mm-266
===
Subject: Paper published by Algebraic and Geometric Topology
Content-Length: 2191
Originator: rusin@vesuvius
The following paper has been published:
Algebraic and Geometric Topology
URL:
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-7.abs.html
Title:
Enrichment over iterated monoidal categories
Author(s):
Stefan Forcey
Abstract:
Joyal and Street note in their paper on braided monoidal
categories
[Braided tensor categories, Advances in Math. 102(1993) 20-78]
that
the 2-category V-Cat of categories enriched over a braided
monoidal
category V is not itself braided in any way that is based upon
the
braiding of V. The exception that they mention is the case in
which V
is symmetric, which leads to V-Cat being symmetric as well. The
symmetry in V-Cat is based upon the symmetry of V. The
motivation
behind this paper is in part to describe how these facts
relating V
and V-Cat are in turn related to a categorical analogue of
topological
delooping. To do so I need to pass to a more general setting
than
braided and symmetric categories -- in fact the k-fold monoidal
categories of Balteanu et al in [Iterated Monoidal Categories,
spaces is a good guide for how to define the concept of
enrichment
over various types of monoidal objects, including k-fold
monoidal
categories and their higher dimensional counterparts. The main
result
is that for V a k-fold monoidal category, V-Cat becomes a
(k-1)-fold
monoidal 2-category in a canonical way. In the next paper I
indicate
how this process may be iterated by enriching over V-Cat,
along the
way defining the 3-category of categories enriched over V-Cat.
In
future work I plan to make precise the n-dimensional case and
to show
how the group completion of the nerve of V is related to the
loop
space of the group completion of the nerve of V-Cat.
Secondary: 18D20
Keywords:
Loop spaces, enriched categories, n-categories, iterated
monoidal
categories
Author(s) address(es):
Department of Mathematics, Virginia Tech
460 McBryde Hall, Blacksburg, VA 24060, USA
Email: sforcey@math.vt.edu
===
Subject: Re: RHom
Content-Length: 1240
Originator: rusin@vesuvius
> Of course the product is supported in S.
> Let U be an open set disjoint to S. Then the restriction to
U of the
> product sheaf is the product of the restrictions which are
zero.
Is the product sheaf quasi-coherent?
> The torsion example is not an example, as , say for Z, you
have to fix
> an integer N and speak of N-torsion. Now the product of
N-torsion
> groups indeed is N-torsion.
Let F be the product sheaf, Q the largest quasi-coherent
subsheaf,
M the product module, and N the largest submodule supported on
S.
What happens, in my opinion, is that N is the module of global
sections of Q. I agree that F is supported on S, but I don't
see
why N would be A-injective.
> You do not need to know that T-injectives are A-njectives.
> All you need is that every object in T has aresolution in T
which is
> A-injective.
This is the same: Assume that every object X in T embeds into
an
object of T which is A-injective. If X is T-injective, the
embedding splits, and X is A-injective.
It seems to me your argument would imply that every
quasi-coherent
sheaf on an affine scheme has a resolution by quasi-coherent
sheaves which are injectives as sheaves. But Verdier gives
a counterexample in SGA6II App. I.
===
Subject: RHom
Epigone-thread: tweldprarpril
Originator: israel@math.ubc.ca (Robert Israel)
Ok, I agree, the product of quasi-coherent sheaves need not be
quasi-coherent.
Take the product of Z/pZ over all primes p as an example.
But perhaps it is even easier. Start with a module M. For each
point p
in the support of M fix an injective A_p module I_p and an
injection
M_p -> I_p.
Then let I be the product of all these I_p.
This then is an injective module and the map M to I is
injective.
What about the support? The only bad thing that could happen
is that
(I_p)_q is non-zero.
Then p must lie in the closure of q and (I_p)_q=I_q.
Can one choose I_p so that this does only happen if M_q is
non-zero?
One should start with an arbitrary injective J_p and then set
I_p
equal to the intersection of all kernels of the maps J_p ->
J_q over
all generalizations q of p for which M_q is zero.
This I_p still contains M_p and I think it still is injective.
Anton
===
Subject: Re: RHom
Originator: israel@math.ubc.ca (Robert Israel)
(Recall that A is a commutative ring, S a set of prime ideals
and
T the category of A-modules supported ans S.)
Since T is Grothendieck, it has injective hulls, and the
following
conditions are equivalent
(1) each object of T embedds into an injective A-module
supported on S,
(2) each injective of T is A-injective,
(3) the injective hull in Mod A of any object of T is in T.
Using Hartshorne, Alg. Geom., Lemma III.3.2 it is easy to see
that
theese conditions hold when A is noetherian. A possible track
to
a counterexample (in the non noetherian case) would be to look
at Exercise III.3.8 (same book).
What's your feeling about this?
===
Subject: quasilinear parabolic equations
Epigone-thread: wherkhirbloy
Content-Length: 467
Originator: rusin@vesuvius
I am aware of the standard local existence proofs for
quasilinear
parabolic equations on R^{n} which use the Sobolev spaces
H^{k}. What
happens if the initial data falls off too slowly as 1/r -> 0
to lie
in H^{k}? Are there existence results available using other
Bananch
spaces?
A related question, under what assumptions will asymptotic
behavior of
intial data be preserve under parabolic evolution?
Any information would be greatly appreciated.
Todd Oliynyk
===
Subject: Finding permutations whose products have known cycle
type
Originator: tchow@markov.mit.edu.mit.edu (Timothy Chow)
Content-Length: 1615
Originator: rusin@vesuvius
Fix positive integers n and k. I secretly pick n permutations
pi_1,
pi_2, ..., pi_n in the symmetric group S_k. Your goal is to
determine
my permutations in time polynomial in n and k. I will answer
queries
of the following form: You may pick any j distinct numbers i_1,
i_2, ..., i_j from 1 to n and ask me for the cycle type (i.e.,
conjugacy class) of the product pi_(i_1) * pi_(i_2) * ... *
pi_(i_j).
Each such query takes constant time. Can you achieve your goal?
Clarifying note: Even if you were to submit all possible
queries,
this might not be enough to determine the secret permutations
uniquely.
So, you may consider yourself successful if you construct
permutations
sigma_1, ..., sigma_n such that the sigmas agree with the pis
on all
possible queries. So in particular, asking all queries and
exhaustively
searching for sigmas will work, but takes exponential time.
I think I can show that if the set pi_1, ..., pi_n includes
enough
transpositions, then the problem is solvable. In general,
though,
I am at a loss.
Perhaps the following related question has been studied? Let x
be an
unknown element of a finite group G and let a_1, ..., a_n be
known
elements of G. Suppose that the conjugacy classes of a_1*x,
..., a_n*x
are known. What are some sufficient conditions for these
equations to
determine x uniquely?
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the
artillery---however great, will
never exceed four of those miles of which as many thousand
separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two
New Sciences
===
Subject: Re: I need help with a SDE
Epigone-thread: strahwhemfrerl
Content-Length: 307
Originator: rusin@vesuvius
If the original SDE is:
dX = (a + b*X)dt + (c + d*X)dW
by following your hint I can define y = c + d*X, apply Ito's
lemma
and transform the SDE so that the diffusion term is d*y*dW.
That
transformed equation is an Orstein-Uhlenbeck process that can
be
solved.
Mariano
===
Subject: Geometry Program, Auckland 2005
Originator: rusin@vesuvius
The New Zealand Institute of Mathematics and its Applications
[http://www.nzima.auckland.ac.nz] is sponsoring a thematic
program on
Geometry: Interactions with Algebra and Analysis
based at The University of Auckland from January-June, 2005.
Program themes:
The program will focus on geometric themes including:
(1) Discrete groups;
(2) Algebraic groups;
(3) Geometric group theory;
(4) Low-dimensional topology and hyperbolic geometry;
(5) Geometric function theory;
(6) Analysis and PDEs.
The two central events of the program are:
1. Summer Workshop, Napier, January 8-15, 2005
The provisional list of principal speakers includes:
* Ben Andrews (Canberra)
* Craig Evans (Berkeley)
* Martin Liebeck (Imperial College)
* Alex Lubotzky (Jerusalem)
* Peter Sarnak (Princeton)
Each will deliver a series of lectures intended for a general
mathematical audience, including senior undergraduate and
graduate students.
The workshop will also include a day of lectures by leading
international researchers to celebrate Fred Gehring's 80th
year.
As usual, we will cover accommodation and some other costs
for NZ-based participants in the workshop.
2. International conference, Auckland, February 14-18, 2005
This meeting will be more traditional in format, featuring a
larger number of invited single research-focused lectures.
The provisional list of principal speakers includes:
* Marston Conder (Auckland)
* Rob Howlett (Sydney)
* Bill Kantor (Oregon)
* Laci Kovacs (Canberra)
* Gus Lehrer (Sydney)
* Martin Liebeck (Imperial College)
* Gunter Malle (Kassel)
* Colin Maclachlan (Aberdeen)
* Chuck Miller (Melbourne)
* Cheryl Praeger (University of Western Australia)
* Peter Schmid (Tuebingen)
* Akos Seress (Ohio State University)
* Aner Shalev (Jerusalem)
There will be opportunities for contributed talks.
Graduate Student Scholarships:
As part of the program, Masters and PhD scholarships are
available for suitably qualified candidates. We particularly
welcome your suggestions of suitable candidates.
Additional information:
The WEB site for the program is
http://www.math.auckland.ac.nz/Conferences/2005/
geometry-program
It contains more information on the program, its activities,
and
on funding opportunities for NZ-based participants and
students.
Some of the international participants will spend additional
time
in New Zealand in conjunction with the program. An up-to-date
register of participants and their visit dates will be
available
later on the WEB site. We welcome enquiries from others
interested
in taking part.
Best wishes.
Gaven Martin and Eamonn O'Brien
Programme directors
Department of Mathematics
University of Auckland
===
Subject: Re: constructing Hadamard Matrices
Originator: israel@math.ubc.ca (Robert Israel)
Will,
I'm not going to have time to finish the program as I have
other more
important work, but I welcome anyone else to try it.
Craig
===
Subject: Re: Compact set
Originator: israel@math.ubc.ca (Robert Israel)
 3. Does `compact separable' imply `metrizable'? I know by
Urysohn's
> theorem that `compact second countable' does, so it is
equivalent to
> ask
> 3'. Does `compact separable' imply `second countable'?
 No. cd space 105, in Counterexamples in Topology
> I^I where I = [0,1].
> I^I isn't first countable, hence not 2nd countable nor
metrizable.
> It's separable by Marcewski-Hewitt theorem.
If by a separable space you mean a topological space with a
countable dence subset then it is easy to show that any
separable
Hausdorff topological space has cardinality less that or equal
to c.
So, by this, I^I is not separable since its cardinality is 2^c.
===
Subject: Re: Compact set
Originator: israel@math.ubc.ca (Robert Israel)
>> 3. Does `compact separable' imply `metrizable'? I know by
Urysohn's
>> theorem that `compact second countable' does, so it is
equivalent to
>> ask
>> 3'. Does `compact separable' imply `second countable'?
>> No. cd space 105, in Counterexamples in Topology
>> I^I where I = [0,1].
>> I^I isn't first countable, hence not 2nd countable nor
metrizable.
>> It's separable by Marcewski-Hewitt theorem.
>If by a separable space you mean a topological space with a
>countable dence subset then it is easy to show that any
separable
>Hausdorff topological space has cardinality less that or
equal to c.
This is false. The best you can get is 2^c.
>So, by this, I^I is not separable since its cardinality is
2^c.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Law of large number in a finite set with distance
measure
Originator: israel@math.ubc.ca (Robert Israel)
(I repost it to sci.math.research as I do not get any response
at sci.stat.math. Hopefully readers of this group can shed
some light
of the problem below. :-) )
===
[ Subject: Law of large number in a finite set with distance
measure
]
Consider a discrete set X in which a distance d(x,y) that
satisfies the
triangle inequality is defined.
Suppose there is a probability measure P of X such that the
mean of P is m. The mean can be defined by, say,
m = argmin_{x in X} sum_{y in X} d(y, x) P(y)
Now, based on P, we get a sample of size D from X,
D = {x[1], x[2], ..., x[n]}.
We then find the mean of D. This can be defined by
hat{m} = argmin_{ x in X } sum_i d( x[i], x )
When n tends to infinity, I expect hat{m} tends to m.
Is this true? If yes, Can you give me some references that
contain the proof? If no, is it possible to impose
additional assumptions on P to make hat{m} go to m?
Also, can anything be said about
the rate of convergence, if it can be defined after all?
===
Subject: Re: Law of large number in a finite set with distance
measure
Originator: israel@math.ubc.ca (Robert Israel)
>(I repost it to sci.math.research as I do not get any response
> at sci.stat.math. Hopefully readers of this group can shed
some light
> of the problem below. :-) )
===
>[ Subject: Law of large number in a finite set with distance
measure
]
>Consider a discrete set X in which a distance d(x,y) that
satisfies the
>triangle inequality is defined.
>Suppose there is a probability measure P of X such that the
>mean of P is m. The mean can be defined by, say,
>m = argmin_{x in X} sum_{y in X} d(y, x) P(y)
This is more like the median than the mean. It may not
be unique. Whether d satisfies the triangle inequality
or is even symmetric is not of great importance, although
it might be in looking at the asymptotics if X is
infinite; the triangle inequality would enable a
somewhat easier proof, but not solve all the problems.
>Now, based on P, we get a sample of size D from X,
>D = {x[1], x[2], ..., x[n]}.
>We then find the mean of D. This can be defined by
>hat{m} = argmin_{ x in X } sum_i d( x[i], x )
>When n tends to infinity, I expect hat{m} tends to m.
Certainly if X is finite, hat{m} will tend to the
set of medians almost surely.
>Is this true? If yes, Can you give me some references that
>contain the proof? If no, is it possible to impose
>additional assumptions on P to make hat{m} go to m?
>Also, can anything be said about
>the rate of convergence, if it can be defined after all?
The probability that hat{m} is not in the set of medians
tends to 0 exponentially in the finite case. If there are
two medians, they are roughly equally likely for large n,
but if there are more, I do not know the asymptotics.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Books on several complex variables?
Originator: israel@math.ubc.ca (Robert Israel)
> I'm looking for suggestions on good introductory books on
the theory
> of several complex variables. I'm mainly interested in the
algebraic
> side of the theory; specifically for its applications to
Algebraic
> Geometry. I've looked at Hormander and Krantz, but they seem
more
> analytic. I looked at Taylor, but I'd really like to see the
theory
> developed for its own sake before I start really getting
into the
> applications. Gunning and Rossi is hard to find, and
Gunning's
> rewrite is ludicrously expensive. Are there any other good
ones I
> should be looking out for?
>
The cited books by Herve and Gunning-Rossi are somewhat
outdated, since
they treat only those aspects which amount to the
consideration of
reduced complex spaces. My favourite is Grauert and Remmert:
Coherent
Analytic Sheaves. Afterwards, have a look at their Analytische
Stellenalgebren for the algebraic aspects (seems to exist only
in
German) and Theory of Stein Spaces for important geometric
applications.
===
Subject: Pointwise Ergodic Theorem
Originator: israel@math.ubc.ca (Robert Israel)
Suppose you have a map T: X -> X for which the probability
measure v
is invariant and ergodic. Also suppose v << m where m is
lebesgue
measure. Is it possible to show convergence in the pointwise
ergodic
theorem for m almost every x in X. Obviously the ergodic
theorem
itself only talks about v almost every convergence.
E