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 |
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https://mathoverflow.net/questions/213196 | 2 | Assume that $\rho$ and $\rho'$ are conformal metrics on the unit disk which is a geodesic disk of radius $1$ w.r.t. both metrics $\rho$ and $\rho'$, and assume that $\rho'$ has a constant Gauss curvature $K'$ which is greater than the curvature of $\rho$. Let $d$ and $d'$ be the corresponding distance functions defined... | https://mathoverflow.net/users/76840 | Comparision theorem for distance function | See [Bob Osserman's 1999 Notices articl](http://www.ams.org/notices/199908/fea-osserman.pdf)e (which is beautifully written, and copiously referenced).
| 0 | https://mathoverflow.net/users/11142 | 213198 | 101,254 |
https://mathoverflow.net/questions/213121 | 4 | Let $\mathbb{A}^3 = \mathbb{F}\_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray\*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) &\mapsto& (y, x, z) \\
v: (x, y, z) &\mapsto& (x, z, y)
\end{eqnarray\*}
It immediately follows that those three morphisms are involutions,... | https://mathoverflow.net/users/nan | What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$? | The experimental observation about the order of $w:(x, y, z) \mapsto (z, y, yz-x)$ isn't hard to verify: By induction, one shows that $w^m(x,y,z)=(a\_mx+b\_mz,y,c\_mx+d\_mz)$ with $\begin{pmatrix}a\_m&b\_m\\c\_m&d\_m\end{pmatrix}=\begin{pmatrix}0&1\\-1&y\end{pmatrix}^m$. Looking at the eigenvalues of $\begin{pmatrix}0&... | 3 | https://mathoverflow.net/users/18739 | 213199 | 101,255 |
https://mathoverflow.net/questions/213057 | 4 | I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there is a theorem that connects the stability of a q-periodic point $(x\_i,y\_i)$ of a symplectomorphism $\psi$ (so we have... | https://mathoverflow.net/users/75290 | Symplectic geometry and stability of orbits | The following paper *Linear stability of periodic orbits in Lagrangian systems* [link](http://www.sciencedirect.com/science/article/pii/0375960183907351) of Mackay and Meiss might be relevant.
| 3 | https://mathoverflow.net/users/11028 | 213212 | 101,259 |
https://mathoverflow.net/questions/213220 | 0 | For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. Are there non-isomorphic lattices $L\not \cong K$ such that ${\cal Id}(L) \cong {\cal Id}(K)$?
| https://mathoverflow.net/users/8628 | Does ${\cal Id}(L) \cong {\cal Id}(K)$ imply $L\cong K$? | No; in fact, we can canonically recover $L$ from $\mathcal{Id}(L)$ as the sublattice of compact elements (that is, elements $x$ such that whenever $x=\bigvee S$, there is a finite subset $F\subseteq S$ such that $x=\bigvee F$). It is clear that any principal ideal is compact; conversely, any compact ideal is a finite j... | 8 | https://mathoverflow.net/users/75 | 213223 | 101,263 |
https://mathoverflow.net/questions/213180 | 6 | In an abelian group, the additive energy between two sets is $$E(A,B)=|\{(a\_1,a\_2,b\_1,b\_2)
\in A\times A\times B\times B:a\_1+b\_1=a\_2+b\_2\}|$$ which is ranges from $|A||B|$ to $(|A||B|)^{3/2}$. What I want to know is, given that this is large, should $A$ look like $B$?
To keep things simple, let's say all set... | https://mathoverflow.net/users/74453 | What pairs of sets have additive energy? | If $\lvert A\rvert \approx \lvert B\rvert$ and $E(A,B)\geq K^{-1}\lvert A\rvert\lvert B\rvert^2$, then the Balog-Szemeredi-Gowers theorem yields large subsets $A'\subset A$ and $B'\subset B$ such that $\lvert A'-B'\rvert\ll K^{O(1)}\lvert A\rvert$. This is essentially the best result possible; from this one can deduce ... | 6 | https://mathoverflow.net/users/385 | 213227 | 101,264 |
https://mathoverflow.net/questions/213225 | -1 | Are there complete lattices $L, K$ such that
* $L\not\cong K$;
* there are injective complete lattice homomorphisms $i:L\to K$ and $j: K\to L$
?
| https://mathoverflow.net/users/8628 | Complete non-isomorphic lattices with injective complete homomorphisms between them? | Let $A=[0,\omega\_0]$ and $K=[0,1]$ be closed intervals of the ordinals and the real numbers with natural orderings. Let $L=A\times K$ be equipped with the lexicographic order where the $A$ coordinate is more important. Then both $L$ and $K$ are complete and the injective homomorphisms are easy to construct. To see tha... | 1 | https://mathoverflow.net/users/16678 | 213229 | 101,265 |
https://mathoverflow.net/questions/213211 | 2 | Let $X\_d^N\subset\mathbb{P}^N$ be a cone over the Grassmannian of lines
$\mathbb{G}(1,d)\subset\mathbb{P}^{d(d+1)/2-1}\subset\mathbb{P}^N$
with vertex a linear space $L\subset\mathbb{P}^N$ of dimension $N-d(d+1)/2-2\geq 0$.
Question: Is $X\_d^N$ a local complete intersection?
Of course, this is true when $d\leq ... | https://mathoverflow.net/users/15606 | Are cones over Grassmannianns of lines local complete intersections? | No, that is not true. Denote $\mathbb{G}=\mathbb{G}(1,d)$. Denote $D=\frac{d(d+1)}{2} - 1$, so that the Plücker embedding of the Grassmannian is $$\mathbb{G} \hookrightarrow \mathbb{P}^D.$$ First consider the case that $N$ equals $D+1$, i.e., $X=X\_d^{D+1}$ is the "usual" projective cone over $\mathbb{G}=\mathbb{G}(1,d... | 4 | https://mathoverflow.net/users/13265 | 213234 | 101,268 |
https://mathoverflow.net/questions/213231 | 12 | Suppose $X\to Y$ is a morphism between varieties with all fibers isomorphic to $\mathbf{P}^n$, is $X$ a projective bundle over $Y$, i.e. $X=\mathbf{P}(E)$ for some vector bundle over $Y$?
| https://mathoverflow.net/users/nan | Is a morphism whose all fibers are $\mathbf{P}^n$ a projective bundle? | No that is not true, and I am confident somebody has written the following example previously on MO. Let $\mathbb{A}^2$ have coordinates $s$ and $t$. Let $Y\subset \mathbb{A}^2$ be the singular plane curve with defining equation $$f(s,t) = t^2-s^2(1-s) = 0.$$ Let $u$ be a coordinate on $\mathbb{A}^1$. The normalization... | 10 | https://mathoverflow.net/users/13265 | 213235 | 101,269 |
https://mathoverflow.net/questions/213226 | 4 | At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom:
>
> For any set $X$, any class $V$ with a surjective map $f : V \twoheadrightarrow X$ there exists a small subclass $V' \subset V$
> such that the restriction of $f$ to $V'$ is already surject... | https://mathoverflow.net/users/22131 | Does this axiom (a weak form of class valued choice) has a name? | In weak set theories, using classical logic and interpreting
"small subclass" as "set", this principle amounts to an
alternative formulation of the collection axiom. For example, in
Zermelo set theory or even much weaker theories, even without the
power set axiom, this principle is equivalent to the collection
axiom sc... | 3 | https://mathoverflow.net/users/1946 | 213247 | 101,273 |
https://mathoverflow.net/questions/213160 | 2 | The following is a question I have asked [here](https://math.stackexchange.com/questions/1381867/when-mb-neq-b-m-is-a-maximal-ideal-of-a-a-subseteq-b) without receiving any comments, therefore I post it here:
Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$.
When $mB \neq B$?
This is true when $... | https://mathoverflow.net/users/72288 | When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$ | Here are a few comments that may help.
1. As abx explains, a good way to think about it is in terms whether or not $m$ lies in image of the map on spectra.
2. As you surmised, either condition of faithful flatness or integrality are sufficient,
3. but not necessary! For example, take $A=k[x,y]$ and the affine blow up... | 1 | https://mathoverflow.net/users/4144 | 213248 | 101,274 |
https://mathoverflow.net/questions/212671 | 8 | This is a [cross-post from Computational Science](https://scicomp.stackexchange.com/questions/20074/bounded-input-boundaed-output-stability-for-heat-equation-proof-or-counter-exam "link").
I am interested in proving or obtaining a counterexample to the following conjecture.
Let $\Omega\subset\mathbb{R}^d$ be a boun... | https://mathoverflow.net/users/76602 | Bounded input Bounded output stability for heat equation | The theory of linear Intial-boundary value problem proceeds the following way. In your case, because of the zero initial condition, you may replace ${\mathbb R}^+$ by $\mathbb R$ and define $u\equiv0$, $u\_d\equiv0$ for $t<0$. Then what matters is the special case where $\Omega$ is a half-space, say $\{x\_d>0\}$. In th... | 1 | https://mathoverflow.net/users/8799 | 213249 | 101,275 |
https://mathoverflow.net/questions/213244 | 6 | Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable partial functions from $\kappa$ to 2, ordered by $\tau \leq \sigma$ iff $1 \Vdash \tau \leq \sigma$. Call this $T(\mathbb P, Add... | https://mathoverflow.net/users/11145 | Reverse of a termspace forcing fact | The answer is no. Forcing with
$\mathbb{P}\*\dot{\text{Add}}(\kappa,1)$ adds no fresh subsets to
$\kappa$, that is, a new subset of $\kappa$ all of whose initial
segments are in $V$. Thus, it cannot add a $V$-generic filter for
$\text{Add}(\kappa,1)^V$.
To see this, note that $\mathbb{P}$ is productively $\kappa$-c.c... | 6 | https://mathoverflow.net/users/1946 | 213255 | 101,277 |
https://mathoverflow.net/questions/213259 | 2 | Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume that $M \otimes\_{\mathbb{Z}G} \mathbb{Q}G$ is a projective $\mathbb{Q}G$-module, can we conclude that $M$ itself is proj... | https://mathoverflow.net/users/13356 | Projectivity of torsion-free modules over integral group rings | That is already false when $G$ equals $\mathbb{Z}$. The group ring $\mathbb{Z}G$ is $\mathbb{Z}[t,t^{-1}]$. Let $p$ be a prime integer, and let $I\subset \mathbb{Z}G$ be the ideal $\langle p, t-1 \rangle$. Then $I\otimes\_{\mathbb{Z}G}\mathbb{Q}G$ is isomorphic to the principal ideal $\langle t-1 \rangle$, which is fre... | 4 | https://mathoverflow.net/users/13265 | 213260 | 101,279 |
https://mathoverflow.net/questions/213258 | 2 | Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/\_{\mathcal D}$ is homeomorphic to $\mathbb S^n$ if $\mathcal D$ is "shrinkable", and there are many conditions on $\mathcal D$ that ens... | https://mathoverflow.net/users/76590 | Shrinkable decompositions with uncountably many non-degenerate elements? | Bing's Dogbone Space stems from a decomposition of $S^3$ unto points
and an uncountable collection of tame arcs. The decomposition space is not a manifold,
so the decomposition is non-shrinkable. W. T. Eaton (Proc. Amer. Math. Soc. 39
(1973), 379--387) presented a higher dimensional analog of the Dogbone Space, involvi... | 4 | https://mathoverflow.net/users/76875 | 213270 | 101,283 |
https://mathoverflow.net/questions/213275 | 11 | The [moment curve](https://en.wikipedia.org/wiki/Moment_curve) is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
[Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s... | https://mathoverflow.net/users/8938 | Tighter Caratheodory on the moment curve? | The answer is **yes** for all dimensions.
An old theorem by [Fenchel](http://link.springer.com/article/10.1007/BF01454836) states that for a compact set $K$ in $\mathbb{R}^n$ every point in the convex hull can be either written as a convex combination of at most $n$ points or $K$ can be separated by a hyperplane. Sin... | 13 | https://mathoverflow.net/users/48084 | 213279 | 101,290 |
https://mathoverflow.net/questions/213251 | 4 | As is well known, the **Cramér-Rao bound** (or **information inequality**) sets a lower bound on the variance of estimators of a parameter.
Consider the case when the parameter is a **scalar**, the estimator is **unbiased**, and sample size is **fixed**. From the [Wikipedia](https://en.wikipedia.org/wiki/Cram%C3%A9r%... | https://mathoverflow.net/users/40432 | Cramér-Rao bound for randomized estimator | I think I got an affirmative answer for the **fixed-size** (i.e. non-sequential) case.
Let $\hat \theta$ be an estimator of the parameter $\theta$. The estimate is obtained as a (deterministic) function of $n$ **observations** $x\_1, \ldots, x\_n$ and $m$ **auxiliary random variables** $y\_1, \ldots, y\_m$. $\hat \th... | 1 | https://mathoverflow.net/users/40432 | 213287 | 101,295 |
https://mathoverflow.net/questions/213011 | 20 | In 1977, Joan Plastiras gave a striking example of two non $\*$-isomorphic C$^\*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M\_2(\mathbb C) \simeq \mathcal B\otimes M\_2(\mathbb C)$$ (<http://www.ams.org/journals/proc/1977-066-02/S0002-9939-1977-0461158-9/S0002-9939-1977-0461158-9.pdf>).
M... | https://mathoverflow.net/users/76593 | C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$ | Such examples do exist. In the paper "Stability of C\*-algebras is not a stable property, Doc. Math. J. DMV , 2, (1997), 375-386.", Rordam gives examples of simple C\*-algebras $A$ such that $M\_2(A)$ is a stable C\*-algebra but $A$ is not. Now take the pair $A$ and $B=M\_2(A)$. These C\*-algebras are not isomorphic, s... | 16 | https://mathoverflow.net/users/13381 | 213292 | 101,298 |
https://mathoverflow.net/questions/202798 | 12 | In the book of Garrett Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and sums). Apparently this is related to the fact that the free modular lattice on three generators has 28 elements. I am no... | https://mathoverflow.net/users/6129 | How many subspaces are generated by three or more subspaces in a Hilbert space? | From four subspaces in general position one can generate an infinite number of other subspaces by closing up under joins and meets. This is true even for subspaces of $\mathbb{R}^3$ (any field of characteristic zero in place of $\mathbb{R}$ would do). This is easy to see in the corresponding projective plane picture: t... | 15 | https://mathoverflow.net/users/2926 | 213299 | 101,301 |
https://mathoverflow.net/questions/213238 | 3 | I am trying to solve the following equation;
$$
U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0
$$
where U is a function of t and C is constant.
The above equation is similar to a form of Riemann equation.
Could anyone please provide any support on how the above equation can... | https://mathoverflow.net/users/76857 | Solution of second order differential equation with singularities at 0,1, and ∞ | I'm not sure if this is more useful than the *Maple* solution given earlier, but Wolfram *Mathematica* finds a slightly different solution in terms of associated Legendre functions:
$$U(t) = \frac{1}{1-t} \left( k\_1 P\_\ell^2(2t-1) + k\_2 Q\_\ell^2(2t-1) \right)$$
Here, $k\_1$ and $k\_2$ are constants of integrati... | 2 | https://mathoverflow.net/users/61479 | 213305 | 101,304 |
https://mathoverflow.net/questions/213246 | 9 | Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod\_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & \text{if } n=0\pmod{6}\\
1 & \text{if } n=1,5\pmod{6}\\
-1 & \text{if } n=2,4\pmod{6}\\
4 & \text{if } n=3\pmod{6}
\end{ca... | https://mathoverflow.net/users/49556 | Product of a Finite Number of Matrices Related to Roots of Unity | The following is a conjectured generalization of the claimed identity which may help in proving it. We prove this generalization (and hence also the identity from the question) in the case that $3$ does not divide $n$, and give a partial result in the remaining case.
The idea of the generalization is to observe that ... | 10 | https://mathoverflow.net/users/18739 | 213320 | 101,313 |
https://mathoverflow.net/questions/64131 | 40 | in the paper
Foundations of the theory of bounded cohomology,
by N.V. Ivanov, the author considers the complex of bounded singular cochains on a simply connected CW-complex $X$, and constructs a chain homotopy between the identity and the null map. The construction of this homotopy involves the description of a Po... | https://mathoverflow.net/users/6206 | Homotopy groups of $S^2$ | SERGEI O. IVANOV, ROMAN MIKHAILOV, AND JIE WU have recently(2nd June 2015) published a paper in arxive giving a proof that for $n\geq2$, $\pi\_n(S^2)$ is non-zero.
You can look at it in the following link.
Sergei O. Ivanov, Roman Mikhailov, Jie Wu, *On nontriviality of homotopy groups of spheres*, [arXiv:1506.00952](... | 47 | https://mathoverflow.net/users/75954 | 214326 | 101,317 |
https://mathoverflow.net/questions/214333 | 6 | Suppose $K$ is a subset of $[0,1]$ with the following property: for almost $x,y \in K$, we have
$$\frac{x+y}{2} \not\in K.$$
(Here, "almost in $K$" means "in $K$ except for a countable subset").
Such a set must have holes, and the Cantor tridiagonal set has this property. What can we say about the Hausdorff dimen... | https://mathoverflow.net/users/19018 | Hausdorff dimension of a Cantor-like set | We can find Cantor-like sets of Hausdorff dimension arbitrarily close to $1$ which satisfy your property.
**Lemma:** If a subset $S \subseteq \{1, 2, \dots, n\}$ of cardinality $|S| = m$ has no length-$3$ arithmetic progressions, then we can find a subset $K \subsetneq [0, 1]$ with a Hausdorff dimension of:
$$ \dfr... | 8 | https://mathoverflow.net/users/39521 | 214336 | 101,322 |
https://mathoverflow.net/questions/214331 | 3 | Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices.
Let $M\in\mathbb{P}(\bigwedge^2V)$, for any choice of $n-k$ zeros on the diagonal of $M$ we can construct a $k$-minor $M\_{i\_1,...,i\_{n-k}}$ of $M$ by deleting form ... | https://mathoverflow.net/users/nan | Varieties parametrizing skew-symmetric matrices | Yes. The variety you denoted by $V\_k$ is the $(k-1)$-secant variety $Sec\_{k-1}(\mathbb{G}(1,n-1))$, where $\mathbb{G}(1,n-1)$ is the Grassmannian of lines in $\mathbb{P}(V)$ parametrizing rank two skew-symmetric matrices.
The ideal $I(Sec\_{k-1}(\mathbb{G}(1,n-1)))$ is generated in degree $k$ by sub-Pfaffians of s... | 4 | https://mathoverflow.net/users/14514 | 214337 | 101,323 |
https://mathoverflow.net/questions/214344 | 1 | Is there a name for a partial order $\preceq$ on a set $X$ with the following property: "there exists a countable set $S \subset X$ such that for all $x \in X$ there exists $y \in S$ with $x \preceq y$"?
(**Remark:** If such a set $S$ exists, then it is possible to choose $S$ to be a chain: letting $(x\_n)$ be an enu... | https://mathoverflow.net/users/15570 | Is there a name for a partial order in which there is a countable chain which "dominates" the whole space? | Such a partial order would be said to have **countable [cofinality](https://en.wikipedia.org/wiki/Cofinality).**
(The set $S$ would be said to be [cofinal](https://en.wikipedia.org/wiki/Cofinal_%28mathematics%29)).
(Strictly speaking, your property could also apply to an order with *finite* cofinality, so if that ... | 4 | https://mathoverflow.net/users/4832 | 214345 | 101,326 |
https://mathoverflow.net/questions/213281 | 3 | I was reading about tetration [here](http://www.tetration.org/Tetration/index.html). The site mentions that the convergence of the expansion for fractional iteration is unproven. However, I was interested in reading more literature about convergence in special cases perhaps.
In addition, is there an explicit form for... | https://mathoverflow.net/users/71104 | Convergence of expansion for fractional iteration | **Convergence of Fractional Iterration**
I looked into the literature on the convergence of fractional iteration and had copied some resent works, but I found I needed a number of supporting papers I had no access to in order to master the papers in my possession. A single source covering the topic would be a great s... | 3 | https://mathoverflow.net/users/nan | 214348 | 101,328 |
https://mathoverflow.net/questions/214355 | 0 | Assume $A$ and $B$ are commutative algebras with $1$.
There is a nice result of Wang, [Corollary 8](http://ac.els-cdn.com/0021869380902331/1-s2.0-0021869380902331-main.pdf?_tid=3e25ad1a-3e2d-11e5-b1dc-00000aab0f6b&acdnat=1439080154_a8c7a76b2c7c860d578180ec8a516365), which says the following: "Let $B = A[z] = A[Z]/(h(Z)... | https://mathoverflow.net/users/72288 | Is it possible to generalize a result of Wang? | I don't see any reason to expect the higher derivatives of $h$ to be important. For a simple counterexample to your conjecture, consider $h(Z)=Z^2$ (for, say, $A$ a field of characteristic $\neq2$). Then $h''$ is a unit, but $B$ has infinite projective dimension over $B\otimes\_A B$. Another counterexample is $h(Z)=0$;... | 1 | https://mathoverflow.net/users/75 | 214356 | 101,330 |
https://mathoverflow.net/questions/214349 | 0 | I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices and no $K\_{s,t}$ subgraph. However, I was unable to find anything about the tripartite equivalent. Is anything known ab... | https://mathoverflow.net/users/75293 | Forbidden Tripartite Graphs | The [Erdos-Stone theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem) is the reference you are looking for.
For every tripartite graph $G$, the number of edges in an $n$-vertex graph that guarantees $G$ as a subgraph is $n^2/4 + o(n^2)$. It is easy to see that $n^2/4$ is a lower bound (for $n$ eve... | 4 | https://mathoverflow.net/users/24076 | 214357 | 101,331 |
https://mathoverflow.net/questions/214361 | 25 | Are there finitely generated groups whose word problem is solvable, but not quickly? It would be great to have specific examples, but existence results would also be helpful.
All of the groups that I know with solvable word problem (linear groups, hyperbolic groups, branch groups) have a low-degree polynomial time a... | https://mathoverflow.net/users/77921 | Groups where word problem is solvable, but not quickly? | There is a very nice recent paper "[Algorithmically complex residually finite groups](http://arxiv.org/abs/1204.6506)" by O. Kharlampovich, A. Myasnikov, and M. Sapir about this issue - containing also many references to earlier results about complex Dehn functions and the complexity of the word problem in specific cas... | 20 | https://mathoverflow.net/users/8176 | 214371 | 101,334 |
https://mathoverflow.net/questions/211679 | 2 | I'm reading a proof of the following theorem
>
> If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category.
>
>
>
In this proof we define $V^\*$ in a natural way as $\mathrm{Lin}\_k(V,k)$. We define action and coaction on $V^\*$ by $\langle h\rhd ... | https://mathoverflow.net/users/75934 | Yetter-Drinfeld modules as rigid category | Not sure if a reply is still needed to this.
This is simply the YD-compatibility condition on V (in the form of e.g. Kassel, IX.5, equation (5.2)) applied to $Sh \otimes v$, and afterwards the output in $V$ is evaluated against the element $f$ of $V^\*$, while $S^{-1}$ is applied to the output in the Hopf algebras $H... | 2 | https://mathoverflow.net/users/33854 | 214372 | 101,335 |
https://mathoverflow.net/questions/214352 | 3 | Let $X$ be a smooth projective variety. Then we have an exact sequence:
$$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$
where $Aut^{o}(X)$ and $H$ are respectively the connected component of the identity and the group of the connected components of $Aut(X)$.
Assume that there is a GIT quotient $Y:... | https://mathoverflow.net/users/nan | GIT quotients and automorphisms | You have not specified anything about the linearizing invertible sheaf $\mathcal{L}$ that you are using to define the GIT quotient. If for every $g\in \text{Aut}(X)$, $g^\*\mathcal{L}$ is isomorphic to $\mathcal{L}$, then there is an induced action of $H$ on $Y$. This follows from Section 5, Chapter 1 of "Geometric Inv... | 2 | https://mathoverflow.net/users/13265 | 214377 | 101,337 |
https://mathoverflow.net/questions/214381 | 2 | I need a computer program, to help me with some very basic group computations.
Specifically, I want to know if some group generated by a few small matrices over a finite field is solvable.
Is there a free program or on-line tool which can do this?
| https://mathoverflow.net/users/6619 | What is a good program for matrix groups computations? | Yes, [GAP](http://www.gap-system.org/). It has an extensive functionality for computing with matrix groups over finite fields.
| 5 | https://mathoverflow.net/users/11142 | 214382 | 101,338 |
https://mathoverflow.net/questions/214378 | 0 | Let $f$ be a non-negative infinitely smooth function on the real line. Is it true that for any constant $\alpha$ the function $f^\alpha$ is infinitely smooth?
| https://mathoverflow.net/users/16183 | Smoothness of a power of smooth non-negative function | No. Let $f(x)=x^2,$ while $a=1/4.$
| 1 | https://mathoverflow.net/users/11142 | 214383 | 101,339 |
https://mathoverflow.net/questions/212886 | 6 | Suppose I have a (possibly infinite) bag of coins with various weights. I select a coin and flip it $n$ times. Averaging over the choice of coins from the bag, there is some probability of seeing exactly $k$ heads, for $k=0,...,n$. Let $r\_k$ be the probability of seeing exactly $k$ heads.
More formally, let $D$ be a... | https://mathoverflow.net/users/8938 | Forbidden coin flips | As fedja noted in the comments, I am essentially asking a classical moment problem. I'm not sure if the $x=p/(1-p)$ change of variables reduction quite works (consider, e.g., the singleton probability distribution with all mass at $p=1$), but the general point is certainly correct.
For the benefit of future readers, ... | 2 | https://mathoverflow.net/users/8938 | 214408 | 101,348 |
https://mathoverflow.net/questions/214412 | 10 | Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with the $G$ action. What can we say about $D$? What more can we say if
(1) $G$ is unitary or
(2) We assume the answer to ... | https://mathoverflow.net/users/297 | Schur's Lemma for Hilbert spaces | To elaborate on my comment, let us suppose that $G$ is closed under taking adjoints (in particular, this holds if $G$ is unitary). Then it is easy to see $D$ is also closed under adjoints, so for any $A\in D$, the self-adjoint operators $\operatorname{Re} A=(A+A^\*)/2$ and $\operatorname{Im} A=(A-A^\*)/2i$ are also in ... | 8 | https://mathoverflow.net/users/75 | 214413 | 101,349 |
https://mathoverflow.net/questions/214406 | 15 | I'm trying to read parts of [McLarty](https://mathoverflow.net/users/38783/colin-mclarty)'s [Grothendieck on Simplicity and Generality](http://www.landsburg.com/grothendieck/mclarty1.pdf). In the article, I read Grothendieck thought of sheaves over some topological space as *meter sticks* measuring it.
>
> What did... | https://mathoverflow.net/users/69037 | Grothendieck - sheaves as meter sticks | Here are two (related) interpretation of this quote I can think of:
A first interpretation is just that Grothendieck was attach to the idea that you can study a 'space' (whatever this mean) by studing certain family of objects indexed by it, for example sheaves over a topological space, vector bundle over a manifold,... | 12 | https://mathoverflow.net/users/22131 | 214424 | 101,355 |
https://mathoverflow.net/questions/214432 | 1 | Let $k⟨X⟩$ be a free associative algebra generated by a set $X$ over a field $k$. Let $S$ be a set of $k$-algebra relations. Then what does it mean by the set of relations are independent ?
| https://mathoverflow.net/users/77963 | Independent set of relations in an algebra | Do you look $k<X>$ as a right module (over $k<X>$ itself) ? or a left-right bimodule ? (which seems to indicate $k$-algebra relations, look at the comment of Todd Trimble).
There is a theorem that I used many times to construct minimal automata (with multiplicities) and reduce the set of relations in a presented alg... | 0 | https://mathoverflow.net/users/25256 | 214434 | 101,359 |
https://mathoverflow.net/questions/214436 | 1 | I would like your help understanding this [article](http://web.mit.edu/kjb/www/Principal_Publications/A_posteriori_Error_Estimation_Techniques_in_Practical_Finite_Element_Analysis.pdf).
Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing.
Things I d... | https://mathoverflow.net/users/77965 | Cea's lemma and norms | About the different norms in (14) and (15) : only the inequality in the *energy* norm $||.||\_V$ stems from Céa's lemma.
The one involving $||.||\_{L^2}$ is said to be proven in Bathe's textbook. It is classical in finite element analysis for the model problem considered here.
In the simplest case of the Dirichlet ... | 2 | https://mathoverflow.net/users/75422 | 214444 | 101,362 |
https://mathoverflow.net/questions/214443 | 6 | I know there are some examples of manifolds which don't admit a PL structure (combinatorial triangulation), and that it has been recently proven that in dimension $n\geq5$ there are manifold which are not triangulable (i.e. which are not homeomorphic to a simplicial complex).
As far as I understand, in dimension $4$ ... | https://mathoverflow.net/users/52926 | Example of a triangulable topological manifold which does not admit a PL structure | The answer is **yes**, see Rudyak's paper [Piecewise linear structures on topological manifolds](http://arxiv.org/abs/math/0105047), Examples 21.4:
>
> *There are topological manifolds that can be triangulated as simplicial complexes but do not admit any* PL *structure.*
>
>
>
Such examples exist in fact in a... | 8 | https://mathoverflow.net/users/7460 | 214447 | 101,363 |
https://mathoverflow.net/questions/214422 | 5 | Let $\mathbb{N}=\{1,2,3,\ldots\}$ be the set of positive integers. For $n,k\in\mathbb{N}$ we define $$\text{Sol}(n,k) = \{(a,b,c)\in \mathbb{N}^3: |a^n + b^n - c^n| \leq k\}.$$ (The set $\text{Sol}(n,k)$ denotes the solutions of the inequality $|a^n + b^n - c^n| \leq k$ for fixed $n,k$.)
Moreover, for $j\in\mathbb{N}... | https://mathoverflow.net/users/8628 | Fermat's Last Theorem "$\pm k$" | A 4-variable version of the infamous ABC Conjecture says the following: Let $a,b,c,d\in\mathbb{Z}$ be non-zero, satisfy $a+b+c+d=0$ and $\gcd(a,b,c,d)=1$, and no subsum of two or three of $a,b,c,d$ equal to $0$. Then for every $\epsilon>0$ there is a constant $K\_\epsilon$ such that
$$ \max\{|a|,|b|,|c|,|d|\} \le K\_\... | 12 | https://mathoverflow.net/users/11926 | 214452 | 101,368 |
https://mathoverflow.net/questions/214340 | 7 | Let $C \rightarrow \mathbb P^6$ be a genus 7 canonically embedded (singular) Gorenstein generically reduced curve. Are there any examples of such a curve so that the conormal sheaf $N^\vee\_{C/\mathbb P^6} \cong \mathscr I\_C/ \mathscr I\_C^2$ is not locally free, or do all genus 7 Gorenstein canonical curves have loca... | https://mathoverflow.net/users/75970 | Example of Genus 7 Curve whose Conormal Sheaf isn't Locally Free | There are examples already beginning in genus $5$, and thus in all higher genera (including genus 7). By Serre's Corollary, Corollary 21.20 of the following, you **cannot** find examples with genus $3$ or $4$ (the problem does not make sense for $g=2$, since the canonical map cannot be a closed immersion). The construc... | 5 | https://mathoverflow.net/users/13265 | 214454 | 101,369 |
https://mathoverflow.net/questions/214335 | 3 | Let $P\_n(t) = p\_0 + p\_1 t + \cdots + p\_n t^n$ be a polynomial (with real coefficients) of degree $n$ in the variable $t$. I am interested in the quantity $$\Phi\_n = \min\_{\sum\_{i=1}^n p\_i^2 = 1} \int\_0^1 P\_n^2(t).$$
In particular: how fast does $\Phi\_n$ approach zero as $n \rightarrow \infty$? Can anything ... | https://mathoverflow.net/users/77908 | Polynomial with the smallest area | This summarizes various comments from above. Write $p=(p\_0,p\_1,\ldots, p\_n)$. Since
$$
\Phi\_n = \min\_{\|p\|\_2 =1} \sum\_{j,k=0}^n \frac{p\_jp\_k}{j+k+1} ,
$$
we can also interpret $\Phi\_n$ as the smallest eigenvalue of the (positive definite) Hilbert matrix $H\_{jk}=1/(j+k+1)$, $j,k=0,1, \ldots , n$.
We find t... | 4 | https://mathoverflow.net/users/48839 | 214458 | 101,372 |
https://mathoverflow.net/questions/201321 | 7 | Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim\_{r\to0}\dim(B(x,r))$; this limit exists because dimension depends monotonously on the set.
What can this dimension function look like?
Are there any res... | https://mathoverflow.net/users/55893 | How can dimension depend on the point? | Lars Olsen **[1]**, **[2]** (2005, 2005) has proved some results about this notion. Let $E \subseteq {\mathbb R}^{n}$ and $x \in {\mathbb R}^{n},$ where $n$ is a fixed positive integer. Let $\dim\_{H}(E,x)$ and $\dim\_{P}(E,x)$ denote the local Hausdorff and local packing dimensions of $E$ at $x,$ defined as in your qu... | 8 | https://mathoverflow.net/users/15780 | 214470 | 101,377 |
https://mathoverflow.net/questions/213182 | 6 | Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$.
If $\xi$ has Euler characteristic 0, then apparently there is a theorem of Hirschowitz and Gottsche that general sheaf in the moduli space has no cohomology.
But now suppose that... | https://mathoverflow.net/users/19088 | Do general sheaves on P^2 have cohomology governed by their Euler characteristic? | See Le Potier's Lectures on Vector Bundles, p 232, Theorem 18.1.1, where he quotes Gottsche and Hirschowitz. Part (ii) is exactly what was requested. The hypothesis needed is $\mu>-3$. (The example $I\_p(-3)$ shows that some hypothesis like this is necessary.)
| 1 | https://mathoverflow.net/users/19088 | 214478 | 101,379 |
https://mathoverflow.net/questions/214479 | 4 | I hope this is not too basic or obvious a question.
Let $d\_1$ and $d\_2$ be metrics on the same set $X$, with $d\_1$ being separable and $d\_2$ not being separable. Is it possible that $d\_1$ and $d\_2$ generate the same Borel $\sigma$-algebra? If so, does the answer change if we require $d\_1$ to be Polish?
Thank... | https://mathoverflow.net/users/15570 | Is it possible for a separable metric and a non-separable metric to have the same Borel $\sigma$-algebra? | It is possible that $(X,d\_{1})$ has the same Borel sets as $(X,d\_{2})$ when $d\_{1}$ is separable and $d\_{2}$ is not by assuming the Martin's axiom and the negation of the continuum hypothesis by the answer <https://mathoverflow.net/a/155527/22277> by Andreas Blass to my question. A set $X$ is said to be a $Q$-set i... | 6 | https://mathoverflow.net/users/22277 | 214482 | 101,381 |
https://mathoverflow.net/questions/214459 | 5 | Let $f$ be an automorphism of the octonions algebra.
Then $f(x)=x$ for $x\in \mathbb R$ and $f$ restricted to $Im \mathbb O$ is in $SO(7)$.
By the properties of the rotations there is an orthonormal basis: $e\_1,e\_2,...,e\_7$ in $\mathbb O$ and real numbers $\phi\_1,\phi\_2,\phi\_3$ such that:
$$
f(e\_1)=e\_1,\\
f(e\_... | https://mathoverflow.net/users/77977 | A question about a form of elements of G2 | You are essentially asking whether or not every element of $G\_2$ is conjugate to an element in its maximal torus. As you remark, every element of $G\_2$ is conjugate to an element in the maximal torus of $\mathrm{SO}(7)$, that much is obvious from, as you say, the properties of rotations in dimension $7$.
Now, for ... | 12 | https://mathoverflow.net/users/13972 | 214483 | 101,382 |
https://mathoverflow.net/questions/214499 | 0 | Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau\_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true?
>
>
> >
> > $(X,\tau)$ is connected if and only if for all $x\_0, x\_1\in X$ there is a continous, injective map $f:\... | https://mathoverflow.net/users/8628 | "Universal" connected spaces | No, there is not, even if you do not require $f$ to be injective. To see this, let $F$ be any ultrafilter on $\kappa$, and consider the following space $X\_F$. The underlying set of $X\_F$ is $\kappa\cup \{x\_0,x\_1\}$, with topology generated by the sets $\{\alpha\}$ for $\alpha\in\kappa$, $\{x\_0\}\cup\kappa$, and $\... | 8 | https://mathoverflow.net/users/75 | 214505 | 101,386 |
https://mathoverflow.net/questions/214481 | 0 | Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu\_u\mathcal H^{N-1}\lfloor S\_u
$$
where $S\_u$ is the jump set of $u$.
Then define set $\mathcal F$ such that
$$
\mathcal F:=\{v\in... | https://mathoverflow.net/users/62560 | Is this set of function belongs to $L^\infty$? | Let $\Omega=[-1,1]$ and $u(x)=u(-x)=0$ for $x\in[2^{-2k+1},2^{-2k+2})$ and $u(x)=u(-x)=2^{-k}$ for $x\in[2^{-2k},2^{-2k+1})$, $k=1,\ldots,n,\ldots$.
Doesn't this $u$ belong to your space $SBV\cap L^\infty$ ? Because if no condition is imposed on the jumps, the set $\mathcal F$ clearly contains piecewise constant func... | 1 | https://mathoverflow.net/users/75422 | 214509 | 101,388 |
https://mathoverflow.net/questions/214507 | 1 | Our situation is following. Assume that we have free product $\star\_{i<n} G\_i$ each $G\_i$ finite group and assume that we have normal subgroup $K$ such that composition of canonical embedding and the quotinet mapping $G\_i\to \star\_{i<n}G\_i\to \star\_{i<n}G\_i/K$ is still embedding. Is there bigger normal subgroup... | https://mathoverflow.net/users/78029 | embedding of finite groups into product | The answer is no, unless you restrict somehow the prime divisors of $|G\_i|$. Take $G\_i = \mathbb{Z}/p\_i$ where $p\_i$ is the $i$-th prime number. Take $K$ to be the trivial subgroup. Now, if you have such a subgroup $L$ then on the one hand $\*\_i G\_i/L$ is finite, and on the other hand, it contains an element of o... | 2 | https://mathoverflow.net/users/41644 | 214511 | 101,389 |
https://mathoverflow.net/questions/214517 | 14 | If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not simplicially enriched, then I could take the hammock localisation first and then take the coherent nerve. I have no technica... | https://mathoverflow.net/users/10366 | Quasicategories for non-simplicial model categories | It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a [recent result of Lennart Meier](http://arxiv.org/abs/1503.02036), a certain "double cosubdivision" of the Rezk classification diagram of a model category is a complete ... | 14 | https://mathoverflow.net/users/11640 | 214522 | 101,392 |
https://mathoverflow.net/questions/214400 | 4 | Let $D$ be a probability distribution on the unit interval $[0,1]$ with moments $\mu\_i=\mathbb{E}\_D [x^i]$. Let $\delta(x)$ be a singleton probability distribution with all weight at $x\in [0,1]$. Let $C(n)=\lfloor (n+2)/2 \rfloor$.
Then we can match the first $n$ moments of any $D$ using a convex combination of $C... | https://mathoverflow.net/users/8938 | Matching moments in even dimensions | The answer to question #2 is no. Consider for example $n = 2$ with the measure $\delta(0)$. You need to choose $x\_1 = 0$, but the choice of $x\_2$ is arbitrary as $\alpha\_2 = 0$.
The answer to question #1 is yes. For a more detailed analysis, I refer to theorem 2.2.3 in Dette/ Studden's *The Theory of Canonical Mom... | 4 | https://mathoverflow.net/users/78041 | 214527 | 101,394 |
https://mathoverflow.net/questions/214526 | 7 | Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
| https://mathoverflow.net/users/8628 | Countable path-connected Hausdorff space | No, a path-connected Hausdorff space is arc-connected, whence it would be of (at least) continuum cardinality provided it has more than one point. This follows from a more general (and deep) result that a Peano space (a compact, connected, locally connected, and metrizable space) is arc-connected if it is path-connecte... | 22 | https://mathoverflow.net/users/2926 | 214530 | 101,395 |
https://mathoverflow.net/questions/214512 | 4 | I have been working with something related to Goldman bracket for $G\_2$ gauge group. There I have something like "$\text{Tr}(M\_{\gamma}O\_i)$", where $M\_{\gamma}$ is a monodromy which takes value in the group $G\_2$ (in the fundamental representation) and $O\_i, i=1,2,\ldots,7$" are the 7x7 skew-symmetric matrices t... | https://mathoverflow.net/users/78032 | Action of G_2 on certain 7x7 skew-symmetric matrices | Yes. This follows from the fact that $G\_2$ acts on the octonions via automorphisms.
Let $e\_i$, $i=1,\dots,7$ be a choice of 7 imaginary octonion units and let $1$ denote the identity. Then by definition of the $O\_i$, we have that
$$
e\_i e\_j = \sum\_k e\_k (O\_i)\_{kj} - \delta\_{ij} 1
$$
where the left-hand side... | 5 | https://mathoverflow.net/users/394 | 214533 | 101,396 |
https://mathoverflow.net/questions/212454 | 21 | I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C\*-algebra of $2\times 2$ complex matrices, let $\mathcal{B}$ be the C\*-subalgebra of $2\times 2$ diagonal matrices, and let $v$ and $w$ b... | https://mathoverflow.net/users/23141 | Separating pure states on the $2\times 2$ matrix algebra | The answer to my question is no. Let $\mathcal{A} = M\_2$ be the algebra of $2\times 2$ complex matrices, let $\mathcal{B}$ be the subalgebra of diagonal matrices, let $\mathcal{A}'$ be a C\*-algebra which unitally contains $\mathcal{A}$, and let $\mathcal{B}'$ be a C\*-subalgebra of $\mathcal{A}'$ which contains $\mat... | 7 | https://mathoverflow.net/users/23141 | 214536 | 101,397 |
https://mathoverflow.net/questions/214531 | 6 | Recall that $O(3,1)$ is the collection of matrices $A\in M\_4(\mathbb R)$ such that
$$A\begin{pmatrix}1 &&&\\&1&&\\&&1&\\&&&-1\end{pmatrix}A^T=\begin{pmatrix}1 &&&\\&1&&\\&&1&\\&&&-1\end{pmatrix}.$$
Let $\psi\in \mathfrak o(3,1)$ be an element in its Lie algebra. Can we find a $Q\in O(3,1)$ such that
$$Q\psi Q^{-1}=\b... | https://mathoverflow.net/users/48006 | A question about $O(3,1)$ | No. E.g. $\psi=\begin{pmatrix}0&0&-1&1\\0&0&0&0\\1&0&0&0\\1&0&0&0\end{pmatrix}$ is nilpotent: $\psi^3=0$. If your $\phi=\begin{pmatrix}0&a&0&0\\-a&0&0&0\\0&0&0&b\\0&0&b&0\end{pmatrix}$ was $Q\psi Q^{-1}$, we would have $\phi^3=\begin{pmatrix}0&-a^3&0&0\\a^3&0&0&0\\0&0&0&b^3\\0&0&b^3&0\end{pmatrix}=0$, whence $a=b=\phi=... | 13 | https://mathoverflow.net/users/19276 | 214540 | 101,401 |
https://mathoverflow.net/questions/214445 | 6 | Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The space of functions that are Dunford integrable, denoted by $\mathbb{D}(\mu,X)$, is a normed space with
$$
\|F\|:=\sup\lef... | https://mathoverflow.net/users/77968 | Is the space of vectorial functions that are Dunford integrable complete? | *This a revised and expanded version of my post.*
**No**, it is not complete. For simplicity suppose that $X$ is reflexive and separable in which case Dunford and Pettis integrals coincide. Suppose also that $\mu$ is finite and non-atomic. In this case the space of Pettis integrable functions is complete if and only ... | 6 | https://mathoverflow.net/users/15129 | 214557 | 101,409 |
https://mathoverflow.net/questions/209634 | 8 | For the purposes of this question, a *dynamical system* means a compact metric space $X$ together with a continuous map $f: X \to X$.
For $x \in X$, the *$\omega$-limit set of $x$*, denoted $\omega(x)$, is the set of limit points of the orbit of $x$. That is,
$$\omega(x) = \bigcap\_{n \in \omega} \overline{\{f^m(x) :... | https://mathoverflow.net/users/70618 | Is there a universal $\omega$-limit set? | The answer is no, there is no universal $\omega$-limit set.
The same is true for the classes of metric minimal dynamical systems and metric dynamical systems in general. I have written up proofs of these facts:
<http://www.math.uni-hamburg.de/home/geschke/papers/NoUniversalMetricOmegaLimitSet.pdf>
The basic idea... | 4 | https://mathoverflow.net/users/7743 | 214558 | 101,410 |
https://mathoverflow.net/questions/214555 | 5 | This must be surely known but I couldn't locate this problem in the literature. It popped out in *a priori* unrelated approximation problem but if true, would help me greatly.
Let $p\in (1,\infty)$.
Informal version:
>
> Do the spaces $\ell\_2^k$ sit well-complemented in all sufficiently large finite-dimeniona... | https://mathoverflow.net/users/15129 | Well-complemented copies of $\ell_p^n$ | If I understand you correctly, much stronger statement holds for any space with nontrivial type, see Theorem 15.10 in the book of Milman and Schechtman on Asymptotic Theory.
| 9 | https://mathoverflow.net/users/37822 | 214561 | 101,411 |
https://mathoverflow.net/questions/129202 | 7 | Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a continuous map on $X$) and a pair of points $x,y \in X$ with $y$ uniformly recurrent and proximal to $x$, such that for som... | https://mathoverflow.net/users/14988 | Are irrational multiples of central sets again central? | The answer is yes, it was pointed out to me by Vitaly Bergelson that it follows from Theorem 6.1 (d) of a paper by Bergelson, Hindman and Kra (Trans. Amer. Math. Soc. 348 (1996), no. 3, 893–912).
The notation $g\_{\alpha,\gamma}[A]$ is defined in the previous section and means
$$g\_{\alpha,\gamma}[A]:=\{\lfloor \alpha ... | 5 | https://mathoverflow.net/users/18698 | 214562 | 101,412 |
https://mathoverflow.net/questions/214594 | 1 | For any topological space $(X,\tau)$ we define $$R\_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R\_{im}(X,\tau)$ is reflexive, and transitivity follows from the fact that the composition of two continuous surjective maps is continuous and surjective... | https://mathoverflow.net/users/8628 | Hausdorff spaces with asymmetric image relation | There are many such spaces.
For example $\omega\_1+1= [0,\omega\_1]$ with
the order topology. A continuous map $f$ with $f(\omega\_1)=0$ must be eventually constant, hence can have only countably many values.
But there is a continuous surjective selfmap $g$ mapping $0$ to $\omega\_1$.
| 4 | https://mathoverflow.net/users/14915 | 214598 | 101,419 |
https://mathoverflow.net/questions/214579 | 2 | It is known that Cramér–Rao\_bound is the lower bound of variance of a parameter. A useful link is <https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound> There is also a term called 'superefficient', mentioned in the link: <https://en.wikipedia.org/wiki/Hodges%27_estimator> which claims that: "In general, any su... | https://mathoverflow.net/users/71105 | Is there any parameter space of Cramér–Rao_bound | The basic form of the Crammer-Rao bound is: if $\hat \theta$ is **unbiased** (or at least locally unbiased around $\theta\_0$) then:
$$ var(\hat \theta | \theta\_0) \geq I\_F^{-1}(\theta\_0) $$
The general form is: define the function $e(\theta) = E(\hat \theta | \theta)$. Then:
$$ var(\hat \theta | \theta\_0) \g... | 2 | https://mathoverflow.net/users/75496 | 214605 | 101,421 |
https://mathoverflow.net/questions/214571 | 7 | If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a regular submanifold of $M$.
*Question*: Is there an analogous result for maps between manifolds modeled off locally conv... | https://mathoverflow.net/users/43445 | Submersion theorem for smooth tame Frechet manifolds | There are a few works which study submersions in the locally convex setting. The most extensive (and recent) is by Helge Glöckner (<http://arxiv.org/abs/1502.05795>). Note that the basic results about submersions and immersions still hold true in this general setting, at least if you use the right notion of sub/immersi... | 7 | https://mathoverflow.net/users/17047 | 214609 | 101,422 |
https://mathoverflow.net/questions/214576 | 2 | Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution.
$$
(f\ast g )(x)=\int\_{-\infty}^{+\infty} f(y)g(x-y)dy.
$$
Assume that $(f\ast g) (x)=0$ on an open interval $x\in (a,b)$. Does this imply that $g(x)$ takes ... | https://mathoverflow.net/users/37987 | Convolution vanishes on an interval | Yes, because $f\*g$ is indeed *real-analytic* (provided $g$ doesn't grow too fast at $\infty$), so that, being $0$ on an interval, it is $0$ everywhere. Then the Fourier transform of $f\*g$, which is a Gaussian times the Fourier transform of $g'$, is $0$, and as the Gaussian is nonzero everywhere this implies $g'=0$ as... | 5 | https://mathoverflow.net/users/75422 | 214612 | 101,423 |
https://mathoverflow.net/questions/209846 | 5 | Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha\_1,\alpha\_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless $\alpha$ is ``simple''. More precisely, suppose that
* f(x,y) vanishes to order at least $0.99\cdot \deg f$ at $(\al... | https://mathoverflow.net/users/806 | Order of vanishing of an integer polynomial at a point | The solution to this question appeared in Section 4 of [my paper on the ranks of matrices with few distinct entries](http://arxiv.org/abs/1508.00145). The solution borrows an idea from the Dracula's answer. Please upvote his answer. Below is a sketch of the solution.
**Step 1)**
Let $V$ be the set of points where $f$... | 1 | https://mathoverflow.net/users/806 | 214628 | 101,428 |
https://mathoverflow.net/questions/214615 | 2 | For any topological space $(X,\tau)$ we define $$R\_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R\_{im}(X,\tau)$ is reflexive. This relation is also transitive because the composition of two continuous surjective maps is continuous and surjective.
... | https://mathoverflow.net/users/8628 | Reconstructing relations with the image relation of a topology | The answer is no to the general question, and also to question (b), for the following simple reason (which works whether or not the space is finite): if $f:X\to X$ is surjective and $f(x)=y$ for some $x\neq y$, then there must be some $w\neq x$ with $f(w)=x$. Thus, the relation $R\_{im}$ must have the property that whe... | 5 | https://mathoverflow.net/users/1946 | 214633 | 101,431 |
https://mathoverflow.net/questions/214626 | 1 | (In the following, a (not necessarily commutative) ring $R$ is *Gorenstein* if it has finite injective dimension as a module over itself on either side, and a finitely generated (right) $R$-module is *maximal Cohen-Macaulay* (MCM) if $\text{Ext}\_R^i(M,R)=0$ for all $i \geqslant1$.)
In the paper [Maximal Cohen-Macaul... | https://mathoverflow.net/users/78095 | Showing that the stable module category of a ring $R$ restricted to maximal Cohen-Macaulay objects is trivial if $\text{gldim } R < \infty$ | If $M$ is a maximal Cohen-Macaulay module for a Noetherian Gorenstein ring $R$, then it follows easily by induction on the projective dimension of $N$ that $\operatorname{Ext}^i\_R(M,N)=0$ for all $i>0$ if $N$ is finitely generated of finite projective dimension.
So if $R$ also has finite global dimension, then $\ope... | 4 | https://mathoverflow.net/users/22989 | 214635 | 101,432 |
https://mathoverflow.net/questions/214627 | 2 | Let $\omega\in L^1\_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$,
$$
\frac{1}{|B(x,r)|}\int\_{B(x,r)}\omega(y)dy\leq C\omega(x)
$$
for any $x\in\mathbb R^N$ and $r>0$.
We define the weighted $L^1\_\omega$ space by, for function $u$,
$$
\int\_{\mathbb R... | https://mathoverflow.net/users/62560 | The convolution between weighted $L^1$ space and normal $L^1$ space | This will never work in the kind of situation you outline. As soon as $\omega$ gets large somewhere, you're doomed. For instance, let's assume that we can find disjoint balls $B\_n$ of radius $1$, such that $\int\_{B\_n} \omega\ge n^2$. Then take $u$ as the characteristic function of the ball of radius $3$ about the or... | 1 | https://mathoverflow.net/users/48839 | 214641 | 101,435 |
https://mathoverflow.net/questions/214625 | 6 | We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
>
> Is it true that for each irreducible $g(X) \in \mathbb{Q}[X]$ there exists some
> nonzero $h(X) \in \mathbb{Q}[X]$ such that $g(X)h(X)$ is a trinomial... | https://mathoverflow.net/users/38889 | Is every polynomial a factor of a trinomial? | To summarize some of the discussion in the comments: a trinomial can have at most four distinct real roots. A random polynomial of degree $d$ (random = all coefficients are iid, there are other models) has $\Omega(\log d)$ real roots. Which means that the probability that a random polynomial divides a trinomial goes to... | 8 | https://mathoverflow.net/users/11142 | 214650 | 101,441 |
https://mathoverflow.net/questions/214603 | 6 | Of course the general answer to the question in the title is: not very simple.
I could not think of a better title, so let me explain my question in more detail.
I have a number field $E/\mathbb{Q}$, and a simple group $H/E$.
Let $G$ denote the Weil restriction of scalars of $H$ from $E$ to $\mathbb{Q}$.
Then $G$ is ... | https://mathoverflow.net/users/21815 | How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$? | I assume that the question is: Is it true that there is always a prime $\ell$ such that $G\_\ell$ has no absolutely simple direct factors?
The answer is YES. There are infinitely many such $\ell$.
>
> **Claim.** Let $\Gamma$ be a finite group acting transitively on a finite set $S$ of cardinality $n>1$.
> Then $... | 3 | https://mathoverflow.net/users/4149 | 214658 | 101,443 |
https://mathoverflow.net/questions/214665 | 3 | Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be specified.
$\begin{equation}
f(n) =\sum\_{k=0}^{n/2}\sum\_{l=0}^{n/2}\sum\_{i=0}^{k}\sum\_{j=0}^{l}\frac{1}{n-k-l}\fr... | https://mathoverflow.net/users/78062 | Closed Form Expression for Nested Series Summation? | Your sum does not appear to have a closed form. However, everything you ever wanted to know about summations like this can be found in ["A=B" by Petkovsek, Wilf, and Zeilberger.](https://www.math.upenn.edu/~wilf/AeqB.html)
| 3 | https://mathoverflow.net/users/11142 | 214666 | 101,448 |
https://mathoverflow.net/questions/214656 | 8 | It's a classical result of Ahlfors that, for any sufficiently nice n-connected domain $\Omega \subset \mathbb C$ there is a holomorphic branched covering $f: \Omega \rightarrow \mathbb D$ to the disk $\mathbb D$, which extends continuously to the boundary and maps the boundary curves of $\Omega$ monotonically onto the ... | https://mathoverflow.net/users/78108 | A Generalization of the Ahlfors function to have varying degrees? | The answer to your question is yes. Indeed, one can replace $\mathbb{D}$ with the right half plane, applying a Mobius transformation. Then the argument is quite simple if you are familiar with the following classical theorem of Bieberbach :
**Theorem**
Let $\Omega$ be a domain bounded by $n$ non-intersecting Jordan... | 4 | https://mathoverflow.net/users/1162 | 214680 | 101,454 |
https://mathoverflow.net/questions/214677 | 45 | The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space $\mathbb{Q}$ with its usual absolute value, or as the Dedekind completion of the ordered set $\mathbb{Q}$ with its usual ordering.
When these two constructions are performe... | https://mathoverflow.net/users/49 | The formal p-adic numbers | Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function with values in $\mathbb{Z}\_p$ corresponds to an element of the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as ... | 26 | https://mathoverflow.net/users/22131 | 214695 | 101,458 |
https://mathoverflow.net/questions/206927 | 15 | Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy between the identity and $Y\rightarrow X\rightarrow Y$, and a couple of less relevant vanishing formulas hold. Moreover... | https://mathoverflow.net/users/12166 | Homotopy transfer in the opposite direction | Let us denote by $p\colon Y\to X$ and $i\colon X\to Y$ the maps of your SDR. Since $pi=\mathop{\mathrm{id}}\nolimits\_X$, the map $i$ is injective, and is an isomorphism with its image. The map $\pi=i\circ p$ is a projector (that is, $\pi^2=\pi$); this projector implements a splitting $Y=\ker(\pi)\oplus\mathop{\mathrm{... | 3 | https://mathoverflow.net/users/1306 | 214697 | 101,459 |
https://mathoverflow.net/questions/214694 | 0 | Let $f$ be a modular form of an even weight $k$ over the modular group $SL\_2(Z).$ Denote $\lambda\_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in finding the expression of the sum $\sum\_{d|n} \lambda\_f\left(\frac{n^2}{d^2}\right)$ in terms of $ \lambda\_f(n)$ o... | https://mathoverflow.net/users/76102 | Expression of a sum of Hecke eigenvalues in terms of one Hecke eigenvalue | From the Hecke relation, we get
$$ \lambda\_f(n)^2 =\sum\_{d|n} \lambda\_f\left(\frac{n^2}{d^2}\right).$$
| 0 | https://mathoverflow.net/users/76102 | 214698 | 101,460 |
https://mathoverflow.net/questions/192014 | 16 | Graph isomorphism problem is one of the longest standing problems that resisted classification into $P$ or $NP$-complete problems. We have evidences that it can not be $NP$-complete. Firstly, Graph Isomorphism can not be $NP$-complete unless the polynomial hierarchy [1] collapses to the second level. Also, the counting... | https://mathoverflow.net/users/8784 | Evidence that Graph Isomorphism problem is not $NP$-complete | A search done for a [related question](https://cstheory.stackexchange.com/questions/32160/what-evidence-is-there-that-graph-isomorphism-is-not-in-p) also turned up (as indicated in a comment) [rjlipton.wordpress.com/2015/03/05/news-on-intermediate-problems](https://rjlipton.wordpress.com/2015/03/05/news-on-intermediate... | 4 | https://mathoverflow.net/users/20781 | 214699 | 101,461 |
https://mathoverflow.net/questions/212803 | 1 | Let $B=\mathcal{O}\_R\left(GL(n)\right)$ be a localization of the algebra $A(R)$ of functions on the quantum formal group corresponding to the matrix $R$ ["Quantization of Lie groups and Lie algebras", Faddev, Reshetikhin, Takhtajan] at $\mathrm{det}$ (quantum determinant), where $\mathrm{det}$ is a central grouplike e... | https://mathoverflow.net/users/75934 | Radicals of co-quasitriangular map | The co-quasitriangular structure $r$ is a bilinear map over the ground field. Hence one can define right and left radicals as in linear algebra.
Note that the quantum determinant is a sum, e.g. in an example presented in the paper by Faddev-Rheshitikhin-Takhtajan,
$
\operatorname{det}\_q T= \sum\_{s∈S\_n} (-q)^{l(s... | 1 | https://mathoverflow.net/users/33854 | 214705 | 101,463 |
https://mathoverflow.net/questions/214651 | 1 | This question is a follow up to this [question](https://mathoverflow.net/questions/212671/bounded-input-bounded-output-stability-for-heat-equation).
Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u\_d:\partial\Omega \to \mathbb{R}$ be in $H^{1/2}(\partial \Omega)$. Let $... | https://mathoverflow.net/users/76602 | Does this time-dependent trace space have a name? | If you insist on "for *all* $t$ " as distinct from "for almost every $t$ ", one possible space is $C^1([0,\infty);H^{-1/2})\cap C^0([0,\infty);H^{1/2})$. A possible extension is the unique harmonic function in $\Omega$, the minimizer of $\int\_\Omega |\nabla v|^2$ : this extension operator $E$ is well known to map $H^{... | 1 | https://mathoverflow.net/users/75422 | 214707 | 101,464 |
https://mathoverflow.net/questions/214683 | 4 | I am wondering what conditions a Lorentzian manifold $(M,g)$ must satisfy to ensure the existence of a global proper-time foliation (i.e. a decomposition of $M$ into spacelike Cauchy hypersurfaces and a global time function whose tangent vector field $\vec{t}$ satisfies $g(\vec{t},\vec{t})=-1$ and $g(\vec{t},\vec{x})=0... | https://mathoverflow.net/users/25490 | Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation? | Global hyperbolicity only gives you a Cauchy time function, whose gradient is past directed timelike. See Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes - Antonio N. Bernal, Miguel Sanchez - Commun. Math. Phys. 257, 43–50 (2005)
| 3 | https://mathoverflow.net/users/47189 | 214712 | 101,467 |
https://mathoverflow.net/questions/214548 | 12 | Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$.
EDIT: I forgot to add that no two vectors should be colinear.
| https://mathoverflow.net/users/41283 | Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$ | The maximum is $cn^{4/3}$ for some constant $c$.
We may as well assume all our points are on the unit sphere $S$. Let $P$ be some plane not containing the origin, which we might think of as being far away. For each point $x\in S$ in our collection let $p\_x$ be the intersection of the line through $0$ and $x$ with $P... | 12 | https://mathoverflow.net/users/20598 | 214715 | 101,469 |
https://mathoverflow.net/questions/214704 | 7 | We got a cubic polynomial which is unexpectedly prime rich.
Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and
$\pi\_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.
Let $F(n)=\frac{\pi\_f(n)}{\frac{n}{\log{n}}}$.
$F(n)$ is greater than one on $F(10^n)$ experimentally increasing
for $n \ge 4$.
... | https://mathoverflow.net/users/12481 | Unexpectedly prime rich cubic polynomial | The expected constant in the [Bateman-Horn conjecture](https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture) is
$$\frac1d \prod\_p\frac{1-\frac{n\_p}{p}}{1-\frac1p},$$ where $n\_p$ is the number of roots of $f(x)$ modulo $p,$ and $d$ is the degree of $f(x).$ For the particular polynomial in question, this con... | 8 | https://mathoverflow.net/users/11142 | 214726 | 101,473 |
https://mathoverflow.net/questions/214727 | 20 | Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in dimension 7.) Yes, the topological category is fine, to avoid the smooth Poincaré conjecture.
| https://mathoverflow.net/users/11142 | Simply-connected rational homology spheres | In dimension 4, we have the following:
Simply-connectedness implies that $H\_1(M)=0$. The condition that $M$ be a rational homology sphere implies that $H\_2(M), H\_3(M)$ are finitely generated torsion groups. It follows that
$H^3(M) = Ext(H\_2(M),\mathbb{Z})$, which is noncanonically isomorphic to $H\_2(M)$ again (tha... | 38 | https://mathoverflow.net/users/39747 | 214730 | 101,475 |
https://mathoverflow.net/questions/87009 | 6 | It is a [known theorem](http://books.google.com.au/books?id=87vtu4HbdawC&lpg=PA114&ots=VtyWCoKPHj&dq=local%20bisections%20lie%20groupoid&pg=PA129#v=onepage&q&f=false) that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functor... | https://mathoverflow.net/users/4177 | Internal equivalence implies weak equivalence for Frechet Lie groupoids? | There is a characterisation of weak equivalences which works in internal categories in a) finitely complete categories and b) finite-dimensional Lie groupoids (and in fact Lie categories). In the case of Lie groupoids, an internal functor $f\colon X\to Y$ is a weak equivalence in the sense of being fully faithful and e... | 0 | https://mathoverflow.net/users/4177 | 214751 | 101,480 |
https://mathoverflow.net/questions/214566 | 20 | The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^\*$-algebras and continuous $\*$-homomorphisms. A well known generalization of $C^\*$-algebras are *pro-$C^\*$-algebras*. Pro-$C^\*$... | https://mathoverflow.net/users/42440 | The Gelfand duality for pro-$C^*$-algebras | The answer is No. Rougly, because it is not a good idea to look at continuous $\mathbb{C}$ valued function on a space which is not completely Haussdorff as completely haussdorf is exactly the hypothesis that says "your space can be understood by looing at function over it"... I really don't think you can obtain somethi... | 6 | https://mathoverflow.net/users/22131 | 214765 | 101,487 |
https://mathoverflow.net/questions/214766 | 16 | What are known estimates for maximal $M$ for which their exists subsets $A\_1,\dots,A\_M$ in $\{1,\dots,n\}$ such that there do not exist different indexes $i,j,k$ for which $A\_i\subset A\_j\cup A\_k$?
It is not hard to prove some exponential bounds $c\_1^n<M<c\_2^n$ for some $1<c\_1<c\_2<2$, but maybe sharp exponen... | https://mathoverflow.net/users/4312 | Maximal number of subsets in $\{1,\dots,n\}$ such that neither is contained in a union of two others | To best of my knowledge, the only bounds on this problem in the literature are in the the [old paper of Erdős, Frankl, and Füredi](http://www.math.uiuc.edu/~z-furedi/PUBS/furedi_erdos_frankl_2cover_free.pdf). See Theorem 2 there. If you look at the [MathSciNet review of the said paper](http://www.ams.org/mathscinet-get... | 6 | https://mathoverflow.net/users/806 | 214776 | 101,491 |
https://mathoverflow.net/questions/214728 | 44 | This will not be altogether unrelated to [this earlier question](https://mathoverflow.net/questions/213046/rearrangements-that-never-change-the-value-of-a-sum).
For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a\_i\}\_{i=1}^\infty$ of real numbers, if $\di... | https://mathoverflow.net/users/6316 | How many rearrangements must fail to alter the value of a sum before you conclude that none do? | **Update.** A research collaboration growing out of this question and some of its answers has now resulted in the following article, providing an account of the rearrangement number:
>
> A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, [The rearrangement number](http://jdh.hamkins.org/the-... | 34 | https://mathoverflow.net/users/1946 | 214779 | 101,493 |
https://mathoverflow.net/questions/214772 | 2 | I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes the angular coorinates taken together), the Laplacian looks like $\Delta = \frac {\partial ^2} {\partial r^2} + H(x,r) \fr... | https://mathoverflow.net/users/54780 | Limited expansion of mean curvature of geodesic spheres | See <http://link.springer.com/article/10.1007%2FBF02395060> Lemma 12.2 (the $\gamma\_i$ terms are defined on page 167, also see $\S2$ for the curvature notations).
An alternative "do-it yourself" approach to what you want might be to consider the expansion of the metric in normal coordinates, e.g. [Riemann's formula... | 3 | https://mathoverflow.net/users/1540 | 214790 | 101,497 |
https://mathoverflow.net/questions/214773 | 2 | This question is related to [a question recently asked](https://mathoverflow.net/questions/214684/does-every-set-x-have-a-topology-for-which-the-only-continuous-self-surjection) by Joel David Hamkins.
Let $(P,\leq)$ be a poset. We call it *surjectively rigid* if the only order-preserving surjective map $f:P\to P$ is ... | https://mathoverflow.net/users/8628 | Surjectively rigid partially ordered sets | I claim that if $X$ is a linear ordering and $f:X\rightarrow X$ is an order preserving surjective mapping which is not the identity mapping, then there is an order preserving injective mapping $g:X\rightarrow X$ which is not the identity mapping.
If $f$ is surjective but not the identity function, the there is some ... | 5 | https://mathoverflow.net/users/22277 | 214796 | 101,499 |
https://mathoverflow.net/questions/214442 | 15 |
>
> Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
>
>
>
Since our polynomial's Galois group over $\mathbb{Q}$ is $S\_5$, Chebotarev's density theorem implies that there are infinitely many such primes, but it is seemingly unclear how to find them.
If the Galois group o... | https://mathoverflow.net/users/38889 | Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes | The object (conjecturally) associated to an Artin representation by Langlands is not a classical modular form except in a very limited number of situations (odd two dimensional representations). Let $K$ be the splitting field of some polynomial over $\mathbf{Q}$ with Galois group $S\_5$. If $V$ is any finite dimensiona... | 12 | https://mathoverflow.net/users/78204 | 214800 | 101,501 |
https://mathoverflow.net/questions/214783 | 3 | Let $X$ and $Y$ be locally compact, second countable spaces, and let $φ:X→Y$ be a measurable function. Let $μ$ be a sigma-finite measure on $X$. In general, the push-forward $\phi\_{\*}\mu$ is not sigma-finite.
Question: what are the causes of this failure?
(naive?) conjecture: The only failure is when there is at ... | https://mathoverflow.net/users/3993 | Failure of a push-forward to be sigma-finite | This is an entirely measure theory question and has nothing to do with topology, continuity etc. Your conjecture is actually true if reformulated in a more appropriate way.
The key example is the following model situation: $X=I^2$ is the unit square, $Y=I$ is the unit interval, and $\phi:(x,y)\to x$ is the vertical c... | 3 | https://mathoverflow.net/users/8588 | 214808 | 101,506 |
https://mathoverflow.net/questions/214342 | 7 | I would like to know if there is any setting where the two notions of
* central charge of 2D conformal field theories,
* Calabi-Yau dimension of fractionally Calabi-Yau categories
can be understood as "**one and the same**".
One motivation (not very serious) is that
* on the one hand there are CY categories of... | https://mathoverflow.net/users/10881 | central charge and Calabi-Yau dimension | Given a $N=(2,2)$ two dimensional superconformal field theory (SCFT), one can construct two topological field theories called the $A$ and $B$ models. To each of these topological field theories, one should be able to associate a ($A\_\infty$) triangulated category of boundary conditions, called category of branes. Thus... | 12 | https://mathoverflow.net/users/25309 | 214817 | 101,508 |
https://mathoverflow.net/questions/214821 | 4 | I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}\_F$. In many articles one introduces the set $\hat{\mathcal{W}}\_F$ of all equivalence classes of irreducible representations of $\mathcal W\_F$.
It seems to me that strictly speaking this does not exist (due to set-... | https://mathoverflow.net/users/56228 | "set of all irreducible representations of a group", set-theoretic issues | It is easy to show that the cardinality of the underlying set of any irreducible representation is bounded in terms of the cardinality of $\mathcal{W}\_F$, so can you just take your favorite set $S$ of sufficiently large cardinality and consider only irreducible representations whose underlying set is a subset of $S$. ... | 11 | https://mathoverflow.net/users/75 | 214824 | 101,510 |
https://mathoverflow.net/questions/214823 | 4 | Let
\begin{align}
\mathfrak{g} = Span\_{\mathbb{C}}\{ e\_1, e\_2, e\_3, e\_4, e\_5: \text{ non-zero brackets are } [e\_1, e\_i]=e\_{i+1}, i=2,3,4, [e\_2, e\_3]=e\_5 \}
\end{align}
be a $5$-dimensional Lie algebra. I want to write $e\_1, \ldots, e\_5$ as matrices. That is, I need to find an injective homomorphism of Li... | https://mathoverflow.net/users/11877 | Faithful linear representation of a nilpotent Lie algebra | The Lie algebra is filiform nilpotent and is generated by $e\_1$ and $e\_2$. It is known that any faithful Lie algebra representation $\rho:\mathfrak{f}\_n\rightarrow \mathbb{gl}(V)$ of a $n$-dimensional filiform Lie algebra $\mathfrak{f}\_n$ is of degree at least $n$. In the above example, we do not need to invoke Ado... | 6 | https://mathoverflow.net/users/32332 | 214825 | 101,511 |
https://mathoverflow.net/questions/214830 | 3 | Let $\Omega$ be an open bounded set of $R^n$, and let $\omega$ be an open subset of $\Omega$ s.t $\overline{\omega} \subset \Omega.$
For $f\in H\_0^1(\omega)$, it is known that the extension of $f$ to $\Omega$ by $0$ is an element of $H\_0^1(\Omega).$
I wonder if the result remains true when we replace $H\_0^1$ wit... | https://mathoverflow.net/users/78215 | Extension by zero in Sobolev spaces | It does not remain true.
If $\omega=B(0,1)$ and $\Omega=B(0,2)$ and $f(x)=1-|x|^2$, then $f\in H^1\_0(\omega)\cap H^2(\omega)$ but the extension by zero is not in $H^2(\Omega)$.
| 6 | https://mathoverflow.net/users/55893 | 214833 | 101,514 |
https://mathoverflow.net/questions/214706 | 0 | Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda\_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to discuss the sign of this sum
$$S=\frac{\lambda\_f(a)\lambda\_f(b)}{(ab)^{3/4}} \sum\_{\substack{l=1\\\gcd(l,a)=\gcd(l,b)=... | https://mathoverflow.net/users/76102 | Discussion for the sign of a specific sum | Using The Euler product, we can express the sum
$$\sum\_{\substack{l=1\\\gcd(l,a)=\gcd(l,b)=1}}^{+\infty}\left(\frac{\lambda\_f(l)}{l^{3/4}}\right)^3$$ as an infinite product over prime numbers so that the sum $S.$
| 1 | https://mathoverflow.net/users/76102 | 214854 | 101,522 |
https://mathoverflow.net/questions/214860 | -2 | Original question:
The symbol looks like a numeral 1 written like an R in $\mathbb{R}$. It has a double vertical line and a serif at the bottom. It represents a function of a parameter: $1\_{\{0,1\}}(x)$. Adding it as a factor to your formula limits your expression to a specific set or range of x values. In my example,... | https://mathoverflow.net/users/43535 | Looking for the name of a mathematical symbol that looks remotely like 1 (answer: indicator function) | The function $\mathbb 1\_A$, whose values are $1$ for arguments in the set $A$ and $0$ for arguments outside $A$, is usually called the characteristic function of the set $A$.
Unfortunately, the same terminology is also used with other meanings. For example, in probability theory, "characteristic function" often mea... | 3 | https://mathoverflow.net/users/6794 | 214861 | 101,525 |
https://mathoverflow.net/questions/214863 | 1 | First let $\mathcal{V}$ be a [closed symmetric monoidal category](http://ncatlab.org/nlab/show/closed+monoidal+category)
and $\mathcal{M}$ be a category [enriched](http://ncatlab.org/nlab/show/enriched+category#InMonoidCat) over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is [cotensored, or powered](http://ncatlab... | https://mathoverflow.net/users/24965 | How to define the internal hom between presheaves valued in cotensored categories? | Yes, if $\mathcal M$ is reasonable. The object-wise internal hom $(s,t) \mapsto G(s)^{F(t)}$ is contravariant in $s \in \mathcal S$ and covariant in $t$. The object you want is the [end](http://ncatlab.org/nlab/show/end) of this functor, i.e. the maximal subobject of $\prod\_{s\in \mathcal S} G(s)^{F(s)}$ satisfying th... | 4 | https://mathoverflow.net/users/78 | 214865 | 101,528 |
https://mathoverflow.net/questions/214847 | 6 | Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$.
>
> What is the isomorphism class of the inertia group $I\_p$, and how do I find it?
>
>
>
I would be more happy with having some procedure for finding inertia groups, rather... | https://mathoverflow.net/users/38889 | Finding the inertia group | This answer produces $I$ somewhat indirectly. So it actually *does not* what the OP asked for.
The decomposition group $D$, which is the Galois group of $h(x)$ over $\mathbb Q\_2$, can be computed as follows: Using resultants, one sees that the minimal polynomial over $\mathbb Q$ of the difference of two distinct roo... | 3 | https://mathoverflow.net/users/18739 | 214874 | 101,531 |
https://mathoverflow.net/questions/214870 | 3 | Cross-posted from [MSE](https://math.stackexchange.com/questions/1398024/is-it-normal-surface-of-general-type-to-have-infinitely-many-positive-rank-ellip).
I am not good at algebraic geometry and almost surely am
misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and
would like to ... | https://mathoverflow.net/users/12481 | Is it normal surface of general type to have infinitely many positive rank elliptic curves? | I am only posting this as an answer because it annoys me to see a question like this listed as "unanswered", thus "hovering" near the top of the list of unanswered questions. If dhy wants to write up his comment as an answer, then I will delete this answer.
The surface given by the OP is as far as possible from being... | 10 | https://mathoverflow.net/users/13265 | 214889 | 101,537 |
https://mathoverflow.net/questions/214893 | 3 | Fix an integer $n>0$. Are there infinite subgroups of $SL\_2(\mathbb{C})$ such that every element is $n$-torsion?
| https://mathoverflow.net/users/791 | Infinite groups of finite exponent inside of SL(2,C) | A theorem of Burnside says that a linear group of finite exponent is finite. So the answer is no.
| 10 | https://mathoverflow.net/users/41644 | 214894 | 101,538 |
https://mathoverflow.net/questions/214888 | 8 | This is in some sense a specialization of the question [integral or rational cohomology of real grassmannians](https://mathoverflow.net/questions/195398/integral-or-rational-cohomology-of-real-grassmannians). Let $G\_3(\mathbb{R}^5)$ denote the real Grassmannian of (unoriented) $3$-planes in $\mathbb{R}^5$, which is a ... | https://mathoverflow.net/users/8103 | Cohomology of $G_3(\mathbb{R}^5)$ | The cohomology groups of the Grassmann manifold are worked out in combinatorial terms in Luis Casian and Yuji Kodama's paper, <http://arxiv.org/pdf/1309.5520v1.pdf>; they make a conjecture at the end about the multiplicative structure. The authors actually do the example you ask about; they compute the groups for $G\_2... | 12 | https://mathoverflow.net/users/3460 | 214896 | 101,539 |
https://mathoverflow.net/questions/214898 | 5 | Let $A$ be a finite dimensional associative algebra (with unity) over a finite field $F$. Let $L$ be a field extension of $F$. Suppose that after extending scalars to $L$, two elements $a,b$ of $A$ are conjugate, i.e. there is an invertible element $u \in A\otimes\_F L$ such that $a=ubu^{-1}$ . Does it follow that ther... | https://mathoverflow.net/users/56010 | Conjugation in associative algebras over finite fields | $\newcommand{\End}{\operatorname{End}}$
$\newcommand{\op}{{\operatorname{op}}}$
The finiteness of $F$ is not needed. I have learnt the idea of the following proof from [Torsten Ekedahl](https://mathoverflow.net/questions/28469/hilbert-90-for-algebras).
In the following, all algebras are associative and with unity. Th... | 3 | https://mathoverflow.net/users/2530 | 214902 | 101,542 |
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