Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{\rho_{0}(t,x)}{(2\pi \theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\theta_{0}(t,x)},\text{\ }\theta_{0}(t,x)=K\rho_{0}^{2/3}(t,x).$Our main result states that if the Hilbert expansion is valid at$t=0$for well-prepared small initial data with irrotational velocity$u_0$, then it is valid for$0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$where$\rho_{0}(t,x)$and$ u_{0}(t,x)$satisfy the Euler-Poisson system for monatomic gas$\gamma=5/3$

A neural network-based estimator for the mixture ratio of the Space Shuttle Main Engine

In order to properly utilize the available fuel and oxidizer of a liquid propellant rocket engine, the mixture ratio is closed loop controlled during main stage (65 percent - 109 percent power) operation. However, because of the lack of flight-capable instrumentation for measuring mixture ratio, the value of mixture ratio in the control loop is estimated using available sensor measurements such as the combustion chamber pressure and the volumetric flow, and the temperature and pressure at the exit duct on the low pressure fuel pump. This estimation scheme has two limitations. First, the estimation formula is based on an empirical curve fitting which is accurate only within a narrow operating range. Second, the mixture ratio estimate relies on a few sensor measurements and loss of any of these measurements will make the estimate invalid. In this paper, we propose a neural network-based estimator for the mixture ratio of the Space Shuttle Main Engine. The estimator is an extension of a previously developed neural network based sensor failure detection and recovery algorithm (sensor validation). This neural network uses an auto associative structure which utilizes the redundant information of dissimilar sensors to detect inconsistent measurements. Two approaches have been identified for synthesizing mixture ratio from measurement data using a neural network. The first approach uses an auto associative neural network for sensor validation which is modified to include the mixture ratio as an additional output. The second uses a new network for the mixture ratio estimation in addition to the sensor validation network. Although mixture ratio is not directly measured in flight, it is generally available in simulation and in test bed firing data from facility measurements of fuel and oxidizer volumetric flows. The pros and cons of these two approaches will be discussed in terms of robustness to sensor failures and accuracy of the estimate during typical transients using simulation data

Spectral stability of Prandtl boundary layers: an overview

In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.Comment: 17 page