Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

Aydın, Muhittin Evren; Mihai, Adela; Yokuş, Asıf

doi:10.1002/mma.7649

Cited by 20 publications

(7 citation statements)

References 35 publications

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“…Last but not least, we expect that relying on a fractional view might allow us to study other curves on Pascal's Surface. One of us has already been working on a related subject by defining the notions of equiaffine arclength and curvature with fractional order [36], a paper which has introduced a classification of the plane curves of constant equiaffine curvature with fractional order, while also providing several explanatory examples. That is why we plan to continue exploring symmetries appearing on such surfaces.…”

Section: Discussionmentioning

confidence: 99%

On a Surface Associated with Pascal’s Triangle

et al. 2022

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An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries.

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Section: Discussionmentioning

confidence: 99%

On a Surface Associated with Pascal’s Triangle

et al. 2022

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“…Beyond its applications in pure mathematics, fractional calculus has gained substantial significance in recent years. In [2], plane curves in equiaffine geometry are examined by considering fractional derivatives. The authors [3] have studied the geometry of curves possessing fractional-order tangent vectors and Frenet-Serret formulas.…”

Section: Introductionmentioning

confidence: 99%

Fractional Curvatures of Equiaffine Curves in Three-Dimensional Affine Space

Öğrenmiş

2024

Journal of New Theory

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This paper presents a method for computing the curvatures of equiaffine curves in three-dimensional affine space by utilizing local fractional derivatives. First, the concepts of $\alpha$-equiaffine arc length and $\alpha$-equiaffine curvatures are introduced by considering a general local involving conformable derivative, V-derivative, etc. In fractional calculus, equiaffine Frenet formulas and curvatures are reestablished. Then, it presents the relationships between the equiaffine curvatures and $\alpha$-equiaffine curvatures. Furthermore, graphical representations of equiaffine and $\alpha$-equiaffine curvatures illustrate their behavior under various conditions.

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“…Initially, Yajima T. and Yamasaki K. conducted studies on surfaces using the Caputo derivative [29]. Following these studies, numerous research works are carried out to investigate the impact of fractional derivatives on geometry [4][5][6]23]. According to the results of these research works, global fractional derivatives are highlighted as not particularly useful in terms of differential geometry by Ayd?n M. E. [7].…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Tzitzeica Curves Using Conformable Frenet Frame

Has,

Yılmaz,

Ayvacı

2023

Preprint

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Tzitzeica curve is a mathematical curve that has been used to describe the behavior of certain physical phenomena. It was first discovered by Romanian mathematician Gheorghe Tzitzeica and has since been used to model many phenomena, including the motion of a pendulum, the oscillations of a vibrating string, and the flow of electricity through a circuit. The curve is an example of what is known as an elliptic curve and can be used to analyze complex systems. Its properties make it useful in many applications, such as cryptography and data compression. In the light of this information, the aim of this study, the conditions for a conformable curve and its spherical indicator curves to be a Tzitzeica curve will be examined. Thus, by understanding the behavior of the Tzitzeica curve, researchers can gain insight into complex systems and make more accurate predictions about their behavior. MSC Classification: 26A33 , 53A04 , 53A55

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Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

Cited by 20 publications

References 35 publications

On a Surface Associated with Pascal’s Triangle

On a Surface Associated with Pascal’s Triangle

Fractional Curvatures of Equiaffine Curves in Three-Dimensional Affine Space

Characterizations of Tzitzeica Curves Using Conformable Frenet Frame

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