Newsgroup sci.math.research 6560

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		Hi!
	I was wondering if someone could give me some bibliographical references on
semi-analytic sets, and mainly the work done by Lojasiewicz in that field. I was
especially looking for something that would give me a good synthetic approach
to the question and that would clearly state the main results.
		thanks,
			Thierry Zell.

In one of his first articles on comparative prime number theory, P.
Turan remarks, that Haselgrove has shown, that for every module q and
every character \chi mod q there is a zero \rho of L(s, \chi), such that
no other L-series mod q vanishes there.
Is there any reference? Or can this theorem be deduced from some other
known theorems on the distribution of zeros?
JCP

418rrb767 wrote:
> 
> Let M be a complex manifold, real dim 2n.  Let D be a 2k dim'l smooth
> (i.e. C^(infinity)) distribution in TM, the real tangent bundle to M.
> Assume D in J invariant, where J is the complex structure M has.  Thus at
> each p in M, J(p):D(p)->D(p).  Now assume D is integrable, [d1,d2] is
> contained in D, for any smooth sections di of D.  Thus D has integral
> manifolds in M through each point.  Each such integral manifold inherits
> an almost complex structure J from M, and in fact this almost complex
> structure is integrable, so each leaf of the foliation defined by D has
> the structure a complex submanifold of M (the integrability of J on each
> leaf, L, occurs because smooth sections of the +i eigenbundle of J on
> TL-tensor-C Lie bracket to vectors in TL-tensor-C and this bracket also
> stays in the +i eigenbundle of J on TM-tensor-C, b/c M is complex).
> Thus, the +i eigenbunbdle of TL-tensor-C, for each leaf L, yields a
> C^(infinity) subbundle, V, of the holomorphic tangent bundle to M.
> 
> Question: can it happen that V is NOT a holomorphic subbundle of the
> holomorphic tangent bundle to M, or does it follow automatically that V
> must be holomorphic?
No, in fact it is a very rigid condition that V be holomorphic.  
> I believe that nonholomorphic smooth subbundles of a holomorphic tangent
> bundle can exist, but can they exist also in this "integrable" setting?
Definitely. 
-- 
David L. Johnson         		dlj0@lehigh.edu, dlj0@netaxs.com
Department of Mathematics		http://www.lehigh.edu/~dlj0/dlj0.html
Lehigh University
14 E. Packer Avenue			(610) 758-3759
Bethlehem, PA 18015-3174

Dear sci.math.research
While implementing a new electronic device, which we call a
"stacatto" device, my colleagues and I discovered a way
to transform real numbers over the range (0-1) into
remarkably intricate (and beautiful) power spectra.
We wonder what these power spectra mean in terms of number
theory?  (We *know* what they mean in practical engineering
terms!)
Here's how our device turns any real number "a" into a 
sequence of zeroes and ones ... the Fourier power spectrum
of this sequence is the output of interest.
(FYI, in our device "zero" means a hardware decision not to
apply a power pulse at the current clock cycle, while "one"
means *do* apply a power pulse at current clock cycle... but
the details of the physical system involved are not
relevant to the mathematics)
Step 1: define a clock function c(t) as follows
              c(t) = \cos(\pi t)
Step 2: define a reference function r(t) which runs
        at a frequency a/2 as follows
              r(t) = \cos(\pi a t)
Step 3: define a coupling function k(t) whose
        value is always either one or minus one.
        Furthermore, k(t) is allowed to change sign
        *only* at decision times t = {0,1,2,3,4,5 ...}.
Step 4: At each decision time t=n, choose the
        sign of k(t) such that the following 
        measure is maximized 
              \int_n^{n+1} dt k(t)r(t)c(t)
Step 5: Take the time derivative of k(t), yielding
        k'(t) as a a sequence of delta functions 
        with coefficients taking values {0,2,-2}
The power spectrum of k'(t) is the quantity of interest.
We are posting this to sci.math.research because the
resulting power spectra are of surpassing beauty!
On a Mathematica simulation of sequences of 32,768
points, averaged over 100 realizations, there is
fractal structure down the the finest detail the
FFT can resolve.  It makes a big difference whether
a is rational, near-rational, or irrational, but
not in any way that is easy for us to quantify.
We post this in the hope that someone can address
the practical problem of identifying values of "a" 
such that the resulting power spectra carries minimum 
power at frequency "a".
If not, I hope you enjoy this funny way that number
theory arises in a practical engineering application.
Sincerely  John Sidles

In article <58kc5o$1qj@hobbes.cc.uga.edu>, kasman@alpha.math.uga.edu (Alex Kasman) writes:
|> Suppose you have a linear ordinary differential equation of order n
|> with variable coefficients.  I will use d to indicate the differential
|> operator d/dx and then we can write this equation by specifying a
|> differential operator
|> 
|> L = d^n + c_{n-1}(x) d^{n-1} + .... + c_1(x) d + c_0(x)
|> 
|> and the equation is L*u=0 for an unknown function u(x).  As is well
|> known, there will be an n-dimensional space of solutions u(x).
|> 
|> I am interested in perturbing the equation by adding a small
|> constant.  In particular, consider the equation (L+z)*u=0 for complex
|> parameter z near 0.  At each fixed z there is an n-dimensional space
|> of solutions u(x).
|> 
|> MY QUESTION IS: can I necessarily find n-functions u_i(x,z)
|> [1<=i<=n] which 
|> 
|> a) are analytic in z at z=0 
|> b) solve the equation (L+z)*u=0 in a neighborhood of z=0
|> 
|> and 
|> 
|> c) are linearly independent functions of x in a neigborhood of z=0?
|> 
|> If you know the answer to this question and can prove it or give me
|> relevant references I would be most grateful.  (By the way, I have
|> already been told that the answer is yes by two reliable
|> mathematicians and that the answer is no by two others, this is why I
|> am insisting on a proof or reference.)
|> 
|> Thank you very much for your help,
|> 
|> Alex Kasman
|> 
I am not much aquainted with your gegeral case. But clearly the case of constant
coefficients is included. In that case the answer is "no" _in general_
If the characteristic polynomial has a multiple root for z=0, the roots of  
the perturbed polynomial are analytic in z**(1/p), p being the multiplicity.
and therefore the solution manifold, now coming from n _different_ roots
cannot be analytic in z. for a proof of the perturbation properties of 
polynomial roots you may have a look e.g. at Ostrowski: solution of nonlinear
equations and systems of equations.
hope this helps
peter

t1207ad@hp22.lrz-muenchen.de wrote:
> 
> I'd like to know:
> 
>   Who discovered the group isomorphy between
> 
>   -  the proper orthochronous Lorentz group 
>   -  the group PSL(2,C) of the complex Moebius tansformations  ?
> 
> Was this fact already known to Poincare or Minkowsky?
> Or was it perhaps found by Dirac ?
> Where was it first published?
	I can't claim to know who first discovered it.  The first
explicit reference that I know of occurs at the end of Cartan's 1914 
paper 'Les groupes reels, simples, finis et continus', where he 
classifies the real forms of the complex simple Lie groups.  I would
be surprised, though, if it wasn't known before in some form.  
Unfortunately, Cartan does not give any historical references in
that paper, although, if you look at his book on spinors, you might
find something.  (I don't have mine handy.)  
	The complex exceptional isomorphism A_1 x A_2 = D_2 (of which the
above is one of the real forms) was known to Lie and Klein sometime
in the 1880's (at least).  I don't know if they knew this particular
real form, or its importance.  It would be surprising if Poincare
and Minkowsky did not know it, but I don't know their works well
enough to know where to look.
> 
> Does this exciting relation find any deeper treatment in current 
> mathematical research?
	Well, there are the other `exceptional isomorphisms', such
as Spin(3) = SU(2), Spin(3,2) = Sp(2,R), etc. Understanding how
these arise requires a deeper treatment.  Beyond that, there are
at least two directions to go:
	(1) Penrose's treatment of relativity using spinors and
twistors takes this isomorphism as fundamental.  See Penroe and
Rindler's "Spinors and space-time", which starts with a discussion
of this isomorphism.
	(2) Taking the isomorphism SL(2,C) = Spin(3,1) as the starting
point, you can go towards spin geometry in general.  A good book
that treats spinors of all signatures in a uniform way is F. R. 
Harvey's "Spinors and calibrations."
	Yours,
	Robert Bryant

Right you are.  Thank you. (And to boot, the poster points out that I
mis-read the question.)  I have cancelled my posting.
			  -Jonathan
israel@math.ubc.ca (Robert Israel) writes:
> 
> In article , Jonathan King
>  writes: 
> |> (Cited posting is below.)
> |>   Yes, even without the hypothesis of the {G_j}_j being decreasing.
> |> Indeed, you can trap the *union* of the  G_j, rather than just the
> |> intersection. 
> |> 
> |>   Letting e mean epsilon, and  X be your Euclidean space, for a set G,
> |> the set 
> |> 	e-ball(G^c)
> |> is an open superset of the complement of G, so
> |> 	X - (e-ball(G^c))
> |> is closed.  Intersecting this with the closed disk (centered at the
> |> origin) of radius 1/e gives a compact subset of G.
> |> 
> |>   Let K_n be the resulting set starting from G=G_n and e=1/n.
> |>   Now consider a point x which is in some G_j; WLOG x is in G_0.  Pick n
> |> large enough that x is in the disk of radius n.  Increase n still
> |> further that the distance from x to the complement of G_0 exceeds 1/n.
> |> (Here is where the openness of the G_j is used.)  Then x is in K_n.
> 
> No.  Obvious counterexample: let G_n be a disjoint union of open balls of
> radius less than 1/n.  Then all your K_n are empty.
>   
> Robert Israel                            israel@math.ubc.ca
> Department of Mathematics             (604) 822-3629
> University of British Columbia            fax 822-6074
> Vancouver, BC, Canada V6T 1Y4
> 

Might anyone know how to effectively compute the monodromy group of
an n-th order, linear, homogeneous, ODE with polynomial coefficients
in the independent variable?
What is the relation between the monodromy group of an ODE and the
differential Galois group?