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| Roger Bagula... |
 Posted: Fri May 30, 2008 3:20 pm |
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First essay:
Why we are behind in Algebraic Geometry
It appears that Grothendieck was well left of center
and French.
For that reason he had a lot of trouble getting anything published in the
US or it's journals (politically he was like Bertold Brecht, blacklisted
in the states)
but was published and translated all over European and China.
He had this troublesome way of making up new terms
and definitions to fit mathematics he discovered / invented.
( schemes, sheaves, stacks... a long list... )
Very few of the Europeans who did/ do study him really get it enough to
make pictures
of the stuff they are talking about and as it is an axiomatic
area of study, many don't even realize they are talking about real
geometry.
So the East Germans, Russians and Chinese got his stuff, and we didn't (
mostly).
What is fabulous about this, is that this puts us something like about
50 years behind.
And not very uniformly either as England accepted a lot of his work.
But very little real technological development has been done,
all the good books in this area are French
and very expensive as translations.
So American think in terms of "Field" domains in vector spaces
and Hausdorff measures/topology ( thanks to Mandelbrot),
while Frenchmen think in terms of Rings and Zariski topology/ measures.
For example Germans are something more than 25 years ahead with Barth's
work
in the Geometry associated with "Implicit" surfaces as algebraic varieties.
What made this gap apparent is the lag that American have in string type
Calabi-Yau
geometry. The papers are nearly unreadable to most American trained people
compared to European trained people. This gap seems to be important in
string and gravity advanced research. Another thing is that the
application of
Grothendieck's main Abelian ring Schemes to Calabi-Yau surfaces
that are from Lie algebras that
aren't commutative seems to have been done
and really may not be workable at all?
It is my one real problem with these sorts of rings: they don't cross
over well to Physics.
Steinbach found a Pisot like "Field" that was Chebyshev;
but it appears that Pisot -Saleem Algebraic varieties actual are a Ring
type Scheme in the Grothendieck algebraic variety sense?
This kind of application of homological algebra to the Penrose type of
crystal doesn't seem to have been investigated even by the French number
theory people. Anything that can make the statistical mechanics
involved simple would be very welcome.
Another area that Grothendieck Algebraic geometry (FGA) might have real
physical application is in the space group crystals and study of these
types of geometric groups. Since just about no one in chemistry has even
heard of it, it hasn't been tried?
I found myself asking why I was trying to learn this area,
and I found that there were answers
and they weren't ones that seemed to have found there way into American
thought processes yet!
Second essay:
Hydrology, Language and genetics
What do "Junk DNA", the development of languages over a long period and
Hurst exponents have in common?
The Zipf's law application of mathematics to of word frequency
of English was one of the first uses of a fractal self-similarity
in a "natural" setting.
The use of Hurst exponents seems to predate even that!
It wasn't until. the 90's that "junk DNA" was found to have a fractal
structure.
In all three of these cases what was called "self-organizing
criticality" by Per Bak
seems to be involved.
DNA is the language of genetic expression and evolves at a much slower
rate that language:
Chaucer's Canterbury tales are in a form of English that unreadable
today without translation ( in less than 1000 years), yet Caesar's
classical Latin is still taught in school boy Latin in England more that
2000 years later!
The geological sedimentary scale of time is much slower.
Even slower is the evolution of galaxies which also seem to show this
same sort of long term process in their distribution.
It turns out it may be too simple to say that a Levy flight distribution
is a basic process of nature, but it does seem to be involved and in
many cases, a limiting model for the
mathematical structures that are observed.
Large earthquakes remind us that sand pile processes are still going on.
Language changes so that even with the "standardization" of English in
American cultural dominance, we see drift as in special communities like
ethic or geekic ( that I can use this new word and be understood show
that language is dynamic!) gatherings. We all know about flash memory
and Ram, for instance. We all pick up some ethnic drift from hip-hop and
rap vocabularies getting used in popular music.
So where is the genetic "drift" that can be expected at scale of maybe
as much as 10000
years? The men in Italy 2000 years ago don't appear to be all that
different than
people of today, but the ice man frozen from the Alps as much as much
8000 years ago
shows very different genetic traits. Again the people in America pre Clovis
are "different" than those currently of that Amerind line.
At 20000 years ago we have those people who had just out survived the
Neanderthals
and they are "different" that those who began farming at about 9000
years ago.
What sand pile collapses cause these genetic differences?
One thing that seems to be different about modern men and those of
Caesar's time is their resistance to diseases like the black plague and
small pox.
Another adaptation has been to alcohol and lately to drugs.
Pollution like chemicals and plastic in the food chain seems to also be
making us "different" than the pre-industrial man of the past.
The people of the future seem to come from "melting pots"
where the races have mixed. There is also selection:
it is said that blue eyes "should" disappear from a gene pool as they
are a recessive trait
and brown eyes dominant. But both men and women will go miles out of
their way to
mate with those with blue eyes.
In the long term language and people get more efficient as
self-organizing processes
of survival and selection change what is currently "human".
In the very long term human have been changing the planet and
some process like a nearly random asteroid can changes thing in the
future very suddenly.
We do, now, have ways to "predict" such long term processes.
Third essay:
On Infinite series on the complex plane:
A designed sequence:
Sum[Exp[Log[2] - 1]/i! - 1/2^i, {i, 0, Infinity}]
The term Exp[Log[2] - 1]/i!
sums to 2.
The term
1/2^i
sums to 2 as well.
They just do it at very different rates!
Mathematica gives:
Sum[Exp[Log[2] - 1]/i! - 1/2^i, {i, 0, Infinity}]==0
After the first six terms the difference gets very small:
aa = Table[N[Exp[Log[2] - 1]/i! - 1/2^i], {i, 0, 16}]
{-0.264241, 0.235759, 0.117879, -0.00237352, -0.0318434, -0.0251187,
-0.0146031, -0.00766652, -0.003888, -0.0019511, -0.00097636,
-0.000488263, -0.000244139, -0.00012207, -0.0000610351, -0.0000305176,
-0.0000152588}
So what kind of use can we make of this?
f[x_] = Sum[Exp[
Log[2] - 1]*(I*x)^i/i!, {i, 0, Infinity}] - Sum[(I*x)^i/2^i, {i,
0, Infinity}]
gives:
f[x_]=2*Exp[-1+I*x]-(2*I)/(2*I+x)
( Circle minus Möbius/ bilinear function of an off center ellipse) which
plots as kind of cochleoid type curve
that goes round and round, but finally locks onto
a single circle.
The scale two curve dies out first leaving the factorial based circle.
ParametricPlot[{Re[f[x]], Im[f[x]]}, {x, -100*Pi, 100*Pi}, PlotPoints ->
2000]
These kinds of functions are actually implicit functions in {i,x}
as variables! Algebraic varieties in the integer infinite variable "i"
and the Complex variable I*x.
The Stirling's approximation:
n!->Exp[n*Log[n]-n]
on this same sort of "arithmetic"?
Substitution of the Stirling approximation gives:
g[x_] = Sum[Exp[Log[2] - 1]*(I*x)^i/(Exp[i*Log[i] - i]), {i, 0,
Infinity}] - Sum[(I*x)^i/2^i, {i, 0, Infinity}]
Which Mathematica fails on ...
Our approximation is:
n!->(2^n)*Exp[Log[2]-1]
which is very bad!
Table[n! - Round[(2^n)*Exp[Log[2] - 1]], {n, 0, 16}]
Here is an alternative way to do a function:
f[x_] = Sum[If[
Mod[n, 2] == 0, (I*x)^(n/2)/2^(n/2), Exp[Log[
2] - 1]*(I*x)^((n - 1)/2)/((n - 1)/2)!], {n, 0, 100}];
ParametricPlot[{Re[f[x]], Im[f[x]]}, {x, -Pi, Pi}]
It is only "faithful" for part of the curve and then it blows up.
Thes curves are not your usually way of looking at analyic function
Working generalized function:
f[x_, n_] = Sum[Exp[Log[ n/(n - 1)] - 1]*(I*x)^i/i!, {i, 0, Infinity}] -
Sum[(I*
x)^i/n^i, {i, 0, Infinity}]
Table[ParametricPlot[{Re[f[x, n]], Im[f[x,
n]]}, {x, -100*Pi, 100*Pi}, PlotPoints -> 2000], {n, 2, 11}]
For higher integer scales it appears to become a torus projection
with center at negative -0.5.
The two infinite series function made me realize that
it was the rate of convergence that made them so different
that they gave a toral geometry.
So I tried a Zeta like series and found it to be very slow compared :
1) fast ->Scaling: associated with self-similar fractal processes
Sum[(I*x)^i/n^i, {i, 0, Infinity}]=n/(n-I*x)
2) intermediate-> Exponental:U(1) symmetry group associated with Fourier
Hilbert space
Sum[Exp[Log[n/(n - 1)] - 1]*(I*x)^i/i!, {i, 0, Infinity}]=Epx[-1+i*x)/(-1+n)
3) slow->Zeta : associated with transcendental constants and the
Riemannian Zeta zero conjecture
Sum[(n/(n - 1))*(I*x)^i/(Zeta[1 + n]*(i + 1)^(1 + n)), {i, 0,
Infinity}]=I*n*PolyLog[1+n,I*x]/((-1+n)*x*Zeta[1+n))
My idea was to "compare" them using a functional subtraction:
( if you take the i sumation as Integration like, you have an Implicit/
Algebraic variaety in variables (x,i,n)
with the "i" being integrated away to invariance).
Mathematica:
Clear[f]
(* three functions: fast ( scaling) and slow ( zeta) minus
twice the expomnential function: arranged so that for x=1, they are
zero*)
f[x_, n_] = 2*Sum[Exp[Log[n/(n - 1)] - 1]*(I*x)^i/i!, {i, 0, Infinity}] -
Sum[(I*x)^i/n^i, {i, 0, Infinity}] -
Sum[(n/(n - 1))*(I*x)^i/(Zeta[1 + n]*(i + 1)^(1 + n)), {i, 0, Infinity}]
s = Table[ParametricPlot[{Re[f[x, n]], Im[f[x, n]]}, {x, -100*Pi, 100*Pi},
PlotPoints -> 2000], {n, 2, 27}]
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: rlbagula at (no spam) sbcglobal.net |
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