A posteriori error estimators for the Stokes equations

Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft     

一类具时滞的Liénard型方程的周期解

考虑一类具时滞的 Liénard型方程x+f[x(t) ]x(t) +g[t,x(t-τ(t) ) ]=p(t) =p(t+T) ,利用重合度理论 ,获得了此方程至少存在一个 T周期解的充分条件     

Asymptotic expansions in sequential estimation for the first-order random coefficient autoregressive model: Regenerative approach

Edgeworth expansions for the distribution of a sequential least squares estimator in the random coefficient autoregressive (RCA) model are derived. The regenerative approach to second-order asymptotic analysis of Markov-type statistical models is developed.     

Limits of parallelism in explicit ODE methods

Numerical methods for ordinary initial value problems that do not depend on special properties of the system are usually found in the class of linear multistage multivalue methods, first formulated by J.C. Butcher. Among these the explicit methods are easiest to implement. For these reasons there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in parallel. Each such group of concurrent function evaluations can be regarded as a stage of the method. However, it turns out that parallelism affords only limited opportunity for reducing the computing time with such methods. This is most evident for the simple linear homogeneous constant-coefficient test problem, whose solution is essentially a matter of approximating the exponential by an algebraic function. For a given number of stages and a given number of saved values, parallelism offers a somewhat enlarged set of algebraic functions from which to choose. However, there is absolutely no benefit in having the degree of parallelism (number of processors) exceed the number of saved values of the method. Thus, in particular, parallel one-step methods offer no speedup over serial one-step methods for the standard linear test problem. Although the implication of this result for general nonlinear problems is unclear, there are indications that dramatic speedups are not possible in general. Also given are some results relevant to the construction of methods.Work supported in part by National Science Foundation grants DMS 89 11410 and DMS 90 15533 and by US Department of Energy grant DOE DEFG02-87ER25026. Work of the second author was completed while at the University of Illinois.     

偶氮苯衍生物三阶非线性的四波混频研究

用皮秒Ｎｄ：ＹＡＧ激光器的倍频光（５３２ｎｍ）对具有离域π－共轭电子云结构的偶氮苯类样品材料作简并四波混频补给，测得三阶非线性电极化率ｘ＾（３）和它们的时间响应分别为１０＾－９ｅｓｕ和２０ｐｓ，并对影响ｘ＾（３）的瞬时光栅作用和ｘ＾（３）的响应时间人了讨论。     

Multi-cracks problem for piezoelectric materials strip subjected to dynamic loading

This paper analyses and models the dynamic interaction among permeable multi-cracks in a piezoelectric strip under anti-plane shear waves by the Schmidt method. The Fourier transform is applied and then two pairs of triple integral equations can be solved using the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on not only the crack length and the piezoelectric coefficient, but also the thickness of the piezoelectric strip, the distance between multi-cracks and the frequency of incident wave.     

Biomagnetic measurements utilising a superparamagnetic marker: a feasibility study

Summary A preliminary study is here reported on a new potential marker for biomagnetic measurements. The marker consists of superparamagnetic polymer microspheres which were detected in the presence of external steady magnetic fields by means of an r.f.-SQUID magnetometer. The particles were prepared in samples differing in the concentration value and immersed in a homogeneous magnetic field of variable intensity. A simple model was taken into account for the distribution of the microspheres in the samples, so that the theoretical values were compared to the marker field values measured by the biomagnetic sensor. The overall sensitivity of the experimental apparatus and the minimum concentration value of the marker were then estimated.     

Algebraic models defined over Q of differential manifolds

Here we prove that every compact differential manifold has a smooth algebraic model defined over Q. In dimension 2 we find an algebraic model (may be singular) defined over Q and birational over Q to the projective plane.     

Density Correlations in the Two-Dimensional Coulomb Gas

We consider a two-dimensional Coulomb gas of positive and negative pointlike unit charges interacting via a logarithmic potential. The density (rather than the charge) correlation functions are studied. In the bulk, the form-factor theory of an equivalent sine-Gordon model is used to determine the density correlation length. At the surface of a rectilinear plain wall, the universality of the asymptotic behavior of the density correlations is suggested. A scaling analysis implies a local form of the compressibility sum rule near a hard wall. A symmetry of the Coulomb system with respect to the Möbius conformal transformation, which induces a gravitational source acting on the particle density, is established. Among the consequences, a universal term of the finite-size expansion of the grand potential is derived exactly for a disk geometry of the confining domain.