From probability to sequences and back

This is a survey covering sequential structures and their applications to the foundations of probability theory. Sequential convergence, convergence groups and the extension of sequentially continuous maps belong to general topology and Trieste for long has been a center of sequential topology. We begin with some personal reflections, con- tinue with topological problems motivated by the extension of probability measures, and close with some recent results related to the categorical foundations of probability theory

THE TWO SCOPES OF FUZZY PROBABILITY THEORY

The aim of this work is to compare between what seems to be entirely different two highly developing “fuzzy probability” theories. The first theory had been developed firstly by statisticians and the other separately by physicists. We start by indicating the needs to develop such theories and what helped to develop each, then we will establish the basis of the two theories and illustrate that each indeed extends classical probability theory. Moreover, we will try to see whether or not any of the two theory can be embedded into the other

Contractive Maps and Complexity Analysis in Fuzzy Quasi-Metric Spaces

En los últimos años se ha desarrollado una teoría matemática con propiedades robustas con el fin de fundamentar la Ciencia de la Computación. En este sentido, un avance significativo lo constituye el establecimiento de modelos matemáticos que miden la "distancia" entre programas y entre algoritmos, analizados según su complejidad computacional. En 1995, M. Schellekens inició el desarrollo de un modelo matemático para el análisis de la complejidad algorítmica basado en la construcción de una casi-métrica definida en el espacio de las funciones de complejidad, proporcionando una interpretación computacional adecuada del hecho de que un programa o algoritmo sea más eficiente que otro en todos su "inputs". Esta información puede extraerse en virtud del carácter asimétrico del modelo. Sin embargo, esta estructura no es aplicable al análisis de algoritmos cuya complejidad depende de dos parámetros. Por tanto, en esta tesis introduciremos un nuevo espacio casi-métrico de complejidad que proporcionará un modelo útil para el análisis de este tipo de algoritmos. Por otra parte, el espacio casi-métrico de complejidad no da una interpretación computacional del hecho de que un programa o algoritmo sea "sólo" asintóticamente más eficiente que otro. Los espacios casi-métricos difusos aportan un parámetro "t", cuya adecuada utilización puede originar una información extra sobre el proceso computacional a estudiar; por ello introduciremos la noción de casi-métrica difusa de complejidad, que proporciona un modelo satisfactorio para interpretar la eficiencia asintótica de las funciones de complejidad. En este contexto extenderemos los principales teoremas de punto fijo en espacios métricos difusos , utilizando una determinada noción de completitud, y obtendremos otros nuevos. Algunos de estos teoremas también se establecerán en el contexto general de los espacios casi-métricos difusos intuicionistas, de lo que resultarán condiciones de contracción menos fuertes. Los resultados obtTirado Peláez, P. (2008). Contractive Maps and Complexity Analysis in Fuzzy Quasi-Metric Spaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/2961Palanci

The Fuzzy Supersphere

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative $Z_{2}$-graded algebras tending in a suitable limit to a dense subalgebra of the $Z_{2}$-graded algebra of ${\cal H}^{\infty}$-functions on the $(2| 2)$-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level".Comment: 33 pages, LaTeX, some typos correcte

Efficient computation of rank probabilities in posets

As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced. A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present. The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven. The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed. The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be

Quasicontinuous derivatives and viscosity functions

In this work we demonstrate how the continuous domain theory can be applied to the theory of nonlinear optimization, particularly to the theory of viscosity solutions. We consider finding the viscosity solution for the Hamilton-Jacobi equation H(x, y) = g(x), with continuous hamiltonian, but with possibly discontinuous right-hand side. We begin by finding a new function space Q(X,L), the space of equivalence classes of quasicontinuous functions from a locally compact set X to a bicontinuous lattice L and we will define on Q(X,L) the qo-topology, which is a variant of classical order topology defined on complete lattices. On this new function space we will show that there exist closed extensions of some differential operators, like the usual gradient and the operator defined by the continuous hamiltonian H. The domain of the closure of the corresponding operator will coincide with the set of viscosity solutions for the Hamilton-Jacobi equation when the hamiltonian is convex in the second argument