Linear Algebra and its Applications最新文献

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Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra 有限维李-山古提代数的普适共作用霍普夫代数

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-10-03 DOI: 10.1016/j.laa.2024.09.017

Saikat Goswami , Satyendra Kumar Mishra , Goutam Mukherjee

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

M.E. Sweedler 首次构造了一个代数的普遍霍普夫代数。众所周知，现有概念的对偶概念在霍普夫代数理论中起着主导作用。尤.马宁（Yu. I. Manin）和丹巴拉（D. Tambara）在不同的著作中介绍了斯韦德勒构造的对偶概念。在本文中，我们为有限维李-山古提代数构造了一个普代数。我们证明了这个普代数具有双代数结构，从而为有限维李-山口组代数建立了一个普协合霍普夫代数。此外，我们还开发了我们结果的表示理论版本。作为应用，我们描述了有限维李-山口组代数的自变群特征，并对其所有无性群分级进行了分类。

引用次数: 0

Semisimple elements and the little Weyl group of real semisimple Zm-graded Lie algebras 实半简单 Zm 级列的半简单元素和小韦尔群

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-30 DOI: 10.1016/j.laa.2024.09.015

Willem de Graaf , Hông Vân Lê

We consider the semisimple orbits of a Vinberg θ-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one corresponding to the restricted roots of a Cartan subspace, the other corresponding to the complex reflections in the (little) Weyl group. The semisimple orbits have representatives in a finite number of sets that correspond to reflection subgroups of the (little) Weyl group. One of the consequences of our technical result is that the elements in a fixed such set all have the same stabilizer in the acting group. Secondly we study what happens when the base field is the real numbers. We look at Cartan subspaces and show that the real Cartan subspaces can be classified by the first Galois cohomology set of the normalizer of a fixed real Cartan subspace. In the real case the orbits can be classified using Galois cohomology. However, in order for that to work we need to know which orbits have a real representative. We show a theorem that characterizes the orbits of homogeneous semisimple elements that do have such a real representative. This closely follows and generalizes a theorem from [6].

我们考虑文伯格 θ 表示的半简单轨道。首先，我们以复数为基域。通过逐例分析，我们展示了一个技术结果，说明两组超平面是相等的，一组对应于 Cartan 子空间的受限根，另一组对应于（小）Weyl 群中的复反射。半简单轨道在对应于（小）韦尔群反射子群的有限数量集合中具有代表。我们的技术结果之一是，固定的此类集合中的元素在作用群中都有相同的稳定子。其次，我们研究了当基域为实数时会发生什么。我们研究了笛卡尔子空间，并证明实笛卡尔子空间可以通过固定实笛卡尔子空间的归一化的第一个伽罗瓦同调集来分类。在实情形中，轨道可以用伽罗瓦同调来分类。然而，要做到这一点，我们需要知道哪些轨道有实数代表。我们展示了一个定理，它描述了具有实代表的同质半简单元素的轨道。这紧跟并推广了 [6] 中的一个定理。

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引用次数: 0

Extreme values of the Fiedler vector on trees 费德勒向量在树木上的极值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-27 DOI: 10.1016/j.laa.2024.09.014

Roy R. Lederman , Stefan Steinerberger

Let G be a tree on n vertices and let

L = D - A

denote the Laplacian matrix on G. The second-smallest eigenvalue

λ_{2} (G) > 0

, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.

让 G 是 n 个顶点上的树，让 L=D-A 表示 G 上的拉普拉斯矩阵。第二最小特征值 λ2(G)>0（也称为代数连通性）以及相关特征向量一直备受关注。我们研究了相关特征向量的最大值和最小值何时假定位于 G 中最长路径端点的问题。我们的结果也适用于更一般的图，这些图的 "全局行为 "类似于树，但可能表现出更复杂的局部结构。关键的新要素是图形特征向量的重现公式。

引用次数: 0

Estimations of Karcher mean by Hadamard product 用 Hadamard 乘积估算 Karcher 平均值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-27 DOI: 10.1016/j.laa.2024.09.013

Masatoshi Fujii , Yuki Seo , Masaru Tominaga

In this paper, we estimate the difference between the Hadamard product and the Karcher mean of n positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of

n = 2

. Moreover, we give ratio inequalities of the operator power means by the Hadamard product.

在本文中，我们用 Specht 比值和 Kantorovich 常量来估计希尔伯特空间上 n 个正可逆算子的 Hadamard 乘积和 Karcher 平均值之间的差异。同时，我们改进了 n=2 情况下的不等式。此外，我们还给出了哈达玛积的算子幂级数的比率不等式。

引用次数: 0

Isomorphisms between lattices of hyperinvariant subspaces 超不变子空间网格之间的同构关系

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-23 DOI: 10.1016/j.laa.2024.09.012

David Mingueza , M. Eulàlia Montoro , Alicia Roca

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.

给定两个零势内同态，我们就能确定它们的超不变子空间网格何时同构。当内变态具有乔丹-切瓦利分解时，超不变子空间网格的研究可以简化为零势情况；例如，如果底层场是复数场，就会出现这种情况。

引用次数: 0

Locally supported, quasi-interpolatory bases for the approximation of functions on graphs 用于逼近图上函数的局部支持准插值基

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-17 DOI: 10.1016/j.laa.2024.09.011

E. Fuselier , J.P. Ward

Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present

ℓ_{\infty}

error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies an assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate

{‖ A ‖}_{\infty} \leq \sqrt{n} {‖ A ‖}_{2}

基于图的近似方法在许多领域越来越受到关注，包括交通、生物和化学网络、金融模型、图像处理、网络流等。在这些应用中，近似空间的基础往往无法通过分析获得，而必须通过计算获得。我们提出了图上拉格朗日基的扰动，其中的拉格朗日函数来自一类类似于经典样条函数的函数。我们考虑的基函数具有局部支持，每个基函数都是通过求解与图上微分算子相关的小能量最小化问题获得的。在底层图满足顶点连接假设、函数未知的情况下，我们提出了局部基函数和相应插值拉格朗日基函数之间的 ℓ∞ 误差估计，并在数值实验中进一步检验了理论边界。我们的分析包括正定矩阵的混合正不等式，它比一般估计值‖A‖∞≤n‖A‖2 更严格。

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引用次数: 0

Degradable strong entanglement breaking maps 可降解的强纠缠断裂图

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-17 DOI: 10.1016/j.laa.2024.09.006

Repana Devendra , Gunjan Sapra , K. Sumesh

In this paper, we provide a structure theorem and various characterizations of degradable strong entanglement breaking maps on separable Hilbert spaces. In the finite-dimensional case, we prove that unital degradable entanglement breaking maps are precisely the $C^{⁎}$ -extreme points of the convex set of unital entanglement breaking maps on matrix algebras. Consequently, we get a structure for unital degradable positive partial transpose (PPT) maps.

本文提供了可分离希尔伯特空间上可降解强纠缠断裂映射的结构定理和各种特征。在有限维情况下，我们证明了单元可降解纠缠断裂映射正是矩阵代数上单元纠缠断裂映射凸集的 C⁎-极值点。因此，我们得到了单元可降解正偏转置（PPT）映射的结构。

引用次数: 0

On the zero forcing number of the complement of graphs with forbidden subgraphs 关于有禁止子图的图的补集的零强制数

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-12 DOI: 10.1016/j.laa.2024.09.009

Emelie Curl , Shaun Fallat , Ryan Moruzzi Jr , Carolyn Reinhart , Derek Young

Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on n vertices is either $n - 3$ or $n - 1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.

零强迫和最大无效性是两个重要的图参数，为了帮助解决逆特征值问题，我们对它们进行了大量研究。观察到 n 个顶点上的树的补集的零强制数是 n-3 或 n-1（在一种特殊情况下），这在一定程度上激励了我们，我们考虑了在某些条件下更一般的图的补集的零强制数，特别是那些不包含完整双方子图的图。我们的研究还远远超出了树的范围，完全研究了单环图和仙人掌图补集的所有可能的零强制数。最后，我们得出了所考虑的几组图补集的最大无效数和零强制数之间的相等关系。

引用次数: 0

Effects on the algebraic connectivity of weighted graphs under edge rotations 边旋转对加权图代数连通性的影响

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-11 DOI: 10.1016/j.laa.2024.09.010

Xinzhuang Chen , Shenggui Zhang , Shanshan Gao , Xiaodi Song

For a weighted graph G, the rotation of an edge $u v_{1}$ from $v_{1}$ to a vertex $v_{2}$ is defined as follows: delete the edge $u v_{1}$ , set $w (u v_{2})$ as $w (u v_{1}) + w (u v_{2})$ if $u v_{2}$ is an edge of G; otherwise, add a new edge $u v_{2}$ and set $w (u v_{2}) = w (u v_{1})$ , where $w (u v_{1})$ and $w (u v_{2})$ are the weights of the edges $u v_{1}$ and $u v_{2}$ , respectively. In this paper, effects on the algebraic connectivity of weighted graphs under edge rotations are studied. For a weighted graph, a sufficient condition for an edge rotation to reduce its algebraic connectivity and a necessary condition for an edge rotation to improve its algebraic connectivity are proposed based on Fiedler vectors of the graph. As applications, we show that, by using a series of edge rotations, a pair of pendent paths (a pendent tree) of a weighted graph can be transformed into one pendent path (pendent edges attached at a common vertex) of the graph with the algebraic connectivity decreasing (increasing) monotonically. These results extend previous findings of reducing the algebraic connectivity of unweighted graphs by using edge rotations.

对于加权图 G，边 uv1 从 v1 到顶点 v2 的旋转定义如下：删除边 uv1，如果 uv2 是 G 的一条边，则设置 w(uv2) 为 w(uv1)+w(uv2)；否则，添加一条新边 uv2，并设置 w(uv2)=w(uv1)，其中 w(uv1) 和 w(uv2) 分别是边 uv1 和 uv2 的权重。本文研究了边旋转对加权图代数连通性的影响。对于加权图，根据图的费德勒向量，提出了边旋转降低其代数连通性的充分条件和边旋转提高其代数连通性的必要条件。作为应用，我们证明了通过使用一系列边旋转，加权图的一对垂径（一棵垂树）可以转化为图的一条垂径（连接在共同顶点的垂边），其代数连通性单调递减（递增）。这些结果扩展了之前利用边旋转降低无权图代数连通性的研究成果。

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引用次数: 0

Eigenbasis for a weighted adjacency matrix associated with the projective geometry Bq(n) 与投影几何 Bq(n) 相关的加权邻接矩阵的特征基础

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-10 DOI: 10.1016/j.laa.2024.09.007

Murali K. Srinivasan

In a recent article Projective geometries, Q-polynomial structures, and quantum groups Terwilliger (arXiv:2407.14964) defined a certain weighted adjacency matrix, depending on a free (positive real) parameter, associated with the projective geometry, and showed (among many other results) that it is diagonalizable, with the eigenvalues and their multiplicities explicitly written down, and that it satisfies the Q-polynomial property (with respect to the zero subspace).

In this note we

•
Write down an explicit eigenbasis for this matrix.
•
Evaluate the adjacency matrix-eigenvector products, yielding a new proof for the eigenvalues and their multiplicities.
•
Evaluate the dual adjacency matrix-eigenvector products and directly show that the action of the dual adjacency matrix on the eigenspaces of the adjacency matrix is block-tridiagonal, yielding a new proof of the Q-polynomial property.

在最近的一篇文章《投影几何、Q-多项式结构和量子群》（Projective geometries, Q-polynomial structures, and quantum groups）中，Terwilliger（arXiv:2407.14964）定义了一个与投影几何相关的、取决于自由（正实数）参数的加权邻接矩阵，并证明（除其他许多结果外）它是可对角的，特征值及其乘数被明确写出，而且它满足 Q-多项式性质（关于零子空间）。在本注释中，我们--为这个矩阵写下了一个明确的特征基础。--评估了邻接矩阵-特征向量乘积，得出了特征值及其乘积的新证明。--评估了对偶邻接矩阵-特征向量乘积，并直接证明了对偶邻接矩阵对邻接矩阵特征空间的作用是块对角的，得出了 Q 多项式性质的新证明。

引用次数: 0

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Linear Algebra and its Applications

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