quantum cohomology of the springer resolution abstract let g denote a complex semi simple simply connected group and b its associated flag variety we identify the equivariant quantum differential equation for the cotangent bundle t b with the affine knizhnik zamolodchikov connection of cherednik and matsuo this recovers kim s description of quantum cohomology of b as a limiting case a parallel result is proven for resolutions of the slodowy slices extension to arbitrary symplectic resolutions is discussed

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

sample size calculations in the presence of competing risks recently with the growth of statistical developments for competing risks analysis some methods have been proposed to compute sample size in this context these methods differ from a modelling approach one is based on the cox regression model for the cause specific hazard while another relies on the fine and gray regression model for the subdistribution hazard of a competing risk in this work we compare these approaches derive a new sample size for comparing cumulative incidence functions when the hazards are not proportional either cause specific or subdistribution and give practical advices to choose the approach best suited for the study question

levinson s theorem and titchmarsh weyl m λ theory for dirac systems on demontre un theoreme de levinson pour un systeme de dirac avec une extremite singuliere on exprime le nombre d etats lies en termes de changement dans la phase asymptotique d une solution appropriee et en termes de facteurs dont les valeurs dependent de la presence des etats demi lies

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

on the titchmarsh weyl m λ coefficient and spectral density for a dirac system for a dirac system on the line we study the jost solutions and the existence of half bound states a characterisation of this phenomenon is given in terms of the m function which supplements a well known spectral characterisation for the system as a corollary we deduce the spectral density behaviour at the spectral gap endpoints which is shown to depend on the possible existence of half bound states

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

one dimensional double hecke algebra and gaussian sums we introduce and describe one dimensional cyclotomic gauss selberg sums generalizing the classical gaussian sums they correspond to irreducible self dual unitary spherical representations of the one dimensional double affine hecke algebra

a class of k sample tests for comparing the cumulative incidence of a competing risk in this paper for right censored competing risks data a class of tests developed for comparing the cumulative incidence of a particular type of failure among different groups the tests are based on comparing weighted averages of the hazards of the subdistribution for the failure type of interest asymptotic results are derived by expressing the statistics in terms of counting processes and using martingale central limit theory it is proposed that weight functions very similar to those for the g p tests from ordinary survival analysis be used simulation results indicate that the asymptotic distributions provide adequate approximations in moderate sized samples

integrability and huygens principle on symmetric spaces the explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found these symmetric spaces have the following characteristic property all multiplicities of their restricted roots are even as a corollary in the odd dimensional case one has that the huygens principle in hadamard s sense for these equations is fulfilled we consider also the heat and laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions heat kernel and green s function this continues our previous investigations 16 of the spherical functions on the same symmetric spaces based on the fact that the radial part of the laplace beltrami operator on such a space is related to the algebraically integrable case of the generalised calogero sutherland moser quantum system in the last section of this paper we apply the methods of heckman and opdam to extend our results to some other symmetric spaces in particular to complex and quaternian grassmannians

levinson s theorem for a fermion monopole system the author obtains an expression for levinson s 1949 theorem for a charged fermion moving in the background field of a dirac monopole

fueter s theorem and its generalizations in dunkl clifford analysis in this paper we give a construction of dunkl monogenic and dunkl harmonic functions starting from holomorphic functions in the plane this construction has the advantage of not needing dunkl s intertwining operator or dunkl spherical harmonics to this end we study vekua type systems and prove a version of fueter s theorem in the case of finite reflection groups important examples such as a dunkl monogenic gaussian distribution or a cauchy kernel will be given at the end

harmonic analysis associated with the jacobi dunkl operator on π 2 π 2 we consider a differential difference operator 1 2 1 2 on 2 2 the eigenfunction of this operator equal to 1 at zero is related to the jacobi polynomials and to their derivatives we give a laplace integral representation for this function called the jacobi dunkl polynomial next we study the harmonic analysis associated with the operator

misspecified regression model for the subdistribution hazard of a competing risk we consider a competing risks setting when evaluating the prognostic influence of an exposure on a specific cause of failure two main regression models are used in such analyses the cox cause specific proportional hazards model and the subdistribution proportional hazards model they are exemplified in a real data example focusing on relapse free interval in acute leukaemia patients we examine the properties of the estimator based on the latter model when the true model is the former an explicit relationship between subdistribution hazards ratio and cause specific hazards ratio is derived assuming a flexible parametric distribution for latent failure times

regression modeling of competing risks data based on pseudovalues of the cumulative incidence function typically regression models for competing risks outcomes are based on proportional hazards models for the crude hazard rates these estimates often do not agree with impressions drawn from plots of cumulative incidence functions for each level of a risk factor we present a technique which models the cumulative incidence functions directly the method is based on the pseudovalues from a jackknife statistic constructed from the cumulative incidence curve these pseudovalues are used in a generalized estimating equation to obtain estimates of model parameters we study the properties of this estimator and apply the technique to a study of the effect of alternative donors on relapse for patients given a bone marrow transplant for leukemia

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

support theorem for the fundamental solution to the schrodinger equation on certain compact symmetric spaces abstract in this paper we construct the fundamental solution to the schodinger equation on a compact symmetric space with even root multiplicities using shift operators of heckman and opdam next we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time moreover we also show that generalized gauss sums appear in the expression of the fundamental solution

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

sekiguchi debiard operators at infinity we construct a family of pairwise commuting operators such that the jack symmetric functions of infinitely many variables x 1 x 2 ldots are their eigenfunctions these operators are defined as limits at n of renormalised sekiguchi debiard operators acting on symmetric polynomials in the variables x 1 ldots x n they are differential operators in terms of the power sum variables p n x 1 n x 2 n cdots and we compute their symbols by using the jack reproducing kernel our result yields a hierarchy of commuting hamiltonians for the quantum calogero sutherland model with an infinite number of bosonic particles in terms of the collective variables of the model our result also yields the elementary step operators for the jack symmetric functions

kaplan meier marginal or conditional probability curves in summarizing competing risks failure time data in the context of competing risks the kaplan meier estimator is often unsuitable for summarizing failure time data we discuss some alternative descriptive methods including marginal probability and conditional probability estimators two sample test statistics are also presented

spherical dunkl monogenics and a factorization of the dunkl laplacian in this paper we consider and study a factorization of the dunkl laplacian in terms of spherical coordinates this allows for the construction of a direct sum decomposition of spherical dunkl harmonics by explicit representation in spherical coordinates of dunkl harmonics one obtains explicit projection operators from dunkl harmonics to inner resp outer dunkl monogenics concrete examples of spherical dunkl monogenics will be given at the end

lectures on affine hecke algebras and macdonald s conjectures this paper gives a review of cherednik s results on the representation theoretic approach to macdonald polynomials and related special functions macdonald polynomials are a remarkable 2 parameter family of polynomials which can be associated to every root system as special cases they include the schur functions the q jacobi polynomials and certain spherical functions on real and p adic symmetric spaces they have a number of elegant combinatorial properties which however are extremely difficult to prove in this paper we show that a natural setup for studying these polynomials is provided by the representation theory of hecke algebras and show how this can be used to prove some of the combinatorial identities for macdonald polynomials

titchmarsh weyl theory and levinson s theorem for dirac operators for a dirac system with a perturbation potential v r obeying integral 0infinity 1 r mod dr infinity the behaviour of the titchmarsh weyl m function near the spectral gap endpoints is examined as a consequence we obtain another demonstration of the levinson theorem 1949

integrability in the theory of schrodinger operator and harmonic analysis the algebraic integrability for the schrodinger equation in n and the role of the quantum calogero sutherland problem and root systems in this context are discussed for the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding sutherland operator is found as an application the explicit formula for the zonal spherical functions on the symmetric spacessu 2n spn type a ii in cartan notations is presented

on the levinson theorem for dirac operators for the dirac equation with potential v r obeying 0 1 r v r dr we prove a relativistic version of levinson s theorem that relates the number of bound states in the spectral gap m m to the variation of an appropriate phase along the continuous part of the spectrum in the process the asymptotic properties of the jost function as e m are analyzed in detail the connection with the nonrelativistic version of levinson s theorem is also established

on elliptic calogero moser systems for complex crystallographic reflection groups abstract to every irreducible finite crystallographic reflection group i e an irreducible finite reflection group g acting faithfully on an abelian variety x we attach a family of classical and quantum integrable systems on x with meromorphic coefficients these families are parametrized by g invariant functions of pairs t s where t is a hypertorus in x of codimension 1 and s g is a reflection acting trivially on t if g is a real reflection group these families reduce to the known generalizations of elliptic calogero moser systems but in the non real case they appear to be new we give two constructions of the integrals of these systems an explicit construction as limits of classical calogero moser hamiltonians of elliptic dunkl operators as the dynamical parameter goes to 0 implementing an idea of v buchstaber g felder and a veselov 1994 bfv and a geometric construction as global sections of sheaves of elliptic cherednik algebras for the critical value of the twisting parameter we also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions

shift operators for the quantum calogero sutherland problems via knizhnik zamolodchikov equation we give a natural interpretation of the shift operators for calogero sutherland quantum problem via kz equation using matsuo cherednik mappings the explicit formulas for the inversions of these mappings and versions of shift operators for kz equations are also found as an application we show that the shift operator can be described via a factorization problem for an appropriate quantum integral discriminant of the calogero system

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

effect of misspecification of regression models in the analysis of survival data the accelerated life and proportional hazards families of models are widely used in the analysis of survival data we examine the question if interest lies in the qualitative effect on failure of various explanatory variables how critical is the choice of model family in assessing the relative importance of the explanatory variables we consider small effects when censoring is independent of the explanatory variables to first order the regression parameters are proportional under the alternative models this is not true when the censoring varies appreciably with the explanatory variables some higherorder theory is examined for the special case of the two sample problem

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

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

a semiparametric random effects model for multivariate competing risks data we propose a semiparametric random effects model for multivariate competing risks data when the failures of a particular type are of interest under this model the marginal cumulative incidence functions follow a generalized semiparametric additive model the associations between the cause specific failure times can be studied through dependence parameters of copula functions that are allowed to depend on cluster level covariates a cross odds ratio type measure is proposed to describe the associations between cause specific failure times and its relationship to the dependence parameters is explored we develop a two stage estimation procedure where the marginal models are estimated in the first stage and the dependence parameters are estimated in the second stage the large sample properties of the proposed estimators are derived the proposed procedures are applied to danish twin data to model the cumulative incidence for the age of natural menopause and to investigate the association in the onset of natural menopause between monozygotic and dizygotic twins

properties of some families of hypergeometric orthogonal polynomials in several variables limiting cases are studied of the koornwinder macdonald multivariable generalization of the askey wilson polynomials we recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable wilson continuous hahn and jacobi type polynomials respectively for each class of polynomials we provide systems of difference or differential equations recurrence relations and expressions for the norms of the polynomials in terms of the norm of the constant polynomial

macdonald polynomials and algebraic integrability we construct explicitly nonpolynomial eigenfunctions of the difference operators by macdonald in the case t qk k z this leads to a new more elementary proof of several macdonald conjectures proved first by cherednik we also establish the algebraic integrability of macdonald operators at t qk k z generalizing the result of etingof and styrkas our approach works uniformly for all root systems including the bcn case and related koornwinder polynomials moreover we apply it for a certain deformation of the an root system where the previously known methods do not work

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

inference for events with dependent risks in multiple endpoint studies abstract kaplan meier and cumulative incidence functions are not sufficient descriptive devices for studies that have multiple time to event endpoints for example in cancer treatment research the probability of tumor recurrence conditional on not having died from treatment related toxicities and the prevalence of graft versus host disease among leukemia free patients surviving a bone marrow transplant are of interest these quantities can be estimated nonparametrically using simple functions of several kaplan meier and cumulative incidence estimates for events with possibly dependent risks we derive asymptotic distribution theory for such functions by representing kaplan meier cumulative incidence and cumulative hazard estimators as sums of iid random variables variance estimation also follows directly from this representation two sample test statistics with asymptotic null distribution theory are presented several examples illustrate the utility of these results

intertwining operators for a degenerate double affine hecke algebra and multivariable orthogonal polynomials several properties of the multivariable hermite and laguerre polynomials associated with the quantum calogero models are investigated by using the operators that intertwine representations of a degenerate version of the double affine hecke algebra as applications raising operators and shift operators for the polynomials are constructed in a unified manner

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

boundary k matrix elliptic dunkl operator and quantum many body systems the infinite dimensional representation of the boundary k operator the solution of the reflection equation is considered the trigonometric k matrix is studied as a degenerate case of the elliptic operator a method to construct the elliptic dunkl operator is proposed and the relationship with the quantum many body problem is also discussed

fermionic limit of the calogero sutherland system we present a construction of an integrable model as a projective type limit of calogero sutherland models of n fermionic particles when n tends to infinity explicit formulas for limits of dunkl operators and of commuting hamiltonians by means of vertex operators are given we present a construction of an integrable model as a projective type limit of calogero sutherland models of n fermionic particles when n tends to infinity explicit formulas for limits of dunkl operators and of commuting hamiltonians by means of vertex operators are given

elliptic quantum many body problem and double affine knizhnik zamolodchikov equation the elliptic matrix quantum olshanetsky perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the dunkl operators its equivalence with the double affine generalization of the knizhnik zamolodchikov equation in the induced representations is established

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

jack laurent symmetric functions we develop the general theory of jack laurent symmetric functions which are certain generalizations of the jack symmetric functions depending on an additional parameter p0

joint inference for competing risks survival data abstractthis article develops joint inferential methods for the cause specific hazard function and the cumulative incidence function of a specific type of failure to assess the effects of a variable on the time to the type of failure of interest in the presence of competing risks joint inference for the two functions are needed in practice because i they describe different characteristics of a given type of failure ii they do not uniquely determine each other and iii the effects of a variable on the two functions can be different and one often does not know which effects are to be expected we study both the group comparison problem and the regression problem we also discuss joint inference for other related functions our simulation shows that our joint tests can be considerably more powerful than the bonferroni method which has important practical implications to the analysis and design of clinical studies with competing risks data we illustrate our method using a hodgkin disease data and a

bispectrality for the quantum ruijsenaars model and its integrable deformation an elementary construction of the eigenfunctions for the quantum rational ruijsenaars model with integer coupling parameter is presented as a by product we establish the bispectral duality between this model and the trigonometric calogero moser model in particular this gives a new way for calculating jack polynomials we propose also a certain one parameter deformation of the ruijsenaars model proving its integrability and bispectrality the generalizations related to other root systems and difference operators by macdonald are considered