Mathematics
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Item Student Choice Among Large Group, Small Group, and Individual Learning Environments in a Community College Mathematics Mini-Course(1986) Baldwin, Eldon C.; Davidson, Neil; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, MD)This study describes the development and implementation of a model for accommodation of preferences for alternative instructional environments. The study was stimulated by the existence of alternative instructional modes, and the absence of a procedure for accommodation of individual student differences which utilized these alternative modes. The Choice Model evolved during a series of pilot studies employing three instructional modes; individual (JM), small group (SGM), and large group (LGM). Three instructors were each given autonomy in designing one learning environment, each utilizing her/his preferred instructional mode. One section of a mathematics course was scheduled for one hundred students. On the first day the class was divided alphabetically into three orientation groups, each assigned to a separate class room. During the first week, the instructors described their respective environments to each group, using video taped illustrations from a previous semester. Environmental preferences were then assessed using take-home student questionnaires. In the final pilot, fifty-five students were oriented to all three environments. Each student was then assigned to his/her preferred learning environment. The distribution of environmental preferences was 24% for IM, 44% for SGM, and 33% for LGM. The following student characteristics were also investigated: 1)sex, 2)age, 3)academic background, 4)mathematics achievement, 5)mathematics attitude, 6)mathematics interest, 7)self-concept, 8)communication apprehension. and 9)interpersonal relations orientation. This investigation revealed several suggestive preference patterns: 1)Females and students with weak academic backgrounds tended to prefer the SGM environment. 2)Students with higher levels of communication apprehension tended to avoid the SGM environment. 3)New college students and students with negative mathematics attitudes tended to avoid the IM environment. 4)Students with higher grades in high school tended to prefer the LGM environment. Student preferences were successfully accommodated, and student evaluations of the Choice Model were generally positive. The literature suggests that opportunities to experience choice in education tend to enhance student growth and development; adaptation and institutionalization of the Model were addressed from this perspective. Additional studies with larger samples were recommended to further investigate environmental preferences with respect t o student and instructor characteristics of gender, age, race, socioeconomic background, academic background, and learning style.Item Weyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces(1989) Walnut, David Francis; Benedetto, John J.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)The theory of wavelets can be used to obtain expansions of vectors in certain spaces. These expansions are like Fourier series in that each vector can be written in terms of a fixed collection of vectors in the Banach space and the coefficients satisfy a "Plancherel Theorem" with respect to some sequence space. In Weyl-Heisenberg expansions, the expansion vectors (wavelets) are translates and modulates of a single vector (the analyzing vector) . The thesis addresses the problem of the existence and stability of Weyl-Heisenberg expansions in the space of functions square-integrable with respect to the measure w(x) dx for a certain class of weights w. While the question of the existence of such expansions is contained in more general theories, the techniques used here enable one to obtain more general and explicit results. In Chapter 1, the class of weights of interest is defined and properties of these weights proven. In Chapter 2, it is shown that Weyl-Heisenberg expansions exist if the analyzing vector is locally bounded and satisfies a certain global decay condition. In Chapter 3, it is shown that these expansions persist if the translations and modulations are not taken at regular intervals but are perturbed by a small amount. Also, the expansions are stable if the analyzing vector is perturbed. It is also shown here that under more general assumptions, expansions exist if translations and modulations are taken at any sufficiently dense lattice of points. Like orthonormal bases, the coefficients in Weyl-Heisenberg expansions can be formed by the inner product of the vector being expanded with a collection of wavelets generated by a transformed version of the analyzing vector. In Chapter 4, it is shown that this transformation preserves certain decay and smoothness conditions and a formula for this transformation is given. In Chapter 5, results on Weyl-Heisenberg expansions in the space of square-integrable functions are presented.Item Analysis of the spectral vanishing method for periodic conservation laws(Copyright: Society for Industrial and Applied Mathematics, 1989-08) Maday, Yvon; Tadmor, EitanItem Convergence of spectral methods for nonlinear conservation laws(Copyright: Society for Industrial and Applied Mathematics, 1989-02) Tadmor, EitanItem Convenient total variation diminishing conditions for nonlinear difference schemes(Copyright: Society for Industrial and Applied Mathematics, 1988-10) Tadmor, EitanItem Stability analysis of finite-difference, pseudospectral and Fourier-Galerkin approximations for time-dependent problems(Copyright: Society for Industrial and Applied Mathematics, 1987-12) Tadmor, EitanItem Stability analysis of spectral methods for hyperbolic initial-boundary value systems(Copyright: Society for Industrial and Applied Mathematics, 1987-04) Gottlieb, David; Lustman, Liviu; Tadmor, EitanItem Convergence of spectral methods for hyperbolic initial-boundary value systems(Copyright: Society for Industrial and Applied Mathematics, 1987-06) Gottlieb, David; Lustman, Liviu; Tadmor, EitanItem Convergence of spectral methods for hyperbolic initial-boundary value systems(Copyright: Society for Industrial and Applied Mathematics, 1987-06) Gottlieb, David; Lustman, Liviu; Tadmor, EitanItem THE WELL-POSEDNESS OF THE KURAMOTO-SIVASHINSKY EQUATION(copyright: Society for Industrial and Applied Mathematics, 1986-07) Tadmor, EitanThe Kuramoto-Sivashinsky equation arises in a variety of applications, among which are modeling reaction-diffusion systems, flame-propagation and viscous flow problems. It is considered here, as a prototype to the larger class of generalized Burgers equations: those consist of quadratic nonlinearity and arbitrary linear parabolic part. We show that such equations are well-posed, thus admitting a unique smooth solution, continuously dependent on its initial data. As an attractive alternative to standard energy methods, existence and stability are derived in this case, by "patching" in the large short time solutions without "loss of derivatives".