By Bloch S.J., et al. (eds.)

ISBN-10: 0821850555

ISBN-13: 9780821850558

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26, 735–747 (2005) 29. : Approximating smallest enclosing balls with applications to machine learning. Int. J. Comput. Geometry Appl. 19(5), 389–414 (2009) 30. : Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27, 919–940 (1990) 31. : Analysis of distance/similarity measures for diffusion tensor imaging. , Weickert, J. ) Visualization and Processing of Tensor Fields: Advances and Perspectives, pp. 113–136. Springer, Berlin (2009) 1 Supremum/Infimum and Nonlinear Averaging 33 32.

24) is proved in [12]. This could be useful in other contexts. Let G = G(A1 , . . , Am ). Then for any point C of P(n) we have m δ22 (G, C) ≤ j=1 1 δ 2 (A j , C) − δ22 (A j , G) . 24). The main argument in [12] is based on the following inequality. Let In be the set of all ordered n-tuples ( j1 , . . , jn ) with jk ∈ {1, 2, . . , m}. This is a set with m n elements. For each element of this set we define, as before, averages Sn ( j1 , . . , jn ; A) inductively as follows: S1 ( j; A) = A j for all j ∈ I1 , 46 R.

1. We observe that for both positive and negative values of P a monotonous convergence to a pair of matrices is achieved. , κ10 (A1 , A2 ) = 41 , and κ−10 (A1 , A2 ) = 14 10 , 01 which is a reasonable value from a numerical viewpoint for the order P of the CHMM. In fact, we can compare these estimations with those obtained by the Minkowski matrix mean of order P, which can be naturally defined as 1/P N ν (A) = P AiP . 85 22 J. Angulo which is coherent with the theoretical results known for scalar values (see Proposition 3), in the sense that the convergence to the maximum/minimum with respect to P is faster (and numerically more stable) for the CHMM than for the Minkowski power mean extended to matrices.

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Applications of algebraic K-theory to algebraic geometry and number theory, Part 1 by Bloch S.J., et al. (eds.)


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