Subject: Re: EXTRAORDINARY PI
From: jawalker@beckman.com (Jack Walker)
Date: Fri, 10 Jan 1997 23:09:05 GMT
bruck@pacificnet.net (Ronald Bruck) wrote:
>In article <32CAACF8.4A48@mindspring.com>, Richard Mentock
> wrote:
>:BLStansbury wrote:
>:
>:> But a square can't have an area of exactly 4pi unless its a circle.
>:
>:If it's side is of length 2 times the square root of pi, it can.
>:
>:Let's use a number system, base pi. Then, pi expressed in this
>:number system is exactly 10.
>:
>:Question: What is the square root of pi, in this number system?
>This entire thread is mathematically silly and historically deviant.
>It (the thread) arose with a question about "squaring the circle", and
>continued with comments that while you couldn't square the circle, you
>could come AS CLOSE AS YOU WANT. Bob Silverman observed that you COULD
>square the circle if you allowed other curves than circles and lines (as
>the Greeks well knew). The comments which followed Bob's showed a
>misapprehension of what MATHEMATICS is all about.
>The first (we know) to write down geometric proofs (in general cases,
>instead of specific instances) were the Greeks, and this thread is
>unhistorical because the Greeks did NOT equate lines to real-number lines.
>With their axiom system one could find the intersection of two lines; the
>intersection of two circles; the intersection of a line and a circle; etc.
>If you extend their methods to allowing the intersection of a line and one
>arch of a sine curve, then you CAN square the circle by geometric methods;
>if you allow other cubes, you can duplicate a cube, too.
>It's not a question of calculating numbers. IT HAS NOTHING TO DO WITH
>NUMBERS. It has entirely to do with staying within the system of
>postulates with which you start. It has to do with "constructions", a
>species of mathematical proof. I can well believe Saunders MacLane's
>comment that proofs are no longer taught in geometry; I can see the proof
>of his claim in this very thread!
>Historically, the representation of real numbers as decimals, or to any
>base, is a very late addition to the question. By the way, how DOES one
>write a real number to base pi? Using WHAT as digits?
Oh sure, bring that up.
>--Ron Bruck
>--
>--Now 100% ISDN from this address
Subject: Re: Good book for Applications of Group Theory?
From: Thomas Kerler
Date: Thu, 09 Jan 1997 23:09:48 -0500
Some of the more classical & mathematical ones:
* The first name that always comes to my mind on
that subject is Hermann Weyl. I think his books
on "Symmetry", "Group Theory & Quantum Mechanics",
"Space, Time, Matter" should still be insprirational
for beginners. After all he was one of the pioneers
looking for applications of group theory in nature.
* As a student I remeber having used lecture notes by
Res Jost (another guy from Zurich), which impressed
me because of their mathematical clarity. Especially
a clean way of describing many particle systems, their
gauge and permutation symmetries, and how those mesh
when considering the representation theory.
I don't know if they have been published, but I would
guess a lot has been taken from Wolgang Pauli's lectures
(yet another guy from Zurich), which I think are published.
In both lectures one should find also a variety of more
concrete applications to special experiments.
Personally I'd also look into the really old stuff by Wigner.
* A few days ago I looked again in Thirring's book on
quantum mechanics in his mathematical physics series.
(this guy now is from Vienna, which is not far from Zurich)
One thing from there, I'd probably couldn't resist teaching
in a course on group theory and QM is the derivation of the
spectrum & degeneracies of the hydrogen atom, using only
the representation theory of SO(3), and no diff. eqs.
(The trick is that with the Runge-Lentz-vector you really
have an SO(4) symmetry). In this and the next book also
many particle systems are nicely introduced and applications
discussed.
* As a sophemore undergrad. I once took a course/seminar on
"GT & QM" organized by mathematicians, which was basically
following the book by Simms, Springer Lecture Notes in Math,
(Vol. 52) It is quite limited in its scope but it's a rather
nice derivation of the general formalism fo relativistic
quantum mechanics from first physical principles (causality,
states & observables, ...) in a really clean way (e.g., via
that Mackey-stuff with induced & small representations.) It
also deals with SU(3), can't remember whether it was color or
flavor though.
The only thing that stuck in my mind from Hamermesh's book
is a rather lengthy explanation of what a projective representation
is, but I think Simms' book not only makes more sense in the
mathematical presentation, there you also have a much clearer
physical reasoning, where the projective things come from.
* Unfortunately I don't know the literature in solid state. Of
course the standard stuff on cristallographic groups should be
explained in a lot of books.( Not very riveting; unfortunately
we are not in hyperbolic 3-space, where that would include, e.g.,
all of the platonic solid groups.) There is of course the more
recent subject of quasi-cristals, which can often be decribed
as "projections" of higher dimensional crystals, and related to
that the stuff with tilings that don't have naive translation
invariance. There must a lots of articles out there, but I don't
know much about serious literature.
The thing on group theory and solid states physics that really
fascinated me as a student (and probably still would) are
classifications of defects (vortices etc.) in crystals and fluids,
given the topology of the "inner configuration space" of the
particles. The latter is usually topological group, G, or more
generally a homogeneous space, G/N, and the singularities are
classfied in terms of the fundamental groups of G/N. I remeber
having a lot of fun talking in a solid state student seminar
about what happens, when, e.g., superfluid helium changes phases
and that configuration space changes. I'm not sure if I can
retrieve the relevant literature, but I could go deep-digging
(it was 9 years ago) if you can't get it from anyone else.
I guess you're right that in view of the wealth and beauty of
the subject, most monographs are rather pale and not even
pedagogically brilliant. There should be a Hermann Weyl IIIrd
or so out there, who can put all what is scattered across the
literature into one book or series about the subject of goups
in nature.
My selection obviouosly reflects the facts that I'm mathematically
inclined, have conservative tastes, and did my PhD in Zurich.
I hope there is still one or the other thing, from which you can draw
ideas for your course.
Thomas
Subject: ** structure of reality ** article 3
From: gary.forbat@hlos.com.au (Gary Forbat)
Date: 11 Jan 97 22:10:14
31-12-96
revised 1-1-97
Notes on the structure of reality - article 3
(first draft)
by Gary Forbat
Copyright (c) G. Forbat 1996
It may now be convenient to extend and qualify some of the main
concepts derived from the theory. In the previous essays I described
a process of material formation which provides the basis for the
observed material reality. The process operates through a building
procedure which involves a relationship between the physical
magnitudes of structures, that is, the volume they occupy, and the
rapidity of their internal cycles. Moreover, the process is universal,
ranging over an infinity of scale tranformations from the most
miniscule sizes to the most gigantic imaginable, in fact infinite in
both directions.
But it is not a single dimensional process involving only scale. What
is peculiar about the sequence is that the smaller structures of
the micro world are highly dynamic due to an extremely rapid internal
cycle operating to hold it together, and the smaller the structure,
the more dynamic it is. Dynamics refers to the rapidity of the
cyclical pulse. As particles break down to the cyclical funtion of a
number of smaller components, those components will have a
significantly more rapid internal cyclical rate than those of the
larger structure they contribute to forming. The atomic structure,
for instance, comes into being due to the cyclical function of the
electron in relation to the nucleus. The composition of the electron
has not yet been penetrated, but the possibilities are few. Either it
is composed of a very large number of tiny parts, or maybe fewer but
of a much higher dynamicity. The nucleus, on the other hand, is known
to break down to combinations of smaller, but much more dynamic parts
known as 'quarks'. Quarks themselves must reduce to even smaller
components, with cyclical rates of increasingly more rapidity. The
many qualities of quarks testify to a variance of configurations.
The quantum proportions testify to this very nature. With the
process of reduction infinite, so with it is the increase in
dynamicity.
We are fortunate enough to be able to observe two vastly different
aspect of the material process. The micro scales of phenomena present
an integrated view of average behaviour over many billions of cycles.
Imagine how the solar system would look if billions of planetary
cycles were pressed into a single second. Theoretically at least, it
would be possible to simulate the effect by taking a long term video
of the solar system in motion over many billions of years, and then
replaying the tape over a matter of seconds. Undoubtedly we could
make computer image simulations of it much more easily.
Then there is the almost static view of the process presented
by the structures of the large scale in their 'real time' cyclical
movements. Our viewpoint of stellar formations is fashioned from the
workings of the atomic structure, and compared to the speed and
capacity of the functioning of our instruments and sensing apparatus,
the stellar structures are both extremely large and so slowly evolving
as to be almost static. But now, let's venture to reconstruct in its
broadest principles the consequences of this infinite sequence of
structuring, not only to determine the status of our own viewpoint
within it, but to attempt to discover general principles that may be
directly affecting us and we are not yet aware of. Firstly, going up
or down in scale, the specific attributes of structure types that
occur depend on the interactive possibilities afforded on each
particular scale. Solar systems of one type or another, whether
binary or planetary are the almost exclusive forms that may be found
at the scale of the direct interaction between the most massive
atomic conglomerations. At this scale of consideration the universe
can be seen to be interspersed with stellar and planetary matter in
mutual interaction as solar systems. But we know that solar systems,
in turn, almost exclusively congregate in the larger massive
formations of galaxies, occuring in a small number of types. Galaxies
themseves form clusters with unique characteristics types of their own.
On the galactic scale of consideration the universe can be seen as
interspersed almost exclusively by galactic formations. Certainly they
are the only long term stable forms to be found at this scale.
In fact we can apply this principle at any level of magnitude. Thus
the universe is interspersed by atoms at the atomic scale of
consideration but with planetary/stellar matter on a larger scale.
So then, as the process builds to infinity, with each structure type
occuring in forms and attributes appropriate to interaction and
formation possibilities at that scale. Each transformation produces
unique structure types, and there is certainly no likelyhood of the
same structure type occuring at different levels either in the micro
and macro scales.
Both the reduction and its reverse process of expansion runs to infinity,
with the roots of each or any structure traceable in infinite steps
toward smaller scales. But this does not work in the reverse toward the
macro. The reason is that not all structures continue to build outward
forever. Large sections of it terminate at a certain level, as in the
case of the structures that intersperse in our seemingly empty spatial
regions. My findings are that these regions are far from empty.
The entire spatiality in fact contains a fine invisible mist of matter,
structured at its highest level to an interactive fabric to form
a micro infrastructure which sets the framework for the workings of
our atomic based matterial environment. But only those elements
which participate in further building processes to form the atomic
base can get through to build outward to form structures on larger
scales. The rest, indeed a very large portion of micro material,
is lost to further structuring. In this infinite chain of
expansions it should be expected that terminal stages are reached
from time to time. Nevertheless, what remains after each of these
mass terminations is still adequete to reconstruct other equally
thickly populated levels of structures on much larger scales.
So what is the status of our material system amid this infinity of
transformation levels ? On the micro end we observe the process through
a very high integration, but on the macro end it tends toward static.
With the two directions reflecting merely different aspects of a
single process, our observational access results from the circumstances
of our evolution as sensing beings and our relation to the material
interaction that brought it about. We are a direct product of our
micro infrastructure and the atomic base. The question remains
whether ours is the only material environment possible or whether
there may be others ? Perhaps other configurational circumstances can
exist among an infinity of types which produces alternative material
bases.
We need firstly to examine the general circumstances which must be
present for a material environment. Obviously the most evident
is the versatility of our atomic structure. It is extremely stable
and durabile with, stability, regularity, as well as variability in
chemical combination. It is truly like a wonder particle which goes on
to create a tremendously varied and interactive world of material
activity. Surely it would be fairly rare to find a scale level of
structuring where such a useful type of particle is found.
Nevertheless it stands to reason that in a infinite chain of
transformations other similarly efficient structure types are bound
to occur. some may indeed be even more flexible than the atom, or
perhaps somewhat less so, but still able to generate a causal
evolution in its conglomerate forms to create an alternative material
environment rivalling ours. Of course on the micro scales a funtional
world would evolve extremely rapidly compared to ours, and on the macro
scales the events would take on gigantic proportions, evolving very
slowly by our way of looking at it.
G. Forbat
to be continued in the next article
Subject: Re: How to determine what values are small enough to be set to zero in SVD?
From: "Hans D. Mittelmann"
Date: Sat, 11 Jan 1997 06:40:58 -0700
Billy Leung wrote:
>
> In SVD solution of linear equation, one often has to zero out certain
> small values to proceed. How do you actually determine a value is small
> enough in reference to a particular problem?
>
> Thanks for your insight
Hi,
you may look at section 5.5.8 in Golub&van; Loan, Matrix Computations.
The section is titled "Numerical Rank and the SVD". Basically, if u is
your unit roundoff quantity
u=.5base^(1-t)
base, the base (2) of the arithmetic, t the mantissa length. Then,
without further errors in the matrix A you set those singular values to
zero which are below u||A||_inf which is the infinity (or row-sum) norm
of A. If, however, you think the elements of A have only two leading
correct decimal digits, then use 10^(-2)||A||_inf etc.
Hope that helps.
Subject: Multi-Dim Numerical Integration Question
From: "W. Bryan Bell"
Date: Sat, 11 Jan 1997 16:34:57 -0600
Hello,
This maybe a somewhat trivial problem but I am trying to
numerically perform a Multi-dimensional integral of a 2D Gaussian
function [ie. Gauss(f(x,y,w,...),g(x,y,w,...))] where numerical
accuracy is required for regions out to 100*sigma (deep in the
tails). Plain rectangular integration works fine but it takes
quite a long time and other funcional results can vary with the
step size. I have also tried Low-order (10th) Multi-dimensional
Gaussian-Quadrature integration but it has a tendency to loose
accuracy in tails of the Gaussian. Should I try and use a
higher-order Gaussian-Quadrature technique? Is there a possible
trick I could employ? [f(*) and g(*) are highly non-linear
functions that vary a great deal so reduction by hand is out
of the question]. Any suggestions or comments would be greatly
appreciated.
Please e-mail any suggestions, I am not a regular visitor of this
newsgroup. [Maybe I should be! :-)]
Bryan Bell
bellwb@lmtas.lmco.com
Subject: Re: Multi-Dim Numerical Integration Question
From: "Hans D. Mittelmann"
Date: Sat, 11 Jan 1997 16:37:44 -0700
W. Bryan Bell wrote:
>
> Hello,
>
> This maybe a somewhat trivial problem but I am trying to
> numerically perform a Multi-dimensional integral of a 2D Gaussian
> function [ie. Gauss(f(x,y,w,...),g(x,y,w,...))] where numerical
> accuracy is required for regions out to 100*sigma (deep in the
> tails). Plain rectangular integration works fine but it takes
> quite a long time and other funcional results can vary with the
> step size. I have also tried Low-order (10th) Multi-dimensional
> Gaussian-Quadrature integration but it has a tendency to loose
> accuracy in tails of the Gaussian. Should I try and use a
> higher-order Gaussian-Quadrature technique? Is there a possible
> trick I could employ? [f(*) and g(*) are highly non-linear
> functions that vary a great deal so reduction by hand is out
> of the question]. Any suggestions or comments would be greatly
> appreciated.
>
> Please e-mail any suggestions, I am not a regular visitor of this
> newsgroup. [Maybe I should be! :-)]
>
> Bryan Bell
> bellwb@lmtas.lmco.com
Hi,
I am not a specialist in quadrature. There may be specialized approaches
to your case. However, a new, high-quality quadrature package is
CUBPACK. You can retrive it through the homepage of its author:
http://www.cs.kuleuven.ac.be/~ronald/
Hope that helps
