Mathematics Commons^™

Pietro Paoli, Italian Algebraist, John L. Cuzzocrea, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Pietro Paoli was a leading Italian mathematician in the late 18th century. His signed letter pertaining to the death of astronomer Giuseppe Antonio Slop is translated from Italian to flowing (American) English.

Joseph Liouville’S ‘Mathematical Works Of Évariste Galois’, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Liouville’s 1846 introduction to the mathematical works of Galois is translated from French to flowing (American) English. It gave an overview of the tragic circumstances of the undergraduate mathematician whose originality led to major advances in abstract Algebra.

Variational Analysis In Nonsmooth Optimization And Discrete Optimal Control, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern methods of variational· analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main attention is paid to the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and. operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the "lack of compactness" in infinite dimensions, which …

Σary, Minnesota State University Moorhead, Mathematics Department

Math Department Newsletters

No abstract provided.

How Have Teachers Affected The Disinterest Towards Mathematics?, Amy Brown

Senior Honors Projects

In our school system today there is a collective disinterest and lack of enthusiasm towards mathematics as a whole. This apathy is prevalent as early as elementary school and continues through higher education. It is disheartening that so many students avoid mathematics because of their misconception that it is too difficult and has little value in their future. How well prepared are our teachers to deal with this? I began my research by looking at the past perceptions of mathematics and how the reform movement has changed this perspective. I also looked at the changing standards and how the Principles …

Lecture Notes: Non-Standard Approach To J.F. Colombeau’S Theory Of Generalized Functions, Todor D. Todorov

Mathematics

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau’ theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau’ theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau’s theory are rings with zero divisors. The …

Operator Semigroups: Definitions, Properties And Applications, Ngoc Nguyen

Masters Theses & Specialist Projects

A large part of contemporary natural science is concerned with investigating the motion of systems in time with a determined state space. To conduct this investigation, we study the theory of one-parameter semigroups. The theory is developed from the simplest scalar case and finite dimensional case to semigroups of linear operators on Banach spaces which started in the first half of the last century. This thesis is designed to give a basic introduction to semigroup theory and its application. Some proofs and illustrative examples are provided.

Teaching Time Savers: A Recommendation For Recommendations, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

I admit it — I enjoy writing recommendation letters for my students. I like
learning about their hopes and dreams, where they have been and where they want to go. A recommendation letter is an opportunity to remind myself how much my students can grow while they are in college, and how much I have grown as an instructor, advisor, and mentor.

A Survey Of The Methods To Find Probability Density Functions, Nancy Picinic Ricca

Theses, Dissertations and Culminating Projects

Various methods are described in this thesis that will approximate probability density functions (PDF), also known as invariant measures for chaotic maps. When studying discrete dynamical systems, the measure of a map is important in describing its behavior because it captures the statistics of long term simulations. Being invariant means that this distribution remains the same, no matter when you observe it. The main goal is to understand how to obtain these PDFs for various one-dimensional maps using alternative methods to time series data.

The methods studied in this thesis include the Z-matrix and the Frobenius-Perron Operator, which are analytic …

Evaluating The Computational Efficiency Of Xfbat And Fbat For Family Based Studies, Yanwei Ouyang

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

Family-based study designs are often employed when investigating the genetic causes of complex disease. While the transmission disequilibrium test (TDT) and its extensions were developed to use family data for assessing linkage between a known genetic marker and a disease-causing gene, the so-called FBAT approach proposed by Rabinowitz and Laird (2000) effectively subsumes these family-based procedures as special cases. FBAT is fully conditional, but its implementation in the freely available FBAT software package uses a large-sample distributional approximation to compute p-values. The exact distribution for FBAT can be enumerated, but doing so explicitly is computationally intensive, particularly for relatively larger …

The Hasse-Minkowski Theorem, Adam Gamzon

Honors Scholar Theses

The Hasse-Minkowski theorem concerns the classification of quadratic forms over global fields (i.e., finite extensions of Q and rational function fields with a finite constant field). Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921. He extended the research of his thesis to quadratic forms over all number fields in 1924. Historically, the Hasse-Minkowski theorem was the first notable application of p-adic fields that caught the attention of a wide mathematical audience. The goal of this thesis is to discuss the Hasse-Minkowski theorem over the rational numbers and over the rational function fields with a …

Periodic Difference Equations, Population Biology And The Cushing-Henson Conjectures, Saber Elaydi, Robert J. Sacker

Mathematics Faculty Research

We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically …

The Elementary Theory Of Normed Linear Spaces And Linear Functionals, Suresh Eswarthasan

Renée Crown University Honors Thesis Projects - All

Abstract not Included

Boundary Behavior Of Laplace Transforms, Timothy Ferguson

Honors Theses

In this thesis, we examine the boundary behavior of Laplace transforms (as analytic functions on the right and left half planes) of certain bounded functions. The types of bounded functions we consider are Fourier transforms of measures and almost periodic functions.

Wavelet Techniques In Time Series Analysis With An Application To Space Physics, Agnieszka Jach

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Several wavelet techniques in the analysis of time series are developed and applied to real data sets.

Methods for long-memory models include wavelet-based confidence intervals for the self-similarity parameter in potentially heavy-tailed observations. Empirical coverage probabilities are used to assess the procedures by applying them to Linear Fractional Stable Motion with many choices of parameters. Asymptotic confidence intervals provide empirical coverage often much lower than nominal and it is recommended to use subsampling confidence intervals. A procedure for monitoring the constancy of the self-similarity parameter is proposed and applied to Ethernet data sets.

A test to distinguish a weakly dependent …

Minimal Nodal Domains For Strictly Elliptic Partial Differential Equations With Homogeneous Boundary Conditions, Charles E. Miller

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This work presents a proof of the dependence of the first eigenvalue for uniformly elliptic partial differential equations on the domain in a less abstract setting than that of Ivo Babušhka and Rudolf Výborný in 1965. The proof contained here, under rather mild conditions on the boundary of the domain, ∂Ω, demonstrates that the first eigenvalue of elliptic partial differential equation

{Lu + λu = 0 in Ω

{u = 0 on ∂Ω

depends continuously on the domain in the following sense. If a sequence of domains is such that Ω_i → Ω in …

Improving Utah State University's Healthcare Plan, Aleece Blake

Undergraduate Honors Capstone Projects

Utah State University provides health insurance for 10,400 people (3,500 contracts). Employees of the university who qualify for this insurance have the option to pick one of 2 plans, Blue or White. Utah State essentially self-insures these plans, and Blue Cross Blue Shield administers them. This means that the university has a reserve set up to pay the medical claims of all of the people covered by these plans and bears most of the risk associated with providing this insurance. Some of the risk is transferred from the university to the covered individuals through deductibles, coinsurance, and copayments. The rest …

A Remark On Conservative Diffeomorphisms, Jairo Bochi, Bassam R. Fayad, Enrique Pujals

Publications and Research

Abstract:

We show that a stably ergodic diffeomorphism can be C¹ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.

Résumé:

On montre qu'un difféomorphisme stablement ergodique peut être C¹ approché par un difféomorphisme ayant des exposants de Lyapunov stablement non-nuls.

Symmetric Powers Of Elliptic Curve L-Functions, Phil Martin, Mark Watkins

Mathematics - All Scholarship

The conjectures of Deligne, Beuilinson, and Bloch-Kato assert that there should be relations between the arithmetic of algebro-geometric objects and the special values of their L-functions. We make a numerical study for symmetric power L-functions of elliptic curves, obtaining data about the validity of their functional equations, frequency of vanishing of central values, and divisibility of Bloch-Kato quotients.

Factoring The Adjoint And Maximal Cohen-Macaulay Modules Over The Generic Determinant, Ragnar-Olaf Buchweitz, Graham J. Leuschke

Mathematics - All Scholarship

A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen-Macaulay modules over the generic determinant, we exhibit all factorizations where one of the factors has determinant equal to the generic determinant. The classification shows not only that the Cohen-Macaulay representation theory of the generic determinant is wild in the tame-wild dichotomy, but that it is quite …

On Conway's Generalization Of The 3x + 1 Problem, Robin M. Givens

Honors Theses

This thesis considers a variation of the 3x+1, or Collatz, Problem involving a function we call the Conway function. The Conway function is defined by letting C3(n)=2k for n=3k and C3(n)=4k±1 for n=3k±1, where n is an integer. The iterates of this function generate a few 'short' cycles, but the s' tructural dynamics are otherwise unknown. We investigate properties of the Conway function and other related functions. We also discuss the possibility of using the Conway function to generate keys for cryptographic use based on a fast, efficient binary implemenation of the function. Questions related to the conjectured tree-like structure …

2nd Annual Undergraduate Research Conference Abstract Book, University Of Missouri--Rolla

Undergraduate Research Conference at Missouri S&T

No abstract provided.

Global Residues For Sparse Polynomial Systems, Ivan Soprunov

Mathematics and Statistics Faculty Publications

We consider families of sparse Laurent polynomials f1, . . . , fn with a finite set of common zeros Z f in the torus Tn = (C − {0})n. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Z f . We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the fi when the Newton polytopes of the fi are full-dimensional. Our results have consequences in sparse polynomial interpolation and lattice point enumeration in Minkowski sums of polytopes.

Some Remarks On Heegner Point Computations, Mark Watkins

Mathematics - All Scholarship

We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004.

On The Growth Of The Betti Sequence Of The Canonical Module, David A. Jorgensen, Graham J. Leuschke

Mathematics - All Scholarship

We study the growth of the Betti sequence of the canonical module of a Cohen-Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen-Macaulay ring possessing a canonical module to be Gorenstein.

Variational Analysis Of Evolution Inclusions, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities. with generally nonsmooth functions. We develop a variational analysis of such roblems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W^1,2-convergence of optimal solutions under consistent perturbations of …

Wavelet Deconvolution In A Periodic Setting Using Cross-Validation, Leming Qu, Partha S. Routh, Kyungduk Ko

Mathematics Faculty Publications and Presentations

The wavelet deconvolution method WaveD using band-limited wavelets offers both theoretical and computational advantages over traditional compactly supported wavelets. The translation-invariant WaveD with a fast algorithm improves further. The twofold cross-validation method for choosing the threshold parameter and the finest resolution level in WaveD is introduced. The algorithm’s performance is compared with the fixed constant tuning and the default tuning in WaveD.

An Extension Of Sharkovsky’S Theorem To Periodic Difference Equations, Ziyad Alsharawi, James Angelos, Saber Elaydi, Leela Rakesh

Mathematics Faculty Research

We present an extension of Sharkovsky’s Theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.

Multiscale Image Registration, Dana C. Paquin, Doron Levy, Eduard Schreibmann, Lei Xing

Mathematics

A multiscale image registration technique is presented for the registration of medical images that contain significant levels of noise. An overview of the medical image registration problem is presented, and various registration techniques are discussed. Experiments using mean squares, normalized correlation, and mutual information optimal linear registration are presented that determine the noise levels at which registration using these techniques fails. Further experiments in which classical denoising algorithms are applied prior to registration are presented, and it is shown that registration fails in this case for significantly high levels of noise, as well. The hierarchical multiscale image decomposition of E. …

New Developments Of The Deformation Method, Jie Liu

Mathematics Dissertations - Archive

New developments of deformation method for grid generation are presented in this work. Theorems for three different cases and different methods for implementing deformation method are presented. One of the new developments is a 3D multi-block moving grid method. In this version, a Poisson equation is solved by finite difference method to get the vector field for moving grid. Special treatment applies to the common boundary of different blocks. Another new development is a numerical method for reconstructing a given differentiable transformations by solving a system of div-curl equation directly formed from each point of the graph. The determinacy and …