Energy conversion by autonomous regulation of chaos: dynamical mechanism of loose coupling

Inspired by recent experiments of molecular motors, a dynamical systems model for a flexible machine is proposed which converts injected energy to output directional motion. The output amount is distributed broadly, and thus the coupling between input energy and output motion is loose, as in the experiments. This energy conversion is shown to be robust against the change of surrounding environment. Stability analysis on the fixed point solutions of the model is presented, which suggests that transient chaotic motion, induced by temporal three-body motion, is relevant to the energy conversion.     

Universal characters, integrable chains and the Painlevé equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painlevé equation of type . Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.     

Folding transformations of the Painlevé equations

New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3, or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations.Acknowledgement The authors wish to thank Prof. Yosuke Ohyama, Prof. Shun Shimomura, and Dr. Yoshikatsu Sasaki for valuable discussions.     

On an Integrable System of q-Difference Equations Satisfied by the Universal Characters: Its Lax Formalism and an Application to q-Painlevé Equations

The universal character is a generalization of the Schur function attached to a pair of partitions. We study an integrable system of q-difference equations satisfied by the universal characters, which is an extension of the q-KP hierarchy and is called the lattice q-UC hierarchy. We describe the lattice q-UC hierarchy as a compatibility condition of its associated linear system (Lax formalism) and explore an application to the q-Painlevé equations via similarity reduction. In particular a higher order analogue of the q-Painlevé VI equation is presented.     

Chiral recognition by cyclic oligosaccharides. Enantioselective complexation of binaphthyl derivatives with cyclodextrins

Abstract

Chiral recognition of binaphthyl derivatives, such as 1,1′-bi-2-naphthol (1), 1,1′-binaphthyl-2,2′-diyl hydrogen phosphate (2), and 2,2′-dihydroxy-1,1′-binaphthyl-3,3′-dicarboxylic acid (3), by cyclodextrins (CDxs) has been studied. The S enantiomers of 1 and 2 are bound to heptakis(2,3,6-tri-O-methyl)-β-CDx (TMe-β-CDx) as well as β-CDx more strongly than the R enantiomers. The molecular mechanics and molecular dynamics calculations for the 1:1 complex of 1 and β-CDx suggest that more effective van der Waals contacts and intermolecular hydrogen bonding stabilize the complex of S-1 compared with that of R-1. Meanwhile the R enantiomer of 3 is the preferable guest for β- and TMe-β-CDxs. Circular dichroism spectroscopy suggests that the complex of S-3 is more unstable than that of R-3 because the dihedral angle of the naphthalene planes of S-3 needs to be reduced for forming the inclusion complex. The enantiomers of the guest binaphthyls are completely separated by means of capillary zone electrophoresis (CZE) when TMe-β-CDx is used as a separating agent. The results of the CZE correspond well with those of the binding constants of the inclusion complexes.     

q-Painlevé VI Equation Arising from q-UC Hierarchy

We study the q-difference analogue of the sixth Painlevé equation (q-P_VI) by means of tau functions associated with the affine Weyl group of type D₅. We prove that a solution of q-P_VI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-P_VI in terms of the universal character which is a generalization of the Schur polynomial attached to a pair of partitions.     

Universal Characters and <Emphasis Type="Italic">q</Emphasis>-Painlevé Systems

We propose an integrable system of q-difference equations satisfied by the universal characters and regard it as a q-analogue of the UC hierarchy; see [10]. Via a similarity reduction of this integrable system, rational solutions of the q-Painlevé systems are constructed in terms of the universal characters.