Expositiones Mathematicae最新文献

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Computation of Lyapunov exponents of matrix products 矩阵乘积的李雅普诺夫指数的计算

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-10-17 DOI: 10.1016/j.exmath.2025.125733

Ai Hua Fan , Evgeny Verbitskiy

For

m

given square matrices

A_{0}, A_{1}, \dots, A_{m - 1}

(

m \geq 2

), one of which is assumed to be of rank 1, and for a given sequence

(ω_{n})

{0, 1, \dots, m - 1}^{N}

, the following limit, if it exists,

L (ω) ≔ lim_{n \to \infty} \frac{1}{n} log ‖ A_{ω_{0}} A_{ω_{2}} \dots A_{ω_{n - 1}} ‖

defines the Lyapunov exponent of the sequence of matrices

{(A_{ω_{n}})}_{n \geq 0}

. It is proven that the Lyapunov exponent

L (ω)

has a closed-form expression under certain conditions. One special case arises when

A_{j}

’s are non-negative and

ω

is generic with respect to some shift-invariant measure; a second special case occurs when

A_{j}

’s (for

1 \leq j < m

) are invertible and

ω

is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of

B

-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.

对于m个给定的方阵A0，A1，…，Am−1 (m≥2)，其中一个设秩为1，对于{0,1，…，m−1}N中的一个给定序列（ω N），如果存在，则L（ω）→∞1nlog‖a ω 0a ω2⋯a ω N−1‖定义矩阵（a ω N） N≥0序列的李雅普诺夫指数。证明了李雅普诺夫指数L（ω）在一定条件下具有闭型表达式。一种特殊情况是Aj是非负的ω对于平移不变测度是泛型的；当Aj（对于1≤j<；m）是可逆的，并且ω是关于平移遍历测度的一个典型点时，出现第二种特殊情况。以无b整数的替换序列和特征序列为例。给出了加权Birkhoff平均多重分形谱计算的一个应用。

引用次数: 0

Galois groups in enumerative geometry and applications 枚举几何中的伽罗瓦群及其应用

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-10-09 DOI: 10.1016/j.exmath.2025.125731

Frank Sottile , Thomas Yahl

As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic study only occurring in the past 15 years. We discuss the current directions of this study, including open problems and conjectures.

正如Jordan在1870年观察到的那样，正如单变量多项式有伽罗瓦群一样，枚举几何问题也有伽罗瓦群。尽管有这样的谱系，伽罗瓦群在计数几何中的研究沉寂了一个世纪，直到最近15年才出现系统的研究。我们讨论了当前的研究方向，包括开放的问题和猜想。

引用次数: 0

Fractional wavelet phase retrieval of bandlimited functions 带限函数的分数小波相位检索

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-09-30 DOI: 10.1016/j.exmath.2025.125732

Bei Liu, Meng Ning, Rui Li, Liping Wu, Qingyue Zhang

In this paper, we demonstrate the real-valued bandlimited signals can be recovered (up to global sign) from the magnitude of their fractional wavelet transforms, also called fractional wavelet phase retrieval. We present a new uniqueness result for fractional wavelet phase retrieval. To be precise, we show that any real-valued bandlimited functions can be recovered uniquely (up to global sign) by fractional wavelets which have finitely many vanishing moments. Furthermore, we show that the real-valued bandlimited function can be uniquely recovered up to global sign from the absolute values of the discrete fractional wavelet transform for fixed

a > 1, b > 0

在本文中，我们证明了实值带限信号可以从分数阶小波变换的幅度中恢复（直到全局符号），也称为分数阶小波相位恢复。给出了分数阶小波相位检索的一个唯一性结果。确切地说，我们证明了任何实值带限函数都可以被具有有限多个消失矩的分数小波唯一地恢复（直到全局符号）。进一步，我们证明了对于固定的a>；1, >0，从离散分数阶小波变换的绝对值可以唯一地恢复到全局符号的实值带限函数。

引用次数: 0

Periodic source detection in discrete dynamical systems via space–time sampling 基于时空采样的离散动力系统周期源检测

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-09-24 DOI: 10.1016/j.exmath.2025.125730

Akram Aldroubi , Carlos Cabrelli , Ursula Molter

In this paper, we examine a discrete dynamical system defined by

x (n + 1) = A x (n) + w (n)

, where

x

takes values in a Hilbert space

H

and

w

is a periodic source with values in a fixed closed subspace

W

H

. Our goal is to identify conditions on some spatial sampling system

G = {g_{j}}_{j \in J}

H

that enable stable recovery of the unknown source term

w

from space–time samples

{〈 x (n), g_{j} 〉}_{n \geq 0, j \in J}

. We provide necessary and sufficient conditions on

G = {g_{j}}_{j \in J}

to ensure stable recovery of any

w \in W

. Additionally, we explicitly construct an operator

R

, dependent on

G

, such that

R {〈 x (n), g_{j} 〉}_{n, j} = w

本文研究了一个由x(n+1)=Ax(n)+w(n)定义的离散动力系统，其中x取Hilbert空间H中的值，w是一个周期源，其值在H的固定闭子空间w中。我们的目标是找出在某个空间采样系统G={gj}j∈j (H)上，能够从时空样本{< x(n),gj >}n≥0，j∈j中稳定恢复未知源项w的条件。我们给出了G={gj}j∈j的充分必要条件，以保证任意w∈w的稳定回收。此外，我们显式构造了一个依赖于G的算子R，使得R{< x(n),gj >}n,j=w。

引用次数: 0

Weaving information packets 编织信息包

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-09-18 DOI: 10.1016/j.exmath.2025.125720

Ole Christensen , Hong Oh Kim , Rae Young Kim

The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames

{f_{k}}_{k \in I}, {g_{k}}_{k \in I}

are woven if the “mixed system”

{f_{k}}_{k \in σ} \cup {g_{k}}_{k \in I ∖ σ}

is a frame for each index set

σ \subset I;

that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to

N

frames, for any integer

N > 2 .

Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in

L^{2} (R) .

Interestingly, we show that certain wavelet systems can be split into

N

woven information packets, for any integer

N \geq 2 .

We finally consider corresponding questions for Gabor system in

L^{2} (R),

and prove that for any fixed

N \in N

we can find a Gabor frame that can be split into

N

woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all

N \in N .

Hilbert空间框架编织的概念由Bemrose等人于2016年提出。如果“混合系统”{fk}k∈σ∪{gk}k∈I∈σ是每个指标集σ∧I的一个框架，则两个框架{fk}k∈I，{gk}k∈I是编织的；也就是说，使用两个编织帧处理一个信号，可以产生一定的稳定性，防止信息丢失。这个概念很容易扩展到N帧，对于任意整数N>；2。不幸的是，构建有用的编织框架并非易事，而且关于明确结构的文献很少。本文引入了所谓的信息包，作为特殊情况，信息包既包含帧，也包含融合帧。编织框架的概念立即推广到信息包，我们演示了如何基于特定的小波系统在L2(R)中构建实用的编织信息包。有趣的是，我们证明了某些小波系统可以被分割成N个编织信息包，对于任意整数N≥2。最后考虑L2(R)中Gabor系统的相应问题，并证明对于任意固定N∈N，我们都可以找到一个可以分割成N个编织信息包的Gabor框架；然而，与小波变换不同的是，Gabor系统的密度条件排除了找到对所有N∈N同时起作用的单一Gabor框架的可能性。

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引用次数: 0

Sums of two squares and the tau-function: Ramanujan’s trail 两个平方和和tau函数：拉马努金轨迹

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-09-06 DOI: 10.1016/j.exmath.2025.125721

Bruce C. Berndt , Pieter Moree

Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.

拉马努金，在他给哈代的著名的第一封信中，声称对可以写成两个平方和的整数的数量有一个非常精确的估计。远不为人所知的是，他还对某些素数的拉马努金τ函数的不可整除性提出了类似性质的进一步主张。在本文中，我们将提供更多的历史细节，并讨论相关的后期发展。这些照片表明，拉马努金像往常一样，是迷人荒野中的探险家，身后留下了一条迷人的小径。

引用次数: 0

Any function I can actually write down is measurable, right? 任何我能写出的函数都是可测量的，对吧？

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-08-11 DOI: 10.1016/j.exmath.2025.125718

James E. Hanson

<div><div>In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert’s tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial <math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>70</mn></mrow></msub><mo>)</mo></mrow></mrow></math> of degree 7 with integer coefficients such that it is independent of <math><mi>ZFC</mi></math> (and much stronger theories) whether the function <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>inf</mo></mrow><mrow><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></munder><munder><mrow><mo>sup</mo></mrow><mrow><mi>z</mi><mo>∈</mo><mi>R</mi></mrow></munder><munder><mrow><mo>inf</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></munder><munder><mrow><mo>sup</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>70</mn></mrow></msup></mrow></munder><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>,</mo><mover><mrow><mi>k</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math>is Lebesgue measurable. We also give similarly defined <math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math> with the property that the statement “<math><mrow><mi>x</mi><mo>↦</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math> is measurable for every <math><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow></math>” has large cardinal consistency strength (and in particular implies the consistency of <math><mi>ZFC</mi></math>) and <math><mrow><mi>h</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> such that <math><mrow><mi>h</mi><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>h</mi><mrow><mo>(</mo><mn>16</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> can consistently be the indicator functions of a Banach–Tarski paradoxical decomposition of the sphere.</div><div>Finally, we discuss some situations in which measurability of analogously defined functions can be concluded by inspection, which touches on model-theoretic o-minimality and the fact that sufficiently strong large cardinal hypotheses (such as Vopěnka’s principle and much weaker assumptions

在这篇针对一般数学读者的说说性论文中，我们讨论了如何将集合论内模论和有效描述集合论的某些经典定理与希尔伯特第十问题和普遍丢芬图方程的工作结合起来，以产生以下惊人的结果：有一个特定的7次多项式p（x,y,z,n,k1，…，k70）具有整数系数，使得它独立于ZFC（以及更强的理论）函数f(x)=infy∈Rsupz∈Rinfn∈Nsupk∈N70p（x,y,z,n,k）是否为勒贝格可测。我们也给出了类似的定义g(x,y)，并给出了“x∑g（x,r）对于每一个r∈r都是可测的”的性质，即命题“x∑g（x,r）对于每一个r∈r都是可测的”具有较大的基一致性强度（特别是暗示了ZFC的一致性）和h(m,x,y,z)，使得h(1,x,y,z)，…，h（16,x,y,z）可以一致地成为球的Banach-Tarski悖论分解的指示函数。最后，我们讨论了可以通过检验得出类似定义函数的可测性的一些情况，这涉及到模型论的o-极小性和足够强的大基数假设（如vopovonka原理和更弱的假设）意味着所有“合理可定义”的函数(包括上述f（x), g（x,y）和h(m,x,y,z)）是普遍可测的事实。

引用次数: 0

Some examples of affine isometries of Banach spaces arising from 1-D dynamics 由一维动力学引起的巴拿赫空间仿射等距的一些例子

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-07-31 DOI: 10.1016/j.exmath.2025.125716

Andrés Navas

We provide a large family of examples of affine isometries of the Banach spaces

C^{0} (S^{1})

L^{1} (S^{1})

and

L^{2} (S^{1} \times S^{1})

that are fixed-point-free despite being recurrent (in particular, they have zero drift). These come from natural cocycles on the group of circle diffeomorphisms, namely the logarithmic, affine and (a variation of the) Schwarzian derivative. Quite interestingly, they arise from diffeomorphisms that are generic in an appropriate context. We also show how to promote these examples in order to obtain families of commuting isometries satisfying the same properties.

我们提供了大量的Banach空间C0(S1), L1（S1）和L2（S1×S1）的仿射等距的例子，尽管它们是循环的（特别是，它们具有零漂移），但它们是不定点的。这些来自于圆微分同态群上的自然环，即对数、仿射和（Schwarzian导数的一种变化）。非常有趣的是，它们起源于在适当的上下文中具有普遍性的微分同构。我们还展示了如何推广这些例子，以获得满足相同性质的交换等距族。

引用次数: 0

Generalizing Hurwitz’s quaternionic proof of Lagrange’s and Jacobi’s four-square theorems 推广赫维茨对拉格朗日定理和雅可比定理的四元数证明

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-07-19 DOI: 10.1016/j.exmath.2025.125715

Matěj Doležálek

A proof of Lagrange’s and Jacobi’s four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm 1 acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of Götzky’s four-square theorem.

赫尔维茨对拉格朗日和雅可比四平方定理的一个证明，利用了四元数代数在有理数上的阶数。为了将这一技术推广到数域上的阶，我们确定了两个关键成分：一个具有良好分解理论的阶，以及所有轨道在范数1的元素群的乘法作用下与所研究的二次型对应的子阶相交的条件。我们利用最近关于四元数阶的类数的结果，找出满足轨道条件的所有子阶。随后，我们得到了相应二次型表示数的通用性和公式。我们也给出了Götzky四平方定理的四元数证明。

引用次数: 0

The law of large numbers for stochastic rumor models 随机谣言模型的大数定律

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2025-07-17 DOI: 10.1016/j.exmath.2025.125713

Elcio Lebensztayn, Lucas Sousa Santos

We investigate the generalization of the Maki–Thompson model for the spreading of a rumor through a finite population in which each spreader stops transmitting the rumor right after being involved in

k

unsuccessful telling interactions. We prove that the proportion of people unaware of the rumor at the end of the process converges in probability to a constant, as the population size goes to

\infty

. The proof relies on an application of the martingale stopping theorem and is based upon the case

k = 1

established by Sudbury (1985), but our approach for proving the convergence is simpler, reducing technicalities.

我们研究了Maki-Thompson模型在有限群体中谣言传播的泛化，其中每个传播者在参与k次不成功的传播互动后立即停止传播谣言。我们证明了在过程结束时不知道谣言的人的比例在概率上收敛于一个常数，随着人口规模趋于∞。证明依赖于鞅停止定理的应用，并基于萨德伯里（1985）建立的k=1的情况，但我们证明收敛性的方法更简单，减少了技术性。

引用次数: 0

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

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全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

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