. In this paper, we study the Poisson cohomology ring of the tensor product of Poisson algebras. Explicitly, it is proved that the Pois- son cohomology ring of tensor product of two Poisson algebras is isomorphic to the tensor product of the respective Poisson cohomology ring of these two Poisson algebras as Gerstenhaber algebras.
{"title":"COHOMOLOGY RING OF THE TENSOR PRODUCT OF POISSON ALGEBRAS","authors":"Can Zhu","doi":"10.4134/JKMS.J180762","DOIUrl":"https://doi.org/10.4134/JKMS.J180762","url":null,"abstract":". In this paper, we study the Poisson cohomology ring of the tensor product of Poisson algebras. Explicitly, it is proved that the Pois- son cohomology ring of tensor product of two Poisson algebras is isomorphic to the tensor product of the respective Poisson cohomology ring of these two Poisson algebras as Gerstenhaber algebras.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"113-129"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70510801","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 be a field of characteristic 0. Let C = Spec(k[x, y]/〈f〉) be a weighted homogeneous plane curve singularity with tangent space πC : TC/k → C . In this article, we study, from a computational point of view, the Zariski closure G (C ) of the set of the 1-jets on C which define formal solutions (in F [[t]]2 for field extensions F of k) of the equation f = 0. We produce Groebner bases of the ideal N1(C ) defining G (C ) as a reduced closed subscheme of TC/k and obtain applications in terms of logarithmic differential operators (in the plane) along C .
{"title":"On the tangent space of a weighted homogeneous plane curve singularity","authors":"J. Sebag, M. Cañón","doi":"10.4134/JKMS.J180796","DOIUrl":"https://doi.org/10.4134/JKMS.J180796","url":null,"abstract":"Let k be a field of characteristic 0. Let C = Spec(k[x, y]/〈f〉) be a weighted homogeneous plane curve singularity with tangent space πC : TC/k → C . In this article, we study, from a computational point of view, the Zariski closure G (C ) of the set of the 1-jets on C which define formal solutions (in F [[t]]2 for field extensions F of k) of the equation f = 0. We produce Groebner bases of the ideal N1(C ) defining G (C ) as a reduced closed subscheme of TC/k and obtain applications in terms of logarithmic differential operators (in the plane) along C .","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"145-169"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70510833","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 introduce a parallel iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in a real reflexive Banach space. Moreover, we give some applications of the main theorem for solving some related problems. Finally, some numerical examples are developed to illustrate the behavior of the new algorithms with respect to existing algorithms.
{"title":"A PARALLEL ITERATIVE METHOD FOR A FINITE FAMILY OF BREGMAN STRONGLY NONEXPANSIVE MAPPINGS IN REFLEXIVE BANACH SPACES","authors":"J. Kim, Truong Minh Tuyen","doi":"10.4134/JKMS.J190268","DOIUrl":"https://doi.org/10.4134/JKMS.J190268","url":null,"abstract":"In this paper, we introduce a parallel iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in a real reflexive Banach space. Moreover, we give some applications of the main theorem for solving some related problems. Finally, some numerical examples are developed to illustrate the behavior of the new algorithms with respect to existing algorithms.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"617-640"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511255","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}
. Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni–Salce [7] and Bazzoni [4], are gener- alized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs–Salce [14], showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost per- fect rings within the class of commutative rings (Theorem 3.7). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair ( P 1 , D ) (Theorem 4.1). Similar characterization is proved concern- ing the existence of divisible envelopes for h -local rings in the same class (Theorem 5.3). In addition, Bazzoni’s characterization via direct sums of weak-injective modules [4] is extended to all commutative rings (Theorem 6.4). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.
{"title":"CHARACTERIZING ALMOST PERFECT RINGS BY COVERS AND ENVELOPES","authors":"L. Fuchs","doi":"10.4134/JKMS.J180793","DOIUrl":"https://doi.org/10.4134/JKMS.J180793","url":null,"abstract":". Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni–Salce [7] and Bazzoni [4], are gener- alized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs–Salce [14], showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost per- fect rings within the class of commutative rings (Theorem 3.7). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair ( P 1 , D ) (Theorem 4.1). Similar characterization is proved concern- ing the existence of divisible envelopes for h -local rings in the same class (Theorem 5.3). In addition, Bazzoni’s characterization via direct sums of weak-injective modules [4] is extended to all commutative rings (Theorem 6.4). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"131-144"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511004","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 S be the collection of the operator matrices ( A C Z B ) where the range of C is closed. In this paper, we study the properties of operator matrices in the class S. We first explore various local spectral relations, that is, the property (β), decomposable, and the property (C) between the operator matrices in the class S and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class S, and as some applications, we provide the conditions for such operator matrices to satisfy a-Weyl’s theorem and a-Browder’s theorem, respectively.
设S是算子矩阵(A C Z B)的集合，其中C的值域是封闭的。本文研究了S类算子矩阵的性质。首先探讨了S类算子矩阵与其组成算子之间的各种局域谱关系，即性质(β)、可分解性和性质(C)。此外，我们研究了S类算子矩阵的Weyl和Browder型谱，并作为一些应用，给出了这类算子矩阵分别满足a-Weyl定理和a-Browder定理的条件。
{"title":"PROPERTIES OF OPERATOR MATRICES","authors":"I. An, E. Ko, J. Lee","doi":"10.4134/JKMS.J190439","DOIUrl":"https://doi.org/10.4134/JKMS.J190439","url":null,"abstract":"Let S be the collection of the operator matrices ( A C Z B ) where the range of C is closed. In this paper, we study the properties of operator matrices in the class S. We first explore various local spectral relations, that is, the property (β), decomposable, and the property (C) between the operator matrices in the class S and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class S, and as some applications, we provide the conditions for such operator matrices to satisfy a-Weyl’s theorem and a-Browder’s theorem, respectively.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"893-913"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511737","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}
. Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly ap- peared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lift- ing of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.
{"title":"ON LIFTING OF STABLE RANGE ONE ELEMENTS","authors":"Meltem Altun-Özarslan, A. C. Ozcan","doi":"10.4134/JKMS.J190382","DOIUrl":"https://doi.org/10.4134/JKMS.J190382","url":null,"abstract":". Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly ap- peared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lift- ing of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"793-807"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511878","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 classify all solutions bounded from below to uniformly elliptic equations of second order in the form of Lu(x) = aij(x)Diju(x) + bi(x)Diu(x) + c(x)u(x) = f(x) or Lu(x) = Di(aij(x) Dju(x)) + bi(x)Diu(x) + c(x)u(x) = f(x) in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term f of the equation.
{"title":"The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders","authors":"Lidan Wang, Lihe Wang, Chunqin Zhou","doi":"10.4134/JKMS.J190836","DOIUrl":"https://doi.org/10.4134/JKMS.J190836","url":null,"abstract":"In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of Lu(x) = aij(x)Diju(x) + bi(x)Diu(x) + c(x)u(x) = f(x) or Lu(x) = Di(aij(x) Dju(x)) + bi(x)Diu(x) + c(x)u(x) = f(x) in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term f of the equation.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"1573-1590"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70513554","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 derive various differential Harnack estimates for positive solutions to the nonlinear backward heat type equations on closed manifolds coupled with the Ricci-Bourguignon flow, which was done for the Ricci flow by J.-Y. Wu [30]. The proof follows exactly the one given by X.-D. Cao [4] for the linear backward heat type equations coupled with the Ricci flow.
{"title":"HARNACK ESTIMATES FOR NONLINEAR BACKWARD HEAT EQUATIONS WITH POTENTIALS ALONG THE RICCI-BOURGUIGNON FLOW","authors":"Jian-hong Wang","doi":"10.4134/JKMS.J190049","DOIUrl":"https://doi.org/10.4134/JKMS.J190049","url":null,"abstract":"In this paper, we derive various differential Harnack estimates for positive solutions to the nonlinear backward heat type equations on closed manifolds coupled with the Ricci-Bourguignon flow, which was done for the Ricci flow by J.-Y. Wu [30]. The proof follows exactly the one given by X.-D. Cao [4] for the linear backward heat type equations coupled with the Ricci flow.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"313-329"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511363","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 p be an analytic function defined on the open unit disk D. We obtain certain differential subordination implications such as ψ(p) := pλ(z)(α+βp(z)+γ/p(z)+δzp′(z)/pj(z)) ≺ h(z) (j = 1, 2) implies p ≺ q, where h is given by ψ(q) and q belongs to P, by finding the conditions on α, β, γ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy | log(zf ′(z)/f(z))| < 1, |(zf ′(z)/f(z))2 − 1| < 1 and zf ′(z)/f(z) lying in the parabolic region v2 < 2u− 1.
{"title":"APPLICATIONS OF DIFFERENTIAL SUBORDINATIONS TO CERTAIN CLASSES OF STARLIKE FUNCTIONS","authors":"S. Banga, Sivaprasad Kumar","doi":"10.4134/JKMS.J190051","DOIUrl":"https://doi.org/10.4134/JKMS.J190051","url":null,"abstract":"Let p be an analytic function defined on the open unit disk D. We obtain certain differential subordination implications such as ψ(p) := pλ(z)(α+βp(z)+γ/p(z)+δzp′(z)/pj(z)) ≺ h(z) (j = 1, 2) implies p ≺ q, where h is given by ψ(q) and q belongs to P, by finding the conditions on α, β, γ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy | log(zf ′(z)/f(z))| < 1, |(zf ′(z)/f(z))2 − 1| < 1 and zf ′(z)/f(z) lying in the parabolic region v2 < 2u− 1.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"331-357"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511459","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 gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Fur- thermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Fi- nally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.
{"title":"GRADIENT ALMOST RICCI SOLITONS WITH VANISHING CONDITIONS ON WEYL TENSOR AND BACH TENSOR","authors":"J. Co., Seungsu Hwang","doi":"10.4134/JKMS.J190201","DOIUrl":"https://doi.org/10.4134/JKMS.J190201","url":null,"abstract":". In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Fur- thermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Fi- nally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"539-552"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511576","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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