nt.number theory - Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.

Let $a$ and $b$ be algebraic numbers, with respective degrees $m$ and $n$. Suppose $m$ and $n$ are coprime. Does the degree of $a+b$ always equal $mn$?

I know that the answer is "yes" in the following particular cases (I can provide details if needed) :

1) The maximum of $m$ and $n$ is a prime number.

2) $(m,n)=(3,4)$.

3) At least one of the fields $mathbf{Q}(a)$ and $mathbf{Q}(b)$ is a Galois extension of $mathbf{Q}$.

4) There exists a prime $p$ which is inert in both fields $mathbf{Q}(a)$ and $mathbf{Q}(b)$ (if $a$ and $b$ are algebraic integers, this amounts to say that the minimal polynomials of $a$ and $b$ are still irreducible when reduced modulo $p$).

I can also give the following reformulation of the problem : let $P$ and $Q$ be the respective minimal polynomials of $a$ and $b$, and consider the resultant polynomial $R(X) = operatorname{Res}_Y (P(Y),Q(X-Y))$, which has degree $mn$. Is it true that $R$ has distinct roots? If so, it should be possible to prove this by reducing modulo some prime, but which one?

Despite the partial results, I am at a loss about the general case and would greatly appreciate any help!

[EDIT : The question is now completely answered (see below, thanks to Keith Conrad for providing the reference). Note that in Isaacs' article there are in fact two proofs of the result, one of which is only sketched but uses group representation theory.]

gr.group theory - What kind group can be realized as a Isometry group of some space?

Every group is the full group of isometries of a connected, locally connected, complete metric space:

de Groot, J. "Groups represented by homeomorphism groups."
Math. Ann. 138 (1959) 80–102.
MR119193
doi:10.1007/BF01369667

Being a group of symmetries is the same thing as being a group.

You may also be interested to know that every group is the full automorphism group of a graph, not just a subgroup. References for this and various refinements are given at the wikipedia page for Frucht's theorem.

gr.group theory - Is a subgroup of a free abelian group free abelian?

A variety of groups $V$ is said to have the Schreier property if every subgroup of a free group in the variety is free. It is a classical theorem of Peter Neumann and James Wiegold that the only varieties of groups with the Schreier property are: the (absolutely) free groups, the free abelian groups, and the free exponent $p$ abelian groups for $p$ prime.

rt.representation theory - Why do Physicists need unitary representation of Kac-Moody algebra?

As others have mentioned, the reasons lie indeed in two-dimensional conformal field theory and in string theory.

The propagation of string on a compact Lie group $G$ is described by the Wess-Zumino-Witten model, whose dynamical variables are maps $g:Sigma to G$ from a riemann surface $Sigma$ to $G$. The quantisation of that model is difficult in terms of $g$ (although see the 1988 papers of Gawedzki and Kupiainen, and also Felder, for a functional integral approach) and one instead chooses to quantise their currents, roughly the (anti)holomorphic components of the pullbacks $g^*theta_L$ and $g^*theta_R$ of the Maurer-Cartan forms on $G$. There is a natural action of two copies of the affine Kac-Moody algebra associated to $G$ on the WZW model which preserves the Poisson structure of the WZW model and gives rise to moment mappings which are, essentially, the currents. In other words, the Poisson bracket of the currents is that of two copies of the affine Kac-Moody algebra of $G$. Hence the quantisation naturally leads one to consider unitary, integrable representations of the affine Kac-Moody algebra. The first "modern" reference for this is a 1986 paper of Doron Gepner and Edward Witten String Theory on Group Manifolds; although there are pioneering papers of Halpern, Bardakci,... in from the late 1960s and early 1970s.

At a more abstract level, we can substitute the group $G$ by any (unitary) two-dimensional (super)conformal field theory with the right central charge. This idea of replacing the geometry by a conformal field theory used to be known as "strings without strings", since one loses the description of strings propagating in some geometry. In this context, it is important to have at one's disposal a number of unitary two-dimensional conformal field theories. The natural ones are those coming from from unitary representations of infinite-dimensional Heisenberg and Clifford algebras (so-called free fields) and unitary integrable representations of affine Kac-Moody algebras, but one can also consider constructions (e.g., orbifolds, coset constructions,...) which generate new unitary CFTs from these ones. The first "modern" reference for the coset construction is perhaps the 1986 paper of Peter Goddard, Adrian Kent and David Olive Unitary representations of the Virasoro and superVirasoro algebras.

Finally, I should say that although it's the affine Kac-Moody algebras which seem to have played the more important rôles thus far, there is also the emergence (in the context of M-theory) of more general Kac-Moody algebras. There's work on this in King's College London (West et al.), Brussels (Henneaux et al.) and Potsdam (Nicolai et al.). I'm not very familiar with this, though.

big list - Computer Algebra Errors

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

in:

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

$A[k]$

$A[z]$

out:

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

$z^2$

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

in:

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

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

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

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

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

out:

2
5
13

2
5
12

2
5
13

gn.general topology - Connectedness of a union regading a proximity

Consider $Xcap A$ and $Ycap A$, starting from a partition $lbrace X,Yrbrace$ of $Acup B$. If both intersections are nonempty we are done, as $(Xcap A)delta(Ycap A)$. Otherwise, $Asubseteq X$, say, but then $Xcap B$ and $Ycap B$ are nonempty and we find $(Xcap B)delta(Ycap B)$.
In either case $Xdelta Y$.

soft question - Why didn't Vladimir Arnold get the Fields Medal in 1974?

As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)

In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, the most prestigious recognition in work in mathematics that is often compared to the Nobel Prize, making him one of the most preeminent mathematicians to never receive the prize.

Since he made some key results before 1974, it seems that the award would have been deserved. Knowing that the Soviets sometimes forced Nobel laureates not to accept their prizes, I thought at first that the same happened here - but noticing that Kantorovich received his Nobel prize the next year, and that Fields laureates both in 1970 and 1978 were Russians (Novikov and Margulis, respectively), I cannot understand why did the Soviets oppose it in case of Arnold. Can someone shed some light?

EDIT: I googled the resources online in English before asking this question here and found no answers. But after posting it here, I googled it in Russian, and found the following:

Владимир Игоревич Арнольд был номинирован на медаль Филдса 1974 году. Далее — изложение рассказа самого Арнольда; надеюсь, что помню его правильно. Всё было на мази, Филдсовский комитет рекомендовал присудить Арнольду медаль. Окончательное решение должен был принять высший орган Международного математического союза — его исполнительный комитет. В 1971 — 1974 годах вице-президентом Исполнительного комитета был один из крупнейших советских (да и мировых) математиков академик Лев Семёнович Понтрягин. Накануне своей поездки на заседание исполкома Понтрягин пригласил Арнольда к себе домой на обед и на беседу о его, Арнольда, работах. Как Понтрягин сообщил Арнольду, он получил задание не допустить присуждение тому филдсовской медали. В случае, если исполком с этим не согласится и всё же присудит Арнольду медаль, Понтрягин был уполномочен пригрозить неприездом советской делегации в Ванкувер на очередной Международный конгресс математиков, а то и выходом СССР из Международного математического союза. Но чтобы суждения Понтрягина о работах Арнольда звучали убедительно, он, Понтрягин, по его словам, должен очень хорошо их знать. Поэтому он и пригласил Арнольда, чтобы тот подробно рассказал ему о своих работах. Что Арнольд и сделал. По словам Арнольда, задаваемые ему Понтрягиным вопросы были весьма содержательны, беседа с ним — интересна, а обед — хорош. Не знаю, пришлось ли Понтрягину оглашать свою угрозу, но только филдсовскую медаль Арнольд тогда не получил — и было выдано две медали вместо намечавшихся трёх. К следующему присуждению медалей родившийся в 1937 году Арнольд исчерпал возрастной лимит. В 1995 году Арнольд уже сам стал вице-президентом, и тогда он узнал, что в 1974 году на членов исполкома большое впечатление произвела глубина знакомства Понтрягина с работами Арнольда.

Translation of the text (combination of Google translate and my knowledge of Russian)

Vladimir I. Arnold was nominated for the Fields Medal in 1974. Next - summary of the story of Arnold, I hope I remember it correctly. Everything was going fine, Fields Committee recommended an award to Arnold medal. The final decision was to be taken by the supreme organ of the International Mathematical Union - its executive committee. In 1971 - 1974 the vice-president of the Executive Committee was one of the greatest Soviet (and world) mathematicians, Academician Lev Semenovich Pontryagin. On the eve of his visit to the meeting of the Executive Committee, Pontryagin invited Arnold to his home for lunch and to talk about Arnold's work. As Pontryagin said then, he was ordered not to allow the award of Fields Medal to Arnold. In case the executive committee wouldn't agree and still try to award the medal to Arnold, Pontryagin was authorized to threaten the Soviet delegation's no-show in Vancouver at the next International Congress of Mathematicians, and Soviets leaving the International Mathematical Union. In order for the judgment of Pontryagin about the work of Arnold to be persuasive, Pontryagin, in his own words, had to know the work very well. Therefore, he invited Arnold to tell him in detail about his work, which Arnold did. According to Arnold, the questions Pontryagin asked him were very thorough, the talk with him was interesting, and the meal was good. I do not know whether Pontryagin had read out his threat, but Arnold did not receive the medal - and only two medals were given instead of the intended three. By the next award of medals Arnold (b. 1937) was too old for awarding the Fields Medal. In 1995, Arnold himself became vice-president, and then he heard that in 1974 the depth of Pontryagin's familiarity with Arnold's work made a great impression on the members of the executive committee.

nt.number theory - Groups related to sum of squares function?

For general odd $k=2m+1$ one can still compute the value of the singular series as Hardy does and obtain a formula similar to those for $k=5,7$, involving the value of an $L$-series with quadratic character at $s=m$. A relation to special groups as in the case $k=3$ is not visible from this, just a relation to the arithmetic of quadratic number fields.

For $k ge 9$, however, the genus of the sum of $k$ squares contains more than one integral equivalence class and by Siegel's Massformel (mass formula) evaluation of the singular series gives the average of the representation numbers for the equivalence classes in the genus and not the representation number of the individual form. (The genus of an integral quadratic form $q$ consists of those forms which have the same signature and are integrally equivalent modulo $m$ for all integral $m$.) Of course this doesn't exclude the possibility of finding a closed formula by other means, as was the case for the special dimensions of the form $4m^2$ or $4(m^2+m)$ in the work of Milne mentioned in Jagy's answer. To my knowledge at present no such formula is known for an odd number $k$ of variables.

Concerning references: The standard reference for sums of squares is still Grosswald's book. Good references for more general questions concerning the arithmetic of quadratic forms are the books of B. Jones, Y. Kitaoka, O. T. O'Meara, G. Shimura and (in german) of M. Eichler and of M. Kneser.

rt.representation theory - Unitary representations of SL(2, R)

I just want to elaborate more on questions 3. and 4. I'll consider the locally isomorphic groups SU(1,1) of SL2(R) and SU(2) of SO(3)

There is an analogy between the discrete series of SU(1,1) and the unitary irreps of SO(3). Both have holomorphic representations on the group's orbit on the flag manifold S^2 = SL(2,C)/B (B is a Borel subgroup). In the case of SU(2), the orbit is the whole of SU(2) while for SU(1,1) its is a noncomapct supspace: The Poicare disc. In both cases the representation space is a reproducing kernel Hilbert space and the group action is throug a Mobius transformation. This analogy generalizes to other non-compact groups having a holomorphic discrete series and it can be considered as a generalization of the Borel-Weil construction for compact groups.

Concerning question 4. I think that you are talking about Wigner's theory of Lie group contraction, in which a Lie group with the same dimension and with more "flat" directions is associated to the original Lie group. For example there is a contraction of SU(2) to Eucledian group in two dimensions and SU(1,1) to the Poncare group in two dimensions. There are interseting connections to the group representations of the contracted versions, and also of the Casimirs.

simplicial stuff - Why is the induced map between pullbacks (of inclusions) by a right fibration a deformation retract?

Let $X$ be a simplicial set. Let $Xto Delta^n$ be a right fibration (has the right lifting property with respect to right horn inclusions), and let $$Delta^{{n-i}}hookrightarrow Delta^{{n-i,dots, n}}hookrightarrow Delta^n$$ (for a fixed $i: 0leq ileq n$) be the obvious inclusion maps.

Then why is the induced map:

$$Xtimes_{Delta^n} Delta^{{n-i}} hookrightarrow Xtimes_{Delta^n}Delta^{{n-i,dots, n}}$$

a deformation retract? It's not like we can apply Whitehead's theorem, since $Delta^n$ is not a Kan complex in general.

(This is from the end of the proof of proposition 2.2.3.1 of HTT. The statement should be true out of the context in the book with the hypotheses I've given, but if not, there's the source.)

lo.logic - How to demonstrate that the union of the singleton of a set is equal to that set?

I'd like to write down a proof of the following (simple) fact: $forall xleft(bigcupleft{xright}=xright)$. I'd like to use the rules of inference of natural deduction. One could show that if $yinbigcupleft{xright}$ then $yin x$ and vice versa. I managed to show the former implication, but I cannot show the latter.

Let me show you what I accomplished, as an example. You suppose that $t_1inbigcupleft{t_0right}$. Then it follows that $exists x_2left(x_2inleft{t_0right}land t_1in x_2right)$. Now you also suppose that $t_2inleft{t_0right}land t_1in t_2$. So $t_2inleft{t_0right}$, $t_1in t_2$, $t_2inleft{t_0,t_0right}$, $t_2=t_0lor t_2=t_0$, $t_2=t_0$, $t_1in t_2leftrightarrow t_1in t_0$, $t_1in t_0$. That's the easy part. The problem now is that if I suppose that $t_1in t_0$ I can't show that $t_1inbigcupleft{t_0right}$.

What can I do? Thanks.

gr.group theory - Brauer's permutation lemma -- extending to some other finite groups?

---

Let $G$ be a finite subgroup of $GL(V)$. I claim that the following are equivalent:

$(1)$ Any two elements of $G$ which are $GL(V)$ conjugate are also $G$ conjugate.

$(2)$ The representation ring $mathbb{Q} otimes mathrm{Rep}(G)$ is spanned by representations $S^{lambda}(V)$, where $S^{lambda}$ ranges over all Schur functors.

Proofs: For $g$ in $G$, let the eigenvalues of $g$ be $e^{2 pi i q_k(g)}$, where $(q_1(g), q_2(g), ldots, q_n(g))$ is a multiset of elements of $mathbb{Q}/mathbb{Z}$. Write $Q(g)$ for this multiset. So $g$ and $g'$ are $GL(V)$ conjugate if and only if $Q(g) = Q(g')$. When necessary, we will write $Q_W(g)$ for the analogous construction for some other representation $W$.

If $Q(g)=Q(g')$ then $Q(g^k) = k Q(g) = k Q(g') = Q((g')^k)$. So, for every Adams operator $psi^k$, the actions of $g$ and $g'$ on the virtual representation $psi^k(V)$ have the same eigenvalues and, in particular, the same trace. Also, $Q_{V_1 oplus V_2}(g)$ and $Q_{V_1 otimes V_2}(g)$ are determined by $Q_{V_1}(g)$ and $Q_{V_2}(g)$. So, if $Q(g)=Q(g')$ then the linera functions $Tr(g)$ and $Tr(g')$ are equal on the sub-$Lambda$-ring of $mathbb{Q} otimes mathrm{Rep}(G)$ generated by $V$.

Assume condition 2, and let $Q(g) = Q(g')$. Then $Tr(g)$ and $Tr(g')$ are equal on all of $mathbb{Q} otimes mathrm{Rep}(G)$, so $g$ and $g'$ are $G$ conjugate.

Conversely, assume condition 1. Fix any $h in G$. Let $S$ be the sub-$Lambda$-ring of $mathbb{Q} otimes mathrm{Rep}(G)$ generated by $V$. We will construct a virtual representation $W$ in $mathbb{C} otimes S$ such that $Tr_W(g)$ is $0$ for $g$ not conjugate to $h$ and $1$ for $g$ conjugate to $h$. Taking such linear combinations of such linear functionals, we can clearly generate all class functions on $G$, so $S$ must be the whole of $mathbb{Q} otimes mathrm{Rep}(G)$.

By condition (1), $Q(h)$ is different from $Q(g)$ for any $g$ not conjugate to $h$. Therefore, we can find a symmetric polynomial $F$, with coefficients in $mathbb{C}$, which vanishes at $e^{2 pi i Q(g)}$ for $g$ not conjugate to $h$, but does not vanish at $e^{2 pi i Q(h)}$. (Here $e^{2 pi i Q(g)}$ is a point of $mathbb{C}^n$, which should be considered as defined only up to the action of $S_n$.) Using the standard relation between symmetric polynomials and $Lambda$-ring operations, $F(V)$ is the desired $W$.

fa.functional analysis - Does this sequence span $L^2$?

Not an answer, but maybe helpful.

These questions can be much harder than they look. There is a simple-looking, explicit set of functions ${f_alpha } subset L^2(0,1; dx/x)$ (see Operators, Functions, and Systems: an Easy Reading: Volume 1 by Nikolai K. Nikolski) with the property: the Riemann Hypothesis is equivalent to the density of the span of ${f_alpha }$.

Of course it doesn't mean your problem is so difficult; but it's certainly interesting and non-trivial (I think). The necessary results should be known and exist somewhere [UPDATE: I've changed my mind, maybe not!!] I haven't found anything yet; but the general problem I describe below is certainly very "natural" and I'm sure I'm not the first to think of it, so if it isn't known then it's an interesting open problem; I would be very interested to know the solution!

The Laplace transform gives (a multiple of) a unitary operator from $L^2(0, infty)$ onto the Hardy space $H^2( { mathrm{Re}(z)>0 })$ (depending on your normalisations).

The Laplace transform of $x^{n} e^{-x/n}$ is, up to a constant, $(z+1/n)^{-(n+1)}$.

So you're asking whether the orthogonal complement of these functions is zero in $H^2$. The scalar product of $F(z)$ with $(z+overline{lambda})^{-k-1}$ is, up to a constant, the derivative $F^{(k)}(lambda)$.

Thus (assuming my algebra is correct), your question is equivalent to asking whether there is any non--trivial $F in H^2$ satisfying

$$
F^{(n)}(1/n) = 0, qquad n=1,2,3,ldots
$$

If you just wanted some non-trivial analytic function $F$ on the half-plane $ { x+iy : x>0 }$ to satisfy this, it's possible (I think, if I remember correctly!) - at each point of a countable set without limit point in the domain, we can prescribe values of finitely many derivatives. The extra condition $F in H^2$ is the difficult part.

More generally we have:

Problem classify all sequences $(z_n)$, $(k_n)$ such that we have a uniqueness result:
$$
F in H^2, quad F^{(k_n)}(z_n) =0 quad (n=1,2,3,ldots) qquad Rightarrow qquad F equiv 0.
$$

Of course there might be a special trick for your particular case $z_n = 1/n$, $k_n = n$.

Special cases are well-known. For example: there is no non-trivial $G in H^2$ satisfying $G(z_n)=0$ if and only if $sum_n frac{mathrm{Re}(z_n)}{|1+z_n|^2} = +infty$ (the Blaschke condition). e.g. consider $z_n$ converging to zero, or out to infinity; if it does this so quickly that the sum is finite, then the set ${ z_n }$ is sparse enough to allow non-trivial $G$. Thus the classification is known if $k_n = 0$ for all $n$.

ag.algebraic geometry - What are the higher homotopy groups of Spec Z ?

$Spec(mathbb{Z})$ should only be considered as $S^3$, if you "compactify" that is add the point at the real place. This is demonstrated by taking cohomology with compcat support.
The étalé homotopy type of $Spec(mathbb{Z})$ is however contractible (indeed what do you get by removing a point form a sphere?) to see this (all results apper in Milne's Arithmetic Dualities Book)

(let $X=Spec(mathbb{Z})$ )

1) $H^r_c(X_{fl},mathbb{G}_m)=H^r_{c}(X_{et},mathbb{G}_m) = 0$ for $r neq 3$.

2) $H^3_c(X_{fl},mathbb{G}_m)=H^3_{c}(X_{et},mathbb{G}_m) = mathbb{Q}/mathbb{Z}$

3) by 2+1, we have:

$H^3_c(X_{fl},mu_n)= mathbb{Z}/n$

$H^r_c(X_{fl},mu_n)= 0$ for $r neq 3$.

4) since we have a duality $$H^r(X_{fl},mathbb{Z}/n)times H^{3-r}_c(X_{fl},mu_n) to mathbb{Q}/mathbb{Z} $$
we have

5) $H^0(X_{fl},mathbb{Z}/n) = H^0(X_{et},mathbb{Z}/n) = mathbb{Z}/n,$

$H^r(X_{fl},mathbb{Z}/n) = H^r(X_{et},mathbb{Z}/n) = 0$, $r >0$

6) Now since $pi_1$ is trivial we have by the Universal Coefficient Theorem, the Hurewicz Theorem and the profiniteness theorem for 'etale homotopy that all homotopy groups are zero.

st.statistics - Why is it so cool to square numbers (in terms of finding the standard deviation)?

If you apply Bessel's correction --- dividing by $5-1$ rather than by $5$ when you have $5$ numbers --- then some of the otherwise right things stated in some of the answers are wrong. Bessel's correction is intended to be used only when the variance one is computing is based on a sample to be used to estimate the variance of the whole population.

I won't be surprised if nobody used the variance and standard deviation before Abraham de Moivre did so in the 18th century. De Moivre considered this question: If you toss a fair coin $1800$ times, what is the probability that the number of heads is in some specified range? You have a binomial distribution, and computing its exact values was not feasible. De Moivre approximated the distribution of the number of heads with a normal distribution with the same mean and the same standard deviation. In so doing, he was the first to introduce the normal distribution, and the first to prove a special case of the central limit theorem. The normal distribution with mean $0$ and variance $1$ is
$$
varphi(x),dx=frac 1 {sqrt{2pi}} e^{-x^2/2},dx
$$
and with mean $mu$ and variance $sigma^2$ it is
$$
varphileft(frac{x-mu}sigmaright), frac{dx}sigma.
$$
It's easy to find the mean and standard deviation for the number of heads when one fair coin is tossed: they're both $1/2$. How do you do it for the sum of $1800$ independent copies of that random variable? De Moivre found that the mean-square deviation is additive: for independent random variables $X_1,ldots,X_{1800}$ one has $operatorname{var}(X_1+cdots+X_{1800})=operatorname{var}(X_1)+cdots+operatorname{var}(X_{1800})$. You cannot do that with mean absolute deviation. If I'm recalling some details correctly, he published these findings in a paper in Latin while he lived in France, and at that time he gave the normal distribution as
$$
C e^{-x^2/2},dx
$$
where he could find $C$ only numerically. Later he went to England to escape the persecution of Protestants and met James Stirling, who showed that $C=1/sqrt{2pi}$. De Moivre wrote a book in English called The Doctrine of Chances, which I think was 18th-century English for the theory of probability. Some have speculated that the Reverend Thomas Bayes may have studied under him, but I don't know that that's gone beyond speculation.

(If you want to know the probability that the number of heads is $ge894$, note that that's the same as $text{“}{>893}text{''}$, and find the probability that the normally distributed random variable with the same mean and variance is $>893.5$. That is a "continuity correction" and works surprisingly well even for fairly small samples.)

On to Bessel's correction: When does one use
$$
frac{(x_1-bar x)^2+cdots+(x_n-bar x)^2}{n-1},
$$
where $bar x=(x_1+cdots+x_n)/n$, with $n-1$ rather than $n$ in the denominator? As you can see from simple examples, that will not serve de Moivre's purpose described above: it's not additive.

If $X_1,ldots,X_n$ are an independent sample from a population with mean $mu$ and variance $sigma^2$, then the expected value of
$$
frac{(X_1-mu)^2+cdots+(X_n-mu)^2} n tag 1
$$
is $sigma^2$. But if one has only the sample and not the whole population, one doesn't know $mu$ and one can use the sample mean $bar X$ as an estimate of $mu$. But the expected value of
$$
frac{(X_1-bar X)^2+cdots+(X_n-bar X)^2} n
$$
is smaller than the expected value of $(1)$. Specifically, a bit of algebra shows that
$$
sum_{i=1}^n (X_i-mu)^2 = left( sum_{i=1}^n (X_i-bar X)^2 right) + n(bar X-mu)^2, tag 2
$$
and since the expectation of the last term is $sigma^2$, that of the first term on the right in $(2)$ must be $(n-1)sigma^2$. Thus Bessel's correction gives an unbiased estimate of the population variance $sigma^2$. (But its square root does not give an unbiased estimate of the population standard deviation. And unbiasedness is at best somewhat overrated anyway, and in some cases is a very very bad thing (I had a paper in the American Mathematical Monthly a few years ago demonstrating how bad it can sometimes be).

lo.logic - How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.

Motivation. This question arises in the context of a certain finite analogue of Borel equivalence relation theory. I explained in
this answer that the purpose of Borel equivalence relation theory is to analyze the complexity of various
naturally occuring equivalence relations in mathematics, such as the isomorphism relations on various types of structures. It turns out that many of the
most natural equivalence relations arising in mathematics are Borel relations on a standard Borel space, and these fit into a hierarchy under Borel reducibility. Thus, this subject allows us make precise the idea that some classification problems are wild and others tame, by fitting them into a precise hierarchy where they can be compared with one another under reducibility. Recently, there has
been some work adapting this research project to other contexts. Last Friday, for example, Sy Friedman gave a
talk for our seminar on an effective analogue of the Borel theory. Part of his analysis provided
a way to think about very fine distinctions in the relative difficulty even of the various problems of classifying finite structures,
using methods from complexity theory, such as considering NP equivalence relations under polytime reductions.
For a part of his application, it turned out that fruitful conclusions could be made when one knows something about
the asymptotic growth rate of the number of isomorphism classes, for the kinds of objects under consideration.

This is where MathOverflow comes in. I find it likely that there are MO people who know about the number of groups.
Therefore, please feel free to ignore all the motivation above, and
kindly tell us all about the values or asymptotics of the
following functions, where n is a natural number:

• G(n) = the number of groups of size at most n.




• R(n) = the number of rings of size at most n.




• F(n) = the number of fields of size at most n.




• Γ(n) = the number of graphs of size at most n.




• P(n) = the number of partial orders of size at most n.

Of course, in each case, I mean the number of isomorphism types of such objects. These particular functions are representative, though of course, there
are numerous variations. Basically I am interested in the number of isomorphism classes of any kind of natural finite structure, limited by size.
For example, one could modify Γ for various specific kinds of graphs, or modify P for various kinds of partial orders, such as trees,
lattices or orders with height or width bounds. And so on.
Therefore, please answer with other natural classes of finite structures, but I shall plan to accept the answer for my favored
functions above. In many of these other cases, there are easy answers. For example, the
number of equivalence relations
with n points is the intensely studied partition number of n. The number of Boolean
algebras of size at most n is just log₂(n), since all finite Boolean algebras are finite power sets.

ag.algebraic geometry - Fontaine-Mazur for GL_1

For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified comes from algebraic geometry (i.e., is a subquotient of the etale cohomology of some variety over $K$, up to Tate twist). As far as I can see, the only cases where any progress has been made concerns the case that $K$ is totally real or CM.

This made me wonder: Is the Fontaine-Mazur conjecture known to be true for $1$-dimensional representations for any number field $K$? For CM fields, the theory of CM abelian varieties gives varieties whose cohomology realizes nontrivial characters (and I guess that easy variations should produce all characters). What are the geometric objects appearing for other fields?

[edit: The word 'geometric' is avoided now, see the comments.]

dg.differential geometry - Least number of charts to describe a given manifold

Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.

E.g. a circle requires at least two charts, and so on (I couldn't manage to get anything relevant neither on wikipedia nor on google, so I guess I'm lacking the correct terminology).

Edit: in the case of an open covering of a topological space by n+1 contractible sets (in that space) then n is called the Lusternik-Schnirelman Category of the space, see Andy Putman's answer. The following book seems to be the standard reference http://books.google.fr/books?id=vMREfNN-L4gC&pg=PP1

Great, now I'm still interested by the initial question: does anybody know of another theory without this contractibility assumption (hoping that it allows more freedom)? e.g. would it lead to different numbers say for genus-g surfaces?

Final edit: yes different numbers for genus-g surfaces (see answers below), but not sure there is a theory without contractibility. Right, really lots of interesting literature on the LS category nevertheless, hence the accepted answer. For example there are estimates for non-simply connected compact simple Lie groups like PU(n) and SO(n) in Topology and its Applications, Volume 150, Issues 1-3, 14 May 2005, Pages 111-123.

soft question - Most helpful math resources on the web

edit by jc: As of May 11, 2010, the work has been completed!

This is a reference that is not yet complete, but it should be very useful when it finally does arrive:

Digital Library of Mathematical Functions (DLMF)
(book and associated website;
will replace Abramowitz & Stegun's Handbook of Mathematical Functions)
NIST / Cambridge University Press
expected 2009/2010
http://dlmf.nist.gov/

This will contain diagrams, tables, properties of, principal values of, and relationships between many important mathematical functions. For example, the trigonometric and other elementary functions are described, with very many formulae relating them.

The Handbook is very good; the Digital Library will be even better.

sheaf theory - vanishing theorems

I would be glad to know about possible generalizations of the following results:

1) (Grothendieck) Let $X$ be a noetherian topological space of dimension $n$. Then for all $i>n$ and all sheaves of abelian groups $cal{F}$ on $X$, we have $H^i(X; cal{F})=$ 0. [See Hartshorne, Algebraic Geometry, III.2.7.]

2) Let $X$ be an $n$-dimensional $C^0$-manifold. Then for all $i>n$ and all sheaves of abelian groups $cal{F}$ on $X$, we have $H^i(X; cal{F})=$ 0 . [See Kashiwara-Schapira, Sheaves on manifolds, III.3.2.2]

More precisely, I'm interested in dropping the "abelian groups" hypothesis: could I take sheaves in any, say, AB5 abelian category?

Apparently, in Grothendieck's theorem, the "abelian groups" hypothesis is necessary -at least in Hartshorne's proof-, because at the end you see a big constant sheaf $mathbf{Z}$. But what happens if we talk about sheaves of $R$-modules, with $R$ any commutative ring with unit, for instance?

Are those generalizations trivial ones? False for trivial reasons?

Any hints or references will be welcome.

finite groups - abelian sylow-p-subgroups

There is also a character-theoretic argument. Suppose $G' cap Z(G)$ has a subgroup $U$ of order $p$. We want a contradiction. Let $lambda$ be a nonprinciipal linear character of $U$. Since $U subseteq P$ and $P$ is abelian, $lambda$ has an extension to $mu$, a linear character of $P$. The induced character $mu^G$ has degree $|G:U|$, which is prime to $p$, so some irreducible constituent $chi$ of $mu^G$ has degree not divisible by $p$. Then $mu$ is a constituent of the restriction $chi_P$ by Frobenius reciprocity, and thus $lambda$ is a constituent of $chi_U$. But $U$ is central, so $chi_U = chi(1)lambda$. Now let $sigma$ be the linear character det$(chi)$. Then $sigma_U = lambda^{chi(1)}$, which is nontrivial since $p$ does not divide $chi(1)$. This is a contradiction, however, since $U subseteq G' subseteq {rm ker}(sigma)$. [Note that transfer proofs can often be replaced by arguments using the determinant of a character.]

abstract algebra - Canonical examples of algebraic structures

I often think of "universal examples". This is useful because then you can actually prove something in the general case - at least theoretically - just by looking at these examples.

Semigroup: $mathbb{N}$ with $+$ or $*$

Group: Automorphism groups of sets ($Sym(n)$) or of polyhedra (e.g. $D(n)$).

Virtual cyclic group: Semidirect products $mathbb{Z} rtimes mathbb{Z}/n$.

Abelian group: $mathbb{Z}^n$

Non-finitely generated group: $mathbb{Q}$

Divisible group: $mathbb{Q}/mathbb{Z}$

Ring: $mathbb{Z}[x_1,...,x_n]$

Graded ring: Singular cohomology of a space.

Ring without unit: $2mathbb{Z}$, $C_0(mathbb{N})$

Non-commutative ring: Endomorphisms of abelian groups, such as $M_n(mathbb{Z})$.

Non-noetherian ring: $mathbb{Z}[x_1,x_2,...]$.

Ring with zero divisors: $mathbb{Z}[x]/x^2$

Principal ideal domain which is not euclidean: $mathbb{Z}[(1+sqrt{-19})/2]$

Finite ring: $mathbb{F}_2^n$.

Local ring: Fields, and the $p$-adics $mathbb{Z}_p$

Non-smooth $k$-algebra: $k[x,y]/(x^2-y^3)$

Field: $mathbb{Q}, mathbb{F}_p$

Field extension: $mathbb{Q}(i) / mathbb{Q}, k(t)/k$

Module: sections of a vector bundle. Free <=> trivial. Point <=> vector space.

Flat / non-flat module: $mathbb{Q}$ and $mathbb{Z}/2$ over $mathbb{Z}$

Locally free, but not free module: $(2,1+sqrt{-5})$ over $mathbb{Z}[sqrt{-5}]$

... perhaps I should stop here, this is an infinite list.

Graph algorithm to find all subgraphs that connect N arbitrary vertices

I have an graph with the following attributes:

• Undirected


• Not weighted


• Each vertex has a minimum of 2 and maximum of 6 edges connected to it.


• Vertex count will be < 100


• Graph is static and no vertices/edges can be added/removed or edited.

I'm looking for all subgraphs between a random subset of the vertices (at least 2).

I've created a (warning! programmer art) animated gif to illustrate what i'm trying to achieve: http://imgur.com/mGVlX.gif

My end goal is to have a set of subgraphs that allow moving from one of the subset vertices (blue nodes) and reach any of the other subset vertices (blue nodes).

pr.probability - Polish spaces in probability

One simple thing that can go wrong is purely related to the size of the space (polish spaces are all size $leq 2^{aleph_0}$). When spaces are large enough product measures become surprisingly badly behaved. Consider Nedoma's pathology: Let $X$ be a measure space with $|X| > 2^{aleph_0}$. The diagonal in $X^2$ is not measurable.

We'll prove this by way of a theorem:

Let $U subseteq X^2$ be measurable. $U$ can be written as a union of at most $2^{aleph_0}$ spaces of the form $A times B$.

Proof: First note that we can find some countable collection $A_i$ such that $U subseteq sigma(A_i times A_j)$ (proof: The set of $V$ such that we can find such $A_i$ is a sigma algebra containing the basis sets).

For $x in {0, 1}^mathbb{N}$ define $B_x = bigcap { A_i : x_i = 1 } cap bigcap { A_i^c : x_i = 0 }$.

Consider all sets which can be written as a (possibly uncountable) union of $B_x times B_y$ for some $y$. This is a sigma algebra and obviously contains all the $A_i times A_j$, so contains $A$.

But now we're done. There are at most $2^{aleph_0}$ of the $B_x$, and each is certainly measurable in $X$, so $U$ can be written as a union of $2^{aleph_0}$ sets of the form $A times B$.

QED

Corollary: The diagonal is not measurable.

Evidently the diagonal cannot be written as a union of at most $2^{aleph_0}$ rectangles, as they would all have to be single points, and the diagonal has size $|X| > 2^{aleph_0}$.

pr.probability - Maximum of Convex combination of random variables

First, $Z$ is distributed like $bX$, with $b>0$ and $b^2=a^2+(1−a)^2$. Second, for every $x$, $u(bx)=sqrt{b}u(x)$. Third, $E(u(X))=-E(sqrt{X};X>0)$ is negative. Hence, $E(u(Z))=sqrt{b}E(u(X))$ is at its maximum when $b$ is at its minimum. This happens when $a=frac12$.

oa.operator algebras - Operator Valued Weights

One of the basic tools in subfactors is the conditional expectation. If $Nsubset M$ is a $II_1$-subfactor (or an inclusion of finite factors), then there is a unique trace-preserving conditional expectation of $M$ onto $N$. This should be thought of as a (Banach space) projection of norm 1. In fact, it is the restriction of the Jones projection $e_N$ on $L^2(M)$ to $M$. In the finite index case, we get another conditional expectation (Jones projection...) from the basic construction $M_1=langle M, e_Nrangle$ onto $M$.

In his thesis, Michael Burns showed that if we iterate the basic construction in the infinite index case, we only get half the conditional expectations (we only get the odd Jones projections). The other half of the time, we get a generalization of the conditional expectation called an operator valued weight, originally defined by Haagerup.

Given an inclusion of semifinite von Neumann algebras $(N, tr_N)subset (M, tr_M)$, there is a unique normal, faithful, semi-finite trace-preserving operator valued weight $Tcolon M_+to widehat{N_+}$, where we must take the "extended part" of the positive cone $N_+$ of $N$.

Edit as per @Dmitri's answer:
Let
$$
n_T={xin M| T(x^ast x)in N_+}
$$
and set
$$
m_T=n_T^ast n_T=span{x^ast y| x,yin n_T}.
$$
There is a natural extension of $T$ to $m_T$. Is there an example of a normal, faithful, semifinite operator valued weight such that

• $N$ is not contained in $T(m_T)$, and/or


• $1notin T(M_+)$?

What about when $M$ and $N$ are factors ($M$ is $II_infty$)?

homological algebra - Can we categorify the equation (1 - t)(1 + t + t^2 + ...) = 1?

I don't know whether or not this satisfies the criterion of "categorification" (what that?), but the equation (1 - t)(1 + t + t² + ...) = 1 and its relation to vector spaces is well-known to differential topologists and geometers. We use it all the time in K-theory and index theory where it becomes the identity Λ_-1V ⊗ S₁V = ℂ. Here, Λ_-1V denotes the alternating sum of the exterior powers of V whilst S₁V is the sum of the symmetric powers.

Of course, the sum of the symmetric powers isn't a class in K-theory as it is an infinite sum. To get round this, we work in K[[t]] and allow parameters, whereupon the equation becomes Λ_-tV ⊗ S_tV = ℂ. Here, the t means formally multiply the kth exterior or symmetric power by t^k.

Where this breaks out of mere formalism and becomes very powerful is in equivariant K-theory. Then the parameter becomes a way of measuring the action of the group, which is usually S¹ or a finite cyclic group for index theory calculations. In particular, in Witten's original adaptation of index theory to loop spaces, we end up with positive energy representations of S¹ which become vector bundles over the original (finite dimensional) manifold with circle actions preserving the fibres. One can decompose these according to the circle action whereupon one has a vector bundle for each k ∈ ℤ. The positive energy criterion means that these are trivial below a certain integer and are always finite dimensional. However, as there are an infinite number of them then the total dimension can be infinite dimensional. Then the identity Λ_-tV ⊗ S_tV = ℂ has real meaning as the power of the t parameter indicates how the circle acts on that component of the vector bundle. That is, if the circle action on V is the standard action then it is multiplication by t^k on Λ^k V and on S^k V.

lie algebras - Representations of reductive Lie group

You need to be over a field of zero characteristic and your representation needs to be rational, i.e. matrix entries need to be algebraic functions on $G$. Then it is completely reducible, see any book on algebraic groups, e.g., Jantzen or Humphreys.

You can always differentiate, so a differential of a map $Grightarrow GL(V)$ is a representation of ${mathfrak g}$. In the opposite direction, a certain care is required. To integrate a vector field, you need exponential function, which is not, in general, algebraic. However, for a semisimple group in characteristic zero, you have enough nilpotent elements $Xin{mathfrak g}$, so that the polynomials $e^{rho (X)}$ define a representation of the group.

Finally, the answer is no. Take ${mathfrak g}$ to be one-dimensional Lie algebra acting on $K^2$ by the nilpotent nonzero transformation.

geometry - Linear transformation takes a polygon to another one.

Jesse Douglas studied linear transformations of polygons on the complex plane in 1930s. He proved, in particular, that a transformation $z_i{}'=sum_{i=1}^na_{ij}z_j$ (all numbers are complex) will transform a polygon $pi=(z_1,cdots,z_n)$ into a polygon $pi'=(z_1{}',cdots,z_n{}')$ if, and only if, the matrix $a_{ij}$ is cyclic, that is, if, and only if, $a_{ij}=alpha_{j-i}$, $alpha_{j-i}=alpha_k$ if $kequiv j-1 (text{mod},n)$. (See his article "On linear polygon transformations", Bull. Amer. Math. Soc. 46, (1940) pp. 551 - 560.)

pr.probability - Suprema of stochastic processes

The answer is, indeed, "No" because there is an unbounded with probability $1$ stochastic process that satisfies the given inequality, namely, $X(t)=0.1log|t-w|$ where $w$ is equidistributed on $[0,1]$. Truncating it at high level $L$, we get a continuous process such that $E|X(t)|$ is uniformly bounded but the supremum is identically $L$. Taking a suitable mixture of such truncations, we see that the tails may decay arbitrarily slowly.

There is essentially only one universal method to gets bounds for the supremum from the bounds for the increments, which is to consider $delta_ k$-nets with diminishing $delta_ k>0$ and bound the supremum by the convergent series of suprema taken over finite sets (differences between points from 2 successive nets). Clever choice of the nets may be crucial for the success but not in this case. Whatever you can get from the standard dyadic nets here is the best you can say.

I'm also tempted to ask whether you, indeed, need the estimate for the supremum (which cannot be made without extra assumptions) rather than for some $L^p$ norm.

pr.probability - probability in number theory

Have you read The Probabilistic Method by Joel Spencer and Noga Alon?

Although originally developed by Erdos, here's an example of the probabilistic method taken from combinatoricist Po-Shen Loh:

$A_1, dots, A_s subseteq { 1, 2, dots, M }$ such that $A_i not subset A_j$ and let $a_i = |A_i|$. Show that $$ sum_{i=1}^s frac{1}{binom{M}{a_i}} leq 1$$
The hint is to consider a random permutation $sigma = (sigma_1, dots, sigma_M)$. Loh defines the event $E_i$ when ${ sigma_1, dots, sigma_{a_i}} = A_i$. Then he observes the events $E_i$ are mutually exclusive and that $mathbb{P}(E_i)$ is relevant to our problem...

There are probably a lot of olympiad combinatorics problems that can be solved this way. Err... you were asking for number theory, but you will find both in Spencer and Alon's book.

ca.analysis and odes - Does the "continuous locus" of a function have any nice properties?

Yes, here's a quick proof that any given $G_delta$ (in $mathbb{R}$) can be realized as the set of continuity points of some real-valued function.

Let $G$ be a given $G_delta$ set in $mathbb{R}$, meaning $G = cap_{i=1}^infty G_i$, each $G_i$ an open set. Define a function $f:mathbb{R} to mathbb{R}$ as follows: $f(x)=0$ if $x$ is in $G$. If x is not in $G$, there is some $k$ such that $x$ is not in $G_k$; let $k$ be minimal with that property. Define $f(x)=1/k$ if $x$ is rational and $f(x)=-1/k$ if $x$ is irrational.

If I'm not very much mistaken, $G$ is precisely the set of continuity points of this $f$. I'm happy to leave this as an exercise for now :-) Let me know if you're not sure how to do it, or - worse - if I'm just wrong about the construction.

nt.number theory - Arithmetic progressions without small primes

I don't know a definite answer either way, but here's one line of questioning: I believe it is a known result that Linnik's constant is at least 2 (although it's less for almost all integers.) Do the methods used to prove that fail to distinguish between primes and composites, or between some set in which the primes have positive density and some set in which they don't?

From the other direction, however, it's apparently possible on GRH to bound the error term in the PNT in arithmetic progressions (as in Brun-Titchmarsh, although I don't think they're that tight) when N > q^{2+epsilon}. This would put some rather tricky constraints on the error term, which one could take as evidence against it.

By the way, do you have a heuristic argument that, for instance, there are infinitely many primes for which the least prime congruent to 1 mod p is at least c p^(2-epsilon)? Or even at least c p^e for some e > 1?

What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?

I think this works:

Consider a topological space consisting of 4 points $A$, $B$, $C$, $D$, where the topology is given by open sets $ABC$, $BCD$, $B$, $C$, $ABCD$, $emptyset$.

Then let the presheaf $mathcal{F}$ be given by:
$$mathcal{F}(ABC)=mathbb{Z}$$
$$mathcal{F}(BCD)=mathbb{Z}$$
$$mathcal{F}(BC)=mathbb{Z}$$
$$mathcal{F}(ABCD)=mathbb{Z}$$
$$mathcal{F}(B)=mathbb{Z}/2mathbb{Z}$$
$$mathcal{F}(C)=mathbb{Z}/2mathbb{Z}$$
$$mathcal{F}(emptyset)=0$$

where all restrictions are what you expect (identity in the case of $mathbb{Z} to mathbb{Z}$ and canonical surjection in the case $mathbb{Z} to mathbb{Z}/2 mathbb{Z}$).

Then if we we get $mathcal{F}^+$ is given by:

$$mathcal{F}^+(ABC)=mathbb{Z}$$
$$mathcal{F}^+ (BCD)=mathbb{Z}$$
$$mathcal{F}^+ (BC)= mathbb{Z}/2mathbb{Z} oplus mathbb{Z}/2mathbb{Z}$$
$$mathcal{F}^+ (ABCD)=mathbb{Z}$$
$$mathcal{F}^+ (B)= mathbb{Z}/2mathbb{Z} $$
$$mathcal{F}^+ (C)=mathbb{Z}/2mathbb{Z}$$
$$mathcal{F}^+ (emptyset)=0$$

where the map from $mathcal{F}^+ (BCD)$ to $mathcal{F}^+ (BC)$ is given by taking the canonical surjection on both copies, and other restrictions are obvious. Then note that if we take 1 over $BCD$ and 3 over $ABC$, these two are compatible over $BC$ but they do not patch.

The key point is that being compatible over a refinement is not the same thing as being compatible. That is, the way the plus construction works is by taking $F^+$ of a space to be some direct limit over open covers of guys on the covers which are compatible on intersections. If we had said instead take direct limit over open covers of guys on the covers which compatible on some refinement of the intersection, then applying just once probably works.

So in our example, 1 and 3, over $ABC$ and $BCD$, in our original presheaf were compatible on a refinement of $BC$ but not on $BC$.

fa.functional analysis - To what extent is convexity a local property?

Claim: Suppose that $G$ is a connected bounded open set in $mathbb R^n$ such that for every $xinpartial G$, $exists r>0$ and a half-space $S$ such that $xinpartial S$ and $Gcap B(x,r)subset S$. Then $G$ is convex.

Proof:

Step 1. Suppose that $f:Gto mathbb R$ is a continuous function such that for every $xin G$, there exists $r>0$ and a linear function $L_x$ satisfying $L_x(x)=f(x)$ and $f(y)<L_x(y)$ for all $yne x$ with $|y-x|<r$. Then $f$ is concave in the sense that if $a,bin G$ and the whole interval $[a,b]$ is contained in $G$, then $f(ta+(1-t)b)ge tf(a)+(1-t)f(b)$ for $tin[0,1]$.

Proof: Suppose not. Then $min_t[f(ta+(1-t)b)-tf(a)+(1-t)f(b)]<0$. Take $sin(0,1)$ to be the point where it is attained and let $x=sa+(1-s)b$. Then the linear function $L_x(ta+(1-t)b)-tf(a)+(1-t)f(b)$ has a strict local minimum at $t=s$, which is impossible.

Step 2. We can replace the strict inequality in the conditions of Step 1 by a nonstrict one keeping the conclusion.

Proof: Just subtract $delta|x|^2$ with small $delta>0$.

Step 3: The distance to the boundary function satisfies the conditions of Step 1.

Proof: Let $xin G$. Let $y$ be the boundary point closest to $x$. Let $r$ and $S$ be the radius and the half-space for $y$. Then $L_x(z)=text{dist}(z,partial S)$ and $r$ work for $x$.

Step 4: $G$ is convex.

Proof: Take any 2 points $a,b$ in $G$. Suppose that the interval $[a,b]$ is not contained in $G$. Start moving $b$ towards $a$ along some path connecting them in $G$. Somewhere on the way, you'll get the situation when $a$ and $b$ are deep inside $G$ (that is true all the time) but $[a,b]$ is just barely inside $G$. Then the distance to the boundary dips on $[a,b]$, which is impossible due to the concavity just proved.

The whole thing is certainly well-known and in good old times all of this would be written in most standard calculus textbooks (possibly, as an exercise). Unfortunately, nowadays we have to teach students to add fractions instead. Nevertheless, the textbooks in convex geometry and analysis written before 1980 would be your best bet if you want a reference. I would try something like Rockafellar's "Convex Analysis" to start with.

big list - Favourite scholarly books?

Anything by John Milnor. His little book Morse Theory is a very clear, concise introduction to certain essential aspects of differential topology and Riemannian
geometry, starting at a fairly elementary level and winding up with Bott periodicity
for unitary groups. In this vein, his Characteristic Classes is similarly clear and
concise. To my mind, Milnor is an extremely gifted expositor. From a more scholarly point of view, Kobiyashi and Nomizu's two volumes on differential geometry (can't recall the exact title right now) are pretty comprehensive, both in material covered and in references. And since theoretical physics is within the purview of MO, I think
The Feynman Lectures on Physics, vols. I,II,III are a work of real genius. When I
was an undergrad at Caltech from 1968-72, we used them for introductory physics;
students jokingly called them "the big red sleeping pills" because the material went down so easily it might make one doze off. His Quantum Electrodynamics provides a beutifully
intuitive introduction to a fairly abstruse subject. Finally, another of may favorites
is Abraham and Marsden's Foundations of Mechanics, both in terms of exposition and
scholarship.

model theory - Are the types of nonstandard natural numbers within a Z-chain identical?

Since the nonstandard numbers believe that every other number is even and every other number is odd, a fact that is expressible in the language you mention, it follows that the types are not the same for every two elements in a $Z$-chain. In fact, more is true: any two nonstandard natural numbers in a common $Z$-chain have distinct types, for if they differ by a finite number $n$, then they will have different residue modulo $n+1$, making their types different.

If one restricts attention only to the order, however, then any two elements in a nonstandard $Z$-chain have the same type, since there are order-automorphisms that shift within this $Z$-chain. And in a countable nonstandard model, the $Z$-chains are ordered like the rationals, and so the order automorphism group acts transitively on these elements. It follows that all the nonstandard elements have the same type in the language containing only the order.

gr.group theory - is amalgamation of groups associative

Given groups $G_1, G_2, G_3$ and injections $A_1 to G_1$ and $A_1
to G_2$ , from $A_2 to G_2$ and $A_2 to G_3$, let $G_1 *_{A_1} *G_2 *_{A_2} G_3$ be the amalgam formed these groups and maps.
Then is it true that $G_1 *_{A_1} *G_2 *_{A_2} G_3$ is the same as (G_1 *_{A_1} G_2 ) *_{A_2} G_3. If yes, how do we see this?

qa.quantum algebra - Quantum Frobenius II

In general, the idea of the Kumar-Littelmann paper is the following: For a semisimple group G, set $V := displaystyle bigoplus_{n geq 0} H^0(lambda)$, where $lambda$ is a fixed regular dominant weight for G. Then $V$ is the projective coordinate ring of $G/B$ under the embedding $G/B hookrightarrow mathbb P( V( lambda ) )$, where $V$ is the Weyl module for $G$ of highest weight $lambda$. In particular, one can obtain the sheaf of regular functions on $G/B$ in the natural way from $V$.

Now, the (absolute) Frobenius morphism on the flag variety $G/B$ induces an automorphism of $V$ as an $mathbb F_p$-vector space, and in fact the converse holds: the appropriate $p^{th}$-power morphism $V to V$ (which is just the morphism of taking $p^{th}$ powers of sections) induces the Frobenius morphism on $G/B$ (this is the process called "sheafification" in their paper, cf section 6). The point of the paper is now that one can define a module (let's call it $V'$) for the quantum group associated to $G$ such that upon base change, $V'$ becomes $V$. Furthermore, Lusztig's Frobenius morphism induces a morphism $V to V'$ (which they call $Fr^*$) which, upon base change, becomes exactly the desired $p^{th}$-power morphism $V to V$.

Let me give an explicit example for $mathbb P^1$. In this case, $mathbb P^1$ is the flag variety of $G = SL_2$. Since the weights of $SL_2$ are parametrized by integers, I'll write $H^0(n)$ for the global sections of the corresponding line bundle on $G/B$ (which is just a complicated way of saying that $H^0(n) = H^0( mathbb P^1, O(n) )$, where that $O$ should be a mathscr O but that doesn't seem to work). Then in this case, we can take $V = displaystyle bigoplus_{n geq 0} H^0(n)$, and $V$ is just $k[x, y]$. The scheme-theoretic Frobenius morphism on $G/B$ is induced by the natural $p^{th}$-power morphism $V to V$, $; s mapsto s^{ otimes p }$ (which is just the natural $p^{th}$-power morphism on the ring $k[x, y]$). We now quantize this picture: Set $$V' := bigoplus_{n geq 0} H^0( X, chi_{n}^xi ) ,$$ where here I'm using their notation from the paper (note that the "X" should be a mathfrak X as in the paper, but somehow I can't do mathfrak here). That is, $H^0( X, chi_{n}^xi )$ is the induction functor from $U_q(b)$ modules to $U_q(g)$ modules, applied to the 1-dimensional $U_q(b)$-module $chi_{n}^xi$ (cf section 2 of the paper). The point is that $V'$ is a quantized version of $V$, and Lusztig's Frobenius morphism induces a morphism $Fr^* : V to V'$ that, upon base change, becomes the $p^{th}$-power morphism $V to V$.

(As for the Podles' q-sphere, I don't know what that is, so I can't speak to that part of your question).

(Edit: I realized that there is a slight white lie in what I wrote above, namely that the morphism Fr* initially isn't quite a morphism from $V$ to $V'$, but from a characteristic-0 version of $V$ to $V'$; one only gets $V$ after base change to positive characteristic. Kumar and Littelmann first construct Fr* in characteristic 0. Morally, though, one can ignore this issue on a first pass; it's a bit confusing because Fr* appears in various incarnations, both before and after base change).

big list - What are some fundamental "sources" for the appearance of pi in mathematics?

As for the normal distribution, you can characterize it as the unique distribution with the following properties:

Let $X_1, X_2, cdots X_n$ be independent identically distributed normal random variables. Then the joint distribution of the vector $X=(X_1, X_2, cdots X_n)$ is the same as that of $AX$ where $A$ is any orthogonal matrix. So the normal distribution is intimately related to the geometry of real inner product spaces.

The $pi$ comes from the fact that you can integrate such a distribution by first integrating over a sphere and then integrating over $[0,infty]$. Because the distribution is orthogonally invariant, you pick up a constant corresponding to the area of the sphere. For $n=2$ you get the circle, and this is the usual calculation for computing the normalization constant for the normal distribution.

So then the mystery becomes: given that the normal distribution is so closely tied to inner product spaces, why does it show up all the time? The central limit theorem tells us that all that really matters in large scale limits are the first and second moments. The first moment can always be eliminated by re-centering. So all that matters is the second moment. But the second moment comes from the covariance, which is an inner product! (technically, only once you restrict to re-centered random variables, but we are doing that)

I'd venture a guess that most, if not all, appearances of $pi$ in statistics boil down to this fact that covariance is an inner product, and the fact that spheres, which are the norm-level sets for inner product spaces, have areas related to $pi$

ag.algebraic geometry - Question on $Ext$

Let $S$ be the polynomial ring $k[x_0,ldots,x_n]$, $x$ one of the variables $x_i$, $Isubseteq S$ a homogeneous ideal which has a generating set $f_1,ldots,f_r$ where $deg_x f_i=0$ for all $i$.

From the short exact sequence
$$0to S/I(-1)xrightarrow{f} S/I to S/(x,I)to 0$$
where the first map $f$ is multiplication with x and the second sends $s+I$ to $s+(x,I)$,
I get the long exact sequence
$$0to Hom(S/(x,I),S)to Hom(S/I,S)to Hom(S/I(-1),S)toldots$$
$$to Ext^{m-1}(S/(x,I),S)to Ext^{m-1}(S/I,S)xrightarrow{f^*} Ext^{m-1}(S/I(-1),S)to Ext^m(S/(x,I),S)toldots$$

My aim is to show that
$$Ext^m(S/(x,I),S)=frac{Ext^{m-1}(S/I,S)}{(x)Ext^{m-1}(S/I,S)},$$
so I thought to get there by showing that the induced map $f^star$ in the above sequence is injective or equivalently:
if $J^bullet$ is an injective resolution of $S$ with boundaries $d^m$ and $varphiin Hom(S/I,J^m)$ with $d^m_star(varphi)=0$ and $f^star(varphi)in d^{m-1}_star(Hom(S/I(-1),J^{m-1})),$
then is $varphiin d^{m-1}_star(Hom(S/I,J^{m-1}))$.

Is it true that $f^star$ on the Ext-modules is an injection? And if yes, how can i show that?
Thanks for any help!

arithmetic scheme - Existence of proper regular models for varieties over Q and other global fields

What is known about regular proper models for smooth projective varieties over Q? Results for other global fields would also be interesting, as well as general comments and suggested references for integral models.

This is a followup to this question on smooth models, and here is part of what I wrote as an answer to the previous question: Nekovar's survey article on the Beilinson conjectures from the early 90s mentions some results for varieties over Q. He says in section 5.3 that given a smooth projective variety over Q, there always exists a proper flat model over Z, but that a regular such model is rarely known to exist. However, in the published version of the same survey, there is an added note at the very end of the article saying that "Spivakovsky recently announced a general result on resolution of singularities, which implies that a regular proper flat model of X mentioned in 5.3 always exist". However, I have never seen this result of Spivakovsky mentioned anywhere else, so I doubt that it is true. Does anyone else know more about this?

The survey is available here. For the published version, google "Serre Jannsen Motives", click at the Google Books link, and then search for "Spivakovsky" within the book.

ac.commutative algebra - Rank of a module

since your profile says you are interested in Algebraic Geometry, here are geometric considerations that might appeal to you.

Consider a projective module $P$ of finite type over a commutative ring $A$. It corresponds to a locally free sheaf $mathcal F $ over $X=Spec(A)$. The rank of $mathcal F $ at the prime ideal $mathfrak p$ is that of the free $A_{mathfrak p}$-module $mathcal F_{mathfrak p}$.
The rank is then a locally constant function on $X$ and if $X$ is connected (this means that the only idempotents in $A$ are $0$ and $1$) it may be seen as an integer.

If $A$ is a domain, then $X$ is certainly connected and has a generic point $eta$ whose local ring is the field of fractions $mathcal O_eta=K=Frac(A)$. The rank of $mathcal F $ or of $P$ is then simply the dimension of the $K$ vector space $Potimes_A K$.

Actually, if $A$ is a domain, this formula can be used to define the rank of any $A$-module $M$ (projective or not, finitely generated or not) : $rank(M)=dim_K ( Motimes_A K) $ .
This is the definition given by Matsumura in his book Commutative Rings, page 84.
It corresponds to the maximum number of elements of $M$ which are linearly independent over $A$.

The minimum number of generators of $M$ (which started this discussion) is quite a different, but interesting invariant, which has been studied by Forster, Swan, Eisenbud, Evans,...
Geometrically it corresponds to the minimum numbers of global sections of $tilde{M}$ which generate this sheaf at each point of $Spec(A)$.
Elementary example: Every non-zero ideal of a Dedekind domain is of rank one, can be generated by at most two elements and can be generated by one element iff it is principal.
If the Dedekind domain is not a PID there always exist non free ideals which thus cannot be generated by less than two elements.

Bibliography
Ischebeck and Rao have published a monograph Ideals and reality: projective modules and number of generators of ideals on exactly this theme

gn.general topology - An example of a space which is locally relatively contractible but not contractible?

A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the path fibration $PX to X$ space of based paths with evaluation at the endpoint as projection) admits local sections iff $X$ is $infty$-well-connected (or locally relatively contractible, or semi-locally contractible), that is, has a basis of neighbourhoods $N$ such that the inclusion maps $Nhookrightarrow X$ are null homotopic. Another use of this concept is by Dold, when he proves a Dold fibration (a map with the Weak Covering Homotopy Property) over an $infty$-well-connected space is locally homotopy trivial.

What, then, is an example of a space which is $infty$-well-connected but not locally contractible?

---

Edit:
Note that the 1-dimensional version of this is a space that is semilocally 1-connected (or 1-well-connected, in my revisionist terminology), but not locally 1-connected.

nt.number theory - Is the smallest primitive root modulo p a primitive root modulo p^2?

The key term here is: Wieferich prime base $a$.

What you observed can be presented to children in the following form: if $p$ is a prime greater than 5 and the fraction $1/p$ has decimal period $d$, numerical tables show $1/p^2$ has decimal period $dp$, $1/p^3$ has decimal period $dp^2$, and generally the decimal period of $1/p^k$ is $dp^{k-1}$. For example, 1/13 has decimal period 6, 1/169 has decimal period $78 = 6 cdot 13$, and 1/2197 has decimal period $1014 = 6 cdot 13^2$.

This works for primes below 100, but if you search far enough you will find a counterexample. The first one is $p = 487$: 1/487 and $1/487^2$ both have decimal period 486. The second counterexample is $p = 56,598,313$. (!!) This list has been Sloaned: http://oeis.org/A045616.

For a general article about this business, see http://www.jstor.org/stable/3219294.

Within algebraic number theory, this phenomenon appears when you compute the ring of integers of ${mathbf Q}(sqrt[n]{2})$, which turns out to be ${mathbf Z}[sqrt[n]{2}]$ for all $n leq 1000$. With that evidence you might guess the ring of integers is always ${mathbf Z}[sqrt[n]{2}]$, just like the ring of integers of ${mathbf Q}(zeta_n)$ is always ${mathbf Z}[zeta_n]$. But in fact it's not always true. There are $n > 1000$ such that ${mathbf Z}[sqrt[n]{2}]$ is not the full ring of integers of ${mathbf Q}(sqrt[n]{2})$. If you search for Wieferich primes to base 2 you will find them.

nt.number theory - Do the base 3 digits of $2^n$ avoid the digit 2 infinitely often -- what is the status of this problem?

I believe this question is due to Erdős and Graham, and I think it is still open: does the base 3 expansion of $2^n$ avoid the digit 2 for infinitely many $n$?

If we concatenate the digits of $2^i$, $i geq 0$, we produce the number $0.110100100010000...$. This number is not simply normal in base 2, so it is not normal. Is it simply normal in base 3? I think even that result would not imply that for sufficiently large $n$, 2 doesn't appear in the base 3 expansion of $2^n$.

The number 20 here is not special:

$2^{20} = 1222021101011_3, ;;;; 2^{21} = 10221112202022_3, ;;;
2^{22} = 21220002111121_3$

Statistically, we seem to be flipping a fair 3-sided coin, and statistical analysis for larger $n$ bears this out (in the past, I did a p-test on the digits, but don't have the data available here). If we actually produced these digits by flipping this 3-sided coin, for fixed $n$ we would have probability about
$$(2/3)^{nln2/ln3}$$
of having no 2s in the base-3 digit expansion.

What is the state of the art for this problem? Is there a good number-theoretic reason why this problem should be very difficult (e.g. an analogy with other supposed-hard problems)? Are there related problems that have been solved?