Algebraische Beschreibung von Symmetrien: das Modell der Nichtkommutativen Multi-Tori

In der Nichtkommutativen Geometrie werden Räume und Strukturen durch Algebren beschrieben. Insbesondere werden hierbei klassische Symmetrien durch Hopf-Algebren und Quantengruppen ausgedrückt bzw. verallgemeinert. Wir zeigen in dieser Arbeit, daß der bekannte Quantendoppeltorus, der die Summe aus einem kommutativen und einem nichtkommutativen 2-Torus ist, nur den Spezialfall einer allgemeineren Konstruktion darstellt, die der Summe aus einem kommutativen und mehreren nichtkommutativen n-Tori eine Hopf-Algebren-Struktur zuordnet. Diese Konstruktion führt zur Definition der Nichtkommutativen Multi-Tori. Die Duale dieser Multi-Tori ist eine Kreuzproduktalgebra, die als Quantisierung von Gruppenorbits interpretiert werden kann. Für den Fall von Wurzeln der Eins erhält man wichtige Klassen von endlich-dimensionalen Kac-Algebren, insbesondere die 8-dim. Kac-Paljutkin-Algebra. Ebenfalls für Wurzeln der Eins kann man die Nichtkommutativen Multi-Tori als Hopf-Galois-Erweiterungen des kommutativen Torus interpretieren, wobei die Rolle der typischen Faser von einer endlich-dimensionalen Hopf-Algebra gespielt wird. Der Nichtkommutative 2-Torus besitzt bekanntlich eine u(1)xu(1)-Symmetrie. Wir zeigen, daß er eine größere Quantengruppen-Symmetrie besitzt, die allerdings nicht auf die Spektralen Tripel des Nichtkommutativen Torus fortgesetzt werden kann.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

Random lattice models

This thesis deals with three different physical models, where each model involves a random component which is linked to a cubic lattice. First, a model is studied, which is used in numerical calculations of Quantum Chromodynamics.In these calculations random gauge-fields are distributed on the bonds of the lattice. The formulation of the model is fitted into the mathematical framework of ergodic operator families. We prove, that for small coupling constants, the ergodicity of the underlying probability measure is indeed ensured and that the integrated density of states of the Wilson-Dirac operator exists. The physical situations treated in the next two chapters are more similar to one another. In both cases the principle idea is to study a fermion system in a cubic crystal with impurities, that are modeled by a random potential located at the lattice sites. In the second model we apply the Hartree-Fock approximation to such a system. For the case of reduced Hartree-Fock theory at positive temperatures and a fixed chemical potential we consider the limit of an infinite system. In that case we show the existence and uniqueness of minimizers of the Hartree-Fock functional. In the third model we formulate the fermion system algebraically via C*-algebras. The question imposed here is to calculate the heat production of the system under the influence of an outer electromagnetic field. We show that the heat production corresponds exactly to what is empirically predicted by Joule's law in the regime of linear response.

Admissibility in Finitely Generated Quasivarieties

Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in Q is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.

A Logical Descriptor for Regular Languages via Stone Duality

In this paper we introduce a class of descriptors for regular languages arising from an application of the Stone duality between finite Boolean algebras and finite sets. These descriptors, called classical fortresses, are object specified in classical propositional logic and capable to accept exactly regular languages. To prove this, we show that the languages accepted by classical fortresses and deterministic finite automata coincide. Classical fortresses, besides being propositional descriptors for regular languages, also turn out to be an efficient tool for providing alternative and intuitive proofs for the closure properties of regular languages.

Admissibility via natural dualities

It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.

Applicative theories for logarithmic complexity classes

We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new two-sorted algebra which is very similar to Cook and Bellantoni's famous two-sorted algebra B for polynomial time [4]. The second theory describes logarithmic space by formalising concatenation- and sharply bounded recursion. All theories contain the predicates WW representing words, and VV representing temporary inaccessible words. They are inspired by Cantini's theories [6] formalising B.

Exact Unification and Admissibility

A new hierarchy of "exact" unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always have a smaller or equal cardinality than those provided by the standard instantiation preordering, and in significant cases a dramatic reduction may be observed. In particular, the classes of distributive lattices, idempotent semigroups, and MV-algebras, which all have nullary unification type, have unitary or finitary exact type. These results are obtained via an algebraic interpretation of exact unification, inspired by Ghilardi's algebraic approach to equational unification.

Eocene calcareous nannofossils in DSDP Holes 95-612 and 95-613

Lower Eocene calcareous nannofossil limestone cored at DSDP Site 612 on the middle slope off New Jersey represents an almost complete biostratigraphic sequence; only the lowest biozone (CP9a; NP10*) was not recovered. The thickness of the strata (198 m), the good preservation of the nannofossils, and the lack of long hiatuses justify the acceptance of this section as a lower Eocene reference for the western North Atlantic margin. The widely recognized and very similar nannofossil zonations of Martini (NP zones) and Bukry-Okada (CP zones) are emended slightly to make their lower Eocene biozones coeval; in addition, five new subzones are erected that subdivide zones CP10 and CPU (NP12 and NP13). Established biozone names are retained as they are altered little in concept, but alphanumeric code systems are changed somewhat by appending an asterisk (*) to identify zones that are emended. Zone CP10* (NP12*) is divided into two parts, the Lophodolithus nascens Subzone (CP10*a; NP12*a) and the Helicosphaera seminulum Subzone (CP10*b; NP12*b). Zone CPU* (NP13*) is divided into three parts, the Helicosphaera lophota Subzone (CP11*a; NP13*a), the Cyclicargolithuspseudogammation Subzone (CP11*b; NP13*b), and the Rhabdosphaera tenuis Subzone (CP11*c; NP13*c). At Site 612, a time-depth curve based on nannofossil datums dated in previous studies reveals a smoothly declining sediment accumulation rate, from 4.9 cm/10**3yr in CP10* (NP12*) to 2.8 cm/103 yr. in CP12* (NP14*). The ages of first-occurrence datums not previously dated are approximated by projection onto this timedepth curve and are as follows: Helicosphaera seminulum, 55.0 Ma; Helicosphaera lophota, 54.5 Ma; Cyclicargolithus pseudogammation, 53.7 Ma; Rhabdosphaera tenuis, 52.6 Ma; and Rhabdosphaera inflata, 50.2 Ma. At nearby Site 613 on the upper rise, strata of similar age, 139 m thick, contain an unconformity representing Subzone CPll*b (NP13*b) and a hiatus of approximately 1.1 m.y. duration. The sediment accumulation rate in the lower part of this section (9.7 cm/10**3yr.) is twice that observed for equivalent strata at Site 612. The hiatus and the heightened sediment accumulation rate at Site 613 probably represent the effects of episodic mass wasting on the early Eocene continental slope and rise.

Examples of PDEs all whose points are characteristic

Let π : FM ! M be the bundle of linear frames of a manifold M. A basis Lijk , j < k, of diffeomorphism invariant Lagrangians on J1 (FM) was determined in [J. Muñoz Masqué, M. E. Rosado, Invariant variational problems on linear frame bundles, J. Phys. A35 (2002) 2013-2036]. The notion of a characteristic hypersurface for an arbitrary first-order PDE system on an ar- bitrary bred manifold π : P → M, is introduced and for the systems dened by the Euler-Lagrange equations of Lijk every hypersurface is shown to be characteristic. The Euler-Lagrange equations of the natural basis of Lagrangian densities Lijk on the bundle of linear frames of a manifold M which are invariant under diffeomorphisms, are shown to be an underdetermined PDEs systems such that every hypersurface of M is characteristic for such equations. This explains why these systems cannot be written in the Cauchy-Kowaleska form, although they are known to be formally integrable by using the tools of geometric theory of partial differential equations, see [J. Muñoz Masqué, M. E. Rosado, Integrability of the eld equations of invariant variational problems on linear frame bundles, J. Geom. Phys. 49 (2004), 119-155]

Relational and Allegorical Semantics for Constraint Logic Programming

El cálculo de relaciones binarias fue creado por De Morgan en 1860 para ser posteriormente desarrollado en gran medida por Peirce y Schröder. Tarski, Givant, Freyd y Scedrov demostraron que las álgebras relacionales son capaces de formalizar la lógica de primer orden, la lógica de orden superior así como la teoría de conjuntos. A partir de los resultados matemáticos de Tarski y Freyd, esta tesis desarrolla semánticas denotacionales y operacionales para la programación lógica con restricciones usando el álgebra relacional como base. La idea principal es la utilización del concepto de semántica ejecutable, semánticas cuya característica principal es el que la ejecución es posible utilizando el razonamiento estándar del universo semántico, este caso, razonamiento ecuacional. En el caso de este trabajo, se muestra que las álgebras relacionales distributivas con un operador de punto fijo capturan toda la teoría y metateoría estándar de la programación lógica con restricciones incluyendo los árboles utilizados en la búsqueda de demostraciones. La mayor parte de técnicas de optimización de programas, evaluación parcial e interpretación abstracta pueden ser llevadas a cabo utilizando las semánticas aquí presentadas. La demostración de la corrección de la implementación resulta extremadamente sencilla. En la primera parte de la tesis, un programa lógico con restricciones es traducido a un conjunto de términos relacionales. La interpretación estándar en la teoría de conjuntos de dichas relaciones coincide con la semántica estándar para CLP. Las consultas contra el programa traducido son llevadas a cabo mediante la reescritura de relaciones. Para concluir la primera parte, se demuestra la corrección y equivalencia operacional de esta nueva semántica, así como se define un algoritmo de unificación mediante la reescritura de relaciones. La segunda parte de la tesis desarrolla una semántica para la programación lógica con restricciones usando la teoría de alegorías—versión categórica del álgebra de relaciones—de Freyd. Para ello, se definen dos nuevos conceptos de Categoría Regular de Lawvere y _-Alegoría, en las cuales es posible interpretar un programa lógico. La ventaja fundamental que el enfoque categórico aporta es la definición de una máquina categórica que mejora e sistema de reescritura presentado en la primera parte. Gracias al uso de relaciones tabulares, la máquina modela la ejecución eficiente sin salir de un marco estrictamente formal. Utilizando la reescritura de diagramas, se define un algoritmo para el cálculo de pullbacks en Categorías Regulares de Lawvere. Los dominios de las tabulaciones aportan información sobre la utilización de memoria y variable libres, mientras que el estado compartido queda capturado por los diagramas. La especificación de la máquina induce la derivación formal de un juego de instrucciones eficiente. El marco categórico aporta otras importantes ventajas, como la posibilidad de incorporar tipos de datos algebraicos, funciones y otras extensiones a Prolog, a la vez que se conserva el carácter 100% declarativo de nuestra semántica. ABSTRACT The calculus of binary relations was introduced by De Morgan in 1860, to be greatly developed by Peirce and Schröder, as well as many others in the twentieth century. Using different formulations of relational structures, Tarski, Givant, Freyd, and Scedrov have shown how relation algebras can provide a variable-free way of formalizing first order logic, higher order logic and set theory, among other formal systems. Building on those mathematical results, we develop denotational and operational semantics for Constraint Logic Programming using relation algebra. The idea of executable semantics plays a fundamental role in this work, both as a philosophical and technical foundation. We call a semantics executable when program execution can be carried out using the regular theory and tools that define the semantic universe. Throughout this work, the use of pure algebraic reasoning is the basis of denotational and operational results, eliminating all the classical non-equational meta-theory associated to traditional semantics for Logic Programming. All algebraic reasoning, including execution, is performed in an algebraic way, to the point we could state that the denotational semantics of a CLP program is directly executable. Techniques like optimization, partial evaluation and abstract interpretation find a natural place in our algebraic models. Other properties, like correctness of the implementation or program transformation are easy to check, as they are carried out using instances of the general equational theory. In the first part of the work, we translate Constraint Logic Programs to binary relations in a modified version of the distributive relation algebras used by Tarski. Execution is carried out by a rewriting system. We prove adequacy and operational equivalence of the semantics. In the second part of the work, the relation algebraic approach is improved by using allegory theory, a categorical version of the algebra of relations developed by Freyd and Scedrov. The use of allegories lifts the semantics to typed relations, which capture the number of logical variables used by a predicate or program state in a declarative way. A logic program is interpreted in a _-allegory, which is in turn generated from a new notion of Regular Lawvere Category. As in the untyped case, program translation coincides with program interpretation. Thus, we develop a categorical machine directly from the semantics. The machine is based on relation composition, with a pullback calculation algorithm at its core. The algorithm is defined with the help of a notion of diagram rewriting. In this operational interpretation, types represent information about memory allocation and the execution mechanism is more efficient, thanks to the faithful representation of shared state by categorical projections. We finish the work by illustrating how the categorical semantics allows the incorporation into Prolog of constructs typical of Functional Programming, like abstract data types, and strict and lazy functions.

Hiord: A type-free higher-order logic programming language with predicate abstraction

A new formalism, called Hiord, for defining type-free higherorder logic programming languages with predicate abstraction is introduced. A model theory, based on partial combinatory algebras, is presented, with respect to which the formalism is shown sound. A programming language built on a subset of Hiord, and its implementation are discussed. A new proposal for defining modules in this framework is considered, along with several examples.

Categorical Structures in Non-Commutative Measure Theory.

In this paper, we investigate effect algebras and base normed spaces from the categorical point of view. We prove that the category of effect algebras is complete and cocomplete as well as the category of base normed spaces is complete, and discuss the contravariant functor from the category of effect algebras to the category of base normed spaces.

Aprendizaje, simulación y desarrollo sobre las tecnologías de voz y datos

El objetivo del Proyecto Fin de Carrera (PFC) es el de conocer, simular y crear una red VoIP sobre una red de datos en un entorno docente, más concretamente, en la asignatura Redes y Servicios de telecomunicación en Grado en Ingeniería de Telecomunicaciones en la Universidad Politécnica de Madrid (UPM). Una vez se adquieran los conocimientos necesarios, se propondrán una serie de prácticas para que los alumnos se vayan familiarizando con el software y hardware utilizados, de manera que, se irá subiendo el grado de dificultad hasta que puedan realizar una auténtica red VoIP por sí mismos. A parte de la realización de las prácticas, los alumnos deberán pasar una prueba de los conocimientos adquiridos al final de cada práctica mediante preguntas tipo test. Los sistemas elegidos para la implantación de una red VoIP en los módulos de laboratorio son: 3CX System Phone y Asteisk-Trixbox. Los cuales, son capaces de trabajar mediante gestores gráficos para simplificar el nivel de dificultad de la configuración. 3CX es una PBX que trabaja sobre Windows y se basa exclusivamente en el protocolo SIP. Esto facilita el manejo para usuarios que solo han usado Windows sin quitar funcionalidades que tienen otras centralitas en otros sistemas operativos. La versión demo activa todas las opciones para poder familiarizarse con este sistema. Por otro lado, Asterisk trabaja en todas las plataformas, aunque se ha seleccionado trabajar sobre Linux. Esta selección se ha realizado porque el resto de plataformas limitan la configuración de la IP PBX, esta es de código abierto y permite realizar todo tipo de configuraciones. Además, es un software gratuito, esto es una ventaja a la hora de configurar novedades o resolver problemas, ya que hay muchos especialistas que dan soporte y ayudan de forma gratuita. La voz sobre Internet es habitualmente conocida como VoIP (Voice Over IP), debido a que IP (Internet Protocol) es el protocolo de red de Internet. Como tecnología, la VoIP no es solo un paso más en el crecimiento de las comunicaciones por voz, sino que supone integrar las comunicaciones de datos y las de voz en una misma red, y en concreto, en la red con mayor cobertura mundial: Internet. La mayor importancia y motivación de este Proyecto Fin de Carrera es que el alumno sea capaz de llegar a un entorno laboral y pueda tener unos conocimientos capaces de afrontar esta tecnología que esta tan a la orden del día. La importancia que estas redes tienen y tendrán en un futuro muy próximo en el mundo de la informática y las comunicaciones. Cabe decir, que se observa que estas disciplinas tecnológicas evolucionan a pasos agigantados y se requieren conocimientos más sólidos. ABSTRACT. The objective of my final project during my studies in university was, to simulate and create a VoIP network over a data network in a teaching environment, more specifically on the subject of telecommunications networks and services in Telecommunication Engineering Degree in Polytechnic University of Madrid (UPM). Once acquiring the necessary knowledge a number of practices were proposed to the students to become familiar with the software and hardware used, so that it would rise to the level of difficulty that they could make a real VoIP network for themselves. Parts of the experimental practices were that students must pass a test of knowledge acquired at the end of each practice by choice questions. The systems chosen for the implementation of a VoIP network in the laboratory modules are: 3CX Phone System and Asteisk - Trixbox. Which were able to work with graphics operators to simplify the difficulty level of the configuration. 3CX is a PBX that works on Windows and is based solely on the SIP protocol. This facilitates handling for users who have only used Windows without removing functionality with other exchanges in other operating systems. Active demo version all options to get to grips with this system. Moreover, Asterisk works on all platforms, but has been selected to work on Linux. This selection was made because other platforms limit the IP PBX configuration, as this is open source and allows all kinds of configurations. Also, Linux is a free software and an advantage when configuring new or solve problems, as there are many specialists that support and help for free. Voice over Internet is commonly known as VoIP (Voice Over IP), because IP (Internet Protocol) is the Internet protocol network. As technology, VoIP is not just another step in the growth of voice communications, but communications of integrating data and voice on a single network, and in particular, in the network with the largest global coverage: Internet. The increased importance and motivation of this Thesis is that the student is able to reach a working environment and may have some knowledge to deal with these technologies that is so much the order of the day. The importances of these networks have and will be of essences in the very near future in the world of computing and communications. It must be said it is observed that these technological disciplines evolve by leaps and bounds stronger knowledge required.