A problem of cauchy type for a fractal string

An asymptotic theory is developed for multivariate regression in cointegrated systems whose variables are moderately integrated or moderately explosive in the sense that they have autoregressive roots of the form ρni = 1 + ci/nα, involving moderate deviations from unity when α ∈ (0, 1) and ci ∈ ℝ are constant parameters. When the data are moderately integrated in the stationary direction (with ci < 0), it is shown that least squares regression is consistent and asymptotically normal but suffers from significant bias, related to simultaneous equations bias. In the moderately explosive case (where ci > 0) the limit theory is mixed normal with Cauchy-type tail behavior, and the rate of convergence is explosive, as in the case of a moderately explosive scalar autoregression (Phillips and Magdalinos, 2007, Journal of Econometrics 136, 115–130). Moreover, the limit theory applies without any distributional assumptions and for weakly dependent errors under conventional moment conditions, so an invariance principle holds, unlike the well-known case of an explosive autoregression. This theory validates inference in cointegrating regression with mildly explosive regressors. The special case in which the regressors themselves have a common explosive component is also considered.

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Global smoothness and uniform convergence of smooth Poisson–Cauchy type singular operators

Applied Mathematics and Computation ◽

10.1016/j.amc.2009.07.030 ◽

2009 ◽

Vol 215 (5) ◽

pp. 1718-1731 ◽

Cited By ~ 5

Author(s):

George A. Anastassiou ◽

Razvan A. Mezei

Keyword(s):

Uniform Convergence ◽

Cauchy Type ◽

Singular Operators ◽

Global Smoothness

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On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

Mathematics and Mechanics of Solids ◽

10.1177/10812865211043152 ◽

2021 ◽

pp. 108128652110431

Author(s):

Rui Cao ◽

Changwen Mi

Keyword(s):

Finite Element ◽

Contact Problem ◽

Integral Equations ◽

Singular Integral ◽

Functional Dependence ◽

Singular Integral Equations ◽

Homogeneous Layer ◽

Cauchy Type ◽

Receding Contact ◽

Contact Pressures

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

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