On estimation of the remainder in the central limit theorem

A nonuniform estimate of the remainder in the central limit theorem is obtained for a sequence of independent, identically distributed random variables. This estimate is a generalization of an earlier result of L. V. Osipov and the author. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 142–146.     

New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author, at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by Golse and Perthame.     

Harmonic measure,L 2-estimates and the Schwarzian derivative

We consider several results, each of which uses some type of “L ²” estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points of a curve in terms of a certain geometric square function. Our next result is anL ^p estimate relating the derivative of a conformal mapping to its Schwarzian derivative. One consequence of this is an estimate on harmonic measure generalizing Lavrentiev’s estimate for rectifiable domains. Finally, we considerL ² estimates for Schwarzian derivatives and the question of when a Riemann mapping ϕ has log ϕ′ in BMO. Supported in part by NSF Grant DMS-91-00671. Supported in part by NSF Grant DMS-86-025000.     

Using successive approximations for improving the convergence of GMRES method

In this paper, our attention is concentrated on the GMRES method for the solution of the system (I–T)x=b of linear algebraic equations with a nonsymmetric matrix. We perform m pre-iterations y _l+1 =T _yl +b before starting GMRES and put y _m for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the mth powers of eigenvalues of the matrix T Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations.     

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn‐Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution u_N(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

On the approximate solution of one-dimensional singular integral equations of first kind

We prove an estimate for the error in approximate solution of one-dimensional singular integral equations. The estimate is obtained by an approximation of the kernel. For a specific problem we give a comparison of numerical results. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

Sections of the Difference Body

Let K be an n -dimensional convex body. Define the difference body by K-K= { x-y | x,y ∈ K }. We estimate the volume of the section of K-K by a linear subspace F via the maximal volume of sections of K parallel to F . We prove that for any m -dimensional subspace F there exists x ∈ \bf R ⁿ , such that for some absolute constant C . We show that for small dimensions of F this estimate is exact up to a multiplicative constant. Received May 6, 1998, and in revised form July 23, 1998.     

A Simple Proof of an Estimate for the Approximation of the Euclidean Ball and the Delone Triangulation Numbers

We give a simple proof of an estimate for the approximation of the Euclidean ball by a polytope with a given number of vertices with respect to the volume of the symmetric difference metric and relatively precise estimate for the Delone triangulation numbers. We also study the same problem for a given number of n−1-dimensional faces.     

An error estimate uniform in time for spectral semi-galerkin approximations of the nonhomogeneous navier-stokes equations

We consider the spectral semi-Galerkin method applied to the non-homogeneous Navier-Stokes equations, which describes the motion of miscibles fluids. Under certain conditions it is known that the aproximate solutions constructed by using this method converge to a global strong solution of these equations. In this paper we prove that these solutions satisfy an optimal uniform in time error estimate in the H ¹-norm for the velocity. We also derive an uniform error estimate in the L ^∞-norm for the density and an improved error estimate in the L ²-norm for the velocity.     

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.     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

A uniform estimate for the ELLAM scheme for transport equations

We prove a priori error estimate in a weighted energy norm for the Eulerian‐Lagrangian localized adjoint method (ELLAM) for the transport equations, without any special refinement or numerical stabilization introduced. The estimate holds uniformly with respect to ?. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Upper bounds on the length of the shortest closed geodesic on simply connected manifolds

The subject of this paper is upper bounds on the length of the shortest closed geodesic on simply connected manifolds with non-trivial second homology group. We will give three estimates. The first estimate will explicitly depend on volume and the upper bound for the sectional curvature; the second estimate will depend on diameter, a positive lower bound for the volume, and on the (possibly negative) lower bound on sectional curvature; the third estimate will depend on diameter, on a (possibly negative) lower bound for the sectional curvature and on a lower bound for the simply-connectedness radius. The technique that we develop in order to obtain the last result will also enable us to estimate the homotopy distance between any two closed curves on compact simply connected manifolds of sectional curvature bounded from below and diameter bounded from above. More precisely, let c be a constant such that any metric ball of radius is simply connected. There exists a homotopy connecting any two closed curves such that the length of the trajectory of the points during this homotopy has an upper bound in terms of the lower bound of the curvature, the upper bound of diameter and c. Received November 10, 1997; in final form June 23, 1998     

An estimate of stability of the reconstruction of the normal multiplicative type

An estimate of the characterization stability of the normal distribution type is obtained for the case of the multiplicative type. Under some conditions, the estimate has the orderɛ ^1/3 L(ɛ), where L(∈) is a slowly varying function. Supported by the Russian Founation for Basic Research (grant Nos. 97-01-00273, 98-01-00621, and 98-01-00926) and by INTAS-RFBR (grant No. IR-97-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Upper Estimate of the Interval of Existence of Solutions of a Nonlinear Timoshenko Equation

Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial data and nonlinear term are given so that solutions to the problem under consideration do not exist for all t > 0. An upper estimate of the t-interval of the existence of solutions is obtained. An estimate of the growth rate of the solutions is given.     

Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

We present an optimal error estimate of the numerical velocity, pressure, and angular velocity for the fully discrete penalty finite element method of the micropolar equations when the parameters ?, Δ t, and h are sufficiently small. In order to obtain this estimate, we present the time discretization of the penalty micropolar equation that is based on the backward Euler scheme; the spatial discretization of the time discretized penalty micropolar equation is based on a finite elements space pair (X _h , M _h ) that satisfies some approximations properties.     

On an equivariant estimate of the density of a matrix normal distribution

An optimal equivariant Bayes estimate of the density of a matrix normal distribution is obtained. This estimate is applied to the construction of the optimal Bayes group classification rule. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 29–39, Perm, 1990.     

0,1 distribution in the highest level sequences of primitive sequences over<Emphasis Type="Italic">Z</Emphasis><Subscript>2e</Subscript>

In this paper, we discuss the 0,1 distribution in the highest level sequence a_e-1 of primitive sequence over Z_2e generated by a primitive polynomial of degreen. First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight fore relatively small ton. We also get an estimate which is suitable fore relatively large ton. Combining the two bounds, we obtain an estimate depending only onn, which shows that the largern is, the closer to 1/2 the proportion of 1 will be.     

An estimate of the deformation of a closed convex surface with a change in its curvature

We consider closed convex surfaces ℱ of the space R³ containing a fixed point 0 in the interior. A central projection from 0 enables us to transfer the curvature ω(u) of the surface ℱ, regarded as a function of a set uɛℱ, onto a sphere with center 0. A. D. Aleksandrov established the fact that the surface ℱ is determined (moreover, uniquely) to·within a homothetic transformation with center 0 by prescribing the curvature transferred in this way onto the sphere. In this paper we give an estimate of the variation of the distances τ _F (B) of points of the surface from 0 as a function of the variation of the curvature transferred onto the sphere. The derivation of this estimate relies substantially on nondegeneracy of the surface ℱ; as a measure of nondegeneracy we take the ratio R/ζ, of the radii ℱ of balls with center ℱ, circumscribed and inscribed, respectively, about 0. Also, in this paper, we introduce and study those characteristics ℒ _F and τ _F of the curvature of the surface ℱ, which make it possible to estimate R/ζ from above and, by the same token, to obtain an estimate of how τ _F (B) varies in terms only of the curvature of the surface and its variation. An analytical treatment shows that basically our result yields an estimate of the maximum of the modulus of the change in the solution of a Monge—Ampere type equation on a sphere in terms of the change in its right-hand side in some integral norm, while the estimate of R/ζ, yields an a priori estimate of the modulus of the solution of this equation. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 83–110, 1974.