hypergeometric functions on complex matrix space the authors had presented the general foundations for a theory of hypergeometric functions of matrix argument over real division algebras here they further develop the fine structure of these functions over the complex field including series expansions integral representations asymptotics differential equations addition formulas multiplication formulas summation theorems transformation properties etc especially important are the operator valued hypergeometric functions required for nonspherical expansions such as addition formulas by the noncommutativity of matrix multiplication

estimating kappa from binocular data a common error in statistical analysis of ophthalmic data is the lack of accounting for the positive correlation generally present between observations made in fellow eyes the alternative of data analysis from only one eye in each patient may lead to loss of power and unrealistically large confidence intervals this paper discusses a method to estimate kappa a measure of agreement between two graders when both graders rate the same set of pairs of eyes the method assumes that the true left eye and right eye kappa values are equal and makes use of the correlated binocular data to estimate confidence intervals for the common kappa simulations show that the new estimators are better than the estimator based on only one eye new confidence intervals had the correct coverage probability but were usually only about 70 per cent as wide as single eye intervals the general methodology described here applies to analysis of grader agreement in rating other paired body structures

sharkovsky s program for the classification of triangular maps is almost completed abstract for a continuous map of the interval there are more than 50 conditions characterizing zero topological entropy some are applicable to the class of triangular maps x y f x g x y of the square but only a few of them are equivalent in this more general setting in 1989 a n sharkovsky posed the problem of proving or disproving all possible implications between them during last 20 years 32 conditions were considered and most of the work was done only 45 relations out of 992 remained not clear in this paper we give a survey of known results provide two new examples disproving another 26 possible implications and spell out the remaining 19 open problems all but one concern distributional chaos

estimating equations for kappa statistics kappa statistics are often used for measuring agreement or association the original measure suggested by cohen for use with nominal data has since been generalized in a number of ways including the incorporation of weights adaptation for use with more than one rater and the application to bivariate data each modification requires a new approximation to the standard error and these have usually been derived by the delta method this paper illustrates the use of estimating equations for kappa which offer a unified approach to many previously considered situations and which is also applicable to more complex designs

spectral decomposition and matrix valued orthogonal polynomials the relation between the spectral decomposition of a self adjoint operator which is realizable as a higher order recurrence operator and matrix valued orthogonal polynomials is investigated a general construction of such operators from scalar valued orthogonal polynomials is presented two examples of matrix valued orthogonal poly nomials with explicit orthogonality relations and three term recurrence relation are pre sented which both can be considered as 2 2 matrix valued analogues of subfamilies of askey wilson polynomials

a note on the kappa statistic for clustered dichotomous data the kappa statistic is widely used to assess the agreement between two raters motivated by a simulation based cluster bootstrap method to calculate the variance of the kappa statistic for clustered physician patients dichotomous data we investigate its special correlation structure and develop a new simple and efficient data generation algorithm for the clustered physician patients dichotomous data based on the delta method and its special covariance structure we propose a semi parametric variance estimator for the kappa statistic an extensive monte carlo simulation study is performed to evaluate the performance of the new proposal and five existing methods with respect to the empirical coverage probability root mean square error and average width of the 95 confidence interval for the kappa statistic the variance estimator ignoring the dependence within a cluster is generally inappropriate and the variance estimators from the new proposal bootstrap based methods and the sampling based delta method perform reasonably well for at least a moderately large number of clusters e g the number of clusters k 50 the new proposal and sampling based delta method provide convenient tools for efficient computations and non simulation based alternatives to the existing bootstrap based methods moreover the new proposal has acceptable performance even when the number of clusters is as small as k 25 to illustrate the practical application of all the methods one psychiatric research data and two simulated clustered physician patients dichotomous data are analyzed

kappa statistic for clustered matched pair data kappa statistic is widely used to assess the agreement between two procedures in the independent matched pair data for matched pair data collected in clusters on the basis of the delta method and sampling techniques we propose a nonparametric variance estimator for the kappa statistic without within cluster correlation structure or distributional assumptions the results of an extensive monte carlo simulation study demonstrate that the proposed kappa statistic provides consistent estimation and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters e g k 50 compared with the variance estimator ignoring dependence within a cluster the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra cluster correlation is fair 0 3 with more pronounced improvement when is further increased to illustrate the practical application of the proposed estimator we analyze two real data examples of clustered matched pair data copyright 2014 john wiley sons ltd

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

on polynomials orthogonal with respect to an inner product involving differences abstract in this paper we consider the inner product r x g x d x c g c where c r and is a distribution function with infinite spectrum such that has no points of increase in the interval c c 1 furthermore 0 and g are functions on r and c c 1 c let qn x be the sequence of monic orthogonal polynomials with respect to this inner product and pn x pnc x the sequences of monic standard orthogonal polynomials 0 with respect to d x and x c x c 1 d x respectively we derive an explicit representation for qn x in terms of pn x and pnc x and we present some results on the distribution of the zeros of qn x in relation to the zeros of pn x finally we treat the special case where pn x are charlier polynomials and c 0

on a class of matrix orthogonal polynomials on the real line abstract it is shown how to construct matrix orthogonal polynomials on the real line provided we have a sequence of scalar orthogonal polynomials over an algebraic harmonic curve we consider the moment matrix associated with a distribution on and related to a basis of the linear space of polynomials obtained through the polynomial which defines the curve this particular moment matrix can be regarded as a hankel block matrix and thus matrix orthogonal polynomials appear in a natural way

a survey on orthogonal matrix polynomials satisfying second order differential equations the subject of orthogonal polynomials cuts across a large piece of mathematics and its applications two notable examples are mathematical physics in the 19th and 20th centuries as well as the theory of spherical functions for symmetric spaces it is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of scalar valued orthogonal polynomials matrix orthogonality on the real line has been sporadically studied during the last half century since krein devoted some papers to the subject in 1949 see ams translations series 2 vol 97 providence rhode island 1971 pp 75 143 dokl akad nauk sssr 69 2 1949 125 in the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case the most recent of these results is the discovery of important examples of orthogonal matrix polynomials many families of orthogonal matrix polynomials have been found that as the classical families of hermite laguerre and jacobi in the scalar case satisfy second order differential equations with coefficients independent of n the aim of this paper is to give an overview of the techniques that have led to these examples a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples

the matrix valued hypergeometric equation the hypergeometric differential equation was found by euler euler l 1769 opera omnia ser 1 11 13 and was extensively studied by gauss gauss c f 1812 comm soc reg sci ii 3 123 162 kummer kummer e j 1836 riene ang math 15 39 83 kummer e j 1836 riene ang math 15 127 172 and riemann riemann b 1857 k gess wiss 7 1 24 the hypergeometric function known also as gauss function is the unique solution of the hypergeometric equation analytic at z 0 and with value 1 at z 0 this function because of its remarkable properties has been used for centuries in the whole subject of special functions in this article we give a matrix valued analog of the hypergeometric differential equation and of gauss function one can only speculate that many of the connections that made gauss function a vital part of mathematics at the end of the 20th century will be shared by its matrix valued version discussed here

estimating kappa from binocular data and comparing marginal probabilities suppose that two graders classify all eyes in a sample of patients for the presence or absence of a specified abnormality in the statistical analysis of the data possible correlation between the observations in the right and left eyes should be taken into account recently general methods have been developed to analyse grader agreement in rating paired body structures in the present paper an alternative procedure is proposed that utilizes existing formulae for the computation of weighted kappa and its standard error in addition the mcnemar test is generalized to compare the probabilities of a positive judgement by the two graders

matrix valued spherical functions associated to the complex projective plane abstract the main purpose of this paper is to compute all irreducible spherical functions on g su 3 of arbitrary type k where k s u 2 u 1 u 2 this is accomplished by associating to a spherical function on g a matrix valued function h on the complex projective plane p2 c g k it is well known that there is a fruitful connection between the hypergeometric function of euler and gauss and the spherical functions of trivial type associated to a rank one symmetric pair g k but the relation of spherical functions of types of dimension bigger than one with classical analysis has not been worked out even in the case of an example of a rank one pair the entries of h can be described by solutions of two systems of ordinary differential equations there is no ready made approach to such a pair of systems or even to a single system of this kind in our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitly in terms of a special class of generalized hypergeometric functions p 1fp

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

nominal scale agreement among observers experiments are considered where each of a sample of subjects is assigned to one of c categories separately by each of a fixed or varying group of observers building on earlier publications general procedures are proposed to analyze agreements and disagreements among observers in the case of a varying group of observers it is shown that it is not necessary to demand a constant number of observers per subject in the case of a fixed group of observers the problem of missing data is considered the procedures are illustrated within the context of two clinical diagnosis examples in the first example it is investigated which categories are relatively hard to distinguish from one another a new theorem is applied that shows a useful property of the statistic kappa in the second example it is investigated if a subgroup of observers can be found with a significantly higher degree of interobserver agreement

a triangular map of type 2 with positive topological entropy on a minimal set we provide a class of triangular maps of the square x y f x gx y of type 2 i e such that the periods of periodic points are the powers of 2 which has a minimal set supporting positive topological entropy this improves the famous example by s kolyada from 1992 and contributes to the solution of an old problem by a n sharkovsky

testing the equality of two dependent kappa statistics procedures are developed and compared for testing the equality of two dependent kappa statistics in the case of two raters and a dichotomous outcome variable such problems may arise when each of a sample of subjects are rated under two distinct settings and it is of interest to compare the observed levels of inter observer and intra observer agreement the procedures compared are extensions of previously developed procedures for comparing kappa statistics computed from independent samples the results of a monte carlo simulation show that adjusting for the dependency between samples tends to be worthwhile only if the between setting correlation is comparable in magnitude to the within setting correlations in this case a goodness of fit procedure that takes into account the dependency between samples is recommended

the analysis of contingency tables preface contingency tables and the chi square test 2 x 2 contingency tables r x c contingency tables multidimensional tables log linear models for contingency table linear logistic models contingency tables with ordered categories some special types of contingency tables appendices percentage points of the chi squared distribution computer software for fitting log linear and logistic models reference index

cyclic tridiagonal pairs higher order onsager algebras and orthogonal polynomials abstract the concept of cyclic tridiagonal pairs is introduced and explicit examples are given for a fairly general class of cyclic tridiagonal pairs with cyclicity n we associate a pair of divided polynomials the properties of this pair generalize the ones of tridiagonal pairs of racah type the algebra generated by the pair of divided polynomials is identified as a higher order generalization of the onsager algebra it can be viewed as a subalgebra of the q onsager algebra for a proper specialization at q the primitive 2 n th root of unity orthogonal polynomials beyond the leonard duality are revisited in light of this framework in particular certain second order dunkl shift operators provide a realization of the divided polynomials at n 2 or q i

matrix elements of irreducible representations of su n 1 x su n 1 and multivariable matrix valued orthogonal polynomials abstract in part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules the spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup so the spherical functions are matrix valued under these assumptions these functions can be described in terms of matrix valued orthogonal polynomials in several variables where the number of variables is the rank of the compact symmetric pair moreover these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators in part 2 we verify that the group case su n 1 meets all the conditions that we impose in part 1 for any k n 0 we obtain families of orthogonal polynomials in n variables with values in the n n matrices where n n k k the case k 0 leads to the classical heckman opdam polynomials of type a n with geometric parameter for k 1 we obtain the most complete results in this case we give an explicit expression of the matrix weight which we show to be irreducible whenever n 2 we also give explicit expressions of the spherical functions that determine the matrix weight for k 1 these expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper triangular matrices this generalizes and gives a new proof of a formula originally obtained by koornwinder for the case n 1 the commuting family of differential operators that have the matrix valued polynomials as simultaneous eigenfunctions contains an element of order one we give explicit formulas for differential operators of order one and two for n k equal to 2 1 and 3 1

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

doubly infinite jacobi matrices revisited resolvent and spectral measure abstract we study the resolvent and spectral measure of certain doubly infinite jacobi matrices via asymptotic solutions of two sided difference equations by finding the minimal or subdominant solutions or calculating the continued fractions for the difference equations we derive explicit formulas for the matrix entries of resolvent of doubly infinite jacobi matrices corresponding to lommel polynomials associated ultraspherical polynomials and al salam ismail polynomials the spectral measures are then obtained by inverting stieltjes transformations

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

on open problems concerning distributional chaos for triangular maps abstract we show that in the class t of the triangular maps x y f x g x y of the square there is a map of type 2 with non minimal recurrent points which is not dc3 we also show that every dc1 continuous map of a compact metric space has a trajectory which cannot be weakly approximated by trajectories of compact periodic sets these two results make possible to answer some open questions concerning classification of maps in t with zero topological entropy and contribute to an old problem formulated by a n sharkovsky

non inferiority test and confidence interval for the difference in correlated proportions in diagnostic procedures based on multiple raters the efficacy of diagnostic procedures is generally evaluated on the basis of the results from multiple raters however there are few adequate methods of performing non inferiority tests with confidence intervals to compare the accuracies sensitivities or specificities when multiple raters are considered we propose new statistical methods for comparing the accuracies of two diagnostic procedures in a non inferiority trial on the basis of the results from multiple independent raters who are also independent of the study centers we consider a study design in which each patient is subjected to two diagnostic procedures and all images are read by all raters by assuming a multinomial distribution for matched pair categorical data arising from the study design we derive a score based full menu that is a non inferiority test confidence interval and sample size formula for inference of the difference in correlated proportions between the two diagnostic procedures we conduct monte carlo simulation studies to examine the validity of the proposed methods which showed that the proposed test has a size closer to the nominal significance level than a wald type test and that the proposed confidence interval has better empirical coverage probability than a wald type confidence interval we illustrate the proposed methods with data from a study of diagnostic procedures for the diagnosis of oesophageal carcinoma infiltrating the tracheobronchial tree copyright 2011 john wiley sons ltd

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points