Generalized commutative quaternions of the Fibonacci type

Quaternions are a four-dimensional hypercomplex number system discovered by Hamilton in 1843 and next intensively applied in mathematics, modern physics, computer graphics and other fields. After the discovery of quaternions, modified quaternions were also defined in such a way that commutative property in multiplication is possible. That number system called as commutative quaternions is intensively studied and used for example in signal processing. In this paper we define generalized commutative quaternions and next based on them we define and explore Fibonacci type generalized commutative quaternions.

Measurement Operators in Conjugate Transformation Structure - Conjugate Hierarchy of Multiple Levels on Logic Constructions, Pairs of 0-1 Feature Vectors and Hamiltonian Dynamics

Hamiltonian dynamics play a key role in the foundation of modern physics and mathematics with wider applications in multiple advanced sciences and technologies.This paper proposes a conjugate transformation structure and its measurement operators on a hierarchy of multiple levels to support intermediate transforming structures from pairs of logic states as micro-ensembles to feature vector transformations as global measurements.Using logic equations and pairs of partitions on phase spaces, conjugate 0-1 vectors provide hypercomplex number systems. Multiple operators can be created and linked with Hamiltonian operators.The main constructions of conjugate transformation structures are described and complex conjugate operators are discussed under a pair of symmetric and antisymmetric parameters with O(2^{2^n}x2^N); 1 =< n =< 2m structural complexity.Using new operators, the Yang-Mills equations are briefly described as an example.

On logarithmic residue of monogenic functions in a three-dimensional commutative algebra with one-dimensional radical

We consider monogenic functions taking values in a three-dimensional commutative algebra A2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A2 into A2. It is shown that the logarithmic residue depends not only on zeros and singular points of a function but also on points at which the function takes values in ideals of A2, and, in general case, is a hypercomplex number.

Applications of the 4D Geometric Algebra to Dimensional Mobility Criteria of Delassus-Parallelogram and Bennett Paradoxical Linkages

Geometric algebra is also termed Clifford-Grassmann algebra or hypercomplex number. It allows studying space geometric problems in an easy and compact way. Transforming three-dimensional (3D) Euclidean geometric entities to actual elements of four-dimensional (4D) geometric algebra (abbreviated to g4) through a methodical approach of geometric algebra, one can describe motion displacements as even elements of g4. This article relies on the combined rotation and translation in g4 to establish the dimensional constraints of two non-exceptional overconstrained paradoxical linkages. Firstly, fundamentals of geometric algebra are recalled. Then, the single finite rotation and the composition of two successive finite rotations are introduced. After that, a general rigid-body motion in g4 is revealed for a possible application in exploring paradoxical chains using the geometric algebra. Finally, the metric or dimensional mobility criteria of Delassus-parallelogram four-screw and Bennett four-revolute paradoxical linkages are algebraically verified.