grid generation techniques in computational fluid dynamics revue des differentes methodes on montre que les regions tridimensionnelles sont mieux traitees par fragmentation du champ en sous regions avec generation de grilles dans chaque sous region interet des grilles adaptatives couplees a des solutions physiques

on some new contiguous relations for the gauss hypergeometric function with applications contiguous relations for hypergeometric series contain an enormous amount of hidden information applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series in this paper a new set of contiguous function relations are established applications of such relations to hypergeometric summation formulas and the theory of jacobi polynomials are presented

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

spectral analysis and the haar functional on the quantum su 2 group the haar functional on the quantumsu 2 group is the analogue of invariant integration on the groupsu 2 if restricted to a subalgebra generated by a self adjoint element the haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both these results by woronowicz and koornwinder have been proved by using the corepresentation theory of the quantumsu 2 group and schur s orthogonality relations for matrix elements of irreducible unitary corepresentations these results are proved here by using a spectral analysis of the generator of the subalgebra the spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of jacobi matrices

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

relative risk trees for censored survival data a method is developed for obtaining tree structured relative risk estimates for censored survival data the first step of a full likelihood estimation procedure is used in a recursive partitioning algorithm that adopts most aspects of the widely used classification and regression tree cart algorithm of breiman et al 1984 classification and regression trees belmont california wadsworth the performance of the technique is investigated through stimulation and compared to the tree structured survival methods proposed by davis and anderson 1989 statistics in medicine 8 947 961 and therneau grambsch and fleming 1990 biometrika 77 147 160

extensions of two q series expansions with applications to biorthogonal rational functions in this paper we derive two most general possible q hypergeometric expansion formulae for 12 11 q and r s z respectively the results are unique in the sense that no such results are available in the literature beyond 4 3 q so far and have the advantage that these results are also applicable to the top level 10 9 q biorthogonal rational functions

survival trees by goodness of split abstract a tree based method for censored survival data is developed based on maximizing the difference in survival between groups of patients represented by nodes in a binary tree the method includes a pruning algorithm with optimal properties analogous to the classification and regression tree cart pruning algorithm uniform convergence of the estimates of the conditional cumulative hazard and survival functions is discussed and an example is given to show the utility of the algorithm for developing prognostic classifications for patients

two dimensional unsteady euler equation solver for arbitrarily shaped flow regions a new technique is described for solving supersonic fluid dynamics problems containing multiple regions of continuous flow each bounded by a permeable or impermeable surface region boundaries are in general arbitrarily shaped and time dependent discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system multiple regions are coupled together through the boundary conditions the technique has been applied to a variety of problems including a shock diffraction problem and supersonic flow over a pointed ogive

on the optimality of conditional expectation as a bregman predictor we consider the problem of predicting a random variable x from observations denoted by a random variable z it is well known that the conditional expectation e x z is the optimal l sup 2 predictor also known as the least mean square error predictor of x among all borel measurable functions of z in this orrespondence we provide necessary and sufficient conditions for the general loss functions under which the conditional expectation is the unique optimal predictor we show that e x z is the optimal predictor for all bregman loss functions blfs of which the l sup 2 loss function is a special case moreover under mild conditions we show that the blfs are exhaustive i e if for every random variable x the infimum of e f x y over all constants y is attained by the expectation e x then f is a blf

contiguous relations basic hypergeometric functions and orthogonal polynomials iii associated continuous dual q hahn polynomials explicit solutions for the three term recurrence satisfied by associated continuous dual q hahn polynomials are obtained a minimal solution is identified and an explicit expression for the related continued fraction is derived the absolutely continuous component of the spectral measure is obtained eleven limit cases are discussed in some detail these include associated big q laguerre associated wall associated al salam chihara associated al salam carlitz i and associated continuous q hermite polynomials

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

contiguous relations basic hypergeometric functions and orthogonal polynomials ii associated big q jacobi polynomials abstract explicit solutions to the recurrence relation for the associated big q jacobi polynomials are obtained using 3 2 contiguous relations a new continued fraction is derived with the help of the minimal solution the orthogonality for polynomial solutions in some special cases is discussed two exceptional cases yield the basic analogues of entry 25 and entry 33 in chapter 12 of ramanujan s second notebook connections of the general case with other associated cases are indicated

contiguous relations and their computations for f12 hypergeometric series the hypergeometric function f12 a 1 a 2 a 3 z plays an important role in mathematical analysis and its application gauss defined two hypergeometric functions to be contiguous if they have the same power series variable if two of the parameters are pairwise equal and if the third pair differs by 1 he showed that a hypergeometric function and any two other contiguous to it are linearly related in this paper we present an interesting formula as a linear relation of three shifted gauss polynomials in the three parameters a 1 a 2 and a 3 more precisely we obtained a recurrence relation including f12 a email protected 1 a 2 a 3 z f12 a 1 a email protected 2 a 3 z andf12 a 1 a 2 a email protected 3 z for any arbitrary integers a 1 a 2 and a 3

estimation of the failure time distribution in the presence of informative censoring we present a method for estimating the survival curve of a continuous failure time random variable from right censored data our method allows adjustment for informative censoring due to measured prognostic factors for time to event and censoring while simultaneously quantifying the sensitivity of the inference to residual dependence between failure and censoring due to unmeasured factors we present the results of a simulation study and illustrate our approach using data from the aids clinical trial group 175 study

regression analysis with randomly right censored data this paper proposes a new estimator of the parameter vector in a linear regression model when the observations are randomly censored on the right and when the error distribution is unknown this estimator is explicitly defined and easily computable the paper contains sufficient conditions under which this estimator is mean square consistent and asymptotically normal a numerical example is given

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

zeros and orthogonality of the askey wilson polynomials for q a root of unity we study some properties of the askey wilson polynomials awp when q is a primitive nth root of unity for general four parameter awp zeros of the nth polynomial and the orthogonality measure are found explicitly special subclasses of the awp e g the continuous q jacobi and big q jacobi polynomials are considered in detail a set of discrete weight functions positive on a real interval is described some new trigonometric identities related to the awp are obtained normalization conditions of some polynomials are expressed in terms of the gauss sums

symmetry groups of bailey s transformations for 10 φ 9 series although most of the symmetry groups or invariance groups associated with two term transformations between basic hypergeometric series have been studied and identified this is not the case for the most general transformation formulae in the theory of basic hypergeometric series namely bailey s transformations for f910 series first we show that the invariance group for both bailey s two term transformations for terminating f910 series and bailey s four term transformations for non terminating f910 series rewritten as a two term transformation of a so called f series is isomorphic to the weyl group of type e 6 we continue our recent research concerning the group structure underlying three term transformations s lievens j van der jeugt invariance groups of three term transformations for basic hypergeometric series j comput appl math 197 2006 1 14 and demonstrate that the group associated with a three term transformation between these f series each admitting bailey s two term transformation is the weyl group of type e 7 we do this by giving a description of the root system of type e 7 that allows to find a transformation between equivalent three term identities in an easy way a computation shows that there are five essentially different three term transformations between these f series we give an explicit form of each of these five transformations in an elegant way to our knowledge only one of these transformations has appeared in the literature

extensions of certain classical integrals of erdelyi for gauss hypergeometric functions it is shown how series manipulation technique and certain classical summation theorems for hypergeometric series can be used to prove erdelyi s integral representations for 2f1 z originally proved using fractional calculus the method not only leads to generalizations but also leads to new integrals of erdelyi type for certain q 1fq z and corresponding pochhammer contour integrals the technique outlined here compared to the method of fractional calculus seems to be more effective as it not only provides transparent elementary proofs of erdelyi s integrals but even leads to various generalizations

interpolating matrix method a finite difference method for arbitrary arrangeme t of mesh points abstract the interpolating matrix method imm is proposed as a finite difference method applicable to the arbitrary arrangement of mesh points in the imm differential coefficients at a mesh point are expressed with linear combinations of a certain number of neighboring mesh points a difference equation is easily constructed by substituting the linear combinations individually into the differential coefficients of the differential equation a computer code for two dimensional incompressible turbulent flows has been developed and some calculation examples are presented to demonstrate the utility of the imm

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

grid evolution in time asymptotic problems the selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration a new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh the technique has been used to solve the simple viscous burgers equation in one and two dimensions substantial error reductions are demonstrated the treatment of boundary points and the effect of using different error criteria in generating grids are discussed

investigating the effects of ties on measures of concordance the concordance between predicted and observed outcomes referred to as the c index or c statistic is frequently used to quantify the discriminatory ability of a prognostic model it also commonly serves as a basis for distinguishing predictive strength between different models two alternative formulations of the c index are widely used one completely excluding ties from the computation c tied out and the other including ties c tied in however there has been little research concerning the effects of ties on these two measures in this paper we characterize changes in the performance of c tied in and c tied out for progressively less coarse or more coarse partitions of the data our theoretical and simulation results show that both measures can be heavily dependent on the number of tied pairs and their results can be substantially divergent we examine potential ambiguities that can occur when the two measures diverge in the presence of a substantial proportion of tied pairs we recommend that both c indices be computed as lower and upper bounds and that when feasible the c indices should be computed with respect to partitions corresponding to the same percentiles for the models being compared when it is desired to estimate the overall concordance the average value of the two c indices might also be considered we use the data from the hemodialysis clinical trial to illustrate our evaluation and methods

q extensions of certain erdelyi type integrals recently we discovered several new erdelyi type integrals in the present paper it is shown how the q extensions of all those integrals involving and representing certain q hypergeometric functions can be developed the well known special cases and applications of these q integrals are also pointed out

contiguous relations continued fractions and orthogonality an 8 p 7 model abstract a contiguous relation for very well poised 8o7 basic hypergeometric functions is used to derive an explict expression for the associated continued fraction when the continued fraction terminates it has a singularity structure of simple poles which can be associated with a system of biorthogonal rational functions having a discrete weight function however in the nonterminating case the continued fraction does not have explicit singularities and thus we are unable to exhibit the associated orthogonality this is the first example of this type that we have encountered

estimating restricted mean treatment effects with stacked survival models the difference in restricted mean survival times between two groups is a clinically relevant summary measure with observational data there may be imbalances in confounding variables between the two groups one approach to account for such imbalances is estimating a covariate adjusted restricted mean difference by modeling the covariate adjusted survival distribution and then marginalizing over the covariate distribution because the estimator for the restricted mean difference is defined by the estimator for the covariate adjusted survival distribution it is natural to expect that a better estimator of the covariate adjusted survival distribution is associated with a better estimator of the restricted mean difference we therefore propose estimating restricted mean differences with stacked survival models stacked survival models estimate a weighted average of several survival models by minimizing predicted error by including a range of parametric semi parametric and non parametric models stacked survival models can robustly estimate a covariate adjusted survival distribution and therefore the restricted mean treatment effect in a wide range of scenarios we demonstrate through a simulation study that better performance of the covariate adjusted survival distribution often leads to better mean squared error of the restricted mean difference although there are notable exceptions in addition we demonstrate that the proposed estimator can perform nearly as well as cox regression when the proportional hazards assumption is satisfied and significantly better when proportional hazards is violated finally the proposed estimator is illustrated with data from the united network for organ sharing to evaluate post lung transplant survival between large volume and small volume centers copyright 2016 john wiley sons ltd

estimating the mean of an increasing stochastic process at a censored stopping time abstract data often come in the form of a longitudinal series say n whose observation has been stopped either by a terminal event u e g death or censoring event c e g lost to follow up when n is not deterministic it is generally not possible to use standard survival analysis tools e g the kaplan meier estimator to consistently estimate functionals of the distribution of n u because of induced informative censoring various seemingly unrelated estimators recently have been proposed to handle this difficulty within the context of specific problems ranging from the analysis of quality adjusted survival data to recurrent event data to lifetime medical cost data in this article i consider a general framework containing each of these specific examples as a special case propose a class of estimators for the mean of the stopped longitudinal process and establish the asymptotic properties of this class of estimators connections between the aforementioned estimators are established

a partitioning deletion substitution addition algorithm for creating survival risk groups summary accurately assessing a patient s risk of a given event is essential in making informed treatment decisions one approach is to stratify patients into two or more distinct risk groups with respect to a specific outcome using both clinical and demographic variables outcomes may be categorical or continuous in nature important examples in cancer studies might include level of toxicity or time to recurrence recursive partitioning methods are ideal for building such risk groups two such methods are classification and regression trees cart and a more recent competitor known as the partitioning deletion substitution addition partdsa algorithm both of which also utilize loss functions e g squared error for a continuous outcome as the basis for building selecting and assessing predictors but differ in the manner by which regression trees are constructed recently we have shown that partdsa often outperforms cart in so called full data settings e g uncensored outcomes however when confronted with censored outcome data the loss functions used by both procedures must be modified there have been several attempts to adapt cart for right censored data this article describes two such extensions for partdsa that make use of observed data loss functions constructed using inverse probability of censoring weights such loss functions are consistent estimates of their uncensored counterparts provided that the corresponding censoring model is correctly specified the relative performance of these new methods is evaluated via simulation studies and illustrated through an analysis of clinical trial data on brain cancer patients the implementation of partdsa for uncensored and right censored outcomes is publicly available in the r package partdsa