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https://mathoverflow.net/questions/108169 | 31 | Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves') says that the image of the representation of $Gal(\bar{K} / K)$ on the $l$-adic Tate module $T\_l(E)$ is open in $GL\... | https://mathoverflow.net/users/16858 | Modern proof of Serre's open image theorem? | Here is Ribet's proof (expanding on Ulrich's comment):
Let $G\_K:=Gal(\bar{K} / K)$ and $V\_l:=T\_l(E)\otimes \mathbb{Q}\_l$.
The image $\rho\_{l,E}(G)$ is a closed subgroup of the $l$-adic Lie group $\text{Aut}(V\_l(E)) \cong \text{GL}\_{2}(\mathbb{Q}\_l)$ and is therefore a Lie subgroup of $\text{Aut}(V\_l(E))$. ... | 23 | https://mathoverflow.net/users/16858 | 108237 | 62,372 |
https://mathoverflow.net/questions/108029 | 4 | Hi !
I'm considering in a general topos $T$ the object $R$ of lower semi-continuous real (one sided lower non-empty Dedekind cuts, as for exemple in <http://ncatlab.org/nlab/show/one-sided+real+number> ).
I want to know if, even if substraction is not possible, there is (internally) some sort of simplification rule... | https://mathoverflow.net/users/22131 | Simplification in Semi-continuous real ? | Ok I think I finally found an internaly valid proof by my self, so I explain it briefly here in case someone is interested some day :
If $U \in \Omega$ is a subterminal object, you can define the element $1\_U \in R$ as :
$q \in 1\_U$ when $q < 0 \cup (U \cap q<1)$.
(this correspond to the indicator function of a... | 3 | https://mathoverflow.net/users/22131 | 108239 | 62,374 |
https://mathoverflow.net/questions/108192 | 16 | I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right?
My question is the following: What is the current state of the art? What is known about the classification of certain families of Hopf algebras? (To be more precise, say for example that I am interested in fi... | https://mathoverflow.net/users/26812 | Classification of Hopf algebras (state of the art) | The general problem about the classification of finite-dimensional Hopf
algebras (over $\mathbb{C}$) is widely open. I mention some general results.
Here
Zhu, Yongchang. Hopf algebras of prime dimension. Internat. Math. Res. Notices 1994, no. 1, 53--59. MR1255253 (94j:16072), [link](https://doi.org/10.1155/S1073792... | 11 | https://mathoverflow.net/users/17845 | 108246 | 62,377 |
https://mathoverflow.net/questions/108145 | 3 | I'm wondering about the following 2 generalizations of Cayley's Theorem (every group embeds in a symmetric group). If these are known to be true/false, references would be appreciated.
1) (Weak Version) Given any finite collection of (not necessarily distinct) finite groups, can we embed them simultaneously in a (fin... | https://mathoverflow.net/users/21857 | 2 Possible Generalizations of Cayley's Theorem? | So the question now is, given two nontrivial finite groups $G,H$, can we embed them both in a finite group $X$ such that the normalizers of $G$ and $H$ in $X$ intersect trivially?
I think we can do that as follows. Suppose that we can find a module $V$ for $G \times H$ over a finite field, such that neither $G$ nor $... | 3 | https://mathoverflow.net/users/35840 | 108250 | 62,379 |
https://mathoverflow.net/questions/108170 | 9 | Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that
1. assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and
2. assigns an infinitesimal--ideally, the same one--to each singleton?
It's (1) that's a problem. The Bernstein-Wattenberg constructio... | https://mathoverflow.net/users/26809 | Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons | Yes, by compactness.
Let $R$ denote your favorite hyperreal ordered field
and let $\delta\in R$ be a positive infinitesimal.
Let $\mathcal{E}$ denote the set of all
(standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$.
For every $A\in\mathcal{E}$,
let $\lambda\_A(I)$ be the (exact) length of $I$
for all... | 7 | https://mathoverflow.net/users/12106 | 108252 | 62,380 |
https://mathoverflow.net/questions/108255 | 6 | Følner's characterization of Amenability says that a group $G$ is amenable if there exists a directed set $(I,\leq)$ and a net {$F\_i:i\in I$} of finite subsets of $G$ such that for every $γ ∈ G$,
$$\lim\_{i\in I}\frac{|γF\_i\Delta F\_i|}{|F\_i|}\ \ \ \rightarrow 0\ ,$$
where $\Delta$ is the symmetric difference of two... | https://mathoverflow.net/users/24891 | Another question about amenability and Følner sequences | Yes - see Lemma 5.1 of Gabor Pete's book on probability and groups: <http://www.math.bme.hu/~gabor/PGG.html> (by the way, these are excellent lecture notes)
| 5 | https://mathoverflow.net/users/1121 | 108258 | 62,382 |
https://mathoverflow.net/questions/108224 | 4 | I have read that if a variety $X$ has KLT singularities, then it has quotient singularities in codimension 2.
>
> Do you know a proof (or where can I find a proof) of this?
>
>
>
| https://mathoverflow.net/users/6430 | KLT singularities are quotient in codimension 2 | (For some reason first I thought that the question was only in dimension $2$...)
The essence of this statement (in dimension $2$) consists of two facts that are useful anyway:
1. The index $1$ cover of a klt
singularity is canonical (in any
dimension): it is relatively easy to
see that it is still klt by looking
at... | 9 | https://mathoverflow.net/users/10076 | 108263 | 62,385 |
https://mathoverflow.net/questions/108254 | 2 | Suppose $A\otimes B$ is the minimal tensor product of two unital $C^\*$ algebras $A$ and $B$.
We know that the set of states, $\{\phi\otimes\psi|\phi\in S(A),\psi\in S(B) \}$ on $A\otimes B$ separates the points of $A\otimes B$. Here $S(A)$ and $S(B)$ are the state spaces of $A$ and $B$ respectively.
My question is:
... | https://mathoverflow.net/users/7360 | Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras | Let $\pi\_A$ and $\pi\_B$ be faithful rep's of $A$ and $B$. Then $\pi=:\pi\_A\otimes\pi\_B$ is a faithful rep of $A\otimes\_{min} B$ (see Thm 4.9(iii) in Chapter 4 in M. Takesaki, Theory of Operator algebras I, Springer-Verlag, 1979). For $x$ a non-zero element in $A\otimes\_{min} B$, use then a vector state associated... | 4 | https://mathoverflow.net/users/14497 | 108264 | 62,386 |
https://mathoverflow.net/questions/108216 | 9 | From $0 = 0.5 - 0.5 = 0.5 - \sqrt{0.25}$, we can adjust the subtrahend slightly to obtain
$$0.5 - \sqrt{0.249} = 0.001\ 001\ 002\ 005\ 014\ 042\ldots$$
where the decimal representation contains the first few Catalan numbers: $1, 1, 2, 5, 14, 42
\ldots$
We can see even more Catalan numbers (albeit spaced apart wi... | https://mathoverflow.net/users/22971 | Intuition Behind a Decimal Representation with Catalan Numbers | To make it less surprising is to fade some of the magic! But, ok.
First let us say that any real sequence $a\_1,a\_2,\cdots$ has an ordinary generating function (ogf) $f(x)=\sum a\_ix^i$ which may be a formal series with radius of convergence $0$ (and still be useful) BUT if the $a\_i$ are positive integers and $f(x... | 11 | https://mathoverflow.net/users/8008 | 108268 | 62,387 |
https://mathoverflow.net/questions/108155 | 15 | It is an easy fact that for a matrix $A \in M\_n(\mathbb C)$, the matrix $A' = (|A(i,j)|)\_{i,j \leq n}$ has a larger operator norm than $A$. By operator norm I mean the norm as an operator on $\ell^2\_n$, or equivalently its largest singular value.
My question is:
>
> What happens when the operator norm is repl... | https://mathoverflow.net/users/10265 | Comparison of the norm of a matrix and its entry-wise absolute value. | Let me answer my own question by contructing, for $p>2$ not an even
integer (say $2k<p<2k+2$), a matrix $A$ such that $\|A\|\_p> \|A'\|\_p$. In fact I construct a family of matrices $A\_n \in M\_n(\mathbb C)$ such that $\|A\_n\|\_p > \|A\_n'\|\_p$ whenever $n-k$ is an even positive integer, and $\|A\_n\|\_p < \|A'\_n\|... | 8 | https://mathoverflow.net/users/10265 | 108275 | 62,390 |
https://mathoverflow.net/questions/108273 | 1 | Hi all
Here is something I have been stuck on for a week. In van der Geer's "The Cohomology of the Moduli Space of Abelian Varieties" he is looking at the subring of $CH\_\mathbb{Q}(\tilde{\mathcal{A}\_g})$ generated by the Chern classes $\lambda\_i$ of the Hodge bundle $\mathbb{E}$ extended to a smooth toroidal comp... | https://mathoverflow.net/users/18844 | van der Geer: Ample on an open dense set implies non-trivial self intersection globally??? | Being "ample on an open set" does not mean that the restriction is ample. It means that the local conditions of being ample (separating points and tangent vectors) hold on an open set, but still using global sections of the original bundle on the original scheme. In particular, $\mathscr O\_{\mathbb P^n}$ is **not** am... | 7 | https://mathoverflow.net/users/10076 | 108283 | 62,394 |
https://mathoverflow.net/questions/108270 | 10 |
> Is the Hasse principle a birational invariant?
>
It is probably a very trivial question, but I am a beginner in arithmetics.
| https://mathoverflow.net/users/4096 | Is the Hasse principle a birational invariant? | In this generality, the answer is no. The projective curve $X$ given by $2y^2z^2 = x^4 - 17z^4$ over the rationals satisfies the HP, since it has local points everywhere (the affine part $z \neq 0$ is given by $2y'^2=x'^4-17$, which is the famous Reichardt-Lind equation which is known to be everywhere locally, but not ... | 25 | https://mathoverflow.net/users/17907 | 108284 | 62,395 |
https://mathoverflow.net/questions/108260 | 4 | If a symplectic manifold $(M,\omega)$ is given,
>
> is there any efficient method to compute $[\omega]$, the cohomology class of the symplectic form?
>
>
>
I do know that efficient is not a good word to use, but for instance: if one knows for sure that the symplectic form is integral -- the class of $\omega$ ... | https://mathoverflow.net/users/16019 | Computing the cohomology class of a symplectic form | As a partial answer I can give you a reference how to describe the cohomology class of the symplectic form for symplectic toric manifolds, i.e. symplectic manifolds $(M^{2n}, \omega)$ with an effective Hamiltonian action of the torus $T^{n}$.
As you probably know, the image of the corresponding moment map is a (Delza... | 8 | https://mathoverflow.net/users/24221 | 108289 | 62,398 |
https://mathoverflow.net/questions/108257 | 8 | Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of *simple* abelian varieties over $k$ and $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-conjugacy classes of algebraic integers in $\mathbb{C}$ all of whose conjugate have absolute value ... | https://mathoverflow.net/users/18238 | Honda-Tate in families | I think your intuition that Weil numbers generalize to functions is wrong, or rather only partially correct. A Weil "number" is in fact a sheaf on $\operatorname{Spec}\mathbb F\_p$: The sheaf corresponding to the Galois representation that sends $Frob\_p$ to a matrix whose characteristic polynomial is the minimal polyn... | 6 | https://mathoverflow.net/users/18060 | 108296 | 62,401 |
https://mathoverflow.net/questions/108301 | 3 | I am trying to verify whether the category of bialgebras and then the category of weak bialgebras are cocomplete.
We know that algebraic categories are cocomplete (Thm. 4.5 of [this book](http://www.iti.cs.tu-bs.de/~adamek/algebraic.theories.pdf&usg=AFQjCNG8SZUmNDUyK5YC6fhL04HYp_9I0g)), so I have been trying to show ... | https://mathoverflow.net/users/59028 | Are the categories of bialgebras and weak biaglebras cocomplete/algebraic? | The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras. (See the edit below for a reference.) Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete... | 10 | https://mathoverflow.net/users/2926 | 108303 | 62,403 |
https://mathoverflow.net/questions/108298 | 3 | Given an explicit description (as an intersection) of an abelian surface $A$ is there an algorithm for computing the period lattice of the surface? For the specific examples that I am interested in, the ideal of $A$ has been obtained by Weil restriction from the affine model of an elliptic curve.
| https://mathoverflow.net/users/3000 | Explicit period lattices for abelian surfaces | It seems to me that it's best to go back up to the quadratic extension and view it as a product of two elliptic curves $E \times E^\sigma$. If you can compute the Weierstrass equations for these elliptic curves, you can compute their $j$ invariants, and then you just need to find $\tau\_1,\tau\_2$ such that $j(\tau\_1)... | 5 | https://mathoverflow.net/users/18060 | 108305 | 62,404 |
https://mathoverflow.net/questions/108307 | 8 | Let $X/\mathbb{C}$ a nonsingular proper variety and $X\_{an}$ it's associated analytic space. Is $X\_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective.
A complex torus is algebraic iff it is projective. Are there Kahler manifolds which are algebraic, but not projective?
| https://mathoverflow.net/users/25854 | Are complex varieties Kahler? - Algebraic, non-projective complex manifolds | Any abstract algebraic compact complex manifold is Moishezon. By Moishezon's theorem, any Kähler Moishezon manifold is projective algebraic. There are non-projective proper complex varieties, so $X\_{an}$ is not necessarily Kähler. This is represented in the diagram at the end of Hartshorne's *Algebraic Geometry* Appen... | 16 | https://mathoverflow.net/users/121 | 108311 | 62,406 |
https://mathoverflow.net/questions/108294 | 1 | Here's the setup: let $N(m)$, $m = 1,2, \cdots$ be a sequence of nonnegative integers for which
$(\*)~~~~~\sum\_{m = 1}^{\infty} \frac{N(m)}{m} < \infty$
Now let $C>0$ be any constant. My question is, does there exist a constant $K < \infty$ such that for all $n \geq 1$,
$\left(1+C \sum\_{m = n+1}^{\infty} \frac{... | https://mathoverflow.net/users/40264 | Question about exponential-type limit | Counterexample: Define $N(m)$ by
$$N(m):=\begin{cases}
\lfloor n!/2\rfloor &\text{if } m=(n+2)!-1\\\\
\lceil n!/2\rceil &\text{if } m=(n+2)!\\\\
0 &\text{otherwise}\\\\
\end{cases}$$
and note that $\sum\limits\_{m=1}^\infty \frac{N(m)}{m}<\sum\limits\_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$, yet
$$\begin{align}
\le... | 1 | https://mathoverflow.net/users/13832 | 108314 | 62,407 |
https://mathoverflow.net/questions/108315 | 1 | I have the following question. Let $A$ be a metrically oriented $n$-dimensional subset of $\mathbb{R}^N$ and $f$ a continuous map from $A$ to $\mathbb{R}^M$. We know that $\operatorname{Lip} f < +\infty$ holds $L^N$-almost everywhere, where $\operatorname{Lip} f$ is the local Lipschitz constant of $f$. Can we then cont... | https://mathoverflow.net/users/26608 | Can we extend an a.e. Lipschitz map defined on a closed subset of R^N to the whole space so that it is still a.e. Lipschitz? | You only need to assume that $A$ is a closed subset of $\mathbb{R}^N$ and then construct an extension of $f$ so that it is locally Lipschitz outside $A$. Something like what I explained in [Can we extend a continuous function with keeping Hausdorff dimension?](https://mathoverflow.net/questions/100693) should work (ext... | 4 | https://mathoverflow.net/users/11716 | 108323 | 62,413 |
https://mathoverflow.net/questions/87162 | 3 | Voevodsky reformulated the Milnor-Bloch-Kato conjecture as a change-of-topology morphism from the Zariski to the étale topology of a *field* with torsion coefficients being an isomorphism. The Beilinson-Lichtenbaum conjecture is more generally about such an isomorphism for *varieties* and integer coefficients. How does... | https://mathoverflow.net/users/nan | Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture | To go from finite coefficients to integral coefficients, one notes that
rationally, Zariski and etale cohomology agree (this boils down to the fact that higher
Galois cohomology is torsion), and compares the two long exact sequences associated to the short exact sequence of coefficients $\mathbb Z(n) \to \mathbb Q(n) \... | 9 | https://mathoverflow.net/users/26735 | 108325 | 62,415 |
https://mathoverflow.net/questions/108319 | 7 | I am probably missing something obvious, but still...
Consider an oriented Riemannian $n$-dimensional vector bundle $\pi: E\rightarrow X$ over compact manifold $X$ with $\omega\_2(E)=0$ so it has spin structures. Let $P\_{SO}(E)$ denote the associated principal $SO\_n$ frame bundle. We would like to count the number ... | https://mathoverflow.net/users/26250 | Number of spin structures | Your first answer is correct, and the second one is almost correct, but the problem is that they count different things:
In the first case, you count all twofold covers of $P = P\_{SO} E$ that restrict to a nontrivial cover of each fibre. In other words, you count $Spin (n)$-bundles $Q$, together with an isomorphism ... | 12 | https://mathoverflow.net/users/9928 | 108328 | 62,417 |
https://mathoverflow.net/questions/108330 | 5 | Let $f: \mathbb R \to \mathbb R$ be any function. When is the graph of $f$ dense in $\mathbb R^2$?
The only examples I know for this are for non-measurable functions, but is that a necessary condition?
| https://mathoverflow.net/users/26844 | When is the graph of a function a dense set? | The [Conway base 13 function](http://en.wikipedia.org/wiki/Conway_base_13_function) is probably a standard example: the graph is dense because any restriction of the function to an open interval is surjective. But meanwhile, since the graph of the base 13 function is easily defined by an arithmetic property of the digi... | 11 | https://mathoverflow.net/users/1946 | 108338 | 62,422 |
https://mathoverflow.net/questions/108310 | 12 | Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. [CH Dowker](http://en.wikipedia.org/wiki/Clifford_Hugh_Dowker) constructed two simplicial complexes $K$ and $L$ associated to $R$:
1. a simplex in $K$ consists of finitely many elements $x \in X$ such that there exists a single $y \in Y$ with $(x,y)... | https://mathoverflow.net/users/25193 | What are the applications of Dowker's theorem? | There is a considerable literature on `applications' of Dowker's result to sociology. This is sometimes doubtful in its depth! The development was started by R. Atkin.
As an example look at
<http://www.ehu.es/ccwintco/uploads/1/11/Blanca-Cases-Q-analysis.pdf>
As an offshoot of this there is fairly recent work i... | 11 | https://mathoverflow.net/users/3502 | 108347 | 62,428 |
https://mathoverflow.net/questions/108352 | 11 | By a theorem of Larson and Sweedler, the antipode of every finite-dimensional Hopf algebra is bijective.
My question is the following:
Is it true that in every noetherian Hopf algebra the antipode is bijective?
| https://mathoverflow.net/users/26812 | Hopf algebras and bijective antipodes | It is conjectured that the antipode is bijective for all noetherian Hopf algebras (Skryabin), but no proof is known. Take a look at this recent short survey, ["Noetherian Hopf Algebras"](http://arxiv.org/abs/1201.4854), by K.R. Goodearl, where this is listed as conjecture 1.9. Skryabin's original paper is:
>
> S. S... | 12 | https://mathoverflow.net/users/2384 | 108356 | 62,431 |
https://mathoverflow.net/questions/108312 | 4 | For the definition of unknotting Number, you can assess <http://www.popmath.org.uk/exhib/pagesexhib/unknum.html>
My question is:
For given a knot K, let n be the crossing number of K, is their any estimate of the unknotting number of K? Of course, the unknotting number is smaller than n-1, but does there exist any ... | https://mathoverflow.net/users/25054 | Problems about the Estimate the Unknotting Number | You're intuition is right: asymptotically, the torus knots have unknotting number roughly half of the crossing number. The [crossing number of a $(p,q)$ torus knot](http://en.wikipedia.org/wiki/Torus_knot) $T\_{p,q}$ is $c(T\_{p,q})=\min\{(p-1)q,(q-1)p\}$, whereas the [unknotting number is $u(T\_{p,q})=(p-1)(q-1)/2$](h... | 11 | https://mathoverflow.net/users/1345 | 108358 | 62,432 |
https://mathoverflow.net/questions/108357 | 0 | I had the following question when I am learning von Neumann algebras:
Let p be a finite projection in a finite von Neumann algebra $M$, let $p>p\_1>p\_2>\cdots$ be a decreasing sequence of projections such that $p\_i\neq p\_{i+1}$. Is such a sequence necessarily of finite length?
Any help or recommendations on books/... | https://mathoverflow.net/users/9858 | Finite projection in Von Neumann algebra | The answer to your question is **no**. Abelian von Neumann algebras are finite, but it is easy to find examples of such algebras with infinite decreasing sequences of projections. For instance, one may take $M = L^\infty(\mathbb{R},\lambda)$ where $\lambda$ is the Lebesgue measure, and let $p\_i \in M$ be the element c... | 5 | https://mathoverflow.net/users/778 | 108360 | 62,433 |
https://mathoverflow.net/questions/78563 | 8 | Let $X$ be a smooth irreducible projective algebraic curve of genus $g\geq 1$ and $S=X^2$ the surface one obtains as the cartesian product of $X$ with itself. Let $\Delta$ be the diagonal in $S$, that is to say, a copy of $X$ embedded diagonally in $X^2$.
Is the quasi-projective surface $U=S\setminus \Delta$ affine?... | https://mathoverflow.net/users/18643 | Is the square of a curve minus its diagonal affine? | This answer is inspired by Rita's second paragraph.
It turns out that this $U$ contains projective curves for all $g\geq 1$ and hence cannot be affine:
Embed $X$ into its Jacobian $X\hookrightarrow J$ and let $Z\subseteq J$ be the image of $X\times X$ via the map $J\times J\to J$, $(x,y)\mapsto x-y$. This contract... | 7 | https://mathoverflow.net/users/10076 | 108361 | 62,434 |
https://mathoverflow.net/questions/96499 | 6 | Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On the character level there is a well-defined "reduction mod $p$", which Brauer showed to be surjective in the sense that e... | https://mathoverflow.net/users/4231 | Algorithm for Brauer lifting via Brauer tree? | If I understand correctly, you have a tree with edges labelled by e1,...,en and vertices labelled
by v0,...,vn,...v(n+m-1), where the last m >= 1 of these label the same (namely the exceptional) vertex.
Relations among Brauer characters are given by: vi = sum of the ej, where ej runs through all the edges
adjacent to v... | 3 | https://mathoverflow.net/users/1729 | 108362 | 62,435 |
https://mathoverflow.net/questions/108350 | 17 | A theorem of Gouvea and Hida (or rather a consequence of it) states that there exist a Galois representation attached to a $p$-adic eigenform $f$ provided the residual representation attached to a classical eigenform $g$ congruent to $f$ is absolutely irreducible. I believe there might be generalizations due to Skinner... | https://mathoverflow.net/users/2081 | Representations attached to p-adic modular forms | I am not sure this answer will satisfy you totally because I am not sure what you mean exactly by $p$-adic modular forms. But at least, for a $p$-adic modular form $f$ in the sense of
Serre, which is an eigenform for almost all the Hecke operators $T\_\ell$ (with eigenvalue $a\_\ell)$ there **always** exists a semi-sim... | 17 | https://mathoverflow.net/users/9317 | 108369 | 62,439 |
https://mathoverflow.net/questions/108354 | 4 |
>
> Let $F$ be a torsionfree subgroup of a commutative group $G$. Are there nontrivial conditions known under which there exists a torsionfree direct summand of $G$ containing $F$?
>
>
>
I would already be happy with such a condition in case $F$ is free or $G$ is of finite type.
*Motivation:*
We consider a p... | https://mathoverflow.net/users/11025 | When is a torsionfree subgroup contained in a torsionfree direct summand? | Here is a counter-example in the case when $G$ is 2-generated. Let $G=<a>\times <b>$ where $a$ has finite order $p>1$ and $b$ has infinite order. Let $H=<c>$, where $c=ab^p$. Then the infinite cyclic group $H$ is not contained in a free factor of $G$. Indeed, otherwise, $c$ admits $p$-th root: $c=x^p$. Then $x=a^n b$, ... | 4 | https://mathoverflow.net/users/21684 | 108375 | 62,443 |
https://mathoverflow.net/questions/108391 | 11 | Suppose we have a de Rham Galois representation $G\_K\rightarrow GL(V)$ for some $p$-adic field $K$ and some finite dimensional vector space $V$ over $\mathbf{Q}\_p$. Then it is a theorem that there is some finite extension $L/K$ such that $D\_{dR}^L(V)\cong L\otimes\_{L\_0}D\_{st}^L(V)$ (this is actually a corollary o... | https://mathoverflow.net/users/88 | Are D_dR and D_st "potentially comparable"? | Actually, I think that Rebecca is right and that the answer is "no". Here's a sketch of the reason why.
Let $V$ be a $p$-adic representation. If $V$ is Hodge-Tate, then $D\_{dR}(V) \neq 0$. So it's enough to find a HT representation such that $D\_{st}^L (V) = 0$ for any $L$. Although I can't think of an explicit one,... | 16 | https://mathoverflow.net/users/5743 | 108401 | 62,454 |
https://mathoverflow.net/questions/108402 | 3 | I asked this question in <https://math.stackexchange.com/questions/204115/decomposition-of-matrices-in-semisimple-and-nilpotent-parts> but remains unanswered.
For any matrix $A\in M\_n(\mathbb F)$, where $\mathbb F$ is an algebraically closed field, there is a matrix $S\in M\_n(\mathbb F)$ such that
$$SAS^{-1}=D... | https://mathoverflow.net/users/20272 | Decomposition of Matrices in Semisimple and Nilpotent Parts | 1. You don't need to conjugate if you want $R$ to be diagonalizable (as opposed to diagonal).
2. I assume you want $R$ and $M$ to have coefficients in $K$, otherwise just work in the algebraic closure.
3. The statement is then true if $K$ is perfect and possibly false otherwise, as you can see by taking $A=[[0, 1], [t,... | 6 | https://mathoverflow.net/users/5743 | 108408 | 62,456 |
https://mathoverflow.net/questions/108399 | 0 | Hi everyone
The question is the following:
A certain event may or may not take place. So we say that if we focus on it one time, it has a probability p of being satisfied (0 <= p < 1)
If we observe it multiple times, and we find out that it occurred zero times among our n observation, what can we say about p? Wha... | https://mathoverflow.net/users/10762 | Probability and events | I'll interpret "most likely" as [Maximum Likelihood Estimation](http://en.wikipedia.org/wiki/Maximum_likelihood), that is, given some observations, what is the value of $p$ that makes the probability of those observations the largest. For example, if we observe something occurring $0$ out of $n$ times, the maximum like... | 0 | https://mathoverflow.net/users/1474 | 108413 | 62,458 |
https://mathoverflow.net/questions/108416 | 2 | This is a somewhat naive question to which I don't know the answer. There is a map of spectra $S \to S$, defined up to homotopy, given by multiplication by $-1$, and it satisfies the relation $(-1)^2 \simeq 1$. Can this be refined to a strict $\mathbb{Z}/2$-action on the sphere $S$ (and thus on all spectra)?
My feel... | https://mathoverflow.net/users/344 | $\mathbb{Z}/2$-action on spectra given by inversion | There's often no self-map of $S$ which induces multiplication by $(-1)$, because it's usually not cofibrant-fibrant. There's no *homotopical* obstruction to it existing, because the map $E \mapsto E \wedge S^1$ is an autoequivalence of the stable homotopy category, and $\mathbb{Z}/2$ acts on $S^1$. Or, if you prefer, w... | 7 | https://mathoverflow.net/users/360 | 108417 | 62,461 |
https://mathoverflow.net/questions/108410 | 2 | Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded operators of $E$ is infinite-dimensional. Nice!
Let us take then an abelian closed subalgebra (feel free to take a max... | https://mathoverflow.net/users/15129 | Subalgebras of $B(E)$ | (Thinking and writing about this in a hurry, so take it with a grain of salt!).
Let $E$ be the space $\ell\_p(\mathbb{Z})$, $1\leq p\leq \infty$, and $T$ the left (or right) shift operator on $E$. Let $\mathscr{A}$ be the (abelian) closed subalgebra of $\mathscr{B}(E)$ generated by $T$ and $\mathscr{I}$ be the closed... | 2 | https://mathoverflow.net/users/848 | 108427 | 62,465 |
https://mathoverflow.net/questions/108409 | 8 | Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a *special* $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say $A=\bigoplus\_{i=0}^\infty A\_i$. In [**Riemann–Roch Algebra**](https://doi.org/10.1007/978-1-4757-1858-4), p. 11, Fulton and Lang defi... | https://mathoverflow.net/users/16046 | $\lambda$-ring structure defined for a graded ring in Fulton–Lang's book | As others have said, the definition of the Chern ring there is wrong. But if memory serves, the only mistake is that they forgot to introduce the right multiplication law on the sets of power series they consider. The usual one in the theory is given by the universal formulas for exterior powers of tensor products $\La... | 9 | https://mathoverflow.net/users/1114 | 108429 | 62,466 |
https://mathoverflow.net/questions/108271 | 20 | Let $A$ be an $n\times n$ symmetric substochastic matrix (i.e. all entries are non-negative and each row adds up to $1$ or less).
Call a vector $v \in \mathbb{R}^n$ an indicator if $v \neq 0$ and each coordinate of $v$ is either $0$ or $1$. Define the indicator spectral radius of $A$ by:
$$r\_I(A) = \max\left\lbrac... | https://mathoverflow.net/users/7631 | Spectral radius on 0-1 vectors. | Here is an example that shows that the bound $f(x) \leq \sqrt x$ does
not work. More precisely $f(x) \geq \sqrt{x(2-x)}$ when $x=2/d$ for an integer $d$.
For an integer $d$, denote by $A$ the Markov operator for the uniform
random walk on an infinite tree with degree $d$ (each vertex has $d$
neighbours). It is well k... | 5 | https://mathoverflow.net/users/10265 | 108443 | 62,471 |
https://mathoverflow.net/questions/108448 | 3 | Suppose I have a set with two metrics, which induce distinct topologies, (so neither is contained in the other). There should exist a sequence which converges in both topologies, but to different points (otherwise all sequences converge to the same point in both metrics, which then implies the topologies are equal). I'... | https://mathoverflow.net/users/7895 | Two metrics and a sequence converging to two points. | It is a "bad question", in that the assumption that neither metric topology is contained in the other does not imply that there is a sequence converging in both topologies, but to different points. Example: on the set $\mathbb{Z} \cup \{+\infty\}\cup\{-\infty\}$ consider the metric topology $\tau\_1$ where all points a... | 5 | https://mathoverflow.net/users/6101 | 108453 | 62,475 |
https://mathoverflow.net/questions/108439 | 9 | Hello,
I usually see the following theorem without seeing any reference or explanation
Theorem: Let X be a blowup of $P^2$ at generic r points. If $r\leq 8$ then there are only finitely many (-1) curves on $X$ , while if $r\geq 9$ there are infinitely many of them.
This is a little mysterious to me. I imagine th... | https://mathoverflow.net/users/26892 | Reference needed for negative curves on blowup of the projective plane at generic points | It is a result of Nagata but it is easy to explain (at least to se the difference between $r\le 8$ and $r\ge 9$):
The Picard group of the blow-up is generated by $L$, the pull-back of a general line, and $E\_1,...,E\_r$ the exceptional curves. Any curve disctinct from the $E\_i$'s is linearly equivalent to $C=dL-\sum... | 14 | https://mathoverflow.net/users/23758 | 108457 | 62,478 |
https://mathoverflow.net/questions/105719 | 5 | Hi,
it is easy to prove the $W^{2,2}(\mathcal S^2)$ regularity for the laplace on the (2 dimensional-) standard sphere $\mathcal S^2:=\lbrace x \in\mathbb R^3: \vert x\vert=1 \rbrace\hookrightarrow\mathbb R^3$ by partial integration, getting
$$\Vert f\Vert\_{W^{2,2}(\mathcal S^2)} \le C ( \Vert \triangle f\Vert\_{L^... | https://mathoverflow.net/users/26017 | $W^{2,p}$ or $W^{1,q}$ regularity for the laplace on a euclidean sphere | ### Sobolev-Norms:
Let $(M,g)$ be a two-dimensional compact remannian-manifold without boundary $r>0$, define for $f\in\mathcal C^\infty(M)$ the sobolev-inequalites
$$ \Vert f\Vert\_{W^{k,p}(M)} := \sum\_{l=0}^k r^l\cdot\left\Vert \vert\nabla^l f\vert\_g\right\Vert\_{L^p(M)}\qquad\forall f\in\mathcal C^1(M),\ k\in\ma... | 3 | https://mathoverflow.net/users/26017 | 108462 | 62,481 |
https://mathoverflow.net/questions/108440 | 2 | Is there anything special about principal $G$-bundles, such that $G$- abelian group? Are there some interesting consequenes of this property, which do not take place in general?
| https://mathoverflow.net/users/26250 | Abelian principal bundles | In easy detail, the classifying space $BG$ for principal $G$-bundles ($G$ a topological abelian group) is itself a topological abelian group, just because the usual classifying space functor preserves products and the multiplication and inverse on $G$ are continuous homomorphisms. Since the set of principal $G$-bundles... | 10 | https://mathoverflow.net/users/14447 | 108470 | 62,482 |
https://mathoverflow.net/questions/108460 | 1 | My question is: "Is it possible to have a sound and rigorous legitimation of the following construction ?". This construction is:
0/ Let ZFC be the usuel set theory, and let us add to the language capital latin letters as names for classes.Let V={x/x=x} be the usual universal class (that is a proper class, directly b... | https://mathoverflow.net/users/30395 | Finite level super classes over ZFC | One of the easiest ways to justify your construction is this:
Take an inaccessible cardinal $\kappa$. Then $V\_\kappa$, the $\kappa$'th iterate of the
powerset operation starting from the empty set and taking unions at limit steps, is a model of ZFC. Now consider a two-sorted structure with the underlying sets $V\_\k... | 6 | https://mathoverflow.net/users/7743 | 108471 | 62,483 |
https://mathoverflow.net/questions/108463 | 5 | Suppose $X$ is a smooth algebraic variety over a field of characteristic $0$. What are the most general conditions under which Hochschild homology and cohomology of $X$ agree?
The existence of a symplectic form will do the job, but is there something more general?
I would be interested in particular into a condition al... | https://mathoverflow.net/users/21710 | When do Hochschild homology and cohomology agree? (Ambidexterity?) | If $X$ is Calabi--Yau (that is $K\_X = 0$) then Hochschild homology and cohomology agree up to a shift of grading by dimension of $X$.
| 5 | https://mathoverflow.net/users/4428 | 108479 | 62,487 |
https://mathoverflow.net/questions/108240 | 2 | Hi everyone,
my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}\_{\infty} ( \Gamma )$, and a convex function $F : A \to [0, + \infty]$ which is lower semicontinuous with respect to poi... | https://mathoverflow.net/users/26825 | Existence of a measure under certain condition | No. Some counterexamples follow: First, fix $V \in \mathcal{B}$, and let $A$ be the set of $\mathcal{B}$-measurable functions which vanish on $V$. Let $F \equiv 0$. Then $F$ satisfies all of the assumptions, since $\inf\_\Gamma f \le 0$ for all $f \in A$.
Stranger examples show that even assuming $F(f) > 0$ for some ... | 2 | https://mathoverflow.net/users/26459 | 108480 | 62,488 |
https://mathoverflow.net/questions/108450 | 1 | Let $ F = C^{\infty}(M, N)$. I wish to give $F$ the structure of a Fréchet manifold. My plan was to emulate the construction of a smooth manifold. I know that for a finite dimensional smooth manifold M, $T\_pM$ will be isomorphic to the model space (i.e if M is m dimensional, then $T\_pM \cong \mathbb{R}^m$). Further m... | https://mathoverflow.net/users/16533 | Fréchet manifold structure of C(M, N) | See Theorem III.1.11 on page 76 of *"Stable mappings and their singularities"* by Golubitsky and Guillemin for a proof that $C^\infty(M,N)$ is a Fréchet manifold when $M$ is compact.
| 3 | https://mathoverflow.net/users/8103 | 108483 | 62,489 |
https://mathoverflow.net/questions/108481 | 3 | I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and let $f\in Hom\_{D^b(X)}(M,N)$. Suppose $i^\*f = 0$ and $j^\*f= 0$.
Then is it true that $f=0$?
My gut answer is yes,... | https://mathoverflow.net/users/23907 | Showing morphism of sheaves is zero | In fact, it is not true. A counterexample is gotten by taking $X = S^1$, $Y$ to be any point on $S^1$, "sheaves" to be, say, sheaves of $\bf{Q}$-vector spaces, and taking $M = \bf{Q}$, $N = \bf{Q} [1]$ and $f$ to be any map classifying a nontrivial first cohomology class on $S^1$. The problem, of course, is that the fi... | 9 | https://mathoverflow.net/users/3931 | 108486 | 62,491 |
https://mathoverflow.net/questions/108494 | 1 | It's obvious that the Lie derivative defined in terms of Lie brackets is anti-symmetric. But what is an intuitive way to visualize the anti-symmetry in the 'differentiating along a flow' definition?
Thanks!
| https://mathoverflow.net/users/25153 | How can I picture antisymmetry of the Lie derivative? | Whichever definition you use that involves differentiating along a flow, it ought to have the property that $L\_X X = 0$; in other words, for reasons that should be clear, the Lie derivative is alternating. In the presence of bilinearity, this is equivalent to antisymmetry.
| 6 | https://mathoverflow.net/users/290 | 108496 | 62,494 |
https://mathoverflow.net/questions/108508 | 7 | Compact Kahler manifolds have the property that surjective maps induce injections on cohomology with coefficents in $\mathbb{Q}$ (That is, if $X,Y$ compact Kahler, then a surjective map $\phi: X \rightarrow Y$ induces injections $\phi^\*: H^i(Y, \mathbb{Q}) \rightarrow H^i(X, \mathbb{Q})$ for all $i$, [Voisen, Hodge Th... | https://mathoverflow.net/users/25854 | Injective maps on cohomology and Kahler manifolds | In order to have this property it is sufficient to require that $\phi^{-1}(y)$ is a non-zero cycle in $H\_\*(X,\mathbb Q)$, where $y$ is a generic point in $Y$. This holds indeed when $X$ and $Y$ are Kahler.
Let me give an example showing that this does not work when $X$ and $Y$ are just complex.
**Example.** Let ... | 15 | https://mathoverflow.net/users/943 | 108511 | 62,498 |
https://mathoverflow.net/questions/100317 | 2 | Consider a smooth metric measure space in which the integral of a gradient is meaningful. For example in the sense of upper gradients of Heinonen, or on a riemannian manifold with the associated measure. Therefore, one can define Sobolev spaces $W^{1,p}$. In the case of a compact riemannian metric, and in particular a ... | https://mathoverflow.net/users/12019 | Rellich-Kondrachov compactness theorem in arbitrary smooth metric measure spaces | Please take a look at Theorem 8.1 (Chapter 8) in the following paper
Hajłasz, Piotr; Koskela, Pekka Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp.
| 3 | https://mathoverflow.net/users/26608 | 108521 | 62,504 |
https://mathoverflow.net/questions/108522 | 2 | I've got two square real matrices ($A$ and $B$) and I want to find a permutation of the rows and columns (same permutation) to best approximate the other matrix. E.g. with $P$ as a permutation matrix and say the Frobenius norm
$\min\_{P \in \Pi} || A - PBP^T ||\_F$
Now, I know (a) if $A = PBP^T$ then you can find $... | https://mathoverflow.net/users/26912 | permute both rols and cols of one matrix to best approximate another? | This problem looks like the [famous Quadratic assignment problem](http://en.wikipedia.org/wiki/Quadratic_assignment_problem), which means that in general you are out of luck as far as an exact solution is concerned.
| 5 | https://mathoverflow.net/users/8430 | 108525 | 62,506 |
https://mathoverflow.net/questions/108523 | 5 | Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum\_j \frac{d^2}{dx\_j^2} + 2\sqrt{-1} \sum\_j \lambda\_j \frac{d}{dx\_j} + \sum\_{ij} a\_{ij}x\_ix\_j$$
on $\mathbb{R}^n$? Here, the matrix $(a\_{ij})$ is supposed to be symmetric and pos... | https://mathoverflow.net/users/16702 | Closed formula for heat kernel | Yes there is. Here is how you do it. First find an orthogonal change in variables
$$ x\_j=\sum\_{jk} s\_{jk}y\_k $$
$(s\_{jk})$ orthogonal matrix, so that in the new coordinates we have
$$ \sum\_{i,j}a\_{ij} x\_ix\_j = \sum\_j \mu\_j^2 y\_j^2, $$
where $\mu\_j^2$ are the eigenvalues of the symmetric matrix $(a... | 14 | https://mathoverflow.net/users/20302 | 108531 | 62,510 |
https://mathoverflow.net/questions/108530 | 13 | Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question [Hadamard-like inequalites for positive definite symmetric matrices](https://mathoverflow.net/questions/105697). If $H$ is a subgroup of $\frak S\_n$
and $\chi$ is an irreducible complex character over $H$, define
$$d\_\... | https://mathoverflow.net/users/8799 | A maximal element, where Schur gives a minimal element | This question is better known as the [**permanental dominance conjecture**](http://math.ecnu.edu.cn/~zhan/papers/ZhanICCM.pdf) and is still an open problem.
According to Zhan's survey, it has been confirmed for every irreducible character of $S\_n$ for $n \le 13$. Another reference cited for this conjecture is [this ... | 18 | https://mathoverflow.net/users/8430 | 108533 | 62,512 |
https://mathoverflow.net/questions/31458 | 39 | Problem.
--------
Let $\{\lambda\_n\}\_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda\_n s)\}\_{n\in\mathbb N}$ $F$-independent (where $F$ is either $\mathbb C$ or $\mathbb R$) iff whenever the series with complex coefficients
$$f(s)=\sum\limits\_{n... | https://mathoverflow.net/users/5371 | On linear independence of exponentials | I have some partial answers.
I. It is not hard to construct a Dirichlet series
$$f(z)=\sum\_{n=1}^\infty a\_ne^{\lambda\_n z}$$
which converges to $0$ absolutely and uniformly on the real line but does not converge at some points
of the complex plane.
It is constructed as a sum of 3 series $f=f\_0+f\_1+f\_2.$ Let $f... | 14 | https://mathoverflow.net/users/25510 | 108547 | 62,518 |
https://mathoverflow.net/questions/108550 | 10 | Hi,
I've been wondering about the following :
Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?
All the examples of inequivalent complete norms I've seen rely on the existence of Hamel bases...
This is most likely well-known, but I'd be glad if someone could... | https://mathoverflow.net/users/1162 | Inequivalent complete norms and the axiom of choice | No, it is not possible. There is a model due to Shelah of ZF+ dependent choice+every set of reals has the Baire property. There is a result of Garnir and Wright that implies that in such model, any two complete norms are equivalent. The [Handbook of Analysis and Its Foundations](http://www.math.vanderbilt.edu/~schectex... | 11 | https://mathoverflow.net/users/35357 | 108554 | 62,521 |
https://mathoverflow.net/questions/108492 | 5 | The question (similar to [MO.96531](https://mathoverflow.net/questions/96531/request-mazur-arithmetic-in-the-geometry-of-symmetric-spaces)) is about the article by Professor Kazuya Kato in this [book](https://rads.stackoverflow.com/amzn/click/com/3540571108).
In this article, Professor Kato indicates the contents of... | https://mathoverflow.net/users/11786 | Request: Kato's article "Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions." Part II | Dear SGP, I hope you (and a few other committed MOers) have received the paper as an email attachment. However, I will make it publicly available here only if Professor Kato authorises me to do so.
| 5 | https://mathoverflow.net/users/2821 | 108557 | 62,523 |
https://mathoverflow.net/questions/108501 | 2 | I was wondering if anyone could give a heuristic (i.e. preferably non-technical) explanation of what is the expected longtime behavior of the periodic KdV equation.
Recall the standard KdV equation is the PDE defined (after suitable normalization) on $\mathbb{R}\_t\times \mathbb{R}\_x$
by
$$
u\_t= u\_{xxx}+6 u u\_x... | https://mathoverflow.net/users/26801 | Longtime behaviour of the periodic KdV equation | Smooth solutions to the periodic KdV equation evolve almost periodically in time in any reasonable topology, due to the existence of action-angle variables that make any given KdV flow equivalent to a linear flow on an infinite-dimensional torus (which, when starting from smooth data, is compact in any reasonable topol... | 5 | https://mathoverflow.net/users/766 | 108559 | 62,525 |
https://mathoverflow.net/questions/108499 | 0 | Consider a positive sequence $x\_n >0$ that satisfy the condition that there exists a constant $0<\alpha<1$ such that $x\_{n+1} \geq \alpha (x\_1+\ldots{} +x\_{n})$.
What can be said about the set of all such sequences? Can one solve for them in terms of a formula for $x\_n$?
I am particularly interested in knowing... | https://mathoverflow.net/users/21269 | Sequences satisfying $x_{n+1} \geq \alpha (x_1+\ldots{} +x_{n})$ for $\alpha<1$ | Not very much can be said about all such sequences. In essence all that can be said is that they grow at least exponentially. As previous comments have noted, for all $n$, we have $x\_n \geq y\_n$, where $y\_1 := x\_1$ and for all $n>1$,
$$
y\_n := \alpha \sum\_{i=1} ^{n-1} y\_{i} = \alpha \sum\_{i=1} ^{n-2} y\_{i} + \... | 3 | https://mathoverflow.net/users/22512 | 108563 | 62,528 |
https://mathoverflow.net/questions/108553 | 2 | Let $v$ be a $C^r$ vector field on a Banach space $V$ such that $0$ is its hyperbolic fixed point, and let $X\_v\subset V$ be its local stable manifold. Does $X\_v$ depend on $v$ smoothly?
| https://mathoverflow.net/users/9390 | smooth dependence of stable manifold on parameters | The answer to your question is yes: please see the subsection "Differentiable dependence of invariant manifolds and foliations on diffeomorphisms" (page 164) of Appendix 1 of Palis-Takens book "Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations" (see, e.g., this link <http://books.google.fr/books/a... | 3 | https://mathoverflow.net/users/1568 | 108568 | 62,530 |
https://mathoverflow.net/questions/108560 | 17 | I am looking for a good book on Topological Groups. I have read Pontryagin myself, and I looked some other in the library but they all seem to go in length into some esoteric topics.
I would love something 250 pages or so long, with good exercises, accessible to a 1st PhD student with background in Algebra, i.e. with... | https://mathoverflow.net/users/5301 | What is a good book on topological groups? | I'm not aware of a book that covers simultaneously Pontryagin duality, property (T) and Tannaka duality. I will refrain from recommending any book on property (T) (guess why?). Apart from Weil's book already mentioned, my favourite ones are:
* for Pontryagin duality: Rudin's "Fourier analysis on groups";
* for functi... | 10 | https://mathoverflow.net/users/14497 | 108571 | 62,533 |
https://mathoverflow.net/questions/108505 | 121 | I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the members of the community share their experience
with these kind of lectures. I was thinking to talk about some theorems... | https://mathoverflow.net/users/7442 | "Mathematics talk" for five year olds | I'm going to quote Bill Thurston from his interview for [More Mathematical People](https://books.google.sk/books?id=Nu_uAAAAMAAJ):
>
> Thurston: ... One thing that is very important is the education of children... In the elementary schools in Princeton that my kids have attended, there is an annual event called Sci... | 83 | https://mathoverflow.net/users/2926 | 108580 | 62,538 |
https://mathoverflow.net/questions/108585 | 3 | This must be known to everyone who might remotely be considered an expert (but not to me...): What are the subgroups of index three of $SL(2, \mathbb{Z}/p \mathbb{Z})?$
**EDIT**
Following up on Qiaochu's answer, the situation for $SL(2, 3)$ is described [here.](http://groupprops.subwiki.org/wiki/Subgroup_structure_of... | https://mathoverflow.net/users/11142 | Subgroups of $SL(2, \mathbb{Z}/p \mathbb{Z})$ | A subgroup $H$ of index $3$ determines a homomorphism $\text{SL}\_2(\mathbb{F}\_p) \to S\_3$ whose kernel $N = \bigcap\_{g \in \text{SL}\_2(\mathbb{F}\_p)} gHg^{-1}$ is a normal subgroup of index either $3$ or $6$. The image of $N$ in $\text{PSL}\_2(\mathbb{F}\_p)$ is therefore also a normal subgroup of index $3$ or $6... | 5 | https://mathoverflow.net/users/290 | 108587 | 62,542 |
https://mathoverflow.net/questions/108589 | 1 | Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
what is the limit of $\frac{\int\_{S^1}|t-a|^{1 + \alpha}.p(z,t)|dt|}{|z-a|}$
as $z \to a, z\in \mathbb{D}$. I am tending ... | https://mathoverflow.net/users/6953 | On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis | The limit is not zero; it does not exist.
FIRST proof. WLOG let $a=1$. Let $-u(z)$ be your Poisson integral,
(in the numerator of your formula) it is a negative harmonic function in the disc, continuous
in the closed disc, $u(1)=0$, and negative at every other point of the circle.
Let $M(r)=max\_{|z|=r}u(z),\; 0\leq ... | 3 | https://mathoverflow.net/users/25510 | 108591 | 62,543 |
https://mathoverflow.net/questions/108578 | 18 | Suppose we fix two Grothendieck universes $\mathcal{U} \in \mathcal{V}.$ Then one has that $\mathcal{U}$-$\mathbf{Set},$ the category of $\mathcal{U}$-small sets, is a locally $\mathcal{U}$-small, $\mathcal{V}$-small category. Grothendieck universes were used often by, well, Grothendieck, in his work with topoi. Part o... | https://mathoverflow.net/users/4528 | How much do universes matter in topos theory? | The change-of-universe construction is faithful but not full.
For example, let X be the topos of sets and let Y be the classifying topos for abelian groups.
The category of geometric morphisms from X to Y is equivalent to the category of abelian groups. If you pass to a larger universe, you get more abelian groups.
| 25 | https://mathoverflow.net/users/7721 | 108595 | 62,546 |
https://mathoverflow.net/questions/108594 | 2 | I would like to know if one can weaken conditions of Proposition 2.8 in
<http://www.jmilne.org/math/xnotes/CA.pdf>
The proposition says that if an ideal $a$ in a ring $A$ is contained in the union of ideals $p\_1,...,p\_r$ with $p\_2,...,p\_r$ prime, then $a$ is contained in one of $p\_i$.
Why do we need to requi... | https://mathoverflow.net/users/13441 | An example ellucidating proposition 2.8 in Milne's notes on commutative algebra | Some variants of prime avoidance (as this property is usually called) are in Eisenbud's "Commutative Algebra with a View..." on p. 114, including an example where it fails: for instance the ideal $(x,y)$ in $\mathbb Z/2\mathbb Z[x,y]/(x,y)^2$ is the union of three (smaller) non-prime ideals.
| 7 | https://mathoverflow.net/users/22873 | 108596 | 62,547 |
https://mathoverflow.net/questions/108605 | 2 | In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following category. An object of $O$-${\mathsf{Cat}}$ is a small category with $O$ as its set of objects. An morphism of $O$-${\mathsf{Cat... | https://mathoverflow.net/users/6301 | Does the following categorial sum preserve weak equivalences? | The general case can be concluded from the finite one as follows. Let $(A\_i \to B\_i \mid i \in I)$ be a family of weak equivalences between simplicial $O$-categories. I'm going to assume that $I = \mathbb{N}$, the general case can be handled similarly, but the notation would be a bit more tedious (you can well-order ... | 3 | https://mathoverflow.net/users/12547 | 108609 | 62,551 |
https://mathoverflow.net/questions/81730 | 6 | Hello. I am trying to give a seminar in my University about the Whitney-Graustein Theorem. There are many elementary proofs for that including Whitney's paper. The conclusion is that the connected components ($π\_0$) of regular immersions $S^1 \rightarrow R^2$ are equal to $Z$ (mod regular homotopies). Is there an elem... | https://mathoverflow.net/users/19480 | regular homotopy | See theorem 2.10 (with elementary proof) for the case of rotation idex $\ne 0$ of the paper:
Peter W. Michor; David Mumford: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48. [pdf](http://www.mat.univie.ac.at/~michor/curves.pdf)
For rotation index $=0$ (with a somewhat surpris... | 3 | https://mathoverflow.net/users/26935 | 108627 | 62,562 |
https://mathoverflow.net/questions/108608 | 3 | If $M$ is a smooth submanifold embedded in $\mathbb{CP}^m$ as a complete intersection, by the adjuction formula, the canonical bundle is given by the restriction to $M$ of $\mathcal{O}(d-m-1)$ where $d$ is the sum of the degrees of the polynomials that define $M$ as a complete intersection.
Now let $M$ be a subcanoni... | https://mathoverflow.net/users/22675 | Is there a relation between the first Chern class of a sub canonical submanifold of the complex projective space and the degrees of the polynomials that define locally the submanifold? | As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ ... | 5 | https://mathoverflow.net/users/10076 | 108641 | 62,569 |
https://mathoverflow.net/questions/108618 | 6 | I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O\_X) such that Hom(X, Spec(A))= Hom(A,Gamma(O\_X)) for every affine scheme Spec(A)? Or does this require some more hypotheses. I have very much a hard time finding... | https://mathoverflow.net/users/9275 | question about higher geometric stacks | Yes -- affinization is defined (as you wrote) as the left adjoint to the inclusion of affine schemes into higher stacks.
This left adjoint exists by the ($\infty$-categorical) adjoint functor theorem, since the inclusion of affines into higher stacks preserves all limits (though it certainly changes colimits). Some ref... | 9 | https://mathoverflow.net/users/582 | 108642 | 62,570 |
https://mathoverflow.net/questions/108643 | 2 | Maybe is a trivial question, but I don't know how to handle it.
***Setting:*** Let $S$ be a semigroup (i.e. has an associative operation with neutral element $e$) and let $(A,+)$ be a commutative group (with neutral element $0$).
A semigroup extension of $S$ with $A$ is any operation on $S \times A$, of the form ... | https://mathoverflow.net/users/7772 | When a semigroup extension $S \times A$ of a semigroup $S$ with a commutative group $A$ contains a copy of $S$? | First of all I would not call what you write a semidirect product. It will be such in your setting precisely when the natural maps to $S$ splits, in which case it will actually be isomorphic to a direct product.
More precisely, you are viewing $A$ here as a trivial module over $\mathbb ZS$. Then $\lambda$ is a 2-cocy... | 4 | https://mathoverflow.net/users/15934 | 108644 | 62,571 |
https://mathoverflow.net/questions/108648 | 6 | I think this should be a 10 minute exercise in a decent computer algebra package - unfortunately I'm hopelessly ignorant of such things, so I'm putting it up here in the hope that someone will be kind enough to do it for me...
Here's the question: partition $n$ into two pieces, $n= p+q$, and let $S\_p\times S\_q \sub... | https://mathoverflow.net/users/2454 | Relations in a particular subgroup of the braid group. | You probably already noticed that, but $B\_{p,q}$ is the fundamental group of
$$
X\_n/(S\_p \times S\_q)
$$
where $X\_n$ is the configuration space of $n$ points in the complex plane. Ths may help to guess some facts about these groups.
So far I know these group are usually called "mixed braid groups" in the litterat... | 7 | https://mathoverflow.net/users/13552 | 108650 | 62,573 |
https://mathoverflow.net/questions/108649 | 9 | I would like to know differential forms representing the cohomology classes of $SU(3)$. I know that there exist a unique bi-invariant form in each class, but I'm not highly motivated by simply putting $R\_g^\*\omega = \omega = L\_g^\*\omega$ and solving it by hand in coordinates. Is there a way to do it less brutally ?... | https://mathoverflow.net/users/21180 | De Rham representatives of the cohomology classes in $H^*(SU(3))$ | The Poincare polynomial of $SU(3)$ is
$$ (1+t^3)(1+t^5). $$
The cohomology algebra is generated by two elements: a generator $x\_3$ in dimension $3$ and one generator $x\_5$ in dimension $5$, given by the formulas mentioned by Alexander Chervov
$$ x\_k ={\rm Tr}\; (g^{-1}dg)^k,\;\; k=3,5. $$
For an explicit de... | 11 | https://mathoverflow.net/users/20302 | 108656 | 62,576 |
https://mathoverflow.net/questions/108119 | 1 | We know the laplacean operator has a Green function which is smooth away from the boundary. Now, consider a linear operator of the form $\partial\_i(a^{ij} \partial\_j u)$.We can prove that this operator has a Green function as well. Now, if we just assume that $a^{ij} \in L^\infty$, and that $a^{ij}$ is uniformly elli... | https://mathoverflow.net/users/12019 | Green's function for a certain elliptic equations with rough coefficients | What you said is the most you can say about regularity away from the origin.
The only thing that can be added is the rate by which it goes to infinity at that point. There is an old result by Littman, Stampacchia and Weimberger which says that this rate is comparable with the corresponding one of the Green's function... | 5 | https://mathoverflow.net/users/26672 | 108658 | 62,578 |
https://mathoverflow.net/questions/108661 | 8 | Edit: we cannot find such an example. It would imply a negative solution to the KS${}\_2$ conjecture which has now been proven by Marcus, Spielman, and Srivastava in [this paper](http://arxiv.org/pdf/1306.3969.pdf).
In fact, their solution implies that there always exists $S$ such that any $f$ supported on $S$ (or $S... | https://mathoverflow.net/users/23141 | approximate uncertainty principle for finite abelian groups | The [restricted isometry property](http://en.wikipedia.org/wiki/Restricted_isometry_property), or RIP (formerly known as the uniform uncertainty principle, or UUP, as per your suspicion that uncertainty principles should be relevant) for random Fourier measurements prohibits $\varepsilon$ from being smaller than about ... | 7 | https://mathoverflow.net/users/766 | 108663 | 62,581 |
https://mathoverflow.net/questions/108519 | 5 | In my research work, I am dealing with a quasi-linear system of first order p.d.e.'s with two independent variables (say $x\_1$ and $x\_2$) and four dependent variables (say $u\_1(x\_1,x\_2)$, $u\_2(x\_1,x\_2)$, $u\_3(x\_1,x\_2)$ and $u\_4(x\_1,x\_2)$) of the form
\begin{equation}
\mathbf{A}(\mathbf{x},\mathbf{u}(\math... | https://mathoverflow.net/users/25516 | Quasi-linear System of First Order P.D.E.s of "Mixed" type | Let me change your notations. You deal with a 1D quasilinear system with size $N=4$: the standard Cauchy problem is
$$
\frac{\partial u}{\partial t}+A(t,x,u)\frac{\partial u}{\partial x}= f(t,x),\quad u(t=0,x)=u\_0(x),
$$
where $t\in \mathbb R$ (time variable) as well as $x$ (this is a 1D problem), $u$ is valued in $\m... | 5 | https://mathoverflow.net/users/21907 | 108666 | 62,583 |
https://mathoverflow.net/questions/108668 | 1 | Let $G$ be a finite group with faithful irreducible representation $\gamma: G \to GL\_n(\mathbb{C})$, $n>1$.
Can we put a bound on the size of $G$? What if $G$ is nilpotent?
| https://mathoverflow.net/users/26947 | Bound on the size of a group given a faithful irrep of a certain dimension | Jordan's theorem says that there is a function $f : \mathbb{N} \to \mathbb{N}$ such that whenever $G$ is a finite subgroup of ${\rm GL}(n,\mathbb{C}),$ there is an Abelian onrmal subgroup $A$ such that $[G:A] \leq f(n)$. Explicit bounds were given later, which can be much improved by invoking the classification of fini... | 11 | https://mathoverflow.net/users/14450 | 108674 | 62,586 |
https://mathoverflow.net/questions/108683 | 0 | Suppose $d\_1, d\_2$ are two fixed coprime integers, $\frac{d\_1}{d\_2} \neq \pm 1$. Given any $n > 0$, can we find a prime number $p$ such that the order of $d\_1d^{-1}\_2$ in the multiplicative group of the field $\mathbb{Z}/p\mathbb{Z}$ be greater than $n$?
| https://mathoverflow.net/users/4760 | Can the order of a rational number in Z/pZ be as large as we want | The answer to your question is "yes" (cf. Douglas Zare's comment). In fact, for all sufficiently large primes $p$, the order of $d\_1 d\_2^{-1}$ is greater than $n$. Here, "sufficiently large" means greater than $|d\_1|^n$ and $|d\_2|^n$.
| 1 | https://mathoverflow.net/users/121 | 108685 | 62,592 |
https://mathoverflow.net/questions/108681 | 4 | I am having some difficulty lining up the definition and my intuition for rational equivalence of cycles. My intuition is based off of the idea that two cycles being rationally equivalent is analogous to the two cycles being homotopic.
If it is requested of me to state the definition of rational equivalence, I will;... | https://mathoverflow.net/users/26954 | Clarification and intuition request for rationally equivalent algebraic cycles | Any codimension 1 cycle that is defined by a regular function is already zero in the Chow group. So your question should not be "Why are the line $x=0$ and the unit circle rationally equivalent to each other?" but rather "Why is the line $x=0$ rationally equivalent to zero?" (and ditto for the unit circle).
The answe... | 11 | https://mathoverflow.net/users/10503 | 108686 | 62,593 |
https://mathoverflow.net/questions/108689 | 3 | In $\mathbb{R}^n$, we say that a linear subspace is *rational* if it admits a basis in $\mathbb{Q}^n$ (or equivalently in $\mathbb{Z}^n$). This means that $E\cap \mathbb{Z}^n$ is a submodule of $\mathbb{Z}^n$ of rank equal to the dimension of the given subspace. By convention, we declare that $\{0\}$ is rational.
Now... | https://mathoverflow.net/users/18938 | Rational subspaces | Given a matrix $M$ whose kernel is $E$, let $\{\alpha\_1, \ldots, \alpha\_k\}$ be a basis for the linear span over $\mathbb Q$ of the entries of $M$. Then $M = \sum\_{j=1}^k \alpha\_j M\_k$ where $M\_j$ are matrices with rational entries. Let $N$ be the matrix obtained by stacking the $M\_j$ vertically.
Any member ... | 3 | https://mathoverflow.net/users/13650 | 108690 | 62,594 |
https://mathoverflow.net/questions/108687 | 5 | One version of Hensel's Lemma is the following statement:
Let $R$ be a commutative ring with a unit. Given a polynomial $Q\in R[X]$ and a root $\alpha$ of $Q$ modulo some ideal $I$ (i.e. $Q(\alpha) \in I$), assuming some non-degeneracy conditions (e.g. $Q$ is square-free), then for every $t > 1$, there exists $\beta... | https://mathoverflow.net/users/5534 | Multivariate Hensel's Lemma, but with only one polynomial | Yes. If the ideal generated by $(I,dQ/dX\_1,..,dQ/dX\_n)$ is the unit ideal at some mod-$I$ solution of $Q$, one can lift to a mod $I^t$ solution of $Q$.
Proof: It is clear that we merely need to check the induction step, that if we have a mod $I^t$ solution we can get a mod $I^{t+1}$ solution. Suppose $Q(X\_1,..,X\_... | 6 | https://mathoverflow.net/users/18060 | 108691 | 62,595 |
https://mathoverflow.net/questions/108468 | 12 | The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently weren't all known until the 1970s.
Start with a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, along with some ... | https://mathoverflow.net/users/4231 | Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$` | In Bourbaki, Ch. VIII, $\S$ 7, Sect. 2, one can find the notion of an $\mbox{$R$-saturated set}$, and Corollary to Prop. 4 in that section proves that for every $R$-saturated set $\mathcal X$ there is a finite dimensional $\mathfrak g$-module whose set of weights coincides with $\mathcal X$. Prop. 6 in the next section... | 7 | https://mathoverflow.net/users/24386 | 108695 | 62,596 |
https://mathoverflow.net/questions/108698 | 3 | I am reading Illusie's lecture notes "topics in algebraic geometry", and I have difficulty in following his proof of unobstructedness of deformation of curves. Here is the statement of the proposition:
**Prop** Let $f\_0: X\_0 \to Y\_0$ be a smooth proper morphism with relative dimension 1, and $i : Y\_0 \to Y$ a fir... | https://mathoverflow.net/users/26460 | The proof of unobstructedness of deformations for curves | Probably Illusie wrote "Zariski's Main Theorem", but he intended the Theorem of Formal Functions (which is the key result needed in the modern proof of Zariski's Theorem).
In fact, the Theorem of Formal Functions implies the following result, see [Hartshorne, Algebraic Geometry, Corollary 11.2 page 279].
>
> **Pr... | 5 | https://mathoverflow.net/users/7460 | 108699 | 62,598 |
https://mathoverflow.net/questions/108485 | 3 | If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two bases is triangular with ones on the "diagonal" (what I mean with this is explained more precisely later, in the case of ab... | https://mathoverflow.net/users/26691 | Linear algebra of finite abelian groups | I think I can prove this in the case when $G$ is a $p$-group. Assume $\phi(h\_1), \ldots, \phi(h\_m)$ are irredundant generators: if any of them is removed they are no longer a generating set for $G$. There must be an element of maximal order, and assume after possibly permuting that it is $\phi(h\_1)$. Then the cyclic... | 1 | https://mathoverflow.net/users/26691 | 108703 | 62,601 |
https://mathoverflow.net/questions/108680 | 1 | We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let $a$, $b$, and $c$ be vertices of a triangle in $\mathbb{R}^2$, and $a'$ be the point on the line $bc$ which is the orthogonal projection of $a$.
Let $t\in (0,1]$ and define the $\mu\_t$ t... | https://mathoverflow.net/users/11100 | sequences of plane measures converging to a singular one: terminology, etc | Whether or not the limit is singular (e.g. with respect to the Lebesgue measure), there are several notions of convergence for measures which can reflect this. Probably the most simple one is [weak convergence of measures](http://en.wikipedia.org/wiki/Convergence_of_measures#Weak_convergence_of_measures) (I prefer the ... | 1 | https://mathoverflow.net/users/9652 | 108705 | 62,602 |
https://mathoverflow.net/questions/108692 | 2 | The set of vertices of $Q(d,n)$ is $\{0,1,\ldots,n-1\}^d$ and every edge is formed by all vertices having $d-1$ coordinates fixed and the last one getting all possible values (so it has $dn^{d-1}$ edges).
I'm trying to prove that $Q(n,n)$ has the following property: if $S$ is some subset of its edges and $U$ is their... | https://mathoverflow.net/users/26959 | Property of cube hypergraph Q(n,n) | **1.** First of all, you can ``shake down'' all your edges along each direction. That is --- choose any direction (say, vertical) and consider all the edges in $S$ of all other directions. Then simultaneously drop all of them as low as possible (preserving the condition that they are distinct). It is easy to see that o... | 0 | https://mathoverflow.net/users/17581 | 108706 | 62,603 |
https://mathoverflow.net/questions/108693 | 4 | Hallo,
I have the following PDE that I am trying to solve via the Cauchy-Kowalewski Theorem. But I have no idea how to do it or if its possible. Maybe one of you has an idea. Here is the problem: Let $U \subset \mathbb{C}^{n}$ be some open subset witch contains zero. Well you can shrink $U$ arbitralily if you wish. L... | https://mathoverflow.net/users/26960 | Solving PDE with Cauchy - Kowalewski Theorem | You don't need the Cauchy-Kowalewski Theorem for your problem. In fact, real-analyticity is a red herring here. What you are asking for is a function $\beta(x,y)$ such that the graph $\bigl(x,y,\beta(x,y)\bigr)$ is an integral manifold of the $1$-form
$$
\theta = d\beta - F\_j(x,y,\beta)\ dx^j - G\_j(x,y,\beta)\ dy^j
$... | 10 | https://mathoverflow.net/users/13972 | 108708 | 62,604 |
https://mathoverflow.net/questions/108696 | 4 | The H-representation of a convex polytope $S$, is just a set of linear inequalities corresponding to the intersection of halfspaces:
$S = ( x | Ax\leq b )$.
One could also represent a convex polytope as the convex-hull of its vertices, called the $V$-representation:
$S = \mathrm{conv}(a\_1,..., a\_m)$.
It is known ... | https://mathoverflow.net/users/26607 | H-representation versus V-representation of polytopes | A generic polytope with $2n$ points in $\mathbb R^n$ has this property. Thus, one can't really characterize such polytopes, unless one can find a way to characterize almost all arrangements of $2n$ points in $\mathbb R^n$
Place $2n$ random points independently identically distributed according to any nontrivial distr... | 6 | https://mathoverflow.net/users/18060 | 108715 | 62,606 |
https://mathoverflow.net/questions/108712 | 0 | An exact Courant algebroid $E$ is one such that the sequence $0\to T^\star M\xrightarrow{\rho^\star} E^\star\simeq E\xrightarrow{\rho} TM\to 0$ is exact. Here $\rho$ is the anchor of the algebroid. Since the sequence is exact, we have a splitting $\varepsilon:TM\rightarrow E$.
However I would like to show that we ca... | https://mathoverflow.net/users/14806 | Isotropic splitting for exact Courant algebroids | This answer, due to Roytenberg, is taken from <http://arxiv.org/abs/math/9910078> page 48 (below Corollary 3.8.4)
"First, remark that $\rho^{\star}T^{\star}M$ is isotropic. Once we have one isotropic
subbundle $T^{\*}M$, transversal isotropic subbundles are sections of a
bundle over $M$ whose fiber is an open cell in... | 2 | https://mathoverflow.net/users/13742 | 108720 | 62,608 |
https://mathoverflow.net/questions/108719 | 13 | If each strict subgroup of a group G is free, must G be free or cyclic of prime order ?
| https://mathoverflow.net/users/21724 | If each strict subgroup of G is free, must G be free or cyclic of prime order ? | No. There is a variation of Tarski monster: a nonabelian group whose each proper nontrivial subgroup is infinite cyclic, see [the book of Olshanskii](http://books.google.ru/books?hl=en&lr=&id=jwNkqFQn-GcC&oi=fnd&pg=PR5&dq=olshanskii+geometry+of+defining&ots=XQQkt5vng4&sig=jNhR97XokHHkZU_qaDboCH6EbDc&redir_esc=y#v=onepa... | 26 | https://mathoverflow.net/users/24165 | 108721 | 62,609 |
https://mathoverflow.net/questions/108601 | 12 | The answer to my question is probably well-known, but I was unable to find a reference.
The Bezout's identity states that for any positive non-zero integers $a\_1, \ldots , a\_n$ there exist integers $x\_1, \ldots , x\_n$ such that
$$
a\_1x\_1+ \cdots + a\_nx\_n=gcd(a\_1, \ldots, a\_n) .
$$
What is the best estimat... | https://mathoverflow.net/users/10251 | Estimates for Bezout coefficients | We can prove $b(a\_1,\ldots,a\_n) \leq a\_1+\cdots+a\_n$ (and thus $f(k) \leq k$) by elementary means as follows.
We may assume $\operatorname{gcd}(a\_1,\ldots,a\_n)=1$ and $1 < a\_1< \cdots <a\_n$.
Start with any Bézout identity $x\_1 a\_1+ \cdots + x\_n a\_n = 1$. Using the transformations $x\_n \leftarrow x\_n +... | 10 | https://mathoverflow.net/users/6506 | 108723 | 62,610 |
https://mathoverflow.net/questions/108711 | 0 | I am investigating whether the following hypergraph is $2$-colorable.
Let $0\le c < d < e$ be fixed natural numbers and consider a graph on $2e$ vertices, with the vertices labelled as $0,1,\cdots 2e-1$. For every vertex $u$, whenever $u+x-y$ and $u+z-y$ make sense as a vertex and $x,y,z$ are such that $\{x,y,z\}=\{c... | https://mathoverflow.net/users/14875 | Hypergraph coloring | As far as I understand, your hyperedges are of the form $\{v+c,v+d,v+e\}$ for all suitable $v$. Hence you can just color the vertices from the left to the right. Color the vertices $u\leq e-c-1$ as you wish; then, when you consider some further vertex $u\geq e-c$, there is exactly one edge with the maximal element $u$;... | 2 | https://mathoverflow.net/users/17581 | 108724 | 62,611 |
https://mathoverflow.net/questions/108728 | 3 | Let $X$ be a Riemann surface and let $E$ be a smooth complex vector bundle on $X$ with a connection $D$. We can write the connection $D$ as the sum $D'+D''$ where $D'$ is the (1,0) part and $D''$ is the (0,1) part of the connection. The integrability theorem for holomorphic structures says that if $D''\circ D''=0$, the... | https://mathoverflow.net/users/11395 | Smoothness of solution to a PDE | The equality
$$f= L(f+\alpha f) $$
implies $\newcommand{\pa}{\partial}$
$$\pa\_{\bar{z}} f-\alpha f =\alpha. $$
The operator with smooth coefficients
$$ T:=\pa\_{\bar{z}}-\alpha $$
is elliptic and the above equation has the form
$$Tf=\alpha. $$
The regularity theorem for elliptic operators with smooth ... | 2 | https://mathoverflow.net/users/20302 | 108731 | 62,614 |
https://mathoverflow.net/questions/108734 | 2 | I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200):
**Theorem 1.** Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: \mathcal{A} \rightarrow \mathcal{B}$, $G: \mathcal{B} \rightarrow \mathcal{C}$ be two additive left exact functors. Let $\m... | https://mathoverflow.net/users/21136 | Theorem on composition of derived functors, question about proof | To show that $E$ is an isomorphism of functors, it suffices to show that $E(A)$ is an isomorphism for each object $A$ of $D^+(\mathcal{A})$. This has been shown for each $K^\bullet$ an object of $\operatorname{Kom}^+(\mathcal{R}\_\mathcal{A})$. For an arbitrary object $A$, choose a quasi-isomorphism $f:A\to K^\bullet$ ... | 4 | https://mathoverflow.net/users/1182 | 108736 | 62,616 |
https://mathoverflow.net/questions/108739 | 4 | All the standard examples for model categories are large categories. Is it possible to have a small model category? Are there any interesting examples?
EDIT:
Since a complete small category is a preorder (proposition V.2.3 in MacLane's Categories), I'd be glad to compromise the limit axioms to be as in Quillen's or... | https://mathoverflow.net/users/17083 | Small model categories? | One of Quillen's original examples was the category of chain complexes of finitely-generated modules over a ring – this is obviously equivalent to a small category, and of course, one has to use Quillen's original definition which only required limits and colimits for *finite* diagrams, rather than the usual definition... | 9 | https://mathoverflow.net/users/11640 | 108752 | 62,626 |
https://mathoverflow.net/questions/108754 | 6 | Suppose we have a 3-manifold $M$ and its respective fundamental group $\pi\_1(M)$. An important question about its fundamental group is to ask if it is linear, i.e. they are isomorphic to a subgroup of the Lie group $GL(n,\mathbb{C})$. What is known about the manifold in such a case, i.e. what are the implications give... | https://mathoverflow.net/users/26949 | What is known about a 3-manifold $M$ when its fundamental group is linear? | If you are considering compact 3-manifolds, then it is conjectured that the fundamental groups are always linear, so there should be no restriction on the topology.
One may as well consider 3-manifolds with indecomposable fundamental group. Then the only remaining case to consider is graph manifolds with a non-trivial ... | 11 | https://mathoverflow.net/users/1345 | 108755 | 62,628 |
https://mathoverflow.net/questions/108756 | 1 | Let $R$ be a commutative ring with unity, $I$ be an ideal and $a\in R$ be an element in $R$. We have the following short exact sequence:$$0\rightarrow R/(I:a)\rightarrow R/I\rightarrow R/(I+(a))\rightarrow 0$$ where the injection is multiplication by $a$, and the surjection is the canonical one. Moreover, it is known t... | https://mathoverflow.net/users/nan | Bounding Castelnuovo-Mumford regularity in a short exact sequence | Of course, after I have been thinking about this for three weeks now, I find the answer right when I post the question. I am sorry.
The problem is that the exact sequence above is not a graded exact sequence. We need to shift the grading, and then the estimation is fine.
| 4 | https://mathoverflow.net/users/nan | 108762 | 62,632 |
https://mathoverflow.net/questions/108725 | 12 | Let $\rho$ denote the irreducible algebraic representation of $GL\_n(\mathbb{C})$ with the highest weight $(2,2,\underset{n-2}{\underbrace{0,\dots,0}})$.
Let $k\leq n/2$ be a non-negative integer. How to decompose into irreducible representations the representation $Sym^k(\rho)$?
More specifically, I am intereste... | https://mathoverflow.net/users/16183 | A question on invariant theory of $GL_n(\mathbb{C})$. | The plethysm $\mathrm{Sym}^k \rho$ contains the irreducible representation with highest weight $(2,\ldots,2,0,\ldots,0)$ exactly once. It looks like a tricky problem to say much about its other irreducible constituents.
Let $\Delta^\lambda$ denote the Schur functor corresponding to the partition $\lambda$, and let $E... | 15 | https://mathoverflow.net/users/7709 | 108767 | 62,634 |
https://mathoverflow.net/questions/108764 | 10 | 1. The dihedral group of order $2n+2$ acts on $K\_n$, the ($n-2$)-dimensional associahedron. Are there any other symmetries? References?
2. Does the answer to 1 change if we restrict to just the 1-skeleton of $K\_n$? References?
3. It is "obvious" that any simple circuit (simple closed walk, simple closed path, whateve... | https://mathoverflow.net/users/15980 | Symmetries and faces of the associahedron | The answer to question 1 is no. A reference for this is:
Carl Lee, The associahedron and triangulations of the $n$-gon, European Journal of Combinatorics, 10 (1989), no. 6, 551--560.
The answer to question 3 is yes. I think this is clear from the viewpoint where you think of vertices of the associahedron as triangu... | 15 | https://mathoverflow.net/users/23408 | 108772 | 62,637 |
https://mathoverflow.net/questions/108773 | 16 | The following is meant to be an axiomatization of differential calculus of a single variable. To avoid complications, let's say that $f$, $g$, $f'$, and $g'$ are smooth functions from $\mathbb{R}$ to $\mathbb{R}$ ("smooth" being defined by the usual Cauchy-Weierstrass definition of the derivative, not by these axioms, ... | https://mathoverflow.net/users/nan | Independence of Leibniz rule and locality from other properties of the derivative? | If you grant that $c'=0$ when $c$ is a constant, you can argue as follows:
EDIT: Actually $c'=0$ follows from $1'=0$ using the chain rule, since $c=c\circ 1$.
Let $I(x)=x$. Since $I=I\circ I$, the chain rule gives $(I')^2=I'$, so $I'$ is the characteristic function of a set $A$ of real numbers. Let $T\_c(x)=x+c$. T... | 16 | https://mathoverflow.net/users/6666 | 108804 | 62,643 |
https://mathoverflow.net/questions/108796 | 5 | Many classical arithmetic functions can be thought of as functions on the set of (non-zero) ideals of $\mathbb{Z}$ rather than as functions on $\mathbb{N}$.
Example: For $n \in \mathbb{N}$ the divisor function $d(n)$ is defined to equal the number of divisors of $n$. Equivalently, we could define $d(I)$ to the number... | https://mathoverflow.net/users/26993 | Extending arithmetic functions (and associated Dirichlet series) to arbitrary rings of integers | The answer to your Question 1 is "yes". It's clear that the number of ideals of $\mathcal{O}$ of norm $\le M$ is bounded above by a polynomial in $M$, so one can manipulate Dirichlet series term-by-term for $Re(s) \gg 0$ and argue that
$$ \zeta\_K(s)^2 = \sum\_{A, B} N(A)^{-s} N(B)^{-s} = \sum\_{C} \#\{ (A, B) : AB =... | 8 | https://mathoverflow.net/users/2481 | 108806 | 62,644 |
https://mathoverflow.net/questions/108635 | 2 | I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak\* topology. Is is a Barreled space, say, a space whose convex, balanced, absorbing and closed subsets are neighborhood of the origin?
| https://mathoverflow.net/users/26941 | A question on Schwartz distributions | If the weak$^\*$ dual of a locally convex space $X$ is barrelled then the bounded sets of $X$ are finite dimensional (because the polar of a bounded set $B$ is a barrel which then contains the polar of a finite set $E$ and
the theorem of bipolars implies that $B$ is contained in the absolutely convex hull of $E$). For ... | 2 | https://mathoverflow.net/users/21051 | 108807 | 62,645 |
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