In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

Orientador: Marta Cilene Gadotti / Banca: Paulo Leandro Dattori da Silva / Banca: Ricardo Parreira da Silva / Resumo: Este trabalho apresenta resultados importantes sobre a Teoria de Integração. Inicialmente é desenvolvida uma parte sobre Teoria da Medida, necessária para introduzir a integral de Lebesgue e suas propriedades. Também é apresentada a integral de Riemann-Stieltjes. Em seguida, são demonstrados resultados importantes sobre converg ência envolvendo as integrais de Lebesgue, resultados estes que não são válidos para integrais de Riemann. Para apresentar tais temas, usa-se mais fortemente as referências [1], [2], [3] e [4] / Abstract: This study presents important results on Integration of Theory. The rst of all part is developed on Measure Theory which is necessary to introduce the Lebesgue integral and its properties and we introduce. It also shows the Riemann-Stieltjes integral. Important results are proved on convergence involving the integrals of Lebesgue, which are not valid for the Riemann integral. Im order to present these themes we strongly use the references [1], [2], [3] and [4] / Mestre

Another extension of Nonlinear Integral, Upper and Lower Nonlinear Integrals, which is a pair of extreme nonlinear integrals to contain all types of Nonlinear Integrals in the same scheme, is also proposed. It can give a set of upper and lower bounds which include all types of Nonlinear Integrals. We tried to find a solution with the smallest distance between the upper and lower bounds and the smallest error which is a NP hard problem. So we use the multi-objective optimization method to find a set of results for the regression model based on the Upper and Lower Nonlinear Integrals. We can just select one or more optimal solution(s) for a specific problem from the set of results. A weather predictor based on this model has been constructed to predict the next days temperature changing trend and range. / Finally, a NI based data mining framework has been established for identifying the chance of developing liver cancer based on the Hepatitis B Virus DNA sequence data. We have shown that the framework obtains the best diagnosing performance amongst many existing classifiers. / Nonlinear Integral (NI) is a useful integration tool. It has been applied to many areas including classification and regression. The classical method relies on a large number of training data, which lead to large time and space complexity. Moreover, the classical Nonlinear Integral has many limitations. For dealing with different situation, we propose Double Nonlinear Integrals and Nonlinear Integrals with Polynomial Kernel to deal with the problems transversely and longitudinally. / The classical Nonlinear Integrals implement projection along a line with respect to the features. But in many cases the linear projection cannot achieve good performance for classification or regression due to the limitation of the integrand. The linear function used for the integrand is just a special type of polynomial functions with respect to the features. We propose Nonlinear Integral with Polynomial Kernel (NIPK) in which a polynomial function is used as the integrand of Nonlinear Integral. It enables the projection to be along different types of curves on the virtual space, so that the virtual values gotten by the Nonlinear Integrals with Polynomial Kernel can be better regularized and easier to deal with. Experiments show that there is evident improvement of performance for NIPK compared to classical NI. / When the data to be classified have special distribution in the data space, the projection may overlap and the classification accuracy will be lowered. For example, when one group of the data is surrounded by the data of another group, or the number of classes for the data is large. To handle this kind of problems; we propose a new classification model based on the Double Nonlinear Integrals. Double Nonlinear Integral means projecting to a 2-Dimensional space by using the Nonlinear Integral twice in succession and classifying the virtual values in the 2-D space corresponding to the original data. Double Nonlinear Integrals can lessen loss of information due to the intersection of different classes on real axis. Accuracy will also be increased accordingly. / Wang, Jinfeng. / Advisers: Kwong Sak Leung; Kin Hong Lee. / Source: Dissertation Abstracts International, Volume: 72-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 139-151). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

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<p>This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by <italic>k</italic> dimensional planes but which are embedded in a <italic>D</italic>>><italic>k</italic> dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.</p> / Dissertation

An arms-race exists between malware authors and system defenders in which defenders develop new detection approaches only to have the malware authors develop new techniques to bypass them. This motivates the need for a formal framework to jointly reason about malware and its detection. This dissertation presents such a formal framework termed the extended Maurer model} (EMM) and then applies this framework to develop a game-theoretic model of the malware authors versus system defenders confrontation. To be inclusive of modern computers and networks, the EMM has been developed by extending to the existing Maurer computer model, a Turing-reducible model of computer operations. The basic components of the Maurer model have been extended to incorporate the necessary structures to enable the modeling of programs, concurrency, multiple processors, and networks. In particular, we show that the proposed EMM remains a Turing equivalent model which is able to model modern computers, computer networks, as well as complex programs such as modern virtual machines and web browsers. Through the proposed EMM, we provide formalizations for the violations of the standard security policies. Specifically, we provide the definitions of the violations of confidentiality policies, integrity policies, availability policies, and resource usage policies. Additionally, we also propose formal definitions of a number of common malware classes, including viruses, Trojan horses, spyware, bots, and computer worms. We also show that the proposed EMM is complete in terms of its ability to model all implementable that could exist malware within the context of a given defended environment. We then use the EMM to evaluate and analyze the resilience of a number of common malware detection approaches. We show that static anti-malware signature scanners can be easily evaded by obfuscation, which is consistent with the results of prior experimental work. Additionally, we also use the EMM to formally show that malware authors can avoid detection by dynamic system call sequence detection approaches, which also agrees with recent experimental work. A measure-theoretic model of the EMM is then developed by which the completeness of the EMM with respect to its ability to model all implementable malware detection approaches is shown. Finally, using the developed EMM, we provide a game-theoretic model of the confrontation of malware authors and system defenders. Using this game model, under game theory's strict dominance solution concept, we show that rational attackers are always required to develop malware that is able to evade the deployed malware detection solutions. Moreover, we show that the attacker and defender adaptations can be modeled as a sequence of iterative games. Hence, the question can be asked as to the conditions required if such a sequence (or arms-race) is to converge towards a defender advantageous end-game. It is shown via the EMM that, in the general context, this desired situation requires that the next attacker adaptation exists as, at least, a computationally hard problem. If this is not the case, then we show via the EMM's measure theory perspective, that the defender is left needing to track statistically non-stationary attack behaviors. Hence, by standard information theory constructs, past attack histories can be shown to be uninformative with respect to the development of the next to be required adaptation of the deployed defenses. To our knowledge, this is the first work to: (i) provide a joint model of malware and its detection, (ii) provide a model that is complete with respect to all implementable malware and detection approaches, (iii) provide a formal bridge between Turing-reducibility and measure theory, and (iv) thereby, allow game theory's strict dominance solution concept to be applied to formally reason about the requirements if the malware versus anti-malware arms-race is to converge to a defender advantageous end-game. / Graduate / melgamal@uvic.ca

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 85-86).

Thesis (M.S.)--University of Texas at El Paso, 2008. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.