Differential quotients in elliptic scator algebra

Fernández‐Guasti, M.

doi:10.1002/mma.4933

Cited by 8 publications

(14 citation statements)

References 15 publications

(18 reference statements)

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“…The notion of scator-holomorphic function was recently introduced by Fernández-Gausti [17]. Here, we propose a different approach to the subject, as following from previously stated remarks and conjectures.…”

Section: Hypercomplex Holomorphic Functionsmentioning

confidence: 94%

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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Section: Hypercomplex Holomorphic Functionsmentioning

confidence: 94%

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Kobus

Cieśliński

2020

Symmetry

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“…Linear affine transformations of the form

\overset{f}{overset} false(\overset{ζ}{overset} false) = α \overset{ζ}{overset} + \overset{β}{overset}

, with constant

α \in double-struckR, \overset{β}{overset} \in S^{1 + n}

have been shown to be scator holomorphic . The identity

\overset{f}{overset} false(\overset{ζ}{overset} false) = \overset{ζ}{overset}

and the constant

\overset{f}{overset} false(\overset{ζ}{overset} false) = \overset{β}{overset}

functions being particular cases of this transformation.…”

Section: Final Remarksmentioning

confidence: 99%

“…Definition The scator derivative of the scator function

\overset{φ}{overset} : U \subseteq R^{1 + n} \to S^{1 + n}

with respect to the scator variable

\overset{ζ}{overset}

U \subseteq R^{1 + n}

is given by the limit

\frac{d \overset{φ}{overset} ((), \overset{ζ}{overset})}{d \overset{ζ}{overset}} = {\overset{φ}{overset}}^{'} ((), \overset{ζ}{overset}) \equiv \lim_{\overset{δ}{overset} \to 0} \frac{\overset{φ}{overset} false(\overset{o}{ζ} + \overset{o}{δ} false) - \overset{φ}{overset} ((), \overset{ζ}{overset})}{\overset{δ}{overset}},

where

\overset{δ}{overset}

S^{1 + n}

is a scator in an open neighbourhood of

\overset{ζ}{overset}

that tends to zero when all its components tend to zero. A less restrictive domain condition in the scator derivative definition has been given here with respect to a previous definition . The necessary condition is that the image of the function must be in

S^{1 + n}

, so that its quotient is defined by the product operation stated in .…”

Section: Scator Algebra Preamblementioning

confidence: 99%

“…In the functions that follow, the argument is dropped for simplicity of notation. The scator differentiability necessary conditions for f 0 , f j real differentiable functions are,, Thm. 3.1

\frac{\partial f_{0}}{\partial z_{0}} = \frac{\partial f_{j}}{\partial z_{j}},

\frac{\partial f_{j}}{\partial z_{0}} = - \frac{\partial f_{0}}{\partial z_{j}},

\frac{\partial f_{0}}{\partial z_{j}} \frac{\partial f_{0}}{\partial z_{l}} = - \frac{\partial f_{j}}{\partial z_{j}} \frac{\partial f_{j}}{\partial z_{l}},

for any j ≠ l , where j , l take values from 1 to n .…”

Section: Scator Algebra Preamblementioning

confidence: 99%

“…In elliptic scator algebra, linear affine transformations involving a scaling and a translation have been shown to be scator holomorphic . Other functions such as the quadratic mapping have been shown not to be scator holomorphic.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Components exponential scator holomorphic function

Fernández‐Guasti

2019

Math Methods in App Sciences

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The components exponential scator function, succinctly labeled "cexp," is a function of scator variable in 1 + n dimensional elliptic scator algebra. This function is introduced here, and its elementary properties are studied. In 1 + 1 dimensions, the cexp function becomes the usual complex exponential function of complex variable. The cexp function is shown to be scator holomorphic in the entire scator set according to the differential quotient criterion. The scator derivative of the cexp function is the cexp function itself. The relationship between the Cartesian and polar forms of the cexp function can be seen as a higher-dimensional extension of Euler formula. The mappings of grids in 1 + 2 dimensional space exhibit ellipses and Lissajous-like figures in addition to circles and radial lines. A cusphere surface is generated for the isometric condition in the function's image. An interesting application of the cexp function as propagators regarding the quantum measurement problem is outlined. KEYWORDS holomorphic functions, hypercomplex variables 2 SCATOR ALGEBRA PREAMBLE Elliptic scator algebra is a 1+n finite dimensional algebra. In the additive form, the elements of this algebra are represented by o = (z 0 ; z 1 , … , z n ) = z 0 + n ∑ =1 zě ,Math Meth Appl Sci. wileyonlinelibrary.com/journal/mma

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Imaginary scators quadratic mapping in 1+2D dynamic space

Fernández‐Guasti

2023

Communications in Nonlinear Science and Numerical Simulation

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Differential quotients in elliptic scator algebra

Cited by 8 publications

References 15 publications

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Components exponential scator holomorphic function

Imaginary scators quadratic mapping in 1+2D dynamic space

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