Mathematics Commons^™

Finite Integration Method Using Chebyshev Polynomials For Solving Time-Dependent Linear Partial Differential Equations And Linear Fractional Order Differential Equations, Arnont Saengsiritongchai

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we devise numerical algorithms base on the finite integration method (FIM) using Chebyshev polynomial to find numerical solutions of time-dependent linear partial differential equations and linear fractional order differential equations. The results show that for time-dependent linear partial differential equations, our algorithm gives a lot better accuracy than the traditional FIMs. Several examples illustrate that our algorithm for linear fractional order differential equations gives good approximate solution as well.

A Taylor Operation Method For Solutions Of Generalized Pantograph Type Delay Differential Equations, Şuayi̇p Yüzbaşi, Nurbol Ismailov

Turkish Journal of Mathematics

In this paper, a new operational matrix method based on the Taylor polynomials is presented to solve generalized pantograph type delay differential equations. The method is based on operational matrices of integration and product for Taylor polynomials. These matrices are obtained by using the best approximation of function by the Taylor polynomials. The advantage of the method is that the method does not require collocation points. By using the proposed method, the generalized pantograph equation problem is reduced to a system of linear algebraic equations. The solving of this system gives the coefficients of our solution. Numerical examples are given …

A Generalization Of The Alexander Polynomial As An Application Of The Delta Derivative, İsmet Altintaş, Kemal Taşköprü

Turkish Journal of Mathematics

In this paper, we define the delta derivative in the integer group ring and we show that the delta derivative is well defined on the free groups. We also define a polynomial invariant of oriented knot and link by carrying the delta derivative to the link group. Since the delta derivative is a generalization of the free derivative, this polynomial invariant called the delta polynomial is a generalization of the Alexander polynomial. In addition, we present a new polynomial called the difference polynomial of oriented knot and link, which is similar to the Alexander polynomial and is a special case …

On The Fekete--Szeg\"{O} Type Functionals For Starlike And Convex Functions, Pawel Zaprawa

Turkish Journal of Mathematics

In the paper we discuss two functionals of the Fekete--Szeg\"{o} type: $\Phi_f(\mu) = a_2 a_4-\mu a_3{}^2$ and $\Theta_f(\mu) = a_4-\mu a_2a_3$ for an analytic function $f(z) = z+a_2z^2+a_3z^3+\ldots$, $z\in\Delta$, ($\Delta = \{z\in\mathbb{C}: z

A Result On The Maximal Length Of Consecutive 0 Digits In $\Beta$-Expansions, Xiang Gao, Hui Hu, Zhihui Li

Turkish Journal of Mathematics

Let $\beta>1$ be a real number. For any $x\in[0,1]$, let $r_{n}(x,\beta)$ be the maximal length of consecutive zero digits in the first $n$ digits of the $\beta$-expansion of $x$. In this note, it is proved that for any $0

On The Unit Index Of Some Real Biquadratic Number Fields, Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous

Turkish Journal of Mathematics

Let $p_1\equiv p_2\equiv1 \pmod 4$ be different prime numbers such that $\left(\dfrac{2}{p_2}\right)=\left(\dfrac{p_1}{p_2}\right)=-\left(\dfrac{2}{p_1}\right)=-1$. Put $\kk=\QQ(\sqrt{2p_1p_2})$ and let $\KK$ be a quadratic extension of $\kk$ contained in its absolute genus field $\kk^{(*)}$. Denote by $k_j$, $1\leq j\leq 3$, the three quadratic subfields of $\KK$. Let $E_{\KK}$ (resp. $E_{k_j}$) be the unit group of $\KK$ (resp. $k_j$). The unit index $\left[E_{\KK}: \prod_{j=1}^3E_{k_j}\right]$ is characterized in terms of biquadratic residue symbols between $2$, $p_1$ and $p_2$ or by the capitulation of $\mathfrak{2}$, the prime ideal of $\QQ(\sqrt{2p_1})$ above $2$, in $\KK$. These results are used to describe the $2$-rank of some CM-fields.

On The Density And Transitivity Of Sets Of Operators, Mohammad Ansari, Bahram Khani Robati, Karim Hedayatian

Turkish Journal of Mathematics

By the well-known result of Yood, every strictly transitive algebra of operators on a Banach space is WOT-dense. This motivated us to investigate the relationships between SOT and WOT largeness of sets of operators and the transitivity behavior of them. We show that, to obtain Yood's result, strict transitivity may not be replaced by the weaker condition of hypertransitivity. We prove that, for a wide class of topological vector spaces, every SOT-dense set of operators is hypertransitive. The general form of SOT-dense sets that are not strictly transitive is presented. We also describe the form of WOT-dense sets that are …

Multivariate Lucas Polynomials And Ideal Classes Inquadratic Number Fields, Ayberk Zeyti̇n

Turkish Journal of Mathematics

In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely related to classical Lucas and Fibonacci polynomial sequences and hence to Lucas and Fibonacci numbers. We use one of these families to give a geometric interpretation of the 200-year-old class number problems of Gauß, which is equivalent to the study of narrow ideal classes in real quadratic number fields.

A Singularly Perturbed Differential Equation With Piecewise Constant Argument Of Generalized Type, Marat Akhmet, Murathan Dauylbaev, Aziza Mirzakulova

Turkish Journal of Mathematics

The paper considers the extension of Tikhonov Theorem for singularly perturbed differential equation with piecewise constant argument of generalized type. An approximate solution of the problem has been obtained. A new phenomenon of humping has been observed in the boundary layer area. An illustrative example with simulations is provided.

On The Numerical Range Of Square Matrices With Coefficients In A Degree $2$ Galois Field Extension, Edoardo Ballico

Turkish Journal of Mathematics

Let $L$ be a degree $2$ Galois extension of the field $K$ and $M$ an $n\times n$ matrix with coefficients in $L$. Let $\langle \ ,\ \rangle : L^n\times L^n\to L$ be the sesquilinear form associated to the involution $L\to L$ fixing $K$. We use $\langle \ ,\ \rangle$ to define the numerical range $\mathrm{Num} (M)$ of $M$ (a subset of $L$), extending the classical case $K=\mathbb {R}$, $L=\mathbb {C}$, and the case of a finite field introduced by Coons, Jenkins, Knowles, Luke, and Rault. There are big differences with respect to both cases for number fields and for all …

Digital Lusternik-Schnirelmann Category, Ayse Borat, Tane Vergi̇li̇

Turkish Journal of Mathematics

In this paper, we define the digital Lusternik-Schnirelmann category cat${}_\kappa$, introduce some of its properties, and discuss how the adjacency relation affects the digital Lusternik-Schnirelmann category.

Conformal Slant Submersions From Cosymplectic Manifolds, Yilmaz Gündüzalp, Mehmet Aki̇f Akyol

Turkish Journal of Mathematics

Akyol [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistics 2017; 462: 177-192] defined and studied conformal antiinvariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from cosymplectic manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, and conformal antiinvariant submersions. More precisely, we mention many examples and obtain the geometries of the leaves of vertical distribution and horizontal distribution, including the integrability of the distributions, …

On The Diophantine Equation $((C+1)M^{2}+1)^{X}+(Cm^{2}-1)^{Y}=(Am)^Z$, Eli̇f Kizildere, Takafumi Miyazaki, Gökhan Soydan

Turkish Journal of Mathematics

Suppose that $c$, $m$, and $a$ are positive integers with $a \equiv 11,\,13 \pmod{24}$. In this work, we prove that when $2c+1=a^{2}$, the Diophantine equation in the title has only solution $(x, y, z)=(1,1,2)$ where $m \equiv \pm 1 \pmod{a}$ and $m>a^2$ in positive integers. The main tools of the proofs are elementary methods and Baker's theory.

$\Mathbb{Q}$-Korselt Numbers, Nejib Ghanmi

Turkish Journal of Mathematics

Let $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{Q}\setminus \{0\}$; a positive integer $N$ is said to be an \emph{$\alpha$-Korselt number} (\emph{$K_{\alpha}$-number}, for short) if $N\neq \alpha$ and $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. In this paper we prove that for each squarefree composite number $N$ there exist finitely many rational numbers $\alpha$ such that $N$ is a $K_{\alpha}$-number and if $\alpha\leq1$ then $N$ has at least three prime factors. Moreover, we prove that for each $\alpha\in \mathbb{Q}\setminus \{0\}$ there exist only finitely many squarefree composite numbers $N$ with two prime factors such that $N$ is a $K_{\alpha}$-number.

On The Coefficient Problem For Close-To-Convex Functions, Katarzyna Trabka Wieclaw, Pawel Zaprawa

Turkish Journal of Mathematics

This paper is concerned with the problem of estimating $ a_4-a_2a_3 $, where $a_k$ are the coefficients of a given close-to-convex function. The bounds of this expression for various classes of analytic functions have been applied to estimate the third Hankel determinant $H_3(1)$. The results for two subclasses of the class $\mathcal{C}$ of all close-to-convex functions are sharp. This bound is equal to 2. It is conjectured that this number is also the exact bound of $ a_4-a_2a_3 $ for the whole class $\mathcal{C}$.

Coefficient Estimates For A Class Containing Quasi-Convex Functions, Osman Altintaş, Öznur Özkan Kiliç

Turkish Journal of Mathematics

In the present study, we introduce the classes $\mathcal {Q_{CV}}\left(\mu, A,B \right)$ and $\mathcal{Q_{ST}}\left(\eta, A,B \right)$. Furthermore, we obtain coefficient bounds of these classes.

Iteration Method Of Approximate Solution Of The Cauchy Problem Fora~Singularly Perturbed Weakly Nonlinear Differential Equation Of An Arbitrary Order, Alexey Alimov, Evgeny Bukhzhalev

Turkish Journal of Mathematics

We construct an iteration sequence converging (in the uniform norm in the space of continuous functions) to the solution of the Cauchy problem for a~singularly perturbed weakly nonlinear differential equation of an arbitrary order (the weak nonlinearity means the presence of a~small parameter in the nonlinear term). The sequence thus constructed is also asymptotic in the sense that the departure of its $n$th element from the solution of the problem is proportional to the $(n+1)$th power of the perturbation parameter.

Approximation By Chlodowsky Type Of Szasz Operators Based On Boas--Buck-Type Polynomials, Mohammad Mursaleen, A.A.H. Al-Abied, Ana Maria Acu

Turkish Journal of Mathematics

A Chlodowsky variant of generalized Szasz-type operators involving Boas-Buck-type polynomials is considered and some convergence properties of these operators by using a weighted Korovkin-type theorem are given. A Voronoskaja-type theorem is proved. The convergence properties of these operators in a weighted space of functions defined on $[0,\infty)$ are studied. The theoretical results are exemplified choosing the special cases of Boas-Buck polynomials, namely Appell-type polynomials, Laguerre polynomials, and Charlier polynomials.

Mental Models And Neutrosophic Cognitive Maps (Modelos Mentales Y Mapas Cognitivos Neutrosóficos), Maykel Leyva-Vazquez, Rebeca Escobar-Jara, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this work, elements related to mental models elicitation and analysis are addressed through causal models. Issues related to the need to include indeterminacy in causal relationships through neutrophic cognitive maps are discussed. A proposal for static analysis in neutrosophic cognitive maps is presented. The following activities are included in the proposal: Calculate, measures of centrality, Classify nodes, De-neutrosification, and Ranking nodes. As future works, the incorporation of new metrics of centrality in neutrosophic cognitive maps is proposed. The inclusion of scenario analysis to the proposal is another area of future work.

On The Composition Of Derivatives, Krzysztof Ciesielski

Faculty & Staff Scholarship

We show that there exists a derivative f : [0, 1] → [0, 1] such that the graph of f ◦ f is dense in [0, 1]2 , so not a Gδ-set. In particular, f ◦ f is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative.

Simultaneous Small Coverings By Smooth Functions Under The Covering Property Axiom, Krzysztof Ciesielski

Faculty & Staff Scholarship

The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every ν ∈ ω ∪ {∞} there exists a family Fν ⊂ C ν (R) of cardinality ω1 < c such that for every g ∈ D ν (R) the set g \ S Fν has cardinality ≤ ω1. Moreover, we show that this result remains true for partial functions g (i.e., g ∈ D ν (X) for some X ⊂ R) if, and only if, ν ∈ {0, 1}. The proof of this result is based on the following theorem of independent interest (which, for ν 6= 0, seems to have been previously unnoticed): for every X ⊂ R with no isolated points, every ν-times differentiable function g : X → R admits a ν-times differentiable extension ¯g : B → R, where B ⊃ X is a Borel subset of R. The presented arguments rely heavily on a Whitney’s Extension Theorem for the functions defined on perfect subsets of R, which short but fully detailed proof is included. Some open questions are also posed.

The Effects Of Different Versions Of A Gateway Stem Course On Student Attitudes And Beliefs, Xiangming Wu, Jessica Deshler, Edgar Fuller

Faculty & Staff Scholarship

Background

Substantial research has been conducted focusing on student outcomes in mathematics courses in order to better understand the ways in which these outcomes depend on the underlying instructional methodologies found in the courses. From 2009 to 2014, the Mathematical Association of America (MAA) studied Calculus I instruction in United States (US) colleges and universities in the Characteristics of Successful Programs of College Calculus (CSPCC). One aspect of this study attempted to understand the impact of these courses on student experience.

Results

In this paper, we describe results from an examination of the effect of course structure on students’ attitudes …

Existence Of Global Solutions For Nonlinear Magnetohydrodynamics With Finite Larmor Radius Corrections, Fariha Elsrrawi, Harumi Hattori

Faculty & Staff Scholarship

We discuss the existence of global solutions to the magnetohydrodynamics (MHD) equations, where the effects of finite Larmor radius corrections are taken into account. Unlike the usual MHD, the pressure is a tensor and it depends on not only the density but also the magnetic field. We show the existence of global solutions by the energy methods. Our techniques of proof are based on the existence of local solution by semigroups theory and a priori estimate.

The Rsa Cryptosystem, Rodrigo Iglesias

Williams Honors College, Honors Research Projects

This paper intends to present an overview of the RSA cryptosystem. Cryptosystems are mathematical algorithms that disguise information so that only the people for whom the information is intended can read it. The invention of the RSA cryptosystem in 1977 was a significant event in the history of cryptosystems. We will describe in detail how the RSA cryptosystem works and then illustrate the process with a realistic example using fictional characters. In addition, we will discuss how cryptosystems worked prior to the invention of RSA and the advantage of using RSA over any of the previous cryptosystems. This will help …

Generalized Characteristics Of A Generic Polytope, Tommy Naugle

Electronic Theses and Dissertations

For a smooth hypersurface S ⊂ R 2n given by the level set of a Hamiltonian function H, a symplectic form ω on R2n induces a vector field XH which flows tangent to S. By the nondegeneracy of ω, there exists a distinguished line bundle LS whose characteristics are the integral curves of XH. When S is the boundary of a smooth convex domain K˜ ⊂ R 2n, then the least action among closed characteristics of LS is equal to the Ekeland-Hofer-Zehnder capacity, a symplectic invariant. From a result due to Artstein-Avidan and Ostrover, there exists a continuous extension of …

Hdg Methods For Dirichlet Boundary Control Of Pdes, Yangwen Zhang

Doctoral Dissertations

"We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems for PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. In this thesis, we use an existing HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control in the high regularity case. We also propose a new HDG method to approximate the solution of a Dirichlet boundary …

Results On The Gold Grabbing Game, Stephen Acampa

Online Theses and Dissertations

In this paper, we will contribute to research on a Graph Theory problem known as the Gold Grabbing Game. The game consists of two players and a tree in which each vertex has a positive integer value of gold. Players take turns removing leaves from the tree and deleting the associated edge until the graph is entirely empty. A winning condition is acquiring at least half of the total gold. Existing research shows that for a tree with an even number of vertices, Player 1 can always win.

It can also be shown via simple examples that for a tree …

Positive Symmetric Solutions Of A Boundary Value Problem With Dirichlet Boundary Conditions, Tek Nath Dhakal

Online Theses and Dissertations

We apply a recent extension of a compression-expansion fixed point theorem of function type to a second order boundary value problem with Dirichlet boundary conditions. We show the existence of positive symmetric solutions of this boundary value problem.

Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

Mathematics Faculty Research

In this paper, we provide a direct and constructive proof of weak factorization of h1 (ℝ×ℝ) (the predual of little BMO space bmo(ℝ×ℝ) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f Є h1 (ℝ×ℝ) there exist sequences {αkj} Є l and functions gjk, hkj Є L2 (ℝ2 ) such that [Equation Unavailable] in the sense of h1 (ℝ×ℝ), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ║fh1║(ℝ×ℝ) is given in terms of ║gjk║ L2(ℝ2) and ║hkj║ L2(ℝ2). By duality, this directly implies a lower bound on the norm of …

Labs For Calculus: Learning Through Collaborative Discovery, Ryan Mckinney

Summer Community of Scholars Posters (RCEU and HCR Combined Programs)

No abstract provided.