46,489 research outputs found
Branching coefficients of holomorphic representations and Segal-Bargmann transform
Let be a complex bounded symmetric domain of tube type in a
Jordan algebra , and let be its real
form in a Jordan algebra . The analytic continuation of
the holomorphic discrete series on forms a family of interesting
representations of . We consider the restriction on of the scalar
holomorphic representations of , as a representation of . The unitary
part of the restriction map gives then a generalization of the Segal-Bargmann
transform. The group is a spherical subgroup of and we find a canonical
basis of -invariant polynomials in components of the Schmid decomposition
and we express them in terms of the Jack symmetric polynomials. We prove that
the Segal-Bargmann transform of those -invariant polynomials are, under the
spherical transform on , multi-variable Wilson type polynomials and we give
a simple alternative proof of their orthogonality relation. We find the
expansion of the spherical functions on , when extended to a neighborhood in
, in terms of the -spherical holomorphic polynomials on , the coefficients being the Wilson polynomials
Bounded decomposition in the Brieskorn lattice and Pfaffian Picard--Fuchs systems for Abelian integrals
We suggest an algorithm for derivation of the Picard--Fuchs system of
Pfaffian equations for Abelian integrals corresponding to semiquasihomogeneous
Hamiltonians. It is based on an effective decomposition of polynomial forms in
the Brieskorn lattice. The construction allows for an explicit upper bound on
the norms of the polynomial coefficients, an important ingredient in studying
zeros of these integrals.Comment: 17 pages in LaTeX2
On the Hodge decomposition in R^n
We prove a version of the hodge decomposition for differential forms in
Euclidean space and a generalization to the class of Lizorkin currents. We also
compute the cohomology of .Comment: 28 page
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
The algebra of the box spline
In this paper we want to revisit results of Dahmen and Micchelli on
box-splines which we reinterpret and make more precise. We compare these ideas
with the work of Brion, Szenes, Vergne and others on polytopes and partition
functions.Comment: 69 page
A Characterization of Reduced Forms of Linear Differential Systems
A differential system , with
is said to be in reduced form if where
is the Lie algebra of the differential Galois group of
. In this article, we give a constructive criterion for a system to be in
reduced form. When is reductive and unimodular, the system is in
reduced form if and only if all of its invariants (rational solutions of
appropriate symmetric powers) have constant coefficients (instead of rational
functions). When is non-reductive, we give a similar characterization via
the semi-invariants of . In the reductive case, we propose a decision
procedure for putting the system into reduced form which, in turn, gives a
constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr
An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method
In this paper we propose a collocation method for solving some well-known
classes of Lane-Emden type equations which are nonlinear ordinary differential
equations on the semi-infinite domain. They are categorized as singular initial
value problems. The proposed approach is based on a Hermite function
collocation (HFC) method. To illustrate the reliability of the method, some
special cases of the equations are solved as test examples. The new method
reduces the solution of a problem to the solution of a system of algebraic
equations. Hermite functions have prefect properties that make them useful to
achieve this goal. We compare the present work with some well-known results and
show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications
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