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

Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$…

Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the…

We use Margulis' construction together with lattice counting arguments to build Cayley graphs on $\mathrm{SL}_{2}\left(\mathbb{F}_{p}\right),\;p\to\infty$ which are d-regular graphs with girth…

This application-oriented study concerns computational musicology, which makes use of grammar systems. We define multi-generative rule-synchronized scattered-context grammar systems (without erasing rules) and demonstrates how to…

Understanding the structural and cognitive underpinnings of musical compositions remains a key challenge in music theory and computational musicology. While traditional methods focus on harmony and rhythm, cognitive models such as the…

We study a construction of the Mathieu group $M_{12}$ using a game reminiscent of Loyd's ``15-puzzle''. The elements of $M_{12}$ are realized as permutations on~12 of the~13 points of the finite projective plane of order~3. There is a…

We introduce a method for analyzing the convex hull of a set in non-positively curved piecewise Euclidean polygonal complexes and we apply this method to prove that, with the usual action of Fm x Zn on the metric product of the Cayley graph…

Traditional approaches to combination tones based on Helmholtz theory encounter essential interpreting difficulties, which the most known example is the anomalous behaviour of the combination tone 2f1-f2. Without doubt the phenomenon of…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

We present an algebraic construction of music notes and show how to associate them inseveral ways to construct music ranges. Then a family of ranges emerge with a fixed number of notes: two, three, five, seven, twelve, seventeen, etc. A…

The world today is experiencing an abundance of music like no other time, and attempts to group music into clusters have become increasingly prevalent. Common standards for grouping music were songs, artists, and genres, with artists or…

This paper, following (Dymetman:1998), presents an approach to grammar description and processing based on the geometry of cancellation diagrams, a concept which plays a central role in combinatorial group theory (Lyndon-Schuppe:1977). The…

Graph transformation is concerned with the manipulation of graphs by means of rules. Graph grammars have been traditionally studied using techniques from category theory. In previous works, we introduced Matrix Graph Grammars (MGGs) as a…

We investigate subsets of the symmetric group with structure similar to that of a graph. The trees of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first…

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory…

The sandpile group of a connected graph $G$, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous…

In this article, a framework for defining and analysing a family of graphs or networks from symbolic music information is discussed. Such graphs concern different types of elements, such as pitches, chords and rhythms, and the relations…

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by…

A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases…

Musical intervals in multiple of semitones under 12-note equal temperament, or more specifically pitch-class subsets of assigned cardinality ($n$-chords) are conceived as positive integer points within an Euclidean $n$-space. The number of…