Плюриполярность графиков квазианалитических функций

Аннотация

OF DISSERTATION OF THE DOCTOR OF PHILOSOPHY (PhD) ON PHYSICAL AND MATHEMATICAL SCIENCES URGENCH-2018 25 The theme of dissertation of doctor of philosophy (PhD) on physical and mathematical sciences was registered at the Supreme Attestation Commission at the Cabinet of Ministers of the Republic of Uzbekistan under number B2018.1.PhD/FM49. Dissertation has been done at Urgench State University. The abstract of the dissertation is posted in three languages (uzbek, russian, english (resume)) on the website (www.ik-mat.urdu.uz) and the “Ziyonet” Information and educational portal (www.ziyonet.uz). Scientific supervisor: Imomkulov Sevdiyor Akramovich Doctor of Physical and Mathematical Sciences Official opponents: Khudaybergenov Gulmirza Doctor of Physical and Mathematical Sciences, professor Prenov Barlikbay Barakbaevich Candidate of Physical and Mathematical Sciences Leading organization: Samarkand State University Defense will take place «____» _____________2018 at _____ at the meeting of Scientific Council number PhD.28.12.2017.FM.55.01 at Urgench State University. (Address: 14, Kh. Olimjan str.,Urgench city, 220100, Uzbekistan, Phone: (+99862)-224-66-11, fax: (+99862)-224-67-00, e-mail: ik-mat.urdu@umail.uz Dissertation is possible to review in Information-resource center at Urgench State University (is registered №_______) (Address: 14, Kh. Olimjan str., Urgench city, 220100, Uzbekistan, Phone: (+99862)-224-66-11, fax: (+99862)-224-67-00, e-mail: ursubox@gmail.com ) Abstract of dissertation sent out on «____» _______________2018 year B. I. Abdullaev Chairman of scientific council on award of scientific degrees, D.F.-M.S. A. A. Atamuratov Scientific secretary of scientific council on award of scientific degrees, C.F.-M.S. A. B. Yakhshimuratov Chairman of scientific Seminar under Scientific Council on award of scientific degrees, D.F.-M.S. 26 INTRODUCTION (abstract of PhD thesis) The aim of research work is to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar; to prove pluripolarity of graphs of quasianalytic functions in the sense of Gonchar; to show pluripolarity of graphs of algebroid functions; to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Denjoy; to define a class of quasiharmonic functions and prove a theorem about thinness of graphs. The object of the research work. The dissertation is devoted to the study of pluripolarity of graphs of quasianalytic functions and uniqueness property of quasianaalytic functions. Scientific novelty of the research work. All the results obtained in the dissertation are new and consist of the following: - Uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar is proved; - Pluripolarity of graphs of quasianalytic functions in the sense of Gonchar is proved; - Pluripolarity of graphs of algebroid functions is proved; - Pluripolarity of graphs of quasianalytic functions of several variables in the sense of Denjoy is proved; - Pluripolarity of graphs of functions from Gevrey class is proved; - The class of quasiharmonic functions is defined and the theorem about thinnes of graphs of functions from this class is proved. Implementation of the research results. The results on pluripolarity of graphs of quasianalytic functions in sense of Gonchar and Denjou and uniqueness property were used in research projects: F4-FA-0-16928: “Property of pluriregularity in pluripotential theory and finite-difference equations with modified sources”. Using methods and results gives opportunity to describe pluripolar envelopes of sets and characterizing geometric properties of pluriregular compacts in complex spaces. The results on pluripolarity of the graphs of quasianalytic functions of the dissertation were used in the implementation of the grant DMS - 1401316 of the National Scientific Foundation of the USA. Results of the research were presented in two workshops held in California State University. The structure and volume of the thesis. The dissertation work consists: introduction, four chapters, conclusion and bibliography. The volume of the dissertation is 85 pages. 27 ЭЪЛОН ҚИЛИНГАН ИШЛАР РЎЙҲАТИ СПИСОК ОПУБЛИКОВАННЫХ РАБОТ LIST OF PUBLISHED WORKS I бўлим (1 часть; part 1) 1. Imomkulov S.A., Ibragimov Z.Sh. The class of quasiharmonic functions admitting best approximations by harmonic polynomials// Узбекский математический журнал, № 2, (2016), С. 72-78. (01.00.00; № 6). 2. Imomkulov S.A., Ibragimov Z.Sh. Pluripolarity of graphs of Denjoy quasianaalytic functions of several variables// Доклады АН РУз. № 2, (2014), С. 9-12. (01.00.00; № 7). 3. Imomkulov S.A., Ibragimov Z.Sh. Pluripolarity of graphs of Denjoy quasianaalytic functions of several variables// Journal of Analysis and Mathematical Physics, Volume 5, Issue 2, (2013), pp. 161-170. (№ 11, Springer. IF=0.632) 4. Имомкулов С.А., Ибрагимов З.Ш. Плюриполярность графико