Saint-Donat Toroidal Embeddings I by G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat

By G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat

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3), we obtain f (x ) = Ex[f (XTr )] = E[J(eSNrx + te rms independent of x )). 5) First , suppose that v > O. Differen t iation of Eq. 2, we get 111'11~ e- r 111'11< 00 . Letting here r -+ + 00 , we conclude that 111'11 = 0 and so If v = 0 then , differenti ating Eq . 6) we get 1" (x ) = E[e2SNr 1" (eSNr x f = const. + .. )), so t hat 111"11~ e- 2 r ll1"11< 00 . 9There is a slight pr oblem if X ,,; probability , so m ay be igno red . 7) but t his only ha pp ens wit h zero 46 L. BOGACHEV, G. DERFEL, S.

P. , An absorption probabi lity problem, J . Math . An al. Appl. 9 (1964 ) , 384-393 . K . GRINTSEVICHYUS, On the continuity of the distribution of a sum of dep endent variables con n ected with independent walks on lines, (Russian) Teor. Vero yatn. i Primenen. 1 9 (1974) , 163-168; (English translation) Theory P roba b. Appl. 19 (1974) , 163-168. [14J A . ISERLES, On th e generalized pantograph fun ctional-differential equation, European J . Appl. Math. 4 (1993) , 1- 38. K. LIU, On pantograph integro-differential equations, J.

2To be more precise, a certain vect or analog of Eq. 1). 29 30 L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J. OCKENDON in a mathematical model for the dynamics of an overhead current collection system on an electric locomotive (with the physically relevant value q < 1) . At about the same time, a systematic analysis of solutions to the pantograph equation was started by Fox et at. [11], where various analytical, perturbation, and numerical techniques were discussed at length (for both q < 1 and q > 1).

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