This research study investigated how professional mathematicians understand and operate with highly-abstract, advanced mathematical concepts in their own work. In particular, this study examined how professional mathematicians operated with mathematical concepts at different levels of understanding. Moreover, this study aimed to capture what factors influence professional mathematicians' level of understanding for particular mathematical concepts. To frame these research goals, three theoretical levels of understanding were proposed, process-level, pseudo-object-level, object-level, leveraging two ways that Piaget (1964) described what it meant to know or understand a mathematical concept. Specifically, he described understanding an object as being able to "act on it," and also as being able to "understand the process of this transformation" (p. 176). Process-level understanding corresponds to only understanding the underlying processes of the concept. Pseudo-object-level understanding corresponds to only being able to act on the concept as a form of object. Object-level understanding corresponds to when an individual has both of these types of understanding. This study is most especially concerned with how professional mathematicians operate with a pseudo-object-level understanding, which is called pseudo-objectification. For this study, six professional mathematicians with research specializing in a subfield of algebra were each interviewed three times. During the first interview, the participants were given two mathematical tasks, utilizing concepts in category theory which were unfamiliar to the participants, to investigate how they operate with mathematical concepts. The second interview utilized specific journal publications from each participant to generate discussion about influences on their level of understanding for the concepts in that journal article. The third interview utilized stimulated recall to triangulate and support the findings from the first two interviews. The findings and analysis revealed that professional mathematicians do engage in pseudo-objectification with mathematical concepts. This demonstrates that pseudo-objectification can be productively leveraged by professional mathematicians. Moreover, depending on their level of understanding for a given concept, they may operate differently with the concept. For example, when participants utilized pseudo-objects, they tended to rely on figurative material, such as commutative diagrams, to operate on the concepts. Regarding influences on understanding, various factors were shown to influence professional mathematicians' level of understanding for the concepts they use in their own work. These included factors pertaining to the mathematical concept itself, as well as other sociocultural or personal factors. / Doctor of Philosophy / In this research study, I investigated how professional mathematicians utilize advanced mathematical concepts in their own work. Specifically, I examined how professional mathematicians utilize mathematical concepts that they do not fully understand. I also examined what factors might influence a professional mathematician to fully understand or choose not to fully understand a mathematical concept they are using. To address these goals, six research-active mathematicians were each interviewed three times. In these interviews, the mathematicians engaged with mathematical concepts that were unfamiliar to them, as well as concepts from one of their own personal research journal publications. The findings demonstrated that professional mathematicians sometimes utilize mathematical concepts in different ways depending on how well they understand the concepts. Moreover, even if mathematicians do not have a full understanding of the concepts they are using, they can still sometimes productively leverage this amount of understanding to successfully reach their goals. I also demonstrate that various factors can and do influence how well a professional mathematician understands a given mathematical concept. Such influences could include the purpose of use for the concept, or what a mathematician's research community values.

Bundle adjustment is the process of simultaneously optimizing camera poses and 3D structure given image point tracks. In structure-from-motion, it is typically used as the final refinement step due to the nonlinearity of the problem, meaning that it requires sufficiently good initialization. Contrary to this belief, recent literature showed that useful solutions can be obtained even from arbitrary initialization for fixed-rank matrix factorization problems, including bundle adjustment with affine cameras. This property of wide convergence basin of high quality optima is desirable for any nonlinear optimization algorithm since obtaining good initial values can often be non-trivial. The aim of this thesis is to find the key factor behind the success of these recent matrix factorization algorithms and explore the potential applicability of the findings to bundle adjustment, which is closely related to matrix factorization. The thesis begins by unifying a handful of matrix factorization algorithms and comparing similarities and differences between them. The theoretical analysis shows that the set of successful algorithms actually stems from the same root of the optimization method called variable projection (VarPro). The investigation then extends to address why VarPro outperforms the joint optimization technique, which is widely used in computer vision. This algorithmic comparison of these methods yields a larger unification, leading to a conclusion that VarPro benefits from an unequal trust region assumption between two matrix factors. The thesis then explores ways to incorporate VarPro to bundle adjustment problems using projective and perspective cameras. Unfortunately, the added nonlinearity causes a substantial decrease in the convergence basin of VarPro, and therefore a bootstrapping strategy is proposed to bypass this issue. Experimental results show that it is possible to yield feasible metric reconstructions and pose estimations from arbitrary initialization given relatively clean point tracks, taking one step towards initialization-free structure-from-motion.

Made available in DSpace on 2016-04-28T20:38:37Z (GMT). No. of bitstreams: 1 Andrea de Davide Ratto Morelli.pdf: 720847 bytes, checksum: f67ae12322be10f9bae642d62c44bd21 (MD5) Previous issue date: 2013-03-15 / This study aims to gather information on some types of pathological organizations of the personality, using psychoanalytic knowledge. Several authors underlie it, starting with Sigmund Freud, Melanie Klein, Joan Riviere, Wilfred Bion, Herbert Rosenfeld, Hanna Segal and getting to John Steiner, Donald Meltzer, Frances Tustin, Judith Mitrani and James Grotstein, whose works in these areas are discussed more deeply. Efforts are made to understand and identify points of convergence, divergence and/or intersection among concepts like claustrum, psychic retreats, autistic capsules and adhesive pseudo-object relations. Discussion of the importance of continence and the development of schizo paranoid and depressive positions, permeate all the work and are fundamental to the approach of the clinical material presented. Analyst's psychic continence is questioned in face of difficulties as the claustrum seduction, attraction of adhesive pseudo-object relations, embarrassment of tenderness and struggles for dominate or exclusion of the analyst. Facing the difficulties of handling complex defensive systems, such as pathological organizations, confidence in the existence of unconscious need of psychic truth remains encouraging and cherishing, both to continue the trajectory of psychoanalytic exercise, and to achieve the needs of patients / Este estudo tem por objetivo recolher informações sobre alguns tipos de organizações patológicas da personalidade, utilizando conhecimentos psicanalíticos. Vários autores embasam-no partindo de Sigmund Freud, Melanie Klein, Joan Riviere, Wilfred Bion, Herbert Rosenfeld, Hanna Segal e chegando a John Steiner, Donald Meltzer, Frances Tustin, Judith Mitrani e James Grotstein, cujos trabalhos nessas áreas são discutidos mais profundamente. Esforços são realizados para compreender e identificar pontos de convergência, divergência e/ou intersecção entre conceitos como claustros, refúgios psíquicos, cápsulas autistas e pseudorrelações objetais adesivas. Discussões da importância da continência e da elaboração das posições esquizoparanoides e depressivas perpassam todo o trabalho e são fundamentais para a abordagem do material clínico apresentado. A continência psíquica do analista é questionada em face de dificuldades como a sedução dos claustros e a atratibilidade de pseudorrelações objetais adesivas, o embaraço diante da ternura e lutas por dominar ou excluir o analista. Diante da dificuldade de manejo com sistemas defensivos complexos, como os das organizações patológicas, a confiança na existência da necessidade inconsciente da verdade psíquica permanece estimulante e acalentadora tanto para continuar a trajetória do exercício psicanalítico, quanto para alcançar as necessidades dos pacientes