Translating solitons of the mean curvature flow

We study some basic properties of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps. Finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. These estimates give rigidity theorems for translating solitons in the Euclidean space in higher codimension.     

A new estimate of the parameters in linear mixed models

In linear mixed models, there are two kinds of unknown parameters: one is the fixed effect, the other is the variance component. In this paper, new estimates of these parameters, called the spectral decomposition estimates, are proposed, Some important statistical properties of the new estimates are established, in particular the linearity of the estimates of the fixed effects with many statistical optimalities. A new method is applied to two important models which are used in economics, finance, and mechanical fields. All estimates obtained have good statistical and practical meaning.     

Positive commutators at the bottom of the spectrum

Bony and Häfner have recently obtained positive commutator estimates on the Laplacian in the low-energy limit on asymptotically Euclidean spaces; these estimates can be used to prove local energy decay estimates if the metric is non-trapping. We simplify the proof of the estimates of Bony-Häfner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincaré inequality. Our main result is the positive commutator estimate

     

Upper Estimates of the Density of States in one Dimension Interpolating between the Wegner and the Lifshitz Regime

We consider the density of states (DOS) of one-dimensional discrete random Schr?dinger operators with an uniformly distributed random potential. The Wegner estimates are uniform estimates of the DOS. They are expected to be inadequate close to the boundary of the spectrum, where Lifshitz behaviour should occur. We prove a result interpolating between the Wegner and the Lifshitz regime. Submitted: May 6, 2008. Accepted: January 20, 2009.     

Lp-Estimates for a Solution to the Dirichlet Problem and to the Neumann Problem for the Heat Equation in a Wedge with Edge of Arbitrary Codimension

The Dirichlet problem and the Neumann problem in a wedge with edge of an arbitrary codimension are studied. On the basis of the Green functions of these problems in a cone, estimates for solutions are obtained. Coercive estimates for the solutions are also obtained in the Kondrat'ev spaces. Bibliography: 14 titles.     

Estimates of solutions of the H^p and BMOA corona problem

We prove new sharper estimates of solutions to the -corona problem in strictly pseudoconvex domains; in particular we show that the constant is independent of the number of generators. We also obtain sharper estimates for solutions to the BMOA corona problem. The proofs also lead to new results about the Taylor spectrum of analytic Toeplitz operators on and BMOA. Received: 28 August 1998 / in final form: 9 February 1999     

On the real accuracy of approximation in the central limit theorem. II

In the present article, we obtain new explicit estimates for accuracy of approximation in the central limit theorem (CLT). We construct these approximations with the use of asymptotic expansions. We compare the estimates with the real accuracy of approximation for a specific distribution. We also discuss the following question: Why the estimate from the Berry–Esseen theorem cannot catch even the order of proximity of distributions in the CLT?     

Weighted estimates of a solution to the linear problem connected with the one-phase Stefan problem in the case where the specific heat tends to zero

We prove estimates in weighted Hölder norms for a solution to the model linear problem related with the one-phase Stefan problem with a small multiplier ε at time derivative in the heat equation. These estimates are uniform with respect to parameter ε and are significant in justification of passage to the limit in the one-phase Stefan problem as the specific heat tends to zero. Bibliography: 8 titles.     

Two-sided estimates for constants in the Marcinkiewicz inequalities

For symmetric independent random values, sharp estimates for constants in the Marcinkiewicz inequalities are derived. Estimates of the left-hand and right-hand constants are roughly equivalent. Similar estimates for selfnormalized sums and new comparative estimates of increasing for some symmetric polynomial are derived as well. Bibliography: 14 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 361, 2008, pp. 45–56.     

Uniform Accuracy of the Maximum Likelihood Estimates for Probabilistic Models of Biological Sequences

Probabilistic models for biological sequences (DNA and proteins) have many useful applications in bioinformatics. Normally, the values of parameters of these models have to be estimated from empirical data. However, even for the most common estimates, the maximum likelihood (ML) estimates, properties have not been completely explored. Here we assess the uniform accuracy of the ML estimates for models of several types: the independence model, the Markov chain and the hidden Markov model (HMM). Particularly, we derive rates of decay of the maximum estimation error by employing the measure concentration as well as the Gaussian approximation, and compare these rates.     

On the quasilinear stationary Ventzel boundary-value problem

A priori estimates for gradients of solutions of a boundary-value problem for a quasilinear nondivergent elliptic equation with the quasilinear Ventzel boundary condition are established. By these estimates, existence theorems in the Hölder and Sobolev spaces are proved. Bibliography:11 titles.     

A posteriori error estimates for the generalized Stokes problem

The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya-Babuška-Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L₂-norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite-dimensional problems. Bibliography: 34 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101.     

Estimates in the kolmogorov theorem on conservation of conditionally periodic motions

A Hamiltonian system which differs from the integrable system by a small perturbation is considered. According to the Kolmogorov theorem /1–3/ the majority of invariant tori present in the unperturbed system do not decompose under a perturbation, and few of them become deformed. The following estimates are obtained below: for the perturbation under the usual conditions of nondegeneracy, the measure of the set of the decomposing tori and the deformation of the remaining tori are both estimate from above by the quantities of the order of √, and these estimates cannot be improved. The proof follows that /2,3/ of the Kolmogorov theorem with the intermediate estimates obtained more accurately. Similar estimates were obtained in the Moser theorem concerning the invariant curves of the mapping of a plane onto itself in /4/, and for the mapping in the multidimensional case, in /5/.     

Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193–219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L ²(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45–61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided. The authors were supported by Ministerio de Ciencia y Tecnología (Spain).     

Gradient estimates below the duality exponent

We show sharp local a priori estimates and regularity results for possibly degenerate non-linear elliptic problems, with data not lying in the natural dual space. We provide a precise non-linear potential theoretic analog of classical potential theory results due to Adams (Duke Math J 42:765–778, 1975) and Adams and Lewis (Studia Math 74:169–182, 1982), concerning Morrey spaces imbedding/regularity properties. For this we introduce a technique allowing for a “non-local representation” of solutions via Riesz potentials, in turn yielding optimal local estimates simultaneously in both rearrangement and non-rearrangement invariant function spaces. In fact we also derive sharp estimates in Lorentz spaces, covering borderline cases which remained open for some while.     

On the existence of consistent estimates and tests

The paper concerns estimates of probability measures (=p-measures) determined from a countable number of independent realizations. The main results are:

1. For suitable topologies, the existence of a consistent sequence of estimates implies the existence of a sequence of estimates which is strongly consistent a. e. with respect to an arbitrary finite prior measure. In the case of a measurable parameter the existence of a consistent sequence of estimates implies strong consistency of the Bayes estimates for the quadratic loss function.
2. If a family ofp-measures is separable with respect to the supremum metric, there exists a sequence of strict estimates which is uniformly consistent with respect to the supremum metric on any totally bounded subset ofp-measures.
3. IfB is totally σ-bounded with respect to the supremum metric there exists a density-consistent sequence of estimates. IfB is separable with respect to the supremum metric there exists a sequence of estimates which is density consistent a. e. with respect to an arbitrary finite prior measure.
4. A sequence of uniformly consistent tests exists for all compact hypotheses in the case of an abstract sample space and for all weakly closed hypotheses in the case of a separable metric sample space.

     

On estimates for potentials related to the problem of stability of a rotating self-gravitating liquid

The paper contains estimates of the volume and surface potentials arising in the evolution free boundary problem for a viscous self-gravitating liquid. The densities and domain where the potentials are defined may depend on time. The estimates are obtained in weighted anisotropic Sobolev-Slobodetskii spaces. Bibliography: 4 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 83–117.     

Contribution to the bandwidth choice for kernel density estimates

In the present paper we focus on the problem of the bandwidth choice for the kernel density estimates. The problem of finding the optimal bandwidth belongs to the crucial problems of the kernel estimates. As a criterion of quality of the estimates the L ₂ type measure is used. A special iterative method based on a relevant estimation of mean integrated square error given in papers Müller and Wang (Prob Theor Relat Fields 85:523–538, 1990), Jones et al. (Ann Stat 19:1919–1932, 1991) is suggested. Moreover the idea of maximal smoothing principle (Terrell in J Am Stat Assoc 85:470–477, 1990) is extended to the higher order kernels. A simulation study brings a comparison of the proposed method and the cross-validation method. Research supported by the GACR:402/04/1308.     

Exponential Decay for the Solutions of Wave Equations of Kirchhoff Type with Strong Damping

We study some decay estimates of the energy of the following nonlinear problem of Kirchhoff type:.....