Error estimates for approximate approximations on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact in- tervals, only. This method, which is based on an approximate partition of unity, was introduced by V. Mazya in 1991 and has mainly been used for functions defied on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed. In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

The empirical study of the price mechanism in the Chinese housing market

In China, the history of the establishment of the private housing market is pretty short. Actually in less then two decades, the market has grown from almost the scratch to playing an important role in the economy. A great achievement! But many problems also exist. They need to be properly addressed and solved. Price problem---simply put, housing price is too high--- is one of them, and this paper is focused on it. Three basic questions are posed, i.e. (1) how to judge the housing affordability? (2) why the housing price is so high? (3) how to solve the housing price problem. The paper pays particular attention to answering the second question. Except the numerous news reports and surveys show that most of the ordinary city dwellers complained about the high housing price, the mathematical means, the four ratios, are applied to judge the housing affordability in Shanghai and Shenzhen. The results are very clear that the price problem is severe. So why? Something is wrong with the price mechanism. This research shows that mainly these five factors contribute to the price problem: the housing reform, the housing development model, the unbalanced housing market, the housing project financing and the poor governmental management. Finally the paper puts forward five suggestions to solve the housing price problem in first-hand private Chinese housing market. They include: the establishment of real estate information system, the creation of specific price management department, the government price regulation, the property tax and the legalization of "cushion money".

Spatio-temporal resolution studies on a highly compact ultrafast electron diffractometer

Time-resolved diffraction with femtosecond electron pulses has become a promising technique to directly provide insights into photo induced primary dynamics at the atomic level in molecules and solids. Ultrashort pulse duration as well as extensive spatial coherence are desired, however, space charge effects complicate the bunching of multiple electrons in a single pulse.Weexperimentally investigate the interplay between spatial and temporal aspects of resolution limits in ultrafast electron diffraction (UED) on our highly compact transmission electron diffractometer. To that end, the initial source size and charge density of electron bunches are systematically manipulated and the resulting bunch properties at the sample position are fully characterized in terms of lateral coherence, temporal width and diffracted intensity.Weobtain a so far not reported measured overall temporal resolution of 130 fs (full width at half maximum) corresponding to 60 fs (root mean square) and transversal coherence lengths up to 20 nm. Instrumental impacts on the effective signal yield in diffraction and electron pulse brightness are discussed as well. The performance of our compactUEDsetup at selected electron pulse conditions is finally demonstrated in a time-resolved study of lattice heating in multilayer graphene after optical excitation.