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/187319 | 18 | While listening to some lecture of Alain Connes about noncommutative geometry, he spoke about various generalizations of the classical concepts from geometry and divided it into "soft" and "hard" part. The audience seemed to know what is all about but for me it was not clear what is the distinction. In particular he me... | https://mathoverflow.net/users/24078 | Soft and hard part of geometry | In the context of geometry, a distinction between "soft" and "hard" was introduced by Gromov, as explained [here](http://books.google.com/books?id=kQtYL7pUWSwC&pg=PA126&lpg=PA126) and applied to [Soft and Hard Symplectic Geometry.](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0081.0098.ocr.pdf)
In Gromov's ... | 13 | https://mathoverflow.net/users/11260 | 187384 | 92,946 |
https://mathoverflow.net/questions/187289 | 2 | What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is **Not** true:
For every vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ with the following properties:
1. $P$ maps the solutions of $\tilde{X}$ to solutions of $X$
2.$Div(\tild... | https://mathoverflow.net/users/36688 | Divergence invariant lifting of a vector field via a submersion | The only thing one really needs to define the operation $\mathrm{Div}:{\frak{X}}(S^3)\to C^\infty(S^3)$, i.e., mapping vector fields on $S^3$ to functions on $S^3$, is a volume form on $S^3$. (One can get away with even less, of course; really, one only needs a flat connection on the top exterior power of the cotangent... | 2 | https://mathoverflow.net/users/13972 | 187391 | 92,950 |
https://mathoverflow.net/questions/186693 | 1 | Consider the following function class: $F={f:R^d\rightarrow [a,b], f(x)=\sigma(w^Tx)}$ where $\sigma(.)$ is Lipschitz, and $w\in R^d$ is a parameter vector. The problem I'm working on is a machine learning problem where $w$ is estimated from from some data, and we can assume that there is enough data to estimate it wel... | https://mathoverflow.net/users/61472 | Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary? | If $w$ is unbounded then any nonzero sum can be dilated to an arbitrary value, so you must have some bounds on it.
If the range of $x$ is not bounded then you will not be able to give a uniform bound on $\hat R\_n(F)$ for the same reason (any bound that holds for a particular $x\in R^d$ will break down when $x$ is mu... | 1 | https://mathoverflow.net/users/12518 | 187392 | 92,951 |
https://mathoverflow.net/questions/187351 | 6 | This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the second question makes sense only if there is a proper and well-known answer to the first one.
**Part 1**
For conven... | https://mathoverflow.net/users/19864 | Decomposing polyhedral cones into "direct sums" and a polynomial | The decomposition of polyhedral cones as mentioned in Part 1 is treated in my article [*Completions of fans*](http://dx.doi.org/10.1007/s00022-011-0073-3), J. Geom. 100 (2011), 147--169. Following Igor's wish I will give a short version here.
(1) Consider a real vector space $V$ and a family $(A\_i)\_{i\in I}$ of pol... | 6 | https://mathoverflow.net/users/11025 | 187395 | 92,953 |
https://mathoverflow.net/questions/187381 | 11 | I am re-posting [a question I asked on math.se](https://math.stackexchange.com/questions/999715/representation-theory-of-the-general-linear-group-over-a-finite-prime-field) here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}\_n(\mathbb C)$ over $\mathbb C$ are comp... | https://mathoverflow.net/users/9947 | Representation theory of the general linear group over a finite prime field | One classical source for the case $n=2$, with somewhat old-fashioned notation for some of the related groups, is a paper by Richard Brauer and his student Cecil Nesbitt: *On the modular characters of groups,* Ann. of Math. (2) 42, (1941). 556–590. In this very special case, the actual modules are not too difficult to d... | 18 | https://mathoverflow.net/users/4231 | 187397 | 92,954 |
https://mathoverflow.net/questions/187404 | 10 | I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest approach is to follow Barry Simon's philosophy in his book ["Functional Integration and Quantum Physics"](http://books.googl... | https://mathoverflow.net/users/7410 | Isomorphisms between spaces of test functions and sequence spaces | Check the following sources:
* [MR0688001](https://mathscinet.ams.org/mathscinet-getitem?mr=0688001) Reviewed Vogt, Dietmar Sequence space representations of spaces of test functions and distributions. Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), pp. 405–443, Lecture Notes in Pure... | 5 | https://mathoverflow.net/users/26935 | 187408 | 92,957 |
https://mathoverflow.net/questions/187411 | 3 | Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed).
Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$,
If I'm not wrong Miller-Lannes,... theory provides tools and criteria to compare and compute the space of fixed points and homotopy fixed points $X^{G}\rightarrow X^{hG}$ (after $p$-c... | https://mathoverflow.net/users/61328 | fixed point and homotopy fixed points | We certainly don't expect to get anything similar to the Sullivan conjecture. In the case where $G$ acts trivially, the Sullivan conjecture tells us that the space of maps from $B(\mathbb{Z}/p)$ to $X$ is homotopy equivalent to the space of constant maps, but that will not be the case if we replace $B(\mathbb{Z}/p)$ by... | 12 | https://mathoverflow.net/users/10366 | 187416 | 92,959 |
https://mathoverflow.net/questions/187241 | 3 | Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commutator subgroup of $\pi\_1(B)$. Given $H\subset H\_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of... | https://mathoverflow.net/users/5259 | Reference request for cohomology of coverings | This is false. It is false for first cohomology and the maximal abelian cover of the classifying space of the integral Heisenberg group. The integral Heisenberg group $H(\mathbb Z)$ of $3\times 3$ upper triangular matrices with integer entries and 1s on the diagonal. The abelianization is $\mathbb Z^2$ given by taking ... | 6 | https://mathoverflow.net/users/4639 | 187425 | 92,962 |
https://mathoverflow.net/questions/187383 | 13 | I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ under the action of $SL(2,\mathbb{Z}) $ look like ?
1) Obviously if the vector space spanned by the coordinates of X ... | https://mathoverflow.net/users/25511 | Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$ | There are very precise results on the repartition of orbits of $SL(2,\mathbb{Z})$ of irrational points in $\mathbb{R}^2$, such as [this one](http://webusers.imj-prg.fr/~antonin.guilloux/articles/diophuni.pdf).
In particular these orbits are dense, and this can be seen easily like this.
If $\theta$ is irrational, t... | 13 | https://mathoverflow.net/users/6451 | 187426 | 92,963 |
https://mathoverflow.net/questions/187439 | 8 | This could as well have been asked in the comments to [this question](https://mathoverflow.net/questions/41167/classifying-stacks-and-homotopy-type-of-a-point), but I prefer to open a new one for the sake of clarity.
Say $G$ is a reductive group over the complex numbers, with compact real form $K$. On the one hand on... | https://mathoverflow.net/users/4721 | Relation between $BG$ in topology and in algebraic geometry | I'm not sure why you're replacing $G$ with $K$. So let me compare $[\*/G]$ and $B(G^{an})$. As you say, for the purposes of homotopy theory, $B(G^{an})$ is as good as $BK$.
One way to understand a stack $\mathcal{X}$ is via a hypercover, which is a simplicial scheme. The easiest way to get a hypercover is to choose a... | 21 | https://mathoverflow.net/users/6950 | 187441 | 92,965 |
https://mathoverflow.net/questions/187436 | 12 | Can anyone point to me where I can find the proof that the Picard group of the product of two curves is isomorphic to the product of the Picard groups times the hom among the Jacobians?
Does the result work on an arbitrary field?
Thank you very much for your time and attention
| https://mathoverflow.net/users/41314 | Picard of the product of two curves | It seems likely that you are assuming the curves are smooth and geometrically connected (e.g., you don't have in mind generalized Jacobians for singular curves), but you have omitted hypotheses on the ground field and have not indicated if you are assuming the existence of rational points.
Let $X$ and $Y$ be geometri... | 12 | https://mathoverflow.net/users/52824 | 187445 | 92,967 |
https://mathoverflow.net/questions/187437 | 5 | **Problem**
Given a C\*-algebra $\mathcal{A}$.
Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
*(More precisely, strongly continuous one-parameter groups.)*
Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D... | https://mathoverflow.net/users/45494 | C*-Algebras: Dynamics vs. Derivations | As was discussed in the comments, it suffices to see that $\delta$ determines $\tau$ uniquely on $\mathrm{dom}(\delta)$ which is dense in $\mathcal{A}$. Suppose that $x\_0 \in \mathrm{dom}(\delta)$. Check that $t \mapsto \tau^t(x\_0)$ is a solution to the initial value problem
\begin{align\*}
\frac{d}{dt} x(t) = \delta... | 6 | https://mathoverflow.net/users/12281 | 187449 | 92,968 |
https://mathoverflow.net/questions/187447 | 0 | Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$ only at $z = 0$. Let $D\_i \subset {\mathbb C}^N$ be the divisor defined by $\partial\_i F$.
My question is: is the... | https://mathoverflow.net/users/2555 | A condition on isolated singularity | Assume that $F$ is quasihomogeneous.
A condition on $F$ would be the following.
For each $k$ write $F=x\_kG+H\_k(x\_1,\dots,x\_{k-1},x\_{k+1},\dots, x\_N)$. Then $H\_k$ is quasihomogeneous.
If for each $k$ the polynomial $H\_k$ has an isolated singularity in $\mathbb{C}^{N-1}$ then $F$ does not vanish identically on... | 1 | https://mathoverflow.net/users/8621 | 187456 | 92,971 |
https://mathoverflow.net/questions/187440 | 17 | Previously I have mentioned the following problem in an addition to the list of [Contest problems with connections to deeper mathematics](https://mathoverflow.net/questions/69737/contest-problems-with-connections-to-deeper-mathematics/178158#178158).
*Is there an infinite bounded sequence $(P\_n) \subset \mathbb{R}^... | https://mathoverflow.net/users/26522 | Is there a bounded sequence of points in the plane with pairwise distances at least $1/\sqrt{|i-j|}$? | It seems that such a sequence exists.
**1.** Firstly, we take an auxiliary sequence $a(n)=\{(n+1)\sqrt2\}$ for $n\geq 0$. For every $m>n$ the standard estimate yields, say,
$$
|a(m)-a(n)|=|(m-n)\sqrt2-p|=\frac{|2(m-n)^2-p^2|}{(m-n)\sqrt2+p}>\frac1{10|m-n|};
$$
here $p=[(m+1)\sqrt2]-[(n+1)\sqrt2]\leq 8(m-n)$.
**2.... | 16 | https://mathoverflow.net/users/17581 | 187463 | 92,973 |
https://mathoverflow.net/questions/186336 | 9 | Define the following,
$$j(\tau) = \Big(\tfrac{E\_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$
$$j\_{2A}(\tau) =\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{12}+2^6 \big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{12}\Big)^2 = \tfrac{1}{q} ... | https://mathoverflow.net/users/12905 | Asymptotic formulas for Monster-related modular functions? | In a recent paper with Ken Ono and John Duncan, we give exact formulas for the coefficients of the Monsterous Moonshine modules. I will give a link to a preprint a few days when I can, but the relevant part relies on prior work of Bringmann and Ono (<http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/115.pdf>) which... | 9 | https://mathoverflow.net/users/61910 | 187472 | 92,976 |
https://mathoverflow.net/questions/187453 | 7 | Complex tori are not associated to projective varieties in general.
But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective variety?
What if we just ask $U$ to be the analytification of a (finite type separated) scheme?
I think that if $U$ com... | https://mathoverflow.net/users/61926 | Do complex tori contain quasi-projective open subsets? | The answer is negative if the complex torus is not algebraic (or equivalently, not Moishezon). More generally, if $U$ is the analytification of a separated scheme of finite type over $\mathbf{C}$ and $Y$ is a proper complex-analytic space admitting an open immersion $j:U \hookrightarrow Y$ onto the complement of a nowh... | 5 | https://mathoverflow.net/users/61939 | 187473 | 92,977 |
https://mathoverflow.net/questions/187193 | 13 | Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f\_n)$ a sequence in $Y^X$ with the following property:
For every $x\_0\in X$ and every $\varepsilon>0$, there exist a neigbourhood $U$ of $x\_0$ and an index $n\_0$ such that we have $d(f\_n(x),f(x))<\varepsilon$ for every $x\in U$ and every ... | https://mathoverflow.net/users/26591 | Between compact and locally uniform: What is the name of this convergence? | This notion of convergence is not often refered to; I think mostly because it does not come from a topology. But this an excellent notion of convergence, probably the best we can put on the space of (continuous) functions. I can tell you a few things about it - *but only in the case where all the functions are continuo... | 14 | https://mathoverflow.net/users/22131 | 187474 | 92,978 |
https://mathoverflow.net/questions/187421 | 6 | I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds:
*Find a pair of complex analytic families $\lbrace M\_t\rbrace$ and $\lbrace N\_t\rbrace$ with $|t|<1$ such that $M\_t=N\_0$ for $t\ne 0$ and $N\_t=M\_0$ for $t\ne0$, with $M\_0\ne N\_0$.* (... | https://mathoverflow.net/users/12310 | Inverted pair of complex analytic families | I'm going to assume that your objects are compact analytic spaces and show that then the situation you describe cannot arise. I don't have access to Morrow and Kodaira and cannot remember whether the "problem" concerns only compact manifolds.
Suppose that $\mathcal M\to S$ is a miniversal deformation of $M\_0$. Then ... | 2 | https://mathoverflow.net/users/61943 | 187478 | 92,981 |
https://mathoverflow.net/questions/187413 | 9 | Bounded Zermelo set theory, and many variants named for MacLane in some way, are used in equiconsistency proofs for Simple Theory of Types plus infinity, and for the Elementary Theory of the Category of Sets starting in 1969 (see [Finite order arithmetic and ETCS](https://mathoverflow.net/questions/115207/finite-order-... | https://mathoverflow.net/users/38783 | When was Bounded Zermelo set theory first formulated? | The Princeton thesis of John Kemeny, written in 1949, was devoted to the relation between Zermelo set theory and type theory. For example, early in the thesis, there is a proof of the consistency of the simple theory of types relative to the consistency of a small fragment (nowadays known as KF) of Bounded Zermelo set ... | 8 | https://mathoverflow.net/users/9269 | 187482 | 92,982 |
https://mathoverflow.net/questions/187486 | 3 | Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.
What is known about group cohomology $H^\*(SL(2,{\mathbb R}),{\mathbb R}^2)$?
And about $$H^\*(\Gamma,{\mathbb R}^2)$$ for a cocompact lattice $$\Gamma\subset SL(2,{\mathbb R})?$$
| https://mathoverflow.net/users/39082 | Cohomology of SL(2,R) with coefficients given by linear action | It is zero. This is an application of the "centre kills" trick, which I will state in homology.
**Trick.** Let $M$ be a $G$-module for which there is an element $z$ in the centre of $G$ which acts as $-1$ on $M$. Then $2H\_\*(G;M)=0$.
In your situation the homology is a real vector space, so if multiplication by 2 ... | 5 | https://mathoverflow.net/users/318 | 187500 | 92,993 |
https://mathoverflow.net/questions/187165 | 10 | I saw the following problem in Mathematical Puzzles from Peter Winkler (very good book, by the way): imagine you infect k cases of a chessboard nxn and the infection spreads to a case if it has at least two neighbors infected. Then k = n is the minimum number such that it is possible to infect the whole chessboard.
I w... | https://mathoverflow.net/users/59249 | Infected square | There is an d-dimensional version of this problem in *The Art of Mathematics - Coffee Time in Memphis* by Bela Bollobas. (Problem 35)
According to it, the answer is $k = \lceil d(n-1)/2 \rceil + 1$.
| 7 | https://mathoverflow.net/users/36579 | 187506 | 92,997 |
https://mathoverflow.net/questions/176164 | 11 | $\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$
In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset \RR^n$ for $m\leq n$ (some of them also for non-integer $m$, but this is not the point here). They all build on Carathéodory'... | https://mathoverflow.net/users/9652 | Geometric measures different from Hausdorff | Hausdorff, spherical Hausdorff and dyadic net measures not only give rise to the same dimension but, for a fixed value of $m$, are comparable up to constants that depend only on the ambient dimension $d$. In particular, the property of having zero, positive and finite, or infinite measure coincides for these three meas... | 7 | https://mathoverflow.net/users/11009 | 187509 | 92,998 |
https://mathoverflow.net/questions/187494 | 13 | Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial complex whose faces are independent subsets of $G$. Said another way, $I(G) = X( \bar{G} )$, where $\bar{G}$ denotes the ... | https://mathoverflow.net/users/4558 | Is there any relationship between the topologies of the clique complex and the independence complex? | Given any graph $G$, let $G'$ denote $G$ with a new vertex $v$
adjacent to every vertex of $G$. The clique complex of $G'$ is
contractible, while the independence complex of $G$ and $G'$ are the
same except for an isolated vertex. Similarly we can adjoin a new
isolated vertex $w$ to $G$, obtaining a graph $G''$ with a
... | 17 | https://mathoverflow.net/users/2807 | 187510 | 92,999 |
https://mathoverflow.net/questions/187516 | 15 | I was reading about the passing of Alexander Grothendieck, and something caught my interest:
>
> Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal mathematical principles that could shed unexpected light on them. Applications of his work are evident in fields as d... | https://mathoverflow.net/users/61965 | Robotics, Cryptography, and Genetics applications of Grothendieck's work? | Here is a guess: first, "Grothendieck's work" is being interpreted as "algebraic geometry," so the real question is what applications of algebraic geometry there are in genetics, cryptography, and robotics.
* Genetics: my guess is that this is a reference to the use of [algebraic statistics](http://en.wikipedia.org/w... | 17 | https://mathoverflow.net/users/290 | 187518 | 93,001 |
https://mathoverflow.net/questions/187519 | 4 | This is related to my previous question [here](https://mathoverflow.net/questions/182027/restricting-the-steinberg-representation-of-sl-2n-over-a-finite-field-to-the).
Let me remind you what that question asked:
---
Let $\text{St}\_n(\mathbb{F}\_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}\_n(... | https://mathoverflow.net/users/61967 | Decomposing representations of finite groups of Lie type via computer | Here's one way to perform this computation in GAP, although I'm sure there are smarter ways.
```
p:=3;;
n:=2;;
G:=SL(2*n,p);;
H:=Sp(2*n,p);;
irrg:=Irr(G);;
irrh:=Irr(H);;
dims:=List(irrg,chi->chi[1]);;
steinberg:=irrg[Position(dims,p^Binomial(2*n,2))];;
restrictedSteinberg:=Restricted(steinberg,H);;
DecomposeCharact... | 7 | https://mathoverflow.net/users/9068 | 187524 | 93,003 |
https://mathoverflow.net/questions/187457 | 2 | Gauss sum is a sum of $p$ roots of unity with magnitude $\sqrt{p}$. Does another sum with such property exist?
More exactly. Let $p$ be a prime number. $\zeta^p=1,\;\zeta\ne 1$. Causs sum: $G=\sum\_{i=0}^{p-1}\zeta^{i^2}$. We know that $|G|=\sqrt p$. I am looking for $a\_0,\ldots,a\_{p-1}\in N\_0$: $∑a\_i=p;|\sum\_{... | https://mathoverflow.net/users/61930 | not Gauss sum with the same magnitude | [A paper by Cavior](http://www.jstor.org/stable/2034027) does just that. It also follows from the method in [Elkies' answer](https://mathoverflow.net/questions/135949/a-question-on-maps-from-mathbbz-p-mathbbz-to-itself) to a previous MO-question.
| 2 | https://mathoverflow.net/users/18739 | 187533 | 93,005 |
https://mathoverflow.net/questions/187537 | 9 | Let $X, Y$ be smooth projective varieties and $f:X \times Y \to Y$ be the natural projetion map. Let $\mathcal{F}$ be a locally free sheaf on $X \times Y$. Is it true that $f\_\*\mathcal{F}$ is locally free on $Y$? If not true in general is there any additional condition on $X, Y$ under which this will hold true?
| https://mathoverflow.net/users/45397 | Push-forward of locally free sheaves | Let $X = P^1$, $Y = P^3$. We will take $F$ to be an extension
$$
0 \to O(-2,1) \to F \to O \oplus O \oplus O \to 0.
$$
Of course $F$ is locally free. Note that
$$
Ext^1(O,O(-2,1)) = H^1(P^1\times P^3,O(-2,1)) = H^1(P^1,O(-2))\otimes H^0(P^3,O(1))
$$
is a 4-dimensional vector space, so we can take $F$ to be the extens... | 21 | https://mathoverflow.net/users/4428 | 187541 | 93,007 |
https://mathoverflow.net/questions/187544 | 1 | Im translating an article about Rauzy fractal and I ran into this sentence:
```
The Rauzy fractal has remarkable properties. Firstly, it is selfsimilar,
more exactly, it is divided into three pieces,
corresponding to the three letters, which are the
solutions of a graph-directed iterated function system.
```
I do ... | https://mathoverflow.net/users/50309 | What is "graph-directed iterated function"? | From the wikipedia article, you have the substition rules $1\to12$, $2\to13$, $3\to1$.
We can write this in matrix form
$$
\begin{pmatrix}
1 & 1 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix}
$$
where the first column indicate what $1$ is mapped to, the second what $2$ is mapped to, and so on.
This makes a bit sense, th... | 2 | https://mathoverflow.net/users/1056 | 187550 | 93,009 |
https://mathoverflow.net/questions/187531 | 12 | Let $X,Y$ be topological spaces. We call a continuous map $u:X\to Y$ *universal* if for every continous map $f:X\to Y$ there is $x\in X$ such that $f(x) = u(x)$.
If $u:X\to Y$ and $v:Y\to Z$ are universal, is $v\circ u: X\to Z$ universal?
(I am thinking about this question because of an inspiration from [Fixed poin... | https://mathoverflow.net/users/8628 | Universal maps between topological spaces | This seems to be answered negatively in "On the composition and products of universal mappings" by W. Holsztyński (Fundamenta Mathematicae 64(2) (1969), 181-188).
I've not looked at the proof or really tried to figure out why it works, but if I understand correctly what is claimed, then one counterexample from the pa... | 8 | https://mathoverflow.net/users/22989 | 187553 | 93,010 |
https://mathoverflow.net/questions/187511 | 2 | There are two questions:
1. How to prove that in general
$[\hat{A}(\mathbb HP^m)]\_{4m} = 0$
It is possible to verify it for low values of $m$.
2. How to prove that in general
$\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]\_{4m} = 0$
where $\hat{M}(\mathbb HP^m)$ is the Mayer class defi... | https://mathoverflow.net/users/61860 | Is true that $\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$? | Since $HP^n$ is a spin manifold and since the homogeneous metric has positive scalar curvature, the $\hat{A}$-genus of $HP^n$ is zero by the index theorem and the Weitzenboeck-Lichenrowicz formula.
I cannot at the moment answer the question for the quotient by the Mayer class. But there is a direct computation for th... | 4 | https://mathoverflow.net/users/9928 | 187560 | 93,013 |
https://mathoverflow.net/questions/187508 | 1 | Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$.
We assume that the map on the level of fomal neighborhoods $X\_{x}\rightarrow Y\_{y}$ is formally smooth, can we find a étale neighborhood $S$ of $x$, such that $S\rightarrow Y$ is smooth at $x$.
| https://mathoverflow.net/users/27398 | Smoothness and smoothness over formal neighborhood | Let $k$ be a field and $Y=\mathrm{Spec}\,R$ where $R=\bigcup\_{n>0}k[[t^{1/n}]]$ is the ring of Puiseux series over $k$. Take $X=\mathrm{Spec}\,(R/tR)$, $f=$ the obvious embedding. The maximal ideal $m$ of $R$ satisfies $m=m^2$, and the same holds in $R/tR$, so both completions are equal to $k$, but of course $f$ is no... | 6 | https://mathoverflow.net/users/7666 | 187563 | 93,015 |
https://mathoverflow.net/questions/187545 | 12 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the
language of <https://en.wikipedia.org/wiki/Free_Lie_algebra>,
the free Lie algebra generated by any choice of basis $e\_1, \ldots , e\_r$ for the vector space $V$. (W... | https://mathoverflow.net/users/2906 | Breaking up the free Lie algebra into GL irreps | The Whitehouse module referred to in one of the other answers is not necessary, since it is related to the *cyclic* operad Lie, that is to the representation of $S\_{n+1}$ in $Lie(n)$.
The decomposition in terms of Young diagrams is, as far as I understand, first done in a paper of Kraskiewicz and Weyman (preprint W.... | 10 | https://mathoverflow.net/users/1306 | 187566 | 93,018 |
https://mathoverflow.net/questions/187549 | 3 | I am thinking a problem: given a subshift of finte type of $\{0,1\}^{\mathbb{N}}$ and $2>q>1$, where $q$ is a real number. Then how can we find the largest and smallest numbers of the projection of this subshift of finite type in base $q$. We know that the projection of the subshift of finite type in base $q$ is a grap... | https://mathoverflow.net/users/58508 | The upper and lower bound of the projection of a subshift of finite type | It's a rational point for each $q$. Suppose the SFT has forbidden words of length at most $\ell$. Then given your current symbol and $\ell-1$ previous symbols, you can decide what symbol to put to maximize(minimize) your projection going forward. This means that your point is eventually periodic with period at most $2^... | 2 | https://mathoverflow.net/users/11054 | 187570 | 93,020 |
https://mathoverflow.net/questions/187575 | 3 | Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them.
In the symplectic world, there are sum formulas for Gromov-Witten invariants of symplectic manfiolds which can be described a symplectic su... | https://mathoverflow.net/users/62001 | Embedded Contact Homology and Manifold Decompositions | Yes, given two contact 3-manifolds $(M\_1,\xi\_1)$ and $(M\_2,\xi\_2)$ we can form their contact sum $(M\_1\# M\_2,\xi\_1\# \xi\_2)$ and then ECH decomposes as the tensor product of the corresponding ECH's of the pieces (at least assuming field coefficients).
See the paper **Sutures and Contact Homology** by Colin-Gh... | 2 | https://mathoverflow.net/users/12310 | 187578 | 93,027 |
https://mathoverflow.net/questions/187602 | 4 | Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse function. However, generally (i.e. ignoring the action of $G$) using Sard's lemma, almost every linear function h will give... | https://mathoverflow.net/users/62014 | Existence of an equivariant Morse function | This is discussed in
MR0250324 (40 #3563) Wasserman, Arthur G. Equivariant differential topology.
Topology 8 1969 127--150.
| 1 | https://mathoverflow.net/users/12156 | 187607 | 93,036 |
https://mathoverflow.net/questions/187415 | 1 | Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X\_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is continuous on $\partial\Omega$ for all $x\in\bar\Omega$. Moreover let suppose that
\begin{equation}
||f||\_{X\_0}:=\sup\_{x\... | https://mathoverflow.net/users/58541 | On the Hölder regularity of an integral function | The condition $\psi(t,x,x)=0$ seems to be irrelevant here, because under the implied conditions on $\varphi$ $\lim\_{x\to y}\psi(x,y,t)=c\_n \varphi(y,t)$. If the limit values of $\psi$ is from $C^{\alpha/2}$ with respect to $t$ one cannot expect it to have uniform estimate in $C^{(1+\alpha)/2}$ up to the boundary.
| 1 | https://mathoverflow.net/users/14551 | 187608 | 93,037 |
https://mathoverflow.net/questions/187609 | 3 | I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^n$ to $\m... | https://mathoverflow.net/users/24563 | Systems of equations in Boolean Algebra | Let $A$ be an $m\times n$ Boolean matrix. Then the mapping $v\mapsto Av$ is 1:1 iff $n\leq m$ and some subset of $n$ rows of gives a permutation matrix.
The reason is duality of modules over the Boolean semiring shows that $A$ is 1:1 iff the transpose is onto. Since the standard basis vectors of a free $\mathbb B$-m... | 5 | https://mathoverflow.net/users/15934 | 187615 | 93,038 |
https://mathoverflow.net/questions/187613 | 0 | Let $H = (V, E)$ be a hypergraph, that is $V$ is a set and $E \subseteq \mathcal{P}(V)$. We say that $H$ is $T\_1$ if for $v\neq w$ there are $e\_v, e\_w \in E$ such that $v\in e\_v, w\notin e\_v, w\in e\_w, v\notin e\_w$.
Fix a set $V$. Let $E\in \mathcal{P}(\mathcal{P}(E))$ be $T\_1$. Does the set
\begin{eqnarray}... | https://mathoverflow.net/users/8628 | Minimal hypergraphs with respect to separation | **Counterexamples:** Let $V=\mathbb R$ and let $E$ be the **base** for the usual topology consisting of the open intervals, or the **subbase** consisting of the open rays of the form $(a,\infty)$ and $(-\infty,a)$.
| 1 | https://mathoverflow.net/users/43266 | 187617 | 93,039 |
https://mathoverflow.net/questions/187598 | 15 | In the paper [Zur Hilbertschen Beweistheorie](http://link.springer.com/article/10.1007%2FBF01475439), John Von Neumann has proposed a consistency proof for
a fragment of first-order arithmetic (the fragment without induction and with
the successor axioms only) by a variant of Hilbert's substitution method.
The paper ... | https://mathoverflow.net/users/11115 | Von Neumann's consistency proof |
>
> I will attempt to answer your first question by describing the context of the von Neumann paper you have asked about.
>
>
>
As is well known, Hilbert had earlier posed the problem of establishing the consistency of first order arithmetic (with full induction), and had suggested the development of the method... | 17 | https://mathoverflow.net/users/9269 | 187619 | 93,040 |
https://mathoverflow.net/questions/187603 | 8 | For $k \geq 2, n \geq 1$ let $$M^{n,k} = \{(x\_1,\dots,x\_k) \in S^n \times \dots \times S^n \ | \ x\_1 + \dots + x\_k = 0\}$$ This is a compact CW-complex and almost, but not quite, a manifold. I generally want to understand this space better. More concretely, I am interested in the following questions:
(1) What is ... | https://mathoverflow.net/users/14233 | Configuration space like subspace of sphere product | Your space $M^{n,k}$ is a moduli space of closed $k$-gons in $\mathbb{R}^{n+1}$ with sides of length 1, viewed up to translation. As such it is a fairly well-studied object in the literature, see for example
Farber, Michael; Fromm, Viktor
*The topology of spaces of polygons,*
Trans. Amer. Math. Soc. 365 (2013), no. ... | 6 | https://mathoverflow.net/users/8103 | 187630 | 93,042 |
https://mathoverflow.net/questions/187624 | 7 | If a set $S$ is endowed with the discrete topology $\mathcal{P}(S)$, then for every normal space $N$ the product $S\times N$ is normal.
**Question**: can we endow a set $S$ with another Hausdorff topology, such that still for all normal spaces $N$ the product $S\times N$ is normal.
| https://mathoverflow.net/users/nan | "Productively normal" space | The answer is no. It was proved by Mary Ellen Rudin in *$\aleph$-Dowker spaces* (1978) that for any non-discrete Hausdorff space $S$ there is a normal Hausdorff space $N$ such that $S\times N$ is not normal, thus solving in the affirmative [Morita´s first conjecture](http://en.wikipedia.org/wiki/Morita_conjectures).
| 12 | https://mathoverflow.net/users/17836 | 187638 | 93,044 |
https://mathoverflow.net/questions/187642 | 0 | An elliptic curve is defined over the field of real numbers:
$y^2=x^3 + ax + b$
A point P and scalar n can be multiplied using a combination of [point doubling and adding](http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_multiplication).
What about point division? Given point P on the curve,... | https://mathoverflow.net/users/62037 | Is elliptic curve point division defined over the field of real numbers? | The multiplication by $n$ map is surjective on the set of complex points, so you can always divide there, and a point has $n^2$ complex inverse images. Over the reals, $E(\mathbb{R})$ is isomorphic, as a real Lie group, to either the circle group $S^1$ or
to two copies $S^1\times\mathbb{Z}/2\mathbb{Z}$. In the former c... | 10 | https://mathoverflow.net/users/11926 | 187646 | 93,046 |
https://mathoverflow.net/questions/187156 | -2 | Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have such an idea:
It's better to find the $\mathbb{P}(\det A \leq 4000)$:
$F\_{\det A} = \mathbb{P}(\det A \leq 4000) = \... | https://mathoverflow.net/users/61783 | Determinant of matrix from set {-1, 1} | Thank you, @Noam, for your idea!
**Statement 1:** Let $A \in \mathbb{R}^{n \times n}$ and $a\_{ij} \in \{-1, 1\}$, then
$\mathbb{E}(\det A) = 0$
**Proof:** It's easy to see, that because of every $a\_{ij}$ is a random value:
$ \mathbb{E}(\det A) = \mathbb{E} \left(
\sum\limits\_{\alpha=(\alpha\_1,\alpha\_2,...,\al... | 1 | https://mathoverflow.net/users/61783 | 187650 | 93,048 |
https://mathoverflow.net/questions/187466 | 9 | I heard from someone that the following problem is an open question.
(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,y|xy^mx^{-1}=y^n\rangle$ and it admits a finite $K(G,1)$, is it a hyperbolic group?
I could not find the relevent st... | https://mathoverflow.net/users/9305 | amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic? | Yes, this problem is as open as Problem 1. There are no feasible obstructions to counter-examples, the main obstacle is lack of tools: *All* known constructions of non-hyperbolic groups with finite $K(G,1)$, yield groups which contain some $BS(p,1)$, $p\ge 1$.
Edit: For elementary amenable groups of finite type, pos... | 7 | https://mathoverflow.net/users/21684 | 187653 | 93,050 |
https://mathoverflow.net/questions/180343 | 8 | Let $G$ be a $n\times n-$symmetric matrix with *integral coefficients* and determinant $1$ (*i.e. unimodular*) such that the associated quadratic form is positive-definite.
I am interested in having an algorithm to find a *rational* basis of a lattice $L$ such that $G$ is the Gram matrix of $L$. Concretely, this cons... | https://mathoverflow.net/users/5239 | Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associated lattice (over $\mathbf Q$)? | Here is a method.
Let $\mathrm{V}$ be the bilinear space $(\mathbf Q^n, b)$ with Gram Matrix $G$ in the canonical basis.
Let $M\_0$ be the lattice $\mathbf Z^n$ on $\mathbf Q^n$. Find a primitive vector $v\_0\in M\_0$ such that $b(v\_0,v\_0)$ is a square, say $b(v\_0,v\_0)=m^2$ in $M\_0$ (note that this can be don... | 3 | https://mathoverflow.net/users/39552 | 187664 | 93,055 |
https://mathoverflow.net/questions/187667 | 2 | First I need some notation (it's all standard I think). For a manifold $M$, let $F\_nM = F\_{0,n}M$ be the space of $n$-tuples of distinct points on $M$ ; let $B\_nM = B\_{0,n}M = F\_nM / \Sigma\_n$. When $M= \mathbb{R}^2$ the fundamental group of $B\_nM$ is the braid group $B\_n$, and that of $F\_nM$ is the pure braid... | https://mathoverflow.net/users/37021 | (Alternative) Presentation for the pure braid group of the sphere | I think this is in Birman's book. Certainly all this appears in many of Fred Cohen's papers.
The bundle $F\_n S^2 \to S^2$ can be understood perhaps best by bringing in the principal bundle for the tangent bundle of $S^2$. This is $SO\_3$. There's the bundle:
$$ SO\_2 \to SO\_3 \to S^2 $$
where the map on the ri... | 3 | https://mathoverflow.net/users/1465 | 187672 | 93,058 |
https://mathoverflow.net/questions/187671 | 4 | If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has
$$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|\_F^2\,d\mathbf{x}\,,$$
where $\Delta$ denotes the Laplacian, $H$ denotes the Hessian (matrix of second derivatives), and $\| \cdot\|\_F^2$ denotes the Frobenius... | https://mathoverflow.net/users/45789 | Relationship between Laplacian and Hessian on compact Lie groups | This has nothing to do with Lie groups, I believe. Let $M$ be a Riemannian manifold. The Bochner formula on $1$-forms states that
$$\nabla^\* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\,\omega.$$
Hence we have for any compactly supported function $f$ (writing round brackets for the $L^2$ scalar product)... | 10 | https://mathoverflow.net/users/16702 | 187673 | 93,059 |
https://mathoverflow.net/questions/187632 | 13 | Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation)
of the Langlands dual group $^LG$.
The irreducible representations of $G$ have additional structure related to tensoring representations: Given re... | https://mathoverflow.net/users/284 | Langlands duality and multiplying cocharacters | This answer will be cheating: you can look at the affine grassmanian of $^LG$, $\mathcal{G}r(^LG)=^LG(\mathcal{K})/^LG(\mathcal{O})$ where $\mathcal{O}=\mathbb{C}[ [t]]$ and $\mathcal{K}$ its field of fractions. The [Geometric Satake isomorphism](http://ncatlab.org/nlab/show/geometric+Satake+equivalence) states that th... | 8 | https://mathoverflow.net/users/17980 | 187674 | 93,060 |
https://mathoverflow.net/questions/187648 | 5 | Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert \_{L^{\infty}} < 1$. It is well known that if $\mu\_n \rightarrow \mu$ in $L^{\infty}$ then $f\_n \rightarrow f$ (the f's are all norm... | https://mathoverflow.net/users/3709 | $L^p$ stability of the Beltrami equation | EDIT: The old answer to the originally asked question (without the additional assumption of convergence of the $L^\infty$ norms) is below.
I am not sure about effective convergence estimates, but the answer to the question is yes, even under slightly weaker assumptions. It directly follows from the Bers-Bojarski conv... | 5 | https://mathoverflow.net/users/26834 | 187676 | 93,061 |
https://mathoverflow.net/questions/187691 | 4 | Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a point, but is that the only case?
Obviously I would be happiest with a general criterion valid for some large clas of... | https://mathoverflow.net/users/39713 | When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$? | This is true only if $X$ has at most one point. Suppose $i:\mathbf{Top}/X\to \mathbf{Top}$ is a full embedding. Write $Id$ for the terminal object of $\mathbf{Top}/X$, the identity map $X\to X$. Then $i(Id)$ must have only one continuous self-map, and hence has at most one point (since any constant map is always contin... | 18 | https://mathoverflow.net/users/75 | 187692 | 93,066 |
https://mathoverflow.net/questions/186357 | 1 | My questions is concerned with the following problem: Given an undirected graph $G = (V, E)$ and (edge costs) $c \in \mathbb{Z}^E$,
$$\min \left\{ \sum\_{e \in E} c\_e x\_e\ \middle|\ x \in \{0,1\}^E \ \wedge\ \forall C \in \mathrm{cycles}(G)\ \forall e \in C:\ x\_e \leq \!\!\!\!\sum\_{e' \in C \setminus \{e\}} \!\! x\... | https://mathoverflow.net/users/43715 | Is it known whether Minimum Cost Multicut is APX-hard? | Yes, it is APX-hard. A proof by Demaine et al. is published [here](http://www.sciencedirect.com/science/article/pii/S0304397506003227).
| 0 | https://mathoverflow.net/users/43715 | 187707 | 93,067 |
https://mathoverflow.net/questions/187514 | 2 | Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function.
I know one can bound sums of the form
$$
\sum\_{x \leq X} e(f(x))
$$
via for example Van der Corputs's result, provided we make assumptions
on the range of the derivatives. I was wondering if there were results of this type
for weighted sums. Can... | https://mathoverflow.net/users/48408 | Bound on exponential sum with weights | You can use van der Corputs's method for weighted sums as well, see Ch. III in
>
> Karatsuba A. A., Voronin S. M. The Riemann zeta-function Walter de Gruyter & Co., 1992.
>
>
>
| 2 | https://mathoverflow.net/users/5712 | 187713 | 93,071 |
https://mathoverflow.net/questions/187681 | 6 | Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable with respect to variations in the metric $g$? I believe the stability problem for minimal surfaces has to be well understoo... | https://mathoverflow.net/users/42326 | Stability of minimal surfaces | Now that your comment has clarified your question, we can answer it: The answer is 'no'. There is the following well-known example:
Consider the following family of circles: $C\_\lambda$ is defined as $x^2+y^2 = 1$ and $z = \lambda$. Let $\lambda>0$ be fixed and orient $C\_{-\lambda}$ counterclockwise and orient $C\... | 12 | https://mathoverflow.net/users/13972 | 187714 | 93,072 |
https://mathoverflow.net/questions/187366 | 3 | This is a cross-post from [math.SE](https://math.stackexchange.com/q/1022034/120628)
Let $E \to M$ be a complex vector bundle with $P$ the associated $GL(n,\mathbb C)$ frame bundle. The group of gauge transformations is the space of sections of $P\times\_{Ad} GL(n,\mathbb C)$ and the space of hermitian metrics is the... | https://mathoverflow.net/users/61887 | Transitivity of the action of the group of gauge transformations on the space of hermitian metrics | View $h\_i$ as conjugate linear bundle isomorphisms $h\_i:E\to E^\*$.
Then $\phi=h\_1^{-1}\circ h\_2$ satisfies $h\_2(x,y)=h\_1(\phi.x,y)$.
Since $h\_2$ is positive hermitian, the endomorphism $\phi$ is positive hermitian with respect to $h\_1$, thus $\sqrt{\phi}$ exists with respect to $h\_1$ and $h\_2(x,y)=h\_1(\sq... | 2 | https://mathoverflow.net/users/26935 | 187720 | 93,074 |
https://mathoverflow.net/questions/187719 | 3 | Suppose $f(z)=a\_0+a\_1z+\cdots+a\_nz^n+\cdots$ is defined in the unit disk and $\|f\|\_{\infty}\leq 1.$ Lets form another series $g$ by interchanging $a\_1$ and $a\_k$ i.e. $g(z)=a\_0+a\_kz+\cdots+a\_1z^k+\cdots$. Is $g$ of norm less than or equal to one?if that is not the case can you provide a counterexample?
| https://mathoverflow.net/users/48438 | Norm of swapped power series in the unit disk | For simplicity, let us assume that the radius of convergence of $f$ is strictly larger than one.
By the maximum principle the absolute value of the analytic function $f:D\to\mathbb C$ obtains its maximum at the boundary $\partial D=S^1$.
Therefore we are interested in the function $g:[0,2\pi]\to\mathbb C$,
$$
h(\phi)
=... | 2 | https://mathoverflow.net/users/55893 | 187724 | 93,075 |
https://mathoverflow.net/questions/187675 | 17 | Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space $A^2(\Omega)$ is called the *Bergman Space* of $\Omega$.
I ask since on page 56 of the second edition of Krantz's "Function The... | https://mathoverflow.net/users/1106 | Can all $L^2$ holomorphic functions on a domain vanish at a particular point? | $\def\CC{\mathbb{C}}$Here is a less trivial example that I think works. Let $U \subset \CC^2$ be
$$\{ (x,y) : |x| \leq \min(1, 1/|y|) \}$$
There are lots of $L^2$ holomorphic functions because the change of variables $(u,v) = (x, xy)$ changes $|x|^2 dx d\bar{x} dy d \bar{y}$ to $du d\bar{u} dv d \bar{v}$, and takes $... | 9 | https://mathoverflow.net/users/297 | 187743 | 93,085 |
https://mathoverflow.net/questions/187740 | 4 | Suppose $A<\_T B$ ($A$ is a set computable from $B$ but not vice versa). Is it always the case that there exists a $B$-computable function which eventually outgrows all $A$-computable functions?
Of my main interest is the case when $A\equiv\_T 0$, and then the problem becomes: does there, for every nonrecursive set, ... | https://mathoverflow.net/users/30186 | Relation between Turing degrees and functions computable with them | *Throughout, "function" means "total function."*
The answer is no! In fact, there are nonzero Turing degrees which only compute functions which are bounded by some computable function. Such degrees are called "hyperimmune-free", or (more understandably) "computably bounded." See "The degrees of hyperimmune sets" by M... | 8 | https://mathoverflow.net/users/8133 | 187746 | 93,087 |
https://mathoverflow.net/questions/187721 | -2 | suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best problem to reduce to this problem is 3-colorable graph and i think X is the set of vertices.But i can't find the best rule for creatin... | https://mathoverflow.net/users/62068 | how to reduce 3-colorable graph to this? | This problem is indeed NP-complete and was in fact one of [Karp's 21 NP-complete problems](http://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems). Googling [exact cover](http://en.wikipedia.org/wiki/Exact_cover) will lead to enlightenment.
| 1 | https://mathoverflow.net/users/2233 | 187752 | 93,091 |
https://mathoverflow.net/questions/186581 | 2 | Suppose $G$ is a finite group and $A$ is the set of all character values of $G$. By character values, I mean entries of the character table of $G$. Let $\Gamma = \operatorname{Gal}({\mathbb{Q}(A)}/{\mathbb{Q}})$. Then $\Gamma$ has an action on the set of all conjugacy classes of $G$. Considering orbits, we get an equiv... | https://mathoverflow.net/users/60642 | Rational Conjugacy Classes of Finite Groups | The answer is yes, and this is not difficult to see, but first you have to show that you have indeed an action of $\Gamma$ on the conjugacy classes. Let $\varepsilon$ be a primitive $|G|$-th roots of unity and let $\widehat{\Gamma}=\operatorname{Gal}(\mathbb{Q}(\varepsilon)/\mathbb{Q})$. Then $\mathbb{Q}(A)\subseteq \m... | 5 | https://mathoverflow.net/users/10266 | 187753 | 93,092 |
https://mathoverflow.net/questions/187657 | 10 | I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T\_p$ to Frobenius at $p$ in characteristic $p$.
Two possible ways to compute $T\_p$ mod $p$ seem to be:
A) Look at the map $Div(X\_0(N)) \to Div(X\_0(N))$ induced by the correspondence $X\_0(N) \leftarrow X\_0(Np) \to X\_0(N)... | https://mathoverflow.net/users/62047 | Eichler-Shimura congruence | There's a little more geometry here that should be accounted for in characteristic $p$. Namely, the curve $X\_0(Np)\_{\mathbb{F}\_p}$ is reducible -- its two components are isomorphic to $X\_0(N)\_{\mathbb{F}\_p}$ and they intersect transversally at the supersingular points (see e.g. Ribet and Stein's online notes). So... | 13 | https://mathoverflow.net/users/949 | 187755 | 93,093 |
https://mathoverflow.net/questions/187385 | 6 | Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the axioms. For example, the octahedron axiom gives an exact triangle $E\to E^{(2)}\to E\xrightarrow[]{\Sigma k\circ j}\Sigma E... | https://mathoverflow.net/users/41291 | What does an endomorphism in a triangulated category give rise to? | Here's an expanded version of my comment, addressing the spectral sequence part of the question. A filtered object in a triangulated category $T$ is simply a sequence
$$ \dots \to X\_{n-1} \to X\_n \to X\_{n+1} \to \dots $$
When you apply a homological functor $H : T\to A$, where $A$ is an abelian category, filtere... | 5 | https://mathoverflow.net/users/20233 | 187768 | 93,099 |
https://mathoverflow.net/questions/187771 | 2 | Suppose you have a deck of $n$ cards; e.g., $n{=}12$:
$$
(1,2,3,4,5,6,7,8,9,10,11,12) \;.
$$
Cut the deck into $k$ equal-sized pieces, where $k|n$;
e.g., for $k{=}4$, the $12$ cards are partitioned into
$4$ piles, each of $m=n/k=3$ cards:
$$
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
10 & 11 &... | https://mathoverflow.net/users/6094 | A perfect $(n,k)$ shuffle function | This is just my comment above, which seems to answer the question. Label the cards from $0$ to $n-1$. Then, with $m=n/k$, the shuffle in the question corresponds to multiplying card $i$ by $m$ (taken mod $n-1$). Thus repeating the shuffle $r$ times amounts to multiplying by $m^r \pmod{n-1}$, which returns us to the ori... | 5 | https://mathoverflow.net/users/38624 | 187773 | 93,101 |
https://mathoverflow.net/questions/187736 | 9 | I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but $S^{n+1}$ is not $r$-generator. What is known about the numbers $n(r,S)$? Could someone give me references to this, pleas... | https://mathoverflow.net/users/32831 | Powers of finite simple groups | See [Collins's thesis](http://www.math.cornell.edu/m/sites/default/files/imported/Research/SeniorTheses/2010/collinsThesis.pdf), Theorem 2.22, page 21.
>
> Theorem 2.22. Let $S$ be a nonabelian simple group and
> $h\_{n-1}(S) < k \le h\_n(S)$. Then $r(S^k)=n$.
>
>
>
Here, $r(G)$ is the minimal number of gener... | 7 | https://mathoverflow.net/users/24165 | 187774 | 93,102 |
https://mathoverflow.net/questions/187759 | 7 | Let $L$ be a free Lie algebra (over $\mathbb{Q}$) on generators $x\_1, x\_2, \ldots, x\_n$, and let $V\_k$ be the subspace spanned by the $k$-fold brackets. Let $U\_1 = \mathrm{span}\{ x\_i | i< n\}\subseteq V\_1$ and let $W\_n \subseteq V\_n$ be the span of the $n$-fold brackets that involve $x\_n$ at least once.
Th... | https://mathoverflow.net/users/3634 | Injectivity of Rewrite Rule in a Free Lie Algebra | Yes. Embed the free Lie algebra in the free associative algebra, then consider the monomial with $x\_n$ as far right as possible (i.e., with the largest number of other $\{x\_i,i\leq n\}$ before it): it won't be cancelled. Let me now give details.
Consider $F$ the free (associative but non-commutative) algebra (over ... | 9 | https://mathoverflow.net/users/36972 | 187777 | 93,104 |
https://mathoverflow.net/questions/187728 | 15 | Let us consider the matrix algebra. $Mat\_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X\_1, X\_2, ..., X\_{2n} \in Mat\_n(\mathbb{C})$ the sum $\Sigma\_{\sigma \in S\_{2n}} sgn(\sigma)X\_{\sigma(1)}...X\_{\sigma(2n)}$ vanishes identically.
Is it true that any identity (noncommutative p... | https://mathoverflow.net/users/33286 | Is the Amitsur-Levitzki identity essentially unique? | The answer to your question is "no", as explained by Anton Klyachko in his answer. Let me refer you to a remarkable statement of Razmyslov and Procesi that describes all identities. They proved (independently) that in fact all identities of $Mat\_n(\mathbb{Q})$ follow, in a sense, from the Cayley--Hamilton theorem. To ... | 17 | https://mathoverflow.net/users/1306 | 187798 | 93,109 |
https://mathoverflow.net/questions/187803 | 1 | Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim\_{n\to\infty} a\_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication.
For $a\in V$ we define the "associated $\zeta$-function" to be \begin{eqnarray} \zeta(a) := \sum\_{n=1}^\infty \frac... | https://mathoverflow.net/users/8628 | Do the sequences with divergent associated $\zeta$-function form a vector space? | Define
$$
a\_n=\begin{cases}\frac{2\log\log n}{\log n}&:&\text{$n$ is odd,}\\0&:&\text{$n$ is even,}\end{cases}\hspace{1cm}b\_n=\begin{cases}0&:&\text{$n$ is odd,}\\\frac{2\log\log n}{\log n}&:&\text{$n$ is even.}\end{cases}
$$
It's not hard to check that $\zeta(a)=\zeta(b)=\infty$, but $\zeta(a+b)<\infty$, so $D$ is ... | 6 | https://mathoverflow.net/users/5263 | 187804 | 93,110 |
https://mathoverflow.net/questions/187806 | 1 | I meet the following problem which I think related to the monodromy:
Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, the fibre $U\_t$ is isomorphic to a fixed variety $X$, then is $U$ isomorphic to the product space $D \times X$?
I thin... | https://mathoverflow.net/users/29730 | Monodromy of a punctured disc | This is well-known to be false. A classical example is given by the Hirzebruch surfaces $\mathbb{F}\_n$. For instance, consider on $\mathbb{P}^1$
all extensions
$$0\rightarrow \mathscr{O}\_{\mathbb{P}^1}(-1)\rightarrow E\_e \rightarrow\mathscr{O}\_{\mathbb{P}^1}(1)\rightarrow 0\ .$$These extensions are parametrized b... | 4 | https://mathoverflow.net/users/40297 | 187809 | 93,112 |
https://mathoverflow.net/questions/187802 | 13 | I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, this particular problem I cannot seem to find a solution to:
Given a line in the plane passing through the origin makin... | https://mathoverflow.net/users/62113 | Conjecture regarding closest point inside a discrete ball to a line | This is true and I would be grateful if someone could make a figure to the argument below.
Your conjecture follows from the following statement.
Let $\ell$ pass through the origin, $O$.
For simplicity, suppose $\ell$ has a positive slope and let $P=(n,m)$ for some $m,n>0$ such that $P$ lies under $\ell$.
Denote by $Q... | 9 | https://mathoverflow.net/users/955 | 187830 | 93,116 |
https://mathoverflow.net/questions/187833 | 2 | Let $M$ be a smooth surface and let $x,y \in M$. Let $d\_{M}(\cdot, \cdot)$ be the geodesic distance metric on $M$, that is the length of the shortest geodesic curve on $M$. Let $\kappa$ be the maximum principle curvature of all points on $M$ along a minimizing geodesic connecting $x$ and $y$. Let $n\_{x}$ and $n\_{y}$... | https://mathoverflow.net/users/62013 | Normal Variation on Manifolds | The maximum principle curvature is the upper bound for the Lipschitz constant of the Gauss map at the point.
Integrating, you get your estimate.
The same can be done in higher dimensions.
If $M$ is $m$-dimensional submanifold in $\mathbb R^n$,
you have Gauss map $\nu\colon M\to \mathrm{Gr}(m,n)$,
where $\mathrm{Gr}(... | 6 | https://mathoverflow.net/users/1441 | 187838 | 93,120 |
https://mathoverflow.net/questions/187819 | 12 | A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as large as possible?
He had something in mind like: if
$$\frac{\mathrm{area}(\mathrm{island})}{\mathrm{area}(\mathrm{wate... | https://mathoverflow.net/users/25590 | How large can you draw an island on a map? | Similar issues come up in studying gerrymandering (drawing political districts with partisan objectives), where it's useful to have a measure of how "irregular" a region is.
You can read about various classical irregularity measures in this political science paper: [Measuring the Compactness of Legislative Districts]... | 21 | https://mathoverflow.net/users/1227 | 187843 | 93,122 |
https://mathoverflow.net/questions/187845 | 11 | Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two discs are tangent iff the corresponding vertices are connected to each other.
Where can I find the/a proof of this theore... | https://mathoverflow.net/users/49822 | Koebe–Andreev–Thurston theorem - where can I find a proof? | There are many proofs, and I'm not claiming that the following list is complete. New references are welcome.
(First proof)
* Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Akad. Leipzig 88 (1936), 141–164 (German)
(Thurston's rediscovery and related)
* Andreev, E. M., Convex polyhedra of... | 12 | https://mathoverflow.net/users/20595 | 187859 | 93,129 |
https://mathoverflow.net/questions/187874 | 5 | Let $k$ be a number field and $X$ be a $k$-scheme. Let $G$ be a linear algebraic group over $k$ and let $f: Z \to X$ be a $G\_X$-torsor ($G\_X = G \times\_k X)$. We can twist the torsor $f$ by 1-cocycles in $Z^1(k,G)$ (see Skorobogatov's "Torsors and rational points" for more details), and we denote the twist of $f$ by... | https://mathoverflow.net/users/62153 | Decomposing adelic points using torsors | The answer is **No.** Here is a counterexample.
Take $k = \mathbb Q$, $X = {\mathbb G}\_{\text{m}}$, $G = \mu\_2$ and $f \colon Z \to X$ the squaring map ${\mathbb G}\_{\text{m}} \to {\mathbb G}\_{\text{m}}$. Then $H^1(k, G) = {\mathbb Q}^\times/\text{squares}$, and its elements can be represented by squarefree integ... | 4 | https://mathoverflow.net/users/21146 | 187876 | 93,134 |
https://mathoverflow.net/questions/187866 | 20 | Suppose that $X$ is a topological space and $\left(U\_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C\_U:\Delta^{op} \to Top.$$ I know that for many good classes of topological spaces, the homotopy colimit of $C\_U$ is $X$ (e.g. for manifolds). How general is this result? Doe... | https://mathoverflow.net/users/4528 | When is a topological space the homotopy colimit of an open covering? | It is true in complete generality that $X$ is the homotopy colimit of $C\_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version of the Seifert-van Kampen theorem. More precisely, Proposition A.3.2 in [Higher Algebra](http://www.math.harvard.edu/~lu... | 20 | https://mathoverflow.net/users/20233 | 187891 | 93,139 |
https://mathoverflow.net/questions/187890 | 3 | In these Lecture Notes <http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf> from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned:
Is the map
$g\_3 \times f\_3 \times f\_5: H^1(-,F\_4) \rightarrow H^3(-,\mathbb{Z}/3\mathbb{Z}) \times H^3(-,\mathbb{Z}/2\mathbb{Z}) \ti... | https://mathoverflow.net/users/51251 | On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants | In the lecture notes [Albert algebras](http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/Alb.-alg.-Ottawa-2012-Vii-new.pdf) by H.P. Petersson, written in $2012$ this is still mentioned as an open problem, see Question $13.2$:
Is an Albert algebra $J$ determined up to isomorphism by its invariants $g\... | 5 | https://mathoverflow.net/users/32332 | 187894 | 93,141 |
https://mathoverflow.net/questions/187787 | 3 | I am looking for a description of the $A\_\infty$ operad in the category of simplicial sets. More specifically, I am looking for a formulation of the loop space recognition principle for simplicial sets. Is such a thing written down anywhere?
| https://mathoverflow.net/users/62105 | Simplicial version of the A-infinity operad | In my opinion the construction in "the geometry of iterated loop spaces" by P. May should carry through the simplicial world. If you want a completely simplicial treatment you can find it in theorem 5.2.6.10 of "Higher Algebra" by J.Lurie. Example 5.1.0.7 in the same book provides you with a small simplicial model for ... | 3 | https://mathoverflow.net/users/43054 | 187895 | 93,142 |
https://mathoverflow.net/questions/187837 | 8 | What is the number of $n$-vertex [median graphs](https://en.wikipedia.org/wiki/Median_graph)? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be interested in asymptotics or lower bounds. For more about median graphs see... | https://mathoverflow.net/users/7252 | Number of median graphs? | The numbers of $m$-edge triangle-free and median graphs are of similar types — at least, the logarithms of these numbers are within a constant factor of each other. In one direction every median graph is triangle-free, and in the other direction the simplex graph of an $m$-edge triangle-free graph is median and has $O(... | 6 | https://mathoverflow.net/users/440 | 187908 | 93,145 |
https://mathoverflow.net/questions/187882 | 6 | Let $C\_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall briefly the construction. Starting with the unit interval, we remote from the center of the interval an interval of length $1-2\la... | https://mathoverflow.net/users/56191 | Precise density estimates for Cantor sets | **Upper densities**
In the following, I freely use the well known fact that $s\_\lambda$-Hausdorff measure gives mass $2^{-k}$ to all intervals that make up the stage $k$ in the construction of $C\_\lambda$.
I don't think it is correct that $\Theta^{\* s\_\lambda}(C\_\lambda,x)\ge c$ for all $x\in C\_\lambda$ and s... | 3 | https://mathoverflow.net/users/11009 | 187913 | 93,147 |
https://mathoverflow.net/questions/187904 | 0 | Suppose I have a 1D advection equation in conservation (divergence) form
$\partial\_t u(x,t) = -\partial\_x [v(x)u(x,t)],$
where $u$ is a conserved quantity in space, and $v$ gives the velocity of the flow of mass in space. I want to know if it is possible to re-write this in a way that describes the rate of change... | https://mathoverflow.net/users/56169 | Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space? | for $h(y,x)=\nu(y)\partial\_y\delta(x-y)$ one has, upon partial integration:
$$\int\_{-\infty}^\infty u(y,t)h(y,x)\,dy=-\int\_{-\infty}^\infty \delta(x-y)\partial\_y[\nu(y)u(y,t)]\,dy=-\partial\_x [\nu(x)u(x,t)]$$
| 1 | https://mathoverflow.net/users/11260 | 187919 | 93,148 |
https://mathoverflow.net/questions/187921 | 1 | Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map
$$Mor\_{Sch}(S,X)\to Mor\_{Sch}(S,Y)$$
is injective.
**Question.** Is it true that $f$ is a closed imbedding?
The simplest case which I... | https://mathoverflow.net/users/16183 | When a proper morphism of schemes is a closed imbedding? | Yes, it is true : this is EGA IV, Cor. 18.12.6. (your condition means by definition that $f$ is a monomorphism).
| 4 | https://mathoverflow.net/users/40297 | 187924 | 93,150 |
https://mathoverflow.net/questions/187842 | 4 | For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)\_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision (rasterised) such that:
* every pixel intersects the Mandelbrot set (or even the boundary of the Mandelbrot set), and
* this ... | https://mathoverflow.net/users/37099 | Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set? | The **Hairiness Conjecture**, formulated by Milnor and proved by Lyubich (["Feigenbaum-Coullet-Tresser universality and Milnor’s Hairiness Conjecture"](http://arxiv.org/abs/math/9903201), Annals of Mathematics, 1999) states that, near any real Feigenbaum parameter, the rescalings of the Mandelbrot set converge to the w... | 6 | https://mathoverflow.net/users/3651 | 187940 | 93,153 |
https://mathoverflow.net/questions/187929 | 6 | Let $G$ be a graph, then we define its *Hadwiger graph* $\textrm{Hadw}(G)$ in the following way:
* $V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$;
* $E(\textrm{Hadw}(G)) = \{\{S,T\}\subseteq V(\textrm{Hadw}(G)): S\cap T = \emptyset \textrm{ and } (\exists s\in S, ... | https://mathoverflow.net/users/8628 | Isomorphic Hadwiger graphs | $\def\Hadw{\mathop{\rm Hadw}}$This is true for finite graphs, and false for (not necessarily connected) infinite graphs. Right now I do not know what happens for infinite connected graphs.
**1.** Each component $G\_1\subseteq G$ corresponds to an isolated vertex $v\_{G\_1}$ in $\Hadw(G)$ and a component $\Hadw(G\_1)\... | 5 | https://mathoverflow.net/users/17581 | 187944 | 93,154 |
https://mathoverflow.net/questions/187963 | 4 | I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an idea of the proof?
I thank you in advance for the help.
| https://mathoverflow.net/users/47294 | Nuclearity noncommutative torus | If $A$ is nuclear, $G$ is a locally compact amenable group, and $\alpha$ is an action of $G$ on $A$, then $A \rtimes\_\alpha G$ is nuclear (see, for instance, Blackadar, *Operator Algebras: Theory of $C^\ast$-Algebras and Von Neumann Algebras*, Corollary IV.3.5.2). In particular, if $\theta$ is irrational, then $C(\mat... | 7 | https://mathoverflow.net/users/6999 | 187966 | 93,161 |
https://mathoverflow.net/questions/187827 | 10 | I asked this on [math.StackExchange](https://math.stackexchange.com/questions/746724/powers-of-traces-integrals-over-spheres-and-class-functions) a while back but got no answers. I hope I'll be forgiven for the double post.
Let $V$ be a complex vector space of dimension $\operatorname{dim}\_{\mathbb C} V = n$, equipp... | https://mathoverflow.net/users/4054 | Powers of traces, integrals over spheres and class functions | Begin by rewriting $\alpha\_k(A\_1,\ldots,A\_k)$ as follows (just move integrals and traces around):
$$
\alpha\_k(A\_1,\ldots,A\_k) = \mathrm{tr}\Big( (A\_1 \otimes \cdots \otimes A\_k) \int\_{S^{2n-1}}(vv^\* \otimes \cdots \otimes vv^\*) d\mu \Big).
$$
Well, the integral on the right is well-known to be $P\_{sym}/\b... | 7 | https://mathoverflow.net/users/11236 | 187970 | 93,163 |
https://mathoverflow.net/questions/187903 | -1 | note: I find this question In stackexchange math, I would be interest to know how I could be answer this kind of question,I pasted it here as I see it appropriate For MO.
check this link: <https://math.stackexchange.com/q/1030616/156150>.
Let $(M,g)$ be a compact Riemannian manifold.
Is there an example of a ge... | https://mathoverflow.net/users/51189 | Are compact complete geodesics closed? | Let $c:\mathbb R\to M$ be a unit speed geodesic. Then $c':\mathbb R\to UM:=\lbrace X\in TM: \|X\|=1\rbrace\subset TM$ is a flow line for the flow of the geodesic spray $S$. Since $UM$ is compact, flow-lines of $S$ are either periodic or non-compact. Thus the same is true for geodesics on $M$.
| 3 | https://mathoverflow.net/users/26935 | 187972 | 93,165 |
https://mathoverflow.net/questions/187949 | 1 | Recently I come cross a question about deficient values of entire functions.
I find that many examples in the book about functions $f$ whose deficient values are singularities of the inverse $f^{-1}$.
I want to know whether there exist an example with the following property (in some sense it aks whether the concep... | https://mathoverflow.net/users/11966 | A question on deficient values of entire functions | The book of Goldberg and Ostrovskii MR2435270 contains several examples of functions whose
deficient value is not asymptotic. And in fact there are such functions
without asymptotic values at all. But deficient value must be in the closure
of the singular set.
This follows from a theorem of E. Collingwood, which has... | 5 | https://mathoverflow.net/users/25510 | 187979 | 93,167 |
https://mathoverflow.net/questions/187975 | 17 | Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states that for every $t \in [0,1]$, there exists a set $S \in \Sigma$ with $\mu(S) = t$.
I would like to use the following str... | https://mathoverflow.net/users/778 | Reference for a strong intermediate value theorem for measures | I would say this is folklore (I proved it and used it many years ago on my undergrad thesis), but here is a concrete reference:
Such a family of measurable sets is called a $[0,1]$-family in *On the Skorokhod representation theorem* by Jean Carlos Cortissoz, PAMS, Vol.135, No. 12, 2007 (see Definition 4.1). A proof t... | 6 | https://mathoverflow.net/users/17836 | 187985 | 93,170 |
https://mathoverflow.net/questions/187961 | 6 | Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi\_!:R-Mod\rightleftarrows S-Mod:\phi^\*,$$
where $\phi\_!=(S\otimes\_R -)$. The right adjoint $\phi^\*$ is easily described if we see $\phi$ as an additive functor between the one-object categories $R$ and $S$ and we view a left $... | https://mathoverflow.net/users/24891 | Exactness of an additive left Kan extension | This is true, even if the ring extension is not necessarily flat. It follows from the fact that $F$ is right exact.
First some generalities: given an additive category $\mathcal{A}$ I will write $\mathcal{PA}$ for the category of additive presheaves (that is, additive functors ${\mathcal{A}}^{\mathrm{op}} \rightarrow... | 6 | https://mathoverflow.net/users/1649 | 187998 | 93,176 |
https://mathoverflow.net/questions/187995 | 35 | Let $A=\{a\_1,\ldots,a\_k\}$ be a fixed, finite set of reals. Let $S\_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ numbers from $A$ (here each element of $A$ can be reused an unlimited number of times). Finally, let $d\_A(n)$ be the mi... | https://mathoverflow.net/users/2575 | Massive cancellations | Let $a\_n$ be an increasing sequence of positive integers which grows really fast, say $a\_{n+1} > \exp(a\_n)$. Take $A = \{10^{-1}, \sum 10^{-a\_n}\}$. Then $d\_A(a\_n) \leq 2\cdot 10^{- a\_{n+1}} \leq 2\cdot 10^{-\exp a\_n}$, so $A$ cannot be tame.
**EDIT.** One could replace $1/10$ by some transcendental $0<x<1$ ... | 36 | https://mathoverflow.net/users/3847 | 188000 | 93,177 |
https://mathoverflow.net/questions/188024 | 0 | Suppose $\mathbb{N}=\bigsqcup\_{i\in\mathbb{N}}E\_i$ with $\#E\_i=\infty$ for each $i$.
1. Is it possible that $\limsup\_{N\to\infty}\frac{1}{N}\#(E\_i\cap\{1,\ldots,N\})=0$ for all $i$, which would mean $\lim\_{N\to\infty}\frac{1}{N}\#(E\_i\cap\{1,\ldots,N\})=0$ for all $i$?
2. Is it possible that $\liminf\_{N\to\i... | https://mathoverflow.net/users/58125 | Are the natural numbers a disjoint union of infinite sets of zero asymptotic density? | Yes, it is possible. We can construct such sets for example as follows: $E\_0$ is going to be the set of all numberss of the form $n^2$. $E\_1$ is going to be the set of numbers of the form $n^2+1$, unless it already appeared in $E\_0$. $E\_2$ is set of numbers of the form $n^2+2$, unless it already appeared in any of ... | 1 | https://mathoverflow.net/users/30186 | 188029 | 93,183 |
https://mathoverflow.net/questions/188043 | 3 | [Asymmetric graphs](http://en.wikipedia.org/wiki/Asymmetric_graph) are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity.
Let's call a graph $G$ strongly asymmetric if the only graph *homomorphism* $h: G\to G$ is the identity (in other ... | https://mathoverflow.net/users/8628 | Strongly asymmetric graphs | I believe the common name for such graphs is *rigid*. In fact, most random graphs are rigid. See this reference: [On the minimal order of a graphs within a semigroup](http://www.sciencedirect.com/science/article/pii/0095895684900212).
| 1 | https://mathoverflow.net/users/934 | 188045 | 93,188 |
https://mathoverflow.net/questions/188032 | -3 | I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem.
Could any one give reference or a simple introduction about such result known in their domain?
| https://mathoverflow.net/users/14024 | Decidable theorem or result that is not weaker than Tarski's theorem | [Ax and Kochen](http://www.jstor.org/stable/1970476) proved decidability for the ring of $p$-adic numbers, and many rings like it. That certainly doesn't follow from Tarski, and I would say it is more difficult.
| 1 | https://mathoverflow.net/users/297 | 188053 | 93,192 |
https://mathoverflow.net/questions/188022 | 0 | Suppose to have a linear irreducible unitary representation $\rho:G\rightarrow U(H)$ on a complex Hilbert space $H$ with $G$ a generic group. Let $A$ be an $\textit{anti}$-linear operator such that
$$
A\rho(g)=\rho(g)A\ \ \ \ \forall g\in G
$$
What can be said about the operator $A$? Does it hold anything like Schur's... | https://mathoverflow.net/users/43915 | Schur's lemma for antiunitary operators on complex Hilbert spaces | The part of Schur's lemma that continues to hold is that any such operator must be invertible or 0, if the representation is irreducible over the reals. I will make no assumption on complex (anti-)linearity from now on, but will assume that all operators are real linear.
The space of real operators commuting with $G$... | 2 | https://mathoverflow.net/users/54311 | 188056 | 93,194 |
https://mathoverflow.net/questions/188077 | 7 | There is a really nice proof of the Cayley-Hamilton Theorem using the generic matrix. I expose it briefly.
One defines the generic matrix $G:=(X\_{ij})\_{ij} \in\mathcal{M}\_n(\mathbb{Z}[X\_{ij}]\_{ij})$.
The discriminant $\Delta\_G$ of $\chi\_G$ (characteristic polynomial of $G$) is an element of $\mathbb{Z}[X\_... | https://mathoverflow.net/users/27767 | Example of proof using the generic matrix | A famous example is the existence and uniqueness of a polynomial ${\bf Pf}$ in the entries of a $2n\times2n$ alternate matrix, called the *Pfaffian*, such that
$${\bf Pf}(A)^2=\det A,\quad\forall A\in{\rm Alt}\_{2n},\qquad{\bf Pf}(J\_{2n})=1,\quad J\_{2n}:=\begin{pmatrix} 0\_n & -I\_n \\\\ I\_n & 0\_n\end{pmatrix}.$$
... | 9 | https://mathoverflow.net/users/8799 | 188079 | 93,201 |
https://mathoverflow.net/questions/187002 | 3 | Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. What is the connectedness of $\mathcal{D}$? Is it simply connected?
| https://mathoverflow.net/users/61712 | Characterizing space that preserves positive-definiteness property | Let $\Gamma$ be a convex subset of the set of symmetric real matrices. Then $\mathcal{D}=\{D\in\Gamma;\Sigma-D\Sigma^{-1}D>0\}$ is a convex set.
Proof: we use the following known result.
(\*) Let $S>0$ and $T$ be real symmetric matrices s.t. $T$ has $k$ positive, $l$ negative and $n-k-l$ zero eigenvalues. Then $ST... | 2 | https://mathoverflow.net/users/9091 | 188095 | 93,210 |
https://mathoverflow.net/questions/188059 | 0 | This might be a basic question, nonetheless I cannot give a proof.
Given an orthogonal matrix $A$ with eigendecomposition $A = Q \Lambda Q^{-1}$ with only non-real eigenvalues. Given also a diagonal real matrix $\Phi$ with $\Phi\_{ii} = \Phi\_{jj}$ if $\Lambda\_{ii} = \overline{\Lambda\_{jj}}$. The following matrix p... | https://mathoverflow.net/users/51478 | Powers of orthogonal matrices is closed | The complex eigenvalues of a real matrix come in conjugate-complex pairs, with conjugate-complex pairs of eigenvectors. But the converse is true: If you associate to conjugate-complex pairs of eigenvectors any conjugate-complex pairs of eigenvalues, the result will be real because it is the sum of two terms which are c... | 2 | https://mathoverflow.net/users/30800 | 188096 | 93,211 |
https://mathoverflow.net/questions/187928 | 3 | Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta\_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito process $X\_t$.
Define the process $Y\_t := \Theta\_t^\* X\_t$. I read that the generator of $Y\_t$ is the time-dependent ... | https://mathoverflow.net/users/16702 | Density for Translated Process | The distribution of the process $Y$ is locally equivalent to the distribution of $X$ if and only if the flow $\Theta$ preserves the principal symbol of $L$.
If $\Theta$ does not preserve the principal symbol of $L$, then the two processes $X$ and $Y$ do not have the same quadratic variation and thus their distributi... | 4 | https://mathoverflow.net/users/48356 | 188100 | 93,212 |
https://mathoverflow.net/questions/187945 | 15 | *(For a formulation of the Mumford–Tate conjecture, see below.)*
The question
============
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate conjecture for Abelian varieties.
That is, we know it for:
* Projective spaces (trivial)
* *[edit]* Othe... | https://mathoverflow.net/users/21815 | Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties? | It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular forms for $SL\_2(\mathbb{Z})$.
Blasius shows in the article "Modular forms and abelian varieties" (Séminaire de Théori... | 11 | https://mathoverflow.net/users/519 | 188103 | 93,214 |
https://mathoverflow.net/questions/188081 | 42 | How I arrived at this question is a rather long story having to do with the honors calculus class I am teaching. At this point it's sheer curiosity on my part. Here is the game.
$\newcommand{\bZ}{\mathbb{Z}}$
We start with a finite collection of stones placed at random somewhere on the set of nodes $\newcommand{\eN... | https://mathoverflow.net/users/20302 | A game of stones | $\newcommand{\bZ}{\mathbb{Z}}$
$\newcommand{\eN}{\mathscr{N}}$
Here is a proof which does not treat overcrowding as a special case. As noted in some comments, it is really enough to prove the desired result for the case of one stone. The actually proofs takes up less space than the statements and all the notation. I ... | 17 | https://mathoverflow.net/users/8008 | 188104 | 93,215 |
https://mathoverflow.net/questions/188093 | 2 | it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality
$$\vert f(t)-f(s)\vert\leq \int\_s^t{m(r) dr},$$
for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely continuous.
On the other hand, if the function satisfies the inequality
$$\vert f(t)-f(s)\vert\leq (g(s)+g(t))\vert t-s\vert,$... | https://mathoverflow.net/users/62252 | Absolutely continuous functions | I think the answer to this question is ``yes" if in the last inequality you assume that $g\in L^1$ and $m\in L^1$. There are two simple ways to see this:
1. In one dimensional case, $f\in W^{1,1}$ if and only if $f$ is absolutely continuous. Both the first inequality and the second inequality implies $f$ is absolutel... | 3 | https://mathoverflow.net/users/26608 | 188106 | 93,216 |
https://mathoverflow.net/questions/188105 | 1 | Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a\_{ij} \in k[x\_1, \ldots, x\_p]\_1$ as entries. Let $I$ be the ideal generated by the maximal minors of $A$.
Is $k[x\_1, \ldots, x\_p]/I$ a Cohen-Macaulay ring?
If no, does the additional assumption that $I$ is a radica... | https://mathoverflow.net/users/36563 | ideal of maximal minors is cohen-macaulay? | No, the ring need not be Cohen-Macaulay. For instance, when $p$ equals $2$, $m$ equals $2$ and $n$ equals $3$, consider the matrix, $$ A = \left[ \begin{array}{rrr} x & 0 & 0 \\ 0 & x & y \end{array} \right].$$ Your ideal is $I = \langle x^2,xy \rangle$, so that $k[x,y]/I$ is not Cohen-Macaulay.
The positive results... | 3 | https://mathoverflow.net/users/13265 | 188127 | 93,221 |
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