Suppose that U(x), x > 0 is differentiable positive, monotone
decreasing regularly varying function with parameter m, i.e. U(x*t) /
U(x) -> t^m, when x -> +oo for any t > 0.
Consider any random variable R, such that P(R > 0) = 1 (where P(.) 
ö
is probability of corresponding event), and such that P(R < 1)*P(R >
1) > 0.
Suppose that function f(b) = E[U(x*(1+b*R))*x*R] (where E[.] ö 
is
mathematical expectation) is finite and equation f(b) = 0 (for some
fixed x > 0) has the solution b*.
Then b* is independent of x when x -> +oo, as E[U(x*(1+b*R))*R] / U(x)
-> E[(1+b*R)^m*R].
Let us now replace x by the sequence of random independent (and
independent of R, as well) variables x1,x2,...,xn,... > 0, such that
E[U(x1*...*xn*(1+b*R))] is finite, but -> +oo, when n -> +oo.
Can we now maintain that the solution to f_n(b) =
E[U(x1*...*xn*(1+b*R))*x1*...*xn*R] = 0 is as well b* (i.e. it is
independent of x) when n -> +oo?
====
 I have three different Partial Differential Equations.
> These PDE have the same initial state, we will call it 
              I(x,y,t=0)
 These pde are:
 1:          d/dt I(x,y,t)= y^2 div( grad (I(x,y,t)) )
> 2:          d/dt I(x,y,t)= y div( y grad (I(x,y,t)) )
> 3:          d/dt I(x,y,t)= div( y^2 grad (I(x,y,t)) )
 Boundary conditions are:
> d/dx I= d/dy I= 0
This doesn't directly answer your question, but may be helpful.
I assume you're dealing with a domain 0 <= x <= L, a <= y <= b,
t >= 0, and the boundary conditions are dI/dx = 0 on x=0 and x=L,
dI/dy = 0 on y=a and y=b.  To avoid singularities, you probably 
want to assume a > 0.
The first pde has separation-of-variables solutions
I = cos(m pi x/L) Y(y) exp(-r t) where Y''(y) + (r/y^2 - n^2 pi^2/L^2) Y = 
0
This equation for Y has fundamental solutions 
sqrt(y) I_(sqrt(1/4-r)(pi n y/L) and
sqrt(y) K_(sqrt(1/4-r)(pi n y/L)
where I_m and K_m are modified Bessel functions of the first and second 
kinds.
Determining for what values of r you have nontrivial solutions satisfying 
the
boundary conditions does not look easy.
For separation-of-variables solutions of the second pde, the equation for Y 
is
Y''(y) + (1/y) Y'(y) + (r/y^2 - n^2 pi^2/L^2) Y = 0
which has fundamental solutions 
I_(i sqrt(r))(n pi y/L) and K_(i sqrt(r))(n pi y/L)
And for the third pde, the equation for Y is
Y''(y) + (2/y) Y'(y) + (r/y^2 - n^2 pi^2/L^2) Y = 0
and the fundamental solutions are
y^(-1/2) I_(sqrt(1/4-r)(pi n y/L) and
y^(-1/2) K_(sqrt(1/4-r)(pi n y/L)
In fact, as this hints, the first equation can be transformed to the third:
if u(x,y,t) satisfies the first, then v(x,y,t) = u(x,y,t)/y satisfies the 
third (of course, with different boundary conditions).
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
====
Julian schrieb:
 
> I have three different Partial Differential Equations.
> These PDE have the same initial state, we will call it
              I(x,y,t=0)
 These pde are:
 1:          d/dt I(x,y,t)= y^2 div( grad (I(x,y,t)) )
> 2:          d/dt I(x,y,t)= y div( y grad (I(x,y,t)) )
> 3:          d/dt I(x,y,t)= div( y^2 grad (I(x,y,t)) )
 Boundary conditions are:
> d/dx I= d/dy I= 0
 Due to modeling, I have a strong asumption that the solution will be
> of the following form:
 I(x,y,t) = integral_R(  integral_R+ ( K((s-x)/y,yu,t) I(s,u,0) ds du
 where K would be some kind of Green function.
> I'm looking for a way to identify the function K.
 If someone knows: A proof of a existance (or non-existance) of
> solutions of the reserached form, a way to identify such function K,
> or more generally information about formal resolutions of heat
> diffusion in anisotropic medium, I'll be very glad.
(my knowledge is somewhat rusty)
You have to specify a domain D for (x,y) in R^2 and boundary conditions, 
e.g.
the von-Neumann conditions given. If the domain is appropriate you get an 
equation
of evolution in the hilbert space L^2 (D)
d/dt I= A I
with some linear operator A. In the case 3 it is obvious that A is a
self-adjoint operator if D is appropriate and the solution exists for 
arbitrary
I(0) in L^2. The cases 1 and 2 can be handled (i.e. A is 
symmetric/selfadjoint) with
a slightly modified scalar product (?) or using the fact that (2)-(3) or
(1)-(3)is a
bounded perturbation.
I would call this an inhomogeneous isotropic heat flow. Anisotropc would 
be,
e.g. 
d/dt I= (d2/dx+ 2*d2/dy)  I
hth
Klaus
 Julien
====
Does anyone know a textbook reference for the construction of the 
linking form of a manifold, and the proof that it is non-singular?  By 
linking form I mean the form defined on torsion homology classes of an 
n-dimensional closed manifold, ie lk: T_k(M) times T_{n-k-1}(M) to 
Q/Z.  The geometric version of lk(a,b) is given by choosing an n-k chain 
C with boundary C = rb, and defining lk(a,b) = (1/r) a cdot C where 
cdot is intersection number.  This is discussed in Seifert and 
Threlfall, but that's a bit out of date (1934, with 1980 translation by 
Joan Birman).
A modern version would phrase this in terms of cup products and the 
Bockstein associated to the exact sequence 0to Z to Q to Q/Z to 0. 
The book of Bredon, Geometry and Topology, gives this construction as an 
exercise, just for dimension 3.  A modern proof that it is nonsingular 
would combine this construction with Poincare duality and universal 
coefficient theorem (where the torsion in cohomology is related to the 
aforementioned Bockstein).   Carrying this out is a good exercise for 
grad students learning topology; on the other hand, I would like to find 
someplace where this proof is written down. 
Any suggestions?  I've looked in all the books I can find, with no 
success.
Daniel Ruberman
====
I've recently become interested in knots, and I was wondering if
anyone could clarify a few points for me...
1. I read on mathworld (http://mathworld.wolfram.com/Knot.html) that
it has been proved that knots cannot exist in dimension greater than
or equal to four. Is this actually saying that 1-manifolds embedded in
R^4 are basically equivalent to the unknot?
2. I read elsewhere that we could generalise the definiton of knot to
an embedding of an n-manifold in an n+2-manifold. Is this dependent at
all on the metric structure of the manifold? e.g. are
pseudo-Riemannian knots any different from conventional ones? The
only reason I ask is in relation to Campbell's theorem, where the
number of dimensions needed to embed Riemannian and pseudo-Riemannian
manifolds in locally flat space are different.
3. Is it the case that in for higher-dimensional knots, embedding them
in yet higher dimensional spaces enables us to untie them (in the
sense of question 1)?
I'd be grateful to anyone who could answer these questions!
from Jonny!
====
dir:
I am looking for a newer book on topological groups.
>Something more recent than Ponrjagin, from the
>1970s, 80s or 90s.
>Can someone provide a reference ?
>
You can try:
- Tammo tom Dieck: Transformation groups, de Gruyter Studies in
Mathematics, 1987.
- Katsuo Kawakubo, The theory of transformation groups, Oxford
University Press, 1991.
- C. Allday & V. Puppe, Cohomological methods in transformation
groups, Cambridge studies in advanced mathematics 32, 1993.
The latter one perhaps is more specialized than the other two.
Agust.92 Roig
====
It depends on what kind of topological groups you are interested in.
Hewitt and Ross is of course a classical monograph, unsurpassed to the
date. I would also recommend in addition to it three more recent
books, dealing with various aspects of `large' (= non locally compact)
topological groups.
(1) Dikranjan, Dikran N.; Prodanov, Ivan R.; Stoyanov, Luchezar N.
Topological groups. Characters, dualities and minimal group
topologies. Monographs and Textbooks in Pure and Applied Mathematics,
130.
Marcel Dekker, Inc., New York, 1990. x+287 pp. $99.75. ISBN
0-8247-8047-7
(2) Neretin, Yu. A.
Categories of symmetries and infinite-dimensional groups.
Translated from the Russian by G. G. Gould. London Mathematical
Society Monographs. New Series, 16. Oxford Science Publications.
The Clarendon Press, Oxford University Press, New York, 1996. xiv+417
pp. $115.00. ISBN 0-19-851186-8
(3) Roelcke, Walter; Dierolf, Susanne
Uniform structures on topological groups and their quotients.
Advanced Book Program.
ar44.95. ISBN 0-07-0543412-8
cheers,
Vladimir Pestov
 I am looking for a newer book on topological groups.
> Something more recent than Ponrjagin, from the
> 1970s, 80s or 90s.
> Can someone provide a reference ?
>