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/195297 | 4 | I'd like to use Corollary 5 of a paper by [Hell & Kirkpatrick on graph packings](http://www.sciencedirect.com/science/article/pii/0012365X8490150X) to obtain an NP-hardness result. They want a 2-vertex-connected graph $F$ such that every matching in $F$ leaves at least 3 vertices unmatched. I would also like the graph ... | https://mathoverflow.net/users/38529 | Are there 2-connected regular graphs whose maximum matching leaves 3 vertices uncovered? | Take an even number $r\ge 4$. Take $r$ copies of a 3-connected graph $G$ which has odd order and is regular of degree $r$. Add two new vertices $x$ and $y$. For each copy of $G$, remove one edge and join one of its former endpoints to $x$ and the other to $y$.
This gives a 2-connected $r$-regular graph whose maximum ... | 4 | https://mathoverflow.net/users/9025 | 195307 | 95,212 |
https://mathoverflow.net/questions/195296 | 3 | I find myself needing to compute (or asymptotically estimate) the following sum over the $2^{S-1}$ compositions of $S$. I am hoping an expert in combinatorics (I am a computer scientist) will recognise this summation.
Let $B\_{S,k}$ denote the set of compositions that have exactly $k$ parts, and define $T\_{k}$ as,
... | https://mathoverflow.net/users/62166 | A multinomial-type sum over compositions of an integer | $$ 1 + \frac{(n+1)^{n-1}-1}{n!}.$$
For the record, I'll mention how I found this formula. First I wrote a Maple procedure for it (about 5 Maple statements). Then I noticed it seemed to be integer$(n)/(n-1)!$. I checked [OEIS](http://oeis.org) for this integer sequence and it wasn't there, so I sent the first 10 terms... | 5 | https://mathoverflow.net/users/9025 | 195311 | 95,214 |
https://mathoverflow.net/questions/195329 | 0 | Let $\pi:X\rightarrow Y$ be a double cover where $X$ and $Y$ are projective smooth curves.
Is it true that $R^1\pi\_\*\mathcal O\_X=0$ ? Why ?
Thanks in advance.
| https://mathoverflow.net/users/66528 | Direct image of structural sheaf | X and Y are proper and irreducible. As a double cover does not crush X to a point, it must be surjective. The fibres are then finite sets which implies that it is a finite morphism. Finite morphisms are affine. Affine morphisms do not have higher direct image (this is just the relative version of the theorem which says... | 6 | https://mathoverflow.net/users/46690 | 195330 | 95,223 |
https://mathoverflow.net/questions/179555 | 13 | **Theorem** ([Øystein Ore, 1938](https://projecteuclid.org/journals/duke-mathematical-journal/volume-4/issue-2/Structures-and-group-theory-II/10.1215/S0012-7094-38-00419-3.short)): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is [distributive](https://en.wikipedia.org/wiki/Distributive_lat... | https://mathoverflow.net/users/34538 | Generalization of a theorem of Øystein Ore in group theory | The answer of the main question is **yes**. Let $G$ be a finite group and $H$ a subgroup.
*Definition*: The group $G$ is called $H$-cyclic if $\exists g \in G$ such that $\langle H,g \rangle = G$.
Note that: $\langle H,g \rangle = G \Leftrightarrow \langle Hg \rangle = G$.
**Ore's theorem for intervals** (1938... | 5 | https://mathoverflow.net/users/34538 | 195331 | 95,224 |
https://mathoverflow.net/questions/195325 | 10 | I'm trying to sum the remainders when dividing N by numbers from $1$ up to $N$
$$\sum\_{i = 1}^{N} N \bmod i$$
It's easy to write a program to evaluate the sum if N is small in $O(N)$ but what if N is large ~ 1e10
so I was wondering if there is a formula or an algorithm to get the sum in a better way than $O(N) $.
... | https://mathoverflow.net/users/66543 | How to calculate the sum of remainders of N? | One can use the [Dirichlet hyperbola method](http://planetmath.org/dirichlethyperbolamethod) to compute $\sum\_{i \leq n} \sigma(i)$ in time $O( n^{1/2} )$ (up to logarithmic factors coming from arithmetic operations such as division):
\begin{align}
\sum\_{i \leq n} \sigma(i) &= \sum\_{i \leq n} \sum\_{d|i} d \\
&= ... | 24 | https://mathoverflow.net/users/766 | 195342 | 95,228 |
https://mathoverflow.net/questions/195339 | 19 | The following open problem was shown to me by Maxim Kontsevich. I
state it in a different but equivalent form. Let $a(n)$ be the sequence
at <http://oeis.org/A131868>, that is,
$$ a(n) =\frac{1}{2n^2}\sum\_{d|n}(-1)^{n+d}\mu(n/d){2d\choose d}. $$
Is it true that $6a(n)/n$ is always an integer? The sequence begins
(sta... | https://mathoverflow.net/users/2807 | A congruence involving binomial coefficients | Let $g(n) = 3(-1)^n\binom{2n}{n}$. The conjecture states $(g \* \mu)(n)$ is divisible by $n^3$. We'll show the divisibility prime-power-wise.
Consider any $p|n$. We can couple summands in $g \* \mu$ as follows: divisors $d\_1,d\_2$ of $n$ are coupled iff $d\_1=pd\_2$ and $\mu(n/d\_1), \mu(n/d\_2)\neq 0$. We can now r... | 13 | https://mathoverflow.net/users/31469 | 195359 | 95,232 |
https://mathoverflow.net/questions/195338 | 3 | Can we summarize string theory (in its actual state) in some principles and fundamental equations like electromagnetism, general relativity, quantum mechanics and classical mechanics ?
I am looking for a book for string theory for mathematicians similar to the book of Von Neumann for QM.
| https://mathoverflow.net/users/65818 | book about string theory a la Von Neumann | Not a book, but an informative and enjoyable summary has been given by Robbert Dijkgraaf:
[The mathematics of string theory](http://www.bourbaphy.fr/dijkgraaf.pdf)
>
> String theory can be considered as a two-parameter deformation of
> classical geometry, where one parameter controls the generalization
> from p... | 4 | https://mathoverflow.net/users/11260 | 195369 | 95,235 |
https://mathoverflow.net/questions/195362 | 4 | Let H be a separable Hilbert space and suppose that H is infinite dimensional. Let B be a closed ball of H-which has a positive radius-and let S be the boundary of B. A non-empty subset C of H is an "inspection set for S" just in case (1) C and B are disjoint and (2) If p is any point of S, there exists a point q of C ... | https://mathoverflow.net/users/4423 | Some questions about "inspecting" the boundary of a closed ball in Hilbert space | (I'll do this for real Hilbert spaces, which seems natural here.) Just as in finitely many dimensions, a point $x\in H$ sees exactly the subset
$$
A(x)=\{ s\in S: \langle x, s\rangle \ge 1\} .
$$
Since we cannot cover $S$ by finitely many sets of the type $\bigcup A(x)$, with the union taken over $x$ from a small ball,... | 3 | https://mathoverflow.net/users/48839 | 195376 | 95,237 |
https://mathoverflow.net/questions/195380 | 14 | I will formalize my question in terms of algebraic theories.
**Background**:
Recall that an *algebraic theory* (in the sense of Lawvere) is a category $\mathcal{C}$ which is closed under taking finite products, and whose set of objects can be identified with the set $\mathrm{Ob}(\mathcal{C})\cong\{T^0,T^1,\ldots\}... | https://mathoverflow.net/users/2811 | Are all smooth functions composites of 0-, 1-, and 2-ary functions? | For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to [this answer to a very similar question](https://mathoverflow.net/a/140862/75), this is due to Vitushkin (at least for $n=3$). Here is a simple but rather inexplicit proof (adapted from [this pape... | 28 | https://mathoverflow.net/users/75 | 195388 | 95,240 |
https://mathoverflow.net/questions/195384 | 3 | I am a physicist with some background in differential geometry and I apologize for any possible unprecise terminology.
Consider the Lie group $SU(2)$ and its tangent space $su(2)$ forming a tangent bundle.
It is well known that SU(2) is noncommutative, thus $ a \circ b \circ
a^{-1} \circ b^{-1} \neq 0$ in general ... | https://mathoverflow.net/users/31722 | SU(2) and differential forms | $SU(2)$ is a compact Lie group, so it has a bi-invariant Riemannian metric, whose Levi-Civita connection and Riemann curvature can be expressed using the Lie bracket on its Lie algebra $su(2)$. See, for example,
[Lie groups with bi-invariant Riemannian metric](https://cuhkmath.wordpress.com/2011/06/30/lie-groups-with... | 8 | https://mathoverflow.net/users/613 | 195390 | 95,241 |
https://mathoverflow.net/questions/195166 | 6 | Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
| https://mathoverflow.net/users/38889 | Do free profinite groups satisfy Howson's theorem? | The following example shows that the answer is **no**.
Let $p$ and $q$ be two different primes. First I want to construct two generated profinite group $G=A\ltimes H$ isomorphic to semi direct product of a infinitely generated free pro-$p$ group $H$ and a cyclic free pro-$q$ group $A=\langle a \rangle$.
It can be ... | 7 | https://mathoverflow.net/users/10482 | 195407 | 95,247 |
https://mathoverflow.net/questions/195381 | 1 | Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
| https://mathoverflow.net/users/38889 | p-groups and 2-generated abelian images | Let $G$ be any finite $p$-group, whose center and derived subgroup both have order $p$. Then every proper quotient of $G$ is abelian. In particular, if $G$ is not generated by 2 elements, it answers the question.
For instance, let $H$ be any non-abelian group of order $p^3$, denote by $Z$ its center. Define $G$ as th... | 7 | https://mathoverflow.net/users/14094 | 195409 | 95,248 |
https://mathoverflow.net/questions/195398 | 4 | I have obtained that the cohomology rings
$$
H^\*(G\_k(\mathbb{R}^\infty);\mathbb{Z}\_2)=\mathbb{Z}\_2[w\_1,\cdots,w\_k].
$$
Also
$$
H^\*(G\_k(\mathbb{R}^m);\mathbb{Z}\_2)=\mathbb{Z}\_2[w\_1,\cdots,w\_k]/(\bar w\_{m-k+1},\cdots,\bar w\_{m}).
$$
The inclusion $i: G\_k(\mathbb{R}^m)\to G\_k(\mathbb{R}^\infty)$ induces a... | https://mathoverflow.net/users/41075 | integral or rational cohomology of real grassmannians | This is just answer in the very special case of integer cohomology of $\mathbb{R}\mathbb{P}^\infty$ and $\mathbb{R}\mathbb{P}^n$ as asked in one of the comments.
$\mathbb{R}\mathbb{P}^\infty$ can be divided into cells $\mathbb{R}\mathbb{P}^k\backslash
\mathbb{R}\mathbb{P}^{k-1}=\mathbb{R}^k$, namely one cell in each ... | 2 | https://mathoverflow.net/users/16183 | 195413 | 95,250 |
https://mathoverflow.net/questions/195404 | 3 | I want to find analog of following two statements.
1. Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi\_1 (BG) =G$). Let $\mathscr{M}$ be a local system corresponding to $M$.
$$ H^{\bullet}\_{Grp} (G, M) = H^{\bullet} (BG, \mathscr{M})... | https://mathoverflow.net/users/62601 | What are cohomology of Lie algebra with coefficients geometrically? | We assume $G$ is a compact connected Lie group with Lie algebra $\mathfrak{g}$. Let $\rho:\mathfrak{g}\to \mathrm{End}(E)$ is a finite representation.
We denote by $\underline{E}=G\times E$ the trivial bundle over $G$. Take $U\in \mathfrak{g}$, then $U$ define a left-invariant vector field $X\_U$ on $G$.
For $s\in... | 2 | https://mathoverflow.net/users/16326 | 195429 | 95,257 |
https://mathoverflow.net/questions/195403 | 3 | If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the *hypercube digraph* of dimension $n$, denoted $H\_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices $U,V\subseteq [n]$ are adjacent and oriented as $\langle U, V\rangle$ if and only if there exists some $q\in [n]$ such t... | https://mathoverflow.net/users/16758 | Directed Hypercube Minimal Cuts | I do not know the answer to your first two questions, but here is what I was able to find relevant to your third question.
>
> Q. Is it anything known on minimal cuts ?
>
>
>
What you call the hypercube digraph is the Hasse diagram of $B\_n$ the Boolean algebra (or Boolean lattice). Using this keyword may help... | 3 | https://mathoverflow.net/users/51668 | 195430 | 95,258 |
https://mathoverflow.net/questions/195424 | 1 | I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano.
Thanks a lot.
| https://mathoverflow.net/users/nan | Reference request: log Fano varieties | Here is the detailed argument. First one has to prove the following.
**Lemma 1**
Let $D = \sum\_i d\_iD\_i$ be a $\mathbb{Q}$-divisor on a normal projective variety $X$ such that $d\_i < 1$ and the pair $(X,\lceil D \rceil)$ is lc. Then $(X,D)$ is klt.
*proof)* Let $f:Y\rightarrow X$ be a log resolution of the pair... | 2 | https://mathoverflow.net/users/14514 | 195436 | 95,260 |
https://mathoverflow.net/questions/195432 | 2 | Let $a,b$ two multiplicatively independant positive integers with $1<a<b$
that is $a^mb^n\ne1$ for all $m,n\in\mathbb N\setminus\{0\}$
We sort out the set $E=\{a^mb^n\mid m,n\in \mathbb N\setminus\{0\}\}$ in a sequence $(u\_n)\_{n\in\mathbb N}$ by ordering $E$ by increasing order.
Can one give an asymptotic for $u\_n$... | https://mathoverflow.net/users/33128 | asymptotic for a sequence coming from multiplicative sets | I found that $ \log u\_n \sim \sqrt{n} $, but I have not determined precisely the costants.
Let $ u\_n := a^{i\_n} b^{j\_n} $, and $\alpha = \log a / \log b, \ \beta = \alpha^{-1}$.
Let's count how many integers of the form $a^rb^s$ are $< u\_n$, i.e. $|A\_n| := | E \cap [1,u\_n [ \, | $.
We partition $A\... | 2 | https://mathoverflow.net/users/66594 | 195438 | 95,261 |
https://mathoverflow.net/questions/195467 | 1 | Hello together,
---------------
I have a rather basic issue on propositional logic: first, consider an arbitrary set of formulas $T$ that is consistent and complete, i.e., for every propositional formula $\varphi$, either there holds $T\vdash\varphi$ or $T\vdash\neg\varphi$.
My Question is: Is there a minimal consi... | https://mathoverflow.net/users/66620 | Propositional logic: Minimal set of formulas, which is consistent and complete | Let $\varphi\_n=A\_0\wedge A\_1\wedge\cdots\wedge A\_{n-1}$ be the assertion that the first $n$ many propositional variables are true. The theory $T=\{\varphi\_n\mid n\in\mathbb{N}\}$ consisting of all these assertions is complete and consistent, but there is no minimal complete consistent subset of $T$, since $\varphi... | 6 | https://mathoverflow.net/users/1946 | 195480 | 95,272 |
https://mathoverflow.net/questions/195009 | 5 | Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension 4, where$\check{R}\_{ab}=R\_{ajkl}R^{jkl}\_b$ and $\mathring{W}$ is Weyl acting on symmetric tensor.
Is there is a sim... | https://mathoverflow.net/users/9253 | Besse p134 Riemann tensor in dimension 4 | I don't have a reference, but there is a conceptual way to see that you only have to check a few constants to get this formula.
Notice that it's a statement about a quadratic mapping from the space of curvature tensors in 4D to the space of traceless quadratic forms in 4D (because both sides are clearly traceless qu... | 9 | https://mathoverflow.net/users/13972 | 195485 | 95,273 |
https://mathoverflow.net/questions/195468 | 6 | There are many results concerning the commutativity of rings satisfying a polynomial equation. I want to know if there is any result/reference about (finite) rings that satisfy the polynomial equation $x^4=x^2$, which seems to be much more complicated than equations like $x^n=x$, etc.
| https://mathoverflow.net/users/40723 | Rings satisfying the polynomial equation $x^4=x^2$ | Alfred Foster introduced a notion of Boolean-like ring in a 1946 paper <http://www.ams.org/tran/1946-059-01/S0002-9947-1946-0015045-5/S0002-9947-1946-0015045-5.pdf> He calls elements of a ring that satisfy $x^4=x^2$ weakly idempotent. Boolean-like ring is a commutative ring of characteristic two with identity in which ... | 14 | https://mathoverflow.net/users/32389 | 195486 | 95,274 |
https://mathoverflow.net/questions/195483 | 0 | Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$M\times \{\ast\}\times\cdots\times \{\ast\}$?
For some particularly nice manifold,i tend to believe,the answer is posi... | https://mathoverflow.net/users/66130 | Ambient isotopy of the diagonal submanifold in product space | The answer is always no for $M$ a closed manifold.
If $\Delta: M\to M\times M$ were isotopic into the first factor, then in particular it would be homotopic to a map $\Delta': M\to M\times M$ with image in $M\times \{ \ast\}$. Then the composition of $\Delta$ with projection onto the second factor $pr\_2\circ \Delta... | 10 | https://mathoverflow.net/users/8103 | 195487 | 95,275 |
https://mathoverflow.net/questions/194832 | 19 | I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (\*). I have looked for such a reference a bit, but didn't find anything. In case there is really nothing, I would be glad if experts ... | https://mathoverflow.net/users/41142 | Open problems in Berkovich geometry | This is a very broad question and it is difficult to know where to start. Remember that Berkovich's theory is a theory of analytic geometry, hence it makes sense to look for the counterpart of anything you have in complex analytic geometry: does there exist a good notion of Kähler manifold, for instance? I will try to ... | 13 | https://mathoverflow.net/users/4069 | 195498 | 95,277 |
https://mathoverflow.net/questions/195494 | -2 | Representation theory (at least the origin of this terminology) aims to exhibit a model (a represetative) in the group of matrices for an abstract group which is known by only its group law. So complex representations is a satisfying beautiful theory with Schur's theorem, Frobenius reciprocity, Artin, Brauer theorems a... | https://mathoverflow.net/users/22878 | Is there any Lefschetz-like principle for representations of finite groups? | I was encouraged to make my comment an answer, so will do so.
If $G$ is a finite group and $\mathbb{K}$ is a field then many interesting results that can be proved using character theory can also be proved by analysing the structure of the group algebra $\mathbb{K}G$. For instance by (27.20) of Curtis and Reiner's "R... | 3 | https://mathoverflow.net/users/22846 | 195502 | 95,279 |
https://mathoverflow.net/questions/195455 | 5 | Let $(p\_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and $j$ belong to the same component of $\pi$.
Is there a random partition $\pi$, such that $\mathbb{P}( i \stackrel{\pi}... | https://mathoverflow.net/users/36687 | Random partitions with prescribed pairwise membership probabilities | The $p \in \mathbb R^{n(n-1)/2}$ corresponding to distributions on partitions form a convex polytope, the extreme points of which correspond to individual partitions. I don't know if there's a simple way to characterize the faces of this polytope in general.
For $n=3$ the polytope is the convex hull of
$[0,0,0], [1,... | 2 | https://mathoverflow.net/users/13650 | 195505 | 95,280 |
https://mathoverflow.net/questions/195489 | 11 | Do there exist finitely presented $C'(1/6)$ small cancellation groups with arbitrarily high asymptotic dimension?
To offer a little more motivation, Roe proves that all hyperbolic groups have finite asymptotic dimension, a result that was particularly interesting at the time as it ensured that such groups satisfy the... | https://mathoverflow.net/users/35269 | Asymptotic dimension of $C'(1/6)$ small cancellation groups | The asymptotic dimension is bounded by 2. I don't know the original proof of this, but I found some references on this. Torsion-free $C'(\frac16)$ small-cancellation groups have [cohomological dimension 2](http://www.math.uiuc.edu/~kapovich/PAPERS/bry1.pdf) (see Theorem 6.5 (5) due to Bestvina and Mess and the followin... | 6 | https://mathoverflow.net/users/1345 | 195507 | 95,281 |
https://mathoverflow.net/questions/195509 | 9 | The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring $H^\*\_T := H^\*(BT)$ (that is, tensoring with its field of fractions), the restriction map
$$H^\*\_T(M) \to H^\*\_T(M^T)$$
... | https://mathoverflow.net/users/5792 | Citation: earliest incidence of the Borel localization theorem | Here is the reference trail, according to this [source](http://www.jstor.org/stable/1971419):
>
> Borel made the key observation [1] that the cohomology of the fixed
> point set was closely related to a torsion-free quotient. In the
> 1960’s, this was formalized as the “localization theorem” of
> Borel-Atiyah-Se... | 9 | https://mathoverflow.net/users/11260 | 195512 | 95,283 |
https://mathoverflow.net/questions/195493 | 7 | In this question *elliptic surface* means a smooth projective complex surface $X$, such that there is an elliptic fibration $\pi \colon X \to C$. (I.e., there is a curve $C$ and a proper map $\pi$, such that almost all fibres are elliptic curves.)
I am aware of [1](http://www.math.ru.nl/~heckman/Heck_14.pdf) and [2](... | https://mathoverflow.net/users/21815 | Is there a description of the moduli space of elliptic surfaces? | For the case with section and $q=0$ see
<http://www.math.colostate.edu/~miranda/preprints/weierstrassfibrations.pdf>
A similar construction should work in the case (with section, $q$ fixed and $p\_g$ sufficiently large) you should get a moduli space together with a morphism to $M\_g$.
In the case (with section; $q=... | 4 | https://mathoverflow.net/users/8621 | 195513 | 95,284 |
https://mathoverflow.net/questions/195484 | 0 | I have the following question:
I know that every holomorphic function $f$ defined on a closed complex submanifold $M$ of the space $\mathbb C^d$ can be extended to a holomorphic function on the total space $\mathbb C^d$.
Now, what about functions which are only defined on an open subset $U$ of $M$? Is it possible t... | https://mathoverflow.net/users/58628 | Continuations of holomorphic functions on submanifolds to the total space | The construction as suggested by Loïc Teyssier can be globalized, with the help of a Stein neighborhood basis.
By Corollary 1 in ''Every Stein subvariety admits a Stein neighborhood'' by Siu, every Stein
manifold $M$, which is a submanifold of some manifold $N$ has a neighborhood $W$ in $N$ such that there exist a hol... | 3 | https://mathoverflow.net/users/49151 | 195521 | 95,287 |
https://mathoverflow.net/questions/193473 | 5 | If $F(v\_1,\dots,v\_k)$ is a $k$-linear form on $\mathbb R^n$, the norm I want to consider is
$$ ||F|| = \sup \frac{ F(v\_1,\dots, v\_k)}{\prod\_{i=1}^k \left|\left|v\_i\right|\right|} $$
where the vector norm is the $2$-norm.
A random $k$-linear form is one given by a $k$-tensor with i.i.d. mean 0 entries.
Kno... | https://mathoverflow.net/users/18060 | Can the method of small moments prove a bound on the norms of random trilinear forms? | No. One cannot prove any bound better than $n^{3/4-\epsilon}$. In case anyone else is interested I am posting my argument here.
My argument for this is slightly unusual - it relies on algebraic geometry to produce a counterexample. I am sure there is an elementary argument that uses a purely combinatorial constructio... | 3 | https://mathoverflow.net/users/18060 | 195522 | 95,288 |
https://mathoverflow.net/questions/195524 | 7 | Consider the language generated by the following context free grammar:
$$
S \to SS \quad S \to () \quad S \to (S) \quad S \to [] \quad S \to [S]
$$
There is a one-to-one correspondence between this language and rooted planar trees where each edge is either dashed or solid ([] corresponds to a dashed edge and () corres... | https://mathoverflow.net/users/4002 | Is this variant of the balanced bracket language context free? | I think it's context-free and generated by roughly speaking the following 7 rules (but see Harry Altman's answer for more precision)
$$S \rightarrow TUT,\quad U\rightarrow [TUT]$$
$$ U\rightarrow e,\quad T\rightarrow e$$
$$T\rightarrow (T)$$
$$T\rightarrow TT,\quad T\rightarrow ()$$
Here $S$ is the start symbol, $T$ re... | 6 | https://mathoverflow.net/users/4600 | 195525 | 95,289 |
https://mathoverflow.net/questions/195528 | 4 | It seems to be a known fact that an algebraic space is a scheme if and only if its associated reduced closed subspace is a scheme. For instance, this is used in Chai-Faltings in proving that the dual abelian scheme always exists. Is there a reference containing a proof of this fact?
| https://mathoverflow.net/users/53197 | A reference for "an algebraic space is a scheme iff its reduction is"? | For arbitrary algebraic spaces (no quasi-separatedness hypothesis) this is Corollary 3.1.12 of "Nagata compactification for algebraic spaces" in Volume 11 (no. 4) of Journal of the Institute of Mathematics of Jussieu (2012), pp. 747--814, deduced from the noetherian case by approximation methods. But for the needs of t... | 4 | https://mathoverflow.net/users/61939 | 195538 | 95,292 |
https://mathoverflow.net/questions/194836 | 4 | I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam.
It was quite easy to study finite-dimensional simple Lie algebras, as there are lots of exercises and solutions, as well as tutorials, available online.
This is the thing that I... | https://mathoverflow.net/users/57464 | Reference for exercises with solutions for affine Lie algebras | You can find some exercises in the following lecture courses: <http://www.maths.ed.ac.uk/~igordon/LA1.pdf> (Infinite-dimensional Lie algebras, by Iain Gordon - no solutions are provided, though)
<http://www.math.uni-hamburg.de/home/runkel/Material/SS12/afflie-overview.pdf> (by Ingo Runkel from this webpage <http://www.... | 2 | https://mathoverflow.net/users/32389 | 195541 | 95,294 |
https://mathoverflow.net/questions/195454 | 5 | Motivated by the [following post](https://math.stackexchange.com/questions/1129897/a-nonmetrizable-image-of-a-metrizable-space), "Gelfand duality" and the fact that "a Hausdorff continuous image of a compact metric space is metrizable", we ask:
>
> What is a counter example of two locally compact Hausdorff spaces $... | https://mathoverflow.net/users/36688 | Proper continuous image of metrizable space | If $X$ is metrisable and $f$ is a perfect surjection onto $Y$, where perfect means continuous, closed and pre-images of singletons are compact, then $Y$ is metrisable. This is proved e.g. in Engelking's General Topology (Thm. 4.4.15), and due independently to K. Morita and S. Hanai ("Closed mappings and metric space", ... | 9 | https://mathoverflow.net/users/2060 | 195546 | 95,296 |
https://mathoverflow.net/questions/195543 | 3 | The following problem is a stumbling block in a research project that I am working on:
>
> **Problem.** Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it true that $ {C\_{c}}(G) $ is a meager subset of $ {L^{2}}(G) $ with respect to the $ L^{2} $-norm topology?
>
>
>... | https://mathoverflow.net/users/50614 | Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $? | Yes. Since $G$ is $\sigma$-compact it is enough to show that $X\_K= \lbrace f\in C(G):$ supp$(f)\subseteq K\rbrace$ is meager (of first category) in $L^2(G)$.
Otherwise, the inclusion map $i\_K: X\_K \to L^2(G)$ between Banach spaces (of course, $X\_K$ endowed with the sup-norm) would have a range of second category an... | 6 | https://mathoverflow.net/users/21051 | 195548 | 95,297 |
https://mathoverflow.net/questions/195555 | 0 | Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?
| https://mathoverflow.net/users/47285 | Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? | There are holomorphic line bundles over a compact Riemann surface $X$ that are topologically trivial, yet not holomorphically trivial. To see this, note that smooth complex line bundles are classified by a complete invariant, called the degree. By contrast, we have the Picard group $Pic(X)$ of isomorphism classes of ho... | 10 | https://mathoverflow.net/users/25358 | 195558 | 95,301 |
https://mathoverflow.net/questions/195515 | 3 | This may be a standard question answered in a book, or article. I don't know. I know that there exist related results with $\lim^1$-sequences (Rosenberg and Schochet).
What is
$KK(C\_0(X),\mathbb{C})$
for $X =$ Cantor set?
What is it more generally when $C\_0(X)$ is an abelian AF-algebra?
My only question is,... | https://mathoverflow.net/users/66649 | K-homology of Cantor set and abelian AF-algebras | As David Handelman already pointed out, by continuity of $K$-theory we obtain that $K\_0(C(X)) \cong \bigoplus\_{\mathbb{N}} \mathbb{Z}$ and $K\_1(C(X)) = 0$. Since $C(X)$ is commutative, it lies in the Bootstrap class of $KK$-theory (see Blackadar, "Operator algebras", section V.1.5.2, p. 414). In particular, the univ... | 2 | https://mathoverflow.net/users/3995 | 195562 | 95,302 |
https://mathoverflow.net/questions/195566 | 2 | As far as I know, log canonical surface singularities were classified. How about higher dimensional case?
I especially want to know whether given 3-fold singularity is log canonical or not.
Let $f$ be a holomorphic function near $0 \in \mathbb C^4$ and $D=\{ f=0\}$. Then how can I check that whether it is log canon... | https://mathoverflow.net/users/66678 | About 3-fold log canonical singularity | Yes, it is log canonical.
Some overarching comments
-------------------------
A quick way is simply to check whether it is F-pure after reduction to characteristic p for some p. (Actually, before [this paper][1], one would have to check infinitely many primes, but now we know that fpt($f$ mod $p$) $\leq$ lct($f$) f... | 6 | https://mathoverflow.net/users/3521 | 195571 | 95,304 |
https://mathoverflow.net/questions/195564 | 7 | Let $X$ be a projective, smooth variety over $\mathbb{C}$, and let $D$ be a irreducible, **big** Cartier divisor(notice, I do not assume nefness). Then is it true that $${\rm{H}}^1(X, K\_X + D) = 0\quad?$$
On the one hand, I doubt it because the result seems to be written nowhere; on the other hand, I feel the condit... | https://mathoverflow.net/users/29730 | Vanishing theorem for big divisors | Let $X\to \mathbb P ^1$ be a family of smooth quadric surfaces degenerating to $X\_0=\mathbb F \_2=\mathbb P (\mathcal O \_{\mathbb P ^1}\oplus \mathcal O \_{\mathbb P ^1} (2))$ where $0\in \mathbb P ^1$. We denote by $E$ the $-2$ curve in $X\_0$, $F$ the fiber of $X\_0\to \mathbb P ^1$ and by $f:X\to Z$ the correspond... | 6 | https://mathoverflow.net/users/19369 | 195577 | 95,305 |
https://mathoverflow.net/questions/194611 | 8 | There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl\_{\mathbb{A}\_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the affine Grassmannian $Gr$ with the ordinary flag variety $G/B$, $Gr \times G/B$. The fiber above $\epsilon = 0$ is the... | https://mathoverflow.net/users/7780 | Global Affine Flag Variety and Affine Flag Variety | Now let me attempt to give an answer myself.
There are very concrete descriptions of the fibers $Fl\_{\epsilon}$ in $Fl\_{\mathbb{A}^1}$ for each $\epsilon \in \mathbb{A}^1$.
$Fl\_{\epsilon} \cong LG/I\_{\epsilon}$, where $LG = G(k((t)))$ is the loop group of the algebraic group $G$, and $I\_{\epsilon}$ is the pr... | 3 | https://mathoverflow.net/users/7780 | 195578 | 95,306 |
https://mathoverflow.net/questions/194492 | 2 | Let $\mathfrak{A}$ be a C${}^\*$ algebra and $\mathbb{R}\ni s \mapsto \alpha\_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha\_s(A)
$$
is well defined as a Bochner integral for any $A\in\mathfrak{A}$ and $f \in L^1(\mathbb{R})$.
| https://mathoverflow.net/users/47256 | Integration in C^* algebra | Yes.
$\alpha\_s(A)$ is a continuous bounded function.
The function
$f(s) \alpha\_s(A)$ is measurable and because of
$$\int\_\mathbb{R} \|f(s) \alpha\_s(A)\| ds \le \int\_\mathbb{R} |f(s)| ds\, \|A\| < \infty$$
in $L^1$.
Quite elementary, look in the book "Serge Lang, Real and Functional Analysis", Chapter Int... | 1 | https://mathoverflow.net/users/66649 | 195593 | 95,310 |
https://mathoverflow.net/questions/195601 | 4 | Is it consistent that there is a model of $\mathsf{ZFC}$ (or $\mathsf{ZF}$) with the following properties:
(1) For all $x \in {}^\omega 2$, $x^\sharp$ exists (or $\mathbf{\Sigma}\_1^1$ determinacy)
(2) For all sets $A$, there exists $x \in {}^\omega 2$ such that $A \in L[x]$, i.e. every set is constructible from a ... | https://mathoverflow.net/users/43354 | Sharps and Every Set is Constructible from a Real | The theory is not consistent, since $\mathbb{R}$ is a set, but if $\mathbb{R}\in L[x]$, then every real is in $L[x]$, and this contradicts the existence of $x^\sharp$.
| 11 | https://mathoverflow.net/users/1946 | 195602 | 95,314 |
https://mathoverflow.net/questions/195392 | 4 | Let $M$ be a manifold.
Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.
Let $B(M,n)=F(M,n)/S\_n$, where $S\_n$ is the symmetric group of order $n$, be the corresponding unordered configuration space.
If $M$ is compact, then the graded vector space structure of $H\_\*(B(M,n);F)$, where $F=\mathbb{Q... | https://mathoverflow.net/users/41075 | homology of configuration spaces of non-compact manifolds | I believe that, although the results of Bodigheimer-Cohen-Taylor are stated for compact manifolds (potentially with boundary), they hold for noncompact manifolds which are homeomorphic to the interior of a compact manifold. See, for instance, the last sentence of 2.1 of that article.
If $M$ is a compact manifold wit... | 5 | https://mathoverflow.net/users/4649 | 195608 | 95,318 |
https://mathoverflow.net/questions/195626 | 4 | Suppose that $X\_1,\ldots,X\_n$ are independent random variables with $\operatorname E X\_k=0$ and $\operatorname E |X\_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the inequalities that establish a lower bound for the $p$-th absolute moment of $S\_n=\sum\_{k=1}^nX\_k$ in terms of the $p$-th abs... | https://mathoverflow.net/users/46211 | Lower bound for the $p$-th absolute moment of a sum of random variables | If the $X\_i$ are i.i.d. Gaussian with variance $1$, then you have
$$ c\_p := \mathbb{E} |X\_k|^p = \frac{2^{p/2} \Gamma(\frac{p+1}{2})}{\sqrt{\pi}}.$$
The variable $S\_n$ is also Gaussian with variance $n$, therefore you have
$$\mathbb{E} |S\_n|^p = c\_p n^{p/2}.$$
Hence, $\frac{\sum\_{k=1}^n \mathbb{E} |X\_k|^p}{\m... | 6 | https://mathoverflow.net/users/39261 | 195627 | 95,323 |
https://mathoverflow.net/questions/195609 | 6 | I'm trying to calculate a table of all simple hurwitz groups of order less than 10^7. None of the tables I found went further than 10^6, so I decided to use the tables of all simple groups up to 10^7 (which is easy to find), and then remove the one which are not hurwitz. Using some of the papers I have read on hurwitz ... | https://mathoverflow.net/users/38744 | Simple Hurwitz Groups of order less than 10^7 | In the paper
M.C. Tamburini and M. Vsemirnov, Irreducible $(2,3,7)$-subgroups of ${\rm PGL}\_n(F)$, $n \le 7$, J. Algebra 300 (2006), 339–362
the Hurwitz groups with absolutely irreducible projective representations of degrees up to $7$ over any field are determined (although the results in the smaller dimensions (... | 6 | https://mathoverflow.net/users/35840 | 195633 | 95,327 |
https://mathoverflow.net/questions/195563 | 5 | Let the Gauss-Bonnet form be $\Omega\propto\text{Pf}(\Omega^i{}\_j)$ with $\Omega^i{}\_j$ the curvature 2-form of an even-dimensional manifold with dim=$n$. The Gauss-Bonnet form is exact, as shown in the explicit construction in [Chern 1944](http://www.jstor.org/stable/1969302) (also available [here](http://www.maths.... | https://mathoverflow.net/users/26762 | Gauss-Bonnet invariant Ω: explicit intrinsic expression for Π in Ω=dΠ? | I see that the OP may not be entirely convinced by my comments, so let me try this, which may help. It's understandable that reading the older literature can be confusing; the classical language is often quite different from ours. Here is a slightly different interpretation of Chern's construction of the transgressed f... | 11 | https://mathoverflow.net/users/13972 | 195635 | 95,328 |
https://mathoverflow.net/questions/195603 | 0 |
>
>
> >
> > I am trying to prove the following statement:
> > $$
> > E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)]
> > $$
> > where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to $X$, and $g(\cdot)$ is a well-behaved,positive-valued, bounded function.
> >
> >
> >
>
>
>
For... | https://mathoverflow.net/users/66693 | An inequality based on expectation of continuous random variables | Converting Yemon Choi's comment to an answer:
If $E[X g(X)] \le 0$, we are done because the left side of your proposed inequality is nonnegative. So assume $E[X g(X)] \ge 0$.
First, by the triangle inequality we have $$E[X g(X)] \le E [|X| g(X)].\tag{1}$$
The Cauchy-Schwarz inequality says that for any random var... | 3 | https://mathoverflow.net/users/4832 | 195644 | 95,334 |
https://mathoverflow.net/questions/195121 | 10 | Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, combinatorics, and number theory)?
| https://mathoverflow.net/users/nan | Surveys of the items of Erdős' "toolbox" | Thank you @Fry, @so-calledfriendDon, and @DanPetersen (see comments) for these interesting references:
>
> Graham, Ronald L., Nešetřil, Jaroslav, Butler, Steve (eds.), *The
> Mathematics of Paul Erdős* I and II, 2nd edition, Springer, 2013.
>
>
> Lovász, László, Ruzsa, Imre, Sós, Vera T. (eds.), *Erdös Centenni... | 6 | https://mathoverflow.net/users/nan | 195656 | 95,336 |
https://mathoverflow.net/questions/195681 | 8 | My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is *actually* a smooth function (not merely a distribution or generalized function).
I'm trying to prove this, but I'm realizing I'm not even sure how we should properly defin... | https://mathoverflow.net/users/32591 | Fourier transform of compactly supported distribution is smooth | The Fourier transform is the same as the FT $\mathcal{S}'\to\mathcal{S}'$ since $\mathcal{E}'$ is canonically contained in $\mathcal{S}'$ (which boils down to $\mathcal{S}$ being dense in $\mathcal{E}$). It is therefore perfectly well-defined to talk about Fourier transforms of compactly supported distributions.
One ... | 7 | https://mathoverflow.net/users/3041 | 195685 | 95,344 |
https://mathoverflow.net/questions/195689 | 4 | I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G\_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial algebra
$$
H^\*(BGL(\mathbb{R}^n);\mathbb{Z}\_2)=\mathbb{Z}\_2[w\_1,\cdots,w\_n].
$$
Let $GL(\mathbb{Z}^n)$ be the group c... | https://mathoverflow.net/users/41075 | classifying space and cohomology of integer general linear group | Let me give a more detailed answer, to give more context to the list of references. Unfortunately, these answers grow to encyclopedic size so easily...
**Stable results:** as mentioned in all the answers, the question becomes easier by stabilizing $GL\_n(\mathbb{Z})$ to $GL\_\infty(\mathbb{Z})$. In this case, one can... | 11 | https://mathoverflow.net/users/50846 | 195702 | 95,350 |
https://mathoverflow.net/questions/195678 | 4 | Let $R$ be an integral domain, and $K$ its field of fractions. It is well known that for a *finitely generated* fractional ideal $I$ of $R$, and $S$ a multiplicative set we have $$(R:\_KI)\_S=(R\_S:\_KI\_S).$$
I suppose that in general the above equation fails, but I don't know such an example.
The usual non-noeth... | https://mathoverflow.net/users/23950 | Example of fractional ideal whose inverse does not commute with localization | Let $R = \mathbb{Z}+X\mathbb{Q}[X]$ and $I = X\mathbb{Q}[X] = (X, X/2, X/3, \ldots)$, and let $S = \{1,2,3,\ldots\}$. Then $K = \mathbb{Q}(X)$, $R\_S = \mathbb{Q}[X]$, and $I\_S = X \mathbb{Q}[X]$, whence $(R\_S:\_K I\_S) = (1/X)\mathbb{Q}[X]$. However, $(R:\_K I) = \mathbb{Q}[X] = R\_S$, so that $(R:\_K I)\_S = (R\_S)... | 3 | https://mathoverflow.net/users/17218 | 195708 | 95,354 |
https://mathoverflow.net/questions/195645 | 5 | I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?
For a primitive Dirichlet character $\chi$ we have:
$$ \sum\limits\_{n=-\infty}^{\infty}\chi(n) f\bigg(\frac{n}{q}x\bigg) =\frac{K}{x} \sum\limits\_{n=-\infty}^{\infty}... | https://mathoverflow.net/users/38290 | Is there a Poisson Summation formula for imprimitive Dirichlet characters? | The reason we can get that (twisted) Poisson summation formula in the first place is that in the primitive case you can interpolate the character to a smooth real function via Gauss sums.
In the imprimitive case this is not the case anymore, and you can't get a function nice enough to anything that resembles a Poisso... | 4 | https://mathoverflow.net/users/43108 | 195717 | 95,357 |
https://mathoverflow.net/questions/195694 | 7 | **Definition:** Let $V\subseteq W$ be two transitive models of $ZFC$. A pseudo-Prikry sequence, $s$, at a cardinal $\kappa$ for $(V, W)$ is an $\omega$-sequence, cofinal at $\kappa$ such that for every club $D\subseteq \kappa$ from $V$, $s\setminus D$ is finite.
In the paper "On squares, outside guessing of clubs an... | https://mathoverflow.net/users/41953 | Pseudo-Prikry sequences vs Prikry sequences | The answer is no, there is no such pseudo-Prikry sequence in the
Prikry extension.
To see this, let's first make some observations.
**Lemma 1.** If $j:V\to M$ is the ultrapower by a normal measure
$\mu$ on $\kappa$, then $$\bigcap\{\ j(C)\mid C\subset\kappa\text{
club }\}=\{\kappa\}.$$
**Proof.** Certainly $\kapp... | 6 | https://mathoverflow.net/users/1946 | 195720 | 95,358 |
https://mathoverflow.net/questions/195714 | 17 | Andre Weil's *Apprenticeship of a Mathematician* (p. 46) tells how he as a student realized that all of Fermat's uses of descent are unified in one principle: "If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials algebraically prime to each other, with integer coefficients, and $x,y$ are integers prime to each other, t... | https://mathoverflow.net/users/38783 | How important is Weil's decomposition theorem today? | Yes, this is the modern statement of Weil's theorem of decomposition. It is a basic component of the theory of heights. For a more recent exposition see 2.7.15 in Bombieri and Gubler's *Heights in Diophantine Geometry*.
If you look for a specific application of the theorem and its point of view, you should be aware ... | 11 | https://mathoverflow.net/users/26522 | 195726 | 95,359 |
https://mathoverflow.net/questions/195707 | 2 | Let $C$ be smooth curve, and let $F$ be a stable rank 2 vector bundle of degree equal to $2c+1$, $c\in\mathbb{N}$, and fix a point $p\in C$:
Can one choose an epi-morphism $u:F\rightarrow \mathbb C\_p$ such that $u$ does not vanish on all the sub-line bundles of $F$ of degree $c$?
(where $\mathbb C\_p$ is the skyscr... | https://mathoverflow.net/users/66528 | Stable Vector bundles | Yes, because $F$ has at most 2 sub-line bundles of degree $c$, so you have plenty of choices. The reason is the following. Twisting by a line bundle of degree $-c$ you reduce to the case $c=0$. You can assume that $F$ contains at least one sub-line bundle of degree $0$, and twisting again that this is $\mathcal{O}\_C$,... | 0 | https://mathoverflow.net/users/40297 | 195730 | 95,361 |
https://mathoverflow.net/questions/195739 | 36 | Recently I gave an interview to local media where I explained some basic open problems in billiard dynamics.
After a 45 min interview the reported asked me what "real life" problems can be solved using billiards...and I gave a really vague answer.
I'm looking for a precise example of a "real life" problem (besides ... | https://mathoverflow.net/users/892 | What "real life" problems can be solved using billiards? |
>
> [The
> billiard-ball computer](https://en.wikipedia.org/wiki/Billiard-ball_computer), also known as a conservative logic
> circuit, is an idealized model of a reversible mechanical computer
> based on Newtonian dynamics, proposed in 1982 by Edward Fredkin and
> Tommaso Toffoli. Instead of using electronic sig... | 27 | https://mathoverflow.net/users/11260 | 195752 | 95,367 |
https://mathoverflow.net/questions/195750 | 8 | This is equivalent to my earlier question [A question about something like "shelling" in a PL manifold](https://mathoverflow.net/questions/159473/a-question-about-something-like-shelling-in-a-pl-manifold), but maybe more comprehensible and to the point.
Given a triangulation of the PL sphere $S^n$, is there always a ... | https://mathoverflow.net/users/6666 | Making spheres shellable | According to the reviewer of
Bruggesser, H.; Mani, P., Shellable decompositions of cells and spheres.
Math. Scand. 29 (1971), 197–205 (1972), MR0328944, "The authors provide a rather ingenious proof of the following proposition: For every triangulation of an n-cell and every triangulation of an n-sphere there exists ... | 8 | https://mathoverflow.net/users/1822 | 195754 | 95,369 |
https://mathoverflow.net/questions/195755 | 8 | Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\pm 1$, and so the spectral radius $\rho(D)$, which is the size of the largest eigenvalue, is at least one. Moreover, if... | https://mathoverflow.net/users/40847 | Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$ | Every monic integer polynomial $f(x) \in \mathbb{Z}[x]$ of degree $n$ is the characteristic polynomial of an $n \times n$ matrix, namely its [companion matrix](http://en.wikipedia.org/wiki/Companion_matrix). The companion matrix is invertible iff the constant term of $f(x)$ is $\pm 1$. Conversely, every characteristic ... | 7 | https://mathoverflow.net/users/290 | 195757 | 95,371 |
https://mathoverflow.net/questions/195770 | 36 | I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct counterexamples.
| https://mathoverflow.net/users/65937 | Maps which induce the same homomorphism on homotopy and homology groups are homotopic | Take the composition of a degree one map $f:T^3\to S^3$ with the Hopf map $g:S^3\to S^2$, where $T^3$ is the 3-torus. This composition is trivial on homotopy groups since $T^3$ is aspherical and $\pi\_1S^2=0$. It is trivial on $H\_i$ for $i>0$ since this is true for $g$. If $gf$ were nullhomotopic we could lift a nullh... | 79 | https://mathoverflow.net/users/23571 | 195773 | 95,377 |
https://mathoverflow.net/questions/195725 | 4 | Let $H\_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S\_g$, the oriented closed surface of genus $g\geq 1$.
Call $F\_g$ be the free group of rank $g$.
Denote by $i:S\_g \to H\_g$ the inclusion map. This map induces an epimorphism
$i\_\*: \pi\_1(S\_g) \to \pi\_1(H\_g) \simeq F\_g$.
*Is ... | https://mathoverflow.net/users/26288 | Geometrisation of inclusion-like epimorphisms to free groups | $\newcommand{\from}{\colon}\newcommand{\Aut}{\rm{Aut}}\newcommand{\bdy}{\partial}$Here is a fairly hands-on proof. I'll use the following notation: $H = H\_g$, $S = S\_g = \bdy H$, and $F = F\_g$. I'll write $g(S) = g(H) = g$. Also, $i \from S \to H$ is the inclusion map.
>
> Suppose that $f\_\* \from \pi\_1(S) \t... | 5 | https://mathoverflow.net/users/1650 | 195774 | 95,378 |
https://mathoverflow.net/questions/195767 | 6 | Let $F$ be a number field and $L=F^{un}$ its maximal unramified extension. By Class Field Theory, $$Gal(L/F)^{ab}\cong Cl(F).$$ It's well-known that we can have $[L:F]=1$ (e.g. $F=\mathbb{Q}$), and $[L:F]=\infty$ (by Golod-Shafarevich we can have infinite towers of Hilbert Class Fields; alternatively see Maire's "On In... | https://mathoverflow.net/users/38495 | Finite Nontrivial Unramified Towers of Number Fields | It is certainly possible that $1 < [L:F] < \infty$, i.e. that the extension $F^{\mathrm{un}}/F$ be finite and non-trivial. The simplest example of this is $F = \mathbb{Q}(\sqrt{-5})$. Its Hilbert class field is $F(\sqrt{-1}) = \mathbb{Q}(\sqrt{-1},\sqrt{-5})$, and it can be shown that this field has a smaller root disc... | 7 | https://mathoverflow.net/users/26522 | 195784 | 95,383 |
https://mathoverflow.net/questions/195611 | 18 | Let $(X,d)$ be a compact metric space and $\sim$ an equivalence relation on $X$ such that the quotient space $X/\sim$ is Hausdorff. It is well known that in this case the quotient is metrizable. My question is, can we choose a compatible metric on $X/\sim$ so that the quotient map does not increase distances?
As in t... | https://mathoverflow.net/users/30721 | Quotient of metric spaces | The following counter-example leaves no doubts :-) (this will answer also some other - more basic - potential similar questions too). This example is related to the *Cantor set*:
Let $\ X:=[0;1]\ $ with the ordinary Euclidean metric $\ |\,.\,|\ \ $ (absolute value). Let $\ f:X\rightarrow Y\ $ be an arbitrary continuo... | 7 | https://mathoverflow.net/users/8385 | 195788 | 95,384 |
https://mathoverflow.net/questions/195797 | 5 | If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
>
> When is it true that $G \cong \operatorname{Aut}(\Gamma)$? In other words, when is a group $G$ (isomorphic to) the automorphi... | https://mathoverflow.net/users/52842 | Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$ | First, some terminology: if $G\cong\mathrm{Aut}(\Gamma)$ then $\Gamma$ is often called a GRR (for graphical regular representation). This may help in looking for references. Determining whether a Cayley graph is a GRR given $G$ and $S$ is very difficult in general.
One necessary criterion is the following: let $\math... | 7 | https://mathoverflow.net/users/22377 | 195799 | 95,389 |
https://mathoverflow.net/questions/195625 | 1 | If you have say an affine variety defined over $\mathbb{R}$, then its image under a morphism (also defined over $\mathbb{R}$) is a constructible set. But presumably there would be no good reason in general why the image of the set of $\mathbb{R}$-rational points under the morphism would have to be equal to the set of $... | https://mathoverflow.net/users/15482 | image under a morphism of a variety defined over R | You are right that in general this is not the case. For example let $X$ be the zero set of $x-y^2$ in $\mathbb{A}^2$ and consider the projection $X \to \mathbb{A}^1, (x,y) \mapsto x$.
Perhaps you are looking for something like that:
Let $f:X \to Y$ be a finite surjective morphism of real smooth varieties. Assume that... | 0 | https://mathoverflow.net/users/36563 | 195804 | 95,391 |
https://mathoverflow.net/questions/195476 | 7 | Let $K:{\rm L}^p({\bf R}^d)\to {\rm L}^p({\bf R}^d)$ be a bounded linear operator for every $p\in(1,\infty)$.
Assume that for some $r\in(2, \infty)$ it holds that $K$ is compact on ${\rm L}^q({\bf R}^d)$ for every $q\in[2,r)$.
My question is: does it hold that $K$ is compact on ${\rm L}^{r}({\bf R}^d)$? I was thinkin... | https://mathoverflow.net/users/66622 | Compact operators on Lebesgue spaces | More is true: if K is compact on $L^{p\_1}$ for some $1\leq p\_1<\infty$ and bounded on $L^{p\_2}$ for some other $1 \leq p\_2 \leq \infty$, then it is compact for all $p$ in the interval $[p\_1,p\_2)$ or $(p\_2,p\_1]$. This is a results of Krasnoselʹskiĭ. See <http://www.ams.org/mathscinet-getitem?mr=119086>
Interes... | 5 | https://mathoverflow.net/users/10265 | 195807 | 95,393 |
https://mathoverflow.net/questions/195765 | 5 | Consider the second order operator
$Lu=\partial\_i(a\_{ij}\partial\_j)u+b\_i\partial\_iu+cu$.
Can we find a functional $I[u]$ such that $Lu$ is the variation of $I[u]$ with respect to $u$? I have successfully dealt with the first and third term, but found difficulties with the second term. Is it even possible to do... | https://mathoverflow.net/users/37103 | Variational formulation of second order equations of the divergence form | Actually, Math604 is closer to the right answer. To see the correct condition, you need to pay attention to the placement of your indices. Your operator should be written in the form
$$
Lu = \partial\_i(a^{ij}\partial\_ju) + b^k\partial\_ku + c u = 0,
$$
where one sums over repeated indices *in opposition* (i.e., `one ... | 7 | https://mathoverflow.net/users/13972 | 195825 | 95,398 |
https://mathoverflow.net/questions/195826 | 2 | I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists.
I'm a student of mathematics, and I understand the notion of a spectral measure (which is somethimes called the resolution of identity) and also have some knowledge in spectral theory ... | https://mathoverflow.net/users/66787 | Quantum Field theory - integral notation | The answer is Yes. The interpretation of the notation is quite straight forward: $dq'|q'\rangle\langle q'| = E(dq')$. We need to presume that $E$ is the spectral measure of an operator $Q' = \int q' E(dq') = \int dq'\, q' |q'\rangle\langle q'|$. The only aspect that doesn't necessarily mesh well with your question, as ... | 3 | https://mathoverflow.net/users/2622 | 195830 | 95,400 |
https://mathoverflow.net/questions/169025 | 6 | Let $X$ be an algebraic variety over a field $\mathbb{K}$ equipped with a right action of a smooth algebraic group $G$.
One can form the quotient stack $[X/G]$. My question is probably quite elementary for algebraic geometers: what is the tangent complex of such a quotient stack over a given $\mathbb{K}$-point $Spec(\m... | https://mathoverflow.net/users/36625 | (Co)tangent complexes of quotient stacks | First of all remember that differentiating the action of $G$ at the identity gives you a Lie algebra morphism $\mathfrak{g}\to\Gamma(T\_X)$, and thus, for any point $x\in X$, a map $\mathfrak g\to T\_xX$.
Now pick a lift $x:Spec(\mathbb{K})\to X$ of your point $[x]:Spec(\mathbb{K})\to [X/G]$.
**Claim**: $\mathbb{... | 13 | https://mathoverflow.net/users/7031 | 195832 | 95,401 |
https://mathoverflow.net/questions/195828 | 2 | The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation.
On the other hand, it's possible to provide a CW-complex with only one 0-cell and no 3-cell on which the knot complement deformation retracts.
Is it reasonable to hope that we should provide suc... | https://mathoverflow.net/users/66789 | Two-bridge knots and CW-complex | Yes, there is in fact a handle decomposition with one 0-handle, two 1-handles, and a single 2-handle. In general, for a k-bridge knot, you'll get k 1-handles and $k-1$ 2-handles. The way that I learned this was to think of the knot (this works for any knot) as a solid wire that you are holding in a bathtub as the water... | 8 | https://mathoverflow.net/users/3460 | 195834 | 95,402 |
https://mathoverflow.net/questions/135926 | 4 | Let $g$ be a nilpotent $L\_{\infty}$-algebra. For every commutative differential graded algebra $A$, one can form the extension $g\otimes A$ and endow it with a nilpotent $L\_{\infty}$-algebra structure. Let us denote by $MC(g)$ the set of Maurer-Cartan elements of $g$.
My question is the following: is there in the lit... | https://mathoverflow.net/users/36625 | Maurer-Cartan elements of the extension of an $L_{\infty}$-algebra | I don't think such a result exists.
Consider the following $L\_\infty$-algebra $\mathfrak g$, which is strict and abelian: $\mathfrak g=\mathfrak g^0=k$ is $1$ dimensional and concentrated in degree $0$.
Let $A=k[\epsilon]$, where $\epsilon$ has degree $1$, with zero differential.
We have that $MC(\mathfrak g)... | 3 | https://mathoverflow.net/users/7031 | 195835 | 95,403 |
https://mathoverflow.net/questions/195732 | 2 | Suppose $D\_1$ and $D\_2$ are two $3\times 3$ diagonal matrices with real positive entries on their diagonal. Let $K$ be a symmetric $3\times 3$ matrix with zeroes on its diagonal but with arbitrary real entries elsewhere. If $D\_1 > K$ and $D\_2 > K$ then is $\sqrt{D\_1 D\_2} > K ?$ (where $A>B$ if $A-B$ is positive-d... | https://mathoverflow.net/users/3709 | Geometric mean of two matrices | EDIT 2. The required result is FALSE. cf. Case 2 below.
Let $D\_1=diag((\lambda\_i)),D\_2=diag((\mu\_i)),K=\begin{pmatrix}0&a&b\\a&0&c\\b&c&0\end{pmatrix}$ where $\lambda\_i,\mu\_i> 0$. Assume that $D\_1-K>0,D\_2-K>0$ ; we want to show $\sqrt{D\_1D\_2}-K>0$. Clearly $\lambda\_1\lambda\_2>a^2,\mu\_1\mu\_2>a^2$ implie... | 1 | https://mathoverflow.net/users/9091 | 195848 | 95,405 |
https://mathoverflow.net/questions/195867 | -1 | If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat connection then $L=M\times\mathbb{R}$?
Thanks.
| https://mathoverflow.net/users/66688 | Fibre bundles and flat connections | No. Any local system (vector bundle with constant coefficient transition matrices) admits a flat connection. You may simply use $d$ in each coordinate of a local trivialization, and the fact that the transitions have zero derivative makes this well-defined.
In fact, local systems are equivalent to representations of... | 4 | https://mathoverflow.net/users/27909 | 195868 | 95,410 |
https://mathoverflow.net/questions/195850 | 1 |
>
> 1. Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected?
>
>
> 2.For a locally compact Hausdorff space $X$, is it true to say that $X$ is path connected if and only if the Stone... | https://mathoverflow.net/users/36688 | Two questions on path connected spaces | For problem 2 the answer is false for most spaces that one wants to consider. If $X$ is a path-connected paracompact space of non-measurable cardinality, then $X$ is a path component of the Stone-Cech compactification $\beta X$. See, for example, Theorem 3 in the paper On fundamental groups of compact Hausdorff spaces ... | 2 | https://mathoverflow.net/users/22277 | 195870 | 95,411 |
https://mathoverflow.net/questions/195869 | 1 | Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E\_p,E\_q, E\_{p,q}$ be respectively the exceptional divisors over $p,q$ and $L$. Therefore $E\_p,E\_q$ are isomorphic to the blow-up of a point in $\mathbb{... | https://mathoverflow.net/users/nan | A question about an intersection number | If $E$ is the exceptional divisor of a smooth blowup, the restriction of $O(E)$ to $E$ is the relative $O(-1)$ for $E = P\_Z(N)$, where $Z$ is the center of the blowup and $N$ is the normal bundle. In your case $Z$ is the proper preimage of the line $L$ in the blowup of two points, consequently $N = O\_L(-1) \oplus O\_... | 2 | https://mathoverflow.net/users/4428 | 195883 | 95,415 |
https://mathoverflow.net/questions/195884 | 2 | I have a secret message which I want to encrypt such that any of several different keys can open it independently. The keys can't know about each other and it has to be able to work completely locally without a network. In short I want two functions
```
encrypt(message, key1, key2, key3, ...) -> code
decrypt(code, ... | https://mathoverflow.net/users/66820 | Future-Proof Encrypt for Multiple Independent Parties | Using a standard encryption method, let
code(i) = encrypt([key(i), message], key(i))
(where [A,B] is concatenation of A and B, with a publicly-known separator), and
code = [code(1), code(2), ..., code(n)]
To decrypt code with a key, you try decrypt(code(i), key) for each i = 1 .. n
and see which result starts with yo... | 5 | https://mathoverflow.net/users/13650 | 195890 | 95,416 |
https://mathoverflow.net/questions/195878 | 3 | Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of a regular local ring by a regular sequence).
The information that I allow myself is the knowledge of the Hilbert series,... | https://mathoverflow.net/users/63877 | A criterion for complete intersection in terms of the Hilbert series? | I believe that this is an open problem and in fact one that is considered to be quite hard. See for instance [this survey](http://www.dima.unige.it/~rossim/Lectures.Rossi.Olinda.pdf) for an idea of what is known (you'll notice that even much easier question than the one you are asking are still wide open).
| 2 | https://mathoverflow.net/users/2284 | 195904 | 95,420 |
https://mathoverflow.net/questions/195871 | 1 | Say $X= \mathbb{P^1}\times \cdots \times \mathbb{P}^1$ is a product of $n\geq3$ lines. Let the group $G=\text{SL}(2)$ act on $X$ diagonally, and let $\mathcal{L} = \mathcal{L}(a\_1,\ldots,a\_n)$ be the $G$-equivariant line bundle corresponding to positive weights $a\_1,\ldots,a\_n$. Suppose $\mathcal{L}$ has $G$-invari... | https://mathoverflow.net/users/66751 | Volume of a GIT quotient of projective lines | I don't know quite what would count as a nice formula. Here are a couple:
1. Consider the polytope of $(n-3)$-tuples $(d\_1,\ldots,d\_{n-3})$ of nonnegative numbers, such that $(a\_1,a\_2,d\_1),(d\_{n-3},a\_{n-1},a\_n)$ and each $(d\_i,a\_{i+2},d\_{i+1})$ are triples satisfying the triangle inequality. Then the volum... | 4 | https://mathoverflow.net/users/391 | 195909 | 95,423 |
https://mathoverflow.net/questions/195877 | 6 | **Question**: What is the largest Beraha number known to be an accumulation point of real zeros of the chromatic polynomial of planar graphs?
**Background**:
The Beraha numbers $B\_n=2+2cos(2\pi/n), n=2,3,\ldots$ are well known in the study of the chromatic polynomial of planar triangulations, and more generally of ... | https://mathoverflow.net/users/25011 | Beraha numbers and zeros of the chromatic polynomial of planar graphs | I don't think that there are known families of graphs with (real) chromatic roots converging to particular Beraha numbers.
Alan Sokal and Jesus Salas wrote a series of papers, the first of which is titled "Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-... | 4 | https://mathoverflow.net/users/1492 | 195910 | 95,424 |
https://mathoverflow.net/questions/195816 | 3 | Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form expression of a bessel function with two exponentials? If so could someone please advise on how I would go about this?
Kind reg... | https://mathoverflow.net/users/66517 | Is it possible to get an equation with two exponentials and a bessel function in closed form? | Maybe you can accept the following result as a closed form. Transforming the variable of integration in your integral
$$
I\_{n}(a,b):= \frac{1}{2 a}\ e^{-a/2}\int\_{0}^{\infty} dx\ x^{(b-2)/4} e^{-x/2}(1-e^{-x/2})^{n} I\_{\frac{b}{2}-1}(\sqrt{a x})
$$
to $y:=\sqrt{a x}$, we get
$$
I\_{n}(a,b)=e^{-a/2}\ a^{-\frac{b}{4}... | 4 | https://mathoverflow.net/users/37436 | 195920 | 95,428 |
https://mathoverflow.net/questions/195813 | 2 | As a novice in algebraic topology, I'm trying to grasp the concept of a spectrum. Let me first sketch two motivations.
One motivation goes like this: for singular cohomology of spaces, we have $H^n(X;G)=[X,K(G,n)]$. We observe that these Eilenberg-Mac Lane spaces are such that the maps $K(G,n)\to \Omega K(G,n+1)$ are... | https://mathoverflow.net/users/6249 | Reconciling two viewpoints for spectra | Your two objectives are not actually that closely tied together, because there is already a representability theorem in the unstable category. In more detail, we can consider contravariant functors $F$ from connected based spaces to sets, such that
* The natural map $F(\bigvee\_{i\in I}X\_i)\to\prod\_iF(X\_i)$ is alw... | 5 | https://mathoverflow.net/users/10366 | 195926 | 95,431 |
https://mathoverflow.net/questions/195927 | 2 | Let $u$ be an (upper semi-continuous) locally bounded subharmonic function in a domain in $\mathbb{R}^n$. Let $\chi\_\epsilon$ be a standard smoothing kernel, namely
$$\chi\_\epsilon(x)=\frac{c\_n}{\varepsilon^n}\chi(\varepsilon^{-1} ||x||),$$
where $\chi\colon \mathbb{R}\to \mathbb{R}\_{\geq 0}$ is a smooth non-negat... | https://mathoverflow.net/users/16183 | Approximation of subharmonic functions | Yes, this is true.
References: Any book on potential theory or subharmonic functions, for example,
N. Landkof, Foundations of modern potential theory,
W. Hayman and P. Kennedy, Subharmonic functions I,
L. Hormander, Notions of convexity.
Sketch of the proof. Let us prove this at $0$. Let $m(r)=(1/2\pi)\int\_{-\pi}^... | 3 | https://mathoverflow.net/users/25510 | 195931 | 95,433 |
https://mathoverflow.net/questions/195923 | 5 | **My question:** Is there an equation connecting the two branches $W\_0(y)$ and $W\_{-1}(y)$ of the [Lambert W function](http://en.wikipedia.org/wiki/Lambert_W_function) for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r\_1(y)$ and $r\_2(y)$ of the equation $x^2=y$ fulfill the equation $r\_1(y)=-r\_2(y)$... | https://mathoverflow.net/users/56668 | Equation between the two branches of the lambert w function | You should specify what do you mean by "equation". If you mean algebraic equation,
then evidently there is none. Because an algebraic equation has finitely many solutions,
and if there is an algebraic equation $F(W\_0,W\_{-1})=0$, analytic continuation
will give you infinitely many solutions. This follows from the desc... | 4 | https://mathoverflow.net/users/25510 | 195932 | 95,434 |
https://mathoverflow.net/questions/195911 | 1 | Do you have an example of an infinite Hausdorff nonabelian topological group $(G,\mathcal T)$ such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we have $\mathcal S = \mathcal T$?
| https://mathoverflow.net/users/47958 | A Hausdorff atom in lattice of group topologies | Any compact group with no nontrivial normal closed subgroups has this property, since there can be no coarser Hausdorff topology and in any coarser non-Hausdorff topology the closure of the identity would be a normal closed subgroup in the original topology. For instance, this includes all (centerless) compact simple L... | 2 | https://mathoverflow.net/users/75 | 195941 | 95,438 |
https://mathoverflow.net/questions/195944 | 12 | We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b\_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its cohomology sheaves. The reason is that the cohomological dimension of $X$ is $1$. We can see Corollary 3.15 of Daniel Huyb... | https://mathoverflow.net/users/24965 | An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal | Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a nonformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to ... | 13 | https://mathoverflow.net/users/66 | 195945 | 95,440 |
https://mathoverflow.net/questions/195907 | 4 | I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C\_p$ such that $\sigma(\mathbb Q\_p)\subset\mathbb Q\_p$.
If $\mathbb C\_p$ were spherically complete, then by Hahn-Banach theorem, that would be impossible. But $\mathbb C\_p$ is not spherically complete, so we can not apply Hahn-Banach. And my proble... | https://mathoverflow.net/users/33128 | Automorphisms of $\mathbb C_p$ | If you strengthen the condition $\sigma(\mathbf Q\_p) \subset \mathbf Q\_p$ to $\sigma$ being the identity on $\mathbf Q\_p$, so $\sigma$ is a $\mathbf Q\_p$-automorphism of $\mathbf C\_p$, then a simple description is possible:
(1) every $\mathbf Q\_p$-isomorphism between two finite extensions of $\mathbf Q\_p$ ext... | 15 | https://mathoverflow.net/users/3272 | 195953 | 95,442 |
https://mathoverflow.net/questions/194986 | 1 | I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:
$$
\partial\_t U - \partial\_x V + [U,V]=0
$$
where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ are matrix-valued functions and $\lambda$ is a parameter.
The motivation behind this question is that the Lax pair... | https://mathoverflow.net/users/49727 | Integrability - conditions of lax pairs | One way to see this, is that you want the zero-curvature representation to be useful and tell you something you didn't know before. Your representation has the problem of being singular, in the sense that the Lax matrices have zero determinant. It would be more eveident if we were really speaking of the Lax representat... | 6 | https://mathoverflow.net/users/27069 | 195956 | 95,443 |
https://mathoverflow.net/questions/158604 | 11 | Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used.
>
> I wonder if there are concrete applications of the theory of characteristic classes in geometric representation theory.
>
>
>
I am mainly interested in the follow... | https://mathoverflow.net/users/32972 | Characteristic Classes in Geometric Representation Theory | Let me attempt a partial answer, trying to estimate the information contained in these characteristic classes. Since the explanations below are a
bit lengthy, I begin with the short version: I think that these
characteristic classes could have representation-theoretic
applications, but they likely do not provide more i... | 7 | https://mathoverflow.net/users/50846 | 195962 | 95,445 |
https://mathoverflow.net/questions/195966 | 7 | Let $a\_i\gt0$ for all $1\le i\le n$. It is well known that
$$
\frac{a\_1+a\_2+\cdots+a\_n}{n}-\frac{n}{\frac{1}{a\_1}+\frac{1}{a\_2}+\cdots+\frac{1}{a\_n}}\ge0,
$$
with the equality when all $a\_i$ are equal. Now let $a\_i$ are not equal but satisfy the following condition $|a\_{i+1}-a\_i|\le \varepsilon$ for some $\... | https://mathoverflow.net/users/47837 | An upper bound for the difference between arithmetic and harmonic mean | a simple upper bound is $(\sqrt{a\_{\rm max}}-\sqrt{a\_{\rm min}})^2$, with $a\_{\rm max}$ and $a\_{\rm min}$ the largest and smallest of the $a\_i$'s. So for $a\_i=a\_1+(i-1)\varepsilon$ this would give as upper bound $(\sqrt{a\_1+(n-1)\varepsilon}-\sqrt{a\_1})^2$.
see theorem 1 of [Some Inequalities for Elementary ... | 10 | https://mathoverflow.net/users/11260 | 195969 | 95,447 |
https://mathoverflow.net/questions/195974 | 2 | Suppose that $\rho : G \longrightarrow U\_n(\mathbb C)$ is an irreducible representation of group $G$. Suppose that $P$ is a projection of $\mathbb C^n$ into a subspace of small codimension (i.e. of codimension $\varepsilon n$ for some small $\varepsilon$). Can you prove that $\mathbb{E}\_{a} \rho(a)^{-1} P \rho(a)$ is... | https://mathoverflow.net/users/18785 | Projection and representation | Assuming $G$ is compact, the answer is yes. The average of $\rho(a)^{-1}P\rho(a)$ over $a$ (this is where I use compactness) will be an operator that commutes with each $\rho(b)$. So by Schur's lemma this average is a multiple of the identity. To find out what multiple, take the trace; this tells us the average is ${\r... | 3 | https://mathoverflow.net/users/763 | 195975 | 95,450 |
https://mathoverflow.net/questions/195952 | 19 | Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.
| https://mathoverflow.net/users/6936 | Mazur secret Bourbaki report "Analyse p-adique" | I have it. Mazur gave me a xerox copy off his shelf when I asked him (in grad school) if a copy exists. It's 56 pages and the first sentence is: L'objet de ce rapport est de construire la série L p-adique de Kubota-Leopold et d'établir quelques propriétés fondamentales. It was in my office and I was going to try scanni... | 71 | https://mathoverflow.net/users/3272 | 195980 | 95,452 |
https://mathoverflow.net/questions/195985 | 5 | Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?
| https://mathoverflow.net/users/14233 | Non-reflexive Banach space s.t. X,X*,X**,... are separable | I believe the James space is an example. It is isomorphic to its double dual (but not by the canonical embedding).
| 7 | https://mathoverflow.net/users/13650 | 195988 | 95,456 |
https://mathoverflow.net/questions/194858 | 13 |
>
> Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism away from codimension 2?
>
>
>
If $X$ is finite type over a field, the answer is yes, by the following argument. Th... | https://mathoverflow.net/users/1 | Do canonical stacks exist over Spec(Z)? | In the smooth case, I think that the answer is positive over an arbitrary regular excellent base. The argument was in my PhD thesis; it was done over a field, but I think it adapts to this case.
Cover your $X$ in the étale topology with schemes of the form $U/G$, where $G$ is prime to all the residue characteristics ... | 7 | https://mathoverflow.net/users/4790 | 195993 | 95,458 |
https://mathoverflow.net/questions/195778 | 3 | Could you point out a comprehensive reference book (or more than one, if it is the case) on *Quantum Probability* that introduces the subject and then gradually builds up to the edges of contemporary research?
| https://mathoverflow.net/users/nan | Reference request: a guide through quantum probability | If you are interested in quantum probability from mathematical perspective, these two books will be helpful:
<http://www.springer.com/mathematics/probability/book/978-3-540-60270-5> (Quantum Probability for Probabilists, by P.A. Meyer).
[http://www.amazon.com/Quantum-Probability-Mathematical-Statistics/dp/012305340... | 1 | https://mathoverflow.net/users/32389 | 195997 | 95,459 |
https://mathoverflow.net/questions/195981 | 0 | In the question "Simple Hurwitz Groups of order less than 10^7", it came up that all the Hurwitz groups with absolutely irreducible projective representations of degrees up to 7 over any field are determined (This later turned out to be false).
I was wondering, exactly what are all the simple groups with absolutely i... | https://mathoverflow.net/users/38744 | The simple groups with an absolutely irreducible projective representations with small degrees | The references for the low-dimensional projective representations of quasisimple groups are:
G. Hiss and G. Malle,
`Low dimensional representations of quasi-simple groups',
LMS J. Comput. Math. 4 (2001) 22-63. [Corrigenda: LMS J. Comput. Math. 5 (2002) 95-126].
F. Lübeck,
`Small degree representations of finite Che... | 2 | https://mathoverflow.net/users/35840 | 196001 | 95,462 |
https://mathoverflow.net/questions/195748 | 2 | (1) The condition that a term $a$ be substitutable for another term in an expression can be given a recursive definition. Who first developed such a definition?
(2) One sometimes see the phrase "$a$ is free for $u$ in E" used with the same meaning as "$a$ is substitutable for $u$ in E"; where does this language use s... | https://mathoverflow.net/users/37385 | Two questions on substitutability | We can browse some of the "early modern" textbooks :
* Stephen Cole Kleene, [Introduction to Metamathematics](https://books.google.it/books?id=HZAjPwAACAAJ) (1952), page 79 :
>
> we say that a term $t$ is *free at the free occurrences of* a variable $x$ *in* a formula $A(x)$ (or $t$ is *free at the substitution p... | 6 | https://mathoverflow.net/users/42676 | 196011 | 95,463 |
https://mathoverflow.net/questions/195955 | 4 | Let $K=\mathbb Q(\xi\_{39})$ be the 39-th cyclotomic field. Pari-GP told me that the prime ideals above $3$ and $13$ are not principal. Is there a way to prove that by hand (no computation made by computer)
| https://mathoverflow.net/users/33128 | Non-principal ideals in cyclotomic fields | The quadratic number field $k = {\mathbb Q}(\sqrt{-39})$ has a cyclic class group of order
$4$. Its genus field is $K = {\mathbb Q}(\sqrt{-3},\sqrt{13})$, which has class number $2$.
The Hilbert class field of $K$ is dihedral over ${\mathbb Q}$, hence not contained in the
cyclotomic field $L = {\mathbb Q}(\zeta\_{39})... | 10 | https://mathoverflow.net/users/3503 | 196013 | 95,464 |
https://mathoverflow.net/questions/196015 | 3 | I think it is well-known that $PA+\neg con(PA)$ is $\Pi\_1$-conservative over $PA$ (for proof see Smorynski's article, 'the incompleteness theorems', in handbook of mathematical logic).
What can we say about $PA+\neg R\_{PA}$ ? is it also $\Pi\_1$-conservative over $PA$ ?($R\_{PA}$ is the Rosser sentence for $PA$).
... | https://mathoverflow.net/users/65878 | Is $PA+ \neg R_{PA}$ $\Pi_1$- conservative over $PA$? | The Rosser sentence $R$ asserts, "for every proof of $R$ in PA,
there is a smaller proof of $\neg R$." So $\neg R$ asserts, "there
is a proof of $R$ in PA, with no smaller proof of $\neg R$."
In particular, we may deduce in $\text{PA}+\neg R$ that "for every
proof of $\neg R$, there is a smaller proof of $R$". This i... | 7 | https://mathoverflow.net/users/1946 | 196020 | 95,465 |
https://mathoverflow.net/questions/196010 | 4 | Is the following statement correct or known to be correct?
**For a 4-dimensional closed orientable
surface bundle $E$ with fiber of genus 2, the signature must be 0 mod 8 (or the Pontryagin number of $E$ must be 0 mod 24).**
I think it is known that for a 4-dimensional closed orientable
surface bundle $E$, the Pont... | https://mathoverflow.net/users/17787 | the Pontryagin number of a 4-dim orientable surface bundle with fiber of genus 2 | In fact, the signature must equal $0$.
Let $\pi: E \rightarrow B$ be the bundle in question, so $B$ is a closed orientable surface and the fibers of $\pi$ are genus $2$ surfaces. Letting $M\_g$ be the genus $g$ mapping class group, the bundle $\pi$ is classified by a monodromy homomorphism $f:\pi\_1(B) \rightarrow M\... | 6 | https://mathoverflow.net/users/317 | 196025 | 95,466 |
https://mathoverflow.net/questions/196033 | 10 | Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
**Question:** Can any finite lattice be realized as an intermediate subgroups lattice?
*Remark*: It's true for all the finite distributive lattices (see theorem 2.... | https://mathoverflow.net/users/34538 | Can any finite lattice be realized as an intermediate subgroups lattice? | This is an open problem. See
* Wikipedia article: <http://en.m.wikipedia.org/wiki/Finite_lattice_representation_problem>
* [Palfy and Pudlak](http://www.ams.org/mathscinet-getitem?mr=593011)'s result (see open-source description in Palfy's article [Intervals in subgroup lattices of finite groups](http://www.math.hawa... | 13 | https://mathoverflow.net/users/4600 | 196036 | 95,468 |
https://mathoverflow.net/questions/196021 | 3 | Denote by $G\_p$ a choice of an absolute Galois group of $Q\_p$, the field of $p$-adic numbers. Consider a continuous representations of $G\_p$ on a $3$-dimensional $Q\_p$ vector space that is a successive extension of the trivial character, the cyclotomic character ($\chi\_p$) and the square of the cyclotomic characte... | https://mathoverflow.net/users/38416 | extensions of crystalline representations | One has a natural exact sequence: $$0 \rightarrow H^1(G\_p, \mathbb{Q}\_p(2)) \rightarrow H^1(G\_p,U) \xrightarrow{s} H^1(G\_p,\mathbb{Q}\_p(1)) \rightarrow 0,$$ where $ H^1(G\_p, \mathbb{Q}\_p(2)) \cong H^1\_{crys}(G\_p,\mathbb{Q}\_p(2))$, $\dim H^1(G\_p,U)=\dim H^1\_{crys}(G\_p,U)+1$, $\dim H^1(G\_p, \mathbb{Q}\_p(1)... | 2 | https://mathoverflow.net/users/58042 | 196041 | 95,471 |
https://mathoverflow.net/questions/196023 | 5 | Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f\_n$ is a sequence of holomorphic functions converging pointwise to 0 on $\gamma$. Does it imply that $f\_n$ converges pointwise on the region enclosed by $\gamma$?
| https://mathoverflow.net/users/37103 | Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior? | For a counterexample, let $\gamma$ be the unit circle. Let $$A\_n = \{z \in \gamma:\; \text{Im}(z) \in [-1,0] \cup [1/n, 1]\}$$ By Runge's theorem there is a polynomial $f\_n$ such that $|f\_n| < 1/n$ on $A\_n$ but $f\_n(0) = (-1)^n$. We then have
$f\_n \to 0$ pointwise on $\gamma$ but $f\_n(0)$ does not converge.
| 10 | https://mathoverflow.net/users/13650 | 196042 | 95,472 |
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