Ilmiy raxbar: Xadjiev Djavvat
Bajarilish muddati: 01.10. 2021 -01.10.2022
Loyiha shifri: UT-AT-2020-2
Loyiha turi: amaliy

Kutilayotgan natijalar va ularning ahamiyati: Geometriyada figura tushunchasi birinchi bo‘lib, Yunonistonda 2350 yil ilgari tariflangan edi. Ular tekislikning yoki uch o‘lchovli fazoning ixtiyoriy qism to‘plamiga figura deb tarif berishgan. Ular tekislikdagi yoki uch o‘lchovli Evklid fazosidagi figuralarning tenglik tarifini xam berganlar. Ular tekislikdagi yoki uch o‘lchovli Evklid fazosidagi uchburchaklarning, to‘rtburchaklarning va boshqa baьzibir ko‘pburchaklarning tenglik shartlarini topishgan.
1850 yildan boshlab Evropada matematikaning yangi soxasi-“invariantlar nazariyasi” nomli soxasi rivojlana boshladi. Shu vaqtga kelib, matematikada ko‘p o‘lchovli chiziqli fazolarda cheksiz ko‘p geometriyalar paydo bo‘ldi. Yangi paydo bo‘lgan xarbir geometriyada figura tushunchasi tariflanib, figuralarning tenglik tushunchasi taьrifi berildi. Invariantlar nazariyasida chekli sondagi elementli figuralarning tenglik problemasini echish bilan shug‘ullanildi.
O‘sha davrda mashhur nemis matematigi Feliks Kleyn aytgan edi: Harqanday geometriya bu geometriyaning o‘zgartirishlari gruppasinig invariantlari nazariyasidir. Invariantlar nazariyasida faqat chekli sondagi elementli figuralarning tenglik problemasini echish bilan shug‘ullanilgani uchun, u vaqtlarda geometriyalarda faqat chekli sondagi elementli figuralar bilan shug‘ullanilgan. Keyinroq geometriyalarda (masalan, differentsial geometriyada) baьzi cheksiz elementli figuralar bilan ham shug‘ullanish boshlangan.
Bizning proektda figura tushunchasi geometriyalardagi figura tushunchasiga ko‘ra ancha keng taьrifi berildi va bu figuralarning invariantlari nazariyasi rivojlantirildi. Bir L chiziqli fa’zo va ixtiyoriy bir T to‘plam berilgan bo‘lsin. T to‘plam L ning ichida bo‘lishi mumkin yoki bo‘lmasligi ham mumkin.
Bizning figuraga oit birinchi ta’rifimiz quyidagichadir: T ixtiyoriy bir to‘plam bo‘lsin. T to‘plamning L chiziqli fazoga aks ettirishiga L da parametrik figura deyiladi.
Figuraga oit ikkinchi ta’rifimiz quyidagichadir: T topologik fazo va L topologik vektor fazo bo‘lsin. T-topologik fazoning L topologik vektor fazoga uzluksiz aks ettirishiga L fazodagi parametrik topologik T-figura deyiladi.
Figuraning bashqa ta’riflari, xususan, silliq figura va polinom xususiyatli figura ta’riflarini ham berdik. Qisqalik uchun ular bu erda berimadi.
Loyihamizning asosiy maqsadlari quyidagilardan iborat:
1). T-bo‘sh bo‘lmagan bir to‘plam va L n-o‘lchovli Evklid fazosi bo‘lgan holda, parametrik figuralarning invariantlari nazariyasini rivojlantirish.
2) 1). T-bo‘sh bo‘lmagan bir topologik fazo va L n-o‘lchovli affin fazo bo‘lgan holda, parametrik topologik figuralarning invariantlari nazariyasini rivojlantirish.
3). T-bo‘sh bo‘lmagan bir topologik fazo va L n-o‘lchovli Evklid fazosi bo‘lgan holda, parametrik topologik figuralarning invariantlari nazariyasini rivojlantirish.
4) 1). T-bo‘sh bo‘lmagan bir topologik fazo va L 2-o‘lchovli psevdo-Evklid fazo bo‘lgan holda, parametrik topologik figuralarning invariantlari nazariyasini rivojlantirish.
5). T, n-o‘lchovli Evklid fazosida bir ochiq to‘plam va L n-o‘lchovli Evklid fazosi bo‘lgan holda, parametrik silliq figuralarning invariantlari nazariyasini rivojlantirish.

Bu maqsadlar loyihada echiladigan asosiy problemalardir.
Olingan natijalarning kelajakda invariantlar nazariyasiga, geometriyaga, mexanikaga i Computer Graphics nazariyasiga tatbiqlar qilinishi ham mo‘ljalanilgan.
Tekshirishning asosiy maqsadlari va masalalri. Parametrik figuralarning invariantlari nazariyasini rivojlantirish va olingan natijalarni invariantlar nazariyasiga, geometriyaga va Computer Graphics nazariyasiga tatbiq qilish.
Bu masalarni bajarish uchun loyihada ushbu 01.10.2020 -01.10.2022 vaqtda quyidagilarni bajarish lozim:
I. Matematikada, mexanikada va obrazlarni tanish nazariyasida figura tushunchalarining analizlari bajarilishi lozim:

1. Invariantlar nazariyasi soxasidagi ishlarni o‘rganish.
2. Farqli geometriyalarda figuralarning tekshirilishiga oid ishlarni o‘rganish.
3. Mexanikada figura tushunchasiga oid va figuralarning harakatiga oid ishlarni o‘rganish.
4. Obrazlarni tanish nazariyasida obrazlarni tekshirishga oid ishlarni o‘rganish.
5. Computer Graphics nazariyasida graphic figuralarni tekshirishga oid ishlarni o‘rganish.
   II. Matematikada, mexanikada va obrazlarni tanish nazariyasida figuraning invarianti, obrazning invarianti va figura harakatining invarianti tushunchalarining analizlari bajarilishi lozim:
6. Invariantlar nazariyasida invariant tushunchasining analizini va figuraning to‘la invariantlari sistemasi tushunchasining analizini qilish.
7. Farqli geometriyalarda invariant tushunchasining analizini va figuraning to‘la invariantlari sistemasi tushunchasining analizini qilish.
8. Mexanikada figura xarakatining invarianti tushunchasining analizini, figura xarakatining to‘la invariantlari sistemasi tushunchasining analizini qilish.
9. Obrazlarni tanish nazariyasida invariant tushunchasining analizini va obrazning to‘la invariantlari sistemasi tushunchasining analizini qilish.
10. Somputer Graphics nazariyasida invariant tushunchasining analizini va figuraning to‘la invariantlari sistemasi tushunchasining analizini qilish.

III. Turli fanlar soxalarida figuraning, obrazning va figura harakatining to‘la invariantlari sistemalarini topish

1. Invariantlar nazariyasida asosiy gruppalar uchun figuraning to‘la invariantlari sistemalarini topish
2. Klassik geometriyalarda asosiy gruppalar uchun figuraning to‘la invariantlari sistemalarini topish
3. Somputer Graphics nazariyasida asosiy gruppalar uchun figuraning to‘la invariantlari sistemalarini topish
   Xisobot davrida olingan muhim natijalar (loyiha bitganda):
4. Invariantlar nazariyasidagi, geometriyadagi va Somputer Graphics nazariyasidagi figura tushunchalarini o‘z ichiga olgan yangi tushunchalar berildi: T-figura, topologik T-figura, silliq T-figura va polinomial T-figura tushunchalarining tariflari berildi.
5. Invariantlar nazariyasida, geometriyalarda va Somputer Graphics nazariyasida T-figuralarning, topologik T-figuralarning, silliq T-figuralarning va polinomial T-figuralarning to‘la invariantlari sistemalari topildi.
6. T-figuralarning, topologik T-figuralarning, silliq T-figuralarning va polinomial T-figuralarning topilgan to‘la invariantlari sistemalarining elementlari orasidagi bog‘lanishlarning to‘la sistemalari topildi.
7. Loyihaga oid ilmiy ishlarni halqaro WoS, Scopus jurnallarida va Uzbek Mathematical Journalda chop etish.
   Loyihaga oid quyidagi maqolalar va tezislar chop etildi:
   Maqolalar
   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.
8. 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.Dj. Xadjiev, G.R. Beshimov, Invariantы posledavatelьnostey dlya gruppы SO(2, Q) dvumernogo bilineyno-metricheskogo prostranstva nad polem ratsionalьnыx chisel, Itogi nauki i texn. Ser. Sovrem. Mat. I eyo pril. Temat. Obz., 2021, tom 197,46-55, DOI:https://doi/org/10/36535/0233-6723-2021-197-46-55.
9. Khadjiev D.,Complete systems of invariants of a parametric figure in the n-dimensional Euclidean space, Uzbek Mathematical Journal
   2022, Volume 66, Issue 2, pp.86-100, DOI: 10.29229/uzmj.2022-2-9.
10. 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. Tezislar 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-
    18 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 x1 y2 – 2×2 y2 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. Tezicы dokladov: Respublikanskaya nauchnaya konferentsiya s uchastiem zarubejnыx uchenыx «Sarыmsakovskie chteniya», 16-18 sentyabrya, 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 x1y1+11x2y2 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 x1y1+7x2y2 over the field of rational numbers, Respublika ilmiy anjumani”Globallashuv davrida matematik ava amaliy matematikaning dolzarb masalalari”, 1-2 Iyun, 2021yil, b.148-150.
6. 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.
7. 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.
8. 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.
9. Khadjiev D., Beshimov G.,Complete systems of T-figure in a two-dimensional bilinear-metric space over the field of rathional numbers, Tezicы dokladov: Respublikanskaya nauchnaya konferentsiya s uchastiem zarubejnыx uchenыx «Sarыmsakovskie chteniya», 16-18 sentyabrya, 2021. NYY, Tashkent.
10. Ayupov Sh. A., Juraev T. F., Rezko ocherchennыe parы (F(X), ƞF(X)) kompaktov vida P(X), Tezicы dokladov: Respublikanskaya nauchnaya konferentsiya s uchastiem zarubejnыx uchenыx «Sarыmsakovskie chteniya», 16-18 sentyabrya, 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, , Tezisы dokladov Respublikanskoy nauchnoy konferentsii s uchastiem zarubejnыx uchenыx “Differentsialьnыe uravneniya i rodstvennыe problemы analiza”, Buxara, Uzbekistan, 04-05 noyabrь, 2021, str.
13.Beshimov G.R. Gafurov I.I., Invariants of m-tuples for the orthogonal group in the Q(√5) with the form x1y1 +5x2y2 over the field of rational numbers, Tezisы dokladov Respublikanskoy nauchnoy konferentsii s uchastiem zarubejnыx uchenыx “Differentsialьnыe uravneniya i rodstvennыe problemы analiza”, Buxara, Uzbekistan, 04-05 noyabrь, 2021, str. 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 Z7, 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 bet.
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.
24.Ayupov Sh.A., Juraev, T.F. Rezko ocherchennыe parы kompaktov vida R(X). Tezisы dokladov konferentsii «Sarыmsakovskie chteniya», 35-37.
25.Gayrat Beshimov, Idris Ören, Djavvat Khadjiev, The concept of the notion of a figure in two-dimensional Euclidean Geometry, 18th 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, 18th International Geometry Symposium in Honor of Prof. Dr. Sadik Keleş, July 12-13 2021, Inönü University, p.140.

Tatbiq qilish
Ilmiy ishlarda, ilmiy maqolalarda, algebra va geometriya darslarida, dissertatsiyalarda.