In this paper, we consider the extension of the analytic functions of two variables by special families of functions — continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function $mathrm{H}_7$ with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.
在本文中，我们考虑通过特殊函数族--续分--来扩展两变量解析函数。特别是，我们利用其续分数表示法，建立了霍恩的汇交超几何函数 $mathrm{H}_7$ 的三个比率的解析延续的新对称域，这些比率对实数和复数参数具有特定条件。我们利用沃皮茨基定理、多重抛物线定理和一种将已知的小域收敛扩展到更大域的技术，得到了连续分数的收敛域，并用 PC 方法证明它们也是解析延续域。
{"title":"On the analytic extension of three ratios of Horn's confluent hypergeometric function $mathrm{H}_7$","authors":"V. Hladun, R. Rusyn, M. Dmytryshyn","doi":"10.15421/242405","DOIUrl":"https://doi.org/10.15421/242405","url":null,"abstract":"In this paper, we consider the extension of the analytic functions of two variables by special families of functions — continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function $mathrm{H}_7$ with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we classify the norming set of a bilinear form on the plane with a certain norm whose unit ball has only four extreme points. We obtain the results of [6, 8] as corollary.
{"title":"The norming set of a bilinear form on a certain normed space $mathbb{R}^2$","authors":"S.G. Kim","doi":"10.15421/242406","DOIUrl":"https://doi.org/10.15421/242406","url":null,"abstract":"In this paper we classify the norming set of a bilinear form on the plane with a certain norm whose unit ball has only four extreme points. We obtain the results of [6, 8] as corollary.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141669839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $b$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $b$-linear functionals in ultrametric 2-normed spaces are given.
{"title":"Some results on ultrametric 2-normed spaces","authors":"J. Ettayb","doi":"10.15421/242404","DOIUrl":"https://doi.org/10.15421/242404","url":null,"abstract":"In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $b$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $b$-linear functionals in ultrametric 2-normed spaces are given.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A virtual endomorphism of a group $G$ is a homomorphism of the form $phi:Hrightarrow G$, where $H
群 $G$ 的虚内变是形式为 $phi:Hrightarrow G$ 的同态，其中 $H
{"title":"Virtual endomorphisms of the group $pg$","authors":"I. Bondarenko, D. Zashkolny","doi":"10.15421/242401","DOIUrl":"https://doi.org/10.15421/242401","url":null,"abstract":"A virtual endomorphism of a group $G$ is a homomorphism of the form $phi:Hrightarrow G$, where $H<G$ is a subgroup of finite index. A virtual endomorphism $phi:Hrightarrow G$ is called simple if there are no nontrivial normal $phi$-invariant subgroups, that is, the $phi$-core is trivial. We describe all virtual endomorphisms of the plane group $pg$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $pg$ admits a transitive self-similar (as well as finite-state) action of degree $d$ if and only if $dgeq 2$ is not an odd prime, and admits a self-replicating action of degree $d$ if and only if $dgeq 6$ is not a prime or a power of $2$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141669266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.
{"title":"Projective tensor products of approximation spaces associated with positive operators","authors":"M. Dmytryshyn, L. Dmytryshyn","doi":"10.15421/242403","DOIUrl":"https://doi.org/10.15421/242403","url":null,"abstract":"In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141669854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show how with the help of nonlinear paired regression, namely cubic regression, having experimental data, it is possible to investigate the relationship of tangential stresses between different layers of a cylindrical part during build-up. Six pairs of dependencies (models) between tangential stresses are considered. For each pair of dependencies, the accuracy of the built model was assessed using the average error of approximation and Fisher's F-criterion, and a comparative analysis was conducted. Despite some contradictions that arose during the evaluation of the models according to various regression parameters it was established that at least four of the six models are optimal and allow to adequately model the process of the formation of the stress-strain state in the details and elements of structures with significantly lower costs even before the stage of manufacturing finished products.
我们展示了如何借助非线性配对回归（即立方回归）和实验数据，研究圆柱形零件在堆积过程中不同层之间的切向应力关系。我们考虑了切向应力之间的六对依赖关系（模型）。对于每一对依存关系，都使用平均近似误差和费雪 F 标准评估了所建模型的准确性，并进行了比较分析。尽管在根据各种回归参数对模型进行评估的过程中出现了一些矛盾，但结果表明，六个模型中至少有四个是最佳模型，可以充分模拟结构细节和元素中应力应变状态的形成过程，甚至在制造成品阶段之前就能大大降低成本。
{"title":"Construction of a non-linear analytical model for the rotation parts building up process using regression analysis","authors":"A.V. Siasiev, R.O. Bilichenko","doi":"10.15421/242413","DOIUrl":"https://doi.org/10.15421/242413","url":null,"abstract":"We show how with the help of nonlinear paired regression, namely cubic regression, having experimental data, it is possible to investigate the relationship of tangential stresses between different layers of a cylindrical part during build-up. Six pairs of dependencies (models) between tangential stresses are considered. For each pair of dependencies, the accuracy of the built model was assessed using the average error of approximation and Fisher's F-criterion, and a comparative analysis was conducted. Despite some contradictions that arose during the evaluation of the models according to various regression parameters it was established that at least four of the six models are optimal and allow to adequately model the process of the formation of the stress-strain state in the details and elements of structures with significantly lower costs even before the stage of manufacturing finished products.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the bilinear forms on the space of measurable $p$-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded $L^q$-elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for $p$-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.
{"title":"A Note on Some Properties of Unbounded Bilinear Forms Associated with Skew-Symmetric $L^q(Omega)$-Matrices","authors":"P.I. Kogut","doi":"10.15421/242407","DOIUrl":"https://doi.org/10.15421/242407","url":null,"abstract":"We study the bilinear forms on the space of measurable $p$-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded $L^q$-elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for $p$-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141669054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The authors study the relations between the properties of torsion groups and their norms of $pd$-subgroups. The norm $N_G^{pdI}$ of $pd$-subgroups of a group $G$ is the intersection of the normalizers of all its $pd$-subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of $pd$-subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a finite extension of its norm of $pd$-subgroups if and only if it is a finite extension of its center. By this fact and the structure of the norm of $pd$-subgroups, we get that any torsion group that is a finite extension of this norm is locally finite.
{"title":"Torsion Groups with the Norm of pd-Subgroup of Finite Index","authors":"T. Lukashova, M. G. Drushlyak, A.V. Pidopryhora","doi":"10.15421/242410","DOIUrl":"https://doi.org/10.15421/242410","url":null,"abstract":"The authors study the relations between the properties of torsion groups and their norms of $pd$-subgroups. The norm $N_G^{pdI}$ of $pd$-subgroups of a group $G$ is the intersection of the normalizers of all its $pd$-subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of $pd$-subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a finite extension of its norm of $pd$-subgroups if and only if it is a finite extension of its center. By this fact and the structure of the norm of $pd$-subgroups, we get that any torsion group that is a finite extension of this norm is locally finite.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141669763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This research article explores some new properties of generalized hypergeometric function and its q-analogue. The connections between ${}_{2}{{R}_{1}}^{upsilon }(mathfrak{z})$, the Wright function, and generalized Mittag-Leffler functions are explored. The authors introduce the q-analogue of generalized hypergeometric function denoted by ${}_{2}{{R}_{1}}^{upsilon ,q}(mathfrak{z})$ and discuss its properties and connections with q-Wright function and q-versions of generalized Mittag-Leffler functions. We get the q-integral transforms such as q-Mellin, q-Euler (beta), q-Laplace, q-sumudu, and q-natural transforms of Wright-type generalized q-hypergeometric function. This article contributes to the understanding of hypergeometric functions in q-calculus.
{"title":"A study on unification of generalized hypergeometric function and Mittag-Leffler function with certain integral transforms of generalized basic hypergeometric function","authors":"K. K. Chaudhary, S.B. Rao","doi":"10.15421/242402","DOIUrl":"https://doi.org/10.15421/242402","url":null,"abstract":"This research article explores some new properties of generalized hypergeometric function and its q-analogue. The connections between ${}_{2}{{R}_{1}}^{upsilon }(mathfrak{z})$, the Wright function, and generalized Mittag-Leffler functions are explored. The authors introduce the q-analogue of generalized hypergeometric function denoted by ${}_{2}{{R}_{1}}^{upsilon ,q}(mathfrak{z})$ and discuss its properties and connections with q-Wright function and q-versions of generalized Mittag-Leffler functions. We get the q-integral transforms such as q-Mellin, q-Euler (beta), q-Laplace, q-sumudu, and q-natural transforms of Wright-type generalized q-hypergeometric function. This article contributes to the understanding of hypergeometric functions in q-calculus.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141670487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbb K$ be a field of characteristic zero, $A := mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(mathbb K)$ the Lie algebra of all $mathbb K$-derivations on $A$. Every polynomial $f in A$ defines a Jacobian derivation $D_fin W_2(mathbb K)$ by the rule $D_f(h)=det J(f, h)$ for any $hin A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(mathbb K)$. It is proved that for a Jordan chain $T(f_1)=lambda f_1+f_2$, ..., $T(f_{k-1})=lambda f_{k-1}+f_k$, $T(f_k)=lambda f_k$ for a derivation $Tin W_2(mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(lambda -mathop{mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(lambda -mathop{mathrm{div}} T)D_{f_{k}}$ in $W_2(mathbb K)$. In case $A:=mathbb K[x_1, ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(mathbb K)$ on the principal ideals $(f)$ is considered.
{"title":"Action of derivations on polynomials and on Jacobian derivations","authors":"O.Ya. Kozachok, A. Petravchuk","doi":"10.15421/242408","DOIUrl":"https://doi.org/10.15421/242408","url":null,"abstract":"Let $mathbb K$ be a field of characteristic zero, $A := mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(mathbb K)$ the Lie algebra of all $mathbb K$-derivations on $A$. Every polynomial $f in A$ defines a Jacobian derivation $D_fin W_2(mathbb K)$ by the rule $D_f(h)=det J(f, h)$ for any $hin A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(mathbb K)$. It is proved that for a Jordan chain $T(f_1)=lambda f_1+f_2$, ..., $T(f_{k-1})=lambda f_{k-1}+f_k$, $T(f_k)=lambda f_k$ for a derivation $Tin W_2(mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(lambda -mathop{mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(lambda -mathop{mathrm{div}} T)D_{f_{k}}$ in $W_2(mathbb K)$. In case $A:=mathbb K[x_1, ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(mathbb K)$ on the principal ideals $(f)$ is considered.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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