discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

two sided generalized hyperbolic qr factorization and its perturbation analysis abstract we first propose the two sided generalized hyperbolic qr factorization of a matrix pair which is a generalization of the generalized qr factorization more explicitly two orthogonal matrices in the generalized qr factorization are replaced with two corresponding j orthogonal matrices where j is a signature matrix then we consider the perturbation analysis of the new matrix factorization some first order normwise perturbation bounds are derived using the refined matrix equation approach and the updated matrix vector equation approach the corresponding perturbation bounds for the generalized qr factorization are also obtained as the special case in comparison these bounds can be much tighter than the previous ones derived from barrlund 1994 a barrlund perturbation bounds for the generalized qr factorization linear algebra appl 207 1994 251 271

on perturbation bounds for the qr factorization abstract certain new perturbation bounds of the orthogonal factor in the qr factorization of a real matrix are derived the bounds of this note improve the known bounds in the literature

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

fixed design regression for linear time series this investigation is concerned with recovering a regression function g x i on the basis of noisy observations taken at uniformly spaced design points x i it is presumed that the corresponding observations are corrupted by additive dependent noise and that the noise is in fact induced by a general linear process in which the summand law can be discrete as well as continuously distributed discreteness induces a complication because such noise is not known to be strong mixing the postulate by which regression estimates are often shown to be asymptotically normal in fact as cited there are processes of this character which have been proven not to be strong mixing the main analytic result of this study is that in general circumstances which include the non strong mixing example the smoothers we propose are asymptotically normal some motivation is offered and a simple illustrative example calculation concludes this investigation the innovative elements of this work mainly consist of compassing models with discrete noise important in practical applications and in dispensing with mixing assumptions the ensuing mathematical difficulties are overcome by sharpening standard arguments

local linear regression estimation under long range dependence strong consistency and rates we consider the nonparametric estimation of multivariate regression functions and their derivatives for a regression model with long range dependent errors we adopt a local linear fitting approach and establish strong consistency and rates for the estimators of the regression function and its derivatives the rates of convergence depend on the amount of smoothing relative to the strength of the long range dependence lrd resulting in distinct rates for small and large bandwidths moreover the conditions determining this dichotomy are different for the estimates of the regression function than for the estimates of its derivatives

strongly asymmetric discrete painleve equations the additive case we examine a class of discrete painleve equations which present a strong asymmetry these equations can be written as a system of two equations the right hand sides of which do not have the same functional form we limit here our investigation to two canonical families of the quispel roberts thompson qrt classification both of which lead to difference equations several new integrable discrete systems are identified

transfer matrix study of the adsorption of a flexible self avoiding polymer chain in two dimensions the authors model the polymer as a self avoiding walk on a semi infinite square lattice with an energy contribution in from each step along the lattice boundary the critical adsorption temperature ta and the ordinary and special surface exponents of the walk above and at the adsorption temperature respectively are estimated with transfer matrix finite size scaling methods for the adsorption temperature they obtain exp in kta 2 041 or 0 002 and for the crossover and susceptibility exponents of the special transition phi sp 0 501 or 0 003 and gamma 1sp 1 454 or 0 004

limits and degeneracies of discrete painleve equations a sequel abstract we present the discrete systems which result from the discrete painleve equations q p vi and d p v associated to the affine weyl group e 7 1 two different procedures limits and degeneracies are used giving rise to a host of new discrete painleve equations but also to some equations which are integrable through linearisation

nonparametric density estimation prediction and regression for markov sequences abstract let xi be a stationary markov sequence having a transition probability density function f y x giving the pdf of x i 1 xi x in this study nonparametric density and regression techniques are employed to infer f y x and m x e x i 1 xi x it is seen that under certain regularity and markovian assumptions the asymptotic convergence rate of the nonparametric estimator mn x to the predictor m x is the same as it would have been had the xi s been independently and identically distributed and this rate is optimal in a certain sense consistency can be maintained after differentiability and even the markovian assumptions are abandoned computational and modeling ramifications are explored i claim that my methodology offers an interesting alternative to the popular arma approach

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

on the construction of approximate solutions for a multidimensional nonlinear heat equation we study three methods based on continuous symmetries to find approximate solutions for the multidimensional nonlinear heat equation delta u delta x0 delta u aun epsilon f u where a and n are arbitrary real constants f is a smooth function and 0 epsilon 1

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

recursive density estimation under dependence recursive estimators of the density of weakly dependent random variables are studied under certain absolute regularity and strong mixing conditions uniform strong consistency of the density estimators is established and their rates of convergence are obtained this study is concerned more with the almost sure uniform consistency of a sequence and its rate of convergence than with pointwise convergence since parameter estimation in time series analysis is often carried out under the gaussian assumption it is useful to check whether or not the density of a time series is gaussian or nearly so the method of proof used here is based on approximations of absolutely regular and strong mixing random variables rvs by independent rvs

uniform consistency of kernel estimators of a regression function under generalized conditions abstract in this article we prove uniform consistency of kernel estimators of a multivariate regression function under various assumptions on the distribution of the data in addition to the usual assumptions that the data are iid and that the distribution of the regressors is absolutely continuous we consider the cases that some regressors are discrete and the data are either stationary mixing themselves or generated by a class of functions of one sided infinite stationary mixing sequences moreover we demonstrate the performance of the kernel estimation method under these generalized conditions by a numerical example

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

proprietes de convergence presque complete du predicteur a noyau let z n be a mixing process which is valued in mathbb e subset mathbb r q g be a real measurable function defined on mathbb e s and k be two positive integers we suppose the existence of a function r satisfying r e g z n s z n k 1 z n n n k and estimate the function r from a sequence z i i 1 n by r n with r n u frac sum limits i k n s g z i s k u z i k 1 cdots z i h n 1 sum limits i k n s k u z i k 1 cdots z i h n 1 forall u in mathbb e k where k is a kernel of kq and h n h n 0 we give various conditions on both the sequences h n and the sequence n which is associated with z n for the two following properties uniform complete convergence to r for the functional estimator r n and complete convergence to 0 for the real random variable r n z n k 1 z n r zn k 1 z n this last result concerns the predictor r n z n k 1 z n of g z n s from z i i 1 n when the process z n is stationary and markovian of order k

generalized qr factorization and its applications abstract the purpose of this paper is to reintroduce the generalized qr factorization with or without pivoting of two matrices a and b having the same number of rows when b is square and nonsingular the factorization implicity gives the orthogonal factorization of b 1 a continuing the work of paige and hammarling we discuss the different forms of the factorization from the point of view of general purpose software development in addition we demonstrate the applications of the gqr factorization in solving the linear equality constrained least squares problem and the generalized linear regression problem and in estimating the conditioning of these problems

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

miura transforms for discrete painleve equations a miura transformation is presented relating the discrete painleve ii equation to what plays the role of the modified d pii equation 34 in the painleve gambier classification the study of this transformation makes possible the derivation of the auto backlund transform of the discrete pii and allows one to obtain particular solutions to the latter moreover the use of this transformation makes it possible to construct a new mapping in the parameter space of pii the latter is a novel method for the construction of integrable discrete systems

boundary critical behavior of d 2 self avoiding walks on correlated and uncorrelated vacancies in this paper we present exact results for the critical exponents of interacting self avoiding walks with ends at a linear boundary effective interactions are mediated by vacancies correlated and uncorrelated on the dual lattice by choosing different boundary conditions several ordinary and special regimes can be described in terms of clusters geometry and of critical and lowtemperature properties of the mathcal o n 1 model in particular the problem of boundary exponents at the point is fully solved and implications for point universality are discussed the surface crossover exponent at the special transition of noninteracting self avoiding walks is also interpreted in terms of percolation dimensions

the discrete painleve ii equations miura and auto backlund transformations we present miura transformations for all discrete painleve ii equations known to date we then use these miuras to derive special solutions in terms of discrete airy functions and to construct auto backlund transformations for the discrete painleve equations these transformations are then used to generate rational solutions some new forms of d pii and d p34 are obtained as well

on a transcendental equation related to painleve iii and its discrete forms we examine the canonical forms of painlev equations and argue that the equation for piii in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard piii our argument is based on the fact that the value of this parameter cannot be modified through auto b cklund transformations we investigate the possible discrete forms of this equation and produce two of them one is of a difference type where the independent variable enters linearly while the second one is of q type where the independent variable enters in a multiplicative way the properties of these discrete equations are also studied

perturbation bounds for the generalized qr factorization abstract let a and b be two matrices with the same number of rows the generalized qr factorization is a way to simultaneously transform these matrices to upper triangular form we present perturbation bounds for this factorization

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

nonparametric function estimation involving time series consider a stationary time series mathbf x t y t t 0 pm 1 ldots with mathbf x t being mathbb r d valued and y t real valued the conditional mean function is given by theta mathbf x 0 e y 0 mid mathbf x 0 under appropriate regularity conditions a local average estimator of this function based on a finite realization mathbf x 1 y 1 ldots mathbf x n y n can be chosen to achieve the optimal rate of convergence n 1 2 d both pointwise and in l 2 norms restricted to a compact and it can also be chosen to achieve the optimal rate of convergence n 1 log n 1 2 d in l infty norm restricted to a compact similar results hold for local median estimators of the conditional median function which is given by theta mathbf x 0 operatorname med y 0 mid mathbf x 0

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

multivariate regression estimation of continuous time processes from sampled data local polynomial fitting approach let y x y t x t spl infin t spl infin be real valued continuous time jointly stationary processes and let t sub j be a renewal point processes on 0 spl infin with a finite mean rate independent of y x we consider the estimation of regression function r x sub 0 x sub 1 x sub m 1 spl tau sub 1 spl tau sub m of spl psi y spl tau sub m given x 0 x sub 0 x spl tau sub 1 x sub 1 x spl tau sub m 1 sub x 1 for arbitrary lags 0 spl tau sub 1 spl tau sub m on the basis of the discrete time observations y t sub j x t sub j t sub j sub j 1 sup n we estimate the regression function and all its partial derivatives up to a total order p spl ges 1 using high order local polynomial fitting we establish the weak consistency of such estimates along with rates of convergence we also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p spl ges 1 and provide explicit expressions for the bias and covariance matrix of the asymptotically normal distribution