<p dir="ltr">The concept of infinity plays a pivotal role in mathematics, yet its precise definition remains elusive. This conceptual ambiguity has given rise to several puzzles in contemporary philosophy of mathematics. In response, this dissertation embarks on a rational reconstruction of Hegels concept of infinity and applies it to resolve two groups of mathematical puzzles, including challenges in applied mathematics, especially the application of differential calculus, and the conceptual ground of set theory, especially Cantors paradox.</p><p dir="ltr">The exploration begins with a historical survey of the concept of infinity in philosophy. It becomes evident that a prevailing interpretation characterizes infinity as the unlimited. In addition, this unlimitedness has taken various forms, including endlessness (Aristotle), all-inclusiveness (Spinoza), and self-sufficiency (Kant).</p><p dir="ltr">The heart of the dissertation lies in reconstructing Hegels concept of genuine infinity. Hegel argues that the unlimited as the negation of the limit entails either the completely indeterminate or another limited entity, neither of which is genuinely infinite. Instead, Hegel points out that genuine infinity is the self-relation of a limited entity. By self-relation, Hegel means that the limited entity alters into another limited entity that is isomorphic to the original one.</p><p dir="ltr">Subsequently, Hegel’s concept of genuine infinity can be translated into a mathematical framework as the intrinsic alteration of quantum (roughly speaking, quantum is Hegel’s term for the variable), which is captured by the corresponding relation among quanta. It is argued that this relation serves as the necessary condition for three mathematical entities traditionally considered infinite: arbitrarily large (small) numbers, infinite sets, and endless sequences. Thus, for Hegel, this intrinsic relation among quanta constitutes the essence of mathematical infinity.</p><p dir="ltr">Hegels concept of mathematical infinity can help us resolve difficulties within contemporary mathematics. First, it addresses the question of why infinite mathematical structures can be applied to describe and predict seemingly finite physical phenomena. The application of mathematics is usually explained by the similarity between mathematical structures and empirical systems, but the lack of apparent empirical counterpart leads one to doubt the application of infinite mathematical structures. Hegels concept of mathematical infinity directs us to focus on the structural similarity between infinite mathematical structures and empirical systems, specifically between the intrinsic alteration of quantum and the change of physical properties with time. With this structural similarity, the application of mathematics can be explained. Second, the dissertation investigates the conceptual ground of set theory, especially the relationship between a set and its members. Hegels analysis of genuine infinity provides a twofold clarification: (1) members of set must be a unit first, which entails that the set of all sets (the Universe) is not a set; (2) members of a set are simultaneously distinct (due to their independent logical content) yet indistinguishable (due to their common structure as a unit). Clarification 1 resolves Cantors paradox as it excludes the Universe; clarification 2 explains arithmetic operations.</p>

This dissertation presents three essays that focus on various aspects pertaining to the management of online platforms, defined as "digital services that facilitate interactions between two or more distinct, but interdependent sets of users (whether firms or individuals) who interact through the service via the Internet" (OECD, 2019). The interactions benefit both the users and the platform. Managing online platforms involves developing strategies for one or more of three value adding functions: (a) lowering search costs for the parties connecting through the platform, (b) providing a technology infrastructure that facilitates transactions at scale by sharing both demand and supply side costs; and (c) locating other audiences or consumers for the output that results from the transaction. The platform manager must manage these value adding functions. Thus, one important management task is to recognize potential asymmetries in the economic and/or psychological motivations of the transacting parties connected through the platform. In this dissertation, I empirically examine these issues in greater detail. The first essay, "Incentivizing User-Generated Content—A Double-Edged Sword: Evidence from Field Data and a Controlled Experiment," addresses the conundrum faced by online platform managers interested in crowdsourcing user-generated content (UGC) in prosocial contexts. The dilemma stems from the fact that offering monetary incentives to stimulate UGC contributions also has a damping effect on peer approval, which is an important source of non-monetary recognition valued by UGC contributors in prosocial contexts. The second essay, "Matching and Making in Matchmaking Platforms: A Structural Analysis," examines matchmaking platforms, focusing specifically on the problem of misaligned incentives between the platform and the agents. Based on data from the Ultimate Fighting Championship (UFC) on fighter characteristics, and pay-per-view revenues associated with specific bouts, we identify the potential for conflicts of interest and examine strategies that may be used to mitigate such problems. The third essay, "Matching and Making in Matching Markets: A Managerial Decision Calculus," extends the empirical model and analytical work to a class of commonly encountered one-sided matching market problems. It provides the conceptual outline of a decision calculus that allows managers to explore the revenue and profitability implications of adaptive changes to the tier structures and matching algorithms. / Doctor of Philosophy / The 21st century has witnessed the rise of the platform economy. Consumers routinely interact with online platforms ways in their day to day activities. For instance, they interact with platforms such as Quora, StackOverflow, Uber, and Airbnb to name only a few. Such platforms address a variety of needs starting from providing users with answers to a variety of questions to matching them with a range of service providers (e.g., for travel and dining needs). However, the rapid growth of the platform economy has created a knowledge gap for both consumers and platforms. The three essays in this dissertation attempt to contribute to the literature in this area. The first essay, "Incentivizing User-Generated Content—A Double-Edged Sword: Evidence from Field Data and a Controlled Experiment," examines how crowdsourcing contests influence the quantity and quality of user-generated content (UGC). Analyzing data from the popular question and answer website Quora, we find that offering monetary incentives to stimulate UGC contributions increases contributions but also has a simultaneous damping effect on peer endorsement, which is an important source of non-monetary recognition for UGC contributors in prosocial contexts. The second essay, "Matching and Making in Matchmaking Platforms: A Structural Analysis," examines matchmaking platforms, focusing on the problem of misaligned incentives between the platform and the agents. Based on data from the Ultimate Fighting Championship (UFC) on fighter characteristics, and pay-per-view revenues associated with specific bouts, we identify the potential for conflicts of interest and examine strategies that may be used to mitigate such problems. The third essay, "Matching and Making in Matching Markets: A Managerial Decision Calculus," extends the empirical model and analytical work to a class of commonly encountered one-sided matching market problems. It provides the conceptual outline of a decision calculus that allows managers to explore the revenue and profitability implications of adaptive changes to the tier structures and matching algorithms.

Yes / Exterior calculus with its three operations meet, join and hodge star complement, is used for the representation of fermion-hole systems and for fermionic analogues of logical gates. Two different schemes that implement fermionic quantum computation, are proposed. The first scheme compares fermionic gates with Boolean gates, and leads to novel electronic devices that simulate fermionic gates. The second scheme uses a well known map between fermionic and multi-qubit systems, to simulate fermionic gates within multi-qubit systems.

This dissertation investigates high-level decision making for agents that are both goal and utility driven. We develop a partially observable Markov decision process (POMDP) planner which is an extension of an agent programming language called DTGolog, itself an extension of the Golog language. Golog is based on a logic for reasoning about action—the situation calculus. A POMDP planner on its own cannot cope well with dynamically changing environments and complicated goals. This is exactly a strength of the belief-desire-intention (BDI) model: BDI theory has been developed to design agents that can select goals intelligently, dynamically abandon and adopt new goals, and yet commit to intentions for achieving goals. The contribution of this research is twofold: (1) developing a relational POMDP planner for cognitive robotics, (2) specifying a preliminary BDI architecture that can deal with stochasticity in action and perception, by employing the planner. / Computing / M. Sc. (Computer Science)

This study was prompted by serious problems regarding specific teaching and learning problems in calculus at the technikon. The general aims were to identify and analyze particular teaching and learning problems relating to 2nd level engineering courses in calculus and to recommend improvements which could increase student performance in engineering calculus courses. An extensive study revealed world wide concern in calculus reform. The empirical research instruments consisted of structured questionnaires given to staff and students from nine technikons plus interviews. Five serious problem areas were identified: student ability in mathematics, content difficulty, background difficulties, timetable pressures and lecturer's presentation. The impact of training technology on calculus was investigated. Recommendations were that routine exercises can be done on computer with extra tutorial time for computer laboratory projects. Background recommendations suggested that schools give more time to trigonometry and coordinate geometry and that bridging courses at technikons for weaker students be developed. / Curriculum and Instructional Studies / M. Ed. (Didactics)

Made available in DSpace on 2016-04-27T16:57:05Z (GMT). No. of bitstreams: 1 Erika Andersen.pdf: 2001632 bytes, checksum: 36ee4401b855c9699524d7159f8bb771 (MD5) Previous issue date: 2011-05-27 / The present study relates the results of a qualitative research that aimed to investigate which mental processes may intervene and be combined by students in the development of activities involving the expression = F( x ) = f ( t )dt . The research was based on the study titled Advanced Mathematical Thinking Processes of Tommy Dreyfus. The survey instrument was developed, implemented and analyzed using some phases of Didactic Engineering. The fourteen participants in this study were students of private university s math course in São Paulo city. The analysis of the student s protocols indicates that the following processes were mobilized: visualization, representation and switching representations, intuition, definition, discovery, validation, generalization, abstraction and synthesis. This allowed many students to conjecture that the derivation and integration are inverse operations of each other. The results of the survey explained that a work of this nature contributes greatly to students to take ownership of interrelationships between concepts involved in the Fundamental Theorem of Calculus / O presente estudo relata os resultados de uma pesquisa qualitativa cujo objetivo era investigar quais processos mentais podem intervir e ser combinados por alunos no desenvolvimento de atividades envolvendo a expressão = F(x) = f ( t )dt . Além disso, verificar se esse tipo de atividade favorece a compreensão das ideias centrais envolvidas no Teorema Fundamental do Cálculo. A pesquisa fundamentou-se no estudo de Tommy Dreyfus intitulado Processos do Pensamento Matemático Avançado. O instrumento de pesquisa foi elaborado, aplicado e analisado, utilizando algumas fases da Engenharia Didática. Os catorze participantes deste estudo eram alunos do curso Licenciatura em Matemática de uma universidade particular da cidade de São Paulo. A análise dos protocolos dos estudantes indica que os processos do PMA mobilizados foram: visualização, representação e mudança entre diferentes representações, intuição, definição, descoberta, validação, generalização, síntese e abstração. O que possibilitou que muitos dos participantes conjecturassem que a derivação e integração são operações inversas uma da outra. Os resultados da pesquisa explicitaram que um trabalho desta natureza muito contribui para que os alunos se apropriem de inter-relações entre conceitos envolvidos no Teorema Fundamental do Cálculo

Made available in DSpace on 2016-04-27T16:57:26Z (GMT). No. of bitstreams: 1 Marcio Vieira de Almeida.pdf: 1460243 bytes, checksum: a4f44f26cc2c378ae48bfea70049311a (MD5) Previous issue date: 2013-06-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The focus on this research is the learning and teaching of Differential and Integral Calculus, through the reading of articles written by the researcher David Tall. It is a bibliographical theoretical research, in the modality of panorama, in which the organization is also based on elements of Content Analysis, according to Bardin. We present information about the biography of the English researcher and his relationship with the community of national research. The CAPES Thesis Database was studied, with the objective of identifying the use of theories developed by Tall in national researches. The material for analysis, used for the development of the panorama, was based on 14 articles, taken from the session Limits, Infinity & Infinitesimals of the academic website of the English researcher. The theoretical elements and the approaches in teaching formulated to the concepts real numbers, infinity, limits, continuity, derivatives, integral and differential equations are highlighted in this material. The panorama brings summaries and analysis of theoretical elements, besides highlighting important information on the learning and teaching of Differential and Integral Calculus under Tall s perspectives. With this research, we hope to have contributed to both research and teaching practice / Esta pesquisa tem por foco a aprendizagem e o ensino do Cálculo Diferencial e Integral. Trata-se da realização de um panorama de artigos de autoria de David Tall relacionados a esse tema. É um estudo teórico de caráter bibliográfico, na modalidade panorama, cuja organização se pautou também em elementos da Análise de Conteúdo, segundo Bardin. São apresentados dados sobre a biografia do pesquisador inglês e a relação dele com a comunidade de pesquisa nacional. É realizado um levantamento, no banco de dissertações e teses da CAPES, com a intenção de identificar a utilização dos elementos teóricos desenvolvidos por Tall, em pesquisas nacionais. O material de análise, utilizado para o desenvolvimento do panorama, constituiu-se de 14 artigos, retirados da seção Limits, Infinity & Infinitesimals do sítio acadêmico do pesquisador. São destacados, nesse material, os elementos teóricos e as abordagens para o ensino formuladas para os conceitos: números reais, infinito, limites, continuidade, derivada, integral e equações diferenciais. O panorama traz sínteses e análises de elementos teóricos, além de colocar em evidência dados importantes sobre a aprendizagem e o ensino do Cálculo Diferencial e Integral, na perspectiva de Tall. Com a apresentação deste trabalho espera-se ter contribuído tanto com a pesquisa quanto com a prática docente

Made available in DSpace on 2016-04-27T17:27:11Z (GMT). No. of bitstreams: 1 Cassia Sampaio Sanctos.pdf: 875084 bytes, checksum: 3baa23ce43e41c748fa70bf983f30e20 (MD5) Previous issue date: 2015-05-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The thesis constitutes a critical review of Gödel´s doctoral dissertation which presents a proof for the completeness of first order logic. The introduction addresses the concepts of formalism, axiomatic method and completeness, thus the proof can be contextualized. The language for the restricted functional calculus is defined, with the corresponding syntax and semantics, and the original Gödel´s demonstration is updated. The appendix contains a translation of the referred dissertation, which is unprecedented in Portuguese / O trabalho constitui um comentário crítico da dissertação de doutorado de Gödel que apresenta uma prova de completude da lógica de primeira ordem. A introdução trata dos conceitos de formalismo, método axiomático e completude, para que seja possível contextualizar a prova. A linguagem para o cálculo funcional restrito é definida, com sua sintaxe e semântica, e a demonstração original de Gödel é atualizada. O apêndice contém a tradução da referida dissertação, que é inédita em língua portuguesa

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2017-08-02T14:32:30Z No. of bitstreams: 1 Marcio Vieira de Almeida.pdf: 5322268 bytes, checksum: 95a05019d55b263aef725a9ef6402f5e (MD5) / Made available in DSpace on 2017-08-02T14:32:30Z (GMT). No. of bitstreams: 1 Marcio Vieira de Almeida.pdf: 5322268 bytes, checksum: 95a05019d55b263aef725a9ef6402f5e (MD5) Previous issue date: 2017-07-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This thesis presents a material for the teaching of Differential and Integral Calculus, composed by seven activities, which were based on theoretical references of Mathematical Education. The concepts of function, continuity, differentiability, solution of a differential equation, integral and limit of sequences were approached in these activities. The intention was to defend that one of the ways to establish the narrowing of the relation of theory and practice in this area of investigation is done through the elaboration of materials for teaching with this goal. The concepts of generic organizer, cognitive root, and Three Worlds of Mathematics by Tall and collaborators and the idea of resource of Documental Genesis of Gueudet and Trouche were used. The use of the computer and the construction of tools on GeoGebra were productive procedures to obtain a material with the planned qualities. The research, which had as a result the material for teaching, followed the methodological orientation of a type of fundamental research, in which the goal is the filling of gaps in knowledge related to the solution of problems through practice. An explanatory, theoretical posture was adopted, the construction of considerations with rigor and logical coherence to validate the obtained results. In the scope of theoretic-methodological references seven activities were elaborated for the teaching of Calculus organized in three components which, compose a resource (mathematics, material and didactics) in the conception of Documental Genesis, incorporating cognitivist ideas of Tall and his associates. Using the components (mathematics, material and didactics) allows that the material may configure itself as an element of the set of resources, according to the Documental Genesis, which a teacher of Calculus can use for the development of a class. As a result it is possible to demonstrate that the way of elaboration proposed for a material for teaching, in which theories of Mathematical Education are elaborated and adequate software is used, may be a powerful way to favor the integration of theory and practice, pursued and necessary for Mathematic Education, besides contributing with learning / Esta tese apresenta um material para o ensino de Cálculo Diferencial e Integral composto por sete atividades que foram embasadas em referenciais teóricos da Educação Matemática. Nelas, foram abordados os conceitos de função, continuidade, diferenciabilidade, solução de uma equação diferencial, integral e limite de sequências. Pretendeu-se defender que uma das formas de se estabelecer o estreitamento da relação teoria e prática nessa área de investigação é feita por meio de elaboração de materiais para o ensino com essa finalidade. Foram utilizadas as noções de organizador genérico, raiz cognitiva e Três Mundos da Matemática de Tall e colaboradores, e a noção de recurso da Gênese Documental de Gueudet e Trouche. O uso do computador e a construção de ferramentas no GeoGebra foram procedimentos férteis para se obter um material com as competências planejadas. A pesquisa, que teve por resultado o material para o ensino, seguiu orientação metodológica de uma do tipo pesquisa fundamental, na qual se objetiva o preenchimento de lacunas no conhecimento relativo à solução de problemas advindos da prática. Adotou-se uma postura teórica exploratória, a da construção de argumentos com rigor e coerência lógica para validar os resultados obtidos. Nesse âmbito de referenciais teórico- metodológicos, foram elaboradas sete atividades para o ensino de Cálculo, organizadas em três componentes, as quais compõem um recurso (matemática, material e didática) na concepção da Gênese Documental, incorporando noções cognitivistas de Tall e seus associados. A utilização das componentes (matemática, material e didática) possibilita que o material possa se configurar em um elemento do conjunto de recursos, conforme a Gênese Documental, de um professor de Cálculo, para o desenvolvimento de uma aula. Como resultado pode-se demonstrar que o modo de elaboração proposto para um material para o ensino, em que se incorporam teorias da Educação Matemática e se utiliza um software adequado, pode ser um meio potente para favorecer a integração teoria e prática, almejada e necessária pela Educação Matemática, além de contribuir com a aprendizagem

Linear Evolution Equations (LEE) have been studied extensively over many years. Their extension in the field of fractional calculus have been defined by Dαu(x, t) = Au(x, t), where α is the fractional order and Dα is a generalized differential operator. Two types of generalized differential operators were applied to the LEE in the state-of-the-art, producing the Riemann-Liouville and the Caputo time fractional evolution equations. However the extension of the new Caputo-Fabrizio derivative (CFFD) to these equations has not been developed. This work investigates existing fractional derivative evolution equations and analyze the generalized linear evolution equations with non-singular ker- nel derivative. The well-posedness of the extended CFFD linear evolution equation is demonstrated by proving the existence of a solution, the uniqueness of the existing solu- tion, and finally the continuous dependence of the behavior of the solution on the data and parameters. Extended evolution equations with CFFD are applied to kinetics, heat diffusion and dispersion of shallow water waves using MATLAB simulation software for validation purpose. / Mathematical Science / M Sc. (Applied Mathematics)