neutron star - Birth and Evolution of Millisecond Pulsars

The standard model for the formation of millisecond pulsars is the recycling scenario.

The starting point of this model is a stellar binary system consisting of a neutron star and a secondary 'regular' star, where the neutron star is a normal pulsar with a rotational period of roughly $sim7$ seconds.

Because a neutron star is such a compact object, its gravitational influence is very large ($F_g propto M/R^2$). If the two stars are close enough, the outer layers of gas may be pulled of the companion star and flow towards the neutron star. All this matter will settle into an disk structure around the neutron star and start orbiting faster and faster as it draws closer to the surface ($v_{orbit} = sqrt{GM/R}$).

By the time this matter touches down on the neutron star surface its orbital period can be as fast as a millisecond, certainly significantly faster than the neutron star. This bit of motion (or more precisely, the angular momentum) is then transferred to the neutron star (this process is called accretion), causing it to rotate slightly faster (spin-up).

After some $0.1sim0.2 M_{odot}$ (where $M_{odot}= 2 times 10^{30}$ kg gives the mass of the Sun) of material is transferred to the neutron star, its rotational period will have decreased from a few second down to milliseconds.

The transfer of this much matter takes a very long time, roughly $10^8$ to $10^9$ years. After this time, and with all the redistribution of mass, the binary system is expected to widen which causes the mass transfer to stop. As the region around the neutron star clear up, the radio emission mechanism turns back on (this mechanism does not work during the accretion stage) and we are left with a millisecond pulsar.

From this point on the neutron star will behave like a regular pulsar: over the course of a very long time it will convert its rotational energy in radio emission and slow down until it becomes to slow to act as a pulsar.

As for the evidence for this model; such binary systems containing an accreting neutron star emit massive amounts of X-ray emission (hence called X-ray binaries) and have been known to exist since the 60's, long before the first millisecond radio pulsar was discovered (in 1982). This entire scenario was proposed quickly after, but could not be confirmed for a very long time.

It was only in the late 90's that a neutron star was found whose X-ray emission was pulsed at millisecond periods (see this, and this), confirming that these neutron stars are indeed spinning up. The final piece of evidence; a millisecond pulsar which switches back and forth between X-ray and radio emission was only found very recently.

sp.spectral theory - Parametrizing eigenvectors

I do not know a reference, but here is an easy argument. Consider the space $overline{M}$ of pairs $(A,lambda)$ where $A$ is a (self-adjoint) $N times N$ matrix and $lambda$ is a root of the characteristic polynomial of $A$. Over the open subset of $overline{M}$ lying above the operators with distinct eigenvalues, the projection onto the first factor is clearly a proper smooth map, with finite fibers; in other words, the space $overline{M}$ is a covering space over this open set. Since everything in sight is differentiable, the result that you want follows from the lifting of paths in the base. You need to choose an eigenvalue for the lifting to be unique, but you seem to know which one to choose!

Btw, you only need the eigenvalue $lambda$ to be simple for the same argument to work. In this case you can use the Implicit Function Theorem to lift the path, since the assumption on the simplicity of the root imply that the differential at the point is surjective.

Why do objects burn when they enter earth's atmosphere?

You'll often hear that it's because of friction, but that's often not the main factor. For larger objects it's more likely the pressure they create.

In both cases the reason is the enormous speed, often tens of kilometers per second. When a larger object enters the atmosphere at these speed the air in front of it gets compressed so much that it becomes extremely hot. (Think of pumping up a tire; you're also compressing air and you can feel the valve becoming hot.) The compressed air will often disintegrate the object in the air, and then the debris may burn because of the heat. This is exactly what happened to the asteroid above Russia last year: it exploded with an enormous flash in the air, and left little traces on the ground.

This happens on other planets as well, if they have a sufficiently dense atmosphere. In 1994 the comet Shoemaker-Levy crashed into Jupiter. It disintegrated before entering Jupiter's atmosphere due to the strong gravitation, but when the fragments entered the atmosphere they could easily be seen lighting up as they burned up.

edit
Remember the Space Shuttle? It had heat resistant tiles on the bottom of the craft to protect it from burning when it entered the atmosphere, even though its speed is only a fraction of a meteorite's speed when that enters the atmosphere.
During the last launch of the Space Shuttle Columbia some material from the external fuel tank damaged this heat shield, and upon re-entry the heat and the highly pressurized air under the craft could enter it, causing the craft to disintegrate and kill all crew.

ag.algebraic geometry - What is the algebraic closure of the field with one element?

There have been several questions on mathoverflow about the field with one element. Of course, such a field doesn't really exist and the discussion must fray sooner or later. So here is a different kind of answer.

Besides finite fields, which are 0-manifolds, there are only two fields which are manifolds, $mathbb{C}$ and $mathbb{R}$. There is a generalization of cardinality for manifolds and similar spaces, namely the geometric Euler characteristic. (This is as opposed homotopy-theoretic Euler characteristic; they are equal for compact spaces.) The geometric Euler characteristic of $mathbb{C}$ is 1, while the geometric Euler characteristic of $mathbb{R}$ is -1. In this sense, $mathbb{C} = mathbb{F}_1$ while $mathbb{R} = mathbb{F}_{-1}$.

It works well for some of the motivating examples of the fictitious field with one element. For instance, the Euler characteristic of the Grassmannian $text{Gr}(k,n)$ over $mathbb{F}_q$ is then uniformly the Gaussian binomial coefficient $binom{n}{k}_q$.

In this interpretation, $mathbb{F}_1$ is algebraically closed. It is also a quadratic extension of $mathbb{F}_{-1}$; the generalized cardinality squares, as it should.

history - What's the origin and culture of funny astronomical terminology?

"Big Bang" was coined by supporters of the alternative 'steady state' cosmological theory, essentially as a form of ridicule, but the name stuck, even after the advocates of 'steady state' were forced to concede that their theory (essentially that, at a grand scale, the universe 'looks' the same at all times and places) was flawed.

The discovery of the Cosmic Microwave Background (CMB) was essentially the nail in the coffin of steady state. CMB is well explained by the "big bang" but not by steady state.
Of course, what we think of as the "big bang" has also changed over the years.

(Fred Hoyle, the principal advocate of the steady state theory wrote some interesting 'hard' science fiction, but I am not aware of any link between this and the term 'big bang'.)

lo.logic - Decidability of matrix algebra

Although Peter Shor gave a proof of the undecidability (as he stated in a comment to the current question), here is another proof. An advantage of this proof is that it gives the undecidability of a very restricted version of the problem.

In an answer to my question, Agol told me that the following problem (which I called the Finite-Dimensional Word Problem for Groups (FWP) in the question) is undecidable by a result of Slobodskoi [Slo81].

Instance: A finite presentation of a group G and an element w of G as a product of generators and their inverses.
Question: Does every matrix representation of G map w to the identity matrix?

(The result in [Slo81] does not literally talk about this problem, but the result there implies the undecidability of this problem. See the answer by Agol linked above and also the discussion linked from my question.)

This problem can be easily translated into a special case of the current problem, which shows that the problem in question is undecidable even if we only allow a sentence of the form:

∃I.((∀X.IX=X)∧(∀X.XI=X)∧(∀X₁…∀X_n.(P₁(X₁,…,X_n)=I∧…∧P_m(X₁,…,X_n)=I→Q(X₁,…,X_n)=I)))

where I, X, X₁, …, X_n are matrix variables and P₁(X₁,…,X_n), …, P_m(X₁,…,X_n), Q(X₁,…,X_n) are products of one or more variables in X₁, …, X_n in some order with repetitions allowed. In particular, the problem is undecidable even if we do not allow scalar variables, vector variables, addition or conjugate transpose!

References

[Slo81] A. M. Slobodskoi. Unsolvability of the universal theory of finite groups. Algebra and Logic, 20(2):139–156, March 1981. http://www.springerlink.com/content/x880g1x17754hq83/

mg.metric geometry - Side-Angle-Side Congruence and the Parallel Postulate

On reflection, SAS tells me that Euclidean geometry has a strong semi-local homogeneity, in that every neighborhood of every point is isotropically isomorphic with some neighborhood of any other point --- once you find a good way to say "neighborhood", that is.

The parallel postulate, on the other hand, can be used to construct *canonical* isomorphisms of point(ed)-neighborhoods --- by parallel translation of course; but since the constructed isomorphisms are all parallel in a good sense, we don't get the isotropy structure without SAS. (edit/add:ed): in the other direction, SAS doesn't give any canonical isomorphisms, which is just as well because hyperbolic and elliptical space both have SAS, but not the parallel postulate. (end edit)

The related postulate that Euclid states properly --- that all right angles are equal --- only gives a pointwise isotropy; it doesn't help much for segments subtended by respectively equal segments at equal angles.

observation - How do astronomers find interesting events?

The answer to your question is yes. Many times discoveries in astronomy are serendipitous in nature - some of the most well-known discoveries fall into this category: the discovery of the cosmic microwave background, the discovery of some of the planets in our solar system, are but two examples of this.

However, many times they are not serendipitous. Astronomers plan to discover things - look at the Kepler satellite which finds nearby exosolar planets by detecting planetary transits. The Sloan Digital Sky Survey was a planned galactic survey which gave us new insights about the properties of galaxies as well as structure formation.

One also has to consider time scales. Many things astronomers care about have timescales which are millions or billions of years (merging clusters, the evolution of main sequence stars, etc..), however, some phenomena are on very short time scales, namely supernovae. Microlensing events can also be somewhat serendipitous in their discovery, though researchers still devise experiments to look for them (see OGLE).

It is certainly a combination of luck and planning. This answer was intended to be a bit vague, since the question doesn't specifically ask about one phenomenon. Software does play a big role in identifying and classifying objects and events in the universe. I can talk a little bit more about this in the context of gravitational lensing if people care to know about it.

the sun - The end of the Earth and human history

The end of the Earth and the end of a habitat that can support life are 2 different things. The Earth will not be able to support life as we know it for 4 billion more years.

The sun is slowly heating up, and estimates are that in 500 million years, the Earth will be too hot to support life. That being said, life is resilient, and who knows what evolutionary changes might occur that would allow life to survive 500 million years from now. After all, the Sun heating up will be such a gradual process that evolution could most definitely keep up with it.

Then there is the fact that in 500 million years, even if we aren't destroyed by cataclysm, humans as we are today won't exist. We will have evolved into something else that wouldn't even remotely resemble humans. Probably Morlocks.

Technology also might allow us to escape our solar system and colonize across the universe.

ac.commutative algebra - When do primes lift uniquely (provided they lift at all)?

Given a ring $R$, a prime ideal $mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $Rto S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) to Spec(R)$ to be injective as a map of topological spaces (completely disregarding any of our sheaf structure)? That is, every prime that lifts lifts to a unique prime? If this problem is intractable as-is, I can add more conditions, so please don't add answers unless they are actual answers.

How far apart is the dust in the Sombrero Galaxy's dust lane?

The sizes of dust particles varies enormously, and thus the density of dust particles depend upon which sizes we are talking about. The dust size distribution can be described by a power law with a slope of roughly $-3.5$; that is, the density of 0.1 µm particles of is $10^{-3.5}$ times that of 0.01 µm particles, or roughly 3000 times smaller.

Moreover, the discussion is complicated by the fact that we don't really have a formal definition of what is dust. But one "definition" takes the minumum size as a conglomeration of a few molecules, and the maximum size as "what has time to grow in a formation process" (these dust particles can later "stick together" and grow to form pebbles, rocks, asteroids, and planets. That's why any definition will be arbitrary).

In a dust-dense region of the interstellar medium, this results in a typical mean density of roughly one dust particle per cubic centimeter.

galaxy - Is the Hubble Ultra-Deep Field image representative of the entire sky?

The Hubble ultra-deep field image is a photograph of an area of sky equivalent to a 1mm x 1mm piece of paper held a metre away from the eye (one thirteen millionth of the entire sky). It contains an estimated 10,000 galaxies.

If galaxies were uniformly distributed, and the HUDF were representative of the entire sky, that would mean there were approximately 130 billion galaxies, which falls between estimates I have read of between 100 and 200 billion as the total number of galaxies.

However, my understanding is that galaxies occur in clusters, which would seem to imply that some areas of the sky should be more densely populated than others.

So is the HUDF representative of the entire sky, or are some areas much more and much less densely populated?

lunar - How significant and accessible is modern Chinese astronomy?

China is pretty well integrated into modern astronomy -- there are regular international conferences held in China (the most recent General Assembly of the International Astronomical Union was held in Beijing in 2012), lots of Chinese astronomers at other international conferences, Chinese participation in international observatory projects (the Joint Institute for VLBI, the Thirty Meter Telescope, the Square Kilometer Array, etc.), and so forth. I'm not sure I'd say they have any really significant facilities yet (with the possible exception of LAMOST), but they have very ambitious plans, as you've noted.

(ITAR has very little to do with astronomy except when it comes to a few things like nuclear physics codes produced by places like Los Alamos.)

I don't know how much international participation is anticipated for the FAST project, though I gather the instrument development has some involvement from British and Australian institutes, so they'd likely get some time.

Yes, results from the Chang’E-3 mission have been published in scientific papers, e.g.
http://www.sciencemag.org/content/347/6227/1226.abstract

There are several Chinese astronomy journals (e.g., Acta Astronomica Sinica; Research in Astronomy and Astrophysics), but like most "national" journals they aren't that important internationally.

Is C/2012 S1 (aka ISON) the fastest comet on record to-date?

ISON is a sungrazer. But there were legendary comets like the Kreutz Family of Sungrazing comets. The Great comet of 1843 is a member of the kreutz family and it passed even more close to the sun at perihelion. It was a time before SOHO was deployed and there are not accurate observations of the speed of the Great Comet.

Since, it was travelling closer to the sun, considering the angular momentum created by the gravity of the sun, The great comet might have been a bit faster than the ISON.

You cannot just term, speed of the comet. Comets travel with higher speeds due to the sun's gravitational pull and can be accelerated to extreme speeds when they catapult around the sun. However, when the comet is moving away from the sun, it'll be travelling slower because of the gravitational pull of the sun.

mg.metric geometry - Triangles, squares, and discontinuous complex functions

With interior: yes. Fix a sequence of squares $Q_1subset Q_2subsetdots$ whose union is the entire plane. Then arrange a map $g:mathbb Rtomathbb R^2$ such that, for every nontrivial segment $[a,b]subsetmathbb R$, its image is one of the squares $Q_i$. To do that, construct countably many disjoint Cantor sets so that every nontrivial interval contains at least one of them. Then send every Cantor set $K$ bijectively onto $Q_n$ where $n$ is the minimum number such that $Kcap [-n,n]neemptyset$. Send the complements of these Cantor sets to a fixed point inside $Q_1$. Then define $f(x,y)=g(y)$.

(This is a detailed version of gowers' answer.)

UPDATE

Without interior: no. Take any triangle $T$ and consider its image $Q$ with vertices $ABCD$. There is a side $I$ of $T$ whose image has infinitely many points on (at least) two sides of $Q$. If these are opposite sides, say $AB$ and $CD$, the image of any triangle containing $I$ must stay within the strip bounded by the lines $AB$ and $CD$. And if these are two adjacent sides of $Q$, say $AB$ and $AD$, the image of any triangle containing $I$ stays within the quarter of the plane bounded by the rays $AB$ and $AD$. In both cases, the images of the triangles containing $I$ do not cover the plane, hence the map is not onto.

gravity - What is the time/size/rigidity ratio for a space object to become mostly round?

Based on observable data, I assume that there is a relationship
between the size, rigidity and the passage of time; that will result
in all objects subject to only their own gravitational influences
(Given: no body is ever truly uninfluenced by others) becoming
spherically shaped (round).

That's called hydrostatic equilibrium. That's one of the main factors that distinguish a dwarf planet from a smaller random piece of rock out there.

https://en.wikipedia.org/wiki/Dwarf_planet

"A dwarf planet is an object the size of a planet (a planetary-mass object) but that is neither a planet nor a moon or other natural satellite. More explicitly, the International Astronomical Union (IAU) defines a dwarf planet as a celestial body in direct orbit of the Sun that is massive enough for its shape to be controlled by gravity, but that unlike a planet has not cleared its orbit of other objects."

There is no fixed size limit, because it depends on the composition.

https://en.wikipedia.org/wiki/Hydrostatic_equilibrium#Planetary_geology

"It had been thought that icy objects with a diameter larger than roughly 400 km are usually in hydrostatic equilibrium, whereas those smaller than that are not. Icy objects can achieve hydrostatic equilibrium at a smaller size than rocky objects. The smallest object that appears to have an equilibrium shape is the icy moon Mimas at 397 km, whereas the largest object known to have an obviously non-equilibrium shape is the rocky asteroid Pallas at 532 km (582×556×500±18 km). However, Mimas is not actually in hydrostatic equilibrium for its current rotation. The smallest body confirmed to be in hydrostatic equilibrium is the icy moon Rhea, at 1,528 km, whereas the largest body known to not be in hydrostatic equilibrium is the icy moon Iapetus, at 1,470 km."

So the range of transition between hydrostatic equilibrium and non-equilibrium is between approx 400 and 1500 km diameter and depends on a number of factors such as composition. There is no simple formula.

Dwarf planets are a good related topic, and I used them in the discussion, because many of them are above the limit of HE but close to it. But any cosmic body is subject to the same laws. Planets proper, moons, stars, etc - these all may be placed above or below the HE limit, depending mostly on size.

E.g. the Earth is clearly above the HE limit. But probably all comets are below it. A rock sitting in the dirt below your feet is clearly below it.

Looking for reference on gauge fields as connections.

Can anyone give me references where I would see a detailed exposition of how to translate gauge field theory as known to physicists into the language of connections. I am looking for a detailed exposition on the mathematical formulation of Yang-Mills field theory. Something which might also give an exposition about Chern-Simons theory and the related whole bag of what get called "topological actions"

I had read a nice long discussion on the geometrical formulation of gauge field theory in a post at Terence Tao's blog namely this article and also probably something on Secret Blogging Seminar (but I can't locate that link)

Along similar lines I had seen a very old book by Atiyah and Hitchin on this.

I would like to know what books/expository papers on this are read by graduate students today when they try entering this field?

Also advanced references on the topic would also be helpful.

nt.number theory - Polynomial representing prime numbers

No. Any such polynomial would have the property that any of its restrictions $f(x)$ to one variable consist only of primes, but this is easily seen to be impossible, since if $p(a)$ is prime then $p(k p(a) + a)$ is divisible by $p(a)$. (Even accounting for the coefficients in $mathbb{Q}$ is straightforward by multiplying by the common denominator and using CRT; in fact, we can show that given an integer polynomial $q(x)$ and a positive integer $n$ there exists $x_n$ such that $q(x_n)$ is divisible by $n$ distinct primes.)

However, there do exist multivariate polynomials with the property that their positive integer outputs consist of the set of primes. See the Wikipedia article.

algebraic k theory - Quillen's Morphism Inverting Functors

Proposition 1 is extremely straightforward to prove (provided you have some facts like the quillen adjunction between SSet and CGWH). Sing(|S|) gives you a simplicial set where all of the edges "forget" their direction, and when you apply the inverse of the nerve functor, you get back a copy of C with all of its arrows as isomorphisms. Covering spaces are equivalent to (etale) bundles (of sets) on a topological space, which by a theorem in Mac Lane (Sheaves in Geometry and Logic) is equivalent to taking sheaves on the space, so by unraveling these equivalences, you get your result. The last equivalence is probably one you're familiar with as the espace 'etal'e. (While in general, the nerve functor does not have an inverse, the nerve of a category has some nice properties that make the total singular complex (the $Sing$ functor) pull back intact, modulo directedness of edges. If you think about the actual graph of the nerve of an ordinary category, it's not hard to see why this is true. This is precisely because the geometric realization "forgets" some information.)

The construction you're describing is generalized by a functor in HTT called the unstraightening functor, which you can read about in HTT Ch 2.2. With a number of more sophisticated results, we can generalize the adjunction between $Sing$ and $| cdot |$ to a Quillen equivalence between SSet-Cat and CGWH-Cat.

HTT is Higher topos theory by J. Lurie.

nt.number theory - Riemann hypothesis generalization names: extended versus generalized?

Hello,

I also agree that the literature is not quite consistent on this topic. I tried to find a published reference on this question and found the following article:

http://www.springerlink.com/content/v563192510820t34/

which is Chapter 5 of a book "The Riemann hypothesis : a resource for the afficionado and virtuoso alike", by Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller, published by the Canadian Mathematical Society.

In the reference above, they use GRH for Dirichlet L-series (section 6.2), and ERH for Dedekind zeta functions (section 6.5). However, to make things more complicated:

1) They mention (in section 6.3) that ERH may be referring to the conjecture for L series of the form
$$sum_{n=1}^infty frac{left(frac{n}{p}right)}{n^s}$$
where $p$ is a prime and $left(frac{n}{p}right)$ is the Legendre symbol. In fact they call this version ERH, and the one for Dedekind zeta functions is called another extended Riemann hypothesis.

2) They mention that the "Grand Riemann hypothesis" (which I had never heard of) refers to L functions of automorphic cuspidal representations.

Alvaro

set theory - Preservation of properties under countable-support iterations

As I indicated in my comment, there will be many silly
examples if we interpret your notion of $Phi$ is
preserved by an iteration $P_delta$'' to mean that
whenever all the proper initial segments of the iteration
$P_alpha$ for $alpha<delta$ have $Phi$, then $P_delta$
has $Phi$. The reason is that if $delta=beta+1$ is a
successor ordinal, then this way of stating the property
doesn't place any requirement on the stage $beta$ forcing.

For example, let $Phi$ hold of a forcing notion $P$ if and
only if $P$ is equivalent to countably closed forcing. This
is preserved by countable support limit ordinal iterations,
since if every initial segment $P_alpha$ for
$alpha<delta$ is countably closed, then so is the whole
iteration $P_delta$, provided $delta$ is a limit. But if
$Q$ is any non-countably forcing notion, consider the
iteration of length $1$, forcing with $Q$ at stage $0$.
Since $P_0$ is trivial, it is countably closed, but the
whole iteration $P_1$ is not, so it violates your property
in the ignoring-the-last-stage manner that you have stated
it.

Another silly example: let $Phi$ hold of proper forcing.
This is preserved by countable support limits, by Shelah's
theorem, but it is not preserved by finite iterations in
the ignoring-the-last-stage sense that you describe.

So I think you don't really want to ignore the last stage
like that. Perhaps you are interested in something like
this: a property $Phi$ of forcing notions (respecting the
equivalence of forcing) is preserved by an iteration of
length $delta$, if whenever each stage $Q_beta$ of an
iteration is forced over $P_beta$ to have the property,
then $P_delta$ also has the property. For example, proper
forcing is preserved by countable support iterations. The
countable chain condition is preserved by finite support
iterations. Countably closed forcing is preserved by
countable support iterations.

Now, you want to ask whether there is a property that is
preserved by all limit iterations, but not by some finite
iterations.

If trivial forcing has property $Phi$, which is the
typical situation (e.g. trivial forcing is c.c.c., proper,
closed, cardinal-preserving, GCH-preserving, etc. etc.),
then the answer is no, again for a silly reason. Suppose
that $P_n$ is a finite iteration that witnesses that $Phi$
is not preserved, so that each stage $Q_m$ of $P_n$ has
$Phi$, but the iteration $P_n$ itself does not. Now let
$P_omega$ simply continue the forcing with trivial forcing
out to stage $omega$, and use countable support. Thus,
every stage of $P_omega$ has property $Phi$, but the
iteration altogether is forcing equivalent to $P_n$, which
does not have property $Phi$.

You can tweak this example to make an iteration $R_omega$
of length $omega$, which is nontrivial at every stage in
the weak sense that no $R_m$ forces with $1$ that $Q_m$ is
trivial, but such that anyway $R_omega$ is forcing
equivalent to the original $P_n$. For example, let the
first stage of forcing generically choose a natural number
$k$, which is interpreted as the place where the actual
iteration $P_n$ will begin. Every stage $m$ has a nonzero
Boolean possibility of being in the nontrivial part of the
iteration, and so no stage of this forcing is forced by $1$
to be trivial.

Perhaps a more interesting question would be to inquire:

Question. Is there a first order property that holds
in all the forcing extensions by limit length countable
support iterations, but not in all forcing extensions by
finite length forcing iterations?

That is, we inquire not about properties of the forcing,
but rather about properties of the universe to which the
forcing leads. In this case, to avoid the silly kind of
example above, let us insist that every stage of the
iteration forces that the next stage of forcing is
nontrivial below every condition.

In this case, the answer is Yes. One easy example is to let
$Phi$ be the assertion: ``the universe is not a forcing
extension of $L$ by adding one Sacks real.'' If I have an
iteration $P_delta$ of any limit length $delta$,
nontrivial at every stage, then it cannot be that
$P_delta$ is equivalent to adding a Sacks real, since that
forcing admits no intermediate models, such as the forcing
extensions arising during the iteration. But the iteration
over $L$ of length $1$, adding a Sacks real at that stage,
fails property $Phi$.

More generally, however, for any class $Gamma$ of forcing
notions, the statement $Psi =$ ``the universe is
obtained by forcing over $L$ with forcing of a certain type
$Gamma$'' is first order expressible. So if you have any
class $Gamma$, such as the class of forcing that is
equivalent to countable support limit length iterations,
the statement $Psi$ will be true exactly in the forcing
extensions of $L$ you are inquiring about, and only those.
This idea generalizes to forcing over $V$, if you allow
parameters, although this is a bit more subtle.

the moon - How do I deduce my latitude and longitude from N obervations of occultations from the same place?

I think it would be an interesting exercise to measure the latiude and longitude of my house using N occultation timings. I would like to see how precisely I can measure it using that technique, and compare it with the GPS readings from my phone.

I'm looking for a reference which will tell me how to do the calculation of inferring my position from occultation timings, as accurately as possible.

nt.number theory - Is there an explicit example of a complex number which is not a period?

I haven't heard the latest about periods, but I want to point out a potential fallacy here. It's said very often that the easy proof of the existence of transcendental numbers (on cardinality grounds) is non-constructive. But, that's false! It is constructive. Given pen, paper and lots of time, I could extract from that argument the decimal expansion $0.a_1 a_2 ldots$ of the transcendental number that the proof constructs.

See, for example, these comments of Joel David Hamkins.

I suspect that the same is true for periods: that there's an effective enumeration of them, so there's an algorithm for generating the decimal digits of a number that isn't a period.

Non-commutative algebraic geometry

In fact, I revised some of problems in Chater II Scheme theory in Hartshorne using Kontsevich-Rosenberg's machine. I have to notice, what you should deal with is probably module category over noncommutative ring. In noncommutative algebraic geometry, this is just category of quasi coherent sheaves on noncommutative affine schemes. But I did not restrict myself to noncommutative ring case. I try to do it in general noncommutative scheme, say a Grothendieck category or an abelian category. Noticed that, one can take grothendieck category as category of quasi coherent sheaves on quasi compact and quasi separated "would be scheme". So one should consider category of grothendieck category as category of "space" and morphism between spaces as iso class of inverse image functor. Rosenberg developed algebraic geometry in this 2-category. He introduced various spectrum for various destination. I should mention, spectrums for abelian category in his sense coincides with prime spectrum of a commutative ring when you take module category over commutative ring. In fact, one can define Zariski topology on this 2-categories using a family of conservative(faithful)exact localization functors(Serre subcategory in dual language). Then one can introduced the associated topology on the spectrum of abelian category. Then one can continue to introduce the "fiber" at each point of spectrum as stack of local category.(as fibered category) This is called geometric realization of an abelian category or Grothendieck category. Then one can take category of quasi coherent sheaves on this fibered category. At last, we get reconstruction theorem for noncommtative scheme. If we take the original category as quasi coherent sheaves of quasi compact(or not in general)quasi separated commutative scheme. Then we get reconstruction theorem for commutative scheme which means commutative algebraic geometry can be fully embedded into noncommutative algebraic geometry.

Beacuse of this "Justify" theorem, we can develop various notions correspondent to commutative algebraic geometry. One can define noncommutative affine scheme (can be seen as category with projective cogenerator, then by Gabriel cheating theorem, equivalent to a module category). One can also define affine morphism, open/closed immersion/coimmersion(for the motivation of representation theory) picard group, vector bundles

One can also define differential operators in abelian category, monoidal category(for motivation of representation theory of quantum group and math phy), in particular, Noncommutative D-modules on noncommutative space, in particular quantum D-module on quantized flag variety. (I think this is related to the problem mentioned by siegels).

As is well known to all, Beilinson Bernstein's framework aiming to representation theory lives in triangulated category. Actually, there is indeed whole abelian picture developed mainly by Rosenberg and Lunts-Rosenberg-Tanisaki later.

In fact,for most(I think it should be all)problems in Hartshorne (facts in commutative algebraic geometry)has correspondence version in noncommutative algebraic geometry(in paticular, what you mentioned, noncommutative ring)

There is indeed noncommutative flat descent theory in Konstevich-Rosenberg's work. I think the more accurate name should be categorical flat descent theory(Beck's theorem)

One more comment: What I mentioned above is ONE framework they developed.(Mainly for representation theory). There is ANOTHER framework introduced by them base on presheave view point(proposed by Gabriel-Grothendieck). They develop algebraic geometry in this view point which is NOT equivalent to the CATEGORICAL GEOMETRY I mentioned above in general. They coincides in affine case and then go to completely different direction.The mainly motivation for this view point is from Konstevich,he wanted to consider noncommutative grassmannian which might be helpful in understanding M-theory in Physics. Along this direction, they define noncommutative algebraic space, stack(DM and Artin) and so on.

last comment: One guy mentioned above, the category of rings did not have good property as commutative ones.But I guess this is not a very big deal. Rosenberg define so called right exact category(say category of rings, category of affine schemes,category of vector bundles). He developed whole homological algebra in this settings and Universal algebraic k theory, algebraic cycles. chow group and so on

I am sorry I have to go back to work instead of typing here. There are various noncommutatve algebraic geometry. If you are dealing with projective scheme, you might be interested in work of Artin's School on NC projective geometry

Several other comments: we have the notion of locally noetherian category whose objects is generated by noetherian object. For example, category of quasi coherent sheaves on noetherian commutative scheme is a locally noetherian category.We can play game in this setting. Then we can get whole commutative algebraic geometry of noetherian scheme

nt.number theory - How to factorize X^n - 1 in Z/pZ?

If you just need a quick answer (to decide if something else is going to work how you need), then you can do this with Wolfram|Alpha. Go there:
http://www.wolframalpha.com/
and input "factor x^26-1" and press the "equal" button. It'll show some info about the polynomial, including the factors mod 2. In many boxes, there's a link for "Show More". Press the one attached to the factors over GF(2), and it'll show you the factors over GF(3). In this case, you get
$$(x+1) (x+2) (x^3+2 x+1) (x^3+2 x+2) (x^3+x^2+2) (x^3+x^2+x+2) (x^3+x^2+2 x+1) (x^3+2 x^2+1) (x^3+2 x^2+x+1) (x^3+2 x^2+2 x+2).$$

Annoying to have "2" instead of "-1" in GF(3), but that's the price of having a machine do your work for you.

star - Metallicity of Celestial Objects: Why "Metal = Non-metal"?

There is no 'history' behind it. Stellar physics is less than a 100 years old. Thus, it is not a terminology that came to be due to some anecdotal reason. This is the way it has been since the start. Why?

Because we don't care. Why, really?

Hydrogen and Helium are more important and abundant, so we need something to measure the others, which can be grouped under one name due to their low significance to us. One fine day, one fine astronomer was sitting in his balcony, drinking coffee, and he thought, "What do I call a group of elements most of which are metals?" Surprise surprise, He decided to call them metals.

I am not kidding with you, although it may have sounded like that. But that is how it is with astronomical terminologies. I am assuming you are new to the field, because this is not the only crazy thing we've done to express how much crap we give in for terms and notations and units.

linear algebra - Endomorphisms of vector bundles

I can't help but repeat the earlier answers but in my own way:

This is really just a question about linear algebra. It is worth remembering that a vector bundle is just a parameterized family of vector spaces (I like to call differential geometry "parameterized linear algebra"). So anything you can do naturally to a vector space, you can do to a vector bundle.

So to extend the definition of a determinant of an endomorphism of vector spaces to one for an endomorphism of vector bundles, you just need a good natural definition of determinant:

An endomorphism of a vector space $V$ naturally induces an endomorphism of $Lambda^nV$, where $n$ is the dimension of $V$. Since $Lambda^nV$ is 1-dimensional, an endomorphism of $Lambda^nV$ must consist of multiplication by a fixed scalar. That scalar is the determinant of the endomorphism.

The extension to vector bundles then becomes obvious, because you just do the same thing to each fiber.

rt.representation theory - What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?

$mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $mathbb{C}$, the isomorphism classes of $n$-dimensional representations of $mathbb{Z}$ are precisely the conjugacy classes of elements of $GL_n(mathbb{C})$; in particular the indecomposable ones are given by Jordan blocks. The representation corresponding to a Jordan block of size $n$ with diagonal entries $lambda$ has the same character as, but is not isomorphic to, the representation corresponding to a diagonal matrix with entries $lambda$. What is an abstract way to describe this relationship that does not refer to characters? (I am mostly interested in how to describe the relationship between an indecomposable representation and a sum of one-dimensional representations with the same character.)

Motivation

A natural way to study an (associative, unital) algebra $A$ over $mathbb{C}$ (to fix ideas) is to study the category $text{Rep}(A)$ of, say, finite-dimensional representations of $A$. However, if $A$ happens to be commutative and Noetherian, then we do something different: we privilege the one-dimensional representations and call them points, and then we analyze higher-dimensional representations as certain structures on the points. What abstract relationship, from the representation-theoretic perspective, between the one-dimensional and higher-dimensional representations lets us do this?

co.combinatorics - Turning Trees into 1-dimensional curves

In the theory of Young diagrams, it's a common move to turn the young diagram 45 degrees and consider it's profile as a function. [ Note for editors: it would be nice to find an image of this... ]

Are there ways to build 1-d (probably piecewise linear) curves from trees or from binary trees or ternary trees?

rt.representation theory - About localization theorem for affine Lie algebra?

Here is my question: how to define global section functor from D-module on affine flag variety to representation of affine Lie algebra?

Let's me explain the difficulty: it seems there doesn't exist global definition of D-module on ind-scheme. For affine flag variety, it is a union of finite dimensional subvarieties, and usually we can't make them smooth. We should think of a D-module on a singular variety as a usual D-module on big smooth space which supports on this singular variety. On the other hand, the global sections of D-module depends on the embedding of singular variety to the other smooth One.

I really don't know how to think of global section functor of D-module on affine flag variety, so I don't know how to formulate the localization theorem.

Maybe I should look at Frenkel-Gaitsgory's paper, but I'm afraid it is a question before reading their papers.

Moreover, I would like to know what is the status of localization theorem for affine Lie algebra?
1. at Critical level
2. at noncritical level

it.information theory - Complete formulas book for Communication System engineer

I'm looking for a formulas book.

I'm currently student in Communication Systems and we have several courses involving mainly complex analysis, fourier analysis, signal processing, information theory and sometimes other principles and I need a lot of books for all these formulas.

Does someone knows a book with these formulas inside ?

It doesn't need to have demonstration or long explanation, I just need something to replace the stack of books I have.

The best one I found so far is the Gieck, but there are more than a half of the book I'll never use. I want something more specific for signal processing and information theory.

rotation - Earth and ferromagnetism

Earth's core is a giant liquid iron ball actually. If I know well, the magnetic field of our planet (that protects the surface from some particles comes from the Sun) can exist because as Earth rotates, the liquid iron core also rotates, and it generates a magnetic field.

However, ferromagnetism has strict conditions. How can the liquid and hot core still be ferromagnetic? How is it possible for Earth to have a magnetic field generated by a body that is not even solid?

Is there any evidence that the Gas Giant planets in our solar system are experiencing orbital migration?

There is no direct evidence of change of orbits of the gaseous planets of the Solar System. But one could wonder about the stability of the Solar System, and if such events could happen again.

Theory:

The Solar System is a chaotic system, as most of gravitational systems involving $N$ bodies. The KAM theorems (for Kolmogorov, Arnold and Moser, the three authors of the theory) show that, for non-degenerate Hamiltonian systems, quasiperiodic trajectories exist, that could ensure the stability of the system. But unfortunatly the KAM theorems cannot be applied to the Solar System, because the planets are too massive.

Simulations:

It leaves us with numerical simulations, that show that the Solar System is indeed chaotic, and that planetary collisions, ejection and migration can happen. But if it is true for the telluric planets, it is not the case for the gaseous planets: one reason could be that they are not much perturbated by the inner planets.

Sources:

rt.representation theory - Relating Deligne-Lusztig virtual representation characters to Green functions

I have 2 questions - the first is what the title refers to, and the second is something I want a reference on (I thought I'd include them in one post since they are very strongly related). Sorry this post is a bit long, I tried to put as much as detail as I could ..

$1$-st question: I'm interested only in the group $GL_n(F_q)$. In Carter's book "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters", in Chapter 7 "The generalized characters of Deligne-Lusztig", the construction of the virtual representations $R_{T, theta}$ as alternating sums of $l$-adic cohomology of Deligne-Lusztig varieties is given in some details, and a series of formulae are proved in the chapter about these ($T$ a torus, and $theta$ a character of $T^{F}$). It says that if $theta in widehat{T^{F}}$ is in general position, then $pm R_{T, theta}$ is irreducible. The following formula is given (also in http://en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory), where $g=su=us$, $s,u$ being the semisimple and unipotent parts, and $Q_{T}(u) = R_{T, 1}(u)$, $C^{0}(s)$ being the identity connected component of the centralizer of $s$, and $F$ the Frobenius endomorphism.

$ R_{T, theta}(g) = frac{1}{ | C^{0}(s)^{F} |} sum_{ x in G^{F}, x^{-1}sx in T^{F} } theta ( x^{-1} s x) Q_{x T x^{-1}}^{C^{0}(s)} (u) $

The book then says that $Q_{T}(u)$ is a Green function, depends only on the torus (I understand it will not change if we conjugate the torus in $G^F$ either so essentially corresponds to an element of $S_n$ for the group general linear group of size $n$, which is what I'm most curious about; unless I'm mistaken). The book does not give an explicit formulae for these $Q_{T}(u)$, but it does give orthogonality relations and such - explicit formulae is what I"m looking for:

Question: What's an explicit formulae for these $Q_{T}(u)$? How does this relate to the Green function that I've been studying from in Macdonald's book "Symmetric Functions and Hall polynomials", in the chapter "Characters of $GL_n$ over a finite field" -i.e., how do I express the character $ pm R_{T, theta}$ as a sum of the irreducible characters described by Green functions in Macdonald's book (or a single irreducible character in the case where $theta$ is in general position)?

In that book, I've learnt that the polynomials correspond to symmetric functions $S_{lambda}$, via a correspondence that maps $A$, the sums of the representation ring of for all $n$, to $B$, an algebra generated by elementary symmetric functions in independent variables $X_{i,f}$ ($f$ ranges over all irreducible polynomials in $mathbb{F}_{q}[t]$). I'm sorry I'm being a bit vague right here - it would take pages to define precisely all the notation that Macdonald uses in his book; feel free to work with any alternative explicit definitions of these Green functions (but please include a reference so I know where to look it up).

$2$-nd question: I have looked through Carter's book and Digne&Michel's book on the same topic, but I have been unable to find a reference which gives the representing matrices for these virtual representations $pm R_{T, theta}$ of these finite Lie type groups (the fact that they are defined with alternating sum complicates matters somewhat). I'm not so interested in the entries of the representing matrices as such, just a construction for the module which enables you to find the representing matrices. Can anyone suggest a good reference for this? The closest I can find is Lusztig's original book "Characters of reductive groups over finite fields", where it mentions that $l$-adic intersection homology can be used as a substitute (this was from what I can see in googlebooks preview); but I hear this book is horrible to learn from, and I'm not entirely certain if what's given there is what I'm looking for (I don't have a copy of the book at present).

at.algebraic topology - Finding cocycles that square to zero

The triple product $langle x,x,xrangle$ has to contain zero.

Indeed, if $a$, $b$, $c$ are odd cohomology classes such that $ab=0$ and $bc=0$, to compute the triple product $langle a, b, crangle$, one picks representative cocycles $alpha$, $beta$ and $gamma$, then picks cochains $delta$ and $eta$ such that $alphabeta=ddelta$ and $betagamma=deta$, and then observes that $tau=alphaeta+deltagamma$ is a cocycle. Then $tau$ is a representative of $langle a,b,crangle$ in an appropriate quotient of the cohomology group which contains the class of $tau$.

In your case, suppose we can represent the class $x$ by a cocycle $xi$ such that $xi^2=0$. Then if we take $a=b=c=x$, we can take $alpha=beta=gamma=xi$ and $delta=eta=0$, so that $tau=0$, that is, $0in langle x,x,xrangle$.

In fact, all Massey products $langle x,x,dots,xrangle$ ("Massey powers"?) have to be zero, by a similar computation---see the book by McCleary on spectral sequences, chapter 8, for a speedy description of these.

amateur observing - Periodic behavior of Venus

Assumed a Venus year is 0.615 198 Earth years.

0.615 198 yr / (1 - 0.615 198) = 1.598739 yr = 583 days 11 hours 24 minutes for one period in mean, or roughly 19.5 months.

That's 1.598739 periods for Earth and 2.598739 periods for Venus:
1.598739 * 365.256 days = 583.95 days = 2.598739 * 224.705 days

The relative accuracy should be somewhere near 1e-5.

Since both orbits are elliptical the actual period oscillates around that mean value.

linear algebra - Number of invertible {0,1} real matrices?

As Michael noted, the conjectured bound for the probability a random $(0,1)$ matrix is singular is conjectured to be $(1+o(1)) n^2 2^{-n} $. This corresponds to the natural lower bound coming from the observation that if a matrix has two equal rows or columns it is automatically singular.

The best bound currently known for this problem is $(frac{1}{sqrt{2}} + o(1) )^n$, and is due to Bourgain, Vu, and Wood. Corollary 3.3 in their paper also gives a bound of $(frac{1}{sqrt{q}}+o(1))^n$ in the case where entries are uniformly chosen from ${0, 1, dots, q-1}$ (here the conjectured bound would be around $n^2 q^{-n})$

Even showing that the determinant is almost surely non-zero is not easy (this was first proven by Komlos in 1967, and the reference is given in Michael's Sloane link).

Is there a theoretical limit on telescope's resolution?

The absolute limit of a telescope resolution is given by diffraction.
No matter how perfectly built and aligned is a telescope, you cannot resolve angles smaller than
$$theta propto frac{lambda}{D}$$
where $lambda$ is the wavelength of interest and $D$ is the diameter of the telescope. This is why a number of millimeters and radio telescope (and also some -antennas) are huge.

The first figure here shows the basic principle. Imagine that you can divide what you observe in tiny squares and consider each one as a point like source. Each one will generate a diffraction pattern when passing into the telescope and if two points are too near, you cannot distinguish between them.

Wisely you put your telescope in space: turbulence in the atmosphere degrade the signal, and the best/biggest telescopes have hard times to go below $0.5 arc seconds$

Interferometry comes to the rescue increasing $D$ from the telescope size to the distance of two (or more) telescopes (called baseline). Interferometry has been used in radio astronomy since decades.
From what I hear optical interferometry is much much more complicated and as far as I know the only large scale attempt is the VLTI project.

So you could imagine to have a constellation of relatively small telescopes spread over hundred thousands or millions of kilometers. But this have the huge problem that you have to know the position and timing of every one of them with an impressive precision (my guess is that the precision in position is of the order of $lambda$).

---

The other problem is light collection. If you want to see something very faint you have two options: 1)you observe for a lot of time or 2) you build a bigger telescope (6 to 40 meters in diameter).
And here also the largest baseline interferometers cannot do much, as the amount of light that they collect is just the sum of the light collected by the single telescopes.

---

To conclude: to observe Pluto or Ceres with high enough accuracy you would need a large number of large space telescopes very far away one from the other with perfect telemetry. It's far easier and cheaper to got there to take pictures.

rt.representation theory - Strata for the nullcone (from Hesselink's paper)

Example of how the multiplicity can be greater than $1$: Let $mathbb{G}_m$ act on $mathbb{A}^2$ by $t: (x,y) mapsto (t^2x, t^3 y)$. Let $v$ be any element of $mathbb{A}^2$ not on the coordinate axes, for example, $(1,1)$. So $mathbb{G}_m$ maps to $mathbb{A}^2$ by $t mapsto (t^2, t^3)$. This extends to a map $mathbb{A}^1 to mathbb{A}^2$, given by the same formula.

I'm not sure what your favorite definition of multiplicity is. Mine is to consider the map of rings in the opposite directions: $k[x,y] to k[t]$ by $x to t^2$, $y to t^3$. The point $(0,0)$ corresponds to the ideal $langle x,y rangle$; this maps to the ideal $langle t^2, t^3 rangle = langle t^2 rangle$ in $k[t]$. Then, by definition, the multiplicity of $f^{-1}(0)$ is the dimension of $k[t]/langle t^2 rangle$, which is $2$.

The geometric point here is that the map $mathbb{A}^1 to mathbb{A}^2$ has vanishing derivative at $0$, so the preimage of zero has multiplicity greater than $1$.

---

You also asked about a Weyl invariant form on $mathrm{Hom}(mathbb{G}_m, T)$, where $T$ is a maximal torus of $GL_n$. Every hom from $mathbb{G}_m$ to $T$ is of the form $t mapsto (t^{w_1}, t^{w_2}, ldots, t^{w_n})$, where the coordinates on the right hand side are the entries in your diagonal matrix. Let's call this $s(w)$.

The obvious choice of Weyl invariant form is
$$langle s(v), s(w) rangle = sum v_i w_i.$$

Technically, I note that $sum v_i w_i + c (sum v_i) (sum w_i)$ would also be Weyl invariant, for any integer $c$. No one ever makes the choice of using nonzero $c$, but I don't know a general principle which would exclude it.

the sun - Given a date obtain latitude and longitude where is the sun zenith

Searching is easy to find terminator line (frontier between day and night) or the position of the sun in the sky given a position on the earth and a time; but I can't find how to obtain where is the zenith of the sun given a date (and time).

I need to obtain the center of the illuminated zone of the earth (latitude and longitude) at a given time. (Well, actually I need the opposite, the Nadir, but with one you can calculate easily the other).

Any knows the function?

Thank you.

homotopy theory - Do the signs in Puppe sequences matter?

There is a specific sort of situation I know about where that sign matters. Suppose you have $f:X rightarrow Y$ and $g:Z rightarrow W$ cofibrations (if the maps are not cofibrations, all the same things work - you just replace the quotient spaces by mapping cones). You extend both maps to their Dold-Puppe sequences, so you get the sequences

$X rightarrow Y rightarrow Y/X rightarrow Sigma X rightarrow Sigma Y ldots$

and

$Z rightarrow W rightarrow W/Z rightarrow Sigma Z rightarrow Sigma W ldots$

Now suppose you have maps $a: X rightarrow W$ and $b: Y rightarrow W/Z$ making the obvious square commute up to homotopy. You can then extend these to make a commutative ladder from the first Dold-Puppe sequence to the second. (Notice that the sequences are deliberately offset from each other by one spot.)

Using the usual parameters and the obvious choices of homotopies you will get a square involving $Y/X, Sigma X, Sigma Z, Sigma W$. This square will commute if it includes the map $-Sigma g: Sigma Z rightarrow Sigma W$, but not generally with the map $Sigma g$. (To check all this, I recommend doing the Dold-Puppe sequences with mapping cones rather than quotient spaces but keeping the homotopy equivalences with the quotient spaces in mind, which is the only way I know to calculate what the right maps should be.)

At this point, if you were feeling stubborn, you could replace the map in your ladder $Sigma X rightarrow Sigma Z$ with $-1$ times that map, and that would allow you to have used $Sigma g$ in the square I mention in the above paragraph, but that creates other issues; if you choose not to simply use suspensions of your original maps to go from one Dold-Puppe sequence to the other then you run into problem when you are mapping between Dold-Puppe sequences without the shift of this example.

I hope this helps unravel Greg's answer (which is correct - you need the sign to get good mapping properties).

Of course one sees exactly the same phenomenon in the category of chain complexes of abelian groups (where homotopy is chain homotopy) and other such categories. I agree with Theo and Mark that one thinks about the suspension as "odd" (in the sense of parity not the sense of peculiar).

The published paper that Mark refers to that has an error of exactly this sort (which is unfortunately fundamental to the paper) is by Lin Jinkun in Topology v. 29, no. 4, pp. 389-407. I read this paper in preprint form in 1988 and missed this error, but discovered it in 1992 when reading another paper by the same author with the same error. In the Topology paper the error is made in diagram 4.4 on the right hand square (proof of Lemma 4.3).

nt.number theory - Trost's Discriminant Trick

T. Nagell in [Norsk Mat. Forenings Skrifter. 1:4 (1921)]
shows that, for an odd prime $q$, the equation
$$
x^2-y^q=1 qquad (*)
$$
has a solution in integers $x>1$, $y>1$, then $y$ is even and $qmid x$.
In his proof of the latter divisibility he uses a similar trick as follows.

Assuming $qnmid x$ write ($*$) as
$$
x^2=(y+1)cdotfrac{y^q+1}{y+1}
$$
where the factors on the right are coprime (they could only have
common multiple $q$). Therefore,
$$
y+1=u^2, quad frac{y^q+1}{y+1}=v^2, quad x=uv,
qquad (u,v)=1, quad text{$u,v$ are odd}.
$$
Using these findings we can state the original equation in the form
$x^2-(u^2-1)^q=1$, or
$$
X^2-dZ^2=1 quadtext{where $d=u^2-1$}.
$$
The latter equation has integral solution
$$
X=uv, quad Z=(u^2-1)^{(q-1)/2},
$$
while its general solution (a classical result for this particular Pell's equation)
is taken the form $(u+sqrt{u^2-1})^n$. It remains to use the binomial theorem in
$$
X+Zsqrt{u^2-1}
=(u+sqrt{u^2-1})^n
$$
(for certain $nge1$) and simple estimates to conclude that this is not possible.

To stress the use of similar trick: instead of showing insolvability of $x^2-(u^2-1)^q=1$,
we assume that a solution exists and then use $d=u^2-1$ to produce a solution $X,Z$ of $X^2-dZ^2=1$;
finally, the pair $X,Z$ cannot solve the resulting Pell's equation.
(Of course, it is hard to claim that this is exactly Trost's trick,
as here is a dummy variable but no discriminants, except the one for
Pell's equation. Trost's trick is less trickier to my taste. $ddotsmile$)

Note that Nagell's result was crucial for showing that ($*$) does not have
integral solutions $x>1$, $y>1$ for a fixed prime $q>3$. This was shown in
a very elegant way, using the Euclidean algorithm and quadratic residues
by Ko Chao [Sci. Sinica 14 (1965) 457--460], and later reproduced
in Mordell's Diophantine equations. The ideas of this proof are in the
heart of Mihailescu's ultimate solution of Catalan's conjecture. A much
simpler proof of Ko Chao's result, based on a completely different (nice!) trick,
was given later by E.Z. Chein [Proc. Amer. Math. Soc. 56 (1976) 83--84].

newtonian telescope - Are Barlow Lenses Good For Deep Sky Observing?

The Failures of High Magnification

Higher magnification doesn't help you observe deep sky objects better. Deep sky objects unlike stars are extended objects. They subtend a finite solid angle on you. This ensures that the surface brightness(brightness per unit solid angle) of extended objects remains constant. Hence, a higher magnification would not make it brighter for you to see.

It is worse for another reason. After a certain magnification, the apparent angular size of the extended object becomes comparable to the field of view, or sometimes even greater. Your eye won't be able to distinguish the object against the background since most of the background is the object itself.

Good Uses of Barlow

Some of the uses that you can put your Barlow:

1. Observe the Moon, Solar System objecs


2. Some of the clusters which seem like point objects really "explode" when using a Barlow. The best example is Omega Centauri.

Experiences

I have a 8" Dobsonian Newtonian Reflector. The viewfinder is 20x80. I really get a better picture of deep sky objects like Ring Nebula, which have a small angular size, when using Barlow. But for cases like Lagoon and Trifid Nebula; they are visible to me from my viewfinder but not through the eyepiece(even in the lowest magnification).

Conclusion

Barlow really helps if you have a deep sky object with small angular size. For larger objects, a binocular may do better, especially for city conditions.

fourier analysis - The maximum of a real trigonometric polynomial

It turns out that it is possible to achieve an arbitrarily small additive error using semidefinite programming. This is from the paper:

J.W. McLean, H.J. Woerdeman. Spectral factorizations and sums of squares representations via semidefinite programming. SIAM J. Matrix Anal. Appl., 23(3):646--655, 2001. (link)

The result can be rephrased as follows. Let $f(x)=F(e^{ix})$ where $F(z)= sum_{n=-N}^N c_n z^n$, with $c_n=frac{1}{2}(a_n-i b_n)$ and $c_{-n}=bar{c}_n$. Then $min_x f(x)$ is equal to $c_0$ minus the value of the following semidefinite program:
$ min_F tr(F) $
such that $F succeq 0$, and $sum_{p=k}^N F_{p,p-k} = c_k$ for $k=1,ldots,N$.

Since semidefinite programming can achieve an arbitrarily small additive error, we can approximate the minimum (and thus, the maximum) of $f$ within the same bound.

How do scientists know there are about 300 billion stars in a galaxy and there are about 100 billion galaxies?

The way it works is as follows. We do detailed studies of stars in the solar neighbourhood. This establishes the local density of stars and the mix of masses they possess (called the stellar mass function). We compare that with the mass function of clusters of stars and note that to first order it appears invariant.

We can then triangulate the problem in various ways: we can make a model for the stellar density of the Galaxy, assume it all has the same mass function and hence get a number of stars. The model may be based on crude light-to-mass conversions, but more often would be based on deep surveys of the sky - either narrow pencil beam surveys from HST, or broader surveys like SDSS, The key is to be able to count stars but also estimate how far away they are. This is highly uncertain and relies on some assumptions about symmetry to cover regions of our Galaxy we cannot probe.

Another method is to count up bright objects that might act as tracers of the underlying stellar population (eg red giants), compare that with the number of giants in our well-studied locale, and from this extrapolate to a total number of stars, again relying on symmetry arguments for those bits of the Galaxy that are distant or obscured by dust.

A third way is to ask, how many stars have lived and died in order to enrich the interstellar medium with heavy elements (a.k.a. metals). For example, it turns out there must have been about a billion core-collapse supernovae to create all the oxygen we see. If we assume the mass function is invariant with time and that supernovae arise from stars above 8 solar masses, then we also know how many long-lived low-mass stars were born with their high-mass siblings and hence estimate how many stars exist today.

The number, whether it be 100 billion or 300 billion is no more accurate than a factor of a few, but probably more accurate than an order of magnitude. The main issue is that the most common stars in the Galaxy are faint M dwarfs,that contribute very little light or mass to the Galaxy, so we really are relying on an extrapolation of our local knowledge of these objects.

The number of galaxies problem is easier, though the number is less well defined. We assume that on large scales the universe is homogeneous and isotropic. We count up how many galaxies we can see in a particular area, multiply it up to cover the whole sky. The number must then be corrected for distant faint galaxies that cannot be seen. The difficult here is that we are looking into the past and the number of galaxies may not be conserved, either through evolution or mergers. So we have to try and come up with a statement like "there are n galaxies in the observable universe today that are more luminous than L". I think this number is certainly only an order of magnitude estimate.

How long will it take Pluto to grow to planet size?

Never.

There are a number of technical papers that give more precise meaning to the concept of "clearing the neighborhood". It's not just now, it's can the object in question clear the neighborhood of its path while the Sun is still a star.

In the case of Pluto, Ceres, Eris, and a host of other not-quite-planet objects, that will not happen. Ever. Or at least so long as the Sun shines. After that, does it matter?

distances - 3D Positions of Nearby Stars

I have come across a number of star catalogs that list stars by right ascension and declination, along with other data such as magnitude.

Is there a star catalog that lists the 3-D position of stars (e.g. includes distance from the sun, or otherwise specifies a 3-D position)?

Would bacteria on incoming meteors burn before impacting?

Any bacteria present in a meteor would not necessarily burn. According to the Bad Astronomer's "Bad Addendum" to another unrelated point, "A Meteoric Rise", meteors sure do get extremely hot, but that outside part usually ablates away, leaving the inner part that was extremely cold for who knows how long, which may not have warmed up much during the fall.

Bad Addendum: many people think that a meteorite, after it hits the ground, is very hot and glows red. Actually, meteorites found shortly after impact tend to be warm, but not hot at all! It turns out that it certainly is hot enough to glow while it is in the part of the atmosphere that decelerates it the strongest, but any part that actually melts will be blown off ("ablated") by the wind of its passage. That leaves only the warm part. Even more, the meteor is slowed down so strongly as it moves through the atmosphere that the impact speed is typically only a few hundred kilometers an hour at most. Only the very large (and we're talking meters across) meteors are still moving at thousands of kilometers an hour or more when they impact. Small ones aren't moving that fast at all. Not to say you'd want to be under one: a car in New York was struck by a small meteorite and had a hole punched through it, and the whole back end crushed in. Ouch!

This American Meteor Society resource seems also to indicate that meteorites can fall to the ground without being extremely hot.

9. Are meteorites “glowing” hot when they reach the ground?

Probably not. The ablation process, which occurs over the majority of the meteorite’s path, is a very efficient heat removal method, and was effectively copied for use during the early manned space flights for re-entry into the atmosphere. During the final free-fall portion of their flight, meteorites undergo very little frictional heating, and probably reach the ground at only slightly above ambient temperature.

For the obvious reason, however, exact data on meteorite impact temperatures is rather scarce and prone to hearsay. Therefore, we are only able to give you an educated guess based upon our current knowledge of these events.

This study, "Bacterial Spores Survive Simulated Meteorite Impact", seems to indicate that it's possible, though unlikely, for a bacterium to survive the pressure shock from such an impact.

So it's plausible that an extremophile bacterium could somehow survive this impact by being on the inside of the meteoroid, surviving the ablation process as a meteor, and not even get warmed up to the surrounding atmospheric temperature when it crashes as a meteorite.

There is nothing to indicate that this has actually happened. But what a discovery it would be if something like this were to happen and be confirmed.

the sun - Will Earth lose the Moon before the Sun goes into supernova?

As HDE 226868 noted in his answer, the Sun is not going to go supernova. That's something only large stars experience at the end of their main sequence life. Our Sun is a dwarf star. It's not big enough to do that. It will instead expand to be a red giant when it burns out the hydrogen at the very core of the Sun. It will continue burning hydrogen as a red giant, but in a shell around a sphere of waste helium. The Sun will start burning helium when it reaches the tip of the red giant phase. At that point it will shrink a bit; a slight reprieve. It will expand to a red giant once again on the asymptotic red giant branch when it burns all the helium at the very core. It will then burn helium in a shell surrounding a sphere of waste carbon and oxygen. Larger stars proceed beyond helium burning. Our Sun is too small. Helium burning is where things stop.

The Sun has two chances as a red giant to consume the Earth. Some scientists say the Sun will consume the Earth, others that it won't. It's all a bit academic because the Earth will be dead long, long before the Sun turns into a red giant. I'll have more to say on this in the third part of my answer.

---

The current lunar recession rate is 3.82 cm/second, which is outside your one to three centimeters per second window. This rate is anomalously high. In fact, it is extremely high considering that dynamics says that
$$frac {da}{dt} = (text{some boring constant})frac k Q frac 1 {a^{11/2}}$$
Here, $a$ is the semi major axis length of the Moon's orbit, $k$ is the Earth-Moon tidal Love number, and $Q$ is the tidal dissipation quality factor. Qualitatively, a higher Love number means higher tides, and a higher quality factor means less tidal friction.

That inverse $a^{5.5}$ factor indicates something seriously funky must be happening to make the tidal recession rate so very high right now, and this is exactly the case. There are two huge north-south barriers to the flow of the tides right now, the Americas in the western hemisphere and Afro-Eurasia in the eastern hemisphere. This alone increases $k/Q$ by a considerable amount. The oceans are also nicely shaped so as to cause some nice resonances that increase $k/Q$ even further.

If something even funkier happens and the Moon recedes at any average rate of four centimeters per second over the next billion years, the Moon will be at a distance of 425,000 km from the Earth (center to center). That's less than 1/3 of the Earth's Hill sphere. Nearly circular prograde orbits at 1/3 or less of the Hill sphere radius should be stable. Even with that over-the-top recession rate, the Moon will not escape in the next billion years.

---

What about after a billion years? I chose a billion years because that's about when the Moon's recession should more or less come to a standstill. If the Earth hasn't already died before this billion year mark, this is when the Earth dies.

Dwarf stars such as our Sun get progressively more luminous throughout their life on the main sequence. The Sun will be about 10% more luminous than it is now a billion years into the future. That should be enough to trigger a moist greenhouse, which in turn will trigger a runaway greenhouse. The Earth will become Venus II. All of the Earth's oceans will evaporate. Water vapor will reach well up into what is now the stratosphere. Ultraviolet radiation will photodissociate that water vapor into hydrogen and oxygen. The hydrogen will escape. Eventually the Earth will not only be bare of liquid water on the surface, it will be bare of water vapor in the atmosphere.

Almost all of the Moon's recession is a consequence of ocean tides. Without oceans, that tunar recession will more or less come to a standstill.

soft question - Mathematics as a hobby

I try to learn and understand as many facts as I can. Of course, many people would like to benefit from the opposite, that is, digging into a certain branch as deep as they can.

I try to do the opposite, which I see as my main advantage, as the opposite to professional mathematicians. This is because they have their own careers and has their professional criteria to fulfill (writing articles in journals, gaining citation points, etc.).

As an amateur I am not obliged to do so, and this is a great freedom. If you want to be creative, you may try to dig here and there, and probably you will be lucky to find certain problems which are not penetrated, or you may find just something interesting enough (for example, your own point of view on a well-known area, maybe you find a surprising connection and, even if it is well known, it is funny to discover it once more, etc.) to write it somewhere, maybe on a blog.

In summary: I read as much as I can, I learn as much as I can, and I ask as much as I can.

As regards to low-level entry (you need of course to be genius to discover it, but nothing more;-), an example is Feigenbaum's famous discovery about chaos, etc. As far as I know, he used only a programmable calculator to discover it. He was just inquisitive, nothing more, nothing less.

ag.algebraic geometry - Two-dimensional quotient singularities are rational: why?

Let's assume the characteristic is $0$.

Let $R$ be a normal $N$-graded ring of dimension 2 with homogenous maximal ideal $m$. Then $R$ has rational singularity if and only if the non-negative degrees part of the graded local cohomology module $H_m(R)$ vanish. That is $H_m(R)_{(i)}=0$ for $igeq 0$. This is due to Watanabe, see Theorem 2.2 in:
http://www.ams.org/tran/2003-355-03/S0002-9947-02-03186-0/home.html

It is safe to work in affine situation, so let $S=k[x,y]$ and $R=S^G$. Then $R$ is normal and generated by forms of positive degree. Since $H_{(x,y)}(S)_{(i)}=0$ for $igeq 0$, it follows that $H_m(R)_{(i)}=0$ for $igeq 0$ (one can compute local cohomology in $S$ by using a system of parameters which are elements in $R$).

In characteristic $p$ one probably has to use Frobenius. Note that Boutot's theorem fails in this case (I think it is still true for finite group though).

A truly easy proof is probably not easy to find unless one has a truly elementary definition of rationality.

EDIT: There are other ways to see this:

II) Again, assume $k$ is algebraically closed of
characteristic $0$.Let $S=k[[x,y]]$ and $R=S^G$. Then the following 2 facts will suffice (using same notation as above):

1) There are only finitely many indecomposable reflexive modules over $R$.
(Proof not hard, they have to be summands of $S$). In particular, the class group of $R$ is finite.

2) Since $R$ is complete, $R$ has rational singularity is equivalent to the class group of $R$ is finite. This is Theorem 17.4 in Lipman paper on rational singularity.

III) Finally, one can quote Prop 5.15 of Kollar-Mori book on birational geometry. It gave the exact statement, but the proof uses general machinery, and probably close to what you already knew.

co.combinatorics - How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

One can sample from the uniform distribution on permutations on $n$ elements with $k$ cycles in expected time $O(nsqrt{k})$.

For large $n$ and $k$ this may be more feasible than
the method Herb Wilf refers to in his answer, which,
if I understand right, requires the generation of the Stirling cycle numbers $S(m,r)$
for $mleq n$ and $rleq k$.

The idea is as follows: consider a Poisson-Dirichlet($theta$) partition of $[n]$. The
blocks of such a partition have the distribution of the cycles of a random permutation
of $[n]$ from the distribution which gives weight proportional to $theta^r$ to any permutation with $r$ cycles. In particular, conditioned on the number of cycles, the permutation is uniformly distributed. (Once one has a partition into blocks corresponding to the cycles, one can just fill in the elements of $[n]$ into the positions in the cycles uniformly at random).

Choose $theta$ in such a way
that the mean number of cycles is around $k$.
One can sample a PD($theta$) partition of $[n]$ in time $O(n)$ (see below).
Keep generating independent samples of the partition until you get one
with exactly $k$ blocks.
The variance of the number of cycles will be $O(k)$ (see below) and the
probability that the number of cycles is precisely $k$ will be on the order of $1/sqrt{k}$, so
one will need to generate
$O(sqrt{k})$ such samples before happening upon one with precisely $m$ cycles.

So this is really not so far from what you suggested in the question
(generate random permutations until you find one that fits) with the twist that
instead of generating from the uniform distribution (which corresponds to PD(1))
you choose a better value of $theta$ and generate from PD($theta$).

Here are two nice ways to sample a PD($theta$) partition of $[n]$:

(1) "Chinese restaurant process" of Dubins and Pitman.
We add elements to the partition one by one. Element 1 starts in a block on its own. Thereafter, when we add element $r+1$,
suppose there are currently $m$ blocks whose sizes are $n_1, n_2, ... n_m$. Add element $r+1$ to block $i$ with probability
$n_i/(r+theta)$, for $1leq ileq m$, and put element $r+1$ into a new block on its own with probability $theta/(r+theta)$.

(2) "Feller representation". Generate independent Bernoulli random variables $B_1, dots, B_n$ with $P(B_i=1)=theta/(i-1+theta)$.
Write a string of length $n$ divided up into blocks, with the rule that we start a new block before position $i$ whenever $B_i=1$.
So for example if $n=10$ with $B_1=B_5=B_6=B_9=1$ and the other $B_i$ equal to 0, then the pattern is

(a b c d)(e)(f g h)(i j).

(Note that always $B_1=1$). Then assign the elements of $[n]$ to the positions in the blocks uniformly at random.

The expected number of blocks is $sum_{i=1}^n mathbb{E}B_i$, which is $sum_{i=1}^n theta/(i+theta)$,
which is approximately $theta(log n-log theta)$. If this is equal to $k$ with $1 << k << n$, then
the number of blocks will be approximately normal with mean $k$ and variance $O(k)$.

For details of some of the things mentioned here
to do with Poisson-Dirichlet partitions, random permutations etc, see e.g. Pitman's lecture notes from
Saint-Flour: http://bibserver.berkeley.edu/csp/april05/bookcsp.pdf

Category of groups = Category of models of group theory?

Let me repeat my above comment as an answer, since I think it is important to the discussion. (I agree that JDH's answer is the answer to the specific question asked, and what I write below assumes that you have read his answer.)

If you remove the word "elementary" from the question, then it is indeed true that the two categories are "the same", in the strongest possible sense: they are canonically isomorphic. (The notion of equality of abstract categories is well known to be sticky and unfruitful, just as for the notion of equality of abstract objects: c.f. Mazur's wonderful essay "When is one thing equal to another?")

Given a language L, one defines L-structures [or slightly more precisely, relational structures] and morphisms between them [often required to be injections, but let's not do so here]. This is certainly a [concrete] category, even though for some reason it is not standard to say so explicitly in Chapter 1 of model theory books. If you have a theory T of that language, then it is natural to consider the full subcategory of models of T. If you do this with the theory of groups [in the language of monoids], what you get is a category in which the objects are groups and the morphisms are homomorphisms of groups. In other words, you get back [up to the provisos of the previous paragraph] the category of groups!

real analysis - What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $sum a_{n} sin(2 pi x * n)$ and its frequencies were $n*f_{B}$ where n is the number of the partial and f_B is the base frequency of the function, then now the frequencies of the partials are $n*f_{B}+f_{o}$. The function becomes an almost-periodic function. The waveform seems to be 'rotating' around the time axis.

Is this effect described somewhere? If so, where, and what is its name?

Here are animations of this happening: rotation.rar, 2MB

By the end of the first animation the function is a bit rippled, those
ripples are only created by errors in the model. It should in fact
look smooth like in the beginning.

In the 2nd movie, at the beginning I raise and then lower the offset
frequency of the partials (f_o), and then set it to 0. On 0:32 I reset
the resynthesis model (this is irrelevant to the effect, but it
explains why the waveform suddenly changes). Later I set f_o to +1 Hz
and 0 Hz, alternating the setting.

In the 3rd and 4th animations I alternate f_o between +1 Hz and 0 Hz.

It is as though the function graph is drawn on a 3-dimensional glass cylinder that rotates around the time axis. To avoid confusion, I define that if f_o is positive, then the direction of the rotation is positive. Note that the points of the graph don't seem to be equidistant from the axis of rotation - some seem to be orbiting at smaller and some at bigger radii. This is not really exposed in the movies, but if you start going up very far with f_o, the graph becomes 'twisted', as if the 'cylinder' were becoming twisted around its axis of rotation.

Motivation to this question, short version: I feel this is a good new way to look at functions while retaining the time-domain toolset of real analysis.

More motivation for this question: I am wondering whether this sort of analysis tool could become a generalization of the time domain. We have a lot of tools for analyzing functions in the frequency domain, and we have a lot of tools in the time domain - however the results they yield are somewhat disjoint. The way this 'spinning graph' effect depends directly on the frequency content of the function at hand. It could make it possible to do time domain based analysis of some form of frequency content, or rather 'slope content': what I am refering to here is the fact that in the time domain it's easier to think in terms of slope rather than the frequency. The two are tied together (a function with a sharp rising edge will probably have a lot of high-frequency content) but not directly (a simple 1 Hz sinusoid with enough amplitude can have higher slope on an interval than a 200 Hz triangle wave). This tool that arises from simply frequency-shifting the function gives a new way to look at time-domain signals, and so I think it could give rise to interesting questions - I am extremely surprised that this is not mentioned anywhere in literature: it feels like a very direct generalization of the time domain display of a signal and can be applied to pretty much every real-valued function out there giving us new information about it. On the one hand it is very useful for periodic functions, on the other hand the way the graph seems to 'coil up' when the frequency shift is increased can make it useful for the analysis of functions which are not periodic. I strongly feel that the analysis of 'where the graph is on that spinning cylinder' can provide new information about functions.

Background: I had come across this effect when studying the musical properties of stretched-harmonic waveform synthesis, motivated by the fact that real instruments tend to have slightly inharmonic rather than harmonic frequency content, whereas electronic synthesizers tend to have purely harmonic timbre. This got me asking some questions not really related to music!

Thanks!

data analysis - Where can I find a catalog of all stars in the Milky Way?

No, such a catalogues does not (yet) exist. There are two reasons.

1 The Milky Way galaxy is about 20kpc (1pc ~= 3 lyr) across and only the very brightest stars are individually identifyable across such large a distance (such bright stars by their nature are very massive and hence young). Astronomers tend to cataloge stars by their apparent brightness, which for stars of identical luminosity declines as $1/d^2$ ($d$=distance). As a consequence, most catalogues contain only stars in the immediate galactic neighbourhood of the Sun. The Hipparcos catalogue (mentioned in another answer), for example, has most stars within a mere 100pc of the Sun.

2 Obtaining distances for individual stars is inherently difficult, in particular the more distant the star in question is. Accurate distances for stare several kpc away can currently only be obtained by indirect methods applicable only to certain types of stars (such as RR Lyrae variables). The classical trigonometric parallax measurement for such distances, however, is subject of ESA's ongoing Gaia mission.

ESA's Gaia satellite launched last year aims at cataloguing about $10^9$ stars across the Milky Way, including their velocity. The first preliminary versions of resulting catalogue, however, will still take some time to appear.

lo.logic - Natural number properties as uninterpreted functions in first order logic

Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so and so" can be expressed in FOL. Also these need to be expressed as uninterpreted functions.

"For every natural number n, there are infinitely many other natural numbers such that the greatest common divisor of n and each of these other numbers is 1"

As I said if you can't express this sentence in FOL at least suggest links/material where I can get leads.

galaxy - What is the Fundamental plane for Elliptical Galaxies?

I am not an expert, but it seems that the fundamental plane is a relation among characteristic quantities of the galaxy, showing a correlation that is the analogous of the Tully-Fisher relation for spiral galaxies.

You can start from the virial theorem:
$frac{M}{R}sim v^2$

($M$ mass contained in the radius $R$, $v$ velocity dispersion of the stars, also indicated with $sigma$ in these cases)
meaning that the stars behave like an isothermal sphere.

Assuming that all galaxies have the same mass-to-light ratio
$L/M$, and that all galaxies have the same surface brightness

$Sigma=L/R^2$

(with $L$ luminosity), you obtain the Tully-Fisher relation:

$Lsim v^4$

Of course, all galaxies do not have the same surface brightness.
So if you take $Sigma=L/R^2$ and substitute that into the virial theorem (keeping the mass-to-light ratio assumption), then a relation between luminosity, surface brightness, and velocity dispersion is obtained:

$L sim sigma^4 Sigma^{-1}$

You can plot the stars distributions in elliptical galaxy according to this relation, and will note that the stars do not distribute randomly, but are more concentrated along the fundamental plane (more or less the same happens for the HR diagram):

(Here $R_e$ is the so called effective radius).

For completeness, another illuminating picture from here:

References here and here.

gravity - Parking a telescope at a Lagrange point: is this a good idea from a debris point of view?

Hmmm no, it wouldn't be cluttered with debris, and yes, it's a good idea to park the JWST (James Webb Space Telescope) at the Sun-Earth L₂ point.

The five Lagrange points are unstable, for one because of the gravitational anomalies of the two massive bodies of the Lagrange system, eccentric orbits, and there are many other factors to their instability. At the same time, they are least gravitationally attractive points around two massive bodies.

Think of L-points as parking your car on a flat space at the top of the hill. You'll have to approach with some control and then try to balance at your parking spot, if you don't plan using your handbrake and also remain there stationary:

    

    ^{Visualisation of the relationship between two massive bodies and their five Lagrange points (Source: Wikipedia)}

There wouldn't be any debris clutter there, or any other matter like smaller particles, at least not any more likely than elsewhere around them, only transient in nature and possibly even less likely than elsewhere since all the other mass particles would gravitate towards the more massive bodies of two Lagrange system centers in their vicinity.

No body with rest mass would stay there on its own accord, not unless it has active attitude control to position itself there and constantly adjust for changes in gravitational attraction vector as the two mass bodies rotate around their axes, change distance while orbiting each other, or L points being influenced by other mass bodies of the same planetary system.

At the same time, JWST will be shielded by the Earth from any Solar activities and also Sun's interference with JWST's sensitive equipment as it starts observing the Universe in the infrared spectrum. Most of the orbital debris from our own space exploration missions is cluttered in the LEO (Low Earth Orbit) belt at roughly an altitude of 500-1500 km above the Earth's surface:

    

             ^{Source: Active Debris Removal: EDDE, the ElectroDynamic Debris Eliminator, Jerome Pearson et al. (PDF)}

Now, the JWST will be deployed to a very large 800,000 kilometres (500,000 mi) radius halo orbit around the Sun-Earth L₂ point, that is 1,500,000 kilometers (930,000 mi) from the Earth, around 4 times farther than the distance between the Earth to the Moon. So not exactly in Earth's shadow and it will still use large deployable sunshield, but that's a long way away from where we're littering our LEO or even GEO/GSO (Geostationary/Geosynchronous orbits) with GEO being the farthest of these frequently used orbits, at 35,786 kilometres (22,236 mi) above the Earth's equator (and its graveyard orbit a bit farther than that). JWST will be placed in halo orbit nearly 40 times as distant from the Earth as you'd expect any space debris cluttering our planet's orbits at.

Yes, it is a very good idea to place JWST around the Sun-Earth L₂ point.

big list - Where have you used computer programming in your career as an (applied/pure) mathematician?

Possibly the most common thing I use a computer for is the Todd-Coxeter algorithm, which enumerating finite index subgroups of a finitely presented group. I can't count how many times I have used it. It comes standard with the computer package gap (or MAGMA which has similar functionality, and is freely available at North American Universities).

I think this algorithm might be worth a mention in your book, because it's cool, but rarely something you would ever want to do by hand. For me, this is often what I want a computer to do, quickly compute some examples, so I can tell how half-baked an idea of mine might be.

Also, both gap and magma give you the option to generate either the complete list of subgroups at a given index or iterate through the list of subgroups if you just are looking for an example with a given property. So it would could serve as a useful way to introduce iterators.

the sun - Will Neptune be visible with the naked eye if I am standing on its satellite

I'm probably misunderstanding the question, but
http://nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html
notes that Neptune's magnitude from Earth is 7.8 at opposition, when it's
4347.31 million km away.

If you get 3 times closer, 1449.1 million km away, Neptune
will appear 9 times brighter, bringing its magnitude to 5.4,
well within our visible range.

As Stellarium notes below, Neptune is almost visible
(magnitude 5.66) from Uranus, an entire planet away.

My version of Stellarium can't simulate the view from Naiad
(Neptune's closest moon), but at a distance of only 48,227 km
(Naiad's semi major axis), Neptune's magnitude would be almost
-2, much fainter than our own moon, but brighter than Sirius
appears to us.

This brightness would be spread out across Neptune's disk, but
the brightness at any point would be about magnitude 3, still
quite visible.

Moreover, Naiad probably has a much thinner atmosphere (and no
light pollution), making Neptune even easier to see.

The other outer planets are visible from Earth (Uranus just
barely), and so would also be visible from their own moons.

I haven't done the calculations for Pluto, which is
technically no longer a planet.