Physical Sciences and Mathematics Commons^™

Anisotropic Classes Of Homogeneous Pseudodifferential Symbols, Árpád Bényi, Marcin Bownik

Mathematics Faculty Publications

We define homogeneous classes of x-dependent anisotropic symbols S˙mγ,δ(A) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander–Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón–Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in S˙01,1(A) yield Calderón–Zygmund kernels, yet their L2 boundedness fails. Finally, we prove boundedness results …

On The Hörmander Classes Of Bilinear Pseudodifferential Operators, Árpád Bényi, Diego Maldonado, Virginia Naibo, Rodolfo H. (Rodolfo Humberto) Torres

Mathematics Faculty Publications

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with …

Sentry Selection In Wireless Networks, Paul Balister, Béla Bollobás, Amites Sarkar, Mark Walters

Mathematics Faculty Publications

Let P be a Poisson process of intensity one in the infinite plane R². We surround each point x of P by the open disc of radius r centred at x. Now let S_n be a fixed disc of area n, and let C_r(S_n) be the set of discs which intersect S_n. Write E_r^k for the event that C_r(S_n) is a k-cover of S_n, and F_r^k for the event that C_r(S_n) …

Secrecy Coverage (Conference Proceeding), Amites Sarkar, Martin Haenggi

Mathematics Faculty Publications

Motivated by information-theoretic secrecy, geometric models for secrecy in wireless networks have begun to receive increased attention. The general question is how the presence of eavesdroppers affects the properties and performance of the network. Previously the focus has been mostly on connectivity. Here we study the impact of eavesdroppers on the coverage of a network of base stations. The problem we address is the following. Let base stations and eavesdroppers be distributed as stationary Poisson point processes in a disk of area n. If the coverage of each base station is limited by the distance to the nearest eavesdropper, …

Bifurcation Of Solutions Of Separable Parameterized Equations Into Lines, Tjalling Ypma, Yun-Qiu Shen

Mathematics Faculty Publications

Many applications give rise to separable parameterized equations of the form A(y,µ)z + b(y, µ) = 0, where y ∈ Rⁿ, z ∈ R^N and the parameter µ ∈ R; here A(y,µ) is an (N + n) × N matrix and b(y, µ) ∈ R^{N +n}. Under the assumption that A(y, µ) has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form f ( …

Zeros Of Some Level 2 Eisenstein Series, Sharon Garthwaite, Ling Long, Holly Swisher, Stephanie Treneer

Mathematics Faculty Publications

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Γ(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation (see formula (1.2)). As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros …

Impulsive Discontinuous Hyperbolic Partial Differential Equations Of Fractional Order On Banach Algebras, Said Abbas, Ravi P. Agarwal, Mouffak Benchohra

Mathematics and System Engineering Faculty Publications

This article studies the existence of solutions and extremal solutions to partial hyperbolic differential equations of fractional order with impulses in Banach algebras under Lipschitz and Carathéodory conditions and certain monotonicity conditions.

Global Caccioppoli-Type And Poincar ´E Inequalities With Orlicz Norms, Ravi P. Agarwal, Shusen Ding

Mathematics and System Engineering Faculty Publications

We obtain global weighted Caccioppoli-type and Poincaré inequalities in terms of Orlicz norms for solutions to the nonhomogeneous A -harmonic equation d A(x,d)=B(x,d).

Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin

Mathematics Faculty Publications

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

Foliations And Global Inversion, Eduardo C. Balreira

Mathematics Faculty Research

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f : M → Rⁿ is bijective if and only if H_n−1(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard-Plastock, including its recent improvement by Nollet-Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well known …

Mathematics In Motion: Linear Systems Of Differential Equations On The Differential Analyzer, Devon A. Tivener

Theses, Dissertations and Capstones

In this work, I will provide an introduction to the dierential analyzer, a machine designed to solve dierential equations through a process called mechanical integration. I will give a brief historical account of dierential analyzers of the past, and discuss the Marshall University Dierential Analyzer Project. The goal of this work is to provide an analysis of solutions of systems of dierential equations using a dierential analyzer. In particular, we are interested in the points at which these systems are in equilibrium and the behavior of solutions that start away from equilibrium. After giving a description of linear systems of …

Modeling Super-Spreading Events For Sars, Thembinkosi P. Mkhatshwa

Theses, Dissertations and Capstones

One of the intriguing characteristics of the 2003 severe acute respiratory syndrome (SARS) epidemics was the occurrence of super spreading events (SSEs). Super-spreading events for a specific infectious disease occur when infected individuals infect more than the average number of secondary cases. The understanding of these SSEs is critical to under- standing the spread of SARS. In this thesis, we present a modification of the basic SIR (Susceptible - Infected - Removed) disease model, an SIPR (Susceptible - Regular Infected - Super-spreader - Removed) model, which captures the effect of the SSEs.

Generalized Complex Hamiltonian Torus Actions: Examples And Constraints, Thomas Baird, Yi Lin

Department of Mathematical Sciences Faculty Publications

Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying rank(T)=n−2, using a surgery procedure on toric manifolds.

Fejér Polynomials And Control Of Nonlinear Discrete Systems, Dmitriy Dmitrishin, Paul Hagelstein, Anna Khamitova, Anatolii Korenovskyi, Alexander M. Stokolos

Department of Mathematical Sciences Faculty Publications

We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f : R → R of the form x(k + 1) = f(x(k)) + u(k) where u(k) = (a₁−1)f(x(k))+a₂f(x(k−T))+· · ·+a_N f(x(k−(N −1)T)) , with a₁ + · · · + a_N = 1. Following an approach of Morgul, we associate to each periodic orbit of f, N ∈ N, and a₁, . . . …

Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Multiresolution Inverse Wavelet Reconstruction From A Fourier Partial Sum, Nataniel Greene

Publications and Research

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities.In previous work [11,12] we described a numerical procedure for overcoming the Gibbs phenomenon called the Inverse Wavelet Reconstruction method (IWR). The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series. However, we only described the method standard wavelet series and …

The Maximum Rectilinear Crossing Number Of The Petersen Graph, Elie Feder, Heiko Harborth, Steven Herzberg, Sheldon Klein

Publications and Research

We prove that the maximum rectilinear crossing number of the Petersen graph is 49. First, we illustrate a picture of the Petersen graph with 49 crossings to prove the lower bound. We then prove that this bound is sharp by carefully analyzing the ten Cs's which occur in the Petersen graph and their properties.

Positive Solutions For A System Of Singular Second Order Nonlocal Boundary Value Problems, Naseer Ahmad Asif, Paul W. Eloe, Rahmat Ali Khan

Mathematics Faculty Publications

Sufficient conditions for the existence of positive solutions for a coupled system of nonlinear nonlocal boundary value problems of the type (see PDF for details) are obtained. The nonlinearities (see PDF) are continuous and may be singular at t = 0, t = 1, x = 0, or y = 0. … An example is provided to illustrate the results.

Linearly Ordered Topological Spaces And Weak Domain Representability, Joe Mashburn

Mathematics Faculty Publications

It is well known that domain representable spaces, that is topological spaces that are homeomorphic to the space of maximal elements of some domain, must be Baire. In this paper it is shown that every linearly ordered topological space (LOTS) is homeomorphic to an open dense subset of a weak domain representable space. This means that weak domain representable spaces need not be Baire.

Coarsening In High Order, Discrete, Ill-Posed Diffusion Equations, Catherine Kublik

Mathematics Faculty Publications

We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order (n=2) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation (n=1) corresponds to another famous model from image processing, namely Perona and Malik's anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the …

Periodic Solutions Of Neutral Delay Integral Equations Of Advanced Type, Muhammad Islam, Nasrin Sultana, James Booth

Mathematics Faculty Publications

We study the existence of continuous periodic solutions of a neutral delay integral equation of advanced type. In the analysis we employ three fixed point theorems: Banach, Krasnosel'skii, and Krasnosel'skii-Schaefer. Krasnosel'skii-Schaefer fixed point theorem requires an a priori bound on all solutions. We employ a Liapunov type method to obtain such bound.

2010 Alumni Presenters, University Of Dayton. Department Of Mathematics

Biennial Alumni Seminar

No abstract provided.

Matching Functions And Graphs At Multiple Levels Of Bloom’S Revised Taxonomy, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

This paper illustrates the power of Bloom's revised taxonomy for teaching, learning and assessing [3] in aligning our curriculum expectations and our assessment tools in multivariable calculus. The particular assessment tool considered involves a common matching problem to evaluate students' abilities to think about functions from graphical and formulaic representations. Through this analysis we gain additional understanding of why students may have difficulty in performing well on certain activities.

Traveling Wave Solutions For A Nonlocal Reaction-Diffusion Model Of Influenza A Drift, Joaquin Riviera, Yi Li

Mathematics and Statistics Faculty Publications

In this paper we discuss the existence of traveling wave solutions for a nonlocal reaction-diffusion model of Influenza A proposed in Lin et. al. (2003). The proof for the existence of the traveling wave takes advantage of the different time scales between the evolution of the disease and the progress of the disease in the population. Under this framework we are able to use the techniques from geometric singular perturbation theory to prove the existence of the traveling wave.

Assessment Of Utah Bankruptcies By Census Tracts: A Spatial Statistical Approach, Kenneth Pena

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

There are two questions raised when looking at the spatial pattern of the rate of bankruptcies in Utah: (i) are there similarities between the bankruptcy data in adjacent census tracts and (ii) can local cluster and outliers be identified within the data? Specifically, are there similar rates of bankruptcies in bordering census tracts and are there any localized areas of interest where we find extremely high or extremely low rates of bankruptcies? This study uses spatial statistics to perform tests for spatial autocorrelation to address these two questions. It also looks at commonalities in the clusters and differences in the …

Improving Accuracy Of Large-Scale Prediction Of Forest Disease Incidence Through Bayesian Data Reconciliation, Ephraim M. Hanks

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Increasing the accuracy of predictions made from ecological data typically involves replacing or replicating the data, but the cost of updating large-scale data sets can be prohibitive. Focusing resources on a small sample of locations from a large, less accurate data set can result in more reliable observations, though on a smaller scale. We present an approach for increasing the accuracy of predictions made from a large-scale eco logical data set through reconciliation with a small, highly accurate data set within a Bayesian hierarchical modeling framework. This approach is illustrated through a study of incidence of eastern spruce dwarf mistletoe …

Numerical Solution Of The Five-Moment Ideal Two-Fluid Equations In One Dimension, Marcus Scott

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Plasmas are frequently treated as a single conducting fluid and modeled using the equations of magnetohydrodynamics. However, this regime works better for low-frequency plasmas. High-frequency plasmas may be modeled using the principles of kinetic theory. For plasmas with frequencies between these two extremes, a two-fluid approach can yield better results. In 2006, Ammar Hakim mathematically modeled a plasma with a set of equations called the five-moment ideal two-fluid equations. An attempt is made reproduce those results. A derivation of this set of equations by taking moments of the Boltzmann equation is presented. Electric and magnetic fields contribute to the source …

Assessing The Precision And Accuracy In A Small Sample Of Actical Devices, Peter Sherick

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Actigraphy is an increasingly popular approach in medicine to assess patient activity levels in a variety of scenarios. The devices are essentially accelerometers encased in a write-watch type assembly. This project sought to determine the device precision and accuracy for the Actical model. In a sample of four Acticals, it was found that intra-device variability was minimal. However, one device was found to be statistically biased in comparison to the other three. This bias could have adverse effects on aggregated or magnitude dependent data analysis. Also, inter-device comparisons may be problematic.

Methods Of Competing Risks Analysis Of End-Stage Renal Disease And Mortality Among People With Diabetes, Hyun J. Lim, Xu Zhang, Roland Dyck, Nathaniel Osgood

Mathematics and Statistics Faculty Publications

Background: When a patient experiences an event other than the one of interest in the study, usually the probability of experiencing the event of interest is altered. By contrast, disease-free survival time analysis by standard methods, such as the Kaplan-Meier method and the standard Cox model, does not distinguish different causes in the presence of competing risks. Alternative approaches use the cumulative incidence estimator by the Cox models on cause-specific and on subdistribution hazards models. We applied cause-specific and subdistribution hazards models to a diabetes dataset with two competing risks (end-stage renal disease (ESRD) or death without ESRD) to measure …

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematics Faculty Publications

Lucy Slater used Bailey's 6Ã6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10Ã10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new …