MULTİPLİKATİF ANALİZDE BİR EĞRİ VEYA YÜZEYİN BAŞKA BİR EĞRİ VEYA YÜZEY ÜZERİNDEN YUVARLANMASI HARAKETİ ÜZERİNE

Öz

Bu çalışma, "Multiplikatif Analizde eğrinin veya yüzeyin başka bir eğri veya yüzey üzerinde yuvarlanması" kavramını yeniden ele alıyor ve keyfi n boyutlu Multiplikatif Öklid uzaylarında homotetik hareketler çerçevesinde daha geniş bir formülasyon geliştiriyor. Makalenin bulguları şu şekilde özetlenebilir: Hareketin davranışı, n’in çift ya da tek tam sayı olmasından bağımsız olarak aynı kalmaktadır. Herhangi bir n boyutu için, homotetik hareketler, ilişkili polar yörüngelerin karşılıklı kayma ve yuvarlanma etkileşimi sergilediği düzenli hareketler sınıfını temsil eder. Her t anında, tek bir anlık kutup noktası benzersiz olarak belirlenebilir. Elde edilen sonuçlar, aynı alanda daha önce yayımlanmış çalışmalarla ilişkilendirilerek tartışılmıştır. Son olarak, homotetik ölçekleme katsayısının sürekli olarak e1=e olduğu durumda afin yapının özel bir durum olarak ortaya çıktığı çıktığı gösterilmiştir.

Anahtar Kelimeler

• Multiplikatif Analiz
• Bir Parametreli Multiplikatif Homotetik Hareketler
• Kutup Noktaları

ON THE MOTION OF A CURVE OR SURFACE ROLLING OVER ANOTHER CURVE OR SURFACE IN MULTIPLICATIVE CALCULUS

Öz

This study revisits the notion of “On the motion of a curve or surface rolling over another curve or surface in multiplicative calculus and develops a broader formulation of it within the framework of homothetic motions in Multiplicative Euclidean spaces of arbitrary dimension n. The findings of the paper may be outlined as follows: The behavior of the motion remains the same regardless of whether n is an even or an odd integer;for any dimension n, homothetic motions represent a class of regular motions, where the associated polar trajectories exhibit a mutual sliding and rolling interaction; at each instant t, a single instantaneous pole point can be uniquely determined. The results obtained here are discussed in relation to previously published investigations in the same area. As a final remark, it is shown that the a¢ ne situation appears as a special case in which the homothetic scaling factor is constantly equal to e1 = e.

Anahtar Kelimeler

• Multiplicative Calculus
• One Parameter Multiplicative Homothetic Motions
• Pole Points

Kaynakça

1. Grossman, M., & Katz, R. (1972). Non-Newtonian calculus. Lee Press.
2. Stanley, D. (1999). A multiplicative calculus. Primus, 9(4), 310–326.
3. Campbell, D. (1999). Multiplicative calculus and student projects. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 9(4), 327–332.
4. Grossman, M. (1979). An introduction to non-Newtonian calculus. International Journal of Mathematical Education in Science and Technology, 10(4), 525–529.
5. Grossman, J., Grossman, M., & Katz, R. (1981). The first systems of weighted differential and integral calculus. University of Michigan.
6. Grossman, J. (1981). Meta-calculus: Differential and integral. University of Michigan.
7. Bashirov, A. E., Kurpınar, E. M., & Ozyapici, A. (2008). Multiplicative calculus and its applications. Journal of Mathematical Analysis and Applications, 337(1), 36–48. (Not: 5. kaynak ile aynıdır).
8. Bashirov, A. E., & Rıza, M. (2011). On complex multiplicative differentiation. TWMS Journal of Applied and Engineering Mathematics, 1(1), 75–85.

9. Bashirov, A. E., Mısırlı, E., Tando˘gdu, Y., & Ozyapıcı, A. (2011). On modeling with multiplicative differential equations. Applied Mathematics-A Journal of Chinese University, 26(4), 425–438.
10. Tekin, S., & Ba¸sar, F. (2013). Certain sequence spaces over the non-Newtonian complex field. Abstract and Applied Analysis, 2013, Article ID 739319.
11. T¨urkmen, C., & Ba¸sar, F. (2012). Some basic results on the sets of sequences with geometric calculus. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 61(2), 17–34.
12. Uzer, A. (2010). Multiplicative type complex calculus as an alternative to the classical calculus. Computers & Mathematics with Applications, 60(10), 2725–2737.
13. Çakmak, A. F., & Ba¸sar, F. (2015). Some sequence spaces and matrix transformations in multiplicative sense. TWMS Journal of Pure and Applied Mathematics, 6(1), 27–37.
14. Boruah, K., & Hazarika, B. (2016). Some basic properties of g-calculus and its applications in numerical analysis. arXiv. https://arxiv.org/abs/1607.07749
15. Boruah, K., & Hazarika, B. (2018). G-calculus. TWMS Journal of Applied and Engineering Mathematics, 8(1), 94–106.
16. Gürefe, Y. (2013). Multiplicative differential equations and its applications [Doctoral dissertation]. Department of Mathematics.
17. Boruah, K., & Hazarika, B. (2016). Application of geometric calculus in numerical analysis and difference sequence spaces. arXiv. https://arxiv.org/abs/1603.09479
18. Çakmak, A. F., & Ba¸sar, F. (2012). On classical sequence spaces and non-Newtonian calculus. Journal of Inequalities and Applications, 2012, Article ID 932734.
19. Es, H. (2022). On the 1-parameter motions with multiplicative calculus. Journal of Science and Arts, 2(59), 295–304.
20. Es, H. (2024). Plane kinematics in homothetic multiplicative calculus. Journal of Universal Mathematics, 7(1), 37–47.
21. Es, H. (2024). Plane kinematics in Lorentzian homothetic multiplicative calculus. Journal of Universal Mathematics, 7(2), 64–74.
22. Es, H. (2024). Geometric interpretation of multiplicative calculus in Lorentzian space. Journal of Science and Arts, 24(3), 613–626.
23. Nurkan, S., G¨urgil, K. ˙I., & Karacan, M. K. (2023). Vector properties of geometric calculus.Mathematical Methods in the Applied Sciences, 1–20.
24. Aslan, S., Bekar, M., & Yaylı, Y. (2023). Geometric 3-space and multiplicative quaternions. International Journal of Geometric Methods in Modern Physics, 20(09), 2350143.
25. Hacısaliho˘glu, H. H. (1971). On the rolling one curve or surface upon another. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 71, 13–17.
26. Has, A., & Yılmaz, B. (2025). On non-Newtonian helices in multiplicative Euclidean space E3. Fundamentals of Contemporary Mathematical Sciences, 6(2), 196–217.
27. Has, A., & Yılmaz, B. (2024). A non-Newtonian conics in multiplicative analytic geometry. Journal of Umm Al-Qura University for Applied Sciences, 10(1).
28. Misirli, E., & Gurefe, Y. (2011). Multiplicative Adams Bashforth-Moulton methods. Numerical Algorithms, 57(4), 425–439.
29. Georgiev, S. G., & Zennir, K. (2022). Multiplicative differential calculus (1st ed.). Chapman and Hall/CRC.