Mathematics Commons^™

Introductory Texts

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

• GAMTE 2015 Officers
• GAMTE 2015 Conference Committee
• GAMTE 2015 Proceeding Committee
• Purposes and Goals of GAMTE
• Letter from the President
• Table of Contents

Quaternionic Hardy Spaces In The Open Unit Ball And Half Space And Blaschke Products, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Hardy spaces H2(B) and H2(H+), where B and H+ denote, respectively, the open unit ball of the quaternions and the half space of quaternions with positive real part, as well as Blaschke products, have been intensively studied in a series of papers where they are used as a tool to prove other results in Schur analysis. This paper gives an overview on the topic, collecting the various results available.

Self-Mappings Of The Quaternionic Unit Ball: Multiplier Properties, Schwarz-Pick Inequality, And Nevanlinna-Pick Interpolation Problem, Daniel Alpay, Vladimir Bolotnikov, Fabrizio Colombo, Irene Sabadini, Fabrizio Colombo

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball B into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space H2(B). In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

Infinite Product Representations For Kernels And Iteration Of Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping R in one complex variable, and its iterations.

Realizations Of Infinite Products, Ruelle Operators And Wavelet Filters, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the system theory notion of state-space realization of matrix-valued rational functions, we describe the Ruelle operator associated with wavelet filters. The resulting realization of infinite products of rational functions have the following four features: 1) It is defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function M, we show that a resulting infinite product realization obtained from M takes the form of an (infinitedimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for M. 3) Starting with a subclass of rational matrix functions, including scalar-valued ones corresponding …

Wiener-Chaos Approach To Optimal Prediction, Daniel Alpay, Alon Kipnis

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this work we combine Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on part of its sample path. This is done by considering special bases for the Gaussian space G generated by the process, which allows us to obtain an orthogonal basis for the Fock space of G such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample …

Spectral Theory For Gaussian Processes: Reproducing Kernels, Random Functions, Boundaries, And L2-Wavelet Generators With Fractional Scales, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: …

Teac 451p: Learning And Teaching Principles And Practices (Secondary Mathematics)—A Peer Review Of Teaching Project Benchmark Portfolio, Lorraine Males

UNL Faculty Course Portfolios

The goal of my peer review portfolio was to better understand how to improve students' learning of how to teach secondary mathematics in reform-oriented ways. Most students that pursue admission into the Secondary Mathematics Teacher Education Program have little to no experience learning mathematics in reform-oriented ways. These preservice teachers (PSTs) were “successful” in mathematics courses in middle and high school, most of them taking honors or accelerated courses. However, many of these PSTs did not have opportunities to engage as active participants in their own learning and develop complex cognitive skills and processes, the focus of reform-oriented instruction. This …

Uniquely Strongly Clean Triangular Matrices, Huanyin Chen, Orhan Gürgün, Handan Kose

Turkish Journal of Mathematics

A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R$ \big($e\in comm(a)$\big) such that $a-e\in U(R)$. We prove, in this note, that a ring $R$ is uniquely clean and uniquely bleached if and only if $R$ is abelian, ${\mathbb{T}}_{n}(R)$ is uniquely strongly clean for all $n\geq 1$, i.e. every $n\times n$ triangular matrix over $R$ is uniquely strongly clean, if and only if $R$ is abelian, and ${\mathbb{T}}_{n}(R)$ is uniquely strongly clean for some $n\geq 1$. In the commutative case, more explicit results are obtained.

$R$-Ideals In Commutative Rings, Rostam Mohamadian

Turkish Journal of Mathematics

In this article we introduce the concept of $r$-ideals in commutative rings (note: an ideal $I$ of a ring $R$ is called $r$-ideal, if $ab\in I$ and ${\rm Ann}(a)=(0)$ imply that $b\in I$ for each $a,b\in R$). We study and investigate the behavior of $r$-ideals and compare them with other classical ideals, such as prime and maximal ideals. We also show that some known ideals such as $z^\circ$-ideals are $r$-ideals. It is observed that if $I$ is an $r$-ideal, then so too is a minimal prime ideal of $I$. We naturally extend the celebrated results such as Cohen's theorem for …

Jacobi-Spectral Method For Integro-Delay Differential Equations With Weakly Singular Kernels, Ishtiaq Ali

Turkish Journal of Mathematics

We present a numerical solution to the integro-delay differential equation with weakly singular kernels with the delay function $\theta (t)$ vanishing at the initial point of the given interval $[0, T]$ ($\theta (t) = qt, 0 < q < 1)$. In order to fully use the Jacobi orthogonal polynomial theory, we use some function and variable transformation to change the intergro-delay differential equation into a new equation defined on the standard interval $[-1, 1]$. A Gauss--Jacobi quadrature formula is used to evaluate the integral term. The spectral rate of convergence is provided in infinity norm under the assumption that the solution of the given equation is sufficiently smooth. For validation of the theoretical exponential rate of convergence of our method, we provide some numerical examples.

T-Spaces, Hassan Maleki, Mohammadreza Molaei

Turkish Journal of Mathematics

In this paper, using generalized groups and their generalized actions, we define and study the notion of $T$-spaces. We study properties of the quotient space of a $T$-space and we present the conditions that imply the Hausdorff property for it. We also prove some essential results about topological generalized groups. As a main result, we show that for each positive integer $n$ there is a topological generalized group $T$ with $n$ identity elements. Moreover, we study the maps between two $T$-spaces and we consider the notion of $T$-transitivity.

Optimality Criteria For Sum Of Fractional Multiobjective Optimization Problem With Generalized Invexity, Deepak Bhati, Pitam Singh

Turkish Journal of Mathematics

The sum of a fractional program is a nonconvex optimization problem in the field of fractional programming and it is difficult to solve. The development of research is restricted to single objective sums of fractional problems only. The branch and bound methods/algorithms are developed in the literature for this problem as a single objective problem. The theoretical and algorithmic development for sums of fractional programming problems is restricted to single objective problems. In this paper, some new optimality conditions are proposed for the sum of a fractional multiobjective optimization problem with generalized invexity. The optimality conditions are obtained by using …

A Note On The Unit Distance Problem For Planar Configurations With $\Mathbb{Q}$-Independent Direction Set, Mark Herman, Jonathan Pakianathan

Turkish Journal of Mathematics

Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must be $\mathbb{Q}$-independent. This is related to the Erd\"os unit distance problem but with a simplifying additional assumption on the direction set that holds ``generically''. We show that $T(n+1)-T(n)$ is the Hamming weight of $n$, i.e. the number of nonzero binary coefficients in the binary expansion of $n$, and find a formula for $T(n)$ explicitly. In particular, $T(n)$ is $\Theta(n log(n))$. Furthermore, we describe …

Some Concrete Operators And Their Properties, Mehmet Gürdal, Mubariz T. Garayev, Suna Saltan

Turkish Journal of Mathematics

We consider integration and double integration operators, the Hardy operator, and multiplication and composition operators on Lebesgue space $L_{p}\left[ 0,1\right] $ and Sobolev spaces $W_{p}^{\left( n\right) }\left[ 0,1\right] $ and $W_{p}^{\left( n\right) }\left( \left[ 0,1\right] \times\left[ 0,1\right] \right) ,$ and we study their properties. In particular, we calculate norm and spectral multiplicity of the Hardy operator and some multiplication operators, investigate its extended eigenvectors, characterize some composition operators in terms of the extended eigenvectors of the Hardy operator, and calculate the numerical radius of the integration operator on the real $L_{2}\left[ 0,1\right] $ space. The main method for our investigation …

Stability Of Perturbed Dynamic System On Time Scales With Initial Time Difference, Coşkun Yakar, Bülent Oğur

Turkish Journal of Mathematics

The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. Sufficient conditions of stability properties are given with the help of Lyapunov-like functions.

Spherically Symmetric Finsler Metrics With Scalar Flag Curvature, Weidong Song, Fen Zhou

Turkish Journal of Mathematics

In this paper, we study spherically symmetric Finsler metrics F= y \phi( x ,\frac{}{ y }), where x \in B^n(r) \subset R^n, y \in T_xB^n(r)\{0} and \phi:[0,r)\times R \rightarrow R. By investigating a PDE equivalent to these metrics being locally projectively flat, we manufacture projectively flat spherically symmetric Finsler metrics in terms of error functions and, using Shen's result, we give its flag curvature.

Symplectic Groupoids And Generalized Almost Subtangent Manifolds, Fulya Şahi̇n

Turkish Journal of Mathematics

We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form, and generalized subtangent maps.

A Decomposition Of Transferable Utility Games: Structure Of Transferable Utility Games, Ayşe Mutlu Derya

Turkish Journal of Mathematics

We define a decomposition of transferable utility games based on shifting the worth of the grand coalition so that the associated game has a nonempty core. We classify the set of all transferable utility games based on that decomposition and analyze their structure. Using the decomposition and the notion of minimal balanced collections, we give a set of necessary and sufficient conditions for a transferable utility game to have a singleton core.

Equivalencies Between Beta-Shifts And S-Gap Shifts, Dawoud Ahmadi Dastjerdi, Somayyeh Jangjoo Shaldehi

Turkish Journal of Mathematics

Let X_{\beta} be a \beta -shift for \beta \in (1, 2] and X(S) a S-gap shift for S\subseteq N \cup {0}. We show that if X_\beta is SFT (resp. sofic), then there is a unique S-gap shift conjugate (resp. right-resolving almost conjugate) to this X_\beta, and if X_\beta is not SFT, then no S-gap shift is conjugate to X_\beta. For any synchronized X_{\beta} , an X(S) exists such that X_{\beta} and X(S) have a common synchronized 1-1 a.e. extension. For a nonsynchronized X_\beta, this common extension is just an almost Markov synchronized system with entropy preserving maps. We then compute …