The Class Of Demi Kb-Operators On Banach Lattices, 2023 TÜBİTAK

#### The Class Of Demi Kb-Operators On Banach Lattices, Hedi Benkhaled, Aref Jeribi

*Turkish Journal of Mathematics*

In this paper, we introduce and study the new concept of demi KB-operators. Let $E$ be a Banach lattice. An operator $T: E\longrightarrow E$ is said to be a demi KB-operator if, for every positive increasing sequence $\{x_{n}\}$ in the closed unit ball $\mathcal{B}_{E}$ of $E$ such that $\{x_{n}-Tx_{n}\}$ is norm convergent to $x\in E$, there is a norm convergent subsequence of $\{x_{n}\}$. If the latter sequence has a weakly convergent subsequence then $T$ is called a weak demi KB-operator. We also investigate the relationship of these classes of operators with classical notions of operators, such as b-weakly demicompact operators …

#### Inverse Nodal Problem For The Quadratic Pencil Of The Sturm$-$Liouville Equations With Parameter-Dependent Nonlocal Boundary Condition, Yaşar Çakmak, Baki̇ Keski̇n

*Turkish Journal of Mathematics*

In this paper, we consider the inverse nodal problem for a quadratic pencil of the Sturm$-$Liouville equations with parameter-dependent Bitsadze$-$Samarskii type nonlocal boundary condition and we give an algorithm for the reconstruction of the potential functions by obtaining the asymptotics of the nodal points.

Bipolar Soft Ideal Rough Set With Applications In Covid-19, 2023 TÜBİTAK

#### Bipolar Soft Ideal Rough Set With Applications In Covid-19, Heba I. Mustafa

*Turkish Journal of Mathematics*

Bipolar soft rough set represents an important mathematical model to deal with uncertainty. This theory represents a link between bipolar soft set and rough set theories. This study introduced the concept of topological bipolar soft set by combining a bipolar soft set with topologies. Also, the topological structure of bipolar soft rough set has been discussed by defining the bipolar soft rough topology. The main objective of this paper is to present some solutions to develop and modify the approach of the bipolar soft rough sets. Two kinds of bipolar soft ideal approximation operators which represent extensions of bipolar soft …

Forming Coupled Dispersionless Equations Of Families Of Bertrand Curves, 2023 TÜBİTAK

#### Forming Coupled Dispersionless Equations Of Families Of Bertrand Curves, Kemal Eren

*Turkish Journal of Mathematics*

In this study, we establish a link of the coupled dispersionless (CD) equations system with the motion of Bertrand curve pairs. Moreover, we find the Lax equations that provide the integrability of these equations. By taking an appropriate choice of variables we show the link of the short pulse (SP) equation with the motion of Bertrand curve pairs via the reciprocal (hodograph) transformation. Finally, we prove that the conserved quantity of the corresponding coupled dispersionless equations obtained from each of these curve pairs is constant.

Some Fractional Dirac Systems, 2023 TÜBİTAK

#### Some Fractional Dirac Systems, Yüksel Yalçinkaya

*Turkish Journal of Mathematics*

In this work, including $\alpha\epsilon(0,1)$; we examined the Dirac system in the frame which includes$\ \alpha$ order right and left Reimann-Liouville fractional integrals and derivatives with exponential kernels, and the Dirac system which includes $\alpha$ order right and left Caputo fractional integrals and derivatives with exponential kernels. Furthermore, we have given some definitions and properties for discrete exponential kernels and their associated fractional sums and fractional differences, and we have studied discrete fractional Dirac systems.

Jackson-Type Theorem In The Weak $L_{1}$-Space, 2023 TÜBİTAK

#### Jackson-Type Theorem In The Weak $L_{1}$-Space, Rashid Aliev, Eldost Ismayilov

*Turkish Journal of Mathematics*

The weak $L_{1}$-space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak $L_{1}$-space. The difficulty of working with the weak $L_{1}$-space is that the weak $L_{1}$-space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak $L_{1}$-space, studied its properties, found a criterion for convergence to zero of the modulus of …

On The Distribution Of Adjacent Zeros Of Solutions To First-Order Neutral Differential Equations, 2023 TÜBİTAK

#### On The Distribution Of Adjacent Zeros Of Solutions To First-Order Neutral Differential Equations, Emad R. Attia, Ohoud N. Al-Masarer, Irena Jadlovska

*Turkish Journal of Mathematics*

The purpose of this paper is to study the distribution of zeros of solutions to a first-order neutral differential equation of the form \begin{equation*} \left[x(t) + p(t) x(t-\tau)\right]' + q(t) x(t-\sigma) = 0, \quad t \geq t_0, \end{equation*} where $p\in C([t_0,\infty),[0,\infty))$, $q \in C([t_0,\infty),(0,\infty))$, $\tau,\sigma>0$, and $\sigma>\tau$. We obtain new upper bound estimates for the distance between consecutive zeros of solutions, which improve upon many of the previously known ones. The results are formulated so that they can be generalized without much effort to equations for which the distribution of zeros problem is related to the study of …

Geodesics And Isocline Distributions In Tangent Bundles Of Nonflat Lorentzian-Heisenberg Spaces, 2023 TÜBİTAK

#### Geodesics And Isocline Distributions In Tangent Bundles Of Nonflat Lorentzian-Heisenberg Spaces, Murat Altunbaş

*Turkish Journal of Mathematics*

Let $(H_{3},g_{1})$ and $(H_{3},g_{2})$ be the Lorentzian-Heisenberg spaces with nonflat metrics $g_{1}$ and $g_{2},\ $and $(TH_{3},g_{1}^{s}),\ (TH_{3},g_{2}^{s})$ be their tangent bundles with the Sasaki metric, respectively. In the present paper, we find nontotally geodesic distributions in tangent bundles by using lifts of contact forms from the base manifold $H_{3}.$We give examples for totally geodesic but not isocline distributions. We study the geodesics of tangent bundles by considering horizontal and natural lifts of geodesics of the base manifold $H_{3}$. We also investigate more general classes of geodesics which are not obtained from horizontal and natural lifts of geodesics.

On The Band Functions And Bloch Functions, 2023 TÜBİTAK

#### On The Band Functions And Bloch Functions, Oktay Veli̇ev

*Turkish Journal of Mathematics*

In this paper, we consider the continuity of the band functions and Bloch functions of the differential operators generated by the differential expressions with periodic matrix coefficients.

Separation, Connectedness, And Disconnectedness, 2023 TÜBİTAK

#### Separation, Connectedness, And Disconnectedness, Mehmet Baran

*Turkish Journal of Mathematics*

The aim of this paper is to introduce the notions of hereditarily disconnected and totally disconnected objects in a topological category and examine the relationship as well as interrelationships between them. Moreover, we characterize each of $T_{2}$, connected, hereditarily disconnected, and totally disconnected objects in some topological categories and compare our results with the ones in the category of topological spaces.

Hahn-Hamiltonian Systems, 2023 TÜBİTAK

#### Hahn-Hamiltonian Systems, Bi̇lender Paşaoğlu, Hüseyi̇n Tuna

*Turkish Journal of Mathematics*

In this paper, we study the basic theory of regular Hahn-Hamiltonian systems. In this context, we establish an existence and uniqueness result. We introduce the corresponding maximal and minimal operators for this system and some properties of these operators are investigated. Moreover, we give a criterion under which these operators are self-adjoint. Finally, an expansion theorem is proved.

Global Bifurcation Of Positive Solutions For A Class Of Superlinear First-Order Differential Systems, 2023 TÜBİTAK

#### Global Bifurcation Of Positive Solutions For A Class Of Superlinear First-Order Differential Systems, Lijuan Yang, Ruyun Ma

*Turkish Journal of Mathematics*

We are concerned with the first-order differential system of the form $$\left\{ \begin{array}{ll} u'(t)+a(t)u(t)=\lambda b(t) f(v(t-\tau(t))), &t\in\mathbb{R},\\ v'(t)+a(t)v(t)=\lambda b(t)g(u(t-\tau(t))), &t\in\mathbb{R},\\ \end{array} \right. $$ where $\lambda\in\mathbb{R}$~is a parameter. $a,b\in C(\mathbb{R},[0,+\infty))$ are two $\omega$-periodic functions such that $\int_0^\omega a(t)\text{d}t>0$,~$\int_0^\omega b(t)\text{d}t>0$,~$\tau\in C(\mathbb{R},\mathbb{R})$ is an $\omega$-periodic function. The nonlinearities~$f,g\in C(\mathbb{R},(0,+\infty))$~are two nondecreasing continuous functions and satisfy superlinear conditions at infinity.~By using the bifurcation theory,~we will show the existence of an unbounded component of positive solutions, which is bounded in positive $\lambda$-direction.

A Note On "Some Properties Of Second-Order Weak Subdifferentials" [Turkish Journal Of Mathematics (2021)45: 955-960], 2023 TÜBİTAK

#### A Note On "Some Properties Of Second-Order Weak Subdifferentials" [Turkish Journal Of Mathematics (2021)45: 955-960], Qilin Wang, Min Liu

*Turkish Journal of Mathematics*

In this note, we provide an example to illustrate that Proposition 2.4 in [Turkish Journal of Mathematics (2021)45: 955-960)] is incorrect, and give a modification of the proposition. Two examples are provided to illustrate the modified result. Meanwhile, we establish a convex function, and correct the proof of Theorem 2.3 in [Turkish Journal of Mathematics (2021)45: 955-960)] by the function.

Geometric Singularities And Regularity Of Solution Of The Stokes System In Nonsmooth Domains, 2023 TÜBİTAK

#### Geometric Singularities And Regularity Of Solution Of The Stokes System In Nonsmooth Domains, Yasir Nadeem Anjam

*Turkish Journal of Mathematics*

This paper deals with the geometrical singularities of the weak solution of the mixed boundary value problem governed by the stationary Stokes system in two-dimensional nonsmooth domains with corner points and points at which the type of boundary conditions changes. The presence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. Moreover, the asymptotic singular representations for the solution which inherently depend on the zeros of certain transcendental functions …

Biharmonic Pnmcv Submanifolds In Euclidean 5-Space, 2023 TÜBİTAK

#### Biharmonic Pnmcv Submanifolds In Euclidean 5-Space, Rüya Şen, Nuretti̇n Cenk Turgay

*Turkish Journal of Mathematics*

In this article, we study 3-dimensional biconservative and biharmonic submanifolds of $\mathbb{E}^5$ with parallel normalized mean curvature vector (PNMCV). First, we prove that the principal curvartures and principal directions of biconservative PNMCV isometric immersions into $\mathbb{E}^5$ can be determined intrinsically. Then, we complete the proof of Chen's biharmonic conjecture for PNMCV submanifolds of $\mathbb{E}^5$.

The Generalized Lyapunov Function As Ao’S Potential Function: Existence In Dimensions 1 And 2, 2023 Missouri University of Science and Technology

#### The Generalized Lyapunov Function As Ao’S Potential Function: Existence In Dimensions 1 And 2, Haoyu Wang, Wenqing Hu, Xiaoliang Gan, Ping Ao

*Mathematics and Statistics Faculty Research & Creative Works*

By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collabora-tors recently. We show that this potential function agrees with the generalized Lyapunov function of the deterministic part of the stochastic dynamical sys-tem. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also …

Complete Study Of An Original Power-Exponential Transformation Approach For Generalizing Probability Distributions, 2023 King Saud University

#### Complete Study Of An Original Power-Exponential Transformation Approach For Generalizing Probability Distributions, Mustafa S. Shama, Farid El Ktaibi, Jamal N. Al Abbasi, Christophe Chesneau, Ahmed Z. Afify

*All Works*

In this paper, we propose a flexible and general family of distributions based on an original power-exponential transformation approach. We call it the modified generalized-G (MGG) family. The elegance and significance of this family lie in the ability to modify the standard distributions by changing their functional forms without adding new parameters, by compounding two distributions, or by adding one or two shape parameters. The aim of this modification is to provide flexible shapes for the corresponding probability functions. In particular, the distributions of the MGG family can possess increasing, constant, decreasing, “unimodal”, or “bathtub-shaped“ hazard rate functions, which are …

Higher Spanier Groups, 2023 West Chester University

#### Higher Spanier Groups, Johnny Aceti

*West Chester University Master’s Theses*

When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …

On The Measure Of Noncompactness In $L_P(\Mathbb{R}^+)$ And Applications To A Product Of $N$-Integral Equations, 2023 TÜBİTAK

#### On The Measure Of Noncompactness In $L_P(\Mathbb{R}^+)$ And Applications To A Product Of $N$-Integral Equations, Mohamed M. A. Metwali, Vishnu Narayan Mishra

*Turkish Journal of Mathematics*

In this article, we prove a new compactness criterion in the Lebesgue spaces $L_p({\mathbb{R}}^+), 1 \leq p < \infty$ and use such criteria to construct a measure of noncompactness in the mentioned spaces. The conjunction of that measure with the Hausdroff measure of noncompactness is proved on sets that are compact in finite measure. We apply such measure with a modified version of Darbo fixed point theorem in proving the existence of monotonic integrable solutions for a product of $n$-Hammerstein integral equations $n\geq 2$.

On The Uniqueness Of Continuation Of A Partially Defined Metric, 2023 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine

#### On The Uniqueness Of Continuation Of A Partially Defined Metric, Evgeniy Petrov

*Theory and Applications of Graphs*

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\colon E\to\mathbb R^+$ on the graph $G$ has a unique continuation to a metric $d\colon V\times V\to\mathbb R^+$ is found.