String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.

<p>This thesis studies computational interpretations of the intuitionistic sequent<br />calculus with implicit and explicit structural rules, with focus on the systems<br />with intersection types. The contributions of the thesis are grouped into three<br />parts. In the first part intersection types are introduced into the lambda<br />Gentzen calculus. The second part presents an extension of the lambda<br />Gentzen calculus to a term calculus with resource control, i.e. with explicit<br />operators for contraction and weakening, and apropriate intersection type<br />assignment system which characterises strong normalisation in the proposed<br />calculus. In the third part both previously studied calculi are integrated into<br />one framework by introducing the notion of the resource control cube.</p> / <p>Ова дисертација се бави рачунским интерпретацијама<br />интуиционистичког секвентног рачуна са имплицитним и експлицитним<br />структурним правилима, са фокусом на типске системе са пресеком.<br />Оригинални резултати тезе су груписани у три целине. У првом делу су<br />типови са пресеком уведени у lambda Gentzen рачун. Други део<br />представља проширење lambda Gentzen рачуна на формални рачун са<br />контролом ресурса, тј. са експлицитним операторима контракције и<br />слабљења, као и одговарајући типски систем са пресеком који<br />карактерише јаку нормализацију у уведеном рачуну. У трећем делу оба<br />рачуна су интегрисана у заједнички оквир увођењем структуре resource<br />control cube.</p> / <p>Ova disertacija se bavi računskim interpretacijama<br />intuicionističkog sekventnog računa sa implicitnim i eksplicitnim<br />strukturnim pravilima, sa fokusom na tipske sisteme sa presekom.<br />Originalni rezultati teze su grupisani u tri celine. U prvom delu su<br />tipovi sa presekom uvedeni u lambda Gentzen račun. Drugi deo<br />predstavlja proširenje lambda Gentzen računa na formalni račun sa<br />kontrolom resursa, tj. sa eksplicitnim operatorima kontrakcije i<br />slabljenja, kao i odgovarajući tipski sistem sa presekom koji<br />karakteriše jaku normalizaciju u uvedenom računu. U trećem delu oba<br />računa su integrisana u zajednički okvir uvođenjem strukture resource<br />control cube.</p>

A covariant Wigner-Weyl quantization formalism on the manifold that uses pseudo-differential operators is proposed. The asymptotic product formula that leads to the symbol calculus in the presence of gauge and gravitational fields is presented. The new definition is used to get covariant differential operators from momentum polynomial symbols. A covariant Wigner function is defined and shown to give gauge-invariant results for the Landau problem. An example of the covariant Wigner function on the 2-sphere is also included.

The utility of dental calculus as a proxy for diet and mouth use is explored for the Middle Holocene Cis-Baikal region of Central Siberia based on two methods: a macroscopic analysis of severity and a microscopic analysis of particles within deposits. The study area was inhabited by two culturally and biologically distinct cultures, the Early Neolithic (EN) Kitoi culture (8,000 to 7,000/6,800 cal B.P.) and the Late Neolithic-Early Bronze Age (LN-EBA) Isakovo-Serovo-Glaskovo (ISG) cultural complex (6,000/5,800 to 4,000 cal B.P.), separated by a period of cultural transition marked by a cessation in formal cemetery use. Data were collected from four cemetery sites, two dating to the EN and two dating to the LN-EBA. Nonparametric testing of calculus severity revealed that, for adult males and juveniles, lakeshore populations displayed greater affinity to each other than to their contemporaneous cultural counterpart populations living along riverine systems in the Angara River Valley. Trends within the EN cemetery Shamanka II contrasted to the other cemetery populations, with noticeably larger deposits in anterior quadrants and significant sexual distinctions. The proportion of protein to carbohydrates consumed is known to influence calculus formation, but both cultural groups lived on a diet based predominately on meat sources so dietary ratios alone do not adequately explain the differences distinguished. A complex multifactorial model involving microregional differences in resources/environment, foraging patterns, individual variation, and dental wear patterns provides at least a partial explanation for the results observed. A wide range of particles were recovered during the microscopic analysis of calculus, albeit in low concentrations. The low starch grain counts were consistent with a diet based predominately on meats but still provide some of the first direct evidence for plant consumption in the Cis-Baikal, including possible plant processing by cooking or grinding based on damage evident on the grains. Other particles recovered may provide evidence of mouth use activities or palaeoenvironmental influences. Together, the two components of this analysis offer strong evidence that dental calculus is a useful tool for reconstructing hunter-gatherer lifeways but also highlight the limitations of conducting this type of research on previously excavated and potentially contaminated material.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / The computation of the greeks of an option is an important aspect of financial mathematics. The information gained from knowing the value of a greek of an option can help investors decide whether or not to hold on to or to sell their options to avoid losses or gain a profit. However, there are technical difficulties that arise from having to do this. Among them is the fact that the mathematical formula for the value some options is complex in nature and evaluating their greeks may be cumber- some. On the other hand the greek might have to be numerically estimated if the option does not posses an explicit evaluation formula. This could be a computationally expensive undertaking. Malliavin calculus offers us a solution to these problems. We can find formula that can be used in combination with Monte Carlo simulations to give results quickly and which are not computationally expensive to obtain and hence give us an degree of accuracy higher that non Malliavin calculus techniques. This thesis will develop the Malliavin calculus tools that will enable us to develop the tools which we will then use to compute the greeks of some known options.

Acompanha: Caderno de tarefas: proposta de tarefas para um estudo inicial de derivadas / CNPq / Esta dissertação apresenta uma proposta de tarefas para o estudo inicial de derivadas no ensino de cálculo diferencial e integral (CDI) no Ensino Superior, em turmas regulares de um curso de Engenharia da Universidade Tecnológica Federal do Paraná (UTFPR) do campus Londrina. Elencou-se como objetivo geral da pesquisa a proposição de tarefas que oportunizem aos estudantes a exploração de ideias necessárias à compreensão do conceito de derivadas, em especial tarefas a serem aplicadas em momentos que iniciam o estudo de derivadas, em sua abordagem mais formal. Por se tratar de um mestrado em âmbito profissional, intencionou-se a construção de caderno de tarefas (o produto educacional), na qual após a aplicação de dois ciclos de pesquisa, elencaram-se três tarefas para compor o produto final da pesquisa, a qual por meio das análises notou-se a necessidade entre os ciclos a adaptação/reformulação das tarefas, e em especial na tarefa 3 a intencionalidade de uma nova reformulação e aplicação em um novo ciclo de pesquisa. / This dissertation presents a proposal to the initial study of derivatives in the teaching of differential and integral calculus (CDI) in higher education, in regular classes of an engineering degree from the Federal University of technology-Paraná (UTFPR) campus. Presented as general purpose of research the proposition of tasks that create opportunities for students to exploration of ideas necessary for the understanding of the concept of derived in particular tasks to be applied at times to begin the study of derived, in your more formal approach. As a master's degree in professional, intended the construction of notebook (the educational product), in which after two cycles of research, bleeding cool is-if three tasks to compose the final research product, which by means of analyses the need was noted between cycles the adaptation/recasting of tasks, and in particular in task 3 the intentionality of a new makeover and application in a new cycle of research.

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-08-25T18:53:26Z No. of bitstreams: 1 DissGP.pdf: 3104844 bytes, checksum: 2e20a531a8e46327359fb4eacc9adf50 (MD5) / Approved for entry into archive by Ronildo Prado (bco.producao.intelectual@gmail.com) on 2018-01-24T12:17:05Z (GMT) No. of bitstreams: 1 DissGP.pdf: 3104844 bytes, checksum: 2e20a531a8e46327359fb4eacc9adf50 (MD5) / Approved for entry into archive by Ronildo Prado (bco.producao.intelectual@gmail.com) on 2018-01-24T12:18:07Z (GMT) No. of bitstreams: 1 DissGP.pdf: 3104844 bytes, checksum: 2e20a531a8e46327359fb4eacc9adf50 (MD5) / Made available in DSpace on 2018-01-24T12:21:46Z (GMT). No. of bitstreams: 1 DissGP.pdf: 3104844 bytes, checksum: 2e20a531a8e46327359fb4eacc9adf50 (MD5) Previous issue date: 2017-05-05 / Não recebi financiamento / The purpose of this work is to propose a method that allow to bring the fundamentals of mathematical logic to high school through the use of set theory, however making the whole approach of the subject through diagrams, making it possible to avoid, for the demonstrations and understanding necessary to the development of the subject, the rigors of writing used in mathematical logic, which, in a first contact, tend to discourage the interest of beginning students. / O objetivo deste trabalho é propor um método que permita levar os fundamentos da lógica matemática para o ensino médio por meio da utilização da teoria dos conjuntos, porém fazendo todo a aproximação do assunto utilizando diagramas, tornando possível evitar assim, para as demonstrações e o entendimento necessários ao desenvolvimento do assunto, os rigores da escrita utilizada na lógica matemática, que, em um primeiro contato, podem desestimular o interesse dos alunos iniciantes

Dans cette thèse, nous nous intéressons à la fois aux aspects théoriques et à la résolution numérique du problème de Mumford-Shah avec anisotropie pour la restauration et la segmentation d'image. Cette fonctionnelle possède en effet la particularité de reconstruire une image dégradée tout en extrayant l'ensemble des contours des régions d'intérêt au sein de l'image. Numériquement, on utilise l'approximation d'Ambrosio-Tortorelli pour approcher un minimiseur de la fonctionnelle de Mumford-Shah. Elle Gamma-converge vers cette dernière et permet elle aussi d'extraire les contours. Les implémentations avec des schémas aux différences finies ou aux éléments finis sont toutefois peu adaptées pour l'optimisation de la fonctionnelle d'Ambrosio-Tortorelli. On présente ainsi deux nouvelles formulations discrètes de la fonctionnelle d'Ambrosio-Tortorelli à l'aide des opérateurs et du formalisme du calcul discret. Ces approches sont utilisées pour la restauration d'images ainsi que pour le lissage du champ de normales et la détection de saillances des surfaces digitales de l'espace. Nous étudions aussi un second problème d'optimisation de forme similaire avec conditions aux bords de Robin. Nous démontrons dans un premier temps l'existence et la régularité partielle des solutions, et dans un second temps deux approximations par Gamma-convergence pour la résolution numérique du problème. L'analyse numérique montre une nouvelle fois les difficultés rencontrées pour la minimisation d'approximations par Gamma-convergence. / In this work, we study both theoretical and numerical aspects of an anisotropic Mumford-Shah problem for image restoration and segmentation. The Mumford-Shah functional allows to both reconstruct a degraded image and extract the contours of the region of interest. Numerically, we use the Amborsio-Tortorelli approximation to approach a minimizer of the Mumford-Shah functional. It Gamma-converges to the Mumford-Shah functional and allows also to extract the contours. However, the minimization of the Ambrosio-Tortorelli functional using standard discretization schemes such as finite differences or finite elements leads to difficulties. We thus present two new discrete formulations of the Ambrosio-Tortorelli functional using the framework of discrete calculus. We use these approaches for image restoration and for the reconstruction of normal vector field and feature extraction on digital data. We finally study another similar shape optimization problem with Robin boundary conditions. We first prove existence and partial regularity of solutions and then construct and demonstrate the Gamma-convergence of two approximations. Numerical analysis shows once again the difficulties dealing with Gamma-convergent approximations.

In this investigation an attempt was made to determine the misconceptions that engineering students have of the idea of a limit. A comprehensive literature study showed that there are a number of common misconceptions that students normally form. The empirical investigation was done in two phases. A questionnaire on the idea of a limit was given to the students during the first phase. During the second phase six interviews were conducted. The findings were grouped according to the nature of a limit and students' views on the relationship between the continuity of a function at a point and the limit at that point. An analysis of these findings led to the identification of the misconceptions that these students have of the idea of a limit. / In hierdie ondersoek is gepoog om die wanbegrippe wat ingenieursstudente van die limietbegrip vorm, bloot te stel. 'n Omvattende literatuurstudie het 'n aantal algemene wanbegrippe aan die lig gebring. Die empiriese ondersoek het in twee fases plaasgevind. Tydens die eerste fase is 'n vraelys aan die studente gegee in 'n poging om meer te wete te kom van hulle begrip van 'n limiet. Die vraelys is opgevolg deur ses onderhoude. Die responsies is gegroepeer in terme van die aard van 'n limiet en studente se sienings van die kontinuiteit van 'n funksie by 'n punt en die limiet by daardie punt. Die analisering van hierdie responsies het die identifisering van 'n aantal wanbegrippe by hierdie groep studente moontlik gemaak. / Educational Studies / M.Ed. (with specialisation in Mathematics Education)

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-07-05T11:22:34Z No. of bitstreams: 1 marcosribeiroraad.pdf: 5949751 bytes, checksum: e73690c146e95040449c3d7e0593ab74 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T16:23:53Z (GMT) No. of bitstreams: 1 marcosribeiroraad.pdf: 5949751 bytes, checksum: e73690c146e95040449c3d7e0593ab74 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T16:24:02Z (GMT) No. of bitstreams: 1 marcosribeiroraad.pdf: 5949751 bytes, checksum: e73690c146e95040449c3d7e0593ab74 (MD5) / Made available in DSpace on 2016-07-13T16:24:02Z (GMT). No. of bitstreams: 1 marcosribeiroraad.pdf: 5949751 bytes, checksum: e73690c146e95040449c3d7e0593ab74 (MD5) Previous issue date: 2012-05-17 / O presente trabalho tem como objetivo o estudo histórico sobre o ensino da disciplina Cálculo Diferencial e Integral na Universidade Federal de Juiz de Fora (UFJF) durante as décadas de 1970 e 1980 com a intenção de identificar traços, vestígios da cultura desse ensino. Dessa forma, a questão norteadora foi: como se caracteriza a cultura do ensino de Cálculo Diferencial e Integral nas décadas de 1970 e 1980? O estudo aqui proposto, ao investigar historicamente os processos de ensino e aprendizagem da Matemática, naturalmente se insere no campo de pesquisa da história da educação matemática com um suporte teórico-metodológico proveniente da História da Educação entendida como especificidade da História. As fontes analisadas foram as notas de aula de um professor de Cálculo do Departamento de Matemática da UFJF do século passado, o caderno de Cálculo de um aluno do referido professor, as atas departamentais, livros texto de Cálculo do período em questão; além de entrevista com o docente autor das notas. As conclusões do estudo delineado apontam para a identificação de elementos da cultura de ensino de Cálculo como: o rigor, os pré-requisitos, a reprovação, as aplicações da matemática, a ênfase no treinamento e a seqüência de ensino função-limite-derivada-integral. / The present work aims to study the teaching of history on Differential and Integral Calculus course at the Federal University of Juiz de Fora (UFJF) during the 1970s and 1980s with the intention of identifying traces, vestiges of the culture of teaching. Thus, the question was: how to characterize the culture of teaching of differential and integral calculus in the 1970s and 1980s? The study proposed here, by taking the history of teaching and learning of mathematics, of course fits into the search field of the history of mathematics education with a theoretical-methodological support from the History of Education understood as specificity of history. The sources were the lecture notes of a professor of the Department of Mathematics Calculus UFJF of the last century, the contract of a Calculus student of that teacher, the departmental proceedings, textbooks Calculation of the period in question, in addition to interview with the faculty author of the notes. The findings outlined point to identify elements of culture as Calculation education: rigor, the prerequisites, the disapproval, the applications of mathematics, the emphasis on training and the sequence teaching function-limit-derivative-integral.