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 p-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 p-envelopes of a Hom-Lie algebra are defined and studied.
本文的目的是将受限性理论应用于 Hom-Lie 对象。受限 Hom-Lie 代数的概念是在 [S. Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11] 中提出的。Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11 2023, 24, Paper No.首先研究 Hom-Lie 代数上 p 结构的存在性和唯一性。然后给出了可限制 Hom-Lie 代数的定义，并构造了可限制 Hom-Lie 代数与可限制 Hom-Lie 代数之间的等价关系。最后，定义并研究了 Hom-Lie 代数的 p 包络。
{"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}
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.
本文旨在介绍左、右 B-韦尔算子这一新类别，它自然地扩展了左、右韦尔算子的传统概念。我们的贡献包括证明左（和右）B-Weyl 算子在微小扰动下的稳定性。我们进一步将左（和右）B-Weyl 算子描述为封闭的左（和右）Drazin 不可逆算子与有限秩算子的直接和。此外，我们还利用左、右德拉津谱作为基本组成部分，提出了左、右 B-Weyl 谱的一些特征。此外，我们所获得的结果对于探索左、右 B-Weyl 光谱与 B-Fredholm 理论领域中其他光谱之间的相互关系起到了关键作用。本文试图加强和扩展最近在 [F. Abdmouleh and T. Abdmouleh and T. Abdmouleh and T. Abdmouleh] 中探索的研究。Abdmouleh 和 T. Ben Lakhal, 左和右 B-Fredholm 算子, 乌克兰数学.J. 74 2023, 10, 1479-1489] 中探索的无界 B-Fredholm 算子理论的更大范畴。
{"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}
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.
{"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}
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.
{"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}
Let 𝒦+(σ)={ℙ(ϑ,σ)(dζ):ϑ∈(0,ϑ+(σ))}{{mathcal{K}_{+}}(sigma)={mathbb{P}_{(vartheta,sigma)}(dzeta):vartheta% in(0,vartheta_{+}(sigma))}} 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 Va{V_{a}}-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 a∈ℝ{ainmathbb{R}}. We prove that Va(ℙ(ϑ,σ))∈
设 𝒦 + ( σ ) = { ϑ , σ ) ( d ζ ) : ϑ ∈ ( 0 , ϑ + ( σ ) ) } {{mathcal{K}_{+}}(sigma)={mathbb{P}_{(vartheta,sigma)}(dzeta):vartheta% in(0,vartheta_{+}(sigma))}} 是由概率度量 σ 生成的 Cauchy-Stieltjes Kernel（CSK）族，该概率度量 σ 是非退化的，且有上界支撑。考虑一下 V a {V_{a}} 的概念。 -中引入的度量的 V a {V_{a}} 变换概念[A.D. Krystek 和 L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin.Dimens.Anal.Quantum Probab.Relat.Top.8 2005, 3, 515-545] for a ∈ ℝ {ainmathbb{R}} . .我们证明 V a ( 𡆙 ( ϑ , σ ) ) ∈ 𝒦 + ( σ ) {V_{a}(mathbb{P}_{(vartheta,sigma)})in{mathcal{K}_{+}}(sigma)} for all a ∈ ℝ ∖ { 0 }. {ainmathbb{R}setminus{0}}，当且仅当度量 σ 是自由高斯（半圆）类型的亲和力法则。
{"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}
In this paper, we consider a class of p(x){p(x)}-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.
在本文中，我们考虑了一类类似于 p ( x ) {p(x)} 的拉普拉斯问题，该问题具有不确定权重，不涉及通量边界条件。利用变分技术以及 Bonanno 和 Marano [4] 的临界点定理，我们证明了该问题在 Sobolev 可变指数空间中至少存在三个弱解。
{"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}
Abla Boulaouad, Youcef Djenaihi, Salah Boulaaras, Hamid Benseridi, Mourad Dilmi
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 ζ{{zeta}} and subsequently we obtain the limit problem and the generalized weak equations of the initial problem.
{"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}
This article discusses the rotations (windings) of solutions to the basic system of equations y′=F1(x,y){y^{prime}=F_{1}(x,y)} and x′=F2(x,y){x^{prime}=F_{2}(x,y)}. 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.
本文讨论基本方程组 y ′ = F 1 ( x , y ) {y^{prime}=F_{1}(x,y)} 和 x ′ = F 2 ( x , y ) {x^{prime}=F_{2}(x,y)} 的旋转（绕组）解。这样，我们就可以从更广阔的视角回到已知极限循环的话题上来，特别是，它使我们有可能描述极限循环存在的条件。
{"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}
In this paper, we investigate the existence of singular directions, singular radii and indirect singular points of holomorphic curves in Pn(ℂ){P^{n}(mathbb{C})}.
本文研究了 P n ( ℂ ) {P^{n}(mathbb{C})} 中全纯曲线的奇异方向、奇异半径和间接奇异点的存在性。
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