Rosário Fernandes, Henrique F. da Cruz, Domingos Salomão
In a previous paper, the authors introduced and studied the Bruhat order in the class of Latin squares of order n. In this paper, we investigate the restriction of the Bruhat order in a class of isotopic Latin squares. We present equivalent conditions for two Latin squares be related by the Bruhat order when one of them is obtained from the other by interchanging rows or columns or symbols. The cover relation is also addressed, and we present orthogonal isotopic Latin squares related by the Bruhat order.
{"title":"The Bruhat order on classes of isotopic Latin squares","authors":"Rosário Fernandes, Henrique F. da Cruz, Domingos Salomão","doi":"10.4171/pm/2046","DOIUrl":"https://doi.org/10.4171/pm/2046","url":null,"abstract":"In a previous paper, the authors introduced and studied the Bruhat order in the class of Latin squares of order n. In this paper, we investigate the restriction of the Bruhat order in a class of isotopic Latin squares. We present equivalent conditions for two Latin squares be related by the Bruhat order when one of them is obtained from the other by interchanging rows or columns or symbols. The cover relation is also addressed, and we present orthogonal isotopic Latin squares related by the Bruhat order.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43359088","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":"Global existence for a one-dimensional non-relativistic Euler model with relaxation","authors":"Shuyang Xiang, Yangyang Cao","doi":"10.4171/pm/2044","DOIUrl":"https://doi.org/10.4171/pm/2044","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"77 1","pages":"45-71"},"PeriodicalIF":0.8,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43037224","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 Note is to specify the links between the three kinds of Lisbon integrals, trace functions and trace forms with the corresponding D−modules.
本注的目的是说明三种里斯本积分、迹函数和迹形式与相应的D -模之间的联系。
{"title":"Note on Lisbon integrals and their associated $D$-modules","authors":"D. Barlet","doi":"10.4171/pm/2071","DOIUrl":"https://doi.org/10.4171/pm/2071","url":null,"abstract":"The aim of this Note is to specify the links between the three kinds of Lisbon integrals, trace functions and trace forms with the corresponding D−modules.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42850618","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 study the existence of solution to the problem begin{equation*} left{begin{array}{l} uin H_{0}^{1}(Omega), [4pt] -textrm{div},(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x), uquad text{in} quadmathcal{D}'(Omega), end{array} right. end{equation*} where $Omega$ is an open bounded set of $mathbb{R}^{2}$, $A(x)$ a coercive matrix with coefficients in $L^infty(Omega)$, $H(x,s,xi)$ a Carath'eodory function satisfying, for some $gamma >0$, $$ -c_{0}, A(x), xixileq H(x,s,xi),{rm sign}(s)leq gamma,A(x),xixi ;;; {rm a.e. }; xin Omega,;;;forall sinmathbb{R},;;; forallxi in mathbb{R}^{2}. $$ Here $f$ belongs to $L^1(log L^1)(Omega)$ and $a_{0} geq 0$ to $L^{q}(Omega )$, $q>1$. For $f$ and $a_{0}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is such that $e^{delta_0 |u|} -1$ belongs to $H_{0}^{1}(Omega)$ for some $delta_0geqgamma$ and satisfies an textit{a priori} estimate.
{"title":"Some existence results for a quasilinear problem with source term in Zygmund-space","authors":"B. Hamour","doi":"10.4171/pm/2035","DOIUrl":"https://doi.org/10.4171/pm/2035","url":null,"abstract":"In this paper we study the existence of solution to the problem \u0000begin{equation*} \u0000left{begin{array}{l} uin H_{0}^{1}(Omega), \u0000[4pt] \u0000-textrm{div},(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x), uquad text{in} \u0000quadmathcal{D}'(Omega), \u0000end{array} \u0000right. \u0000end{equation*} \u0000where $Omega$ is an open bounded set of $mathbb{R}^{2}$, \u0000$A(x)$ a coercive matrix with coefficients in \u0000$L^infty(Omega)$, $H(x,s,xi)$ a Carath'eodory function \u0000satisfying, for some $gamma >0$, \u0000$$ \u0000 -c_{0}, A(x), xixileq H(x,s,xi),{rm sign}(s)leq gamma,A(x),xixi ;;; \u0000{rm a.e. }; xin Omega,;;;forall sinmathbb{R},;;; \u0000 forallxi in mathbb{R}^{2}. \u0000$$ \u0000Here $f$ belongs to $L^1(log L^1)(Omega)$ and $a_{0} geq 0$ to $L^{q}(Omega )$, $q>1$. \u0000For $f$ and $a_{0}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is such that $e^{delta_0 |u|} -1$ belongs to $H_{0}^{1}(Omega)$ for \u0000some $delta_0geqgamma$ and satisfies an textit{a priori} estimate.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46118963","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 study the existence and regularity of solutions to the following Dirichlet problem −div(a(x)|∇u| p−2 ∇u) + u|u| r−1 = f (x) u θ in Ω, u > 0 in Ω, u = 0 on ∂Ω proving that the lower order term u|u| r−1 has some regularizing effects on the solutions. Mathematics Subject Classification (2010). 35J66, 35J75
本文研究了以下Dirichlet问题−div(a(x)|∇u| p−2∇u) + u|u| r−1 = f (x) u θ在Ω中，u >在Ω中，u = 0在∂Ω中，证明了低阶项u|u| r−1对解有一定的正则化作用。数学学科分类(2010)。35 j66 35 j75
{"title":"The impact of a lower order term in a Dirichlet problem with a singular nonlinearity","authors":"L. Boccardo, G. Croce","doi":"10.4171/PM/2041","DOIUrl":"https://doi.org/10.4171/PM/2041","url":null,"abstract":"In this paper we study the existence and regularity of solutions to the following Dirichlet problem −div(a(x)|∇u| p−2 ∇u) + u|u| r−1 = f (x) u θ in Ω, u > 0 in Ω, u = 0 on ∂Ω proving that the lower order term u|u| r−1 has some regularizing effects on the solutions. Mathematics Subject Classification (2010). 35J66, 35J75","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46517568","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}
. This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [9], we propose a more generalized algorithm for such systems. The algorithm is based on a block decomposition for a block tridiagonal quasi-Toeplitz matrix and the Sherman-Morrison-Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method. A theoretical accuracy and error analysis is also considered. All algorithms have been implemented in Matlab. Numerical experiments performed with a wide variety of test problems show the effectiveness of our algorithm in terms of efficience, stability and robustness.
{"title":"A fast method for solving a block tridiagonal quasi-Toeplitz linear system","authors":"S. Belhaj, Fahd Hcini, Yulin Zhang","doi":"10.4171/pm/2036","DOIUrl":"https://doi.org/10.4171/pm/2036","url":null,"abstract":". This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [9], we propose a more generalized algorithm for such systems. The algorithm is based on a block decomposition for a block tridiagonal quasi-Toeplitz matrix and the Sherman-Morrison-Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method. A theoretical accuracy and error analysis is also considered. All algorithms have been implemented in Matlab. Numerical experiments performed with a wide variety of test problems show the effectiveness of our algorithm in terms of efficience, stability and robustness.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43472574","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 generalized Bregman nonspreading mappings and zero points of maximal monotone operator in a reflexive Banach space","authors":"L. Jolaoso, O. Mewomo","doi":"10.4171/pm/2034","DOIUrl":"https://doi.org/10.4171/pm/2034","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"76 1","pages":"229-258"},"PeriodicalIF":0.8,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49528854","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}
I. Boutaayamou, Salah-Eddine Chorfi, L. Maniar, O. Oukdach
We consider the heat equation with dynamic bounary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.
{"title":"The cost of approximate controllability of heat equation with general dynamical boundary conditions","authors":"I. Boutaayamou, Salah-Eddine Chorfi, L. Maniar, O. Oukdach","doi":"10.4171/PM/2061","DOIUrl":"https://doi.org/10.4171/PM/2061","url":null,"abstract":"We consider the heat equation with dynamic bounary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49377281","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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