<p> Various models of computability of partial functions f on the real numbers are studied: two abstract, based on approximable computation w.r.t high level programming languages; two concrete, based on computable tracking functions on the rationals; and two based on polynomial approximation. It is shown that these six models are equivalent, under the assumptions: (1) the domain of f is a union of an effective sequence of rational open intervals, and (2) f is effectively locally uniformly continuous. This includes the well-known functions of elementary real analysis (rational, exponential, trigonometric, etc., and their inverses) and generalises a previously know equivalence result for total functions on the reals. </p> / Thesis / Master of Science (MSc)

Several results from classical computability theory (computability over discrete structures such as the natural numbers and strings over finite alphabets, due to Turing, Church, Kleene and others) have been shown to hold for a generalisation of computability theory over total abstract algebras, using for instance the model of \While\ computation. We present a number of results relating to computation on topological partial algebras, again using \While\ computation. We consider several results from the classical theory in the context of topological algebra of the reals: closure of semicomputable sets under finite union (Chapter \ref{chap:results1} Theorem \ref{thm:union_While_scomp_not_While_scomp}, p.\pageref{thm:union_While_scomp_not_While_scomp}), the equivalence of semicomputable and projectively (semi)computable sets (Chapter \ref{chap:results2} Theorem \ref{thm:proj_while_equivalents}, p.\pageref{thm:proj_while_equivalents}), and Post's Theorem (i.e.~a set is computable iff both it and its complement are semicomputable) (Appendix \ref{appendix:posts_theorem} Theorem \ref{thm:posts_general}, p.\pageref{thm:posts_general}). This research has significance in the field of scientific computation, which is underpinned by computability on the real numbers. We will consider a ``continuity principle", which states that computability should imply continuity; however, equality, order, and other total boolean-valued functions on the reals are clearly discontinuous. As we want these functions to be basic for the algebras under consideration, we resolve this incompatibility by redefining such functions to be partial, leading us to consider topological partial algebras. / Thesis / Master of Computer Science (MCS) / We investigate to what extent certain well-known results of classical computability theory on the natural numbers hold in the context of generalised computability theories on the real numbers.

<p>This thesis develops a formalism of function tables, inspired by the work of D. Parnas. It adapts that formalism so as to apply to topological partial algebras, involving continuous partial functions on the reals. In particular, it studies semantics-preserving transformations between two classes of tables: normal and inverted. This leads to a 3-valued logic different from that used by Wei Lei (2007) who investigated the application of function tables to "error algebras".</p> / Master of Science (MSc)

No description available.

La presente tesis parte de la hipótesis de que los derechos reales son necesarios para la consecución de las libertades comunitarias y del mercado interior. No obstante, lo cierto es que los derechos reales no constituyen hoy por hoy una parte importante del acervo comunitario vigente, cuyas normas regulan primordialmente el derecho contractual. Una iniciativa importante en este ámbito es la del grupo de investigación de la eurohipoteca, que ha elaborado unas Basic Guidelines en torno a una garantía real inmobiliaria útil, segura y flexible para el mercado hipotecario europeo. Por lo tanto, la situación actual de los derechos reales en la UE no permite alcanzar un auténtico mercado interior. En consecuencia, la presente tesis pretende: a) en primer lugar, proponer las bases de un sistema de propiedad europeo en base al status quo de la armonización de los derechos reales en Europa. Este análisis es útil para ver en qué contexto se podría regular la eurohipoteca; y b) en segundo lugar, analizar un modelo de derecho real consolidado, como el de garantía real inmobiliaria previsto en las BG (la eurohipoteca), y su incidencia en estas libertades. / The hypothesis of the current dissertation takes as a starting point that property rights are necessary for the achievement of community freedoms and the internal market. Nevertheless, property rights are not today an important part of the current acquis communautaire, which rules focus mainly on contract law. An important initiative in this field is the Basic guidelines, which have been created by the Eurohypothec research group and foresee a flexible, safe and useful security right over immovables.Therefore, the current situation of property rights in Europe is insufficient to achieve a genuine internal market. For these reasons, the present dissertation tries to: a) firstly, propose the basis for a European system of property law on the basis of the status quo of the harmonisation of property rights in Europe. This analysis is useful to see in what context the Eurohypothec could be regulated; and b) secondly, analyze a model of property rights, such as the eurohypothec (a security right over immovable property foreseen in the BG), and its impact on community freedoms.

Exact real arithmetic systems can specify any amount of precision on the output of the computations. They are used in a wide variety of applications when a high degree of precision is necessary. Some of these applications include: differential equation solvers, linear equation solvers, large scale mathematical models, and SMT solvers. This dissertation proposes a new exact real arithmetic system which uses lazy list of floating point numbers to represent the real numbers. It proposes algorithms for basic arithmetic computations on these structures and proves their correctness. This proposed system has the advantage of algorithms which can be supported by modern floating point hardware, while still being a lazy exact real arithmetic system.

Alguns problemes clàssics sobre teoria analítica de polinomis estan relacionats amb un problema més general: determinar com estan ordenades les arrels reals d'un polinomi a coeficients reals i les arrels reals de totes les seves derivades. Si ens restringim a l'ordenació entre arrels de derivades consecutives d'un polinomi, aquest problema pot formular-se de la següent manera. Sigui n un nombre natural no nul. Per a cada j=0,1,.,n-1 considerem variables indeterminades xj,1,xj,2,...,xj,m(j), que anomenarem variables de derivació j, i que considerarem lligades per les desigualtats xj,1<xj,2<···<xj,m(j). Definir un ordre entre variables de derivacions consecutives significa especificar, per a dues variables qualssevol de derivacions consecutives, diguem xj,k i xj+1,s, una de les tres ordenacions següents: (i) xj,k<xj+1,s, (ii) xj,k=xj+1,s, o (iii) xj,k>xj+1,s. Llavors, el problema consisteix en determinar per a quines ordenacions entre variables de derivacions consecutives existeix un polinomi P(x), de grau n, de manera que si les arrels reals de cada derivada P(j), 0&#8804;j&#8804;n-1, són els nombres yj,1<yj,2<···<yj,r(j), aleshores r(j)=m(j) i entre arrels de derivades consecutives es verifiquen els lligams proposats. És a dir, si (i) xj,k<xj+1,s, (ii) xj,k=xj+1,s, o (iii) xj,k>xj+1,s, aleshores s'ha de complir (a) yj,k<yj+1,s, (b) yj,k=yj+1,s, o (c) yj,k>yj+1,s respectivament. Si tal polinomi existeix aleshores es diu que l'ordenació proposada és representable per un polinomi. El teorema de Rolle imposa restriccions a l'ordenació de les variables en el cas que aquesta ordenació sigui representable per polinomis. Concretament, si xj,k<xj,k' són dues variables de derivació j, aleshores ha d'existir una variable de derivació j+1, xj+1,s, tal que xj,k<xj+1,s<xj,k'. No obstant, les restriccions imposades pel teorema de Rolle no són suficients per a que una ordenació de les variables sigui representable per un polinomi.En aquest sentit, ens proposem assolir els tres objectius següents:(1) Caracteritzar les ordenacions entre variables de derivacions consecutives que són representables per polinomis.(2) Classificar els polinomis en base a l'ordenació de les arrels de derivades consecutives i trobar certs nombres d'interès relacionats amb aquesta classificació, com per exemple el nombre de classes en que queden classificats els polinomis de grau n i el nombre de classes obertes de grau n (classes estables per pertorbacions).(3) Estudiar què succeeix quan es consideren ordenacions que inclouen lligams entre variables de derivacions no consecutives.L'objectiu (1) s'ha assolit establint que les ordenacions entre variables de derivacions consecutives representables per polinomis coincideixen amb les ordenacions que satisfan les restriccions imposades per un resultat que generalitza el teorema de Rolle. Essencialment, s'ha obtingut el recíproc del teorema que diu que entre cada dues arrels reals consecutives d'un polinomi hi ha un nombre senar d'arrels de la derivada comptant multiplicitats.L'objectiu (2) s'ha assolit classificant els polinomis segons l'ordenació que presenten les arrels de les seves derivades consecutives. Els nombres d'interès relacionats amb aquesta classificació s'han obtingut a partir de fórmules recurrents.L'objectiu (3) s'ha assolit determinant els nombres n per als quals la mencionada generalització del teorema de Rolle és suficient per a que una ordenació de les variables que inclogui lligams entre variables de derivacions no consecutives sigui representable per un polinomi. / Some classical problems in analytic theory of polynomials are related to a more general one that consists in determining how the real roots of a real polynomial and the roots of all its derivatives are ordered.If we restrict our attention to the ordering amongst the roots of consecutive derivatives of a polynomial, this problem can be stated as follows: Let n be a nonzero natural number. For each j=0,1,.,n-1 we consider some indeterminate variables xj,1,xj,2,...,xj,m(j), called variables of derivative j, which will be linked by the inequalities xj,1<xj,2<···<xj,m(j). To define an order amongst variables of consecutive derivatives means to specify, for any two variables of consecutive derivatives, say xj,k and xj+1,s, one of the following three relations: (i) xj,k<xj+1,s, (ii) xj,k=xj+1,s, or (iii) xj,k>xj+1,s. Then, the problem consists in determining for which of those orderings amongst variables of consecutive derivatives there exists a polynomial of degree n, say P(x), so that if the real roots of each derivative P(j), 0&#8804;j&#8804;n-1, are the numbers yj,1<yj,2<···<yj,r(j), then r(j)=m(j) and between roots of consecutive derivatives the suggested connections hold. That is, if (i) xj,k<xj+1,s, (ii) xj,k=xj+1,s, or (iii) xj,k>xj+1,s, then (a) yj,k<yj+1,s, (b) yj,k=yj+1,s, or (c) yj,k>yj+1,s must hold respectively. If such a polynomial exists, then we say that the suggested ordering is represented by a polynomial.Rolle's theorem sets up restrictions to the ordering of the variables in the case when this ordering is represented by polynomials. More precisely, if xj,k<xj,k+1 are two consecutive variables of the same derivative j, then there must exist a variable of derivative j+1, namely xj+1,s, such that xj,k<xj+1,s<xj,k+1. However, the restrictions imposed by Rolle's theorem are not sufficient to ensure that an ordering of the variables is represented by a polynomial.In this sense, we intend to achieve the following goals:(1) To characterize the orderings amongst variables of consecutive derivatives that are represented by polynomials.(2) To classify the polynomials according to the ordering of the roots of consecutive derivatives and to find certain numbers of interest related to this classification, such as the number of classes of equivalence in which polynomials of degree n are classified and the number of classes of equivalence which are open as subsets of the space of polynomials of degree at most n.(3) To study what happens when we consider orderings that include connections between variables of non-consecutive derivatives.Goal (1) has been achieved by showing that the orderings amongst variables of consecutive derivatives that are represented by polynomials coincide with the orderings that satisfy the restrictions imposed by a result which generalizes Rolle's theorem. Essentially, we have obtained the inverse of the theorem that states that between every two consecutive real roots of a polynomial, there is an odd number of roots of its derivative counting their multiplicities.Goal (2) has been attained by classifying the polynomials according to the ordering of the roots of their consecutive derivatives. The numbers of interest related to this classification have been obtained by means of recurrent formulae.Goal (3) has been attained by determining all numbers n for which Rolle's theorem generalization, mentioned above, is sufficient to ensure that an ordering of the variables that include connections between variables of non-consecutive derivatives, be represented by a polynomial.

Aquesta tesi doctoral té per objecte l'estudi de les "pactiones et stipulationes" com a mode de constitució inter vivos de les servituds predials, en el dret romà clàssic. La principal font que ens informa de la seva existència és Gai. 2, 31, segons la qual les pactiones et stipulationes serien el mecanisme generalment emprat per a la constitució de servituds i d'usdefruit a les províncies romanes. Aquest treball intenta ser una contribució a l'estudi d'aquest negoci constitutiu de drets reals, a partir d'una revisió de les fonts que s'allunyi de la tendència a considerar-les interpolades i ens permeti observar la intervenció del pretor en aquest camp, així com la precisa resposta dela jurisprudència clàssica, davant problemes com ara la trasnformació de les servituds en res incorporales i la progressiva decadència de la mancipatio i de la in iure cessio —agreujada per la impossibilitat d'adquisició d'aquests drets a través d'usucapio (lex Scribonia). / The goal of this doctoral thesis is to study the pactiones et stipulationes as a method to constitute servitudes inter vivos, in Roman classical law. The main source, which informs us about its existance, is Gai. 2, 31, according to it, the pactiones et stipulationes would be the method generally used to constitute such rights in the Roman provinces. This work wants to be a contribution to the study of this method to constitute real rights, with a review of the sources that goes beyond the tendency to consider most of the sources as interpolated; that allows us to observe the praetor's intervention in this area and also the answer of the jurisprudence in front of such problems as the transformation of servitudes into res incorporales and the progressive decay of the mancipatio and the in iure cessio —with the further difficulty of the impossibility to acquire these real rights by means of the usucapio (lex Scribonia).

En este trabajo se definen y estudian los conceptos de H-convexidad y H-conjugación, siendo H una familia de funciones reales de variable real cerrada para el supremo puntual de tal manera que coinciden con los clásicos al considerar la familia H de las traslaciones de R. Mediante ellos se construye una teoría de la dualidad en programación matemática y se estudian los Lagrangieros que se derivan. Entre las aplicaciones de estas nociones figura la interpretación de algunas teorías previas sobre conjugación quasi-convexa que se obtienen al considerar ciertas familias H de funciones crecientes. También se aborda la conjugación de multiaplicaciones en conjuntos abstractos, generalizando así las ya conocidas en los que se requieren estructuras algebraicas y de orden.

La compréhension et la prévision des phénomènes d'éboulements rocheux sont des éléments déterminants pour la gestion des risques dans l'aménagement des zones et des infrastructures de montagnes, qui nécessitent, entre autre, la mise au point d'outils numériques opérationnels permettant d'anticiper la trajectoire des épisodes rocheux et les zones susceptibles d'être impactées. L'approche numérique proposée dans le cadre de la thèse, est basée sur une modélisation numérique discrète qui permet de simuler le comportement collectif de blocs rocheux lors d'un éboulement, en considérant des blocs de géométrie réaliste et une topographie tridimensionnelle du versant de propagation. Pour faciliter son utilisation en bureau d'études, le modèle numérique de contact, qui gère les interaction entre les blocs et entre les blocs et le versant de propagation, fait appel à un nombre restreint de paramètres, pouvant être estimés par des mesures in situ ou par des rétro-analyses de cas existants. Les paramètres les plus influant sur les mécanismes de propagation sont : la géométrie des blocs et la topographie de surface, les paramètres dissipatifs de contact tels que le coefficient de restitution normale qui agit dans la direction normale au contact, le coefficient de frottement qui agit dans la direction tangentielle au plan de contact, et la résistance au roulement entre la pente et les blocs rocheux dans l'hypothèse d'un impact rocheux sur un sol meuble.La thèse présente des nombreuses données de terrain qui ont servies de base à l'analyse des mécanismes de propagation sur d'anciens cas d'éboulements et sur des falaises potentiellement instables. Des méthodes permettant la reconstruction d'un volume instable pour la modélisation de la propagation ont été développées sur la base des modèles tridimensionnels de terrain. Après des tests de validation et de compréhension du modèle numérique, ce dernier a été appliqué à des cas réels. Trois sites, très différents en termes de géométrie, ont été retenus pour tester le modèle numérique. Les paramètres d'études sont suivant les cas, les mécanismes de propagation, les mécanismes de dissipation d'énergie, les zones de dépôt, les forces d'interactions et les énergies d'impact sur des structures de protection.Ce travail a permis d'établir une méthodologie de modélisation des éboulements rocheux adaptée à l'ingénierie. Un protocole spécifique permettant de déterminer ou d'estimer les paramètres a été proposé et des logiciels de pré et post traitements permettant à la fois une analyse pertinente et rapide des résultats ont été développés. La corrélation entre les résultats numériques et les observations de terrain sont satisfaisantes et les résultats en termes de prévision encourageants. / Understanding and predicting rock avalanches are key elements in risk management when developing mountainous areas. Due to the complexity of the mechanisms involved, developments of numerical and operational tools are useful and necessary to properly estimate block trajectories and define riskiest areas. The numerical approach proposed in this thesis is based in the discrete element method which allows simulating the collective behavior of a group of rocks by using realistic block geometries and three-dimensional slope topographies. The numerical contact model handles interaction between the blocks and between the blocks and the slope by using a limited number of parameters that can be estimated by in situ measurements or feed-back analysis of ancient rockfalls events. The most influential parameters in the propagation phenomena are: the geometry and shape of blocks and topography, the contact parameters such as the normal restitution coefficient which acts in the normal direction to the contact, the friction coefficient acting in the tangential direction to the contact plane and the rolling resistance coefficient of blocks with the slope that somehow accounts of the impact mechanisms in case of soft soil.Various field results of past rockfall events or instable rocky cliffs are presented in the thesis. In addition, different procedures allowing the modeling of the whole unstable volume involved in the propagation motion are presented through real applications by using three-dimensional terrain models as input data. The numerical model was applied to three real rockfalls events, very different in terms of geometry. The parameters analyzed concerned mainly the propagation distances and energy dissipation mechanisms, deposit zones, interaction forces and impact energies on protective structures.A specific protocol for determining and estimating the contact parameters was proposed and some pre and post processing software enabling both relevant and rapid analysis of the results have been developed. The correlation between the numerical results and field observations are satisfactory with very encouraging results in terms of forecasting.