{"title":"Random polynomials in several complex variables","authors":"Turgay Bayraktar, Thomas Bloom, Norm Levenberg","doi":"10.1007/s11854-023-0316-x","DOIUrl":"https://doi.org/10.1007/s11854-023-0316-x","url":null,"abstract":"<p>We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials <span>({H_n}(z): = sumnolimits_{j = 1}^{{m_n}} {{a_j}{p_j}} )</span> that are linear combinations of basis polynomials {<i>p</i><sub><i>j</i></sub>} with i.i.d. complex random variable coefficients {<i>a</i><sub><i>j</i></sub>} where {<i>p</i><sub><i>j</i></sub>} form an orthonormal basis for a Bernstein-Markov measure on a compact set <span>(K subset {mathbb{C}^d})</span>. Here <i>m</i><sub><i>n</i></sub> is the dimension of <span>({{cal P}_n})</span>, the holomorphic polynomials of degree at most <i>n</i> in <span>({mathbb{C}^d})</span>. We consider more general bases {<i>p</i><sub><i>j</i></sub>}, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow <span>({H_n}(z): = sumnolimits_{j = 1}^{{m_n}} {{a_{nj}}{p_{nj}}(z)} )</span>, i.e., we have an array of basis polynomials {<i>p</i><sub><i>j</i></sub>} and random coefficients {<i>a</i><sub><i>nj</i></sub>}. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of <span>({1 over n}log |{H_n}|)</span> in <span>(L_{{rm{loc}}}^1({mathbb{C}^d}))</span> to the (weighted) extremal plurisubharmonic function for <i>K</i>. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139030289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New vector solutions for the cubic nonlinear schrödinger system","authors":"Lipeng Duan, Xiao Luo, Maoding Zhen","doi":"10.1007/s11854-023-0315-y","DOIUrl":"https://doi.org/10.1007/s11854-023-0315-y","url":null,"abstract":"<p>In this paper, we construct a family of new solutions for the following nonlinear Schrödinger system: </p><span>$$left{{matrix{{- Delta u + P(y)u = mu {u^3} + beta u{upsilon ^2},} & {u > 0,,,{rm{in}},{mathbb{R}^3},} cr {- Delta upsilon + Q(y)upsilon = v{upsilon ^3} + beta {u^2}upsilon ,} & {upsilon > 0,,,{rm{in}},{mathbb{R}^3},} cr}} right.$$</span><p> where <i>P</i>(<i>y</i>), <i>Q</i>(<i>y</i>) are positive radial potentials, <i>μ > 0, v > 0</i> and <span>(beta in mathbb{R})</span>. Motivated by the doubling construction of the entire finite energy sign-changing solution for the Yamabe equation in M. Medina and M. Musso (J. Math. Pures Appl. 2021), by using another type of building blocks, which are not equal to the ones adopted in S. Peng and Z.-Q. Wang (Arch. Ration. Mech. Anal. 2013), we successfully construct new segregated and synchronized vector solutions for the nonlinear Schrödinger system with more complex concentration structure. Our results extend the main results of S. Peng and Z.-Q. Wang (Arch. Ration. Mech. Anal. 2013), and in particular, for the segregated case, we well complement the previous works when the potentials <i>P</i>(<i>y</i>) and <i>Q</i>(<i>y</i>) decay in different rates.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139025403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Universal eigenvalue statistics for dynamically defined matrices","authors":"Arka Adhikari, Marius Lemm","doi":"10.1007/s11854-023-0314-z","DOIUrl":"https://doi.org/10.1007/s11854-023-0314-z","url":null,"abstract":"<p>We consider dynamically defined Hermitian matrices generated from orbits of the doubling map. We prove that their spectra fall into the GUE universality class from random matrix theory.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139030290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Infinite partial sumsets in the primes","authors":"","doi":"10.1007/s11854-023-0323-y","DOIUrl":"https://doi.org/10.1007/s11854-023-0323-y","url":null,"abstract":"<h3>Abstract</h3> <p>We show that there exist infinite sets <em>A</em> = (<em>a</em><sub>1</sub>, <em>a</em><sub>2</sub>, …} and <em>B</em> = {<em>b</em><sub>1</sub>, <em>b</em><sub>2</sub>, …} of natural numbers such that <em>a</em><sub><em>i</em></sub> + <em>b</em><sub><em>j</em></sub> is prime whenever 1 ≤ <em>i</em> < <em>j</em>.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139023951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The size of wild Kloosterman sums in number fields and function fields","authors":"","doi":"10.1007/s11854-023-0325-9","DOIUrl":"https://doi.org/10.1007/s11854-023-0325-9","url":null,"abstract":"<h3>Abstract</h3> <p>We study <em>p</em>-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter <em>k</em> that recovers the classical Kloosterman sum when <em>k</em> = 2, over general <em>p</em>-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when <em>k</em> is not divisible by <em>p</em>, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when <em>k</em> is divisible by <em>p</em>. This gives both an upper bound and a lower bound showing the upper bound is essentially sharp. This generalizes previously known bounds [3] in the case of ℤ<sub><em>p</em></sub>. The lower bounds in the equal characteristic case have two applications to function field number theory, showing that certain short interval sums and certain moments of Dirichlet <em>L</em>-functions do not, as one might hope, admit square-root cancellation.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139025410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples","authors":"Rena Chu, Lillian B. Pierce","doi":"10.1007/s11854-023-0335-7","DOIUrl":"https://doi.org/10.1007/s11854-023-0335-7","url":null,"abstract":"","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"22 1","pages":"59-114"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139341187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：