Abstract. Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra Uq(X (1) n ) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary Xn.

We introduce a fermionic formula associated with any quantum affine algebra Uq(X (r) N ). Guided by the interplay between corner transfer matrix and Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowski-Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one dimensional sums conjecturally give rise to branching functions of an integrable G

SummaryWe have developed a novel gain-of-function system that we have named the FOX hunting system (Full-length cDNA Over-eXpressing gene hunting system). We used normalized full-length cDNA and introduced each cDNA into Arabidopsis by in planta transformation. About 10 000 independent full-length Arabidopsis cDNAs were expressed independently under the CaMV 35S promoter in Arabidopsis. Each transgenic Arabidopsis contained on average 2.6 cDNA clones and was monitored under various categories such as morphological changes, fertility and leaf color. We found 1487 possible morphological mutants from 15 547 transformants. When 115 pale green T 1 mutants were analyzed, 59 lines represented the mutant phenotypes in more than 50% of the T 2 progeny. Characterization of two leaf color mutants revealed the significance of this approach. We also document mutants from several categories and their corresponding full-length cDNAs.

The chemical composition of the cell wall of Sz. pombe is known as b-1,3-glucan, b-1,6-glucan, a-1,3-glucan and a-galactomannan; however, the three-dimensional interactions of those macromolecules have not yet been clarified. Transmission electron microscopy reveals a three-layered structure: the outer layer is electron-dense, the adjacent layer is less dense, and the third layer bordering the cell membrane is dense. In intact cells of Sz. pombe, the high-resolution scanning electron microscope reveals a surface completely filled with a-galactomannan particles. To better understand the organization of the cell wall and to complement our previous studies, we set out to locate the three different types of b-glucan by immuno-electron microscopy. Our results suggest that the less dense layer of the cell wall contains mainly b-1,6-branched b-1,3-glucan. Occasionally a line of gold particles can be seen, labelling fine filaments radiating from the cell membrane to the a-galactomannan layer, suggesting that some of the radial filaments contain b-1,6-branched b-1,3-glucan. b-1,6-glucan is preferentially located underneath the a-galactomannan layer. Linear b-1,3-glucan is exclusively located in the primary septum of dividing cells. b-1,6-glucan only labels the secondary septum and does not co-localize with linear b-1,3-glucan, while b-1,6-branched b-1,3-glucan is present in both septa. Linear b-1,3-glucan is present from early stages of septum formation and persists until the septum is completely formed; then just before cell division the label disappears. From these results we suggest that linear b-1,3-glucan is involved in septum formation and perhaps the separation of the two daughter cells. In addition, we frequently found b-1,6-glucan label on the Golgi apparatus, on small vesicles and underneath the cell membrane. These results give fresh evidence for the hypothesis that b-1,6-glucan is synthesized in the endoplasmic reticulum-Golgi system and exported to the cell membrane.

We formulate the inverse scattering method for a periodic boxball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansätze at q = 1 and q = 0, which provides the ultradiscrete analogue of quasi-periodic solutions in soliton equations, e.g., action-angle variables, Jacobi varieties, period matrices and so forth. As an application we establish explicit formulas counting the states characterized by conserved quantities and the generic and fundamental period under the commuting family of time evolutions. from the following diagram:

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebraIt is a crystal theoretic formulation of the generalized box-ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of U ′ q (AM −1 ). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete KP equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.(1) M ) corresponding to the k-fold symmetric tensor representation. As a set it consists of the single row semistandard tableaux of length k on letters {1, 2, . . . , M + 1}:where we have omitted the k − 1 vertical lines separating the entries. We also represent the elements by the multiplicities of their contents. Namely, b = m 1 · · · m k ∈ B k is also denoted by b = (x 1 , x 2 , · · · , x M +1 ) with x i = #{l | m l = i}.

The box-ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.

Cytokinesis in many eukaryotes involves the contraction of an actomyosin-based contractile ring. However, the detailed mechanism of contractile ring contraction is not fully understood. Here, we establish an experimental system to study contraction of the ring to completion in vitro. We show that the contractile ring of permeabilized fission yeast cells undergoes rapid contraction in an ATP- and myosin-II-dependent manner in the absence of other cytoplasmic constituents. Surprisingly, neither actin polymerization nor its disassembly is required for contraction of the contractile ring, although addition of exogenous actin-crosslinking proteins blocks ring contraction. Using contractile rings generated from fission yeast cytokinesis mutants, we show that not all proteins required for assembly of the ring are required for its contraction in vitro. Our work provides the beginnings of the definition of a minimal contraction-competent cytokinetic ring apparatus.