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https://mathoverflow.net/questions/46343 | 4 | Hello!
This is a very short question:
**Given a local graded Noetherian ring $R\_{\bullet}$, is it true that any graded projective module over $R\_{\bullet}$ is free?**
In the ungraded case, this is true, but I do not know where the graded case is considered. Are there any references?
Thank you!
Hanno
| https://mathoverflow.net/users/3108 | Kaplansky's theorem for graded local rings | There is a proof of the ungraded result on page 10 of Matsumura's "Commutative Ring Theory", and you can insert gradings everywhere in a straightforward way to prove the graded result.
The main reason why you cannot just appeal to the ungraded result is as follows: a local graded ring has (by definition) precisely o... | 9 | https://mathoverflow.net/users/10366 | 46349 | 29,329 |
https://mathoverflow.net/questions/46362 | 6 | The Gaussian algorithm tells us, that for any field $k$ a $n\times n$-matrix over $k$ can written as a product of at most $C$ elementary matrices ($C\sim n^2$).
I am wondering, whether such a constants also exists for other rings - like $\mathbb{Z}$.
Given a matrix $A\in SL\_2(\mathbb{Z})$, one can basically use the Eu... | https://mathoverflow.net/users/3969 | length of decompositions into elementary matrices | Yes. The group $SL\_2(\mathbb Z)$ is virtually free and is not boundedly generated by any finite generating set because of that. You can look at the (very nice) slides of Dave Witte-Morris' talks on bounded generation [here.](http://people.uleth.ca/~dave.morris/talks.shtml)
| 4 | https://mathoverflow.net/users/nan | 46365 | 29,338 |
https://mathoverflow.net/questions/46367 | 0 | Let $X$ be a smooth projective variety and $E\longrightarrow X$ a vector bundle of rank $n$. For any $0\leq k\leq n$ the associated Grassmann bundle $G\_k(E)\longrightarrow X$ yields and we have the so-called "basis theorem" (see Fulton "Intersection theory", Proposition 14.6.5) which asserts that for any $s\geq 0$,
$$... | https://mathoverflow.net/users/10541 | Chow group of a fiber product of grassmann bundles | You just need to repeat the same procedure twice. Indeed, let $E'$ be the pullback of $E$ from $X$ to $G\_{k\_1}(E)$. Then $G\_{k\_1}(E)\times\_X G\_{k\_2}(E) \cong G\_{k\_2}(E')$.
| 2 | https://mathoverflow.net/users/4428 | 46372 | 29,343 |
https://mathoverflow.net/questions/46388 | 5 | Does anyone know of the link to an online library of of unlabeled, connected graphs on n vertices? I remember looking at such an archive a few years ago while at a Macaulay 2 workshop, but I've been unable to find it (or any other one) since then.
The page I remember seeing only had enumerated unlabeled graphs up to ... | https://mathoverflow.net/users/5002 | Online Library of Unlabeled Connected Graphs on n Vertices | Is this it? It is a searchable database of
"[Small Simple Graphs:
Connected, undirected, and unlabeled](http://gfredericks.com/sandbox/graphs)."
And an explicit list of the vertices, edges, and faces of the 600-cell can be found
at [Paul Bourke's site](http://local.wasp.uwa.edu.au/~pbourke/geometry/platonic4d/).
| 4 | https://mathoverflow.net/users/6094 | 46389 | 29,349 |
https://mathoverflow.net/questions/46404 | 0 | Given a flat and projective morphism of noetherian schemes, $f: X \rightarrow Y$ and $F$, $G$ two coherent $O\_X$-modules, flat over $Y$. Furthermore given a morphism $u: Y' \rightarrow Y$ of noetherian schemes and a quasi coherent $O\_{Y'}$-module $\mathcal{M}$. Then we have $X':=X\times\_YY'$ with $p\_i$ the i-th pro... | https://mathoverflow.net/users/3233 | Subtleties in the construction of base change morphisms | It is easy, $p\_1^\*G = G\otimes\_A A'$ is flat over $A'$ since $G$ is flat over $A$. Hence $Tor\_1^{A'}(p\_1^\*G,M) = 0$, hence your sequence is exact after tensoring with $M$. The same argument works for the first question as well.
| 1 | https://mathoverflow.net/users/4428 | 46408 | 29,360 |
https://mathoverflow.net/questions/46413 | 2 | What is the best reference for someone (i.e. me) trying to learn Instanton Floer homology? Assume I already know symplectic Floer homology.
| https://mathoverflow.net/users/9052 | Instanton homology - reference request | Donaldson's book
<http://books.google.com/books?id=CbMq-dh8nEoC&pg=PA106&dq=simon+donaldson+floer+yang&hl=en&ei=hFPkTMWtM4WnnAep1LzXDg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEoQ6AEwBQ#v=onepage&q&f=false>
| 4 | https://mathoverflow.net/users/3874 | 46416 | 29,364 |
https://mathoverflow.net/questions/46414 | 7 | Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic [billiard trajectories](http://en.wikipedia.org/wiki/Dynamical_billiards)
inside a convex polygon. Assume they all have the same speed.
Define an *$\epsilon$-cluster* as a configuration of the particles
in which they all simultaneously lie with... | https://mathoverflow.net/users/6094 | How quickly will billiard trajectories cluster? | I'll offer a few partial answers, which may eventually lead to a complete answer.
As observed in Saussol's paper (Theorem 3, Kac's lemma), if you have an ergodic invariant measure $\mu$ then the mean return time to a set $A$ is equal to $1/\mu(A)$. (Note, however, that this is the mean return time *for trajectories t... | 5 | https://mathoverflow.net/users/5701 | 46422 | 29,368 |
https://mathoverflow.net/questions/46230 | 2 | What does it exactly mean to say that in a certain category pushouts and pullbacks "commute"? Is it the same to say that they "distribute"?
| https://mathoverflow.net/users/2625 | Distributivity / commutativity of pushouts and pullbacks | The condition Martin and Todd mention is indeed a sort of distributivity condition. It is also often called {\em stability} of pushouts. I think that it should not be called commutativity.
Let D and C be small categories, and A a category with D-shaped limits and C-shaped colimits. Then D-limits commute in A with C-... | 3 | https://mathoverflow.net/users/10862 | 46426 | 29,371 |
https://mathoverflow.net/questions/46399 | 13 | Let R be the 2-periodic complex K-theory spectrum, or any other naturally occuring 2-periodic E-infty ring spectrum. The suspend-once functor gives an autoequivalence of the category of R-module spectra, and since R is 2-periodic applying it twice is isomorphic to the identity functor. In some coarse sense this defines... | https://mathoverflow.net/users/1048 | How often does suspension define an action of Z/2 on a category of module spectra? | At least for complex K-theory, I believe the answer is yes. If you want an explicit
construction, you can use the fact that there's a symmetric monoidal functor from
the 2-groupoid of Clifford algebras and Morita equivalences to the groupoid of invertible K module spectra. So it suffices to construct a monoidal functo... | 15 | https://mathoverflow.net/users/7721 | 46443 | 29,382 |
https://mathoverflow.net/questions/46299 | 9 | it is motivated by [Density of congruence classes covered by a set](https://mathoverflow.net/questions/46249/density-of-congruence-classes-covered-by-a-set/46277#46277)
Let say just "$n$-gon" for the set of vertices of a regular $n$-gon inscribed in a unit circle, "2-gon" for the set of two opposite points.
is it t... | https://mathoverflow.net/users/4312 | union of regular polygons | Let me attempt a proof using the group-theoretic formulation. I will use the additive notation for the group operation.
The proof is by induction on $n=|G|$, with the base being trivial. Let $n=p^rm$ for some prime $p$ with $\gcd(p,m)=1$.
Consider $G' = pG$. Our goal is to reduce the problem for $G$ to its instance f... | 4 | https://mathoverflow.net/users/8733 | 46451 | 29,385 |
https://mathoverflow.net/questions/46448 | 0 | Let x,y,z be points taken exclusively from the positive orthant.
For the scaling transformation
x'=x/(x+y+z)
y'=y/(x+y+z)
z'=z/(x+y+z)
where each function is a linear fractional transform
how can I interpret this?
That is, does this sort of componentwise LFT enjoy all the usual LFT properties?
I am trying t... | https://mathoverflow.net/users/10917 | geometric interpretation of componentwise linear fractional transformation(LFT) | As mentioned, this is a central projection to the $x+y+z=1$ plane, or, in the positive orthant, you can also view it as an $L\_1$-norm normalization.
Preservation of circles:
* Circles in planes parallel to $x+y+z=1$ are preserved.
* Circles (and any other shapes) in planes ~~normal to $x+y+z=1$~~ *passing through ... | 1 | https://mathoverflow.net/users/10876 | 46456 | 29,390 |
https://mathoverflow.net/questions/25411 | 33 | There are two books by Matsumura on commutative algebra. The earlier one is called *Commutative Algebra* and is frequently cited in Hartshorne. The more recent version is called *Commutative Ring Theory* and is still in print. In the preface to the latter, Matsumura comments that he has replaced a section from a previo... | https://mathoverflow.net/users/5094 | Matsumura: "Commutative Algebra" versus "Commutative Ring Theory" | By comparing the tables of contents, the two books seem to contain almost the same material, with similar organization, with perhaps the omission of the chapter on excellent rings from the first, but the second book is considerably more user friendly for learners. There are about the same number of pages but almost twi... | 40 | https://mathoverflow.net/users/9449 | 46457 | 29,391 |
https://mathoverflow.net/questions/46418 | 8 | Hello,
I'm trying to understand the relation between the points of view of
log geometry (monoids) and toric geometry (fans).
Suppose that $k$ is a field and $P$ is a finitely generated monoid.
Then $k[P]$ has a natural log structure and furthermore, any choice
of generators $\mathbf N^r\to P$ induces a closed embed... | https://mathoverflow.net/users/10580 | relation between toric geometry and log geometry | Not all finitely generated monoids $P$ will come from the cone construction. You need to assume that:
1. $P^{gp}$ is torsion-free: If $x \in P^{gp}$ and $n\cdot x = 0$ then $x = 0$.
2. $P$ is cancellative: If $x + y = x + y'$, then $y = y'$. This is equivalent to saying that the map $P \rightarrow P^{gp}$ is injectiv... | 4 | https://mathoverflow.net/users/8914 | 46460 | 29,394 |
https://mathoverflow.net/questions/46433 | 1 | Let $\pi:E\to M$ be a vector bundle over a closed smooth manifold and supose $\Pi:F\to E$ is a fibre bundle over the total space of $\pi$. I'd like to know if, restricted to $E\_p$, the second bundle $Pi$ is trivializable (moreover, homogeneous) as a fibre bundle over the vector space $E\_p$. I belive this is true if $... | https://mathoverflow.net/users/10328 | Is a fibre bundle over a vector bundle trivializable on each fibre? | If $E \to M$ is a vector bundle then it is a weak equivalence. (being a vector bundle means it is a fibration and then look at the LES in homotopy and as Somnath points out the fiber is contractible.) So having a bundle on E is the same as having a bundle on M up to homotopy.
But as for your question, up to isomorphi... | 2 | https://mathoverflow.net/users/3901 | 46461 | 29,395 |
https://mathoverflow.net/questions/46358 | 2 | Assume that $G$ is a semi-simple linear algebraic group defined over $\mathbb{Q}$, which is $\mathbb{Q}$-simple, and that $G(\mathbb(R)$ is non-compact, without $\mathbb{R}$-factors of rank 1. Then by Margulis's works, the arithmetic subgroups of $G(\mathbb{R})$ are the same as discrete lattice in $G(\mathbb{R})$. Here... | https://mathoverflow.net/users/9246 | arithmetic groups VS. Zariski dense discrete subgroups? | Arbitrary Zariski-dense subgroups in a semisimple group can be very small from a real-analytic point of view. It seems that algebra cannot distinguish between "small" and "large" Zariski-dense subgroups, so most criteria to distinguish between the two have a strong non-algebraic flavour. (Of course one can also charact... | 5 | https://mathoverflow.net/users/9927 | 46468 | 29,398 |
https://mathoverflow.net/questions/46485 | 4 | Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
| https://mathoverflow.net/users/10621 | Parametrization of O(3) | The general element is $\pm\exp(A)$ where $A$ is skew-symmetric.
(This gives each element infinitely often). This trick essentially
works for all compact Lie groups.
There is also the Cayley parameterization: $(I+A)(I-A)^{-1}$
for skew-symmetric $A$
is the general element of $SO(3)$ which lacks an eigenvalue $-1$
(so... | 11 | https://mathoverflow.net/users/4213 | 46487 | 29,410 |
https://mathoverflow.net/questions/46508 | 3 | Let $\mathbb{Z}/2\mathbb{Z}$ act on $\mathbb{A}^1$ as $x \mapsto -x$, and let $\mathscr{X}$ be the quotient stack. It has coarse moduli space $\mathbb{A}^1$ and a residual gerbe $B\mathbb{Z}/2\mathbb{Z}$ at 0. There is then a surjective map from cartesian powers $\mathscr{X}^n$ to $\mathbb{A}^n$. What are the closed su... | https://mathoverflow.net/users/2234 | closed substacks of cartesian powers of a stack | A substack of $\mathcal{X}$ is given by a $Z/2Z$ invariant subscheme of $\mathbb{A}^1$ and hence given by a $Z/2Z$ invariant ideal $I\subset k[x]$. So for example, the ideals $(x^n)$ give substacks, but they only pullback back from subschemes of the coarse space for $n$ even: the coarse space is given by $Spec(k[x^2])$... | 4 | https://mathoverflow.net/users/9617 | 46511 | 29,422 |
https://mathoverflow.net/questions/46502 | 17 | Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)?
I have seen a couple of algebraic discussions but no true proof. Also, I am looking more at trying to prove it topologically, but for now, any resource will help.\*
\*I worked on this proj... | https://mathoverflow.net/users/10918 | On the number of Archimedean solids | A proof of the enumeration theorem for the Archimedean solids (which basically dates back to Kepler) can be found in the beautiful book [*"Polyhedra"*](http://books.google.co.uk/books?id=OJowej1QWpoC&printsec=frontcover&dq=cromwell+polyhedra&source=bl&ots=R2YEvXquUw&sig=eujgkMXZQRwcRiv9DIimWU7tn90&hl=en&ei=B2TlTM-zNsid... | 18 | https://mathoverflow.net/users/5371 | 46512 | 29,423 |
https://mathoverflow.net/questions/46480 | 5 | Let $p(x)$ be the chromatic polynomial of a special graph. Performing a certain type of operation on the graph changes $p$ by shifting it and adding a constant, say to: $q(x)=p(x+a) + b$.
I have noticed that this operation always results in $q$ having the same discriminant and splitting field as $p$. This would of c... | https://mathoverflow.net/users/4078 | polynomials with the same discriminant | [Substantial edit: As I mentioned previously, cubic fields can have the same discriminant but not be isomorphic; but I've revised my answer to better address the author's question.]
There are a lot of polynomials that will generate the same cubic field. Here is the basic idea, due to Delone and Faddeev. Write down th... | 5 | https://mathoverflow.net/users/1050 | 46513 | 29,424 |
https://mathoverflow.net/questions/46431 | 6 | This is kind of a spin-off of the question asked [here](https://mathoverflow.net/questions/46414/how-quickly-will-billiard-trajectories-cluster). Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The Poincaré Recurrence theorem tells... | https://mathoverflow.net/users/934 | Poincaré Recurrence and Dense Sets | If $f$ is minimal, i.e. every orbit is dense, then $E$ is either empty or dense. So it remains to decide if your set is empty. Clearly it is non-empty for positive measure sets. If $\mu(E) = 0$. Then also
$$
\mu(\bigcup\_{n} f^{-n} E) =0
$$
and for this set the exceptional set is non-empty (and trivially dense).
| 1 | https://mathoverflow.net/users/3983 | 46525 | 29,431 |
https://mathoverflow.net/questions/46524 | 18 | Are there examples of non-Kahler complex manifolds with holomorphically trivial canonical bundle?
| https://mathoverflow.net/users/10934 | Non-Kahler "Calabi-Yau"? | Yes, you might look at the following paper by J. Fine and D. Panov: <http://arxiv.org/abs/0905.3237>
| 16 | https://mathoverflow.net/users/10675 | 46528 | 29,433 |
https://mathoverflow.net/questions/46503 | 1 | I'm reading a paper on complex semi-simple algebraic group geometry at the moment, but finding the going a bit tough since I'm missing alot of the prerequisites. Firstly, the author refers to a *principal embedding* of one Lie group into another. I guess that this is an embedding of one group into another that is maxim... | https://mathoverflow.net/users/1095 | Lie Group Principal Embedding | This doesn't answer your question completely, but at least it's a start.
If $G$ is a complex Lie group, then its quotient $G/H$ by a complex subgroup $H$ is always a complex manifold. In case $G$ is semisimple (and connected), then the *compact* quotients $G/H$ come from what are called parabolic subgroups $H$. For s... | 2 | https://mathoverflow.net/users/430 | 46532 | 29,435 |
https://mathoverflow.net/questions/46507 | 5 | I am looking for a reference for the following fact.
Let $G$ be simple undirected connected planar graph with $\geq 2$ vertices. Then $G$ contains an edge $\{u,v\}$ such that $|N(u) \cap N(v)| \leq 2$, in other words the number of common neighbors of $\{u,v\}$ is at most $2$.
Here's a sketch of the proof. Any edg... | https://mathoverflow.net/users/5200 | Every connected planar graph contains adjacent vertices with at most 2 common neighbors | it is not what you are asking for (not a reference), but just maybe slightly easier proof, without searching for $K\_{3,3}$. We may take any vertex $x$ of degree $d$, if any of $d$ vertices adjacent to $x$ has at least three neighbours between those $d$ vertices, we get at least $d+3d/2$ edges between $d+1$ vertices, h... | 3 | https://mathoverflow.net/users/4312 | 46536 | 29,438 |
https://mathoverflow.net/questions/46495 | 6 | In the vector space $V$ of $3\times 3$ symmetric real matrices, we can define a nonassociative algebra structure by the multiplication
$$A \bullet B = \frac12 (AB +BA).$$
This turns $V$ into a Jordan algebra.
### Question
>
> What is the minimum number of generators of this Jordan algebra? And could you give me o... | https://mathoverflow.net/users/3945 | Generators for the Jordan algebra of symmetric 3-by-3 matrices | The minimum number of generators is 2. First, it is easy to check that one generator is not enough: every symmetric matrix is diagonalizable, so the subalgebra it generates has dimension at most 3.
Next, the claim is that $$A:=S\_{11}+2S\_{22}+3S\_{33}$$ and $$B:=S\_{12}+2S\_{13}$$ generate the algebra, where $S\_{i... | 9 | https://mathoverflow.net/users/10675 | 46537 | 29,439 |
https://mathoverflow.net/questions/45602 | 18 | One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no nonidentity morphisms). In terms of this definition, my question is:
>
> Can one define a particular locally presentable cate... | https://mathoverflow.net/users/49 | Can Vopenka's principle be violated definably? | **Update.** My new article grows out of and extends my 2010 answer to this question. The new part is the conservativity result, showing that the Vopěnka principle has the same first-order consequences as the strictly weaker Vopěnka scheme.
* Joel David Hamkins, [The Vopěnka principle is inequivalent to but conservati... | 21 | https://mathoverflow.net/users/1946 | 46538 | 29,440 |
https://mathoverflow.net/questions/44850 | 12 | A continuous representation $\hat{\mathbb{Z}} \rightarrow GL\_n(\mathbb{Q}\_p)$ is determined by the image of $1$. But the image of $1$ does not always defines such a representation (consider for example the representation which sends $1$ on $p$ from $\mathbb{Z}$ to $GL\_1(\mathbb{Q}\_p)$). So my question is : what are... | https://mathoverflow.net/users/10427 | What are the $p$-adic representations of $\hat{\mathbb{Z}}$ ? | I am going to write a community wiki answer here which people can vote up.
(See [this meta thread](http://mathoverflow.tqft.net/discussion/780) concerning the Mathoverflow user,
which bumps questions with no voted-up answer.)
Main result: A homomorphism $f: \mathbb Z \to GL\_n(\mathbb Q\_p)$ extends continuously to
$... | 19 | https://mathoverflow.net/users/2874 | 46542 | 29,442 |
https://mathoverflow.net/questions/46534 | 8 | The Lovász $\vartheta$-function of a graph $G$, $\vartheta(G)$, is well-known to be "sandwiched" between the independence number of the graph, $\alpha(G)$, and the chromatic number of its complement, $\chi (\overline{G})$.
When $G$ is perfect then $$\alpha(G)=\vartheta(G)=\chi (\overline{G}).$$
I would like to kno... | https://mathoverflow.net/users/nan | When the Lovász theta-function saturates its upper bound | Suppose $G$ is a $k$-regular graph on $n$ vertices, with least eigenvalue $\tau$.
Lovasz proved that
$$
\theta(G) \le \frac{n}{1-\frac{k}{\tau}}.
$$
Further if the automorphism group of $G$ acts arc-transitively, then equality holds.
In fact equality holds if G is a single class in a homogeneous coherent configura... | 14 | https://mathoverflow.net/users/1266 | 46547 | 29,446 |
https://mathoverflow.net/questions/45855 | 37 | In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture:
>
> If for coprime integers $n$ and $r$ the equality $(X-1)^n = X^n - 1$ holds in $\mathbb{Z}\_n[X]/(X^r-1)$ then either $n$ is prime or $n^2 = 1 \pmod{r}$.
>
>
>
If true this would give a beautiful characterizati... | https://mathoverflow.net/users/10792 | What is the current status of Agrawal's conjecture? | I had some students look at this problem during an REU. A group proved that the conjecture is true if $r > n/2$, that's not too hard and can certainly be improved. Another group tried to find a counterexample using the computer for $r=5$, without success. I agree with Lenstra and Pomerance that the conjecture should be... | 19 | https://mathoverflow.net/users/2290 | 46564 | 29,457 |
https://mathoverflow.net/questions/46561 | 4 | I am currently a mathematics graduate student at Western Kentucky University in Bowling Green, KY. I am looking for some kind of summer opportunity to participate in during summer 2011.
Does anyone have any suggestions of good opportunities or a good list of opportunities?
I would really appreciate it!
EDIT: I am... | https://mathoverflow.net/users/10918 | Mathematics Graduate Student Summer Opportunities | NSA has a summer program. Should be somewhere at <http://www.nsa.gov>
| 2 | https://mathoverflow.net/users/2290 | 46565 | 29,458 |
https://mathoverflow.net/questions/46566 | 32 | The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a [1992 paper of Banaschewski](http://www3.interscience.wiley.com/journal/113463992/articletext?DOI=10.1002%2Fmalq.19920380136), which I don't have access to, asserting that the proof only requires ... | https://mathoverflow.net/users/290 | Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma? | Qiaochu, using the [link](http://consequences.emich.edu/file-source/htdocs/conseq.htm) I provided in my answer to [this question](https://mathoverflow.net/questions/45928/does-arzela-ascoli-require-choice), you find that this question is still open (or was, as of the mid 2000s, and I haven't heard of any recent results... | 26 | https://mathoverflow.net/users/6085 | 46568 | 29,459 |
https://mathoverflow.net/questions/46541 | 70 | In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this subject and keeping my own lecture notes. So I feel relatively prepared to teach the course in the sense that I know more than ... | https://mathoverflow.net/users/1149 | How to introduce notions of flat, projective and free modules? | Hi Pete, this sounds like a lot of fun! I wish I could be there (-:
Here is a concrete and useful property of flatness, you can explain it without using Tor. Suppose $R\to S$ is a flat extension.
Then if $I$ is an ideal of $R$, tensoring the exact sequence:
$$ 0 \to I \to R \to R/I \to 0$$
with $S$ gives that $I\ot... | 21 | https://mathoverflow.net/users/2083 | 46570 | 29,461 |
https://mathoverflow.net/questions/46291 | 3 | Let $\bar{\mathcal{P}}$ denote the closed $n$-dimensional convex polytope subtended by the origin and the lattice points {$b\_{i} \textbf{e} \_ {i}$}, where {$\mathbf{e}\_{i}$} is the standard basis of $\mathbb{R}^{n}$. Define the Ehrhart function $L \_{\bar{\mathcal{P}}}(t) = | t \bar{\mathcal{P}} \cap \mathbb{Z} ^{n}... | https://mathoverflow.net/users/10280 | Rational dilates of integral convex polytopes | R. Diaz, S. Robins, and I studied your question for the inequality $b\_1 x\_1 + \dots + b\_d x\_d \le t$ for integral $t$ (which gives a rational polytope) in
* *The Frobenius problem, rational polytopes, and Fourier-Dedekind Sums*, Journal of Number Theory 96 (2002), 1–21, doi:[10.1006/jnth.2002.2786](https://doi.or... | 5 | https://mathoverflow.net/users/3193 | 46595 | 29,478 |
https://mathoverflow.net/questions/46387 | 3 | (Sorry I'm outsider in this field.)
I need to count the number of integral points in a convex polytope in $\mathbf{R}^3$. The cones in the dual fan are not necessarily regular (does it create any problem?)
So, I need the $c\_1$ coefficient of the Ehrhart polynomial. There're some formulas (complicated enough for me... | https://mathoverflow.net/users/2900 | Counting integral points of a polytope in R^3 (the c_1 coefficient of Ehrhart polynomial) | Jamie Pommersheim gave a general formula for $c\_1$ in his 1993 paper in Math. Ann. for the case of a tetrahedron. So if you can easily triangulate your polytope, this might be useful.
If you're "only" interested in bounds, you might want to look at the Ehrhart series instead, i.e., the generating function of the Ehr... | 10 | https://mathoverflow.net/users/3193 | 46596 | 29,479 |
https://mathoverflow.net/questions/46597 | 15 | I am under the impression that in the definition of the Grothendieck group $K\_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K\_0$ construction to the exact category of all finitely generated projective *left* $R$-modules, or if we apply the construction to the category of all finitely... | https://mathoverflow.net/users/1148 | Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules? | First of all, right $R$-modules are the same as left $R^{op}$-modules. Hence you are asking whether the $K$-theory changes if you pass form $R$ to $R^{op}$. The answer is: It does not change.
The reason for the independence is that one has a very concrete picture.
$$K\_0(R) = Gr(V(R))$$
where $Gr$ denotes the Gr... | 17 | https://mathoverflow.net/users/8176 | 46598 | 29,480 |
https://mathoverflow.net/questions/46574 | 5 | Does there exist a field $k$ and a subring $R$ of $S = M\_2(k)$ such that $R$ is not finitely generated over its center, $S=kR$ and $1\_R = 1\_S$? ($S$ is the algebra of $2 \times 2$ matrices over $k$.)
| https://mathoverflow.net/users/10943 | M_2(k) as a central extension | I think the answer is "yes". Let $A$ be a non-Noetherian integral domain (for example a polynomial ring in infinitely many variables over a field), let $I$ denote a non-finitely-generated ideal, and let $k$ be the field of fractions of $A$. Let $R$ denote the ring of $2\times 2$ matrices with coefficients in $A$ and wi... | 7 | https://mathoverflow.net/users/1384 | 46600 | 29,481 |
https://mathoverflow.net/questions/46576 | 8 | Write $M\_n = S^n \cup\_2 D^{n+1}$. I know, as a matter of folklore, that the identity map $\mathrm{id}\_M$, considered as an element of the group $[M\_n, M\_n]$, has order $4$ (for $n > 3$, let's say).
What is a reference for this? And is there a simple and pretty argument?
| https://mathoverflow.net/users/3634 | Order of the identity map of a Moore space. | Alternatively: if the order were $2$ then $M\_n\wedge M\_n$ would be $M\_{2n}\vee M\_{2n+1}$. The mod $2$ cohomology of $M\_n$ has a generator $a$ in degree $n$ and a generator $b=Sq^1(a)$ in degree $n+1$ and nothing else. It follows that the cohomology of $M\_n\wedge M\_n$ has generators $a\otimes a$, $a\otimes b$, $b... | 5 | https://mathoverflow.net/users/10366 | 46603 | 29,484 |
https://mathoverflow.net/questions/46577 | 13 | If $\Gamma=\Gamma\_1(N)$, or $\Gamma=\Gamma\_0(N)$, the Hecke operator $[\Gamma diag(1,l) \Gamma]$ for $l$ a prime (acting on the space of cusp forms of level $\Gamma$ and some weight $k$) is in general denoted $T\_l$ when $l$ does not divide $N$, but $U\_l$ otherwise. It is well-known that the Hecke operators $T\_l$ a... | https://mathoverflow.net/users/9317 | Adjoint of Atkin-Lehner's U_p | The following is essentially taken from Miyake's "On Automorphic Forms on GL2 and Hecke Operators", with some help interpreting notation by looking at Gelbart's "Automorphic forms on adele groups".
From the point of view of representation theory, you can define Hecke operators at any prime (well, up to a normalizing ... | 6 | https://mathoverflow.net/users/6753 | 46606 | 29,486 |
https://mathoverflow.net/questions/46605 | 1 | randomized SVD decomposes a matrix by extracting the first k singular values/vectors using k+p random projections.
my question concerns the singular values that are output from the algorithm. why aren't the values equal to the first k-singular values if you do the full SVD?
Below I have a simple implementation in... | https://mathoverflow.net/users/10950 | randomized SVD singular values | They should be approximation to the true singular values (with suitable hypotheses, with good probability...).
Intuitively, it is like trying to infer the singular values of a $n\times n$ matrix by looking at its leading $(k+p)\times (k+p)$ submatrix only and replacing the rest with zeros: it is cheaper to compute, but... | 4 | https://mathoverflow.net/users/1898 | 46612 | 29,490 |
https://mathoverflow.net/questions/46505 | 28 | What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.
Usual completion of square breaks. For a finite field there is [Chen Formula](http://www.ams.org/mathscinet-getitem?mr=... | https://mathoverflow.net/users/5301 | How to solve a quadratic equation in characteristic 2 ? | I think this solves $X^2+X+c=0$ over $F((t))$:
I want to assume that $c\in F[[t]]$. If not, say $c=at^{-m}+...$, then the quadratic has no solutions when $m$ is odd or $a$ is not a square, and otherwise the substitution $X\mapsto X+\sqrt{a}t^{-m/2}$ gives a new equation with smaller $m$. So, after finitely many steps... | 17 | https://mathoverflow.net/users/3132 | 46616 | 29,492 |
https://mathoverflow.net/questions/46553 | 9 | Which would be the most efficient way (in computational time) to compute tr(inv(H)), where H is a (dense) square matrix?
In my particular problem I also have a LU decomposition of H already available, which was used in a previous context to solve a system of linear equations. My current approach is to simply use the ... | https://mathoverflow.net/users/4430 | Fast trace of inverse of a square matrix | Given that the poster has specified that his matrix is symmetric, I offer a general solution and a special case:
1. Eigendecomposition actually becomes more attractive here: the bulk of the work is in reducing the symmetric matrix to tridiagonal form, and finding the eigenvalues of a tridiagonal matrix is an O(n) pro... | 7 | https://mathoverflow.net/users/7934 | 46620 | 29,495 |
https://mathoverflow.net/questions/46625 | 4 | Is there an infinite (finite degree) transitive amenable hyperbolic graph ?
| https://mathoverflow.net/users/10957 | hyperbolic amenable graph | I think the answer is that every such graph should be quasi-isometric to the infinite line. Look at [this paper.](http://arxiv.org/PS_cache/math/pdf/9806/9806129v1.pdf) Theorem 9 there states that even a quasi-transivite amenable graph should have trivial Poisson boundary (no non-constant harmonic functions). On the o... | 1 | https://mathoverflow.net/users/nan | 46629 | 29,499 |
https://mathoverflow.net/questions/46628 | 5 | Let $\Omega\subset \mathbb{R}^d$ be an open and bounded domain with Lipschitz smooth boundary. Let $\delta>0$ and
$
\Omega\_\delta = \{ x\in\Omega : \inf\_{y\in\partial\Omega} \left\|x-y\right\|\_{L\_2(\Omega)}\geq \delta \}
$
Is there a $\hat{\delta}>0$ such that $\forall 0<\delta<\hat{\delta}$ the space $\Omega\_{... | https://mathoverflow.net/users/2011 | Shrinking a Lipschitz smooth domain. | My answer is **Yes**. Of course, I presume that you assume $\Omega$ is on one side only of its boundary.
The boundary $\partial\Omega$ is compact. By assumptions, it has an atlas with finitely many charts, each one corresponding to a piece $\Gamma\_j$ which is the graph of a Lipschitz function: in appropriate coordin... | 4 | https://mathoverflow.net/users/8799 | 46631 | 29,501 |
https://mathoverflow.net/questions/46567 | 11 | I'm wondering if a classification of analytic functions, $f\,$ (it may be that $C^1$ is enough, but I'm not taking any chances, if you have a reason why I only need to consider a larger class of functions, I would enjoy that as well) with the property $F(x):=\int f(x) dx = \mathcal{O}(xf(x))$ as $x\to\infty$ AND $xf(x)... | https://mathoverflow.net/users/4701 | Functions whose antiderivative behaves like xf(x) | Note: This is a major rewrite of my earlier answer, to include necessary and sufficient conditions applicable to an even wider class of functions.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be ... | 13 | https://mathoverflow.net/users/2926 | 46635 | 29,504 |
https://mathoverflow.net/questions/46643 | 8 | In the book "profinite groups, arithmetic, and geometry" of Shatz, the index $(G:H)$ of a closed subgroup $H$ of a profinite group $G$ is defined to be the supernatural number $lcm\big((G/U):(H/(H\cap U))\big)$ where $U$ runs over the open normal subgroups of $G$. There is an exercise following this definition saying t... | https://mathoverflow.net/users/10469 | index of a closed subgroup of a profinite group | (The first time around I had read your question too quickly and not properly appreciated it. Sorry about that.)
You are right: the exercise on p. 12 of Shatz's book is false, because of the example you suggest. You asked if there were also counterexamples among infinite profinite groups. Certainly: let $n \geq 5$, le... | 8 | https://mathoverflow.net/users/1149 | 46644 | 29,508 |
https://mathoverflow.net/questions/46648 | 3 | I have a few really basic questions about minimal surfaces. Does a smooth or piecewise smooth injection $S^1$ into $\mathbb R^3$ always give a unique minimal surface or are there instances with discrete distinct solutions? Can it not be the case that a 1-parameter family of minimal surfaces exists for a given "frame"? ... | https://mathoverflow.net/users/2031 | minimal surfaces | In $\mathbb{R}^3$ there is always some minimal surface spanning a given connected simple curve (assuming the curve is not too horrible, say Lipschitz). Here surface needs to be broadly understood as being possibly immersed and of arbitrary topology. This can be seen using the machinery of geometric measure theory and t... | 11 | https://mathoverflow.net/users/26801 | 46651 | 29,513 |
https://mathoverflow.net/questions/29638 | 3 | What I'm really looking for is a good set of examples for a semi-direct product of two (coprime) cyclic groups, with a non-trivial (not $1$ and not everything) center. What resource would you recommend for that purpose?
| https://mathoverflow.net/users/5309 | List of centers of finite groups | If you would like a complete classification following Jack's strategy, say your group is $G=C\_n\rtimes C\_m$ with $n,m$ coprime, $C\_n=\langle c\rangle$, $C\_m=\langle x\rangle$ and $xcx^{-1}=c^j$. The order of $j$ in $({\mathbb Z}/n{\mathbb Z})^\times$ has to divide $m$, and every such triple $(n,m,j)$ gives a valid ... | 5 | https://mathoverflow.net/users/3132 | 46652 | 29,514 |
https://mathoverflow.net/questions/46633 | 3 | Let $f \colon X \to Y$ be a proper morphism of (Noetherian) schemes, $\mathcal{F} \in \mathop{Coh}(X)$. Let $i\_Z \colon Z \hookrightarrow Y$ be a closed subscheme and take the inverse image $W := X \times\_Y Z \overset{i\_W}{\hookrightarrow} X$.
If we assume that $\dim X\_y \leq k - 1$ for all $y \in Y \setminus Z$,... | https://mathoverflow.net/users/828 | A form of cohomology and base change | Here are more details for the second example from comments. Set $Y={\mathbb A}^2$, $Z=0$ (the origin), $X=$ blow-up of $Y$ and $Z$, so that $W$ is the exceptional divisor. Take $F\_n=O\_X(nW)$. Since the self-intersection of $W$ is $-1$, we see that $F\_n/F\_{n-1}=F\_n|\_W\simeq O\_W(-n)$
(more properly, the direct ima... | 4 | https://mathoverflow.net/users/2653 | 46654 | 29,516 |
https://mathoverflow.net/questions/35736 | 17 | I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical bundle is defined:
$$\mathcal{K}=\mathcal{O}\_{X,-\sum D\_i}$$
where the $D\_i$ are representatives of all divisors... | https://mathoverflow.net/users/1887 | The canonical line bundle of a normal variety |
>
> **Edit** (11/12/12): I added an explanation of the phrase "this is essentially equivalent to $X$ being $S\_2$" at the end to answer aglearner's question in the comments.
> [See also [here](https://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-hartogs-phenomenon/45354#45354... | 51 | https://mathoverflow.net/users/10076 | 46663 | 29,519 |
https://mathoverflow.net/questions/46650 | 1 | Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>0$ and that $f$ decays rapidly to $0$ on the boundary.
Let $$ \Omega\_{\delta} = \{ x\in\Omega : \inf\_{y\in\partial\Omega} \left\|x-y\right\... | https://mathoverflow.net/users/2011 | Bounding a smooth function near the boundary | For sure $\|f\|\_ {L^2(\Omega\setminus\Omega\_\delta)}=o(1)$ as $\delta\to0$ for any $f\in L^2(\Omega)$ (this, even if $\Omega$ was not bounded). For $f\in L^\infty(\Omega)$ you have $\|f\|\_ {L^2(\Omega\setminus\Omega\_\delta)}=O(\delta)$, for the Lebesgue measure of $\Omega\setminus\Omega\_\delta$ is bounded by $\del... | 3 | https://mathoverflow.net/users/6101 | 46665 | 29,521 |
https://mathoverflow.net/questions/46662 | 5 | As is well known (see Kassel), when $q$ is not a root of unity, the centre or the quantum enveloping algebra $U\_q({\mathfrak sl}\_2)$ of ${\mathfrak sl}\_2$ is generated by the element
$$
C\_q = EF + \frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^2}.
$$
The element is called the quantum Casimir. My questions are as follows:
(i)... | https://mathoverflow.net/users/2612 | Does There Exists a General Quantum Casimir Extending the $U_q({\mathfrak sl}_2$ Case? | The centers of the Drinfeld-Jimbo quantum groups $U\_q(\mathfrak{g})$ are well-understood and quite analogous to the classical case. See the book by Klimyk and Schmüdgen, Section 6.3, where in particular the quantum Casimirs are constructed.
| 8 | https://mathoverflow.net/users/10756 | 46668 | 29,524 |
https://mathoverflow.net/questions/46660 | 7 | Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?
If the cost of reusing $L$ an... | https://mathoverflow.net/users/4430 | Adding a multiple of the Identity to a LU factorized matrix | Such an efficient computation is unlikely. Suppose you can do it, with factors $U\_\lambda$ and $L\_\lambda$. Then by $\det(A+\lambda I)=\det U\_\lambda\cdot\det L\_\lambda$, you obtain the characteristic polynomial for free. Thus the cost of the calculation you are looking for cannot be smaller than the cost of the ca... | 2 | https://mathoverflow.net/users/8799 | 46670 | 29,526 |
https://mathoverflow.net/questions/41102 | 10 | The mutual information $I(\mathfrak A\_1;\mathfrak A\_2)$ of two complete $\sigma$-algebras $\mathfrak A\_1$ and $\mathfrak A\_2$ in a Lebesgue probability space $(X,m)$ is the integral of the logarithm of the Radon-Nikodym derivative $dP/d(P\_1\otimes P\_2)$, where $P\_i$ are the quotient measures on the factor-spaces... | https://mathoverflow.net/users/8588 | Continuity of the mutual information | The last proof of this continuity property (and of its analogue for increasing sequences) is given in [this paper](http://www.ams.org/mathscinet-getitem?mr=2472012) by Harremöes and Holst. It also contains a pretty comprehensive list of references to earlier work. Apparently, first this property was established by Pins... | 5 | https://mathoverflow.net/users/8588 | 46672 | 29,528 |
https://mathoverflow.net/questions/46686 | 3 | I am told that finite groups have unique factorization under direct product. That is, call a nontrivial group "indivisible" if it is not isomorphic to a direct product of nontrivial groups. Then every finite group can be "factored" (by direct product) into a unique collection of indivisible groups.
In particular, if ... | https://mathoverflow.net/users/1079 | Unique factorization of finite groups under direct sum? | About the first fact see [this](https://groupprops.subwiki.org/wiki/Direct_product_is_cancellative_for_finite_groups) page (the Krull–Remak–Schmidt theorem). For infinite (even finitely generated) groups the situation is different because there exists an infinite f.g. group isomorphic to its [direct square.](https://gr... | 7 | https://mathoverflow.net/users/nan | 46690 | 29,537 |
https://mathoverflow.net/questions/46701 | 16 | I have heard references to "hard" vs. "soft" analysis. What is the difference? It seems to do with generality versus nitty-gritty estimates, but I haven't gotten any responses more clear than that.
| https://mathoverflow.net/users/10946 | What is the difference between hard and soft analysis? | Disclaimer: I'm no expert-this is really a question for the analysts and historians of mathematics.
As far as I know,the terminology came into existence in the early 20th century distinguish the classical "calculus" type analysis (hard analysis) from the new point set topology/functional analysis approach (soft analy... | 12 | https://mathoverflow.net/users/3546 | 46705 | 29,547 |
https://mathoverflow.net/questions/46700 | 19 | ### Background
One of my favourite elementary results in group theory is [Goursat's Lemma](http://en.wikipedia.org/wiki/Goursat%27s_lemma). This lemma characterises the subgroups of a direct product of groups in terms of fibred products.
Indeed, let $L$ and $R$ be groups and let $G < L \times R$ be a subgroup of th... | https://mathoverflow.net/users/394 | For which categories does one have a Goursat Lemma? | To make life simple, suppose that finite limits and finite colimits exist. If we work with regular epimorphisms rather than epimorphisms then your condition is equivalent to saying that for any two (regular) epimorphisms $G\to L$ and
$G\to R$, if you form the pushout $F$ then the canonical comparison from $G$ to the p... | 16 | https://mathoverflow.net/users/10862 | 46709 | 29,550 |
https://mathoverflow.net/questions/46687 | 4 | It is known for Hilbertian fields that all groups that are abelian, solvable, $A\_n$ or $S\_n$ are realizable over them. $\mathbb{Q}(x)$ is one such field, but it's not obvious that the extensions that are guaranteed by this theorem will be regular extensions.
$A\_n$ and $S\_n$ are regularly realizable through the sy... | https://mathoverflow.net/users/5309 | Are all solvable groups *regularly* realizable over Q(x)? | It is known that all finite abelian groups are regularly realisable over $\mathbb{Q}(x)$. See e.g. B.H. Matzat, Konstruktive Galoistheorie, p. 224, M. Fried and M. Jarden, Field Arithmetic, Lemma 24.46, or J.P Serre, Topics in Galois Theory, p. 36. It is also known that, e.g., the set of regularly realisable groups is ... | 6 | https://mathoverflow.net/users/35416 | 46710 | 29,551 |
https://mathoverflow.net/questions/46736 | 1 | The question is in subject.
**Update:** See Andreas Thom's answer.
| https://mathoverflow.net/users/4807 | Do separable $C^*$-algebras form a set? | It is not so clear what you mean.
However, every separable $C^\ast$-algebra embeds in $B(\ell^2 \mathbb N)$. Hence, the isomorphism classes of separable $C^\ast$-algebras form a set.
| 7 | https://mathoverflow.net/users/8176 | 46737 | 29,566 |
https://mathoverflow.net/questions/46752 | 13 | Let me be more specific. Let $M$ be a Kahler manifold with Riemannian metric $g$ and complex structure $I$. Then $T^\ast M$ will also be Kahler with metric and complex structure induced from $M$ (I will give them the same name). It is also holomorphic symplectic, with canonical holomorphic symplectic form $\Omega \_\ma... | https://mathoverflow.net/users/7762 | Is the cotangent bundle to a Kahler manifold hyperkahler? | Such hyper Kaehler metrics do exist near the zero section, e.g. in a formal or an analytic tubular neighborhood of the zero section. After that one can use some homogeneity to spread them on the whole cotangent bundle but typically the resulting metrics are non-complete. One gets nice global metrics on the cotangent bu... | 15 | https://mathoverflow.net/users/439 | 46755 | 29,575 |
https://mathoverflow.net/questions/46748 | 21 | This is not actually a question asked by me. But since I do not know the answer, I would love to know if someone here could answer it.
| https://mathoverflow.net/users/4760 | Is every locally connected subset of Euclidean space R^n locally path connected ? | No. There are locally connected subsets of $\mathbb{R}^2$ which are totally path disconnected. See my answer to this old MO question "[Can you explicitly write R2 as a disjoint union of two totally path disconnected sets?](https://mathoverflow.net/questions/156/can-you-explicitly-write-r2-as-a-disjoint-union-of-two-tot... | 27 | https://mathoverflow.net/users/1004 | 46758 | 29,576 |
https://mathoverflow.net/questions/46757 | 3 | Suslin's problem is:
>
> Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
>
>
>
The answer is that it's independent of ZFC. The related question:
>
> Given a complete dense linear order without endpoints, if it's separable must it be isomorphic... | https://mathoverflow.net/users/7521 | A problem about posets similar to Suslin's problem | Amit:
If ${\mathbb P}$ is a non-trivial separative partial order, and it is countable, an easy argument (back-and-forth) shows that it is forcing isomorphic to Cohen forcing. (This is an exercise in Chapter VII of Kunen's book, I believe, and it can be found in a few other sources, such as the appropriate chapter of ... | 9 | https://mathoverflow.net/users/6085 | 46766 | 29,581 |
https://mathoverflow.net/questions/46626 | 4 | Let $c\_0$ be the Banach space of doubly infinite sequences $$\lbrace
a\_n: -\infty\lt n\lt \infty, \lim\_{|n|\to \infty} a\_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable on $[0,2\pi]$.
Let $$S=\lbrace \lbrace a\_n\rbrace \in c\_0: a\_n=\hat{f}(n) \forall n \mbox{ for some function } f... | https://mathoverflow.net/users/10583 | Characterizations of a linear subspace associated with Fourier series | This is to summarize what were discussed in the comments, so the title will not be listed as unanswered.
The linear subspace $S$ of $c\_0(\mathbb{Z})$ is equal to the convolution product of two copies of $\ell^2(\mathbb{Z})$.
More precisely, $\lbrace a\_n \rbrace$ is in $S$ if and only if there exist two sequences... | 2 | https://mathoverflow.net/users/10583 | 46773 | 29,584 |
https://mathoverflow.net/questions/46742 | 2 | $A$ a Cohen-Macaulay ring (not necessarily local), $M$ a Cohen-Macaulay $A$-module. Then does it necessarily follow that $\mbox{ann}(M)$ is height-unmixed?
| https://mathoverflow.net/users/5292 | CM module is height-unmixed? | This is true if $A$ is local but fails in general.
First, a counterexample. Let $A=\mathbb Z[X]$ and $M= A/p\oplus A/q$ with $p=(2)$, $q=(3,X)$. Since any maximal ideal $m$ of $A$ can not contain both $2$ and $3$, when you localize at $m$ only one of the summands can survive at most, so $M\_m$ will be CM. The annihi... | 5 | https://mathoverflow.net/users/2083 | 46783 | 29,592 |
https://mathoverflow.net/questions/46769 | 17 | Define the *length* of a set of arithmetic progressions
of natural numbers
$A=\lbrace A\_1, A\_2, \ldots \rbrace$
to be $\min\_i | A\_i |$: the length of the shortest sequence
among all the progressions.
Say that $A$ *exactly covers* a set $S$
if $\bigcup\_i A\_i = S$.
Let $P'$ be the primes excluding 2.
>
> What i... | https://mathoverflow.net/users/6094 | Covering the primes by arithmetic progressions | Despite the comments to the question (including mine), this is a bit easier than it seems at first sight. We can show that $L\_{\max}=2$ or $3$. Almost certainly we have $L\_\max=3$. However, determining which of these is actually the case seems to be beyond current technology, according to this MO answer "[Are all pri... | 10 | https://mathoverflow.net/users/1004 | 46784 | 29,593 |
https://mathoverflow.net/questions/46523 | 7 | If $X$ is a smooth, projective variety over $\mathbb{F}\_q$, the [Weil conjectures](http://en.wikipedia.org/wiki/Weil_conjectures) tell us:
$$\prod \mathrm{det} (I - TF|\_{H^i\_c(X)})^{(-1)^{i+1}} = \mathrm{exp}\left(\sum\_{m=1}^{\infty} \frac{N\_m}{m} T^m \right)$$
here, $T$ is a formal variable, $H^i\_c(X)$ is an... | https://mathoverflow.net/users/4707 | Zeta function of monodromy and counting points over C((t)) | Hi Vivek. You should have a look to the following papers :
>
> J. Nicaise and J. Sebag (2007).
> *Motivic Serre invariants, ramification, and the analytic Milnor
> fiber*. Inventiones mathematicae,
> **168** (1), p. 133-173.
>
>
> J. Nicaise (2009). *A trace formula
> for rigid varieties, and motivic Weil
> ... | 9 | https://mathoverflow.net/users/10696 | 46789 | 29,594 |
https://mathoverflow.net/questions/46785 | 12 | Hello,
I'd love to learn more about the field of additive combinatorics. From what I've understand, there's a book by Tao and Vu out on the subject, and it looks fun, but I think I lack the prerequisites. Right now, I've had basic real analyis (Rudin), read the first volume of Stanley's "Enumerative combinatorics", and... | https://mathoverflow.net/users/10984 | A learning roadmap for Additive combinatorics. | Some portions of their book should be accessible without too much background. Take a look at their sections on additive geometry, graph-theoretic methods, and algebraic methods, for example. For the bulk of the book, though, knowing some probability theory will make a big difference.
A recent book that I like and you... | 7 | https://mathoverflow.net/users/6085 | 46790 | 29,595 |
https://mathoverflow.net/questions/46794 | 1 | Let $A$ be a commutative ring with identity. If $A$ is a ring with only a finite set of prime ideals $p\_1...p\_n$ and moreover $\prod\_{i=1}^n p\_i^{k\_i}=0$ for some k\_i. Is $A$ then isomorphic to $\prod\_{i=1}^nA\_{(p\_i)}$?
| https://mathoverflow.net/users/10988 | Generalization of the Structure theorem for artinian rings? | No. Let $A$ be a DVR. It has two prime ideals: the maximal ideal $p\_1=\mathfrak m\subset A$ and $p\_2=(0)\subset A$. So, $p\_1p\_2=0$, but $A$ is not a product (of two local rings).
| 8 | https://mathoverflow.net/users/10076 | 46796 | 29,599 |
https://mathoverflow.net/questions/46779 | 1 | Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. Then the graph of $s$ could be considered both as a finite correspondence from $Y$ to $X$ and as a finite correspondence ... | https://mathoverflow.net/users/2191 | A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree? | Let $f:Y \to X\times Y$ be the graph of $s$ and $f^T:Y \to Y\times X$ the transpose of the graph. By definition you should take the fiber product of $(1\times f^T):X\times Y \to X\times Y\times X$ and $(f\times 1):Y\times X \to X\times Y\times X$. It is easy to see that the fiber product is $Y$. Then we should take the... | 1 | https://mathoverflow.net/users/4428 | 46803 | 29,604 |
https://mathoverflow.net/questions/46787 | 24 | A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: $H\_\ast(X^\mu) \cong H\_{\ast-n}(X)$ and $H^\ast(X^\mu) \cong H^{\ast-n}(X)$.
This isomorphism can be proven in many w... | https://mathoverflow.net/users/798 | Is there a map of spectra implementing the Thom isomorphism? | There is a construction for both Thom isomorphisms, homological and cohomological, via classical stable homotopy theory. You find the details in Rudyaks book "On Thom spectra, orientability, and cobordism", chapter V, §1. The Thom class is a map $X^{\mu} \to\Sigma^{n} H \mathbb{Z}$. Moreover, there is a map of spectra ... | 23 | https://mathoverflow.net/users/9928 | 46809 | 29,609 |
https://mathoverflow.net/questions/46804 | 37 | I was writing up some notes on harmonic analysis and I thought of a question that
I felt I should know the answer to but didn't, and I hope someone here can help me.
Suppose I have a compact Riemannian manifold $M$ on which a compact Lie group $G$
acts isometrically and transitively---so you can think of $M$ as $G/K... | https://mathoverflow.net/users/7311 | When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations? | The Peter-Weyl theorem tells you that $L^2(G)$ is isomorphic to $\bigoplus\_{\pi}\pi\otimes\pi^\*$ as $G\times G$ representation, where $\pi$ runs through all irreducible unitary representations.
It follows that
$$
L^2(G/K)\cong L^2(G)^K\cong\bigoplus\_\pi \pi\otimes(\pi^\*)^K.
$$
So, the first thing you absolutely nee... | 23 | https://mathoverflow.net/users/nan | 46814 | 29,611 |
https://mathoverflow.net/questions/46815 | 13 |
>
> Question: Assuming finiteness of the Tate-Shafarevich group, is there an algorithm to determine whether a curve $C$ defined over a number field $K$ has infinitely many $K$-rational points?
>
>
>
I believe that this is (a) true and (b) sufficiently important that it has been carefully explained somewhere, but... | https://mathoverflow.net/users/3132 | Which curves have infinitely many rational points | I believe the following is an algorithm, albeit a horrible one.
First, as the OP surely knows, it comes down entirely to curves of genus one. Indeed, if the genus is at least $2$ then by Faltings' Theorem there are only finitely many $K$-rational points, whereas if the genus is zero, there are infinitely many rationa... | 16 | https://mathoverflow.net/users/1149 | 46819 | 29,613 |
https://mathoverflow.net/questions/36085 | 11 | A Hausdorff space $(X,\tau)$ is said to be **minimal Hausdorff** if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff.
Every compact Hausdorff space is minimal Hausdorff.
I would like to know:
1) Is every minimal Hausdorff space compact?
2) Does every Hausdor... | https://mathoverflow.net/users/8628 | Minimal Hausdorff | The answer to both questions is no - see 7.5 in Porter and Woods book, "Extensions and Absolutes of Hausdorff Spaces", Springer-Verlag, 1988. The space of rational numbers with the usual topology has no coarser minimal Hausdorff topology. Every Hausdorff space can be embedded in a minimal Hausdorff space; in particular... | 12 | https://mathoverflow.net/users/10995 | 46825 | 29,619 |
https://mathoverflow.net/questions/46802 | 5 | Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E\_1\to\dots\to E\_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E\_1\to\dots\to E\_n \to B\to 0.$$
A morphism of $n$-extensions of $Y$ by $X$ is defined to be a hammock
$$\begin{matrix}
&&A\_1&\to&A\_2&\to&A\_3&\to&\ldots&\to &A\_{n-2}&\... | https://mathoverflow.net/users/1353 | Simplicial "universal extensions", the hammock localization, and Ext | For your first question:
>
> Given an $n$-extension $\sigma$ of $B$ by $M$, how can we produce a morphism in the
> derived category $M\to N[n]$ that generates an n-extension in the same connected
> component
> of $n{-}ext(B,M)$?
>
>
>
Do you mean a morphism $B\to M[n]$? In which case, isn't this a the s... | 7 | https://mathoverflow.net/users/437 | 46829 | 29,621 |
https://mathoverflow.net/questions/46834 | 1 | Given a flat and projective morphism $f:X\rightarrow Y$ of noetherian schemes over some algebraically closed $k$ and $F$, $G$ coherent $O\_X$-modules, flat over $Y$.
Then the base change theroem for the relative $Ext$ sheaves reads:
Let $y\in Y$ and assume $\tau^i(y): \mathcal{E}xt\_f^i(F,G)\otimes k(y)\rightarrow Ex... | https://mathoverflow.net/users/3233 | Application of the base change theorem | It is always useful to look from the derived category point of view. What you want is the object of $D(Y)$ which is $Rf\_\*R{\mathcal H}om(F,G)$ --- the sheaves ${\mathcal E}xt^i\_f$ are its sheaf cohomology. The base change tells you that $Li^\*Rf\_\*R{\mathcal H}om(F,G) \cong RHom(F\_{y\_0},G\_{y\_0})$, where $i$ is ... | 4 | https://mathoverflow.net/users/4428 | 46850 | 29,629 |
https://mathoverflow.net/questions/46844 | 1 | The Hodge star operator $\ast$ acts on the differential forms of a differential manifold sending $\Omega^{k}$ to $\Omega^{N-k}$. If the manifold is complex, then for $p+q=k$, does $\ast$ map $\Omega^{p,q}$ into some $\Omega^{a,b}$, where $a+b=N-k$.
| https://mathoverflow.net/users/1867 | Does the Hodge star operator respect complex structure? | As Abtan requested, I'm converting my comments to an answer:
Suppose that $X$ is an $N$ (complex) dimensional complex manifold endowed with a Hermitean metric, or equivalently a Riemannian metric g satisfying $g(JX,JY)=g(X,Y)$, where $J$ is the complex structure.
Let $\*$ denote the $\mathbb{C}$-antilinear extension ... | 5 | https://mathoverflow.net/users/4144 | 46853 | 29,631 |
https://mathoverflow.net/questions/46854 | 15 | Let $f: \mathbb R^n \rightarrow \mathbb R^n$, where $n> 1$ be a bijective map such that the image of every line is a line.
Is $f$ continuous?
I think it is, but the proof isn't immediately obvious to me.
Related to [this question on math.SE](https://math.stackexchange.com/q/11232/3638).
Feel free to retag.
| https://mathoverflow.net/users/10876 | Continuity in terms of lines | This is called the fundamental theorem of affine geometry. Let $f : E \to E'$ be a map between affine spaces over a field $K$. Suppose that
1. $f$ is bijective;
2. $\dim E=\dim E'\ge 2$;
3. If $a, b, c\in E$ are aligned, then so are $f(a), f(b), f(c)$.
Then $f$ is semi-affine: fix some $a\_0\in E$, then there exis... | 22 | https://mathoverflow.net/users/3485 | 46860 | 29,635 |
https://mathoverflow.net/questions/46855 | 9 | I know how to generate all Abelian groups of order n, but how would I generate the others? I can't seem to find anything about this.
By "generate", I mean produce the Cayley tables for all groups of order n.
| https://mathoverflow.net/users/866 | How to generate all finite groups of order n? | See
Hans~Ulrich Besche, Bettina Eick, and E.A. O'Brien.
A millennium project: constructing small groups.
*Internat. J. Algebra Comput.*, 12:623-644, 2002.
for a description of the construction of groups of order up to 2000. (I believe they narrowly failed to achieve this before the end of the year 2000.) In fact th... | 18 | https://mathoverflow.net/users/35840 | 46861 | 29,636 |
https://mathoverflow.net/questions/46866 | 52 | The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure with perimeter $P'$.
**Napkin conjecture**: You always have $P' \leq P$.
In other words, you cannot increase the ... | https://mathoverflow.net/users/3005 | Is the "Napkin conjecture" open? (origami) | There is a general version of this question which is known as *"the rumpled dollar problem"*. It was posed by V.I. Arnold at his seminar in 1956. It appears as the very first problem in [*"Arnold's Problems"*](http://books.google.co.uk/books?id=LqvLnu8c3ToC&printsec=frontcover&dq=arnold%2527s+problems&source=bl&ots=l0U... | 47 | https://mathoverflow.net/users/5371 | 46867 | 29,638 |
https://mathoverflow.net/questions/46863 | 6 | Assuming the existence of enough large cardinals (I'm not sure whether I mean in the original V or in L(R), do whatever is standard), is the partially ordered class of cardinals order-isomorphic to something simpler?
If so, what is the weakest large cardinal assumption that gives this result?
Is L(R) ≠ L sufficient... | https://mathoverflow.net/users/nan | What does the partially ordered class of cardinals look like in L(R)? | Ricky:
I assume you mean to ask your question in $L({\mathbb R})$.
In general, without choice, the ordering of cardinals tends to be rather pathological, although we do not yet know by how much. Here is an example: It is open whether ZF proves that, if there is no infinite set all of whose members are pairwise inc... | 7 | https://mathoverflow.net/users/6085 | 46868 | 29,639 |
https://mathoverflow.net/questions/46871 | 6 | Background/motivation: I'm investigating the construction of models for a first-order modal system (S5) as products of classical models. Since ultraproducts are all classical models and I need non-classical ones as well, I need to look at reduced products where the filter is not an ultrafilter. This leads me to ask abo... | https://mathoverflow.net/users/8224 | Which properties of ultrafilters on countable sets hold for filters in general? | If you use literally the definitions in Bell and Slomson, only changing "ultrafilter" to "filter," and if, as in the lemma you cited, you're interested only in flters on a countable set, then I believe non-principal is equivalent to $\omega$-incomplete, while "regular" is strictly stronger and "uniform" is strictly wea... | 3 | https://mathoverflow.net/users/6794 | 46876 | 29,641 |
https://mathoverflow.net/questions/46874 | 9 | The intuitive idea is that the sphere connected the two manifolds is not contractible, which implies the (n-1)th homotopy group is not zero. Another argument, which I am not totally understand, uses the fact that the universal covering of an aspherical manifold has only one end. I am wondering if someone here could cla... | https://mathoverflow.net/users/4760 | How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical? | If $N$ is an open contractible manifold of dimension at least two, then it has one end. For instance, you can compute the zeroth reduced homology of $N$ minus a closed ball $B$ by using the long exact sequence for $(N,B)$ by using excision and the fact that $N$ is contractible.
If $M$ is a closed aspherical manifold ... | 20 | https://mathoverflow.net/users/7021 | 46878 | 29,642 |
https://mathoverflow.net/questions/46862 | 1 | Consider the 1-torus $\mathbb{T}$. Let $k$ be a smooth function on $\mathbb{T}^2$ and $K$ be the integral operator on $L^2(\mathbb{T})$ with kernel $k$. One can show that $K$ is of trace class, hence $|K|^{1/2}$ is a Hilbert Schmidt operator=integral operator. But what is the kernel of $|K|^{1/2}$?
| https://mathoverflow.net/users/9401 | Square root of integral operator | It seems to me that you are looking for a formula for the kernel of $|K|^{1/2}$. But, as Gerald, mentioned, such a formula (in the case where the space $\mathbb{T}$ is replaced by a finite set) would give you a formula for the entries of the square root of an arbitrary positive matrix. And I don't think such a thing ex... | 4 | https://mathoverflow.net/users/3698 | 46898 | 29,656 |
https://mathoverflow.net/questions/46907 | 37 | I attended a talk given by W. Hugh Woodin regarding the Ultimate L axiom and I wanted to verify my current understanding of what the search for this axiom means. I find it to be a fascinating topic but the details are so far beyond my grasp.
Given the language of set theory, one can write down a multitude of first-or... | https://mathoverflow.net/users/7154 | Completion of ZFC | I am not sure which statement you heard as the "Ultimate $L$ axiom," but I will assume it is the following version:
>
> There is a proper class of Woodin cardinals, and for all sentences $\varphi$ that hold in $V$, there is a universally Baire set $A\subseteq{\mathbb R}$ such that, letting $\theta=\Theta^{L(A,{\mat... | 54 | https://mathoverflow.net/users/6085 | 46920 | 29,668 |
https://mathoverflow.net/questions/46888 | 6 | Let $P:=P(a\_1,\dots,a\_n)$ be a Pretzel link ( <https://en.wikipedia.org/wiki/Pretzel_link> ).
For every permutation $\sigma\in S\_n$ we can consider the link $$\sigma P:=P(a\_{\sigma(1)},\dots,a\_{\sigma(n)})$$. If $\sigma=(12\dots n)^k$ for some k then $P$ and $\sigma P$ are equivalent.
My questions are:
* Are the... | https://mathoverflow.net/users/5001 | How to distinguish Pretzel links with the same coefficients? | [Richard Bedient](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-112/issue-2/Double-branched-covers-and-pretzel-knots/pjm/1102709604.full) has proved in 1984 that two Pretzel knots $P$ and $\sigma P$ are equivalent if and only if $\sigma$ is a cyclic permutation, an order reversing permutation... | 4 | https://mathoverflow.net/users/6205 | 46925 | 29,671 |
https://mathoverflow.net/questions/46856 | 7 | Is the subgroup of $GL(2,\mathbb Z)$ generated by the matrices
$$ \left( \begin{array}{cc}
1 & 1 \\\
1 & 0 \end{array} \right) \ \ \text{and} \ \
\left( \begin{array}{cc}
2 & 1 \\\
1 & 0 \end{array} \right)
$$
free of exponential growth? More generally, how does one find all the relations between two matrices?
I am... | https://mathoverflow.net/users/8131 | A free subgroup of GL(2,Z)? | I interpret the question "how does one find all the relations between the matrices" as "find a set of defining relations for the group generated by the two matrices".
To do that we need a presentation of $GL(2,\mathbb{Z})$. I found one in the paper:
T. Brady, Automatic structures on Aut$F\_2)$, *Arch. Math.* 63, 97... | 13 | https://mathoverflow.net/users/35840 | 46929 | 29,673 |
https://mathoverflow.net/questions/46926 | 2 | In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ we have
$
u\in C^{\alpha, \gamma}(\overline{U})
$
where $\alpha = k-\left[n/p\right]-1$ and $\gamma = \left[n/p\right]... | https://mathoverflow.net/users/2011 | General Sobolev Inequalities | 1- The Sobolev embedding is proved first in the case $U=\mathbb R^n$, and then for a general $U$ by using a right inverse $j$ of the restriction operator $W^{s,p}(\mathbb R^n)\rightarrow W^{k,p}(U)$. When $U$ is a half-space, a convenient $j$ is Babitch's extension. When the boundary is smooth, use an atlas to reduce t... | 1 | https://mathoverflow.net/users/8799 | 46931 | 29,674 |
https://mathoverflow.net/questions/46932 | 1 | Each book on algebraic geometry write I^2 when it deal with nongsingular varieties, here I
is a ideal sheaf. But no one give the definition. I guess it's the sheafification. It's right?
Thanks.
| https://mathoverflow.net/users/3525 | What is the definition of product of ideal sheaves? | It is the image of the map $I \otimes I \to O \otimes O = O$.
| 2 | https://mathoverflow.net/users/4428 | 46933 | 29,675 |
https://mathoverflow.net/questions/46936 | 10 | Exist simply connected CW complexes $X$, $Y$ and a mapping $f:X\to Y$ with the property that the reduced suspension $\Sigma f:\Sigma X\to\Sigma Y$ is a homotopy equivalence but $f$ is not?
| https://mathoverflow.net/users/10997 | Is a map a homotopy equivalence if its suspension is so? | Whitehead's Theorem (it is Corollary 4.33 in Allen Hatcher's [book](http://www.math.cornell.edu/~hatcher/#ATI)) says that a map between simply connected CW-complexes is a homotopy equivalence if and only if the induced map on homology (with $\mathbb Z$-coefficients) is an isomorphism. If $\Sigma f : \Sigma X \to \Sigma... | 17 | https://mathoverflow.net/users/8176 | 46937 | 29,677 |
https://mathoverflow.net/questions/46934 | 6 | I would like to show that any Zariski-closed subsemigroup of $SL\_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of <http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF> .
Is there a more elementary proof? For $SL\_2(\mathbb{C})$, the result is quite easy to show directly, or using t... | https://mathoverflow.net/users/408 | Zariski-closed subsemigroups of SL_n(C) are groups | It is quite elementary. Let $S$ be the semi-group in question. Then for any $g \in S$, the set
$g^kS$ for $k=1,2, \dots$ is a decreasing sequence of closed sets, hence it has to stabilize. So, $g^kS=g^{k+1}S$ implies that $gS=S$. Hence $S$ is closed with respect to taking inverse, and therefore is a group.
| 24 | https://mathoverflow.net/users/3635 | 46943 | 29,682 |
https://mathoverflow.net/questions/46942 | 13 | Away from the hyperelliptic locus, the moduli of curves immerses
in the moduli of principally polarized abelian varieties. The
ambient space has a riemannian metric, so one can ask about the
second fundamental form, the first-order deviation of the
submanifold from being totally geodesic. What is this second
fundamenta... | https://mathoverflow.net/users/4639 | What is the second fundamental form of moduli space? | The following papers might be useful:
$(1)$ E. Colombo- G. Pirola- A. Tortora
"Hodge-Gaussian maps"
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 125–146.
$(2)$ E. Colombo - P. Frediani
"Siegel metric and curvature of the moduli space of curves"
Trans. Amer. Math. Soc. 362 (2010), no. 3, 1231–... | 3 | https://mathoverflow.net/users/7460 | 46947 | 29,685 |
https://mathoverflow.net/questions/46949 | 15 | **Background**
One of my friends told me the following story: A child must walk from his home at point A = (1,0) to his school at point B = (0,1). The laws in his country state that you can only walk parallel to the horizontal and vertical axis. No matter how he tries to get to school, he finds that he must walk at l... | https://mathoverflow.net/users/1106 | Justifying the definition of arclength | Yes it is true and quite elementary too. The definition of arclength you gave is also known as the (classical) total variation, and you can define it for any curve $\gamma:[a,b]\to M$ in any metric space $(M,d)$ as
$\mathrm{V}(\gamma,[a,b]):=\sup\_P \mathrm{V}(\gamma, P)$
where the supremum is taken over all subdivisio... | 5 | https://mathoverflow.net/users/6101 | 46953 | 29,686 |
https://mathoverflow.net/questions/46957 | 9 | My question: Is the natural filtration of a right-continuous process also right-continuous?
I would say yes, but don't know where to start proving it.
Thanks for your help/ideas!
| https://mathoverflow.net/users/11011 | Right-continuity of natural filtrations | Right continuity fails even for canonical continuous processes.
The natural filtration on $C([0,\infty))$ is not right continuous.
For example, the event $\{\omega: {d^+\over dt}\ \omega\_t\mbox{ exists at }t=0\}$ belongs
to ${\cal F}\_{0+}$ but not ${\cal F}\_0$. In words, you can tell whether
the function $\ome... | 9 | https://mathoverflow.net/users/nan | 46962 | 29,691 |
https://mathoverflow.net/questions/46900 | 42 | The complex irreps of a finite group come in three types: self-dual by a
symmetric form, self-dual by a symplectic form, and not self-dual at all.
In the first two cases, the character is real-valued, and in the third
it is sometimes only complex-valued. The cases can be distinguished by
the value of the Schur indicato... | https://mathoverflow.net/users/391 | Are there "real" vs. "quaternionic" conjugacy classes in finite groups? | It's a great question! Disappointingly, I think the answer to (2) is **No** :
The only restriction on a `good' division into "symmetric" vs. "symplectic" conjugacy classes that I can see is that it should be intrinsic, depending only on $G$ and the class up to isomorphism. (You don't just want to split the self-dual ... | 22 | https://mathoverflow.net/users/3132 | 46964 | 29,693 |
https://mathoverflow.net/questions/46966 | 7 | Consider the circle map $\times d:x\mapsto dx \mod 1$. The lebesgue measure is the only absolutely continuous invariant probability measure, but this map has many other invariant measures. Of course, one can take barycentric combinations of invariant measures to get a new one, so let us restrict to the extremal points,... | https://mathoverflow.net/users/4961 | What do singular, atomless invariant measures of $\times d$ look like? | If you do the Cantor measure construction for d=2, you just get Lebesgue measure... so it's a little bit special.
There are lots of fully supported invariant measures for the map $\times d$: the thermodynamic formalism that you mention gives you a whole zoo of them. In particular, if $\phi\colon [0,1]\to \mathbb{R}$ ... | 3 | https://mathoverflow.net/users/5701 | 46969 | 29,694 |
https://mathoverflow.net/questions/46976 | 0 | a short question: Is every 3-Sasakian manifold a Sasaki-Einstein manifold? If not, do you have an example? If yes, how can I prove this?
Thanks and best regards
| https://mathoverflow.net/users/7028 | Is every 3-Sasakian a Sasakian-Einstein manifold? | Yes, because a manifold is Sasaki-Einstein if and only if its metric cone is Ricci-flat Kähler, whereas the cone of a 3-sasakian manifold is hyperkähler.
See, for instance, Bär's "[Real Killing spinors and holonomy](https://doi.org/10.1007/BF02102106)" published in CMP.
| 3 | https://mathoverflow.net/users/394 | 46978 | 29,699 |
https://mathoverflow.net/questions/46970 | 96 | Recently, I learnt in my analysis class the proof of the uncountability of the reals via the [Nested Interval Theorem](http://personal.bgsu.edu/%7Ecarother/cantor/Nested.html) ([Wayback Machine](http://web.archive.org/web/20180402194030/http://personal.bgsu.edu/%7Ecarother/cantor/Nested.html)). At first, I was excited ... | https://mathoverflow.net/users/5627 | Proofs of the uncountability of the reals | Mathematics isn't yet ready to prove results of the form, "Every proof of Theorem T *must use* Argument A." Think closely about how you might try to prove something like that. You would need to set up some *plausible system for mathematics* in which Cantor's diagonal argument is blocked and *the reals are countable*. N... | 78 | https://mathoverflow.net/users/3106 | 46979 | 29,700 |
https://mathoverflow.net/questions/46971 | 2 | When I took model theory is an undergraduate, early on we wrestled with trying to state the fundamental theorem of arithmetic in the first order language of arithmetic. The problem was that we needed and unbounded and variable number of quantifiers Depending on each natural numbers factorization in the statement. I.e. ... | https://mathoverflow.net/users/10579 | FTA in first order setting | Possibly you will find [this](http://people.cs.uchicago.edu/~laci/REU09/tr9.pdf) somewhat useful.
| 2 | https://mathoverflow.net/users/2926 | 46982 | 29,702 |
https://mathoverflow.net/questions/46732 | 10 | This question grew out of this [post](https://mathoverflow.net/questions/46714/profinite-topologies).
>
> **Question:** Is there a finitely generated, infinite, residually finite group such that every finite index subgroup has $p$-power index for a fixed prime $p$?
>
>
>
The $p$-adic integers $\mathbb Z\_p$ gi... | https://mathoverflow.net/users/8176 | Finitely generated, infinite, residually finite groups whose finite quotients are $p$-groups. | Yes, there are torsion free examples. I do not know who constructed them first, but some examples can be found in papers by Grigorchuk and his co-authors. For example Bartoldi and Grigorchuk proved that a certain Fabrykowski-Gupta group $\Gamma $ has the following properties (see Propositions 6.4 and 6.5 in arXiv:math/... | 11 | https://mathoverflow.net/users/10251 | 46999 | 29,710 |
https://mathoverflow.net/questions/46613 | 0 | For any universal cover p of the wedge S1 V S1 is it true that the two actions of π1(X, x\_0) on the fiber p^-1(x0) given by lifting loops at x0 and the action given by restricting deck transformations to the fiber coincide. I'm not sure but I think you need for π1(X , x0) to be abelian no?
| https://mathoverflow.net/users/10951 | wedge sum deck transformation | It's indeed true that the action by lifting loops and the action by deck transformations agree exactly when the fundamental group is abelian. This statement is a bit vague, so let me be precise.
Let $X$ be a space with universal cover $Y\stackrel{p}{\longrightarrow} X$, and choose basepoints $x\_0 \in X$, $y\_0 \in Y... | 3 | https://mathoverflow.net/users/4042 | 47012 | 29,718 |
https://mathoverflow.net/questions/47005 | 11 |
>
> Let $R$ be a commutative regular local ring. Is it true that for every $\mathfrak p \in \mathrm{Spec}(R)$ there is a finitely generated $R$-module $M$ such that $\mathrm{projdim}(M)=\mathrm{ht}(\mathfrak p)$ and $\mathrm{Ass}(M)=\{\mathfrak p\}$?
>
>
> Or is there some family of commutative noetherian rings whe... | https://mathoverflow.net/users/10931 | Homologically nice commutative rings | My feeling is that this is really about set-theoretic complete intersections.
Let $X={\rm Spec} A$ be a noetherian affine scheme such that every irreducible subscheme of $X$ is a set-theoretic complete intersection. In other words, for any prime $\mathfrak p\subset A$, there exist a set of elements $x\_1,\dots,x\_t\i... | 3 | https://mathoverflow.net/users/10076 | 47018 | 29,722 |
https://mathoverflow.net/questions/47015 | 14 | A polygon $P\_k$ divided by $k-2$ diagonals into triangles is called a polygonal triangulation. These are the vertices of the triangulation graph $\mathcal P\_k$. Two vertices are connected by an edge if one triangulation is obtained from another by the *diagonal flip*, i.e. we take two triangles of the triangulation t... | https://mathoverflow.net/users/nan | The flip graph of triangulations | Just to add a little to Joseph's nice answer, for part 1 of your question: although the problem of computing the flip distance in polynomial time is wide open for triangulations of convex polygons, it can be solved in polynomial time for triangulations of certain highly nonconvex point sets (such as the intersection of... | 9 | https://mathoverflow.net/users/440 | 47026 | 29,728 |
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