交换代数

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[1] arXiv:2508.02709 (交叉列表自 math.RA) [中文pdf, pdf, html, 其他]

标题： 用于分析交换超复数代数的计算工具的发展 显示英文标题

标题： Advancing Computational Tools for Analyzing Commutative Hypercomplex Algebras

José Domingo Jiménez-López, Jesús Navarro-Moreno, Rosa María Fernández-Alcalá, Juan Carlos Ruiz Molina

主题： 环与代数 (math.RA) ; 交换代数 (math.AC) ; 谱理论 (math.SP) ; 统计理论 (math.ST)

交换超复数代数由于其与线性代数技术的兼容性和高效的计算实现，相较于传统的四元数具有显著优势，这对于广泛的应用至关重要。 本文探讨了一类新颖的交换超复数代数，称为(alpha,beta)-张量数，它们扩展了广义Segre四元数系统，从而也扩展了椭圆四元数。 本工作的主要贡献是开发了该代数系统中矩阵的理论和计算工具，包括求逆、平方根计算、带部分选主元的LU分解以及行列式计算。 此外，建立了(alpha,beta)-张量数的谱理论，涵盖特征值和特征向量分析、幂法、奇异值分解、秩-k近似以及伪逆。 还提出了经典最小二乘问题的解。 这些结果不仅增强了对超复数代数的基本理解，还为研究人员提供了以前研究中尚未广泛探索的新矩阵运算。 理论成果得到了实际例子的支持，包括图像重建和彩色人脸识别，这些例子展示了所提出技术的潜力。 显示英文摘要

Commutative hypercomplex algebras offer significant advantages over traditional quaternions due to their compatibility with linear algebra techniques and efficient computational implementation, which is crucial for broad applicability. This paper explores a novel family of commutative hypercomplex algebras, referred to as (alpha,beta)-tessarines, which extend the system of generalized Segre's quaternions and, consequently, elliptic quaternions. The main contribution of this work is the development of theoretical and computational tools for matrices within this algebraic system, including inversion, square root computation, LU factorization with partial pivoting, and determinant calculation. Additionally, a spectral theory for (alpha,beta)-tessarines is established, covering eigenvalue and eigenvector analysis, the power method, singular value decomposition, rank-k approximation, and the pseudoinverse. Solutions to the classical least squares problem are also presented. These results not only enhance the fundamental understanding of hypercomplex algebras but also provide researchers with novel matrix operations that have not been extensively explored in previous studies. The theoretical findings are supported by real-world examples, including image reconstruction and color face recognition, which demonstrate the potential of the proposed techniques.

替换提交 (展示 2 之 2 条目 )

[2] arXiv:2304.00527 (替换) [中文pdf, pdf, html, 其他]

标题： G-维数对于交换DG环上的DG模 显示英文标题

标题： G-dimensions for DG-modules over commutative DG-rings

Jiangsheng Hu, Xiaoyan Yang, Rongmin Zhu

评论： 20页，将发表于《爱丁堡数学学会会刊》

主题： 交换代数 (math.AC) ; K理论与同调 (math.KT) ; 环与代数 (math.RA)

我们定义并研究了在非正分次交换诺特DG环$A$上的DG模的G-维数概念。通过应用Minamoto引入的DG投影分解的DG版本[以色列数学杂志 245 (2021) 409-454]，给出了DG模的G-维数有限性的某些条件。此外，证明了G-维数的有限性刻画了$A$的局部Gorenstein性质。应用分为三个方向。第一个是建立G-维数与 &\mathcal{A}& 的小有限维数之间的联系。第二个是通过最大局部-Cohen-Macaulay DG模类与DG模的一个特殊G类之间的关系来刻画Cohen-Macaulay和Gorenstein DG环。第三个是将经典的Buchweitz-Happel定理及其逆从交换诺特局部环扩展到交换诺特局部DG环的设置中。我们的方法与经典交换环的方法有些不同。 显示英文摘要

We define and study a notion of G-dimension for DG-modules over a non-positively graded commutative noetherian DG-ring $A$. Some criteria for the finiteness of the G-dimension of a DG-module are given by applying a DG-version of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021) 409-454]. Moreover, it is proved that the finiteness of G-dimension characterizes the local Gorenstein property of $A$. Applications go in three directions. The first is to establish the connection between G-dimensions and the little finitistic dimensions of &\mathcal{A}&. The second is to characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of DG-modules. The third is to extend the classical Buchwtweiz-Happel Theorem and its inverse from commutative noetherian local rings to the setting of commutative noetherian local DG-rings.Our method is somewhat different from classical commutative ring.

[3] arXiv:2507.22408 (替换) [中文pdf, pdf, html, 其他]

标题： 坐标代数的生成元的仿射ind-概形 显示英文标题

标题： On the generators of coordinate algebras of affine ind-varieties

Alexander Chernov

评论： 6页，0图

主题： 代数几何 (math.AG) ; 交换代数 (math.AC)

在本文中，我们研究仿射 ind-流形的坐标环的结构。 我们证明，任何不是同构于仿射代数簇的仿射 ind-流形的坐标环都不具有可数生成元集。 此外，我们证明仿射 ind-流形的坐标环具有一个处处稠密的可数维子空间。 显示英文摘要

In this paper we study the structure of the coordinate ring of an affine ind-variety. We prove that any coordinate ring of an affine ind-variety which is not isomorphic to an affine algebraic variety does not have a countable set of generators. Also we prove that coordinate rings of affine ind-varieties have an everywhere dense subspace of countable dimension.