Mathematics Commons^™

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.

The Kumaraswamy Weibull Geometric Distribution With Applications, Mahdi Rasekhi, Morad Alizadeh, Gholamhossein G. Hamedani

Mathematical and Statistical Science Faculty Research and Publications

In this work, we study the kumaraswamy weibull geometric (Kw-WG) distribution which includes as special cases, several models such as the kumaraswamy weibull distribution, kumaraswamy exponential distribution, weibull geometric distribution, exponential geometric distribution, to name a few. This distribution was monotone and non-monotone hazard rate functions, which are useful in lifetime data analysis and reliability. We derive some basic properties of the Kw-WG distribution including non-central rth-moments, skewness, kurtosis, generating functions, mean deviations, mean residual life, entropy, order statistics and certain characterizations of our distribution. The method of maximum likelihood is used for estimating the model parameters and a simulation …

On Modified Burr Xii-Inverse Exponential Distribution: Prop¬Erties, Characterizations And Applications, Fiaz Ahmad Bhatti, Gholamhossein Hamedani, Haitham M. Yousof, Azeem Ali, Munir Ahmad

Mathematical and Statistical Science Faculty Research and Publications

In this paper, a flexible lifetime distribution with increasing, increasing and decreasing and modified bathtub hazard rate called Modified Burr XII-Inverse Exponential (MBXII-IE) is introduced. The density function of MBXII-IE has exponential, left-skewed, right-skewed and symmetrical shapes. Descriptive measures such as moments, moments of order statistics, incomplete moments, inequality measures, residual life function and reliability measures are theoretically established. The MBXII-IE distribution is characterized via different techniques. Parameters of MBXII-IE distribution are estimated using maximum likelihood method. The simulation study is performed to illustrate the performance of the Maximum Likelihood Estimates (MLEs) of the parameters of the MBXII-IE distribution. The …

Neutrosophic Optimization And Its Application On Structural Designs, Florentin Smarandache, Mridula Sarkar, Tapan Kumar Roy

Branch Mathematics and Statistics Faculty and Staff Publications

In the real world, uncertainty or vagueness is prevalent in engineering and management computations. Commonly, such uncertainties are included in the design process by introducing simplified hypothesis and safety or design factors. In case of structural and pavement design, several design methods are available to optimize objectives. But all such methods follow numerous monographs, tables and charts to find effective thickness of pavement design or optimum weight and deflection of structure calculating certain loop of algorithm in the cited iteration process. Most of the time, designers either only take help of a software or stop the cited procedure even after …

Special Issue: Algebraic Structures Of Neutrosophic Triplets, Neutrosophic Duplets, Or Neutrosophic Multisets, Vol. I, Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity (i.e., element, concept, idea, theory, logical proposition, etc.), is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor [1]. Based on neutrosophy, the neutrosophic triplets were founded; they have a similar form: (x, neut(x), anti(x), that satisfy some axioms, for each element x in a given set [2–4]. This book contains the successful invited submissions [5–56] to a special issue of Symmetry, reporting on state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic …

Neutrosophic Linear Programming Problems, Florentin Smarandache, Abdel-Nasser Hussian, Mai Mohamed, Mohamed Abdel-Baset

Branch Mathematics and Statistics Faculty and Staff Publications

Smarandache presented neutrosophic theory as a tool for handling undetermined information. Wang et al. introduced a single valued neutrosophic set that is a special neutrosophic sets and can be used expediently to deal with real-world problems, especially in decision support. In this paper, we propose linear programming problems based on neutrosophic environment. Neutrosophic sets are characterized by three independent parameters, namely truth-membership degree (T), indeterminacy-membership degree (I) and falsity-membership degree (F), which are more capable to handle imprecise parameters. We also transform the neutrosophic linear programming problem into a crisp programming model by using neutrosophic set parameters. To measure the …

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.

The Encyclopedia Of Neutrosophic Researchers - Vol. 2, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This is the second volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to my invitation. The introduction contains a short history of neutrosophics, together with links to the main papers and books. The authors who have published neutrosophic papers, books, or defended neutrosophic master theses or PhD dissertations and are not included in the two ENR volumes, are kindly invited to send their self-presentations or their CVs, a photo, and a list of neutrosophic publications to [email protected] and [email protected] to be part of a third volume.

Florentin Smarandache