By F Defever; J M Morvan; et al (eds.)

This quantity offers a scientific and unified method of the research, identity and optimum regulate of continuous-time dynamical platforms through orthogonal polynomials (such as Legendre, Laguerre, Hermite, Tchebycheff, Jacobi and Gegenbauer) and through orthogonal services akin to sine-cosine, block-pulse, and Walsh. This booklet concentrates at the software of orthogonal polynomials in structures and keep an eye on and goals to set up the prevalence of orthogonal polynomials over different orthogonal services

**Read or Download Geometry and topology of submanifolds, IX : dedicated to Prof. Radu Rosca on the ocasion of his 90th birthday, Valenciennes, France, 26-27 March, Lyon, France, 17-18 May, Leuven, Belgium, 19-20 September, 1997 PDF**

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**Extra info for Geometry and topology of submanifolds, IX : dedicated to Prof. Radu Rosca on the ocasion of his 90th birthday, Valenciennes, France, 26-27 March, Lyon, France, 17-18 May, Leuven, Belgium, 19-20 September, 1997**

**Example text**

Let x, x* be a polar pair and n = 3. Then the following statements are equivalent: (i) (M, I) is curvature homogeneous and H\ = constant; 42 (ii) (M, I*) is curvature homogeneous and H* = constant. 5. Corollary. Let x,x* : M -t S " + 1 ( l ) be polar hypersurfaces of dimension n > 3. Then the following statements are equivalent; (i) x is umbilical; (ii) x" is umbilical; (iii) (M, I) has constant sectional curvature; (iv) (M, f ) has constant sectional curvature. 6. Proposition. Let x, x" be polar.

1. Definition. Let (M, g) be a semi-Riemannian manifold and $ a Codazzi tensor. If the eigenvalues of $ are constant on M, $ is called an isoparametric Codazzi tensor. 2. Theorem. , en. ,n} befixed;then we have the Cartan identity J2 (Xi-\j)-lIC(ei,ej) =0 3 for g{ej,ek) = 6ik. 3. Corollary. Let x : M -> M(«) be a hypersurface in a space form M(K) of constant curvature K. ,fir. Then, for i € { 1 , . . 5. Weitzenboeck formulas. We recall two different versions of a Weitzenboeck for mula for traceless Codazzi tensors; see [WEG], [SI-1], [LIU-S-W]; [KOHL] contains other Weitzenboeck formulas.

3. Curvature relations b e t w e e n x and x'. ,n. For a polar pair x,x* : M -*• S " + 1 ( l ) we have: (i) The metric I is flat if and only if the polarized metric I* is flat; (ii) let 0 < r < n — 1 and r even; then H* is g-intrinsic (iii) let 0 < r < n - 1 and r odd; then {H*)2 is g-intrinsic and Hr is g*-intrinsic; and {Hr)2 is g'-intrinsic; (iv) if n=2 or 3