An important property for finite difference schemes designed on curvilinear meshes is the exact preservation of free-stream solutions. This property is difficult to fulfill for high order conservative essentially non-oscillatory (WENO) finite difference schemes. In this paper we explore an alternative flux formulation for such finite difference schemes which can preserve free-stream solutions, based on the numerical technique for the metric terms, which can be applied to this alternative flux formulation but is difficult to be applied to the standard finite difference formulation. Free-stream and vortex preservation properties are investigated, and comparison with standard finite difference WENO schemes is made. Theoretical derivation and numerical results show that the finite difference WENO schemes based on the alternative flux formulation can preserve free-stream and vortex solutions on both stationary and dynamically generalized coordinate systems, hence giving much better performance than the standard finite difference WENO schemes for such problems.

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We investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate p/n regime, where the number of covariates p grows proportionally with the sample size n. Under appropriate regularity conditions, we establish the coordinate-wise asymptotic normality of regression M-estimates assuming a fixed-design matrix. Our proof is based on the second-order Poincare inequality (Chatterjee, 2009) and leave-one-out analysis (El Karoui et al., 2011). Some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices. We also show a counterexample, namely the ANOVA-type design, to emphasize that the technical assumptions are not just artifacts of the proof. Finally, the numerical experiments confirm and complement our theoretical results.

In this paper we study the generalization of the Brocard congurations. The denition of q-period broken line is given and then the new necessary and sucient condition about the existence of two conjugate Brocard points of the q􀀀period loop broken line is investigated. Based on the past research, a geometric method for the construction of all possible loop broken lines with conjugate Brocard points is developed.

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field.