Mathematics Commons^™

New Techniques In Celestial Mechanics, Ali Abdulrasool Abdulhussein

Graduate Theses, Dissertations, and Problem Reports (ETD)

It is shown that for the classical system of the N body problem ( Newtonian Motion), if the motion of the N particles starts from a planar initial motion at t=t_{0}, then the motion of the N particles continues to be planar for every t\in[t_{0},t_{1}], assuming that no collisions occur between the N particles. Same argument is shown about the linear motion, namely, for the classical system of the N body problem, if the motion of the N particles starts from a linear initial motion at t=t_{0}, then the motion of the N particles continues to be linear for every …

Cycle Decomposition For Integral Current Homology, Kristin Julia Duling

Graduate Theses, Dissertations, and Problem Reports (ETD)

A standard graph theoretical result states that every element of the cycle space of a graph has a cycle decomposition. Georgakopoulos expands this result to a primitive decomposition and minimal representation of each element in a modified 1-dimensional singular homology. We modify the m-dimensional integral current homology in order to ensure a primitive decomposition for each element.

On The Classification Of Generalized Pseudo-Orthogonal Lie Groups Via Curvature, Cohomology, And Algebraic Structure, Adam C. Fletcher

Graduate Theses, Dissertations, and Problem Reports (ETD)

The study of Lie groups has yielded a rich catalogue of mathematical spaces that, in some sense, provide a theoretical and computational framework for describing the “world in which we live.” In particular, these topological groups that represent the rigid motions of a space, the behavior of subatomic particles, and the shape of the expanding universe consist of specialized matrices. In what follows, we define a new collection of matrices with a very specific transposition relation and attempt to classify this Lie group algebraically, geometrically, and topologically. We consider fields, $\Bbb{F},$ of characteristic zero and define the group of pseudo-orthogonal …

Inverse Boundary Value Problems For Polyharmonic Operators With Non-Smooth Coefficients, Landon Gauthier

Theses and Dissertations--Mathematics

We consider inverse boundary problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.

An Integrated Computational Pipeline To Construct Patient-Specific Cancer Models, Daniel Plaugher

Theses and Dissertations--Mathematics

Precision oncology largely involves tumor genomics to guide therapy protocols. Yet, it is well known that many commonly mutated genes cannot be easily targeted. Even when genes can be targeted, resistance to therapy is quite common. A rising field with promising results is that of mathematical biology, where in silico models are often used for the discovery of general principles and novel hypotheses that can guide the development of new treatments. A major goal is that eventually in silico models will accurately predict clinically relevant endpoints and find optimal control interventions to stop (or reverse) disease progression. Thus, it is …

Improving College Students’ Views And Beliefs Relative To Mathematics: A Systematic Literature Review Followed By A Multiple Case Mixed Methods Exploration Of The Experiences That Underpin Community College Students’ Attitudes, Self-Efficacy, And Values In Mathematics, Marquita H. Sea

Theses and Dissertations

Mathematics is particularly important due to its relevance in our daily lives. It is a general requirement throughout schooling. Unfortunately, many students openly declare negative views/beliefs regarding math in their personal and academic lives. These in turn, negatively influence students’ achievement related behaviors and outcomes. First, a systematic literature review was conducted to determine what types of studies/initiatives have aimed to enhance students’ views/beliefs relative to mathematics, including domain general and specific perceptions of math as well as their judgements of who is successful in mathematics and if they themselves can be successful. Specifically, the review centered on the components …

Estimating The Statistics Of Operational Loss Through The Analyzation Of A Time Series, Maurice L. Brown

Theses and Dissertations

In the world of finance, appropriately understanding risk is key to success or failure because it is a fundamental driver for institutional behavior. Here we focus on risk as it relates to the operations of financial institutions, namely operational risk. Quantifying operational risk begins with data in the form of a time series of realized losses, which can occur for a number of reasons, can vary over different time intervals, and can pose a challenge that is exacerbated by having to account for both frequency and severity of losses. We introduce a stochastic point process model for the frequency distribution …

Mathematical Reasoning: Writing And Proof (Pretext Edition), Ted Sundstrom

Open Textbooks

Mathematical Reasoning: Writing and Proof is a text for the first college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.

Mathematical Reasoning: Writing and Proof (PreTeXt Edition) was developed as part of the Accelerating Open Educational Resources Initiative at Grand Valley State University, with support from the University Libraries and the President's Innovation Fund.

Mathematical Reasoning: Writing and Proof was written by Ted Sundstrom, Professor Emeritus of Mathematics at Grand Valley State University. This textbook was converted into PreTeXt by Ian Curtis, Editorial Assistant for the …

Automated Conjecturing On The Independence Number And Minimum Degree Of Diameter-2-Critical Graphs, Joshua R. Forkin

Theses and Dissertations

A diameter-2-critical (D2C) graph is a graph with diameter two such that removing any edge increases the diameter or disconnects the graph. In this paper, we look at other lesser-studied properties of D2C graphs, focusing mainly on their independence number and minimum degree. We show that there exist D2C graphs with minimum degree strictly larger than their independence number, and that this gap can be arbitrarily large. We also exhibit D2C graphs with maximum number of common neighbors strictly greater than their independence number, and that this gap can be arbitrarily large. Furthermore, we exhibit a D2C graph whose number …

A Computationally Efficient Framework For Vector Representation Of Persistence Diagrams, Kit C. Chan, Umar Islambekov, Alexey Luchinsky, Rebecca Sanders

Mathematical and Statistical Science Faculty Research and Publications

In Topological Data Analysis, a common way of quantifying the shape of data is to use a persistence diagram (PD). PDs are multisets of points in R² computed using tools of algebraic topology. However, this multi-set structure limits the utility of PDs in applications. Therefore, in recent years efforts have been directed towards extracting informative and efficient summaries from PDs to broaden the scope of their use for machine learning tasks. We propose a computationally efficient framework to convert a PD into a vector in Rⁿ, called a vectorized persistence block (VPB). We show that our representation …

Optimizing Model Observer Performance In Learning-Based Ct Reconstruction, Gregory Ongie, Emil Y. Sidky, Ingrid S. Reiser, Xiaochuan Pan

Mathematical and Statistical Science Faculty Research and Publications

Deep neural networks used for reconstructing sparse-view CT data are typically trained by minimizing a pixel- wise mean-squared error or similar loss function over a set of training images. However, networks trained with such losses are prone to wipe out small, low-contrast features that are critical for screening and diagnosis. To remedy this issue, we introduce a novel training loss inspired by the model observer framework to enhance the detectability of weak signals in the reconstructions. We evaluate our approach on the reconstruction of synthetic sparse-view breast CT data, and demonstrate an improvement in signal detectability with the proposed loss.

Question 1: Dinosaur Killer Crater; Question 2: Graduation Speeches, Larry Weinstein

Physics Faculty Publications

The Fermi Questions: In 2003, the "Dinosaur Killer" asteroid crater was imaged for the first time (see National Geographic News, March 7, 2003). The crater is believed to be 100 km in radius and 1 km deep. What was the kinetic energy of the asteroid that made the crater? Express your answer in joules and in megatons (1 MT = 4 ·10 J).

Question 1: Space Telescope; Question 2: Jwst2, Larry Weinstein

Physics Faculty Publications

[Introduction] Space telescope

Question 1: If the James Webb Space Telescope could be turned to face Earth, how small an object could it resolve? How does that compare to the Hubble Space Telescope? ...

Question 2: How much heat from Earth is blocked by the Sun and Earth shade of the James Webb Space Telescope? ...

Question: While I was at a Norfolk Tides baseball game, a foul ball landed in the section above me and showered some of my friends with beer. What is the probability of a foul ball landing directly in a cup of beer during one …

Methods For Computing The Global Optimum Of Non-Convex Objectives, Isaac Michael Hawn

Graduate Research Theses & Dissertations

\begin{abstract}In this thesis, we concern ourselves with solving the unconstrained optimization problem % \begin{gather*} \text{Minimize}\; f(x)\\\text{subject to}\; x\in X \end{gather*} % where $f\colon\mathbb{R}^N\to \mathbb{R}$ is a non-convex function, possibly with infinitely many local minima. Solving such a problem, especially in higher dimensions often proves to be an extraordinarily difficult task, either in time complexity or in the methodology itself. Indeed, mathematicians must often resort to algorithms which make use of problem structure and which may not generalize well. In this thesis, we present two algorithms which solve this problem, albeit with their own shortcomings.

First, we present a new, $N$-dimensional …

Conductors For Maximal Orders Of Group Rings Over Local Fields, Brooke Randazzo

Graduate Research Theses & Dissertations

This dissertation examines conductors for maximal orders of group rings, $FD$, contained in integral group rings, $\mathcal{O}_F D$, where $D$ is a finite group, $F$ a local field, and $\mathcal{O}_F$ its discrete valuation ring. (Given two rings $R \subseteq S$, the conductor of $S$ in $R$, if it exists, is the largest ideal of $S$ contained in $R$.) First, we use a theorem of Jacobinski to make this conductor explicit where $F$ is a particular extension of $\mathbb{Q}_p$ of varying ramification index and $D$ is any elementary abelian $p$-group. Next, we examine this conductor problem for this same $F$ within …

Numerical Solution Of A Singularly Perturbed Fredholm Integro Differential Equation With Robin Boundary Condition, Muhammet Enes Durmaz, Gabi̇l Ami̇rali̇, Mustafa Kudu

Turkish Journal of Mathematics

In this paper, we deal with singularly perturbed Fredholm integro differential equation (SPFIDE) with mixed boundary conditions. By using interpolating quadrature rules and exponential basis function, fitted second order difference scheme has been constructed on a Shishkin mesh. The stability and convergence of the difference scheme have been analyzed in the discrete maximum norm. Some numerical examples have been solved and numerical outcomes are obtained.

Scattering Properties Of Impulsive Difference Dirac Equations, Şeyda Solmaz, Elgi̇z Bayram

Turkish Journal of Mathematics

In this paper, we explore the Jost solutions and the scattering matrix of the impulsive difference Dirac systems (IDDS) on the whole axis and study their analytic and asymptotic properties. Furthermore, characteristic properties of the scattering matrix of the IDDS have been examined.

Scattering Theory Of The Quadratic Eigenparameter Depending Impulsive Sturm-Liouville Equations, Güler Başak Öznur, Elgi̇z Bairamov

Turkish Journal of Mathematics

We handle an impulsive Sturm-Liouville boundary value problem. We find the Jost solution, Jost function, and scattering function of this problem and examine the properties of scattering function. We also study eigenvalues and resolvent operator of this problem. Finally, we exemplify our work by taking a different problem.

Global Existence And Energy Decay For A Coupled System Of Kirchhoff Beam Equations With Weakly Damping And Logarithmic Source, Ducival Carvalho Pereira, Carlos Alberto Raposo Da Cunha, Adriano Pedreira Cattai

Turkish Journal of Mathematics

This paper deals with the global solutions and exponential stability for a coupled system of Kirchhoff beam weakly damping and with a logarithmic source. We apply the potential well and establish the global well-posedness by using the Faedo--Galerkin approximations, taking into account that the initial data is located in a suitable set of stability created from the Nehari manifold. Moreover, by using Nakao's lemma, we prove the exponential stability of the solution.

On A Solvable System Of Rational Difference Equations Of Higher Order, Merve Kara, Yasi̇n Yazlik

Turkish Journal of Mathematics

In this paper, we present that the following system of difference equations $$ x_{n}=\frac{x_{n-k}z_{n-l}}{b_{n}x_{n-k}+a_{n}z_{n-k-l}}, \ y_{n}=\frac{y_{n-k}x_{n-l}}{d_{n}y_{n-k}+c_{n}x_{n-k-l}}, \ z_{n}=\frac{z_{n-k}y_{n-l}}{f_{n}z_{n-k}+e_{n}y_{n-k-l}}, $$ where $n\in \mathbb{N}_{0}$, $k,l\in\mathbb{N}$, the initial values $x_{-i},y_{-i},z_{-i}$ are real numbers, for $i \in \overline{1,k+l}$, and sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are non-zero real numbers, for all $n\in \mathbb{N}_{0}$, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case …

Multiple Positive Solutions For Nonlinear Fractional $Q$-Difference Equation With $P$-Laplacian Operator, Zhongyun Qin, Shurong Sun, Zhenlai Han

Turkish Journal of Mathematics

In this paper, we investigate a class of four-point boundary value problems of fractional $q$-difference equation with $p$-Laplacian operator which is the first time to be studied and is extended from a bending elastic beam equation. By Avery-Peterson theorem and the method of lower and upper solutions associated with monotone iterative technique, we obtain some sufficient conditions for the existence of multiple positive solutions. As applications, examples are presented to illustrate the main results.

Existence And Transportation Inequalities For Fractional Stochastic Differential Equations, Abdelghani Ouahab, Mustapha Belabbas, Johnny Henderson, Fethi Souna

Turkish Journal of Mathematics

In this work, we establish the existence and uniqueness of solutions for a fractional stochastic differential equation driven by countably many Brownian motions on bounded and unbounded intervals. Also, we study the continuous dependence of solutions on initial data. Finally, we establish the transportation quadratic cost inequality for some classes of fractional stochastic equations and continuous dependence of solutions with respect Wasserstein distance.

Composition Laws On The Fricke Surface And Markov Triples, Abdurrahman Muhammed Uludağ, Esra Ünal Yilmaz

Turkish Journal of Mathematics

We determine some composition laws related to the Fricke surface and the "double" Fricke surface. This latter surface admits the squares of Markov triples as its solutions.

Necessary Conditions For Extended Spectral Decomposable Multivalued Linear Operators, Yosra Barkaoui, Maher Mnif

Turkish Journal of Mathematics

In this paper, we use subsets of the Riemann sphere and specific types of invariant linear subspaces to introduce the extended spectral decomposable multivalued linear operators (linear relations) in Banach spaces. We also introduce the extended Bishop's property, the extended relatively single-valued extension property and the extended Dunford's property. More importantly, we show that these properties are three necessary conditions for a linear relation to be extended spectral decomposable.

The Arens-Michael Envelopes Of Laurent Ore Extensions, Petr Kosenko

Turkish Journal of Mathematics

For an Arens-Michael algebra $A$ we consider a class of $A$-$\hat{\otimes}$-bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over $A$. Given a Frechet-Arens-Michael algebra $A$ and a topologically invertible Frechet $A$-$\hat{\otimes}$-bimodule $M$, we construct an Arens-Michael algebra $\widehat{L}_A(M)$ which serves as a topological version of the Laurent tensor algebra $L_A(M)$. Also, for a fixed algebra $B$ we provide a condition on an invertible $B$-bimodule $N$ which allows us to explicitly describe the Arens-Michael envelope of $L_B(N)$ as a topological Laurent tensor algebra. In particular, we provide an explicit description of …

On The Blow-Up Of Solutions To A Fourth-Order Pseudoparabolic Equation, Mustafa Polat

Turkish Journal of Mathematics

In this note, we consider a fourth-order semilinear pseudoparabolic differential equation including a strong damping term together with a nonlocal source term. The problem is considered under the periodic boundary conditions and a finite time blow-up result is established. Also a lower bound estimate for the blow-up time is obtained.

Second Main Theorem For Meromorphic Mappings Intersecting Moving Targets On Parabolic Manifolds, Jiali Chen, Qingcai Zhang

Turkish Journal of Mathematics

In this paper, we establish a new second main theorem for meromorphic mappings from $M$ into $\mathbb{P}(V)$ intersecting moving targets $g_{j}:M\rightarrow\mathbb{P}(V^{\ast}),\ 1\leq j\leq q,$ where $M$ is a parabolic manifold and $V$ is a Hermitian vector space. As an application, we prove the algebraic dependence problem for meromorphic mappings with moving targets in general position.

Ricci-Yamabe Solitons And 3-Dimensional Riemannian Manifolds, Uday Chand De, Arpan Sardar, Krishnendu De

Turkish Journal of Mathematics

In this paper, we classify 3-dimensional Riemannian manifolds endowed with a special type of vector field if the Riemannian metrices are Ricci-Yamabe solitons and gradient Ricci-Yamabe solitons, respectively. Finally, we construct an example to illustrate our result.

A Parametric Family Of Ternary Purely Exponential Diophantine Equation $A^X+B^Y=C^Z$, Yasutsugu Fujita, Maohua Le

Turkish Journal of Mathematics

Let $a,b,c$ be fixed positive integers such that $a+b=c^2$, $2 \nmid c$ and $(b/p)\ne 1$ for every prime divisor $p$ of $c$, where $(b/p)$ is the Legendre symbol. Further let $m$ be a positive integer with $m>1$. In this paper, using the Baker method, we prove that if $m>\max\{10^8,c^2\}$, then the equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$ has only one positive integer solution $(x,y,z)=(1,1,2)$.

Inverse Coefficient Identification Problem For A Hyperbolic Equation With Nonlocal Integral Condition, Azizbayov Elvin

Turkish Journal of Mathematics

This paper is concerned with an inverse coefficient identification problem for a hyperbolic equation in a rectangular domain with a nonlocal integral condition. We introduce the definition of the classical solution, and then the considered problem is reduced to an auxiliary equivalent problem. Further, the existence and uniqueness of the solution of the equivalent problem are proved using a contraction mapping principle. Finally, using equivalency, the unique existence of a classical solution is proved.