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In the long-time scale, we consider the fluid dynamical limits for the kinetic equations when the fluctuation is decomposed into even and odd parts with respect to the microscopic velocity with different scalings. It is shown that when the background state is an absolute Maxwellian, the limit fluid dynamical equations are the incompressible Navier-Stokes equations with viscous heating. This is different from the case when the even and odd parts of the fluctuation have the same scaling where the standard incompressible Navier-Stokes equations without viscous heating are obtained. On the other hand, when the background is a local Maxwellian, it is shown that the above even-odd decomposition leads to a non-classical fluid dynamical system without viscous heating which has been used to describe the ghost effect in the kinetic theory. In addition, the above even-odd decomposition is justified rigorously for the Boltzmann equation for the former case when the background is an absolute Maxwellian.

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This article introduces the large portfolio selection using gross-exposure constraints. It shows that with gross-exposure constraints, the empirically selected optimal portfolios based on estimated covariance matrices have similar performance to the theoretical optimal ones and there is no error accumulation effect from estimation of vast covariance matrices. This gives theoretical justification to the empirical results by Jagannathan and Ma. It also shows that the no-short-sale portfolio can be improved by allowing some short positions. The applications to portfolio selection, tracking, and improvements are also addressed. The utility of our new approach is illustrated by simulation and empirical studies on the 100 FamaFrench industrial portfolios and the 600 stocks randomly selected from Russell 3000.

The analysis of the vestibular system (VS) is an important research topic in medical image analysis. VS is a sensory structure in the inner ear for the perception of spatial orientation. It is believed several diseases, such as the Adolescent Idiopathic Scoliosis (AIS), are due to the impairment of the VS function. The morphology of the VS is thus of great research significance. A major challenge is that the VS is a genus-3 surface. The high-genus topology of the VS poses great challenges to find accurate pointwise correspondences between the surfaces and whereby perform accurate shape analysis. In this paper, we present a method to obtain the landmark constrained diffeomorphic registration between the VS surfaces based on the quasi-conformal theory. Given a set of corresponding landmarks on the VS surfaces, a diffeomorphism between the VS surfaces that matches the features consistently can be obtained. The basic idea is to iteratively search for an admissible Beltrami coefficient, which is associated to our desired landmark matching registration. With the obtained surface registrations, vertex-wise morphometric analysis can be carried out. Two types of geometric features are used for shape comparison. One is the collection of homotopic loops on each canals of the VS, which can be used to measure the local thickness of the canals. From the homotopic loops, centerlines can be extracted. By examining the deviations of the centerlines from the best fit planes, bendings of the canals can be detected. The second geometric feature is the minimal surface enclosed by the homotopic loop. From the minimal surfaces of each homotopic loops, cross-sectional area of the canals can be evaluated. To study the local shape difference more comprehensively, a complete shape index, which is defined using the Beltrami coefficients and surface curvatures, is used. We test proposed registration method on 15 VS of normal control subjects and 12 VS of patients suffering from AIS. Experimental results show the efficacy and accuracy of the proposed algorithm to compute the VS surface registration. Shape analysis has also been carried out using the proposed geometric features and shape index, which reveals shape differences in the posterior canal between normal and diseased AIS groups.