Hydrogen atom in space with a compactified extra dimension and potential defined by Gauss' law

We investigate the consequences of one extra spatial dimension for the stability and energy spectrum of the non-relativistic hydrogen atom with a potential defined by Gauss' law, i.e. proportional to 1 /| x | 2 . The additional spatial dimension is considered to be either infinite or curled-up in a circle of radius R. In both cases, the energy spectrum is bounded from below for charges smaller than the same critical value and unbounded from below otherwise. As a consequence of compactification, negative energy eigenstates appear: if R is smaller than a quarter of the Bohr radius, the corresponding Hamiltonian possesses an infinite number of bound states with minimal energy extending at least to the ground state of the hydrogen atom.

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.

Numerical and experimental study of the parametric rolling of a fisihing vessel in regular head waves

The dynamic behaviour of a fishing vessel in waves is studied in order to reveal its parametric rolling characteristics. This paper presents experimental and numerical results in longitudinal regular waves. The experimental results are compared against the results of a time-domain non-linear strip theory model of ship motions in six degrees-of-freedom. These results contribute to the validation of the parametric rolling prediction method, so that it can be used as an assessment tool to evaluate both the susceptibility and severity of occurrence of parametric rolling at the early design stage of these types of vessels.

La deformación del tipo Construcción de bóvedas no-canónicas en España (siglos XVI-XVIII)

Una bóveda no canónica es una bóveda que se adapta a una forma distinta de aquella para la que ha sido inicialmente concebida. Bóvedas raras, anormales, no convencionales, habitualmente consideradas excepciones o casos particulares, resultan ser más frecuentes de lo inicialmente esperado. El interés por este tipo de bóvedas surge a raíz de una investigación inicial sobre las bóvedas empleadas para cubrir espacios de planta anular, como en el caso de las girolas de las iglesias. Sin embargo, el problema de la bóveda anular no puede ser abordado directamente, sino como parte de una investigación más general sobre bóvedas que se deforman para adaptarse a una situación anómala. El análisis de las posibilidades que un determinado tipo de bóveda brinda para resolver el abovedamiento de espacios de planta irregular, trascendiendo el problema de la planta anular, es lo que da origen a esta investigación. La cuestión de las bóvedas deformadas forma parte de un contexto mayor, el de la deformación en arquitectura abovedada. Ante una contradicción, la deformación de la bóveda es sólo una de las posibles opciones que esta arquitectura ofrece para resolver un problema de deformación. La tesis se estructura en dos partes: en la primera parte se analizan los conceptos de forma y deformación en el contexto de la arquitectura abovedada con objeto de sentar las bases para una teoría de las bóvedas no canónicas. El objetivo es establecer un punto de partida para la investigación en un campo que todavía no había sido abordado. En la segunda parte se analizan tres tipos de bóveda desde la perspectiva de las bóvedas no canónicas, a partir de un estudio de casos de bóvedas en España entre los siglos XVI y XVIII. El estudio de la deformación en arquitectura abovedada se centra en el problema de la girola, por tratarse de un caso generalizado de deformación, directamente relacionado con el problema de las bóvedas irregulares y cuyo estudio, llamativamente, no había sido llevado a cabo hasta la fecha. Se propone una primera aproximación al problema de la girola, desde un punto de vista puramente morfológico, al margen de consideraciones históricas. En el caso de las bóvedas deformadas, el análisis se centra en tres tipos de bóveda: la bóveda de crucería, la bóveda de arista y la bóveda baída. Estos tres tipos de bóveda, aunque basadas en criterios formales distintos, están íntimamente relacionados entre sí. Por un lado permiten resolver el mismo problema –planta cuadrada delimitada por arcos–, por otro lado es posible establecer una relación formal entre la bóveda de arista y la bóveda baída a través de la bóveda de crucería. El estudio de casos recogido en la segunda parte de la tesis se fundamenta en dos líneas de investigación, la primera sobre soluciones teóricas de bóvedas no convencionales propuestas en los manuscritos y tratados de cantería, y la segunda sobre bóvedas efectivamente construidas, tratado de establecer una comparación entre teoría y práctica, confrontando el grado de relación entre ambas. Sin embargo este doble análisis sólo se ha podido llevar a cabo en contadas ocasiones. Constatamos que las bóvedas no canónicas reflejadas en los tratados son pocas y apenas se han llevado a la práctica, mientras que las soluciones construidas no responden a modelos teóricos propuestos, manifestando un divorcio entre teoría y práctica. El estudio de estas bóvedas permite poner en cuestión la definición tradicional que relaciona los conceptos de ‘bóveda’ y ‘superficie’. Al iniciar el trabajo nos encontramos con un modelo teórico extremadamente rígido que deja fuera un gran número de bóvedas, obligando a agruparlas bajo el término «no canónicas». El trabajo realizado pone en evidencia lo limitado del modelo. El problema no está en la presencia de bóvedas anómalas, que no se adaptan al modelo tradicionalmente propuesto, sino en la extrema rigidez del modelo. ABSTRACT A non canonical vault is a vault adapted to a different form from that for which was originally conceived. These rare, abnormal, unconventional vaults are usually considered as exceptions or special cases. However they prove to be more frequent than it was initially expected. Interest in this type of vaults arises from an initial research on the vaults used to roof annular spaces, such as ambulatories. Nevertheless, the annular vault question cannot be addressed directly, but as a part of a broader research on distorted vaults; a research on vaults deformed to conform an anomalous layout. The analysis of the possibilities that a particular type of vault provides to solve the vaulting of an irregular layout, beyond the problem of the annular plan is the origin of this research. The argument of deformed vaults is part of a greater context, the context of deformation in vaulted architecture. Facing a contradiction, deforming a vault is just one of the options that vaulted architecture offers to solve a problem of deformation. This dissertation is organised in two parts: in the first part we analyse the concepts of form and deformation in the context of vaulted architecture in order to lay the foundations for a non canonical vaults theory. The objective is to establish a starting point for future research in a field that has not been addressed yet. In the second part, we analyse three types of vault from the perspective of non canonical vaults, based on a case study of Spanish vaults between the 16th and 18th Centuries. The analysis of deformation in vaulted architecture focuses on the question of the ambulatory, because it is a generalized example of deformation, directly related to the problem of irregular vaults. Remarkably, the analysis of these spaces had not been conducted to date. We propose a first approach to the question of the ambulatory, from a purely morphological point of view, setting aside historical considerations. The analysis of deformed vaults focuses on three types of vault: the groin vault, the ribbed vault and the sail vault. These three types of vault, although based on different formal criteria, are closely related between them. On the one hand, they allow to solve the same problem –a square perimeter limited by arcs-; on the other hand, it is possible to establish a formal relationship between the groin vault and the sail vault through the ribbed vault. The case study presented in the second part of this dissertation is based on two research lines: theoretical non conventional vaults solutions proposed on stonecutting treatises; and currently built vaults. The aim of this double analysis was to establish a comparison between theory and practice, comparing the degree of relationship between them. Nevertheless, this double analysis has only been carried out on rare occasions. It is noted that non canonical vaults reflected in treaties are few and hardly been employed, while the built solutions do not meet proposed theoretical models, expressing a divorce between theory and practice. The analysis of these vaults allows us to question the traditional definition that connects the concepts of 'vault' and 'surface'. When we began this research, we found an extremely rigid theoretical model that leaved out many vaults, forcing to group them under the term of «non canonical vaults». This research evidences the limitations of the model. The problem is not the presence of abnormal vaults, which cannot adapt to the traditional model, but in the very high stiffness of the model.

Stability of nonrelativistic quantum mechanical matter coupled to the (ultraviolet cutoff) radiation field

We announce a proof of H-stability for the quantized radiation field, with ultraviolet cutoff, coupled to arbitrarily many non-relativistic quantized electrons and static nuclei. Our result holds for arbitrary atomic numbers and fine structure constant. We also announce bounds for the energy of many electrons and nuclei in a classical vector potential and for the eigenvalue sum of a one-electron Pauli Hamiltonian with magnetic field.

Communicating continuous quantum variables between different Lorentz frames

We show how to communicate Heisenberg-limited continuous (quantum) variables between Alice and Bob in the case where they occupy two inertial reference frames that differ by an unknown Lorentz boost. There are two effects that need to be overcome: the Doppler shift and the absence of synchronized clocks. Furthermore, we show how Alice and Bob can share Doppler-invariant entanglement, and we demonstrate that the protocol is robust under photon loss.

Exits, entrances and entrepreneurship : an exploration of economic theories of entrepreneurship through new independent entry and infant exit

The thesis began as a study of new firm formation. Preliminary research suggested that infant death rate was considered to be a closely related problem and the search was for a theory of new firm formation which would explain both. The thesis finds theories of exit and entry inadequate in this respect and focusses instead on theories of entrepreneurship, particularly those which concentrate on entrepreneurship as an agent of change. The role of information is found to be fundamental to economic change and an understanding of information generation and dissemination and the nature and direction of information flows is postulated to lead coterminously to an understanding of entrepreneurhsip and economic change. The economics of information is applied to theories of entrepreneurhsip and some testable hypotheses are derived. The testing relies on etablishing and measuring the information bases of the founders of new firms and then testing for certain hypothesised differences between the information bases of survivors and non-survivors. No theory of entrepreneurship is likely to be straightforwardly testable and many postulates have to be established to bring the theory to a testable stage. A questionnaire is used to gather information from a sample of firms taken from a new micro-data set established as part of the work of the thesis. Discriminant Analysis establishes the variables which best distinguish between survivors and non-survivors. The variables which emerge as important discriminators are consistent with the theory which the analysis is testing. While there are alternative interpretations of the important variables, collective consistency with the theory under test is established. The thesis concludes with an examination of the implications of the theory for policy towards stimulating new firm formation.

A perturbative analysis of dispersion-managed solitons

We study soliton solutions of the path-averaged propagation equation governing the transmission of dispersion-managed (DM) optical pulses in the (practical) limit when residual dispersion and nonlinearity only slightly affect the pulse dynamics over one compensation period. In the case of small dispersion map strengths, the averaged pulse dynamics is governed by a perturbed form of the nonlinear Schrödinger equation; applying a perturbation theory – elsewhere developed – based on inverse scattering theory, we derive an analytic expression for the envelope of the DM soliton. This expression correctly predicts the power enhancement arising from the dispersion management. Theoretical results are verified by direct numerical simulations.

A perturbative analysis of dispersion-managed solitons

We study soliton solutions of the path-averaged propagation equation governing the transmission of dispersion-managed (DM) optical pulses in the (practical) limit when residual dispersion and nonlinearity only slightly affect the pulse dynamics over one compensation period. In the case of small dispersion map strengths, the averaged pulse dynamics is governed by a perturbed form of the nonlinear Schrödinger equation; applying a perturbation theory – elsewhere developed – based on inverse scattering theory, we derive an analytic expression for the envelope of the DM soliton. This expression correctly predicts the power enhancement arising from the dispersion management. Theoretical results are verified by direct numerical simulations.

Wave kinetics of random fibre lasers

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example—random fibre laser—we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

The Operators Aγ = γA + -γA for a Class of Nondissipative Operators A with a Limit of the Corresponding Correlation Function

2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12

Triangular Models and Asymptotics of Continuous Curves with Bounded and Unbounded Semigroup Generators

2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12.

Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator

We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant ϰ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker [11]. Next, we describe sets of perturbations V for which the Fermi Golden Rule is valid at each embedded eigenvalue of H; these sets turn out to be dense in various suitable topologies. Finally, we assume that V decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair (H+V, H), and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator H.

Particle Trajectory Computation for Accelerators and Related Technologies

This work reports an alternative method for single non-relativistic charged particle trajectory computation in 2D electrostatic or magnetostatic fields. This task is approached by analytical computation of particle trajectory, by parts, considering the constant fields within each finite element. This method has some advantages over numerical integration ones: numerical miscomputation of trajectories, and stability problems can be avoided. Among the examples presented in this paper, an interesting alternative approach for positive ion extraction from cyclotrons is shown, using strip-foils. Other particle optics devices can benefit of a method such the one proposed in this paper, as beam bending devices, spectrometers, among others. This method can be extended for particle trajectory computation in 3D domains.

Holographic Tools for Probing the Dynamics of Strongly Coupled Field Theories

Thesis (Ph.D.)--University of Washington, 2016-08