2,364 research outputs found
Berezin Kernels and Analysis on Makarevich Spaces
Following ideas of van Dijk and Hille we study the link which exists between
maximal degenerate representations and Berezin kernels. We consider the
conformal group of a simple real Jordan algebra . The
maximal degenerate representations () we shall study
are induced by a character of a maximal parabolic subgroup of . These representations can be realized on a space of
smooth functions on . There is an invariant bilinear form
on the space . The problem we consider is to diagonalize this bilinear
form , with respect to the action of a symmetric subgroup
of the conformal group . This bilinear form can be written as an
integral involving the Berezin kernel , an invariant kernel on the
Riemannian symmetric space , which is a Makarevich symmetric space in the
sense of Bertram. Then we can use results by van Dijk and Pevzner who computed
the spherical Fourier transform of . From these, one deduces that the
Berezin kernel satisfies a remarkable Bernstein identity : where is an invariant differential operator on
and is a polynomial. By using this identity we compute a Hua
type integral which gives the normalizing factor for an intertwining operator
from to . Furthermore we obtain the diagonalization of the
invariant bilinear form with respect to the action of the maximal compact group
of the conformal group
Projective Pseudodifferential Analysis and Harmonic Analysis
We consider pseudodifferential operators on functions on which
commute with the Euler operator, and can thus be restricted to spaces of
functions homogeneous of some given degree. Their symbols can be regarded as
functions on a reduced phase space, isomorphic to the homogeneous space
, and the resulting calculus is a
pseudodifferential analysis of operators acting on spaces of appropriate
sections of line bundles over the projective space : these spaces are
the representation spaces of the maximal degenerate series
of . This new approach to the quantization of
, already considered by other authors, has several advantages: as an
example, it makes it possible to give a very explicit version of the continuous
part from the decomposition of under the quasiregular action of
. We also consider interesting special symbols, which arise from the
consideration of the resolvents of certain infinitesimal operators of the
representation
Finite automata for testing uniqueness of Eulerian trails
We investigate the condition under which the Eulerian trail of a digraph is
unique, and design a finite automaton to examine it. The algorithm is
effective, for if the condition is violated, it will be noticed immediately
without the need to trace through the whole trail
On a Speculated Relation Between Chv\'atal-Sankoff Constants of Several Sequences
It is well known that, when normalized by n, the expected length of a longest
common subsequence of d sequences of length n over an alphabet of size sigma
converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele
regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to
do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma
and d are small integers.Comment: 13 pages. To appear in Combinatorics, Probability and Computin
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