regression models for relative survival four approaches to estimating a regression model for relative survival using the method of maximum likelihood are described and compared the underlying model is an additive hazards model where the total hazard is written as the sum of the known baseline hazard and the excess hazard associated with a diagnosis of cancer the excess hazards are assumed to be constant within pre specified bands of follow up the likelihood can be maximized directly or in the framework of generalized linear models minor differences exist due to for example the way the data are presented individual aggregated or grouped and in some assumptions e g distributional assumptions the four approaches are applied to two real data sets and produce very similar estimates even when the assumption of proportional excess hazards is violated the choice of approach to use in practice can therefore be guided by ease of use and availability of software we recommend using a generalized linear model with a poisson error structure based on collapsed data using exact survival times the model can be estimated in any software package that estimates glms with user defined link functions including sas stata s plus and r and utilizes the theory of generalized linear models for assessing goodness of fit and studying regression diagnostics

a legendre spectral element method for the one dimensional fourth order equations abstract a legendre galerkin spectral element method is proposed to solve the one dimensional fourth order equations c 1 continuity between the elemental faces is imposed by constructing appropriate basis functions the method leads to linear systems with sparse matrices for the discrete variational formulations rigorous error analysis is carried out to establish the convergence of the method several numerical examples are provided to confirm the theoretical results

an overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies relative survival provides a measure of the proportion of patients dying from the disease under study without requiring the knowledge of the cause of death we propose an overall strategy based on regression models to estimate the relative survival and model the effects of potential prognostic factors the baseline hazard was modelled until 10 years follow up using parametric continuous functions six models including cubic regression splines were considered and the akaike information criterion was used to select the final model this approach yielded smooth and reliable estimates of mortality hazard and allowed us to deal with sparse data taking into account all the available information splines were also used to model simultaneously non linear effects of continuous covariates and time dependent hazard ratios this led to a graphical representation of the hazard ratio that can be useful for clinical interpretation estimates of these models were obtained by likelihood maximization we showed that these estimates could be also obtained using standard algorithms for poisson regression

sinc galerkin method for solving biharmonic problems there are many techniques available to numerically solve the biharmonic equation in this paper we show that the sinc galerkin method is a very effective tool in numerically solving this equation hermite interpolation is used to treat the nonhomogeneous boundary conditions our method is tested on examples and comparisons with other methods are made it is shown that the sinc galerkin method yields good results even when singularities occur at the boundaries

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

riesz distributions and laplace transform in the dunkl setting of type a abstract we study riesz distributions in the framework of rational dunkl theory associated with root systems of type a as an important tool we employ a laplace transform involving the associated dunkl kernel which essentially goes back to macdonald 18 but was so far only established at a formal level we give a rigorous treatment of this transform based on suitable estimates of the type a dunkl kernel our main result is a precise analogue in the dunkl setting of a well known result by gindikin stating that a riesz distribution on a symmetric cone is a positive measure if and only if its exponent is contained in the wallach set for riesz distributions in the dunkl setting we obtain an analogous characterization in terms of a generalized wallach set which depends on the multiplicity parameter on the root system

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

semiparametric analysis of the additive risk model in contrast to the proportional hazards model the additive risk model specifies that the hazard function associated with a set of possibly time varying covariates is the sum of rather than the product of the baseline hazard function and the regression function of covariates this formulation describes a different aspect of the association between covariates and the failure time than the proportional hazards model and is more plausible than the latter for many applications in the present paper simple procedures with high efficiencies are developed for making inference about the regression parameters under the additive risk model with an unspecified baseline hazard function

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

a legendre petrov galerkin method for fourth order differential equations in this paper we present a legendre petrov galekin method for one dimensional linear fourth order differential equations a legendre petrov galerkin and chebyshev collocation method is developed for the nonlinear kuramoto sivashinsky equation numerical results are presented to demonstrate the efficiency of the proposed schemes and optimal rates of convergence in the l 2 norm are rigorously derived

additive risk models for survival data with high dimensional covariates as a useful alternative to cox s proportional hazard model the additive risk model assumes that the hazard function is the sum of the baseline hazard function and the regression function of covariates this article is concerned with estimation and prediction for the additive risk models with right censored survival data especially when the dimension of the covariates is comparable to or larger than the sample size principal component regression is proposed to give unique and numerically stable estimators asymptotic properties of the proposed estimators component selection based on the weighted bootstrap and model evaluation techniques are discussed this approach is illustrated with analysis of the primary biliary cirrhosis clinical data and the diffuse large b cell lymphoma genomic data it is shown that this methodology is numerically stable and effective in dimension reduction while still being able to provide satisfactory prediction and classification results

global well posedness and scattering for a class of nonlinear dunkl schrodinger equations in this paper we introduce a class of nonlinear schrodinger equations associated with the dunkl operators in this regard we study local and global well posedness and the scattering theory associated with these equations

confidence bands for cumulative incidence curves under the additive risk model in the context of competing risks the cumulative incidence function is often used to summarize the cause specific failure time data as an alternative to the proportional hazards model the additive risk model is used to investigate covariate effects by specifying that the subject specific hazard function is the sum of a baseline hazard function and a regression function of covariates based on such a formulation we present an approach to constructing simultaneous confidence intervals for the cause specific cumulative incidence function of patients with given risk factors a melanoma data set is used for the purpose of illustration

model building in nonproportional hazard regression recent developments of statistical methods allow for a very flexible modeling of covariates affecting survival times via the hazard rate including also the inspection of possible time dependent associations despite their immediate appeal in terms of flexibility these models typically introduce additional difficulties when a subset of covariates and the corresponding modeling alternatives have to be chosen that is for building the most suitable model for given data this is particularly true when potentially time varying associations are given we propose to conduct a piecewise exponential representation of the original survival data to link hazard regression with estimation schemes based on of the poisson likelihood to make recent advances for model building in exponential family regression accessible also in the nonproportional hazard regression context a two stage stepwise selection approach an approach based on doubly penalized likelihood and a componentwise functional gradient descent approach are adapted to the piecewise exponential regression problem these three techniques were compared via an intensive simulation study an application to prognosis after discharge for patients who suffered a myocardial infarction supplements the simulation to demonstrate the pros and cons of the approaches in real data analyses

confluent hypergeometric orthogonal polynomials related to the rational quantum calogero system with harmonic confinement two families type a and type b of confluent hypergeometric polynomials in several variables are studied we describe the orthogonality properties differential equations and pieri type recurrence formulas for these families in the one variable case the polynomials in question reduce to the hermite polynomials type a and the laguerre polynomials type b respectively the multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum calogero models with harmonic confinement for the classical root systems and are closely connected to the symmetric generalized spherical harmonics investigated by dunkl

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

goodness of fit of relative survival models additive regression models are preferred over multiplicative models in the analysis of relative survival data such preferences are mainly grounded in practical experience with mostly cancer registries data where the basic assumption of the additivity of hazards is more likely to be met also the interpretation of coefficients is more meaningful in additive than in multiplicative models nonetheless the question of goodness of fit of the assumed model must still be addressed and while there is an abundance of methods to check the goodness of fit of multiplicative models the respective arsenal for additive models is almost empty we propose here a variety of procedures for testing the null hypothesis of a good fit these are based on partial residuals defined similarly to schoenfeld residuals familiar for cox model diagnostics the tests have appropriate sizes under the null hypothesis and good power under different alternatives we investigate their performance through simulations and apply the methods to data from a study into survival of colon cancer patients

radon transform on symmetric matrix domains let k r c h be the field of real complex or quaternionic numbers and mp q k the vector space of all p q matrices let x be the matrix unit ball in mn r r k consisting of contractive matrices as a symmetric space x g k o n r r o n r o r u n r r u n r u r and respectively sp n r r sp n r sp r the matrix unit ball y0 in mr r r with r n 1 is a totally geodesic submanifold of x and let y be the set of all g translations of the submanifold y0 the set y is then a manifold and an affine symmetric space we consider the radon transform rf y for functions f c 0 x defined by integration of f over the subset y and the dual transform rt f x x x for functions f y on y for 2r n 2r r with a certain evenness condition in the case k r we find a g invariant differential operator m and prove it is the right inverse of rtr rtr mf cf for f c 0 x c 0 the operator f rtr f is an integration of f against a singular function determined by the root systems of x and y0 we study the analytic continuation of the powers of the function and we find a bernstein sato type formula generalizing earlier work of the author in the set up of the berezin transform when x is a rank one domain of hyperbolic balls in kn 1 and y0 is the hyperbolic ball in kr 1 1 r n we obtain an inversion formula for the radon transform namely mrtr f cf this generalizes earlier results of helgason for non compact rank one symmetric spaces for the case r n 1

flexible modeling of the effects of continuous prognostic factors in relative survival relative survival methods permit separating the effects of prognostic factors on disease related excess mortality from their effects on other causes natural mortality even when individual causes of death are unknown as in conventional crude survival accurate assessment of prognostic factors requires testing and possibly modeling of non proportional effects and for continuous covariates of non linear relationships with the hazard we propose a flexible extension of the additive hazards relative survival model in which the observed all causes mortality hazard is represented by a sum of disease related excess and natural mortality hazards in our flexible model the three functions representing i the baseline hazard for excess mortality ii the time dependent effects and iii for continuous covariates non linear effects on the logarithm of this hazard are all modeled by low dimension cubic regression splines non parametric likelihood ratio tests are proposed to test the time dependent and non linear effects the accuracy of the estimated functions is evaluated in multivariable simulations to illustrate the new insights offered by the proposed model we apply it to re assess the effects of patient age and of secular trends on disease related mortality in colon cancer

yangian gelfand zetlin bases gl n jack polynomials and computation of dynamical correlation functions in the spin calogero sutherland model we consider the invariant calogero sutherland models with n 1 2 3 in the framework of symmetric polynomials

additive and multiplicative models for relative survival rates the relative survival rate is the ratio of the overall survival rate to that expected for demographically similar individuals in a reference population it is commonly used to estimate the effect of a particular disease on mortality when the cause of death is not reliably known the effect of the disease on the underlying hazards may be multiplicative or additive the former case has been considered elsewhere but seems biologically less plausible than the latter both models are examined in this paper the effect of the disease is assumed to be constant throughout the follow up period or to be piecewise constant within k follow up intervals maximum likelihood estimates and related statistics are presented as are simpler statistics based on moment estimators of the disease effect the moment based statistic for testing the homogeneity of r groups may be expressed as a sum of individual scores which are shown to be closely related to the logrank scores when follow up intervals are made arbitrarily small

estimating multilevel regional variation in excess mortality of cancer patients using integrated nested laplace approximation models of excess mortality with random effects were used to estimate regional variation in relative or net survival of cancer patients statistical inference for these models based on the markov chain monte carlo mcmc methods is computationally intensive and therefore not feasible for routine analyses of cancer register data this study assessed the performance of the integrated nested laplace approximation inla in monitoring regional variation in cancer survival poisson regression model of excess mortality including both spatially correlated and unstructured random effects was fitted to the data of patients diagnosed with ovarian and breast cancer in finland during 1955 2014 with follow up from 1960 through 2014 by using the period approach with five year calendar time windows we estimated standard deviations associated with variation i between hospital districts and ii between municipalities within hospital districts posterior estimates based on the inla approach were compared to those based on the mcmc simulation the estimates of the variation parameters were similar between the two approaches variation within hospital districts dominated in the total variation between municipalities in 2000 2014 the proportion of the average variation within hospital districts was 68 95 posterior interval 35 93 and 82 60 98 out of the total variation in ovarian and breast cancer respectively in the estimation of regional variation the inla approach was accurate fast and easy to implement by using the r inla package

jacobi and laguerre quasi orthogonal approximations and related interpolations in this paper we investigate jacobi quasi orthogonal approximation and generalized jacobi gauss lobatto interpolation we also propose laguerre quasi orthogonal approximation and generalized laguerre gauss radau interpolation a series of sharp results on these approximations are established which are applicable to spectral and pseudospectral methods for mixed nonhomogeneous boundary value problems of high order

additive hazards regression with current status data current status data arise when the only knowledge about the failure time of interest is whether the failure occurs before or after a random monitoring time we propose to analyse such data by the semiparametric additive hazards model which specifies that the hazard function for the failure time associated with a set of possibly time dependent covariates is the sum of an arbitrary baseline hazard function and a regression function of covariates under certain conditions on the monitoring time one can make inferences about the regression parameters of the additive hazards model by using the familiar asymptotic theory and software for the proportional hazards model with right censored data an application to a carcinogenicity experiment is provided

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

legendre galerkin spectral element method for the biharmonic equations and its applications abstract a spectral element method based on the legendre galerkin approximation is presented to solve the two dimensional biharmonic equations rigorous error analysis is carried out to establish the convergence of the method by constructing appropriate basis functions which satisfy the boundary conditions of the differential equations the discrete variational formulation is reduced to linear system with sparse and symmetric matrices which can be efficiently solved by a fast schur complement method accuracy test is provided to confirm the convergence rate of the theoretical results finally the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero reynolds number flow in a driven cavity the results compare well with established results

a multilevel excess hazard model to estimate net survival on hierarchical data allowing for non linear and non proportional effects of covariates the excess hazard regression model is an approach developed for the analysis of cancer registry data to estimate net survival that is the survival of cancer patients that would be observed if cancer was the only cause of death cancer registry data typically possess a hierarchical structure individuals from the same geographical unit share common characteristics such as proximity to a large hospital that may influence access to and quality of health care so that their survival times might be correlated as a consequence correct statistical inference regarding the estimation of net survival and the effect of covariates should take this hierarchical structure into account it becomes particularly important as many studies in cancer epidemiology aim at studying the effect on the excess mortality hazard of variables such as deprivation indexes often available only at the ecological level rather than at the individual level we developed here an approach to fit a flexible excess hazard model including a random effect to describe the unobserved heterogeneity existing between different clusters of individuals and with the possibility to estimate non linear and time dependent effects of covariates we demonstrated the overall good performance of the proposed approach in a simulation study that assessed the impact on parameter estimates of the number of clusters their size and their level of unbalance we then used this multilevel model to describe the effect of a deprivation index defined at the geographical level on the excess mortality hazard of patients diagnosed with cancer of the oral cavity copyright 2016 john wiley sons ltd

nonparametric testing for multiple survival functions with non inferiority margins new nonparametric tests for the ordering of multiple survival functions are developed with the possibility of right censorship taken into account the motivation comes from non inferiority trials with multiple treatments the proposed tests are based on nonparametric likelihood ratio statistics which are known to provide more powerful tests than wald type procedures but in this setting have only been studied for pairs of survival functions or in the absence of censoring we introduce a novel type of pool adjacent violator algorithm that leads to a complete solution of the problem the limit distributions can be expressed as weighted sums of squares involving projections of certain gaussian processes onto the given ordered alternative a simulation study shows that the new procedures have superior power to a competing combined pairwise cox model approach we illustrate the proposed methods using data from a three arm non inferiority trial

additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time dependent effects relative survival is used to estimate patient survival excluding causes of death not related to the disease of interest rather than using cause of death information from death certificates which is often poorly recorded relative survival compares the observed survival to that expected in a matched group from the general population models for relative survival can be expressed on the hazard mortality rate scale as the sum of two components where the total mortality rate is the sum of the underlying baseline mortality rate and the excess mortality rate due to the disease of interest previous models for relative survival have assumed that covariate effects act multiplicatively and have thus provided relative effects of differences between groups using excess mortality rate ratios in this paper we consider i the use of an additive covariate model which provides estimates of the absolute difference in the excess mortality rate and ii the use of fractional polynomials in relative survival models for the baseline excess mortality rate and time dependent effects the approaches are illustrated using data on 115 331 female breast cancer patients diagnosed between 1 january 1986 and 31 december 1990 the use of additive covariate relative survival models can be useful in situations when the excess mortality rate is zero or slightly less than zero and can provide useful information from a public health perspective the use of fractional polynomials has advantages over the usual piecewise estimation by providing smooth estimates of the baseline excess mortality rate and time dependent effects for both the multiplicative and additive covariate models all models presented in this paper can be estimated within a generalized linear models framework and thus can be implemented using standard software copyright 2005 john wiley sons ltd

dynamic regression hazards models for relative survival a natural way of modelling relative survival through regression analysis is to assume an additive form between the expected population hazard and the excess hazard due to the presence of an additional cause of mortality within this context the existing approaches in the parametric semiparametric and non parametric setting are compared and discussed we study the additive excess hazards models where the excess hazard is on additive form this makes it possible to assess the importance of time varying effects for regression models in the relative survival framework we show how recent developments can be used to make inferential statements about the non parametric version of the model this makes it possible to test the key hypothesis that an excess risk effect is time varying in contrast to being constant over time in case some covariate effects are constant we show how the semiparametric additive risk model can be considered in the excess risk setting providing a better and more useful summary of the data estimators have explicit form and inference based on a resampling scheme is presented for both the non parametric and semiparametric models we also describe a new suggestion for goodness of fit of relative survival models which consists on statistical and graphical tests based on cumulative martingale residuals this is illustrated on the semiparametric model with proportional excess hazards we analyze data from the trace study using different approaches and show the need for more flexible models in relative survival

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

timely communication efficient algorithms for solving a fourth order equation with the spectral galerkin method we show that one can derive an o n 3 spectral galerkin method for fourth order biharmonic type elliptic equations based on the use of chebyshev polynomials the use of chebyshev polynomials provides a fast transform between physical and spectral space which is advantageous when a sequence of problems must be solved e g as part of a nonlinear iteration this improves the result of shen siam j sci comput 16 1995 pp 74 87 which reported an o n 4 algorithm inferior to the o n 3 method developed earlier shen siam j sci comput 15 1994 pp 1440 1451 based on legendre polynomials but less practical in the case of multiple problems we further compare our method with an improved implementation of the legendre galerkin method based on the same approach

a model combining excess and relative mortality for population based studies considering expected mortality provides an attractive approach to analyse mortality of population based cohorts of patients presenting with a chronic disease two classes of methods are available either modelling the excess mortality using an additive hazard model or modelling the relative mortality using a multiplicative hazard model because these two models are informative to look for factors associated with mortality related to a chronic disease we developed an alternative model modelling both the excess and the relative mortality we generalised andersen and vaeth s model to fit covariates and obtain directly an estimation of the excess mortality ratio and relative mortality ratio for each covariate we assessed the performances of the combined model by using simulations and it appeared satisfactorily we illustrate the combined model by data collected in patients presenting with end stage renal disease and treated by dialysis the combined model offers the possibility of performing pure additive and multiplicative models and thus to compare their log likelihoods the combined model appears useful to select one of these pure models or to conclude to the need of modelling both excess and relative mortality in this latter case our model enabled better describing the effect of covariates on the excess and relative mortality

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

orthogonal polynomials on the sphere with octahedral symmetry for any finite reflection group g acting on rn there is a family of g invariant measures h2 dw where h is a certain product of linear functions whose zero sets are the reflecting hyperplanes for g on the unit sphere and an associated partial differential operator lh f a fh fa h a is the laplacian analogously to spherical harmonics there is an orthogonal with respect to h2 do decomposition of homogeneous polynomials that is if p is of degree n then i n 21 p x j ix12jpn 2 x 0 where lh p 0 and deg p i for each i but with the restriction that p and p must all be g invariant the main topic is the hyperoctahedral group with h x x ix2 i n u x xj i j the special case n 2 leads to jacobi polynomials a detailed study of the case n 3 is made an important result is the construction of a third order differential operator that maps polynomials associated to h with indices a i to those associated with a 2 fi 1 introduction regular figures have long been objects of fascination we study here the interplay between them and the theory of orthogonal polynomials this will also touch on finite groups generated by reflections and their polynomial invariants as well as a generalization of spherical harmonics the underlying situation throughout will consist of spaces of homogeneous polynomials on rn that satisfy a second order differential equation and whose restrictions to the unit sphere have an orthogonality relation with respect to a weight function and of a finite reflection group for which all these objects are invariant the weight measure h2 x dw x on the unit sphere s in r3 where h x xlx2x3 a x 23 3 a 2 dw is surface area and the associated polynomials will be studied most intensely some of the highlights of the theory are as follows define the differential operator lhf f fah received by the editors april 19 1982 and in revised form december 15 1982 1980 mathematics subject classification primary 33a45 33a65 secondary 20h15 this research was partially supported by nsf grant mcs 81 02581 d 984 american mathematical society 0002 9947 84 1 00 25 per page

stokes eigenmodes in square domain and the stream function vorticity correlation the stokes eigenmodes in the square are numerically determined and their symmetry properties are identified the spectra evolution laws are in excellent qualitative agreement with the theoretical asymptotic predictions proposed by constantin and foias in navier stokes equations university of chicago press 1988 k k o k the slopes are reported here and are found to be specific to the eigenmodes symmetry family the dynamic equilibria are analyzed and show a linear relationship between the vorticity and the stream function in the core of the eigenmodes these features of the stokes eigenmodes confined in the square are shared by the fully periodic stokes eigenmodes

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

flexible parametric models for relative survival with application in coronary heart disease relative survival is frequently used in population based studies as a method for estimating disease related mortality without the need for information on cause of death we propose an extension to relative survival of a flexible parametric model proposed by royston and parmar for censored survival data the model provides smooth estimates of the relative survival and excess mortality rates by using restricted cubic splines on the log cumulative excess hazard scale the approach has several advantages over some of the more standard relative survival models which adopt a piecewise approach the main being the ability to model time on a continuous scale the survival and hazard functions are obtained analytically and it does not use split time data

a general framework for parametric survival analysis parametric survival models are being increasingly used as an alternative to the cox model in biomedical research through direct modelling of the baseline hazard function we can gain greater understanding of the risk profile of patients over time obtaining absolute measures of risk commonly used parametric survival models such as the weibull make restrictive assumptions of the baseline hazard function such as monotonicity which is often violated in clinical datasets in this article we extend the general framework of parametric survival models proposed by crowther and lambert journal of statistical software 53 12 2013 to incorporate relative survival and robust and cluster robust standard errors we describe the general framework through three applications to clinical datasets in particular illustrating the use of restricted cubic splines modelled on the log hazard scale to provide a highly flexible survival modelling framework through the use of restricted cubic splines we can derive the cumulative hazard function analytically beyond the boundary knots resulting in a combined analytic numerical approach which substantially improves the estimation process compared with only using numerical integration user friendly stata software is provided which significantly extends parametric survival models available in standard software

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

weighted least squares estimation for aalen s additive risk model abstract cox s proportional hazards model has so far been the most popular model for the regression analysis of censored survival data however the additive risk model of aalen can provide a useful and biologically more plausible alternative aalen s model stipulates that the conditional hazard function for a subject whose covariates are y y 1 yp has the form h t y y t where 1 p is an unknown vector of hazard functions this article discusses inference for based on a weighted least squares wls estimator of the vector of cumulative hazard functions the asymptotic distribution of the wls estimator is derived and used to obtain confidence intervals and bands for the cumulative hazard functions both a grouped data and a continuous data version of the estimator are examined an extensive simulation study is carried out the method is applied to grouped data on the incidence of cancer mortality among japanese atomic bomb survivors

biorthogonal polynomials associated with reflection groups and a formula of macdonald abstract dunkl operators are differential difference operators on r n which generalize partial derivatives they lead to generalizations of laplace operators fourier transforms heat semigroups hermite polynomials and so on in this paper we introduce two systems of biorthogonal polynomials with respect to dunkl s gaussian distributions in a canonical way these systems called appell systems admit many properties known from classical hermite polynomials and turn out to be useful for the analysis of dunkl s gaussian distributions in particular these polynomials lead to a new proof of a generalized formula of macdonald due to dunkl the ideas for this paper are taken from recent works on non gaussian white noise analysis and from the umbral calculus

a legendre spectral galerkin method for the biharmonic dirichlet problem a legendre spectral galerkin method is presented for the solution of the biharmonic dirichlet problem on a square the solution and its laplacian are approximated using the set of basis functions suggested by shen which are linear combinations of two legendre polynomials a schur complement approach is used to reduce the resulting linear system to one involving the approximation of the laplacian of the solution on the two vertical sides of the square the schur complement system is solved by a preconditioned conjugate gradient method or the cholesky method the total cost of the algorithm is o n3 numerical results demonstrate the spectral convergence of the method