Details of a Researcher - NOBE, Atsushi

写真a

NOBE, Atsushi

Scopus Paper Info

Paper Count: 21 Citation Count: 902 h-index: 6

Click to view the Scopus page. The data was downloaded from Scopus API in April 19, 2024, via http://api.elsevier.com and http://www.scopus.com .

Google Scholar Information (Citations per year)

Citation Count: 1472 h-index: 9 i10-index: 9

Click to view the Google Scholar page.

Scopus Information

Affiliation

Faculty of Political Science and Economics, School of Political Science and Economics

Job title

Professor

Degree

博士（数理科学） ( 2002.03 東京大学 )
修士（数理科学） ( 1999.03 東京大学 )
修士（工学） ( 1996.03 東京大学 )
学士（理学） ( 1994.03 名古屋大学 )

Research Experience

2021.04

-

Now

Waseda University Faculty of Political Science and Economics Professor
2010.04

-

2021.03

Chiba University Graduate School of Science
2009.10

-

2021.03

Chiba University Faculty of Education
2007.04

-

2009.09

Osaka University Graduate School of Engineering Science
2004.04

-

2007.03

Osaka University Graduate School of Engineering Science
2003.09

-

2004.03

The University of Tokyo Graduate School of Mathematical Sciences Postdoctoral Researcher
2011.04

-

Now

気象大学校大学部非常勤講師

線形代数学、数学演習
2018.11

-

2021.03

津田塾大学学芸学部非常勤講師
2013.04

-

2021.03

日本大学生産工学部非常勤講師

線形代数学、微分積分学

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Education Background

1999.04

-

2002.03

The University of Tokyo Graduate School of Mathematical Sciences
1997.04

-

1999.03

The University of Tokyo Graduate School of Mathematical Sciences
1994.04

-

1996.03

The University of Tokyo The Graduate School of Engineering Department of Quantum Engineering and Systems Science
1990.04

-

1994.03

Nagoya University School of Science

Committee Memberships

2015.04

-

Now

日本応用数理学会 JSIAM Letters編集委員
2020.04

-

2022.03

日本応用数理学会応用可積分系研究部会主査
2016.04

-

2022.03

日本応用数理学会応用可積分系研究部会幹事
2009.10

-

2011.09

日本物理学会領域11役員

Professional Memberships

　

　

　

THE JAPAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS
　

　

　

日本数学会
　

　

　

日本物理学会

Research Areas

Mathematical analysis / Applied mathematics and statistics / Basic mathematics

Research Interests

Ultradiscrete system
ソリトン
クラスター代数
セルオートマトン
トロピカル幾何
可積分系
Cellular automata
Tropical geometry
Integrable systems

▼display all

Papers

The Volterra lattice, Abel's equation of the first kind, and the SIR epidemic models

Atsushi Nobe

arXiv 2402.11888 2024.02

Authorship：Lead author, Corresponding author

　View Summary

The Volterra lattice, when imposing non-zero constant boundary values, admits
the structure of a completely integrable Hamiltonian system if the system size
is sufficiently small. Such a Volterra lattice can be regarded as an epidemic
model known as the SIR model with vaccination, which extends the celebrated SIR
model to account for vaccination. Upon the introduction of an appropriate
variable transformation, the SIR model with vaccination reduces to an Abel
equation of the first kind, which corresponds to an exact differential
equation. The equipotential curve of the exact differential equation is the
Lambert curve. Thus, the general solution to the initial value problem of the
SIR model with vaccination, or the Volterra lattice with constant boundary
values, is implicitly provided by using the Lambert W function.

DOI
Exact solutions to SIR epidemic models via integrable discretization

Atsushi Nobe

arXiv 2303.17198 2023.03

Authorship：Lead author, Corresponding author

　View Summary

An integrable discretization of the SIR model with vaccination is proposed. The conserved quantities of the continuous model are inherited to the discrete model through the discretization, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. The discretization procedure similarly applied to the original SIR model leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is moreover shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.
Periodicity, linearizability and integrability in seed mutations of type $A^{(1)}_N$

Atsushi Nobe, Junta Matsukidaira

Journal of Mathematical Physics 62 ( 1 ) 013510 - 013510 2021.01 [Refereed] [International journal]

Authorship：Lead author, Corresponding author

　View Summary

In the network of seed mutations arising from a certain initial seed, an
appropriate path emanating from the initial seed is intendedly chosen, noticing
periodicity of the exchange matrices in the path each of which is assigned to
the generalized Cartan matrix of type $A^{(1)}_N$. Then dynamical property of
the seed mutations along the path, which is referred to as of type $A^{(1)}_N$,
is intensively investigated. The coefficients assigned to the path form certain
$N$ monomials that posses periodicity with period $N$ under the seed mutations
and enable to obtain the general terms of the coefficients. The cluster
variables assigned to the path of type $A^{(1)}_N$ also form certain $N$
Laurent polynomials possessing the same periodicity as the monomials generated
by the coefficients. These Laurent polynomials lead to sufficiently number of
conserved quantities of the dynamical system derived from the cluster mutations
along the path. Furthermore, by virtue of the Laurent polynomials with
periodicity, the dynamical system is non-autonomously linearized and its
general solution is concretely constructed. Thus the seed mutations along the
path of type $A^{(1)}_N$ exhibit discrete integrability.

DOI

Scopus
A family of integrable and non-integrable difference equations arising from cluster algebras

Atsushi Nobe, Junta Matsukidaira

RIMS Kokyuroku Bessatsu B78 99 - 120 2020.04 [Refereed] [Invited]

Authorship：Lead author, Corresponding author

　View Summary

The one-parameter family of second order nonlinear difference equations each
of which is given by $$ x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{\beta-1}+x_{n+1}
\qquad(\beta\in\mathbb{N}) $$ is explored. Since the equation above is arising
from seed mutations of a rank 2 cluster algebra, its solution is periodic only
when $\beta\leq3$. In order to evaluate the dynamics with $\beta\geq4$,
algebraic entropy of the birational map equivalent to the difference equation
is investigated; it vanishes when $\beta=4$ but is positive when $\beta\geq5$.
This fact suggests that the difference equation with $\beta\leq4$ is integrable
but that with $\beta\geq5$ is not. It is moreover shown that the difference
equation with $\beta\geq4$ fails the singularity confinement test. This fact is
consistent with linearizability of the equation with $\beta=4$ and reinforces
non-integrability of the equation with $\beta\geq5$.
Generators of rank 2 cluster algebras of affine types via linearization of seed mutations

Atsushi Nobe

Journal of Mathematical Physics 60 ( 7 ) 072702 - 072702 2019.07 [Refereed] [International journal]

Authorship：Lead author, Corresponding author

DOI

Scopus

1

Citation

(Scopus)
Group actions on the tropical Hesse pencil

Atsushi Nobe

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 33 ( 3 ) 537 - 556 2016.12 [Refereed]

Authorship：Lead author, Corresponding author

　View Summary

Addition of points on the tropical Hesse curve is realized via its intersections with two tropical lines. Then the addition formula for the points on the curve is reduced from the one for the level-three theta functions through the ultradiscretization procedure. In addition, a tropical analogue of the Hessian group , the group of linear automorphisms acting on the Hesse pencil, is investigated; it is shown that the dihedral group of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

DOI

Scopus
Mutations of the cluster algebra of type $A^{(1)}_1$ and the periodic discrete Toda lattice

Atsushi Nobe

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 49 ( 28 ) 2016.07 [Refereed] [International journal]

Authorship：Lead author, Corresponding author

　View Summary

A direct connection between two sequences of points, one of which is generated by seed mutations of the cluster algebra of type A(1)((1)) and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map, and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations of the cluster algebra of type A(1)((1)) as an addition of points on an elliptic curve.

DOI

Scopus

3

Citation

(Scopus)
A geometric realization of the ultradiscrete periodic Toda lattice via tropical plane curves

NOBE Atsushi

RIMS Kokyuroku Bessatsu B47 55 - 76 2014.12 [Refereed] [Invited]

Authorship：Lead author, Corresponding author

　View Summary

This is a review article on a tropical geometric realization of the 
ultradiscrete periodic Toda lattice (UD-pTL). Time evolution of the UD-pTL is 
translated into an addition on the Picard group of its spectral curve, which is 
a tropical hyperelliptic curve of arbitrary genus depending on the system size. 
The addition on the Picard group can be realized by using intersection of 
several tropical plane curves, one of which is the spectral curve. In addition, 
the tropical eigenvector map, which maps a point in the phase space of the 
UD-pTL into a set of points on the spectral curve, can also be realized by 
using intersection of tropical curves. Thus, if the initial values are given 
then the time evolution of the UD-pTL is completely translated into a motion of 
intersection points of tropical plane curves. Moreover, all tropical plane 
curves appearing in the curve intersection are explicitly given in terms of the 
conserved quantities of the UD-pTL.
A geometric realization of the periodic discrete Toda lattice and its tropicalization

Atsushi Nobe

Journal of Physics A: Mathematical and Theoretical 46 ( 46 ) 2013.11 [Refereed]

　View Summary

An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice (pdTL) is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the pdTL, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdTL. Finally, the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box-ball system is constructed via tropical curve intersections. © 2013 IOP Publishing Ltd.

DOI

Scopus

1

Citation

(Scopus)
Addition in Jacobians of tropical hyperelliptic curves

NOBE Atsushi

RIMS Kokyuroku Bessatsu B30 25 - 51 2012.11 [Refereed] [Invited]

Authorship：Lead author, Corresponding author

　View Summary

We show that there exists a surjection from the set of effective divisors of 
degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical 
version of the Riemann-Roch theorem. We then show that the restriction of the 
surjection is reduced to the bijection on an appropriate subset of the set of 
effective divisors of degree $g$ on the curve. Thus the subset of effective 
divisors has the additive group structure induced from the Jacobian. We finally 
realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ 
via the intersection with a tropical curve of degree $3g/2$ or $3(g-1)/2$.
An ultradiscrete integrable map arising from a pair of tropical elliptic pencils

Atsushi Nobe

PHYSICS LETTERS A 375 ( 47 ) 4178 - 4182 2011.11 [Refereed]

　View Summary

We present a tropical geometric description of a piecewise linear map whose invariant curve is a concave polygon. In contrast to convex polygons, a concave one is not directly related to tropical geometry; nevertheless the description is given in terms of the addition formula of a tropical elliptic curve. We show that the map arises from a pair of tropical elliptic pencils, each member of which is the invariant curve of an ultradiscrete QRT map. (C) 2011 Elsevier B.V. All rights reserved.

DOI

Scopus

2

Citation

(Scopus)
Constructing two-dimensional integrable mappings that possess invariants of high degree (Expansion of Integrable Systems)

TANAKA Hironori, MATSUKIDAIRA Junta, NOBE Atsushi, TSUDA Teruhisa

RIMS Kokyuroku Bessatsu B13 75 - 84 2009.10 [Refereed] [Invited]

CiNii
ULTRADISCRETIZATION OF A SOLVABLE TWO-DIMENSIONAL CHAOTIC MAP ASSOCIATED WITH THE HESSE CUBIC CURVE

Kenji Kajiwara, Masanobu Kaneko, Atsushi Nobe, Teruhisa Tsuda

KYUSHU JOURNAL OF MATHEMATICS 63 ( 2 ) 315 - 338 2009.09 [Refereed]

　View Summary

We present a solvable two-dimensional piecewise linear chaotic mail that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the non-trivial ultradiscrete limit of the solution in spite of a problem known as 'the minus-sign problem'

DOI J-GLOBAL

Scopus

5

Citation

(Scopus)
Ultradiscretization of solvable one-dimensional chaotic maps

Kenji Kajiwara, Atsushi Nobe, Teruhisa Tsuda

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 41 ( 39 ) 2008.10 [Refereed]

　View Summary

We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed.

DOI J-GLOBAL

Scopus

5

Citation

(Scopus)
Ultradiscrete QRT maps and tropical elliptic curves

Atsushi Nobe

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 41 ( 12 ) 2008.03 [Refereed]

　View Summary

It is shown that the polygonal invariant curve of the ultradiscrete QRT (uQRT) map, which is a two-dimensional piecewise linear integrable map, is the complement of the tentacles of a tropical elliptic curve on which the curve has a group structure in analogy to classical elliptic curves. Through the addition formula of a tropical elliptic curve, a tropical geometric description of the uQRT map is then presented. This is a natural tropicalization of the geometry of the QRT map found by Tsuda. Moreover, the uQRT map is linearized on the tropical Jacobian of the corresponding tropical elliptic curve in terms of the Abel- Jacobi map. Finally, a formula concerning the period of a point in the uQRT map is given, and an exact solution to its initial-value problem is constructed by using the ultradiscrete elliptic theta function.

DOI J-GLOBAL

Scopus

15

Citation

(Scopus)
Linearizable cellular automata

Atsushi Nobe, Fumitaka Yura

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 40 ( 26 ) 7159 - 7174 2007.06 [Refereed]

　View Summary

The initial value problem for a class of reversible elementary cellular automata with periodic boundaries is reduced to an initial- boundary value problem for a class of linear systems on a finite commutative ring Z(2). Moreover, a family of such linearizable cellular automata is given.

DOI

Scopus

3

Citation

(Scopus)
Soliton-antisoliton collision in the ultradiscrete modified KdV equation

S. Isojima, M. Murata, A. Nobe, J. Satsuma

PHYSICS LETTERS A 357 ( 1 ) 31 - 35 2006.08 [Refereed]

　View Summary

The discrete modified Korteweg-de Vries equation admits exact solutions with nondefinite sign, which describe interaction among solitons with positive and negative amplitude. In this Letter a transformation of hyperbolic sine type is proposed in order to ultradiscretize this equation and solutions. (c) 2006 Elsevier B.V. All rights reserved.

DOI J-GLOBAL

Scopus

6

Citation

(Scopus)
Ultradiscretization of elliptic functions and its applications to integrable systems

A Nobe

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 39 ( 20 ) L335 - L342 2006.05 [Refereed]

　View Summary

It is shown that there exist three kinds of ultradiscrete analogues of Jacobi's elliptic functions. In this process, the asymptotic behaviour of the poles and the zeros of the functions plays a crucial role. Using the ultradiscrete analogues and an addition formula, exact solutions to the ultradiscrete KP equation are constructed, and their relation to the ultradiscrete QRT system is discussed.

DOI J-GLOBAL

Scopus

5

Citation

(Scopus)
Exact solutions for discrete and ultradiscrete modified KdV equations and their relation to box-ball systems

M Murata, S Isojima, A Nobe, J Satsuma

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 39 ( 1 ) L27 - L34 2006.01 [Refereed]

　View Summary

A new class of solutions is proposed for discrete and ultradiscrete modified KdV equations. These are directly related to solutions of the box and ball system with a carrier. Moreover, an extended box and ball system and its exact solutions are discussed.

DOI J-GLOBAL

Scopus

13

Citation

(Scopus)
Periodic multiwave solutions to the Toda-type cellular automaton

A Nobe

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 38 ( 43 ) L715 - L723 2005.10 [Refereed]

　View Summary

An ultradiscretization of the Riemann theta function is proposed. The ultradiscretization satisfies an addition formula, which is an ultradiscrete analogue of an addition formula for the Riemann theta function. Using the addition formula, periodic multiwave solutions to the Toda-type cellular automaton are obtained.

DOI J-GLOBAL

Scopus

2

Citation

(Scopus)
An ultradiscretization of the sine-Gordon equation

S Isojima, M Murata, A Nobe, J Satsuma

PHYSICS LETTERS A 331 ( 6 ) 378 - 386 2004.11 [Refereed]

　View Summary

An ultradiscrete system corresponding to the sine-Gordon equation is proposed. A new dependent variable for the discrete sine-Gordon equation is introduced in order to apply the procedure of ultradiscretization. The ultradiscrete system possesses exact solutions which are directly related to soliton solutions of the discrete equation. (C) 2004 Elsevier B.V. All rights reserved.

DOI J-GLOBAL

Scopus

15

Citation

(Scopus)
On reversibility of cellular automata with periodic boundary conditions

A Nobe, F Yura

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 37 ( 22 ) 5789 - 5804 2004.06 [Refereed]

　View Summary

Reversibility of one-dimensional cellular automata with periodic boundary conditions is discussed. It is shown that there exist exactly 16 reversible elementary cellular automaton rules for infinitely many cell sizes by means of a correspondence between elementary cellular automaton and the de Bruijn graph. In addition, a sufficient condition for reversibility of three-valued and two-neighbour cellular automaton is given.

DOI J-GLOBAL

Scopus

26

Citation

(Scopus)
On periodic mappings arising from the QRT system

NOBE Atsushi

Theoretical and Applied Mechanics Japan 52 229 - 237 2003.12 [Refereed]

　View Summary

An eight-parameter family of two-dimensional piecewise linear mappings is discussed. Since the dynamical system is obtained from the QRT system through the ultradiscretization, the dynamical system is called the ultradiscrete QRT system. The ultradiscrete QRT system is considered to be integrable because it has an eight-parameter family of invariant curves which fills the plane. It is shown that, for particular parameters, the dynamical system can be regarded as a dynamical system on a fan associated with the conserved quantity. It is also shown that such a dynamical system has periodic solutions for any initial value. Therefore we call such a dynamical system the ultradiscrete periodic QRT system. From the ultradiscrete periodic QRT system, the periodic QRT system is obtained in terms of the inverse ultradiscretization.

DOI J-GLOBAL
Stable Difference Equations Associated with Elementary Cellular Automata

Atsushi Nobe, Junkichi Satsuma, Tetsuji Tokihiro

Japan Journal of Industrial and Applied Mathematics 18 ( 2 ) 293 - 305 2001 [Refereed]

　View Summary

We construct a difference equation which preserves any time evolution pattern of the rule 90 elementary cellular automaton. We also demonstrate that such difference equations can be obtained for any elementary cellular automata.

DOI

Scopus

2

Citation

(Scopus)
From cellular automaton to difference equation: A general transformation method which preserves time evolution patterns

A. Nobe, J. Satsuma, T. Tokihiro

Journal of Physics A: Mathematical and General 34 ( 25 ) L371 - L379 2001 [Refereed]

　View Summary

We propose a general method to construct a partial difference equation which preserves any time evolution patterns of a cellular automaton. The method is based on inverse ultradiscretization with filter functions. © 2001 IOP Publishing Ltd.

DOI J-GLOBAL

Scopus

9

Citation

(Scopus)
Numerical analysis of breaking waves using the moving particle semi-implicit method

S Koshizuka, A Nobe, Y Oka

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 26 ( 7 ) 751 - 769 1998.04 [Refereed]

　View Summary

The numerical method used in this study is the moving particle semi-implicit (MPS) method, which is based on particles and their interactions. The particle number density is implicitly required to be constant to satisfy incompressibility. A semi-implicit algorithm is used for two-dimensional incompressible non-viscous flow analysis. The particles whose particle number densities are below a set point are considered as on the free surface. Grids are not necessary in any calculation steps. It is estimated that most of computation time is used in generation of the list of neighboring particles in a large problem. An algorithm to enhance the computation speed is proposed. The MPS method is applied to numerical simulation of breaking waves on slopes. Two types of breaking waves, plunging and spilling breakers, are observed in the calculation results. The breaker types are classified by using the minimum angular momentum at the wave front. The surf similarity parameter which separates the types agrees well with references. Breaking waves are also calculated with a passively moving float which is modelled by particles. Artificial friction due to the disturbed motion of particles causes errors in the flow velocity distribution which is shown in comparison with the theoretical solution of a cnoidal wave. (C) 1998 John Wiley & Sons, Ltd.

DOI J-GLOBAL
$A^{(1)}_N$型ミューテーションの可積分性について

野邊厚, 松木平淳太

応用力学研究所研究集会報告 No. 2019 ( AO-S2 ) 45 - 50 2020.03

Authorship：Lead author
Cluster algebras and cellular automata

野邊厚

数理解析研究所講究録 2071 ( 2071 ) 141 - 159 2018.04 [Invited]

　View Summary

連続的なクイバーの変異とある群の台グラフへの作用が可換になる場合, 対応するクラスター変数の列はその台グラフ上の離散力学系とみることができる. このような離散力学系の解は正値性とLaurent性をもつため, 超離散化の手法を適用してセルオートマトンを導出することが可能である. 本稿においては, 平行移動群T(2)の作用と可換なクイバーの変異から離散KdV方程式や離散戸田格子などの離散可積分系を導出し, これらに超離散化を適用して箱玉系を構成する. さらに, 同様の手法を用いてA_{infty}型クイバーの変異から離散力学系を導出し, 適当な仮定の下で, その時間発展はルール204 ECAと見なせることを示す. また, このような離散力学系の一般解を構成する. 本研究は, 黒田謙吾氏(千葉大学), 問田潤氏(日本大学), 中田庸一氏(東京大学)との共同研究に基づいている.

CiNii
ランク2ミューテーションの不変曲線について

野邊厚

応用力学研究所研究集会報告 No.29AO-S7 69 - 74 2018.03 [Refereed]
Birational maps conjugate to the rank 2 cluster mutations of affine types and their geometry

Atsushi Nobe

1801.10320v1 2018.01

　View Summary

Mutations of the cluster variables generating the cluster algebra of type
$A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a
quartic birational map. The invariant curve of the map is a singular quartic
curve, and its resolution of the singularity induces a discrete integrable
system on a conic governed by a cubic birational map conjugate to the cluster
mutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also the
invariant curve of the quadratic birational map arising from the cluster
mutations of type $A^{(1)}_1$ and the two birational maps on the conic are
commutative. Finally, the commutative birational maps are reduced as singular
limits of additions of points on an elliptic curve arising as the spectral
curve of the discrete Toda lattice of type $A^{(1)}_1$.
$A^{(1)}_1$型クラスター代数の変異と離散戸田格子

野邊厚

応用力学研究所研究集会報告 No.28AO-S6 25 - 30 2017.03 [Refereed]
一般化戸田格子の超離散化

野邊厚

応用力学研究所研究集会報告 No.26AO-S 1 - 7 2015.03 [Refereed]
超離散周期戸田格子の幾何学的実現

野邊厚

応用力学研究所研究集会報告 No.25AO-S2 114 - 120 2014.03 [Refereed]
超楕円曲線に付随する倍角写像力学系

野邊厚

応用力学研究所研究集会報告 No.24AO-S3 77 - 82 2013.03 [Refereed]
Addition in Jacobians of hyperelliptic curves and the periodic discrete Toda lattice

NOBE Atsushi

RIAM Report No.23AO-S7 84 - 89 2012.03 [Refereed]
The group law on the tropical Hesse pencil

Atsushi Nobe

1111.0131v1 2011.11

　View Summary

We show that the addition of points on the tropical Hesse curve can be
realized via the intersection with a tropical line. Then the addition formula
for the tropical Hesse curve is reduced from those for the level-three theta
functions through the ultradiscretization procedure. A tropical analogue of the
Hessian group, the group of linear automorphisms acting on the Hesse pencil, is
also investigated; it is shown that the dihedral group of degree three is the
group of linear automorphisms acting on the tropical Hesse pencil.

DOI

Scopus
On the addition formula for the tropical Hesse pencil (Diversity of the Theory of Integrable Systems)

NOBE Atsushi

RIMS Kokyuroku 1765 188 - 208 2011.09 [Invited]

Authorship：Lead author, Corresponding author

　View Summary

We give the addition formula for the tropical Hesse pencil, which is the 
tropicalization of the Hesse pencil parametrized by the level-three theta 
functions, via those for the ultradiscrete theta functions. The ultradiscrete 
theta functions are reduced from the level-three theta functions through the 
procedure of ultradiscretization by choosing their parameters appropriately. 
The parametrization of the level-three theta functions firstly introduced in 
\cite{KKNT09} gives an explicit correspondence between the amoeba of the real 
part of the Hesse cubic curve and the tropical Hesse curve. Moreover, through 
the parametrization, we obtain the subtraction-free forms of the addition 
formulae for the level-three theta functions, which lead to the addition 
formula for the tropical Hesse pencil in terms of the ultradiscretization. 
Using the parametrization, the tropical counterpart of the Hesse configuration 
is also given.

CiNii
A tropical analogue of the Hessian group

Atsushi Nobe

RIAM Report No.22AO-S8 (2011) 37-42 2011.04

　View Summary

We investigate a tropical analogue of the Hessian group $G_{216}$, the group
of linear automorphisms acting on the Hesse pencil. Through the procedure of
ultradiscretization, the group law on the Hesse pencil reduces to that on the
tropical Hesse pencil. We then show that the dihedral group $\mathcal{D}_3$ of
degree three is the group of linear automorphisms acting on the tropical Hesse
pencil.
周期写像を用いた高次保存量を持つ可積分方程式の生成

田中宏典, 津田照久, 野邊厚, 松木平淳太

応用力学研究所研究集会報告 No.21ME-S7 166 - 172 2010.03
QRT系から生成される高次保存量を持つ2階差分方程式

田中宏典, 野邊厚, 松木平淳太

応用力学研究所研究集会報告 No.20ME-S7 175 - 181 2009.03
ブラウン写像について

応用力学研究所研究集会報告 20ME-S7 35-42 2009
トロピカル楕円曲線と超離散QRT系

九州大学応用力学研究所研究集会報告 19ME-S2 Article No.6 2008
可逆エレメンタリーセルオートマトンの可積分性について(可積分系数理の眺望)

野邊厚, 由良文孝

数理解析研究所講究録 1541 178 - 191 2007.04 [Invited]

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リーマンテータ関数の超離散化とその可積分系への応用

九州大学応用力学研究所研究集会報告 No.17ME-S2 2006
Sine-Gordon方程式のある超離散化

九州大学応用力学研究所研究集会報告 15ME-S3 88-93 2004
周期境界をもつセルオートマトンの可逆性について

野邊厚

九州大学応用力学研究所研究集会報告 15ME-S3 82-87 2004
超離散QRT系と扇について

九州大学応用力学研究所研究集会報告 14ME-S7 165-170 2003
Stable dynamical systems associated with cellular automata

Doctoral thesis, Univ. of Tokyo 2002
セルオートマトンと微分方程式 (離散可積分系の応用数理)

野辺厚

数理解析研究所講究録 1098 14 - 22 1999.04 [Invited]

Authorship：Lead author, Corresponding author

　View Summary

RIMS研究集会「離散可積分系の応用数理」

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Books and Other Publications

``Reversibility of Cellular Automata'' in Cellular Automata, ed. Thomas M. Li

Atsushi Nobe, Fumitaka Yura( Part： Contributor)

Nova Science Publishers 2011
Expansion of Integrable Systems, RIMS Kokyuroku Bessatsu B13

Atsushi Nobe( Part： Edit)

Research Institute for Mathematical Sciences, Kyoto University 2009 [Refereed]

Presentations

Integrable discretization of the SIR model with vaccination

Atsushi Nobe [Invited]

Presentation date： 2023.09

Event date：
2023.09

　

　
Integrable discretization of the SIR model with vaccination and the parametric solution

Atsushi Nobe

Presentation date： 2023.03

Event date：
2023.03

　

　
Integrability of the SIR model with vaccination and its discretization

Atsushi Nobe

Presentation date： 2022.11

Event date：
2022.11

　

　
Discretization of the SIR epidemic models preserving conserved quantities

Atsushi Nobe

Presentation date： 2022.09

Event date：
2022.09

　

　
Integrable discretization of SIR epidemic models

Atsushi Nobe [Invited]

FPSE Faculty Workshop

Presentation date： 2022.05
A family of integrable and non-integrable difference equations arising from cluster algebras

Atsushi Nobe [Invited]

Cluster Algebras 2019 (Research Institute for Mathematical Science)

Presentation date： 2019.06

Event date：
2019.06

　

　
クラスター代数とカオス

野邊厚 [Invited]

研究集会「離散力学系と組合せ論」 (津田塾大学小平キャンパス) 津田塾大学数学・計算機科学研究所

Presentation date： 2019.02

Event date：
2019.02

　

　
Chaos in cluster algebras

NOBE Atsushi [Invited]

Presentation date： 2018.09

Event date：
2018.09

　

　
Mutations of cluster algebras and discrete integrable systems

NOBE Atsushi [Invited]

The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory (Georgia Center for Continuing Education University of Georgia, Athens, GA)

Presentation date： 2017.03

Event date：
2017.03

-

2017.04
Tropical elliptic curves and discrete integrable systems

NOBE Atsushi [Invited]

Workshop on Tropical geometry - basic and applications

Presentation date： 2017.01
Ultradiscretization of generalized Toda lattices

NOBE Atsushi [Invited]

Workshop on Integrable and nonintegrable lattice models: theory and computation

Presentation date： 2016.07
セルオートマトンの幾何学 -- トロピカル超楕円曲線とソリトンセルオートマトン --

野邊厚 [Invited]

早稲田大学理工学術院数物コロキウム

Presentation date： 2013.10
周期離散戸田格子の幾何学的実現とそのトロピカル化

野邊厚 [Invited]

RIMS研究集会「非線形離散可積分系の新展開」

Presentation date： 2013.09
A geometric realization of the periodic discrete Toda lattice and its tropicalization

NOBE Atsushi

China-Japan Joint Workshop on Integrable Systems 2013

Presentation date： 2013.03
Integrable maps arising from the addition on tropical elliptic curves

NOBE Atsushi [Invited]

Mini-Workshop: Tropical geometry and ultradiscrete integrable systems, NEEDS 2012

Presentation date： 2012.07
Tropical Jacobian of a hyperelliptic curve

NOBE Atsushi [Invited]

International Workshop on Tropical and Quantum Geometries

Presentation date： 2012.02
トロピカル超楕円曲線のJacobi多様体における加法

野邊厚 [Invited]

RIMS研究集会「可積分系数理の進化」

Presentation date： 2011.08
セルオートマトンとトロピカル幾何学

野邊厚 [Invited]

明治大学第9回現象数理若手シンポジウム「セルオートマトンは現象数理学の武器となりうるか？」

Presentation date： 2011.02
トロピカルHesse曲線の加法公式について

野邊厚 [Invited]

RIMS研究集会「可積分系数理の多様性 - Diversity of the Theory of Integrable Systems -」

Presentation date： 2010.08
Tropicalization of a solvable chaotic map associated with the Hesse cubic curve

野邊厚 [Invited]

研究集会「離散・超離散系の課題」

Presentation date： 2010.03
non-QRT mappingについて

野邊厚 [Invited]

龍谷数理科学セミナー

Presentation date： 2009.12
トロピカル楕円曲線の加法の定める力学系について

野邊厚 [Invited]

語ろう数理解析第７４回セミナー

Presentation date： 2009.12
超離散可積分系とトロピカル楕円曲線

野邊厚 [Invited]

デジタル解析学セミナー

Presentation date： 2009.07
Hesseの３次曲線のトロピカル化と可解カオス写像

野邊厚 [Invited]

研究集会「トロピカル幾何と超離散系の展開」

Presentation date： 2009.03
Tropical curves, integrable maps and solvable chaos

NOBE Atsushi [Invited]

Workshop ``Crystals and Tropical Combinatorics''

Presentation date： 2008.08
トロピカル楕円曲線と超離散QRT系

野邊厚 [Invited]

九州可積分系セミナー

Presentation date： 2007.12
トロピカル楕円曲線と超離散QRT系

野邊厚 [Invited]

古典解析セミナー

Presentation date： 2007.11
可逆エレメンタリーセルオートマトンの可積分性について

野邊厚 [Invited]

RMS研究集会「可積分系数理の眺望」

Presentation date： 2006.08
離散非線形Schr\"odinger方程式のソリトン解について

野邊厚 [Invited]

研究集会「シミュレーションの数理科学」

Presentation date： 2004.05
セルオートマトンと微分方程式

[Invited]

短期共同研究「離散可積分系の応用数理」

Presentation date： 1998.07
アフィン型ミューテーションの周期性と線形化

野邊厚, 松木平淳太

日本応用数理学会2020年度年会 (オンライン) 日本応用数理学会

Presentation date： 2020.09

Event date：
2020.09

　

　
$A^{(1)}_N$型ミューテーションの線形化

野邊厚, 松木平淳太

第16回日本応用数理学会研究部会連合発表会 (中央大学) 日本応用数理学会

Presentation date： 2020.03

Event date：
2020.03

　

　
$A^{(1)}_N$型ミューテーションの可積分性について

野邊厚, 松木平淳太

応用力学研究所研究集会「非線形波動研究の多様性」 (九州大学応用力学研究所)

Presentation date： 2019.11

Event date：
2019.10

-

2019.11
アフィン型ミューテーションの可積分性について

野邊厚, 松木平淳太

日本応用数理学会2019年度年会 (東京大学駒場キャンパス) 日本応用数理学会

Presentation date： 2019.09

Event date：
2019.09

　

　
クラスター代数とカオス

野邊厚

第15回日本応用数理学会研究部会連合発表会

Presentation date： 2019.03
$A^{(2)}_2$型クラスター代数の生成元について

野邊厚

日本応用数理学会 2018年度年会

Presentation date： 2018.09
ランク2ミューテーションの不変曲線について

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の新潮流 -- 理論とその応用 -- 」 (九州大学応用力学研究所)

Presentation date： 2017.11
A₂⁽²⁾型マトリックスミューテーションの幾何学

野邊厚

日本応用数理学会2017年度年会 (武蔵野大学有明キャンパス) 日本応用数理学会

Presentation date： 2017.09

Event date：
2017.09

　

　
$A^{(1)}_1$型クラスター代数の変異と離散戸田格子

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の深化と展開」 (九州大学応用力学研究所)

Presentation date： 2016.11

Event date：
2016.11

　

　
$A^{(1)}_1$型クラスター代数の変異と離散戸田格子

野邊厚

日本数学会 2016年度秋季総合分科会

Presentation date： 2016.09
クラスター代数とセルオートマトン

野邊厚, 間田潤

Presentation date： 2016.09

Event date：
2016.09

　

　
クラスター代数とセルオートマトン

野邊厚 [Invited]

RIMS研究集会「可積分系数理の現状と展望」 (京都大学数理解析研究所)

Presentation date： 2016.09

Event date：
2016.09

　

　
クラスター代数とQRT系

野邊厚, 間田潤

第12回日本応用数理学会研究部会連合発表会 (神戸学院大学) 日本応用数理学会

Presentation date： 2016.03

Event date：
2016.03

　

　
戸田格子とクラスター代数

野邊厚

2015年度応用数学合同研究集会

Presentation date： 2015.12
Toda lattice, QRT maps, and cluster algebras

野邊厚

MIMS共同研究集会「可積分系が拓く現象数理モデル」

Presentation date： 2015.11
戸田格子とクラスター代数

野邊厚 [Invited]

数学教室談話会 (東京理科大学理工学部)

Presentation date： 2015.11
離散戸田格子とQRT系

野邊厚

Presentation date： 2015.09

Event date：
2015.09

　

　
周期戸田格子を用いて実現可能な一般化戸田格子について

野邊厚

第11回日本応用数理学会研究部会連合発表会

Presentation date： 2015.03
一般化戸田格子の超離散化

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の現状—課題と展望を探る—」

Presentation date： 2014.10
C⁽¹⁾_N型超離散戸田格子と戸田型セルオートマトン

野邊厚

Presentation date： 2014.09

Event date：
2014.09

　

　
$C^{(1)}_N$型離散戸田格子の幾何学的実現

野邊厚

第10回日本応用数理学会研究部会連合発表会

Presentation date： 2014.03
周期離散戸田格子の幾何学的実現とトロピカル化

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の拡がり」

Presentation date： 2013.11
周期箱玉系の幾何学的実現

野邊厚

Presentation date： 2013.09

Event date：
2013.09

　

　
周期箱玉系の幾何学的実現

野邊厚

日本数学会 2013年度秋季総合分科会

Presentation date： 2013.09
周期箱玉系の幾何学的実現

野邊厚

日本応用数理学会 2013年度年会 (アクロス福岡) 日本応用数理学会

Presentation date： 2013.09

Event date：
2013.09

　

　
超楕円曲線に付随する可解カオス系

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の最前線 -- 構造と現象の多様性 -- 」

Presentation date： 2012.11
離散周期戸田格子の幾何学的実現

野邊厚

第9回日本応用数理学会研究部会連合発表会

Presentation date： 2012.03
超楕円曲線の加法と可積分系 -- 差分および超離散 --

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の進展 -- 現象と数理の相互理解 -- 」

Presentation date： 2011.10
トロピカルHesse曲線の加法公式について

野邊厚

日本数学会2011年度年会

Presentation date： 2011.03
The group law on the tropical Hesse pencil

野邊厚

九州大学応用力学研究所研究集会「非線形波動研究の新たな展開 -- 現象とモデル化 -- 」

Presentation date： 2010.10
Hesseの3次曲線に附随する可解カオス力学系の超離散化

梶原健司, 金子昌信, 野邊厚, 津田照久

超幾何方程式研究会2010

Presentation date： 2010.01
高次保存量を持つ2階可積分方程式について

松木平淳太, 田中宏典, 野邊厚, 津田照久

九州大学応用力学研究所研究集会「非線形波動研究の現状と将来 -- 次の１０年への展望」 (九州大学応用力学研究所)

Presentation date： 2009.11

Event date：
2009.11

　

　
カオス系の超離散化

梶原健司, 金子昌信, 野邊厚, 津田照久

九州大学応用力学研究所研究集会「非線形波動研究の現状と将来 -- 次の１０年への展望」

Presentation date： 2009.11
周期写像を用いた高次保存量を持つ可積分方程式の生成

田中宏典, 津田照久, 野邊厚, 松木平淳太

日本応用数理学会2009年度年会 (大阪大学豊中キャンパス) 日本応用数理学会

Presentation date： 2009.09

Event date：
2009.09

　

　
高次保存量をもつ2次元可積分写像

野邊厚, 松木平淳太, 田中宏典, 津田照久

Presentation date： 2009.09

Event date：
2009.09

　

　
Hesseの3次曲線に附随する2次元可解カオス系の超離散化

梶原健司, 金子昌信, 野邊厚, 津田照久

日本数学会2009年秋季総合分科会 (大阪大学豊中キャンパス) 日本数学会

Presentation date： 2009.09

Event date：
2009.09

　

　
Ultradiscretization of solvable chaotic maps and the tropical geometry

Kenji Kajiwara, Atsushi Nobe, Teruhisa Tsuda

Workshop ``Geometric aspects of discrete and ultra-discrete integrable systems'' (University of Glasgow, Glasgow)

Presentation date： 2009.04
QRT系から生成される高次保存量を持つ2階差分方程式

田中宏典, 松木平淳太, 野邊厚, 津田照久

第5回日本応用数理学会研究部会連合発表会 (京都大学) 日本応用数理学会

Presentation date： 2009.03
Hesseの３次曲線のトロピカル化と可解カオス写像

野邊厚

第4回非線形テクノサイエンス講演会

Presentation date： 2009.03
QRT系から生成される高次保存量を持つ2階差分方程式

田中宏典, 野邊厚, 松木平淳太

九州大学応用力学研究所研究集会「非線形波動の数理と物理」

Presentation date： 2008.11
ブラウン写像について

野邊厚

九州大学応用力学研究所研究集会「非線形波動の数理と物理」

Presentation date： 2008.11
1次元可解カオス系の超離散化とトロピカル幾何

梶原健司, 野邊厚, 津田照久

日本数学会2008年秋季総合分科会

Presentation date： 2008.09
Ultradiscretization of one-dimensional solvebale chaotic systems and tropical geometry

KAJIWARA Kenji, NOBE Atsushi, TSUDA Teruhisa

Presentation date： 2008.09

Event date：
2008.09

　

　
超離散QRT写像とトロピカル楕円曲線

野邊厚

日本物理学会2008年春季大会

Presentation date： 2008.03
トロピカル楕円曲線と超離散QRT系

野邊厚

研究集会「ソリトンの数理とその応用」 (湯田簡易保険保養センター)

Presentation date： 2007.12
トロピカル楕円曲線と超離散QRT系

野邊厚

九州大学応用力学研究所研究集会「戸田格子４０周年非線形波動研究の歩みと展望」

Presentation date： 2007.11
線形化可能セルオートマトン

野邊厚, 由良文孝

日本物理学会2007年春季大会

Presentation date： 2007.03
楕円関数の超離散化とその可積分系への応用

野邊厚

日本物理学会2006年春季大会

Presentation date： 2006.03
リーマンテータ関数の超離散化とその可積分系への応用

野邊厚

九州大学応用力学研究所研究集会「非線形波動および非線形力学系の現象と数理」

Presentation date： 2005.11
Ultradiscrete Theta Functions

野邊厚

研究集会「離散・超離散可積分系の数理とその応用」 (ホテル大佐渡)

Presentation date： 2005.10
戸田型セルオートマトンの周期波解

野邊厚

日本数学会秋季総合分科会

Presentation date： 2005.09
非正値変数の超離散化

礒島伸, 村田実貴生, 野邊厚, 薩摩順吉

日本数学会秋季総合分科会

Presentation date： 2004.09
周期境界をもつセルオートマトンの可逆性について

野邊厚

九州大学応用力学研究所研究集会「非線形波動および非線形力学系の数理とその応用」

Presentation date： 2003.11
Sine-Gordon方程式のある超離散化

礒島伸, 村田実貴生, 野邊厚, 薩摩順吉

九州大学応用力学研究所研究集会「非線形波動および非線形力学系の数理とその応用」

Presentation date： 2003.11
Ultradiscrete QRT system and fans

NOBE Atsushi

National Congress of Theoretical and Applied Mechanics, Japan

Presentation date： 2003.01

　View Summary

超離散QRT系とは、８個のパラメータを含む区分線形写像で与えられる二次元の力学系である。この力学系はQRT系とよばれる二次元の離散可積分系から超離散化により得られ、一つの保存量をもつので可積分である。保存量を与える区分線形方程式から平面の分割が定まるが、パラメータがある関係を満たすときにはその分割から扇を得ることが出来る。このような場合、時間発展は扇の写像となるので超離散QRT系は扇の力学系とみなすことができ、任意の初期値に対して一定周期の解をもつことが示せる。扇の力学系となる場合は全部で２４通りあり、それらの解の周期は２，３，４，５，６，７，８および無限大のいずれかである。
超離散QRT系と扇について

野邊厚

九州大学応用力学研究所研究集会「非線形波動および非線形力学系に関する最近の話題」

Presentation date： 2002.11
超離散QRT系

野邊厚

日本物理学会2002年秋季大会 (中部大学) 日本物理学会

Presentation date： 2002.09
Stable difference equations associated with elementary cellular automata

NOBE Atsushi, SATSUMA Junkichi, TOKIHIRO Tetsuji

4th International Interdisciplinary Meeting on ``Symmetries and Integrability of Difference Equations (SIDE4)"

Presentation date： 2000.11
フラクタル構造を示すある安定な差分方程式

野邊厚, 薩摩順吉, 時弘哲治

日本応用数理学会2000年秋の分科会

Presentation date： 2000.10
Cellular Automata and Differencial Equations

SATSUMA Junkichi, TOKIHIRO Tetsuji, NOBE Atsushi

Meeting Abstracts of the Physical Society of Japan

Presentation date： 1998.09
Cellular Automata and Diffution Equations

NOBE A, SATSUMA J, TOKIHIRO T, OHTA Y

Meeting Abstracts of the Physical Society of Japan

Presentation date： 1998.09
セルオートマトンと微分方程式の対応

野辺厚

日本応用数理学会1998年度年会 (早稲田大学) 日本応用数理学会

Presentation date： 1998.09

Event date：
1998.09

　

　
粒子法を用いた海岸での砕波の数値解析

野邊厚, 越塚誠一, 岡芳明

第9回数値流体力学シンポジウム

Presentation date： 1995.12

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Research Projects

Integrability of dynamical systems that exhibit Laurent phenomena and positivity and their algebraic entropy

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

Project Year :

2020.04

-

2025.03

Atsushi Nobe
Studies on discrete integrable systems via tropical algebraic curves

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

Project Year :

2014.04

-

2018.03

Nobe Atsushi

　View Summary

First we considered Lie-algebraic generalizations of the Toda lattice. We realized the generalized Toda lattices of types A(2)2N, C(1)N and D(2)N as the sub-dynamical systems of the Toda lattices of types A(1)2N-1, A(1)2N, A(1)2N+1, respectively. We also obtained their tropical analogues.
Next we studied a tropical analogue of the Hessian group, which is the group of linear automorphisms acting on the Hesse pencil. We then obtained the dihedral group of degree 3 as the group of linear automorphisms acting on the tropical analogue of the Hesse pencil.
We moreover investigated the cluster algebras of rank 2 from the view point of discrete integrable systems. We gave the conserved quantities of the dynamical systems arising from the cluster algebras of types A1*A1, A2, B2, G2, A(1)1 and A(2)2. We also showed direct connections between the dynamical systems and the Mordell-Weil groups of the elliptic curves arising via the conserved quantities of the dynamical systems.
Studies on the geometry of discrete integrable systems and solvable chaotic systems

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

Project Year :

2010.04

-

2014.03

NOBE Atsushi

　View Summary

We establish a geometric realization of the periodic discrete Toda lattice of an arbitrary system size by using the intersection of its spectral curve and other two curves. This realization is reduced from the addition of points on the symmetric product of the spectral curve, and the curves in the realization are concretely given by using the conserved quantities of the periodic discrete Toda lattice.
We also establish a geometric realization of the ultradiscrete periodic Toda lattice of an arbitrary system size, which is an ultradiscrete analogue of the periodic discrete Toda lattice, via tropical plane curves. The tropical curves in the realization, one of which is the spectral curve of the ultradiscrete periodic Toda lattice (a tropical hyperelliptic curve), are also given by using the conserved quantities of the ultradiscrete periodic Toda lattice.
Tropical geometry and integrable systems

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

Project Year :

2007.04

-

2010.03

NOBE Atsushi

　View Summary

We construct an 8-parameter family of two-dimensional bi-piecewise linear maps in terms of the addition of the points on tropical elliptic curves, and obtain the general solution for each member of the family. Through the ultradiscretization procedure, we associate the family with an 18-parameter family of two-dimensional birational maps called the QRT map including its general solutions. Similarly, applying such technique to solvable chaotic maps induced from the duplication of points on tropical elliptic curves, we obtain their general solution and clarify the correspondence to the rational maps induced from the duplication of points on elliptic curves. Moreover, we show that there exists a family of cellular automata each of which has a property called the linearizability. Then we obtain a formula concerning the fundamental period with respect to the time evolution of the family imposing periodic boundary conditions.
ソリトン方程式の超離散解析

日本学術振興会科学研究費助成事業若手研究(B)

Project Year :

2005.04

-

2007.03

野邊厚

　View Summary

平成18年度は次に挙げる研究を行った.
(1)Jacobiの楕円関数の超離散化およびその可積分系への応用
(2)双曲型sin関数を用いた変形KdV方程式の超離散化
(3)可逆エレメンタリーセルオートンマトンの可積分性
Jacobiの楕円関数の超離散化については既に先行する研究があったが,そこでは母数の超離散極限値が1の場合しか考慮されていなかった.そこで,(1)において,母数の超離散極限値全て(0,1,∞)について考察し,それぞれの場合におけるJacobiの楕円関数の超離散化を求めた.いずれの場合にもいても三つの楕円関数のうちの二つのみ周期関数となり,残りの一つは定数関数となることが示された.また,このようにして得られた超離散sn関数を用いて超離散QRT系の解を構成した.
これまでの超離散化手法では零点を含むような関数を直接超離散化することはできなかった.しかし,(2)において,双曲型sin関数を用いた新しい超離散化手法を導入し,零点を含むような関数も超離散化可能であることを示した.さらに,この超離散化手法を用いて,変形KdV方程式のソリトンと反ソリトンの衝突が超離散系においても再現されることを示した.
これまで箱玉系以外の可積分セルオートマトンはほとんど発見されていなかった.そこで,可逆セルオートマトンの中から可積分セルオートマトンを発見すべく,(3)において,可逆エレメンタリーセルオートマトンについて詳しく調べ,その中に線形化可能なセルオートマトンが存在することを示した.すなわち,初期状態を適切に分割することにより,周期境界をもつある非線形可逆セルオートマトンの初期値問題が,ある線形系の初期値境界値問題に帰着されることを示した.また,そのような線形化可能セルオートマトンの基本周期の明示公式を得た.
グラフ上のchip-firingと可積分系

千葉大学研究費獲得促進プログラム（多様型Ｂ）

Project Year :

2019.04

-

2020.03
Representation Theory of Quantum Groups and Integrable Systems

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2008.04

-

2011.03

OKADO Masato, NOBE Atsushi, SAKAMOTO Reiho, NAKASHIMA Toshiki, FOURIER Ghislain, LECOUVER Cedric, MISRA Kailash C., SCHILLING Anne, SHIMOZONO Mark

　View Summary

We proved the existence of the crystal bases in the sense of Kashiwara for KR modules of the quantum groups of nonexceptional affine type, and determined their crystal structure. We then settled affirmatively the conjecture on the perfectness of KR crystals. By examining the combinatorial structure of the KR crystal, we solved the X=K conjecture by Shimozono-Zabrocki. On the other hand, we showed that a similar relation holds for the generating functions of rigged configurations, thereby solving the X=M conjecture for all nonexceptional affine types when the rank is sufficiently large.
8th International Conference on ``Symmetry and Integrability of Difference Equations (SIDE8)'

日本学術振興会国際学会等派遣事業

Project Year :

2008.04

-

2009.03

野邊厚
Integrable Systems and Combinatorial Representation Theory

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2006.04

-

2009.03

OKADO Masato, YAMADA Yasuhiko, KUNIBA Atsuo, NOBE Atsushi

　View Summary

During the period of research project, we mainly obtained the following results.
1. [Affine geometric crystal]
In collaboration with M. Kashiwara and T. Nakashima, we constructed geometric crystals associated to nonexceptional affine Lie algebras. We confirmed that the ultra-discrete limit of these geometric crystals coincide with the limit of previously known perfect crystals. Moreover, except type C, we obtained explicit formulas for birational maps, called tropical R maps, that satisfy the Yang-Baxter equation.
2. [Existence of crystal bases of the KR modules for nonexceptional types]
There was a conjecture saying that any finite-dimensional representation of a quantum affirm algebra that has an integer multiple of a level 0 fundamental weight as highest weight (KR module) has a crystal base. We solved this conjecture for all affine Lie algebras of nonexceptional types. In collaboration with A. Schilling, we also proved that the crystals of type B^<(1)>_n, D^<(1)>_n, and A^<(2)>_<2n-1> are isomorphic to the combinatorial crystals recently constructed by Schilling.
3. [Construction of the coherent family of perfect crystals for exceptional types]
In collaboration with M. Kashiwara, K.C. Misra and D. Yamada, we revealed the crystal structure of the perfect crystals associated to the exceptional affine lie algebra D^<(3)>_4 at any level.
Mathematical Study of the Nonlinear Partial Differential Equations Arising in the Statistical Mechanics

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2004.04

-

2009.03

SUZUKI Takashi, MATSUMURA Akitaka, ODANAKA Shinji, NAWA Hayato, WADA Takeshi, NOBE Atsushi

　View Summary

In this research project, we provided a unified analysis to the critical phenomena, arising in the solution to the partial differential equation provided with the nonlinearity due to the self-interaction and the non-equilibrium. We formulate the mathematical principle across the hierarchy to control several phenomena common to many nonlinear problems. These problems are the mean field of stationary turbulence in high energy and of gauge field, Ricci flow, nonlinear parabolic equations, self-interacting fluids, material-energy transport, tumor growth, and nonlinear thermodynamics. Among them, we formulated the system of chemotaxis, derived in the context of th the formation of spores of the cellular slime molds, as the fundamental equation of the material transport subject to the mass conservation and the decrease of the free energy in the thermodynamics called Smoluchowski-Poisson equation and clarified the quantized blowup mechanism by developing various new methods of analysis. Then, we obtained the notions of the blowup envelope, formulation of the stationary and non-stationary states by the dual variation, hierarchical control of the stationary states upon the non-stationary states, which motivates the study on the structure and the stability of the set of stationary solutions of phenomenological equations concerning the critical phenomena arising in the non-equilibrium thermodynamics, formation of sub-collapses and the collision of collapses in the mean field equation arising in the gauge theory and turbulent theory and the semilinear parabolic equation with the critical Sobolev exponent, deformed quantization in the nonlinear parabolic equation with non-local term and the normalized Ricci flow, and mass quantization in higher dimensions.
脈管形成の数理モデルに関する解析的研究

日本学術振興会科学研究費補助金萌芽研究

Project Year :

2004.04

-

2006.03

鈴木貴

　View Summary

輸送現象に関して、質量保存と自由エネルギー減衰を実現する数値スキームを開発し、数値解析によってその正当性を数学的に確立した。これに基づいて血管新生に関するRascle方程式の数値シミュレーションを行い、解の集中や減衰する進行波が現れることを発見し、これらを視覚化することに成功した。次に脈管形成に関するOthmer-Stevens方程式について数学解析を行い、空間1次元で解が常に時間大域的に存在すること、原論文で提起されていた問題であるcollapseの形成が空間多次元で起こりうることを証明した。3種の連立系であるAnderson-Chaplain方程式も取り上げ、ヘルダー空間の中で基本定理を確立するとともに、近平衡でLyapunov関数が存在することを見出した。この点に関連して高分子科学の研究者と研究討論を行い、Prigogineの非平衡散逸系での自己組織化を数学的に確立する原理として、進行波とLyapunov関数が混在することがキーポイントであることが明確となり、今後の研究の基本的な指針が得られた。最後に固形腫瘍の数理モデルをとりあげ、強最大原理を用いて解の基本定理を確立するアイデアを得た。計測医療に関わる数値解法アルゴリズムでは、引き続き平行最適化によるプログラミングと数値実験を行い、特に逆源探索問題において逆源の数を時系列データや先験的情報なしに、正確に決める点で強力な汎用性があることを明らかにした。これらの成果は日本応用数理学会年会での数理医学研究部会オーガナイズドセッション、研究部会連合発表会、また藤田保健衛生大学での医学数学シンポジウム等により報告し、関連する研究者と研究討論を行った。また生体間で生命のとる戦略の数理的な意味付けを図るため、ゲーム理論を適用する研究の調査を行い、樹木成長に関する数理モデルの開発と解析、数値シミュレーションを行うための準備的な研究を行った。
Combinatorial Study of Crystal Bases and its Application to Discrete Integrable Systems

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research

Project Year :

2004.04

-

2005.03

OKADO Masato, OGAWA Toshiyuki, NOBE Atsushi, KUNIBA Atsuo

　View Summary

(i) Geometric crystal
Geometric crystal, indtoduced axiomatically by Berenstein and Kazhdan, was consttructed explicitly for type D^(1)_n of Kac-Moody Lie algebras. By using a matrix realization of this geometric crystal, we constructed a birational mapping on the product (tropical R) commuting with the action of the geometric crystal and also,showed that it satisfies the Yang-Baxter equation. It is believed that a geometric crystal exists arrogated to each vertex of the Dynkin diagram corresponding to the Lie algebra. For type D, there are n vertices, so it is expected that there are n distinct geometric crystals. During the last yeah we calculated the geometric crystal for k=2 by Mathematics in collaboration with Masaki Kashiwara at RIMS, Kyoto University. Data is huge and no way to print it out. However to write it down in a meaningful manner is, besides with the extension to the case when k is greater than 2, becomes a future problem.
(ii)Crystal and soliton cellular automaton associated to an exceptional acne Lie algebra
Coordinate representation of a series of finite crystals for an exceptional affine Lie algebra D_4^(3) is given and the zero action is explicitly obtained. Moreover we constructed a cell automaton corresponding to this series of crystals and determined the internal degree of freedom of the solitons appearing in the system and the scattering rule of two solitons.
(iii)Box-ball system with reflecting end
We have extended the box-ball system, an important example of ultra discrete integrable systems, to the case with one reflecting end. Similar to the usual box-ball system, it has an infinite family of commuting time evolutions and conserved quantities associated to each time evolution. We also defined soliton states and described the reflection rule of one soliton and the scattering rule of two solitons in terms of combinatorics of crystals.

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Misc

アフィン型ミューテーションの可積分性について

野邊厚, 松木平淳太

日本応用数理学会年会講演予稿集(CD-ROM) 2019 560‐561 2019.08

J-GLOBAL
A₂⁽²⁾型クラスター代数の生成元について

野邊厚

日本応用数理学会年会講演予稿集(CD-ROM) 2018 241‐242 2018.09

J-GLOBAL
A₂⁽²⁾型マトリックスミューテーションの幾何学

野邊厚

日本応用数理学会年会講演予稿集(CD-ROM) 2017 467‐468 2017.09

J-GLOBAL
クラスター代数とセルオートマトン

野邊厚, 間田潤

日本応用数理学会年会講演予稿集(CD-ROM) 2016 ROMBUNNO.9GATSU13NICHI,09:30,1A,3 2016

J-GLOBAL
離散戸田格子とQRT系

野邊厚

日本応用数理学会年会講演予稿集(CD-ROM) 2015 ROMBUNNO.9GATSU9NICHI,09:30,E,4 2015.09

J-GLOBAL
C⁽¹⁾_N型超離散戸田格子と戸田型セルオートマトン

野邊厚

日本応用数理学会年会講演予稿集(CD-ROM) 2014 ROMBUNNO.9GATSU4NICHI,09:30,D,2 2014.08

J-GLOBAL
周期箱玉系の幾何学的実現

野邊厚

日本物理学会講演概要集 68 ( 2 ) 238 2013.08

J-GLOBAL
周期箱玉系の幾何学的実現

野邊厚

日本応用数理学会年会講演予稿集(CD-ROM) 2013 ROMBUNNO.9051 2013

J-GLOBAL
A geometric realization of the periodic box-ball system

Nobe Atsushi

Meeting Abstracts of the Physical Society of Japan 68 ( 0 ) 238 - 238 2013

CiNii
周期写像を用いた高次保存量を持つ可積分方程式の生成

田中宏典, 津田照久, 野邊厚, 松木平淳太

日本応用数理学会年会講演予稿集 2009 251 - 252 2009.09

J-GLOBAL
On the Brown map

Reports of RIAM symposium 20ME-S7 35-42 2009
27aQC-3 Two-dimensional integrable maps which possess invariants of high degree

Nobe A., Matsukidaira J., Tanaka H., Tsuda T.

Meeting Abstracts of the Physical Society of Japan 64 ( 0 ) 201 - 201 2009

DOI CiNii
Tropical elliptic curves and ultradiscrete QRT maps

Reports of RIAM Symposium 19ME-S2 Article No.6 2008
24pWD-4 超離散QRT写像とトロピカル楕円曲線(古典・量子可積分系(数値計算アルゴリズムを含む),領域11,統計力学,物性基礎論,応用数学,力学,流体物理)

野邊厚

日本物理学会講演概要集 63 ( 0 ) 294 - 294 2008

CiNii
18aWB-3 Linearizable cellular automata

Nobe Atsushi, Yura Fumitaka

Meeting Abstracts of the Physical Society of Japan 62 ( 0 ) 238 - 238 2007

CiNii
Ultradiscretization of the Riemann theta functions and its applications to integrable systems

RIAM Symposium "Phenomena and Mathematical Theory of Nonlinear Waves and Nonlinear Dynamical Systems" No.17ME-S2 2006
27aXE-5 Ultradiscretization of elliptic functions and its applications to integrable systems

Nobe Atsushi

Meeting Abstracts of the Physical Society of Japan 61 ( 0 ) 254 - 254 2006

CiNii
東京大学大学院数理科学研究科平成15年度研究成果報告書

野邊厚

133 - 134 2004.03

Authorship：Lead author, Corresponding author
Ultradiscrete QRT system and fans

NOBE Atsushi

NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan 52 ( 0 ) 149 - 149 2003

　View Summary

超離散QRT系とは、８個のパラメータを含む区分線形写像で与えられる二次元の力学系である。この力学系はQRT系とよばれる二次元の離散可積分系から超離散化により得られ、一つの保存量をもつので可積分である。保存量を与える区分線形方程式から平面の分割が定まるが、パラメータがある関係を満たすときにはその分割から扇を得ることが出来る。このような場合、時間発展は扇の写像となるので超離散QRT系は扇の力学系とみなすことができ、任意の初期値に対して一定周期の解をもつことが示せる。扇の力学系となる場合は全部で２４通りあり、それらの解の周期は２，３，４，５，６，７，８および無限大のいずれかである。

CiNii
Ultradiscrete QRT System

Nobe Ataushi

Meeting Abstracts of the Physical Society of Japan 57 ( 0 ) 275 - 275 2002

CiNii
An attempt to relate cellular automata and differential equations by means of ultradiscretization

NOBE A.

Master Thesis of University of Tokyo 1999

CiNii
Cellular Automata and Differencial Equations

SATSUMA Junkichi, TOKIHIRO Tetsuji, NOBE Atsushi

Meeting Abstracts of the Physical Society of Japan 53 ( 0 ) 914 - 914 1998

CiNii
Cellular Automata and Diffution Equations

NOBE A, SATSUMA J, TOKIHIRO T, OHTA Y

Meeting Abstracts of the Physical Society of Japan 53 ( 0 ) 914 - 914 1998

CiNii

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Syllabus

Fresh-Start Seminar [J] 49

School of Political Science and Economics

2024 spring semester
Fresh-Start Seminar [J] 51

School of Political Science and Economics

2024 spring semester
Mathematical Analysis (Differential Equations) [J] 01

School of Political Science and Economics

2024 fall semester
Mathematical Analysis (Multivariate Analysis) [J] 01

School of Political Science and Economics

2024 spring semester
Advanced Calculus [E] 01

School of Political Science and Economics

2024 fall quarter
Calculus (Multivariable Calculus) [E] 01

School of Political Science and Economics

2024 spring semester
Introduction to Mathematical Analysis B (Calculus) [E] 01

School of Political Science and Economics

2024 winter quarter
Introduction to Mathematical Analysis 2 (Calculus)

School of Social Sciences

2024 winter quarter
Convex Analysis (Jpn) (Nobe, A)

Graduate School of Economics

2024 spring semester
Research Guidance (Seminar) on Analysis B (Nobe, A)

Graduate School of Economics

2024 fall semester
Mathematics for Economics III (Optimization Theory) (Nobe, A)

Graduate School of Economics

2024 fall quarter
Mathematics for Economics I (Calculus) (Nobe, A)

Graduate School of Economics

2024 fall quarter
Research Guidance (Seminar) on Analysis A (Nobe, A)

Graduate School of Economics

2024 spring semester

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Teaching Experience

Analysis (Differential Equations)

Waseda University, School of Political Science and Economics
2021.04

-

Now
Convex Analysis

Waseda University
2021.04

-

Now
Introduction to Mathematics for Economics

Waseda University
2021.04

-

Now
数学演習

気象大学校大学部
2011.04

-

Now
線形代数学

気象大学校大学部
2011.04

-

Now
幾何学発展

千葉大学教育学部
2020.10

-

2021.03
解析学発展

千葉大学教育学部
2020.10

-

2021.03
離散数学c

津田塾大学学芸学部
2018.11

-

2021.03
解析学研究II

千葉大学教育学部
2011.10

-

2021.03
大域解析学

千葉大学大学院教育学研究科
2011.10

-

2021.03
機械計算論

千葉大学教育学部
2010.10

-

2021.03
数値解析

千葉大学教育学部
2009.10

-

2021.03
微分積分学

日本大学生産工学部
2017.09

-

2020.11
解析学

千葉大学教育学部
2020.04

-

2020.09
解析学研究I

千葉大学教育学部
2011.04

-

2020.09
小学校算数

千葉大学教育学部
2011.04

-

2020.09
情報数学

千葉大学大学院教育学研究科
2011.04

-

2020.09
線形代数学

日本大学生産工学部
2017.04

-

2020.06
長さ・面積・体積

千葉大学教育学部
2010.04

-

2019.08
数値計算法

千葉大学教育学部
2010.10

-

2018.02
コンピュータ概論

千葉大学教育学部
2010.04

-

2017.08
線形代数学II

日本大学生産工学部
2013.09

-

2017.01
線形代数学I

日本大学生産工学部
2013.04

-

2016.08
図形と画像の処理

千葉大学教育学部
2010.10

-

2016.02
数理モデルの解析

千葉大学教育学部
2010.10

-

2011.02
微積分学A

千葉大学普遍教育
2010.04

-

2010.08
計算数理A

大阪大学基礎工学部
2004.04

-

2009.10
基礎数理演習C

大阪大学始祖工学部
2004.04

-

2009.09
基礎数理演習A

大阪大学基礎工学部
2004.04

-

2009.09
Advanced Mathematical Science A

Graduate School of Engineering Scicence, Osaka University
2008.04

-

2008.09

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Social Activities

逆関数について

東京都立城東高等学校ジョイントセミナー（東京都立城東高等学校）
2019.06

　

　
算数

千葉県教育委員会教員免許法認定講習（千葉大学）
2016.08

　

　
暦の数理〜木星のカレンダーを作ろう〜

千葉大学高大連携講座（千葉県立木更津高等学校）
2016.06

　

　
算数

千葉県教育委員会教員免許法認定講習（千葉大学）
2013.08

　

　
審査委員

千葉大学第６回高校生理科研究発表会
2012.09

-

　
楕円曲線のはなし

千葉大学高大連携講座（千葉県立木更津高等学校）
2012.05

　

　
箱と玉の数理実験～ソリトンを作ろう〜

千葉大学未来の科学者養成講座（千葉大学）
2012.04

　

　
数学の最先端

千葉県教育委員会教員免許状更新講習（千葉大学）
2011.08

　

　
ソリトンのはなし

千葉大学高大連携講座（千葉県立千葉女子高等学校）
2010.06

　

　
箱と玉の数理実験～究極の波を調べる〜

千葉大学未来の科学者養成講座（千葉大学）
2010.04

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Academic Activities

reviewer

Peer review

Japan Journal of Industrial and Applied Mathematics

2017.02

-

Now
reviewer

Peer review

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

2010.11

-

Now
reviewer

Peer review

Journal of the Physical Society of Japan

2010.09

-

Now
reviewer

Peer review

Journal of Physics A: Mathematical and Theoretical

2005.04

-

Now
日本応用数理学会2021年研究部会連合発表会実行委員

Academic society, research group, etc.

日本応用数理学会法政大学（オンライン）

2021.03

　

　
日本応用数理学会2020年度年会応用可積分系オーガナイズドセッション主催者

Academic society, research group, etc.

日本応用数理学会愛媛大学（オンライン）

2020.09

　

　
Organizer, RIMS Conference ``Expansion of Integrable systems"

Academic society, research group, etc.

京都大学数理解析研究所

2008.08

　

　
reviewer

Peer review

Journal of Nonlinear Mathematical Physics

2020.03

-

Now
reviewer

Peer review

Journal of Algebraic Combinatorics

2019.06

-

Now
reviewer

Peer review

Hokkaido Mathematical Journal

2018.12

-

Now
研究集会「非線形波動から可積分系へ」世話人

Academic society, research group, etc.

津田塾大学数学計算機科学研究所オンライン

2020.11

　

　
reviewer

Peer review

Integrable Systems and Algebraic Geometry (Cambridge University Press)

2018.11

-

2019.02
reviewer

Peer review

IEICE Transactions

2015.12

-

2016.04
日本応用数理学会2009年度年会実行委員

Academic society, research group, etc.

日本応用数理学会大阪大学豊中キャンパス

2009.09

　

　
Organizer, 5th East Asia Partial Differential Equations Conference

Academic society, research group, etc.

Takashi Suzuki 大阪大学中之島センター

2005.01

-

2005.02

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Sub-affiliation

Faculty of Political Science and Economics Graduate School of Economics
Faculty of Social Sciences School of Social Sciences

Internal Special Research Projects

非代数的保存量をもつ可積分系の離散化とその感染症数理モデルへの応用

2023

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周期的境界条件のもとで可積分系であることが広く知られているVolterra格子に対して、境界での値が一定という非周期的境界条件を課したHamilton系は一般に可積分ではない。しかし、そのような系の中でとくに可積分となる場合に、代表的な感染症数理モデルであるSIRモデルおよびそのワクチン接種を考慮した拡張モデル（SIRワクチン接種モデル）が含まれることを示した。また、微分幾何学的視点からそれらの感染症モデルのシンプレクティック多様体としての構造を明らかにした。さらに、SIRワクチン接種モデルは第1種Abel方程式へ変換可能であり、その変換を援用するとLambertのW函数を用いて一般解を陰的に構成できることを示し、Lamber曲線と適当な直線との交叉を用いて、SIRワクチン接種モデルの可積分な離散化を構成した。
非代数的保存量をもつ可積分系の離散化

2023

　View Summary

代表的な感染症数理モデルであるSIRモデルおよびそのワクチン接種を考慮した拡張モデル（SIRワクチン接種モデル）に対して、微分幾何学的視点からそれらのシンプレクティック多様体としての構造を明らかにした。Volterra格子に境界での値が一定という特別な境界条件を課したHamilton系は一般に可積分ではないが、そのような系の中でとくに可積分となる場合にSIRワクチン接種モデルが含まれることを示した。さらに、SIRワクチン接種モデルは第1種Abel方程式へ変換可能であり、その変換を援用すると、LambertのW函数を用いて一般解を陰的に構成できることを示した。また、Lamber曲線と適当な直線との交叉を用いて、SIRワクチン接種モデルの可積分な離散化を構成した。
クラスター代数のアフィン型ミューテーションの可積分性について

2022

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クラスター代数に関連する離散力学系の研究によって得られた成果を用いて、感染症数理モデルの一つであるSIR with vaccination modelの可積分離散化について研究した。連続モデルの保存量から得られる不変曲線と他の曲線との交叉を用いて離散モデルを導出し、その離散モデルは（１）連続モデルと同じ保存量をもち、（２）順方向／逆方向の時間発展がいずれも一意に定まることを示した。これらの性質をもつことから、得られた離散化は連続モデルの可積分性を保つ離散化（可積分離散化）であることが結論づけられる。さらに、離散モデルの解は連続モデルと共有する不変曲線のパラメータ表示を与えることを用いて、連続モデルの厳密解を構成した。
正値ローラン性をもつ力学系の可積分性と代数的エントロピー

2021 松木平淳太

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報告者らはこれまでにクラスター代数のアフィン型ミューテーションから双有理写像力学系を導出し、その可積分性について考察してきた。ランクN+1の$A^{(1)}_N$型ミューテーションから導かれる双有理写像力学系は、対応するクイバーミューテーションの周期性を利用して、力学系の保存量を構成することが可能であり、それらを用いて一般解が得られる。このような先行研究を踏まえて、$D^{(1)}_N$型ミューテーションから導かれるN+1次元双有理写像力学系の可積分性について考察した。$A^{(1)}_N$型と同様に、対応するクイバーミューテーションの周期性を利用して、双有理写像力学系の保存量を構成しその可積分性を示すことができた。$A^{(1)}_N$型の不変曲線は2次曲線であるため容易に一般解を得ることができるが、$D^{(1)}_N$型の不変曲線は高次の代数曲線であるため、力学系の一般解を構成することは今後の課題である。