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Abstract  

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y i Y:= {y i } i∈ℤ such that for x ∈ [y i , y i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each nN(Y) we construct a trigonometric polynomial P n of order ≦ n, changing its sign at the same points y i Y as f, and

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {f - P_n } \right\| \leqq c(s)\omega _3 \left( {f,\frac{\pi } {n}} \right),$$ \end{document}
where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω 3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.

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Abstract  

The work is aimed to develop the diagnostic method for testing the state of surface coated with the wear-resistant films. Thin wear-resistant ceramic films based on titanium such as TiN, TiCN, TiAlN are deposited on working surface of cutting tools or machine elements in order to improve their tribological properties. The operation life depends mainly on the residual stresses occurring in films and the kinetics of their relaxation as a function of temperature and time. The value of the stresses is influenced by the technological conditions of film deposition and the physical and chemical properties of the substrate and film. The paper has demonstrated the usability of the modulated-temperature dilatometry (MT DIL) for recording the changes in mechanical effects of the adhesive film on the substrate as a function of temperature and time. The substrates where in the shape of cylindrical rod, 30 mm length and 3 mm diameter and of the ribbon 30 mm in length, 2 mm in wide and 120 μm thick. The thickness of the coatings was from 2 to 3 μm. The films deposition were performed using the physical vapour deposition (PVD) technique.

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Analysis Mathematica
Authors:
G. A. Dzyubenko
,
J. Gilewicz
, and
I. A. Shevchuk

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