Mathematics Commons^™

On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Scholarship

This paper deals with the convergence of the summation of power series of the form Σ_{a ≤ k ≤ b}f(k)x^k, where 0 ≤ a ≤ b < ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) a differentiable function defined on [a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler's transformation series will also be given. In addition, we will compare the convergence of the given series transforms.

On The Generalized Möbius Inversion Formulas, Tian-Xiao He, Peter Shiue3, Leetsch Hsu

Scholarship

We provide a wide class of M¨obius inversion formulas in terms of the generalized M¨obius functions and its application to the setting of the Selberg multiplicative functions.

Construction Of Rational Points On Elliptic Curves Over Finite Fields, Andrew Shallue, Christiaan Van De Woestijne

Scholarship

We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero.

Problem Solving In The Fourth Grade Classroom, Lisa Lefevre

Senior Honors Theses and Projects

No abstract provided.

Enumerations Of The Kolmogorov Function, Richard Beigel, Harry Buhrman, Peter Fejer, Lance Fortnow, Piotr Grabowski, Luc Longpré, Andrej Muchnik, Frank Stephan, Leen Torenvliet

Computer Science Faculty Publication Series

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A.

We determine exactly how hard it is to enumerate the Kolmogorov function, which assigns to each string x its Kolmogorov complexity:

• For every underlying universal machine U, there is a constant a …

Discourse On The Interface Of Matheatics And Physics: A Panel Discussion Sponsored By Dit And The Ria., Brendan Goldsmith

Articles

No abstract available

The Asymptotics Of Neutral Curve Crossing In Taylor–Dean Flow, C. P. Hills, A. P. Bassom

Articles

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. This work considers the small gap, large wavenumber limit for linear perturbations when the onset of the Taylor and Dean instabilities is concurrent. A consistent, matched asymptotic solution is found across the whole annular domain and identifies five regions of interest: two …

Hamiltonian Formulation And Integrability Of A Complex Symmetric Nonlinear System, Rossen Ivanov

Articles

The integrability of a complex generalisation of the ’elegant’ system, proposed by D. Fairlie and its relation to the Nahm equation and the Manakov top is discussed.

The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity …

Continued Fractions And Generalizations With Many Limits: A Survey, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

There are infinite processes (matrix products, continued fractions, (r, s)-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a survey of results in this area, focusing on recent results of the authors.

Further Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

In this paper we use a formula for the n-th power of a 2×2 matrix A (in terms of the entries in A) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if m and n are positive integers and s ∈ {0, 1, 2, . . . , b(mn − 1)/2c}, then X i,j,k,t 2 1+2t−mn+n (−1)nk+i(n+1) 1 + δ(m−1)/2, i+k m − 1 − i i ! m − 1 − 2i k ! × n(m − 1 − 2(i + k)) 2j ! j t − n(i + k) ! n …

A Q-Continued Fraction, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi

Mathematics Faculty Publications

Let a, b, c, d be complex numbers with d 6= 0 and |q| < 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq2n+1 + cqn (a + b)q n+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d) j (−c/bd)j q j(j+3)/2 (q)j (−aq2/d)j P∞ j=0 (b/d) j (−c/bd)j q j(j+1)/2 (q)j (−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 3 1 + q 2 − q 5 1 + q 3 − · · · − q 2n−1 1 + q n − · · · = (q 2 ; q 3 )∞ (q; q 3)∞ , (−aq)∞ X∞ j=0 (bq) j (−c/b)j q j(j−1)/2 (q)j (−aq)j = (−bq)∞ X∞ j=0 (aq) j (−c/a)j q j(j−1)/2 (q)j (−bq)j , and the Rogers-Ramanujan identities, X∞ n=0 q n 2 (q; q)n = 1 (q; q 5)∞(q 4; q 5)∞ , X∞ n=0 q n 2+n (q; q)n = 1 (q 2; q 5)∞(q 3; q 5)∞.

The Convergence Behavior Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of qcontinued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, YG, on the unit circle such that if y ∈ YG then G(y) does not converge to a finite value. We discuss …

The Convergence And Divergence Of Q-Continued Fractions Outside The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

We consider two classes of q-continued fraction whose odd and even parts are limit 1-periodic for |q| > 1, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.

Fade Statistics For A Lasercom System And The Joint Pdf Of A Gamma-Gamma Distributed Irradiance And Its Time Derivative, Frida Stromqvist Vetelino

Electronic Theses and Dissertations

The performance of lasercom systems operating in the atmosphere is reduced by optical turbulence, which causes irradiance fluctuations in the received signal. The result is a randomly fading signal. Fade statistics for lasercom systems are determined from the probability density function (PDF) of the irradiance fluctuations. The expected number of fades per second and their mean fade time require the joint PDF of the fluctuating irradiance and its time derivative. Theoretical integral expressions, as well as closed form, analytical approximations, were developed for the joint PDF of a gamma-gamma distributed irradiance and its time derivative, and the corresponding expression for …

Frames In Hilbert C*-Modules, Wu Jing

Electronic Theses and Dissertations

Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*-modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*-modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both …

Mathematical Modeling Of Smallpox Withoptimal Intervention Policy, Niwas Lawot

Electronic Theses and Dissertations

In this work, two differential equation models for smallpox are numerically solved to find the optimal intervention policy. In each model we look for the range of values of the parameters that give rise to the worst case scenarios. Since the scale of an epidemic is determined by the number of people infected, and eventually dead, as a result of infection, we attempt to quantify the scale of the epidemic and recommend the optimum intervention policy. In the first case study, we mimic a densely populated city with comparatively big tourist population, and heavily used mass transportation system. A mathematical …

Modeling Inter-Plant Interactions, Jessica Larson

Electronic Theses and Dissertations

The purpose of this paper is to examine the interactions between two plant species endemic to Florida and develop a model for the growth of one of the plant species. An equation for the growth of Hypericum cumulicola is developed through analyzing how the distance to and the height of the nearest Ceratiola ericoides (Florida rosemary) affects the growth of Hypericum cumulicola. The hypericums were separated into five separate regions according to the distance to the nearest rosemary plant. The parameters for a basic growth equation were obtained in each of the five regions and compared to each other along …

A Comparative Study Of Ant Colony Optimization, Matthew Becker

Electronic Theses and Dissertations

Ant Colony Optimization (ACO) belongs to a class of biologically-motivated approaches to computing that includes such metaheuristics as artificial neural networks, evolutionary algorithms, and artificial immune systems, among others. Emulating to varying degrees the particular biological phenomena from which their inspiration is drawn, these alternative computational systems have succeeded in finding solutions to complex problems that had heretofore eluded more traditional techniques. Often, the resulting algorithm bears little resemblance to its biological progenitor, evolving instead into a mathematical abstraction of a singularly useful quality of the phenomenon. In such cases, these abstract computational models may be termed biological metaphors. Mindful …

Some General Notions Of Stochastic Orderings For Weighted Reliability And Uncertainty Measures With Applications, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this note, stochastic comparisons of reliability measures and related functions are presented. Inequalities for uncertainty of a residual life distribution and certain modified cross-entropy or discrimination information measures under weighted models are established. Comparisons of the expected uncertainty about the remaining lifetime of a component for weighted conditional distributions and unweighted conditional distributions are presented.

Presentations Of Finitely Generated Cancellative Commutative Monoids And Nonnegative Solutions Of Systems Of Linear Equations, Scott T. Chapman, Pedro A. García Sánchez, David Llena, José Carlos Rosales

Mathematics Faculty Research

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143–172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.

Brownian Motion And Its Applications In The Stock Market, Angeliki Ermogenous

Undergraduate Mathematics Day: Past Content

Wilfrid Kendall notes on the complexity of the paths of Brownian motion: If you run Brownian motion in two dimensions for a positive amount of time, it will write your name. The oddness and complexity of Brownian motion reveal a really deep subject in the field of mathematics that cannot be fully understood and explained even until now. The purpose of this paper is to introduce the Brownian motion with its properties and to explain how it is applied in an everyday but totally unpredictable environment like the stock market.

Methods Of Solution Of Second Order Linear Equations On Time Scales, Ashley Askew

Undergraduate Mathematics Day: Past Content

A time scale, T, is a nonempty, closed subset of the real numbers, R. Several methods of solution exist for second order linear equations on a time scale. An advantage of these methods is that we can obtain solutions on a system comprising of continuous and/or discrete elements. After restricting the time scale to be R, these solutions are equivalent to those obtained using differential equations methods.

A time scale, T, is a nonempty, closed subset of the real numbers, R. Several methods of solution exist for second order linear equations on a time scale. An advantage of these methods …

Shuffle Up And Deal: Should We Have Jokers Wild?, Kristen Lampe

Undergraduate Mathematics Day: Past Content

In the neighborhood poker games, one often hears of adding the Jokers as wild cards, or declaring deuces wild. In this talk, we’ll explore what happens mathematically when two Jokers are added to the deck. The probabilities for different hands change, in ways that might surprise you. Further, we will explore the mathematical consequences of playing poker with more or fewer than five cards. Finally, we will look at the historical beginnings of poker, and a mathematical anomaly that occurred along the way.

2006 Alumni Presenters, University Of Dayton. Department Of Mathematics

Biennial Alumni Seminar

No abstract provided.

2006 (Winter), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2006 Winter Colloquium.

A Qualitative Analysis On Nonconstant Graininess Of The Adaptive Grids Via Time Scales, Paul W. Eloe, Stefan Hilger, Qin Sheng

Mathematics Faculty Publications

Calculus on time scales plays a crucial role in unifying the continuous and discrete calculus. In this paper, we apply the time scales calculus methods to study qualitatively properties of the numerical solution of second order ordinary differential equations via different finite difference schemes. The properties become particularly interesting in the case when the computational grids are nonuniform, on which the finite difference operators do not commute. To investigate the solution properties, we introduce the graininess function, and express the numerical solution as functions of the variable grid steps, that is, functions of the graininess and its dynamic derivatives implemented …

The Role Of Biostatistics In Medical Devices: Making A Difference In People’S Lives Every Day (Abstract), Gregory Campbell

Kenneth C. Schraut Memorial Lectures

Statistics plays a key role in society in general and, in particular, in the fields of biology and medicine.

On Doubly Periodic Solutions Of Quasilinear Hyperbolic Equations Of The Fourth Order, T. Kiguradze, T. Smith

Publications

The problem on doubly periodic solutions is considered for a class of quasilinear hyperbolic equations. Effective sufficient conditions of solvability and unique solvability of this problem are established.

Pure Extensions Of Locally Compact Abelian Groups, Peter Loth

Mathematics Faculty Publications

In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every …