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# Approximation by rational functions with prescribed numerator degree in L p spaces for 1

Acta Mathematica Hungarica
Authors: X.F. Mei and F.P. Zhou

## Abstract

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in L pspaces for 1 < p < ∞ and proves that if f(x)∈L p [-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈R n l such that

\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} $$\left\| {f(x) - r(x)} \right\|_{L^p } \leqq C_{p,l,b} \omega (f,n^{ - 1} )_{L^p } ,$$ \end{document}
where R n l indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and C p , l,b is a positive constant depending only on p, l and b which relates to the distance among the sign change points of f(x) and will be given in 3.

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# The asymptotic property of approximation to |x| by Newman's rational operators

Acta Mathematica Hungarica
Authors: T.F. Xie and S.P. Zhou

## Abstract

Let
\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} $$a = e^{ - 1/\sqrt n } ,p(x) = \Pi _{k = 1}^{n - 1} (a^k + x),r_n (x) = x\frac{{p(x) - p( - x)}} {{p(x) + p( - x)}}$$ \end{document}
. The present note gives the asymptotoc formula of max
\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} $$\mathop {\max }\limits_{|x| \leqq 1} \left| {|x| - r_n (x)} \right|$$ \end{document}
.
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# A remark on approximation by monotone sequences of polynomials

Acta Mathematica Hungarica
Authors: T. F. Xie and S. P. Zhou
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# On simultaneous approximation to a differentiable function and its derivative by Pál-type interpolation polynomials

Acta Mathematica Hungarica
Authors: T. F. Xie and S. P. Zhou
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# On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Acta Mathematica Hungarica
Authors: F. J. Feng and S. P. Zhou

## Abstract

Let fL 2π be a real-valued even function with its Fourier series , and let S n(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {a n} is decreasing and , then if and only if . Later, the monotonicity condition set on {a n} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM7 condition in real space. In this paper, we give a complete generalization to the complex space.

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# Spatial pattern of water controlled wetland communities on the Sanjiang Floodplain, Northeast China

Community Ecology
Authors: D. Zhou, H. Gong, Z. Luan, J. Hu, and F. Wu

The study site is the Honghe National Nature Reserve, a Ramsar designated site on the Sanjiang Plain in Northeast China. We present results regarding the spatial pattern and structure of plant communities in these most important natural but continually diminishing freshwater wetlands of China to help promote both protection and restoration. By investigating three ecological levels (landscape, ecosystem and community), this paper quantifies the characteristics of spatial pattern with the aim to identify specific ecological correlations with different hydrogeomorphic features. Specifically, the research involves hierarchical mapping of vegetation types by use of remote sensed data, and the coupling of landscape indices with fluvial topographic zones that have been deduced by GIS from DEM. Statistics from historical survey data are also used to measure the degradation of marshes as well as the historical change of the hydrological regime. We found that dominant is the Calamagrostis angustifolia — Carex spp. community type, a wet meadow and marsh complex within the prevailing landscape mosaic of shrubland and meadow. The results suggest that the sites’ hydro-geomorphic character has decisive influence on plant community structure and composition. There is only limited direct human interference in the sites and, as a consequence, the spatial pattern of vegetation distribution is natural. However, changes to the hydrological regime as the result of extensive irrigation activity in the surrounding area has led to rapid degradation of marsh wetlands within the sites, which threatens the ecological status in this storehouse of “Natural Genes” in the reserve.

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# Ultrahigh Molecular mass Polyethylene/ Carbon Nanotube CompositesCrystallization and melting properties

Journal of Thermal Analysis and Calorimetry
Authors: X. Xie, K. Aloys, X. Zhou, and F. Zeng

## Abstract

Ultrahigh molecular mass polyethylene (UHMMPE) is filled with carbon nano-tubes (CNTs) by solution in the presence of maleic anhydride grafted styrene-(ethylene-co-butylene)-styrene copolymer (MA-SEBS) as a compatibilizer. The UHMMPE/CNT composites crystallized from melt were prepared at a cooling rate of 20C min-1. The melting and crystallization behaviors of UHMMPE/ CNT composites were investigated by differential scanning calorimetry. The results showed that onset melting temperature (T m) and degree of crystallinity (X c) of UHMMPE/CNT composites crystallized from solution are higher than those from melt due to the larger crystalline lamellar thickness. The onset crystallization temperature (T c) of UHMMPE/CNT composites tends to shift to higher temperature region with increasing CNT content in the composites. Tm and Tc of UHMMPE phase in UHMMPE/CNT composites decrease with the addition of MA-SEBS. Moreover, the crystallization rate of UHMMPE phase in UHMMPE/CNT composite is increased due to the introduction of CNTs. MA-SEBS acts as compatilizer, enhances the dispersion of CNTs in the UHMMPE matrix. Thereby, the crystallization rate of UHMMPE phase in UHMMPE/CNT composite is further increased with the addition of MA-SEBS.

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# Non-isothermal crystallization and crystalline structure Of PP/POE blends

Journal of Thermal Analysis and Calorimetry
Authors: J. Ying, S. Liu, F. Guo, X. Zhou, and X. Xie

## Abstract

Polypropylene (PP) /ethylene-octene copolymer (POE) blends with different content of POE were prepared by mixing chamber of a Haake torque rheometer. The crystallization behaviors and crystal structure of PP/POE blends were systematically investigated by differential scanning calorimetry (DSC), wide angle X-ray diffraction (WAXD) and polarized optical microscopy (POM). The results showed that PP spherulites became defective and the crystallization behavior was influenced intensely with the introduction of POE. At the low content of POE, the addition of POE decreases the apparent incubation period (Δt i) and the apparent total crystallization period (Δt c) of PP in blends due to the heterogeneous nucleation of POE, and small amount of β-form PP crystals form because of the existence of POE. However, at high content of POE, the addition of POE decreases the mobility of PP segments due to their strong intermolecular interaction and chain entanglements, resulting in retarding the crystallization of PP, decreasing in the amount of β-form PP crystals, and increasing in Δt i and Δt c of PP in blends.

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# Effect of nucleating agents on the crystallization of Li2O-Al2O3-SiO2 system glass

Journal of Thermal Analysis and Calorimetry
Authors: A.-M. Hu, K.-M. Liang, G. Wang, F. Zhou, and F. Peng

## Abstract

The processes of nucleation of Li2O-Al2O3-SiO2 glasses with TiO2 and TiO2+ZrO2 as nucleating agents were discussed. The DTA peak temperature and DTA peak height shown a strong dependence on the nucleation temperature in the glass with TiO2, while in the glass with TiO2+ZrO2 this tendency was small. The optimum nucleation temperatures were 745 and 760°C for two glasses. It suggested that with TiO2+ZrO2 as nucleating agents, the crystallization had lower sensitivity for nucleation temperature, and the glass had higher nucleation efficiency than with TiO2.

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# DSC Studies of the Effects of Cisplatin and Transplatin on G-Actin

Journal of Thermal Analysis and Calorimetry
Authors: X.-L. Zhou, Y. Yang, Z.-F. Li, B.-H. Wang, and Y.-M. Zhang

## Abstract

The effects of cisplatin and its trans isomer transplatin on the thermal denaturation of G-actin were studied with a Micro DSC-III differential scanning calorimeter. The denaturation enthalpy of G-actin was found to be 12 J g–1, and the denaturation temperature was 328 K. The thermal denaturation curve showed that increasing cisplatin concentration decreased the enthalpy change. However, after the ratio of cisplatin to G-actin attained 8:1 (mol:mol), the denaturation enthalpy no longer decreased. Transplatin decreased the enthalpy change more rapidly. In contrast with cisplatin, the denaturation peak at 328 K disappeared, and a strong exothermic peak appeared at 341 K when the ratio of transplatin to G-actin was 8:1 (mol:mol). The enthalpy change was 75 J g–1, which is far in excess of the range of weak interactions. This strong exothermic phenomenon probably reflects the agglutination of protein. The effects of cisplatin and transplatin on the number of the free thiol groups of G-actin are discussed.

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