some observations on the distribution of values of continued fractions there have been many studies of the values taken on by continued fractionsk a n 1 when its elements are all in a prescribed sete the set of all values taken on is the limit regionv e it has been conjectured that the values inv e are taken on with varying probabilities even when the elementsa n are uniformly distributed overe in this article we present the first concrete evidence that this is indeed so we consider two types of element regions a e is an interval on the real axis our best results are for intervals 1 1 0 1 2 b e is a disk in the complex plane defined bye z z 1 0 1 2

on almost global existence for nonrelativistic wave equations in 3d almost global solutions are constructed to three dimensional quadratically nonlinear wave equations the proof relies on generalized energy estimates and a new decay estimate the method applies to equations that are only classically invariant such as the nonlinear system of hyperelasticity 1996 john wiley sons inc

global existence for dirichlet wave equations with quadratic nonlinearities in high dimensions we prove global existence of solutions to quasilinear wave equations with quadratic nonlinearities exterior to nontrapping obstacles in spatial dimensions 4 and higher this generalizes a result of shibata and tsutsumi in spatial dimensions greater than or equal to 6 the technique of proof would allow for more complicated geometries provided that an appropriate local energy decay exists for the associated linear wave equation

truncation error bounds for limit periodic continued fractionsk an 1 with liman 0 truncation error bounds are developed for continued fractionsk a n 1 where a n 1 4 for alln sufficiently large the bounds are particularly suited some are shown to be best for the limit periodic case when lima n 0 among the principal results is the following if a n n p for alln sufficiently large with constants 0 p 0 then f f m c d m 2 p m 2 for allm sufficiently large for some constantsc 0 d 0 heref denotes the limit assumed finite ofk a n 1 andf m denotes itsmth approximant applications are given for continued fraction expansions of ratios of kummer functions1 f 1 and of ratios of hypergeometric functions0 f 1 it is shown thatp 1 for1 f 1 andp 2 for0 f 1 wherep is the parameter determining the rate of convergence numerical examples indicate that the error bounds are indeed sharp

space time l 2 estimates for elastic waves and applications abstract we establish a kind of weighted space time l 2 estimate which belongs to keel smith sogge type estimates for perturbed linear elastic wave equations in r n n 3 we also apply this estimate to give an alternative and short proof for john s almost global existence theorem for 3 d nonlinear elastic waves john 1988 10 and show the global existence of small smooth solutions for nonlinear elastic waves in high space dimensions n 4

on bandwidth choice for density estimation with dependent data we address the empirical bandwidth choice problem in cases where the range of dependence may be virtually arbitrarily long assuming that the observed data derive from an unknown function of a gaussian process it is argued that unlike more traditional contexts of statistical inference in density estimation there is no clear role for the classical distinction between short and long range dependence indeed the boundaries that separate different modes of behaviour for optimal bandwidths and mean squared errors are determined more by kernel order than by traditional notions of strength of dependence for example by whether or not the sum of the covariances converges we provide surprising evidence that even for some strongly dependent data sequences the asymptotically optimal bandwidth for independent data is a good choice a plug in empirical bandwidth selector based on this observation is suggested we determine the properties of this choice for a wide range of different strengths of dependence properties of cross validation are also addressed

on nonlinear elastic waves in 2 d abstract for the cauchy problem of 2 d nonlinear elastic waves of isotropic homogeneous and hyperelastic materials satisfying null conditions assuming the smallness of h 3 r 2 h 2 r 2 norm and the boundedness of h 4 r 2 h 3 r 2 norm for radially symmetric initial data we show the global existence of classical solutions similar results for exterior domain problems are also proved

truncation error bounds for continued fractions k a_n 1 with parabolic element regions value regions for continued fractions are intimately related to such questions as convergence truncation error analysis and numerical stability the present paper introduces a new family of value regions corresponding to parabolic element regions for continued fractions k a n 1 the value regions are used to obtain lens shaped inclusion regions and truncation error bounds the results generalize and bring into focus connections between some earlier work by thron and others on parabolic convergence regions and by henrici and pfluger on stieltjes fractions

on the computation of incomplete gamma functions in the complex domain abstract some new continued fractions for incomplete gamma functions a z and a z with a and z complex are derived for many of these expansions it is shown that the approximants can be evaluated by a numerically stable backward recurrence algorithm truncation error bounds for these approximations are discussed with applicable references cited consideration is given to important special cases such as the error and complementary error functions and exponential integrals in connection with the error function we discuss the anomalous convergence of general t fractions two point pade approximants first pointed out and studied by gautschi

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

effect of dependence on stochastic measures of accuracy of density estimations in kernel density estimation those data values that make a nondegenerate contribution to the estimator computed at a given point tend to be spaced well apart this property has the effect of suppressing many of the conventional consequences of long range dependence for example slower rates of convergence which might otherwise be revealed by a traditional lossor risk based assessment of performance from that viewpoint dependence has to be very long range indeed before a density estimator experiences any first order effects however an analysis in terms of the convergence rate for a particular realization rather than the rate averaged over all realizations reveals a very different picture we show that from that viewpoint and in the context of functions of gaussian processes effects on rates of convergence can become apparent as soon as the boundary between short and long range dependence is crossed for example the distance between ise and mise optimal bandwidths is generally of larger order for long range dependent data we shed new light on cross validation too in particular we show that the variance of the cross validation bandwidth is generally larger for long range dependent data and that the first order properties of this bandwidth do not depend on how many data are left out when constructing the cross validation criterion moreover for long range dependent data the cross validation bandwidth is usually perfectly negatively correlated in the limit with the optimal stochastic bandwidth

global existence of null form wave equations in exterior domains we provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains in particular our proof does not require estimation of the fundamental solution for the free wave equation we instead rely upon a class of keel smith sogge estimates for the perturbed wave equation using this a notable simplification is made as compared to previous works concerning wave equations in exterior domains one no longer needs to distinguish the scaling vector field from the other admissible vector fields

a convergence question inspired by stieltjes and by value sets in continued fraction theory abstract let v be a subset of the complex plane c let fn n 1 be a sequence of self mappings of v i e fn v v the question is then under what conditions will the sequence fn w f1 f2 fn w n 1 2 3 of composite maps converge to a constant function in v in this paper we give a survey of some of the answers and open problems connected with this question such answers have applications in dynamical systems schur analysis continued fractions and other similar structures like infinite exponentials infinite radicals

reflections on value regions limit regions and truncation errors for continued fractions for continued fractionsk a n 1 the concept of limit region is discussed and its use for obtaining modified truncation error estimates is illustrated on examples a certain strategy for numerical computation of limit regions is presented and illustrated on examples

wavelet regression in random design with heteroscedastic dependent errors we investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of besov scales f b s r and for a variety of l p error measures we consider error distributions with long range dependence parameter 0 2 it is seen that there are three rate phases namely the dense sparse and long range dependence phase depending on the relative values of s p and furthermore we show that long range dependence does not come into play for shape estimation f f the theory is illustrated with some numerical examples

global existence for nonlinear elastic waves in high space dimensions abstract in georgiev et al 2005 the authors proved the global existence of solutions to the cauchy problem of nonlinear elastic waves with memory for space dimensions n 3 in this manuscript we show that in high space dimensions n 4 even without such memory effect global solutions can be also constructed the main tool we used is the vector fields method adapted to elastic waves in sideris s work sideris 2000

almost global existence for quasilinear wave equations in three space dimensions this article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions the approach here uses only the classical invariance of the wave operator under translations spatial rotations and scaling using these techniques we can handle wave equations in minkowski space or dirichlet wave equations in the exterior of a smooth star shaped obstacle we can also apply our methods to systems of quasilinear wave equations having different wave speeds this extends our work 11 for the semilinear case previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non obstacle case in 9 john and klainerman proved almost global existence on minkowski space for quadratic quasilinear equations using the lorentz invariance of the wave operator in addition to the symmetries listed above subsequently in 14 klainerman and sideris obtained the same result for a class of quadratic divergence form nonlinearities without relying on lorentz invariance this line of thought was refined and applied to prove global in time results for null form equa tions related to the theory of elasticity in sideris 22 23 and for multiple speed systems of null form quasilinear equations in sideris and tu 24 and yokoyama 29 the main difference between our approach and the earlier ones is that we ex ploit the 0 x 1 decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0 t l decay here of course x x x2 x is the spatial component and t the time component of a space time vector t x g m x e3 establishing 0 x 1 decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation a weighted l2 space time estimate for inhomogeneous wave equations proposition 3 1 below from 11 is important in making the spatial decay useful for the long time existence argument for semilinear systems one can show almost global existence from small data using only this spatial decay 11 for quasilinear systems however we also have to show that both first and second derivatives of u decay like 1 t fortunately we can do this using a variant of some l1 l estimates of john h rmander

nonresonance and global existence of prestressed nonlinear elastic waves this article considers the existence of global classical solutions to the cauchy problem in nonlinear elastodynamics the unbounded elastic medium is assumed to be homogeneous isotropic and hyperelastic as in the theory of 3d nonlinear wave equations in three space dimensions global existence hinges on two basic assumptions first the initial deformation must be a small displacement from equilibrium in this case a prestressed homogeneous dilation of the reference conflguration and equally important the nonlinear terms must obey a type of nonresonance or null condition the omission of either of these assumptions can lead to the breakdown of solutions in flnite time in particular nonresonance complements the genuine nonlinearity condition of f john under which arbitrarily small spherically symmetric displacements develop singularities although one expects this to carry over to the nonsymmetric case as well 4 john also showed that small solutions exist almost globally 5 see also 10 formation of singularities for large displacements was illustrated by tahvildar zadeh 16 the nonresonance condition introduced here represents a substantial improvement over our previous work on this topic 13 to explain the difierence roughly our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones here only the cancellation of nonlinear wave interactions among individual wave families is required the di culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal helmholtz projection which is ill suited to nonlinear analysis however our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous and this permits the usage of approximate local decompositions