Research Article
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Değiştirilmiş tek bölen fonksiyonları ve bölen yaprak modeli

Year 2023, Volume: 25 Issue: 1, 233 - 248, 16.01.2023
https://doi.org/10.25092/baunfbed.1203159

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.

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.

The shifted odd divisor functions and divisor leaves model

Year 2023, Volume: 25 Issue: 1, 233 - 248, 16.01.2023
https://doi.org/10.25092/baunfbed.1203159

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.
There are 10 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Nazlı Yıldız İkikardeş 0000-0001-8756-8085

Daeyeoul Kim 0000-0002-2970-1666

Esra Çolak Aktaş 0000-0001-7864-8550

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

Cite

APA Yıldız İkikardeş, N., Kim, D., & Çolak Aktaş, E. (2023). The shifted odd divisor functions and divisor leaves model. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 25(1), 233-248. https://doi.org/10.25092/baunfbed.1203159
AMA Yıldız İkikardeş N, Kim D, Çolak Aktaş E. The shifted odd divisor functions and divisor leaves model. BAUN Fen. Bil. Enst. Dergisi. January 2023;25(1):233-248. doi:10.25092/baunfbed.1203159
Chicago Yıldız İkikardeş, Nazlı, Daeyeoul Kim, and Esra Çolak Aktaş. “The Shifted Odd Divisor Functions and Divisor Leaves Model”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 25, no. 1 (January 2023): 233-48. https://doi.org/10.25092/baunfbed.1203159.
EndNote Yıldız İkikardeş N, Kim D, Çolak Aktaş E (January 1, 2023) The shifted odd divisor functions and divisor leaves model. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 25 1 233–248.
IEEE N. Yıldız İkikardeş, D. Kim, and E. Çolak Aktaş, “The shifted odd divisor functions and divisor leaves model”, BAUN Fen. Bil. Enst. Dergisi, vol. 25, no. 1, pp. 233–248, 2023, doi: 10.25092/baunfbed.1203159.
ISNAD Yıldız İkikardeş, Nazlı et al. “The Shifted Odd Divisor Functions and Divisor Leaves Model”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 25/1 (January 2023), 233-248. https://doi.org/10.25092/baunfbed.1203159.
JAMA Yıldız İkikardeş N, Kim D, Çolak Aktaş E. The shifted odd divisor functions and divisor leaves model. BAUN Fen. Bil. Enst. Dergisi. 2023;25:233–248.
MLA Yıldız İkikardeş, Nazlı et al. “The Shifted Odd Divisor Functions and Divisor Leaves Model”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 25, no. 1, 2023, pp. 233-48, doi:10.25092/baunfbed.1203159.
Vancouver Yıldız İkikardeş N, Kim D, Çolak Aktaş E. The shifted odd divisor functions and divisor leaves model. BAUN Fen. Bil. Enst. Dergisi. 2023;25(1):233-48.