In this talk, we discuss the maximum number of n × n pure imaginary quaternionic solutions to the Hurwitz matrix equations given by T i T * j + T j T Abstract Let T be a bounded linear operator on a complex Hilbert space H. For 0 ≤ q ≤ 1, the

It is shown that elementary indistinguishability properties of partially polarized mixtures are consistent only with the conventional Hilbert space model of quantum mechanics and a few exotic alternatives. This applies even in low dimensions where quantum logic and Gleason’s theorem give either weak or no constraints. Experimental methods for eliminating the exotic cases (which include quaterni...

We consider the logistic map over quaternions H ∼ R 4 and different 2D projections of Mandelbrot set in 4D quaternionic space. The approximations (for finite number of iterations) of these 2D projections are fractal circles. We show that a point process defined by radiuses Rj of those fractal circles exhibits pure 1/f noise.

We consider complete nearly Kähler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly Kähler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly Kähler manifold to be a twistor space over a quaternionic Kähler manifold is given. This enables us to give classifica...

In this paper we prove a version of Mordell’s inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that 16dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structu...

Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Bryant's Weierstrass type representation.

We consider the logistic map over quaternions H ∼ R 4 and different 2D projections of Mandelbrot set in 4D quaternionic space. The approximations (for finite number of iterations) of these 2D projections are fractal circles. We show that a point process defined by radiuses Rj of those fractal circles exhibits pure 1/f noise.

In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their holomorphic functions. Mathematics Subject Classification (2000). 53C28, 32L25,...

Properties of the Cauchy–Riemann–Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3–surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy– Riemann–Fueter equation are established.

In this paper we prove an analogue of Mordell’s inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that 16dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz struc...