By Björn Gustafsson, Alexander Vasiliev

Our wisdom of gadgets of complicated and strength research has been more desirable lately by way of principles and buildings of theoretical and mathematical physics, resembling quantum box idea, nonlinear hydrodynamics, fabric technological know-how. those are many of the issues of this refereed number of papers, which grew out of the 1st convention of the ecu technology starting place Networking Programme 'Harmonic and complicated research and functions' held in Norway 2007.

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**Example text**

Now the notations U± b , Ur , Ug , db , dr , dg have the obvious meaning. 3. The Image of the Pants. 4. 5. The image p(P(R3 )) of the pants lies in the intersection of annuli α and α ¯ – see Fig. 10b). Proof. We refer to the four sectors: C \ ε±1 R as to ‘top’, ‘down’, ‘left’ and ‘right’. It is a matter of notation to say that the disc U+ b is mapped to the ‘top’ and ‘left’ sectors while the disc U− is mapped to the ‘down’ and ‘right’ sectors. b The disc Ug covers either the ‘top’ or the ‘left’ sector and both are covered + − by the disc U+ b .

3. Intersection of circles. 1. The circle C does not intersect the two other circles ε±1 R. the spectral parameter 1 < λ < 2 when the projective structure p(y) branch point is inside the pants. ˆ Proof. We know that the point 0 lies in the intersection of two of our circles: εR 2ˆ −1 p (0)} of the pre-images of this point (counting the and ε R. The total number {˜ multiplicities) is N and cannot be less than db + dg – the number of pre-images on the blue and green boundary components of the pants.

10a). 44 A. Bogatyrev (∂M)∗ ✬ ∗ * ✩ * (a) U+U− ✫∗ M C (b) ✪ Figure 10. (a) Filling in the hole bounded by (∂M)∗ (b) The shaded area is the image of P(R3 ). 2. 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.