Abstract The purpose of this work is to prove the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces S{N}{M}(ℝn) {S^{{M}}_{{N}}(mathbb{R}^{n})} , and to establish the inclusion between them.
{"title":"On the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces and inclusion between them","authors":"M’Hamed Bensaid, Rachid Chaïli","doi":"10.1515/gmj-2023-2087","DOIUrl":"https://doi.org/10.1515/gmj-2023-2087","url":null,"abstract":"Abstract The purpose of this work is to prove the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>S</m:mi> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {S^{{M}}_{{N}}(mathbb{R}^{n})} , and to establish the inclusion between them.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"136 S237","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135776403","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}
Abstract Recently, Freud and Rodriguez proposed a new counting process which is called the Bell–Touchard process and based on the Bell–Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are “degenerate versions” of the Bell–Touchard probability distributions and the Bell–Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process.
{"title":"Study on discrete degenerate Bell distributions with two parameters","authors":"Taekyun Kim, Dae San Kim, Hye Kyung Kim","doi":"10.1515/gmj-2023-2084","DOIUrl":"https://doi.org/10.1515/gmj-2023-2084","url":null,"abstract":"Abstract Recently, Freud and Rodriguez proposed a new counting process which is called the Bell–Touchard process and based on the Bell–Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are “degenerate versions” of the Bell–Touchard probability distributions and the Bell–Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"57 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136159052","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}
摘要 让ℳ {mathcal{M}} 是一个希尔伯特 C * {mathrm{C}^{*}} -模块。-模块。在本文中，我们证明了所有希尔伯特 C * {mathrm{C}^{*} -模块之间存在一一对应关系。｝-模块高阶导数 { φ n : ℳ → ℳ } n = 0 ∞ {{varphi_{n}：φ 0 = I {varphi_{0}=I} 满足 φ n ( 〈 x , y 〉 z ) = ∑ i + j + k = n 〈 φ i ( x ) , φ j ( y ) 〉 φ k ( z ) ( x , y 、z ∈ ℳ , n ∈ ℕ ∪ { 0 } ) varphi_{n}(langle x,yrangle z)=sum_{i+j+k=n}langlevarphi_{i}(x)、varphi_% {j}(y)ranglevarphi_{k}(z)quad(x,y,zinmathcal{M},,ninmathbb{N}cup{0}) and all Hilbert C * {mathrm{C}^{*}}.-模块派生 { ψ n : ℳ → ℳ } n = 1 ∞ {{psi_{n}：满足 ψ n ( 〈 x , y 〉 z ) = 〈 ψ n ( x ) 、y 〉 z + 〈 x , ψ n ( y ) 〉 z + 〈 x , y 〉 ψ n ( z ) ( x , y , z ∈ ℳ , n ∈ ℕ ) 、 psi_{n}(angle x,yrangle z)=langlepsi_{n}(x),yrangle z+langle x,psi_{n% }(y)rangle z+langle x、yranglepsi_{n}(z)quad(x,y,zinmathcal{M},,nin% mathbb{N})，并且我们证明了对于每一个希尔伯特 C * {mathrm{C}^{*}-上的每一个希尔伯特 C * {mathrm{C}^{*}} 模块的高阶导数 { φ n } n = 0 ∞ {{varphi_{n}}_{n=0}^{infty}} ，都存在唯一的序列。，存在一个唯一的希尔伯特 C * {mathrm{C}^{*}} 序列。-模块派生 { ψ n } n = 1 ∞ {{psi_{n}}_{n=1}^{infty}} 在 ℳ {{mathcal{M}} 上，使得 ψ n = ∑ k = 1 n ( ∑ ∑ j = 1 k r j = n ( - 1 ) k - 1 r 1 φ r 1 φ r 2 ... φ r k ) psi_{n}=sum_{k=1}^{n}biggl{(}sum_{sum_{j=1}^{k}r_{j}=n}(-1)^{k-1}~{}r_{1}% varphi_{r_{1}}varphi_{r_{2}}dotsvarphi_{r_{k}}biggr{)} 适用于所有正整数 n 、其中内求和取所有正整数 r j {r_{j}}，∑ j = 1 k r j = n {sum_{j=1}^{k}r_{j}=n} 。
Abstract Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 𝐏(i∂x)=(P1(i∂x))m1⋯(Pl(i∂x))ml mathbf{P}(ipartial_{x})=(P_{1}(ipartial_{x}))^{m_{1}}cdots(P_{l}(ipartial% _{x}))^{m_{l}} with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 𝐏(i∂x)u=f {mathbf{P}(ipartial_{x})u=f} in ℝn {mathbb{R}^{n}} . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.
摘要研究了具有实数零的特征多项式的准椭圆偏微分算子𝐏(i∂x)=(p1¹¹(i∂x)) m 1¹⋯⋯(p1¹¹(i∂x)) m l mathbf{P}(ipartial_{x})=(P_{1}(ipartial_{x}))^{m_{1}}cdots(P_{l}(ipartial% _{x}))^{m_{l}}的基本解在无穷远处的渐近行为。基于基本解的渐近展开式，给出了函数的渐近类，并建立了方程𝐏(i∑∂x)∑u=f {mathbf{P}(ipartial_{x})u=f}在∈n {mathbb{R}^{n}}中的解的存在唯一性。所得结果特别暗示了经典亥姆霍兹方程的一个新的唯一性定理。
{"title":"Asymptotic analysis of fundamental solutions of hypoelliptic operators","authors":"George Chkadua, Eugene Shargorodsky","doi":"10.1515/gmj-2023-2072","DOIUrl":"https://doi.org/10.1515/gmj-2023-2072","url":null,"abstract":"Abstract Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>𝐏</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:msub> <m:mo>∂</m:mo> <m:mi>x</m:mi> </m:msub> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:msub> <m:mo>∂</m:mo> <m:mi>x</m:mi> </m:msub> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:msub> <m:mi>m</m:mi> <m:mn>1</m:mn> </m:msub> </m:msup> <m:mo></m:mo> <m:mi mathvariant=\"normal\">⋯</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mi>l</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:msub> <m:mo>∂</m:mo> <m:mi>x</m:mi> </m:msub> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:msub> <m:mi>m</m:mi> <m:mi>l</m:mi> </m:msub> </m:msup> </m:mrow> </m:mrow> </m:math> mathbf{P}(ipartial_{x})=(P_{1}(ipartial_{x}))^{m_{1}}cdots(P_{l}(ipartial% _{x}))^{m_{l}} with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>𝐏</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:msub> <m:mo>∂</m:mo> <m:mi>x</m:mi> </m:msub> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> {mathbf{P}(ipartial_{x})u=f} in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {mathbb{R}^{n}} . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"16 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136159054","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}
Amal S. Alali, Hafedh Alnoghashi, Junaid Nisar, Nadeem ur Rehman, Faez A. Alqarni
Abstract According to Posner’s second theorem, a prime ring is forced to be commutative if a nonzero centralizing derivation exists on it. In this article, we extend this result to prime rings with antiautomorphisms and nonzero skew derivations. Additionally, a case is shown to demonstrate that the restrictions placed on the theorems’ hypothesis were not unnecessary.
{"title":"On skew derivations and antiautomorphisms in prime rings","authors":"Amal S. Alali, Hafedh Alnoghashi, Junaid Nisar, Nadeem ur Rehman, Faez A. Alqarni","doi":"10.1515/gmj-2023-2082","DOIUrl":"https://doi.org/10.1515/gmj-2023-2082","url":null,"abstract":"Abstract According to Posner’s second theorem, a prime ring is forced to be commutative if a nonzero centralizing derivation exists on it. In this article, we extend this result to prime rings with antiautomorphisms and nonzero skew derivations. Additionally, a case is shown to demonstrate that the restrictions placed on the theorems’ hypothesis were not unnecessary.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"53 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318651","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}
Abstract The numerical solution of the nonlinear system of equations resulting from a real engineering problem is discussed. We use the approximate solution of a system of two nonlinear integrodifferential equations to build the nonlinear system of equations. This system can be solved by Newton’s method if the solution is differentiable, or using some derivative-free methods, such as Steffensen’s method. Here we show that Steffensen’s method does not always converge and secant method requires more iterations than Traub’s method and Newton’s method. We recommend Traub’s method in case the solution is not differentiable.
{"title":"Comparison of several numerical solvers for a discretized nonlinear diffusion model with source terms","authors":"Beny Neta","doi":"10.1515/gmj-2023-2078","DOIUrl":"https://doi.org/10.1515/gmj-2023-2078","url":null,"abstract":"Abstract The numerical solution of the nonlinear system of equations resulting from a real engineering problem is discussed. We use the approximate solution of a system of two nonlinear integrodifferential equations to build the nonlinear system of equations. This system can be solved by Newton’s method if the solution is differentiable, or using some derivative-free methods, such as Steffensen’s method. Here we show that Steffensen’s method does not always converge and secant method requires more iterations than Traub’s method and Newton’s method. We recommend Traub’s method in case the solution is not differentiable.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234073","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}
Ljubiša D. R. Kočinac, Farkhod G. Mukhamadiev, Anvar K. Sadullaev
Abstract In this paper, we study the behavior of some classes of topological spaces under the influence of the functor of G -permutation degree 𝖲𝖯Gn {operatorname{sf SP}^{n}_{G}} . We prove: (a) if a space X is an r -space, then so is 𝖲𝖯GnX {operatorname{sf SP}_{G}^{n}X} , (b) if X is a cosmic space, then so is 𝖲𝖯GnX {operatorname{sf SP}_{G}^{n}X} , (c) if a space X is a C(κ) {C(kappa)} -cosmic, then so is 𝖲𝖯GnX {operatorname{sf SP}_{G}^{n}X} , (d) if a space X is an α-space, then so is 𝖲𝖯GnX {operatorname{sf SP}_{G}^{n}X} .
{"title":"Some classes of topological spaces and the space of <i>G</i>-permutation degree","authors":"Ljubiša D. R. Kočinac, Farkhod G. Mukhamadiev, Anvar K. Sadullaev","doi":"10.1515/gmj-2023-2080","DOIUrl":"https://doi.org/10.1515/gmj-2023-2080","url":null,"abstract":"Abstract In this paper, we study the behavior of some classes of topological spaces under the influence of the functor of G -permutation degree <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>𝖲𝖯</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> {operatorname{sf SP}^{n}_{G}} . We prove: (a) if a space X is an r -space, then so is <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>𝖲𝖯</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo></m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {operatorname{sf SP}_{G}^{n}X} , (b) if X is a cosmic space, then so is <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>𝖲𝖯</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo></m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {operatorname{sf SP}_{G}^{n}X} , (c) if a space X is a <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>κ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {C(kappa)} -cosmic, then so is <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>𝖲𝖯</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo></m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {operatorname{sf SP}_{G}^{n}X} , (d) if a space X is an α-space, then so is <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>𝖲𝖯</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo></m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {operatorname{sf SP}_{G}^{n}X} .","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135766917","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}
Abstract Let G be a finite non-abelian group. The commuting conjugacy class graph Γ(G) {Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ(G) {Gamma(G)} are connected by an edge if there exist elements x∈X {xin X} and y∈Y {yin Y} such that xy=yx {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that GZ(G) {frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or p2q {p^{2}q} , is determined.
{"title":"The commuting conjugacy class graphs of finite groups with a given property","authors":"Mehdi Rezaei, Zeinab Foruzanfar","doi":"10.1515/gmj-2023-2069","DOIUrl":"https://doi.org/10.1515/gmj-2023-2069","url":null,"abstract":"Abstract Let G be a finite non-abelian group. The commuting conjugacy class graph <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {Gamma(G)} are connected by an edge if there exist elements <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {xin X} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:math> {yin Y} such that <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:math> {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mi>G</m:mi> <m:mrow> <m:mi>Z</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:math> {frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi>q</m:mi> </m:mrow> </m:math> {p^{2}q} , is determined.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549063","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}
Abstract In this study, the concept of ρ-statistical convergence with respect to the neutrosophic norm in the neutrosophic normed spaces is introduced. Some properties and some inclusion theorems related to this concept are investigated.
{"title":"On ρ-statistical convergence in neutrosophic normed spaces","authors":"Sibel Ersan","doi":"10.1515/gmj-2023-2076","DOIUrl":"https://doi.org/10.1515/gmj-2023-2076","url":null,"abstract":"Abstract In this study, the concept of ρ-statistical convergence with respect to the neutrosophic norm in the neutrosophic normed spaces is introduced. Some properties and some inclusion theorems related to this concept are investigated.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547570","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}
Abstract Under GCH , there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the card(G) {{rm card}(G)} -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G . Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group 𝐒1 {{bf S}_{1}} .
{"title":"Quasi-invariant measures on topological groups and ω-powers","authors":"Alexander Kharazishvili","doi":"10.1515/gmj-2023-2073","DOIUrl":"https://doi.org/10.1515/gmj-2023-2073","url":null,"abstract":"Abstract Under GCH , there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>card</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {{rm card}(G)} -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G . Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>𝐒</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {{bf S}_{1}} .","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"221 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549145","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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