By Bailin Hao
This ebook is a monograph on chaos in dissipative platforms written for these operating within the actual sciences. Emphasis is on symbolic description of the dynamics and diverse features of the attractors, and written from the view-point of sensible functions with out going into formal mathematical rigour. the writer used user-friendly arithmetic and calculus, and trusted actual instinct at any time when attainable. mammoth consciousness is paid to numerical innovations within the examine of chaos. a part of the e-book is predicated at the courses of chinese language researchers, together with these of the author's collaborators.
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Extra resources for Elementary symbolic dynamics and chaos in dissipative systems
Evaluation and Self-reflection In the pre-composition phase of composing it is not yet the piece, which takes the centre, but it is about probing and exhausting the respective selected material as completely as possible with respect to its intrinsic potential. Conceptual procedures, which are done at the desk, are as equally important as the practical trying out of sounds—either by myself if I have the instrument available or with friends that are musicians. This experimentation goes along with the intellectual considerations and often continues into the actual composition process if it is about unusual and selected sounds.
1 without number 4 as partial sum resp. without major third was developing a catalogue of criteria for (3), those concerning tonality being the crucial ones. When enumerating larger frame intervals the number of possible chords rapidly increases. Hereby many chords with large partial intervals were occurring which didn’t fit Nachtmann’s needs. So we restricted to a maximal interior interval of a major seventh, the first additional constraint: (3a) Define minimal and maximal interior intervals (finally minor second—but this one only if beneath a major second—and major seventh).
The technical procedures, which I utilise, are of course not fully invented by me but the result of self-observation, listening and reading of other music and reading of theoretical literature; the essays of Pierre Boulez, in particular the ones from the 50s and 60s [4–7], have strongly influenced me therein, maybe even more than his music; this is true similarly for the considerations of Stockhausen about musical time from the 50s, which I still consider fundamental. Much of the technical procedures I have adopted, others I have rejected and finally some I have indeed myself invented.