Given an integer $g geq 0$ and a weight vector $w in mathbb{Q}^n cap (0, 1]^n$ satisfying $2g - 2 + sum w_i>0$, let $Delta_{g, w}$ denote the moduli space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We calculate the automorphism group $mathrm{Aut}(Delta_{g, w})$ for $g geq 1$ and arbitrary $w$, and we calculate the group $mathrm{Aut}(Delta_{0, w})$ when $w$ is heavy/light. In both of these cases, we show that $mathrm{Aut}(Delta_{g, w}) cong mathrm{Aut}(K_w)$, where $K_w$ is the abstract simplicial complex on ${1, ldots, n}$ whose faces are subsets with $w$-weight at most $1$. We show that these groups are precisely the finite direct products of symmetric groups. The space $Delta_{g, w}$ may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space $overline{mathcal{M}}_{g, w}$. Following the work of Massarenti and Mella on the biregular automorphism group $mathrm{Aut}(overline{mathcal{M}}_{g, w})$, we show that $mathrm{Aut}(Delta_{g, w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.
给定一个整数$g geq 0$和一个满足$2g - 2 + sum w_i>0$的权向量$w in mathbb{Q}^n cap (0, 1]^n$，令$Delta_{g, w}$表示$n$标记的、$w$稳定的$g$属热带曲线和卷1的模空间。我们计算了$g geq 1$和任意$w$的自同构群$mathrm{Aut}(Delta_{g, w})$，当$w$是重/轻时，我们计算了$mathrm{Aut}(Delta_{0, w})$。在这两种情况下，我们都证明了$mathrm{Aut}(Delta_{g, w}) cong mathrm{Aut}(K_w)$，其中$K_w$是${1, ldots, n}$上的抽象简单复合体，其面是$w$ -权值最多为$1$的子集。我们证明了这些群正是对称群的有限直积。空间$Delta_{g, w}$也可以看作是代数哈塞特空间$overline{mathcal{M}}_{g, w}$中奇异曲线的因子的对偶复形。继Massarenti和Mella对双正则自同构群$mathrm{Aut}(overline{mathcal{M}}_{g, w})$的研究之后，我们证明了$mathrm{Aut}(Delta_{g, w})$是保留奇异曲线的因子的自同构子群。
{"title":"Automorphisms of tropical Hassett spaces","authors":"S. Freedman, J. Hlavinka, S. Kannan","doi":"10.4171/pm/2075","DOIUrl":"https://doi.org/10.4171/pm/2075","url":null,"abstract":"Given an integer $g geq 0$ and a weight vector $w in mathbb{Q}^n cap (0, 1]^n$ satisfying $2g - 2 + sum w_i>0$, let $Delta_{g, w}$ denote the moduli space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We calculate the automorphism group $mathrm{Aut}(Delta_{g, w})$ for $g geq 1$ and arbitrary $w$, and we calculate the group $mathrm{Aut}(Delta_{0, w})$ when $w$ is heavy/light. In both of these cases, we show that $mathrm{Aut}(Delta_{g, w}) cong mathrm{Aut}(K_w)$, where $K_w$ is the abstract simplicial complex on ${1, ldots, n}$ whose faces are subsets with $w$-weight at most $1$. We show that these groups are precisely the finite direct products of symmetric groups. The space $Delta_{g, w}$ may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space $overline{mathcal{M}}_{g, w}$. Following the work of Massarenti and Mella on the biregular automorphism group $mathrm{Aut}(overline{mathcal{M}}_{g, w})$, we show that $mathrm{Aut}(Delta_{g, w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46755860","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}
The Talbot effect is usually modeled using the Helmholtz equation, but its main experimental features are captured by the solution to the free Schr"odinger equation with the Dirac comb as initial datum. This simplified description is a consequence of the paraxial approximation in geometric optics. However, it is a heuristic approximation that is not mathematically well justified, so K. I. Oskolkov raised the problem of"mathematizing"it. We show that it holds exactly in the sense of distributions.
塔尔博特效应通常是用亥姆霍兹方程来模拟的，但它的主要实验特征是用狄拉克梳作为初始基准的自由薛定谔方程的解来捕捉的。这种简化的描述是几何光学中近轴近似的结果。然而，它是一种启发式近似，在数学上没有很好的证明，所以K. I. Oskolkov提出了“数学化”它的问题。我们证明了它在分布的意义上是正确的。
{"title":"The Talbot effect as the fundamental solution to the free Schrödinger equation","authors":"Daniel Eceizabarrena","doi":"10.4171/PM/2068","DOIUrl":"https://doi.org/10.4171/PM/2068","url":null,"abstract":"The Talbot effect is usually modeled using the Helmholtz equation, but its main experimental features are captured by the solution to the free Schr\"odinger equation with the Dirac comb as initial datum. This simplified description is a consequence of the paraxial approximation in geometric optics. However, it is a heuristic approximation that is not mathematically well justified, so K. I. Oskolkov raised the problem of\"mathematizing\"it. We show that it holds exactly in the sense of distributions.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46485188","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}
In this work, we consider the nonlocal obstacle problem with a given obstacle $psi$ in a bounded Lipschitz domain $Omega$ in $mathbb{R}^{d}$, such that $mathbb{K}_psi^s={vin H^s_0(Omega):vgeqpsi text{ a.e. in }Omega}neqemptyset$, given by [uinmathbb{K}_psi^s:langlemathcal{L}_au,v-uranglegeqlangle F,v-uranglequadforall vinmathbb{K}^s_psi,] for $Fin H^{-s}(Omega)$, the dual space of $H^s_0(Omega)$, $0
在这项工作中，我们考虑了在$mathbb｛R｝^｛d｝$中有界Lipschitz域$Omega$中给定障碍物$psi$的非局部障碍问题，使得$mathbb{K}_psi^s={vin H^s_0（Omega）：vgeqpsitext｛a.e.in｝Omega｝neqpemptyset$，由[uinmathbb给出{K}_psi ^s:langlemathcal{L}_au，v-uranglegeqlangle F，v-u ranglequad for all vinmathbb｛K｝^s_psi，]对于$Fin H^｛-s｝（Omega）$，$H^s_0（Ome茄）$的对偶空间，$0
{"title":"On a class of nonlocal obstacle type problems related to the distributional Riesz fractional derivative","authors":"Catharine Lo, J. Rodrigues","doi":"10.4171/PM/2100","DOIUrl":"https://doi.org/10.4171/PM/2100","url":null,"abstract":"In this work, we consider the nonlocal obstacle problem with a given obstacle $psi$ in a bounded Lipschitz domain $Omega$ in $mathbb{R}^{d}$, such that $mathbb{K}_psi^s={vin H^s_0(Omega):vgeqpsi text{ a.e. in }Omega}neqemptyset$, given by [uinmathbb{K}_psi^s:langlemathcal{L}_au,v-uranglegeqlangle F,v-uranglequadforall vinmathbb{K}^s_psi,] for $Fin H^{-s}(Omega)$, the dual space of $H^s_0(Omega)$, $0<s<1$. The nonlocal operator $mathcal{L}_a:H^s_0(Omega)to H^{-s}(Omega)$ is defined with a measurable, bounded, strictly positive singular kernel $a(x,y)$, possibly not symmetric, by [langlemathcal{L}_au,vrangle=P.V.int_{mathbb{R}^d}int_{mathbb{R}^d}v(x)(u(x)-u(y))a(x,y)dydx=mathcal{E}_a(u,v),] with $mathcal{E}_a$ being a Dirichlet form. Also, the fractional operator $tilde{mathcal{L}}_A=-D^scdot AD^s$ defined with the distributional Riesz $s$-fractional derivative and a bounded matrix $A(x)$ gives a well defined integral singular kernel. The corresponding $s$-fractional obstacle problem converges as $snearrow1$ to the obstacle problem in $H^1_0(Omega)$ with the operator $-Dcdot AD$ given with the gradient $D$. We mainly consider problems involving the bilinear form $mathcal{E}_a$ with one or two obstacles, and the N-membranes problem, deriving a weak maximum principle, comparison properties, approximation by bounded penalization, and the Lewy-Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in $L^infty(Omega)$, local H\"older regularity when $a$ is symmetric, and local regularity in $W^{2s,p}_{loc}(Omega)$ and $C^1(Omega)$ for fractional $s$-Laplacian obstacle-type problems. These novel results are complemented with the extension of the Lewy-Stampacchia inequalities to the order dual of $H^s_0(Omega)$ and some remarks on the associated $s$-capacity for general $mathcal{L}_a$.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47723559","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}
{"title":"On the discriminant locus of a rank $n-1$ vector bundle on $mathbb P^{n-1}$","authors":"Hirotachi Abo","doi":"10.4171/pm/2053","DOIUrl":"https://doi.org/10.4171/pm/2053","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42989518","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}
{"title":"Endowing evolution algebras with properties of discrete structures","authors":"R. González-López, J. Núñez","doi":"10.4171/pm/2058","DOIUrl":"https://doi.org/10.4171/pm/2058","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41961669","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}
{"title":"Convergence of a full discrete finite element method for the Korteweg–de Vries equation","authors":"Pengzhan Huang","doi":"10.4171/pm/2043","DOIUrl":"https://doi.org/10.4171/pm/2043","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"77 1","pages":"31-43"},"PeriodicalIF":0.8,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47359789","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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