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In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem. This model well describes the behavior at the far-field for many physical and engineering systems such as fluid dynamical equation and electro-magnetic equation. The results obtained here should provide some theoretical guidance for designing effective far-field boundary conditions.

The Okounkov body is a construction which, to an effective divisor $D$ on an $n$-dimensional algebraic variety $X$, associates a convex body $\Delta(D)$ in the $n$-dimensional Euclidean space $\RR^n$. It may be seen as a generalization of the moment polytope of an ample divisor on a toric variety, and it encodes rich numerical information about the divisor $D$. When constructing the Okounkov body, an intermediate product is a lattice semigroup $\Gamma(D)\subset \NN^{n+1}$, which we will call the Okounkov semigroup. Recently it was discovered that finite generation of the Okounkov semigroup has interesting geometric implication for $X$ regarding toric degenerations and integrable systems, however the finite generation condition is difficult to establish except for some special varieties $X$. In this article, we show that smooth projective Fano varieties of coindex${}\le 2$ have finitely generated Okounkov semigroups, providing the first family of nontrivial higher dimensional examples that are not coming from representation theory. Our result also gives a partial answer to a question of Anderson, K\"uronya, and Lozovanu.

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the FischerBurmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function _FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan <i>P</i>-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (<i>An unconstrained smooth minimization reformulation of the second-order cone complementarity problem</i>, Math. Program. 104(2005), pp

Maximum likelihood ratio theory contributes tremendous success to parametric inferences, due to the fundamental theory of Wilks (1938). Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to nd and can not be optimal as shown in this paper. In this paper, we introduce the sieve likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks' phenomenon is unveiled. We demonstrate that the sieve likelihood statistics are asymptotically distribution free and follow 2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and generalized varying coefficient models. We further demonstrate that sieve likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster (1993). They can even be adaptively optimal in the sense of Spokoiny (1996) by using a simple choice of adaptive smoothing parameter. Our work indicates that the sieve likelihood ratio statistics are indeed general and powerful for nonparametric inferences based on function estimation.