molecular biology - Is the eukaryotic nucleus composed of a single or double membrane?

The nucleus is topologically single membrane but functionally, and as visualized, double membrane. Naturally, this is a bit confusing.

First consider the mitochondria and plastids. One of these organelles has two entirely separate lipid bilayers, one of which is nested inside the other. If you start inside the organelle and draw a line to an external point, the line will pass through two lipid bilayers (assume there are no membrane folds to be encountered), first through the nested one and then through the entirely separate outer bilayer.

The nucleus at a gross level appears to be structured similarly. A line from the interior to the exterior will usually pass through two lipid bilayers, one that is the inner wall of the nucleus, then through the outer wall. This may seem like the inner wall is separate from, and nested inside, the outer wall, but it is not. As nicely illustrated in your reference (Martin 2005), the two walls are continuous with each other and not nested. Because of this, a line from nuclear interior to exterior may in fact not pass through any lipid bilayers; it may pass though a nuclear pore. Topologically (mathematically) speaking, the two walls are one surface, and the "inside" of the nucleus is on the same side of the surface as the outside. (The topological interior is perinuclear space and the interior of the endoplasmic reticulum.) Functionally though, the structure acts as a double membrane since it physically constrains its contents.

The distinction is not very important for most biological contexts, but for a few purposes needs to be considered. The main reason to pay attention to the distinction is that it probably is related to the evolutionary history of the nucleus: it is likely to have evolved from a single bilayer somehow. In contrast, the mitochondria and plastids, with nested bilayers, evolved from two bilayers, one from a separate cell and one from an enveloping piece of a host cell. This is why the distinction is important in the cited paper, which concerns the evolution of the nucleus.

sequences and series - Seeking for a formula or an expression to generate non-repeatative random number ..

When we hear random permutations, we bring in our intuition about permutations, and try to give a method which could generate a complicated permutation. Thus, I think we didn't pay enough attention to your examples like n*3 mod N, which for most situations would not be an acceptable way of generating random numbers. The only problem is what to do if N is divisible by 3. As far as I can tell, divisibility by 10 is irrelevant, so I'm not sure why you mentioned it.

You say you don't want to write a program, just a simple formula in Excel. This is reasonable, and even something which makes sense mathematically: There are a few operations available in Excel formulas such as addition, exponentiation, factorial, conditional evaluation based on whether a statement is true or false (characteristic functions), etc. Can one create a formula with fixed complexity which takes in n and N, and which is a permutation of {1,...,N] for a fixed N? Trivially returning n works, but can one produce a permutation other than (+-n+k mod N)+1?

I suggest creating a formula which is equivalent to the following:

If N is not divisible by 71, return (71*n mod N) + 1.
Otherwise N is divisible by 71. Permute the last digit base 71: return a + (3*b mod 71) +1
where n-1 = a + b and a is divisible by 71 and $0 le b lt 93$, i.e.,
b = n-1 mod 71.
a = n-1 - (n-1 mod 71).

IF(MOD(N,71)!=0,n-(MOD(n-1,71)) + MOD(3*(MOD(n-1,71),71),MOD(71*n,N)+1).

(Debugging left to the reader.)

This would be lousy as a random permutation, but it may be acceptable for some purposes.

A better random permutation might be based on f(n), where f reverses the lowest binary digits of n if n is at most than the greatest power of 2 less than N, and does nothing if n is greater. Try f(N+1-f(n)). This can be done using the DEC2BIN and StrReverse functions, but you need a little Excel expertise to use those.

Once you have a few ways to generate random permutations, you can compose them, and even using unsatisfactory random permutations like adding floor(sqrt(N)) can improve the appearance of the resulting permutation.

biochemistry - What is the correct model for enzyme-substrate complementarity?

Both models are true depending on how you frame the mechanisms of catalysis. As mentioned by @Blues, proteins are highly dynamic. In that manner, a protein will adopt both the unbound active state shown in the induced fit model and the complementary shape shown in the lock and key model.

(apologies since this is the only figure that I could find to explain this concept). Using the above description the induced fit model (E) will change its structure to the E*S model. In the lock and key model, the E state will be equivalent to the E*S state. According to the below figure, this would imply that the E*S state always exists but as it is a few kcals higher in free energy, the state is rarely seen. Thermodynamically, this means that the "lock" always exists but it is an unstable configuration. When the substrate is added to the system, it will stablize the lock and thermodynamically favor an E*S state.

Long story short, the induced fit model is a good explanation of how enzymes morph into an active state but depending on how you frame the mechanism, you are always seeing a lock-key model (at least according to my enzymology professor). Unfortunately, the majority of biochemistry textbook continue to teach using the induced fit model since it is a much easier concept to understand given the majority of undergrads; and 1st year graduates' understanding of statistical thermodynamics.

The induced fit model is more appropriately used to understand the mechanisms of substrate specificity. As hinted by your professor, enzymes will perform their function in the lock-key mechanism. This is true for many serine proteases which all do the exact same reaction. However, substrate specificity can be incorporated by unstabilizing the E*S complex which largely has to do with the E state.

ag.algebraic geometry - Algebraic versus Analytic Brauer Group

Let $X$ be a smooth projective algebraic variety over $mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic Brauer Group $H^{2}(X,mathcal{O}^{times})$. This latter group is calculated in the classical (metric) topology on the associated complex manifold with the sheaf of nowhere vanishing holomorphic functions.

However there can easily be non-torsion elements in $H^{2}(X,mathcal{O}^{times})$: for instance consider the image in $H^{3}(X,mathbb{Z}) cap (H^{(2,1)}(X) oplus H^{(1,2)}(X))$.

Could there be a topology more refined than etale but defined algebraically which can see these non-torsion classes? Notice that one can also ask the question for any $H^{i}(X,mathcal{O}^{times})$. For $i=0,1$ the Zariski and etale work fine.

Why do things break down for $i>1$?

algebraic groups - "Eigenvalue characters"

This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group over an algebraically closed field $k$. Let $G_s$ be the semisimple part of $G$. Let $rho:G rightarrow GL_n(V)$ be an embedding. Then $rho(G_S)$ is a set of commuting diagonalizable endomorphisms and I know from yesterday that I have unique morphisms of algebraic groups $chi_i: rho(G_s) rightarrow mathbb{G}_m$, $1 leq i leq r$, and a decomposition $V = bigoplus _{i=1}^r E _{chi_i}$, where $E_{chi_i} = lbrace v in V mid fv = chi_i(f)v forall f in rho(G_s) rbrace$. Now, my question is: are the morphisms $chi_i$ independent of $rho$ so that I get well-defined morphisms $chi_i:G_s rightarrow mathbb{G}_m$?

If somebody knows what I'm talking about, then please change the title appropriately! :)

pr.probability - What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?

The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name for a slightly different definition, then I want to know about it.

Let $(X,mu)$ be a measure space, and let $rho$ be a probability measure on $X$. I call a subset $A$ of $X$ special if for all measurable $Bsubseteq X$,

1. $mu(B)leqmu(A)$ implies $rho(B)leqrho(A)$, and


2. $mu(B)=mu(A)$ and $rho(B)=rho(A)$ implies $B=A$ up to measure zero (with respect to both $mu$ and $rho$).

What is the standard name for my "special" sets? Equivalently, one could stipulate $mu(A)leqbeta$ and call $A$ "special" if it is essentially the unique maximizer of $rho(A)$ given that constraint.

Also equivalently, we could stipulate a particular $rho$-measure and consider sets achieving that $rho$-measure having the smallest possible $mu$-measure. That's probably the most intuitive way to think about this: we're looking for sets that contain a certain (heuristically: large) fraction of of the mass of $rho$ but are as small as possible (with respect to $mu$). That seems like a completely natural and obvious concept, which is why I think it should have a standard name. But I have almost no training in statistics, so I don't know what the name is.

This example might be far-fetched, but just to illustrate: suppose the FBI has knowledge that somebody is going to attempt a terrorist attack in a certain huge city at a particular hour. They might not know where, but they might have (some estimate of) a probability distribution for the location of the attack. They want to distribute agents strategically throughout the city, but they probably don't have enough agents to cover the entire city. Let's say every agent can forestall an attack if it occurs within a certain radius of his/her position (which is unrealistic, since the number of nearby agents surely also matters, but ignore that); then, to maximize the probability that the attack will be stopped, to an approximation, they should distribute their agents uniformly over a special subset of the city's area. To approach this from the other perspective, it could be the case that 99% of the mass of their probability distribution is contained in a region with very small area. (The one with the smallest area will be a special set.) Then, to save resources, if they're okay with 99:1 odds (c'est la vie), they might only distribute a relatively small number of agents to that small special region.

If $rho$ has a density $f$ with respect to $mu$ (when it makes sense to talk about such), then special sets are closely related to the superlevel sets of $f$, i. e., sets of the form ${x:f(x)geq c}$ for $cgeq 0$. (I think they're basically the same, but specialness of $A$ is unaffected by changing $A$ by a set of measure zero, so a superlevel set actually corresponds to an equivalence class of special sets.) I mention this here because (1) the connection to superlevel sets is one of my reasons for caring about specialness, and (2) "superlevel sets of the density" is not the answer I'm looking for.

---

Example 1

Here's a very simple example in which special sets can be completely characterized. Let $X={x_1,ldots,x_n}$ be a finite set, and let $mu$ be counting measure on $X$. Let $rho$ be any probability distribution on $X$, which necessarily has a density function $f:Xtomathbb{R}_+$, so by definition, $f(x_1) + ldots + f(x_n) = 1$ and $rho(A) = sum_{xin A} f(x)$. Suppose that no two points have the same $f$-value; then, without loss of generality, $f(x_1) > f(x_2) > ldots > f(x_n)$. It's easy to see that the special sets in this setup are exactly the sets $A_k = {x_1,x_2,ldots,x_k}$, i. e., which contain the largest $k$ points as measured by $rho$, for $k=0,ldots,n$. (Why: if you have some other candidate special set $B$, then $A_{\#B}$ has the same $mu$-measure as $B$ but higher $rho$-measure, so $B$ can't be special.) It's easy to generalize this example to the case in which $f$ isn't necessarily one-to-one: you have to treat all points with the same $f$-value as a block: either all of them are in the special set, or none of them are. (Otherwise, there's no way to satisfy the "uniqueness" part (point 2) of the definition.)

---

Example 2

Here's a generalization of the first example that hopefully clarifies what I said above. Let $(X,mu)$ be some nice measure space on which integration of functions makes sense (like a Riemannian manifold, or just $mathbb{R}^d$). Let $f:Xtomathbb{R}_+$ be a nonnegative integrable function with $int_X f(x) dmu = 1$, and let $rho$ be the probability measure $rho(Y) = int_Y f(x) dmu$, so $f$ is the density of $rho$ with respect to $mu$. Fix some $cgeq 0$ and let $A={x:f(x)geq c}$.

Claim: $A$ is a special set.

Proof: It suffices to show that if $mu(B) = mu(A)$, then $rho(B)leq rho(A)$, with equality if and only if $B$ and $A$ differ by a set of measure zero. If $mu(B) = mu(A)$, then $mu(B-A) = mu(A-B)$. Now we write
$begin{align*} rho(A) - rho(B) &= int_A f(x) dmu - int_B f(x) dmu \\
&= int_{A-B} f(x) dmu - int_{B-A} f(x) dmu \\
&= int_{A-B} f(x) dmu - int_{A-B}c\,dmu - int_{B-A} f(x) dmu + int_{B-A}c\,dmu\\
&= int_{A-B} (f(x)-c) dmu - int_{B-A} (f(x)-c) dmu.
end{align*}$

By construction, $f(x) geq c$ on $A$ and $f(x) < c$ on $B-A$, so the first integral is nonnegative and the second integral is nonpositive, and is in fact negative unless $mu(B-A)=0$, in which case $mu(A-B)=0$ as well. Thus, $rho(A)-rho(B)geq 0$, with strict inequality unless $A$ and $B$ differ by measure zero, QED.

evolution - Which came first: The Chicken or the Egg?

Dunno about any serious scientific inquiries into the answer but I always thought the answer is egg. At some point the modern chicken has come into being as the progeny of two pre-modern chickens, however it had to be an egg before it could be a chicken and its parents couldn't have been modern chickens.

However, all that presupposes that you can draw a line in the evolutionary history of the modern (extant) chicken and say this is modern and what goes before is not. I dunno if that really can be done.

Edit: came across this guardian article: http://www.guardian.co.uk/science/2006/may/26/uknews

"Whether chicken eggs preceded chickens hinges on the nature of chicken eggs," said panel member and philosopher of science David Papineau at King's College London.

"I would argue it's a chicken egg if it has a chicken in it. If a kangaroo laid an egg from which an ostrich hatched, that would surely be an ostrich egg, not a kangaroo egg. By this reasoning, the first chicken did indeed come from a chicken egg, even though that egg didn't come from chickens."

And from Prof. Brookfield of the University of Nottingham:

The first chicken must have differed from its parents by some genetic change, perhaps a very subtle one, but one which caused this bird to be the first ever to fulfil our criteria for truly being a chicken; Thus the living organism inside the eggshell would have had the same DNA as the chicken that it would develop into, and thus would itself be a member of the species of chicken

ag.algebraic geometry - What properties "should" spectrum of noncommutative ring have?

I know almost nothing about noncommutative rings, but I have thought a bit about what the general concept of spectra might or should be, so I'll venture an answer.

One other property you might ask for is that it has a good categorical description. I'll explain what I mean.

The spectrum of a commutative ring can be described as follows. (I'll just describe its underlying set, not its topology or structure sheaf.) We have the category CRing of commutative rings, and the full subcategory Field of fields. Given a commutative ring $A$, we get a new category $A/$Field: an object is a field $k$ together with a homomorphism $A to k$, and a morphism is a commutative triangle. The set of connected-components of this category $A/$Field is $mathrm{Spec} A$.

There's a conceptual story here. Suppose we think instead about algebraic topology. Topologists (except "general" or "point-set" topologists) are keen on looking at spaces from the point of view of Euclidean space. For example, a basic thought of homotopy theory is that you probe a space by looking at the paths in it, i.e. the maps from $[0, 1]$ to it. We have the category Top of all topological spaces, and the subcategory Δ consisting of the standard topological simplices $Delta^n$ and the various face and degeneracy maps between them. For each topological space $A$ we get a new category Δ$/A$, in which an object is a simplex in $A$ (that is, an object $Delta^n$ of Δ together with a map $Delta^n to A$) and a morphism is a commutative triangle. This new category is basically the singular simplicial set of $A$, lightly disguised.

There are some differences between the two situations: the directions have been reversed (for the usual algebra/geometry duality reasons), and in the topological case, taking the set of connected-components of the category wouldn't be a vastly interesting thing to do. But the point is this: in the topological case, the category Δ$/A$ encapsulates

how $A$ looks from the point of view of simplices.

In the algebraic case, the category $A/$Field encapsulates

how $A$ looks from the point of view of fields.

$mathrm{Spec} A$ is the set of connected-components of this category, and so gives partial information about how $A$ looks from the point of view of fields.

pr.probability - How many trial picks expectedly sufficient to cover a sample space?

The expected number of picks needed equals the sum of the probabilities that at least $t$ picks are needed, which means that $t-1$ subsets left at least one value uncovered. We can use inclusion-exclusion to get the probability that at least one value is uncovered.

The probability that a particular set of $k$ values is uncovered after $t-1$ subsets are chosen is

$$Bigg(frac{n-k choose r}{n choose r}Bigg)^{t-1}$$

So, by inclusion-exclusion, the probability that at least one value is uncovered is

$$ sum_{k=1}^n {n choose k}(-1)^{k-1}Bigg(frac{n-k choose r}{n choose r}Bigg) ^{t-1} $$

And then the expected number of subsets needed to cover everything is

$$ sum_{t=1}^infty sum_{k=1}^n {n choose k}(-1)^{k-1} Bigg(frac{n-k choose r}{n choose r}Bigg)^{t-1} $$

Change the order of summation and use $s=t-1$:

$$ sum_{k=1}^n {n choose k}(-1)^{k-1} sum_{s=0}^infty Bigg( frac{n-k choose r}{n choose r}Bigg)^s$$

The inner sum is a geometric series.

$$ sum_{k=1}^n {n choose k} (-1)^{k-1}frac{n choose r}{{n choose r}-{n-k choose r}}$$

$$ {n choose r} sum_{k=1}^n (-1)^{k-1}frac{n choose k}{{n choose r}-{n-k choose r}}$$

I'm sure that should simplify further, but at least now it's a simple sum. I've checked that this agrees with the coupon collection problem for $r=1$.

Interestingly, Mathematica "simplifies" this sum for particular values of $r$, although what it returns even for the next case is too complicated to repeat, involving EulerGamma, the gamma function at half-integer values, and PolyGamma[0,1+n].

differential topology - Every Manifold Cobordant to a Simply Connected Manifold

Assume that $M^n$ has $pi_1$ finitely generated (Edit: and n>3). Choose a generator. We will construct (using surgery) a cobordism to $M'$ which kills that generator, and by induction we can kill all of $pi_1$. Choose an embedded loop which represents the generator, and choose a tubular neighborhood of the loop. We can view this as a (n-1)-dimensional vector bundle over $S^1$, the normal bundle. Since $M$ is oriented, this is a trivial vector bundle so we can identify this tubular neighborhood with $S^1 times D^{n-1}$.

Now we build the cobordism. We take $M times I$, which is a cobordism from $M$ to itself. To one end we glue $D^2 times D^{n-1}$ along the boundary piece $S^1 times D^{n-1}$ via its embedding into $M$. This is just attaching a handle to $M times I$. This new manifold is a cobordism from $M$ to $M'$, where $M'$ is just $M$ where we've done surgery along the given loop.

A van Kampen theorem argument shows that we have exactly killed the given generator of $pi_1$. Repeating this gives us a cobordism to a simply connected manifold.

Note that it is essential that our manifold was oriented. $mathbb{RP}^2$ is a counter example in the non-oriented setting, as all simply connected 2-manifolds are null-cobordant, but $mathbb{RP}^2$ is not.

---

[I was implicitly thinking high dimensions. Thanks to Tim Perutz for suggesting something was amiss when n=3]

If n=3 then this is "surgery in the middle dimension" and it is more subtle. First of all the normal bundle is an oriented 2-plane bundle over the sphere, so there are in fact $mathbb{Z} = pi_1(SO(2))$ many ways to trivialize the bundle (these are normal framings). Ignoring this, if you carry out the above construction, you will see that (up to homotopy) M' is the union of $M - (S^1 times D^2)$ and $D^2 times S^1$ along $S^1 times S^1$. This can (and does) enlarge the fundamental group.

However a different argument works in dimensions n=1,2,3. The oriented bordism groups in those dimensions are all zero (see the Wikipedia entry on cobordism), so in fact every oriented 3-manifold is cobordant to the empty set (a simply connected manifold). The fastest way to see this is probably a direct calculation of the first few homotopy groups of the Thom spectrum MSO.

ag.algebraic geometry - Uniqueness/motivation for the Suslin-Voevodsky theory of relative cycles.

I will just sum up the situation as I see it (too big for the comment box).

One important goal is to set up a good intersection theory for cycles without quotienting by rational equivalence, and using it to get a composition product for finite correspondences, which are by definition elements of groups of the form $c_{equi}(Xtimes_S Y/X,0)$

It is true that the variety of definitions of cycle groups in the paper is somewhat confusing. There are 16 possible groups because starting from the "bare" notion of relative cycles (def. 3.1.3) there are 4 binary conditions : being effective, being equidimensional, having compact support (c, PropCycl), and being "special", i.e satisfying the equivalent conditions of lemma 3.3.9 (everything except Cycl and PropCycl). So you have

1)$z_{equi}(X/S,r)subset z(X/S,r)subset Cycl(X/S,r) supset Cycl_{equi}(X/S,r)$

and their effective counter-parts.

2)$c_{equi}(X/S,r)subset c(X/S,r)subset PropCycl(X/S,r) supset PropCycl_{equi}(X/S,r)$

and their effective counter-parts.

(1) is then a "subline" of 2))

In a sense, the most satisfying definition would be to use only cycles which are flat over $S$ (the $mathbb{Z}Hilb$-groups, or the closely related $z_{equi}$) but pullbacks along arbitrary morphisms are not defined there in general.

With the groups Cycl, thanks to the relative cycle condition built in Cycl, you have pullbacks along arbitrary morphism, but only with rational coefficients (thm 3.3.1, the denominators of the multiplicities are divisible by residue characteristics)

The main interest of the "special" relative cycles $z(-,-)$ is in their definition : they admit integral pullbacks ! Then you have the small miracle that this condition is stable by those pullbacks and you get a subpresheaf. This means that using them you can set up intersection theory with integral coefficients even on singular car p schemes.

All this zoology simplifies when $S$ is nice : there are some results when $S$ is geometrically unibranch, but the nicest case is $S$ regular, in which the chains of inclusions I wrote down collapse, you are left with two distinctions which are reasonable from the point of view of classical intersection theory : effective/non-effective, general/with compact support. Furthermore, the intersection multiplicities are computed by the Tor multiplicity formula, so the Suslin-Voevodsky theory is really an extension of local intersection theory of regular rings as in Serre's book.

fa.functional analysis - Way to memorize relations between the Sobolev spaces?

Sobolev norms are trying to measure a combination of three aspects of a function: height (amplitude), width (measure of the support), and frequency (inverse wavelength). Roughly speaking, if a function has amplitude $A$, is supported on a set of volume $V$, and has frequency $N$, then the $W^{k,p}$ norm is going to be about $A N^k V^{1/p}$.

The uncertainty principle tells us that if a function has frequency $N$, then it must be spread out on at least a ball of radius comparable to the wavelength $1/N$, and so its support must have measure at least $1/N^d$ or so:

$V gtrsim 1/N^d.$

This relation already encodes most of the content of the Sobolev embedding theorem, except for endpoints. It is also consistent with dimensional analysis, of course, which is another way to derive the conditions of the embedding theorem.

More generally, one can classify the integrability and regularity of a function space norm by testing that norm against a bump function of amplitude $A$ on a ball of volume $V$, modulated by a frequency of magnitude $N$. Typically the norm will be of the form $A N^k V^{1/p}$ for some exponents $p$, $k$ (at least in the high frequency regime $V gtrsim 1/N^d$). One can then plot these exponents $1/p, k$ on a two-dimensional diagram as mentioned by Jitse to get a crude "map" of various function spaces (e.g. Sobolev, Besov, Triebel-Lizorkin, Hardy, Lipschitz, Holder, Lebesgue, BMO, Morrey, ...). The relationship $V gtrsim 1/N^d$ lets one trade in regularity for integrability (with an exchange rate determined by the ambient dimension - integrability becomes more expensive in high dimensions), but not vice versa.

These exponents $1/p, k$ only give a first-order approximation to the nature of a function space, as they only inspect the behaviour at a single frequency scale N. To make finer distinctions (e.g. between Sobolev, Besov, and Triebel-Lizorkin spaces, or between strong L^p and weak L^p) it is not sufficient to experiment with single-scale bump functions, but now must play with functions with a non-trivial presence at multiple scales. This is a more delicate task (which is particularly important for critical or scale-invariant situations, such as endpoint Sobolev embedding) and the embeddings are not easily captured in a simple two-dimensional diagram any more.

I discuss some of these issues in my lecture notes

http://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/

EDIT: Another useful checksum with regard to remembering Sobolev embedding is to remember the easy cases:

1. $W^{1,1}({bf R}) subset L^infty({bf R})$ (fundamental theorem of calculus)


2. $W^{d,1}({bf R}^d) subset L^infty({bf R}^d)$ (iterated fundamental theorem of calculus + Fubini)


3. $W^{0,p}({bf R}^d) = L^p({bf R}^d)$ (trivial)

These are the extreme cases of Sobolev embedding; everything else can be viewed as an interpolant between them.

EDIT: I decided to go ahead and draw the map of function spaces I mentioned above, at

http://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/

reference request - What is the equivariant cohomology of a group acting on itself by conjugation?

For any interested latecomers who somehow discover this question in the future, I've found a very low-tech answer, bootstrapping from the low-tech answer to Is a Lie group equivariantly formal under conjugation by a maximal torus?.

Anyway, once you believe that the conjugation action upon a compact Lie group $G$ of its maximal torus $T$ is equivariantly formal, it follows that the action of $G$ by conjugation is as well. This argument will actually work for any reasonably good space $M$ on which $G$ acts and the restricted $T$-action is equivariantly formal, because one has $$H_G(M) = H_T(M)^W = (H(M) otimes H(BT))^W = H(M) otimes H(BT)^W = H(M) otimes H(BG)$$ as $H(BT)$-modules, where $W = N_G(T)/T$ is the Weyl group of $G$.

If, say, you don't buy that the action of $W$ on $H(M)$ is trivial, a longer proof goes like this.

$require{AMScd}$

The homotopy quotient $M_G$ is a further quotient of $M_T$, and the projection $EG times M to EG$ then induces a commutative diagram

begin{CD}
M @= M\
@VVV @VVV\
M_T @>>> M_G\
@VVV @VVV\
BT @>>> BG
end{CD}

where the upper vertical maps are fiber inclusions.

The projection $BT = EG/T to EG/G = BG$ induces an inclusion $H(BG) cong H(BT)^W hookrightarrow H(BT)$ in cohomology, and there are induced maps both in cohomology and on the Serre spectral sequences for the equivariant cohomologies, starting with this $E_2$ page:

begin{CD}
H(M) @= H(M)\
@AAA @AAA\
H(M) otimes H(BT) @<<< H(M) otimes H(BG)\
@AAA @AAA\
H(BT) @<<< H(BG)
end{CD}

Because the top and bottom horizontal maps are injective, so is the middle one, so the differentials for the spectral sequence converging to $H_G(M)$ are restrictions of those for $H_T(M)$. But the differentials for $H_T(M)$ are all zero, by equivariant formality, so the spectral sequence for $H_G(M)$ collapses as well.

ct.category theory - Is there a tricategory of bicategories and biprofunctors?

Depending on what you actually need for your application, there might be something useful already known. Do you really need a tricategory, as opposed to some other model of weak 3-categories? Even if you had that, would you be able to do much with it?

There are several equivalent ways to think of profunctors, and some of them lend themselves to categorification quite easily. One is this:

A profunctor from $C$ to $D$ is the same a a colimit-preserving ordinary functor between the presheaf categories $PSh(C)$ and $PSh(D)$.

(See here for details: http://ncatlab.org/nlab/show/profunctor#FuncsOnPresheaves)

That's good, because a lot is known about categorifications of categories of presheaves and of morphisms between them.

For instance it is straightforward to set this up over bicategories: take objects bicategories, and hom-bicategories to be the full sub-bicategory on bicolimit-preserving bifunctors between their bipresheaf categories.

In case that your application is such that you only need higher categories whose higher morphisms are all invertible, one can go much further and consider the (oo,1)-category of (oo,1)-profunctors. By the above, this is simply the gadget whose objects are small (oo,1)-categories and whose hom-oo-groupoids are the full sub-oo-groupoids on the (oo,1)-colimit preserving (oo,1)-functors between the corresponding (oo,1)-presheaf categories.

More generally, one can generalize here (oo,1)-presheaf categories and all in addition all their reflective sub-(oo,1)-categories. That structure of (oo,1)-profunctors has proven to play a major role as kind of categorification of the category of vector spaces: one thinks of a presentable (oo,1)-category as vector space, of colimits as being sums of vectors, as colimit preserving (oo,1)-functors as linear maps.

More on this is here: http://ncatlab.org/nlab/show/Pr(infinity,1)Cat .

reference request - Stokes theorem for manifolds with corners?

Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The Stokes theorem tells us that, for a $n+1$-dimensional manifold $M$ with boundary $partial M$ and any differentiable $n$-form $omega$ on $M$, we have $int_{partial M} omega = int_M domega $.

But Stokes theorem is also true, say, for a cone $M = {(x,y,z) in mathbb{R}^3 vert x^2 + y^2 = z^2, 0leq z leq 1 }$, or a square in the plane, $M ={(x,y) in mathbb{R}^2 vert 0 leq x, yleq 1 }$ which are not manifolds. So my questions are:

1. Are these cone and square examples of what I think are called "manifold with corners"?


2. If this is so, where can I find a reference for a version of Stokes' theorem for manifolds with corners?


3. If "manifold with corners" is not, which is the appropriate setting (and a reference) for a Stokes' theorem that includes those examples?

Any hints will be appreciated.

EDIT: Since thanking individually everyone would be too long, let me edit my question to acknowledge all of your answers. Thank you very much: I've found what I was looking for and more.

nt.number theory - How do I calculate the discriminant of a galois closure and its other subfields?

According to my hand computation, the claim is not true for $K = mathbb{Q}(sqrt[3]{3})$. Have you checked this case?

It is true that a prime ramifies in $K$ if and only if it ramifies in the Galois closure of $K$. There might be a formula like the one you are describing that is true for primes greater than $d$.

---

Warning: this part of the answer was written without access to my usual number theory references, so there may be errors. It isn't meant to be read by itself anyway; it is meant to be a guide to a good book on algebraic number theory, like Neukirch or Janusz.

Your fundamental strategy to be to relate the order of ramification of $p$ in $L$ and in the Galois closure of $L$. This does not precisely determine the power of $p$ dividing the discriminant, but we have the following

Fact: Let $L/K$ be an extension of number fields and $p$ a prime of $K$ with ramification indices $e_1$, $e_2$, ... $e_r$ and residue field extensions of degree $f_1$, $f_2$, ..., $f_r$. Then the power of $p$ dividing $D_{L/K}$ is at least $sum (e_i-1) f_i$, with equality if and only if all the $e_i$ are less than relatively prime to the characteristic of $p$.

From now on, I will address the question of how to relate the orders of ramification in $L$ and in the Galois closure. If you need to deal with the case that some of the $e_i$ are greater than or equal to the characteristic of $p$, you should read about higher ramification groups.

Let $M$ be the Galois closure of $L$, let $G$ be $mathrm{Gal}(M/K)$, and let $H$ be the fixed field of $L$. Fix a prime $p$ of $K$, and¹ $mathfrak{p}$ a prime over $p$. Let $D subseteq G$ be the decomposition group of $mathfrak{p}$ and $I subset D$ the inertia group. $I$ is normal in $D$; the quotient $D/I$ is cyclic and has a canonical generator $F$ called the Frobenius.

I believe that FC's computation was for the case $G=S_n$, $D=I={ e, (12) }$.

In $M$, the primes lying over $p$ are in bijection with $G/D$. Each of them has $f=|D/I|$ and $e=|I|$.

Let $X=G/H$ (a set with $G$-action). In $L$, the primes above $p$ are in bijection with the $D$-orbits in $X$. Let $O$ be such a $D$-orbit, corresponding to a prime $q_O$. Because $I$ is normal, $O$ breaks up as a union of $I$-orbits all of the same cardinality. Then $q_O$ has $e$ equal to the cardinality of these $I$-orbits, and $f$ equal to the number of them.

Using these ideas, you should be able to relate ramification in $L$ and $M$ for any $G$ and $H$ which interest you.

¹ I fix $mathfrak{p}$ only for expositional purposes. Changing $mathfrak{p}$ will conjugate $(D,I,F)$. Probably the right way to think of all of this stuff is working up to conjugacy.

big list - Computer Algebra Errors

This error affects all versions of Mathematica from 6 to 8. The result of a function depends on what letter is chosen for argument when calling it. In the simplest case it can be illustrated as follows:

in:

$A[text{x_}]text{:=}sum _{k=0}^{x-1} x $

$A[k]$

$A[z]$

out:

$1/2 (-1 + k) k$

$z^2$

The correct answer is evidently, the later. This behavior affects not only sums but also integrals, so one have to check so that the letter user for the argument not to coincide with the index variable used for definition. In the case of recursion this becomes very difficult. The following example shows that moving a factor not dependent on the index variable out of the sum sign changes the result:

in:

A1[0,x_]:=1
A2[0,x_]:=1

A1[n_,x_]:=Sum[A1[-1 - j + n, x]*Sum[A1[j, k], {k, 0, -1 + x}], {j, 0, -1 + n}]
A2[n_,x_]:=Sum[Sum[A2[j, k]*A2[-1 - j + n, x], {k, 0, -1 + x}], {j, 0, -1 + n}]

A1[1,x]/.x->2
A1[2,x]/.x->2
A1[3,x]/.x->2

A2[1,x]/.x->2
A2[2,x]/.x->2
A2[3,x]/.x->2

A2[1,2]
A2[2,2]
A2[3,2]

out:

2
5
13

2
5
12

2
5
13

real analysis - About the Riemann integrability of composite functions

Warning: not an answer. Rather, some comments and some links.

I came upon this issue myself when I was teaching an undergraduate real analysis course some years ago. The point is that in the development of the Riemann/Darboux integral, a standard technical result is that if $f: [a,b] rightarrow [c,d]$ is integrable and $varphi: [c,d] rightarrow mathbb{R}$ is continuous, then $varphi circ f$ is integrable. It follows easily that the product of two integrable functions is integrable (which is not so obvious otherwise). This result appears, for instance, as Theorem 6.11 in Rudin's Principles of Mathematical Analysis.

It is easy to see that the composition of integrable functions need not be integrable. So it is natural to ask whether it works the other way around. Remarkably, I know of no standard text which addresses this question. Rudin immediately asks a much more ambitious question and then moves right on to something else.

At the time I convinced myself of the existence of continuous $f$ and integrable $varphi$
such that $varphi circ f$ was not integrable. However, in order to do so I needed to use ideas which were more advanced than I could explain in my course. At least, this is what it says on p. 7 of my lecture notes:

http://math.uga.edu/~pete/243integrals2.pdf

Unfortunately I didn't write down the counterexample that I had in mind (I suppose back then I was clinging naively to the idea that the lecture notes were for the students and not to preserve my own knowledge of the material in the coming years), so I don't know now what it was.

The example in Jitan Lu's 1999 Monthly article that Qiaochu referred to seems elementary enough so that it should at least be referenced in texts and courses, and possibly included explicitly. For those who couldn't get the whole paper from the previous link, it is now also available here:

http://math.uga.edu/~pete/Lu99.pdf

Of course, I don't believe for a second that an example of this type (i.e., to show that integrable $circ$ continuous need not be integrable) was first constructed in 1999. Can anyone supply an earlier reference? (I never know how to go about solving math history problems like this.) I should say that I am impressed that Qiaochu was even able to track down this paper. The MathSciNet review is quite unhelpful. It says:

In this note the following result is given. If $f$ is a Riemann integrable function defined on $[a,b], g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $fcirc g$ is Riemann integrable on $[c,d]$.

This is not the main result given in the paper; rather it is a proposition stated (without proof!) at the very end.

ca.analysis and odes - On the existence of a sequence of positive continuous functions

See the paper, First Class Functions, in the American Mathematical Monthly 98 (March, 1991) 237-240.

EDIT: Let $f_n(x)=n$ if $x=p/q$ with $qle n$, $f_n(x)=0$ if $x=(p/q)pm n^{-4}$ with $qle n$, and let $f_n$ be piecewise linear between these points. So $f_n$ is continuous, mostly zero, but with a sharp spike at each rational. Clearly $f_n(x)$ goes to infinity with $n$ at all rational $x$. If $x$ is irrational and has only finitely many rational approximations $p/q$ such that $|x-(p/q)|le q^{-4}$ (and this is all $x$ save a set of measure zero), then $f_n(x)=0$ for all $n$ sufficiently large. If $x$ has infinitely many rational approximations with $|x-(p/q)|le q^{-4}$, then $f_n(x)=0$ for most $n$ (those that are far from a $q$ which gives a good approximation, and those $q$ are guaranteed to be few and far between), but is occasionally quite large, so $f_n(x)$ has no limit, finite or infinite.

cloning - Deletion errors with Phusion Polymerase?

In my own personal lab experience, I got some unexpected results using Phusion polymerase. However, I have not seen deletions, only insertions (ranging from single bases to 3 tandem copies of a primer sequence). This is not an answer to your deletion question, but it does suggest unusual things might happen.

I have seen deletions with T4 DNA polymerase during "blunting" reactions. According to the NEB website:

Q10: Is T4 DNA Polymerase active at room temperature?

A10: We suggest 12°C. The DNA ends "breathe" at higher temperatures allowing the exonuclease to remove nucleotides past blunt.

lo.logic - In set theories where Continuum Hypothesis is false, what are the new sets?

The question of what happens when CH fails is, of course, intensely studied in set theory. There are entire research areas, such as the area of cardinal characteristics of the continuum, which are devoted to studying what happens with sets of reals when the Continuum Hypothesis fails.

The lesson of much of this analysis is that many of the most natural open questions turn out to be themselvesd independent of ZFC, even when one wants ¬CH. For example, the question of whether all sets that are intermediate in size between the natural numbers and the continuum should be Lebesgue measure 0, is independent of ZFC+¬CH. The question of whether only the countable sets have continuum many subsets is independent of ZFC+¬CH. There are a number of cardinal characteristics that I mention here, whose true nature becomes apparant only when CH fails. For example, must every unbounded family of functions from ω to ω have size continuum? It is independent of ZFC+¬CH. Must every dominating family of such functions have size continuum? It is independent of ZFC+¬CH. Those question are relatively simple to state and could easily be considered part of "ordinary" mathematics.

However, much of the rest of what you might think of as ordinary mathematics is simply not affected by CH or not CH. In particular, the existence of non-measurable sets that you mentioned is provable in ZFC, whether or not CH holds. (This proof requires the use of the Axiom of Choice, however, unless large cardinals are inconsistent, a result proved by Solovay and Shelah.)

Nevertheless, there is a growing body of research on some sophisticated axioms in set theory called forcing axioms, which have powerful consequences, and many of these new axioms imply the failure of CH. This topic began with Martin's Axiom MA_ω1, and has continued with the Proper Forcing Axiom, Martin's Maximum and now many other variations.

Lastly, in your title you asked what are the new sets like. The consistency of the failure of the Continuum Hypothesis was proved by Paul Cohen with the method of forcing. This highly sophisticated and versatile method is now used pervasively in set theory, and is best thought of as a fundamental method of constructing models of set theory, sharing many affinities with construction methods in algebra, such as the construction of algebraic or transcendental field extensions. Cohen built a model of ZFC+ ¬CH by starting with a model V of ZFC+CH, and then using the method of forcing to add ω₂ many new real numbers to construct the forcing extension V[G]. Since V and V[G] have the same cardinals (by a detailed combinatorial argument), it follows that the set of reals in V[G] has size at least (in fact, exactly) ω₂. In particular, the old set of reals from V, which had size ω₁, is now one of the sets of reals of intermediate size. Thus, these intermediate sets are not so mysterious after all!

computational complexity - Completeness, easiest, hardest problems

One says that a language $L$ is complete for a complexity class $mathcal{C}$ if $L$ is in $mathcal{C}$ and every language in $mathcal{C}$ is reducible to $L$. Thus, in a sense, $L$ is the hardest language in $mathcal{C}$. For example, $3SAT$ is the hardest language in $mathcal{NP}$.

Now recall Rice's theorem: any nontrivial property of a language is undecidable. The proof of this theorem essentially rests on the fact that if any such property were decidable, then it could be used to decide the $HALTING$ language. Thus, in a sense, $HALTING$ is the easiest property of all properties.

And then here is a peculiar thing: the proof of Rice's theorem---at least the one I've seen---amounts to constructing a reduction from $HALTING$ to every property. Contrast this with the definition of completeness, and one realizes that these are ``opposites'' of each other in a sense.

Question: Is there more to this observation? Is there work done on this? If there are experts in the audience, could you elaborate on my observation? Does one ever consider ``easiest'' problems in complexity theory as in my example above, and if so, does it lead to interesting results (like it does to Rice's theorem above)?

Thank you in advance for your comments and your effort in reading and thinking my question.

Rev. 1:

Note that when I say $HALTING$ is the easiest property, I don't mean to imply that it is easy. Of course, it is an undecidable language. It is ``easy'' in a relative sense: every other property is at least as hard as $HALTING$, because if any other property can be decided, then it yields a decider for $HALTING$.

When we talk of an complete problem, we use similar terminology: if a language has a reduction from every other language in its class, then we call it complete and rightly refer to it as being one of the hardest languages in the class.

This is the subtle similarity / difference I am trying to point out. These two are in a sense opposites. In one case, we get reductions from all languages in the class. In the other, we get reductions from one language to all other languages in the class. In the first case, we call the target hardest. In the second, we (could) call the source the easiest.

I hope that this provides a clearer context for my question. Thanks.

dg.differential geometry - A topological consequence of Riemann-Roch in the almost complex case

In general, are there topological consequences of the existence of the Dolbeault resolution that would be difficult to prove (or, more ambitiously, would fail) for arbitrary pseudo-complex manifolds?

Since an almost complex manifold has a tangent bundle like that of a complex manifold, the place to measure the difference is not, I think, in things involving characteristic classes of bundles and indices of elliptic operators. David's answer illustrates this, and I'll say something more philosophical.

The topological restrictions imposed by integrability are stark in real dimension 4. Almost complex structures on 4-manifolds $X$ are cheap: all you need for existence is a candidate $c$ for the first Chern class, which should satisfy $w_2=c mod 2$ and $c^2[X]=2chi+3sigma$ (where $sigma$ is the signature, and the second equation rewrites $p_1=c_1^2-2c_2$). Integrable complex structures are hard to come by. So as to avoid recourse to Kaehler methods, let's say that $b_1$ should be odd. Complex surfaces of this kind are still not completely classified, but they have been hunted down to a few specific topological "locations", one of them being $pi_1=mathbb{Z}$ and $H^2$ negative-definite (Class VII surfaces).

To get that far, one uses nearly all the complex geometry one can think of. The story starts with Dolbeault and the degeneration of the Hodge to de Rham spectral sequence (which uses Serre duality), but it invokes many further arguments (see Barth et al., Compact complex surfaces). It's expected that Class VII surfaces contain non-separating 3-spheres; to prove this when $b_2=1$, A. Teleman carefully analysed compactified moduli spaces of stable rank 2 bundles.

Your question was inspired by Dmitri's quotation of Gromov, who asserted in the quoted passage from Spaces and Questions that "complex manifolds have not stood up to their fame!" In the case of non-Kaehler surfaces, he might be right; a great deal of work turns up only a handful of quirky specimens which it is hard to fall in love with.

lo.logic - Horn clauses and satisfiability

In the paper The complexity of satisfiability problems MR0521057, Tom Schaefer characterizes exactly which general classes of satisfiability problems are in P and which are NP-complete. Those problems which are in P fall into six cases:

• Every relation in S is satisfied when all variables are 0.




• Every relation in S is satisfied when all variables are 1.




• Every relation in S is definable by a CNF formula in which each conjunct has at most one negated variable.




• Every relation in S is definable by a CNF formula in which each conjunct has at most one unnegated variable.




• Every relation in S is definable by a CNF formula having at most 2 literals in each conjunct.




• Every relation in S is the set of solutions of a system of linear equation over the two-element field {0,1}.

Here, S is a set of boolean relations that one takes as primitives for the language; the associated satisfiability problem is then deciding the satisfiability of a finite conjunction of such primitives. Schaefer moreover shows that any set of relations which does not fall into one of the above has a NP-complete satisfiability problem. In your example, S would be a set of boolean relations definable by a CNF formulas in which each conjunct has at most two unnegated variables. This is not in the above list, so the corresponding satisfiability problem is NP-complete.

soft question - Graphical representation of mathematical structures (in the spirit of unified modeling language)

In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are about ten diagrams for different views on the system. These diagrams have tremendously improved the ability to construct large systems by large teams, as one can look at the system at different levels and as one can profit from visualization.
Furthermore the UML models contains all the system's information needed for implementation.

I'm wondering whether a similar method could be useful for doing (and commuicating about) mathematical structures and theories.
Often so many definitions are built one upon the other and so many properties are introduced that it seems to be difficult to have an overview of the "architecture" of a theory.

For example one could have one sort of diagram showing how the mathematical structures are build form each other (e.g. a field built by two groups - with the respective axioms "inherited" - and further "compatibility conditions" between them). In priciple you would be able to track back all structures to the "mother structures" algebraic structure, order structure, topolological structure.) Interestingly the object oriented paradigm used by UML, i.e. encapsulating attributes and methods into classes is somehow similar to the categorical approach in mathematics (encapsulating objects and morphisms).

Another sort of diagram could represent a (part of a) theory by an annotated graph, the nodes/edges of which are the definitions, theorems and proofs and by navigating you see exactly which property/definition is used at which point.

I apologize for the "fuzziness" of the question but I feel a discussion about a sort of visual notation / graphical representation in mathematics could be of interest (perhaps the categorical viewpoint with its unifying force and its diagramms is already what can be achieved, but I think there could be other, complementary ways).
Does anybody know of attempts in this direction? Would you consider such a graphical representation of mathematical structures (in addition to the standard, more linear way of representing things) helpful for communication in research and/or in education?

Just starting with [combinatorial] game theory

Winning Ways for your Mathematical Plays (in four volumes) has an enormous amount of stuff about combinatorial games. But most of it you probably won't be interested in for a while. There are a few quickly diverging directions one could study in combinatorial games. Here are some that come to mind immediately, and a possible list of topics to study in each:

1) Impartial games. Read a bit of On Numbers and Games so that you know how to read the notation and understand game equivalence and addition. Then learn the winning strategy for Nim, read the relevant bits of Chapter 3 and all of Chapter 4 of Winning Ways. After that, if you like the infinite theory, Lenstra has a paper called "On the algebraic closure of two" which is really nice. If you like the finite theory, learn about nim multiplication from ONAG, and then read Conway and Sloane's paper "Lexicographic codes: error-correcting codes from game theory." I think this part of the theory is the most interesting.

2) (Surreal) numbers. Again, learn how to read the notation and about game equivalence and addition. (You will need this for everything.) Then read the first part of ONAG. Then, perhaps learn about real-closed fields in general; you can make most of real analysis work over the Field of surreal numbers. (A Field is something like a field, but it has a proper class of objects instead of a set.)

3) Weird games, for example from Hackenbush and Domineering. Read Volume 1 of Winning Ways. The stuff on thermography and all-small games is quite interesting.

at.algebraic topology - What is the difference between homology and cohomology?

On a closed, oriented manifold, homology and cohomology are represented by similar objects, but their variance is different and there is an important change in degrees. For simplicity, consider homology or cohomology classes represented by submanifolds. Then if $f : M to N$ is a smooth map between manifolds of dimension $m$ and $n$ respectively and $W$ is a submanifold of $M$ representing a homology class then $f(M)$ represented (really, $f_*([M])$ is) a homology class of $N$, in the same dimension. On the other hand, if $V$ is a submanifold of $N$, then we can consider $f^{-1}(V)$, which is a manifold if $f$ is transverse to $F$. Its codimension is the same as that of $V$ (that is, $n - dim(V) = m - dim(f^{-1}(V))$).

Note that this preimage is generically at least as "nice" as $V$ (smooth is $V$ is, with reasonable singularities if $V$ has, etc.) whereas little can be said about the geometry of $f(W)$. That's one reason I think that cohomology is of more use in algebraic geometry.

If you want to relate this view of cohomology to the standard one, a codimension $d$ submanifold of $M$ (that is, one of dimension $m-d$) generically intersects a dimension $d$ one in a finite number of points which can be counted with signs. The former defines a class in $H^d(M)$ while the latter a class in $H_d(M)$, and this intersection count is evaluation of cohomology on homology.

This point of view is more applicable than it might seem since in a manifold with boundary cohomology classes are similarly defined by submanifolds whose boundary lies on the boundary of the ambient manifold. Since any finite CW complex is homotopy equivalent to a manifold with boundary, one can view cohomology in this way for finite CW complexes and often infinite ones as well.

zoology - Are there any pre-Holocene venomous animals?

Squamates

Extant venomous snakes do have venomous ancestors. Fry et al. (2006) reported on finding venom toxins more broadly within Reptilia, beyond the well-known venomous snakes and the helodermatid lizards. They show that varanid and iguanid lizards also have venom toxins.

The same group of authors (Fry et al., 2009) then reported on the use of venom in the extant varanid lizard, Varanus komodoensis (Komodo monitor). This is a different hypothesis than the usual "toxic bacteria" hypothesis for predation in these animals. They then reinterpret the skull of the giant (~5.5 m) varanid Varanus (Megalania) priscus. They conclude, based on anatomical similarity and the close phylogenetic relationship between it and V. komodoensis, that V. priscus' predatory mode was more similar to other varanid lizards, likely involving venom delivery.

Archosaurs

Discovery of a venomous clade within squamates, including snakes and several clades of lizards, does not imply venom more broadly within Reptilia. Specifically, it would not parsimoniously reconstruct the presence of venom in Archosauria, the clade that includes dinosaurs. However, Gong et al. (2009) interpret the teeth of the dromaeosaurid theropod dinosaur, Sinornithosaurus as having venom grooves. These grooves are similar to grooves in the teeth of Uatchitodon, a Triassic squamate from Arizona.

There is currently no solid evidence that archosaurs could spit, nor is there ever likely to be.

Mammals

Venomous extant mammals include the platypus (Ornithorhynchus), Solenodon, and some species of shrews. The venom gland in the platypus is a modified sweat gland on the hind limb, which includes an ossified spur. A similar structure has also been found in the basal mammalian taxa Gobiconodon and Zhangheotherium (Hurum et al., 2006). Hurum et al. (2006) described a similar structure in the multituberculates. As the most basal clade of Mammalia, the authors conclude that an ossified spur is apomorphic for Mammalia and that the ancestral mammal may have used it for venom delivery. This structure would have been retained more-or-less unmodified in platypus and lost in other mammalian lineages. The authors' conclusions are supported by the presences of a non-venomous spur in the other clade of monotremes, the echidnas.

Fox and Scott (2005) report that the Palaeocene (~60 Ma) pantolestid Bisonalveus browni also possessed teeth with venom delivery grooves.

Non-gnathostomes

Szaniawski, 2009 used similar logic based on grooved "tooth" elements to conclude that some extinct jawless chordate conodonts may also have been venomous.

co.combinatorics - What does the generating function $x/(1 - e^{-x})$ count?

In the comments, Tom Copeland asked if, six years later, I had any further insights into the diagrammatics I proposed near the end of my question. So I figured I'd mention what I know, which is yet another (marginally) diagrammatic description of BCH multiplication and in particular the map $mathcal U mathfrak g to mathcal S mathfrak g$. What I'll describe is some part of http://dx.doi.org/10.1090/pspum/088 (also available at http://arxiv.org/abs/1307.5812). Some pictures are available in the last chapter of my thesis, but note that that chapter has a subtle error (the details of which I haven't yet sussed out) which I have corrected in the linked paper; the error does not affect the case of (duals of) Lie algebras, but does affect Poisson structures whose Taylor expansions include quadratic or higher terms.

I will describe some homological algebra, and then unpack into diagrams. Let $mathfrak G = mathfrak g otimes Omega_{mathrm{cpt}}(mathbb R)[1]$ denote the cochain complex of $mathfrak g$-valued compactly supported de Rham forms on $mathbb R$, shifted so that its cohomology is concentrated in degree $0$. Then $mathrm H^0(mathfrak G) = mathfrak g$, and the projection to homology is given by integrating de Rham forms — there is a cohomological-degree-$0$ map $int : mathfrak G to mathfrak g$. Note also that $Omega_{mathrm{cpt}}(mathbb R)$ has a non-unital graded-commutative multiplication ($wedge$), through which $mathfrak G$ picks up a graded-symmetric (!) cohomological-degree-$(+1)$ map $delta : mathfrak G otimes mathfrak G to mathfrak G$ by $delta(xotimes alpha,yotimes beta) = [x,y] otimes (alpha wedge beta)$ (up to a sign that depends on a choice of conventions about how to handle elements of shifted complexes). Consider the (graded-commutative) symmetric algebra $mathcal S mathfrak G$. I will denote its usual differential (which is the extension of de Rham-form differentiation as a derivation) by $partial_{mathrm{dR}}$.
Integration of de Rham forms extends to an algebra homomorphism $int : (mathcal Smathfrak G, partial_{mathrm{dR}}) to mathcal S mathfrak g$.

The map $delta$ has a canonical extension to a second-order differential operator on $mathcal S mathfrak G$ which I will also call $delta$. (A second order differential operator on a symmetric algebra is unique determined by its values on constant, linear, and quadratic terms. We declare that $delta$ vanishes on constants and linears, and set it to be the original $delta$ on quadratics.) Because $wedge$ plays well with $partial_{mathrm{dR}}$, $delta$ and $partial_{mathrm{dR}}$ graded-commute. The Jacobi identity implies $delta^2 = 0$. So we have a new differential $partial_q = partial_{mathrm{dR}} + delta$ on $mathcal S mathfrak G$.

Pick any $alpha in Omega_{mathrm{cpt}}^1(mathbb R)$ with $intalpha = 1$. (Henceforth I will call such one-forms bumps.) The map $x mapsto x otimes alpha$ from $mathfrak g to mathfrak G$ extends to an algebra homomorphism $mathcal S mathfrak g to mathcal S mathfrak G$ splitting $int$. There is a unique contracting homotopy $eta_alpha : Omega_{mathrm{cpt}}(mathbb R) to Omega_{mathrm{cpt}}(mathbb R)$ (of cohomological degree $-1$) such that the graded commutator $[partial_{mathrm{dR}},eta_{alpha}] = mathrm{id} - (alpha otimes) circ int$; it vanishes on $Omega_{mathrm{cpt}}^0$ and on $Omega_{mathrm{cpt}}^1$ satisfies $eta_alpha(beta) = partial_{mathrm{dR}}^{-1}(beta - (int beta)alpha)$; note that $int(beta - (int beta)alpha) = 0$, so the one-form $beta - (int beta)alpha$ has a unique antiderivative among compactly-supported functions.

We can extend $eta_alpha$ to $mathcal Smathfrak G$ in many ways, and the choice can be proven not to matter. To make a choice, declare that on the constants $mathcal S^0mathfrak G$ we have $eta_alpha = 0$, and on $mathcal S^nmathfrak G$ we have
$$ eta_alpha(beta_1 odot dots odot beta_n) = frac1n sum_i beta_1 odot dots odot eta_alpha(beta_i) odot dots odot beta_n, $$
where $odot$ denotes the symmetric multiplication in $mathcal S$. Note that this is not the extension of $eta_alpha$ as a derivation.

Since $eta_alpha$ always attaches 1-forms and $delta$ involves wedge multiplication, $eta_alpha : mathcal S mathfrak g to (mathcal S mathfrak G,partial_q)$ is a chain map. The integration map $int : mathcal S mathfrak G to mathcal S mathfrak g$ splitting this map is not a chain map from $(mathcal S mathfrak G,partial_q)$. But with the above choices we can choose a different splitting, namely $int circ (mathrm{id} - delta eta_alpha)^{-1}$. (I have a 50% chance of getting that minus sign wrong.) I will leave checking that this is a chain map splitting $eta_alpha$ to you. Note that $(mathrm{id} - delta eta_alpha)^{-1} = sum_N (delta eta_alpha)^N$ converges on $mathcal S mathfrak G$, since $delta eta_alpha$ drops polynomial degree by $1$.

Pick bumps $alpha_1, dots,alpha_n$ such that the support of $alpha_i$ is in $[i-1,i]$, and pick one final bump $alpha$ arbitrarily. One can prove that the map $mathcal U mathfrak g to mathcal S mathfrak g$ is given on monomials $x_1 dots x_n$ (with multiplication in $mathcal U$) by
$$ mathcal U mathfrak g ni x_1 dots x_n mapsto int circ (mathrm{id} - delta eta_alpha)^{-1} bigl( (alpha_1 otimes x_1) odot dots odot (alpha_n otimes x_n) bigr) in mathcal S mathfrak g$$
(or I might be off by a sign somewhere). In general, similar formulas describe the entire product on $mathcal Smathfrak g$ given by transporting the one from $mathcal U mathfrak g$ along the symmetrization isomorphism.

Let me now unpack this formula, or rather give the answer after some unpacking. (Proving that this is a valid unpacking is straightforward: you need to track the numerical factors coming from $eta_alpha$, understand how to apply a second-order differential operator to a monomial, and also include a brief "degree reasons" argument to get $eta_alpha$ and $delta$ to apply always to the same things at the same time.)

Define an $n$-leaf binary heighted forest, abbreviated forest, to be set of binary rooted trees whose leaves are put in bijection with the set ${1,dots,n}$ and whose nodes are totally ordered (I mean: totally order the set of all nodes) such that in a given tree, and path from root to leaf is increasing for the total ordering. Arbitrarily choose for each node which of its two branches is left and which right (the choice will cancel out).

Given a forest and the list $x_1,dots,x_n$ of elements in $mathfrak g$, there is an obvious element of $mathcal S mathfrak g$ given by putting the $x_i$s at the leaves and reading the forest as instructions of who to bracket with whom (then multiply the "root" outputs).

Now I will describe, for each forest, how to compute a number. Consider the map $Omega_{mathrm{cpt}}^1(mathbb R) otimes Omega_{mathrm{cpt}}^1(mathbb R) to Omega_{mathrm{cpt}}^1(mathbb R)$ given by $beta_1 otimes beta_2 mapsto beta_1 wedge eta_alpha (beta_2) - beta_2 wedge eta_alpha(beta_1) $. Place this map at each vertex, and $alpha_i$ at the $i$th leaf, and let the forest tell you how to apply this map to end up with a bunch of $1$-forms at the roots. Then integrate all these $1$-forms to get numbers, and multiply those numbers together.
Finally, suppose there are $k$ roots (and hence $(n-k)$ nodes). Then multiply by $frac1 n frac1{n-1} dots frac1{n-k}$.

Note that neither the number nor the element of $mathcal S mathfrak g$ determined by a forest depends on the height ordering. But I now want you to sum over all forests with total node-ordering of the product of these two numbers. That sum computes the map $mathcal U mathfrak g to mathcal S mathfrak g$ above. (If I had a good way to count the number of total orderings of the nodes for a given un-heighted forest, I would have used it.)

This is similar to, but not the same as, Kontsevich's star product. In particular, note that my forests have no wheels, whereas Kontsevich does not describe the symmetrization map $mathcal S mathfrak g cong mathcal U mathfrak g$, but rather this map twisted by some traces in the adjoint representation.

ag.algebraic geometry - Singular points of an irreducible polynomial

What do you mean by irreducible and what do you mean by $S(f)$?

Does irreducible mean absolutely irreducible (ie irreducible over the algebraic closure of $k$)? Is $S(f)$ considered as a scheme or as a set of rational points? If the latter, then is $S(f) := { (a,b) in k^2: f(a,b) = 0 = frac{partial f}{partial x}(a,b) = frac{partial f}{partial y}(a,b) }$? Or is it the set of singular points over the algebraic closure of $k$?

If by irreducible, you mean absolutely irreducible, then as Douglas Zare suggests, you can pass to the algebraic closure and prove that $S(f)$ is finite.

If irreducible is to be read over $k$, but you are considering $S(f)$ scheme theoretically or are evaluating the points in the algebraic closure of $k$, then the assertion is false. Consider for instance $k$ of characteristic $p$ with $a in k$ a non-$p^mathrm{th}$ power and $f(x,y) = x^p + y^p + a$.

Finally, if by $S(f)$ you mean the $k$-rational points, then if $S(f)(k)$ were infinite, then the set of $k$-rational points on the curve defined by $f$ would be infinite and $f$ would then be absolutely irreducible so that by the first case considered, $S(f)$ would be finite.

cv.complex variables - How can I calculate the characteristic function of these distributions? [previously: difficult integral]

If $q$ is a positive integer, then I think one can find this in any one of several undergraduate textbooks on complex analysis, where it's usually one of the standard examples to show the power of contour integration. I dimly remember something like this in Priestley's little OUP book, for instance. For arbitrary positive real values of $q$, I can't remember how this works I'm afraid.

(This is probably the sort of question which you could try out on fellow colleagues/students first, in my view.)

ra.rings and algebras - Is a left invertible element of a group ring also right invertible?

Given a group $G$ we may consider its group ring $mathbb C[G]$ consisting of all finitely supported functions $fcolon Gtomathbb C$ with pointwise addition and convolution. Take $f,ginmathbb C[G]$ such that $f*g=1$. Does this imply that $g*f=1$?

If $G$ is abelian, its group ring is commutative, so the assertion holds. In the non-abelian case we have $f*g(x)=sum_y f(xy^{-1})g(y)$, while $g*f(x)=sum_y f(y^{-1}x)g(y)$, and this doesn't seem very helpful.

If $G$ is finite, $dim_{mathbb C} mathbb C[G]= |G|<infty$, and we may consider a linear operator $Tcolon mathbb C[G]tomathbb C[G]$ defined by $T(h) = f*h$. It is obviously surjective, and hence also injective. Now, the assertion follows from $T(g*f)=f=T(1)$.

What about infinite non-abelian groups? Is a general proof or a counterexample known?

ag.algebraic geometry - What, precisely, is the relationship between "fields of moduli" and "moduli spaces"?

Notation

The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ¹ be a G-Galois cover, where everything is over the algebraic closure of some field L. Assume that X->ℙ¹ descends (without group action -- as a cover) to X_L->ℙ_L¹. Then I define the field of moduli to be the intersection of all finite extensions of L for which base change of X_L->ℙ_L¹ becomes G-Galois.

Question

There is the saying that the field of moduli is the function field of the (coarse?) moduli space of when you let the branch points vary. What is the precise statement of that? (and why is it true?)

Thoughts

It would seem that we should fix a dedekind ring whose quotient field is L (ℤ if L is ℚ), and call it D. Then descend to a D-model of ℙ¹ (for a D-model of X take the integral closure of ℙ¹ in the function field of X). Then do something like look at the moduli space of all covers of ℙ¹ with that number of (distinct) branch points, and in it look at the subscheme of all covers that can be achieved by deforming any of the fibers of our X_D->ℙ_D¹ (pick a fiber such that there's no coalescence of branch points) by a family. But there's a lot missing here, even in terms of making this precise. For example: IS there a coarse moduli space of all covers with n branch points over ℙ_D¹ (where by n branch points, I mean n branch point on each geometric fiber)? What does it look like? Why should the function field of said subscheme be the field of moduli?

Thanks in advance.

botany - How does the sensitive plant detect vibrations?

The sensitive plant (Mimosa pudica) is a remarkable little plant whose characteristic feature is its ability to droop its leaves when disturbed:

Apparently, this ability to droop rests on the cells in the leaves of the sensitive plant being able to draw water out of themselves through changes in intracellular ion concentrations, which makes the leaves less turgid.

What I'm hazy about is how the plant "senses" vibrations. Plants don't really have a nervous system to speak of; how then does the sensitive plant "know" to droop when disturbed?

set theory - What are the Martin's Maximum consequences of Namba forcing?

I think that I may have found a suitable candidate; namely, the result of
Konig and Yoshinobu that $MM$ implies that there are no $omega_{1}$-regressive $omega_{2}$-Kurepa trees. The proof seems to have the same relatively direct flavor as those in Baumgartner's $PFA$ article.

polymath5 - Improving a sequence of 1s and -1s

This is part of topological dynamics (a close cousin of ergodic theory, aka measurable dynamics, in which the underlying space on which the dynamics takes place is a topological space rather than a measure space). The relationship with combinatorics is roughly as follows: topological dynamics is to colouring Ramsey theorems (such as van der Waerden) as measurable dynamics is to density Ramsey theorems (such as Szemeredi).

For simplicity, let's work on the integers (there is a trick to then deal with the natural numbers, that I will talk about later).

Consider the cube $Omega := {-1,+1}^{{bf Z}}$, with the standard right shift T. We give this cube the product topology, making it a compact metrisable space. Every +-1 sequence is then a point x in $Omega$, and defines an orbit $T^{bf Z} x = { T^n x: n in {bf Z} }$, and then an orbit closure $overline{T^{bf Z} x}$. This is a closed, T-invariant subset of $Omega$ (which a topological dynamicist would call a subsystem of $Omega$).

Note that if y appears in this orbit closure, then every finite substring of y appears somewhere in x. So it is natural to try to look for what is in this orbit closure.

A simple application of Zorn's lemma tells us that every orbit closure contains at least one minimal non-empty closed T-invariant subset; these are known as minimal systems. (The notion of minimality in topological dynamics is broadly equivalent to the notion of ergodicity in ergodic theory.) Every element of a minimal system is almost periodic, which means that every finite block in the element appears syndetically (with bounded gaps). So one can always get an almost periodic element (this is a special case of the Birkhoff recurrence theorem).

All this is discussed in these lecture notes of mine.

Now, one can go beyond minimality and obtain further classification of such systems, and the subject gets rather interesting at this point. For instance, we have isometric systems (analogous to the compact systems in ergodic theory), with the property that if two points $x,y$ in the system are close, then all their shifts $T^n x, T^n y$ are uniformly close as well. Orbit closures of quasiperiodic sequences fall in this category. At the other extreme, we have topologically mixing systems (analogous to the mixing systems in ergodic theory), in which given any two non-empty open sets U, V in the system, the shift $T^n U$ of one of them will intersect the other V for all sufficiently large n. Orbit closures of random sequences (i.e. all of $Omega$) fall into this category. Then there are various intermediate systems between these, for instance one can take isometric extensions of isometric systems (analogous to things like nilsystems in ergodic theory), and so forth. This is all discussed to some extent in these later lecture notes of mine.

If one is working on the natural numbers rather than the integers, then it may seem prima facie that the shift T is now not invertible, but it is not difficult to convert the natural number situation to the integer situation, by starting with a sequence x on the integers and looking at the set of strings on the integers with the property that every finite substring of those integers appears infinitely often in the original sequence. This is a closed T-invariant subset of $Omega$; a simple compactness argument shows that it is nonempty. So one can basically reduce to subsystems of $Omega$ as before.

Finally, I should mention that there is an approach to this subject via ultrafilters. Given any non-principal ultrafilter $p in {Bbb Z}$, one can take the ultrashift $T^p x$ of a sequence $x$, defined as the ultralimit of the shifts $T^n x$ along the ultrafilter p (this is well-defined because $Omega$ is compact metrisable). One can then reduce a significant fraction of topological dynamics to the algebraic properties of ultrafilters. For instance, Hindman's theorem is a quick consequence of the existence of an idempotent ultrafilter. This is discussed at the first set of notes above, and also in its sequel.

lo.logic - What are some results in mathematics that have snappy proofs using model theory?

Plane geometry is decidable. That is, we have a computable algorithm that will tell us the truth or falsity of any geometrical statement in the cartesian plane.

This is a consequence of Tarski's theorem showing that the theory of real closed fields admits elimination of quantifiers. The elimination algorithm is effective and so the theory is decidable. Thus, we have a computable procedure to determine the truth of any first order statement in the structure (R,+,.,0,1,<). The point is that all the classical concepts of plane geometry, in any finite dimension, are expressible in this language.

Personally, I find the fact that plane geometry has been proven decidable to be a profound human achievement. After all, for millennia mathematicians have struggled with geometry, and we now have developed a computable algorithm that will in principle answer any question.

I admit that I have been guilty, however, of grandiose over-statement of the situation---when I taught my first logic course at UC Berkeley, after I explained the theorem some of my students proceeded to their next class, a geometry class with Charles Pugh, and a little while later he came knocking on my door, asking what I meant by telling the students "geometry is finished!". So I was embarrassed.

Of course, the algorithm is not feasible--its double exponential time. Nevertheless, the fact that there is an algorithm at all seems amazing to me. To be sure, I am even more surprised that geometers so often seem unaware of the fact that they are studying a decidable theory.

ct.category theory - Is the category commutative monoids cartesian closed?

Here's a couple of useful references:

P. Freyd: Algebra valued functors in general and tensor products in particular (MR0195920)

This paper shows that a commutative variety of algebras (such as commutative monoids) is closed monoidal. This is the construction that Peter mentions in his answer.

I don't have access to MathSciNet right now, but if you go to the MSN page for the above and click on the "in reviews" button, then you will (I think) get a paper that considers when such an algebraic theory has a cartesian closed structure. I'm not sure whether or not the condition is merely sufficient, or is an if-and-only-if, but if I remember correctly, the condition had to do with diagonals. That is, you take an arbitrary operation and feed the same thing in to every input. But I'm not entirely sure what the condition was, off the top of my head, except that it was very strong and not satisfied by commutative monoids!

(Apologies for the vagueness of the above; when I get MathSciNet access again I'll fill in the details)

---

Update: A long bus ride with little to do provided ample opportunity to fill in the details. We want an algebraic theory that is cartesian closed. Let $mathcal{V}$ be our theory. Then we need an internal hom, $underline{mathcal{V}}(B,C)$, and an adjunction $mathcal{V}(A,underline{mathcal{V}}(B,C)) cong mathcal{V}(A times B,C)$. Now, having the internal hom is, via the paper of Freyd above, tantamount to the algebraic theory being commutative. That means that all the operations of the theory commute with each other (so, for example, a binary operation commuting with itself has to satisfy $(ab)(cd) = (ac)(bd)$, more on the nlab page http://ncatlab.org/nlab/show/commutative+theory). This implies, as Peter says above and Freyd proves in his paper, that $mathcal{V}$ is closed monoidal, but the monoidal structure is not necessarily the cartesian product. So we want to look at when it is the cartesian product. So assume that it is. Then a $mathcal{V}$-morphism $A to underline{mathcal{V}}(B,C)$ corresponds to a morphism $A times B to C$.

Consider an operation in the algebraic theory, say $nu$, and suppose that it has arity $n$. Let $f colon A times B to C$ be a $mathcal{V}$-morphism. As this is a morphism, we have that

$$
nubig(f(a_1,b_1),...,f(a_n,b_n)big) = f(nubig((a_1,b_1),...,(a_n,b_n)big)) = f(nu(a_1,...,a_n),nu(b_1,...,b_n))
$$

We also insist that the map $a mapsto (b mapsto f(a,b))$ is a $mathcal{V}$-morphism. Now the $mathcal{V}$-structure on $underline{mathcal{V}}(B,C)$ is given as follows: $nu$ applied to $g_1,...,g_n$ is the map $b mapsto nu(g_1(b),...,g_n(b))$. That is, it factors as $B to B^n to C^n to C$ where the first map is the diagonal, the second the product of the $g_i$, and the third is $nu$. So to say that $a mapsto (b mapsto f(a,b))$ is a $mathcal{V}$-morphism, we mean that it commutes with our typical operation, $nu$. Thus

$$
b mapsto f(nu(a_1,...,a_n),b)
$$

is the same as
$$
nuBig(b mapsto f(a_1,b), b mapsto f(a_2,b), ..., b mapsto f(a_n,b)Big)
$$

This latter is

$$
b mapsto nu(f(a_1,b), ..., f(a_n,b))
$$

But since $f$ itself was a morphism of $mathcal{V}$-algebras, this simplifies to

$$
b mapsto f(nu(a_1, ..., a_n), nu(b,...,b))
$$

So we conclude that

$$
f(nu(a_1,...,a_n),nu(b,...,b)) = f(nu(a_1,...,a_n),b)
$$

As everything was generic, we conclude that we must have the identity

$$
nu(b,...,b) = b
$$

for every $mathcal{V}$-operation. This is a necessary condition. I am not sure if it is sufficient (I have a vague recollection that it is not, but still don't have MathSciNet access so can't check).

For commutative monoids, we have two operations: $0$ and $+$. For addition, we get the condition $b + b = b$ which is pretty strong! For $0$, we get the even stronger condition that $0 = b$. So if we take "commutative monoids" and impose identities to try to get a cartesian closed category, we end up with a pretty trivial algebraic theory!

smooth manifolds - On the $mathbb R$-algebra structure on $C^infty(M)$.

You can determine the R-algebra structure of $C^infty(M)$ purely from its ring structure. As Robin Chapman mentions, the constant function 1_M is uniquely determined by the fact that it is the identity element, and multiplication by rationals is uniquely defined, so the functions equal to a constant rational value are uniquely determined.

Actually, the ring homomorphism $Fcolonmathbb{R}to C^infty(M)$ is unique, which also uniquely defines the R-algebra structure.

The positive elements $xinmathbb{R}$ are squares, so $F(x)$ must be a square in $C^infty(M)$, hence nonnegative everywhere. Then, for any $xinmathbb{R}$ and rational numbers $ale xle b$ we have $F(x)-a1_M=F(x-a)ge0$ and $b1_M-F(x)=F(b-x)ge0$, so $F(x)in C^infty(M)$ takes values in the interval $[a,b]$. This shows that, in fact, $F(x)=x1_M$.

Thinking about it, this works because $mathbb{R}/mathbb{Q}$ has trivial Galois group. You can see this by asking if the C-algebra structure on $Aequiv C^{infty}(M,mathbb{C})$ is uniquely determined by its ring structure, for which the answer is no. For any $sigmain{rm Gal}(mathbb{C}/mathbb{Q})$ it is not possible to distinguish a constant $fin A$ from $sigma(f)$ in terms of ring operations [edit: if $sigma$ is continuous, that is. So, only considering the identity element and complex conjugation]. Instead, you could ask if it is possible to determine the C-algebra structure up to the action of the Galois group. If the manifold is connected then the answer to this is yes. The constant function taking the value $pm i$ everywhere is given by $i_M^2+1_M=0$, and the constant functions $fin A$ are those for which $f-lambda1_M-mu i_M$ are units for all but at most one choice of $lambda,muinmathbb{Q}$. The constant functions are isomorphic to $mathbb{C}$, which is determined up to the action of the Galois group. If it is not connected, then we can't even say that much. For any locally constant map $sigmacolon Mto{rm Gal}(mathbb{C}/mathbb{Q})$, it is not possible to distinguish $fin A$ from $f_sigma(P)equivsigma(P)(f(P))$ using ring operations. The C-algebra structure is uniquely determined up to the action of such a locally constant $sigma$ though, which should still be enough to tell you everything about the manifold. Working over the reals, none of this matters, because of the triviality of the Galois group.

ds.dynamical systems - Almost complex 4-manifolds with a "holomorphic" vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?

The following sub question is rewritten thanks to the comment of Robert Bryant:

Is it true that if $(M,J)$ admits a vector field that preserves $J$ then there is $J'$ on $M$ homotopic to $J$ that is preserved by an $S^1$-action on $M$?

After three years I don't think indeed that the following is such a reasonable motivation

POSSIBLE MOTIVATION. Claire Voisin gave a construction of the Hilbert scheme of points for every almost complex 4-fold by an analogy with the Hilbert scheme of points of a complex surface. The first calculation of the Euler charactericstics of Hilbert scheme of points of complex surfaces was done via localisation techniques, for $CP^2$. Now, if we have a "holomorphic" vector field on an almost complex manifold, this could potentially help to reduce the calculation of the Euler charachteristics of its Hilbert scheme to the study of fixed points of the manifold. So the question is how flexible this notion is... But in its nature this question seems to be more a question (maybe not a hard one) on dynamical systems.

complex geometry - Riemann surface disconnected at infinity

This question may be trivial, I did not think hard about it.

A friend of mine is looking for an irreducible (reduced) analytic subspace $C subset mathbb{C}^2$ with the following property. Let $f colon C rightarrow mathbb{C}$ be the projection on the first factor. He wants that

1) All singular points of $C$ and all ramification points for $f$ lie in a limited set, so removing that set we obtain a topological covering from some open set of $C$ to $mathbb{C}$ with a ball removed.

2) That covering should be trivial (even better if it is finitely-sheeted).

So the curve $C$ is connected, but only if one passes near the origin. Sufficiently far from that ther should be no way to jump between sheets. Is it possible to find such a $C$?

measure theory - Are there sigma-algebras of cardinality $kappa>2^{aleph_0}$ with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $sigma$-algebras. The only proof that I know is via a contradiction argument which yields no estimate on the minimum cardinality of an infinite $sigma$-algebra.

Given an a set $X$ of infinite cardinality $kappa$, the $sigma$-algebra of all co-countable subsets of $X$ is of cardinality $2^kappa$ $kappa^{aleph_0}$. This example doesn't tell me whether there are $sigma$-algebras of cardinality below $2^{aleph_0}$, if I don't assume the Continuum Hypothesis.

My question is as the title says: Are there $sigma$-algebras of every uncountable cardinality?

Edit: The combined answer with Stephen, Matthew proves that the cardinality of a $sigma$-algebra is necessarily at least $2^{aleph_0}$. Further, for each cardinality $kappage 2^{aleph_0}$ with uncountable cofinality, the $sigma$-algebra of countable (or cocountable) subsets of a set $X$ with cardinality $kappa$, is of cardinality $kappa$.

What is left is whether for $kappage 2^{aleph_0}$ with $cf(kappa)=aleph_0$ are there $sigma$-algebras of cardinality $kappa$. (I changed the title to reflect this.)

Thanks Stephen, Matthew, Apollo, for the combined work!

splicing - What are limiting factors for intron length?

If you examine the human genome ~99% of the introns are under 500 kb. I would assume that a limit between 250 kb - 500 kb is reasonable for gene prediction. You may incorrectly predict the proper structure of a small number of genes that have these very large introns but this should be a small number. Furthermore, most popular sequence aligners tend to set an intron length limit between 500 kb and 750 kb.

Just keep in mind that you may increase the number of false positive introns you detect if you set this limit to high. Therefore, it may be worthwhile to try a few settings and evaluate the results.

EDIT:

My guess is that larger introns are constrained in mammals for two reasons

1. They are more difficult for the spliceosome to excise properly. The spliceosome may have reduced assembly at the proper 5' / 3' splice sites and the branch point site. There is also a higher chance that there will be cryptic / decoy splice sites witihin the intron.


2. They increase the time it takes to transcribe the gene

nt.number theory - Locally profinite fields ?

This is a question about terminology and should be taken lightly.

The expression local field is used in at least three different senses :

1) For a locally compact totally disconnected field. These are the finite extensions of $mathbb{Q}_p$ or of $mathbb{F}_p((T))$, where $p$ is a prime number and $T$ is an indeterminate.

2) For a field complete with respect to a discrete valuation whose residue field is merely perfect, not necessarily finite as in 1). This is how Fontaine uses the expression.

3) For locally compact fields which are not discrete. These are the fields in 1), but also $mathbb{R}$ and $mathbb{C}$ in addition. People who adopt this definition refer to the fields in 1) as non-Archimedean local fields.

(Nobody insists that a local field is a local ring which happens to be a field, but the "logic" is impeccable.)

Question. Would locally profinite field be a good piece of terminology for the fields in 1) ?

This would certainly avoid the confusion with the other fields in 2) and 3).

The expression locally profinite group is already in use (for example in the book Bushnell-Henniart). The additive and the multiplicative groups of a locally profinite field would be locally profinite groups in their sense.

mathematics education - Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings,

I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar transitional-curriculum courses would be interesting as well.

The context is that we are creating a 2nd year analysis course which would be our students 1st exposure to analysis. This is to be followed up by a 3rd year analysis course which would be something of a "rigorous multi-variable calculus course". Next year the 3rd year course is going to be offered but the students will not have the 2nd year course as background (in future years the 2nd year analysis course will be a prerequisite).

What they will have is a fairly extensive "service calculus" background, consisting of four courses: a more or less standard 2-course 1st year single-variable calculus sequence (text is Edwards and Penney), plus a 2nd year multi-variable calculus course (also Edwards and Penney - this is a standardish calculus in $mathbb R^n$ for $n leq 3$ text). They follow that up with a a 3rd year course on multi-variable calculus in $mathbb R^n$, the text is Folland. This course also covers some material that is traditionally taught in analysis classes, things like uniform convergence, Fourier transforms and such. But very little time is spent fussing about with open subsets of $mathbb R^n$, they don't get to see bump functions, what a function is isn't discussed in much detail, what numbers are isn't dwelled on (not even axiomatically).

So, that's the setup. It can be safely assumed these students are motivated to study analysis, as they're taking this course to transition into our upper-level analysis courses. But I can't do too much too fast. And I don't want to bore them. So things of importance for this course to dwell on are things like what numbers are (at least axiomatically), maybe even a bit of set theory, completeness, fussy details about continuous functions like when extreme values exist, the various formulations of continuity for functions on open subsets of $mathbb R^n$. Bump functions. And fussy details from multi-variable calculus, such as the inverse and implicit function theorem.

Are there texts out there that are designed for situations like this?

One initial inclination would be to supplement something like Hubbard's calculus text with some additional foundational notes. I suppose there are many standard intro analysis courses but some of these might be oddly paced for this group of students. I presume it's unlikely anyone has written a text for just this situation but who knows?

Thanks in advance for your comments.

ag.algebraic geometry - Is every homogeneous G-variety of the form G/H?

It depends on what you mean by "closed subgroup". If you mean a Zariski closed subset which forms a subgroup then the answer is no. If you mean a closed subgroup scheme, then the answer is yes. An example where you need to use the second definition is the Frobenius map $Fcolon G to G^{(p)}$. If we let $G$ act on $G^{(p)}$ through $F$ then the action is transitive and indeed $G^{(p)}$ is isomorphic to $G/Ker F$. However, unless $G$ is zero-dimensional $Ker F$ is a non-trivial finite group scheme whose $k$-points consist of just the identity.

Note however that $G/H$ is always quasi-projective even when $H$ is a subgroup scheme so all homogeneous $G$-spaces are quasi-projective.

na.numerical analysis - What is the time complexity of computing sin(x) to t bits of precision?

Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference?

Long version of the question:
I'm sort of surprised to be asking this, because it's such an extremely basic sounding question. Here are some variants on it:

1. How much time does it take to compute $pi$ to $t$ bits of precision?


2. How much time does it take to compute $sin(x)$ to $t$ bits of precision?


3. How much time does it take to compute $e^x$ to $t$ bits of precision?


4. How much time does it take to compute $mathrm{erf}(x)$ to $t$ bits of precision?


5. How much time and how many random bits does it take to generate a (discrete) random variable $X$ such that there is a coupling of $X$ with a standard Gaussian $Z sim N(0,1)$ for which $|X - Z| < delta$ except with probability at most $epsilon$?

In my area of theory of computer science, no one seems to pay much attention to such questions; an algorithm description might typically read

"Generate two Gaussian random variables $Z$ and $Z'$ and check if $sin(Z cdot Z') > 1/pi$"

or some such thing. But technically, one should worry about the time complexity here.

One colleague of mine who's more of an expert on these things assured me that all such "calculator functions" take time at most poly$(t)$. I well believe that this is true. But again, what polynomial (out of curiosity, at least), and what is the reference?

I kind of assumed that the answers would be in every single numerical analysis textbook, but I couldn't find them there. It seems (perhaps reasonably) that numerical analysis cares mainly about getting the answers to within a fixed precision like 32 or 64 bits or whatever. But presumably somebody has thought about getting the results to arbitrary precision, since you can type

Digits := 5000; erf(1.0);

into Maple and it'll give you an answer right away. But it seemed hard to find. After much searching, I hit upon the key phrase "unrestricted algorithm" which led me to the paper "An unrestricted algorithm for the exponential function", Clenshaw-Olver-1980. It's pretty hard to read, analyzing the time complexity for $e^x$ in terms of eight (??!) parameters, but its equation (4.55) seems to give some answers: perhaps $tilde{O}(t^2)$ assuming $|x|$ is constant?

And really, all that work for little old $e^x$? As for erf$(x)$, I found the paper "The functions erf and erfc computed with arbitrary precision" by Chevillard in 2009. It was easier to read, but it would still take me some time to extract the answer; my first impression was $tilde{O}(t^{3/2})$. But again, surely this question was not first investigated in 2009, was it?!

(By the way, question #5 is the one for which I really want to know the answer, but I can probably work it out from the answer to question #4.)