Differential quotients in elliptic scator algebra

The quotient of the function difference over the variable increment is evaluated in 1+n finite dimensional imaginary scator algebra. A distinctive property of the scator product is that it is commutative albeit not distributive over addition. In S1+n, a subset of R1+n where the product is defined, a...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2018-08, Vol.41 (12), p.4827-4840
Main Author: Fernández‐Guasti, M.
Format: Article
Language:English
Subjects:
Affine transformations
Algebra
Analytic functions
differential calculus
hypercomplex analysis
hyperholomorphic functions
Mathematical analysis
Partial differential equations
Quaternions
Quotients
scator algebra
Set theory
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Summary:The quotient of the function difference over the variable increment is evaluated in 1+n finite dimensional imaginary scator algebra. A distinctive property of the scator product is that it is commutative albeit not distributive over addition. In S1+n, a subset of R1+n where the product is defined, all the elements have inverse provided that zero is excluded. The quotient of scators and their differential limit can thus be defined in this subset, establishing the notion of scator differentiability. The necessary conditions that a scator holomorphic function must satisfy are then derived in terms of an extended set of partial differential equations. It is shown that affine transformations involving a scaling and a translation are scator holomorphic. These results are compared with holomorphy in quaternion and Cℓ0,n Clifford algebras. Functions, such as the quadratic mapping, are shown not to satisfy the scator holomorphic conditions.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.4933