some recursive formulas for selberg type integrals a set of recursive relations satisfied by selberg type integrals involving monomial symmetric polynomials are derived generalizing previous results in aomoto 1987 siam j math anal 18 545 49 and iguri 2009 lett math phys 89 141 58 these formulas provide a well defined algorithm for computing selberg schur integrals whenever the kostka numbers relating schur functions and the corresponding monomial polynomials are explicitly known we illustrate the usefulness of our results discussing some interesting examples

low rank scale invariant tensor product smooths for generalized additive mixed models summary a general method for constructing low rank tensor product smooths for use as components of generalized additive models or generalized additive mixed models is presented a penalized regression approach is adopted in which tensor product smooths of several variables are constructed from smooths of each variable separately these marginal smooths being represented using a low rank basis with an associated quadratic wiggliness penalty the smooths offer several advantages i they have one wiggliness penalty per covariate and are hence invariant to linear rescaling of covariates making them useful when there is no natural way to scale covariates relative to each other ii they have a useful tuneable range of smoothness unlike single penalty tensor product smooths that are scale invariant iii the relatively low rank of the smooths means that they are computationally efficient iv the penalties on the smooths are easily interpretable in terms of function shape v the smooths can be generated completely automatically from any marginal smoothing bases and associated quadratic penalties giving the modeler considerable flexibility to choose the basis penalty combination most appropriate to each modeling task and vi the smooths can easily be written as components of a standard linear or generalized linear mixed model allowing them to be used as components of the rich family of such models implemented in standard software and to take advantage of the efficient and stable computational methods that have been developed for such models a small simulation study shows that the methods can compare favorably with recently developed smoothing spline anova methods

random matrix theory the exceptional lie groups and l functions there has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number theoretical l functions one example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of l functions these connections are extended here to non classical groups we focus on an explicit example the exceptional lie group g2 the value distributions for characteristic polynomials associated with the 7 and 14 dimensional representations of g2 defined with respect to the uniform invariant haar measure are calculated using two of the macdonald constant term identities a one parameter family of l functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the seven dimensional representation of g2 the random matrix calculations extend to all exceptional lie groups

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

on the prediction of stationary functional time series this article addresses the prediction of stationary functional time series existing contributions to this problem have largely focused on the special case of first order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology it is shown here how standard multivariate prediction techniques can be used in this context the connection between functional and multivariate predictions is made precise for the important case of vector and functional autoregressions the proposed method is easy to implement making use of existing statistical software packages and may therefore be attractive to a broader possibly nonacademic audience its practical applicability is enhanced through the introduction of a novel functional final prediction error model selection criterion that allows for an automatic determination of the lag structure and the dimensionality of the model the usefulness of the proposed methodology is demonst

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

a note on penalized spline smoothing with correlated errors we investigate the behavior of data driven smoothing parameters for penalized spline regression in the presence of correlated data it has been shown for other smoothing methods that mean squared error minimizers such as generalized cross validation or the akaike information criterion are extremely sensitive to misspecifications of the correlation structure resulting in over or under fitting the data in contrast to this we show that a maximum likelihood based choice of the smoothing parameter is more robust and that for a moderately misspecified correlation structure over or under fitting does not occur this is demonstrated in simulations and data examples and is supported by theoretical investigations

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

asymptotic properties of penalized spline estimators we study the class of penalized spline estimators which enjoy similarities to both regression splines without penalty and with fewer knots than data points and smoothing splines with knots equal to the data points and a penalty controlling the roughness of the fit depending on the number of knots sample size and penalty we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines with a clear breakpoint distinguishing the cases we prove that using fewer knots results in better asymptotic rates than when using a large number of knots we obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

testing for trends in high dimensional time series the article considers statistical inference for trends of high dimensional time series based on a modified l2 distance between parametric and nonparametric trend estimators we propose a de diagon

quantum group structure in the fock space resolutions of widehat sl n representations we describe a complex of wakimoto type fock space modules for the affine kac moody algebra widehat sl n the intertwining operators that build the complex are obtained from contour integrals of so called screening operators we show that a quantum group structure underlies the algebra of screening operators this observation greatly facilitates the explicit determination of the intertwiners we conjecture that the complex provides a resolution of an irreducible highest weight module in terms of fock spaces

recurrence equations for the computation of correlations in the 1 r 2 quantum many body system in a previous paper the two particle distribution function and one particle density matrix for the quantum many body system with the 1 r2 pair potential have been expressed as limiting cases of selberg correlation integrals recurrence equations are derived which allow rapid evaluation of these multidimensional integrals the exact results for the two particle distribution are compared with the harmonic approximation

nonparametric trend estimation in functional time series with application to annual mortality rates this research was supported by the king abdullah university of science and technology kaust

chern simons matrix models and stieltjes wigert polynomials employing the random matrix formulation of chern simons theory on seifert manifolds we show how the stieltjes wigert orthogonal polynomials are useful in exact computations in chern simons matrix models we construct a biorthogonal extension of the stieltjes wigert polynomials not available in the literature necessary to study chern simons matrix models when the geometry is a lens space we also study the relationship between stieltjes wigert and rogers szego polynomials and the corresponding equivalence with a unitary matrix model finally we give a detailed proof of a result that relates quantum dimensions with averages of schur polynomials in the stieltjes wigert ensemble

testing stationarity of functional time series economic and financial data often take the form of a collection of curves observed consecutively over time examples include intraday price curves yield and term structure curves and intraday volatility curves such curves can be viewed as a time series of functions a fundamental issue that must be addressed before an attempt is made to statistically model such data is whether these curves perhaps suitably transformed form a stationary functional time series this paper formalizes the assumption of stationarity in the context of functional time series and proposes several procedures to test the null hypothesis of stationarity the tests are nontrivial extensions of the broadly used tests in the kpss family the properties of the tests under several alternatives including change point and i 1 are studied and new insights present only in the functional setting are uncovered the theory is illustrated by a small simulation study and an application to intraday price curves

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

thin plate regression splines discuss the production of low rank smoothers for d greater than or equal to 1 dimensional data which can be fitted by regression or penalized regression methods the smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother by making use of lanczos iteration the basis change and truncation are computationally efficient the smoothers allow the use of approximate thin plate spline models with large data sets avoid the problems that are associated with knot placement that usually complicate modelling with regression splines or penalized regression splines provide a sensible way of modelling interaction terms in generalized additive models provide low rank approximations to generalized smoothing spline models appropriate for use with large data sets provide a means for incorporating smooth functions of more than one variable into non linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines given that the approach produces spline like models with a sparse basis it also provides a natural way of incorporating unpenalized spline like terms in linear and generalized linear models and these can be treated just like any other model terms from the point of view of model selection inference and diagnostics

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

matrix pade via lanczos solutions for vibrations of fluid structure interaction for multiple frequency full field solutions of the boundary value problem describing small fluid structure interaction vibration superimposed on a nominal state with prestress we propose an efficient reduced order method by constructing the full field matrix pade approximant of its finite element matrix function exploiting the matrix pade via lanczos connection the pade coefficients are computed in a stable and efficient way via an unsymmetric banded lanczos process the full field pade type approximant is the result of one sided projection onto krylov subspace we established its order of accuracy which is not maximal the superiority of this method in terms of various problem dimensions and parameters is established by complexity analysis via flop counts numerical examples obtained by using a model problem verified the accuracy of this full field matrix pade approximant copyright 2010 john wiley sons ltd

fast bivariate p splines the sandwich smoother national science foundation dms 0805975 national institutes of health r01 ns060910 national center for research resources ul1 rr024996 national natural science foundation of china 11201390

blood flow velocity field estimation via spatial regression with pde penalization we propose an innovative method for the accurate estimation of surfaces and spatial fields when prior knowledge of the phenomenon under study is available the prior knowledge included in the model derives from physics physiology or mechanics of the problem at hand and is formalized in terms of a partial differential equation governing the phenomenon behavior as well as conditions that the phenomenon has to satisfy at the boundary of the problem domain the proposed models exploit advanced scientific computing techniques and specifically make use of the finite element method the estimators have a penalized regression form and the usual inferential tools are derived both the pointwise and the areal data frameworks are considered the driving application concerns the estimation of the blood flow velocity field in a section of a carotid artery using data provided by echo color doppler this applied problem arises within a research project that aims at studying atherosclerosis pathogenesis supplementa