In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operator$D$(α)=$D$_{0}−α$V$1 acting in$L$_{2}($R$^{3};$C$^{4}) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.

Let ($A$_{$0, A$}_{$1$}) be a compatible pair of quasi-Banach spaces and 1et$A$be a corresponding space of real interpolation type such that$A$_{$0$}∩$A$_{$1$}is not dense in$A$. Upper and lower estimates are obtained for the distance of any element$f$of$A$from$A$_{$0$}∩$A$_{$1$}. These lead to formulae for the distance in a large number of concrete situations, such as when$A$_{$0$}∩$A$_{$1$}=$L$^{$∞$}and$A$is either weak-$L$^{q}, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image, when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.

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In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton-Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.