User talk:Ramon Roca
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http://openwetware.org/wiki/Molecular_computing
http://openwetware.org/wiki/User_talk:Ramon_Roca
http://openwetware.org/wiki/Talk:Molecular_computing
http://www.youtube.com/user/rmnDeliriaLegitur
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
(i.e., one reality comfomed of differente complex systems dynamically interacting by its prime evolution)
|
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 two-point correlations go as.... (turning to write on a nearby blackboard):
Dyson: Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix? It's also a model of the energy levels in a heavy nucleus [erbium-166 (protons neutrons)].
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"Performing good theory implies to attend to contexts, that are always particular. And these require interdisciplinarity, the use of induction and some epistemological pretensions more modest than those supposed by abstract theories of illustrated root. Criticism from Frankfurt School was referred to social character questions. [...] In front of the greek intellectualism, that admitted persuassion as daughter of good reasoning, Gorgias proclaim that its efficacy was rather within the words employed in reasoning. Humans, said Benjamin, tend to "overname" things by means of abstractions and generalizations." Ferran Requejo (ferran.requejo@upf.edu), LaVanguardia 23/VI/2007 |
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NUMBER THEORETICS AND INFORMATION LEVELS
Levin and Chaitin definition of algorithmic entropy: A program is self-delimiting 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 coarse-grained 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<sup>jθ</sup>] 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 surfaces-defining equations (e.g., quadratic, cubic, etc) [tesis/biblio.html#Ayers (Ayers, 1994)] - [tesis/fig3_4.html see table] -:
symmetry information embodied in simple shapes |
Hsym (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; n-th level, n-ary 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 component-system which produces them by complexity-increasing procedures. [tesis/biblio.html#Kampis (Kampis, 1991)].
Golay codes are based upon prime numbers: Golay codes G23 (binary, [23,12,7]2, 3-correcting) and G11 (ternary, [11,6,5]3, 2-correcting) 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 even-odd 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 self-organizing 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 equal-tempered 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 Ca2 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 self-organized (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":
"Erbium-like":
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)
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wav files (5 seconds each prime wave)
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Slow motion rendering (only phases): [Unregistered] Screen Movie Studio (MandSoft) |
avi files (1000 frames per second)
|
"Erbium-like" series (local strong attractors: the Riemann series)
erbium-like 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 {anti-matter/matter, to be or not?}.
Volume is a function of distribution of mass-energy 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)
|
H2 |
1:1 |
2 |
CH4 |
11:1 |
15 |
NH3 |
13:1 |
16 |
NH4 |
13:1 |
17* |
H2O |
1:17 |
19 |
N2 |
13:13 |
26 |
CO |
11:17 |
28 |
CH5N |
11:1:13 |
29 |
O2 |
17:17 |
34 |
C3H8 |
11:1 |
41 |
N2O , C2H4O |
|
43 |
CO2 |
11:17 |
45 |
NaOH |
29:17:1 |
47 |
Acetic (C2H4O2) |
11:1:17 |
60 |
ClOH |
53:17:1 |
71 |
C3H6O2 |
|
73 |
Gly |
|
74 |
SO2H2 |
47:17:1 |
83 |
Ala |
|
87 |
C6H6O , CaC2 |
|
89 |
PO3H3 |
43:17:1 |
97 |
Ca(OH)2 |
67:17:1 |
103 |
Ser |
|
104 |
Guanine (C5H5N5O) |
|
105 |
Pro |
|
111 |
Val |
|
113 |
Thr |
|
117 |
Thymine (C5H6N2O2) |
11:1:13:17 |
121 |
Adenine (C5H5N5) |
11:1:13 |
125 |
Leu |
|
126 |
Orn |
|
127 |
Asn |
|
129 |
Asp |
|
132 |
Cys |
|
134 |
Lys |
|
140 |
Gln |
|
142 |
Arg |
|
143 |
Glu |
|
145 |
His |
|
148 |
C5H10O5 |
|
150 |
Phe |
|
157 |
Met |
|
160 |
olivine (Mg2SiO4) |
31:41:17 |
171 |
Cl2Ca |
53:67 |
173 |
Tyr |
|
174 |
Glucose (C6H12O6) |
|
180 |
Trp |
|
193 |
Adenosine (C10H13N5O4) |
|
256 |
olivine (Fe2SiO4) |
97:41:17 |
303 |
chromite (Cr2O4Fe) |
83:17:97 |
331 |
Fe3O4 |
97:17 |
359 |
3Fe 4H2O = Fe3O4 4H2 |
[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 (Mg3Si2O5 - 2H2O Fe Ni) |
[253]::[19]2:: [97]::[103] |
491 |
Porphyirine (C34H34N4O4) |
|
528 |
ATP (C10H16N5O13P3) |
11:1:13:17:43 |
541 |
(CrO4)3Fe2 |
|
647 |
Lecitine (C42H82NPO8) |
|
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 e-error correcting code" Theorem (Tietäväinen 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 Lempel-Ziv codes. The well-known 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 finite-alphabet source generates a sequence which, when encoded using the Lempel-Ziv algorithm, yields a compression rate equal to the entropy rate, asymptotically as the number of source samples goes to infinity. The Lempel-Ziv 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 Lempel-Ziv 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 July-August, 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), 493-499.
Data
(audible spectrum: 16.4 - 21096 Hz; ~ 10 scales)
Standard La (A4) is usually tuned at 438-440 Hz Dynamics: Unstable/Instable & Hyperstable
|
FREQUENCIES AND WAVELENGTHS FOR EQUAL-TEMPERED 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)2SiO4 Al Ca Ti |
(from La historia de la Tierra. Un estudio global de la materia, M.J. Mediavilla-Pérez; McGrawHill 1999) |
Stable nuclides(neutrons minus protons vs. atomic number) |
Upupa epops (hoopoe, abubilla, puput), bird song:
Definitions:
Cardinal Numbers are positive integers (counting numbers) that represent "how many?"
Ordinal Numbers are numbers that describe position: first, second, third, fourth,... last.
(Cardinal numbers -real space-:) Set of the atomic elements and their atomic/electronic -integer- numbers (and, by extension, molecules): {H1, He2, ..., C6, N7, O8, ..., Na11, Mg12, ..., K19, Ca20, ..., Fe26, ..., Zn30,...}. (Ordinal numbers -reciprocal space-:) Gödelization, in prime numbers (nuclear types -harmonic fundamental-): {H1, He2, ..., C11, N13, O17, ..., Na29, Mg31, ..., K61, Ca67, ...,Fe97, ...,Zn109,...}
See atomic typology graph and table: Representation in log10 is explicit (it also should result interesting log2 [log3 , log5 , log7 ], ln or logp ).
Waves & Atoms:
Melody or tone: Succession (atom). Harmony or accord: Simultaneous combination (molecule). Rythm: Relative regular groups (Mendeleiev: a code of infinite period , or, better, an infinite series of finite sets/rings-).
Musical language considers an octave as 7 notes (12 semitones), while harmony is constructed classically (major scale or ionian mode) from 3rd and 5th chords (the 2nd is equal to a whole note -step-); jazz added 11th and 13th, i.e., some additional complexity appears as considering 2 octaves -24 semitones-, because within a 7-based code, 11 and 13 are equal to 4th and 6th steps. Modes result from different step-patterns or sequences of the notes (modality, its own sound characteristics). Then, relationships between notes (symbols) conform different phrases and motives (codes), and their interactions (grammar), including environmental noise, can result onto characteristic (sounds, music, words, silence) languages. It is not coincidence the true relationship between music and prime numbers: waves and their interactions need some "harmony" (proportions between attractor types or prime levels).
Atoms and molecules can be described as concrete sets of electrons and their respective nuclei, i.e., as integer numbers (atomic or electronic numbers at Mendeleiev table). Moreover, because of "volume is a function of distribution of mass-energy quanta", let's consider each atom as a "perfect code of a characteristic type" -a prime Gödel? number- (due to specific volume/surface relationships, n interacting electrons conform different wave configurations or information/entropy classes). Here, it is intented to describe interactions as "fract[ion]al relationships" of different nuclear and molecular types. So, each atom or molecule is represented as a binary vector (n protons, p-order -information/entropy level-); for each electronic (atomic or molecular) wave equation, there is a typical wave [prime] value and their "aritmethic operations" (physico-chemical interactions or level transitions) are due through mass/energy quanta (as Bethe cycle is a "set of integer operations"). Stability and dynamics are then referred to a "concrete set of prime levels" (the components of the complex system), as progressive "harmonic chords" (or augmented, disminished, sustained, ...).
Computability, constructibility, self-reproduction and evolvability are concepts easily applied to biomolecular systems. Complexity-increasing procedures (physics, chemistry, biology) and "privileged zero" {l} states (quark, foton, electron, energy, temperature, noise) have digitized interactions, although at different levels. Cell metabolism transferres the energy from 1 proton (hydrogen) onto a molecule with 260 protons (ATP), both quantic wave states with different information/entropy architectures ("magnitude orders" or level codes). Proteins or nucleic acids also have "quantizable" descriptions and interactions (H transferences, ...). As biopolymers (DNA, proteins) can be stabilized by helix conformations, it is clear that this characteristic property (self organization) is related to the electronic wave nature of atoms and molecules. The genetic code permits to connect thermodynamical, morphological and control information (as well as symbolic), by combining DNA sequences (4 nucleotides) and proteins (20 aminoacids) plus environment (water, ions, other molecules and physicochemical conditions) as a whole (cell metabolism). Each organism is defined as a set of genetic information; each protein is a 3D structure of a concrete sequence of aminoacids; each molecule is a peculiar set of atoms; each atom is a characteristic string of subatomic particles.
(hard) Hypotheses:
Bethe cycle (H / He / C / N / O) |
lambda cycle:λ λ*=1
|
C12 H = N13 + ΔG |
{11 1*=13* + Δλ} |
N13 = C<sup>13</sup> + <nowiki>ΔG |
{13*=11* + Δλ} |
C13 H = N14 + ΔG |
{11*+1*=13 + Δλ} |
N14 H = O15 + ΔG |
{13+1*=17* + Δλ} |
O15 = N15 + ΔG |
{17*=13* + Δλ} |
N15 H = C12 He | {13*+1*=11+2} |
Stable nuclear forms (protons neutrons, protons, neutrons) |
He (4,2,2) |
C (12,6,6) |
Mg (24,12,12) |
Si (32,14,18) |
Fe (56,26,30) |
(dimension, length, minimum distance) G12: An extended ternary Golay code is a self-dual (12,6,6) 'linear code over Galois Field with 3 elements -GF(3)- G11: The ternary (11,6) Golay code 'is the only known perfect nonbinary code, 'has a minimum distance of 5 and can correct up to 2 errors |
(2,2,2) |
(11,6,6) |
(31,12,12) |
(41,14,18) |
(97,26,30) |
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 {1826-1866}: 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 fórmula, 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}}