We introduce and study a class of determinantal probability measures generalising the discrete point processes. These live on Grassmannian real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by positive self-adjoint contraction space, in way equivariant under action group isometries preserve splitting.

A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J,K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of A.D. Aleksandrov and Chern-Levine-Niren...

The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in H. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].

We show that noncompact simply connected harmonic manifolds with volume density Θ p (r) = sinh n−1 r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θ p (r) = sinh 2n−1 r cosh r is isometric to the complex hyperbolic space. A similar result is also proved for Quaternionic Kähler manifolds. Using our methods we get an alterna...

In this text, we prove that every quaternionic-contact structure can be embedded in a quaternionic manifold.

Analysis of the logical foundations of quantum mechanics indicates the possibility of constructing a theory using quaternionic Hilbert spaces. Whether this mathematical structure reflects reality is a matter for experiment to decide. We review the only direct search for quaternionic quantum mechanics yet carried out and outline a recent proposal by the present authors to look for quaternionic e...

Ostrowski type and Brauer type theorems are derived for the left eigenvalues of quaternionic matrix. We see that the above theorems for the left eigenvalues are also true for the case of right eigenvalues, when the diagonals of quaternionic matrix are real. Some distribution theorems are given in terms of ovals of Cassini that are sharper than the Ostrowski type theorems, respectively, for the ...