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A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$

Year 2025, Volume: 18 Issue: 1, 97 - 110

Aykut Has , Beyhan Yılmaz

Abstract

The aim of this article is to characterize pairs of curves within multiplicative (non-Newtonian) spaces. Specifically, we investigate how famous curve pairs such as Bertrand partner curves, Mannheim partner curves, which are prominent in differential geometry, are transformed under the influence of multiplicative analysis. By leveraging the relationships between multiplicative Frenet vectors, we introduce multiplicative versions of Bertrand, Mannheim curve pairs. Subsequently, we characterize these curve pairs using multiplicative arguments. Examples are provided, and multiplicative graphs are presented to enhance understanding of the subject matter. Through this analysis, we aim to elucidate the behavior and properties of these curve pairs within the context of multiplicative geometry.

Keywords

Multiplicative Frenet frame multiplicative Euclidean space non-Newtonian calculus special partner curves

References

  • Afrouzi, H. H., Ahmadian, M., Moshfegh, A., Toghraie, D., Javadzadegan, A.: Statistical analysis of pulsating non-Newtonian flow in a corrugated channel using Lattice-Boltzmann method. Physica A: Statistical Mechanics and its Applications. 535, 122486 (2019).
  • Aydin, M. E., Has, A., Yılmaz, B.: A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curves. Bulletin of the Korean Mathematical Society. 61(3), 849–66 (2024).
  • Bashirov, A. E., Kurpınar, E. M., Özyapıcı, A.: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337, 36–48 (2008).
  • Bertrand, J.: Mémoire sur la théorie des courbes à double courbure. Comptes Rendus 36; Journal de Mathématiques Pures et Appliquées. 15, 332–350 (1850).
  • Boruah, K., Hazarika, B.: Some Basic Properties of Bigeometric Calculus and its Applications in Numerical Analysis. Afrika Matematica. 32, 211-227 (2021).
  • Boyer, C.: A History of Mathematics. Wiley. New York. (1968).
  • Carmo, M. P. D.: Differential Geometry of Curves and Surfaces. Prentice Hall Inc. New Jersey. (1976).
  • Ceylan, H., Özdemir, Z., Gök, ˙I.: Multiplicative generalized tube surfaces with multiplicative quaternions algebra. Mathematical Methods in the Applied Sciences. 47(11), 9157-9168 (2024).
  • Durmaz, H., Özdemir, Z., ¸Sekerci, Y.: Fractional approach to evolution of the magnetic field lines near the magnetic null points. Physica Scripta. 99(2), 025239 (2024).
  • Es, H.: Plane kinematics in homothetic multiplicative calculus. Journal of Universal Mathematics. 7(1), 37-47 (2024).
  • Es, H.: On the 1-parameter motions with multiplicative calculus. Journal of Science and Arts. 22(2), 395-406 (2022).
  • Georgiev, S. G., Zennir, K.: Multiplicative Differential Calculus (1st ed.). Chapman and Hall/CRC. New York. (2022).
  • Georgiev, S. G.: Multiplicative Differential Geometry (1st ed.). Chapman and Hall/CRC. New York. (2022).
  • Georgiev, S. G., Zennir, K., Boukarou, A.. Multiplicative Analytic Geometry (1st ed.). Chapman and Hall/CRC. New York. (2022).
  • Grossman, M., Katz, R.: Non-Newtonian Calculus, 1st ed. Lee Press. Pigeon Cove Massachussets. (1972).
  • Grossman, M.: Bigeometric Calculus: A System with a Scale-Free Derivative. Archimedes Foundation. Massachusetts. (1983).
  • Gulsen, T., Yılmaz, E., Goktas, S.: Multiplicative Dirac system. Kuwait J.Sci. 49(3), 1-11 (2022).
  • Has, A., Yılmaz, B., Baleanu, D.: On the Geometric and Physical Properties of Conformable Derivative. Mathematical Sciences and Applications E-Notes. 12(2), 60-70 (2024).
  • Has, A., Yılmaz, B.: Measurement and Calculation on Conformable Surfaces. Mediterr. J. Math. 20, 274 (2023).
  • Has, A., Yılmaz, B.: Effect of fractional analysis on some special curves. Turkish Journal of Mathematics. 47(5), 1423-1436 (2023).
  • Has, A., Yılmaz, B., Ayvacı, K. H.: Cα- ruled surfaces respect to direction curve in fractional differential geometry. J. Geom. 115, 11 (2024).
  • Has, A., Yılmaz, B.: On non-Newtonian Helices in Multiplicative Euclidean Space $E^3_*$. Preprint arxiv: 2403.11282 (2024).
  • Has, A., Yılmaz, B.: A non-Newtonian Conics in Multiplicative Analytic Geometry. Turkish Journal of Mathematics. 48(5), 976-994 (2024).
  • Mannheim, A.: De l’emploi de la Courbe Représentative de la Surface des Normales Principales d’une Courbe Gauche Pour la Démonstration de Propriétés Relatives à Cette Courbure. Comptes Rendus des Séances de l’Académie des Sciences, Paris C.R. 86, 1254–1256 (1878).
  • Mısırlı, E., Gürefe, Y.: Multiplicative Adams Bashforth–Moulton methods. Numerical Algorithms, 57 , 425–439 (2011).
  • Nurkan, S. K., Gurgil, I., Karacan, M. K.: Vector properties of geometric calculus. Math. Meth. Appl. Sci. 46(17), 17672-17691 (2023).
  • Rybaczuk, M., Stoppel, P.: The fractal growth of fatigue defects in materials. International Journal of Fracture. 103, 71-94 (2000).
  • Samuelson, W. F., Mark, S. G.: Managerial Economics. Wiley. New York. (2012).
  • Sernesi, E.: Gaussian maps of plane curves with nine singular points. Ann Univ Ferrara. 63, 201–210 (2017).
  • Şenyurt, S., Ayvacı, K. H., Canlı, D.: Smarandache Curves According to Flc-frame in Euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences. 4(1), 16-30 (2023).
  • Takahashi M.: Equi-affine plane curves with singular points. J. Geom. 113, 16 (2022).
  • Taşdemir, M., Canfes, E. Ö., Uzun, B.: On Caputo fractional Bertrand curves in $E^3$ and $E^3_1$. Filomat. 28(5), 1681–1702 (2024).
  • Volterra, V., Hostinsky, B.: Operations Infinitesimales Lineares. Herman. Paris. (1938).

Year 2025, Volume: 18 Issue: 1, 97 - 110

Aykut Has , Beyhan Yılmaz

Abstract

References

  • Afrouzi, H. H., Ahmadian, M., Moshfegh, A., Toghraie, D., Javadzadegan, A.: Statistical analysis of pulsating non-Newtonian flow in a corrugated channel using Lattice-Boltzmann method. Physica A: Statistical Mechanics and its Applications. 535, 122486 (2019).
  • Aydin, M. E., Has, A., Yılmaz, B.: A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curves. Bulletin of the Korean Mathematical Society. 61(3), 849–66 (2024).
  • Bashirov, A. E., Kurpınar, E. M., Özyapıcı, A.: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337, 36–48 (2008).
  • Bertrand, J.: Mémoire sur la théorie des courbes à double courbure. Comptes Rendus 36; Journal de Mathématiques Pures et Appliquées. 15, 332–350 (1850).
  • Boruah, K., Hazarika, B.: Some Basic Properties of Bigeometric Calculus and its Applications in Numerical Analysis. Afrika Matematica. 32, 211-227 (2021).
  • Boyer, C.: A History of Mathematics. Wiley. New York. (1968).
  • Carmo, M. P. D.: Differential Geometry of Curves and Surfaces. Prentice Hall Inc. New Jersey. (1976).
  • Ceylan, H., Özdemir, Z., Gök, ˙I.: Multiplicative generalized tube surfaces with multiplicative quaternions algebra. Mathematical Methods in the Applied Sciences. 47(11), 9157-9168 (2024).
  • Durmaz, H., Özdemir, Z., ¸Sekerci, Y.: Fractional approach to evolution of the magnetic field lines near the magnetic null points. Physica Scripta. 99(2), 025239 (2024).
  • Es, H.: Plane kinematics in homothetic multiplicative calculus. Journal of Universal Mathematics. 7(1), 37-47 (2024).
  • Es, H.: On the 1-parameter motions with multiplicative calculus. Journal of Science and Arts. 22(2), 395-406 (2022).
  • Georgiev, S. G., Zennir, K.: Multiplicative Differential Calculus (1st ed.). Chapman and Hall/CRC. New York. (2022).
  • Georgiev, S. G.: Multiplicative Differential Geometry (1st ed.). Chapman and Hall/CRC. New York. (2022).
  • Georgiev, S. G., Zennir, K., Boukarou, A.. Multiplicative Analytic Geometry (1st ed.). Chapman and Hall/CRC. New York. (2022).
  • Grossman, M., Katz, R.: Non-Newtonian Calculus, 1st ed. Lee Press. Pigeon Cove Massachussets. (1972).
  • Grossman, M.: Bigeometric Calculus: A System with a Scale-Free Derivative. Archimedes Foundation. Massachusetts. (1983).
  • Gulsen, T., Yılmaz, E., Goktas, S.: Multiplicative Dirac system. Kuwait J.Sci. 49(3), 1-11 (2022).
  • Has, A., Yılmaz, B., Baleanu, D.: On the Geometric and Physical Properties of Conformable Derivative. Mathematical Sciences and Applications E-Notes. 12(2), 60-70 (2024).
  • Has, A., Yılmaz, B.: Measurement and Calculation on Conformable Surfaces. Mediterr. J. Math. 20, 274 (2023).
  • Has, A., Yılmaz, B.: Effect of fractional analysis on some special curves. Turkish Journal of Mathematics. 47(5), 1423-1436 (2023).
  • Has, A., Yılmaz, B., Ayvacı, K. H.: Cα- ruled surfaces respect to direction curve in fractional differential geometry. J. Geom. 115, 11 (2024).
  • Has, A., Yılmaz, B.: On non-Newtonian Helices in Multiplicative Euclidean Space $E^3_*$. Preprint arxiv: 2403.11282 (2024).
  • Has, A., Yılmaz, B.: A non-Newtonian Conics in Multiplicative Analytic Geometry. Turkish Journal of Mathematics. 48(5), 976-994 (2024).
  • Mannheim, A.: De l’emploi de la Courbe Représentative de la Surface des Normales Principales d’une Courbe Gauche Pour la Démonstration de Propriétés Relatives à Cette Courbure. Comptes Rendus des Séances de l’Académie des Sciences, Paris C.R. 86, 1254–1256 (1878).
  • Mısırlı, E., Gürefe, Y.: Multiplicative Adams Bashforth–Moulton methods. Numerical Algorithms, 57 , 425–439 (2011).
  • Nurkan, S. K., Gurgil, I., Karacan, M. K.: Vector properties of geometric calculus. Math. Meth. Appl. Sci. 46(17), 17672-17691 (2023).
  • Rybaczuk, M., Stoppel, P.: The fractal growth of fatigue defects in materials. International Journal of Fracture. 103, 71-94 (2000).
  • Samuelson, W. F., Mark, S. G.: Managerial Economics. Wiley. New York. (2012).
  • Sernesi, E.: Gaussian maps of plane curves with nine singular points. Ann Univ Ferrara. 63, 201–210 (2017).
  • Şenyurt, S., Ayvacı, K. H., Canlı, D.: Smarandache Curves According to Flc-frame in Euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences. 4(1), 16-30 (2023).
  • Takahashi M.: Equi-affine plane curves with singular points. J. Geom. 113, 16 (2022).
  • Taşdemir, M., Canfes, E. Ö., Uzun, B.: On Caputo fractional Bertrand curves in $E^3$ and $E^3_1$. Filomat. 28(5), 1681–1702 (2024).
  • Volterra, V., Hostinsky, B.: Operations Infinitesimales Lineares. Herman. Paris. (1938).
There are 33 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry, Pure Mathematics (Other)
Journal Section Research Article
Authors

Aykut Has 0000-0003-0658-9365

Beyhan Yılmaz 0000-0002-5091-3487

Early Pub Date April 20, 2025
Publication Date
Submission Date May 2, 2024
Acceptance Date December 18, 2024
Published in Issue Year 2025 Volume: 18 Issue: 1

Cite

APA Has, A., & Yılmaz, B. (2025). A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$. International Electronic Journal of Geometry, 18(1), 97-110.
AMA Has A, Yılmaz B. A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$. Int. Electron. J. Geom. April 2025;18(1):97-110.
Chicago Has, Aykut, and Beyhan Yılmaz. “A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$”. International Electronic Journal of Geometry 18, no. 1 (April 2025): 97-110.
EndNote Has A, Yılmaz B (April 1, 2025) A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$. International Electronic Journal of Geometry 18 1 97–110.
IEEE A. Has and B. Yılmaz, “A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$”, Int. Electron. J. Geom., vol. 18, no. 1, pp. 97–110, 2025.
ISNAD Has, Aykut - Yılmaz, Beyhan. “A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$”. International Electronic Journal of Geometry 18/1 (April 2025), 97-110.
JAMA Has A, Yılmaz B. A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$. Int. Electron. J. Geom. 2025;18:97–110.
MLA Has, Aykut and Beyhan Yılmaz. “A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$”. International Electronic Journal of Geometry, vol. 18, no. 1, 2025, pp. 97-110.
Vancouver Has A, Yılmaz B. A Non-Newtonian Some Partner Curves in Multiplicative Euclidean Space $\:E_*^3$. Int. Electron. J. Geom. 2025;18(1):97-110.
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