exact operator solution of the calogero sutherland model the wave functions of the calogero sutherland model are known to be expressible in terms of jack polynomials a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived the creation operators that enter in this formula of rodrigues type for the jack polynomials involve dunkl operators

intertwining operators associated to the group for any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values containing only certain negative rational numbers this paper constructs an integral transform implementing the intertwining operator for the group s3 the symmetric group on three objects for parameter value the transform is realized as an absolutely continuous measure on a compact subset of m2 r which contains the group as a subset of its boundary the construction of the integral formula involves integration over the unitary group u 3 associated to any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g it has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values this singular set contains only negative values and is closely linked to the zero set of the poincare series of g this paper constructs an integral transform implementing the intertwining map for the group s3 the symmetric group on three objects for positive parameter values previously this had been done only for the group z2 acting by sign change on r where the transform is a classical fractional integral the transform in this paper has its origin in the adjoint action of the unitary group u 3 on the linear space of real diagonal 3 x 3 matrices the complexification of the maximal torus this will lead to a transform realized as an absolutely continuous measure on a certain compact subset of m2 r here is a concise statement of the main result rephrased from formulas 5 1 5 6 the operator v intertwines the differential difference operators received by the editors august 22 1994 1991 mathematics subject classification primary 22e30 33c80 secondary 33c50 20b30

a transference theorem for the dunkl transform and its applications for a family of weight functions invariant under a finite reflection group we show how weighted lp multiplier theorems for dunkl transform on the euclidean space rd can be transferred from the corresponding results for h harmonic expansions on the unit sphere sd of rd 1 the result is then applied to establish a hormander type multiplier theorem for the dunkl transform and to show the convergence of the bochner riesz means of the dunkl transform of order above the critical index in weighted lp spaces

on shape preserving quadratic spline interpolation in this paper we discuss the design of algorithms for interpolating discrete data using c 1 quadratic splines in such a way that the monotonicity and or convexity of the data is preserved the analysis culminates in an interactive algorithm which takes full advantage of the flexibility which quadratic splines permit

strichartz estimates for the dunkl wave equation and application in this paper we prove dispersive and strichartz estimates associated for the dunkl wave equation

convex cubic hermite spline interpolation we present a method to construct convex cubic c1 splines which interpolate a given convex data set the problem is reduced to the solution of a system of linear inequalities the existence of such convex interpolation splines is assured if the data fulfill slight additional conditions for stronger conditions some easier methods are developed finally error estimations are given

unramified class field theory of arithmetical surfaces let k be an algebraic number field 9k its ring of integers and v a non empty open subscheme of spec ck let x be a projective smooth geometrically irreducible scheme over k and x a regular proper flat scheme over v such that x x vk x the purpose of this paper is to give an explicit description of the abelian fundamental groups ab x and 7t b x using some idele class groups attached to x and x by a k theoretical method its outline is as follows in general for a noetherian scheme z and i 0 or 1 we define the group ski z to be the cokernel of

riesz transforms for dunkl hermite expansions abstract in the present paper we establish that riesz transforms for dunkl hermite expansions introduced by nowak and stempak are singular integral operators with hormander s type condition we prove that they are bounded on l p r d d for 1 p and from l 1 r d d into l 1 r d d

monotonicity preserving c2 rational cubic spline for monotone data abstract designers in industries need to generate splines which can interpolate the data points in such a way that they preserve the inherited shape characteristics positivity monotonicity convexity of data among the properties that the spline for curves and surfaces need to satisfy smoothness and shape preservation of given data are mostly needed by all the designers in this paper a rational cubic function with three shape parameters has been developed data dependent sufficient constraints are derived for one of these shape parameters to preserve the inherited shape feature like monotonicity of data remaining two shape parameters are left free for designer to refine the shape of the monotone curve as desired numerical examples and interpolation error analysis show that the interpolant is not only c 2 local computationally economical and visually pleasant but also smooth the error of rational cubic function is also calculated when the arbitrary function being interpolated is c 3 in an interpolating interval the order of approximation of interpolant is o h i 3

nonlinear generalized dunkl wave equations and applications abstract in this paper we introduce a class of nonlinear wave equations associated with the generalized dunkl laplace operator we study local and global well posedness next we establish the linearization of bounded energy solutions in the spirit of gerard 1996 9 the proof uses strichartz type inequalities and the energy estimate

a finiteness theorem for zero cycles over p adic fields let r be a henselian discrete valuation ring let x be a regular projective flat scheme over spec r with generalized semistable reduction we prove a bijectivity theorem for etale cycle class maps of the chow group of 1 cycles on x as an application we prove a finiteness theorem for the chow group of 0 cycles on a projective smooth variety over a p adic field

shape preserving piecewise rational interpolation with higher order continuity abstract a united form of the classical hermite interpolation and shape preserving interpolation is presented in this paper the presented interpolation method provides higher order continuous shape preserving interpolation splines the given interpolants are explicit piecewise rational expressions without solving a linear or nonlinear system of consistency equations by setting parameter values the interpolation curve can be generated by choosing the classical piecewise hermite interpolation polynomials or the presented piecewise rational expressions for monotonicity preserving and convexity preserving interpolation the appropriate values of a parameter are given on each subinterval numerical examples indicate that the given method produces visually pleasing curves

radial mollifiers mean value operators and harmonic functions in dunkl theory abstract in this paper we show how to use mollifiers to regularize functions relative to a set of dunkl operators in r d with coxeter weyl group w multiplicity function k and weight function k in particular for a w invariant open subset of r d for d r d a radial function and u l loc 1 k x d x we study the dunkl convolution product u k and the action of the dunkl laplacian and the volume mean operators on these functions the results are then applied to obtain an analog of the weyl lemma for dunkl harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators

convexity preserving piecewise rational quartic interpolation a family of piecewise rational quartic interpolants is given identified uniquely by the value of a tension parameter lambda i each interpolant of the family can be c 2 spline without solving a linear or nonlinear system of consistency equations for the derivative values at the knots the interpolant can preserve the local convexity concavity properties of the given data a proper choice of lambda i to guarantee shape preservation is given a convergence analysis establishes an error bound in terms of lambda i and shows that o h 3 accuracy is obtained for c 2 continuity several examples are supplied to support the practical value of the method

convexity preserving approximation by univariate cubic splines a convexity preserving approximation method is presented similar to the cubic spline interpolation the given approximation function is univariate cubic spline with c 2 continuity a very simple computing method is described the approximation order and polynomial reproducing property of this convexity preserving approximant are discussed the computing method for interpolating a small quantity of data is given some numerical examples are given to show the effectivity

fefferman stein inequalities for the mathbb z _2 d dunkl maximal operator in this article we establish the fefferman stein inequalities for the dunkl maximal operator associated with a finite reflection group generated by the sign changes similar results are also given for a large class of operators related to dunkl s analysis

weak approximation over function fields of curves over large or finite fields let k k c be the function field of a curve over a field k and let x be a smooth projective separably rationally connected k variety with x k neq emptyset under the assumption that x admits a smooth projective model pi mathcal x to c we prove the following weak approximation results 1 if k is a large field then x k is zariski dense 2 if k is an infinite algebraic extension of a finite field then x satisfies weak approximation at places of good reduction 3 if k is a nonarchimedean local field and r equivalence is trivial on one of the fibers mathcal x p over points of good reduction then there is a zariski dense subset w subseteq c k such that x satisfies weak approximation at places in w as applications of the methods we also obtain the following results over a finite field k 4 if k 10 then for a smooth cubic hypersurface x k the specialization map x k longrightarrow prod p in p mathcal x p kappa p at finitely many points of good reduction is surjective 5 if mathrm char k neq 2 3 and k 47 then a smooth cubic surface x over k satisfies weak approximation at any given place of good reduction

riesz transform and riesz potentials for dunkl transform analogues of riesz potentials and riesz transforms are defined and studied for the dunkl transform associated with a family of weight functions that are invariant under a reflection group the lp boundedness of these operators is established in certain cases

hormander s multiplier theorem for the dunkl transform abstract for a normalized root system r in r n and a multiplicity function k 0 let n n r k denote by d w x r x k d x the associated measure in r n let f stand for the dunkl transform given a bounded function m on r n we prove that if there is s n such that m satisfies the classical hormander condition with the smoothness s then the multiplier operator t m f f 1 m f f is of weak type 1 1 strong type p p for 1 p and is bounded on a relevant hardy space h 1 to this end we study the dunkl translations and the dunkl convolution operators and prove that if f is sufficiently regular for example its certain schwartz class seminorm is finite then the dunkl convolution operator with the function f is bounded on l p d w for 1 p we also consider boundedness of maximal operators associated with the dunkl convolutions with schwartz class functions

zero cycles on homogeneous varieties abstract in this paper we study the group a 0 x of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety x to do this we translate rational equivalence of 0 cycles on a projective variety into r equivalence on symmetric powers of the variety for certain homogeneous varieties we then relate these symmetric powers to moduli spaces of etale subalgebras of central simple algebras which we construct this allows us to show a 0 x 0 for certain classes of homogeneous varieties for groups of each of the classical types extending previous results of swan karpenko of merkurjev and of panin

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

zero cycles sur les fibrations au dessus d une courbe de genre quelconque resume soit x une variete propre et lisse sur un corps de nombres k des conjectures sur l image du groupe de chow des zero cycles de x dans le produit des memes groupes sur tous les completes de k ont ete proposees par colliot thelene kato et saito nous demontrons ces conjectures pour l espace total de fibrations en varietes rationnellement connexes verifiant l approximation faible au dessus de courbes dont le groupe de tate shafarevich est fini sous une hypothese d abelianite sur les fibres singulieres

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

translation operator and maximal function for the k 1 generalized fourier transform abstract in this paper we study a translation operator associated with the n dimensional k 1 generalized fourier transform where k is a multiplicity function for the dunkl operators in particular we prove that the translation is a positivity preserving operator acting on a suitable space of radial functions on r n we then use it to define a hardy littlewood type maximal operator where weak type 1 1 and strong type p p estimates for the maximal operator are established with a precise behavior in n and k

shape preserving piecewise rational interpolant with quartic numerator and quadratic denominator an explicit representation of a piecewise rational interpolant with quartic numerator and quadratic denominator is presented for positivity data monotone data and convex data the shape preserving properties of the interpolant are given the interpolant is c 2 continuous spline with a shape parameter w i on each subinterval the values of w i to guarantee shape preservation are estimated a convergence analysis establishes an error bound in terms of w i and shows that the interpolant is o h 2 or o h 3 accurate several examples are supplied to support the practical value of the given interpolation method

singular polynomials for finite reflection groups the dunkl operators involve a multiplicity function as parameter for generic values of this function the simultaneous kernel of these operators acting on polynomials is equal to the constants for special values however this kernel is larger we determine these singular values completely and give partial results on the representations of g that occur in this kernel

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

quadratic families of elliptic curves and unirationality of degree 1 conic bundles we study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t all such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers we prove that such conic bundles are unirational as a consequence one obtains that for infinitely many values of t the resulting elliptic curve has rank at least 1

common algebraic structure for the calogero sutherland models we investigate a common algebraic structure for the rational and trigonometric calogero sutherland models by using the exchange operator formalism we show that the set of jack polynomials whose arguments are dunkl type operators provides an orthogonal basis for the rational case

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

fundamental solutions for partial differential equations with reflection group invariance the dunkl differential difference operator associated with a finite reflection group is used to extend the weyl heisenberg algebra the subalgebra of mastersymmetries graded by the characters of the coxeter group w is constructed it provides us with a new class of intertwining operators for appropriate w invariant differential equations these intertwining operators turn out to be suitable for constructing fundamental solutions the remarkable example is given by the iterated wave operator with a calogero type potential for which the fundamental solution can be derived through the relevant intertwining operator in an explicit form it is found that under appropriate conditions such an equation satisfies huygens s principle in the sense that its fundamental solution possesses a nontrivial inner lacuna

rationally connected varieties over finite fields let x be a geometrically rational or more generally separably rationally connected variety over a finite field k we prove that if k is large enough then x contains many rational curves defined over k as a consequence we prove that r equivalence is trivial on x if k is large enough these results imply the following conjecture of j l colliot thelene let y be a rationally connected variety over a number field f for a prime p let y p denote the corresponding variety over the local field f p then for almost all primes p the chow group of 0 cycles on y p is trivial and r equivalence is trivial on y p

a positive radial product formula for the dunkl kernel it is an open conjecture that generalized bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated dunkl operators in this paper a partial result towards this conjecture is proven namely a positive radial product formula for the non symmetric counterpart of the generalized bessel function the dunkl kernel radial here means that one of the factors in the product formula is replaced by its mean over a sphere the key to this product formula is a positivity result for the dunkl type spherical mean operator it can also be interpreted in the sense that the dunkl type generalized translation of radial functions is positivity preserving as an application we construct dunkl type homogeneous markov processes associated with radial probability distributions

towards the brauer manin obstruction on varieties fibred over the projective line abstract recently dasheng wei proved that the brauer manin obstruction is the only obstruction to the hasse principle for 0 cycles of degree 1 on some fibrations over the projective line defined by bi cyclic normic equations in the present paper we prove the exactness of the global to local sequence for chow groups of 0 cycles of such varieties this signifies that the brauer manin obstruction is also the only obstruction to weak approximation for 0 cycles of arbitrary degree our main theorem also generalizes several existing results