estimates, we improve, for large values of k, the results of Chapter 2 and the results of Chapter 3 on additive forms. Also using Hua's estimates, we look, in Chapter 6 at polynomials of any degree (&le; k). In the course of this work we improve some results of Liu and Cook. Birch, Davenport and Ridout proved that if Q is an indefinite quadratic form in n( &ge; 21) variables, the inequality |Q (X)| < epsilon, has an integral solution X with |x| &ge;. In the second part of the thesis we investigate the inequality [equation] where Q is an indefinite quadratic form in n (&ge;21) variables of rank r, and k = min (r, n-r). We find values for f(n, k) and show that (i) for k &ge; 6, lim(N->infinity) f(n, k) = 1/2, (ii) for k &ge; 7, 10k 3n lim f(n, k) = &frac12;, and (iii) for 10k &ge; 3n, lim(N->infinity) f(n, k) = 1/2, k->infinity.

This thesis looks at finite subgroups of the projective group of 2 x 2 matrices over a skew field and the invariants of these subgroups. Chapter 0 recalls most of the preliminary results needed in subsequent chapters. In particular the construction of K_k(x) is. outlined briefly. Chapter 1 establishes an isomorphism between the group of tame automorphisms in one variable over the skew field K and the projective group of 2 x 2 matrices over K, PGL₂(K). It shows that if K is of suitable characteristic, then any element A of PGL₂(K) of finite order has either two or else infinitely many fixed points in some extension of K. In particular this means that such A can be diagonalized. Chapter 2 is divided into three sections. The first section deals with finite subgroups of PGL₂(K) whose elements may have infinitely many fixed points. The second section analyses finite cyclic subgroups whose elements have only two fixed points. The third section finds the finite non-diagonal groups in PGL₂(K) whose elements have exactly two fixed points. In particular a complete classification is given of the finite subgroups of PGL₂(K) when the centre k of K is algebraically closed. Chapter 3 shows that if the centre k of K is algebraically closed, then, any finite subgroup of PGL₂(K) is infact conjugate to one in PGL₂(k). It finds the fixed fields in K_k(x) of the finite subgroups of PGL₂(k) and shows that their respective generators are the same as in the commutative case.

We study the -adic completion dD[1] of Berthelot's differential operators of level one on the projective line over a complete discrete valuation ring of mixed characteristic (0; p). The global sections are shown to be isomorphic to a Morita context whose objects are certain fractional ideals of primitive factor rings of the -adic completion of the universal enveloping algebra of sl2(R). We produce a bijection between the coadmissibly primitive ideals of the Arens Michael envelope of a nilpotent finite dimensional Lie algebra and the classical universal enveloping algebra. We make limited progress towards characterizing the primitive ideals of certain a noid enveloping algebras of nilpotent Lie algebras under restrictive conditions on the Lie algebra. We produce an isomorphism between the primitive factor rings of these affinoid enveloping algebras and matrix rings over certain deformations of Berthelot's arithmetic differential operators over the a fine line.

In this thesis we obtain new results on the structures of real C*-algebras and nonsurjective isometries between them. Some of the results have been published in [1]. We prove a spectral inequality for real Banach*-algebras and give characterisations of real C*-algebras among Banach*-algebras. We study the ideal and facial structures in real C*-algebras and show that there is a bijection from the class of norm-closed left ideals I of a real C*-algebra A to the class of weak*-closed faces F of the state space S(A). The bijection is given by I 7! F = f 2 S(A) : (a a) = 0 for all a 2 Ig, with inverse F 7! I = fa 2 A : (a a) = 0 for all 2 Fg. As an application, we use the structures of faces to show an algebraic property of linear maps on real C*-algebras. We prove that if T : A ! B is a linear contraction between real C*-algebras A and B, then there is a projection p in the second dual B00 of B such that T(aa a)p = T(a)T(a) T(a)p (a 2 A). If T is an isometry, not necessarily surjective, we obtain a stronger result which also extends a celebrated result of Kadison on surjective isometries between complex C*-algebras. We prove the following theorem. Let T be a linear isometry between two real C*-algebras A and B, which can be non-surjective. Then for each a 2 A there exists a partial isometry u 2 B00 and a projection p 2 B00 such that (i) fu; T(ff; g; hg); ug = fu; fT(f); T(g); T(h)g; ug; (ii) T(ff; g; hg)p = fT(f); T(g); T(h)gp, for all f; g; h in the real JB*-triple A(a) generated by a 2 A, where ff; g; hg is the triple product defined by 2ff; g; hg = fg h + hg f. Moreover, fu; T( ):ug : A(a) ! B00 and T( )p : A(a) ! B00 are isometries. This theorem cannot be proved by simple complexification. We give an example of a real linear isometry which cannot be complexified to a complex isometry. We conclude by proving a theorem which states that a Jordan*-homomorphism T : A ! B between real C*-algebras A and B is a sum of a C*-homomorphism and a C*-antihomomorphism, extending a well-known result for complex C*- algebras.

In 1965 J. Ginibre introduced an ensemble of random matrices with no symmetry conditions imposed as the mathematical counterpart to hermitian random matrix theory. In his original paper he treats the case of matrices with i.i.d. normally distributed real, complex or quaternion entries. Since then, mainly due to interest from applications, the development of non-hermitian random matrix theory has further evolved, though the eigenvalue statistics of non-hermitian random matrices are far from being as thoroughly understood as their hermitian counterpart. A characteristic of non-hermitian random matrices are eigenvalue distributions in the complex plane. Real asymmetric random matrices have the additional caveat of having real and complex eigenvalues and thus are technically more challenging. In the following work a new three-fold family of non-hermitian random matrices is introduced via a quadratization procedure. As a consequence the entries of these matrices are highly dependent. For all three ensembles the joint eigenvalue probability density functions and eigenvalue correlations are derived for β = 1, 2. In the limit of large matrix dimensions a classification of eigenvalue correlation functions for different asymptotic regimes is undertaken. In tune with the title of this work for all three ensembles there exists an asymptotic regime, in which the eigenvalues are supported on an annulus around the origin. Thus the induced family of non-hermitian random matrix ensembles serves as an example, for ensembles of the Feinberg-Zee type with logarithmic potential.

L'importance des opérateurs non auto-adjoints dans la physique moderne augmente chaque jour, car ils commencent à jouer un rôle plus important dans la mécanique quantique. Cependant, la signification de leur examen est beaucoup plus récente que l'intérêt pour l'examen des opérateurs auto-adjoints. Ainsi, étant donné que de nombreuses techniques auto-adjointes ne sont pas généralisées à ce contexte, il n’existe pas beaucoup de méthodes bien développées pour examiner leurs propriétés. Cette thèse vise à contribuer à combler cette lacune et démontre plusieurs modèles non auto-adjoints et les moyens de leur étude. Les sujets comprennent le pseudo-spectre comme un analogue approprié du spectre, un modèle d'une guide d'onde avec un gain et une perte équilibrés à la frontière et l'équation de Kramers-Fokker-Planck avec un potentiel à courte distance. / The importance of non-self-adjoint operators in modern physics increases every day as they start to play more prominent role in Quantum mechanics. However, the significance of their examination is much more recent than the interest in the examination of their selfadjoint counterparts. Thus, since many selfadjoint techniques fail to be generalized to this context, there are not many well-developed methods for examining their properties. This thesis aims to contribute to filling this gap and demonstrates several non-self-adjoint models and the means of their study. The topics include pseudospectrum as a suitable analogue of the spectrum, a model of a quantum layer with balanced gain and loss at the boundary, and the Kramers-Fokker-Planck equation with a short-range potential.

SUMARYA group is said to be locally finite if every finite subset of G generates a fi-nite subgroup. The class of locally finite groups is placednear the cross-roads of finite group theory and the general theory of infinite groups. Many theoremsabout finite groups can be phrased in such a way that their statements still make sense for locally finite groups. However, in general, Sylow's Theorems do not hold in the class of locally finite groups and there are a numberof generic examples which show that locally finite groups can be very varied and complex. If we restrict our attention to locally finite-soluble groups with min-p for all primes p then the Sylow ¼-subgroups are very well behavedif ¼ or its complementary in the set of all primes is finite. The conjugacy of Sylow p-subgroups in these groups is a very strong condition which have guaranteed the successful development of formation theory and interestingresults on Fitting classes in the universe c¯L of all radical locally finite groups with min-p for all primes p. Moreover, using an extension of the Frattini subgroup introduced by Tomkinson, it has been proved a Gasch¨utz-Lubeseder type theorem characterizing saturated formations in this universe.It is therefore appropriate to study the class c¯L of all radical locally finitegroups with min-p for all primes p in more detail. In this thesis we haveobtained results which help to understand better the groups in this class.Consequently, the unspoken rule is that all groups considered in the threechapters of this thesis belong to the class c¯L. The work is organized as follows.In Chapter 1, we explore the class B of generalized nilpotent groups inthe universe c¯L. We obtain that this class behaves in the universe c¯L as thenilpotent groups in the finite universe and we determine the structure of B-groups explicitly. Moreover, we show that the largest normal B-subgroup ofa c¯L-group is the Fitting subgroup. This fact allows us to prove some results1concerning the Fitting subgroup of a c¯L-group which are extensions of thefinite ones. The aim of the last section is to study the injectors associatedto the class B. In fact, we obtain a description of the B-injectors similar tothe characterization of nilpotent injectors of a finite soluble group.Chapter 2 is devoted to study the local version of the class B. This isa natural generalization of the class of finite p-nilpotent groups. We extendsome results of finite groups to the above universe using a local version ofa Frattini-like subgroup. In particular, some properties appear relating theFrattini and Fitting subgroups. The injectors associated to this class ofgeneralized p-nilpotent groups are also characterized.Finally, Chapter 3 is concerned with the structure of a radical locallyfinite group with min-p for all p, G = AB, factorized by two subgroups Aand B in the class B. We extend the well-known results of finite productsof nilpotent groups to the above universe.We have introduced a Chapter 0 establishing the notation and terminology.It also presents many of the well-known results that will be usedthroughout this thesis.