Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Cieśliński, Jan L.; Zhalukevich, Dzianis

doi:10.3390/sym12091550

Cited by 5 publications

(8 citation statements)

References 12 publications

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“…cos(mϕ k ) sin mϕ j ěj . (19) Proof. From the multiplicative to additive transformation (5) applied to both sides of Equation ( 17), Equation ( 19) is obtained.…”

Section: Victoria Equationmentioning

confidence: 99%

“…From the multiplicative to additive transformation (5) applied to both sides of Equation ( 17), Equation ( 19) is obtained. While ( 17) is valid for any arguments, the possible lack of associativity in the additive representation could mar (19), because the left side of the equation is product associativity dependent while the right side is not. Theorem 1 establishes the associativity sufficient condition in the additive representation, namely, that none of the possible product associations give a scator with zero scalar additive component for n > 1.…”

Section: Victoria Equationmentioning

confidence: 99%

“…, where the prime denotes differentiation with respect to o ζ. The set of holomorphic functions according to the differential quotient criterion [18] is somewhat larger than their counterpart in the quaternion set [19]. The additive and multiplicative (rectangular/polar) scator representations allow for a generalization of De Moivre's formula to 1 + n dimensions in scator algebra.…”

Section: Introductionmentioning

confidence: 99%

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Powers of Elliptic Scator Numbers

Fernández‐Guasti

2021

Axioms

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Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

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“…cos(mϕ k ) sin mϕ j ěj . (19) Proof. From the multiplicative to additive transformation (5) applied to both sides of Equation ( 17), Equation ( 19) is obtained.…”

Section: Victoria Equationmentioning

confidence: 99%

Section: Victoria Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Powers of Elliptic Scator Numbers

Fernández‐Guasti

2021

Axioms

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“…The set of solutions to this system is not as rich as the set of holomorphic functions. In a recent paper, all solutions to this system were found and classified; see [18].…”

Section: Hypercomplex Holomorphic Functionsmentioning

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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“…Scator algebra has been successfully applied to several problems: time-space description in a deformed Lorentz metric [5], a wave-function evolution and collapse unified description in quantum mechanics [3] and three dimensional fractal structures [6]. Explicit formulae for scator holomorphic functions recently published, will very likely expand applications to other areas [7].…”

Section: Introductionmentioning

confidence: 99%

Roots of Second Order Polynomials with Real Coefficients in Elliptic Scator Algebra

Fernández‐Guasti

2021

Journal of New Theory

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The roots of second order polynomials with real coefficients are obtained in the S 1+2 scator set. Explicit formulae are computed in terms of the polynomial coefficients. Although the scator product does not distribute over addition, the lack of distributivity is surmountable in order to find the zeros of the polynomial. The structure of the solutions and their distribution in 1+2 dimensional scator space are illustrated and discussed. There exist six, two, or eight solutions, depending on the value of polynomial coefficients. Four of these roots only exist in the hypercomplex S 1+2 \ S 1+1 set.

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Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Cited by 5 publications

References 12 publications

Powers of Elliptic Scator Numbers

Powers of Elliptic Scator Numbers

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Roots of Second Order Polynomials with Real Coefficients in Elliptic Scator Algebra

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