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

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;
Computation. A fundamental physical view.

Phys. Scr.35, 88-95 (1987).

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:
(inverse sine waveform) Image:331w.jpg

[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):

1 – ((sin πx) / πx)2

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)].




[In Heisenberg's formulation of quantum mechanics, the internal state of an atom or a nucleus is represented by a Hermitian matrix whose eigenvalues are the energy levels of the spectrum.]

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] 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):
2 (He)
10 (Ne)
18 (Ar)
36 (Kr)
54 (Xe)
86 (Rn)
118 (Uuo)
Number of elements per period (Mendeleiev table):
2 (K) 8 (L) 8 (M) 18 (N) 18 (O) 32 (P) 32 (Q)
Types of electronic orbitals or (sub)shells:
s2 p6
d10
f14
3x2=6 5x2=10 7x2=14


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.

Image:log.jpg




Results and other conjectures

Atom prime chart:

Image:atomchart.jpg

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":

2
Image:2wave.jpg

23
Image:23wave.jpg

59
Image:59w.jpg

149
Image:149w.jpg

241
Image:241w.jpg

439
Image:439w.jpg



"Erbium-like":

37
Image:37w.jpg

73
Image:73w.jpg

109
Image:109w.jpg

223
Image:223w.jpg

331
Image:331w.jpg

439
Image:439w.jpg



More prime phases: part I (661 to 1427); part II (1429 to 7919); part III (7927 to 21059).

661
Image:661w.jpg

769
Image:769w.jpg

881
Image:881w.jpg

991
Image:991w.jpg

1103
Image:1103w.jpg

1213
Image:1213w.jpg


Dynamics, transitions and polarization

Realtime data:

Adobe Audition

  • Waveform view (relative volume of the signal vs. time).
  • Frequency spectrum (measured in Hz, vs. amplitude, measured in dB).
    • Six types of FFT windows are available for frequency graph:
            • Triangular
            • Hanning
            • Hamming
            • Blackmann
            • Welch (Gaussian)
            • Blackmann-Harris
  • Phase analysis (Lissajous Plot graph):
Gibbs Energy = Enthalpy Entropy (or Information)
Image:439Adobe.jpg
(Phase modulation and primes kaleidoscope: supernovae, crystals, viruses)

wav files (5 seconds each prime wave)
at YOUTUBE

  • periods KLM (1'30'')
  • periods N, O (1'30'')
  • periods P, Q (2'40'')
  • periods R, S (4'30'')
  • periods T, U [6'40'']
  • periods V, W [9'30'']
  • periods X, Y [12'40'']
  • periods Z1, Z2 (16'30'')
  • periods Z3 Z4 (21'10'')
  • periods Z5, Z6 (26'30'')

Slow motion rendering (only phases):

[Unregistered] Screen Movie Studio (MandSoft)
(download FrontPlayer: simple viewer for avi files)

Image:screenmoviestudio.jpg

avi files (1000 frames per second)
at YOUTUBE

  • periodsKLMOPQ (10'00'' - 296MB -)
  • periodsRS (9'00'' - 243MB -)
  • periodT (6'48'' - 208MB -)
  • periodU (6'47'' - 193MB -)
  • periodV (9'37'' - 247MB -)
  • periodW (9'36'' - 225MB -)
  • periodX
    • part a (6'29'' - 169MB -)
    • part b (6'29'' - 166MB -)
  • periodY
    • part a (6'29'' - 166MB -)
    • part b (6'29'' - 154MB -)
  • periodZ1
    • part a (8'24'' - 182MB -)
    • part b (8'24' - 175MB -)
  • periodZ2
    • part a (8'27'' - 169MB -)
    • part b (8'21' - 195MB -)
  • periodZ3
    • part a (10'42'' - 223MB -)
    • part b (10'41'' - 237MB -)
  • periodZ4
    • part a (10'47'' - 179MB -)
    • part b (10'47'' - 179MB -)
  • periodZ5
    • part a (13'29'' - 226MB -)
    • part b (13'28'' - 248MB -)
  • periodZ6
    • part a (13'23'' - 247MB -)
    • part b plus (14'06'' - 244MB -)
      -extended up to maximum audio signal: 22039 Hz-


"Erbium-like" series (local strong attractors: the Riemann series)


erbium-like prime
nearby note

Image:erbiumlike1.jpg

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.

Atomic typology (prime vs. integer):

Image:atomos_graph0.jpg


Prime series of atomic elements

Image:group_numbers.jpg


Periods K to Z20 [n/p (log p - log n)]

p=prime
H 1
Medeleev period
(n =atomic number)
additional protons= 1 [1 0]
(n'=1)
zz=n/p
log p
log n log p - log n zz Δlog

zz Δln
He 2
K (2) 1s2
= 2 (n'=2)
[1 1<==>2]
1 0.30
0.30
0
0
0
Ne 23
L (10) 1s22s22p6

= 8 (n'3, n'=2 Σ2p)
[2 (3x2)]

0.435 1.36
1
0.36
0.157273
0.362134
Ar 59
M(18) = 8
0.305 1.77
1.26
0.52
0.157295
0.362186
Kr 149
N (36) d10 = 18
[2 (3x2) (5x2)]
0.242 2.17
1.56
0.62
0.1490
0.3432
Xe 241
O (54) = 18
0.224 2.38
1.73
0.65
0.1456
0.3352
Rn 439
P (86) f14 = 32
[2 (3x2) (5x2) (7x2)]
0.196 2.64
1.93
0.71
0.1387
0.3193
Uuo 643
Q (118) = 32
0.184 2.81
2.07
0.74
0.1351
0.3111
etz1 1019 R (172) = 54
[2 (3x2) (5x2) (7x2) (11x2)]
0.169 3.01
2.24
0.77
0.1304

0.3003
etz2 1427 S (226) = 54
0.158 3.15
2.35
0.80
0.1267
0.2919
etz3 2011 T (306) = 80
[2 (3x2) (5x2) (7x2) (11x2) (13x2)]
0.152
3.30
2.49
0.82
0.1244
0.2865
etz4 2659 U (386) = 80
0.145
3.42
2.59
0.84
0.1217
0.2802
3559
4517
5827
7177
8969
10781
13183
15647
18787
21961
V (500)
'W
(614)
X (766)
Y (918)
Z1 (1116)
Z2 (1314)
Z3(1570)
Z4 (1826)
Z5 (2144)
Z6 (2462)
80 34= 114
80 34= 114
114 38= 152
114 38= 152
152 46= 198

198 58= 256

256
62= 318
0.140
0.136
0.131
0.127





25933
29989
34843
39877
45887
51929
59023
66383
74857
83401
93377
103573
2854
3246
3720
4194
4754
5314
5968
6622
7382
8142
9020
9898
318 74= 392

392
82= 474

474
86= 560

560
94= 654

654
106= 760

760 118= 878











0.0956











5.02











4.00











1.097











0.0974









0.2244
115303
126943
10898
11898
878 122= 1000
0.0945
0.0937
5.06
5.10
4.04
4.08
1.025
1.024
0.0968
0.0964
0.2230
0.2219
p = Σ n n = Π p Image:zzlog.jpg
a pictorial view of Theorem of Best for Golay codes
e= Σ (1+1/n)1/n

[e**=Σ(1+1/p)1/p]


"Prime interactions"


molecule fract[ion]al 

interaction
(magnitude orders)

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

Image:atomsA.jpg








Atoms vs. molecules

H
1
He2 2
O8 17
H2O 19
Ne
23
Ar
59
Ca20 67
Kr
149
Sn50 227
Xe
241
Yb
347
Fe3O4 359
Pb82 419
Rn
439
ATP
541
Uuo 643
(stable nucleids)

Image:molecules.jpg






NOTES

Lissajous figure

[from wims]

Lissajous curve is a parametric curve in dimension 2, described by a pair of functions x = cos(nt) , y = sin(mt a) .


[from "Collins English Dictionary", 1984]

A curve traced out by a point that undergoes two simple harmonic motions in mutually perpendicular directions. The shape of these curves is characteristic of the relative phase and frequencies of the motion; they are used to determine the frequencies and phases of alternating voltages [C19: named after A. Lissajous (1822-1880), French physicist].

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 the Lloyd algorithm. An N-level vector quantizer for k-dimensional blocks of real-valued source samples is designed by determining the N codevectors in k-dimensional Euclidean space into which the source blocks are to be quantized. The goal in vector quantizer design is to find a vector quantizer for which the expected squared-error quantization noise achieves a value close to the minimum. Stuart Lloyd, in an unpublished technical report written in 1957 (eventually published in 1982 [14]) proposed an algorithm for accomplishing this goal. (Although Lloyd stated his algorithm for the scalar quantization case in which k = 1, it trivially extends to the k > 1 case.) The Lloyd algorithm starts with an initial quantizer and modifies it through a sequence of iterations--on each iteration, the codevectors from the preceding iteration are replaced with the centroids of the nearest neighbor regions corresponding to these codevectors. As long as successive iterations of the Lloyd algorithm continue to generate new quantizers, the quantization noise strictly decreases. (This is because of the Lloyd-Max necessary conditions for an optimal quantizer, which appear to have been first discovered not by Lloyd or Max but by the Polish mathematicians Lukaszewicz and Steinhaus [15].) In the scalar quantization case, if the density function of the source samples is log-concave and has finite variance, it is known that the N-level quantizers generated by iterates of the Lloyd algorithm have asymptotically optimal performance, independently of which N-level quantizer is chosen at the start of the iteration process. Unfortunately, the Lloyd algorithm is sensitive to the choice of initial quantizer if k > 1: for some choices, the quantization noise may not decrease to the minimum value. Recent research (cited in [11]) has focused on modifications of the Lloyd algorithm through simulated annealing or stochastic relaxation techniques that avoid this problem. It should be mentioned that there have been other significant contributions to the area of quantization in addition to the Lloyd algorithm--there have been numerous advances in high rate quantization theory and in lattice quantization, for example. The reader is referred to the recent text [7] for a thorough discussion of quantizer theory and design.

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).

"Are the prime numbers in a self-organized critical state?"

M. Wolf, "1/f noise in the distribution of prime numbers", Physica A 241 (1997), 493-499.

P. Bak, C. Tang, and K. Wiesenfeld, "Self-organized criticality", Physical Review A 38 (1988), 364-374.


Data

(audible spectrum: 16.4 - 21096 Hz; ~ 10 scales)

Standard La (A4)
is usually tuned at 438-440 Hz
Image:438a.jpgImage:440a.jpg
Dynamics: Unstable/Instable & Hyperstable


(439 is prime)
Image:439wA.jpg
Metastable



FREQUENCIES AND WAVELENGTHS FOR EQUAL-TEMPERED SCALE
from C0 to D#8 / Eb8
Image:scalevsprimes.jpg

Image:natural_and_temperated.jpg




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)

Image:atomic_abundance.jpg


Stable nuclides

(neutrons minus protons vs. atomic number)
Image:neutrons.jpg
-Isotopes chart-



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]


An Algorithm for Discovery
David Paydarfar and William J. Schwartz

1. Slow down to explore.
2. Read, but not too much.
3. Pursue quality for its own sake.
4. Look at the raw data.
5. Cultivate smart friends.

from http://www.sciencemag.org/
Volume 292, Number 5514, Issue of 6 Apr 2001, p. 13.


  • Fermat {1601-1665}
  • Euler {1707-1783}
  • Fourier {1768-1830}
  • Gauss {1777-1855}
  • Hamilton {1805-1865}
  • Russell (1872-1970}
  • von Neumann {1903-1957}
  • Gödel {1906-1978}
  • Turing {1912-1954}
  • Church (1941): Lambda calculus
  • Shannon (1949): Mathematical theory of communication
  • Kolmogorov (1965): Three approaches to the quantitative definition of information
  • Markov (1951): Theory of algorithms
Bernhard Riemann {1826-1866}: Zeta function (1859).
Dimitri Mendeleiev {1834-1907}: Periodic table of the chemical elements (1869).
En Milán, Eckermann recapacita taciturno sobre el efímero valor de escrutar vidas ajenas y la importancia de contemplar las propias manos vacías. "Cuál debe ser la naturaleza de mi existencia?", se pregunta. Y escribe a Goethe, de retorno: "Vuestra excelencia dice en broma que viajar sería 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}}



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