co.combinatorics - Convert a confusion matrix to a distance/covariance matrix

I'm not sure how the specifics of your confusion matrix can help, but as far as I know, there is no general way of mapping dissimilarity functions (or matrices) to metrics (which is, probably, a bit more general than what you're asking). There are, however, empirically quite useful ways of doing so.

For example, you may wish to retain the dissimilarity ordering, so that objects/points ordered by distance from a reference point will retain their ordering under the new dissimilarity/distance. This is possible.

In your weighted directed graphs question, you've had the symmetry question answered (e.g., take the minimum of both directions; sum or average will also work). Non-negativity can easily be fixed by shifting all distances by the same constant. Positive-definiteness (i.e., $d(x,y)=0$ iff $x=y$), can be fixed (if it matters) as long as there is a minimum distance between non-identical objects. (For a finite matrix, you should be able to fix this anyway, by shifting by a positive $varepsilon$ and setting the diagonal to zero, or the like.)

The main challenge is enforcing triangularity, and this can be done by composing your function (or, in your case, matrix) with a strictly increasing, concave function $f$ for which $f(0)=0$.

It should be obvious that such a function will not change the dissimilarity ordering (given that your measure of dissimilarity is already non-negative). What it will do, however, is magnify the smaller dissimilarities more than the larger ones, moving the measure in the direction of triangularity. It's only a matter of finding a function that is "concave enough."

An example of such a function would be $f(x) = x^frac{1}{1+w}$. Now it's just a matter of choosing a large enough $w$, and you can find that by bisection, for example. (For more details on this approach, see the paper on the subject by Tomáš Skopal.)

As I said, this doesn't really address the specific properties of your matrix, but deals with the general problem. Maybe there are better solutions in your case; I don't know.

By the way, a few years ago, I had a student working on the problem of making substitution weight matrices for string edit distance metric—also quite similar to what you're asking. He explored several algorithms, and his Master's thesis ("Making substitution matrices metric") is is available online.

career - Switching Research Fields

I think our questioner is aware of the difficulties of switching fields and if not he or she soon will be, so let me be naive and try to be constructive.

For Quantum Computation, Isaac Chuang and Michael Nielsen's "Quantum Computation and Quantum Information" has become a standard introduction to the subject, suitable for a graduate student in either mathematics, physics or computer science.

Since I have no idea what background you have in PDEs (you could be a specialist in D-modules for all I know and find these suggestions childish), here are some texts I've become acquainted with:

-V.I. Arnold's "Lectures on Partial Differential Equations" gives a beautifully geometric and intuitive understanding of PDEs, introducing and weaving together contact and symplectic geometry. The table of contents looks quite basic, but it contains the depth you should expect from Arnold.

-Lawrence C. Evans "Partial Differential Equations" is nice and contains the basic notions from Functional Analysis, Sobolev Spaces, Weak Theory and Regularity Theory. It does a good job of being self-contained and trying to give physical interpretations of various PDEs.

-Gilbarg and Trudinger have the classic "Elliptic PDEs of Second Order", which is dense, but a classic nonetheless.

As a mathematician you don't need to learn how physicists think in the next year. Physicists have different ways of looking at problems and are constrained to their own paradigms just as mathematicians are. It is often quicker to pick up advanced physics if you know advanced mathematics, with many excellent bridge texts by world-class mathematicians. Examples that come to mind are Bott's "Morse Theory Indomitable" which includes an exposition of some of Witten's ideas for a mathematician. Atiyah's "Geometry and Physics of Knots" is also an excellent example of this. Feynman's Lectures are great, but won't advance you to research. It's more like a Caltech undergraduate degree bound in 3 volumes.

Finally, as a note of inspiration, I have heard of at least two new faculty who self-studied PDEs in their post-doctoral years. One was supplanting a thesis in deformation theory and integrable systems, the other in knot theory and Floer homology. It is definitely a hard path to follow, but is sometimes necessary for growth. Also, bear in mind that Ed Witten was a history major as an undergrad, dropped out of economics grad school before applying for Princeton applied math and then switching to physics. Raoul Bott switched from electrical engineering to mathematics after his PhD (a much harder path, one might argue). Finally, my personal hero, Douglas Hofstadter, after quitting his Berkeley math PhD and finishing a 7+ year physics PhD in Oregon, then lived at home for a few years re-tooling himself as an AI researcher. Now he has a Pulitzer and tenure at a university -- not too shabby.

Good luck!

at.algebraic topology - Free action of SL_2(F_p) on a sphere

Apparently, a linear free action exists only for $p=5$ (if $pge 5$), see paper by C. Thomas "Almost linear actions by $SL_2(p)$ on $S^{2n-1}$". There is a weaker notion of an "almost linear" action, and it seems that constructing such actions is a fairly complicated business, using state-of-the-art differential geometry and topology; see arXiv:math/9911250. It seems that simple explicit actions for higher $p$ are not expected (also, of course, the sphere must be odd dimensional, since an orinentation preserving self-map of an even dimensional sphere has a fixed point by the Lefschetz theorem).

radio astronomy - Has SETI data been used for astrophysics? How, or why not?

Well, that might depend on exactly what definitions you use. SETI itself is a broad program. In this answer I'll focus solely on what I consider the most iconic part of modern SETI, which is SETI@home.

We have the following quote from SETI@home's webpage (emphasis their own):

SETI (Search for Extraterrestrial Intelligence) is a scientific area whose goal is to detect intelligent life outside Earth. One approach, known as radio SETI, uses radio telescopes to listen for narrow-bandwidth radio signals from space. Such signals are not known to occur naturally, so a detection would provide evidence of extraterrestrial technology.

SETI@home's specific purpose deals with the radio SETI data. So they are implicitly asserting that the data they collect is expected to be useless outside of their namesake: they're specifically looking for things that are not known to occur naturally.

That being said, it turns out that the SETI@home data collection is almost never active (see here on their site). It is done passively, as a piggyback collection when the Arecibo radio telescope is being used for other science projects. So if your question is "is any data useful outside of SETI actually collected while SETI(@home) is gathering data?", then the answer is "yes".

Now, we can also note that one of the original goals of the SETI@home project was to prove the viability and practicality of distributed computing for scientific projects. This we can say was a resounding success, as it spawned BOINC, which serves as a distributed computing platform for a wide range of non-SETI science projects: Folding@home and Rosetta@home deal with protein folding, Einstein@home is for gravity wave detection, the now-defunct Mesenne@home was for finding Mersenne primes, and many others (my copy of BOINC specifically states that there are over 30 currently active).

So if you are willing to consider the advances they spurred in distributed computing in scientific projects, and how this has impacted scientific projects inside astrophysics (or outside), then the answer is again "yes".

---

EDIT: Einstein@Home recently made the following announcement about a gamma ray pulsar that was detected via the program; it includes links to a published article on the detection, including an arxiv version link if you find the official publication is behind a paywall. So that's a concrete, published, and current discovery in astrophysics. If you're willing to give SETI some of the credit for having spawned the @Home style of distributed computing projects in science, as suggested above, then there you go.

ct.category theory - nontrivial isomorphisms of categories

This is really a comment, not an answer. But since it is a not-so-short comment to many
answers together, it had to become an answer.

It has been observed that (first-order) definitional equivalences give categorical
isomorphisms, at least for categories of first order structures with isomorphisms as
their only morphisms. In my opinion the fact that two equivalent definitions of a
mathematical structure give the same isomorphisms but possibly different morphisms
(which maps between [complete] lattices should one consider: isotone? meet-semilattice
morphisms? join-semilattice morphisms? lattice morphisms? [complete join, or meet, or both,
morphisms?]) is a big virtue: it means that two definitions give really different points
of view on the same kind of structure (in a way, they formalize a kind of non-triviality
of the equivalence). This also happens for second-order structures (complete lattices,
uniform and topological spaces); the definitional equivalences are expressed in the natural
language of Bourbaki's "scale of sets" (or natural model of type theory) above the base
sets of the (multisorted) structure (detractors of Bourbaki and/or lowers of category
theory would instead speak of the topos "somewhat freely" generated by the (sorts for the)
base stets;
when the equivalence of definitions is completely constructive one can really take a free
topos, but depending of the principles of classical logic which are needed to prove the
equivalence of definitions, one considers the topos freely generated in more restricted
classes).

So in summary: syntactically defined equivalences induce isomorphisms between categories
of structures. As Hodges notes (for example in his book "model theory"), pratically
everything which in mathematics can "really" be considered a "construction" is
formalizable as a interpretation or at least a "word-construction" (and moreover it
is the syntactical form itself which shows what kind of morphisms more general than
isomorphisms are "preserved" by the construction. I understand that few lovers of
category theory would approve such a extreme syntactical view, but note that even
the "categories, allegories" book by Freyd and Scedrov insists on the
"Galois correspondence" between syntactical
and semantical aspects; I simply happen to prefer the syntactical side). From this
point of view, Hodges'remarks about (cases slightly more general than) adjunctions
among quasivarieties (and universal
Horn classes) induced by forgetful functors are related to the already given remark
about monadic adjunctions.

Besides, the book "abstract and concrete categories" by Adameck, Herrlich, Strecker
conains many examples of "concrete isomorphisms"; some of them shoulb be interesting
(and all of them, if I remember correctly, can be seen as syntactically defined as
above).

Incidentally, the three authors say that non reasonable concept of "concrete
equivalence" can be given; I disagree since cases exist where two categories can be
concretely reflected on full subcateories of objects "in normal form", and the
subcategories are concretely isomorphic [for example, take affine geometry of dimension
at least three: form affine spaces algebraically
defined by points,
group of translations, sfield of scalars one "normalizes"
to the particular case where translations
are a subgroup of the group of permutations of the points and scalars are a subring of
the ring of endomorphisms of the group of translations. For affine spaces geometrically
defined in Hilbert's Grundlagen style, the general case can be reflected onto the
"normal" case with the same set of points where lines and planes are sets of points and
incidence is the set-theoretic one]

It has already been observed that, in presence of choice, "isomorphic categories"
means "equivalent categories where corresponding isomorphism classes of objects have
the same cardinality". Freyd and Scedrov observe that, even in absence of choice,
the "correct" notion of equivalence is: to have isomorphic inflations. This means that
all usual examples of equivalence of categories induce examples of isomorphisms
(without the trick with arbitrary choices to consider skeletons, but instead using
canonical "inflations" of the isomorphism classes)

naming - How were the designations of "North" and "South" applied to the hemispheres of Mars?

North and south are defined according to the planet's rotation about its axis. A "right hand" rule is used: If you make a fist with your right hand, and orient your curled fingers in the direction the planet rotates, your thumb points north. Alternatively: Standing on the equator, facing the direction the planet rotates, North is to your left.

Axial Tilt on wikipedia has this information (and more.)

update

hey! Don't up vote this answer; It's wrong... see the comments where the OP has actually researched and discovered the real answer. (I don't want to delete this answer because we'll lose the comments too!)

ca.analysis and odes - How many ways can we characterize gamma function?

maybe I can give you some help.
Gamma function is also called the second Euler integral.

Here comes some characterizations.

a f(s)= $$t(x)=int_{0}^{+infty}{t^(s-1)}{exp(-t)}dt$$ s>0

b f(s)=$$lim n!n^s/[s(s+1)...(s+n)] $$ $$nrightarrow +infty$$

c $$B（p，q）=Gamma(p)Gamma(q)/Gamma(pq）$$ p>0 q>0

d $$Gamma(2s)=2^(2s-1）Gamma(s)Gamma(s+1/2)/sqrt(2pi) $$ s>0

e $$Gamma(s)Gamma(1-s)=pi/sin(spi)$$ 0

May it help!

elliptic curves - Alternate expresion of L-series coefficients

There are two different recursions involved here, one for the points of $E$ over ${mathbb F}_{p^n}$, and the other for the coefficients of the $L$-function.

If we write $a_p = alpha + beta,$ where $alphabeta = p$ (so $alpha$ and $beta$ are
the two roots of the char. poly. of Frobenius), then

$$1 + p^n - E({mathbb F}_{p^n}) = alpha^n + beta^n.$$

On the other hand, the Euler factor at $p$ for the $L$-function of $E$ is
$$(1 - alpha p^{-s})^{-1}(1-beta p^{-s})^{-1}$$
$$= (1 + alpha p^{-s} + alpha^2 p^{-2s} + cdots )(1 + beta p^{-s} + beta^2 p^{-2s} +
cdots )$$
$$= 1 + (alpha + beta) p^{-s} + (alpha^2 + alphabeta + beta^2) p^{-2s} +
cdots ,$$
and so we conclude that $a_{p^n}$ (the coefficient of $p^{-ns}$ in the $L$-function)
equals
$$alpha^n + alpha^{n-1} beta + cdots + alphabeta^{n-1} + beta^n.$$

These formulas are simply different, as soon as $n > 1.$ The recursion given in
the question describes the second, and not the first.

co.combinatorics - Discrete harmonic function on a planar graph

The answer is no.

I first describe the graph $G$. Let $N_i$ be a sequence of positive integers; we will choose $N_i$ later. Let $T$ be an infinite tree which has one root vertex, the root has $N_1$ children; the children of that root have $N_2$ children, those children have $N_3$ children and so forth. Let $V_0$ be the set containing the root, $V_1$ be the set of children of the root, $V_2$ the children of the elements of $V_1$, and so forth. To form our graph, take $T$ and add a sequence of cycles, one going through the vertices of $V_1$, one through $V_2$ and so forth. (In the way which is compatible with the obvious planar embedding of $T$.)

Every face of $G$ is either a triangle or a quadrilateral.

We will build a harmonic function $f$ on $G$ as follows: On the root, $f$ will be $0$. On $V_1$, we choose $f$ to be nonzero, but average to $0$. On $V_i$, for $i geq 2$, we compute $f$ inductively by the condition that, for every $u in V_{i-1}$, the function $f$ is constant on the children of $u$. Of course, we may or may not get a bounded function depending on how we choose the $N_i$. I will now show that we can choose the $N_i$ so that $f$ is bounded. Or, rather, I will claim it and leave the details as an exercise for you.

Let $a_i$ be a decreasing sequence of positive reals, approaching zero. Take $N_i = 6/(a_{i+1} - a_i)$. Exercise: If $f$ on $V_1$ is taken between $-1+a_1$ and $1-a_1$, then $f$ on $V_i$ will lie between $-1+a_i$ and $1-a_i$. In particular, $f$ will be bounded between $-1$ and $1$ everywhere.

cv.complex variables - Inversion of Laurent series

The Lagrange inversion formula is meant to give you the Taylor series expansion of $f^{-1}$ at the point $f(0)$. If $f$ has a Laurent series instead, then it means that $f(0) = infty$ and that $f$ is meromorphic. The Taylor series at $infty$ of $f^{-1}$ then doesn't particularly mean anything unless you change to a different coordinate patch on the Riemann sphere, for instance $zeta = 1/z$. So you can first switch to the function $1/f$, which has a usual Taylor series, and then use the standard Lagrange inversion formula for $(1/f)^{-1}$.

(If I have understood the question correctly. Maybe this answer is too straightforward to address the real question.)

rt.representation theory - Representations in characteristic p

Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes of elements of G. A paper I'm reading says that if the characteristic of F is p>0, then the number of F-irreps of G is the same as the number of conjugacy classes of elements whose order is not divisible by p.

If G is abelian, it seems to me that this should say that the p-sylow subgroup of G acts trivially on every characteristic p irrep. This is because I can split G into G'x P (non-p and p-sylow subgroups), and then any irrep of G' extends to one of G by letting P act trivially. Since the formula mentioned above would say that they have the same number of irreps these must be all of them.

My question is: Is this true? If not, then where is my reasoning going off track?

co.combinatorics - How to optimize student happiness in group work?

This is a generalization of the stable roommate problem (which is the same thing where $k = n/2$, ie, groups of 2). In general, there exist groups in which under any pair of groups contain members who would both like to switch teams.

From wikipedia:

For a minimal counterexample, consider 4 people A, B, C and D where all prefer each other to D, and A prefers B over C, B prefers C over A, and C prefers A over B (so each of A,B,C is the most favorite of someone). In any solution, one of A,B,C must be paired with D and the other 2 with each other, yet D's partner and the one for whom D's partner is most favorite would each prefer to be with each other.

solar system - How did scientists determine an estimate of the number of planets greater than Sedna's size to exist in the Inner Oort Cloud?

Unfortunately, the paper is not available on ArXiv (oh, what hardships we must overcome!), but I have found it here. In it, where the "900" figure is mentioned (2nd page, I believe), the authors (Trujillo and Sheppard) say that they ran simulations with the data already found and their additional findings, and found that 900$^{+800}_{-500}$ bodies larger than 1,000 km in diameter could exist. They also found that 430$^{+400}_{-240}$ relatively "bright" bodies could exist, as well. Notice that there is quite a lot of uncertainty there; they are rather cautious (and rightfully so) when announcing the results of their simulations.

Interestingly enough, they also mention a paper in that section, written partly by none other than Mike Brown (!) that also used simulation methods. The authors, Schwamb et al, concluded (in 5.4, as well as the conclusion) that 393$^{+1286}_{-264}$ and 74$^{+249}_{-47}$ objects greater than or equal to Sedna in brightness, not size, could exist (393 is for the "hot" population, while 74 is for the "cold" population). The reason Trujillo and Sheppard cited this paper is that their simulation results fit the results of the simulations by Schwamb et al., which is encouraging.

Summary: Trujillo and Sheppard arrived at 900 via simulations based on prior observations. There is a large amount of uncertainty in this number, but it does agree with prior simulations by Schwamb et al. While zibadawatimmy 's comment is perhaps the funniest I have seen, it is, fortunately, inaccurate. Those statisticians must be relieved.

I hope this helps.

ag.algebraic geometry - Have people successfully worked with the full ring of diferential operators in characteristic p?

This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here by the full ring of differential operators I mean the same thing as the ring of divided-power differenial operators, which is the terminology used in the cited question.)

My question is:

Do people have experience using the full ring of differential operators successfully in characteristic $p$ (for localization, or other purposes)?

I always found this ring somewhat unpleasant (its sections over affines are not Noetherian,
and, if I recall correctly a computation I made a long time ago, the structure sheaf
${mathcal O}_X$ is not perfect over ${mathcal D}_X$). Are there ways to get around
these technical defects? (Or am I wrong in thinking of them as technical defects, or am
I even just wrong about them full stop?)

EDIT: Let me add a little more motivation for my question, inspired in part by Hailong's answer and associated comments. A general feature of local cohomology in char. p is that you don't have the subtle theory of Bernstein polynomials that you have in char. 0. See e.g. the paper of Alvarez-Montaner, Blickle, and Lyubeznik cited by Hailong in his answer. What I don't understand is whether this means that (for example) localization with the full ring of differential ops is hopeless (because the answers would be too simple), or a wonderful prospect (because the answers would be so simple).

Companion Sun the cause of comet impacts with Earth

Ahh, I see you've met our friend, the hypothetical star known as Nemesis. I don't know which scientist(s) you're referring to, but the idea goes back quite a while. The original theory was created by Raup and Sepkoski way back in 1984. Their paper analyzed mass extinctions in the past and concluded that there was a pattern. Note, though, that the paper does not specifically say that a star orbiting the Sun is the cause; it merely favors general impacts of "extraterrestrial" origin.

However, two more teams analyzed the data and posited that there was a celestial body responsible. Whitmire and Jackson and Davis, Hut, and Muller (Regrettably, you must pay to see the papers, which I have not done) suggested that the Sun had a companion in the far reaches of the solar system, and it was perturbing comets and other bodies, which came towards Earth.

Honestly, I (and quite a few others) don't think there's much truth to the idea. Why? Because we haven't found any evidence of its existence. As Wikipedia explains,

Searches for Nemesis in the infrared are important because cooler stars shine in infrared light. The University of California's Leuschner Observatory failed to discover Nemesis by 1986. The Infrared Astronomical Satellite (IRAS) failed to discover Nemesis in the 1980s. The 2MASS astronomical survey, which ran from 1997 to 2001, failed to detect a star, or brown dwarf, in the Solar System.

and, later on,

In particular, if Nemesis is a red dwarf star or a brown dwarf, the WISE mission (an infrared sky survey that covered most of our solar neighborhood in movement-verifying parallax measurements) was expected to be able to find it. WISE can detect 150 kelvin brown dwarfs out to 10 light-years. But the closer a brown dwarf is the easier it is to detect. Preliminary results of the WISE survey were released on April 14, 2011. On March 14, 2012, the entire catalog of the WISE mission was released. In 2014 WISE data ruled out a Saturn or larger-sized body in the Oort cloud out to ten thousand AU.

In fact,

According to NASA, "recent scientific analysis no longer supports the idea that extinctions on Earth happen at regular, repeating intervals, and thus, the Nemesis hypothesis is no longer needed." And, indeed, a recent sky survey by NASA's WISE mission found no star or brown dwarf orbiting the Sun.

NASA's conclusions are laid out in this page, which talks about how WISE found no brown dwarf in the Oort Cloud . . . but found other brown dwarfs further out.

pr.probability - 'Focusing' the mass of the Probability Density Function for a Random Walk

Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability density (for the position of the walker) near the midpoint of the circle than near its contour.

Given this example, my question is - for a discrete/continuous random walk in a two-dimensional (or higher dimensional) space, now with arbitrary reflecting boundary conditions, how 'well' can one restrict/focus the mass of the probability density function to the smallest possible area relative to the total surface area available to the walker?

In other words, how effectively can one construct a 'trap' (I'm using this term very loosely) for such a walker, given random initial conditions?

(I obviously welcome any help to ask this question in a more appropriate manner.)

quantum mechanics - Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?

To answer your first question:

Actually the assumption is not that the wave function and its derivative are continuous. That follows from the Schrödinger equation once you make the assumption that the probability amplitude $langle psi|psirangle$ remains finite. That is the physical assumption. This is discussed in Chapter 1 of the first volume of Quantum mechanics by Cohen-Tannoudji, Diu and Laloe, for example. (Google books only has the second volume in English, it seems.)

More generally, you may have potentials which are distributional, in which case the wave function may still be continuous, but not even once-differentiable.

To answer your second question:

Once you deduce that the wave function is continuous, the equation itself tells you that the wave function cannot be twice differentiable, since the second derivative is given in terms of the potential, and this is not continuous.

telescope - Tools for creating a multiwavelength view of the sky

Not sure that this is the correct place to ask this question, but here goes.

I am trying to find tools to accumulate image data (preferably in FITS format) on a given (RA, Dec)-coordinate of sky. Right now there are several image services that offer cutouts, for example:

• irsa.ipac.caltech.edu holds data from Spitzer, Planck, WMAP, IRAS, WISE


• hla.stsci.edu holds Hubble Space Telescope data


• skyserver.sdss.org/dr7/en/tools/chart/list.asp holds SDSS data


• etc.

I wonder if anyone has written any code for accumulating these images given an argument of RA and Dec? Or if they can point me in the right direction of tools that they know exist? Any help would be appreciated.

differential operator in noncommutative geometry?

The Ginzburg's paper quoted above has a specific setup which is related to the examples like the "preprojective algebras" related to quivers. There is a recipe attaching an algebra to a noncommutative algebra by generators and relations reminiscent of the Weyl algebra. This recipe is local not only under flat localization, but even under stably flat maps. Yuri Berest has studied these aspects with hopes to globalize that definition to the nonaffine situations.

The usual Grothendieck definition may work in some noncommutative cases, e.g. possibly when the only noncommutative variables are nilpotent and alike. I do not know which definition in Kapranov, Oren means in his comment, but I think he points to similar cases related to nilpotent thickenings. Already for quantum groups this does not suffice.

Lunts and Rosenberg have tried to find a definition which would go along the Grothendieck's geometric picture: dealing with resolutions of diagonal. This has been studied in two of their Max Planck Bonn preprints, in very abstract categorical framework, and the results are global in the language of categories of quasicoherent sheaves on noncommutative schemes so to speak. Then they wrote two other papers on the same topic with down-to-earth recipes in the case of rings, modules and graded case. These differential operators correspond to filtrations reminding the Grothendieck's case of filtration by order, but being corrected in an improtant subtle detail. Their basic property is standard behaviour under flat localization functors as you suggest. There is also an arxiv paper by Tomasz Maszczyk who uses a variant of nc algebraic geometry based on bimodules and monoidal categories, and he rederives the same definition of the ring of regular differential operators as Lunts and Rosenberg do, with different geometric insight based on the duality between the infinitesimals and differential operators.

For the references look at nlab page on differential operators in nc geometry which I just started writing

nlab:regular differential operator in noncommutative geometry

fa.functional analysis - Odd element of L^1 group algebra of the integers

I'm a bit uncertain what is meant in the third condition: is this the supremum norm of the Gelfand/Fourier transform of $a^m$, or the norm of $a*a*dots*a$ in $ell^infty(mathbb Z)$?

In the first interpretation, it would clearly suffice to find an element satisfying the first two conditions, and then multiply it by an appropriate scalar betwen $0$ and $1$.
In the second interpretation, as Matthew says, it would suffice to find $a$ satisfying conditions 1) and 2) with the additional property that the Fourier transform of $a$ has modulus $<1$ at all but finitely many points of the unit circle.

Anyway, it turns out that we can find (trigonometric) polynomials satisfying conditions 1) and 2), which should by the previous remarks be enough. I don't know where the first examples are in the literature, but you can find explicit quadratic examples in this paper of D. J. Newman:

MR0241980 (39 #3315)
Newman, D. J.
Homomorphisms of $l_{+}$.
Amer. J. Math. 91 1969 37--46

which I personally think is a gem.

orbit - Apogalacticon and Perigalacticon

This question was once valid, but as of 1994 we found out that our Solar System is part of the Sagittarius Dwarf Elliptical Galaxy, or Sag-DEG (M54), which is the largest of the 30 or so Dwarf galaxies orbiting the Milky Way. We are in a 500 million year long circum-polar orbit. The two apsides are very far above the equatorial plane of the Milky Way. We just passed through the disc and are now about 50 light years above it, and rising towards our northern apsis (north with respect to the Galactic Coordinate System). Our journey will take us 125 million years to reach the top, and 125 million years to reach the disc of the Milky Way again. Ironically, if the Milky Way experiences a 250 million year rotation with respect to our current position in the Orion Arm (remembering that every location of the MW disc experiences a differential rotation with different orbital speeds), then by the time we reach the disc again 250 million years from now, we will basically pierce through the Orion arm very near to where we are right now! Isn't that amazing?

As for your question, and the answer provided by Gerald, we know that the stars orbiting the MW's core do oscillate up and down. This was predicted in 1978 and is due to the Galactic Current Sheet, which is why all of the stars follow each other in a sine wave pattern as they bob up and down. Because of this, we can predict the apsides for most of these stars, meaning the high and low point above the disc. However, these are not "apogalacticon" or "perigalacticon", which technically represent the furthest and closest and points in a satellites orbit, not as it oscillates through the disc, because Satellites don't do that, but around the Galactic core in an elliptical orbit much like a Sun-Grazing comet gets close to and then very far from the Sun. Think the Large Magellanic Cloud, or the 30 known Dwarf Galaxies, or many of the 158 known Globular Clusters.

ag.algebraic geometry - Reference request for a proof of Ramanujan's tau conjecture

Following the discussion at meta.MO, I'm going to post a good answer from the comments (made by JT) as a "community wiki" answer. I should mention that the Rogawski article mentioned by Tommaso says almost nothing about the proof of Ramanujan's conjecture, but it seems to be a very nice introduction to Jacquet-Langlands.

Deligne reduced Ramanujan's conjecture about the growth of tau to the Weil conjectures (in particular, the Riemann hypothesis) applied to a Kuga-Sato variety, in his paper Formes modulaires et representations l-adiques, Seminaire Bourbaki 355. I believe Jay Pottharst has made an English translation available.

Deligne then proved the Weil conjectures in his paper La conjecture de Weil. I.

As far as I know, all known proofs of this conjecture involve the use of cohomology of varieties over finite fields in an essential way.

Added by Emerton: One point to make is that the Weil conjectures (in their basic form,
saying that the eigenvalues of Frobenius on the $i$th etale cohomology of a variety over
$mathbb F_q$ have absolute value $q^{i/2}$) apply only to smooth proper varieties. On the other hand, the Kuga-Sato variety is the symmeteric power of the universal elliptic curve over a modular curve, which is not projective. Thus one has to pass to a smooth compactification in order to apply the Weil conjectures, and then hope that this does not mess anything up in the rest of the argument. A certain amount of Deligne's effort in
his Bourbaki seminar is devoted to dealing with this issue. If you don't worry about this
(i.e. you accept that it all works out okay) then the proof is essentially just Eichler--Shimura theory (i.e. the relation between modular forms and cohomology of modular curves), but done with etale cohomology, combined with the Eichler--Shimura congruence relation that connects the $p$th Hecke operator to Frobenius mod $p$. (The latter was treated in the
following question.)

distances - What's the furthest object observable by the naked eye from earth?

This is partly a question of 'limiting magnitude' - the faintest magnitude that is visible using a particular instrument, or in this case, just the eye.

+6 - maybe +6.5 - magnitude is sometimes used as a baseline expectation of what can be seen in dark skies sites.

The most distant star that can easily be seen is Deneb. It is actually quite bright, at brighter than +2 Magnitude, but is (probably) 1425ly away. So though it is at quite some distance, it's luminosity means that actually, it isn't limiting magnitude that is at play.

There are stars much further away that are at the barely perceivable limit, and seeing them will depend on conditions. The +4 mag mu Cephei is probably fairly achievable, at 5900ly, and V762 in Cassiopeia is about +6 mag and maybe 16000ly, which would probably be the record if you can actually see it, but you'll need special conditions.

When it comes to galaxies, Andromeda is 2500000ly away, and +3.4, very visible and much further away than any individual star. If you have good conditions, the +5.7 mag M33 is slightly further away at 2900000ly, but it would be much harder to see.

While the generally accepted limit is maybe +6.5, some people claim to have seen to +8. If it is possible, M83 is 14700000ly away at +8.2, with naked-eye claims reported.

galaxy - Using Tully Fisher to measure Distance Problem

Can anyone explain by looking at the solution in the pic, how did the cos (i) came about?
I guess they are getting this from the major-minor axis info in the question, but I am not sure about the derivation.

Also v=300/sin(i), what is this formula?

Many thanks :)

pr.probability - Pennies on a carpet problem

Rota (on page $viii$ of his introduction, page 10 of the pdf file) is talking about the difficulty of having an analytic solution for statistical mechanics in the 2-dimensional and 3-dimensional cases, while it is possible to attack the problem somewhat in the one-dimensional case.

He also mentions how stochastic methods and simulation can be used to come up with a quick-and-dirty approximation of the answers by modeling the physical sysyem and using Monte Carlo methods: iterating the system with random steps.

Topics to research would be Monte Carlo methods, stochastic models, random walks, etc. Can you say a little more about exactly what it is that you wish to study or examine?

Do you recognise this variant of the cubes operad?

In the subject of operads one comes across many quirky variants of more or less the same operad. The cubes operad has many incarnations with various interesting properties. In a recent paper I came across one such variant. It's so simple I presume other people have come across it before, so it would be nice to have a reference if that's the case. I'll give a sketch of this rather simple construction, below.

Let me describe the operad, what I think it's good for, and how it relates to other operads. First, I'll set up some notation convention with the cubes operad.

Def'n: (cubes) An increasing affine-linear function $[-1,1] to [-1,1]$ is a little interval. A product of little intervals $[-1,1]^n to [-1,1]^n$ is a little $n$-cube. The space $mathcal C_n(j)$ is the collection of $j$-tuples of
little $n$-cubes whose images are required to have disjoint interiors, $mathcal C_n(0)={*}$ is the empty cube. The collection $mathcal C_n = sqcup_{j=0}^infty mathcal C_n(j)$ is the operad of little $n$-cubes, it is a $Sigma$-operad with
structure maps

$$mathcal C_n(k) times left( mathcal C_n(j_1) times cdots times mathcal C_n(j_k) right) to mathcal C_n(j_1+cdots+j_k)$$

defined by

$$(L,J_1,cdots,J_k) longmapsto (L_1 circ J_1, cdots, L_k circ J_k)$$

and $mathcal C_n(j) times Sigma_j to mathcal C_n(j)$ given by $(L, sigma) longmapsto Lcirc sigma$.

Def'n: (overlapping cubes) A collection of $j$ overlapping $n$-cubes is an equivalence class of pairs $(L, sigma)$ where $L=(L_1,cdots,L_j)$, each $L_i$ is a little $n$-cube and $sigma in Sigma_j$. Two collections of
$j$ overlapping $n$-cubes $(L,sigma)$ and $(L',sigma')$ are taken to be equivalent provided $L = L'$ and
whenever the interiors of $L_i$ and $L_k$ intersect $sigma^{-1}(i) < sigma^{-1}(k) Longleftrightarrow
sigma'^{-1}(i) < sigma'^{-1}(k)$. Given $j$ overlapping $n$-cubes $(L_1,cdots,L_j,sigma)$ say the $i$-th cube $L_i$ is at height $sigma^{-1}(i)$. $sigma(1)$ is the index of the bottom cube, and $sigma(j)$ is the index of the top cube. Let $mathcal C_n'(j)$ be the space of all $j$ overlapping $n$-cubes, with the quotient topology induced by the equivalence relation.

The structure map
$$mathcal C_n'(k) times left( mathcal C_n'(j_1) times cdots times mathcal C_n'(j_k) right) to mathcal C_n'(j_1 + cdots + j_k)$$

is defined by

$$left((L,sigma), (J_1,alpha_1), cdots, (J_k, alpha_k)right) longmapsto ((L_1circ J_1, cdots, L_kcirc J_k), beta)$$

the permutation $beta$ is given for $1 leq a leq k$, $1 leq b leq j_a$ by

$$beta^{-1}left(sum_{i<a} j_i + bright) = left( sum_{i<sigma^{-1}(a)} j_{sigma(i)}right) +alpha^{-1}_a(b)$$

This permutation is obtained by taking the lexicographical order on the set ${(a,b) : a in {1,cdots,k}, b in {1,cdots,j_a}}$ and then identifying with ${1, 2, cdots,j_1+cdots+j_k}$ in the order-preserving way.

== The point ==

So there is a map of operads $mathcal C_{n+1} to mathcal C'_n$ given by sending $(L_1, cdots, L_j)$ to $(L_1^pi, cdots, L_j^pi, sigma)$ where we write $L_i = L_i^pi times L_i^nu$ where $L_i^pi$ is an $n$-cube and $L_i^nu$ a $1$-cube. The permutation $sigma$ is any element $sigma in Sigma_j$ such that $L_{sigma(j)}^nu(1) geq L_{sigma(j-1)}^nu(1) geq cdots geq L_{sigma(1)}^nu(1)$.

Some of the nice things about this operad are:

• (1) it's a multiplicative operad, the inclusion of the associative operad is given by the elements $(Id_{mathcal [-1,1]^n}, cdots, Id_{mathcal [-1,1]^n}, Id_{{1,cdots,j}}) in mathcal C'_n(j)$.




• (2) The map above $mathcal C_{n+1} to mathcal C'_n$ is an equivalence of operads.




• (3) While $mathcal C_{n+1}$ acts on spaces such as $Omega^{n+1} X$, $mathcal C'_n$ does not. $mathcal C'_n$ acts on spaces of the form $Omega^n M$ where $M$ is a topological monoid.

The operad of overlapping intervals $mathcal C'_1$ has a certain affinity to the cactus operad. For example, imagine $[-1/2,1/2]$ as an element of $mathcal C'_1(1)$ as being represented by $[-1,1]$ with a $1$-cell attached at the points $-1/2$ and $1/2$.

And there are all kinds of variants of this idea -- overlapping discs, or overlapping framed discs, etc. So you can get cyclic multiplicative operads out of these types of constructions.

The criterion for getting the answer "right" is either showing me an occurance of this operad in the literature, or coming up with some convincing argument it's a new construction.

ho.history overview - Good books on problem solving / math olympiad

Hello,
I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would be the books from The art of problem solving, Engels book and Paul Zeits book. Books on certain topics, say geometry is also appreciated!

ag.algebraic geometry - Rational Hilbert modular surfaces

The answer is (probably) yes.

A theorem of Margulis et al. shows that an irreducible lattice in a Lie group is arithmetic unless the group is isogenous to SO(1,n)x(compact) or SU(1,n)x(compact).

A theorem of Mumford shows that the quotient of a hermitian symmetric domain by a neat arithmetic subgroup is of logarithmic general type in the sense of Iitaka (hence not rational). (An arithmetic subgroup is neat if the subgroup generated by the eigenvalues of any element of the subgroup is torsion-free. Every sufficiently small subgroup is neat).

For a discussion of the first theorem, see Section 5B of Witte Morris, Introduction to Arithmetic Groups,
http://front.math.ucdavis.edu/0106.5063

For the second theorem, see: Mumford, D. Hirzebruch's proportionality theorem in the noncompact case. Invent. Math. 42 (1977), 239--272. MR0471627

Added: As @moonface points out, this doesn't prove it.

The congruence subgroup problem is known for $SL_{2}$ over totally real fields $F$ other than $Q$. Let $Gamma$ be an irreducible lattice in $SL_{2}(mathbb{R})times SL_{2}(mathbb{R})$. By Margulis, it is arithmetic, hence congruence. Moreover, after conjugating we may suppose that $GammasubsetGamma(1)=SL_{2}(O)$. Now $Gammabackslashmathbb{H}timesmathbb{H}$ covers $Gamma(1)backslash mathbb{H}timesmathbb{H}$, and so if the first is rational, so is the second (Zariski 1958; requires dimension 2). Thus, we have a finite list of the possible totally real quadratic extensions $K$ to work with. For some $N$, $GammasupsetGamma(N)$, and we may suppose that $Gamma(N)$ is neat. Since $Gamma(1)/Gamma(N)$ is finite, for any integer $N$ we get (in sum) only finitely many rational Hilbert modular surfaces of level $N$.

The problem remains to show that, for each totally real quadratic field on our list, every rational Hilbert modular surface is of level $N$ for some fixed $N$. Apart from looking case by case, I don't see how to do this (but it is surely true).

set theory - Finite axiomatizability and constructible sets

In the development of constructible sets via the Gödel operations, one of the main point seems to be that for $Delta_0$ formulas, one can construct the sets of of elements verifying them with a finite number of Gödel operations, called $G_1$,...,$G_{10}$.

My questions are : does this means that Set theory with separation restricted to $Delta_0$ formulas is finitely axiomatizable since a set is closed under a finite number of these operations iff it is closed under $Delta_0$ formulas? Also since $Delta_0$ formulas are absolute in transitive models, does this mean that if we consider $Th(M)$ with $M$ the fragment generated only by $Delta_0$ formulas, then it is finitely axiomatizable? Now if you had $Delta_1$ formulas is it still finitely axiomatizable? Finally, does this imply that a reflection theorem can't hold in Set Theory with $Delta_0$ separation?

I hope my questions were accurate.

co.combinatorics - Looking for cubic, bipartite graphs with girth at least six and no cycles of length 8.

Desargues graph, also known as the generalized Petersen graph $G(10,3)$ has girth 6 and also contains cycles of length 8. There exist three 10-cages, smallest cubic graphs of girth 10. They have 70 vertices and none of them is vertex-transitive. For more information about the motivation
see here.

Therefore it is quite surprising that Gordon's example is so small. By the way, it is also arc-transitive. One may consider it as a Levi graph of a self-dual, flag-transitive 3-configuration of 19 points and 19 lines. Configuration contains triangles but has no quadrangles.

Here is a geometric realization of this graph.

Configuration with triangles but no quadrangles

1. A natural question is, whether this is the smallest graph with required properties.




2. Another related question is, whether there exist smaller bipartite graphs if the vertex-transitivity condition is dropped.

na.numerical analysis - Convergence of iterative algorithm.

For quite a long time I'm trying to prove convergence of an iterative algorithm in case of a particular system of nonlinear equations.

Here are some characteristics of this system:
It consists of n equation and n variables.
Every equation is in similar form - sum of products = constant.
The lenght of every product is the same (it will be denoted as k).
The number of elements in sum might be different in each equation.
In every product in equation $i$ one of elements is $x_{i}$ - this is very important.
This system is "symmetrical", it means that if $x_{i} cdot x_{j} cdot ...$ is one of elements of equation i, then it is also in equaton j.
$b_{i} > 0$ - where $b_{i}$ is intercept in equation i.

I'll write an example of such equation for k=3 and n=6:
$x_{1} cdot x_{3} cdot x_{6} + x_{1} cdot x_{2} cdot x_{4} = b_{1}$
$x_{2} cdot x_{1} cdot x_{4} + x_{2} cdot x_{5} cdot x_{6} = b_{2}$
$x_{3} cdot x_{1} cdot x_{6} + x_{3} cdot x_{4} cdot x_{6} = b_{3}$
$x_{4} cdot x_{1} cdot x_{2} + x_{4} cdot x_{3} cdot x_{6} = b_{4}$
$x_{5} cdot x_{2} cdot x_{6} = b_{5}$
$x_{6} cdot x_{1} cdot x_{3} + x_{6} cdot x_{2} cdot x_{5} + x_{6} cdot x_{3} cdot x_{4} = b_{6}$

It is very easy to transform this equation to following form (it just need to be divided once).
$x_{i} = b_{i} / something$ , $x_{i}$ is only on left-hand side of ith equation.

If we have all equation in such form then the fixed point is solution of it.
I've experimentally checked that algorithm analogic to Gauss-Seidel is covergent (i've checked ~100 random examples, and in every case it was convergent).
By analogic to Gauss-Seidel algorithm I mean:
1) Choose any initial solution $[x_{1}^{0} , ... , x_{n}^{0}]$
2.1) Calculate value of $x_{1}^{i+1}$ using $[x_{2}^{i} , ... , x_{n}^{i}]$
2.2) Calculate value of $x_{2}^{i+1}$ using $[x_{1}^{i+1} , ... , x_{n}^{i}]$
...
2.n) Calculate value of $x_{n}^{i+1}$ using $[x_{1}^{i+1} , ... , x_{n-1}^{i+1}]$
3) If solution is good enough stop, otherwise go to 2.1

I've tried Banach fixed point theorem, but is hard to say anything about spectral radius.
Does anyone have a clue how to prove convergence of this algorithm?

Edited 7.02.2009 11:09
I've found another restriction.
If we denote by $m$ number of all products (in this example it would be 12, becuase in 1st,2nd,3th,4th we have 2 products, in 5th we have 1, and in 6th we have 3).
Then following equation is true:
$m = k cdot sum_{i=1}^{n}{b_{i}}$
Which also implies thah $sum_{i=1}^{n}{b_{i}}$ is a natural number (from the symmetry), but it doesn't imply that any of $b_{i}$ is natural.

terminology - What's the name of this 2D cellular automaton?

Does this 2D cellular automaton have a known name and history?

• n colors (numbered 1 to n), assigned randomly at the start.


• For each generation, every cell that has at least one neighbour cell with a color that is one higher changes its color to that "next higher" color. Additionally, the "lowest" color is considered "next higher" to the highest one.


• Emergent behaviour shows up best around n=16, disappears for much higher or much lower n

I have it implemented on my website so you can see it in action.

I saw this ages ago and always remembered it as a great example of emergent behaviour, but can't remember what it was called, and couldn't find it on Wikipedia or Wolfram Mathworld.

distances - Do we make predictions in our time, or local time?

I was scrolling idly through the Wikipedia article on Orion when I read:

Orion will still be recognizable long after most of the other constellations—composed of relatively nearby stars—have distorted into new configurations, with the exception of a few of its stars eventually exploding as supernovae, for example Betelgeuse, which is predicted to explode sometime in the next million years.

Given that Betelgeuse is only 640 light-years away from Earth, the question of whether we're talking about Betelgeuse going supernova in the next million years, or about news of the supernova reaching us in the next million years, is irrelevant, since 640 years is hardly noticeable when you're talking about ten thousand centuries.

But what about a star further away? Say, something in the Andromeda Galaxy, 2.5 million light-years from us? If I read an article stating that a star in the Andromeda Galaxy was going nova in a million years, would that mean:

a) That we think the star will go nova one million years from now, and its light will reach us in 3.5 million years; or
b) That we believe the star went nova 1.5 million years ago, and its light will reach us in one million years

In other words, are we making estimates in our time, or in celestial-body-local time?

soft question - Magic trick based on deep mathematics

Here is a trick much in the spirit of the original number-adding example; moreover I'm sure Richard will appreciate the type of "deep mathematics" involved.

On a rectangular board of a given size $mtimes n$, Alice places (in absence of the magician) the numbers $1$ to $mn$ (written on cards) in such a way that rows and columns are increasing but otherwise at random (in math term she chooses a random rectangular standard Young tableau). She also chooses one of the numbers say $k$ and records its place on the board. Now the she removes the number $1$ at the top left and fills the empty square by a "jeu de taquin" sequence of moves (each time the empty square is filled from the right or from below, choosing the smaller candidate to keep rows and columns increasing, and until no candidates are left). This is repeated for the number $2$ (now at the top left) and so forth until $k-1$ is gone and $k$ is at the top left. Now enters the magician, looks at the board briefly, and then points out the original position of $k$ that Alice had recorded. For maximum surprise $k$ should be chosen away from the extremities of the range, and certainly not $1$ or $mn$ whose original positions are obvious.

All the magician needs to do is mentally determine the path the next slide (removing $k$) would take, and apply a central symmetry with respect to the center of the rectangle to the final square of that path.

In fact, the magician could in principle locate the original squares of all remaining numbers (but probably not mentally), simply by continuing to apply jeu de taquin slides. The fact that the tableau shown to the magician determines the original positions of all remaining numbers can be understood from the relatively well known properties of invertibility and confluence of jeu de taquin: one could slide back all remaining numbers to the bottom right corner, choosing the slides in an arbitrary order. However that would be virtually impossible to do mentally. The fact that the described simple method works is based on the less known fact that the Schútzenberger dual of any rectangular tableau can be obtained by negating the entries and applying central symmetry (see the final page of my contribution to the Foata Festschrift).

rt.representation theory - irreducible representations of O(2) - reference?

I just see now, that the issue is appearently real representations. I consider complex representations. I not experienced with real representations and whether my strategy works there as well.

You can induce from $SO(2)$. Define on $SO(2)$ the rep $epsilon_n: theta mapsto e^{i theta n}$. Let $rho_n$ be the induced one, then $rho_n$ is irreducible if $n neq 0$. You have $rho_n cong rho_{-n}$ and $rho_{0} = 1 oplus det$. These are up to isomorphism all irreducible representations.

Reference: Traces of Hecke operator by Knightly and Li.

A proof also is in my thesis: http://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-000D-F074-7/palm.pdf?sequence=1 on pg 101.

mg.metric geometry - Rectifying texture from image

By picking orthogonal coordinates in the given plane you can make an angle preserving projective map $mathbb{P}^2tomathbb{P}^3$ whose image is the given plane. Composing with your camera mapping, you now have a mapping $Gcolonmathbb{P}^2tomathbb{P}^2$ that does not preserve angles. Let $H=G^{-1}$. The composition $HG=I$ clearly preserves angles; hence so does $HP$ when restricted to the given plane.

(Reverse the order of composition if you follow the usual computer graphics convention of letting matrices act on the right.)

linear algebra - Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space

So I'm trying to get the marginal density of a multivariate normal over an affine space
if $A$ is a matrix in $mathbb{R}^p times mathbb{R}^n$ for $p < n$ and $B in mathbb{R}^n$, $Sigma$ is a positive definite matrix.

We're looking at...
$$lim_{epsilon rightarrow 0}frac{1}{epsilon}int_{||Ax-B||<epsilon} e^{-frac{1}{2} x^t Sigma^{-1} x } dx$$

Edit
The space over which I integrate is wrong, see edit at the end... I should be controlling the distance to the projection, not something that depends on the scale of $A$ and $B$.

The first step is to get $$x_0 = hbox{argmin}_x left[ x^t Sigma^{-1} x | Ax-B = 0 right]$$

If one writes the Cholesky decomposition $Sigma^-1 = L^T L$ and $hbox{}^+$ is the Moore-Penrose pseudo inverse, then $$x0 = (A L^{-1})^{+} B$$

We substitute $x = x_0 + y$ in the integral, $y$ is orthogonal to $x_0$ with respect to $Sigma^-1$ so the $x Sigma^{-1} y$ terms disappear and we get something like

$$e^{-frac{1}{2} x_0^t Sigma^{-1} x_0} lim_{epsilon rightarrow 0}frac{1}{epsilon}int_{||Ay||<epsilon} e^{-frac{1}{2} y^t Sigma^{-1} y} dy$$

setting $z = Ly$

Edit
urrrr...
$$e^{-frac{1}{2} x_0^t Sigma^{-1} x_0} lim_{epsilon rightarrow 0}frac{1}{epsilon}int_{||A L^{-1}z||<epsilon} e^{-frac{1}{2} z^t z} frac{dy}{dz} dz$$

hum the $frac{dy}{dz}$ wouldn't be very nice...

At this point, I've gotten rid of the affine part by factoring the $x_0$ out ( yay.. ) but I can't quite get that last cut... It's likely going to involve the determinant of $Sigma$ and the rank of $A$ ( assume it's $p$ if needed ) but I can't figure out how.

a) Do you notice anything obviously wrong in the derivation of $x_0$ ?

b) What's the obviously right answer that has been eluding me ?

Edit
c) Possible way, orthonormalize the kernel of $A$ then the rest of the space. Then the problem boils down to finding the density of a multivariate normal conditional on some elements of the vector being known... that's not a nice formula, it involves slicing $Sigma$ and taking the Schur complement see
Wikipedia, Multivariate normal, Conditional distributions

Edit
It occurs to me that the $||Ay||<epsilon$ criterion isn't the right one, otherwise, the answer would depend on the scale of $A$ which is dumb. I guess the right criterion is
$||A^{t}(A A^{t})^{-1}A y||<epsilon$, which is the distance of vector $y$ to its orthogonal projection on the subspace $A. = $

Thanks!

Category and homotopy theoretic methods in set theory

There is an interaction between category theory and set theory. In 1965, one year after Cohen's proof of the independence of the continuum hypothesis, Vopenka gave a proof using sheaf theory, see here.

This is nowadays systematized in the topos theoretic interpretation of set theory, for which you should look up MacLane/Moerdijk, as Dylan Wilson pointed out. The authors give a proof of the independence of the continuum hypothesis.

Marta Bunge has given a topos theoretic proof of the independence of the Suslin hypothesis from ZFC in: Marta Bunge, Topos Theory and Souslin's Hypothesis. J.Pure & Applied Algebra 4 (1974) 159-187.

As for the reformulation of Vopenka's principle: It is equivalent to the statement that a locally presentable category can not have a large full discrete subcategory. This, and more, is nicely explained in Adamek/Rosicky's Locally Presentable and Accessible Categories

On the other hand I know of no worked out connection between homotopy theory and set theory, just the indirect one via type theory mentioned by Andrej.

ag.algebraic geometry - Is there a Whitney theorem type theorem for projective schemes?

This is a reply to a question posed in Georges' answer. It started as a comment, but I was worried about space limitations. Upon reflection, it's better to have it as an answer, because -- unlike the comment I posted to Georges' answer -- I will be automatically notified of any responses to it.

Since Georges asked for it, the [rough!] lecture notes where I discuss the facts that any smooth curve over an infinite field can be embedded in $mathbb{P}^3$ and "immersed" in $mathbb{P}^2$ with only ordinary double point singularities are available here (see Section 5):

http://www.math.uga.edu/~pete/8320notes6.pdf

As you will see, I am merely repeating the argument in Hartshorne -- omitting the trickier details of the immersion result -- and explaining why the ground field need not be algebraically closed but does need to be infinite.

Concerning Horrocks-Mumford and Van de Ven: I was not familiar with these results until Georges' post. But all the non-embeddability statements carry over immediately: if you have an embedding into $mathbb{P}^n$ over the ground field, then the base change to the algebraic closure is still an embedding, of course.

This leaves the question of the positive part of the Horrocks-Mumford result. In strongest form, the question is: is it true that for any field $k$, there is an abelian surface over $k$ that can be embedded in $mathbb{P}^4$? [I can certainly do it with $mathbb{P}^2 times mathbb{P}^2$ -- take a product of two elliptic curves -- and it is conceivable to me that one might be able to get from this an embedding into $mathbb{P}^4$ by composing with a well-chosen birational isomorphism, but I haven't even tried to decide whether this would work.]

I would have to see the proof of H-M to see whether it can be adapted to answer this question. Can you post a link to the paper? Or, if you need to know ASAP, ask Bjorn Poonen -- he eats questions like this for breakfast.

Finally, let me remark that over a non-algebraically closed field, a principal homogeneous space under an abelian variety may have higher embedding dimension than the (Albanese) abelian variety itself. The easy example of this is that if a smooth curve of genus one can be emedded in $mathbb{P}^2$, then for geometric reasons it must embed as a cubic and therefore has a rational point of degree at most $3$. [Actually, it is possible that this is the only example. By the same theorems Georges quoted above, the only other possibility is a phs which does not embed in $mathbb{P}^4$ while its Albanese abelian surface does.]

ag.algebraic geometry - Serre duality and low dimensional cohomology groups

Given a locally free sheaf $M$ on $mathbb{P}^2$ with $h^0(M)=1$.
Is it true that we have $h^2(M)=0$ in this case?

I got this idea from Friedman's book "Algebraic Surfaces and holomorphic vector bundles".
In Chapter 4, p.109, Ex. 4 he wrote: $h^0(Hom(V,V))=1$, by Serre duality $h^2(Hom(V,V))=h^0(Hom(V,V)otimes K)=0$ since $K=O(-3)subset O$.

But I don't see why $h^0(Hom(V,V)otimes K)=0$ follows from $K=O(-3)subset O$. I mean it could stil have dimension one or is it because $O(-3)$ has no global sections? Would $h^0(Hom(V,V)otimes O(-i))=0$ still be true for $i=1,2$? Can one generalize to arbitrary locally free sheaves or is this only correct in this special case, i.e. $V$ stable?

${bf Edit:}$ Given a simple sheaf $V$ on $mathbb{P}^2$, Bjorn's answer shows $H^0(Hom(V,V)otimes O(-i))=0$ for $i>0$ which can be written as $Hom(V,V(-i))=0$ for $i>0$. Can this be generalized?

For example given a sheaf of rings or algebras $R$ and a simple left $R$-module M, do we always have $Hom_R(M,M(-i))=0$ for $i>0$? Or do I need to be more careful in this case?

I remembered this question reading arxiv.org/abs/0810.0067, page 8, where such an equality shows up, without further explanation, so i thought the argumentation should carry over to this more general case.

iwasawa theory - When does a p-adic function have a Mahler expansion?

Let $f: mathbb{Z}_p rightarrow mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n in mathbb{C}_p$ with

$$
f(x) = sum^{infty}_{n=0} a_n {x choose n}.
$$

This is known as the Mahler expansion of $f$. Here, to make sense of $x choose n$ for $x notin mathbb{Z}$ define

$$
{xchoose n} = frac{x(x-1)ldots(x-n+1)}{n!}.
$$

There's an important application to number theory: expressing a function in terms of its Mahler expansion is one step in translating the old fashioned interpolation-based language of $p$-adic $L$-functions into the more modern measure-theoretic language. This application is explained in Coates and Sujatha's book Cyclotomic Fields and Zeta Values.

However, when the $p$-adic $L$-function is more complicated than Kubota-Leopoldt's, it seems to me that this "translation" really requires one to be able to write down a Mahler expansion of a function $f$ with much larger domain, e.g. the ring of integers of the completion of the maximal unramified extension of $mathbb{Q}_p$ or some finitely ramified extension. (See, for instance, line (8), p. 19 of de Shalit's Iwasawa Theory of Elliptic Curves with Complex Multiplication).

I can't find a published reference that these functions really have Mahler expansions. Mahler's paper uses in an essential way that the positive integers are dense in $mathbb{Z}_p$, so it doesn't instantly generalize.

So is it true or false that for a ring of integers $mathcal{O}$ in a finitely ramified complete extension of $mathbb{Q}_p$, and a function $f: mathcal{O} rightarrow mathbb{C}_p$, there is a Mahler expansion as above?

convexity - Existence of an extreme point of a compact convex set

Here is Greg's proof of the Krein-Milman theorem (only the existence part).

Proposition 1: Let $Asubseteq V$ be a non-empty compact convex set of an hausdorff locally convex semi-topological vector space (over some field which contains the reals topologically). Then $A$ has an extreme point.

Specifically we shall show that every non-empty convex open-in-$A$ proper subset $U^{-}$, has an extension to a convex open-in-$A$ proper subset $U^{+}supseteq U^{-}$, for which $Asetminus U^{+}={e}$ some extreme point $e$.

Proof (Prop 1): If $A$ is a singleton, we are done. So suppose otherwise. First notice we can separate two points by a convex open set, $U$, yielding $U^{-}=Ucap A$ a convex proper subset open in $A$. So fix such a set $U^{-}$.

Now let $mathbb{P}:={W:U^{-}subseteq Wmbox{ convex proper subset open in }A}$. We claim that the p.o. $(mathbb{P},subseteq)$ has the Zorn property. Clearly the union of any chain of convex subsets open in $A$, is again a convex subset open in $A$. Suppose it{per contra} that a union of a chain $mathcal{C}subseteqmathbb{P}$ is equal to $A$. Then since $A$ is compact, there must be a finite sub(chain) $mathcal{C}'subseteqmathcal{C}$ for which $operatorname{max}(mathcal{C}')=bigcupmathcal{C}'=A$, which contracts the properties of members of $mathbb{P}$. Thus $mathbb{P}$ is non-empty and has the Zorn property. So fix $U^{+}$ open such that $U^{+}cap Ainmathbb{P}$ maximal.

Sub-claim 1: $U^{+}cap Asubsetoperatorname{cl}_{A}U^{+}$ strictly.

Proof (sub-claim 1): Let $xin U^{+}cap A$ and $yin Asetminus U^{+}$. Consider the map (here we need the underlying field to contain the reals topologically)

$$begin{array}[t]{lrclll}
&f &: &mathbb{F} &to &V\
&&: &s &mapsto &s~x+(1-s)~y\
end{array}$$

which is continuous and affine. Thus the set

$$S:={sin[0,1]:smapsto s~x+(1-s)~yin U^{+}cap A} =[0,1]cap f^{-1}U^{+}cap f^{-1}A)$$

is a convex. The set $f^{-1}A$ is closed convex and contains $0,1$, thus contains $[0,1]$. So $S=[0,1]cap f^{-1}U^{+}$ which is convex open in $[0,1]$. Since $0notin S$ and $1in S$, then $S=(a,1]$ some $ain[0,1)$. Clearly $f(a)=lim_{ssearrow a}f(s)$ which is a limit of vectors from $U^{+}cap A$, and thus lies in $operatorname{cl}U^{+}$. It also clearly lies in $A$, since $f^{-1}Asupseteq[0,1]$. We also have $f(a)notin U^{+}$. Thus $operatorname{cl}_{A}U^{+}setminus U^{+}=Acapoperatorname{cl}U^{+}setminus U^{+} supseteq{f(a)}$. Thus the containment is strict. QED (sub-claim 1)

Sub-claim 2: If $Wsubseteq A$ is convex, then $U^{+}cup W$ is convex.

Proof (sub-claim 2): Fix $xin U^{+}cap A,tin(0,1)$. Consider the map

$$begin{array}[t]{lrclll}
&T &: &V &to &V\
&&: &y &mapsto &t~x+(1-t)~y\
end{array}$$

This is continuous and affine. By the proposition below, we also see that $T(operatorname{cl}(U^{+}))subseteq U^{+}$. So $Acap T^{-1}U^{+}$ is convex, open in $A$, and contains $Acapoperatorname{cl}(U^{+})$ which strictly contains $U^{+}cap A$ by Claim 1. Thus by maximality of $U^{+}$ in $mathbb{P}$, $Acap T^{-1}U^{+}overset{mbox{must}}{=}A$. In particular, $T^{-1}U^{+}supseteq A$, and so $TWsubseteq TAsubseteq U^{+}$. Utilising such maps as $T$, we see that a convex-linear combination of any pair of elements from $U^{+}cup W$ is contained in $U^{+}cup W$. QED (sub-claim 2)

Sub-claim 3: $Asetminus U^{+}$ is a singleton.

Proof (sub-claim 3): Else, let $x_{1},x_{2}in Asetminus U^{+}$ distinct. Let $W$ be a convex open set separating $x_{1}$ from $x_{2}$. Then by Claim 3, $U^{+}cap Acup Wcap A$ is convex open in $A$. Since $x_{1}in Wsetminus U^{+}$, this convex set is strictly large than $U^{+}cap A$. By maximality of $U^{+}$ in $mathbb{P}$, we have that $U^{+}cap Acup Wcap A=Ani x_{2}$. But this contradicts the fact that $x_{2}notin U^{+}cup W$. QED (sub-claim 3)

At last, we claim that the point $ein Asetminus U^{+}$ is extreme in A. Consider $x,yin A$ and $tin(0,1)$ for which $tx+(1-t)y=e$. Case by case, we see that if $x,yin U^{+}$ then $e=tx+tyin U^{+}$. If $xin Asetminus U^{+}$, $y=frac{e-tx}{1-t}=frac{e-te}{1-t}=e=x$. If $yin Asetminus U^{+}$, then $x=e=y$ similarly. Thus $e$ is extreme. QED(Prop 1)

Proposition 2: Let $Asubseteq V$ be a convex subset of a semi-topological space, and $x,y$ be vectors lying respectively in the closure and interior of $A$. Then for any $tin(0,1)$, we have $tx+(1-t)yin A$.

Proof: Let $U$ be an open neighbourhood of $y$, contained in $A$. Observe that $x-t^{-1}(1-t)(U-y)$ is an open neighbourhood of $x$, and thus meets $A$, say at $x'$. Thus $xin x'+t^{-1}(1-t)(U-y)$ implying $tx+(1-t)yin tx'+(1-t)Usubseteq tA+(1-t)Asubseteq A$, since $A$ is convex. QED(Prop 2)

Representation of $*$-automorphism on finite dimensional matrix algebras

As an alternative to Robin Chapman's solution, I would like to state Exercise 7.8 from Rørdam's, Larsen's and Laustsen's "Introduction to the K-theory of C*-algebras":

For every unital AF-algebra $A$ there is a short exact sequence
$$
1tooverline{mathrm{Inn}}(A)tomathrm{Aut}(A)tomathrm{Aut}(K_0(A))to 1,
$$
where $overline{mathrm{Inn}}(A)$ denotes approximately inner automorphisms and $mathrm{Aut}(K_0(A))$ denotes group automorphisms preserving the unit class and the positive cone in $K_0(A)$.

If $A$ is the matrix ring, then $mathrm{Aut}(K_0(A))$ is trivial and hence every automorphism of $A$ is approximately inner. Since $A$ is separable, every approximately inner automorphism is the pointwise limit of a sequence of inner automorphisms. And I think the finite-dimensionality of $A$ implies that the pointwise limit of a sequence of inner automorphisms is again inner.

Using the statement above, one immediately sees that, for instance, $mathbb Coplusmathbb C $ possesses an automorphism which is not approximately inner.

nt.number theory - Isogenies between Tate curves

Let $q$ and $q'$ be complex numbers with $0<|q|,|q'|<1$, and let $m$ and $n$ be positive integers.
Suppose that $q^m={q'}^n$. Then the map
$$
f:mathbb{C}^times/q^{mathbb{Z}} to mathbb{C}^times/{q'}^{mathbb{Z}}qquad text{defined by}qquad umapsto u^m
$$
gives an isogeny of (analytic) elliptic curves over $mathbb{C}$.

The Tate curve $mathrm{Tate}(q)$ is an (algebraic) elliptic curve over the Laurent series ring $mathbb{Z}((q))$ which can be used to give a uniformization of the curve $mathbb{C}^times/q^mathbb{Z}$ by means of certain well known explicit formulae.

My question is:

Does there exist an isogeny $mathrm{Tate}(q)to mathrm{Tate}(q')$ of elliptic curves defined over $mathbb{Z}((q,q'))/(q^m-{q'}^n)$ which "lifts" the map $f$ above, and if so, how do you prove it exists?

It should suffice to construct such an isogeny for $(m,n)=(m,1)$, and use dual isogenies and composition to get the general case.

(I'm being vague about "lifts", because one has to worry about convergence somewhere. Probably you want to say that the isogeny is defined over some subring of $mathbb{Z}((q,q'))/(q^m-{q'}^n)$ of power series which are analytically convergent near $q=0$, or something like that.)

I presume (though I probably can't prove) that the existence of the analytic isogenies means that such a map of schemes is defined over $mathbb{C}((q,q'))/(q^m-{q'}^n)$, so that this is just a question about integrality.

This is very closely related to exercise 5.10 in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. There, he apparently asks us to show that for a $p$-adic field $K$, if $q,q'in K$, $0<|q|,|q'|<1$, and $q^m={q'}^n$, then the function $overline{K}^times/q^mathbb{Z}to overline{K}^times/{q'}^{mathbb{Z}}$ defined by $umapsto u^m$ lifts to an isogeny $E_qto E_{q'}$ of elliptic curves over $K$, where $E_q$ and $E_{q'}$ are defined by the Tate curve equations. (An answer to my question solves his exercise, right?)

Unfortunately, I have no idea how to do Silverman's exercise either (he marks it as difficult). Any hints?

sg.symplectic geometry - Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?

I've scanned the paper I linked above for a few minutes(I am saying that so people judge what I say critically) and I'd like to expand on the comment that some category of modules over the deformation quantization is a sort of classical limit of the Fukaya category(e.g. a version without pseudo-holomorphic disks). I am sure there have been advances since this paper, since it goes back to 2002, and I'd be really interested to hear what they were. First of all, Bressler and Soibelman have the lemma that I was alluding to above that if $A_X$ is the quantized algebra of smooth functions, $HH^*(A_X-mod)= H^*(X)otimes C((t))$(I guess Bressler et. al. work with complex valued smooth functions.) with the ordinary product. Meanwhile (assuming one has symplectic form that is integral??) the quantum cup product can be defined on the same vector space.The difference is that the product in the first vector space is the ordinary product, while the product in the second one is deformed by pseudo-holomorphic discs.

This can be some kind of closed string version of the classical limit analogy. Now, the observation that is really cool in the paper however, is that if one restricts to the category of what they call holonomic modules, one can get a much more precise version of this analogy(a categorical one). Let Hol(X), be the full subcategory of modules whose support(of M/tM) consists of modules are Lagrangian Submanifolds. Bressler and Soibelman claim that the data of a langrangian submanifold and a local system (L,p) can be used to produce a holonomic module in a canonical way. Furthermore, one has the following analogy with the Fukaya category... Let $M_{(L,p)}$ be the module produced above... One has $Ext^*(M_{(L,p)},M_{(L,p)})=H^*(L,p)otimes C((t))$. The obvious proposition about two Lagrangian's with transversal support is correct too, see Prop 2. on page 12. So again, this category has the right Hom's as vector spaces, but not as an $A(infty)$ category (the Hom's are again the Hom's in the absence of holomorphic disks).

Bressler and Soibelman then continue with some speculation about algebraic ways to put in the discs, their basic idea being to find a functor $phi : hol(X) mapsto hol(X)$ such that $Hom(A,phi(B)$ agrees as an A-infinity category with the Fukaya category. By the dg category yoga, this should come in the form of a bi-module. They then explain roughly how to do it in the cotangent bundle case. Their vision seems to be that maybe one could have a purely algebraic approach to the Fukaya category via holonomic modules and thereby avoid some of the technical issues with moduli spaces of discs. This also gives an approach to the co-isotropic branes people have been talking about.

gr.group theory - Connections between properties of a group and local symmetries of its Cayley graph

I think there is very little that can be said, even when $Gamma$ is finite. Obviously
any automorphism of $Gamma$ that fixes the connection set of the Cayley graph gives
rise to an automorphism of the ball of radius $n$ about $e$; hence if the ball is
asymmetric then there is no automorphism of $Gamma$ that fixes the connection set.
(There can be automorphisms of the Cayley graph that do not arise from automorphisms of the group, but I cannot see that these are relevant to your question.)

To get a result of the type you are asking about, you would need to assume that $2n$
is larger than the girth (or your ball would be a "regular" tree, and tell you nothing
about the group). But now the balls of radius $n$ will be complex structures, and it
is getting difficult to even formulate a result.

The balls do provide information about the spectrum of the adjacency operator, but it
is not clear how to relate this to the group structure in general.

In the finite case there is a lot of work devoted to studying automorphism groups of Cayley
graphs, but I am not aware of any results of the type you describe.

the sun - How big would a disc have to be to blot out the sun covering 5% of the land mass of earth?

The angular diameter of the sun is approximately 30 arcminutes (roughly .0087 radians), or half of a degree.

The formula for angular diameter of a round object in radians is:
$$
delta = 2arctangfrac{diameter}{(2)distance}
$$

Given this, we should also think about position. Assuming we want to continually block the same spot on Earth we need to the orbital period to one year. (I'm not sure if there is a technical difference between orbiting the Earth once per year such that you're always between the Sun and Earth, and just plain orbiting the Sun)

According to this calculator, our sun blocker would need at a height of 2,152,050km to orbit Earth once per year. So, given that we can set up our formula with...

$$
.0087 = 2arctangfrac{diameter}{(2)2152050}
$$

...and the needed diameter of an object should come out to be about 18,722km in diameter. Given that, Mr. Burns's terrestrial solution is far more practical, though certainly a nice orbital one is far more devious. Being evil isn't cheap.

Edit:

This answer is a bit rough as it doesn't calculate for the 5% of the Earth's land mass stipulation. The answer which would meet this would depend on whether only an umbral shadow qualifies, or if a penumbral shadow would also suffice. I suspect probably the prior. The shadow would certainly reach the Earth, but probably not cover 5% of it, which is a shame considering its diameter is actually larger than Earth's. Regardless, it's not a very elegant solution.

formation - Is there some upper limit in the moon size distribution?

Small point, but Ganymede is slightly larger than Titan as moons are measured by their solid surface, though with atmosphere, Titan appears slightly bigger. The drawing is probably accurate, but slightly misleading.

It's worth adding that Titan is out-gassing so it may have, quite some time ago, been larger than Ganymede.

Source

According to wikipedia, the larger, close to Jupiter and Saturn moons are unlikely to have been captured moons.

https://en.wikipedia.org/wiki/Formation_and_evolution_of_the_Solar_System#Moons

Jupiter and Saturn have several large moons, such as Io, Europa,
Ganymede and Titan, which may have originated from discs around each
giant planet in much the same way that the planets formed from the
disc around the Sun.[76] This origin is indicated by the large sizes
of the moons and their proximity to the planet. These attributes are
impossible to achieve via capture, while the gaseous nature of the
primaries also make formation from collision debris unlikely. The
outer moons of the giant planets tend to be small and have eccentric
orbits with arbitrary inclinations. These are the characteristics
expected of captured bodies

and the 3 listed methods of moon formation are:

Co-formation
Impact
Capture.

The largest co-formation ratio is Titan to Saturn, about 1 to 4,000 in terms of mass. Ganymede to Jupiter, less than 1 to 12,000. The wiki article points out that ratios of 100 to 1 are likely giant impacts and you're unlikely to see that kind of ratio in co formation, so a conservative estimate, the largest co-formation you mgiht see might be around a few hundred to 1. If you figure the largest planets are about 12-12.5 Jupiters (more than that, they become brown dwarfs, though a cool brown dwarf orbiting a sun might get called a planet too at greater than 13 Jupiter masses), but lets say, 12 Jupiters. That's about 4,000 earth masses. If we look at a co-formation ratio of a few hundred to perhaps over a thousand to 1 as about the maxiumum, there's a possible ballpark answer of a handul of earth masses might be about as big as a co-formation moon is likely to get in a perfect formation situation, And those would probably be pretty rare.

Giant impacts are different. Pluto-Charon could be an impact generated dwarf planet/moon system and Charon is 1/8th the mass of Pluto. How likely that kind of impact is to happen on a super earth is hard to say, but theoretically a super-earth with a giant impact could have a large moon too - a bigger earth with a bigger impact and bigger moon, but there's problems here. Giant planets are likely to form close to the sun and the orbital sphere of influence gets smaller closer to the sun. Also, giant planets might tend to form thick gaseous atmospheres, retaining all their hydrogen and helium, so too big, it might begin to resemble a gas giant, which aren't as good at creating debris when impacted, so there's a few limiting factors here and we might be limited to, certainly several times larger than our moon, but perhaps not as large as co-formation moons, further from the sun.

Finally, moon capture. While there's no examples of this in our solar system, I see no reason why a large planet couldn't capture another planet, though you then have to determine whether it's a co-planet orbit, which I think a captured planet should be considered, or whether the captured object is a moon. - personally I think a captured planet that orbits a planet would be a co-planet system, so I don't like the capture as a method for getting as large a moon as possible, though I suppose it could be looked at either way.

We know almost nothing about moons in other solar systems (exomoons). The observations are just barely possible with current technology, but very difficult. That's likely going to change in the next few years, but for now we know virtually nothing in other solar systems, so, any present predictions on the largest possible moons would have to be modeled or estimated.

A few articles on exomoons:

http://www.astrobio.net/news-exclusive/new-exomoon-hunting-technique-could-find-solar-system-like-moons/

http://discovermagazine.com/2012/jul-aug/06-hunting-moons-outside-the-solar-system

http://www.space.com/25438-exomoon-around-alien-planet-discovery.html

http://phys.org/news/2014-10-exomoons-abundant-sources-habitability.html

http://blogs.scientificamerican.com/life-unbounded/has-an-exomoon-been-found2/

rt.representation theory - A relative Noether number for invariants

EDIT: Wrong definition of $betaleft(G,Hright)$ fixed. One of the results is open (i. e., I cannot prove it).

---

In "Finite Groups and invariant theory" (a paper in Malliavin's LNM #1478 which can be found on Springerlink and in other sources), Barbara Schmid defines a positive integer $betaleft(Gright)$ for any finite group $G$ and any field $K$ of characteristic $0$ as the smallest integer $k$ such that for any representation $V$ of $G$ over $K$, the invariant ring $Kleft[Vright]^G$ is generated by the invariants of degree $leq k$.

Here is my very first question: Is there a simple reason why $betaleft(Gright)$ is stable under field extension of $K$ ? I can prove that it is stable under field extension of $K$ for any single representation of $G$ (because being invariant under $G$ means satisfying a family of linear equations, and being generated by invariants of smaller degree means satisfying another family of linear equations), but the problem is that some representations only pop up when the field is sufficiently extended. The problem can be avoided by referring to Larry Smith's note which shows that the regular representation is the worst case (and it is, of course, defined over every field), but this altogether leads to a not particularly direct proof. (EDIT: There is a simpler proof: whenever $V$ is a subrepresentation of another representation $W$, the $betaleft(Gright)$ number evaluated at $V$ is smaller or equal to that evaluated at $W$. Now use some basic Galois theory.)

Schmid proves some nice results on $beta$:

• Noether's bound says that $betaleft(Gright)leq left|Gright|$. (Larry Smith showed that this actually holds in the $mathrm{char} Knmid left|Gright|$ case, but for me the $mathrm{char} K=0$ case is enough at the moment.)




• If $A$ is a commutative filtered $K$-algebra with filtration $A_0subseteq A_1subseteq A_2subseteq ...$ such that $A_0=K$ and such that $A$ is generated by $A_1$ over $K$, and if $G$ acts on $A$ by $K$-algebra automorphisms respecting the filtration, then the ring of invariants $A^G$ is generated (as a $K$-algebra) by the invariants in $A_{betaleft(Gright)}$. (This is trivial (by the very definition of $betaleft(Gright)$) whenever $A$ is the coordinate ring of a representation, but here we don't require this.)




• If $H$ is a subgroup of $G$, then $betaleft(Gright)leq left[G:Hright]betaleft(Hright)$.




• If $N$ is a normal subgroup of $G$, then $betaleft(Gright)leq betaleft(Gdiagup Nright)betaleft(Nright)$.

Now I think this $beta$ notion has a relative version. My question is: Is it any good? Has it been studied?

For a group $G$ and a subgroup $Hsubseteq G$, we define a number $betaleft(G,Hright)$ as the smallest integer $k$ such that for any representation $V$ of $G$ over $K$, and for any system $left(u_1,u_2,...right)$ of $K$-algebra generators of the invariant ring $Kleft[Vright]^H$, the invariant ring $Kleft[Vright]^G$ is generated by invariants which can be written as polynomials of degree $leq k$ in the variables $gu_i$ for $gin G$ and $iinleftlbrace 1,2,3,...rightrbrace$.

Similarly to Noether's bound, we can show that $betaleft(G,Hright)leq left[G:Hright]$.

I can prove the following facts (in a rather ugly way):

• If $A$ is a commutative filtered $K$-algebra with filtration $A_0subseteq A_1subseteq A_2subseteq ...$ such that $A_0=K$ and such that $A$ is generated by $A_1$ over $K$, and if $G$ acts on $A$ by $K$-algebra automorphisms respecting the filtration, and if the ring of invariants $A^H$ is generated (as a $K$-algebra) by the invariants in $A_1$, then the ring of invariants $A^G$ is generated (as a $K$-algebra) by the invariants in $A_{betaleft(G,Hright)}$.




• If $H^{prime}subseteq Hsubseteq G$ is a subgroup tower, then $betaleft(G,H^{prime}right)leq betaleft(G,Hright)betaleft(H,H^{prime}right)$.




• If $N$ is a normal subgroup of $G$, then $betaleft(G,Nright)=betaleft(Gdiagup Nright)$.

I am wondering whether any subgroup tower $H^{prime}subseteq Hsubseteq G$ must satisfy $betaleft(G,Hright)leq betaleft(G,H^{prime}right)$. And I am wondering whether the above results are any good, except of giving a rather involved generalization of the standard $beta$. Is there a reasonable way to compute $betaleft(G,Hright)$ ? (It sounds way harder than computing $betaleft(Gright)$. Can this relative $beta$ be of any use in proving results about the absolute $beta$ ?