Pub Date : 2018-07-30DOI: 10.4310/arkiv.2019.v57.n2.a2
T. Bayraktar, T. Bloom, N. Levenberg, C. H. Lu
We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge-Amp`ere equation in an appropriate finite energy class. This is achieved using a variational approach.
{"title":"Pluripotential theory and convex bodies: large deviation principle","authors":"T. Bayraktar, T. Bloom, N. Levenberg, C. H. Lu","doi":"10.4310/arkiv.2019.v57.n2.a2","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a2","url":null,"abstract":"We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge-Amp`ere equation in an appropriate finite energy class. This is achieved using a variational approach.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393646","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}
Pub Date : 2018-07-11DOI: 10.4310/ARKIV.2019.V57.N1.A7
P. Ivanisvili, T. Tkocz
We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete cube).
{"title":"Comparison of moments of Rademacher chaoses","authors":"P. Ivanisvili, T. Tkocz","doi":"10.4310/ARKIV.2019.V57.N1.A7","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A7","url":null,"abstract":"We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete cube).","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393581","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}
Pub Date : 2018-05-04DOI: 10.4310/arkiv.2019.v57.n2.a12
S. Zelditch, Peng Zhou
In a recent series of articles (arXiv:1604.06655, arXiv:1708.09267), the authors have studied the transition behavior of partial Bergman kernels $Pi_{k, [E_1, E_2]}(z,w)$ and the associated DOS (density of states) $Pi_{k, [E_1, E_2]}(z)$ across the interface $ccal$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H: M to R$ on a kahler manifold. The allowed region is $H^{-1}([E_1, E_2])$ and the interface $ccal$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-half} $ tube around $ccal$ for regular interfaces, one obtains $delta$-asymptotics in $k^{-frac{1}{4}}$-tubes around singular points of a critical interface. In $k^{-half}$ tubes, the transition law is given by the osculating metaplectic propagator.
{"title":"Interface asymptotics of Partial Bergman kernels around a critical level","authors":"S. Zelditch, Peng Zhou","doi":"10.4310/arkiv.2019.v57.n2.a12","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a12","url":null,"abstract":"In a recent series of articles (arXiv:1604.06655, arXiv:1708.09267), the authors have studied the transition behavior of partial Bergman kernels $Pi_{k, [E_1, E_2]}(z,w)$ and the associated DOS (density of states) $Pi_{k, [E_1, E_2]}(z)$ across the interface $ccal$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H: M to R$ on a kahler manifold. The allowed region is $H^{-1}([E_1, E_2])$ and the interface $ccal$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-half} $ tube around $ccal$ for regular interfaces, one obtains $delta$-asymptotics in $k^{-frac{1}{4}}$-tubes around singular points of a critical interface. In $k^{-half}$ tubes, the transition law is given by the osculating metaplectic propagator.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48049073","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}
Pub Date : 2018-04-14DOI: 10.4310/ARKIV.2019.V57.N1.A3
Ela Celikbas, Olgur Celikbas, S. Goto, Naoki Taniguchi
In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fr"oberg concerning Arf numerical semigroup rings.
{"title":"Generalized Gorenstein Arf rings","authors":"Ela Celikbas, Olgur Celikbas, S. Goto, Naoki Taniguchi","doi":"10.4310/ARKIV.2019.V57.N1.A3","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A3","url":null,"abstract":"In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fr\"oberg concerning Arf numerical semigroup rings.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49305400","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}
Pub Date : 2018-04-01DOI: 10.4310/ARKIV.2018.V56.N1.A5
C. Costoya, J. Scherer, A. Viruel
Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.
{"title":"A torus theorem for homotopy nilpotent loop spaces","authors":"C. Costoya, J. Scherer, A. Viruel","doi":"10.4310/ARKIV.2018.V56.N1.A5","DOIUrl":"https://doi.org/10.4310/ARKIV.2018.V56.N1.A5","url":null,"abstract":"Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393050","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}
Pub Date : 2018-03-12DOI: 10.4310/arkiv.2021.v59.n1.a4
S. Janson
We consider P'olya urns with infinitely many colours that are of a random walk type, in two related version. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014--2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).
{"title":"A.s. convergence for infinite colour Pólya urns associated with random walks","authors":"S. Janson","doi":"10.4310/arkiv.2021.v59.n1.a4","DOIUrl":"https://doi.org/10.4310/arkiv.2021.v59.n1.a4","url":null,"abstract":"We consider P'olya urns with infinitely many colours that are of a random walk type, in two related version. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014--2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393677","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}
Pub Date : 2018-02-25DOI: 10.4310/ARKIV.2019.V57.N1.A10
R. Pandharipande, D. Zvonkine
A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.
{"title":"Cohomological field theories with non-tautological classes","authors":"R. Pandharipande, D. Zvonkine","doi":"10.4310/ARKIV.2019.V57.N1.A10","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A10","url":null,"abstract":"A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393162","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}
Pub Date : 2018-01-10DOI: 10.4310/arkiv.2019.v57.n2.a6
Alexander J. Izzo, N. Levenberg
It is shown that there exists a Cantor set X in C^4 whose polynomially convex hull is strictly larger than X but contains no analytic discs.
证明了C^4中存在一个多项式凸包严格大于X但不包含解析盘的康托集X。
{"title":"A Cantor set whose polynomial hull contains no analytic discs","authors":"Alexander J. Izzo, N. Levenberg","doi":"10.4310/arkiv.2019.v57.n2.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a6","url":null,"abstract":"It is shown that there exists a Cantor set X in C^4 whose polynomially convex hull is strictly larger than X but contains no analytic discs.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393697","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}
Pub Date : 2018-01-01DOI: 10.4310/ARKIV.2018.V56.N2.A1
D. M. Almeida, E. Falbel
In this paper we start the classification of strongly bracket generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.
本文开始对强括号生成的子对称空间进行分类。我们证明了一个结构定理，并分析了幂零的情况。
{"title":"Fat sub-Riemannian symmetric spaces: the nilpotent case","authors":"D. M. Almeida, E. Falbel","doi":"10.4310/ARKIV.2018.V56.N2.A1","DOIUrl":"https://doi.org/10.4310/ARKIV.2018.V56.N2.A1","url":null,"abstract":"In this paper we start the classification of strongly bracket generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393089","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}
Pub Date : 2017-12-29DOI: 10.4310/ARKIV.2019.V57.N1.A8
S. Larson
We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of $(mathbb{N}-tfrac12)times(mathbb{N}-tfrac12)$ inside triangles with vertices $(0, 0), (0, lambda sqrt{beta})$ and $(lambda/{sqrt{beta}}, 0)$ with respect to $beta>0$, for fixed $lambdageq 0$. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice $(mathbb{N}+sigma)times(mathbb{N}+tau)$, for $sigma, tau >-1$. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, $sigma=tau=-tfrac12$, is a critical case.
研究平面上各向异性谐振子特征值的渐近极小化问题。我们特别研究了特征值的Riesz均值和相应热核的迹线。特征值最小化问题可以被重新表述为一个点阵问题，在这个点阵问题中，对于固定的$lambdageq 0$，人们希望最大化具有顶点$(0, 0), (0, lambda sqrt{beta})$和$(lambda/{sqrt{beta}}, 0)$的$(mathbb{N}-tfrac12)times(mathbb{N}-tfrac12)$内部三角形相对于$beta>0$的点的数量。这个问题的晶格点公式自然会引出一系列的广义问题，人们转而考虑位移晶格$(mathbb{N}+sigma)times(mathbb{N}+tau)$，对于$sigma, tau >-1$。我们表明，这些问题的性质是相当不同的取决于移位参数，特别是问题对应于谐波振荡器，$sigma=tau=-tfrac12$，是一个临界情况。
{"title":"Maximizing Riesz means of anisotropic harmonic oscillators","authors":"S. Larson","doi":"10.4310/ARKIV.2019.V57.N1.A8","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A8","url":null,"abstract":"We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of $(mathbb{N}-tfrac12)times(mathbb{N}-tfrac12)$ inside triangles with vertices $(0, 0), (0, lambda sqrt{beta})$ and $(lambda/{sqrt{beta}}, 0)$ with respect to $beta>0$, for fixed $lambdageq 0$. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice $(mathbb{N}+sigma)times(mathbb{N}+tau)$, for $sigma, tau >-1$. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, $sigma=tau=-tfrac12$, is a critical case.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48676740","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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