A Solution of the Linear Matrix Equation by Double by Hitchcock F. L.

By Hitchcock F. L.

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Additional info for A Solution of the Linear Matrix Equation by Double Multiplication

Example text

Lemma. Let Yo, YI be two loops in 1[;* with base point Zoo We have PROOF. In fact, if ea = Zo and Co, CI are the respective liftings of Yo, Y1 starting at a, then co(2t) 0 ::;; t ::;; t coo:= { c (2t -1) + c (1)---- a "2::;; 1 t::;; 1 o 1 is a lifting ofYoYI starting at a. 26. Lemma.

0 if z E 0' and there are disjoint disks Bj centered at aj , non vanishing functions IjIj E ,ff(Bj ) and integers kj with the property that J(z) = (z - aj)kjljlj(z) in Bj { Show that aJ. dJ kj dz • + -dljlj In B\ {a·} . z - aj IjIj } J (ii) Given any 9 E 0)(0), we can find go such that 9 = go + gl + ... j) d - dg J = Conclude that for any 9 E = O. = 9 near aj 1\ dg 0)(0): f. n dJ -j' = L kjg(aj). n j~1 xd~ - y~X in 1R12\{O}. ). 1 = gj E IjIj -2' 14. Let fI 0)(0'), O. ) = Bj 1\ E = O. Compute (xdx + ydy) 1\ fl.

Moreover, if F is compact then

0 everywhere and Id(x) - d(Y)1 ::;; IIx - YII· Using Zorn's lemma we can now construct a maximal sequence of points Xm in IR n such that the balls B(x m, lod(xm)) are pairwise disjoint. d(xm)) = IRn. In fact, if Xo If. U B(xm, lod(xm}) then there must exist PROOF. m~l m~l an integer m such that B( X m , 110d (X m ») n B( X ») #- 0· o, 110d (xo Otherwise the sequence (xm)m would not be maximal.

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