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Counterexamples to maximal regularity for operators in divergence form 发散形式算子最大正则性的反例
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-20 DOI: 10.1007/s00013-024-02014-9
Sebastian Bechtel, Connor Mooney, Mark Veraar
{"title":"Counterexamples to maximal regularity for operators in divergence form","authors":"Sebastian Bechtel,&nbsp;Connor Mooney,&nbsp;Mark Veraar","doi":"10.1007/s00013-024-02014-9","DOIUrl":"10.1007/s00013-024-02014-9","url":null,"abstract":"<div><p>In this paper, we present counterexamples to maximal <span>(L^p)</span>-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal <span>(L^2)</span>-regularity on <span>(H^{-1})</span> under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal <span>(L^p)</span>-regularity on <span>(H^{-1}(mathbb {R}^d))</span> or <span>(L^2)</span>-regularity on <span>(L^2(mathbb {R}^d))</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02014-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On interpretation of Fourier coefficients of Zagier type lifts 关于扎吉尔式升降机傅里叶系数的解释
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-13 DOI: 10.1007/s00013-024-02005-w
Vaibhav Kalia
{"title":"On interpretation of Fourier coefficients of Zagier type lifts","authors":"Vaibhav Kalia","doi":"10.1007/s00013-024-02005-w","DOIUrl":"10.1007/s00013-024-02005-w","url":null,"abstract":"<div><p>Jeon, Kang, and Kim defined the Zagier lifts between harmonic weak Maass forms of negative integral weights and half integral weights. These lifts were defined by establishing that traces related to cycle integrals of harmonic weak Maass forms of integral weights appear as Fourier coefficients of harmonic weak Maass forms of half integral weights. For fundamental discriminants <i>d</i> and <span>(delta ,)</span> they studied <span>(delta )</span>-th Fourier coefficients of the <i>d</i>-th Zagier lift with respect to the condition that <span>(ddelta )</span> is not a perfect square. For <span>(ddelta )</span> being a perfect square, the interpretation of coefficients in terms of traces is not possible due to the divergence of cycle integrals. In this paper, we provide an alternate definition of traces called <i>modified trace</i> in the condition that <span>(ddelta )</span> is a perfect square and interpret such coefficients in terms of the <i>modified trace</i>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141350095","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}
引用次数: 0
On the number of conjugacy classes of a finite solvable group 论有限可解群的共轭类数
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-13 DOI: 10.1007/s00013-024-01989-9
Yong Yang, Mengtian Zhang
{"title":"On the number of conjugacy classes of a finite solvable group","authors":"Yong Yang,&nbsp;Mengtian Zhang","doi":"10.1007/s00013-024-01989-9","DOIUrl":"10.1007/s00013-024-01989-9","url":null,"abstract":"<div><p>Let <i>p</i> be a prime that divides the order of the group <i>G</i>. We show that a finite solvable group has class number at least <i>f</i>(<i>p</i>) where <span>(f(p):=min {x+frac{p-1}{x}: xin mathbb {N}, x mid (p-1)})</span>. We also obtain some applications to character degrees.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141349812","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}
引用次数: 0
Pólya-type estimates for the first Robin eigenvalue of elliptic operators 椭圆算子第一个罗宾特征值的波利亚型估计值
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-13 DOI: 10.1007/s00013-024-02012-x
Francesco Della Pietra
{"title":"Pólya-type estimates for the first Robin eigenvalue of elliptic operators","authors":"Francesco Della Pietra","doi":"10.1007/s00013-024-02012-x","DOIUrl":"10.1007/s00013-024-02012-x","url":null,"abstract":"<div><p>The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic <i>p</i>-Laplace operator, namely: </p><div><div><span>$$begin{aligned} lambda _F(beta ,Omega )= min _{psi in W^{1,p}(Omega ){setminus }{0} } frac{displaystyle int _Omega F(nabla psi )^p dx +beta int _{partial Omega }|psi |^p F(nu _{Omega }) d{mathcal {H}}^{N-1} }{displaystyle int _Omega |psi |^p dx}, end{aligned}$$</span></div></div><p>where <span>(pin ]1,+infty [,)</span> <span>(Omega )</span> is a bounded, convex domain in <span>({mathbb {R}}^{N},)</span> <span>(nu _{Omega })</span> is its Euclidean outward normal, <span>(beta )</span> is a real number, and <i>F</i> is a sufficiently smooth norm on <span>({mathbb {R}}^{N}.)</span> We show an upper bound for <span>(lambda _{F}(beta ,Omega ))</span> in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on <span>(beta )</span> and on the volume and the anisotropic perimeter of <span>(Omega ,)</span> in the spirit of the classical estimates of Pólya (J Indian Math Soc (NS) 24:413–419, 1961) for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity </p><div><div><span>$$begin{aligned} tau _p(beta ,Omega )^{p-1} = max _{begin{array}{c} psi in W^{1,p}(Omega ){setminus }{0} end{array}} dfrac{left( displaystyle int _Omega |psi | , dxright) ^p}{displaystyle int _Omega F(nabla psi )^p dx+beta int _{partial Omega }|psi |^p F(nu _{Omega }) d{mathcal {H}}^{N-1} } end{aligned}$$</span></div></div><p>when <span>(beta &gt;0.)</span> The obtained results are new also in the case of the classical Euclidean Laplacian.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02012-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Conformal geometry of complete quasi Yamabe gradient solitons 完整准山叶梯度孤子的共形几何
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-13 DOI: 10.1007/s00013-024-02016-7
Joao Francisco da Silva Filho, Larissa Braga Fernandes
{"title":"Conformal geometry of complete quasi Yamabe gradient solitons","authors":"Joao Francisco da Silva Filho,&nbsp;Larissa Braga Fernandes","doi":"10.1007/s00013-024-02016-7","DOIUrl":"10.1007/s00013-024-02016-7","url":null,"abstract":"<div><p>The purpose of this work is to study the conformal geometry of complete quasi Yamabe gradient solitons, which correspond to an interesting generalization for gradient Yamabe solitons. In this sense, we present a rigidity result for complete quasi Yamabe gradient solitons with constant scalar curvature. Moreover, we prove that quasi Yamabe gradient solitons can be conformally changed to constant scalar curvature.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524878","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}
引用次数: 0
Characterization of self-majorizing elements in Archimedean unital f-algebras 阿基米德单元f数组中自约化元素的特征
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-12 DOI: 10.1007/s00013-024-02007-8
Mohamed Ali Toumi
{"title":"Characterization of self-majorizing elements in Archimedean unital f-algebras","authors":"Mohamed Ali Toumi","doi":"10.1007/s00013-024-02007-8","DOIUrl":"10.1007/s00013-024-02007-8","url":null,"abstract":"<div><p>We give a complete description of self-majorizing elements of Archimedean unital <i>f</i>-algebras. As an application, we furnish a new characterization of self-majorizing elements of Archimedean vector lattices.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141351160","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}
引用次数: 0
Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements 有素数元素的 Krull 域上整值多项式因式分解的长度
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-07 DOI: 10.1007/s00013-024-02001-0
Victor Fadinger-Held, Daniel Windisch
{"title":"Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements","authors":"Victor Fadinger-Held,&nbsp;Daniel Windisch","doi":"10.1007/s00013-024-02001-0","DOIUrl":"10.1007/s00013-024-02001-0","url":null,"abstract":"<div><p>Let <i>D</i> be a Krull domain admitting a prime element with finite residue field and let <i>K</i> be its quotient field. We show that for all positive integers <i>k</i> and <span>(1 &lt; n_1 le cdots le n_k)</span>, there exists an integer-valued polynomial on <i>D</i>, that is, an element of <span>({{,textrm{Int},}}(D) = { f in K[X] mid f(D) subseteq D })</span>, which has precisely <i>k</i> essentially different factorizations into irreducible elements of <span>({{,textrm{Int},}}(D))</span> whose lengths are exactly <span>(n_1, ldots , n_k)</span>. Using this, we characterize lengths of factorizations when <i>D</i> is a unique factorization domain and therefore also in case <i>D</i> is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02001-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141372923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Virtual Morita equivalences and Brauer character bijections 虚拟莫里塔等价和布劳尔特征双射
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-29 DOI: 10.1007/s00013-024-02010-z
Xin Huang
{"title":"Virtual Morita equivalences and Brauer character bijections","authors":"Xin Huang","doi":"10.1007/s00013-024-02010-z","DOIUrl":"10.1007/s00013-024-02010-z","url":null,"abstract":"<div><p>We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Galois compatible bijections between Brauer characters to virtual Morita equivalences. As a corollary, we obtain that the block version of Navarro’s refinement of Alperin’s weight conjecture holds for blocks with cyclic and Klein four defect groups, blocks of symmetric and alternating groups with abelian defect groups, and <i>p</i>-Blocks of <span>(textrm{SL}_2(q))</span> and <span>(textrm{GL}_2(q))</span>, where <i>p</i>|<i>q</i>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141187936","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}
引用次数: 0
Sylow intersections and Frobenius ratios Sylow 交集和 Frobenius 比率
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-29 DOI: 10.1007/s00013-024-01995-x
Wolfgang Knapp, Peter Schmid
{"title":"Sylow intersections and Frobenius ratios","authors":"Wolfgang Knapp,&nbsp;Peter Schmid","doi":"10.1007/s00013-024-01995-x","DOIUrl":"10.1007/s00013-024-01995-x","url":null,"abstract":"<div><p>Let <i>G</i> be a finite group and <i>p</i> a prime dividing its order |<i>G</i>|, with <i>p</i>-part <span>(|G|_p)</span>, and let <span>(G_p)</span> denote the set of all <i>p</i>-elements in <i>G</i>. A well known theorem of Frobenius tells us that <span>(f_p(G)=|G_p|/|G|_p)</span> is an integer. As <span>(G_p)</span> is the union of the Sylow <i>p</i>-subgroups of <i>G</i>, this <i>Frobenius ratio</i> <span>(f_p(G))</span> evidently depends on the number <span>(s_p(G)=|textrm{Syl}_p(G)|)</span> of Sylow <i>p</i>-subgroups of <i>G</i> and on <i>Sylow intersections</i>. One knows that <span>(s_p(G)=1+kp)</span> and <span>(f_p(G)=1+ell (p-1))</span> for nonnegative integers <span>(k, ell )</span>, and that <span>(f_p(G)&lt;s_p(G))</span> unless <i>G</i> has a normal Sylow <i>p</i>-subgroup. In order to get lower bounds for <span>(f_p(G))</span> we, study the permutation character <span>({pi }={pi }_p(G))</span> of <i>G</i> in its transitive action on <span>(textrm{Syl}_p(G))</span> via conjugation (Sylow character). We will get, in particular, that <span>(f_p(G)ge s_p(G)/r_p(G))</span> where <span>(r_p(G))</span> denotes the number of <i>P</i>-orbits on <span>(textrm{Syl}_p(G))</span> for any fixed <span>(Pin textrm{Syl}_p(G))</span>. One can have <span>(ell ge kge 1)</span> only when <i>P</i> is irredundant for <span>(G_p)</span>, that is, when <i>P</i> is not contained in the union of the <span>(Qne P)</span> in <span>(textrm{Syl}_p(G))</span> and so <span>(widehat{P}=bigcup _{Qne P}(Pcap Q))</span> a proper subset of <i>P</i>. We prove that <span>(ell ge k)</span> when <span>(|widehat{P}|le |P|/p)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01995-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multiplicativity of linear functionals on function spaces on an open disc 开放圆盘上函数空间线性函数的乘法性
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-28 DOI: 10.1007/s00013-024-02002-z
Jaikishan, Sneh Lata, Dinesh Singh
{"title":"Multiplicativity of linear functionals on function spaces on an open disc","authors":"Jaikishan,&nbsp;Sneh Lata,&nbsp;Dinesh Singh","doi":"10.1007/s00013-024-02002-z","DOIUrl":"10.1007/s00013-024-02002-z","url":null,"abstract":"<div><p>This paper presents a fairly general version of the well-known Gleason–Kahane–<span>(dot{text {Z}})</span>elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials <span>(mathcal {P})</span>. We do not make any topological assumptions on <span>(mathcal {P})</span>. We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141171009","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}
引用次数: 0
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