Journal of Computational Algebra最新文献

英文中文

An innovative graph polynomial numerical strategy for the time-fractional Kuramoto-Sivashinsky model based on the complete bipartite graph's independence polynomials 基于完全二部图无关多项式的时间分数型Kuramoto-Sivashinsky模型的一种创新图多项式数值策略

Journal of Computational Algebra

Pub Date : 2026-02-04 DOI: 10.1016/j.jaca.2026.100044

A.N. Nirmala, S. Kumbinarasaiah

The time-fractional Kuramoto-Sivashinsky equation (TF-K-SE) models chaotic dynamics with memory effects, necessitating advanced computational techniques for stability and accuracy. Conventional methods frequently encounter instability, excessive computational costs, and dependence on transformation techniques, limiting their effectiveness in handling fractional derivatives. This study introduces the Independence Polynomial Collocation Method (ICCM), leveraging graph-theoretic independence polynomials to enhance spectral accuracy and computational efficiency. Constructing sparse operational matrices, ICCM optimizes spectral discretization while preserving nonlinear characteristics, ensuring a more stable numerical framework. Caputo fractional derivatives capture memory-dependent dynamics and provide smooth transitions between fractional and integer orders without sacrificing physical fidelity. Unlike conventional orthogonal polynomials, which require increasing degrees for refinement and dense matrix representations, ICCM introduces a graph-theoretic spectral basis, improving smoothness, maintaining linear independence, and enhancing stability. A key advantage of ICCM is its independence from controlling parameters, distinguishing it from conventional semi-analytical methods that rely on stabilization terms or artificial tuning. ICCM achieves intrinsic numerical stability without external adjustments, making it a computationally efficient alternative while accurately preserving the chaotic nature of TF-K-SE. Numerical validation across four test cases demonstrates robust spectral stability, with error norms confirming precision and smooth transitions across fractional orders. ICCM establishes a computationally efficient framework for fractional PDE modeling and nonlinear system analysis, offering a novel approach distinct from existing techniques.

时间分数Kuramoto-Sivashinsky方程（TF-K-SE）模拟具有记忆效应的混沌动力学，需要先进的计算技术来保证稳定性和准确性。传统的方法经常遇到不稳定、计算成本过高以及对变换技术的依赖，限制了它们处理分数阶导数的有效性。本研究引入独立多项式搭配法（ICCM），利用图论独立多项式来提高谱精度和计算效率。通过构建稀疏运算矩阵，ICCM在保持非线性特性的同时优化了谱离散化，确保了更稳定的数值框架。卡普托分数阶导数捕获依赖于内存的动态，并在不牺牲物理保真度的情况下提供分数阶和整数阶之间的平滑转换。与传统的正交多项式不同，ICCM引入了图论谱基，提高了平滑性，保持了线性独立性，增强了稳定性。ICCM的一个关键优势是它不依赖于控制参数，与依赖于稳定项或人工调谐的传统半解析方法不同。ICCM在没有外部调整的情况下实现了内在的数值稳定性，使其成为一种计算效率高的替代方案，同时准确地保留了TF-K-SE的混沌性质。四个测试用例的数值验证显示了强大的光谱稳定性，误差规范确认了分数阶的精度和平滑过渡。ICCM为分数阶偏微分方程建模和非线性系统分析建立了一个计算效率高的框架，提供了一种不同于现有技术的新方法。

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引用次数: 0

The dynamics of the Hesse derivative on the j-invariant j不变量上的Hesse导数的动力学

Journal of Computational Algebra

Pub Date : 2026-02-03 DOI: 10.1016/j.jaca.2026.100043

Jake Kettinger

In this paper, we study the Hesse derivative of a cubic curve on the set of j-invariants, which can be viewed as a rational function on the Riemann sphere. This effect comes in the form of a rational function on the Riemann sphere. We then analyze the dynamics of this rational function, including counting the number of orbits of a given size. We proceed to investigate when a cubic curve is isomorphic to its n-fold Hesse derivative, showing that when an elliptic curve has a j-invariant that is periodic under this rational function, the curve itself must be periodic under the Hesse derivative. We finish with some data collected comparing the sizes of the orbits of elliptic curves to those of their j-invariants, and some further questions about this dynamical system.

本文研究了j不变量集上的三次曲线的Hesse导数，该曲线可以看作是Riemann球上的有理函数。这种效应以黎曼球上的有理函数的形式出现。然后我们分析这个有理函数的动力学，包括计算给定大小的轨道的数量。我们进一步研究了当一个三次曲线与它的n次Hesse导数同构时，证明了当一个椭圆曲线在这个有理函数下具有周期的j不变量时，曲线本身在Hesse导数下必须是周期的。最后，我们收集了一些数据，将椭圆曲线的轨道大小与j不变量的轨道大小进行了比较，并对这一动力系统提出了一些进一步的问题。

引用次数: 0

Stable Andrews-Curtis trivialization of AK(3) revisited. A case study using automated deduction 对稳定的Andrews-Curtis AK(3)琐琐化进行了重新审视。使用自动推理的案例研究

Journal of Computational Algebra

Pub Date : 2025-10-13 DOI: 10.1016/j.jaca.2025.100041

Alexei Lisitsa

Recent work by Shehper et al. (2024) [13] proposed that the well-known Akbulut–Kirby presentation

AK (3)

is stably Andrews–Curtis (AC) equivalent to the trivial presentation, based on a reduction to a previously studied presentation P. In this paper, we present an independently obtained reduction of

AK (3)

to P, using a fully automated theorem proving approach. Subsequent to our initial submission, it has emerged that the earlier theoretical claim about the stable AC-trivializability of P—on which the Shehper et al. result relied—was based on an incorrect theorem, and thus the status of

AK (3)

remains unresolved. While this paper does not establish the stable AC-triviality of

AK (3)

, it contributes a reproducible and verifiable case study in the use of automated reasoning to analyze deep problems in combinatorial group theory. We conclude by suggesting future directions for computational exploration of stable AC-transformations.

Shehper等人（2024）[13]最近的工作提出，基于对先前研究的表示P的约简，著名的Akbulut-Kirby表示AK(3)与平凡表示稳定地Andrews-Curtis （AC）等价。在本文中，我们使用全自动定理证明方法，提出了一个独立获得的AK(3)到P的约简。在我们最初的提交之后，我们发现早先关于稳定ac的理论主张——p的可琐碎性（Shehper等人的结果基于此）是基于一个不正确的定理，因此AK(3)的地位仍未得到解决。虽然本文没有建立AK(3)的稳定ac -平凡性，但它为使用自动推理分析组合群论中的深层问题提供了一个可重复和可验证的案例研究。最后，我们提出了稳定交流变换计算探索的未来方向。

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引用次数: 0

Machine learning mutation-acyclicity of quivers 机器学习突变-颤振的不周期性

Journal of Computational Algebra

Pub Date : 2025-09-01 DOI: 10.1016/j.jaca.2025.100040

Kymani T.K. Armstrong-Williams , Edward Hirst , Blake Jackson , Kyu-Hwan Lee

Machine learning (ML) has emerged as a powerful tool in mathematical research in recent years. This paper applies ML techniques to the study of quivers—a type of directed multigraph with significant relevance in algebra, combinatorics, computer science, and mathematical physics. Specifically, we focus on the challenging problem of determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. Although this classification is known for quivers with at most 3 vertices, little is known about quivers on more than 3 vertices. We give a computer-assisted proof of a theorem to prove that mutation-acyclicity is decidable for quivers on 4 vertices with edge weight at most 2. By leveraging neural networks (NNs) and support vector machines (SVMs), we then accurately classify more general 4-vertex quivers as mutation-acyclic or non-mutation-acyclic. Our results demonstrate that ML models can efficiently detect mutation-acyclicity, providing a promising computational approach to this combinatorial problem, from which the trained SVM equation provides a starting point to guide future theoretical development.

近年来，机器学习（ML）已成为数学研究中的一个强大工具。本文将机器学习技术应用于颤栗的研究，颤栗是一种与代数、组合学、计算机科学和数学物理有重要关联的有向多图。具体来说，我们专注于确定4个顶点上颤振的突变-不环性这一具有挑战性的问题，这是一个关键的性质，因为突变-不环性通常是涉及路径代数和簇代数的定理的必要条件。虽然这种分类对于最多3个顶点的箭囊是已知的，但对于超过3个顶点的箭囊却知之甚少。对于边权不超过2的4个顶点上的颤振，给出了突变-不环性可判定的一个定理的计算机辅助证明。通过利用神经网络（nn）和支持向量机（svm），我们准确地将更一般的4顶点颤振分类为突变-无环或非突变-无环。我们的研究结果表明，ML模型可以有效地检测突变-非周期性，为这一组合问题提供了一种有前途的计算方法，由此训练的SVM方程为指导未来的理论发展提供了一个起点。

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引用次数: 0

Computing quadratic subfields of number fields 计算数字域的二次子域

Journal of Computational Algebra

Pub Date : 2025-07-23 DOI: 10.1016/j.jaca.2025.100039

Andreas-Stephan Elsenhans , Jürgen Klüners

Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, we can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information, we give algorithms to determine all quadratic and cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.

给定一个数域，如何确定其所有子域是算法数论中的一个重要问题。如果搜索仅限于阿贝尔子域，我们可以尝试使用类场论来确定它们。为此，有必要知道派生素数。我们证明了子域的分支素数可以有效地计算。利用这些信息，给出了确定初始域的所有二次子域和循环三次子域的算法。该方法推广到素数次的循环子域。在二次子域的情况下，我们的方法比其他方法快得多。

引用次数: 0

A 3-skeleton for a classifying space for the symmetric group 对称群分类空间的3-骨架

Journal of Computational Algebra

Pub Date : 2025-06-23 DOI: 10.1016/j.jaca.2025.100038

Matthew B. Day , Trevor Nakamura

We construct a 3-dimensional cell complex that is the 3-skeleton for an Eilenberg–MacLane classifying space for the symmetric group

S_{n}

. Our complex starts with the presentation for

S_{n}

with

n - 1

adjacent transpositions with squaring, commuting, and braid relations, and adds seven classes of 3-cells that fill in certain 2-spheres bounded by these relations. We use a rewriting system and a combinatorial method of K. Brown to prove the correctness of our construction. Our main application is a computation of the second cohomology of

S_{n}

in certain twisted coefficient modules; we use this computation in a companion paper to study splitting of extensions related to braid groups. As another application, we give a concrete description of the third homology of

S_{n}

with untwisted coefficients in

Z

我们构造了一个三维细胞复合体，它是对称群Sn的Eilenberg-MacLane分类空间的3-骨架。我们的复合体从Sn的n−1相邻转置的平方，交换和编织关系开始，并添加了7类3-细胞，这些3-细胞填充在以这些关系为界的特定2-球体中。我们用一个改写系统和K. Brown的组合方法来证明我们构造的正确性。我们的主要应用是计算某些扭曲系数模中Sn的二次上同调；我们在另一篇论文中使用这种计算方法来研究与辫状群相关的扩展的分裂。作为另一个应用，我们给出了Sn在Z中具有未扭系数的第三同调的具体描述。

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引用次数: 0

An algorithm to compute the Hausdorff dimension of regular branch groups 正则分支群的Hausdorff维数计算算法

Journal of Computational Algebra

Pub Date : 2025-06-19 DOI: 10.1016/j.jaca.2025.100037

Jorge Fariña-Asategui

An explicit algorithm is given for the computation of the Hausdorff dimension of the closure of a regular branch group in terms of an arbitrary branch structure. We implement this algorithm in GAP and apply it to a family of GGS-groups acting on the 4-adic tree.

给出了一种计算任意分支结构下正则分支群闭包的Hausdorff维数的显式算法。我们在GAP中实现了该算法，并将其应用于作用于四进树的一组ggs群。

引用次数: 0

Nested fifth root radical identities from elliptic curves 椭圆曲线的嵌套五根根恒等式

Journal of Computational Algebra

Pub Date : 2025-03-01 DOI: 10.1016/j.jaca.2025.100032

Joseph Tonien

Ramanujan discovered the following elegant identity involving cube roots:

\sqrt{m \sqrt[3]{4 (m - 2 n)} + n \sqrt[3]{4 m + n}} = \pm \frac{1}{3} (\sqrt[3]{{(4 m + n)}^{2}} + \sqrt[3]{4 (m - 2 n) (4 m + n)} - \sqrt[3]{2 {(m - 2 n)}^{2}}) .

The goal of this paper is to derive nested fifth root radical identities in the form of

\sqrt{P_{1} \sqrt[5]{Q_{1}} + P_{2} \sqrt[5]{Q_{2}}} = p_{1} \sqrt[5]{q_{1}} + p_{2} \sqrt[5]{q_{2}} + p_{3} \sqrt[5]{q_{3}} + p_{4} \sqrt[5]{q_{4}} + p_{5} \sqrt[5]{q_{5}} .

We show that these identities can be derived from a specific family of elliptic curves. Furthermore, we include the SageMath code for computing these expressions.

拉马努金发现了一个简洁的立方根等式：m4(m−2n)3+n4m+n3=±13((4m+n)23+4(m−2n)（4m+n)3−2(m−2n)23）。本文的目标是推导出嵌套的五根根恒等式，其形式为p1q15+ P2Q25=p1q15+ P2Q25 +p3q35+p4q45+p5q55。我们证明了这些恒等式可以从特定的椭圆曲线族中导出。此外，我们还包含了用于计算这些表达式的SageMath代码。

引用次数: 0

Ideals of generic forms 一般形式的理想

Journal of Computational Algebra

Pub Date : 2025-03-01 DOI: 10.1016/j.jaca.2025.100033

Ralf Fröberg

We determine the Hilbert series of some classes of ideals generated by generic forms of degree two and three, and investigate the difference to the Hilbert series of ideals generated by powers of linear generic forms of the corresponding degrees.

我们确定了由二阶和三次一般形式生成的若干类理想的希尔伯特级数，并研究了它们与由相应次的线性一般形式幂生成的希尔伯特级数的区别。

引用次数: 0

Listing words in free groups 在自由组中列出单词

Journal of Computational Algebra

Pub Date : 2025-03-01 DOI: 10.1016/j.jaca.2025.100034

Colin Ramsay

Lists of equivalence classes of words under rotation or rotation plus reversal (i.e., necklaces and bracelets) have many uses, and efficient algorithms for generating these lists exist. In combinatorial group theory elements of a group are typically written as words in the generators and their inverses, and necklaces and bracelets correspond to conjugacy classes and relators respectively. We present algorithms to generate lists of freely and cyclically reduced necklaces and bracelets in free groups. Experimental evidence suggests that these algorithms are CAT – that is, they run in constant amortized time.

在旋转或旋转加反转（即，项链和手镯）下的等效类的单词列表有许多用途，并且存在生成这些列表的有效算法。在组合群论中，群的元素通常在生成器及其逆中写成单词，项链和手镯分别对应于共轭类和关系类。我们提出算法，以产生自由和循环约简项链和手镯在自由组列表。实验证据表明这些算法是CAT -也就是说，它们在常数平摊时间内运行。

引用次数: 0

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Journal of Computational Algebra

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