Georgian Mathematical Journal最新文献

{"title":"Modular structure theory on Hom-Lie algebras","authors":"Dan Mao, Baoling Guan, Liangyun Chen","doi":"10.1515/gmj-2024-2048","DOIUrl":"https://doi.org/10.1515/gmj-2024-2048","url":null,"abstract":"The aim of this paper is to transfer the restrictedness theory to Hom-Lie algebras. The concept of restricted Hom-Lie algebras, which is introduced in [S. Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11 2023, 24, Paper No. 4955], will be used in this paper. First, the existence and uniqueness of <jats:italic>p</jats:italic>-structures on a Hom-Lie algebra is studied. Then the definition of a restrictable Hom-Lie algebra is given and the equivalence relation between restrictable Hom-Lie algebras and restricted Hom-Lie algebras is constructed. Finally, the <jats:italic>p</jats:italic>-envelopes of a Hom-Lie algebra are defined and studied.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224752","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":"Insights into a new class of unbounded operators","authors":"Aymen Bahloul","doi":"10.1515/gmj-2024-2047","DOIUrl":"https://doi.org/10.1515/gmj-2024-2047","url":null,"abstract":"The aim of this paper is to introduce the new class of left and right B-Weyl operators, which naturally extends the conventional concepts of left and right Weyl operators. Our contributions encompass demonstrating the stability of the left (and right) B-Weyl operators under small perturbations. We further characterize the left (and right) B-Weyl operators as the direct sum of a closed left (and right) Drazin invertible operator and a finite rank operator. Additionally, we present some characterizations of the left and right B-Weyl spectra, utilizing the left and right Drazin spectra as essential components. Furthermore, our obtained results play a pivotal role in exploring the interrelations between the left and right B-Weyl spectra and other spectra integral to the realm of B-Fredholm theory. This paper seeks to enhance and extend the recent research explored in [F. Abdmouleh and T. Ben Lakhal, Left and right B-Fredholm operators, Ukrainian Math. J. 74 2023, 10, 1479–1489] to a larger class in the unbounded B-Fredholm operators theory.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224753","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":"Dynamical mixed boundary-transmission problems of the generalized thermo-electro-magneto-elasticity theory for composed structures","authors":"Tengiz Buchukuri, Otar Chkadua, David Natroshvili","doi":"10.1515/gmj-2024-2051","DOIUrl":"https://doi.org/10.1515/gmj-2024-2051","url":null,"abstract":"In the present paper we investigate three-dimensional dynamical mixed boundary-transmission problems for composed body consisting of two adjacent anisotropic elastic components having a common interface surface. The two contacting elastic components are subject to different mathematical models: Green–Lindsay’s model of generalized thermo-electro-magneto-elasticity in one component and Green–Lindsay’s model of generalized thermo-elasticity in the other one. We assume that the composed elastic structure under consideration contains an interfacial crack. We prove the uniqueness and existence theorems in appropriate function spaces.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192110","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 a nonlinear general eigenvalue problem in Musielak–Orlicz spaces","authors":"Soufiane Kassimi, Hajar Sabiki, Hicham Moussa","doi":"10.1515/gmj-2024-2050","DOIUrl":"https://doi.org/10.1515/gmj-2024-2050","url":null,"abstract":"In this paper, we concern the existence result of the following general eigenvalue problem: &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo&gt;{&lt;/m:mo&gt; &lt;m:mtable columnspacing=\"0pt\" displaystyle=\"true\" rowspacing=\"0pt\"&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"right\"&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒜&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mrow&gt; &lt;m:mi/&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi mathvariant=\"script\"&gt;ℬ&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;m:mtd/&gt; &lt;m:mtd columnalign=\"right\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mtext&gt;in &lt;/m:mtext&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Ω&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"right\"&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;D&lt;/m:mi&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mrow&gt; &lt;m:mi/&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;m:mtd/&gt; &lt;m:mtd columnalign=\"right\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mtext&gt;on &lt;/m:mtext&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∂&lt;/m:mo&gt; &lt;m:mo&gt;⁡&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Ω&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;/m:mtable&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2050_eq_0066.png\"/&gt; &lt;jats:tex-math&gt;left{begin{aligned} displaystyle{}mathcal{A}(u)&amp;displaystyle={lambda}% mathcal{B}(u)&amp;&amp;displaystylephantom{}text{in }{Omega}, displaystyle D^{alpha}(u)&amp;displaystyle=0&amp;&amp;displaystylephantom{}text{on }% {partialOmega},end{aligned}right.&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt; in an arbitrary Musielak–Orlicz spaces, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒜&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2050_eq_0280.png\"/&gt; &lt;jats:tex-math&gt;{mathcal{A}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℬ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2050_eq_0281.png\"/&gt; &lt;jats:tex-math&gt;{mathcal{B}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; are quasilinear operators in divergence form of order &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" x","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192109","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":"Existence result for a Steklov problem involving a singular nonlinearity and variable exponents","authors":"Haikel Ouerghi, Khaled Ben Ali, A. Drissi","doi":"10.1515/gmj-2024-2049","DOIUrl":"https://doi.org/10.1515/gmj-2024-2049","url":null,"abstract":"\u0000 In this paper, we use the variational method to study some singular Steklov-type problem with variable exponents. More precisely, we use the min-max method in order to prove the existence of a solution to such a problem.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141929693","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":"A property of the free Gaussian distribution","authors":"Raouf Fakhfakh, Fatimah Alshahrani","doi":"10.1515/gmj-2024-2037","DOIUrl":"https://doi.org/10.1515/gmj-2024-2037","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">𝒦</m:mi> <m:mo>+</m:mo> </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:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:msub> <m:mi>ℙ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϑ</m:mi> <m:mo>,</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>ζ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mi>ϑ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:msub> <m:mi>ϑ</m:mi> <m:mo>+</m:mo> </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:mo stretchy=\"false\">)</m:mo> </m:mrow> </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-2024-2037_eq_0162.png\"/> <jats:tex-math>{{mathcal{K}_{+}}(sigma)={mathbb{P}_{(vartheta,sigma)}(dzeta):vartheta% in(0,vartheta_{+}(sigma))}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the Cauchy–Stieltjes Kernel (CSK) family generated by a probability measure σ which is non degenerate and has support bounded from above. Consider the concept of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2037_eq_0072.png\"/> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-transformation of measures introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545] for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>a</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-2024-2037_eq_0131.png\"/> <jats:tex-math>{ainmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>ℙ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϑ</m:mi> <m:mo>,</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886874","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":"Triple weak solution for p(x)-Laplacian like problem involving no flux boundary condition","authors":"Khaled Kefi, Nguyen Thanh Chung, Walid Abdelfattah","doi":"10.1515/gmj-2024-2043","DOIUrl":"https://doi.org/10.1515/gmj-2024-2043","url":null,"abstract":"In this paper, we consider a class of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <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:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2043_eq_0156.png\"/> <jats:tex-math>{p(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Laplacian like problems with indefinite weight involving no flux boundary condition. Using variational techniques and the critical point theorem of Bonanno and Marano [4], we prove the existence of at least three weak solutions to the problem in Sobolev variable exponent spaces.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886864","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 of a boundary value problem governed by the general elasticity system with a new boundary conditions in a thin domain","authors":"Abla Boulaouad, Youcef Djenaihi, Salah Boulaaras, Hamid Benseridi, Mourad Dilmi","doi":"10.1515/gmj-2024-2044","DOIUrl":"https://doi.org/10.1515/gmj-2024-2044","url":null,"abstract":"The aim of this work is the study of a nonlinear boundary value problem which theoretically generalizes the Lamé system with disturbance in a thin 3D domain with friction and a generalized boundary condition. For the resolution of the considered problem and after the variational formulation, we construct an operator from the variational problem. Then we prove that this operator has certain properties which allows us to apply the theorem of existence and uniqueness of the solution of variational inequalities of the 2nd kind. Finally, using a change of scale, we transport the variational problem to an equivalent problem defined on a domain independent of the parameter <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ζ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2044_eq_0343.png\"/> <jats:tex-math>{{zeta}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and subsequently we obtain the limit problem and the generalized weak equations of the initial problem.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886868","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 rotations and limit cycles of solutions to the basic system of equations","authors":"Grigor Barsegian","doi":"10.1515/gmj-2024-2042","DOIUrl":"https://doi.org/10.1515/gmj-2024-2042","url":null,"abstract":"This article discusses the rotations (windings) of solutions to the basic system of equations <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>y</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</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:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2042_eq_0102.png\"/> <jats:tex-math>{y^{prime}=F_{1}(x,y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</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:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2042_eq_0099.png\"/> <jats:tex-math>{x^{prime}=F_{2}(x,y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This allows us to return to the topic of known limit cycles from a much broader point of view, in particular, it makes it possible to describe the conditions for the existence of limit cycles.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886865","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 singular directions of a holomorphic mapping in P n(ℂ)","authors":"Nan Wu","doi":"10.1515/gmj-2024-2045","DOIUrl":"https://doi.org/10.1515/gmj-2024-2045","url":null,"abstract":"In this paper, we investigate the existence of singular directions, singular radii and indirect singular points of holomorphic curves in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>P</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ℂ</m:mi> <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-2024-2045_eq_0098.png\"/> <jats:tex-math>{P^{n}(mathbb{C})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886859","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}