We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Expressions for the fluid velocity, density and temperature are derived using equilibrium statistical mechanics. The results show that the number of samples needed to adequately resolve the flow-field scales as the inverse square of the Mach number. The theoretical results are verified for a dilute gas using direct Monte Carlo simulations. The agreement between theory and simulation verifies that the use of equilibrium theory is justified. / Singapore-MIT Alliance (SMA)

International academia expressed doubt that china¡¦s statistical surveys reflect actual economic performance in 2001, in regard to the China¡¦s statistical accuracy, many scholars put forward different opinions. And compared the Chinese government's official figures with the countries of the world, lie in the question of investigating information on¡§ the degree of public¡¨ and the statistical surveys of¡§ report to a higher level officials¡¨. The main purpose of this research is probing into the problem of the Chinese government's official figures via the surveys of Chinese household income to prove that the distribution income of China to the initial data inaccuracy making China¡¦s household income unable to reflect the true. Therefore, we should doubt that China's statistics for lacking credibility on household income. Through the analysis structure of this text, we find out the following¡Gfirst of all, under the centralization type statistical system, the local statistical organization undertake the job of¡§survey¡¨ and¡§analysis¡¨ as the role of ¡§the sportsman and concurrently referee¡¨ make the question of statistical standard unable to improve. Secondly, with the Chinese non-sampling error proportion is far too larger. China statistical year book reported Chinese household income date merely conform to the ¡§salaried family¡¨¡F those who worked in the state-owned enterprise, but surveys date are not suitable for rural family and non state-owned enterprises worker¡¦s family. Moreover, through analyzing China¡¦s statistical education; the protection of the relent law and the impetus of the civil statistical work, my research discovered that the Chinese government's official figures are less accurate than those of industrialized countries and the supervision function are very weak. Finally, China has transformed its planned economy system into an initial socialist market economy system that interwove by the various forms and multiple interests relation, therefore strengthened the difficulty of statistical surveys of the professional. Especially under the turn rail environment deepening the degree of high-level and low-level officials¡§ the individual information asymmetric¡¨, and after the dismantling of the old supervise management system, the new statistical supervisal system is unable to confine yet. Contemporary the implemented statistical surveys of socialist market economy are more error then those of planned economy. On the above analyzes, I suggested that the Chinese government¡¦s official statistics should strengthened the sophisticated data compiling systems such as American statistics system that adopted the statistics system of social feedback and the statistics of decentralize, unable china to increase its social statistics and reduce the question of statistical surveys of¡§ report to a higher level officials¡¨, and heightening the Chinese statistical data a little more accurately.

Uncertainty analysis is an important part of system design. The formula for error propagation through a system model that is most-often cited in literature is based on a first-order Taylor series. This formula makes several important assumptions and has several important limitations that are often ignored. This thesis explores these assumptions and addresses two of the major limitations. First, the results obtained from propagating error through nonlinear systems can be wrong by one or more orders of magnitude, due to the linearization inherent in a first-order Taylor series. This thesis presents a method for overcoming that inaccuracy that is capable of achieving fourth-order accuracy without significant additional computational cost. Second, system designers using a Taylor series to propagate error typically only propagate a mean and variance and ignore all higher-order statistics. Consequently, a Gaussian output distribution must be assumed, which often does not reflect reality. This thesis presents a proof that nonlinear systems do not produce Gaussian output distributions, even when inputs are Gaussian. A second-order Taylor series is then used to propagate both skewness and kurtosis through a system model. This allows the system designer to obtain a fully-described non-Gaussian output distribution. The benefits of having a fully-described output distribution are demonstrated using the examples of both a flat rolling metalworking process and the propeller component of a solar-powered unmanned aerial vehicle.