[Fwd: Various (thanks, computer, dynamical systems...)]

[Nguyen Tien Zung <tienzung@darboux.math.univ-montp2.fr>]

Message-ID: <33277FAB.2D32@math.univ-montp2.fr>
Date: Wed, 12 Mar 1997 20:32:25 -0800
From: Nguyen Tien Zung <tienzung@math.univ-montp2.fr>
Organization: CNRS & Universite Montpellier II
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Subject: Various  (thanks, computer, dynamical systems...)
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Cha`o ca'c ba'c

Ca'm o+n ba'c Tuan Pham chi? cho ca'c subscribe digest mode cua?
vnforum.
 
Ca'm o+n ba'c Thanh Tung gia?ng gia?i cho ddo^i ddie^`u ve^` fuzzy. 
Ba'c vie^'t la^`n sau thi` de^~ hie^?u la('m.

Tui gia` ro^`i ma` pha?i ddi ho.c computer, kho^? qua'. Co'
ba'c na`o co' bie^'t kie^'m dda^u ddu+o+.c online help
ve^` ma^'y thu+' sau: using UNIX, PERL, fuzzy logic hay ba^'t cu+' gi`
kha'c (tru+` html vi` tui dda~ kie^'m ddu+o+.c ru`i) thi` chi? cho
vo+'i, xin dda ta.

Ba'c AiViet o+i, tui co' the^? du`ng sai chu+~ khi vie^'t independent,
dda'ng nhe~ ba'c cu+' thay chu+~ kha'c va`o ma` ddo.c (orthogonal ?).
Inpedendent cu?a to^i kho^ng co' nghia~ la` kho^ng giao nhau ddu+o+.c
ba'c a. . A` to^i  hie^?u ba'c ddi.nh no'i "small is beautiful" ddu'ng
kho^ng? Nhu+ng ma` ca'i ca^u gi` ddo' nhu+ "big is not just to be big"
cha('c cu~ng ddu'ng?

Ca'i vu. quaternion ma` ba'c HLVu~ ho?i dda~ co' ba'c Vie^.t tra? lo+`i
ro^`i. Chi? xin go'p the^m 1 chi tie^'t: ca'i spin ddu+o+.c ddi.nh
nghia~
ve^` ma(.t toa'n du+.c tre^n clifford algebras va` spin groups. Ti`nh
co+` thi` clifford algebra la.i co' components tu+o+ng ddu+o+ng vo+'i
quaternion field (khi chie^`u tha^'p, khi chie^`u cao nhie^`u khi cu~ng
la` dda(?ng ca^'u vo+'i some algebras over quaternion field). Ca'i su+.
ti`nh co+` tre^n cu~ng kho^ng ti`nh co+` la('m vi` so^' "primitive
objects" trong toa'n kho^ng nhie^`u la('m ne^n de^~ co' su+. tru`ng
ho+.p
Xin lo^~i ma^'y ca^u tre^n vie^'t cha('c ho+i fuzzy.

Co`n dda^y la` symbolic dynamics vie^'t cho ba'c AiViet va` nhu+~ng ai
quan ta^m (ne^'u vie^'t sai dda^u xin ca'c ba'c su+?a cho) :
La^'y 1 ta^.p hu+u~ ha. S = {1,..., n} ro^`i cho 1 ddo^. ddo sa'c xua^'t
tre^n S.  (p_1,..., p_n, sum of p_i = 1). La^'y S^N (N= set of natural
numbers) = the set of infinite series (k_1, k_2,...) where each k_i
has one of the value 1,...n. Now the Bernoulli shift is simply the shift

(k_1, k_2, k_3,...)  ->  (k_2,k_3,...)

(throw away the first element). It is a map from S^N onto itself, and is
a (discrete) dynamical system (a symbolic one - the elements of S are
simbols). 

Important: Such symbolic systems can actually be found as subsystems
(say of an attractor) of smooths dynamical systems on manifolds. 
(a simple model is the very ubiquitous "Smale's horseshoe", 
wich leads to an actractor which is topologically a Cantor set, and a
Cantor set is equivalent to {1,2}^N ).

S^N is also a measured space (it inherits the measure from S).
The (measure theoretic) 
entropy (defined by Kolmogorov, similar to that of Shannon) of the
above Bernoulli shift is
sum p_i ln (1/p_i). It is obvious that if two dynamical systems are
measure-theoretically equivalent then they have the same entropy. 
Orstein (?) proved the inverse for Bernoulli shifts: two Bernoulli
shifts
with the same entropy are measure-theoretically equivalent.

You may find that Bernoulli shifts are also Markov processes.
(there is something called ergodic theory: it is = dynamical systems +
probability theory).

So long

Zung