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Archive for May, 2012

Nietzsche in Beyond Good and Evil accuses Stoics of imposing their ideal on nature.  “For all your love of truth, you have forced yourself so long, so persistently, so rigidly-hypnotically, to see nature the wrong way, namely Stoically, that you are no longer able to see her differently.  And some abysmal arrogance still inspires you with the hope that because you know how to tyrannize yourselves–Stoicism is self-tyranny–nature, too, lets herself be tyrannized.  But this is an ancient, eternal story: what happened formerly with the Stoics still happens today, too, as soon as any philosophy begins to believe itself.  It always creates the world in its own image; it cannot do otherwise.  Philosophy is this tyrannical drive itself, the most spiritual will to power, to the “creation of the world”, to the causa prima.

The recognition of the ideals of Republic as fundamental ideals necessary for a eudaimonia on Earth that gripped me from 2008 which was followed a a series of adventures including dispossession and destitution was partly due to my faith that these ideals originate not from human thought but from the heart of nature, which, because I had discovered that the universe had four rather than three macroscopic spatial dimensions, seemed that it should be directly provable.  Only in the last few days did I obtain a clear explanation of the redshift phenomenon, so the project of attempting to elucidate how ideals can be explained from first principles in a metaphysical four dimensional universe is still an ambitious task.

Nietzsche’s criticism of the Stoics contains extremely potent insights that I can verify from my own metaphysical adventures as well as material plane experiences since 2008.  Conscious metaphysical universe do contain dramas of conflicts between contending ideals, and thus the “most spiritual will to power” is quite literal.  Strong philosophies, and religions are not different in this way, are products of literal metaphysical wars, and they do attempt to “create the world” in their own image.  Agents adhering to particular sets of ideals are continually attempting to reorder the metaphysical universe according to these, and Stoics represent the faithful partisans in such wars.

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The redshift is the phenomenon where the analysis of light spectrum of distant galaxies or stars shows characteristic features such as the absorption line of hydrogen H-$alpha$ which has been measured at 656 nm to shift to a longer wavelength.  Hubble in 1929 considered objects in the range from 3.392 parsecs to 212 parsecs and found that the redshift increases linearly with distance.  Subsequent studies had increased the distance ranges for the validity of this relation.  Hubble’s interpretation of this was a Doppler effect from the velocity of the objects moving away from us.  Fritz Zwicky had immediately reacted with the suggestion that light loses energy in their passage but this hypothesis was rejected and the recession velocity hypothesis was accepted, and with it an expansionary universe model.

In order for the S4 theory, which posits that the universe is a four dimensional sphere of fixed radius to be valid, we must find an explanation of how a redshift can occur in a stationary non-expanding universe.  My initial approach to the problem was to consider that the redshift might occur as an effect of the cosmic background radiation, which has a peak at around 6 cm but has nonvanishing tails for all higher frequencies, with the idea that there could be interference with small probability for higher frequencies.  This approach seems to be a failure as the cosmic background radiation in the visible frequencies in the order 400 THz produces terms like exp(-7000) which cannot be compensated by the long distances.

A more promising approach is to use the fact that gravitational fields affect light and then treat the uniform curvature of the four dimensional sphere as a uniform pseudo-gravitational field which affects all photons regardless of their frequency.  The idea is that photons feel this ambient curvature as a gravitational effect that dampens their energy uniformly per unit time, and thus there is a redshift that is uniform in the distance traveled by the photons.  As for conservation of energy, the natural candidate for where the energy goes is the cosmic background radiation, but the mechanism for this energy transfer remains unclear to me.

Another possible explanation of redshift in a stationary universe is from extrapolation of the fact that in transparent dense media both the speed of light as well as the wavelenghts shrink.  Thus it is possible that the speed of light in the solar system, which is gravitationally denser than intersteller space has shorter wavelengths and slower speed than intersteller space.  Then redshift can be explained as the expansion of wavelengths in less dense interstellar space through which light travels to reach us.

Update:  Last night I was able to find an concrete explanation of the redshift that produced numbers in the order of magnitude of the redshift that is observed even in a stationary 4-sphere universe.  The idea is the following.  The wave equation can be solved on a four sphere and yield solutions of the type \Phi_n(x) \exp(-i \sqrt{n(n+3)} t) where the first factor is a spherical harmonic, ignoring some constant.  So essentially for a spherical wave when the true wavelength is \lambda=c/n the frequency is not n but \sqrt{n(n+3)}.  This error is per second, and increases with time linear in distance.

Consider the H-alpha line of 656.3 nanometers.  If we take n to be c/lambda and then consider the shift over 106 parsecs, which is the scale used by Hubble, using the slope

1/n - 1/\sqrt{n(n+3)}

then we find a shift of 0.023 nanometers for the wavelength which corresponds to a radial velocity of approximately 10,500 km/s in Hubble terms, which is in the right order of magnitude.

In other words, treating spherical waves on a 4-sphere as linear waves will produces an error in frequency which will produce a linear redshift of the type that Hubble interpreted as a Doppler shift.  This explanation of the redshift suggests that the redshift is not even a physical phenomenon.

In order to test whether the explanation of the redshift as an artifact of mistaking spherical waves as linear waves has merit, I downloaded a modern dataset of redshift and distance from this site from table1.dat which has the ‘radial velocity’ in the second column and distance in megaparsecs in the fifth column.  Then I model redshift as the linear function of distance by which the 656.3 nanometer would move as the error exacerbates.  Fitting observed redshifts by this model produces an R-squared of 0.998.

Here is python code for this test.

import struct
from scipy import stats
import re

cSpeedOfLight = 299792458.
# Estimated in nanometers for H-alpha
cExpectedRedshift = 0.2200922

def getFieldWidths(line):
fw = []
lline = list(line)
N = len(lline)
pc = lline[0]
ilast = 0
for i in range(1,N):
c = lline[i]
if pc != ‘ ‘ and (c == ‘ ‘ or c==’\n’):
fw.append(i – ilast)
ilast=i
pc = c
return fw

def getFields(line, fmtString):
parse = struct.Struct(fmtString).unpack_from
fields =parse(line)
return fields

file = ‘amiga1.dat’

fd = open( file, ‘rb’)
line  = fd.readline()
line = fd.readline()
LineLen = len(str(line))
fieldWidths = getFieldWidths(line)
fmtString = ”.join(‘%ds’ % f for f in fieldWidths)

vel = []
dist = []

for line in fd:
s = str(line)
if len(s) < LineLen:
s = s + ‘ ‘ * (LineLen-len(s))
fields = getFields(s,fmtString)
# Field 1 is velocity and Field 4 is distance

if re.match( r’ +\d+\.\d*’, fields[1], re.M):
vel.append( float(fields[1]))
dist.append( float(fields[4]))

expected_redshift = []
for d in dist:
expected_redshift.append( d * cExpectedRedshift )

observed_redshift = []
for i in range(len(vel)):
observed_redshift.append( (vel[i]/cSpeedOfLight)*656.3*1000 )

grad,intercept,r_value,p_value,std_err = stats.linregress(expected_redshift,
observed_redshift)
print ‘Gradient’, grad
print “R-squared”, r_value**2

and the output is

MacBooks-MacBook:Hydrogen macbook$ python proc-amiga1.py
Gradient 0.747194701348
R-squared 0.998190028796

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I had previously given my intuitive reasoning why quantization can be expected in a compact manifold based on the idea that the spectrum of the Laplacian is discrete, but have to step back from the general claim and focus on the sphere.  On a sphere of radius R, every geodesic is closed and the geodesics are great circles of length 2*pi*R.  On a spherical universe, the only possible wavelengths are 2*pi*R/N for integral N, and therefore the frequencies N*(1/(2*pi*R)) are quantized.  This sort of argument requires that every geodesic is closed, which holds for a sphere but not for all compact manifolds.  On a torus, for example, this argument fails when one takes the image from R^2 of a line with an irrational slope which produces a geodesic which is not closed.

Imagine now going back to 1900 during the time Planck hypothesized the quantization of energy.  Planck hypothesized that energy is quantized to explain the observed intensity distribution of a blackbody which could not be explained by the Rayleigh-Jeans law.  The reasoning in the last paragraph would have given an explanation of quantization of energy that is global rather than local.  If the universe is spherical, then quantization of energy, at least electromagnetic energy, is a consequence of the spherical shape of the universe.

The observations of crystals, or what have been called quasicrystals, since early 1980s, with 5, 8, 10 and 12 fold rotational symmetries show that the universe has at least four macroscopic spatial dimensions.  No crystals have been found in this period to my knowledge which would require us to consider that the universe has more than four macroscopic spatial dimensions.  There are other reasons for considering the universe to be four dimensional, such as the fact that in four dimensions there is special structure for SO(4) which is locally SU(2)xSU(2) — that Spin(4)=SU(2)xSU(2) is the double cover of SO(4) — and that the two factors represent the duality of electromagnetism.

Additionally, one obtains formally the gravitational field equations when calculating the Ricci curvature of any three dimensional submanifold of a four-dimensional sphere.  For the radius of C/h for a 4-sphere with |C| <= 10, one obtains the gravitational field equations with a cosmological constant term that matches the measured cosmological constant with agreement in the order of magnitude.

In four dimensions, one can construct a dynamical system of an electron and a proton which do not have the problem of stability.  One can see this by considering circular orbits in orthogonal planes.  The problem of stability of such a system was one of the two reasons for abandoning classical mechanics in the early part of twentieth century.

Electromagnetism can be formally studied on any compact four dimensional manifold X by considering a principal G-bundle over X and then considering the electromagnetic potential to be a connection 1-form.  In the case of a four dimensional sphere, there is a natural principal SU(2)-bundle which occurs from considering S4 as a quaternionic projective line.  The projection map restricted to the unit vectors in pairs of quaternions, the 7-sphere, produce the Hopf fibration S^7->S^4 with SU(2)=S^3 fibers.

So in the above we have a very simple “unification” of electromagnetism and gravity as represented by the gravitational field equations.  But quantization of energy now is a natural part of the description of the system without the need to abandon classical mechanics.

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While I am still optimistic about my claim that one can hear the shape of the universe in the hydrogen energy spectrum, this note is a curiosity rather than a result because I realized that the error of fit to data is much higher than the fit of the Bohr model to the hydrogen energy spectrum, which is quite tight:
> B=1/(n*n)
> D<-lm(diff(Level)~diff(B))
> summary(D)

Call:
lm(formula = diff(Level) ~ diff(B))

Residuals:
Min        1Q    Median        3Q       Max
-0.012714 -0.001076  0.001384  0.002820  0.003612

Coefficients:
Estimate Std. Error    t value Pr(>|t|)
(Intercept) -4.432e-03  1.701e-03 -2.606e+00   0.0285 *
diff(B)     -1.097e+05  3.769e-02 -2.910e+06   <2e-16 ***

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.004913 on 9 degrees of freedom
Multiple R-squared:     1,    Adjusted R-squared:     1
F-statistic: 8.468e+12 on 1 and 9 DF,  p-value: < 2.2e-16

I want to say that we can hear the shape of the universe in the energy spectrum of the hydrogen atom.  So we pull up the measured hydrogen energy spectrum from the NIST site, obtaining

Config Level
2 82259.158
3 97492.304
4 102823.904
5 105291.657
6 106632.1681
7 107440.4508
8 107965.0568
9 108324.7253
10 108581.9945
11 108772.3445
12 108917.1209
13 109029.7913

These are averages over energy level values over the orbitals.  For example, the n=2 value of 82259.158 is obtained from averaging the three values

82 258.9191133, 82 259.2850014, 82 258.9543992821

for each of which, according to the NIST site, “Uncertainties of the level values vary between 3 and 23 in the units of the last given decimal place.”  Thus the value in the table is past the measurement error bar.

We assume that these occur from the spherical harmonics of some dimension D.  The eigenvalues of the Laplacian for dimension D are k(k + D – 1) for any integral dimension D, but using this formula we can extrapolate to fractional dimensions as well.  We can compute the sum of absolute residuals for fitting the model

Level = a + b * I

where I = 1/(n-th eigenvalue of Laplacian) – 1/((n+1)-th eigenvalue of Laplacian) using the R function

rsbydim<-function(D) {
+ I=1/(n*(n+D-1))-1/((n+1)*(n+D))
+ m1=lm(Level~I)
+ sum(abs(m1$residuals))
+ }

and then plot sum of absolute value of residuals versus dimension.

This suggests the conclusion that if the energy spectrum of hydrogen atom does arise from eigenvalues of the Laplacian of a sphere of some dimension, the best candidate for that dimension is 4.

If we were to be able to hear the shape of the universe where the pure tones are represented by the energy spectrum of the hydrogen atom, then we would be hearing the shape to be a four dimensional sphere.

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The following exercise is not strong enough to show that the energy spectrum of hydrogen encapsulate the pure tones of the universe.  That is because the fits below are much weaker than the 1/n^2 fit of the Bohr model, which is:
> B=1/(n*n)
> D<-lm(diff(Level)~diff(B))
> summary(D)

Call:
lm(formula = diff(Level) ~ diff(B))

Residuals:
Min        1Q    Median        3Q       Max
-0.012714 -0.001076  0.001384  0.002820  0.003612

Coefficients:
Estimate Std. Error    t value Pr(>|t|)
(Intercept) -4.432e-03  1.701e-03 -2.606e+00   0.0285 *
diff(B)     -1.097e+05  3.769e-02 -2.910e+06   <2e-16 ***

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.004913 on 9 degrees of freedom
Multiple R-squared:     1,    Adjusted R-squared:     1
F-statistic: 8.468e+12 on 1 and 9 DF,  p-value: < 2.2e-16

A very simple exercise can be used to compare fit to data of hydrogen energy spectrum by S^2 versus S^4 harmonics.  The eigenvalues of the Laplacian for S^2 are k(k+1) while those of S^4 are k(k+3).  So we can obtain the measured energy spectrum from NIST.  The averages per orbital level are:

Config Level
2 82259.158
3 97492.304
4 102823.904
5 105291.657
6 106632.1681
7 107440.4508
8 107965.0568
9 108324.7253
10 108581.9945
11 108772.3445
12 108917.1209
13 109029.7913

We can pull in the data into R and compare fits as follows:
> I1=1/((Config-1)*(Config))-1/((Config)*(Config+1))
> I2=1/((Config-1)*(Config+2))-1/((Config)*(Config+3))
> m1<-nls(Level~a + b*I1,start=list(a=1,b=1))
> m2<-nls(Level~a + b*I2,start=list(a=1,b=1))
> summary(m1)

Formula: Level ~ a + b * I1

Parameters:
Estimate Std. Error t value Pr(>|t|)
a   107756        525  205.27  < 2e-16 ***
b   -79967       5258  -15.21 3.06e-08 ***

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1656 on 10 degrees of freedom

Number of iterations to convergence: 1
Achieved convergence tolerance: 1.258e-07

> summary(m2)

Formula: Level ~ a + b * I2

Parameters:
Estimate Std. Error t value Pr(>|t|)
a  108128.5      406.9  265.74  < 2e-16 ***
b -179496.5     8903.7  -20.16 1.99e-09 ***

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1261 on 10 degrees of freedom

Number of iterations to convergence: 1
Achieved convergence tolerance: 2.54e-07

> m1res = Level – ( 107756 – 79967*I1)
> m2res = Level – ( 108128.5 – 179496.5*I2)
> sum(abs(m1res))
[1] 14723.02
> sum(abs(m2res))
[1] 11296.49
> Z=(m2res-m1res)/m2res
> t.test(Z)

One Sample t-test

data:  Z
t = -4.3618, df = 11, p-value = 0.001133
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
-0.5556485 -0.1829476
sample estimates:
mean of x
-0.369298

This is an elementary exercise, but the point is the argument that the numerical result supports the idea that although quantum mechanics is numerically the most successful physical theory, we can expect a different physics where the universe is modeled as a four dimensional sphere of fixed radius 1/h to produce numerically superior fit to data.

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The protein shape determination problem is to determine the three dimensional shape of folded proteins given their amino acid sequence.  We encapsulate the shape of a folded protein by a twist decomposition.  This is performed by choosing the nitrogen atom of each amino acid on the backbone of the protein as location proxies for the entire amino acid and then considering the shape of the protein to be a sequence of elements of SO(3).  Elements of SO(3) can be written as an axis and a rotation angle.  We provide some results for prediction of protein twist angle sequences using an algorithm that has linear complexity in the number of amino acids.

The algorithm is the following: for each triple of amino acids, we consider the angle distribution for that triple over a set of approximately 37,000 protein shapes.  We observe that these angle spectra are similar to mass spectra.  We take a square root of the spectrum to stabilize variances as these occur from count data, then assume they arise from convolution of a sparser spectrum by a Gaussian of fixed width.  We deconvolve the spectrum after a soft threshold to remove baseline noise, and then use the mean location of the deconvolved spectrum as the estimate for the angle belonging to the amino triple.  The convolution operator is not invertible and I used the approach to deconvolution I learned from Prof. David Donoho´s work which suggests that sparsity can result from minimizing the L1 norm among all the solutions to Ax=y, where y is the observed signal and A is the Gaussian convolution operation. This processing can be done off-line and the angle estimates simply looked up in the simplest incarnation of our angle prediction algorithm.  This is a sensible approach because the conditional mean is the optimal prediction in various statistical idealizations.

We evaluate results of this algorithm by considering an angle prediction within 0.45 radians of the actual angle which is quite wide but provides us with a reasonable estimate of the efficacy of simple linear complexity algorithms for angle prediction.

Protein      Pred in 0.45 radians         Total                Percentage “Correct”

2G4Z         224                                         314                    71.34%

2GMN       173                                          262                   66.03%

3CC0           68                                          104                  65.38%

2VJM        302                                          425                  71.06%

2XEG         482                                          692                  69.65%

3GDQ         256                                          374                   68.45%

3CDZ           316                                          628                  50.32%

2RFH          212                                           305                  69.51%

2YSS              86                                           127                   67.72%

3C60            102                                           232                  43.97%

3GHC             97                                           184                   52.72%

3GT6              49                                            99                    49.49%

2HLN            174                                           306                    56.86%

3GYE             214                                           297                    72.05%

2KSQ             130                                            178                     73.03%

3CSX               54                                              59                      91.53%

2VHL              251                                           390                    64.36%

2R6M              130                                           176                     73.86%

3EML              233                                           286                     81.47%

2ZF0                162                                           240                     67.50%

2O5M                131                                          150                     87.33%

2P0G                 50                                            77                       64.94%

2GDE               160                                           247                      64.78%

3AIR                 362                                          489                    74.03%

3AO1                  103                                         140                      73.57%

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