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https://mathoverflow.net/questions/69174 | 3 | Let $k,m\in\mathbb{N}$ be given. Let $M:=\{0,... , m-1\}$. How to find a subset $T\subset M$, $|T|=k$ such that $|T+T|$ is maximal, where $T+T=\{ (a+b)\mathbin\%m \mid a\in T,b\in T \}$ (“%” means modulo)?
I tried to construct a sequence of numbers which maximize $|T+T|$. But I couldn’t figure out:
* Is it possible... | https://mathoverflow.net/users/16100 | Optimize / simple Set Covering Problem | A relevant keyword in this context is Sidon set (however Sidon is also used in a different context, so when searching do not be surprised if you stumble also over unrelated things).
A subset $T$ of an abelian group $G$, written additively, in your case $\mathbb{Z}/m \mathbb{Z}$ with addition, is called a Sidon set if... | 5 | https://mathoverflow.net/users/nan | 69208 | 42,490 |
https://mathoverflow.net/questions/69206 | 1 | Let $ \; \langle X,\mathcal{T} \hspace{.06 in} \rangle \; $ be a second-countable Hausdorff space.
Let $ \; \phi : 2^X \to [0,+\infty] \; $ be an outer regular [outer measure](http://en.wikipedia.org/wiki/Outer_measure#Formal_definitions).
Does it follow that all open subsets of $X$ are [Caratheodory-measurable](ht... | https://mathoverflow.net/users/nan | Do outer regular outer measures always measure open sets? | No. Suppose there is a set $E$ that is not measurable. Take the topology that is generated by $E$ and the original topology. This is also second countable and $\phi$ is regular for it.
| 3 | https://mathoverflow.net/users/2554 | 69210 | 42,491 |
https://mathoverflow.net/questions/69209 | 9 | It is well-known that Seifert fibered $3$--manifolds are geometric: they admit one of the Thurston geometries $S^2 \times R$, $R^3$, $H^2 \times R$, $S^3$, $Nil$, and $PSL(2,R)$. Furthermore, the converse is also true: every $3$--manifold admitting one of these 6 geometries has a Seifert fibration over a $2$--orbifold.... | https://mathoverflow.net/users/8183 | 3-orbifolds with a Seifert geometry that are not actually Seifert fibered | From *Three-dimensional orbifolds and their geometric structures*, by Boileau, Maillot and Porti (p. 15):
>
> Remark. There are orbifolds with geometry $E^3$ and $S^3$ which are not orbifold-Seifert fibered [52, 54].
>
>
>
[54] is this article of Dunbar's you quoted. [52] is [*Geometric orbifolds*](http://www.... | 8 | https://mathoverflow.net/users/9114 | 69211 | 42,492 |
https://mathoverflow.net/questions/69187 | 3 | Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of course be obtained if $A$, $B$ have a common eigenspace of multiplicity at least $m$.
My question is: is it the only pos... | https://mathoverflow.net/users/3465 | Multiplicity of eigenvalues in 2-dim families of symmetric matrices | This is false. Let $A\_0$ and $B\_0$ be $k \times k$ symmetric matrices with no common eigenspace. Let $A$ and $B$ be the $2k \times 2k$ matrices with block forms $\left( \begin{smallmatrix} A\_0 & 0 \\ 0 & A\_0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} B\_0 & 0 \\ 0 & B\_0 \end{smallmatrix} \right)$. ... | 9 | https://mathoverflow.net/users/297 | 69214 | 42,493 |
https://mathoverflow.net/questions/69213 | 3 | Let $G = \prod\_p \mathbb{Z}/p\mathbb{Z}$, where $p$ ranges over all primes, considered as an abelian group. What does $\text{Aut}(G)$ (or even $\text{End}(G)$) look like?
I know that that if we take $t(G) = \oplus\_p \mathbb{Z}/p\mathbb{Z}$, then $\text{Aut}(t(G)) = \prod\_p (\mathbb{Z}/p\mathbb{Z})^\times$ (can be ... | https://mathoverflow.net/users/16107 | Automorphisms of an infinite direct product of abelian groups | Put $A\_p=\mathbb{Z}/p$ and $B\_p=\prod\_{q\neq p}A\_q$ so $G=A\_p\times B\_p$. Multiplication by $p$ acts as zero on $A\_p$ and as an automorphism on $B\_p$ so $pG=0\times B\_p$ and $G/pG\simeq A\_p$. Put $U\_p=\text{Aut}(A\_p)=(\mathbb{Z}/p)^\times$. There is an evident homomorphism $\phi:\prod\_pU\_p\to\text{Aut}(G)... | 10 | https://mathoverflow.net/users/10366 | 69216 | 42,494 |
https://mathoverflow.net/questions/69218 | 26 | What are the axioms that a good notion of entropy must satisfy? Please note that I am *not* asking for the definitions of various types of entropy such as *topological entropy* or *measure-theoretic entropy* or etc. My question is, if $X$ is a 'space', in a broad sense (topological, measure, algebraic, etc.) and $\varp... | https://mathoverflow.net/users/16046 | If you were to axiomatize the notion of entropy | This isn't a full axiomatisation, partly because it's a little vague, and partly because I only am really familiar with the notion of entropy in two contexts: topological space and measure space. Nevertheless, there's a commonality to the procedure in both those cases.
1. Start with a space $X$ and a map $f\colon X\t... | 19 | https://mathoverflow.net/users/5701 | 69231 | 42,502 |
https://mathoverflow.net/questions/69222 | 45 | The Monster group $M$ acts on the moonshine vertex algebra $V^\natural$.
Because $V^\natural$ is a holomorphic vertex algebra (i.e., it has a unique irreducible module), there is a corresponding cohomology class $c\in H^3(M;S^1)=H^4(M;\mathbb Z)$
associated to this action.
>
> Roughly speaking, the construction o... | https://mathoverflow.net/users/5690 | $H^4$ of the Monster | There is some evidence from characters that $H^4(M,\mathbb{Z})$ contains $\mathbb{Z}/12\mathbb{Z}$. In particular, the conjugacy class 24J (made from certain elements of order 24) has a character of level 288, and the corresponding irreducible twisted modules have a character whose expansion is in powers of $q^{1/288}$... | 29 | https://mathoverflow.net/users/121 | 69233 | 42,503 |
https://mathoverflow.net/questions/69230 | 6 | 1. Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.)
2. More generally, let $G$ be the **free** group on a finite set $S$ of generators, and consider the integral group ring $R := \mathbb{Z}G$. Are... | https://mathoverflow.net/users/16109 | Projective modules over free groups | The answer to your questions is 'yes' and, as you feel, it is a result of Bass:
MR0178032 (31 #2290)
Bass, Hyman
Projective modules over free groups are free.
J. Algebra 1 1964 367–373.
| 8 | https://mathoverflow.net/users/12166 | 69235 | 42,505 |
https://mathoverflow.net/questions/69207 | 4 | **Update:** From Clinton's comment below follows that I made some mistakes (that I'm going to correct) and that the question is completely answered by Arzhantseva, Guba and Guyot. Besides giving a precise definition of what I meant with $\alpha(G)$, they proved that for any $n$, there is an $n$-generated amenable group... | https://mathoverflow.net/users/13809 | On growth rate of finitely generated groups | I suppose I should convert the comment into an answer so that the question doesn't appear unanswered.
Given a group $G$ with finite generating set $S$, one can define its *rate of growth* (matching as much as possible the notation of the question) $\alpha(G,S)$ by
$$\alpha(G,S) = \lim\_{r \to \infty} \sqrt[r]{|B\_r|}... | 8 | https://mathoverflow.net/users/14913 | 69237 | 42,507 |
https://mathoverflow.net/questions/69239 | 28 | Consider a hypersurface $X=V(f) \subset \mathbb A^n\_{\mathbb C}$, where $f(T\_1, T\_2,\ldots,T\_n)\in \mathbb C[T\_1,Y\_2,\ldots, T\_n]$ is a polynomial .
Assume that $X$ is smooth, i.e. that $df(x)\neq 0 \;$ for all $x\in X$ . My question is simply whether $X $ is parallelizable i.e. whether its tangent bundle $T\... | https://mathoverflow.net/users/450 | Is every algebraic smooth hypersurface of affine space parallelizable? | Yes. Suslin has proved that every stably trivial vector bundle of rank $n$ on an affine variety of dimension $n$ over an algebraically closed field is trivial. See:
Suslin, A. A. Stably free modules. Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537–550, 632.
Note that this is not true over arbitrary fields; for example... | 29 | https://mathoverflow.net/users/519 | 69240 | 42,509 |
https://mathoverflow.net/questions/69232 | 3 | For a givien partial order, how many generic extensions might exist? In other words, for a boolean valued model class which dreams of a generic extension, how many unique generic objects exist for a given partial order (which gives rise to a boolean algebra naming such an extension)?
| https://mathoverflow.net/users/nan | For a given partial order, how many generic extensions? | It is often the case that when we force with a partial order $\mathbb{P}$, then in the resulting forcing extension $V[G]$ there may be other $V$-generic filters and indeed many other $V$-generic filters $G'$, even giving rise to the same extension $V[G]=V[G']$. Thus, although from a the filter $G$ we may construct the ... | 5 | https://mathoverflow.net/users/1946 | 69246 | 42,512 |
https://mathoverflow.net/questions/68842 | 41 | I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school.
*Caveat*: I'm not a symplectic geometer, nor a differential topologist in the 'classical' sense, so my questions might have a well-known answer,... | https://mathoverflow.net/users/13119 | Can cotangent bundles see exotic smooth structures? | I wrote a little expository piece about this and related matters in the Newsletter of the European Mathematical Society:
<http://www.ems-ph.org/journals/newsletter/pdf/2010-03-75.pdf>
The classical topology of $X:=T^\ast L$ can be taken to include a little more than its diffeomorphism type: there's also an almost c... | 25 | https://mathoverflow.net/users/2356 | 69249 | 42,515 |
https://mathoverflow.net/questions/69253 | 14 | Are there any positive integer solutions to $2^n-3^m=1$ with $n,m>2$ ?
By way of justifying the question, I've found lots of info on what happens when $m=n$ (mostly FLT variations, Darmon + Merel,...), but I don't really know where to look for $m\not=n$.
Also it's pretty obvious that you can't have solutions to sim... | https://mathoverflow.net/users/16114 | Are there any solutions to $2^n-3^m=1$? | Here is the proof of Gersonides [Levi ben Gershon] (1343) for $2^n-3^m=1$. It uses nothing more that arithmetic modulo $8$.
Case I: $m$ is even. Then $3^m$ is 1 mod 4, so $2^n$ is 2 mod 4, implying $n=1$ and $m=0$.
Case II: $m$ is odd. Then $3^m$ is 3 mod 8, so $2^n$ is 4 mod 8, implying $n=2$ and $m=1$.
The alte... | 49 | https://mathoverflow.net/users/5267 | 69256 | 42,520 |
https://mathoverflow.net/questions/69272 | 2 | By the way, does anyone know how to prove in an elementary way (i.e. expanding) that $\prod\_1^n (1+a\_i r)$ tends to $e^r=\sum \frac{r^k}{k!}$ as you let $\max|a\_i|\to 0$ with $0\leq a\_i \leq 1$ and $\sum a\_i = 1$? An easy solution goes by writing the product with the exponential function so that you get the expone... | https://mathoverflow.net/users/7193 | A limiting product formula for the exponential function | $\prod\_{i=1}^n (1+a\_ir)=1+\sum\_{k=1}^n r^k\sum\_{i\_1 < \ldots < i\_k}a\_{i\_1}\ldots a\_{i\_k}$.
Notice that $1^k=\left(\sum\_{i=1}^n a\_i\right)^k =k!\sum\_{i\_1 < \ldots < i\_k}a\_{i\_1}\ldots a\_{i\_k}+\text{other terms}$, where the other terms are (positive) terms with a repeated $a\_i$. It follows that the $... | 4 | https://mathoverflow.net/users/11054 | 69280 | 42,534 |
https://mathoverflow.net/questions/69225 | 4 | Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?
I'm working on a research project on Poisson approximations and am looking to perform a coupling on the unknown distribution. However, I cannot find much material on how to perform a coupling and the general calculat... | https://mathoverflow.net/users/10473 | Reading Material on Couplings | My friend Marty suggests the Lindvall book as well as
H. Thorisson, Coupling, Stationarity, and Regeneration. Springer, New York, 2000.
<http://www.springer.com/mathematics/probability/book/978-0-387-98779-8>
and points out that coupling is used now in basic textbooks in stochastic processes to prove the ergodic... | 6 | https://mathoverflow.net/users/3324 | 69282 | 42,536 |
https://mathoverflow.net/questions/69144 | 8 | Let $u:A \to \prod\_{\mathcal U} M\_n$ be a unital completely positive map (ucp) from a unital separable $C^\*$algebra into the von Neumann algebra ultraprodut $\prod\_{\mathcal U} M\_n$.
Here $\mathcal U$ is an ultrafilter on $\mathbb N$ and $\prod\_{\mathcal U} M\_n$ is the quotient of $B=\{(x\_n)\_{n \in \mathbb ... | https://mathoverflow.net/users/10265 | Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras | The answer is no, in general there is no lifting. A lifting exists if the $C^{\ast}$-algebra has the so-called lifting property (LP), and local liftings exist if it has the local lifting property (LLP).
I constructed in
Andreas Thom, *Examples of hyperlinear groups without factorization property*, Groups Geom. Dyn... | 4 | https://mathoverflow.net/users/8176 | 69284 | 42,537 |
https://mathoverflow.net/questions/69278 | 4 | This question comes from Proposition 2.6 in Chapter 2 of Hartshorne's *Algebraic Geometry*. In my edition, that's on page 78.
For a variety $V$, Hartshorne defines the topological space $t(V)$ to consist of the nonempty closed irreducible subsets of $V$, where the closed sets of $t(V)$ are of the form $t(Y)$ for $Y$ ... | https://mathoverflow.net/users/16120 | Hartshorne's associated scheme for a variety | To show that $(t(V),\alpha\_\*\mathcal{O}(V))$ is a scheme, you must show that $t(V)$ has an open cover on which this ringed space is isomorphic to an affine scheme.
Take an affine open cover $\{U\_i\}$ of $V$. Since you believe the affine case, it suffices to show that $\{t(U\_i)\}$ is an open cover of $t(V)$, and
... | 7 | https://mathoverflow.net/users/12107 | 69285 | 42,538 |
https://mathoverflow.net/questions/69226 | 17 | The following inequality is from page 125 of D.S. Mitrinovic, J. Pecaric, A.M. Fink, Classical and new inequalities in analysis, Kluwer
Academic Publishers, Dordrecht/Boston/London, 1993.
If $a\_i>0$, $b\_i>0$ for $i=1,\cdots, n$ and $A=\frac{\max a\_k}{\min a\_k}$, $B=\frac{\max b\_k}{\min b\_k}$ with $\frac{1}{p}+\... | https://mathoverflow.net/users/3818 | How to prove a known inequality from a book | The following proof was inspired by Fedor Petrov's and Gjergji's Zaimi's argument, but it is simpler.
By a scaling argument we may assume $a\_i\in[1,A]$, $b\_i\in[1,B]$.
The inequality can be rewritten as
$$x^{1/p}y^{1/q} \leq (A^pB^q-1)\sum\_{i=1}^n a\_ib\_i,$$
where
$$x:=p(AB^q-B)\sum\_{i=1}^na\_i^p\qquad\text{and}... | 16 | https://mathoverflow.net/users/11919 | 69302 | 42,552 |
https://mathoverflow.net/questions/55878 | 13 | Let $A$ and $B$ be two closed, 2-dimensional, non-positively-curved Riemannian disks (not necessarily with convex boundary). Suppose that their boundaries $\partial A$ and $\partial B$ have the same length, and identify $\partial A$ with $\partial B$ with a length-preserving diffeomorphism of circles. Suppose also that... | https://mathoverflow.net/users/1450 | A comparison question for non-positively curved disks | The answer is yes. Moreover you don't need to assume that $B$ is nonpostively curved. (And, if you are not interested in the equality case or can afford a convex boundary, the nonpositive curvature of $A$ can be replaced by a weaker assumption that the geodesics in $A$ have no conjugate points).
This follows from a f... | 13 | https://mathoverflow.net/users/4354 | 69303 | 42,553 |
https://mathoverflow.net/questions/69307 | 21 | I assume it is partially because they are good generalizations of polynomial rings, but what makes this generalization better than graded algebras or other generalizations of polynomial rings?
| https://mathoverflow.net/users/4692 | Why are noetherian rings such natural objects in algebraic geometry? | The best answer I've ever been able to come up with is that the class of noetherian rings contains the classical number rings $\mathbf{Z}$ and $\mathbf{R}$ and is closed under the formation of polynomial rings, localization, completion, and quotients. So it contains many of the rings you will come across in ordinary si... | 38 | https://mathoverflow.net/users/1114 | 69309 | 42,556 |
https://mathoverflow.net/questions/69306 | 0 | Reading [this question](https://mathoverflow.net/questions/54818/consistency-strength-needed-for-applied-mathematics), and the [Wikipedia page](http://en.wikipedia.org/wiki/Reverse_mathematics) on reverse mathematics, I wonder whether one needs more than the subfield $\mathcal{P} \subset \mathbb{C}$ of periods for appl... | https://mathoverflow.net/users/4177 | Do we need more than the periods? | If my knowledge is sufficiently up to date, it is not known whether the periods form a field at all. So I would say yes! We need more than the periods. Even applied mathematics benefits from the stuctural simplicity of certain objects. And as Qiaochu pointed out, it is not known whether $e$ is a period.
In general I... | 3 | https://mathoverflow.net/users/7743 | 69314 | 42,560 |
https://mathoverflow.net/questions/69312 | 3 | Given a real-valued data set $ x\_1, \dots, x\_n $, what do you call the quantity
$$\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits\_{i=1}^n |x\_i - x|$$
This seems like a pretty basic thing to ask for. For example, in a game in which I have to guess the number you're thinking of, and I have to pay you... | https://mathoverflow.net/users/16127 | name for $\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$ | The value of $x$ that satisfies this is the median of $x\_i$. It minimizes $L\_1$ loss.
Note that the mean of $x\_i$ is the value of $x$ minimizing $\sum(x\_i - x)^2$, the $L\_2$ loss.
| 7 | https://mathoverflow.net/users/9896 | 69319 | 42,564 |
https://mathoverflow.net/questions/69317 | 1 | I am looking for an answer to the following questions:
Are there infinite-dimensional Banach spaces $X$ and $Y$ for which there are non-split extensions $0 \to X \to E\_1 \to Y \to 0$ and $0 \to X \to E\_2 \to Y \to 0$ such that $X$ is complemented in $E\_1$ but non-complemented in $E\_2$?
Also, let $\Delta \subset... | https://mathoverflow.net/users/16129 | Extensions of Banach spaces | I don't understand the questions for the following reason: If the image of $X$ is complemented in $E\_1$ then the extension is split. Indeed, if $P$ is a projection of $E\_1$ onto the image of $X$ then $1-P$ is a projection onto an isomorph of $Y$ by the open mapping theorem (see e.g. [Nicolas Monod's thesis](http://dx... | 3 | https://mathoverflow.net/users/11081 | 69321 | 42,565 |
https://mathoverflow.net/questions/69323 | 2 | Reading about the geometrical theory of systems of first order pdes, I have met a result from symplectic geometry, that is easy to prove, but I am unable to give a reference for it. So my question is:
>
> Where could I find a reference to the following easy result?
>
>
>
Let $W$ be a coisotropic embedded subma... | https://mathoverflow.net/users/12617 | On the Complete integrability of a tangent distribution | This is an exercise in McDuff and Salamon's *Introduction to Symplectic Topology*. In the second edition, it's exercise 3.29. I don't know if an exercise is a great reference, but I've seen them used before.
| 2 | https://mathoverflow.net/users/12412 | 69327 | 42,568 |
https://mathoverflow.net/questions/69328 | 9 | Given a morphism $f:X\rightarrow Y$ between smooth complex varieties, one can define functors from the bounded derived category with holonomic cohomology on $Y$ to the same category on $X$. The easiest one is $Lf^{\*}$ which can be obtained by putting a $D$-module structure on the inverse image of $\mathcal O$-modules ... | https://mathoverflow.net/users/2837 | Confusion about D-modules and functors | Just to clear up some notational confusion (there doesn't seem to be any completely standard notation):
**In Bernstein's notes**:
The "easy" pullback (i.e. the one that coincides with the pullback of the underlying $\mathcal O$-module) is denoted $Lf^\Delta$.
$f^! = Lf^\Delta [dim X - dim Y]$ (right adjoint to $f... | 12 | https://mathoverflow.net/users/7762 | 69333 | 42,571 |
https://mathoverflow.net/questions/69331 | 2 | Let $X\subset\mathbb{P}^N\_{\mathbb{C}}$ be a projective irreducible variety and $p\in X$. I put
$$
A= \{ x\in X:\langle x,p\rangle\subseteq X \},
$$
where I denotes with $\langle x,p\rangle$ the line through $x$ and $p$. Is it true that $A$ is constructible?
Thank you
| https://mathoverflow.net/users/15606 | On the constructability of a particular set. | Yes. Moreover, $A$ is closed. Indeed, assume that $X$ is given by equations $f\_1,\dots,f\_n$. First, consider the set $B$ of lines through $p$ contained in $X$. Note that the set of all lines through $p$ is $P^{N-1}$ and $B$ is a closed subset of $P^{N-1}$. Indeed, assume for example that $p = (1,0,\dots,0)$. Then a p... | 5 | https://mathoverflow.net/users/4428 | 69338 | 42,573 |
https://mathoverflow.net/questions/69337 | 12 | I have some background in set theory and automata and I am looking for a good place to start with lambda calculus.
| https://mathoverflow.net/users/16132 | What is some good introduction to lambda calculus? | There is, of course, the very famous book by Barendregt,
* The Lambda Calculus, Its Syntax and Semantics (Studies in Logic and the Foundations of Mathematics, Volume 103). Revised Edition, North-Holland, 1985. [(link to vendor)](http://rads.stackoverflow.com/amzn/click/0444875085)
which doesn't require much backgr... | 7 | https://mathoverflow.net/users/2926 | 69340 | 42,574 |
https://mathoverflow.net/questions/69062 | 11 | Let $E\_2$, $E\_4$, and $E\_6$ denote the standard Eisenstein series.
The usual variables $q=e^{2\pi i\tau}$ allow us to regard the
$E\_n$'s as functions on either the upper half plane or the unit
disk and we can define $E\_n'=\frac{1}{2\pi
i}\frac{d}{d\tau}E\_n(\tau)=q\frac{d}{dq}E\_n(q)$. I had cause to
calculate a f... | https://mathoverflow.net/users/13377 | Binomial coefficients and derivatives of modular forms | The constant $\alpha$ in your question can be in fact written explicitly as $(k)\_n/12^n$, where $(a)\_n=\Gamma(a+n)/\Gamma(n)$ is the Pochhammer symbol (shifted factorial) and $k$ denotes the (even) weight of the corresponding Eisenstein series.
Your observation is indeed related to the Rankin--Cohen brakets; see Se... | 16 | https://mathoverflow.net/users/4953 | 69348 | 42,579 |
https://mathoverflow.net/questions/69352 | 27 | $U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like to understand its topology.
By one of the tables [here](http://en.wikipedia.org/wiki/Table_of_Lie_groups) $SU(3)$ is a ... | https://mathoverflow.net/users/16019 | Topology of SU(3) | Apart from jokes, an answer which may satisfy you is the following: $SU(3)$ is a $S^3$-bundle over $S^5$. To see this just consider the defining representation of $SU(3)$ on $\mathbb{C}^3$; this induces a transitive action of $SU(3)$ on the unit sphere of $\mathbb{C}^3$, which is $S^5$. Since the stabilizer of a point ... | 47 | https://mathoverflow.net/users/8320 | 69355 | 42,582 |
https://mathoverflow.net/questions/69344 | 13 | Does anyone know an example of a smooth hyperbolic surface bundle over a hyperbolic surface (surface = compact two-manifold) which does *not* have a complex structure? Is there any decision procedure to tell, given such a bundle, whether it has a complex structure, or is it more of a black art?
| https://mathoverflow.net/users/11142 | A four-dimensional counterexample? | This [paper by Hillmann](http://www.ams.org/mathscinet-getitem?mr=1733043) addresses this question. He proves that a surface bundle over a surface which is a complex surface has a holomorphic fibration over the base, for some choice of complex structure on the base. He uses this to prove that when the fiber has genus 2... | 12 | https://mathoverflow.net/users/1345 | 69370 | 42,591 |
https://mathoverflow.net/questions/69335 | 2 | Hi,
I am interested in the following question. Let $F(x\_1, x\_2, x\_3, x\_4)$ be a quadratic form in four variables with integer coefficients. Let $B > 0$ be a parameter. Define $N\_1(F,B)$ to be the number of rational solutions to the equation $F(x\_1, x\_2, x\_3, x\_4) = 0$, such that $|x\_i| \leq B$ for $i = 1,2,... | https://mathoverflow.net/users/10898 | Rational roots to quadratic forms in 4 variables | I'm afraid that, even if we assume (as suggested in A.Meyerowitz's comment) that the intention is *integer* solutions of $F(x\_1,x\_2,x\_3,x\_4) = 0$, the number of solutions with $\max\_i |x\_i| \leq B$ will grow at least as $B^2$ as long as there's any nonzero solution, and sometimes the growth will even be a bit fas... | 10 | https://mathoverflow.net/users/14830 | 69379 | 42,595 |
https://mathoverflow.net/questions/69243 | 5 | Consider diffusion:$$d\eta^x\_{t}=\sigma(\eta^x\_{t})dW\_{t}+\mu(\eta^x\_{t})dt,\quad \eta\_0^x =x,$$ where $W$ is a Wiener process. We assume that $\sigma,\mu$ are such that the diffusion is well-defined and that it converges to the invariant measure $\eta\_{\infty}$ (e.g. the Ornstein-Uhlenbeck process with $\sigma(\... | https://mathoverflow.net/users/1302 | Diffusion convergence | Preliminaries and Notation
--------------------------
I can proof your result without using the explicit transition density. The approach could generalize to the case, where the drift $\mu$ is a potential, so set $\mu(x)= -\nabla H(x)$. Then in the case of the OU-Process $H(x)=\frac{1}{2}x^2$. So we consider a proces... | 3 | https://mathoverflow.net/users/13400 | 69387 | 42,601 |
https://mathoverflow.net/questions/68039 | 4 | My question is: Are the orbits of the geodesic flow on $S^n$ determined as the fibers of the momentum map for its $SO(n+1)$ symmetry?
I started by considering the analog problem for the orbits of the hamiltonian flow of the n-dimensional harmonic oscillator and the fibers of the momentum map for its $U(n)$-symmetry. ... | https://mathoverflow.net/users/12617 | The fibers of the momentum map for the $SO(n+1)$ symmetry of the geodesic flow on $S^n$ | You may want to look into the notion of a dual pair in symplectic and Poisson geometry
to place your examples in a more general context.
Your first example is one of the canonical examples of a dual pair. Weinstein
defined dual pairs in symplectic/Poisson geometry as a symplectic analogue
of Howe's dual pairs importa... | 2 | https://mathoverflow.net/users/2906 | 69388 | 42,602 |
https://mathoverflow.net/questions/58374 | 3 | Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of ${}\_{2}F\_{1}[-n,-n+1;2;x]$ and ${}\_{2}F\_{1}[-n-1,-n+3;2;x]$ hoping to obtain it in some form ${}\_{p}F\_{q}$.
I've found some stuff in Bailey (1928,1935), bu... | https://mathoverflow.net/users/12418 | Product of hypergeometric functions/Jacobi Polynomials | Of course, there is no general formula of the type you wanted
but a whole bunch of the formulae expressing the product of
two $\_2F\_1$ by hypergeometric (or nearly hypergeometric) means.
They are known as Orr-type theorems and can be found in
Slater's book "Generalized hypergeometric functions", Section 2.5
(there are... | 3 | https://mathoverflow.net/users/4953 | 69392 | 42,603 |
https://mathoverflow.net/questions/69353 | 6 | Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely generated, rank one projective modules which are not invertible. Can someone help me out by providing a specific example? Ad... | https://mathoverflow.net/users/9015 | An example of a rank one projective R-Module that is not invertible | A rank one projective module $M$ over a commutative noetherian ring is necessarily finitely generated. Indeed assume otherwise. Let $a\_i$ be the minimal idempotents of $R$ (there are finitely many since $R$ is commutative noetherian), so that $R=\bigoplus\_i a\_iR$. Then $a\_iM$ is projective over the connected (=inde... | 10 | https://mathoverflow.net/users/14094 | 69414 | 42,614 |
https://mathoverflow.net/questions/69412 | 7 | In ZFC, every construction of a Lebesgue or Borel non-measurable set uses the axiom of choice. None of them that I've seen use choice to define a *unique* set, even though it's entirely possible to do so (e.g. under the AoC, if $\kappa = |A|$ is the cardinality of set $A$, then $\kappa$ is unique). So I've been wonderi... | https://mathoverflow.net/users/10828 | Is Lebesgue/Borel non-measurability actually caused by non-uniqueness? | **The answer below has been edited in light of other answers and comments.**
There are all sorts of models of $ZFC$ in which *every* set is definable without parameters, including nonmeasurable sets; indeed a [recent paper](https://arxiv.org/abs/1105.4597) of Hamkins, Linetsky, and Reitz is devoted to such "pointwise... | 10 | https://mathoverflow.net/users/9269 | 69423 | 42,619 |
https://mathoverflow.net/questions/69391 | 7 | I'm writing up a paper now where I'm the only author and have a stylistic question.
Should I write ''I'' or ''we'' as in ''I/we recall the definition...'' etc. I think this simple example will make everyone understand what I'm talking about.
Or should/can I mix? Is this too confusing. Or simply bad? Or ok?
I fe... | https://mathoverflow.net/users/2147 | Stylistic question | In the interest of having an answer and since this is CW anyway:
English is not my native language and (thus) I read several articles/books/chapeters on mathematical writing in English, among others by Halmos, Krantz, and Knuth (et al.)
I do not have the references handy and, since it's been a while, do not recall ... | 2 | https://mathoverflow.net/users/nan | 69424 | 42,620 |
https://mathoverflow.net/questions/69432 | 5 | I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking for an explicit/implicit solution (general or particular, given the context). Flow lines are an example of this, but I'... | https://mathoverflow.net/users/15856 | Looking for ideas concerning the teaching of lower-division differential equation courses... | One thing you can try (and if you decide to do it, I'd like to hear how it goes), is to discuss the differential equation $\frac{dz}{dt} = \alpha z$ with $\alpha \in {\mathbb C}$. The initial condition $z(0) = 1$, of course, corresponds to $e^{\alpha t}$, but what's nice is that you can draw the flow lines and see with... | 3 | https://mathoverflow.net/users/7193 | 69435 | 42,624 |
https://mathoverflow.net/questions/69428 | 6 | This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to be what Conway, Sloane, and in particular Gabriele Nebe refer to as "covering radius less than $\sqrt 2,$" see
<http:... | https://mathoverflow.net/users/3324 | Is the square of the covering radius of an integral lattice/quadratic form always rational? | Yes, the square $R^2$ of the covering radius is always rational; and in small dimensions its denominator is always a factor of $2^{n+1} \Delta$ where $\Delta$ is the lattice discriminant, but possibly not for all $n$.
[I see that David Speyer just posted a very similar answer...]
A point $P$ at maximal distance $R$... | 8 | https://mathoverflow.net/users/14830 | 69438 | 42,626 |
https://mathoverflow.net/questions/69422 | 10 | Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the category of locally constant sheaves (of sets) on $X$ and the category of covers (sous-entendu local homoemorphism) of $X$.
... | https://mathoverflow.net/users/11964 | locally constant constructible sheaves and finite etale coverings | Consider your functor from étale coverings to locally constant constructible sheaves. It is fully faithful, by Yoneda's lemma. The fact that it is essentially surjective follows from descent theory. If $F$ is a locally constant constructible sheaf, take an étale cover $\{U\_i \to X\}$ such that the restriction of $F$ t... | 9 | https://mathoverflow.net/users/4790 | 69442 | 42,629 |
https://mathoverflow.net/questions/69456 | 7 | The following (very simply looking!) problem occurs in regularization
of the harmonic series
which can be formally thought of as the limit as $q\to1$, $|q|<1$, of
$$
h(q):=(1-q)\sum\_{n=1}^\infty\frac{q^n}{1-q^n}.
$$
I can show (with some effort) that
$$
h(q)=-\log(1-q)+f(q) \qquad\text{as}\quad q\to1, \ |q|<1,
$$
wher... | https://mathoverflow.net/users/4953 | Asymptotics of the $q$-harmonic series as $q\to1$ | Andrew is right, the following limit seems to be what you are looking for
$$\lim\_{q\uparrow 1}\left(\log(1-q)-\log q \sum\_{n\geq 0}\frac{q^{n+1}}{1-q^{n+1}}\right)=\gamma$$
See , for example theorem 1 in ["Summations for Basic Hypergeometric Series Involving a $q$-Analogue of the Digamma Function"](http://citeseerx.i... | 9 | https://mathoverflow.net/users/2384 | 69460 | 42,637 |
https://mathoverflow.net/questions/69462 | 10 | I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following question, which seems natural to me and that I haven't seen
answered in the literature (but this is perhaps a consequence of m... | https://mathoverflow.net/users/9317 | Motives from the fundamental group made nilpotent | As you say, in general the representation should not be semi-simple even if $X$ is smooth projective. One can construct explicit examples as follows:
Let $X$ be any smooth projective curve of genus $g=2$ over $\mathbb{Q}$ with a rational point such that for some prime $p$, $X$ has a regular model over $\mathbb{Z}\_p$... | 9 | https://mathoverflow.net/users/519 | 69466 | 42,639 |
https://mathoverflow.net/questions/69427 | 12 | It is well known that there exists three and four manifolds that do not admit an Einstein metric, but I wonder if this question is still open for manifolds of dimension higher than four. That is, does anyone know of a compact n-manifold, $n>4$, that does not admit an Einstein metric?
| https://mathoverflow.net/users/15856 | Obstructions to Einstein metrics in high dimensions | No one knows this. Here is a citation from Gromov's beautiful article: <http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf>
Page 19.
*Following Alex we (I speak for myself) are lead to the pessimistic conclusion that there is no chance for a distinguished* $g\_{best}$ *for* $n\ge 5$ *and "natural" metrics, e.... | 14 | https://mathoverflow.net/users/943 | 69473 | 42,642 |
https://mathoverflow.net/questions/69471 | 6 | Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute the ring of invariants? The case of personal interest is when $K$ is a finite field but the the answer over $\mathbb{C}... | https://mathoverflow.net/users/1199 | What is the ring of invariants of GL acting on quaternary cubic forms? | According to Dolgachev in
<http://arxiv.org/abs/math/0408283>
the ring of invariants is generated by 6 invariants of degrees 6, 16, 24, 32, 40 and 100
the last one being a polynomial in the other ones.
In addition to the references to Salmon and Clebsch indicated by Dolgachev, I would also
look up the book by Salmon "A... | 5 | https://mathoverflow.net/users/7410 | 69478 | 42,644 |
https://mathoverflow.net/questions/69476 | 20 | Hello everybody !
I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it.
Hence... | https://mathoverflow.net/users/1715 | Fast evaluation of polynomials | If the polynomial is given as $\alpha\_0x^0+\dots+\alpha\_nx^n$ and you do not know a priori anything about the $\alpha\_i$’s, then you can’t do better than [Horner’s scheme](http://en.wikipedia.org/wiki/Horner_scheme) (which takes $n$ additions and multiplications). If you know that the polynomial is sparse and you ar... | 21 | https://mathoverflow.net/users/12705 | 69480 | 42,646 |
https://mathoverflow.net/questions/35594 | 10 | Littlewood's well-known conjecture about simultaneous rational approximation is that for all $x, y \in \mathbb{R}$, $\liminf\_{n \to \infty} n \Vert nx \Vert \Vert ny \Vert = 0$ (where $\Vert x \Vert$ denotes the distance from $x$ to the nearest integer).
A heuristic argument for this (mentioned in this [survey artic... | https://mathoverflow.net/users/3755 | Simultaneous rational approximation of two reals using their continued fractions | Yes, there are badly approximable numbers $x$ and $y$, in fact quadratic irrationals, such that $||q\_n(x)y||$ and $||q\_n(y)x||$ are bounded away from zero. Specifically, we can take $x = \sqrt{2}/2$ and $y = \sqrt{2} + 1/2$. It's straightforward to check that $q\_n(\sqrt{2}/2)$ is always odd, from which it follows th... | 5 | https://mathoverflow.net/users/4720 | 69486 | 42,651 |
https://mathoverflow.net/questions/69458 | 2 | What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups with respect to the usual addition. One example of a non-split extension is the Heisenberg group $H\_3(\mathbb{R})$ (Pleas... | https://mathoverflow.net/users/16129 | Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$ | Central extensions
$$
0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0
$$
in which $G$ is a principal $\mathbb{R}$-bundle over $\mathbb{R}^2$ (I suppose you mean that by "topological")
are classified by continuous maps
$$
f: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}
$$
satisfying
$$
f(x,y)f(y,z) = f(x,z).
$$
The a... | 2 | https://mathoverflow.net/users/3473 | 69489 | 42,652 |
https://mathoverflow.net/questions/68803 | 13 | I am trying to understand how all the players in the title relate, but with all the grading shifts,and difficult isomorphisms involved in the subject I am having a hard time being sure that I have the picture right. I am going to write what I think is true, and if someone would confirm or deny it, that would be really ... | https://mathoverflow.net/users/6986 | How to relate equivariant symplectic cohomology, Contact Homology, Cyclic Homology and String Topology? | **Some blah on symplectic homology vs. cohomology.** There's an invariant $SH(M)$ of Liouville domains $M$ which some people call symplectic homology and some symplectic cohomology. This is the direct limit of Hamiltonian Floer groups associated with functions of increasing eventual slope. The dual theory has two rathe... | 11 | https://mathoverflow.net/users/2356 | 69490 | 42,653 |
https://mathoverflow.net/questions/69440 | 10 | I refer to the mathematician described here:
<http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=19612>
I am interested in learning, e.g., his full name.
| https://mathoverflow.net/users/658 | Who is L. A. Balashov? | It is fantastic to learn how easy people are forgotten.
I made several calls to the Faculty of Mechanics and Mathematics
of the Moscow Lomonosov State University (including the Human
Resources and the Chair of Function Theory and Functional Analysis
where Balashov worked till his sudden death) without success.
I have j... | 19 | https://mathoverflow.net/users/4953 | 69491 | 42,654 |
https://mathoverflow.net/questions/69431 | 6 | To follow up on [A four-dimensional counterexample?](https://mathoverflow.net/questions/69344/a-four-dimensional-counterexample), I am probably being dense, but are there examples of spaces which are homotopy equivalent to bundles of surfaces over surfaces (or three-manifolds over the circle, or circle over three-manif... | https://mathoverflow.net/users/11142 | More four-dimensional counterexamples | Borel's conjecture predicts that any homotopy equivalence of closed aspherical manifolds is homotopic to a homeomorphism. The conjecture has been proved for many fundamental groups, see e.g.
[The Borel Conjecture for hyperbolic and CAT(0)-groups](https://arxiv.org/abs/0901.0442) by Bartels-Lueck.
Basic ingredients ar... | 8 | https://mathoverflow.net/users/1573 | 69498 | 42,655 |
https://mathoverflow.net/questions/69179 | 5 | I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some symm. pos. def matrix A. Here assume that differentiation $\nabla = (D\_i)\_{i=1,..,n}$ is a skew-adjoint operator densel... | https://mathoverflow.net/users/4047 | Integration by parts for a general negative-definite self-adjoint operator. | It is possible to make sense of $T^{1/2}$ without some of the particulars mentioned, when $T$ is a positive self-adjoint (densely-defined) operator on a Hilbert space. Namely, Friedrichs' argument (as in Riesz-Nagy, for example) shows that the resolvent $(T-\lambda)^{-1}$ exists and is a bounded operator for $\lambda$ ... | 4 | https://mathoverflow.net/users/15629 | 69501 | 42,658 |
https://mathoverflow.net/questions/69420 | 7 | This question is motivated by [this one](https://mathoverflow.net/questions/69353/an-example-of-a-rank-one-projective-r-module-that-is-not-invertible). The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps it is st... | https://mathoverflow.net/users/10076 | The rank of a not necessarily finitely generated module. | Here are a few comments, too long to fit in the comments box.
1) One can modify Tom Goodwillie's example as follows: for simplicity lets pick $R$ to be a local domain of dimension $1$ (so there are only $2$ prime ideals, $0$ and $\mathfrak m$, with residues $K$, the quotient field and $k$, the residue field of $R$ r... | 5 | https://mathoverflow.net/users/2083 | 69504 | 42,659 |
https://mathoverflow.net/questions/69510 | 4 | Hi, my question is :
Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. $ Does there exist a complex analytic diffeomorphism $F$ ( analytic in two complex variables ) whose domain is either ... | https://mathoverflow.net/users/6953 | Is complex analytic extension of real-analytic diffeomorphism a diffeomorphism ? | Indeed, a real-analytic map can always be extended to some complex neighbourhood.
The problem is, the neighbourhood may be very small. Consider, for example,
the map
$$f:I\to I,\, I=[-1,1],$$
defined by
$$f:x\mapsto x+\frac{a^3x(x-1)^2}{x^2+a}.$$
For small $a>0$ this is a diffeomorphism of $I$, but it cannot be exte... | 7 | https://mathoverflow.net/users/9833 | 69511 | 42,660 |
https://mathoverflow.net/questions/69516 | 3 | Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of irreducible with finite multiplicity?
I know that this is true, for $\pi$ trivial and $H$ unimodular (=> G unimodular). Is u... | https://mathoverflow.net/users/10400 | Inducing from cocompact subgroups | Yes it is necessary. For instance take $G=GL(2, {\mathbb Q}\_p )$ and $H$ the non-unimodular subgroup of upper triangular matrices. Then the (smoothly) induced representation ${\rm Ind}\_H^G {\mathbf 1}$ is not semisimple. It is of length $2$. It has the trivial representation as a subrepresentation and the Steinberg r... | 5 | https://mathoverflow.net/users/4767 | 69518 | 42,662 |
https://mathoverflow.net/questions/69507 | 3 | My question(s) relate(s) to pp51-52 of Local Representation Theory by JL Alperin -- the relevant pages are contained in the Google Books preview <http://books.google.com/books?id=p7ylsZUmK3MC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false>. In these pages he is dealing with representations of $SL(2,p)$ over a f... | https://mathoverflow.net/users/15632 | Projective modules and tensor products | He claims that by the proof of Lemma 5 (which is not in google preview) it follows that $V\_2\otimes V\_p$ has a submodule isomorphic to $V\_{p-1}$, but this is the socle of $P\_{p-1}$ since $P\_{p-1}$ is the projective cover of $V\_{p-1}$ and a group algebra is always symmetric.
For the next question he first claims... | 3 | https://mathoverflow.net/users/15887 | 69524 | 42,663 |
https://mathoverflow.net/questions/69523 | 6 | This should be an elementary question for anyone who knows SGA by heart (alas, not for me). It smells a lot like a descent problem. All schemes are supposed to be noetherian, and all morphisms to be locally of finite presentation.
Let $X$ be a scheme of finite type over a (base) scheme $S$, and let $R \subseteq X(S)$... | https://mathoverflow.net/users/5952 | When does Zariski closure commute with base change? | It seems that the condition you need is that the generic points of $S'$ go to generic points of $S$; this is much weaker than flatness.
Assuming this condition, we can reduce to the case that $S$ and $S'$ are both $Spec$ of some field and we can also assume that $\overline{R} = X$. If $X$ is a variety over a field $k... | 6 | https://mathoverflow.net/users/519 | 69527 | 42,664 |
https://mathoverflow.net/questions/69526 | 4 | Suppose $Y$ is an algebraic variety and $\mathcal{E}$ a coherent sheaf on $Y$. Suppose $f:X=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E})) \to Y$ is a morphism of algebraic varieties with all fibres scheme theoretically projective spaces.
If the fibres all had the same dimension, I would have $\mathrm{R}f\_\* \mathbb{C}\_X... | https://mathoverflow.net/users/4707 | Cohomology of projective space bundles | I think what you want is true if $X$ and all the $Y\_k$ are smooth (or have some very mild singularities e.g. quotient singularities) but I don't know many such examples. In general it appears to be false as shown by the following example:
Let $Y$ be the quadric cone given by $x\_1x\_2 - x\_3x\_4 = 0$ in $\mathbb{A}^... | 6 | https://mathoverflow.net/users/519 | 69535 | 42,669 |
https://mathoverflow.net/questions/69543 | 11 | I refrained from asking the technical questions; maybe everyone didn't like my attitude. At least, help me finding good books.
Can anyone suggest a good book that gives a complete reference to "Dirichlet's class number formula" and Class number theory, and explaining each nook and corner of it? Or any reference mater... | https://mathoverflow.net/users/nan | Good books on Dirichlet's class number formula | Where you might want to start: The classical approach is based on special functions, and given e.g. here: <http://www-math.mit.edu/~kedlaya/Math254B/zetafunction.pdf> (I found this directly with google). I think the standard reference for such things is Neukirch "Algebraic Number Theory" and the later chapters on $L$ f... | 4 | https://mathoverflow.net/users/10400 | 69544 | 42,671 |
https://mathoverflow.net/questions/69484 | 1 | Let $(A,\mathfrak{m})=k[[x,y]]$ with $char(k)=0$ and $K=Quot(A)$. Set $X=Spec(A)$, $U=Spec(A)\backslash \lbrace \mathfrak{m} \rbrace$ the pointed spectrum. Furthermore given an $A$-algebra $B$, which can be embedded in $C=M\_n(A)$, where $B$ is free $A$-module of rank $n^2$. One can see the algebra as a sheaf on $X$ re... | https://mathoverflow.net/users/3233 | Cohomology of the general linear group on punctured spectra of 2-dim. power series rings | a): An element of $C^\times$ can be thought of as a pair $(a,b)$ of elements of $C$ with $ab=1$. This gives a) by applying of existence extension to $a$ and $b$ and unicity to $ab$ and $1$.
b): The relation $F=C^\times/B^\times$ is a relation of a sheaf quotient for the flat topology (even if one defines $F$ as schem... | 2 | https://mathoverflow.net/users/4008 | 69549 | 42,673 |
https://mathoverflow.net/questions/69545 | 15 | Most proofs of undecidability for various theories (pure logic with binary relation, group theory, etc.) show that the natural numbers and Robinson's $Q$, in one form or another, can be encoded appropriately. Hence the decision problem for these theories is as hard as $K$, the halting set.
Are there are recursively a... | https://mathoverflow.net/users/9896 | Undecidable theories easier than $Q$ | When I was looking around trying to find some inspiration to answer your question, I found the following result of Feferman from 1957:
>
> For any set $X$ of natural numbers there is a theory $T(X)$ such that:
>
>
> * The set $X$ and the set of Gödel numbers of consequences of $T(X)$ have the same degree of unsol... | 18 | https://mathoverflow.net/users/5442 | 69555 | 42,676 |
https://mathoverflow.net/questions/69554 | 1 | Suppose that $f : Y \to X$ is a birational morphism, and $E$ is some effective exceptional divisor on $Y$. The negativity lemma says that $E \cdot C \leq 0$ for any curve $C$ contracted by $f$. I'm wondering about a sort of converse: suppose $C$ is a curve such that $E \cdot C < 0$. Must $C$ be contracted by $f$? Would... | https://mathoverflow.net/users/12992 | Curves negative for an exceptional divisor | No. $E\cdot C<0$ implies that $C\subseteq E$, but it may not be contracted.
In fact, if $E$ is not contracted to a point, then many curves in $E$ will be like that.
Assume that $f$ is the blow up of a smooth subvariety of a smooth variety $\Sigma\subset X$ and $E=f^{-1}(\Sigma)$. Then $f|\_E:E\to\Sigma$ is a $\mathbb... | 8 | https://mathoverflow.net/users/10076 | 69557 | 42,678 |
https://mathoverflow.net/questions/69542 | 8 | My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill). I'm interested in constructing a uniformly convex Banach space with conditional structure without using interpolation. The constructions of Ferenczi and Maurey-Rosenthal both use interpolation.
Using existing metho... | https://mathoverflow.net/users/15388 | Uniformly Convex spaces | Kevin, there are non reflexive spaces with non trivial type--even of type 2. James constructed the first one; his argument is very complicated. Later Pisier-Xu did it much more simply using interpolation between $\ell\_1$ and $\ell\_\infty$, but using the universal non weakly compact operator instead of the formal iden... | 12 | https://mathoverflow.net/users/2554 | 69559 | 42,679 |
https://mathoverflow.net/questions/69560 | 4 | Let $X$ be a finite CW-complex and $A$ an abelian group, and consider the space $Maps(X,K(A,n))$ of continuous maps from $X$ to $K(A,n)$ endowed with the compact-open topology, so that it represents the functor $Y\mapsto C(X\times Y,K(A,n))$. Let $Maps\_0(X,K(A,n))$ be the path-connected component of $Maps(X,K(A,n))$ c... | https://mathoverflow.net/users/8320 | Fundamental groups of spaces of maps to Eilenberg-MacLane spaces | In general, the homotopy groups (based at the trivial map) of a **based** mapping space $Map(X,Y)$ are $\pi\_nMap(X,Y)=[\Sigma^n,Y]$, $n\geq 0$, where the brackets denote sets of homotopy classes of maps. If $Y=K(A,n)$ then you get $\pi\_1Map(X,K(A,n))=[\Sigma X,K(A,n)]=H^n(\Sigma X,A)\cong H^{n-1}(X,A)$.
| 10 | https://mathoverflow.net/users/12166 | 69561 | 42,680 |
https://mathoverflow.net/questions/69563 | 8 | Assume that $\beta:\tilde{X}\to X$ is the blow-up of a nonsinular $\Bbbk$-variety $X$ along a sheaf of ideals $\mathcal{I}$. Let $Y:=Z(\mathcal{I})$. Given nonsingular, closed subvarieties $Z\_1,\ldots,Z\_r\subseteq X$ such that $\bigcap\_i Z\_i \subseteq Y$, is it true that $\bigcap\_i \tilde{Z}\_i=\emptyset$, where $... | https://mathoverflow.net/users/9947 | Blow-up removes intersections? | As Sasha and Ramsey point out, this isn't true in the generality requested. However, the following is true, see Hartshorne, Chapter II, Exercise 7.12.
**Statement:** Suppose that $X$ is a Noetherian scheme and let $Y, Z$ be closed subschemes, neither one containing the other. Let $\widetilde{X}$ be the blowing up of ... | 7 | https://mathoverflow.net/users/3521 | 69570 | 42,685 |
https://mathoverflow.net/questions/69274 | 2 | In R.P. Stanley's book, Enumerative Combinatorics, Vol.2, paragraph 5.6, there is an intuitive proof of the BEST therem, which states that the number of eulerian tours in a balanced digraph $D$ with vertices in $V$ is given by
$\epsilon(D) = t(D) \prod\_{v' \in V} (\mathrm{out}\_{v'}(D)-1)!$
Here $\mathrm{out}\_u(D... | https://mathoverflow.net/users/3441 | BEST theorem for Eulerian paths with open ends | The question seems to be sinking into the depths of Lethe; here is a styrofoam noodle for it.
Every Eulerian path on G can be completed to an Eulerian tour on G' which is G augmented with the edge (v,u). This correspondence is easily seen to be 1-1, so the number of desired paths on G is the formula you mention above... | 1 | https://mathoverflow.net/users/3402 | 69574 | 42,687 |
https://mathoverflow.net/questions/69582 | 9 | Let $C,C',D,D'$ be chain complexes of $R$-modules (let's say with upper indexing, so perhaps I should call them cochain complexes, though they're not duals of anything). Let $f\in Hom^\ast(C,C')$ and $g\in Hom^\*(D,D')$. Then the standard convention is that $$(f\otimes g)(x\otimes y)=(-1)^{|g||x|}f(x)\otimes g(y),$$ wh... | https://mathoverflow.net/users/6646 | Signs and functoriality of tensor products | There are two options. If you just want an ordinary category of cochain complexes, then you have to take the morphisms to be cochain maps of degree zero. In that context we have $(-1)^{|k||f|}=1$ so there is no problem.
Alternatively, you can have an enriched category of cochain complexes. In more detail, for any sy... | 14 | https://mathoverflow.net/users/10366 | 69584 | 42,693 |
https://mathoverflow.net/questions/69595 | 2 | How transversally elliptic pseudo-differential operator naturally induces a K-homology class in KK(A, C), where the algebra A is the crossed product algebra C(M) ⋊ G, where M is compact manifold and G is compact Lie group. Do you have any reference paper about this work? Thanks.
And what if M is no longer compact?
| https://mathoverflow.net/users/15970 | transversally elliptic operator, fundamental class, K-homology | Since $G$ is compact, averaging over $G$ you may assume that your operator is $G$-invariant. If you assume that $G$ acts freely on your manifold, then $C\_0(M)\rtimes G$ is Morita equivalent to $C\_0(M/G)$, and what you want boils down to the standard fact that an elliptic pseudo-differential operator defines a $K$-hom... | 4 | https://mathoverflow.net/users/14497 | 69596 | 42,698 |
https://mathoverflow.net/questions/69581 | 5 | Hi all, I'm trying to formulate an infinite linear program to prove optimality (via duality) for the Continuous Transportation Problem, e.g. the Kantorovich-Wasserstein distance. This is the continuous version of the discrete transformation of the Transportation Problem commonly solved by the transportation simplex alg... | https://mathoverflow.net/users/16200 | Continuous Transportation Problem | If I understand your confusion correctly, you are trying to relate $f\_1$ to $p\_i$ and $f\_2$ to $p\_2$. Those are not related. The measure $\rho$ corresponds to $\{f\_{ij}\}\_{ij}$, the measure $\mu\_1$ to $\{p\_i\}\_i$ and $\mu\_2$ to $\{q\_j\}\_j$. The functions $f\_1$ and $f\_2$ are used as test functions to say t... | 2 | https://mathoverflow.net/users/11716 | 69601 | 42,701 |
https://mathoverflow.net/questions/69573 | 1 | **Preparation.**
Let $R$ be a noetherian ring (perhaps we have to assume $2 \in R^\*$), $1 \leq d \leq n$ and consider the Plücker embedding $\omega : G \hookrightarrow P$, where $G = \text{Grass}\_d(R^n)$ and $P = \mathbb{P}(R^{\binom{n}{d}}) = \text{Proj}(R[\{x\_{i\_1,...,i\_d}\}])$. On $X$-valued points points, th... | https://mathoverflow.net/users/2841 | Calculation of twisted presentation of universal locally free sheaf on Grassmannian under Plücker embedding | To get a simpler answer it is better to fix a free $R$-module $V$ of rank $n$ so that $G = Gr(d,V)$ and $P = P(\Lambda^dV)$. First we have a tautological epimorphism $V\otimes O\_G \to F$ on $G$ which by adjunction gives an epimorphism $V\otimes O\_P \to i\_\*F$. Denote the kernel by $K$. Twisting the short exact seque... | 0 | https://mathoverflow.net/users/4428 | 69603 | 42,702 |
https://mathoverflow.net/questions/60633 | 3 | In a paper of A. Weinstein on the geometry of Poisson manifolds, he relates the formal linearization around a zero, p, of the Poisson bivector to extensions of the Lie algebra induced by the bivector on the tangent space over p.
I wanted to know if this is part of a big picture, possibly relating deformations of Lie... | https://mathoverflow.net/users/13981 | Is there any relation between deformation and extension of Lie algebras? | In fact the picture is extremely simple and works indeed for any type of algebra as follows :
Let $\mu$ be a Lie algebra on a vector space $V$, $c$ a two-cocycle $c\in CE^2(V;M)$ (Chevalley Eilenberg cohomology) where $M$ is a module over the Lie algebra. The extension of $\mu$ by $c$ is nothing else than a deformati... | 5 | https://mathoverflow.net/users/13742 | 69612 | 42,709 |
https://mathoverflow.net/questions/69611 | 5 | Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)?
The reason I am asking is because in quantum mechanics, the wavefunctions of spin 1/2 particles are invariant under 720° / 4$\pi$ rotations, but not under rotations of 3... | https://mathoverflow.net/users/16246 | Space with 720° / not 2$\pi$ rotational symmetry? | This is quite classical.
The point is that the group $\operatorname{SO}(3)$ is *not* symply connected, in fact
$$\pi\_1(\operatorname{SO}(3)) \cong \mathbb{Z}/2 \mathbb{Z}.$$
Its universal cover is the group
$$\operatorname{Spin}(3) \cong \operatorname{SU(2)}.$$
Hence we have an exact sequence
$$1 \to \mathbb{Z}/2 ... | 14 | https://mathoverflow.net/users/7460 | 69613 | 42,710 |
https://mathoverflow.net/questions/69620 | 1 | In "On definable susbsets of p-adic fields", Macintyre 1976, he states the following fact that, let $x\in Q\_p^\*$ such that $v(x) = 0$ (where $v$ is the p-adic valuation) and $k\in N$ then there exists $n\in Z$ such that $x/n\in (Q\_p)^k$ (ie the $k$-th powers).
This property is evident if $k$ is prime to $p$ by Hen... | https://mathoverflow.net/users/15587 | k-th powers in the field of p-adics | The basic idea is that it is possible to extract $k^\mathrm{th}$-roots of elements of $\mathbb{Z}\_p$ that are close enough to 1. This is because you can compute explicitely the radius of convergence of the series $(1+X)^{1/k}$ (because you know the $p$-adic valuation of the factorials, hence of the binomial coefficien... | 3 | https://mathoverflow.net/users/4069 | 69626 | 42,713 |
https://mathoverflow.net/questions/69624 | 3 | Hi, I am interested in the following question:
Fix $n$.
Let $M\_n$ be matrix algebra over the field of complex numbers with normalized trace $tr\_n$. Let $M\_n^{\omega}$ be an ultrapover of $M\_n$, namely we consider the algebra of all bounded (in norm) sequences in $M\_n$,say $l\_{\infty}(M\_n)$, and take a quotient... | https://mathoverflow.net/users/16259 | Ultraproduct of n-dimensional Banach spaces and algebras | If the space $F$ is $k$-dimensional with basis $b\_1,\ldots b\_k$ then the ultraproduct with respect to the ultrafilter $U$ will also be $k$-dimensional. To see this let $x\_i\in \ell\_\infty(F\_i)$ and let $q\_{i,j}$ be scalars such that $\sum\_{j=1}^kq\_{i,j}b\_j =x\_i $. By the compactness of the ball in finite dime... | 4 | https://mathoverflow.net/users/13878 | 69629 | 42,716 |
https://mathoverflow.net/questions/69569 | 37 | I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for centuries.
The present mathematical tecnology allows one, according to the context, to formalize this notion in several w... | https://mathoverflow.net/users/4721 | Various flavours of infinitesimals | I don't know of the Connes calculus, but the others (including nonstandard analysis à la Robinson) have been brought under a common framework using models of synthetic differential geometry. However: it is important to point out that the infinitesimals used in algebraic geometry (for jet bundles, etc.) are *nilpotent* ... | 12 | https://mathoverflow.net/users/2926 | 69630 | 42,717 |
https://mathoverflow.net/questions/69637 | 8 | Let $A$, $B$ and $C$ be discrete countable groups. Let $\alpha$ be an action of $A$ on $B$ and let $\beta$ be an action of $B$ on $C$.
>
> **Question** Does there always exist a group $G$ which has $A$, $B$ and $C$ as subgroups and such that the group generated by $A$ and $B$ is $A\ltimes B$ and the group generated... | https://mathoverflow.net/users/2631 | "double" semidirect product | The answer to the specific question is "yes". Let $U$ be the semidirect product of $C$ and $B$. Then as $G$, you take the HNN extension of $U$ with the free letter $t$ and associated subgroups equal to $B$, with the automorphism provided by the shift of generators. The result then follows from the usual properties of H... | 12 | https://mathoverflow.net/users/nan | 69644 | 42,724 |
https://mathoverflow.net/questions/69614 | 4 | Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.
Is there any possibility that $J\subset I$ or even a subgroup?
| https://mathoverflow.net/users/9842 | cuspidal types and Iwahori subgroup for $SL(2,F)$ | Here is a novel, explicit answer to your question :
Benedict Gross and Mark Reeder have recently discovered a family of supercuspidal representations, called "simple supercuspidal representations", of simply connected, split, almost simple, reductive groups. These representations are examples of what you are looking... | 5 | https://mathoverflow.net/users/8891 | 69645 | 42,725 |
https://mathoverflow.net/questions/69615 | 26 | It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set,
a set of representatives for the equivalence classes of the relation on the reals "the same modulo a rational number". Is it known whether... | https://mathoverflow.net/users/7743 | Axiom of choice: ultrafilter vs. Vitali set | Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set.
More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E\... | 16 | https://mathoverflow.net/users/6085 | 69651 | 42,729 |
https://mathoverflow.net/questions/69631 | 32 | A *smooth* curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. These are in fact all (see Hartshorne IV.3.9).
For this reason, in any characteristic we do not use lines to get embed... | https://mathoverflow.net/users/1887 | Which 'well-known' algebraic geometric results do not hold in characteristic 2? | I am not going to add any new examples but suggest a systematic way of looking at examples. If one looks at special phenomena in characteristic $2$ one can classify them as follows (though this division is far from clear cut):
1. They are really special to positive characteristic and not only characteristic $2$.
2. T... | 25 | https://mathoverflow.net/users/4008 | 69665 | 42,734 |
https://mathoverflow.net/questions/69654 | 3 | Background: I've been studying some flows on vector spaces associated with linear actions of Lie groups, with the hope of using these flows to prove things about quotient spaces. The most successful tool so far has been a certain kind of inequality for functions associated with the action. I'm trying to generalize this... | https://mathoverflow.net/users/14628 | Subspaces of commuting matrices | There is no hope of a general result. Look at the vector space
$$ V = \left \lbrace t \begin{pmatrix} i & 0 \newline 0 & 0\end{pmatrix} + s\begin{pmatrix} 0 & 1 \newline -1 & 0 \end{pmatrix} \right \rbrace . $$
So $V$ is two dimensional and not closed under the commutator. Any one-dimensional subspace is maximal co... | 7 | https://mathoverflow.net/users/6781 | 69666 | 42,735 |
https://mathoverflow.net/questions/69671 | 15 | It is well known that, for square matrix $x$ and $y$, we have $\operatorname{tr}(xy)=\operatorname{tr}(yx)$. Here of course the trace of a matrix is just the sum of the elements of the diagonal.
The notion of trace has a lot of generalization. As I know, the most general definition is the following: let $(\mathcal C,... | https://mathoverflow.net/users/7845 | trace(xy)=trace(yx) in full generality | Yes. The string diagram chase can be found on page 8 of [Ponto and Shulman - Traces in symmetric monoidal categories](https://arxiv.org/abs/1107.6032).
| 17 | https://mathoverflow.net/users/2926 | 69674 | 42,740 |
https://mathoverflow.net/questions/69680 | 3 | Hello!
I am interested in the asymptotic behavior of the function $p\_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - <http://oeis.org/A099773> .
I couldn't find any paper or book studying the mentioned quantity but the amount of literature available to me is quite limited and I am w... | https://mathoverflow.net/users/1737 | Asymptotics for the number of partitions of $n$ into odd prime parts | Flajolet and Sedgewick, [Analytic Combinatorics](http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf) (link goes to free, legal downloadable PDF of book), section VIII.6 treats the asymptotics of various types of partitions. They get that the number of partitions of $n$ into prime parts, which they denote $P\_... | 7 | https://mathoverflow.net/users/143 | 69682 | 42,745 |
https://mathoverflow.net/questions/69683 | 5 | I heard that there is a theorem due to Rosenlicht which says the following:
>
> **Theorem.** Let $X$ be a complex projective manifold and $V$ a non-trivial holomorphic vector field on $X$. Then $X$ is uniruled,ie,can be covered by rational curves,if $V$ has a zero.
>
>
>
I have thought for a few days and faile... | https://mathoverflow.net/users/15882 | Rosenlicht theorem about uniruledeness and zeroes of holomorphic vector field on complex projective manifold | You can look at Lieberman's paper [Holomorphic Vector Fields on Projective Manifolds](http://books.google.com.br/books?id=pa1I1gG3eyUC&pg=PA273&lpg=PA273&dq=Holomorphic+vector+fields+on+projective+varieties.&source=bl&ots=QxbsauUatN&sig=8gFvKiAuur3UwDrP0eGPu94DLH4&hl=pt-BR&ei=KhIVTrmQKOPx0gGd-pRT&sa=X&oi=book_result&ct... | 3 | https://mathoverflow.net/users/605 | 69688 | 42,748 |
https://mathoverflow.net/questions/69643 | 2 | Let $D$ be a reduced projective scheme over $\mathbb{C}$ such that $H^1(D,\mathcal{O}\_D) = 0$ and $D$ is Gorenstein.
There is a map
\begin{equation}
r:= \frac{d \log}{2 \pi i }: H^1(D, \mathcal{O}\_D^{\ast}) \otimes \mathbb{C} \rightarrow H^1(D,\Omega\_D^1)
\end{equation}
locally defined by $f \mapsto \frac{df}{f}... | https://mathoverflow.net/users/12390 | Question about arguments using Du Bois complex | Let me convert my obscure comment into an (obscure?) answer, with some corrections.
The problem with your argument as it stands is that a map is not well defined:
$H^1(D,\underline{\Omega}\_D^1)$ is only a subquotient of $H^2(D,\mathbb{C})$.
But the problem is minor. To fix things observe that $H^2(D)$ carries a mixe... | 3 | https://mathoverflow.net/users/4144 | 69711 | 42,762 |
https://mathoverflow.net/questions/69715 | 8 | I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a short elegant method by Zagier. Where?
In short, I know how to deduce the Selberg trace formula from Arthur's trace fo... | https://mathoverflow.net/users/10400 | What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula? | Sorry to give a reference to my own paper, but perhaps what you are looking for is contained in section 2 of [this paper](http://arxiv.org/abs/1011.5486); see also Theorem 1.3. The basic idea is that the Selberg and Kuznetsov trace formulae both involve spectral sums but with different weights. To get Selberg weights f... | 9 | https://mathoverflow.net/users/2627 | 69723 | 42,768 |
https://mathoverflow.net/questions/69733 | 1 | The situation is similar to [this question](https://mathoverflow.net/questions/69643/question-about-arguments-using-du-bois-complex)
Let $D$ be a reduced projective scheme over $\mathbb{C}$ whose associated analytic space $D\_{an}$ is simply connected as a topological space i.e. $\pi\_1 (D\_{an}) = 1$.
Let $\Omega^1... | https://mathoverflow.net/users/12390 | Vanishing of $H^0( D, \hat{\Omega}^1_D)$ on simply connected surface $D$? | Sorry, but it's not true. Let $D$ be a cone over an elliptic curve with vertex $p$.
It is simply connected, but $H^0(D,(\Omega^1\_D)^{\*\*})= H^0(D-p, \Omega\_{D-p}^1)\not=0$.
**Added** to address your question below: $H^0(D,\Omega\_D^1)=0$
for any degree $d$ surface in $P=\mathbb{P}^3$, and so in particular for the ... | 4 | https://mathoverflow.net/users/4144 | 69736 | 42,774 |
https://mathoverflow.net/questions/69741 | 4 | Today in a talk, it has been mentioned that there exists algebraic groups over the local field $\mathbb{R}$ such that the finite central extension can not be defined algbraically over $\mathbb{R}$ or its algebraic closure $\mathbb{C}$. I guess already covers of $SL(2)$, which is even defined over $\mathbb{Z}$, and the ... | https://mathoverflow.net/users/10400 | Why is the double cover of $Sl(2,\mathbb{R})$ not algebraic? | The double cover of $SL(2,\mathbb R)$ is not algebraic.
>
> This can be blamed on the fact that the map
> $$\pi\_1\big(SL(2,\mathbb R)\big)\cong \mathbb Z\quad\longrightarrow\quad \pi\_1\big(SL(2,\mathbb C)\big)=0$$
> is not injective.
>
>
>
If the double cover of $\pi\_1(SL(2,\mathbb R))$ were algebraic, it... | 28 | https://mathoverflow.net/users/5690 | 69748 | 42,781 |
https://mathoverflow.net/questions/69746 | 14 | A goal which I have been pursuing is to understand how number fields are distributed with respects to their invariants. To be more precise I was captivated by the following question:
Let $N(X,n,G)$ be the number of number fields of dimension $n$ where $G$ is the Galois group of its Galois closure, and their discriminan... | https://mathoverflow.net/users/8419 | Statistics of Number fields | There are general conjectures and heuristics; see for example a [recent survey](http://math.stanford.edu/~akshay/research/evicm.pdf) by Ellenberg and Venkatesh on this and the function field analog.
Roughly, there is a (modified) conjecture of Malle asserting that given $G$ the number of respective extensions of dis... | 11 | https://mathoverflow.net/users/nan | 69749 | 42,782 |
https://mathoverflow.net/questions/69729 | 3 | Let $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X = \operatorname{Spec}(A)$. Let $B$ denote a free $A$-algebra of rank $e^2$; actually, we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv = \xi\_e vu$, where $\xi\_e$ is an $e$-th root of unity. Now we have $M\_n(B)$ and the subalgebra $D$, where entries under the diagonal ... | https://mathoverflow.net/users/3233 | Sequences of groups, exact not just in étale but also in the Zariski topology | Let me elaborate on Torstens comment.
The map $GL\_n(B) \to i\_\*F$ in your example is an fppf $D^\times$-torsor over $i\_\*F$,
a fact which can be checked easily directly from the definition of torsor which you can find in
any of the referenced texts. A priori it is a torsor for the fppf topology, but it can be
show... | 3 | https://mathoverflow.net/users/1084 | 69757 | 42,788 |
https://mathoverflow.net/questions/69578 | 19 | **First question**: For a semisimple invertible $n \times n$ matrix with entries over a field *K*, its characteristic polynomial completely describes the similarity class of the matrix. For non-semisimple elements, the characteristic polynomial is no longer a complete description of the similarity class, but there exis... | https://mathoverflow.net/users/3040 | What invariants of a matrix or representation can be used to find its GL(n,Z)-conjugacy class? | Here is the requested example of two representations of the Klein 4 group over $\mathbb{Z}$, locally conjugate but not conjugate.
Let $K:= \mathbb{Z}/2 \times \mathbb{Z}/2$ act on $\mathbb{Z}^4$ by permuting the coordinates. Inside $\mathbb{Z}^4$, consider the following two lattices:
$$L\_1 := \{ (a,b,c,d) \in \ma... | 11 | https://mathoverflow.net/users/297 | 69761 | 42,790 |
https://mathoverflow.net/questions/69760 | 7 | The classical Schauder estimates (see the link)
<http://en.wikipedia.org/wiki/Schauder_estimates>
Requires $f\in C^\alpha$ in order to get a solution $u\in C^{2+\alpha}$ of the equation
$$\Delta u=f$$
In fact, we can construct a continuous function f, which is not Hölder of any order on a positive meausre set.
... | https://mathoverflow.net/users/15214 | The Hölder continuity condition of the Schauder estimates | Merely $f\in C$ cannot guarantee $u\in C^2$, as there are examples of $u\not\in C^2$ with $\Delta u\in C$, cf. an exercise in Gilbarg-Trudinger. This failure can be formulated in terms of non-closedness of the range of the Laplacian, or unboundedness of the inverse between certain spaces. The spaces $C^k$ (as well as $... | 11 | https://mathoverflow.net/users/824 | 69762 | 42,791 |
https://mathoverflow.net/questions/69758 | 2 | One can define products on the K-theory of graded C\*-algebras as in
[http://web.me.com/ndh2/math/Papers\_files/Higson,%20Guentner%20-%202004%20-%20Group%20C\*-algebras%20and%20K-theory.pdf](http://web.me.com/ndh2/math/Papers_files/Higson,%2520Guentner%2520-%25202004%2520-%2520Group%2520C%2a-algebras%2520and%2520K-theo... | https://mathoverflow.net/users/9401 | Products on the K-theory of graded C*-algebras | The cup-cap product of Kasparov, in bivariant $KK$-theory, is defined for graded $C^\*$-algebras; for ungraded algebras, and when the first argument is $\mathbb{C}$, it restricts to the product you describe. See: Kasparov, G. G. Equivariant $KK$-theory and the Novikov conjecture. Invent. Math. 91 (1988), no. 1, 147–201... | 2 | https://mathoverflow.net/users/14497 | 69768 | 42,796 |
https://mathoverflow.net/questions/69730 | 4 | Let $X$ be a smooth Calabi-Yau 3-fold with Kahler form $w$,
It is true that $\int c\_2(TX) \wedge w \geq 0$ (for any Kahler form $w$ on $X$).
Proof via algebraic geometry is rather difficult. Some where It was saying that for such $X$
$$\int c\_2(TX) \wedge w = \int \left\| R \right\|^2 dvol $$
where $R$ is th... | https://mathoverflow.net/users/5259 | wedge product of second chern class and kahler form on Calabi-Yau 3-folds. | The calculation in local coordinates is not too hard and it works in any dimension $n$, namely if $c\_1(M)=0$ then $\int\_M c\_2(M)\wedge\omega^{n-2}\geq 0$ for any Kahler metric $\omega$. Of course you are free to assume that the Kahler metric $\omega$ is Ricci-flat (by Yau's theorem), and in this case the integral is... | 5 | https://mathoverflow.net/users/13168 | 69771 | 42,798 |
https://mathoverflow.net/questions/69754 | 3 | I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta characteristic) $\kappa$ on $C$ via the Abel-Jacobi map $\alpha\_c$ based at $c \in C$.
I know (from looking at Birkenhake and Lan... | https://mathoverflow.net/users/16298 | Pulling back a line bundle on the Jacobian to a spin bundle on the curve | For a general curve $C$ of genus $g$, it is a fact that the Neron-Severi group of the Jacobian $J$ of $C$ is generated by the class $\theta$ corresponding to the divisor $\Theta$. (I am not very strong in algebraic geometry, so I guess that I would rather prefer to work with the probably equivalent statement: The subgr... | 1 | https://mathoverflow.net/users/83 | 69776 | 42,801 |
https://mathoverflow.net/questions/69765 | 2 | What is the holonomy group of the 1-dimensional octonionic projective space ?
| https://mathoverflow.net/users/14576 | Holonomy group of $\mathbb{O}P^1$ | Following David Roberts' comment, and using the fact that the holonomy of the round sphere $S^n$ is $SO(n)$, you get $SO(8)$ as the answer to your question.
| 4 | https://mathoverflow.net/users/5690 | 69781 | 42,805 |
https://mathoverflow.net/questions/69700 | 4 | Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$ for $n$, $k$, $N$ $\in\mathbb{N}$, greater than 2.
This is related to a previous answered question: [Are there any solutions to $2^n-3^m=1$](https://mathoverflow.net/questions/69253/are-there-any-solutions-to-2n-3m1)
Things I already know:
There are no a... | https://mathoverflow.net/users/16114 | Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$ | It is not too hard to show that there are no solutions. The assumption that
$2^k-3^n$ divides $3^n-2^n$ implies that the former quantity divides $2^{k-n}-1$
and, further, that $2^k < 3^{n+1}$, so that $k < (n+1) \log (3)/\log(2)$. We thus have
$$
\left| 2^k - 3^n \right| < 2^{(n+1) \log (3)/\log(2)-n}.
$$
On the other ... | 9 | https://mathoverflow.net/users/7302 | 69794 | 42,811 |
https://mathoverflow.net/questions/69792 | 11 | Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on $X$ by $\tilde \omega = f \omega$. If $f$ is non-constant, then can this new metric ever be Kahler?
If $\dim\_{\mathbb C... | https://mathoverflow.net/users/4054 | Can a metric conformal to a Kahler metric be Kahler? | You don't have to use hard Lefschetz to conclude $df=0$ from $\omega\wedge df=0$.
This is a linear algebra fact, valid pointwise : if $\alpha \in T\_x^\*X$ satisfies $\omega\_x \wedge \alpha=0$, then $\alpha=0$ (of course, assuming $\dim\_R X \geq 4$.
The short argument is that, $\omega\_x^{n-1}\wedge : T^\*\_x X... | 10 | https://mathoverflow.net/users/6451 | 69798 | 42,813 |
https://mathoverflow.net/questions/69805 | 5 | I've been working with a collaborator (Arek Goetz) on a dynamics problem involving
piecewise isometries (a map $T$ on a domain $X$ (say a subset of the plane)
such that $X$ is divided into a finite number of polygonal regions and a
separate isometry is applied to each).
In the case where the defining isometries are a... | https://mathoverflow.net/users/11054 | Algorithm to determine sign of a polynomial | See
<http://cgi.di.uoa.gr/~et/papers/et-computations-alg-numbers2.pdf>
particularly section 3.3
| 1 | https://mathoverflow.net/users/11142 | 69812 | 42,820 |
https://mathoverflow.net/questions/69786 | 14 | I got my copy of Computational Aspects of Modular Forms and Galois Representations in the mail yesterday. The goal of the book is "How one can compute in polynomial time the value of Ramanujan's tau at a prime", well, or any other modular form of level 1. It's all very thrilling!
The following fact is essential: for ... | https://mathoverflow.net/users/2024 | Why is there a weight 2 modular form congruent to any modular form | By "level $\ell$" I assume you mean "level $\Gamma\_1(\ell)$".
Here's a proof. By the Eichler-Shimura theorem, the system of eigenvalues associated to the modular form shows up in $H^1(SL(2,\mathbf{Z}),Symm^{k-2}(\mathbf{C}))$. Hence (by some easy commutative algebra) the mod $\ell$ reduction of the system of eigenva... | 17 | https://mathoverflow.net/users/1384 | 69816 | 42,821 |
https://mathoverflow.net/questions/69814 | 4 | Let $f:X \to Y$ be a proper morphism between smooth algebraic varieties (say over $\mathbb{C}$), let me write $A\_X$ for the constant sheaf on $X$ with coefficients of the appropriate type. Then the decomposition theorem of [Beilinson-Bernstein-Deligne] tells me that $f\_\* A\_X$ splits as a direct sum of shifted semis... | https://mathoverflow.net/users/4707 | Decomposition theorem and virtual Poincare polynomial | (I'll just note beforehand that is this is a "I don't know the answer to your actual question, perhaps you'd rather know the answer to a slightly different one" answer). The answer is yes if you replace virtual Betti number with Frobenius trace; these are also motivic, if admittedly slightly harder to calculate, as the... | 3 | https://mathoverflow.net/users/66 | 69823 | 42,824 |
https://mathoverflow.net/questions/69824 | 11 | Dear MO,
Let $p$ be a prime and let $E/\mathbb{Q}\_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}\_p$ has good ordinary reduction at $p$. In his 1972 paper ``*Propriétés galoisiennes des points d'ordre fini des courbes elliptiques*'' (more specifically, see *Corollaire*, in p. 274), Serre shows along the way tha... | https://mathoverflow.net/users/4180 | Ramification in p-division fields associated to elliptic curves with good ordinary reduction | Assume $p \ne 2$. The condition for the representation to be tamely ramified (i.e $\* = 0$ in the upper right entry of the matrix) is that $j(E) \equiv j\_0 \mod p^2$ where $j(E)$ is the $j$-invariant of $E$ and $j\_0$ is the $j$-invariant of the canonical lift of the reduction of $E$. This is proved in Gross "A tamene... | 12 | https://mathoverflow.net/users/2290 | 69825 | 42,825 |
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