Journal of Computational Algebra最新文献

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symmetry emerges in the quiver exchange graph embedding. The ratio between number of seeds and number of quivers associated to this symmetry is computed for finite Dynkin type algebras up to rank 5, and conjectured for higher ranks. Simple machine learning techniques successfully learn to classify cluster algebras using the data of seeds. The learning performance exceeds 0.9 accuracies between algebras of the same mutation type and between types, as well as relative to artificially generated data.

In this work, we propose a new invariant for 2D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a 2D persistence module M, we propose an “interval-decomposable replacement” ${\delta }^{⁎}\left(M\right)$ (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of interval-decomposable modules, that is, its positive and negative parts. We show that M is interval-decomposable if and only if ${\delta }^{⁎}\left(M\right)$ is equal to M in the split Grothendieck group. Furthermore, even for modules M not necessarily interval-decomposable, ${\delta }^{⁎}\left(M\right)$ preserves the dimension vector and the rank invariant of M. In addition, we provide an algorithm to compute ${\delta }^{⁎}\left(M\right)$ (a high-level algorithm in the general case, and a detailed algorithm for the size $2\times n$ case).

We present an algorithm to compute the 9th Dedekind number: 286386577668298411128469151667598498812366. The key aspects are the use of matrix multiplication and symmetries in the free distributive lattice, that are determined with techniques from Formal Concept Analysis.

Suppose that a finite solvable permutation group G acts faithfully and primitively on a finite set Ω. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$ and the rank of G be the number of distinct orbits of $G_0$ in Ω (including the trivial orbit $\left\{\alpha \right\}$). Then G always has rank greater than four except for in a few cases. We completely classify these cases in this paper.

In this paper, we first generate a family of rational surfaces in affine 3-space from three rational space curves by dual quaternion multiplication utilizing dual quaternions as a tool to represent rigid transformations. We provide an algorithm to compute all the base points of the homogeneous tensor product parametrization of this family of surfaces. Our main focus is the syzygies of these surfaces. We discover two sets of special syzygies, and show that the syzygy module and a μ-basis of this surface can be extracted from either set of special syzygies. Finally, we describe the structure of a free resolution of the module generated by these special syzygies, and use this free resolution to classify the minimal free resolutions of this module.

We study the Modular Isomorphism Problem for groups of small order based on an improvement of an algorithm described by Eick. Our improvement allows to determine quotients $I\left(kG\right)/I{\left(kG\right)}^m$ of the augmentation ideal without first computing the full augmentation ideal $I\left(kG\right)$. Our computations yield a positive answer to the MIP for groups of order 3⁷ and strongly reduce the cases that need to be checked for groups of order 5⁶. We also show that the counterexamples to the Modular Isomorphism Problem found recently by García-Lucas, Margolis and del Río are the only 2- or 3-generated counterexamples of order 2⁹. Furthermore, we provide a proof for an observation of Bagiński, which is helpful in eliminating computationally difficult cases.