Expositiones Mathematicae最新文献

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Rings of tautological forms on moduli spaces of curves 曲线模空间上的同义形式环

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2023.02.008

Robin de Jong, Stefan van der Lugt

引用次数: 0

Krull-Remak-Schmidt decompositions in Hom-finite additive categories 有限加性范畴中的Krull-Remak-Schmidt分解

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.12.003

Amit Shah

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A $Hom$ -finite category is an additive category $A$ for which there is a commutative unital ring $k$ , such that each $Hom$ -set in $A$ is a finite length $k$ -module. The aim of this note is to provide a proof that a $Hom$ -finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.

其中每个对象都有一个Krull-Remak-Schmidt分解的加性范畴，即由具有局部自同态环的对象组成的有限直接和分解，称为Krull-Schmidt范畴。一个荷有限范畴是一个加性范畴A，它存在一个可交换的单位环k，使得A中的每个荷有限集是一个有限长度的k模。本文的目的是证明一个有限范畴是Krull-Schmidt，当且仅当它有分裂幂等，当且仅当每个不可分解对象有一个局部自同态环。

引用次数: 4

A theory of composites perspective on matrix valued Stieltjes functions 矩阵值Stieltjes函数的复合透视理论

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.12.005

Graeme W. Milton , Mihai Putinar

A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.

在复合材料研究中出现的一系列物理驱动运算，用矩阵值的有理Stieltjes函数的初等连分式变换来解释。

引用次数: 0

Splitting fields of Xn− Xn−<mm的拆分字段

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2023.02.007

Chandrashekhar B. Khare, Alfio Fabio La Rosa, Gabor Wiese

引用次数: 0

Noncommutative Ck functions and Fréchet derivatives of operator functions 非交换Ck函数与算子函数的Fréchet导数

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.12.004

Evangelos A. Nikitopoulos

<div>Fix a unital <math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math>-algebra <math><mi>A</mi></math>, and write <math><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub></math> for the set of self-adjoint elements of <math><mi>A</mi></math>. Also, if <math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>ℂ</mi></mrow></math> is a continuous function, then write <math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>→</mo><mi>A</mi></mrow></math> for the operator function <math><mrow><mi>a</mi><mo>↦</mo><mi>f</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math> defined via functional calculus. In this paper, we introduce and study a space <math><mrow><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math> of <math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math> functions <math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>ℂ</mi></mrow></math> such that, no matter the choice of <math><mi>A</mi></math>, the operator function <math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>→</mo><mi>A</mi></mrow></math> is <math><mi>k</mi></math>-times continuously Fréchet differentiable. In other words, if <math><mrow><mi>f</mi><mo>∈</mo><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math>, then <math><mi>f</mi></math> “lifts” to a <math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math> map <math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>→</mo><mi>A</mi></mrow></math>, for any (possibly noncommutative) unital <math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math>-algebra <math><mi>A</mi></math>. For this reason, we call <math><mrow><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math> the space of noncommutative <math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math> functions. Our proof that <math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>;</mo><mi>A</mi><mo>)</mo></mrow></mrow></math>, which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estim

固定一个单位C*-代数a，并为a的自伴随元素集写Asa。此外，如果f:R→ℂ 是连续函数，则写fA:Asa→A用于操作员功能A↦f（a）通过函数演算定义。本文介绍并研究了Ck函数f:R的一个空间NCk（R）→ℂ 这样，无论选择A，运算符函数fA:Asa→A是连续的k次Fréchet可微的。换句话说，如果f∈NCk（R），则f“提升”到Ck映射fA:Asa→A、对于任何（可能是非对易的）单位C*-代数A。因此，我们称NCk（R）为非对易Ck函数的空间。我们的证明fA∈Ck（Asa；A），只需要知道多项式的Fréchet导数和“多算子积分”（MOI）的算子范数估计，比标准方法更基本；然而，NCk（R）包含已知可比较结果的所有函数。具体地，我们证明了NCk（R）包含齐次Besov空间Ḃ1k，∞（R）和Hölder空间Clock（R）。然而，我们强调，本文中的结果是第一个被证明适用于任意单位C*-代数的结果，并且对这种一般设置的扩展利用了作者最近对MOI定义中某些“可分性问题”的解决方案。最后，我们通过展示具体的例子证明了Wk（R）loc⊊NCk（R。

引用次数: 0

Rational points on X0+( X0+上的有理点（</mml:

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2023.02.009

V. Arul, J. Müller

引用次数: 0

Approximations of the Riley slice Riley切片的近似

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.12.002

Alex Elzenaar , Gaven Martin , Jeroen Schillewaert

Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify ‘half-space’ neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.

采用L. Keen和C. Series在研究由两条抛物线产生的肖特基群的Riley切片时所使用的思想，我们明确地确定了完全位于Riley切片中的褶皱射线的“半空间”邻域。这给出了一个可证明的方法来确定一个点是否在Riley切片中。讨论了决定有理褶线的Farey多项式族及其决定Riley切片的根集;这导致了对切片的动态系统解释。将这些方法应用于由两个椭圆元生成的肖特基群的情况，使程序能够识别所有有限多个算术广义三角形群及其同类群。

引用次数: 3

Thresholds for the monochromatic clique transversal game 单色集团横向博弈的阈值

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.11.001

Csilla Bujtás , Pakanun Dokyeesun , Sandi Klavžar

We study a recently introduced two-person combinatorial game, the $(a, b)$ -monochromatic clique transversal game which is played by Alice and Bob on a graph $G$ . As we observe, this game is equivalent to the $(b, a)$ -biased Maker–Breaker game played on the clique-hypergraph of $G$ . Our main results concern the threshold bias $a_{1} (G)$ that is the smallest integer $a$ such that Alice can win in the $(a, 1)$ -monochromatic clique transversal game on $G$ if she is the first to play. Among other results, we determine the possible values of $a_{1} (G)$ for the disjoint union of graphs, prove a formula for $a_{1} (G)$ if $G$ is triangle-free, and obtain the exact values of $a_{1} (C_{n} □ C_{m})$ , $a_{1} (C_{n} □ P_{m})$ , and $a_{1} (P_{n} □ P_{m})$ for all possible pairs $(n, m)$ .

最近介绍二人组合游戏,我们研究(a, b)单色集团横向游戏由爱丽丝和鲍勃在一个图G .我们观察,这个游戏相当于(b, a)偏见Maker-Breaker游戏的clique-hypergraph G .我们的主要结果担心阈值偏差a1 (G)是最小的整数,爱丽丝可以赢得(a, 1)单色集团横向游戏G如果她是第一次玩。在其他结果中，我们确定了图的不相交并的a1(G)的可能值，证明了如果G是无三角形的a1(G)的一个公式，并获得了所有可能对(n,m)的a1(Cn□Cm)， a1(Cn□Pm)和a1(Pn□Pm)的精确值。

引用次数: 1

The product of lattice covolume and discrete series formal dimension: p-adic GL(2) 格协体积与离散级数形式维数的乘积:p进GL(2)

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.09.001

L.C. Ruth

Let $F$ be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in $P G L (2, F)$ and the formal dimension of a discrete series representation of $G L (2, F)$ . The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

设F为特征为0的非阿基米德局部域和不能被2整除的阶剩余域。我们展示了如何计算PGL(2,F)中无扭转晶格的协体积与GL(2,F)的离散级数表示的形式维数的乘积。协体积来自Ihara的一个定理，形式维数包含在Corwin、Moy和Sally的结果中。通过一个可以追溯到Atiyah的定理，以及一个自由群的第二个上同调群的平凡性，得到的乘积是一个离散级数表示的冯·诺依曼维，被认为是一个自由群因子的表示。

引用次数: 0

An analogue of Furstenberg–Sárközy’s theorem and an alternative solution to Waring’s problem over finite fields Furstenberg-Sárközy定理的一个类比和有限域上韦林问题的一个替代解

IF 0.7 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae

Pub Date : 2023-03-01 DOI: 10.1016/j.exmath.2022.10.003

Yeşi̇m Demi̇roğlu Karabulut

In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field $F_{q}$ can be written as a sum of $m$ many $k th$ powers as long as $q > k^{\frac{2 m}{m - 1}}$ ; and we also compute the smallest positive integers $m$ such that every element of $F_{q}$ can be written as a sum of $m$ many $k th$ powers for all $q$ too small to be covered by the above mentioned results when $2 ⩽ k ⩽ 37$ .

In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset $E$ of a finite field $F_{q}$ for which $| E | > \frac{q k}{\sqrt{q - 1}}$ must contain at least two distinct elements whose difference is a $k th$ power.

本文利用Cayley有向图给出了有限域上Waring问题的一些新的自包含证明，证明了有限域上的任意元素Fq可以写成m个k次幂的和，只要q>k2mm−1;并且我们还计算了最小的正整数m，使得Fq的每个元素都可以写成m个k次幂的和，当2≤k≤37时，所有的q都太小而不能被上述结果覆盖。在发展上述证明的过程中，我们得到了另一个结果(提供Furstenberg-Sárközy定理的有限域模拟)，表明|E|>qkq−1的有限域Fq的任何子集E必须包含至少两个不同的元素，其差值为k次幂。

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引用次数: 0

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