663 research outputs found
Cubature formulas of multivariate polynomials arising from symmetric orbit functions
The paper develops applications of symmetric orbit functions, known from
irreducible representations of simple Lie groups, in numerical analysis. It is
shown that these functions have remarkable properties which yield to cubature
formulas, approximating a weighted integral of any function by a weighted
finite sum of function values, in connection with any simple Lie group. The
cubature formulas are specialized for simple Lie groups of rank two. An optimal
approximation of any function by multivariate polynomials arising from
symmetric orbit functions is discussed.Comment: 19 pages, 4 figure
ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS
The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra C2
Cubature formulas, geometrical designs, reproducing kernels, and Markov operators
Cubature formulas and geometrical designs are described in terms of
reproducing kernels for Hilbert spaces of functions on the one hand, and Markov
operators associated to orthogonal group representations on the other hand. In
this way, several known results for spheres in Euclidean spaces, involving
cubature formulas for polynomial functions and spherical designs, are shown to
generalize to large classes of finite measure spaces and
appropriate spaces of functions inside . The last section
points out how spherical designs are related to a class of reflection groups
which are (in general dense) subgroups of orthogonal groups
The Numerical Sausage
The renormalization group equation describing the evolution of the metric of
the non linear sigma models poses some nice mathematical problems involving
functional analysis, differential geometry and numerical analysis. We describe
the techniques which allow a numerical study of the solutions in the case of a
two-dimensional target space (deformation of the --model. Our
analysis shows that the so-called sausages define an attracting manifold in the
U(1) symmetric case, at one-loop level. The paper describes i) the known
analytical solutions, ii) the spectral method which realizes the numerical
integrator and allows to estimate the spectrum of zero--modes, iii) the
solution of variational equations around the solutions, and finally iv) the
algorithms which reconstruct the surface as embedded in .Comment: 15 pages, uuencoded postscript fil
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