G=S(1), G=S(2) ve alt Grubları için G- Yörüngeler

Abstract

(G, * )  bir grup, X bir küme olmak üzere G:X  etkisi verilsin. Bir x Î C noktası için Gx = {gx: gÎG}  kümesine  x elemanının G- yörüngesi denir. (G, * )  bir grup olmak üzere bir x Î C elemanının kendisini içeren en küçük G-invaryant altküme  x’in  G-yörüngesidir. Bu çalışmada Benzerlik grubu G = S(n)   ve  tüm alt grupları için n=1 ve n=2 durumlarında G- invaryant alt uzaylar olan G- yörüngeler elde edilmiştir.

Keywords

• Benzerlik grubu,G- invariant altküme,G- yörüngeler

The G- orbits for G=S(1), G=S(2) and their Subgroups

Abstract

Let (G, * ) is a group and X is a nonempty set and let group action  are given. For any point  the set is called G- orbits of the element x.  Let  (G, * ) is a group then, the smallest G- invariant subset  containing  is G- orbit of x. In this paper G-orbits of the similarity group S(n) and all subgroups of it in case n=1 and n=2, which are G- invariant subspaces therewithal, are obtained.     

Keywords

• G- invariant subsets,G- orbits

References

1. [1] G. Sartori, A theorem on orbit structures (strata) of compact linear Lie groups, Journal of Mathematical Physics 24, 765 (1983)
2. [2] Peter Symonds, The orbit space of the p-subgroup complex is contractible, Commentarii Mathematici Helvetici, 73 (1998) 400–405
3. [3] Kenzi Odani , Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc. 110 (1990), 281-284
4. [4] S. Mukhi, SL(2,R) conformal field theory, minimal models and two dimensional gravity, Proceedings of the International Colloquium on Modern Quantum Field Theory, 8-14 Jan 1990, TIFR, Bombay, India.
5. [5] R.Gatto, G.Sartori, Zeros of the D-term and complexification of the gauge group in supersymmetric theories, Physics Letters B, Volume 157, Issues 5–6, 25 July 1985, Pages 389-392.
6. [6] R. W. Richardson, Affine Coset Spaces of Reductive Algebraic Groups, Bulletin of the London Mathematical Society, Vol.9, Issue 1 , March 1977, Pages 38-41.
7. [7] Ozeki, Ikuzō. On the microlocal structure of the regular prehomogeneous vector space associated with $\mathrm{SL}\left( 5 \right) \times \mathrm{GL} \left( 4 \right)$, I. Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 2, 37-40. doi:10.3792/pjaa.55.37.
8. [8] SA Kirillov MI Kuznetsov and NG Chebochko , ON DEFORMATIONS OF THE LIE ALGEBRA OF TYPE G2 OF CHARACTERISTIC 3, Russian Mathematics ( Iz. VUZ), Vol 44. No. 3 pp 31-36, 2000

9. [9] Robert M. Guralnick (1997) Invertible Preservers and Algebraic Groups II: Preservers of Similarity Invariants and Overgroups of PSLn (F), Linear and Multilinear Algebra, 43:1-3, 221-255
10. [10] D. Hinrichsen and J. O’Halloran, A Complete Characterization of Orbit Closures of Controllable Singular Systems under Restricted System Equivalence , SIAM J. Control Optim., 28(3), 602–623
11. [11] INCESU, M. The complete system of point invariants in the similarity geometry. 2008. PhD Thesis. Phd. Thesis, Karadeniz Technical University, Trabzon.
12. [12] A. N. KOLMOGOROFF, Interpolation und Extrapolation, Bull. Acad. Sci. U.S.S.R. Ser. Math., (1941), pp. 3-14
13. [13] N. WIENER, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, New York, 1949
14. [14] N. ARONSZAJN AND K. T. STMITH INVARIANT SUBSPACES OF COMPLETELY CONTINUOUS OPERATORS, ANNALS OF MATHEMATICS Vol. 60, No. 2, 1954
15. [15] Victor A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proceedings of the Royal Society of Edinburgh, 125A, 225-246,1995.
16. [16] Weyl H., The Classical Groups, Their Invariants and Representations, 2nd ed., with suppl.. Princeton, Princeton University Press, 1946.
17. [17] Khadjiev Dj., An Application of the Invariant Theory to the Differential Geometry of Curves, Fan, Tashkent, 1988. ( in Russian )
18. [18] Cassier E. T., The concept of group and the theory of perception, Philosophy and Phenomenological Research, 5 (1944) 1-35. (original French version published in 1938)
19. [19] Hoffman W. C., The Lie algebra of visual perception, Journal of Mathematical Psychology, 3 (1966) 65-98.
20. [20] Hoffman W. C., The Lie transformation group approach to visual neuropsychology, in E.L.J. Leewenberg & H. F. J. M. Buffart, Formal theories of visual perception, 27-66, Chichester, UK. Wiley, 1978.
21. [21] Chan &Chan, A transformational analysis of form recognition under plane isometries, Journal of Mathematical Psychology, 26, 3 (1982) 237-251.
22. [22] Chan &Chan, A mental space similarity Group model of Shape constancy, Journal of Mathematical Psychology, 43 (1999) 410-432.
23. [23] Leyton M. A theory of information structure: II.A theory of perceptual organization, Journal of Mathematical Psychology, 30 (1986) 257-305.
24. [24] V. Asil, T. Körpınar, S. Baş, New Parametric Representation of a Surfaces B-Pencil with a Common Line of Curvature, Siauliai Math. Semin., 9 (17) (2014), 5-14.
25. [25] T. Körpınar, S. Baş, Characterization of Quaternionic Curves by Inextensible Flows, Prespactime Journal 7(12) (2016), 1680-1684.
26. [26] Ören İ., Khadjiev D., Pekşen Ö., "Global invariants of paths and curves with respect to similarity groups in the two-dimensional Euclidean space and their applications to mechanics", INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS, vol.15, pp.1-32, 2018
27. [27] Khadjiev D., Ören İ., Pekşen Ö., "Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry", TURKISH JOURNAL OF MATHEMATICS, vol.37, no.1, pp.80-94, 2013
28. [28] Sağiroğlu Y., "Equi-affine differential invariants of a pair of curves", TWMS Journal of Pure and Applied Mathematics, vol.6, pp.238-245, 2015
29. [29] Yapar Z., Sağiroğlu Y., "Curvature Motion On Dual Hyperbolic Unit Sphere H_0^2", Journal of Applied Mathematics and Physics, vol.2, no.8, pp.828-826, 2014