We show that a generalized cellular automaton, exhibiting solitonic interactions, can be explicitly solved by means of techniques first introduced in the context of the scattering problem for the KdV equation. We apply this method to calculate the phase-shifts caused by interactions between the solitonic and non-solitonic parts into which arbitrary initial states separate in time.

Abstract. We propose an ultradiscrete analogue of the vertex operator in the case of the ultradiscrete KdV equation, which maps N -soliton solutions to N + 1-soliton ones.

We propose an ultradiscrete analogue of the vertex operator in the case of the ultradiscrete KP equation-several other ultradiscrete equations-which maps Nsoliton solutions to N + 1-soliton ones.

We define a function by means of the minimum weight flow on a planar graph and prove that this function solves the ultradiscrete Toda molecule equation, its Bäcklund transformation and the two dimensional Toda molecule equation. The method we employ in the proof can be considered as fundamental to the integrability of ultradiscrete soliton equations.

Technology which can monitor states of granular systems will become a key to optimize manufacturing systems for developing functional materials such as for battery and medicine. Several methods are already available including process tomography and PIV. This paper focuses and discusses mixing index by Shannon Entropy, Shannon mixing index termed in the paper, by comparing the Lacey mixing known as standard mixing index based on statistics and experimental data. The following results are obtained: 1) Shannon mixing index can handle multi-component systems, instead of Lacey mixing index being limited to binary systems. 2) Shannon mixing index can evaluate really small amount of component quantitatively as well as qualitatively, which is not available by statistically based methods. 3) Shannon mixing index can be calculated experimentally by material sampling method, which is comparable to predictions by DEM (Discrete Element Method) simulation. The results obtained through the research indicates that Shannon Entropy can be qualified to the standard mixing index.