spectral collocation solutions to multiparameter mathieu s system our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of mathieu s system as a multiparameter eigenvalue problem mep the reduced wave equation for small deflections is solved directly without approximations introduced by the classical mathieu functions we show how for moderate values of the cut off collocation parameter the qr algorithm and the arnoldi method may be applied successfully while for larger values the jacobi davidson method is the method of choice with respect to convergence accuracy and memory usage

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

on the quantum filtration of the khovanov rozansky cohomology abstract we prove the quantum filtration on the khovanov rozansky link cohomology h p with a general degree n 1 monic potential polynomial p x is invariant under reidemeister moves and construct a spectral sequence converging to h p that is invariant under reidemeister moves whose e 1 term is isomorphic to the khovanov rozansky sl n cohomology h n then we define a generalization of the rasmussen invariant and study some of its properties we also discuss relations between upper bounds of the self linking number of transversal links in standard contact s 3

the classification of topologically expansive lorenz maps we show that topologically expansive lorenz maps can be described up to topological conjugacy by their kneading invariants we also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive lorenz map a simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described

prime and renormalisable kneading invariants and the dynamics of expanding lorenz maps abstract we study the dynamics and kneading theory of topologically expansive piecewise increasing maps of the interval with a single discontinuity lorenz maps we characterise those maps which are locally eventually onto locally onto and neither of the above in terms of the renormalisability or otherwise of the kneading invariant we also show how the decomposition of the non wandering set is related to the renormalisability of the kneading invariant we discuss our results in the context of other authors results on similar classes of maps

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

splitting words and lorenz braids abstract a canonical way to split lorenz periodic orbits is presented this allows one to recognize the braids on the lorenz template which can occur in non chaotic semi flows on this surface we first present a combinatorial version of our main result and then merely reinterpret it in topological terms

on the unknotting number of minimal diagrams answering negatively a question of bleiler we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent

spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems in numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables in several orthogonal coordinate systems separation of variables applied to the helmholtz laplace or schrodinger equation leads to a multiparameter eigenvalue problem mep important cases include mathieu s system lame s system and a system of spheroidal wave functions although multiparameter approaches are exploited occasionally to solve such equations numerically meps remain less well known and the variety of available numerical methods is not wide the classical approach of discretizing the equations using standard finite differences leads to algebraic meps with large matrices which are difficult to solve efficiently the aim of this paper is to change this perspective we show that by combining spectral collocation methods and new efficient numerical methods for algebraic meps it is possible to solve such problems both very efficiently and accurately we improve on several previous results available in the literature and also present a matlab toolbox for solving a wide range of problems

chebyshev spectral methods and the lane emden problem the three dimensional spherical polytropic lane emden problem is yrr 2 r yr y m 0 y 0 1 yr 0 0 where m 0 5 is a constant parameter the domain is r 0 where is the first root of y r we recast this as a non linear eigenproblem with three boundary conditions and as the eigenvalue allow ing imposition of the extra boundary condition by making the change of coordinate x r yxx 2 x yx 2 y m 0 y 0 1 yx 0 0 y 1 0 we find that a newton kantorovich iteration always converges from an m independent starting point y 0 x cos 2 x 0 3 we apply a chebyshev pseudospectral method to discretize x the lane emden equation has branch point singularities at the endpoint x 1 whenever m is not an integer we show that the chebyshev coefficients are a n constant n 2m 5 as n however a chebyshev truncation of n 100 always gives at least ten decimal places of accuracy much more accuracy when m is an inte ger the numerical algorithm is so simple that the complete code in maple is given as a one page table ams subject classifications 65l10 65d05 85a15

some flesh on the skeleton the bifurcation structure of bimodal maps abstract one of the main tools in the numerical study of two parameter families of one dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits we use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points then we explain how the bifurcation structure hangs on this skeleton

braids and the jones polynomial an important new invariant of knots and links is the jones polynomia and the subsequent generalized jones polynomial or two variable polynomial we prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends we also relate an exponent in the polynomial to the number of components of this link in j vaughan jones introduced a new polynomial invariant of oriented knots and links in 3 space subsequently a number of researchers freyd and yetter hoste lickorish and millett and ocneanu see fyhlmo independently realized that thlis could be generalized to produce an invariant which is a laurent polynomial of two variables and which specializes to give both the invariant of j and the classical alexander polynomial see r more precisely there is a function p from isotopy classes of oriented links to z x x 1 y y l if l is an oriented link we will write p l j x y by abuse of notation we will also write p b for b an oriented braid meaning of course the invariant associated to the oriented link determined by closing up b in the usual way several equivalent ways of codifying this invariant are described in fyhlmo we have chosen the approach of lickorish and millett hence our p k j x y is precisely their pk with x and y substituted for i and m respectively see fyhlmo and l m there appears as yet to be no consensus on a name for this two variable invariant in order to avoid awkward circumlocutions and being unable to amalgamate the names of the six or eight individuals who actually discovered it we have chosen to refer to it as the generalized jones polynomial or jones polynomial for short when there is no likelihood of confusion with the original jones polynomial this polynomial which we will denote j x y is characterized by figure 1 the interpretation of this is as follows given a regular projection of a link k with a crossing pictured as below one can form two new links k and ko which are identical to k except the one crossing is changed as shown when this is done the jones polynomials of the three links are related as in the formula it is a remarkable fact that this together with the normalization that p unknot 1 un quely determines a laurent polynomial j x y and that this polynomial is a link invariant i e is independent of the projection this result can be found in l m this formula in fact gives a good method for recursively computing j x y one received by the editors january 13 1986 1980 mathematics subject classification 1985 revision primary 57m25 research supported in part by nsf grant mcs 83

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

circuit partitions and the homfly polynomial of closed braids we present an expansion of the homfly polynomial p d z a of a braid diagram d in terms of its circuit partitions another aspect of this result is an expression of p d z a as the trace of a matrix associated to d in a simple way we show how certain degree properties of the homfly polynomial can be derived easily from this model in particular we obtain that if d is a positive braid diagram on n strings with w crossings the maximum degree of p d z a in the variable a equals n 1 w resume nous presentons une expansion pour le polyn6me homfly p d z a d un diagramme de tresse d en termes de ses partitions en circuits un autre aspect de ce resultat consiste en une expression de p d z a comme trace d une matrice associee de facon simple a d nous montrons comment certaines proprietes de degre du polyn6me homfly derivent simplement de ce modele en particulier nous obtenons que pour un diagramme de tresse positif d a n brins et uw croisements le degre maximum de p d z a en la variable a est egal a n 1 w

a genealogy for finite kneading sequences of bimodal maps on the interval we generate all the finite kneading sequences of one of the two kinds of bimodal map on the interval building each sequence uniquely from a pair of shorter ones there is a single pair at generation 0 with members of length 1 concomitant with this genealogy of kneading sequences is a unified genealogy of all the periodic orbits see figure 0

effective computation of the multivariable alexander polinomial of lorenz links given two different representations of a lorenz link we compare how they affect the computation of the multivariable alexander polynomial we also compare the alexander polynomial with the trip number and genus our experimental results lead us to conjecture that for lorenz knots the alexander polynomial is an equivalent invariant to the pair trip number genus finally we give a counterexample in the case of lorenz links

asymptotic formulae for lorenz and horseshoe knots we derive various asymptotic formulae for the numbers of closed orbits in the lorenz and horseshoe templates with given knot invariants for example braid index and genus we indicate how these estimates can be applied to more complicated flows by giving a bound for the genus of knotted periodic orbits in the figure of eight template

on the numerical construction of ellipsoidal wave functions the ellipsoidal wave equation which is the most general equation derived by separation of the helmholtz equation in confocal coordinates presents unusual computational difficulties and its solutions despite their importance for physical applications have not hitherto been effectively computed this paper describes a successful technique which involves the solution of a four term recursion and the simultaneous handling of two eigenparameters 1 introduction the ellipsoidal wave equation or lame wave equation has been known in mathematical literature for a considerable time certainly since 1926 when ince 12 listed it among the equations derivable from the general second order linear differential equation with five regular singularities more significantly it is the equation which emerges when the technique of separation of variables is applied to the helmholtz equation in ellipsoidal coordinates 3 1 6 physically therefore its solution is a matter of considerable interest there are applications among others to problems of scalar and electromagnetic scattering by an ellipsoid and diffraction by an elliptic plate or through an elliptic aperture the equation has been the subject of some fairly extensive theoretical analysis and quite a lot is known about the general properties of its solutions e g 2 remarkably however computed results are almost wholly lacking and indeed the usual computational techniques which serve for most other special functions are simply not applicable this is because of mathematical difficulties to be mentioned shortly the only numerical results or formulae which can give such results hitherto published are in the form of perturbation series or asymptotic series corresponding roughly to low frequency and high frequency approximations respec tively 1 7 10 the purpose of this paper is to report on a general method for computing the eigenvalues and the solutions there is an interesting parallel between this problem and the corresponding problem posed by mathieu s equation the latter was first formulated in 1868 and used in connection with an astronomical problem 15 11 considerable progress was made regarding its theoretical solution but it was not until some 60 years after mathieu s original memoir that ince accomplished the first systematic tabulation 13 14 for ellipsoidal wave functions the corresponding gap has been similar the greater problems of tabulation have been compensated by the availability of modern computing facilities in fact the difficulties are not so much computational as analytic the limitation has been imposed not by any lack of speed or storage

topological invariants and renormalization of lorenz maps we prove that the invariants of the topological semiconjugation of lorenz maps with transformations remains constant on the renormalization archipelagoes and analyze how the dynamics on the archipelagoes depends on its structure

a hierarchical multivariate two part model for profiling providers effects on health care charges procedures for analyzing and comparing health care providers effects on health services delivery and outcomes have been referred to as provider profiling in a typical profiling procedure patient level responses are measured for clusters of patients treated by providers that in turn can be considered statistically exchangeable thus a hierarchical model naturally represents the structure of the data when provider effects on multiple responses are profiled a multivariate model rather than a series of univariate models can capture associations among responses at both the provider and patient levels when responses are in the form of charges for health care services and sampled patients include nonusers of services charge variables are a mix of 0 s and highly skewed positive values that present a modeling challenge for analysis of covariate effects on charges for a single service a frequently used approach is a two part model that combines logistic or probit regression on any use of the service and li

the automatic solution of two parameter sturm liouville eigenvalue problems in ordinary differential equations an algorithm is described for the numerical solution of quite general two parameter eigenvalue problems whether singular or not the scheme is completely automatic even the usual requirement of an initial guess is not needed the scheme is based on the solution of suitable initial value problems or shooting

bennequin s inequality and the positivity of the signature we use an algorithm for special diagrams to prove a bennequin type inequality tor the signature of an arbitrary link diagram related to its murasugi sum decomposition we apply this inequality to show that the signature of a non trivial positive 3 braid knot is greater than its genus and that the signature of a positive braid link is minorated by an increasing function of its negated euler characteristic the latter property is conjectured to extend to positive links

on the crossing number of positive knots and braids and braid index criteria of jones and morton williams franks we give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations in particular we show that positive braid knots may not have positive minimal strand number braid representations giving a counterpart to results of franks williams and murasugi other examples answer partially questions of cromwell on homogeneous and of adams on almost alternating knots we give a counterexample to and a corrected version of a theorem of jones on the alexander polynomial of 4 braid knots we also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply from below the braid index a relation between generalizations of such examples and a conjecture of jones that a minimal braid representation has unique writhe is discussed

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

theory of acoustic eigenmodes in parabolic cylindrical enclosures we present an exact method for calculating eigenmodes and eigenfrequencies for an acoustic enclosure ae defined by two confocal parabolic cylinders and two plane surfaces using parabolic cylinder coordinates rigid wall boundary conditions are assumed however the model can easily be extended to include the more general case with mixed boundary conditions a discussion of eigenmode symmetry properties and mode frequency dependence on ae shape is given the latter problem is addressed by considering a series of symmetrical and asymmetrical aes having the same volume

multiparameter spectral theory and separation of variables multiparameter spectral theory generalizes the classical spectral theory of linear operators to n linear operators linked by n spectral parameters the spectral parameters arise as separation constants when for example separation of variables techniques are used to solve boundary value problems for partial differential equations in this paper we discuss some of the fundamental aspects of the theory such as solvability commutativity and abstract representation theory many of these concepts are of importance to modern developments in separation of variables techniques the ideas are illustrated in application to differential equations and special functions of mathematical physics

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

knotted periodic orbits in suspensions of annulus maps we consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3 manifold t 2 x i using results of birman williams topology 22 47 82 1983 contemp math 20 1 60 1983 we construct a knotholder or template that carries the set of periodic orbits of the flow we define rotation numbers and show that any orbit of period q and rotation number p q can be arranged as a positive braid on p strands this yields existence and uniqueness results for families of resonant torus knots p braids that are p q torus knots of period q p which correspond to order preserving birkhoff periodic orbits of the diffeomorphism we show that all other q periodic p braids have higher genus and we establish bounds on the genera of such knots we obtain existence and uniqueness results for a number of other non resonant torus knots including non order preserving q s q torus knots of rotation number 1

on 2d bisection method for double eigenvalueproblems the two dimensional bisection method presented in siam j matrix anal appl 13 4 1085 1992 is efficient for solving a class of double eigenvalue problems this paper further extends the 2d bisection method to full matrix cases and analyses its stablity as in a single parameter case the 2d bisection method is very stable for the tridiagonal matrix triples satisfying the symmetric definite condition since the double eigenvalue problems arise from two parameter boundary value problems an estimate of the discretization error in eigenpairs is also given some numerical examples are included

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

monte carlo methods of inference for implicit statistical models a prescribed statistical model is a parametric specification of the distribution of a random vector whilst an implicit statistical model is one defined at a more fundamental level in terms of a generating stochastic mechanism this paper develops methods of inference which can be used for implicit statistical models whose distribution theory is intractable the kernel method of probability density estimation is advocated for estimating a log likelihood from simulations of such a model the development and testing of an algorithm for maximizing this estimated log likelihood function is described an illustrative example involving a stochastic model for quantal response assays is given possible applications of the maximization algorithm to ad hoc methods of parameter estimation are noted briefly and illustrated by an example involving a model for the spatial pattern of displaced amacrine cells in the retina of a rabbit

braids transversal links and the khovanov rozansky theory we establish some inequalities for the khovanov rozansky cohomologies of braids these give new upper bounds of the self linking numbers of transversal links in standard contact s 3 which are sharper than the well known bound given by the homfly polynomial we also introduce a sequence of transversal link invariants and discuss some of their properties

a genealogy for the periodic orbits of a class of 1d maps abstract we describe a genealogy for the kneading sequences of one of the two kinds of bimodal map of the interval it provides a unified genealogy for all the periodic orbits and a complete road map for the periodic dynamics in the parameter plane

a multi level two part random effects model with application to an alcohol dependence study two part random effects models j am statist assoc 2001 96 730 745 statist methods med res 2002 11 341 355 have been applied to longitudinal studies for semi continuous outcomes characterized by a large portion of zero values and continuous non zero positive values examples include repeated measures of daily drinking records monthly medical costs and annual claims of car insurance however the question of how to apply such models to multi level data settings remains in this paper we propose a novel multi level two part random effects model distinct random effects are used to characterize heterogeneity at different levels maximum likelihood estimation and inference are carried out through gaussian quadrature technique which can be implemented conveniently in freely available software aml the model is applied to the analysis of repeated measures of the daily drinking record in a randomized controlled trial of topiramate for alcohol dependence treatment

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

on cobordisms between knots braid index and the upsilon invariant we use ozsvath stipsicz and szabo s upsilon invariant to provide bounds on cobordisms between knots that contain full twists in particular we recover and generalize a classical consequence of the morton franks williams inequality for knots positive braids that contain a positive full twist realize the braid index of their closure we also establish that quasi positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure finally we provide inductive formulas for the upsilon invariant of torus knots and compare it to the levine tristram signature profile

partial classification of lorenz knots syllable permutations of torus knots words abstract we define families of aperiodic words associated to lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots an algorithm to construct symbolic words of satellite lorenz knots is defined we prove subject to the validity of a previous conjecture that lorenz knots coded by some of these families of words are hyperbolic by showing that they are neither satellites nor torus knots and making use of thurston s theorem infinite families of hyperbolic lorenz knots are generated in this way to our knowledge for the first time the techniques used can be generalized to study other families of lorenz knots

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

gaussian estimation for correlated binomial data summary the mean variance structure of correlated binomial data is discussed and whittle s 1961 gaussian estimation is suggested as a useful general method in this context comparisons are made with other monrent methods in current use

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea