# Geometric and kinematic invariants of T-shapes in Euclidean and pseudo-Euclidean spaces.

**Scientific adviser****:** Khadjiev Djavvat

** Due date:** 01.10. 2021 -01.10.2022

** Project password: **UТ-АТ-2020-2

** Project type: **applied

**Expected results and their relevance:** The concept of a figure in geometry was defined in ancient Greece 2350 years ago. They defined a figure as an arbitrary subset on a plane or in three-dimensional space. They also defined the equality of figures in Euclidean geometry. They also obtained conditions for the equality of triangles, quadrilaterals and some other polygons in Euclidean geometry.

Since 1850, a new field of mathematics called "Invariant Theory" began to develop in Europe. By this time, an infinite number of geometries in multidimensional spaces had emerged in mathematics. In each of the geometries that have arisen, the concept of equality of figures is naturally defined. In the theory of invariants, the problem of equality of figures with a finite number of elements was solved.

At one time, the famous German mathematician Felix Klein said: “Every geometry is a theory of invariants of a group of transformations of this geometry.” Since only invariants of figures defined by a finite number of elements have been studied in invariant theory, Felix Kleina's statement does not cover geometric figures that are not completely determined by a finite number of elements.

In our project, we have significantly generalized the concept of a figure. It covers both figures defined by a finite number of elements and figures that are not completely defined by a finite number of elements. Let be given an arbitrary linear space L and an arbitrary set T. The set T may lie in L, but it may not lie in L.

Our first definition of a figure is as follows: The mapping of a set T to L is called a parametric T-figure in L.

Our second definition of the figure is as follows: Let T be a topological space and L be a topological vector space. A continuous mapping of a set T in L is called a parametric topological T-figure in L.

We also give definitions of a smooth figure and a polynomial figure.

In our definition, the set T can be a finite set and can be an infinite set. In invariant theory, T is a finite set.

In the classical definition in invariant theory , T lies in L. In our definition, T may lie in L, but it may not lie in L. In our definition, a T-shape is a mapping of a set T to L.

It is easy to see that it contains the definition of the ancient Greeks and our definition is much broader than the definition of the ancient Greeks, the definition of figures in geometry and the definition of a figure in the theory of invariants.

The objectives of this project are to develop the theory of invariants of parametric T-figures for the following cases: 1). T is an arbitrary nonempty set and L is an n-dimensional linear space over some field;

2) T is an arbitrary nonempty set and Lisa ann-dimensional Euclidean space; 3) T is an arbitrary nonemptyset and Lisa 2-dimensional pseudo-Euclidean space.

The second goal of this project was to develop the theory of invariants of parametric topological T-figures for the following cases:1). T is an arbitrary topological space and L is an n-dimensional affine space;

2) T is an arbitrary topological space and L is an n-dimensional Euclidean space; 3) T is an arbitrary topological space and L is a 2-dimensional pseudo-Euclidean space.

The third goal of this project is to develop the theory of invariants of parametric smooth T-figures for the following cases: T is an open subset in an n-dimensional Euclidean space and L is an n-dimensional Euclidean space; in this case, the mapping of T to L is a smooth mapping.

These goals were planned for the reporting period.

Applications of the obtained results in invariant theory, geometry, mechanics and Computer Graphics will be given.

**The purpose and objectives of the study.** Development of the invariant theory of parametric figures and their applications in invariant theory, Geometry, Mechanics and Computer Graphics.

**To solve the tasks set in the project for the period 01.10.2020 - 01.10.2022, it is necessary:**

**I. To analyze the concepts of figures, images and movements of figures in various sections of mathematics, physics and other sciences:**

**1. Research of works in the field of invariant theory.**

**2. Study of works on the study of figures in various geometries.**

**3. Study of works on the concept of figures and movements of figures in the field of physics.**

**4. The study of works on the study of images in the theory of pattern recognition.**

**5. Study of works on the study of figures in Computer Graphics.**

**II. Analysis of the concept of the invariant of a figure, image and movement of figures in various branches of mathematics, physics and other sciences:**

**1. To analyze the concept of an invariant and a complete system of invariants of a figure in the theory of invariants.**

**2. To analyze the concept of an invariant and a complete system of invariants of a figure in various geometries.**

**3. To analyze the concept of an invariant and a complete system of invariants of a figure in the field of mechanics.**

4. To analyze the concept of an image invariant and a complete system of image invariants in the theory of pattern recognition.

5. To analyze the concept of an invariant and a complete system of invariants of a figure in Computer Graphics.

**III. Finding a complete system of invariants of figure, image and movement in various fields of sciences**

**1. Finding a complete system of invariants for fundamental transformation groups in invariant theory;**

**2. Finding a complete system of invariants for fundamental transformation groups in classical geometries;**

**3. Finding a complete system of invariants for fundamental transformation groups in Computer Graphics.**

**Important results achieved during the reporting period (upon completion of the project):****1. Days of defining arid new concepts, a topological figure of a smooth figure and a polynomial figure in mathematics, covering the basic ideas of figures in invariant theory, in Geometry and Somputer Graphics.****2. The obtained complete systems of invariant figures of topological figures in invariant theory, in geometry and Somputer Graphics.**

4. Complete systems of relations between elements of complete systems of invariants of figures and topological figures in invariant theory, geometry and Computer Graphics are found.

Within the framework of the project to publish international scientific papers based on WoS, Scopus and Uzbek Mathematical Journal.

The following articles and abstracts have been published:

**Papers **

**1**.Djavvat Khadjiev, Shavkat Ayupov, Gayrat Beshimov, Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidian space, Uzbek Mathematical Journal, 2020, 1, pp.57-84,DOI: 10.29229/uzmj.2020-1-6.

**2**.İdris Őren, Djavvat Khadjiev , Ömer Pekşen, Identifications of paths and curves under the plane similarity transformations and their applications to mechanics, Journal of Geometry and Physics, 151, (2020), 103619,1-17. Sc.

**3**.Khadjiev Dj., Bekbaev U. Aripov, R. , . On equivalence of vector-valued maps, arXiv: 2005.08707v1 [math GM] 13 May 2020.

**4**.Ören I., Khadjiev D., Recognition of plane paths and plane curves under linear pseudo-similarity transformations. J.Geom. 111, 38. (2020). https:// doi. Org/10.1007/s 00022-020-00551-6. (published 26 August 2020). Sc

**5.Idris Ören** **and Djavvat Khadjiev,**** **Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry**, **INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY, VOLUME 13 NO. 2 PAGE 116–134 (2020) DOI: HTTPS://DOI.ORG/10.36890/IEJG.768821

**6.**U.Bekbaev, On equivalence of polygons in finite dimensional vector spaces , Proceedings of scientific conference "Actual problems of stochastic analysis"- February 20-21,2021, Tashkent, pp. 375- -379.

**7.**U. Bekbaev, Sh. Eshmirzaev. Complete classification of two-dimensional algebras over the field of rational numbers. Vol. 11 No. 1 (2021): Acta of Turin Polytechnic University in Tashkent, pp 49—54.

** 8.** U. Bekbaev, Sh. Eshmirzaev. On classification of two-dimensional algebras over the field of rational numbers, Proceedings of scientific conference "Actual problems of stochastic analysis",pp.379--382, February 20-21, 2021, Tashkent.

** 9. Sh. Ayupov, A. Jalilov, Asymptotic Distribution of Hitting Times for Critical Maps of the Circle. Vestnik Udmurtskogo Universiteta. MATHEMATICS. 31(2021) no.3****, 1****-19.**

**10**.Khadjiev D., Ayupov Sh., Beshimov G., Affine invariants of a parametric figure for fundamental groups of n-dimensional affine space, Uzbek Mathematical Journal, 2021, Volume 65, Issue 4, pp. 27-47, DOI, 10, 29229/uzmj.2021-4-3.

**11**.Дж. Xаджиев, Г.Р. Бешимов, Инварианты последавательностей для группы SO(2, Q) двумерного билинейно-метрического пространства над полем рациональных чисел, Итоги науки и техн. Сер. Соврем. Мат. И её прил. Темат. Обз., 2021, том 197,46-55, DOI:https://doi/org/10/36535/0233-6723-2021-197-46-55.

**12**. Khadjiev D.,Complete systems of invariants of a parametric figure in the n-dimensional Euclidean space, Uzbek Mathematical Journal, 2022, Volume 66, Issue 2.

**13**. Khadjiev D., Ayupov Sh., Beshimov G., Complete systems of invariants of a parametric figure in the n-dimensional pseudo-Euclidean spaces, Uzbek Mathematical Journal, 2022, Volume 66, Issue 3.

** Theses**

**1**.Khadjiev Dj., Bekbaev U., Aripov R. On equivalence of vector valued maps, Book of abstracts of the National scientific conference with foreign participants “Modern Methods of Mathematical Physics and its Applications”, Tashkent, 17-

- november-2020. V-2, pp. 87-91.

**2**.Bekbaev U. Dj. On equivalence of vector valued maps with respect to some motion groups and change of variable, Book of abstracts of the National scientific conference with foreign participants “Modern Methods of Mathematical Physics and its Applications”, Tashkent, 17-18 November-2020. V-2, pp 75-79.

**3**.Beshimov G.R. Invariants of m-points in the two-dimensional bilinear-metric space with the form x_{1} y_{2} - 2x_{2} y_{2} over the field of rational numbers, Abstracts of the International Online Conference Frontier in mathematics and computer science, Tashkent, October 12–15, 2020, pp. 36-38.

**4.** U. Bekbaev. On classification of two-dimensional algebras over any basic field. Teзиcы доkлaдов: Республиканскaя научнaя конференция с участием зарубежных ученых "Сарымсаковские чтения", 16-18 сентября, 2021. NYY, Tashkent.

**5**. Beshimov G., Khadjiev D., Gafurov I. Evident forms of elements of the orthogonal group of the two-dimensional bilinear-metric space with the form x_{1}y_{1}+11x_{2}y_{2} over the field of rational numbers, Respublika ilmiy anjumani”Globallashuv davrida matematik ava amaliy matematikaning dolzarb masalalari”, 1-2 Iyun, 2021yil,b.147-148.

**6**. Beshimov G., Khadjiev D., Sadullaeva M. On the orthogonal group of the two-dimensional bilinear-metric space with the form x_{1}y_{1}+7x_{2}y_{2 }over the field of rational numbers, Respublika ilmiy anjumani”Globallashuv davrida matematik ava amaliy matematikaning dolzarb masalalari”, 1-2 Iyun, 2021yil, b.148-150.

**7**. Khadjiev D., Beshimov G., Solieva M. Descriptions of elements of the orthogonal group of the two-dimensional bilinear-metric space with the form over the field of rational numbers, Respublika ilmiy anjumani”Globallashuv davrida matematik ava amaliy matematikaning dolzarb masalalari”, 1-2 Iyun, 2021yil,b.165-166.

**8**. Gayrat Beshimov, İdris ÖREN, Djavvat Khadjiev, The concept of the notion of a figure in two-dimensional Euclidean geometry and its Euclidean invariants, 18th International Geometry Symposium in honor of Prof. Dr. Sadık KELEŞ July 12-13, 2021 İnönü University, Malatya-TURKEY.

**9.** İdris ÖREN, Gayrat Beshimov, Djavvat Khadjiev, Euclidean invariants of plane paths, 18th International Geometry Symposium in honor of Prof. Dr. Sadık KELEŞ July 12-13, 2021 İnönü University, Malatya-TURKEY.

**10**. Khadjiev D., Beshimov G.,Complete systems of T-figure in a two-dimensional bilinear-metric space over the field of rathional numbers, Teзиcы доkлaдов: Республиканскaя научнaя конференция с участием зарубежных ученых "Сарымсаковские чтения", 16-18 сентября, 2021. NYY, Tashkent.

**11.** Аюпов Ш. А., Жураев Т. Ф., Резко очерченные пары (F(X), ƞ_{F}(X)) компактов вида P(X), Teзиcы доkлaдов: Республиканскaя научнaя конференция с участием зарубежных ученых "Сарымсаковские чтения", 16-18 сентября, 2021. NYY, Tashkent.

**12.**Sadullayeva M. S., Beshimov G.R. Invariants of m-tuples for the group of special-orthogonal in the two-dimensional bilinear-metric space with the form x1y1 + 13x2y2 over the field of rational numbers, , Teзисы докладов Республиканской научной конференции с участием зарубежных ученых “Дифференциальные уравнения и родственные проблемы анализа”, Бухара, Узбекистан, 04-05 ноябрь, 2021, стр.

**13**.Beshimov G.R. Gafurov I.I., Invariants of m-tuples for the orthogonal group in the Q(√5) with the form x_{1}y_{1} +5x_{2}y_{2} over the field of rational numbers, Teзисы докладов Республиканской научной конференции с участием зарубежных ученых “Дифференциальные уравнения и родственные проблемы анализа”, Бухара, Узбекистан, 04-05 ноябрь, 2021, стр. 102-104.

**14**.Beshimov G.R., Soliyeva M. A description of all non-congruent symmetric bilinear forms on the two-dimensional vector space over the field Z_{7}, Modern problems of applied mathematics and information technologies al-Khwarizmi 2021, VII International Scientific Conference, Fergana,Uzbekistan, 15-17 November, 2021,

**15**.Khadjiev D., Beshimov G., Joraeva Z. Complete systems of ınvarıants of polynomıal parametrıc curves for groups SO(2,R), O(2,R) of the two-dımensıonal Euclıdean space**, **Modern problems of applied mathematics and information technologies al-Khwarizmi 2021, VII International Scientific Conference, Fergana,Uzbekistan, 15-17 November, 2021, 15-17 November, 2021, Fergana, Uzbekistan, 250 бет.

**16**.Gafforov I., Khadjiev D., Beshimov G. A description of elements of the orthogonal group of the two-dimensional bilinear-metric space with the form x1y1 − 5x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 20-21.

**17.**Joraeva Z.,Khadjiev D. A description of all orthogonal matrices of the two- dimensional bilinear-metric space with the form x1y1−3x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 22-23.

**18.**Uktamov Sh.,Khadjiev D.,Beshimov G. A description of all orthogonal transformations of the two-dimensional bilinear-metric space with the form x1y1+19x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 43-44.

**19**.Otaqulova F.,Beshimov G. A description of elements of the orthogonal group of the two-dimensional bilinear-metric space with the form x1y1 + 11x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 44-45.

**20**.Qodirova D., Khadjiev D.,Beshimov G. A description of elements of the orthogonal group of the two-dimensional bilinear-metric space with the form x1y1 + 17x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 45-46.

**21**.Khadjiev D.,Beshimov G.R., Sadullayeva M.S. Invariants of m-tuples for the group of special-orthogonal in the two-dimensional bilinear-metric space with the form x1y1 + 13x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 51-52.

**22.**Shayimova F., Khadjiev D. A description of all orthogonal transformations of the two-dimensional bilinear-metric space with the form x1y1+19x2y2 over the field of rational numbers. Abstracts of the conference of young scientists MATHEMATICS, MECHANICS AND INTELLECTUAL TECHNOLOGIES., 21-22 April 2022, Tashkent, Uzbekistan. pp. 56-57.

**23**.Sh. A. Ayupov , Yusupov B. B. **LOCAL DERIVATIONS ON NILPOTENT LEIBNIZ ALGEBRAS NFn + F1m**. Contemporary mathematics and its application. Abstracts of the international scientific conference (19-21 November 2021, Tashkent, Uzbekistan). Tashkent. 2021. p. 52-54.

**2**4.Аюпов Ш.А., Жураев, Т.Ф**. Резко очерченные пары компактов вида Р(Х). **Тезисы докладов конференции «Сарымсаковские чтения», 35-37.

**25**.Gayrat Beshimov, Idris Ören, Djavvat Khadjiev, The concept of the notion of a figure in two-dimensional Euclidean Geometry, 18^{th }International Geometry Symposium in Honor of Prof. Dr. Sadik Keleş, July 12-13 2021, Inönü University, p.139.

**26**.Idris Ören, Gayrat Beshimov, Djavvat Khadjiev, Euclidean invariants of plane curves, 18^{th }International Geometry Symposium in Honor of Prof. Dr. Sadik Keleş, July 12-13 2021, Inönü University, p.140.

** Integration**

**They are used in scientific works, in teaching algebraic and geometric disciplines at the National University of Uzbekistan, in dissertations.**