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/210540 | 13 | Is there a transparent explanation of why the singular cochain complex of a topological space X is an $E\_\infty$ algebra. There are combinatorial proofs using, say, the surjection operad, but is there a topological picture behind those? If we wanted to restrict,say, to the level of little (2-)discs, could we describe ... | https://mathoverflow.net/users/11051 | E-infinity structure on singular cochains | If you wrote $E(n)$ for the chain complex of natural transformations $C\_\*(-) \to C\_\*(-)^{\otimes n}$, the $E(n)$ collectively form an operad, parametrizing all natural "one-to-many" transformations on chains, called the Eilenberg-Zilber operad. The defining co-action on chains turns into a natural action on cochain... | 12 | https://mathoverflow.net/users/360 | 210545 | 100,250 |
https://mathoverflow.net/questions/210526 | 1 | Let $X$ be a compact Riemann surface of genus $g\geq 1$. If $f$ is a meromorphic function on $X$ then, the meromorphic differential $\omega=\frac{df}{f}$ is a differential
of the third kind with integral residues.
**Q** Let $\omega$ be a meromorphic differential of the third kind on $X$ with integral
residues. Is the... | https://mathoverflow.net/users/11765 | criterion for a differential of the third kind to be a logarithmic derivative of a function | Yes, in principle. If the curve is given by $F(x,y)=0$ and the differential by $D(x,y)dx$
(every curve and differential can be described like this), then we look for a function in
the form $R(x,y)$ where $R$ is rational. The degrees of the numerator and denominator of $R$
are bounded by the residues of the differential... | 2 | https://mathoverflow.net/users/25510 | 210548 | 100,252 |
https://mathoverflow.net/questions/210552 | -1 | For a given number of nodes how many non-isomorphic graphs are available? Might be this is an open problem. For less number of vertices some computational statistics available.
I want to get all non-isomorphic graphs of order $5, 6, \dots 12$. Is there any complete list of all non isomorphic graphs available for a g... | https://mathoverflow.net/users/36977 | How to generate computational data in graph theory? | Use Brendan McKay's program geng, which is distributed with the nauty/Traces package and is available from <http://pallini.di.uniroma1.it/>.
There are about 165091172592 graphs on 12 vertices, so it might take you a while.
| 6 | https://mathoverflow.net/users/1492 | 210553 | 100,253 |
https://mathoverflow.net/questions/210550 | 3 | The following question was asked on math.stackexchange.com with no reply for the past week or so. Let $f : X \to Y$ be a morphism of smooth (integral) varieties over $\Bbb{C}$ with generic fiber equal to $\Bbb{P}^1$.
>
>
> >
> > Is it true that $R^if\_\ast \mathcal{O}\_X = 0$ for all $i > 0$?
> >
> >
> >
>
>... | https://mathoverflow.net/users/21278 | Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$ | No. Take $Y=\mathbb{A}^2\_\mathbb{C}$ and $X=U\times \mathbb{P}^1$ where $j:U\hookrightarrow Y$ is the complement of the origin. One immediately finds $R^1 f\_\*\mathscr{O}\_X=R^1 j\_\*\mathscr{O}\_U$, which is nonzero.
| 7 | https://mathoverflow.net/users/7666 | 210554 | 100,254 |
https://mathoverflow.net/questions/210108 | 1 | The spherical harmonics are given by
$$Y^m\_l(\phi,\theta):=N^m\_l e^{im\theta}P^m\_l(\cos \phi) $$
where $P^m\_l$ are the associated Legendre Polynomials and $N^m\_l$ is the normalisation.
From inspecting the first few it seems
$$\left\Vert Y^m\_l \right\Vert \leq C <\infty.$$
**Is this true?** If not is there an ea... | https://mathoverflow.net/users/19874 | Are spherical harmonics uniformly bounded? | As mentioned above, your question cannot be answered without knowledge of which norming sequence you are using. However, there are uniform bounds for the derivatives of the Legendre polynomials in the literature (see, e.g., pp. 523, 524 of the article reviewed in MR 0306897 by Guillemont-Tessier, which is available on ... | 1 | https://mathoverflow.net/users/75303 | 210561 | 100,256 |
https://mathoverflow.net/questions/210318 | 4 | Consider a probability distribution on $\mathbb{R}^k$, say $\mu$. Then there is a sequence of probability measures $\mu\_n$ that converge weakly to $\mu$ so that each of them is discrete (takes finitely values).
Question: Assume we replace $\mathbb{R}^k$ by any any locally compact Hausdorf group $G$. Is the previous ... | https://mathoverflow.net/users/24494 | Discretizing probability measures | Since the answer above addresses the problem for signed measures rather than probabilities let me add to it by noting that both versions are true and are direct consequences of versions of the Hahn-Banach theorem (bipolar theorem). The appropriate duality is that between the space of bounded, continuous functions on a ... | 2 | https://mathoverflow.net/users/75303 | 210563 | 100,257 |
https://mathoverflow.net/questions/210567 | 0 | A space $(X,\tau)$ is said to be $R\_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$.
If $X$ is compact and $R\_1$, does this imply that it is regular?
| https://mathoverflow.net/users/nan | Compact $R_1$-spaces | The answer is Yes.
Suppose $(X,\tau)$ is $R\_1$ and compact. Pick $V$ open and let $x\in V$. For each $y\in X\setminus V$ there are open neighborhoods $U(y)$ of $x$ and $V(y)$ of $y$ such that $U(y)\cap V(y)=\emptyset$. The family $\{V(y): y\in X\setminus V\}$ is an open cover of $X\setminus V$ which is closed, there... | 1 | https://mathoverflow.net/users/8628 | 210568 | 100,258 |
https://mathoverflow.net/questions/210562 | 0 | I have a question about weak derivatives.
Let $u,v \in L^{1}\_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is the $\alpha$-th ($\alpha$ is a multi index) weak derivative of $u$ if the following equation holds:
\begin{align}
\int\_... | https://mathoverflow.net/users/68463 | About weak derivatives | If $\partial U$ has measure zero (which follows from $C^1$ regularity, for example), then $L^1(U)=L^1(\bar U)$.
Since $L^1(U)\subset L^1\_{\text{loc}}(U)$, $L^1(\bar U)$ is just a special case.
Derivatives in $L^1(\bar U)$ can be defined exactly the same way, taking test functions in $C\_0^\infty(U)$.
The support of ... | 2 | https://mathoverflow.net/users/55893 | 210578 | 100,262 |
https://mathoverflow.net/questions/210569 | 4 | Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$.
Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$
of exponent $p$ has a maximal finite quotient
$\overline{B}=\overline{B(d,p)}=B/N$ where $N$ is the intersection of all
the subgroups of $B$ of finite index. I understand that Ko... | https://mathoverflow.net/users/23827 | Restricted Burnside Problem: Lower bound nilpotency class | For Questions 1, 2.
We know $c(d,2)=1$ for all $d$ and $c(1,3)=1$, $c(2,3)=2$ and $c(d,3)=3$ for all $d>2$.
So suppose if necessary $p\geq 5$.
Take $G(d,p)$ to be the unitiriangular matrices of size $(d+1)\times (d+1)$ over the field of size $p$, where $p>d$. Then $G(d,p)$ is nilpotent of class $d$ and the exponent ... | 3 | https://mathoverflow.net/users/19075 | 210583 | 100,264 |
https://mathoverflow.net/questions/210586 | 3 | I've encountered the following question in my research:
Let $A$ be a subset of
$\mathbb{Z}/n\mathbb{Z}$. Let me call $A$ "sum-free" if there is no solution to
$x+y=z$ for $x,y,z \in A$ with **distinct** $x$ and $y$.
The addition is of course mod $n$.
**Question:** For which $n$ is the set $A = \{2^x \!\!\mod n: x>0... | https://mathoverflow.net/users/75584 | When are the powers of 2 sum-free mod n? | This question is very similar to the one [here](https://mathoverflow.net/questions/172706/primes-dividing-2a2b-1/172714), and the heuristic should apply equally well. In particular, $A$ is sum-free if and only if there does not exist a $k$ with $k \ne \frac{n+1}{2}$ so that $k$ and $1-k$ are both in $A$, and the chance... | 3 | https://mathoverflow.net/users/48142 | 210589 | 100,268 |
https://mathoverflow.net/questions/210587 | 2 | Suppose $F$ is a primitive element of the Selberg class and $\displaystyle{\prod\_{j=1}^{r}\Gamma(\lambda\_{j}s+\mu\_{j})}$ with $r>1$ the product of Gamma functions appearing in the gamma factor $\gamma\_{F}$ of $F$. Can there exist a couple $(i,j)$ with $i\neq j$ such that $(\lambda\_{i},\mu\_{i})=(\lambda\_{j},\mu\_... | https://mathoverflow.net/users/13625 | gamma-factor of a primitive element of the Selberg class | The answer is yes if we accept the Ramanujan conjecture for principal automorphic $L$-functions.
Indeed, the $L$-function of an even Maass form of Laplacian eigenvalue $1/4$ for some congruence subgroup of $\mathrm{SL}\_2(\mathbb{Z})$ has two gamma factors, each equal to $\Gamma(s/2)$. Moreover, by a result of Ram Mu... | 3 | https://mathoverflow.net/users/11919 | 210595 | 100,269 |
https://mathoverflow.net/questions/210601 | 3 | A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$.
Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$.
However, can all these paths be constructed so that they don't cross?.
**Note**: If $S$ is simply connected, then by the Riemann mapping theorem... | https://mathoverflow.net/users/49537 | Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$? | Consider a double-sided topologist's comb, with teeth $\{1/n\} \times I$ for $n \in \mathbb{Z}, n \ne 0$ together with $[-1,1] \times {0}$ and ${0} \times I$. This is closed. The complement is path-connected. There is no path in the complement to $(0,1/2)$.
While this set is not simply connected, that's not important... | 3 | https://mathoverflow.net/users/2954 | 210606 | 100,272 |
https://mathoverflow.net/questions/177484 | 4 | Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W\_{dom}:A\to W$ and $i^W\_{cod}:B\to W$ to denote the boundary-component inclusions. Let $\pi\_0:{\bf Man}\to{\bf Set}$ denote the connected components fun... | https://mathoverflow.net/users/2811 | An orthogonal factorization system on 1-Cob? | This question was answered in the affirmative by Joseph Abadi, [here](http://arxiv.org/pdf/1506.03119v1.pdf).
| 1 | https://mathoverflow.net/users/2811 | 210608 | 100,274 |
https://mathoverflow.net/questions/210603 | 1 | Let two real matrices $A$ and $B$ be unitarily equivalent. How to determine (computationally or theoretically) the unitary operator $U$ s.t. $A = UBU^\dagger$? Is it possible for some special class of matrices? Please give me some references.
| https://mathoverflow.net/users/36977 | How to determine an unitary operator involved in an unitary transformation? | An algorithm for arbitrary matrices is given by Heydar Radjavi in 1962 (On unitary equivalence of arbitrary matrices, TAMS). The "arbitrary" in the title is there because the problem is trivial for normal matrices (you diagonalize both matrices, and check that the diagonalizations are the same up to order).
| 2 | https://mathoverflow.net/users/11142 | 210609 | 100,275 |
https://mathoverflow.net/questions/210516 | 1 | Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group.
Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X\_H(x(t))$ and for a any $t$ there is a $g(t) \in G\_x:=\{g \in G; Ad^\*\_h(x)=x\}$ (here $x \in \mathfrak{g}^\*$) such that $x(t) = \Phi(g(t... | https://mathoverflow.net/users/75290 | Non Hamiltonian vector field | I'm afraid I'm not entirely clear on what you are asking - you say you "found" a result, but then ask how it can be derived. Here's a comment that might be helpful.
>
> In this situation I found that $d$ already satisfies the Hamiltonian equation
> $$\dot{d}(t)+Z\_{\zeta(t)}(d(t)) = X\_H(d(t)),$$
>
>
>
From y... | 1 | https://mathoverflow.net/users/17945 | 210616 | 100,277 |
https://mathoverflow.net/questions/209958 | 11 | For all $M \in \mathbb{Z}$, is there a finite sequence of positive integers (not necessarily distinct) $(n\_i)\_{i \in I}$, s.t. $\sum\_{i \in I} \frac{1}{n\_i} = M$, and there is no subsequence $(n\_i)\_{i \in J}$ of $(n\_i)\_{i \in I}$ such that $\sum\_{i \in J} \frac{1}{n\_i}$ is an integer ?
| https://mathoverflow.net/users/18659 | Subsequence and integers as a sum of $\frac{1}{n}$ | You have answered your question in the comments, just construct your sequence inductively.
Suppose you have a sequence $n\_1,n\_2,\ldots,n\_k$ of pairwise relatively prime numbers. Put $q=n\_1n\_2\ldots n\_k$. You put $q\_i={q\over n\_i}$ and find $d\_i<n\_i$ such that $d\_iq\_i+1=0$ mod $n\_i$. This implies that $d\... | 4 | https://mathoverflow.net/users/16678 | 210623 | 100,280 |
https://mathoverflow.net/questions/210621 | 4 | Denote by $A\_n$ the number of prime numbers between $n$ and $n + \log n$.
Is it true that $A\_n < const$?
UPD: Is it true that $A\_n > \log \log n$ (or something another) for infinite number $n$?
| https://mathoverflow.net/users/31356 | Number of prime numbers in a range | Theorem 3.2 in [Maynard](http://arxiv.org/pdf/1405.2593.pdf) shows that there are many values $x$ for which the interval $[x,x+\log x]$ contains $\gg \log \log x$ primes. This is a quantification of his earlier breakthrough work where he showed that there are intervals of length $\ll e^{(4+o(1))m}$ containing $m$ prime... | 11 | https://mathoverflow.net/users/38624 | 210626 | 100,283 |
https://mathoverflow.net/questions/210612 | 6 | Is there an exposition and development of ordinals theory separate from set theory? That is, some first-order theory where terms are interpreted as ordinals, with constant $0$ (and maybe $\omega$), axioms for taking successor and limit (this might be easier to express in a second-order theory), an axiom schema for tran... | https://mathoverflow.net/users/31055 | Ordinals separate from set theory | I think the following article by Takeuti does what you want:
Takeuti, Gaisi, A formalization of the theory of ordinal numbers. J. Symbolic Logic 30 1965 295–317.
The Mathematical Reviews / MathSciNet label for it is MR0197302 (33 #5467).
| 5 | https://mathoverflow.net/users/6794 | 210632 | 100,286 |
https://mathoverflow.net/questions/210634 | 1 | In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.
Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive integer, we have
$$
{{M} \choose {p^a}} \equiv {\rm floor}\left(\frac M{p^a} \right) \mod p.
$$
[The thing on the left... | https://mathoverflow.net/users/42278 | binomial/factorial identity mod p | Sorry, don't know a reference, but here is a quick argument.
If $M=p^ab+c$ with $0\leq c\leq p^a-1$, then
$$(1+x)^M=(1+x)^{p^ab}(1+x)^c
=(1+x^{p^a})^b(1+x)^c
\mod p.
$$
In turn, this equals
$$
(1+bx^{p^a} + ...)(1+x)^c \mod p
$$
where $...$ means higher degree terms. Since $c<p^a$, there is no further
correction af... | 6 | https://mathoverflow.net/users/38468 | 210635 | 100,288 |
https://mathoverflow.net/questions/210636 | 3 | In the paper *[Homology fibrations and group completion theorem, McDuff-Segal](http://www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf)* (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi\_0M$ is generated by $s\_1,s\_2, \cdots,s\_k$. Let $s=s\_1s\_2\cdots s\_k$. Then
... | https://mathoverflow.net/users/65800 | group completion theorem by using homology fibrations | The submonoid (isomorphic to $\mathbb{Z}\_{\geq 0}$) generated by the product $s = s\_1 \cdots s\_k$, while not equal to $\pi\_0(M)$, is cofinal in it, and so has the same localization. More directly: the mapping telescope for $s$ and the iterated mapping telescope for each of the $s\_i$ have isomorphic homologies.
... | 5 | https://mathoverflow.net/users/4649 | 210642 | 100,290 |
https://mathoverflow.net/questions/210641 | 1 | We know that the Vandermonde determinant of order $n$ is the determinant defined as follows:
$$\begin{vmatrix}
1&x\_1&x\_1^2&\dots&x\_1^{n-1}\\
1&x\_2&x\_2^2&\dots&x\_2^{n-1}\\
\ldots&\ldots&\ldots&\ldots&\ldots\\
1&x\_n&x\_n^2&\dots&x\_n^{n-1}\\
\end{vmatrix}=\prod\limits\_{1\leq i<j\leq n}(x\_j-x\_i).$$
For any o... | https://mathoverflow.net/users/58096 | A question about the Vandermonde determinant | It is not what you are looking for, I'm sure, but there is a matrix which is $N \times N$ for $N=2|S|.$ It is almost entirely zeros except that on the main diagonal there are $2 \times 2$ blocks $$\begin{vmatrix}
1&x\_j\\
1&x\_i \end{vmatrix},$$ one for each desired term $x\_i-x\_j.$
There are certainly more compact ... | 1 | https://mathoverflow.net/users/8008 | 210648 | 100,292 |
https://mathoverflow.net/questions/210655 | -1 | The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: \lambda(\mathbb{N}\setminus A) = 1 - \lambda(A)\}.$$
Do both ${\cal C}$ and ${\cal P}(\mathbb{N})\setminus {\cal C}$ have cardinality $2^{... | https://mathoverflow.net/users/8628 | Subsets of $\mathbb{N}$ whose lower density respects complements | The answer is *yes*: Given a set $A$ in $\mathcal{C}$ or in
$\mathcal{P}(\mathbb{N}) \setminus \mathcal{C}$, you can take
any subset $B \subset \mathbb{N} \setminus A$ of lower density $0$,
and $A \cup B$ is still in $\mathcal{C}$ or in
$\mathcal{P}(\mathbb{N}) \setminus \mathcal{C}$, depending on
which of the two clas... | 5 | https://mathoverflow.net/users/28104 | 210659 | 100,297 |
https://mathoverflow.net/questions/210666 | 1 | If $A$ is chosen uniformly at random over all possible $m \times n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all arithmetic is over the reals.
In [Expected determinant of a random NxN matrix](https://mathoverflow.net/questions/13008/expect... | https://mathoverflow.net/users/45564 | Expected size of determinant of $AA^T$ for non-square random $A$ | By the Cauchy-Binet theorem, $\det AA^T=\sum (\det B)^2$, where $B$
ranges over all $m\times m$ submatrices of $A$. The expected value of
$(\det B)^2$ is $(m+1)!/4^m$, so the expected value of $\det AA^T$ is
${n\choose m}(m+1)!/4^m$. This is also the expected value of $|\det
AA^T|$ since $\det AA^T\geq 0$.
Incidental... | 15 | https://mathoverflow.net/users/2807 | 210668 | 100,300 |
https://mathoverflow.net/questions/210481 | 0 | Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, suppose the kernel is $K(x,y)$, since $K(x,\cdot) \in \mathcal{K} $, $K$ is continuous seperately in $x$ and $y$. But it need ... | https://mathoverflow.net/users/75471 | Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together | Let $1=a\_0=a\_1>\ldots >a\_n>\ldots >0$, and let $e\_n$ be the "triangle" function that vanishes outside $(a\_{n+1},a\_{n-1})$, equals $1$ at $a\_n$, and interpolates linearly from $0$ to $1$ on $[a\_{n+1},a\_n]$ and from $1$ to $0$ on $[a\_n,a\_{n-1}]$.
Put $K(x,y)=\sum\_1^\infty e\_n(x)e\_n(y)$. This is well defin... | 1 | https://mathoverflow.net/users/75422 | 210685 | 100,301 |
https://mathoverflow.net/questions/210656 | 4 | Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x\_{0}\in \Gamma$, we denote $S^{1}(x\_{0})$ the circle osculating $\Gamma$ at $ x\_0$.
Denote also $ g = \lbrace g\_{ij}\rbrace $ and $ h = \lbrace h\_{ij}\rbrace $ the induced me... | https://mathoverflow.net/users/75614 | Differential Operators On A Curve And On Osculating Circle | The one-dimensional case is actually quite confusing so I will jump right to the general case.
Consider hypersurfaces $M\_1,M\_2\subset \mathbb{R}^{n+1}$ which make "order-two" contact at $0 \in \mathbb{R}^{n+1}$. Let me define this in the following way:
>
>
> >
> > Order-two contact at $0$ means that (after r... | 5 | https://mathoverflow.net/users/1540 | 210692 | 100,305 |
https://mathoverflow.net/questions/210644 | 24 | My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens?
For example, was it before Grothendieck introduced schemes to generalize the notion of covering spaces and algebraic curves to include primes in rings? Today we call p-adic fields ... | https://mathoverflow.net/users/58443 | History of Geometric Analogies in Number Theory | Treating number and function fields on the same footing or (for instance) the idea that ramification in algebraic number theory and in the theory of covering of Riemann or analytic surfaces are two incarnations of the same mathematical phenomenon are classical ideas of the German school of the second half of the 19th c... | 48 | https://mathoverflow.net/users/2284 | 210700 | 100,307 |
https://mathoverflow.net/questions/210706 | 8 |
>
> Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve?
>
>
>
Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem.
My bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are suffic... | https://mathoverflow.net/users/49537 | When is the boundary of an open planar set a Jordan curve? | This was addressed in a recent paper of R.L. Moore:
```
A Characterization of Jordan Regions by Properties Having no Reference to their Boundaries
Robert L. Moore
Proceedings of the National Academy of Sciences of the United States of America
Vol. 4, No. 12 (Dec. 15, 1918), pp. 364-370
```
| 8 | https://mathoverflow.net/users/11142 | 210707 | 100,308 |
https://mathoverflow.net/questions/210689 | 1 | Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ be the ring of integers of $F$. Assume that the residue field of $R$ is algebraically closed. Does there exist a model o... | https://mathoverflow.net/users/54369 | Model over DVR for smooth projective curves | By Lemma 1.12 in the Deligne-Mumford paper, the answer is no. Any isomorphism of generic fibers extends uniquely to an isomorphism over $R$. This is one way of saying that the stack $\mathcal{M}\_g$ of genus $g$ stable curves is proper.
| 1 | https://mathoverflow.net/users/121 | 210721 | 100,315 |
https://mathoverflow.net/questions/210725 | 1 | I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert\_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert\_F$.
| https://mathoverflow.net/users/75659 | Comparison of Lp norm of matrix and its entry wise norm. | If $A$ is $n\times n$, then
$$\frac1{\sqrt n}\|A\|\_F\le\|A\|\_1\le\sqrt n\,\|A\|\_F,\qquad \|A\|\_2\le\|A\|\_F\le\sqrt n\,\|A\|\_2.$$
More generally, if $A$ is $n\times m$, then
$$\frac1{\sqrt m}\|A\|\_F\le\|A\|\_1\le\sqrt n\,\|A\|\_F,\qquad \|A\|\_2\le\|A\|\_F\le\min(\sqrt n\,,\sqrt m)\,\|A\|\_2.$$
To see that these ... | 2 | https://mathoverflow.net/users/8799 | 210727 | 100,317 |
https://mathoverflow.net/questions/210460 | 1 | Let $R$ be a complete dvr and $k$ its residue field of positive characteristic.
Let $H$ be a finite subgroup of $PGL\_2(k)$ such that the order of $H$ is prime with $char(k)$.
Is there some elementary way to show that $H$ is a subgroup of $PGL\_2(R)$?
| https://mathoverflow.net/users/75536 | Subgroup of Projective general linear group on complete discrete valuation ring | I don't know if the following qualifies as "elementary", but here is how I would prove this (for $2$ replaced by some $d\in \mathbb{N}$):
$\DeclareMathOperator{\gl}{GL}$
First, choose some finitely generated subgroup
$U \leq \gl(d, k)$ with $U/(U\cap k^\*) = H$.
Then $U\cap k^\*$ is finitely generated abelian and t... | 4 | https://mathoverflow.net/users/10266 | 210755 | 100,328 |
https://mathoverflow.net/questions/206585 | 26 | Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is that the existence of such a group would answer the question in [this 2011 MO post](https://mathoverflow.net/questions... | https://mathoverflow.net/users/14094 | Group with finite outer automorphism group and large center | Theorem B of [this paper](http://dx.doi.org/10.1017/S1446788700000252) implies that we can take any two nontrivial involution-free groups $A$ and $B$ and construct a
complete simple groups $D$ with a (diagrammatically) aspherical presentation $D=A\*B/\langle\!\langle w\_1, w\_2,\dots\rangle\!\rangle$ (though such use o... | 8 | https://mathoverflow.net/users/24165 | 210759 | 100,330 |
https://mathoverflow.net/questions/210764 | 7 | I am currently thinking about a physics model related to framed bordism $\Omega\_3^{fr}=\mathbb{Z}/24=\pi^s\_3$, and the first stable example is $\pi\_8(S^5)$, so I was curious about the generator, and happened to see $\pi\_8(SO(6))=\mathbb{Z}/24$ as well.
So my question is: is there a homomorphism sending $\pi\_8(SO... | https://mathoverflow.net/users/24094 | Is there a homomorphism between $\pi_8(S^5)$ and $\pi_8(SO(6))$? | There is a fibration $p : SO(m+1) \to S^m$ with fibre $SO(m)$ which induces a long exact sequence in homotopy
$$\dots \to \pi\_n(SO(m)) \to \pi\_n(SO(m+1)) \xrightarrow{p\_\*} \pi\_n(S^m) \to \pi\_{n-1}(SO(m)) \to \dots$$
For your particular question, we have the fibration $p : SO(6) \to S^5$ with fibre $SO(5)$ whi... | 22 | https://mathoverflow.net/users/21564 | 210766 | 100,333 |
https://mathoverflow.net/questions/210736 | 6 | Let $\Bbb{S}\_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices.
The Log-Euclidean distance between two points of $\Bbb{S}\_{++}^n$, i.e. between two SPD matrices $A,B\in\Bbb{S}\_{++}^n$, is given by
$$
d(A,B)=\lVert\log(A)-\log(B)\rVe... | https://mathoverflow.net/users/42645 | Prove that the Log-Euclidean distance is negative-definite | First, recall that if $\psi$ is negative definite, then $\exp(-\gamma \psi)$ is positive definite for all $\gamma >0$. Now, from Corollary 2.10 of [*Harmonic analysis on Semigroups* (Berg, Christensen, Ressel)](http://link.springer.com/book/10.1007%2F978-1-4612-1128-0) we also know that if $\psi$ is negative definite, ... | 4 | https://mathoverflow.net/users/8430 | 210769 | 100,335 |
https://mathoverflow.net/questions/210760 | 2 | Let $V$ a vector space of dimension $2$ over a field $k$ of characteristic different from $2$ and $3$. Let $S^{3}V$ the third symmetric power of $V$.
How to construct a symplectic form on $S^{3}V$ such that elements coming from the linear group of $V$ are similitudes for this form?
I believe it's some miraculous ... | https://mathoverflow.net/users/48893 | Symplectic form on the third symmetric power of a plane | There's nothing miraculous about this: Here is an explicit formula: Let $x$ and $y$ be a basis for $V$. Then $A,B\in S^3(V)$ can be written in the form
$$
A = a\_{-3}\,x^3+3a\_{-1}\,x^2y+3a\_1\,xy^2+a\_3\,y^3
\quad\text{and}\quad
B = b\_{-3}\,x^3+3b\_{-1}\,x^2y+3b\_1\,xy^2+b\_3\,y^3
$$
where $a\_i$ and $b\_i$ are in $k... | 5 | https://mathoverflow.net/users/13972 | 210801 | 100,344 |
https://mathoverflow.net/questions/210818 | 4 | Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the diagram
$$0 \leftarrow C \rightarrow D$$
in the $\infty$-category of presentable stable $\infty$-categories and continuous ... | https://mathoverflow.net/users/75707 | What is the cokernel of a map of presentable stable $\infty$-categories? | Yes. Colimits in $Pr^L$ are the same as limits in $Pr^R$, which are created by the forgetful functor to $Cat\_\infty$ (Higher Topos Theory, 5.5.3.18). So the pushout $E$ of your diagram is the pullback, in $Cat\_\infty$, of the diagram composed of the right adjoint functors. This means an object in $E$ is an object in ... | 8 | https://mathoverflow.net/users/20233 | 210826 | 100,351 |
https://mathoverflow.net/questions/210816 | 5 | Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?
| https://mathoverflow.net/users/75707 | Is the $\infty$-category of presentable $\infty$-categories presentable? | When you pass to a larger universe, all categories that were in your old universe become small. A small category which is not a poset cannot be closed under colimits (think about taking coproducts with index sets larger than your category), and so cannot be presentable. All this applies equally well to $(\infty,1)$-cat... | 5 | https://mathoverflow.net/users/75 | 210828 | 100,352 |
https://mathoverflow.net/questions/210768 | 7 | Theorem 3.2 in <http://arxiv.org/pdf/1405.2593.pdf> shows that for any $x$ there are $\gg x\exp(-\sqrt{\log x})$ integers $x\_0 \in [x; 2x]$ such that $\pi(x\_0 + \log x) - \pi(x\_0) \gg \log\log x$.
Is there an upper bound for number of such $x\_0$? I think it must be $<x(\log x)^{-c}$ for any $C$.
UPD: It is int... | https://mathoverflow.net/users/31356 | Upper bound for number of prime numbers in a range | If $x\_0<x$ satisfies that $[x\_0, x\_0+\log x]$ contains $\log\log x$ primes, then for a parameter $r$ we have that this interval contains $\binom{\log\log x}{r}$ different $r$-tuples $p, p+d\_1, p+d\_2, \ldots, p+d\_{r-1}$ of primes, such that $0<d\_1<\dots<d\_{r-1}\leq\log x$. The number of possible choices for $d\_... | 5 | https://mathoverflow.net/users/37555 | 210830 | 100,353 |
https://mathoverflow.net/questions/210815 | 1 | Let $X$ be a smooth, projective complex variety and $j \colon D \hookrightarrow X$ a smooth divisor. Then we have a Gysin morphism in singular cohomology
$$
j\_\ast \colon H^{\bullet}(D) \to H^{\bullet+2}(X)
$$
Now assume that $X$ is acted upon by a finite group $G$ and that $D$ is stable under this action. Then we... | https://mathoverflow.net/users/75706 | Is the Gysin morphism equivariant? | Let me expand my comment slightly into an answer. To simplify matters, suppose that the coefficients are $\mathbb{Q}$ or $\mathbb{C}$. The Gysin map is Poincaré dual to the restriction map $j^\*$. In more detail, we have isomorphisms
$H^i(X)\cong H^{2n-i}(X)^\vee$ and $H^i(D)\cong H^{2n-2-i}(D)^\vee$, where $n=\dim X$,... | 6 | https://mathoverflow.net/users/4144 | 210831 | 100,354 |
https://mathoverflow.net/questions/210835 | 2 | Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}\_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not jointly), and satisfies $g(x,y)=g(y,x)$.
Alternating optimization on $g$ in this case takes a nice form: $x\_{i+1}\gets \... | https://mathoverflow.net/users/25311 | Fixed point iteration on symmetric biconvex function | The paper cited in [my answer here](https://mathoverflow.net/a/179167/8430) provides a detailed proof of the two-block case of alternating minimization (block coordinate descent). In particular, as mentioned in my comment, the convergence follows thanks ultimately to having unique subproblem solutions.
| 3 | https://mathoverflow.net/users/8430 | 210845 | 100,357 |
https://mathoverflow.net/questions/210803 | 1 | Considering,
1. the set of all n dim. vectors $\{x\_i\}\_{i=1,...,n} $ such that $x\_i \geq 0 $ and $\sum\_{i=1}^{n}x\_i = K$
2. Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to \mathbb R^+ $ (i.e. where $f^+(x+\epsilon)> f^+(x) \quad \forall \epsilon >0 \quad x \in \mathbb R^+$)
**My qu... | https://mathoverflow.net/users/75698 | Lower bound for $ \sum_{i=1}^n x_i f(x_i)$ when $\sum_{i=1}^{n}x_i = K$ | Actually,the conclusion is negative.
By **chebyshev's theorem**, we have
$$
\sum\_{i=1}^{n}{x\_{i}\cdot f(x\_{i})}\geq \frac{\sum\_{i}^{n}{f(x\_{i} ) } }{n} \times \sum\_{1}^{n}{x\_{i} }
$$
$$
\forall x\_{1}<x\_{2}<...<x\_{n}\in \left\{ x\_{i} \right\} ,
f^{+}(x+\epsilon ) >f^{+}(x+ ) ,\forall \epsilon >0,x\in R^+... | 1 | https://mathoverflow.net/users/75701 | 210846 | 100,358 |
https://mathoverflow.net/questions/210838 | 5 | [Marc's answer](https://mathoverflow.net/questions/210818/what-is-the-cokernel-of-a-map-of-presentable-stable-infty-categories/210826#210826) to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to th... | https://mathoverflow.net/users/75707 | Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits? | It doesn't. For instance, the $\infty$-category of spectra is the colimit of the tower
$$ \mathcal{S}\_\* \stackrel\Sigma\to \mathcal{S}\_\* \stackrel\Sigma\to ... $$
in $Pr^L$, but its colimit in $Cat\_\infty$ is the Spanier-Whitehead category whose objects are formal desuspensions $\Sigma^{-n}X$ of pointed spaces... | 15 | https://mathoverflow.net/users/20233 | 210852 | 100,361 |
https://mathoverflow.net/questions/210658 | 4 | Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G\_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes\_{\mathbb{Q}} F)$. Let $K\_{\infty}$ be a maximal compact subgroup of $G\_{\infty}$. We fix an embedding $G\hookrightarrow GL\_N$ for some $N$. Let $\mathfrak{P}$ denote a... | https://mathoverflow.net/users/69289 | Volume of arithmetic quotients of symmetric spaces | For a unimodular topological group $G$, and for discrete subgroups $\Theta\subset\Gamma\subset G$, for $f\in C^o\_c(\Theta\backslash G)$, it is true that
$$
\int\_{\Theta\backslash G} f \;=\; \int\_{\Gamma\backslash G}\sum\_{\gamma \in \Theta\backslash\Gamma} f\circ \gamma
$$
in the sense that choice of right $G$-invar... | 2 | https://mathoverflow.net/users/15629 | 210864 | 100,365 |
https://mathoverflow.net/questions/210855 | 14 | Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f\_\*:\pi\_qX\to \pi\_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, then $f$ is a homotopy equivalence (e.g. Theorem (Whitehead) on p.75 of May's "Concise Course in Algebraic Topology"... | https://mathoverflow.net/users/50409 | Counterexamples for strengthening Whitehead's theorem? | It seems to me that the stronger statement is true. If $f$ is only an isomorphism on homotopy in degrees $\* \leq n$ then the homotopy fibre $F$ is $(n-1)$-connected and its Hurewicz map is an isomorphism in degree $n$. Considering the Serre spectral sequence of the fibration seqeuence $F \to \widetilde{X} \to \widetil... | 9 | https://mathoverflow.net/users/318 | 210874 | 100,368 |
https://mathoverflow.net/questions/157084 | 16 | By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of every operation defines an algebra homomorphism, or (ii) the hom-sets have pointwise algebraic structure, or (iii) $\ma... | https://mathoverflow.net/users/5152 | Does every commutative variety of algebras have a cogenerator? | The answer is no.
Let $A$ be the algebra with universe $\{0,1\}$ and fundamental operations $f(x,y,z)=x+y+z \pmod{2}$ and $g(x)=x+1\pmod{2}$. Then $f$ and $g$ commute with each other and with themselves, so the variety generated by $A$ is commutative. This variety has a weird property: on every $B\in \mathcal V(A)$ ... | 12 | https://mathoverflow.net/users/75735 | 210882 | 100,374 |
https://mathoverflow.net/questions/210823 | 20 |
>
> Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
>
>
>
This question was proposed (problem A.611)
some time ago at [KoMaL](http://www.komal.hu/verseny/feladat.cgi?a=feladat&f=A611&l=en).
The minimal v... | https://mathoverflow.net/users/70464 | Minimum value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ | [*edited* to explain a few steps and connect with the Hahn polynomials]
The answer is
$$
\frac{(2n+1)(2n+3) n!^4}{(2n)!}
$$
assuming that I did the algebra right, which seems likely because this formula
agrees with the previously computed values $3,5,14,324/5$ for $n=0,1,2,3$.
Consider first the minimum of $\sum\_{... | 33 | https://mathoverflow.net/users/14830 | 210892 | 100,375 |
https://mathoverflow.net/questions/210885 | 0 | Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions.
For all $f\in L^2(\Omega)$, we denote by $S(t)f$ the solution of the equation
$$
dy/dt=\Delta y,\; y(0)=f.
$$
We say that $f\ge 0$ iff $f(x)\ge 0,\; \forall x\in \Omega$.
I have two questions :
... | https://mathoverflow.net/users/75742 | Maximum principle for the heat equation with Dirichlet conditions | The answer to all these questions is "no". Think of $S$ as an averaging operator. Initial temperature can be mostly positive on most of $\Omega$ but somewhere slightly negative. After some time $t\_1$ it will be positive everywhere. This answers the first question. Second one is similar.
| 2 | https://mathoverflow.net/users/25510 | 210901 | 100,380 |
https://mathoverflow.net/questions/168338 | 3 | Suppose one has two locally finite quasivarieties $\mathcal{V}$ and $\mathcal{W}$.
Further suppose that:
1. $\mathcal{V}$ is a variety.
2. The finite algebras $\mathcal{V}\_f$ are dually equivalent to $\mathcal{W}\_f$.
>
>
> >
> > Does it follow that $\mathcal{W}$ is also a variety?
> >
> >
> >
>
>
>
... | https://mathoverflow.net/users/5152 | Dualities between varieties and quasivarieties at the finite level | Here is a counterexample.
Let $A = \langle \{0,1\}; d, s\rangle$ be the algebra on $\{0,1\}$ whose defining operations are the discriminator operation $d$ on $\{0,1\}$ and the "switch" ($s(0)=1$, $s(1)=0$). According to the NU Duality Theorem of natural duality theory, $A$ is dualizable, and the topological quasivari... | 2 | https://mathoverflow.net/users/75735 | 210907 | 100,384 |
https://mathoverflow.net/questions/210903 | 6 | Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is *admissible* if the evaluation map $$e: \text{Cont}(X,Y)\times X\to Y; \ (f,x)\mapsto f(x)$$ is continuous.
Is there an example of spaces $X,Y$ such... | https://mathoverflow.net/users/8628 | Coarsest admissible topology on $\text{Cont}(X,Y)$ | Here's a simple example. Let $X=\mathbb{N}^2\cup\{\infty\}$ topologized such every subset of $\mathbb{N}^2$ is open and the neighborhood filter at $\infty$ is generated by the sets $\mathbb{N}\times[n,\infty)\cup\{\infty\}$ for $n\in\mathbb{N}$. Let $Y=\{0,1\}$ topologized such that $\{1\}$ is open but $\{0\}$ is not. ... | 9 | https://mathoverflow.net/users/75 | 210912 | 100,385 |
https://mathoverflow.net/questions/210916 | 7 | I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book *Rigid Geometry and Its Applications*. It claims that all quasi $p$-group $G$ could be characterized as certain Galois group of a Galois covering $Y $to $\mathbf{P}\_1$, only ramified at infinity (ove... | https://mathoverflow.net/users/75756 | About Abhyankar's conjecture | For a bit of context, recall that the affine line over an algebraically closed field of characteristic zero is simply connected in the sense that it has no nontrivial étale covers. (You can see this using Riemann-Hurwitz or by reducing to the corresponding topological statement over $\mathbb{C}$.) In characteristic $p>... | 13 | https://mathoverflow.net/users/4144 | 210919 | 100,388 |
https://mathoverflow.net/questions/210926 | 3 | I had this question up on Math stackexchange: <https://math.stackexchange.com/questions/1349247/number-of-non-isomorphic-models/1350763#1350763> . While it was answered partially there, I'm posting here in the hope that I might get a more detailed answer.
Let $C$ be the class of cardinals. Define by recursion $C\_0 =... | https://mathoverflow.net/users/nan | Number of non-isomorphic models | Under the global choice principle (which asserts that there is a class well-ordering of the set-theoretic universe), then every theory $T$ has rank $0$ as you define it, regardless of the number of models of the various uncountable sizes. The reason is that the well-ordering allows us to select a unique representative ... | 3 | https://mathoverflow.net/users/1946 | 210930 | 100,390 |
https://mathoverflow.net/questions/210933 | 0 | For a fixed Cantor set $K\subset [0,1]$ and a continuous function $g:[0,1]\to \mathbb R.$ Is it always possible to find a $C^{\infty}$ map $f:[0,1]\to \mathbb R$ such that $g$ and $f$ coincide in $K?$
The case $g=0$ (the constant function $0$) is covered in [Non-zero smooth functions vanishing on a Cantor set](https:... | https://mathoverflow.net/users/39115 | On Cantor sets every map is $C^{\infty}$ | Let's see. $1/3 \in K$ is a limit point from the left of points in $K$. So try
$$
g(x) = \left|x - \frac{1}{3}\right|^{1/2}
$$
No function $f$ that agrees with $g$ on $K$ can have a finite derivative at $1/3$.
| 13 | https://mathoverflow.net/users/454 | 210936 | 100,393 |
https://mathoverflow.net/questions/210891 | 18 | Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}\_A(M, N)$. Differential operators of order zero should be the $B$-linear maps from $M$ to $N$, i.e. $\text{Hom}\_B(M, N)$.
First, note that the co... | https://mathoverflow.net/users/nan | Equivalence of "Weyl Algebra" and "Crystalline" definitions of rings of differential operators between modules? | It's straightforward to check that the definitions agree for order $0$ differential operators.
To show that they agree for higher order DOs it suffices to show that $D$ is a crystalline DO of order $\leq n$ iff $[b,D]$ is a crystalline DO of order $\leq n-1$ for all $b\in B$.
Keeping in mind that the kernel $I$ o... | 6 | https://mathoverflow.net/users/745 | 210943 | 100,397 |
https://mathoverflow.net/questions/210947 | 2 | If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|\_{\Lambda '}$ is a subnet of $x$. We call subnets of this form *strict subnets*.
Of course, not all subnets are of this form, and indeed, the definition of a su... | https://mathoverflow.net/users/16639 | A quasicompact space with a net that contains no convergent strict subnet | This phenomenon is very typical in compact spaces whose construction involves taking an uncountable product. For instance, consider $X=\{0,1\}^{\mathcal{P}(\mathbb{N})}$ with the product topology. Consider the sequence $(x\_n)$ in $X$ where $x\_n(A)=1$ iff $n\in A$. If a subsequence were to converge, that would mean th... | 4 | https://mathoverflow.net/users/75 | 210950 | 100,401 |
https://mathoverflow.net/questions/155678 | 2 | I apologize if this question is meaningless or trivial:
What are examples of **Algebras** admitting quantifier elimination? Especially are there **Groups** admitting quantifier elimination?
I need to say some words about my motivation: last week I was proved some results concerning relatively free algebras in varie... | https://mathoverflow.net/users/44949 | Algebras admitting quantifier elimination | The finite groups with quantifier elimination are classified in
Cherlin, Gregory; Felgner, Ulrich, Homogeneous finite groups.
J. London Math. Soc. (2) 62 (2000), no. 3, 784–794.
| 6 | https://mathoverflow.net/users/75735 | 210953 | 100,403 |
https://mathoverflow.net/questions/210880 | 2 | Recently in a paper we get the following result:
Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of $G\rtimes\Gamma$ on a Hilbert space $H$ satisfies $\dim{H\_\Gamma}\leq 1$. Here $H\_\Gamma$ is the space of $\Gamma$-invariant ve... | https://mathoverflow.net/users/7360 | Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group | I would think that this result would go all the way back to Frobenius. Anyway, the proof seems easy enough:
Let $V$ be the trivial $\Gamma$-module. We want to show that $\dim H\_\Gamma=\dim\mathrm{Hom}\_{\Gamma}(V,H)\leq 1$. To this end, note that by Frobenius reciprocity,
$$
\mathrm{Hom}\_{\Gamma}(V,H)\cong\mathrm{H... | 0 | https://mathoverflow.net/users/4366 | 210959 | 100,406 |
https://mathoverflow.net/questions/209489 | 8 | Let $$f(x,y)=\sum\_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}
$$
Is it true that $\|f\|\_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
| https://mathoverflow.net/users/4519 | Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded? | The answer is yes. Fix $x,y$, and write $e(\alpha) := e^{2\pi i \alpha}$.
Using a Littlewood-Paley partition of unity and the triangle inequality, we may bound
$$ |f(x,y)| \leq \sum\_N a\_N$$
where $N$ ranges over powers of two,
$$ a\_N := \left|\sum\_{n \in {\bf Z} \backslash 0} \psi( \frac{n}{N}) \frac{1}{n} e(x ... | 16 | https://mathoverflow.net/users/766 | 210960 | 100,407 |
https://mathoverflow.net/questions/210957 | 11 | Braverman & Yampolsky have shown that there exist noncomputable [Julia sets](https://en.wikipedia.org/wiki/Julia_set),
i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$
is not computable.
"A set is computable, if, roughly
speaking, its image can be generated by a computer with an arbitrar... | https://mathoverflow.net/users/6094 | Is an explicit $c$ known to lead to a noncomputable Julia set? | Constructing such polynomials is the topic of a [follow-up paper by Braverman and Yampolsky.](http://www.math.toronto.edu/yampol/preprints/STOC07.pdf) (which, apparently, appeared in STOC '07). They don't give an explicit example, but give an algorithm to construct (arbitrarily close) approximations thereto.
| 7 | https://mathoverflow.net/users/11142 | 210961 | 100,408 |
https://mathoverflow.net/questions/210952 | 2 | Take $ui = pt\_i +j\_i$ where $p$ is a prime number and $u(p-r) \equiv 1 $ $(\mbox{mod p})$ for positive integers $1 \le i, r, j\_i\le p-1$ and $t\_i \ge 0$. How can I prove that:
\begin{equation}
it\_r - rt\_i + i \le r
\end{equation}
I guess that an other way to look at it is:
\begin{equation}
i\left \lceil{\... | https://mathoverflow.net/users/75773 | Proving inequation with ceilings in Finite Field of characteristic $p$ | We have $1\equiv u\left( p-r\right) \equiv u\left( -r\right)
=-ur\operatorname{mod}p$, so that $p\mid1+ur=ur+1$. Hence, $\left\lceil
\dfrac{ur}{p}\right\rceil =\dfrac{ur+1}{p}$.
We need to prove that $i\left\lceil \dfrac{ur}{p}\right\rceil \leq
r\left\lceil \dfrac{ui}{p}\right\rceil $. We transform this inequality eq... | 3 | https://mathoverflow.net/users/2530 | 210964 | 100,409 |
https://mathoverflow.net/questions/157974 | 25 | I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to the Elienberg-Moore category is an equivalence. For example, every object is free in
* The category of sets
* The categ... | https://mathoverflow.net/users/2362 | Varieties where every algebra is free | If there are no constant symbols in the language, $\mathcal L$, then the nontrivial varieties where every algebra is free are exactly the varieties term equivalent to sets or to affine spaces over a division ring. Here is why.
1. If there are no constants in the language, $F\_{\mathcal V}(\emptyset)$ is empty, so $F\... | 19 | https://mathoverflow.net/users/75735 | 210979 | 100,414 |
https://mathoverflow.net/questions/210982 | 2 | For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$.
Let $L$ be a finite lattice. Is there a finite set such that there is an injective lattice homomorphism $\varphi:L\to\text{Part}(S)$ fo... | https://mathoverflow.net/users/8628 | Embedding finite lattices into the lattice of partitions of a finite set | Yes, this is apparently a fairly hard theorem of [Pudlak and Tuma](http://link.springer.com/article/10.1007%2FBF02482893) (or at least I assume it is hard, because it seems to have been an open problem for decades before they finally proved it in 1980).
| 10 | https://mathoverflow.net/users/75 | 210987 | 100,418 |
https://mathoverflow.net/questions/210758 | 6 | Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that can give bounds or estimates for the probability that the kernel of $M$ contains at least one short non-zero vector $v\in \m... | https://mathoverflow.net/users/45564 | Probability a random matrix contains a short integer vector in its kernel | For $m$ around $n/log n$ the probability goes to $0$, with $(0,1)$ or $(-1,1)$, it doesn't matter.
Combinatorial Lemma: Any set of subsets of $\{1,\dots, n\}$ which contains no pair of subsets $A, B$ with $A \subset B$ has size at most $\begin{pmatrix} n \\\lfloor n/2 \rfloor \end{pmatrix}$ elements.
Proof: Conside... | 2 | https://mathoverflow.net/users/18060 | 210993 | 100,421 |
https://mathoverflow.net/questions/210969 | 7 | Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested in proving the following inequality relating even moments of $X$ with even moments of $Y:$
For non-negative integers ... | https://mathoverflow.net/users/7576 | Moments of a random variable and of its conditional expectation | So I think this is false. I want to consider the case $i=1$, $j=\infty$ first. Let $Y$ take values in $\{\pm 1,\pm\frac 12\}$ with equal probability of $\frac 14$ each. Now let $X$ take values $\pm 1$ with probability $\frac 14$ each and $\pm \frac 34,\pm\frac 14$ with probability $\frac 18$ each. The $\sigma$-algebra ... | 4 | https://mathoverflow.net/users/11054 | 210998 | 100,423 |
https://mathoverflow.net/questions/208957 | 4 | Let $X$ be a normal projective surface with just two singular points $x\_1,x\_2\in X$, where $X$ has rational quotient singularities.
Assume that both the singularities in $x\_1$ and in $x\_2$ admit a non-trivial first order infinitesimal deformation.
Can we conclude that $X$ itself admit a non-trivial first order... | https://mathoverflow.net/users/nan | Infinitesimal deformations of a singular projective surface | You need to know that the natural map
$$Ext^{1}(\Omega\_{X},\mathcal{O}\_{X})\rightarrow H^{0}(X,\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X}))$$
is surjective. For instance, this is true when the deformations of $X$ are unobstructed that is $H^{2}(X,T\_X)=0$.
The vector space of locally trivial first order infinit... | 1 | https://mathoverflow.net/users/14514 | 211003 | 100,424 |
https://mathoverflow.net/questions/196113 | 3 | It is well known that $\overline{M}\_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret $\overline{M}\_{0,5}$ as the blow-up of a smooth quadric surface $Q\subset\mathbb{P}^3$ in three general points.
Now let $Q... | https://mathoverflow.net/users/nan | Quadrics and Moduli Spaces | The moduli space $\overline{M}\_{0,6}$ can be realized by blowing-up in $\mathbb{P}^3$ five points $p\_i$ and then the strict transforms of the ten lines $l\_{i,j} = \left\langle p\_i,p\_j\right\rangle$ through two of them. This is the Kapranov's construction of $\overline{M}\_{0,n}$.
Now, consider in $\mathbb{P}^3$ ... | 2 | https://mathoverflow.net/users/14514 | 211004 | 100,425 |
https://mathoverflow.net/questions/210999 | 2 | Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial first order infinitesimal deformation, that is $Ext^1(\Omega\_Y,\mathcal{O}\_Y)\neq 0$, and $Ext^1(\Omega\_{f^{-1}(y)},\m... | https://mathoverflow.net/users/nan | Infinitesimal deformations of a fibration | A candidate for a counterexample is the *Cartwright-Steger surface*, see <http://arxiv.org/abs/1412.4137>.
It is complex surface $X$ of general type with $$p\_g(X)=q(X)=1, \quad K\_X^2=9,$$ hence $K\_X^2=9 \chi(\mathcal{O}\_X)$. Since equality is attained in the Miyaoka-Bogomolov-Yau inequality, $X$ is a ball quotien... | 3 | https://mathoverflow.net/users/7460 | 211015 | 100,428 |
https://mathoverflow.net/questions/210858 | 18 | This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
>
> **Question 1:** Are the fibres of a family of complex varieties isomorphic *as schemes* to the geometric generic fibre, outside of a union of countably many subfamilies?
>
>
>
... | https://mathoverflow.net/users/75616 | Geometric generic fibre | This statement is indeed pretty remarkable. A reference is Lemma 2.1 in C. Vial. [Algebraic cycles and fibrations](https://www.dpmms.cam.ac.uk/~cv248/algcycles-fibrations.pdf). Documenta Math. 18 (2013). The statement there is given for varieties, but it seems likely that it holds more generally.
| 7 | https://mathoverflow.net/users/3996 | 211023 | 100,432 |
https://mathoverflow.net/questions/210985 | 7 | I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper ["Infinite Irreducible Representations of the Lorentz Group"](http://www.jstor.org/stable/97833). I'm interested in particular in the *non-unitary* o... | https://mathoverflow.net/users/75321 | Infinite-dimensional admissible representations of SL(2,C) | Though I'm not at all a specialist in this area, I'd certainly urge you to look into some of the relatively modern mathematical textbooks. For example, David Vogan's 1981 book *Representations of Real Reductive Lie Groups* (Progress in Mathematics, Birkhauser, Boston) might still be a useful source even though he focus... | 2 | https://mathoverflow.net/users/4231 | 211030 | 100,433 |
https://mathoverflow.net/questions/211007 | 3 | Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms of $a$. For instance if $a$ is a prime which is $1$ mod $4$ then by quadratic reciprocity, we are really asking about th... | https://mathoverflow.net/users/74453 | Least prime for which a square-free integer is a non-residue | For fixed $a$, the function $\big( \frac ap \big)$ defines a Dirichlet character (mod $4a$) (and often modulo a smaller modulus). More precisely, the Jacobi symbol $\big( \frac an \big)$ defines such an extension of the Legendre symbol to all odd $n$ (simply by multiplicativity), and that extension is a Dirichlet chara... | 4 | https://mathoverflow.net/users/5091 | 211031 | 100,434 |
https://mathoverflow.net/questions/211022 | 24 | Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states:
For every irrational number $\alpha$, there are infinitely many coprime integers $p$ and $q$ such that:
$$\left|\alpha - \frac{p}{q} \right| < \frac{1}{\sqrt{5}q^2} $$
It turn... | https://mathoverflow.net/users/75293 | Why is there a $\sqrt{5}$ in Hurwitz's Theorem? | As Terry mentions in the comments, the reason for the $\sqrt{5}$ is that the limiting case, the golden ratio, forces it. There is a very neat explanation of all of this in the classic number theory book by Hardy and Wright, pages 209 to 212. I give a brief sketch of the ideas.
* Why $\phi$ is the worst case.
As Har... | 26 | https://mathoverflow.net/users/43108 | 211037 | 100,436 |
https://mathoverflow.net/questions/211041 | 0 | This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
$H=I\_{3}-2v\cdot v^{T}$,
where $I\_{3}$ is the $3\times3$ identity matrix. Then we have
$Hv=-v$,
$Hw=w$.
If we now multiply... | https://mathoverflow.net/users/75811 | SO(3) transformation that produces a reflection | What's the problem?
$\Omega$ only coincides with the parity transformation on the orthogonal complement of $v$; on the other hand $\Omega v = v$.
If, for example, $v = \pmatrix{1\cr 0\cr 0}$, then $\Omega = \pmatrix{1 & 0 & 0\cr 0 & -1 & 0\cr 0 & 0 & -1\cr}$
which is certainly in $SO(3)$.
| 3 | https://mathoverflow.net/users/13650 | 211042 | 100,439 |
https://mathoverflow.net/questions/210970 | 6 | Given the ***j-function*** $j:=j(\tau)$, and $q=e^{2\pi i\tau} = \exp(2\pi i\tau)$ where, for convenience, we set $\tau=\sqrt{-n}$.
>
> **I.** $\frac{A\_2(q)}{A\_1(q)} = \text{q-cfrac}:\;$ Icosahedral group
>
>
>
$$\begin{aligned}
A\_1(q) &= q^{-1/60} \prod\_{n=1}^\infty \frac{1}{(1-q^{5n-1})(1-q^{5n-4})} = j\... | https://mathoverflow.net/users/12905 | Relations between modular functions of certain $q$-continued fractions | Still thinking about (2).
As for (3), there are many such identities. For example, try the three level seven functions and the exponent 3. I can't at the moment find the reference that I was thinking of, but [this](https://www.ideals.illinois.edu/bitstream/handle/2142/18485/Gugg_Chadwick.pdf?sequence=1) seems to have s... | 2 | https://mathoverflow.net/users/nan | 211044 | 100,441 |
https://mathoverflow.net/questions/201736 | 5 | If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, then decode it to create an estimate $\hat{X}$, rate-distortion theory tells us that the lowest mean-squared-error we can achieve through the encoder-decoder is $\sigma^2 2^{-2R}.$ That is, using an optimal encoder and decoder, we can a... | https://mathoverflow.net/users/10668 | Rate-Distortion theory: What is the distribution of distortion on an optimal Gaussian encoder? | Yes, you can design a countable-alphabet quantizer for Gaussian RVs where quantization noise approaches Gaussian noise in relative entropy, as codeword length increases.
'On Lattice Quantization Noise' by Zamir and Feder in Transactions on Information Theory Vol. 42 No. 4, July 1996:
<http://www.eng.tau.ac.il/~zam... | 5 | https://mathoverflow.net/users/10668 | 211054 | 100,445 |
https://mathoverflow.net/questions/210779 | 27 |
>
> **Is the Solar System stable?**
>
>
>
You can see [this](https://en.wikipedia.org/wiki/Stability_of_the_Solar_System) Wikipedia page.
In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and eternity (stabilization and long-time beha... | https://mathoverflow.net/users/nan | Stability of the Solar System | Due to chaotic behaviour of the Solar System, it is not possible to precisely predict the evolution of the Solar System over 5 Gyr and the question of its long-term stability can only be answered in a statistical sense. For example, in
<http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html> (Existence o... | 32 | https://mathoverflow.net/users/32389 | 211057 | 100,446 |
https://mathoverflow.net/questions/211060 | 5 | If $X, Y$ are topological spaces, let $C(X,Y)$ denote the collection of continuous maps $f: X\to Y$, endowed with the [compact-open topology](https://en.wikipedia.org/wiki/Compact-open_topology).
Assume that we are given topological spaces $X,Y$ such that for all spaces $Z$ we have $C(X,Z) \cong C(Y,Z)$. Does this im... | https://mathoverflow.net/users/8628 | Does "$\forall Z(C(X,Z) \cong C(Y,Z))$" imply $X\cong Y$? | Let $Z$ be the Sierpinski 2-point space. Then the underlyying set of $C(X,Z)$ is naturally identified with the collection of open sets of $X$, and the specialization order from the compact-open topology is the inclusion order. Thus we can recover the poset of open sets of $X$ from $C(X,Z)$. So if you restrict your ques... | 13 | https://mathoverflow.net/users/75 | 211065 | 100,447 |
https://mathoverflow.net/questions/211066 | 3 | My question is as follows:
It is known that a closed smooth curve in $\mathbb{R}^2$ is convex iff its (signed) curvature has a constant sign. I wonder if one can characterize smooth convex cones in $\mathbb{R}^3$ in a similar way.
Here is the precise statement. I say that an open set $V$ is a cone if $tV \subset V$... | https://mathoverflow.net/users/75826 | The sign of the mean curvature on convex cones in three dimensions | The second fundamental form $II$ of $\partial V$ vanishes in the radial direction (along any line that goes through the origin), so one eigenvalue of $II$ is zero.
There are only two eigenvalues since $\dim(\partial V)=2$ and the mean curvature is (up to a factor of 2) the trace of $II$.
Therefore the nonzero eigenvalu... | 4 | https://mathoverflow.net/users/55893 | 211068 | 100,448 |
https://mathoverflow.net/questions/211063 | 0 | Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be *incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee\_L S) \neq \bigvee\_K f(S).$
Suppose that $L$ contains a [prime ideal](https://en.wikipedia.org/wiki/Ideal_%28order_theory%29#Prime_ideals) $P$ such that ther... | https://mathoverflow.net/users/8628 | Incomplete lattice homomorphisms between complete lattices | Let $h:\mathbb R\to \mathbb R$ be an incomplete lattice homomorphism, e.g. the identity below $0$, and $x\mapsto x+1$ on $\mathbb R\_{\ge 0}$.
Let $X\_1$ and $X\_2$ be disjoint copies of the reals, and let $X:=X\_1\cup X\_2 \cup \{\infty, -\infty\}$ be the horizontal sum of those two linear orders, with new last and... | 4 | https://mathoverflow.net/users/14915 | 211081 | 100,452 |
https://mathoverflow.net/questions/211077 | 1 | The proof is trivial in the Abelian case by the Stokes' theorem.How to prove it in the non-Abelian case?
| https://mathoverflow.net/users/75830 | Prove that the holonomies along any two homotopic paths are the same if the curvature of the connection vanishes | A possible answer would be to invoke *Ambrose-Singer Holonomy theorem*:
**Theorem (Ambrose-Singer)** Let $M$ be a (smooth) connected manifold, $E\to M$ a vector bundle over $M$, and $\nabla$ a connection on $E$. Then, for each $x\in M$, $\mathfrak{hol}\_x(\nabla)$ is a Lie subalgebra of $\text{End}(E\_x)$ which, as a... | 2 | https://mathoverflow.net/users/75818 | 211085 | 100,453 |
https://mathoverflow.net/questions/161989 | 2 | Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they have the same equational theory in a generalized sense.
Now suppose each variety $\mathcal{V}\_i$ has cogenerator $K\... | https://mathoverflow.net/users/5152 | When does a cogenerator determine a variety? | I understand the question to be: Let $\mathcal V$ be a commutative variety with an injective cogenerator $K$ and assume that $\mathcal V$-epis are surjective. Does it follow that the clone of operations on $K$ contains precisely those functions which are preserved by all homomorphisms $K^n\to K$? (I.e., does the clone ... | 3 | https://mathoverflow.net/users/75735 | 211089 | 100,455 |
https://mathoverflow.net/questions/210218 | 4 | I'm currently trying to prove or disprove the following claim. First let me set up some notation.
Let $G$ be a connected reductive group over a field $K$, let $S \leq Z \leq N \leq G$ be respectively a maximal split torus, its centralizer and its normalizer. Let $\Phi$ be the root system of the pair $(G,S)$, and for ... | https://mathoverflow.net/users/3824 | Braid relations $n_\alpha n_\beta n_\alpha \ldots = n_\beta n_\alpha n_\beta \ldots $ in arbitrary reductive groups | Despite what I wrote earlier **the claim is not true**, despite Proposition 6.1.8 of Bruhat-Tits (first volume). Said proposition states that
>
> **Proposition 6.1.8**: Let $\Phi$ be a root system of rank $2$ and let $(T, (U\_a,M\_a)\_{a \in \Phi})$ be a *generative* root datum. Order the set $\Phi^{\text{red}} = \... | 1 | https://mathoverflow.net/users/3824 | 211093 | 100,457 |
https://mathoverflow.net/questions/211095 | 5 | The Mertens function $M(x)$ is the summatory Möbius function i.e.
$$M(x) = \sum\_{k=1}^{x} \mu (k)$$
The conjecture that $M(x) = \mathcal{O}\left(x^{\frac{1}{2} + \epsilon}\right)$ was shown to be equivalent to the Riemann hypothesis but a stronger version of the same was disproved by Odlyzko and te Reile in 1985. ... | https://mathoverflow.net/users/75842 | Approximations to the Mertens function | The best known bound under the Riemann Hypothesis is due to Soundararajan, see [here](http://arxiv.org/abs/0705.0723). This paper and its MathSciNet review (MR2542220) contain further information about what is known unconditionally and what is expected to be the precise order of magnitude.
| 4 | https://mathoverflow.net/users/11919 | 211113 | 100,462 |
https://mathoverflow.net/questions/211112 | 1 | Let $A$ be an $m$ by $n$ $(0,1)$-matrix with $m < n$. If we maximize $\operatorname{det}(AA^T)$ then what property of $A$ are we optimizing?
This isn't simply maximizing the rank of $A$ and nor is it exactly ensuring that the rows of $A$ are orthogonal.
**July 9 2015**
Fixed typo and changed $A^TA$ to $AA^T$ in ... | https://mathoverflow.net/users/45564 | What are you maximizing when you maximize the determinant of $A^TA$? | Of course if $A$ is $m$ by $m$ then this is the square of $det (A)$, so the square of the factor by which $A:\mathbb R^m\to \mathbb R^m$ multiplies $m$-dimensional volume.
More generally if $A$ is a linear map from $\mathbb R^m$ to $\mathbb R^n$ with $m\le n$, viewed as a map to an $m$-dimensional subspace of $\mathb... | 6 | https://mathoverflow.net/users/6666 | 211117 | 100,464 |
https://mathoverflow.net/questions/211064 | 4 | This question is related to the answer I gave to [this MO question](https://mathoverflow.net/questions/210999/infinitesimal-deformations-of-a-fibration). What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By [Mostow Rigidity Theorem](https:/... | https://mathoverflow.net/users/7460 | Infinitesimal deformations of fake projective planes (or ball quotients) | Yes, this is true. It follows from a more general theorem of Calabi and Vesentini on the vanishing of $H^i(X, T\_X)$ (in a suitable range) for $X$ a compact quotient of an irreducible Hermitian symmetric space of non-compact type
(which predates Mostow rigidity).
The precise reference is Theorem 2 of their paper "On... | 4 | https://mathoverflow.net/users/519 | 211134 | 100,474 |
https://mathoverflow.net/questions/211091 | 3 | Let $G = \{(x; y) : x \in \mathbb{R}, y > 0\}$. With $(x, y)(u, v) = (x + yu, yv)$, $G$
is a group. If we topologize $G$ as a subset of $\mathbb{R}^2$, it is known that $G$ is a locally compact group that is not unimodular (see (15.17) of Hewitt-Ross). Is there another
topological structure of the group $G$ such that $... | https://mathoverflow.net/users/75572 | Exotic group topologies on the affine group $ax+b$ | [I realize I had misread the question, as I understood that the real subgroup required to be compact is the normal one. Since asking the question with the normal subgroup ($X$ in Dave's post) required to be compact seems much less trivial than the original question (answered by Dave), I'll include the full answer to th... | 3 | https://mathoverflow.net/users/14094 | 211143 | 100,479 |
https://mathoverflow.net/questions/208413 | 12 | I previously asked this on [Math.SE](https://math.stackexchange.com/questions/1264251/pull-back-of-a-fibration-along-a-homotopy-equivalence) but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ an... | https://mathoverflow.net/users/68660 | Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections | Here is one way of proving the conjecture is true in general, using the modern method of [weak factorization systems](http://ncatlab.org/nlab/show/weak+factorization+system).
A weak factorization system has at its core two classes of maps the left class and the right class and they satisfy lifting properties with re... | 5 | https://mathoverflow.net/users/184 | 211163 | 100,485 |
https://mathoverflow.net/questions/211149 | 0 | We have a set of positive random variables $\boldsymbol X=\{X\_1, X\_2,\ldots\}$, where $X\_1, X\_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X\_i$ are known in advance.
Define $S\_n=\sum\_{i=1}^nX\_i$.
As we can see $\boldsymbol X$ can be viewed as inter-arr... | https://mathoverflow.net/users/41220 | Distribution of bounded summation of i.i.d random variables | The CDF of $K$ is
$$ P(K \le n) = P(S\_n > T) = \int\_{T}^\infty dt\; f\_n(t)$$
The CDF of $S\_K$ (for $s > T$) is
$$\eqalign{P(S\_K \le s) &= \sum\_{n=1}^\infty P(K = n, S\_n \le s) = \sum\_{n=1}^\infty P(S\_{n-1} \le T, T < S\_n \le s)\cr &= \sum\_{n=1}^\infty \int\_0^T dt\; f\_{n-1}(t) \int\_T^s dr\; f(r-t)}$$
ED... | 1 | https://mathoverflow.net/users/13650 | 211166 | 100,486 |
https://mathoverflow.net/questions/211128 | 11 | Consider the following question: Let $X$ be a compact complex manifold
and $\beta \in H\_2(X, \mathbb{Z})$ a fixed homology class. Let
$\mu\_1, \mu\_2, \ldots, \mu\_k$ denote certain generic submanifolds (of the
right dimensions) in $X$.
What is $N\_{g,\beta}(\mu\_1, \ldots, \mu\_k)$, the number of genus $g$ curves... | https://mathoverflow.net/users/4463 | Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space? | Here is an attempt at an overview of tropical curve counts by someone who has been involved in the story for a while but certainly hasn't followed everything that has happened. I look forward to being corrected by others.
**Toric surfaces are done**: That's [Mikhalkin](http://arxiv.org/abs/math/0312530)'s work and h... | 11 | https://mathoverflow.net/users/297 | 211167 | 100,487 |
https://mathoverflow.net/questions/211160 | 17 | I have a curiosity about homology spheres: I was wondering if they were uniquely characterized by their fundamental group. I.e. given two $n-$dimensional (integral) homology spheres with isomorphic fundamental groups, are they homeomorphic? If not, how many homeomorphism classes corresponds to a given fundamental group... | https://mathoverflow.net/users/52926 | Homology spheres and fundamental group | See edit at bottom for further information answering the question in all dimensions.
In all odd dimensions $2k -1 > 3$, there are non-homeomorphic homology spheres with fundamental group G = the binary icosahedral group (fundamental group of the Poincaré homology sphere). This follows from basic surgery theory; the W... | 20 | https://mathoverflow.net/users/3460 | 211170 | 100,490 |
https://mathoverflow.net/questions/211131 | 3 | Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be *incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee\_L S) \neq \bigvee\_K f(S).$
Consider the following statement:
>
>
> >
> > If $L$ contains an ideal $J$ such that $\bigvee\_L J \notin J$, then there is... | https://mathoverflow.net/users/8628 | Incomplete lattice homomorphisms between complete lattices (2) | Yes. Suppose that $J$ is an ideal in $L$ such that $\bigvee^{L}J\not\in J$. Let $\mathcal{Id}(L)$ be the set of all ideals of $L$. Then $\mathcal{Id}(L)$ is a complete lattice. Define a mapping $V:L\rightarrow\mathcal{Id}(L)$ by letting $V(x)=\downarrow x$ for each $x\in L$. Then $V$ is a lattice homomorphism. However,... | 5 | https://mathoverflow.net/users/22277 | 211172 | 100,492 |
https://mathoverflow.net/questions/211109 | 7 | For a Polish space $X$, let $C\_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped with the topology of weak convergence, i.e. the weakest topology such that the map $\mu \mapsto \int f \,d\mu$ is contin... | https://mathoverflow.net/users/4832 | Is taking the product of signed measures weakly continuous? | It seems that it is not continuous, even in the compact case. I think this is a proof, loosely following Anthony Quas's outline.
Take $X = Y = [0,1]$ and set $f(x,y) = e^{xy}$. Let $$A = \left\{(\mu, \nu) \in M([0,1]) \times M([0,1]) : \left| \int f\,d (\mu \times \nu) \right| < 1\right\}.$$
If the product map is to ... | 2 | https://mathoverflow.net/users/4832 | 211173 | 100,493 |
https://mathoverflow.net/questions/211159 | 66 | This question asks for intuition, not a proof.
An earlier question,
[Measures of non-abelian-ness](https://mathoverflow.net/q/125501/6094)
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
the probability that two randomly selected
elements commute is at most $5/8$, a... | https://mathoverflow.net/users/6094 | Why can't a nonabelian group be 75% abelian? | (In response to suggestion of Johannes Hahn): As indicated in my comment, and developed further in j.p.'s comment, a fairly intuitive "explanation" is provided by the fact that Lagrange's Theorem tells us that an element $x$ of a finite group $G$ is already central in $G$ once it commutes with more than half the elemen... | 66 | https://mathoverflow.net/users/14450 | 211175 | 100,494 |
https://mathoverflow.net/questions/211165 | 30 | Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open intervals?
Some related ideas/constructions were explored in [this paper](https://wrbrian.files.wordpress.com/2012/01/sl... | https://mathoverflow.net/users/70618 | Is there a subset of the plane that meets every line in two open intervals? | Let $E$ be a set of the claimed form. Call a direction $\omega \in S^1$ a *limit direction* of $E$ if there exists a sequence $p\_n$ of points in $E$ going to infinity whose argument goes to $\omega$, or equivalently if $E$ does not avoid an infinite open sector containing the direction $\omega$ in the limit. I can sho... | 16 | https://mathoverflow.net/users/766 | 211180 | 100,496 |
https://mathoverflow.net/questions/211183 | 1 | If we observe the correspondence
$$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g\_{2}(\Lambda)}{4}X - \frac{g\_{3}(\Lambda)}{4},$$
we see the relationship between weight 4 and weight 6 modular forms, the coefficients of an elliptic curve, and a lattice $\Lambda$, i.e. a free $\mathbb{Z}$ module of rank 2.
... | https://mathoverflow.net/users/20343 | Complex plane mod lattice to elliptic curve correspondence generalization | There is a higher dimensional correspondence that works for some full-rank lattices $\Lambda \subseteq \mathbb{C}^{g}$, but not all! The reference I know for this is Hindry and Silverman's book "Diophantine Geometry". Theorem A.5.0.1 states that the complex torus $\mathbb{C}^{g}/\Lambda$ is an abelian variety if and on... | 8 | https://mathoverflow.net/users/48142 | 211184 | 100,497 |
https://mathoverflow.net/questions/211129 | 1 | By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A *graph homomorphism* between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\}\in E(G)$ implies $\{f(v), f(w)\} \in E(H)$.
Given graphs $G,H$, we denote by $\text{Hom}(G, H)$ the set of all grap... | https://mathoverflow.net/users/8628 | "Canonical" graph structure on $\text{Hom}(G, H)$ | It is trivial that such a maximal $E$ exists, and consists of pairs $\{f,g\}$ such that whenever $x$ and $y$ are adjacent, $f(x)$ and $g(y)$ are adjacent. However, it does *not* make $\operatorname{Hom}(G,H)$ an exponential object, essentially because morally (according to the definition above) it should have a loop at... | 3 | https://mathoverflow.net/users/75 | 211193 | 100,501 |
https://mathoverflow.net/questions/211145 | 1 | I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able to find such a presentation online. Is it known? And, if so, is it 'basic/common' knowledge in the field?
For example,... | https://mathoverflow.net/users/72317 | Geometric interpretation of Chern classes over flag manifolds | You mentioned the degeneracy subset definition. First lets re-establish the Grassman case as an example. Let $BU(n)=Gr\_n(\mathbb{R}^{\infty})$ and take $e\_i=(0,\dots 1, \dots )$ in the $i$-th position. Then define vectorfields of the canonical bundle by $v\_i=pr\_{V}(e\_i)$ at point $V$. Then the degenerancy set is $... | 3 | https://mathoverflow.net/users/41103 | 211196 | 100,502 |
https://mathoverflow.net/questions/142318 | 3 | Can anyone please help me with this problem.
I must let you know from the beginning that it's not an easy one.
"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
with $|u(t)|\leq c<b$. Knowing that the bandwidth of $u$ is $\nu$, i.e., $(\mathcal{F}u)(s)=0
, \forall s\in \mathbb{R}... | https://mathoverflow.net/users/40074 | Bandwidth approximation for a nonlinear problem | This may be too crude for what you need, but you could expand y in a geometric series:
$$y=\sum\_{n=1}^\infty (-1)^{n+1}({u\over b})^n.$$
Clearly every finite truncation of the series has finite bandwidth. Then you can combine this with an estimate of the error of the series.
| 2 | https://mathoverflow.net/users/12120 | 211200 | 100,504 |
https://mathoverflow.net/questions/211207 | 0 | Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be *join-incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee\_L S) \neq \bigvee\_L f(S).$
Suppose $L$ contains an ideal $J$ such that $\bigvee\_L J \notin J$. Does this imply that there is a join-incomplete lattice ... | https://mathoverflow.net/users/8628 | Join-incomplete lattice endomorphisms | Here is a way to build examples of complete lattices with no join-incomplete endomorphisms. Suppose $L$ is
1. complete,
2. simple, and
3. each proper principal ideal of $L$ has ACC, but $L$ does not have ACC.
Then $L$ has a nonprincipal ideal but no join-incomplete endomorphism.
Reason: If $L$ does not have ACC,... | 5 | https://mathoverflow.net/users/75735 | 211213 | 100,506 |
https://mathoverflow.net/questions/211210 | 5 | I'm looking for some good references about *Forcing with Side Conditions*, including expository papers that explain the main ideas with some details in order to give me a fairly clear insight of those set theoretic problems that are solvable using this approach. Any other reference is also welcome.
| https://mathoverflow.net/users/nan | References for Forcing with Side Conditions | 1. S. Todorcevic, Notes on Forcing Axioms, Chapter 7.
2. Itay Neeman, Forcing with side conditions. Oberwolfach, 2011. <http://www.math.ucla.edu/~ineeman/>
3. B. Velickovic, G. Venturi, Proper forcing remastered. <http://www.math.cmu.edu/~eschimme/Appalachian/Pfr.pdf>
| 5 | https://mathoverflow.net/users/57583 | 211216 | 100,507 |
https://mathoverflow.net/questions/211214 | 1 | *([The question](https://math.stackexchange.com/q/1351154) was originally posted on Math StackExchange.)*
**Preliminaries:** Given ${s \geq 0}$, let ${\mathrm{H}\_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm
\begin{equation\*}
\| u \|\_{\math... | https://mathoverflow.net/users/68629 | A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces | Let me expand my comment into an answer.
Let $s>\frac12$ and pick any $r\in(\frac12,s)$.
Interpolating between Sobolev spaces gives $\|u\|\_{H^r\_P}\lesssim\|u\|\_{H^0\_P}^\delta\|u\|\_{H^s\_P}^{1-\delta}$ for some $\delta>0$.
As was shown to you in [the MSE answer](https://math.stackexchange.com/a/1354391/166535) to t... | 0 | https://mathoverflow.net/users/55893 | 211217 | 100,508 |
https://mathoverflow.net/questions/211046 | 0 | Consider a short exact sequence of partially ordered groups
$$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a convex subgroups of $G.$ In content of ordered groups how do we define the spliting condition of this sequence? Suppose if th... | https://mathoverflow.net/users/30267 | Spliting of short exact exact sequences of partially ordered groups | Let $G$ be a (Abelian) po-group and remember that the partial order is completely determined by the positive cone $G^+=\{g\in G:g\geq 0\}$. Furthermore, a subset $P\subseteq G$ is a positive cone if and only if it has the following properties:
-- $0 \in P$;
-- if $a \in P$ and $b \in P$, then $a+b \in P$;
-- if $... | 1 | https://mathoverflow.net/users/24891 | 211218 | 100,509 |
https://mathoverflow.net/questions/211211 | 4 | This is a re-post [from MSE](https://math.stackexchange.com/q/1341836/72031) (because I did not get the kind of answer I wanted even after offering a bounty).
At the outset I must mention that I don't have a fairly working knowledge of Galois Theory (but do have some idea of group theory in the sense that I can under... | https://mathoverflow.net/users/15540 | Unsolvability of a Quintic and its link with "Simplicity" of $A_{5}$ | I think this is very like the "usual" proof of the unsolvability of $A\_5$. I put usual in quotes because most books skip directly to proving that $A\_5$ is simple, without proving unsolvability first.
Let $G$ be a finite group. Recall that the abelianization, $G^{ab}$, of $G$ is the largest abelian quotient of $G$. ... | 7 | https://mathoverflow.net/users/297 | 211222 | 100,511 |
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