Intrinsic Geometry of Surfaces by A D Aleksandrov, V A Zalgaller, J M Danskin

By A D Aleksandrov, V A Zalgaller, J M Danskin

Publication via A D Aleksandrov, V A Zalgaller

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An iterated function system (IFS) in X is a quadruplet I (X,:F, V, h,), where: = • X is the underlying metric space, • :F is a set of contractions in X, • V is an alphabet of contraction labels, • h : V -+ :F is a labeling function, taking the letters of alphabet V to the contractions from :F. The labeling function h is extended to words and languages over V using the equations: h(at) 0 h(a2) h(ala2 ... an ) U h(w), h(L) 0 ••• 0 h(a n ), wEL where 0 denotes function composition, The attract or of an IFS I is the smallest nonempty set A eX, closed with respect to all transformations of :F.

The length of this lines is equal to the loudness (amplitude) of the explosion or so called etJent. The mathematical modellation of this system is quite easy. Thus we can write a small computer-programme to generate a sample with the specified behaviour which we can then listen to. 1 How to generate a Sample If we want to make something audible by digital computers we should firstly study how we can bring the computer to play a sound. A signal can be described as a timeseries (1) This timeseries S = r(l) will be translated by a digital-analog converter.

Hedlung: Endomorphisms and automorphisms of the shift dynamical systems, Math. Syst. E. Hutchinson-: Fractals and selfsimilarity, Indiana Math. , 30 (1981), 713-747 [JUE91] H. -O. Peitgen, D. E. Kummer: Uber die Ergiinzungssiitze zu den allgemeinen Reciprocitiitsgesetzen, J. reine. angew. Math. 44(1852), 93-146. [MAN82] B. H. , New York, 1982 [MAN85] B. Mandelbrot, Y. Gefen, A. Aharony, J. Peyriere: Fractals, their transfer matrices and their eigen-dimensional sequences, J. Phys. A: Math. , 18 (1985), 335-354 [MAR84] O.

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