Pub Date : 2018-11-17DOI: 10.1142/s1664360720500162
Michael Ruzhansky, Bolys Sabitbek, D. Suragan
In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.
{"title":"Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups","authors":"Michael Ruzhansky, Bolys Sabitbek, D. Suragan","doi":"10.1142/s1664360720500162","DOIUrl":"https://doi.org/10.1142/s1664360720500162","url":null,"abstract":"In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"11 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77255287","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 : 2018-10-02DOI: 10.1142/S1664360719500164
G. Kozma, A. Lubotzky
We show that for a fixed [Formula: see text], Gromov random groups with any density [Formula: see text] have no nontrivial degree [Formula: see text] representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when [Formula: see text] such groups have a faithful linear representation over [Formula: see text], a.a.s.
{"title":"Linear representations of random groups","authors":"G. Kozma, A. Lubotzky","doi":"10.1142/S1664360719500164","DOIUrl":"https://doi.org/10.1142/S1664360719500164","url":null,"abstract":"We show that for a fixed [Formula: see text], Gromov random groups with any density [Formula: see text] have no nontrivial degree [Formula: see text] representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when [Formula: see text] such groups have a faithful linear representation over [Formula: see text], a.a.s.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"9 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85667576","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 : 2018-09-25DOI: 10.1007/s13373-018-0131-3
S. Hsu
{"title":"Minimizer of an isoperimetric ratio on a metric on $${mathbb {R}}^2$$R2 with finite total area","authors":"S. Hsu","doi":"10.1007/s13373-018-0131-3","DOIUrl":"https://doi.org/10.1007/s13373-018-0131-3","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"124 2-3 1","pages":"603-617"},"PeriodicalIF":1.2,"publicationDate":"2018-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78151519","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 : 2018-09-19DOI: 10.1142/s1664360720500113
Jiakun Liu, G. Loeper
We study an optimal transport problem where, at some intermediate time, the mass is either accelerated by an external force field or self-interacting. We obtain the regularity of the velocity potential, intermediate density, and optimal transport map, under the conditions on the interaction potential that are related to the so-called Ma–Trudinger–Wang condition from optimal transport [X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problems, Arch. Ration. Mech. Anal. 177 (2005) 151–183.].
{"title":"Optimal transport with discrete long-range mean-field interactions","authors":"Jiakun Liu, G. Loeper","doi":"10.1142/s1664360720500113","DOIUrl":"https://doi.org/10.1142/s1664360720500113","url":null,"abstract":"We study an optimal transport problem where, at some intermediate time, the mass is either accelerated by an external force field or self-interacting. We obtain the regularity of the velocity potential, intermediate density, and optimal transport map, under the conditions on the interaction potential that are related to the so-called Ma–Trudinger–Wang condition from optimal transport [X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problems, Arch. Ration. Mech. Anal. 177 (2005) 151–183.].","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"117 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88422863","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 : 2018-09-02DOI: 10.1142/s1664360719500231
A. Olshanskii, M. Sapir
We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.
构造了具有二次Dehn函数和不可定共轭问题的有限呈现群。这就解决了Rips在1994年提出的问题。
{"title":"Conjugacy problem in groups with quadratic Dehn function","authors":"A. Olshanskii, M. Sapir","doi":"10.1142/s1664360719500231","DOIUrl":"https://doi.org/10.1142/s1664360719500231","url":null,"abstract":"We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"5 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89665226","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 : 2018-08-20DOI: 10.1007/S13373-018-0129-X
M. Ragusa, V. Shakhmurov
{"title":"Embedding of vector-valued Morrey spaces and separable differential operators","authors":"M. Ragusa, V. Shakhmurov","doi":"10.1007/S13373-018-0129-X","DOIUrl":"https://doi.org/10.1007/S13373-018-0129-X","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"15 1","pages":"1-23"},"PeriodicalIF":1.2,"publicationDate":"2018-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88442001","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 : 2018-08-06DOI: 10.1007/S13373-018-0128-Y
Gerardo Ariznabarreta, J. C. García-Ardila, Manuel Mañas, F. Marcellán
{"title":"Matrix biorthogonal polynomials on the real line: Geronimus transformations","authors":"Gerardo Ariznabarreta, J. C. García-Ardila, Manuel Mañas, F. Marcellán","doi":"10.1007/S13373-018-0128-Y","DOIUrl":"https://doi.org/10.1007/S13373-018-0128-Y","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"165 7 Suppl 2 1","pages":"1-66"},"PeriodicalIF":1.2,"publicationDate":"2018-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83327530","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 : 2018-08-01DOI: 10.1007/S13373-017-0104-Y
F. Marcellán, Yamilet Quintana, José M. Rodríguez
{"title":"Weighted Sobolev spaces: Markov-type inequalities and duality","authors":"F. Marcellán, Yamilet Quintana, José M. Rodríguez","doi":"10.1007/S13373-017-0104-Y","DOIUrl":"https://doi.org/10.1007/S13373-017-0104-Y","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"43 1","pages":"233-256"},"PeriodicalIF":1.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86647577","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 : 2018-07-20DOI: 10.1142/S1664360719500115
N. Papageorgiou, V. Rǎdulescu, Dušan D. Repovš
We consider a nonlinear parametric Dirichlet problem driven by the p-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ($$p-1$$p-1)-linear near $$+infty $$+∞. The problem is uniformly nonresonant with respect to the principal eigenvalue of $$(-Delta _p,W^{1,p}_0(Omega ))$$(-Δp,W01,p(Ω)). We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter $$lambda >0$$λ>0.
{"title":"Positive solutions for nonlinear parametric singular Dirichlet problems","authors":"N. Papageorgiou, V. Rǎdulescu, Dušan D. Repovš","doi":"10.1142/S1664360719500115","DOIUrl":"https://doi.org/10.1142/S1664360719500115","url":null,"abstract":"We consider a nonlinear parametric Dirichlet problem driven by the p-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ($$p-1$$p-1)-linear near $$+infty $$+∞. The problem is uniformly nonresonant with respect to the principal eigenvalue of $$(-Delta _p,W^{1,p}_0(Omega ))$$(-Δp,W01,p(Ω)). We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter $$lambda >0$$λ>0.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"42 1","pages":"1-22"},"PeriodicalIF":1.2,"publicationDate":"2018-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82427083","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 : 2018-07-05DOI: 10.1142/S1664360719500206
Nikola Advzaga
A Diophantine [Formula: see text]-tuple is a set of [Formula: see text] distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that [Formula: see text]. Our proof is relatively simple compared to the proofs of similar results in positive integers.
{"title":"On the size of Diophantine m-tuples in imaginary quadratic number rings","authors":"Nikola Advzaga","doi":"10.1142/S1664360719500206","DOIUrl":"https://doi.org/10.1142/S1664360719500206","url":null,"abstract":"A Diophantine [Formula: see text]-tuple is a set of [Formula: see text] distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that [Formula: see text]. Our proof is relatively simple compared to the proofs of similar results in positive integers.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"29 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86214081","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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