Let X be a spanning point set of PG(N, K), N [member of] N, N [greater than or equal to] 3, with K any skew field, and let [SIGMA] be a collection of planes of PG(N, K) such that, for any [pi] [member of] [SIGMA], the intersection [pi] [intersection] X is an oval X([pi]) in [pi] (and then, for x [member of] X([pi]), we sometimes denote [T.

To be more precise, let V = (X, [SIGMA]) be a Veronesean cap, where X is a set of points in PG(N, K), for some skew field K, and [SIGMA] its collection of planes.

Ten chapters discuss the

skew field of quaternions; elements of the geometry of S3, Hopf bundles, and spin representations; internal variables of singularity free vector fields in a Euclidean space; isomorphism classes, Chern classes, and homotopy classes of singularity free vector fields in 3-space; Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras, and special linear groups; the Heisenbreg group and natural C*-algebras of a vector field in 3-space; the Schrodinger representation and the metaplectic representation; the Heisenberg group as a basic geometric background of signal analysis and geometric optics; quantization of quadratic polynomials; and field theoretic Weyl quantization of a vector field in 3-space.

In the past, researches into the quaternion skew field had a more theoretical importance, but now a growing number of investigations give wide practical applications of quaternions.

In [29], we have obtained new determinantal representations of the W-weighted Drazin inverse over the quaternion skew field without any auxiliary matrices.

The theory of the row-column determinants over the quaternion skew field has been introduced in [30-32], and later it has been applied to research generalized inverses and generalized inverse solutions of matrix equations.

The following theorem gives determinantal representations of the Moore-Penrose inverse over the quaternion skew field H.

Lei, "Least squares Hermitian solution of the matrix equation (AXB; CXD) = (E; F) with the least norm over the skew field of quaternions," Mathematical and Computer Modelling, vol.

Song, "Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications," Electronic Journal of Linear Algebra, vol.

Kyrchei, "Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field," Applied Mathematics and Computation, vol.

Kyrchei, "Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer's rules," Linear and Multilinear Algebra, vol.

Kyrchei, "Determinantal representation of the Moore-Penrose inverse matrix over the quaternion skew field," Journal of Mathematical Sciences, vol.