arXiv subjects

Yoshihiro Mizoguchi

Publications and source records attributed to Yoshihiro Mizoguchi.

7 recordsLinked to original sources

arXiv2017

A linear algorithm for Brick Wang tiling

The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2015 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

cs.DM↗

arXiv2015

A Property of Random Walks on a Cycle Graph

We analyze the Hunter vs Rabbit game on graph, which is a kind of model of communication in an adhoc mobile network. Let $G$ be a cycle graph with $N$ nodes. The hunter can move from a vertex to another vertex on the graph along an edge. The rabbit can move to any vertex on graph at once. We formalized the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over $\mathbb{Z}$. We classify strategies using the order $O(k^{-β-1})$ of their Fourier transformation. We investigate lower bounds and upper bounds of a probability that the hunter catches the rabbit. We found a constant lower bound if $β\in (0,1)$. That is there is not depend on the size $N$ of the graph. We show the order is equivalent to $O(1/\log N)$ if $β=1$ and a lower bound is $1/N^{(β-1)/β}$ if $β\in (1,2]$. Those results assist to choose the parameter $β$ of a rabbit strategy according to the size $N$ of the given graph. We introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the hunter catches the rabbit, and we also show computing simulation results.

math.PR↗

arXiv2014

Uniqueness of Butson Hadamard matrices of small degrees

For positive integers $m$ and $n$, we denote by $\mathrm{BH}(m,n)$ the set of all $H\in M_{n\times n}(\mathbb{C})$ such that $HH^\ast=nI_n$ and each entry of $H$ is an $m$-th root of unity where $H^\ast$ is the adjoint matrix of $H$ and $I_n$ is the identity matrix. For $H_1,H_2\in \mathrm{BH}(m,n)$ we say that $H_1$ is \textit{equivalent} to $H_2$ if $H_1=PH_2 Q$ for some monomial matrices $P, Q$ whose nonzero entries are $m$-th roots of unity. In this paper we classify $\mathrm{BH}(17,17)$ up to equivalence by computer search.

math.CO↗

arXiv2012

Bipartition of graphs based on the normalized cut and spectral methods

In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices and present an alternative proof for finding eigenvalues of the adjacency matrix of paths and cycles using Chebyshev polynomials. Even though each results is separately well known, we unite them, and provide uniform proofs in a simple manner. The main objective of this study is to solve the problem of finding graphs, on which spectral clustering methods and normalized cuts produce different partitions. First, we derive a formula for a minimum normalized cut for graph classes such as paths, cycles, complete graphs, double-trees, cycle cross paths, and some complex graphs like lollipop graph $LP_{n,m}$, roach type graph $R_{n,k}$, and weighted path $P_{n,k}$. Next, we provide characteristic polynomials of the normalized Laplacian matrices ${\mathcal L}(P_{n,k})$ and ${\mathcal L}(R_{n,k})$. Then, we present counter example graphs based on $R_{n,k}$, on which spectral methods and normalized cuts produce different clusters.

math.CO↗

arXiv2005

Statistical Properties of a Quantum Cellular Automaton

We study a quantum cellular automaton (QCA) whose time-evolution is defined from global transition function of classical cellular automata (CA). In order to investigate natural transformations from CA to QCA, the present QCA includes CA with Wolfram's rule 150 and 105 as special cases. We firstly compute the time-evolution of the QCA and examine its statistical properties. As a basic statistical value, the probability of finding an active cell averaged over a spatial-temporal space is introduced, and the difference between CA and QCA is considered. In addition, it is shown that statistical properties in QCA are related to the classical trajectory in the configuration space.

quant-ph↗

arXiv2005

On Reversible Cellular Automata with Triplet Local Rules

Bijections between sets may be seen as discrete (or crisp) unitary transformations used in quantum computations. So discrete quantum cellular automata are cellular automata with reversible transition functions. This note studies on 1d reversible cellular automata with triplet local rules.

nlin.CG↗

arXiv2004

Generalized Partitioned Quantum Cellular Automata and Quantization of Classical CA

In this paper, in order to investigate natural transformations from discrete CA to QCA, we introduce a new formulation of finite cyclic QCA and generalized notion of partitioned QCA. According to the formulations, we demonstrate the condition of local transition functions, which induce a global transition of well-formed QCA. Following the results, extending a natural correspondence of classical cells and quantum cells to the correspondence of classical CA and QCA, we have the condition of classical CA such that CA generated by quantumization of its cells is well-formed QCA. Finally we report some results of computer simulations of quantumization of classical CA.

quant-ph↗