http://www.nnseek.com/e/sci.math.research/ Posts for sci.math.research Tue, 19 Aug 2008 11:30:02 PDT http://www.nnseek.com/ http://www.nnseek.com/img/64.png 64 64 NNSeek http://www.nnseek.com/e/sci.math.research/eigenvalue_142591838t.html http://www.nnseek.com/e/sci.math.research/eigenvalue_142591838t.html
A = (1/(i*beta+r-s))_{r,s}

where I am indexing things by r,s rather than i,j just so as to avoid
conflicts with the imaginary unit i.
(Here beta is a positive real.) The indices r,s range from -N to N,
where N is some large integer.

What is the norm of A? In other words: can you bound |alpha| of every
eigenvalue alpha of A?

I have a feeling that we must have

|alpha| <= pi coth(pi beta),

but I cannot prove it. (After some effort, I have been able to prove
an upper bound that is worse than this by a factor of about 1.5 in the
range I care most about (0.05 < beta < 0.5, roughly).)

* * *

On the importance of this: if we had a good upper bound on the norm of
A, we would have a good large sieve with exponential smoothing.

The form of the large sieve seen most often bounds a quantity of the
form

\sum_{a,q} |\sum_{n<=N} a_n e(an/q)|^2

Now, we all know that sharp truncations are a thing of the past and
should not be used.
As if to illustrate this philosophy, good bounds were proven for the
following form before they were known for the form I stated three
lines above:

\sum_{a,q} |\sum_n g(n) a_n e(an/q)|^2

where g(n) = (N-n)/N if n<=N,
0 otherwise.

This g is a nice smoothing function for reasons known since Fejer.
However, sometimes a different sort of smoothing is best for other
purposes, and carries over to the large-sieve inequality we want to
estimate. It would be very nice if we had an optimal or near-optimal
bound for the quantity

\sum_{a,q} |\sum_n a_n e(an/q) e^{-n/N}|^2

This would follow if we had an optimal or near-optimal bound for the
norm of the matrix I defined at the beginning. (I actually need a
bound for the matrix (1/(i beta + lambda_r - lambda_s)),
where lambda_n are reals separated by at least a distance of 1 from
each other; given our current state of knowledge, this shouldn't be
much harder than the special case lambda_n=n.)

Note that Hilbert's inequality is a bound on the norm of the matrix

(a_{r,s})_{r,s},

where a_{r,s} = 1/(r-s) if r and s are distinct
0 otherwise.

It seems to me that the question I asked at the beginning of this
email is a little more natural, since the definition of the matrix
entries does not require the LaTeX commands \begin{cases} and
\end{cases}.

Harald

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]]> Tue, 19 Aug 2008 11:30:02 PDT http://www.nnseek.com/e/sci.math.research/rank_theorem_142591582t.html http://www.nnseek.com/e/sci.math.research/rank_theorem_142591582t.html and Riemannian geometry, 2nd ed, Pure and Applied Mathematics, Vol.
120, Academic Press, Orlando, 1986, contains the following two
results:

(7.1) Theorem (Rank Theorem). Let $A_0 \subset R^n$, $B_0 \subset R^m$
be open sets, $F:A_0 \rightarrow B_0$ be a $C^r$ mapping, and suppose
the rank of $F$ on $A_0$to be equal to $k$. If $a \in A_0$ and $b=F(a)
$, then there exist open sets $A \subset A_0$ and $B \subset B_0$ with
$a \in A$ and $b \in B$, and there exist $C^r$ diffeomorphisms $G:A
\rightarrow U \subset R^n$, $H:B \rightarrow V \subset R^m$ ($U$, $V$
open) such that $H \circ F \circ G^{-1}(U) \subset V$ and such that
this map has the simple form $H \circ F \circ G^{-1}(x^1,\ldots,x^n) =
(x^1,\ldots,x^k,0,\ldots,0)$.

(7.2) Corollary.We may choose the neighbourhoods $U$ and $V$ in either
of the following two ways: (i) $U = B^n_\epsilon(0)$ and $V = B^m_
\epsilon(0)$, or (ii) $U = C^n_\epsilon(0)$ and $V = C^m_\epsilon(0)$,
with the same $\epsilon > 0$ for both $U$ and $V$. Then, if $\pi$
denotes the projection of $R^m = R^k \times R^{m-k}$ to $R^k$ and
$i:R^k \rightarrow R^k \times R^{n-k}$ is the injection to the subset
$R^k \times {0}$, we have $\pi \circ H \circ F \circ G^{-1} \circ i$
is the identity on $B^k_\epsilon(0)$ in case (i) or on $C^k_
\epsilon(0)$ in case (ii). (Please note, $B^n_\epsilon(0)$ is the
euclidean open ball of centre $0$ and radius $\epsilon$, while $C^n_
\epsilon(0)$ is the open ball of centre $0$ and radius $\epsilon$ but
with respect to the sup norm distance, et cetera.)

This corollary seems to me untrue. Does anybody know a counterexample,
or if true, a proof or a reference thereof? Thanks.

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]]> Tue, 19 Aug 2008 11:30:01 PDT http://www.nnseek.com/e/sci.math.research/hadamard_levy_theorem_141787230t.html http://www.nnseek.com/e/sci.math.research/hadamard_levy_theorem_141787230t.html inverse function theorem) in Abraham, Marsden & Ratiu, Manifolds,
tensor analysis, and applications, 2nd ed. I find it very unclear.
Does anybody know other references for a proof of this theorem?

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]]> Fri, 15 Aug 2008 10:54:44 PDT http://www.nnseek.com/e/sci.math.research/markov_models_or_markov_chains_141497950t.html http://www.nnseek.com/e/sci.math.research/markov_models_or_markov_chains_141497950t.html
I have an interesting problem here. I am not sure if I have posted
this question before, but have had no responses so far and so trying
again.


My problem is related to markov models.


My scenario is:


1) I have an observation matrix of million events N...this is usually
inter arrival times of packets of particular protocols in random
networks, so depending on the protocols they will have some
distributions.


2) If i plot N+1 over N and then use 3d Histogram or density plots
for
the million events, this will allow me to see the distributions
shapes, whether poisson, gaussian or continous random distributions.
Irrespective of the distributions, I can shrink the number of bins
blindly and using the counts density of each bin [nXn], I can
normalise the values across the columns and build the state
transition
matrix, and hence I have a Markov model with states which are related
to the number of bins I shrink down to. But doing this will give me
only a very basic fit of the model.


3) To get a better model with optimum fit, I need to replace the
distributions I find in the graphs, with relevant appropriate state
transitions.....and this is where my problem is. I have not been able
to understand how to replace these distributions with unique state
transitions.


Anybody who can help me in this, I will be really grateful. If you
have not understood my problem, I am willing to clarify further.....


But in general, I need help with a mathematical relationship between
a
distribution in an observation matrix and an appropriate state
transition.


Many thanks in advance,


Ads.

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]]> Thu, 14 Aug 2008 05:15:49 PDT http://www.nnseek.com/e/sci.math.research/map_on_homotopy_group_of_bg_141153630t.html http://www.nnseek.com/e/sci.math.research/map_on_homotopy_group_of_bg_141153630t.html
My thesis involves working with the classifying space of the Poincare
group, P. For those of you not familiar with the Poincare group, here is
a reference: <http://en.wikipedia.org/wiki/Binary_icosahedral_group>. It
is the fundamental group of the Poincare sphere (Poincare's
counter-example to his first formulation of the Poincare Conjecture (now
Theorem?)).

The classifying space for P, BP, can be found from the action of P on
S^3. For more information on how to do this, see _Cohomology of Groups_
by Brown, pp 20-21. For the actual application of this to P, see
_Cohomology of Finite Groups_ by Adem and Milgram, page 279.

Now, P is perfect, so one may apply the Plus Construction to BP to form
the simply connected space BP+. The higher homotopy group of BP+ are now
no longer 0. As is alluded to in "Manifolds with a Given Homology and
Fundamental Group" by Hausmann and completed in _Cohomology of Finite
Groups_, pp 279-280, \pi_n(BP+) = \pi_n(S^3)/120 x \pi_{n-1}(S^3)_120
(where the latter group is all elements in \pi_n(S^3) whose order
divides 120).

Also, P is isomorphic to SL(2,5) and the automorphism group of P,
Aut(P), is isomorphic to PGL(2,5). If P = (st)^2*t^(-5)> and we choose s = [[1,4][1,0]] and t = [[4,0][4,4]], then
one specific outer automorphism (i.e., not an inner automorphism) is
given by congugation by the projective class of [[0,1][3,0]]; denote
this automorphism by \Phi.

Note that \Phi induces a map from BP to itself and then from BP+ to
itself; finally, it induces a map from \pi_7(BP+) (= Z_2 x Z_12) to
itself.

My question is, what explicitly is this map on Z_2 x Z_12? It is quite
important for my thesis, and I fear I don't have the tools to compute
it. I spoke to some of the above-mentioned authors, but they didn't know
the answer, either.

Any assistance you can provide is greatly appreciated.

Sincerely,
--
Jeffrey Rolland
hotmail.com>

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]]> Tue, 12 Aug 2008 14:57:52 PDT http://www.nnseek.com/e/sci.math.research/seven_papers_published_by_geometry_topology_public_140496478t.html http://www.nnseek.com/e/sci.math.research/seven_papers_published_by_geometry_topology_public_140496478t.html
(1) Algebraic & Geometric Topology 8 (2008) 1281-1293
On asymptotic dimension of amalgamated products
and right-angled Coxeter groups
by Alexander Dranishnikov
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p045.xhtml
DOI: 10.2140/agt.2008.8.1281

(2) Algebraic & Geometric Topology 8 (2008) 1295-1332
Surgery description of colored knots
by R A Litherland and Steven D Wallace
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p046.xhtml
DOI: 10.2140/agt.2008.8.1295

(3) Algebraic & Geometric Topology 8 (2008) 1333-1370
Fundamental groups of topological stacks with the slice property
by Behrang Noohi
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p047.xhtml
DOI: 10.2140/agt.2008.8.1333

(4) Algebraic & Geometric Topology 8 (2008) 1371-1402
Actions of certain arithmetic groups on Gromov hyperbolic spaces
by Jason Fox Manning
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p048.xhtml
DOI: 10.2140/agt.2008.8.1371

Three papers have been published by Geometry & Topology

(5) Geometry & Topology 12 (2008) 2095-2171
Closed quasi-Fuchsian surfaces in hyperbolic knot complements
by Joseph D Masters and Xingru Zhang
URL: http://www.msp.warwick.ac.uk/gt/2008/12-04/p048.xhtml
DOI: 10.2140/gt.2008.12.2095

(6) Geometry & Topology 12 (2008) 2173-2201
Covering link calculus and iterated Bing doubles
by Jae Choon Cha and Taehee Kim
URL: http://www.msp.warwick.ac.uk/gt/2008/12-04/p049.xhtml
DOI: 10.2140/gt.2008.12.2173

(7) Geometry & Topology 12 (2008) 2203-2247
Orbifold string topology
by Ernesto Lupercio, Bernardo Uribe and Miguel A Xicotencatl
URL: http://www.msp.warwick.ac.uk/gt/2008/12-04/p050.xhtml
DOI: 10.2140/gt.2008.12.2203

Abstracts follow

(1) On asymptotic dimension of amalgamated products and right-angled
Coxeter groups by Alexander Dranishnikov

We prove that the asymptotic dimension of A and B amalgamated over C
is bounded above by the maximum of the asymptotic dimensions of A, B
and C+1. Then we apply this inequality to show that the asymptotic
dimension of any right-angled Coxeter group does not exceed the
dimension of its Davis complex.


(2) Surgery description of colored knots
by R A Litherland and Steven D Wallace

The pair (K,rho) consisting of a knot K in S^3 and a surjective map
rho from the knot group onto a dihedral group of order 2p for p an odd
integer is said to be a p-colored knot. In [Algebr. Geom. Topol. 6
(2006) 673-697] D Moskovich conjectures that there are exactly p
equivalence classes of p-colored knots up to surgery along unknots in
the kernel of the coloring. He shows that for p=3 and 5 the
conjecture holds and that for any odd p there are at least p distinct
classes, but gives no general upper bound. We show that there are at
most 2p equivalence classes for any odd p. In [Math. Proc. Cambridge
Philos. Soc. 131 (2001) 97-127] T Cochran, A Gerges and K Orr define
invariants of the surgery equivalence class of a closed 3-manifold M
in the context of bordism. By taking M to be 0-framed surgery of S^3
along K we may define Moskovich's colored untying invariant in the
same way as the Cochran--Gerges--Orr invariants. This bordism
definition of the colored untying invariant will be then used to
establish the upper bound as well as to obtain a complete invariant of
p-colored knot surgery equivalence.


(3) Fundamental groups of topological stacks with the slice property
by Behrang Noohi

The main result of the paper is a formula for the fundamental group of
the coarse moduli space of a topological stack. As an application, we
find simple formulas for the fundamental group of the coarse quotient
of a group action on a topological space in terms of the fixed point
data. In particular, we recover, and vastly generalize, results of
Armstrong, Bass, Higgins and Taylor and Rhodes.


(4) Actions of certain arithmetic groups on Gromov hyperbolic spaces
by Jason Fox Manning

We study the variety of actions of a fixed (Chevalley) group on
arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain
a complete classification. In rank one, we obtain some partial results
and give a conjectural picture.


(5) Closed quasi-Fuchsian surfaces in hyperbolic knot complements
by Joseph D Masters and Xingru Zhang

We show that every hyperbolic knot complement contains a closed
quasi-Fuchsian surface.


(6) Covering link calculus and iterated Bing doubles
by Jae Choon Cha and Taehee Kim

We give a new geometric obstruction to the iterated Bing double of a
knot being a slice link: for n>1 the (n+1)-st iterated Bing double of
a knot is rationally slice if and only if the n-th iterated Bing
double of the knot is rationally slice. The main technique of the
proof is a covering link construction simplifying a given link. We
prove certain similar geometric obstructions for n < 2 as well. Our
results are sharp enough to conclude, when combined with algebraic
invariants, that if the n-th iterated Bing double of a knot is slice
for some n, then the knot is algebraically slice. Also our geometric
arguments applied to the smooth case show that the Ozsvath--Szabo and
Manolescu--Owens invariants give obstructions to iterated Bing doubles
being slice. These results generalize recent results of Harvey,
Teichner, Cimasoni, Cha and Cha--Livingston--Ruberman. As another
application, we give explicit examples of algebraically slice knots
with nonslice iterated Bing doubles by considering von Neumann
rho-invariants and rational knot concordance. Refined versions of
such examples are given, that take into account the
Cochran--Orr--Teichner filtration.


(7) Orbifold string topology
by Ernesto Lupercio, Bernardo Uribe and Miguel A Xicotencatl

In this paper we study the string topology (a la Chas-Sullivan) of an
orbifold. We define the string homology ring product at the level of
the free loop space of the classifying space of an orbifold. We study
its properties and do some explicit calculations.

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]]> Sat, 09 Aug 2008 06:13:39 PDT http://www.nnseek.com/e/sci.math.research/hidden_markov_model_please_kindly_help_me_139988574t.html http://www.nnseek.com/e/sci.math.research/hidden_markov_model_please_kindly_help_me_139988574t.html
Dear Everybody,

Please kindly help me in a problem that I am trying to understand
here.

My research problem that I have to solve is:

I have a sequence of million elements[1:1000000]. This can be
typically anything of randomly varying elements...[Although Usually I
will be considering Interarrival times of packets in a network stream,
so there will be many repititions inside it.

My objective:

What I want is that with this sequence of events, I want to estimate a
hidden Markov Model.

I think in my opinion estimating a hidden Markov Model means, that I
am estimating how many states are there in the model, what is the
state transition matrix although I am not really sure how much I have
to worry about
I have been trying to use matlab to my favour to solve these problems.

Now I have been trying to use Matlab help solve my problem.

I began by looking the following command and also its relatives.


[TR, E] = HMMESTIMATE(SEQ,STATES)

I know Sequence is my observation matrix which is the million samples,
how do I know the STATES sequence? How can I estimate this? how do I
know how may states are there?

Please kindly help me in this regard and I will be really grateful to
you.

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]]> Thu, 07 Aug 2008 04:40:28 PDT http://www.nnseek.com/e/sci.math.research/two_permutations_generating_the_symmetric_group_139988318t.html http://www.nnseek.com/e/sci.math.research/two_permutations_generating_the_symmetric_group_139988318t.html
Is it of interest to know that two permutations of order 2 cannot
generate the symmetric group Sn for values n>3?

Has this already been proven?

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]]> Thu, 07 Aug 2008 04:40:28 PDT http://www.nnseek.com/e/sci.math.research/statistica_sinica_july_issue_139684190t.html http://www.nnseek.com/e/sci.math.research/statistica_sinica_july_issue_139684190t.html
I am sending you the table of contents for the July 2008 issue of
Statistica Sinica. In this issue, we have a collection of seven
articles on the theme topic of "FDR and ROC", with invited editorials
by Professors Bradley Efron and Peter Westfall. Please click on the
current issue at http://www.stat.sinica.edu.tw/statistica/ or click on
the "membership only area" at http://www.icsa.org/ (for ICSA members)
for accessing these articles.

Sincerely,
Karen Li---on behalf of the Co-editors
--------------------------------------------------------------------
Editor

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]]> Tue, 05 Aug 2008 14:00:03 PDT http://www.nnseek.com/e/sci.math.research/eigenvalues_and_eigenvectors_of_perfectly_balanced_139683678t.html http://www.nnseek.com/e/sci.math.research/eigenvalues_and_eigenvectors_of_perfectly_balanced_139683678t.html formulas
for the eigenvalues and more importantly eigenvectors of perfectly
balanaced binary trees.

Thanks,
Petar

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]]> Tue, 05 Aug 2008 14:00:02 PDT