On The Relation Between Oscillation Of Solutions Of Differential Equations And Corresponding Equations On Time Scales, 2023 TÜBİTAK

#### On The Relation Between Oscillation Of Solutions Of Differential Equations And Corresponding Equations On Time Scales, Olexandr Stanzhytskyi, Roza Uteshova, Victoriia Tsan, Zoia Khaletska

*Turkish Journal of Mathematics*

This paper studies oscillatory properties of solutions of a dynamic equation on the set of time scales $\mathbf{T}_\lambda$ provided that the graininess function $\mu_\lambda$ approaches zero as $\lambda\to 0$. We derived the conditions under which oscillation of solutions of differential equations implies that of solutions of the corresponding equations defined on time scales with the same initial data, and vice versa.

A Note On $Ss$-Supplement Submodules, 2023 TÜBİTAK

#### A Note On $Ss$-Supplement Submodules, Emi̇ne Önal Kir

*Turkish Journal of Mathematics*

In this paper, we describe $ss$-supplement submodules in terms of a special class of endomorphisms. Let $R$ be a ring with semisimple radical and $P$ be a projective $R-$module. We show that there is a bijection between ss-supplement submodules of $P$ and ss-supplement submodules of $End_{R}(P)$. Moreover, we define radical-s-projective modules as a generalization of projective modules. We prove that every $ss$-supplement submodule of a projective $R-$module is radical-s-projective over the ring $R$ with semisimple radical. We show that over $SSI$-ring $R$, every radical-s-projective $R-$module is projective. We provide that over a ring $R$ with semisimple radical, every $ss$-supplement submodule …

Qualitative Study Of A Second Order Difference Equation, 2023 TÜBİTAK

#### Qualitative Study Of A Second Order Difference Equation, Messaoud Berkal, Juan Francisco Navarro

*Turkish Journal of Mathematics*

In this paper, we study a second order rational difference equation. We analyze the stability of the unique positive equilibrium of the equation and prove the existence of a Neimark-Sacker bifurcation, validating our theoretical analysis via a numerical exploration of the system.

Notes On Totally Geodesic Foliations Of A Complete Semi-Riemannian Manifold, 2023 TÜBİTAK

#### Notes On Totally Geodesic Foliations Of A Complete Semi-Riemannian Manifold, An Sook Shin, Hyelim Han, Hobum Kim

*Turkish Journal of Mathematics*

In this paper, we prove that the orthogonal complement $\mathcal{F}^{\perp}$ of a totally geodesic foliation $\mathcal{F}$ on a complete semi-Riemannian manifold $(M,g)$ satisfying a certain inequality between mixed sectional curvatures and the integrability tensor of $\mathcal{F}^{\perp}$ is totally geodesic. We also obtain conditions for the existence of totally geodesic foliations on a complete semi-Riemannian manifold $(M,g)$ with bundle-like metric $g$.

Numerical Solution For Benjamin-Bona-Mahony-Burgers Equation With Strang Time-Splitting Technique, 2023 TÜBİTAK

#### Numerical Solution For Benjamin-Bona-Mahony-Burgers Equation With Strang Time-Splitting Technique, Meli̇ke Karta

*Turkish Journal of Mathematics*

In the present manuscript, the Benjamin-Bona-Mahony-Burgers (BBMB) equation will be handled numerically by Strang time-splitting technique. While applying this technique, collocation method based on quintic B-spline basis functions is applied. In line with our purpose, after splitting the BBM-Burgers equation given with appropriate initial boundary conditions into two subequations containing the derivative in terms of time, the quintic B-spline based collocation finite element method (FEM) for spatial discretization and the suitable finite difference approaches for time discretization is applied to each subequation and hereby two different systems of algebraic equations are obtained. Four test problems are utilized to test the …

Bi-Periodic Incomplete Horadam Numbers, 2023 TÜBİTAK

#### Bi-Periodic Incomplete Horadam Numbers, Eli̇f Tan, Mehmet Dağli, Amine Belkhir

*Turkish Journal of Mathematics*

In this paper, we introduce bi-periodic~incomplete~Horadam numbers as a natural generalization of incomplete Horadam numbers. We study their basic properties and provide recurrence relations. In particular, we derive the generating function of these numbers.

Some Remarks On Parameterized Inequalities Involving Conformable Fractional Operators, 2023 TÜBİTAK

#### Some Remarks On Parameterized Inequalities Involving Conformable Fractional Operators, Ci̇han Ünal, Fati̇h Hezenci̇, Hüseyi̇n Budak

*Turkish Journal of Mathematics*

In this paper, we prove an identity for differentiable convex functions related to conformable fractional integrals. Moreover, some parameterized inequalities are established by using conformable fractional integrals. To be more precise, parameterized inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, previous and new results are presented by using special cases of the obtained theorems.

#### On Conditions Of Regular Solvability For Two Classes Of Third-Order Operator-Differential Equations In A Fourth-Order Sobolev-Type Space, Araz R. Aliev, Nazila L. Muradova

*Turkish Journal of Mathematics*

In this paper, we study two classes of operator-differential equations of the third order with a multiple characteristic, considered on the whole axis. We introduce the concept of a smooth regular solution of order 1 and obtain sufficient conditions for the "smoothly" regular solvability of these equations.

Curves And Stick Figures Not Contained In A Hypersurface Of A Given Degree, 2023 TÜBİTAK

#### Curves And Stick Figures Not Contained In A Hypersurface Of A Given Degree, Edoardo Ballico

*Turkish Journal of Mathematics*

A stick figure $X\subset \mathbb{P}^r$ is a nodal curve whose irreducible components are lines. For fixed integers $r\ge 3$, $s\ge 2$ and $d$ we study the maximal arithmetic genus of a connected stick figure (or any reduced and connected curve) $X\subset \mathbb{P}^r$ such that $\deg (X)=d$ and $h^0(\mathcal{I}_X(s-1))=0$. We consider Halphen's problem of obtaining all arithmetic genera below the maximal one.

Contiguity Distance Between Simplicial Maps, 2023 TÜBİTAK

#### Contiguity Distance Between Simplicial Maps, Ayşe Borat, Mehmetci̇k Pamuk, Tane Vergi̇li̇

*Turkish Journal of Mathematics*

For simplicial complexes and simplicial maps, the notion of being in the same contiguity class is defined as the discrete version of homotopy. In this paper, we study the contiguity distance, $SD$, between two simplicial maps adapted from the homotopic distance. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion. Moreover, we present the behaviour of $SD$ under the barycentric subdivision, and its relation with strong collapsibility of a simplicial complex.

Novel Fano Type Lower Bounds On The Minimum Error Probability Of List $M$-Ary Hypothesis Testing, 2023 TÜBİTAK

#### Novel Fano Type Lower Bounds On The Minimum Error Probability Of List $M$-Ary Hypothesis Testing, Berkan Dülek

*Turkish Journal of Mathematics*

he problem of list $M$-ary hypothesis testing with fixed list size $L< M$ is considered. Based on some random observation, the test outputs a list of $L$ candidates out of $M$ possible hypotheses. The probability of list error is defined as the probability of the event that the list output by the test does not contain the true hypothesis that has generated the observation. An identity is derived that relates the minimum average probability of error of the optimal list hypothesis test to the minimum average probability of error of an optimal maximum a posteriori probability decision rule. The latter decides among an alternative set of hypotheses corresponding to all possible $L$-component mixtures of the distributions that characterize the observation under the original $M$ candidate hypotheses. As an application, the proposed identity is employed to obtain novel Fano type lower bounds on the minimum error probability of list $M$-ary hypothesis testing.

On The Hilbert Series Of The Tangent Cones For Some 4-Generated Pseudosymmetric Monomial Curves, 2023 TÜBİTAK

#### On The Hilbert Series Of The Tangent Cones For Some 4-Generated Pseudosymmetric Monomial Curves, Ni̇l Şahi̇n

*Turkish Journal of Mathematics*

In this article, we study Hilbert series of non-Cohen-Maculay tangent cones for some 4-generated pseudosymmetric monomial curves. We show that the Hilbert function is nondecreasing by explicitly computing it. We also compute standard bases of these toric ideals.

Numerical Solutions Of Differential Equations Having Cubic Nonlinearity Using Boole Collocation Method, 2023 TÜBİTAK

#### Numerical Solutions Of Differential Equations Having Cubic Nonlinearity Using Boole Collocation Method, Kübra Erdem Bi̇çer, Hale Gül Dağ

*Turkish Journal of Mathematics*

The aim of the study is to develop a numerical method for the solution of cubic nonlinear differential equations in which the numerical solution is based on Boole polynomials. That solution is in the form of the truncated series and gives approximate solution for nonlinear equations of cubic type. In this method, firstly, the matrix form of the serial solution is set and the nonlinear differential equation is converted into a matrix equation system. By adding the effect of both the conditions of the problem and the collocation points to this system of equations, we obtain the new system of …

Clairaut Riemannian Maps, 2023 TÜBİTAK

#### Clairaut Riemannian Maps, Kiran Meena, Akhilesh Yadav

*Turkish Journal of Mathematics*

In this paper, first we define Clairaut Riemannian map between Riemannian manifolds by using a geodesic curve on the base space and find necessary and sufficient conditions for a Riemannian map to be Clairaut with a nontrivial example. We also obtain necessary and sufficient condition for a Clairaut Riemannian map to be harmonic. Thereafter, we study Clairaut Riemannian map from Riemannian manifold to Ricci soliton with a nontrivial example. We obtain scalar curvatures of $rangeF_\ast$ and $(rangeF_\ast)^\bot$ by using Ricci soliton. Further, we obtain necessary conditions for the leaves of $rangeF_\ast$ to be almost Ricci soliton and Einstein. We also …

Lyapunov-Type Inequalities For $(\Mathtt{N},\Mathtt{P})$-Type Nonlinear Fractional Boundary Value Problems, 2023 TÜBİTAK

#### Lyapunov-Type Inequalities For $(\Mathtt{N},\Mathtt{P})$-Type Nonlinear Fractional Boundary Value Problems, Paul W. Eloe, Muralee Bala Krushna Boddu

*Turkish Journal of Mathematics*

This paper establishes Lyapunov-type inequalities for a family of two-point $(\mathtt{n},\mathtt{p})$-type boundary value problems for Riemann-Liouville fractional differential equations. To demonstrate how the findings can be applied, we provide a few examples, one of which is a fractional differential equation with delay.

#### Generating Functions For Reciprocal Catalan-Type Sums: Approach To Linear Differentiation Equation And ($P$-Adic) Integral Equations, Damla Gün, Yilmaz Şi̇mşek

*Turkish Journal of Mathematics*

This article is inspired by the reciprocal Catalan sums associated with problem 11765, proposed by David Beckwith and Sag Harbor. For this reason, partial derivative equations, the first-order linear differentiation equation and integral representations for series and generating functions for reciprocal Catalan-type sums containing the Catalan-type numbers are constructed. Some special values of these series and generating functions, which are given solutions of problem 11765, are found. Partial derivative equations of the generating function for the Catalan-type numbers are given. By using these equations, recurrence relations and derivative formulas involving these numbers are found. Finally, applying the $p$-adic Volkenborn integral …

Locally Product-Like Statistical Submersions, 2023 TÜBİTAK

#### Locally Product-Like Statistical Submersions, Kazuhi̇ko Takano, Esra Erkan, Mehmet Gülbahar

*Turkish Journal of Mathematics*

In this paper, the main identities on locally product-like statistical submersions are obtained with the aid of statistical structures and their Riemannian curvature tensors. Some examples of locally product-like statistical submersions are presented. Some results on $F$-invariant, $F^{\ast}$-invariant and antiinvariant locally product-like statistical submersions are given.

On Generalized Darboux Frame Of A Spacelike Curve Lying On A Lightlike Surface In Minkowski Space $\Mathbb{E}^{3}_{1}$, 2023 TÜBİTAK

#### On Generalized Darboux Frame Of A Spacelike Curve Lying On A Lightlike Surface In Minkowski Space $\Mathbb{E}^{3}_{1}$, Jelena Djordjevic, Emilija Nesovic, Ufuk Öztürk

*Turkish Journal of Mathematics*

In this paper we introduce generalized Darboux frame of a spacelike curve $\alpha$ lying on a lightlike surface in Minkowski space $\mathbb{E}_{1}^{3}$. We prove that $\alpha$ has two such frames and obtain generalized Darboux frame's equations. We find the relations between the curvature functions $k_g$, $k_n$, $\tau_g$ of $\alpha$ with respect to its Darboux frame and the curvature functions $\tilde{k}_{g}$, $\tilde{k}_{n}$, $\tilde{\tau}_{g}$ with respect to generalized Darboux frames. We show that such frames exist along a spacelike straight line lying on a ruled surface which is not entirely lightlike, but contains some lightlike points. We define lightlike ruled surfaces on …

Duality Approach To The Regularity Problems For The Navier-Stokes Equations, 2023 TÜBİTAK

#### Duality Approach To The Regularity Problems For The Navier-Stokes Equations, Grigory Seregin

*Turkish Journal of Mathematics*

In this note, we describe a way to study local regularity of a weak solution to the Navier-Stokes equations, satisfying the simplest scale-invariant restriction, with the help of zooming and duality approach to the corresponding mild bounded ancient solution.

Global Regularity For The 3d Axisymmetric Incompressible Hall-Mhd System With Partial Dissipation And Diffusion, 2023 TÜBİTAK

#### Global Regularity For The 3d Axisymmetric Incompressible Hall-Mhd System With Partial Dissipation And Diffusion, Meilin Jin, Quansen Jiu, Huan Yu

*Turkish Journal of Mathematics*

In this paper, we study the Cauchy problem for the 3D incompressible axisymmetric Hall-MHD system with horizontal velocity dissipation and vertical magnetic diffusion.We obtain a unique global smooth solution of which in the cylindrical coordinate system the swirl velocity fields, the radial and the vertical components of the magnetic fields are trivial. This type of solution has been studied for the MHD system in [17][16] and [15] and for the Hall-MHD system with total dissipation and diffusion in [11]. Some new and fine estimates are obtained in this paper to overcome the difficulties raised from the Hall term and the …