We show that the Kazhdan-Lusztig category KL_k of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1)ˆ has vertex algebraic braided tensor supercategory structure, and that its full subcategory O_k^fin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)ˆ-module in KL_k has a projective cover in O_k^fin, and we determine all fusion rules involving simple and projective objects in O_k^fin. Then using Knizhnik-Zamolodchikov equations, we prove that KL_k and O_k^fin are rigid. As an application of the tensor supercategory structure on O_k^fin, we study certain module categories for the affine Lie superalgebra sl(2|1)ˆ at levels 1 and −1/2. In particular, we obtain a tensor category of sl(2|1)ˆ-modules at level −1/2 that includes relaxed highest-weight modules and their images under spectral flow.

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Building on [BB1] we prove a general criterion for convergence of (possibly singular) Bergman measures towards pluripotential-theoretic equilibrium measures on complex manifolds. The criterion may be formulated in terms of the growth properties of the unit-balls of certain norms on holomorphic sections, or equivalently as an asymptotic minimization property for generalized Donaldson$L$-functionals. Our result settles in particular a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points and it gives the convergence of Bergman measures towards the equilibrium measure for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.

Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).

We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure, in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.