By Björn Gustafsson, Alexander Vasiliev

Our wisdom of gadgets of advanced and strength research has been improved lately by means of principles and structures of theoretical and mathematical physics, corresponding to quantum box thought, nonlinear hydrodynamics, fabric technology. those are a number of the subject matters of this refereed selection of papers, which grew out of the 1st convention of the ecu technological know-how starting place Networking Programme 'Harmonic and intricate research and purposes' held in Norway 2007.

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**Example text**

4). Therefore it gives a solution of Poincar´e–Steklov integral equation with parameter R3 (x). 9) we get the reconstruction rule u(x) = (2πi)−1 W1 (R3 (x) + i0) − W1 (R3 (x) − i0) , x ∈ [−1, 1]. 2. 7) in the ˆ \ {−1, 1, a1, a2 , a3 , a4 }. 3. Monodromy invariant. The following statement is proved by direct computation. 3. , D2 = D2 2 = D3 2 = 1) and (ii) conserve the quadratic form 3 3 Wk2 − δ J(W ) := Wj Ws . 11) j~~
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~~Special coordinate coverings of Riemann surfaces. Math. : Branched structures and aﬃne and projective bundles on Riemann surfaces. Trans. : Monodromy groups and linearly polymorphic functions. : On the periods of quadratic diﬀerentials. Russian Math. : The monodromy groups of Schwarzian equations on closed Riemann surfaces// Ann. of Math. : Poincar´e–Steklov integral equations and the Riemann monodromy problem. Funct. Anal. Appl. : PS-3 integral equations and projective structures on Riemann surfaces. ~~

The chain of equivalent transformations of PS-3 equation described here in a somewhat sketchy fashion is given in [10, 11] with more details. 1. 1) where Q(t) is the denominator in an irreducible representation of R(t) as the ratio of two polynomials; x1 (x) = x, x2 (x), x3 (x) – are all solutions (including multiple and inﬁnite) of the algebraic equation R3 (xs ) = R3 (x). 1) as a certain relationship for the Cauchy-type integral u(t) ˆ \ [−1, 1]. 1). For a known Φ(x), the eigenfunction u(t) may be recovered by the Sokhotskii-Plemelj formula: u(t) = (2πi)−1 [Φ(t + i0) − Φ(t − i0)] , t ∈ I.

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