Georgian Mathematical Journal最新文献

{"title":"Reproducing kernel Banach space defined by the minimal norm property and applications to partial differential equation theory","authors":"Tomasz Łukasz Żynda","doi":"10.1515/gmj-2023-2095","DOIUrl":"https://doi.org/10.1515/gmj-2023-2095","url":null,"abstract":"It it well known that a Hilbert space <jats:italic>V</jats:italic> of functions defined on <jats:italic>U</jats:italic> is a reproducing kernel Hilbert space if and only if for any <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:mi>U</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2095_eq_0240.png\" /> <jats:tex-math>{zin U}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, in the set <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>z</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mi>V</m:mi> </m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2095_eq_0132.png\" /> <jats:tex-math>{V_{z}:={fin Vmid f(z)=1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>z</m:mi> <m:mo>,</m:mo> <m:mi>w</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mover accent=\"true\"> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>w</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>¯</m:mo> </m:mover> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2095_eq_0105.png\" /> <jats:tex-math>{K(z,w)=overline{K(w,z)}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> does not have to be true without assumptions that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>z</m:mi> <m:mo>,</m:mo> <m:mi>w</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo ","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504238","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}

{"title":"Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform","authors":"Fethi Soltani, Ibrahim Maktouf","doi":"10.1515/gmj-2023-2077","DOIUrl":"https://doi.org/10.1515/gmj-2023-2077","url":null,"abstract":"Abstract We define and study the Stockwell transform <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> mathscr{S}_{g} associated to the Dunkl–Weinstein operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msub> </m:math> Delta_{k,beta} and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>f</m:mi> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:msub> </m:math> f_{Delta} and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> mathscr{S}_{g} . Moreover, we define the localization operators <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">L</m:mi> <m:mi>g</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> mathcal{L}_{g}(sigma) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>F</m:mi> <m:mrow> <m:mi>η</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>∗</m:mo> </m:msubsup> <m:mo lspace=\"0.278em\" rspace=\"0.278em\">:=</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mi>I</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msubsup> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>g</m:mi> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>g</m:mi> </m:msub> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>g</m:mi> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>k</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> F^{ast}_{eta,smash{k}}:=(eta I+mathscr{S}^{ast}_{g}mathscr{S}_{g})^{-1}mathscr{S}^{ast}_{g}(k) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> mathscr{S}_{g} on the Sobolev space <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi mathvariant=\"script\">H</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:m","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340610","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Characterization of Hilbert C^*-module higher derivations","authors":"S. Kh. Ekrami","doi":"10.1515/gmj-2023-2085","DOIUrl":"https://doi.org/10.1515/gmj-2023-2085","url":null,"abstract":"Abstract Let <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℳ</m:mi> </m:math> {mathcal{M}} be a Hilbert <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> {mathrm{C}^{*}} -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> {mathrm{C}^{*}} -module higher derivations <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:msub> <m:mi>φ</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mi mathvariant=\"script\">ℳ</m:mi> <m:mo>→</m:mo> <m:mi mathvariant=\"script\">ℳ</m:mi> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msubsup> </m:math> {{varphi_{n}:mathcal{M}rightarrowmathcal{M}}_{n=0}^{infty}} with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>φ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>=</m:mo> <m:mi>I</m:mi> </m:mrow> </m:math> {varphi_{0}=I} satisfying <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>φ</m:mi> <m:mi>n</m:mi> </m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> <m:mi>z</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:munder> <m:mo largeop=\"true\" movablelimits=\"false\" symmetric=\"true\">∑</m:mo> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>+</m:mo> <m:mi>j</m:mi> <m:mo>+</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>n</m:mi> </m:mrow> </m:munder> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:msub> <m:mi>φ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi>φ</m:mi> <m:mi>j</m:mi> </m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> <m:msub> <m:mi>φ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo mathvariant=\"italic\" separator=\"true\"> </m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"script\">ℳ</m:mi> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mi>n</m:mi> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> 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 al","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136159059","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}