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Noncommutative algebras related with Schubert calculus on Coxeter groups
For any finite Coxeter system 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 We prove this conjecture for classical Coxeter groups and .
We define a ``quantization'' and a multiparameter deformation of our
construction and show that for Lie groups of classical type and 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 -, - and -bracket algebras, and as an application, discover
{\it Pieri type formula} in the -bracket algebra. As a corollary, we
obtain Pieri type formula for multiplication of arbitrary -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 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 CaSrRuO near the
metal-insulator transition. The hopping exponent shows a systematic
evolution from a value of deeper in the insulator to the
conventional Mott value 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
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