Abstract. We prove that if f is increasing on [ -1,1], then for each n = 1, 2 ..... there is an increasing algebraic polynomial P. of degree n such that {f(x) -P.(x){ < cw2( f, V/I -x 2/n), where w2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.

Abstract.We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.

Based on previous work for the static problem, in this paper, we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

ABSTRACT. We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.

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Diamonds like carbon (Ag-DLC) films of nine Ag contents were deposited on (100) Si substrate in a mid-frequency dual-magnetron system. The influence of Ag content and nanograin size on the microstructure, mechanical properties and sliding tribological behaviors of the films has been investigated. It is found that (i) the Ag nanocrystallites were dispersed in the amorphous DLC matrix, and increasing Ag content resulted in an upward tendency of the Ag nanocrystallite size; (ii) the films with Ag content ranging between 3.3-11.4 at% and with Ag nanograin sizes in range of 5.4-16.8 nm exhibited higher hardness, lower intrinsic stress, better adhesion, and lower friction coefficient and wear rate as compared with those of 14.7-23.6 at% Ag content and 19.7-34.7 nm nanograin sizes; and (iii) the best combined properties were achieved for the film deposited with 8.7 at% Ag and with grain size 12.9 nm.

We study the behavior of moduli of smoothness of splines s of order r with equally spaced knots {xi}, x,+l xi h. The main results are as follows.(1) For each 0 <_ rn < r, all quantities hJa,_j(s(J),h)p, 0 <_ j <_ rn, are equivalent and can be measured by a discrete norm of the ruth differences of the B-spline coefficients of s, which we call the rnth discrete modulus of smoothness of s.(2) All quantities hJwm_j(s(J), h)p, m >_ r, 0 _< j <_ r-1, are equivalent to cot(s, h)p, which can be measured by the rth discrete modulus of s.(3) When h is replaced by t, 0 _< _< h, in the results above, all the quantities can still be measured by the corresponding discrete modulus multiplied by a power of t/h. The results generalize the notion of a discrete norm of B-spline series in the case of equal spacing.As an application, we use these results to prove that w3 is the best rate of convex approximation by such splines.