The real representation of the quaternionic matrix is definited and studied. The relations between the positive (semi)define quaternionic matrix and its real representation matrix are presented. By means of the real representation, the relation between the positive (semi)definite solutions of quaternionic matrix equations and those of corresponding real matrix equations is established. Keywords...

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null are spacelike with null principal normals, and their can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur space. The first result paper is more precisely, straight lines. Furthermore, a method reconstruction curve from given line its involute provi...

In [14] Matsuda and Yorozu.explained that there is no special Bertrand curves in Eⁿ and they new kind of Bertrand curves called (1,3)-type Bertrand curves Euclidean space. In this paper , by using the similar methods given by Matsuda and Yorozu [14], we obtain that bitorsion of the quaternionic curve is not equal to zero in semi-Euclidean space E4^2. Obtain (N,B2) type quaternionic Bertrand cur...

Abstract The genus Operculina , a large symbiont-bearing benthic foraminifer, is characterized by high morphological variability showing thick involute to intermediate semi-involute flat evolute tests. Different morphotypes are either considered as ecophenotypes or distinct species. In order test the hypothesis of versus different species, single cell throughput sequencing approach was applied ...

Consider a 2D smooth closed curve evolving in time, the skeleton (medial axis) of the figure bounded by the curve, and the evolute of the curve. A new branch of the skeleton can appear / disappear when an evolute cusp intersects the skeleton. In this paper, we describe exact conditions of the skeleton bifurcations corresponding to such intersections. Similar results are also obtained for 3D sur...

Since the mode of Pythagorean-Hodograph curve’s derivate is a polynomial, its involute should be a rational polynomial. We study the expression form of PH cubic’s involute, discovering that its control points and weights are all constructed by corresponding PH cubic’s geometry property. Moreover, we prove that cubic PH curves have neither cusp nor inflection; there are two points coincidence in...

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of Z/2 by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.