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«Algebraic and Geometric Methods of Analysis» International scientific conference «Algebraic and geometric methods of analysis» Book of abstracts May 31 - June 5, 2017 Odessa Ukraine http://imath.kiev.ua/~topology/conf/agma2017/ LIST OF TOPICS • Algebraic methods in geometry • Differential geometry in the large • Geometry and topology of differentiable manifolds • General and algebraic topology • Dynamical systems and their applications • Geometric problems in mathematical analysis • Geometric and topological methods in natural sciences • History and methodology of teaching in mathematics ORGANIZERS • The Ministry of Education and Science of Ukraine • Odesa National Academy of Food Technologies • The Institute of Mathematics of the National Academy of Sciences of Ukraine • Taras Shevchenko National University of Kyiv • The International Geometry Center PROGRAM COMMITTEE Chairman: Prishlyak A. Maksymenko S. Rahula M. (Kyiv, Ukraine) (Kyiv, Ukraine) (Tartu, Estonia) Balan V. Matsumoto K. Sabitov I. (Bucharest, Romania) (Yamagata, Japan) (Moscow, Russia) Banakh T. Mashkov O. Savchenko A. (Lviv, Ukraine) (Kyiv, Ukraine) (Kherson, Ukraine) Fedchenko Yu. Mykytyuk I. Sergeeva А. (Odesa, Ukraine) (Lviv, Ukraine) (Odesa, Ukraine) Fomenko A. Milka A. Strikha M. (Moscow, Russia) (Kharkiv, Ukraine) (Kyiv, Ukraine) Fomenko V. Mikesh J. Shvets V. (Taganrog, Russia) (Olomouc, Czech Republic) (Odesa, Ukraine) Glushkov A. Mormul P. Shelekhov A. (Odesa, Ukraine) (Warsaw, Poland) (Tver, Russia) Haddad М. Moskaliuk S. Shurygin V. (Wadi al-Nasara, Syria) (Wien, Austri) (Kazan, Russia) Herega A. Panzhenskiy V. Vlasenko I. (Odesa, Ukraine) (Penza, Russia) (Kyiv, Ukraine) Khruslov E. Pastur L. Zadorozhnyj V. (Kharkiv, Ukraine) (Kharkiv, Ukraine) (Odesa, Ukraine) Kirichenko V. Plachta L. Zarichnyi M. (Moscow, Russia) (Krakov, Poland) (Lviv, Ukraine) Kirillov V. Pokas S. Zelinskiy Y. (Odesa, Ukraine) (Odesa, Ukraine) (Kyiv, Ukraine) Konovenko N. Polulyakh E. (Odesa, Ukraine) (Kyiv, Ukraine) 1 2 ADMINISTRATIVE COMMITTEE • Egorov B., chairman, rector of the ONAFT; • Povarova N., deputy chairman, Pro-rector for scientific work of the ONAFT; • Mardar M., Pro-rector for scientific-pedagogical work and international communications of the ONAFT; • Fedosov S., Director of the International Cooperation Center of the ONAFT; • Volkov V., Director of the Educational Research Institute of Mechanics, Automation and Com- puter Systems named after P. M. Platonov; • Bukaros A., Dean of the Faculty of automation, mechatronics and robotics ORGANIZING COMMITEE Kirillov V. Hladysh B. Maksymenko S. Konovenko N. Nuzhnaya N. Khudenko N. Fedchenko Yu. Osadchuk E. Cherevko E. 3 Про біортогональні сітки ліній пари поверхонь Л. Л. Безкоровайна (Одеський національний університет імені І. І. Мечникова) E-mail: [email protected] Нехай дві поверхні S і S∗ тривимірного евклідового простору задані векторно - параметричними рівняннями. Припустимо, що встановлено відображення цих поверхонь (за допомогою рівнянь, що однозначно виражають криволінійні координати однієї поверхні через координати іншої поверхні). Віднесемо ці поверхні до спільних координат u; v. Тоді, за теоремою Тіссо [1], існує одна і лише одна система ліній, що є ортогональною і на поверхні S, і на поверхні S∗, яка визначається рівнянням Edu + F dv F du + Gdv = 0; (1) E0du + F 0dv F 0du + G0dv де E; F; G і E0;F 0;G0 - коефіцієнти перших квадратичних форм заданих поверхонь. Розгорнемо рівняння (1) до вигляду (EF 0 − E0F )du2 + (EG0 − E0G)dudv + (FG0 − F 0G)dv2 = 0: (2) Отже, диференціальне рівняння (2) визначає дійсну ортогональну регулярну сітку ліній, спільну для двох різних поверхонь. Такі сітки ліній в даній роботі називаються біортогональними. Дослі- джуються властивості біортогональних сіток для деяких пар поверхонь, віднесених до спільних координат. Насамперед виникає необхідність у явному вираженні сіткового тензора для біортогональної сітки. Має місце Теорема 1. Сітковий тензор біортогональної сітки ліній для пари поверхонь можна подати у вигляді γδ Jαβ = (cαγaβδ + cβγaαδ) g ; де cαγ - дискримінантний тензор поверхні S з компонентами p − − 2 c11 = c22 = 0; c12 = c21 = g; g = g11g22 g12: Для того, щоб здобути тензор біортогональної сітки, передусім необхідно від гаусових позначень 1 2 геометричних величин в рівнянні (2) перейти до індексних позначень u = x ; v = x ;E = g11;F = 0 0 0 g12;G = g22;E = a11;F = a12;G = a22: Безпосередньою перевіркою можна переконатися, що рівняння (2) набуває інваріантного вигляду α β Jαβdx dx = 0 або, що те ж саме, γδ α β (cαγaβδ + cβγaαδ) g dx dx = 0: Доведено, що біортогональна сітка є спільною як для пар паралельних поверхонь, так і для сімейства паралельних поверхонь у цілому. При цьому вона збігається з сіткою ліній кривини. Знайдено рівняння біортогональної сітки деформованої поверхні S та здеформованої поверхні S∗ за умови, що їх радіус-вектори пов’язані рівністю (t ! 0) r∗(x1; x2; t) = r(x1; x2) + tU(x1; x2): Встановлено, що у випадку ареальної нескінченно малої деформації біортогональна сітка поверхонь S і S∗ збігається з сіткою головних ліній деформації. Ряд властивостей цієї сітки при ареальній нескінченно малій деформації сформульовано в [2]. ЛІТЕРАТУРА [1] M. Tissot. Memoire sur la representation des surfaces et les projections des eartes geographiques.Paris, 1881, p.337. [2] Л. Л. Безкоровайна. Головна сітка нескіченно малих деформацій із стаціонарною площею.Тези доповідей 5-ої між- народної конференції з геометрії та топології. Пам’яті О. В. Погорєлова, Черкаси 2003, ст. 12-13. 4 Про ізотопність функцій леми Морса Бондар О. П. (КЛА НАУ, Кропивницький) E-mail: [email protected] В. В. Шарко [1] дав означення ізотопних функцій Морса, за допомогою яких вивчались вла- стивості многовидів, на яких було задано ці функції. З метою розширення можливостей вивчення зв’язку топології многовидів із заданими на них функціями було узагальнено поняття ізотопних функцій Морса, а саме, введено означення ізотопних функцій, [2]. Це означення, зокрема, дозво- лило побудувати шлях, [1], що поєднує функції леми Морса, показавши їх ізотопність. n ! 1 n Твердження 1. Нехай f0 : R R — диференційовна функція і x0 = (x0; : : : ; x0 ) — невироджена критична точка цієї функції. Тоді можна вказати координатні подання ізотопії H : U0 × [0; k] ! Uk ! [0; k]; k 2 N; n околу U0 точки x0 на деякий окіл Uk початку координат 0 простору R та ізотопії h : V0 × [0; k] ! Vk ! [0; k]; k 2 N; околу V0 точки f0(x0) на окіл Vk початку координат 0 простору R, такі, що диференційовні відображення ⊂ Hk Iso0(Uk);H0 = idUk ; ⊂ + hk Iso0 (Vk); h0 = idVk ; 1 n i і для всіх точок y = (y ; : : : ; y ) 2 Uk, для яких y (x0) = 0; i = 1; : : : ; n, функція 1 2 λ 2 λ+1 2 n 2 fk = −(y ) − ::: − (y ) + (y ) + ::: + (y ) буде локально ізотопною функції f0: ◦ ◦ −1 fk = hk f0 Hk ; тобто можна вказати такі локальні ізотопні перетворення систем координат, що функція f0 буде локально диференційовно ізотопна функції fk. Існування локальної системи координат (y1; : : : ; yn), в якій справедлива тотожність 1 n 1 n f0(x ; : : : ; x ) = f0(x0) + fk(y ; : : : ; y ); є лемою Морса. Координатне подання необхідних ізотопій полягає у побудованій послідовності елементарних ізотопій i H : Ui−1 × [0; 1] ! Ui × [0; 1]; i = 1; : : : ; k; Ui ⊆ Ui−1; i = 2; : : : ; k i ⊂ i 2 Ht Iso0(Uk);H0 = idUi−1 ; для всіх t [0; 1]; і елементарних ізотопій i h : Vi−1 × [0; 1] ! Vi × [0; 1]; i = 1; : : : ; k; Vi ⊆ Vi−1; i = 2; : : : ; k; i ⊂ + i 2 ht Iso0 (Vk); h0 = idVi−1 ; для всіх t [0; 1]; для яких кінцеве відображення попередньої елементарної ізотопії є початковим — тотожним — від- ображенням наступної, а композиції відповідних елементарних ізотопій є потрібними ізотопіями Hk і hk. ЛІТЕРАТУРА [1] В. В. Шарко. Функции на многообразиях (алгебраические и топологические аспекты). Киев: Наук. думка, 1990. [2] О. П. Бондарь. Об определении изотопных функций. Тези доповідей міжнародної конференції ”Геометрія в Одесі- 2015, (2015), С.67. 5 Інфінітезимальні деформації кругового циліндра зі стаціонарною рімановою зв’язністю Вашпанова Н.В. (Одеська національна академія харчових технологій, Одеса, Україна) E-mail: [email protected] Потапенко І.В. (Одеський національний університет ім.І.І. Мечникова, Одеса, Україна) E-mail: [email protected] У теперішній час нерідко можна спостерігати застосування кругового циліндра як в техніці, так і в медицині. Саме від технічного стану магістральних труб (мають форму циліндра) в першу чергу залежить стабільне транспортування газу, нафти та різних нафтопродуктів. Особливо це стосується тих ділянок, де труби деформуються під деяким зовнішнім навантаженням. Відомо [1], що течія крові у великих кровоносних судинах характеризується відносно слабким впливом реологічних властивостей разом із сильним впливом механічних характеристик судиної стінки, в якості якої розглядають гіперпружну ізотропну трубку, яка в початковий момент часу має циліндричну форму, а потім деформується з часом. У роботі [2] задача про існування інфінітезимальної деформації певного класу поверхонь оберта- ння з фіксованою рімановою зв’язністю зведена до дослідження і розв’язування диференціального рівняння другого порядку в частинних похідних відносно невідомої функції '(x1; x2) класу C2 (в лініях кривини): @2' @2' @' b + b + a + b' = F; 11 (@x2)2 22 (@x1)2 @x1 де bij(i = 1; 2) - коефіцієнти другої квадратичної форми поверхні, a; b; F - відомі функції. Розглянемо нескінченно малі деформації кругового циліндра зі стаціонарною рімановою зв’язністю. Нехай круговий циліндр заданий рівнянням r = fR cos v; R sin v; ug; де R - радіус основи циліндра. Слід відзначити, що циліндр не належить до класу поверхонь, що розглядалися
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