Numbers

Periodic solutions of quaternionic-valued ordinary differential equations

“…Moreover, q commutes with any quaternion r if and only if q ∈ R. With this definition H is a noncommutative field or a division ring that contains the real numbers as a subfield [9]. It is also true that the complex numbers can be included in it; however, there are several ways to do it.…”

Finite-dimensional absolute-valued algebras

“…The work [R04] contains a fundamental compilation of results in this subject. According to the information collected in pages 243 and 245 of [EH90], A. Ostrowsky in his 1918 paper [O18] seems to have been the first mathematician considering absolute-valued algebras as abstract objects which are worth being studied. Since this pioneering work, there was a series of results on absolute-valued algebras culminating in Albert's paper [A47] asserting that any finite-dimensional absolute-valued real algebra is of dimension n = 1, 2, 4 or 8 and isotopic to one of the classical absolute-valued algebras R, C, H or O.…”

Hopf Fibrations and Hurwitz-Radon Numbers

“…Note also that a similar formula of size [2,2,2], that corresponds to multiplication of complex numbers, was known to Diophantus, and also appeared in the early VII century in a book of an Indian mathematician Brahmagupta; Fibonacci also used it in his ''Book of Squares.'' There exists also a square identity of size [8,8,8], found by a Dutch mathematician Ferdinand Degen in 1818, that corresponds to multiplication of octonions.…”

“…In a sense, contemporary algebraic topology has grown up with the Hopf fibration: the development of the theory of characteristic classes, homotopy theory, and K-theory was much influenced by the study of Hopf fibration; see [4,5,8,13,14]. Hopf fibration appears in other areas of mathematics and physics, including fluid dynamics, gauge theories, cosmology, and elementary particles [22].…”