Physical Sciences and Mathematics | Open Access Articles

Logarithmic Dimension And Bases In Whitney Spaces, Alexander Goncharov, Yasemi̇n Şengül Tezel

Turkish Journal of Mathematics

We give a formula for the logarithmic dimension of the generalized Cantor-type set $K$. In the case when the logarithmic dimension of $K$ is smaller than $1$, we construct a Faber basis in the space of Whitney functions $\mathcal{E}(K)$.

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On Congruences Related To Trinomial Coefficients And Harmonic Numbers, Neşe Ömür, Si̇bel Koparal, Laid Elkhiri

Turkish Journal of Mathematics

In this paper, we establish some congruences involving the trinomial coefficients and harmonic numbers. For example, for any prime $p>3,$ \begin{equation*} \sum\limits_{k=0}^{p-1}\left( -1\right) ^{k}\binom{p-1}{k}_{2}H_{k}\equiv 0 \ \pmod {p}. \end{equation*}

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Relative Conics And Their Brianchon Points, Magdalena Lampa-Baczynska, Daniel Wojcik

Turkish Journal of Mathematics

The purpose of this paper is to study some additional relations between lines and points in the configuration of six lines tangent to the common conic. One of the most famous results concerning with this configuration is Brianchon theorem. It says that three diagonals of a hexagon circumscribing around conic are concurrent. They meet in the so called Brianchon point. In fact, by relabeling the vertices of hexagon, we obtain $60$ distinct Brianchon points. We prove, among others, that, in the set of all intersection points of six tangents to the same conic, there exist exactly $10$ sextuples of points …

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A Refinement Of The Bergström Inequality, Gülteki̇n Tinaztepe, İlknur Yeşi̇lce, Gabi̇l Adi̇lov

Turkish Journal of Mathematics

In this paper, the Bergstrom inequality is studied, and a refinement of this inequality is obtained by performing the optimality conditions based on abstract concavity. Some numerical experiments are given to illustrate the efficacy of the refinement.

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Ranks Of Nilpotent Subsemigroups Of Order-Preserving And Decreasing Transformation Semigroups, Emrah Korkmaz, Hayrullah Ayik

Turkish Journal of Mathematics

Let $\mathcal{C}_{n}$ be the semigroup of all order-preserving and decreasing transformations on $X=\{1,\ldots ,n\}$ under its natural order, and let $N(\mathcal{C}_{n})$ be the subsemigroup of all nilpotent elements of $\mathcal{C}_{n}$. For $1\leq r \leq n-1$, let \begin{eqnarray*} N(\mathcal{C}_{n,r})&=&\{ \alpha\in N(\mathcal{C}_{n}) : \lvert im(\alpha)\rvert \leq r\} ,\\ N_{r}(\mathcal{C}_{n})&=&\{\alpha\in N\mathcal({C}_{n}):\alpha\mbox{ is an } m\mbox{-potent for any } 1\leq m\leq r\} . \end{eqnarray*} In this paper we find the cardinality and the rank of the subsemigroup $N(\mathcal{C}_{n,r})$ of $\mathcal{C}_{n}$. Moreover, we show that the set $N_{r}(\mathcal{C}_{n})$ is a subsemigroup of $N(\mathcal{C}_{n})$ and then, we find a lower bound for the rank of $N_{r}(\mathcal{C}_{n})$.

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On The Spectra Of Generalized Fibonomial And Jacobsthal-Binomial Graphs, Hati̇ce Topcu, Nurten Yücel

Turkish Journal of Mathematics

In this work, we first give a more general form of the binomial, Fibonomial, and balance-binomial graphs that is called generalized Fibonomial graph. We also argue the spectra of generalized Fibonomial graph. Next, we introduce a new type of graph on Jacobsthal numbers that is called Jacobsthal-binomial graph and denoted by $JB_{n}$. We obtain the adjacency, Laplacian and signless Laplacian characteristic polynomials of $JB_{n}$, respectively. We lastly give inequalities for the adjacency, Laplacian and signless Laplacian energies of $JB_{n}$.

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Decompositions Of Complete Symmetric Directed Graphs Into The Oriented Heptagons, Uğur Odabaşi

Turkish Journal of Mathematics

The complete symmetric directed graph of order $v$, denoted by $K_{v}$, is the directed graph on $v$~vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and~$y$. For a given directed graph $D$, the set of all $v$ for which $K_{v}$ admits a $D$-decomposition is called the spectrum of~$D$-decomposition. There are 10 nonisomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.

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Conformal Bi-Slant Submersions, Sezi̇n Aykurt Sepet

Turkish Journal of Mathematics

In this paper, we study conformal bi-slant submersions from almost Hermitian manifolds onto Riemannian manifolds as a generalized of conformal anti-invariant, conformal semi-invariant, conformal semi-slant, conformal slant, and conformal hemi-slant submersions. We investigate the integrability of distributions and obtain necessary and sufficient conditions for the maps to have totally geodesic fibers. Also, we consider some decomposition theorems for the new submersion and study the total geodesicity of such maps. Finally, we find curvature relations between the base space and the total space.

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Global Attractivity Of Delay Difference Equations In Banach Spaces Via Fixed-Point Theory, Abdullah Kalkan

Turkish Journal of Mathematics

We formulate initial value problems for delay difference equations in Banach spaces as fixed-point problems in sequence spaces. By choosing appropriate sequence spaces various types of attractivity can be described. This allows us to establish global attractivity by means of fixed-point results. Finally, we provide an application to delay integrodifference equations in the space of continuous functions over a compact domain.

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A Regularized Trace Formula For "Weighted" Sturm-Liouville Equation With Point Delta-Interaction, Manaf Manafli

Turkish Journal of Mathematics

In this study, we obtain a formula for the regularized trace formula for "weighted" Sturm--Liouville equation with point $\delta $- interaction. At the begining, for the correct determination of solutions of analyzed equation at the point of discontinuty, the matching conditions are required. As a result, an equation is derived for the eigenvalues of the differential operator given in this study.

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Some Properties Of The Semigroup $Pg_Y(X)$: Green's Relations, Ideals, Isomorphism Theorems And Ranks, Worachead Sommanee

Turkish Journal of Mathematics

Let $T(X)$ be the full transformation semigroup on the set $X$. For a fixed nonempty subset $Y$ of $X$, let \begin{equation*} PG_Y(X) = \{\alpha\in T(X) : \alpha _Y\in G(Y)\} \end{equation*} where $G(Y)$ is the permutation group on $Y$. It is known that $PG_Y(X)$ is a regular subsemigroup of $T(X)$. In this paper, we give a simpler description of Green's relations and characterize the ideals of $PG_Y(X)$. Moreover, we prove some isomorphism theorems for $PG_Y(X)$. For finite sets, we investigate the cardinalities of $PG_Y(X)$ and of its subsets of idempotents, and we also calculate their ranks.

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New Results On Derivatives Of The Shape Operator Of A Real Hypersurface In A Complex Projective Space, Juan De Dios Perez, David Pérez-López

Turkish Journal of Mathematics

We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull real number $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ we can define a tensor field of type (1,2) on $M$, $L^{(k)}_F$, related to both connections. We study symmetry and skewsymmetry of the tensor $A^{(k)}_F$ associated to the shape operator$A$ of $M$.

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General Characteristics Of A Fractal Sturm-Liouville Problem, Fatma Ayça Çeti̇nkaya, Alireza Khalili Golmankaneh

Turkish Journal of Mathematics

In this paper, we consider a regular fractal Sturm-Liouville boundary value problem. We prove the self-adjointness of the differential operator which is generated by the $F^\alpha$-derivative introduced in [32]. We obtained the $F^\alpha$-analogue of Liouville's theorem, and we show some properties of eigenvalues and eigenfunctions. We present examples to demonstrate the efficiency and applicability of the obtained results. The findings of this paper can be regarded as a contribution to an emerging field.

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Half Inverse Problems For The Impulsive Quadratic Pencil With The Discontinouty Coefficient, Rauf Ami̇rov, Sevi̇m Durak

Turkish Journal of Mathematics

In this paper, we study the inverse spectral problem for the quadratic differential pencils with discontinuity coefficient on $\left[ 0,\pi\right] $ with separable boundary conditions and the impulsive conditions at the point $x=\dfrac{\pi}{2}$. We prove that two potential functions on the interval $\left[ 0,\pi\right] $, and the parameters in the boundary and impulsive conditions can be determined from a sequence of eigenvalues for two cases: (i) The potentials are given on $\left( 0,\dfrac{\pi}{4}\left( 1+\alpha\right) \right) ,$ (ii) The potentials are given on $\left( \dfrac{\pi}{4}\left( 1+\alpha\right) ,\pi\right) $, where $0

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On Sense Of Yamakawa Family Of Meromorphic Bi-Univalent And Bi-Subordinate Functions, Fethi̇ye Müge Sakar

Turkish Journal of Mathematics

This study offers three different univalent function families of bi-meromorphic and bi-subordinate functions defined on $\Delta=\{z:z\in\mathbb{C}, 1

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Normalized Null Hypersurfaces In Non-Flat Lorentzian Space Forms Satisfying $L_R X =\Mathcal U X +B$, Hans Fotsing Tetsing

Turkish Journal of Mathematics

In the present work, we classify normalized null hypersurfaces $x:(M,g,N)\to Q^{n+2}_1(c)$ immersed into one of the two real standard nonflat Lorentzian space-forms and satisfying the equation $L_r x=\mathcal U x+b$ for some field of screen constant matrices $\mathcal U$ and some field of screen constant vectors $b\in\mathbb{R}^{n+2}$, where $L_r$ is the linearized operator of the $(r+1)-$mean curvature of the normalized null hypersurface for $r=0,...,n$. We show that if the immersion $x$ is a solution of the equation $L_r x=\mathcal U x+b$ for $1\leq r\leq n$ and the normalization $N$ is quasi-conformal, then $M$ is either an $(r+1)-$maximal null hypersurface, or …

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A Contiguous Extension Of Dixon's Theorem For A Terminating ${}_4f_3(1)$ Series With Applications, Mohammad Idris Qureshi, Richard Bruce Paris, Shakir Hussain Malik

Turkish Journal of Mathematics

We derive a summation formula for the terminating hypergeometric series \[{}_4F_3\left[\!\!\begin{array}{c}-m,a,b,1+c\\1+a+m,1+a-b,c\end{array}\!\!;1\right],\] where $m$ denotes a nonnegative integer. Using this summation formula, we establish a reduction formula for the Srivastava-Daoust double hypergeometric function with arguments $z$ and $-z$. Special cases of this reduction formula lead to several reduction formulas for the hypergeometric functions ${}_{p+1}F_p$ with quadratic arguments when $p=2,3$ and 4 by employing series rearrangement techniques. A general double series identity involving a bounded sequence of arbitrary complex numbers is also given.

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2-Absorbing $\Phi$-$\Delta$-Primary Ideals, Sanem Yavuz, Serkan Onar, Bayram Ali̇ Ersoy, Ünsal Teki̇r, Suat Koç

Turkish Journal of Mathematics

This paper aims to introduce 2-absorbing $\phi$-$\delta$-primary ideals over commutative rings which unify the concepts of all generalizations of 2-absorbing and 2-absorbing primary ideals. Let $A $be a commutative ring with a nonzero identity and $\mathcal{I(A)}$ be the set of all ideals of $A$. Suppose that $\delta:\mathcal{I(A)}\rightarrow\mathcal{I(A)}$ is an expansion function and $\phi:\mathcal{I(A)}\rightarrow\mathcal{I(A)}% \cup\left\{ \emptyset\right\} $ is a reduction function. A proper ideal $Q\ $of $A\ $is said to be a 2-absorbing $\phi$-$\delta$-primary if whenever $abc\in Q-\phi(Q)$,\ where $a,b,c\in R,\ $then either $ab\in Q$ or $ac\in\delta(Q)$ or $bc\in\delta(Q). $Various examples, properties, and characterizations of this new class of ideals are …

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Weak C-Ideals Of A Lie Algebra, Zeki̇ye Çi̇loğlu Şahi̇n, David Anthony Towers

Turkish Journal of Mathematics

A subalgebra $B$ of a Lie algebra $L$ is called a weak c-ideal of $L$ if there is a subideal $C$ of $L$ such that $L=B+C$ and $B\cap C\leq B_{L} $ where $B_{L}$ is the largest ideal of $L$ contained in $B.$ This is analogous to the concept of weakly c-normal subgroups, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also note that one-dimensional weak c-ideals are c-ideals.

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Bilinear Multipliers Of Small Lebesgue Spaces, Öznur Kulak, Ahmet Turan Gürkanli

Turkish Journal of Mathematics

Let $G$ be a compact abelian metric group with Haar measure $\lambda $ and $% \hat{G}$ its dual with Haar measure $\mu $. Assume that$~1 < p_{i}

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Some Qualitative Results For Functional Delay Dynamic Equations On Time Scales, Hali̇s Can Koyuncuoğlu

Turkish Journal of Mathematics

Let $\mathbb{T}$ be a nonempty, closed, and arbitrary set of real numbers, namely a time scale, and consider the following delay dynamical equation% \[ x^{\Delta}\left( t\right) =a\left( t\right) x\left( t\right) +f\left( t,x\left( \mathbb{\vartheta}\left( t\right) \right) \right) ,\text{ }t\in\mathbb{T}, \] where $\mathbb{\vartheta}$ stands for the abstract delay function. The main goal of this study is three-fold: obtaining the existence of an equi-bounded solution, proving the asymptotic stability of the zero solution, and showing the existence of a periodic solution based on new periodicity concept on time scales for the given delayed equation under certain conditions. In our analysis, we propose an …

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Some Applications Of Fractional Calculus For Analytic Functions, Nesli̇han Uyanik, Shi̇geyoshi̇ Owa

Turkish Journal of Mathematics

For analytic functions $f\left( z\right) $ in the class $A_{n},$ fractional calculus (fractional integrals and fractional derivatives) $D_{z}^{\lambda }f\left( z\right) $ of order $\lambda $ are introduced. Applying $% D_{z}^{\lambda }f\left( z\right) $ for $f\left( z\right) \in A_{n},$ we introduce the interesting subclass $A_{n}\left( \alpha _{m},\beta ,\rho ,\lambda \right) $ of $A_{n}.$ The object of this paper is to discuss some properties of $f\left( z\right) $ concerning $D_{z}^{\lambda }f\left( z\right) .$

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Optimization Of Mayer Functional In Problems With Discrete And Differential Inclusions And Viability Constraints, Gülseren Çi̇çek, Eli̇mhan N. Mahmudov

Turkish Journal of Mathematics

This paper derives the optimality conditions for a Mayer problem with discrete and differential inclusions with viable constraints. Applying necessary and sufficient conditions of problems with geometric constraints, we prove optimality conditions for second order discrete inclusions. Using locally adjoint mapping, we derive Euler-Lagrange form conditions and transversality conditions for the optimality of the discrete approximation problem. Passing to the limit, we establish sufficient conditions to the optimal problem with viable constraints. Conditions ensuring the existence of solutions to the viability problems for differential inclusions of second order have been studied in recent years. However, optimization problems of second-order differential …

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Sparse Polynomial Interpolation With Bernstein Polynomials, Erdal İmamoğlu

Turkish Journal of Mathematics

We present an algorithm for interpolating an unknown univariate polynomial $f$ that has a $t$ sparse representation ($t

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Reduced Limit Approach To Semilinear Pdes Involving The Fractional Laplacian With Measure Data, Ratan Kumar Giri, Debajyoti Choudhuri

Turkish Journal of Mathematics

We study the following partial differential equation (PDE) \begin{align} \begin{split} (-\Delta)^s u + g(x,u) & = \mu\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\,\mbox{in}\,\,\mathbb{R}^N\setminus\Omega,\label{eqn_abs} \end{split} \end{align} where $(-\Delta)^s$ is the fractional Laplacian operator, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $\partial\Omega$ being the boundary of $\Omega$, $g(.,.)$ is a nonlinear function defined over $\Omega\times\mathbb{R}$. Let $(\mu_n)_n$ be a sequence of measure in $\Omega$. Assume that there exists a solution $u_n$ with data $\mu_n$, i.e. $u_n$ satisfies the equation (0.1) with $\mu=\mu_n$. We further assume that the sequence of measures weakly converges to $\mu$, while $(u_n)_n$ converges to $u$ in $L^1(\Omega)$. In general, …

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Indecomposable Vector Bundles Via Monads On A Cartesian Product Of Projective Spaces, Damian Maingi

Turkish Journal of Mathematics

The existence of monads on products of projective spaces $P^{a_1}\times\cdots\times\ P^{a_n}$ is nontrivial. In this paper, we construct monads over the polycyclic variety $P^{2n+1}\times\ P^{2n+1}$, we prove that cohomology vector bundle associated to these monads is simple. We also construct a monad on $P^1\times P^1\times\ P^2$. We also study the vector bundles associated to monads and prove stability and simplicity.

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(Co)Limits Of Hom-Lie Crossed Module, Ali̇ Ayteki̇n

Turkish Journal of Mathematics

In this paper, we give categorical properties such as pullback, finite product, finite limit, coproduct, colimit and pushout in $\boldsymbol{XHom-Lie/A}$ of the category of Hom-Lie crossed modules.

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Explicit Formulas And Recurrence Relations For Generalized Catalan Numbers, Muhammet Ci̇hat Dağli

Turkish Journal of Mathematics

In this paper, we present an explicit formula and recurrent relation for generalized Catalan numbers, from which we can give corresponding formulas for Schröder numbers, large and small generalized Catalan numbers for the special cases of our results.

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The Kernel Spaces And Fredholmness Of Truncated Toeplitz Operators, Xiaoyuan Yang, Ran Li, Yufeng Lu

Turkish Journal of Mathematics

In this paper, we study some conditions about invertible and Fredholm truncated Toeplitz operators which have unique symbols. For $f\in L^\infty$, if $A_f$ is a Fredholm operator, then $f _E\neq 0$ for any $E\subset \mathbb{T}$ with $ E >0$. Moreover \textnormal {ind} $(A_f)=0.$ In particular, if $A_f$ is invertible in $\mathfrak{L}(K_u^2)$, then $f$ is invertible in $L^\infty$. Besides, we give some results about the kernel spaces of truncated Toeplitz operators. For $f \in L^\infty$, we obtain the necessary and sufficient condition that the defect operator $I-A_f^*A_f$ of truncated Toeplitz operator $A_f$ meeting some conditions is compact on the model space …

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Bounded Invertibility And Separability Of A Parabolic Type Singular Operator In The Space $L_{2}(R^{2})$, Mussakan Muratbekov, Madi Muratbekov, Ainash Suleimbekova

Turkish Journal of Mathematics

In this paper, we consider the operator of parabolic type $$ Lu=\frac{\partial u}{\partial t}-\frac{\partial^{2}u}{\partial x^{2}}+q(x)u, $$ in the space $L_{2}(R^{2})$ with a greatly growing coefficient at infinity. The operator is originally defined on $C_{0}^{\infty}(R^{2})$, where $C_{0}^{\infty}(R^{2})$ is the set of infinitely differentiable and compactly supported functions. \noindent Assume that the coefficient $q(x)$ is a continuous function in $R=(-\infty, \infty)$, and it can be a strongly increasing function at infinity. \noindent The operator $L$ admits closure in space $L_{2}(R^{2})$, and the closure is also denoted by $L$. \noindent In the paper, we proved the bounded invertibility of the operator $L$ in …

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Full-Text Articles in Physical Sciences and Mathematics

Logarithmic Dimension And Bases In Whitney Spaces, Alexander Goncharov, Yasemi̇n Şengül Tezel

Turkish Journal of Mathematics

On Congruences Related To Trinomial Coefficients And Harmonic Numbers, Neşe Ömür, Si̇bel Koparal, Laid Elkhiri

Turkish Journal of Mathematics

Relative Conics And Their Brianchon Points, Magdalena Lampa-Baczynska, Daniel Wojcik

Turkish Journal of Mathematics

A Refinement Of The Bergström Inequality, Gülteki̇n Tinaztepe, İlknur Yeşi̇lce, Gabi̇l Adi̇lov

Turkish Journal of Mathematics

Ranks Of Nilpotent Subsemigroups Of Order-Preserving And Decreasing Transformation Semigroups, Emrah Korkmaz, Hayrullah Ayik

Turkish Journal of Mathematics

On The Spectra Of Generalized Fibonomial And Jacobsthal-Binomial Graphs, Hati̇ce Topcu, Nurten Yücel

Turkish Journal of Mathematics

Decompositions Of Complete Symmetric Directed Graphs Into The Oriented Heptagons, Uğur Odabaşi

Turkish Journal of Mathematics

Conformal Bi-Slant Submersions, Sezi̇n Aykurt Sepet

Turkish Journal of Mathematics

Global Attractivity Of Delay Difference Equations In Banach Spaces Via Fixed-Point Theory, Abdullah Kalkan

Turkish Journal of Mathematics

A Regularized Trace Formula For "Weighted" Sturm-Liouville Equation With Point Delta-Interaction, Manaf Manafli

Turkish Journal of Mathematics

Some Properties Of The Semigroup $Pg_Y(X)$: Green's Relations, Ideals, Isomorphism Theorems And Ranks, Worachead Sommanee

Turkish Journal of Mathematics

New Results On Derivatives Of The Shape Operator Of A Real Hypersurface In A Complex Projective Space, Juan De Dios Perez, David Pérez-López

Turkish Journal of Mathematics

General Characteristics Of A Fractal Sturm-Liouville Problem, Fatma Ayça Çeti̇nkaya, Alireza Khalili Golmankaneh

Turkish Journal of Mathematics

Half Inverse Problems For The Impulsive Quadratic Pencil With The Discontinouty Coefficient, Rauf Ami̇rov, Sevi̇m Durak

Turkish Journal of Mathematics

On Sense Of Yamakawa Family Of Meromorphic Bi-Univalent And Bi-Subordinate Functions, Fethi̇ye Müge Sakar

Turkish Journal of Mathematics

Normalized Null Hypersurfaces In Non-Flat Lorentzian Space Forms Satisfying $L_R X =\Mathcal U X +B$, Hans Fotsing Tetsing

Turkish Journal of Mathematics

A Contiguous Extension Of Dixon's Theorem For A Terminating ${}_4f_3(1)$ Series With Applications, Mohammad Idris Qureshi, Richard Bruce Paris, Shakir Hussain Malik

Turkish Journal of Mathematics

2-Absorbing $\Phi$-$\Delta$-Primary Ideals, Sanem Yavuz, Serkan Onar, Bayram Ali̇ Ersoy, Ünsal Teki̇r, Suat Koç

Turkish Journal of Mathematics

Weak C-Ideals Of A Lie Algebra, Zeki̇ye Çi̇loğlu Şahi̇n, David Anthony Towers

Turkish Journal of Mathematics

Bilinear Multipliers Of Small Lebesgue Spaces, Öznur Kulak, Ahmet Turan Gürkanli

Turkish Journal of Mathematics

Some Qualitative Results For Functional Delay Dynamic Equations On Time Scales, Hali̇s Can Koyuncuoğlu

Turkish Journal of Mathematics

Some Applications Of Fractional Calculus For Analytic Functions, Nesli̇han Uyanik, Shi̇geyoshi̇ Owa

Turkish Journal of Mathematics

Optimization Of Mayer Functional In Problems With Discrete And Differential Inclusions And Viability Constraints, Gülseren Çi̇çek, Eli̇mhan N. Mahmudov

Turkish Journal of Mathematics

Sparse Polynomial Interpolation With Bernstein Polynomials, Erdal İmamoğlu

Turkish Journal of Mathematics

Reduced Limit Approach To Semilinear Pdes Involving The Fractional Laplacian With Measure Data, Ratan Kumar Giri, Debajyoti Choudhuri

Turkish Journal of Mathematics

Indecomposable Vector Bundles Via Monads On A Cartesian Product Of Projective Spaces, Damian Maingi

Turkish Journal of Mathematics

(Co)Limits Of Hom-Lie Crossed Module, Ali̇ Ayteki̇n

Turkish Journal of Mathematics

Explicit Formulas And Recurrence Relations For Generalized Catalan Numbers, Muhammet Ci̇hat Dağli

Turkish Journal of Mathematics

The Kernel Spaces And Fredholmness Of Truncated Toeplitz Operators, Xiaoyuan Yang, Ran Li, Yufeng Lu

Turkish Journal of Mathematics

Bounded Invertibility And Separability Of A Parabolic Type Singular Operator In The Space $L_{2}(R^{2})$, Mussakan Muratbekov, Madi Muratbekov, Ainash Suleimbekova

Turkish Journal of Mathematics