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By Eschenburg J. H., Tribuzy R.

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ZnJ Ln fnzn 39 Walks with Unbounded Jumps Finally, as K(z, u)F(z, u) = R(z, u), we obtain F(z u) , - u b(l - u) -fIb (u-u o) <=0< + zu b - zu b(l - u)P(u) 1 Setting u = 1 and u = 0 gives formulae (4) and decomposition of the rightmost part of (8) gives (6). (8) (5) and a partial fraction 0 The "kernel method" has been part of the folklore of combinatorialists for some time and is related to the what is known as ''the quadratic method" in enumeration of planar maps [10]. Earlier references (see [17] Ex.

22] Elisa Pergola, Renzo Pinzani, and Simone Rinaldi. Towards an algebra of succession rules. In Mathematics and computer science (Versailles, 2000), pages 141- 152, Basel, 2000. Birkha,user. [23] Elisa Pergola and Robert A. Sulanke. Schroder triangles, paths, and parallelogram polyominoes. 7, 1998. [24] Jim Pitman. Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electronic Journal of Probability, 4:no. 11 , 33 pp. (electronic), 1999. [25] J.

Ub(Z)) , (10) where the expressions involve all the small branches uo, ... ,Ub of the characteristic curve (3). Generally, the generating function Wk of paths terminating at altitude k is, for -00 < k < b, (11) and for -a < k < +00, Wk(z) = -z where V1, ... ,Va Vj z d L vo(z)k+1 = k dz L Vj(z) ' (Z) a )=1 ) ( a -k ) , (12) )=1 are the large branches. ) The proof of an identity similar to (10) for walks with bounded jumps is given in [4] and holds verbatim for walks with unbounded jumps: Consider a bridge and let m (with m :<:; 0) be the minimal altitude of any vertex.

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### (1,1)-geodesic maps into grassmann manifolds by Eschenburg J. H., Tribuzy R.

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