: In this article, quaternions, which is a preferred and elegant method for expressing spherical rotations, are generalized with the help of generalized scalar product spaces, and elliptical rotations on any given ellipsoid are examined by them. To this end, firstly, we define the generalized elliptical scalar product space which accepts the given ellipsoid as a sphere and determines skew symmetric matrices, and the generalized vector product related to this scalar product space. Then we define the generalized elliptical quaternions by using these notions. Finally, elliptical rotations on any ellipsoid in the space are examined by using the unit generalized elliptical quaternions. The formulas and results obtained are supported with numerical examples.
{"title":"Generalized elliptical quaternions with some applications","authors":"H. B. Çolakoğlu, M. Özdemir","doi":"10.55730/1300-0098.3364","DOIUrl":"https://doi.org/10.55730/1300-0098.3364","url":null,"abstract":": In this article, quaternions, which is a preferred and elegant method for expressing spherical rotations, are generalized with the help of generalized scalar product spaces, and elliptical rotations on any given ellipsoid are examined by them. To this end, firstly, we define the generalized elliptical scalar product space which accepts the given ellipsoid as a sphere and determines skew symmetric matrices, and the generalized vector product related to this scalar product space. Then we define the generalized elliptical quaternions by using these notions. Finally, elliptical rotations on any ellipsoid in the space are examined by using the unit generalized elliptical quaternions. The formulas and results obtained are supported with numerical examples.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41731256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: In this paper, we define a two-variable p olynomial i nvariant o f r egular i sotopy, M K f or a d isoriented link diagram K . By normalizing the polynomial M K using complete writhe, we obtain a polynomial invariant of ambient isotopy, N K , for a disoriented link diagram K . The polynomial N K is a generalization of the expanded Jones polynomial for disoriented links and is an expansion of the Kauffman polynomial F to the disoriented l inks. Moreover, the polynomial M K is an expansion of the Kauffman p olynomial L to the disoriented links.
：在本文中，我们定义了一个双变量多项式变函数，M K f或d等向链接图K。通过使用完全扭体对多项式MK进行归一化，我们得到了定向链接图K的环境同构的多项式不变量NK。多项式N K是定向链路的扩展Jones多项式的推广，并且是Kauffman多项式F对定向链路的扩张。此外，多项式MK是Kauffman多项式L对定向链路的扩展。
{"title":"An invariant of regular isotopy for disoriented links","authors":"Ismet Altintas, H. Parlatıcı","doi":"10.55730/1300-0098.3345","DOIUrl":"https://doi.org/10.55730/1300-0098.3345","url":null,"abstract":": In this paper, we define a two-variable p olynomial i nvariant o f r egular i sotopy, M K f or a d isoriented link diagram K . By normalizing the polynomial M K using complete writhe, we obtain a polynomial invariant of ambient isotopy, N K , for a disoriented link diagram K . The polynomial N K is a generalization of the expanded Jones polynomial for disoriented links and is an expansion of the Kauffman polynomial F to the disoriented l inks. Moreover, the polynomial M K is an expansion of the Kauffman p olynomial L to the disoriented links.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46737543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: In this paper, we introduce and study the new concept of demi KB-operators. Let E be a Banach lattice. An operator T : E −→ E is said to be a demi KB-operator if, for every positive increasing sequence { x n } in the closed unit ball B E of E such that { x n − Tx n } is norm convergent to x ∈ 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 and demi Dunford-Pettis operators.
{"title":"The class of demi KB-operators on Banach lattices","authors":"Hedi Benkhaled, A. Jeribi","doi":"10.55730/1300-0098.3366","DOIUrl":"https://doi.org/10.55730/1300-0098.3366","url":null,"abstract":": In this paper, we introduce and study the new concept of demi KB-operators. Let E be a Banach lattice. An operator T : E −→ E is said to be a demi KB-operator if, for every positive increasing sequence { x n } in the closed unit ball B E of E such that { x n − Tx n } is norm convergent to x ∈ 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 and demi Dunford-Pettis operators.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49215909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: 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
{"title":"Separation, connectedness, and disconnectedness","authors":"M. Baran","doi":"10.55730/1300-0098.3360","DOIUrl":"https://doi.org/10.55730/1300-0098.3360","url":null,"abstract":": 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","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49594432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on the jumps of the argument is found which is sufficient for solvability of these problems, and, in case of solvability, a general solution is constructed. It is established the basicity of the parts of exponential system in Hardy-Orlicz classes.
{"title":"On solvability of homogeneous Riemann boundary value problems in Hardy-Orlicz classes","authors":"Y. Zeren, Fi̇dan A. Ali̇zadeh, Feyza Eli̇f Dal","doi":"10.55730/1300-0098.3379","DOIUrl":"https://doi.org/10.55730/1300-0098.3379","url":null,"abstract":": This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on the jumps of the argument is found which is sufficient for solvability of these problems, and, in case of solvability, a general solution is constructed. It is established the basicity of the parts of exponential system in Hardy-Orlicz classes.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46628035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: 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 loss of vertical velocity dissipation and horizontal magnetic diffusion. Finally we can show that the estimates ∫ T 0 ∥∇ u ( t ) ∥ L ∞ dt and ∫ T 0 ∥∇ b ( t ) ∥ L ∞ dt are finite in a priori way and hence obtain the global well-posedness to the system under considered.
{"title":"Global regularity for the 3D axisymmetric incompressible Hall-MHD system with partial dissipation and diffusion","authors":"Meilin Jin, Q. Jiu, Huan Yu","doi":"10.55730/1300-0098.3403","DOIUrl":"https://doi.org/10.55730/1300-0098.3403","url":null,"abstract":": 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 loss of vertical velocity dissipation and horizontal magnetic diffusion. Finally we can show that the estimates ∫ T 0 ∥∇ u ( t ) ∥ L ∞ dt and ∫ T 0 ∥∇ b ( t ) ∥ L ∞ dt are finite in a priori way and hence obtain the global well-posedness to the system under considered.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47008895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: Using ( p, q ) -Lucas polynomials and bi-Bazilevic̆ type functions of order ρ + iξ, we defined a new subclass of biunivalent functions. We obtained coefficient inequalities for functions belonging to the new subclass. In addition to these results, the upper bound for the Fekete-Szegö functional was obtained. Finally, for some special values of parameters, several corollaries were presented
{"title":"On a new subclass of biunivalent functions associated with the $(p,q)$-Lucas polynomials and bi-Bazilevic type functions of order $rho+ixi$","authors":"H. Orhan, İ. Aktaş, H. Arikan","doi":"10.55730/1300-0098.3348","DOIUrl":"https://doi.org/10.55730/1300-0098.3348","url":null,"abstract":": Using ( p, q ) -Lucas polynomials and bi-Bazilevic̆ type functions of order ρ + iξ, we defined a new subclass of biunivalent functions. We obtained coefficient inequalities for functions belonging to the new subclass. In addition to these results, the upper bound for the Fekete-Szegö functional was obtained. Finally, for some special values of parameters, several corollaries were presented","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47051463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: This work aims to construct the Titchmarsh-Weyl M ( λ ) − theory for an even-dimensional left-definite Hamiltonian system. For this purpose, we introduce a suitable Lagrange formula and selfadjoint boundary conditions including the spectral parameter λ . Then we obtain circle equations having nesting properties. Using the intersection point belonging to all the circles we share a lower bound for the number of Dirichlet-integrable solutions of the system
本文旨在建立偶维左定哈密顿系统的Titchmarsh-Weyl M (λ)−理论。为此，我们引入了一个合适的拉格朗日公式和包含谱参数λ的自伴随边界条件。然后得到具有嵌套性质的圆方程。利用所有圆的交点，我们得到了方程组狄利克雷可积解个数的下界
{"title":"Left-definite Hamiltonian systems and corresponding nested circles","authors":"Ekin Uğurlu","doi":"10.55730/1300-0098.3427","DOIUrl":"https://doi.org/10.55730/1300-0098.3427","url":null,"abstract":": This work aims to construct the Titchmarsh-Weyl M ( λ ) − theory for an even-dimensional left-definite Hamiltonian system. For this purpose, we introduce a suitable Lagrange formula and selfadjoint boundary conditions including the spectral parameter λ . Then we obtain circle equations having nesting properties. Using the intersection point belonging to all the circles we share a lower bound for the number of Dirichlet-integrable solutions of the system","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45042646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
O. Stanzhytskyi, Roza E. Uteshova, Victoriia Tsan, Zoia Khaletska
{"title":"On the relation between oscillation of solutions of differential equations and corresponding equations on time scales","authors":"O. Stanzhytskyi, Roza E. Uteshova, Victoriia Tsan, Zoia Khaletska","doi":"10.55730/1300-0098.3373","DOIUrl":"https://doi.org/10.55730/1300-0098.3373","url":null,"abstract":"","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43851580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: 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
{"title":"Bi-periodic incomplete Horadam numbers","authors":"E. Tan, M. Daǧlı, Amine Belkhir","doi":"10.55730/1300-0098.3378","DOIUrl":"https://doi.org/10.55730/1300-0098.3378","url":null,"abstract":": 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","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43956714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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