By Stancho Dimiev, Kouei Sekigawa

This quantity provides the state-of-the-art contributions to the 7th foreign Workshop on advanced constructions and Vector Fields, which used to be equipped as a continuation of the excessive winning previous workshops on related study. the quantity contains works treating bold themes in differential geometry, mathematical physics and expertise resembling Bezier curves in house varieties, strength and catastrophy of a cleaning soap movie, computer-assisted experiences of logistic maps, and robotics.

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**Example text**

In the decomposition H'(M, O(End(e))) ~_ Hi(M, O(End°(C))) + HI(M, O) of the tangent space at D" E He(E, h), the holomorphic sectional curvature vanishes in the direction of Hi(M, 0). The question remains whether it is positive in the direction of Hl( M, O( End°( ED") )). Although we can compute the curvature of 2~I(E, h) explicitly using the Gauss equation for CR-submersions [8], the fact that the holomorphic sectional curvature is nonnegative can be understood intuitively from the general principle that the holomorphic curvature decreases with a holomorphic subbundle and increases with a quotient bundle.

REFERENCES 1. W. Casselman, Canonical eztensions of Harish-Chandra modules to representations of G, Preprint. 2. W. -S. I. Piatetski-Shapiro and P. Sarnak, Poincard series for SO(n, 1), In preparation. 35 3. R. S.. 4. V. Kuznetsov, Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture. Sums of Kloosterman sums~ Math. USSR Sbornik 39 (1981)~ 299-342. 5. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. I. (1965), 1-15. 6. R. Wallach, Asymptotic ezpansions of generalized matriz entries of representations of real reductive groups.

38 2 Deformation of Surfaces Infinitesimal and Punctured Compact Riemann Deformation Assume t h a t X, B are real analytic manifolds and that ,r : X --* B is a surjective, proper, smooth real analytic map. th. th. for two such charts the w-components depend hotomorphically on each other for fixed z-components. T h e atlas `4 makes every fiber Xt := ~r-i(t), t E B a compact complex submanifold of X . We assume that every X~ is a Riemann surface of genus p. th. ,s,~(t) are disjoint. 1 X is called a real analytic deformation of Riemanu surfaces of genus p with m distinguished points.