arXiv subjects

Fumio Hazama

Publications and source records attributed to Fumio Hazama.

6 recordsLinked to original sources

arXiv2022

Construction of smooth rhythms through a monotone invariant measure

The present article introduces the notion of quasi-smoothness of marked rhythms through a certain transformation $Ref$, called reformation map. A marked rhythm consists of a rhythm together with a marker, and the map $Ref$ modifies the marked onset of the rhythm. It is shown that an iteration of the map $Ref$ transforms an arbitrary marked rhythm into a quasi-smooth one. A numerical criterion for a marked rhythm to be quasi-smooth is given in terms of the difference of its rhythm part. Through this criterion, the rhythm part of any quasi-smooth marked rhythm is shown to be smooth.

math.CO↗

arXiv2021

Boolean polynomial expression of the discrete average on the space of rhythms

An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence formulas among them are derived.

math.CO↗

arXiv2017

Iterative method of construction of good rhythms

On the space of rhythms of arbitrary length with a fixed number of onsets, a self map $F$ is constructed. It is shown that for any rhythm $\mathbf{r}$ of the space there exists a nonnegative integer $k$ such that $F^k(\mathbf{r})$ falls into the category of "maximally even rhythms" investigated in [1].

math.CO↗

arXiv2014

Mathematical Analysis of Melodies: Slope and Discrete Frechet Distance

A directed graph, called an M-graph, is attached to every melody. Our chief concern in this paper is to investigate (1) how the positivity of the slope of the M-graph is related to singability of the melody, (2) when the M-graph has a symmetry, and (3) how we can detect a similarity between two melodies. For the third theme, we introduce the notion of transposed discrete Frechet distance, and show its relevance in the study of similarity detection among an arbitrary set of melodies.

math.HO↗

arXiv2014

Fixed-Point Problems in Discrete Tomography: Case of Square Windows

A kind of fixed-point problem in the area of discrete tomography is proposed and investigated. Our chief concern in this paper is the case of square windows in the plane. Dealing with the arrays which are bounded, of polynomial growth, and finite-ring-valued, one comes across several interesting phenomena of combinatorial and arithmetic nature.

math.CO↗

arXiv2013

The moduli space of polygons with area center

A point $p$ is said to be an area center of a polygon if all of the triangles composed of $p$ and its edges have one and the same area. We construct a moduli space $AC_n$ of such $n$-gons and study its geometry and arithmetic. For every $n\geq 5$, the moduli space is proved to be a rational complete intersection subvariety in $\mathbb{A}^n$. With the help of some subvarieties of low degree in $AC_n$, we also find a unified method of construction of good-looking polygons with area center.

math.AG↗