9 Vectors and Quaternions 51 9.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 51 9.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 52 9.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 53 9.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 54...

4 Vectors and Quaternions 47 4.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 49 4.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 51...

The anti-self-adjoint operators of coordinate and momentum are introduced applied to the study tunnelling through potential barrier, in which imaginary value unavoidably appears. Tunnelling a temporal barrier is treated similarly it shown that quantum system can tunnel, while classical systems destroyed by such barrier. observables used novel treatment passage event horizon. By considering hori...

Let V be a finite dimensional complex vector space and W ⊆ GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. We prove that V reg is a K(π, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after...

Quaternions 1.1 Quaternions are a class of hypercomplex numbers with four real components [1]. By analogy with the complex numbers being representable as a sum of real and imaginary parts (z  a + bi), quaternions can also be written as a linear combination: q  a + bi + cj + dk, (1) where 1, i, j, k make a group and satisfy the noncommutative rules: i2  j2  k2  -1, ij  ji  k, jk  -kj ...

an involution or anti-involution is a self-inverse linear mapping. in this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. moreover, properties and geometrical meanings of these matrices will be given as reflections in r^3.

Remarkably simple closed-form expressions for the elements of the groups SU(n), SL(n,R), and SL(n,C) with n = 2, 3, and 4 are obtained using linear functions of biquaternions instead of n × n matrices. These representations do not directly generalize to SU(n > 4). However, the quaternion methods used are sufficiently general to find applications in quantum chromodynamics and other problems whic...