We investigate a soliton cellular automaton (Box-Ball system) with periodic boundary conditions. Since the cellular automaton is a deterministic dynamical system that takes only a finite number of states, it will exhibit periodic motion. We determine its fundamental cycle for a given initial state. PrefaceA cellular automaton (CA) is a discrete dynamical system consisting of a regular array of cells [1]. Each cell takes only a finite number of states and is updated in discrete time steps. Although the updating rules are simple, CAs often exhibit very complicated time evolution patterns which resemble natural phenomena such as chemical reactions, turbulent flow, nonlinear dispersive waves and solitons. A typical CA exhibiting a solitonic behaviour is the box and ball system (BBS) which is a reinterpretation of the CA proposed by Takahashi and Satsuma [2,3]. The BBS is integrable in the sense that it is obtained from the KdV equation through a limiting procedure called ultradiscretization [4]. It can also be obtained from a two dimensional integrable lattice model and its relation to combinatorial R matrices of U ′ q (A (1) N ) is well established [5,6]. The original box-ball system is defined as a dynamical system of a finite number of balls in an infinite one dimensional array of boxes. However, it is possible to extend the time evolution rule to a system consisting of a finite number of boxes with periodic boundary conditions [7]. The BBS with periodic boundary condition (pBBS) is also connected to the combinatorial R matrix of U ′ q (A (1) N ) and, its time evolution rule is represented as a Boolean recurrence formula related to the algorithm for calculating the 2N th root. As a CA, the pBBS is composed of a finite number of cells, and it can only take on a finite number of patterns. Hence the time evolution of the pBBS is necessarily periodic. In the present article, we investigate the fundamental cycle, i.e., the shortest period of the discrete periodic motion of the pBBS.1 In section 2, we review the pBBS and present its conserved quantities. The formula for the total number of patterns for a given set of conserved quantities is also presented. In section 3, we define several notions which are necessary to prove the formula for * Graduate school of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan † Imai quantum computing and information project, ERATO, JST, Daini Hongo White Bldg. 201, 5-28-3 Hongo, Bunkyo, Tokyo 113-0033, Japan 1 Part of the present work was already announced in Ref. [8].

The 2:2 complex of Co-carbene is generated by photolysis of [Co(hfpip)2(D2py2(TBA))]2, 1, having a cyclic structure functioning as a single-molecule magnet with Ueff = 96 K and Hc = 10 kOe at 1.9 K after annealing at 70 K.

A novel tetraphenylethene-based fluorescence (FL) chemosensor exhibits nonlinear turn-on FL switching though cooperative binding of L-tartarate, where its convergent binding to form cyclic substructures is responsible for the FL increase. This binding scheme achieves selective detection of dicarboxylates over monocarboxylates, thus is potentially applicable to the preliminary screening for metabolic disorders.

Although self-assembly has realized the spontaneous formation of nanoarchitectures, the nanoscopic expression of chemical structural information at the molecular level can alternatively be regarded as a tool to translate molecular structural information with high precision. We have found that a newly developed guanidinium-tethered oligophenylenevinylene exhibits characteristic fluorescence (FL) responses toward L- and meso-tartarate, wherein the different self-assembly modes, termed J- or H-type aggregation, are directed according to the molecular information encoded as the chemical structure. This morphological difference originates from the geometric anti versus gauche conformational difference between L- and meso-tartarate. A similar morphological difference can be reproduced with the geometric C=C bond difference between fumarate and maleate. In the present system, the dicarboxylate structural information is embodied in the inherent threshold concentration of the FL response, the signal-to-noise ratio, and the maximum FL wavelength. These results indicate that self-assembly is meticulous enough to sense subtle differences in molecular information and thus demonstrate the potential ability of self-assembly for the expression of a FL sensory system.

The newly developed oligophenylenevinylene (OPV)-based fluorescent (FL) chiral chemosensor (OPV-Me) for the representative enantiomeric guest, 1,2-cyclohexanedicarboxylic acid (1,2-CHDA: RR- and SS-form) showed the high chiral discrimination ability, resulting in the different aggregation modes of OPV-Me self-assembly: RR-CHDA directed the fibrous supramolecular aggregate, whereas SS-CHDA directed the finite aggregate. The consequent FL intensity toward RR-CHDA was up to 30 times larger than that toward SS-CHDA. Accordingly, highly enantioselective recognition was achieved. Application to the chirality sensing was also possible: OPV-Me exhibited a linear relationship between the FL intensity and the enantiomeric excess through the morphological development of stereocomplex aggregates. These results clearly show that the chiral recognition ability is manifested by the amplification cascade of the chirality difference through self-assembly.