Mathematics Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2793
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Item THE UNCERTAINTY PRINCIPLE IN HARMONIC ANALYSIS AND BOURGAIN'S THEOREM(2003) Benedetto, John J.; Powell, Alexander M.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)We investigate the uncertainty principle in harmonic analysis and how it constrains the uniform localization properties of orthonormal bases. Our main result generalizes a theorem of Bourgain to construct orthonormal bases which are uniformly well-localized in time and frequency with respect to certain generalized variances. In a related result, we calculate generalized variances of orthonormalized Gabor systems. We also answer some interesting cases of a question of H. S. Shapiro on the distribution of time and frequency means and variances for orthonormal bases.Item BESOV WE11-POSEDNESS FOR HIGH DIMENSIONAL NON-LINEAR WAVE EQUATIONS(2003) Sterbenz, Jacob; Machedon, Matei; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)Following work of Tataru, [13] and [11], we solve the division problem for wave equations with generic quadratic non-linearities in high dimensions. Specifically, we show that non-linear wave equations which can be written as systems involving equations of the form Φ = Φ∇Φ and Φ = |∇Φ|^2 are well-posed with scattering in (6+1) and higher dimensions if the Cauchy data are small in the scale invariant ℓ^1 Besov space B^sc,1.Item The Uncertainty Principle in Harmonic Analysis and Bourgain's Theorem(2003) Powell, Alexander M.; Benedetto, John J.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)We investigate the uncertainty principle in harmonic analysis and how it constrains the uniform localization properties of orthonormal bases. Our main result generalizes a theorem of Bourgain to construct orthonormal bases which are uniformly well-localized in time and frequency with respect to certain generalized variances. In a related result, we calculate generalized variances of orthonormalized Gabor systems. We also answer some interesting cases of a question of H. S. Shapiro on the distribution of time and frequency means and variances for orthonormal bases.Item Error Control for the Mean Curvature Flow(2002) Lakkis, Omar; Nochetto, Ricardo H.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study the equation describing the motion of a nonparametric surface according to its mean curvature flow. This is a nonuniformly parabolic equation that can be discretized in space via a finite element method. We conduct an aposteriori error analysis of the semidiscrete scheme and derive upper bounds to the error in terms of computable quantities called estimators. The reliability of the estimators is practically tested through numerical simulations.Item An Exposition of Stochastic Integrals and Their Application to Linearization Coefficients(2009) Kuykendall, John Bynum; Slud, Eric V; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Stochastic integration is introduced as a tool to address the problem of finding linearization coefficients. Stochastic, off-diagonal integration against a random spectral measure is defined and its properties discussed, followed by a proof that two formulations of Ito's Lemma are equivalent. Diagonals in R<\bold>n<\super> are defined, and their relationship to partitions of {1, ..., n} is discussed. The intuitive notion of a stochastic integral along a diagonal is formalized and calculated. The relationship between partitions and diagonals is then exploited to apply Moebius inversion to stochastic integrals over different diagonals. Diagonals along which stochastic integrals may be nonzero with positive probability are shown to correspond uniquely to diagrams. This correspondence is used to prove the Diagram Formula. Ito's Lemma and the Diagram Formula are then combined to calculate the linearization coefficients for Hermite Polynomials. Finally, future work is suggested that may allow other families of linearization coefficients to be calculated.Item Sequential Search With Ordinal Ranks and Cardinal Values: An Infinite Discounted Secretary Problem(2009) Palley, Asa Benjamin; Cramton, Peter; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We consider an extension of the classical secretary problem where a decision maker observes only the relative ranks of a sequence of up to N applicants, whose true values are i.i.d. U[0,1] random variables. Applicants arrive according to a homogeneous Poisson Process, and the decision maker seeks to maximize the expected time-discounted value of the applicant who she ultimately selects. This provides a straightforward and natural objective while retaining the structure of limited information based on relative ranks. We derive the optimal policy in the sequential search, and show that the solution converges as N goes to infinity. We compare these results with a closely related full information problem in order to quantify these informational limitations.Item MEANS AND AVERAGING ON RIEMANNIAN MANIFOLDS(2009) Afsari, Bijan; Krishnaprasad, P.S.; Grove, Karsten; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Processing of manifold-valued data has received considerable attention in recent years. Standard data processing methods are not adequate for such data. Among many related data processing tasks finding means or averages of manifold-valued data is a basic and important one. Although means on Riemannian manifolds have a long history, there are still many unanswered theoretical questions about them, some of which we try to answer. We focus on two classes of means: the Riemannian $L^{p}$ mean and the recursive-iterative means. The Riemannian $L^{p}$ mean is defined as the solution(s) of a minimization problem, while the recursive-iterative means are defined based on the notion of Mean-Invariance (MI) in a recursive and iterative process. We give a new existence and uniqueness result for the Riemannian $L^{p}$ mean. The significant consequence is that it shows the local and global definitions of the Riemannian $L^{p}$ mean coincide under an uncompromised condition which guarantees the uniqueness of the local mean. We also study smoothness, isometry compatibility, convexity and noise sensitivity properties of the $L^{p}$ mean. In particular, we argue that positive sectional curvature of a manifold can cause high sensitivity to noise for the $L^{2}$ mean which might lead to a non-averaging behavior of that mean. We show that the $L^{2}$ mean on a manifold of positive curvature can have an averaging property in a weak sense. We introduce the notion of MI, and study a large class of recursive-iterative means. MI means are related to an interesting class of dynamical systems that can find Riemannian convex combinations. A special class of the MI means called pairwise mean, which through an iterative scheme called Perimeter Shrinkage is related to cyclic pursuit on manifolds, is also studied. Finally, we derive results specific to the special orthogonal group and the Grassmannian manifold, as these manifolds appear naturally in many applications. We distinguish the $2$-norm Finsler balls of appropriate radius in these manifolds as domains for existence and uniqueness of the studied means. We also introduce some efficient numerical methods to perform the related calculations in the specified manifolds.Item Regularized Variable Selection in Proportional Hazards Model Using Area under Receiver Operating Characteristic Curve Criterion(2009) Wang, Wen-Chyi; Yang, Grace L; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The goal of this thesis is to develop a statistical procedure for selecting pertinent predictors among a number of covariates to accurately predict the survival time of a patient. There are available many variable selection procedures in the literature. This thesis is focused on a more recently developed “regularized variable selection procedure”. This procedure, based on a penalized likelihood, can simultaneously address the problem of variable selection and variable estimation which previous procedures lack. Specifically, this thesis studies regularized variable selection procedure in the proportional hazards model for censored survival data. Implementation of the procedure requires judicious determination of the amount of penalty, a regularization parameter λ, on the likelihood and the development of computational intensive algorithms. In this thesis, a new criterion of determining λ using the notion of “the area under the receiver operating characteristic curve (AUC)” is proposed. The conventional generalized cross-validation criterion (GCV) is based on the likelihood and its second derivative. Unlike GCV, the AUC criterion is based on the performance of disease classification in terms of patients' survival times. Simulations show that performance of the AUC and the GCV criteria are similar. But the AUC criterion gives a better interpretation of the survival data. We also establish the consistency and asymptotic normality of the regularized estimators of parameters in the partial likelihood of proportional hazards model. Some oracle properties of the regularized estimators are discussed under certain sparsity conditions. An algorithm for selecting λ and computing regularized estimates is developed. The developed procedure is then illustrated with an application to the survival data of patients who have cancers in head and neck. The results show that the proposed method is comparable with the conventional one.Item The Multivariate Variance Gamma Process and Its Applications in Multi-asset Option Pricing(2009) Wang, Jun; Madan, Dilip B; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Dependence modeling plays a critical role in pricing and hedging multi-asset derivatives and managing risks with a portfolio of assets. With the emerge of structured products, it has attracted considerable interest in using multivariate Levy processes to model the joint dynamics of multiple financial assets. The traditional multidimensional extension assumes a common time change for each marginal process, which implies limited dependence structure and similar kurtosis on each marginal. In this thesis, we introduce a new multivariate variance gamma process which allows arbitrary marginal variance gamma (VG) processes with flexible dependence structure. Compared with other multivariate Levy processes recently proposed in the literature, this model has several advantages when applied to financial modeling. First, the multivariate process built with any marginal VG process is easy to simulate and estimate. Second, it has a closed form joint characteristic function which largely simplifies the computation problem of pricing multi-asset options. Last, it can be applied to other time changed Levy processes such as normal inverse Gaussian (NIG) process. To test whether the multivariate variance gamma model fits the joint distribution of financial returns, we compare the model performance of explaining the portfolio returns with other popular models and we also develop Fast Fourier Transform (FFT)-based methods in pricing multi-asset options such as exchange options, basket options and cross-currency foreign exchange options.Item Abundance of escaping orbitsin a family of anti-integrable limitsof the standard map(2009) De Simoi, Jacopo; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We give quantitative results about the abundance of escaping orbits in a family of exact twist maps preserving Lebesgue measure on the cylinder T × R; geometrical features of maps of this family are quite similar to those of the well-known Chirikov-Taylor standard map, and in fact we believe that the techniques presented in this work can be further improved and eventually applied to studying ergodic properties of the standard map itself. We state conditions which assure that escaping orbits exist and form a full Hausdorff dimension set. Moreover, under stronger conditions we can prove that such orbits are not charged by the invariant measure. We also obtain prove that, generically, the system presents elliptic islands at arbitrarily high values of the action variable and provide estimates for their total measure.