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Atomic functions 40 years behind

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Vladimir
Logvinovich
Rvachev
and
ATOMIC FUNCTIONS
O. Kravchenko, A. Safin
Moscow, Russia, 2010
1971,
V.L. and V.A. Rvachev’s and Wolfgang Hilberg first papers
1971 - 1991, a research group of atomic functions appeared in Kharkov
1991, Wolfgang Hilberg opened a link between fractal Morse-Thue sequence and atomic functions
University of
Wisconsin-Madison
01
0110
01101001
... ...
(
)
(
)
1
k
i
y k x i
=
=
0
1 0
2 0
3 0
4 0
5 0
6 0
- 8
- 4
0
4
8
0
1 0
2 0
3 0
4 0
5 0
6 0
- 8
- 4
0
4
8
Morse-Thue Sequence
Axel Thue
Marston Morse
“ ” Cb1 c3 Cb8 c6 Cc3 b1 Cc6 b8.
“ ” Cg1 f3 Cg8 f6 Cf3
1
g1 Cf g.
0
6 8
−−−−
−−−−
Max Euwe and infinite chess game
Max Euwe
0 0 00 00 0
11 1 11 1 11
0
...
1. NC3 Nc6 2. CB8 Cb1 3. Nf3 Nf6 4. CG1 Cg8 5. Nf3 Nf6 6. CG1 Cg8 ....
МТ Sequence and Self-Organization
Victor Korotkikh
Papers and books about atomic functions
Todor Cooclev
Uwe Meyer-Baese
Jackie Shen
Tsuyoshi Yoneda
Willi Freeden
Gerald Edgar
Hrvoe Gotovac
R.-Q. Jia
Vladimir Rvachev
Victor Kravchenko
Interesting features of atomic functions.
Table. Comparison of wavelets (Cooclev’s paper)
Figure
. Logical “One” and “Zero” (Hilberg’s patent)
Figure. FDE solution (Yoneda’s paper)
Animation. FDE solution
Why Atomic Functions are Nonanalytic?
Masao Iri: Because of support fractal structure.
Why Atomic Functions are Nonanalytic?
Fabius example.
J. Fabius: 1965, A Probabilistic Example of a Nowhere Analytic C-Infty Function.
Probability properties of atomic functions
Weighted sum of random variables
.
Further investigation:
•
Physical nature of atomic functions
•
Finite solutions of FDE with variable coefficients
•
Distributional solutions of FDE
THANK YOU FOR ATTENTION
Vladimir Logvinovich Rvachev
21.10.1926 – 26.04.2005
Автор
olevkcom
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