Tratamiento N-categorial de la lógica de la programación

En el contexto de las técnicas de validación de programas, de las estructuras de axiomatización de la programación planteadas por Hoare y del desarrollo de la Lógica de la Programación que Dijkstra realiza con el operador wp, hemos dotado a los elementos esenciales que intervienen en dicho desarrollo de una estructura que permite su estudio en el ámbito de las categorias. En este trabajo se demuestra que el conjunto de precondiciones de un fragmento de programa anotado tiene estructura de N-categoría y que sucede lo mismo con el conjunto de postcondiciones. Más aún, se ha puesto de manifiesto (y probado) que el operador wp actúa como un funtor entre estos pares de N-categorías. Además, los conjuntos de guardas surgen de un modo natural en la lógica de la programación, han sido tratados desde el punto de vista de la semántica denotacional (Wlrth para el PASCAL y Scott y otros, después, en un enfoque más general) y presentados por Manes y Arbib en su semántica parcialmente aditiva. Pues bien, en este trabajo se demuestra que los conjuntos de guardas tienen también estructura de N-categoría y gue cualguier N-categoría dotada de una suma definida adecuadamente tiene estructura de conjunto de guardas, de tal modo, además, gue el preorden inducido por la suma en el conjunto de guardas coincide con la flecha de la N-categoría. Esta suma es, en concreto, la disyunción exclusiva, lo que adicionalmente supone una sorprendentemente sencilla definición alternativa a la suma de las categorías parcialmente aditivas definida por Manes y Arbib. Con todo esto, se aportan herramientas conceptuales para entender mejor y resolver más eficientemente los problemas que tiene planteados la lógica de la programación, pues se dispone de un punto de vista distinto y nuevo y de toda una familia de instrumentos adicionales.---ABSTRACT---In the context of program validation techniques, Hoare's systems for programming and Dijkstra's development of the logic of programming, based on the operator wp, we have endowed the essential features of this development with a structure that permits to study them in the frame of category theory. In this thesis we show that the set of preconditions of an annotated program segment is an N-category, and the same happens for the set of postconditions. Even more, it is shown that the operator wp acts as a functor between those pairs of Ncategories. Furthermore, guard sets come out in a natural way in the logic of programming, they have been considered from a denotational semantics point of view (Wirth for Pascal and afterwards Scott and al. in a more general setting) and they have been embodied by Manes and Arbib in their partially additive semantics. Then, it is shown in this thesis that the above mentioned guard sets also have the structure of an N-category and that any N-category with an appropiately defined sum has the structure of a guard set in such a way that, besides, the preorder defined in the guard set by the sum operation coincides with the arrows of the N-category. This sum is just the exclusive or of Boolean Logic and this fact adds a surprisingly simple alternative definition for the sum operation in Manes and Arbib partially additive categories. The present work, in summary, makes a contribution of conceptual tools for a better understanding and a more efficient solution of the problems posed to the logic of programming and it does so by providing a new different point of view and a whole family of additional techniques.

Numerical CFD modelling of non-neutral atmospheric boundary layers for offshore wind resource assessment based on Monin-Obukhov theory

The presented works aim at proposing a methodology for the simulation of offshore wind conditions using CFD. The main objective is the development of a numerical model for the characterization of atmospheric boundary layers of different stability levels, as the most important issue in offshore wind resource assessment. Based on Monin-Obukhov theory, the steady k-ε Standard turbulence model is modified to take into account thermal stratification in the surface layer. The validity of Monin-Obukhov theory in offshore conditions is discussed with an analysis of a three day episode at FINO-1 platform.

Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed

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.

Adaptive Pointing Theory (APT) Artificial Neural Network

The choice value and the testing process against the vigilance parameter, characteristic of ART Neural Network, are merged. Only, a single unique test is required to determine if a committed category node can represent the current input or not. Advantages of APT over ART are: 1-Avoid testing every committed category node before deciding to train a committed category node or a new node must be committed, 2-The vigilance parameter is fixed during training, and 3-The choice value parameter is eliminated.

On multiple-path sonic anemometer measurement theory

In this paper, a model of the measuring process of sonic anemometers with more than one measuring path is presented. The main hypothesis of the work is that the time variation of the turbulent speed field during the sequence of pulses that produces a measure of the wind speed vector affects the measurement. Therefore, the previously considered frozen flow, or instantaneous averaging, condition is relaxed. This time variation, quantified by the mean Mach number of the flow and the time delay between consecutive pulses firings, in combination with both the full geometry of sensors (acoustic path location and orientation) and the incidence angles of the mean with speed vector, give rise to significant errors in the measurement of turbulence which are not considered by models based on the hypothesis of instantaneous line averaging. The additional corrections (relative to the ones proposed by instantaneous line-averaging models) are strongly dependent on the wave number component parallel to the mean wind speed, the time delay between consecutive pulses, the Mach number of the flow, the geometry of the sensor and the incidence angles of mean wind speed vector. Kaimal´s limit k W1=1/l (where k W1 is the wave number component parallel to mean wind speed and l is the path length) for the maximum wave numbers from which the sonic process affects the measurement of turbulence is here generalized as k W1=C l /l, where C l is usually lesser than unity and depends on all the new parameters taken into account by the present model.