Quaternion Algebras and Generalized Fibonacci–Lucas Quaternions

Flaut, Cristina; Savin, Diana

doi:10.1007/s00006-015-0542-0

Cited by 22 publications

(11 citation statements)

References 6 publications

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“…The reader shall see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some works on number theory and its applications.…”

Section: Resultsmentioning

confidence: 99%

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Mamombe¹

2019

JAMCS

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The proportiones perfectus law states that σ_x^y=(x+√(x^2+4y))/2 is a proportione perfectus if 1≤y≤x such that for an arbitrary positive integer , there exists an integer sequence defined simultaneously by the quasigeometric relation h_(n+1)=round(σ_x^y h_n ),n≥1 and the arithmetic relation h_(n+2)=〖xh〗_(n+1)+〖yh〗_n,n≥1. When x=y=1 σ_x^y is the golden mean. When x=2,y=1,σ_x^y is the silver mean. In previous works we introduced the theory of number genetics – a framework of logic within which the golden section is studied. In this work we apply the concept to all proportiones perfectus. Let be defined as above. Furthermore, let h1 satisfy {█(round(h_1/(σ_x^y ))=w@round(wσ_x^y)≠h_1 ). ┤Now let H'nbe Hn with h1=1 Again let be defined as A robust universal computing machine is herein developed for the purpose of establishing the relationship h_i=g_(n+i-1)±h_i^',i,n≥1, which relationship is key to the logic protocol. It is clear therefore that this system of logic presents as the building block of every (including itself) within a particular σ_x^y regime. Clearly number genetics takes central place in the proportiones perfectus theory.

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“…The reader shall see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some works on number theory and its applications.…”

Section: Resultsmentioning

confidence: 99%

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Mamombe¹

2019

JAMCS

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“…The norm of

a

bold-italicn ((), a) = a \cdot ā = a_{1}^{2} - α a_{2}^{2} - β a_{3}^{2} + α β a_{4}^{2}

and the trace of the element

a

boldt false(a false) = a + ā

. If, for

x \in double-struckH ((), α, β)

, the relation

boldn false(x false) = 0

implies

x = 0

, then the algebra

double-struckH ((), α, β)

is called a division algebra, otherwise the quaternion algebra is called a split algebra, (see Flaut and Savin 16 ).…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 99%

“…Let H ( , ) be the generalized quaternion algebra over an arbitrary field K, ie, the algebra of the elements of the form a = a 1 • 1 + a 2 e 2 + a 3 e 3 + a 4 e 4 , where a i ∈ K, i ∈ {1, 2, 3, 4}, and the elements of the basis {1, e 2 , e 3 , e 4 } satisfying the following rules, given in the below multiplication table: and the trace of the element a is t(a) = a +ā. If, for x ∈ H ( , ), the relation n(x) = 0 implies x = 0, then the algebra H ( , ) is called a division algebra, otherwise the quaternion algebra is called a split algebra, (see Flaut and Savin 16 ).…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 99%

Properties and applications of some special integer number sequences

Flaut

Savin

Zaharia

2020

Math Methods in App Sciences

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“…Generalized noncommutative Fibonacci quaternions were introduced by Horadam in [15]. In [2,[11][12][13] the authors presented some properties of the generalized noncommutative Fibonacci quaternions.…”

Section: Introductionmentioning

confidence: 99%

Generalized commutative quaternions of the Fibonacci type

Szynal‐Liana

Włoch

2021

Bol. Soc. Mat. Mex.

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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.

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Quaternion Algebras and Generalized Fibonacci–Lucas Quaternions

Abstract: In this paper, we introduce the generalized Fibonacci-Lucas quaternions and we prove that the set of these elements is an order-in the sense of ring theory-of a quaternion algebra. Moreover, we investigate some properties of these elements.

Cited by 22 publications

References 6 publications

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Properties and applications of some special integer number sequences

Generalized commutative quaternions of the Fibonacci type

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