threats are one of the most fundamental problems of our modern societies that daily will affect citizens in urban spaces. this becomes more problematic when the people in their everyday living spaces encounter with behaviors that cause lack of security and should have been feel comfort and their families used these spaces freely. this causes many problems for citizens. in today cities because o...

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g. [9, 10]. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H–planar curves coincides with the clas...

The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group RSO(4) appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoidal structure we indicate the existence of a canonical quaternionic gerbe associ...

We discuss the Schur complement formula for quaternionic matrices, M , and give an efficient method to calculate the matrix inverse. We also introduce the functional D[M ] which extends to quaternionic matrices the non-negative number |det[M ]|. 1. Introduction. Much of the spectral theory of complex matrices does not extend to quaternion matrices without further modifications [1, 2, 3]. In par...

Every lower semi-continuous closed-and-convex valued mapping Φ : X → 2 , where X is a Σ-product of metrizable spaces and Y is a Hilbert space, has a single-valued continuous selection. This improves an earlier result of the author.

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension 1 or dimension (codimension and 2 2) in the product complex (quaternionic) projective space with any other Riemannian manifold. characterize as only CSMI two spaces. As an application, we rank one symmetric space.

We relate the moduli space of Yang-Mills instantons to quaternionic manifolds. For instanton number one, the Wolf spaces play an important role. We apply these ideas to instanton calculations in N = 4 SYM theory. Instantons in Yang-Mills theories are defined by the solutions to the self-duality equation in four-dimensional Euclidean space, Fμν = Fμν = 1 2ǫμνρσFρσ , (1) and are characterized by ...