By Björn Gustafsson, Alexander Vasiliev

ISBN-10: 3764399058

ISBN-13: 9783764399054

Our wisdom of items of complicated and capability research has been stronger lately by means of principles and structures of theoretical and mathematical physics, resembling quantum box concept, nonlinear hydrodynamics, fabric technological know-how. those are the various issues of this refereed selection of papers, which grew out of the 1st convention of the ecu technological know-how beginning Networking Programme 'Harmonic and complicated research and purposes' held in Norway 2007.

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**Extra resources for Analysis and Mathematical Physics **

**Example text**

Branched covering of a sphere. At the moment we do not know which of the three boundary ovals of the pants P(R3 ) contains the critical points of p(y). Therefore we introduce the ‘nicknames’ {1, 2, 3} for the set of colors {r, g, b} so that the critical points will be on the oval number 3. 1 allows us to glue two discs U± 3 to the latter boundary. 1 ﬁlls in the remaining two holes with two discs U1 and U2 . Positive integers arising in those constructions are denoted by d± 3 , d1 , d2 respectively.

With removed segment −ε2 [h1 , h2 ] (resp. −ε[h1 , h2 ]), 0 < h1 < h2 < ∞. As in the previous case those pants may be modiﬁed by sewing in several annuli α, α. The scheme of cutting and gluing is shown in Fig. 3 _ α _ α _ α α α * α α Figure 2. The scheme for sewing pants P1 (λ, h1 , h2 |3, 2). Asterisk is the critical point of p(y). α α α α * * _ α α * _ α * α _ α _ α _ α Figure 3. The scheme for sewing pants P2 (λ, h1 , h2 |4, 1) (left); and P3 (λ, h1 , h2 |1, 3) (right). Asterisks are the critical points of the mapping p(y).

Vector W (y) is holomorphic and bounded in the pants P(R3 ) as all three points xs , s = 1, 2, 3, remain in the holomorphicity domain of the function Φ(x). We claim that the boundary values of the vector W (y) are related via constant matrices: W (y + i0) = D∗ W (y − i0), when y ∈ {slot∗ }. 7) The matrix D∗ assigned to the “green” [a3 , a4 ], “blue” [a1 , a2 ], and “red” [−1, 1] slot respectively is 0 D2 := 0 1 0 1 1 0 0 0 ; 0 D3 := 1 0 1 0 0 0 0 1 ; D := −1 δ δ 0 1 0 0 0 1 . 6) is a solution of a certain Riemann monodromy problem.

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