Certain versions of the formulation of problems of non-linear elasticity in terms of stresses

New versions of the formulation of static non-linear elasticity theory problems are elucidated in terms of stresses for materials with locally reversible state laws (for instance, for a semilinear John material /1/), that reduce to the solution of nine equations in the Piola stress tensor component for six or three boundary conditions and three integral conditions. This paper is related to the investigations in /2–7/ devoted to analogous problems of the mechanics of a solid linearly deformable body. Examples are presented of the realization of one of the versions.     

Superconvergence of mixed finite element semi-discretizations of two time-dependent problems

We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation for the error in the space discretization can be performed in the same way as for the underlying elliptic problem.     

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein?s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed.     

Asymptotics of the spectral function of an elliptic differential operator of second order

In the work the Riesz means of the spectral function of a boundary-value problem for an elliptic differential operator of second order are studied for the case of a geodesically concave boundary. Asymptotic formulas for the Riesz means with an estimate of the remainder term that is uniform over the entire domain are obtained in terms of the formal asymptotics of Green's function of the problem considered. An explicit expression is obtained for the spectral function in a neighborhood of the diagonal from which there follows, in particular, a precise estimate of the remainder term in the Weyl formula.The present work contains a detailed treatment of the results published in [1],Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 152–203, 1979.The author thanks V. M. Babich for posing the problem and for major resistance in the work and V. B. Filippov for useful discussions.     

Toward determination of the eigenvalues of problems on description of the rotation of deep-water drill columns

This paper presents a method for computing the complex eigenvalues for a boundary-value problem describing the rotation of deep-water drill columns for drilling at depths of thousands of meters. A solution is sought for a singularly perturbed boundary-value problem in the form of an asymptotic expansion in a small parameter ε for two parts of the of the drill column: a lower compressed length, and the upper length. The method can be used to investigate the stability of drill columns used for deep-water drilling. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 29–36, 1993.     

Acceleration of Convergence of Static and Dynamic Iterations

A new technique for acceleration of convergence of static and dynamic iterations for systems of linear equations and systems of linear differential equations is proposed. This technique is based on splitting the matrix of the system in such a way that the resulting iteration matrix has a minimal spectral radius for linear systems and a minimal spectral radius for some value of a parameter in Laplace transform domain for linear differential systems.This revised version was published online in October 2005 with corrections to the Cover Date.     

流动稳定性弱非线性理论的重新考虑*

弱非性理论已被广泛用于流动稳定性理论及其它领域.然而其应用对某些问题虽是成功的,但对另一些问题,其结果却常不令人满意,特别是对转捩或自由剪切流中涡的演化这类问题,这时理论研究的目的不是寻找稳态解,而是预测演化过程.在本文中,我们将研究不成功的原因并建议一些改进的办法.     

Numerical modelling of the plane problem of the stress state of a tube immersed in a liquid

A numerical method for solving the plane problem of determining the stress state of a tube of arbitrary section immersed in a homogeneous incompressible liquid is proposed. The change from the boundary conditions for this problem to the boundary conditions for a biharmonic stress function is carried out, which enables the algorithm for solving boundary value problems in the case of a polyharmonic function developed earlier to be used to solve the problem under consideration. It is shown that the boundary conditions for doubly-connected domains contain three unknown constants. The conditions for finding these constants in a form that is convenient for the implementation of a numerical algorithm are obtained. Tubes with sections in the form of concentric, eccentric and elliptic rings are considered as examples.     

The properties of solutions of a class of isoperimetric problems of stability optimization

A class of isoperimetric problems of stability optimization is considered. These arise, for example, when maximizing the Euler force in the destabilization of a column (rod) of varying cross-section and given volume (Lagrange's problem). It is well known that an extremum which depends on the form of the boundary conditions can be achieved for both simple and double eigenvalues. A class of problems is identified for which a global maximum is found at a simple eigenvalue. The possibility of achieving a local extremum for the first (simple) eigenvalue at stationary points is analysed qualitatively in terms of the parameter values and the form of the boundary conditions.     

Analysis of critical motions of floating structures

Validation of numerical methods for describing the motion of a ship in sea conditions by adequate experiments is a major research field in ocean engineering. For the development of a method for the systematic determination of critical and safe operational conditions and for the classification of capsize scenarios bifurcation analyses are performed. The computational effort for these analyses is enormous using a full model describing the nonlinear dynamics of a floating body. Therefore, a method for model reduction is currently being developed at the Institute of Mechanics and Ocean Engineering at TUHH. Bases for the validation of this new method are experiments conducted in the institute's wave tank. The determination of position and attitude of the body is performed with an integrated measurement system: An inertial measurement unit and a video system are combined using an extended Kalman Filter. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On one problem of dynamics of thermoviscoelastic medium of Oldroyd type

We establish nonlocal existence theorem for the weak solution for an initial-boundary value problem for the dynamic model of thermoviscoelasticity of Oldroyd type in the planar case.     

New Estimates for the Rate of Convergence of the Method of Subspace Corrections

We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections.     

BATE curve in assessment of clinical utility of predictive biomarkers

In this paper,for time-to-event data,we propose a new statistical framework for casual inference in evaluating clinical utility of predictive biomarkers and in selecting an optimal treatment for a particular patient.This new casual framework is based on a new concept,called Biomarker Adjusted Treatment Effect (BATE) curve.The BATE curve can be used for assessing clinical utility of a predictive biomarker,for designing a subsequent confirmation trial,and for guiding clinical practice.We then propose semi-parametric methods for estimating the BATE curves of biomarkers and establish asymptotic results of the proposed estimators for the BATE curves.We also conduct extensive simulation studies to evaluate finite-sample properties of the proposed estimation methods.Finally,we illustrate the application of the proposed method in a real-world data set.     

模糊层次分析法在矩阵论教材评价方面的应用

基础学科的教材直接影响学生的基本功,尤其是几乎每个学科都会涉及的数学类教材,如矩阵论.矩阵论是研究生的基础课程,在对学生以后的学术道路有举足轻重的作用.所以选择一本合适的教材,对学生和教师来说都有不小的帮助.然而,对教材的评价而言,不能单单从一个方面入手,因此将模糊综合分析法与层次分析法结合,在矩阵论教材的评价方面建立评价体系,为高校选择合适的教材提供依据.     

On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudosolutions

Here we examine a method of selective smoothing in image restoration in BV space using a variable exponent functional of linear growth. Variable exponent growth was suggested in P. Blomgren et al. (1997) [6], and in Y. Chen et al. (2006) [8] the notion of a pseudosolution of the minimization problem is discussed, as well as the proof of existence and uniqueness of the solution of the time flow for the associated time dependent PDE. Here we prove stability for pseudosolutions, while also discussing the case for more general convex functionals of linear growth and variable exponent for use as an approximation for the time flow other than that which is used in Y. Chen et al. (2006) [8].     

Effective moduli of elasticity of a composite composed of anisotropic layers

Relations are obtained for the effective moduli of elasticity and Poisson's ratios of a laminated fiber-reinforced composite, each layer of which has at least orthorhombic symmetry. The elastic properties of the composite in terms of the elastic constants of the layer are expressed exactly, and the elastic constants of the individual layer in terms of the values for the fiber and the matrix are expressed approximately. Two approximations are considered: one corresponds to the Hashin-Shtrikman variational approach, while in the second the comparison material is assigned elastic properties equal to the Voigt or Reuss means of the values for each layer. A numerical example is worked for the combination boron fibers-epoxy resin. The results of the calculation are compared with the exact solution of the problem for a composite composed of alternating layers of boron and epoxy resin.     

Control of length in problems on nonlinear dynamics of flexible connections of variable length in a flow

We investigate the equations of models describing the change in length of a connection in time for a dynamic process. We introduce three conditions for variation of the distance between nodes in the computational grid that must be satisfied for a specific combination of coefficients in the model, and we introduce coefficients for approximation of equations themselves on a finite-difference grid. We give numerical results and analysis for three cases. Translated fromDinamicheskie Sistemy. Vol. 12, pp. 51–55, 1993.     

On the eigenvalues of a class of saddle point matrices

We study spectral properties of a class of block 2 × 2 matrices that arise in the solution of saddle point problems. These matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system. We give conditions for having a (positive) real spectrum and for ensuring diagonalizability of the matrix. In particular, we show that these properties hold for the discrete Stokes operator, and we discuss the implications of our characterization for augmented Lagrangian formulations, for Krylov subspace solvers and for certain types of preconditioners. The work of this author was supported in part by the National Science Foundation grant DMS-0207599 Revision dated 5 December 2005.     

The effect of receptor site nonuniformity on the measurement of rate constants

The BIAcore is an instrument for measuring rate constants in real time by using a surface–volume geometry. Though current models for the resulting reaction include transport effects for the reactant in solution, they do not account for spatial nonuniformities in the reactant attached to the wall. This work accounts for such nonuniformities and establishes that in the limit of small Damköhler number, such effects are negligible due to the averaging characteristics of the instrumentation.     

Adjoints of composition operators on Hilbert spaces of analytic functions

We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.