parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/110429 | 4 | Let $H$ denote a Hilbert space and let $\cal A$ be a subalgebra of the algebra ${\cal B}(H)$ of all bounded operators on $H$ such that $\cal A$ consists of compact operators only and such that each vector $v\in H$ lies in the closure of ${\cal A}v$.
Is it true that there must exist an irreducible subspace for $\cal ... | https://mathoverflow.net/users/nan | Existence of irreducible subspace | Well, if ${\cal A}$ can be non self-adjoint then I think there are easy counterexamples. Let $(e\_n)$ be the standard basis of $l^2({\bf N})$ and let ${\cal A}$ be the closed span of the operators $e\_n \otimes e\_m$ for $n \leq m$ (where $(e\_n \otimes e\_m) v = \langle v, e\_n\rangle e\_m$). The condition that any $v... | 7 | https://mathoverflow.net/users/23141 | 110479 | 63,460 |
https://mathoverflow.net/questions/110491 | 14 | My question is rather straight forward. I am currently applying for postdoctoral positions in (pure) mathematics. I am almost done writing my research statement and it seems to come out to about seven pages. Is that too long?
A similar question has been asked [here](https://mathoverflow.net/questions/27001/research-s... | https://mathoverflow.net/users/nan | Seven pages -- too long for a research statement for postdoc application? | Unsurprisingly, your research statement addresses at least two very-different audiences, and in your specific situation, maybe three or more. The usual entirely-true cliche is that most people on the postdoc-hiring committee will not themselves read beyond the first page or so of the whole thing, so you should be sure ... | 34 | https://mathoverflow.net/users/15629 | 110493 | 63,468 |
https://mathoverflow.net/questions/110496 | 2 | Consider the following optimization problem:
Maximize $\|X\|\_2$, subject to $X$ being Hermitian (or symmetric) and a bunch of semidefinite constraints on $X$. Here, $\|X\|\_2$ is the spectral norm of $X$, i.e., the largest eigenvalue of $X$ by magnitude (since $X$ is Hermitian).
Can this be written as a semidefin... | https://mathoverflow.net/users/8075 | Can one maximize the spectral norm of a matrix via semidefinite programming? | No: maximizing the norm makes it a non-convex problem.
| 5 | https://mathoverflow.net/users/13650 | 110497 | 63,470 |
https://mathoverflow.net/questions/110505 | 5 | It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely many points.
My question is, given the above representation in terms of hyperplanes, one can determine (easily) whet... | https://mathoverflow.net/users/11674 | Is there a simple test to determine whether a polytope is integral? | An incomplete solution:
There is a polynomial-time test for total unimodularity, and if $A$ is totally unimodular, then the polytope is integral.
| 1 | https://mathoverflow.net/users/19029 | 110508 | 63,477 |
https://mathoverflow.net/questions/110487 | 2 | This is just a question about terminology. I had thought that "enumerable" is a synonym for "countable," and you could call a set "enumerated" to mean it comes with some specific ordering of type $\omega$ (or an initial segment, if finite). Is that standard or is there another concise terminology for the distinction?
... | https://mathoverflow.net/users/38783 | Terminology for sequences and countability | Might as well add it:
A *countable* set is one for which there exists a bijection with $\omega$, and a *counted* set is one equipped with a bijection with $\omega$. Thus the statement true in ZF is
>
> A countable union of counted sets is countable.
>
>
>
| 2 | https://mathoverflow.net/users/4177 | 110513 | 63,480 |
https://mathoverflow.net/questions/103696 | 1 | As we know, solution to $dX\_t=\mu dt+\sigma dW\_t$ is normal distributed and is light tailed; solution to $dX\_t=\mu X\_tdt+\sigma X\_t dW\_t$ is log-normal distributed and is heavy tailed. Is there any reference on discussion of criteria to identify the tail behavior from the driven SDE?
| https://mathoverflow.net/users/22694 | Tail of solutions of a stochastic differential equation | You want to consider:
$$dx\_t=f(x\_t)dt + \sigma(x\_t)dW\_t$$
In dimension 1, you can express analytically the stationary probability density (if it exists) as:
$$P(x)=\frac{1}{Z}\exp(-\int^x \frac{f(u)}{\sigma(u)^2} du)$$
where $Z$ is a normalizing constant.
Then you can express the various stationary moments ... | 2 | https://mathoverflow.net/users/16215 | 110518 | 63,482 |
https://mathoverflow.net/questions/91627 | 7 | Let $p \in \mathbb R [x,y,z]$ be a homogeneous irreducible polynomial of degree $d$. From Dickson in 1920 we know that there exists $A$, $B$ and $C$ such that
$$\det (Ax + By + Cz) = c p(x,y,z)$$
where $c$ is some constant.
Vinnikov in 1988 was able to describe all the non-equivalent determinantal representatio... | https://mathoverflow.net/users/2011 | Computing a determinantal representation of a bivariate polynomial | This is discussed in a recent work of Plaumann, Sturmfels and Vinzant:
<http://arxiv.org/abs/1011.6057>
| 5 | https://mathoverflow.net/users/15506 | 110524 | 63,485 |
https://mathoverflow.net/questions/110492 | 6 | I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to define a notion of "continuous truth" in a topological structure.
Suppose I have a topological structure $(M, \tau)$ - th... | https://mathoverflow.net/users/8133 | A continuous notion of realizability | Kleene defined a continuous realizability over Baire space, i.e. $\mathbb N^{\mathbb N}$ with the product topology. In this model $\forall x\exists y\phi(x,y)$ is valid, if there is a continuous function $f$ such that $\forall x\phi(x,f(x))$ is valid. That sounds like what you are looking for. A realizability model ass... | 13 | https://mathoverflow.net/users/3603 | 110525 | 63,486 |
https://mathoverflow.net/questions/110540 | 8 | I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I was told this was a common feature of localisations (i.e. they are often reflective), but then I was told that not every... | https://mathoverflow.net/users/25442 | Fpqc sheafification and localisation | $\newcommand\Set{\mathit{Set}}\newcommand\Aff{\mathit{Aff}}$An fpqc sheaf is exactly what you think it is: a functor from the opposite of the category of schemes (or relative schemes) to $\Set$ (i.e. a presheaf) such that the usual glueing conditions hold for fpqc covers.
You may be thinking of the fact remarked on h... | 9 | https://mathoverflow.net/users/4177 | 110542 | 63,494 |
https://mathoverflow.net/questions/110535 | 13 | What are largest betti numbers $b\_2$ and $b\_3$ of three-dimensional Calabi-Yau manifolds that are discovered for today?
Is there some nice reference?
| https://mathoverflow.net/users/13441 | Today's world record on the Betti numbers of Calabi-Yau three-folds. | Since mirror symmetry exchanges the Hodge numbers $h^{1,1} = b\_2$ and $h^{2,1} = \frac{1}{2}(b\_3 - 1)$, it is perhaps more natural (and of course equivalent) to discuss these. The record-holders all come from the [list of hypersurfaces](http://arxiv.org/abs/hep-th/0002240) in toric fourfolds, constructed by Kreuzer a... | 21 | https://mathoverflow.net/users/22975 | 110547 | 63,497 |
https://mathoverflow.net/questions/110552 | 5 | In a comment on question 110345 I made a claim that might be incorrect. I claimed that if
f(z) is a non-constant analytic function defined by a power series whose circle of convergence C
has a positive radius, then f(z) cannot be bounded at all points in the interior of C. But is
this really true? Or am I just imagini... | https://mathoverflow.net/users/4423 | Is this a "folk theorem" about analytic functions of a complex variable? | This is not true. For a counterexample take
$$\sum\_{n=1}^{\infty} \frac{x^n}{n^2} $$
The radius of convergence is one and this is bounded by
$$\sum\_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6 $$
You are likely confusing this with the [maximum modulus principle](http://en.wikipedia.org/wiki/Maximum_modulus_princi... | 18 | https://mathoverflow.net/users/nan | 110555 | 63,499 |
https://mathoverflow.net/questions/109660 | 13 | Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi\_{n,1}(an)$ the number of primes
$p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let
$$Q\_a(n) = \frac{\pi\_{n,1}(an)}{\pi(an)} \phi(n),$$
where $\phi(n)$ is Euler's phi function.
If instead of fixing $a$ we f... | https://mathoverflow.net/users/9317 | Small primes in arithmetic sequences | When $a=2$, the sum you want the asymptotics of is basically
$$
\frac12 \frac{\log n}n \sum\_{p\le n} \frac{\phi(p-1)}{p-1} \sim \frac12 \frac1{\pi(n)} \sum\_{p\le n} \frac{\phi(p-1)}{p-1},
$$
which is half the average value of the multiplicative function $\phi(n)/n$ on shifted primes $p-1$. The heuristic for evaluatin... | 12 | https://mathoverflow.net/users/5091 | 110566 | 63,503 |
https://mathoverflow.net/questions/110529 | 7 | I would like to find an appropriate reference for the following statement:
**Statement.** Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi\_1(M)$ is virtually abelian.
It seems to me that the statement should follow from the article of Cheeger and Gromoll
"The spletting theorem... | https://mathoverflow.net/users/13441 | Fundamental groups of compact manifolds with non-negative Ricci curvature. | The following paper has more than you want.
[Wilking, Burkhard On fundamental groups of manifolds of nonnegative curvature. Differential Geom. Appl. 13 (2000), no. 2, 129–165.](http://wwwmath.uni-muenster.de/sfb/about/publ/wilking3.ps)
| 7 | https://mathoverflow.net/users/10330 | 110567 | 63,504 |
https://mathoverflow.net/questions/110456 | 7 | The following are questions of Don Hadwin:
If $A$ is a unital continuous trace C\*-algebra, is there an upper bound on the dimension of all the irreducible representations?
It is known that all irreducible representations are finite-dimensional?
| https://mathoverflow.net/users/6269 | Is there an upper bound on the dimension for irreducible representations of a continuous trace $C^{*} $-algebra? | There is an upper bound. Here is why (this is a simpler way of arguing than the one I used below):
In 4.5.2 of "C\*-algebras", Dixmier defines a continuous trace C\*-algebra as one such that
the set `\[\{ a\in A^+ \mid \pi\mapsto\mathrm{Tr}(\pi(a))\mbox{ is continuous}\}\]` spans a dense two-sided ideal of $A$. Here ... | 5 | https://mathoverflow.net/users/13381 | 110582 | 63,508 |
https://mathoverflow.net/questions/110530 | 6 | Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by left-multiplication).
My vague question is: "How to measure the lack of cocompactness of
$\Gamma$"?.
**Edit:** My question was in... | https://mathoverflow.net/users/10265 | Measuring how far from being cocompact a lattice is | It seems to be that what you are asking for is roughly the measure of a neighborhood of the "cusp" of $G/\Gamma$.
For the case of $\Omega(n) = SL(n,\mathbb{R})/SL(n,\mathbb{Z})$ there is a classical calculation of a closely related quantity. Recall that $\Omega(n)$ is the moduli space of unimodular lattices in $\mat... | 7 | https://mathoverflow.net/users/16143 | 110588 | 63,511 |
https://mathoverflow.net/questions/110585 | 1 | Hi everyone, please consider the following problem:
Let $(M\_t)\_{t\geq 0}$ be a continuous and positive submartingale and $S\_t=\sup\_{0\leq s\leq t}M\_s$. Please prove that for any $\lambda>0$ we have
$$\lambda P(S\_t>2\lambda)\leq E[M\_t1\_{\{M\_t>\lambda\}}]$$
This inequality makes me remember the Doob inequa... | https://mathoverflow.net/users/25005 | modification of Doob inequality | Nice little exercise, it should go to math.stackexchange.com, though. I'll give you a hint: Let $T$ be the hitting time of $2\lambda$. Then, $E[M\_t 1\_{M\_t\ge \lambda}] \ge E[M\_t 1\_{M\_t\ge \lambda,\ T\le t}] \ge E[M\_t1\_{T\le t}]- E[M\_t1\_{M\_t\le \lambda}1\_{T\le t}].$
| 1 | https://mathoverflow.net/users/18032 | 110593 | 63,514 |
https://mathoverflow.net/questions/110559 | 2 | Consider a strongly convex potential $U: \mathbb{R}^d \to \mathbb{R}$ and the Langevin diffusion $$dX = -\nabla U(X) dt + dW \qquad (\*)$$ where $W$ is a standard Brownian motion. If $(X\_t)\_{t \geq 0}$ and $(Y\_t)\_{t \geq 0}$ are two solutions driven by the same Brownian motion, one can check that $t \mapsto \mathbb... | https://mathoverflow.net/users/1590 | contraction property for conditioned SDEs | I don't think so. In the limit, it would mean that if $U$ is merely convex, then we still must have this inequality with $\beta=\frac 12$. Now take any function $U$ on $\mathbb R$ that is $0$ on $[-1,1]$ and is strictly convex beyond the ends. Choose two distinct starting points in $(-0.5,0.5)$. Then, if what you said ... | 2 | https://mathoverflow.net/users/1131 | 110600 | 63,517 |
https://mathoverflow.net/questions/110623 | 5 | Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U\_{n}$. Let $\Omega$ be a constant ($\nabla \Omega = 0$) holomorphic non-vanishing (n,0)-form. Can one say that the holonomy group is now cotained in $SU\_{n}$ ? Is i... | https://mathoverflow.net/users/27486 | Holonomy of a Kähler manifold | The answer is yes.
The holonomy principle states that a given a riemannian manifold $(M,g)$ and a point $x\in M$, the datum of a parallel tensor field of a given type is equivalent to the datum of a tensor of the same type at the point $x$ which is invariant under the action of the holonomy group.
Now, $SU(n)$ is t... | 5 | https://mathoverflow.net/users/9871 | 110624 | 63,525 |
https://mathoverflow.net/questions/110621 | 2 | I understand that this question would be trivial for experts, sorry for that, I just need to clarify things.
So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W\_p$, $W\_q$ are the Sobolev spaces, or in the other words completions of $S(\mathbb{R}^n)$ with respect to $p$ and $q$ - Sobolev norm... | https://mathoverflow.net/users/26250 | Easy question on Sobolev spaces | I would like to expand a bit what Delio said. Your question is a bit confusing as it is, but we may assume that you mean that $p$ and $q$ are the parameters representing the derivatives. Then what you need is
$$ \|f\|\_p \leq C\|f\|\_q.$$
If you write out the definition of the Sobolev norm on $S$, then you see immedi... | 2 | https://mathoverflow.net/users/12898 | 110631 | 63,529 |
https://mathoverflow.net/questions/110511 | 33 | Let $V$ be a smooth connected algebraic variety over an algebraically closed field $k$. Let $W\_1, W\_2$ be closed subvarieties of $V$ of positive codimension whose intersection $W\_1 \cap W\_2$ has codimension at least 2, and let $p$ be a point in $V \backslash (W\_1 \cup W\_2)$. Then we can form the four etale fundam... | https://mathoverflow.net/users/766 | An etale version of the van Kampen theorem | To simplify notation, let me write $U\_i$ for $V\smallsetminus W\_i$, and $U\_{12}$ for $U\_1\cap U\_2=V\smallsetminus(W\_1\cup W\_2)$.
**Fact**: The obvious functor
$$(\mathrm{Sch}/V)\longrightarrow (\mathrm{Sch}/U\_1) \times\_{(\mathrm{Sch}/U\_{12})} (\mathrm{Sch}/U\_2)$$
is an equivalence. In other words, a $V$-s... | 30 | https://mathoverflow.net/users/7666 | 110638 | 63,533 |
https://mathoverflow.net/questions/110646 | 5 | Is there anything known about when two links have the same Jones polynomial (beyond a calculated list of small actual examples)? The first thing *I* would try is to compute the (formal - you would have something\*split-horizontal+something\*split-vertical) Jones polynomial of (4-)tangles and look *there* for two tangle... | https://mathoverflow.net/users/11504 | Links with same Jones polynomial | Probably the best way to produce infinite families of links with same Jones polynomial is by Conway mutation, this operation does not alter the HOMFLY polynomial either. A good example
of this is given by the family of pretzel links. Take a look at the answer of a question of mine here:
[How to distinguish Pretzel li... | 7 | https://mathoverflow.net/users/5001 | 110648 | 63,536 |
https://mathoverflow.net/questions/110630 | 0 | Consider probability distribution which is unimodal, symmetric and mean value exists. (Of course due to symmetry mean will coincide with mode (position of the maximum)).
**Question** Is sample mean always the "best" estimate of the mean or may be median, whatever can be better ?
Situation: assume the "tails" are h... | https://mathoverflow.net/users/10446 | Estimation of mean of the unimodal symmetric distrubution - is "sample mean" the best estimate ? | First consider a non-bayesian setting in which $X\_1,X\_2,...,X\_n$ are i.i.d. with a [Laplace distribution](http://en.wikipedia.org/wiki/Laplace_distribution) with unknown mean $\mu$. Clearly, the maximum-likelihood estimator here is the sample median and not the sample mean. While in general the ML estimator is non-o... | 1 | https://mathoverflow.net/users/27261 | 110652 | 63,538 |
https://mathoverflow.net/questions/110654 | 12 | I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical background does it need? books or papers? (I only learned Nakahara's book on geometry)
Thanks in advance.
| https://mathoverflow.net/users/25715 | About MF Atiyah and R Bott's 1983 paper | This is one very beautiful, influential and very challenging paper. You need to know differential geometry, some topology (cohomology, characteristic classes a bit of Morse theory), some algebraic geometry, and a bit of analysis (elliptic operators and complexes).
There is no one book that contains all of these, tho... | 9 | https://mathoverflow.net/users/20302 | 110667 | 63,545 |
https://mathoverflow.net/questions/110655 | 6 | *(I asked this question at math stackexchange 4 months ago, but received no answers)*
Let $\{e\_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q\_k$ by
$$
q\_k=\sum\_{n=1}^ke\_{nn}.
$$
Let $\{p\_1,p\_2,\ldots\}\subset B(H)$ be a sequence of orthogonal projections in $B(H)$ w... | https://mathoverflow.net/users/3698 | Strong convergence of projections in $B(H)$ | $Q\_k$ is just the orthogonal projection to the linear span $L\_k$ of $e\_1,\dots,e\_k$, right?
Let $u\_m$ be some sequence of unit vectors. Let $P\_mx=(x,u\_m)u\_m$. Now, $Q\_kP\_mQ\_k x=(Q\_kx,u\_m)Q\_ku\_m=(x,Q\_ku\_m)Q\_ku\_m$, so the condition is that $Q\_ku\_m$ stabilize for each $k$. Let's stabilize them to $\f... | 7 | https://mathoverflow.net/users/1131 | 110672 | 63,548 |
https://mathoverflow.net/questions/110658 | 10 | I wanted to ask the following question,
Suppose $\mathbf{M}$ a cofibrantly generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. Is it true that the map
$\mathrm{hocolim}\_{J}(F(i))\rightarrow (\mathrm{hocolim}\_J~F)(i)$ is a weak equivalence in $\... | https://mathoverflow.net/users/21369 | Fubini theorem for hocolim | This property holds actually for right derivable categories in the sense of:
MR2729017 Reviewed Cisinski, Denis-Charles Catégories dérivables. (French) [Derivable categories] Bull. Soc. Math. France 138 (2010), no. 3, 317–393.
At least under suitable finiteness assumptions on $I$ and $J$. It also holds for arbitrar... | 10 | https://mathoverflow.net/users/12166 | 110683 | 63,555 |
https://mathoverflow.net/questions/110669 | 2 | I would like to know if the following is true :
Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure.
Let $q$ be the orthogonal projection on the subspace of $\mathcal{H}$ spammed by the $(\exp(\pm 2 i \pi 2^k x ))\_{k \in \mathbb{N}}$.
Let $f\_n$ be a sequence of elements of $\mathc... | https://mathoverflow.net/users/22131 | Non-perfect type one C^*-algebra, and a lemma in Fourier analysis | It is true. More precisely, if $P\_n$ is the orthogonal projection on the space of functions supported in $[0,1/2^n]$, I claim that $\| P\_n q P\_n \| \leq (2n+2) 2^{-n}$. This implies what you are asking for.
Here is a proof. For $k \in \mathbb Z$, let $e\_k$ be the function $e^{sgn(k) 2i\pi 2^{|k|}}$ and $q(k)$ the... | 4 | https://mathoverflow.net/users/10265 | 110684 | 63,556 |
https://mathoverflow.net/questions/110602 | 6 | Let $\mathcal{X}\to\Delta$, $\Delta \subset \mathbb{C}$ is the unit disk, be a smooth family of varieties whose fibers over $t\neq 0$ are smooth and the central fiber $\mathcal{X}\_0$ is a nice simple normal crossing divisor (in $\mathcal{X}$).
Let $\mathcal{X}\_0=\cup X\_i$, and define $X\_I=\cap\_{i\in I} X\_i$.
... | https://mathoverflow.net/users/5259 | Degeneration of varieties to simple normal crossings | The more modern approach to the question adressed by Friedman is via
logarithmic geometry. Most relevant for your question is the paper of
Kawamata and Nammikawa, "Logarithmic deformations of normal crossing
varieties and smoothing of degenerate Calabi-Yau varieties," Invent. Math.,
118, (1994) 395-409. However, even t... | 11 | https://mathoverflow.net/users/23917 | 110689 | 63,560 |
https://mathoverflow.net/questions/110608 | 2 | It is well known that Kirwan's injection theorem gives an ring injection from $H^{\ast}\_T(M)$ to $H^{\ast}\_T(M^T)$ which is induced by the inclusion $M^T \to M$, where $T$ is a torus acting on manifold $M$ and $M^T$ is the fixed point set of this torus action.
I came across a problem when my professor tried to use... | https://mathoverflow.net/users/27040 | How to calculate the equivariant cohomology ring of $P^2$? | It sounds like you're talking about GKM (=Goresky–Kottwitz–MacPherson) theory, in which case it's better to think of tori as complex tori, i.e. as products of copies of $\mathbb C^\times$ and not of $S^1$. The triangle to which you're referring is the so-called moment graph of $\mathbb{CP}^2 = SL\_3(\mathbb C)/P$, wher... | 2 | https://mathoverflow.net/users/430 | 110694 | 63,564 |
https://mathoverflow.net/questions/109007 | 9 | In an attempt to write a proof by contradiction, I end up with a space $X$ with the following properties:
(0) $X$ is nonempty,
(1) $X$ is Hausdorff,
(2) $X$ has no isolated points,
(3) every subspace of $X$ is constructible (finite union of locally closed subsets).
>
> Is this indeed a contradiction?
> ... | https://mathoverflow.net/users/7666 | Constructible sets in Hausdorff spaces | No, it is not a contradiction.
A space $X$ is called *submaximal* if every subset of $X$ is locally closed (and hence constructible). It is easy to see that spaces with only finitely many non-isolated points are submaximal. Finding a submaximal Hausdorff space with no isolated points seems harder but there are plenty... | 6 | https://mathoverflow.net/users/17836 | 110698 | 63,566 |
https://mathoverflow.net/questions/110680 | 9 | Let $H < G$ be finite groups with $|G:H|=n$, and let $M$ be an irreducible $FH$-module for some field $F$. Is it always true that all irreducible constituents of the induced $FG$-module $M^G$ have dimension at least $\dim M$? If not, then is it true that the composition length of $M^G$ is at most $n$?
For complex rep... | https://mathoverflow.net/users/35840 | Constituents of induced representation | I am assuming you mean "compositon factor" when you speak of irreducible constituent. In the algebraically closed case you seem to be asking the following question in the first part, phrased in terms of Brauer characters: Let $\phi$ be a Brauer character of $H,$ and $\psi$ be a Brauer character of $G.$ Let $\alpha$ be ... | 4 | https://mathoverflow.net/users/14450 | 110700 | 63,568 |
https://mathoverflow.net/questions/110695 | 7 | Background
----------
Let $\mathcal{A} = \lbrace A\_0, \ldots, A\_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X\_\mathcal{A}$ may be constructed so that its $m$-th homology group $H\_m(X\_\mathcal{A})$ equals $A\_m$ for all $0 \leq m \leq M$. In... | https://mathoverflow.net/users/18263 | Which Abelian Group sequences arise as the Homology of Embedded CW Complexes? | This is simpler than that other question, because it's only the homology and not the homotopy type that is being prescribed.
A Moore space with $H\_1$ finite cyclic can be embedded in $4$-space, so by suspending this a Moore space with $H\_m$ finite cyclic can be embedded in $n$-space if $1\le m\le n-3$. And of cours... | 10 | https://mathoverflow.net/users/6666 | 110714 | 63,574 |
https://mathoverflow.net/questions/110713 | 2 | Hello,
I've been wondering today around the following exercise in J.Heinonen's "Lectures on Analysis on Metric Spaces" (see below for terminology): *prove that the statement of Vitali Covering Theorem for bounded subsets $A\subset X$ implies the statement for all subsets $A\subset X$*.
Here $X$ is a metric space eq... | https://mathoverflow.net/users/66825 | Vitali Covering Theorem for Arbitrary Subsets of Doubling Metric Spaces | The case where $A$ is unbounded does not really need much modifications. First you use the basic covering theorem to get a disjointed collection of balls $B\_i$ with $A \subset \bigcup\_i 5B\_i$. Then, for example, fix a point $x\_0 \in X$ and take $R>0$ and consider $A\_R = A \cap B(x\_0,R)$. Now
$$
\sum\_{5B\_i \cap ... | 4 | https://mathoverflow.net/users/11716 | 110729 | 63,580 |
https://mathoverflow.net/questions/110702 | 3 | The factor rings of the ordinary integers $\mathbb Z$ are the well-known residual classes $\mathbb Z\_n$. For the Gaussian integers $\mathbb Z[i]$ the factor rings are studied in
1) J. T. Cross, The Euler ϕ-function in the Gaussian integers, Amer. Math. Monthly 90 (1983) 518–528.
2) Dresden, Greg(1-WLEE); Dymàček,... | https://mathoverflow.net/users/24864 | Factor Rings of the Ring of integers in a Number field | The determination of the structure of the multiplicative group $({\mathfrak o}/{\mathfrak p}^n)^\times$, which could not be done with Dedekind's theory of ideals, was one of the first successes of Hensel's $\mathfrak p$-adic numbers. You can read all about it in Hasse's *Number Theory*, Chapter 15, completed by Nakagos... | 4 | https://mathoverflow.net/users/2821 | 110732 | 63,582 |
https://mathoverflow.net/questions/110727 | 19 | I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois :
*Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by radicals if and only if the splitting field of $P$ is generated by any two roots of $P$.*
I was asked by a student wh... | https://mathoverflow.net/users/6506 | On a theorem of Galois | The question asks about the relation of the properties 1. and 3., though possibly the intended meaning of 1. was 2.:1. The splitting field of $P$ is generated by two roots of $P$.
2. The splitting field of $P$ is generated by *any* two roots of $P$.
3. The Galois group $G$ of $P$ is solvable.
In the prime degree case... | 23 | https://mathoverflow.net/users/18739 | 110740 | 63,586 |
https://mathoverflow.net/questions/35872 | 7 | Let $\theta \in \mathbb{R}\backslash\mathbb{Q}$. The irrational rotation C\*-algebra $\mathcal{A}\_{\theta}$ is the universal C\*-algebra generated by unitary elements $u$ and $v$ with $vu=e^{2\pi i \theta}uv$.
What is the spectrum of u+v?
(Note: in the case where the unitary elements u and v are the standard gener... | https://mathoverflow.net/users/6269 | Spectrum of the sum of generators for irrational rotation algebra | The question is nicely resolved here: <http://arxiv.org/abs/1210.4771>
| 3 | https://mathoverflow.net/users/6269 | 110750 | 63,591 |
https://mathoverflow.net/questions/110682 | 2 | Hello,
I am trying to learn about Chow ring on smooth projective manifolds (over an algebraic closed field). Do any one know of good references for this? Thank you.
Edit: Thank you for the suggestions. I am working in complex geometry, and am familiar with Griffiths-Harris book. I also did detail reading of some ch... | https://mathoverflow.net/users/27301 | References on Chow ring | Bonho,
Let me turn my comment into an answer. My initial suggestion was Fulton's *Intersection Theory*. This certainly has the most complete treatment of Chow groups, but it is certainly
not easy reading, as Michael Joyce pointed out.
He gives some suggestions in his comments which
should be more suitable as an intr... | 5 | https://mathoverflow.net/users/4144 | 110753 | 63,594 |
https://mathoverflow.net/questions/110744 | 6 | Let $p,q\in \beta \mathbb{N}\setminus \mathbb{N}$. Must always the spaces $\beta \mathbb{N}\setminus \{p\}$ and $\beta \mathbb{N}\setminus \{q\}$ be homeomorphic? If no, can we for each point $p\in \beta \mathbb{N}\setminus \mathbb{N}$ find $q\in \beta \mathbb{N}\setminus \mathbb{N}$ ($q\neq p$) such that $$\beta \math... | https://mathoverflow.net/users/27523 | How much $\beta \mathbb{N}$ is homogenous? | As Emil pointed out in his comment, there is a correspondence between permutations of $\mathbb{N}$ and self-homeomorphisms of $\beta\mathbb{N}$, and that pretty much answers the OP´s original questions.
If we restrict ourselves to $\mathbb{N}^\ast=\beta\mathbb{N} \setminus \mathbb{N}$ the questions get more interesti... | 5 | https://mathoverflow.net/users/17836 | 110759 | 63,596 |
https://mathoverflow.net/questions/110741 | 2 | Suppose that $u\_k$ is a sequence of $L^1$ functions defined on a compact $K\subset R^n$ and a function $f:[0, \infty)\to[0, \infty)$ with the following properties
* $u\_k\ge 0$
* $\|u\_k\|\_{L^1}=\int u\_k=1$
* $u\_k\to u$ strongly in $L^1$
* $f$ is convex, $f(0)=0$ and has superlineair growth at $+\infty$ (that is:... | https://mathoverflow.net/users/1969 | Weak convergence of the image of an $L^1$ converging sequence under a convex function | With the extra assumption it is true, and only continuity on $f:[0,\infty)\rightarrow [0,\infty)$ is needed. Of course, it is sufficient to show that some subsequence of $f(u\_k)$ converges. So we can also assume w.l.o.g. that $u\_k$ converges a.e. to $u$.
Consider the sequence of non-negative measurable functions o... | 1 | https://mathoverflow.net/users/6101 | 110760 | 63,597 |
https://mathoverflow.net/questions/110709 | 13 | By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if they have the same image in $(M,g)$.
Manifolds with constant curvature $\leq 0$,
by Cartan's theorem,
cannot have any cl... | https://mathoverflow.net/users/20557 | A riemannian manifold with finitely many closed contractible geodesics | I think if you take the metric on $\mathbb{R}^2$ obtained by rotating a curve which is $\sqrt{1-x^2}$ for $-1\leq x\leq 0$, and $x^2+1$ for $x\geq 0$ around the $x$-axis, then I think there will be a single closed contractible geodesic obtained by rotating the point $(0,1)$ around the $x$-axis.
| 12 | https://mathoverflow.net/users/1345 | 110764 | 63,601 |
https://mathoverflow.net/questions/110722 | 13 | 1) Many Mathematics departments ask to send a "list of publications" while applying for research postdoctoral jobs. My question is: how important is it to post my papers in arXiv. I know, posting on arXiv is always good, because people might search for the arXiv -ed papers, but how much difference is publication on arX... | https://mathoverflow.net/users/6953 | Question on "publication List" for applying to post-doctoral jobs | While usually I don't like answering questions like this on MO, there is actually an important fact here specific to the mathematics community which would probably be missed on academia.stackexchange, or another non-mathematical site.
The answer to your 2) is:
>
> There is no logic behind asking for separate pub... | 15 | https://mathoverflow.net/users/66 | 110766 | 63,602 |
https://mathoverflow.net/questions/110337 | 8 | It's obvious why any graph containing K(5) wouldn't be 4-colourable, but what about graphs containing only instances of K(3,3) to assert their non-planarity?
(Edit: By a graph "containing" another graph, I mean having it as a subgraph. Sorry for being unclear. Although now that I think about it, perhaps the word "min... | https://mathoverflow.net/users/27459 | Are there any non-planar graphs containing only K(3,3) as a subgraph that are not 4-colourable? | There are a couple different answers to this question, depending on what question you're actually asking. You talk about a copy of $K\_{3,3}$ to "assert their non-planarity," but it's unclear whether you mean this in the context of Kuratowski's Theorem (*a graph is planar if and only if it does not contain a subdivisio... | 15 | https://mathoverflow.net/users/785 | 110774 | 63,606 |
https://mathoverflow.net/questions/110772 | 5 | Let G be a word-hyperbolic group acting on its boundary, which is homeomorphic to $S^n$ (n-sphere), effectively. Does this imply that G acts on the boundary as a convergence group of $S^n$?
If this is true in general, is it easy to see, at least for n = 1 or 2?
| https://mathoverflow.net/users/27570 | convergence action on the boundary of hyperbolic groups | Bowditch proved much more. Namely, if a group $\Gamma$ acts properly discontinuously on a $\delta$-hyperbolic space $X$, then $\Gamma$ acts as a convergence group on $\partial X$. See Lemma 1.11 of his paper
B.H. Bowditch,
Convergence groups and configuration spaces,
in ``Geometric Group Theory Down Under, proceedin... | 5 | https://mathoverflow.net/users/317 | 110775 | 63,607 |
https://mathoverflow.net/questions/110771 | 7 | Any characterization based on the adjacency matrix for directed acyclic graphs (DAG)?
An undirected graph could be simply characterized by saying that its adjacency matrix is symmetric. What about a DAG?
| https://mathoverflow.net/users/27571 | Algebraic characterisation of directed acyclic graphs | Given a finite, directed graph, it will be a DAG if and only if you can conjugate its adjacency matrix $A$ by a permutation matrix to get an upper triangular matrix. The idea is to index the rows and likewise the columns of $A$ by the vertices of the graph. Conjugating by a permutation matrix amounts to simultaneously ... | 9 | https://mathoverflow.net/users/23408 | 110777 | 63,608 |
https://mathoverflow.net/questions/110755 | 0 | Let me use the notation from Maple <http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG> for the Meijer G-function. Then let me define,
$f\_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$
$f\_-(x) = MeijerG( [[-1/2],[]], [[0,0],[]], x )$
Then by numerical evaluation I was able to show that
$\lim\_{... | https://mathoverflow.net/users/4526 | Series representation of ratio of two Meijer G-functions | According to Maple 16, both $f\_+(x)$ and $f\_-(x)$ can be expressed in terms of BesselK:
$$\eqalign{f\_+(x) &= \sqrt {\pi }\;{{\rm e}^{x/2}}\;{K\_0 \left(x/2\right)}\cr
f\_-(x) &= \frac{\sqrt {\pi }}{2}{{\rm e}^{x/2}}\left( \left( 1+x \right) { K\_0\left(x/2\right)}-x\;
{ K\_1\left(x/2\right)} \right)
\cr}$$
We t... | 2 | https://mathoverflow.net/users/13650 | 110778 | 63,609 |
https://mathoverflow.net/questions/110781 | 1 | $[n]$ is the set $\{0,1\}^n$ equipped with the product order $(\epsilon\_1,\dots,\epsilon\_n) \leq (\eta\_1,\dots,\eta\_n)$ if and only if $\forall i=1,\dots,n$, $\epsilon\_i \leq \eta\_i$. Let $$d((\epsilon\_1,\dots,\epsilon\_n),(\eta\_1,\dots,\eta\_n)) = \sum\_{i=1}^{i=n}|\epsilon\_i-\eta\_i|.$$ A set map $f$ from $[... | https://mathoverflow.net/users/24563 | About the $n$-cube | Identify elements of $[n]$ with subests of an $n$-element set. $f$ has to be level preserving because a maximal chain must be sent to a chain. If $f$ commutes with permutations, then the image is symmetric, so the image of $f$ contains all singletons. $f$ must be the identity on sets of size $1$ since $S\_n$ has no cen... | 4 | https://mathoverflow.net/users/2954 | 110782 | 63,611 |
https://mathoverflow.net/questions/73817 | 0 | I want to test the hypothesis that a group of vectors in 3D space, say given by a long list of xyz coordinates from some experiment, have no preferred direction. Is it sufficient to pick some direction in space, say the x-axis, and calculate the cosine angle between each data vector and this direction, and look at the ... | https://mathoverflow.net/users/17417 | A test for randomness of direction of vector data | <http://onlinelibrary.wiley.com/doi/10.1111/j.1365-246X.1956.tb05561.x/abstract>
| 1 | https://mathoverflow.net/users/27574 | 110785 | 63,613 |
https://mathoverflow.net/questions/110784 | 5 | The following is not exactly a research question (it was originated from manufacturing of exercises for calculus), and has no other motivation than explaining a phenomenon. I apologize if it is inappropriate (and will quickly remove it).
Consider the sequence of real numbers defined recursively as follows
$$u\_0:=\l... | https://mathoverflow.net/users/6101 | Computing the limit of a certain recursively defined sequence | Multiplying top and bottom by the conjugate and simplifying (assuming $u\_n \neq 0$) we get: $$u\_{n+1}=2^{n+1}(\sqrt{1+2^{-n}u\_n}-1).$$ Calling $v\_n:=\frac{u\_n}{2^n}+1$ we have the recursion: $v\_{n+1}=\sqrt{v\_n}$ and therefore $v\_n=v\_0^{2^{-n}}$. Going back to $u$´s we have $u\_n=2^n[(1+\lambda)^{2^{-n}}-1]$. F... | 7 | https://mathoverflow.net/users/17836 | 110789 | 63,616 |
https://mathoverflow.net/questions/110534 | 7 | I investigate certain stationary charged perfect fluid solutions to the Einstein-Maxwell equations in general relativity. The classification of these solutions has led me to the following question:
Does hyperbolic 3-space admit a non-constant harmonic function (i.e., with vanishing Laplacian) that has a gradient of c... | https://mathoverflow.net/users/27516 | Harmonic function with gradient of constant norm in hyperbolic 3-space | The main paper you want to consult is
É. Cartan, *Familles de surfaces isoparamétriques dans les espaces à courbure constante*,
Annali di Mat. 17 (1938), 177–191.
In this paper, Cartan considers the problem of studying the functions $f$ defined on an open set in a space $M$ of constant curvature that satisfy two e... | 7 | https://mathoverflow.net/users/13972 | 110796 | 63,621 |
https://mathoverflow.net/questions/110800 | 1 | For instance, lets say we have a set $S = (0,1)$ containing $n = 2$ distinct elements.
The [multiset](http://mathworld.wolfram.com/Multiset.html) $M = (1,1)$ has rank $5$ because there are $4$ multisets less than it based on lexicographic ordering: $(0), (1), (0,0), (0,1)$.
If we insert $0$, we get $(0,1,1)$ which ... | https://mathoverflow.net/users/27579 | Is there a function that determines the rank of a multiset after inserting another element? | Start by attempting the problem for ordered multisets; once you have found a formula, go back and adjust for non-ordered multisets (if you so desire).
**First, re-stating your ranks for ordered multisets:** The rank of (1,1) is 6, since it's what you have and (1,0). Similarly, (0,1,1)'s rank is 10, because it has 9 m... | 1 | https://mathoverflow.net/users/22971 | 110804 | 63,626 |
https://mathoverflow.net/questions/110812 | 6 | Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex.
My question is: When is a (homological or cohomological) spectral sequence coming from a double complex?
| https://mathoverflow.net/users/39742 | What kind of spectral sequences come from double complexes? | There are two different ways to understand the question:
1. If I see an abstract spectral seqeunce, is there a double complex such that its spectral sequence is isomorphic to the given spectral sequence? I do not have an answer to that question and, to be honest, do not believe it is an interesting question.
2. For ... | 16 | https://mathoverflow.net/users/9928 | 110820 | 63,633 |
https://mathoverflow.net/questions/110808 | 16 | As an honest question (probably with some subjectivity), **how many smooth oriented 4-manifolds are actually symplectic?** Can I say half (perhaps under some mild assumptions)? I ask this question because every compact smooth oriented 4-manifold with $b^2\_+\ge 1$ admits a *near-symplectic* form, i.e. a closed 2-form w... | https://mathoverflow.net/users/12310 | How Many 4-Manifolds are Symplectic? | *I have to apologize, in fact the answer to the second question is still unknown. Namely, up to now all known symplectic manifolds of dimension 4 that have negative Euler characteristic are blow ups of ruled surfaces. However it is not known if there are no other examples. I have corrected the answer accordingly.*
It... | 14 | https://mathoverflow.net/users/943 | 110821 | 63,634 |
https://mathoverflow.net/questions/110799 | 16 | It is consistent with ZFC that the universe is well-ordered, e.g. in $V=L$ where global choice holds. I also know that it is consistent that global choice fails (although I have no immediate example from the top of my head).
However one can try and ask a slightly weaker question, much like the axiom of choice implies... | https://mathoverflow.net/users/7206 | Does ZFC prove the universe is linearly orderable? | *Update.* I've repaired the argument. The idea was to
use the analogue of the usual non-AC arguments, but using a class forcing instead of just Cohen reals.
**Theorem.** Every model of ZFC has a class forcing extension
that is a model of ZFC, in which there is no class global linear
ordering of the universe that is d... | 14 | https://mathoverflow.net/users/1946 | 110823 | 63,635 |
https://mathoverflow.net/questions/110805 | 6 | Let $X$ be a closed subset of $\mathbb{R}^2$. What restrictions are there on $\pi\_1(X)$ and on the homology groups of $X$ (both singular and Cech)? This is elementary if $X$ has reasonable local properties, but the example of the Hawaiian Earring shows that things can be very complicated indeed.
| https://mathoverflow.net/users/27246 | Fundamental groups and homology groups of closed subsets of the plane | **Fundamental Group:** The fundamental group of a planar set naturally injects into the first Cech homotopy group, which is an inverse limit of free groups. In particular, the algebraic restrictions gained from this fact are: the fundamental group must be locally free, fully residually free (and thus torsion free), and... | 10 | https://mathoverflow.net/users/5801 | 110830 | 63,642 |
https://mathoverflow.net/questions/110841 | 1 | Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k.
Let G a finite group acting on X and suppose that his order is divisible by the caracteristic of p.
Do I have that the fixed point scheme X^{G} is flat over A^{n}?
| https://mathoverflow.net/users/27398 | fixed point scheme in caracteristic p | No. Take $X \to \mathbb A^{1} =\operatorname {Spec} k[x,y] \to \operatorname{Spec} k[x]$. Let $G=\mathbb Z/p$ act by $y \to y+x$. Then the fixed point scheme is just a copy of $\mathbb A^1$ over the point $y=0$.
| 3 | https://mathoverflow.net/users/18060 | 110843 | 63,648 |
https://mathoverflow.net/questions/110846 | 6 | The Lyness 5-cycle is the map that sends $(x,y)$ to $(y,z)$ with $z=(y+1)/x$. Leaving aside the set on which the map is not well-defined, the map is of order 5 (hence its name). Is there an algebraic map that conjugates the map to a rotation by 72 degrees?
| https://mathoverflow.net/users/3621 | Conjugating the Lyness 5-cycle into a rotation of the plane | Yes, this map is conjugate to an automorphism of $\mathbf{P}^2$. See
A. Beauville, J. Blanc, [*On Cremona transformations of prime order*](http://arxiv.org/abs/math/0402037), C.R. Acad. Sci. Paris 339 (2004), no4, 257-259.
See also T. de Fernex, *On planar Cremona maps of prime order*, Nagoya Math. Journal, Vol. 17... | 3 | https://mathoverflow.net/users/6506 | 110847 | 63,649 |
https://mathoverflow.net/questions/110848 | 3 | Let $k$ be a fixed algebraically closed field and $X/k$ an irreducible scheme smooth and proper over $k$. Can there exist a line bundles $\mathcal{L}, \mathcal{M}$ and an integer $m > 0$ so that
1.) $\dim\_k \Gamma(\mathcal{L}) = 0$
2.) $\dim\_k \Gamma(\mathcal{M}) > 0$
With $\mathcal{L}^m \cong \mathcal{M}^m$.... | https://mathoverflow.net/users/25854 | Roots of line bundles | Yes. Take $\mathcal L$ the trivial line bundle, with a one-dimensional space of global sections, and $\mathcal M$ a nontrivial torsion line bundle, so $\mathcal M^k=\mathcal L$.
Then $\Gamma(\mathcal M)$ is certainly zero-dimensional, since otherwise $\mathcal M$ would have a nonvanishing section and be trivial or a ... | 7 | https://mathoverflow.net/users/18060 | 110850 | 63,650 |
https://mathoverflow.net/questions/110855 | 8 | Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. [MO 62088](https://mathoverflow.net/questions/62088/products-of-conjugacy-classes-in-s-n)). So we get ring with a basis and structure constants are natu... | https://mathoverflow.net/users/10446 | Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ? | The answer to your first question is negative. For a concrete example, you can show that the conjugacy class rings of the nonisomorphic groups $Q\_8$ and $D\_8$ are isomorphic, via an isomorphism that pairs off the bases as follows: $[1] \leftrightarrow [1]$, $[-1] \leftrightarrow [r^2]$, $[i] \leftrightarrow [r]$, $[j... | 6 | https://mathoverflow.net/users/430 | 110863 | 63,655 |
https://mathoverflow.net/questions/110869 | 0 | Why a projective module is a projective cover for its largest semisimple quotient? That is - why the projection on the quotient is an essential morphism in this case?
| https://mathoverflow.net/users/27018 | why a projective module is a projective cover for its largest semisimple quotient? | Answer: if Q is a submodule of a projective module P which projects surjectively on the largest semisimple quotient of P, then Q projects surjectively on each simple quotient of P, and hence Q lies outside of any maximal submodule of P - contradiction.
| 0 | https://mathoverflow.net/users/27018 | 110873 | 63,657 |
https://mathoverflow.net/questions/110877 | 7 | I have three three questions, the first two of which probably have the same answer and the third of which is more vague.
For a set $A$ let $L\_\alpha(A)$ be the constructible universe up to $\alpha$, built from $A$ as a set (and not a predicate). Further let $X = (B, f)$ where $B$ is a transitive set and $f$ is a bi... | https://mathoverflow.net/users/8106 | Effect of large cardinals on the value of $\omega_1^L$ in $L$ | The answer is no. If there is a transitive set model $M$ of set theory (and this is all you need), then if $\alpha$ is its height (that is, if $\alpha=\mathsf{ORD}^M$), then $L\_\alpha$ is a model of $\mathsf{ZFC}+V=L$. Note that the assumption is strictly stronger than the existence of an $\omega$-model of $\mathsf{ZF... | 7 | https://mathoverflow.net/users/6085 | 110880 | 63,661 |
https://mathoverflow.net/questions/110872 | 6 | Suppose we want to find the root of the equation $f(x)=\phi(x) - d = 0$, where d is a real constant and $f$ is continuously differentiable function.
The problem is well posed if the inverse $\phi^{-1}$ exists, since in that case $\phi^{-1} (d) = x$.
Now most numerical analysis books I read on the topic of solving... | https://mathoverflow.net/users/27611 | Condition Number related to Root finding problems | In addition to the convergence speed (and radius) mentioned in Pietro Majer's answer, there is another factor: if the problem is ill-conditioned, the solution is sensitive to perturbations.
If you make an error of magnitude $\varepsilon$ in computing the iteration or the parameters data, then the solution is perturb... | 6 | https://mathoverflow.net/users/1898 | 110895 | 63,668 |
https://mathoverflow.net/questions/110871 | 18 | The Lévy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be generalized to *all* forcings, but I cannot think of a counterexample.
Is there a forcing notion that creates a $\kappa$-complete (... | https://mathoverflow.net/users/1682 | Can measures be added by forcing? | Your title question is asking whether the measurability of a measurable
cardinal is downwards absolute to ground models: if $\kappa$ is
measurable in a forcing extension $V[G]$, must it be measurable in
$V$? This is a question that makes sense for any of the large
cardinal notions.
The answer is that, although the sm... | 16 | https://mathoverflow.net/users/1946 | 110898 | 63,671 |
https://mathoverflow.net/questions/110893 | 8 | Hi to all!
Perhaps it is a silly question, if so i'll delete this post.
Suppose we have a compact Kahler manifold $(M,g)$ of complex dimension $m$ with constant scalar curvature with respect to its metric $g$. My question is: does the condition of constant scalar curvature imply that the metric $g$ automatically real... | https://mathoverflow.net/users/4971 | If a compact Kahler manifold $(M,g)$ has constant scalar curvature, is the metric $g$ real analytic? | It's not a silly question, but there's a standard answer, and it's a purely local result: If the Kähler metric is $C^2$ and has constant scalar curvature, then it is real-analytic with respect to the real-analytic structure that underlies the complex-analytic structure. The reason is that setting the scalar curvature e... | 18 | https://mathoverflow.net/users/13972 | 110900 | 63,672 |
https://mathoverflow.net/questions/110912 | 0 | Dear mathoverflow.
This is a question to a proof in a graduate text. I have asked two professors at my university without help, so I hope it suffices in difficulty for this forum otherwise I appologize.
I can't add an image but the theorem is at scribd page 336.
[The book](http://de.scribd.com/doc/58071878/Proba... | https://mathoverflow.net/users/27584 | transition probability convergence for Harris chains - Durrett. | You already know that the random walk $S\_m-T\_m$ is recurrent. It sounds like the piece that you are missing is that recurrent random walks (especially on $\mathbb{R}$) not only visit their initial value infinitely often but any other finite value as well as long as it has a positive probability of reaching this other... | 0 | https://mathoverflow.net/users/11332 | 110913 | 63,678 |
https://mathoverflow.net/questions/110915 | 1 | We know following theorem by Schur:
Suppose that $f(x) \in \mathbb{Z}[x]$ is a polynomial such that exists an integer $m$ such that $f(n) = m^2$ for
every integer $n$. Then $f(x) = g(x)^2$ for some $g(x) \in \mathbb{Z}[x]$.
Is this true if I using $3,4,5 \cdots$ instead of $2$?
| https://mathoverflow.net/users/22954 | Another Number Theory Question about Polynomial | I think the proposer tried to say that *if for every integer $n$ there is an integer $m$ such $f(n)=m^2$, then $f(x)=g(x)^2\dots$*.
The generalization from exponent $2$ to higher exponents $e$ is an exercise in the second volume of the classical Polya/Szegö problem book. I don't have it at hand now, so I cannot give ... | 2 | https://mathoverflow.net/users/18739 | 110928 | 63,683 |
https://mathoverflow.net/questions/110757 | 2 | When ever I hear noncommutative geometers talking about quantum groups, it is usually $q-SU(2)$ that they are discussing. As a result there are many good and explicit generator and relation presentations of this Hopf algebra. For an easy example take this other M.O. [question](https://mathoverflow.net/questions/10581/k... | https://mathoverflow.net/users/11206 | Explicit Descriptions of $q-SO(2)$, and $q-Sp(2)$? | If I am not mistaken, $SO(2)\approx U(1)$ has no nontrivial quantum deformation, but $SO(3)$ does; this is explicitly constructed in:
[Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups](http://xxx.lanl.gov/abs/hep-th/9402069), P. Podles (1994)
[Quantum SO(3) groups](http:... | 2 | https://mathoverflow.net/users/11260 | 110937 | 63,687 |
https://mathoverflow.net/questions/110936 | 2 | Let $X^\nu,Y^\nu$ be normalizations of affine varieties $X$ and $Y.$
If a morphism $f:X\to Y$ is a bijection, does it imply that its lift $f^\nu: X^\nu\to Y^\nu$ is an isomorphism?
| https://mathoverflow.net/users/23935 | normalization of a bijection | We should be able to construct a counterexample as follows:
Let $Y$ be an affine curve which is smooth away from a single node. We obtain $X$ from the normalization of $Y$ by removing one of the points mapping to the node of $Y$. The map from $X$ to $Y$ is a bijection, but the map $X^\nu\to Y^\nu$ is not an isomorphi... | 4 | https://mathoverflow.net/users/5263 | 110938 | 63,688 |
https://mathoverflow.net/questions/110448 | 2 | Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.
We say that $\sigma$ has *weak divergence* if there exists a function $w \in L^2(\Omega)$ such that for all $\varphi \in C\_c^\infty (\Omega)$ we have
$$ \int\_\Omega \sigma ... | https://mathoverflow.net/users/13093 | Weak divergence implies weak differentiability of components? | So $\sigma=\sum\_{1\le j\le n}\sigma\_j(x)\frac{\partial}{\partial x\_j}$ is a vector field with distributions coefficients $\sigma\_j$ and divergence in $L^2$:
$$
\sum\_{1\le j\le n}\frac{\partial \sigma\_j}{\partial x\_j}\in L^2.
$$
If I understand your question correctly, you ask if this implies that each $\sigma\_j... | 2 | https://mathoverflow.net/users/21907 | 110940 | 63,689 |
https://mathoverflow.net/questions/110918 | 3 | Let $f \in H^{-1}(U)$ and $u \in H^1(U)$. I know that we write $f(u)$ as the pairing $$\langle f, u \rangle\_{H^{-1}, H^1}$$.
Suppose that $v$ is the weak/distributional derivative of $u$. So
$$\int\_0^T u\phi' = -\int\_0^T v\phi$$
holds for all $\phi \in C\_c^\infty(0,T)$. What does it mean to write $\langle v, w ... | https://mathoverflow.net/users/27547 | Dual space pairing question (Sobolev space, Bochner space) | Maybe [Thm. 1.5.5. of these notes](http://www.nd.edu/~lnicolae/Pseudo.pdf) is the answer to your question.
| 3 | https://mathoverflow.net/users/20302 | 110941 | 63,690 |
https://mathoverflow.net/questions/110948 | 2 | Is there a natural *partial order* and/or *lattice structure* on the set of *closed symmetric* or *self-adjoint extensions* of a densely defined, unbounded, symmetric operator on a Hilbert space? Any reference where such order structures are discussed?
One sometimes encounters references to the "minimal" or "maximal"... | https://mathoverflow.net/users/2622 | Partial order on self-adjoint extensions? | Yes. First define $D\_{\max} = \{u \in H: Au \in H\}$ and $D\_{\min}$ as the graph closure
in $H \times H$ of the graph of $A$ over some "core domain". In the usual PDE examples, one should think of $D\_{\max}$ as consisting of all $L^2$ functions such that $Au$, defined as a distribution, happens to lie in $L^2$, and ... | 3 | https://mathoverflow.net/users/17969 | 110951 | 63,694 |
https://mathoverflow.net/questions/97049 | 18 | In their 1987 paper "[Balanced Tableaux](https://mathscinet.ams.org/mathscinet-getitem?mr=871081)", Edelman and Greene construct a bijection between standard young tableaux with staircase shape $(n-1,n-2, \dots , 1)$ and reduced decompositions of the reverse permutation $(n,n-1, \dots, 1)$. The bijection is constructed... | https://mathoverflow.net/users/7717 | Comparing the Edelman-Greene bijection to David Little's bijection | Question one now has an answer. [Benjamin Young](https://mathoverflow.net/users/20281/benjamin-young) and I have proved that the two bijections are the same in this case. A reference is available on the [arXiv](http://arxiv.org/abs/1210.7119). More generally, we have shown the Little map is the same map as the recordin... | 12 | https://mathoverflow.net/users/7717 | 110952 | 63,695 |
https://mathoverflow.net/questions/110327 | 13 | Hi,
Consider $\theta\_n = (\theta\_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta\_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take this process to hit $(0,\alpha)$ ? From the ergodic theorem I know that, if I denote $N(n)$ the number of times $\theta... | https://mathoverflow.net/users/19334 | What time does it take for irrational rotations to hit an interval? | There is a theorem of Kesten, which roughly says, that if you take $(\theta, \theta\_0)$ random, and the number of times you hit $(0, \alpha)$ in the first $N$ iterations, subtract the expected $N \times \alpha$, and normalize by $\rho \times ln(n)$, the result will converge to Cauchy distribution. This can be viewed a... | 10 | https://mathoverflow.net/users/21005 | 110958 | 63,697 |
https://mathoverflow.net/questions/110890 | 1 | For the definition of Coxeter System, you can see: <http://en.wikipedia.org/wiki/Coxeter_group>
Given a chamber Q, given a Coxeter System $(\Gamma,V)$, we can defined a set M by the following way: defined $M=\Gamma\times X/$~, the ~ is defined by $(g,x)$~$(h,y)$ if and only if x=y and $g^{-1}h\in \Gamma\_{V(x)}$
V(... | https://mathoverflow.net/users/25054 | How to judge a manifold generated by Coxeter system smooth? | See:
Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Reflection groups on Riemannian manifolds. Annali di Matematica 186 (2006), 25-58.
[(pdf)](http://www.mat.univie.ac.at/~michor/reflec.pdf)
The idea there is to put a Riemannian metric on the chamber with the walls totally geodesic and with the right... | 2 | https://mathoverflow.net/users/26935 | 110974 | 63,701 |
https://mathoverflow.net/questions/110966 | 2 | Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple reflexions that appear in a reduced decomposition of w.
Let w an element such that supp(w)=I and length(w)>r+1, is it true... | https://mathoverflow.net/users/27398 | elements in the weyl group | I agree with the answer below, but to turn it into a rigorous proof one could argue in a slightly different fashion:
Let $l$ be the length of $w$ and let $Red(w)$ be the set of $all$ reduced expressions for $w$. Given ${\bf r}=(i\_1,\ldots, i\_l)\in Red(w)$ denote by $k({\bf r})$ the smallest $k\le l$ such that $i\_k... | 5 | https://mathoverflow.net/users/24386 | 110989 | 63,707 |
https://mathoverflow.net/questions/110987 | 9 | As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a holomorphic immersion into the complex plane. Another example is given by the theorem of Behnke and Stein, which says tha... | https://mathoverflow.net/users/13244 | Interesting results for open Riemann surfaces | The results on open Riemann surfaces are not "rare". They are just well forgotten.
I only list a few books which deal with open Riemann surfaces:
MR0114911, MR0228671, MR0159935, MR0264064, MR1973182.
It is true that there are "too many" Riemann surfaces, and not too much can be said about
"all of them". However, there... | 11 | https://mathoverflow.net/users/25510 | 110997 | 63,709 |
https://mathoverflow.net/questions/110993 | 2 | Let $f$ be a periodic $L^1$ function, and $S\_n[f]$ the $n$-th partial sum of its Fourier series. I [am aware](https://mathoverflow.net/questions/28428/convergence-of-fourier-series-of-l1-functions) that $S\_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at least converge weakly? In othe... | https://mathoverflow.net/users/17064 | Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$? | No. If the partial sum projections $S\_n$ converged in the weak operator topology, they would be pointwise weakly bounded hence pointwise norm bounded whence uniformly bounded. That would give convergence pointwise strongly.
| 6 | https://mathoverflow.net/users/2554 | 111001 | 63,711 |
https://mathoverflow.net/questions/110998 | 6 | Consider an extension $R\subseteq S$ of commutative rings, and suppose that $R$ is principal (i.e., $0$ is the only zero-divisor of $R$ and every ideal of $R$ has a generating set of cardinality $1$). By means of scalar restriction we consider $S$ as an $R$-module. Let $M$ be a sub-$R$-module of finite type of $S$ cont... | https://mathoverflow.net/users/11025 | Linear algebra over principal rings | I think that I have a counterexample. Let $R=\mathbb Z$, $S=\mathbb Q$ and $M= 1/2 \mathbb{Z}$. The induced map $R/2 R \rightarrow M/2M$ is zero. If $R$ is a direct summand of some $N$ the $R/2R\rightarrow N/2N$ is non-zero. Hence, $R\subset M\subset N$ implies that
$R$ cannot be a direct summand of $N$, since $R/2R \... | 6 | https://mathoverflow.net/users/19229 | 111003 | 63,712 |
https://mathoverflow.net/questions/110955 | 4 | There are a couple of statements that I have read which are made as though they were trivial, but I am doubtful about them.
1. One is related to an example showing that the s-invariant of an ample line bundle on a projective variety X is an algebraic integer of degree $\leq dim X$. Recall that, given an ideal sheaf $... | https://mathoverflow.net/users/18013 | Amplitude and bigness issues | (1) was handled in the comments. Regarding (2): since $\epsilon(L;x) > 2n$ with strict inequality, in fact $\nu^\ast \left( \frac{1}{2} L \right) - (n+c) E\_x$ is nef for some $c> 0$, where $\nu$ is the blow-up at just $x$. Pulling this back to $X^\prime$, we have $\mu^\ast \left( \frac{1}{2} L \right) - (n+c) E\_x$ ne... | 2 | https://mathoverflow.net/users/27655 | 111007 | 63,714 |
https://mathoverflow.net/questions/111009 | 1 | Usually disk packing problems require that no two disks of the packing intersect.
Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the center of any other disk?
| https://mathoverflow.net/users/22237 | Disks Packing Variant | Yes. Here is an example: [Rigidity of infinite disk patterns](http://arxiv.org/pdf/math/9901148.pdf) by Zheng-Xu He.
| 1 | https://mathoverflow.net/users/27659 | 111013 | 63,718 |
https://mathoverflow.net/questions/111030 | 0 | A group $G$ has Serre's property $FA$ if any isometric action of $G$ on a simplicial tree has a global fixed point. Let $n\geq 3$. It is well-known that $SL\_n(\mathbb{Z} )$ has property $FA$. Now my question is that are there nontrivial group actions of $SL\_n(\mathbb{Z} )$ on a simplicial tree by isometries? Here "no... | https://mathoverflow.net/users/1546 | Non-trivial action of $SL_n(\mathbb{Z} )$ on a simplicial tree | Any non-trivial group $G$ has a non-trivial action on a "star" tree $T$ whose vertex set is $G\cup\{\infty\}$ (where $\infty\notin G$) and edges are $\{\infty,g\}$ for $g\in G$. Thus any group admits a faithful action on a tree.
Any residually finite countable group has a faithful action on a *locally finite* tree. I... | 5 | https://mathoverflow.net/users/14094 | 111032 | 63,725 |
https://mathoverflow.net/questions/110894 | 23 | After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}\_0$ denote the Hilbert space of square integrable random variables with zero mean. A stochastic process $X$ is called a second order process if $\mathbf{E}X(t)^2 < \i... | https://mathoverflow.net/users/24955 | Bochner integral of stochastic process = path by path Lebesgue integral? | Yes, the Bochner integral does agree with the Lebesgue integral of the sample paths of the process. We can prove this in a slightly more general situation than that asked for in the question.
For a probability space $(\Omega,\mathcal{F},\mathbb{P})$, let $X\colon[0,T]\to L^p(\mathbb{P})$ ($1\le p\le\infty$) be Bochne... | 17 | https://mathoverflow.net/users/1004 | 111037 | 63,727 |
https://mathoverflow.net/questions/111033 | 0 | What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
For example for a text which is generated by a multinomial distribution over words, this distribution of distance will be a Geometric distribution. Thanks.
| https://mathoverflow.net/users/19294 | What is the distribution of the distance between a specific word in a Text which is generated by a markov process? | With infinitely many states, any distribution is possible, as demonstrated by a Markov chains on $N$ which jump from $n$ only to $n+1$ or $0$.
If the Markov chain is has finitely many states, or is reversible, many distributions are possible, and there is no complete characterization of which distributions can or can... | 3 | https://mathoverflow.net/users/9422 | 111038 | 63,728 |
https://mathoverflow.net/questions/111039 | 4 | I'm looking for articles describing or proving nonexistence of isometric embeddings of $m$-dimensional space $\ell\_q^m$ into $L\_p$ and $\ell\_p$ for $q,p\in[1,+\infty]$.
Since $\ell\_q^m$ is finite dimensional some (not necessary isometric) embedding always exist. Here is my progress.
Since every separable Bana... | https://mathoverflow.net/users/19593 | Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$ | See
$$
$$
MR0417756 (54 5804) 46B05 Dor, L. E. Potentials and isometric embeddings in $L\_1$. Israel J. Math. 24 (1976), no. 3-4, 260–268
$$
$$
for a complete answer to your question.
| 5 | https://mathoverflow.net/users/2554 | 111041 | 63,729 |
https://mathoverflow.net/questions/111025 | 8 | This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model category of spaces, i.e. obtained by inverting the suspension functor, or something along those lines, what is this process c... | https://mathoverflow.net/users/11546 | Monoidal Model Categories with Suspension Functor | Fernando Muro's comment is right on the money. I'm interested in this type of question, too, and have read Hovey's paper closely. I'm going to summarize the paper here. I believe Marc Hoyois is correct that you don't need much machinery to make this work in $(\infty,n)$ categories (see Dylan Wilson's comment [here](htt... | 3 | https://mathoverflow.net/users/11540 | 111043 | 63,731 |
https://mathoverflow.net/questions/111053 | 0 | Any help in this problem?
Suppose U and V are independent random variables with density f(u) and g(v) respectively.
The domain of U is the interval (0, 1) and the domain of V is v > 0. After the transformation
X = V sin(2U)
Y = V cos(2U)
X and Y are independent, each following the standard normal distribution N(... | https://mathoverflow.net/users/27677 | Transformation problem involving 2 random variables | One obvious answer to (a) is:
g(v) is density of Rayleigh distribution and f(u) is density of uniform distribution on [0,1].
For (b), you need only noticed that V could be generated by
V = \sqrt(-ln(W))
Here W is also uniform distributed on [0,1], the same as U. Nevertheless, W is independent of U.
This is well-kn... | 0 | https://mathoverflow.net/users/27681 | 111054 | 63,734 |
https://mathoverflow.net/questions/111029 | 11 | In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find any
information about prime power gaps. So, what I'm asking is:
>
> What is the asymptotically largest function $f(n)... | https://mathoverflow.net/users/27663 | Prime Power Gaps | The [paper](http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf) shows that for $x \gt x\_0$ there are primes in the interval $[x-x^{0.525},x]$ where the value of $x\_0$ could be found effectively "with enough effort." The result fails for $x=126$ but I can't immediately rule out that $x\_0=127$ suffices. Th... | 3 | https://mathoverflow.net/users/8008 | 111056 | 63,735 |
https://mathoverflow.net/questions/111050 | 4 | W.Rudin in *Real and Complex Analysis*(262) mentioned that
>
> **Theorem** Suppose $M$ is a vector space of continuous complex functions
> on the closed unit disc $\bar U$,with
> the following properites:
>
>
> (a) $1 \in M$
>
>
> (b)If $f \in M$,then also $jf \in
> > M$,where $j$ denote the identity
> func... | https://mathoverflow.net/users/27670 | A converse of the maximum modulus Theorem | In the notes and comments section at the end of Real and Complex Analysis (page 397 in my copy), Rudin attributes this result "in a slightly different form" to:
W. Rudin, [Analyticity, and the maximum modulus principle](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077465335). ... | 2 | https://mathoverflow.net/users/630 | 111060 | 63,736 |
https://mathoverflow.net/questions/111059 | 34 | as far as I know, there are two main ways to have a relative version of De Rham Cohomology for a pair (M,N), where M and N are smooth manifolds and N is a closed (as a topological subspace) submanifold of M:
1) Godbillon, Elements de topologie algébrique: $\Omega^p(M,N)$ is the space of all forms on $M$ whose restric... | https://mathoverflow.net/users/18938 | Relative De Rham cohomologies | A chain map $\Theta$ from the Godbillon theory to the Bott-Tu version is given by $\omega \mapsto (\omega,0)$ (note that is a chain map only on $\Omega^{p} (M;N)\_{G}$).
I claim that this induces an isomorphism on cohomology. A couple of special cases is obvious: if $N=\emptyset$, then both theories agree with absolute... | 36 | https://mathoverflow.net/users/9928 | 111063 | 63,739 |
https://mathoverflow.net/questions/111081 | 2 | Let $X^k$ be a complete intersection in $\mathbb CP^n$ of dimension $k>1$. Is it true that a smooth variety obtained by resolutions
of singularities of $X$ is simply-connected?
Note that in the case $X^k$ is smooth itself, $\pi\_1(X^k)=0$.
| https://mathoverflow.net/users/13441 | Fundamental groups of resolutions of complete intersections | No. For example, take $X$ to be the projective cone in $\mathbb P^3$ of a smooth plane curve $C$ of degree at least $3$. Then blowing up the vertex gives a resolution that is a $\mathbb P^1$-bundle over $C$, and the fundamental group is that of $C$.
| 5 | https://mathoverflow.net/users/8726 | 111083 | 63,745 |
https://mathoverflow.net/questions/111085 | 6 | I was reading chapter 6 in the book of Harris on Algebraic geometry and came to the following puzzle.
It seems to me that every Schubert cell in a Grassmanian is obtaining by cutting the Grassmanian by a certain plane in its Plucker embedding (hope this is correct, I got this idea from Harris' book)
Now, I was thin... | https://mathoverflow.net/users/13441 | Little puzzle about intersections of Schubert cells. | Precisely as you say, the intersection is almost always not transverse.
For example, think about $G(2,4)$. It is the quadric $p\_{12} p\_{34} - p\_{13} p\_{24} + p\_{14} p\_{23} =0$ in $\mathbb{P}^{\binom{4}{2}-1}$. There are two codimension $2$ Schubert cells: One of them is given by $p\_{12}=p\_{13} = p\_{23}=0$ an... | 10 | https://mathoverflow.net/users/297 | 111091 | 63,746 |
https://mathoverflow.net/questions/111066 | 15 | I am trying to solve the following differential equation
$$f ' (x) = f( f( x ) ),$$
but I have no idea how. I don't think the chain rule is useful for this.
Although I don't think this differential equation is solvable, I'd like to know if there is any interesting approach to solve a differential equation of this... | https://mathoverflow.net/users/27692 | Are there any techniques for solving a differential equation of the form $f ' (x) = f( f( x ) )$? | Nothing is new under the Moon...
<http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=321705>
| 19 | https://mathoverflow.net/users/1131 | 111092 | 63,747 |
https://mathoverflow.net/questions/111089 | 10 | In Lurie's paper [*Tannaka Duality for Geometric Stacks*](http://www.math.harvard.edu/~lurie/papers/Tannaka.pdf), it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback morphism, which is a symmetric monoidal functor
$$ f^\* \... | https://mathoverflow.net/users/19801 | Recovering classical Tannaka duality from Lurie's version for geometric stacks | Tannaka duality makes two complementary statements: first, that an affine algebraic group can be recovered from its category of representations (the "reconstruction problem"), and second, that certain categories are the categories of representations of an affine algebraic group (the "recognition problem").
Lurie's pa... | 15 | https://mathoverflow.net/users/396 | 111099 | 63,751 |
https://mathoverflow.net/questions/111101 | 9 |
>
> Is there a surface in $\mathbb R^3$ which is a closed subset and whose curvature is negative and bounded away from zero?
>
>
>
And the small-print...
* By surface I mean smooth surface without boundary, and by smooth I mean at least $C^2$. If one allows a boundary the question becomes silly, as a closed di... | https://mathoverflow.net/users/1409 | Surfaces in $\mathbb R^3$ with negative curvature bounded away from zero | Efimov proved that there are no $C^2$ isometric immersions of complete surfaces with negative Guassian curvature bounded away from zero.
N.V. Efimov, "Imposibility of a complete regular surface in euclidean 3-space whose Gaussian curvature has a negative upper bound" Soviet Math. Dokl. , 4 : 3 (1963) pp. 843–846 Dokl... | 16 | https://mathoverflow.net/users/353 | 111125 | 63,762 |
https://mathoverflow.net/questions/53647 | 16 | I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a more general picture. (The commutative algebra course that I had mentioned in an earlier question has recently begun.) ... | https://mathoverflow.net/users/1149 | Questions about spectra of rings of continuous functions | For a commutative ring $A$, $MaxSpec(A)$ is Hausdorff if and only if $A/JacobsonRadical(A)$ is a Gelfand ring (i.e. all equations of the form $(1-xb)\cdot (1-y(1-b))=0$, with $b$ from the ring, are solvable in that ring). $MaxSpec(A)$ is boolean if and only if every element of $A/JacobsonRadical(A)$ is a sum of a unit ... | 8 | https://mathoverflow.net/users/27714 | 111137 | 63,766 |
https://mathoverflow.net/questions/111055 | 2 | In a first-order logic, L, it is possible to transform a sentence, S, containing variables into a sentence S' in so-called 'clause form' - i.e. with a matrix in conjunctive normal form and a prefix consisting of only universal quantifiers - such that S is valid iff S' is. If L is expanded to a second-order language con... | https://mathoverflow.net/users/27684 | Normal form in second-order logic | First, if you don’t place any restrictions on the transformation, you can actually make $S'$ as simple as you want in any logic: if $S$ is valid, let $S'$ be any fixed tautology, otherwise let $S'$ be a fixed non-tautology. In order to rule
out such a trivial answer, I will henceforth assume that the transformation $S... | 4 | https://mathoverflow.net/users/12705 | 111138 | 63,767 |
https://mathoverflow.net/questions/110922 | 9 | This must be an elementary question, as I couldn't find any proofs on the Internet, but I still can't do it. And yes, Hatcher says that the join is not actually associative for general topological spaces, but it is for locally Hausdorff ones. My definition of the join is a quotient of a product.
My problem is that I ... | https://mathoverflow.net/users/27433 | Associativity of topological join and join of spheres | Actually there are *two useful topologies* on the join $X \* Y$ using initial topologies, or final topologies. The former is useful for maps into the join, and the latter for maps out of the join. With initial topology, the join is associative. With final topologies, the standard problem is that the product of identifi... | 14 | https://mathoverflow.net/users/19949 | 111139 | 63,768 |
https://mathoverflow.net/questions/111104 | 1 | As explained in [Singularities of pairs](https://mathoverflow.net/questions/42010/singularities-of-pairs) by Karl Schwede, log canonical center is quite useful for induction. For example, to study effective freeness by induction on dimension of log canonical centers. In general, log canonical centers are not smooth.
... | https://mathoverflow.net/users/2348 | Dimension and singularities of the minimal log canonical center | If $(X, D)$ is a log canonical pair and $Z$ is the minimal log canonical center, then for some appropriate $D\_Z$, the pair $(Z, D\_Z)$ is KLT. In particular, $Z$ always has rational singularities. Thus I'd say the singularities of minimal LC centers are not so bad. See the papers of Kawamata on subadjunction. Florin A... | 4 | https://mathoverflow.net/users/3521 | 111143 | 63,770 |
https://mathoverflow.net/questions/111040 | 3 | I know this is a stupid question, but I can't find a reference, or the result stated in in full generality online. I was hoping somebody knew:
Let $X = \textrm{Spec }A$ a noetherian scheme, $Z\hookrightarrow Y \hookrightarrow X$ a sequence of closed immersions with $Z,Y$ corresponding to the ideals $I,J$ respectively... | https://mathoverflow.net/users/27666 | Ideal of strict transform | Let me fix some notation, let's set $\pi : \widetilde{X} \to X$ be the blowup and set $\bar{I}$ to be the ideal sheaf $(\pi^{-1} I) \cdot O\_{\widetilde{X}}$ (note this is an invertible sheaf) and set $\bar{J} := (\pi^{-1} J) \cdot O\_{\widetilde{X}}$ (this is probably not invertible). Finally, for clarity, let's set $... | 2 | https://mathoverflow.net/users/3521 | 111146 | 63,773 |
https://mathoverflow.net/questions/111147 | 1 | I posted the following question also [here](https://math.stackexchange.com/questions/225297/linear-maps-between-the-l1-spaces-of-two-singular-measures), but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu\_1$, $\nu\_2$ two probability measures on it. For $i=1,2$, let $L\... | https://mathoverflow.net/users/27715 | Linear Maps between $L^1$-spaces of singular measures | How about $M \colon L^1(\nu\_1)\rightarrow L^1(\nu\_2) \colon g \mapsto (x \mapsto \int g~d\nu\_1)$?
| 1 | https://mathoverflow.net/users/11716 | 111152 | 63,778 |
https://mathoverflow.net/questions/110911 | 8 | While I was studying the book [*Variation et Optimisation de formes*](https://doi.org/10.1007/3-540-37689-5) by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for every compact by:
$$ \operatorname{cap}(K)=\inf \lbrace \|u\|\_{H^1(\Bbb{R}^N)... | https://mathoverflow.net/users/13093 | Books about capacity theory | Maz'ya's book contains a fruitful treatment of Capacity and Weighted capacity and its relation with Sobolev spaces theory, in particular the (weighted) Sobolev inequality or Poincare inequality. Heinonen's book contains the treatment of modulus and capacity in metric setting.
1. Maz'ya, Vladimir Sobolev spaces with ... | 5 | https://mathoverflow.net/users/26608 | 111155 | 63,779 |
https://mathoverflow.net/questions/107629 | 1 | Assume you have a graph with an [equitable partition](https://mathoverflow.net/questions/96858/complexity-of-equitable-partitions) with respect to cells $V\_1,\ldots,V\_n$. Accordingly, you can take the cellwise average value of a function on the node set - in other words, the projection of the node space onto the susb... | https://mathoverflow.net/users/26039 | a variation on the theory of equitable partitions for graphs | The column space of $I-P$ is $A$-invariant and therefore there is a matrix $D$ such that
$A(I-P)=(I-P)D$. The difficulty is that $D$ will generally not be non-negative and so
it it less useful to interpret it as a weighted adjacency matrix. Additionally, in
practice $C$ is small (which is what makes it useful) and so ... | 2 | https://mathoverflow.net/users/1266 | 111158 | 63,780 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.