Science > Physics > Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao
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Science > Physics |
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"OsherD" |
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28 Feb 2006 11:43:21 PM |
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Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao |
From Osher Doctorow
Bau-Sen Du of Institute of Mathematics Academica Sinica, Taipei Taiwan
in "On the nature of chaos," math.DS/0602585 v1 26 Feb 2006, redefins
chaos in terms of both infinity and 0 based on newly discovered
properties of the shift map (his Theorem 1).
The 0 property is lim inf d(f^n(x), f^n(y)) = 0 and the infinity
property is lim sup d(f^n(x), f^n(y)) > = delta for fixed y and any x
and any nonempty open set V where y is in V and f maps infinite compac
metric space (X, d) to itself. Here lim inf and lim sup are understood
to be for n --> infinity. The continuous map f is called chaotic
under the above circumstances.
Du has refereed publications including J. Diff. Equ. Appl. 11 (2005).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao |
01 Mar 2006 12:12:17 AM |
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From Osher Doctorow
On page 4 of Bau-Sen Du is Theorem 3, which has 3 parts, part (c) of
which says that F_u is chaotic on LAMBDA_u for any u > = 4 where F_u(x)
is:
1) F_u(x) = ux(1 - x)
and LAMBDA_u is the intersection from n = 0 to infinity of the inverse
images of F_u^(-n([0, 1]) for u > 4 and F_u(x) = [0, 1] for u = 4.
The right hand side of (1) is the right hand side of the logistic
differential equation and is the 0-infinity expression which I code as
0-1 in this thread provided that (in the case of (1)) y = x.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao |
01 Mar 2006 12:26:01 AM |
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From Osher Doctorow
Theorem 1 of Bao-Sen Du says that for any given countably infinite
subset X of SIGMA_2, the lim sup and lim inf conditions of 1 and 0
respectively hold on some dense uncountable invariant 1-scrambled set
of transitive points which are in SIGMA_2 for every x in X and y in Y
where now f is replaced by sigma where sigma is the shift map defined
by:
1) sigma(bob1...) = (b1b2...), bi = 0 or 1
and SIGMA_2 is the set of b's equal to bob1... with bi equal to 0, 1,
that is to say it is the set of such sigma's. A delta-scrambled set is
roughly defined by a > = delta and > = delta/2 condition where in the
second condition y is replaced by a periodic point p.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao |
01 Mar 2006 01:34:56 AM |
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From Osher Doctorow
From Bau-Sen Du's reference section, M. Thaler (reference 9) gives us
some clues as to where some of the other researchers are besides Du.
M. Thaler in that reference coauthored with S. Zeller, but Maximilian
Thaler coauthored with Jon Aaronson and Roland Zweimueller. Following
up on Jon Aaronson, we find 7 papers in Front for the Mathematics
ArXiv. The coauthors are:
1) Jon Aaronson, Tel Aviv U.
2) Tom Meyerovich, Tel Aviv U.
3) Mariusz Lemanczyk Nicolaus Copernicus U. Poland
4) H. Nakada, Keio U., Yokohama, Japan
5) O. Sarig, Penn State U. USA
6) Maximilian Thaler, U. Salzburg Institut fur Mathematik, Austria
7) Roland Zweimuller, Imperial College, London
Most of these authors specialize on topics related to Du's paper
including subshifts, measures, infinity, ergodic transformations, sets
of infinite measure, exchangeable measures, invariant measures, etc.
It's my impression that Taiwan, Israel, Japan, the U.K., and the USA
and Austria are way ahead of the other nations in Nonconformist
Physics-related Mathematics based on these references. It wouldn't
surprise me if Australia, Italy, South Korea, Denmark join this list
based on their past research in somewhat different directions.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao |
01 Mar 2006 01:46:18 AM |
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From Osher Doctorow
I should have mentioned that Poland is also way ahead of the other
nations.
Poland has produced some surprising people historically, including the
mathematical logician Lukaciewicz and the Romantic Music Creative
Genius Chopin. Some of the toughest Israelis have a Polish background
or ancestry. Pope John Paul II was Polish.
Osher Doctorow
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| User: "Khadiija Faatima" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 6: Taiwan on 0-Infinity Chao |
01 Mar 2006 02:58:33 PM |
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"OsherD" <> wrote in message
news:1141199178.931045.233000@v46g2000cwv.googlegroups.com...
From Osher Doctorow
I should have mentioned that Poland is also way ahead of the other
nations.
Poland has produced some surprising people historically, including the
mathematical logician Lukaciewicz and the Romantic Music Creative
Genius Chopin. Some of the toughest Israelis have a Polish background
or ancestry. Pope John Paul II was Polish.
Osher Doctorow
That is an Old Joke, "Is the Pope Polish?" Good one for you Kosher.
But you are intentionally leaving out the Huge contribution to Mathematics
and Physics by Arabs.
Ibrahim ibn Sinan was a grandson of Thabit ibn Qurra and studied geometry
and in particular tangents to circles. He also studied the apparent motion
of the Sun and the geometry of shadows. There is no doubt that had he not
died at the young age of thirty-eight, he would have achieved a degree of
fame for his mathematical works going even beyond the opinion of Sezgin (see
[5] and [6]) that he was:-
.... one of the most important mathematicians in the medieval Islamic world.
Perhaps his early death robbed him of the chance to make a contribution even
more important than that of his famous grandfather.
Ibrahim's most important work was on the quadrature of the parabola where he
introduced a method of integration more general than that of Archimedes. His
grandfather Thabit ibn Qurra had started to view integration in a different
way to Archimedes but Ibrahim realised that al-Mahani had made improvements
on what his father had achieved. To Ibrahim it was unacceptable that (see
for example [1]):-
.... al-Mahani's study should remain more advanced than my grandfather's
unless someone of our family can excel him.
Ibrahim is also considered the foremost Arab mathematician to treat
mathematical philosophy. He wrote (see for example [1]):-
I have found that contemporary geometers have neglected the method of
Apollonius in analysis and synthesis, as they have in most of the things I
have brought forward, and that they have limited themselves to analysis
alone in so restrictive a manner that they have led people to believe that
this analysis did not correspond to the synthesis effected.
We know of Ibrahim's works through his own work Letter on the description of
the notions Ibrahim derived in geometry and astronomy in which Ibrahim lists
his own works. This is one of seven treatises by Ibrahim given with full
Arabic text and English summaries in [2]. Among the works published in [2]
are On drawing the three conic sections in which Ibrahim give a pointwise
construction for the ellipse, the parabola and the hyperbola. Although based
on ideas due to Apollonius there are aspects of this work which illustrate
the changed point of view of Arabic mathematicians. For example Ibrahim uses
an arithmetical term to denote the product of two geometrical lines.
In On the measurement of the parabola Ibrahim ibn Sinan gives a beautiful
proof that the area of a segment of the parabola is four-thirds of the area
of the inscribed triangle. Another work is On the method of analysis and
synthesis, and the other procedures in geometrical problems which contains a
systematic exposition of analysis, synthesis and related subjects, with many
easy examples. This is in contrast to The selected problems in which 41
difficult geometrical problems are solved, usually by analysis only, without
a discussion of the number of solutions or conditions which make the
solutions possible.
On the motions of the sun is an astronomical work which discusses of the
motion of the solar apogee. It also provides a critical analysis of the
observations underlying Ptolemy's solar theory, and Ibrahim ibn Sinan
provides his own theory of the sun. The work On the astrolabe includes work
on map projections. Ibrahim proves in this work that the stereographic
projection maps circles which do not pass through the pole of projection
onto circles.
In fact geometric transformations figure a great deal in Ibrahim's works and
this interesting aspect is discussed in detail in [4]. Examples are given
which illustrate how Ibrahim applied an orthogonal compression to transform
a circle into an ellipse, and an oblique compression to map a hyperbola into
a second hyperbola. In a different work Ibrahim uses a transformation which
maps figures keeping invariant the ratio between their areas.
Ibrahim's contribution is summed up in [1] as follows:-
Considering both the problem of infinitesimal determinations and the history
of mathematical philosophy, it is obvious that the work of ibn Sinan is
important in showing how the Arab mathematicians pursued the mathematics
that they had inherited from the Hellenistic period and developed it with
independent minds. That is the dominant impression left by his work.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ibrahim.html
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