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The spectral gap of graphs and Steklov eigenvalues on surfaces 图的谱隙和曲面上的Steklov特征值
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-10-10 DOI: 10.3934/era.2014.21.19
B. Colbois, A. Girouard
{"title":"The spectral gap of graphs and Steklov eigenvalues on surfaces","authors":"B. Colbois, A. Girouard","doi":"10.3934/era.2014.21.19","DOIUrl":"https://doi.org/10.3934/era.2014.21.19","url":null,"abstract":"Using expander graphs, we construct a sequence \u0000 ${Omega_N}_{Ninmathbb{N}}$ of smooth compact surfaces with boundary of \u0000 perimeter $N$, and with the first non-zero Steklov \u0000 eigenvalue $sigma_1(Omega_N)$ uniformly bounded away from \u0000 zero. This answers a question which was raised in [10]. The \u0000 sequence $sigma_1(Omega_N) L(partialOmega_n)$ grows linearly with the genus of \u0000 $Omega_N$, which is the optimal growth rate.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"19-27"},"PeriodicalIF":0.0,"publicationDate":"2013-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232716","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}
引用次数: 12
Compactly supported Hamiltonian loops with a non-zero Calabi invariant 具有非零Calabi不变量的紧支持哈密顿循环
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-10-06 DOI: 10.3934/era.2014.21.80
A. Kislev
{"title":"Compactly supported Hamiltonian loops with a non-zero Calabi invariant","authors":"A. Kislev","doi":"10.3934/era.2014.21.80","DOIUrl":"https://doi.org/10.3934/era.2014.21.80","url":null,"abstract":"We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"80-88"},"PeriodicalIF":0.0,"publicationDate":"2013-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233829","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}
引用次数: 3
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies 二次和三次频率的须状环面分离矩阵分裂的指数小渐近估计
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-06-04 DOI: 10.3934/ERA.2014.21.41
A. Delshams, M. Gonchenko, P. Gutiérrez
{"title":"Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies","authors":"A. Delshams, M. Gonchenko, P. Gutiérrez","doi":"10.3934/ERA.2014.21.41","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.41","url":null,"abstract":"We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector ! = (1; ), where is a quadratic irrational number, or a 3-dimensional torus with a frequency vector ! = (1; ; 2 ), where is a cubic irrational number. Applying the Poincar e{Melnikov method, we nd exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fullled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which is the so-called cubic golden number (the real root of x 3 +x 1 = 0), obtaining also asymptotic estimates. We point out the similitudes and dierences between the results obtained for both the quadratic and cubic cases.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"41-61"},"PeriodicalIF":0.0,"publicationDate":"2013-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232728","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}
引用次数: 11
A gradient estimate for harmonic functions sharing the same zeros 具有相同零点的调和函数的梯度估计
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-06-03 DOI: 10.3934/era.2014.21.62
D. Mangoubi
{"title":"A gradient estimate for harmonic functions sharing the same zeros","authors":"D. Mangoubi","doi":"10.3934/era.2014.21.62","DOIUrl":"https://doi.org/10.3934/era.2014.21.62","url":null,"abstract":"Let $u, v$ be two harmonic functions in ${|z|<2}subsetmathbb{C}$ \u0000which have exactly the same set $Z$ of zeros. \u0000We observe that $big|nablalog |u/v|big|$ is bounded in the unit disk \u0000by a constant which depends on $Z$ only. In case $Z=emptyset$ this goes back \u0000to Li-Yau's gradient estimate for positive harmonic functions. \u0000The general boundary Harnack principle gives \u0000only Holder estimates on $log |u/v|$.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"62-71"},"PeriodicalIF":0.0,"publicationDate":"2013-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233322","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}
引用次数: 8
New results on fat points schemes in $mathbb{P}^2$ $mathbb{P}^2$中肥点格式的新结果
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-04-01 DOI: 10.3934/ERA.2013.20.51
M. Dumnicki, T. Szemberg, H. Tutaj-Gasinska
{"title":"New results on fat points schemes in $mathbb{P}^2$","authors":"M. Dumnicki, T. Szemberg, H. Tutaj-Gasinska","doi":"10.3934/ERA.2013.20.51","DOIUrl":"https://doi.org/10.3934/ERA.2013.20.51","url":null,"abstract":"The purpose of this note is to announce two results, Theorem A and Theorem B below, concerning geometric and algebraic properties of fat points in the complex projective plane. Their somewhat technical proofs are available in [10] and will be published elsewhere. Here we present only main ideas which are fairly transparent.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"20 1","pages":"51-54"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232874","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}
引用次数: 0
Nullspaces of conformally invariant operators. Applications to $boldsymbol{Q_k}$-curvature 共形不变算子的零空间。$boldsymbol{Q_k}$-曲率的应用
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-03-01 DOI: 10.3934/ERA.2013.20.43
Y. Canzani, A. Gover, D. Jakobson, Raphael Ponge
{"title":"Nullspaces of conformally invariant operators. Applications to $boldsymbol{Q_k}$-curvature","authors":"Y. Canzani, A. Gover, D. Jakobson, Raphael Ponge","doi":"10.3934/ERA.2013.20.43","DOIUrl":"https://doi.org/10.3934/ERA.2013.20.43","url":null,"abstract":"We study conformal invariants that arise from functions in the \u0000nullspace of conformally covariant differential operators. \u0000The invariants include nodal sets and the topology of nodal domains \u0000of eigenfunctions in the kernel of GJMS operators. We establish \u0000that on any manifold of dimension $ngeq 3$, there exist many metrics \u0000for which our invariants are nontrivial. We discuss new applications \u0000to curvature prescription problems.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"20 1","pages":"43-50"},"PeriodicalIF":0.0,"publicationDate":"2013-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232846","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}
引用次数: 2
The structure theorems for Yetter-Drinfeld comodule algebras yeter - drinfeld模代数的结构定理
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-03-01 DOI: 10.3934/ERA.2013.20.31
Ling Jia
{"title":"The structure theorems for Yetter-Drinfeld comodule algebras","authors":"Ling Jia","doi":"10.3934/ERA.2013.20.31","DOIUrl":"https://doi.org/10.3934/ERA.2013.20.31","url":null,"abstract":"In this paper, we first introduce the notion of a Yetter-Drinfeld comodule algebra and give examples. Then we give the structure theorems of Yetter-Drinfeld comodule algebras. That is, if $L$ is a Yetter-Drinfeld Hopf algebra and $A$ is a right $L$-Yetter-Drinfeld comodule algebra, then there exists an algebra isomorphism between $A$ and $A^{coL} mathbin{sharp} H$, where $A^{coL}$ is the coinvariant subalgebra of $A$.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"20 1","pages":"31-42"},"PeriodicalIF":0.0,"publicationDate":"2013-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232833","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}
引用次数: 0
Segre classes of monomial schemes 单项式方案的分离类
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-02-28 DOI: 10.3934/era.2013.20.55
P. Aluffi
{"title":"Segre classes of monomial schemes","authors":"P. Aluffi","doi":"10.3934/era.2013.20.55","DOIUrl":"https://doi.org/10.3934/era.2013.20.55","url":null,"abstract":"We propose an explicit formula for the Segre classes of monomial \u0000subschemes of nonsingular varieties, such as schemes defined by \u0000monomial ideals in projective space. The Segre class is expressed as \u0000a formal integral on a region bounded by the corresponding Newton \u0000polyhedron. We prove this formula for monomial ideals in two variables \u0000and verify it for some families of examples in any number of \u0000variables.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"20 1","pages":"55-70"},"PeriodicalIF":0.0,"publicationDate":"2013-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232885","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}
引用次数: 6
$alpha$-concave functions and a functional extension of mixed volumes $alpha$-凹函数和混合体的功能扩展
Electronic Research Announcements in Mathematical Sciences Pub Date : 2013-02-04 DOI: 10.3934/era.2013.20.1
V. Milman, Liran Rotem
{"title":"$alpha$-concave functions and a functional extension of mixed volumes","authors":"V. Milman, Liran Rotem","doi":"10.3934/era.2013.20.1","DOIUrl":"https://doi.org/10.3934/era.2013.20.1","url":null,"abstract":"Mixed volumes, which are the polarization of volume with respect to \u0000the Minkowski addition, are fundamental objects in convexity. In this \u0000note we announce the construction of mixed integrals, which are functional \u0000analogs of mixed volumes. We build a natural addition operation $oplus$ \u0000on the class of quasi-concave functions, such that every class of \u0000$alpha$-concave functions is closed under $oplus$. We then define \u0000the mixed integrals, which are the polarization of the integral with \u0000respect to $oplus$. \u0000 \u0000We proceed to discuss the extension of various classic inequalities \u0000to the functional setting. For general quasi-concave functions, this \u0000is done by restating those results in the language of rearrangement \u0000inequalities. Restricting ourselves to $alpha$-concave functions, \u0000we state a generalization of the Alexandrov inequalities in their \u0000more familiar form.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"20 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2013-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232803","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}
引用次数: 17
On Totally integrable magnetic billiards on constant curvature surface 常曲率曲面上的完全可积磁台球
Electronic Research Announcements in Mathematical Sciences Pub Date : 2012-08-12 DOI: 10.3934/ERA.2012.19.112
M. Biały
{"title":"On Totally integrable magnetic billiards on constant curvature surface","authors":"M. Biały","doi":"10.3934/ERA.2012.19.112","DOIUrl":"https://doi.org/10.3934/ERA.2012.19.112","url":null,"abstract":"We consider billiard ball motion in \u0000a convex domain of a constant curvature surface influenced by the \u0000constant magnetic field. We prove that if the billiard map is \u0000totally integrable then the boundary curve is necessarily a circle. \u0000This result shows that the so-called Hopf rigidity phenomenon which \u0000was recently obtained for classical billiards on constant curvature \u0000surfaces holds true also in the presence of constant magnetic field.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"19 1","pages":"112-119"},"PeriodicalIF":0.0,"publicationDate":"2012-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232244","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}
引用次数: 11
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