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On The Higher Derivatives Of The Inverse Tangent Function, Mohamed Amine Boutiche, Mourad Rahmani
On The Higher Derivatives Of The Inverse Tangent Function, Mohamed Amine Boutiche, Mourad Rahmani
Turkish Journal of Mathematics
In this paper, we find explicit formulas for higher-order derivatives of the inverse tangent function. More precisely, we study polynomials that are induced from the higher-order derivatives of $\arctan(x)$. Successively, we give generating functions, recurrence relations, and some particular properties for these polynomials. Connections to Chebyshev, Fibonacci, Lucas, and matching polynomials are established.
A Taylor Operation Method For Solutions Of Generalized Pantograph Type Delay Differential Equations, Şuayi̇p Yüzbaşi, Nurbol Ismailov
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
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
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
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
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
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
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
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
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
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
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
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
$\Mathbb{Q}$-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
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
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
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
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.
Restriction Of A Quadratic Form Over A Finite Field To A Nondegenerate Affine Quadric Hypersurface, Edoardo Ballico
Restriction Of A Quadratic Form Over A Finite Field To A Nondegenerate Affine Quadric Hypersurface, Edoardo Ballico
Turkish Journal of Mathematics
Let $h, h_M: \mathbb {F} _q^n\to \mathbb {F} _q$ be quadratic forms with $h$ not degenerate. Fix $k\in \mathbb {F} _q$ and set $C_n(k,h)_{\mathbb {F} _q}:= \{h(x_1,\dots ,x_n)=k\}\subset \mathbb {F}_q^n$. We compute (in many cases) the image of $h_{M C_n(k,h)_{\mathbb {F} _q}}$. This question is related to a question on the numerical range of matrices over a finite field.
A Coanalytic Menger Group That Is Not $\Sigma$-Compact, Seçi̇l Tokgöz
A Coanalytic Menger Group That Is Not $\Sigma$-Compact, Seçi̇l Tokgöz
Turkish Journal of Mathematics
Under $V=L$ we construct coanalytic topological subgroups of reals, demonstrating that even for definable groups of reals, selection principles may differ.
The Equation ${Dd'+D'D=D^2}$ For Derivations On C$^*$-Algebras, Sayed Khalil Ekrami, Madjid Mirzavaziri
The Equation ${Dd'+D'D=D^2}$ For Derivations On C$^*$-Algebras, Sayed Khalil Ekrami, Madjid Mirzavaziri
Turkish Journal of Mathematics
Let $\mathcal{A}$ be an algebra. A linear mapping $d:\mathcal{A}\to\mathcal{A}$ is called a derivation if $d(ab)=d(a)b+ad(b)$ for each $a,b\in\mathcal{A}$. Given two derivations $d$ and $d'$ on a C$^*$-algebra $\mathcal{A}$, we prove that there exists a derivation $D$ on $\mathcal A$ such that $dd'+d'd=D^2$ if and only if $d$ and $ d' $ are linearly dependent.
Boolean Differential Operators, Luis Hernandez Encinas, Ángel Martín Del Rey
Boolean Differential Operators, Luis Hernandez Encinas, Ángel Martín Del Rey
Turkish Journal of Mathematics
In this work the notion of boolean differential operator is revisited and some new properties are stated.
Analysis Of Periodic And Asymptotically Periodic Solutions In Nonlinear Coupled Volterra Integro-Differential Systems, Youssef Raffoul
Analysis Of Periodic And Asymptotically Periodic Solutions In Nonlinear Coupled Volterra Integro-Differential Systems, Youssef Raffoul
Turkish Journal of Mathematics
In this note, we investigate the existence of periodic and asymptotically periodic solutions of a system of coupled nonlinear Volterra integro-differential equations with infinite delay. We will make use of Schauder fixed point theorem to prove our maps have fixed points.
Regularity And Projective Dimension Of The Edge Ideal Of A Generalized Theta Graph, Seyyede Masoome Seyyedi, Farhad Rahmati
Regularity And Projective Dimension Of The Edge Ideal Of A Generalized Theta Graph, Seyyede Masoome Seyyedi, Farhad Rahmati
Turkish Journal of Mathematics
Let $k\geq 3$ and $G=\theta_{n_1,\ldots, n_k}$ be a graph consisting of $k$ paths that have common endpoints. In this paper, we show that the projective dimension of $R/I(G)$ equals $bight I(G)$ or $ bight I(G)+1$. For some special cases, we explain $depth(R/I(G))$ in terms of invariants of graphs. Moreover, we prove the regularity of $R/I(G)$ equals $c_G$ or $c_G+1$, where $c_G$ is the maximum number of 3-disjoint edges in $G$.
On A Class Of Unitary Operators On The Bergman Space Of The Right Half Plane, Namita Das, Jitendra Kumar Behera
On A Class Of Unitary Operators On The Bergman Space Of The Right Half Plane, Namita Das, Jitendra Kumar Behera
Turkish Journal of Mathematics
In this paper, we introduce a class of unitary operators defined on the Bergman space $L_a^2(\mathbb{C}_+)$ of the right half plane $\mathbb{C}_+$ and study certain algebraic properties of these operators. Using these results, we then show that a bounded linear operator $S$ from $L_a^2(\mathbb{C}_+)$ into itself commutes with all the weighted composition operators $W_a, a \in \mathbb{D}$ if and only if $\widetilde{S}(w)=\langle Sb_{\overline{w}},b_{\overline{w}}\rangle, w \in \mathbb{C}_+ $ satisfies a certain averaging condition. Here for $a=c+id \in \mathbb{D}, f \in L_a^2(\mathbb{C}_+), W_af=(f \circ t_a) \frac{M^{\prime}}{M^{\prime} \circ t_a}, Ms=\frac{1-s}{1+s}, t_a(s)=\frac{-ids +(1-c)}{(1+c)s + id}$, and $b_{\overline{w}}(s)=\frac{1}{\sqrt{\pi}} \frac{1+w}{1+\overline{w}} \frac{2 \mbox {Re} w}{(s+w)^2}, w=M\overline {a}, …
The Cauchy-Kowalevski Theorem Applied For Counting Connections With A Prescribed Ricci Tensor, Barbara Opozda, Wlodzimierz M. Mikulski
The Cauchy-Kowalevski Theorem Applied For Counting Connections With A Prescribed Ricci Tensor, Barbara Opozda, Wlodzimierz M. Mikulski
Turkish Journal of Mathematics
How many linear connections are there with a prescribed Ricci tensor? The question is answered in the analytic case by using the Cauchy-Kowalevski theorem
Construction Of Some New Families Of Apostol-Type Numbers And Polynomials Via Dirichlet Character And $P$-Adic $Q$-Integrals, Yilmaz Şi̇mşek
Construction Of Some New Families Of Apostol-Type Numbers And Polynomials Via Dirichlet Character And $P$-Adic $Q$-Integrals, Yilmaz Şi̇mşek
Turkish Journal of Mathematics
In this paper, by applying the $p$-adic $q$-integrals to a family of continuous differentiable functions on the ring of $p$-adic integers, we construct new generating functions for generalized Apostol-type numbers and polynomials attached to the Dirichlet character of a finite abelian group. By using these generating functions with their functional equations, we derive various new identities and relations for these numbers and polynomials. These results are generalizations of known identities and relations including some well-known families of special numbers and polynomials such as the generalized Apostol-type Bernoulli, the Apostol-type Euler, the Frobenius-Euler numbers and polynomials, the Stirling numbers, and other …
An Inequality On The Hodge Number $H^{1,1}$ Of A Fibration And The Mordell-Weil Rank, Cheng Gong, Hao Sun
An Inequality On The Hodge Number $H^{1,1}$ Of A Fibration And The Mordell-Weil Rank, Cheng Gong, Hao Sun
Turkish Journal of Mathematics
In this paper, we establish some formulas on the Mordell-Weil rank and the Hodge number $h^{1,1}$ for a fibration.
An Efficient Technique To Solve Nonlinear Equations Usingmultiplicative Calculus, Muhammad Waseem, Muhammad Aslam Noor, Farooq Ahmed Shah, Khalida Inayat Noor
An Efficient Technique To Solve Nonlinear Equations Usingmultiplicative Calculus, Muhammad Waseem, Muhammad Aslam Noor, Farooq Ahmed Shah, Khalida Inayat Noor
Turkish Journal of Mathematics
In this paper, we develop an efficient technique in the framework of multiplicative calculus and suggest a new class of numerical methods for solving multiplicative nonlinear equations \textit{g}(\textit{x}% ) = 1. We also develop the convergence criteria of the proposed methods. We solve the population growth model and minimization problem, which demonstrate the implementation and efficiency of the new techniques. We also show that these techniques perform much better as compared to the similar ordinary methods for solving ordinary nonlinear equations \textit{f}(\textit{x}) = 0.
An Application Of $Q$-Sumudu Transform For Fractional $Q$-Kinetic Equation, Sunil Dutt Purohit, Faruk Uçar
An Application Of $Q$-Sumudu Transform For Fractional $Q$-Kinetic Equation, Sunil Dutt Purohit, Faruk Uçar
Turkish Journal of Mathematics
The aim of this paper is to give an alternative solution for the $q$-kinetic equation involving the Riemann--Liouville fractional $q$-integral operator. The solution is obtained in terms of the $q$-Mittag--Leffler functions using inverse $q$-Sumudu transform. As applications, some corollaries are presented to illustrate the main results.