Mathematics Commons^™

Positivity-Preserving, Energy Stable Numerical Schemes For The Cahn-Hilliard Equation With Logarithmic Potential, Wenbin Chen, Cheng Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

In this paper we present and analyze finite difference numerical schemes for the Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both first and second order accurate temporal algorithms are considered. in the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the logarithmic arguments, i.e., the phase variable is always between −1 and 1, at a point-wise level. in particular, our …

Closing Banquet Eulogies, Russell Howell, C. Ray Rosentrater

ACMS Conference Proceedings 2019

A tribute to David Lay; A tribute to John Roe

On The Bures–Wasserstein Distance Between Positive Definite Matrices, Rajendra Bhatia, T. Jain, Yongdo Lim

Journal Articles

The metric d(A,B)=trA+trB−2tr(A1∕2BA1∕2)1∕21∕2 on the manifold of n×n positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to …

Generalized Bagley-Torvik Equation And Fractional Oscillators, Mark Naber, Lucas Lymburner

Applications and Applied Mathematics: An International Journal (AAM)

In this paper the Bagley-Torvik Equation is considered with the order of the damping term allowed to range between one and two. The solution is found to be representable as a convolution of trigonometric and exponential functions with the driving force. The properties of the effective decay rate and the oscillation frequency with respect to the order of the fractional damping are also studied. It is found that the effective decay rate and oscillation frequency have a complex dependency on the order of the derivative of the damping term and exhibit properties one might expect of a thermodynamic Equation of …

Generalized Differential Transform Method For Solving Some Fractional Integro-Differential Equations, S. Shahmorad, A. A. Khajehnasiri

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we use a generalized form of two-dimensional Differential Transform (2D-DT) to solve a new class of fractional integro-differential equations. We express some useful properties of the new transform as a proposition and prove a convergence theorem. Then we illustrate the method with numerical examples.

On The Lucas Difference Sequence Spaces Defined By Modulus Function, Murat Karakaş, Tayfur Akbaş, Ayşe M. Karakaş

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, firstly, we define the Lucas difference sequence spaces by the help of Lucas sequence and a sequence of modulus function. Besides, we give some inclusion relations and examine geometrical properties such as Banach-Saks type p, weak fixed point property.

Functional Dimension Of Solution Space Of Differential Operators Of Constant Strength, Morteza Shafii-Mousavi

Applications and Applied Mathematics: An International Journal (AAM)

A differential operator with constant coefficients is hypoelliptic if and only if its solution space is of finite functional dimension. We extend this property to operators with variable coefficient. We prove that an equally strong differential operator with variable coefficients has the same property. In addition, we extend the Zielezny’s result to operators with variable coefficients; prove that an operator with analytic coefficients on ℝⁿ is elliptic if and only if locally the functional dimension of its solution space is the same as the Euclidean dimension n.

Integral Inequalities Of Hermite-Hadamard Type Via Green Function And Applications, Tuba Tunç, Sümeyye Sönmezoğlu, Mehmet Z. Sarıkaya

Applications and Applied Mathematics: An International Journal (AAM)

In this study, we establish some Hermite- Hadamard type inequalities for functions whose second derivatives absolute value are convex. In accordance with this purpose, we obtain an identity using Green's function. Then using this equality we get our main results.

Study Of Specially And Temporally Dependent Adsorption Coefficient In Heterogeneous Porous Medium, Dilip K. Jaiswal, Gulrana _

Applications and Applied Mathematics: An International Journal (AAM)

One-dimensional advection-dispersion equation (ADE) is studied along unsteady longitudinal flow through a semi-infinite heterogeneous medium. Adsorption coefficient is considered temporally and spatially–dependent function i.e., expressed in degenerate form. The dispersion parameter is considered as inversely proportional to adsorption coefficient. The input source is of pulse type. The Laplace Transformation Technique (LTT) is used to obtain the analytical solution by introducing certain new independent variables through separate transformations. The effects of adsorption, heterogeneity and unsteadiness are investigated and discussed with the help of various graphs.

On Nonlinear Contractions In New Extended 𝒃-Metric Spaces, Hassen Aydi, Abdelbasset Felhi, Tayyab Kamran, Erdal Karapınar, Muhammad U. Ali

Applications and Applied Mathematics: An International Journal (AAM)

Very recently, the notion of extended 𝑏-metric spaces was introduced by replacing the modified triangle inequality with a functional triangle inequality and the analog of the renowned Banach fixed point theorem was proved in this new structure. In this paper, continuing in this direction, we further refine the functional inequality and establish some fixed point results for nonlinear contractive mappings in the new setting. A nontrivial example for the new extended 𝑏-metric space is given.

Transient Thermal Stresses Due To Axisymmetric Heat Supply In A Semi-Infinite Thick Circular Plate, S. D. Warbhe, K. C. Deshmukh

Applications and Applied Mathematics: An International Journal (AAM)

The present paper deals with the determination of thermal stresses in a semi-infinite thick circular plate of a finite length and infinite extent subjected to an axisymmetric heat supply. A thick circular plate is considered having constant initial temperature and arbitrary heat flux is applied on the upper and lower face. The governing heat conduction equation has been solved by using integral transform technique. The results are obtained in terms of Bessel’s function. The thermoelastic behavior has been computed numerically and illustrated graphically for a steel plate.

Generalized Growth And Faber Polynomial Approximation Of Entire Functions Of Several Complex Variables In Some Banach Spaces, Devendra Kumar, Manisha Vijayran

Applications and Applied Mathematics: An International Journal (AAM)

In this paper the relationship between the generalized order of growth of entire functions of many complex variables m(m ≥ 2) over Jordan domains and the sequence of Faber polynomial approximations in some Banach spaces has been investigated. Our results improve the various results shown in the literature.

The Martingale Approach To Financial Mathematics, Jordan M. Rowley

Master's Theses

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In …

Increasing C-Additive Processes, Nadjib Bouzar

Communications on Stochastic Analysis

No abstract provided.

A Limiting Process To Invert The Gauss-Radon Transform, Jeremy J. Becnel

Communications on Stochastic Analysis

No abstract provided.

Spectral Theorem Approach To The Characteristic Function Of Quantum Observables, Andreas Boukas, Philip J. Feinsilver

Communications on Stochastic Analysis

No abstract provided.

Fuchsian Groups, Bob Anaya

Electronic Theses, Projects, and Dissertations

Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.

Analogues Between Leibniz's Harmonic Triangle And Pascal's Arithmetic Triangle, Lacey Taylor James

Electronic Theses, Projects, and Dissertations

This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the …

Pascal's Triangle, Pascal's Pyramid, And The Trinomial Triangle, Antonio Saucedo Jr.

Electronic Theses, Projects, and Dissertations

Many properties have been found hidden in Pascal's triangle. In this paper, we will present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.

Geodesics On Generalized Plane Wave Manifolds, Moises Pena

Electronic Theses, Projects, and Dissertations

A manifold is a Hausdorff topological space that is locally Euclidean. We will define the difference between a Riemannian manifold and a pseudo-Riemannian manifold. We will explore how geodesics behave on pseudo-Riemannian manifolds and what it means for manifolds to be geodesically complete. The Hopf-Rinow theorem states that,“Riemannian manifolds are geodesically complete if and only if it is complete as a metric space,” [Lee97] however, in pseudo-Riemannian geometry, there is no analogous theorem since in general a pseudo-Riemannian metric does not induce a metric space structure on the manifold. Our main focus will be on a family of manifolds referred …

Calculus Remediation As An Indicator For Success On The Calculus Ap Exam, Ty Stockham

Electronic Theses, Projects, and Dissertations

This study investigates the effects of implementing a remediation program in a high school Advanced Placement Calculus AB course on student class grades and success in passing the AP Calculus AB exam.

A voluntary remediation program was designed to help students understand the key concepts and big ideas in beginning Calculus. Over a period of eight years the program was put into practice and data on student participation and achievement was collected. Students who participated in this program were given individualized recitation activities targeting their specific misunderstandings, and then given an opportunity to retest on chapter exams that they had …

Smoothing Parameters For Recursive Kernel Density Estimators Under Censoring, Yousri Slaoui

Communications on Stochastic Analysis

No abstract provided.

Strong Convergence Rate In Averaging Principle For The Heat Equation Driven By A General Stochastic Measure, Vadym Radchenko

Communications on Stochastic Analysis

No abstract provided.

Unified Ball Convergence Of Inexact Methods For Finding Zeros With Multiplicity, Ioannis K. Argyros, Santhosh George

Applications and Applied Mathematics: An International Journal (AAM)

We present an extended ball convergence of inexact methods for approximating a zero of a nonlinear equation with multiplicity m; where m is a natural number. Many popular methods are special cases of the inexact method.

Mirror Symmetry For Non-Abelian Landau-Ginzburg Models, Matthew Michael Williams

Theses and Dissertations

We consider Landau-Ginzburg models stemming from non-abelian groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group G*, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors in general.

Secondary Preservice Mathematics Teachers' Curricular Reasoning, Kimber Anne Mathis

Theses and Dissertations

Researchers have found that teachers' decisions affect students' opportunity to learn. Prior researchers have investigated teachers' decisions while planning, implementing, or reflecting on lessons, but few researchers have studied teachers' decisions and their reasoning throughout the teaching process. It is important to study teachers' reasoning for why they make the decisions they do throughout the teaching process. Furthermore, because inservice and preservice teachers differ in experience and available resources that they draw on while making decisions, it is helpful to consider the resources PSTs' draw on while reasoning. Curricular reasoning is a framework that describes teachers' thinking processes when making …

New Approach Of Deterministic Key Pre-Distribution Scheme Using Triangle Free Quasi Symmetric Designs, Debashis Ghosh, Joydeb Pal

Applications and Applied Mathematics: An International Journal (AAM)

A wireless sensor network (WSN) consists of tiny autonomous sensor nodes with some constraints. There are organizations having moderately necessitates of these kind of networks. So, security become an indispensable concern in WSN, due to potential adversaries. To overcome the security problem, keys are pre-loaded to the nodes before deployment. Among all key distribution schemes, deterministic key pre-distribution scheme (KPS) using combinatorial design is efficient regarding security aspect. In this paper, a deterministic approach, based on combinatorial design, for key assignment before the network deployment has been presented. Here the quasi-symmetric design which is of triangle-free is being used to …

Cubic Interior Ideals In Semigroups, G. Muhiuddin

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we apply the cubic set theory to interior ideals of a semigroup. The notion of cubic interior ideals is introduced, and related properties are investigated. Characterizations of (cubic) interior ideals are established, and conditions for a semigroup to be left (right) simple are provided.

Optimal Quantization For The Cantor Distribution Generated By Infinite Similutudes, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that P=Σ∞j=112jP∘S−1j, where for each j ∈ ℕ and x ∈ ℝ, Sj(x)=13jx+1−13j−1. Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector (12,122,...), for all n ∈ ℕ, we …

Tribonacci Convolution Triangle, Rosa Davila

Electronic Theses, Projects, and Dissertations

A lot has been said about the Fibonacci Convolution Triangle, but not much has been said about the Tribonacci Convolution Triangle. There are a few ways to generate the Fibonacci Convolution Triangle. Proven through generating functions, Koshy has discovered the Fibonacci Convolution Triangle in Pascal's Triangle, Pell numbers, and even Tribonacci numbers. The goal of this project is to find inspiration in the Fibonacci Convolution Triangle to prove patterns that we observe in the Tribonacci Convolution Triangle. We start this by bringing in all the information that will be useful in constructing and solving these convolution triangles and find a …