splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

denoising enhancing images on elastic manifolds the conflicting demands for simultaneous low pass and high pass processing required in image denoising and enhancement still present an outstanding challenge although a great deal of progress has been made by means of adaptive diffusion type algorithms to further advance such processing methods and algorithms we introduce a family of second order in time partial differential equations these equations describe the motion of a thin elastic sheet in a damping environment they are also derived by a variational approach in the context of image processing the new operator enables better edge preservation in denoising applications by offering an adaptive lowpass filter which preserves high frequency components in the pass band better than the adaptive diffusion filter while offering slower error propagation across edges we explore the action of this powerful operator in the context of image processing and exploit for this purpose the wealth of knowledge accumulated in physics and mathematics about the action and behavior of this operator the resulting methods are further generalized for color and or texture image processing by embedding images in multidimensional manifolds a specific application of the proposed new approach to superresolution is outlined

young measure solutions for the wave equation with p x t laplacian existence and blow up abstract we consider the dirichlet problem u t t l u f x t x t q t 0 t l u div a x t u p x t 2 u b x t u x t 2 u u x 0 u 0 x u t x 0 u 1 x x u t 0 t 0 t where the coefficients a x t b x t f x t and the exponents of nonlinearities p x t x t are given functions we prove local and global existence and blow up of young measure solutions we construct young measure solutions as the limit of the sequence of solutions of the regularized equations u t t l u div e u t f x t

global nonexistence of solutions for a nonlinear wave equation with time delay and acoustic boundary conditions abstract we consider a quasilinear wave equation u t t u t d i v u 2 u d i v u t 2 u t a u t m 2 u t 1 u t x t 2 u t x t b u p 2 u a b 0 associated with initial and dirichlet boundary conditions at one part and acoustic boundary conditions at another part respectively we prove under suitable conditions on m p and for negative initial energy a global nonexistence of solutions

blow up of solutions of nonlinear parabolic inequalities we study nonnegative solutions u x t of the nonlinear parabolic inequalities au u t u u in various subsets of r n x r where a n 2 n and a 0 1 are constants we show that changing the value of a in the open interval 0 1 can dramatically affect the blow up of these solutions

continuous convolution with hydrologic data the integral form of the convolution equation can be used for hydrologic data sampled at equal time increments if simple functional forms are assumed for the change in the hydrologic variables between sampling points the forms studied here are a step function linear segment and sliding polynomial and algorithms are presented for convolution of all combinations of these functions the effect of the different assumptions is demonstrated by a sample data set these algorithms have three advantages an appropriate functional form can be selected for the two variables to be convolved different time increments may be used for the two variables and the output may be calculated at any point in continuous time not just at multiples of the sampling increments the algorithms may also be applied to deconvolution but suffer many of the difficulties of matrix inversion techniques when the data contain errors or are derived from a system which is only approximately linear

initial blow up of solutions of semilinear parabolic inequalities abstract we study classical nonnegative solutions u x t of the semilinear parabolic inequalities 0 u t u u p in 0 1 where p is a positive constant and is a bounded domain in r n n 1 we show that a necessary and sufficient condition on p for such solutions u to satisfy a pointwise a priori bound on compact subsets k of as t 0 is p 1 2 n and in this case the bound on u is max x k u x t o t n 2 as t 0 if in addition is smooth u satisfies the boundary condition u 0 on 0 1 and p 1 2 n then we obtain a bound for u on the entire set as t 0

global existence and exponential stability of small solutions to nonlinear viscoelasticity the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn dimensional euclidean space with smooth boundary dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described time periodic solutions are also studied as an application of our main theorem nonlinear viscoelasticity strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated

global existence of solutions for quasi linear wave equations with viscous damping in this paper the global existence of solutions to the initial boundary value problem for a class of quasi linear wave equations with viscous damping and source terms is studied by using a combination of galerkin approximations compactness and monotonicity methods

nonexistence results for a semilinear heat equation with bounded potentials abstract we study the liouville type theorem for positive solutions to the semilinear heat equation u t u v u p in r n r we show that the nonexistence of positive solutions holds for any p 1 and for any v bounded from below by a positive constant in the superlinear case p 1 we give some sufficient conditions on the potential and the exponent which guarantee the nonexistence of positive solutions in the whole space

unit graph estimation and stabilization using quadratic programming and difference norms for a rainfall runoff event quadratic programming is particularly suited to the estimation of unit graph as linear and positivity constraints on unit graph ordinates can be naturally implemented in this paper the quadratic programming framework is invoked in a novel way to stabilize the unit graph estimation procedure via the use of difference norms the advantage of the latter over standard ridge regression is that penalties are placed on oscillations of the unit graph rather than on the size of its ordinates application of the methodology to real rainfall runoff data is provided and comparisons with existing approaches are made the latter indicate that our approach generally yields unit graph estimates of more realistic shape and smaller variance resulting in a better fit to the runoff data

cauchy problem for quasi linear wave equations with nonlinear damping and source terms abstract the paper studies the existence and non existence of global weak solutions to the cauchy problem for a class of quasi linear wave equations with nonlinear damping and source terms it proves that when max m p where m 1 1 and p 1 are respectively the growth orders of the nonlinear strain terms the nonlinear damping term and the source term under rather mild conditions on initial data the cauchy problem admits a global weak solution especially in the case of space dimension n 1 the weak solutions are regularized and so generalized and classical solution both prove to be unique on the other hand if 0 1 and the initial energy is negative then under certain opposite conditions any weak solution of the cauchy problem blows up in finite time and an example is shown

initial blow up rates and universal bounds for nonlinear heat equations we establish a universal upper bound on the initial blow up rate for all positive classical solutions of the dirichlet problem for the nonlinear heat equation ut u upon 0 t where p 1 and is a smoothly bounded domain in rn namely we show that u t c p t t on 0 t 2 for some n p 0 this is proved for all subcritical p i e p n 2 n 2 if n 4 and under a stronger assumption on p if n 5 as a consequence we improve the known results on universal bounds for global solutions furthermore if p n 3 n 1 then we may take n 1 2 and we show that this value of is optimal interestingly the rate can be faster than the previously known maximal initial blow up rate of the cauchy problem applications to universal blow up estimates at t t are given the neumann problem is also considered and we obtain the estimate on the initial rate for all subcritical p up to dimension n 6

uniqueness and eventual uniqueness of optimal designs in some time series models abstract earlier results on the uniqueness and eventual uniqueness of optimal designs for certain time series models are extended to a wider class of processes which includes those with covariance structures such as that of multiple integrals of brownian motion and brownian bridge processes the relationship between the problems of regression design for time series and piecewise polynomial approximation with free breakpoints is discussed and consequently asymptotic results obtained by sacks and ylvisaker 1970 are seen to hold under weaker assumptions for these processes

the initial boundary value problem for quasi linear wave equation with viscous damping in this paper the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi linear wave equation with viscous damping are proved the nonexistence of the global solution for this problem is discussed by an ordinary differential inequality finally an example is given

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

cauchy problem for quasi linear wave equations with viscous damping abstract the paper studies the global existence and asymptotic behavior of weak solutions to the cauchy problem for quasi linear wave equations with viscous damping it proves that when p max m where m 1 1 and p 1 are respectively the growth orders of the nonlinear strain terms the nonlinear damping term and the source term the cauchy problem admits a global weak solution which decays to zero according to the rate of polynomial as t as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition especially in the case of space dimension n 1 the solutions are regularized and so generalized and classical solution both prove to be unique comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms the states of the initial energy and the existence asymptotic behavior and nonexistence of global solutions of the cauchy problem

viscous solutions on some nonlinear wave equations abstract the paper studies the existence asymptotic behavior and non existence of global solutions to the initial boundary value problem and the cauchy problem for some nonlinear wave equations with nonlinear strain terms viscous damping and nonlinear source terms some results on the global existence decay and blowup of solutions to the above mentioned problems are derived and a few examples are shown

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a

isolated singularities of nonlinear parabolic inequalities we study c 2 1 nonnegative solutions u x t of the nonlinear parabolic inequalities 0 le u t delta u le u lambda in a punctured neighborhood of the origin in bf r n times 0 infty when n ge 1 and lambda 0 we show that a necessary and sufficient condition on for such solutions u to satisfy an a priori bound near the origin is lambda le frac n 2 n and in this case the a priori bound on u is u x t o t n 2 quad text as x t to 0 0 t 0 this a priori bound for u can be improved by imposing an upper bound on the initial condition of u