1,944 research outputs found
Some Operations on Quaternion Numbers
In this article, we give some equality and basic theorems about quaternion numbers, and some special operations.Li Bo - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaWang Pan - Qingdao University of Science and Technology, ChinaZhuang Yanping - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Fuguo Ge. Inner products, group, ring of quaternion numbers. Formalized Mathematics, 16(2):135-139, 2008, doi:10.2478/v10037-008-0019-x.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Xiquan Liang and Fuguo Ge. The quaternion numbers. Formalized Mathematics, 14(4):161-169, 2006, doi:10.2478/v10037-006-0020-1.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Similar sublattices of the root lattice
Similar sublattices of the root lattice are possible, according to a
result of Conway, Rains and Sloane, for each index that is the square of a
non-zero integer of the form . Here, we add a constructive
approach, based on the arithmetic of the quaternion algebra and the existence of a particular involution of the
second kind, which also provides the actual sublattices and the number of
different solutions for a given index. The corresponding Dirichlet series
generating function is closely related to the zeta function of the icosian
ring.Comment: 17 pages, 1 figure; revised version with several additions and
improvement
Split Quaternions and Particles in (2+1)-Space
It is known that quaternions represent rotations in 3D Euclidean and
Minkowski spaces. However, product by a quaternion gives rotation in two
independent planes at once and to obtain single-plane rotations one has to
apply by half-angle quaternions twice from the left and on the right (with its
inverse). This 'double cover' property is potential problem in geometrical
application of split quaternions, since (2+2)-signature of their norms should
not be changed for each product. If split quaternions form proper algebraic
structure for microphysics, representation of boosts in (2+1)-space leads to
the interpretation of the scalar part of quaternions as wavelength of
particles. Invariance of space-time intervals and some quantum behavior, like
noncommutativity and fundamental spinor representation, probably also are
algebraic properties. In our approach the Dirac equation represents the
Cauchy-Riemann analyticity condition and the two fundamental physical
parameters (speed of light and Planck's constant) appear from the requirement
of positive definiteness of quaternionic norms.Comment: The version published in Eur. Phys. J.
- …