相关论文: Musical Systems with $\mathbb{Z}_n$ -- Cayley Grap…

Chords in musical harmony can be viewed as objects having shapes (major/minor/etc.) attached to base sets (pitch class sets). The base set and the shape set are usually given the structure of a group, more particularly a cyclic group. In a…

We investigate Cayley graphs of graph products by showing that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs.

We propose a new graph convolutional block, called MusGConv, specifically designed for the efficient processing of musical score data and motivated by general perceptual principles. It focuses on two fundamental dimensions of music, pitch…

We leverage a correspondence between group actions and edge-labelled graphs in two ways. First, we give a unified presentation of several folklore results connecting weak containment, local-global convergence, and continuous model theory.…

We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example…

We construct a new large family of finitely generated groups with continuum many values of the following monotone parameters: spectral radius, critical percolation, and asymptotic entropy. We also present several open problems on other…

A quandle is an algebraic system originating in knot theory, which can be regarded as a generalization of the conjugation of groups. This structure naturally defines two subgroups of its automorphism group, which are called the inner…

We have previously proposed a study of arrangements of small circles which also surround regions in the plane realized as the images of natural real algebraic maps yielding Morse-Bott functions by projections. Among studies of arrangements,…

Mathematics is a far reaching discipline and its tools appear in many applications. In this paper we discuss its role in music and signal processing by revisiting the use of mathematics in algorithms that can extract chord information from…

Digital advances have transformed the face of automatic music generation since its beginnings at the dawn of computing. Despite the many breakthroughs, issues such as the musical tasks targeted by different machines and the degree to which…

This is the third paper of the CayleyPy project applying artificial intelligence to problems in group theory. We announce the first public release of CayleyPy, an open source Python library for computations with Cayley and Schreier graphs.…

We consider Cayley sum graphs over the cyclic group $\mathbb{Z}_n$ and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the…

For an integer $n\geq 2$, let $X_n$ be the Cayley graph on the symmetric group $S_n$ generated by the set of transpositions ${(1 2),(1 3),...,(1 n)}$. It is shown that the spectrum of $X_n$ contains all integers from $-(n-1)$ to $n-1$…

We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish…

We review some aspects of gauged WZW models. By choosing a nilpotent subgroup as gauge group, one is lead to three main applications: the construction of field theories with an extended conformal symmetry, the construction of the effective…

Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, powers of cycles, Cayley graphs on some non-abelian groups, and…

Based on the rules of magic cubes, a game of two-dimensional magic cube was deliberately designed. This essay will explore its properties with the assistance of group theory and computer programming. It will first elaborate the rules of…

We use the matrix model -- gauge theory correspondence of Dijkgraaf and Vafa in order to construct the geometry encoding the exact gaugino condensate superpotential for the N=1 U(N) gauge theory with adjoint and symmetric or anti-symmetric…

Cadences are complex structures that have been driving music from the beginning of contrapuntal polyphony until today. Detecting such structures is vital for numerous MIR tasks such as musicological analysis, key detection, or music…

We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's…