Download PDF by Goodwin: Dynamic system identification. Experiment design and data

By Goodwin

ISBN-10: 0122897501

ISBN-13: 9780122897504

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The representation U is said to be primary if the center of @ ( U ) consists only of the homotheties of E, and i m t y p i c if it is primary and if there exists a nontrivial irreducible subrepresentation of U . 36 XXI COMPACT LIE GROUPS AND SEMISIMPLE LIE GROUPS (a) For U to be primary, it is necessary and sufficient that the center of a(U) contain no orthogonal projection other than 0 and 1,. ) (b) For U to be isotypic, it is necessary and sufficient that U should be equivalent to a (finite or infinite) Hilbert sum of equivalent irreducible representations.

Show that for each s E C,, either x,(s) = 0 or Ix,(s)l = np. ) (c) Suppose that C, # {e}and that the number of elements in C, is a prime power. Show that there exists p # po in R such that 1 x,(s) I = np for s E C,. ) Deduce that G cannot be a noncommutative simple group. 7. Let G be a finite group of order #q", where p and q are prime numbers. Then G is solvable (Burnside's theorem). (Argue by induction on the order of G, by considering a Sylow p-subgroup of G, which has a nontrivial center.? ) 4.

7). 3), it follows that the V(f) . x already span a dense subspace of E as x runs through E andfruns through the space %,-(G) of continuous functions on G. 8), and afortiori 0 i. c Ij i. P c i, j. P V(j-1- 1 i 1 clP)V(m:p)) 5 N, i. j , P f- 1 1ciy'mip) i. j . p 5 E. ),we have V(mi5))= V(u,,)V(mir)),and therefore the c$)V(m)p)) x belongs to the sum of the E,. This shows that the vector i. j . p sum of the E, is dense in E. (ii) That each E, is stable under V follows from the fact that the up belong to the center of the algebra M,(G).

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Dynamic system identification. Experiment design and data analysis by Goodwin

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