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https://mathoverflow.net/questions/106437 | 1 | I am reading de la Harpe's book "Topics in Geometric Group Theory".
On page 145, there is a theorem:
Let $V$ be a complete $n$-Riemannian manifold with sectional curvature satisfying $K\ge 0$. Then there exist a finite normal subgroup $F\_1$ of the fundamental group $\pi\_1(V)$ and a short exact sequence of the form:
$... | https://mathoverflow.net/users/3922 | Extension of groups in Bieberbach's theorem | There is a ``quasi-isometric rigidity'' context in which a finite kernel arises naturally. I am not sure how general this is so let me state it in the narrower context of constant sectional curvature equal to zero.
Theorem (quasi-isometric extension of the Bieberbach Theorem): If $\Gamma$ is an abstract group quasi-... | 3 | https://mathoverflow.net/users/20787 | 106443 | 61,479 |
https://mathoverflow.net/questions/106430 | 0 | Given three vectors $v\_1,v\_2$ and $v\_3$ in a Hilbert space Hthe follwing is true
$$\angle(v\_1,v\_2)+\angle(v\_2,v\_3)\geq \angle(v\_1,v\_3).$$
It tried to substitute $\angle(v\_1,v\_2) = cos^{-1}\frac{v\_1 \cdot v\_2}{\Vert v\_1 \Vert \Vert v\_2 \Vert}$ but I could not show the resulting inequality. What is the n... | https://mathoverflow.net/users/19874 | Sum of Angles in a Hilbert space | WLOG we can assume that we are in three space as mentioned above and that the vectors have length $ 1$. With the origin $O$ they span a triangular prism $OABC$. Then the area of say $OAB$ is less than the sum of that of $OAC$ and that of $OCB$. If we express these areas in term of the sines of the angles, we get an ine... | 2 | https://mathoverflow.net/users/26013 | 106449 | 61,483 |
https://mathoverflow.net/questions/106448 | 11 | Ostrowski's theorem give the answer for valuations, but is there a complete classification of (at least separated) topologies on Q (compatible with the field operations, obviously)?
| https://mathoverflow.net/users/17164 | Topologies on the field of rationals | According to [this link](http://www.encyclopediaofmath.org/index.php/Topological_field), there are as many field topologies on $\mathbb{Q}$ as there are subsets of $\mathbb{R}$, so I doubt there is a classification. A reference seems to be Wieslaw's book "topological fields" (it doesn't seem to be on google books, unfo... | 19 | https://mathoverflow.net/users/6451 | 106454 | 61,485 |
https://mathoverflow.net/questions/106388 | 1 | Given a curve $\gamma$ in a Banach space $X$ and a function f defined along the curve s.t.
$$\big\Vert f(\gamma(t))-f(\gamma(s))\big\Vert\\leq L\big\Vert\gamma(t)-\gamma(s)\big\Vert$$
is it possible to extend the Lipschitz functions to the whole of $X$?
| https://mathoverflow.net/users/19874 | Extension of lipschitz functions along a curve | It is not always possible to extend when $X$ is a Banach space. Take a Banach space $Y\_n$ which contains an $n$ dimensional subspace $E\_n$ such that every projection from $Y\_n$ onto $E\_n$ has norm at least $C\_n$ with $C\_n\to \infty$. ($Y\_n$ can e.g. be $L\_1$ and $E\_n$ the span of $n$ IID gaussian random variab... | 4 | https://mathoverflow.net/users/2554 | 106459 | 61,487 |
https://mathoverflow.net/questions/106426 | 14 | I am sorry but I am reposting this question because I wasn't logged in when I first asked it.
Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which fibers over the circle with pseudo-Anosov monodromy of the fiber through periodic splitting of train tracks. Such triang... | https://mathoverflow.net/users/24768 | Comparing layered triangulations of 3-manifolds which fiber over the circle. | **Revision 10/3/12**
1. is true. First of all, by a [result of Fried](https://wikis.uit.tufts.edu/confluence/display/~dmarga01/FLP), associated to a fibered face of the Thurston norm on homology, there is a unique (up to isotopy) pseudo-Anosov flow which is transverse to every fibration in (the interior of) the face.... | 9 | https://mathoverflow.net/users/1345 | 106462 | 61,489 |
https://mathoverflow.net/questions/106472 | 4 | Let $F\_\infty$ be the free group on infinitely many generators, and $\phi: \mathbb{Z} \rightarrow$ Aut$(F\_\infty)$ be any group homomorphism.
My question is: If we form the semi-direct product $F\_\infty \rtimes Z$ could this group be free- in particular could it be isomorphic to $F\_\infty$?
Thanks
-Kevin
| https://mathoverflow.net/users/26246 | Could $F_\infty \rtimes Z$ be isomorphic to $F_\infty$? | It's certainly not free for every group homomorphism $\phi: \mathbb{Z} \to Aut(F\_\infty)$; for example, if $\phi$ is trivial, then the semidirect product would be the cartesian product $F\_\infty \times \mathbb{Z}$. This cannot be a free group, because every subgroup of a free group is free, but $F\_\infty \times \mat... | 16 | https://mathoverflow.net/users/2926 | 106473 | 61,493 |
https://mathoverflow.net/questions/106463 | 1 | The Gauss-Newton algorithm optimizes functions
$$
E(x) = \sum f(x)^2
$$
by approximating f as (locally) linear, in which case the Hessian of $E$ is approximated as
$$
H = 2 \sum {J\_f}^T J\_f
$$
Now if I introduce a robust cost function in place of the squared cost above, I can similarly approximate the Hessian... | https://mathoverflow.net/users/19899 | Nonstandard Hessian approximations in Gauss-Newton | You could certainly try this, but you'd have to do a lot of careful analysis to derive any convergence results. Among other things to consider:
1. Your objective E(x) is more likely to have local minima due to the nonconvexity introduced by the logarithms. This could also happen with the sum of squares objective, but... | 1 | https://mathoverflow.net/users/9022 | 106477 | 61,495 |
https://mathoverflow.net/questions/94435 | 3 | I want to know how to find an upper bound of the following expectation taken for both $t$ and $y$ as
$$\mathbb{E}\sup\_{x \in D} \left|\sum\_{k=1}^n t\_k x^T y\_k\right|,$$
where $D$ is the set of vectors defined by
$$D = ( x \in \mathbb{R}^m \mid 0\leq x\_i \leq 1, \forall 1\leq i\leq m ),$$
$\left(t\_k\right)\_{k... | https://mathoverflow.net/users/11870 | How to bound the sup norm of a Rademacher process or equivalently a Gaussian process? | The quantity is of the order $n$ (at least without any additional restriction on the $p\_i$'s).
To see the upper bound:
$$\mathbb{E}\sup\_{x \in D} \left|\sum\_{k=1}^n t\_k x^T y\_k\right| \leq \sum\_{k=1}^n \mathbb{E} \left| 1^T y\_k\right| \leq n$$
where $1^T$ denotes the $1$'s vector.
To see the lower boun... | 0 | https://mathoverflow.net/users/630 | 106487 | 61,500 |
https://mathoverflow.net/questions/105321 | 4 | Hi all,
Currently I'm reading a paper about the geometry of Grassmannians:
www.omup.jp/modules/papers/riemann/04Nagatomo.pdf
In there, the author regards the second fundamental form of the k-dimensional tautological bundle $\tau \rightarrow Gr:=Gr\_k(\mathbb{C}^N)$ over a grassmannian
given by
$H=\pi\_{\tau^c} ... | https://mathoverflow.net/users/21779 | second fundamental form | Don't know if anyone is interested in the (quite simple) answer:
The bundles $Hom(\tau,\tau^c)$ and $TGr$ are not only isomorphic as bundles but also as bundles with their natural connection, i.e. the induced Hom-connection from $\nabla^{\tau}$ respectively $\tau^{\tau^c}$ and den Levi-Civita-Connection.
We have to sho... | 2 | https://mathoverflow.net/users/21779 | 106488 | 61,501 |
https://mathoverflow.net/questions/106482 | 2 | There are well known bounds for $W\_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W\_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, and associated big-O asymptotics of $W\_0$ as $x\rightarrow\infty$.
What about bounds for $W\_{-1}(x)$, the other real-... | https://mathoverflow.net/users/26248 | Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior | First, if $x \in ]-1/e,0[$, one has $1 < -W\_{-1}(x) \leq -\frac{1}{x}$ (it is easy to prove that these inequalities are equivalent to $-1/e < x \leq \frac{\exp(1/x)}{x}$, and that is true here). Then $0 < \ln(-W\_{-1}(x)) \leq -\ln(-x)$.
Since $W\_{-1}(x)\exp(W\_{-1}(x)) = x$, the above inequalities yield $1 < -W\_{... | 4 | https://mathoverflow.net/users/8786 | 106489 | 61,502 |
https://mathoverflow.net/questions/106490 | 0 | Hi all,
I read recently that for any three continuous random variables, X,Y and Z, the conditional densities are related by the following formula:
$p(x|y) = \int g(x| z) h(z | y ) dz $
where $p(x|y)$ is conditional density of X given Y = y and so on.
I have never come across this relation before. It seems sim... | https://mathoverflow.net/users/26249 | Conditional Density of Random Variables | This is not true in general since you would need the additional condition $g(x|z)=p(x|y,z)$ on the conditional densities for that formula to hold true.
As a counter-example, consider $y,z,\epsilon$ all independent standard normal and let $x=y+z+\epsilon$. Then $x|y$ is a normal $N(y,2)$ (conditional expectation $y$ and... | 1 | https://mathoverflow.net/users/25326 | 106493 | 61,504 |
https://mathoverflow.net/questions/106452 | 19 | I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a love of Bitcoin. The point of a senior thesis is to get a student to teach themselves a subject and learn to find and read ... | https://mathoverflow.net/users/10206 | Bitcoin Research | Added: There are financial aspects of bitcoin too.
[With Bitcoin At $60k, Satoshi Nakamoto Is Now One Of The 20 Richest People On The Planet](https://www.celebritynetworth.com/articles/billionaire-news/bitcoin-satoshi-nakamoto-20-richest/) on March 13, 2021.
Satoshi Nakamoto is the **anonymous** "entity" who create... | 17 | https://mathoverflow.net/users/12481 | 106494 | 61,505 |
https://mathoverflow.net/questions/106460 | 5 | On page 3 of [this paper](http://www.imsc.res.in/%7Ekapil/papers/topol.pdf), the authors give a Bruhat cell decomposition of a quadric hypersurface $Q$ of complex dimension $n$. This may be a stupid question, but it doesn't seem clear to me what exactly the decomposition is. They only explicitly describe the real dimen... | https://mathoverflow.net/users/4566 | Bruhat decomposition of a quadric hypersurface | Let $Q$ live in $\mathbb{P}^{2n-1}$, so $Q$ defines a symmetric bilinear form on $\mathbb{C}^{2n}$, and the complex dimension of $Q$ is $2n-2$. Take a flag of isotropic subspaces $0 \subset F\_1 \subset F\_2 \subset \cdots \subset F\_n$. So $\mathbb{P}(F\_i) \subset Q$. For $1 \leq i \leq n-1$, the class of $\mathbb{P}... | 10 | https://mathoverflow.net/users/297 | 106495 | 61,506 |
https://mathoverflow.net/questions/106510 | 4 | Let $S$ be a scheme.
**Definition.** Let $X$ be an $S$-scheme and $G$ a smooth affine group $S$-scheme acting on $X.$ An $S$-scheme $Y$ is a *geometric quotient* of $X$ by $G$ if there exists a morphism $\pi\_X\colon X\rightarrow Y$ such that
1. $\pi\_X$ is $G$-invariant,
2. the geometric fibers of $\pi\_X$ are or... | https://mathoverflow.net/users/nan | Lift of a morphism between geometric quotients | It is not true : for instance, $X$ and $X'$ might be two non-isomorphic $G$-torsors on the same $S$-scheme $T$.
Here is a precise example. Consider $S=T=Spec(\mathbb{R})$ and $G=\mathbb{Z}/2\mathbb{Z}$, let $X=Spec(\mathbb{C})$ and $X'=Spec(\mathbb{R})\cup Spec(\mathbb{R})$ viewed as $S$-schemes, and let $G$ act on ... | 6 | https://mathoverflow.net/users/2868 | 106511 | 61,511 |
https://mathoverflow.net/questions/106481 | 2 | In Jantzen's AMS text 'Lectures on Quantum Groups' he makes the following remark (p.187, preface to Chapter 9):
"For general (complex semisimple f.d. Lie algebra) $\frak{g}$ we can consider for each simple root $\alpha$... a Lie subalgebra (of $\frak{g}$) isomorphic to $\frak{sl}\_{2}$ (ie, the Lie subalgebra $\frak{... | https://mathoverflow.net/users/9970 | A remark in Jantzen's 'Lectures on Quantum Groups' | Your argument in the second to last paragraph is wrong. The basis of the adjoint representation given by any basis vector in each root space, and the basis of $\mathfrak{h}$ given by the simple coroots $H\_i=[E\_i,F\_i]$ has this property for any semi-simple Lie algebra.
The mistake you made was thinking that the bas... | 2 | https://mathoverflow.net/users/66 | 106519 | 61,516 |
https://mathoverflow.net/questions/106528 | 2 | (I find it easier to think in terms of [Grothendieck pretopologies](http://ncatlab.org/nlab/show/Grothendieck+pretopology), instead of topologies. If this annoys any experts, please forgive me.)
Suppose that $C$ is a (full) subcategory of a category $D$. Furthermore, let $J$ be a Grothendieck pretopology on $C$ and l... | https://mathoverflow.net/users/778 | Is there a name for this "weak compatibility" between Grothendieck (pre)topologies? | I would just say that the inclusion preserves covering families (in the naïve sense). You don't need Grothendieck pretopologies to make sense of this – just plain coverages (in the sense of Johnstone; see [*Sketches of an elephant*, §C2.1]) are enough. It's not a good notion though – a better one would say something li... | 1 | https://mathoverflow.net/users/11640 | 106532 | 61,521 |
https://mathoverflow.net/questions/106533 | 1 | It is a well known result from Galois theory that, given a Galois extension $L$ of a field $K$, an element $x$ of $L$ is in $K$ if for all $\sigma$ in $Gal(L/K)$, one has $\sigma(x)=x$.
My question is: is it true that every element $y$ of $L$, such that there exists at least one field homomorphism that doesn't prese... | https://mathoverflow.net/users/13625 | About some kind of "converse" of a theorem from Galois theory | No. Take $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$, $L=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$, $y=\sqrt{3}$, and $s$ the automorphism of $L$ that fixes $\sqrt{3}$ and $\sqrt{5}$, but sends $\sqrt{2}$ to $-\sqrt{2}$.
| 6 | https://mathoverflow.net/users/22989 | 106534 | 61,522 |
https://mathoverflow.net/questions/106536 | 1 | Given a (non-multi)graph $G$ let $N\_G$ be the least number of nodes that must be colored (by a single color) such that every other node in $G$ shares an edge with at least one colored node. (I am only interested in the case where $G$ is an $n \times n$ grid, where such an $N\_G$ obviously always make sense - I don't k... | https://mathoverflow.net/users/13753 | Name for minimum size of a set of vertices S such that every other vertex has a neighbor in S | This is the domination number of the graph.
Here's a wikipedia article:
[en.wikipedia.org/wiki/Dominating\_set](http://en.wikipedia.org/wiki/Dominating_set)
| 5 | https://mathoverflow.net/users/23408 | 106538 | 61,523 |
https://mathoverflow.net/questions/106543 | 5 | Hi,
is there a possibility to classify irreducible representations of (finite) groups over $ \mathbb{Z\_p} $
with $p\neq 2$ a prime? Can one get the number of the irred. representations here? I tried to solve this for "easy" groups like the cyclic or symmetric groups, but did not really get a result. Maybe there is a ... | https://mathoverflow.net/users/26269 | p-adic representations of groups | Usually, "irreducible" means having no subrepresentations. In the integral context, there is no such thing, since for any $\mathbb{Z}\_p[G]$-module $M$, $pM$ is a proper submodule. The right question is to try and classify the indecomposable representations, i.e. those that cannot be written as direct sums of subrepres... | 10 | https://mathoverflow.net/users/35416 | 106548 | 61,528 |
https://mathoverflow.net/questions/106498 | 16 | Let $p$ be a prime number and $N$ a positive integer not divisible by $p$.
For some easy choices of $p$ and $N$, can anybody provide me with explicit examples of collections $$\{f\_k,\quad 2\leq k \leq k\_0 \}$$ of q-expansions such that, for each $k$, $f\_k$ is the $q$-series of a classical eigenform of weight $k$ ... | https://mathoverflow.net/users/18643 | Examples of q-expansions in a Hida family | In the case when the Hida algebra is simply $\Lambda$, one can use *families* of overconvergent modular symbols to compute the $q$-expansion of the Hida family where the coefficients are functions of the weight.
The idea is the following: form a "random" family of overconvergent modular symbols -- that is modular sy... | 16 | https://mathoverflow.net/users/86179 | 106549 | 61,529 |
https://mathoverflow.net/questions/106497 | 25 | The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of:
Non trivial vector bundles over a paracompact non-Hausdorff contractible space
Non trivial vector bundles over a Hausdorff non-paraco... | https://mathoverflow.net/users/26252 | Non trivial vector bundle over non-paracompact contractible space | Let $U$ and $V$ be two copies of the real line and make a space $X$ by gluing them by the identity along the strictly positive half-line: $x\in U$ equals $x\in V$ for $x>0$. Now make a rank one vector bundle over this space by taking a trivial bundle over each of the lines: glue $U\times\mathbb R$ to $V\times \mathbb R... | 26 | https://mathoverflow.net/users/6666 | 106563 | 61,533 |
https://mathoverflow.net/questions/106540 | 2 | A 3-plane field $D$ on a 6-manifold $M$ is **generic** if $[D, D] = TM$. I'd like to do some explicit computations with a general local plane field of this type, and so I want to find a local (quasi-)normal form as possible for these plane fields, given, say, as an explicit triple of 1-forms on $M$, which is as simple ... | https://mathoverflow.net/users/26266 | Local (quasi-)normal form for 3-plane fields on 6-manifolds | You probably can't do much better than this: Given a point $u$, there always exist $u$-centered coordinates $x^1,x^2,x^3,y^1,y^2,y^3$ and functions $F^i(x,y)$ vanishing at $u$ (i.e., at $(x,y)=(0,0)$) such that the plane field $D$ near $u$ is defined by three equations of the form
$$
dy^i - F^i(x,y)\ dx^i = 0,\qquad i ... | 5 | https://mathoverflow.net/users/13972 | 106566 | 61,535 |
https://mathoverflow.net/questions/106568 | 4 | A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth\_{\mathcal{O}\_x}(\mathcal{F}\_x) = \dim\mathcal{O}\_x$ for any $x\in X$, where $\mathcal{O}\_x$ is the local ring of $X$ at $x$.
Is there a simple example of $(X, \mathcal{F})$ such that $\mathcal{F}$ is Cohen-Ma... | https://mathoverflow.net/users/2348 | Cohen-Macaulay sheaves which are not locally free | 1
-
You are right. The Auslander-Buchsbaum-Serre theorem implies that the projective dimension of a CM module over a regular local ring is $0$ and hence a CM sheaf over a regular scheme is locally free.
2
-
It is quite easy to give examples of non-locally free CM sheaves.
**(a)**
It is relatively easy to prove... | 7 | https://mathoverflow.net/users/10076 | 106575 | 61,539 |
https://mathoverflow.net/questions/106570 | 1 | What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself by previously visiting all nearest-neighbor vertices?
Note of clarification (in response to Vincent Beffara's answer): ... | https://mathoverflow.net/users/25881 | The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself | That will depend on your exact setup, *i.e.* on what you mean exactly by a SAW ...
* If the question is "take a simple random walk, how long will it remain self-avoiding", this will behave like a geometric variable because the number of self-avoiding walks is exponential, like $\mu^n$: the probability to survive for ... | 7 | https://mathoverflow.net/users/9430 | 106576 | 61,540 |
https://mathoverflow.net/questions/106564 | 2 | May I ask whether there are good references for computing blowups of the Du Val Singularities? Also, how are these singularities related to the Dynkin diagrams?
| https://mathoverflow.net/users/7780 | Du Val Singularities and Dynkin diagrams references | See [Durfee, Alan H. (1979), "Fifteen characterizations of rational double points and simple critical points" , L'Enseignement Mathématique. Revue Internationale. IIe Série 25 (1): 131–163](http://retro.seals.ch/digbib/view?rid=ensmat-001%3A1979%3A25%3A%3A300)
| 9 | https://mathoverflow.net/users/10076 | 106577 | 61,541 |
https://mathoverflow.net/questions/106585 | 8 | I asked the following question over at [math.stackexchange](https://math.stackexchange.com/questions/191032/when-is-an-orbit-spherical), but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, algebraic group $G$ acting algebraically on... | https://mathoverflow.net/users/9947 | When is an orbit spherical? | This was an active problem 20 years ago. I don't know if it's completely resolved, but more can be said for *affine* spherical homogeneous spaces. (Also---in general, a spherical variety need not be homogeneous, so searching for "spherical homogeneous variety might yield better results.) Try Brion, "Classification des ... | 7 | https://mathoverflow.net/users/5081 | 106587 | 61,549 |
https://mathoverflow.net/questions/106589 | 7 | Is every distance-regular graph vertex-transitive?
| https://mathoverflow.net/users/26286 | Is every distance-regular graph vertex-transitive? | I don't think so. There are examples of even strongly regular graphs with a trivial automorphism group. See a detailed discussion [here](https://mathoverflow.net/questions/41194/are-almost-all-strongly-regular-graphs-rigid).
| 14 | https://mathoverflow.net/users/22051 | 106595 | 61,551 |
https://mathoverflow.net/questions/106594 | 1 | Let $x$ be a positive scalar variable whose time derivative satisfies
$$|\dot{x}(t)|\leq \exp \left\{\left(-\int\_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$
where $|\cdot|$ denotes the absolute value.
Does the above inequality imply that $x(t)$ is a bounded function? (Note that we do not need to know a... | https://mathoverflow.net/users/12734 | Does this variable have an upper bound? | Integrate a few times:
Using that $x > 0$ we have that $|\dot{x}(t)| \leq \exp (0) = 1$ which implies that $x(t) \leq M + t$ for some $M > 0$. Plug it back in we have
$$ |\dot{x}(t)| \leq \exp \left( - \int\_0^t \frac{1}{M+\tau} \mathrm{d}\tau \right) = \exp \left( - \ln (M+t) + \ln M\right) = \frac{M}{M+t} $$
th... | 6 | https://mathoverflow.net/users/3948 | 106599 | 61,555 |
https://mathoverflow.net/questions/106590 | 14 | Let $M$ be a manifold (variety, scheme, your favorite object) and let $N\_1,N\_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N\_1 \cap N\_2 \neq \emptyset$. Denote by $\text{BL}\_{N\_i}M$ the blowup of $M$ along $N\_i, i=1,2$ and let $\text{BL}\_{N\_1}\text{BL}\_{N\_2}M$ be th... | https://mathoverflow.net/users/675 | When do blowups ''commute''? | Sándor is exactly right, this generally isn't true. However, there is a related statement that is always true, instead of blowing up strict transforms of $N\_1$ and $N\_2$, you should blow up total transforms.
---
Blowing up total transforms
---------------------------
Suppose that $I\_1$ and $I\_2$ are the ide... | 14 | https://mathoverflow.net/users/3521 | 106601 | 61,557 |
https://mathoverflow.net/questions/106521 | 4 | Finite [p-groups](http://en.wikipedia.org/wiki/P-group) - have p^n elements by definition. According to WP there is rich structure theory.
**Question:** How far is representation theory of p-groups is understood?
In case this question is too broad let me restrict it to upper triangular matrices over F\_q.
What is ... | https://mathoverflow.net/users/10446 | Representation theory of p-groups in particular upper tringular matrices over F_p | (This answer is only about *representation theory over $\mathbb{C}$*, that is, character theory.) You should take a look at §26: *Characters of $p$-groups* of Bertram Huppert's book *Character Theory of Finite Groups*. In particular, it contains a proof of Isaacs' results that the set of character degrees of a $p$-grou... | 9 | https://mathoverflow.net/users/10266 | 106605 | 61,559 |
https://mathoverflow.net/questions/106609 | 1 | I have a Lipschitz function $X=X(t)$ with the property that, at all points $t$, the right derivative $\lim\_{\epsilon \downarrow 0} \epsilon^{-1}(X(t+\epsilon)-X(t))$ exists and is given by $f(X)$ for some (discontinuous) function $f$. (Of course, at all regulat points, i.e. almost everywhere, this is then the ordinary... | https://mathoverflow.net/users/17883 | Uniqueness of Lipschitz function satisfying differential equation | I suppose you want uniqueness assuming some initial condition.
Even when $F$ is continuous and $X$ is $C^1$ you don't have uniqueness. For $t\geq 0$ $X(t)=0$ and $X(t)=t^2$ are both solution of $X'=2\sqrt{X}$.
Maybe with stronger assumptions on $F$ you can do something assuming only that $X$ is Lipschitz but I'm n... | 2 | https://mathoverflow.net/users/8887 | 106610 | 61,561 |
https://mathoverflow.net/questions/106560 | 292 | Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?
| https://mathoverflow.net/users/5756 | Philosophy behind Mochizuki's work on the ABC conjecture | I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki's claimed proof, his other work, and I... | 185 | https://mathoverflow.net/users/3545 | 106619 | 61,564 |
https://mathoverflow.net/questions/106616 | 0 | Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int\_{\Omega}K(x,y)g(y)dy$
$K:L^2(\Omega)\rightarrow L^2(\Omega) , f , g \in L^2(\Omega) $
1. $ K(x,y) : \Omega\times\Omega\rightarrow R $ where $\Omega$ ... | https://mathoverflow.net/users/26265 | A special Integral Kernel | The function
$$
k(x,y)=\frac{H(x)H(y)}{\pi(x+y)},\quad \text{with}\quad H=\mathbf 1\_{\mathbb R\_+}
$$
is the kernel of the Hardy operator which is bounded on $L^2(\mathbb R)$ with operator norm 1.
| 0 | https://mathoverflow.net/users/21907 | 106621 | 61,565 |
https://mathoverflow.net/questions/106612 | 3 | The geometric mean of two positive definite matrices $A, B$ is defined by $A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$. The following inequality holds true $$\left(\sum\_{i=1}^n A\_i\right)\sharp \left(\sum\_{i=1}^n B\_i\right)\ge\sum\_{i=1}^n A\_i\sharp B\_i$$ for positive definite matrices $A\_i, B\_i$, $i=1\l... | https://mathoverflow.net/users/24492 | Concavity of Spectral mean | Some experiments reveal that the said inequality **does not** hold for the spectral mean.
Here is a random (i.e., mindless) counterexample:
\begin{equation\*}
A\_1=\begin{bmatrix}
29 & 15\\\\
15 & 26
\end{bmatrix},\quad A\_2=\begin{bmatrix}
5 & 0\\\\
0 & 5
\end{bmatrix},\quad B\_1=\begin{bmatrix}
4 & -16\\\\
-16... | 5 | https://mathoverflow.net/users/8430 | 106625 | 61,567 |
https://mathoverflow.net/questions/106634 | 2 | $\DeclareMathOperator\SL{SL}$Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and left invariant metric (like $\SL(n,\mathbb Z)\backslash \SL(n,\mathbb R)$). Then for fixed $q$ the distance function $d(q,.... | https://mathoverflow.net/users/26297 | Sobolev norm of distance function on Riemannian manifold | You should definitely have a look at [this paper of Mantegazza-Mennucci](https://doi.org/10.1007/s00245-002-0736-4), which gives you quite a lot of information on how the regularity of the set $V$ influences the regularity of $d(V,\cdot)$ and $d(V,\cdot)^2$.
For example, in Proposition 4.2 they show that if $V$ is a ... | 3 | https://mathoverflow.net/users/13168 | 106639 | 61,574 |
https://mathoverflow.net/questions/106614 | 4 | Part of the definition of a sc Banach space $E=E\_0 \supset E\_1 \supset \cdots$ is that $E\_\infty = \bigcap\_m E\_m$ is dense in each $E\_m$. Unfortunately, this rules out scales of Hölder spaces $E\_m=C^{m,\alpha}$, since $C^\infty$ is not dense in $C^{m,\alpha}$.
I am wondering: How important is the density requi... | https://mathoverflow.net/users/26291 | Density in sc Banach spaces and polyfold theory | Preliminary response...
My understanding is that, strictly speaking, everything breaks down because sc-Banach spaces are defined (with the density requirement) in the first several pages of HWZ's first polyfold paper, and the hundreds of pages that follow are all predicated on that definition. In other words, it seem... | 2 | https://mathoverflow.net/users/26300 | 106644 | 61,577 |
https://mathoverflow.net/questions/106642 | 3 | I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi\_{|D|}(X)\geq 2$ and even someone just say a pencil of curves etc. It seems these terms have different meanings. My question is the following.
... | https://mathoverflow.net/users/3525 | What's the meaning of pencils in birational geometry? | Well, I think this is kind of an antiquated terminology and I think it is going out of use (hopefully). I believe the classical meaning of a pencil was a 1-dimensional linear system of curves (whose members are connected) on a surface. In other words, calling the equivalent notion a pencil (in arbitrary dimensions) is ... | 5 | https://mathoverflow.net/users/10076 | 106651 | 61,582 |
https://mathoverflow.net/questions/21673 | 26 | It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the secound-countable-condition.
>
> How do you prove that every connected $1$-dimensional manifold homeomorphic to $\ma... | https://mathoverflow.net/users/2841 | Classification of 1-dimensional manifolds (not second-countable) | Here is a response to the first boxed question (the second was already answered by Robin Chapman). (Much belated of course, but I only just saw this question.)
Suppose that $Y$ is a connected (nonempty) topological 1-manifold without boundary; let $y$ be a point. Unless $Y$ is a circle, the complement $Y - \{y\}$ ha... | 17 | https://mathoverflow.net/users/2926 | 106653 | 61,584 |
https://mathoverflow.net/questions/105150 | 4 | Given $A$ a matrix with spectral radius smaller than 1 and a matrix $C$ symmetric.
It can be shown that $U=\sum\_{k=0}^\infty (A^T)^k C A^k$ converges, is symmetric and is the solution of the equation above.
Is it possible to show that if $C$ is non-negative also $U$ is non-negative?
| https://mathoverflow.net/users/24440 | non negative solution of the matrix equation $A^T U A = U - C$ if C is non-negative | Here is an alternative approach to solving the original equation.
The desired equation subject to the constraint $U \ge 0$ (elementwise nonnegativity) can be written as
\begin{equation\*}
U = C + A^TUA,\qquad U \ge 0.
\end{equation\*}
Define now the iteration:
\begin{equation\*}
U\_{k+1} = [C + A^TU\_kA]\_+,\... | 3 | https://mathoverflow.net/users/8430 | 106664 | 61,590 |
https://mathoverflow.net/questions/106660 | 12 | Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was the first realisation of this: i.e. $\mathbb{Q}^{ab}=\mathbb{Q}^{cycl}=\mathbb{Q}(e^{2\pi i \mathbb{Q}})$.
If $K$ is ... | https://mathoverflow.net/users/16858 | Kronecker's Jugendtraum for real quadratic fields? | For totally real quadratic fields, Stark's conjecture (still a conjecture) gives an answer. I quote:
In the case that $k$ is
totally real, Tate (1984, 3.8) determines the subfield they generate; for
example, when $[K:Q]=2$, they generate the abelian closure of
$k$ in $\mathbb{R}$. This has implications for Hilbert's ... | 16 | https://mathoverflow.net/users/22295 | 106667 | 61,592 |
https://mathoverflow.net/questions/106623 | 5 | Let $M$ be $C^{\infty}$-manifold and $x\in M$. We define $(k,r)$-velocity space at x as
$(T\_k^rM)\_x:=J\_0^{r}(\mathbb{R}^k,M)\_x$.Can we define vector space structure on $(T\_k^rM)\_x$?
| https://mathoverflow.net/users/26296 | Vector space structure on velocity space of manifold | Nice answer BS. I was about to post something similar but I didn't have a proof of the non-linearity of the action of $k$-jets of diffeomorphisms. One additional remark that might help the OP is that you can put a vector space structure on $J^k\_0({\mathbb R}^n, M)\_x$ if you have additional structure on $M$. Basically... | 2 | https://mathoverflow.net/users/13762 | 106668 | 61,593 |
https://mathoverflow.net/questions/100008 | 2 | **Heuristic Background**
Consider a set of states labeled $n=1,2,...$ in order of non-increasing probability $p(n)$.
The standard Shannon argument gives meaning to the entropy $S$ of $p$ in terms of the number of states needed to encode the distribution with small error in the limit of many iid copies.
Roughly spea... | https://mathoverflow.net/users/23953 | Error bounds for truncating a probability distribution based on the entropy? | I was able to answer my own question with a little help from my fellow physicists :)
---
**Similar result for Renyi entropy**
I stumbled upon a weaker version of this result before finding Lemma 2 of <http://arxiv.org/abs/cond-mat/0505140> which also gives a simple proof using majorization.
Let $p\_\chi$ deno... | 2 | https://mathoverflow.net/users/23953 | 106671 | 61,596 |
https://mathoverflow.net/questions/106654 | 3 | Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ does there exists an IRREDUCIBLE family of curves in $\mathbb{P}^3$ such that $C$ is a general fiber and there exists a... | https://mathoverflow.net/users/9164 | Deformation of space curves to union of lines | I thought somebody had already answered this question for you in an earlier post. Assuming that by "union of lines (not necessarily reduced)" you mean "connected one-dimensional scheme whose irreducible components are lines", then then answer is yes. Take any codimension $2$ linear subspace $M$ that is disjoint from $C... | 3 | https://mathoverflow.net/users/13265 | 106673 | 61,598 |
https://mathoverflow.net/questions/106682 | 5 | Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two prime factors. (It seems that this would be easier because it removes the parity obstacle.)
I am interested in both uncondit... | https://mathoverflow.net/users/6043 | Large gaps between P2s | Here is a result, due to Halberstam, Heath-Brown, and Richert that seems to be of the flavor you are looking for (*Almost-primes in short intervals*. Recent progress in analytic number theory, Vol. 1 (Durham, 1979), pp. 69–101, Academic Press, London-New York, 1981.)
**Theorem.** *For all sufficiently large $x$ the i... | 10 | https://mathoverflow.net/users/3659 | 106686 | 61,601 |
https://mathoverflow.net/questions/106684 | 9 | Serre duality and the Hirzebruch-Riemann-Roch formula are usually stated for $X$ a smooth projective algebraic variety. Do you know of a reference which proves these results for $X$ smooth and proper?
(1) Since Hirzebruch-Riemann-Roch is true for compact complex manifolds, and Serre duality holds for compact Kähler m... | https://mathoverflow.net/users/3847 | Serre duality and Hirzebruch-Riemann-Roch in the non-projective case | Yes, sure.
1. See Theorem 15.2 (at least in my edition) of Fulton's *Intersection Theory* for Grothendieck-Riemann-Roch for a proper map of smooth varieties. Now take the target to be a point to obtain HRR.
2. This is certainly overkill, but you can find a proof of Grothendieck duality for proper
maps (with finite To... | 10 | https://mathoverflow.net/users/4144 | 106687 | 61,602 |
https://mathoverflow.net/questions/40452 | 8 | The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is that one takes the 2-category of internal groupoids and localises with respect to internal functors $f:X\to Y$ such that
$$X\_1 \simeq X\_0^2\t... | https://mathoverflow.net/users/4177 | Who first came up with the idea of essential/Morita equivalence of internal groupoids/categories? | The stacks studied in the reference [1] are fibrations over a base topos **S** not internal to a site S. The notion of an intrinsic stack originates with Lawvere, where one replace a site S with the site consisting of the class epimorphisms in the topos **S**. In this sense, the reference given is the first one in the ... | 11 | https://mathoverflow.net/users/26318 | 106688 | 61,603 |
https://mathoverflow.net/questions/106695 | 0 | Recall that (for $1\le p<\infty$), $\ell^p = \{\{a\_n\}\_{n=1}^\infty:\sum\limits\_{i=1}^\infty|a\_i|\lt\infty\}$, with norm $||\{a\_n\}||=(\sum\limits\_{i=1}^\infty|a\_i|^p)^{\frac{1}{p}}$.
It is well known that $(\ell^p)^\*\cong\ell^q$ where $\frac{1}{q}+\frac{1}{p}=1$, and so
$$(\ell^p)^{\*\*}=(\ell^q)^\*=\ell^p=(... | https://mathoverflow.net/users/26320 | Double duals characteristic | Note that there exists such a $Y$ iff the sequence
$$0\to X\overset{\varphi}{\to} (X^\\*)^\\*\overset{\eta}{\to} \mathrm{coker}\:\varphi\to 0$$
splits, where $\varphi$ is the canonical injection and $\eta$ the canonical projection. This happens iff there exists an injection $j:\mathrm{coker}\:\varphi\to (X^\\*)^\\*$ su... | 3 | https://mathoverflow.net/users/13832 | 106697 | 61,607 |
https://mathoverflow.net/questions/106678 | 6 | Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell\_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m \circ \lambda\_g = m$ where $\lambda\_g: \ell\_\infty(G) \to \ell\_\infty(G)$ is the left-regular action of $G$) into a... | https://mathoverflow.net/users/18974 | left- and right- Folner sets | Yes, it is obvious. If you have a sequence of probability measures which is approximately invariant on the left, by convolving these measures on the right with the reflected ones you get an approximately invariant sequence of symmetric measures. By applying to this new sequence the standard "slicing" argument (the same... | 6 | https://mathoverflow.net/users/8588 | 106702 | 61,610 |
https://mathoverflow.net/questions/106694 | 8 | Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures. These are $J:TM\to TM$ with $J^2=-\text{id}$. A given $J$ is **$\omega$-tame** when $\omega(v,Jv)>0$, and $J$ is **$\omega$-compatible** when it is $\omega$-tame and $\omega(J\cdot,J\cdot)=\omega(\cdot,\cdot)$. The set of either suc... | https://mathoverflow.net/users/12310 | Almost Complex Structures: 'Tame' versus 'Compatible' | If you assume that $M$ is compact and of dimension $4$ then Donaldson has conjectured that if $J$ is $\omega$-tame then $J$ is $\omega'$-compatible for some symplectic form $\omega'$, see Question 2 [here.](http://arxiv.org/abs/math/0607083) By classical results of Gromov and Taubes this is known to hold if $M=\mathbb{... | 12 | https://mathoverflow.net/users/13168 | 106708 | 61,612 |
https://mathoverflow.net/questions/106705 | 22 | It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex exponentials), or we compute a power series' radius of convergence by thinking in the complex plane (or use complex analy... | https://mathoverflow.net/users/20757 | 2D problems which are easier to solve in 3D | Of course, there is one such problem! This is the Cauchy problem for the wave equation
$$\frac{\partial^2u}{\partial t^2}=\Delta u,\qquad u(x,0)=f(x),\quad \frac{\partial u}{\partial t}(x,0)=g(x),$$
where $x\in{\mathbb R}^d$. To solve it, it is enough to know the case where $f\equiv0$.
If $d=3$, this problem is solve... | 26 | https://mathoverflow.net/users/8799 | 106711 | 61,614 |
https://mathoverflow.net/questions/106677 | 4 | Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$.
Denote by $L\_{[p]} $ the minimal $p$-envelope of $L$ and let ${\mathfrak t}$ be a 2-dimensional torus of $L\_{[p]}$.
For a nontrivial irreducible restricted $L\_{[p]}$-mo... | https://mathoverflow.net/users/17582 | Weights of restricted modules of some Cartan type Lie algebras | Such modules do exist for $p>3$. The Lie algebra $L\_p={\rm Der}\ H(2;\underline{1};\Phi(\tau))^{(1)}$ has a filtration of depth $1$ such that ${\rm gr}(L)$ is a graded Lie algebra of type $\rm H$. This can be seen by embedding it into $W(2;\underline{1})$ as explained in Strade's paper in Can. J. Math. , vol. 43, 1991... | 5 | https://mathoverflow.net/users/24386 | 106718 | 61,618 |
https://mathoverflow.net/questions/106721 | 6 | Hello. I'm thinking about where does the basic quantum mechanics things comes from. I mean the forms of operators and a Shroedinger equation. The more intuitive explanation is better.
To get forms of operators and Shroedinger eq., we can start from assumption that in our representation square of absolute value of wav... | https://mathoverflow.net/users/23834 | Quantum mechanics basics | Probably the explanation via Heisenberg picture is more intuitive (at least for mathematically minded person like me).
Equations of motion in classical mechanics can be described as Hamilton form as follows:
d\dt f = {H, f}
(here { } are Poisson brackets and these equations reduces to standard [Hamilton equation... | 7 | https://mathoverflow.net/users/10446 | 106723 | 61,620 |
https://mathoverflow.net/questions/106719 | 7 | It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally.
My question is, is there a quantitative version of the above result? By this, I mean for example, given $\varepsilon>0$... | https://mathoverflow.net/users/26330 | Nearly constant curvature implies "nearly isometric" to a space form? | This question is two-sided, and I'm not sure what you mean by "local diffeomorphism" so I'll treat both aspects. There is a local and a global version :
Local version :
---------------
>
> **Q1 :** Given a point $x$ in a Riemannian manifold $(M,g)$, can we find a constant curvature metric on a neighborhood of $x$... | 6 | https://mathoverflow.net/users/8887 | 106728 | 61,625 |
https://mathoverflow.net/questions/106676 | 0 | Recently when I read a paper about Fatou component, I met the following theorem which cited in Professor
Eremenko's paper "on the iteration of entire functions"
If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded.
it is really known theorem proved by I.N Baker in his pap... | https://mathoverflow.net/users/11966 | If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded | The correct statement is this. If the Fatou set $F(f)$ of an ENTIRE function has a multiply connected
component, then all components of $F(f)$ are bounded. The idea of the proof is the following.
Let $D$ be a multiply connected component. Then Maximum Principle implies that that the iterates
tend to infinity in $D$, an... | 1 | https://mathoverflow.net/users/25510 | 106733 | 61,628 |
https://mathoverflow.net/questions/106736 | 2 | I have an undirected, unweighted graph representing a network.
I have a starting node and an end one.
My 'network' is reliable if there is no node such that without that node s and t are not reachable i.e. no node is necessary and all nodes have at least one redundant path.
how can I formally verify this?
thank you
... | https://mathoverflow.net/users/26335 | number of totally different path between two nodes in graph theory | If you mean that the network remains connected upon removing any one node, the magic words are "2-vertex-connected graph", or "biconnected graph". An algorithm for determining biconnectivity is described [here](http://garryowen.csisdmz.ul.ie/~cs4115/resources/lect24.pdf), though I am sure there are plenty of other plac... | 4 | https://mathoverflow.net/users/11142 | 106744 | 61,633 |
https://mathoverflow.net/questions/106745 | 14 | Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known to be false in characteristic zero?
| https://mathoverflow.net/users/nan | Characteristic zero and characteristic $p$ in algebraic geometry | Here are two examples.
The moduli space of dimension $g$ principally polarized abelian varieties $A\_g$ contains complete codimension $g$ subvarieties in any positive characteristic $p$ (for instance, the locus of abelian varieties with no nontrivial $p$-torsion points), but not in characteristic $0$ (by Keel and Sad... | 15 | https://mathoverflow.net/users/2868 | 106747 | 61,634 |
https://mathoverflow.net/questions/106729 | 10 | If finite group G has a center how does it influence the representations of this group ?
And vice versa - can we see somehow the center (or some of its properties) from representations (from character table, from ring structure, ... whatever) ?
One has a natural map Z(G)-> G-> G/Z(G),
so we can pull-back representati... | https://mathoverflow.net/users/10446 | Center and representations of finite group - how are related ? | The center of a group and its isomorphism type can be seen from the character table: The center is the set $\newcommand{\Irr}{\operatorname{Irr}}$
$$ Z(G) = \{ g\in G \mid \: |\chi(g)| = \chi(1) \text{ for all } \chi \in \Irr(G)\} .$$
Since $\chi\_{Z(G)}=\chi(1)\lambda$, you see the linear characters of $Z(G)$ in the c... | 6 | https://mathoverflow.net/users/10266 | 106750 | 61,635 |
https://mathoverflow.net/questions/106681 | 5 | An ideal on $\omega\_1$ is *strong* if it is precipitous and the associated generic elementary embedding always maps $\omega\_1$ to $\omega\_2$. This definition is from Baumgartner and Taylor, *Saturation Properties of Ideals in Generic Extensions II*
(available online at <http://www.jstor.org/stable/1998900>.) Every p... | https://mathoverflow.net/users/1682 | Strong ideals that are not pre-saturated | It is consistent that the nonstationary ideal on $\omega\_1$ is strong but not pre-saturated. Baumgartner and Taylor proved in the aforementioned paper that strong ideals are preserved by c.c.c. forcing and asked whether the same is true for pre-saturated ideals. The answer to this question is negative, implying a nega... | 2 | https://mathoverflow.net/users/1682 | 106755 | 61,637 |
https://mathoverflow.net/questions/106765 | 7 | Let $V$ be a finite-dimensional irreducible representation of the Lie group $\text{SL}\_n(\mathbb{R})$. Must $V$ remain irreducible when you restrict the action to $\text{SL}\_n(\mathbb{Z})$? More generally, when you restrict it to other lattices in $\text{SL}\_n(\mathbb{R})$?
| https://mathoverflow.net/users/26059 | Restricting representations to lattices | The result is essentially the statement of Borel's stability theorem for $\mathrm{SL}\_n(\mathbf{R})$, see for example
Theorem 4.39 of the following:
<http://people.uleth.ca/~dave.morris/books/IntroArithGroups.pdf>
Sometimes Borel's stability theorem is phrased in terms of one of the corollaries, which in this case... | 8 | https://mathoverflow.net/users/nan | 106770 | 61,647 |
https://mathoverflow.net/questions/106773 | 2 | This is mostly a reference request. I have integrals of the type
\begin{equation}
\int\_C f(A) (dA)
\end{equation}
where $f$ is a real-valued function of a positive-(semi)definite matrix argument, and the integration region $C$ is an interval in the cone of positive-definite matrices, such as
$C=[0.I]$, where this con... | https://mathoverflow.net/users/6494 | Numerical integration for functions of symmetric matrices | I think that you could transform the problem through the use of tools from Chapter 5 in the following book
* [Eaton (2007). Multivariate Statistics: A Vector Space Approach](http://projecteuclid.org/euclid.lnms/1196285102)
There are explicit examples there for integration over spaces of symmetric positive definite ... | 1 | https://mathoverflow.net/users/25326 | 106778 | 61,652 |
https://mathoverflow.net/questions/106738 | 3 | I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $\mathrm{gpf}(x) \le p$ where $p$ is any prime.
Clearly, as $x$ increases, the distance $d$ between an integer where $\mathrm{gpf}(x) \le p$ and $\mathrm{gpf}(x+d) \le p$ increases at a seem... | https://mathoverflow.net/users/15915 | For any prime $p$, is there $C$ such that if $x\ge C$, then all but one integer among $x+1, x+2, \dots, x+p$ has Greatest Prime Factor $> p$ | This is a summary of Gerhard Paseman's comment.
Suppose that $A$ and $B$ are two numbers with $A-B = c$ small such that $A$ and $B$ are only
divisible by prime numbers less than or equal to $p$. Then one can write
$$A = a x^3, \qquad B = b y^3,$$
where all the prime factors of $a$ and $b$ are less than or equal t... | 4 | https://mathoverflow.net/users/26186 | 106783 | 61,656 |
https://mathoverflow.net/questions/106785 | 3 | Let $X$ be an elliptic curve over a complete local field.
The definition of semi-abelian reduction is: "the Neron model of $X$ is a semi-abelian scheme". On the other hand, the definition of semi-stable reduction is: "the minimal regular model of $X$ is semi-stable."
For elliptic curves, are the two definitions equ... | https://mathoverflow.net/users/5274 | reduction of elliptic curves | The Néron model of $X$ is the smooth locus of the minimal regular model of $X$ (see Bosch-Lütkebohmert-Raynaud: Néron models, §1.5). The equivalence is then clear using the classification of Kodaira-Néron of the types of reduction of the minimal regular model.
Note that if $g(X)\ge 2$, then it is also true that $X$ ... | 12 | https://mathoverflow.net/users/3485 | 106787 | 61,657 |
https://mathoverflow.net/questions/106777 | 0 | In H. Halbertsam and H. Richert's book "Sieve Methods" (Academic Press 1974) the authors go about illustrating the proof of Chen's theorem in Chapter 11 (Pages 321-337). They use a function $S\_0$ defined on line (2.4) at the bottom of page 323. I simply want to know how this $S\_0$ relates to the number of Chen Primes... | https://mathoverflow.net/users/10920 | Chen Primes in Chen's theorem | No, they are different. This sum $S\_0$ is just the contribution of numbers with three prime factors to the counting function used by Chen.
Roughly speaking, Chen first proves a lower bound for a weighted sum (say $S$, which may be different notation from Halberstam-Richert) over primes $p\leq X$ such that $p-2$ has... | 1 | https://mathoverflow.net/users/26352 | 106789 | 61,659 |
https://mathoverflow.net/questions/106775 | 1 | A subset A of a compact metric M is called a $\epsilon$ net if it satisfies the following conditions
(1)$\epsilon$ dense: the neighborhood of A is the entire M
(2)$\epsilon$ separate: $\forall x, y \in A$, $d(x,y)>\epsilon$
It is a well known fact that for any $\epsilon$, there is a fintie $\epsilon$ net.
And... | https://mathoverflow.net/users/20225 | a question about $\epsilon$ net of a compact metric space. | Are you asking whether there is always an upper bound on the cardinality of an $\epsilon$-separated set of points in a compact metric space $X$? If so, the answer is yes.
Find a finite $\epsilon/2$-net $N$. Let $S$ be an $\epsilon$-separated set of points. Then every point of $S$ is in $B\_{\epsilon/2}(x)$ for some $... | 6 | https://mathoverflow.net/users/20598 | 106790 | 61,660 |
https://mathoverflow.net/questions/106786 | 9 | It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):
$$S = \int\_M R \mu\_g,$$
is given by the Einstein's field equation:
$$Ric -\frac{1}{2}R g = 0, $$
where $\mu\_g$ is the canonical volume form given by ... | https://mathoverflow.net/users/16464 | Coordinate-free derivation of the Einstein's field equation from the Hilbert action. | This can be found in Besse "Einstein Manifolds", in chapter 4.
The idea is to use Koszul formula for the Levi-Civitta connection to compute the derivative of the curvature with respect to the metric. Bianchi identities also help.
| 10 | https://mathoverflow.net/users/8887 | 106792 | 61,661 |
https://mathoverflow.net/questions/106767 | 5 | Let $P\_1, P\_2, ..., P\_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:
$D=\displaystyle \prod\_{1\leq i < j \leq n} \overline{P\_iP\_j}$
I wonder...
1) What is the probability density function for $D$? What is the expected value of ... | https://mathoverflow.net/users/25889 | Product of random diagonals on the unit circle | To 2, the answer is yes. This is easy. Re-enumerate points according to their cyclic order. For every fixed $k$ ($1\le k<n$), the product
$$
D\_k:= \prod\_i |P\_i P\_{i+k}|
$$
(where the indices are taken modulo $n$), equals
$$
D\_k=\prod\_i 2\sin (t\_i/2) = 2^n\exp \left(\sum\_i \log\sin(t\_i/2) \right)
$$
where $t\... | 12 | https://mathoverflow.net/users/4354 | 106797 | 61,663 |
https://mathoverflow.net/questions/106793 | 10 | Let $(R, \mathfrak{m})$ be a Noetherian local ring. It is well know that
$R$ is regular iff $pd(R/\mathfrak{m}) < \infty$ (i.e. $R/\mathfrak{m}$ has finite projective dimension).
Assume that $\dim R > 0$. Is $R$ regular, if $pd(R/\mathfrak{m}^2)< \infty$?
| https://mathoverflow.net/users/17901 | regularity of local ring | [Levin-Vasconcelos](http://www.ams.org/mathscinet-getitem?mr=230715) ([journal link](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102986272)): for $R$ a local ring with maximal ideal $\mathfrak{m}$, the existence of a finitely generated $R$-module $M$ such that $\mathfrak{m}M$ h... | 19 | https://mathoverflow.net/users/460 | 106802 | 61,666 |
https://mathoverflow.net/questions/106804 | 7 | It is well-known that the $n$th Catalan number is equal to $(n+1)^{-1}\binom{2n}{n}$. A long time ago I had wondered what happens if you look at the sequence generated by $(n+k)^{-1}\binom{pn}{n}$ - for which $p,k$ is it integral?
I found no other integral-producing values except for $(p,k)=(2,1)$ but then I'm probab... | https://mathoverflow.net/users/22051 | A generalization of Catalan numbers | Yes, $(p,k)=(2,1)$ is the only case when the sequence is integral.
We always need $p\geq 2$. First let's pick a prime $q$ which is $\geq k$ and which divides one of the numbers in $\lbrace pk-1,pk-2,\dots,pk-k\rbrace$. This always exists for $k\geq 2$ by a famous result of Sylvester.
Now taking $n=q-k$, it is easy ... | 16 | https://mathoverflow.net/users/2384 | 106806 | 61,669 |
https://mathoverflow.net/questions/106696 | 2 | Let $\phi(x)$ be a convex polynomial of degree $m$ at least two. Note that for $x,q \in \mathbb{R}$
$$\phi(x) + \phi(q) - 2\phi(\frac{x+q}{2}) =
\sum\_{l=1}^{m/2}\frac{\phi^{(2l)}(\frac{x+q}{2})}{2^{2l-1}(2l)!}|x-q|^{2l}$$
is strictly positive unless $x=q$, because the slopes of secant lines to $\phi$ are increasing.... | https://mathoverflow.net/users/12968 | Best constant in a convex polynomial inequality. | This is just an answer to the new question added in the edit: no, $p\geq q$ need not hold everywhere. Let $p(x) = x^2 + (1+\alpha) x^6$ for some $\alpha\in (0,\frac{1}{4})$ and $q(x) = x^4 + x^6$. Then $p(x)\geq q(x)$ if and only if $\lvert x\rvert\in [0,M\_0]\cup[M\_1,\infty)$ where
\[
M\_0 = \sqrt{\frac{1-\sqrt{1-4\a... | 0 | https://mathoverflow.net/users/5963 | 106811 | 61,673 |
https://mathoverflow.net/questions/106756 | 25 | The Johnson-Lindenstrauss lemma implies that any set of $k$ points in $\mathbb{R}^d$ can be randomly projected into $d' \approx \log(k)/\epsilon^2$ dimensions such that the distances between each pair of points are approximately preserved, up to a multiplicative factor of $(1 \pm \epsilon)$.
My question is whether s... | https://mathoverflow.net/users/26343 | Do random projections (approximately) preserve convexity? | Yes, if the convex body is "sufficiently round". If it is not, the resulting "closeness" to the boundary of a convex set is in absolute terms rather than relative. I don't know whether it can be improved, but Bill Johnson's remarks suggest that it can't.
Let $X=\{x\_i\}$, $i=1,\dots,k$, be the set in question and $K$... | 17 | https://mathoverflow.net/users/4354 | 106828 | 61,681 |
https://mathoverflow.net/questions/106827 | 1 | The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by
$$
\phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu
$$
(or, so says Wikipedia). How does one recover the pdf from $\phi$, i.e., what is the Fourier inversion formula for measures on $\mathbb{C}$? The $\mu$ I am working with is a... | https://mathoverflow.net/users/26362 | Fourier inversion formula for complex-valued random variables? | you have to take the real part of $\bar{t}X$ in the exponent, in order for the integral to make sense; if you then decompose $t=t\_1+it\_2$ and $X=X\_1+iX\_2$ into real and imaginary parts, you just have a conventional two-dimensional Fourier transform
$\phi(t\_1,t\_2)=\int\_{-\infty}^{\infty}dX\_1 \int\_{-\infty}^{\... | 1 | https://mathoverflow.net/users/11260 | 106832 | 61,682 |
https://mathoverflow.net/questions/106818 | 1 | Do there exist results concerning preservation or not of the Morse index of a symmetric matrix $A$, after permuting its diagonal entries, and keeping fixed the off--diagonal ones?
Thanks!
| https://mathoverflow.net/users/24998 | Morse index and permutation of diagonal entries of a symmetric matrix | You can see that the signature is not preserved by this operation considering a matrix $A$ which is a direct sum of two matrices respectively of orders 2 and 1. Say that the off-diagonal entries are all zero but $a\_{12}=a\_{21}=1$.For instance, it is definite positive iff $a\_{11}+a\_{22} > 0$ $a\_{11}a\_{22} > 1,$ an... | 1 | https://mathoverflow.net/users/6101 | 106837 | 61,684 |
https://mathoverflow.net/questions/106826 | 11 | Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl\_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "*Algorithms in Invariant Theory*", [Bernd Sturmfels](http://math.berkeley.edu/~bernd/index.html) describes an algorithm he calls *Hilbert's Algori... | https://mathoverflow.net/users/26361 | Algorithms in Invariant Theory | [Magma](http://magma.maths.usyd.edu.au/magma/) (not free, unfortunately) is capable of this. The method [`FundamentalInvariants(R) : RngInvar -> RngMPol`](http://magma.maths.usyd.edu.au/magma/handbook/text/1197#12744) should do what you need.
| 3 | https://mathoverflow.net/users/6263 | 106840 | 61,685 |
https://mathoverflow.net/questions/106839 | 5 | Dear collegues, I am writing an overview paper for an academic journal, where I also need to state theorems proved by other authors. Usually I cite the source and then rephrase the theorem. However in a few cases rephrasing seems counterproductive and the optimal formulation was already given by the author. Is it OK to... | https://mathoverflow.net/users/26365 | proper quoting of theorems | Depending on the style of your exposition, one possibility for distinguishing rephrased from verbatim results might be to say something along the lines of "In [reference], So-and-So proved a theorem that can be stated as follows: ..." on the one hand, and "In [reference], So-and-So stated and proved the following theor... | 0 | https://mathoverflow.net/users/15837 | 106842 | 61,687 |
https://mathoverflow.net/questions/106836 | 5 | I'm a computer vision student, and I'm looking for some symmetric group literature guidance. I'm going to provide some context, and finally ask two questions.
The Cayley distance and other distances on permutations ($S\_n$) are quite useful in applied machine learning; see the beginning of [this paper](http://www.mat... | https://mathoverflow.net/users/24144 | Distances on generalizations of the symmetric group | I think the generalization you want is a total preorder, or equivalently a [strict weak ordering](http://en.wikipedia.org/wiki/Strict_weak_ordering). On finite sets, these correspond to the orderings of values of a function from the set to $\mathbb{R}$. In a total preorder, if $f(x) = f(y)$ then you say $x \le y$ and $... | 4 | https://mathoverflow.net/users/2954 | 106843 | 61,688 |
https://mathoverflow.net/questions/106835 | 1 | A corollary of Euler's formula tells us that the edge-vertex graph of every convex 3-polyhedron must have a face with either 3, 4, or 5 edges, using an argument about the average degree of vertices.
Does it imply anything further about the exact degree of the vertices of these faces?
In particular, is at least one ... | https://mathoverflow.net/users/26363 | Does Euler's formula imply bounds on the degree of vertices in a 3-polytopal graph? | It does follow from Euler's theorem that every graph of a 3-polytope has either a triangle face or a vertex of degree 3. To see this note that if every face has 4 or more edges then $4F \le 2E$ (double count pairs (e,f) where e is an edge f is a face and f contains e.) and now look at Euler's theorem: $2V-2E+2F=4$ so $... | 3 | https://mathoverflow.net/users/1532 | 106850 | 61,692 |
https://mathoverflow.net/questions/106848 | 29 | While prime numbers are central objects in mathematics it looks that they were ignored and forgotten for long periods of time. I am interested to get some facts and insights about this matter, in particular:
1) Were prime numbers studied in ancient times only by the ancient Greeks? At what periods were they studied b... | https://mathoverflow.net/users/1532 | At what times were people interested in prime numbers | The [Liber Abaci](http://en.wikipedia.org/wiki/Liber_Abaci) (1202) of Fibonacci contains a chapter on perfect numbers and Mersenne primes (of course Mersenne came much later, but possibly slightly before Fermat; he is born slightly before Fermat but is essentially a contemporary).
I do not know if there are any new ... | 13 | https://mathoverflow.net/users/nan | 106851 | 61,693 |
https://mathoverflow.net/questions/106816 | 10 | My question derives from reading a recent preprint (arXiv:1209.0827v1, in particular Section 4.1), but it can be phrased quite independently from that paper. The setup is as follows.
Let $A$ be the tridiagonal $n\times n$ matrix with $a\_{ii} = -1$ on the main diagonal and $a\_{i,i+1} = a\_{i+1,i} = 2$ on the other t... | https://mathoverflow.net/users/26359 | Linear system of equations with nonnegative solutions and a recursion rule | Yes there are infinitely many such values of $n$ and the sequence satisfies the rule you observed. The proof is straightforward but technical.
Let $x\_1,\dots,x\_n$ be the solution. Add $x\_0=0$ and $x\_{n+1}=0$, then $2x\_{k-1}-x\_k+x\_{k+1}=1$ for all $k=1,\dots,n$. Introduce $y\_k=3x\_k-1$, then $y\_k$ satisfies a... | 9 | https://mathoverflow.net/users/4354 | 106857 | 61,696 |
https://mathoverflow.net/questions/106743 | 3 | Tian has defined bisectional curvature for unit and perpendicular tangent vectors $X,Y$ as follow
$$R(X,Y,X,Y)+R(X,JY,X,JY).$$
If bisectional curvature be constant, is there any relationship between sectional curvature and bisectional curvature?
In this case (bisectional curvature is constant) what we can say about ... | https://mathoverflow.net/users/15466 | Relationship between sectional curvature, bisectional curvature and conjugate points | Since you seem to have some basic misunderstandings about these concepts, I will try to clarify the definitions.
Let $(M,J)$ be a complex manifold, with a Riemannian metric $g$ which is Hermitian, i.e. $g(JX,JY)=g(X,Y)$ for all $X,Y\in TM$ real tangent vectors. Assume furthermore that $g$ is Kähler, i.e. that the re... | 22 | https://mathoverflow.net/users/13168 | 106861 | 61,699 |
https://mathoverflow.net/questions/106875 | 2 | I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem):
If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and complete) and we had $\pi\_1(S)\cong \mathbb{Z}\ $ then the surface would have an infinite volume which is a contradiciton. Hen... | https://mathoverflow.net/users/26369 | Volume of a Riemannian manifold and its relation to fundamental group | Hyperbolic surfaces are of the form $\mathbb{H}^2/\Gamma$, where $\Gamma$ is a Fuchsian group which is isomorphic to the fundamental group. If $\Gamma \cong \mathbb{Z}$, then it is generated by a single element $\gamma$. The order of $\gamma$ is infinite, so it cannot be elliptic. It is thus either parabolic (and hence... | 6 | https://mathoverflow.net/users/26059 | 106879 | 61,708 |
https://mathoverflow.net/questions/106876 | 2 | If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?
I know that in this case the universal cover is the hyperbolic plane and it has a complete metric. Do we project this metric to the puntured surface? If so, why is it complete?
I will dee... | https://mathoverflow.net/users/26369 | Complete metric on a Riemann surface with punctures | A good reference for this is, say, Kobayashi and Nomizu "Foundations of Differential Geometry". The result you are looking for is: If $M$ is a complete Riemannian manifold and
$p: M\to M'$ is a (locally) isometric covering map to another Riemannian manifold, then $M'$ is also complete. To prove this note that every ge... | 6 | https://mathoverflow.net/users/21684 | 106890 | 61,714 |
https://mathoverflow.net/questions/87674 | 102 | Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? The argument there is extraordinarily complicated and a simpler proof would be desirable.
Is there evidence from any o... | https://mathoverflow.net/users/21179 | Independent evidence for the classification of topological 4-manifolds? | After 7 months, over 800 MO views and (as suggested) emails to experts, the answer to the question is "No": other than Freedman's 1982 paper, there is no evidence what-so-ever that topological 4-manifolds are so much simpler (i.e. determined up to homeomorphism by their intersection form) than smooth 4-manifolds. Subse... | 54 | https://mathoverflow.net/users/21179 | 106894 | 61,716 |
https://mathoverflow.net/questions/106878 | 10 | The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved (a set of $n$ points in high
dimensional Euclidean space can be mapped into an $O(\log (n)/ \varepsilon^... | https://mathoverflow.net/users/24193 | Non-probabilistic proof of the Johnson–Lindenstrauss lemma | You might be interested in something Jelani Nelson wrote me in an email on Oct. 13, 2011:
"Another notion of derandomizing JL is the following: come up with a
distribution over embeddings that can be sampled using as few random
bits as possible so that, for any vector $x$ in $R^d$, a random vector has
its $\ell\_2$ n... | 11 | https://mathoverflow.net/users/2554 | 106906 | 61,719 |
https://mathoverflow.net/questions/106895 | 0 | Seeley's extension operator it is a linear continuous mapping $E: C^\infty([0, \infty)) \rightarrow C^\infty(\mathbb R) $ with the property $E(g)|\_{[0, \infty))}=g$ for $g \in C^\infty([0,\infty))$.
Let $(a\_n), (b\_n)$ are real sequences
s.t.
* $b\_n<0$, $b\_n \rightarrow -\infty$,
* $ \sum\_{n=1}^\infty |a\_n|... | https://mathoverflow.net/users/26382 | Extending operator | Why isn't this trivial? For example, take a smooth cutoff function $\chi(t)$ which equals $1$ for $t \leq c/2$ and vanishes for $t \geq 3c/4$. Then let $E\_{c,\chi}(g) = E (\chi g) + (1-\chi)g$ where $E$ is Seeley's original operator.
| 3 | https://mathoverflow.net/users/17969 | 106912 | 61,722 |
https://mathoverflow.net/questions/106909 | 4 | Suppose that $G$ is a finite group such that $(|G|, 15)=1$. Why $G$ is solvable group?
| https://mathoverflow.net/users/25390 | Finite solvable group | We note that the Suzuki groups are only non-Abelian simple groups of order
prime to $3$ and $5$ is a prime divisor of the Suzuki groups. Let $G$ be
unsolvable group, then $G$ has the following normal series:
$1\unlhd K\lhd M\unlhd G$ such that $M/K$ is a non-Abelian simple group (and or $M/K\cong S\times $ $
S\cdot \cd... | 5 | https://mathoverflow.net/users/26052 | 106914 | 61,724 |
https://mathoverflow.net/questions/106838 | 48 | Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it necessarily true that $K \cong L$?
I'm also interested in the corresponding question for the special linear group in place of t... | https://mathoverflow.net/users/3040 | Isomorphic general linear groups implies isomorphic fields? | The answer is "yes", see below.
Dieudonné in his book "La géométrie des groupes classiques"
considers the abstract group $SL\_n(K)$ for a field $K$, not necessarily commutative,
and writes $PSL\_n(K)$ for $SL\_n(K)$ modulo the center.
In Ch. IV, Section 9, he considers the question whether $PSL\_n(K)$ can be isomorph... | 41 | https://mathoverflow.net/users/4149 | 106916 | 61,726 |
https://mathoverflow.net/questions/106905 | 4 | Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows:
1. build two finite automata $M\_R$ and $M\_S$ such that $L(R) = L(M\_R)$ and $L(S) = L(M\_S)$
2. build an automaton $M$ such that $L(M) = (L(M\_R) - L(M\_S)) \cup (L(M\_S) - L(M\_R))$
3. test empt... | https://mathoverflow.net/users/24769 | Deciding equivalence of regular languages | If $M\_R$ and $M\_S$ have $m,n$ states respectively, then one has that $L(M\_R)=L(M\_S)$ iff they have the same words of length at most $mn-1$. This is essentially the content of your algorithm. Suppose that they are different and let $w$ be a minimal length word accepted by one of the machines and not the other. If $w... | 2 | https://mathoverflow.net/users/15934 | 106932 | 61,732 |
https://mathoverflow.net/questions/106933 | 11 | Given open map f: $R^n$ to $R^n$ such that each open set $U\in R^n$, $f(U)$ is also open. Are Borel sets in $R^n$ preserved under f?
Motivation: Pre-image of Borel sets under continuous map is a Borel set in $R^n$.
The problem of the analogous statement above is when $U$ and $V$ are open, $f(U\cap V) \subset f(U)\c... | https://mathoverflow.net/users/23835 | Borel sets preserved under open maps? | Every [analytic set](http://en.wikipedia.org/wiki/Analytic_set) ($\Sigma^1\_1$ set) of reals is the projection of a Borel subset of $\mathbb{R}\times\mathbb{R}$, and the projection map $p(x,y)\mapsto x$ is an open map. So the standard examples of non-Borel $\Sigma^1\_1$ sets are also examples where Borel sets are not p... | 14 | https://mathoverflow.net/users/1946 | 106936 | 61,735 |
https://mathoverflow.net/questions/106931 | 1 | Hi all
I've the following non-linear PDE
$-\Delta Y + Y^3 =U$ on $\Omega \subset R^n $, open, bounded, Lipschitz boundary domain
$Y=0 , $ on $\partial\Omega$
**1**.Let $Y\in H\_0^1 $ and as $H\_0^1 \hookrightarrow \hookrightarrow L^5 $ can we define a compact operator $T:L^5 \times [0,1] \rightarrow L^5 $ ... | https://mathoverflow.net/users/26265 | Existence of solution of a Non-linear PDE via Fixed point theorem | The Leray-Schauder is a fixed point theorem in the spirit of Brouwer's. When it gives an existence result, it says nothing about uniqueness. Your problem is much better than that, because it does have a unique solution in $H^1\_0(\Omega)\cap L^4(\Omega)$, whenever $U$ belongs to the dual space $X=H^{-1}(\Omega)+L^{4/3}... | 4 | https://mathoverflow.net/users/8799 | 106950 | 61,742 |
https://mathoverflow.net/questions/104595 | 12 | $\DeclareMathOperator\Ext{Ext}$This question is related to [Are the supports of $Ext^i(M,N)$ eventually periodic?](https://mathoverflow.net/questions/32310/are-the-supports-of-extim-n-eventually-periodic) .
Let $(R, \mathfrak{m}, k)$ be a Noetherian local ring of dimension $d$. It well known that
**Theorem** $R$ is... | https://mathoverflow.net/users/17901 | Vanishing of $\mathrm{Ext}^i_R(N, R)$ | No.
Let's work with an Artinian local ring $R$. Let $X$ be a module satisfying $\mathrm{Ext}\_R^i(X,R)\neq 0$ for all $i$. (These are plentiful; for example, assume $R$ is non-Gorenstein and let $X$ be the residue field.) Define a module $Q$ by "Serre's trick": take generators $\chi\_1, \dots, \chi\_r$ for $\mathrm{E... | 5 | https://mathoverflow.net/users/460 | 106954 | 61,744 |
https://mathoverflow.net/questions/106945 | 3 | Consider two sets M, N such that C[M] is isomorphic to C[N] as representations of G, somewhat surprisingly it does not imply M, N are isomorphic as G sets. (Everything is finite.)
However it does imply that 1) |M| = |N| 2) number of orbits in M = number of orbits in N, because orbits correspond to invariant functions... | https://mathoverflow.net/users/10446 | Sets M,N with G action such that C[M] = C[N] as G modules, how are they related ? | An old paper of mine [On exponentiation of G-sets, Discrete Math. 135 (1994) 69-79] contains, among other things, the result that the free complex-linear actions generated by two $G$-sets $X$ and $Y$ are isomorphic if and only if the power sets, $P(X)$ and $P(Y)$, with the obvious induced actions of $G$, are isomorphic... | 5 | https://mathoverflow.net/users/6794 | 106958 | 61,746 |
https://mathoverflow.net/questions/106943 | 23 | What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.
| https://mathoverflow.net/users/15819 | Who named it the Snake Lemma? | I suspect the name just arose naturally (for obvious reasons) but that it would be tough to trace back to any single person. After Cartan-Eilenberg proved it in 1956 (Homological Algebra, p.40) the first mention I see in English is by Tate in 1966/67 (p-divisible groups, p.178) followed by Hartshorne in 1968 (Cohomolog... | 24 | https://mathoverflow.net/users/22971 | 106962 | 61,749 |
https://mathoverflow.net/questions/106941 | 2 | Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta\_M)$ and $(N,\Delta\_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor product $M \otimes N$, defined by
$$
(\Delta\_M \otimes \Delta\_N)(m \otimes n):= m\_{(-1)}n\_{(-1)} \otimes m\_{(0)} \otim... | https://mathoverflow.net/users/3787 | Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products | Unless I am mistaken, what's going on is the following. As vector spaces, there is a canonical isomorphism $M \otimes N \cong N \otimes M$. But this is not generally a map of $H$-comodules. Given left $H$-comodules $M$ and $N$, I will write $\overline M$ and $\overline N$ for the corresponding right $H$-comodules that ... | 3 | https://mathoverflow.net/users/78 | 106964 | 61,750 |
https://mathoverflow.net/questions/106965 | 6 | May assume field $k=\mathbb{C}$.
Let $X$ be an affine variety and $G$ be a reductive group (may assume connected).
Is the ring of invariants $k[X]^G$ integral closed in $k[X]$?
The claim may not true in general, probably we may assume $X$ is normal and the affine quotient $X/G$ also normal.
Any reference?
| https://mathoverflow.net/users/16762 | Is $k[X]^G$ integral closed in $k[X]$. | Yes, if $G$ is connected. Let $f$ be a function in $k[X]$ which satisfies a polynomial equation over $k[X]^G$. Everything in the $G$-orbit of $f$ must satisfy that polynomial equation. Thus the $G$-orbit is finite, because it is contained in the set of roots of a polynomial, and is connected because it is the orbit of ... | 21 | https://mathoverflow.net/users/18060 | 106967 | 61,752 |
https://mathoverflow.net/questions/106975 | 1 | Let $X$ be a regular, irreducible projective scheme, of finite type over an arbitrary field $k$. Both Weil divisors and Cartier divisors are defined on $X$ and naturally correspond. If $k$ is algebraically closed (but of arbitrary characteristic), then we know (Hartshorne, Prop II.7.7) that global sections of $\mathcal... | https://mathoverflow.net/users/25854 | Linear systems over non-algebraically closed field | Dear LMN, I think the answer to all your questions is *yes*. Everything is true.
See for example, Proposition 2.9 in Hartshorne's *Generalized divisors on Gorenstein schemes* and for some corrections, see Remark 2.9 Hartshorne's *Generalized divisors and biliaison*.
Here's a statement (more general versions can be... | 3 | https://mathoverflow.net/users/3521 | 106977 | 61,755 |
https://mathoverflow.net/questions/106923 | 2 | Let a gaussian circle $C\_R$ be any circle defined by the equation:
$$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mathbb{Z}^2$). IF $R$ contains $n$ prime factors, it is not hard to show that $C\_R$ contains $2^{n+2}$ integer points organized in $2^n$ cla... | https://mathoverflow.net/users/22279 | Groups of Rational Points on Gaussian Circles | Answering mainly about "any nontrivial rational point".
Don't think factoring becomes easy even if an oracle gave you one *integral* point.
For some numbers such oracle come for free, e.g. for $R=u^2+1$ the point is $(u,1)$.
Fermat numbers are of such form and they are hard to factor.
Since your problem is a gen... | 3 | https://mathoverflow.net/users/12481 | 106987 | 61,758 |
https://mathoverflow.net/questions/106893 | 3 | Let $C$ be a curve of genus 3 and suppose that it admits a branched cover $\varphi:C\rightarrow E$ with $E$ elliptic and such that $\varphi$ does not factor through any \' etale cover. Then the degree of the ramification divisor $R$ is $4$ and the branch divisor $B\subseteq E$ is supported in a set of cardinality betwe... | https://mathoverflow.net/users/6951 | Configuration of the branch locus of a branched covering of an elliptic curve | According to Sofia's and my comments to the original question, we look for instance for elements $a,b\in S\_n$ which generate a primitive subgroup of $S\_n$ and where the commutator $[a,b]=a^{-1}b^{-1}ab$ is a $5$-cycle.
This is possible for each $n\ge5$: Set $a=(1,2\,3,\cdots,n)$, $b=(1,2,\cdots,n-4,n-3,n-1,n-2)$. T... | 2 | https://mathoverflow.net/users/18739 | 107002 | 61,768 |
https://mathoverflow.net/questions/107007 | 4 | [CHSH inequality](http://en.wikipedia.org/wiki/CHSH_inequality) requires both *locality* and *realism*. I will equate here realism with [counterfactual definiteness](https://en.wikipedia.org/wiki/Counterfactual_definiteness).
Now counterfactual definiteness tells us that given two different measurements on the same o... | https://mathoverflow.net/users/14976 | Tracking down locality assumption in CHSH inequality | Suppose that:
1) $A\_2$ and $B\_2$ have some fixed joint probability distribution.
2) When $A\_2$ is measured, we have $A\_1=B\_1=A\_2$. This gives a well-defined joint probability distribution for $(A\_1,B\_1)$.
3) When $B\_2$ is measured, we have $A\_1=-B\_1=B\_2$. This also gives a well-defined joint probabili... | 5 | https://mathoverflow.net/users/10503 | 107014 | 61,774 |
https://mathoverflow.net/questions/106940 | 3 | Assume $C\subset \mathbb{P}^2$ is a smooth cubic curve. Then there is a cyclic triple cover $\pi: S\rightarrow \mathbb{P}^2$ ramified in $C$. Let $\sigma$ be the covering automorphism (sheet interchange automorphism) associated to $\pi$, i.e. $S/<\sigma>\cong\mathbb{P}^2$, here $\sigma^3=id$. If $H:=\pi^{\*}l$ denotes ... | https://mathoverflow.net/users/3233 | How to find the action of an automorphism on the 27 lines on a cubic surface? | It is true that the surface obtained is a special one (for example, a general cubic surface admits no automorphism). In fact, taking coordinates $X,Y,Z$ on $\mathbb{P}^2$, the cubic curve is given by $F(X,Y,Z)$ for some polynomial of degree $3$ and the equation of the surface is $W^3=F(X,Y,Z)$. The automorphism corresp... | 5 | https://mathoverflow.net/users/23758 | 107025 | 61,779 |
https://mathoverflow.net/questions/107017 | 3 | Let $Y \to X$ be a finite morphism of schemes of dimension $n$. The induced map on the top cohomology $H^n(X, \mathcal O\_X) \to H^n(Y, \mathcal O\_Y)$ is always surjective. When is it injective? Can one achieve this by adding reasonable conditions? We may restrict ourselves to projective varieties over an algebraicall... | https://mathoverflow.net/users/22823 | When does a finite morphism induce isomorphism on cohomology? | I don't think that map is always surjective. For example, suppose that $Y = X$ is a supersingular elliptic curve. Then the Frobenius map $H^1(X, O\_X) \to H^1(Y, O\_Y)$ is the zero map (and both are 1-dimensional vector spaces). See for example Hartshorne's chapter on elliptic curves.
Alternately, suppose that $X$ i... | 11 | https://mathoverflow.net/users/3521 | 107028 | 61,782 |
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