mm-115
Here is a pb Id like to submit, which I guess is
close to the Metric > TSP and wonder whether some of you may
know a solution (exact or approx).> Given an interdistance
matrix (Aij=d(Xi,Xj) for some XiOs), what is > the best
permutation so that given r>0 the sum, for every line of the
> matrix, of the r terms around the diagonal is minimum?Could
you be a bit more explicit about what the problem is?What are
you permuting? What are you minimizing?One possibility that
comes to mind:Find P, a permutation of 0..N-1 to minimizeN-1
(i+r)%NSUM SUM A[P[i],P[j]]i=0 j=i+1For r=1, this is a
TSP.For r> 1, I donOt know how to characterize it, but if d
satis'esthe triangle inequality, I would expect the above TSP
to be a decentapproximation.Improvement algorithms of the type
applied to TSPs should also work.As noted by another poster,
removing the modular arithmeticwould change things.
=Let Z
be a positive integer. Let M = Z/2 if Z is even and (Z+1)/2
if Z is odd.For each integer k, 1<= k <= M+1 letZ = kq(k) +
r(k), where q(k) and r(k) are integers and0<=r(k) < k.Let p
be a real number , 0 < p < 1.De'ne V(k,p) =
(k-r(k))q(k)p^q(k) + r(k)(q(k)+1)p^(q(k)+1). whenever1<=k<=
M+1.De'ne f_k(p) = V(k,p) - V(k-1,p), for 2<= k <= M+1so
f_k(p) is a polynomial in p.The polynomials f_k(p) have some
interesting properties:1. Whenever j > k , fO_j(1) >=
fO_k(1), with equality iff f_j = f_k.2. if f_i != f_j, then
there is unique p(i,j) in (0,1) such thatf_i(p(i,j)) =
f_j(p(i,j)), and moreover p(i,j) is a simple root off_i -
f_j.3. if j > i and 0 < p < 1 then f_j(p) = f_i(p) > 0 if and
only if 0 <p < p(i,j).these 3 properties are not dif'cult to
show.can one show the following?4. if p(i,j) and p(i,k) are
de'ned, and k > j, thenp(i,k) <= p(i,j) .Are there other
families of polynomials that have this decreasingroots
property?tom foregger
Journal of Knot Theory and Its Rami'cations (JKTR)is out.
Articles are available in electronic format
fromhttp://www.worldscinet.com/jktr.htmlTrivial Double-Torus
Knotby Chuichiro HayashiOn Weight Systems Derived from
Heisenberg Lie AlgebrasLink Invariant for the Spinor
Representation ofby Bertrand Patureau-MirandAlexander
Polynomial of Sexticsby Mutsuo OkaDiagrammatic Unknotting of
Knots and Links in the Projective Spaceby Maciej
MroczkowskiQuantum Invariants of Templatesby Louis H.
Kauffman, Masahico Saito and Michael C. SullivanThinning
Genus Two Heegaard Spines in S3by Martin Scharlemann and
Abigail ThompsonThe Linear Growth in the Lengths of a Family
of Thick Knotsby Y. Diao, C. Ernst and M. ThistlethwaiteA
Move on Diagrams that Generates S-Equivalence of Knotsby
Swatee Naik and Theodore StanfordIf you are interested in
submitting your work to JKTR forpublication, please
visithttp://www.worldscinet.com/jktr/mkt/guidelines.shtml to
=Arturo
MagidinOs example is very instructive. It
perfectlydemonstrates the difference between a prime element
and an irreducibleelement. (I knew such examples only for
non-integral domains.)The lack of uniqueness of factorization
is very unusual phenomenon.Still thinking over it...
=I
cannot 'nd a reference for the follwing result which I was
told to be true; it should hold over any (say perfect or char
0) 'eld:consider a curve C inside its Jacobian J, and an
endomorphismT:J-->J of the Jacobian J. Then the preimage
T^(-1)(C) is irreducible.Would it be true if instead of curve
we take a variety of anydimension ? Then we refumulate this
asIf W lies in an Abelian variety A and W does not lie in any
properAbelian subvariety of A, and T is an endomorphism of the
Abelianvariety A, then, as before, the preimage T^(-1) W is
irreducible.
=I want to notify the forum of two papers on
the riemann hypothesis:http://maa.org/features/chaitin.html2)
My brief note which discusses and extends ChaitinOs idea to
attemptto prove the RH by using probablistic methods
athttp://arXiv.org/abs/math.GM/0309148Enjoy!Craig
=It seems
that in the study of random matrices (GUE, GSE, GOE), thatthe
main matter of interest is the pairwise correlation function
ofthe eigenvalues. Has anyone 'gured out any kind of
correlation function of theelements of the eigenvectors? A
paper by Chalker and Mehlig on thexxx.lanl.gov server
indicates that the random matrix eigenvectorstatistics for
the GUE, for example, are determined by the Haarmeasure that
leaves the ensemble invariant. I have some idea of whata Haar
measure is, but I found this to be a very cryptic remark.Is
there any kind of explicit formula concerning the statistics
of theeigenvector of a random Hermitian matrix? Any help
would beappreciated.
=It is known any knot can be
represented by a 2n braid.Does the following hold?To each n,
there is a knot that canOt be representedby a 2i braid with
i<n. (I.e., knots can get arbitrarily complicated.)-- Hauke
Reddmann <:-EX8 For our chemistry workgroup,remove math from
It is known
any knot can be represented by a 2n braid.Does your
restriction to even numbers mean you want touse the plat
representation, rather than the closedbraid representation?
The answer doesnOt really matter,because on either
interpretation the answer to your question below is yes.>Does
the following hold?>>To each n, there is a knot that canOt
be
represented>by a 2i braid with i<n. (I.e., knots can get
>arbitrarily complicated.)Phrases to search on are bridge
number (originallyde'ned independently of braids, later
shown--see JoanBirmanOs book on braids--to be half the
minimum [even]number of strings in a plat representation) and
braid index (the minimum number of strings in aclosed braid
reprsentation; sometimes called string index or braid number,
but I prefer braid index and would like to popularize brin(K)
as a standard notation for it, since itOs mnemonic in
English
and `brinO is the French word for `stringO [in
the context of
braids, at least]).Lee Rudolph
=Can somebody suggest a
good book for a novice on representation theoryof groups -
particularly over characteristic zero 'elds. Most of thebooks
i tried looking at, assume algebraically closed 'elds for
someimportant results. I am interested in representations
over anarbitrary number 'eld.Kiran
Is there a term for a
category whose morphisms can be expressed in>terms of
composion through a single object? Namely, thereOs an
object>A in the category such that for objects B and C,
Hom(B,C) is Hom(B,A)>composed with Hom(A,C). Ie, the
morphisms to and from A generate all>morphisms in the
category. IOm also looking for a more general>version, a
category where the morphisms to and from a 'nite set
of>objects generate the morphisms in the category. Any help
included>you.by the way was my answer of any help? it was
sort of a joke answer(because i thought that the way that you
described the property turnedout to be equivalent to a rather
differently stated property relatedto karoubi envelope aka
splitting idempotents completion akageneralized cauchy
completion of a category) but it was alsointended to be
serious and hopefully helpful. anyway i hope thatmy answer
was at least factually correct.--
=|to karoubi envelope
aka splitting idempotents completion aka|generalized cauchy
completion of a category) but it was also2. now that i think
about it i think the name generalized cauchycompletion is the
name of a different but related completion processon
categories of some kind, but i donOt remember the details.--
Is there a term for a category whose morphisms can be
expressed in>terms of composion through a single object?
Namely, thereOs an object>A in the category such that for
objects B and C, Hom(B,C) is Hom(B,A)>composed with
Hom(A,C). Ie, the morphisms to and from A generate all>
>morphisms in the category. IOm also looking for a more
general>version, a category where the morphisms to and from
a 'nite set of>objects generate the morphisms in the
category. Any help included>you.> by the way was my
answer of any help? it was sort of a joke answer> (because i
thought that the way that you described the property turned>
out to be equivalent to a rather differently stated property
related> to karoubi envelope aka splitting idempotents
completion aka> generalized cauchy completion of a category)
but it was also> intended to be serious and hopefully
helpful. anyway i hope that> my answer was at least factually
correct.I think so. Currently, IOm brushing up on my math
for
a comprehensiveexam at UC Davis in a couple of weeks so I
really wonOt be able tolook at this till after that exam.
But
the little I was able to dodoes indicate that this is similar
in interesting ways to what I wasrely at some points on this
property) for his dissertation and waskind enough to point me
to that work.Karl Hallowell
=with three-and-a-half handles,
(projective plane with 3 handles), admitsholocontiguous maps
of exactly 10 regions. (Dually, an embedded K_10 graph.)Exact
in the sense that each region borders each other one *exactly*
once;(just as the torus so admits 7 regions).I am keen to see
examples:- I have many of my own, but would like to seeothers
(coded into any obvious database), to get some idea of how
manymight be the total number of distinct such maps. (It must
be 'nite.)Does anyone have any?Maps with 9 regions on the
surface with EC = -5 would also be welcome;though IOm fairly
sure there are only two of these, (closely related).Anyway,
if anyone has data for maps of either sort, please let me
know.--------------------------------------------------------
---------------------- Bill Taylor
W.Taylor@math.canterbury.ac.nz-------------------------------
----------------------------------------------- Every nation
ridicules other nations, and they are all
right.-------------------------------------------------------
-----------------------
=I am trying to understand the
proof of the following result:Let K < E < F be 'eld
extensions. Let M be a transcendence basis of Eover K, and N
be a transcendence basis of F over E. Then Mcap N isempty,
and Mcup N is a transcendence basis of F over K.In section 75
of van der WaerdenOs Algebra, there is a proof ofsimilar
statement. In the proof there is a paragraph
(notationslightly changed):The 'eld F is algebraic over E(N),
E is algebraic over K(M); HENCE Fis algebraic over K(M,N).I
canOt understand this hence. We see that there is the
followingseries of 'elds:K -- K(M) -- E -- E(N) -- FAll we
need to establish, is that any element from F is a root of
somepolynomial with coef'cients from K(M,N). How one can do
it?In BourbakiOs Algebra, Ch. V, S.5 the exposition is more
eloquent, butalso leaves some discontent:E(N) is algebraic
over K(Mcup N)=K(M)(N), since 1) E(N)=K(Mcup N)(E) and2)
every element of E is algebraic over K(M) hence over K(Mcup
N)[reference].Thus, F is algebraic over K(Mcup N)
[reference].Here I managed to persuade myself that all the
reasoning is valid.Namely, from 1) E(N) is being obtained
from K(Mcup N) by adjoiningroots of polynomials (since E is
algebraic over K(Mcup N)), henceE(N) is algebraic over K(Mcup
N). But is it possible to showdirectly, taken an arbitrary
element x from E(N), that x is a root ofsome polynomial over
K(Mcup N)?
solutions to Det(xy(x+y+I)) > 0. Does itexist a homeomorphism
that map (X,Y) to a simple set (say some subset of(phi, theta)
such that Det(phi)*Det(theta) > 0) ?Any suggestion on how to
approach this problem will be greatly appreciated.K.
Yam
=IOm looking to visualize a modular form. Might there
be a link to
=Suppose that f(x,y) is a polynomial of degree
n over the reals, and suppose that the line y=a meets the
curve f(x,y)=0 in n points. It is elementary that y=a meets
f_x (the partial derivative of f with respect to x) in n-1
points.Question: What conditions on f imply that the line y=a
also meets f_y (the partial derivative of f with respect to y)
in n-1 points?Steve
=I have the following basic linear
system:dot{x} = A times x + B times f(t)with x a vector, A
and B matrices, and f(t) a time-varying signal.ItOs easy to
'gure out what the signal f has to be in order to force
thesystem to minimize an optimality conditionintegral{x^{T}
times Q times x + f^{T} times R times F dt}However, I want to
do the reverse, i.e. IOm interested in the conditionsunder
which one can construct a cost functional which the linear
systemnaturally minimizes. Is this possible in general, or
even under somerestricted set of
conditions?Glen______________________________________________
______________Dr. Glen HenshawNaval Center for Space
TechnologyU.S. Naval Research Laboratory(202) 767-1196
=I
have some problems with this question. Can anybody help
me?IOm trying to recover signal from Wigner-Ville
distribution aftertime-frequency 'ltering. I base on Spatial
and Time-FrequencySignature Estimation of Nonstationary
Sources paper [Moeness] (foundin internet).The procedures
from this paper work as follows:1. Calculate Wigner-Ville
distribution; WVD(t,f) = sum[ x(t+k/2)*xO(t-k/2)*exp(-jkf)
]2. Filter WVD as you need WVDO(t,f) = WVD(t,f)*G(t,f) where
G(t,f) - binary mask (0,1) 3. Take the inverse Fast Fourier
Transform of WVDO P(t,n) = IFFT( WVDO ) 4.
Construct the
matrix Q with Q(i,j) = P( (i+j)/2 , i-j )5. Apply
eigen-decomposition to the matrix Q and obtain the
maximumeigenvalue and associated eigenvector. Now recovered
signal should be based on this eigenvector. And now my
computation: i.e. signal: sig=[1 2 3 4 5 6 7 1]but I want to
make it easier) WVD = Columns 1 through 7 1.0000 10.0000
35.0000 84.0000 119.0000 114.0000 61.0000 1.0000 8.2426
20.3137 27.3137 56.1127 85.4975 57.4853 1.0000 4.0000 -1.0000
-8.0000 -17.0000 28.0000 49.0000 1.0000 -0.2426 -2.3137 4.6863
-6.1127 -13.4975 40.5147 1.0000 -2.0000 3.0000 -4.0000 15.0000
-26.0000 37.0000 1.0000 -0.2426 -2.3137 4.6863 -6.1127
-13.4975 40.5147 1.0000 4.0000 -1.0000 -8.0000 -17.0000
28.0000 49.0000 1.0000 8.2426 20.3137 27.3137 56.1127 85.4975
57.4853 Column 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.0000 1.00002. IOm going to do this without mask to make it
easier3. My inwerse transform P= Columns 1 through 7 1.0000
4.0000 9.0000 16.0000 25.0000 36.0000 49.0000 0 3.0000 8.0000
15.0000 24.0000 35.0000 6.0000 0 0 5.0000 12.0000 21.0000
4.0000 0 0 0 0.0000 7.0000 2.0000 0 0.0000 0 0 0 0 0 0 0 0 0
0.0000 7.0000 2.0000 0 0.0000 0 0 5.0000 12.0000 21.0000
4.0000 0 0 3.0000 8.0000 15.0000 24.0000 35.0000 6.0000
Column 8 1.0000 0 0 0 0 0 0 04. And in the end Q-matrix Q=
Columns 1 through 7 2.0000 0 3.0000 0 5.0000 0 7.0000 0
8.0000 0 8.0000 0 12.0000 0 3.0000 0 18.0000 0 15.0000 0
21.0000 0 8.0000 0 32.0000 0 24.0000 0 5.0000 0 15.0000 0
50.0000 0 35.0000 0 12.0000 0 24.0000 0 72.0000 0 7.0000 0
21.0000 0 35.0000 0 98.0000 0 2.0000 0 4.0000 0 6.0000 0
Column 8 0 2.0000 0 4.0000 0 6.0000 0 2.0000 5. After that
eigenvectors/eigenvalues are copmputed by Matlab, soitOs no
problem The results obtained as above are not satis'ed for me
because whenI try to use 'ltering the recovered signal is not
correct in timedomain.I think WVD and P matrix are computed
correctly but IOm not sure aboutQ-matrix.Has anybody use
this
method and can help to solve my problem?Maybe you know another
method that work in signal synthesis fromWigner-Ville
distribution?IOll be grateful for any
help.Krzyskidzkows@elka.pw.edu.pl
=Readers of this
newsgroup may 'ng this preprint by of interest an worthy
ofdiscussion:Resource Bounded Unprovability of Computational
Lower Bounds Tatsuaki Okamoto and Ryo KashimaAbstract. This
paper shows that no polynomial-time Turing machine can
produce aproof (based on a reasonable theoryincluding Peano
Arithmetic) of a super-polynomial-time lower bound of an NP
(ormore generally, PSPACE) problem. Inother words, no
polynomial-time Turing machine can produce a proof of
``P$not=$ NPOO. Therefore, {it to prove
``P$not=$ NPOO} (by
any technique and any reasonable theory) {it
requiressuper-polynomial-time computational power}.This
result is a kind of generalization of the result of ``Natural
ProofsOO byRazborov and Rudich, who showed
that toprove ``P
$not=$ NPOO by a class of techniques called
``NaturalOO
impliescomputational power that can break a typical
cryptographic primitive, apseudo-random generator. This
result also implies that {it there is no('nite-size) formal
proof for ``P $not=$ NPOO} in any reasonable
theory. Thisis
considered to be a generalization of the result by Baker,
Gill andSolovay,*@ who showed that there is no relativizable
proof for ``P $not=$NPOO. Based on this
result, we show that
{it the security of any computationalcryptographic scheme is
unprovable} in the standard setting of moderncryptography,
where an adversary is modeled as a polynomial-time
Turingmachine. Category / Keywords. foundations / Contact
author: Tatauaki Okamoto (okamoto@isl.ntt.co.jp)Available
formats: Postscript (PS) | Compressed Postscript (PS.GZ) |
BibTeXCitation
_____________________________________________________________
_______________________________________________Professor
Michael AnshelDepartment of Computer Sciences R8/206The City
College of New YorkNew York,New York
10031http://www-cs.engr.ccny.cuny.edu/~csmma/csmma@
cs.ccny.cuny.edu MikeAt1140@aol.com
=The LambertW can be
used to solve a couple of interesting problems,including some
relating to the 'xed points of the hyperpower
sequencex^x^x^...Turns out there is another bonus in there,
thatOs again using theLambertW, this time to
de'ne the
analytic continuation of the realhyperpower function F(x) =
x^x^x^...investigation, but the idea that it could be used as
the analyticcontinuation of F(x) hadnOt occured to me, until
two days ago.There is also a nice investigation of the two
components of the Hopfbifurcation using Maple, and some code
to sketch the branches
in(0,e^(-e))<http://users.forthnet.gr/ath/jgal/math/
hyperpower.html>,<http://users.forthnet.gr/ath/jgal/math/
exponents.html>in my math
section:<http://users.forthnet.gr/ath/jgal/math/>Enjoy.--
Ioannishttp://users.forthnet.gr/ath/jgal/____________________
_______________________Eventually, _everything_ is
understandable.
=anyone who have a glimpse into the point
set structure theory willknow that Cantor set is endowed with
the following properties:1.nowhere dense in R^1;2.of Lebesgue
measure zero;3.is a (nonempty)complete set;4.If we have the
following subset of real numbers for a given set A:D(A)={s:
s=d(a.b), a,b as members of A}then the Cantor set G satis'es
D(G)=[0,1], i.e. for any 0<=s<=1, there exist a,b as members
of G,s.t.s=d(a,b).I have paid some effort on 'nding another
subset of [0,1] satisfyingthe above four properties but
fails. Can anyone offer one?Bill
anyone who have a
glimpse into the point set structure theory will> know that
Cantor set is endowed with the following properties:>
1.nowhere dense in R^1;> 2.of Lebesgue measure zero;> 3.is a
(nonempty)complete set;> 4.If we have the following subset of
real numbers for a given set A:> D(A)={s: s=d(a.b), a,b as
members of A}> then the Cantor set G satis'es >
D(G)=[0,1], i.e. for any 0<=s<=1, there exist a,b as members
of G,> s.t.> s=d(a,b).> I have paid some effort on 'nding
another subset of [0,1] satisfying> the above four properties
but fails. Can anyone offer one?> Bill> How about the set
of all numbers in [0,1] which have a base 5expansions using
only the digits {0,2,4}?-- G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
the advancement of science and civilization, there is always
theproblem of creativity and education. Without creativity,
the sciencesand civilization would stagger and decline.
Without widespreadeducation effort, human talents would be
wasted and the creativeprocesses would halt.The aim of the
Series on University Mathematics is to publisheducational
books written by creative mathematicians for undergraduateand
graduate students as well as the general public, helping them
tounderstand and enjoy modern and advanced mathematics.or
visit http://www.wspc.com/books/series/scor_series.shtmlthank
you,The EditorsSeries on University Mathematics
=Here are
this weekOs titles in the mathematics arXiv, available at:
http://front.math.ucdavis.edu/
http://front.math.ucdavis.edu/submissionsThis week in the
mathematics arXiv may be freely redistributedwith attribution
and without modi'cation.Titles in the mathematics arXiv (25
Aug - 29
Aug)-------------------------------------------------AC:
Commutative Algebra-----------------------math.AC/0308272
<http://front.math.ucdavis.edu/math.AC/0308272> Jooyoun Hong:
Rees Algebras of Conormal Modulesmath.AC/0308264
<http://front.math.ucdavis.edu/math.AC/0308264> Sara Faridi:
Simplicial Trees are Sequentially
Cohen-Macaulaymath.AC/0308263
<http://front.math.ucdavis.edu/math.AC/0308263> Samuel
Wuthrich: Homology of powers of regular idealsAG: Algebraic
Geometry----------------------math.AG/0308266
<http://front.math.ucdavis.edu/math.AG/0308266> Kiumars
Kaveh: Fixed Points of Torus Action and Cohomology Ring of
Toric Varietiesmath.AG/0308247
<http://front.math.ucdavis.edu/math.AG/0308247> Thomas
Keilen: Smoothness of Equisingular Families of
Curvesmath.AG/0308233
<http://front.math.ucdavis.edu/math.AG/0308233> John Hubbard,
Victor H. Moll: A geometric view of rational Landen
transformationsmath.AG/0308221
<http://front.math.ucdavis.edu/math.AG/0308221> Philip
Boalch: The Klein solution to PainleveOs sixth
equationmath.AG/0308218
<http://front.math.ucdavis.edu/math.AG/0308218> Megumi
Harada, Nicholas J. Proudfoot: Hyperpolygon spaces and their
coresmath.AG/0308216
<http://front.math.ucdavis.edu/math.AG/0308216> Tom Braden:
Koszul duality for toric varietiesmath.AG/0308212
<http://front.math.ucdavis.edu/math.AG/0308212> StOephane
Druel: CaractOerisation de lOespace
projectifmath.AG/0308209
<http://front.math.ucdavis.edu/math.AG/0308209> Kefeng Liu,
Andrey Todorov, Shing-Tung, Kang Zuo: ShafarevichOs
Conjecture for CY Manifolds Imath.AG/0308208
<http://front.math.ucdavis.edu/math.AG/0308208>
arithmetically Gorenstein schemesAP: Analysis of
PDEs--------------------math.AP/0308278
<http://front.math.ucdavis.edu/math.AP/0308278> Andrew
Hassell, Jared Wunsch: On the structure of the Schrodinger
propagatormath.AP/0308220
<http://front.math.ucdavis.edu/math.AP/0308220> Steve
Zelditch: Billiards and boundary traces of
eigenfunctionsmath.AP/0308214
<http://front.math.ucdavis.edu/math.AP/0308214> N. Burq, P.
Gerard, N. Tzvetkov: Bilinear Eigenfunction Estimates and the
Nonlinear Schroedinger Equation on SurfacesAT: Algebraic
Topology----------------------cond-mat/0308530
<http://front.math.ucdavis.edu/cond-mat/0308530> Lucjan
Jacak, Piotr Sitko, Konrad Wieczorek, Arkadiusz WOojs:
Quantum Hall Systems: Braid groups, composite fermions, and
fractional chargemath.AT/0308253
<http://front.math.ucdavis.edu/math.AT/0308253> Matthias
Franz: On the integral cohomology of smooth toric
varietiesmath.AT/0308246
<http://front.math.ucdavis.edu/math.AT/0308246> Regis
Pellissier: Weak enriched categories - Categories enrichies
faiblesmath.AT/0308243
<http://front.math.ucdavis.edu/math.AT/0308243> Barbu
Berceanu, Martin Markl, Stefan Papadima: Multiplicative
models for con'guration spaces of algebraic varietiesCA:
Classical Analysis and
ODEs-------------------------------math.CA/0308211
<http://front.math.ucdavis.edu/math.CA/0308211> A.A.
Korenovskyy, A.K. Lerner, A.M Stokolos: On multidimensional
F. RieszOs Rising Sun LemmaCO:
Combinatorics-----------------math.CO/0308280
<http://front.math.ucdavis.edu/math.CO/0308280> Mike Develin,
Seth Sullivant: Markov bases of binary graph
modelsmath.CO/0308265
<http://front.math.ucdavis.edu/math.CO/0308265> Thomas Lam:
Growth diagrams, Domino insertion and
Sign-imbalancecond-mat/0308515
<http://front.math.ucdavis.edu/cond-mat/0308515> Sergio
Caracciolo, Andrea Sportiello: General duality for
abelian-group-valued statistical-mechanics
modelsmath.CO/0308234
<http://front.math.ucdavis.edu/math.CO/0308234> Marcos Kiwi,
Martin Loebl, Jiri Matousek: Expected length of the longest
common subsequence for large alphabetsDG: Differential
Geometry-------------------------math.DG/0308283
<http://front.math.ucdavis.edu/math.DG/0308283> Joseph A.
Wolf: Complex Forms of Quaternionic Symmetric
Spacesmath.DG/0308279
<http://front.math.ucdavis.edu/math.DG/0308279> Anna
Pratoussevitch: Fundamental Domains in Lorentzian
Geometryhep-th/0308141
<http://front.math.ucdavis.edu/hep-th/0308141> JosOe
Figueroa-OOFarrill, Teruhiko Kawano, Satoshi Yamaguchi:
Parallelisable Heterotic Backgroundsmath.DG/0308249
<http://front.math.ucdavis.edu/math.DG/0308249> Xianzhe Dai:
A Positive Mass Theorem for Spaces with Asymptotic SUSY
Compacti'cationmath.DG/0308244
<http://front.math.ucdavis.edu/math.DG/0308244> C. Bartocci,
I. Mencattini: Hyper-symplectic structures on integrable
systemsmath.DG/0308241
<http://front.math.ucdavis.edu/math.DG/0308241> Vestislav
Apostolov, Tedi Draghici, Andrei Moroianu: The
odd-dimensional Goldberg Conjecturemath.DG/0308235
<http://front.math.ucdavis.edu/math.DG/0308235> Jouko
Mickelsson: Gerbes on quantum groupsmath.DG/0308232
<http://front.math.ucdavis.edu/math.DG/0308232> Denis Kochan:
Pseudodifferential forms and supermechanicsmath.DG/0308227
<http://front.math.ucdavis.edu/math.DG/0308227>
D.D.Porosniuc: A locally symmetric Kaehler Einstein structure
on the cotangent bundle of a space formmath.DG/0308217
<http://front.math.ucdavis.edu/math.DG/0308217> Aristide
Tsemo: Af'ne manifolds, lagrangian manifoldsmath.DG/0308215
<http://front.math.ucdavis.edu/math.DG/0308215> Brian Dean:
Compact Embedded Minimal Surfaces of Positive Genus Without
Area BoundsDS: Dynamical
Systems---------------------math.DS/0308252
<http://front.math.ucdavis.edu/math.DS/0308252> Toshiaki
Fujiwara, Richard Montgomery: Convexity in the Figure Eight
Solution to the Three-Body Problemmath.DS/0308223
<http://front.math.ucdavis.edu/math.DS/0308223> Nguyen T.
Thao, C. Sinan Gunturk: Ergodic Dynamics in Sigma-Delta
Quantization: Tiling Invariant Sets and Spectral Analysis of
ErrorFA: Functional
Analysis-----------------------math.FA/0308273
<http://front.math.ucdavis.edu/math.FA/0308273> Sever
Silvestru Dragomir: Reverses of Schwarz, Triangle and Bessel
Inequalities in Inner Product Spacesmath.FA/0308270
<http://front.math.ucdavis.edu/math.FA/0308270> S.S.
Dragomir, Y.J. Cho, S.S. Kim, A. Sofo: Some Boas-Bellman Type
Inequalities in 2-Inner Product Spacesmath.FA/0308251
<http://front.math.ucdavis.edu/math.FA/0308251> Eric Weber:
The Geometry of Sampling on Unions of Latticesmath.FA/0308250
<http://front.math.ucdavis.edu/math.FA/0308250> Akram
Aldroubi, David Larson, Wai-Shing Tang, Eric Weber: Geometric
Aspects of Frame Representations of Abelian
Groupsmath.FA/0308240
<http://front.math.ucdavis.edu/math.FA/0308240> Victor
Shulman, Lyudmila Turowska: Operator synthesis. I. Synthetic
sets, bilattices and tensor algebrasmath.FA/0308226
<http://front.math.ucdavis.edu/math.FA/0308226> Thomas
William Dawson: Extensions of Normed AlgebrasGN: General
Topology--------------------math.GN/0308236
<http://front.math.ucdavis.edu/math.GN/0308236> A.
Chigogidze, V. Valov: Extraordinary dimension of
mapsmath.GN/0308219
<http://front.math.ucdavis.edu/math.GN/0308219> H. Murat
Tuncali, E. D. Tymchatin, Vesko Valov: Extensional dimension
and completion of mapsGT: Geometric
Topology----------------------math.GT/0308277
<http://front.math.ucdavis.edu/math.GT/0308277> periodicity
of the open booksmath.GT/0308276
<http://front.math.ucdavis.edu/math.GT/0308276> Tolga Etgu,
B. Doug Park: Symplectic tori in rational elliptic
surfacesmath.GT/0308274
<http://front.math.ucdavis.edu/math.GT/0308274> Koji
Fujiwara, Takashi Shioya, Saeko Yamagata: Parabolic
isometries of CAT(0) spaces and
CAT(0)-dimensionsmath.GT/0308268
<http://front.math.ucdavis.edu/math.GT/0308268> Francis
Bonahon, Xiaodong Zhu: The metric space of geodesic
laminations on a surface II: small surfacesmath.GT/0308267
<http://front.math.ucdavis.edu/math.GT/0308267> Xiaodong Zhu,
Francis Bonahon: The metric space of geodesic laminations on a
surface. Imath.GT/0308222
<http://front.math.ucdavis.edu/math.GT/0308222> Riccardo
Piergallini, Daniele Zuddas: A universal ribbon surface in
B^4hep-th/0308152
<http://front.math.ucdavis.edu/hep-th/0308152>MG: Metric
Geometry-------------------math.MG/0308262
<http://front.math.ucdavis.edu/math.MG/0308262> Joseph
Corneli, Paul Holt, George Lee, Nicholas Leger, Eric
Schoenfeld, Benjamin Steinhurst: The Double Bubble Problem on
the Flat Two-Torusmath.MG/0308254
<http://front.math.ucdavis.edu/math.MG/0308254> Mike Develin,
Bernd Sturmfels: Tropical Convexitymath.MG/0308239
<http://front.math.ucdavis.edu/math.MG/0308239> Igor Rivin:
Some observations on the simplexMP: Mathematical
Physics------------------------math-ph/0308040
<http://front.math.ucdavis.edu/math-ph/0308040> Raymond
Brummelhuis, Mary Beth Ruskai: One-dimensional models for
atoms in strong magnetic 'elds, II: Anti-Symmetry in the
Landau Levelsmath-ph/0308039
<http://front.math.ucdavis.edu/math-ph/0308039> V. V.
Varlamov: Hyperspherical Functions and Harmonic Analysis on
the Lorentz Groupmath-ph/0308038
<http://front.math.ucdavis.edu/math-ph/0308038> V. V.
Varlamov: Relativistic wavefunctions on the Poincare
groupmath-ph/0308037
<http://front.math.ucdavis.edu/math-ph/0308037> R. F.
Streater: Duality in quantum information
manifoldsmath-ph/0308036
<http://front.math.ucdavis.edu/math-ph/0308036> P.J.
Forrester, N.S. Witte: Discrete PainlevOe equations,
Orthogonal Polynomials on the Unit Circle and $N$-recurrences
for averages over U(N) -- PVI $tau$-functionsmath-ph/0308035
<http://front.math.ucdavis.edu/math-ph/0308035> Guowu Meng:
Legendre Transform, Hessian Conjecture and Tree
Formulamath-ph/0308034
<http://front.math.ucdavis.edu/math-ph/0308034> Liudmila
Joukovskaya, Yaroslav Volovich: Energy Flow from Open to
Closed Strings in a Toy Model of Rolling
Tachyonmath-ph/0308033
<http://front.math.ucdavis.edu/math-ph/0308033> F. Benatti,
V. Cappellini, F. Zertuche: Quantum Dynamical Entropies in
Discrete Classical Chaosmath-ph/0308032
<http://front.math.ucdavis.edu/math-ph/0308032>
operatorsnlin.CD/0308026
<http://front.math.ucdavis.edu/nlin.CD/0308026> J. Kaidel, M.
Brack: Uniform approximations for pitchfork bifurcation
sequencesmath-ph/0308031
<http://front.math.ucdavis.edu/math-ph/0308031> Soeren
Koester: Structure of Coset Modelscond-mat/0308466
<http://front.math.ucdavis.edu/cond-mat/0308466> Malte
Henkel, Gunter Schutz: On the universality of the
§uctuation-dissipation ratio in non-equilibrium critical
dynamicsmath-ph/0308030
<http://front.math.ucdavis.edu/math-ph/0308030> J.E. Avron:
Colored Hofstadter butter§iesmath-ph/0308029
<http://front.math.ucdavis.edu/math-ph/0308029> Yasuyuki
Kawahigashi: Classi'cation of operator algebraic conformal
'eld theories in dimensions one and twomath-ph/0308028
<http://front.math.ucdavis.edu/math-ph/0308028> Bergthor
Hauksson, Jakob Yngvason: Asymptotic Exactness of Magnetic
Thomas-Fermi Theorycond-mat/0112325
<http://front.math.ucdavis.edu/cond-mat/0112325> Y. M. Cho:
Creation of Knots in Two-component Bose-Einstein
Condensatescond-mat/0308182
<http://front.math.ucdavis.edu/cond-mat/0308182> Y. M. Cho,
H. J. Khim: Non-Abelian Vortices in Condensed
Mattermath-ph/0308027
<http://front.math.ucdavis.edu/math-ph/0308027> Shigeki
Matsutani: Relations in a Loop Soliton as a Quantized
Elasticamath-ph/0308026
<http://front.math.ucdavis.edu/math-ph/0308026> Bindu A.
Bambah: Polynomial Algebras and their ApplicationsNT: Number
Theory-----------------math.NT/0308213
<http://front.math.ucdavis.edu/math.NT/0308213> James McKee,
Chris Smyth: There are Salem numbers of every traceOA:
Operator Algebras---------------------math.OA/0308271
<http://front.math.ucdavis.edu/math.OA/0308271> Kenley Jung:
A hyper'nite inequality for free entropy
dimensionmath.OA/0308261
<http://front.math.ucdavis.edu/math.OA/0308261> Massoud
Amini: Tannaka-Krein duality for compact groupoids III,
duality theorymath.OA/0308260
<http://front.math.ucdavis.edu/math.OA/0308260> Massoud
Amini: Tannaka-Krein duality for compact groupoids II,
Fourier transformmath.OA/0308259
<http://front.math.ucdavis.edu/math.OA/0308259> Massoud
Amini: Tannaka-Krein duality for compact groupoids I,
representation theorymath.OA/0308258
<http://front.math.ucdavis.edu/math.OA/0308258> Massoud
Amini, Alireza Medghalchi: Restricted algebras on inverse
semigroups III, Fourier algebramath.OA/0308257
<http://front.math.ucdavis.edu/math.OA/0308257> Massoud
Amini, Alireza Medghalchi: Restricted algebras on inverse
semigroups II, positive de'nite functionsmath.OA/0308256
<http://front.math.ucdavis.edu/math.OA/0308256> Massoud
Amini, Alireza Medghalchi: Restricted algebras on inverse
semigroups I, representation theorymath.OA/0308255
<http://front.math.ucdavis.edu/math.OA/0308255> Gero Fendler:
Simplicity of the reduced C-*-algebras of certain Coxeter
groupsmath.OA/0308245
<http://front.math.ucdavis.edu/math.OA/0308245> Michael
Skeide: Independence and Product Systemsmath.OA/0308231
<http://front.math.ucdavis.edu/math.OA/0308231> Michael
Skeide: Commutants of von Neumann Modules, Representations of
B^a(E)math.OA/0308230
<http://front.math.ucdavis.edu/math.OA/0308230> Michael
Skeide: Von Neumann Modules, Intertwiners and
Self-Dualitymath.OA/0308207
<http://front.math.ucdavis.edu/math.OA/0308207> Marc A.
Rieffel: Compact Quantum Metric SpacesPR: Probability
Theory----------------------math.PR/0308282
<http://front.math.ucdavis.edu/math.PR/0308282> Vlada Limic,
Robin Pemantle: More rigorous results on the Kauffman-Levin
model of evolutionmath.PR/0308242
<http://front.math.ucdavis.edu/math.PR/0308242> Patrik L.
Ferrari, Herbert Spohn: Constrained Brownian motion:
§uctuations away from circular and parabolic
barriersmath.PR/0308238
<http://front.math.ucdavis.edu/math.PR/0308238> Peter Friz,
Nicolas Victoir: Approximations of the Brownian Rough Path
with Applications to Stochastic Analysiscond-mat/0308508
<http://front.math.ucdavis.edu/cond-mat/0308508> D.
Gabrielli, A. Galves, D. Guiol: Fluctuations of the Empirical
Entropies of a Chain of In'nite Ordermath.PR/0308237
<http://front.math.ucdavis.edu/math.PR/0308237> Veronique
Ladret: Asymptotic hitting time for a simple evolutionary
model of protein foldingQA: Quantum
Algebra-------------------math.QA/0308281
<http://front.math.ucdavis.edu/math.QA/0308281> Stephen F.
Sawin: Quantum Groups at Roots of Unity and
Modularitymath.QA/0308275
<http://front.math.ucdavis.edu/math.QA/0308275> Alain Connes,
Michel Dubois-Violette: Moduli space and structure of
noncommutative 3-spheresmath.QA/0308269
<http://front.math.ucdavis.edu/math.QA/0308269> Edward
Frenkel: Opers on the projective line, §ag manifolds and
Bethe Ansatzmath.QA/0308248
<http://front.math.ucdavis.edu/math.QA/0308248> Yi-Zhi Huang,
Liang Kong: Open-string vertex algebras, tensor categories and
operadsmath.QA/0308229
<http://front.math.ucdavis.edu/math.QA/0308229> product
algebras IImath.QA/0308228
<http://front.math.ucdavis.edu/math.QA/0308228> Nicolas
Andruskiewitsch, Sonia Natale: Double categories and quantum
groupoidsSG: Symplectic
Geometry-----------------------math.SG/0308225
<http://front.math.ucdavis.edu/math.SG/0308225> Cheol-Hyun
Cho, Yong-Geun Oh: Floer cohomology and disc instantons of
Lagrangian torus 'bers in Fano toric manifoldsmath.SG/0308224
<http://front.math.ucdavis.edu/math.SG/0308224> Cheol-Hyun
Cho: Holomorphic disc, spin structures and Floer cohomology
of the Clifford torusmath.SG/0308210
<http://front.math.ucdavis.edu/math.SG/0308210> Andrey
Todorov: Large Radius Limit and SYZ Fibrations of
Hyper-Kahler Manifolds-- / Greg Kuperberg (UC Davis) / /
Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
/ * All the math thatOs 't to e-print *
=A recent
discussion - triggered by M J MurphyOs post of 20th August
onThe Goedel sentence in the Foundations of Mathematics (FOM)
Forum -highlighted a wide range of foundational issues that
should be o'nterest to some readers of this group.As seems
almost inevitable when the standard interpretation
ofGoedelOs
reasoning in his famous 1931 paper, On formally
undecidablepropositions of Principia Mathematica and related
systems I, is thesubject of a multi-disciplinary discussion,
issues tend to gravitatefrustratingly into a semantic
cul-de-sac. However, if one persiststill the diversionary
element - invariably, the apparent incongruityof true but
unprovable sentences - clears away, then signi'cantissues
emerge that may remain obscured otherwise.It is fascinating
to note that these issues overwhelmingly indicatethe need for
an overdue resolution of a number of concepts,
centeredessentially around the need for a more effective
veri'ability of theconcept of mathematical truth than that
offered currently by classicaltheory. The need for such
resolution seems to be increasingly felt,and demanded from
logicians, by disciplines that look to mathematicsnot only
for expressing their concepts precisely, but more
importantlyfor communicating these concepts effectively, and
unambiguously, toothers - both inside and outside a
discipline.But 'rst, to avoid the most common semantic
diversion, it isessential to note that - despite the current
lack of an effectivemethod for verifying the truth of an
arbitrary Arithmetic propositionsuch as (Ax)F(x) - there is
nothing particularly mysterious oresoteric about the fact
that a particular Arithmetic proposition, say(Ax)R(x), can be
constructively asserted as Tarskian-true from theaxioms of the
Arithmetic, but not as proof-sequentially provable fromthe
axioms of the Arithmetic by the classical deduction rules of
theArithmetic.It simply means that some Arithmetic
propositions can be meaningfully,and constructively, assigned
one each of two, not entirely unrelated,sets of properties -
truth/falsity and provability/unprovability -that are not,
however, equivalent.More precisely, what Goedel proved (by an
intuitionisticallyunobjectionable meta-proof) was that there
is a relation R(x),constructively de'nable in any recursively
de'ned axiomaticArithmetic of the natural numbers (whether
this is considered as aformal system or the standard
interpretation of such a system) suchthat R(n) is provable
for any natural number n (in the sense that,given any natural
number n, there is always a classical, constructive,proof
sequence/deduction for R(n) from the axioms of the Arithmetic
bythe classical deduction rules of the Arithmetic). By
classicalde'nitions of the truth of number-theoretic
relations (formallyexpressed by TarskiOs
de'nitions), this
property of R(x) isconventionally expressed symbolically as
the meta-assertion that(Ax)R(x) is a constructively
established true arithmetic assertion.However, since Goedel
also proved, again constructively, that there isno classical,
constructive, proof sequence - consisting of onlystatements
within the Arithmetic - from which the symbolic
expression(Ax)R(x) can be mechanically (i.e. without
interpreting any of thestatements using TarskiOs
de'nitions)
deduced, and so termed asprovable from the axioms of the
Arithmetic by the classical deductionrules of the Arithmetic,
he (arguably unfortunately) expressed thisasymmetry as the
meta-assertion that (Ax)R(x) is a true, butunprovable,
proposition of the Arithmetic.Now, the wider signi'cance of
GoedelOs reasoning does not lie inwhether, or even why, the
particular Arithmetic proposition (Ax)R(x) -that he de'ned
constructively - is true but unprovable in theArithmetic in
question.It lies, rather, in the fact that, 'rstly, if any
proposition(Ax)F(x) were provable in the Arithmetic, then
there would be auniform effective method, say EM_Alpha,
necessarily independent of n,that, given any natural number
n, would constructively prove that F(n)holds in the
Arithmetic; and, secondly, in the fact that, if (Ax)F(x)were
Tarskian-true, then, given any natural number n, there
wouldalways be some individual effective method, say
EM_Beta(n) - notnecessarily independent of n - that would
constructively prove thatF(n) holds in the Arithmetic
(assuming that TarskiOs de'nitionsimplicitly
imply the
existence of an effective method - notnecessarily algorithmic
- for verifying the truth of a propositionunder the standard
interpretation).Clearly, whilst the former implies the
latter, the 'rst half ofGoedelOs proof of
Theorem VI in his
1931 paper can be reasonablyinterpreted as asserting,
essentially, that the converse does not holdif we assume the
consistency of the Arithmetic in question.Now the interesting
foundational issue here is, 'rstly, whether, andunder what
conditions, EM_Alpha can be taken to be, say, a
Markovalgorithm; and, secondly, whether, and under what
conditions, theexistence of EM_Alpha for a given F(x) would
imply that (Ax)F(x) isprovable.Of interest, similarly, would
be the question of whether theTarskian-truth of a proposition
(Ax)F(x) under any interpretation canbe quali'ed as holding
if, and only, if, given any value s in therange of the
interpretation, there is an effective method in
theinterpretation for determining that F(s) holds. In other
words, can,and if so how and under what conditions,
Tarskian-truth be madeeffectively veri'able in any
interpretation of an Arithmetic (moreparticularly, in the
standard interpretation of an Arithmetic such as,say,
PA).Another interesting foundational issue raised in the
discussion is thequestion of when, and under what conditions,
we may assert that anumber-theoretic relation, and the
standard interpretation of itsformal expression in an
Arithmetic such as PA, can be asserted ashaving the same
meaning.A related, albeit obliquely raised, question is
whether the PA-formulaCon(PA), under the standard
interpretation, asserts that PA isconsistent by an arbitrary
convention, or whether it can be shown tobe equivalent to the
formal meta-de'nition of classical consistencyin a
constructive, and intuitionistically unobjectionable way
(inGoedelOs sense).Yet another intriguing, also obliquely
raised, foundational issue, iswhether, and under what
conditions, a number-theoretic relation - andits de'ning
expressions - can be introduced axiomatically into aformal
Arithmetic such as PA, so that it directly supports
referenceto its own syntax. In other words, can we
effectively, andmeaningfully, de'ne a mathematical object;
and under whatconditions can we introduce such objects into a
formal system such asPA, or into a set theory such as ZFC,
etc. without invitinginconsistency?Another, again obliquely
raised, issue is that if PA and its standardinterpretation SI
are to be treated as different languages, how arethey related?
Since, despite the misleading semantics, PA isessentially
intended to formalise SI, which should be considered themore
fundamental? Are there any conditions under which PA could
beconsidered a constructive interpretation of SI?Bhupinder
Singh Anand