Pub Date : 2021-03-13DOI: 10.1142/s1664360722500138
J. Giacomoni, Lais Moreira dos Santos, C. Santos
A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $lambda-$direction and bifurcates from infinity at $lambda=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.
{"title":"Multiplicity for a strongly singular quasilinear problem via bifurcation theory","authors":"J. Giacomoni, Lais Moreira dos Santos, C. Santos","doi":"10.1142/s1664360722500138","DOIUrl":"https://doi.org/10.1142/s1664360722500138","url":null,"abstract":"A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $lambda-$direction and bifurcates from infinity at $lambda=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89790121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-05DOI: 10.1142/S1664360721500053
Simion Breaz, Tomasz Brzezi'nski
It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.
{"title":"The Baer–Kaplansky Theorem for All Abelian Groups and Modules","authors":"Simion Breaz, Tomasz Brzezi'nski","doi":"10.1142/S1664360721500053","DOIUrl":"https://doi.org/10.1142/S1664360721500053","url":null,"abstract":"It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75347203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-02DOI: 10.1142/s1664360722500035
D. Borisov, P. Exner
We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.
{"title":"Approximation of point interactions by geometric perturbations in two-dimensional domains","authors":"D. Borisov, P. Exner","doi":"10.1142/s1664360722500035","DOIUrl":"https://doi.org/10.1142/s1664360722500035","url":null,"abstract":"We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80036499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-09DOI: 10.1142/s1664360720500125
Manuel Cortés-Izurdiaga, P. A. G. Asensio, Berke Kalebog̃az, A. K. Srivastava
We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.
{"title":"Ziegler partial morphisms in additive exact categories","authors":"Manuel Cortés-Izurdiaga, P. A. G. Asensio, Berke Kalebog̃az, A. K. Srivastava","doi":"10.1142/s1664360720500125","DOIUrl":"https://doi.org/10.1142/s1664360720500125","url":null,"abstract":"We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75461590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-25DOI: 10.1142/S1664360720500046
N. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš
We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map [Formula: see text]. We do not assume that [Formula: see text], incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.
{"title":"Relaxation methods for optimal control problems","authors":"N. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš","doi":"10.1142/S1664360720500046","DOIUrl":"https://doi.org/10.1142/S1664360720500046","url":null,"abstract":"We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map [Formula: see text]. We do not assume that [Formula: see text], incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80235813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-31DOI: 10.1142/s1664360720500101
A. Lucchini
For a finite group [Formula: see text] denote by [Formula: see text] the genus of the subgroup graph of [Formula: see text] We prove that [Formula: see text] tends to infinity as either the rank of [Formula: see text] or the number of prime divisors of [Formula: see text] tends to infinity.
{"title":"The genus of the subgroup graph of a finite group","authors":"A. Lucchini","doi":"10.1142/s1664360720500101","DOIUrl":"https://doi.org/10.1142/s1664360720500101","url":null,"abstract":"For a finite group [Formula: see text] denote by [Formula: see text] the genus of the subgroup graph of [Formula: see text] We prove that [Formula: see text] tends to infinity as either the rank of [Formula: see text] or the number of prime divisors of [Formula: see text] tends to infinity.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91354083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-25DOI: 10.1142/s166436072050006x
O. Bezushchak, B. Oliynyk
We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number [Formula: see text] there exists a unital locally matrix algebra [Formula: see text] having the Steinitz number [Formula: see text] and not isomorphic to a tensor product of finite-dimensional matrix algebras.
构造了一个维数不可数的一元局部矩阵代数，其(1)不允许初等分解，(2)具有无限的局部有限Steinitz数。它对[V.]的问题给出了否定的答案。M. Kurochkin，关于局部简单代数和局部正规代数的理论，vol . b. vol . 22(64)(3) (1948) 443-454;刘建军，单元局部矩阵代数与Steinitz数，代数学报。(2020)，在线准备]。我们还证明了对于任意无限Steinitz数[公式:见文]，存在具有Steinitz数[公式:见文]且与有限维矩阵代数的张量积不同构的一元局部矩阵代数[公式:见文]。
{"title":"Primary decompositions of unital locally matrix algebras","authors":"O. Bezushchak, B. Oliynyk","doi":"10.1142/s166436072050006x","DOIUrl":"https://doi.org/10.1142/s166436072050006x","url":null,"abstract":"We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number [Formula: see text] there exists a unital locally matrix algebra [Formula: see text] having the Steinitz number [Formula: see text] and not isomorphic to a tensor product of finite-dimensional matrix algebras.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79030251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-30DOI: 10.1142/s1664360720500150
Evgeny Khukhro, P. Shumyatsky
An Engel sink of an element [Formula: see text] of a group [Formula: see text] is a set [Formula: see text] such that for every [Formula: see text] all sufficiently long commutators [Formula: see text] belong to [Formula: see text]. (Thus, [Formula: see text] is an Engel element precisely when we can choose [Formula: see text].) It is proved that if every element of a compact (Hausdorff) group [Formula: see text] has a countable (or finite) Engel sink, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. This settles a question suggested by J. S. Wilson.
一个群[公式:见文]的元素[公式:见文]的恩格尔汇是一个集合[公式:见文]，使得对于每一个[公式:见文]，所有足够长的换向子[公式:见文]都属于[公式:见文]。(因此，当我们可以选择[Formula: see text]时，[Formula: see text]就是一个恩格尔元素。)证明了如果紧(Hausdorff)群[公式:见文]中的每一个元素都有一个可数的(或有限的)Engel sink，则[公式:见文]有一个有限的正规子群[公式:见文]使得[公式:见文]局部幂零。这就解决了威尔逊提出的一个问题。
{"title":"Compact groups with countable Engel sinks","authors":"Evgeny Khukhro, P. Shumyatsky","doi":"10.1142/s1664360720500150","DOIUrl":"https://doi.org/10.1142/s1664360720500150","url":null,"abstract":"An Engel sink of an element [Formula: see text] of a group [Formula: see text] is a set [Formula: see text] such that for every [Formula: see text] all sufficiently long commutators [Formula: see text] belong to [Formula: see text]. (Thus, [Formula: see text] is an Engel element precisely when we can choose [Formula: see text].) It is proved that if every element of a compact (Hausdorff) group [Formula: see text] has a countable (or finite) Engel sink, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. This settles a question suggested by J. S. Wilson.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84155811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-19DOI: 10.1142/S166436072050023X
M. Acosta, M. Soleimani-Mourchehkhorti
We prove that the class of positive operators from [Formula: see text] to [Formula: see text] has the Bishop-Phelps-Bollobás property for any positive measure [Formula: see text], whenever [Formula: see text] is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair [Formula: see text] for any uniformly monotone Banach lattice [Formula: see text] Further we show that these results are optimal in case that [Formula: see text] is strictly monotone.
{"title":"Bishop-Phelps-Bollobás property for positive operators when the domain is L∞","authors":"M. Acosta, M. Soleimani-Mourchehkhorti","doi":"10.1142/S166436072050023X","DOIUrl":"https://doi.org/10.1142/S166436072050023X","url":null,"abstract":"We prove that the class of positive operators from [Formula: see text] to [Formula: see text] has the Bishop-Phelps-Bollobás property for any positive measure [Formula: see text], whenever [Formula: see text] is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair [Formula: see text] for any uniformly monotone Banach lattice [Formula: see text] Further we show that these results are optimal in case that [Formula: see text] is strictly monotone.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87127824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-12DOI: 10.1142/S1664360719500036
Tianyi Ren
We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].
{"title":"(Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n","authors":"Tianyi Ren","doi":"10.1142/S1664360719500036","DOIUrl":"https://doi.org/10.1142/S1664360719500036","url":null,"abstract":"We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85878935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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