| > I have found that explaining and discussing mathematics helps me to
> understand the substance more. I am thinking of writing a few
> expository essays on topology, algebra and perhaps analysis. I wonder
> if there is any interest here.
I would certinly be interested in reading such things. I am
currently very busy, and I couldn't promise to be a very responsive
correspondent. It does seem to me that the topics you raise above are
rather broad (to put it mildly) and it might be better to focus on
something much more tightly. In this way, you'd be able to get beyond
the trivial into some interesting stuff. Especially, it would be nice to
take an area which integrates a whole range of different areas.
As a random example of something that I would like to know more about,
there is a theory which does for differential equations what Galois theory
does for polynomial equations. There is a corresponding result to the
fundamental theorem of Galois theory, but in order to get it, you need to
define a topology on the group, and look at the component containing the
identity under this topology. To give a flavour, I quote the main theorem:
Suppose that M:K is a Picard-Vessiot extension, such that K has characteristic
zero and the field of constants C is aglebraically closed. Suppose that M
can be embedded in a differential field obtained from K by a finite series of
simple algebraic extensions, adjunctions of intgrals or adjunctions of
exponentials of integrals. Then the component of the identity of the
differential Galois group is soluble. Conversely, if the differential Galois
group of M:K has soluble component of the identity, then M can be obtained
from K by a finite normal extension followed by a Liouville extension.
What this means, for instance, is that if you take the differential
equation:
y" - xy = 0
over the field C(x) of rational complex functions, then the differential
Galois group is the full group of unimodular 2x2 matrices over C. This does
not have soluble component of the identity, so you can't start with rational
functions and perform algebraic operations, integrals or exponentiations
of integrals. This means that you can't solve the equation above with
any nice or even moderately nasty formula involving standard functions of
analysis.
However, the equation (airy's equation) does have a simple power-
series solution.
Reference: Kaplanskjy I, (1969) An introduction to differential algebra
Interested?
Regards,
Andrew.
|
| Drat, a typo slipped in: it's "Kaplansky".
Maybe you find this an uninteresting topic, Eugene, but I like
the concrete, and in pursuing this you would learn a lot of topology,
algebra *and* analysis, I think.
Up to you,
Regards,
Andrew.
|
| If you write something about group theory, chaos, or first order logic I'd be
interested in reviewing them.
I've recently started to read a few books on these areas:
'Group Theory', Philip Hall
'Groups, Rings, Modules', Auslander & Buchsbaum
'First Order Logic and Automated Theorem Proving', Melvin Fitting
'Introduction to Chaotic Dynamic(al?) Systems', (some professor @ B.U.)
The second one is quite difficult (very theorem-dense with almost no examples),
and I've been thinking of giving up on it until I finish the first one, which
is more readable.
The first order logic stuff has some applicability to my work (although I'm
not currently trying to get a machine to prove things).
Topology apparently pops up in chaos studies (or at least that's the impression
I'm getting from the 4th book).
|
| re .1, .2,
Oh, I am very interested in such topics. I think you are right that it
is important to get beyond the trivial, but I also think it shouldn't
be too specialized; otherwise, we will lose the readers. I have a
suggestion for the avid participants of this notefile to do some
research and write espository essays in the field that interests them.
This way, we can really start something interesting and dynamic in this
file. Sort of like everyone has to give a report once in a while. To
begin with, Andrew should write a more detailed essay on the subject he
discussed in .1. :-)
Eugene
|
| > Andrew should write a more detailed essay on the subject he
> discussed in .1. :-)
Splutter! :-)
Even if my waves and waves of multiple deadlines were to stop still,
I am currently zapped by (hopefully requited :-) love, and the probability
of being able to bend my mind round Piccard-Vessiot extensions is low at best.
Regards,
Andrew-the-newly-romantic.
|