We consider generic degenerate subvarieties $X_isubsetmathbb{P}^n$. We determine an integer $N$, depending on the varieties, and for $ngeq N$ we compute dimension and degree formulas for the Hadamard product of the varieties $X_i$. Moreover, if the varieties $X_i$ are smooth, their Hadamard product is smooth too. For $n
{"title":"On the Hadamard product of degenerate subvarieties","authors":"G. Calussi, E. Carlini, G. Fatabbi, A. Lorenzini","doi":"10.4171/pm/2029","DOIUrl":"https://doi.org/10.4171/pm/2029","url":null,"abstract":"We consider generic degenerate subvarieties $X_isubsetmathbb{P}^n$. We determine an integer $N$, depending on the varieties, and for $ngeq N$ we compute dimension and degree formulas for the Hadamard product of the varieties $X_i$. Moreover, if the varieties $X_i$ are smooth, their Hadamard product is smooth too. For $n<N$, if the $X_i$ are generically $d_i$-parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45700213","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":"A new class of monohedral pentagonal spherical tilings with GeoGebra","authors":"A. Breda, José Manuel Dos Santos Dos Santos","doi":"10.4171/PM/2006","DOIUrl":"https://doi.org/10.4171/PM/2006","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"74 1","pages":"257-266"},"PeriodicalIF":0.8,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46068240","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 describe some group theory which is useful in the classification of combinatorial objects having given groups of automorphisms. In particular, we show the usefulness of the concept of a friendly subgroup: a subgroup H of a group K is a friendly subgroup of K if every subgroup of K isomorphic to H is conjugate in K to H. We explore easy-to-test sufficient conditions for a subgroup H to be a friendly subgroup of a finite group K, and for this, present an algorithm for determining whether a finite group H is a Sylow tower group.
{"title":"On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups","authors":"L. H. Soicher","doi":"10.4171/PM/2004","DOIUrl":"https://doi.org/10.4171/PM/2004","url":null,"abstract":"We describe some group theory which is useful in the classification of combinatorial objects having given groups of automorphisms. In particular, we show the usefulness of the concept of a friendly subgroup: a subgroup H of a group K is a friendly subgroup of K if every subgroup of K isomorphic to H is conjugate in K to H. We explore easy-to-test sufficient conditions for a subgroup H to be a friendly subgroup of a finite group K, and for this, present an algorithm for determining whether a finite group H is a Sylow tower group.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"74 1","pages":"233-242"},"PeriodicalIF":0.8,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46475633","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":"Corrigendum to ‘‘Positive integers divisible by the product of their nonzero digits’’, Portugaliae Math. 64 (2007), 1: 75–85","authors":"J. Koninck, F. Luca","doi":"10.4171/PM/1999","DOIUrl":"https://doi.org/10.4171/PM/1999","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"74 1","pages":"169-170"},"PeriodicalIF":0.8,"publicationDate":"2017-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/1999","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45116581","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":"Approximation rates for regularized level set power mean curvature flow","authors":"H. Kröner","doi":"10.4171/PM/1995","DOIUrl":"https://doi.org/10.4171/PM/1995","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"74 1","pages":"115-126"},"PeriodicalIF":0.8,"publicationDate":"2017-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/1995","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49332925","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}
Víctor Hernández-Santamaría, L. Teresa, A. Poznyak
{"title":"Corrigendum and addendum to ‘‘Hierarchic control for a coupled parabolic system’’, Portugaliae Math. 73 (2016), 2: 115–137","authors":"Víctor Hernández-Santamaría, L. Teresa, A. Poznyak","doi":"10.4171/PM/1998","DOIUrl":"https://doi.org/10.4171/PM/1998","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"74 1","pages":"161-168"},"PeriodicalIF":0.8,"publicationDate":"2017-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/1998","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45927190","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 bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.
{"title":"Bitangents of non-smooth tropical quartics","authors":"H. Lee, Yoav Len","doi":"10.4171/PM/2011","DOIUrl":"https://doi.org/10.4171/PM/2011","url":null,"abstract":"We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2017-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46573226","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}
A new scheme for proving pseudoidentities from a given set {Sigma} of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when {Sigma} defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when {Sigma} defines a pseudovariety V which is {sigma}-reducible for the equation x=y, provided {Sigma} is enough to prove a basis of identities for the variety of {sigma}-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.
{"title":"Towards a pseudoequational proof theory","authors":"J. Almeida, Ondvrej Kl'ima","doi":"10.4171/PM/2012","DOIUrl":"https://doi.org/10.4171/PM/2012","url":null,"abstract":"A new scheme for proving pseudoidentities from a given set {Sigma} of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when {Sigma} defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when {Sigma} defines a pseudovariety V which is {sigma}-reducible for the equation x=y, provided {Sigma} is enough to prove a basis of identities for the variety of {sigma}-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2017-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47125792","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 present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time–dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from [26, 27]. It is in particular well suited for problems with time–dependent PDE constraints. Instead of the quasi-normal step in a classical SQP method which results in solving the linearized PDE sufficiently well, in this algorithm a (nonlinear) solver is applied to the current discretization of the PDE. Moreover, different discretizations and solvers for the PDE and the adjoint PDE may be applied. The resulting inexactness of the reduced gradient in the current discretization is controlled within the algorithm. Thus, highly efficient PDE solvers can be coupled with the proposed optimization framework. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process implementable criteria for an adaptive refinement strategy of the current discretization based on error estimators. We prove global convergence to a stationary point of the infinitedimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using a posteriori error estimators for the state and the adjoint equation. Numerical results for a semilinear parabolic PDE–constrained problem with pointwise control constraints are presented. Mathematics Subject Classification (2010). 90C55 49M05 49M25 49M37
{"title":"Adaptive multilevel trust-region methods for time-dependent PDE-constrained optimization","authors":"S. Ulbrich, J. Ziems","doi":"10.4171/PM/1992","DOIUrl":"https://doi.org/10.4171/PM/1992","url":null,"abstract":"We present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time–dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from [26, 27]. It is in particular well suited for problems with time–dependent PDE constraints. Instead of the quasi-normal step in a classical SQP method which results in solving the linearized PDE sufficiently well, in this algorithm a (nonlinear) solver is applied to the current discretization of the PDE. Moreover, different discretizations and solvers for the PDE and the adjoint PDE may be applied. The resulting inexactness of the reduced gradient in the current discretization is controlled within the algorithm. Thus, highly efficient PDE solvers can be coupled with the proposed optimization framework. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process implementable criteria for an adaptive refinement strategy of the current discretization based on error estimators. We prove global convergence to a stationary point of the infinitedimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using a posteriori error estimators for the state and the adjoint equation. Numerical results for a semilinear parabolic PDE–constrained problem with pointwise control constraints are presented. Mathematics Subject Classification (2010). 90C55 49M05 49M25 49M37","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"74 1","pages":"37-67"},"PeriodicalIF":0.8,"publicationDate":"2017-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45636374","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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