Physical Sciences and Mathematics Commons^™

Power Series Solutions To Ordinary Differential Equations, John Lagrange

Masters Theses & Specialist Projects

In this thesis, the reader will be made aware of methods for finding power series solutions to ordinary differential equations. In the case that a solution to a differential equation may not be expressed in terms of elementary functions, it is practical to obtain a solution in the form of an infinite series, since many differential equations which yield such a solution model an actual physical situation. In this thesis, we introduce conditions that guarantee existence and uniqueness of analytic solutions, both in the linear and nonlinear case. Several methods for obtaining analytic solutions are introduced as well. For the …

Elgenvalues Of Fibonacci-Like Sequences, Elyssa Hurst

Masters Theses & Specialist Projects

The familiar Fibonacci sequence 1,1,2,3,5,8,13,... can be described by the recurrence relation x(0) = 1, x(1) = 1, x(n) = x(n-1) + x(n-2). For this relation, as n → oo, x(n+1) → 1 +√5 x(n) 2 ' which is the familiar golden ratio. This value is also the dominant eigenvalue of the above recurrence relation. In this series, we consider the dominant eigenvalue of some Fibonacci-like sequence of the form x(n) = ∑n-1/k+1 ak Zk (n-k) where the Zk's are independent random variables with Zk = {+1 with probability p - 1 with probability q, with p + q = …

The Modern Mathematics Classroom: A Collection Of Virtual Manipulatives For Teachers And Students, David Stowell

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

An important assumption in the implementation of the National Council of Teachers of Mathematics (NCTM) Standards 2000 is that the mathematics classroom is a place where students are actively involved in the learning process. One way to foster such a learning environment is by using manipulatives. By their nature, manipulatives make the learning of mathematics a discovery-based activity. As computer use increases in the classroom, virtual manipulatives will become more important as instructional tools. Virtual manipulatives offer advantages over their traditional versions. Most important is their dynamic nature. Their dynamic capabilities provide two main benefits. First, the number of potential …

Some Problem In Homogenization., M. Rajesh Dr.

Doctoral Theses

No abstract provided.

Engineering Production Function And Choice Of Technique., N. S. S. Narayana Dr.

Doctoral Theses

This study deala wi th the selection of technological al ternaid ves of production, a problom referred to as choice of techninues" in ecommie theory. Selection of an appropriate technique (or a technolo gical al ternative) haa to be Bede with ro spe et to speci fied objectives and avail phle reDurces. Selection of proper techniques from smong many avall ablo el ternatives ha boen one of the min tasks whilu planni ng for econonio development. Generally develooping co untries like India are handi capped by problema of BCerci ty of on-lebour reaources, abundant labour force, lini ted foreign-aid, 1inited …

Blowup In A Mass-Conserving Convection-Diffusion Equation With Superquadratic Nonlinearity, Todd L. Fisher, Christopher P. Grant

Faculty Publications

A nonlinear convection-diffusion equation with boundary conditions that conserve the spatial integral of the solution is considered. Previous results on nite-time blowup of solutions and on decay of solutions to the corresponding Cauchy problem were based on the assumption that the nonlinearity obeyed a power law. In this paper, it is shown that assumptions on the growth rate of the nonlinearity, which take the form of weak superquadraticity and strong superlinearity criteria, are suffcient to imply that a large class of nonnegative solutions blow up in nite time.

Abstracting Aristotle’S Philosophy Of Mathematics, John J. Cleary

Research Resources

In the history of science perhaps the most influential Aristotelian division was that

between mathematics and physics. From our modern perspective this seems like an unfortunate deviation from the Platonic unification of the two disciplines, which guided Kepler and Galileo towards the modern scientific revolution. By contrast, Aristotle’s sharp distinction between the disciplines seems to have led to a barren scholasticism in physics, together with an arid instrumentalism in Ptolemaic astronomy. On the positive side, however, astronomy was liberated from commonsense realism for the conceptual experiments of Aristarchus of Samos, whose heliocentric hypothesis was not adopted by later astronomers because …

Computational Geometry Column 41, Joseph O'Rourke

Computer Science: Faculty Publications

The recent result that n congruent balls in R^d have at most 4 distinct geometric permutations is described.

Topics In Extremal Graph Theory: Ramsey Numbers And The Turan Function, Damon J. (Damon John) Gulczynski

WWU Honors College Senior Projects

"Topics in Extremal Graph Theory: Ramsey Numbers and the Turan" Function by Damon J. Gulczynski

Proof With Words: 2 + 11 - 1 = 12, Arthur T. Benjamin

All HMC Faculty Publications and Research

Proof with words: 2 + 11 – 1 = 12

TWo ELeVEn

A New Error Analysis For Brun's Constant, Thomas R. Nicely

Virginia Journal of Science

Enumeration of the twin primes, and the sum of their reciprocals, is extended to 3 x 10¹⁵, yielding the count π₂(3 x 10¹⁵) = 3310517800844. A more accurate estimate is obtained for Brun’s constant, B2 = 1.90216 05823 ± 0.00000 00008 . Error analysis is presented to support the contentioli that this estimate produces a 95 % confidence interval for B₂. In addition, published values of the count π(x) of primes, obtained previously by indirect means, are verified by direct coiiIit to x = 3 x 10¹⁵

Separability Of Tilings, Nicholas Baeth, Jason Deblois, Lisa Powell

Mathematical Sciences Technical Reports (MSTR)

A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflection is cyclic or abelian. A complete classification of those surfaces with separating …

Stochastic Comparision And Dependence Among Order Statistics,Spacing And Concomitants Of Order Statistics., Baha-Eldin Khaledi Dr.

Doctoral Theses

The simplest and the most common way of comparing two random variables is through their means and variances. It may happen that in some cases the median of X is larger than the median of Y, while the mean of X is smaller than the mean of Y. However, this confusion will not arise if the random variables are stochastically ordered. Similarly, the same may happen if one would like to compare the variability of X with that of Y based only on numerical measures of variability. Besides, these characteristics of distributions might not exist in some cases. In most …

Canonical Thurston Obstructions, Kevin M. Pilgrim

Mathematics and Statistics Faculty Research & Creative Works

We refine Douady and Hubbard's proof of Thurston's topological characterization of rational functions by proving the following theorem. Let f: S2→S2 be a branched covering with finite postcritical set Pf and hyperbolic orbifold. Let Γc denote the set of all homotopy classes γ of nonperipheral, simple closed curves in S2-Pf such that the length of the unique geodesic homotopic to γ tends to zero under iteration of the Thurston map induced by f on Teichmüller space. Then either Γc is empty, and f is equivalent to a rational function, or else Γc is a Thurston obstruction. © 2001 Academic Press.

The Stable Manifold Theorem For Stochastic Differential Equations (Dynamical Systems And Probability Seminar, Loughborough University), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

A New Cohomology Theory Of Orbifold, Weimin Chen Chen

Weimin Chen

No abstract provided.

The Stable Manifold Theorem For Stochastic Differential Equations (Analysis And Probability Seminar, University Of Hull), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

Numerics Of Stochastic Systems With Memory (Applied Mathematics And Numerical Analysis Seminars, University Of Manchester), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

Norms Of Sums Of Squares, Robert W. Fitzgerald

Articles and Preprints

For a finite separable extension K/F of fields of characteristic not 2, the norm of a sum of 2ⁿ squares in K is a sum of 2ⁿ squares in F. We find explicit identities.

Isotropy And Factorization In Reduced Witt Rings, Robert W. Fitzgerald

Articles and Preprints

We consider reduced Witt rings of finite chain length. We show there is a bound, in terms of the chain length and maximal signature, on the dimension of anisotropic, totally indefinite forms. From this we get the ascending chain condition on principal ideals and hence factorization of forms into products of irreducible forms.

Branching Exponent Heterogeneity And Wall Shear Stress Distribution In Vascular Trees, Kelly Lynn Karau, Gary S. Krenz, Christopher A. Dawson

Mathematics, Statistics and Computer Science Faculty Research and Publications

A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D ₁)^z = (D ₂)^z + (D ₃)^z; D ₁ is the parent vessel diameter and D ₂ and D ₃ are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance …

Hermite Collocation Solution Of Partial Differential Equations Via Preconditioned Krylov Methods, Stephen Brill

Stephen H. Brill

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi-CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red-black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi-CGSTAB …

Mapping Quantitative Trait Loci In Humans: Some Statistical Contributions., Saurabh Ghosh Dr.

Doctoral Theses

Maty qualitative tralts - such an, milk yield la cows, blood pressure in lumans --are known to be determined primarly, though zot exclusively, by inherited genetic luctora. It ls the of coasklerable impartance to identify chromosontal locations of tho genes that control a quantitative character. Linkage analysis (Ou 1990), which deals with the deduction of linkagn and estimation of recombination fractions among the loci controlling a qualitative/quantitative character and major loci wkoo poertions are knows aprfori, is widely used for localisation of gens. Although statistical methodologies for magplag gemen determining dichotomos qualitative charactes in humans aro well-developed, the demicrant of …

Boolean Functions With Important Cryptographic Properties., Subhamoy Maitra Dr.

Doctoral Theses

In this thesis we concentrate on properties of cryptographically significant Boolean functions.The techniques are mainly combinstorial and provide new resulta on enumeration and construction of such functions. Initially we concentrate on a partieular subset of Boolean functions called the symmetric Boolean functions. A closed form expression for the Walsh transform of an arbitrary symmetric Boolean function is presented. We completely characterize the symmetric functions with maximum nonlinearity and show that the maximum nonlinearity of n-variable symmetrie function can be 2n-1-2[n-1l2], Moreover, new classes of symmetric balanced and symmetric correlation immune functions are considered.We provide a randomised heuristic to construct balanced …

Contributions To Random Interactions And Dynamical Systems., Santanu Chakraborty Dr.

Doctoral Theses

In recent years random iterations of maps on Polish spaces has gained prominence. They are. nice examples of Markov processes whose invariant measures can be used in Computer imaging (see Berger ( 1). They also arise as random perturabations of deterministic dynamical systems.Let S be a Polish space with its Borel a-field. Let r be a collection of Borel maps from S to S. Let P be a probability on r. Then starting with a point z in S, we choose a map yn er according to the law P and move to ya(x). Then we choose 2 € r …

Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot

Mathematical Sciences Technical Reports (MSTR)

Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups …

Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods

Mathematical Sciences Technical Reports (MSTR)

The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL₂(13). These tilings can be distinguished by the lengths of their systoles.

Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su

All HMC Faculty Publications and Research

We analyze the drunkard's walk on the unit sphere with step size θ and show that the walk converges in order C/sin²(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.

Multi-Mode Cavity Effects In The Microwave Heating Of A Ceramic Slab, Stuart J. Walker

Dissertations

In order to gain insight into hot spot development in microwave heated ceramics, a partially insulated, two dimensional ceramic slab situated in a TE_M01 cavity is modeled in the small Biot number limit. If the electrical conductivity is an exponential function of temperture and E₀ is the strength of the incident mode, then the relationship between the spatially uniform, steady state leading order temperature, v₈, and E₀¹ is characterized by the well known bi-stable, or S shaped, response curve. The steady state second order temperature, v₁, is described by a boundary …

Analysis Of Discrete Dynamical System Models For Competing Species, Jerry J. Chen

Dissertations

A discrete version of the Lotka-Volterra (LV) differential equations for competing population species is analyzed in detail, much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. Another related system, namely, the Exponentially Self Regulating (ESR) population model, is also thoroughly analyzed. It is found that in addition to logistic dynamics - ranging from the very simple to manifestly chaotic regimes in terms of the governing parameters - the discrete LV model and the ESR model exhibit their own brands of bifurcation and chaos that are …