We develop Hilbert-Kunz theory in a combinatorial setting namely for binoids. We show that the Hilbert-Kunz multiplicity for commutative, finitely generated, semipositive, cancellative and reduced binoids exists and is a rational number. This implies that the corresponding Hilbert-Kunz multiplicity for the binoid algebras does not depend on the characteristic.

Indiana University-Purdue University Indianapolis (IUPUI) / We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

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Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*-modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*-modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both research fields can benefit from achievements of the other field. Our purpose of this dissertation is to work on several basic problems on frames for Hilbert C*-modules. We first give a very useful characterization of modular frames which is easy to be applied. Using this characterization we investigate the modular frames from the operator theory point of view. A condition under which the removal of element from a frame in Hilbert C*-modules leaves a frame or a non-frame set is also given. In contrast to the Hilbert space situation, Riesz bases of Hilbert C*-modules may possess infinitely many alternative duals due to the existence of zero-divisors and not every dual of a Riesz basis is again a Riesz basis. We will present several such examples showing that the duals of Riesz bases in Hilbert $C^*$-modules are much different and more complicated than the Hilbert space cases. A complete characterization of all the dual sequences for a Riesz basis, and a necessary and sufficient condition for a dual sequence of a Riesz basis to be a Riesz basis are also given. In the case that the underlying C*-algebra is a commutative W*-algebra, we prove that the set of the Parseval frame generators for a unitary group can be parameterized by the set of all the unitary operators in the double commutant of the unitary group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also prove the existence and uniqueness of the best Parseval multi-frame approximations for multi-frame generators of unitary groups on Hilbert C*-modules when the underlying C*-algebra is commutative. For the dilation results of frames we show that a complete Parseval frame vector for a unitary group on Hilbert C*-module can be dilated to a complete wandering vector. For any dual frame pair in Hilbert C*-modules, we prove that the pair are orthogonal compressions of a Riesz basis and its canonical dual basis for some larger Hilbert C*-module. For the perturbation of frames and Riesz bases in Hilbert C*-modules we prove that the Casazza-Christensen general perturbation theorem for frames in Hilbert spaces remains valid in Hilbert C*-modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains a Riesz basis. However, this no longer holds for Riesz bases in Hilbert C*-modules. We also give a complete characterization on all the Riesz bases for Hilbert C*-modules such that the perturbation (under Casazza-Christensen's perturbation condition) of a Riesz basis still remains a Riesz basis.

<p> It is known that when two subspaces of a Hilbert space are in some sense close to each other, then there exists a unitary operator which is called the direct rotation. This operator maps one of the subspaces onto the other while being as close to identity as possible. In this thesis we study such a pair of subspaces, and the application of the angles between them to the invariant subspace perturbation theory We also develop an efficient algorithm for computing the direct rotation for pairs of subspaces of relatively small dimension. </p> / Thesis / Master of Science (MSc)

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Esta tesis tiene dos objetivos fundamentales. El primer objetivo es presentar y desa- rrollar una representación y dualidad topológica para variedades de álgebras que corres- ponden a los reductos {→} y {→, ∨} de la variedad de las álgebras de Heyting. Estas representaciones están basadas en un clase particular de espacios topológicos conocidos como espacios sober. Es un hecho bien conocido que toda álgebra de Heyting es repre- sentable como subálgebra del álgebra de Heyting de todos los subconjuntos crecientes de un conjunto ordenado. También es sabido que un álgebra de Heyting es representa- ble como una subálgebra del conjunto de todos los abiertos de un espacio topológico T0. Estas representaciones tienen muchas aplicaciones tanto en el estudio algebraico de es- tas estructuras como en las aplicaciones de la l ógica intuicionista Int y algunas de sus extensiones. Además, estas representaciones son la base para las conocidas dualidades topológicas de Priestley y de Stone para las álgebras de Heyting. Cuando miramos algún subreducto de las álgebras de Heyting, como por ejemplo, en las álgebras que correspon- den al fragmento implicativo, conocidas como álgebras de Hilbert, la teoría de represen- tación y dualidad desarrollada para las álgebras de Heyting no es directamente aplicable a estos fragmentos. El primer resultado que conocemos sobre representación de un álgebra de Hilbert se encuentra en la tesis de A. Diego [29]. En dicha tesis aparece un teorema de representación tipo Stone, pero este resultado no tuvo un impacto muy significativo ya que es insuficiente para desarrollar una dualidad categórica. El primer objetivo de esta tesis es, justamente, presentar una dualidad topológica completa para las álgebras de Hil- bert y extender esta dualidad a la variedad de las álgebras de Hilbert con supremo. Estos resultados están basados en los espacios topológicos conocidos como espacios sober y extienden a los dados por M. Stone [67]. Primero probamos que la categor´ıa formada por álgebras de Hilbert con semi-homomorfismos como morfismos es dualmente equivalente a la categoría de espacios de Hilbert con ciertas relaciones binarias. También obtenemos una dualidad para las álgebras de Hilbert con homomorfismos. Aplicamos estos resulta- dos para demostrar que el retículo de sistemas deductivos de un álgebra de Hilbert y el ret´ıculo de subconjuntos abiertos de su espacio de Hilbert dual, son isomorfos. Explo- ramos cómo esta dualidad está relacionada con la dada en [18] para álgebras de Hilbert finitas, y con la dualidad topológica desarrollada en [19] para álgebras de Tarski. Todos estos resultados son presentados en el Capítulo 3. La otra variedad asociada a un fragmento de la lógica Int que estudiamos es la va- riedad de las álgebras de Hilbert con supremo, i.e., álgebras de Hilbert donde el orden asociado es un supremo-semiret´ıculo. Extendemos la dualidad encontrada para las álge- bras de Hilbert al caso de las álgebras de Hilbert con supremo. Probamos que el conjunto ordenado de todos los ideales de un álgebra de Hilbert con supremo tiene estructura de retículo. Demostramos que en este retículo es posible definir una implicación, pero la estructura resultante no es un álgebra de Heyting ni tampoco es un semiretículo implica- tivo. Damos una descripción dual para el retículo de ideales de un álgebra de Hilbert con supremo. Estos resultados son presentados en el Capítulo 5. El segundo objetivo fundamental de esta memoria está centrado en estudiar algunas extensiones modales de las álgebras de Hilbert y de las álgebras de Hilbert con supre- mo. Estas extensiones corresponden a fragmentos de algunas extensiones modales de la l ógica intuicionista Int. En esta memoria nos hemos centrado únicamente en dos frag- mentos. Primero introducimos la variedad de álgebras de Hilbert con un operador mo- dal , llamadas H -álgebras. La variedad de H -álgebras es la contraparte algebraica del {→, }-fragmento de la lógica modal intuicionista IntK , al cual denotamos con IntK→. Estudiamos la teoría de representación y damos una dualidad topológica para la variedad de H -álgebras. Aplicamos estos resultados para probar que la l ógica modal implicativa IntK→ es canónica y por lo tanto es completa. Determinamos las álgebras simples y subdirectamente irreducibles en algunas subvariedades de H -álgebras. Tam- bien estudiamos una interesante variedad de álgebras, llamadas álgebras de Hilbert Lax. Todos estos resultados son presentados en el Caíıtulo 4. El otro fragmento que investigamos es el fragmento {→, ∨, ♦} de la lógica modal in- tuicionista IntK♦. Introducimos y estudiamos la variedad de H∨ ♦ -álgebras, las cuales son álgebras de Hilbert con supremo enriquecidas con un operador modal ♦. Damos una re- presentación topológica para estas álgebras usando la representación topológica obtenida para las álgebras de Hilbert con supremo. Consideramos algunas subvariedades particula- res de H∨ ♦ -álgebras. Estas variedades son la contraparte algebraica de algunas extensiones del fragmento implicativo de la l ógica modal intuicionista IntK♦. Usamos la representa- ción topológica obtenida para lasH∨ ♦ -álgebras para probar que la l ógica modal implicativa IntK→ ♦ es canónica, y en consecuencia la lógica IntK→ ♦ es completa. Tambi´en determi- namos las congruencias de las H∨ ♦ -álgebras en términos de ciertos subconjuntos cerrados del espacio asociado, y en términos de una clase particular de sistemas deductivos. Es- tos resultados nos permitieron caracterizar las H∨ ♦ -álgebras simples y subdirectamente irreducibles. Estos resultados son presentados en el Capítulo 6. / This thesis has two main objectives. The first objective is to present and develop a representation and a topological duality for some varieties of algebras corresponding to the reducts {→} and {→, ∨} of the variety of Heyting algebras. These representations are based on a particular class of topological spaces known as sober spaces. It is well- known that every Heyting algebra is representable as a subalgebra of Heyting algebra of all increasing subsets of a poset. Also, a Heyting algebra is representable as a subalgebra of the set of all open subsets of a topological space T0. These representations have many applications the algebraic study of these structures and applications of intuitionistic lo- gic Int and some of its extensions. Moreover, these representations are the basis for the known topological dualities of Priestley and Stone for Heyting algebras. When we look at some subreduct of Heyting algebras, for example, algebras corresponding to the implica- tive fragment, known as Hilbert algebras, representation theory and duality developed for Heyting algebras is not directly applicable to these fragments. The first result we know about representation of a Hilbert algebra is the thesis of A. Diego [29]. In this thesis a theorem of Stone representation type appears, but this result did not have a significant impact because this theorem is insufficient to develop a categorical duality. The first ob- jective of this thesis is precisely present a complete topological duality for Hilbert algebras and extend this duality to the variety of Hilbert algebras with supremum. These results are based on topological spaces known as sober spaces and extend those given by M. Stone [67]. First we prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations.We also obtained a duality for Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic.We explore how this duality is related to the duality given in [18] for finite Hilbert algebras, and with the topological duality developed in [19] for Tarski algebras. All these results are presented in Chapter 3. The other variety associated to a fragment of the logic Int that we study is the variety of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. We extend the duality for Hilbert algebras to the case of Hilbert algebras with supremum. We prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We also see that in this lattice it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. We give a dual description of the lattice of ideals of a Hilbert algebra with supremum. These results are presented in Chapter 5. The second main objective of this memory is centered on studying some modal exten- sions of Hilbert algebras and Hilbert algebras with supremum. These extensions corres- pond to fragments of some modal extensions of intuitionistic logic Int. In this memory we have focused on only two fragments. First we introduce the variety of Hilbert algebras with a modal operator , called H -algebras. The variety of H -algebras is the alge- braic counterpart of the {→, }-fragment of the intuitionitic modal logic IntK , which we denoted by IntK→ . We study the theory of representation and we give a topological duality for the variety of H -algebras. We use these results to prove that the basic impli- cative modal logic IntK→ is canonical and therefore is complete. We also determine the simple and subdirectly irreducible algebras in some subvarieties of H -algebras. These results are presented in Chapter 4. The other fragment investigated is the fragment {→, ∨, ♦} of intuitionistic modal lo- gic IntK♦.We introduce and study the variety ofH∨ ♦ -algebras, which are Hilbert algebras with supremum endowed with a modal operator ♦. We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in [22].We consider some particular varieties of H∨ ♦ -algebras. The- se varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic IntK♦. We use the topological representation for H∨ ♦ -algebras to prove that the implicative modal logic IntK→ ♦ is canonical, and consequently the logic IntK→ ♦ is complete. We also determine the congruences of H∨ ♦ -algebras in terms of cer- tain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible H∨ ♦ -algebras. These results are presented in Chapter 6.

No / Finite quantum systems in which the position and momentum take values in the Galois field GF(p), are studied. Ideas from the subject of field extension are transferred in the context of quantum mechanics. The Frobenius automorphisms in Galois fields lead naturally to the "Frobenius formalism" in a quantum context. The Hilbert space splits into "Frobenius subspaces" which are labeled with the irreducible polynomials associated with the yp¿y. The Frobenius maps transform unitarily the states of a Galois quantum system and leave fixed all states in some of its Galois subsystems (where the position and momentum take values in subfields of GF(p)). An analytic representation of these systems in the -sheeted complex plane shows deeper links between Galois theory and Riemann surfaces. ©2006 American Institute of Physics

No / Unitary transformations in an angular momentum Hilbert space H(2j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU(2j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H(2j + 1) in terms of smaller qudits in Hilbert spaces H(2ji + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.