Let σ be the geodesic inversion on a Heisenberg type group$N$with homogeneous dimension$Q$, and denote by$S$the jacobian of σ. We prove that, for $$ - \frac{1}{2}Q< \alpha< \frac{1}{2}Q$$ , the operators $$T_\alpha :f \mapsto S^{1/2 - \alpha /Q} (f \circ \sigma )$$ are bounded on certain homogeneous Sobolev spaces $$\mathcal{H}^\alpha (N)$$ if and only if$N$is an Iwasawa$N$-group.

We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless, if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.

Let F be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from GLn+1(F) to GLn(F). A main result shows that each Bernstein component of an irreducible smooth representation of GLn+1(F) restricted to GLn(F) is indecomposable. We also classify all irreducible representations which are projective when restricting from GLn+1(F) to GLn(F). A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space X such that every run of the K_X-MMP immediately terminates with a Mori fibre space, yet X does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint ($\nabla \cdot \B = 0$) of its magnetic field $\B$. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements \cite{Li2005}, and another approach of the divergence cleaning technique given by Dedner et al. \cite{Dedner2002}. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.