{"title":"Non-coercive Neumann Boundary Control Problems","authors":"Thomas Apel, Mariano Mateos, Arnd Rösch","doi":"10.1007/s00025-024-02255-8","DOIUrl":"https://doi.org/10.1007/s00025-024-02255-8","url":null,"abstract":"<p>The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem’s data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"73 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Certain Observations on Tightness and Topological Games in Bornology","authors":"Debraj Chandra, Pratulananda Das, Subhankar Das","doi":"10.1007/s00025-024-02256-7","DOIUrl":"https://doi.org/10.1007/s00025-024-02256-7","url":null,"abstract":"<p>This article is a continuation of our investigations in the function space <i>C</i>(<i>X</i>) with respect to the topology <span>(tau ^s_mathfrak {B})</span> of strong uniform convergence on <span>(mathfrak {B})</span> in line of (Chandra et al. in Indag Math 31:43-63, 2020; Das et al. in Topol Appl 310:108005, 2022) using the idea of strong uniform convergence (Beer and Levi in J Math Anal Appl 350:568-589, 2009) on a bornology. First we focus on the notion of the tightness property of <span>((C(X),tau ^s_mathfrak {B}))</span> and some of its variations such as the supertightness, the Id-fan tightness and the <i>T</i>-tightness. Certain situations are discussed when <i>C</i>(<i>X</i>) is a k-space with respect to the topology <span>(tau ^s_mathfrak {B})</span>. Next the notions of strong <span>(mathfrak {B})</span>-open game and <span>(gamma _{mathfrak {B}^s})</span>-open game on <i>X</i> are introduced and some of its consequences are investigated. Finally, we consider discretely selective property and related games. On <span>((C(X),tau ^s_mathfrak {B}))</span> several interactions between topological games related to discretely selective property, the Gruenhage game on <span>((C(X),tau ^s_mathfrak {B}))</span> and certain games on <i>X</i> are presented.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"82 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Class of Projectively Flat Finsler Metrics","authors":"Huaifu Liu, Xiaohuan Mo, Ling Zhu","doi":"10.1007/s00025-024-02252-x","DOIUrl":"https://doi.org/10.1007/s00025-024-02252-x","url":null,"abstract":"<p>Projectively flat Finlser metrics on a convex domain <i>U</i> in <span>(mathbb {R}^n)</span> are regular solutions to Hilbert’s Fourth Problem. In this paper, we study projectively flat Finlser metrics on <i>U</i>. We find equations that characterize these metrics with weakly orthogonal invariance, refining a theorem due to Sol<span>(acute{o})</span>rzano-Le<span>(acute{o})</span>n. As its application, we obtain infinitely many <i>new</i> projectively flat Finlser metrics on <span>(mathbb {S}^{n+1})</span> and determine their scalar flag curvature. These metrics contain Bryant’s projective spherically symmetric Finsler metric of constant flag curvature 1.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"23 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Thomas Kneib, Jan-Christian Schlüter, Benjamin Wacker
{"title":"Revisiting Maximum Log-Likelihood Parameter Estimation for Two-Parameter Weibull Distributions: Theory and Applications","authors":"Thomas Kneib, Jan-Christian Schlüter, Benjamin Wacker","doi":"10.1007/s00025-024-02258-5","DOIUrl":"https://doi.org/10.1007/s00025-024-02258-5","url":null,"abstract":"<p>In this article, we reexamine properties of maximum log-likelihood parameter estimation for two-parameter Weibull distributions which have been applied in many different sciences. Finding reasons for this popularity is a key question. Our main contribution is a thorough existence and uniqueness proof for a global maximizer with respect to the parameter space. We first provide existence and uniqueness of local maximizers by Schauder’s first fixed point theorem, monotony arguments and local concavity of its Hessian matrix. Thus, we can prove our main result of existence and uniqueness of a global maximizer by considering all limiting cases with respect to the parameter space. We finally strengthen our theoretical findings on four data sets. On the one hand, two synthetic data sets underline our need for our data assumptions while, on the other hand, we choose two data sets from wind engineering and reliability engineering to demonstrate usefulness in real-world applications.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"164 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Trajectories of a Particle in a Translation Invariant Involutive Field","authors":"Cristian Cobeli, Alexandru Zaharescu","doi":"10.1007/s00025-024-02240-1","DOIUrl":"https://doi.org/10.1007/s00025-024-02240-1","url":null,"abstract":"<p>We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct partitions of the set of lattice points in the plane. Our object is to analyze these trajectories and to point out a few special arithmetic properties of the integers they represent. We also introduce and study the parabolic-taxicab distance, which measures the fast traveling on the steps of the stairs defined by points on the parabolic trajectories whose coordinates are based on triangular numbers.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"2012 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Volterra Operators on Dales–Davie Algebras","authors":"A. H. Gholizadeh, Z. S. Hosseini, A. H. Sanatpour","doi":"10.1007/s00025-024-02249-6","DOIUrl":"https://doi.org/10.1007/s00025-024-02249-6","url":null,"abstract":"<p>We investigate the boundedness and compactness properties of integral operators, Volterra operators, and generalized Volterra operators between the Dales–Davie algebras. Additionally, we study the analogous properties of these operators when applied to the Lipschitz versions of Dales–Davie algebras.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"92 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Reversed Hardy-Littlewood-Sobolev Type Inequality on $${mathbb {R}}^{n-m}times {mathbb {R}}^{n}$$","authors":"Xiang Li, Minbo Yang","doi":"10.1007/s00025-024-02227-y","DOIUrl":"https://doi.org/10.1007/s00025-024-02227-y","url":null,"abstract":"<p>In this paper, we are going to establish a reversed Hardy–Littlewood–Sobolev inequality on different dimensional space and prove the existence of extremal functions for the best constant. Furthermore, we investigate the regularity of extremal functions and provide a necessary conditions for the existence of solutions of the integral systems. Finally, we classify the extremal functions.\u0000</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"34 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Strong Consistency of Wavelet Estimator for Biased Nonparametric Regression Function Under Strong Mixing","authors":"Yuncai Yu","doi":"10.1007/s00025-024-02248-7","DOIUrl":"https://doi.org/10.1007/s00025-024-02248-7","url":null,"abstract":"<p>This paper focuses on the function estimation problem in nonparametric regression model based on biased samples under strong mixing. We propose a wavelet estimator by using wavelet kernel and investigate the consistency properties of the wavelet estimator. The mean consistency, strong consistency and convergence rate are obtained and the convergence rate is similar as that of wavelet estimator in the standard nonparametric model even although with the presence of bias and strong mixing dependence.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"78 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Gevrey Versus q-Gevrey Asymptotic Expansions for Some Linear q-Difference–Differential Cauchy Problem","authors":"Alberto Lastra, Stéphane Malek","doi":"10.1007/s00025-024-02250-z","DOIUrl":"https://doi.org/10.1007/s00025-024-02250-z","url":null,"abstract":"<p>The asymptotic behavior of the analytic solutions of a family of singularly perturbed <i>q</i>-difference–differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to the time variable are provided: one of Gevrey nature, and another of mixed type Gevrey and <i>q</i>-Gevrey. These asymptotic phenomena are observed due to the modification of the norm established on the space of coefficients of the formal solution. The techniques used are based on the adequate path deformation of the difference of two analytic solutions, and the application of several versions of Ramis–Sibuya theorem.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"34 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dual Framelets Transform on Manifolds and Graphs","authors":"Radhakrushna Sahoo","doi":"10.1007/s00025-024-02247-8","DOIUrl":"https://doi.org/10.1007/s00025-024-02247-8","url":null,"abstract":"<p>In this paper, the concept of dual framelets on manifolds and its characterization are introduced. The accuracy of the proposed dual framelets transform is determined by sparse representation on graphs. If any pair of the framelet system is associated with filter-bank transform, then compactly supported refinable functions can have vanishing moments at most one and framelet approximation is the order of at most two. An algorithm of decomposition and reconstruction for the dual framelets transform on graph is presented. A new method called dual framelets filter-bank transform (DFFT) is employed, which is faster than the existing method spectral graph wavelet transform (SGWT). The theoretical results along with algorithms for accurate and efficient computation of the DFFT on discrete data sets are provided. Subsequently, some numerical examples are provided to show the importance of DFFT over SGWT on graphs.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"221 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}