Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2007. / Includes bibliographical references (p. 27-28).

In this research report we explore some applications of symmetry methods for boundary value problems in the pricing of barrier options. Various nancial instruments satisfy the Black-Scholes partial di erential equation (pde) but with di erent domain, maturity date and boundary conditions. We nd Lie symmetries that leave the Black-Scholes (pde) invariant and will guarantee that the relevant solutions satisfy the boundary conditions. Using these sym- metries, we can thus generate group-invariant solutions to the boundary value problem.

Thesis (Ph.D.)--University of Western Sydney, 2008. / A thesis presented to the University of Western Sydney, College of Health and Science, School of Computing and Mathematics, in fulfilment of the requirements for the degree of Doctor of Philosophy - Science. Includes bibliographies.

Reprint of one article by the author included in back of publication / 155 leaves ; 26 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematical Physics, 1973

Thesis (Ph.D.) -- University of Adelaide, Dept. of Mathematical Physics, 1973. / Reprint of one article by the author included in back of publication.

Thesis (Ph. D.)--University of Oregon, 2005. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 44-45). Also available for download via the World Wide Web; free to University of Oregon users.

Made available in DSpace on 2014-06-11T19:24:52Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-11-16Bitstream added on 2014-06-13T20:13:17Z : No. of bitstreams: 1 neves_prv_me_rcla.pdf: 1296931 bytes, checksum: b45ea16895bf9e20db063325be68a349 (MD5) / Este trabalho teve o objetivo de produzir um conjunto de atividades para analisar como o uso do caleidoscópio associado ao estudo dos ornamentos planos pode contribuir no ensino de grupos de simetria e transformações geométricas em um curso de graduação em Matemática. Esta pesquisa tem caráter qualitativo e foi desenvolvida segundo a proposta metodológica de Romberg. Elaborou-se uma proposta de ensino baseada na metodologia de Resolução de Problemas que foi aplicada a um grupo de professores (alguns em fase de formação) de matemática. As atividades tiveram a finalidade de fazer com que os alunos usassem o caleidoscópio para reproduzir ornamentos planos e, a partir de então, discutissem, com base em argumentos geométricos e algébricos, quais as possibilidades (e impossibilidades) que esse instrumento oferece para obtenção desses ornamentos e suas respectivas justificativas. A coleta de dados ocorreu, essencialmente, por observação participante em sala de aula por meio do uso de questionários, anotações e registros fotográficos. Após a coleta de dados, foi feita uma análise das possibilidades e limitações do material desenvolvido para o ensino de grupos de simetria e transformações geométricas, bem como o uso do caleidoscópio enquanto recurso didático / The purpose of this work was to develop a set of activities to analyze how the use of kaleidoscope associated to the study of ornaments can contribute to the teaching of symmetry groups and geometric transformations on a undergraduate course in Mathematics. This is a qualitative research and it was developed according to the methodological proposal of Romberg. A teaching proposal was drafted and was applied to a group of mathematics teachers. Activities were designed following the methodology of problem-solving and intended to make students to use the kaleidoscope to reproduce some ornaments and thereafter, discuss, based on geometric and algebraic arguments, the possibilities and impossibilities that this tool provides to obtain ornaments and their respective justifications. Data collection occurred primarily by participant observation in the classroom through the use of questionnaires, notes and photographic records. After the end of the course a viability analysis of the activities was done (possibilities and limitations) for teaching symmetry groups and geometric transformations as well as the use of Kaleidoscope as a didactic tool

A system has been implemented which is able to detect symmetrical groupings in edge images. The initial stages of the algorithm consist of edge detection, curve smoothing, and the extension of the perceptual grouping phase of the SCERPO [Low87] vision system to enable detection of instances of endpoint proximity and curvilinearity among curved segments. The symmetry detection stage begins by first locating points along object boundaries which are significant in terms of curvature. These key points are then tested against each other in order to detect locally symmetric pairs. An iterative grouping procedure is then applied which matches these pairs together using a more global definition of symmetry. The end result of this process is a set of pairs of key points along the boundary of an object which are bilaterally symmetric, along with the axis of symmetry for the object or sub-object. This paper describes the implementation of this system and presents several examples of the results obtained using real images. The output of the system is intended for use as indexing features in a model-based object recognition system, such as SCERPO, which requires as input a set of spatial correspondences between image features and model features. / Science, Faculty of / Computer Science, Department of / Graduate

Mean Field Games (MFG) are the infinite-population analogue of symmetric stochastic differential games with interacting players. By considering a limiting model with a continuum of players, the theory of MFG provides a more tractable representation and can effectively approximate a broad class of perfectly symmetric stochastic dynamic games. This thesis studies games with heterogeneous players, the heterogeneity being expressed either through a type parameter or through asymmetric interactions among players, and aims at understanding under which condition the MFG approximation remains valid for such games and, if it fails, to find a substitute model. In many real-life settings, players do not view themselves as exchangeable and accurate models should incorporate this heterogeneity. We first adapt the MFG paradigm to model more heterogeneous agents by introducing a type parameter in a financial problem that has gained huge interest in the recent years: the competitive Merton problem under relative performance criteria. By deriving a closed-form solution for the finitely many player investment-consumption problem, we show how the risk tolerance and competitivity of the investors influence their optimal strategy in equilibrium. Moreover, this thesis contributes to a very recent line of work bridging MFG theory and network games by studying n-player stochastic dynamic games in which interactions are governed by a graph. For games with perfectly symmetric players, the MFG approximation can be rigorously justified under suitable assumptions for two main reasons: On the one hand, the equilibria of n-player games can be shown to converge to the MFG limit. On the other hand, a solution of the continuum model may be used to construct approximate equilibria for the corresponding n-player model. This thesis extends these results in two cases: first, for games on general graph sequences in the setting of a specific yet rich linear-quadratic model and second, for general games on dense graph sequences. For linear-quadratic games, we show that the MFG is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. We show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence. Finally, since graphons have been shown to be the correct limit object for converging dense graph sequences, we develop the theory of graphon-based analogues of MFG. We propose a new formulation of graphon games based on a single typical player's label-state distribution. We show how our notion of graphon equilibrium can be used to construct approximate equilibria for large finite games set on any (weighted, directed) graph which converges in cut norm. The lack of players' exchangeability necessitates a careful definition of approximate equilibrium, allowing heterogeneity among the players' approximation errors, and we show how various regularity properties of the model inputs and underlying graphon lead naturally to different strengths of approximation.

This thesis constructs several groups entirely by hand via their symmetric presentations. In particular, the technique of double coset enumeration is used to manually construct J₃ : 2, the automorphism group of the Janko group J₃, and represent every element of the group as a permutation of PSL₂ (16) : 4, on 120 letters, followed by a word of length at most 3.