The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.     

Feasible criterion for designs based on fixed effect ANOVA model

A design is called feasible if all the ANOVA parameters in it are estimable. This paper provides a simple method to judge the feasibility of a design using matrix image. Furthermore, a broad family of feasible designs is obtained.     

A note on the Schrödinger smoothing effect

The Kato–Yajima smoothing estimate is a smoothing weighted L² estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey–Campanato weights. In this paper we make this generalization more sharp in terms of the so‐called Kerman–Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted L² space. Obtained results are also extended to the fractional Schrödinger propagator.     

The analytic smoothing effect of solutions for the nonlinear spatially homogeneous Landau equation with hard potentials

In this work, we study the Cauchy problem of the nonlinear spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem with the initial datum in L² enjoys an analytic regularizing effect, and the evolution of the analytic radius is the same as that of heat equations.

     

The fixed points of the multivariate smoothing transform

Let \((\mathbf {T}_1, \mathbf {T}_2, \ldots )\) be a sequence of random \(d\times d\) matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors \(X\) with nonnegative entries, satisfying

$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$

(*)

where \(\mathop {=}\limits ^{{\mathcal {L}}}\) denotes equality of the corresponding laws, \((X_i)_{i \ge 1}\) are i.i.d. copies of \(X\) and independent of \((Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )\). For \(d=1\), this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions \(X\) to (*) in the non-critical case, and existence results in the critical case.

     

The smoothing effect of a simultaneous directions parallel method as applied to Poisson problems

By using local Fourier analysis, a simultaneous directions parallel method, which is a particular instance of the parallel fractional step algorithm, is shown to possess smoothing effects when applied to Poisson problems. The specific smoothing factor is determined and the expected factor values are found to be consistent with those obtained. The simultaneous directions approach is an advantageous alternative to other existing smoothers in the multigrid environment. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Remarks on analytic smoothing effect for the Schrödinger equation

We study analytic smoothing effect of solutions to the Schrödinger equation with Cauchy data decaying exponentially at infinity. The domain of analyticity in the space variables of solutions is described under weight conditions on the data in terms of the corresponding supporting function. The domain of analyticity in the time variable is characterized by means of weight conditions of Gaussian type on the data. A generalization of various isometrical identities related to the analytic smoothing effect is introduced.     

Combining Lagrangian decomposition and excessive gap smoothing technique for solving large-scale separable convex optimization problems

A new algorithm for solving large-scale convex optimization problems with a separable objective function is proposed. The basic idea is to combine three techniques: Lagrangian dual decomposition, excessive gap and smoothing. The main advantage of this algorithm is that it automatically and simultaneously updates the smoothness parameters which significantly improves its performance. The convergence of the algorithm is proved under weak conditions imposed on the original problem. The rate of convergence is $O(\frac {1}{k})$ , where k is the iteration counter. In the second part of the paper, the proposed algorithm is coupled with a dual scheme to construct a switching variant in a dual decomposition framework. We discuss implementation issues and make a theoretical comparison. Numerical examples confirm the theoretical results.     

Local smoothing effect on the Schrödinger equation with harmonic potential

We consider the linear Schrödinger equation with repulsive harmonic potential. We establish the local smoothing effect of this type of equations. Our work extends the related results obtained by L. Vega and N. Visciglia for the free Schrödinger equation.     

A smoothing effect and polynomial growth of the Sobolev norms for the KP-II equation

We prove that the eventual growth in time of the Sobolev norms of the solutions of the KP-II equation is at most polynomial.     

The hat matrix for smoothing splines

The matrix which transforms the data vector to the vector of fitted values for smoothing splines is termed the hat matrix. This matrix is shown to have many of the same properties, and is seen to play the same role in the variances and covariances of the residuals, as its regression analysis counterpart. This fact is utilized to propose several possible diagnostic measures for use with smoothing splines. The extension of these results to include multivariate Laplacian smoothing spline is also indicated.     

Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations

In this paper the authors consider the Cauchy problem of weakly dissipative Klein-Gordon-Schrödinger equations through Yukawa coupling in . Making use of a Strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions.     

Testing for no effect in nonparametric regression via spline smoothing techniques

We propose three statistics for testing that a predictor variable has no effect on the response variable in regression analysis. The test statistics are integrals of squared derivatives of various orders of a periodic smoothing spline fit to the data. The large sample properties of the test statistics are investigated under the null hypothesis and sequences of local alternatives and a Monte Carlo study is conducted to assess finite sample power properties.     

The problem of optimal smoothing for convex functions

A procedure is described for smoothing a convex function which not only preserves its convexity, but also, under suitable conditions, leaves the function unchanged over nearly all the regions where it is already smooth. The method is based on a convolution followed by a gluing. Controlling the Hessian of the resulting function is the key to this process, and it is shown that it can be done successfully provided that the original function is strictly convex over the boundary of the smooth regions.     

Fixed points of the smoothing transformation

Summary Let W ₁,..., W _N be N nonnegative random variables and let be the class of all probability measures on [0, ∞). Define a transformation T on by letting Tμ be the distribution of W ₁X₁+ ... + W _N X _N , where the X _i are independent random variables with distribution μ, which are independent of W ₁,..., W _N as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the W _i are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some γ>1, EW _i ^γ <∞ for all i, we determine exactly when T has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, T always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett. The research of both authors was supported in part by NSF Grant MCS 80-02732. The first author is an Alfred P. Sloan fellow