Banking is a risk and return business; in order to obtain the desired returns, banks are required to take on risks. Following the demise of Lehman Brothers in September 2008, the Basel III Accord proposed considerable increases in capital charges for banks. Whilst this ensures greater economic stability, banks now face an increasing risk of becoming capital inefficient. Furthermore, capital analysts are not only required to estimate capital requirements for individual business lines, but also for the organization as a whole. Copulas are a popular technique to model joint multi-dimensional problems, as they can be applied as a mechanism that models relationships among multivariate distributions. Firstly, a review of the Basel Capital Accord will be provided. Secondly, well known risk measures as proposed under the Basel Accord will be investigated. The penultimate chapter is dedicated to the theory of copulas as well as other measures of dependence. The final chapter presents a practical illustration of how business line losses can be simulated by using the Gaussian, Cauchy, Student t and Clayton copulas in order to determine capital requirements using 95% VaR, 99% VaR, 95% ETL, 99% ETL and StressVaR. The resultant capital estimates will always be a function of the choice of copula, the choice of risk measure and the correlation inputs into the copula calibration algorithm. The choice of copula, the choice of risk measure and the conservativeness of correlation inputs will be determined by the organization’s risk appetite. / Sc (Applied Mathematics), North-West University, Potchefstroom Campus, 2014

We present a scalable information-optimal compressive imager optimized for the target classification task, discriminating between two target classes. Compressive projections are optimized using the Cauchy-Schwarz Mutual Information (CSMI) metric, which provides an upper-bound to the probability of error of target classification. The optimized measurements provide significant performance improvement relative to random and PCA secant projections. We validate the simulation performance of information-optimal compressive measurements with experimental data.

Master of Science / Department of Statistics / Paul Nelson / Trimmed means are robust estimators of location for distributions having heavy tails. Theory and simulation indicate that little efficiency is lost under normality when using appropriately trimmed means and that their use with data from distributions with heavy tails can result in improved performance. This report uses the principle of equivariance applied to trimmed means sampled from a Cauchy distribution to form a discrepancy function of the data and parameters whose distribution is free of the unknown median and scale parameter. Quantiles of this discrepancy function are estimated via asymptotic normality and simulation and used to construct confidence intervals for the median of a Cauchy distribution. A nonparametric approach based on the distribution of order statistics is also used to construct confidence intervals. The performance of these intervals in terms of coverage rate and average length is investigated via simulation when the data are actually sampled from a Cauchy distribution and when sampling is from normal and logistic distributions. The intervals based on simulation estimation of the quantiles of the discrepancy function are shown to perform well across a range of sample sizes and trimming proportions when the data are actually sampled from a Cauchy distribution and to be relatively robust when sampling is from the normal and logistic distributions.

A novel image reconstruction algorithm from edges (image gradients) follows from the Sokhostki-Plemelj Theorem of complex analysis, an elaboration of the standard Cauchy (Singular) Integral. This algorithm demonstrates the use of Singular Integral Equation methods to image processing, extending the more common use of Partial Differential Equations (e.g. based on variants of the Diffusion or Poisson equations). The Cauchy Integral approach has a deep connection to and sheds light on the (linear and non-linear) diffusion equation, the retinex algorithm and energy-based image regularization. It extends the commonly understood local definition of an edge to a global, complex analytic structure - the analytic edge - the contrast weighted kernel of the Cauchy Integral. Superposition of the set of analytic edges provides a "filled-in" image which is the piece-wise analytic image corresponding to the edge (gradient data) supplied. This is a fully parallel operation which avoids the time penalty associated with iterative solutions and thus is compatible with the short time (about 150 milliseconds) that is biologically available for the brain to construct a perceptual image from edge data. Although this algorithm produces an exact reconstruction of a filled-in image from the gradients of that image, slight modifications of it produce images which correspond to perceptual reports of human observers when presented with a wide range of "visual contrast illusion" images.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 119-121. / Ph.D.

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-02-22Bitstream added on 2014-06-13T19:48:31Z : No. of bitstreams: 1 amaral_js_me_sjrp.pdf: 356567 bytes, checksum: 85048dd943d0aa273e0a6151db09d116 (MD5) / SejamAum operador fechado e densamente definido em um espa¸co de BanachX ef∈L 1 ([0,τ];X). O objetivo deste trabalho e apresentar uma condição necessária e suficiente para a existência de solução weak, dada por J. Ball, do problema { d dt u(t) = Au(t) +f(t), t > 0 u(0) = x. Neste caso, a solução weak coincide com a solução mild (dada pela Fórmula da Variação das Constantes). Como aplicação, estudaremos um problema de valor inicial e de fronteira para equações parabólicas de segunda ordem e concluiremos que sua solução fraca, no sentido usual de EDP’s, coincide com a solução mild do problema de Cauchy abstrato associado / LetAbe a closed linear operator densely defined on a Banach spaceXand f∈L 1 ([0,τ];X). The purpose of this work is to present a necessary and sufficient condition to the existence of weak solution, introduced by J. Ball, for the problem { d dt u(t) = Au(t) +f(t), t > 0 u(0) = x. In this case, the weak solution coincides with the mild solution (given by the Variation of the Constants Formula) As an application we study an initial boundary value problem for a second order parabolic and conclude that its weak solution, coincides with the mild solution of the associated Abstract Cauchy Problem

Finally, we prove that weak solutions to the compressible Navier-Stokes equations with the Navier boundary condition stabilize to static equilibrium states under a fair condition. / First, we show that the most general class of weak solutions to one-dimensional full compressible Navier-Stokes equations do not exhibit vacuum states in a finite time provided that no vacuum is present initially with the minimum physical assumptions on the data. Moreover, two initially non interacting vacuum regions will never meet each other in the future. / Secondly, we construct the local classical solutions to the compressible Navier-Stokes equations for initial vacuum far fields. In this case, we describe the blow-up phenomena of two-dimensional compact support smooth spherically symmetric solutions. When the far field of the initial state is away from vacuum, we obtain the global classical solutions and show the large time blow-up behavior of the gradient of the density. / This thesis deals with some important problems of compressible Navier-Stokes equations, including the well-posedness of the Cauchy problem, the regularity of the weak solutions constructed by Lions and Feireisl, and the dynamics of vacuum states, etc.. / Luo, Zhen. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 152-161). / 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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The Cauchy–Riemann equations admit a bilinear multiplication of solutions, since the product of two holomorphic functions is again holomorphic. This multiplication plays the role of a nonlinear superposition principle for solutions, allowing for construction of new solutions from already known ones, and it leads to the exceptional property of the Cauchy–Riemann equations that all solutions can locally be built from power series of a single solution z = x + iy ∈ C. In this thesis we have found a differential algebraic characterization of linear first order systems of partial differential equations admitting a bilinear ∗-multiplication of solutions, and we have determined large new classes of systems having this property. Among them are the already known quasi-Cauchy–Riemann equations, characterizing integrable Newton equations, and the gradient equations ∇f = M∇g with constant matrices M. A systematic description of linear systems of PDEs with variable coefficients have been given for systems with few independent and few dependent variables. An important property of the ∗-multiplication is that infinite families of solutions can be constructed algebraically as power series of known solutions. For the equation ∇f = M∇g it has been proved that the general solution, found by Jodeit and Olver, can be locally represented as convergent power series of a single simple solution similarly as for solutions of the Cauchy–Riemann equations.

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