Диссертация (1103157), страница 25
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Ïðåäëîæåíèå 5.2 äîêàçàíî.Äëÿ äîêàçàòåëüñòâà ïðåäëîæåíèÿ 5.3 íàì ïîíàäîáèòñÿ îäíî âñïîìîãàòåëüíîå óòâåðæäåíèå.Ëåììà 5.4. Ïóñòü E > 0 çàôèêñèðîâàíî. Ïðåäïîëîæèì, ÷òî u(λ, λ0 , E),(λ, λ0 ) ∈ T 2 , êîìïëåêñíîçíà÷íàÿ ôóíêöèÿ, èíòåãðèðóåìàÿ ïî ìåðå |dλ||dλ0 |è óäîâëåòâîðÿþùàÿ ñîîòíîøåíèþu(λ, λ0 , E) = u(−λ0 , −λ, E),(λ, λ0 ) ∈ T 2 .(5.112)√Îïðåäåëèì ôóíêöèþ g(p, E), p = (p1 , p2 ) ∈ R2 , |p| ≤ 2 E , ôîðìóëîé√gE Re(λ − λ0 ),√E Im(λ − λ0 ), E= u(λ, λ0 , E),(5.113)ãäå (λ, λ0 ) ∈ T 2 . Òîãäà ñïðàâåäëèâî ðàâåíñòâîZ√|p|≤2 EEe−ipx g(p, E) dp =2Z 0 λλ1 |dλ| |dλ0 |,u(λ, λ0 , z, E)−02i λλ (5.114)T2ãäå x = (x1 , x2 ) ∈ R2 , z = x1 +ix2 , à u(λ, λ0 , z, E) îïðåäåëÿåòñÿ â ñîîòâåòñòâèèñ ôîðìóëîé (5.104).151Ôîðìóëà (5.114) ïîëó÷àåòñÿ â ðåçóëüòàòå ñëåäóþùåé çàìåíû ïåðåìåííûõèíòåãðèðîâàíèÿ:√√E(cos φ − cos φ0 ),√√0p2 = E Im(λ − λ ) = E(sin φ − sin φ0 ),p1 =0E Re(λ − λ ) =(5.115)0ãäå λ = eiφ , λ0 = eiφ .
Ïåðåéä¼ì ê äîêàçàòåëüñòâó ïðåäëîæåíèÿ 5.3.Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 5.3. Ïóñòü λ, λ0 ∈ T îïðåäåëåíû ôîðìóëîé (5.10).Èç ôîðìóë (5.12a) è (5.12b) âûòåêàþò ñëåäóþùèå ñîîòíîøåíèÿ:√E λ + λ−1 + λ0 + λ0−1 ,√2(k2 + l2 ) = −i E λ − λ−1 + λ0 − λ0−1 ,√√k1 ± ik2 = Eλ±1 , l1 ± il2 = Eλ0±1 ,√k1 + l2 ± i(k2 + l2 ) = E λ±1 + λ0±1 ,2(k1 + l1 ) =(5.116)|k + l|2 = E|λ + λ0 |2 ,ãäå (k, l) ∈ ME . Èñïîëüçóÿ ëåììó 5.4 è ôîðìóëû (5.45)(5.46b), (5.48)(5.49b)è 5.116, ìû ïîëó÷àåì ñëåäóþùèå ðàâåíñòâà: 0 1λλ0ediv,0 (λ, λ0 , z, E)Aappr,j 2i λ0 − λ |dλ| |dλ |,T2 Z0 Eλλ1div,0div,0 |dλ| |dλ0 |,Vappr(x, E) =Veappr(λ, λ0 , z, E)−022i λλ 2T Z0 1 λEλ±,0±,00e |dλ| |dλ0 |,Aappr,1 (x, E) =A−appr,1 (λ, λ , z, E)022i λλ T2 Z0 1 λEλ±,0±,00 |dλ| |dλ0 |,Vappr (x, E) =Veappr (λ, λ , z, E)−022i λλ EAdiv,0(x,E)=appr,j2Z(5.117)T2ediv,0 , Ve div,0 èãäå x ∈ R2 , j = 1, 2, z îïðåäåëåíî â ôîðìóëå (5.9), à ôóíêöèè Aapprappr,j±,0±,0eeAappr,j , Vappr çàäàþòñÿ ñëåäóþùèì îáðàçîì:elin (λ, λ0 , z, E) f11e√+,A(λ, λ , z, E) =λ−1 + λ0−1 λ + λ04 Eelin (λ, λ0 , z, E) f11div,00e√A−,appr,2 (λ, λ , z, E) =λ−1 + λ0−1 λ + λ04i Ediv,0appr,10(5.118a)(5.118b)152f lin (λ, λ0 , z, E) + f lin (−λ0 , −λ, z, E)div,00eVappr (λ, λ , z, E) =,21 felin (λ, λ0 , z, E)e±,0 (λ, λ0 , z, E) = √A,appr,12 E λ±1 + λ0±1λ0±1 f lin (λ, λ0 , z, E) + λ±1 f lin (−λ0 , −λ, z, E)±,00e,Vappr (λ, λ , z, E) =λ±1 + λ0±1(5.118c)(5.118d)(5.118e)ãäå f lin (λ, λ0 , z, E) îïðåäåëåíî â ñîîòâåòñòâèè ñ ôîðìóëîé (5.104) èfelin (λ, λ0 , z, E) = f lin (λ, λ0 , z, E) − f lin (−λ0 , −λ, z, E).Èñïîëüçóÿ ðàâåíñòâà (5.118a)(5.118e), ìû ïðåäñòàâëÿåì êàæäûé èç èíòåãðàëîâ â ôîðìóëå (5.117) êàê ñóììó èíòåãðàëà, ñîäåðæàùåãî f lin (λ, λ0 , z, E) èèíòåãðàëà, ñîäåðæàùåãî f lin (−λ0 , −λ, z, E).Äåëàÿ çàìåíó ïåðåìåííûõ (λ, λ0 ) → (−λ0 , −λ) â êàæäîì èç ïîëó÷èâøèõñÿc± ,èíòåãðàëîâ, ñîäåðæàùåì f lin (−λ0 , −λ, z, E), è ó÷èòûâàÿ ôîðìóëó (5.48) äëÿ A2ìû ïîëó÷àåì ôîðìóëû (5.56a) è (5.56b).Ïðåäëîæåíèå 5.3 äîêàçàíî.153Çàêëþ÷åíèåÎñíîâíûå ðåçóëüòàòû, ïîëó÷åííûå â äèññåðòàöèè, ìîãóò áûòü êðàòêî ñôîðìóëèðîâàíû ñëåäóþùèì îáðàçîì:1.
Ïîëó÷åíû òåîðåìû õàðàêòåðèçàöèè äëÿ èíòåãðàëüíûõ ïðåîáðàçîâàíèéòèïà Ðàäîíà, âîçíèêàþùèõ, â ÷àñòíîñòè, â îáîáù¼ííîé ìîäåëè ÕàóòåêêåðàÈîõàíñåíà ïðè ïðåäïîëîæåíèè, ÷òî ëèáî ðåñóðñû áëèçêè ê âçàèìíî äîïîëíèòåëüíûì, ëèáî ýëàñòè÷íîñòü èõ çàìåùåíèÿ ïîñòîÿííà.2. Ïîëó÷åíû ÿâíûå ôîðìóëû îáðàùåíèÿ è êðèòåðèè îáðàòèìîñòè äëÿ èíòåãðàëüíûõ îïåðàòîðîâ òèïà Ðàäîíà.3. Ïîëó÷åíû òåîðåìû åäèíñòâåííîñòè äëÿ îáðàòíîé çàäà÷è ÄèðèõëåÍåéìàíà,âîçíèêàþùåé â ìîäåëè àêóñòè÷åñêîé òîìîãðàôèè äâèæóùåéñÿ æèäêîñòè.Ïðèâåäåíû ïðèìåðû íååäèíñòâåííîñòè.4. Ïîëó÷åíû ôîðìóëû è óðàâíåíèÿ, ïîçâîëÿþùèå ñâåñòè çàäà÷ó ÄèðèõëåÍåéìàíà ê îáðàòíîé çàäà÷å ðàññåÿíèÿ ïðè ôèêñèðîâàííîé ýíåðãèè äëÿêàëèáðîâî÷íî-êîâàðèàíòíîãî îïåðàòîðà Øð¼äèíãåðà, ÷àñòíûìè ñëó÷àÿìèêîòîðîãî ÿâëÿþòñÿ îïåðàòîð Øð¼äèíãåðà âî âíåøíåì ïîëå ßíãàÌèëëñàè îïåðàòîð Øð¼äèíãåðà â ìàãíèòíîì ïîëå.5.
Ïîëó÷åí àëãîðèòì ðåøåíèÿ îáðàòíîé çàäà÷è ðàññåÿíèÿ ïðè ôèêñèðîâàííîé ýíåðãèè äëÿ äâóìåðíîãî ñêàëÿðíîãî êàëèáðîâî÷íî-êîâàðèàíòíîãî îïåðàòîðà Øð¼äèíãåðà, ÷àñòíûìè ñëó÷àÿìè êîòîðîãî ÿâëÿþòñÿ îïåðàòîð Øð¼äèíãåðà â ìàãíèòíîì ïîëå è îïåðàòîð, îïèñûâàþùèé ïîãëîùàþùóþ äâèæóùóþñÿ æèäêîñòü.Ìîæíî âûäåëèòü ñëåäóþùèå îñíîâíûå ïåðñïåêòèâû äàëüíåéøåé ðàçðàáîòêèòåìû äèññåðòàöèè:1.
Ïðèìåíèòü îáîáù¼ííóþ ìîäåëü ÕàóòåêêåðàÈîõàíñåíà ê èññëåäîâàíèþïðîèçâîäñòâà â ðåàëüíûõ îòðàñëÿõ, ôóíêöèîíèðóþùèõ â óñëîâèÿõ ãëîáàëèçàöèè. Èññëåäîâàòü îáúÿñíèòåëüíûé ïîòåíöèàë ìîäåëè, å¼ ñëàáûå èñèëüíûå ñòîðîíû. Ìîæíî îæèäàòü, ÷òî êàê è êëàññè÷åñêàÿ ìîäåëü ÕàóòåêêåðàÈîõàíñåíà, îáîáù¼ííàÿ ìîäåëü õîðîøî ïðèñïîñîáëåíà äëÿ ó÷¼òàèçìåíåíèé â ïðîèçâîäñòâå â ðåçóëüòàòå íàó÷íî-òåõíè÷åñêîãî ïðîãðåññà.1542. Àêòóàëüíîé ïðîáëåìîé ðîññèéñêîé ýêîíîìèêè ÿâëÿåòñÿ ïðîáëåìà ñîñòàâëåíèÿ ìåæîòðàñëåâîãî áàëàíñà â óñëîâèÿõ ñèëüíîãî çàìåùåíèÿ íà ìèêðîóðîâíå. Îäíîé èç ïåðñïåêòèâ ðàçâèòèÿ òåìû äèññåðòàöèè ÿâëÿåòñÿ èññëåäîâàíèå ïîòåíöèàëà ôîðìàëèçìà ðàñïðåäåëåíèÿ ìîùíîñòåé ïî òåõíîëîãèÿì â óñëîâèÿõ çàìåùåíèÿ äëÿ ðåøåíèÿ ýòîé çàäà÷è.3.
Àäàïòèðîâàòü àëãîðèòì àêóñòè÷åñêîé òîìîãðàôèè äëÿ ñëó÷àÿ íåïîëíûõäàííûõ. Ýòà çàäà÷à î÷åíü âàæíà ñ òî÷êè çðåíèÿ ïðèëîæåíèé: â òîìîãðàôèè îêåàíà êîëè÷åñòâî äåòåêòîðîâ ÿâëÿåòñÿ îòíîñèòåëüíî ìàëûì ïîñðàâíåíèþ ñ êîëè÷åñòâîì ðàáî÷èõ ÷àñòîò. Âîçíèêàåò âîïðîñ î âîçìîæíîñòè èñïîëüçîâàòü áîëüøåå ÷èñëî ÷àñòîò, ÷òîáû êîìïåíñèðîâàòü ìàëîå÷èñëî äåòåêòîðîâ.4.  ðåàëüíûõ òîìîãðàôè÷åñêèõ ýêñïåðèìåíòàõ, èñïîëüçóþùèõ ýëåìåíòàðíûå ÷àñòèöû, ïðîùå èçìåðÿòü íå ïîëíóþ àìïëèòóäó ðàññåÿíèÿ, à ëèøü å¼ìîäóëü, òåñíî ñâÿçàííûé ñ âåðîÿòíîñòüþ ðàññåÿíèÿ ÷àñòèö â òîì èëè èíîìíàïðàâëåíèè. Âàæíîé àêòóàëüíîé çàäà÷åé ÿâëÿåòñÿ îáîáùåíèå àëãîðèòìà ðåøåíèÿ îáðàòíîé çàäà÷è ðàññåÿíèÿ äëÿ êàëèáðîâî÷íî-êîâàðèàíòíîãîîïåðàòîðà Øð¼äèíãåðà íà ñëó÷àé, êîãäà èçâåñòíà ëèøü àáñîëþòíàÿ âåëè÷èíà àìïëèòóäû ðàññåÿíèÿ (îòñóòñòâóåò èíôîðìàöèÿ î ôàçå).
Ñîèñêàòåëåì áûëè íà÷àòà ðàáîòà íàä ýòîé òåìîé âî âðåìÿ ñòàæèðîâêè â óíèâåðñèòåò ãîðîäà üòòèíãåí îñåíüþ 2015 ãîäà ïîä ðóêîâîäñòâîì ïðîôåññîðà T.Hohage. Ïî ðåçóëüòàòàì ñòàæèðîâêè áûë ïîäãîòîâëåí ïðåïðèíò [6], åù¼îäíà ñòàòüÿ íàõîäèòñÿ â ïðîöåññå íàïèñàíèÿ.5. Îáîáùåíèå àëãîðèòìà ðåøåíèÿ îáðàòíîé çàäà÷è ÄèðèõëåÍåéìàíà äëÿêàëèáðîâî÷íî-êîâàðèàíòíîãî îïåðàòîðà Øð¼äèíãåðà íà ñëó÷àé îáëàñòåéíåòðèâèàëüíîé ãåîìåòðèè.  ýòîì íàïðàâëåíèè ñîèñêàòåëåì îïóáëèêîâàíàðàáîòà [5], ñâÿçàííàÿ ñ âîññòàíîâëåíèåì ðèìàíîâîé ïîâåðõíîñòè ïî îïåðàòîðó ÄèðèõëåÍåéìàíà äëÿ îïåðàòîðà Ëàïëàñà.155Ñïèñîê îáîçíà÷åíèéÌíîæåñòâàZd+ = {(α1 , . .
. , αd ) | αj ∈ Z, αj ≥ 0, j = 1, . . . , d}Rd+ = {(x1 , . . . , xd ) | xj > 0, j = 1, . . . , d}Rd+ = {(x1 , . . . , xd ) | xj ≥ 0, j = 1, . . . , d}Mn (C) ìíîæåñòâî êîìïëåêñíûõ ìàòðèö ðàçìåðà n × nGLn (C) ìíîæåñòâî íåâûðîæäåííûõ êîìïëåêñíûõ ìàòðèö ðàçìåðà n × nÌíîæåñòâî {a} èç îäíîãî ýëåìåíòà a îáîçíà÷àåòñÿ a (áåç ñêîáîê)Ïðîñòðàíñòâà ôóíêöèéLp (Rd ) ïðîñòðàíñòâî âåùåñòâåííîçíà÷íûõ èçìåðèìûõ ôóíêöèé íà Rd ñ èíòåãðèðóåìîé p-îé ñòåïåíüþLpr,c (Rd+ ) ìíîæåñòâî èçìåðèìûõ ôóíêöèé íà Rd+ ñ êîíå÷íîé íîðìîé k · kr,c , ñì.ôîðìóëó (1.15)H p (D), H p (∂D), ãäå D ⊂ Rd ñòàíäàðòíûå ñîáîëåâñêèå ïðîñòðàíñòâàCcN,σ (Rd+ ) ñì. ôîðìóëó (2.1)j(D, Mn (C)), ãäå D ⊂ Rd ìíîæåñòâî j -ðàç íåïðåðûâíî äèôôåðåíöèðóåCcompìûõ Mn (C)-çíà÷íûõ ôóíêöèé â Rd ñ íîñèòåëåì â Dj,αjCcomp(D, Mn (C)), ãäå D ⊂ Rd ìíîæåñòâî ôóíêöèé èç Ccomp(D, Mn (C)), îáëàäàþùèõ ïîêîìïîíåíòíî ã¼ëüäåð-íåïðåðûâíûìè ñ ïîêàçàòåëåì α ÷àñòíûìèïðîèçâîäíûìè ïîðÿäêà j∞L (Rd , Mn (C)) ìíîæåñòâî Mn (C)-çíà÷íûõ ïîêîìïîíåíòíî ñóùåñòâåííî îãðàíè÷åííûõ ôóíêöèé â RdW 1,∞ (Rn , Mn (C)) ìíîæåñòâî Mn (C)-çíà÷íûõ ôóíêöèé íà Rd , ïðèíàäëåæàùèõ âìåñòå ñ îáîáù¼ííûìè ïðîèçâîäíûìè ïåðâîãî ïîðÿäêà ïðîñòðàíñòâóL∞ (Rd , Mn (C))Îïåðàòîðû è ïðî÷èå îáîçíà÷åíèÿqp (x) = q(p1 x1 , .
. . , pn xn ), ãäå p = (p1 , . . . , pn ), x = (x1 , . . . , xn )ab = ab11 · · · abnn , ãäå a = (a1 , . . . , an ), b = (b1 , . . . , bn )I = (1, . . . , 1)|α| = α1 + · · · + αd , ãäå α = (α1 , . . . , αd ) ∈ Zd+p|x| = x21 + · · · + x2d , ãäå x = (x1 , . . . , xd ) ∈ Rd∂2∂2∆ = ∂x2 + · · · + ∂x2 îïåðàòîð Ëàïëàñà1d∇ = ∂x∂ 1 , .
. . , ∂x∂ d ãðàäèåíòsupp f íîñèòåëü ôóíêöèè f156Ñïèñîê ëèòåðàòóðû[1] Ablowitz M. J., Yaarov D. B., Fokas A. S. On the inverse scattering trans-form for the Kadomtsev-Petviashvili equation // Stud. Appl. Math. ––1983. –– Vol. 69. –– P. 135–143.[2] Agaltsov A. D. On the reconstruction of parameters of a moving fluid fromthe Dirichlet-to-Neumann map // Eurasian Journal of Mathematical andComputer Applications. –– Vol. 4, no. 1.
–– P. 4–11.[3] Agaltsov A. D. Finding scattering data for a time-harmonic wave equationwith first order perturbation from the Dirichlet-to-Neumann map // Journalof Inverse and Ill-Posed Problems. –– 2015. –– Vol. 23, no. 6. –– P. 627–645.[4] Agaltsov A. D. A global uniqueness result for acoustic tomography of movingfluid // Bulletin des Sciences Mathematiques.
–– 2015. –– Vol. 139, no. 8. ––P. 937–942.[5] Agaltsov A. D., Henkin G. M. Explicit reconstruction of Riemann surfacewith given boundary in complex projective space // The Journal of Geometric Analysis. –– 2015. –– Vol. 25, no. 4. –– P. 2450–2473.[6] Agaltsov A. D., Novikov R. G. Error estimates for phaseless inverse scattering in the Born approximation at high energies. ––http://arxiv.org/abs/1604.06555.[7] Agaltsov A. D., Novikov R. G. Riemann-Hilbert problem approach for twodimensional flow inverse scattering // J. Math. Phys. –– 2014.
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