groupoids - Double Category of Topological Stacks

There are two equivalent ways of describing topological stacks.

1. One is the "stacky" definition, that is, a topological stack is a stack $mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if you'd like) equipped with the topology generated by open covers, such that $mathbb{X}$ admits an atlas (representable epimorphism) $X to mathbb{X}$ from a topological space. This is equivalent to saying that $mathbb{X}$ is 2-iso to the stackification of a pseudofunctor $Hom(blank,G)$ for some topological groupoid $G$. Topological stacks are then the full sub-2-category all stacks on $Top$ consisting of those stacks with an atlas.




2. One is a "groupoidy" definition. The bicategory $BunGpd$ has topological groupoids as objects, and a morphism $H to G$ is a principal $G$-bundle over $H$ (and biequivariant maps of as 2-cells). This bicategory is equivalent to that of topological stacks.

However, 2.) can be naturally strengthened to a weak double category by declaring vertical morphisms to be continuous functors and horizontal arrows to be principal bundles.

Now, I have a preference to working in 1.) as the stacky-language is quite useful. However, 2.) manifestly has "more structure". My question is, is there a way to "beef up" topological stacks into a weak double category in a natural way? I'm not really satisfied with taking the objects to be topological stacks with a preferred atlas and declaring the vertical arrows to be maps which factor through these atlases.

tag removed - The role of the mean value theorem (MVT) in first-year calculus.

My view is that there are essentially two strands in a first calculus course.

The first is not really concerned with a rigorous presentation; rather it tries to get the main ideas, their interrelations, and uses across.

The second is concerned with the technicalities, showing how abstract mathematics can lead to very useful, interesting, and important results.

This means that we are really working with two different definitions of the derivative. The first is to draw the tangent line and measure its slope. The second is to compute a certain limit. To be sure, the limit is motivated by the tangent approach, but no attempt is ever made to show that the two approaches give the same answer (indeed this can't be proved using the usual definitions, since ultimately the definition of tangent line is based on the derivative).

The MVT is the basis for all proofs that geometric intuition about slopes of tangent lines holds for the limit definition. That is, the metamathematical content of the MVT is that the intuition definition matches the formal definition.

When I realized this, I decided that this point was so subtle that I either have to make a big point of explaining the question or else drop it. This choice varies from class to class.

EDIT: You use the MVT to show that positive derivative corresponds to increasing function.
This is obvious from the intuitive point of view, but not from the formal point of view.

complex multiplication - Class Field Theory for Imaginary Quadratic Fields

Here is a case where it is non-Abelian. I use $K$ of class number 3. If I use the Gross curve, it is Abelian. If I twist in $Q(sqrt{-15})$, it is Abelian for every one I tried, maybe because it is one class per genus. My comments are not from an expert.

> K<s>:=QuadraticField(-23);                                                    
> jinv:=jInvariant((1+Sqrt(RealField(200)!-23))/2);                             
> jrel:=PowerRelation(jinv,3 : Al:="LLL");                                      
> Kj<j>:=ext<K|jrel>;                                                           
> E:=EllipticCurve([-3*j/(j-1728),-2*j/(j-1728)]);
> HasComplexMultiplication(E);
true -23                 
> c4, c6 := Explode(cInvariants(E)); // random twist with this j                
> f:=Polynomial([-c6/864,-c4/48,0,1]);                                          
> poly:=DivisionPolynomial(E,3); // Linear x Linear x Quadratic                 
> R:=Roots(poly);                                                               
> Kj2:=ext<Kj|Polynomial([-Evaluate(f,R[1][1]),0,1])>;                          
> KK:=ext<Kj2|Polynomial([-Evaluate(f,R[2][1]),0,1])>;                          
> assert #DivisionPoints(ChangeRing(E,KK)!0,3) eq 3^2; // all E[3] here         
> f:=Factorization(ChangeRing(DefiningPolynomial(AbsoluteField(KK)),K))[1][1];  
> GaloisGroup(f); /* not immediate to compute */                                
Permutation group acting on a set of cardinality 12
Order = 48 = 2^4 * 3
> IsAbelian($1);                                                                
false

This group has $A_4$ and $Z_2^4$ as normal subgroups, but I don't know it's name if any.

PS. 5-torsion is too long to compute most often.

gt.geometric topology - Orbifold fundamental group in terms of loops?

There is also a natural interpretation of the orbifold fundamental group in terms of loops, using an extended version of a Wirtinger presentation. Let's start out with the closed case and mention the cusped case at the end.

As a warm-up, consider an orbifold that has singular locus a link with underlying space $S^3$. To compute the fundamental group of the orbifold can be computed first from the Wirtinger presentation of the link and then by introducing torsion relations for each meridian for example if the cone angle along the link is $2pi/3$ everywhere, then each meridian $mu_i$ should have the relation $mu_i^3$. A loop around a link component does not bound a (smooth immersed) disk in the orbifold, instead it bounds a disk with a cone point of order 3. However $mu_i^3$ does bound a disk in the orbifold, and is trivial.

To make this interpretation general, we need to consider 3-orbifolds more generally. A geometric 3-orbifold is really a trivalent graph with edges decorated by torsion (or cone angles, depending on taste) embedded in a 3-manifold, and so the underlying manifold, the embedding, and the trivalent points need to be accounted for.

In reverse order, the trivalent points are introduce relations abc=1 (compare to finite subgroups of SO(3) which are actually the isotropy subgroups that fix these types of points). The next two conditions really have to be considered together. Really what needs to be computed is a Wirtinger presentation of the complement of a trivalent graph in the underlying space. Then quotient out by the torsion relations and relations coming from the trivalent points.

For cusped (geometric) manifolds, there are extra ways to decorate the graph, namely there can be trivalent points where the orders of torsion along edges incident to the these vertices are (2,4,4), (2,3,6) or (3,3,3) and there could be 4-valent points where the torsion orders of the edges are (2,2,2,2).

It should be pointed out that such a computation can also be done by Damian Heard's ORB, which is extremely useful if a large example needs to be considered.

ca.analysis and odes - Fourier transform of Analytic Functions

Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.

I'm trying to construct a function according to some conditions in the frequency domain of the Fourier transformation. I want the function to be analytic and real when I transform it back to the time domain. The Fourier transformation of $f$ has of course some symmetry criteria to make $f$ real. But what about the Analytic property. As an analytic function imply some convergent power series expansion, and the Fourier transform of a polynomial is a sum of derivatives of Delta functions, I assume that there is a corresponding criteria of the Fourier transformation.

So the question is:
If a function $f:mathbb{R}rightarrow mathbb{R}$ is assumed to be analytic, what is the corresponding criteria for the Fourier transform of the function $mathcal{F}[f] (k)$?

Edit: what I am trying to construct is probability distribution with the following condition

$f(x/mu)/mu=frac{2}{3} f(x) + frac{1}{3} (fast f)(x)quad$
where $ast$ mark the convolution, and $mu=frac{4}{3}$. $f$ is positive and real for $xin [0,infty)$

Taking the fourier transformation make the condition simpler:

$tilde f(mu k) = frac{2}{3}tilde f(k) + frac{1}{3}tilde f^2(k)$

So my problem is to construct $f$ (I am in particular interested in the tail behavior) and I try to use the properties of $tilde f$. I posted a similar problem a while ago (see here). Julián Aguirre answered how to construct $tilde f$ if it is analytic. But the inverse transformation of the power expansion is an infinite sum of derivatives of Delta functions, and is of little help.

ca.analysis and odes - Splines, harmonic analysis, singular integrals.

Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)

One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiable functions: $varphi: mathbb{R} rightarrow mathbb{R}$ such that for all natural numbers $n$ and $r$,

$lim_{xrightarrowpminfty} |x^n varphi^{(r)}(x)|$

What I would like to know is why is necessary or important for test functions to decay rapidly in this manner? i.e. faster than powers of polynomials. I'd appreciate an explanation of the intuition behind this statement and if possible a simple example.

Thanks.

EDIT: the OP is actually interested in a particular 1994 paper on "Spatial Statistics" by
Kent and Mardia,
1994 Link between kriging and thin plate splines (with J. T. Kent). In Probability, Statistics and Optimization (F. P. Kelly ed.). Wiley, New York, pp 325-339.

Both are in Statistics at Leeds,

http://www.amsta.leeds.ac.uk/~sta6kvm/

http://www.maths.leeds.ac.uk/~john/

http://www.amsta.leeds.ac.uk/~sta6kvm/SpatialStatistics.html

Scanned article:
http://www.gigasize.com/get.php?d=90wl2lgf49c

FROM THE OP: Here is motivation for my question: I'm trying to understand a paper that replaces an integral $$int f(omega) domega$$ with $$int frac{|omega|^{2p + 2}}{ (1 + |omega|^2)^{p+1}} ; f(omega) ; domega$$ where $p ge 0$ ($p = -1$ yields to the unintegrable expression) because $f(omega)$ contains a singularity at the origin i.e. is of the form $frac{1}{omega^2}.$

LATER, ALSO FROM THE OP:
I understand some parts of the paper but not all of it. For example, I am unable to justify the equations (2.5) and (2.7). Why do they take these forms and not some other form?

at.algebraic topology - Hopf fibration inside the retraction of R^4 minus line -> S^2?

First, note that $mathbb R^4 setminus mathbb R simeq S^2times mathbb R^2$.
So you are asking for an immersion $S^3to S^2times mathbb R^2$
representing the generator $eta$ of $pi_3(S^2times mathbb R^2)=pi_3(S^2)=mathbb Z$.

I'm guessing that your immersion doens't exist, and that you need to consider maps $S^3to S^2times mathbb R^n$ with larger $n$, in order to represent $eta$ by an immersion.

But if you are willing to go a little bit up in dimension, and consider maps $S^3to S^2times mathbb R^4$, then you can even find an embedding representing $eta$.
It is given by $(H,I):S^3to S^2times mathbb R^4$, where $H:S^3to S^2$ is the Hopf map, and $I: S^3 rightarrow {mathbb R}^4$ is the standard inclusion.

nt.number theory - Polynomial bijection from $mathbb Qtimesmathbb Q$ to $mathbb Q$?

Jonas Meyer's answer:

Quote from arxiv.org/abs/0902.3961, Bjorn Poonen, Feb. 2009: "Harvey Friedman asked whether there exists a polynomial $f(x,y)in Q[x,y]$ such that the induced map $Q × Qto Q$ is injective. Heuristics suggest that most sufficiently complicated polynomials should do the trick. Don Zagier has speculated that a polynomial as simple as $x^7+3y^7$ might already be an example. But it seems very difficult to prove that any polynomial works. Our theorem gives a positive answer conditional on a small part of a well-known conjecture." – Jonas Meyer

quantum groups - Classical Calculi as Universal Quotients

In the classical case, if $Omega(A)$ is the kernel of the multiplication map $m:Aotimes Ato A$, then—since $A$ is commutative, so that $m$ is not only a map of $A$-bimodules but also a morphism of $k$-algebras,—it turns out that $Omega(A)$ is an ideal of $Aotimes A$, not only a sub-$A$-bimodule. In particular, you can compute its square $(Omega(A))^2$. Then the classical module of Kähler differentials $Omega^1_{A/k}$ is the quotient $Omega(A)/Omega(A)^2$.

(This is the construction used by Grothendieck in EGA IV, for example)

advice - How do you make a good math research poster for a non-mathematical audience?

I agree, somewhat, with Jon Yard. Make individual "pages" with LaTeX and/or beamer. Save the pages as pdf, and arrange using Illustrator. You can preassign the page size in illustrator, and take the file to Kinkos to have it made. Or find out where the cool color copier is in the chemistry, biology, physics, or administrative department is. The cool copier will render a single page glossy poster.

Since the author of the question has very cool graphics of his work, many of which work better in color, these should be included.

If you don't want to use beamer, you can import eps files (and I think, pdf) into xfig. I have seen amazing xfig posters.

ac.commutative algebra - Lifting results from smooth maps to essentially smooth maps.

Recall that a morphism of rings $Rto S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $Rto S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general technique to lift results from the smooth case to the essentially smooth case?

Edit: According to Mel, every essentially smooth morphism is a localization of a smooth morphism. However, this direction is much more involved than the other direction, which is immediate from the definitions. Anyway, this would be the answer to the question.

ct.category theory - Topological homotopy category as derived category

In the Introduction of his
Derived Categories for the working mathematician
Richard Thomas mentions the following theorem of Whitehead.

Suppose that $X,Y$ are simplicial complexes, then
the underlying topological spaces |X|, |Y| (both simply connected) are homotopy equivalent
if and only if there are maps of simplicial complexes
$X leftarrow Z rightarrow X$ inducing quasi-isomorphims
$C_X leftarrow C_Z rightarrow C_Y$ of simlicial-chain complexes.

This suggest that there is a relation between the homotopy category of
simplicial complexes and the derived cagtegory of abelian groups.
E.g. the functor

$ Ho( SimplicialSpaces ) longrightarrow mathcal{D}(Mod-mathbb{Z}), X mapsto C_X$.

induces an injection on the level of isomorphy classs when restricted to simply-connected
spaces.

Does this functor have any good properties?
What about homomorphisms (fullness/faithfulness)?
How does it help to study topological spaces?

More generally one can ask:
What kind of topological homotopy categories are related to "algebraic" derived-categories?

The above example can be interpreted as $C_x = Rpi_* (mathbb{Z}_X)$
where $pi: X rightarrow {pt}$ is the projetion to a point, and $mathbb{Z}-Mod$ occures as abelian sheaves on $pt$.
One could try to generalize using sheaved spaces $X rightarrow B$ over a scheme $B$.

This seem to be quiet obvious questions. However I have seen nobody elaborating
on this so far. Or maybe I am missing a link to something well known?

ca.analysis and odes - l^p space inequality related to compressed sensing

I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).

He makes the sparsity assumption on $theta in mathbb{R}^m$ that for some $0<p<2$ and $R>0$ we have $|theta|_pleq R$. Then if $theta_N$ denotes $theta$ with everything except the $N$ largest coefficients set to $0$ he claims that $| theta-theta_N |_2 leq zeta_{2,p} cdot | theta |_p cdot (N+1)^{1/2-1/p}$ for $N=0,1,2,ldots$ where $zeta_{2,p}$ depends only on $p$.

I've tried writing out the definitions of various things. I've noticed that the $N$th largest coefficient must satisfy $midtheta_imid leq RN^{-1/p}$ but I can't figure out how the result above follows.

I'm also having some difficulty thinking about $ell^p$ spaces with $0<p<1$, in particular knowing what results from the $p>1$ theory apply. Does anyone know some good notes or a book that covers this?

soft question - Dimension Leaps

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" dimensions, with occasional "dimension leaps" marking the boundary from one type of behaviour to another. Sometimes there is just one dimension that has is markedly different from others. Examples of this behaviour can be good provokers of the "That's so weird, why does that happen?" reaction that can get people hooked on mathematics. I want to know examples of this behaviour.

My instinct would be that as "dimension" increases, there's more room for strange behaviour so I'm more surprised when the opposite happens. But I don't want to limit answers so jumps where things get remarkably more different at a certain point are also perfectly valid.

ct.category theory - Exactness of filtered colimits

Here's a dumb counterexample. If C is an abelian category, so is C^op. In C^op, filtered colimits are filtered limits in C. And, of course, there are many examples of abelian categories (such as abelian groups) where filtered limits aren't exact.

Of course, your question is really: when is an abelian category C sufficiently close to Set, so that we can ratchet up the fact that filtered colimits are exact in Set to a proof for C.

Any category of sheaves of abelian groups on a space (or on a Grothendieck topos) will have exact filtered colimits, for instance.

In what sense are fields an algebraic theory?

Fields are not algebraic. An algebraic theory, for example, has free objects: there are free rings, free groups, a free monoids. The free functor is left adjoint to the forgetful functor to sets (okay, I'm talking about models in Set). There are, though, no free fields.

One can extend one's idea of an "algebraic theory" to an "essentially algebraic theory" in which partially defined operations are allowed (it's not clear to me that fields satisfy those since you need to specify the domain in terms of other operations whereas it seems that one can only specify the domain of the inverse as the complement of such a subset). Or, (maybe, but I doubt it), one could define a field as a Z₂-graded algebraic theory where 0 is in degree 0 and everything else is in degree 1. Here, a grading should be regarded simply as a labelling system.

Alternatively, one can talk about meadows. Meadows are algebraic theories which are modified versions of fields. Instead of multiplicative inverses, there is a unary operation ι:M → M which satisfies the identity xι(x)x = x. Defining ι(x) = x^-1 for non-zero x, and ι(0) = 0 turns any field into a meadow. The relationship between meadows and fields is quite strong.

An arXiv search throws up 68 references (at time of writing; for some reason google doesn't turn up anything particularly relevant, even when combined with the word "field"). One prominent name is that of Jan Bergstra.

ag.algebraic geometry - Hodge numbers of compactifications

Let $E$ and $F$ be two elliptic curves and let the involution $sigma$ act on
$Etimes F$ by $sigma(e,f)=(-e,f+alpha)$, $alpha$ is an element of order two
of $F$. Finally let $overline{X}=(Etimes F)/sigma$ (this is a so called hyperelliptic
surface). We have an inclusion $F':=0times F/langlealpharanglesubseteq S$ and
put $X:=overline{X}setminus F'$. Then $X$ is Hodge-Tate but all other good
compactifications of $X$ are obtained by blowing ups and downs of $overline{X}$
which means that you can never get rid of $F'$ (alternatively any good
compactification $X'$ has $H^1(X')=H^1(X)$ and you need something non-Hodge-Tate
at the boundary to kill that off).

Addendum: This example is all wrong it took care of $H^3(X)$ but not (the more interesting) $H^1(X)$. At the moment I am less sure than I was that the answer to 1) is no.

As for 2) you can just look at $mathbb A^3subseteqmathbb P^3$ which is a good
Hodge-Tate compactification with $mathbb P^2$ as divisor at infinity and then
blow up something non-Hodge-Tate in $mathbb P^2$. This gives a good
compactification with two components one of which (the exceptional divisor for
the blowing up) is non-Hodge-Tate (as is the intersection of these two
divisors).

reference request - Learning statistical mechanics for non-particle phenomena

I'm interested in various areas of complex systems, and I often come across articles like these:

http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106096v1.pdf

http://arxiv.org/abs/cond-mat/9804180

The main points are accessible in each (much less so the 2nd one though), but I'd like to be able to understand this sort of writing deeply, or even be able to do it myself.

What sort of studies would I need to undertake? Would a standard thermal/statistical physics class do it, or do I need something more drastic? Are there any resources along the lines "statistical physics for the social scientist" that are still rigorous and high-level?

(There's a question about "statistical physics for the mathematician", but this is almost exactly the opposite of what I need, funnily enough").

ag.algebraic geometry - Existence of zero cycles of degree one vs existence of rational points

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$,
with connected geometric stabilizers.
Assume that $X$ has a zero cycle of degree 1.
If $k$ is a either a $p$-adic field or a number field,
then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer)
and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof.
If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0.
If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point.
If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a
zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because
$k_v$ is isomorphic to $mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that both assumptions of the theorem,
namely that geometric stabilizers are connected
and that the base field $k$ is either a $p$-adic field or a number field, are important.

Mathieu Florence in the paper Zéro-cycles de degré un sur les espaces homogènes, Int. Math. Res. Not. 2004, no. 54, 2897–2914,
http://alg-geo.epfl.ch/~florence/esphomog.pdf,
constructed homogeneous spaces $X$ over $p$-adic and number fields with non-connected (finite) geometric stabilizers,
such that $X$ has a zero cycle of degree 1, but neither $X$ nor any smooth compactification of $X$ has rational points.

Parimala in the paper Homogeneous varieties — zero cycles of degree one versus rational points,
Asian J. Math. 9 (2005), 251–256, see the link in Artie's answer,
constructed a projective homogeneous space $X$ (hence with connected geometric stabilizers)
over the Laurent series field over a $p$-adic field,
such that again $X$ has a zero cycle of degree 1, but no rational points.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if
a principal homogeneous space $X$ of a connected linear group $G$
over a field $k$ of virtual cohomological dimension $le 2$ has a zero cycle of degree 1,
and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.

soft question - What subfields of mathematics better lend themselves to visualization?

To start with something of an anti-answer, when I took courses in algebra (not linear algebra - I mean groups, rings, modules etc.) or representation theory as an undergraduate, I found it practically impossible to get anywhere by trying to visualize what was going on. I suppose, concurrent with other answers, the important thing is that I understood the material some other way.

By contrast, and as you say, some parts of combinatorics are certainly very visual. And (obviously?) topology (not so much first-course point-set stuff, but the real stuff) and differential (at least) geometry can both be very visual subjects. It can be a lot of fun trying to find ways to use geometric inituition to attack something that is ostensibly out of reach visually (e.g. in 4 dimensions or something not embedded in $R^3$ etc.)

At the moment I'm interested in geometric analysis, where I have come across some of the most pleasingly visual things yet.

lo.logic - Cardinality: Why is there no "ℵ½"?

The point is that without the Axiom of Choice, cardinalities are not linearly ordered, and it is possible under $neg AC$ that there are additional cardinalities to the side of the $aleph$'s. Thus, the issues is not additional cardinalities between $aleph_0$ and $aleph_1$, but rather additional cardinalities to the side, incomparable with these cardinalities.

Let me explain. We say that two sets $A$ and $B$ are equinumerous or have the same cardinality if there is a bijection $f:Ato B$. We say that $A$ has smaller-or-equal cardinality than $B$ if there is an injection $f:Ato B$. It is provable (without AC) that $A$ and $B$ have the same cardinality if and only if each is smaller-or-equal to the other (this is the Cantor-Shroeder-Bernstein theorem).

Under AC, every set is bijective with an ordinal, and so we may use these ordinals to select canonical representatives from the equinumerosity classes. Thus, under AC, the $aleph_alpha$'s form all of the possible infinite cardinalities.

But when AC fails, the cardinalities are not linearly ordered (the linearity of cardinalities is equivalent to AC). Let me mention a few examples:

• It is a consequence of the Axiom of Determinacy that there is no $omega_1$ sequence of distinct reals. Thus, in any model of AD, the cardinality of the reals is uncountable, but incomparable to $aleph_1$. Thus, in such a model, it is no longer correct to say that $aleph_1$ is the smallest uncountable cardinal. One should say instead that $aleph_1$ is the smallest uncountable well-orderable cardinal.




• A more extreme example is provided by the Dedekind finite infinite sets. These sets are not finite, but also not bijective with any proper subset. It follows that they can have no countably infinite subsets. In particular, they are uncountable sets, but their cardinality is incomparable with $omega$. Thus, in a model of $neg AC$ having a Dedekind finite infinite set, it is no longer correct to say that $aleph_0$ is the smallest infinite cardinal.

Thus, the issue isn't whether there is something between $aleph_0$ and $aleph_1$, but rather, whether there are additional cardinalities to the side of these cardinalities.

computational complexity - How hard is it to solve SAT if the promise is that it has an odd number of solutions?

UPDATED (after the question got changed): All right... your new questions are open questions in complexity theory, as far as I know. There has been some work on derandomizing the Valiant-Vazirani theorem, under reasonable hardness assumptions. A reference:

Adam Klivans, Dieter van Melkebeek: Graph Nonisomorphism Has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses. SIAM J. Comput. 31(5): 1501-1526 (2002)

So, under some plausible circuit lower bound assumptions, there is a deterministic polynomial time reduction from SAT to USAT. This would give a deterministic reduction from SAT to "Odd-or-Zero-SAT" as well as a deterministic reduction from "Odd-or-Zero-SAT" to USAT.

--

(UPDATE: Some stuff got deleted here, as it is no longer relevant to the current version of the question)

--

Despite all this, there is an extremely related problem that should be of interest to you. The problem "Parity-SAT" (often written as $oplus SAT$ in the literature) is the problem of determining whether or not a given Boolean formula has an odd number of assignments. It is well-studied, and is complete for the class $oplus P$ which contains all languages of the form {$x ~|~ text{there are an odd number of accepting computation paths in}~N(x)$}, where $N$ is a nondeterministic polynomial time machine.

Now, by the Valiant-Vazirani Theorem (which I suspect you know, since you mentioned USAT) we have
$$SAT ~leq_R~ oplus SAT,$$ where $leq_R$ denotes a randomized polytime reduction. Hence $oplus SAT$ is "hard" under randomized reductions.
It is not known if $NP = oplus P$, or $UP = oplus P$. But, as the Valiant-Vazirani Theorem suggests, you can do a hell of a lot with randomized polynomial time and an oracle for $oplus P$. We are still figuring out everything you can do. Toda's Theorem tells us that the entire polynomial time hierarchy is in $BPP^{oplus P}$. It could be that even $PSPACE$ is in $BPP^{oplus P}$. Another interesting fact due to Papadimitriou and Zachos is that $oplus P^{oplus P} = oplus P$. That is, an oracle for $oplus P$ is superfluous if you already have the power of $oplus P$. This follows from the observation that the XOR of a bunch of XORs is still an XOR. (Similarly, $P^{P} = P$, but it is not known or believed that $NP^{NP} = NP$.)

ag.algebraic geometry - Is it possible to classify all Weil cohomologies?

If we believe in the standard conjectures (or something similar) so that the category of motives is Tannakian, then a Weil cohomology theory is just a fibre functor and as such twists of each other (that does not quite give the multiplicative structure however) which gives, more or less, a classification. Note that your question on Grothendieck sites is only vaguely related, the whole point of the notion of Weil cohomology theory is that there is no site in sight (pun intended).

Addendum: This is a little bit simple minded as the requirements for a Weil cohomology theory to be a fibre functor go beyond the standard conjectures I think. Also one would need to define motives using cohomological equivalence. However, I think the statements I made are philosophically OK and one cannot hope to get anything in the way of a more precise classification.

ag.algebraic geometry - Is there an Abelian surface such that every effective divisor is ample? (Together with a boil down version to a question in Complex Lie group theory)

The Nakai-Moishezon criterion states that a line bundle $L$ over a surface $X$ is ample iff $L cdot L > 0$ and $L cdot C > 0$ for every curve $C$.
We can use this criterion to check that if $X$ is the product of two elliptic curves, then lots of divisors of $X$ are not ample. The fibers of the projection maps of $X$ to its factors have zero self intersection and hence cannot be ample.

Question: is there an Abelian surface such that everyone of its curves is ample?

This is what I attempted. I don't believe it leads anywhere, tough...
Suppose $X$ is an Abelian surface that is not the product of two elliptic curves. Suppose that $C_1$ and $C_2$ are two curves in $X$ representing different homology classes. Then, they must intersect [fix an element $theta in X$ such that $theta$ sends $C_1$ to a curve that intersects $C_2$...]. So, all that matters is to check that $C_1 cdot C_1 > 0$.

We do it by contradiction. Assume that $C_1 cdot C_1 = 0$. By acting with the inverse of a point of $C_1$ on $C_1$, we can assume that the identity element of $X$ is in $C_1$. Since $C_1 cdot C_1 = 0$, $C_1$ is a subgroup of $X$, furthermore, it is smooth [just act on $C_1$ with $C_1$ itself]. So, we have a mapt $X rightarrow X/C_1$, a elliptic fibration of $X$ with elliptic fibers. If this was a trivial HOLOMORPHIC bundle then we would get the contradiction we sought. But that is very unlikely to be the case.

co.combinatorics - Does War have infinite expected length?

Dear Joel David,
I will try to explain it, but I have to note that article is quite primitive, and is written in a readable English. Moreover there are many figures. But I will try:
I will make a list of statements and then You can mention the number of the non clear one:

1. By our assumption (players do not have strategy and do not have fixed rules how to return cards) the game is a Markov chain.




2. Absorbing (final) state is a state where you stay forever :)
   For us it means the end of the game i.e. the state when one of players has got all cards.

3A. In finite Markov chain, assuming arbitrary initial state, you are absorbed with probability ONE If And Only If "for each vertex of the Markov chain graph there is a way to an absorbing state."

3B. So we have to prove that for the graph of our game of war, there no exists such initial state that players do not have any chance to reach the end.

1. To prove it we should consider first the simplification. Consider the game with cards {1,...,n} i.e. every value meets only once.

We call a vertex attaining if it has got terminal states as its descendants, and wandering otherwise. It is obvious that a descendant of a wandering vertex is again wandering, and an ancestor
of attaining is again attaining.
For an arbitrary oriented graph it is possible that an attaining vertex has got wandering vertices among its descendants. We show that for our graph G it is not so. For that, we need to understand some properties of the graph G.

LEMMA 1.
A: Let state be such that one of the players has got only one
card in his hand, then this state has got exactly one ancestor.

B: If both players have got at least two cards, then this state has got exactly two ancestors.

LEMMA 2. For the graph of the game it holds that a descendant of an attaining vertex is again an attaining vertex.
(Page 5 of the article)

Lemma 3. The states in which one of the players has got only one card are attaining. (page 6)

Lemma 4. Every vertex has got an ancestor that corresponds to the state in which one of the players has got only one card.

Therefore, we have shown that each vertex has got an ancestor that corresponds to the state in which each player has got exactly one card. This state is attaining by Lemma 3. By Lemma 2 descendants of attaining vertices are again attaining, therefore, the initial state is again attaining, and we have proved

Theorem: Graph G does not have any wandering vertices.

---

Now how to apply it to the standard GAME:
We use the following obvious fact: If a subgraph of an oriented graph does not have wandering vertices, then the original graph does not have any wandering vertices either.

Now the proof is similar.

I hope it is better to read the article, I am sorry.
and I want to note once more time, that question of strategy is never been discussed.

---

[Added by J.O'Rourke:]
The paper has appeared: "On Finiteness in the Card Game of War,"
Evgeny Lakshtanov and Vera Roshchina,
The American Mathematical Monthly,
Vol. 119, No. 4 (April 2012) (pp. 318-323).
JSTOR link.

dg.differential geometry - When does a submersion have connected fibers?

If $M$ and $N$ are both compact, then the submersion $F$ can be thought of as a fiber bundle map with fiber $F^{-1}(p)$ for any $pin N$. Then one can apply the long exact sequence of homotopy groups of a fiber bundle to learn that if, for example, $M$ is connected and $N$ is 1-connected, that the fibers must be connected.

These sufficient conditions may be too specific, though.

rt.representation theory - Existence of supercuspidal representations

Let $G$ be an unramified reductive group over $Q_p$. I want to prove that the group $G(Q_p)$ has a supercuspidal representation (complex coefficients).

I have been looking in many parts of the literature, and it seems that many people are convinced that it is true; however up to now I never saw it stated explicitly.

By the works of L. Morris I reduce to showing that a reductive group $M$ over $F_p$ has a cuspidal representation (L. Morris, level zero G-types, p 140).

So then, I should prove that $M(F_p)$ has a cuspidal representation. The article of Deligne–Lusztig provides such a representation when given a minisotropic torus $T$ in $M$ and a character $chi$ of $T(F_p)$ which is in general position.

Let me recall that "character in general position" means that the rational Weil group acts freely on the character.

So now comes my doubt and question. Is it true that such a pair $(T, chi)$ can always be found for all reductive groups $M$ over $F_p$ ?

I am "afraid" of "small" groups that have tori with "large" Weyl groups.

The supercuspidal representations that come from the above construction are of "level 0".

In the book of Carter (Finite Groups of Lie Type) I found a result pointing in this direction. Lemma 8.4.2 p. 281 (with an easy proof) shows that for $T$ given and $q$ sufficiently large, the torus $T(F_q)$ has a character in general position.

oc.optimization control - Convergence to a (unique?) fixed point?

Consider a given $Ntimes P$ matrix $X$ (full rank with columns ${bf x}_p$, $p=1,ldots,P$), a given vector ${bf y}in R^N$ and a thresholding function $eta_lambda(|x|)=(|x|-lambda)_+$ with $lambda>0$.

Start with a vector ${boldsymbol alpha in R^P}$, define ${bf r}_{-p}({boldsymbol alpha})={bf y}-X {boldsymbol alpha}+ alpha_p {bf x}_p$, and define a sequence of vectors that changes only one entries to the current iterate, say the $p$th one, to
$$
alpha_p^{rm new} =frac{{bf r}_{-p}({boldsymbol alpha}^{rm old})^{rm T} {bf x}_p}{ |{bf x}_p |_2^2 } left {frac{eta_{lambda}(|{bf r}_{-p}({boldsymbol alpha}^{rm old})^{rm T} {bf x}_p|)}{ |{bf r}_{-p}({boldsymbol alpha}^{rm old})^{rm T} {bf x}_p|}right }^gamma.
$$
Repeat with a cycling rule, successively letting $p=1,2,ldots, P, 1,2 ldots$

Note that the case $gamma=0$ amounts to solving the least squares problem
$$
min_{boldsymbol alpha in R^P} | {bf y}- X {boldsymbol alpha}|_2^2,
$$
by a coordinate descent algorithm; also the case $gamma=1$ amounts to solving the $ell_1$-penalized least squares problem
$$
min_{boldsymbol alpha in R^P} frac{1}{2}| {bf y}- X {boldsymbol alpha}|_2^2 + lambda | {boldsymbol alpha}|_1,
$$
by a coordinate descent algorithm. Fact: both algorithms converge to a unique point: the minimum of its corresponding optimization problem.

Questions: (1) For a given starting vector ${boldsymbol alpha}$, does the sequence converge to a fixed point for all $gamma$'s? (2) If so, is the fixed point unique (regardless of the starting vector)?

ac.commutative algebra - Ring objects in the category of cocommutative coalgebras (aka Hopf rings).

I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite a while - initially by Milgram and Ravenel-Wilson since they arise in studying homology of ring spectra - and go by the name "Hopf rings" (for better or worse).

I'd like to make the calculations more "meaningful" if possible so my question is: how much of commutative algebra has been reproduced for ring objects in categories other than vector spaces sets? For some suitably nice categories C, hopefully including cocommutative coalgebras, do ring objects in C have analogues of ideals/modules, dimension, Spec, localization,...? I realize this is open-ended, but if there is existing work along any of these lines it would be nice to look and see if the ring objects I am considering fit in with and are illuminated by such a framework. Since (an interesting subset of) cocommutative Hopf algebras are given by group objects in the category of cocommutative coalgebras, one would suspect that this category's ring objects are also particularly nice but I am not aware of any such general development.

dg.differential geometry - Global description of the Levi-Civita connection

I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.

I'm not looking for a description of this object as a differential operator.

Instead, I'm looking for a splitting of the natural map
$alpha = (pi_{T(TX)}, Dpi_{TX}): T(TX) to TX oplus TX$, where $pi_{T(TX)}$ is the structure map of the double tangent bundle and $Dpi_{TX}$ is the map on tangent bundles induced by the structure map $pi_{TX} : TX to X$.

Noting that $TX oplus TX= (pi_{TX})^* (TX)$, a splitting of $alpha$ is the analogue, for the vector bundle $TX$, of the standard notion of a connection on a principle bundle: it's a way of lifting tangent vectors on the manifold $X$ up to tangent vectors on the bundle $TX$.

Lang (in GTM 160, Differential and Riemannian Manifolds) explains how to obtain this splitting using the metric spray, which is a map $F: TX to T(TX)$ that splits both of the above maps $T(TX) to TX$ (and satisfies another "quadratic" condition). Lang gives a global description of $F$ as the vector field on $TX$ corresponding, under the metric, to the 1-form -dK, where $K(v) = (1/2)langle v,vrangle$ is the kinetic energy functional on $TX$. However, he doesn't really give a coordinate-free extension of F to the desired splitting. From studying the discussion in Lang, it seems to me that there is a unique splitting $H: T(TX) to TX oplus TX$ satisfying $F(v+w) = (F(v) + H(w,v)) + (F(w) + H(v,w))$ and such that in any local chart U on X, H has the form $H(x, v, w) = (x, v, w, B(x, v, w))$ (as a map $Utimes E times E to (U times E) times (Etimes E)$) with $B(x, -, -)$ a symmetric bilinear mapping. Here E is the Hilbert space on which X is modeled.

The parentheses in the expression $(F(v) + H(w,v)) + (F(w) + H(v,w))$ are important: inside the parentheses, + means addition in the fibers of the map $Dpi_{TX}$, whereas outside the parentheses, + means addition in a fiber of $pi_{T(TX)}$. Note that H itself is definitely not symmetric, so I don't think it's clear from the global formula that H exists.

Establishing existence of the map H seems to depend on the rather ugly change-of-coordinate formulas for the "quadratic part" of the spray F, given by Lang.

Lang mentions that the book Symmetric Spaces (Loos, 1969) gives some discussion of this material in terms of second-order jet bundles, and I suspect that may be what I'm looking for. However, this book is hard to come by. I can't find any previews on-line, and it's not in our library. Lang also mentions Pohl's paper "Differential geometry of higher order" (Topology 1 1962 169--211) but I couldn't see anything about the Levi-Civita connection in there.

Does anyone know if Loos has what I'm looking for? Are there other discussions of these ideas in the literature? Does anyone have other suggestions for how to think about the splitting H?

I'll point out, as motivation, that the splitting $H$ gives a decomposition of $T(TX)$ as a direct sum $pi^* (TX) oplus pi^* (TX)$ of bundles over $TX$ (because the kernel of $alpha$ is isomorphic to $pi^* (TX)$, and so this is one way to think about the standard fact that $TX$ is an orientable manifold, with a Riemannian metric inherited from the one on $X$.

co.combinatorics - Number of A Subset of Monomials

Let $S_i$ be the set of monic monomials $m in mathbb{Z}[x_1, dots,
x_k]$ which are divisible by $x_i$ but not $x_i^2$. If I am reading
your question correctly, you are looking for $|S_1 cup cdots cup
S_k|$.

Note that for $1 leq i_1 < dots < i_m leq k$, the intersection
$S_{i_1} cap cdots cap S_{i_m}$ is the set of monomials of degree
$n$ divisible by $x_{i_1} cdots x_{i_m}$ but not by $x^2_{i_1} cdots
x^2_{i_m}$. If $m < n$ and $m < k$, then there is a bijection between
$S_{i_1} cap cdots cap S_{i_m}$ and the set of monic monomials of
degree $n-m$ in $mathbb{Z}[x_1, dots, x_{k-m}]$. (If $m = n leq k$,
then the intersection has one element, $x_{i_1} cdots x_{i_m}$. In
any other case, the intersection is empty.) Hence, for $1 leq i_1 <
cdots < i_m leq k$,
$$|S_{i_1} cap cdots cap S_{i_m}| = begin{cases} left(matrix{n +
k - 2m -1 cr k - m - 1}right), & text{if $m < n$ and $m < k$} cr
1, & text{if $m = n leq k$} cr 0, & text{otherwise.}end{cases}$$

So, by the principle of inclusion-exclusion,
$$|S_1 cup cdots cup S_k| = sum_{m =1}^{min(n, k)-1} (-1)^{m-1}
left(matrix{k cr m}right)left(matrix{n + k - 2m - 1 cr k - m -
1}right) + (-1)^{n-1} left(matrix{kcr n}right)a,$$
where $$a = begin{cases} 1, & text{if $k geq n$} cr 0, &
text{otherwise.}end{cases}$$

nt.number theory - Constructing coherent sheaves on Shimura varieties.

Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper half plane by the congruence subgroup $Gamma=Gamma_1(N)$, then there are two kinds of sheaves that one often sees showing up in the theory of automorphic forms in this setting:

1) Locally constant sheaves. The ones showing up typically come from representations of $Gamma$ of the form $Symm^{k-2}(mathbf{C}^2)$, with $Gamma$ acting in the obvious way on $mathbf{C}^2$. These sheaves---call them $V_k$---are related to classical modular forms of weight $k$ via the Eichler-Shimura correspondence. They only exist for $kgeq2$ (weight 1 forms are not cohomological) and representation-theoreticically the sheaves are associated to representations of the algebraic group $SL(2)$ (the reason one starts in weight 2 rather than weight 0 is that there is a correction factor of "half the sum of the positive roots").

2) Coherent sheaves. The ones showing up here are powers $omega^k$ of a canonical line bundle $omega$ coming from the universal elliptic curve. The global sections of $omega^k$ (which are bounded at the cusps) are classical modular forms of weight $k$. Although there are no classical modular forms of negative weight, the sheaf $omega^k$ still makes sense for $k<0$ (in contrast to case 1 above). I am much vaguer about what is conceptually going on here. I have it in my mind that here $k$ is somehow a representation of the group $SO(2,mathbf{R})$.

Now my question: what is the generalisation of this to arbitrary, say, PEL Shimura varieties? Part (1) I understand: I can consider algebraic representations of the reductive group I'm working with and for each such gadget I can make a locally constant sheaf. But Part (2) I understand less. I am guessing I can construct a big vector bundle on my moduli space coming from the abelian variety. Now, given some representation of some group or other, I can build coherent sheaves somehow, possibly by "changing the structure group" somehow. For which representations of which group does this give me a coherent sheaf on the moduli space?

Basically---what is the general yoga for supplying natural coherent sheaves on Shimura varieties, which specialises to the construction of $omega^k$ in the modular curve case, and which explains why $omega^k$ exists even for $k<0$?

at.algebraic topology - Is the complement of a strong deformation retract of a manifold M homotopic equivalent with the boundary of M?

The Alexander horned sphere furnishes an example of a subspace of the interior of a compact $3$-ball that is contractible (being a $3$-ball itself) and whose complement is not simply connected.

There are also less pathological examples in high dimensions: If $n$ is at least $5$ or so then it is not hard to make a smooth compact contractible $n$-manifold whose boundary is not simply connected, whereas the complement of a point in a simply connected manifold of dimension $3$ or more must be simply connected.

EDIT: I retract (heh) my first example. I overlooked the requirement that the subset should be a deformation retract; I was using the weaker requirement that the inclusion should have an inverse up to homotopy.

EDIT: Doh! My first example was correct. If i:A-->B is an inclusion of compact metric spaces and A is homeomorphic to a ball, then by the Tietze extension theorem there is a retraction r:B-->A. If B is also a ball, then the resulting "straight-line" homotopy from ir to the identity gives a deformation retraction.

differential topology - Checking whether the image of a smooth map is a manifold

The specific $F(M)$ is not a smooth submanifold. Here is an argument.

To simplify formulas, I renormalize the sphere: let it be the set of $(z_1,z_2)inmathbb C^2$ such that $|z_1|^2+|z_2^2|=2$ rather than 1. Then, as Gregory Arone pointed out, $F(M)$ is the set of $(b,c)inmathbb C^2$ such that the roots $z_1,z_2$ of the equation $z^2-bz+c$ satisfy $|z_1|^2+|z_2^2|=2$. I claim that it is not a smooth manifold near the point $p:=(2,1)in F(M)$.

Let us intersect $F(M)$ with two planes: ${b=2}$ and ${c=1}$. If it is is a smooth submanifold, at least one of the intersections should be locally (near $p$) a 1-dimensional smooth submanifold of the respective plane (otherwise both planes are tangent to $F(M)$ at $p$, but this is impossible since they span the whole space). But this is not the case:

If $b=2$, the equation is $z^2-2z+c=0$, hence $z_1+z_2=2$, then $|z_1|+|z_2|ge 2$ and therefore $|z_1|^2+|z_2|^2ge 2$. Equality is attained only for $z_1=z_2=1$, thus the intersection is a single point $c=1$, not a 1-dimensional submanifold.

If $c=1$, the equation is $z^2-bz+1$, hence $z_1z_2=1$, then $|z_1|cdot|z_2|=1$ and therefore $|z_1|^2+|z_2|^2ge 2$. The equality is attained if and only if $|z_1|=|z_2|=1$, so $z_1$ and $z_2$ are conjugate to each other. The set of $b$'s for which this happens is the real line segment $[-2,2]$ which is not a submanifold near 2.

co.combinatorics - Symmetric colorings of regular tesselations

Given a regular tesselation, i.e. either a platonic solid (a tesselation of the sphere), the tesselation of the euclidean plane by squares or by regular hexagons, or a regular tesselation of the hyperbolic plane.

One can consider its isometry group $G$. It acts on the set of all faces $F$. I want to define a symmetric coloring of the tesselation as a surjective map from $c:Frightarrow C$ to a finite set of colors $C$, such that for each group element $G$ there is a permutation $p_g$ of the colors, such that $c(gx)=p_gcirc c(x)$. ($p:Grightarrow $Sym$(C)$ is a group homomorphism).

Examples for such colorings are the trivial coloring $c:Frightarrow {1}$ or the coloring of the plane as an infinite chessboard.
The only nontrivial symmetric colorings of the tetrahedron, is the one, that assigns a different color to each face. For the other platonic solids there are also those colorings that assign the same colors only to opposite faces.

So my question is: Does every regular tesselation of the hyperbolic plane admit a nontrivial symmetric coloring?

I wanted to write a computer program, that visualizes those tesselations, but i didnt find a good strategy which colors should be used. So i came up with this question.

nt.number theory - Geometry for Anderson's motives?

Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category with a tensor product, we have a well behaved weight filtration, and there are realisation functors with good properties. (After all, the name "motive" was not chosen at random.)

However, a classical motive is something that one wants to associate with a scheme or variety, thinking of it as its universal cohomology, and now it bugs me that I have no geometric objects at hand to which I could associate $t$-motives. I don't even know whether these, if any, should be varieties or something else.

To make this a halfway real question: Let $C$ be a smooth proper curve over $mathbb{F}_p(t)$ of genus $g$. Is there a natural, functorial way of associating with $C$ a pure $t$-motive $M(C)$ of rank $g$, in such a way of course that the cohomology of $C$ is related, via a functor morphism, to the realisations of $M(C)$? So I'm asking here for some kind of Jacobian construction.

But as I said, I don't even know whether varieties over $mathbb{F}_p(t)$ are the right geometric objects to look at.

dg.differential geometry - The invariant 3-form on a compact Lie group

Let $G$ be a compact Lie group. We have the well-known Maurer-Cartan left-invariant and right-invariant $1$-forms $theta$ and $bartheta$ in $Omega^1(G, mathfrak{g})$, probably discussed in every Lie theory lectures.

However the canonical bi-invariant closed $3$-form $chi = frac{1}{12} (theta, [theta, theta]) = frac{1}{12} (bartheta, [bartheta, bartheta])$ in $Omega^3(G)$ may be a little bit less-known. And when I heard of it I had some questions in mind...

1) Are there canonical invariant $5$-forms and higher invariant forms on a $G$?

2) How are they related to Lie algebra cohomology and equivariant cohomology?

3) The construction looks a little bit like $tr(A wedge dA)$, if we regard $theta$ as a connection $A$ on the frame bundle of $G$, and use the Maurer-Cartan equation $dtheta = -frac{1}{2}[theta, theta]$.

Now there is another famous $3$-form: the Chern-Simons form $tr(A wedge dA + frac{2}{3} A wedge A wedge A)$, and I wonder if these two are somehow related. Does the $tr(A wedge A wedge A)$ part vanish here, due to Jacobi identity?

(and, note there are Chern-Simons 5-forms etc.)

Thank you very much.

pr.probability - When does a pointwise CLT hold?

Feller states the Berry-Esseen theorem in the following way. Let the $X_k$ be independent variables with a common distribution $F$ such that $$E[X_k]=0, E[X_k^2]=sigma^2>0, E[|X_k|^3]=rho<infty,$$ and let $F_n$ stand for the distribution of the normalized sum $$(X_1+ dots X_n)/(sigma sqrt{n}).$$ Then for all $x$ and $n$ $$|F_n(x)-N(x)| leq frac{3rho}{sigma^3 sqrt{n}}.$$

The expression you are interested in is
$$left|frac{F_n(epsilon)-F_n(-epsilon)-N(epsilon)+N(-epsilon)}{epsilon}right|,$$
which is less than
$$left| frac{F_n(epsilon)-N(epsilon)}{epsilon} right| + left| frac{F_n(-epsilon)-N(-epsilon)}{epsilon} right|,$$
which by Berry-Esseen is bounded by
$$2frac{3rho}{epsilon sigma^3 sqrt{n}}.$$
So, if $epsilonsqrt{n}$ goes to infinity, then you are good.

I realize this isn't what you asked, in that you wanted conditions on $X$, and this instead gives you conditions on $epsilon_n$. Still, perhaps it'll help.

Reference: Feller, An Introduction to Probability Theory and Its Applications, Volume II, Chapter XVI.5.

reference request - Rallis inner product formula for U(2,2) and U(3)

Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are:

1. "A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998.


2. "An Application of the Regularized Siegel-Weil Formula on Unitary Groups to a Theta Lifting Problem", Proceedings of the AMS, 1999.

One natural thing to look for after obtaining such a result is a Rallis inner product formula. Tan doesn't prove such a result in either of the two references mentioned above, though he seems to come close at the end of the second one.

Does anyone know if such a formula is written down anywhere?

gt.geometric topology - How big can the Hausdorff dimension of a function graph get?

This question is inspired by How kinky can a Jordan curve get?

What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the least upper bound attained by some function?

It may be noted that the area (2-dimensional Hausdorff measure) of a function graph is zero. However, this does not rule out the possible existence of a function graph of dimension two.

reference request - Do separable and normal have topological meanings for fields?

The terminology would suggest that a separable field extension is so because the resulting field extension has some sort of separable topology, and that a normal extension corresponds to one with a normal topology.

I imagine this is true, or else they wouldn't have named them in such a way.

Also, I'm not sure what subfield this falls under, so if you could suggest additional tags, that would be great as well.

nt.number theory - other examples of composition of functions

Another example: there are 3x3 matrices which, when applied to a vector representing a pythagorean triple, produce other pythagorean triples. I think it is even a way to
produce all primitive pythagorean triples from (3,4,5).

In general, you are looking for a finite number of operations which produce through
composition a number of objects, and it seems that you want no overlap, i.e. if
h and k are two composition series for which h(obj)=k(obj) for some initial object
obj, then h=k as composition series, or in other words they are built up the same way.

Universal algebra has some means for the study of the generation of objects through
operation composition. In particular, the operations form a clone (semigroup if you look at just the operations which take one argument) which, given the uniqueness requirement
above, is relatively free on the set of generators, as there will be no nontrivial
equations satisfied by the compostions.

There are other areas and examples as well, but until the question becomes more specific,
I will stop here.

Gerhard "Ask Me About System Design" Paseman, 2010.06.08

nt.number theory - Are there two non-isomorphic number fields with the same degree, class number and discriminant?

John Jones has computed tables of number fields of low degree with prescribed ramification. Though the tables just list the defining polynomials and the set of ramified primes, and not any other invariants, it's not hard to search them to find, e.g., that the three quartic fields obtained by adjoining a root of x^4 - 6, x^4 - 24, and x^4 - 12*x^2 - 16*x + 12 respectively all have degree 4, class number 1, and discriminant -2^11 3^3. On the other hand these three fields are non-isomorphic (e.g. the regulators distinguish them, the splitting fields distinguish them...).

nt.number theory - Distinct numbers in multiplication table

This is the multiplication table problem of Erdos. According to Kevin Ford, Integers with a divisor in
$(y,2y]$, Anatomy of integers, 65-80, CRM Proc. Lecture Notes, 46, Amer Math Soc 2008, MR 2009i:11113, the number of positive integers $nle x$, which can be written as $n=m_1m_2$, with each
$m_ilesqrt x$, is bounded above and below by a constant times $x(log x)^{-delta}(loglog x)^{-3/2}$, where $delta=1-(1+loglog2)/log2$.

Erdos' work on this problem can be found (in Russian) in An asymptotic inequality in the theory of numbers, Vestnik Leningrad Univ. Mat. Mekh. i Astr. 13 (1960) 41-49.

Another reference is http://oeis.org/A027424 where a PARI program is given.

deformation theory - Kodaira-Spencer map in a concrete instance

Here is an attempt. Based on your comment to Kevin Lin's post, I think that you know the first part of what I have written, but I included this for the sake of completeness.

• Some Generalities on $phi$: Any deformation of an affine hyperelliptic curve such as

$$
y^2 = prod (x - lambda_i(epsilon))
$$

is trivial and hence corresponds to the zero cohomology class. Indeed, any deformation of a smooth, affine scheme (separated and of finite type over a field?) is trivial. Given a deformation $X_{epsilon} to Delta$ as you describe, the Kodaria-Spencer map is computed by fixing an open affine cover $U_i$ of $X_0$ and isomorphisms $phi_i colon X_{epsilon}|_{U_i} to U_{i} otimes k[epsilon]$ of the restriction of $X_{epsilon}$ to $U_i$ with the trivial deformation of $U_{i}$. The automorphism $phi_{i} circ phi_{j}^{-1}$ of determines an explicit Cech cocyle that represents a class in $H^{1}(X_0, TX_0)$, and one checks that this class is independent of the choices made. The main point: the Kodaira-Spencer class comes from deforming the gluing data NOT from deforming the equations.

• Computation of $phi$: As you wrote, it is not clear from that description how everything works in a concrete cases. Here is how it works out in the case of a general genus $2$ hyperelliptic curve. Working over the field $k$, this curve
  can be described as the curve obtained by gluing the two affine schemes

$$
U_1 := operatorname{Spec}(k[x_1, y_1]/(y_1^2 = prod_{i=1}^{6} (x_1-r_i)),
$$
$$
U_2 := operatorname{Spec}(k[x_2, y_2]/(y_2^2 = prod_{i=1}^{6} (1-r_i x_2)),
$$
over the usual opens via the isomorphism $g$ defined by the rules
$$
x_1 mapsto x_2^{-1},
$$
$$
y_1 mapsto y_2 x_2^{-3}.
$$
Here $r_1, dots, r_6$ are general scalars.

Associated to the affine open cover ${U_1, U_2}$
is the usual Cech complex, and we can use this complex
to compute $H^{1}(X, TX)$. Some elements of this cohomology
group are given by the Cech cocycles
$$
y_1/x_1 frac{partial}{partial x_1}, y_1/x_1^{2} frac{partial}{partial x_1}, y_1/x_1^{3} frac{partial}{partial x_1} in H^{0}(U_{12}, TX).
$$
Here $U_{12}$ denotes the intersection of $U_1$ and $U_2$. Note: one needs to check that these vector fields are regular on $U_{12}$.
The vector field $frac{partial}{partial x_1}$ has simple poles at ramification points of the degree $2$ to $mathbb{P}^1$, and the $y_1$ terms
are needed to cancel these poles. I think these elements form a basis, but
you just asked for an example so I guess we don't care about this.

Let's compute the 1st order deformation of $X$ associated to $D:= y_1/x_1 frac{partial}{partial x_1}$. To construct the deformation, we take
the trivial deformations of $U_1$ and $U_2$ and deform the gluing automorphism. The trivial deformations are
$$
operatorname{Spec}(k[epsilon, x_1, y_1]/(y_1^2 = prod_{i=1}^{6} (x_1-r_i)),
$$
$$
operatorname{Spec}(k[epsilon, x_2, y_2]/(y_2^2 = prod_{i=1}^{6} (1-r_i x_2)).
$$

The general rule is that the deformed gluing map $tilde{g}$ is given by $tilde{g}(a) = g(a) + epsilon cdot g(D(a))$. For our particular choice of
$D$, I think this yields:
$$
x_1 mapsto x_2^{-1} + y_2 x_2^{-2} epsilon,
$$
$$
y_1 mapsto y_2 x_2^{-3} + y_2 x_2^{-2} frac{-x_2^{-1} q'(x_2) + 6 x_2^{-2} q(x_2)}{2 y_2} epsilon.
$$
Here $q(x_2) = prod_{i=1}^{6} (1-r_i x_2)$.

The expression for the image of $y_1$ is quite complicated, but it hopefully is just
$g(y_1/x_1 frac{partial y_1}{partial x_1})$.

One can work our a similar description for the deformations coming from the other cohomology classes that I wrote down. Assuming these form a basis, this completely describes the map $phi$.

It is easy to reverse this construct as well. Every deformation arises by deforming the map $g$ to a map $tilde{g}$ as we have done. The associated cohomology class can be described by writing $tilde{g} = g + epsilon cdot D$ for some function $D$. One can show that $D$ defines a regular vector field on $U_{12}$ and hence represents an element of $H^{1}(X, TX)$.

ct.category theory - is localization of category of categories equivalent to |Cat|

There are set theoretic issues that will hinder any proof of this statement. In particular I don't believe you can construct the adjunction in part (2) without applying the axiom of choice to CAT, and it's not clear the localization S^(-1)CAT makes sense either.

That said, let's ignore these issues and pretend that CAT is a small category of (small) categories.

I was originally confused by this. I believe that there are some problems in the formulation of this question. In particular if S consists of those functors which are equivalences of categories, then it does not form a multiplicative system. Specifically, the right Ore condition fails. I'll give an example later. What this means though, is that the property your are trying to check in part (1) actually fails. To see this pick a group G and view it as a category with one object. Then the functors from H to H are the homomorphisms, and the equivalence classes of this functors are the orbits of Hom(H,H) under the conjugation action. Choose H such that there are distinct automorphisms of H which are equivalent under conjugation.

Claim: If F,G: H --> H are two such automorphisms, then there is no equivalence s:H --> C such that sF = sG. I think this is easy to check, so I'll leave it at that.

So the second part of (1) is impossible to prove.

I first thought that we can answer this question be going to skeletal categories. This is not correct, but let's look at why it is not correct. Let SKEL be the full subcategory of CAT consisting of the Skeletal categories, i.e. those categories in which $a cong b$ implies $a = b$, i.e. those categories which have a single object in each isomorphism class.

Every object in CAT is equivalent to one in SKEL. This is an easy exercise you should work out for yourself. By applying the axiom of choice to all of CAT we can construct an equivalence between CAT and SKEL, and in particular we have L: CAT ---> SKEL with an equivalence $C cong L(C)$ for every category C.

Notice that a functor between skeletal categories is an equivalence if and only if it is an isomorphism on the nose. So we are part way there. This made me think that SKEL was the localizing subcategory we were after.

However two functors into a skeletal category can be naturally isomorphic without being equal. The group example above is actually an example of this. The functors F and G are naturally isomorphic, but not equal.

Let J denote the "Joyal interval", the category with two objects and an isomorphism between them. J can be used to say when two functors are naturally isomorphic. F,G: C --> D are nat. isomorphic if they extend to a functor $C times J to D$. There is a quotient of J where we identify the two objects. This is in fact the category Z (the group of integers viewed as a category). Let W be the set of morphisms consisting of the single morphism J --> Z. Notice that SKEL consists of exactly the "W-local" objects. But this is not quite what we want.

---

Okay, now for the rest. First, I promised an example where the right Ore condition fails. This is given by considering the two inclusions of pt into J. Both of these are equivalences, but this cannot be completed to an appropriate square. So the class of equivalences in not a multiplicative system.

Next, Toen doesn't assume that S is a multiplicative system. He (correctly) defines the localization in terms of a (weak) universal property. Namely, there exists a locization functor:

$ L: CAT to S^{-1}CAT$

such that fro any other category D, $L^* : Fun( S^{-1}CAT, D) to Fun(CAT, D)$ is fully-faithfull and the essential image consists of those functors which send elements of S to isomorphisms in D. In particular the quotient functor from CAT to |CAT| does this, so this gives us functor from S^(-1)CAT to |CAT| defined up to unique natural isomorphism.

On the other hand, |CAT| is also defined by a universal property. This gives us maps between |CAT| and S^(-1)CAT and by the usual sort of general nonsense this is an equivalence. The details aren't too hard, so I'll leave them to you.

Now the second part of your question asks about whether we can realize |CAT| as a full subcategory of CAT such that the quotient $ q:CAT to |CAT|$ is left adjoint to the inclusion. The answer is no. Let's first look at whether q can have a right adjoint at all and what this would mean.

Suppose that R: |CAT| --> CAT is right adjoint to the quotient q. Be abuse, we will identify the objects of CAT and |CAT|. This means that for all categories X and Y we have:

$CAT(X, R(Y)) = |CAT|(X, Y) = |CAT|(X times J, Y) = CAT( X times J, R(Y))$

So $R(Y)$ must be cartesian local with respect to the map J --> pt. In particular we see, taking X=pt, that R(Y) has no non=identity automorphisms, i.e. R(Y) is rigid.

Now let's look at an example. Consider the group Z viewed as a category with one object. Then the endomorphisms of Z in |CAT| is the monoid Z (under multiplication). Now if R is fully-faithful then we need:

$ mathbb{Z} = |CAT| (mathbb{Z}, mathbb{Z}) = CAT (R(mathbb{Z}), R(mathbb{Z}))$

but this last is equal to the automorphism of some set R(Z), hence we have a contradiction. So there is no such fully-faithful right adjoint.

geometry - How to pick a random direction in n-dimensional space

The easiest way to do this efficiently is to rely on the fact that a gaussian distribution is spherically symmetric and also separable. So, what you need to do is :

1) Build a vector V where each element is a Gaussian distributed value of mean 0, choose any width that makes sense.

2) Normalize the vector V

This vector now is a random unit vector uniformly distributed across the hypersphere of the vector V. This algorithm is both fast and is linear in the dimension of V.

oc.optimization control - Recommendations for a large scale bounded variable least squares (BVLS) solver for sparse matrices

This is such a well-solved problem that there are many software packages that have built in functions for this.

Here are a selection of built-in functions in different software packages that can be used:

In Matlab: lsqlin (type help lsqlin into Matlab and it tells you exactly what to type. I have just (approximately) solved your problem with random sparse matrices and it works great.)

KNITRO for Mathematica this package also solves this exact problem but I don't have this software so I can't tell you which exact function.

For a free solver I have found this: http://sourceforge.net/projects/quadprog/
However it assumes that $A$ has full column rank. This is just because this algorithm uses the dual problem which exists when the Hessian $A^TA$ is positive definite.

abstract algebra - about state-field correspondence

I want to elaborate a little on Pavel's excellent answer.

We can think (very schematically) of local operators in an n-dimensional
field theory the following way. We have an n-1 manifold M with some additional
structures (topological, conformal, metric etc), to which our field theory assigns a vector space Z(M) of states. Given x in M and a time t in the interval,
we can ask for local operators on Z(M) at the point x and time t. This
can be visualized (following field theory axiomatics) as follows: we cross
M with an interval, and cut out a tiny ball around the point (x,t) in
this cylinder. We obtain a cobordism (with additional structure) between
M times $S^{n-1}$ and M. We can then use the field theory axioms to turn states $Z(S^{n-1})$ into operators on $Z(M)$. Physically we think of inserting measurements on fields
on spacetime M times interval that only ask about the value of fields in a small (punctured) neighborhood of (x,t).

In general this is a very complicated structure. But if we're in a topological field theory, then this picture is independent of lots of things - such as most importantly the size and shape of the ball we removed (as well as its position). In a 2d CFT at least we know this structure is independent of size and shape of the disc we've cut out, and depends holomorphically on z=(x,t) a point in a Riemann surface.

For simplicity though let's stick to TFT, since this picture works equally well in any dimension. If we apply this idea to the case $M=S^{n-1}$ itself, we find that $Z(S^{n-1})$ has an algebra structure --- in fact an algebra structure parametrized by cutting a ball out of a cylinder (if you look at this carefully you find the topologist's notion of $E_n$ algebra -- for $n=1$ it's simply associative, for $n=2$ it's "braided" (commutative in a coarse sense) and it gets more and more commutative as n increases).
Moreover Z(M) for ANY M is now a module over this algebra.

This is how I think of state-field correspondence: states on the n-1 sphere are equivalent to local operators in the field theory acting on any space (these are the fields). In chiral CFT we find the notion of vertex algebra immediately from this -- it's a conformal refinement of the abstract notion of $E_2$ (or "braided") algebra we derived above from TFT, where now things depend holomorphically rather than locally constantly on parameters..

EDIT: One more piece of data here is the unit - there's a canonical state on the (n-1) sphere, given by considering it as the boundary of the ball we cut out (ie doing the path integral on the ball with boundary conditions on the sphere..) This is the vacuum state. It's easy to see it corresponds to the identity operator on $Z(M)$ for any $M$, and is the unit for the algebra structure on $Z(S^{n-1})$. We now recover the injectivity of the state-field correspondence: we consider the pair of pants (punctured cylinder) picture above for $M=S^{n-1}$ itself, and apply a given operator $vin Z(S^{n-1})$ to the vacuum incoming state, obtaining an outgoing state which is again v. Saying this more carefully in the 2d CFT case recovers the vacuum axiom of a vertex algebra, which Pavel explains gives injectivity of the state-field correspondence.