User talk:Ramon Roca
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Contents 
A SET THEORY OF INFORMATION: MODES OF VIBRATION
bibliography: http://openwetware.org/wiki/Talk:Molecular_computing
E. Lubkin; Keeping the entropy of measurement: Szilard revisited. 
Reality is not subdivided

R. Landauer; 
There really is no software, in the strict sense of disembodied information, but only inactive and relatively static hardware. Thus, the handling of information is inevitably tied to the physical universe. Evolution and the origin of life can be viewed as an optimization process, in which fluctuations (e.g., mutations) take us from one metastable ecology, to a new one. We might, with equal justice, refer to the revolution of an electron around a hydrogen nucleus, or the rotation of a water wheel, as self organization. 
Erbium (Er) n=68 integer (atomic number protons) p=331 prime (atomic information quanta) Fourier phase analysis: 
[1972: Hugh Montgomery has just been introduced to Freeman Dyson] Montgomery: [the distribution of the zeros of the Riemann zeta function] It seems the twopoint correlations go as.... (turning to write on a nearby blackboard):
Dyson: Extraordinary! Do you realize that's the paircorrelation function for the eigenvalues of a random Hermitian matrix? It's also a model of the energy levels in a heavy nucleus [erbium166 (protons neutrons)].


NUMBER THEORETICS AND INFORMATION LEVELS
Levin and Chaitin definition of algorithmic entropy: A program is selfdelimiting it if needs no special symbol, other than the digits 0 and 1 of which it is composed, to marks its end; discrete objects other than binary strings, e.g., integers, or coarsegrained cells in a continous phase space, may be defined by indexing them by binary strings in a standard way (as Monte Carlo program describes distributions of macrostates).
Fundamental Theorem of Arithmetic: Every natural number is uniquely decomposable into a product of prime powers. Primes are the building blocks (factors) of the positive integers: The prime integers of an integer determine its properties.
Euler's function [cos θ + j sen θ = e^{jθ}] describes vibrations (waves) connecting geometry (trigonometric functions the integer values n, in real space) and algebra (exponential function the prime factors p, in reciprocal space) consider: n + n' = p .
Any arbitrary shape can be regarded as a locus of intersecting surfaces of nth order (generated in terms of Fourier descriptors). The total amount of information embodied in simple shapes is determined by the shape category, the number of surfaces and the form or order of the surfacesdefining equations (e.g., quadratic, cubic, etc) [tesis/biblio.html#Ayers (Ayers, 1994)]  [tesis/fig3_4.html see table] :
symmetry information embodied in simple shapes  H_{sym} (bits) 
rectangle  1 
square  2 
cone  10 
sphere  30 
Badii utilized the idea that chaos is made up of combinations of the periodic orbits. A 'primitive' is defined by two conditions: it is periodic (i.e., in the infinity sequence there are arbitrarily long repetitions of it) and it cannot be broken down to other primitives. The [hierarchical] tree [structure] is constituted by the primitives and their admissible combinations (the first level, the primitive combination; 2nd level, pairwise combination; nth level, nary combinations, and so on). [tesis/biblio.html#Kampis (Kampis, 1991)].
With Gödel numbering (by instance, using the original symbols as exponents of a prime factorization) it becomes possible to encode and decode state transformations to and from states directly. This procedure, which is itself algorithmic, ensures the existence of a dynamics without the use of any further information. All we need is a componentsystem which produces them by complexityincreasing procedures. [tesis/biblio.html#Kampis (Kampis, 1991)].
Golay codes are based upon prime numbers: Golay codes G23 (binary, [23,12,7]_{2}, 3correcting) and G11 (ternary, [11,6,5]_{3}, 2correcting) are perfect, systematic and linear. Theorem of Best: All perfects codes over any alphabet, with p≥3 and p≠6,8 are equivalent to Rep2(n) or G23. [tesis/biblio.html#Brunat (Brunat, 2001)]
This theorem implies two basic levels of information (2 and 23). Similarly, in music, it is possible to observe these same levels: first one, ternary, it would be formed by values 0, 1 and 2 and they correspond to silence, semitone and tone; second level depends on "musical temperament", so, considering temperated scale, it is formed by 12 notes. Another example results from comparing binary (2 elements, first level) and decimal (10 elements, second level) systems; in this case, adding a third level (20, 100, 1000), although related to logarithmic scales, it is usually done on an arbitrary and subjective manner.
Atomic periodicity (n protons of noble gases): 
 
Number of elements per period (Mendeleiev table): 
 
Types of electronic orbitals or (sub)shells: 
 

Reinterpreting theorem of Best, i.e., if G23 is a prototypical perfect code, it could be stablished the next periods for prime numbers:
period I:  1        2 
2 elements; n=2 ^{ } 
period II:  3  5  7  11  13  17  19  23  8 elements; n=10 
period III:  29  31  37  41  43  47  53  59  8 elements; n=18 
(because of "technical" reasons, it is considered 1 as the "first" prime, i.e., n=1)
First period (elements 1 and 2) defines the evenodd concepts, equivalent in signal transmition, to renormalized [1, 0, 1], or in musical terms to semitone and tone. Second period (8 elements, primes from 3 to 23: increment=20) defines consolidates the primity concept and is equivalent, for instance, to 2 musical scales, or to binary Golay code G23. Third period (8 elements, primes from 29 to 59: increment=30) determines the selforganizing character of the progression of the prime numbers: 29 is (almost 30) a "prime multiple" of 3.
With only 18 elements, it is incremented the "informative value" from a signal type 2 to another signal type 59 (almost 60). In music, considering equaltempered scales, it is possible to divide the octave into 59 intervals to approximate the frequency ratios from just intonation; so, it is also a good choice the standard division of the octave in 12 intervals, as 5 octaves of 12 notes (i.e., 60 notes is quite near of 59: just a semitone) permit to "construct a quasiperfect code" (as extended Golay codes G24 and G12 can be generators of perfect codes G23 and G11).
From an "evolutionary" point of view (as increment of information complexity), it must be considered the three first elements as critical (and concentrical):
1 defines unity and appears as "opposition" to 0 (algebraically, a "trivial" vectorial subspace);
2 defines pair/even, as first distinctive of symmetry (1 1);
3 defines "primity" concept in its orthodox sense, considered as another distinctive of (a/self)symmetry (2 1), conditioning asymmetrical interactions.
The most important detail is just the increase of complexity: as the number of elements grows, "the growing itself becomes faster". Essentially, this increasing is due to main basic relationships of the three first elements (2 1, 3 2), that, altogether with the fourth element (5, just a consequence of the previous relationships), conform the first "extended period". Another approach is to consider "3 2" interaction as the combination of ternary and binary systems, a basic subset of complexity in terms of information.
prime (p)  1  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59 
integer (n)  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18 
Analogous to Golay code G11, parameters [11,6] (respectively, dimension and length) could be extended to other concrete/discrete pairs [p,n], being p prime and n its integer (the number of protons); the third parameter of Golay codes, related to error correction, is named "Hamming minimum distance" [could be equivalent to the number of neutrons?]. Then, Carbon is represented as a "ternary code" with parameters C [11,6]; first fourth elements are defined as the next pairs (vectors, matrices, codes): Hydrogen, H [1,1]; Helium, He [2,2]; Lithium, Li [3,3]; and Beryllium, Be [5,4].
'
From these premises:
a. It is represented, by FFT phases of generated inverse sine waves, "p orbitals" (p spheres of information) of the first 118 elements and others, so some elements, like Er (atomic number: n=68; p= 331), Rn (86,439) and predicted Ac' (121,659), "reorganize their orbitals into N beams".
b. Simple arithmetical rules are used for different molecules (from water to ATP; resp., p=19 and p=541). My initial question was if hydrogen and ATP could be compared "quantitatively" at entropy and information levels; within "primity scale", H and ATP information values differe in a "logarithmic order 1:101".
My proposal implies that different levels of information are organized in a complex system within the basis of prime numbers, as a result from considering as "Golay codes" the atomic elements, and then, distributed by "Medeleiev periods ".
COMPLEX SYSTEMS: MOLECULAR COMPUTATION
In biology, different molecules interact by means of their composing elements and determine diverse informative values by their respective surfaces and volumes. For example, H^{ } and Ca^{2 } are two types of signals at the subcellular level; at protein level, 20 aminoacids are classified depending on their characteristics (analysable through Markov models or equivalence matrices). So, it seems logical to consider each atom as a characteristic signal type, quantizable, normalized by "Gödel numbers". Evolutionary, increment of complexity appears because of the periodical progression (Mendeleiev) of prime numbers, and because of interactions that appear due to increasing possibilities and elements.
Prime integers are hierarchical strange attractors (characteristic or typical tones) of complex selforganized (harmonized) systems.
Results and other conjectures
Atom prime chart:
n=19                  n=36 
n=37                  n=54 
n=55                                n=86 
n=87                                n=118 
n=119                                                      n=172 
n=173                                                      n=226 
n=227                                                                                n=306 
n=307                                                                                n=386 
Digital signal processing
Phase analysis (Lissajous Plot graphs ***) of prime generated tones:
 Inverse sine (pure tone: fundamental, no harmonics);
 Audio signals in Hz.
Non prime: Control data and Dynamics
Prime phases of atomic elements (p orbitals): [original size] & [reduced view]
"Nobles":
"Erbiumlike":
More prime phases: part I (661 to 1427); part II (1429 to 7919); part III (7927 to 21059).
Dynamics, transitions and polarization
Realtime data:
Gibbs Energy = Enthalpy Entropy (or Information)

wav files (5 seconds each prime wave)

Slow motion rendering (only phases): [Unregistered] Screen Movie Studio (MandSoft) 
avi files (1000 frames per second)

"Erbiumlike" series (local strong attractors: the Riemann series)
erbiumlike prime  nearby note  
1429  F#6  
1321  E6  
1213  D#6  
1103  G#6  
991  B5  
881  A5  
769  G5  
661  E5  
547  C#5  
439  A4  
331  E4  
223  A3  
109  A2  
73  D2  
37  D1  
erbiumlike.avi  1'22'' (97.4Mb) 
Uncoupled and coupled terms are related to even and odd (1/2 or N/N 1), determining the L (levo) or R (dextro) character {antimatter/matter, to be or not?}.
Volume is a function of distribution of massenergy quanta
Integers: Atomic numbers (electrons and protons) represent mass/energy quanta. Primes: Every prime is associated to its integer, and represents a standarization of atomic volume/surface relationship, defining atomic information quanta.
Prime series of atomic elements
Periods K to Z20 [n/p (log p  log n)]
"Prime interactions"
molecule  fract[ion]al
interaction  enharmonic number (prime)

H_{2} 
1:1 
2 
CH_{4} 
11:1 
15 
NH_{3} 
13:1 
16 
NH_{4}^{ } 
13:1 
17* 
H_{2}O 
1:17 
19 
N_{2} 
13:13 
26 
CO 
11:17 
28 
CH_{5}N 
11:1:13  29 
O_{2} 
17:17 
34 
C_{3}H_{8} 
11:1 
41 
N_{2}O , C_{2}H_{4}O 

43 
CO_{2} 
11:17 
45 
NaOH 
29:17:1 
47 
Acetic (C_{2}H_{4}O_{2}) 
11:1:17 
60 
ClOH 
53:17:1  71 
C_{3}H_{6}O_{2} 
 73 
Gly 

74 
SO_{2}H_{2} 
47:17:1 
83 
Ala 

87 
C_{6}H_{6}O , CaC_{2} 

89 
PO_{3}H_{3} 
43:17:1  97 
Ca(OH)_{2} 
67:17:1  103 
Ser 

104 
Guanine (C_{5}H_{5}N_{5}O) 

105 
Pro 

111 
Val 

113 
Thr 

117 
Thymine (C_{5}H_{6}N_{2}O_{2}) 
11:1:13:17 
121 
Adenine (C_{5}H_{5}N_{5}) 
11:1:13 
125 
Leu 

126 
Orn 

127 
Asn 

129 
Asp 

132 
Cys 

134 
Lys 

140 
Gln 

142 
Arg 

143 
Glu 

145 
His 

148 
C_{5}H_{10}O_{5} 

150 
Phe 

157 
Met 

160 
olivine (Mg_{2}SiO_{4}) 
31:41:17 
171 
Cl_{2}Ca 
53:67  173 
Tyr 

174 
Glucose (C_{6}H_{12}O_{6}) 

180 
Trp 

193 
Adenosine (C_{10}H_{13}N_{5}O_{4}) 

256 
olivine (Fe_{2}SiO_{4}) 
97:41:17 
303 
chromite (Cr_{2}O_{4}Fe) 
83:17:97 
331 
Fe_{3}O_{4} 
97:17 
359 
3Fe 4H_{2}O = Fe_{3}O_{4} 4H_{2} 
[97]_{3}::[19]_{4} == [359]::[2]_{4} 
367 
2Fe 3Cr 
[97]_{2}::[83]_{3} 
443 
3Fe 2Cr 
[97]_{3}::[83]_{2} 
457 
mFe nNi   
serpentine (Mg_{3}Si_{2}O_{5}  2H_{2}O Fe Ni) 
[253]::[19]_{2}:: [97]::[103] 
491 
Porphyirine (C_{34}H_{34}N_{4}O_{4}) 

528 
ATP (C_{10}H_{16}N_{5}O_{13}P_{3}) 
11:1:13:17:43 
541 
(CrO_{4})_{3}Fe_{2} 

647 
Lecitine (C_{42}H_{82}NPO_{8}) 

736 
Atoms vs. molecules

NOTES
Lissajous figure
[from wims]
[from "Collins English Dictionary", 1984]
CODING THEORY
Ternary Golay code G11.
Extended ternary Golay code G12.
"Perfect eerror correcting code" Theorem (TietÃ¤vÃ¤inen and Van Lint, 1971)
and proof tools (sphere packing condition & Lloyd's theorem). From "Ten milestones in the history of source coding theory":
Discovery of LempelZiv codes. The wellknown universal noiseless source coding technique due to Jacob Ziv and Abraham Lempel was announced in 1977 [26], although the method is based upon a notion of string complexity that had been proposed by these two authors in a paper the year before. With probability one, a stationary ergodic finitealphabet source generates a sequence which, when encoded using the LempelZiv algorithm, yields a compression rate equal to the entropy rate, asymptotically as the number of source samples goes to infinity. The LempelZiv algorithm is the most important noiseless source coding technique in the entire history of source coding. A spate of papers has been devoted to the theoretical and practical aspects of LempelZiv coding. On the theoretical side, perhaps the most significant of these is the recent paper by Ornstein and Weiss [18].
Digital Signal Processing: LWZ Compression
The spectrum of Riemannium
Brian Hayes (American Scientist JulyAugust, 2003; vol. 91 (4), 296:300)
Prime numbers not so random? (Phillip Ball, 2003).
Surprising connections between number theory and physics (M. Watkins, 2004).
M. Wolf, "1/f noise in the distribution of prime numbers", Physica A 241 (1997), 493499.
Data
(audible spectrum: 16.4  21096 Hz; ~ 10 scales)
Standard La (A4) is usually tuned at 438440 Hz Dynamics: Unstable/Instable & Hyperstable

FREQUENCIES AND WAVELENGTHS FOR EQUALTEMPERED SCALE
from C0 to D#8 / Eb8

Major constituents (and fraction of total mass) of a heavy star, at the end of its evolution, just prior to a supernova explosion  (from The Natural Selection of the Chemical Elements: ''''The Environment and Life's Chemistry, by R. J. P. Williams & J. R. R. Frausto Da Silva): 
~40%  H He 
~20%  He 
~20%  C O Ne Mg 
~10%  Si S Cl Ar K Ca 
~10%  Ti V Cr Mn Fe Co Ni 
Rare elements in Nature  Li Rb Cs Sr Ba Ra Ga In Tl Ge Sn Se Te 
Stable nuclear forms (protons neutrons)  He(4), C(12), Mg(24), Si(32), Fe(56) 
********************  ******************** 
Comet "Wild2"  olivine: (Mg,Fe)_{2}SiO_{4} Al Ca Ti 
(from La historia de la Tierra. Un estudio global de la materia, M.J. MediavillaPÃ©rez; McGrawHill 1999) 
Stable nuclides(neutrons minus protons vs. atomic number) 
Euler: La naturaleza de la radiación que nos permite ver un objeto [...] depende del movimiento vibratorio de los átomos de su superficie, como cuerdas tensadas afinadas con cierta frecuencia, con la radiación incidente, emitiendo sus propias ondas. [Oliver Sacks; El Tío Tungsteno (Recuerdos de un químico precoz). Anagrama, 2003]  
Wittgenstein: Logic when already stablished may be used for describing formal implications, but the rules themselves do not follow any logic. [from Kampis] 

Bernhard Riemann {18261866}: Zeta function (1859).  En Milán, Eckermann recapacita taciturno sobre el efímero valor de escrutar vidas ajenas y la importancia de contemplar las propias manos vacÃas. "Cuál debe ser la naturaleza de mi existencia?", se pregunta. Y escribe a Goethe, de retorno: "Vuestra excelencia dice en broma que viajar serÃa gran cosa si no hubiera que volver. Yo vuelvo ya". En Weimar, lo recibe Goethe y halla la repuesta a sus tres necesidades: "Aumentar mis conocimientos, mejorar mi existencia y sobre todo hacer algo". En esta sencilla fÃ³rmula, se encierra el secreto de las Conversaciones: la trama de complicidades entre el narrador y su oyente. [from an undocumented newspaper clipping] 
{{las Álgebras del ritmo}}