Mathematics Commons^™

Importance Sampling For Dispersion-Managed Solitons, Elaine T. Spiller, Gino Biondini

Mathematics, Statistics and Computer Science Faculty Research and Publications

The dispersion-managed nonlinear Schrödinger (DMNLS) equation governs the long-term dynamics of systems which are subject to large and rapid dispersion variations. We present a method to study large, noise-induced amplitude and phase perturbations of dispersion-managed solitons. The method is based on the use of importance sampling to bias Monte Carlo simulations toward regions of state space where rare events of interest—large phase or amplitude variations—are most likely to occur. Implementing the method thus involves solving two separate problems: finding the most likely noise realizations that produce a small change in the soliton parameters, and finding the most likely way that …

Maximal Class Numbers Of Cm Number Fields, Ryan C. Daileda, Raju Krishnamoorthy, Anton Malyshev

Mathematics Faculty Research

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large …

2010 (Spring), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2010 Spring Colloquium.

Analysis Of Boolean Functions With High Second Order Nonlinearity, Corneliu A. Bodea

Honors Theses

Highly nonlinear Boolean functions play a central role in the design and security analysis of high speed stream cyphers and block cyphers. We focus on analyzing the structure of Boolean functions that exhibit high second order nonlinearity. We commence with a theoretical overview of Boolean functions and Reed- Muller codes. We then introduce a new equivalence relation, 2-equivalence, for which we prove a number of important properties. Finally, we analyze the second order nonlinearity of concatenations of two Boolean functions.

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.

Detection Of Outliers In Time Series Data, Samson Sifael Kiware

Master's Theses (2009 -)

This thesis presents the detection of time series outliers. The data set used in this work is provided by the GasDay Project at Marquette University, which produces mathematical models to predict the consumption of natural gas for Local Distribution Companies (LDCs). Flow with no outliers is required to develop and train accurate models. GasDay is using statistical approaches motivated by normally distributed samples such as the 3 -sigma rule and the 5 -sigma rule to aid the experts in detecting outliers in residuals from the models. However, the Jarque-Bera statistical test shows that the residuals from the GasDay models are …

Non Bayesian Conditioning And Deconditioning, Jean Dezert, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we present a Non-Bayesian conditioning rule for belief revision. This rule is truly Non-Bayesian in the sense that it doesn’t satisfy the common adopted principle that when a prior belief is Bayesian, after conditioning by X, Bel(X|X) must be equal to one. Our new conditioning rule for belief revision is based on the proportional conflict redistribution rule of combination developed in DSmT (Dezert-Smarandache Theory) which abandons Bayes’ conditioning principle. Such Non-Bayesian conditioning allows to take into account judiciously the level of conflict between the prior belief available and the conditional evidence. We also introduce the deconditioning problem …

Rao's Quadratic Entropy And Some New Applications, Yueqin Zhao

Mathematics & Statistics Theses & Dissertations

Many problems in statistical inference are formulated as testing the diversity of populations. The entropy functions measure the similarity of a distribution function to the uniform distribution and hence can be used as a measure of diversity. Rao (1982a) proposed the concept of quadratic entropy. Its concavity property makes the decomposition similar to ANOVA for categorical data feasible. In this thesis, after reviewing the properties and providing a modification to quadratic entropy, various applications of quadratic entropy are explored. First, analysis of quadratic entropy with the suggested modification to analyze the contingency table data is explored. Then its application to …

Towards A Formal Theory Of Interoperability, Saikou Y. Diallo

Computational Modeling & Simulation Engineering Theses & Dissertations

This dissertation proposes a formal theory of interoperability that explains 1) what interoperability is as opposed to how it works, 2) how to tell whether two or more systems can interoperate and 3) how to identify whether systems are interoperating or merely exchanging bits and bytes. The research provides a formal model of data in M&S that captures all possible representations of a real or imagined thing and distinguishes between existential dependencies and transformational dependencies. Existential dependencies capture the relationships within a model while transformational dependencies capture the relationships between interactions with a model. These definitions are used to formally …

Statistical Learning And Behrens-Fisher Distribution Methods For Heteroscedastic Data In Microarray Analysis, Nabin K. Manandhr-Shrestha

USF Tampa Graduate Theses and Dissertations

The aim of the present study is to identify the di®erentially expressed genes be- tween two di®erent conditions and apply it in predicting the class of new samples using the microarray data. Microarray data analysis poses many challenges to the statis- ticians because of its high dimensionality and small sample size, dubbed as "small n large p problem". Microarray data has been extensively studied by many statisticians and geneticists. Generally, it is said to follow a normal distribution with equal vari- ances in two conditions, but it is not true in general. Since the number of replications is very small, …

Randomized Kaczmarz Solver For Noisy Linear Systems, Deanna Needell

CMC Faculty Publications and Research

The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method converges with expected exponential rate, independent of the number of equations in the system. Here we analyze the case where the system Ax=b is corrupted by noise, so we consider the system where Ax is approximately b + r where r is an arbitrary error vector. We prove that in this noisy version, …

Applications Of Descriptive Set Theory In Homotopy Theory, Samuel M. Corson

Theses and Dissertations

This thesis presents new theorems in homotopy theory, in particular it generalizes a theorem of Saharon Shelah. We employ a technique used by Janusz Pawlikowski to show that certain Peano continua have a least nontrivial homotopy group that is finitely presented or of cardinality continuum. We also use this technique to give some relative consistency results.

Reconstructability Analysis Of Elementary Cellular Automata, Martin Zwick, Hui Shi

Systems Science Friday Noon Seminar Series

Reconstructability analysis is a method to determine whether a multivariate relation, defined set- or information-theoretically, is decomposable with or without loss (reduction in constraint) into lower ordinality relations. Set-theoretic reconstructability analysis (SRA) is used to characterize the mappings of elementary cellular automata. The degree of lossless decomposition possible for each mapping is more effective than the λ parameter (Walker & Ashby, Langton) as a predictor of chaotic dynamics.

Complete SRA yields not only the simplest lossless structure but also a vector of losses of all decomposed structures, indexed by parameter, τ. This vector subsumes λ, Wuensche’s Z parameter, and Walker …

An Algorithm For Wavelet–Based Elemental Spectrum Analysis, Bruce Kessler

Mathematics Faculty Publications

At the previous Approximation Theory XII meeting, I discussed some preliminary work with the Applied Physics Institute at Western Kentucky University in using multiwavelets to provide an objective analysis of gamma-ray spectrum generated from fast neutron bombardment of objects, for the purpose of identifying the elemental composition of the object. The method discussed at the time worked moderately well with the limited amount of data provided, but subsequent use with data sets of different compounds and with different detectors brought to light serious flaws with its implementation.

This talk will illustrate those issues and will address how they have been …

Planar Cat(K) Subspaces, Russell M. Ricks

Theses and Dissertations

Let M_k^2 be the complete, simply connected, Riemannian 2-manifold of constant curvature k ± 0. Let E be a closed, simply connected subspace of M_k^2 with the property that every two points in E are connected by a rectifi able path in E. We show that E is CAT(k) under the induced path metric.

Verification Of Kam Theory On Earth Orbiting Satellites, Christian L. Bisher

Theses and Dissertations

This paper uses KAM torus theory and Simplified General Perturbations 4 (SGP4) orbit prediction techniques compiled by Dr. William Wiesel and compares it to Analytical Graphics ® Incorporated (AGI) Satellite Toolkit ® (STK) orbit data. The goal of this paper is to verify KAM torus theory can be used to describe and propagate an Earth satellite orbit with similar accuracy to existing general perturbation techniques. Using SGP4 code including only truncated geopotential effects, KAM torus generating code, and other utilities were used to describe a particular satellite orbit as a torus and then propagate the satellite using traditional and KAM …

Numerical Investigation Of Pre-Detonator Geometries For Pde Applications, Robert T. Fievisohn

Theses and Dissertations

A parametric study was performed to determine optimal geometries to allow the successful transition of a detonation from a pre-detonator into the thrust tube of a pulse detonation engine. The study was performed using a two-dimensional Euler solver with progress variables to model the chemistry. The geometrical configurations for the simulations look at the effect of shock reflections, flow obstructions, and detonation diffraction to determine successful geometries. It was observed that there are success and failure rates associated with pre-detonators. These success rates appear to be determined by the transverse wave structure of a stably propagating detonation wave and must …

Validation Of A Novel Approach To Solving Multibody Systems Using Hamilton's Weak Principle, Ashton D. Hainge

Theses and Dissertations

A novel approach for formulating and solving for the dynamic response of multibody systems has been developed using Hamilton’s Law of Varying Action as its unifying principle. In order to assure that the associated computer program is sufficiently robust when applied across a wide range of dynamic systems, the program must be verified and validated. The purpose of the research was to perform the verification and validation of the program. Results from the program were compared with closed-form and numerical solutions of simple systems, such as a simple pendulum and a rotating pendulum. The accuracy of the program for complex …

Simulation Of A Diode Pumped Alkali Laser, A Three Level Numerical Approach, Shawn W. Hackett

Theses and Dissertations

This paper develops a three level model for a continuous wave diode pumped alkali laser by creating rate equations based on a three level system. The three level system consists of an alkali metal vapor, typically Rb or Cs, pumped by a diode from the ²S_1/2 state to the ²P_3/2 , a collisional relaxation from ²P_3/2 to ²P_{1/ 2} , and then lasing from ²P_1/2 to ²S_1/2 . The hyperfine absorption and emission cross sections for these transitions are developed in detail. Differential equations for intra-gain pump attenuation …

Stability Of Choice In The Honey Bee Nest-Site Selection Process, Andrew L. Nevai, Kevin M. Passino, Parthasarathy Srinivasan

Mathematics and Statistics Faculty Publications

We introduce a pair of compartment models for the honey bee nest-site selection process that lend themselves to analytic methods. The first model represents a swarm of bees deciding whether a site is viable, and the second characterizes its ability to select between two viable sites. We find that the one-site assessment process has two equilibrium states: a disinterested equilibrium (DE) in which the bees show no interest in the site and an interested equilibrium (IE) in which bees show interest. In analogy with epidemic models, we define basic and absolute recruitment numbers (R0R0 and B0B0) as measures of the …

Finite Element Approximations For Stokes-Darcy Flow With Beavers-Joseph Interface Conditions, Yanzhao Cao, Max Gunzburger, Xiaolong Hu, Fei Hua, Xiaoming Wang, Weidong Zhao

Mathematics and Statistics Faculty Research & Creative Works

Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits. Such situations arise, for example, for groundwater flows in karst aquifers. the coupled flow is modeled by the Darcy equation in a porous medium and the Stokes equations in the conduit domain. on the interface between the matrix and conduit, Beavers-Joseph interface conditions, instead of the simplified Beavers-Joseph-Saffman conditions, are imposed. Convergence and error estimates for finite element approximations are obtained. Numerical experiments illustrate the validity of the theoretical results. © 2010 Society for Industrial and Applied Mathematics.

Finite-State Markov Chains Obey Benford’S Law, Babar Kaynar, Arno Berger, Theodore P. Hill, Ad Ridder

Research Scholars in Residence

A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (Pⁿ - P*) and (Pⁿ⁺¹-Pⁿ) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 …

Dynamical Laws Of The Coupled Gross-Pitaevskii Equations For Spin-1 Bose-Einstein Condensates, Weizhu Bao, Yanzhi Zhang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we derive analytically the dynamical laws of the coupled Gross- Pitaevskii equations (CGPEs) without/with an angular momentum rotation term and an external magnetic field for modelling nonrotating/rotating spin-1 Bose-Eintein condensates. We prove the conservation of the angular momentum expectation when the external trapping potential is radially symmetric in two dimensions and cylindrically symmetric in three dimensions; obtain a system of first order ordinary differential equations (ODEs) governing the dynamics of the density of each component and solve the ODEs analytically in a few cases; derive a second order ODE for the dynamics of the condensate width and …

First-Order And Second-Order Optimality Conditions For Nonsmooth Constrained Problems Via Convolution Smoothing, Andrew C. Eberhard, Boris S. Mordukhovich

Mathematics Research Reports

This paper mainly concerns deriving first-order and second-order necessary (and partly sufficient) optimality conditions for a general class of constrained optimization problems via smoothing regularization procedures based on infimal-like convolutions/envelopes. In this way we obtain first-order optimality conditions of both lower subdifferential and upper subdifferential types and then second-order conditions of three kinds involving, respectively, generalized second-order directional derivatives, graphical derivatives of first-order subdifferentials, and secondorder subdifferentials defined via coderivatives of first-order constructions.

On Sumudu Transform And System Of Differential Equations, Adem Kiliçman, Hassan Eltayeb, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

The regular system of differential equations with convolution terms solved by Sumudu transform.

Preface

Communications on Stochastic Analysis

No abstract provided.

On The Existence Of Weak Variational Solutions To Stochastic Differential Equations, L Gawarecki, V Mandrekar

Communications on Stochastic Analysis

No abstract provided.

Some Solvable Classes Of Filtering Problem With Ornstein-Uhlenbeck Noise, Zhicheng Liu, Jie Xiong

Communications on Stochastic Analysis

No abstract provided.

Commutativity Properties Of Conditional Distributions And Palm Measures, Olav Kallenberg

Communications on Stochastic Analysis

No abstract provided.

Some Asymptotic Results For Near Critical Branching Processes, Amarjit Budhiraja, Dominik Reinhold

Communications on Stochastic Analysis

No abstract provided.