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.

Falsification and corroboration of conceptual hydrological models using geophysical data

Geophysical data may provide crucial information about hydrological properties, states, and processes that are difficult to obtain by other means. Large data sets can be acquired over widely different scales in a minimally invasive manner and at comparatively low costs, but their effective use in hydrology makes it necessary to understand the fidelity of geophysical models, the assumptions made in their construction, and the links between geophysical and hydrological properties. Geophysics has been applied for groundwater prospecting for almost a century, but it is only in the last 20 years that it is regularly used together with classical hydrological data to build predictive hydrological models. A largely unexplored venue for future work is to use geophysical data to falsify or rank competing conceptual hydrological models. A promising cornerstone for such a model selection strategy is the Bayes factor, but it can only be calculated reliably when considering the main sources of uncertainty throughout the hydrogeophysical parameter estimation process. Most classical geophysical imaging tools tend to favor models with smoothly varying property fields that are at odds with most conceptual hydrological models of interest. It is thus necessary to account for this bias or use alternative approaches in which proposed conceptual models are honored at all steps in the model building process.

Coarse-graining and thermodynamics in far-from-equilibrium systems

Lying at the core of statistical physics is the need to reduce the number of degrees of freedom in a system. Coarse-graining is a frequently-used procedure to bridge molecular modeling with experiments. In equilibrium systems, this task can be readily performed; however in systems outside equilibrium, a possible lack of equilibration of the eliminated degrees of freedom may lead to incomplete or even misleading descriptions. Here, we present some examples showing how an improper coarse-graining procedure may result in linear approaches to nonlinear processes, miscalculations of activation rates and violations of the fluctuation-dissipation theorem.

Reliable computation of robust response tori on the verge of breakdown

We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy-Hammel route for the creation of strange non- chaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computer- assisted and are based on a tailored version of the Newton-Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques.

On the Analysis and Control of a Linear Synchronous Servomotor with a Flexible Load

Industry's growing need for higher productivity is placing new demands on mechanisms connected with electrical motors, because these can easily lead to vibration problems due to fast dynamics. Furthermore, the nonlinear effects caused by a motor frequently reduce servo stability, which diminishes the controller's ability to predict and maintain speed. Hence, the flexibility of a mechanism and its control has become an important area of research. The basic approach in control system engineering is to assume that the mechanism connected to a motor is rigid, so that vibrations in the tool mechanism, reel, gripper or any apparatus connected to the motor are not taken into account. This might reduce the ability of the machine system to carry out its assignment and shorten the lifetime of the equipment. Nonetheless, it is usually more important to know how the mechanism, or in other words the load on the motor, behaves. A nonlinear load control method for a permanent magnet linear synchronous motor is developed and implemented in the thesis. The purpose of the controller is to track a flexible load to the desired velocity reference as fast as possible and without awkward oscillations. The control method is based on an adaptive backstepping algorithm with its stability ensured by the Lyapunov stability theorem. As a reference controller for the backstepping method, a hybrid neural controller is introduced in which the linear motor itself is controlled by a conventional PI velocity controller and the vibration of the associated flexible mechanism is suppressed from an outer control loop using a compensation signal from a multilayer perceptron network. To avoid the local minimum problem entailed in neural networks, the initial weights are searched for offline by means of a differential evolution algorithm. The states of a mechanical system for controllers are estimated using the Kalman filter. The theoretical results obtained from the control design are validated with the lumped mass model for a mechanism. Generalization of the mechanism allows the methods derived here to be widely implemented in machine automation. The control algorithms are first designed in a specially introduced nonlinear simulation model and then implemented in the physical linear motor using a DSP (Digital Signal Processor) application. The measurements prove that both controllers are capable of suppressing vibration, but that the backstepping method is superior to others due to its accuracy of response and stability properties.

Monoidal functors, acyclic models and chain operads

We prove that for a topological operad $P$ the operad of oriented cubical singular chains, $C^{\ord}_\ast(P)$, and the operad of simplicial singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{\ord}_\ast(P\nsemi\mathbb{Q})$ is formal if and only if $S_\ast(P\nsemi\mathbb{Q})$ is formal, thus linking together some formality results which are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by $R$-simplicial differential graded algebras.

Classification supervisée multi-étiquette en actes de dialogue: analyse discriminante et transformations de Schoenberg

Abstract This work studies the multi-label classification of turns in simple English Wikipedia talk pages into dialog acts. The treated dataset was created and multi-labeled by (Ferschke et al., 2012). The first part analyses dependences between labels, in order to examine the annotation coherence and to determine a classification method. Then, a multi-label classification is computed, after transforming the problem into binary relevance. Regarding features, whereas (Ferschke et al., 2012) use features such as uni-, bi-, and trigrams, time distance between turns or the indentation level of the turn, other features are considered here: lemmas, part-of-speech tags and the meaning of verbs (according to WordNet). The dataset authors applied approaches such as Naive Bayes or Support Vector Machines. The present paper proposes, as an alternative, to use Schoenberg transformations which, following the example of kernel methods, transform original Euclidean distances into other Euclidean distances, in a space of high dimensionality. Résumé Ce travail étudie la classification supervisée multi-étiquette en actes de dialogue des tours de parole des contributeurs aux pages de discussion de Simple English Wikipedia (Wikipédia en anglais simple). Le jeu de données considéré a été créé et multi-étiqueté par (Ferschke et al., 2012). Une première partie analyse les relations entre les étiquettes pour examiner la cohérence des annotations et pour déterminer une méthode de classification. Ensuite, une classification supervisée multi-étiquette est effectuée, après recodage binaire des étiquettes. Concernant les variables, alors que (Ferschke et al., 2012) utilisent des caractéristiques telles que les uni-, bi- et trigrammes, le temps entre les tours de parole ou l'indentation d'un tour de parole, d'autres descripteurs sont considérés ici : les lemmes, les catégories morphosyntaxiques et le sens des verbes (selon WordNet). Les auteurs du jeu de données ont employé des approches telles que le Naive Bayes ou les Séparateurs à Vastes Marges (SVM) pour la classification. Cet article propose, de façon alternative, d'utiliser et d'étendre l'analyse discriminante linéaire aux transformations de Schoenberg qui, à l'instar des méthodes à noyau, transforment les distances euclidiennes originales en d'autres distances euclidiennes, dans un espace de haute dimensionnalité.

Impact of a cis-associated gene expression SNP on chromosome 20q11.22 on bipolar disorder susceptibility, hippocampal structure and cognitive performance.

BackgroundBipolar disorder is a highly heritable polygenic disorder. Recent enrichment analyses suggest that there may be true risk variants for bipolar disorder in the expression quantitative trait loci (eQTL) in the brain.AimsWe sought to assess the impact of eQTL variants on bipolar disorder risk by combining data from both bipolar disorder genome-wide association studies (GWAS) and brain eQTL.MethodTo detect single nucleotide polymorphisms (SNPs) that influence expression levels of genes associated with bipolar disorder, we jointly analysed data from a bipolar disorder GWAS (7481 cases and 9250 controls) and a genome-wide brain (cortical) eQTL (193 healthy controls) using a Bayesian statistical method, with independent follow-up replications. The identified risk SNP was then further tested for association with hippocampal volume (n = 5775) and cognitive performance (n = 342) among healthy individuals.ResultsIntegrative analysis revealed a significant association between a brain eQTL rs6088662 on chromosome 20q11.22 and bipolar disorder (log Bayes factor = 5.48; bipolar disorder P = 5.85×10(-5)). Follow-up studies across multiple independent samples confirmed the association of the risk SNP (rs6088662) with gene expression and bipolar disorder susceptibility (P = 3.54×10(-8)). Further exploratory analysis revealed that rs6088662 is also associated with hippocampal volume and cognitive performance in healthy individuals.ConclusionsOur findings suggest that 20q11.22 is likely a risk region for bipolar disorder; they also highlight the informative value of integrating functional annotation of genetic variants for gene expression in advancing our understanding of the biological basis underlying complex disorders, such as bipolar disorder.

Dismissal of the illusion of uncertainty in the assessment of a likelihood ratio

The use of the Bayes factor (BF) or likelihood ratio as a metric to assess the probative value of forensic traces is largely supported by operational standards and recommendations in different forensic disciplines. However, the progress towards more widespread consensus about foundational principles is still fragile as it raises new problems about which views differ. It is not uncommon e.g. to encounter scientists who feel the need to compute the probability distribution of a given expression of evidential value (i.e. a BF), or to place intervals or significance probabilities on such a quantity. The article here presents arguments to show that such views involve a misconception of principles and abuse of language. The conclusion of the discussion is that, in a given case at hand, forensic scientists ought to offer to a court of justice a given single value for the BF, rather than an expression based on a distribution over a range of values.

Clasificación automática de textos y explotación BI

El presente proyecto tiene como objetivo desarrollar una tecnología que permita codificar grandes cantidades de texto de manera automática para posteriormente ser visualizada y analizada mediante una aplicación diseñada en Qlikview. El motor de la investigación e implementación de este proyecto se ha encontrado en la incipiente presencia de tecnologías informáticas en los procesos de codificación para ciencias políticas. De esta manera, el programa creado tiene como objetivo automatizar un proceso que se desarrolla comúnmente de manera manual y, por ende, las ventajas de introducir técnicas informáticas son notablemente valiosas. Estas automatizaciones permiten ahorrar tanto en tiempo de codificación, como en recursos económicos o humanos. Se ha elaborado una revisión teórica y metodológica que han servido como instrumentos de estudio y mejora, con el firme propósito de reducir al máximo el margen de error y ofrecer un instrumento de calidad con salida de mercado real. El método de clasificación utilizado ha sido Bayes, y se ha implementado utilizando Matlab. Los resultados de la clasificación han llegado a índices del 99.2%. En la visualización y análisis mediante Qlikview se pueden modificar los parámetros referentes a partido político, año, categoría o región, con lo que se permite analizar numerosos aspectos relacionados con la distribución de las palabras repartidas entre las diferentes categorías y en el tiempo.

$E_{1}$-Formality of complex algebraic varieties

Let $X$ be a smooth complex algebraic variety. Morgan showed that the rational homotopy type of $X$ is a formal consequence of the differential graded algebra defined by the first term $E_{1}(X,W)$ of its weight spectral sequence. In the present work, we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. In particular, our results generalize the formality theorem of Deligne, Griffiths, Morgan and Sullivan for morphisms of compact Kähler varieties, filling a gap in Morgan"s theory concerning functoriality over the rationals. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.

A Cartan-Eilenberg approach to Homotopical Algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.

Deformation of Gabor systems

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames 'without inequalities' from lattices to non-uniform sets.

Highly eccentric hip-hop solutions of the 2N

We show the existence of families of hip-hop solutions in the equal-mass 2N-body problem which are close to highly eccentric planar elliptic homographic motions of 2N bodies plus small perpendicular non-harmonic oscillations. By introducing a parameter ϵ, the homographic motion and the small amplitude oscillations can be uncoupled into a purely Keplerian homographic motion of fixed period and a vertical oscillation described by a Hill type equation. Small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise, via a Bolzano argument, to resonant periodic solutions of the uncoupled system in a rotating frame. For small ϵ ≠ 0, the topological transversality persists and Brouwer's fixed point theorem shows the existence of this kind of solutions in the full system

Parametric sensitivity analysis for the helium dimers on a model potential

Potential parameters sensitivity analysis for helium unlike molecules, HeNe, HeAr, HeKr and HeXe is the subject of this work. Number of bound states these rare gas dimers can support, for different angular momentum, will be presented and discussed. The variable phase method, together with the Levinson's theorem, is used to explore the quantum scattering process at very low collision energy using the Tang and Toennies potential. These diatomic dimers can support a bound state even for relative angular momentum equal to five, as in HeXe. Vibrational excited states, with zero angular momentum, are also possible for HeKr and HeXe. Results from sensitive analysis will give acceptable order of magnitude on potentials parameters.