An Elliptic Non Distributive Algebra

Fernández‐Guasti, M.; Zaldívar, Felipe

doi:10.1007/s00006-013-0406-4

Cited by 13 publications

(13 citation statements)

References 12 publications

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“…That is, real algebra is embedded in scator algebra. Although in general, the product does not distribute over addition, in the particular case of a scalar times an arbitrary scator, the scalar does distribute over the scator components [Fernández-Guasti & Zaldívar, 2013b]. This result can be seen from (1), by letting a 0 = λ, a 1 = a 2 = 0.…”

Section: Iterated Quadratic Mappingmentioning

confidence: 96%

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

Fernández‐Guasti

2016

Int. J. Bifurcation Chaos

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The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

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Section: Iterated Quadratic Mappingmentioning

confidence: 96%

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

Fernández‐Guasti

2016

Int. J. Bifurcation Chaos

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“…16, Lemma 2.2 For this reason, it cannot be expressed as a matrix-matrix product. 10 The S 1+1 sets represent scators with arbitrary real scalar and single, possibly nonvanishing, j th director component. In 1 + 1 dimensions, for any one director componentě , imaginary scator algebra is identical to the algebra of complex numbers.…”

Section: Definition the Product Of Two Scatorsmentioning

confidence: 99%

“…Elliptic or imaginary scator algebra is a commutative algebra in 1+n dimensions endowed with a second-order involution and an order parameter. 10 The salient property that is not fulfilled in this algebraic structure is distributivity of the product over addition. Nonetheless, as we shall see, there are precise rules that state how to accomplish the product of a scator over the sum of scators.…”

Section: Introductionmentioning

confidence: 99%

Differential quotients in elliptic scator algebra

Fernández‐Guasti

2018

Math Methods in App Sciences

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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 S 1+n , a subset of R 1+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 C 0,n Clifford algebras. Functions, such as the quadratic mapping, are shown not to satisfy the scator holomorphic conditions. KEYWORDS differential calculus, hypercomplex analysis, hyperholomorphic functions, scator algebra On sabbatical leave from Universidad Autónoma Metropolitana -Iztapalapa.

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“…Thus, the hyperbolic case is associated with a generalization of the special relativity. The eliptic case can be considered as a new extension of the complex numbers to higher dimensions [8,9]. Algebraic properties of scators suffer from many inconvenient perplexities including, most importantly, lack of distributivity.…”

Section: Introductionmentioning

confidence: 99%

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Kobus

Cieśliński

2020

Symmetry

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The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

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An Elliptic Non Distributive Algebra

Cited by 13 publications

References 12 publications

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

Differential quotients in elliptic scator algebra

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

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