mm-1047
===
Subject: Re: Comma category
> and while were at it we might as well cast this in the
> language of weighted limits: take V = Cat (where
V-categories
> are now *2-categories*), let J be the category (-) -> (0)
<- (+)
> (considered as a 2-category by viewing the hom-sets as
discrete
> categories), and let F: J --> V be the functor (or
2-functor)
> into Cat which sends (+) and (-) to the terminal category 1,
> (0) to the category 2 = (0 -> 1), the arrow (-) -> (0) to
the
> inclusion i_0: 1 --> 2 valued at 0, and the arrow (+) -> (0)
> to the inclusion i_1: 1 --> 2 valued at 1.
> Then given a 2-category X, and a 2-functor G: J --> X (viz.,
> a diagram of 1-cells C -F-> E <-G- D in X), a limit of G
> w.r.t. the weight F defined above is called a *comma object*
> F|G in X. I believe it is also called a *lax pullback* of F
> and G, although I tend not to like such terminology, as lax
> is overused, sometimes confusingly so.
I knew this definition of comma category as a lax pullback. In
fact, In
my first post starting this thread, I tried to reformulate it
as a natural
isomorpphism in order to see it as a representation.
stupid mistake in my previous post ^_^.
David.
===
Subject: Re: a non-linear second-order PDE
Epigone-thread: gromstunpand
>>I consider the following non-linear second-order partial
differential
>>equation with two non-negative variables x and y:
>>
>>(a+ b x) [ df/dx d^2f/(dx dy) - df/dy d^2f/(dx^2) ] + b y [
df/dx
d^2
>>f/d(y^2)- df/dy d^2f/(dx dy)] =0
>>
>>where a >0 and b is a real number. Additional conditions are
>>
>>f(x,0) = (a + b x)^(1-1/b) /(1/b (1-1/b))
>>
>>and
>>
>>df(x,0)/dy =0.
>> I assume you mean df/dy = 0 when y = 0.
>>for all x.
>>
>>(For b=1, f(x,0) converges to logarithmic function ln(a+x),
but
that
>>should not really matter).
>>
>>One solution to the PDE is:
>>
>>f(x,y) = 1/(1/b (1-1/b)) [ (a + b x)^(1- 1/b) + c2 b
y^(1-1/b) ]
>> This doesnt satisfy your second additional condition in
general,
however:
>> df/dy has a singularity at y=0 if b > 0. It does work if
c2 = 0 or
if
>> b < 0. Of course, in general theres also a singularity
when
a+bx=0.
>>with constants c1, c2.
>> You didnt have a c1.
>>My question: Are there other solutions? If so, which? If
no, how
can
>>one prove uniqueness?
>> Solutions of your PDE include
>> f(x,y) = c_0 + sum_{j=1}^n c_j (a+bx)^(k_j) y^(p-k_j)
>> for any constants c_j, k_j and p. For example, you could
add to
>> your solution a term in (a+bx)^(-1-1/b) y^2 and get another
solution
>> with the same values of f(x,0) and df/dy (x,0). So, no
uniqueness
>> there.
>> BTW, an interesting feature of the PDE is that if f(x,y)
is a
solution, so
>> is f(x,y)^k for any k, or ln(f(x,y)), or exp(f(x,y)).
>I believe Ive found the general solution to the PDE:
>f(x,y) = G((a+bx) H(y/(a+bx)))
>where G and H are arbitrary (twice differentiable) functions.
>Wlog we can take H(0) = 1, and then f(x,0) = G(a+bx)
determines G,
>while df/dy (x,0) = 0 as long as H(0) = 0.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics <a
href=http://www.math.ubc.ca/~israel>http://www.math.ubc.ca/~
israel</a University of British Columbia Vancouver, BC, Canada
I believe Ive got a solution to a simpler equation that may
help solving this one:
d^2f/dx^2*df/dy-d^2/dx dy*df/dx = 0 (1) possess a solution:
f(x,y)=h(x+g(y)); (2)
permuting x and y in (1) gives f1(x,y)=h(y+g(x)).
Combine with my observation d/dx (a*(c+y)+bxy) = by
d/dy (a*(c+y)+bxy) = a +bx
===
Subject: Re: a non-linear second-order PDE
>I believe Ive got a solution to a simpler equation that 
may
>help solving this one:
> d^2f/dx^2*df/dy-d^2/dx dy*df/dx = 0 (1) possess a solution:
> f(x,y)=h(x+g(y)); (2)
> permuting x and y in (1) gives f1(x,y)=h(y+g(x)).
> Combine with my observation d/dx (a*(c+y)+bxy) = by
> d/dy (a*(c+y)+bxy) = a +bx
Yes, this equation (with variables renamed X and Y) is
equivalent to the
original one under the transformation a + b x = exp(X+Y), b y
= exp(X-Y).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: provability and truth
|If P is a statement of arithmetic, define the hermeneutics of
P (I
|choose this word for no particular reason) to be the
statement
|
| <PA|-P> => P
|
|(in other words, if P is provable in PA, then P is true).
As has been noted, this is usually called a reßection
principle. One of my fondest college memories is of studying
the paper in vol. 27 of the Journal of Symbolic Logic on
reßection principles, on my own. There is also a recent
book by Torkel Franzen aimed at making some results on
iterated reßection principles more accessible.
I have an amusing tangent to this. Theres a famous theorem.
I thought I knew its name, but it appears that was the name
of a completely different theorem. It says that for each P,
<PA |- (<PA|-P> => P)> <=> <PA|-P>.
In other words, the only cases of this reßection principle
provable in PA are the trivial ones, where P is already a
theorem of PA. (Theres nothing very special about PA here;
the result applies to a wide range of formal systems; they
only have to satisfy the assumptions implicit below.)
This can be proven using reasoning parallel to the reasoning
in Goedels proof of his incompleteness theorems. It is in
fact a direct generalization; Goedels second incompleteness
theorem is the special case of the above where P is some
sentence such as 1=0. In that case PA|-P is equivalent to
PA being inconsistent, and (<PA|-P> => P) is equivalent to PA
being consistent, and the second incompleteness theorem
says that a system like PA can only prove its own consistency
if it is inconsistent.
We can concoct a sentence G such that G is true if and only
if G or P is not a theorem of PA, and such that this fact
is provable in PA. We rely on the fact that if a sentence is
a theorem of PA, then the fact that it is a theorem of PA
is also a theorem of PA. We also rely on the fact described
in the last sentence being a theorem of PA!
Assume the left hand side above, that <PA|-P> => P is a
theorem of PA. The following argument could then be
formalized in PA:
Suppose G is false. Then the falsity of G is a theorem
of PA (because G is false is equivalent to the
provability of something). But also G or P is provable
in PA. Hence P is a theorem of PA. {...} Hence P.
Where I have {...}, we insert the proof in PA of <PA|-P>=>P
that we are assuming exists. The above argument would be
a proof in PA that if G is false then P is true, or
G or P. Since G denies this, G would be false.
So G is false. Then the falsity of G is a theorem
of PA (because G is false is equivalent to the
provability of something). But also G or P is provable
in PA. Hence P is a theorem of PA.
A second special case of this theorem is for sentences
that have been concocted in sort of the opposite way from
the Goedel sentence, i.e. sentences H that are equivalent
to their own provability in PA. The theorem whose proof I
just sketched implies that they are all theorems of PA. They
are, but their proofs in PA, as constructed by the proof
above, come off like doubletalk.
Because H is equivalent by construction to H is provable
in PA, a proof of H in PA must show there exists a proof
of H in PA. There is a metatheorem in this case that implies
any such proof can be rewritten to make it plainly
constructive, which roused my curiosity. Its not especially
hard to write it in a plainly constructive form, which then
raises the question, whats the proof of H that the proof of
H constructs? I traced through this once and found that the
proof it constructs is-- essentially-- itself.
|For
|example, the hermeneutics of the False statement is the
statement
|Consis(PA) that PA is consistent (and the hermeneutics of
the True
|statement is again the True statement, of course); the
hermeneutics of
|~Consis(PA) is essentially equivalent, modulo Consis(PA), to
the
|consistency of PA+Consis(PA). And so on.
|
|Define the Grand Hermeneutical Axiom Scheme (Totally
Ludicrous,
|Yuck) (Ghastly for short) to be the axiom scheme consisting
of the
|hermeneutics of every arithmetical statement P (so, Ghastly
allows us
|to deduce, if any P is provable from PA, that P is true). As
|explained above, Ghastly implies Consis(PA) and
Consis(PA+Consis(PA)),
|and so on - with a reasonable definition of and so on.
A reßection principle is sometimes used as a function on
formal systems of some type, in this case going from a system
S to the system S augmented by an axiom scheme.
Youre showing here that the reßection principle you 
described
before is, in a certain sense, a lot more powerful than the
one
merely going from S to S+Consis(S).
[...]
|Finally I come to my question (but Im interested more
generally in
|anything that could be said about Ghastly, including any more
|reasonable name for it): is Ghastly equivalent to some
(finite)
|statement about arithmetic? Or is it, at least, implied by
some
|(finite) statement about arithmetic that is consistent
(provably in
|ZF)?
By finite, perhaps you mean a statement in 
first-order
arithmetic, which is the language of PA.
No, any such sentence is disprovable in PA. Suppose that Q
is such a sentence. The reßection principle as applied to
Q is false is <PA|-Q is false>=>Q is false. If it was a
theorem in PA that Q implied it, then we would have
PA |- Q => (<PA|-Q is false> => Q is false),
which is equivalent to
PA |- <PA|-Q is false> => Q is false.
By the above theorem whose name I seem to have been
misremembering, this implies that Q is false is a theorem
of PA.
For each n, one can formalize in first-order arithmetic the
property of an n-quantifier sentence being true, and thus
the reßection principle as restricted to n-quantifier
sentences can be formalized. But it needs more than n
quantifiers to be formalized. So the reßection principle
in general can only be formalized in a stronger language;
it doesnt need to be very much stronger, however.
Note that ZF in this context is overkill. You could have
just as well used a standard elementary theory of
natural numbers and sets of natural numbers such as the
formal system PA_2. Thats strong enough to do everything
youve mentioned doing with ZF.
Keith Ramsay
===
Subject: Re: provability and truth
Aatu Koskensilta in litteris
<ckr1ju$3cp$1@dizzy.math.ohio-state.edu>
scripsit:
> There are numerous complications here. One of the more
obvious ones is
> that if we accept e.g. Fefermans analysis that RA_Gamma_0
or IR
> contains exactly those principles acceptable on basis of
acceptance of
> PA, we are in fact asserting the acceptability of
RA_Gamma_0. In this
> case we can convincingly argue that the analysis is not
acceptable on
> basis of PA, since the notion of a well-ordering isnt. 
But
in case of
> e.g. ZFC there is no reason to consider the theory obtained
by
> autonomously iterating addition of reßection principles
(Aut(ZFC)) to
> comprise everything that is implicit in acceptance of ZFC.
This is
> because it seems that basic observations about what
acceptance means and
> what well-orderings are lead one to accept Aut(ZFC) on
basis of
> acceptance of ZFC.
suggestions that could serve as introductory material to these
questions? (Assuming I have basic background on logic and set
theory,
e.g., the contents of Jechs *Set Theory* and 
Poizats *Model
Theory*.)
For example, where can I find a definition of 
Aut(ZFC)?
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
===
Subject: Re: provability and truth
Originator: tchow@maclaurin.mit.edu.mit.edu (Timothy Chow)
>suggestions that could serve as introductory material to
these
>questions?
One possibility is Torkel Franzens new book, which I 
believe
has just been
published. http://www.sm.luth.se/~torkel/eget/progps.html
--
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: provability and truth
>>suggestions that could serve as introductory material to
these
>>questions?
> One possibility is Torkel Franzens new book, which I
believe has just
been
> published. http://www.sm.luth.se/~torkel/eget/progps.html
I dont know about introductory - I suspect 
Franzens book
here would be
the perfect choice although I havent read it yet - , but 
the
following
Solomon Feferman: Transfinite recursive progressions of
arithmetic
theories, JSL 27(3) 1962
Solomon Feferman: Arithmetization of metamathematics in
general
settings
Fundamenta Mathematica 49 196(1?)
Solomon Feferman: Systems of Predicative Analysis I
JSL 29(1) 1964
Solomon Feferman: Reßecting on incompleteness
JSL 56(1) 1991
Solomon Feferman: Turing in the land of Oz
in The Universal Turing Machine : a Half Century
Summary
Alan Turing: Systems of logic based on ordinals
in M. Davis: The Undecidable
Torkel Franzen: Transfinite progressions: a second look at
Solomon Feferman has many papers available on-line at his
home page,
including those on so called unfoldings of theories, which
are an
attempt to obtain more perspicious formulation of what sort
of things
are implicit in the acceptance of a theory.
For basic information on recursion theory, constructive
ordinals,
recursive well-founded trees and such like, I suggest
Odifreddis
Classical Recursion Theory and Rogers Theory of Recursive
Functions and
Effective Computability. I dont know whether more modern
treatments
would be better, but these two books are where I learned this
material
(and what I keep with me as references).
(Im writing these references from memory, so they might be
slightly
inaccurate)
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: provability and truth
tchow@lsa.umich.edu in litteris
<ckpjtd$2m2$1@dizzy.math.ohio-state.edu>
scripsit:
>>If P is a statement of arithmetic, define the hermeneutics
of P (I
>>choose this word for no particular reason) to be the
statement
>> <PA|-P> => P
> This is more commonly known as a reßection principle.
>>Define the Grand Hermeneutical Axiom Scheme (Totally
Ludicrous,
>>Yuck) (Ghastly for short) to be the axiom scheme consisting
of the
>>hermeneutics of every arithmetical statement P (so, Ghastly
allows us
>>to deduce, if any P is provable from PA, that P is true).
> I dont think you mean to say *every* arithmetical
statement, because
there
> are arithmetical statements that are false, including those
that are
> disprovable in PA.
I dont see how that is a problem. Even if P is the False
statement
(0=1, if you will), the <PA|-P>=>P statement makes good sense
(it is
Consis(PA)).
So I *do* mean every P (every closed statement in the
language of
arithmetic), even those that are provably false in PA (in
which case
we just get Consis(PA)).
> What it sounds like youre try to get at is the concept of
the
reßective
> closure of PA. Other keywords are iterated consistency
extension
and
> Feferman-Schuette ordinal. The idea behind these concepts
is to add
to
> PA any arithmetical statements that we implicitly must
accept if we
accept
> PA as being true. Those keywords should get you started and
help you
> make your questions more precise.
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
===
Subject: Re: supplementary subsets of (Z/pZ)^2
> Ive stumbled upon the following provokingly
innocent-sounding problem
> which has kept me bafßed:
> Let p be a prime. Let U and V be two subsets of (Z/pZ)^2
each of
> cardinality p, and assume that U+V = (Z/pZ)^2 (in other
words,
> pairwise addition forms a bijection between the cartesian
product U*V
> and (Z/pZ)^2). Is it true that projection on the first
coordinate
> must define a bijection of one of the two sets (either U or
V) onto
> Z/pZ? (It is of course equivalent to ask whether, for each
linear
> form l from (Z/pZ)^2 to Z/pZ, l forms a bijection from one
of the two
> sets onto Z/pZ.)
I hope the following is correct:
1) first, rephrase geometrically (as you rightly pointed out,
it is a
tiling problem): we have a finite square of side length p, so
p^2
points. We assume that we can find a subset U of p points (the
set of
origins), and another subset V of p points (the pattern) such
that we
can can tile the square (modulo identifications) when we
translate our
pattern in the various reference frames. Your question is
wether we
can find a pattern which moreover avoids at least one row
(projection
on first coordinate) and/or at least one column (projection on
second
coordinate). Lets assume you actually meant 
Ôand.
2) second, count: the total number of possible patterns is N
= {p^2
choose p}. The number of patterns which avoid exactly one row
and one
column is M = {(p^2-p+1) choose p}.
Now, tiling means no overlap, so lets count the 
constraints.
We
start with the first origin u_1 and construct the p points
w(1,i)=u_1+v_i. Then we take u_2 and the v_i must now
satisfy: w(1,i)
!= w(2,j) for i<=j, ie thats n(1,2) = p+p.(p-1)/2
constraints. With
u_3 all the constraints are now : [w(1,i) != w(2,j) AND
w(1,i) !=
w(3,k) AND w(2,j) != w(3,k)]. So thats n(1,2,3) = {3 choose
2}.n(1,2)
. So when we reach u_p we have n(1,...,p) = {p choose
2}.n(1,2) . All
we need to do now is compare M with n(1,...,p), which I leave
you do.
3) remark: the question made sense for p prime only, since
when p is
composite there are obvious counter-examples. Namely, with
p=4 and
noting by Ô* the points of U and V and by 
Ô% Ôthe other
points of the
square, we have (use a font of fixed width):
% % % % % % % %
* % * % % % % %
U = % % % % and V = * * % %
* % * % * * % %
If all the above is correct then the case of (Z/p^2Z) could
be studied
along those lines too.
--
thomas. ...by natural selection our mind has adapted itself
to the
conditions of the external world. It has adopted the geometry
most
advantageous to the species or, in other words, the most
convenient.
Geometry is not true, it is advantageous. H. Poincare.
===
Subject: Re: supplementary subsets of (Z/pZ)^2
|Let p be a prime. Let U and V be two subsets of (Z/pZ)^2
each of
|cardinality p, and assume that U+V = (Z/pZ)^2 (in other
words,
|pairwise addition forms a bijection between the cartesian
product U*V
|and (Z/pZ)^2).
Let f and g be the characteristic functions of U and V. The
condition
you have is equivalent to f*g, the convolution, being the
constant 1
function. Take the Fourier transform. The Fourier transform
of 1 is
supported only at the identity. Moreover 1^=(f*g)^=f^ g^.
|Is it true that projection on the first coordinate
|must define a bijection of one of the two sets (either U or
V) onto
|Z/pZ?
So one of f^ or g^ has to be 0 on the character chi which
takes
(m, n) to e^(2*pi*i*m/p). For it to be 0, the sum of the
values of
this character on the points of U or V has to be 0. But the
only
way to get 0 as a sum of p p-th roots of 1 is by taking all
of the
p-th roots of 1. The fact that the value of chi for each
element
is different is equivalent to projection on the first
coordinate
being a bijection with Z/pZ.
I could have avoided the reference to the Fourier transform,
but
it seems to me that any set U of p elements such that the
Fourier transform of its characteristic function is 0 on at
least
half of the nonzero elements of the dual group has to be
pretty
special, and I want to ask someone to figure out what such a
subset has to look like. Does it have to be a coset of a
subgroup of order p?
The value is nonzero at any chi whose kernel contains the
difference between distinct elements of U, and conversely,
so this property is equivalent to the differences between
elements of U lying in at most half of the subgroups of
order p.
Keith Ramsay
===
Subject: Re: On Hodge and Betti numbers
Epigone-thread: sherdphendspoy
Marcel
===
Subject: HELP: A problem about Turing Machine
I am puzzled by a problem about Turing machine:
Given a Turing Machine M, is the following problem decidable?
M moves its head to the left more than 1,000 times on input 0
Best,
Tao
===
Subject: Re: A problem about Turing Machine
> I am puzzled by a problem about Turing machine:
> Given a Turing Machine M, is the following problem
decidable?
> M moves its head to the left more than 1,000 times on input
0
> Best,
> Tao
This is simpler than I first thought.
I will assume M is a Busy Beaver type TM and must move left
or right (or
halt) on every step.
Assume M has m states and is given a blank tape.
M can not move right m or more times without going into an
infinite loop.
This is because blank is the only input on the tape that M
can read without
moving left.
Now, we know M must move left in (m-1) or less steps or it
will never move
left.
Define x be the step when M moves left and let y be the tape
position M
read on step x.
We know the input tape must be blank beyond position y.
If M moves right two positions we are now in an area of blank
tape.
Again, M must move left in less than m steps or it will never
move left.
We can build a finite series that is a limit to the number of
steps M can
peform without moving left at least n times:
(m-1) + (m) + (m+1) + ... + (m+n-2)
Russell
- 2 many 2 count
===
Subject: Re: HELP: A problem about Turing Machine
> I am puzzled by a problem about Turing machine:Given a
Turing Machine M, is the following problem decidable?M moves
its head to the left more than 1,000 times on input 0
> Best,
> Tao
I think that any problem must be presented as a question.
I do not see a problem that a head moves to the left more
than n times on input 0. Where is the case for decide
something?
===
Subject: Re: HELP: A problem about Turing Machine
> I am puzzled by a problem about Turing machine:
> Given a Turing Machine M, is the following problem
decidable?
> M moves its head to the left more than 1,000 times on input
0
I think the answer is Yes. If the states of the machine are M
and the
alphabet of the tape is A, we can turn such a machine into a
push-down
stack automaton with states M*X, where X is the set of all
strings in A
with length <= 1000, and represents all that accessible in
the remaining
left
moves to the left of the head.
In turn, a push-down stack automaton can be solved by setting
up a set
of equations for the values of the relation s1 -> s2, where
s1 and s2 are
states (in this case, elements of M*X), and s1 -> s2 is true
if there is
a program path from s1 to s2 which does not look at the
current stack and
pops everything it pushes off the stack, so that when we get
to s2 the
stack is exactly the same as it was at s1.
The equations are of the form:
(1) s -> s (for all s);
(2) If at s1 there is a change of state to s2 which doesnt
push or pop
anything,
then for s3 /= s1, s1 -> s3 iff s2 -> s3;
(3) If at s1 there is a change of state to s2 which also
pushes a symbol
(a),
at s3 there is a change of state to s4 which pops the same
symbol
(a),
and s1 /= s5, then s1 -> s5 iff s2 -> s3 and s4 -> s5
We can then solve these equations to find the /minimum/
solution for the
relation
===
Subject: Re: HELP: A problem about Turing Machine
> Given a Turing Machine M, is the following problem
decidable?
> M moves its head to the left more than 1,000 times on input
0
No. If M halts on input 0, then youll know how many times M
moved its
head to the left and you can answer yes or no
If M doesnt halt on input 0, then youll have 
to wait until
a tally of
1,000 left head moves. If such occures, then you can answer
yes. If such
doesnt occure, you still have to wait longer to see if
perhaps itll
occure. So youll never be able to say no. Problem is
undecidable.
===
Subject: Re: HELP: A problem about Turing Machine
>> Given a Turing Machine M, is the following problem
decidable?
>> M moves its head to the left more than 1,000 times on
input 0
> No. If M halts on input 0, then youll know how many times
M moved its
> head to the left and you can answer yes or no
> If M doesnt halt on input 0, then youll 
have to wait
until a tally of
> 1,000 left head moves. If such occures, then you can answer
yes. If
such
> doesnt occure, you still have to wait longer to see if
perhaps itll
> occure. So youll never be able to say no. Problem is
undecidable.
I dont think thats obvious at all.
The question is, could you find a _different_ Turing machine
which will tell you for given input
if the first machine would move backwards 1000 times.
Actually, the 1000 moves seems irrelevant to me,
since one could construct a TM with 1000 times as many states
corresponding to each step to the left.
So it seems to me the problem is equivalent to:
Is the halting problem for a 1-stack machine decidable?
I dont know the answer, but I would have guessed it was 
Yes.
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
===
Subject: Re: Internal language of a category
>>There are different notions of internal languages such as
the one for
>>Cartesian closed categories or the one for toposes. But,
each time, the
>>definition of the internal language is given ad hoc. Is
there an abstract
>>definition of what is an (or the?) internal language of a
category?
> Not quite sure what you mean ad hoc, or even what you are
looking
> for really, since the language of category theory stands on
its
> own, but if I had to guess, you might be looking for
general ways
> of arguing as if objects had elements.
Given a category, Id like to have a textual (not graphical)
formal language which I could use to express the objects,
arrows,
commutative diagrams... of this category. For instance, in
the case of the
category of directed graphs, Id like a textual language to
describe directed graphs. But I am not sure that internal
language is the
right notion for this purpose. Indeed, the category of
directed graphs is
a topos but I do not think the internal language of a topos
is convenient
to describe directed graphs.
David.
===
Subject: Fixed point free finite group actions on n-disks
Epigone-thread: kherdfeifreh
Let D^n denote the usual n-disk as a smooth
manifold-with-boundary.
I believe it was Robert Oliver who first gave an example of a
smooth
fixed point free action of a finite group G on a 
D^n; if I
recall
correctly, n = 8 in this example.
Can someone please say what is the least known n for which a
smooth
fixed point free action of a finite group G on D^n 
is known to
exist?
Also, Id like to know for what n > 2, if any, such an 
action
has been
shown to be impossible.
References to the literature would be appreciated.
--Dan Asimov
===
Subject: 2 questions on finitely generated solvable groups
1) Can a solvable, finitely generated group act 2-transitively
on an
infinite set X? Or, at least, with a finite number 
of orbits in
X[Times]X?
2) If every solvable, finitely generated group a quotient of
some
finitely presented solvable group?
--
Yves de Cornulier
===
Subject: Re: 2 questions on finitely generated solvable groups
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> 1) Can a solvable, finitely generated group act
2-transitively on an
> infinite set X? Or, at least, with a finite 
number of orbits
in
X[Times]X?
> 2) If every solvable, finitely generated group a quotient of
some
> finitely presented solvable group?
I thank people who answered to these questions. I found that
the
answer is known to be negative for the second one:
Thm (Bieri, Strebel, 1979)
If G is a finitely presented solvable group and H a metabelian
quotient of G, then H is finitely presented.
And as Ignat noticed, free metabelian groups are not finitely
presented, so they are counterexamples to (2).
I have still no answer to (1); I am almost sure that it is,
in fact,
an open question.
--
Yves de Cornulier
===
Subject: Re: 2 questions on finitely generated solvable groups
>Can a solvable, finitely generated group act 2-transitively
on an
>infinite set X?
There is a construction of Philip Hall of a finitely generated
soluble
group G of derived length 3 which is not residually finite.
This group
has a maximal subgroup H of infinite index, so the action of G
on the
cosets of H is primitive - you might try and check whether it
is
actually 2-transitive.
The group was constructed in
AUTHOR = {Hall, P.},
TITLE = {On the finiteness of certain soluble groups},
JOURNAL = {Proc. London Math. Soc. (3)},
VOLUME = {9},
YEAR = {1959},
PAGES = {595--622},
MRCLASS = {20.00},
MRNUMBER = {MR0110750 (22 #1618)},
MRREVIEWER = {K. Gruenberg},
}
@book {MR986732,
AUTHOR = {Hall, Philip},
TITLE = {The collected works of {P}hilip {H}all},
SERIES = {Oxford Science Publications},
NOTE = {Compiled and with a preface by K. W. Gruenberg and J.
E.
Roseblade,
With an obituary by Roseblade},
PUBLISHER = {The Clarendon Press Oxford University Press},
ADDRESS = {New York},
YEAR = {1988},
PAGES = {xii+776},
ISBN = {0-19-853254-7},
MRCLASS = {01A75 (20-06)},
MRNUMBER = {MR986732 (90b:01108)},
MRREVIEWER = {Annabelle McIver},
}
The group is also described in exercise 15.4.6 in
@book {MR648604,
AUTHOR = {Robinson, Derek John Scott},
TITLE = {A course in the theory of groups},
SERIES = {Graduate Texts in Mathematics},
VOLUME = {80},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1982},
PAGES = {xvii+481},
ISBN = {0-387-90600-2},
MRCLASS = {20-01},
MRNUMBER = {MR648604 (84k:20001)},
}
I suspected that Halls group G could be a possible example,
but its
Francesco de Giovanni who pointed out to me that the exercise
in
Robinsons book states that G has indeed a maximal subgroup 
H
of
infinite index.
I am curious to know whether the action of G on the cosets of
H is
indeed 2-transitive; I may give it a try myself, time
permitting.
Andreas
===
Subject: Re: 2 questions on finitely generated solvable groups
>>Can a solvable, finitely generated group act 2-transitively
on an
>>infinite set X?
>There is a construction of Philip Hall of a finitely
generated soluble
>group G of derived length 3 which is not residually finite.
This group
>has a maximal subgroup H of infinite index, so the action of
G on the
>cosets of H is primitive - you might try and check whether
it is
>actually 2-transitive.
[...]
>I am curious to know whether the action of G on the cosets
of H is
>indeed 2-transitive; I may give it a try myself, time
permitting.
I worked out the exercise in Robinsons book together with 
my
colleague Francesca Dalla Volta. Alas, it seems that the
action of G
on the set X of the cosets of H in G is not 2-transitive.
>>Or, at least, with a finite number of orbits in X[Times]X?
Also, it seems to us that G has infinitely many orbits on X x
X. With
hindsight, I guess the original poster was well aware that
this group
did not provide an example.
Andreas
===
Subject: Re: 2 questions on finitely generated solvable groups
Epigone-thread: colsmymeu
>1) Can a solvable, finitely generated group act
2-transitively on an
>infinite set X? Or, at least, with a finite 
number of orbits in
X[Times]X?
>2) If every solvable, finitely generated group a quotient of
some
>finitely presented solvable group?
>Yves de Cornulier
1) The simplest example is G = Z x Z acting on X = Z by
translations:
(a,b).x = x + a + b
Clearly, it is 2-transitive.
2) No, the free n-generated solvable group of solvable length
d is not
finitely presented for n,d > 1. This was proved by A.L.
Shmelkin:
Zbl 0134.02801
Shmelkin, A.L.
Uber außosbare Produkte von Gruppen (Russian)
[J] Sib. Mat. Zh. 6, 212-220 (1965). [ISSN 0037-4474]
See review here:
http://www.emis.de/cgi-bin/Zarchive?an=0134.02801
Ignat Soroko,
Minsk, Belarus
===
Subject: Re: 2 questions on finitely generated solvable groups
YdC:
>>1) Can a solvable, finitely generated group act
2-transitively on an
>>infinite set X? Or, at least, with a finite 
number of orbits
in
X[Times]X?
Ignat Soroko:
> 1) The simplest example is G = Z x Z acting on X = Z by
translations:
> (a,b).x = x + a + b
> Clearly, it is 2-transitive.
Clearly, it is NOT 2-transitive (it preserves the
distance...). More
generally, an abelian group cannot act 2-transitively on an
infinite set.
YdC:
> 2) If every solvable, finitely generated group a quotient of
some
>>finitely presented solvable group?
Ignat Soroko:
> 2) No, the free n-generated solvable group of solvable
length d is not
> finitely presented for n,d > 1. This was proved by A.L.
Shmelkin:
And then? This does not prove that this group is not a
quotient of a
finitely presented solvable group (with more generators or
higher
solubility length)...
--
Yves
===
Subject: Re: 2 questions on finitely generated solvable groups
Epigone-thread: colsmymeu
>YdC:
>Clearly, it is NOT 2-transitive (it preserves the
distance...). More
>generally, an abelian group cannot act 2-transitively on an
infinite
set.
> ...
>YdC:
>And then? This does not prove that this group is not a
quotient of a
>finitely presented solvable group (with more generators or
higher
>solubility length)...
You are right, I goofed in 1 and was too hasty in 2.
1.
There is some description of all solvable 2-transitive groups
given in
the book:
Suprunenko D.A., Groups of Substitutions, Minsk, Navuka i
Tekhnika,
1996. (Russian)
First, any 2-transitive group G must be primitive. (Since any
two
elements from a nontrivial block can be put to different
blocks by the
action.)
The author proves that a primitive permutaion group G acting
on a set
X is similar (i.e. isomorphic as a transformation group) to a
group of
the type:
G = A.T acting on the set X = V, where
V is a vector space over a prime field F (GF(p) or Q)
T is the full group of translations over vectors of V
A is a solvable subgroup of GL(V) acting transitively on V{0}.
A.T means the semidirect product of A and T
So G is a subgroup of Aff(V).
Since our group in question is infinite, the field 
F is either
Q (the
rationals), or F=GF(p) and dim V = infinity. All that remains
is to
show that such groups are never finitely generated.
If you are interested, I can provide you with the details of
the proof
Hope this helps,
Ignat Soroko
Minsk, Belarus
===
Subject: Re: Multiplication and Division in Triplets;
division by zero-sized
vectors
snip
> More generally one can consider a general group algebra C G
> over a finite group, and define a Moore-Penrose 
type inverse
> by decomposing CG as a product of matrix algebras and
applying
> Moore-Penrose inversion in each. I was going to suggest
this with
> C replaced by an arbitrary characteristic zero field K, but
that
> would require Moore-Penrose inverses in matrix algebras
over skew
> fields. I dont know if there are such things 
.... anyone?
> Of course there are no problems if, as here G is abelian,
snip
Triplet polar-dual, powers, roots.
In Triplet (or, my preferred term, Terplex) algebra, the
functions Xs
& Xp combine with Xa =ArcTan[2x1 -x2 -x3, -Root[3](x2-x3)] to
give a
Polar Dual for X. (Mathematica ArcTan convention,
Arctan[opposite,
adjacent]). The quadratic Xp releases a degree of freedom,
which is
taken up by the angle Xa. Angles add on multiplication, so the
polar-dual of AB is {AsBs, ApBp, Aa+Ba}. With A={3,1,4} and
B={1,5,2}
as in my original posting, the angles are Aa
=ArcTan[3Root[3]]=1.3807,
Ba =ArcTan[0.6 Root[3]-Pi]=-2.3370, ABa =ArcTan[9Root[3]/11]
=-.9563.
Procedures toPolar, toVector, and genPower are available.
genPower[A]
give SQRT[A] ={1.7789, -.07326, 1.12278}. You can easily
write your
own procedures from the definitions; care has to be taken with
roots
of powers when the power takes the angles across the cut for
ArcTan.
Robin Chapman wondered about non-abelian algebras. The
simplest is D3.
I use the following protoloop (preferred isomorph):-
1 2 3 4 5 6
2 1 6 5 4 3
3 4 5 6 1 2
4 3 2 1 6 5
5 6 1 2 3 4
6 5 4 3 2 1
with multiplication rule and conserved properties:-
AB={a1 b1+a2 b2+a5 b3+a4 b4+a3 b5+a6 b6,
a2 b1+a1 b2+a4 b3+a5 b4+a6 b5+a3 b6,
a3 b1+a4 b2+a1 b3+a6 b4+a5 b5+a2 b6,
a4 b1+a3 b2+a6 b3+a1 b4+a2 b5+a5 b6,
a5 b1+a6 b2+a3 b3+a2 b4+a1 b5+a4 b6,
a6 b1+a5 b2+a2 b3+a3 b4+a4 b5+a1 b6}
Xs=x1+x2+x3+x4+x5+x6, Xt=x1-x2+x3-x4+x5-x6,
Xp=((x1-x3)^2-(x2-x4)^2+(x3-x5)^2-(x4-x6)^2+(x5-x1)^2-(x6-x2)
^2)/2.
Demonstration of conservation on multiplication:-
A={3,1,4,-2,5,0};B={2,7,1,0,2,4};
Multiplication gives AB->{26,37,11,46,12,44} &
BA->{26,39,13,48,10,40}
Shapes A B AB BA
{13,11,-4},{-6,16,-36},{-78,176,144},{-78,176,144}
I do not give the inverse procedure here; it gives a left
inverse.
Demonstrations of division:-
Ainv->{159/572,48/143,-127/572,-237/572,4/143,49/572};
Binv->{-23/864,113/864,-23/864,-55/864,1/864,41/864};
genTimes[Binv,BA]=genTimes[AB,Binv]->A
genTimes[Ainv,AB]=genTimes[BA,Ainv]->B
I have not found a polar form with additive angles, because
there is
only one squared radius and so 3 degrees of freedom are lost.
However, splitting Xp gives a polar form that reverts
correctly and
provides positive powers (as a continuous function of the
power):-
Xra=((x1-x3)^2+(x3-x5)^2+(x5-x1)^2)/2;
Xa=ArcTan[2x1-x3-x5,SQRT[3](x5-x3)];
Xrb=((x2-x4)^2+(x4-x6)^2+(x6-x2)^2)/2;
Xb=ArcTan[2x2-x4-x6,SQRT[3](x6-x4)]-Xa;
Demonstration:-toPolar[A]->{13,11,3,2.618,7,-1.904}
toVector[A]->{.1118,-0.9537,1.9941,1.1507,-0.1059,-1.197}(
treating A
as a polar form) with toPolar[toVector[A]]->{3,1,4,-2,5,0} and
toVector[toPolar[A]] ->{3,1,4,-2,5,0};
rootA=genPower[A,.5]->{1.3808,0.9679,0.3062,-0.8842,1.7741,-
0.2281}
genpower[rootA,2]->{3,1,4,-2,5,0}.
But genTimes[rootA,rootA]-> {4.76, 0.359, 3.12, -0.903, 4.112,
-0.456};
That last line demonstrates something that upsets moderators.
If A has
any negative sizes (-4 in the example) A.A is not the same as
A^2.
A.A = {54, -12, 47, -3, 44, -9}
A^1.99= {48.09,-7.07,48.18,-12.07,45.15,-4.15}
A^2 = {49.33,-7.33,49.33,-12.37,46.33,-4.33}
A^2.01= {50.60,-7.60,50.15,-12.61,47.54,-4.52}
This is because A.A must have the third size=16; it is -16 in
the
power function.
If anyone wishes to follow-up Robins comment about matrix
formulations of non-abelian tables, the dot-products of
{{0,-i},{i,0}}
and {{j,0},{0,j j}}, where i^4=1,j^3=1, give the elements of a
different D3 isomorph.
Roger Beresford.
I apologise for the ungracious tailquote in my previous
posting. I had
passed my (low) concentration-span limit, and did not notice
how rude
it looked. Sorry.
And more, much more than this. I did it my way. (P. Anka.)
===
Subject: help: limit of exponential series
Ive gone through the algebra of a series of exponential
growth and
decays and come up with the following form of the solution
x = 1-exp(-A/tau)*[1-exp(-B/tau)*[1-exp(-A/tau)*[.....]]]
if the text gets scrambled the above is supposed to be a
nested
infinite product.
Now A + B = constant and A,B >= 0.
This seems like it should have a limit as n goes to infinity
but I
dont know how to go about finding it. Any hints 
or is this a
common
solution.
It arises from a heat transfer problem Im working on where
power is
applied for A seconds then off for B seconds. So that at the
end of
every A cycle the temp will rise and then cool off for the B
cycle.
It has a limit if A is equal to the constant and B is zero.
That
solution is easy (and of course the solution for A=0 is
trivial).
Im familiar with the old tricks like what is the sqrt of a
nested
sqrt(5 + sqrt(5+sqrt(5+ .....
Is there something similar here.
Bill
===
Subject: Re: help: limit of exponential series
>Ive gone through the algebra of a series of exponential
growth and
>decays and come up with the following form of the solution
>x = 1-exp(-A/tau)*[1-exp(-B/tau)*[1-exp(-A/tau)*[.....]]]
>if the text gets scrambled the above is supposed to be a
nested
>infinite product.
>Now A + B = constant and A,B >= 0.
I hope you mean A and B both constants.
>This seems like it should have a limit as n goes to infinity
but I
>dont know how to go about finding it. Any 
hints or is this a
common
>solution.
There isnt any n in your formula, but I suppose you mean
something
like this:
x_0 arbitrary,
x_{n+1} = 1 - exp(-A/tau)(1 - exp(-B/tau) x_n)
= 1 - exp(-A/tau) + exp(-(A+B)/tau) x_n
which Ill write as
x_{n+1} = a + b x_n
If A + B > 0 you have 0 < b < 1, and then its elementary 
that
x_n converges to the fixed point of the iteration, namely
a/(1-b), since
x_{n+1} - a/(1-b) = b (x_n - a/(1-b))
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: limit of exponential series
[Bill Schuh]
> Ive gone through the algebra of a series of exponential
growth and
> decays and come up with the following form of the solution
> x = 1-exp(-A/tau)*[1-exp(-B/tau)*[1-exp(-A/tau)*[.....]]]
> if the text gets scrambled the above is supposed to be a
nested
> infinite product.
> Now A + B = constant and A,B >= 0.
> This seems like it should have a limit as n goes to infinity
but I
> dont know how to go about finding it. Any 
hints or is this
a common
> solution.
Lets simplify the daunting appearance by letting
a=exp(-A/tau), and
b=exp(-B/tau). Then its just:
x = 1-a*(1-b*(1-a*(1-b*(...))))
^^^^^^^^^^^^^^^
which is x
So if it has a limit at all, it must satisfy:
x = 1-a*(1-b*x)
which is easily solved as:
x = (1-a)/(1-a*b) = [1-exp(-A/tau)]/[1-exp(-(A+B)/tau)]
> It arises from a heat transfer problem Im working on 
where
power is
> applied for A seconds then off for B seconds. So that at
the end of
> every A cycle the temp will rise and then cool off for the
B cycle.
> It has a limit if A is equal to the constant and B is zero.
That
> solution is easy (and of course the solution for A=0 is
trivial).
Too practical for me <wink>.
===
Subject: Re: Finite Automata and Category Theory
> I am interested in Finite Automata from the categorical
> point of view.
OK.
Let M be a monoid. Let A = (Q,I,F,H) be a finite automaton 
over
M, defined by:
I, F subsets of Q
H subset of Q x M x Q
|= relation over the free extension M[Q] defined by:
* a |= a
* a |= b, b |= c --> a |= c
* f in F --> f |= 1
* (q,m,r) in F --> a q b |= a m r b
[A], for A subset of M[Q] defined by
[A] = { m in M: a |= m for some a in A }
define
L(A) = [I] = the subset of M recognized by the automaton A
Category:
Objects: Q union { I, F }
Arrows: q -- m --> r whenever q |= m r; for m in M; q,r in Q
Identity = monoid identity: q -- 1 --> q
Composition = monoid product:
q -- m --> r; r -- n --> s ==> q -- mn --> s
Some authors define a category such that no 2 arrows have the
same
label. This is easily accomodated by writing q -- (q,m,r) -->
r
as the arrow, and is therefore not an essential issue here.
A particular case is that the set H allows for lambda
transitions,
(q,1,r) in H: q -- 1 --> r.
Allowing for multiple arrows per label makes the following
easier
to state. The automaton is deterministic if no two arrows from
the same state have the same label.
A functor is restricted to those that map the labels
consistently.
That is: if 2 arrows have the same label, they map to arrows
with
the same label.
This means, among other things, that the underlying monoid is
mapped via a monoid homomorphism.
All the above is quite general, much more so than merely an
account of finite state automata. For one, the underlying
monoid need not be free. For instance, it may be a monoid
product, such as M = X* x Y*, of two free monoids on sets X
and Y.
This yields the usual definition of a finite state 
MACHINE, if
outputs.
The set Q need not be finite. For instance, it may be
factorable
into a form:
Q = Q0 x S*; I, F subsets of Q0 x {1}.
for some set S, with Q0 finite. If the transitions in H 
satisfy
the properties:
(1) Symmetry Property
if (q,sa) --> m --> (r,ba) for some s in S, b in S*, then
(q,sa) --> m --> (r,ba) for all 
a in S*.
(2) Selection Rules
transitions restricted to the forms
(q,sa) --> m --> (r,ba) for q,r in Q0, s in S, a, b in S*, m
in M
(q,1) --> m --> (r,a) for q,r in Q0, a in S*, m in M
These 2 restrictions carve out the subfamily of infinite state
automata that correspond to push down automata (generalized,
here,
to PDAs over general monoids M, not just over free monoids
X*).
In the case M = X*, one has the usual definition of a PDA; if
A similar set of symmetries and selection rules will give you
a
definition for the equivalent of a turing machine. Automata
families, in general, are carved out by the appropriate
selection
of symmetries and selection rules. The transition set H plays
the role of a discrete analogue of the systems Hamiltonian;
if one attaches probabilities to the labels (and to elements
of F)
so that the sum of all probabilities for all outgoing labels
plus
the probability attached to the state itself (if it is in F)
add
to 0; then one has a definition for a stochastic automaton. In
that case, H plays the role of the integral of the state
evolution operator over a finite time interval -- a discrete
analogue to a kind of S-matrix, if one is thinking of quantum
field theoretic applications. The symmetries and selection
rules then play the analogue of Noetherian symmetries and
superselection rules.
===
Subject: Chebyshev polynomials
Epigone-thread: slalsmoißen
I meet equations about Chebyshev polynomials.Tn(Xn)is n degree
Chebyshev polynomials about Xn. Mi is const (known value).
T1(x1)+T2(x2)+...+T1(xn)=M1
T3(x1)+T3(x2)+...+T3(xn)=M3
T5(x1)+T5(x2)+...+T5(xn)=M5
.......................
T2n+1(x1)+.......+T2n+1(xn)=Mn
How obtain xi?
===
Subject: Re: Chebyshev polynomials
>I meet equations about Chebyshev polynomials.Tn(Xn)is n
degree
>Chebyshev polynomials about Xn. Mi is const (known value).
>T1(x1)+T2(x2)+...+T1(xn)=M1
>T3(x1)+T3(x2)+...+T3(xn)=M3
>T5(x1)+T5(x2)+...+T5(xn)=M5
>.......................
>T2n+1(x1)+.......+T2n+1(xn)=Mn
>How obtain xi?
Since no one else is stating the obvious, I guess I will.
You could expand out the polynomials T1, T3, ... as powers of
x;
then linear combinations of these n equations tell you exactly
what the quantities sum (x_i)^(2k+1) are, in terms of
M1, M3, ..., M_(2k+1). But those quantities are in turn
expressible in terms of the elementary symmetric polynomials
S_i
in the x_i. Youve got enough equations then to determine 
the
S_i in terms of the M_i. Finally of course the x_i can be
obtained as roots of the polynomial P(X) = X^n - S1 X^(n-1) +
...
Its not clear to me why this must have been the case, but 
in
fact the S_i are _rational_ functions of the M_i, at least up
through the case n=5, so that the x_i may be obtained as the
roots of a polynomial in Z[M1, M2, ... ][X].
The polynomials P for the cases n <= 5 are attached below.
I confess that I see no particular pattern or simplification,
and so I cant suggest a way to generalize this to higher n,
but maybe Im just not being very creative.
dave
n=1:
X - M1
n=2:
(12*M1) * X^2 - (12*M1^2) * X + (4*M1^3 - 3*M1 - M2)
n=3:
(240*M1^3-180*M1-60*M3)*X^3 - 60*M1*(-3*M1+4*M1^3-M3)*X^2
+ (90*M1+96*M1^5-180*M1^3-60*M1^2*M3+45*M3+9*M5)*X
+
(-45*M1^2-15*M1*M3-16*M1^6+60*M1^4+5*M3^2+20*M1^3*M3-9*M1*M5)
n=4:
(-8400*M3^2+75600*M1^2-33600*M1^3*M3+25200*M1*M3+15120*M1*M5-
100800*M1^4+268
80
*M1^6)*X^4-1680*M1*(-60*M1^4+15*M1*M3+45*M1^2+16*M1^6-20*M1^3
*M3+9*M1*M5-5*M
3^
2)*X^3+(-15120*M1*M5+1260*M3*M5+50400*M1^3*M3+6300*M3^2+
100800*M1^4-25200*M1
*
M3-2700*M7*M1+10080*M1^3*M5-20160*M1^5*M3+11520*M1^8-56700*M1
^2-60480*M1^6)*
X^
2-(25200*M1^4*M3-700*M3^3-6720*M1^6*M3+6300*M1*M3^2+5040*M1^4
*M5-2700*M7*M1^
2-
11340*M1^2*M5+2520*M3*M1*M5+2560*M1^9-37800*M1^3+50400*M1^5-
12600*M1^2*M3-
20160*M1^7)*X+(-6300*M1^3*M3+3150*M1*M3+1260*M1^2*M3*M5-2880*
M1^8+256*M1^10
+675*M7*M1-2520*M1^3*M5-189*M5^2+5040*M1^5*M3-700*M3^3*M1-960
*M1^7*M3+945*M1
*M5
-315*M3*M5+1008*M1^5*M5-900*M7*M1^3+225*M7*M3+4725*M1^2+10080
*M1^6-12600*M1^
4)
n=5:
(152409600*M1^6-190512000*M1^4+71442000*M1^2+19051200*M1^2*M3
*M5+14288400*M1
*M5
-43545600*M1^8-38102400*M1^3*M5+47628000*M1*M3-95256000*M1^3*
M3-4762800*M3*M
5+
76204800*M1^5*M3+3870720*M1^10-13608000*M7*M1^3-14515200*M1^7
*M3+15240960*M1
^5
*M5-10584000*M3^3*M1+10206000*M7*M1-2857680*M5^2+3402000*M7*
M3) * X^5
-15120*M1*(-6300*M1^3*M3+3150*M1*M3+1260*M1^2*M3*M5-2880*M1^8
+256*M1^10+675*
M7
*M1-2520*M1^3*M5-189*M5^2+5040*M1^5*M3-700*M3^3*M1-960*M1^7*
M3+945*M1*M5-315
*
M3*M5+1008*M1^5*M5-900*M7*M1^3+225*M7*M3+4725*M1^2+10080*M1^6
-12600*M1^4) *
X^4
+(134946000*M1^3*M3-1984500*M3^2-53581500*M1*M3-19051200*M1^2
*M3*M5+10523520
0*
M1^8-22579200*M1^10-12757500*M7*M1+52390800*M1^3*M5+3572100*
M5^2-139708800*
M1^5*M3-10584000*M1^4*M3^2+11466000*M3^3*M1+56851200*M1^7*M3+
7938000*M1^2*M3
^2
-1984500*M9*M1-17860500*M1*M5+4762800*M3*M5-38102400*M1^5*M5+
23814000*M7*M1^
3-
8164800*M7*M1^5+1587600*M1*M5*M3^2+2646000*M9*M1^3-2857680*M1
^2*M5^2+7983360
*
M1^7*M5+6350400*M1^4*M3*M5-7526400*M1^9*M3-4704000*M1^3*M3^3+
2822400*M1^6*M3
^2
+1720320*M1^12+294000*M3^4-661500*M9*M3+510300*M7*M5-3402000*
M7*M3-71442000*
M1
^2-222264000*M1^6+214326000*M1^4) * X^3 +
(-79380000*M1^4*M3+76204800*M1^6*M3-1984500*M1*M3^2-283500*M3
^2*M7+1701000*M
1*
M3*M7-510300*M7*M1*M5+396900*M5*M3^2+238140*M3*M5^2-38102400*
M1^4*M5+1020600
0*
M7*M1^2+3628800*M7*M1^6+7096320*M1^11-430080*M1^13+14288400*
M1^2*M5-4762800*
M3
*M1*M5-17010000*M7*M1^4-2646000*M9*M1^4-2857680*M1*M5^2+
21591360*M1^6*M5-
24192000*M1^8*M3-8820000*M1^2*M3^3+1905120*M1^3*M5^2-2903040*
M1^8*M5-1612800
*
M1^7*M3^2+2365440*M1^10*M3+1176000*M1^4*M3^3+588000*M3^4*M1+
4233600*M1^5*M3^
2+
1984500*M9*M1^2+2268000*M1^3*M3*M7+15876000*M1^3*M3*M5-
1270080*M1^5*M5*M3-
3175200*M1^2*M5*M3^2+661500*M9*M1*M3+53581500*M1^3-142884000*
M1^5+35721000*
M1^2*M3+120657600*M1^7-43545600*M1^9) * X^2 +
(-35721000*M1^3*M3+1488375*M3^2+11907000*M1*M3+3572100*M1^2*
M3*M5-36288000*M
1^8
+10080000*M1^10+2551500*M7*M1-11907000*M1^3*M5-893025*M5^2+
41277600*M1^5*M3+
5292000*M1^4*M3^2-2425500*M3^3*M1-23889600*M1^7*M3-3969000*M1
^2*M3^2+992250*
M9
*M1+3572100*M1*M5-595350*M3*M5+14288400*M1^5*M5+567000*M3^2*
M7*M1+510300*M7*
M1
^2*M5-7654500*M7*M1^3-1036800*M7*M1^7+6123600*M7*M1^5-1190700
*M1*M5*M3^2-
476280*M3*M5^2*M1-1134000*M1^4*M3*M7-1984500*M9*M1^3+1058400*
M9*M1^5+2143260
*
M1^2*M5^2-44100*M3^3*M5-5987520*M1^7*M5-4762800*M1^4*M3*M5+
423360*M1^6*M5*M3
+
2116800*M1^3*M5*M3^2-661500*M9*M1^2*M3+5644800*M1^9*M3+
3528000*M1^3*M3^3-
952560*M1^4*M5^2+645120*M1^9*M5+403200*M1^8*M3^2-430080*M1^11
*M3-470400*M1^5
*
M3^3-588000*M3^4*M1^2-2116800*M1^6*M3^2-1290240*M1^12-220500*
M3^4+61440*M1^1
4+
496125*M9*M3+99225*M9*M5-382725*M7*M5+637875*M7*M3+13395375*
M1^2-91125*M7^2+
61916400*M1^6-47628000*M1^4) * X +
(7938000*M1^4*M3+165375*M3^3-7938000*M1^6*M3+496125*M1*M3^2+
70875*M3^2*M7+
212625*M1*M3*M7+127575*M7*M1*M5-99225*M5*M3^2-59535*M3*M5^2+
3572100*M1^4*M5-
637875*M7*M1^2-907200*M7*M1^6-1048320*M1^11+107520*M1^13-
1786050*M1^2*M5+
1701000*M7*M1^4+661500*M9*M1^4+178605*M1*M5^2-2540160*M1^6*M5
+3326400*M1^8*M
3+
220500*M1^2*M3^3-476280*M1^3*M5^2+725760*M1^8*M5+403200*M1^7*
M3^2-591360*M1^
10
*M3-294000*M1^4*M3^3-147000*M3^4*M1-1058400*M1^5*M3^2-496125*
M9*M1^2-567000*
M1
^3*M3*M7-396900*M1^3*M3*M5+317520*M1^5*M5*M3+793800*M1^2*M5*
M3^2-165375*M9*M
1*
M3+55125*M9*M3^2-85050*M5*M7*M3+35721*M5^3-24500*M3^5-64512*
M1^10*M5+190512*
M1
^5*M5^2-170100*M1^3*M5*M7+35840*M1^12*M3+78400*M1^6*M3^3+
196000*M1^3*M3^4-
44800*M1^9*M3^2-529200*M1^4*M3^2*M5+238140*M1^2*M3*M5^2-60480
*M1^7*M3*M5+
226800*M1^5*M3*M7-283500*M1^2*M3^2*M7-4096*M1^15+44100*M3^3*
M1*M5+220500*M9*
M1
^3*M3-99225*M9*M1*M5-176400*M9*M1^6+91125*M1*M7^2+129600*M1^8
*M7-4465125*M1^
3+
11907000*M1^5-2976750*M1^2*M3-11113200*M1^7+4838400*M1^9)
===
Subject: Re: Chebyshev polynomials
>I meet equations about Chebyshev polynomials.Tn(Xn)is n
degree
>Chebyshev polynomials about Xn. Mi is const (known value).
>T1(x1)+T2(x2)+...+T1(xn)=M1
^^
I assume that should be T1
>T3(x1)+T3(x2)+...+T3(xn)=M3
>T5(x1)+T5(x2)+...+T5(xn)=M5
>.......................
>T2n+1(x1)+.......+T2n+1(xn)=Mn
^^
... and that should be M_{2n+1}.
>How obtain xi?
The odd powers of x can be expressed as linear combinations of
T_j(x) for odd j:
x^{2m+1} = 4^(-m) sum_{j=0}^m (2m+1 choose m-j) T_{2j+1}(x)
So from your equations you can get
x_1^(2m+1) + ... + x_n^(2m+1) = 4^(-m) sum_{j=0}^m (2m+1
choose m-j)
M_{2j+1}
Call this b_{2m+1}.
Now if you had b_j for even j as well as odd j, I would
suggest using
Newtons identities to get the elementary symmetric 
functions
of
x_1,...,x_n, and thus the coefficients of a polynomial whose
roots
(counting multiplicity) are x_1,...,x_n.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Approximation of Stirling numbers of 2. kind
Epigone-thread: ßangputul
>> Does anyone know of a closed form approximation/upper
bound of
the
>> stirling numbers of the second kind?
>I dont believe one exists. But if you notice that there 
are
>k!S(n,k) onto functions from an n element set to a k element
set,
>and count the same number using inclusion-exclusion, you get
>the reasonable closed form
> [S(n,k) = frac{1}{k!}sum_{j=0}^{k}(-1)^j{k choose
j}(k-j)^n.]
>e.g., S(3,2) = (1/2!)[C(2,0)2^3 - C(2,1)1^3 + C(2,2)0^3]
> = (1/2) [ 8 - 2 + 0 ]
> = 3.
>Rick
Does anyone know of a formula (can be approximate) for
computing the
logarithm of a Stirling number of the second directly? I.e.
Not
computing the number and then taking its log.
===
Subject: Re: Approximation of Stirling numbers of 2. kind
<cl35ot$7rs$1@dizzy.math.ohio-state.edu>
In my paper
Asymptotic Estimates of Stirling Numbers,
Studies in Applied Mathematics, 89, 1993, 233 -- 243,
I give an estimate for large n that holds uniformly with
respect to m.
Let x0 be the positive solution of
m x / n = 1 - exp(-x).
Let t0 = (n-m)/m.
Then the Stirling number of the second kind Snm has the
S(n,m) (n ge m) has the asymptotic estimate
S(n,m) sim (m/(n-m x0))^m (n-m)^{n-m} e^{m-n} {n choose m}
sqrt{t0/(1+t0)/(x0-t0)}, n to infty.
Nico Temme
Nico M. Temme, http://homepages.cwi.nl/~nicot/
C W I: Centrum voor Wiskunde en Informatica
Kruislaan 413, NL-1098 SJ Amsterdam Tel +31 20 592 4240
P.O. Box 94079, NL-1090 GB Amsterdam Fax +31 20 592 4199
===
> Subject: Re: Approximation of Stirling numbers of 2. kind
>> Does anyone know of a closed form approximation/upper
bound of
> the
>> stirling numbers of the second kind?
>I dont believe one exists. But if you notice that there 
are
>k!S(n,k) onto functions from an n element set to a k element
set,
>and count the same number using inclusion-exclusion, you get
>the reasonable closed form
> [S(n,k) = frac{1}{k!}sum_{j=0}^{k}(-1)^j{k choose
j}(k-j)^n.]
>e.g., S(3,2) = (1/2!)[C(2,0)2^3 - C(2,1)1^3 + C(2,2)0^3]
> = (1/2) [ 8 - 2 + 0 ]
> = 3.
>Rick
> Does anyone know of a formula (can be approximate) for
computing the
> logarithm of a Stirling number of the second directly? I.e.
Not
> computing the number and then taking its log.
===
Subject: Re: determinant inequality
> Does anyone know a proof (or a reference) for this
inequality?
> det(I+xA+B) det(I+A+yB) > det(I+A+B) det(I+xA+yB)
> where
> 0 < x,y < 1
> A,B are positive definite n x n matrices
> I is the n x n identity matrix
For two symmetric matrices C and D, lets say that C >>D iff
(C-D) is
positive definite, and lets write C >> 0 if C 
is positive
definite.
One easily verifies that if C >> D >> 0, then det(C) > det(D)
and
also C^(-1) << D^(-1), and GCG >> GDG whenever G is symmetric
and
positive definite.
Also, lets note the formula
(1) det[(C+D)D^(-1)] = det(I + C^(1/2)D^(-1)C^(1/2)],
for positive definite matrices C and D.
Lets call a function F , defined on a real 
interval and with
values in
the set of symmetric mxm matrices _increasing_ if x1 < x2
implies
F(x1) << F(x2), and _decreasing_ if -F is increasing.
Let now A and B be given symmetric matrices of the same size,
A >>0
and B >>0. Let 0 < y < 1 be fixed. For x in [0,1], consider 
the
function
F(x) = det[(I + xA + yB)(I + xA + B)^(-1)].
The assertion is equivalent to the inequality F(x) < F(1) for
0 < x <
1, so lets show that F is increasing:
(I + xA + B) is increasing in x,
thus (I + xA + B)^(-1) is decreasing,
thus I + (y-1)B^(1/2)(I + xA + B)^(-1)B^(1/2) is increasing
(since y <
1),
thus det[I + (y-1)B^(1/2)(I + xA + B)^(-1)B^(1/2)] is
increasing.
By formula (1) this equals F(x), qed.
</x-ßowed>
===
Subject: Paper published by Geometry and Topology
The following paper has been published:
URL:
http://www.maths.warwick.ac.uk/gt/GTVol8/paper36.abs.html
Title:
On groups generated by two positive multi-twists:
Teichmueller curves and Lehmers number
Author(s):
Christopher J Leininger
Abstract:
several interesting facts about subgroups of the mapping
class group
generated by two positive multi-twists. In particular, we
identify all
configurations of curves for which the corresponding groups
fail to be
free, and show that a subset of these determine the same set
of
Teichmueller curves as the non-obtuse lattice triangles which
were
classified by Kenyon, Smillie, and Puchta. We also identify a
pseudo-Anosov automorphism whose dilatation is Lehmers
number, and
show that this is minimal for the groups under consideration.
In
addition, we describe a connection to work of McMullen on
Coxeter
groups and related work of Hironaka on a construction of an
interesting class of fibered links.
Secondary: 20H10, 57M25
Keywords:
Coxeter, Dehn twist, Lehmer, pseudo-Anosov, mapping class
group,
Teichmueller
Proposed: Benson Farb
Seconded: Walter Neumann, Joan Birman
Author(s) address(es):
Department of Mathematics, Columbia University
2990 Broadway MC 4448, New York, NY 10027, USA
Email: clein@math.columbia.edu
===
Subject: Category Theory: Representable functors
Hi
Let C be a category and X, Y be two objects of C. The product
of X and Y
can be defined as the representation of a functor which
returns a product
in Set.
C(_, XxY) =~ C(_, X) x C(_, Y)
A limit can also be defined as the representation of a 
functor:
C(_, Lim F) = D^C(Diagonal(_), F)
Similarly to the case of the product, is the represented
functor
D^C(Diagonal(_), F) a functor which returns a limit in Set?
David.
===
Subject: Re: Category Theory: Representable functors
Epigone-thread: toiplendchy
>Let C be a category and X, Y be two objects of C. The
product of X
and Y
>can be defined as the representation of a functor which
returns a
product
>in Set.
> C(_, XxY) =~ C(_, X) x C(_, Y)
>A limit can also be defined as the representation of a
functor:
> C(_, Lim F) = D^C(Diagonal(_), F)
>Similarly to the case of the product, is the represented
functor
>D^C(Diagonal(_), F) a functor which returns a limit in Set?
>David.
Yes, certainly. For a functor F: J --> C, if the limit exists
in C, we have
C(A, lim Fj) ~= lim C(A, Fj)
since representables C(A, -) preserve limits (essentially by
definition of limit). So C(-, lim Fj) ~= lim C(-, Fj).
Todd Trimble
===
Subject: Re: Category Theory: Representable functors
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>>Hi
>>Let C be a category and X, Y be two objects of C. The
product of X
> and Y
>>can be defined as the representation of a functor which
returns a
> product
>>in Set.
>> C(_, XxY) =~ C(_, X) x C(_, Y)
>>A limit can also be defined as the representation of a
functor:
>> C(_, Lim F) = D^C(Diagonal(_), F)
>>Similarly to the case of the product, is the represented
functor
>>D^C(Diagonal(_), F) a functor which returns a limit in Set?
>>David.
> Yes, certainly. For a functor F: J --> C, if the limit
exists
> in C, we have
> C(A, lim Fj) ~= lim C(A, Fj)
> since representables C(A, -) preserve limits (essentially by
> definition of limit). So C(-, lim Fj) ~= lim C(-, Fj).
> Todd Trimble
How does it extend to weighted limits?
begin{delirium}
Let C be a category. Let RepFun be the category of
representable funtors
from C^op to Set. Let Rep be the function from objets of
RepFun to
objects of C such that Rep(F) is the object of C (up to
isomorphism) such that
C(_, Rep(F)) =~ F
Rep surely extends to a unique functor, doesnt it? 
Dont we
have that:
C(A, Rep(F)) =~ Rep(C(A, F)) ?
end{delirium}
David.
===
Subject: Re: Category Theory: Representable functors
Epigone-thread: toiplendchy
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>Hi
>Let C be a category and X, Y be two objects of C. The
product of X
>> and Y
>can be defined as the representation of a functor which
returns a
>> product
>in Set.
> C(_, XxY) =~ C(_, X) x C(_, Y)
>A limit can also be defined as the representation of a
functor:
> C(_, Lim F) = D^C(Diagonal(_), F)
>Similarly to the case of the product, is the represented
functor
>D^C(Diagonal(_), F) a functor which returns a limit in Set?
>David.
>> Yes, certainly. For a functor F: J --> C, if the limit
exists
>> in C, we have
>> C(A, lim Fj) ~= lim C(A, Fj)
>> since representables C(A, -) preserve limits (essentially
by
>> definition of limit). So C(-, lim Fj) ~= lim C(-, Fj).
>> Todd Trimble
>How does it extend to weighted limits?
The same way. Work in V-Cat (assume V complete cocomplete
closed
symmetric monoidal) and suppose F: J --> V is a weight and
G: J --> C is a V-functor. If the weighted limit {F, G}
exists in C, then C(A, -) preserves this limit in the sense
that C(A, {F, G}) is the limit of the composite
G C(A, -)
J --> C -------> V
w.r.t. the weight F. I.e. {F, C(A, G-)} ~ C(A, {F, G}).
Proof: V(v, {F, C(A, G-)}) ~ V^J(F-, V(v, C(A, G-)))
~ V(v, V^J(F-, C(A, G-)))
~ V(v, C(A, {F, G}))
and the desired isomorphism follows from the Yoneda lemma.
The second isomorphism follows from the fact that ultimately
the enriched hom V^J(F-, C(A, G-)) may be constructed in terms
of ends and cotensors in V; V(v, -) preserves ends as ordinary
limits in V, and preserves cotensors,
V(v, V(w, -)) ~ V(w, V(v, -)),
using the symmetry of the symmetric monoidal structure. For
further details, I recommend the book by Kelly.
Todd Trimble
===
Subject: p-adic transcendentals
It is well-known (and follows from the Gelfond-Scheinder
theorem) that
a real quantity such as log(3)/log(2) (more generally, a
quotient of
the logs of two nonzero rationals, provided it is not
rational) is
transcendental.
Now in fact, such a quantity - lets stick to log(3)/log(2)
for
simplicity - can be defined over Q_p for all p other than 2
and 3 (the
usual power series for log(1+x) defines the log of a p-adic
integer
which is congruent to 1 mod p; but we can extend to all
invertible
p-adic integers by declaring that the log of a root of unity
must be
zero and writing an arbitrary p-adic integer as a (p-1)-th
root of
unity times a p-adic integer congruent to 1).
I dont see any reason a priori why the transcendence of
log(3)/log(2)
over the reals should be related to the transcendence of the
same
quantity over the p-adics (even merely for infinitely many 
ps
or
something); specifically: while an algebraic equation (over Q)
for
log(3)/log(2) in one of the Q_ps would define 
an algebraic
number in
infinitely many Q_ps, I dont 
see why that same algebraic
should also
be log(3)/log(2) in another Q_p.
So, is there a known way of proving that log(3)/log(2) is
transcendental in Q_p?
More generally, is there any kind of way to relate (in a vague
sense) the value of log(3)/log(2) (or similarly defined
quantities) in
the Q_ps and in R (the reals)?
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
===
Subject: Re: p-adic transcendentals
David Madore a ecrit:
> So, is there a known way of proving that log(3)/log(2) is
> transcendental in Q_p?
I dont know the answer, but your question reminds me the
following,
[roughly translated from Exercice 4 of Chapter IV 5 of
Borevitch-Chafarevitch Thorie des Nombres] :
Find a mistake in the following proof of the irrationality of
pi.
The number pi is the smallest number >0 such that sin(pi)=0.
Suppose that pi is rational. Then since pi>3, the numerator
of pi
has to be divisible by some odd prime p, or by 2^2. In that
case
set p=2. It follows that the series sin(x) and cos(x) are
convergent
in Q_p at x=pi. but since
sin(x+y)=sin(x)cos(y)+sin(y)cos(x)
it follows that sin(n pi)=0 for any integer n. Hence the
function
sin(x) has infinitely many zeros inside its convergence area,
hence it is identically zero (from Exercice 1 at the same
page).
This is a contradiction.
Serge.
===
Subject: Re: p-adic transcendentals
> David Madore a ecrit:
So, is there a known way of proving that log(3)/log(2) is
> transcendental in Q_p?
>
Is there a sense in which this p-adic number x satisfies 2^x =
3 ?
> I dont know the answer, but your question reminds me the
following,
> [roughly translated from Exercice 4 of Chapter IV 5 of
> Borevitch-Chafarevitch Theorie des Nombres] :
> Find a mistake in the following proof of the irrationality
of pi.
> The number pi is the smallest number >0 such that sin(pi)=0.
> Suppose that pi is rational. Then since pi>3, the numerator
of pi
> has to be divisible by some odd prime p, or by 2^2. In that
case
> set p=2. It follows that the series sin(x) and cos(x) are
convergent
> in Q_p at x=pi. but since
> sin(x+y)=sin(x)cos(y)+sin(y)cos(x)
> it follows that sin(n pi)=0 for any integer n. Hence the
function
> sin(x) has infinitely many zeros inside its convergence 
area,
> hence it is identically zero (from Exercice 1 at the same
page).
> This is a contradiction.
I guess the error is: A series of rationals that convervges
both in
the usual absolute value and in a p-adic absolute value need
not
converge to the same thing in the two cases, even if both
limits
are rationals.
===
Subject: Re: p-adic transcendentals
Epigone-thread: kanßarcho
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>> David Madore a ecrit:
>>
>> So, is there a known way of proving that log(3)/log(2) is
>> transcendental in Q_p?
>>
>Is there a sense in which this p-adic number x satisfies 2^x
= 3 ?
I believe the OP covered this, if we define 2^x to be
exp(x*log(2)).
(We suppose p is neither 2 nor 3.) The point is that the usual
MacLaurin expansions for exp(x), log(1+x) give well-defined
mutually
inverse homomorphisms
exp: pZ^_p --> 1 + pZ^_p
log: 1 + pZ^_p --> pZ^_p
Now of course 2 does not belong to 1 + pZ^_p, but there is a
unique root of unity w such that 2w belongs to 1 + pZ^_p, and
log(2) is to be defined as log(2w). In other words, there is
an identification
1 + pZ^_p ~ (Z^_p)*/((p-1)th roots of unity)
so we may say 2^{(log 3)/(log 2)} and 3 differ by a factor
which is a root of unity.
Todd Trimble
===
Subject: Re: p-adic transcendentals
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
G. A. Edgar a .8ecrit:
> I guess the error is: A series of rationals that convervges
both in
> the usual absolute value and in a p-adic absolute value
need not
> converge to the same thing in the two cases, even if both
limits
> are rationals.
That was also my guess, concerning this exercise. But I also
guess
I would have accepted this proof as correct if the mistake
had not
been explicitly announced in the exercice : this proof really
looks correct at first sight !:-)
Serge.
===
Subject: Re: p-adic transcendentals
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
G. A. Edgar in litteris
<clb27v$20f$1@dizzy.math.ohio-state.edu>
scripsit:
>> David Madore a ecrit:
>> So, is there a known way of proving that log(3)/log(2) is
>> transcendental in Q_p?
> Is there a sense in which this p-adic number x satisfies 2^x
= 3 ?
Unless Im mistaken:
Write x as the limit, for the p-adic topology, of a sequence
of
integers x_i all congruent mod p-1 (I mean all congruent to
some fixed
quantity mod p-1). Then (2^{x_i}) converges (again, in the
p-adics)
to some number which is equal to 3 up to multiplication by a
(p-1)-th
root of unity, the root in question being determined by the
class mod
p-1 of the x_i (and for one particular class we do get
exactly 3).
In particular, x cannot be rational (I believe Im not 
making
the
error which Serge points out, here), otherwise some power of
2 would
be equal to some power of 3 (up to multiplication by a root
of unity,
but that cannot be) *in the rationals*.
>> Find a mistake in the following proof of the irrationality
of pi.
>> The number pi is the smallest number >0 such that
sin(pi)=0.
>> Suppose that pi is rational. Then since pi>3, the
numerator of pi
>> has to be divisible by some odd prime p, or by 2^2. In
that case
>> set p=2. It follows that the series sin(x) and cos(x) are
convergent
>> in Q_p at x=pi. but since
>> sin(x+y)=sin(x)cos(y)+sin(y)cos(x)
>> it follows that sin(n pi)=0 for any integer n. Hence the
function
>> sin(x) has infinitely many zeros inside its convergence
area,
>> hence it is identically zero (from Exercice 1 at the same
page).
>> This is a contradiction.
> I guess the error is: A series of rationals that convervges
both in
> the usual absolute value and in a p-adic absolute value
need not
> converge to the same thing in the two cases, even if both
limits
> are rationals.
Yes, theres no reason that because sin(a/b)=0 in the reals
we should
also have sin(a/b)=0 in the p-adics. But perhaps theres a
way to
prove this anyway, if the coefficients of the sin power series
expansion are tame enough in some sense.
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
===
Subject: Re: p-adic transcendentals
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Just for the record:
David Madore in litteris <clbg31$u26$1@news.ks.uiuc.edu>
scripsit:
> Write x as the limit, for the p-adic topology, of a
sequence of
> integers x_i all congruent mod p-1 (I mean all congruent to
some fixed
> quantity mod p-1). Then (2^{x_i}) converges (again, in the
p-adics)
> to some number which is equal to 3 up to multiplication by
a (p-1)-th
> root of unity, the root in question being determined by the
class mod
> p-1 of the x_i
I believe the above is correct. However, this
> (and for one particular class we do get exactly 3).
might not be true unless we assume that 2 is primitive mod p
(or, more
generally, that 3 belongs to the multiplicative subgroup
generated by
2 in the finite field with p elements).
For example, there is no way to approach 3 with powers of 2
in the
7-adics (the powers of 2 in the finite field with 
7 elements
are 1, 2
and 4), but log(3)/log(2) still makes sense in the way I
suggested
(and it is precisely 3 + 3*7 + 49 + 0 + 5*7^4 + 6*7^5 +
O(7^6)).
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
===
Subject: Expected longest run of heads
I was told by email that there is a theorem of Erdos et al
that in a
series of n Bernoulli trials with P = p, the expected longest
run is
log to the base 1/p of n.
I expressed surprise, not at the result, but at the
involvement
of Erdos. My correspondent couldnt cite a written source. I
had
a look at Math Reviews, and found this:
MR1139688 (92j:60010)
Kopocinski, B.
On the distribution of the longest success-run in Bernoulli
trials.
Mat. Stos. 34 (1991), 3--13.
60C05
The author considers the random variable Z_n, defined to be
the length
of the longest head run in n coin tosses. Recurrence
relations are
obtained for the probability and cumulative probability
functions of
Z_n, and the corresponding generating functions are found.
The expected
value of Z_n is estimated.
Reviewed by Anant P. Godbole
This suggests to me that no exact formula is known for the
expected
longest run (although I have only looked at this review, not
at the
actual paper).
Would someone like to clear up the situation for me?
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Expected longest run of heads
>I was told by email that there is a theorem of Erdos et al
that in a
>series of n Bernoulli trials with P = p, the expected
longest run is
>log to the base 1/p of n.
Clearly this cant be an exact result. In n trials there are
only
finitely many possible outcomes. If p is a rational number, so
is
the probability of each outcome, and so is the expected value
of
an integer-valued random variable. But log_{1/p}(n) is
generally
irrational.
I think the reference is to Erdos and Renyi, On a new law of
large
numbers, J. Anal. Math. 22 (1970), 103-111.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Expected longest run of heads
> I was told by email that there is a theorem of Erdos et al
that in a
> series of n Bernoulli trials with P = p, the expected
longest run is
> log to the base 1/p of n.
This must be an asymptotic result, not an exact one. In
particular, it
fails for n=1 where the expected longest run E[Z_1]=p rather
than
-ln(n)/ln(p)=0. Other cases with small n can also be solved
by hand,
e.g. E[Z_2]=2p and E[Z_3]=2p^3-2p^2+3p. The formula
-ln(n)/ln(p), on
the other hand, is transcendental in p, not a polynomial.
In fact, its not hard to prove that E[Z_n] is a polynomial
in p with
integer coefficients of degree at most n. For
X=(X_1,X_2,...,X_n) the
n trials, Z_n=Z(X) the maximal success length,
E[Z_n] = SUM_{x=(x_1,...,x_n)} Z(x) * P[X=x].
Now, Z(x) is an integer and P[X=x]=p^i*(1-p)^(n-i) where
i=number of
successes in x, which proves the claim.
Einar
===
Subject: Re: Expected longest run of heads
> I was told by email that there is a theorem of Erdos et al
that in a
> series of n Bernoulli trials with P = p, the expected
longest run is
> log to the base 1/p of n.
This must be an asymptotic result, not an exact one. In
particular, it
fails for n=1 where the expected longest run E[Z_1]=p rather
than
-ln(n)/ln(p)=0. Other cases with small n can also be solved
by hand,
e.g. E[Z_2]=2p and E[Z_3]=2p^3-2p^2+3p. The formula
-ln(n)/ln(p), on
the other hand, is transcendental in p, not a polynomial.
In fact, its not hard to prove that E[Z_n] is a polynomial
in p with
integer coefficients of degree at most n. For
X=(X_1,X_2,...,X_n) the
n trials, Z_n=Z(X) the maximal success length,
E[Z_n] = SUM_{x=(x_1,...,x_n)} Z(x) * P[X=x].
Now, Z(x) is an integer and P[X=x]=p^i*(1-p)^(n-i) where
i=number of
successes in x, which proves the claim.
Einar
===
Subject: total positivity question
I recently asked for a proof/reference of a determinant
inequality
the following question:
A,B are positive definite matrices
I is the identity matrix
define f(x,y) = 1/det(I+xA+yB)
Problem:
Show that f(x,y) is a strictly totally positive function on
the positive
reals.
This means:
for any positive integer n
for all sequences
0 < x1 < ... < xn,
0 < y1 < ... < yn
the n by n matrix whose i-j entry is f(xi,yj) has positive
determinant.
If f(x,y)=x+y the positivity is a consequence of Cauchys
double
alternant identity; if f(x,y) = (x+y)^2 then it is a
consequence of
Borchardts identity.
===
Subject: Re: total positivity question
Epigone-thread: plumkroobrimp
Originator: israel@math.ubc.ca (Robert Israel)
>I recently asked for a proof/reference of a determinant
inequality
of
>the following question:
>A,B are positive definite matrices
>I is the identity matrix
>define f(x,y) = 1/det(I+xA+yB)
>Problem:
>Show that f(x,y) is a strictly totally positive function on
the
positive
>reals.
>This means:
>for any positive integer n
>for all sequences
>0 < x1 < ... < xn,
>0 < y1 < ... < yn
>the n by n matrix whose i-j entry is f(xi,yj) has positive
determinant.
>If f(x,y)=x+y the positivity is a consequence of Cauchys
double
>alternant identity; if f(x,y) = (x+y)^2 then it is a
consequence of
>Borchardts identity.
Hello ,
your problem seems to be very nice , but I dont understand
exactly :
what are the matrices A and B if f(x,y)=x+y or (x+y)^2 ?
with best wishes
Emmanuel Preissmann
===
Subject: Re: Free software for Homotopy Continuation Methods
Epigone-thread: praxkhingglou
[
Moderators Note. The post in question is dated 1997, from
David
Harney, Chemical Engineering Dept., University of Missouri -
Rolla.
Is anything current known about David Harney or the HOMES
software?
]
I am a engineering student at India . I was searching the
fortran
program to solve Non linear systems of equations by homotopy.
I got
your mail ID from the link
http://mathforum.org/epigone/sci.math.research/praxkhingglou
I was trying to resolve the turning point issue but i could
not do it.
I will be very grateful if you can send me the fortran source
code for
it along with all the details of Subroutines.
I will be very grateful to you for this.
Sonal jain
>HOMES 2.0, a fortran program that solves systems of algebraic
>nonlinear systems using homotopy continuation methods, has
been
>developed here at the University of Missouri - Rolla.
>A choice of homotopy functions are available as is a
selection
>of path tracking algorithms. Bifurcations into the complex
domain
>at turning points and real domain bounding algorithms are
among
>other selections available.
>If you would like a copy of the code, please email me at
>harney@umr.edu
>It will eventually be available on a website, as soon as I
muster up
>the motivation to learn how to create one!
>David Harney
>Chemical Engineering Dept.
>University of Missouri - Rolla
===
Subject: Re: Free software for Homotopy Continuation Methods
Epigone-thread: praxkhingglou
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
I abandoned the academic world in 1998 for the grubby moolah
offered
by the industrial one.
However, I still have this software if interested. Please
contact me
if you would still like a copy.
David Harney
Process Engineer, Sr.
BASF Corp.
harneyd@basf.com
>Moderators Note. The post in question is dated 1997, from
David
>Harney, Chemical Engineering Dept., University of Missouri -
Rolla.
>Is anything current known about David Harney or the HOMES
software?
>I am a engineering student at India . I was searching the
fortran
>program to solve Non linear systems of equations by
homotopy. I got
>your mail ID from the link
href=http://mathforum.org/epigone/sci.math.research/
praxkhingglou>http:/
/mathforum.org/epigone/sci.math.research/praxkhingglou</aI
was trying to resolve the turning point issue but i could not
do
it.
>I will be very grateful if you can send me the fortran
source code
for
>it along with all the details of Subroutines.
>I will be very grateful to you for this.
>Sonal jain
>>HOMES 2.0, a fortran program that solves systems of
algebraic
>>nonlinear systems using homotopy continuation methods, has
been
>>developed here at the University of Missouri - Rolla.
>>A choice of homotopy functions are available as is a
selection
>>of path tracking algorithms. Bifurcations into the complex
domain
>>at turning points and real domain bounding algorithms are
among
>>other selections available.
>>If you would like a copy of the code, please email me at
>>harney@umr.edu
>>It will eventually be available on a website, as soon as I
muster up
>>the motivation to learn how to create one!
>>David Harney
>>Chemical Engineering Dept.
>>University of Missouri - Rolla
===
Subject: Re: Free software for Homotopy Continuation Methods
>Moderators Note. The post in question is dated 1997, from
David
>Harney, Chemical Engineering Dept., University of Missouri -
Rolla.
>Is anything current known about David Harney or the HOMES
software?
>I am a engineering student at India . I was searching the
fortran
>program to solve Non linear systems of equations by
homotopy. I got
>your mail ID from the link
>http://mathforum.org/epigone/sci.math.research/praxkhingglou
>I was trying to resolve the turning point issue but i could
not do it.
>I will be very grateful if you can send me the fortran
source code for
>it along with all the details of Subroutines.
>I will be very grateful to you for this.
>Sonal jain
I know nothing about HOMES 2.0 but
http://plato.la.asu.edu/topics/problems/zero.html
lists several good sources for continuation methods (in
fortran)
capable of continuation beyond turning points
hth
peter
>>HOMES 2.0, a fortran program that solves systems of
algebraic
>>nonlinear systems using homotopy continuation methods, has
been
>>developed here at the University of Missouri - Rolla.
>>A choice of homotopy functions are available as is a
selection
>>of path tracking algorithms. Bifurcations into the complex
domain
>>at turning points and real domain bounding algorithms are
among
>>other selections available.
>>If you would like a copy of the code, please email me at
>>harney@umr.edu
>>It will eventually be available on a website, as soon as I
muster up
>>the motivation to learn how to create one!
>>David Harney
>>Chemical Engineering Dept.
>>University of Missouri - Rolla
===
Subject: Fermat Quartics and Chord-Tangent process using
Matrices
Introduction (Questions below)
------------
After fixing my method for reducing F(x,y,z,t) = G(x,y,z,t) =
0,
where F, G are homogenous quadratics in x,y,z,t, I end up
with,
in general, one equation of the following form:
| a1 - X b1 c1 d1 |
| a2 b2 - X c2 d2 | = 0
| a3 b3 c3 - Y d3 |
| a4 b4 c4 d4 - Y |
in which ai, bi, are rational functions of the original
(rational)
coefficients of F and G. and X, Y are unknowns for which
rational
values are sought.
(If anyone is interested I can email them the reduction
method. This
requires only one non-trivial rational solution to each of F
= 0 and
G = 0 separately, and it works the same for a system of n
homogenous
quadratics in n+2 variables. But for now Im interested in
the above
equation in its own right.)
Obviously the above is equivalent to a Fermat quartic f(Z) =
T^2
where f is a quartic polynomial in Z.
In fact it can be expressed as H = f1(X).Y^2 + f2(X).Y +
f3(X) = 0
where the fi are quadratic, and this means that in general
(i.e.
if the coefficients dont satisfy a special 
condition) any
single
rational solution implies a two-dimensional net of solutions
obtained by recursively hopping between roots and expressing H
as a quadratic in one or other of X and Y successively.
Questions
---------
1. Id first like to understand how, if at all, 
the 2d
solution net[s]
relate to the Mordell-Weill group of the associated
Weierstrass
equation.
If more than one net is possible then these must obviously be
disjoint. So is the number of nets related to the rank of the
M-W group?
Also, can a net be periodic, in one or both directions. (In
the
latter case it could have a torus structure, which is
obviously
fitting for an elliptic curve!)
2. Can anyone think of a way to use the above determinental
form
in expressing a chord-tangent style composition formula, i.e.
given two rational solutions (X1, Y1) and (X2, Y2) to express
a third rational solution in terms of these?
Despite the tantalizingly simple appearance of the
determinant,
I cant make any headway with this.
However, it can easily be reduced to an equivalent 2*2 form
(with reassigned unknowns X, Y) where i,j = 1,2 independently:
M(X,Y) = | a_ij.XY + b_ij.X + c_ij.Y + d_ij | = 0
So I wonder if there exist suitable 2*2 matrices A, B, C such
that
the following, with X3 and Y3 each as a function of X1, Y1,
X2, Y2,
is an identity:
A . M(X1,Y1) . B . M(X2,Y2) . C = M(X3,Y3)
(Whether elements of A, B, C would also need to be functions
of
X1, Y1, X2, Y2 I have no idea; but this seems likely.)
This looks vaguely reminiscent of Gausss composition of
quadratic forms, and if the parallel is closer than 
superficial
appearance then perhaps all three Xi,Yi would themselves need
to
be constrained by some (bilinear?) parametrization for this
relation to work.
John R Ramsden (jr@adslate.cam) com not cam
===
Subject: curvature invariants
Let (M,g) be a (pseudo-)Riemannian manifold, and Lambda^2 the
space
of antisymmetric tensors of second rank on M. On Lambda^2
there is a
metric G induced by
G(X,Y,A,B) = g(X,A) g(Y,B) - g(X,B) g(Y,A)
as G shares the Riemann-Christoffel symmetries
G(X,Y,A,B) = -G(Y,X,A,B) = -G(X,Y,B,A)
G(X,Y,A,B) = G(A,B,X,Y)
The geometrical significance of G is that G(X,Y,X,Y) measures
the area
squared of the parallelogram defined by X and Y. Knowing only
G, one
apparently cannot reconstruct the metric g.
Question: Are there curvature invariants of (M,g) that can be
written
in terms of only G and its first and second derivatives, i.e.
without
using the metric g?
===
Subject: This week in the mathematics arXiv (11 Oct - 15 Oct)
Here are this weeks titles in the mathematics arXiv,
available at:
http://front.math.ucdavis.edu/
http://front.math.ucdavis.edu/submissions
This week in the mathematics arXiv may be freely redistributed
with attribution and without modification.
Titles in the mathematics arXiv (11 Oct - 15 Oct)
-------------------------------------------------
AC: Commutative Algebra
-----------------------
math.AC/0410304
<http://front.math.ucdavis.edu/math.AC/0410304>
Emanoil Theodorescu: Bivariate Hilbert Functions for the
Torsion Functor
math.AC/0410303
<http://front.math.ucdavis.edu/math.AC/0410303>
Emanoil Theodorescu: Derived Functors and Hilbert Polynomials
math.AC/0410265
<http://front.math.ucdavis.edu/math.AC/0410265>
Marcel Morales, Apostolos Thoma: Complete Intersection
Lattice Ideals
math.AC/0410264
<http://front.math.ucdavis.edu/math.AC/0410264>
Anargyros Katsabekis: Projections of cones and the
arithmetical rank of
toric varieties
math.AC/0410257
<http://front.math.ucdavis.edu/math.AC/0410257>
David Jorgensen, Liana Sega: Independence of the total
reßexivity
conditions for modules
math.AC/0410253
<http://front.math.ucdavis.edu/math.AC/0410253>
Mitsuhiro Miyazaki: On the discrete counterparts of
Cohen-Macaulay
algebras
with straightening laws
math.AC/0410220
<http://front.math.ucdavis.edu/math.AC/0410220>
Rouchdi Bahloul: Generic and comprehensive standard bases
AG: Algebraic Geometry
----------------------
math.AG/0410327
<http://front.math.ucdavis.edu/math.AG/0410327>
Victor Przyjalkowski: Gromov-Witten invariants of Fano
threefolds of
genera
6 and 8
math.AG/0410323
<http://front.math.ucdavis.edu/math.AG/0410323>
Brian Osserman: Mochizukis crys-stable bundles: a lexicon 
and
applications
math.AG/0410313
<http://front.math.ucdavis.edu/math.AG/0410313>
Jean Michel: Hurwitz action on tuples of Euclidean reßections
math.AG/0410309
<http://front.math.ucdavis.edu/math.AG/0410309>
Euisung Park: Noncomplete embeddings of rational surfaces
math.AG/0410306
<http://front.math.ucdavis.edu/math.AG/0410306>
Tomohide Terasoma: Rational convex cones and cyclotomic
multiple zeta
values
math.AG/0410295
<http://front.math.ucdavis.edu/math.AG/0410295>
Michael Lonne: On bifurcation braid monodromy of elliptic
fibrations
math.AG/0410283
<http://front.math.ucdavis.edu/math.AG/0410283>
Alexander Polishchuk: Holomorphic bundles on 2-dimensional
noncommutative
toric orbifolds
math.AG/0410281
<http://front.math.ucdavis.edu/math.AG/0410281>
Samuel Boissiere: On the McKay correspondences for the
Hilbert scheme of
points on the affine plane
math.AG/0410269
<http://front.math.ucdavis.edu/math.AG/0410269>
Frederic Paugam: Quelques bords irrationnels de varietes de
Shimura
math.AG/0410268
<http://front.math.ucdavis.edu/math.AG/0410268>
Dominic Joyce: Configurations in abelian categories. IV.
Changing
stability
conditions
math.AG/0410267
<http://front.math.ucdavis.edu/math.AG/0410267>
Dominic Joyce: Configurations in abelian categories. III.
Stability
conditions and invariants
math.AG/0410262
<http://front.math.ucdavis.edu/math.AG/0410262>
Martin Moeller: Periodic points on Veech surfaces and the
Mordell-Weil
group
over a Teichmueller curve
math.AG/0410259
<http://front.math.ucdavis.edu/math.AG/0410259>
Kenichiro Kimura: A rational map between some threefolds
math.AG/0410258
<http://front.math.ucdavis.edu/math.AG/0410258>
David B. Massey: L^e Modules and Traces
math.AG/0410255
<http://front.math.ucdavis.edu/math.AG/0410255>
Kai Behrend: On the de Rham Cohomology of Differential and
Algebraic
Stacks
math.AG/0410254
<http://front.math.ucdavis.edu/math.AG/0410254>
Frederic Paugam: Three examples of noncommutative boundaries
of Shimura
varieties
math.AG/0410252
<http://front.math.ucdavis.edu/math.AG/0410252>
Ivan Cheltsov: Factorial nodal threefolds in $mathbb{P}^{5}$
math.AG/0410240
<http://front.math.ucdavis.edu/math.AG/0410240>
Michel Brion: Lectures on the geometry of ßag varieties
math.AG/0410224
<http://front.math.ucdavis.edu/math.AG/0410224>
Carlos T. Simpson: Formalized proof, computation, and the
construction
problem in algebraic geometry
math.AG/0410223
<http://front.math.ucdavis.edu/math.AG/0410223>
R. Cluckers, F. Loeser: Ax-Kochen-Er{v{s}}ov Theorems for
$p$-adic
integrals and motivic integration
math.AG/0410221
<http://front.math.ucdavis.edu/math.AG/0410221>
S.Subramanian: Notes on abelian class field theory
AP: Analysis of PDEs
--------------------
math.AP/0410330
<http://front.math.ucdavis.edu/math.AP/0410330>
Adrien Blanchet, Jean Dolbeault, Regis Monneau: On the
one-dimensional
parabolic obstacle problem with variable coefficients
math.AP/0410287
<http://front.math.ucdavis.edu/math.AP/0410287>
Li Ma, DeZhong Chen: Radial Symmetry and Monotonicity Results
for an
Integral Equation
math.AP/0410279
<http://front.math.ucdavis.edu/math.AP/0410279>
W. J. Golz: On the Convection-Dispersion Equation for a
Finite Domain:
Third-Type Boundaries as a Necessary Condition of the
Conservation Law
math.AP/0410229
<http://front.math.ucdavis.edu/math.AP/0410229>
Alexander Vasilev: Evolution dynamics of conformal maps 
with
quasiconformal
extensions
CA: Classical Analysis and ODEs
-------------------------------
math.CA/0410320
<http://front.math.ucdavis.edu/math.CA/0410320>
A. Martinez-Finkelshtein, R. Orive: Riemann-Hilbert analysis
for Jacobi
polynomials orthogonal on a single contour
math.CA/0410284
<http://front.math.ucdavis.edu/math.CA/0410284>
Vivina Barutello, Susanna Terracini: A bisection algorithm
for the
numerical
Mountain Pass
math.CA/0410250
<http://front.math.ucdavis.edu/math.CA/0410250>
George Gasper, Mizan Rahman: Some Systems of Multivariable
Orthogonal
q-Racah polynomials
math.CA/0410249
<http://front.math.ucdavis.edu/math.CA/0410249>
George Gasper, Mizan Rahman: Some Systems of Multivariable
Orthogonal
Askey-Wilson Polynomials
math.CA/0410248
<http://front.math.ucdavis.edu/math.CA/0410248>
George Gasper, Mizan Rahman: q-Analogues of Some Multivariable
Biorthogonal
Polynomials
math.CA/0410228
<http://front.math.ucdavis.edu/math.CA/0410228>
Stephen Semmes: Potpourri, 5
CO: Combinatorics
-----------------
math.CO/0410335
<http://front.math.ucdavis.edu/math.CO/0410335>
Sonja Lj. Cukic, Dmitry N. Kozlov: Higher connectivity of
graph coloring
complexes
math.CO/0410334
<http://front.math.ucdavis.edu/math.CO/0410334>
Raymond Hemmecke: Exploiting Symmetries in the Computation of
Graver
Bases
math.CO/0410319
<http://front.math.ucdavis.edu/math.CO/0410319>
L. Friess: Die Anzahl der Faerbungen ebener Graphen
math.CO/0410308
<http://front.math.ucdavis.edu/math.CO/0410308>
Elena Fuchs: Longest Induced Cycles on Cayley Graphs
math.CO/0410301
<http://front.math.ucdavis.edu/math.CO/0410301>
Peter McNamara: Cylindric skew Schur functions
math.CO/0410289
<http://front.math.ucdavis.edu/math.CO/0410289>
Raymond Hemmecke: Computation of Atomic Fibers of Z-Linear
Maps
math.CO/0410222
<http://front.math.ucdavis.edu/math.CO/0410222>
William Y.C. Chen, Qing-Hu Hou, Yan-Ping Mu: Applicability of
the
$q$-Analogue of Zeilbergers Algorithm
math.CO/0410218
<http://front.math.ucdavis.edu/math.CO/0410218>
B. Bollobas, V. Nikiforov: The sum of degrees in cliques
math.CO/0410217
<http://front.math.ucdavis.edu/math.CO/0410217>
B. Bollobas, V. Nikiforov: Joints in graphs
math.CO/0410216
<http://front.math.ucdavis.edu/math.CO/0410216>
V. Nikiforov: The smallest eigenvalue of K_p-free graphs
cs.DM/0410013 <http://front.math.ucdavis.edu/cs.DM/0410013>
Alex Vinokur: Fibonacci connection between Huffman codes and
Wythoff
array
CT: Category Theory
-------------------
math.CT/0410328
<http://front.math.ucdavis.edu/math.CT/0410328>
Toby Bartels: Categorified gauge theory: Two-bundles
math.CT/0410230
<http://front.math.ucdavis.edu/math.CT/0410230>
Mark Weber: Operads within monoidal pseudo algebras
CV: Complex Variables
---------------------
math.CV/0410296
<http://front.math.ucdavis.edu/math.CV/0410296>
Pengfei Guan: Extremal function of intrinsic norms
math.CV/0410294
<http://front.math.ucdavis.edu/math.CV/0410294>
Andrew McIntyre, Leon A. Takhtajan: Holomorphic factorization
of
determinants of laplacians on Riemann surfaces and a higher
genus
generalization of Kroneckers first limit 
formula
math.CV/0410274
<http://front.math.ucdavis.edu/math.CV/0410274>
Y.Peterzil, S.Starchenko: Subanalytic sets and complex
analytic geometry
DG: Differential Geometry
-------------------------
math.DG/0410332
<http://front.math.ucdavis.edu/math.DG/0410332>
D. Auroux, S. K. Donaldson, L. Katzarkov: Singular Lefschetz
pencils
math.DG/0410314
<http://front.math.ucdavis.edu/math.DG/0410314>
Wayne Rossman: Infinite Periodic Discrete Minimal Surfaces
Without
Self-Intersections
math.DG/0410312
<http://front.math.ucdavis.edu/math.DG/0410312>
Mikhail G. Katz, Stephane Sabourau: Entropy of systolically
extremal
surfaces and asymptotic bounds
math.DG/0410260
<http://front.math.ucdavis.edu/math.DG/0410260>
Yat-Ming Chan: Desingularizations of Calabi-Yau 3-folds with
a conical
singularity
math.DG/0410239
<http://front.math.ucdavis.edu/math.DG/0410239>
Toshiki Mabuchi: An energy-theoretic approach to the
Hitchin-Kobayashi
correspondence for manifolds, II
math.DG/0410232
<http://front.math.ucdavis.edu/math.DG/0410232>
Gabriela P. Ovando: Invariant pseudo Kaehler metrics in
dimension four
math.DG/0410215
<http://front.math.ucdavis.edu/math.DG/0410215>
D. Kotschick: Entropies, volumes, and Einstein metrics
DS: Dynamical Systems
---------------------
math.DS/0410316
<http://front.math.ucdavis.edu/math.DS/0410316>
John Franks: Erratum to Generalizations of the
Poincare-Birkhoff
Theorem
math.DS/0410310
<http://front.math.ucdavis.edu/math.DS/0410310>
A. J. Roberts, I. G. Kevrekidis: Higher order accuracy in the
gap-tooth
scheme for large-scale solutions using microscopic simulators
math.DS/0410299
<http://front.math.ucdavis.edu/math.DS/0410299>
Minoru Ogawa: A sufficient condition for pseudointegrable
systems with
weak
mixing property
math.DS/0410237
<http://front.math.ucdavis.edu/math.DS/0410237>
M.F. Kondratieva, S.Yu. Sadov: Two-system of a Hamiltonian
system
math.DS/0410231
<http://front.math.ucdavis.edu/math.DS/0410231>
Stefano Luzzatto, Ian Melbourne, Frederic Paccaut: The Lorenz
attractor
is
mixing
FA: Functional Analysis
-----------------------
math.FA/0410286
<http://front.math.ucdavis.edu/math.FA/0410286>
D. Q. Cao, Dongsheng Liu, Charles H.-T. Wang: Three
Dimensional Nonlinear
Dynamics of Slender Structures: Cosserat Rod Element Approach
math.FA/0410256
<http://front.math.ucdavis.edu/math.FA/0410256>
Jesus M. F. Castillo, Yolanda Moreno: Extensions by spaces of
continuous
functions
GM: General Mathematics
-----------------------
cs.GT/0410018 <http://front.math.ucdavis.edu/cs.GT/0410018>
Petra Berenbrink, Leslie Ann Goldberg, Paul Goldberg, Russell
Martin:
Utilitarian resource assignment
math.GM/0410241
<http://front.math.ucdavis.edu/math.GM/0410241>
Sebastian Martin Ruiz: A congruence with the Euler totient
function
math.GM/0410234
<http://front.math.ucdavis.edu/math.GM/0410234>
Sergey Sadov: On a necessary and sufficient cyclicity
condition for a
quadrilateral
GN: General Topology
--------------------
math.GT/0410329
<http://front.math.ucdavis.edu/math.GT/0410329>
Louis H. Kauffman: Knot Diagrammatics
GT: Geometric Topology
----------------------
math.GT/0410326
<http://front.math.ucdavis.edu/math.GT/0410326>
Clifford Henry Taubes: Minimal surfaces in germs of hyperbolic
3--manifolds
math.GT/0410321
<http://front.math.ucdavis.edu/math.GT/0410321>
J. O. Button: Fibred and Virtually Fibred hyperbolic
3-manifolds in the
censuses
math.GT/0410300
<http://front.math.ucdavis.edu/math.GT/0410300>
Peter Ozsvath, Zoltan Szabo: Knot Floer homology and integer
surgeries
math.GT/0410288
<http://front.math.ucdavis.edu/math.GT/0410288>
F. Deloup, D. Garber, S. Kaplan, M. Teicher: Palindromic
Braids
math.GT/0410278
<http://front.math.ucdavis.edu/math.GT/0410278>
Martin Scharlemann: Proximity in the curve complex: boundary
reduction
and
bicompressible surfaces
math.GT/0410275
<http://front.math.ucdavis.edu/math.GT/0410275>
Florian Deloup: Involutive braids
math.GT/0410272
<http://front.math.ucdavis.edu/math.GT/0410272>
Julien Marche: Surgery on a single clasper and the 2-loop
part of the
Kontsevich integral
math.GT/0410233
<http://front.math.ucdavis.edu/math.GT/0410233>
Jessica S. Purcell: Cusp Shapes of Hyperbolic Link
Complements and Dehn
Filling
KT: K-Theory and Homology
-------------------------
math.KT/0410315
<http://front.math.ucdavis.edu/math.KT/0410315>
Denis Perrot: The equivariant index theorem in entire cyclic
cohomology
math.KT/0410261
<http://front.math.ucdavis.edu/math.KT/0410261>
Moritz C. Kerz: The complex of words and Nakaoka stability
MG: Metric Geometry
-------------------
math.MG/0410324
<http://front.math.ucdavis.edu/math.MG/0410324>
Oleg R. Musin: The kissing problem in three dimensions
math.MG/0410251
<http://front.math.ucdavis.edu/math.MG/0410251>
D.Siersma, M. van Manen: The nine Morse generic tetrahedra
MP: Mathematical Physics
------------------------
math-ph/0410038
<http://front.math.ucdavis.edu/math-ph/0410038>
Alexander Elgart, Laszlo Erdos, Benjamin Schlein, Horng-Tzer
Yau:
Gross-Pitaevskii Equation as the Mean Field Limit of Weakly
Coupled
Bosons
math-ph/0410037
<http://front.math.ucdavis.edu/math-ph/0410037>
Joseph V. Pule, Andre F. Verbeure, Valentin A. Zagrebnov: A
Dicke Type
Model
for Equilibrium BEC Superradiance
math-ph/0410027
<http://front.math.ucdavis.edu/math-ph/0410027>
Naqing Xie: A Generalized Positive Energy Theorem for Spaces
with
Asymptotic
SUSY Compactification
cond-mat/0410320
<http://front.math.ucdavis.edu/cond-mat/0410320>
Luis Morales-Molina, Franz G. Mertens, Angel Sanchez: Ratchet
behavior in
nonlinear Klein-Gordon systems with point-like inhomogeneities
nlin.SI/0410016
<http://front.math.ucdavis.edu/nlin.SI/0410016>
Fabio Musso, Matteo Petrera, Orlando Ragnisco: Algebraic
extensions of
Gaudin models
math-ph/0410036
<http://front.math.ucdavis.edu/math-ph/0410036>
Hellmut Baumgaertel: On Lax-Phillips semigroups
math-ph/0410035
<http://front.math.ucdavis.edu/math-ph/0410035>
Richard L. Hall, Qutaibeh D. Katatbeh, Nasser Saad: A basis
for
variational
calculations in d dimensions
cond-mat/0410192
<http://front.math.ucdavis.edu/cond-mat/0410192>
Kazumitsu Sakai, Andreas Klumper: Non-dissipative Thermal
Transport and
Magnetothermal Effect for the Spin-1/2 Heisenberg Chain
quant-ph/0410079
<http://front.math.ucdavis.edu/quant-ph/0410079>
D. B. Cervantes, S. L. Quiroga, L. J. Perissinotti, M.
Socolovsky:
Improper
transformations of non relativistic spin 1/2 spinors
math-ph/0410034
<http://front.math.ucdavis.edu/math-ph/0410034>
R. Aldrovandi, A. L. Barbosa: Wu-Yang ambiguity in connection
space
math-ph/0410033
<http://front.math.ucdavis.edu/math-ph/0410033>
Marcel Griesemer: Non-relativistic Matter and Quantized
Radiation
hep-th/0408079 <http://front.math.ucdavis.edu/hep-th/0408079>
Davide Fioravanti: Geometrical Loci and CFTs via the Virasoro
Symmetry of
the mKdV-SG hierarchy: an excursus
quant-ph/0410050
<http://front.math.ucdavis.edu/quant-ph/0410050>
Gerardo Adesso, Fabrizio Illuminati: Multipartite
Entanglement and its
Polygamy in Continuous Variable Systems
nlin.SI/0409050
<http://front.math.ucdavis.edu/nlin.SI/0409050>
P.M. Santini, M. Nieszporski, A. Doliwa: An integrable
generalization of
the
Toda law to the square lattice
math-ph/0410032
<http://front.math.ucdavis.edu/math-ph/0410032>
T. Spencer, M.R. Zirnbauer: Spontaneous symmetry breaking of
a hyperbolic
sigma model in three dimensions
math-ph/0410031
<http://front.math.ucdavis.edu/math-ph/0410031>
A. Wereszczynski: Nested Multi-Soliton Solutions with
Arbitrary Hopf
Index
math-ph/0410030
<http://front.math.ucdavis.edu/math-ph/0410030>
Guido Gentile, Daniel A. Cortez, Joao C. A. Barata: Stability
for
quasi-periodically perturbed Hills equations
math-ph/0410029
<http://front.math.ucdavis.edu/math-ph/0410029>
Roland Friedrich: On Connections of Conformal Field Theory
and Stochastic
L{oe}wner Evolution
math-ph/0410028
<http://front.math.ucdavis.edu/math-ph/0410028>
Mark Naber: Time fractional Schrodinger equation
math-ph/0410026
<http://front.math.ucdavis.edu/math-ph/0410026>
Jining Gao: The Maurer-Cartan structure of BRST differential
math-ph/0410025
<http://front.math.ucdavis.edu/math-ph/0410025>
Hayriye Tutunculer, Ramazan Koc: Differential Realizations of
the
Two-Mode
Bosonic and Fermionic Hamiltonians: A unified Approach
hep-th/0410083 <http://front.math.ucdavis.edu/hep-th/0410083>
Antonino Flachi, Alan Knapman, Wade Naylor, Misao Sasaki:
Zeta Functions
in
Brane World Cosmology
cond-mat/0410159
<http://front.math.ucdavis.edu/cond-mat/0410159>
L. Diago-Cisneros, H. Rodriguez-Coppola, R. Perez-Alvarez, P.
Pereyra:
Symmetries and General Principies in the Multiband Effective
Mass Theory:
A
Transfer Matrix Study
math-ph/0410024
<http://front.math.ucdavis.edu/math-ph/0410024>
Xu-Dong Luo, Han-Ying Guo, Yu-Qi Li, Ke Wu: Difference
Discrete
Variational
Principle in Discrete Mechanics and Symplectic Algorithm
NT: Number Theory
-----------------
math.NT/0410333
<http://front.math.ucdavis.edu/math.NT/0410333>
Lev A. Borisov: Holomorphic Eisenstein series with Jacobian
twists
math.NT/0410297
<http://front.math.ucdavis.edu/math.NT/0410297>
Bernd C. Kellner: A conjecture about numerators of Bernoulli
numbers
related
to Integer Sequence A092291
math.NT/0410292
<http://front.math.ucdavis.edu/math.NT/0410292>
Alexander Schmidt: Tame class field theory for arithmetic
schemes
math.NT/0410270
<http://front.math.ucdavis.edu/math.NT/0410270>
T. W. Hilberdink, M. L. Lapidus: Beurling Zeta Functions,
Generalised
Primes, and Fractal Membranes
math.NT/0410266
<http://front.math.ucdavis.edu/math.NT/0410266>
John Voight: Binary quadratic forms that represent almost the
same primes
math.NT/0410246
<http://front.math.ucdavis.edu/math.NT/0410246>
Yuri F. Bilu, Florian Luca: Divisibility of class numbers:
enumerative
approach
math.NT/0410245
<http://front.math.ucdavis.edu/math.NT/0410245>
A.C. de la Maza: Classes of forms Witt equivalent to a second
trace form
over fields of characteristic two
math.NT/0410244
<http://front.math.ucdavis.edu/math.NT/0410244>
A.C. de la Maza: Generalization of the second trace form of
central
simple
algebras in characteristic two
OA: Operator Algebras
---------------------
math.OA/0410337
<http://front.math.ucdavis.edu/math.OA/0410337>
Matthew Neal, Bernard Russo: Representation of contractively
complemented
Hilbertian operator spaces and the Fock space
math.OA/0410305
<http://front.math.ucdavis.edu/math.OA/0410305>
Marcelo Laca, Machiel van Frankenhuijsen: Phase transitions
on Hecke
C*-algebras and class-field theory over Q
math.OA/0410290
<http://front.math.ucdavis.edu/math.OA/0410290>
Benton L. Duncan: Explicit construction and uniqueness for
universal
operator algebras of directed graphs
math.OA/0410235
<http://front.math.ucdavis.edu/math.OA/0410235>
Marius Junge: Embedding of OH and logarithmic `little
Grothendieck
inequality
math.OA/0410219
<http://front.math.ucdavis.edu/math.OA/0410219>
Ilwoo Cho: Random Variables in Graph W*-Probability Spaces
OC: Optimization and Control
----------------------------
math.OC/0410277
<http://front.math.ucdavis.edu/math.OC/0410277>
Michael Malisoff, Mikhail Krichman, Eduardo Sontag: Global
Stabilization
for
Systems Evolving on Manifolds
math.OC/0410225
<http://front.math.ucdavis.edu/math.OC/0410225>
Raymond Hemmecke, Robert Weismantel: Integral Function Bases
PR: Probability
---------------
math.PR/0410336
<http://front.math.ucdavis.edu/math.PR/0410336>
Bela Bollobas, Oliver Riordan: The critical probability for
random
Voronoi
percolation in the plane is 1/2
math.PR/0410331
<http://front.math.ucdavis.edu/math.PR/0410331>
Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Russell
Martin: Markov
chain
comparison
math.PR/0410318
<http://front.math.ucdavis.edu/math.PR/0410318>
B. Chauvin, A. Rouault: Connecting Yule Process, Bisection
and Binary
Search
Tree via Martingales
math.PR/0410311
<http://front.math.ucdavis.edu/math.PR/0410311>
Geoffrey Grimmett, Svante Janson: Branching Processes, and
Random-Cluster
Measures on Trees
math.PR/0410298
<http://front.math.ucdavis.edu/math.PR/0410298>
Jianjun Tian, Xiao-Song Lin: Continuous Time Markov Processes
on Graphs
math.PR/0410285
<http://front.math.ucdavis.edu/math.PR/0410285>
Huyen Pham: On the smooth-fit property for one-dimensional
optimal
switching
problem
math.PR/0410282
<http://front.math.ucdavis.edu/math.PR/0410282>
Itai Benjamini, Oded Schramm, David B. Wilson: Balanced
Boolean functions
that can be evaluated so that every input bit is unlikely to
be read
math.PR/0410276
<http://front.math.ucdavis.edu/math.PR/0410276>
A. Ruzmaikina, M. Aizenman: Characterization of invariant
measures at the
math.PR/0410236
<http://front.math.ucdavis.edu/math.PR/0410236>
Davar Khoshnevisan, David A. Levin, Pedro J.
Mendez-Hernandez: Capacities
in
Wiener Space, Quasi-Sure Lower Functions, and Kolmogorovs
Epsilon-Entropy
math.PR/0410227
<http://front.math.ucdavis.edu/math.PR/0410227>
R. Ferriere, A. Guionnet, I. Kurkova: Timescales of
population rarity and
commonness in random environments
QA: Quantum Algebra
-------------------
math.QA/0410322
<http://front.math.ucdavis.edu/math.QA/0410322>
Gaetano Fiore: New approach to Hermitian q-differential
operators on
R_q^N
math.QA/0410291
<http://front.math.ucdavis.edu/math.QA/0410291>
Hiroshige Kajiura, Jim Stasheff: Homotopy algebras inspired
by classical
open-closed string field theory
hep-th/0410084 <http://front.math.ucdavis.edu/hep-th/0410084>
Yas-Hiro Quano: Form factors, correlation functions and
vertex operators
in
the eight-vertex model at reßectionless points
math.QA/0410263
<http://front.math.ucdavis.edu/math.QA/0410263>
Julien Bichon, Giovanna Carnovale: Lazy cohomology: an
analogue of the
Schur
multiplier for arbitrary Hopf algebras
math.QA/0410247
<http://front.math.ucdavis.edu/math.QA/0410247>
Jining Gao: $L_{infty}$ algebra structures of Lie algebra
deformations
math.QA/0410238
<http://front.math.ucdavis.edu/math.QA/0410238>
Marta M. Asaeda, Jozef H. Przytycki, Adam S. Sikora: A
categorification
of
the skein module of tangles
hep-th/0410086 <http://front.math.ucdavis.edu/hep-th/0410086>
N.A. Gromov, V.V. Kuratov: Quantum kinematics
RA: Rings and Algebras
----------------------
math.RA/0410317
<http://front.math.ucdavis.edu/math.RA/0410317>
Heide Gluesing-Luerssen, Wiland Schmale: On doubly-cyclic
convolutional
codes
math.RA/0410226
<http://front.math.ucdavis.edu/math.RA/0410226>
Bilbo Baggins, Laurent Bartholdi: Branch Rings, Thinned
Rings, Tree
Enveloping Rings
RT: Representation Theory
-------------------------
math.RT/0410339
<http://front.math.ucdavis.edu/math.RT/0410339>
Volodymyr Mazorchuk, Catharina Stroppel: On functors
associated to a
simple
root
math.RT/0410325
<http://front.math.ucdavis.edu/math.RT/0410325>
K. Baur: A normal form for admissible characters in the sense
of Lynch
math.RT/0410302
<http://front.math.ucdavis.edu/math.RT/0410302>
Toshihiko Matsuki: Equivalence of domains arising from
duality of orbits
on
ßag manifolds III
math.RT/0410293
<http://front.math.ucdavis.edu/math.RT/0410293>
I. Gordon, J.T.Stafford: Rational Cherednik algebras and
Hilbert schemes
II:
representations and sheaves
math.RT/0410273
<http://front.math.ucdavis.edu/math.RT/0410273>
Franc{c}ois Rodier: Errata `a ``Sur les representations non
ramifiees
des groupes reductifs $p$-adiques; lexemple 
de ${rm
GSp}(4)$
math.RT/0410242
<http://front.math.ucdavis.edu/math.RT/0410242>
Yuri A. Neretin: On compression of Bruhat-Tits buildings
SG: Symplectic Geometry
-----------------------
math.SG/0410338
<http://front.math.ucdavis.edu/math.SG/0410338>
Michael Entov, Leonid Polterovich: Quasi-states and symplectic
intersections
math.SG/0410243
<http://front.math.ucdavis.edu/math.SG/0410243>
D. Kotschick: Free circle actions with contractible orbits on
symplectic
manifolds
SP: Spectral Theory
-------------------
math.SP/0410307
<http://front.math.ucdavis.edu/math.SP/0410307>
R.F. Efendiev: The characterization problem for one class
high order
ordinary differential operators with periodic coefficients
math.ST/0410280
<http://front.math.ucdavis.edu/math.ST/0410280>
Olivier Catoni: Improved Vapnik Cervonenkis bounds
ST: Statistics
--------------
q-bio.GN/0410012
<http://front.math.ucdavis.edu/q-bio.GN/0410012>
Satya N. Majumdar, Sergei Nechaev: Exact Asymptotic Results
for a Model
of
Sequence Alignment
math.ST/0410271
<http://front.math.ucdavis.edu/math.ST/0410271>
J.J. Lok: Statistical modelling of causal effects in
continuous time
--
/ Greg Kuperberg (UC Davis)
/ Home page: http://www.math.ucdavis.edu/~greg/
/ Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
/ * All the math thats fit to e-print *
===
Subject: new products in geometric algebra
Ive been studying some of the additional operations 
defined
in the
literature on geometric algebra, AKA Clifford algebra. There
are some
very interesting constructions, but I am having a hard time
mapping them
to more common operations on the exterior algebra.
One question I have: has anyone come up with a clean
definition in terms
of Clifford (geometric) multiplication for the inner product
as
defined on the exterior algebra to define the 
Hodge star? For A
and B
k-blades (i.e. the exterior products of k vectors A_i and
B_j) this is
the construction
<A, B> := det(<A_i, B_j>)
with the result defined to be zero for blades of different
grade
(composed of different numbers of vectors). Even an operation
that
coincided with this for two k-blades would be of interest.
Ive looked
into various operations including:
- inner product A.B := <AB>_|j-k| where A and B are j- and
k-vectors
- commutator product A x B := (AB - BA)/2
- contraction A J B := (A / (Bw))w^-1 where w is the volume
element
(pseudoscalar) and J is supposed to be a backwards L
but none seem to do the job.
A related item: Ive been putting the definition 
of the Hodge
star in
terms of geometric algebra operations
*A = ((w^-1)A)^+ where w^-1 is the inverse of the volume
element
(pseudovector), and ^+ indicates reversion
into some more usable (for what Im doing) forms. Can anyone
verify
these results? Here A is a k-blade, w is the volume element
(pseudoscalar), and s is the number of negative signs in the
signature
of the inner product defining the Clifford algebra.
*A = (-1)^(k(k-1)/2 + s) Aw
*A = (w^+w)(A^+)w
===
Subject: Re: new products in geometric algebra
Mark Adams <markjadams@lycos.com> schrieb im Newsbeitrag
> One question I have: has anyone come up with a clean
definition in terms
> of Clifford (geometric) multiplication for the inner
product as
> defined on the exterior algebra to define the 
Hodge star? For
A and B
> k-blades (i.e. the exterior products of k vectors A_i and
B_j) this is
> the construction
> <A, B> := det(<A_i, B_j>)
> with the result defined to be zero for blades of different
grade
> (composed of different numbers of vectors). Even an
operation that
> coincided with this for two k-blades would be of interest.
Ive looked
> into various operations including:
> - inner product A.B := <AB>_|j-k| where A and B are j- and
k-vectors
> - commutator product A x B := (AB - BA)/2
> - contraction A J B := (A / (Bw))w^-1 where w is the volume
element
> (pseudoscalar) and J is supposed to be a backwards L
> but none seem to do the job.
I think what you are looking for is <A^+,B>_0 ,
(where ^+ is reversion).
You can check that this gives the desired result by looking
at a convenient
orthonormal basis.
You can explicitly derive this expression by noting that the
exterior bundle
supports two mutually supercommuting copies of the Clifford
algebra that you
have in mind, constructed from sums and differences of
operators of exterior
and interior multiplication, respectively. Noting that the
latter are mutual
adjoints with respect to the Hodge inner product and using
the symbol map
you can then derive the above (up to a global sign, depending
on some
conventions) from the Hodge inner product.
If you want to see the details have a look at pp.279 of
http://www-stud.uni-essen.de/~sb0264/sqm.html .
> A related item: Ive been putting the 
definition of the
Hodge star in
> terms of geometric algebra operations
> *A = ((w^-1)A)^+ where w^-1 is the inverse of the volume
element
> (pseudovector), and ^+ indicates reversion
> into some more usable (for what Im doing) forms. Can
anyone verify
> these results? Here A is a k-blade, w is the volume element
> (pseudoscalar), and s is the number of negative signs in
the signature
> of the inner product defining the Clifford algebra.
> *A = (-1)^(k(k-1)/2 + s) Aw
> *A = (w^+w)(A^+)w
This is also discussed at the above reference.
I have found it very helpful to pass between Clifford
calculus and exterior
calculus this way when dealing with supersymmetric quantum
systems. For
instance hep-th/0311064 and math-ph/0407005 makes use of this.
===
Subject: Re: new products in geometric algebra
Originator: israel@math.ubc.ca (Robert Israel)
for the pointer, lots of interesting stuff! I have several
questions on
the details of forming a Clifford algebra from the linear
operators on
the exterior algebra, and the relationship of this to the
quantum
harmonic oscillator and thus to QFT. Is there a main
reference that you
used for this...or in general, can anyone recommend a book
that goes
through related relationships in detail from a more
mathematical
> Mark Adams <markjadams@lycos.com> schrieb im Newsbeitrag
> One question I have: has anyone come up with a clean
definition in
terms
> of Clifford (geometric) multiplication for the inner
product as
> defined on the exterior algebra to define the 
Hodge star? For
A and
B
> k-blades (i.e. the exterior products of k vectors A_i and
B_j) this
is
> the construction
> <A, B> := det(<A_i, B_j>)
> with the result defined to be zero for blades of different
grade
> (composed of different numbers of vectors). Even an
operation that
> coincided with this for two k-blades would be of interest.
Ive
looked
> into various operations including:
> - inner product A.B := <AB>_|j-k| where A and B are j- and
k-vectors
> - commutator product A x B := (AB - BA)/2
> - contraction A J B := (A / (Bw))w^-1 where w is the volume
element
> (pseudoscalar) and J is supposed to be a backwards L
> but none seem to do the job.
> I think what you are looking for is <A^+,B>_0 ,
> (where ^+ is reversion).
> You can check that this gives the desired result by looking
at a
convenient
> orthonormal basis.
> You can explicitly derive this expression by noting that
the exterior
bundle
> supports two mutually supercommuting copies of the Clifford
algebra
that you
> have in mind, constructed from sums and differences of
operators of
exterior
> and interior multiplication, respectively. Noting that the
latter are
mutual
> adjoints with respect to the Hodge inner product and using
the symbol
map
> you can then derive the above (up to a global sign,
depending on some
> conventions) from the Hodge inner product.
> If you want to see the details have a look at pp.279 of
> http://www-stud.uni-essen.de/~sb0264/sqm.html .
> A related item: Ive been putting the 
definition of the
Hodge star
in
> terms of geometric algebra operations
> *A = ((w^-1)A)^+ where w^-1 is the inverse of the volume
element
> (pseudovector), and ^+ indicates reversion
> into some more usable (for what Im doing) forms. Can
anyone verify
> these results? Here A is a k-blade, w is the volume element
> (pseudoscalar), and s is the number of negative signs in the
signature
> of the inner product defining the Clifford algebra.
> *A = (-1)^(k(k-1)/2 + s) Aw
> *A = (w^+w)(A^+)w
> This is also discussed at the above reference.
> I have found it very helpful to pass between Clifford
calculus and
exterior
> calculus this way when dealing with supersymmetric quantum
systems.
For
> instance hep-th/0311064 and math-ph/0407005 makes use of
this.
===
Subject: Library for Subgraph matching and graph dissimilarity
Hi all...
I have two graphs G1 and G2 labeled along nodes and edges. I
have to
get a pattern/ subgraph from these two graphs such that this
subgraph
captures those nodes and edges in G1 which are not included
in G2. i.e
there is a subgraph of G1 which is changed in G2. Rest of G1
in G2 is
not changed and I need to capture this subgraph. These are NP
complete
problems as far as I know.
I request you to suggest me an optimum way to do this. Is
there any
library or tool which does this.
Vipindeep
===
Subject: Integral equations of first kind
Good Morning,
I am working with the integral equations of first kind on an
annular
in R^3.
I have to prove the unicity of the solution,
Numerically, i have a well conditionned matrix, but
theorically i
dont know how to prove it.
In fact, i dont know which methods use for the equations of
first
kind. ive tried with the alternative of Fredholm, it helps
me to
prove the unicity in the quotient space but not in the space
itself
(what i want to prove) !
So, i would like to ask if someone can help me ,
maybe with some theorems of fixed point .....
Zimar.
===
Subject: Re: Category of categories axiomatically
>> How to define the category of (locally small) categories
axiomatically?
> Maybe my question is on a highly sensitive issue in
Category Theory,
which
> explains there are no replies ^_^. Indeed it is not clear
that the notion
> of category of categories makes sense.
The question does make sense. But you have to be careful.
There is a lots of things known about the category Cat of
small
categories: it is locally small, complete, cocomplete,
cartesian
closed, etc...
I believe that considering three Grothendieck universes U
subset V
subset W is sufficient to solve most of the problems of size.
The
unique rule is then: instead of talking about sets,
collections, and I
dont know what next, use the term U-small set, V-small set,
W-small
set. A U-small set has a U-small cardinal. etc... In other
terms, at
each step, specify cleary to which universe the set under
consideration belongs.
Instead of talking about small category, use the term U-small
category, i.e. a U-small set of U-small objects with U-small
homsets :
denote by Cat_U the corresponding V-small and locally U-small
category. Instead of talking about locally small category,
use the
term locally U-small category, i.e. a V-small set of U-small
objects
with U-small homsets. Denote by CAT_U the corresponding
category: if I
dont make any mistake, CAT_U is certainly not V-small, but is
W-small
because CAT_U subset Cat_V and the latter is W-small.
Since Cat_V is V-complete, it is U-complete. So I believe
(but I did
not check that) that CAT_U is U-complete as well: the only
thing you
have to check is that a U-small limit in CAT_U stays inside
CAT_U. CAT_U is probably U-cocomplete as well. CAT_U is
probably not
cartesian closed however because the obvious candidat for
HOM(C,D) is
the category of functors from C to D. And it is well-known
that
HOM(C,D) is in CAT_U iff C is essentially U-small (i.e.
equivalent to
a U-small category). etc...
pg.
===
Subject: Re: Category of categories axiomatically
<clb27u$20b$1@dizzy.math.ohio-state.edu>
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> CAT_U. CAT_U is probably U-cocomplete as well. CAT_U is
probably not
> cartesian closed however because the obvious candidat for
HOM(C,D) is
> the category of functors from C to D. And it is well-known
that
> HOM(C,D) is in CAT_U iff C is essentially U-small (i.e.
equivalent to
> a U-small category). etc...
A small mistake : D=Set is understood. For over category D,
the
statement above may be false. The very-known statement I had
in mind
is the: a category of presheaves is locally small iff the base
category (C here) is essentially small.
pg.
===
Subject: Re: Category of categories axiomatically
> I dont believe Lawveres axioms were 
completely successful
for
> foundational purposes; IIRC, Isbell pointed out that the
category
> of categories whose only endomorphisms are identities
satisfied
> AFAIK, no one has followed up on this particular direction.
Blanc, Georges and Donnadieu, Marie-Rene
Axiomatisation de la catgorie des
catgories,
Cahiers Topologie Gom. Diffrentielle 17, no.
2, 135--170, 1976
David.
===
Subject: Re: Category of categories axiomatically
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> I dont believe Lawveres axioms were 
completely successful
for
> foundational purposes; IIRC, Isbell pointed out that the
category
> of categories whose only endomorphisms are identities
satisfied
> AFAIK, no one has followed up on this particular direction.
> Blanc, Georges and Donnadieu, Marie-Ren?e
> Axiomatisation de la cat?gorie des cat?gories,
> Cahiers Topologie G?om. Diff?rentielle 17, no. 2, 135--170,
1976
> David.
Eventually, Lawvere added a new axiom to get rid of Isbells
objection. But it still wasnt overly successful. He was
really
trying to get at set theory without membership and in this
toposes
were completely successful.
Part of the problem with the question is that you havent
said what
axioms you are using for a category. If you go back to General
theory of natural equivalences, the Eilenberg-Mac Lane paper
that
started it all, there were no hom sets. Just a bucket of
arrows
along with a law of partical composition and identities,
subject to
the obvious identities. Notice that this is (or anyway can be)
carried out in a pre-set-theoretic universe and has exactly
the same
epistomological status as axioms for set theory which,
unavoidably,
must be carried out in a pre-set-theoretic universe. In that
setting,
there is no problem with the categories of categories. The
arrows are
functors and the identities (or objects, if you insist) are
categories.
If you are doing it within set theory, you have some choices.
There
is the category of small categories, which is no more
problematic than
the category of groups. You can use Grothendieck universes,
which a
previous writer mentioned. That ought to be banned on pure
esthetic
grounds, especially as it never plays the slightest role
except that
of uglification. Or you can ignore the whole question and if
someone
calls you on it, you say it is a metaphorical way of using
universes
without mentioning them explicitly. You are not going to make
any
construction--believe me on this--that cannot be coded into
universes.
This last thing is what I have always done.
===
Subject: Does this definition of dim 2 n-free poset exterior
ring any
bells with anyone?
Let H be the Hasse diagram of any dim 2 n-free poset
and let Hp be any natural plane embedding of H. Then
inasmuch as Hp is a DAG whose edges are defined by the
immediate covering relations in H, one can assign
the integer d to each vertex v of Hp, where d is the
depth of v in Hp interpreted as a DAG (i.e. the number
of vertices in Hp which are ancestors of v, excluding v
itself.)
Further, since Hp is embedded in the plane relative
to the geometry of the paper, a left-right or
horizontal ordering of the vertices of Hp results
from its embedding in the plane. That is, we can
assign the numbers p and t to each vertex v of Hp, where:
p is the number of vertices to the left of v in Hp
t is the number of vertices to the right of v in Hp.
And it is readily shown that because Hp is both dim 2
and n-free:
a) the values (p0+d0),...,(pi+di),...,(pn+dn) for
the n+1 vertices of Hp are 0,...,n+1
b) the values (t0+d0),...,(ti+di),...,(tn+dn) for
the n+1 vertices of Hp are a permutation of 0,...,n+1.
Suppose now, however, that one uses the integers
pi, ti, di to assign each vertex v of Hp the triple
of integers (pi,ti,di).
And further, suppose that one defines the exterior
of Hp as just those vertices in Hp for which at least
one of pi,ti,di is 0.
And finally, suppose that one defines the 
exterior
union of Hp as the union of all the exteriors of all
the subgraphs of Hp which are themselves interpretable
as Hasse diagrams of dim 2 n-free posets.
Does this notion of exterior union ring any bells
with anyone ??
considering this matter.
===
Subject: TOC / Index for the 3 Notebooks of Ramanujan
Epigone-thread: doichayßon
[Firstly, I would like to know if there is a facility in this
User
Group to upload a document.] Pl read on ...
published by TIFR in 2 Volumes. Then, as I wanted to read the
proofs
of Ramanujans Theorems as published by Bruce C Berndt in 
his
5-Part
books, I noticed that there is an obvious strong refernce
made to the
3 Notebooks and the 2 Volumes of TIFR. I then thought that it
is very
essential at least for a beginner like me to have some form
of a
quick-reference TOC / index to these creations of Ramanujan.
This effort was relatively easier wrt the so-called organized
parts,
the 21 Chapters ; however, I need some help to update (and
correct)
parts that I have described as Misc in my prepared document.
I have
categorized Misc itself into 3 main types. It would be great
if
someone could help me state the split of appropriate
subject(s) for
these Misc Pages.
For eg, in Volume I, ie, NB 1, I could count a total of 121
pages -
marked *under my so-called Misc category. Since it is claimed
that
there are 100 Unorganized pages in NB 1, I would like to know
which
100 pages of these 121 pages fall in the Unorganized category.
Now, how do I upload my document (SR_NBs_TOC_Index.doc) ?
If there is no such facility to upload documents in this User
Group,
interested readers can request me by sending me a mail to my
email id
: sundarkrishnan@hotmail.com
Sundar Krishnan
[
Moderators Note. sci.math.research newsgroup does not 
include
documents. Offering to send by email is one way. Another is
to post it on a web page, then tell us the URL here.
]
===
Subject: HyperGeometric Series
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
I am not an expert in HyperGeometric Series and Functions
[from now
on, I use the short form HG below.]
Along with other basic Maths books, I have been trying to
learn this
subject also by reading the famous book : A Course of Modern
Analysis
(Cambridge Mathematical Library), by E. T. Whittaker, G. N.
Watson.
[from now on, I call it the W & W book below.]
Also, I realised that many of Srinivasa Ramanujans works
relate to HG
Fns and Series, as can also be seen in the 5-Part book
authored by
Bruce C Berndt.
At many places, I have been able to follow the derivation of
complex
expressions of the W & W book, but often with slightly
different
expressions like say, an extra snippet of expression, a
change of sign
etc. And even the so-called (simple, but very appropriate to
the
subject at that point) Examples are sometimes difficult to
answer.
I would therefore like to know if there is a Solutions Manual
to this
book so that it is easier to learn and a little faster too -
something
similar to Donald Knuths books with Solutions on The Art of
Computer
Programming.
My brief background :
After working for many years in a number of companies like
ABB, HP,
Motorola, I have now taken up some studies in Video
Compression, Error
Correction Codes, DSP and Maths - Elliptical Integrals,
HyperGeometric Series etc.
So, I am neither a student nor a Professor, but very much
keen on the
above subjects.
Sundar Krishnan
...
PS : Interested readers may also refer to my other query :
TOC /
Index for the 3 Notebooks of Ramanujan.
...
===
Subject: Solving Quintics Using One Fifth Root Extraction
Epigone-thread: cymingßu
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Hello all,
For those who like solvable quintics, heres another paper:
Solving Solvable Quintics Using One Fifth Root Extraction
ABSTRACT: We prove that all irreducible but solvable equations
of degree n can be transformed in radicals into the binomial
form
y^n+c=0 using a Tschirnhaussen transformation of degree n-1.
The
resulting equation is then solvable by a single nth root
extraction. In particular, we illustrate the method using the
solvable quintic.
Mathematics Subject Classification. Primary: 12E12; Secondary:
11D09
http://www.geocities.com/titus_piezas/morequintics.html
Just click at the link above. Its the 2nd pdf 
file.
-Titus (tpiezasIII@uap.edu.ph -> remove III for email)
===
Subject: Paper published by Algebraic and Geometric Topology
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
The following paper has been published:
Algebraic and Geometric Topology
URL:
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-41.abs.html
Title:
Partition complexes, duality and integral tree representations
Author(s):
Alan Robinson
Abstract:
We show that the poset of non-trivial partitions of 1,2,...,n
has a
fundamental homology class with coefficients in a Lie
superalgebra. Homological duality then rapidly yields a range
of known
results concerning the integral representations of the
symmetric
groups S_n and S_{n+1} on the homology and cohomology of this
partially-ordered set.
Secondary: 17B60
Keywords:
Partition complex, Lie superalgebra
Author(s) address(es):
Mathematics Institute, University of Warwick, Coventry CV4
7AL, UK
Email: car@maths.warwick.ac.uk
===
Subject: The extent of a plane curve
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Call an arc in the plane convex if it is simple (i.e., 1-1
except that
its
endpoints may coincide) and it lies on the boundary of its
convex hull.
Define the extent of a curve c of length L in the plane to be
the
length of the shortest convex arc whose convex hull contains
the (range of
the) curve c.
So, for example, the extent of a triangle is the sum of its
two longer
sides, and the extent of a circle of radius r is (2 + pi)r.
This notion
has a number of interesting properties, but it seems awkward
to investigate.
My inquiry here is whether anyone knows of any mention of the
notion in the
literature.
--J. Wetzel
===
Subject: Re: The extent of a plane curve
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> Call an arc in the plane convex if it is simple (i.e., 1-1
except that
its
> endpoints may coincide) and it lies on the boundary of its
convex hull.
> Define the extent of a curve c of length L in the plane to
be the
> length of the shortest convex arc whose convex hull
contains the (range of
> the) curve c.
> So, for example, the extent of a triangle is the sum of its
two longer
> sides, and the extent of a circle of radius r is (2 + pi)r.
This
notion
> has a number of interesting properties, but it seems
awkward to
investigate.
> My inquiry here is whether anyone knows of any mention of
the notion in
the
> literature.
I dont know of any mention, but it seems like the extent 
can
always be
determined as follows: find a tangent line L to the convex
hull of c,
find two line segments perpendicular to L and tangent to the
convex hull
of c, and form an arc with these two segments and the
remaining portion
of the boundary of the convex hull.
The only question is where to place the tangent line but (if
cs convex
hull is a convex polygon) this can probably be done
efficiently using a
rotating calipers algorithm.
--
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/
===
Subject: Re: The extent of a plane curve
Received-SPF: Received-SPF: none (mailbox9.ucsd.edu: domain of
poster@giganews.com does not designate permitted sender hosts)
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> Call an arc in the plane convex if it is simple (i.e., 1-1
except
that
> its
> endpoints may coincide) and it lies on the boundary of its
convex hull.
Define the extent of a curve c of length L in the plane to be
the
> length of the shortest convex arc whose convex hull
contains the (range
of
> the) curve c.
So, for example, the extent of a triangle is the sum of its
two longer
> sides,
Why isnt the extent of a triangle the sum of lengths of it
two
SHORTER sides?
===
Subject: Re: The extent of a plane curve
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Oops! Well, it was late at night, and ... ... ...
--J. Wetzel
> Why isnt the extent of a triangle the sum of lengths of 
it
two
> SHORTER sides?
===
Subject: algebraic sets
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
topology and that the following question comes from
optimization:
Let p_i be a finite set of k relatively prime polynomials,
each of
which maps R^n into R. I am interested in describing the
boundary of
the set
A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k}
as a union of manifolds. Can I do this? Specifically, I would
like to
say the following:
Let I be an index set and let
A_I = {x in R^N : p_i(x) = 0 for i in I and p_i(x) > 0 for i
notin
I}.
Then A_I is the finite union of manifolds of dim <=
cardinality(I).
I have been told that a semialgebraic subset of R^n (like A)
can be
written as a union of manifolds of decreasing dimension, but
I have
not seen this result stated anywhere. I have seen a result
that says
a semialgebraic set is composed of a disjoint union of
connected
semialgebraic sets, but these are not quite what I want.
Could someone point out some results for me in this direction
and
where I might find them written down? Even results requiring
additional hypotheses would be welcome.
===
Subject: Re: algebraic sets
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Hi all,
I will make another post after absorbing all of this, which
may take a
little while. One point I want to make now is that it is
important to
me that I can describe the boundary of my set A as a union of
sets of
the form A_I, and that the relationship between the
codimension of A_I
and the cardinality of I is very important.
Cory
===
Subject: Re: algebraic sets
Received-SPF: Received-SPF: none (mailbox5.ucsd.edu: domain of
mod-submit@uni-berlin.de does not designate permitted sender
hosts)
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> topology and that the following question comes from
optimization:
> Let p_i be a finite set of k relatively prime polynomials,
each of
> which maps R^n into R. I am interested in describing the
boundary of
> the set
> A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k}
> as a union of manifolds. Can I do this? Specifically, I
would like to
> say the following:
> Let I be an index set and let
> A_I = {x in R^N : p_i(x) = 0 for i in I and p_i(x) > 0 for
i notin
> I}.
> Then A_I is the finite union of manifolds of dim <=
cardinality(I).
> I have been told that a semialgebraic subset of R^n (like
A) can be
> written as a union of manifolds of decreasing dimension,
but I have
> not seen this result stated anywhere. I have seen a result
that says
> a semialgebraic set is composed of a disjoint union of
connected
> semialgebraic sets, but these are not quite what I want.
> Could someone point out some results for me in this
direction and
> where I might find them written down? Even results requiring
> additional hypotheses would be welcome.
more or less (read: dim <= n=N) ... and with some handwaving
(i always
feel a little bit unsure switching to the real case, so feel
free to
correct ...) it should go like this:
First you need the notion of Ôreduced as {x: 
p(x)=0} = {x:
p(x)^2=0},
it means you can ignore multiple zeros if you are only
interested in
the algebraic set, just accept that this can be done through
algebra.
Then you need: for a reduced space the set of singularities
is algebraic
and has lower dimension. Here Ôsingularities 
means, that the
defining
polynomials P have a Hessian which has not maximal rank. The
complement
are the regular points - they are open and dense (but need
not have the
full dimension or are connected over the reals, think of the
transversal
union of a line and a plane in R^3) and are manifolds.
With that you can proceed by induction (giving a
stratification):
Reduce your X={x in R^n: p_i(x)=0, all i} which does not
change the space.
For Reg(X) you are done and for Sing(X) you know it by
induction on dim(X).
Not correct is ... dim <= cardinality(I).
What you possibly do not want to have for optimization is: if
you Ôclose
that sub-manifolds (by coming from interior points) they may
become
singular
again (look at Neils parabola x1^3=x2^2 which homeomorph is
just R^1 ...
you loose a lot of Ôglobal informations of 
embedding in R^n
by that
stratification) & there should be other 
Ôbad examples
(Whitneys
umbrella?).
--
use mail for mail not nonail
===
Subject: Re: algebraic sets
Epigone-thread: derlyoltrand
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>topology and that the following question comes from
optimization:
>Let p_i be a finite set of k relatively prime polynomials,
each of
>which maps R^n into R. I am interested in describing the
boundary of
>the set
>A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k}
>as a union of manifolds. Can I do this?
In a word, yes. This comes under the heading stratification
theory
where it is a well-known theorem that a semi-algebraic set
admits
a Whitney stratification, i.e., admits a partition into smooth
connected manifolds, called strata, which fit together nicely,
meaning roughly that if M and N are strata and M lies in the
boundary of N, then M admits a tubular neighborhood in N which
projects onto M in a locally trivial way. In the
semi-algebraic
case, the strata may be taken to be real analytic manifolds.
One useful (but densely written) reference for this general
area is
Masahiro Shiota, Geometry of Subanalytic and Semialgebraic
Sets
(Birkhauser, 1997).
See especially section I.2 on Whitney stratifications for
semi-algebraic sets, and the existence of the so-called
canonical
stratification.
A great deal of this has been generalized to other settings,
within
axiomatic frameworks with important ties to model theory.
Besides
Shiotas axiomatics, one should mention o-minimal 
structures.
An accessible reference is
van den Dries, Tame Topology and O-minimal Structures
(London Math. Soc. Lecture Note Series 248, 1998)
where so-called cell decompositions play an important role for
stratification theory. (A key point made in this work is that
much of the structure theory of semi-algebraic sets ßows from
the Tarski-Seidenberg theorem.)
Todd Trimble
===
Subject: Re: algebraic sets
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> and that the following question comes from optimization:
> Let p_i be a finite set of k relatively prime polynomials,
each of which
> maps R^n into R. I am interested in describing the boundary
of the set
> A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k}
> as a union of manifolds. Can I do this?
yes, I believe so. It can be described as a stratified set,
which is
what you are aiming at.
> Specifically, I would like to say
> the following:
> Let I be an index set and let
> A_I = {x in R^N : p_i(x) = 0 for i in I and p_i(x) > 0 for
i notin
> I}.
> Then A_I is the finite union of manifolds of dim <=
cardinality(I).
[Later corrected to co-dimension rather than dimension]
Well, the second part of your condition does not seem to add
anything,
other than restricting to the appropriate open set in the
ambient space.
The first part should be the important part of the statement.
> Could someone point out some results for me in this
direction and where
> I might find them written down?
Look for references to stratified sets. I remember reading
about this
years ago, but I dont have any references handy. I used 
this
idea at the
time to deal with a max/min problem on a set defined by a
collection of
(real) polynomials. The tangent space at each point is still
determined
by the various gradients of the defining functions, for each
of the strata.
--
David L. Johnson
__o | Arguing with an engineer is like mud wrestling with a
pig... You
_`(,_ | soon find out the pig likes it!
(_)/ (_) |
===
Subject: Re: algebraic sets
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Sorry, a few clarifications. First, when I say relatively
prime, I
mean that the polynomials do not share a commom factor. (I
dont know
if this definition is standard.) Second, I want the set A_I to
be
composed of manifolds with CO-dimension at least card(I) or,
equivalently, of dimension at most N-card(I).
> topology and that the following question comes from
optimization:
> Let p_i be a finite set of k relatively prime polynomials,
each of
> which maps R^n into R. I am interested in describing the
boundary of
> the set
> A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k}
> as a union of manifolds. Can I do this? Specifically, I
would like to
> say the following:
> Let I be an index set and let
> A_I = {x in R^N : p_i(x) = 0 for i in I and p_i(x) > 0 for
i notin
> I}.
> Then A_I is the finite union of manifolds of dim <=
cardinality(I).
> I have been told that a semialgebraic subset of R^n (like
A) can be
> written as a union of manifolds of decreasing dimension,
but I have
> not seen this result stated anywhere. I have seen a result
that says
> a semialgebraic set is composed of a disjoint union of
connected
> semialgebraic sets, but these are not quite what I want.
> Could someone point out some results for me in this
direction and
> where I might find them written down? Even results requiring
> additional hypotheses would be welcome.
===
Subject: Re: algebraic sets
Epigone-thread: derlyoltrand
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>Sorry, a few clarifications. First, when I say relatively
prime, I
>mean that the polynomials do not share a commom factor. (I
dont
know
>if this definition is standard.) Second, I want the set A_I
to be
>composed of manifolds with CO-dimension at least card(I) or,
>equivalently, of dimension at most N-card(I).
>> topology and that the following question comes from
optimization:
>> Let p_i be a finite set of k relatively prime polynomials,
each of
>> which maps R^n into R. I am interested in describing the
boundary
of
>> the set
>> A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k}
so we talk about the set
B= {x in R^n : p_i(x) = 0 for all i, 1 <= i <= k}
There are two problems you run into here.
The first one is that you are only interested in real points.
A large part of algebraic geometry is known only over
algebraically
closed fields. I cant comment on you situation 
without knowing
something on your polynomials. However, generally speaking
you need
to find some topological invariant to do this for you. You can
see
spectacular examples in:
1) Mikhalkins paper in the Annals math.AG/0010018
2) B. Gross and J. Harris, Real algebraic curves, Ann. scient.
&#769;Ec. Norm. Sup. 14 (1981), 157182, MR 83a:14028, Zbl
0533.1401
The second problem is that the dimension of an intersection
is the
expected dimension only generically.
Example: the set {(x,y)| x=y=x+y=0}.
What you want to look at to analyse this problem is the excess
intersection formula.
memoirs notebook.
HTH
D.L.
>> as a union of manifolds. Can I do this? Specifically, I
would like
to
>> say the following:
>> Let I be an index set and let
>> A_I = {x in R^N : p_i(x) = 0 for i in I and p_i(x) > 0 for
i
notin
>> I}.
>> Then A_I is the finite union of manifolds of dim <=
cardinality(I).
>> I have been told that a semialgebraic subset of R^n (like
A) can be
>> written as a union of manifolds of decreasing dimension,
but I have
>> not seen this result stated anywhere. I have seen a result
that
says
>> a semialgebraic set is composed of a disjoint union of
connected
>> semialgebraic sets, but these are not quite what I want.
>> Could someone point out some results for me in this
direction and
>> where I might find them written down? Even results 
requiring
>> additional hypotheses would be welcome.
===
Subject: Open problem
Epigone-thread: vayspunquem
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Let X be a normed space and let M denote the set of all unit
bounded
linear functionals on X.
I would like to know if for every a and b in X, there exists
f and g
in M such that the modulus of
f(a)g(b)+f(b)g(a)
is >= to 1.
===
Subject: Re: Open problem
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> for every a and b in X, there exists f and g
> in M such that the modulus of
> f(a)g(b)+f(b)g(a)
> is >= to 1.
Unless I misunderstood you completely, that fails for a=0.
===
Subject: Re: Open problem
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>Let X be a normed space and let M denote the set of all unit
bounded
>linear functionals on X.
>I would like to know if for every a and b in X, there exists
f and g
>in M such that the modulus of
>f(a)g(b)+f(b)g(a)
>is >= to 1.
I assume you mean to have ||a|| = ||b|| = 1, since without
some such
condition the question is trivial. Then the answer is no.
We may assume wlog that X is spanned by a and b.
Note that f(a)g(b) + f(b)g(a) = g(f(a)b+f(b)a). This < 1 for
all g in M iff ||f(a)b + f(b)a|| < 1. So an equivalent
question is:
Does there exist f in M such that ||f(a)b + f(b)a|| >= 1?
Consider X = R^2 with the infinity norm, ||[x,y]||_infty =
max(|x|,|y|).
The dual space is identified as R^2 with the 1-norm
||[x,y]||_1 = |x|+|y|.
Let a = [sqrt(2)-1,1] and b = [1,1-sqrt(2)].
Then if f = [x,y] with |x|+|y|<=1,
f(a) b + f(b) a = 2 (sqrt(2)-1) [x+y, x-y]
and so ||f(a) b + f(b) a||_infinity <= 2 (sqrt(2)-1) (|x|+|y|)
< 1.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: have you seen this sequence?
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Essentially, Im wondering whether the following sequence,
which I
encountered by chance, seems sufficiently interesting to be
added to
Sloanes encyclopaedia, and incidentally whether someone has
met it
before or has anything worth while to say about it.
It is probably easier to show how it is constructed rather
than to try
to describe it in words (here for the first 30 terms):
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3
1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3
1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 1 2 1 2 3 3
1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 6 6 1 2 6 3
start with a sequence of 1s on the first line; 
then replace
every
other 1 by a 2 to form the second line; then replace every
third 1 by
a 3 and every third 2 by a 3 to form the third line; then
replace
every fourth 1, every fourth 2 and every fourth 3 by a 4; and
so on.
The sequence I speak of is the limit of these: it is obvious
that,
whatever the (finite) number of terms that are to be produced,
after a
certain number of lines the terms in question arent 
affected
any
more.
<URL: http://www.eleves.ens.fr:8080/home/madore/.misc/seq.png
> gives
an idea of what the sequence looks like for a largish number
of terms
(2520, chosen because it is the LCM of the integers from 1
through
10). Experimentally it seems that the supremum of the n first
terms
of the sequence grows like sqrt(4n/3). One could also ask for
the
density of 1s among the n first term, which can 
probably be
computed
since the 1s are selected from a kind of sieving process
(remove
every other 1, then every third, then every fourth, and so
on),
remeniscent to that for primes or lucky numbers.
So, is this a worthy candidate for inclusion in Sloane?
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
===
Subject: Re: have you seen this sequence?
Epigone-thread: thoochahprou
Originator: israel@math.ubc.ca (Robert Israel)
Experimentally it seems that the supremum of the n first terms
of the sequence grows like sqrt(4n/3).
Seems true. Let (a(n))_{n>=1} denotes the considered sequence
and let
M(n) be the supremum sequence :
M(n)=Max{a(k):1<=k<=n}
Then I believe there is this exact rule:
M(3k^2)=2k for k>=1
Max{a(k):1<=k<=3}=2
Max{a(k):1<=k<=12}=4
Max{a(k):1<=k<=27}=6
Max{a(k):1<=k<=48}=8
...
Does this inequality holds ?
abs(M(n)-sqrt(4n/3))<1
Regarding the density of 1s :
letting b(n)=#{1<=k<=n : a(k)=1} I observed :
0<=M(n)-b(n)<=1
I was wrong regarding the sequence of indices of occurrences
of 1s in
(a(n)) : it is not the Favius sieve but it looks like.
This is truly an interesting sequence and I hope someone will
like to
explore it further and prove above facts or other facts.
B. Cloitre
===
Subject: Re: have you seen this sequence?
Epigone-thread: thoochahprou
Originator: israel@math.ubc.ca (Robert Israel)
I found a related sequence but with a simpler rule :
http://www.research.att.com/projects/OEIS?Anum=A055881
There is a comment in order to construct A055881 which can be
rewritten as follows :
to construct A055881 sequence start with a line of 1s :
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ....
second step : replace 1 by 2 every 2 1s :
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
third step : replace 2 by 3 every 3 2s :
1 2 1 2 1 3 1 2 1 2 1 3 1 2 1 2
etc.
Your sequence is also a k! periodic sequence at k-th step.
But the
structure is very reach and more complicate. I cant have
access to
your link. Could you provide 1000 first terms in my email?
If Im not wrong the sequence of Occurrence of 1s in your
sequence
(1,3,7,13,19,27,) is given by the Flavius Josephus sieve :
http://www.research.att.com/projects/OEIS?Anum=A000960
Should be interesting to know what is the sequence of
occurrences of
2s (2,4,8,14,20,28,) divided by 2.
B. Cloitre
>Essentially, Im wondering whether the following sequence,
which I
>encountered by chance, seems sufficiently interesting to be
added to
>Sloanes encyclopaedia, and incidentally whether someone 
has
met it
>before or has anything worth while to say about it.
>It is probably easier to show how it is constructed rather
than to
try
>to describe it in words (here for the first 30 terms):
>1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
>1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
>1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3
>1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3
>1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 1 2 1 2 3 3
>1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 6 6 1 2 6 3
>start with a sequence of 1s on the first line; 
then replace
every
>other 1 by a 2 to form the second line; then replace every
third 1 by
>a 3 and every third 2 by a 3 to form the third line; then
replace
>every fourth 1, every fourth 2 and every fourth 3 by a 4;
and so on.
>The sequence I speak of is the limit of these: it is obvious
that,
>whatever the (finite) number of terms that are to be
produced, after
a
>certain number of lines the terms in question arent
affected any
>more.
><URL: <a
href=http://www.eleves.ens.fr:8080/home/madore/.misc/seq.png>
http://www.
eleves.ens.fr:8080/home/madore/.misc/seq.png</a gives
>an idea of what the sequence looks like for a largish number
of terms
>(2520, chosen because it is the LCM of the integers from 1
through
>10). Experimentally it seems that the supremum of the n first
terms
>of the sequence grows like sqrt(4n/3). One could also ask
for the
>density of 1s among the n first term, which 
can probably be
computed
>since the 1s are selected from a kind of sieving process
(remove
>every other 1, then every third, then every fourth, and so
on),
>remeniscent to that for primes or lucky numbers.
>So, is this a worthy candidate for inclusion in Sloane?
>--
> David A. Madore
> (david.madore@ens.fr,
> <a
href=http://www.dma.ens.fr/~madore/>http://www.dma.ens.fr/~
madore/</a>
)
===
Subject: Re: have you seen this sequence?
Epigone-thread: thoochahprou
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
As you said this looks like lucky numbers but I never seen
it. Clearly
it is an interesting sequence. You should submit it directly
via the
automatic form avalaible at :
http://www.research.att.com/~njas/sequences/Submit.html
Neil Sloane will decide and in this case Im certain he will
accept
your sequence. Check first whether it is not already in the
database
(check it with superseeker too). A related sequence could be
the
first occurence of n which seems begin :
1,2,5,9,15,...
The behaviour of the supremum is interesting. It looks like
this one :
http://www.research.att.com/projects/OEIS?Anum=A080096
B. Cloitre