Lógica Fuzzy aplicada à geologia

Pós-graduação em Matemática em Rede Nacional - IBILCE

Considerações históricas sobre o infinito e alguns de seus paradoxos

This research aims to elucidate some of the historical aspects of the idea of infinity during the creation of calculus and set theory. It also seeks to raise discussions about the nature of infinity: current infinite and potential infinite. For this, we conducted a survey with a qualitative approach in the form of exploratory study. This study was based on books of Mathematics' History and other scientific works such as articles, theses and dissertations on the subject. This work will bring the view of some philosophers and thinkers about the infinite, such as: Pythagoras, Plato, Aristotle, Galilei, Augustine, Cantor. The research will be presented according to chronological order. The objective of the research is to understand the infinite from ancient Greece with the paradoxes of Zeno, during the time which the conflict between the conceptions atomistic and continuity were dominant, and in this context that Zeno launches its paradoxes which contradict much a concept as another, until the theory Cantor set, bringing some paradoxes related to this theory, namely paradox of Russell and Hilbert's paradox. The study also presents these paradoxes mentioned under the mathematical point of view and the light of calculus and set theory

Considerações históricas sobre o infinito e alguns de seus paradoxos

This research aims to elucidate some of the historical aspects of the idea of infinity during the creation of calculus and set theory. It also seeks to raise discussions about the nature of infinity: current infinite and potential infinite. For this, we conducted a survey with a qualitative approach in the form of exploratory study. This study was based on books of Mathematics' History and other scientific works such as articles, theses and dissertations on the subject. This work will bring the view of some philosophers and thinkers about the infinite, such as: Pythagoras, Plato, Aristotle, Galilei, Augustine, Cantor. The research will be presented according to chronological order. The objective of the research is to understand the infinite from ancient Greece with the paradoxes of Zeno, during the time which the conflict between the conceptions atomistic and continuity were dominant, and in this context that Zeno launches its paradoxes which contradict much a concept as another, until the theory Cantor set, bringing some paradoxes related to this theory, namely paradox of Russell and Hilbert's paradox. The study also presents these paradoxes mentioned under the mathematical point of view and the light of calculus and set theory

On universal Banach spaces of density continuum

We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density at most continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that a""(a)/c (0) is such a space if we assume the continuum hypothesis. Some additional set-theoretic assumption is indeed needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory. We also prove that it is consistent that there are universal Banach spaces of density c, but a""(a)/c (0) is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into a""(a)/c (0).

Maximal Subgroups of Compact Lie Groups

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to: (2) the classification of the finite maximal Sigma-invariant subgroups of centerfree connected compact simple Lie groups and (3) the classification of the Sigma-primitive subalgebras of compact simple Lie algebras, where Sigma is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive subalgebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups.

Periódicos de Ciência da Informação nas bases de dados internacionais

PO artigo demonstra as interações entre algumas bases de dados internacionais na disponibilização de periódicos da área de Ciência da Informação. Para isso, foram investigadas seis bases de dados com acesso a texto completo dentro da área de Ciência da Informação. Foram elas: Web of Science, Scopus, Library, Information Science & Technology Abstracts, Information Science & Technology Abstracts, Library Literature Information Science – full text e Library Information Science Abstract. No desenvolvimento da pesquisa foram utilizados conceitos, princípios e equações da ‘teoria ingênua dos conjuntos’, o que permitiu observar as bases de dados como conjuntos compostos por elementos, aqui considerados os títulos de periódicos, bem como conferir consistência aos resultados. A análise dos dados foi dividida em três etapas, a saber: a) a primeira priorizou o cruzamento entre as duas maiores bases de dados utilizadas para levantamento bibliográfico (Web of Science e Scopus), seguido do cruzamento com a mais utilizada na Ciência da Informação, a Library and Information Science Abstracts; b) a segunda etapa tratou somente do cruzamento das bases especialistas da área; c) a terceira e última etapa tratou da interseção de todas as bases de dados do estudo. O resultado mostra que 10 títulos de periódicos constituem a interseção dos seis conjuntos de bases de dados. Foi possível identificar, também, o número de periódicos disponibilizados exclusivamente por uma base de dados, demonstrando que as necessidades de pesquisas bibliográficas exaustivas passam, obrigatoriamente, pelo levantamento em todas as bases de dados aqui elencadas.

A GIS based model for solving the planar Huff problem considering different demand distributions and forbidden regions

[EN] In this paper, we have used Geographical Information Systems (GIS) to solve the planar Huff problem considering different demand distributions and forbidden regions. Most of the papers connected with the competitive location problems consider that the demand is aggregated in a finite set of points. In other few cases, the models suppose that the demand is distributed along the feasible region according to a functional form, mainly a uniform distribution. In this case, in addition to the discrete and uniform demand distributions we have considered that the demand is represented by a population surface model, that is, a raster map where each pixel has associated a value corresponding to the population living in the area that it covers...

Interactive theorem provers: issues faced as a user and tackled as a developer

Interactive theorem provers (ITP for short) are tools whose final aim is to certify proofs written by human beings. To reach that objective they have to fill the gap between the high level language used by humans for communicating and reasoning about mathematics and the lower level language that a machine is able to “understand” and process. The user perceives this gap in terms of missing features or inefficiencies. The developer tries to accommodate the user requests without increasing the already high complexity of these applications. We believe that satisfactory solutions can only come from a strong synergy between users and developers. We devoted most part of our PHD designing and developing the Matita interactive theorem prover. The software was born in the computer science department of the University of Bologna as the result of composing together all the technologies developed by the HELM team (to which we belong) for the MoWGLI project. The MoWGLI project aimed at giving accessibility through the web to the libraries of formalised mathematics of various interactive theorem provers, taking Coq as the main test case. The motivations for giving life to a new ITP are: • study the architecture of these tools, with the aim of understanding the source of their complexity • exploit such a knowledge to experiment new solutions that, for backward compatibility reasons, would be hard (if not impossible) to test on a widely used system like Coq. Matita is based on the Curry-Howard isomorphism, adopting the Calculus of Inductive Constructions (CIC) as its logical foundation. Proof objects are thus, at some extent, compatible with the ones produced with the Coq ITP, that is itself able to import and process the ones generated using Matita. Although the systems have a lot in common, they share no code at all, and even most of the algorithmic solutions are different. The thesis is composed of two parts where we respectively describe our experience as a user and a developer of interactive provers. In particular, the first part is based on two different formalisation experiences: • our internship in the Mathematical Components team (INRIA), that is formalising the finite group theory required to attack the Feit Thompson Theorem. To tackle this result, giving an effective classification of finite groups of odd order, the team adopts the SSReflect Coq extension, developed by Georges Gonthier for the proof of the four colours theorem. • our collaboration at the D.A.M.A. Project, whose goal is the formalisation of abstract measure theory in Matita leading to a constructive proof of Lebesgue’s Dominated Convergence Theorem. The most notable issues we faced, analysed in this part of the thesis, are the following: the difficulties arising when using “black box” automation in large formalisations; the impossibility for a user (especially a newcomer) to master the context of a library of already formalised results; the uncomfortable big step execution of proof commands historically adopted in ITPs; the difficult encoding of mathematical structures with a notion of inheritance in a type theory without subtyping like CIC. In the second part of the manuscript many of these issues will be analysed with the looking glasses of an ITP developer, describing the solutions we adopted in the implementation of Matita to solve these problems: integrated searching facilities to assist the user in handling large libraries of formalised results; a small step execution semantic for proof commands; a flexible implementation of coercive subtyping allowing multiple inheritance with shared substructures; automatic tactics, integrated with the searching facilities, that generates proof commands (and not only proof objects, usually kept hidden to the user) one of which specifically designed to be user driven.

Friedrich Riesz' Beiträge zur Herausbildung des modernen mathematischen Konzepts abstrakter Räume

Der ungarische Mathematiker Friedrich Riesz studierte und forschte in den mathematischen Milieus von Budapest, Göttingen und Paris. Die vorliegende Arbeit möchte zeigen, daß die Beiträge von Riesz zur Herausbildung eines abstrakten Raumbegriffs durch eine Verknüpfung von Entwicklungen aus allen drei mathematischen Kulturen ermöglicht wurden, in denen er sich bewegt hat. Die Arbeit konzentriert sich dabei auf den von Riesz 1906 veröffentlichten Text „Die Genesis des Raumbegriffs". Sowohl für seine Fragestellungen als auch für seinen methodischen Zugang fand Riesz vor allem in Frankreich und Göttingen Anregungen: Henri Poincarés Beiträge zur Raumdiskussion, Maurice Fréchets Ansätze einer abstrakten Punktmengenlehre, David Hilberts Charakterisierung der Stetigkeit des geometrischen Raumes. Diese Impulse aufgreifend suchte Riesz ein Konzept zu schaffen, das die Forderungen von Poincaré, Hilbert und Fréchet gleichermaßen erfüllte. So schlug Riesz einen allgemeinen Begriff des mathematischen Kontinuums vor, dem sich Fréchets Konzept der L-Klasse, Hilberts Mannigfaltigkeitsbegriff und Poincarés erfahrungsgemäße Vorstellung der Stetigkeit des ‚wirklichen' Raumes unterordnen ließen. Für die Durchführung seines Projekts wandte Riesz mengentheoretische und axiomatische Methoden an, die er der Analysis in Frankreich und der Geometrie bei Hilbert entnommen hatte. Riesz' aufnahmebereite Haltung spielte dabei eine zentrale Rolle. Diese Haltung kann wiederum als ein Element der ungarischen mathematischen Kultur gedeutet werden, welche sich damals ihrerseits stark an den Entwicklungen in Frankreich und Deutschland orientierte. Darüber hinaus enthält Riesz’ Arbeit Ansätze einer konstruktiven Mengenlehre, die auf René Baire zurückzuführen sind. Aus diesen unerwarteten Ergebnissen ergibt sich die Aufgabe, den Bezug von Riesz’ und Baires Ideen zur späteren intuitionistischen Mengenlehre von L.E.J. Brouwer und Hermann Weyl weiter zu erforschen.

Knots and links in lens spaces

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

GROUPS THAT INVOLVE FINITELY MANY PRIMES AND HAVE ALL SUBGROUPS SUBNORMAL II

It is shown that if G is a hypercentral group with all subgroups subnormal, and if the torsion subgroup of G is a pi-group for some finite set pi of primes, then G is nilpotent. In the case where G is not hypercentral there is a section of G that is much like one of the well-known Heineken-Mohamed groups. It is also shown that if G is a residually nilpotent group with all subgroups subnormal whose torsion subgroup satisfies the above condition then G is nilpotent.

A Comparative Study of Two Semantic Systems: Body Russian and Russian Phraseology

Grigorij Kreidlin (Russia). A Comparative Study of Two Semantic Systems: Body Russian and Russian Phraseology. Mr. Kreidlin teaches in the Department of Theoretical and Applied Linguistics of the State University of Humanities in Moscow and worked on this project from August 1996 to July 1998. The classical approach to non-verbal and verbal oral communication is based on a traditional separation of body and mind. Linguists studied words and phrasemes, the products of mind activities, while gestures, facial expressions, postures and other forms of body language were left to anthropologists, psychologists, physiologists, and indeed to anyone but linguists. Only recently have linguists begun to turn their attention to gestures and semiotic and cognitive paradigms are now appearing that raise the question of designing an integral model for the unified description of non-verbal and verbal communicative behaviour. This project attempted to elaborate lexical and semantic fragments of such a model, producing a co-ordinated semantic description of the main Russian gestures (including gestures proper, postures and facial expressions) and their natural language analogues. The concept of emblematic gestures and gestural phrasemes and of their semantic links permitted an appropriate description of the transformation of a body as a purely physical substance into a body as a carrier of essential attributes of Russian culture - the semiotic process called the culturalisation of the human body. Here the human body embodies a system of cultural values and displays them in a text within the area of phraseology and some other important language domains. The goal of this research was to develop a theory that would account for the fundamental peculiarities of the process. The model proposed is based on the unified lexicographic representation of verbal and non-verbal units in the Dictionary of Russian Gestures, which the Mr. Kreidlin had earlier complied in collaboration with a group of his students. The Dictionary was originally oriented only towards reflecting how the lexical competence of Russian body language is represented in the Russian mind. Now a special type of phraseological zone has been designed to reflect explicitly semantic relationships between the gestures in the entries and phrasemes and to provide the necessary information for a detailed description of these. All the definitions, rules of usage and the established correlations are written in a semantic meta-language. Several classes of Russian gestural phrasemes were identified, including those phrasemes and idioms with semantic definitions close to those of the corresponding gestures, those phraseological units that have lost touch with the related gestures (although etymologically they are derived from gestures that have gone out of use), and phrasemes and idioms which have semantic traces or reflexes inherited from the meaning of the related gestures. The basic assumptions and practical considerations underlying the work were as follows. (1) To compare meanings one has to be able to state them. To state the meaning of a gesture or a phraseological expression, one needs a formal semantic meta-language of propositional character that represents the cognitive and mental aspects of the codes. (2) The semantic contrastive analysis of any semiotic codes used in person-to-person communication also requires a single semantic meta-language, i.e. a formal semantic language of description,. This language must be as linguistically and culturally independent as possible and yet must be open to interpretation through any culture and code. Another possible method of conducting comparative verbal-non-verbal semantic research is to work with different semantic meta-languages and semantic nets and to learn how to combine them, translate from one to another, etc. in order to reach a common basis for the subsequent comparison of units. (3) The practical work in defining phraseological units and organising the phraseological zone in the Dictionary of Russian Gestures unexpectedly showed that semantic links between gestures and gestural phrasemes are reflected not only in common semantic elements and syntactic structure of semantic propositions, but also in general and partial cognitive operations that are made over semantic definitions. (4) In comparative semantic analysis one should take into account different values and roles of inner form and image components in the semantic representation of non-verbal and verbal units. (5) For the most part, gestural phrasemes are direct semantic derivatives of gestures. The cognitive and formal techniques can be regarded as typological features for the future functional-semantic classification of gestural phrasemes: two phrasemes whose meaning can be obtained by the same cognitive or purely syntactic operations (or types of operations) over the meanings of the corresponding gestures, belong by definition to one and the same class. The nature of many cognitive operations has not been studied well so far, but the first steps towards its comprehension and description have been taken. The research identified 25 logically possible classes of relationships between a gesture and a gestural phraseme. The calculation is based on theoretically possible formal (set-theory) correlations between signifiers and signified of the non-verbal and verbal units. However, in order to examine which of them are realised in practice a complete semantic and lexicographic description of all (not only central) everyday emblems and gestural phrasemes is required and this unfortunately does not yet exist. Mr. Kreidlin suggests that the results of the comparative analysis of verbal and non-verbal units could also be used in other research areas such as the lexicography of emotions.

Admissibility in Finitely Generated Quasivarieties

Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in Q is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.

Stochastic search for semiparametric linear regression models

This paper introduces and analyzes a stochastic search method for parameter estimation in linear regression models in the spirit of Beran and Millar [Ann. Statist. 15(3) (1987) 1131–1154]. The idea is to generate a random finite subset of a parameter space which will automatically contain points which are very close to an unknown true parameter. The motivation for this procedure comes from recent work of Dümbgen et al. [Ann. Statist. 39(2) (2011) 702–730] on regression models with log-concave error distributions.

Functional and motor neuronal analysis of defensive siphon responses in {\it Aplysia californica}

One of the central goals of neuroscience research is to determine how networks of neurons control and modify behavior. One of the most influential model systems for this kind of analysis is the siphon and gill withdrawal reflex of the marine mollusc A. californica. In response to tactile stimulation, the siphon displays 3 different responses: (1) a posterior pointing and leveling (flaring) of the siphon in response to tail stimulation (the siphon T response), (2) constriction and anterior pointing to head stimulation (the siphon H response) and (3) constriction and withdrawal between the animal's parapodia (the siphon S response). The siphon S response is pseudoconditioned by a noxious tail stimulus to resemble the siphon T response. Behavioral and combined behavioral/intracellular studies were conducted to determine the motor neuronal control of these behaviors and to search for mechanisms of siphon response transformation following pseudoconditioning. The present studies have found that the flaring component of pseudoconditioned siphon S responses occurs during mantle pumping (MP) triggered by noxious tail stimulation. Siphon stimulation also triggers MP, as recorded in neurons of the Interneuron II pattern generator which commands MP. The 4 LF$\rm\sb{SB}$ siphon motor neurons (SMNs) were found necessary and sufficient for the siphon T response, while SMNs RD$\rm\sb S$ and LD$\rm\sb{S1}$ were found necessary and sufficient for the siphon H response. Following pseudoconditioning, there is an increase in the number of evoked spikes to the test stimulus for the LF$\rm\sb{SB}$ cells and a decreased number for RD$\rm\sb S.$ Siphon flaring occurring during the pseudoconditioned response correlates with increased LF$\rm\sb{SB}$ activity during triggered MP cycles. This suggests that psuedoconditioning is in part due to reconfiguration of the motor outputs of the Interneuron II network. These results suggest that these defensive responses are controlled and patterned by a well-defined, finite set of motor neurons and interneurons (Interneuron II) that are dedicated to specific behavioral functions, but also have parallel distributed properties. ^