Mathematics Commons^™

A New Approach To Word Standardization And Some Of Its Applications, Wesam Talab

Turkish Journal of Mathematics

In this article, we study word standardization in comparison to Young tableau standardization. We count the number of words (respectively Young tableau) standardized to a given permutation (respectively to a given standard Young tableau). We prove that both rectification and standardization applications commute and show that the standardization commutes with the insertion of Robinson--Schensted. We show that the standardizations of Knuth-equivalent two words are also Knuth equivalent. Finally, using word standardization we establish a proof for the following well-known equality: $$ \forall l \in \left\lbrace 0,1,\ldots,n-1\right\rbrace ,~~\left \langle {n\atop l} \right \rangle=d_{n,l}=a_{n,l}= \sum_{0\leq k \leq l}(-1)^k { n+1 \choose k …

Hermite-Hadamard-Mercer Type Inclusions For Interval-Valued Functions Via Riemann-Liouville Fractional Integrals, Hasan Kara, Muhammad Aamir Ali, Hüseyi̇n Budak

Turkish Journal of Mathematics

In this research, we first establish some inclusions of fractional Hermite-Hadamard-Mercer type for interval-valued functions. Moreover, by special cases of our main results, we show that our main results reduce several inclusions obtained in the earlier works.

Formulas For Special Numbers And Polynomials Derived From Functional Equations Of Their Generating Functions, Nesli̇han Kilar

Turkish Journal of Mathematics

The main purpose of this paper is to investigate various formulas, identities and relations involving Apostol type numbers and parametric type polynomials. By using generating functions and their functional equations, we give many relations among the certain family of combinatorial numbers, the Vieta polynomials, the two-parametric types of the Apostol-Euler polynomials, the Apostol-Bernoulli polynomials, the Apostol-Genocchi polynomials, the Fibonacci and Lucas numbers, the Chebyshev polynomials, and other special numbers and polynomials. Moreover, we give some formulas related to trigonometric functions, special numbers and special polynomials. Finally, some remarks and observations on the results of this paper are given.

Initial Value Problem For Elastic System In Transversely Isotropic Inhomogeneous Media, Meltem Altunkaynak

Turkish Journal of Mathematics

In this paper, we consider an initial value problem (IVP) for three dimensional elasticity system in a transversely isotropic inhomogeneous media. We will rewrite the problem in the form of Fourier images by means of Fourier transform method. After some arrangements, the problem is reduced to integral equations in the vector form. Using the properties of the vector integral equation and successive approximations method, an explicit formula for the solution of the IVP in transversely isotropic inhomogeneous media is constructed, and existence and uniqueness of the solution is stated. By a computational example, we illustrate the robustness of the method.

Nonunique Best Proximity Point Results With An Application To Nonlinear Fractional Differential Equations, Mustafa Aslantaş

Turkish Journal of Mathematics

In this paper, we point out an error in proving famous Achari type nonunique fixed point results. Also, we prove some best proximity point results in $b$ -metric spaces by introducing new concepts. Hence, we develop some results existing in the literature. Finally, we give a result for the existence of the solution of nonlinear fractional differential equations.

Existence And Uniqueness Of Mild Solutions For Mixed Caputo And Riemann-Liouville Semilinear Fractional Integrodifferential Equations With Nonlocal Conditions, Ashraf H. A. Radwan

Turkish Journal of Mathematics

The purpose of this paper is to investigate the existence and uniqueness of the mild solution to a class of semilinear fractional integrodifferential equations with state-dependent nonlocal fractional conditions. Our problem includes both Caputo and Riemann-Liouville fractional derivatives. Continuous dependence of solutions on initial conditions and $\epsilon$-approximate mild solutions of the considered problem will be discussed.

New Form Of Laguerre Fractional Differential Equation And Applications, Zahra Kavooci, Kazem Ghanbari, Hanif Mirzaei

Turkish Journal of Mathematics

Laguerre differential equation is a well known equation that appears in the quantum mechanical description of the hydrogen atom. In this paper, we aim to develop a new form of Laguerre Fractional Differential Equation (LFDE) of order $2\alpha$ and we investigate the solutions and their properties. For a positive real number $\alpha$, we prove that the equation has solutions of the form $L_{n,\alpha}(x)=\sum_{k=0}^na_kx^k$, where the coefficients of the polynomials are computed explicitly. For integer case $\alpha=1$ we show that these polynomials are identical to classical Laguerre polynomials. Finally, we solve some fractional differential equations by defining a suitable integral transform.

The Fourier Spectral Method For Determining A Heat Capacity Coefficient In A Parabolic Equation, Durdimurod Durdiev, Dilshod Durdiev

Turkish Journal of Mathematics

In this paper, the comparison of finite difference and Fourier spectral numerical methods for an inverse problem of simultaneously determining an unknown coefficient in a parabolic equation with the usual initial and boundary conditions is proposed. We represent the detailed description of the methods and their algorithms. The research work conducted in this paper shows that the Fourier spectral method is highly accurate.

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 …

Does Bias Have Shape? An Examination Of The Feasibility Of Algorithmic Detection Of Unfair Bias Using Topological Data Analysis, Ansel Steven Tessier

Senior Projects Spring 2022

Artificial intelligence and machine learning systems are becoming ever more prevalent; at every turn these systems are asked to make decisions that have lasting impacts on peoples’ lives. It is becoming increasingly important that we ensure these systems are making fair and equitable decisions. For decades we have been aware of biased and unfair decision making in many sectors of society. In recent years a growing body of evidence suggests these biases are being captured in data that are then used to build artificial intelligence and machine learning systems, which themselves perpetuate these biases. The question is then, can we …

Hidden Symmetries Of The Kepler Problem, Julia Kathryn Sheffler

Senior Projects Spring 2022

The orbits of planets can be described by solving Kepler’s problem which considers the motion due to by gravity (or any inverse square force law). The solutions to Kepler’s problem, for energies less then 0, are ellipses, with a few conserved quantities: energy, angular momentum and the Laplace-Runge-Lenz (LRL) vector. Each conserved quantity corresponds to symmetries of the system via N ̈other’s theorem. Energy conservation relates to time translations and angular momentum to three dimensional rotations. The symmetry related to the LRL vector is more difficult to visualize since it lives in phase space rather than configuration space. To understand …

The Kepler Problem On Complex And Pseudo-Riemannian Manifolds, Michael R. Astwood

Theses and Dissertations (Comprehensive)

The motion of objects in the sky has captured the attention of scientists and mathematicians since classical times. The problem of determining their motion has been dubbed the Kepler problem, and has since been generalized into an abstract problem of dynamical systems. In particular, the question of whether a classical system produces closed and bounded orbits is of importance even to modern mathematical physics, since these systems can often be analysed by hand. The aforementioned question was originally studied by Bertrand in the context of celestial mechanics, and is therefore referred to as the Bertrand problem. We investigate the qualitative …

A Drop In The Bucket?: Solutions For Fermi Questions, December 2022, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Valuation Of American Commodity Options, Piyapoom Nonsoong

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this dissertation, we present analytical option pricing formulas for European and American options in which the price dynamics of a risky asset follows a meanreverting process with time-dependent parameter. The process can be adapted to describe both nonseasonal and seasonal variation in price, especially, in commodity markets such as agricultural commodities. The formulas are derived based on the solutions of partial differential equations showing that the values of both European and American options can be decomposed into two parts: the payoff of the option at initial time and the time-integral over the lifetime of the option, which is driven …

Grade K-5 Teachers’ Perceptions Of Professional Development That Supports Mathematics Instruction, Shannon Annette Manley

Walden Dissertations and Doctoral Studies

Many Grade K-5 teachers in the United States do not receive the mathematics support they need from the professional development (PD) activities offered by their school districts. The purpose of this qualitative research was to explore the perceptions of Grade K-5 teachers on the PD activities they received from their school district to support mathematics instruction. The conceptual framework that supported this study was andragogy, an adult learning theory that takes the learner’s needs into account and values the connection to real-world situations. The research question addressed how Grade K-5 teachers perceive the PD that they were offered by their …

Integral Representations Of Sl_2(Z/Nz), Yatin Dinesh Patel

Wayne State University Dissertations

The aim of this work is to determine for which commutative rings integral representations of SL_2(Z/nZ) exist and to explicitly compute them. We start with R = Z/pZ and then consider Z=p^\lambda Z. A new approach will be used to do this based on the Weil representation. We then consider general finite rings Z/nZ by extending methods described in [26]. We make extensive use of group theory, linear representations of finite groups, ring theory, algebraic geometry, and number theory. From number theory we will employ results regarding modular forms, Legendre symbols, Hilbert symbols, and quadratic forms. We consider the works …

The Parker Problem In Hall Magnetohydrodynamics Analytical And Numerical Solutions, Chad Malott

Electronic Theses and Dissertations, 2020-

In this thesis we follow on the mathematical aspects of the previous work of Shivamoggi (2009) on the Parker problem in Hall magnetohydrodynamics (MHD). We will present an analysis involving detailed analytical and numerical solutions to the Parker problem in Hall MHD. We give an analytical formulation for the Parker problem in Hall MHD, involving an initial value problem (IVP) associated with a first order Riccati equation (RE). We present Mathematica software exact solutions directly with special functions and more straightforward solutions that use the change of variables and power series methods without special functions. We give an asymptotic formulation …

There From The Beginning: The Women Of Los Alamos National Laboratory Supporting National And International Nuclear Security, Olga Martin, Laura Mcclellan, Octavio Ramos, Heather Quinn

International Journal of Nuclear Security

From the beginning of the Manhattan Project in the early 1940s, the women of what would become Los Alamos National Laboratory (LANL) worked in technical positions alongside their male counterparts, played a key role as computers, and worked in administrative jobs as secretaries, phone operators, bookkeepers, and on behalf of the U.S. Army in the Women’s Army Corps.

Throughout the history of the Laboratory, women experts at LANL helped establish and lead important national and international security programs, with careers in science, technology, engineering, and mathematics. Over time, the women of Los Alamos have come together under various Employee Resource …

Making Upper-Level Math Accessible To A Younger Audience, Allyson Roller

WWU Honors College Senior Projects

Symmetry is all around us. It appears on fabrics and on the buildings that surround us. Believe it or not, there is actually quite a bit of math that goes into generating these patterns, which are known as the seven frieze patterns. In my work, I explain how each unique pattern is generated using different types of symmetries. I also created a PDF of a children’s book about frieze patterns to ensure that people of all ages have the opportunity to learn about seemingly complex patterns.

Middle School Teacher Perceptions On Implementation Of The Math Workshop Model, Donna Mack

Walden Dissertations and Doctoral Studies

Mathematics achievement levels in the middle school grades have reportedly been below proficiency for students in the United States. The math workshop model has been identified as a possible approach to increase student achievement in mathematics. The purpose of this generic qualitative study, guided by Mezirow’s transformative learning theory, focused on understanding middle school teacher perceptions of the implementation of the math workshop model. The five research questions addressed middle school teacher perceptions of the implementation and influence of the math workshop and how it had transformed their understanding of teaching mathematics. Data were collected through semistructured interviews with nine …