mm-232
===
Subject: Paper published by Algebraic and Geometric
TopologyThe following paper has been published:Algebraic and
Geometric
TopologyURL:http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-
27.abs.htmlTitle:Generalized orbifold Euler characteristics of
symmetric orbifolds and covering spaces Author(s):Hirotaka
Tamanoi Abstract:Let G be a finite group and let M be a
G-manifold. We introduce theconcept of generalized orbifold
invariants of M/G associa to anarbitrary group Gamma, an
arbitrary Gamma-set, and an arbitrarycovering space of a
connec manifold Sigma whose fundamental groupis Gamma. Our
orbifold invariants have a natural and simple geometricorigin
in the context of locally constant G-equivariant maps
fromG-principal bundles over covering spaces of Sigma to the
G-manifoldM. We calculate generating functions of orbifold
Euler characteristicof symmetric products of orbifolds associa
to arbitrary surfacegroups (orientable or non-orientable,
compact or non-compact), in bothan exponential form and in an
infinite product form. Geometrically,each factor of this
infinite product corresponds to an isomorphismclass of a
connec covering space of a manifold Sigma. The
essentialingredient for the calculation is a structure theorem
of thecentralizer of homomorphisms into wreath products
described in termsof automorphism groups of Gamma-equivariant
G-principal bundles overfinite Gamma-sets. As corollaries, we
obtain many identities incombinatorial group theory. As a
byproduct, we prove a simple formulawhich calculates the
number of conjugacy classes of subgroups of givenindex in any
group. Our investigation is motiva by orbifoldconformal field
theory.Secondary: 57S17, 57D15, 20E22, 37F20,
05A15Keywords:Automorphism group, centralizer, combinatorial
group theory, coveringspace, equivariant principal bundle,
free group, Gamma-sets,generating function, Klein bottle
genus, (non)orientable surfacegroup, orbifold Euler
characteristic, symmetric products, twissector, wreath
===
lot of fixed-record length files fromall kinds of sources, and
and I am looking for away to determine the record length. In
some casesthe records are termina by several combinationsof
<CR> <NL>; those are pretty easy. The hard files,however, come
from IBM mainframes, in EBCDIC, with no line(or record)
terminator. The first thing I do is convertthose to ASCII
(with the Unix dd conv=ascii) command).The files contain
personnel information, such as SSN,employee's name, phone
number, and in many cases thereare several fixed consecutive
blanks which were reservedbut not used. I figure that if I
divide the charactersin 3 main classes: alphabetic, numeric
and blanks it wouldbe much mor eefficient than trying an
exhaustive bruteforce approach.I one took a digital design
class in which the bigrecent discovery was spectral techniques
to minimizethe chip's real state use. I guess I am looking for
analgorithm like those, but much simpler, since we aredealing
with one-dimension only.Any suggestion or comments are welcome
===
locally compact groupLet $G$ be a locally compact group, and
let $wG$ be its weakly almostperiodic compactification. The
kernel $K(wG)$ of $wG$ is then a groupisomorphic to the almost
periodic compactification of $G$.There are (non-compact) groups
such that $wG = K(wG) cup G$ (the unionis necessarily
disjoint). For such groups, the closed ideal $wGsetminus G$ of
$wG$ then has an identity.Question: Let $G$ be a locally
compact group (not compact) such that $wGsetminus G$ has an
identity. Does that necessarily mean that $wGsetminus G =
K(wG)$?Any pertinent hints are apprecia.Volker
Runde.begin:vcard n:Runde;Volkertel;fax:+1 780 492
6826tel;home:+1 780 480 1181tel;work:+1 780 492
/org:University of Alberta;Mathematical and Statistical
Sciencesadr:;;CAB 632;Edmonton;Alberta;T6G
2G1;Canadaversion:2.1email;internet:vrunde@
math.ualberta.catitle:Associate
Professorx-mozilla-cpt:;1856fn:Volker
===
dense in L^2Originator: baez@math-cl-n01.math.ucr.edu (John
Baez)What's an easy low-brow way to prove that
polynomialfunctions times the Gaussian exp(-x^2) are dense in
L^2(R)?I know a general-purpose L^2 Stone-Weierstrass
theoremthat does the job, but it requires knowing the
Fouriertransform is unitary, which takes a while to prove.I
also know the eigenfunctions of the harmonic
oscillatorHamiltonian are an orthonormal basis of L^2(R), and
thisalso does the job, but again it relies on some stuff
thattakes work to prove.For a class I plan to teach, I'd like
a quick proofthat doesn't use much machinery, if one
===
in L^2> What's an easy low-brow way to prove that polynomial>
functions times the Gaussian exp(-x^2) are dense in
L^2(R)?Suppose that f is in L^2(R) and is orthogonal to all
such products. Expanding exp(zx) in a power series and using
Fubini's theorem, check thatint_R f(x) exp(zx) exp(-x^2) dx =
0,first for all complex z of sufficiently small modulus, then
for all complex z by analytic continuation. In
particular,int_R f(x) exp(-itx) exp(-x^2) dx = 0,for all real
t. As f(x)exp(-x^2) is clearly an element of L^1(R),this last
display means that the Fourier transform of f vanishes. Thus f
===
Gaussian are dense in L^2> What's an easy low-brow way to prove
that polynomial> functions times the Gaussian exp(-x^2) are
dense in L^2(R)?See chapter 1 of Igusa's (Springer Ergebnisse,
70's) bookTheta Functions for a low-brow proof. I suspect that
thisbook will also have other things you may find useful
===
are dense in L^2 > What's an easy low-brow way to prove that
polynomial functions times > the Gaussian exp(-x^2) are dense
in L^2(R)? I know a general-purpose L^2 Stone-Weierstrass
theorem that does > the job, but it requires knowing the
Fourier transform is unitary, > which takes a while to prove.
I also know the eigenfunctions of the harmonic oscillator
Hamiltonian > are an orthonormal basis of L^2(R), and this
also does the job, but > again it relies on some stuff that
takes work to prove. For a class I plan to teach, I'd like a
quick proof that doesn't use > much machinery, if one
exists.For each compact interval I, polynomials times
exp(-x^2) are dense inthe continuous functions on I with the
sup norm; you can appeal toStone-Weierstrass, or its special
case for polynomials. And continuousfunctions with compact
support are dense in L^2(R).The above does not use Fourier
===
Re: Polynomials times a Gaussian are dense in L^2What's an
easy low-brow way to prove that polynomial functions times>
the Gaussian exp(-x^2) are dense in L^2(R)?I know a
general-purpose L^2 Stone-Weierstrass theorem that does> the
job, but it requires knowing the Fourier transform is
unitary,> which takes a while to prove.I also know the
eigenfunctions of the harmonic oscillator Hamiltonian> are an
orthonormal basis of L^2(R), and this also does the job, but>
again it relies on some stuff that takes work to prove.For a
class I plan to teach, I'd like a quick proof that doesn't
use> much machinery, if one exists.>>For each compact
interval I, polynomials times exp(-x^2) are dense in>the
continuous functions on I with the sup norm; you can appeal
to>Stone-Weierstrass, or its special case for polynomials. And
continuous>functions with compact support are dense in
L^2(R).>>The above does not use Fourier transform being
unitary.>>Is this low-brow enough?Possibly a little too
low-brow, since it doesn't _quite_ give aproof. Say f is in
L^2. You can choose g continuous withcompact support so
||f-g||_2 < epsilon. And now you showthat one can approximate
g _on_ the support of g by afunction of the required type, but
you actually have toapproximate g in
===
times a Gaussian are dense in L^2>What's an easy low-brow
way to prove that polynomial functions times>the Gaussian
exp(-x^2) are dense in L^2(R)?>>I know a general-purpose L^2
Stone-Weierstrass theorem that does>the job, but it requires
knowing the Fourier transform is unitary,>which takes a while
to prove.>>I also know the eigenfunctions of the harmonic
oscillator Hamiltonian> are an orthonormal basis of L^2(R),
and this also does the job, but>again it relies on some stuff
that takes work to prove.>>For a class I plan to teach, I'd
like a quick proof that doesn't use>much machinery, if one
exists.>>For each compact interval I, polynomials times
exp(-x^2) are dense in>>the continuous functions on I with the
sup norm; you can appeal to>>Stone-Weierstrass, or its special
case for polynomials. And continuous>>functions with compact
support are dense in L^2(R).>>The above does not use Fourier
transform being unitary.>>Is this low-brow enough?> Possibly
a little too low-brow, since it doesn't _quite_ give a> proof.
Say f is in L^2. You can choose g continuous with> compact
support so ||f-g||_2 < epsilon. And now you show> that one can
approximate g _on_ the support of g by a> function of the
required type, but you actually have to> approximate g in
===
of Jaynes' _Probability_Theory_I have pos this to amazon.com,
and readers of this forumshould also find it to be of
interest. Review of Edwin T. Jaynes' book, _Probability
Theory: The Logic of Science_ (Five stars (out of five))To
pure mathematicians, probability theory is measure theory
inspaces of measure 1. To the extent to which you remain a
puremathematician, this book will be incomprehensible to
you.To frequentist statisticians, probability theory is the
study ofrelative frequencies or of proportions of a
population; those areprobabilities.To Bayesian statisticians,
probability theory is the study ofdegrees of belief. Bayesians
may assign probability 1/2 to theproposition that there was
life on Mars a billion years ago;frequentists will not do that
because they cannot say that therewas life on Mars a billion
years ago in precisely half of allcases -- there are no such
cases.To _subjective_ Bayesians, probability theory is about
subjectivedegrees of belief. A subjective degree of belief is
merely howsure you happen to be.Noninformative _objective_
Bayesians assign noninformativeprobability distributions when
they deal with uncertainpropositions or uncertain quantities,
and replace them withinformative distributions only when they
update them because ofdata. Data, in this sense, consists of
the outcomes of randomexperiments.Informative _objective_
Bayesians -- a rare species -- ask whatdegree of belief in an
uncertain proposition is logicallynecessita by whatever
information one has, and they don'tnecessarily require that
information to consist of outcomes ofrandom experiments.Jaynes
is an informative objective Bayesian. This book is hisdefense
of that position and his account of how it is to be used.Pure
mathematicians will not find that this book resembles
thatbranch of pure mathematics that they call probability
theory.Jaynes rails against those he disagrees with at great
length.Often he is right. But often he simply misunderstands
them. Forexample, writing in the 1990s, he said that pure
mathematiciansreject the use of Dirac's delta function and its
derivatives, andrela topics. That is nonsense; the delta
function has longbeen considered highly respectable, and
required material in thegraduate curriculum. Unfortunately
Jaynes's misunderstandings maycause some others to
misunderstand him when he is right.Statisticians are more
informed than pure mathematicians andwill disagree with Jaynes
for better reasons. _Some_statisticians will agree with
him.This book many flaws, made all the more annoying by the
factthat we need to overlook them in order to understand
===
Goldbach-Idea - probably dumb!Just thought I'd throw in my 2
cents on Goldbach. Not my area, but thethread made me think of
the following.If you were to generate a set of false primes by
insisting that inany interval they are distribu amongst the
odd numbers with thesame frequency as the real ones (we can
forget about 2),What is the probability that such a set has
the Goldbach property(i.e. any even number can be expressed as
the sum of two falseprimes)? Is it non-zero? What would the
limit of this probability beif one only had to start looking
after an arbitrarily large n?My really stupid idea was that if
(subject to some reasonableverification method for sets of
false primes being sta) theprobability was non-zero, then the
Goldbach conjecture could be trueby accident.If however the
probability does tend to zero, then surely theconjecture is
equally fascinating!I'm no number theorist, but would love to
===
shift theoremsHi all.Let $L$ be a second-order strongly
elliptic operator over a region $Omega$.Consider the problem
of finding the solution $u$ of the problem $Lu=f$in $Omega$
(with $u=0$ on $partialOmega$), where $f$ is areal-valued
function defined over $Omega$.Roughly speaking, the usual
shift theorem for such problems says thatif $fin H^r(Omega)$,
then $uin H^{r+2}(Omega)$, with$$|u|_{H^{r+2}(Omega)} le C
|f|_{H^r(Omega)}.$$Can anybody point me to information about
how the shift constant $C$depends on the coefficients of the
differential operator $L$? Forexample, suppose that
appropriate norms of these coefficients arebounded; is there
Werschulz (8-{)} Metaphors be with you. -- bumper stickerGCS/M
(GAT): d? -p+ c++ l u+(-) e--- m* s n+ h f g+ w+ t++ r- y?
Internet: agw@cs.columbia.edu<a
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