Entire Approximations for a Class of Truncated and Odd Functions

We solve the problem of finding optimal entire approximations of prescribed exponential type (unrestricted, majorant and minorant) for a class of truncated and odd functions with a shifted exponential subordination, minimizing the $L^{1}(\mathbb {R})$ -error. The class considered here includes new examples such as the truncated logarithm and truncated shifted power functions. This paper is the counterpart of the works (Carneiro and Vaaler in Trans. Am. Math. Soc. 362:5803–5843, 2010) and (Carneiro and Vaaler in Constr. Approx. 31(2):259–288, 2010) where the analogous problem for even functions was treated.     

Bandlimited Approximations to the Truncated Gaussian and Applications

In this paper, we extend the theory of optimal approximations of functions f:?→? in the L ¹(?)-metric by entire functions of prescribed exponential type (bandlimited functions). We solve this problem for the truncated and the odd Gaussians using explicit integral representations and properties of truncated theta functions obtained via the maximum principle for the heat operator. As applications, we recover most of the previously known examples in the literature and further extend the class of truncated and odd functions for which this extremal problem can be solved, by integration on the free parameter and the use of tempered distribution arguments. This is the counterpart of the work (Carneiro et al. in Trans. Am. Math. Soc., 2012), where the case of even functions is treated.     

Approximations to the Stochastic Burgers Equation

This article is devoted to the numerical study of various finite-difference approximations to the stochastic Burgers equation. Of particular interest in the one-dimensional case is the situation where the driving noise is white both in space and in time. We demonstrate that in this case, different finite-difference schemes converge to different limiting processes as the mesh size tends to zero. A theoretical explanation of this phenomenon is given and we formulate a number of conjectures for more general classes of equations, supported by numerical evidence.     

An Entire Holomorphic Function Associated to an Entire Harmonic Function

Let h be a harmonic function on ^N. Then there exists a holomorphic function f on such that f(t)=h(t, 0, …, 0) for all real t. Precise inequalities relating the growth rate of f to that of h are proved. These results are applied to deduce uniqueness theorems for harmonic functions of sufficiently slow growth that vanish at certain lattice points. Another application concerns the rate at which a harmonic function of finite order can decay along a ray.     

Approximations relative to Hausdorff distance

The present paper is an abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Physical and Mathematical Sciences. The dissertation was defended on November 30, 1967 before the faculty of the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR. The official opponents were Professors N. P. Korneichuk, P. P. Korovkin, and S. B. Stechkin, Doctors of Physical and Mathematical Sciences.     

Approximations to solutions of the zakai filtering equation

Mean square estimates are obtained for approximations to solutions of Zakai's equation which depend on the values of the observation process at the points of a regular partition in time     

A Probabilistic Approach to Semi-classical Approximations

We study in this paper the semi-classical expansion of the Schrödinger equation, using a probabilistic approach based on the Wiener measure. Using almost-analytic extensions, we exhibit a probabilistic ansatz for the wave function. We show that this ansatz approximates very well the wave function in the semi-classical regime, and gives the semi-classical expansion under mild hypothesis on the potential at infinity, and no analyticity conditions. In this paper, the study takes place before the caustics.     

Composite Approximations to the Solutions of the Orr-Sommerfeld Equation

The method of matched asymptotic expansions is used to derive composite approximations to the solutions of the Orr-Sommerfeld equation which satisfy Olver's completeness requirement. It is shown that the inner expansions can be obtained to all orders in terms of a certain class of generalized Airy functions, and these expansions are then used to derive approximations to the connection formulae. Because of the linearity of the problem it is possible and convenient to fix the normalization of the inner and outer expansions separately and then to relate them through the central matching coefficients. The Stokes multipliers can then be expressed in terms of the central matching coefficients and the coefficients which appear in the connection formulae. Once the inner and outer expansions have been matched they can be combined, if desired, to form composite approximations of either the additive or multiplicative type. For example, the ‘modified’ viscous solutions of Tollmien emerge in a natural way as first-order composite approximations obtained by multiplicative composition; similarly, the form of the ‘viscous correction’ to the singular inviscid solutions which I conjectured some years ago emerges as a first-order additive composite approximation. Because of the completeness requirement, however, these composite approximations are valid only in certain wedge shaped domains; approximations which are valid in the complementary sectors can then be obtained by the use of the connections formulae. The theory thus provides a relatively simple and explicit method of obtaining higher approximations, and its structure permits a direct comparison of the present results not only with the older heuristic theories but also with the comparison equation method.     

Approximations to the distribution of the sample correlation matrix

In this article, multivariate density expansions for the sample correlation matrix R are derived. The density of R is expressed through multivariate normal and through Wishart distributions. Also, an asymptotic expansion of the characteristic function of R is derived and the main terms of the first three cumulants of R are obtained in matrix form. These results make it possible to obtain asymptotic density expansions of multivariate functions of R in a direct way.     

Approximations to permutation and exchangeable processes

We obtain weighted approximations by a Brownian bridge to permutation and exchangeable processes and to appropriately defined inverse processes. Our results provide as special cases useful weighted approximations to the uniform empirical and quantile processes and to generalized bootstrapped versions of these processes. A number of other applications are discussed. Our approach is based on the Skorokhod embedding for martingales.     

Entire solutions to the Monge-Ampère equation

We consider the Monge-Ampère equation det(D²u)=Ψ(x,u,Du) in Rn, n?3, where Ψ is a positive function in C²(Rn×R×Rn). We prove the existence of convex solutions, provided there exist a subsolution of the form and a superharmonic bounded positive function φ satisfying: .     

Approximations to the distributions of ordered distance random variables

Summary In the present note we give short proofs of asymptotic theorems for the distributions of extreme and intermediate ordered distance random variables. Moreover, a quick goodness-of-fit test is proposed which is based on a single intermediate ordered distance random variable.     

Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model

Abstract

Nonlinear mixed-effects models have received a great deal of attention in the statistical literature in recent years because of the flexibility they offer in handling the unbalanced repeated-measures data that arise in different areas of investigation, such as pharmacokinetics and economics. Several different methods for estimating the parameters in nonlinear mixed-effects model have been proposed. We concentrate here on two of them—maximum likelihood and restricted maximum likelihood. A rather complex numerical issue for (restricted) maximum likelihood estimation in nonlinear mixed-effects models is the evaluation of the log-likelihood function of the data, because it involves the evaluation of a multiple integral that, in most cases, does not have a closed-form expression. We consider here four different approximations to the log-likelihood, comparing their computational and statistical properties. We conclude that the linear mixed-effects (LME) approximation suggested by Lindstrom and Bates, the Laplacian approximation, and Gaussian quadrature centered at the conditional modes of the random effects are quite accurate and computationally efficient. Gaussian quadrature centered at the expected value of the random effects is quite inaccurate for a smaller number of abscissas and computationally inefficient for a larger number of abscissas. Importance sampling is accurate, but quite inefficient computationally.     

Approximations and Randomization to Boost CSP Techniques

In recent years we have seen an increasing interest in combining constraint satisfaction problem (CSP) formulations and linear programming (LP) based techniques for solving hard computational problems. While considerable progress has been made in the integration of these techniques for solving problems that exhibit a mixture of linear and combinatorial constraints, it has been surprisingly difficult to successfully integrate LP-based and CSP-based methods in a purely combinatorial setting. Our approach draws on recent results on approximation algorithms based on LP relaxations and randomized rounding techniques, with theoretical guarantees, as well on results that provide evidence that the runtime distributions of combinatorial search methods are often heavy-tailed. We propose a complete randomized backtrack search method for combinatorial problems that tightly couples CSP propagation techniques with randomized LP-based approximations. We present experimental results that show that our hybrid CSP/LP backtrack search method outperforms the pure CSP and pure LP strategies on instances of a hard combinatorial problem.     

关于无穷维算子的有限维最佳逼近

A method of approaching to the infinite-dimensional linear operators by the finite-dimensional operators is discussed. It is shown that,for every infinite-dimensional operator A and every natural number n,there exists an n-dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n-dimensional optimal approximations to be unique is obtained.