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 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 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 Sonic Limit Singularities in the Hodograph Method(1958) Schot, Steven H.; Ludford, Geoffrey S.S.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)In the hodograph transformation, introduced to linerize the equations governing the two-dimensional inviscid potential flow of a compressible fluid, there may appear so-called limit-points and limit-lines at which the Jacobian J = ∂(x,y)/ ∂(q,θ) of the transformation vanishes. This thesis investigate these singularities when they occur at points or segments of arc of the sonic line (Mach number unity). Assuming the streamfunction to be regular in the hodograph variables, it is show that sonic limit points cannot be isolated but must lie on a supersonic limit line or form a sonic limit line [cf. H. Geiringer, Math. Zeitschr., 63, (1956), 514-524]. Using this dichotomy a classification of sonic limit points is set up and certain geometrical properties of the mapping in the neighborhood of the singularity are discussed. In particular the general sonic limit line is shown to be an equipotential and an isovel; an envelope of both families of characteristics; and the locus of cusps of the streamlines and the isoclines. Flows containing sonic limit lines may be constructed by forming suitable linear combinations of the Chaplygin product solutions for any value of the separation constant n ≥ 0. For n less than a certain value n0 and greater than zero (n = 0 corresponds to the well-known radial flow), these flows represent a compressible analogue of the incompressible corner flows and may be envisaged as taking place on a quadruply-sheeted surface. The sheets are joined at a super-sonic limit line and at the sonic limit line which has the shape of a hypocycloid (n >1), cycloid (n = 1), or epicycloid (n <1). To exemplify the general behavior, the flows are constructed explicitly for n = 1/2, 1, and 2. The shape of the sonic limit line is also discussed when solutions corresponding to different n are superposed, and it is shown how then the supersonic limit line can be eliminated so that an isolated sonic limit line is obtained. A flow containing such an isolated sonic limit line is presented. An appendix derives the asymptotic solution for large values of n which corresponds to the sonic limit solution. The above results have been published in part in Math. Zeitschr., 67, (1957), 229-237. Other portions of this thesis will appear in two papers in Archive Rational Mech. and Anal., 2, (1958).Item A COMBINATORIAL REPRESENTATION FOR ORIENTED POLYHEDRAL SURFACES(1960) Edmonds, John Robert Jr; Reinhart, Bruce; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)Item The Axiom of Choice for Collections of Finite Sets(1969) Gauntt, Robert James; Karp, Carol R.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)Some implications among finite versions of the Axiom of Choice are considered. In the first of two chapters some theorems are proven concerning the dependence or independence of these implications on the theory ZFU, the modification of ZF which permits the existence of atoms. The second chapter outlines proofs of corresponding theorems with "ZFU" replaced by "ZF" . The independence proofs involve Mostowski type permutation models in the first chapter and Cohen forcing in the second chapter. The finite axioms considered are C^n , "Every collection of n-element sets has a choice function"; W^n, "Every well-orderable collection of n-element sets has a choice function"; D^n, "Every denumerable collection of n-element sets has a choice function"; and A^n (x), "Every collection Y of n-element sets, with Y ≈ X, has a choice function". The conjunction C^nl &...& C^nk is denoted by CZ where Z = {nl ,...,nk}. Corresponding conjunctions of other finite axioms are denoted similarly by Wz, Dz and Az (X). Theorem: The following are provable in ZFU: W^k1n1+...+krnr ➔ W^n1 v...v W^nr, D^k1n1+...+krnr ➔ D^n1 v...v D^nr, and C^k1n1+...+krnr ➔ C^n1 v W^n2 v...v W^nr