Mathematics Commons^™

On The Cotorsion Images Of The Baer-Specker Group, Brendan Goldsmith, T. Kelly, S, Wallutis

Articles

No abstract available

Quasi-Minimal Abelian Groups, Brendan Goldsmith, S. O. Hogain, S. Wallutis

Articles

An abelian group $G$ is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as $G$. Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.

On The Empirical Balanced Truncation For Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.

The Spectral Function For Sturm-Liouville Problems Where The Potential Is Of Wigner-Von Neumann Type Or Slowly Decaying, Daphne Gilbert, B.J. Harris, S.M. Riehl

Articles

We consider the linear, second-order, differential equation (∗) with the boundary condition (∗∗)

We suppose that q(x) is real-valued, continuously differentiable and that q(x)→0 as x→∞ with q∉L1[0,∞). Our main object of study is the spectral function ρα(λ) associated with () and (). We derive a series expansion for this function, valid for λ⩾Λ0 where Λ0 is computable and establish a Λ1, also computable, such that () and () with α=0, have no points of spectral concentration for λ⩾Λ1. We illustrate our results with examples. In particular we consider the case of the Wigner–von Neumann potential.

Analysis Of A Multigrid Algorithm For Time Harmonic Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a "backslash" multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the …

A Characterization Of Hybridized Mixed Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart–Thomas and the Brezzi–Douglas–Marini methods of similar order are identical.

Quasioptimality Of Some Spectral Mixed Methods, Jay Gopalakrishnan, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectralmixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.

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

Mathematics Faculty Publications

In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = (a b c d) be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define yn : = b X n/2c i=0 (n − i i )T n−2i (−D) i . Then, for n ≥ 1, A n = (yn − d yn−1 b yn−1 c yn−1 yn − a yn−1) . We use this formula together with an existing formula …

A Theorem On Divergence In The General Sense For Continued Fractions, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of q continued fraction to show, that if G(q) is one of these continued fractions and |q| > 1, then either G(q) converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction K∞n=1an/1 converge to …

On The Divergence Of The Rogers-Ramanujan Continued Fraction On The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φ di(t) infinitely often}, where φ = (√ 5 + 1)/2. Let YS = {exp(2πit) : t ∈ S}. It is shown that if y ∈ YS then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an …

The Wave Structure Function And Temporal Frequency Spread In Weak To Strong Optical Turbulence, Aaron J. Masino

Electronic Theses and Dissertations

This paper presents analytic expressions for the wave structure function, frequency spread of the temporal frequency spectrum, and the temporal frequency spectrum of optical signals propagating through a random medium, specifically the Earth’s atmosphere. The results are believed to be valid for all optical turbulence conditions. These expressions are developed using the Rytov approximation method. Generally, the validity of statistical quantities obtained via this method is restricted to conditions of weak optical turbulence. However, in this work, by using a modification of the effective atmospheric spectral model presented by Andrews et al. for scintillation index, wave structure function expressions have …

The Effects Of The Cognitive Tutor Algebra On Student Attitudes And Achievement In A 9th Grade Algebra Course, Gary S. Plano

Seton Hall University Dissertations and Theses (ETDs)

.

Applications Of Stochastic Calculus To Finance, Scott Stelljes

UNF Graduate Theses and Dissertations

Stochastic Calculus has been applied to the problem of pricing financial derivatives since 1973 when Black and Scholes published their famous paper "The Pricing of Options and Corporate Liabilities" in the Joumal of Political Economy. The purpose of this thesis is to show the mathematical principles underlying the methods applied to finance and to present a new model of the stock price process.

As part of this paper, we present proofs of Ito's Formula and Girsanov's Theorem which are frequently used in financial applications. We demonstrate the application of these theorems to calculating the fair price of a European call …

Algebras With Inner Mb-Representation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We investigate algebras of sets, and pairs (A,I) consisting of an algebra A and an ideal I, which is a subset of A, that possess an inner MB-representation. We compare inner MB-representability of (A,I) with several properties of (A,I) considered by Baldwin. We show that A is inner MB-representable if and only if A =S(A \ H (A)), where S(^.) is a Marczewski operation defined below and H consists of sets that are hereditarily in A. We study uniqueness issue of the ideal in that representation.

Continuous Images Of Big Sets And Additivity Of S0 Under Cpaprism, Krzysztof Ciesielski

Faculty & Staff Scholarship

We prove that the Covering Property Axiom CPA_prism, which holds in the iterated perfect set model, implies the following facts.

• There exists a family G of uniformly continuous functions from R to [0,1] such that G has cardinality \omega₁ < \continuum and for every subset S of R of cardinality \continuum there exists a g in G with g[S]=[0,1].
• The additivity of the Marczewski's ideal s₀ is equal to \omega₁ < \continuum.

Palindrome-Polynomials With Roots On The Unit Circle, John Konvalina, Valentin Matache

Mathematics Faculty Publications

Given a polynomial f(x) of degree n, let fr(x) denote its reciprocal, i.e., fr(x) = xnf(1=x). If a polynomial is equal to its reciprocal, we call it a palindrome since the coefficients are the same when read backwards or forwards. In this mathematical note we show that palindromes whose coefficients satisfy a certain magnitude-condition must have a root on the unit circle...

Simultaneous Data Perturbations And Analytic Center Convergence, Allen G. Holder

Mathematics Faculty Research

The central path is an infinitely smooth parameterization of the non-negative real line, and its convergence properties have been investigated since the middle 1980s. However, the central "path" followed by an infeasible-interior-point method relies on three parameters instead of one, and is hence a surface instead of a path. The additional parameters are included to allow for simultaneous perturbations in the cost and righ-hand side vectors. This paper provides a detailed analysis of the perturbed central path that is followed by infeasible-interior-point methods, and we characterize when such a path converges. We develop a set (Hausdorff) convergence property and show …

Newton-Raphson Versus Fisher Scoring Algorithms In Calculating Maximum Likelihood Estimates, Andrew Schworer, Peter Hovey

Undergraduate Mathematics Day: Past Content

In this work we explore the difficulties and the means by which maximum likelihood estimates can be calculated iteratively when direct solutions do not exist. The Newton-Raphson algorithm can be used to do these calculations. However, this algorithm has certain limitations that will be discussed. An alternative algorithm, Fisher scoring, which is less dependent on specific data values, is a good replacement. The Fisher scoring method converged for data sets available to the authors, that would not converge when using the Newton-Raphson algorithm. An analysis and discussion of both algorithms will be presented. Their real world application on analysis of …

Pebbling On Directed Graphs, Gayatri Gunda, Aparna Higgins

Undergraduate Mathematics Day: Past Content

Consider a finite connected graph G whose vertices are labeled with non-negative integers representing the number of pebbles on each vertex. A pebbling move on a graph G is defined as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number f(G) of a connected graph is the least number of pebbles such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified but arbitrary vertex. We consider pebbling on directed graphs and study what configurations of directed graphs allow for pebbling …

Newton’S Unfinished Business: Uncovering The Hidden Powers Of Eleven In Pascal’S Triangle, Robert Arnold, Tom Attenweiler, Christopher Brockman, Bethany Lesko, Christine Martinek, Colleen Mccormick, Jessica Mcquiston, Jessica Parker, Amy Rohmiller

Undergraduate Mathematics Day: Past Content

Sir Isaac Newton once observed that the first five rows of Pascal’s Triangle, when concatenated, yield the corresponding powers of eleven. He claimed without proof that subsequent rows also generate powers of eleven. Was he correct? While not all rows can simply be concatenated, the powers of eleven can still be easily derived from each. We have uncovered an algorithm the supports Newton’s claim and will prove its validity for all rows of the Triangle.

2004 Vol. 1 Table Of Contents, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Past Content

No abstract provided.

Some Interesting Multiples Of Nine: Use Your Digits To Get The Digits!, Kevin Hurley

Undergraduate Mathematics Day: Past Content

We have unraveled two neat and powerful algorithms for calculating certain multiples of nine. These discussions might make for an interesting introduction to a number theory course, or a supplemental project in calculus or advanced algebra. The mathematics involved is within a student’s grasp, and the results are quite startling.

Ramanujan Graphs In The Construction Of Ldpc Codes, Walter H. Chen

Undergraduate Mathematics Day: Past Content

Low-density parity-check (LDPC) codes have recently become a popular interdisciplinary area of research. Widely unknown after their invention by Gallager in 1965, the existence of efficient encoding and decoding algorithms coupled with performance that operates near theoretical limits has led to the rediscovery of LDPC codes. This paper will address the reasoning and construction of LDPC codes with Ramanujan graphs.

How Not To Get Lost While On A Random Walk, Robert Lewand

Undergraduate Mathematics Day: Past Content

What happens if you go on a random walk? Will you ever return home? Well, sometimes yes (probably) and sometimes no (probably). During this talk we will derive some elementary identities in favor you're not getting lost while on a random walk.

2004 Alumni Presenters, University Of Dayton. Department Of Mathematics

Biennial Alumni Seminar

No abstract provided.

2004 (Winter), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2004 Winter Colloquium

Conversations Among Women In Mathematics (Program), University Of Dayton. Department Of Mathematics

Biennial Alumni Seminar

No abstract provided.

A Comparison Of Modified Reconstructability Analysis And Ashenhurst‐Curtis Decomposition Of Boolean Functions, Anas Al-Rabadi, Marek Perkowski, Martin Zwick

Complex Systems Faculty Publications and Presentations

Modified reconstructability analysis (MRA), a novel decomposition technique within the framework of set‐theoretic (crisp possibilistic) reconstructability analysis, is applied to three‐variable NPN‐classified Boolean functions. MRA is superior to conventional reconstructability analysis, i.e. it decomposes more NPN functions. MRA is compared to Ashenhurst‐Curtis (AC) decomposition using two different complexity measures: log‐functionality, a measure suitable for machine learning, and the count of the total number of two‐input gates, a measure suitable for circuit design. MRA is superior to AC using the first of these measures, and is comparable to, but different from AC, using the second.

State-Based Reconstructability Analysis, Martin Zwick, Michael S. Johnson

Complex Systems Faculty Publications and Presentations

Reconstructability analysis (RA) is a method for detecting and analyzing the structure of multivariate categorical data. While Jones and his colleagues extended the original variable‐based formulation of RA to encompass models defined in terms of system states, their focus was the analysis and approximation of real‐valued functions. In this paper, we separate two ideas that Jones had merged together: the “g to k” transformation and state‐based modeling. We relate the idea of state‐based modeling to established variable‐based RA concepts and methods, including structure lattices, search strategies, metrics of model quality, and the statistical evaluation of model fit for analyses based …

Conjugacy Classes Of Finite Subgroups Of Certain Mapping Class Groups, Michal Stukow

Turkish Journal of Mathematics

We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that, for a fixed genus g, there are at most five such classes.