A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Adding similarity-based reasoning capabilities to a Horn fragment of possibilistic logic with fuzzy constants

PLFC is a first-order possibilistic logic dealing with fuzzy constants and fuzzily restricted quantifiers. The refutation proof method in PLFC is mainly based on a generalized resolution rule which allows an implicit graded unification among fuzzy constants. However, unification for precise object constants is classical. In order to use PLFC for similarity-based reasoning, in this paper we extend a Horn-rule sublogic of PLFC with similarity-based unification of object constants. The Horn-rule sublogic of PLFC we consider deals only with disjunctive fuzzy constants and it is equipped with a simple and efficient version of PLFC proof method. At the semantic level, it is extended by equipping each sort with a fuzzy similarity relation, and at the syntactic level, by fuzzily “enlarging” each non-fuzzy object constant in the antecedent of a Horn-rule by means of a fuzzy similarity relation.

Performance characteristics of high performance liquid chromatography, first order derivative UV spectrophotometry and bioassay for fluconazole determination in capsules

The bioassay, first order derivative UV spectrophotometry and chromatographic methods for assaying fluconazole capsules were compared. They have shown great advantages over the earlier published methods. Using the first order derivative, the UV spectrophotometry method does not suffer interference of excipients. Validation parameters such as linearity, precision, accuracy, limit of detection and limit of quantitation were determined. All methods were linear and reliable within acceptable limits for antibiotic pharmaceutical preparations being accurate, precise and reproducible. The application of each method as a routine analysis should be investigated considering cost, simplicity, equipment, solvents, speed, and application to large or small workloads.

The relationship between first-order capabilities and online innovations. A comparative case study on the publishing sector.

The objective of this thesis was to study the relationship between firstorder capabilities and online innovations. First-order capabilities can be divided into market and technology capabilities, and they play an important role in the production of innovations. The study was carried out in publishing industry, where many changes have taken place in the online environment during the last few years. In the empirical research, four companies were studied, two magazine publishers and two newspaper publishers. The analysis was done in two phases; first every case was analyzed alone and then the cases were compared in cross-case analysis. The most important finding was the positive impact of market capability to the production of online innovations. The study also increased understanding about the relationship between market and technology capabilities and online innovations in general.

A System for Models of First Order Theories

If you want to know whether a property is true or not in a specific algebraic structure,you need to test that property on the given structure. This can be done by hand, which can be cumbersome and erroneous. In addition, the time consumed in testing depends on the size of the structure where the property is applied. We present an implementation of a system for finding counterexamples and testing properties of models of first-order theories. This system is supposed to provide a convenient and paperless environment for researchers and students investigating or studying such models and algebraic structures in particular. To implement a first-order theory in the system, a suitable first-order language.( and some axioms are required. The components of a language are given by a collection of variables, a set of predicate symbols, and a set of operation symbols. Variables and operation symbols are used to build terms. Terms, predicate symbols, and the usual logical connectives are used to build formulas. A first-order theory now consists of a language together with a set of closed formulas, i.e. formulas without free occurrences of variables. The set of formulas is also called the axioms of the theory. The system uses several different formats to allow the user to specify languages, to define axioms and theories and to create models. Besides the obvious operations and tests on these structures, we have introduced the notion of a functor between classes of models in order to generate more co~plex models from given ones automatically. As an example, we will use the system to create several lattices structures starting from a model of the theory of pre-orders.

First-order molecular descriptors for molecular steric similarity

En aquest treball s'analitza la contribució estèrica de les molècules a les seves propietats químiques i físiques, mitjançant l'avaluació del seu volum i de la seva mesura de semblança, a partir d'ara definits com a descriptors moleculars de primer ordre. La difeèsncia entre aquests dos conceptes ha estat aclarida: mentre que el volum és la magnitud de l'espai que ocupa la molècula com a entitat global, la mesura de semblança ens dóna una idea de com està distribuïda la densitat electrònica al llarg d'aquest volum, i reflecteix més les diferències locals existents. L'ús de diverses aproximacions per a l'obtenció d'ambdós valors ha estat analitzat sobre diferents classes d'isòmers

The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

First-order turbulence closure for modelling complex canopy flows

Simple first-order closure remains an attractive way of formulating equations for complex canopy flows when the aim is to find analytic or simple numerical solutions to illustrate fundamental physical processes. Nevertheless, the limitations of such closures must be understood if the resulting models are to illuminate rather than mislead. We propose five conditions that first-order closures must satisfy then test two widely used closures against them. The first is the eddy diffusivity based on a mixing length. We discuss the origins of this approach, its use in simple canopy flows and extensions to more complex flows. We find that it satisfies most of the conditions and, because the reasons for its failures are well understood, it is a reliable methodology. The second is the velocity-squared closure that relates shear stress to the square of mean velocity. Again we discuss the origins of this closure and show that it is based on incorrect physical principles and fails to satisfy any of the five conditions in complex canopy flows; consequently its use can lead to actively misleading conclusions.