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Invariants of a mapping of a set to the two-dimensional Euclidean space

Yıl 2023, Cilt: 72 Sayı: 1, 137 - 158, 30.03.2023
https://doi.org/10.31801/cfsuasmas.1003511

Öz

Let $E_{2}$ be the $2$-dimensional Euclidean space and $T$ be a set such that it has at least two elements. A mapping $\alpha : T\rightarrow E_{2}$ will be called a $T$-figure in $E_{2}$. Let $O(2, R)$ be the group of all orthogonal transformations of $E_{2}$. Put $SO(2, R)=\left\{ g\in O(2, R)|detg=1\right\}$, $MO(2, R)=\left\{F:E_{2}\rightarrow E_{2}\mid Fx=gx+b, g\in O(2,R), b\in E_{2}\right\}$,
$MSO(2, R)= \left\{F\in MO(2, R)|detg=1\right\}$.
The present paper is devoted to solutions of problems of $G$-equivalence of $T$-figures in $E_{2}$ for groups $G=O(2, R), SO(2, R)$, $MO(2, R)$, $MSO(2, R)$. Complete systems of $G$-invariants of $T$-figures in $E_{2}$ for these groups are obtained. Complete systems of relations between elements of the obtained complete systems of $G$-invariants are given for these groups.

Destekleyen Kurum

The Ministry of Innovative Development of the Republic of Uzbekistan and The Scientific and Technological Research Council of Turkey

Proje Numarası

UT-OT-2020-2 and 119N643

Teşekkür

This work is supported by The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan) under Grant Number UT-OT-2020-2 and The Scientific and Technological Research Council of Turkey (T\"{U}B{\.I}TAK) under Grant Number 119N643.

Kaynakça

  • Aripov, R., Khadjiev, D., The complete system of global differential and integral invariants of a curve in Euclidean geometry, Izvestiya Vuzov, Ser. Mathematics, 542 (2007), 114, http://dx.doi.org/10.3103/S1066369X07070018.
  • Berger, M., Geometry I, Springer-Verlag, Berlin, Heidelberg, 1987.
  • Dieudonne, J. A. ,Carrell, J.B. , Invariant Theory, Academic Press, New-York, London, 1971.
  • Greub, W. H. , Linear Algebra, Springer-Verlag, New York Inc., 1967.
  • İncesu, M., Gürsoy, O., LS(2)-equivalence conditions of control points and application to planar Bezier curves, New Trends in Mathematical Sciences, 5(3) (2017), 70-84., http://dx.doi.org/10.20852/ntmsci.2017.186.
  • Höfer, R., m-Point invariants of real geometries, Beitrage Algebra Geom., 40 (1999), 261-266.
  • Ören, İ., Khadjiev, D., Pekşen, Ö., Identifications of paths and curves under the plane similarity transformations and their applications to mechanics, Journal of Geometry and Physics, 151 (2020), 1-17, 103619, https://doi.org/10.1016/j.geomphys.2020.103619.
  • Khadjiev, D.,Application of the Invariant Theory to the Differential Geometry of Curves, Fan Publisher, Tashkent, 1988, [in Russian].
  • Khadjiev, D., Pekşen, Ö., The complete system of global integral and differential invariants for equi-affine curves, Differential Geometry and its Applications, 20 (2004), 167-175, https://doi.org/10.1016/j.difgeo.2003.10.005.
  • Khadjiev, D., Complete systems of differential invariants of vector fields in a Euclidean space, Turkish Journal of Mathematics, 34(2010), 543-559,https://doi.org/10.3906/mat-0809-10
  • Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space, Journal of Nonlinear Mathematical Physics, 17(1) (2010), 49-70, https://doi.org/10.1142/S1402925110000799.
  • 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, 37 (2013), 80-94, https://doi.org/10.3906/mat-1104-41.
  • Khadjiev, D., Göksal, Y., Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-Euclidean space, Adv. Appl. Clifford Algebras, 26 (2016) 645-668, https://doi.org/10.1007/s00006-015-0627-9
  • Khadjiev, D., Ören, İ., Pekşen, Ö., Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space, International Journal of Geometric Methods in Modern Physics, 15(6) (2018), 1850092, https://doi.org/10.1142/S0219887818500925.
  • Khadjiev, D., Projective invariants of m-tuples in the one-dimensional projective space, Uzbek Mathematical Journal, 1 (2019) 61-73.
  • Khadjiev, D., Ayupov, Sh., Beshimov, G., Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidian space, Uzbek Mathematical Journal, 1 (2020), 57-84.
  • Khadjiev, D., Bekbaev, U., Aripov, R., On equivalence of vector-valued maps, arXiv:2005.08707v1 [math GM] 13 May 2020.
  • Khadjiev, D., Ayupov, Sh., Beshimov, G., Affine invariants of a parametric figure for fundamental groups of n-dimensional affine space, Uzbek Mathematical Journal, 65(4)(2021), 27-47.
  • Mundy, J. L. , Zisserman, A., Forsyth , D.(Eds.), Applications of Invariance in Computer Vision, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
  • Mumford, D., Fogarty, J., Geometric Invariant Theory, Springer-Verlag, Berlin, Heidelberg, 1994.
  • O’Rourke, J.,Computational Geometry in C , Cambridge University Press, 1997.
  • Ören, İ., Equivalence conditions of two Bezier curves in the Euclidean geometry, Iranian Journal of Science and Technology, Transactions A: Science, 42(3) (2018), 1563-1577., http://dx.doi.org/10.1007/s40995-016-0129-1.
  • Ören, İ., Invariants of m-vectors in Lorentzian geometry, International Electronic Journal of Geometry, 9(1)(2016), 38-44.
  • Pekşen, Ö., Khadjiev, D., Invariants of curves in centro-affine geometry, J. Math. Kyoto Univ., 44(3)(2004), 603-613.
  • Pekşen, Ö., Khadjiev, D., On invariants of null curves in the pseudo-Euclidean geometry, Differential Geometry and its Applications 29 (2011), 183-187, https://doi.org/10.1016/j.difgeo.2011.04.024.
  • Pekşen, Ö., Khadjiev, D., Ören, İ., Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry, Turkish Journal of Mathematics , 36 (2012), 147-160, http://dx.doi.org/10.3906/mat-0911-145.
  • Reiss, T. H. ,Recognizing Planar Objects Using Invariant Image Features, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
  • Sağıroğlu, Y., Khadjiev, D., Gözütok, U., Differential invariants of non-degenerate surfaces, Applications and Applied Mathematics, Special issue, 3 ( 2019), 35-57.
  • Sibirskii, K. S., Introduction to the Algebraic Invariants of Differential Equations, Manchester University Press, New York, 1988.
  • Springer, T. A., Invariant Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
  • Weyl, H., The Classical Groups: Their Invariants and Representations, Princeton Univ. Press, Princeton, New Jersey, 1946.
Yıl 2023, Cilt: 72 Sayı: 1, 137 - 158, 30.03.2023
https://doi.org/10.31801/cfsuasmas.1003511

Öz

Proje Numarası

UT-OT-2020-2 and 119N643

Kaynakça

  • Aripov, R., Khadjiev, D., The complete system of global differential and integral invariants of a curve in Euclidean geometry, Izvestiya Vuzov, Ser. Mathematics, 542 (2007), 114, http://dx.doi.org/10.3103/S1066369X07070018.
  • Berger, M., Geometry I, Springer-Verlag, Berlin, Heidelberg, 1987.
  • Dieudonne, J. A. ,Carrell, J.B. , Invariant Theory, Academic Press, New-York, London, 1971.
  • Greub, W. H. , Linear Algebra, Springer-Verlag, New York Inc., 1967.
  • İncesu, M., Gürsoy, O., LS(2)-equivalence conditions of control points and application to planar Bezier curves, New Trends in Mathematical Sciences, 5(3) (2017), 70-84., http://dx.doi.org/10.20852/ntmsci.2017.186.
  • Höfer, R., m-Point invariants of real geometries, Beitrage Algebra Geom., 40 (1999), 261-266.
  • Ören, İ., Khadjiev, D., Pekşen, Ö., Identifications of paths and curves under the plane similarity transformations and their applications to mechanics, Journal of Geometry and Physics, 151 (2020), 1-17, 103619, https://doi.org/10.1016/j.geomphys.2020.103619.
  • Khadjiev, D.,Application of the Invariant Theory to the Differential Geometry of Curves, Fan Publisher, Tashkent, 1988, [in Russian].
  • Khadjiev, D., Pekşen, Ö., The complete system of global integral and differential invariants for equi-affine curves, Differential Geometry and its Applications, 20 (2004), 167-175, https://doi.org/10.1016/j.difgeo.2003.10.005.
  • Khadjiev, D., Complete systems of differential invariants of vector fields in a Euclidean space, Turkish Journal of Mathematics, 34(2010), 543-559,https://doi.org/10.3906/mat-0809-10
  • Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space, Journal of Nonlinear Mathematical Physics, 17(1) (2010), 49-70, https://doi.org/10.1142/S1402925110000799.
  • 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, 37 (2013), 80-94, https://doi.org/10.3906/mat-1104-41.
  • Khadjiev, D., Göksal, Y., Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-Euclidean space, Adv. Appl. Clifford Algebras, 26 (2016) 645-668, https://doi.org/10.1007/s00006-015-0627-9
  • Khadjiev, D., Ören, İ., Pekşen, Ö., Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space, International Journal of Geometric Methods in Modern Physics, 15(6) (2018), 1850092, https://doi.org/10.1142/S0219887818500925.
  • Khadjiev, D., Projective invariants of m-tuples in the one-dimensional projective space, Uzbek Mathematical Journal, 1 (2019) 61-73.
  • Khadjiev, D., Ayupov, Sh., Beshimov, G., Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidian space, Uzbek Mathematical Journal, 1 (2020), 57-84.
  • Khadjiev, D., Bekbaev, U., Aripov, R., On equivalence of vector-valued maps, arXiv:2005.08707v1 [math GM] 13 May 2020.
  • Khadjiev, D., Ayupov, Sh., Beshimov, G., Affine invariants of a parametric figure for fundamental groups of n-dimensional affine space, Uzbek Mathematical Journal, 65(4)(2021), 27-47.
  • Mundy, J. L. , Zisserman, A., Forsyth , D.(Eds.), Applications of Invariance in Computer Vision, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
  • Mumford, D., Fogarty, J., Geometric Invariant Theory, Springer-Verlag, Berlin, Heidelberg, 1994.
  • O’Rourke, J.,Computational Geometry in C , Cambridge University Press, 1997.
  • Ören, İ., Equivalence conditions of two Bezier curves in the Euclidean geometry, Iranian Journal of Science and Technology, Transactions A: Science, 42(3) (2018), 1563-1577., http://dx.doi.org/10.1007/s40995-016-0129-1.
  • Ören, İ., Invariants of m-vectors in Lorentzian geometry, International Electronic Journal of Geometry, 9(1)(2016), 38-44.
  • Pekşen, Ö., Khadjiev, D., Invariants of curves in centro-affine geometry, J. Math. Kyoto Univ., 44(3)(2004), 603-613.
  • Pekşen, Ö., Khadjiev, D., On invariants of null curves in the pseudo-Euclidean geometry, Differential Geometry and its Applications 29 (2011), 183-187, https://doi.org/10.1016/j.difgeo.2011.04.024.
  • Pekşen, Ö., Khadjiev, D., Ören, İ., Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry, Turkish Journal of Mathematics , 36 (2012), 147-160, http://dx.doi.org/10.3906/mat-0911-145.
  • Reiss, T. H. ,Recognizing Planar Objects Using Invariant Image Features, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
  • Sağıroğlu, Y., Khadjiev, D., Gözütok, U., Differential invariants of non-degenerate surfaces, Applications and Applied Mathematics, Special issue, 3 ( 2019), 35-57.
  • Sibirskii, K. S., Introduction to the Algebraic Invariants of Differential Equations, Manchester University Press, New York, 1988.
  • Springer, T. A., Invariant Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
  • Weyl, H., The Classical Groups: Their Invariants and Representations, Princeton Univ. Press, Princeton, New Jersey, 1946.
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Djavvat Khadjiev 0000-0001-7056-5662

Gayrat Beshimov Bu kişi benim 0000-0002-5394-2179

İdris Ören 0000-0003-2716-3945

Proje Numarası UT-OT-2020-2 and 119N643
Yayımlanma Tarihi 30 Mart 2023
Gönderilme Tarihi 3 Ekim 2021
Kabul Tarihi 19 Eylül 2022
Yayımlandığı Sayı Yıl 2023 Cilt: 72 Sayı: 1

Kaynak Göster

APA Khadjiev, D., Beshimov, G., & Ören, İ. (2023). Invariants of a mapping of a set to the two-dimensional Euclidean space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(1), 137-158. https://doi.org/10.31801/cfsuasmas.1003511
AMA Khadjiev D, Beshimov G, Ören İ. Invariants of a mapping of a set to the two-dimensional Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Mart 2023;72(1):137-158. doi:10.31801/cfsuasmas.1003511
Chicago Khadjiev, Djavvat, Gayrat Beshimov, ve İdris Ören. “Invariants of a Mapping of a Set to the Two-Dimensional Euclidean Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, sy. 1 (Mart 2023): 137-58. https://doi.org/10.31801/cfsuasmas.1003511.
EndNote Khadjiev D, Beshimov G, Ören İ (01 Mart 2023) Invariants of a mapping of a set to the two-dimensional Euclidean space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 1 137–158.
IEEE D. Khadjiev, G. Beshimov, ve İ. Ören, “Invariants of a mapping of a set to the two-dimensional Euclidean space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 72, sy. 1, ss. 137–158, 2023, doi: 10.31801/cfsuasmas.1003511.
ISNAD Khadjiev, Djavvat vd. “Invariants of a Mapping of a Set to the Two-Dimensional Euclidean Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/1 (Mart 2023), 137-158. https://doi.org/10.31801/cfsuasmas.1003511.
JAMA Khadjiev D, Beshimov G, Ören İ. Invariants of a mapping of a set to the two-dimensional Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:137–158.
MLA Khadjiev, Djavvat vd. “Invariants of a Mapping of a Set to the Two-Dimensional Euclidean Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 72, sy. 1, 2023, ss. 137-58, doi:10.31801/cfsuasmas.1003511.
Vancouver Khadjiev D, Beshimov G, Ören İ. Invariants of a mapping of a set to the two-dimensional Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(1):137-58.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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