Research Article
Equivalence conditions of two systems of vectors in the taxicab plane and its applications to taxicab polygons
Abstract
This study presents the conditions of M_{T}(2)-equivalence for two systems of vectors {x₁,x₂,x₃} and {y₁,y₂,y₃} in R_{T}², where M_{T}(2) is the group of all isometries of the 2-dimensional taxicab space R_{T}². Firstly a minimal complete system of M_{T}(2)-invariants of {x₁,x₂,x₃} is obtained. Then, using the conditions of M_{T}(2)-equivalence, an answer is given to the open problem posed in sch. Furthermore, an algorithm is given for constructing taxicab regular polygons in terms of M_{T}(2)-invariants. This algorithm is general and useful to construct the taxicab regular 2n-gons and gives a tool to solve special cases of the open problem posed in col. Besides, both the conditions of the taxicab regularity of Euclidean regular polygons and Euclidean regularity of taxicab regular polygons are given in terms of M_{T}(2)-invariants.
Keywords
References
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- Çolakoğlu, H. B., Kaya, R., Regular polygons in the taxicab plane, KoG, 12 (2008), 27-33.
- Ekici, Kocayusufoğlu, İ., Akça, Z. The norm in taxicab geometry, Turk. J. Math, 22(3) (1998), 295-307.
- Goulson, D., Darvill, B., Niche overlap and diet breadth in bumblebees; are rare species more specialized in their choice of flowers?, Apidologie, 35 (2004), 55-63.
- Greub, W., Linear algebra, Berlin-Heidelberg-New York:Springer-Verlag, 1967.
- Hanson, J. R., Regular polygons in taxicab geometry, Int. J. Math. Educ. Sci. Technol., 45(7) (2014), 1084-1095.
- Legendre, P., Gallagher, E., Ecologically meaningful transformations for ordination of species data, Oecologia, 129 (2001), 271-280.
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- Schattschneider, D. J., The taxicab group, Am. Math. Mon., 91 (1984), 423-428.
- Slobodchikoff, C. N., Schulz, W. C. Measures of niche overlap, Ecology, 61(7) (1980), 1051-1055.
- Sohn, M. W., Distance and cosine measures of niche overlap, Soc. Networks, 23 (2001), 141-165.
- Struble, R. A., Non-Linear differential equations, New York:Mc Graw-Hill, 1962.
- Sturmfels, B. Algorithms in invariant theory, Wien:Springer, 2008.
- Yüksel, S., Özcan, M., Some regular polygons in the taxicab 3-space, KoG, 19 (2005), 32-41.
- Warren, D. L., Glor, R. E., Turelli, M., Environmental niche equivalency versus conservatism: quantitative approaches to niche evolution, Evolution, 62(11) (2008), 2868-2883.
- Weyl, H., The classical groups, their invariants and representations, New Delhi: Princeton University Press, 2013.
Year 2020,
Volume: 69 Issue: 1, 413 - 430, 30.06.2020
Abstract
References
- Abrams, P., Some comments on measuring niche overlap, Ecology, 61(1) (1980), 44-49.
- Çolakoğlu, H. B., Kaya, R., Regular polygons in the taxicab plane, KoG, 12 (2008), 27-33.
- Ekici, Kocayusufoğlu, İ., Akça, Z. The norm in taxicab geometry, Turk. J. Math, 22(3) (1998), 295-307.
- Goulson, D., Darvill, B., Niche overlap and diet breadth in bumblebees; are rare species more specialized in their choice of flowers?, Apidologie, 35 (2004), 55-63.
- Greub, W., Linear algebra, Berlin-Heidelberg-New York:Springer-Verlag, 1967.
- Hanson, J. R., Regular polygons in taxicab geometry, Int. J. Math. Educ. Sci. Technol., 45(7) (2014), 1084-1095.
- Legendre, P., Gallagher, E., Ecologically meaningful transformations for ordination of species data, Oecologia, 129 (2001), 271-280.
- Ören, İ, Çoban, H. A., Some invariant properties of curves in the taxicab geometry, Missouri J. Math. Sci., 26 (2014), 107-114.
- Ören,İ., Equivalence conditions of two Bézier Curves in the Euclidean geometry, Iran. J. Technol. A. Sci., 42(3) (2018), 1563-1577.
- Schattschneider, D. J., The taxicab group, Am. Math. Mon., 91 (1984), 423-428.
- Slobodchikoff, C. N., Schulz, W. C. Measures of niche overlap, Ecology, 61(7) (1980), 1051-1055.
- Sohn, M. W., Distance and cosine measures of niche overlap, Soc. Networks, 23 (2001), 141-165.
- Struble, R. A., Non-Linear differential equations, New York:Mc Graw-Hill, 1962.
- Sturmfels, B. Algorithms in invariant theory, Wien:Springer, 2008.
- Yüksel, S., Özcan, M., Some regular polygons in the taxicab 3-space, KoG, 19 (2005), 32-41.
- Warren, D. L., Glor, R. E., Turelli, M., Environmental niche equivalency versus conservatism: quantitative approaches to niche evolution, Evolution, 62(11) (2008), 2868-2883.
- Weyl, H., The classical groups, their invariants and representations, New Delhi: Princeton University Press, 2013.
There are 17 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 30, 2020 |
| Submission Date | October 10, 2017 |
| Acceptance Date | December 23, 2019 |
| Published in Issue | Year 2020 Volume: 69 Issue: 1 |
Cite
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.
