Uma análise dos livros de André Perez Y Marin: um momento da história da matemática escolar brasileira no início do século XX

Pós-graduação em Educação Matemática - IGCE

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

A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2D

A direct reconstruction algorithm for complex conductivities in W-2,W-infinity(Omega), where Omega is a bounded, simply connected Lipschitz domain in R-2, is presented. The framework is based on the uniqueness proof by Francini (2000 Inverse Problems 6 107-19), but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.

Periodontal Manifestations and Ambulatorial Management in a Patient With Sturge Weber Syndrome

Sturge-Weber syndrome is a nonhereditary congenital condition characterized by leptomeningeal and facial skin angiomatous malformation following the trigeminal nerve path. The intraoral angiomatosis are presented in 40% of cases and results in an important periodontal alteration, increasing the risk of bleeding during dental procedures. A 43-year-old male patient presented with port wine stain on the right side of the face, the entire hard and soft palates, the alveolar ridge, and buccal mucosa, and had an excessive accumulation of calcified masses in both supragingival and subgingival sites, with swelling and generalized inflammation throughout the gingiva and alveolar mucosa. He reported not having sanitized the area for years for fear of bleeding. Periodontal management, to remove calculus and to control gingivitis initiated in the supragingival region and gradually reaching the subgingival region to control oral microbiota, was performed with mild bleeding. The redness of the staining greatly diminished with time and the extreme halitosis of the patient also improved sharply leading to a dramatic improvement in quality of life. Ambulatory care is a feasible alternative for periodontal management that within safety limits for bleeding risks reduces the operational cost.

Fractal descriptors based on Fourier spectrum applied to texture analysis

This work proposes the development and study of a novel technique lot the generation of fractal descriptors used in texture analysis. The novel descriptors are obtained from a multiscale transform applied to the Fourier technique of fractal dimension calculus. The power spectrum of the Fourier transform of the image is plotted against the frequency in a log-log scale and a multiscale transform is applied to this curve. The obtained values are taken as the fractal descriptors of the image. The validation of the proposal is performed by the use of the descriptors for the classification of a dataset of texture images whose real classes are previously known. The classification precision is compared to other fractal descriptors known in the literature. The results confirm the efficiency of the proposed method. (C) 2012 Elsevier B.V. All rights reserved.

Shape analysis using fractal dimension: A curvature based approach

The present work shows a novel fractal dimension method for shape analysis. The proposed technique extracts descriptors from a shape by applying a multi-scale approach to the calculus of the fractal dimension. The fractal dimension is estimated by applying the curvature scale-space technique to the original shape. By applying a multi-scale transform to the calculus, we obtain a set of descriptors which is capable of describing the shape under investigation with high precision. We validate the computed descriptors in a classification process. The results demonstrate that the novel technique provides highly reliable descriptors, confirming the efficiency of the proposed method. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4757226]

A model of road effect using line integrals and a test of the performance of two new road indices using the distribution of small mammals in an Atlantic Forest landscape

Effects of roads on wildlife and its habitat have been measured using metrics, such as the nearest road distance, road density, and effective mesh size. In this work we introduce two new indices: (1) Integral Road Effect (IRE), which measured the sum effects of points in a road at a fixed point in the forest; and (2) Average Value of the Infinitesimal Road Effect (AVIRE), which measured the average of the effects of roads at this point. IRE is formally defined as the line integral of a special function (the infinitesimal road effect) along the curves that model the roads, whereas AVIRE is the quotient of IRE by the length of the roads. Combining tools of ArcGIS software with a numerical algorithm, we calculated these and other road and habitat cover indices in a sample of points in a human-modified landscape in the Brazilian Atlantic Forest, where data on the abundance of two groups of small mammals (forest specialists and habitat generalists) were collected in the field. We then compared through the Akaike Information Criterion (AIC) a set of candidate regression models to explain the variation in small mammal abundance, including models with our two new road indices (AVIRE and IRE) or models with other road effect indices (nearest road distance, mesh size, and road density), and reference models (containing only habitat indices, or only the intercept without the effect of any variable). Compared to other road effect indices, AVIRE showed the best performance to explain abundance of forest specialist species, whereas the nearest road distance obtained the best performance to generalist species. AVIRE and habitat together were included in the best model for both small mammal groups, that is, higher abundance of specialist and generalist small mammals occurred where there is lower average road effect (less AVIRE) and more habitat. Moreover, AVIRE was not significantly correlated with habitat cover of specialists and generalists differing from the other road effect indices, except mesh size, which allows for separating the effect of roads from the effect of habitat on small mammal communities. We suggest that the proposed indices and GIS procedures could also be useful to describe other spatial ecological phenomena, such as edge effect in habitat fragments. (C) 2012 Elsevier B.V. All rights reserved.

Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students' problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.

Odd homoclinic orbits for a second order Hamiltonian system

The aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.

Formalizing languages for service oriented computing

Service Oriented Computing is a new programming paradigm for addressing distributed system design issues. Services are autonomous computational entities which can be dynamically discovered and composed in order to form more complex systems able to achieve different kinds of task. E-government, e-business and e-science are some examples of the IT areas where Service Oriented Computing will be exploited in the next years. At present, the most credited Service Oriented Computing technology is that of Web Services, whose specifications are enriched day by day by industrial consortia without following a precise and rigorous approach. This PhD thesis aims, on the one hand, at modelling Service Oriented Computing in a formal way in order to precisely define the main concepts it is based upon and, on the other hand, at defining a new approach, called bipolar approach, for addressing system design issues by synergically exploiting choreography and orchestration languages related by means of a mathematical relation called conformance. Choreography allows us to describe systems of services from a global view point whereas orchestration supplies a means for addressing such an issue from a local perspective. In this work we present SOCK, a process algebra based language inspired by the Web Service orchestration language WS-BPEL which catches the essentials of Service Oriented Computing. From the definition of SOCK we will able to define a general model for dealing with Service Oriented Computing where services and systems of services are related to the design of finite state automata and process algebra concurrent systems, respectively. Furthermore, we introduce a formal language for dealing with choreography. Such a language is equipped with a formal semantics and it forms, together with a subset of the SOCK calculus, the bipolar framework. Finally, we present JOLIE which is a Java implentation of a subset of the SOCK calculus and it is part of the bipolar framework we intend to promote.

User interaction widgets for interactive theorem proving

Matita (that means pencil in Italian) is a new interactive theorem prover under development at the University of Bologna. When compared with state-of-the-art proof assistants, Matita presents both traditional and innovative aspects. The underlying calculus of the system, namely the Calculus of (Co)Inductive Constructions (CIC for short), is well-known and is used as the basis of another mainstream proof assistant—Coq—with which Matita is to some extent compatible. In the same spirit of several other systems, proof authoring is conducted by the user as a goal directed proof search, using a script for storing textual commands for the system. In the tradition of LCF, the proof language of Matita is procedural and relies on tactic and tacticals to proceed toward proof completion. The interaction paradigm offered to the user is based on the script management technique at the basis of the popularity of the Proof General generic interface for interactive theorem provers: while editing a script the user can move forth the execution point to deliver commands to the system, or back to retract (or “undo”) past commands. Matita has been developed from scratch in the past 8 years by several members of the Helm research group, this thesis author is one of such members. Matita is now a full-fledged proof assistant with a library of about 1.000 concepts. Several innovative solutions spun-off from this development effort. This thesis is about the design and implementation of some of those solutions, in particular those relevant for the topic of user interaction with theorem provers, and of which this thesis author was a major contributor. Joint work with other members of the research group is pointed out where needed. The main topics discussed in this thesis are briefly summarized below. Disambiguation. Most activities connected with interactive proving require the user to input mathematical formulae. Being mathematical notation ambiguous, parsing formulae typeset as mathematicians like to write down on paper is a challenging task; a challenge neglected by several theorem provers which usually prefer to fix an unambiguous input syntax. Exploiting features of the underlying calculus, Matita offers an efficient disambiguation engine which permit to type formulae in the familiar mathematical notation. Step-by-step tacticals. Tacticals are higher-order constructs used in proof scripts to combine tactics together. With tacticals scripts can be made shorter, readable, and more resilient to changes. Unfortunately they are de facto incompatible with state-of-the-art user interfaces based on script management. Such interfaces indeed do not permit to position the execution point inside complex tacticals, thus introducing a trade-off between the usefulness of structuring scripts and a tedious big step execution behavior during script replaying. In Matita we break this trade-off with tinycals: an alternative to a subset of LCF tacticals which can be evaluated in a more fine-grained manner. Extensible yet meaningful notation. Proof assistant users often face the need of creating new mathematical notation in order to ease the use of new concepts. The framework used in Matita for dealing with extensible notation both accounts for high quality bidimensional rendering of formulae (with the expressivity of MathMLPresentation) and provides meaningful notation, where presentational fragments are kept synchronized with semantic representation of terms. Using our approach interoperability with other systems can be achieved at the content level, and direct manipulation of formulae acting on their rendered forms is possible too. Publish/subscribe hints. Automation plays an important role in interactive proving as users like to delegate tedious proving sub-tasks to decision procedures or external reasoners. Exploiting the Web-friendliness of Matita we experimented with a broker and a network of web services (called tutors) which can try independently to complete open sub-goals of a proof, currently being authored in Matita. The user receives hints from the tutors on how to complete sub-goals and can interactively or automatically apply them to the current proof. Another innovative aspect of Matita, only marginally touched by this thesis, is the embedded content-based search engine Whelp which is exploited to various ends, from automatic theorem proving to avoiding duplicate work for the user. We also discuss the (potential) reusability in other systems of the widgets presented in this thesis and how we envisage the evolution of user interfaces for interactive theorem provers in the Web 2.0 era.

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.

The social construction of cooperation: exchange, identification and collective intentions within the organization

My aim is to develop a theory of cooperation within the organization and empirically test it. Drawing upon social exchange theory, social identity theory, the idea of collective intentions, and social constructivism, the main assumption of my work implies that both cooperation and the organization itself are continually shaped and restructured by actions, judgments, and symbolic interpretations of the parties involved. Therefore, I propose that the decision to cooperate, expressed say as an intention to cooperate, reflects and depends on a three step social process shaped by the interpretations of the actors involved. The first step entails an instrumental evaluation of cooperation in terms of social exchange. In the second step, this “social calculus” is translated into cognitive, emotional and evaluative reactions directed toward the organization. Finally, once the identification process is completed and membership awareness is established, I propose that individuals will start to think largely in terms of “We” instead of “I”. Self-goals are redefined at the collective level, and the outcomes for self, others, and the organization become practically interchangeable. I decided to apply my theory to an important cooperative problem in management research: knowledge exchange within organizations. Hence, I conducted a quantitative survey among the members of the virtual community, “www.borse.it” (n=108). Within this community, members freely decide to exchange their knowledge about the stock market among themselves. Because of the confirmatory requirements and the structural complexity of the theory proposed (i.e., the proposal that instrumental evaluations will induce social identity and this in turn will causes collective intentions), I use Structural Equation Modeling to test all hypotheses in this dissertation. The empirical survey-based study found support for the theory of cooperation proposed in this dissertation. The findings suggest that an appropriate conceptualization of the decision to exchange knowledge is one where collective intentions depend proximally on social identity (i.e., cognitive identification, affective commitment, and evaluative engagement) with the organization, and this identity depends on instrumental evaluations of cooperators (i.e., perceived value of the knowledge received, assessment of past reciprocity, expected reciprocity, and expected social outcomes of the exchange). Furthermore, I find that social identity fully mediates the effects of instrumental motives on collective intentions.

Expressiveness of Concurrent Languages

The aim of this thesis is to go through different approaches for proving expressiveness properties in several concurrent languages. We analyse four different calculi exploiting for each one a different technique. We begin with the analysis of a synchronous language, we explore the expressiveness of a fragment of CCS! (a variant of Milner's CCS where replication is considered instead of recursion) w.r.t. the existence of faithful encodings (i.e. encodings that respect the behaviour of the encoded model without introducing unnecessary computations) of models of computability strictly less expressive than Turing Machines. Namely, grammars of types 1,2 and 3 in the Chomsky Hierarchy. We then move to asynchronous languages and we study full abstraction for two Linda-like languages. Linda can be considered as the asynchronous version of CCS plus a shared memory (a multiset of elements) that is used for storing messages. After having defined a denotational semantics based on traces, we obtain fully abstract semantics for both languages by using suitable abstractions in order to identify different traces which do not correspond to different behaviours. Since the ability of one of the two variants considered of recognising multiple occurrences of messages in the store (which accounts for an increase of expressiveness) reflects in a less complex abstraction, we then study other languages where multiplicity plays a fundamental role. We consider the language CHR (Constraint Handling Rules) a language which uses multi-headed (guarded) rules. We prove that multiple heads augment the expressive power of the language. Indeed we show that if we restrict to rules where the head contains at most n atoms we could generate a hierarchy of languages with increasing expressiveness (i.e. the CHR language allowing at most n atoms in the heads is more expressive than the language allowing at most m atoms, with mcalculus is a formalism for modelling molecular biology where molecules are terms with internal state and sites, bonds are represented by shared names labelling sites, and reactions are represented by rewriting rules. Depending on the shape of the rewriting rules, several dialects of the calculus can be obtained. We analyse the expressive power of some of these dialects by focusing on decidability and undecidability for problems like reachability and coverability.

Higher-Order Concurrency: Expressiveness and Decidability Results

Higher-order process calculi are formalisms for concurrency in which processes can be passed around in communications. Higher-order (or process-passing) concurrency is often presented as an alternative paradigm to the first order (or name-passing) concurrency of the pi-calculus for the description of mobile systems. These calculi are inspired by, and formally close to, the lambda-calculus, whose basic computational step ---beta-reduction--- involves term instantiation. The theory of higher-order process calculi is more complex than that of first-order process calculi. This shows up in, for instance, the definition of behavioral equivalences. A long-standing approach to overcome this burden is to define encodings of higher-order processes into a first-order setting, so as to transfer the theory of the first-order paradigm to the higher-order one. While satisfactory in the case of calculi with basic (higher-order) primitives, this indirect approach falls short in the case of higher-order process calculi featuring constructs for phenomena such as, e.g., localities and dynamic system reconfiguration, which are frequent in modern distributed systems. Indeed, for higher-order process calculi involving little more than traditional process communication, encodings into some first-order language are difficult to handle or do not exist. We then observe that foundational studies for higher-order process calculi must be carried out directly on them and exploit their peculiarities. This dissertation contributes to such foundational studies for higher-order process calculi. We concentrate on two closely interwoven issues in process calculi: expressiveness and decidability. Surprisingly, these issues have been little explored in the higher-order setting. Our research is centered around a core calculus for higher-order concurrency in which only the operators strictly necessary to obtain higher-order communication are retained. We develop the basic theory of this core calculus and rely on it to study the expressive power of issues universally accepted as basic in process calculi, namely synchrony, forwarding, and polyadic communication.