Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system $(W,S)$ we construct a certain noncommutative algebra, so-called {\it bracket algebra}, together with a familiy of commuting elements, so-called {\it Dunkl elements.} Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the group $W.$ We prove this conjecture for classical Coxeter groups and $I_2(m)$. We define a ``quantization'' and a multiparameter deformation of our construction and show that for Lie groups of classical type and $G_2,$ the algebra generated by Dunkl elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define {\it quantum Bruhat representation} of the corresponding bracket algebra. We study in more detail relations and structure of $B_n$-, $D_n$- and $G_2$-bracket algebras, and as an application, discover {\it Pieri type formula} in the $B_n$-bracket algebra. As a corollary, we obtain Pieri type formula for multiplication of arbitrary $B_n$-Schubert classes by some special ones. Our Pieri type formula is a generalization of Pieri's formulas obtained by A. Lascoux and M.-P. Sch\"utzenberger for flag varieties of type $A.$ We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements which describes ``noncommutative differential geometry on a finite Coxeter group'' in a sense of S. Majid

Characteristics of the wavelength of ripples on icicles

It is known that the wavelength of the ripples on icicles in nature is of centimeter-scale. Such study on morphological instability of ice-water interface during ice growth from flowing supercooled water film with one side being a free surface has recently been made [K. Ueno, Phys. Rev. E 68, 021603 (2003)]. This is a first theoretical study taking into account the influence of the shape of the water-air surface on the growth condition of infinitesimal disturbances of the ice-water interface. A simpler formula to determine the wavelength of the ripples than that in the previous paper is derived. It seems that the wavelength of ripples is insensitive to the water supply rates, diameters of the icicles and surrounding air temperatures. The details of dependence of the wavelengh of ripples on these parameters are investigated.Comment: 15 pages, 6 figure

Mechanism of hopping transport in disordered Mott insulators

By using a combination of detailed experimental studies and simple theoretical arguments, we identify a novel mechanism characterizing the hopping transport in the Mott insulating phase of Ca$_{2-x}$Sr$_x$RuO$_4$ near the metal-insulator transition. The hopping exponent $\alpha$ shows a systematic evolution from a value of $\alpha=1/2$ deeper in the insulator to the conventional Mott value $\alpha=1/3$ closer to the transition. This behavior, which we argue to be a universal feature of disordered Mott systems close to the metal-insulator transition, is shown to reflect the gradual emergence of disorder-induced localized electronic states populating the Mott-Hubbard gap.Comment: 5 pages, 3 figures, To be published in Physical Review Letter