{"title":"Editorial","authors":"Diogo Gomes, J. M. Urbano","doi":"10.4171/pm/2015","DOIUrl":"https://doi.org/10.4171/pm/2015","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44665997","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":"Rigidity for perimeter inequalities under symmetrization: State of the art and open problems","authors":"F. Cagnetti","doi":"10.4171/PM/2022","DOIUrl":"https://doi.org/10.4171/PM/2022","url":null,"abstract":"We review some classical results in symmetrization theory, some recent progress in understanding rigidity, and indicate some open problems.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47233637","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}
We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.
{"title":"On the Hilbert vector of the Jacobian module of a plane curve","authors":"Armando Cerminara, A. Dimca, G. Ilardi","doi":"10.4171/pm/2038","DOIUrl":"https://doi.org/10.4171/pm/2038","url":null,"abstract":"We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49085556","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}
We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical semigroups with given multiplicity and genus.
{"title":"The set of numerical semigroups of a given multiplicity and Frobenius number","authors":"M. B. Branco, I. Ojeda, J. Rosales","doi":"10.4171/pm/2064","DOIUrl":"https://doi.org/10.4171/pm/2064","url":null,"abstract":"We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical semigroups with given multiplicity and genus.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70897297","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}
We give combinatorial generalizations of the Cayley-Bacharach theorem and induced map.
给出了Cayley-Bacharach定理和归纳映射的组合推广。
{"title":"Sparse versions of the Cayley–Bacharach theorem","authors":"L. Matusevich, B. Reznick","doi":"10.4171/pm/2063","DOIUrl":"https://doi.org/10.4171/pm/2063","url":null,"abstract":"We give combinatorial generalizations of the Cayley-Bacharach theorem and induced map.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44319452","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 aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $Omega$ be an odd-symplectic form on an oriented closed manifold $Sigma$ of odd dimension. We say that $Omega$ is Zoll if the trajectories of the flow given by $Omega$ are the orbits of a free $S^1$-action. After defining the volume of $Omega$ and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided $Omega$ is Zoll. This builds the equality case of a conjectural systolic inequality for odd-symplectic forms close to a Zoll one. We prove the conjecture when the $S^1$-action yields a flat $S^1$-bundle or $Omega$ is quasi-autonomous. In particular the conjecture is established in dimension three. This new inequality recovers the contact systolic inequality as well as the inequality between the minimal action and the Calabi invariant for Hamiltonian isotopies $C^1$-close to the identity on a closed symplectic manifold. Applications to the study of periodic magnetic geodesics on closed orientable surfaces is given in the companion paper available at arXiv:1902.01262.
{"title":"On a local systolic inequality for odd-symplectic forms","authors":"G. Benedetti, Jungsoo Kang","doi":"10.4171/pm/2039","DOIUrl":"https://doi.org/10.4171/pm/2039","url":null,"abstract":"The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $Omega$ be an odd-symplectic form on an oriented closed manifold $Sigma$ of odd dimension. We say that $Omega$ is Zoll if the trajectories of the flow given by $Omega$ are the orbits of a free $S^1$-action. After defining the volume of $Omega$ and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided $Omega$ is Zoll. This builds the equality case of a conjectural systolic inequality for odd-symplectic forms close to a Zoll one. We prove the conjecture when the $S^1$-action yields a flat $S^1$-bundle or $Omega$ is quasi-autonomous. In particular the conjecture is established in dimension three. This new inequality recovers the contact systolic inequality as well as the inequality between the minimal action and the Calabi invariant for Hamiltonian isotopies $C^1$-close to the identity on a closed symplectic manifold. Applications to the study of periodic magnetic geodesics on closed orientable surfaces is given in the companion paper available at arXiv:1902.01262.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42411068","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 paper, we consider a coupling mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart-Thomas space for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable while we consider the backward Euler method for the time discretization. This numerical scheme appears to be one common approach applied to existing reservoir engineering simulators. Theoretical convergence error estimates are derived in a discrete-in-time setting. Previous a priori error estimates described in the literature e.g. [2][19], which are optimal, show first order convergency with respect to the L-norm for the pressure and for the average fluid velocity approximation errors and with respect to the H-norm for the displacement approximation error. Here we prove one extra order of convergence for the displacement approximation with respect to the L-norm. We also demonstrate that, by including a post-processing step in the scheme, the order of convergence for the approximation of pressure can be improved. Even though this result is critical for deriving the Lnorm error estimates for the approximation of the deformation variable, surprisingly the corresponding gain of one convergence order holds independently of including or not the post-processing step in the method.
{"title":"Accuracy of a coupled mixed and Galerkin finite element approximation for poroelasticity","authors":"S. Barbeiro","doi":"10.4171/pm/2018","DOIUrl":"https://doi.org/10.4171/pm/2018","url":null,"abstract":"In this paper, we consider a coupling mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart-Thomas space for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable while we consider the backward Euler method for the time discretization. This numerical scheme appears to be one common approach applied to existing reservoir engineering simulators. Theoretical convergence error estimates are derived in a discrete-in-time setting. Previous a priori error estimates described in the literature e.g. [2][19], which are optimal, show first order convergency with respect to the L-norm for the pressure and for the average fluid velocity approximation errors and with respect to the H-norm for the displacement approximation error. Here we prove one extra order of convergence for the displacement approximation with respect to the L-norm. We also demonstrate that, by including a post-processing step in the scheme, the order of convergence for the approximation of pressure can be improved. Even though this result is critical for deriving the Lnorm error estimates for the approximation of the deformation variable, surprisingly the corresponding gain of one convergence order holds independently of including or not the post-processing step in the method.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70896792","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}
We obtain large deviations estimates for systems with stretched exponential decay of correlations, which improve the ones obtained in cite{AFLV11}. As a consequence we obtain better large deviations estimates for Viana maps and get large deviations estimates for a class of intermittent maps with stretched exponential loss of memory.
{"title":"Large deviations for dynamical systems with stretched exponential decay of correlations","authors":"R. Aimino, J. Freitas","doi":"10.4171/pm/2030","DOIUrl":"https://doi.org/10.4171/pm/2030","url":null,"abstract":"We obtain large deviations estimates for systems with stretched exponential decay of correlations, which improve the ones obtained in cite{AFLV11}. As a consequence we obtain better large deviations estimates for Viana maps and get large deviations estimates for a class of intermittent maps with stretched exponential loss of memory.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46662865","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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