parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/120197 | 7 | Does anyone know of any computer calculations of the E2-term of the Adams-Novikov spectral sequence at p=2?
I'd love to get my hands on this data.
| https://mathoverflow.net/users/13856 | Adams-Novikov spectral sequence at p = 2 | I don't think anybody knows how to compute this $E\_2$-term efficiently (not just at the prime $2$). I would love to be proved wrong on this, of course.
So far the only documented, algorithmic method that has any chance to be computationally succesful seems to be the method described by Zahler in 1969/1970.
You co... | 11 | https://mathoverflow.net/users/8824 | 120204 | 67,688 |
https://mathoverflow.net/questions/120177 | 8 | A friend and I were reading up on the classification of compact surfaces when we realized that the minimal number of triangles in a triangulation of the sphere, torus, and projective plane are 4 (routine), 7 (an exercise in several places, such as in Katok - Lectures on Surfaces), and 6 (using a similar counting trick ... | https://mathoverflow.net/users/81883 | coincidence between minimal triangulation numbers and chromatic numbers | The answer to your "is it obvious" is "no". But it is a theorem in many cases. Wikipedia + one click takes to you "Solution of the Heawood map coloring problem" by Ringel and Youngs. There they completed the proof that Heawood's bound on the chromatic number was tight for surfaces
with positive genus. As they explain i... | 9 | https://mathoverflow.net/users/1266 | 120205 | 67,689 |
https://mathoverflow.net/questions/120178 | 2 | Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X\_i$ bounded in $L\_1$, that is $\mathbb E\left|X\_i\right|\leq M$ for all $i$. Suppose that they converge in distribution to $X$ (which by Fatou's lemma will also be in the $M$-ball of $L\_1$). This is not enou... | https://mathoverflow.net/users/30746 | Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable? | You need uniform integrability. To change notation a bit, suppose $X\_i$ and $X$ have distributions $F\_i$ and $F$, respectively. Suppose for now only that $F\_i(x) \rightarrow F(x)$ at each continuity point $x$ of $F$. The following are from Billingsley's Convergence of Probability Measures, Theorems 3.5 and 3.6: (1) ... | 1 | https://mathoverflow.net/users/26459 | 120207 | 67,691 |
https://mathoverflow.net/questions/120187 | 5 | I have an optimization problem where I need to select $k$ integers over the interval $[0, N]$ s.t. I maximize the minimum difference between any pairwise sum of the $k$ integers (where we also include the sum of one selected integer with itself). For example, if $k = 3$, $N = 3$, and we select the set of integers $(1, ... | https://mathoverflow.net/users/30997 | Selecting $k$ integers from an interval $[0, N]$ to maximize the minimum difference between pairwise sums | This is in part a bit informal, but I hope it is still of interest.
First, let us recall a somewhat related property: a subset $S$ of $\lbrace 1, \dots, n \rbrace =: [[1,N]]$ is called a Sidon set if all pairwaise sums of elements of $A$ are distinct, i.e., if your minimal difference is non-zero.
This is a very we... | 4 | https://mathoverflow.net/users/nan | 120208 | 67,692 |
https://mathoverflow.net/questions/120209 | 2 | Well, I need someone here with programming skills (because I have none of it) to check if this problem that I am proposing is at least true for the known Mersenne primes, and here is the list of the exponents of the known Mersenne primes :
<http://wwwhomes.uni-bielefeld.de/achim/mersenne.html>
And the problem is:
... | https://mathoverflow.net/users/30730 | Mersenne primes problem | No programming skills are needed, not even a computer, or even pocket-calculator ;)
Look, $M\_p = 2^p - 1$ is congruent $1$ or $13$ modulo $18$ (which is what you are asking) if and only if $2^p$ is $2$ or $14$ modulo $18$.
Now, modulo $18$, one has $2^1=2$, $2^2=4$, $2^3= 8$, $2^4= 16$, $2^5= 14$, $2^6=10$, $2^7... | 3 | https://mathoverflow.net/users/nan | 120213 | 67,693 |
https://mathoverflow.net/questions/120150 | 21 | I will ask the question first and then explain.
QUESTION: FPA can prove its own consistency in the Godelian sense. But can it really prove its consistency?
FPA is a multi-sorted first-order theory, with lower-case or small letters (for numbers) and upper-case or big letters for relations of n-arity (n >= 1). (Pract... | https://mathoverflow.net/users/20716 | Can FPA really prove its consistency? | In principle, the answer can depend on the proof system, but as long as you stick to some of the usual Hilbert-style or sequent proof systems, this shouldn’t matter.
First, as explained in <https://mathoverflow.net/questions/120106>, the question is equivalent to provability of the consistency of FPA in $I\Delta\_0+\... | 20 | https://mathoverflow.net/users/12705 | 120216 | 67,695 |
https://mathoverflow.net/questions/87230 | 7 | Let $P\_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly distributed *on* a sphere;
(2) the convex hull of $N>n$ points randomly and uniformly distributed *in* a sphere;
(3) analogous... | https://mathoverflow.net/users/6094 | Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$ | The answer is YES. (I am assuming you mean the angle between two adjacent edges on a common face. (The dihedral angles all go to $\pi$.)) The easy and brief reason is that, in a large random point set, everything (that depends on *local* conditions) happens almost surely.
Here is a sketch of a proof for your model (1... | 1 | https://mathoverflow.net/users/30800 | 120246 | 67,711 |
https://mathoverflow.net/questions/120238 | 11 | Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued random variables $X$ and $Y$, $X$ is $\sigma(Y)$-measurable iff there exists a Borel-measurable function $f:\mathbb{R}\t... | https://mathoverflow.net/users/30364 | Good examples of random variables whose image is not a measurable set? | An analytic set that is not a Borel set...see [this post](https://groups.google.com/forum/#!topic/sci.math/oEh4FzmmGxQ)\* from long ago.
Such an analytic set is a continuous image of $[0,1] \setminus \mathbb Q$, and thus a Borel image of $[0,1]$.
>
> \*From: e...@math.ohio-state.edu (Gerald Edgar)
> Newsgroups: ... | 7 | https://mathoverflow.net/users/454 | 120249 | 67,713 |
https://mathoverflow.net/questions/119736 | 4 | Let $R$ be a DVR, and $k$ residue field of $R$. We suppose that $X\_{0}$ is a stable curve over Spec$k$.
Dose there exist a stable model $X$ over $R$ such that the special fiber isomorphic to $X\_{0}$ ?
If we assume $R=C[[t]]$, where C is complex number field, how to find a deformation which make the generic fiber ... | https://mathoverflow.net/users/5274 | deformation of stable curve | The comments above fully answer the OP's question. I am simply collecting some of these into an answer.
First, the answer to the original question is "no" if one does not impose some additional hypothesis on $R$ such as being complete or Henselian. As with many similar such questions, one negative answer comes from t... | 5 | https://mathoverflow.net/users/13265 | 120256 | 67,715 |
https://mathoverflow.net/questions/120163 | 14 | I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.
As an example, let $X\_i$ be the $\{0,1\}$-random variable that represents whether a uniformly random permutation $\sigma \in S\_m$, has that $\sigma(i)$ is even (assuming permutations in $S\_... | https://mathoverflow.net/users/8846 | Concentration bounds for sums of random variables of permutations | One useful trick that comes in handy sometimes (I originally saw it in [this paper](http://arxiv.org/pdf/math/9406212.pdf) of Talagrand, though it may go further back): We can view a random permutation in $S\_n$ as being generated as follows: Start with the identity permutation, and successively perform transpositions ... | 12 | https://mathoverflow.net/users/405 | 120257 | 67,716 |
https://mathoverflow.net/questions/120235 | 13 | An extension of the Dold-Kan equivalence gives an adjunction between the stable homotopy category and the (unbounded) derived category of abelian groups $SH \rightleftarrows D(Ab)$.
Question 1: Is the right adjoint $D(Ab) \to SH$ faithful?
Question 2: If not, is there a class of objects on which it is faithful (for... | https://mathoverflow.net/users/12914 | Is the derived category of abelian groups a subcategory of the stable homotopy category? | I've found the following somewhat intricate way of answering Q1 in the affirmative. Any complex in $D(Ab)$ quasi-isomorphic to a graded abelian group. Hence, it is enough to consider complexes concentrated in a single degree. Given an abelian group $A$ and $n\in\mathbb Z$, let $(A,n)$ be the abelian group $A$ concentra... | 8 | https://mathoverflow.net/users/12166 | 120268 | 67,722 |
https://mathoverflow.net/questions/120210 | 2 | I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in <http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf>: Namely the author on page 32 says that the Arnold conjecture would immediately follow from a theorem 3.7 on the same page. But as far as I see, the t... | https://mathoverflow.net/users/29973 | Proof of Arnold Conjecture for monotone symplectic manifolds | Note that for the Floer boundary map to be well-defined, we require a Floer regular *pair* $(H\_t, J\_t)$ (we want a family of $\omega$-compatible $J\_t$ so we have a nice metric with which to take gradients for Floer's equation). So if we're given an $H\_t$ to define the Floer homology groups with, we can hope that ma... | 1 | https://mathoverflow.net/users/21375 | 120272 | 67,724 |
https://mathoverflow.net/questions/120255 | 2 | I'm working on a project and I've used the Picard-Fuchs equation at a maximally unipotent monodromy point for a certain 1-dimensional family of Calabi-Yau 3-folds to calculate the A-model Yukawa coupling, and thus the generating function of the genus 0 GW invariants on the suspected mirror. I'm wondering if it's possib... | https://mathoverflow.net/users/13139 | Zero and Negative Gromov-Witten invariants in genus 0 | It is certainly possible to have $n\_d$ be negative or zero. For example if $X=K\_{\mathbb{P}^2}$, the total space of the canonical bundle over $\mathbb{P}^2$, then $n\_2=-6$. Your $n\_d$ are genus 0 Gopakumar-Vafa invariants (or BPS numbers) and they have a description in terms of sheaves on $X$ which is (conjecturall... | 5 | https://mathoverflow.net/users/9617 | 120274 | 67,725 |
https://mathoverflow.net/questions/120254 | 1 | For the integer sequence {1,13,314,9368,312411,11163022, ...}, each term is given by the function
$f(n)=\sum\_{k=0}^n{1\over2n+3k-1}{2n+3k-1\choose k}{6n-6k-3\choose2n-2k-2}$. Is there a method to determine the exact value of $\lim\_{n\rightarrow\infty}f(n+1)/f(n)$? The approximate value of $f(10001)/f(10000)$ is 47.72... | https://mathoverflow.net/users/14207 | Growth constant limit for sum of products of two binomial coefficients | Let $\alpha\approx 0.0493932733732$ be the positive zero of $621\alpha^3+1242\alpha^2+585\alpha-32$. Then the limit you want is probably
$$ \rho = \frac{729(2+3\alpha)^2}{64(1+\alpha)^2}
\left(\frac{4(8+9\alpha)}{621\alpha(1+\alpha)^2}\right)^{\textstyle\alpha}
\approx 47.7322896460174547.
$$
The reason why I say ... | 6 | https://mathoverflow.net/users/9025 | 120284 | 67,730 |
https://mathoverflow.net/questions/120289 | 3 | Is there an elementary example of a function f, such that $|f(x+t)+f(x-t)-2f(x)|/|t|^a\le C$, where $a>1$, such that $f$ is not $C^1$?
| https://mathoverflow.net/users/19544 | Second difference | Yes, i think.
Pick an integer $n>1$ and let $f$ be the function given by $f(x) = x^n\sin(x^{-n})$ for $x>0$ and $f(x)=0$ for $x\leq 0$. It is continuous, and smooth except at $x=0$. It is not $C^1$ at zero, because the derivative for $x>0$ is $nx^{n-1}\sin(x^{-n}) - nx^{-1}\cos(x^{-n})$, which enters in a state of f... | 0 | https://mathoverflow.net/users/5952 | 120294 | 67,735 |
https://mathoverflow.net/questions/120298 | 2 | On [Wikipedia](http://en.wikipedia.org/wiki/Amoeba_%28mathematics%29) there some pictures of two dimensional amoebas (Thanks to [Oleg Alexandrov](http://en.wikipedia.org/wiki/User%3AOleg_Alexandrov/Pictures) for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures of... | https://mathoverflow.net/users/29928 | Picture of a 3 dimensional amoeba. | [There is now a 3-dimensional amoeba on wikipedia.](http://en.wikipedia.org/wiki/Amoeba_%28mathematics%29)
You are welcome.
The main reason for the difficulty of making such amoebas is that it is the projection
of a 4-dimensional surface (the zero set of the polynomial). Finding a parametrization of the zeros set of... | 3 | https://mathoverflow.net/users/1056 | 120310 | 67,744 |
https://mathoverflow.net/questions/120309 | 2 | Hello
I know that there are abelian subgroups of order (q^2-1)/d (cyclic group) and (q-1)^2/d and q^2(elementary abelian) and (q-1)q/d in the group PSL(3,q), where d=(3,q-1).
Also I saw in a text that
1) The orders of maximal abelian subgroups which divide (q-1)^2(q+1) are equal to (q-1)^2/d.
and 2) The orders of... | https://mathoverflow.net/users/30252 | Abelian subgroup of PSL(3,q) | Here's a rough answer as I have very little time. I'm going to work in $K=SL(3,q)$ rather than $PSL(3,q)$ as it amounts to the same thing.
(1) Observe first that the centralizer of a $p$-element of $K$ has order $q^3$ or $q(q-1)$. This is just a matter of working with matrices. Then you can check that the centralizer... | 5 | https://mathoverflow.net/users/801 | 120320 | 67,746 |
https://mathoverflow.net/questions/120219 | 7 | Fix a positive integer $k>0$. For $p>k$ a prime, let $A\_p$ be a subset of the finite field $\mathbb{Z}/p\mathbb{Z}$ of size $k$ which contains a primitive element.
Define $G\_p$ to be the (di)graph whose vertices are elements of $\mathbb{Z}/p\mathbb{Z}$, with two vertices $i,j$ joined by an edge provided $j=ia$ or $... | https://mathoverflow.net/users/801 | Constructing expanders in Z/pZ | No, because solvable groups are amenable. You're asking: Is is there a set in Z/pZ almost invariant by x->x+1 and 2x? Here's one: take the union of I, I/2, .., I/2^n, where I is an interval of length much bigger than 2^n.
| 2 | https://mathoverflow.net/users/31048 | 120328 | 67,748 |
https://mathoverflow.net/questions/120333 | 4 | How can one characterize monotonic bijections from $\mathbb{Q}$ to
$\mathbb{Q}$? It is easy to see that piecewise linear functions which are
strictly monotonic and surjective will do the trick, but are these functions
already all monotonic bijections of the rationals, or are there also "curved"
ones? Classical curved b... | https://mathoverflow.net/users/24306 | Monotonic bijections of rational numbers | A simple example of a monotonic homeomorphism from $\mathbb{Q}$ to itself that is not piecewise linear is given by $f(x)=x$ if $x<0$, $f(x)=\frac{x}{1-x}$ if $x\in[0,\frac{1}{2}]$ and $f(x)=x+\frac{1}{2}$ if $x>\frac{1}{2}$.
| 11 | https://mathoverflow.net/users/22989 | 120339 | 67,753 |
https://mathoverflow.net/questions/120337 | -1 | Let $G$=PSU$(3,q)$ be projective special unitary group where $q$ is prime
power. I would like to know why there is not any prime $r$ such that the
number of Sylow $r$-subgroups of $G$ is $r+1$?
| https://mathoverflow.net/users/30203 | Question on the projective special unitary group | This can be answered on a case-by-case basis. Work in $SU(3,q)$ because it's easier, and observe that if $r>3$ then $r$ divides one of $q, q+1, q-1, q^2-q+1$. Now go through these one at a time.
E.g. if $r$ divides $q$, then $r=p$. Now either calculate the size of the normalizer of a Sylow $p$ or just observe $r+1$ i... | 3 | https://mathoverflow.net/users/801 | 120342 | 67,755 |
https://mathoverflow.net/questions/120260 | 5 | Let $X$ be a simply connected smooth projective variety, whose Picard group is generated by the classes of the irreducible codimension 1 loci $D\_1, \ldots, D\_k$. Let $E\_1, \ldots, E\_r$ be other irreducible codimension 1 loci, and suppose that $X^0$ is the complement in $X$ of the divisors $D\_i$ and $E\_j$.
Suppo... | https://mathoverflow.net/users/23194 | A question on the Picard group | I'm going to assume that $X^0$ is the same as $X\_0$ and that the Picard group of $X$ is *freely* generated by the $D\_i$, since without that the question doesn't make much sense. In this cases, the answer is yes. This is the same as Mohan's answer, but with a little more detail.
The relevant tool is that if $Z$ is a... | 2 | https://mathoverflow.net/users/8914 | 120343 | 67,756 |
https://mathoverflow.net/questions/120336 | 3 | I have a graph with multiple connected components, and its adjacency matrix. I form the `Laplacian matrix` ([wiki Laplacian matrix](http://en.wikipedia.org/wiki/Laplacian_matrix)), and from the 1K nodes there around 100 eigenvalues of value zero. (I use `eig` in `matlab` and the first 5 have negative values which I ass... | https://mathoverflow.net/users/19684 | Can the first non-zero eigenvalue of a Laplacian matrix with more than 1 zero valued eigenvalue be used to reorder an adjacency matrix? | In fact, the multiplicity of the eigenvalue 0 matches the number of connected components in the graph. However, I doubt that in the case of multiple components the eigenvector corresponding to the first non-zero eigenvalue has a similar clear meaning as the Fiedler vector for a single connected component. My doubts ari... | 6 | https://mathoverflow.net/users/27164 | 120344 | 67,757 |
https://mathoverflow.net/questions/120338 | 9 | I want to ask what should be a nice way to build C\*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C\*-algebras out of those objects; but I want to understand what philosophy lies under the recipe.
Say for discrete groups I know group... | https://mathoverflow.net/users/13739 | General recipe for building C*-algebras out of combinatorial object | Let me begin with inverse semigroup $C^{\ast}$-algebras. Inverse semigroups are semigroups $S$ with the property that for all $s\in S$, there is a unique element $s^\*$ with $ss^\ast s=s$ and $s^\ast ss^\ast=s^\ast$. The key example of an inverse semigroup is the symmetric inverses semigroup of all partial bijections o... | 14 | https://mathoverflow.net/users/15934 | 120347 | 67,759 |
https://mathoverflow.net/questions/120340 | 0 | Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes.
Let $\mathbf{d} := (d\_1, d\_2, ..., d\_n) \in \mathbb{R}\_{>0}^n$ be a vector of arbitrary given *node weights*. Now, I want to find symmetric, positive *edge weights* $W := [w\_{ij}]\_{i,j=1,...,n}$ that fit the given nod... | https://mathoverflow.net/users/27164 | Find edge weights that fit given node weights | You can solve it by using maximum network flow: First you duplicate every vertex $i$, creating a twin $i'$, which inherits the same degree $d\_{i'}:= d\_i$. Each edge $ij$ becomes two edges $i,j'$ and $i',j$. If you solve your problem on this new bipartite graph $G'$, you can recover a solution for $G$ by averaging the... | 2 | https://mathoverflow.net/users/30800 | 120348 | 67,760 |
https://mathoverflow.net/questions/120286 | 6 | It is well-known that the Weitzenböck formula for the real Laplacian is
$$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$
where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the gradient vector of $f$, $Ricci$ denotes the Ricci curvature of the manifold $M$.
If $\Delta\_{\bar\partial}$ denotes the ... | https://mathoverflow.net/users/31034 | What is the Weitzenböck formula for the $\bar\partial$-Laplacian | You can just prove it yourself directly in local holomorphic coordinates. Indeed, the $\overline{\partial}$ Laplacian on functions is equal to $\Delta\_{\overline{\partial}}f=g^{i\overline{j}}\partial\_i \partial\_{\overline{j}}f$. Apply this to $|\partial f|^2=g^{k\overline{\ell}}\partial\_k f \partial\_{\overline{\el... | 7 | https://mathoverflow.net/users/13168 | 120352 | 67,762 |
https://mathoverflow.net/questions/119506 | 4 | Let $\kappa$ be a singular cardinal, and let $\langle \kappa\_i \mid i<\mathrm{cf}(\kappa) \rangle$ be an increasing sequence of regular cardinals cofinal in $\kappa$. Recall that a scale on $\Pi\_{i<\mathrm{cf}(\kappa)} \kappa\_i$ is a sequence $\langle f\_\alpha \mid \alpha < \kappa^+ \rangle$ such that:
1. For eve... | https://mathoverflow.net/users/26002 | Existence of scales with special properties | I have a negative answer assuming some mild cardinal arithmetic assumptions. Namely, if $(\kappa\_i)^i < \kappa$ for every $i<\mathrm{cf}(\kappa)$, then there can be no scale with the desired property. This is true, for example, whenever $\mathrm{cf}(\kappa) = \omega$ or $\kappa$ is strong limit. We also make the harml... | 1 | https://mathoverflow.net/users/26002 | 120353 | 67,763 |
https://mathoverflow.net/questions/120335 | 13 | Suppose we are given the Fourier coefficients of an $L^2$ function on the circle. Are there necessary and sufficient conditions on the coefficients that allow us to determine that $f$ is Hölder continuous of order $\alpha$?
Note that the necessary condition $|\hat{f}(n)| \leq C\_f|n|^{-\alpha}$ is not sufficient. Fo... | https://mathoverflow.net/users/30983 | Fourier Coefficients and Hölder Continuity | There is an excellent characterization of Hölder spaces via the Fourier transform, using Besov spaces. Let $\alpha\in (0,1)$: a function $u$
defined on $\mathbb R^n$ belongs to $L^\infty\cap C^\alpha$ if and only if it belongs to $B^\alpha\_{\infty,\infty}$, i.e.
$$
\sup\_{\nu\in \mathbb N}2^{\nu\alpha}\Vert\phi\_\nu(D... | 11 | https://mathoverflow.net/users/21907 | 120358 | 67,767 |
https://mathoverflow.net/questions/120351 | 5 | Let $X$ be a reasonable topological space (let's say it has the homotopy type of a CW complex) and let $G$ be a topological group acting on that space. Let $E\_G \rightarrow B\_G$ be the universal bundle. Then the equivariant cohomology $H^\*\_G(X)$ is defined as $H^\*(E\_G \times\_G X)$. We call the space $E\_G \times... | https://mathoverflow.net/users/42661 | Equivariant Cohomology for actions with finite stabilizers | The argument in the quoted paper is a bit too sketchy. An actual proof will have two parts:
1. Let $f:X \to Y$ be map. Assume that for each $y \in Y$, $\tilde{H}^{\ast} (f^{-1}(y);\mathbb{Q})=0$ (this is what the phrase ''$\mathbb{Q}$-acyclic'' means). Find conditions that guarantee that $f$ induces an isomorphism in... | 9 | https://mathoverflow.net/users/9928 | 120361 | 67,769 |
https://mathoverflow.net/questions/120291 | 23 | **Edit:** Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, this is the dual space of $C(\Omega)$, the space of all continuous real valued functions on $\Omega$. Denote by $$\mathcal{P}(\Omega) ... | https://mathoverflow.net/users/9652 | Metrization of weak convergence of signed measures | Of course, there are many ways of metrizing the weak topology on $\mathcal M(\Omega)$ by using various tools of functional analysis. However, as it has already been pointed out by Dan, the most natural way is to use the transportation metric on the space of measures. [It is much more natural than the Prokhorov metric. ... | 14 | https://mathoverflow.net/users/8588 | 120364 | 67,771 |
https://mathoverflow.net/questions/120367 | 1 | Take the usual Brownian motion on $R^d$ and project it to $T^d$, for almost every individual trajectory, will it be equidistributed on the torus? Does this depend on $d$?
| https://mathoverflow.net/users/21929 | Is a typical path of a Brownian motion on a torus equidistributed? | Yes - it is equidistributed (which is the case for the Brownian motion on any compact Riemannian manifold).
| 1 | https://mathoverflow.net/users/8588 | 120368 | 67,773 |
https://mathoverflow.net/questions/120326 | 5 | In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite order) which take algebraic values. He calls them, characters of type (A) (the (A) probably stands for algebraic, I gue... | https://mathoverflow.net/users/11765 | On Weil's characters of type (A) | Now that I've had the chance to look at Fargues's proof carefully, it seems incomplete for what it claims to prove, but it might answer the question under consideration.
Suppose that you have a collection of integers $f\_{\iota}$ indexed by embeddings $\iota:K\hookrightarrow\mathbb{C}$ and satisfying for some intege... | 7 | https://mathoverflow.net/users/7868 | 120375 | 67,777 |
https://mathoverflow.net/questions/120317 | 10 | I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I wrong?
*Question.* Let $\varphi : \mathbb{A}^r \to \mathbb{A}^r$ be a regular map defined over $\bar{\mathbb{Q}}$, and let... | https://mathoverflow.net/users/26522 | The height of an orbit under rational self-maps | I thank Mahdi for the pointer to the paper. It was my first paper on this subject. It considers especially the case of monomial maps. It contains some references to a small number of papers by other people who have studied the growth rate of $h(\phi^n(x\_0))$.
Dynamical Degrees, Arithmetic Degrees, and Canonical Heig... | 9 | https://mathoverflow.net/users/11926 | 120379 | 67,780 |
https://mathoverflow.net/questions/120190 | 10 | Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities.
We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by $\textrm{Pic}(X)$ the group of Cartier divisors modulo linear equivalence. We also write $G(X):=\textrm{Cl}(X)/\textrm{Pic}(... | https://mathoverflow.net/users/7460 | Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singularities | The answer to Question 2 is true. If $(R,m)$ is a local complete intersection of equicharacterstic $0$ and of dimension $3$ then the Picard group of $Y = \text{Spec} R-\{m\}$ is torsion-free. This group agrees with the class group of $R$ when $R$ has isolated singularity. The proof in this case essentially follows from... | 5 | https://mathoverflow.net/users/2083 | 120384 | 67,781 |
https://mathoverflow.net/questions/120220 | 0 | Let 0<=p<=1, I want the value of q1 and q2 where
$q1=\sum\_{k=0}^n [C(n,k)p^k(1-p)^{n-k}\*\sum\_{i=0}^{k-1} C(m,i)p^i(1-p)^{m-i}]$
$q2=\sum\_{k=0}^n [C(n,k)p^k(1-p)^{n-k}\*\sum\_{i=k}^{m} C(m,i)p^i(1-p)^{m-i}]$
where C(m,i) is the number of i-combination of a set of m elements.
Obviously q1+q2=1.
For special... | https://mathoverflow.net/users/29936 | asymptotic or approximate formula for a combination expression | If independent variables $X,Y$ are distributed Binom$(n,p)$, Binom$(m,p)$, respectively, then $q\_1$ is the probability that $X>Y$. If $mp,np$ are large and the line $X=Y$ is not too far from the point $(np,mp)$, then the normal approximation of $X$ and $Y$ will give a reasonable answer since $X-Y$ has a 1-dimensional ... | 1 | https://mathoverflow.net/users/9025 | 120398 | 67,786 |
https://mathoverflow.net/questions/120371 | 34 | In their treatise *Groupes et algebres de Lie*, Bourbaki (no doubt heavily influenced by Tits) devoted Chapter IV (1968) to the general theory of what they dubbed "Coxeter systems" $(W,S)$ along with "Tits systems" (BN-pairs). Here $S$ is an arbitrary set and $W$ a group generated by a subset $S$ consisting of elements... | https://mathoverflow.net/users/4231 | Is there any need to study Coxeter systems (W,S) with S infinite? | This does not precisely answer your question, but whenever I see or write a result on Coxeter groups, I wonder whether it extends to infinite rank. In a number of cases, results on infinite rank formally follow from the study in finite rank, e.g. existence of a certain type of finitely generated subgroups, or the exist... | 7 | https://mathoverflow.net/users/14094 | 120406 | 67,788 |
https://mathoverflow.net/questions/120403 | 11 | It is known that $A[[X]]$ is flat if $A$ is noetherian (see for example Bourbaki, Algèbre commutative, Ch. III, §3, Cor. 3 p. 146).
What happens if A is not noetherian? Is there an easy counter-example to the flatness of $A[[X]]$?
| https://mathoverflow.net/users/4763 | flatness of power series rings | As a module, $A[[X]]$ is the product of a countable family of copies of $A$. It is known that the product of flat $A$-modules is flat if and only if the ring $A$ is *coherent*, that is, every finitely generated ideal is finitely presented ( <http://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0120260-3/S0002-9... | 15 | https://mathoverflow.net/users/4790 | 120408 | 67,789 |
https://mathoverflow.net/questions/120377 | 5 | Agol's recent VHC paper gave a characterization of virtually special groups in terms of being $\mathcal{QVH}$. He remarks that this may be taken as the defining property of virtually special groups which can be used in this paper (and so presumably, in the proof of the the main theorem).
**My concern:** Recall that ... | https://mathoverflow.net/users/29775 | QVH characterization of virtually special groups | Being virtually special is actually a group-theoretic property, independent of the cube complex (at least in the word-hyperbolic case of interest to Agol). More precisely, Haglund and Wise, in their seminal GAFA paper 'Special cube complexes', proved the following.
**Theorem:** Let $X$ be a finite cube complex with $... | 8 | https://mathoverflow.net/users/1463 | 120413 | 67,792 |
https://mathoverflow.net/questions/120400 | 1 | For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see <http://arxiv.org/pdf/math/0103156.pdf> and <http://arxiv.org/pdf/math/0603151.pdf>.
Is there some sense, or some class of examples (that include true orbifolds, namely orbifolds that are not schemes), in which these numbers have som... | https://mathoverflow.net/users/23194 | Enumerativity of Gromov-Witten invariants of orbifolds | One simple example (although it is a genus 0 example) is the following.
Consider a global quotient $\mathscr{X} = [X/(\mathbb{Z}/2)]$. Then if we look at the genus 0 GW theory of $\mathscr{X}$ where the source curve has $2g + 2$ stacky $\mathbb{Z}/2$ points whose evaluations lie in the twisted sector of $\mathscr{X}$... | 3 | https://mathoverflow.net/users/1703 | 120416 | 67,794 |
https://mathoverflow.net/questions/120423 | 2 | Given a unimodular Lie group $G$ and a discrete subgroup $\Gamma\subseteq G$, under what conditions does there exists a discrete subgroup $H$ s.t. $\Gamma\subseteq H$ and $G/H$ has finite volume? Also, can someone give an example of a discrete subgroup $\Gamma$ of a unimodular Lie Group which cannot be extended to a la... | https://mathoverflow.net/users/31071 | Extending a discrete sub group to a lattice in unimodular Lie groups | The same argument as in Yves' comment applies whenever you have a discrete subgroup of a semisimple Lie group without factors locally isomorphic to $SL(2,{\mathbb R})$, since Mostow Rigidity (or even local rigidity) suffices here (this observation is due to Selberg). The best result in the positive direction is due to ... | 3 | https://mathoverflow.net/users/21684 | 120428 | 67,798 |
https://mathoverflow.net/questions/120405 | 5 | Hello,
I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b\_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from $L^p(\mathbb{T}^d)$ to itself for all finite $p>1$ (*via* the multiplication of the Fourier coefficients of course).
In fact my ... | https://mathoverflow.net/users/27767 | Sufficient condition for $L^p$ multiplier on the torus | $L^p$ boundedness (for $p\in(1,+\infty)$) of your discrete Riesz transform can be derived applying De Leeuw's theorem (Theorem 3.8 in Chapter VII of Introduction to Fourier Analysis on Euclidean Spaces by E. Stein). The multiplier $m(\xi):=\frac{\xi\_i}{|\xi|}$ gives rise to a bounded operator on $L^p(\mathbb{R}^n)$ ($... | 5 | https://mathoverflow.net/users/1049 | 120429 | 67,799 |
https://mathoverflow.net/questions/88718 | 3 | The Tverberg Theorem states the following: Let $x\_1,x\_2,\dots, x\_m$ be points in $R^d$ with $m \ge (r-1)(d+1)+1$. Then there is a partition $S\_1,S\_2,\dots, S\_r$ of $\{1,2,\dots,m\}$ such that $\cap \_{j=1}^rconv (x\_i: i \in S\_j) \ne \emptyset$.
The bound of $(r-1)(d+1)+1$ in the theorem is sharp because there... | https://mathoverflow.net/users/21482 | Tverberg partitions with less than (r-1)(d+1)+1 points | This is an excellent question but we know very little about such conditions. As Boris Bukh remarked the issue is about points in special positions, because for points in sufficiently general position, even the affine hulls of parts for every partition to r parts will have an empty intersection. However, configurations ... | 4 | https://mathoverflow.net/users/1532 | 120431 | 67,801 |
https://mathoverflow.net/questions/120437 | 0 | Consider $X= \left( X\_t \right)\_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A \sum\_{n=1} ^{\infty} p^n \delta\_{-n} \left( dx \right) + Bx^{\beta-1}\left( 1+x \right)^{-\alpha -\beta}e^{-\lambda x } \ma... | https://mathoverflow.net/users/24538 | Concerning Jump process (Lévy process) | I don't think there should be any problem with the definition of $Z\_t$, but of course showing that moments are finite can be tricky. You are trying to take exponential moments of $X\_t$ and so you need fast enough exponential decay for the (right) tails of $X\_t$. I'd need to review some stuff on Levy triples to be su... | 1 | https://mathoverflow.net/users/7813 | 120441 | 67,804 |
https://mathoverflow.net/questions/120442 | 27 | Is it true that every smooth rational variety X is simply connected? How is the proof?
Would it be still true if X has mild (for example orbifold) singularities?
| https://mathoverflow.net/users/5259 | Are rational varieties simply connected? | Yes! (I assume it was implicit in your question that the variety be *projective*?)
More generally: any smooth, complex, rationally connected projective variety is simply connected. See Debarre's book ("Higher dimensional algebraic geometry").
Alternatively, take a look at Debarre's Bourbaki talk ("Varietes rationn... | 34 | https://mathoverflow.net/users/26522 | 120444 | 67,805 |
https://mathoverflow.net/questions/120453 | 1 | Let $\pi$ be a cuspidal automorphic representation of $GL\_2(\mathbb{A}\_\mathbb{Q})$ with trivial central character.
Then we can attach an L-function to $\pi$ which can be written as a Dirichlet series
$ L(s,\pi) = \sum\_{m\in\mathbb{N}} \frac{\lambda\_\pi(m)}{m^s}. $
Using Hecke operators one can prove the follow... | https://mathoverflow.net/users/936 | Hecke Relations on Fourier Coeficients for GL(n), n>2 | Have you checked Sections 9.3 and 9.4 in Goldfeld: Automorphic forms and L-functions for the group GL(n,R)? It might have what you need.
Note also that the Euler product translates into the Hecke relations, even over GL(n). Here it is good to know that at an unramified prime $p$ the Euler factor is of the form $H\_p(... | 6 | https://mathoverflow.net/users/11919 | 120457 | 67,810 |
https://mathoverflow.net/questions/120447 | 7 | Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^\*\_W(X) = H^\* (X)^W$? We always have a map $H^\*\_W(X) \rightarrow H^\* (X)^W$, but it is certainly not an isomorphism with $\mathbb{Z}$-coefficients (take $X = E\_W$, the total space of a universal bundle). But is it true with $\m... | https://mathoverflow.net/users/42661 | Equivariant cohomology of finite group actions and invariant cohomology classes | These results follow from the *Cartan-Leray spectral sequence*, which for a regular covering map $X\to X/W$ and a commutative ring $k$ of coefficients has
$$
E\_2^{p,q}=H^p(W,H^q(X;k))
$$
(cohomology of the group $W$ with coefficients in the $kW$-module $H^\ast(X,k)$) and converges to a graded group associated to $H^\... | 9 | https://mathoverflow.net/users/8103 | 120458 | 67,811 |
https://mathoverflow.net/questions/120421 | 3 | Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of
>
> Every continuous linear map $A\colon C\prime(X\_2) \to C(X\_1)$ is given by a kernel $k \in C(X\_1 \times X\_2)$?
>
>
>
Here $C(\cdot)$ is the space of continuous functions, equipped with a suitable t... | https://mathoverflow.net/users/13356 | Schwartz kernel theorem for topological spaces | As has been pointed out by Peter Michor above, this is false, even for compact spaces. However, it is perhaps of interest that one can characterise in a natural way those operators from $C(K\_1)'$ to $C(K\_2)$ which are represented by kernels of the above type. They are the linear operators whose restrictions to the un... | 4 | https://mathoverflow.net/users/26013 | 120460 | 67,812 |
https://mathoverflow.net/questions/120433 | 3 | Let $\mathcal{G}$ be a coherent 2-group. Following [Baez:HDA5](http://arxiv.org/abs/math/0307200) $\mathcal{G}$ is uniquely determined up to 2-equivalence by the following data: The fundamental group $\pi\_1(\mathcal{G})=G$, the second homotopy group $\pi\_2(\mathcal{G})=H$, the action of $G$ on $H$ by group autos and ... | https://mathoverflow.net/users/27923 | Groups lying horizontally in 2-groups | Let $\partial\colon C\_1\rightarrow C\_0$ be a crossed module, i.e. a group homomorphism together with a right action of $C\_0$ on $C\_1$, denoted exponentially $c\_1^{c\_0}$, satisfying the following laws:
$$\partial(c\_1^{c\_0})=-c\_0+c\_1+c\_0$$
$$c\_1^{\partial(c\_1')}=-c\_1'+c\_1+c\_1'.$$
Here I denote the group ... | 3 | https://mathoverflow.net/users/12166 | 120462 | 67,814 |
https://mathoverflow.net/questions/120470 | 0 | for any coprime numbers a and b, all natural numbers >= (a-1)(b-1) are linear combinations (with integer coefficients) of a and b.
[1] is this conjecture true?
if so:
[2] what is it called? (couldn't find it online)
[3] are/can the linear combination coefficients be restricted to the natural numbers?
| https://mathoverflow.net/users/31079 | coprime numbers, number theory | For 1 and 3, yes. (For 3 assuming both are positive, technically one positive would suffice but then it is essentially reduced to 1, while of course both negative will not work.)
And, note that this is only interisting for natural coefficients, with integers you can write any integer whatsoever, and this is very cla... | 9 | https://mathoverflow.net/users/nan | 120472 | 67,819 |
https://mathoverflow.net/questions/120445 | 5 | This is probably simple. Is there a finite non-commutative ring $R$ with identity in which all of its ideals are two-sided !?
| https://mathoverflow.net/users/nan | non-commutative finite rings | A "natural" example is given by the group ring $\mathbb{F}\_2[Q]$ of the Quaternion group of order 8.
For, we have to show that each left ideal is also a right ideal, and conversely, each right ideal is also a left ideal. The first half (i.e. left is right) is shown in [this paper](http://www.ams.org/journals/proc/1... | 13 | https://mathoverflow.net/users/10194 | 120479 | 67,824 |
https://mathoverflow.net/questions/120395 | 10 | Let $A$ be a small simplicial category. The category $Fun(A,s\mathrm{Set})$ of simplicial functors from $A$ to simplicial sets can be given the projective model structure in which fibration and weak equivalences are objectwise.
Now assume further that $A$ is a symmetric monoidal category with respect to some binary o... | https://mathoverflow.net/users/10707 | Monoidal model category structure on a functor category. | [This should probably be a comment, since it is so short. Nevertheless, it is an answer to the question.]
The result you ask for is a consequence of proposition 2.2.15 in [Sam Isaacson's Ph.D. thesis](http://www.math.uwo.ca/~sisaacso/PDFs/diss.pdf) (Harvard University, 2009). That proposition is stated for combinator... | 11 | https://mathoverflow.net/users/21095 | 120487 | 67,829 |
https://mathoverflow.net/questions/120365 | 3 | Let $M$ be a Kaehler manifold of complex dimension $n$. Let $\Delta$ be the real Laplacian of the underline Riemannian manifold. Let's assume the Ricci curvature of $M$ satisfies $\text {Ric}\ge k>0$. Denoted the first eigenvalue of $\Delta$ by $\lambda\_1$. Then we have
$$
\lambda\_1\ge 2k.
$$
Notice that this estimat... | https://mathoverflow.net/users/30176 | First eigenvalue of $\Delta$ on Kaehler manifold with $Ricci\ge k$. | I am not sure about the history of this result. You can find in Thierry Aubin's book "Some nonlinear problems in Riemannian geometry", Theorem 4.20. He attributes it to [his own 1978 paper.](http://www.ams.org/mathscinet-getitem?mr=0494932)
The proof is really simple, though. If you write $\Delta f=g^{i\overline{j}}f... | 1 | https://mathoverflow.net/users/13168 | 120489 | 67,830 |
https://mathoverflow.net/questions/120499 | 1 | Recall that the group $Ext^1(F'',F')$ parametrizes extensions $$0 \rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0$$ as follows: given one such extension, consider the long exact cohomology sequence arising from the functor $Hom(F'',\bullet)$. If $\delta$ is the connecting coboundary map $$\delta:Hom(F'',F'')... | https://mathoverflow.net/users/18013 | Extension class and cup product | This follows from a general fact concerning the behaviour of coboundary maps on cup products. Let $$
0\to M'\to M\to M''\to 0
$$ be a short exact sequence, and let $N$ be an object such that the sequence
$$
0\to M'\otimes N \to M\otimes N \to M''\otimes N\to 0$$
is exact. Then $\delta(u\cup v)=\delta u\cup v$ for any $... | 3 | https://mathoverflow.net/users/8103 | 120513 | 67,839 |
https://mathoverflow.net/questions/120511 | 35 | I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p\_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I have found that there are only five primes with this property:
$$
p\_1 = 2
$$
$$
p\_3 = 5
$$
$$
p\_{20} = 71
$$
$... | https://mathoverflow.net/users/23388 | Why do primes dislike dividing the sum of all the preceding primes? | Here is a heuristic argument that there is nothing to explain:
The probability that $p$ divides the sum of the preceding primes is $1/p$. So the expected number of primes less than $10^9$ with this property is $\sum\_{p \leq 10^9} \frac{1}{p}$. Using [Mertens' second theorem](https://en.wikipedia.org/wiki/Mertens%27_... | 127 | https://mathoverflow.net/users/297 | 120514 | 67,840 |
https://mathoverflow.net/questions/120495 | 2 | I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in [0, B]$, and place my point at the coordinates $(x, y)$.
Let $R$ be the radius of some circle I place somewhere on the ... | https://mathoverflow.net/users/30997 | The largest circle that encloses no points on a plane with points placed at $N$ random coordinates | The largest empty ball in a point set is known in the literature as the *dispersion*; see, for example, [this definition](http://planning.cs.uiuc.edu/node203.html). This was explored a bit in a previous MO question, "[Finding the most-isolated point in a high-dimensional cube](https://mathoverflow.net/questions/98682/)... | 2 | https://mathoverflow.net/users/6094 | 120515 | 67,841 |
https://mathoverflow.net/questions/120521 | 8 | For $n$ a natural integer congruent to $7$ modulo $8$, one has seemingly always
$$\sum\_{k=1}^{(n-1)/2}\left(\frac{k}{n}\right)k=0$$
where $\left(\frac{k}{n}\right)$ denotes the Jacobi symbol.
First cases:
$n=7$: $1+2-3$
$n=15$: $1+2+4-7$
$n=23$: $1+2+3+4-5+6-7+8+9-10-11$
I do not see any reason for this. Did... | https://mathoverflow.net/users/4556 | A curious sum for integers $\equiv 7\pmod 8$. | There are probably lots of ways to see this, but here's one: let $S\_1$ be the sum above, and let $$S\_2 = \sum\_{k=(n+1)/2}^{n-1} k \left( \frac{k}{n} \right).$$ Then $$S\_1 + S\_2 = S = \sum\_{k=1}^{n-1} k \left( \frac{k}{n} \right).$$ Now you can rewrite $S$ (since $x \to 2x$ is a bijection mod $n$) as $$S = \sum\_{... | 16 | https://mathoverflow.net/users/2698 | 120524 | 67,847 |
https://mathoverflow.net/questions/120485 | 6 | While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I know of the fourier analytic approach used by combinatorialists like Ben Green. But what are the other fourier analytic pr... | https://mathoverflow.net/users/30081 | fourier analytic proofs | Hermann Weyl's delightful proof that for irrational $\alpha$ the sequence of values $k\alpha$ mod $1$, $k \in {\bf N}$, is uniformly distributed in $[0,1]$ deserves a mention. It's so simple I can summarize it here. First we check that for any nonzero $n \in {\bf Z}$ we have $$\frac{1 + e^{2\pi i n\alpha} + \cdots + e^... | 11 | https://mathoverflow.net/users/23141 | 120531 | 67,851 |
https://mathoverflow.net/questions/120438 | 3 | How to show that
$$ \lim\_{\alpha \rightarrow \infty} \sup\_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int \_ 0 ^t e^{\alpha s} ~ dB\_s \right | =0, \ \ \text{a.e.}$$
where $\left (B\_s \right)\_{s\geq 0}$ is a real standard brownian motion starting from zero ?
I'd like to have some ideas to deal with t... | https://mathoverflow.net/users/24538 | Limit of a Wiener integral | Here's one way of dealing with it. Integrate by parts to see that the expression under the sup is
$$
\Bigl|B(t)-\alpha\int\_0^t e^{\alpha(s-t)}B(s)ds\Bigr|\le\alpha\int\_0^t e^{\alpha(s-t)}|B(t)-B(s)|ds +e^{-\alpha t}|B(t)|.
$$
Now the result follows since $B$ is a.s.-bounded and a.s.-Holder on [0,T].
| 9 | https://mathoverflow.net/users/2968 | 120537 | 67,854 |
https://mathoverflow.net/questions/120539 | 1 | Let $\Sigma$ be a finite set. Let $F\_\Sigma$ be the free group over $\Sigma$. Let $G$ and $H$ be finite index subgroups of $F\_\Sigma$. Consider the sets $GH$ and $HG$. Is it always true that $GH=HG$? If not, could you provide a counter-example?
The motivation for this question is automata theory. The subgroups G an... | https://mathoverflow.net/users/31092 | Let G and H be finite index subgroups of a free group. Does GH=HG? | No, ths is not true.
Let $E$ be a finite group and let $\phi:F\_\Sigma\to E$ be a surjective group homomorphism. Let $A,B\subset E$ any subgroups and let $G=\phi^{-1}(A)$ and $H=\phi^{-1}(B)$.
If $GH=HG$ holds, then by applying $\phi$ we get $AB=BA$.
So if the claim was true, then for every pair of subgroups $A,B$ of a... | 3 | https://mathoverflow.net/users/nan | 120540 | 67,855 |
https://mathoverflow.net/questions/120522 | 1 | I have adjacency matrices which have nearly connected components. That is partitions with a dense number of edges between nodes in the same group and few edges acting as bridges between these groups. I have clustered them so that they form a band matrix.
What ways exist to identity these nodes linking groups? These ... | https://mathoverflow.net/users/19684 | How to identify bridge nodes between nearly connected graph components in partitioned adjacency matrices? | As i commented, you need to be more specific about the details. Here is an idea though which might work well if within groups there are lots (or at least a reasonable number of) triangles but no triangles involve bridges:
Take the adjacency matrix $A.$ and compare it to $A^2.$ If the $u,v$ position is positive in both ... | 1 | https://mathoverflow.net/users/8008 | 120541 | 67,856 |
https://mathoverflow.net/questions/120536 | 33 | This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed.
The question is the following.
>
> How many spanning trees does the empty graph $E$ have?
>
>
>
According to Sage it ... | https://mathoverflow.net/users/1737 | Is the empty graph a tree? | In a paper "[Is the null-graph a pointless concept?](https://doi.org/10.1007/BFb0066433)" Harary and Read examine reasons for
assigning certain properties to the empty graph. They observe that from the enumeration perspective it appears to be convenient to consider the empty graph as a forest, but not a tree.
| 53 | https://mathoverflow.net/users/8733 | 120543 | 67,858 |
https://mathoverflow.net/questions/120525 | 4 | I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities".
Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" divisor on $X$, say $D$ has strictly normal crossings. (Maybe we could just take $D$ to be a closed subscheme?)
Then,... | https://mathoverflow.net/users/31091 | Embedded resolution of singularities | What embedded resolution does achieve is this: if $Z\subset X$ is a Zariski closed set in a variety, then there is a smooth variety $Y\to X$, obtained from a series of blow ups along smooth centers, such that the preimage of $Z$ is a divisor with normal crossings. However, there is no reason to expect that the inverse ... | 3 | https://mathoverflow.net/users/4144 | 120554 | 67,863 |
https://mathoverflow.net/questions/120534 | 14 | I apologize in advance if this the answer to this question is standard or well-known. I am not in any way an algebraic topologist.
$\newcommand{\s}{\mathscr}$Let $\s T$ be the category of topological spaces, $\s T\_\*$ the category of pointed topological spaces. We can construct the homotopy category $\s H$ and the "... | https://mathoverflow.net/users/6856 | Basic questions on the homotopy category | For products and coproducts, this is covered on [page 67-68](http://books.google.com/books?id=5nSCRSN51aoC&pg=PA67&lpg=PA67&dq=%2522homotopy+category%2522+%2522coproduct%2522&source=bl&ots=nYG-7fQ8L-&sig=96P788qaRX9odI5hoXt4GjltWI4&hl=en&sa=X&ei=f0wMUdvxF8jO0QGDo4GgAw&ved=0CEIQ6AEwAg#v=onepage&q=%2522homotopy%2520categ... | 19 | https://mathoverflow.net/users/11540 | 120559 | 67,866 |
https://mathoverflow.net/questions/120529 | 4 | Is the following inequality known? I believe it's true, but I could find no reference.
>
> For any convex body $C$ in the plane
> we have
> $$\left(4-\frac{8}{\pi}\right)area(C)\leq
> > diam(C)(per(C)-2diam(C)).$$
>
>
>
If true, this would be tight, with equality when $C$ is a disk. If it turns out not to b... | https://mathoverflow.net/users/5572 | a diameter-perimeter-area inequality for convex figures | This inequality is not true. Consider the rectangle on $\mathbb R^2$ with vertices $(\pm 1, 0)$, $(0, \pm \varepsilon)$. Then on the left you have $2\varepsilon(4-8/\pi)$ on the right you have approximatively $4\varepsilon^2$.
| 6 | https://mathoverflow.net/users/943 | 120563 | 67,867 |
https://mathoverflow.net/questions/120568 | 7 | Let $f\_1, f\_2, \ldots, f\_m \in \mathbb{Z}[x\_1, \ldots, x\_n]$. Assume $f\_1(X) = f\_2(X) = \ldots = f\_m(X) = 0$ have no solutions over $\mathbb{C}^n$, then by Hilbert's Nullstellensatz, there exists polynomials $g\_1, \ldots, g\_m \in \mathbb{C}[x\_1, \ldots, x\_n]$ such that $1 = f\_1 g\_1 + \dots f\_m g\_m$.
I... | https://mathoverflow.net/users/26659 | Hilbert's Nullstellensatz on polynomials with integer coefficients | Yes. Given the degrees of the $g\_i$, the equation $1 = \sum\_i f\_i g\_i$
is tantamount to a system of linear equations in the coefficients of the $g\_i$,
and *those* linear equations have rational coefficients. Once such a system has
a complex solution it automatically has a rational solution.
| 14 | https://mathoverflow.net/users/14830 | 120574 | 67,872 |
https://mathoverflow.net/questions/120424 | 7 | Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal and $x, y$ elements. Is it true that
$$\mathfrak{a}(\mathfrak{a}:x) \cap \mathfrak{a}(\mathfrak{a}:y) = \mathfrak{a}(\mathfrak{a}:(x,y))?$$
| https://mathoverflow.net/users/17901 | ideal operations | No it is not true.
Let $R=\mathbb{K}[a,b,x,y]/\langle abx-aby\rangle$ and $\mathfrak{a}=\langle ax,by\rangle$.
Now $abx\in\mathfrak{a}(\mathfrak{a}:x)\cap\mathfrak{a}(\mathfrak{a}:y)$ but $abx\notin\mathfrak{a}(\mathfrak{a}:\langle x,y\rangle)$.
| 8 | https://mathoverflow.net/users/6066 | 120577 | 67,874 |
https://mathoverflow.net/questions/120565 | 4 | Let $X$ be a nonsingular projective algebraic variety over $\mathbb{C}$. Let $L$ be a base point free line bundle. Bertini's theorem asserts that general divisors in the complete linear system $|L|$ are nonsingular.
An example is $X=\mathbb{P}^n$, and $L=\mathcal{O}(m)$, for $m\ge 2$. The singular divisors of $|L|$ a... | https://mathoverflow.net/users/14854 | singular divisors in a complete linear system | The answer is yes. Here is an example generalising the one of Serge and giving arbitrary codimension.
Consider line bundle $L=O(1)\boxtimes O(1)$ on $\mathbb CP^1\times \mathbb CP^n$. A section of such a bundle has shape $$\sum\_{i=0,j=0}^{i=1,j=n}a\_{ij}x\_iy\_j=0,$$ i.e. $|L|=\mathbb C^{2n+2}$. A section is singul... | 5 | https://mathoverflow.net/users/943 | 120584 | 67,878 |
https://mathoverflow.net/questions/120592 | 0 | That is, is it true that the bound
$$\phi(mn)\leq m\phi(n)$$ holds for all pairs of natural numbers $m$ and $n$?
It is true on average, in the sense that
$$\sum\_{mn=k}m\phi(mn)\leq\sum\_{mn=k}mn\phi(n),$$ and it holds for every pair I have computed. If it does not hold, what is the smallest counter example?
| https://mathoverflow.net/users/10980 | Is Euler's totient sub-homogeneous of degree 1? | First note that, since $\phi$ is a multiplicative arithmetic function, i.e. $\phi(ab)=\phi(a)\phi(b)$ for coprime $a,b$, and since $\phi(a) \le a$ for every $a$, it suffices to consider the problem for prime powers.
And $$\phi(p^k p^t) = p^{k+t-1}(p-1) = p^k (p^{t-1}(p-1))= p^k\phi(p^t).$$
| 2 | https://mathoverflow.net/users/nan | 120594 | 67,882 |
https://mathoverflow.net/questions/81607 | 2 | Let
$P(t)$ be a polynomial in $t$ of degree $n$,
with some contiguous coefficients (not the first or last) being $x\_1,\dots,x\_k$
and the rest of the coefficients are fixed.
(E.g. $p(t)=1+2t+x\_1t^2+x\_2t^3+(5i+7)t^6$ is ok).
Consider the set $S$ defined as $(x\_1,\dots,x\_k) \in \mathbb{C}^k$
with the property ... | https://mathoverflow.net/users/1056 | Discriminant on boundary of semi-algebraic surface | Let $P(t)=(t-t\_1)(t-t\_2)\dots(t-t\_k)(t-a)$ where $(-1)^{k+1}t\_1t\_2\dots t\_k a = b$
which has leading coefficient 1 and constant coefficient $b$. If you fix $b$, then $a$ is a rational function of the $t\_i$. Treat the case where some $t\_i=0$ with care.
The other coefficients are
$$x\_i = (-1)^i \sigma\_i(t\_1,\d... | 2 | https://mathoverflow.net/users/26935 | 120595 | 67,883 |
https://mathoverflow.net/questions/120567 | 8 | Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi\_0$ of this category, i.e. objects of $\mathcal{X}^s$ are the objects in $\mathcal{X}$ and morphisms of $\mathcal{X}^s$ are the morphisms in $\mathcal{X}$ modulo automorphisms... | https://mathoverflow.net/users/30850 | On the coarse moduli space of a stack | Yes, this would imply that $\newcommand{\X}{\mathcal X}\X^s$ is the coarse moduli space, but I don't think this is the "right" question to ask -- I believe that $\X^s$ will not even form a sheaf unless $\X$ happens to be a scheme/algebraic space to begin with.
Anyway, any morphism from a groupoid to a set factors th... | 8 | https://mathoverflow.net/users/1310 | 120603 | 67,886 |
https://mathoverflow.net/questions/118643 | 20 | Hello, this is a question regarding Reineke's paper "Cohomology of non-commutative Hilbert schemes", <http://arxiv.org/abs/math/0306185>, and more precisely the formula on page 4 there (at $n=1$), namely:
$$\chi(H\_{d,1}^{(m)})=\frac{1}{(m-1)d+1}\binom{md}{d}$$
Here at left we have the cohomological Euler characteris... | https://mathoverflow.net/users/29333 | Fuss-Catalan algebras and non-commutative Hilbert schemes | Not really an answer, but too long for a comment. The question is certainly quite interesting, theoretically speaking, but, in practice, who might be *really interested* in doing that? It's a cat-mouse game. What you're asking for is to connect an extremely down-to-earth subarea of OA - the one centered around the TL a... | 0 | https://mathoverflow.net/users/31111 | 120606 | 67,888 |
https://mathoverflow.net/questions/120553 | 8 | $\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$
Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ already seems to be interesting enough). For some $\ell\neq \char K$, $n>0$, should the $n$-th $\mathbb Q\_\ell$-adic Galois c... | https://mathoverflow.net/users/2191 | Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why? | For a base field finitely generated over the prime field, the Tate conjecture implies that the $l$-adic realization gives an equivalence from the category of pure motives tensor $\mathbb{Q}\_l$ to the category of $l$-adic Galois representations generated by the cohomology of smooth projective algebraic varieties over t... | 8 | https://mathoverflow.net/users/31112 | 120608 | 67,890 |
https://mathoverflow.net/questions/120611 | 4 | Let $x\_1,\ldots,x\_k \in \mathbb{R}^d$ be $k$ unit vectors in $d$ dimensional Euclidean space, and let $S = \mathrm{span}(x\_1,\ldots,x\_k)$ be a linear subspace defined by these points. Let $P \in \mathbb{R}^{\ell\times d}$ be a $\ell\times d$ random projection matrix as specified by the Johnson-Lindenstrauss lemma, ... | https://mathoverflow.net/users/31113 | How well do random projections preserve the distance between a point and a linear subspace? | Gideon Schechtman and I thought through this. First, you need to multiply the random projection $P$ by a factor that is of order $(d/\ell)^{1/2}$ to have the distances preserved; i.e., where you wrote $P$ you should have written this constant times $P$. Secondly, you need $\ell \ge k$ else the image of the $k$ dimensio... | 1 | https://mathoverflow.net/users/2554 | 120630 | 67,902 |
https://mathoverflow.net/questions/120535 | 7 | This question is related to the question [The Higman group](https://mathoverflow.net/questions/87347/the-higman-group) (with a nice answer by M. Sapir). So for background, please,
see the above cited question.
The Higman group has an automorphism $h(a\_j)=a\_{j+1}$ ($j+1$ is mod 4). Does the Higman group have a nontr... | https://mathoverflow.net/users/22112 | The Higman group II | I think that Higman's group H has plenty of such normal subgroups. Indeed, let G be the extension of H with the automorphism h. Then H has index 4 in G. By Schupp's theorem, H is SQ-universal, hence the same is true about G (that SQ-universality is stable under a passage to finite index sub/over groups was proved by Pe... | 11 | https://mathoverflow.net/users/7644 | 120633 | 67,904 |
https://mathoverflow.net/questions/120399 | 1 | *Recall a definition*. Let $V\subset \mathbb CP^n$ be a projective variety
and $E$ be a holomorphic vector bundle on it. We call $E$ *linearly trivial* if the restriction of $E$ to any projective line in $V$ is trivial.
It is well known that any linearly trivial bundle on $\mathbb CP^n$ itself is trivial (see Okonek,... | https://mathoverflow.net/users/13441 | Linearly trivial bundles on hypersufaces in $\mathbb CP^n$ | I didn't want to comment, since it might take longer. I need more than the fact that through a general point of $X\subset \mathbb{P}^N$ of degree $n\leq N-2$, there is at least $N-n-1$ dimensional family of lines, but exactly of that dimension. Once we have that (and it is proved in Kollar's book, and this is where I n... | 2 | https://mathoverflow.net/users/9502 | 120641 | 67,907 |
https://mathoverflow.net/questions/120369 | 10 | A friend recently told me the following two facts, for which he cannot recall a proof or a reference
(but he remembers seeing them in the literature):
1. Let $f$ be a holomorphic function mapping the upper half-plane $H$ into itself. Let $G$ be the
group of fractional linear transformations $(az+b)/(cz+d)$ where $ad-... | https://mathoverflow.net/users/25510 | Analytic function avoiding elements of the modular group | This is an answer to part of question 1. Such a function $f:H \rightarrow H$ does exist. To see this, let $z\in H$ and $P(w)$ be the meromorphic function on the plane which is the Weierstrass $P$ function corresponding to the lattice $L\_z= {\mathbb Z}\oplus {\mathbb Z}z\quad$. I wish to add that $P(w)=P(w,z)\quad$ is ... | 5 | https://mathoverflow.net/users/23291 | 120644 | 67,910 |
https://mathoverflow.net/questions/120635 | 2 | Context: This question arose as I was reading the proof of Application 6.1 in Mumford's *Abelian Varieties*. However, I have extracted all of the relevant information below so this question should stand alone.
Let $X$ be a complete variety over an algebraically closed field $k$. Let $D$ be an effective divisor on $X$... | https://mathoverflow.net/users/31118 | On morphisms to projective space arising from a linear system | $E$ can't intersect $C$ in a finite number of points because otherwise the restriction of $\phi$ to $C$ would be a finite degree morphism, which you assume is not. $E$ is the pull-back of the hyperplane divisor, as Vesselin remarked.
| 2 | https://mathoverflow.net/users/4096 | 120658 | 67,917 |
https://mathoverflow.net/questions/120681 | 2 | I'd like to know the syntax for describing a number of elements in a set, and that each of them are distinct. e.g.
{$x,y,z$} $\in A$
I would like to know how I can succinctly express the following, without having to write it out as such:
$ x \neq y \;\;\;\; x \neq z \;\;\;\; y \neq z$
| https://mathoverflow.net/users/31130 | Elementary question: distinct elements in a set | "Let $x,y,z,$ be pairwise distinct", is perfectly fine.
| 4 | https://mathoverflow.net/users/1056 | 120688 | 67,932 |
https://mathoverflow.net/questions/93182 | 5 | Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha\_0<\alpha\_1\cdots <\alpha\_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha\_m$ is the $m+1$-th admissible ordinal relative to $x$.
Anybody knows the proof? Or where to find it?
| https://mathoverflow.net/users/14340 | Countable admissible ordinals | There is a model theoretic proof of the generalization to countable sequences which appears in a paper by Simpson and Weitkamp.
High and low Kleene degrees of coanalytic sets
Stephen G. Simpson & Galen Weitkamp
Journal of Symbolic Logic 48 (2):356-368 (1983)
I believe that this proof is due to Harrington, based on... | 7 | https://mathoverflow.net/users/31026 | 120698 | 67,935 |
https://mathoverflow.net/questions/120694 | 14 | I am currently taking a graduate logic course on Modal Logic and I can't help notice that there are a certain class of graphs characterized by the modal axioms such as (4) $\Box p \rightarrow \Box \Box p$, (5) $\Diamond p \rightarrow \Box \Diamond p$, or (B) $p \rightarrow \Box \Diamond p$ which can characterize frames... | https://mathoverflow.net/users/20343 | How are Modal Logic and Graph Theory related? | The relationship between modal logic and graph theory has, indeed, been studied before. Peter mentioned sheaf models in the comments; I want to mention a more classical-logic-y perspective.
(First, let me note that when we say that $\phi$ characterizes a class of frames $V$, we mean that for every frame in $V$, *and ... | 15 | https://mathoverflow.net/users/8133 | 120700 | 67,936 |
https://mathoverflow.net/questions/120683 | 2 | A finite group $G$ is called rational if and only if $N\_G(\langle x\rangle)/C\_G(x)\cong Aut(\langle x\rangle)$ for all $x\in G$.
The word ``rational" is because there is an equivalent definition in group representation theoretic terms: A finite group $G$ is rational if and only if for any complex irreducible characte... | https://mathoverflow.net/users/19075 | Rational groups | No, this is not true, and the smallest counterexample is the dihedral group of order 24. The condition is trivially satisfied for all $x\in G$ of order $2$, and it also holds for rotations of order $3$ and $4$, since these are conjugate to their inverses. But rotations of order $12$ have $N/C$ of size 2, rather than 4.... | 8 | https://mathoverflow.net/users/3132 | 120716 | 67,946 |
https://mathoverflow.net/questions/120721 | 16 | Let $X$ be a (let us say smooth to obscure any confusions I have between $H(X)$ and $H\_c(X)$) algebraic variety defined over some subfield of $\mathbb{C}$. I have occasionally overheard the expression "$H^\*(X)$ is Hodge-Tate" used to mean something which, as far as I could tell from context, resembled one of the foll... | https://mathoverflow.net/users/4707 | What does "$H^*(X)$ is Hodge-Tate" mean? | 2 does imply 1 (for smooth projective varieties) via $p$-adic Hodge theory and perhaps a simpler argument.
1 does not imply 2. Indeed, blow up $\mathbb P^2$ at the Galois orbit of some point that is not $\mathbb Q$-rational but is rational over some quadratic field extension, say $(1: \sqrt{-1} : 0)$ . Mod a prime $p... | 10 | https://mathoverflow.net/users/18060 | 120725 | 67,951 |
https://mathoverflow.net/questions/120674 | 1 | I have attempted to calculate the number of unlabelled bipartite graphs as follows:
>
> Let $G = (V\_1, V\_2, E)$ be a bipartite graph on $n$ vertices with $|V\_1| = m$ and $|V\_2| = n-m$. Assume without loss of generality that $|V\_1| \leq |V\_2|$ so $m \leq \left\lfloor \frac{n}{2} \right\rfloor$. If $G$ is compl... | https://mathoverflow.net/users/31128 | What is the cardinality of the family of unlabelled bipartite graphs on n vertices? | It seems that what Andrew wants to count are what are called in enumerative contexts
"bicolored graphs". A bicolored graph is a graph in which the vertices have been colored black and white so that every edge joins two vertices of different colors. A bipartite (or bicolorable) graph is a graph that has a bicoloring. A ... | 3 | https://mathoverflow.net/users/10744 | 120726 | 67,952 |
https://mathoverflow.net/questions/120687 | 1 | Let $V$ be a finite dimensional vector space over $\mathbb{Z}\_2$ with a linear map $f\_i : V \to V$ for each $i$ in some finite index set $I$.
Then one can always find some subset $G \subseteq V$ of minimal cardinality such that the set of all elements:
$f\_{i\_1} \circ \dots \circ f\_{i\_n}(g)$ where $g \in G$, $... | https://mathoverflow.net/users/5152 | Minimal generating sets of monoids acting on finite vector spaces. | Things can differ as much as you like even if all maps are idempotents. Let F be any field. Let M be the monoid of all constant maps on {1,...n} (acting on the left) together with the identity. In otherwords, M has n left zeroes and an identity. Let V be the regular left FM-module. It is a cyclic module generated by th... | 1 | https://mathoverflow.net/users/15934 | 120727 | 67,953 |
https://mathoverflow.net/questions/120738 | 8 | Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
---
**Edit:** My original question said "measurable function" instead of the more accurate "measurable homomorphism." Marc Palm and other people answered both q... | https://mathoverflow.net/users/238 | Is a measurable homomorphism on a Lie group smooth? | No, take a proper closed set $X$ and define $f|X = 1$ and $f|G-X = g \neq 1$.
If you consider $f$ to be a group homomorphism, then the answer is yes in general.
| 10 | https://mathoverflow.net/users/10400 | 120747 | 67,961 |
https://mathoverflow.net/questions/120740 | 3 | Let $S$ be a smooth, projective surface over $\mathbb{C}$ and $L\in\mathrm{Pic}(S)$ be globally generated. Then, a general curve $C\in\vert L\vert$ is smooth. Now, let $I\_\xi$ be the ideal of a (possibly non-reduced) $0$-dimensional subscheme $\xi\subset S$ and assume that $L\otimes I\_\xi$ is still globally generated... | https://mathoverflow.net/users/33841 | Smooth curves in a linear system satisfying certain conditions. | $\xi$ has to be "curvilinear", i.e., contained in some smooth curve of $S$, or equivalently, has to be of multiplicity one. It is clear that this is necessary.
Let me now show that it is sufficient. The simplest case is when $\xi$ is reduced. Then you consider the blow up $\pi\_\xi:\tilde S\rightarrow S$ at all poin... | 2 | https://mathoverflow.net/users/1939 | 120755 | 67,965 |
https://mathoverflow.net/questions/120756 | 3 | I have the following sum
$\sum\_{j=1}^K {K \choose j} (-1)^{j+1}/j$. Now I can write this as the integral $\int\_{-1}^0 \frac{(1+x)^K - 1}{x} dx$. However, I wonder whether there is a closed form expression for that integral? Thanks.
| https://mathoverflow.net/users/31151 | Sum involving binomial coefficients | The closed form of the integral is
$$
\int\_{-1}^{0}\frac{(1+x)^k - 1}{x} = \frac{s(k+1,2)}{k!}
$$
where $s(k+1,2)$ denote the [Stirling number](http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind) of the first kind.
| 6 | https://mathoverflow.net/users/23388 | 120758 | 67,966 |
https://mathoverflow.net/questions/120757 | 4 | I am interested in a description of orbits of the natural $GL(V)$ action on $V^{\otimes d}$. I know this is a classical problem but I tried to find some "good" reference and I couldn't. I'm also interested in orbits of the natural $S\_d\times GL(V)$ action on $V^{\otimes d}$. I'm assuming characteristic $0$ but referen... | https://mathoverflow.net/users/31152 | orbits in tensor representations of GL(V) | See, for instance, Chapter 5 of Goodman and Wallach's book "Symmetry, Representations and Invariants". Another good reference is Procesi's "Lie Groups: An approach through Invariants and Representations".
| 9 | https://mathoverflow.net/users/2698 | 120764 | 67,968 |
https://mathoverflow.net/questions/120777 | 3 | Suppose that $Q$ is a quaternion division algebra with center $k$, where $k$ is an arbitrary commutative field (let's say with $\operatorname{char}(k) \neq 2$ if necessary). Assume that $D$ is an arbitrary skew field (which a priori has nothing to do with $Q$ nor with the base field $k$), and assume that there is an in... | https://mathoverflow.net/users/12858 | Skew fields inside quaternion division algebras | First note that if $A$ is the $2\times 2$ matrix algebra over a field, and $z\in A$ has trace zero, then by the Cayley Hamilton Theorem, $z^2=-det (z)$ is a scalar. Hence, if $x,y \in A$ then $(xy-yx)^2$ is a scalar matrix. Secondly, if $A$ is the $n\times n$ matrix algebra over a field, with $n\geq 3$, then it is easy... | 5 | https://mathoverflow.net/users/23291 | 120790 | 67,976 |
https://mathoverflow.net/questions/120787 | 2 | I'm asking a question following the reading of Set Theory from Jech (page 126).
For the record, a $\kappa$-sequence of functions $\langle f\_\alpha : \alpha < \kappa\rangle$ is called
a $\kappa$-scale if $f\_\beta < f\_\alpha$ whenever $\beta < \alpha$, and if for every $g\colon \omega \rightarrow \omega$ there exist... | https://mathoverflow.net/users/41060 | Building a scale by diagonalization and transfinite induction | The inequality relation between functions $\omega\to\omega$ that is written $f<g$ in the question must mean that $f$ is *eventually* (not everywhere) below $g$, i.e., that $f(n)<g(n)$ holds for all sufficiently large $n$. (If it meant "for *all* $n$, then the induction would indeed break down at the first limit ordinal... | 5 | https://mathoverflow.net/users/6794 | 120796 | 67,979 |
https://mathoverflow.net/questions/120771 | 1 | I considered the problem of a form of CTMC evolving in a graph:
Consider a graph of $G(V,E)$ with $|V|=N$ nodes. Each node has a 1-0 CTMC associated with it:
* There is a vertex dependent rate $\mu\_i$ such that node $i$ moves from 1 to 0 at a Poisson rate $\mu\_i$.
* There is an edge-dependent Poisson rate $\lambd... | https://mathoverflow.net/users/42371 | First hit time in a graph setting | This model (at least when all the $\lambda\_{ij}$ are the same and all the $\mu\_i$ are the same) is called the [Contact Process](https://en.wikipedia.org/wiki/Contact_process_(mathematics)).
| 2 | https://mathoverflow.net/users/1061 | 120797 | 67,980 |
https://mathoverflow.net/questions/120748 | 2 | Let $q(y, z) = u\_1 + u\_2y + u\_3 z + u\_4y^2 + u\_5yz + u\_6z^2 + u\_7y^3 + u\_8y^2z + u\_9yz^2 +$ $\hspace{2.55cm}u\_{10}y^3z + u\_{11}y^2z^2 + u\_{12}y^3z^2$
Can $q(y, z)$ be factorized as
\begin{equation}
q(y, z) =(v\_1+v\_2y)(v\_3+v\_4y+v\_5z+v\_6yz)(v\_7+v\_8y+v\_9z+v\_{10}yz)?
\end{equation}
Here, {$u$} an... | https://mathoverflow.net/users/31145 | Factorization of bivariate polynomial | I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these [lecture notes](http://people.csail.mit.edu/madhu/FT98/lect7.ps).
For a computer algebra implementation, you could try [Sage](http://www.sagemath.o... | 1 | https://mathoverflow.net/users/11260 | 120800 | 67,981 |
https://mathoverflow.net/questions/120769 | 7 | Let $X$ be a finite-type scheme over a field $k$. Let $G$ be a finite-type group scheme over $k$; we write $G\_X$ for the base-change of $G$ from $\operatorname{Spec}(k)$ to $X$.
Suppose $f : Y \rightarrow X$ is a $G\_X$-torsor for the fppf topology (i.e. we have an $X$-group scheme action of $G\_X$ on $Y$ such that ... | https://mathoverflow.net/users/17907 | Do torsors give a long exact sequence of cohomology? | It actually more like the ending of a long exact sequence, rather than the beginning. To see what's going on consider the analogous case in topology. For this you replace the Galois group of $k$ with a discrete group $\Gamma$ and the category of $k$-schemes with the category of $\Gamma$-spaces. Instead of an algebraic ... | 9 | https://mathoverflow.net/users/24559 | 120805 | 67,983 |
https://mathoverflow.net/questions/120795 | 3 | Does there exist an infinite locally finite group of finite rank and bounded exponent?
| https://mathoverflow.net/users/21566 | Locally finite groups of finite rank and bounded exponent | I'll expand on Derek Holt's comment, which answers your question. Suppose one has a group $G$ of the type you describe, so that finitely generated subgroups are generated by $r$ elements and have exponent $n$. Consider a finitely generated subgroup $K< G$. By the [restricted Burnside problem](http://en.wikipedia.org/wi... | 5 | https://mathoverflow.net/users/1345 | 120807 | 67,985 |
https://mathoverflow.net/questions/120202 | 8 | Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal O\_{Y,Q}=\mathcal O\_{X,P}[T\_1,\ldots,T\_k]/(T\_1^n-x\_1,\ldots,T\_k^n-x\_k)$$
for certain $x\_i\in\mathcal O\_{X,P}$. In o... | https://mathoverflow.net/users/9947 | Question about local description of the branch locus | There are two issues. You presumably mean that $x\_i$ are themselves reduced and have no common divisors, or else the answer would be trivially no: just take $x\_1=x\_2$ and then the reduced branching ideal is generated by $x\_1,$ not $x\_1x\_2.$ You also need to be careful what you mean in characteristic $p$, as then ... | 4 | https://mathoverflow.net/users/7108 | 120813 | 67,989 |
https://mathoverflow.net/questions/120822 | 0 | If $\pi:E\to M$ is a vector bundle then the set of sections $\Gamma(E)$ is naturally a vector space under fibrewise addition and scalar multiplication on the bundle $E$. This holds similarily for bundles of algebras or modules, thought I'm not sure if it holds for bundles of groups (certainly not for principal bundles)... | https://mathoverflow.net/users/10328 | When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)? | A sheaf ${\cal O}$ is *a fortiori* a presheaf. This means it's a functor that takes the category of open sets in $X$ to your fixed category ${\cal C}$. So, by definition, $\Gamma(X,{\cal O})={\cal O}(X)$ is an object of ${\cal C}$.
| 6 | https://mathoverflow.net/users/10503 | 120824 | 67,993 |
https://mathoverflow.net/questions/120819 | 16 | What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or review of them would be welcome.
| https://mathoverflow.net/users/30081 | open problems in Seiberg-Witten Theory on 4-Manifolds | One basic structural problem about the SW invariants is the question of *simple type*: suppose that $X$ is a simply connected 4-manifold with $b^+>1$, and $\mathfrak{s}$ a $\mathrm{Spin}^c$-structure such that $SW\_X(\mathfrak{s})\neq 0$. Must $\mathfrak{s}$ arise from an almost complex structure? This is true when $X$... | 17 | https://mathoverflow.net/users/2356 | 120827 | 67,995 |
https://mathoverflow.net/questions/120829 | 2 | While reading about the Burnside problem, I thought of the following question:
```
If every proper subgroup of G is finite, does it follow that G is also finite?
```
Despite extensive searching (and thinking), I am unable to find a solution. (I suspect that the answer is no)
| https://mathoverflow.net/users/nan | An infinite group such that every proper subgroup is finite? | No. The direct limit of the cyclic groups of order $p^n$ is infinite, but every proper subgroup of it is finite.
| 18 | https://mathoverflow.net/users/4053 | 120830 | 67,997 |
https://mathoverflow.net/questions/120697 | 3 | I've been learning a bit about the Hitchin fibration, and I wanted to ask about how it works outside of type A.
**Background:** In type A, the Hitchin fibration is reviewed on pg 14 of [this paper of Bezrukavnikov and Braverman](http://arxiv.org/abs/math/0602255). Fix a curve $C$; let $\text{Bun}\_n$ be the stack of ... | https://mathoverflow.net/users/2623 | Hitchin fibration outside of type A | Here is a very brief explanation of what is going on. You can find more details in the papers that Ana lists and in arxiv.org/abs/math/0604617 as you say.
If $G$ is a reductive group, the Hitchin base $\text{Hitch}\_{G}$ is defined to be the moduli of cameral covers of $X$. Concretely
$$ \tag{1} \text{Hitch}\_{G} ... | 8 | https://mathoverflow.net/users/439 | 120834 | 67,999 |
https://mathoverflow.net/questions/120778 | 3 | Let $X$ be a "nice" scheme and $Z \hookrightarrow X$ closed of codimension $\geq 2$. Let $Y$ over $X \setminus Z$ be a torsor for a finite flat group scheme $G/X$.
Does $Y$ spread out to a $G$-torsor over the whole of $X$?
For $G/X$ finite étale one can use Zariski-Nagata purity to spread $Y$ out to a scheme and th... | https://mathoverflow.net/users/nan | purity for finite flat group schemes | It is true if $X$ is regular. This is stated (as Lemme 2) in my CRAS 1985 note on purity for families of curves:
<http://gallica.bnf.fr/ark:/12148/bpt6k5495813c/f45.image>
where, unfortumately, the proof is missing (I only claim that it extends Auslander's proof of the étale case).
| 5 | https://mathoverflow.net/users/7666 | 120841 | 68,001 |
https://mathoverflow.net/questions/120734 | 3 | There are many applications of "pairwise", for instance different, disjunct, orthogonal, independent, intersecting, connected, and many more. Some of them like "pairwise intersecting" or "pairwise connected" seem meaningful. But most of them appear to express no more information than with "pairwise" deleted. Who introd... | https://mathoverflow.net/users/nan | Who invented the expression "pairwise different" and what is its advantage over "different" | After some pondering about my question (and after finding out that this expression turns up in one of my books) I would like to revise my position a bit: "Pairwise orthogonal" seems redundant, but that may depend on the implicit understanding of quantifiers that have to be added to colloquial speech. "A set of orthogon... | 2 | https://mathoverflow.net/users/nan | 120846 | 68,004 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.