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Bound For The Cocharacters Of The Identities Of Irreducible Representations Of $\Mathfrak{Sl}_2(\Mathbb{C})$, Mátyás Domokos
Bound For The Cocharacters Of The Identities Of Irreducible Representations Of $\Mathfrak{Sl}_2(\Mathbb{C})$, Mátyás Domokos
Turkish Journal of Mathematics
For each irreducible finite dimensional representation of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ of $2\times 2$ traceless matrices, an explicit uniform upper bound is given for the multiplicities in the cocharacter sequence of the polynomial identities satisfied by the given representation.
Invariants Of Symplectic And Orthogonal Groups Acting On $Gl(N,Cc)$-Modules, Vesselin Drensky, Elitza Hristova
Invariants Of Symplectic And Orthogonal Groups Acting On $Gl(N,Cc)$-Modules, Vesselin Drensky, Elitza Hristova
Turkish Journal of Mathematics
Let $GL(n) = GL(n, CC)$ denote the complex general linear group and let $G \subset GL(n)$ be one of the classical complex subgroups $OO(n)$, $SO(n)$, and $Sp(2k)$ (in the case $n = 2k$). We take a finite dimensional polynomial $GL(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=SL(n)$, we develop a method for determining the Hilbert series $H(S(W)^G, t)$ of the algebra of invariants $S(W)^G$. Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing $H(S(W)^G, t)$. As an application, we …
On The Mesyan Conjecture, Pedro Souza Fagundes, Thiago Castilho De Mello, Pedro Henrique Da Silva Dos Santos
On The Mesyan Conjecture, Pedro Souza Fagundes, Thiago Castilho De Mello, Pedro Henrique Da Silva Dos Santos
Turkish Journal of Mathematics
The well-known Lvov--Kaplansky conjecture states that the image of a multilinear polynomial $f$ evaluated on $n\times n$ matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if $m=\deg f$ and $n\geq m-1$ then its image contains the set of trace zero matrices. Such conjecture has been proved for polynomials of degree $m \leq 4$. The proof of the case $m=4$ contains an error in one of the lemmas. In this paper, we correct the proof of such lemma and present some evidence which allows us to state the Mesyan conjecture for …
Symmetric Polynomials In The Free Metabelian Associative Algebra Of Rank 2, Şehmus Findik
Symmetric Polynomials In The Free Metabelian Associative Algebra Of Rank 2, Şehmus Findik
Turkish Journal of Mathematics
Let $F$ be the free metabelian associative algebra generated by $x$ and $y$ over a field of characteristic zero. We call a polynomial $f\in F$ symmetric, if $f(x,y)=f(y,x)$. The set of all symmetric polynomials coincides with the algebra $F^{S_2}$ of invariants of the symmetric group $S_2$. In this paper, we give the full description of the algebra $F^{S_2}$.
$\Mathbb{Z}$-Graded Identities On The Infinite-Dimensional Grassmann Algebra And Arithmetic Tools: Revisited, Claudemir Fidelis
$\Mathbb{Z}$-Graded Identities On The Infinite-Dimensional Grassmann Algebra And Arithmetic Tools: Revisited, Claudemir Fidelis
Turkish Journal of Mathematics
The main purpose of this paper is to provide a survey of results concerning the $\mathbb{Z}$-gradings on the infinite-dimensional Grassmann algebra $E$ over a field of characteristic zero. First, we provide graded identities and central polynomials for $E$ equipped with fine gradings on $E$ by the semigroup $(\mathbb{Z}^\ast,\times)$. We also describe briefly techniques in order to illustrate some important methods to exhibit graded identities and central polynomials of $E$ for other abelian groups. In particular, over a field of characteristic zero, so-called $2$-induced gradings of full support were considered. In order to obtain these descriptions, we strongly use elementary number …
Nonlinear Evolution Equations Related To Kac-Moody Algebras $A_R^{(1)}$: Spectral Aspects, Vladimir S. Gerdjikov
Nonlinear Evolution Equations Related To Kac-Moody Algebras $A_R^{(1)}$: Spectral Aspects, Vladimir S. Gerdjikov
Turkish Journal of Mathematics
We analyze three types of integrable nonlinear evolution equations (NLEE) related to the Kac-Moody algebras $A_r^{(1)}$. These are $\mathbb{Z}_h$-reduced derivative NLS equations (DNLS), multicomponent mKdV equations and 2-dimensional Toda field theories (2dTFT). We outline the basic tools of this analysis: i) the gradings of the simple Lie algebras using their Coxeter automorphisms; ii) the construction of the relevant Lax representations; and iii) the spectral properties of the Lax operators and their reduction to Riemann-Hilbert problems. We also formulate the minimal set of scattering data which allow one to recover the asymptotics of the fundamental analytic solutions to $L$ and its …
Computing Forgotten Topological Index Of Zero-Divisor Graphs Of Commutative Rings, Ari̇f Gürsoy, Necla Kircali Gürsoy, Alper Ülker
Computing Forgotten Topological Index Of Zero-Divisor Graphs Of Commutative Rings, Ari̇f Gürsoy, Necla Kircali Gürsoy, Alper Ülker
Turkish Journal of Mathematics
The main objective of this paper is to calculate the forgotten topological index of the zero-divisor graph of $\mathbb{Z}_n$. Let $p$, $q$ and $r$ be distinct prime numbers. We calculate the forgotten topological index of the ring $\Gamma(\mathbb{Z}_n)$ where $n=p^\alpha, pq, p^2q, p^2q^2, pqr$. Also, we study the forgotten topological index of the product of rings of integers modulo $n$. We construct a polynomial algorithm to compute the forgotten topological index of $\Gamma(\mathbb{Z}_n)$.
Polynomial Identities In Matrix Algebras With Pseudoinvolution, Antonio Ioppolo
Polynomial Identities In Matrix Algebras With Pseudoinvolution, Antonio Ioppolo
Turkish Journal of Mathematics
Let $F$ be an algebraically closed field of characteristic zero. In this paper we deal with matrix superalgebras (i.e. algebras graded by $\mathbb{Z}_2$, the cyclic group of order $2$) endowed with a pseudoinvolution. The first goal is to present the classification of the pseudoinvolutions that it is possible to define, up to equivalence, in the full matrix algebra $M_n(F)$ of $n \times n$ matrices and on its subalgebra $UT_n(F)$ of upper-triangular matrices. Along the way we shall give the generators of the $T$-ideal of identities for the algebras $M_2(F)$, $UT_2(F)$ and $UT_3(F)$, endowed with all possible inequivalent pseudoinvolutions.
Depth And Stanley Depth Of The Quotient Rings Of Edge Ideals Of Some Lobster Trees And Unicyclic Graphs, Adnan Iqbal, Muhammad Ishaq
Depth And Stanley Depth Of The Quotient Rings Of Edge Ideals Of Some Lobster Trees And Unicyclic Graphs, Adnan Iqbal, Muhammad Ishaq
Turkish Journal of Mathematics
We compute depth and Stanley depth of the quotient rings of the edge ideals associated with different classes of graphs. These classes include some lobster trees and unicyclic graphs. We show that the values of depth and Stanley depth are equal for the classes we considered.
A Note On The $\Mathcal{A}$-Generators Of The Polynomial Algebra Of Six Variables And Applications, Tin Nguyen Khac
A Note On The $\Mathcal{A}$-Generators Of The Polynomial Algebra Of Six Variables And Applications, Tin Nguyen Khac
Turkish Journal of Mathematics
Let $ \mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra of $n$ generators $x_1, x_2, \ldots, x_n$ with the degree of each $x_i$ being 1. We investigate the Peterson hit problem for the polynomial algebra $ \mathcal P_{n},$ regarded as a module over the mod-$2$ Steenrod algebra, $ \mathcal{A}.$ For $n>4,$ this problem remains unsolvable, even with the aid of computers in the case of $n=5.$ In this article, we study the hit problem for the case $n=6$ in degree $d_s=6(2^s -1)+3.2^s,$ with $s$ an arbitrary nonnegative integer. By considering $ \mathbb Z_2$ as a trivial $ \mathcal …
On Almost S-Weakly Regular Rings, Kanchan Jangra, Dinesh Udar
On Almost S-Weakly Regular Rings, Kanchan Jangra, Dinesh Udar
Turkish Journal of Mathematics
An element $a$ of $R$ is called s-weakly regular (SWR) if $a\in aRa^2R$. A ring $R$ is called an almost SWR if for any $a\in R$, either $a$ or $1-a$ is SWR. In this paper, we introduce almost SWR rings as the generalization of abelian von Neumann local (VNL) rings and SWR rings. We provide various properties and characterizations of almost SWR rings. We discuss various extension rings to be almost SWR. Further, we discuss SWR group rings and almost SWR group rings.
Spectral Theory Of B-Weyl Elements And The Generalized Weyl's Theorem In Primitive C*-Algebra, Yingying Kong, Yanxun Ren, Lining Jiang
Spectral Theory Of B-Weyl Elements And The Generalized Weyl's Theorem In Primitive C*-Algebra, Yingying Kong, Yanxun Ren, Lining Jiang
Turkish Journal of Mathematics
Let $\mathcal{A}$ be a unital primitive C*-algebra. This paper studies the spectral theories of B-Weyl elements and B-Browder elements in $\mathcal{A}$, including the spectral mapping theorem and a characterization of B-Weyl spectrum. In addition, we characterize the generalized Weyl's theorem and the generalized Browder's theorem for an element $a\in\mathcal{A}$ and $f(a)$, where $f$ is a complex-valued function analytic on a neighborhood of $\sigma(a)$. What's more, the perturbations of the generalized Weyl's theorem under the socle of $\mathcal{A}$ and quasinilpotent element are illustrated.
Structure Of Annihilators Of Powers, Jongwook Baeck, Nam Kyun Kim, Tai Keun Kwak, Yang Lee
Structure Of Annihilators Of Powers, Jongwook Baeck, Nam Kyun Kim, Tai Keun Kwak, Yang Lee
Turkish Journal of Mathematics
We study the following two conditions in rings: (i) the right annihilator of some power of any element is an ideal, and (ii) the right annihilator of any nonzero element $a$ contains an ideal generated by some power of any right zero-divisor of the element $a$. We investigate the structure of rings in relation to these conditions; especially, a ring with the condition (ii) is called right APIP. These conditions are shown to be not right-left symmetric. For a prime two-sided APIP ring $R$ we prove that every element of $R$ is either nilpotent or regular, and that if $R$ …
On Central Polynomials And Codimension Growth, Fabrizio Martino
On Central Polynomials And Codimension Growth, Fabrizio Martino
Turkish Journal of Mathematics
Let $A$ be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of $A.$ In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.
Codimensions Of Algebras With Additional Structures, Daniela La Mattina
Codimensions Of Algebras With Additional Structures, Daniela La Mattina
Turkish Journal of Mathematics
Let $A$ be an associative algebra endowed with an automorphism or an antiautomorphism $\varphi$ of order $\leq 2.$ One associates to $A,$ in a natural way, a numerical sequence $c^\varphi_n(A),$ $n=1, 2, \ldots$, called the sequence of $\varphi$-codimensions of $A$ which is the main tool for the quantitative investigation of the polynomial identities satisfied by $A$. In \cite{GLM} it was proved that such a sequence is eventually nondecreasing in case $\varphi$ is an antiautomorphism. Here we prove that it still holds in case $\varphi$ is an automorphism and present some recent results about the asymptotics of $c^\varphi_n(A)$.
On $(A,D)$-Edge Local Antimagic Coloring Number Of Graphs, Rajkumar Sundaramoorthy, Nalliah Moviri Chettiar
On $(A,D)$-Edge Local Antimagic Coloring Number Of Graphs, Rajkumar Sundaramoorthy, Nalliah Moviri Chettiar
Turkish Journal of Mathematics
For any graph $G=(V,E),$ the order and size of G are $p$ and $q$. A bijection $l$ from $V(G)$ to $\{1,2,..,p\}$ is called $(a,d)$-edge local antimagic labeling if for any two adjacent edges are not received the same edge-weight (color) and the set of all edge-weights are formed an arithmetic progression $\{a,a+d,a+2d,\dots,a+(c-1)d\}$, for some integers $a,d>0$ and $c$ is the number of distinct colors used in the proper coloring.} An edge-weight (color) $w(uv)$ is the sum of two end vertices labels, $w(uv)=f(u)+f(v),uv\in E(G).$ The $(a,d)$-edge local antimagic coloring number is the least color (edge-weight) used in any $(a,d)$-edge local …
A New Approach To Word Standardization And Some Of Its Applications, Wesam Talab
A New Approach To Word Standardization And Some Of Its Applications, Wesam Talab
Turkish Journal of Mathematics
In this article, we study word standardization in comparison to Young tableau standardization. We count the number of words (respectively Young tableau) standardized to a given permutation (respectively to a given standard Young tableau). We prove that both rectification and standardization applications commute and show that the standardization commutes with the insertion of Robinson--Schensted. We show that the standardizations of Knuth-equivalent two words are also Knuth equivalent. Finally, using word standardization we establish a proof for the following well-known equality: $$ \forall l \in \left\lbrace 0,1,\ldots,n-1\right\rbrace ,~~\left \langle {n\atop l} \right \rangle=d_{n,l}=a_{n,l}= \sum_{0\leq k \leq l}(-1)^k { n+1 \choose k …
On $ Mj $-Clean Ring And Strongly $ Mj $-Clean Ring, Gülşen Ulucak, Arda Kör
On $ Mj $-Clean Ring And Strongly $ Mj $-Clean Ring, Gülşen Ulucak, Arda Kör
Turkish Journal of Mathematics
In this paper, we introduce the concepts of $ mj $-clean and strongly $ mj $-clean rings which are generalizations of $ j $-clean ring and strongly $ j $-clean ring, respectively. Let $ R $ be a ring with a nonzero identity and $ m\geq 2 $ a positive integer. We call the ring $ R $ as $ mj $-clean if each element of $ R$ can be written as a sum of an $ m $-potent and an element of $J(R)$ and also if these elements are commute, then we call $R$ as strongly $ mj $-clean …
Nilpotent Varieties And Metabelian Varieties, Angela Valenti, Sergey Mishchenko
Nilpotent Varieties And Metabelian Varieties, Angela Valenti, Sergey Mishchenko
Turkish Journal of Mathematics
We deal with varieties of nonassociative algebras having polynomial growth of codimensions. We describe some results obtained in recent years in the class of left nilpotent algebras of index two. Recently the authors established a correspondence between the growth rates for left nilpotent algebras of index two and the growth rates for commutative or anticommutative metabelian algebras that allows to transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.
On $S$-Comultiplication Modules, Eda Yildiz, Ünsal Teki̇r, Suat Koç
On $S$-Comultiplication Modules, Eda Yildiz, Ünsal Teki̇r, Suat Koç
Turkish Journal of Mathematics
Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is a multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion of an $S$-prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules,\ $S$-Noetherian modules and etc. Afterwards, in \cite{AnArTeKo}, Anderson et al. defined the concepts of $S$-multiplication modules and $S$-cyclic modules which are $S$-versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to …
Analyzing Bifurcation, Stability, And Chaos Control For A Discrete-Time Prey-Predator Model With Allee Effect, Fi̇gen Kangalgi̇l, Ni̇lüfer Topsakal, Ni̇hal Öztürk
Analyzing Bifurcation, Stability, And Chaos Control For A Discrete-Time Prey-Predator Model With Allee Effect, Fi̇gen Kangalgi̇l, Ni̇lüfer Topsakal, Ni̇hal Öztürk
Turkish Journal of Mathematics
In this paper, the qualitative behavior of a discrete-time prey-predator model with Allee effect in prey population is discussed. Firstly, the existence of the fixed points and their topological classification are analyzed algebraically. Then, the conditions of existence for both period-doubling and Neimark--Sacker bifurcations arising from coexistence fixed point with the help of the center manifold theorem and bifurcation theory are investigated. OGY feedback control method is implemented to control chaos in the proposed model due to the emergence of bifurcations. Finally, numerical simulations are performed to support the theoretical findings.
Solvability In The Small Of $M$-Th Order Elliptic Equations In Weighted Grand Sobolev Spaces, Bilal Bilalov, Yusuf Zeren, Sabina Sadigova, Şeyma Çeti̇n
Solvability In The Small Of $M$-Th Order Elliptic Equations In Weighted Grand Sobolev Spaces, Bilal Bilalov, Yusuf Zeren, Sabina Sadigova, Şeyma Çeti̇n
Turkish Journal of Mathematics
In this work we consider the Sobolev spaces generated by the norm of the power weighted grand Lebesgue spaces. It is considered $m$-th order elliptic equation with nonsmooth coefficients on bounded domain in $R^{n} $. This space is nonseparable and by using shift operator we define the separable subspace of it, in which infinitely differentiable functions are dense. The investigation needs to establish boundedness property of convolution regarding weighted grand Lebesgue spaces. Then on scheme of nonweighted case we establish solvability (strong sense) in the small of $m$-th order elliptic equations in power weighted grand Sobolev spaces. Note that in …
Distortion Bound And Growth Theorems For A Subclass Of Analytic Functions Defined By $Q$-Derivative, Osman Altintaş, Ni̇zami̇ Mustafa
Distortion Bound And Growth Theorems For A Subclass Of Analytic Functions Defined By $Q$-Derivative, Osman Altintaş, Ni̇zami̇ Mustafa
Turkish Journal of Mathematics
In this study, we introduce and examine a subclasses of analytic and univalent functions defined by $q$-derivative. Here, we give necessary conditions for the functions to belong to these subclasses, and distortion bound and growth theorems for the functions belonging to these classes.
On The Geodesics Of Deformed Sasaki Metric, Nour Elhouda Djaa, Abderrahim Zagane
On The Geodesics Of Deformed Sasaki Metric, Nour Elhouda Djaa, Abderrahim Zagane
Turkish Journal of Mathematics
We define in this note a natural metric over the tangent bundle $TM$ by using a vertical deformation of Sasaki metric. First we present the geometric result concerning the Levi-Civita connection and all forms of Riemannian curvature tensors of this metric. Secondly, we study the geodesics on the tangent bundle $TM$ and unit tangent bundle $T_{1}M$. Finally, we characterize the geodesic curvatures on $T_{1}M$.
Vanishing Ideals Of Parameterized Subgroups In A Toric Variety, Esma Baran Özkan
Vanishing Ideals Of Parameterized Subgroups In A Toric Variety, Esma Baran Özkan
Turkish Journal of Mathematics
Let $K$ be a finite field and $X$ be a complete simplicial toric variety over $K$ with split torus $T_X\cong (K^*)^n$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which can be used to study algebraic geometric codes arising from $Y_Q$. We give a method to compute the lattice $L$ whose ideal $I_L$ is exactly $I(Y_Q)$ under a mild condition. As applications, we give precise descriptions for the lattices corresponding to some special subgroups. We also prove a Nullstellensatz type theorem valid over finite …
A New Kind Of $F$-Contraction And Some Best Proximity Point Results For Such Mappings With An Application, Hakan Şahi̇n
A New Kind Of $F$-Contraction And Some Best Proximity Point Results For Such Mappings With An Application, Hakan Şahi̇n
Turkish Journal of Mathematics
In this paper, we aim to present a new and unified way, including the previously mentioned solution methods, to overcome the problem in [7] for closed and bounded valued $F$-contraction mappings. We also want to obtain a real generalization of fixed point results existing in the literature by using best proximity point theory. Further, considering the strong relationship between homotopy theory and various branches of mathematics such as category theory, topological spaces, and Hamiltonian manifolds in quantum mechanics, our objective is to present an application to homotopy theory of our best proximity point results obtained in the paper. In this …
Inverse Nodal Problem For Sturm-Liouville Operator On A Star Graph With Nonequal Edges, Sevi̇m Durak
Inverse Nodal Problem For Sturm-Liouville Operator On A Star Graph With Nonequal Edges, Sevi̇m Durak
Turkish Journal of Mathematics
In this study, Sturm-Liouville operator was investigated on a star graph with nonequal edges. First, the behaviors of sufficiently large eigenvalues were learned, then the solution of the inverse problem was given to determine the potantial functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points and obtain a constructive solution to the inverse problems of this class.
Hermite-Hadamard-Mercer Type Inclusions For Interval-Valued Functions Via Riemann-Liouville Fractional Integrals, Hasan Kara, Muhammad Aamir Ali, Hüseyi̇n Budak
Hermite-Hadamard-Mercer Type Inclusions For Interval-Valued Functions Via Riemann-Liouville Fractional Integrals, Hasan Kara, Muhammad Aamir Ali, Hüseyi̇n Budak
Turkish Journal of Mathematics
In this research, we first establish some inclusions of fractional Hermite-Hadamard-Mercer type for interval-valued functions. Moreover, by special cases of our main results, we show that our main results reduce several inclusions obtained in the earlier works.
Global Differential Invariants Of Nondegenerate Hypersurfaces, Yasemi̇n Sağiroğlu, Uğur Gözütok
Global Differential Invariants Of Nondegenerate Hypersurfaces, Yasemi̇n Sağiroğlu, Uğur Gözütok
Turkish Journal of Mathematics
Let $\{g_{ij}(x)\}_{i, j=1}^n$ and $\{L_{ij}(x)\}_{i, j=1}^n$ be the sets of all coefficients of the first and second fundamental forms of a hypersurface $x$ in $R^{n+1}$. For a connected open subset $U\subset R^{n}$ and a $C^{\infty }$-mapping $x:U\rightarrow R^{n+1}$ the hypersurface $x$ is said to be $d$-\textit{nondegenerate}, where $d\in \left\{1, 2, \ldots n\right\}$, if $L_{dd}(x)\neq 0$ for all $u\in U$. Let $M(n)=\{F:R^{n}\longrightarrow R^{n}\mid Fx=gx+b, \; g\in O(n), \; b\in R^{n}\}$, where $O(n)$ is the group of all real orthogonal $n\times n$-matrices, and $SM(n)=\{F\in M(n)\mid g\in SO(n)\}$, where $SO(n)=\left\{g\in O(n)\mid \det(g)=1\right\}$. In the present paper, it is proved that the set $\left\{g_{ij}(x), …