Pub Date : 2024-01-27DOI: 10.1007/s00605-023-01935-x
Maxwell Forst, Lenny Fukshansky
A lattice (Lambda ) is said to be an extension of a sublattice L of smaller rank if L is equal to the intersection of (Lambda ) with the subspace spanned by L. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.
如果 L 等于 (Lambda )与 L 所跨子空间的交集,那么一个网格 (Lambda )就可以说是一个秩较小的子网格 L 的扩展。我们首先证明给定网格的小确定性扩展的存在,然后研究连续最小值和覆盖半径。为此,我们研究了保留给定网格的连续最小值的扩展(在环境网格内),以及保留覆盖半径的扩展。我们还展示了平面网格深洞的一些有趣的算术性质。
{"title":"On lattice extensions","authors":"Maxwell Forst, Lenny Fukshansky","doi":"10.1007/s00605-023-01935-x","DOIUrl":"https://doi.org/10.1007/s00605-023-01935-x","url":null,"abstract":"<p>A lattice <span>(Lambda )</span> is said to be an extension of a sublattice <i>L</i> of smaller rank if <i>L</i> is equal to the intersection of <span>(Lambda )</span> with the subspace spanned by <i>L</i>. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-25DOI: 10.1007/s00605-023-01938-8
Wenguang Cheng, Ji Lin
We consider the Cauchy problem of a Camassa–Holm type equation with quadratic and cubic nonlinearities. We establish a new sufficient condition on the initial data that leads to the wave-breaking for this equation. Moreover, we obtain the persistence results of solutions for the equation in weighted (L^p) spaces.
{"title":"Wave-breaking and persistence properties in weighted $$L^p$$ spaces for a Camassa–Holm type equation with quadratic and cubic nonlinearities","authors":"Wenguang Cheng, Ji Lin","doi":"10.1007/s00605-023-01938-8","DOIUrl":"https://doi.org/10.1007/s00605-023-01938-8","url":null,"abstract":"<p>We consider the Cauchy problem of a Camassa–Holm type equation with quadratic and cubic nonlinearities. We establish a new sufficient condition on the initial data that leads to the wave-breaking for this equation. Moreover, we obtain the persistence results of solutions for the equation in weighted <span>(L^p)</span> spaces.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-25DOI: 10.1007/s00605-023-01941-z
Egle Bettio, Enrico Jabara
Let G be a group which is generated by the set of its involutions, and assume that the set of integers which occur as orders of products of two involutions in G is ({1,2,3,4}). It is shown that (Gsimeq textrm{PSL}(2,7)) or (Gsimeq textrm{PSU}(3,3)) or G is a ({2,3})-group and (G/O_{2}(G) simeq S_{3}).
让 G 是一个由它的渐开线集合生成的群,并假设在 G 中作为两个渐开线乘积的阶出现的整数集合是 ( ( ({1,2,3,4}))。证明了(Gsimeq textrm{PSL}(2,7)) 或(Gsimeq textrm{PSU}(3,3)) 或 G 是一个({2,3})-群,并且(G/O_{2}(G) simeq S{3}/)。
{"title":"Groups in which all involutions are 3-transvections","authors":"Egle Bettio, Enrico Jabara","doi":"10.1007/s00605-023-01941-z","DOIUrl":"https://doi.org/10.1007/s00605-023-01941-z","url":null,"abstract":"<p>Let <i>G</i> be a group which is generated by the set of its involutions, and assume that the set of integers which occur as orders of products of two involutions in <i>G</i> is <span>({1,2,3,4})</span>. It is shown that <span>(Gsimeq textrm{PSL}(2,7))</span> or <span>(Gsimeq textrm{PSU}(3,3))</span> or <i>G</i> is a <span>({2,3})</span>-group and <span>(G/O_{2}(G) simeq S_{3})</span>.\u0000</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-24DOI: 10.1007/s00605-023-01932-0
Adam Osękowski
Let ({mathcal {M}}_{mathcal {D}}) be the dyadic maximal operator on ({mathbb {R}}^n). The paper contains the identification of the best constant in the two-weight estimate
under the assumption that the pair ((sigma ,w)) of weights satisfies an appropriate bump condition. The result is shown to be true in the larger context of abstract probability spaces equipped with a tree-like structure.
{"title":"A sharp two-weight estimate for the maximal operator under a bump condition","authors":"Adam Osękowski","doi":"10.1007/s00605-023-01932-0","DOIUrl":"https://doi.org/10.1007/s00605-023-01932-0","url":null,"abstract":"<p>Let <span>({mathcal {M}}_{mathcal {D}})</span> be the dyadic maximal operator on <span>({mathbb {R}}^n)</span>. The paper contains the identification of the best constant in the two-weight estimate </p><span>$$begin{aligned} Vert {mathcal {M}}_{mathcal {D}}fVert _{L^p(w)}le C_{p,sigma ,w}Vert fVert _{L^p(sigma ^{1-p})} end{aligned}$$</span><p>under the assumption that the pair <span>((sigma ,w))</span> of weights satisfies an appropriate bump condition. The result is shown to be true in the larger context of abstract probability spaces equipped with a tree-like structure.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-24DOI: 10.1007/s00605-023-01933-z
Eleftherios Tachtsis
We provide answers to open questions from Banerjee and Gopaulsingh (Bull Pol Acad Sci Math 71: 1–21, 2023) about the relationship between the Erdős–Dushnik–Miller theorem ((textsf{EDM})) and certain weaker forms of the Axiom of Choice ((textsf{AC})), and we properly strengthen some results from Banerjee and Gopaulsingh (2023). We also settle a part of an open question of Lajos Soukup (stated in Banerjee and Gopaulsingh (2023) [Question 6.1]) about the relationship between the following two order-theoretic principles, which [as shown in Banerjee and Gopaulsingh (2023)] are weaker than (textsf{EDM}): (a) “Every partially ordered set such that all of its antichains are finite and all of its chains are countable is countable” (this is known as Kurepa’s theorem), and (b) “Every partially ordered set such that all of its antichains are countable and all of its chains are finite is countable”. In particular, we prove that (b) does not imply (a) in (textsf{ZF}) (i.e., Zermelo–Fraenkel set theory without (textsf{AC})). Moreover, with respect to (b), we answer an open question from Banerjee and Gopaulsingh (2023) about its relationship with the following weak choice form: “Every set is either well orderable or has an amorphous subset”; in particular, we show that (b) follows from, but does not imply, the latter weak choice principle in (textsf{ZFA}) (i.e., Zermelo–Fraenkel set theory with atoms).
{"title":"On the deductive strength of the Erdős–Dushnik–Miller theorem and two order-theoretic principles","authors":"Eleftherios Tachtsis","doi":"10.1007/s00605-023-01933-z","DOIUrl":"https://doi.org/10.1007/s00605-023-01933-z","url":null,"abstract":"<p>We provide answers to open questions from Banerjee and Gopaulsingh (Bull Pol Acad Sci Math 71: 1–21, 2023) about the relationship between the Erdős–Dushnik–Miller theorem (<span>(textsf{EDM})</span>) and certain weaker forms of the Axiom of Choice (<span>(textsf{AC})</span>), and we properly strengthen some results from Banerjee and Gopaulsingh (2023). We also settle a part of an open question of Lajos Soukup (stated in Banerjee and Gopaulsingh (2023) [Question 6.1]) about the relationship between the following two order-theoretic principles, which [as shown in Banerjee and Gopaulsingh (2023)] are weaker than <span>(textsf{EDM})</span>: (a) “Every partially ordered set such that all of its antichains are finite and all of its chains are countable is countable” (this is known as Kurepa’s theorem), and (b) “Every partially ordered set such that all of its antichains are countable and all of its chains are finite is countable”. In particular, we prove that (b) does not imply (a) in <span>(textsf{ZF})</span> (i.e., Zermelo–Fraenkel set theory without <span>(textsf{AC})</span>). Moreover, with respect to (b), we answer an open question from Banerjee and Gopaulsingh (2023) about its relationship with the following weak choice form: “Every set is either well orderable or has an amorphous subset”; in particular, we show that (b) follows from, but does not imply, the latter weak choice principle in <span>(textsf{ZFA})</span> (i.e., Zermelo–Fraenkel set theory with atoms).</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-24DOI: 10.1007/s00605-023-01929-9
Abstract
Main results of the paper are: the Lebesgue Density Theorem does not hold for the uniform density points and the uniform density topology is completely regular.
摘要 本文的主要结果是:对于均匀密度点,Lebesgue 密度定理不成立,均匀密度拓扑完全正则。
{"title":"Uniform density topology","authors":"","doi":"10.1007/s00605-023-01929-9","DOIUrl":"https://doi.org/10.1007/s00605-023-01929-9","url":null,"abstract":"<h3>Abstract</h3> <p>Main results of the paper are: the Lebesgue Density Theorem does not hold for the uniform density points and the uniform density topology is completely regular. </p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-23DOI: 10.1007/s00605-023-01930-2
Abstract
We study the best coapproximation problem in Banach spaces, by using Birkhoff–James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets–Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.
{"title":"On some special subspaces of a Banach space, from the perspective of best coapproximation","authors":"","doi":"10.1007/s00605-023-01930-2","DOIUrl":"https://doi.org/10.1007/s00605-023-01930-2","url":null,"abstract":"<h3>Abstract</h3> <p>We study the best coapproximation problem in Banach spaces, by using Birkhoff–James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets–Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-22DOI: 10.1007/s00605-023-01928-w
Abstract
We show that the power-law decay exponents in von Neumann’s Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value 1. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann’s Ergodic Theorem depend on sequences of time going to infinity.
{"title":"On spectral measures and convergence rates in von Neumann’s Ergodic theorem","authors":"","doi":"10.1007/s00605-023-01928-w","DOIUrl":"https://doi.org/10.1007/s00605-023-01928-w","url":null,"abstract":"<h3>Abstract</h3> <p>We show that the power-law decay exponents in von Neumann’s Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value 1. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann’s Ergodic Theorem depend on sequences of time going to infinity.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-22DOI: 10.1007/s00605-023-01931-1
Jinlu Li, Yanghai Yu, Weipeng Zhu
In this short note, we prove that given initial data (u_0in H^s(mathbb {R})) with (s>frac{3}{2}) and for some (T>0), the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in (L^infty (0,T;H^s(mathbb {R}))) to the inviscid Burgers equation as the filter parameter (alpha ) tends to zero. This is a complement of our recent result on the zero-filter limit.
{"title":"Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit","authors":"Jinlu Li, Yanghai Yu, Weipeng Zhu","doi":"10.1007/s00605-023-01931-1","DOIUrl":"https://doi.org/10.1007/s00605-023-01931-1","url":null,"abstract":"<p>In this short note, we prove that given initial data <span>(u_0in H^s(mathbb {R}))</span> with <span>(s>frac{3}{2})</span> and for some <span>(T>0)</span>, the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in <span>(L^infty (0,T;H^s(mathbb {R})))</span> to the inviscid Burgers equation as the filter parameter <span>(alpha )</span> tends to zero. This is a complement of our recent result on the zero-filter limit.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01Epub Date: 2023-03-29DOI: 10.1007/s00605-023-01849-8
Diksha Tiwari, Akbarali Mukhammadiev, Paolo Giordano
This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.
本文是本刊论文 Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers 的自然延续。我们通过分析收敛半径的概念和证明超幂级数的代数运算、组成和倒数等经典结果来研究一变超幂级数。然后,我们定义并研究一变广义实解析函数,考虑它们的求导、积分、同一性定理的适当表述以及导数的局部均匀上界的特征。与哥伦布实解析函数理论中经典使用的序列相反,我们可以在非无限收敛集中恢复几个经典例子。广义实解析函数的概念相对于经典实解析函数和科伦坡理论都不那么僵化,例如,它包括具有平点的经典非解析光滑函数和几种分布,如 Dirac delta。另一方面,每个科伦坡实解析函数也是广义实解析函数。
{"title":"Hyper-power series and generalized real analytic functions.","authors":"Diksha Tiwari, Akbarali Mukhammadiev, Paolo Giordano","doi":"10.1007/s00605-023-01849-8","DOIUrl":"10.1007/s00605-023-01849-8","url":null,"abstract":"<p><p>This article is a natural continuation of the paper Tiwari, D., Giordano, P., <i>Hyperseries in the non-Archimedean ring of Colombeau generalized numbers</i> in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11427534/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88654685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}