Mathematics
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Item A Modern Overview of Local Sections of Flows(1990) Colston, Helen Marie; Markley, Nelson; Mathematics; University of Maryland (College Park, Md); Digital Repository at the University of MarylandThis paper examines local cross sections of a continuous flow on a locally compact metric space. Sane of the history of the study of local cross sections is reviewed, with particular attention given to H. Whitney's work. The paper presents a modern proof that local cross sections always exist at noncritical points of a flow. Whitney is the primary source for the key idea in the existence proof; he also gave characterizations of local cross sections on 2- and 3-dimensional manifolds. We show various topological properties of local cross sections, the most important one being that local cross sections on the same orbit are locally homeomorphic. A new elementary proof using the Jordan Curve Theorem shows that when a flow is given on a 2-manifold, a local cross section will be an arc. Whitney is cited for a similar result on 3-maniforlds. Finally, the so-called "dob=bone" space of R. Bing is used to construct a flow on a 4-manifold with a point at which every local cross section is not homeomorphic to a 3-dimensional disk.Item Some Solutions to Overdetermined Boundary Value Problems on Subdomains of Spheres(1990) Karlovitz, Max A.; Berenstein, Carlos; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)For n an open domain contained in a Riemannian manifold M, various researchers have considered the problem of finding functions u : Ω → R which satisfy overdetermined boundary value problems such as Δu + αu = 0 in Ω and u = 0 and ∂u/∂n = constant on ∂Ω. (Here Δ is the Laplace-Beltrami operator on M.) Their results demonstrate the relative difficulty of finding such solutions. It has been shown for various choices of M (e.g., M = R^n or S+n) that the only domains Ω with ∂Ω connected and sufficiently regular which admit solutions to problems such as the one above are metric balls (see, e.g., [Be1] or [Se]) . The first result of this thesis is a set of domains contained in S^n which are not metric balls but which do admit solutions to various overdetermined boundary value problems. In the case of the problem stated above, solutions are found for infinitely many choices of α. It is observed that the solutions found are isoparametric functions. (A function g is isoparametric if ~g and the le ngth of the gradient of g are both functions of g, see [Ca].) In some cases, it is shown that these functions are restrictions of spherical eigenfunctions. In some cases, they are not. Next, for these same domains, an original choice of variables is developed under which the Laplace operator can be separated. This separation of variables is used to find a complete set of Dirichlet eigenfunctions for the domains. Initial sequences of Dirichlet eigenvalues for some of the domains are computed numerically. Finally, some comments are made about the connection between solutions to overdetermined problems and isoparametric functions.Item WIENER AMALGAM SPACES IN GENERALIZED HARMONIC ANALYSIS AND WAVELET THEORY(1990) Heil, Christopher Edward; Benedetto, John J.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)This thesis is divided into four parts. Part I, Introduction and Notation, describes the results contained in the thesis and their background. Part II, Wiener Amalgam Spaces, is an expository introduction to Feichtinger's general amalgam space theory, which is used in the remainder of the thesis to formulate and prove results. Part III, Generalized Harmonic Analysis, presents new results in that area. Part IV, Wavelet Theory, contains exposition and miscellaneous results on Gabor ( also known as Weyl-Heisenberg) wavelets. Amalgam, or mixed-norm, spaces are Banach spaces of functions determined by a norm which distinguishes between local and global properties of functions. Specific cases were introduced by Wiener. Feichtinger has developed a far-reaching generalization of amalgam spaces, which allows general function spaces norms as local or global components. We use Feichtinger's amalgam theory, on d-dimensional Euclidean space under componentwise multiplication, to prove that the Wiener transform (introduced by Wiener to analyze the spectra of infinite-energy signals) is an invertible mapping of the amalgam space with local L2 and global Lq. components onto an appropriate space defined in terms of the variation of functions, for each q between one and infinity. As corollaries, we obtain results of Beurling on the Fourier transform and results of Lau and Chen on the Wiener transform. Moreover, our results are carried out in higher dimensions. In addition, we prove that the higher-dimensional variation spaces are complete by using Masani's helices; this generalizes a one-dimensional result of Lau and Chen. In wavelet theory, we present a survey of frames in Hilbert and Banach spaces and the use of the Zak transform in analyzing Gabor wavelets. Frames are an alternative to unconditional bases in these spaces; like bases, they Provide representations of each element of the space in terms of the frame elements, and do so in a way in which the scalars in the representation are explicitly known. However, unlike bases, the representations need not be unique. We then discuss the specific case of Gabor frames in the space of square-integrable functions, concentrating on the role of the Zak transform in the analysis of such frames.Item Retention of Concepts and Skills in Traditional and Reformed Applied Calculus(1998) Garner, Bradley Evan; Fey, James T.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)A fundamental question is currently being asked throughout the collegiate mathematics education community: "How can we help students understand and remember calculus better?" There seems to be general dissatisfaction with the knowledge and abilities of students who have completed calculus courses. Reformers in the calculus arena are striving to change instruction to help students understand better and remember longer what they have learned. The Calculus Consortium based at Harvard (CCH) recently published new textbooks for applied calculus which embody a major switch in the philosophy of calculus teaching. The CCH texts, in which applications are the central motivation and not coincidental afterthoughts, emphasize concepts more than symbol manipulation and encourage student-driven discovery of fundamental ideas. Is this reformed way of teaching applied calculus more effective than the traditional method? Which method leads to better long-term understanding and ability? The purpose of this study was to shed light on these questions by characterizing and comparing the skills and conceptual understandings of students of traditional and reformed methods several months after they completed their applied calculus course. A sample of 108 students of applied calculus (57 reformed, 51 traditional) who completed their course in April of 1997 were given a written test in November of 1997 to assess their conceptual understandings and computational skills. Sixteen of these students (8 traditional, 8 reformed) were interviewed to ascertain more about their conceptual understandings as well as their motivation, commitment and attitudes with respect to their applied calculus courses. Test results indicate that although there was no significant difference in overall performance between the two groups, students of the reformed method performed better on conceptual problems, while students of the traditional method performed better on computational problems. Interview results indicate that of the two groups, reformed course students were more confident in their ability to explain derivatives. Reformed course students mentioned graphs and applications more, and they also were more inclined to use estimation techniques than traditional course students. The traditional course students had a clearer idea of the connection between the derivative and the integral.Item The Metaplectic Case of the Weil-Siegel Formula(1990) Sweet, William Jay Jr.; Kudla, Stephen S.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)The Weil-Siegel formula, in the form developed by Weil, asserts the equality of a special value of an Eisenstein series with the integral of a related theta series. Recently, Kudla and Rallis have extended the formula into the range in which the Eisenstein series fails to converge at the required special value, so that Langlands' meromorphic analytic continuation must be used. In the case addressed by Kudla and Rallis, both the Eisenstein series and the integral of the theta series are automorphic forms on the adelic symplectic group. This thesis concentrates on extending the Weil-Siegel formula in the case in which both functions are automorphic forms on the two-fold metaplectic cover of the adelic symplectic group. First of all, a concrete model of the global metaplectic cover mentioned above is constructed by modifying the local formulas of Rao. Next, the meromorphic analytic continuation of the Eisenstein series is shown to be holomorphic at the special value in question. In the course of this work, we develop the functional equation and find all poles of an interesting family of local zeta-integrals similar to those studied in a paper of Igusa. Finally, the Weil-Siegel formula is proven in many cases by the methods of Kudla and Rallis.Item Nonparametric Estimation of a Distribution Function in Biased Sampling Models(1993) Xu, Jian-Lun; Yang, Grace L.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)The nonparametric maximum likelihood estimator (NPMLE) of a distribution function F in biased sampling models have been studied by Cox (1969), Vardi (1982, 1985), and Gill, Vardi, and Wellner (1988). Their approaches are based on the assumption that the observations are drawn from biased distributions of F and biasing functions do not depend on F. These assumptions have been used in Patil and Rao (1978). This thesis extends the biased sampling model by making the biasing functions depend on the distribution function F in a variety of ways. With this extension, many of the existing models, including the ranked-set sampling model and the nomination sampling model, become special cases of the biased sampling model. The statistical inference about F becomes to a large extent the study of the biasing function. We develop conditions under which the generalized model is identifiable. Under these conditions, an estimator of the underlying distribution F is proposed and its strong consistency and asymptotic normality are established. In certain situation, estimation of Fin a biased sampling model is in fact a problem of estimating a monotone decreasing density. Several density estimators are studied. They include the nonparametric maximum likelihood estimator, a kernel estimator, and a modified histogram type estimator. The strong consistency, the asymptotic normality, and the bounds on average error for the estimators are studied in detail. In summary, this thesis is a generalizations of the estimation results available for the ordinary s-biased sampling model, the ranked-set sampling model, the nomination sampling model, and a monotone decreasing density.Item Microfunctions for Sheaves of Holomorphic Functions with Growth Conditions(1994) Cheah, Sin-Chnuah; Berenstein, Carlos A.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Mikio Sato devised microfunctions as a means of measuring the singularities of hyperfunctions. In 1970, Kawai and Sato introduced Fourier hyperfunctions in their study of partial differential operators. The class of Fourier hyperfunctions has been generalized by Saburi, Nagamachi, and Kaneko, among others, and most recently by Berenstein and Struppa. Berenstein and Struppa introduced Fourier p-hyperfunctions, where p is a plurisubharmonic function satisfying certain smoothness and growth conditions. p(z) = |z|^s, s \ge 1 are the cases studied by Sato, Kawai, Nagamachi, and Kaneko. Following the methods of Sato, Kawai and Kashiwara, this dissertation introduces Fourier p-microfunctions functorially, though under very severe conditions on p. These restrictions on p are satisfied when, for instance, p(z) = log^+ |f| where f is a product of 1 variable holomorphic functions with zeroes uniformly bounded away from the real axis. Kaneko has introduced Fourier microfunctions for p(z) = |Rez|^s, s > 0, using tubes. When s < 1 these p's are not plurisubharmonic. Thus the results here complement his.Item SPECTRAL METHODS FOR HYPERBOLIC PROBLEMS(1994-01) Tadmor, EitanWe review several topics concerning spectral approximations of time-dependent problems, primarily | the accuracy and stability of Fourier and Chebyshev methods for the approximate solutions of hyperbolic systems. To make these notes self contained, we begin with a very brief overview of Cauchy problems. Thus, the main focus of the first part is on hyperbolic systems which are dealt with two (related) tools: the energy method and Fourier analysis. The second part deals with spectral approximations. Here we introduce the main ingredients of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, differentiation matrices ... The third part is devoted to Fourier method for the approximate solution of periodic systems. The questions of stability and convergence are answered by combining ideas from the first two sections. In this context we highlight the role of aliasing and smoothing; in particular, we explain how the lack of resolution might excite small scales weak instability, which is avoided by high modes smoothing. The forth and final part deals with non-periodic problems. We study the stability of the Chebyshev method, paying special attention to the intricate issue of the CFL stability restriction on the permitted time-step.Item High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws(Copyright: Society for Industrial and Applied Mathematics, 1998-12) JIANG, G.-S.; LEVY, D.; LIN, C.-T.; OSHER, S.; TADMOR, E.We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408{463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397{425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892{1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.Item FROM SEMIDISCRETE TO FULLY DISCRETE: STABILITY OF RUNGE-KUTTA SCHEMES BY THE ENERGY METHOD(Copyright: Society for Industrial and Applied Mathematics, 1998-03) LEVY, DORON; TADMO, EITANThe integration of semidiscrete approximations for time-dependent problems is encountered in a variety of applications. The Runge{Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of lines are governed by possibly ill-conditioned systems with a growing dimension; consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for well-posed semidiscrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong stability of general fully discrete RK methods governed by coercive approximations. We conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection-diffusion equations with variable coeficients. A straightforward implementation of our results verify the strong stability of RK methods for local finite-difference schemes as well as global spectral approximations. Since our approach is based on the energy method (which is carried in the physical space), and since it avoids the von Neumann analysis (which is carried in the dual Fourier space), we are able to easily adapt additional extensions due to nonperiodic boundary conditions, general geometries, etc.