By Gardner M.

ISBN-10: 1716721881

ISBN-13: 9781716721885

Gardner M. Fractal tune, hypercards and more.. mathematical recreations from Sci.Am. (Freeman, 1992)(ISBN 01716721880)(400dpi)(T)(334s)_MPop_

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Example text

I n s t e a d o f ( * ) one has . ) be an (LFf-sDace, F a B a i r e space a l i n e a r napping w i t h c l o s e d graph i n ExF, which i s onto. Then f restricted to E P i s open f o r sow p o s i t i v e i n t e g e r p. 31. L e t us t r y a d i f f e r e n t approach which i s s u i t a b l e f o r g e n e r a l i z a t i o n s . ) u(An: n=l,2,. ) . ) i s o f f i r s t cateqory. f o r n o t a t i o n . It s u f f i c e s t o Take x(O(A) and U a n o n - v o i d open x - -nghb. ). UAAn i s o f second c a t e g o r y .

Then t h e r e e x i s t s b>O 1 such t h a t y+bx(R and y - b x C R and t h e r e f o r e 2 b ~ ~ s - ~ U + s - ~ U VC fso- r each s . tained i n Since V i s balanced we a r r i v e t o a c o n t r a d i c t i o n f o r a l l t h o s e s s a t i s f y i n q 2bs 71 1. CHAPTER I 15 A------------------i s a b s o r b i n g i n E: l e t y be a v e c t o r i n E and s e t L:=sp(y). Since U i s a 0-nghb i n E, L A U i s a B a i r e space and hence t h e r e e x i s t s a c e r t a i n D, a r e a l b)O and a complex number z such t h a t (1) ay(A f o r every a w i t h la-zlLb P On t h e o t h e r hand, t h e f u n c t i o n e x p ( i t ) i s u n i f o r m l y continuous on O , L t , L Z x and t h e r e f o r e t h e r e e x i s t s q;r/p such t h a t (2) /exp(it) - exp(ir)l 5 b/21zl if I t - r l c Z n l q Suppose n q q .

L e t ( z ( n ) :n=1,2,. ) be a seauence o f balanced, open 0-nghbs i n E such t h a t z ( n ) n f o r each n . L e t Q:E-+F be t h e canodoes n o t b e l o n g t o Un and Un+l+Un+lCUn n i c a l s u r j e c t i o n o n t o F:=E/N, S e t t i n g Vn:=Q(Un) N b e i n g t h e c l o s e d subspace A(Un:n=l,2,. ). f o r each n, F i s a separable i n f i n i t e - d i m e n s i o n a l B a i r e space ( 1 . 2 . )=(0) and Vn+l+V t h e r e e x i s t s a t o p o l o g y t on F such t h a t ( F , t ) cVn f o r each n.

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Fractal music, hypercards and more.. mathematical recreations from Sci.Am by Gardner M.


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