The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.
{"title":"On the representation of solution for the perturbed quasi-linear controlled neutral functional-differential equation with the discontinuous initial condition","authors":"Abdeljalil Nachaoui, Tea Shavadze, Tamaz Tadumadze","doi":"10.1515/gmj-2023-2122","DOIUrl":"https://doi.org/10.1515/gmj-2023-2122","url":null,"abstract":"The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"53 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139422507","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}
Let <jats:italic>R</jats:italic> be a prime ring, let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≠</m:mo> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:mi>R</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2121_eq_0112.png" /> <jats:tex-math>{0neq bin R}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let α and β be two automorphisms of <jats:italic>R</jats:italic>. Suppose that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>F</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2121_eq_0147.png" /> <jats:tex-math>{F:Rrightarrow R}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2121_eq_0155.png" /> <jats:tex-math>{F_{1}:Rrightarrow R}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are two <jats:italic>b</jats:italic>-generalized <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2121_eq_0103.png" /> <jats:tex-math>{(alpha,beta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivations of <jats:italic>R</jats:italic> associated with the same <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2121_eq_0103.png" /> <jats:tex-math>{(alpha,beta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2121_eq_0098.png" /> <jats:tex-math>d:Rrightarrow R</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi>
设 R 是素环，设 0≠b∈R {0neq bin R} ，设 α 和 β 是 R 的两个自变量。 假设 F : R → R {F:Rrightarrow R} , F 1 : R → R {F_{1}:Rrightarrow R} 是 R 的两个自变量。 , F 1 : R → R {F_{1}:Rrightarrow R} 是 R 的两个 b-generalized ( α , β ) {(alpha,beta)} -derivation ，与同一个 ( α , β ) {(alpha,beta)} -derivation d 相关联： R → R d:Rrightarrow R ，让 G : R → R G:Rrightarrow R 是 R 的一个 b-generalized ( α , β ) (alpha,beta) -derivation ，与 ( α , β ) (alpha,beta) -derivation g 相关联： R → R g:Rrightarrow R 。本文的主要目的是研究以下代数等式：(1) F ( x y ) + α ( x y ) + α ( y x ) = 0 {F(xy)+alpha(xy)+alpha(yx)=0} ，(2) F ( x y ) + α ( x y ) + α ( y x ) = 0 {F(xy)+alpha(xy)+alpha(yx)=0} 。 (2) F ( x y ) + G ( x ) α ( y ) + α ( y x ) = 0 {F(xy)+G(x)alpha(y)+alpha(yx)=0} (3) F ( x y ) + G ( y x ) + α ( x y ) + α ( y x ) = 0 {F(xy)+G(yx)+alpha(xy)+alpha(yx)=0} , (4) F ( x ) + G ( x ) + α ( y x ) = 0 {F(xy)+G(x)+alpha(yx)=0} (4) F ( x ) F ( y ) + G ( x ) α ( y ) + α ( y x ) = 0 {F(x)F(y)+G(x)alpha(y)+alpha(yx)=0} , (5) F ( x y ) + G ( yx ) + α ( y x ) = 0 {F(x)F(y)+G(x)alpha(yx)=0} (5) F ( x y ) + d ( x ) F 1 ( y ) + α ( x y ) = 0 {F(xy)+d(x)F_{1}(y)+alpha(xy)=0} (6) F ( x y ) + d ( x ) F 1 ( y ) = 0 {F(xy)+d(x)F_{1}(y)=0} , (7) F ( x y ) + d ( x ) F 1 ( y ) = 0 {F(xy)+d(x)F_{1}(y)=0} (7) F ( x y ) + d ( x ) F 1 ( y ) + α ( y x ) = 0 {F(xy)+d(x)F_{1}(y)+alpha(yx)=0} , (8) F ( x y ) + d ( x ) F 1 ( y ) = 0 {F(xy)+d(x)F_{1}(y)+alpha(yx)=0} (8) F ( x y ) + d ( x ) F 1 ( y ) + α ( x y ) + α ( y x ) = 0 {F(xy)+d(x)F_{1}(y)+alpha(xy)+alpha(yx)=0} , (9) F ( x y ) + d ( x ) F 1 ( y ) + α ( y x ) = 0 {F(xy)+d(x)F_{1}(y)+alpha(yx)=0} (9) F ( x y ) + d ( x ) F 1 ( y ) + α ( y x ) - α ( x y ) = 0 {F(xy)+d(x)F_{1}(y)+alpha(yx)-alpha(xy)=0} , (10) [ F ( x y ) +d(x)F_{1}(y)+alpha(yx)-alpha(xy)=0} (10) [ F ( x ) , x ] α , β = 0 {[F(x),x]_{alpha,beta}=0} , (11) ( F ( x ) , x ] α , β = 0 {[F(x),x]_{alpha,beta}=0} (11) ( F ( x ) ∘ x ) α , β = 0 {(F(x)circ x)_{alpha,beta}=0} , (12) F ( [ x ] α , β = 0 {[F(x)circ x]_{alpha,beta}=0} (12) F ( [ x , y ] ) = [ x , y ] α , β {F([x,y])=[x,y]_{alpha,beta}} (13) F ( x ∘ y ) = ( x ∘ y ) α , β {F(xcirc y)=(xcirc y)_{alpha,beta}} for all x , y {x,y} in some suitable subset of R.
{"title":"A note on b-generalized (α,β)-derivations in prime rings","authors":"Nripendu Bera, Basudeb Dhara","doi":"10.1515/gmj-2023-2121","DOIUrl":"https://doi.org/10.1515/gmj-2023-2121","url":null,"abstract":"Let <jats:italic>R</jats:italic> be a prime ring, let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>0</m:mn> <m:mo>≠</m:mo> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:mi>R</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0112.png\" /> <jats:tex-math>{0neq bin R}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let α and β be two automorphisms of <jats:italic>R</jats:italic>. Suppose that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>F</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0147.png\" /> <jats:tex-math>{F:Rrightarrow R}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0155.png\" /> <jats:tex-math>{F_{1}:Rrightarrow R}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are two <jats:italic>b</jats:italic>-generalized <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0103.png\" /> <jats:tex-math>{(alpha,beta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivations of <jats:italic>R</jats:italic> associated with the same <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0103.png\" /> <jats:tex-math>{(alpha,beta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>d</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>→</m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0098.png\" /> <jats:tex-math>d:Rrightarrow R</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi>","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"130 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139103439","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 present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T1mathbb{T}^{1}. The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.
本文研究了圆 T 1 mathbb{T}^{1} 上双曲方程的非局部边界值问题。本文提出了用于圆上双曲方程非局部边界值问题数值求解的一阶修正差分方案。建立了差分方案解在各种赫尔德规范下的稳定性和矫顽力估计。此外，还提供了数值示例。
{"title":"Numerical approaches for solution of hyperbolic difference equations on circle","authors":"Allaberen Ashyralyev, Fatih Hezenci, Yasar Sozen","doi":"10.1515/gmj-2023-2103","DOIUrl":"https://doi.org/10.1515/gmj-2023-2103","url":null,"abstract":"The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"double-struck\">T</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2103_ineq_0001.png\" /> <jats:tex-math>mathbb{T}^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"705 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139092302","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}
Under some suitable conditions, we show that at least three weak solutions exist for a system of differential equations involving the (p(x),q(x)){(p(x),q(x))} Laplacian-like with indefinite weights. The proof is related to the Bonanno–Marano critical theorem (Appl. Anal. 89 (2010), 1–10).
在一些合适的条件下，我们证明了涉及 ( p ( x ) , q ( x ) ) 的微分方程系统至少存在三个弱解 {(p(x),q(x))} 的微分方程系统至少存在三个弱解。证明与 Bonanno-Marano 临界定理有关（Appl.89 (2010), 1-10).
{"title":"Multiplicity result for a (p(x),q(x))-Laplacian-like system with indefinite weights","authors":"Khaled Kefi, Chaima Nefzi","doi":"10.1515/gmj-2023-2107","DOIUrl":"https://doi.org/10.1515/gmj-2023-2107","url":null,"abstract":"Under some suitable conditions, we show that at least three weak solutions exist for a system of differential equations involving the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2107_eq_0108.png\" /> <jats:tex-math>{(p(x),q(x))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Laplacian-like with indefinite weights. The proof is related to the Bonanno–Marano critical theorem (<jats:italic>Appl. Anal.</jats:italic> 89 (2010), 1–10).","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"124 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077826","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}
Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2113_eq_0076.png" /> <jats:tex-math>{Phi:[0,infty)rightarrow[0,infty)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Young’s function satisfying the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2113_eq_0070.png" /> <jats:tex-math>{Delta_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-condition and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>M</m:mi> <m:mi mathvariant="script">ℬ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2113_eq_0063.png" /> <jats:tex-math>{M_{mathcal{B}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the geometric maximal operator associated to a homothecy invariant basis <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℬ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2113_eq_0098.png" /> <jats:tex-math>{mathcal{B}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> acting on measurable functions on <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2113_eq_0090.png" /> <jats:tex-math>{mathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:italic>Q</jats:italic> be the unit cube in <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2113_eq_0090.png" /> <jats:tex-math>{mathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">Φ</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:
设 Φ : [ 0 , ∞ ) → [ 0 , ∞ ) {Phi:[0,infty)rightarrow[0,infty)}是满足Δ 2 {Delta_{2}} 的杨氏函数。 -条件，并让 M ℬ {M_{mathcal{B}} 是与作用于ℝ n {mathbb{R}^{n} 上可测函数的同神不变基 ℬ {mathcal{B}} 相关联的几何最大算子。｝ .设 Q 是 ℝ n {mathbb{R}^{n}} 中的单位立方体，设 L Φ ( Q ) {L^{Phi}(Q)} 是与Φ 相关的奥利兹空间，其规范为 ∥ f ∥ L Φ ( Q ) := inf { c > 0 : ∫ Q Φ ( | f | c ) ≤ 1 } . . |f|_{L^{Phi}(Q)}:=infBiggl{{}c>0:int_{Q}Phibigg{(}frac{|f|}{c}bigg{% )}leq 1Bigg{}}. 我们证明 M ℬ {M_{mathcal{B}}} 满足弱类型估计 | { x ∈ ℝ n : M ℬ f ( x ) > α }。 | ≤ C 1 ∫ ℝ n Φ ( | f | α ) |{xinmathbb{R}^{n}:M_{{mathcal{B}}kern 1.422638ptf(x)>alpha}|leq C_{1}% int_{mathbb{R}^{n}}Phibigg{(}frac{|f|}{alpha}bigg{)} for all measurable functions f on ℝ n {mathbb{R}^{n}} and α >；0 {alpha>0} 当且仅当 M ℬ {M_{mathcal{B}}} 满足弱类型估计 | { x∈ Q ： M ℬ f ( x ) > α } ≤ C 2 ∥ f ∥ L Φ ( Q ) α |{xin Q:M_{mathcal{B}}kern 1.422638ptf(x)>alpha}|leq C_{2}frac{|f|{% L^{Phi}(Q)}}{alpha} for all measurable functions f supported on Q and α > 0 {alpha>0} .由于这一等价性，我们证明，如果 Φ 满足上述条件，且 ℬ {mathcal{B}} 是一个同神不变基，微分 ℝ n {mathbb{R}^{n} 上所有可测函数 f 的积分，使得 ∫ ℝ n Φ ( | f | ) <；∞ {int_{mathbb{R}^{n}}Phi(|f|)<infty} 那么相关的最大算子 M ℬ {M_{mathcal{B}}} 满足上述两个弱类型估计。
{"title":"Two presentations of a weak type inequality for geometric maximal operators","authors":"Paul Hagelstein, Giorgi Oniani, Alex Stokolos","doi":"10.1515/gmj-2023-2113","DOIUrl":"https://doi.org/10.1515/gmj-2023-2113","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0076.png\" /> <jats:tex-math>{Phi:[0,infty)rightarrow[0,infty)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Young’s function satisfying the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0070.png\" /> <jats:tex-math>{Delta_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-condition and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mi mathvariant=\"script\">ℬ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0063.png\" /> <jats:tex-math>{M_{mathcal{B}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the geometric maximal operator associated to a homothecy invariant basis <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℬ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0098.png\" /> <jats:tex-math>{mathcal{B}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> acting on measurable functions on <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0090.png\" /> <jats:tex-math>{mathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:italic>Q</jats:italic> be the unit cube in <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0090.png\" /> <jats:tex-math>{mathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">Φ</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077895","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 notion of (real-valued) almost measurable functions on probability spaces is introduced and some of their properties are considered. It is shown that any almost measurable function may be treated as a quasi-random variable in the sense of [A. Kharazishvili, On some version of random variables, Trans. A. Razmadze Math. Inst. 177 2023, 1, 143–146].
引入了概率空间上（实值）几乎可测函数的概念，并考虑了它们的一些性质。研究表明，任何几乎可测函数都可被视为[A. Kharazishvili, On some version of random variable, Trans.Kharazishvili, On some version of random variables, Trans.A. Razmadze Math.177 2023, 1, 143-146].
{"title":"Almost measurable functions on probability spaces","authors":"Alexander Kharazishvili","doi":"10.1515/gmj-2023-2120","DOIUrl":"https://doi.org/10.1515/gmj-2023-2120","url":null,"abstract":"The notion of (real-valued) almost measurable functions on probability spaces is introduced and some of their properties are considered. It is shown that any almost measurable function may be treated as a quasi-random variable in the sense of [A. Kharazishvili, On some version of random variables, Trans. A. Razmadze Math. Inst. 177 2023, 1, 143–146].","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077822","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 paper introduces and studies the ϕ-u-S-flat (resp., nonnil-u-S-injective) modules, which are a generalization of both ϕ-flat modules and u-S-flat modules (resp., both nonnil-injective modules and u-S-injective modules). We give the Cartan–Eilenberg–Bass theorem for nonnil-u-S-Noetherian rings. Finally, we offer some new characterizations of the ϕ-von Neumann regular ring.
{"title":"On φ-u-S-flat modules and nonnil-u-S-injective modules","authors":"Hwankoo Kim, Najib Mahdou, El Houssaine Oubouhou","doi":"10.1515/gmj-2023-2117","DOIUrl":"https://doi.org/10.1515/gmj-2023-2117","url":null,"abstract":"This paper introduces and studies the ϕ-u-<jats:italic>S</jats:italic>-flat (resp., nonnil-u-<jats:italic>S</jats:italic>-injective) modules, which are a generalization of both ϕ-flat modules and u-<jats:italic>S</jats:italic>-flat modules (resp., both nonnil-injective modules and u-<jats:italic>S</jats:italic>-injective modules). We give the Cartan–Eilenberg–Bass theorem for nonnil-u-<jats:italic>S</jats:italic>-Noetherian rings. Finally, we offer some new characterizations of the ϕ-von Neumann regular ring.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077993","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}
Hari M. Srivastava, Bhawna Gupta, Mohammad Idris Qureshi, Mohd Shaid Baboo
Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.
{"title":"Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava","authors":"Hari M. Srivastava, Bhawna Gupta, Mohammad Idris Qureshi, Mohd Shaid Baboo","doi":"10.1515/gmj-2023-2114","DOIUrl":"https://doi.org/10.1515/gmj-2023-2114","url":null,"abstract":"Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"173 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077828","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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