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On lattice extensions 关于晶格扩展
Pub Date : 2024-01-27 DOI: 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。
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引用次数: 0
Wave-breaking and persistence properties in weighted $$L^p$$ spaces for a Camassa–Holm type equation with quadratic and cubic nonlinearities 具有二次和三次非线性的卡马萨-霍姆型方程在加权$L^p$$$空间中的破波和持久特性
Pub Date : 2024-01-25 DOI: 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.

我们考虑了具有二次方和三次方非线性的卡马萨-霍姆型方程的考奇问题。我们在初始数据上建立了一个新的充分条件,该条件导致了该方程的破波。此外,我们还得到了该方程在加权(L^p)空间中的解的持久性结果。
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引用次数: 0
Groups in which all involutions are 3-transvections 所有渐开线都是 3 变换的群
Pub Date : 2024-01-25 DOI: 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}/)。
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引用次数: 0
A sharp two-weight estimate for the maximal operator under a bump condition 凹凸条件下最大算子的尖锐两重估计值
Pub Date : 2024-01-24 DOI: 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

$$begin{aligned} Vert {mathcal {M}}_{mathcal {D}}fVert _{L^p(w)}le C_{p,sigma ,w}Vert fVert _{L^p(sigma ^{1-p})} end{aligned}$$

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.

让 ({mathcal {M}}_{mathcal {D}}) 是 ({mathbb {R}}^n) 上的二元最大算子。本文包含对两重估计 $$begin{aligned} 中最佳常数的识别。Vert {mathcal {M}}_{mathcal {D}}fVert _{L^p(w)}le C_{p,sigma ,w}Vert fVert _{L^p(sigma ^{1-p})}end{aligned}$$假设权重对((sigma ,w))满足适当的碰撞条件。结果表明,在具有树状结构的抽象概率空间的更大范围内,该结果是正确的。
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引用次数: 0
On the deductive strength of the Erdős–Dushnik–Miller theorem and two order-theoretic principles 关于厄尔多斯-杜什尼克-米勒定理的演绎强度和两个秩序理论原则
Pub Date : 2024-01-24 DOI: 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).

我们回答了 Banerjee 和 Gopaulsingh (Bull Pol Acad Sci Math 71: 1-21, 2023) 提出的关于厄尔多斯-杜什尼克-米勒定理 ((textsf{EDM})) 和选择公理 ((textsf{AC})) 的某些较弱形式之间的关系的公开问题,并适当加强了 Banerjee 和 Gopaulsingh (2023) 的一些结果。我们还解决了拉约什-苏库普(Lajos Soukup)提出的一个未决问题的一部分(在班纳吉和戈帕辛格(2023)[问题6.1]中提出),即下面两个有序理论原则之间的关系,这两个原则[如班纳吉和戈帕辛格(2023)所示]比(textsf{EDM})弱:(a) "每一个部分有序集合,如果它的所有反链都是有限的,而且它的所有链都是可数的,那么它就是可数的"(这被称为库雷帕定理),以及 (b) "每一个部分有序集合,如果它的所有反链都是可数的,而且它的所有链都是有限的,那么它就是可数的"。特别是,我们证明了(b)在(textsf{ZF})中并不意味着(a)(即没有(textsf{AC})的Zermelo-Fraenkel集合论)。此外,关于(b),我们回答了班纳吉和戈珀辛格(2023)提出的一个开放问题,即它与下面弱选择形式的关系:"每个集合要么是有序的,要么有一个无定形子集";特别是,我们证明了(b)是从(textsf{ZFA})(即有原子的泽梅洛-弗伦克尔集合论)中的后一个弱选择原则得出的,但并不意味着后一个弱选择原则。
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引用次数: 0
Uniform density topology 均匀密度拓扑
Pub Date : 2024-01-24 DOI: 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。
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引用次数: 0
On some special subspaces of a Banach space, from the perspective of best coapproximation 从最佳协同逼近的角度看巴拿赫空间的某些特殊子空间
Pub Date : 2024-01-23 DOI: 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.

摘要 我们利用伯克霍夫-詹姆斯正交技术研究巴拿赫空间中的最佳逼近问题。我们引入了两种特殊类型的子空间,分别称为反逼近子空间和强反逼近子空间。我们得到了反向巴拿赫空间中强反oproximinal子空间的必要条件,其对偶空间满足卡德茨-克利性质(Kadets-Klee Property)。另一方面,我们为一般巴拿赫空间中的强反oproximinal子空间提供了一个充分条件。我们还描述了光滑巴拿赫空间的反oproximinal子空间的特征。此外,我们还研究了有限维多面体巴拿赫空间中的这些特殊子空间,并发现了与之相关的一些有趣的几何结构。
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引用次数: 0
On spectral measures and convergence rates in von Neumann’s Ergodic theorem 论冯-诺依曼遍历定理中的谱量和收敛率
Pub Date : 2024-01-22 DOI: 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.

摘要 我们证明了 von Neumann Ergodic Theorem(针对离散系统)中的幂律衰减指数是光谱量在光谱值为 1 时的点式缩放指数。在这项工作中,我们还证明了在弱收敛的假设下,在没有谱差距的情况下,von Neumann Ergodic Theorem 中的时间平均值的收敛率取决于无穷大的时间序列。
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引用次数: 0
Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit 卡马萨-霍尔姆方程零滤波极限解的非均匀收敛性
Pub Date : 2024-01-22 DOI: 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.

在这篇短文中,我们证明了给定初始数据(u_0in H^s(mathbb {R}))与(s>frac{3}{2})并且对于某个(T>;0)时,卡马萨-霍尔姆方程的解并不会随着滤波参数(α )趋于零而均匀地收敛于(L^infty (0,T;H^s(mathbb {R}))中的初始数据。这是对我们最近关于零滤波极限结果的补充。
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引用次数: 0
Hyper-power series and generalized real analytic functions. 超幂级数和广义实解析函数
Pub Date : 2024-01-01 Epub Date: 2023-03-29 DOI: 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。
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Monatshefte für Mathematik
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