Plane-strain propagation of a fluid-driven fracture during injection and shut-in: Asymptotics of large toughness

This paper considers the problem of plane-strain fluid-driven fracture propagating in an impermeable elastic medium under condition of large toughness or, equivalently, of low fracturing fluid viscosity. We construct an explicit solution for a fracture propagating in the toughness-dominated regime when the energy dissipated in the viscous fluid flow inside the fracture is negligibly small compared to the energy expended in fracturing the solid medium. The next order corrections in viscosity to this limiting solution are then derived, allowing the range of problem parameters corresponding to the toughness-dominated regime to be established. The first-order small viscosity (large toughness) solution is shown to provide an excellent approximation of the solution for the crack length in the wide range of the viscosity parameter. Furthermore, this solution, when combined with the first-order small-toughness solution of Garagash and Detournay [Journal of Applied Mechanics, 2005], provides a simple analytical approximation of the crack length solution in practically the entire range of viscosity (toughness). It is also shown that the established method of asymptotic expansion in small parameter is equally applicable to study other small effects (e.g., fluid inertia) on the otherwise toughness-dominated solution. A solution for the fracture evolution during shut-in (i.e., after fluid injection rate is suddenly stopped) is also obtained. This solution, which corresponds to a slowing fracture evolving towards the toughness-dominated steady state, draws attention to the possibility of substantial fracture growth after fluid injection is ceased especially under conditions when the fracture propagation during injection phase is dominated by viscous dissipation.     

Coating flows of non-Newtonian fluids: weakly and strongly elastic limits

This paper presents an asymptotic analysis of the thickness of the liquid film that coats a smooth solid substrate when it is withdrawn from a bath of non-Newtonian fluid, and compares the results with experimental measurements. The film thickness is, to a good approximation, uniform above the point where the film is withdrawn from the fluid bath, and depends on the rotation rate, the fluid properties and the substrate geometry. Theoretical predictions of the film thickness for a number of different substrate geometries (an inclined plate, roller and fiber) are presented, and are compared with experimental measurements in a single roller geometry. Results are obtained for two different limits of the Criminale–Ericksen–Filbey constitutive equation in which the fluid rheology is either weakly elastic and dominated by shear thinning, or strongly elastic and dominated by elastic stresses. A lubrication analysis yields a thin-film equation which characterizes the film thickness as a function of spatial position. The rheological properties of the test fluids are measured independently using steady and oscillatory shearing deformations. The viscometric parameters are then used, in conjunction with the governing thin-film equation, which is solved using matched asymptotics, to give a quantitative prediction of the thickness of the fluid coating. The onset of an instability which causes the film thickness to vary with axial position along the roller is also observed experimentally.     

Lyapunov stability analysis for nonlinear delay systems

Sufficient conditions ensuring that a nonlinear system with disturbances having a delay is globally asymptotically stable independent of delay are given. The proof carried out relies extensively on a characterization of the stability property in terms of Lyapunov function. The result is applied to some biological systems and neural networks. A stabilizing memoryless controller for a second-order system with state-delay is also proposed.     

Consistency and Asymptotic Property of a Weighted Least Squares Method for Networked Control Systems

In this paper,we study the problems related to parameter estimation of a single-input and single-output networked control system,which contains possible network-induced delays and packet dropout in both of sensor-to-controller path and controller-to-actuator path. A weighted least squares (WLS) method is designed to estimate the parameters of plant,which could overcome the data uncertainty problemcaused by delays and dropout. ThisWLSmethod is proved to be consistent and has a good asymptotic property. Simulation examples are given to validate the results.     

[[2n,n]] Additive quantum error correcting codes

The self-orthogonal condition is analyzed with respect to symplectic inner product for the binary code that generated by [B1 I B2 B3],where Bi are the binary n ×n matrices,I is an identity matrix.By the use of the binary codes that generated by [B1 I B2 B2B1^T],asymptotic good[[2n ,n ]]additive quantum codes are obtained.     

Risk-efficient sequential estimation of multivariate random coefficient autoregressive process

A vector-valued autoregressive time series model is considered. The autoregressive coefficients of the model are random with possible dependencies among them. Estimation of the large number of parameters in such models becomes costly with an increase in dimension. A sequential procedure is proposed that promises a significant gain in the sample size thus reduction in the cost of implementation. The procedure is also risk efficient in the sense that as the cost of sampling becomes negligible the asymptotic predictive risk of the proposed procedure reaches the oracle predictive risk corresponding to the best fixed sample size procedure that assumes the values of the nuisance parameters to be known. Extensive simulation results are presented to illustrate the properties of the proposed procedure in a finite sample.     

非正态过程能力指数研究中的几个问题

非正态分布条件下过程能力指数的研究是目前关于过程能力分析中的热点之一。针对该领域中存在的若干问题,分析了正态性假设的重要性,指出导致质量特性呈现非正态分布的原因及其后果,以及如何进行质量特性分布的一维和多维正态性检验。对非正态过程能力指数的估计方法进行了分析和归类,简要指出在非正态过程能力指数研究中需要继续深入的几个方向。     

半变系数模型约束PLS估计的渐近正态性

半变系数模型已经获得了广泛的研究和应用,近几年,人们提出许多方法来估计其函数系数和常系数.在PLS方法基础上,本文给出半变系数模型模型在线性随机约束条件下的估计,并证明了常系数和函数系数估计的渐近正态性.     

Asymptotic solution of thermocapillary convection in thin annular two-layer system with upper free surface

The steady laminar two-dimensional thermocapillary convection in the thin annular two superposed horizontal liquid layers with one free surface, one liquid/liquid interface subjected to a radial temperature gradient was investigated using asymptotical analysis. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the lower layer thickness to the gap width, goes to zero. The numerical experiments are also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. The asymptotic results indicate that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.     

随机删失下半参数回归模型中估计的渐近性质

考虑半参数回归模型Yi=xiβ g(ti) iσiε,i=1,2,…,n,其中2iσ=f(ui).当Yi因受某种随机干扰而被随机右删失时,就删失分布未知的情形,利用所获得的删失数据定义了β与g(t)的估计^βn和^gn(t),在适当的条件下,证明了^nβ的渐近正态性,同时得到了^gn(t)的最优收敛速度.