Abstract
Bu araştırmada yaprakların bölen fonksiyonları yardımıyla modellenmesi üzerinde çalışılmıştır. Bir pozitif tamsayı k (1 ≤ k ≤ 100) verildiğinde, σ(n) = σ(n + 2k) denkleminin tek tam kare olmayan tamsayı n ile çözümlerini araştırıyoruz. Ayrıca, pozitif bir tamsayı l ve tek asal q için, σ2i(n) = σ2i(q) denkleminin hiçbir sonucu yoktur. Bir uygulama olarak, kaydırılmış tek bölen fonksiyonlarının denkleminden türetilen gerçek zamanlı sanal ekosistem inşası için yaprak modelinin temel yapısını oluşturuyoruz. Ayrıca eliptik, flabellate ve beş loblu yaprakların alanı ve yaprakların büyüme süreci bölen fonksiyonları yardımıyla modelleme yapılmıştır.
Keywords
Yaprak modeli değiştirilmiş tek bölen fonksiyonları modelleme yöntemi. Modelling of the leaves shifted odd divisor functions modelling.
Supporting Institution
Tübitak
Project Number
TUBITAK the 2221-Science Fellowships and Grant Programmes (BIDEB)
References
- Sierpi´nski, W., Elementary Theory of Numbers, Trans. by A. Hulanicki, Polska Akademia Nauk, Monografic Matematyczne Tom Panstwowe Wydawnictwo Naukowe, Warszawa, Poland: 1964, 42, p.166.
- Makowski, A., On Some Equations Involving Functions ϕ(n) and σ(n), American Mathematical Monthly, 1960, correction ib:dem 1961; 67 (68): 668-670.
- Mientka, W. E., Vogt, R. L., Computational Results Relating to Problems Concerning σ(n), Matematnykn Bechnk, 1970; 7 crp.(22): 35-36.
- Hunsucker, J. L., Nebb, J., Stearns, R. E., Computational Results Concerning Some Equations Involving σ(n), Math. Student, 1973; 285-289.
- Weingartner, A., On The Solutions of σ(n) = σ(n + k), J. Integer Sequences, 2011; 14: 7.
- Guy, R., Unsolved Problems in Number Theory, Unsolved Problems in Intuitive Mathematics, Volume I. Springer-Verlag New York, 1994.
- De Koninck, J. M., On The Solutions of σ2(n) = σ2(n+l), Annales Univ. Sci. Budapest. Sect. Comp., 2004; 21: 127-133.
- Kim, J., Kim, D., Cho, H., Procedural modeling of trees based on convolution sums of divisor functions for real-time virtual ecosystems, Computer Animation Virtual Worlds, 2013; 24: 237-246.
- Adam, J. A., Mathematics in Nature: Modeling Patterns in the Natural World, Princeton University Press, 2003.
- Burton, D. M., The History of Mathematics; An Introduction, McGraw Hill Companies, 2011, pp. 205.
Abstract
In this research, modelling of the leaves with the help of divisor functions are worked on. Given a positive integer k (1 ≤ k ≤ 100), we investigate solutions of the equation σ(n) = σ(n + 2k) with odd square-free integer n. Further, for a positive integer l and odd prime q, there are no results of the equation σ2i(n) = σ2i(q). As an application, we pose the basic structure of the leaves model for real-time virtual ecosystem construction derived from the equation of shifted odd divisor functions. Also, the elliptic, flabellate and five-lobes leaves’s area and the growth process of the leaves were made modelling with the help of divisor functions.
Project Number
TUBITAK the 2221-Science Fellowships and Grant Programmes (BIDEB)
References
- Sierpi´nski, W., Elementary Theory of Numbers, Trans. by A. Hulanicki, Polska Akademia Nauk, Monografic Matematyczne Tom Panstwowe Wydawnictwo Naukowe, Warszawa, Poland: 1964, 42, p.166.
- Makowski, A., On Some Equations Involving Functions ϕ(n) and σ(n), American Mathematical Monthly, 1960, correction ib:dem 1961; 67 (68): 668-670.
- Mientka, W. E., Vogt, R. L., Computational Results Relating to Problems Concerning σ(n), Matematnykn Bechnk, 1970; 7 crp.(22): 35-36.
- Hunsucker, J. L., Nebb, J., Stearns, R. E., Computational Results Concerning Some Equations Involving σ(n), Math. Student, 1973; 285-289.
- Weingartner, A., On The Solutions of σ(n) = σ(n + k), J. Integer Sequences, 2011; 14: 7.
- Guy, R., Unsolved Problems in Number Theory, Unsolved Problems in Intuitive Mathematics, Volume I. Springer-Verlag New York, 1994.
- De Koninck, J. M., On The Solutions of σ2(n) = σ2(n+l), Annales Univ. Sci. Budapest. Sect. Comp., 2004; 21: 127-133.
- Kim, J., Kim, D., Cho, H., Procedural modeling of trees based on convolution sums of divisor functions for real-time virtual ecosystems, Computer Animation Virtual Worlds, 2013; 24: 237-246.
- Adam, J. A., Mathematics in Nature: Modeling Patterns in the Natural World, Princeton University Press, 2003.
- Burton, D. M., The History of Mathematics; An Introduction, McGraw Hill Companies, 2011, pp. 205.
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Project Number | TUBITAK the 2221-Science Fellowships and Grant Programmes (BIDEB) |
| Publication Date | January 16, 2023 |
| Submission Date | November 12, 2022 |
| Published in Issue | Year 2023 Volume: 25 Issue: 1 |