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On an extension of Fuglede’s theory of pseudo-balayage and its applications 论福格莱德伪balayage理论的拓展及其应用

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2026-02-01 Epub Date: 2024-11-08 DOI: 10.1016/j.exmath.2024.125630

Natalia Zorii

For suitable function kernels

κ

on a locally compact space, we develop a theory of inner pseudo-balayage

{\hat{ω}}^{A}

of signed (Radon) measures

ω

of finite energy onto a quasiclosed set

A

{\hat{ω}}^{A}

being defined as the solution to the problem of minimizing the Gauss functional

\int κ (x, y) d (μ \otimes μ) (x, y) - 2 \int κ (x, y) d (μ \otimes ω) (x, y),

where

μ

ranges over all positive measures of finite energy concentrated on the set

A

. If

A

is Borel, the concept of inner pseudo-balayage

{\hat{ω}}^{A}

is shown to coincide with that of outer pseudo-balayage, introduced in Fuglede’s work (Fuglede, 2016), which was however only concerned with

ω ⩾ 0

, whereas the investigation of signed

ω

requires essentially different methods and approaches. The theory of pseudo-balayage thereby established enables us to improve substantially our recent results on the well-known inner Gauss variational problem (Zorii, 2024), by strengthening their formulations and/or by extending the area of their validity. This study covers many interesting kernels in classical and modern potential theory, which looks promising for further applications.

对于局部紧化空间上合适的函数核κ，我们建立了准闭集a上有限能量的有符号（Radon）测度ω的内伪度量ω的理论，ω ω被定义为最小化高斯泛函∫κ(x,y)d(μ⊗μ)(x,y)−2∫κ(x,y)d(μ⊗ω)（x,y）问题的解，其中μ的范围超过集中在集合a上的所有有限能量的正测度。内部伪balayage ω³A的概念被证明与Fuglede的工作（Fuglede, 2016）中引入的外部伪balayage的概念相吻合，然而，该工作仅与ω小于或等于0有关，而对带符号ω的调查则需要本质上不同的方法和途径。由此建立的伪平衡理论使我们能够通过加强其公式和/或扩展其有效性的范围，大大改进我们最近在著名的内高斯变分问题（Zorii, 2024）上的结果。本研究涵盖了经典和现代势理论中许多有趣的问题，具有广阔的应用前景。

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引用次数: 0

A complete Bernstein function related to the fractal dimension of Pascal’s pyramid modulo a prime 与帕斯卡金字塔模数素数分形维度相关的完整伯恩斯坦函数

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2026-02-01 Epub Date: 2024-08-02 DOI: 10.1016/j.exmath.2024.125601

Christian Berg

Let

f_{r} (x) = log (1 + r x) / log (1 + x)

for

x > 0

. We prove that

f_{r}

is a complete Bernstein function for

0 \leq r \leq 1

and a Stieltjes function for

1 \leq r

. This answers a conjecture of David Bradley that

f_{r}

is a Bernstein function when

0 \leq r \leq 1

设为 .我们证明，，是一个完整的伯恩斯坦函数，并且，是一个斯蒂尔杰斯函数。这回答了戴维-布拉德利（David Bradley）的一个猜想，即当 . 是伯恩斯坦函数时， .

引用次数: 0

Reprint of: Extremal length and duality 转载：极限长度和二元性

IF 0.9 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2026-02-01 Epub Date: 2025-12-01 DOI: 10.1016/j.exmath.2025.125746

Kai Rajala

Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In “Extremal length and functional completion”, Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings:

(1) Extremal length and uniformization of metric surfaces.

(2) Extremal length of families of surfaces and quasiconformal maps between

n

-dimensional spaces.

(3) Schramm’s transboundary extremal length and conformal maps between multiply connected plane domains.

经典极值长度（或共形模量）是一个涉及黎曼球上路径族的共形不变量。Fuglede在《极值长度与功能补全》中提出了一种抽象的极值长度理论，并得到了广泛的应用。聚焦对偶性质及其在拟共形分析中的应用，我们展示了该理论的灵活性，并介绍了在三种不同情况下的最新进展：(1)度量曲面的极值长度和均匀化。(2) n维空间间曲面族和拟共形映射的极值长度。(3) Schramm的跨界极值长度和多连通平面域间的保角映射。

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 : 2025-12-01 Epub 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

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 : 2025-12-01 Epub 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

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 : 2025-12-01 Epub 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

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 : 2025-12-01 Epub 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

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 : 2025-12-01 Epub 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

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 : 2025-12-01 Epub 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

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 : 2025-12-01 Epub 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。

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