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In geometry we will consider n-dimensional generalizations of the Power of a Point Theorem and of Pascal's Hexagon Theorem. In generalizing the Power of a Point Theorem, we will consider collections of cones determined by the intersections of an (n-1)-sphere and a pair of hyperplanes. We will then use these constructions to produce an n-dimensional generalization of Pascal's Hexagon Theorem, a classical plane geometry result which states that "Given a hexagon inscribed in a conic section, the three pairs of continuations of opposite sides meet on a straight line." Our generalization of this theorem will consider a pair of n-simplices intersecting an (n-1)-sphere, and will conclude with the intersections of corresponding faces lying in a hyperplane. In graph theory we will explore the interaction between zero forcing and cut-sets. The color change rule which lies at the center of zero forcing says "Suppose that each of the vertices of a graph are colored either blue or white. If u is a blue vertex and v is its only white neighbor, then u can force v to change to blue." The concept of zero forcing was introduced by the AIM Minimum Rank - Special Graphs Work Group in 2007 as a way of determining bounds on the minimum rank of graphs. Later, Darren Row established results concerning the zero forcing numbers of graphs with a cut-vertex. We will extend his work by considering graphs with arbitrarily large cut-sets, and the collections of components they yield, to determine results for the zero forcing numbers of these graphs.

My dissertation focuses on mathematical explanation found in proofs looked at from a historical point of view, while stressing the importance of mathematical practices. Current philosophical theories on explanatory proofs emphasize the structure and content of proofs without any regard to external factors that influence a proof’s explanatory power. As a result, the major philosophical views have been shown to be inadequate in capturing general aspects of explanation. I argue that, in addition to form and content, a proof’s explanatory power depends on its targeted audience. History is useful here, because from it, we are able to follow the transition from a first-generation proof, which is usually non-explanatory, into its explanatory version. By tracking the similarities and differences between these proofs, we are able to gain a better understanding of what makes a proof explanatory according to mathematicians who have the relevant background to evaluate it as so. My first chapter discusses why history is important for understanding mathematical practices. I describe two kinds of history: one that presents a narrative of events, which influenced developments in mathematics both directly and indirectly, and another, typically used in mathematical research, which concentrates only on technical developments. I contend that both versions of the past benefit the philosopher. History used in research gives us an idea of what mathematicians desire or find to be important, while history written by historians shows us what effects these have on mathematical practices. The next two chapters are about explanatory proofs. My second chapter examines the main theories of mathematical explanation. I argue that these theories are short-sighted as they only consider what appears in a proof without considering the proof’s purported audience or background knowledge necessary to understand the proof. In the third chapter, I propose an alternative way of analyzing explanatory proofs. Here, I suggest looking at a theorem’s history, which includes its successive proofs, as well as the mathematicians who wrote them. From this, we can better understand how and why mathematicians prove theorems in multiple ways, which depends on the purposes of these theorems. The last chapter is a case study on the computer proof of the Four Color Theorem by Appel and Haken. Here, I compare and contrast what philosophers and mathematicians have had to say about the proof. I argue that the main philosophical worry regarding the theorem—its unsurveyability—did not make a strong impact on the mathematical community and would have hindered mathematical development in computer-assisted proofs. By studying the history of the theorem, we learn that Appel and Haken relied on the strategy of Kempe’s flawed proof from the 1800s (which, obviously, did not involve a computer). Two later proofs, also aided by computer, were developed using similar methods. None of these proofs are explanatory, but not because of their massive lengths. Rather, the methods used in these proofs are a series of calculations that exhaust all possible configurations of maps.

This work presents a novel technique for data compression based on multi-residue number systems. The basic theorem is that an under-determined system of congruences could be solved to accomplish data compression for a signal satisfying continuity of its information content and bounded in peak-to -peak amplitude by the product of relatively prime moduli,. This thesis investigates this property and presents quantitative results along with MATLAB codes. Chapter 1 is introductory in nature and Chapter 2 deals in more detail with the basic theorem. Chapter 3 explicitly mentions the assumptions made and chapter 4 shows alternative solutions to the Chinese remainder theorem. Chapter 5 explains the experiments in detail whose results are mentioned in chapter 6. Chapter 7 concludes with a summary and suggestions for future work.

In this paper we will investigate the connection between a random walk and a continuous time stochastic process. Donsker's Theorem states that a random walk under certain conditions will converge to a Wiener process. We will provide a detailed proof of this theorem which will be used to prove that a geometric random walk converges to a geometric Brownian motion.

The notions of partial Lagrangians, partial Noether operators and partial Euler-Lagrange equations are used in the construction of first integrals for ordinary differential equations (ODEs) that need not be derivable from variational principles. We obtain a Noetherlike theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler-Lagrange equations. In order to investigate the effectiveness of the partial Lagrangian approach, some models such as the oscillator systems both linear and nonlinear, Emden and Ermakov-pinnery equations and the Hamiltonian system with two degrees of freedom are considered in this work. We study a general linear system of two second-order ODEs with variable coefficients. Note that, a Lagrangian exists for the special case only but, in general, the system under consideration does not have a standard Lagrangian. However, partial Lagrangians do exist for all such equations in the absence of Lagrangians. Firstly, we classify all the Noether and partial Noether operators for the case when the system admits a standard Lagrangian. We show that the first integrals that result due to the partial Noether approach is the same as for the Noether approach. First integrals are then constructed by the partial Noether approach for the general case when there is in general no Lagrangian for the system of two second-order ODEs with variable coefficients. We give an easy way of constructing first integrals for such systems by utilization of a partial Noether’s theorem with the help of partial Noether operators associated with a partial Lagrangian. Furthermore, we classify all the potential functions for which we construct first integrals for a system with two degrees of freedom. Moreover, the comparison of Lagrangian and partial Lagrangian approaches for the two degrees of freedom Lagrangian system is also given. In addition, we extend the idea of a partial Lagrangian for the perturbed ordinary differential equations. Several examples are constructed to illustrate the definition of a partial Lagrangian in the approximate situation. An approximate Noether-like theorem which gives the approximate first integrals for the perturbed ordinary differential equations without regard to a Lagrangian is deduced. We study the approximate partial Noether operators for a system of two coupled nonlinear oscillators and the approximate first integrals are obtained for both resonant and non-resonant cases. Finally, we construct the approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling. Since the system mentioned above does not satisfy a standard Lagrangian, the approximate first integrals are still constructed by invoking an approximate Noether-like theorem with the help of approximate partial Noether operators. This approach can give rise to further studies in the construction of approximate first integrals for perturbed equations without a variational principle.

>Magister Scientiae - MSc / We introduce the concept of an algebraically closed field with emphasis of the basic model-theoretic results concerning the theory of algebraically closed fields. One of these nice results about algebraically closed fields is the quantifier elimination property. We also show that the theory of algebraically closed field with a given characteristic is complete and model-complete. Finally, we introduce the beautiful Ax-Grothendieck theorem and an application to it.

After a detailed discussion of the variation theorem for upper bound calculation of eigenvalues, many standard procedures for determining lower bounds to eigenvalues are presented with chemical applications in mind. A new lower bound method, arbitrary choice without trunctation is presented and tested on the helium atom. This method is attractive because it does not require knowledge of the eigenvalues or eigenvectors of the base problem. In application, however, it is shown that the method is disappointing for two reasons: 1) the method does not guarantee improved bounds as calculational effort is increased; and 2) the method requires some a priori information which, in general, may not be available. A possible direction for future work is pointed out in the end. An extension of a lower bound method by Calogero and Marchioro has been developed and is presented in appendix G along with comments on the effective field method in appendix H for Virginia Tech access only. / Ph. D. / To avoid copyright infringements, access to these three appendices (G, H, and I) has been permanently limited to the Virginia Tech campus. In the case that Virginia Tech places these appendices freely on the internet, Virginia Tech is solely responsible for copyright violations.

Consider the class of polynomials P (z) = (Formula Presented) with 0 ≤ a0 ≤ a1 ≤ · · · ≤ an. The classical Eneström–Kakeya Theorem states that any polynomial in this class has all its zeros in the unit disk |z| ≤ 1 in the complex plane. We introduce new classes of polynomials by imposing a monotonicity-type condition on the coefficients with all indices congruent modulo m for some given m ≤ n. We give the inner and outer radii of an annulus containing all zeros of such polynomials. We also give an upper bound on the number of zeros in a disk for polynomials in these classes.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20546号 / 理博第4304号 / 新制||理||1618(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 早川 尚男, 教授 川上 則雄, 教授 佐々 真一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM