437 research outputs found
On A Curious Linear Relationship Between Rainfall Averages
Empirical evidence points to the fact that the area average rain rate and the fraction of the area. where rain rate exceeds a gi ven threshold tend to be highly correlated, provided the area is large enough and the threshold is chosen optimally as to increase the correlation. This fact has an important application in rainfall estimation from space using satellite borne instruillents. A statistical explanation is provided for the observed linearity, and a IIlct.hod for optimal thresholds is discussed
A path model for Whittaker vectors
40 pages, 2 figuresIn this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra . This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the -difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way
Q-systems, Heaps, Paths and Cluster Positivity
We consider the cluster algebra associated to the -system for as a
tool for relating -system solutions to all possible sets of initial data. We
show that the conserved quantities of the -system are partition functions
for hard particles on particular target graphs with weights, which are
determined by the choice of initial data. This allows us to interpret the
simplest solutions of the Q-system as generating functions for Viennot's heaps
on these target graphs, and equivalently as generating functions of weighted
paths on suitable dual target graphs. The generating functions take the form of
finite continued fractions. In this setting, the cluster mutations correspond
to local rearrangements of the fractions which leave their final value
unchanged. Finally, the general solutions of the -system are interpreted as
partition functions for strongly non-intersecting families of lattice paths on
target lattices. This expresses all cluster variables as manifestly positive
Laurent polynomials of any initial data, thus proving the cluster positivity
conjecture for the -system. We also give an alternative formulation in
terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure
Fusion products, Kostka polynomials, and fermionic characters of su(r+1)_k
Using a form factor approach, we define and compute the character of the
fusion product of rectangular representations of \hat{su}(r+1). This character
decomposes into a sum of characters of irreducible representations, but with
q-dependent coefficients. We identify these coefficients as (generalized)
Kostka polynomials. Using this result, we obtain a formula for the characters
of arbitrary integrable highest-weight representations of \hat{su}(r+1) in
terms of the fermionic characters of the rectangular highest weight
representations.Comment: 21 pages; minor changes, typos correcte
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