Mathematics Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2793
Browse
493 results
Search Results
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 The Effect of Behavioral Objectives on Measures of Learning and Forgetting on High School Algebra(1972) Loh, Elwood Lockert; Walbesser, Henry H.; Mathematics and Education; Digital Repository at the University of Maryland; University of Maryland (College Park, MD)During the past decade, the number of educators who advocate the use of behavioral objectives in education has increased. The increase in the number of advocates of behavioral objectives has been followed by an increasing awareness of the need for empirical research to give credence to such a viewpoint. At present, there is not a substantial number of research studies in which behavioral objectives have been used as a manipulated variable. In previously reported learning studies in which behavioral objectives have been used as an experimental variable, measures of learning and measures of forgetting have been derived from achievement scores. The results obtained in the learning studies have not been singular in support of the use of behavioral objectives, however, the results obtained in forgetting studies have consistently supported their use. This two part study investigated the effect of presenting behavioral objectives to students during the initial phase of a learning program. There were six criterion variables observed: index of learning, rate of learning, index of forgetting, rate of forgetting, index of retention, and index of efficiency. Two 2-year algebra one classes with a total of 52 students were randomly partitioned into two treatment groups for the learning phase of the study. The classes were further randomly partitioned into three retention groups for the forgetting phase of the study. The instructional materials were programmed within the framework of a learning hierarchy. The use of the learning hierarchy facilitated the use of a procedure for separating behaviors not yet possessed by a student from behaviors previously acquired. This was accomplished by presenting students with preassessment tasks prior to instruction for a behavior in the learning hierarchy. If the subject's response to the preassessment task indicated that he possessed the behavior, instruction was not given for that behavior. If the response indicated that the subject had not previously acquired the behavior, instruction was presented. The measures of the time needed to acquire the behavior were subsequently used to compute the six experimental measures. Three retention periods of 7 calendar days, 14 calendar days, and 15 to 21 calendar days were used for the forgetting phase of the study. The results of the three retention periods were pooled for the two forgetting measures, the index of retention, and the index of efficiency. The data collected in the study were analyzed by six separate tests using a one-way analysis of variance. A 0.05 level of significance was used for each of the six tests. The following results were obtained: 1. The index of learning for students who were informed of behavioral objectives during the initial phases of the learning program was not greater than the index of learning for students who were not so informed. 2. The rate of learning for students who were informed of behavioral objectives during the initial phases of the learning program was not greater than the rate of learning for students who were not so informed. 3. The index of forgetting for students who were informed of behavioral objectives during the initial phases of the learning program was not less than the index of forgetting for students who were not so informed. 4. The rate of forgetting for students who were informed of behavioral objectives during the initial phases of the learning program was not less than the rate of forgetting for students who were not so informed. 5. The index of retention for students who were informed of behavioral objectives during the initial phases of the learning program was not greater than the index of retention for students who were not so informed. 6. The index of efficiency for students who were informed of behavioral objectives during the initial phases of the learning program was not greater than the index of efficiency for students who were not so informed. It was concluded that the results of the study do not support the use of behavioral objectives as a procedure for improving either measures of learning or measures of forgetting which are functions of the time needed to reach criterion in a learning program using programmed instruction for teaching an algebraic topic to below average mathematics students in senior high school. It was recommended that further research is needed to determine a reliable and valid procedure for measuring learning and forgetting. It was also recommended that alternatives to programmed instruction be considered for learning and forgetting studies.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 Submaximal Function Algebras(1971) Van Meter, Garrett Oliver II; Gulick, Denny; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)Let X be a compact Hausdorff space. A function algebra on X is a complex Banach subalgebra of C(X) which separates the points of X and contains the constants. Moreover, a function algebra on X is maximal if it is contained properly in no proper subalgebra of C(X). We mention that maximal function algebras are large enough to have a goodly amount of structure. In order that we be able to state the ideas and results simply let us assume that for each algebra A the underlying space X is so adjusted that A contains no non-trivial ideals of C(X). Generally if A is a maximal function algebra on X, then the topological dimension of X is at most one. The idea of this thesis is to extend the notion of maximal function algebra so that on the one hand features of maximal algebras would be retained, while on the other hand the topological dimension of the underlying space could be forced to be arbitrarily large. Thus our introduction of the notion of submaximal function algebra. We prove that all maximal algebras are submaximal. A submaximal, non-maximal algebra is A(Tn), the completion of the polynomials in n-complex variables on the unit n-torus in Cn. However, if A is submaximal on X, then each proper function algebra between A and C(X) is contained in a proper maximal function algebra on X. Moreover, we show by example that the converse to this last statement is false. If A is a submaximal function algebra on X, then every point in X has a compact neighborhood in X such that the algebra of restrictions of functions in A is dense in the continuous functions on the neighborhood. This is the (natural) analogue of the "pervasive" property of maximal function algebras. It turns out that maximal function algebras are antisymmetric, which means that they contain no non-constant real-valued functions. This is not true in general for submaximal function algebras. However, if we render the antisymmetric property in the following way, then it holds true for submaximal algebras: if the real-valued continuous functions f1,...,fn on X along with A together generate a dense subalgebra of C(X), then the continuous real-valued functions h1,...,hn on X and A together generate a dense subalgebra of C(X), provided only that each hj is sufficiently close to fj. In addition, we show that if A is submaximal on X, then there are always exist finitely many real-valued continuous functions on X which together with A generate a dense subalgebra of C(X). Finally we discuss tensor products of submaximal algebras. In particular, we prove that under certain restrictions, the tensor product of two submaximal algebras is submaximal.Item Completions(1964) Nielsen, Robert Maurice; Brace, John W.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)This paper presents a new approach to the theory of completions. The treatment is based on the concept of convergence on filters and related topologies. For a given uniform Hausdorff space Xu and a collection S of Cauchy filters in Xu, the basic result is the construction of a uniform Hausdorff space. Xu having the properties that Xu is isomorphic to a dense subspace of Xu and every filter in S converges to a point in S. As a special case, the completion of Xu of Xu is obtained. The construction is so given as to prove the existence of the space Xu. The technique involves embedding the object X to be "completed" in a space of functions F which has as its domain a space of continuous functions C(X) defined on X. The procedure is analogous to the process of taking the bidual E" of a locally convex topological vector space. Indeed, E" is obtained as a special case. In the absence of sufficient structure on X, the Xu is obtained as the closure of X in F. In a locally convex space or an abelian topological group having enough character to separate points, Xu is obtained as a bidual or a second character group of the object X.Item Markov multi-state models for survival analysis with recurrent events(2019) Zhang, Tianhui; Yang, Grace; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Markov models are a major class within the system of multi-state models for the analysis of lifetime or event-time data. Applications abound, including the estimation of lifetime of ultra-cold neutrons, the bias correction of the apparent magnitude distribution of the stars in a certain area of the sky, and the survival analysis of clinical trials. This thesis addresses some of the problems arising in the analysis of right-censored lifetime data. Clinical trials are used as examples to investigate these problems. A Markov model that takes a patient's disease development into account for the analysis of right-censored data was first constructed by Fix and Neyman (1951). The Fix-Neyman (F-N) model is a homogeneous Markov process with two transient and two absorbing states that describes a patient's status over a period of time during a cancer clinical trial. This thesis extends the F-N model by assuming the transition rates (hazard rates) to be both state and time dependent. Recurrent transitions between relapse and recovery are allowed in the extended model. By relaxing the condition of time-independent hazard rates, the extension increases the applicability of the Markov models. The extended models are used to compute the model survival functions, cumulative hazard functions that take into consideration of right censored observations as it has been done in the celebrated Kaplan-Meier estimator. Using the Fix-Neyman procedure and the Kolmogorov forward equations, closed-form solutions are obtained for certain irreversible 4-state extended models while numerical solutions are obtained for the model with recurrent events. The 4-state model is motivated by an Aplastic Anemia data set. The computational method works for general irreversible and reversible models with no restriction on the number of states. Simulations of right-censored Markov processes are performed by using a sequence of competing risks models. Simulated data are used for checking the performance of nonparametric estimators for various sample sizes. In addition, applying Aalen's (1978) results, estimators are shown have asymptotic normal distributions. A brief review of some of the literature relevant to this thesis is provided. References are readily available from a vast literature on the survival analysis including many text books. General Markov process models for survival analysis are described, e.g., in Andersen, Borgan, Gill and Keiding (1993).Item Regression Analysis of Recurrent Events with Measurement Errors(2019) Ren, Yixin; Smith, Paul J; He, Xin; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Recurrent event data and panel count data are often encountered in longitudinal follow-up studies. The main difference between the two types of data is the observation process. Continuous observations will result in recurrent event data; and discrete observations will lead to panel count data. In statistical literature, regression analysis of the two types of data have been well studied; and a typical assumption of those studies is that all covariates are accurately recorded. However, in many applications, it is common to have measurement errors in some of the covariates. For example, in a clinical trial, a medical index might have been measured multiple times. Then dealing with the differences among those measurements is an essential topic for statisticians. For recurrent event data, we present a class of semiparametric regression models that allow correlations between censoring time and recurrent event process via frailty. An estimating equation based approach is developed to account for the presence of measurement errors in some of the covariates. Both large and finite sample properties of the proposed estimators are established. An example from the study of gamma interferon in chronic granulomatous disease is provided. For panel count data, we consider two situations in which the observation process is independent or dependent of covariates. Estimating equations are developed for the estimation of the regression parameters for both cases. Simulation studies indicate that the proposed inference procedures perform well for practical situations. An example of bladder cancer study is used to demonstrate the value of the proposed method.Item Modeling Imatinib-Treated Chronic Myelogenous Leukemia and the Immune System(2019) Peters, Cara Disa; Levy, Doron; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Chronic myelogenous leukemia can be considered as a chronic condition thanks to the development of tyrosine kinase inhibitors in the early 2000s. Most CML patients are able to manage the disease, but unending treatment can affect quality of life. The focus of much clinical research has thus transitioned to treatment cessation, where many clinical trials have demonstrated that treatment free remission is possible. While there are a lot of existing questions surrounding the criteria for cessation candidates, much evidence indicates the immune system plays a significant role. Mathematical modeling provides a complementary component to clinical research. Existing models well-describe the dynamics of CML in the first phase of treatment where most patients experience a biphasic decline in the BCR-ABL ratio. The Clapp model is one of the first to incorporate the immune system and capture the often-seen oscillations in the BCR-ABL ratio that occur later in therapy. However, these models are far from capable of being used in a predictive manner and do not fully capture the dynamics surrounding treatment cessation. Based on clinical research demonstrating the importance of immune response, we hypothesize that a mathematical model of CML should include a more detailed description of the immune system. We therefore present a new model that is an extension of the Clapp model. The model is then fit to patient data and determined to be a good qualitative description of CML dynamics. With this model it can be shown that treatment free remission is possible. However, the model introduces new parameters that must be correctly identified in order for it to have predictive power. We next consider the parameter identification problem. Since the dynamics of CML can be considered in two phases, the biphasic decline of and oscillations in the BCR-ABL ratio, we hypothesize that parameter values may differ over the course of treatment and look to identify which parameters are most variable by refitting the model to different windows of data. It is determined that parameters associated with immune response and regulation are most difficult to identify and could be key to selecting good treatment cessation candidates. To increase the predictive power of our model, we consider data assimilation techniques which are successfully used in weather forecasting. The extended Kalman filter is used to assimilate CML patient data. Although we determine that the EKF is not the ideal technique for our model, it is shown that data assimilation methods in general hold promising value to the search for a predictive model of CML. In order to have the most success, new techniques should be considered, data should be collected more frequently, and immune assay data should be made available.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.