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.

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.

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

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.

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

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.

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'

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

We establish a on-to-one correspondence between the configurations in the Wolfram cellular automaton, which is abbreviated to the WCA, and the paths in the de Bruijn quiver. Extending the correspondence to that between the associative algebra whose underlying vector space is generated by the configurations in the WCA and the path algebra of the de Bruijn quiver, we obtain the global transition of the associative algebra associated with the WCA. Thus we translate the problem concerning reversibility of the WCA into that concerning surjectivity of the endomorphism on the associative algebra. We then show that the induced problem concerning the endomorphism can be solved in terms of the adjacency matrix of the WCA, which is defined from that of the de Bruijn quiver through the one-to-one correspondence. Indeed, we give a necessary and sufficient condition for reversibility of the WCA. By virtue of the necessary and sufficient condition, we classify all 16 reversible rules in the ECA imposing periodic boundary conditions. © 2011 Nova Science Publishers, Inc.

RIMS研究集会「離散可積分系の応用数理」