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https://mathoverflow.net/questions/96558 | 3 | Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system, with $T$ invertible, where the $\sigma$-algebra $\mathcal{B}$ is a Borel algebra arising from a topology which makes $T$ continuous, and such that $X$ is compact Hausdorff. Then there's a well-defined action of the semigroup $\beta \mathbb{Z}$ on $X$ :
$ ... | https://mathoverflow.net/users/21724 | Does the "measure-preserving property" commute with ultralimits ? | Looks false to me. Let $X$ be the product of countably many copies of $\{0,1\}$ with the standard Bernoulli measure (each point has measure $1/2$). Index these copies with ${\bf Z}$, and let $T$ be the shift operator. I claim that evaluating along any free ultrafilter won't be measure-preserving. To see this, let $A\_0... | 5 | https://mathoverflow.net/users/23141 | 96580 | 56,437 |
https://mathoverflow.net/questions/96300 | 1 | Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL\_2(\mathbb{Q}\_p)$ will be sufficient with $B$ upper triangular characters
Let $\Delta\_B$ be the modulus-character of Borel subgroup $B(F)$. We define
$$ H (bk) = \Delta\_B(b), \qquad ... | https://mathoverflow.net/users/10400 | Heights in reductive groups | Yes, thinking of the Iwahori decomposition (as a refined sort of p-adic Cartan decomposition), $GL\_n=\bigcup\_w I\cdot w\cdot I$ for $w$ in the affine Weyl group, here "monomial" matrices modulo diagonal units-entry matrices. At least for $GL\_n$, it is easy to see the semi-direct product decomposition of affine Weyl ... | 2 | https://mathoverflow.net/users/15629 | 96587 | 56,439 |
https://mathoverflow.net/questions/96584 | 15 | One of the motivation of the theory of Lie Algebras is that every associative algebra $A$ is a LA when the bracket is defined by $[a,b]=ab-ba$ : this is skew-symmetric and satisfies the Jacobi identity $[[a,b],c]+[[b,c],a]+[[c,a],b]=0$. Conversely, every abstract LA can be embedded into an associative algebra (its *env... | https://mathoverflow.net/users/8799 | Ternary "Lie structure" | ["Identities for the ternary commutator"](http://www.sciencedirect.com/science/article/pii/S0021869398974336) by Bremner classifies all such identities up to degree 7. A recent exposition can also be found in ["Ternutator Identities"](http://arxiv.org/abs/0908.1738) by C. Devchand, D. Fairlie, J. Nuyts, G. Weingart. Si... | 16 | https://mathoverflow.net/users/2384 | 96588 | 56,440 |
https://mathoverflow.net/questions/96493 | 4 | I'm studying information theory right now and I'm reading about channel capacities.
I know that there are known expressions for computing the capacities for some well known simple channels such as BSC, the Z channel.
Could you show me or point me to the source showing how to derive the channel capacity for a bina... | https://mathoverflow.net/users/11105 | Computing channel capacities for non-symmetric channels | A quick google search for "capacity of binary asymmetric channel" gives a few papers stating a closed form solution (for example [this paper of Stefan Moser](http://moser.cm.nctu.edu.tw/docs/papers/smos-2012-4.pdf)). I've never personally seen a derivation of this, so following on from Dinesh's very nice answer we can ... | 4 | https://mathoverflow.net/users/3547 | 96590 | 56,442 |
https://mathoverflow.net/questions/95726 | 11 | [This MathOverflow question](https://mathoverflow.net/questions/36358/computing-the-mertens-function) seems to indicate that the state of the art in computing
$$
M(x)=\sum\_{n\le x}\mu(n)
$$
takes time $\Theta(n^{2/3}(\log\log n)^{1/3}),$ which matches my understanding. Recently I came across a paper [1] which gives, a... | https://mathoverflow.net/users/6043 | Mertens' function in time $O(\sqrt x)$ | This is really a response to Dror Speiser's comment (but it is too long to give as a comment), and gives an analytic $O(n^{1/3+\epsilon})$ time and $O(n^{1/6+\epsilon})$ space algorithm to count the number of square free numbers less than $x$ (thus improving on the complexity of the algorithm given in the Jakub Pawlewi... | 10 | https://mathoverflow.net/users/10811 | 96602 | 56,450 |
https://mathoverflow.net/questions/92529 | 1 | How does one show that the polynomial system $F(x) = 0,$ where $F:\mathbb{C}^n \rightarrow \mathbb{C}^n,$ has only isolated roots?
| https://mathoverflow.net/users/22435 | Polynomial System has only isolated solutions | I'm a bit late to the party, but here are five answers, not using Groebner basis:
(1), if you can show that the zero set of $F$ is local complete intersection, then the local dimension must all be zero, i.e., the zero sets are geometrically isolated sets of points. But the converse is not true.
(2), if you weaken y... | 1 | https://mathoverflow.net/users/21522 | 96605 | 56,451 |
https://mathoverflow.net/questions/96583 | 15 | So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$.
For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney classes, one can prove when $n$ not of the form $2^k - 1$, it can not be embedded in $\mathbb{R}^{n+1}$. I would appreciate i... | https://mathoverflow.net/users/4760 | Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1 | I'll use cohomology with coefficients $\mathbb{Z}/2$ everywhere.
Suppose that the space $P=\mathbb{R}P^{n-1}$ embeds in $S^{n}$ (where $n>2$). Recall that
$$ H^\*(P)=(\mathbb{Z}/2)[x]/x^{n} = (\mathbb{Z}/2)\{1,x,\dotsc,x^{n-1}\} $$
By examining the top end of the long exact sequence of the pair $(S^{n},P)$ we find ... | 29 | https://mathoverflow.net/users/10366 | 96606 | 56,452 |
https://mathoverflow.net/questions/96591 | 1 | Let $X$ be a compact Hausdorff space. If $X$ is extremally disconnected then the complement of every point $x$ in $X$ is C\*-embedded in $X$ (i.e every continuous bounded real-valued function on $X\setminus\{x\}$ extends to a continuous function on $X$) because every open subset of $X$ is C\*-embedded [Gillman and Jeri... | https://mathoverflow.net/users/22260 | F-spaces and points whose complements are C*-embedded | Consider the $F$-space $\omega^\* $ (the Cech-Stone remainder of $\omega$).
Under the Continuum Hypothesis there is no point whose complement is $C^\* $-embedded. On the other hand it is also consistent that the complement of *every* point of $\omega^\* $ is $C^\* $-embedded. See [this paper](http://dutiaw37.twi.tudelf... | 0 | https://mathoverflow.net/users/5903 | 96616 | 56,458 |
https://mathoverflow.net/questions/96620 | 6 | Let $X \to Y$ be a family of hypersurfaces in a constant $\mathbb{P}^n$, i.e. $X \subset Y \times \mathbb{P}^n$ is locally on $Y$ given by one equation of degree $d$ in $\mathbb{P}^n$.
Is $X \to Y$ automatically flat? I know that it is so if $Y$ is reduced, since in this case the fact that the Hilbert polynomial of $... | https://mathoverflow.net/users/23194 | Flatness for family of hypersurfaces | (Of course, you have the implicit assumption that the equation of degree $d$ is not $0$.) The answer is yes. In the case where $Y$ is locally Noetherian, it is true by the "slicing criterion for flatness on the source", as $\mathbb{P}^n\_Y \rightarrow Y$ is flat. See Exercise 25.6.F in the May 12 2012 version of <http:... | 11 | https://mathoverflow.net/users/299 | 96624 | 56,459 |
https://mathoverflow.net/questions/96548 | 7 | Grothendieck seemed to try to eliminate Noetherian conditions as much as possible in EGA.
For example, he developed the cohomology theory of schemes without Noetherian conditions.
On the other hand, the Hartshorne's book assumes Noetherian conditions to do it.
If one's main interest is in the geometry of varieties or s... | https://mathoverflow.net/users/37646 | On elimination of Noetherian conditions in the geometry of schemes | For the most part, the answer is "no"; most people who work with varieties will never need to worry about non-Noetherian rings. But there are reasons to be open to the non-Noetherian setting. First, they can actually come up (as pointed out in the comments). As just one example, the normalization of a Noetherian ring c... | 19 | https://mathoverflow.net/users/299 | 96625 | 56,460 |
https://mathoverflow.net/questions/96642 | 11 | Does anyone know the current progress in showing the Riemann hypothesis? I was only able to find [this](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-20/issue-1/At-least-two-fifths-of-the-zeros-of-the-Riemann/bams/1183554910.full) paper of Conrey that says at least 4... | https://mathoverflow.net/users/22095 | Current Status of the Riemann Hypothesis | In terms of fraction of zeros on the critical line (which seems to be your question), the best result to date is 41.05% by Bui, Conrey and Young ("[More than 41% of the zeros of the zeta function are on the critical line](https://arxiv.org/abs/1002.4127)" arXiv:1002.4127). Of course, this is only one measure of progres... | 14 | https://mathoverflow.net/users/630 | 96643 | 56,468 |
https://mathoverflow.net/questions/96649 | 0 | Let $k$ be a $p$-adic field with ring of integers $\mathcal{O}\_K$ and residue field $\mathbb{F}$. Say I have a (projective) quadric $Q$ which is smooth over $\mathcal{O}\_K$, such that the reduction $\bar{Q}$ (smooth over $\mathbb{F}$) contains a line $\cong \mathbb{P}^1$ defined over $\mathbb{F}$. Does it follow that... | https://mathoverflow.net/users/1107 | Smooth quadric over p-adic integers | Yes, for the reason you give. The Hilbert scheme of lines in $Q$ is smooth over $\mathcal O\_K$ (the obstruction to smoothness lies in $H^1$ of a normal bundle $\mathcal N$; the homogeneity of $Q$ shows that $\mathcal N$ is generated by $H^0$, so has $H^1=0$ from the classification of bundles on $\mathbb P^1$). Now app... | 8 | https://mathoverflow.net/users/8726 | 96653 | 56,473 |
https://mathoverflow.net/questions/96659 | 3 | A finite $p$-group $G$ always has a non-trivial center. However, there exist infinite $p$-groups with trivial center, see for example this question [Simple(st) example of an infinite $p$-group with trivial center](https://mathoverflow.net/questions/49349/simplest-example-of-an-infinite-p-group-with-trivial-center). My ... | https://mathoverflow.net/users/12402 | Infinite p-groups with no normal subgroups. | The group $G$ described in " [Simple(st) example of an infinite $p$-group with trivial center](https://mathoverflow.net/questions/49349/simplest-example-of-an-infinite-p-group-with-trivial-center) " is in fact an ICC group. This means that every non-trivial element $g\in G$ has an infinite conjugacy class. As a consequ... | 7 | https://mathoverflow.net/users/2055 | 96660 | 56,478 |
https://mathoverflow.net/questions/96671 | 3 | The homology cylinder group is the monoid of homology cylinders modulo homology cobordism. I wonder whether there is any known finite order element in this group.
| https://mathoverflow.net/users/6569 | Is there a torsion element in the homology cylinder group? | Certainly there is a lot of $2$-torsion. One easy way to construct such an example is to look at the embedding of the string link cobordism group on $n$ strands into the homology cobordism group of genus $n$. (Levine explains this embedding in his paper "Homology cylinders: an enlargment of the mapping class group." Se... | 7 | https://mathoverflow.net/users/9417 | 96672 | 56,483 |
https://mathoverflow.net/questions/96667 | 8 | I am trying to figure out why the transversal matroid is a matroid.
Specifically, if $L$ and $M$ are two independent sets s.t. $|L| < |M|$, why is there an $i$ in $M \setminus L$ s.t. $L \cup \{ i \} $ is independent?
thank you in advance!
| https://mathoverflow.net/users/23624 | Transversal Matroid | Let me check that we have the same definition of transversal matroids. For me, the input is a bipartite graph $\Gamma$, with white vertex set $W$ and black vertex set $B$. The ground set of the matroid is $B$, with a subset of $B$ being independent if it can be matched to a subset of $W$.
**Direct Proof that this is ... | 9 | https://mathoverflow.net/users/297 | 96676 | 56,484 |
https://mathoverflow.net/questions/96219 | 9 | I suppose most of you are familiar with the [Mathematics Subject Classification (MSC)](http://www.ams.org/mathscinet/msc/msc2010.html). Particularly, when submitting an article for publication one has to
choose appropriate classification codes.
But I am wondering if it does play a significant role in searching for
l... | https://mathoverflow.net/users/23509 | Do you use the Mathematics Subject Classification (MSC) when searching for literature? | At first, I thought this was a silly question, and it seemed highly implausible that searching by MSC code could actually be good for anything. Certainly it's a terrible way to find specific information, but I just gave it a try and it's quite a bit more useful than I thought. I put 52C17 in MathSciNet and got a list o... | 22 | https://mathoverflow.net/users/4720 | 96683 | 56,490 |
https://mathoverflow.net/questions/90831 | 5 | For a Lie algebra $L$ of dimension $n$ over a field ${\mathbb F}$ we denote by $\beta(L)$ the maximal dimension of abelian ideals of $L$. In general, $\beta(L)$ is not preserved under extensions of the ground field (see e.g. Example 2.7 in <http://homepage.univie.ac.at/dietrich.burde/papers/burde_39_max_ab.pdf>).
Do y... | https://mathoverflow.net/users/14653 | Maximal dimension of abelian ideals of a Lie algebra and extensions of the ground field | There's no such example.
Since this is convenient, I denote by $L$ the Lie algebra over the algebraic closure. Let $A$ be a codimension 1 abelian ideal and let us show that some (possibly other) abelian ideal $A'$ is defined over the ground field, i.e. is a hyperplane that can be defined by a linear equation with coe... | 4 | https://mathoverflow.net/users/14094 | 96712 | 56,507 |
https://mathoverflow.net/questions/96711 | 5 | Suppose $f\_1,f\_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x\_1^\*,x\_2^\*$ be their respective (not necessarily unique) minima, and let y be a minima of $f$ (once again need not be unique). Can we prove that there exists a version of $x\_1^\*,x\_2^\*,\ldots$ such that... | https://mathoverflow.net/users/12586 | Does the minima of a sequence of convex convergent functions converge? | No; here's a counterexample: let $f = 0$ and consider the minimizer $y = 0.$ Then you can construct convex functions which converge to $0$ pointwise but whose minima are always moving away from $y =0,$ e.g. $f\_n(x) = (x - n)^2/n^n.$
| 10 | https://mathoverflow.net/users/2586 | 96716 | 56,510 |
https://mathoverflow.net/questions/96703 | 8 | Consider a Brownian particle in the plane with a circular trap at the origin. If we give the particle enough time it falls into the trap (since Brownian motion is space filling in 2D). However, suppose we only let the particle evolve for a finite interval of time. What is the probability it is trapped?
Scaling time ... | https://mathoverflow.net/users/6781 | Finite time hitting probabilities for Brownian motion in the plane | Instead of a concrete answer, I will give what appears to be the most useful reference. I quote the first paragraph of [Wendel, J. G. "Hitting spheres with Brownian motion". *Ann. Prob.* **8**, 164 (1980)](https://projecteuclid.org/journals/annals-of-probability/volume-8/issue-1/Hitting-Spheres-with-Brownian-Motion/10.... | 10 | https://mathoverflow.net/users/1847 | 96719 | 56,512 |
https://mathoverflow.net/questions/96720 | 6 | Informally, Löb's theorem ([Wikipedia](https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem), [PlanetMath](https://planetmath.org/lobstheorem)) shows that:
>
> a mathematical system cannot assert its own soundness without becoming inconsistent [[Yudkowsky]](https://www.lesswrong.com/posts/ALCnqX6Xx8bpFMZq3/the-cartoo... | https://mathoverflow.net/users/20215 | Non-trivial consequences of Löb's theorem | Löb's theorem provides the essential ingredient for a complete axiomatization of propositional provability logic. In detail: Work with the usual notation of propositional modal logic, which has propositional variables, the usual Boolean connectives, and the unary modal operator $\square$. The usual reading of $\square ... | 13 | https://mathoverflow.net/users/6794 | 96722 | 56,514 |
https://mathoverflow.net/questions/96715 | 3 | In the complex plane, say $a\_n \rightarrow \infty$ and $d\_n$ and $A\_n$ are arbitrary complex numbers. can we find an entire function with $f(a\_n)=A\_n$ with order $d\_n$? (here "order" means $f(z)-A\_n$ has a zero with order $d\_n$)
If without restrictions on the order, it is an exercise from Ahlfors 3rd edition... | https://mathoverflow.net/users/12904 | entire functions of one complex variable with prescribed value and order. | Yes this is possible. See Theorem 15.13 in Rudin: Real and Complex Analysis:
>
> If $\Omega \subseteq \mathbb C$ is open and $A \subseteq \Omega$ has no limit point in $\Omega$, and to each $a \in A$ there is an associated integer $m(a)$ and complex numbers
> $w\_{n,a}\,(0 \le n \le m(a))$. Then there exists a f... | 7 | https://mathoverflow.net/users/10194 | 96725 | 56,516 |
https://mathoverflow.net/questions/96701 | 10 | In 1958, Smale proved that a $2$-sphere can be "turned inside out", and throughout the 60s, 70s, and 80s, explicit constructions such as Thurston Corrugations, and Minimax eversions were developed to visualize this sphere eversion. I just encountered a recent youtube video: [Torus Eversion](http://www.youtube.com/watch... | https://mathoverflow.net/users/20343 | Surface Eversions: Generalizing from Sphere and Torus Eversions | I'll expand on my comment above. Consider an eversion of a sphere, namely a path of immersions $e\_t: S^2\to \mathbb{R}^3$ such that $e\_0$ and $e\_1$ are embeddings, and which exchanges the two sides. By possibly reparameterizing the maps $e\_t$, we may assume that there exists $D^2\subset S^2$ such that $e\_{t|D^2}: ... | 10 | https://mathoverflow.net/users/1345 | 96731 | 56,521 |
https://mathoverflow.net/questions/96675 | 1 | Hello!
Let $G=A\underset{C}\star B$ be an amalgamated Product.
Let $a\in A$. If a is conjugated to an Element $b\in B$, then $a$ is conjugated to an Element $c\in C$. The Question is: Why is that true? It is clear, when $a\in A\cap C=C$.
It seems to be very easy. But at the moment, i think, i make a fault while im c... | https://mathoverflow.net/users/23627 | Conjugated elements in amalgameted Product | There is also a simple geometric proof using Bass-Serre theory, which does not require much in the way of calculation. Let $G$ act on the Bass-Serre tree $T$ of the amalgamated free product $A\*\_CB$. The stabilizer of every edge of $T$ is conjugate to $C$, and the vertices of the tree $T$ are partitioned into two subs... | 3 | https://mathoverflow.net/users/20787 | 96733 | 56,523 |
https://mathoverflow.net/questions/96386 | 14 | As from this website
<http://math.uchicago.edu/~lxiao/workshop_site/>
My question is: What does it mean by **"purely local"**?
Also, I heard about this phrase "purely local" in other problems as well, mostly with the phrase "a purely local proof".
The other question is, for GL(1) and GL(2), are there already a "p... | https://mathoverflow.net/users/1238 | "Purely local" proof of local Langlands | The short answer to the question is that all currently known proofs of the local Langlands correspondence (and I'm just referring to GL(n) here) are "global" in the sense that they involve embedding the local problem into a global one. That is, the local field in question is realized as the completion of a global field... | 19 | https://mathoverflow.net/users/271 | 96734 | 56,524 |
https://mathoverflow.net/questions/96680 | 1 | subj: etale covers of line bundles on an abelian variety
Is there an explicit decryption of finite
*etale covers* of a line bundle $L$ on an abelian variety and its associated C\*-bundles
$L^o = L \setminus A\times {0}$ (i.e. the C\*-bundle $L^o$ is $L$ without the zero section) ?
Pull-back along multiplication by... | https://mathoverflow.net/users/22247 | etale covers of line bundles on an abelian variety | For clarity, the best way to work with this is complex-analytically. I am sure there is a, probably more involved, algebraic proof.
Lemma: Let $M$ be a complex manifold and let $X$ be a $\mathbb C^\times$-bundle on $X$. Let $Y$ be a finite etale cover of $X$. Then $Y$ is a $\mathbb C^\times$ bundle on an etale cover ... | 1 | https://mathoverflow.net/users/18060 | 96737 | 56,526 |
https://mathoverflow.net/questions/96745 | 0 | Correct me if I'm wrong but can't the nearest neighbor algorithm be used to find a Hamiltonian Circuit in an arbitrary graph and hence proved P = NP?
| https://mathoverflow.net/users/22495 | Finding a Hamiltonian Circuit using Nearest-neighbor algorithm | The nearest neighbor algorithm as I understand it (repeatedly select a neighboring vertex that hasn't been visited yet and travel to that vertex) does not guarantee that you will find a circuit even if one exists. For example consider the graph with vertices A,B,C,D with edges AB, AC, AD, BC and CD (a complete graph on... | 2 | https://mathoverflow.net/users/3669 | 96746 | 56,529 |
https://mathoverflow.net/questions/96743 | 2 | Let a group $G$ act on a finite set $\Omega$. Suppose that the corresponding permutation character has a regular component. Does it follow that $\Omega$ has a regular $G$-orbit?
(The converse is obviously true.)
| https://mathoverflow.net/users/12961 | Permutation characters and regular orbits | No, this is false in general, and the smallest counterexample is the Klein 4-group $G=C\_2\times C\_2$. It can act transitively on a set of order 2 in three different ways, with 3 different $C\_2$'s in the kernel. Call these sets $\Omega\_1, \Omega\_2, \Omega\_3$, and their permutation characters $1+\chi\_1, 1+\chi\_2,... | 6 | https://mathoverflow.net/users/3132 | 96747 | 56,530 |
https://mathoverflow.net/questions/96750 | 3 | Does every element of the derived series of a pro-p group is also a pro-p group?
The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true?
Hope you'll be able to help me
Thanks !
| https://mathoverflow.net/users/20568 | Derived Series of Pro-p groups | It seems this can't be expected. From [Simons' thesis](http://ora.ouls.ox.ac.uk/objects/uuid%253A01075c36-c7e6-4def-9647-86b4346e4726/datastreams/THESIS01) (p. xii):
>
> The derived group of any finitely generated profinite group is closed, but Roman'kov [29] has provided an example of a finitely generated pro-p g... | 7 | https://mathoverflow.net/users/10194 | 96756 | 56,534 |
https://mathoverflow.net/questions/96760 | 3 | Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. (This last assumption might not be necessary.)
Now, suppose that $F(R)$ is a finitely generated $R$-module for all $k$... | https://mathoverflow.net/users/22189 | Schemes associated to vector spaces | Since $F(k) \cong k^n$ for some $n$, we have $F(R) \cong R^n$, naturally in $R$. But $R^n \cong (\mathbb{A}^n)(R)$, so that $F$ is just the scheme $\mathbb{A}^n$. This is also isomorphic to $\mathrm{Spec}(\mathrm{Sym}(F(k))$.
EDIT: More generally and coordinate free: When $S$ is some base scheme and $\mathcal{E}$ is ... | 7 | https://mathoverflow.net/users/2841 | 96762 | 56,535 |
https://mathoverflow.net/questions/96726 | 7 | Suppose $G$ is a finitely generated group, with given generating set $S={g\_1, \dots, g\_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (**EDIT:** Also assume that $S$ is minimal in the sense that no proper subset of $S$ is generating.)) Given complex numbers $a\_1, \dots , a\_n$, we can form the element $\sum a... | https://mathoverflow.net/users/13360 | Sums of unitaries with small norm in full group $C^*$-algebras | Here is an algebraic characterization of $\alpha(S)=1$: if $S=\{g\_1,\dots,g\_n\}$, $\alpha(S)=1$ if and only if $\{g\_1^{-1}g\_k,k=2\dots n\}$ generate a free abelian group in the abelianization $G^{ab}$ of $G$.
In fact, since $\alpha(S)$ is unchanged under the action of $G$ by left (or right) multiplication, it is ... | 6 | https://mathoverflow.net/users/10265 | 96764 | 56,537 |
https://mathoverflow.net/questions/96742 | 13 | I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be turned into group schemes, allowing for a standard treatment with deformation theory.
Are there any results from (the de... | https://mathoverflow.net/users/11084 | Lie groups vs. algebraic groups and deformations | If you take, say, set of real points of the group-scheme $O(n)$, i.e., $O(n, {\mathbb R})$, then you recover the usual orthogonal (real Lie) group, which you know as $O(n)$. Same applies to $SL(n)$, etc. There is one case when this does not work well, namely when you deal with character varieties. For instance, take $\... | 15 | https://mathoverflow.net/users/21684 | 96767 | 56,539 |
https://mathoverflow.net/questions/96777 | 20 | Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)?
Q2: Do we have a sufficient criterion for a general field $K$ of characteristic $0$ which guarantees that if $K(x\_1,\ldots,x\_n)\simeq L(x\_1,\ldots, x\_n)$ (he... | https://mathoverflow.net/users/11765 | non-isomorphic stably isomorphic fields | I don't think that there are any really easy examples. In the famous paper of Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer "Variétés stablement rationnelles non rationnelles" they construct surfaces $S$ over $\mathbb Q$ that are not rational, but such that the products $S \times \mathbb P^3$ are rational. You... | 14 | https://mathoverflow.net/users/4790 | 96781 | 56,545 |
https://mathoverflow.net/questions/96780 | 8 | By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom.
I have heard (but never seen written) that these assumptions imply paracompactness (and thus the existence of a Riemannian metric by the well-known construction using Partition of unity).
Does anybody know a refer... | https://mathoverflow.net/users/39082 | Manifolds are paracompact | Theorem: A countable atlas of charts for a Hausdorff $n$-manifold $M$ can be refined to a locally finite atlas. In fact, each chart only needs to be trimmed.
Proof: Let $U\_1,U\_2,\ldots$ be the charts. Each $U\_i$, as a subset of $\mathbb{R}^n$, is the limit of a nested sequence of compact subsets $K\_{i,1} \subsete... | 27 | https://mathoverflow.net/users/1450 | 96783 | 56,546 |
https://mathoverflow.net/questions/96787 | 4 | I can try to define an averaging operator for functions, namely let
$$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$
by
$$Af = \lim\_{N\to\infty} \frac{1}{N}\int\_0^N f(x)dx$$
whenever the limit converges. My question is: is there any intuitive description of the domain $D$ (other than the above limit converging)... | https://mathoverflow.net/users/23655 | What is the domain of the "average operator"? | I would say that you can define $A$ in all $L^\infty$. Indeed, let $V$ be the set of essentially bounded functions such that your limit exists. $V$ is a linear subspace of $L^\infty$ and the operator $A$ is linear and continuous and then it admits an extension to all of $L^\infty$ by the Hahn-Banach theorem. This exten... | 6 | https://mathoverflow.net/users/13809 | 96790 | 56,548 |
https://mathoverflow.net/questions/96789 | 9 | Assume we have a Riemann surface, the underlying topological surface of which is a sphere with (possibly uncountably many) points removed. Can we always conformally embed this Riemann surface in the Riemann sphere? If not, can someone suggest a counter example?
| https://mathoverflow.net/users/22338 | Embedding a Riemann surface in the sphere | See e.g. [here](http://www.math.tifr.res.in/~pablo/download/book/chp3.ps):
Theorem 3.2.7. Any planar connected Riemann surface is biholomorphic to an open subset of $S^2$.
The proof is very straightforward: Exhaust a genus $0$ surface $S$ by relatively compact domains $D\_n$ each of which necessarily has genus $0$... | 10 | https://mathoverflow.net/users/21684 | 96795 | 56,550 |
https://mathoverflow.net/questions/96806 | 5 | I am interested in the collection of possible values for permanents over square binary matrices. Consider $n \times n$ 0-1 matrices. The possible permanents for $n=1$ are $\{0,1\}$. For $n=2$ the possible permanents are $\{0,1,2\}$, and for $n=3$ the permanents are $\{0,1,2,3,4,6\}$. Note that $5$ is missing for $n=3$.... | https://mathoverflow.net/users/23662 | Set of permanents over binary square matrices | That set consists of all natural numbers. Consider the $n\times n$-matrix $A=(a\_{ij})$ where all numbers on the diagonal, first row and first column are equal to 1, all the other entries are 0. Then the permanent of that matrix is $n$. Indeed, each non-zero summand in the definition of permanent should start with some... | 8 | https://mathoverflow.net/users/nan | 96808 | 56,555 |
https://mathoverflow.net/questions/96749 | 1 | Let $\hat{R}$ is $m$-adic completion of a local ring $(R,m)$.What is the relation between $Min R$ and $Min \hat{R}$. we know that $\hat{R}$ is faithfully flat $R$-module.
$Min R$=set of all minimal prime divisors of zero.
I think
$p\in Min(R)$ iff $\hat{p}=p\hat{R}\in Min\hat{R}$.
| https://mathoverflow.net/users/18970 | relation between Min(R) and Min(R^) | What you are sayinng is not true in general. For example, the completion of a domain may not be a domain. An example is given in exercises of Bourbaki's Commutative Algebra.
| 1 | https://mathoverflow.net/users/16046 | 96811 | 56,557 |
https://mathoverflow.net/questions/96779 | 3 | This is a question on the proof of this fact in chapter 3 of the book "Functional Analysis: Surveys and Recent Results II". There at the end the proof is outlined as follows:
Let $\mu$ be a shift invariant measure on $X=\{0,1\}^\mathbb{Z}$. For each subset $\Lambda\subset\mathbb{Z}$ define $C\_\Lambda$ to be the subs... | https://mathoverflow.net/users/23644 | Understanding a proof that the simplex of shift invariant probability measures on $\{0,1\}^\mathbb{Z}$ is Poulsen? | The language of the proof given in the book you refer to is a little different from the language I'm accustomed to, but I'll give what I believe is the exact same argument using a slightly different language, and hopefully do it in such a manner that the issue you point out doesn't arise. (Since I'm not quite sure how ... | 3 | https://mathoverflow.net/users/5701 | 96816 | 56,560 |
https://mathoverflow.net/questions/96818 | 0 | We know that for an immersion $j:U \to X$ the restriction functor $j^\*:{\cal O}\_X-mod \to {\cal O}\_u-mod$ has a left adjoint $j!$.
I am looking for some condition to deduce that $j!$ takes its values in Qco(X) that is to be a left adjoint for the functor $j^\*:Qco(X) \to Qco(u)$.
| https://mathoverflow.net/users/23200 | left adjoint to restriction functor | The restriction functor $\mathrm{Qcoh}(X) \to \mathrm{Qcoh}(U)$ doesn't preserve infinite products in general (which always exist, by the way). Therefore it cannot have a left adjoint.
| 2 | https://mathoverflow.net/users/2841 | 96823 | 56,563 |
https://mathoverflow.net/questions/96812 | 1 | A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.
In particular, let $G$ be a Lie group acting on a manifold $M$. Let $\underline G$ be the associated group scheme acting on the scheme $\underline M$, where the $\mathb... | https://mathoverflow.net/users/11084 | Proper morphisms: Lie groups vs. group schemes | A morphism of schemes over $\mathbb C$ is proper iff the map on topological spaces is proper, so the whole thing works for complex points if the fibred products are over $\mathbb C$ as well.
Over the reals it goes wrong as the following example shows:
Let $G$ be the group scheme $SO(2)$ and let $M$ be a single point. T... | 3 | https://mathoverflow.net/users/nan | 96831 | 56,568 |
https://mathoverflow.net/questions/96830 | 5 | Let $f:X \to Y$ be a finite map of (smooth, compact) complex algebraic varieties.
Then a map $f^\*$ is defined at the level of Chow and cohomology rings. Say that for simplicity we work with rational coefficients.
Question: is $f^\*$ injective? I know this is true when $f$ is the natural map onto the quotient via t... | https://mathoverflow.net/users/23434 | injectivity of the pull-back via a finite map | Let $f:X\to Y$ be a finite surjective map of degree $d$ between smooth projective varieties of dimension $n$. Then $f$ is flat (apply EGA IV 2 Prop 6.1.5), so Example 1.7.4 of Fulton's Intersection Theory shows that $f\_{\ast}\circ f^{\ast}$ is multiplication by $d$ on the Chow groups $CH^\*(Y)$.
Hence the kernel of $f... | 13 | https://mathoverflow.net/users/2868 | 96834 | 56,570 |
https://mathoverflow.net/questions/96817 | 4 | Suppose we have a graph on $n$ nodes. We would like to assign to each node either a $+1$ or a $-1$. Call this a configuration $\sigma \in \{+1,-1\}^n$. The number of $+1$s that we have to assign is exactly $s$ (hence the number of $-1$s is $n-s$.) Given a configuration $\sigma$, we look at each node $i$ and sum the val... | https://mathoverflow.net/users/36687 | An optimization involving (random) graphs | OK. Let me make a wild guess. I'll assume that you're looking at a random graph with $p=(\log n)/n$ and $s=n/4$ for concreteness.
Since for $p$ in that range, there are very few small cycles, the graph is behaving quite a lot like a tree with branching number $\log n$ at each vertex.
An easy lower bound for the nu... | 1 | https://mathoverflow.net/users/11054 | 96836 | 56,572 |
https://mathoverflow.net/questions/96794 | 2 | Can anyone tell me the origin &/or original applications of Smirnov's Deleted Sequence topology? ([This is #64 in Steen & Seebach's *Counterexamples in Topology*](https://i.stack.imgur.com/tCCAD.png).) Thanks.
| https://mathoverflow.net/users/12606 | Smirnov's Deleted Sequence topology | The example appears already in Alexandroff and Hopf's Topologie (1935) in Chapter 1, section 1, number 4, Example 1.
It served as an example of a Hausdorff non-regular space.
| 1 | https://mathoverflow.net/users/5903 | 96848 | 56,580 |
https://mathoverflow.net/questions/96850 | 8 | Let $L$ be a Lie algebra over a field $F$ and denote by $U(L)$ the universal enveloping algebra of $L$. Regard $U(L)$ as a Lie algebra with respect to the Lie bracket $[a,b]=ab-ba$ for any $a,b\in U(L)$. If $U(L)$ is solvable as a Lie algebra, is $L$ necessarily abelian?
| https://mathoverflow.net/users/23674 | Enveloping algebras which are solvable as Lie algebras | If $F$ has characteristic different from 2 the answer is yes. This follows from Corollary 6.1 in the paper by D. Riley - A.Shalev: The Lie structure of enveloping algebras, J. Algebra 162, 46-61 (1993).
On the other hand, in characteristic 2 this conclusion is false. For instance, if $L$ is a 2-dimensional nonabelian... | 12 | https://mathoverflow.net/users/14653 | 96851 | 56,581 |
https://mathoverflow.net/questions/96586 | 13 | A type $II\_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x\_{1}, x\_{2},..., x\_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $M$ with $\tau (u)=0$ and $||ux\_{j}-x\_{j}u||\_{2}<\epsilon$ for all $1 \leq j \leq n$. (Here $||T||\_2=(\tau(T^{\*}T))... | https://mathoverflow.net/users/6269 | Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$? | Yes, it has property $(\Gamma)$. This follows from Stalder's proof of inner amenability plus a fact that the semigroup $\langle T\_m, T\_n \rangle$ admits an approximately invariant subsets having proportional measures. Here, $T\_m$ is the $m$-times map on $[0,1)$, $T\_m x = mx \mod 1$. As Stalder proves, $\sum\_{1\le ... | 11 | https://mathoverflow.net/users/7591 | 96857 | 56,582 |
https://mathoverflow.net/questions/96864 | 2 | I am looking for an algorithm for checking the satisfiability (with natural values) of a set of inequalities made of variables and natural numbers, for example: $u < v, u \leq z, 3 \leq v$.
In this paper, <http://boole.stanford.edu/pub/sefnp.pdf> , there's the following claim in the second page:
>
> "Then the con... | https://mathoverflow.net/users/22861 | Algorithm for satisfiability of inequalities. | This looks very much like the "difference constraints" that are explored in the big white textbook "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. The graph seems about the same and there is a theorem about negative weight cycles. I suggest you will find what you need there.
| 3 | https://mathoverflow.net/users/9025 | 96865 | 56,586 |
https://mathoverflow.net/questions/96868 | 5 | I'm seeking a reference or a sketch for any sort of normal form that would enable rapid enumeration without redundancies of the elements of hyperbolic triangle groups and/or von Dyck groups.
| https://mathoverflow.net/users/10909 | Normal form(s) for the elements of hyperbolic triangle groups | [This](http://www.math.rwth-aachen.de/~Gerhard.Hiss/Students/DiplomarbeitPfeiffer.pdf) thesis contains a complete rewriting system for every triangle group (see Section C at the end of the thesis). Then normal forms are just words that do not contain left parts of the rewriting rules (which is just a finite set of word... | 9 | https://mathoverflow.net/users/nan | 96869 | 56,587 |
https://mathoverflow.net/questions/96854 | 8 | I am trying to derive a variational formulation for the following problem $$\left\{ \begin{array}{ll} \Delta^2u=f, & \Omega \\ \Delta u+\rho \partial\_{\nu}u=0, & \partial \Omega \end{array}\right.$$
where $\rho>0$ is constant. I intend to show that the right functional setting is $H^2(\Omega)\cap H^1\_0(\Omega)$ and... | https://mathoverflow.net/users/18013 | Variational formulation for bilaplacian | To begn with, your Boundary-Value Problem (BVP) is under-determined, because it lacks *one* boundary condition: because the PDE is elliptic and *fourth*-order, you need **two** boundary conditions, not only one. Because you insist on working with $H^2\cap H^1\_0$, it seems that the hidden BC is
$$u=0\qquad\hbox{on }\pa... | 9 | https://mathoverflow.net/users/8799 | 96870 | 56,588 |
https://mathoverflow.net/questions/96867 | 9 | I just finished a first course in Descriptive Set Theory using Kechris' "Classical Descriptive Set Theory" and was hoping to find a good source for learning some of the Effective DST. Kechris doesn't mention it at all, and after looking at Moschovakis' "Descriptive Set Theory", it doesn't seem like the easiest thing to... | https://mathoverflow.net/users/6342 | Good source for Effective Descriptive Set Theory | A nice source is "Recursive aspects of descriptive set theory" by Mansfield and Galen Weitkamp, as mentioned by Yu in a comment. A problem with it is that it leaves out all the details of admissibility and its relatives. I also find it a bit more elementary than one would want, but it is a good initial source.
An exc... | 14 | https://mathoverflow.net/users/6085 | 96873 | 56,590 |
https://mathoverflow.net/questions/96849 | 2 | I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:
>
> "We shall denote by $f\_\pi$ a pseudo-coefficient for $\pi$, although it is highly non
> unique. But as regards invariant harmonic analysis this plays no role. In
> particular... | https://mathoverflow.net/users/10400 | Pseudo coefficients and orbital integrals | Existence of pseudo-coefficients for square-integrable representations (and the link with character values of the representations) is stated and proved in
D. Kazhdan, Cuspidal geometry of $p$-adic groups.
J. Analyse Math. 47 (1986), 1–36.
| 3 | https://mathoverflow.net/users/4767 | 96881 | 56,595 |
https://mathoverflow.net/questions/96875 | 3 | Let $G = (V, E)$ be a simple, undirected graph. We consider the following two definitions of graph connectedness:
(1) $G$ is connected if for $x,y \in V$ there is a finite path connecting $x$ and $y$.
(2) If $A \subseteq V$ such that $A \neq \emptyset$ and $A \neq V$ there is $e\in E$ such that $e \cap A \neq \empt... | https://mathoverflow.net/users/8628 | Two definitions of graph connectedness | To be more explicit, fix a vertex a and let A0 be the singleton set having a as a member. Using the mechanism of (2) and some version of the axiom of choice, define An+1 by adding the one vertex that is guaranteed to be adjacent to but not in the vertex set An. Take the union of these sets and call it A. Now either A i... | 1 | https://mathoverflow.net/users/3568 | 96911 | 56,609 |
https://mathoverflow.net/questions/96924 | 6 | For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further details, Sharpe refers the reader to B. Perron's "Pseudo-Isotopies de plongements en codimension 2", another reference ... | https://mathoverflow.net/users/20516 | Reference request: embedded Morse theory | I found the treatment in Goresky-MacPherson's book "Stratified Morse theory" (available from Goresky's webpage [here](http://www.math.ias.edu/~goresky/pdf/SMT.djvu)) very enlightening. The focus is not on low-dimensional topology, but the treatment is very geometric.
| 3 | https://mathoverflow.net/users/317 | 96929 | 56,622 |
https://mathoverflow.net/questions/96887 | 2 | I have read few textbooks and papers about the $K$-theory groups $K\_{0}$ and $K\_{1}$ of (reduced) $C^{\*}$-algebra and most of them didn't give a clear simple way to define these groups.
Just wondering if anybody can give me good sources for that?
| https://mathoverflow.net/users/9842 | The definition of the $K$-theory groups $K_{0}$ and $K_{1}$. | I also think this question would benefit from some more detail, however...
1. A very down-to-earth account of K-theory for operator algebras is in the book "K-Theory and C\*-algebras" by N.E. Wegge-Olsen.
2. A more sophisticated approach is in the book "Analytic K-Homology" by Nigel Higson and John Roe.
If you pref... | 4 | https://mathoverflow.net/users/703 | 96937 | 56,626 |
https://mathoverflow.net/questions/96793 | 4 | Let us consider the Dirac complex
\begin{equation}
D\_{\rm Dirac}:S^+\to S^-
\end{equation}
where $S^{\pm}$ are the chiral-spinor bundles on $\mathbb{R}^4$.
Using the fact that the bundle $S^+$ is given by $\Omega^{0,0} \oplus \Omega^{0,2}$ twisted by $K^{1/2}$ while $S^-$ is given by $\Omega^{0,1}$ twisted by $K^{1/2}... | https://mathoverflow.net/users/17644 | The equivariant index of Dirac operator | To see why the spinor bundle is the bundle $\Omega^{0,\* }\otimes K^{1/2}$, you need to understand the relation between the spinor representation $S$ of $Spin(2n)$ and the exterior algebra representations $\Lambda^\* (\mathbf C^n)$ of $U(n)$.
If you choose an orthogonal complex structure $J$ on $\mathbf R^{2n}$, this... | 3 | https://mathoverflow.net/users/11670 | 96939 | 56,627 |
https://mathoverflow.net/questions/96891 | 2 | It is a basic fact of set theory that the following holds:
Let $(X,\leq)$ be a linear ordered set. Then it holds that, for each finite sequence $y\_{1}$,...,$y\_{n}$ of sets and each formula $\phi(x,y\_{1},...,y\_{n})$ of set theory, we have
$$\forall{y\_{1},...,y\_{n}}(\forall{x\in X}(\forall{z}\lt x\phi(z,y\_{1},... | https://mathoverflow.net/users/15814 | "Ill-founded Recursion" without Fund | Define $F(x,f,p)$ to be the smallest ordinal not in the range of $f$ (so the variables $x$ and $p$ are just dummy variables, to match the notation in the question). Suppose there exists a solution $f$ of the recursion $f(x)=F(x,f|X\_x,p)$. Then $f$ embeds $X$ (with its given ordering) strictly monotonically into the or... | 4 | https://mathoverflow.net/users/6794 | 96944 | 56,630 |
https://mathoverflow.net/questions/96940 | 0 | Suppose one has $n$ real random variables $X\_1, X\_2, \dots, X\_n$ from a certain distribution. Sort these random variables to get a sequence $Y\_1, Y\_2, \dots, Y\_n$. What is known about the distribution, mean, variance, higher moments of the random variables $Y\_i$?
To be more specific:
1) Is it true that there i... | https://mathoverflow.net/users/23712 | Rank $k$ of a sequence of random variables | The following papers estimate the variances of order statistics:
Yang, H. (1982) "On the variances of median and some other order statistics." Bull. Inst. Math. Acad. Sinica, 10(2) pp. 197-204
Papadatos, N. (1995) ["Maximum variance of order statistics."](http://www.ism.ac.jp/editsec/aism/pdf/047_1_0185.pdf) Ann. ... | 2 | https://mathoverflow.net/users/2954 | 96945 | 56,631 |
https://mathoverflow.net/questions/96914 | 17 | What are good/interesting examples of theorems than can be proven classically, but not constructively, and have applications in e.g. physics?
| https://mathoverflow.net/users/13729 | Applications of nonconstructive mathematics | In general it is very difficult to be sure that a theorem cannot be constructivised in some form that preserves its applicability. As you will notice most of the answers offered have comments attesting this fact.
One reason for this is that many non-constructive theorems in analysis become constructive when they are ... | 39 | https://mathoverflow.net/users/1176 | 96947 | 56,633 |
https://mathoverflow.net/questions/96931 | 6 | Let $K$,$L$,$M$ be convex lattice polytopes (so their vertices are in $\mathbb{Z}^n$) in $\mathbb{R}^n$ satisfying $K+L\subseteq M+M$ (Minkowski sum). Do we always have
$$|K\cap\mathbb{Z}^n|\cdot|L\cap\mathbb{Z}^n|\le |M\cap\mathbb{Z}^n|^2 ?$$
I looked at books/papers on Erhart polynomial and Brunn-Minkowski inequali... | https://mathoverflow.net/users/10332 | Inequality of arithmetic and geometric means for the lattice polytopes? | This inequality does not necessarily hold, at least for $n\geq 3$. It is somehow connected with the fact that there is no Pick's formula in more than two dimensions since there exists a convex lattice polytope with a large volume but containing a small number of lattice points.
So, for instance, for $n=3$ let $M$ be... | 6 | https://mathoverflow.net/users/17581 | 96956 | 56,641 |
https://mathoverflow.net/questions/96912 | 1 | Hello,
working on some machine learning problem I end up facing a problem which looks like generalizing the notion of [Cauchy product](http://en.wikipedia.org/wiki/Cauchy_product).
I briefly go back to Cauchy products before exposing my question. Consider, two sequences $(a\_n)\_{n \in \mathbb N}$ and $(b\_n)\_{n \... | https://mathoverflow.net/users/23705 | Generalization of Cauchy product | If the $\theta\_i$ are different (but still all $|\theta\_i| < 1 - \epsilon < 1$ for some fixed $\epsilon$), then
$$ \sum\_{n=0}^\infty \sum\_{k=0}^n \left|a\_k \prod\_{i=k}^n \theta\_i \right|
< \sum\_{k=0}^\infty \sum\_{n=k}^\infty |a\_k| (1-\epsilon)^{n-k+1} = \sum\_{k=0}^\infty |a\_k| \frac{1-\epsilon}{\epsilon}$... | 1 | https://mathoverflow.net/users/13650 | 96965 | 56,645 |
https://mathoverflow.net/questions/96963 | 1 | How can I translate a Büchi automaton A to LTL(linear temporal logic) if $L(A)$ is definable in the LTL?
MY idea is : Büchi automaton $A$ ===> QPTL ===> LTL
I know that given any Buchi automaton, we can translate it to QPTL(Quantified Propositional Temporal Logic), formally speaking, For every B¨uchi automaton A o... | https://mathoverflow.net/users/23716 | Translate a buchi automaton to LTL | You can always go through the $\omega$-semigroup. It might not be the most straightforward algorithm, but at least it should work.
You can find details in <http://www.lsv.ens-cachan.fr/Publis/PAPERS/PDF/DG-WT08.pdf>
The principle is to translate your automaton into an $\omega$-semigroup, via the transition matrices... | 3 | https://mathoverflow.net/users/21059 | 96994 | 56,656 |
https://mathoverflow.net/questions/72101 | 8 | Hello,
I want to ask if anyone can tells us what is known (consistently) about $Ded(\kappa)$, $\kappa$ an infinite cardinal.
**Definition** If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then we write $D(\kappa,\lambda)$. Then $Ded(\kappa)=\sup\_\lambda \{D(\k... | https://mathoverflow.net/users/13694 | Is $Ded(\kappa)<Ded(\kappa)^\omega$ consistent? | I just found the following paper on arXiv: "On non-forking spectra" by Artem Chernikov, Itay Kaplan and Saharon Shelah ( <http://arxiv.org/abs/1205.3101> ). They claim that it is consistent that $Ded(\kappa)< Ded(\kappa)^\omega$, therefore answering this question positively.
| 5 | https://mathoverflow.net/users/13694 | 96996 | 56,658 |
https://mathoverflow.net/questions/97000 | 2 | Hi guys just a quick questions
What are the real life application of catalan numbers?
Thanks a lot!
| https://mathoverflow.net/users/23731 | Application of Catalan number | In all seriousness, this type of enumerative combinatorics is very useful e.g. in figuring out expected running time for computer programs. An example involving Catalan numbers (which may seem a little contrived, but I think things like this probably do really come up) is: count how many times you go through the inner ... | 7 | https://mathoverflow.net/users/23408 | 97003 | 56,661 |
https://mathoverflow.net/questions/97005 | 4 | Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has cardinal $|A|=3$, there exists a unique circle (possibly a line) containing $A$; let me denote it $\Gamma\_A$.
Let me say th... | https://mathoverflow.net/users/8799 | Planar sets closed under intersection of circles | Send one of the points to infinity by a Mobius tranformation. Your set of points now has the property that the intersection of any two lines passing from pairs of points in the set is also in the set. Such configurations are either all collinear, dense in the plane, or one of these two exceptional cases:
* A point to... | 12 | https://mathoverflow.net/users/2384 | 97010 | 56,663 |
https://mathoverflow.net/questions/96984 | 1 | Let $\Phi: M \times \mathbf{R} \rightarrow M$ be a smooth dynamical system having no periodic orbits, i.e. such that the canonical map $\pi:M \rightarrow \mathbf{R}\backslash M$ is a principal $\mathbf{R}$-bundle. Is $\pi$ always locally trivial? If not, are there any nice (not too contrived/complicated) counterexample... | https://mathoverflow.net/users/3824 | Example of dynamical system $M$ such that $M \rightarrow \mathbf{R}\backslash M$ is not locally trivial? | The answer is negative. Take, for instance, the irrational foliation of the flat 2-torus by geodesics with the obvious ${\mathbb R}$ action via translations along leaves.
Note that every fiber bundle is locally trivial (by definition), so this should not have been one of the assumptions, only nonexistence of periodi... | 4 | https://mathoverflow.net/users/21684 | 97012 | 56,665 |
https://mathoverflow.net/questions/97002 | 4 | An element x in a noncommutative ring R is strongly nilpotent if any chain
$x\_1=x, x\_2, ... $, with $x\_{n+1}\in x\_n R x\_n$ terminates at zero. It becomes clear
why this is a good definition once one has shown that the set of all such elements
is the semi-prime radical (the intersection of all prime ideals).
Howeve... | https://mathoverflow.net/users/23732 | strong nilpotent elements | Consider the infinite word $W=xx\_1xx\_2xx\_3x....$. Let $R=R(W)$ be the ring which consists of all integral linear combinations of finite subwords of $W$. The product of two subwords $u,v$ is either the concatenation $uv$ if $uv$ is a subword of $W$ or $0$ otherwise. It is clearly an associative ring. The element $x$ ... | 7 | https://mathoverflow.net/users/nan | 97013 | 56,666 |
https://mathoverflow.net/questions/94113 | 4 | What does it mean to say that a scheme $X$ is **simple** over $Spec(A)$ ?
I stumbled on this terminology in a paper of S. Lubkin entitled "On a conjecture of Andre Weil".
| https://mathoverflow.net/users/11765 | A scheme simple over Spec(A)? | I have copied A. Stasinsky's comment who quoted a passage in the introduction of SGA1:
"/.../ et de faire un ajustage terminologique, le mot **morphisme simple** ayant notamment \'et\'e remplac\'e entre-temps par celui de **morphisme lisse**, qui ne pr\^ete pas aux m\^emes confusions."
| 2 | https://mathoverflow.net/users/11765 | 97016 | 56,668 |
https://mathoverflow.net/questions/96670 | 37 | A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories **TOP**, **DIFF** and **PL**. Well known proofs (e.g. via triangulations, or Morse theory) yield the *same* classification because of results that *connect* these categories... | https://mathoverflow.net/users/23204 | Classification of surfaces and the TOP, DIFF and PL categories for manifolds | I have come to believe that answering the questions I posted would be more enlightening if I try to provide an overview of the larger context that they are part of.
The literature treating and generalizing the topics mentioned in the post for surfaces is as extensive as it is interesting. The 1960's and 70's were tim... | 48 | https://mathoverflow.net/users/23204 | 97019 | 56,670 |
https://mathoverflow.net/questions/97017 | 1 | Is a family of bounded bi-harmonic functions defined in the unit disk an equicontinuous family of functions on compacts? A bi-harmonic function $u$ is a solution of the equation $\Delta^2 u =0$.
| https://mathoverflow.net/users/19544 | Biharmonic function | **Yes**. First, $\Delta u\in C^\infty$ because it is harmonic, thus $u\in C^\infty$, because $\Delta$ is elliptic. Next, a harmonic function $f$ is its own mean; for instance
$$2\pi f(x)=\int\_0^{2\pi}f(x+re^{i\theta})d\theta,$$
for every $r<1-|x|$. From this we deduce that if $\phi\in C^\infty(|x|-1,1-|x|)$ is even, h... | 1 | https://mathoverflow.net/users/8799 | 97024 | 56,674 |
https://mathoverflow.net/questions/96999 | 8 | Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial).
The valuation $v:K^{\times}\to\Gamma$ decomposes $K^{\times}$ as a disjoint union of nonempty open subsets, indexed by $\Gamma$. Each of these is homeomorphic... | https://mathoverflow.net/users/7666 | When is a valued field second-countable? | I now think that the answer is no. Let $K$ be the field of Hahn series for the group $\mathbb{Q}$ over a countable field $k$, in the variable $t$ (so $\kappa=\omega$). For any sequence $a=(a\_i)$ of elements of $k$, and any increasing sequence $\gamma=(\gamma\_i)$ of rational numbers between $0$ and $1$, we have an ele... | 3 | https://mathoverflow.net/users/10174 | 97029 | 56,675 |
https://mathoverflow.net/questions/97041 | 19 | The basic question that I have is in the title, but let us make it more rigorous below.
Let $N=\{1, 2, ..., n\}$, and put the (normalized) counting measure, $\mu\_n$, on $N\times N$.
Let $\mathcal{S}\_n= \{ (a, b)\in N\times N: gcd(a, b)=1\}$
and $x\_n=\mu\_n(\mathcal{S}\_n).$
Then what is the assymptotic beha... | https://mathoverflow.net/users/5732 | What is the probability that two numbers are relatively prime? | The probability tends to $\frac{1}{\zeta(2)}=\frac{6}{\pi^2}$ as was mentioned by Qiaochu. This actually generalizes to arbitrary number fields, and is a less commonly known fact.
In fact in any number field, the probability that two ideals are relatively prime is given by $1/\zeta\_K(2)$, where $\zeta\_K$ is the [De... | 24 | https://mathoverflow.net/users/2384 | 97045 | 56,679 |
https://mathoverflow.net/questions/96445 | 6 | [Elsewhere](https://mathoverflow.net/questions/95699/iterating-ultrapowers-of-c-algebras) I asked about ultrapowers of the C\*-algebra $A$ of compact operators on separable infinite-dimensional Hilbert space. My question was whether the process of taking ultrapowers of ultrapowers ever stabilizes.
It was pointed out... | https://mathoverflow.net/users/22260 | iterating ultrapowers of C*-algebras: the Calkin algebra | The statement does not depend on separability of A. Actually in the Ge-Hadwin paper they explicitly state the result for C\*-algebras of cardinality continuum. So under CH you have
that the ultrapowers of C^1 are isomorphic to C^1.
It is not completely trivial that C is not isomorphic to C^1.
This was proved in
arX... | 11 | https://mathoverflow.net/users/23737 | 97050 | 56,681 |
https://mathoverflow.net/questions/97059 | 2 | I am interested in conjugacy classes in connected reductive groups over a non-archimedean field $F$ of characteristic $0$, or its algebraic closure. On this topic it is often required that the group in question have simply connected derived group. Whether a group of type $G\_2$ has simply connected derived group may be... | https://mathoverflow.net/users/23740 | Is the derived group of $G_2$ simply connected? | This is basically a comment, but got too long. All the algebraic groups mentioned in the question are "simple" (also called "quasi-simple") over an algebraically closed field in the sense of having no closed connected normal subgroups except the entire group and the trivial group. Groups of type $G\_2$ are in fact simp... | 4 | https://mathoverflow.net/users/4231 | 97062 | 56,686 |
https://mathoverflow.net/questions/97044 | 3 | What is known about unstable bundles on curves ?
What is "maximally unstable bundle" and what means it corresponds to Schwarzian differential equation ?
Consider the P^1 with pairs of points glued p1=q1, p2=q2, ...,
what can be said for such curves ?
Let me mention that for such curves bundles can be described just... | https://mathoverflow.net/users/10446 | Unstable bundles on curves. "The maximally unstable ones correspond to Schwarzian differential equation on Riemann surfaces" ? | Vector bundles on curves come equipped with a canonical filtration, the Harder Narasimhan filtration, defined (iteratively) maximal destabilizing subbundles. Maximally unstable just means that the HN filtration is a filtration by line bundles, ie is as refined as possible (so bundle is as far from stable as possible). ... | 7 | https://mathoverflow.net/users/582 | 97064 | 56,688 |
https://mathoverflow.net/questions/97048 | 3 | Let $X$ and $Y$ be random variables
with cumulative distribution functions $F\_X(t)$ and $F\_Y(t)$ respectively.
Suppose that
$\forall t \in \mathbb{R} \ F\_X(t) \geq F\_Y(t)$.
Does it imply that $E(X) \leq E(Y)$?
Note that $X \leq Y$ implies $F\_X(t) \geq F\_Y(t)$ but not vise versa:
1. $X \leq Y$ $\ \Rightarro... | https://mathoverflow.net/users/23736 | "Generalized" monotonicity of the expected value | Note that $E(X) = \int\_{-\infty}^\infty x f(x) d x = -x (1-F(x))\vert\_{-\infty}^\infty +\int\_{-\infty}^\infty (1-F(x))d x.$
From this you should be able to deduce conditions under which the answer to your question is in the affirmative.
Using monotone convergence for the integrals on $(-\infty, 0)$ and $(0, \inf... | 2 | https://mathoverflow.net/users/11142 | 97066 | 56,689 |
https://mathoverflow.net/questions/97036 | 3 | Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c\_1,c\_2]=c\_1q\_2-c\_2q\_1$
(2) $[q\_1,q\_2]=0 $
(3) $ [c\_1,q\_2]=[c\_2,q\_1] $
**Question** May be any two elements in any ring are a... | https://mathoverflow.net/users/10446 | Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)... | The answers to the two parts of the first part of the question are "no" and "probably yes". Take the free associative algebra (over any ring, say, $\mathbb{Z}$) with free generators $x,y$. Then $x,y$ do not commute in your sense. The main obstacle is that $q\_1,q\_2$ in your definition must really commute and commuting... | 3 | https://mathoverflow.net/users/nan | 97067 | 56,690 |
https://mathoverflow.net/questions/97073 | 11 | In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere bundles over $S^4$. Also, Kervaire and Milnor proved that there are exactly 28 h-cobordism (therefore diffeomorphism) classes ... | https://mathoverflow.net/users/5069 | representatives of the group of homotopy 7-spheres | This is done in the paper ["An invariant for certain smooth manifolds"](http://www.springerlink.com/content/75m7x61uq2966987/) by James Eells and Nicolaas Kuiper. They introduce and study the so called $\mu$-invariant which is strong enough to classify homotopy $7$-spheres up to oriented diffeomorphism. A theorem on pa... | 16 | https://mathoverflow.net/users/1573 | 97075 | 56,694 |
https://mathoverflow.net/questions/97077 | 12 | These days, Peano Arithmetic ($PA$) refers to the first-order version of the axioms, where induction is only over formulas referring to natural number variables. Peano's original version of the induction axiom was actually second-order, allowing reference to sets-of-natural-numbers variables. This latter version corres... | https://mathoverflow.net/users/4177 | Z_2 versus second-order PA | $Z\_2$ as it is usually viewed is a first-order theory with two sorts, and as such is not categorical. The difference (apart from terminological issues) is entirely in the semantics that are used. In "full" second-order semantics, the set variables quantify over all subsets of the domain, while in first-order "Henkin" ... | 12 | https://mathoverflow.net/users/5442 | 97078 | 56,696 |
https://mathoverflow.net/questions/97083 | 16 | It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic?
(By isospectral, I mean that the Laplace-Beltrami operator on functions has the same spectrum on both manifolds.)
Th... | https://mathoverflow.net/users/23743 | Are isospectral manifolds necessarily homeomorphic? | There's an example due to Doyle and Rossetti ("Tetra and Didi, the cosmic spectral twins"; <http://arxiv.org/abs/math.DG/0407422>) of 3-manifolds that are isospectral but not even homeomorphic. I don't know if this was the first such example.
| 20 | https://mathoverflow.net/users/4720 | 97085 | 56,700 |
https://mathoverflow.net/questions/97082 | 0 | I want to know that whether follow equality holds:
$ |N\_G(C\_G(a)):C\_G(a)|=|a^G\cap C\_G(a)|.$
It is easy to see that the left hand is no more than the right hand. I think this equality does not hold, but I can't find a counter example.
| https://mathoverflow.net/users/22049 | The number in the join of conjugate class and centralizer | Take $G$ to be a symmetric group of degree at least 5, and $a$ a transposition.
| 6 | https://mathoverflow.net/users/22989 | 97087 | 56,701 |
https://mathoverflow.net/questions/97070 | 1 | I am currently working on research involving packing problems and am finding myself needing the tools from Combinatorial Geometry (in particular, I've been reading Pach and Agarwal's book on the subject) and I am in the dark on what I think should be a very simple point. I apologize if the question is too elementary fo... | https://mathoverflow.net/users/20343 | Generalizing the internal angle of a graph in $\mathbb{E}^2$ to $\mathbb{S}^2$ | If vertex $C$ is adjacent to vertices $X$ and $Y$, then the distances $CX$ and $CY$ are 1, and the distance $XY$ is at least 1, so the angle $XCY$ is at least $\pi/3$. If $C$ is adjacent to $X\_1,X\_2,\dots,X\_d$, where the $X\_i$ are taken in, say, clockwise order with respect to $C$, then each angle $X\_iCX\_{i+1}$ i... | 1 | https://mathoverflow.net/users/3684 | 97089 | 56,703 |
https://mathoverflow.net/questions/97061 | 9 | For a Riemannian manifold $(M,g)$ with exterior derivative d, the *codifferential* d$^\ast$ is defined to be the unique map for which
$$
g(\omega,d\omega') = g(d^\* \omega,\omega'), ~~~ \omega,\omega' \in \Omega^{\bullet}.
$$
Now if $\ast$ is the Hodge map for $g$, then it is not too difficult to show that d$= (-1)^k\a... | https://mathoverflow.net/users/3787 | Adjoint of a Connection Using the Hodge Map? | Yes, even in a more general situation: Let $E\to M$ be a vector bundle with metric and metric connection $\nabla.$ Then there exists $d^\nabla\colon\Omega^k(M,E)\to\Omega^{k+1}(M,E)$ satisfying $d^\nabla(s\otimes\omega)=\nabla s\wedge\omega+s\otimes d\omega,$ and also the Hodge star extends to $\Lambda T^\*M\otimes E.$... | 10 | https://mathoverflow.net/users/4572 | 97093 | 56,705 |
https://mathoverflow.net/questions/97086 | 22 | Does smooth and proper over $\mathbb Z$ imply rational?
I think someone told me that this is a standard conjecture. Is it a widely held? held at all? Did someone in particular make this conjecture? write it down somewhere? give evidence for it? Would you like to provide evidence? Why rational and not, say, unirationa... | https://mathoverflow.net/users/4639 | Does smooth and proper over $\mathbb Z$ imply rational? | (This answer has been edited -- it used to say that a finite cover of $\overline M\_{g,n}$ gives a counterexample, which no longer seems obvious.)
If you had written "Deligne-Mumford stack" instead of "scheme", then a counterexample would be given by the spaces $\overline M\_{g,n}$, which are certainly smooth and pro... | 10 | https://mathoverflow.net/users/1310 | 97094 | 56,706 |
https://mathoverflow.net/questions/97100 | 3 | How we can define quaternionic projective space and metric on it using Jordan algebra?
Thank you in advance!
| https://mathoverflow.net/users/22841 | Jordan algebra and quaternionic projective space | I am not much educated in this subject so please take my answer with a grain of salt.
The appropriate Jordan algebra is $J\_n(\mathbb{H}) := \{ A \in M(n,\mathbb{H})\\,|\\, \overline{A}^T=A \}$ with multiplication defined as $A\circ B = (AB+BA)/{2}$. The metric is defined by $\mathrm{Tr}(A^2)$. The quaternionic proj... | 5 | https://mathoverflow.net/users/6818 | 97101 | 56,709 |
https://mathoverflow.net/questions/97105 | 4 | Let $G$ be a finite group, and let $K$ be a finite field whose characteristic does not divide $|G|$. I am interested in the theory of finitely generated modules over $K[G]$. Of course many problems are not present here because $K[G]$ is semisimple and all modules are projective. My case is partly covered by Section 15.... | https://mathoverflow.net/users/10366 | Modular representations with unequal characteristic - reference request | Your last statement is not true in general. Let $G=C\_3$ and take your favourite finite field that does not contain the cube roots of unity, e.g. $K=\mathbb{F}\_5$. Then the two non-trivial one-dimensional representations over $\bar{K}$ are not defined over $K$, but their sum is, since it's the regular representation m... | 8 | https://mathoverflow.net/users/35416 | 97106 | 56,712 |
https://mathoverflow.net/questions/97092 | 17 | I would like to get a better idea of how badly compactness fails in $\mathcal{L}\_{\omega\_1\omega}$.
Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}\_{\omega\_1\omega}$. Let the underlying signature $\tau$ also have arbitrary cardinality. Is there some cardinal $\kappa $ such that if every $\Delta\su... | https://mathoverflow.net/users/9324 | How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$? | I like the question very much.
First, let me mention briefly that the question has a flaw in the
quantifier order, since you have first fixed the theory $\Gamma$
and then ask for a cardinal $\kappa$ such that if all subtheories
$\Delta\subset\Gamma$ of size at most $\kappa$ are consistent,
then $\Gamma$ is consistent... | 23 | https://mathoverflow.net/users/1946 | 97117 | 56,717 |
https://mathoverflow.net/questions/97123 | 7 | Let $A$ be a C\*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes\_\alpha G$. I am looking for references where I can read up on what is known about relations between the state spaces of $A$ and $A\rtimes\_\alpha G$. (The case $... | https://mathoverflow.net/users/23753 | States/functionals on crossed product C*-algebras | In the special case, where $A$ is the group vNa algebra of a type 1 group $H$, and $\alpha$ acts on $H$ via automorphism, then the extremal states on $A \rtimes\_\alpha G$ correspond roughly to irreducible representations of $H \rtimes\_\alpha G$ (assume that for instance that $H$ is compact), and these can be computed... | 0 | https://mathoverflow.net/users/10400 | 97126 | 56,720 |
https://mathoverflow.net/questions/97131 | 4 | I have the following problem:
I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n\_1,d\_1), (n\_2,d\_2),...,(n\_k,d\_k)]$ such that a point $p \in \Omega$ if $n\_i^T p \geq d\_i \quad \forall (n\_i,d\_i) \in S $. Now I have a "joining" hyperplane $(n\_{k+1},d\_{k+1})$ and I want to kn... | https://mathoverflow.net/users/23734 | How to find the minimum number of hyperplanes to define a convex hull? | If you search for *detection of redundant constraints in linear programming* you will find many hits, including one to an MO question, "[Detection of Redundant Constraints](https://mathoverflow.net/questions/69662/)."
One source paper is
>
> J. Gondzio. Presolve analysis of linear programs prior to applying an inte... | 6 | https://mathoverflow.net/users/6094 | 97133 | 56,723 |
https://mathoverflow.net/questions/96259 | 3 | How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1\_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.
More specifically, I was reading a paper that said roughly the following:
Let C be a curve and $\Gamma$ (with maps $\pi\_i: \Gamma \rightarrow C$ for $... | https://mathoverflow.net/users/7313 | Correspondences on curves and their induced maps on differentials? | This was pretty much answered in the comments by Jason Starr and Emerton, but to elaborate a bit, the simplest type of correspondence is (the graph of) a map $f:C\_1\to C\_2$.
In this case the action on $H^0(\Omega^1)$, or anything else, is by $f^\*$. In general,
given $C\_1\leftarrow \Gamma\to C\_2$, with maps labeled... | 5 | https://mathoverflow.net/users/4144 | 97136 | 56,726 |
https://mathoverflow.net/questions/97098 | 8 | This question, is a slightly different disguise (see below), came up in discussions of [this question about equitable partitions](https://mathoverflow.net/questions/96858)
A *$0,1$ vector* in $\mathbb{Z}^n$ is any vector with all entries $0$ and $1$ (at least one of each.) This excludes (for temporary convenience) th... | https://mathoverflow.net/users/8008 | Complexity of finding a 0-1 vector in a subspace or showing that there is none | It is NP-complete. Consider the case of $n$ vectors of length $n+1$, where the vectors are the rows of a matrix $[I | x]$ where $I$ is an $n\times n$ identity matrix and $x$ is a column vector of even integers. Any linear combination of the rows that is 0-1 must have multipliers 0 or 1 (due to the identity matrix part)... | 10 | https://mathoverflow.net/users/9025 | 97140 | 56,728 |
https://mathoverflow.net/questions/97138 | 12 | Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is $f$ necessarily holomorphic on $X$?
| https://mathoverflow.net/users/22338 | Functions holomorphic on a region minus a Cantor set | This belongs to the subject of holomorphic removability. See [this Wiki article](http://en.wikipedia.org/wiki/Analytic_capacity#Removable_sets_and_Painlev.C3.A9.27s_problem) for more references. In particular, the article implies that any set with Hausdorff dimension smaller than $1$ is holomorphically removable, and i... | 13 | https://mathoverflow.net/users/11142 | 97143 | 56,730 |
https://mathoverflow.net/questions/97139 | 1 | I see many times the words "conjugacy" and "conjugation", and I don't really get the difference between the two. Especially, when we take an element of a group and want to say that this has some property "up to conjugacy/tion", which one is better, and why?
| https://mathoverflow.net/users/23758 | What is the difference between "up to conjugacy" and "up to conjugation" ? | To expand upon Gerhard's comment, I would add the following:
In English, the -acy suffix tends to denote the nounification of an *adjective* and the -ation suffix tends to denote the nounification of a *verb*.
Obviously, 'conjugate' can act as both an adjective and a verb e.g.
'$x$ and $y$ are conjugate' or 'One ... | 6 | https://mathoverflow.net/users/14352 | 97149 | 56,735 |
https://mathoverflow.net/questions/97156 | 2 | is the localisation of the ring $$A:=\mathbb{Z}\_p[T]/(pT^2+T+1)$$ at the prime ideal (p) isomorphic to $\mathbb{Z}\_p$?
If not, how to understand this ring very explicitly?
| https://mathoverflow.net/users/3848 | about the local ring of $\mathbb{Z}_p[T]/(pT^2+T+1)$ at the prime p | That equation has a solution in $\mathbb Z\_p$. Proof: $f(a)=-pa^2-1$ is a contraction mapping for the $p$-adic metric, and a fixed point must be a solution.
Let $\alpha$ be that solution, then $(T-\alpha)$ divides your equation. The other factor must be $pT-\alpha^{-1}$. So we have:
$A = \mathbb Z\_p[T]/(T-\alpha)... | 8 | https://mathoverflow.net/users/18060 | 97159 | 56,740 |
https://mathoverflow.net/questions/96705 | 24 | Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S\_7$):
(1) $V \otimes W$
(2) $S\_\lambda V$, where $S\_\lambda$ is a Schur functor, or even just $\wedge^s V$,
where $V$ and $W$ are input as sums of irreducible representations, i.e.... | https://mathoverflow.net/users/2267 | Computer package for representation theory of the symmetric group | In the years since I left the answer below, Sage has improved dramatically, especially its symmetric function theory. In order to compute the third exterior power of the $S\_7$-irrep corresponding to the partition $3+3+1$, one need only type the following:
```
s = SymmetricFunctions(QQ).schur()
s[1,1,1].inner_plethy... | 16 | https://mathoverflow.net/users/9068 | 97176 | 56,748 |
https://mathoverflow.net/questions/97174 | 12 | Let $N,a\in\mathbf{Z}\_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum\_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either the trivial character or the sign character i.e. $x\mapsto sign(x)$. Define a partial $\Psi$-function as
$$
\Psi(s,\omega;... | https://mathoverflow.net/users/11765 | non-trivial zeros of partial zeta functions | The answer to question 1 is classical: Any Dirichlet series which has a finite abscissa of absolute convergence has a zero-free half-plane.
Suppose the Dirichlet series
$$ A(s)=\sum\_{n=1}^\infty \frac{a\_n}{n^s}$$
has an abscissa of absolute convergence $\sigma\_a$. If $a\_m$ is the first non-zero coefficient, then ... | 11 | https://mathoverflow.net/users/3659 | 97179 | 56,751 |
https://mathoverflow.net/questions/97163 | 3 | I am looking for ways to improve my mathematical French while learning more material about either finite group theory or geometric group theory. In particular, I would love to find a French equivalent to Rotman's group theory book, if possible. As far as geometric group theory goes, I am open to about anything, but I l... | https://mathoverflow.net/users/23761 | French resources for (Geometric) Group Theory | I suggest the survey articles of [Séminaire Bourbaki](http://numdam.org/numdam-bin/feuilleter?j=SB), such as
Ghys, Étienne
Groupes aléatoires (d'après Misha Gromov,…).
Astérisque No. 294 (2004), viii, 173–204.
Ghys, Étienne
Les groupes hyperboliques. Séminaire Bourbaki, 32 (1989-1990), Exposé No. 722, [36 p.](ht... | 5 | https://mathoverflow.net/users/2821 | 97185 | 56,753 |
https://mathoverflow.net/questions/97190 | 1 | I feel like there should either be a straightforward example of such a thing, or a homological reason why it can't exist. The reason why I ask is that if the Hessian matrix of a Lagrangian is non-singular then its dynamics are well defined, and (locally) you can pass to a Hamiltonian via the Legendre transformation, bu... | https://mathoverflow.net/users/5463 | Can a function from R^n to R^n fail to be one-to-one, and also lack critical points? | $f(x,y) = (e^x \cos y, e^x \sin y)$?
| 8 | https://mathoverflow.net/users/23141 | 97192 | 56,757 |
https://mathoverflow.net/questions/97191 | 2 | Is the following statement true: If $A\succeq 0$ and $B \preceq 0$ are two $n\times n$ real-valued matrices, then $AB \preceq 0$.
If not, is it true that $\forall x\ge 0$ (i.e., all vectors in the positive orthant), $x^TABx \le 0$?
If not, is the second statement true if all entries of $B$ are non-positive.
I'd r... | https://mathoverflow.net/users/14358 | Product of positive semidefinite and negative semidefinite matrices | No, $AB$ won't even be Hermitian in general. The correct formulation is that $A^{1/2}BA^{1/2} \preceq 0$.
| 9 | https://mathoverflow.net/users/23141 | 97193 | 56,758 |
https://mathoverflow.net/questions/97180 | 3 | In January, 1949, Shannon publishes the paper *Communication in the Presence of Noise*, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available [here](http://www.stanford.edu/class/ee104/shannonpaper.pdf), which establishes the Information Theory. In this paper, the sampling theorem is presented.
Any references about the **... | https://mathoverflow.net/users/22714 | History of the Sampling Theorem | As a start to a more comprehensive search, some notes on interpolation using the Dirichlet and Fejer kernels, close cousins of the sinc kernel, can be found in "[Leopold Fejer: In Memoriam 1880-1959](https://www.ams.org/journals/bull/1960-66-05/S0002-9904-1960-10441-7/S0002-9904-1960-10441-7.pdf)" by Gabor Szego.
And... | 4 | https://mathoverflow.net/users/12178 | 97202 | 56,764 |
https://mathoverflow.net/questions/97203 | 7 | In their paper "[New lower bounds for the border rank of matrix multiplication](http://arxiv.org/abs/1112.6007)", Landsberg and Ottaviani make use of the fact that
$$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong \bigoplus\nolimits\_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda\_1\le m }} \m... | https://mathoverflow.net/users/9947 | Question about decomposition of exterior product | This is well-known in the theory of symmetric functions and is one of two Cauchy identities.
You can find this in most books, for example, MacDonald Chapter I, Section 4. Orthogonality
equation (4.3').
Knuth's extension of the Robinson-Schensted correspondence gives bijective proofs of both Cauchy identities, for exa... | 2 | https://mathoverflow.net/users/3992 | 97209 | 56,766 |
https://mathoverflow.net/questions/97204 | 4 | Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these.
In particular I'm trying to understand MMP for real surfaces.
Let $X$ be a smooth projective surface over $\mathbb{R}$.
We can consider its com... | https://mathoverflow.net/users/6430 | Real vs complex surfaces | I think a lot of your questions apply to arbitrary Galois field extensions, not just the field extension $\mathbb{R} \subset \mathbb{C}$. This might help clarify the problems you are having by putting them into a general context.
Let $E \subset F$ be a finite Galois extension of fields and let $X$ be a smooth variet... | 5 | https://mathoverflow.net/users/5101 | 97218 | 56,770 |
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