Using expander graphs, we construct a sequence ${Omega_N}_{Ninmathbb{N}}$ of smooth compact surfaces with boundary of perimeter $N$, and with the first non-zero Steklov eigenvalue $sigma_1(Omega_N)$ uniformly bounded away from zero. This answers a question which was raised in [10]. The sequence $sigma_1(Omega_N) L(partialOmega_n)$ grows linearly with the genus of $Omega_N$, which is the optimal growth rate.
{"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}
We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds.
给出了一类开辛流形上具有非零Calabi不变量的紧支持哈密顿环的例子。
{"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}
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.
{"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}
Let $u, v$ be two harmonic functions in ${|z|<2}subsetmathbb{C}$ which have exactly the same set $Z$ of zeros. We observe that $big|nablalog |u/v|big|$ is bounded in the unit disk by a constant which depends on $Z$ only. In case $Z=emptyset$ this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives only Holder estimates on $log |u/v|$.
{"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}
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.
{"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}
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $ngeq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
{"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}
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 $A$.
本文首先引入了yeter - drinfeld模代数的概念，并给出了实例。然后给出了叶特-德林菲尔德模代数的结构定理。即，如果$L$是一个yeter - drinfeld Hopf代数，$ a $是一个右$L$- yeter - drinfeld模代数，则$ a $与$ a ^{coL} mathbin{sharp} H$之间存在代数同构，其中$ a ^{coL}$是$ a $的协不变子代数。
{"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}
We propose an explicit formula for the Segre classes of monomial subschemes of nonsingular varieties, such as schemes defined by monomial ideals in projective space. The Segre class is expressed as a formal integral on a region bounded by the corresponding Newton polyhedron. We prove this formula for monomial ideals in two variables and verify it for some families of examples in any number of variables.
{"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}
Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation $oplus$ on the class of quasi-concave functions, such that every class of $alpha$-concave functions is closed under $oplus$. We then define the mixed integrals, which are the polarization of the integral with respect to $oplus$. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to $alpha$-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.
{"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}
We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrable then the boundary curve is necessarily a circle. This result shows that the so-called Hopf rigidity phenomenon which was recently obtained for classical billiards on constant curvature surfaces holds true also in the presence of constant magnetic field.
{"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}
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