Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster connection in complex CR geometry, we construct a canonical connection associated with a quaternionic pseudohermitian structure, when the underlying quaternionic CR structure satisfies the ultra-pseudoconvexity which is stronger than the strong pseudoconvexity. Comparison to Biquard's quaternionic contact structure [4] is also made.

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds isometrically immersed in Euclidean space ℝⁿ in terms of an integral representation of Weierstraß type. Restricting to the case of immersions in ℝ⁴, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.

In this paper, we give a new class of rigid Coxeter groups, which is an extension of [9] and a result of D. Radcliffe in [10].

In this paper we introduce three types of inequalities related to the Lorentz spaces on a measure space (M,m).

We consider random perturbations of non-singular measurable transformations S on [0,1]. By using the spectral decomposition theorem of Komorník and Lasota, we prove that the existence of the invariant densities for random perturbations of S. Moreover the densities for random perturbations with small noise strongly converges to the deinsity for Perron-Frobenius operator corresponding to S with respect to L¹([0,1])-norm.

This is the third in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin quivers. This paper is the companion to [W12] and deals with dihedral groups G = $\mathbb{D}$_n,q which have rank two special CM modules. We show that such reconstruction algebras are described by combining a preprojective algebra of type $\tilde{D}$ with some reconstruction algebra of type A.