On the boundedness and unboundedness of external polyhedral estimates for reachable sets of linear differential systems

The boundedness and unboundedness properties of external polyhedral (paralle-lepiped-valued) estimates are investigated for reachable sets of linear differential systems with a stable matrix. Boundedness and unboundedness criteria on an infinite time interval are presented for two types of estimates (“touching” estimates, which were introduced earlier, and estimates with constant orientation matrix). Conditions for the system matrix and bounding sets are given under which there are bounded estimates among the estimates of the mentioned types, under which there are unbounded estimates, and under which all the estimates are bounded or all the estimates are unbounded. In terms of the exponents of the estimates, the possible rate of their growth is described. For two-dimensional systems, the classification and comparison of possible situations of the boundedness or unboundedness for estimates of both types are given and boundedness criteria for estimates with special (orthogonal and “quasi-orthogonal”) constant orientation matrices are found. Results of numerical modeling are presented.     

Asymptotic normality of cumulant spectral estimates

The paper considers higher-order cumulant spectral estimates obtained by directly Fourier transforming weighted cumulant estimates. Such estimates computationally are different from those based on the finite Fourier transform. These estimates can be looked at continuously as well as directly on submanifolds. The estimates of cumulants are based on unbiased moment estimates. Asymptotic normality is obtained for these estimates and is based on a strong mixing condition and only a finite number of cumulant summability conditions.     

Nonparametric estimation of the location and scale parameters based on density estimation

Summary By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals. Using the properties of well-known methods of density estimates, it is shown that the proposed estimates possess nice large sample properties and it is indicated that they are also robust against dependence in the sample. The estimates perform well against other estimates of location and scale parameters.     

增长曲线模型回归系数线性估计的泛容许性

本文讨论增长曲线模型回归系数的线性估计的容许性.我们给出了回归系数线性估计的泛容许性定义,并在某些线性估计类中得到了泛容许估计的充要条件.     

The Construction and Properties of Boundary Kernels for Smoothing Sparse Multinomials

In recent years several authors have investigated the use of smoothing methods for sparse multinomial data. In particular, Hall and Titterington (1987) studied kernel smoothing in detail. It is pointed out here that the bias of kernel estimates of probabilities for cells near the boundaries of the multinomial vector can dominate the mean sum of squared error of the estimator for most true probability vectors. Fortunately, boundary kernels devised to correct boundary effects for kernel regression estimators can achieve the same result for these estimators. Properties of estimates based on boundary kernels are investigated and compared to unmodified kernel estimates and maximum penalized likelihood estimates. Monte Carlo evidence indicates that the boundary-corrected kernel estimates usually outperform uncorrected kernel estimates and are quite competitive with penalized likelihood estimates.     

Error analysis of variable degree mixed methods for elliptic problems via hybridization

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.

     

Window estimates of location and scale with applications to the cauchy distribution

Location and scale parameters are estimated via “window estimates”. The consistency and asymptotic normality of the estimates are established. The special case of the Cauchy distribution is considered, where the estimates are shown to have the same asymptotic distribution as the maximum-likelihood estimates. Additional applications are given for the Pearson type-VII distributions. The estimates have the advantages of ease of computation and high asymptotic efficiencies for certain heavy-tailed distributions.     

Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.     

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.     

Functions of commuting contractions under perturbation

The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, Hölder type estimates, Schatten–von Neumann estimates are obtained. The results generalize earlier known results for functions of self‐adjoint operators, normal operators, contractions and dissipative operators.     

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.     

L~2 estimates for oscillatory singular integral operators with analytic phrases

L2 estimates are obtained for some oscillatory singular integral operators with analytic phrases by using the technology of almost orthogonality, oscillatory estimates and size estimates.     

Multivariate k-nearest neighbor density estimates

Under appropriate assumptions, expressions describing the asymptotic behavior of the bias and variance of k-nearest neighbor density estimates with weight function w are obtained. The behavior of these estimates is compared with that of kernel estimates. Particular attention is paid to the properties of the estimates in the tail.     

Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension

Summary. I derive a posteriori error estimates for two-point boundary value problems and parabolic equations in one dimension based on interpolation error estimates. The interpolation error estimates are obtained from an extension of the error formula for the Lagrange interpolating polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems and parabolic equations. Computational results demonstrate the convergence of a posteriori error estimates and their effectiveness when combined with an hp-adaptive code for solving parabolic systems. Received April 17, 2000 / Revised version received September 25, 2000 / Published online May 30, 2001     

Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle

Second-order elliptic operators are transformed into second-order elliptic operators of a higher dimensionality acting on differences of functions. Applying the maximum principle to the resulting operators yields various a-priori pointwise estimates to difference-quotients of solutions of elliptic differential, as well as finite-difference, equations. We derive Schauder estimates, estimates for equations with discontinuous coefficients, and other estimates.     

Error estimates in elastoplasticity using a mixed method

In this paper, a mixed formulation and its discretization are introduced for elastoplasticity with linear kinematic hardening. The mixed formulation relies on the introduction of a Lagrange multiplier to resolve the non-differentiability of the plastic work function. The main focus is on the derivation of a priori and a posteriori error estimates based on general discretization spaces. The estimates are applied to several low-order finite elements. In particular, a posteriori estimates are expressed in terms of standard residual estimates. Numerical experiments are presented, confirming the applicability of the a posteriori estimates within an adaptive procedure.     

Solution Estimates for Nonlinear Nonautonomous Evolution Equations in Spaces with Normalizing Mappings

Nonlinear nonautonomous evolution equations in a space with a normalizing mapping (a generalized norm) are considered. Solution estimates are established. In particular cases these estimates generalize the Wazewski and Lozinskii estimates from the theory of ordinary differential equations. By the obtained estimates, the following problems are investigated: asymptotic stability, boundedness of solutions, input-output stability, existence of periodic solutions. Applications to integro-differential equations are discussed.     

变系数的一般型发展方程的色散估计

本文研究的是带变系数的一般型线性发展方程.首先建立了其基本解的一系列色散估计:Kato光滑型估计,极大函数估计及Strichartz估计.最后应用这些估计研究了一些非自治非线性色散方程的初值问题在H~s(R)空间中的局部可解性.     

Generalized estimates for performance sensitivities of stochastic systems

Three kinds of estimates for performance sensitivities (gradients, Hessians etc.) of stochastic systems are introduced. These estimates are given in general operator form. Their convergence conditions and rate of convergence are presented. Particular attention is given to estimates obtained from a single sample path. Various examples of estimates are considered.     

不同步交易模型的参数估计

对金融证券中高频数据的不同步交易模型,讨论了参数的矩估计,极大似然估计及估计的有关性质.