ct.category theory - internal version of a flat functor?

I'm working out of Sheaves in geometry and logic, for reference.

There is a characterisation of flat functors $A:C to Set$ as those such that the Grothendieck construction $int_C A$ is a filtering category. There are more general versions of this result, in which $Set$ is replaced by a more general topos. One should also be able to characterise those discrete opfibrations that arise from flat functors (up to iso/equiv?). How about if we replace $C$ by an internal category, in a topos $E$ say? Then functors out of $C$ are replaced by discrete opfibrations over $C$ in $E$.

My question is this:

What sort of thing should be considered as the analogue of a flat functor in the internal setting?

pr.probability - Coupling of Wiener processes

If there is no coupling s.t. the distance goes to 0 in $L^{1}$ (which I agree seems likely), you might want to look up an introduction to the Wasserstein-1 distance (which is exactly expected $L^{1}$ distance after an optimal coupling). This is the language that I've seen this type of problem most often discussed in. The field of finding optimal couplings for given metrics is 'optimal transport'.

I vaguely recall that there are theorems about optimal couplings (in only certain $L^{p}$ only, of course!) never giving rise to crossing lines. In a 1-dimensional problem, such as the one you have, this would tell you what the optimal coupling is explicitly (in this case, if you construct your Brownian motion via Donsker's theorem, it says: whenever $W^{0}$ takes a move in the $alpha$ percentile, make $W^{x}$ also move in the $alpha$ percentile... in other words, my probably-misremembered theorem would imply coupling doesn't help your $L^{1}$ distance at all in this case).

Cedric Villani has two excellent books on the subject, at least one of which was available for free download the last time I checked, and you should be able to find 'this sort' of theorem. Please don't take my word for the statements.

Edit: Here is (I believe) a proof... though it might fit in the category of so-simple-its-wrong. First of all, we have starting points x,0 and add two normal (0,a) random variables X and Y to them. Plugging in the obvious cost functions, the "Kantorovitch Duality" formula tells us that the $L^{1}$ distance between x+X and 0+Y is at least x (while plugging in the independent coupling to the standard way of writing this metric tells us it is at most x). So, at time a, the $L^{1}$ distance between the brownian motions must be at least x (since at time a they have the same distribution as x+X and 0+Y, and we have found this lower bound for ALL couplings, and in particular all couplings that come from them both being brownian motions). In particular, the $L^{1}$ distance can't go to 0.

Coefficients from Stone Weierstrass versus Fourier Transform

Re-reading your question, I think that I see what you are asking.

Per @Andrea Ferretti's comments, you have to be careful to distinguish between ${e^{inx}}$ and $span {e^{inx}}$. You certainly are interested the latter. Sorry if my comments were sloppy and confusing above.

So, I think that the it goes like this:

From some corollary of Stone-Weirstrauss you can show that $span {e^{inx}}$ is dense in $C(mathbb{S}^1)$ with the supremum norm. Because we know that $C(mathbb{S}^1)hookrightarrow L^2([0,1])$ has its image a dense subset of $L^2([0,1])$ and we know that if $f_n to f$ in the supremum topology on $C(mathbb{S}^1)$, then the images also converge in $L^2([0,1])$.

Thus, by this reasoning, for $fin L^2([0,1])$ we can find $f_n in span {e^{inx}}$ such that $f_n = L^2([0,1])$. Lets write
$$
f_n = sum_{kin mathbb{Z}} c_k^{(n)} e^{ikx}
$$
where all but finitely many of the $c_k^{(n)}$ are zero (this is because in the span of infinitely many objects we only take a finite number of them to add together)

Now, what I think you are asking is: what can we say about the coefficients $c_k^{(n)}$? The answer is that they converge to the $k$-th Fourier coefficient of $f$ as $ntoinfty$ because
$$
hat f(k) = langle f, e^{ikx} rangle = lim_{ntoinfty} langle f_n ,e^{ikx}rangle = lim_{ntoinfty} c_k^{(n)}
$$

In fact if $c_k^{(n)}$ are arbitrary complex numbers, defining $f_n$ as above, we see that
$$
Vert f - f_n Vert_{L^2} = sum_{kin mathbb{Z}} |hat f(k) - c_k^{(n)}|^2
$$
assuming convergence. Thus, if $(c_k^{(n)})_k to (hat f(k))_k$ as $ntoinfty$ in $ell^2(mathbb{Z})$ then $f_nto f$ in $L^2$, which is a pretty weak condition.

intuition - Why is addition of observables in quantum mechanics commutative?

The introduction you outlined is basically reminiscent of the old quantum mechanics, anyway in the approach you depicted it is the culmination and not the premise of the construction, and the comment was surely intended to be explanatory of the far origin of these choice. I now try to resume the history.

There are basically two approaches to mathematical quantum mechanics. The first one very complex and stratified in its development, but simple in the premise was discussed by John Von Neumann in a lot of papers after the "Foundation of quantum mechanics", the second one is basically conceveid to be an extension for the first, and is this second approach you are referring to: the GNS approach.

Anyway both of them are surely derived after an abstraction process very far beginning on the methods of classical mechanics, joint to the newest evidence from atomic and particle physics of the first quantum mechanics.

Just as in classical mechanics we define functions of observables dynamical quantity so the founders of quantum mechanics conceived it is possible in quantum mechanics, anyway we need to clarify in which sense this is possible and explanation isn't fully depleted from the naive extension of classical theory of the measure, based on real numbers, but it need of a clear axiomatic and this was furnished from John Von Neumann (and in some way from Heisenberg, Dirac, and Schroedinger before him formulated this axiomatic)

Anyway, just as in classical mechanics there is a notion of repeatibility and regularity, so there is in quantum mechanics. The true difference is in the outcome of the measures, deterministic in classical, probabilistic in quantum mechanics. So that measure processes are conceived deterministic in a statistical sense, and, for example, the component energies of the isotropic harmonic oscillator sums exactly in mean value, but the variance is zero if the considered states are eigenstates. Old quantum mechanics can be founded on few axioms about the measures and led Von Neumann, in a natural way to linear operators acting, like a non commutative algebra, on Hilbert spaces.

In order to grant correspondence principle we, following the founders of quantum mechanics, need to hypothesize the existence of intrinsically deterministically evolving observable, and just the measure process make the difference, because these dynamical "quantities" with respect to the measures doesn't appear as real numbers, this point was the first time realized some time after the Copenaghen interpretation was developed.

So they are assumed, after Heisenberg (speaking of non commutative numbers) and Jordan (speaking of matrices), and Schroedinger (speaking of operators acting on functional space of probability) all these three point of view were showed to be in a certain strict framework to be equivalent, from Dirac assuming they are algebraic elements obeying to canonical commutation relation generalizing the Poisson algebra.

In brief the Dirac point can be summarized in assuming an Hilbert space structure for the states, and in developing step by step a theory of observables compatible with the Copenaghen interpretation spirit and with the correspondence principle.

Anyway Von Neumann felt the need to obtain an axiomatic foundation based on more general operators algebras, and an axiomatic of measure, unifying from scratch the theoretical
framework, in fact obtaining a more general theory with respect to the Heisenberg and Dirac theoretical "prejudices". The Von Neumann point was in fact based on the general representation theory in the geometrical framework of Banach operator algebras of operators in Hilbert space, and in particular on the CCR irreducible representation theory, but from this point the research of Von Neumann continued in search of an intrinsic point of view based on the geometry of observable.

After time and time was in fact recognized that part of quantum theory of measure is nothing else then a generalized probabilistic theory in a Banach algebra and the general setting of Gelfand Najmark Segal construction rebuild intrinsically the Hilbert spaces. Anyway the field extension of this setting is very problematic and a hierarchy of Hilbert spaces appears. Anyway in this way a circle is closed and a new loop is opened: in the GNS approach to quantum mechanics we postulate that operators are living in an abstract algebra, obeying familiar rules for an algebra with an involution (the * operation). Via Gelfand theorem the commutative case led to the algebra of complex valued continuous functions in an Hausdorf space, the spectrum of the algebra (which will led the ordinary numerical set of coordinates of classical mechanics), and more in general to a spectral theory, culminating in the GNS construction, which associate to a given linear form an Hilbert space and a representation for the algebra.

Anyway the true achievement of this approach is the net of algebras, that is very more general with respect to the Hilbert space interpretation of quantum mechanics,this achievement is useful in relativistic field theory and leads to very far reaching results firstly partially discovered by Von Neumann in some papers, and after then developed from Araky, Haag, Kastler. In this full setting is now possible to address in more precise terms the question of the cluster decomposition principle implicit in the deterministic evolutionary scheme, and the question of repeatability principle of classical and quantum mechanics, and to understand quantitatively something about the limitation, arising from the change of the state of the universe, to this principle, which can be espressed, for example, in term of a change of representation, becaused from the change of the linear form representing the thermokinetic state of "universe", without any change in the postulates of quantum field theory and the derived quantum mechanics. This is perhaps the perspective of the search about KMS theorem.

I'm not very satisfied from this resume, anyway I think you can correct and integrate it, and I hope to read and write something else more precise and delimited.

derived category - Sebastiani-Thom isomorphism for D-modules

Considering $f:Xto mathbb{C}$, $g:Xto mathbb{C}$ and $foplus g:(x,y)mapsto f(x)+g(y)$.
The Sebastiani-Thom isomorphism is an isomorphism $Phi_{foplus g}(Mboxtimes N) = Phi_{f}(M) otimes Phi_{g}(N)$ compatible with monodromies.

The original theorem was for constant coefficient $M = mathbb{C}_X$, $N = mathbb{C}_Y$. David Massey gave a proof for general constructible coefficients.
Is there an algebraic proof for D-modules?

All proofs use topological arguments that don't seem to translate. In his article "On microlocal b-funtions" Saito mentions a result to be published but I couldn't find it.

human biology - Is there an evolutionary advantage to crying when sad?

First, here is a link to an article that discussion the answer to your question:
Why Cry? Evolutionary Biologists Show Crying Can Strengthen Relationships

The original article to which it refers can be found here:
Emotional Tears as Biological Signals

To speculate a bit on Jez's question, crying in adults could be a result of our possible Neoteny. Crying in infant animals is extremely common as a way to elicit attention and sympathy from the mother. If we are neotenous apes, crying could be a left-over behavior that was seized by evolution, instead of being discarded.

nt.number theory - Mod l local Galois representations (l different from p)

My question is referred to the statement and proof of Prop. 2.4 of Diamond's
article "An extension of Wiles' Results", in Modular Forms and Fermat Last
Theorem, page 479.

More precisely: fix $l$ and $p$ two distinct primes, with $l$ odd. Let $sigma$ be an
irreducible, continuous, degree 2 representation of the absolute Galois group
$G_{p}$ of $Q_{p}$, with coefficients in $k$, an algebraic closure of the finite
field with $l$ elements.
Proposition 2.4 states that if the restriction of $sigma$ to the inertia
subgroup of $G_{p}$ is irreducible and $p$ is odd, then $sigma$ is isomorphic to the
representation induced from a character of the Galois group of a quadratic
ramified extension $M$ of $Q_{p}$.
The proof given works if the restriction of $sigma$ to the wild inertia of $G_{p}$ is
reducible (I think there's a typo in the first line of the proof). What if
$sigma$ is irreducible on wild inertia (and $p$ is always odd)? It seems to me that this case is not covered in the proof of the Proposition, but maybe I'm not seeing something obvious.. If such a representation exists, it cannot be induced from a quadratic extension $M$ as above, so how does it fit in the description given by the Proposition? Can one say something about such a representation (for example something about its projective image?).

Thanks

st.statistics - Corruption and Recovery

Take $B$ to be a "parity-check matrix", i.e. an $n times (n-m)$ matrix with $BA=0$ which exists by linear algebra as long as $n > m$ (but maybe your $m$'s and $n$'s are mixed up, the input should be from a lower dimensional space then the code vector). So, putting $g(x) = Bx$ will have the property that $g(Af+epsilon)=g(epsilon)$ but $g$ takes values in $mathbb{R}^{n-m}$. Now, syndrome decoding requires you to have, for each syndrome $s in mathbb{R}^{n-m}$ a choice of $v_s in mathbb{R}^n, Bv_s=s$ usually by minimizing the norm of $v_s$ among all choices. In your situation, $y-v_{g(epsilon)}$ will be a codeword which, hopefully is $f$.

These things are not usually done in $mathbb{R}^n$ because there is no good way to choose the $v_s$. If you work over a finite field, then it is possible for all $s$ to choose $v_s$ minimizing the Hamming norm and so maximizing the changes of correct decoding (assuming few errors in the Hamming norm). Now making this list of the $v_s$ for all $s$ is not efficient in the sense of complexity theory as there might be a lot of $s$'s. But, if you can afford the time to do it once and for all, then using the syndromes is an efficient way of decoding.

analytic number theory - orthogonality relation for quadratic Dirichlet characters

So I think I solved half of the problem. Suppose that $n$ is a perfect square. Then $chi_d(n) = 1$ unless $gcd(d,n) > 1$, in which case its $0$. So, for $gcd{d,n} = 1$, we are simply pulling out the subset of fundamental discriminants having no common divisor with $n$. To quantify the size of this subset, we must first count fundamental discriminants.

The set of fundamental discriminants consists of all square-free integers congruent to $1$ modulo $4$ (i.e. odd fundamental discriminants) and all such numbers multiplied by $-4$ and $pm 8$ (i.e. even fundamental discriminants). The odd fundamental discriminants may be counted by considering the series $$sum_{text{$d$ odd}} frac{1}{|d|^s}.$$ In fact, this is a Dirichlet series. For observe that $$sum_{text{$d$ odd}} frac{1}{|d|^s} = 1 + frac{1}{3^s} + frac{1}{5^s} + frac{1}{7^s} + cdots = prod_{p>2} left(1 + frac{1}{p^s}right) = frac{zeta(s)}{zeta(2s)} left(1 + frac{1}{2^s}right)^{-1},$$
where $$frac{zeta(s)}{zeta(2s)} = sum_{n=1}^infty frac{|mu(n)|}{n^s},$$ is the Dirichlet series which generates the square-free numbers. So, by following the definition of fundamental discriminants given above, we can count fundamental discriminants by using the Dirichlet series $$left(1 + frac{1}{4^s} + frac{2}{8^s}right) frac{zeta(s)}{zeta(2s)} left(1 +frac{1}{2^s}right)^{-1} = left(1 + frac{1}{4^s} + frac{2}{8^s}right) underbrace{prod_{p>2} left(1 + frac{1}{p^s}right)}_{l_p(s)}.$$

Now, to omit those discriminants with $gcd(d,m) > 1$, we just omit the corresponding factors in $l_p(s)$. What's missing is $$prod_{substack{p>2 \ p|m}} left(1 + frac{1}{p^s}right),$$ so the relative density (compared to all fundamental discriminants $d$) is can be quantified by the expression $$frac{1}{D} cdot prod_{substack{p>2 \ pnmid m}} left(1 + frac{1}{p^s}right) = prod_{substack{p>2 \ p|m}} left(1 + frac{1}{p^s}right)^{-1}.$$ As in the proof of the prime number theorem, the main contribution here comes from the simple pole at $s=1$ (of $zeta(s)$). In fact, $p=2$ also fits at $s=1$ since $1 + frac{1}{4} + frac{2}{8} = 1 + frac{1}{2}$. Thus, in the end we obtain $$lim_{Xrightarrow infty} frac{1}{D} sum_{0 < |d| leq X} chi_d(m) = prod_{p|m} left(1 + frac{1}{p}right)^{-1},$$ as desired.

Now, does anyone know how to explain the other case. I assume it just follows from the symmetry of the roots of unity. But can anyone validate and perhaps clarify this. Thank you.

ag.algebraic geometry - Degrees of subvarieties of projective space

I've always thought of the degree of a subvariety of projective space as the degree of the divisor that defines the (given) embedding into projective space. It's been pointed out to me that this works only for curves. Now I'm confused: is there a similar characterization of the degree of a general subvariety of some projective space?

dna - Primer Dimer / Hairpin Algorithms

It is important visualize more faces to design primer or oligo and PCR experiments:

- calculate thermodynamic parameters of DNA Hybridization (Oligo/Template)

• compute hairpin-loop, dimer, bases penality and melting temperature about primer or pair primer




• calculate statistics about melting temperature with the change in composition and mix PCR concentration

with google you could find more tools but you must choose a tool of a sequencing team.
almost all tools use mfold algorithm, it's simple and effective:

M. Zuker, D. H. Mathews & D. H. Turner. Algorithms and Thermodynamics for RNA Secondary Structure Prediction: A Practical Guide In A Biochemistry and Biotechnology, 11-43, J. Barciszewski and B. F. C. Clark, eds. , NATO ASI Series, Kluwer Academic Publishers, Dordrecht, NL, 1999.

I worked in a sequencing team and there was a programmer. he developed a fantastic tools. he is a little english, but he could support you: http://promix.cribi.unipd.it/cgi-bin/promix/melting/melting_main.exe

soft question - Your favorite surprising connections in Mathematics

My favorite connection in mathematics (and an interesting application to physics) is a simple corollary from Hodge's decomposition theorem, which states:

On a (compact and smooth) riemannian manifold $M$ with its Hodge-deRham-Laplace operator $Delta,$ the space of $p$-forms $Omega^p$ can be written as the orthogonal sum (relative to the $L^2$ product) $$Omega^p = Delta Omega^p oplus cal H^p = d Omega^{p-1} oplus delta Omega^{p+1} oplus cal H^p,$$ where $cal H^p$ are the harmonic $p$-forms, and $delta$ is the adjoint of the exterior derivative $d$ (i.e. $delta = text{(some sign)} star dstar$ and $star$ is the Hodge star operator).
(The theorem follows from the fact, that $Delta$ is a self-adjoint, elliptic differential operator of second order, and so it is Fredholm with index $0$.)

From this it is now easy to proof, that every not trivial deRham cohomology class $[omega] in H^p$ has a unique harmonic representative $gamma in cal H^p$ with $[omega] = [gamma]$. Please note the equivalence $$Delta gamma = 0 Leftrightarrow d gamma = 0 wedge delta gamma = 0.$$

Besides that this statement implies easy proofs for Poincaré duality and what not, it motivates an interesting viewpoint on electro-dynamics:

Please be aware, that from now on we consider the Lorentzian manifold $M = mathbb{R}^4$ equipped with the Minkowski metric (so $M$ is neither compact nor riemannian!). We are going to interpret $mathbb{R}^4 = mathbb{R} times mathbb{R}^3$ as a foliation of spacelike slices and the first coordinate as a time function $t$. So every point $(t,p)$ is a position $p$ in space $mathbb{R}^3$ at the time $t in mathbb{R}$. Consider the lifeline $L simeq mathbb{R}$ of an electron in spacetime. Because the electron occupies a position which can't be occupied by anything else, we can remove $L$ from the spacetime $M$.

Though the theorem of Hodge does not hold for lorentzian manifolds in general, it holds for $M setminus L simeq mathbb{R}^4 setminus mathbb{R}$. The only non vanishing cohomology space is $H^2$ with dimension $1$ (this statement has nothing to do with the metric on this space, it's pure topology - we just cut out the lifeline of the electron!). And there is an harmonic generator $F in Omega^2$ of $H^2$, that solves $$Delta F = 0 Leftrightarrow dF = 0 wedge delta F = 0.$$ But we can write every $2$-form $F$ as a unique decomposition $$F = E + B wedge dt.$$ If we interpret $E$ as the classical electric field and $B$ as the magnetic field, than $d F = 0$ is equivalent to the first two Maxwell equations and $delta F = 0$ to the last two.

So cutting out the lifeline of an electron gives you automagically the electro-magnetic field of the electron as a generator of the non-vanishing cohomology class.

nt.number theory - Fiddling with p-adics

1) The field K_0 is the fraction field of the Witt vectors of F_p-bar (my non-TeX notation for the algebraic closure of F_p). Call that ring W. It is a complete DVR with an alg. closed residue field F_p-bar.

Any finite extension field of K_0 is naturally a finite-dimensional K_0-vector space.
Since K_0 is complete, all vector space norms on a fin. dim. K_0-vector space V (say, the sup-norm w.r.t. some basis, but truly any vector space norm can be used) are equivalent to each other: each is bounded above and below by a constant multiple of any other norm. So all these norms provide V with the same topology. It's just the product topology on a direct sum of copies of K_0 (as many as the dimension), i.e. the topology of componentwise convergence if you pick a basis and look at the norm attached to that choice of basis.

Since the abs. value on K_0 is complete, it has a unique extension to an abs. value on any finite extension field K. That extended absolute value on K is a special type of K_0-vector space norm (namely it is multiplicative, going beyond the strict conditions of a vector space norm), but it's a vector space norm all the same, so the topology it puts on K is just coefficientwise convergence in a basis. Thus visibly it is complete. This all has nothing to do with K containing a completion of Q_p-unram.

The residue field of a finite extension of a complete valued field is a finite extension of the residue field of the initial field. So if the bottom residue field is F_p-bar, so alg. closed, then certainly any finite extension of your field will still have residue field F_p-bar. It's similar to why residue fields at points on Riemann surfaces are all C, since C is alg. closed.

To address the second part of question 1, completing and taking alg. closure doesn't produce a complete field in general. Try passing from Q to Q_p to its alg. closure.

2) Since the Witt vectors W = W(F_p-bar) is/are a complete DVR with alg. closed residue field, the finite extensions of its fraction field involve no unram. extensions (the res. field can't grow anymore). All finite extensions are totally ramified, so generated by the root of an Eisenstein polynomial with coeff. in W. By Krasner's lemma, you can get the same extension by adjoining a root of a poly. with coeff. close to those of your original polynomial. Therefore by choosing nearby coeff. in W(F_q) for large q, you get an Eisenstein poly. with coeff. in Q_p-unram, and adding a root of that to Q_p-unram gives you a finite extension of Q_-unram which is dense in your finite extension of K_0.

pr.probability - Probabilistic Proofs of Analytic Facts

Here's something that's pretty neat: find a measurable subset $A$ of $[0,1]$ such that for any subinterval $I$ of $[0,1]$, the Lebesgue measure $mu(Acap I)$ has $0 < mu(Acap I) < mu(I)$. There's an explicit construction of such a set in Rudin, who describes such sets as "well-distributed". Balint Virag (and maybe others) found a very slick probabilistic construction.

Let $X_1, X_2, ldots$ be i.i.d. coin flips, i.e. $X_1$ is $1$ with probability $1/2$ and $-1$ with probability $1/2$. Consider the (random) series

$$S:=sum_{n=1}^infty X_n/n.,,,$$

By the Kolmogorov three-series theorem, it converges almost surely. However, it's a simple exercise to see that for any $a$, the event ${S > a}$ has non-trivial measure: for $a>0$, there's a positive chance of the first $e^a$ terms of the series being positive, so the $e^a$-th partial sum is positive, and the tail is independent and positive or negative with equal probability, due to symmetry. For $aleq 0$, it's trivial, again because of symmetry.

A common way of realizing i.i.d. coin flips on the unit interval is as Rademacher functions: for $xin[0,1]$, let ${b_n}$ be its binary expansion, and $X_n(x) = (-1)^{b_n}$. Realized this way, the random sum $S$ becomes an almost everywhere finite measurable function from $[0,1]$ to $mathbb R$. It only takes a bit more work to see that the set ${S>a}$ is exactly a well-distributed set.

Alex Bloemendal has written this up in a short note, but I'm not sure if he's published it anywhere.

nt.number theory - Similarly Ordered

A theorem of Erdos states:

"There exists an absolute constant c such that, if n>ck, and if a₁/b₁, a₂/b₂, ... are the Farey fractions of order n, then a_x/b_x and a_x+k/b_x+k are similarly ordered."

Can someone provide a definition of "similarly ordered" as used here?

Thanks for any insight.

Cheers, Scott

@ARTICLE{Erdos:1943,
author={Erd{"o}s, Paul},
title={A note on {F}arey series},
journal={Quart. J. Math., Oxford Ser.},
fjournal={The Quarterly Journal of Mathematics. Oxford. Second Series},
volume={14},
year={1943},
pages={82--85},
issn={0033-5606},
mrclass={40.0X},
mrnumber={MR0009999 (5,236b)},
mrreviewer={G. Szeg{"o}}

entomology - What are the white dots on the tree in this photo?

I'm not 100% sure, but I think they look like scale insects (Coccoidea). In particular it looks at bit like a hermaphrodite cottony cushion scale insect (Icerya purchasi)...

The white fluffy thing underneath the insects is the ootheca (egg case). The mature insect migrates to the main trunk of its host tree and attaches to the bark. It then secretes the cottony sack between itself and the bark. Hundreds of eggs are laid into the case, which eventually hatches and the nymphs disperse to the tips of branches where they feed on leaves. As they get older they gradually move first to twigs, then to branches and finally back to the main trunk to breed again.

set theory - Montague's Reflection Principle and Compactness Theorem

As Sridhar already explained, Lévy–Montague Reflection is a theorem scheme and not a single theorem which resolves the apparent contradiction, but here are a few additional cool facts.

First, note that ZFC is not finitely axiomatizable (otherwise we would indeed have a contradiction) but there is a recursive listing of the axioms of ZFC. Let's fix such a listing $phi_0$,$phi_1$,$phi_2$,... If $M$ is a model of ZFC, then either $M$ is an $omega$-model (i.e. the finite ordinals of $M$ are truly finite) or it is not (i.e. $M$ has some nonstandard finite ordinals). Let's see what happens in each case.

Suppose first that $M$ is an $omega$-model. The recursive listing $phi_0$,$phi_1$,$phi_2$,... exists in $M$ and, by Lévy–Montague, people living in $M$ believe that ${phi_0,ldots,phi_n}$ has a model for each $n < omega$. Since people living in $M$ also believe in the Compactness Theorem, they also believe that there is a model of ZFC. This is surprising, but note that the hypothesis that $M$ is an $omega$-model is essential since without it we there is no reason for $M$'s notion of finite to agree with ours. This is where your initial reasoning strayed, you naturally assumed that every model of ZFC was an $omega$-model.

Suppose now that $M$ is not an $omega$-model. The recursive listing $phi_0$,$phi_1$,$phi_2$,... makes sense in $M$, but since $M$ has nonstandard finite ordinals this listing continues beyond the true $omega$ and people who live in $M$ believe that these nonstandard $phi_N$'s are real axioms of ZFC! By Lévy–Montague, $M$ believes that ${phi_0,ldots,phi_n}$ has a model for every standard $n$, but since Lévy–Montague Reflection doesn't say anything about nonstandard axioms, there may be some nonstandard finite ordinal $N$ in $M$ such that people living in $M$ do not believe that the nonstandard finite set ${phi_0,ldots,phi_N}$ has a model.

Now here is a funny thing that was pointed out by Joel David Hamkins in answer to another question. Suppose $M$ is a model of ZFC + ¬Con(ZFC). Since people in $M$ believe that their finite ordinals are wellordered, there must be a first finite ordinal $N$ in $M$ such that ${phi_0,ldots,phi_N}$ has no model in $M$. This $N$ must be nonstandard finite ordinal, and so must its predecessor $N-1$. By minimality of $N$, people in $M$ believe that ${phi_0,ldots,phi_{N-1}}$ does have a model. Let $M'$ be such a model. Note that $M' models phi_n$ for every standard axiom $phi_n$ since $n < N-1$. Therefore, although people living in $M$ certainly don't believe it, this $M'$ is in fact a model of ZFC!!!

Thus, Lévy–Montague Reflection does imply that every model of ZFC contains another model of ZFC, but the models are not necessarily aware of that fact...

polymath5 - Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1

As a matter of fact, it isn't hard to construct a multiplicative sequence $a_n$ such that $f(z)$ is an entire function without zeroes. Unfortunately, it is completely useless for the questions that you brought up as "motivation".

Here is the construction.

Claim 1: Let $lambda_jin [0,1]$ ($j=0,dots,M$). Assume that $|a_j|le 1$ and $sum_ja_jlambda_j^p=0$ for all $0le ple P$. Then, if $P>2eT$, we have $left|sum_j a_je^{lambda_j z}right|le (M+1)(eT/P)^P$

Proof: Taylor decomposition and a straightforward tail estimate.

Claim 2: Let $P$ be large enough. Let $Delta>0$, $M>P^3$ and $(M+1)Delta<1$. Let $I_j$ ($0le jle M$) be $M+1$ adjacent intervals of length about $Delta$ each arranged in the increasing order such that $I_0$ contains $0=lambda_0$. Suppose that we choose one $lambda_j$ in each interval $I_j$ with $jge 1$. Then for every $|a_0|le 1$, there exist $a_jinmathbb C$ such that $|a_j|le 1$ and $sum_{jge 0} a_jlambda_j^p=0$ for $0le ple P$.

Proof: By duality, we can restate it as the claim that $sum_{jge 1}|Q(lambda_j)|ge |Q(0)|$ for every polynomial $Q$ of degree $P$. Now, let $I$ be the union of $I_j$. It is an interval of length about $MDelta$, so, by Markov's inequality, $|Q'|le CP^2(MDelta)^{-1} Ale CP^{-1}Delta^{-1}A$ where $A=max_I |Q|ge |Q(0)|$. But then on the 5 intervals $I_j$ closest to the point where the maximum is attained, we have $|Q|ge A-5Delta CP^{-1}Delta^{-1}Age A/2$. The rest is obvious.

Claim 3: Suppose that $a_0$ is fixed and $a_j$ ($jge 1$) satisfy $sum_{jge 0} a_jlambda_j^p=0$ for $0le ple P$. Then we can change $a_j$ with $jge 1$ so that all but $P+1$ of them are exactly $1$ in absolute value and the identities still hold.

Proof: As long as we have more than $P+1$ small $a_j$, we have a line of solutions of our set of $P+1$ linear equations. Moving along this line we can make one of small $a_j$ large. Repeating this as long as possible, we get the claim.

Now it is time to recall that the logarithm of the function $f(z)$ is given by
$$
L(z)=sum_{ninLambda}a_ne^{-zlog n}
$$
where $Lambda$ is the set of primes and prime powers and $a_n=m^{-1}a_p^m$ if $n=p^m$. We are free to choose $a_p$ for prime $p$ in any way we want but the rest $a_n$ will be uniquely determined then. The key point is that we have much more primes than prime powers for unit length.

So, split big positive numbers into intervals from $u$ to $e^Delta u$ where $Delta$ is a slowly decaying function of $u$ (we'll specify it later). Formally we define the sequence $u_k$ by $u_0=$something large, $u_{k+1}=e^{Delta(u_k)}u_k$ but to put all those backward apostrophes around formulae is too big headache, so I'll drop all indices. Choose also some slowly growing functions $M=M(u)$ and $P=P(u)$ to be specified later as well.

We need a few things:

1) Each interval should contain many primes. Since the classical prime number theorem has the error term $ue^{-csqrt{log u}}$, this calls for $Delta=exp{-log^{frac 13} u}$
Then we still have at least $u^{4/5}$ primes in each interval (all we need is to beat $u^{1/2}$ with some margin).

2) We should have $MDeltall 1$, $M>P^3$, and $uleft(frac{eT}Pright)^Ple (2u)^{-T-3}$ for any fixed $T>0$ and all sufficiently large $u$. This can be easily achieved by choosing $P=log^2 u$ and $M=log^6 u$.

3) At last, we'll need $M(P+sqrt u)ll u^{4/5}$, which is true for our choice.

Now it is time to run the main inductive construction. Suppose that $a_n$ are already chosen for all $n$ in the intervals up to $(u,e^{Delta}u)$ and we still have almost all primes in the intervals following the current interval free (we'll check this condition in the end). We want to assign $a_p$ for all $p$ in our interval for which the choice hasn't been made yet or was made badly. We start with looking at all $a_p$ that are not assigned yet or assigned in a lame way, i.e., less than one in absolute value. Claim 3 (actually a small modification of it) allows us to upgrade all of them but $P+1$ to good ones (having absolute value $1$) at the expense of adding an entire function that in the disk of radius $T$ is bounded by $(2u)^{T} uleft(frac{eT}Pright)^Ple u^{-3}$ to $L(z)$. Now we are left with at most $sqrt u$ powers of primes and $P+1$ lame primes to take care of. We need the prime powers participate in small sums as they are and we need the small coefficients to be complemented by something participating in small sums too. For each of them, we choose $M$ still free primes in the next $M$ intervals (one in each) and apply Claim 2 to make a (lame) assignment so that the corresponding sum is again bounded by $u^{-3}$ in the disk of radius $T$. We have at most $u$ such sums, so the total addition will be at most $u^{-2}$. This will finish the interval off. Now it remains to notice that we used only about $sqrt u+P$ free primes in each next interval and went only $M$ intervals ahead. This means that in each interval only $M(sqrt u+P)$ free primes will ever be used for compensating the previous intervals, so we'll never run out of free primes. Also, the sum of the blocks we constructed will converge to an entire function. At last, when $Re z>1$, we can change the order of summation and exponentiate finally getting the Dirichlet series representation that we need.

The end.

genomics - Percentage of genome devoted to regulating gene expression

Okay, I'll take this out of the comments and put in an answer for all of us to work on.

To directly answer your question:

"Is there an estimate for the percentage of these genes whose primary
function is related to regulation of gene expression?"

It depends on how you define "gene expression." And what cellular processes you want to include in that definition.

Larry's answer is the usual standard response, especially for people (such as me, ha) that have spent significant time studying transcription factors. About 1% of human genes have DNA binding domains and are thought to be directly involved in regulating the transcription of genes into mRNA - these are transcription factors (TFs). Closely related are cofactors, which regulate expression by binding to TFs or RNA polymerase machinery, but not directly to DNA.

Regulation of gene expression could also include modifications at the chromatin level - here you would include chromatin remodelers, histone acetylases, deacetylases, methylases and the histones themselves.

mRNA transcripts can also be regulated by miRNAs: post-transcriptional regulators that bind to complementary sequences on target mRNAs, which leads to translational repression or target degradation and gene silencing. So you would also include the proteins involved in this process, most notably the RNA-induced silencing complex, which includes Dicer.

There are also proteins involved in mRNA stabilization and turnover, which effects gene expression.

I'm not sure if anyone has added up all of the genes above to determine an overall percentage of the genome.

If you include in your definition of "gene regulation" post-transcriptional modification, folding chaperones, intra-cellular transport, extra-cellular and intra-cellular signaling, and so on - then Shigeta is right, you begin approaching 100%. In the most basic sense, life itself is gene regulation.

at.algebraic topology - Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize.

Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe periods.

Let $G$ be a complex reductive (connected) Lie group and let $T$ be a maximal torus of $G$. Set $mathfrak{g}=Lie(G)$ and $mathfrak{t}=Lie(T)$. Notice that $mathfrak{t}$ has a natural integral structure: $mathfrak{t}=mathfrak{t}(mathbf{Z})otimes_mathbf{Z}mathbf{C}$ where $mathfrak{t}(mathbf{Z})$ is formed by all $x$ such that $exp(2pi ix)$ is the unit $e$ of $G$.

All there is to know about $G$ can be extracted from the (covariant) root diagram of $G$, which is formed by $mathfrak{t}(mathbf{Z})$, the sublattice $M$ corresponding to the connected component of the center of $G$ (this is a direct summand) and the coroot system $R$ of $G$, which is included in some complementary sublattice of $mathfrak{t}(mathbf{Z})$. For example, $pi_1(G)$ is the quotient of $mathfrak{t}(mathbf{Z})$ by the subgroup spanned by $R$. See e.g. Bourbaki, Groupes et alg`ebres de Lie IX, 4.8-4.9 (Bourbaki gives a classification in terms of compact groups, but this is equivalent).

The question is how to extract information on the cohomology of $G$ (as a topological space) from the above.

For the complex cohomology there are no problems whatsoever. We only need $mathfrak{g}$: restricting the complex formed by the left invariant forms to the unit of $G$ we get the standard cochain complex of $mathfrak{g}$.

The next step would be the rational cohomology. One possible guess on how to get it would be to notice that $mathfrak{g}$ is in fact defined over $mathbf{Q}$. So one can find an algebra $mathfrak{g}(mathbf{Q})$ such that $mathfrak{g}=mathfrak{g}(mathbf{Q})otimes_mathbf{Q}mathbf{C}$. We can identify $H^{bullet}(mathfrak{g},mathbf{C})=H^{bullet}(mathfrak{g}(mathbf{Q}),mathbf{Q})otimesmathbf{C}$ and so we get two rational vector subspaces in the complex cohomology of $G$. One is the image of of $H^{bullet}(G,mathbf{Q})$ and the other is the image of $H^{bullet}(mathfrak{g}(mathbf{Q}),mathbf{Q})$ under $$H^{bullet}(mathfrak{g}(mathbf{Q}),mathbf{Q})to H^{bullet}(mathfrak{g},mathbf{C})to H^bullet(G,mathbf{C})$$ where the last arrow is the comparison isomorphism mentioned above.

1). What, if any, is the relationship between these subspaces? More precisely, apriori the second subspace denends on the choice of $mathfrak{g}(mathbf{Q})$ (I don't see why it shouldn't, but if in fact it doesn't, I'd be very interested to know) and the question is if there is a $mathfrak{g}(mathbf{Q})$ such that the relationship between the above subspaces of $H^{bullet}(G,mathbf{C})$ is easy to describe.

Notice that this is somewhat similar to what happens when we compare the cohomology of the algebraic de Rham complex with the rational cohomology. Namely, suppose we have a smooth projective or affine algebraic variety defined over $mathbf{Q}$; its algebraic de Rham cohomology (i.e. the (hyper)cohomology of the de Rham complex of sheaves) sits inside the complex cohomology, but this is not the same as the image of the topological rational cohomology. Roughly speaking, the difference between the two is measured by periods, e.g. as defined by Kontsevich and Zagier.

2). If question 1 has a reasonable answer, then what about the integral lattice in $H^{bullet}(G,mathbf{C})$? Again, a naive guess would be to take a $mathfrak{g}(mathbf{Z})$ such that $mathfrak{g}(mathbf{Z})otimesmathbf{C}=mathfrak{g}$. At present, I'm not sure whether such an integral form exist for any reductive $G$ (and I'd be very interested to know that), but in any case it exists for $SL(n,mathbf{C})$. By taking the standard complex of $mathfrak{g}(mathbf{Z})$ and extending the scalars we get a lattice in $H^{bullet}(mathfrak{g},mathbf{C})cong H^{bullet}(G,mathbf{C})$. Is there a choice of $mathfrak{g}(mathbf{Z})$ for which this lattice is related to the image of the integral cohomology in $H^bullet(G,mathbf{C})$ in some nice way?

3). If even question 2. has a reasonable answer, then what about the integral cohomology itself? Here, of course, the answer is interesting even up to isomorphism. A very naive guess would be to take an appropriate integral form $mathfrak{g}(mathbf{Z})$ as in question 2 and compute the integral cohomology of the resulting standard complex.

nt.number theory - Does the image of a p-adic Galois representation always lie in a finite extension?

I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $rho:mbox{Gal}(bar{mathbb Q}/mathbb Q)to mbox{GL}(n,V).$ In some references (eg Serre's book on $ell$-adic representations), $V$ is a vector space over a finite extension of $mathbb Q_p$. However, in more recent papers (eg Buzzard, Diamond, Jarvis) $V$ is a vector space over $bar{mathbb Q_p}$. It is easy to show that the former definition is a special case of the latter, but I suspect, and would like to prove that they are actually the same. That is, I would like to show that the image of any any continuous Galois representation over $bar{mathbb{Q}_p}$ actually lies in a finite extension of $mathbb Q_p$.

Is this the case?

I think that a proof should use the fact that $G_{mathbb Q}$ is compact and that $bar{mathbb Q}_p$ is the union of finite extensions. I have tried to mimic the proof that $bar{mathbb Q}_p$ is not complete, but have not been able to find an appropriate Cauchy sequence in an arbitrary compact subgroup of GL($n,V$).

(This is my first question, so please feel free to edit if appropriate. Thanks!)

ac.commutative algebra - Simple example of a ring which is normal but not CM

Another family of examples is given by the homogeneous coordinate rings of
irregular surfaces (ie 2-dimensional $X$ such that $H^1({mathcal O}_X) neq 0$);
these surfaces cannot be embedded in any way so that their homogeneous coordinate rings
become Cohen-Macaulay. Elliptic scrolls (such as the one in the previous answer)
and Abelian surfaces in P4, made from the sections of the Horrocks-Mumford bundle, are such examples.

The point is that sufficiently positive, complete embeddings of any smooth variety (or somewhat more generally) will have normal homogeneous coordinate rings, and they will be Cohen-Macaulay iff the intermediate cohomology of the variety vanishes. All the examples above fall into this category. It's an interesting general question to ask how positive is "sufficiently positive".

ag.algebraic geometry - Complex Projective Space as a $U(1)$ quotient

My original answer was unsalvegable so I've deleted it and am posting a new "answer". As with the first one, I don't rate this as particularly an answer but more just trying to understand what's going on.

I was initially having trouble understanding Scott's answer, but now I think I do and I think it gives the matrix representation wanted which isn't quite what David wrote.

We have $SU(n+1)$ and inside this we have $SU(n)$ and quotient out to get $S^{2n+1}$. We also have a slightly larger subgroup which is $S(U(n) times U(1))$, which contains $SU(n)$, such that the quotient is $mathbb{CP}^n$.

Now, $S(U(n) times U(1))$ is $U(n)$ via $A mapsto (det A^{-1},A)$ and the inclusion $SU(n) to S(U(n) times U(1))$ goes over to the standard inclusion. Here, $SU(n)$ is a normal subgroup and $U(n)$ is the semi-direct product of $SU(n)$ and $U(1)$ with the map $U(1) to U(n)$ given by $lambda mapsto (lambda, 1,dots,1)$ (diagonal matrix). When taken over to $S(U(n) times U(1))$ this becomes $lambda mapsto (lambda^{-1},lambda,1,dots,1)$.

So then $SU(n+1)/S(U(n) times U(1)) cong (SU(n+1)/SU(n))/U(1)$ where $U(1) to SU(n+1)$ is the map $lambda to (lambda^{-1},lambda,1,dots,1)$.

This isn't the same as David's, I know, so it may not be what you want (since that answer's been accepted). Presumably only one satisfies the condition that you want and presumably it's David's since that answer's been accepted. Still, I was confused and I think I've straightened myself out now.

Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?

This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step.

Suppose I have a huge system of linear equations, say ~10^6 equations in ~10^4 variables, and I have some external knowledge that suggests there's a small solution space, ~100 dimensional. Moreover, the equations are sparse; in fact, the way I produce the equations gives me an upper bound on the number of variables appearing in each equation, ~10. (These numbers all come form the latest instance of our problem, but we expect to want to try even bigger things later.) Finally, all the coefficients are in some number field.

Which computer algebra system should I be using to solve such a system? Everyone knows their favourite CAS, but it's often hard to get useful comparisons. One significant difficulty here is that even writing down all the equations occupies a big fraction of a typical computer's available RAM.

I'll admit that so far I've only tried Mathematica; it's great for most of our purposes, but I'm well aware of its shortcomings, hence this question. A previous slightly smaller instance of our problem was within Mathematica's range, but now I'm having trouble.

(For background, this problem is simply finding the "low weight spaces" in a graph planar algebra. See for example Emily Peter's thesis for an explanation, or our follow-up paper, with Noah Snyder and Stephen Bigelow.)

convexity - If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $Ainmathbb{R}^{n times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A x geq 0$ for all $x in mathbb{R}^n$.

Now suppose we have a convex subset $Phi$ of $mathbb{R}^n$ such that $x in Phi$ implies $x^T A x geq 0$. Is $x^T A x$ a convex function on $Phi$ (even if $A$ is not positive definite)? Of course, the answer in general is "no," but we can still ask about the most inclusive conditions under which convexity holds for a given $A$ and $Phi$. In particular I'm interested in the question:

Suppose we have a quadratic form $Q:mathbb{R}^{n times n} rightarrow mathbb{R}$. What is the weakest condition on $Q$ that guarantees it will be convex when restricted to the set of positive semidefinite matrices?

How are graph automorphisms are affected by transformations?

I also have the sense that the problem is, in general, extremely hard. Actually, I'll give an example which hopefully will convince you that there's not a lot we can say, in general. (Unless you mean "symmetric" in the technical sense, to mean "arc-transitive," or really even "vertex-transitive," in which case my example breaks down, although it may well be possible to do something similar in those cases.) For concreteness, let's think about the delta-Y transformation.

There are certainly graphs with large automorphism group where you can perform a delta-Y tranformation and get back another graph with, um, large automorphism group. For instance, take a complete graph, or, if you're squeamish about the whole "it's not simple anymore" thing, subdivide each edge of a complete graph. But we'll take a higher level of abstraction and just say that Aut(G) is the automorphism group of our graph.

So if you're doing algebraic graph theory at all, you know about the Cayley graph of a group. The salient details here are just that it's got a vertex for each element of the group, a unique automorphism for each element corresponding to right multiplication, and no other automorphisms. The standard construction of the Cayley graph is directed and has edges assigned to color classes, but there's a combinatorial trick to make a "vanilla" Cayley graph, which is to attach little gadgets to your edges to indicate color and direction. This graph is no longer vertex-transitive, which sucks, but its automorphism group is still completely described (at least if you're careful about building your gadgets...)

Now consider an (undirected uncolored) Cayley graph $Gamma$ of Aut(G). You can see that if we perform a delta-Y transformation in one of our gadgets, NONE of the original nontrivial automorphisms of $Gamma$ remain, since the colored edge the gadget's standing in for couldn't stay in the same place. So depending on how your gadget is constructed, the new graph might have trivial automorphism group.

journals - Editors in peer-review systems

This is a strictly technical question on peer-review systems currently employed in the mathematical literature, not a subjective discussion of merits/drawbacks of such systems, so I think/hope it's suitable for MO.

I have noticed that some journals (e.g. PNAS, CRAS, Nonlinearity...) always publish papers with the name of the editor who supervised the refereeing process ("Presented by X", "Recommended by X", "Communicated by X"). Most other journals, while having editors listed explicitly for each area (and hence in theory one could also know in most cases who supervised what), do not make this explicit.

I was wondering was difference it makes, as a junior author, to have an editor's name on a paper:

• is that a strong endorsement of the paper?


• a way to say that the journal is ultra-strict about the refereeing process?


• simply a full-disclosure practise of the journal?


• an incentive to publish there (when some editor is a "big name", or to make sure a specific person read your paper) ?

It's really not clear to me.

gr.group theory - Finite groups with centerless quotients

The quotients of a finite group, G/N have minimal normal subgroups K/N. These are called chief factors. Chief factors are divided into two kinds, central chief factors and eccentric chief factors. The central ones are precisely those such that K/N ≤ Z(G/N). You are therefore asking for a classification of groups all of whose chief factors are eccentric.

Another way to say this that sounds clever is that G has no chief factor of prime order.

Such a group cannot have any central "top" factors, and in particular G must be a perfect group. Perfect groups are not entirely easy to classify, but one standard view of them is to build them layer by layer, and under this view you simply never adjoin a central factor.

However, such a classification may not be very useful in concrete circumstances. The things you can adjoin are any (fixed point free, that is, noncentral) G-module repeatedly to construct the solvable radical, and any wreath product of G with a non-abelian simple group.

For every permutation representation of A5, you can take A5 wr A5 to get such examples. For any A5-module V (with no central A5-composition factors), you can take the semi-direct product of A5 with V.

There are 26 such groups (not counting G=1) of order at most 10,000; most are just the simple groups of those orders. There are also 2^4:A5, 4^2:A5, 2^3:L3(2), 2^3.L3(2), A5 x A5, 3^4:A5, 3^4.A5, 2^4:A6, 5^3:A5, 5^3.A5. The first wreath product example is A5 wr A5 of order 60^6.

ag.algebraic geometry - Smooth proper scheme over Z

Does every smooth proper morphism $X to operatorname{Spec} mathbf{Z}$ with $X$ nonempty have a section?

EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- Pete L. Clark]

Here are some special cases, according to the relative dimension $d$. If $d=0$, a positive answer follows from Minkowski's theorem that every nontrivial finite extension of $mathbf{Q}$ ramifies at at least one prime. If $d=1$, it is a consequence (via taking the Jacobian) of the theorem of Abrashkin and Fontaine that there is no nonzero abelian scheme over $mathbf{Z}$, together with (for the genus $0$ case) the fact that a quaternion algebra over $mathbf{Q}$ split at every finite place is trivial.

gt.geometric topology - Seiberg-Witten theory on 4-manifolds with boundary

Hi Fabian! Kronheimer and Mrowka's book Monopoles and three-manifolds lays out comprehensively the construction of a Seiberg-Witten TQFT, called monopole Floer homology.
It is conjectured to be isomorphic to the Heegaard Floer homology TQFT of Ozsváth-Szabó.
There are also beautiful constructions due to Froyshov and to Manolescu which do not apply in quite so much generality.

The structure of monopole Floer homology is as follows. The TQFT is a functor on the cobordism category $COB_{3+1}$ whose objects are connected, smooth, oriented 3-manifolds. In fact, the TQFT consists of a trio of functors, denoted $widehat{HM}_{bullet}$, $overline{HM_bullet}$ and $check{HM}_{bullet}$ (the last of these is such a sophisticated invariant that you have to download a special LaTeX package just to typeset it properly). These are $mathbb{Z}[U]$-modules; there's a story about gradings that's too long to be worth summarising here. There are natural transformations which, for any connected 3-manifold $Y$, define the maps in a long exact sequence
$$ cdotsto widehat{HM} _{bullet}(Y) to overline{HM_bullet}(Y)to check{HM}_{bullet}(Y) to widehat{HM}_{bullet}(Y) to cdots
$$
Why this structure? Well, the theory is based on the Chern-Simons-Dirac functional $CSD$ on a global Coulomb gauge slice through a space of (connection, spinor) pairs. $CSD$ is a $U(1)$-equivariant functional, and $check{HM}_{bullet}$ is, philosophically, its $U(1)$-equivariant semi-infinite Morse homology. $overline{HM}_bullet$ is the part coming from the restriction of $CSD$ to the $U(1)$-fixed-points, and $widehat{HM}_bullet$ is the equivariant homology relative to the fixed point set.

Now here's a subtlety for the TQFT enthusiasts out there to get your teeth into (axiomatize, explain...)! The invariant of a closed 4-manifold $X$ in any of the three theories is... zero. The famous SW invariant of a 4-manifold with $b_+>0$ comes about via a secondary operation, not part of the TQFT itself. Delete two balls from $X$ to get a cobordism from $S^3$ to itself. When $b_+(X)>0$, there are generically no reducible SW monopoles on this cobordism, and this implies that the TQFT-map $widehat{HM}_bullet(S^3) to widehat{HM}_bullet(S^3)$ lifts canonically to a map $widehat{HM}_bullet(S^3) to check{HM}_bullet(S^3)$; it is this lift that carries the SW invariant.

metabolism - How long does it take for E. coli to shift feedstocks?

This must depend upon the conditions in question, but I think it would not be very long.
The length of a generation for e coli can be 12 minutes or 24 hours, so that gives some idea of a typical time.

I did find this interesting case in the literature. They found that even as you subject e coli to minimal nutrients they are expressing genes that prepare them for a richer nutrient mix. The bacteria are primed for a flood of carbohydrates succinate or amino acids.

This study incubated the e coli overnight before taking their readings, so this puts an upper bound of about 12 hours. The implication is that if you are moving the bacteria to rich media it might be just a short time before they are in log phase growth.

In practice moving from LB plates to log phase growth is usually just like 4 hours. This paper interestingly shows that cold adapted E coli moving to other media react in a few hours, but can show behavioral changes days out.

I would say that 4 hours might be a typical expectation. 12 is typically used in the literature.

rt.representation theory - Permutation representation inner product

Much as I like Burnside's Lemma, induced (permutation) representations, and other parts of group theory, I can't resist pointing out an alternative argument that uses essentially no group theory but relies on the fact that expectation (of random variables) is linear. Since $chi(g)$ is the number of fixed-points of $g$, its square is the number of fixed ordered pairs $(x,y)$, where of course fixing a pair means fixing both its components. So the $langlechi,chirangle$ in the question is the average number of fixed pairs of a permutation $g$, in other words the expectation (with respect to the uniform probability measure on $S_n$) of the random variable "number of fixed pairs." That random variable is the sum, over all pairs $(x,y)$, of the indicator variable $F_{x,y}$ whose value at any permutation $g$ is 1 or 0 according to whether $g$ fixes $x$ and $y$ or not. So $langlechi,chirangle$ is the sum, over all $x,y$, of the expectations of these $F_{x,y}$, and these expectations are just the probabilities that a random permutation fixes $x$ and $y$. For each of the $n$ pairs where $x=y$, that probability is $1/n$, so all these together contribute 1 to the sum. For each of the remaining $n^2-n$ pairs, the probability is $(1/n)(1/(n-1))$ (namely, probability $1/n$ to fix $x$ and conditional probability $1/(n-1)$ to fix $y$ given that $x$ is fixed). So these pairs also contribute 1 to the sum, for a total of 2.

graph theory - Reconstruction puzzles

For your question #5, this can be generalized to: given two categories $A$ and $B$, describe the functors $Ato B$. For a closer match, you can restrict to the functors which map irreducible maps (those that cannot be written as a composition of two non-identity maps) to irreducible maps.

You example corresponds to letting $A$ be the poset you drew considered as a category, and $B$ the category of graphs and embeddings. This provides a formal statement of your example.

You can generalize a bit differently by asking: given a category $A$, describe the pairs $(B,F)$ with $B$ a category and $F:Ato B$ a functor (preserving irreducibility of maps, if you want...). It would not be without interest to know of another such functor from your poset to a category which is not graphs---as that would, I think, give a category equivalent to (Finite graphs and embeddings).

From this point of view, we get the following answer to your question #3: this is just representation theory.

It is clear that your puzzle can be solved algorithmically (in so far as infinite puzzles can be...): starting from the root, and proceeding level by level (where levels are defined by counting vertices) just try assigning graphs to vertices, and backtracking when you hit an inconsistency. Provided you know the puzzle is solvable (and in this case you do know!)

As Harrison notes, your question #4 is equivalent to Harary's Set Reconstruction Conjecture: the levels up to 4 vertices you work out by hand, and then use the conjecture to check that a node in a level is determined by those in the level right below it from which there is an arrow coming in.

ag.algebraic geometry - Interdependence between A^1 homotopy theory and algebraic cobordism

The two topics are logically, if not morally, independent of one another. $mathbb{A}^1$-homotopy encodes objects like motivic cohomology & it's relatives which are of interest regardless of the framework. There's no way for algebraic cobordism to supersede that -- algebraic cobordism is more directly comparable to Chow theory and $K^0$ than to motivic cohomology.

Conversely, algebraic cobordism provides a more geometric viewpoint on (a piece of) $MGL$ -- surely a valuable thing to have around as well as being of independent geometric interest. That said, if your interests are more motivic than geometric, you could get by without knowing the details of algebraic cobordism provided that you know all the classical statements in complex cobordism that inspired it.

Motivic vs algebraic cobordism.

The $mathbb{P}^1$-spectrum $MGL$, or "motivic cobordism," enjoys a privileged role in the world of $mathbb{A}^1$-homotopy similar to that of $MU$ in classical homotopy. There is a relationship between this "motivic cobordism" and "algebraic cobordism." The former is a bigraded theory, and Levine showed that $MGL^{2n,n}(X) = omega^n(X)$. (This bigrading issue is analogous to how Chow theory occurs as the $(2n,n)$-graded piece of motivic cohomology, and explains why you don't get a long exact sequence in algebraic cobordism, etc...)

So one can view Morel-Levine's (or Levine-Pandharipande's) algebraic cobordism as giving an axiomatic (or geometric) viewpoint on the motivic theory, like we had for $MU$. Unlike the case of complex cobordism, where one can directly compare it to $MU$-cohomology using transversality results, here the comparison is much more difficult and computational. The proof of this comparison relies on a (currently unpublished) spectral sequence due to Hopkins-Morel. It should be noted that constructing this spectral sequence is hard, and by the time you've constructed it you've had to independently check lots of things that you might've wanted to deduce from the comparison with algebraic cobordism (for instance you pretty much end up computing $MGL^{2*,*}(Spec k)$, you can see the comparison of cobordism to Chow theory, etc.).

Degree formula (or, application to B-K)

The reference to open problems likely refers to the use of cobordism and Rost's degree formula in the final steps of proving Bloch-Kato for $ell neq 2$. Cobordism is a tool in the proof, but introducing algebraic cobordism is not strictly necessary. (One can get by with explicit computations with the characteristic numbers of interest. It'd certainly be fair to like the Levine-Morel proof of the degree formula, though.)

The Bloch-Kato conjecture is concerned with the "Galois symbol" map
$$ K^n_M(k)/ell = H^n(k, mathbb{Z}/ell(n)) to H_{et}^n(k, mathbb{Z}/ell(n)) = H_{et}^n(k, mu_ell^{otimes n}) $$
No cobordism in sight yet. Suslin-Merkurjev's proof for $n=2$ and Voevodsky's proof for $ell=2$ made use of "splitting varieties" that one could write down pretty much explicitly and then proceed to study: Brauer-Severi varieties and Pfister quadrics, respectively. This doesn't seem to work for the general case, and instead one writes down a minimalist wishlist for splitting varieties and then has to show that they exist --- it is in this step where cobordism (or really, characteristic numbers) play a role.

A "splitting variety" for a non-zero symbol $0 neq u = u_1 otimes cdots otimes u_n in K^n_M(k)/ell$ should be a smooth variety $X/k$ such that $u$ pulls back to zero in $H^n(k(X), mathbb{Z}/ell(n))$ ("$X$ splits $u$"), with $dim X = ell^{n-1}-1$, and some more technical conditions, including a partial "universality" for this property: $X'$ splits $u$ iff there is a rational map from (a degree prime-to-$ell$ cover of) $X'$ to $X$; it follows that $X$ must have no degree prime-to-$ell$ zero-cycles (or else $Spec k$ splits $u$, i.e., $u = 0$).

Cobordism (of whatever flavor you like: complex cobordism suffices) enters when relating this to characteristic numbers: namely, to the property of being a $v_n$-variety (=representing a $v_n$ class in complex or algebraic cobordism, up to decomposables). Here, one needs something like "Rost's degree formula", which implies for instance that the property of being a $v_n$-variety with no prime-to-$ell$ zero-cycles is invariant under prime-to-$ell$ degree covers.

What is the tRNA gene copy variation between different yeast strains?

I'd take that computational angle and run the sequence data through tRNAscan-SE (Lowe & Eddy, Nucl Acids Res 25: 955-964). Ideally, you'd install this locally. This tool is what the UCSC folks use and it has been the best known, most widely used tRNA predictor for years. It's what we all used on Arabidopsis thaliana genome annotation back in the late 1990's.

There is also a genomic tRNA database that may have many of the predictions you seek, at least for some of your species/strains.

analytic number theory - complete estimates of the error for a well-known asymptotic expression of partition p(n,m)

I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved here that
$$p(n,m)sim frac{n^{m-1}}{m!(m-1)!}$$ holds for $m=o(n^{1/3})$. In generality for any finite set $A$, with $|A|=m$ and $p(n,A)$ denoting the number of partitions of $n$ with parts from $A$, one has
$$p(n,A)=frac{1}{prod_{ain A}a}frac{n^{m-1}}{(m-1)!}+O(n^{m-2}).$$

Such estimations can be deduced from the generating function of $p$ by using methods that are described in many books, for example "Analytic Combinatorics" by Flajolet and Sedgewick.

intuition - Etale cohomology and l-adic Tate modules

$newcommand{bb}{mathbb}DeclareMathOperator{gal}{Gal}$
Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off hand remarks and reading encyclopedia type articles. So I'm looking for an answer that will have some meaning to an etale cohomology naif. I welcome corrections to any evident misconceptions below.

Let $E/bb Q$ be an elliptic curve the rational numbers $bb Q$: then to $E/bb Q$, for each prime $ell$, we can associate a representation $gal(bar{bb Q}/bb Q) to GL(2n, bb Z_ell)$ coming from the $ell$-adic Tate module $T_ell(E/bb Q)$ of $E/bb Q$ (that is, the inverse limit of the system of $ell^k$ torsion points on $E$ as $kto infty$). People say that the etale cohomology group $H^1(E/bb Q, bb Z_ell)$ is dual to $T_ell(E/bb Q)$ (presumably as a $bb Z_ell$ module) and the action of $gal(bar{bb Q}/bb Q)$ on $H^1(E/bb Q, bb Z_ell)$ is is the same as the action induced by the action of $gal(bar{bb Q}/bb Q)$ induced on $T_ell(E/bb Q)$.

Concerning this coincidence, I could imagine two possible situations:

(a) When one takes the definition of etale cohomology and carefully unpackages it, one sees that the coincidence described is tautological, present by definition.

(b) The definition of etale cohomology (in the case of an elliptic curve variety) and the action of $gal(bar{bb Q}/bb Q)$ that it carries is conceptually different from that of the dual of the $ell$-adic Tate module and the action of $gal(bar{bb Q}/bb Q)$ that it carries. The coincidence is a theorem of some substance.

Is the situation closer to (a) or to (b)?

Aside from the action $gal(bar{bb Q}/bb Q)$ on $T_ell(E/bb Q)$, are there other instances where one has a similarly "concrete" description of representation of etale cohomology groups of varieties over number fields and the actions of the absolute Galois group on them?

Though I haven't seen this stated explicitly, I imagine that one has the analogy [$gal(bar{bb Q}/bb Q)$ acts on $T_ell(E/bb Q)$: $gal(bar{bb Q}/bb Q)$ acts on $H^1(E/bb Q; bb Z_ell)$]::[$gal(bar{bb Q}/bb Q)$ acts on $T_ell(A/K)$: $gal(bar{bb Q}/bb Q)$ acts on $H^1(A/K; bb Z_ell)$] where $A$ is an abelian variety of dimension $n$ and $K$ is a number field: in asking the last question I am looking for something more substantively different and/or more general than this.

I've also inferred that if one has a projective curve $C/bb Q$, then $H^1(C/bb Q; bb Z_ell)$ is the same as $H^1(J/bb Q; bb Z_ell)$ where $J/bb Q$ is the Jacobian variety of $C$ and which, by my above inference I assume to be dual to $T_ell(J/bb Q)$, with the Galois actions passing through functorially. If this is the case, I'm looking for something more general or substantially different from this as well.

The underlying question that I have is: where (in concrete terms, not using a reference to etale cohomology as a black box) do Galois representations come from aside from torsion points on abelian varieties?

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[Edit (12/09/12): A sharper, closely related question is as follows. Let $V/bb Q$ be a (smooth) projective algebraic variety defined over $bb Q$, and though it may not be necessary let's take $V/bb Q$ to have good reduction at $p = 5$. Then $V/bb Q$ is supposed to have an attached 5-adic Galois representation to it (via etale cohomology) and therefore has an attached (mod 5) Galois representation. If $V$ is an elliptic curve, this Galois representation has a number field $K/bb Q$ attached to it given by adjoining to $bb Q$ the coordinates of the 5-torsion points of $V$ under the group law, and one can in fact write down a polynomial over $bb Q$ with splitting field $K$. The field $K/bb Q$ is Galois and the representation $gal(bar{bb Q}/bb Q)to GL(2, bb F_5)$ comes from a representation $gal(K/bb Q) to GL(2, bb F_5)$. (I'm aware of the possibility that knowing $K$ does not suffice to recover the representation.)

Now, remove the restriction that $V/bb Q$ is an elliptic curve, so that $V/bb Q$ is again an arbitrary smooth projective algebraic variety defined over $bb Q$. Does the (mod 5) Galois representation attached to $V/bb Q$ have an associated number field $K/bb Q$ analogous to the (mod 5) Galois representation attached to an elliptic curve does? If so, where does this number field come from? If $V/bb Q$ is specified by explicit polynomial equations is it possible to write down a polynomial with splitting field $K/bb Q$ explicitly? If so, is a detailed computation of this type worked out anywhere?

I'm posting a bounty for a good answer to the questions succeeding the "Edit" heading.

ag.algebraic geometry - The Jouanolou trick

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with fibers being affine spaces. This was used e.g. by D. Arapura to (re)prove that the Leray spectral sequence of any morphism of quasi-projective varieties is equipped from the second term on with a natural mixed Hodge structure.

Here is a proof when $X$ is $mathbf{P}^n$ over a field $k$: take $Y$ to be the affine variety formed by all $n+1 times n+1$ matrices which are idempotent and have rank 1. This is indeed affine since it is given by the equations $A^2=A$, the characteristic polynomial of $A$ is $x^n(x-1)$. Moreover, $Y$ is mapped to $mathbf{P}^n(k)$ by taking a matrix to its image. The preimage of a point of $mathbf{P}^n(k)$ is "the set of all hyperplanes not containing a given line", which is isomorphic to an affine space.

The general (quasi-projective) case follows easily from the above. However, it is not clear how to generalize Jouanolou's trick for arbitrary varieties. Nor is it clear (to me) that this is impossible.

1. Is there an analogue of the Jouanolou lemma for arbitrary (not necessarily quasi-projective) varieties (i.e. reduced separated schemes of finite type over say an algebraically closed field)?




2. (weaker version of 1 over complex numbers) Is there, given a complex algebraic variety $X$, an affine variety $Y$ that maps surjectively to $X$ and such that all fibers are contractible in the complex topology? A negative answer would be especially interesting.




3. (the following question is a bit vague, but if it has a reasonable answer, then it would probably imply a positive answer to 2.) Is there a quasi-projective analog of the topological join of two projective spaces? I.e., if $P_1$ and $P_2$ are two complex projective spaces, is there a quasi-projective variety $X$ which "contains the disjoint union of $P_1$ and $P_2$ and is formed by all affine lines joining a point in $P_1$ with a point in $P_2$"?

Edit 1: in 1. and 2. the varieties are required to be connected (meaning that the set of closed points is connected in the Zariski topology; in 2 one could use the complex topology instead).

Edit 2: as Vanya Cheltsov explained to me, the answer to question 3 is most likely no.

discrete geometry - Combinatorial distance ≡ Euclidean distance

I wonder if the class of polytopes I am going to define might have property X:

Consider the regular n-simplex $Delta^n$.

Let $F_k^n$ be the set of k-dimensional faces of $Delta^n$:

• $F_0^n$ = the set of vertices


• $F_1^n$ = the set of edges


• ...


• $F_n^n$ = $Delta^n$

Let $P_k^n$ be the polytope the vertices of which are the centers of the elements of $F_k^n$.

$P_k^n$ represents in a natural way the subsets of [n+1]={0,1,..,n} with exactly k+1 elements.

$P_1^3$ (= $P_2^3$) is the octahedron.

$P_1^4$ (= $P_3^4$) is the rectified 4-simplex (with the triangular prism for vertex figure).

Claim:
$P_1^n$ (= $P_{n-1}^n$) is the rectified n-simplex.

Claim:
For any vertex v of the regular hypercube $C^n$ the vertices with combinatorial distance k to v are the vertices of $P_k^n$.

Conjecture:
*For all n, k, the polytope $P_k^n$ has property X.*

Question: Is there a standard name for the polytopes $P_k^n$?

Question: Can anyone canonically name some other $P_k^n$ for 1 < k < n-1 (like "rectified n-simplex" for k=1)?

Question: Does a proof of the above conjecture seem to be (i) feasible, (ii) trivial, or - if (i) but not (ii) - does anyone (iii) could sketch a proof?

rt.representation theory - Non-smooth algebra with smooth representation variety

A not necessarily commutative algebra A (over C, say) is called formally smooth (or quasi-free) if, given any map $f:A to B/I$, where $I subset B$ is a nilpotent ideal, there is a lifting $F:A to B$ that commutes with the projection. (The reason for the terminology is that if we restrict to the category of finitely generated commutative algebras, this condition is equivalent to Spec(A) being smooth. For more info see the paper "algebra extensions and nonsingularity" by Cuntz and Quillen, 1995.) It isn't hard to see that if A is formally smooth, then the representation varieties $Rep_mathbb{C}(A,V)$ are smooth (V is finite dimensional). Does anyone know of an example of an algebra that is not formally smooth, but whose representation varieties are smooth?

One almost-answer is the Weyl algebra $A = mathbb{C}langle x,yrangle/(xy - yx = 1)$. This isn't formally smooth, but its representation varieties are all empty. (To see this, take the trace of $xy - yx = 1$ to get $0 = n$.) This doesn't seem like it should count as answer, does anyone know a better one?

notation - What does ! above = mean

Can someone please explain what the symbol $stackrel{!}{=}$, consisting of an exclamation mark (!) above an equals sign (=) means? Below is the example I'm trying to decipher:

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The normalization factor is chosen such that in average, Dynamic Θ Time passes as fast as physical time. In practice it is determiend by the condition that the interval in Dynamic Θ Time corresponding to a 4-year reference period [$T_0$, $T_1$] should be of exactly the same length:

$T_1 - T_0 stackrel{!}{=} int_{T_0}^{T_1} a(t) dt$

linear algebra - A generalization of Boolean matrix multiplication for order-3 tensors

The Boolean matrix product of two 0-1 $n times n$ matrices $A$ and $B$ is the matrix $C$ defined as
$$C[i,j] = vee_{k=1}^n (A[i,k] wedge B[k,j]).$$ If $A = B$ and the matrix is an adjacency matrix of a graph $G$, then $C$ determines for all pairs of vertices in $G$ whether or not there is a path of length two from one vertex to the other. This operation can be used to detect if a graph has a 3-clique: we check for an $(i,j)$ such that $C[i,j] wedge A[i,j] = 1$.

I am interested in a natural generalization of this to 4-cliques in 3-uniform hypergraphs. Let $i,j,k$ be vertices of such a hypergraph $H$. We say that a triangle on $i,j,k$ in $H$ is a set of three edges $e_1,e_2,e_3$ from $H$ of the form $e_1$ = {$ell, i,j$}, $e_2$ = {$ell, j,k$}, $e_3$ = {$ell, k,i$} for some vertex $ell$.

Let $T(i,j,k) = 1$ iff there is a triangle on $i,j,k$ in $H$. We can detect if there is a 4-clique in $H$ by checking for $(i,j,k)$ such that $T(i,j,k) wedge A(i,j,k) = 1$, where $A$ is the "adjacency tensor" of $H$.

More formally, we can define $$T(i,j,k) = vee_{ell=1}^n (A(i,j,ell)wedge A(j,ell,k) wedge A(ell,k,i)).$$

Is there a name for this type of operation in the literature? I would expect the following to have a name: Given three order-3 tensors $A,B,C$ of dimensions $n times n times n$, define the order-3 tensor $$T(i,j,k) = sum_{ell = 1}^n A(i,j,ell) cdot B(j,ell,k) cdot C(ell,k,i).$$ However I can't seem to find anything related. (Pardon me for the lack of subscripts, but I find the above notation easier to read.)

gt.geometric topology - Hurwitz Encoding

What it means for a covering of a sphere to be branched: Let $f:X to Y$ be a map of Riemann surfaces. We are particularly interested in the case that $Y$ is $mathbb{CP}^1$; in this case, $Y$ has the topology of a sphere. At most points $y$ in $Y$, there will be a neighborhood $V$ of $y$ so that $f^{-1}(V)$ is just a union of $n$ disjoint copies of $U$, each mapping isomorphically to $V$. These are the points where there is no branching.

At a few points of $y$, something different will happen. Let $x$ be a preimage of $y$, let $V$ be a small neighborhood of $y$ and let $U$ be the connected component of $f^{-1}(V)$ containing $x$. At these points, the map $f$ looks like $t mapsto t^e$, as a map from the unit disc in $mathbb{C}$ to itself. This is called branching.

The generic situation is that, at finitely many points of $Y$, one of the preimages is branched with $e=2$ and the other $n-2$ preimages are unbranched.

Algebraically, if $t$ is a local coordinate on $X$, then we have branching where $partial f/partial t$ vanishes. Even if we can't explicitly write $f$ as a function of $t$, if we can find a polynomial relation $P(f,t)=0$, then we have $(partial P/partial f)(partial f/partial t) = partial P/partial t$, so branching will occur when $partial P/partial t=0$.

The relation between branched covers and the symmetric group: Let $f: X to mathbb{CP}^1$ be a branched cover. Let $R subset mathbb{CP}^1$ be the points over which branching occurs. Then $f^{-1}(mathbb{CP}^1 setminus R) to mathbb{CP}^1 setminus R$ is a cover, in the sense of algebraic topology.

As you probably know, connected covers of a space $U$ are classified by subgroups of $pi_1(U)$. In particular, degree $n$ covers are classified by index $n$ subgroups. I like to recast this and say that degree $n$ connected covers of $U$ are classified by transitive actions of $pi_1(U)$ on an $n$-element set. This has the advantage that, more generally, we can say that degree $n$ covers of $U$ are classified by actions of $pi_1(U)$ on an $n$-element set.

Now, in this case, $pi_1(mathbb{CP}^1 setminus R)$ is isomorphic to the group generated by $R$, modulo the relation $prod{r in R} [r]=1$. You should be warned that this isomorphism depends on some choices. First of all, we need to choose a base point $y$ in $mathbb{CP}^1 setminus R$! Even once we've done that, we need to choose loops based at $y$, circling each of the elements of $R$, and disjoint from each other away from $y$. These will then be the classes $[r]$. The order that the product above is taken is related to the circular order at which these loops come in to $y$.

So, to covers of $mathbb{CP}^1 setminus R$ correspond to maps from this group to $S_n$. To give such a map, we choose an element $b(r)$ of $S_n$ for each $r$ in $R$; these must obey $prod_{r in R} b(r)=1$. (I am being very sloppy about when two such maps give isomorphic covers, and, indeed, what it means to say two covers are isomorphic.)

The relation between the geometry of the cover, and the map $b(r)$ is the following: If the permutation $b(r)$ has cycles of lengths $e_1$, $e_2$, ..., $e_k$, then $f^{-1}(r)$ contains $k$ points, which are branched with degrees $e_1$, $e_2$, ..., $e_k$.

Useful facts to know: The cover is connected if and only if the action of the $b(r)$ on $[n]$ is transitive.

The genus of $X$ is given by, the Riemmann-Hurwitz formula:
$$2g-2 = -2n+sum_{r in R} (n-#mbox{cycles of $r$}).$$

It is very difficult to obtain an explicit equation for $X$ from the data of $R$ and $b:R to S_n$.

gel electrophoresis - How do Proteins migrate in MES vs. MOPS

Well, to rationalize everyone's comments, I think @leonardo is right.

This is a denaturing SDA PAGE gel. The migration of the SDS Micelles which are negatively charged, depends upon the shielding of the solution around it. The difference in mobility is because the SDS micelles will experience a slightly different field at pH ~6.2 (MES) vs 7.2 (MOPS).

The thought that these have the same charge would be right at exactly the pH corresponding to the pKa. For these two ions the proportions will not be the same when running 0.8 pH units from their respective pKas. the ion concentration goes as +/- log ([BH]/[B-]) and the charge environment of the buffer should not be the same. I think that the higher pH for MOPS running will tend to create more negative charge in the solution from MOPS- but also OH- in solution, slowing the mobility of the micelles and favoring the resolution of the larger proteins with MOPS.

This is all given even if the buffering ion concentration is the same.

I'm sure the gory details are buried in the musty tomes of some physical biochemistry journals deep in the library, but this is how it sounds to me.