molecular genetics - Do single crossovers occur in circular polynucleotides?

Although crossover events can be observed in mitosis (mitotic recombination), they most frequently occur in prophase I of meiosis (crossing over in bivalents). Circular chromosomes are common in prokaryotes, but eukaryotes have linear chromosomes. Remember what meiosis accomplishes and that prokaryotes reproduce asexually. Prokaryotes don't have a need for -- or engage in -- meiotic division, so crossover events in circular chromosomes can really only be observed in instances of mitotic recombination, and I don't know the frequency of this in prokaryotes.

In prokaryote DNA replication, if any entangling of the two product circular chromosomes occurs, it is resolved by DNA topoisomerase. So perhaps a crossover would present a similar situation and the cell would resolve it by similar means.

zoology - Small worm living in some kind of cocoon, what are these animals?

This little dude is not a Caddisfly, but a true moth, Tinea pellionella, a case making clothing moth. Confusing because they do resemble caddisfly larvae cocoons more so than those of their own family (Tineidae). Sort blurb Here:

"The brown-headed larva spins a silken case that is open at both ends. The case in the above image is covered with fine sand and debris, and superficially resembles a caddisfly case. The flattened case is about 10-11 mm long (3/8 to 1/2 inch). When crawling, the larva's head, thorax, and three pairs of legs protrude out of the case, and drag it along. According to Internet sources, the larva feeds on a variety of material, including hair, fur, silk, felt, feathers, woolen clothing, upholstered furniture and carpets. It apparently prefers darkness and soiled clothing, and is not fond of synthetic fabrics, such as nylon and polyesters."

additive combinatorics - Cauchy-Davenport strengthening?

I believe that your statement follows from Cauchy-Davenport via matroid intersection theorem. (Matroid intersection theorem is stated in Chapter 41 of Alexander Schrijver's "Combinatorial optimization" book and can be also found here.)

You want to find a "rainbow" spanning tree in a complete bipartite graph you define, where colors correspond to edgesums. "Rainbow" spanning trees, in fact, seem to be commonly used as an example of matroid intersection.

By matroid intersection it suffices to show that for any set of edges $U$ in your graph

$r_1(U)+r_2(E setminus U) geq |A|+|B|-1,$

where:

$E$ is the set of all $|A||B|$ edges,

$r_1(U)$ is the rank of $U$ in the cycle matroid, and is equal to $|A|+|B|-c(U)$ where $c(U)$ is the number of connected components in the graph induced by $U$, and

$r_2(E setminus U)$ is the number of edgesums obtained by the edges not in $U$.

If $c(U)=1$ then we are done. Otherwise, let $A' subseteq A$, $B' subseteq B$ be obtained from $A$ and $B$ by choosing one element from each component of the graph induced by $U$, so that both are non-empty. Then the edges between $A'$ and $B'$ are not in $U$ and thus by Cauchy-Davenport

$r_2( Esetminus U) geq c(U)-1$,

as desired.

pr.probability - order statistics for components of a random unit vector

The distribution should be obtainable by integrating over the section of the simplex segment of the surface of the hypersphere bounded by the points (1,0,0,0,...), (1,1,0,0,0...)/sqrt(2), (1,1,1,0,0...)/sqrt(3) etc. along the ith axis.

All the distributions (n,m) have support contained within the unit interval, are piecewise smooth and share the same set of non-smooth points at the reciprocals of the square roots of the natural numbers.

latex - Tools for collaborative paper-writing

I use bzr (any particular reason why git, by the way?) and I've ended up using it for just about everything: papers, seminars, teaching, configuration files, my entire website, just about everything I do on a computer is in a bzr repository. Although I've yet to convince any collaborator to use it as well, I still find that it makes life easier since I can easily keep a record of when I sent something to someone else and merge in changes against that particular revision. I can also publish a repository and make it easy for a collaborator to have access to the files without needing to use bzr themselves.

Within a paper, I use the changes.sty package for sharing comments back and forth between myself and a collaborator.

Bzr has "nice" frontends so it might be possible to persuade a non-technologically minded person to use it (I'm a commandline junkie so have no experience of the available GUIs).

I also use a wiki (nlab, naturally) but that is (at the moment) for less focussed projects than a specific paper. However, when writing anything substantial there then I do it "offline" (even so far as to "compiling" and viewing it) and only sending it into the ether when I'm happy with it.

I find it completely incomprehensible that people want everything to be "in the cloud". I have access to several high-powered computers which are capable of running whatever software I'm using incredibly fast. Why would I swap that for a slow, crackly internet link which is guaranteed to be down the one time that I really need it? By using a DVCS (distributed version control system), I only need to be connected to the internet at the start of a given session and I can get my files off any one of a number of machines so it doesn't matter if one or other is down. In the worst case scenario that I can't connect, I can work offline on something and then merge my changes back again later. Indeed, my entire DVCS currently takes up a mere 71Mb (of which 25Mb consists of my local copies of Instiki and xournal) so I could easily carry it around with me on a memory stick (encrypted, of course).

If I really did want to do some "real time" collaboration, I would use either gobby (for working on files or papers) or jarnal (for working on maths). Gobby has real-time editing (and has had for quite some time) whilst with jarnal I can use my graphics tablet to actually write the mathematics for the other person to see just as if we were at a blackboard together. After all, if I'm doing real-time collaboration then I don't want to bother with getting the LaTeX syntax exactly right. I'm not bad at TeX, but if I'm in "Math Mode" then I don't want to be bothering with it.

soft question - What does it take to run a good learning seminar?

I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might expect, they can be rather successful. However I haven't been able to quite put my finger on what makes a good seminar good.

Certainly there are obvious necessary conditions for success, such as having sufficiently many (dedicated) participants and at least some semblance of a goal. But in my experience these conditions aren’t at all sufficient.

And on the other hand there are clear pitfalls that should be avoided, such as going too fast or not going fast enough, or scheduling the seminar at 8 in the morning. But there are also more subtle pitfalls that aren't as easily avoided: for example, having consecutive speakers of a certain style that might put off or discourage other participants. (Of course a plausible solution to this specific problem is to not have such people speak one after another, but often this is infeasible.)

So I turn to the collective wisdom of MO: In your experience, what has made a specific learning seminar feel successful to you? Feel free to interpret the word "successful" any way you want. Anecdotes and horror stories welcome.

(Aside: The seminar I'm planning is a "classics in geometry and topology" type of deal. By this I mean, each participant will select a classic paper at the beginning of the summer and then briefly discuss its contents sometime during the course of the seminar. I would consider this seminar successful if, at the end, each participant walks away with a set of their own notes on each paper, explaining why it's important, and containing a sketch of its main ideas and how it fits in the grand scheme of things; the hope is that such a set of notes might prove useful if one were to take a closer look at the paper down the road. If anyone has any experience about running a seminar of this type, then I'd be especially interested in hearing their comments!)

linear algebra - Sarrus determinant rule: references, extensions

The logic behind this does extend to general $ntimes n$ determinants, though probably not as nicely as you wish. Note that I am taking the liberty to replace "last" by "first" in "where R cyclically permutes the last three rows of the matrix A". It doesn't matter, because Sarrus' rule is invariant under cyclic shift, and a simple cyclic shift turns the last three rows to the first three rows.

Consider the alternating group $A_{n-1}$ embedded into the symmetric group $S_n$: every element of $A_{n-1}$ is a permutation of the set $leftlbrace 1,2,...,n-1rightrbrace$, and thus can be seen as a permutation of the set $leftlbrace 1,2,...,nrightrbrace$ which leaves $n$ fixed.

Also consider the dihedral group $D_n$ defined as the subgroup of $S_n$ generated by the cyclic shift $left(xmapsto x+1mod nright)$ and the reflection $left(xmapsto n+1-xright)$.

Then, every element $piin S_n$ can be uniquely written as $pi=sigmaxi$ with $sigmain A_{n-1}$ and $xiin D_n$. In fact, $xi$ is uniquely determined by the conditions $left(pixi^{-1}right)left(nright)=n$ and $mathrm{sign}left(pixi^{-1}right)=1$, and then $sigma$ results.

Now, write the determinant of an $ntimes n$ matrix in the form $sum_{piin S_n}mathrm{sign}picdotprod ...=sum_{sigmain A_{n-1}}sum_{xiin D_n}mathrm{sign}xicdotprod ...$. Each inner sum $sum_{xiin D_n}mathrm{sign}xicdotprod ...$ is the naive "Sarrus determinant" of some permutation of the matrix; which permutation it actually is is decided by the $sigma$.

For $n=3$, we have $A_{n-1}=A_2=1$, so the outer sum $sum_{sigmain A_{n-1}}$ has only one term, and the "Sarrus determinant" is the real determinant.

For $n=4$, we have $A_{n-1}=A_3=C_3$ (the cyclic group with $3$ elements), so the outer sum $sum_{sigmain A_{n-1}}$ has three terms, and it follows that the determinant of a $4times 4$ matrix can be written as a sum of three "Sarrus determinants". A closer look at the sum shows which ones.

ag.algebraic geometry - Conditions for "bootstrapping" a smooth DM stack?

Notation: Let $x$ be a point of a smooth separated finite type DM stack $mathcal X$ over a field. Suppose
• $G$ is the stabilizer of $x$,
• $V$ is the tangent space of $x$ (which comes equipped with an action of $G$),
• $G^textrm{triv}subseteq G$ is the subgroup which acts trivially on $V$,
• $H = G/G^textrm{triv}$ (note $H$ acts on $V$),
• $K$ is the subgroup of $H$ generated by pseudoreflections on $V$, and
• $K'$ the commutator subgroup of $K$.

$mathcal X$ can be expressed as you described (in an étale neighborhood of $x$) if and only if $K'$ is trivial.

I'll now unpack that answer. Any smooth separated finite type DM stack over a field can be (canonically!) obtained from its coarse space with the following steps (this is basically the main Theorem of my paper with Matt Satriano, A "bottom up" characterization of smooth Deligne-Mumford stacks):

1. take the canonical stack of the coarse space,


2. do a root stack construction along the ramification divisor of the coarse space map, rooting each component of the ramification divisor by the degree of ramification,


3. take the canonical stack again (the root stack may not be smooth any more, but it will have quotient singularities!), and


4. add a gerbe.

Your question is "when can we skip step 3?" That is, when is the root stack from step 2 already smooth?

Using the notation above, and looking formally locally around $x$ (so we can assume $mathcal X=[V/G]$; you can describe it étale locally too, but it's clearer this way), the above steps are:

1. $bigl[(V/K)/(H/K)bigr]$ is the canonical stack of the coarse space $V/H = V/G$,


2. $bigl[(V/K')/(H/K')bigr]$ is a root stack of $bigl[(V/K)/(H/K)bigr]$,


3. $[V/H]$ is the canonical stack of $bigl[(V/K')/(H/K')bigr]$, and


4. $mathcal X = [V/G]$ is a $G^textrm{triv}$-gerbe over $[V/H]$.

Note that step 1 is the familiar way of building the canonical stack of a space with quotient singularities (in this case $V/H$) by expressing it as a quotient by a finite group somehow, and then quotienting out the subgroup generated by pseudoreflections, with the Chevalley-Shephard-Todd theorem ensuring that you don't lose smoothness. This description tells us that the canonical stack is any description of the space as a quotient where the group acts without pseudoreflections.

Note that in step 3, we're quotienting out by $K'$, which has no pseudoreflections since it's a commutator subgroup (all its elements must therefore act with determinant 1, and there are no pseudoreflections of determinant 1). So it makes sense that this step is a canonical stack.

It's pretty clear that step 4 is a (trivial) gerbe.

Seeing that step 2 is a root stack is more complicated; you can find the details in the section titled "A local description of Theorem 1" in the paper linked above.

sg.symplectic geometry - Cotangent bundle of a submanifold

Maybe this is a silly question (or not even a question), but I was wondering whether the cotangent bundle of a submanifold is somehow canonically related to the cotangent bundle of the ambient space.
To be more precise:
Let $N$ be a manifold and $iota:M hookrightarrow N$ be an embedded (immersed) submanifold. Is the cotangent bundle $T^ast M$ somehow canonical related to the cotangent bundle $T^ast N$. Canonical means, without choosing a metric on $N$. The choice of a metric gives an isomorphism of $TN$ and $T^ast N$ and therefore a "relation", since the tangent bundle of the submanifold $M$ can be viewed in a natural way as a subspace of the tangent bundle of the ambient space $N$ ($iota$ induces an injective linear map at each point $iota_ast : T_pM rightarrow T_pN$). I think this is not true for the cotangent space (without a metric)
Moreover, the cotangent bundle $T^ast N$ of a manifold $N$ is a kind of "prototype" of a symplectic manifold. The symplectic structure on $T^ast N$ is given by $omega_{T^ast N} = -dlambda$, where $lambda$ is the Liouville form on the cotangent bundle. (tautological one-form, canonical one-form, symplectic potential or however you want). The cotangent bundle of the submanifold $T^ast M$ inherits in the same way a canonical symplectic structure. So, is there a relation between $T^ast N$ and $T^ast M$ respecting the canonical symplectic structures. (I think the isomorphism given by a metric is respecting (relating) these structures, or am I wrong?) As I said, this question is perhaps strange, but the canonical existence of the symplectic structure on the cotangent bundle is "quite strong". For example:
A given diffeomorphism $f:X rightarrow Y$ induces a canonical symplectomorphism $T^ast f : T^ast Y rightarrow T^ast X$ (this can be proved by the special "pullback cancellation" property of the Liouville form).
So in the case of a diffeomorphism the symplectic structures are "the same".
Ok, a diffeomorphism has more structure than an embedding, but perhaps there is a similar relation between $T^ast M$ and $T^ast N$?

---

EDIT: Sorry fot the confusion, but Kevins post is exactly a reformulation of the problem, I'm interested in. To clarify things: with the notation of Kevin's post:

When (or whether) are the pulled back symplectic structures the same ? Under what circumstances holds $a^ast omega_{T^ast N} = b^ast omega_{T^ast N}$

I think this isn't true for any submanifold $M subset N$, but what is a nice counterexample? Is it true for more restricted submanifolds as for example embedded submanifolds which are not just homoeomorphisms onto its image, but diffeomorphisms (perhaps here the answer is yes, using the diffeomorphism remark above?)?

qa.quantum algebra - Solutions of the Quantum Yang-Baxter Equation

The general theory (due to Jimbo) is that each irreducible finite dimensional representation
of the quantised enveloping algebra of a Kac-Moody algebra (not of finite type) gives a trigonometric R-matrix. There is substantial information on these representations but the R-matrices are not explicit. There is a special case which is explicit and is given by the
"tensor product graph" method (this was worked out by Niall MacKay and Gustav Delius).

I used this in my paper:
R-matrices and the magic square. J. Phys. A, 36(7):1947–1959, 2003.
and you can find the references there.

If you want to go beyond this special case and be explicit then you can use
"cabling" a.k.a "fusion".

The only papers which deal with R-matrices not covered by the tensor product graph method that I know of are

Vyjayanthi Chari and Andrew Pressley. Fundamental representations
of Yangians and singularities of R-matrices. J. Reine Angew. Math.,
417:87–128, 1991.

G´abor Tak´acs. The R-matrix of the Uq(d(3)4 ) algebra and g(1)2 affine
Toda field theory. Nuclear Phys. B, 501(3):711–727, 1997.

Bruce W. Westbury. An R-matrix for D(3) 4 . J. Phys. A, 38(2):L31–L34, 2005

Deepak Parashar, Bruce W. Westbury
R-matrices for the adjoint representations of Uq(so(n))
arXiv:0906.3419

The Chari & Pressley paper deals with rational R-matrices.
The last preprint was an incomplete attempt to try and find the trigonometric analogues of these R-matrices.

rt.representation theory - Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not products of Eilenberg-MacLane spaces. This rules out obvious constructions using crossed modules (at least, obvious to me).

One idea is this: take a representation $rho:Gamma_g to SO(3)$ of $Gamma_g = pi_1(Sigma_g)$, the fundamental group of a compact, connected, orientable surface or genus $g$. Then form the associated sphere bundle $X=widetilde{Sigma_g} times_rho S^2 to Sigma_g$.

Then, unless my calculation is wrong, $X$ is a 2-type space with $pi_2(X) = mathbb{Z}$, $pi_1(X) = Gamma_g$ and $k$-invariant $ain H^3(Gamma_g,mathbb{Z})$.

So my question is,

are there any non-trivial representations $rho$ or cohomology classes $a$?

and secondarily,

would a non-trivial representation $rho$ give rise to a non-trivial $k$-invariant in the above situation?

soft question - Taking lecture notes in lectures

Note taking is a bit of a religious dogma for me.

As a chemistry major,I was trained by Dr.Robert Engel something I've found to be very true as a student and has been confirmed by educational psychologists: "There's a connection between your hand and your brain." i.e. writing something out in detail forces your brain to process it.If the notes are good and informative,I find this is very true. Indeed,a good measure of how instructive lecture notes on a subject are is how well you learned from them by taking them down in detail!

That's why to be honest,I'm a little shocked by the responses here to the effect that note taking distracts them from a lecture. How can you be distracted from what the speaker in a graduate level mathematics colloquia is saying if you're forcing yourself to take notes on it?!? Yes,speakers in real time go quickly and of course,we're usually not completely alert and awake-but doesn't that force the mind to pay attention more?

I take very detailed notes along with my commentary. And later-I dissect the notes exhaustively. Or to use Paul Halmos' words;"Don't just read it,FIGHT IT!" And I do. Fiercely. I can think of no better advice to give your students.

There's also a more personal reason for taking notes in a detailed way:Each set of notes is a living record of an experience in your life. It's also a documentation of a personal style of a lecturer-each set of notes is like a personal fingerprint of the author. That relic will remain with the note taker long after the lecture ends.Your memories of the experience will be forever intertwined with those notes.

Now a lot of people here-such as Anton and Theo-are making the case for detailed TeX-ing of notes.Looking at their creations at thier websites as well as the notes made up by my other fellow graduate students,I have to admit-they're making a very compelling case for it. I just wonder about whether or not that personalized element so conducive to learning and nostalgia will be lost once everyone does this.

But once again-the results are VERY impressive. So it's certainly worth a good hard thought.

Those are my 2 cents on the issue.

ag.algebraic geometry - Model theoretic applications to algebra and number theory(Iwasawa Theory)

I'd just like to expand on John Goodrick's mention of Zilber's work on exponential fields and mention that `Categoricity' of is an active area of research. In particular, model theory may be used to give some justification as to why theorems of classical mathematics should hold.

In general it's an interesting question to see what good model theoretic behaviour translates to in the world of classical mathematics.

One way of viewing things (and this is Zilber's point of view) is that if a mathematical structure is useful, and therfore well studied by the mathematical community, then it will be complicated enough to be interesting, but nice enough to be analysed. One aspect of model theory is involved with trying to classify structures with respect to how nice (or wild) they are (e.g. a structure could be strongly minimal, O-minimal, stable, categorical etc...).

At the top of the logical hierarchy sit categorical theories. A theory is $kappa$-categorical if it has one model up to isomorphism in cardinality $kappa$. The stereotypical example of a categorical theory is the theory of algebraically closed fields of characteristic $0$. The unique model of cardinality continuum is $langle mathbb{C}, + , cdot , 0,1 rangle$. This mathematical structure has pretty much every nice model theoretic property that you'd want in a structure - it's strongly minimal (definable sets are very simple i.e. either finite or cofinite), $omega$-stable (there aren't many types of elements around), homogeneous (you can extend partial automorphisms to automorphisms of the whole structure), saturated (you can realise types - i.e. solutions to polynomials are in there). This theory is also complete and `categorical in powers' i.e. $kappa$-categorical for every uncountable cardinal.

An amazing theorem of Morley actually says that if a first order theory is $kappa$-categorical for one uncountable cardinal, then it is categorical for every uncountable cardinal. Morley's theorem (1965) kick started stability theory, and from there Shelah has developed an unbelievable amount of abstract model theoretic technology.

However, after initiating stability theory in the first place it seemed that the study of categorical structures had run its course (Baldwin-Lachlan theorem completely categorises theories which are categorical in powers). But recently Zilber realised that some of Shelah's abstract model theoretic technology regarding infinitary logics can be used to study concrete, well known, and very interesting mathematical structures.

For example, as John Goodrick mentions, if you try and axiomatize the interaction of the exponential function with the complex field i.e. you try and capture the theory of $langle mathbb{C},+ cdot ,0,1, e^x rangle$, and you want it to be categorical, then you need things like Schanuel's conjecture and the conjecture on the intersections of Tori (CIT) to hold. Along similar lines is the Zilber-Pink conjecture.

So model theory can give us some kind of justification as to why certain results should hold. For example, if you look the theory of the universal cover of a non CM elliptic curve over a number field, and you ask it to be $aleph_1$-categorical, then it turns out that a famous theorem of Serre saying that the image of the Galois representation on the Tate module is open must be true.

ac.commutative algebra - Characterization of a certain class of modules-broader than Noetherian

Let R be a commutative ring with 1.

An R-module K has the 'S' property if K/T = K implies that the submodule T is trivial.

By Fitting's Lemma any Noetherian module has the 'S' property. There exist non-Noetherian modules with this property. For example the infinite product of Z_{2}xZ_{3}xZ_{5}x... running over all of the primes has the 'S' property, but is not Noetherian.

I am curious if there is a characterization of these kinds of modules out there.

set theory - Set theories that do require the existence of urelements?

Your question is equivalent to asking whether the urelements, or atoms, can form a proper class. This axiom is consistent with ZFA, but usually ZFA is introduced so as to not insist on this (and indeed, not insist on any atoms at all). I believe that many (or most) of the other standard set-theories-with-urelements also allow this.

Andreas Blass has an article here, where he investigates the connection between some theorems in homological algebra and the Axiom of Choice. In his introduction, he states:

In Section 3, we construct a model of set theory with no nontrivial injective abelian groups. It is a permutation model in which the atoms (= urelements) form a proper class;

In contrast, sometimes it is useful to have only a set of atoms, as witnessed by Eric Hall's article, which contains the following remark.

Definitions and Conventions. The theory ZFA is a modification of ZF allowing atoms, also
known as urelements. See Jech [4] for a precise definition. A model of ZFA may have a proper
class of atoms; however, for this paper we redefine ZFA to include an axiom which says that
the class of atoms is a set (always denoted by A).

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

ag.algebraic geometry - Open affine subscheme of affine scheme which is not principal

For a simple, really concrete example you can also look at:

$A=k[x,y,u,v]/(xy+ux^2+vy^2)$, $X =Spec(A)$, $I=(x,y)$, $U = D(I)$.

Then the functions $f=frac{-v}{x}=frac{y+ux}{y^2}$ and $g=frac{-u}{y}=frac{x+vy}{x^2}$ are defined on $U$. But $yf+xg=1$, so $U$ is affine!

Cheers,

How might I break down bread into glucose in a model of the human digestive system?

Generally speaking and as suggested by Rory, mechanical force probably plays a role in most instances (crustless bread may be different):

http://www.modelgut.com/dmg.html

I was hoping to post more about this but the websites of one of the institutes involved in developing the model gut doesn't provide much accessible information on first read.

Maybe this still gives you something of a starting point:

http://www.ifr.ac.uk/info/about/index.htm

co.combinatorics - Number of metric spaces on N points

My analysis leads to different answers from the above, and to some references. Edit: I get different answers because I changed the question without realizing it. I'm working from the collection of pointed metric balls, i.e., metric balls with a distinguished center. Gabe asked the question about the collection of unpointed metric balls, noting that the same set can be a metric ball with two different centers. That seems more complicated, although I would suggest working from the pointed solution.

Let's let ${1,2,ldots,n}$ be the points in $X$.
The distances from $i$ to the other points in the set $X$ induce a strict weak ordering of those points by their distance from $i$. This has the same information as the set of metric balls with center $i$. Thus the information in the metric is given by all of the comparisons between the distances $x_{(i,j)} = d(i,j)$ and $x_{(i,k)} = d(i,k)$. You can express these relations by a hyperplane arrangement in $mathbb{R}^{n(n-1)/2}$, where the hyperplanes are given by the equations $x_{(i,j)} = x_{(i,k)}$.

You might also think about the triangle inequalities satisfied by all of the distances, and the fact that the distances are all positive numbers. The set of feasible distance vectors is called the "metric cone". Although the metric has an interesting combinatorial structure, it looks like it matters for nothing in this particular question: The hyperplanes all meet at an interior point of the cone in which the distances are all equal. In other words, you can always add a constant distance $h gg 0$ to all of the distances without changing any of the metric balls, so that the triangle inequality and positivity of distance become irrelevant.

If a hyperplane arrangement has the property that all hyperplanes are given by setting two coordinates equal, then the arrangement is called "graphical". The coordinates correspond to the vertices of a graph $G$, and the hyperplanes correspond to the edges. The hyperplane arrangement gives you a partially ordered set of chambes and other faces, ordered by inclusion. This poset has a lot of properties and there are techniques to compute the number of faces from $G$. In our case, $G$ is the line graph $T_n$ of the complete graph $K_n$, which is sometimes confusingly called a triangular graph.

For $n=3$, my answer is that there are 13 different types of metrics, not 7, corresponding to the hyperplane arrangement $x=y$, $y=z$, $x=z$ in $mathbb{R}^3$. In other words, I count 13 types of triangles: 6 scalene, 3 short-base isosceles, 3 long-base isosceles, and 1 equilateral.

Some of the types of metrics lie in chambers, meaning generic metrics in which $d(i,j) ne d(i,k)$ for all $i$, $j$, and $k$. It is a theorem that the number of chambers of the graphical arrangment $A(G)$ of a graph $G$ is $|chi_G(-1)|$, where $chi_G$ is the chromatic polynomial of $G$. (See this excellent review by Richard Stanley.) So I asked Maple to compute

ChromaticPolynomial(LineGraph(CompleteGraph(n)),q)

for small values of $n$, and I got the following answers:

$$chi_{T_3}(q) = q(q-1)(q-2)$$
$$chi_{T_4}(q) = q(q-1)(q-2)(q^3-9^2+29q-32)$$
$$chi_{T_5}(q) = q(q-1)(q-2)(q-3)(q-4)(q^5-20q^4+170q^3-765q^2+1804q-1764).$$

Evaluating at $q=-1$, I get that there are 1, 6, 426, and 542880 generic types of metrics on 2, 3, 4, and 5 points. This sequence is not in the Encyclopedia of Integer Sequences, although possibly it should be. I think that you can obtain the total number of faces of a graphical arrangement $A(G)$ from the Tutte polynomial of $G$, and therefore the total number of types of metrics, but I did not do the calculation.

Approximation of a Normal Distribution function

This is the (divergent) asymptotic development for $
f(x)=int_x^{infty} e^{-{1over 2}t^2} dt
$ given by

$$f(x) sim
e^{-{x^2over 2}} ( {1over x} +
Sigma_{k=1}^{infty} {(-1)^k(2k-1)!over 2^{k-1}(k-1)!} {1over x^{2k+1}} )
$$
or
$$f(x) sim
e^{-{x^2over 2}} ( {1over x} - {1over x^2} + {3over x^5} - {15over x^7} + {105over x^9} - {945over x^{11}}...)$$

It is easily obtained from an integration by parts. The rest is given by an explicit integral. From its expression, one can check that f(x) is in fact squeezed between two consecutive sums of the series. As a result, we have the bound, for all x>0,

$$0 geq f(x) -
e^{-{x^2over 2}} ( {1over x} - {1over x^2} + {3over x^5} - {15over x^7} + {105over x^9}) geq - e^{-{x^2over 2}} {945over x^{11}}$$

Divergent series were standard tools at the beginning of the XXe century. I can't provide a reference, but the book of Hardy "Divergent series" may be a starting point for a bibliographic search.

The divergent asymptotics was chosen because it is "good" at infinity. When truncated to some power (say 9), it has the correct behavior when x goes to infinity, that is, it goes to zero, just like f(x). So it gives an interesting approximation of f for all sufficiently big value of x. This of course is not the case of a development obtained by truncating a converging series in positive powers of x.

ecology - Is there a standard format for plant community composition data?

There does not appear to be a current standard, but Veg-X is a standard that has been developed to facilitate exchange of plot-based vegetation data, and may provide the closest to what you are looking for.

Veg-X is described in Wiser et al 2011 and the project home page is http://wiki.tdwg.org/Vegetation/

From the abstract:

The exchange standard for plot-based vegetation data (Veg-X) allows
for observations of vegetation at both individual plant and aggregated
observation levels. It ensures that observations are fixed to physical
sample plots at specific points in space and time, and makes a
distinction between the entity of interest (e.g. an individual tree)
and the observational act (i.e. a measurement). The standard supports
repeated measurements of both individual organisms and plots, allows
observations of entities to be grouped following predefined or
user-defined criteria, and ensures that the connection between the
entity observed and taxonomic concept associated with that observation
are maintained.

---

Wiser, S.K., N. Spencer, M. De Caceres, M. Kleikamp, B. Boyle, & R.K. Peet. 2011. Veg-X -- An international exchange standard for plot-based vegetation data. Journal of Vegetation Science 22:598-609.

homology - disagreement between two definitions of the singular boundary map

The point Rotman is trying to make is the following: if you have a singular $q$-simplex $sigma:Delta^qto X$, then for example the restriction $sigma|_{[e_0,e_2,dots,e_q]}$ is not a singular $(q-1)$-simplex, simply because its domain $[e_0,e_2,dots,e_q]$ is not the standard simplex $Delta^{q-1}$, which is instead $[e_0,e_1,dots,e_{q-1}]$.

He fixes this by composing with the face maps $varepsilon$, so as to get the domains right.

dg.differential geometry - What is meant by smooth orbifold?

To be quick, just like manifolds, orbifolds have a fixed dimension. This does not vary point to point. This is also true of their tangent spaces. This is actually true for any etale differentiable stack. Here is more explanation:

As mentioned in many of the comments, orbifolds are actually instance of differentiable stacks. To make this jump, we have to really make sense of what a smooth map between orbifolds should be. The correct notion is NOT the one first introduced by Satake, but is slightly more refined, so called- strong (or good) maps between orbifolds. These are precisely those maps which induce geometric morphisms between the associated categories of sheaves (See "Orbifolds, Sheaves and Groupoids" by Pronk and Moerdijk). From now on, I will only consider strong maps.

If O is an orbifold, and f,g:M->O are smooth maps from an manifold, then, because of the group-structure in the charts, it makes sense to consider when two such maps are isomorphic. So, to every manifold M, we can assign the category Hom(M,O)- which is a groupoid (every map between two smooth maps is an isomorphism). This assignment "M mapsto Hom(M,O)" is a weak presheaf in groupoids over the category of differentiable manifolds (this is just fancy talk for saying that it's nearly a contravariant functor, but (g^)(f^) and (fg)^* need only be naturally isomorphic rather than equal, and then some needed coherence conditions needed to make things consistent after this). The point is, given another orbifold, L, the category of weak natural transformations between Hom( ,O) and Hom( ,L) is naturally equivalent to Hom(O,L). This means, orbifolds embed fully-faithfully into stacks, so instead of studying the orbifolds themselves, we can study the functors which they represent.

The fuctor Hom( ,O) is not just a functor, but it's actually a stack (it's a "sheaf of groupoids", so it satisfies some gluing conditions).

To do this in practice, if you start with an orbifold given to you as a topological space with a chart, then from this chart, you can construct an etale proper Lie groupoid G (See "Orbifolds, Sheaves and Groupoids"). If instead, you are given a presentation of a Lie group acting on a manifold with finite stabilizers, simply take G to be the action groupoid. Then the functor Bun_G which assigns every manifold M the groupoid of principal G-bundles over M is the same as the functor Hom( ,O). (In general, a differentiable stack is a weak functor from manifolds to groupoids of the form Bun_G for some Lie groupoid G.)

Now, every single stack has a "tangent stack". This can be shown by abstract nonsense (I can elaborate, or you can look at "Vector Fields and Flows on Differentiable stacks" by Richard Hepworth). When the the stack is X=Bun_G, the tangent stack turns out to simply be TX=Bun_TG. (By TG I mean if you look at the diagram expressing G as a groupoid object in manifolds, apply the tangent functor to get a groupoid object in vector bundles, and then apply the forgetful functor to get another groupoid object in manifolds, you get TG). There is a canonical map from TX->X. A vector bundle over a stack X is defined to be a map Y->X of stacks such that if M->X is any map from a manifold, the pullback $Y times_X M to M$ is a vector bundle. In general, TX->X is not a vector bundle but a 2-vector bundle, BUT, if G is etale (or Morita equivalent to an etale guy), it IS a vector bundle (See "Vector Fields and Flows on Differentiable stacks"). In particular, the dimension does not change.

To be concrete, if G acts on M, and x is in M, then the Lie algebra of G_x acts on T_xM. If p is the image of x in M//G (where here I mean the stacky quotient) then T_p (M//G)=T_x(M)//Lie(G_x). But, in the case of orbifolds, the stabilizers are finite, so, they have no Lie algebras.

homological algebra - How do I know the derived category is NOT abelian?

I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel that should be there but isn't, or something to that effect (for that matter, I could ask the same question just about the homotopy category).

Unless I'm mistaken, the derived category of a semisimple category is just a ℤ-graded version of the original category, which should still be abelian. So even though I have no reason to doubt that this is a really special case, it would still be nice to have an illustrative counterexample for, say, abelian groups.

st.statistics - Is there a tool for finding probability distributions given some samples?

I'm looking for a tool that does "probability distribution fitting" given a set of data points. Sort of like curve fitting, but tries to fit to standard density distributions.

For example if I input

(0, 0.0497871), (1, 0.149361), (2, 0.224042), (3, 0.224042), (4,0.168031), (5, 0.100819), (6, 0.0504094)

I would hope that it would tell me these data points fit a Poisson distribution.

cohomology - Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable tensoring the decomposition of the algebraic $H_{rm dR}^{1}(E,K)$ in eigenspaces for the natural $F^times$-action coincides with the Hodge decomposition of $H_{rm dR}^{1}(E,{Bbb C})$ and (for ordinary good reduction at $p$) with the Dwork-Katz decomposition of $H_{rm dR}^{1}(E)otimes B$ for $p$-adic algebras $B$.

Then, he asks for a converse statement. Namely, is it true that if the Hodge decomposition of $H_{rm dR}^{1}(E,{Bbb C})$, where $E_{/K}$ is an elliptic curve, is induced by a splitting of the algebraic de Rham, then $E$ has complex multiplications?

The question is left unanswered in that paper. Does anyone know if the question has been answered since?

algebraic k theory - Symplectic Steinberg group

Let me do a bit of necromancy here and address the third question.

A Note on Milnor–Witt K-Theory and a Theorem of Suslin by K. Hutchinson and L. Tao provides a description for $H_2left(Sp(F),mathbb{Z}right)=H_2left(SL(2,F),mathbb{Z}right)$ for an infinite field $F$ as Milnor—Witt K-theory $K_2^{MW}(F)$, introduced by F. Morel in 2003 in his study of $mathbb{A}^1$-homotopy theory.

$K_*^{MW}(F)$ is a graded associative ring generated by the symbols $[u]$, $uin F^*$ of degree $+1$ and one symbol $eta$ of degree $-1$ modulo the following relations:

• For $ain Fsetminus{0,1}$, $[a]cdot[1-a]=0$;


• For $a,b,in F^*$, $[ab]=[a]+[b]+eta[a][b]$;


• For $uin F^*$, $[u]eta=eta[u]$;


• $eta^2[-1]+2eta=0$.

The proof is based on Matsumoto—Moore presentation for $H_2left(Sp(F),mathbb{Z}right)$ and the coincidence of $K_2^{MW}(F)$ with $K_2^{MM}(F)$.

PS. The equality $H_2left(Sp(F),mathbb{Z}right)=H_2left(SL(2,F),mathbb{Z}right)$ has something to do with $mathsf{A}_1=mathsf{C}_1$ (see this MO question).

co.combinatorics - Pascal Triangle and Prime Numbers

I'm not convinced this is MO-appropriate, but I'm posting an answer 'cause what I'd have to say is probably too long for a comment.

Expanding on Reid's comment. Yeah, Lucas' theorem is nice. Lucas' theorem is one of a fair number of combinatorial results which can be thought of as "first steps towards p-adic numbers." What's that mean? There's a "different" absolute value that you can define on rational numbers, which has a lot of the same properties as the usual absolute value, but in other ways behaves totally differently. Actually there are an infinite number of these guys, one for every prime p! It's called the p-adic absolute value, and you can read about it here.

What the p-adic numbers do is help you get around the following obstacle: Say you want to tell whether a quotient of two numbers, a/b, is divisible by p. (We'll assume for now that a/b is definitely an integer, although this ends up not mattering at all. But "divisibility" is a trickier notion for non-integers.) If a is divisible by p and b isn't, then it's obvious that a/b is; if a isn't divisible by p, then of course a/b isn't. But things get tough if both a and b are divisible by p; it could happen that a is divisible by p^2, and b is divisible by p but not p^2. Or that a is divisible by p^17, and b is divisible by p^14 but not p^15. You see how this gets confusing! The p-adic absolute value encodes this sort of information for you.

This also mentions why we don't work with, say, 10-adic numbers in mathematics; it's because if you take the integers but consider two integers to be the same if they have the same remainder when you divide by n, you can still multiply and add and subtract perfectly well. So you get something called a ring. And if n is prime, you can also divide numbers! (Well, you can't divide by 0, or by a multiple of the prime, which is "the same as" 0. But this is true no matter what, so it's not a real problem.) But this isn't true for composites.

Anyway, the patterns for primes in Pascal's triangle are pretty well known. Google "Pascal's triangle modulo" (without quotes, probably) to find more stuff. Composites don't behave as nicely, for the reasons Wikipedia and I both briefly mentioned, but powers of primes do have interesting patterns, which you can read about in this wonderfully-titled paper.

Hope this helps!

evolution - Does becoming martyr have an evolutionary advantage?

There isn an effect called "Indirect reciprocity" where individuals just give to everyone they meet without direct requirement of reciprocity.

This sort of benefit to others is common - hospitality to strangers, general politeness, good customer service all fall along these lines. You hope they will come back and benefit you again, but maybe they will tell someone else who will know you are a good community member.

It is only sustainable in a system where the cost/benefit ratio is less than the reputation benefit of the act. It sounds as if this is only good for public acts but if the benefit is transferred to a social entity that outlasts the individual (like your children, a relative's children, a religion or a corporation say), the result could still hold.

If you think about typical morality/ethics really it still makes sense to think that what we call altruism must still have a net positive benefit. If there is no benefit long term or to anyone, it really isn't useful or even good, its random. What we usually call altruism is usually some sort of reciprocal cooperation.

A soldier who dies in combat or someone who dies for their beliefs but everyone knows about it as a public statement benefits from their act indirectly. I don't think its altruism in the pure sense of the word. Defending the nation, ones' beliefs or whatever is, in its sense its own reward. Veterans come back from a war are hopefully respected for their work. Having a purple heart can be a good thing to show people. I'm not saying these people are adequately compensated for what they have been through, but just trying to draw a distinction between pure biological altruism and 'indirect reciprocity'.

Examples of Indirect reciprocity might be the use of tax money to build highways and build power and water infrastructure. Its important - its the glue that holds a nation or a group together. If you got punished for doing these things we wouldn't be hanging as a nation very long!

A martyr with no family at all who would benefit would still count as an altruism I think, but most acts of public piety and sacrifice do benefit the individual by reputation. Something to think about.

ag.algebraic geometry - Algebraic equivalence VS Numerical Equivalence - An Example.

This question is arose from the question
Difference between equivalence relations on algebraic cycles
and the example 3 in lecture 1 in Mumford's book Lectures on curves on an algebraic surface.

Here is the example.

Let $E$ be an elliptic curve. A curve $C$ on $mathbb{P}^1times E$ is said has projection degree $(d, e)$, if the projections $Ctomathbb{P}^1$ and $Cto E$ are of degree $d$ and $e$ respectively. In that example Mumford shows that the curves with projection degree (d,e) and $d>0$ forms a $d(e+1)$ irreducible family of curves. Clearly, these curves are algebraic equivalence. But this family is not a linear system.

I can understand why the dimension is $d(e+1)$ with the help that numerical and algebraic equivalence coincide for divisors. But why the equivalences coincide for divisors? Where is the place discussing these equivalence of algebraic cycles? Also what is the irreducible family in this example?

st.statistics - Conjugate prior of the Dirichlet distribution?

Neil sent me an email asking:

===

I read your post at http://www.stat.columbia.edu/~cook/movabletype/archives/2009/04/conjugate_prior.html and I was wondering if you could expand on how to update the Dirichlet conjugate prior that you provided in your paper:

S. Lefkimmiatis, P. Maragos, and G. Papandreou,
Bayesian Inference on Multiscale Models for Poisson Intensity Estimation: Applications to Photon-Limited Image Denoising,
IEEE Transactions on Image Processing, vol. 18, no. 8, pp. 1724-1741, Aug. 2009

In other words, given in your paper's notation the prior hyper-parameters (vector $mathbf{v}$, and scalar $eta$), and $N$ Dirichlet observations (vectors $mathbf{theta}_n, n=1,dots,N$), how do you update $mathbf{v}$ and $eta$?

===

Here is my response:

Conjugate pairs are so convenient because there is a standard and simple way to incorporate new data by just modifying the parameters of the prior density. One just multiplies the likelihood with its conjugate prior; the result has the same parametric form as the prior, and the new parameters can be readily "read-off" by comparing the likelihood-prior product with the prior parametric form. This is described in detail in all standard texts in Bayesian statistics such as Gelman et al. (2003) or Bernardo and Smith (2000).

In the case of the Dirichlet and its conjugate prior described in our paper and using its notation, after observing $N$ Dirichlet vectors $mathbf{theta}_n$, $n=1,dots,N$, where each vector $mathbf{theta}_n$ is $D$ dimensional with elements $theta_n[t]$, $t=1,dots,D$, the $D+1$ hyper-parameters should be updated as follows:

• $eta_N = eta_0 + N$


• $v_N[t] = v_0[t] - sum_{n=1}^N ln theta_n[t], quad t=1,dots,D$, where $eta_0$, $mathbf{v}_0$ and $eta_N$, $mathbf{v}_N$ are the initial and updated model parameters, respectively.

You can verify this in a few lines of equations by following the previously described general rule.

Hope this helps!

ac.commutative algebra - Locally square implies square

OK, I've got it. There is no such local criterion for squareness.

Let $k$ be a field of characteristic not $2$. Take the ring of triples $(f,g,h) in k[t]^3$, subject to the conditions that $f(1)=g(-1)$, $g(1)=h(-1)$ and $h(1)=f(-1)$. Consider the element $(t^2,t^2,t^2)$. If this were a square, its square root would have to be $(pm t, pm t, pm t)$. But two of those $pm$'s would be the same sign, and $t$ evaluated at $1$ and at $-1$ are not equal.

Now, to check that $(t^2, t^2, t^2)$ is everywhere locally a square. Geometrically, we are talking about three lines glued into a triangle. Any prime ideal has a neighborhood which is contained in the union of two neighboring lines, say the first two. On the first two lines, $(-t, t, 1)$ is a square root of $(t^2, t^2, t^2)$.

For the suspicious, an algebraic proof. Set $u_1=(0, (1+t)/2, (1-t)/2)$ and let $u_2$ and $u_3$ be the cyclic permutations thereof. We have $u_1+u_2+u_3=1$ so, in any local ring, one of the $u_i$ must be a unit. WLOG, suppose that $u_1$ is a unit. Notice that $u_1 (1, -t, t)^2 = u_1 (t^2, t^2, t^2)$. So, in a local ring where $u_1$ is a unit, $(1,t,-t)$ is a square root of $(t^2, t^2, t^2)$.

pr.probability - Is there a way to analytically compute the recurrence time of a finite Markov process?

This is a response to a comment.

The coupon collector's problem is elementary. I don't have a particular scholarly reference in mind, but rather the technique of the proofs. There are a few proofs of the $n H_n$ expected time to collect all coupons. One possibility is that you can compute the expected time to collect the $k$th new coupon,
$n/(n-k+1)$. That uses a lot of symmetry you don't have for a general Markov process. Here, you have transition probabilities and times on (current location, subset visited so far).

Analogous to what I did here, you can use inclusion-exclusion. The expected time to cover everything (with discrete time) is the sum of the probability that you haven't covered everything by time $t-1$, which you can express as

$$sum_t sum_{Ssubset V} -1^{|S|+1}Prob({X_i}_{ilt t}cap S = emptyset) $$

where $V$ = ${1,...,n}$. You can switch the order of summation to get about $2^n$ analytically solvable problems about avoiding particular subsets.

$$sum_{Ssubset V} -1^{|S|+1} A(S)$$

where $A(S)$ = expected time before you first enter $S$.

The same holds for continuous time.

big list - "Oldest" bug in computer algebra system?

The goal of this question is to find an error in a computation by a computer algebra system where the 'correct' answer (complete with correct reasoning to justify the answer) can be found in the literature. Note that the system must claim to be able to perform that computation, not implementing a piece of (really old) mathematics is sad, but is a different topic.

From my knowledge of the field, there are plenty of examples of 19th century mathematics where today's computer algebra system get the wrong answer. But how far back can we go?

---

Let me illustrate what I mean. James Bernoulli in letters to Leibniz (circa 1697-1704) wrote that [in today's notation, where I will assume that $y$ is a function of $x$ throughout] he could not find a closed-form to $y' = y^2 + x^2$. In a letter of Nov. 15th, 1702, he wrote to Leibniz that he was however able to reduce this to a 2nd order LODE, namely $y''/y = -x^2$. Maple can find (correct) closed-forms for both of these differential equations, in terms of Bessel functions.

An example that is 'sad' but less interesting is
$$r^{n+1}int_0^{pi}cos(rrho cos (omega))sin(omega)^{2n+1}domega$$
with $n$ assumed to be a positive integer, $r>0$ and $rho$ real; this can be evaluated as a Bessel functions but, for example, Maple can't. Poisson published this result in a long memoir of 1823.

One could complain that (following Schloemilch, 1857) that he well knew that
$$J_n(z) = sum_{0}^{infty} frac{(-1)^m(z/2)^{n+2*m}}{m!(n+m)!}$$
Maple seems to think that this sum is instead $J_n(z)frac{Gamma(n+1)}{n!}$, which no mathematician would ever write down in this manner.

Another example which gets closer to a real bug is that Lommel in 1871 showed that the Wronskian of $J_{nu}$ and $J_{-nu}$ was $-2frac{sin(nupi)}{nu z}$. Maple can compute the Wronskian, but it cannot simplify the result to $0$. This can be transformed into a bug by using the resulting expression in a context where we force the CAS to divide by it.

For a real bug, consider
$$int_{0}^{infty} t^{-lambda} J_{mu}(at) J_{nu}(bt)$$
as investigated by Weber in 1873. Maple returns an unconditional answer, which a priori looks fine. If, however, the same question is asked but with $a=b$, no answer is returned! What is going on? Well, in reality that answer is only valid for one of $0lt alt b$ or $0lt b lt a$. But it turns out (as Watson explains lucidly on pages 398-404 of his master treatise on Bessel functions, this integral is discontinuous for $a=b$. Actually, the answer given is also problematic for $lambda=mu=0, nu=1$. And for the curious, the answer given is
$$frac{2^{-lambda}{a}^{lambda-1-nu}{b}^{nu}
Gamma left( 1/2nu+1/2mu-1/2lambda+1/2 right)} {
Gammaleft( 1/2mu+1/2lambda+1/2-1/2nuright) Gamma left( nu+1 right)}
{F(1/2-1/2mu-1/2lambda+1/2nu,1/2nu+1/2mu-1/2lambda+1/2;nu+1;{frac {{b}^{2}}{{a}^{2}}})}
$$

---

EDIT: I first asked this question when the MO community was much smaller. Now that it has grown a lot, I think it needs a second go-around. A lot of mathematicians use CASes routinely in their work, so wouldn't they be interested to know the 'age' gap between human mathematics and (trustable) CAS mathematics?

chronobiology - Why is maintaining a circadian rhythm important?

In theory, having a circadian rhythm should help anticipating daily environmental changes (light, temperature etc.) so that the metabolic performance is maximized.

In practice, mice chronically exposed to environmental light-dark cycles with a period length dramatically shorter or longer than that of their circadian clock are prone to become obese. Moreover, animals kept under short cycles die earlier than mice kept under normal or extended cycles. So, chronic circadian disturbance by a shortened light-dark cycle increases mortality (at least in mice). In humans, shift workers are likely the population that is more susceptible to circadian cycle derangements.

Ref: Park et al., Neurobiol Aging. 2011

ac.commutative algebra - Is (relatively) algebraically closed stable under finite field extensions?

Counterexample:

let $F$ be a non-perfect field of characteristic $p$.

Let $L$ be an extension of $F$ of degree $p^2$ such that $L=F(a,b)$ with $a^p,b^pin F$.

The polynomial $f(Y):=Y^p-(a^px^p+b^p)in F(x)[Y]$ is irreducible, where $F(x)$ is the rational function field in the variable $x$.

Consider $F^prime := F(x,y)$, where $y$ is a root of $f$.

Then $F$ is algebraically closed in $F^prime$: let $K$ be the algebraic closure of $F$ in $F^prime$. Then $[K:F]=[K(x):F(x)]leq [F^prime :F(x)]=p$. Hence $Kneq F$ implies $F^prime =K(x)$ and thus $y=g(x)in K[x]$ with $[K:F]=p$ -- in contradiction to the choice of $y$.

The tensor product $F^primeotimes_F L$ is not a field: the tensor product $F^primeotimes_K L$ equals $L(x)[Y]/(f)$. However $f$ is a $p$-th power in $L(x)[Y]$.

H

mg.metric geometry - Feasibility of a list of prescribed distances in R^3

This isn't a complete answer, but it might shed some light on what is involved.

Let $n=m(m-1)/2$, and let's say you've decided which pairs of the $m$ points should get which of the $n$ distances. You should form the $mtimes m$ matrix $M$ whose entries are the squared distances, i. e., $M_{ij}=||x_i-x_j||^2$, where the $m$ (unknown) points are $x_1,ldots,x_m$. Suppose the $m$ points all belong to $mathbb{R}^k$. Let $X$ be the $ktimes m$ matrix whose $j$-th column is $x_j$. Without loss of generality, the points have mean zero, so $x_1 + ldots + x_m = Xalpha^T = 0$, where $alpha=(1,ldots,1)$. It's easy to show that if $P=I-frac1malpha alpha^T$ is the orthogonal projection onto the orthogonal complement of $alpha$, then $PMP = -2X^TX$. Thus, once you know $M$, you can attempt to find $X$ by a modification of Cholesky decomposition applied to $-frac12PMP$. You need to modify the procedure to make it return a $ktimes m$ matrix, not an $mtimes m$ matrix. (Since you want $mathbb{R}^3$, you should take $k=3$.) If it succeeds, then you have a solution; otherwise, no solution exists.

Unfortunately, that assumes that you've assigned the $n$ distances to pairs of points already. If the above procedure fails, then it's still completely possible that some other assignment of distances to pairs will yield a solution. I'm not seeing any easy way to solve this; checking all possible assignments is out of the question even for pretty modest values of $m$ even if you can break all the symmetries, since I think you would need to check $(m(m-1)/2)!/m!$ assignments in the worst case. Maybe there's a way to reduce the number of permutations you need to check, but I'm not seeing it.

homotopy theory - Explicit classifying spaces for crossed complexes

It is not clear to me what you need / want. The classifying space of a cyclic group is constructed using a presentation and then killing off higher identities that may be around (there aren't any!). From that viewpoint the question you seem to ask is related to the combinatorial group theory of the group in question (or am I misderstanding the question.) There are examples that might help due to Loday in his paper on higher syzygies, but that may not quite fit the bill as he does not explicitly give the link with classifying spaces.

If you are happy with simplicial methods then you can build a simplicial T-complex from a crossed complex of groups by a modified Dold-Kan construction. The classifying space of that simplicial group (its Wbar) is something that has the same properties as Ronnie's classifying space. It is feasible if you know the crossed complex reasonably fully to construct this explicitly.

It depends what you need the construction for? How is your crossed complex arising precisely (and incidently what do you mean by the `topology', will a simplicial model do)?

human biology - What are possible health risks to women having large numbers of children?

One health risk from going through multiple pregnancies is the risk of a uterine prolapse, also called a prolapsed uterus. The weakening of ligaments basically leads to the cervix and uterus "sagging". In the most severe cases, the uterus ends up in the vagina.

Parity and obesity were strongly associated with increased risk for uterine prolapse, cystocele, and rectocele.

Pelvic organ prolapse in the women's health initiative: Gravity and gravidity

Parity in this context is the number of pregnancies that led to a viable gestational age (pregnancies excluding miscarriages)

Uterine prolapse makes subsequent pregnancies high-risk and can lead to fetal and maternal problems.

The general risk of anything happening increases due to going through multiple pregnancies in a short time span, though. Without sufficient recovery between pregnancies, women are at a higher risk of placental abruptions and placenta previa. For the baby, this leads to a higher risk of premature birth with the associated issues of low birth weight and small size.

dg.differential geometry - Does Ricci flow with surgery come from sections of a smooth Riemannian manifold?

More precisely, is Ricci flow with surgery on a 3-dimensional Riemannian manifold M given by the "constant-time" sections of some canonical smooth 4-dimensional Riemannian manifold?

There would be a discrete set of times corresponding to the surgeries, but the 4-dimensional manifold might still be smooth at these points even though its sections would have singularities. The existence of such a 4-manifold is well known if there are no singularities: the problem is whether one can still construct it in the presence of singularities.

Background: Ricci flow on M in general has finite time singularities. These are usually dealt with by a rather complicated procedure, where one stops the flow just before the singularities, then carefully cuts up M into smaller pieces and caps off the holes, and constructs a Riemannian metric on each of these pieces by modifying the metric on M, and then restarts the Ricci flow. This seems rather a mess: my impression is that it involves making several choices so is not really canonical, and has a discontinuity in the metric and topology on M. If the flow were given by sections of a canonical smooth 4-manifold as in the question this would give a cleaner way to look at the surgeries of Ricci flow.

(Presumably if the answer to the question is "yes" this is not easy to show, otherwise people would not spend so much time on the complicated surgery procedure. But maybe Ricci flow experts know some reason why this does not work.)

gr.group theory - What does the typical non-solvable group look like?

I don't have reasonable access to internet at the moment, but I will edit this and add references when I can.

There is an old paper called "Almost Every group is solvable" where one considers a finite group and its jordan holder decomposition. Ignoring all the factors which are cyclic groups, one multiplies the size of the remaining factors and divides by the size of the group. This gives a number which is <=1, and is equal to 1 only for nonabelian simple groups. They show in that paper that the "average" over all groups of this statistic is 0. In other words, most simple composition factors are cyclic abelian groups.

I do not know enough about PSL_2(F_p) to say whether this fits the bill (in other words, as p increases, what is the chart of this statistic).

ag.algebraic geometry - Fundamental groups of the spaces of rational functions

Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.

Let $X$ be a smooth complete complex curve (=a compact Riemann surface) of genus $g$ and let $Rat(X,d)$ be the space of all regular (=holomorphic) maps from $X$ to $mathbf{P}^1(mathbf{C})$ of degree $d$. In this question I'm interested in the fundamental group of the open subset $U(X,d)$ of $Rat(X,d)$ formed by all $f$ such that all critical points of $f$ are simple and all critical values are distinct. (A critical point is a point at which the derivative of $f$ vanishes; a critical value is the image of a critical point.) To be more specific, let's say I'd like to

1. find a "nice" system of generators of $pi_1(U(X,d))$;




2. to describe, for each of these generators, its image under the map induced by the map $G$ from $U(X,d)$ to the configuration space $B(mathbf{P}^1(mathbf{C}),k)$ of unordered subsets of $mathbf{P}^1(mathbf{C})$ of cardinality $k=2(d+g-1)$ that takes $f$ to its branch divisor (i.e. the divisor of the critical points).

Here are some remarks that may be useful (or may not):

First, here is how one can think of the fundamental group of $Rat(X,d)$. By associating to every function its divisor of poles we get a map $F$ from $Rat(X,d)$ to the $d$-th symmetric power $S^d(X)$ of $X$.

Assume $d> 2g-2$. By the Riemann-Roch theorem, for any degree $d$ divisor $D$ the linear space ${cal{L}}(D)=H^0(X,{cal{O}}(D))$ (which is formed by all rational functions $f$ such that for any $xin X$ the order of the pole of $f$ at $x$ is at most the multiplicity of $x$ in $D$) is $d-g+1$. So $F$ is surjective and a fiber of $F$ is $mathbf{C}^{d-g+1}$ minus some number of hyperplanes (these are given by the condition that order the pole of $f$ at a point $x$ of $D$ is less then the multiplicity of $x$ in $D$).

The map $F$ is probably not a fibration. However, the fundamental group of $Rat(X,d)$ is spanned by the loops in a general fiber of $F$ going around one of the hyperplanes, and lifts of the loops in $S^d(X)$ (these are all of the form "one of the points moves along a loop in $X$ and the other stand still").

Second, recall that the Jacobian $J(X)$ of $X$ is defined as follows. Integration along cycles gives an injective map $H_1(X,mathbf{Z})tomathbf{C}^g=Hom(H^0(X,Omega_X),mathbf{C})$ and the Jacobian of $X$ is the quotient. Moreover, once we have chosen a base point $x$ in $X$, we get a natural map $j:Xto J(X)$ defined as follows: for any $x'in X$ take a path $gamma$ from $x$ to $x'$ and set $j(x')$ to be the image in $J(X)$ of the "integration along $gamma$ function". This is well defined map that can be extended by $mathbf{Z}$-linearity to $S^d(X)$.

Abel's theorem says that two disjoint effective divisors are the divisors of the zeros and the poles of a rational function if and only if their images under $j$ coincide. This may be useful in this problem, but I don't see how.

ag.algebraic geometry - Modules and Square Zero Extensions

If $X$ is a scheme and $mathcal M$ is a quasi-coherent sheaf on $X$ then we can
form a sheaf of rings $mathcal A := mathcal O_X oplus mathcal M$, on which multiplication
of sections is given just by the same formula as for $R oplus M$.

The pair $(X,mathcal A)$ is then a scheme which is an infinitesimal thickening of
$X$, and this is precisely how you pass from a quasi-coherent sheaf to the corresponding
thickening; it is just a sheafified version of the construction in your posting.

(Regarding cohomology, in your question you seemed most interested in the case when
$X =$ Spec $R$ is affine, in which case quasi-coherent sheaves have vanishing higher cohomology, so I'm not sure there is much to say about this.)

Added in response to comment below: To see how these come up geometrically,
consider for example a $k$-scheme $X$ embedded diagonally into $X times X$.
(Here $k$ is a field, and everything is happening over Spec $k$.)

Let $mathcal I_X$ be the ideal sheaf on $X times X$ cutting out the diagonal,
and consider the square-zero thickening
$mathcal O_{Xtimes X}/mathcal I_X^2$ of $X$.

This sits in the short exact sequence
$$0 to Omega^1_X = mathcal I_X/mathcal I_X^2
to mathcal O_{Xtimes X}/mathcal I_X^2 to mathcal O_{Xtimes X}/I_X = mathcal O_X
to 0.$$ The projection $p_1:Xtimes X to X$ gives a spliting of this short exact
sequence, and so we find that $mathcal O_{Xtimes X}/mathcal I_X^2 = mathcal O_X oplus
Omega^1_X$.

Recapitulating, we see that in the special case $mathcal M = Omega^1_X$, then
$(X, mathcal O_X oplus Omega^1_X)$ is equal to the first order infinitesimal neighbourhood of $X$ in $Xtimes X$.

Suppose for example that $X$ is a smooth curve, so that $Omega^1_X$ is a line-bundle.
Then $(X,mathcal O_X oplus Omega^1_X)$ is locally like the dual numbers (as you observe
in your comment) but is globally twisted (unless $X$ is an elliptic curve, i.e. the genus is 1, which is the one case when $Omega^1_X$ is actually trivial).

This should give you some sense of how these kinds of objects arise geometrically (and
why one would consider other examples rather than just the dual numbers).

linear algebra - Geometric interpretation of singular values

The singular values of a matrix A can be viewed as describing the geometry of AB, where AB is the image of the euclidean ball under the linear transformation A. In particular, AB is an elipsoid, and the singular values of A describe the length of its major axes.

More generally, what do the singular values of a matrix say about the geometry of the image of other objects? How about the unit L1 ball? This will be some polytope: is there some natural way to describe this shape in terms of singular values, or other properties of matrix A?

rt.representation theory - Are there interesting monoidal structures on representations of quantum affine algebras?

The fusion product for affine Lie algebras is closely related to the existence of "evaluation homomorphisms" from the loop algebra to the finite-dimensional semisimple Lie algebra g, which split the natural inclusion of g as the subalgebra of constant loops. In the quantum case there is no evaluation map from the quantum affine algebra to the finite-type quantum algebra outside of type A (this is proved - at least for Yangians - in Drinfeld's original paper I'm pretty sure).

You see consequences of this in lots of places: e.g. for representations of g, the evaluation homomorphisms mean any irreducible representation for g can be lifted to an irreducible representation of the affine Lie algebra Lg. On the other hand, irreducible representations of the associated quantum groups do not (necessarily) lift to representations of quantum affine algebras, and so one asks about "minimal affinizations" -- irreducible finite dimensional representations of the quantum affine algebra which have the given irreducible as a constituent when restricted to the finite-type quantum group.

That said, the "ordinary" tensor product for finite dimensional representations of quantum affine algebras is pretty interesting -- it's not braided any more for example.