arXiv subjects

Shigeki Matsutani

Publications and source records attributed to Shigeki Matsutani.

At least 37 records · Page 2Linked to original sources

arXiv2019

Jacobi inversion formulae for a curve in Weierstrass normal form

We consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)$ where $x$ is an affine coordinate on $\mathbb{P}^1$, the point $\infty$ on $X$ is mapped to $x=\infty$, and each $A_j$ is a polynomial in $x$ of degree $\leq js/r$ for a certain coprime positive integers $r$ and $s$ ($r<s$) so that its Weierstrass non-gap sequence at $\infty$ is a numerical semigroup. It is a natural generalization of Weierstrass' equation in the Weierstrass elliptic function theory. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian using the result of Jorgenson (Israel J. Math (1992) 77 pp 273-284).

math.AG↗

arXiv2018

An algebraic description of screw dislocations in SC and BCC crystal lattices

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately.

math-ph↗

arXiv2017

The sigma function for trigonal cyclic curves

A recent generalization of the "Kleinian sigma function" involves the choice of a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$. This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at $P$. We exhibit the Riemann constant for a Weierstrass semigroup at $P$ with minimal set of generators $\{3, 2r+s,2s+r\}$, $r<s$, equivalently, non-symmetric, we construct a basis of $H^1(X, \mathbb{C})$ and a fundamental 2-differential on $X\times X$, we give the order of vanishing for sigma on Wirtinger strata of the Jacobian of $X$, and a solution to the Jacobi inversion problem.

math.AG↗

arXiv2017

New theory of diffusive and coherent nature of optical wave via a quantum walk

We propose a new theory on a relation between diffusive and coherent nature in one dimensional wave mechanics based on a quantum walk. It is known that the quantum walk in homogeneous matrices provides the coherent property of wave mechanics. Using the recent result of a localization phenomenon in a one-dimensional quantum walk (Konno, Quantum Inf. Proc. (2010) 9, 405-418), we numerically show that the randomized localized matrices suppress the coherence and give diffusive nature.

quant-ph↗

arXiv2017

A novel conductivity mechanism of highly disordered carbon systems based on an investigation of graph zeta function

In the previous report [Phys. Rev. B {\bf{62}} 13812 (2000)], by proposing the mechanism under which electric conductivity is caused by the activational hopping conduction with the Wigner surmise of the level statistics, the temperature-dependent of electronic conductivity of a highly disordered carbon system was evaluated including apparent metal-insulator transition. Since the system consists of small pieces of graphite, it was assumed that the reason why the level statistics appears is due to the behavior of the quantum chaos in each granular graphite. In this article, we revise the assumption and show another origin of the Wigner surmise, which is more natural for the carbon system based on a recent investigation of graph zeta function in graph theory.

cond-mat.dis-nn↗

arXiv2017

Mathematics in Caging of Robotics

It is a crucial problem in robotics field to cage an object using robots like multifingered hand. However the problem what is the caging for general geometrical objects and robots has not been well-described in mathematics though there were many rigorous studies on the methods how to cage an object by certain robots. In this article, we investigate the caging problem more mathematically and describe the problem in terms of recursion of the simple euclidean moves. Using the description, we show that the caging has the degree of difficulty which is closely related to a combinatorial problem and a wire puzzle. It implies that in order to capture an object by caging, from a practical viewpoint the difficulty plays an important role.

math.MG↗

arXiv2016

The Riemann constant for a non-symmetric Weierstrass semigroup

The zero divisor of the theta function of a compact Riemann surface $X$ of genus $g$ is the canonical theta divisor of Pic${}^{(g-1)}$ up to translation by the Riemann constant $Δ$ for a base point $P$ of $X$. The complement of the Weierstrass gaps at the base point $P$ given as a numerical semigroup plays an important role, which is called the Weierstrass semigroup. It is classically known that the Riemann constant $Δ$ is a half period $\frac{1}{2}Γ_τ$ for the Jacobi variety $\mathcal{J}(X)=\mathbb{C}^g/Γ_τ$ of $X$ if and only if the Weierstrass semigroup at $P$ is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor $D_0$, we show a relation between the Riemann constant $Δ$ and a half period $\frac{1}{2}Γ_τ$ of the non-symmetric case. We also identify the semi-canonical divisor $D_0$ for trigonal curves, and remark on an algebraic expression for the Jacobi inversion problem using the relation

math.AG↗

arXiv2015

From Euler's elastica to the mKdV hierarchy, through the Faber polynomials

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.

math-ph↗

arXiv2014

On homogenized conductivity and fractal structure in a high contrast continuum percolation model

In the previous article (S. Matsutani and Y. Shimosako and Y. Wang, Physica A \bf{391} (2012) 5802-5809) we numerically investigated an electric potential problem with high contrast local conductivities ($γ_0$ and $γ_1$, $0<γ_0 \ll γ_1$) for a two-dimensional continuum percolation model (CPM). As numerical results, we showed there that the equipotential curves exhibit the fractal structure around the threshold $p_c$ and gave an approximated curve representing a relation between the homogenized conductivity and the volume fraction $p$ over $[p_c,1]$. In this article, using the duality of the conductivities and the quasi-harmonic properties, we re-investigate these topics to improve these results. We show that at $γ_0\to0$, the quasi-harmonic potential problem in CPM is quasiconformally equivalent to a random slit problem, which leads us to an observation between the conformal property and the fractal structure at the threshold. Further we extend the domain $[p_c,1]$ of the approximated curve to $[0,1]$ based on the these results, which is partially generalized to three dimensional case. These curves represent well the numerical results of the conductivities.

math-ph↗

arXiv2013

The al function of a cyclic trigonal curve of genus three

A cyclic trigonal curve of genus three is a $\mathbb{Z}_3$ Galois cover of $\mathbb{P}^1$, therefore can be written as a smooth plane curve with equation $y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. Following Weierstrass for the hyperelliptic case, we define an ``$\mathrm{al}$'' function for this curve and $\mathrm{al}^{(c)}_r$, $c=0,1,2$, for each one of three particular covers of the Jacobian of the curve, and $r=1,2,3,4$ for a finite branchpoint $(b_r,0)$. This generalization of the Jacobi $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ functions satisfies the relation: $$ \sum_{r=1}^4 \frac{\prod_{c=0}^2\mathrm{al}_r^{(c)}(u)}{f'(b_r)} = 1 $$ which generalizes $\mathrm{sn}^2u + \mathrm{cn}^2u = 1$. We also show that this can be viewed as a special case of the Frobenius theta identity.

math.AG↗

arXiv2013

The sigma function for Weierstrass semigroups <3,7,8> and <6,13,14,15,16>

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular form have pointed to the relevance of $τ$-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) $σ$-function. This paper proposes a construction of $σ$-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction of plane affine models of the Riemann surface. Because our definition is algebraic, we are able to consider the properties of the $σ$-functions including their Jacobi inversion formulae, and to give an observation of their properties to those of a Norton basis for replicable functions, in turn relevant to the Monstrous Moonshine.

math.AG↗

arXiv2013

Sigma functions for a space curve of type (3, 4, 5)

In this article, a generalized Kleinian sigma function for an affine (3,4,5) space curve of genus 2 was constructed as the simplest example of the sigma function for an affine space curve, and in terms of the sigma function, the Jacobi inversion formulae for the curve are obtained. An interesting relation between a space curve with a semigroup generated by (6,13,14,15,16) and Norton number associated with Monster group is also mentioned with an Appendix by Komeda.

math-ph↗

arXiv2013

Measuring Agglomeration of Agglomerated Particles Pictures

In this article, we introduce a novel geometrical index $δ_{agg}$, which is associated with the Euler number and is obtained by an image processing procedure for a given digital picture of aggregated particles such that $δ_{agg}$ exhibits the degree of the agglomerations of the particles. In the previous work (Matsutani, Shimosako, Wang, Appl.Math.Modeling {\bf{37}} (2013), 4007-4022), we proposed an algorithm to construct a picture of agglomerated particles as a Monte-Carlo simulation whose agglomeration degree is controlled by $γ_{agg} \in (0,1)$. By applying the image processing procedure to the pictures of the agglomeration particles constructed following the algorithm, we show that $δ_{agg}$ statistically reproduces the agglomeration parameter $γ_{agg}$.

cs.CE↗

arXiv2012

Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r = f(x)$. II

Previous work by the authors (this journal, \vol{60} (2008), 1009-1044) produced equations that hold on certain loci of the Jacobian of a cyclic $C_{rs}$ curve. A curve of this type generalizes elliptic curves, and the equations in question are given in terms of (Klein's) generalization of Weierstrass' $σ$-function. The key tool is a matrix with entries that are polynomial in the coordinates of the affine plane model of the curve, thus can be expressed in terms of $σ$ and its derivatives. The key geometric loci on the Jacobian of the curve give a stratification of Brill-Noether type. The results are of the type of Riemann-Kempf singularity theorem, the methods are germane to those used by J.D. Fay, who gave vanishing tables for Riemann's $θ$-function and its derivatives. The main objects we use were developed by several contemporary authors, aside from the classical definitions: meromorphic differentials were expressed in terms of the coordinates mainly by V.M. Buchstaber, J.C. Eilbeck, V.Z. Enolski, D.V. Leykin, and Taylor expansions for $σ$ in terms of Schur polynomials also contributed by A. Nakayashiki, in terms of Sato's $τ$-function. Within this framework, following specific results for $σ$-derivatives given by Y. Ônishi, we arrive at our main results, namely statements on the vanishing on given strata of the partial derivatives of $σ$ indexed by Young-diagrams subsets that can be worked out in terms of the Weierstrass semigroup of the curve at its point at infinity. The combinatorial statements hold not only for Jacobians but for the stratification of Sato's infinite-dimensional Grassmann manifold as well.

math.AG↗

arXiv2012

Prime form and sigma function

In this article, we study some cyclic $(r,s)$ curves $X$ given by $y^r =x^s + λ_{1} x^{s-1} +...+ λ_{s-1} x + λ_s$. We give an expression for the prime form $\cE(P,Q)$, where $(P, Q \in X)$, in terms of the sigma function for some such curves, specifically any hyperelliptic curve $(r,s) = (2, 2g+1)$ as well as the cyclic trigonal curve $(r,s) = (3,4)$, $$ \cE(P,Q) =\frac{σ_{\natural_{r}}(u - v)}{\sqrt{du_1}\sqrt{d v_1}}, $$ where $\natural_r$ is a certain index of differentials. Here $u_1$ and $v_1$ are respectively the first components of $u = w(P)$ and $v = w(Q)$ which are given by the Abel map $w: X \to \CC^g$, where $g$ is the genus of $X$.

math.AG↗

arXiv2012

Numerical Computations of Conductivities over Agglomerated Continuum Percolation Models

In order to clarify how the percolation theory governs the conductivities in real materials which consist of small conductive particles, e.g., nanoparticles, with random configurations in an insulator, we numerically investigate the conductivities of continuum percolation models consisting of overlapped particles using the finite difference method as a sequel of our previous article (Int. J. Mod. Phys. 21 (2010), 709). As the previous article showed the shape effect of each particle by handling different aspect ratios of spheroids, in this article we numerically show influences of the agglomeration of the particles on conductivities after we model the agglomerated configuration by employing a simple numerical algorithm which simulate an agglomerated configuration of particles by a natural parameter. We conclude that the dominant agglomeration effect on the conductivities can be interpreted as the size effect of an analyzed region. We also discuss an effect of shape of the agglomerated clusters on its universal property.

cond-mat.mtrl-sci↗

arXiv2012

Fractal Structure of Equipotential Curves on a Continuum Percolation Model

We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear the {\it{quasi-equipotential clusters}} which approximately and locally have the same values like steps and stairs. Since the quasi-equipotential clusters has the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over $[0,1]$. The fractal dimension in [1.00, 1.257] has a peak at the percolation threshold $p_c$.

math.NA↗

arXiv2012

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23}) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and soliton solutions. The Toda lattice, as it came to be known, was then extensively studied as one of the completely integrable (differential-difference) non-linear equations which admit exact solutions in terms of theta functions of hyperelliptic curves. In this paper, we extend Toda's original approach to give hyperelliptic solutions of the Toda lattice in terms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The key identities are given by generalized addition formulae for the hyperelliptic sigma functions (J.C. Eilbeck \textit{et al.}, {\it J. reine angew. Math.} {\bf 619}, 2008). We then show that periodic (in the discrete variable, a standard term in the Toda lattice theory) solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi's division polynomials, and note these are related to solutions of Poncelet's closure problem. One feature of our solution is that the hyperelliptic curve is related in a non-trivial way to the one previously used.

math.AG↗