high energy astrophysics - meaning of p-wave charmonia

You're correct that "$p$-wave" in this context means that the charmonium has orbital angular momentum $L=1$. The principal observable effect is that two-particle states with even $L$ have even parity, while those with odd $L$ change sign under parity.

If you were to skim the Particle Data Group's list of charmonia (or the $cbar c$ section of the short list) you would find that each particle has a value listed for spin, $J$, and parity, $P$. The lightest few states are

• $eta_c(1S)$, with $J^P = 0^-$. This is a charm-anticharm bound state where the two quarks have opposite spins and the angular momentum $L=0$. (The total parity is negative because the intrinsic parities of the $c$ and $bar c$ have opposite sign.)




• $J/psi(1S)$, with $J^P=1^-$. This has the same parity as the spinless $eta_c$, but a unit of spin. So it must be that here the two quarks have the same spin and $L=0$.




• $chi_{c0}(1P)$, with $J^P=0^+$. The change in parity suggests that this state must have a different $L$ than the $eta_c$ and $J/psi$, and it's the first with a $P$ in its name. There's also a $chi_{c1}$ and $chi_{c2}$ with $J^P=1^+, 2^+$, respectively. That suggests these are the three spin combinations of a charm-anticharm pair with spin $S=1$ and orbital angular momentum $L=1$.

There are also some charmonia with $(2S)$ and $(2P)$ in their names. There don't appear to be any definitively labeled $d$-wave charmonia.

How do we know an event is a gamma-ray burst?

Yes, on the right track where the energy produced in a supernova can indicate if it was a LGRB or SGRB.

An indicative tools stems from the light curves of the remnants, both the afterglow brightness and it's region of explosion can indicatively be traced to how long the event lasted. Large gamma-ray remnants are typically associated with rapid star formation regions and are a more luminous remnant, where short gamma-rays are typically associated with little or no star formation regions and are a less luminous remnant.

It's not an exact science, as reproducing the effects and duplicating results have proven vastly difficult, however using distribution plots can definitively separate the differences and provide a theoretical model to work on.

Some minor information can be found at Wikipedia, however further reading and a more in depth view can be found at Cornel University Library. It is a bit of a read and attempts to explain the physics behind emission through mathematics, I'm not entirely sure whether is will provide the answer you are seeking.

the sun - Broadband spectrum of Sun

The point here is, that the kink you're pointing out is not belonging to the BB-spectrum of the sun.
This part of the solar spectrum is a classic example for a broad class of radiation, called nonthermal radiation.
The physical origin of this is in general that, every accelerated particle will radiate part of its energy away, with the experienced acceleration a, radiated power P and their relation $P sim a^2$ in a wavelength-range depending on the exact process.
The origins of those accelerations are usually thought to be strong magnetic interactions on the solar surface plasma, creating the famous sunspots and coronal loops. In those loops, electrons and protons are being shot into space, following the loop, and then return back onto the solar 'surface'. Upon return they release Bremsstrahlung, as the interactions between loop-particles and surface plasma create strong braking accelerations.

Spectrae for such processes are derived in every standard textbook about theoretical astrophysics and are given by roughly $frac{dP}{dt} sim exp(-frac{h nu}{k_BT})$, which should fit the UV-part in the spectrum you gave pretty well.
This now also explains the variability of this part of the solar spectrum: As the number of loops and sunspots correlates with solar activity, it is clear that the UV-bremsstrahlung-production should too.
There is however much more to this, if you are interested. Given certain violent outbreaks that happen upon release of magnetic energy, there are modulations of the emitted radio flux (also indicated in your graph) up higher-energetic regions of x-rays.

co.combinatorics - Tractably Partitioning the possible vertex k-colorings of a graph by local stability and instability.

If you limit it to specific classes of graphs, say for example star graphs, you can come up with some answers. For a star graph $S_m$, with a vertex at the center and $m$ vertices connected to the center, yielding a graph $G$ with $n=m+1$ vertices and $m$ edges, it can be calculated that for $k=2$

If the center vertex is labeled black, then the only "0-unstable" coloring is where all of the leaves are white. If any of the leaves are also black, say $j$ of the $m$ leaves are black while the center is also black, then the center and those $j$ black leaves are unstable, leaving $m-j$ leaves as stable nodes. There are $m choose j$ = ($m$ choose $j$) ways to color $j$ of the $m$ leaves as black.

The same is true with the color labels reversed if the center is labeled white.
Thus for $k=2$, for two-color labeling of a star-graph $S_m$ with $m$ leaves and $n=m+1$ vertices, the sizes of the partitions of all of the possible two-colorings are as follows:

|{0 unstable}| = 2

|{1 unstable}| = 0

|{r unstable}| = $2 times$ ${m}choose{r-1}$ for $ 2 le r le n$, with $r in Z $

The size of the 1-unstable partition is always zero for this family of graphs. The size of the 1-unstable partition is always zero for any graph and for any $k$ because instablity occurs over an edge linking two vertices with the same color label, thus always creating two unstable vertices if there are any unstable vertices at all.

The sum of all of these partitions sizes is $2^n$, thus all of the possible $2^n$ colorings of the $n=m+1$ vertex star graph $S_m$ have been accounted for. A similar calculation can be made for star graphs for $k>2$.

Apurva

linear algebra - spectral radius of a matrix as one element changes

Here's my question --

Let $A$ be an $n times n$ real matrix, and suppose that the spectral radius $rho(A)$ is less than one (spectral radius = max eigenvalue). Let's choose some $1 leq i leq N$ and look at $A_{N,i}$. Namely, let's replace $A_{N,i}$ with some new value, $a$, to give us a new matrix $hat A$. I want to characterize the set $lbrace a : rho(hat A) < 1 rbrace$. It pretty clear that this set is of the form $[0, a_{max})$, but I want to be able to compute $a_{max}$ analytically, given $A$ and $i$. (Also clearly $a_{max} geq A_{N,i}$, since $rho(A) < 1$ by assumption.)

This seems like it should be a fairly easy exercise but I haven't been able to make any useful progress on it.

Thanks!

-h

Stable polar solar orbit with the Earth continuously observable

As ganbustein says, this is not too difficult to imagine. The simplest case (approximating with circular orbits and only the Sun, Earth and Satellite) would have the satellite orbit orthogonally to the Earth with a 1 year orbit. The Satellite will pass the Earth orbit plane in two places, call these "down crossing" and "up crossing" points.

To minimize Earth - Satellite iterations, keep them 90 degrees apart. Have the Satellite over the north solar pole when the Earth is at the "down crossing" point. Then when the satellite gets to the "down crossing point" the Earth will be furthest from the Satellite plane. When the earth is at the "up crossing point" the satellite will under the Suns south pole. And so on.

This would not be completely stable when we include Jupiter and the mutual interactions, but I think they should be small, allowing this to work generally. If someone "does the math" and proves me wrong, I will accept that.

rt.representation theory - What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra

If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of quantized enveloping algebra. Some others generalized to triangulated(derived category of coherent sheaves on projective spaces,so on so forth)

I wonder know whether there exists some explicit geometric explanation to these works. I think we can consider the quantized enveloping algebra as noncommutative affine scheme. Therefore, it seems these work are doing some sort of reconstruction of schemes in abelian category or derived category.

I wonder whether Bondal-Orlov's work has any relationship(explicit)to these stuff. Therefore, what I am really interested in is what is the real meaning of Ringel-Hall algebra

At last, it is well known that Belinson-Bernstein's theorem established the equivalence of category of U(g)-module and category of D-modules on flag variety of Lie algebra. And some others, say Bezrukavnikov,Frenkel,gaitsgory generalized these results to Kac-Moody algebra.On the other hand, Van den Berg used generalized Ringel Hall algebra to realize quantum group of Kac-Moody algebra. I wonder whether anybody here can say something about these stuff.

All the other comments are welcomed

ca.analysis and odes - Is there a topology on growth rates of functions?

There is some fascinating work in the subject of cardinal
characteristics of the continuum in set theory that
directly relates to the concept of growth rates of
functions. I believe that it is the ideas in this subject
that are ultimately fundamental to your question. I explain
a little about the general subject of cardinal
characteristics in my answer
here.

Much of the interest of your question is already present
for functions on the natural numbers. The two main orders on such functions that
one considers in cardinal characteristics are

• almost-less-than, where $f lt^ast g$ means that
  $f(n) lt g(n)$ for all $n$ except finitely often, and


• domination, where $f lt g$ means that $f(n) lt g(n)$ for
  all $n$.

A family $F$ of functions is said to be unbounded if there
is no function $g$ that has $flt^ast g$ for all $fin F$. That is, an unbounded family is a family that is not
bounded with respect to $lt^ast$. A family $F$ is
dominating if every function $f$ is dominated by some
function $gin F$. The corresponding cardinal characteristics
of these two types of families are:

• The bounding number $frak{b}$ is the size of the smallest
  unbounded family.


• The dominating number $frak{d}$ is
  the size of the smallest dominating family.

It is easy to see that $frak{b}leqfrak{d}$, simply
because any dominating family is also unbounded. Also, both
$frak{b}$ and $frak{d}$ are at most the continuum $frak{c}$,
the size of the reals. It is not difficult to see that both
of these numbers must be uncountable, since for any
countable family of functions $f_0,f_1, f_2,ldots$, we can build the function $g(k) = sup_{nleq k}f_n(k)+1$, which eventually exceeds any particular
$f_n$. In other words, any countable family of
functions is bounded with respect either to
almost-less-than or with respect to domination. Thus,
$omega_1leqfrak{b},frak{d}leqfrak{c}$.

It follows from these simple observations that if the
Continuum Hypothesis holds, then both the bounding number
and the dominating number are equal to $omega_1$, which under CH is the same as the continuum $frak{c}$.

Now, the amazing thing is that ZFC independence abounds
with these concepts. First, it is relatively consistent
with ZFC that the Continuum Hypothesis fails, and both the
dominating number and the bounding number are as large as
they could possibly be, the continuum itself, so that $frak{b}=frak{d}=frak{c}$.
Second, it is
also consistent that both are strictly intermediate between
$omega_1$ and the continuum $frak{c}$, but still
equal. Next, it is also consistent with $text{ZFC}+negtext{CH}$ that
the bounding number $frak{b}$ is as small as it could be,
namely $omega_1$, but the domintating number is
much larger, with value $frak{c}$. The tools for proving all
these results and many others involve the method of
forcing.

Now, let me get to the part of my answer that directly
relates to the idea of rates-of-growth. A slalom is
defined to be a sequence of natural number pairs
$(a_0,b_0),
(a_1,b_1), ldots$ with $a_nlt b_n$. Each slalom s corresponds to the collection
of functions $f:omegatoomega$ such that $f(n)$ is in the
interval $(a_n,b_n)$ for all but
finitely many $n$. That is, imagine an olympic athlete on
skiis, who must pass through (all but finitely many of) the
slalom posts. An $h$-slalom is a slalom such that
$|b_n-a_n|leq h(n)$.

Thus, a slalom is a growth rate of functions, in a very
precise sense. With suitably chosen (countable collections
of) slaloms, it is possible to express the concept of
growth rate that you mentioned in your question.

The set theory gets quite interesting. For example, a major
question is: how many slaloms suffice to cover all the
functions? This is particularly interesting when one
restricts the size of the slaloms by considering $h$-slaloms.
A fat slalom is a $2^n$-slalom, where the
$n^{rm th}$ interval has size at most $2^n$.

It turns out that this is connected with ideas involving
meagerness, otherwise known as category. For example,
Bartoszynski
proved that every set of reals of size less than $kappa$ is
meager if and only if for every function $h$ and every family
of $h$-slaloms $F$ of size less than $kappa$, there is a
function $g$ eventually missing every slalom in $F$. In other
words, the possibility of a family of fewer than $kappa$
many $h$-slaloms covering all the functions is equivalent to
every set of size less than $kappa$ being meager.

And so on. There is a large amount of work on these and
similar ideas. An article particularly focused on slaloms
would be
this.
And there is a survey article by
Brendle on
cardinal characteristics.

How many planets are there in this solar system?

In addition to Undo's fine answer, I would like to explain a bit about the motivation behind the definition.

When Eris was discovered, it turned out to be really, really similar to Pluto. This posed a bit of a quandary: should Eris be accepted as a new planet? Should it not? If not, then why keep Pluto? Most importantly, this pushed to the foreground the question

what, exactly, is a planet, anyway?

This had been ignored until then because everyone "knew" which bodies were planets and which ones were not. However, with the discovery of Eris, and the newly-realized potential of more such bodies turning up, this was no longer really an option, and some sort of hard definition had to be agreed upon.

The problem with coming up with a hard definition that decides what does make it to planethood and what doesn't is that nature very rarely presents us with clear, definite lines. Size, for example, is not a good discriminant, because solar system bodies come in a continuum of sizes from Jupiter down to meter-long asteroids. Where does one draw the line there? Any such size would be completely arbitrary.

There is, however, one characteristic that has a sharp distinction between some "planets" and some "non-planets", and it is the amount of other stuff in roughly the same orbit. This is still slightly arbitrary, because it's hard to put in numbers exactly what "roughly" means in this context, but it's more or less unambiguous.

Consider, then a quantity called the "planetary discriminant" µ, equal to the ratio of the planet's mass to the total mass of other bodies that cross its orbital radius and have periods up to a factor of 10 longer or shorter. This is still a bit arbitrary (why 10?) but it's otherwise quite an objective quantity.

Now take this quantity and calculate it for the different bodies you might call planets:

Suddenly, a natural hard line emerges. There's a finite set of bodies that have "cleared their orbits", and some other bodies which are well, well behind in that respect. Note also that the vertical scale is logarithmic: Neptune's planetary discriminant is ~10,000 bigger than Ceres'.

This is the main reason that "clearing its orbital zone" was chosen as a criterion for planethood. It relies on a distinction that is actually there in the solar system, and very little on arbitrary human decisions. It's important to note that this criterion need not have worked: this parameter might also have come out as a continuum, with some bodies having emptier orbits and some others having slightly fuller ones, and no natural place to draw the line, in which case the definition would have been different. As it happens, this is indeed a good discriminant.

For further reading, I recommend the Wikipedia article on 'Clearing the neighbourhood', from which I took the data for the image. If you don't mind skipping over some technical bits, go for the original paper where this was proposed,

What is a planet? S Soter, The Astronomical Journal 132 no.6 (2006), p. 2513. arXiv:astro-ph/0608359.

which is in general very readable.

soft question - Famous mathematical quotes

Like many people, I am fascinated by the quote from Weyl (already listed
here), that

In these days the angel of topology and the devil of abstract algebra
fight for the soul of each individual mathematical domain.

But I can see why people are puzzled by the quote, so I'd like to add some
more information (too much to put in a comment) as another answer.

First, what is the context? The quote occurs in Weyl's paper Invariants
in Duke Math. J. 5 (1939), pp. 489--502, the first page of which can be seen
here. This page includes most of what Weyl has to say on algebra v.
geometry, though the quote itself does not occur until p.500. Then on p.501
Weyl explains his discomfort with algebra as follows

In my youth I was almost exclusively active in the field of analysis;
the differential equations and expansions of mathematical physics were
the mathematical things with which I was on the most intimate footing.
I have never succeeded in completely assimilating the abstract
algebraic way of reasoning, and constantly feel the necessity of translating
each step into a more concrete analytic form.

Second, why the image of angel and devil? According to V.I Arnold,
writing here, Weyl had a particular image in mind, namely, the
Uccello painting "Miracle of the Profaned Host, Episode 6", which can be
viewed here.

Arnold describes this painting as "representing an event that happened in
Paris in 1290." "Legend" is probably a better word than "event," but in
any case it is a very strange origin for a famous mathematical quote.

exoplanet - How do scientists know if an Earth-like planet is really Earth-like?

The earth similarity index (ESI) is a weighted geometric mean of four similarities.

The formula documented on
http://phl.upr.edu/projects/earth-similarity-index-esi
(as of march 23, 2014) should be adjusted a bit, since
$n$ should be the weight sum instead of the number of planetary properties.
With the weights provided on the site we get

$$ESI =left( left(1-left | frac{r_E - r_P}{r_E + r_P} right |right)^{0.57}cdot
left(1-left | frac{rho_E - rho_P}{rho_E + rho_P} right |right)^{1.07}cdot\
left(1-left | frac{v_E - v_P}{v_E + v_P} right |right)^{0.70}cdot
left(1-left | frac{288mbox{K} - vartheta_P}{288mbox{K} + vartheta_P} right |right)^{5.58}right)^{frac{1}{7.92}},$$
with $r_E=6,371 mbox{ km}~~$ Earth's radius, $r_P$ the radius of the planet,
$rho_E=5.515mbox{ g}/mbox{cm}^3$ Earth's bulk density, $rho_P$ the bulk density of the planet,
$v_E=11.2 mbox{ km}/mbox{s}~~$ the escape velocity on the surface of Earth, $v_P$ the escape velocity on the surface of the planet, and
$vartheta_P$ the surface temperature of the planet; the weight sum is $0.57+1.07+0.70+5.58=7.92$.

Mars as an example:
With $r_P=0.53 r_E$, $rho_P=0.71 rho_E$, $v_P=0.45v_E$, $vartheta_P=227mbox{ K}$ we get
$$ESI_s =left( left(1-left | frac{0.47}{1.53} right |right)^{0.57}cdot
left(1-left | frac{0.29}{1.71} right |right)^{1.07}cdot\
left(1-left | frac{0.55}{1.45} right |right)^{0.70}cdot
left(1-left | frac{61mbox{ K}}{515mbox{ K}} right |right)^{5.58}right)^{frac{1}{7.92}}=\
(0.811241627cdot 0.819676889cdot 0.716163454cdot 0.494865663)^{frac{1}{7.92}}=
0.833189885$$ as surface similarity. (Some of the data used from here.)
Global similarity combines surface similarity with interior similarity.
Global similarity of Mars with Earth is about 0.7.

Surface gravity can be calculated from radius and bulk density of a planet.
The radius of an expoplanet can be estimated by the transit method, relating the estimated diameter of the star to the brightness change of the
star during planet transit. The mass of the planet can be estimated by the wobble of the radial velocity of the star (using Doppler shift).
By mass estimate of the star and the orbital period of the planet the distance of the planet to the star can be estimated. An estimate of the
absolute brightness of the star can then be used to estimate the surface temperture of the planet. There exist more methods.
The accuracy of these estimates vary with the quality of the observations.

ESI values between 0.8 and 1.0 are considered as Earth-like.

Details of the planet's atmosphere, amount of surface water, and other details are not considered in the formula. So it's just a very rough prioritization. With future spectroscopic data further refinement could be possible.

ag.algebraic geometry - Classification of simply connected smooth projective varieties?

The best kind of classification of smooth projective varieties would be a list of deformation types. We have this for curves, and something close to it for surfaces of Kodaira dimension less than 2. We do not have such a list for smooth projective surfaces of Kodaira dimension 2, not even for the simply connected ones, and the situation in higher dimensions is worse.

The next best kind might be a complete invariant, algorithmically computable from the defining equations. So far as I know, we don't have this either. For instance, one might ask whether the canonical ring $bigoplus_n{H^0(K^n)}$ is algorithmically computable. A recent triumph of the minimal model program has been to prove that this is finitely generated for non-singular varieties of general type; its Proj is then the canonical model of the variety. I presume that it's not effectively computable at present; perhaps someone else can comment on positive or negative results in this direction (say for surfaces?).

A still weaker request would be for a theorem which says "there exists an algorithm to decide whether these two varieties are deformation-equivalent". I think we probably do have this, by virtue of Grothendieck's theorem that, after you fix the Hilbert polynomial, there's a proper Hilbert scheme, which in particular has only finitely many connected components. As a matter of logic (logicians, please correct me if necessary!), if there are only finitely many possibilities, an algorithm exists to test which one you have - because there exists a finite list of those possibilities, encoded as numbers, and you just have to check yours against each of them. What we don't have is a practical method to produce that list.

This assertion does have content; by contrast, there's no algorithm to decide diffeomorphism of compact smooth manifolds (given as real semi-analytic sets, say) because one can't compute $pi_1$. There is an algorithm to check diffeomorphism of simply connected manifolds of dimension $>4$ (compute the cohomology groups, compare the finite list of $k$-invariants specifying the Postnikov tower and hence the homotopy type, compare the Pontryagin classes, appeal to a finiteness result from surgery theory - see Nabutovsky-Weinberger, "Algorithmic aspects of homeomorphism problems", MR1707346).

co.combinatorics - Consensus clustering using set union

Problem statement

Let $P$, $Q$ and $R$ be three partitions into $p$ nonempty parts (denoted by $P_h$'s, $Q_i$'s and $R_j$'s) of the set {$1,2,ldots,n$}. Find two permutations $pi$ and $sigma$ that minimise $$sum_{i=1}^pleft|P_icup Q_{pi_i}cup R_{sigma_i}right|.$$

Questions

1) Is there a polynomial time algorithm to solve this problem, or is it NP-hard to do so? What is the complexity of this problem (or of the corresponding decision problem)?

2) If the problem is indeed solvable in polynomial time, does it remain true for any number $kgeq 4$ of partitions?

Previous work

Berman, DasGupta, Kao and Wang study the same problem for $k$ partitions, but using pairwise $Delta$'s instead of $cup$ in the above sum. They prove that the problem is MAX-SNP-hard for $k=3$, even when each part has only two elements, by reducing MAX-CUT on cubic graphs to a special case of their problem, and give a $(2-2/k)$-approximation for any $k$. So far, I have not been able to find my problem in the literature, or to adapt their proof.

Easy subcases

Here are some subcases I've found to be solvable in polynomial time, I'll update this section as I go until the question is resolved:

• the case $k=2$;


• the case $p=2$, for any $k$;


• when $k=3$: when no two parts are equal and all parts have size $2$, we have the lower bound $3p+1$ (I don't know if it's tight).

mp.mathematical physics - Clifford Algebra in Dirac Equation

I am slightly confused by this question. The fact that one can formulate the Dirac equation in either (3,1) or (1,3) signature, which have non-isomorphic Clifford algebras and hence Clifford modules of different type (real for (1,3) and quaternionic for (3,1) in my naming conventions), does not mean that one is complexifying the Clifford algebra in formulating the Dirac equation.

Unitarity of the time evolution -- a physical requirement independent of choices -- requires that $i D$, where $D$ is the Dirac operator, be hermitian, and in turn this forces a certain hermiticity condition on the "gamma" matrices, in essence choosing a real form of the complex Clifford algebra. It is the spinor representation which can be taken to be complex since after all wave functions live in the tensor product of the Clifford module with a complex Hilbert space. This is not the same thing as complexifying the Clifford algebra, though.

So to summarise, in the Dirac equation the Clifford algebra is real, but the pinor representation can be taken to be complex.

Could supermassive black holes form in dwarf galaxies?

The prediction was that dwarf galaxies have SMBHs, but with smaller masses than regular galaxies. This was based on the idea that regular galaxies were built from dwarf galaxies and the BHs need to have started growing early on. Also the correlation between bulge mass and BH mass requires the BHs to come from the building blocks. A recent paper at arxiv detects 28 AGN in dwarf galaxies with BHs in the $10^3 - 10^4 M_{odot}$ mass range.

pr.probability - Probability of one binomial variable being greater than another.

Edit: I've filled in a few more details.

The Hoeffding bound from expressing $Y-X$ as the sum of $n$ differences between Bernoulli random variables $B_q(i)-B_p(i)$ is

$$Prob(Y-X ge 0) = Prob(Y-X + n(p-q) ge n(p-q)) le expbigg(-frac{2n^2 (p-q)^2}{4n}bigg)$$

$$Prob(Y-X ge 0) le expbigg(-frac{(p-q)^2}{2}nbigg)$$

I see three reasons you might be unhappy with this.

• The Hoeffding bound just isn't sharp. It's based on a Markov bound, and that is generally far from sharp.


• The Hoeffding bound is even worse than usual on this type of random variable.


• The amount by which the Hoeffding bound is not sharp is worse when $p$ and $q$ are close to $0$ or $1$ than when they are close to $frac12$. The bound depends on $p-q$ but not how extreme $p$ is.

I think you might address some of these by going back to the proof of Hoeffding's estimate, or the Bernstein inequalities, to get another estimate which fits this family of variables better.

For example, if $p=0.6$, $q=0.5$, or $p=0.9$, $q=0.8$, and you want to know when the probability is at most $10^{-6}$, the Hoeffding inequality tells you this is achieved with $nge 2764$.

For comparison, the actual minimal values of $n$ required are $1123$ and $569$, respectively, by brute force summation.

One version of the Berry-Esseen theorem is that the Gaussian approximation to a cumulative distribution function is off by at most

$$0.71 frac {rho}{sigma^3 sqrt n}$$
where $rho/sigma^3$ is an easily computed function of the distribution which is not far from 1 for the distributions of interest. This only drops as $n^{-1/2}$ which is unacceptably slow for the purpose of getting a sharp estimate on the tail. At $n=2764$, the error estimate from Berry-Esseen would be about $0.02$. While you get effective estimates for the rate of convergence, those estimates are not sharp near the tails, so the Berry-Esseen theorem gives you far worse estimates than the Hoeffding inequality.

Instead of trying to fix Hoeffding's bound, another alternative would be to express $Y-X$ as a sum of a (binomial) random number of $pm1$s by looking at the nonzero terms of $sum (B_q(i)-B_p(i))$. You don't need a great lower bound on the number of nonzero terms, and then you can use a sharper estimate on the tail of a binomial distribution.

The probability that $B_q(i)-B_p(i) ne 0$ is $p(1-q) + q(1-p) = t$. For simplicity, let's assume for the moment that there are $nt$ nonzero terms and that this is odd. The conditional probability that
$B_q(i)-B_p(i) = +1$ is $w=frac{q(1-p)}{p(1-q)+q(1-p)}$.

The Chernoff bound on the probability that the sum is positive is $exp(-2(w-frac 12)^2tn)$.

$$ exp(-2bigg(frac{q(1-p)}{p(1-q)+q(1-p)} - frac 12bigg)^2 big(p(1-q) + q(1-p)big) n)$$

is not rigorous, but we need to adjust $n$ by a factor of $1+o(1)$, and we can compute the adjustment with another Chernoff bound.

For $p=0.6, q=0.5$, we get $n ge 1382$. For $p=0.9, q=0.8$, we get $n ge 719$.

The Chernoff bound isn't particularly sharp. Comparison with a geometric series with ratio $frac{w}{1-w}$ gives that the probability that there are more $+1$s than $-1$s is at most

$${nt choose nt/2} w^{nt/2} (1-w)^{nt/2} frac {1-w}{1-2w}$$

This gives us nonrigorous bounds of $nt gt 564.4, n ge 1129$ for $p=0.6,q=0.5$ and
$ntgt 145.97, nge 562$ for $p=0.9,q=0.8$. Again, these need to be adjusted by a factor of $1+o(1)$ to get a rigorous estimate (determine $n$ so that there are at least $565$ or $146$ nonzero terms with high probability, respectively), so it's not a contradiction that the actual first acceptable $n$ was $569$, greater than the estimate of $562$.

I haven't gone through all of the details, but this shows that the technique I described gets you much closer to the correct values of $n$ than the Hoeffding bound.

gn.general topology - Topological spaces that resemble the space of irrationals

(This question actually arose in some research on number theory.)

I once learned that any countable dense subspace of any Euclidean space ℝⁿ is homeomorphic to the rationals ℚ.

Now I wonder if something similar is true for the irrationals J := ℝ - ℚ (with the subspace topology from ℝ).

Let c denote the cardinality of the continuum.

I. Is each cartesian power Jⁿ homeomorphic to J ?

Also, how far can this be pushed?

II. Let X be a dense totally disconnected subspace of ℝⁿ such that every neighborhood of each point of X contains c points. Is X homeomorphic to J ?

What about for such subspaces of fairly nice subspaces of ℝⁿ ?

IIa. Let X be any subspace of ℝⁿ as described in II., and let B denote any subspace of ℝⁿ homeomorphic to [the open unit ball in ℝⁿ union any subset of its boundary]. Then is X ∩ B homeomorphic to J ?

And what about greater generality ?

III. Is there a simple set of conditions that describe exactly all spaces (or subspaces of ℝⁿ) that are homeomorphic to J ? What about Jⁿ ? (Perhaps the word homogeneous or metric needs to be included.)

(I found nothing relevant via Google, in MathSciNet, or here on MathOverflow.)

galaxy - Which spiral arm of the Milky Way is Kepler-62 in?

I am trying to find out which spiral arm of the Milky Way the star Kepler-62 and its planetary system are in. All I can find by Google searching is that it is part of the Lyra constellation, but I don't know if that entire constellation is within 1 spiral arm or not, nor which arm it is in.

Is Kepler-62 in the Carina–Sagittarius Arm, or in the Orion Arm, or...?

The only map picture I can find that is somewhat related is this (I assume that the yellow light area is the area in which Kepler-62 was sighted by the Kepler spacecraft): http://upload.wikimedia.org/wikipedia/commons/b/be/LombergA1024.jpg

Thank you!

Categories of Geometry

there are a couple of subtle differences. Some of the concepts are only relevant when talking about geometry from an axiomatic perspective.

When one is talking about geometry from an axiomatic perspective ( you want to talk about points, lines, planes, angles etc.) you are really looking at a model for your axioms. Here we might talk about Euclidean, Riemannian, Hyperbolic, Projective, Spherical and (perhaps) Elliptic geometries. Usually the main difference is whether or not we choose to take the parallel postulate as an axiom or one of its negations. If we take the parallel postulate then we are working in a model of Euclidean geometry, this is sort of flat geometry where things are what you expect. In Spherical geometry (sometimes called Riemannian, and maybe elliptical but i am not sure about that) is where we have no parallel lines, a model of this is the sphere where we take lines to be great circles. Note though that typically Riemannian geometry is about manifolds with a riemannian structure, but that we can save until later. Hyperbolic geometry is when we have infinitely many parallel lines through a given point, a model of this is the Poincare disk. Projective geometry has every two parallel lines meet... at the point at infinity. There are much better references for this stuff but so far we have just been looking at models of axiom systems that one could work with and "do geometry" in.

Algebraic, differential, and Riemannian geometry are more complicated. Here are some "one line slogans" which i am sure others can improve upon. DIfferential geometry is about curves, surfaces and homogeneous objects that you want to study via calculus, there is a priori some smooth structure on that object. Algebraic geometry wants to study similar objects but only when you are concerned with what you can tell about an object via rational functions. For me, this starts with the Gelfand-Naimark result. it gets much much richer. RIemannian geometry is differential geometry except you have a well behaved notion of distance between points, distance ON the hypersurface itself! All of these could be fixed by someone with more knowledge then I on the respective subjects.

I didnt mention affine geometry, but i will take a stab and say it is like a geometry in any one of the above models except you only care about "flat" things, lines, planes, etc.

The above comment suggests looking at some good books. It mentions Klein's Erlangen program, which is where Klein proposed studying a geometry by understanding the group of symmetries that preserve it. So you can think of the first big paragraph as looking at groups and the geometries you get from them. The second big paragraph you can get by looking at different types of sheaves on various topological spaces (i think) where the sheaf keeps track of the type of structure you care about. The two classes are sort of a bit different, but they are similar in that you can look at objects that act as receptacles for the geometric content: groups and sheaves.

any suggestions are very welcome!

solar system - Why are gas giants colored the way they are?

You could just have Googled this question.

Post #5 from the first hit:

First off, by "wondering what colors different gas giants can be", you
are presumably asking about their light spectra through the visible
range of wavelengths (380-720 nm), right?**

Light interacts primarily with electrons. It is scattered or absorbed
in the presence of electrons, which come in a variety of "phases".
Here are the most relevant:

1) free within an ionized gas (can absorb in the presence of the
electric field of an ion)

2) attached to atoms and ions

3) attached to molecules

4) attached to molecules that have condensed into solid state aerosols and grains, or liquid droplets.

The most important thing to take away from this is that every type of
material, in terms of composition and "phase", absorbs and scatters
light uniquely.

The prevalence and importance of each of the above four "phases"
depend on (a) the elemental composition of the giant planet's
atmosphere (defined as that layer responsible for light
reflected/emitted by planet), and (b) its equation of state (how
pressure changes as a function of density and temperature). The first
of these provides the raw materials, and the second arranges them in
"phase". Very roughly speaking, one may assign decreasing temperatures
(T) to the above 4 "phases" moving down the list 1-->4. Pressure (P)
also plays a role, and in general one may place the above phases on a
P-T diagram. Physics and Chemistry are at work to determine what kind
of "stuff" is present as a function of depth through the giant planet
atmosphere. One rule of thumb is that chemistry is much more effective
at higher temperatures (to a point) and/or in the presence of
moderately energetic light.

Next, before proceeding, go back and read the bold statement, above.

To finish off this overly long post: Two giant planets of equal bulk
compositions will almost certainly appear differently if their P vs. T
profiles differ, or if their atmospheric compositions differ (e.g.,
due to convective mixing from the interior, mixing due to wind
currents, heterogeneous settling of heavier matter towards the center
over time). Two giant planets of equal bulk compositions, but
differing ages will appear differently, since a planet's interior
cools over time, affecting P-T relation within the planet as well as
its thermally emitted spectrum. The intensity and spectral shape of
the light incident from the parent star will affect the P-T diagram,
the chemistry and phase of the matter, the thermally emitted spectrum,
as well as the distribution of photons available for scattering.

From the second hit:

Jupiter is a giant gas planet with an outer atmosphere that is mostly
hydrogen and helium with small amounts of water droplets, ice
crystals, ammonia crystals, and other elements. Clouds of these
elements create shades of white, orange, brown and red. Saturn is also
a giant gas planet with an outer atmosphere that is mostly hydrogen
and helium. Its atmosphere has traces of ammonia, phosphine, water
vapor, and hydrocarbons giving it a yellowish-brown color. Uranus is a
gas planet which has a lot of methane gas mixed in with its mainly
hydrogen and helium atmosphere. This methane gas gives Uranus a
greenish blue color Neptune also has some methane gas in its mainly
hydrogen and helium atmosphere, giving it a bluish color

the sun - the metal distribution in our solar system

Production of Emission Lines of Iron in the Sun

The visible Emission Spectrum of Iron contains around two hundred emission lines from violet to red which represent the large number of Electron energy losses when Electrons drop from a higher Quantum Energy Level to a lower Quantum Energy Level

A Neutral 56Fe atom has an Electron Configuration of 2, 8, 14, 2

Within the Sun Iron Nuclei above 7,250K start collecting Electrons at the lowest energy level (2) and in the process Photon Energy at Wavelengths less than 4,000A are emitted as Ultra-Violet Light

As the Iron Nuclei cool further, the Electrons progressively drop into the second (8)and third (14) energy levels progressively producing Photons at (5,800K 5,000A Green), (4,800K 6,000A Orange) and (4,150K 7,000A Red)

The final two valence Electrons produce Infra-Red radiation at temperatures below 4,150K

The Energy density of the Blue, Indigo, Violet emission lines is much larger than the density of the Red, Orange, Yellow emission lines which is due to the higher Temperature at the Blue, Indigo, and Violet emission lines. The Energy density of the Green emission lines lies between those of Violet and Red

gn.general topology - What is a metric space?

A classic category-theoretic view of metric spaces says that the "correct" maps are the distance-decreasing ones:
$$
d(f(a), f(a')) leq d(a, a')
$$
where $A$ and $B$ are metric spaces, $f: A to B$, and $a, a' in A$. Then all maps are continuous, and the isomorphisms are the isometries onto.

This comes from viewing metric spaces as enriched categories, as proposed by Lawvere. The enriched functors are then exactly the distance-decreasing maps.

Edit Let me add some detail. Consider the set $V=[0, infty]$ of non-negative reals. (The inclusion of $infty$ isn't important here.) It's ordered by $geq$, and so can be regarded as a category: there's one map $x to y$ if $x geq y$, and there are no maps $x to y$ otherwise. It becomes a monoidal category under $+$ and $0$.

A $V$-enriched category is then a set $A$ of objects (or points) together with, for each pair $(a, b)$ of points, an object $A(a, b)$ of $V$ --- that is, a non-negative real, which you might prefer to call $d(a, b)$. Composition then becomes the triangle inequality, and identities the assertion that the distance from a point to itself is $0$. So, a $V$-enriched category is a "generalized metric space": there's no requirement of symmetry (so you could take distance to be the work done in moving between points of a mountainous region) or that points distance $0$ apart are equal (which is just like not asking for isomorphic objects of a category to be equal).

You should then be able to see that $V$-enriched functors are what I said they were.

Edit re Lipschitz maps I don't want to evangelize this point
of view too much. But it's a matter of fact that Lipschitz maps do
arise naturally in this framework.

To explain this I first need to explain a little about 'change of
base' for enriched categories. Any lax monoidal functor $Phi:
mathcal{V} to mathcal{W}$ induces a functor $Phi_*:
mathcal{V}-mathbf{Cat} to mathcal{W}-mathbf{Cat}$, in an obvious
way. For example, if $Phi: mathbf{Vect} to mathbf{Set}$ is the
forgetful functor, then $Phi_*$ sends a linear category to its
underlying ordinary category.

This means that given a lax monoidal $Phi: mathcal{V} to mathcal{W}$,
a $mathbf{V}$-enriched category $mathbf{A}$, and a
$mathbf{W}$-enriched category $mathbf{B}$, we can define a
$Phi$-enriched functor $mathbf{A} to mathbf{B}$ to be a
$mathcal{W}$-enriched functor $Phi_*(mathbf{A}) to mathbf{B}$. One
might also call this a 'functor over $Phi$'.

That's completely general enriched category theory. Now let's
apply it to $mathcal{V} = mathcal{W} = [0, infty]$. For any $M
geq 0$, multiplication by $M$ defines a (strict) monoidal functor
$Mcdot -: [0, infty] to [0, infty]$. Let $A$ and $B$ be metric
spaces. Then an $(Mcdot -)$-enriched functor from $A$ to $B$ is
precisely a function $f: A to B$ such that
$$
d(f(a), f(a')) leq Mcdot d(a, a')
$$
for all $a, a' in A$. In other words, it's a Lipschitz map.

A bit more can be squeezed out of this. The maps $Mcdot -$ are the
strict monoidal endofunctors of $[0, infty]$. But we can talk
about $phi$-enriched maps of metric spaces for any lax
monoidal endofunctor of $[0, infty]$. `Lax monoidal' means that
$$
phi(0) = 0,

phi(x + y) leq phi(x) + phi(y),
$$
which is a kind of concavity property (satisfied by $phi(x) = sqrt{x}$,
for instance). Then a $phi$-enriched map from $A$ to $B$ is a
function $f: A to B$ such that
$$
d(f(a), f(a')) leq phi(d(a, a'))
$$
for all $a, a' in A$. Is that kind of map found useful?

linear algebra - Infinite Tensor Products

Let $A$ be a commutative ring and $M_i, i in I$ be a infinite family of $A$-modules. Define their tensor product $bigotimes_{i in I} M_i$ to be a representing object of the functor of multilinear maps defined on $prod_{i in I} M_i$ (this exists by the usual construction). Thus there is a universal multilinear map $otimes : prod_{i in I} M_i to bigotimes_{i in I} M_i$. Some years ago, I wanted to examine this infinite tensor product, but in the literature I could not find anything going beyond some natural isomorphisms (e.g. associativity) or the submodule consisting of tensors which become eventually constant a specific element of $prod_{i in I} M_i$ which yields a colimit of finite tensor products (denoted $U_x$ below). In general, it seems to be quite hard to describe $bigotimes_{i in I} M_i$. For example, for a field $K$, $K otimes_K otimes_K ...$ has dimension $|K^*|^{aleph_0}$ (see below) and you cannot write down a basis, which might be scary when you see it the first time. The point is that multilinear relations cannot be applied infinitely many times at once: For example in $K otimes_K otimes_K ...$, we have $x_1 otimes x_2 otimes ... = y_1 otimes y_2 otimes ...$ if and only if $x_i = y_i$ for almost all $i$ and for the rest we have $prod_i x_i = prod_i y_i$.

Before posing my question, I provide some results.

1.1. Assume that $M_i$ are torsionfree $A$-modules (meaning $am=0 Rightarrow a=0 vee m=0$). In this case, we may decompose $bigotimes_{i in I} M_i$ as follows: Define $X = prod_{i in I} M_i setminus {0}$ and let $x sim y Leftrightarrow {i : x_i neq y_i}$ finite. Then $sim$ is an equivalence relation on $X$. Let $R$ be a set of representatives (this makes this description ugly!). Then there is a canonical map

$H : text{Mult}(prod_{i in I} M_i,-) to prod_{x in R} lim_{E subseteq I text{~finite}} text{Mult}^x(prod_{i in E} M_i,-)$,

where $text{Mult}^x$ indicates that the transition maps of the limit are given by inserting the entries of $x$, and it is not hard to show that $H$ is bijective. Thus

$bigotimes_{i in I} M_i = bigoplus_{x in R} U_x$,

where $U_x = cup_{E subseteq I text{~finite}} bigotimes_{i in E} M_i otimes otimes_{i notin E} x_i$ is the colimit of the finite tensor products $otimes_{i in E} M_i$ (the transition maps given by tensoring with entries of $x$). The canonical maps $otimes_{i in E} M_i to U_x$ don't have to be injective; at least when $A$ is a PID, this is the case. Remark that $U_x$ only depends on the equivalence class of $x$, so that the decomposition into the $U_x$ is canonical, whereas the representation of $U_x$ as direct limit (including the transition maps!) depends really on $x$.

1.2 If $M_i = A$ is an integral domain, we get $bigotimes_{i in I} A = oplus_{x in R} U_x$, where $U_x$ is the direct limit of copies $A_E$ of $A$, for every finite subset $E subseteq I$, and transition maps $prod_{i in E' setminus E} x_i : A_E to A_{E'}$ for $E subseteq E'$. $U_x$ is just the localization of $A$ at the $x_i$.

1.3 If $A$ is a field, and $M_i$ has basis $B_i$, then $B_x = cup_{E subseteq I} bigotimes_{i in E} B_i otimes otimes_{i notin E} x_i$ is a basis of $U_x$ and thus $cup_{x in R} B_x$ is a basis of $bigotimes_{i in I} M_i$. According to this question, this has cardinality $max(|X|,|I|,max_i(dim(M_i)))$.

1.4 If $A_i$ are $A$-algebras, then $bigotimes_{i in I} A_i$ is a $A$-algebra. If the $A_i$ are integral domains, then it is a graded algebra by the monoid $X/sim$ with components $U_x$.

If $A=A_i=K$ is a field with $U=K^x$, then there is a vector space isomorphism between $bigotimes_{i in I} K$ and the group algebra $K[U^I / U^{(I)}]$. A sufficient, not neccessary, condition for the existence of a $K$-algebra isomorphism is that $U^{(I)}$ is a direct summand of $U^I$, which is quite rare (see this question). Nevertheless, we can ask if these $K$-algebras isomorphic. In some sense I have proven this already locally (subalgebras given by finitely generated subgroups of the group $U^I / U^{(I)}$ are isomorphic, in a terribly uncanonical way). Many questions I'm currently posing here are addressed to this problem.

2.1 What about interchanging tensor product with duals? Let $(V_i)_{i in I}$ be a family of vector spaces over a field $K$. For elements $lambda_i in K$, define their infinite product $prod_{i in I} lambda_i$ to be the usual product if $lambda_i=1$ for almost all $i$, and otherwise to be $0$. This yields a multilinear map $prod : K^I to K$ and thus a linear map
$delta : bigotimes_{i in I} V_i^* to (bigotimes_{i in I} V_i)^*, otimes_i f_i mapsto (otimes_i x_i mapsto prod_{i in I} f_i(x_i)).$
Then it can be shown that $delta$ is injective, but the proof is pretty fiddly.

2.2 Let $W_i$ be another family of vector spaces over a field $K$. Then there is a canonical map
$alpha : bigotimes_{i in I} Hom(V_i,W_i) to Hom(bigotimes_{i in I} V_i, bigotimes_{i in I} W_i).$
Is $alpha$ injective? This is known when $I$ is finite.

3 What about other properties of finite tensor products, do they generalize? For example let $J_i, i in I$ be a family of index sets and $M_{i,j}$ be a $A$-module where $i in I, j in J_i$. Then there is a canonical homomorphism
$delta : bigoplus_{k in prod_{i in I} J_i} bigotimes_{i in I} M_{i,k(i)} to bigotimes_{i in I} oplus_{j in J_i} M_{i,j}$.
It can be shown that $delta$ is injective, but is it also bijective (as in the finite case)?

4 The description of the tensor product given in 1.1 depends on a set of representatives and is not handy when you want to prove something. Are there better descriptions?

Remark that in this question I'm not interested in infinite tensor products defined in functional analysis or just colimits of finite ones. I'm interested in the tensor product defined above (which probably every mathematician regards as "the wrong one"). Any hints about their structure or literature about it are appreciated.

the sun - Hour angle sunrise calculation problem

I am trying to program a sunrise/sunset calculator using this formula tutorial.

If you follow the steps, eventually you get to this formula for "H" (the "hour angle"):

$H = arccos( [sin(-0.83) - sin(ln) * sin(δ)] / [cos(ln) * cos(δ)] )$

Which comes with a note attached:

Note: If H is undefined, then there is either no sunrise (in winter) or no sunset (in summer) for the supplied latitude.

Now, as it turns out, this happens quite often (because inverse cosine has a domain of -1 to 1). I ran a program and here are all of the dates where the H is defined from Jan 2008 through Dec 2014 (using a longitude west of -77.0368710 and latitude north of 38.9071920 [Washington D.C.]):

January 1 2008
January 3 2008
January 4 2008
January 11 2008
January 30 2008
******************************
February 5 2008
February 24 2008
******************************
March 2 2008
March 24 2008
March 25 2008
March 29 2008
March 31 2008
******************************
April 3 2008
April 4 2008
April 6 2008
April 11 2008
April 12 2008
April 13 2008
April 14 2008
April 17 2008
April 19 2008
April 22 2008
April 23 2008
April 25 2008
April 26 2008
April 30 2008
******************************
May 2 2008
May 3 2008
May 5 2008
May 6 2008
May 8 2008
May 14 2008
May 15 2008
May 19 2008
May 21 2008
May 24 2008
May 25 2008
May 27 2008
******************************
June 1 2008
June 2 2008
June 3 2008
June 4 2008
June 7 2008
June 9 2008
June 12 2008
June 13 2008
June 15 2008
June 16 2008
June 20 2008
June 22 2008
June 23 2008
June 25 2008
June 26 2008
June 28 2008
******************************
July 4 2008
July 5 2008
July 9 2008
July 11 2008
July 14 2008
July 15 2008
July 17 2008
July 22 2008
July 23 2008
July 24 2008
July 25 2008
July 28 2008
July 30 2008
******************************
August 2 2008
August 3 2008
August 5 2008
August 6 2008
August 10 2008
August 12 2008
August 13 2008
August 15 2008
August 16 2008
August 18 2008
August 24 2008
August 25 2008
August 29 2008
August 31 2008
******************************
September 3 2008
September 4 2008
September 6 2008
September 11 2008
September 12 2008
September 13 2008
September 14 2008
September 17 2008
September 19 2008
September 22 2008
September 23 2008
September 25 2008
September 26 2008
September 30 2008
******************************
October 2 2008
October 3 2008
October 5 2008
October 6 2008
October 8 2008
October 14 2008
October 15 2008
October 19 2008
October 21 2008
October 24 2008
October 25 2008
October 27 2008
******************************
November 1 2008
November 2 2008
November 3 2008
November 4 2008
November 7 2008
November 9 2008
November 12 2008
November 13 2008
November 15 2008
November 16 2008
November 20 2008
November 22 2008
November 23 2008
November 25 2008
November 26 2008
November 28 2008
******************************
December 4 2008
December 5 2008
December 9 2008
December 11 2008
December 14 2008
December 15 2008
December 17 2008
December 22 2008
December 23 2008
December 24 2008
December 25 2008
December 28 2008
December 30 2008
******************************
January 2 2009
January 3 2009
January 16 2009
January 23 2009
******************************
February 11 2009
February 17 2009
******************************
March 8 2009
March 15 2009
March 23 2009
March 25 2009
March 26 2009
March 28 2009
March 29 2009
March 31 2009
******************************
April 6 2009
April 7 2009
April 11 2009
April 13 2009
April 16 2009
April 17 2009
April 19 2009
April 24 2009
April 25 2009
April 26 2009
April 27 2009
April 30 2009
******************************
May 2 2009
May 5 2009
May 6 2009
May 8 2009
May 9 2009
May 13 2009
May 15 2009
May 16 2009
May 18 2009
May 19 2009
May 21 2009
May 27 2009
May 28 2009
******************************
June 1 2009
June 3 2009
June 6 2009
June 7 2009
June 9 2009
June 14 2009
June 15 2009
June 16 2009
June 17 2009
June 20 2009
June 22 2009
June 25 2009
June 26 2009
June 28 2009
June 29 2009
******************************
July 3 2009
July 5 2009
July 6 2009
July 8 2009
July 9 2009
July 11 2009
July 17 2009
July 18 2009
July 22 2009
July 24 2009
July 27 2009
July 28 2009
July 30 2009
******************************
August 4 2009
August 5 2009
August 6 2009
August 7 2009
August 10 2009
August 12 2009
August 15 2009
August 16 2009
August 18 2009
August 19 2009
August 23 2009
August 25 2009
August 26 2009
August 28 2009
August 29 2009
August 31 2009
******************************
September 6 2009
September 7 2009
September 11 2009
September 13 2009
September 16 2009
September 17 2009
September 19 2009
September 24 2009
September 25 2009
September 26 2009
September 27 2009
September 30 2009
******************************
October 2 2009
October 5 2009
October 6 2009
October 8 2009
October 9 2009
October 13 2009
October 15 2009
October 16 2009
October 18 2009
October 19 2009
October 21 2009
October 27 2009
October 28 2009
******************************
November 1 2009
November 3 2009
November 6 2009
November 7 2009
November 9 2009
November 14 2009
November 15 2009
November 16 2009
November 17 2009
November 20 2009
November 22 2009
November 25 2009
November 26 2009
November 28 2009
November 29 2009
******************************
December 3 2009
December 5 2009
December 6 2009
December 8 2009
December 9 2009
December 11 2009
December 17 2009
December 18 2009
December 22 2009
December 24 2009
December 27 2009
December 28 2009
December 30 2009
******************************
January 4 2010
January 10 2010
January 17 2010
January 29 2010
******************************
February 5 2010
February 24 2010
******************************
March 2 2010
March 9 2010
March 21 2010
March 23 2010
March 25 2010
March 28 2010
March 29 2010
March 31 2010
******************************
April 1 2010
April 5 2010
April 7 2010
April 8 2010
April 10 2010
April 11 2010
April 13 2010
April 19 2010
April 20 2010
April 24 2010
April 26 2010
April 29 2010
April 30 2010
******************************
May 2 2010
May 7 2010
May 8 2010
May 9 2010
May 10 2010
May 13 2010
May 15 2010
May 18 2010
May 19 2010
May 21 2010
May 22 2010
May 26 2010
May 28 2010
May 29 2010
May 31 2010
******************************
June 1 2010
June 3 2010
June 9 2010
June 10 2010
June 14 2010
June 16 2010
June 19 2010
June 20 2010
June 22 2010
June 27 2010
June 28 2010
June 29 2010
June 30 2010
******************************
July 3 2010
July 5 2010
July 8 2010
July 9 2010
July 11 2010
July 12 2010
July 16 2010
July 18 2010
July 19 2010
July 21 2010
July 22 2010
July 24 2010
July 30 2010
July 31 2010
******************************
August 4 2010
August 6 2010
August 9 2010
August 10 2010
August 12 2010
August 17 2010
August 18 2010
August 19 2010
August 20 2010
August 23 2010
August 25 2010
August 28 2010
August 29 2010
August 31 2010
******************************
September 1 2010
September 5 2010
September 7 2010
September 8 2010
September 10 2010
September 11 2010
September 13 2010
September 19 2010
September 20 2010
September 24 2010
September 26 2010
September 29 2010
September 30 2010
******************************
October 2 2010
October 7 2010
October 8 2010
October 9 2010
October 10 2010
October 13 2010
October 15 2010
October 18 2010
October 19 2010
October 21 2010
October 22 2010
October 26 2010
October 28 2010
October 29 2010
October 31 2010
******************************
November 1 2010
November 3 2010
November 9 2010
November 10 2010
November 14 2010
November 16 2010
November 19 2010
November 20 2010
November 22 2010
November 27 2010
November 28 2010
November 29 2010
November 30 2010
******************************
December 3 2010
December 5 2010
December 8 2010
December 9 2010
December 11 2010
December 12 2010
December 16 2010
December 18 2010
December 19 2010
December 21 2010
December 22 2010
December 24 2010
December 30 2010
December 31 2010
******************************
January 17 2011
January 23 2011
January 30 2011
******************************
February 11 2011
February 18 2011
******************************
March 9 2011
March 15 2011
March 22 2011
March 23 2011
March 25 2011
March 30 2011
March 31 2011
******************************
April 1 2011
April 2 2011
April 5 2011
April 7 2011
April 10 2011
April 11 2011
April 13 2011
April 14 2011
April 18 2011
April 20 2011
April 21 2011
April 23 2011
April 24 2011
April 26 2011
******************************
May 2 2011
May 3 2011
May 7 2011
May 9 2011
May 12 2011
May 13 2011
May 15 2011
May 20 2011
May 21 2011
May 22 2011
May 23 2011
May 26 2011
May 28 2011
May 31 2011
******************************
June 1 2011
June 3 2011
June 4 2011
June 8 2011
June 10 2011
June 11 2011
June 13 2011
June 14 2011
June 16 2011
June 22 2011
June 23 2011
June 27 2011
June 29 2011
******************************
July 2 2011
July 3 2011
July 5 2011
July 10 2011
July 11 2011
July 12 2011
July 13 2011
July 16 2011
July 18 2011
July 21 2011
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You'll notice there are quite a lot of dates where H (the hour angle) is undefined. I understand from a trigonometric standpoint why it is incalculable, but is there a scientific reason for this? Also, how do they calculate when the sun will set and when the sun will rise on those days? Is it an average of the last known date and the next known date?

What is the hour angle, exactly, and how do you calculate the sunrise/sunset if you can't calculate it?

planet - What is the orientation of planetary orbits?

I'm working on a planetary motion simulator. I've been working through the equations anomaly, eccentricity, etc. The one thing I'm curious about is if all the ellipses are oriented the exact same way (the direction of the periapsis of all planets are pointing the same way), or is it different for each planet?

What would this be called? I've only seen eccentricity and inclination.

linear algebra - Matrix decomposition problem

As Sergei Ivanov pointed out in his first comment, it is necessary and sufficient, to solve your (ii) and (iii), to have $$ sum_{i = 1}^{n} x_i = sum_{i=1}^{n} y_i ; ; .$$ If this is true then take $ M = sum_{i = 1}^{n} x_i = sum_{i=1}^{n} y_i ; ; . $ The most natural solution to (ii) and (iii) is the rank-one matrix $C^0$ given by $$ c_{ij}^{0} = frac{x_i y_j}{M} $$

Now, there is a kernel involved next of dimension $(n-1)^2,$ these being matrices $F$ satisfying $F 1 = 0$ and $F^t 1 = 0.$ One may specify any entries desired in the upper left square $n-1$ by $n-1$ block of $F$, then fill in the final column and row. Any solution of (ii) and (iii) must be of the form $$ C^0 + F ; ; .$$

Progress: for your purpose it is better to specify the matrix $F$ as shown below for $n=4,$ the other entries of $F$ are forced by the condition that all row sums and all column sums are zero.
$$ F = left( begin{array}{cccc} & r & s & t \
& & u & v \
a & & & w \
b & c & &
end{array} right). $$
As a result, $C^0 + F$ can be arranged to have all zeroes above the diagonal, then zeroes below a single layer alongside the main diagonal. The result is slightly better than what is called tridiagonal in that the entries above the diagonal are also 0.

http://en.wikipedia.org/wiki/Tridiagonal_matrix

We have arranged
$$ C^0 + F = left( begin{array}{cccc}
a_1 & & & \
r_1 & b_1 & & \
& s_1 & c_1 & \
& & t_1 & d_1
end{array} right) .$$

Now that we know that we can insist on this shape, we can just start out with this and a simple scheme involving your (ii) and (iii) defines the values for all the nonzero positions. Furthermore,
if in addition $x = y,$ then it follows from (ii) and (iii) that
$C^0 + F$ is actually diagonal. Done.

Does the Earth have another moon?

One kilometer, no way! That would've been known since long ago. Most asteroids of that size have already been found, all the way out to the asteroid belt beyond Mars. Earth has no second Moon. But there are always some tiny asteroids around, which are temporarily captured by Earth's gravity. Here's a funny illustration of such an orbit, it is not what we would like to call a moon. I think that only one of them have been found, the 5 meter diameter 2006RH120. Smaller objects would not be detectable today, but there are at any moment likely many meter and submeter sized asteroids visiting the Sun-Earth Lagrange points. A paper about it.

Space is not yet being scanned, but telescopes are now being built to do that. It will be a new kind of astronomy, looking for the unexpected, and who knows what will be found out there in the blackness. Still today amateur astronomers can discover asteroids with the telescope in their backyard. I'm afraid that space telescopes like Gaia will kill their hobby and that this classic astronomical mapping of the sky will finally be finished.

set theory - Is it possible to decrease the rank of known structures?

Recall that for a set $x$ its rank $alpha$ is the least ordinal such that $x in V_{alpha+1}$. Or in other words: $x$ is built up out of $alpha$ levels of braces and the empty set.

I think with the usual constructions of numbers (cartesian products, sets of equivalence classes, Dedekind cuts, etc.), we have

$rank(mathbb{N})=omega, rank(mathbb{Z})=omega+4, rank(mathbb{Q})=omega+8, rank(mathbb{R})=omega+10$

Now it is possible to find a bijection $mathbb{Z} cong mathbb{N}$, so that there is a copy of $mathbb{Z}$ of smaller rank, namely $omega$. But this is, of course, nonsense. We should also consider the ring structure on $mathbb{Z}$, which is given by two maps $mathbb{Z} times mathbb{Z} to mathbb{Z}$. Therefore we might ask the following:

Is there a ring $(R,+,*)$, which is isomorphic to $(mathbb{Z},+,*)$, but $rank(R,+,*) < rank(mathbb{Z},+,*)$? What about the other rings above?

Of course, this question is just out of curiosity. I doubt that anybody cares about these bounds of ranks (if not, please let me know).

gravity - Has someone looked the other way?

What I ask myself is: Couldn't there be "negative" bundles of mass just the other way that pushes matter away instead of invisible dark matter that pulls it?

The galaxy rotation curve indicates a (positively massed) dark matter distribution that is close to spherically symmetric; cf. dark matter halo. I take it that you are asking whether instead of modeling it as a positive-mass distribution around the galaxy, we can model it instead as a negative-mass distribution surrounding the galaxy pushing the stars inward.

No, that's not possible. Since the effect is spherically symmetric, the hypothetical negative-mass distribution would need to be spherically symmetric as well. However, Newton's shell theorem guarantees that everywhere inside a void surrounded by such a distribution with this symmetry, whatever the sign of the mass is, the gravitational force vanishes. So a negatively-massed dark matter distribution outside the galaxy would do nothing at all.

In general relativity, the analogous statement is guaranteed by Birkhoff's theorem, and although GTR is nonlinear (and there are stars in the way), in our galaxy gravity is weak enough to be handled by the linearized theory.

What is the proof for the fact "Gravity is always attractive" beside that fact that it seems to be?

In general relativity, gravity doesn't have to be attractive. For example, inside a perfect fluid with density energy $rho$ and pressure $p$, $rho+3p<0$ would imply that gravity is locally repulsive in the sense that a ball of test particles initially at rest expands rather than contracts under gravitational freefall. More generally, the strong energy condition characterizes whether or not gravity is attractive in this sense.

However, there is no known material that violates the strong energy condition, besides possibly dark energy, which is a cosmological constant in the standard ΛCDM model, as Gerald says.

Has someone tried to find something like that?

There are many studies and simulations that consider dark matter with a nonzero pressure component, sometimes even anisotropic pressure components. For example, "hot" dark matter would have $psimrho/3$, while "cold" dark matter would have $psim 0$, "warm" dark matter somewhere inbetween. The CDM in the ΛCDM model stands for "cold dark matter", naturally.

There are more exotic possibilities with anisotropic pressure components, etc. But I don't know of any paper that looked specifically for $rho<0$ dark matter, and for the above reasons I'd be surprised if one existed.

lie algebra cohomology - Algebraic/Categorical motivation for the Chevalley Eilenberg Complex

Both (Lie) group and Lie algebra cohomology are essentially part of a more general procedure. Namely, we take an abelian category $C$ with enough projectives or enough injectives, take a (say, left) exact functor $F$ from $C$ to abelian groups (or modules over a commutative ring) and compute the (right) derived functors of $F$ using projective or injective resolutions.

For example, if $mathfrak{g}$ is a Lie algerba over a field $k$, we can take the category of $U(mathfrak{g})$-modules as $C$ and $$Mmapstomathrm{Hom}_{mathfrak{g}}(k,M)$$

as $F$ (here $k$ is a trivial $mathfrak{g}$-module). Notice that this takes $M$ to the set of all elements annihilated by any element of $mathfrak{g}$; this is not a $mathfrak{g}$-module, only a $k$-module, so the target category is the category of $k$-vector spaces.

In the category of $U(mathfrak{g})$-modules there are enough projectives and enough injectives, so in principle to compute the Exts from $k$ to $M$ we can use either an injective resolution of $M$ or a projective resolution of $k$ as $U(mathfrak{g})$-modules.
I've never seen anyone considering injective $U(mathfrak{g})$-modules, probably because they are quite messy. So most of the time people go for the second option and construct a projective resolution of $k$.
One of the ways to choose such a resolution is the "standard" resolution with

$$C_q=U(mathfrak{g}otimesLambda^q(mathfrak{g})$$ but any other resolution would do, e.g. the bar-resolution (which is "larger" then Chevalley-Eilenberg, but more general, it exists for arbitrary augmented algebras). Applying $mathrm{Hom}_{mathfrak{g}}(bullet,M)$ to the standard resolution we get the Chevalley-Eilenberg complex.

All the above holds for group cohomology as well. We have to replace $U(mathfrak{g})$ by the group ring (or algebra) of a group $G$. The only difference is that there is no analogue of the Chevalley-Eilenberg complex, so one has to use the bar resolution. Probably, Van Est cohomology of a topological group can also be described in this way.

Theories of Noncommutative Geometry

In accordance with the suggestion of Yemon Choi, I am going to suggest some further delineation of the approaches to "Non-commutative Algebraic Geometry". I know very little about "Non-commutative Differential Geometry", or what often falls under the heading "à la Connes". This will be completely underrepresented in this summary. For that I trust Yemon's summary to be satisfactory. (edit by YC: BB is kind to say this, but my attempted summary is woefully incomplete and may be inaccurate in details; I would encourage anyone reading to investigate further, keeping in mind that the NCG philosophy and toolkit in analysis did not originate and does not end with Connes.)

Also note that much of what I know about these approaches comes from two sources:

1. The paper by Mahanta




2. My advisor A. Rosenberg.

Additionally, much useful discussion took place at Kevin Lin's question (as Ilya stated in his answer).

I think a better break down for the NCAG side would be:

A. Rosenberg/Gabriel/Kontsevich approach

Following the philosophy of Grothendieck: "to do geometry, one needs only the category of quasi-coherent sheaves on the would-be space" (edit by KL: Where does this quote come from?)

In the famous dissertation of Gabriel, he introduced the injective spectrum of an abelian category, and then reconstructed the commutative noetherian scheme, which is a starting point of noncommutative algebraic geometry. Later, A. Rosenberg introduced the left spectrum of a noncommutative ring as an analogue of the prime spectrum in commutative algebraic geometry, and generalized it to any abelian category. He used one of the spectra to reconstruct any quasi-separated (not necessarily quasi-compact), commutative scheme. (Gabriel-Rosenberg reconstruction theorem.)

In addition, Rosenberg has described the NC-localization (first observed also by Gabriel) which has been used by him and Kontsevich to build NC analogs of more classical spaces (like the NC Grassmannian) and more generally, noncommutative stacks. Rosenberg has also developed the homological algebra associated to these 'spaces'. Applications of this approach include representation theory (D-module theory in particular), quantum algebra, and physics.

References in this area are best found through the MPIM Preprint Series, and a large collection is linked here. Additionally, a book is being written by Rosenberg and Kontsevich furthering the work of their previous paper. Some applications of these methods are used here, here, here, and here. The first two are focusing on representation theory, the second two on non-commutative localization.

Kontsevich/Soibelman approach

They might refer to their approach as "formal deformation theory", and quoting directly from their book

The subject of deformation theory can be defined as the "study of moduli spaces of structures...The subject of this book is formal deformation theory. This means $mathcal{M}$ will be a formal space(e.g. a formal scheme), and a typical category $mathcal{W}$ will be the category of affine schemes..."

Their approach is related to $A_{infty}$ algebras and homological mirror symmetry.
References that might help are the papers of Soibelman. Also, I think this is related to the question here. (Note: I know hardly anything beyond that this approach exists. If you know more, feel free to edit this answer! Thanks for your understanding!)

(Some comments by KL: I am not sure whether it is appropriate to include Kontsevich-Soibelman's deformation theory here. This kind of deformation theory is a very general thing, which intersects some of the "noncommutative algebraic geometry" described here, but I think that it is neither a subset nor a superset thereof. In any case, I've asked some questions related to this on MO in the past, see this and this.

However, there is the approach of noncommutative geometry via categories, as elucidated in, for instance, Katzarkov-Kontsevich-Pantev. Here the idea is to think of a category as a category of sheaves on a (hypothetical) non-commutative space. The basic "non-commutative spaces" that we should have in mind are the "Spec" of a (not necessarily commutative) associative algebra, or dg associative algebra, or A-infinity algebra. Such a "space" is an "affine non-commutative scheme". The appropriate category is then the category of modules over such an algebra. Definitively commutative spaces, for instance quasi-projective schemes, are affine non-commutative schemes in this sense: It is a theorem of van den Bergh and Bondal that the derived category of quasicoherent sheaves on a quasi-projective scheme is equivalent to a category of modules over a dg algebra. (I should note that in my world everything is over the complex field; I have no idea what happens over more general fields.) Lots of other categories are or should be affine non-commutative in this sense: Matrix factorization categories (see in particular Dyckerhoff), and probably various kinds of Fukaya categories are conjectured to be so as well.

Anyway I have no idea how this kind of "noncommutative algebraic geometry" interacts with the other kinds explained here, and would really like to hear about it if anybody knows.)

Lieven Le Bruyn's approach

As I know nearly nothing about this approach and the author is a visitor to this site himself, I wouldn't dare attempt to summarize this work.

As mentioned in a comment, his website contains a plethora of links related to non-commutative geometry. I recommend you check it out yourself.

Approach of Artin, Van den Berg school

Artin and Schelter gave a regularity condition on algebras to serve as the algebras of functions on non-commutative schemes. They arise from abstract triples which are understood for commutative algebraic geometry. (Again edits are welcome!)

Here is a nice report on Interactions between noncommutative algebra and algebraic geometry. There are several people who are very active in this field: Michel Van den Berg, James Zhang, Paul Smith, Toby Stafford, I. Gordon, A. Yekutieli. There is also a very nice page of Paul Smith: noncommutative geometry and noncommutative algebra, where you can find almost all the people who are currently working in the noncommutative world.

References: This paper introduced the need for the regularity condition and showed the usefulness. Again I defer to Mahanta for details.
Serre's FAC is the starting point of noncommutative projective geometry. But the real framework is built by Artin and James Zhang in their famous paper Noncommutative Projective scheme.

Non-commutative Deformation Theory by Laudal

Olav Laudal has approached NCAG using NC-deformation theory. He also applies his method to invariant theory and moduli theory. (Please edit!)

References are on his page here and this paper seems to be a introductory article.

Apologies

Without a doubt, I have made several errors, given bias, offended the authors, and embarrassed myself in this post. Please don't hold this against me, just edit/comment on this post until it is satisfactory. As it was said before, the nlab article on noncommutative geometry is great, you should defer to it rather than this post.

Thanks!

knot theory - slice-ribbon for links (surely it's wrong)

The slice-ribbon conjecture asserts that all slice knots are ribbon.

This assumes the context:

1) A `knot' is a smooth embedding $S^1 to S^3$. We're thinking of the 3-sphere as the boundary of the 4-ball $S^3 = partial D^4$.

2) A knot being slice means that it's the boundary of a 2-disc smoothly embedded in $D^4$.

3) A slice disc being ribbon is a more fussy definition -- a slice disc is in ribbon position if the distance function $d(p) = |p|^2$ is Morse on the slice disc and having no local maxima. A slice knot is a ribbon knot if one of its slice discs has a ribbon position.

My question is this. All the above definitions have natural generalizations to links in $S^3$. You can talk about a link being slice if it's the boundary of disjointly embedded discs in $D^4$. Similarly, the above ribbon definition makes sense for slice links. Are there simple examples of $n$-component links with $n geq 2$ that are slice but not ribbon? Presumably this question has been investigated in the literature, but I haven't come across it. Standard references like Kawauchi don't mention this problem (as far as I can tell).