mm-78
===

Suppose that U(x), x > 0 is differentiable positive,
monotonedecreasing 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(.) 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[.] is mathematical expectation)
is ?ite and equation f(b) = 0 (for some?ed 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 (andindependent of R,
as well) variables x1,x2,...,xn,... > 0, such
thatE[U(x1*...*xn*(1+b*R))] is ?ite, 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 isindependent 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= 0This doesnt directly answer your question, but
may be helpful.I assume youre 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 ?st pde
has separation-of-variables solutionsI = cos(m pi x/L) Y(y)
exp(-r t) where Y(y) + (r/y^2 - n^2 pi^2/L^2) Y = 0This
equation for Y has fundamental solutions sqrt(y)
I_(sqrt(1/4-r)(pi n y/L) andsqrt(y) K_(sqrt(1/4-r)(pi n
y/L)where I_m and K_m are modi?d Bessel functions of the
?st and second kinds.Determining for what values of r you
have nontrivial solutions satisfying theboundary conditions
does not look easy.For separation-of-variables solutions of
the second pde, the equation for Y isY(y) + (1/y) Y(y) +
(r/y^2 - n^2 pi^2/L^2) Y = 0which 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 isY(y) + (2/y) Y(y) +
(r/y^2 - n^2 pi^2/L^2) Y = 0and the fundamental solutions
arey^(-1/2) I_(sqrt(1/4-r)(pi n y/L) andy^(-1/2)
K_(sqrt(1/4-r)(pi n y/L)In fact, as this hints, the ?st
equation can be transformed to the third:if u(x,y,t) satis?s
the ?st, then v(x,y,t) = u(x,y,t)/y satis?s the third (of
course, with different boundary conditions).Robert Israel
israel@math.ubc.caDepartment 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.> Im 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,
Ill 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 equationof evolution in the hilbert
space L^2 (D)d/dt I= A Iwith some linear operator A. In the
case 3 it is obvious that A is aself-adjoint operator if D is
appropriate and the solution exists for arbitraryI(0) in L^2.
The cases 1 and 2 can be handled (i.e. A is
symmetric/selfadjoint) witha slightly modi?d scalar product
(?) or using the fact that (2)-(3) or(1)-(3)is abounded
perturbation.I would call this an inhomogeneous isotropic
heat ?nisotropc would be,e.g. d/dt I= (d2/dx+ 2*d2/dy)
IhthKlaus> > 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 de?ed 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 de?ing lk(a,b) = (1/r) a
cdot C where cdot is intersection number. This is
discussed in Seifert and Threlfall, but thats 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 coef?ient
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 ?d someplace where this proof is
written down. Any suggestions? Ive looked in all the books I
can ?d, with no success.Daniel Ruberman===Ive recently
become interested in knots, and I was wondering ifanyone
could clarify a few points for me...1. I read on mathworld
(http://mathworld.wolfram.com/Knot.html) thatit has been
proved that knots cannot exist in dimension greater thanor
equal to four. Is this actually saying that 1-manifolds
embedded inR^4 are basically equivalent to the unknot?2. I
read elsewhere that we could generalise the de?iton of knot
toan embedding of an n-manifold in an n+2-manifold. Is this
dependent atall on the metric structure of the manifold? e.g.
arepseudo-Riemannian knots any different from conventional
ones? Theonly reason I ask is in relation to Campbells
theorem, where thenumber of dimensions needed to embed
Riemannian and pseudo-Riemannianmanifolds in locally ?ce
are different.3. Is it the case that in for
higher-dimensional knots, embedding themin yet higher
dimensional spaces enables us to untie them (in thesense of
question 1)?Id 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 inMathematics, 1987.- Katsuo Kawakubo, The
theory of transformation groups, OxfordUniversity Press,
1991.- C. Allday & V. Puppe, Cohomological methods in
transformationgroups, 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 thedate. I would also recommend in addition to
it three more recentbooks, 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
grouptopologies. Monographs and Textbooks in Pure and Applied
Mathematics,130.Marcel Dekker, Inc., New York, 1990. x+287 pp.
$99.75. ISBN0-8247-8047-7(2) Neretin, Yu. A.Categories of
symmetries and in?ite-dimensional groups.Translated from the
Russian by G. G. Gould. London MathematicalSociety Monographs.
New Series, 16. Oxford Science Publications.The Clarendon
Press, Oxford University Press, New York, 1996. xiv+417pp.
$115.00. ISBN 0-19-851186-8(3) Roelcke, Walter; Dierolf,
SusanneUniform structures on topological groups and their
quotients.Advanced Book Program.ar44.95. ISBN
0-07-0543412-8cheers,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 ?>