An m charge in the n dimensional Euclidean space is a linear functional acting on m dimensional polyhedral chains and satisfying the following continuity condition. The value of the linear functional approaches zero on chains whose normal masses are bounded and whose flat norms asymptotically vanish. Our main theorem relates m charges to pairs of continuous differential forms. Luzin's theorem states that every measurable function on the line is the derivative of a continuous, almost everywhere differentiable function. We show this can be improved in several dimensions. Finally we prove that a compact subset C of the n dimensional Euclidean space does not support the distributional divergence of a bounded measurable vector field if and only if C has vanishing (n-1) dimensional Hausdorff measure.

We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.

Modeling of radio wave propagation is indispensable for the design and analysis of wireless communication systems. The use of the Finite-Difference Time-Domain (FDTD) method for wireless channel modeling has gained significant popularity due its ability to extract wideband responses from a single simulation. FDTD-based techniques, despite providing accurate channel characterizations, have often employed point sources in their studies, mainly due to the large amounts of resources required for modeling fine geometrical details or features inherent in antennas into a discrete spatial domain. The underlying influences of the antenna on wave propagation have thus been disregarded. This work presents a possible approach for the efficient space-time analysis of antennas by deducing FDTD-compatible macromodels that completely encapsulate the electromagnetic behaviour of antennas and then incorporating them into a standard FDTD formulation for modeling their interactions with a general environment.

Modeling of radio wave propagation is indispensable for the design and analysis of wireless communication systems. The use of the Finite-Difference Time-Domain (FDTD) method for wireless channel modeling has gained significant popularity due its ability to extract wideband responses from a single simulation. FDTD-based techniques, despite providing accurate channel characterizations, have often employed point sources in their studies, mainly due to the large amounts of resources required for modeling fine geometrical details or features inherent in antennas into a discrete spatial domain. The underlying influences of the antenna on wave propagation have thus been disregarded. This work presents a possible approach for the efficient space-time analysis of antennas by deducing FDTD-compatible macromodels that completely encapsulate the electromagnetic behaviour of antennas and then incorporating them into a standard FDTD formulation for modeling their interactions with a general environment.

We give a detailed proof for two discrete analogues of Courant's Nodal Domain Theorem. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

The Ambarzumyan Theorem states that for the classical Sturm-Liouville problem on $[0,1]$, if the set of Neumann eigenvalue $sigma_N={(npi)^2: nin { f N}cup { 0}}$, then the potential function $q=0$. In this thesis, we study the analogues of Ambarzumyan Theorem for the Sturm-Liouville operators on star-shaped graphs with 3 edges of different lengths. We first solve the direct problem: to find out the set of eigenvalues when $q=0$. Then we use the theory of transformation operators and Raleigh-Ritz inequality to prove the inverse problem. Following Pivovarchik's work on star-shaped graphs of uniform lengths, we analyze the Kirchoff condition in detail to prove our theorems. In particular, we study the cases when the lengths of the 3 edges satisfy $a_1=a_2=frac{1}{2}a_3$ or $a_1=frac{1}{2}a_2=frac{1}{3}a_3$. Furthermore, we work on Neumann boundary conditions as well as Dirichlet boundary conditions. In the latter case, some assumptions about $q$ have to be made.

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The aim of this study is to analyze students¡¦ mathematics concepts in solving Pythagorean Theorem problems presented in two different representations (word problems and word problems with diagrams). The investigators employed the mathematics competence indicators in Grade 1-9 Integrated Curriculum in developing such problems. In analyzing data, the investigator used Schoenfeld¡¦s method in depicting their problem-solving processes, with attention to students¡¦ sequence and difference in time consumption. Four eight grade students with good competence in mathematics and expressions from a secondary school were selected as research subjects. Problems related to Pythagorean Theorem were divided into three types: Shape, Area, and Number. Data were collected using thinking aloud method and semi-structured interview, and triangulation was further applied in protocol analysis. The research results revealed 3 findings: (1) For the ¡§Shape¡¨ type problems, students¡¦ problem-solving concepts varied with different problem representation. For the ¡§Area¡¨ and ¡§Number¡¨ types of problems (without diagram), students were required to use their geometric concept when processing word problems. Students¡¨ use of problem-solving concepts would not significantly vary with problem representation types. However, students¡¦ use of problem-solving methods would affect the types and priorities of concepts used. Generally, the types of mathematics concepts could be made up by the frequency of concepts used, and more types of problem-solving concepts would be used for word problems representation than for word problems with diagrams representation. (2) In terms of the time consumed in the first three problem-solving stages of Schoenfeld, the time required to solve word problems was 1.6 times of that required to solve word problems with diagrams. In terms of the total time consumed, the time required to solve word problems was 1.25 times of that required to solve word problems with diagrams. In the problem-solving stages, students needed to explore the problem first when dealing with word problems before they could go on to solve the problem, and such repetition was more frequent when they dealt with word problems. (3) For both type of problem representations, there is a higher number of correctly-answered problems. This finding indicated that a higher frequency of problem-solving concepts and less repetition in the problem-solving stage were required; and vice versa. As to the sequence of Pythagorean Theorem concepts to be taught, the investigator suggest teachers to start with the concept of area filling in the ¡§Shape¡¨ type of problems to derive Pythagorean Theorem, and further apply the formula to - III - solving ¡§Number¡¨ problems. After students have acquired basic competency in ¡§Shape¡¨ and ¡§Number¡¨ Pythagorean Theorem problems, teachers could explain and introduce this theorem from the perspective of ¡§Area¡¨. Finally, in problem posing, teachers were also advised to apply various contexts; covering all kinds of representations of problems that enhance students¡¦ utilization of mathematics concepts; and to cater for various needs of students.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 131-133).