parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/142321 | 7 | Suppose that $V$ is a universe of $\sf ZFC$, and $c$ is a Cohen generic real over $V$. Is it possible that $c$ is also generic in other senses? I know that it can't be random or Sacks or whatnot because those reals have different properties with regards to preservation of measure and category, or minimality properties.... | https://mathoverflow.net/users/7206 | How Random is Cohen? | The central facts governing the relation between generic filters arising from different forcing notions are the following:
**Theorem.** Suppose that $G\subset\mathbb{P}$ and $H\subset\mathbb{Q}$ are $V$-generic.
* If $V[G]=V[H]$, then there are conditions $p\in G$ and $h\in H$ such that $\text{RO}(\mathbb{P}\uphar... | 8 | https://mathoverflow.net/users/1946 | 142331 | 77,507 |
https://mathoverflow.net/questions/142295 | 4 | Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by
$$S(t)y=\displaystyle\sum\_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) \rangle \sin(n\pi x).$$
For $t>0, y\in L^1(0,1),$ we have $\|S(t)y\|\_{L^\infty(0,1)}\le C t^{-\frac{1}{2}} \|y\|\... | https://mathoverflow.net/users/40060 | A Laplacian semi-group estimation | A rough estimate is obtained as follows. The $L^{+\infty}$−norm of $S(t)y$ is majorized by $\sum\_{n=1}^\infty e^{-n^2\pi^2 t}|\langle y,\sin(n\pi x)\rangle|$ which in turn is majorized by $\|y\|\_{L^1}\sum\_{n=1}^\infty e^{-n^2\pi^2 t}$ *assuming that* $\langle y,\sin(n\pi x)\rangle$ is intended to mean $\int\_0^1y(s)... | 0 | https://mathoverflow.net/users/12643 | 142343 | 77,512 |
https://mathoverflow.net/questions/142330 | 2 | Are there any known examples of lattice automorphisms of finite order in indefinite lattices being classified up to conjugacy?
| https://mathoverflow.net/users/40080 | Lattice automorphisms of finite order | Yes, there are such examples. You can refer to [this question on MO](https://mathoverflow.net/questions/141284/integral-orthogonal-group-for-indefinite-ternary-quadratic-form/141300#141300) where the automorphism goup of an indefinite lattice is described. In this example, torsion elements are of order 2, and an elemen... | 4 | https://mathoverflow.net/users/39552 | 142350 | 77,515 |
https://mathoverflow.net/questions/142353 | 3 | Let $G = \langle x,y,q ~|~ xy=qyx,~ qx = xq,~ qy=yq\rangle$ is an abstract group.
1) Does this group have a common name? ("quantum tori" is the name of a certain related algebra)
2) What properties it has?
3) In particular, what subgroups it has?
Thanks.
| https://mathoverflow.net/users/8381 | Properties of quantum torus group | This is the integral Heisenberg group, the group of upper unitriangular integer $3\times 3$-matrices. It is nilpotent of class 2, and has many other properties. All subgroups can be described completely.
| 9 | https://mathoverflow.net/users/nan | 142354 | 77,518 |
https://mathoverflow.net/questions/142261 | 6 | What is the relation between omitting types theorems in model theory and the baire category theorem?
| https://mathoverflow.net/users/nan | Omitting types and Baire category | OK, found the thesis.
What you can prove is that the omitting types theorem follows from the Baire category theorem. I haven't thought about whether the other direction holds in the sense of reverse mathematics.
One starts by fixing a signature $\tau$, and assigning an appropriate topology to the space $\mathcal E\... | 6 | https://mathoverflow.net/users/6085 | 142356 | 77,519 |
https://mathoverflow.net/questions/142365 | 0 | I want to know what techniques are known to present a diffeomorphism on a surface with boundary (the diffeomorphism is not necessarily the identity restricted to the boundary) as product of Dehn twists (of course, up to isotopy). I know there are a few books I can look at but can someone briefly describe the methods, o... | https://mathoverflow.net/users/31475 | mapping class group of a surface | $\newcommand{\RR}{\mathbb{R}}$Dehn twists are orientation preserving, so you can never write a reflection as a product of Dehn twists. On the other hand writing an orientation preserving, periodic element $\phi$ as a product of Dehn twists is a non-trivial problem -- that is to say, there are mechanical ways to get the... | 6 | https://mathoverflow.net/users/1650 | 142367 | 77,523 |
https://mathoverflow.net/questions/142312 | 3 | If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order?
The answer is affirmative on abelian groups, where being torsion-free is necessary and sufficient both for having a linear order and for partial orders to extend to linear orders ([Fuchs, 1950](http:... | https://mathoverflow.net/users/26809 | Extensions of partial orders to linear orders on (nonabelian) groups | There are necessary and sufficient conditions in the literature for a (left) partial order $\le$ on $G$ to extend to a (left) linear order $\le^{\ast}$ on $G$. This shows in particular, that not every partial left order extends to a linear left order in the non-abelian case, even though the group is orderable.
In t... | 2 | https://mathoverflow.net/users/32332 | 142373 | 77,526 |
https://mathoverflow.net/questions/142357 | 2 | I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems:
Let $\Omega$ be an open connected subset of $\mathbb{C}^n$ ($n$-tuples of complex numbers) and let $\mathbf{A}(z)$ be a $n\times n$ complex matrix for every $z$ ... | https://mathoverflow.net/users/40090 | How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω | Here the complex structure does not play a special role. It is true more generally that, for an open subset $\Omega \subset \mathbb{R}^m$ and a $C^r$ family of $n\times n$ real matrices
$\mathbf{A}:\Omega\to \mathrm{M} \_ n (\mathbb{R})$ with constant rank, the distribution $\operatorname{ker}\mathbf{A}(x)$ is a $C^r$... | 1 | https://mathoverflow.net/users/6101 | 142377 | 77,527 |
https://mathoverflow.net/questions/142389 | 9 | Since [my question](https://math.stackexchange.com/questions/494476/measures-whose-projections-are-absolutely-continuous) was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projecti... | https://mathoverflow.net/users/38629 | Measures whose projections are absolutely continuous | The answer to the question is that it *almost* exists a characterization, but (as far as I know) there is a gap.
There are a number of *projection theorems* in geometric measure theory, which have the following flavor: if a measure or set in $\mathbb{R}^2$ satisfies some property, then almost all orthogonal projectio... | 9 | https://mathoverflow.net/users/11009 | 142397 | 77,533 |
https://mathoverflow.net/questions/142395 | 30 | By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, in the language of 0,1, + and $\times$, is recursive. Is this field either known to be decidable, or known not to be?
A... | https://mathoverflow.net/users/38783 | Is the field of constructible numbers known to be decidable? | According to the following paper:
Carlos R. Videla, On the constructible numbers, Proceedings of the American Mathematical Society Vol. 127, No. 3 (Mar., 1999), pp. 851-860.
The problem has remained open at least until 1999. I think the problem is still open. In the above paper the author proves that the ring of co... | 26 | https://mathoverflow.net/users/27034 | 142401 | 77,536 |
https://mathoverflow.net/questions/142347 | 2 | $$\displaystyle\min\_{\mathbf{P}} \text{trace}(\mathbf{APP^HA^H}) \quad{} \text{subject to} \quad{} \text{trace}(\mathbf{(I-P)(I-P)^H})=\alpha, \alpha \geq0$$ Can also be rewritten as $$\displaystyle\min\_{\mathbf{P}} \text{trace}(\mathbf{APP^HA^H}) \quad{} \text{subject to} \quad{} \mathbf{\|I-P\|}\_F^2=\alpha, \alpha... | https://mathoverflow.net/users/40087 | Find the transformation $P$ that minimizes the following: | Over the real numbers:
Ok, assume that $\alpha<L$ and the matrices are real.
Let $f(P)=tr(P^TA^TAP), g(P)=K-2tr(P)+tr(PP^T)-\alpha$. By the Lagrange's method, we seek $\lambda\in\mathbb{R}$ s.t. $Df\_P+\lambda Dg\_P=0$, that is, for any matrix $H$,
$tr(H^TA^TAP+P^TA^TAH)+\lambda(-2tr(H)+tr(PH^T+HP^T))=0$, that is
... | 1 | https://mathoverflow.net/users/9091 | 142404 | 77,538 |
https://mathoverflow.net/questions/142371 | 22 | The [law of iterated logarithm](http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm) asserts that if $x\_1,x\_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S\_n=x\_1+x\_2+\cdots+x\_n$, then
$$\limsup\_{n \to \infty} S\_n/\sqrt {n \log \log n} = \sqrt 2, $$ almost surely.
**1. Random matrices**
-------... | https://mathoverflow.net/users/1532 | Laws of Iterated Logarithm for Random Matrices and Random Permutation | My initial answer was wrong. Instead of completely deleting it, I left it at the bottom of the post. I use notation now that slightly differ from my original post.
As before, I give only a partial answer related to the max eigenvalue question and the limsup. Recall that the tail estimates for the max eigenvalue is
$$... | 15 | https://mathoverflow.net/users/35520 | 142405 | 77,539 |
https://mathoverflow.net/questions/142391 | 8 | It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (*i.e.* the singularity of the analytic variety defined by the zero locus of the function), is locally conjugate (by right composition with a local biholomorphism) to a polynomial (in fact, a finite jet o... | https://mathoverflow.net/users/24309 | Local polynomial form of holomorphic functions | The following example of non-algebraic germ is due to Whitney, see
H. Whitney: *Local properties of analytic varieties*, Differential and Combinatorial Topology 205–244, Princeton University Press (1965).
Take the germ of singularity in $\mathbb{C}^3$ given by $f(x,y,z)=0$, where $$f=xy(x+y)(x- zy)(x-e^zy).$$
The... | 11 | https://mathoverflow.net/users/7460 | 142409 | 77,541 |
https://mathoverflow.net/questions/140064 | 5 | What is the Dehn twist factorization of the hyperelliptic involution on an oriented surface of genus g (with one boundary component)?
| https://mathoverflow.net/users/31475 | hyperelliptic involution on a surface | This is almost covered by Proposition 4.12 of Farb and Margalit's book "The primer on mapping class groups". Another useful reference is the paper "Presentations for the punctured mapping class groups in terms of Artin groups" by Labruere and Paris. See the second displayed formula in Proposition 2.12 of that paper.
... | 10 | https://mathoverflow.net/users/1650 | 142410 | 77,542 |
https://mathoverflow.net/questions/142415 | 9 | I think the title speaks for itself. Thus I just explain the story behind the question. Of course, you may want to skip the story.
**Story**: Currently, I teach a course in linear algebra and matrices with a mathematician colleague of mine. In preparing for the class, we discussed together about one of the standard t... | https://mathoverflow.net/users/29316 | Is there any monoid in which the product of two non-invertible elements could be invertible? | Well, why not?
Let $\oplus\_{n \in \mathbb{N}} k$ be a direct sum of countably many copies of a 1-dimensional space over a field $k$; the direct sum affords a standard basis $e\_i = (0, \ldots, 0, 1, 0, \ldots)$ with $1$ in the $i^{th}$ place. Define endomorphisms $A$, $B$ by $A(e\_i) = e\_{i-1}$ for $i \gt 1$, $A(e... | 12 | https://mathoverflow.net/users/2926 | 142416 | 77,544 |
https://mathoverflow.net/questions/142422 | 5 | Suppose $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ a proper subscheme, there is a formal duality isomorphism (here we consider the Zariski topology) due to Hartshorne:
$$ tr: R^n\Gamma\_{Z}(\Omega\_X^n) \cong \mathbb{C} $$
given in proposition 5.2 of his paper,"On the de Rham cohomology of a... | https://mathoverflow.net/users/36931 | Comparison of two traces | The issue of equality of maps in the derived category is always a delicate issue. In the algebraic setting the commutativity is explained in Lipman's Asterisque 117. When you get into the analytic setting, then there is a comparison between the algebraic map and the one coming from De Rham theory that involves a *perio... | 5 | https://mathoverflow.net/users/6348 | 142424 | 77,548 |
https://mathoverflow.net/questions/142426 | 2 | It is conjectured that a divisor on $\overline{\mathcal{M}}\_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}\_{g,n}$. Does the intersection degree with all F-curves identify completely a divisor $D$? I.e.: if $D$ and $D'$ have the same degree once restricted to any F-curve, do w... | https://mathoverflow.net/users/4096 | F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$ | Yes, this is true.
First of all, the map $\newcommand{\MM}{\mathcal{\overline M}\_{g,n}}A^1(\MM) \to H^2(\MM)$ is an isomorphism (say with $\mathbf Q$-coefficients on both sides), so by Poincaré duality it will be enough to prove that $H^{6g-6+2n-2}(\MM)$ is spanned by classes of F-curves.
We do this using the spe... | 3 | https://mathoverflow.net/users/1310 | 142433 | 77,551 |
https://mathoverflow.net/questions/142434 | 7 | How can I find the general solution of $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)?
And how did Euler find the solution $158^4+59^4=133^4+134^4$?
| https://mathoverflow.net/users/40133 | Diophantine equation - $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$) | Euler wrote $a^4-d^4=c^4-b^4$ with $a=p+q, d=p−q, c=r+s$ and $b=r−s$, obtaining
$pq(pp + qq) = rs(rr + ss)$, and then did several other special transformations, until
he arrived at the special solution
$$
a=p+q=2219449, c=r+s=1584749,
$$
$$
b=r−s=−555617, d=p−q=−2061283
$$
which satisfies $a^4 + b^4 = c^4+d^4$.
A... | 8 | https://mathoverflow.net/users/32332 | 142439 | 77,554 |
https://mathoverflow.net/questions/142418 | 7 | Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say *the* standard linear programming problem: we are determining whether $p = \sum\_{i=1}^{V(p)} \lambda\_i v\_i,$ with $\lambda\_i \geq 0,$ $... | https://mathoverflow.net/users/11142 | Is a given point in the interior of the convex hull of a given finite collection of points? | Take your linear program and add the objective function **max** $x$, and the inequalities $\lambda\_i - x \geq 0$. If the point is on the exterior, the optimum solution has $x=0$. Otherwise, there is a solution with $x > 0$.
| 13 | https://mathoverflow.net/users/2294 | 142440 | 77,555 |
https://mathoverflow.net/questions/142454 | 14 | The [Hall marriage theorem](https://en.wikipedia.org/wiki/Hall%27s_marriage_theorem) has several relatively easy combinatorial proofs. Are there short analytical or topological reformulations and proofs of that theorem?
| https://mathoverflow.net/users/nan | An analysis proof of the Hall marriage theorem | At Gil Kalai's [blog](http://gilkalai.wordpress.com/2012/11/25/happy-birthday-ron-aharoni/), Hall’s theorem for hypergraphs (Ron Aharoni and Penny Haxell, 1999) is given, and then it says, "Ron Aharoni and Penny Haxell described special type of triangulations, and then miraculously deduced their theorem from Sperner’s ... | 16 | https://mathoverflow.net/users/3684 | 142458 | 77,560 |
https://mathoverflow.net/questions/33770 | 13 | Let $cd(G)$ be the set of degrees of the irreducible complex characters of the finite group $G$.
It is conjectured that if $G$ is solvable then $dl(G)\leq |cd(G)|$ and it is a result by Gluck that $dl(G) \leq 2|cd(G)|$.
I have managed to show that $dl(G)\leq 2|cd(G)|-3$ and I was wondering if this is a well-known b... | https://mathoverflow.net/users/4614 | Is this a well-known bound for the derived length of a finite group? | In order to try getting this question off the unanswered list, here is an answer:
The result turned out not to be previously known, and has now been published in Communications in Algebra 40 (2012), no. 5, 1856–1859, "Bounding the derived length of a solvable group: an improvement on a result by Gluck" (<http://ams.o... | 12 | https://mathoverflow.net/users/4614 | 142461 | 77,561 |
https://mathoverflow.net/questions/142380 | 1 | It is very well know that groupoids, considered as spaces via the nerve construction, are homotopy 1-types, i.e. aspherical. Here is a sketch of proof: Consider the canonical functor $f:C\rightarrow \pi\_1(C)$ for a category $C$ and its fundamental groupoid $\pi\_1(C)$. The Quillen-fiber based at some object $Y$, i.e. ... | https://mathoverflow.net/users/27923 | Standard way to prove that groupoids are homotopy 1-types | A low tech-argument:
-a groupoid is a disjoint union of connected groupoids
-the nerve construction preserves disjoint unions
-a connected groupoid is equivalent to a group, and thus its nerve is aspherical.
| 3 | https://mathoverflow.net/users/9928 | 142463 | 77,562 |
https://mathoverflow.net/questions/140121 | 3 | Let $cf(\alpha) > \omega$, and $P\_\alpha := \langle P\_{\beta}, \dot{Q}\_{\beta} : \beta < \alpha \rangle$ be a countable support iterated forcing construction (so for each $\beta$, $P\_{\beta} = P\_{\beta - 1} \ast \dot{Q}\_{\beta}$ if $\beta$ is a successor ordinal, $P\_{\beta} =$ inverse limit of $\langle P\_{\gamm... | https://mathoverflow.net/users/29231 | Countable support iterated forcing of length $\alpha$ which forces $|\alpha| > \aleph_2$ | Let each $\dot{Q}\_{\beta}$ be $\{f | f : \gamma \rightarrow \{0, 1\}$ and $\gamma < \omega\_1\}$, ordered by inclusion, as defined in $V[G\_{\beta}]$. Let the length of the iteration, $\alpha$, be $> (2^{\omega})^{++}$, with $cf(\alpha) > \omega$. Then $P\_{\beta}$ forces $|\dot{Q}\_{\beta}| = \aleph\_1$ for all $\bet... | 2 | https://mathoverflow.net/users/29231 | 142464 | 77,563 |
https://mathoverflow.net/questions/142474 | 4 | This is unlikely a research level question... one that would be answered in a blink of an eye, rather...it is an (early) exercise from the book "Analytic Pro-p groups". But since no reply was received when posted on the sibling site, I guess I might try my chances here, risking an immediate closure of the thread (nope,... | https://mathoverflow.net/users/4000 | Layman question: A dense subgroup with completion not isomorphic to the big (pro-p) group? | Let $G = \mathbb{Z}\_p$, and let $H$ be a densely embedded copy of $\mathbb{Z} \oplus \mathbb{Z}$, e.g., $(1,0) \mapsto 1$ and $(0,1) \mapsto \alpha$ for some irrational $\alpha$. Assuming $\hat{H}$ means the pro-$p$ completion of $H$, it is isomorphic to $\mathbb{Z}\_p \times \mathbb{Z}\_p$. This is not isomorphic to ... | 8 | https://mathoverflow.net/users/121 | 142482 | 77,569 |
https://mathoverflow.net/questions/142483 | 5 | Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that $Y=\mathbb{A}^{n}=k[f\_{1},\dots,f\_{n}]$ with $f\_{i}\in K[X]^{G}$.
Then we have a finite flat surjective morphism $\pi:X\rightarrow Y$ generically... | https://mathoverflow.net/users/27398 | quotient by finite group actions that are smooth | No, there is no reason that $X$ should be a complete intersection. For instance, begin with $Z=\mathbb{A}^2\_k$ with coordinates $(z\_0,z\_1)$. Consider the action on $Z$ of the group of $n^{\text{th}}$ roots of unity, $\mu\_n$, by $\zeta\cdot(z\_0,z\_1) = (\zeta z\_0,\zeta z\_1)$. The ring of invariants, $k[z\_0,z\_1]... | 15 | https://mathoverflow.net/users/13265 | 142492 | 77,571 |
https://mathoverflow.net/questions/142459 | 4 | In a profinite group: Does the existence of a countable generating (topologically) set imply the existence of a countable basis for the topology.
| https://mathoverflow.net/users/38889 | Is every countably generated profinite group countably based? | This depends on what you mean by countably generated. If you mean by a countable set of generators converging to 1 then yes, see the Ribes and Zalesskii book.
If you mean a countable set generating a dense subgroup the answer is no. If you take the free profinite group on a countably infinite discrete space X (not i... | 6 | https://mathoverflow.net/users/15934 | 142495 | 77,572 |
https://mathoverflow.net/questions/142505 | 10 | Is there a known characterization of the rings $R$ (containing $1$) with this property: **every element of $R$ is a sum of two commuting idempotents**.
| https://mathoverflow.net/users/nan | rings in which every element is a sum of two commuting idempotents | If $x$ and $y$ are two commuting idempotents, then $(x+y)^3 = x+y+6xy$, $(x+y)^2=x+y+2xy$, so $z=x+y$ satisfies the equation $z^3-3z^2+2z=0$. Plugging $z=3$ into the equation, we obtain $6=0$. So the ring is a product of a ring of characteristic $3$ and a ring of characteristic $2$.
In the ring of characteristic $2$,... | 15 | https://mathoverflow.net/users/18060 | 142506 | 77,576 |
https://mathoverflow.net/questions/141918 | 12 | Are there existing theories of integration in which $I\_0 = \int\_0^{\infty} dx$ and $I\_1 = \int\_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can think of at least two different ways to set up such a theory; in one, $I\_1 = I\_0^2$, and in the other, $I\_1 = I\_... | https://mathoverflow.net/users/3621 | "Values" of divergent integrals | For an attempt of such a theory, see <http://carlossicoli.free.fr/B/Burgin_M.-Hypernumbers_and_Extrafunctions__Extending_the_Classical_Calculus-Springer(2012).pdf>
(Hypernumbers and Extrafunctions: Extending the Classical Calculus, by Mark Burgin).
Link in amazon:
[http://www.amazon.com/Hypernumbers-Extrafunctions-Ex... | 4 | https://mathoverflow.net/users/32389 | 142507 | 77,577 |
https://mathoverflow.net/questions/142511 | 7 | Define the "model theoretic" notion of a closure function as follows:
**Definition (1):** Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the following properties:
$(1)~\forall A\subseteq D~~~~A\subseteq cl(A)$
$(2)~\forall A,B\subseteq D~~~~A\subseteq B\... | https://mathoverflow.net/users/nan | What are the essential properties of algebraic closure on an arbitrary structure? | $\textbf{Question 1}$ The answer to this question is affirmative. The closure operators that you call good closure operators are normally called algebraic closure operators. Furthermore, each algebraic closure operator is of the from $\mathrm{scl}\_{\mathcal{A}}$ for some algebra $\mathcal{A}$. A proof of this fact is ... | 3 | https://mathoverflow.net/users/22277 | 142515 | 77,580 |
https://mathoverflow.net/questions/142493 | 2 | Question:
>
> Given a finite simplicial complex $K$, what general techniques allow one to efficiently compute (a presentation of) the group $\text{Aut}(K)$ of $K$'s automorphisms?
>
>
>
Since this is strictly harder than the corresponding problem for graphs (often solved using [NAUTY](http://www.cs.sunysb.edu... | https://mathoverflow.net/users/18263 | Finding automorphism groups of simplicial complexes | In Sage, at least, [the documentation](http://www.sagemath.org/doc/reference/homology/sage/homology/simplicial_complex.html#sage.homology.simplicial_complex.SimplicialComplex.automorphism_group) suggests what algorithm is used:
```
This is done by creating a bipartite graph, whose vertices are
vertices and facets of... | 5 | https://mathoverflow.net/users/4194 | 142517 | 77,582 |
https://mathoverflow.net/questions/142512 | 3 | *Preamble:*
This question is similar to the one in [total variation distance between two solutions of SDE](https://mathoverflow.net/questions/115090/total-variation-distance-between-two-solutions-of-sde) . The difference is that in my case the drift is the same but there are different diffusion coefficients.
*Questio... | https://mathoverflow.net/users/40173 | Total variation distance between diffusion processes with different volatility coefficient | No. the total variation distance is always one, since the quadratic variation of the processes is different, and so the measures are mutually absolutely singular.
| 4 | https://mathoverflow.net/users/35520 | 142522 | 77,585 |
https://mathoverflow.net/questions/142447 | 2 | In Carter's book (Finite groups of Lie type), he reviews the truncated induction procedure (called j-operation in the text) of Macdonald-Lusztig-Spaltenstein in great detail for the classical Weyl groups. But, as far as I can see, he doesn't say much about how this operation works in the exceptional cases. My questions... | https://mathoverflow.net/users/26208 | Truncated induction for exceptional cases | Maybe I can partially answer your second question by refocusing it somewhat. Carter's chapters 11-13 cover a lot of ground and were hard to organize in a straight line fashion, but the main theme is the study of *unipotent* characters of a finite group of Lie type (consisting of fixed points in a suitable algebraic gro... | 3 | https://mathoverflow.net/users/4231 | 142525 | 77,586 |
https://mathoverflow.net/questions/142481 | 5 | 1. Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
2. I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M are needed to this end... | https://mathoverflow.net/users/2191 | On triangulated categories of pro-objects | 1. See the comment by Karol. Hovey's book on model categories is the standard reference.
2. For triangulation on Ho(pro-M) the relevant reference is [Fausk-Isaksen paper](http://www.intlpress.com/HHA/v9/n1/a16/). See also an earlier [preprint by Isaksen](http://math.wayne.edu/~isaksen/Research/prospectra.pdf).
The em... | 5 | https://mathoverflow.net/users/30641 | 142528 | 77,587 |
https://mathoverflow.net/questions/142532 | 11 | I've seen texts that talk about topological spaces being essentially locales, like *Topology via Logic* by Vickers, and texts related to homotopy theory that talk about topological spaces being essentially infinity-groupoids.
However, I've never seen any treatment of topology that combines these perspectives or talks... | https://mathoverflow.net/users/38334 | What's the link between topological spaces as locales and topological spaces as infinity-groupoids? | I don't think it's a good idea to mix the slogans "topological spaces are locales" and "topological spaces are $\infty$-groupoids." I think the former slogan encapsulates what we ended up defining as topological spaces while the latter slogan encapsulates what we should've defined as topological spaces, at least if we'... | 7 | https://mathoverflow.net/users/290 | 142534 | 77,591 |
https://mathoverflow.net/questions/142540 | 2 | **Preliminaries** A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But a Priestley space also has [connected components as a poset](http://planetmath.org/connectedposet), i.e. the maxima... | https://mathoverflow.net/users/20781 | Are the connected components of a Priestley space closed? | Let $\mathbb{N}\cup\{\infty\}$ denote the one-point compactification. Give $\mathbb{N}\cup\{\infty\}$ the partial ordering where $m>n$ if and only if $m$ is even, $n$ is odd and $|m-n|=1$. In other words, we give $\mathbb{N}$ a zig-zag ordering and we let $\infty$ be a point which is not comparable to any point of $\ma... | 4 | https://mathoverflow.net/users/22277 | 142542 | 77,592 |
https://mathoverflow.net/questions/142546 | 5 | Let $A \rightarrow B$ and $C \rightarrow B$ be two maps of schemes. How can I compute the derived fiber product $A \otimes^L\_B C$? I'm guessing this is a dg-scheme.
For instance - let $B=\mathbb{A}^1$ and $A = 0 \in \mathbb{A}^1$, and $C \in \mathbb{A}^1$ some point (either $0$ or $1$).
**Question**: The example ... | https://mathoverflow.net/users/2623 | $A \otimes^L_B C$ computing the derived fiber product of schemes | As usual, you replace $A$ (or $C$) with a (sheaf of) DG-algebras which is flat over $B$ and compute the usual tensor product of it with $O\_A$ over $O\_B$. In case when $A$ is a complete intersection subscheme a good choice for such DG-algebra is the Koszul complex. Then the derived fiber product is given by the pullba... | 7 | https://mathoverflow.net/users/4428 | 142550 | 77,595 |
https://mathoverflow.net/questions/142485 | 7 | Maybe the answer to my question is obvious.
Let $p$ be a prime $\geq 3$. Let $D$ be an étale $(\varphi, \Gamma)$-module over $A\_{\mathbb{Q}\_p} = \{ \sum\_{n \in \mathbb{Z}} a\_n X^n \, \vert \, a\_n \in \mathbb{Z}\_p, a\_n \to 0 \mbox{ as } n\to -\infty \}$ of finite rank.
Recall that the action of $\varphi$ is g... | https://mathoverflow.net/users/38010 | $(\varphi, \Gamma)$-modules of finite height | *Proposition*: If $D$ is not of finite height, then nor is $D \otimes \delta$ for any $\delta$ of rank 1.
*Proof*: It suffices to show the contrapositive: if $D$ is of finite height so is $D \otimes \delta$ for any rank 1 $\delta$. This follows easily from the fact that every continuous character of $G\_{\mathbf{Q}\... | 9 | https://mathoverflow.net/users/2481 | 142575 | 77,601 |
https://mathoverflow.net/questions/142566 | 2 | Am I right in thinking that a set theory $T$ may interpret a set theory $U$ while it is the case that $T$ has models which are not models of $U\ ?$
In particular, it seems to me that this can happen in case $M$ is a minimal model of $T$.
| https://mathoverflow.net/users/37385 | A question on models and interpretations of set theories | ZFC can interpret ZFC + $V=L$ since the constructible sets in any model of ZFC form a model of ZFC + $V=L$. However, if there is a model of ZFC then there is a model of ZFC + $V\neq L$.
More strikingly, ZFC can interpret [Aczel's AFA](http://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axiom) but since the latter ... | 5 | https://mathoverflow.net/users/2000 | 142582 | 77,603 |
https://mathoverflow.net/questions/142567 | 1 | I'm reading some paper recently. I find a notion which I can not find the exact definition. What is the numerically positive cone in the Neron-Severi group?
| https://mathoverflow.net/users/40042 | What is numerically positive cone | One of the definitions would be the set of real (1,1)-cohomology classes $\alpha$ such that for any analytic cycle $Y\subset X$ of dimension $\dim Y=d$ one has $\int\_Y\alpha^d>0$. A result of Demailly-Paun shows that the Kähler cone is nothing but a connected component of the numerically positive one.
| 1 | https://mathoverflow.net/users/10941 | 142584 | 77,605 |
https://mathoverflow.net/questions/142580 | 1 | Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let $x\in X$. To avoid missing any important conditions, let us for simplicity also assume that $M=\mathbb C^{n\times n}$ i... | https://mathoverflow.net/users/9947 | Are orbits of an affine algebraic monoid affine? | The orbits of a monoid aren't even varieties. Let $M$ be $\mathbb{A}^2$ with the monoid structure $(x\_1, y\_1) \cdot (x\_2, y\_2) = (x\_1 x\_2, y\_1 y\_2)$. Let $M$ act on $\mathbb{A}^2$ by $(x,y) \cdot (t,u) = (xt, xyu)$. Then $M \cdot (1,1)$ is the image of the map $(x,y) \mapsto (x,xy)$, which isn't a variety at al... | 6 | https://mathoverflow.net/users/297 | 142600 | 77,611 |
https://mathoverflow.net/questions/142588 | 5 | Sorry if the question is trivial - are there closed form expressions or good approximations for the sum of a symmetric function taken over all integer compositions (into given number of parts) of a number?
More precisely, I'm interested in:
$$
S(n,k) = \sum\_{a1+ \cdots +a\_k = n, \ \ a\_i \geq 1} \phi\_k(a\_1,\dots... | https://mathoverflow.net/users/7368 | Sum over integer compositions | Since the question is about compositions, there is a generating function approach when $\phi(a\_1,\ldots,a\_n)=\prod\_i f(a\_1)$ for some function $f$. Namely, $S(n,k)$ is the coefficient of $x^n$ in $\left(\sum\_{i\ge 1} f(i)x^i\right)^k$. For example if $f(i)=i$ then $\sum\_{i\ge 1} f(i)x^i=x/(x-1)^2$ so $S(k,n)$ is ... | 7 | https://mathoverflow.net/users/9025 | 142606 | 77,616 |
https://mathoverflow.net/questions/142587 | 17 | (1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a\_j,b\_j|\prod[a\_j,b\_j]=1>$, and given an element in this group, can we determine whether this element can be represented by a simple closed curve?
(2) More specifically, if we consider only a commutative element, whose abelizaions is tri... | https://mathoverflow.net/users/39167 | How to detect a simple closed curve from the element in the fundamental group? | There is no simple necessary and sufficient condition for whether an element of the fundamental group can be realized by a simple closed curve. However, there are a variety of algorithms known. I believe that the first such algorithm is given in
Reinhart, Bruce L.
Algorithms for Jordan curves on compact surfaces.
An... | 10 | https://mathoverflow.net/users/317 | 142607 | 77,617 |
https://mathoverflow.net/questions/142111 | 9 | [I asked this question on StackExchange](https://math.stackexchange.com/questions/490449/what-are-some-characterizations-of-the-strong-and-total-variation-topologies-on) a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on [convergence of measures](http://en.wikipedia.or... | https://mathoverflow.net/users/39080 | What are some characterizations of the strong and total variation convergence topologies on measures? | To summarize the situation. Let $(X,\mathcal{F})$ a measurable space.
The space $M(X,\mathcal{F})$ of all real-valued signed measures on $(X,\mathcal{F})$ is a Banach space wrto the total variation norm $\|\mu\|:=|\mu|(X)$ where the (non-negative) measure $|\mu|:= \mu\_+ +\mu\_-$ is the variation of $\mu$. The space $M... | 10 | https://mathoverflow.net/users/6101 | 142624 | 77,622 |
https://mathoverflow.net/questions/69699 | 5 | Let $p:X \to S$ and $q:Y\to S$ be two objects in the category of ringed spaces over the ringed space
$S$, and let $f:X \to Y$ be a morphism over $S$.
Given a sheaf $\mathcal{F}$ of $\mathcal{O}\_Y$-modules, there are at least
two different ways to produce a morphism
$$
q\_\*\mathcal{F} \to p\_\*f^\*\mathcal{F}
$$
of ... | https://mathoverflow.net/users/1084 | Coherence for pull-backs and push-forwards | I would like to revive this old question by answering it with a pointer to my recent preprint [Obvious natural morphisms of sheaves are unique](http://arxiv.org/abs/1307.4678), which addresses precisely this problem. In particular, I give an example in the first section that discuses the kind of natural morphism you as... | 3 | https://mathoverflow.net/users/6545 | 142628 | 77,624 |
https://mathoverflow.net/questions/141214 | 9 | The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G\_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric situation: The exceptional Lie-group $G\_2$ can be regarded as the stabilizer of a vector in $S^7$ under the action of $Spin(7)... | https://mathoverflow.net/users/39536 | Construction of Thom-Spectrum for G_2-Structures | The second version of the arXiv post of the Crowley-Nordström paper goes into some detail in giving an answer to this question: see Section 7 of version 2. The out-line is as follows:
1) First define stable $G\_2$-structures and stable tangential $G\_2$-bordism.
2) Then make the standard identification of stable ta... | 6 | https://mathoverflow.net/users/8264 | 142639 | 77,628 |
https://mathoverflow.net/questions/142609 | 23 | We consider the field of "usual" linear algebra.
**Q.** Which aspects of it can be carried out without the Axiom of Choice?
**Q.** Do interesting "exotic" phenomena appear in presence of (some instance of) the negation of the Axiom of Choice?
Without Choice, vector spaces may have a basis (hence, in particular, ... | https://mathoverflow.net/users/4721 | Linear Algebra without Choice | Some things about vector spaces which are *consistent* with the failure of choice:
1. Vector spaces may have bases of different cardinality. In particular, this means that the notion of "dimension" is not well-defined. It follows from the Boolean Prime Ideal theorem (which is strictly weaker than $\sf AC$ itself) tha... | 36 | https://mathoverflow.net/users/7206 | 142644 | 77,631 |
https://mathoverflow.net/questions/142647 | 1 | To fix the ideas all curves are supposed to be defined over $\mathbb{C}$. Let $C$ be a rational connected projective curve. Note that we don't assume the curve to be smooth. Let $Aut(C)$ be the group of automorphisms of $C$ in the category of algebraic varieties.
**Q1** When is $Aut(C)$ infinite?
For example, if $C... | https://mathoverflow.net/users/11765 | Automorphisms of rational (connected) projective curves | Let $D$ be the normalization of $C$. The map $D \to C$ comes from a canonical construction, so automorphisms of $C$ lift uniquely to automorphisms of $D$.
So $Aut(C) \subseteq PGL\_2$.
Next note that $Aut(C)$ is contained in the group of automorphism of $D$ that fix the inverse image of the singular points of $C$. ... | 6 | https://mathoverflow.net/users/18060 | 142650 | 77,632 |
https://mathoverflow.net/questions/142565 | 2 | **Question** : Letting $k,n$ be positive integers, let's define a sequence $\{a\_i\}\ (i=0,1,\cdots, kn)$ as
$$(1+x+\cdots+x^k)^n=\sum\_{i=0}^{kn}a\_ix^i.$$
Then, is the 'special' difference-sequence $\{d^Na\_i\}$ a unimodal sequence for every non-negative integer $N$? If the answer is yes, then prove that. If the answ... | https://mathoverflow.net/users/34490 | Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' difference-sequence $\{d^Na_i\}$ a unimodal sequence? | **UPDATE** I now have a complete proof. I'll start with the Pascal's triangle case.
For any sequence of real numbers $r=(r\_1, r\_2, \ldots, r\_k)$, define the number of sign changes of $r$ as follows: Delete all zeroes from $r$, then count how many times the signs of the resulting sequence changes. Recall that Desca... | 8 | https://mathoverflow.net/users/297 | 142657 | 77,633 |
https://mathoverflow.net/questions/141400 | 5 | I've found a paper of Spanier's ([Higher Order Operations](http://www.ams.org/journals/tran/1963-109-03/S0002-9947-1963-0158399-1/S0002-9947-1963-0158399-1.pdf)) where he uses the theory of "carriers" to study $n$-th order operations. The set-up is rather general; for example a particular case defines the Massey triple... | https://mathoverflow.net/users/16785 | Toda brackets and factorisation of a sequence of spectra | Just to close this off - thanks to Mike-Doherty it appears that the answer is yes (and in fact for spaces this goes back to the paper "The decomposition of stable homotopy" by Joel Cohen.)
Using the terminology of Shipley's paper, given a sequence of maps
$$
A\_{n-1} \xrightarrow{f\_{n-1}} A\_{n-2} \xrightarrow{f\_... | 3 | https://mathoverflow.net/users/16785 | 142658 | 77,634 |
https://mathoverflow.net/questions/142654 | 6 | It is easy to find binary quadratic form parameterizations $F(x,y)$ to,
$$a^3+b^3+c^3+d^3 = 0\tag{1}$$
(See the identity (5) described in this MSE [post](https://math.stackexchange.com/questions/381111/).) To solve,
$$x\_1^3+x\_2^3+x\_3^3 = 1\tag{2}$$
in the integers, all one has to do is to check if one term ... | https://mathoverflow.net/users/12905 | On integers as sums of three integer cubes revisited | A parametrization with quadratic forms like you want defines a conic inside the projective cubic surface with equation (3). Such a conic is contained in a plane and the residual intersection of the plane with the surface is a line. Conversely any plane containing a line inside the cubic surface will have as residual in... | 5 | https://mathoverflow.net/users/2290 | 142663 | 77,635 |
https://mathoverflow.net/questions/142213 | 2 | This question is a follow-up to [Galois classes of L-functions](https://mathoverflow.net/questions/122026/galois-classes-of-l-functions). My goal here is to make things clearer.
Definition 1
Let $A$ be a subclass of the Selberg class containing $s\mapsto 1$, closed under products, and such that every element of $A... | https://mathoverflow.net/users/13625 | "good" automorphisms of Galois classes of L functions | The size of the automorphism group of a nontrivial Selberg $L$-function is at most $2$.
The reason is the distribution of zeroes of elements of the Selberg class. Such functions have no zeroes for $Re s>1$. For $Re s<0$, they have only the trivial zeroes. These live in a horizontal strip on linear progressions. In th... | 1 | https://mathoverflow.net/users/18060 | 142665 | 77,637 |
https://mathoverflow.net/questions/142670 | 1 | Is there any general rule to find the number of linearly independent equations such that
$$L\_i(T\_{\mu\nu},\partial\_\eta T\_{\mu\nu},\partial\_\omega\partial\_\eta T\_{\mu\nu},...)=0$$
where $L\_i$ is a multi-linear function.
Here is a simple example:
Given equations
$$\partial\_i b\_{jk}+\partial\_j b\_{ki}+\parti... | https://mathoverflow.net/users/39246 | Number of linear independent equations | In this generality, it is not at all clear what you mean by 'any general rule'. Of course, there *is* a general rule: Compute the rank of the appropriate matrix or linear map. However, computing that rank can easily be nontrivial, and there is no universal way to simplify this problem.
In your specific case, of the s... | 1 | https://mathoverflow.net/users/13972 | 142681 | 77,649 |
https://mathoverflow.net/questions/142581 | 3 | This is a very general question-- I'm really after any references to anything similar...
Let $A$ be a $C^\*$-algebra equipped with a continuous one-parameter group of automorphisms $(\alpha\_t)\_{t\in\mathbb R}$ and let $\pi:A\rightarrow B(H)$ be a (non-degenerate) \*-representation. I'm after conditions which give a... | https://mathoverflow.net/users/406 | Passing automorphism group through a representation |
>
> Borchers, H.-J.
> On the implementability of automorphism groups.
> Comm. Math. Phys. 14 1969 305–314.
> [MathSciNet](http://www.ams.org/mathscinet-getitem?mr=259631) [Project Euclid](http://projecteuclid.org/euclid.cmp/1103841819)
>
>
>
Borchers defines $\pi$ to be *covariant extendible* if $\pi$ is (uni... | 1 | https://mathoverflow.net/users/406 | 142683 | 77,650 |
https://mathoverflow.net/questions/142623 | 5 | If we have two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ over a field $k$, and if we have a Lie algebra homomorphism $\mathfrak{g}\rightarrow \text{Der}\_k(\mathfrak{h})$, then we can define the semi-direct product $\mathfrak{g}\ltimes \mathfrak{h}$: as a $k$-linear space it is just $\mathfrak{g}\oplus\mathfrak{h}... | https://mathoverflow.net/users/24965 | Could we define the semi-direct product of two universal enveloping algebras? | I guess this is what is usually called the cross-product of Hopf algebras, restricted to the case of cocommutative Hopf algebras. The starting relevant paper should be this one, by Susan Montgomery:
<http://link.springer.com/chapter/10.1007/978-94-009-2985-2_22>
and of course also her book "Hopf algebras and their ... | 3 | https://mathoverflow.net/users/6032 | 142688 | 77,652 |
https://mathoverflow.net/questions/142687 | 8 | If $q=p^a$ , where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$ .
| https://mathoverflow.net/users/25674 | The number of conjugacy classes of the simple group PSL(2,q) | There is one conjugacy class of elements of order 2, and if $p$ is odd then there are two conjugacy classes of elements of order $p$. This goes back to Dickson's 1901 book on Linear Groups.
Added later: for order $p$ elements this can be seen as follows. All order-$p$ elements of $PSL(2,q)$ are conjugate to elements ... | 11 | https://mathoverflow.net/users/30412 | 142691 | 77,653 |
https://mathoverflow.net/questions/142519 | 1 | The Rodgers polynomials $C\_{\alpha,q}$ are a particular family of well-known $q$-hypergeometric function. For example, a description can be found here on [Wikipedia](http://en.wikipedia.org/wiki/Rogers_polynomials). For the special case of $q=1$, we get back the famous ultra-spherical polynomials $C\_{\alpha}$ (a desc... | https://mathoverflow.net/users/36946 | $q$-differential equation for the Rodgers polynomials? | Yes.
See <http://arxiv.org/pdf/0704.3123v1.pdf> and references therein.
This is known since long, I guess Koelink's paper was one of the first on the argument...
<http://www.jstor.org/discover/10.2307/2161278?uid=3738296&uid=2129&uid=2&uid=70&uid=4&sid=21102660095777>
Finally let me mention that the famous wor... | 1 | https://mathoverflow.net/users/6032 | 142695 | 77,655 |
https://mathoverflow.net/questions/142685 | 7 | Observe that for any $\epsilon > 0$ there are infinitely many triples of
$c^\epsilon$-[smooth](http://en.wikipedia.org/wiki/Smooth_number) coprime positive integers $a$, $b$ and $c$ such
that $a + b = c$. -- Considering triples of the form $(2^n-1,1,2^n)$
and the factorizations of the polynomials $x^n-1 \in \mathbb{Z}[... | https://mathoverflow.net/users/28104 | Smooth sums of coprime smooth integers | Balog and Sarkozy (Stud. Sci. Math. Hungarica 1984) showed that large $N$ may be written as $x+y+z$ where $x$, $y$, and $z$ are all $\exp(3\sqrt{\log N \log \log N})$ smooth. An analogous result applies to $a+b=c$, answering your question with a bound of the form
$\exp((\log c)^{1/2+\epsilon})$. Much more is expected ... | 10 | https://mathoverflow.net/users/38624 | 142701 | 77,657 |
https://mathoverflow.net/questions/142705 | 0 | I have found similar results here and mathematics stack exchange but they all imposed specific conditions that don't suit this problem in particular. The problem is as follows.
Let A,B be square $n\times n$ matrices such that
1) $[A,B]=0$
2) $AB\neq0$
3) $A^2=B^2=0$
4) $\mbox{Ker}(A)\cap\mbox{Ker}(B)\neq\{0\... | https://mathoverflow.net/users/35099 | Kernel of $AB$ if $[A,B]=0$ and $AB\neq0$? | The result does not hold. Here is a counterexample:
take $X=\begin{bmatrix}
1 \\
0
\end{bmatrix}$, $E=\begin{bmatrix}
1 & 0 \\
0 & 0
\end{bmatrix}$, $F=\begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}$, and consider the matrices
$$A:=\begin{bmatrix}
[0]\_{2 \times 2} & [0]\_{2 \times 2} & I\_2 & [0]\_{2 \times 1} \\
[0]\_{... | 4 | https://mathoverflow.net/users/34951 | 142711 | 77,661 |
https://mathoverflow.net/questions/142713 | 2 | I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows.
Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n \to C^\infty(X)^n$ is a second order vector valued differential operator with elliptic symbol (in my case the second order... | https://mathoverflow.net/users/17965 | vector valued pde's good reference | There is another famous criterion which is used a lot in geometry, which is the Bochner technique. If the system is symmetric (and self-adjoint with Dirichlet conditions) and
there is a good integration by parts formula
$\int Pu \cdot u = \int |\nabla u|^2 + A |u|^2 $
(which would require u = 0 on the boundary) an... | 1 | https://mathoverflow.net/users/17969 | 142718 | 77,665 |
https://mathoverflow.net/questions/142706 | 3 | We know this important fact from A.A.Kirillov that :
Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a [central extension](http://en.wikipedia.org/wiki/Group_extension) of it.
But when can we precisely say that a homogeneous symplectic ... | https://mathoverflow.net/users/nan | Coadjoint orbits and homogeneous symplectic $G$-manifolds | Certainly, if $G$ has a central extension, $G$ will act on the coadjoint orbits of the central extension. So I think your question might be "When can we be sure that our group has no central extensions?"
The central extensions of the Lie algebra are given by elements of $H^2({\mathfrak g})$. So you want that to vanis... | 4 | https://mathoverflow.net/users/391 | 142720 | 77,666 |
https://mathoverflow.net/questions/142715 | 9 | Let $H\_1$ and $H\_2$ be Hilbert spaces.
Let $A\subset B(H\_1)$ be a [factor](http://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors) and $A'$ its commutant.
If a von Neumann algebra $M\subset B(H\_1\otimes H\_2)$ contains $A\otimes 1$ and commutes with $A'\otimes 1$, is it then necessarily of the form $M=A\otim... | https://mathoverflow.net/users/5690 | is this von Neumann algebra a tensor product? | What about $A = A\_1 \oplus A\_2$ and $M = (A\_1\otimes 1) \oplus (A\_2 \otimes B(H\_2))$?
It seems like your condition just says that $A\otimes 1 \subseteq M \subseteq A \otimes B(H\_2)$, did you leave something out?
Edit: in case $A$ is a factor, the answer is yes. Ge and Kadison, On tensor products of von Neuman... | 14 | https://mathoverflow.net/users/23141 | 142723 | 77,668 |
https://mathoverflow.net/questions/142625 | 20 | In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces $L^2(0,T;H^1)$ can be used in linear parabolic PDEs. If we weaken this to consider certain parabolic PDEs with monotone operators... | https://mathoverflow.net/users/35613 | When to use more exciting function spaces than ordinary Sobolev spaces? | Spatial weights would be relevant in non-homogeneous settings in which one expects the behaviour at different regions of space to be different. For instance, if there is an obstacle or a boundary, a weight that depends on the distance to the boundary would be natural in order to capture boundary effects. If the initial... | 19 | https://mathoverflow.net/users/766 | 142726 | 77,669 |
https://mathoverflow.net/questions/142724 | 3 | Let $\left(g\_i\right)$ be a sequence of $N$ elements of a Lie algebra. Let $s$ be a cyclic permutation of $N$ elements of order $N$: $(1,2,...,N)\to(2,...,N,1)$.
Let
$c\_N=\sum\limits\_{k=1}^N[g\_{s^k(1)},[g\_{s^k(2)},[...,[g\_{s^k(N-1)},g\_{s^k(N)}]...]]]$.
We know that $c\_2=0$ by skew-symmetry and $c\_3=0$ by... | https://mathoverflow.net/users/38448 | Jacobi identity for circular permutations | The condition $c\_N=0$ does impose a restriction on a Lie algebra $L$ in general (with the exception of cases where, say, $L$ is nilpotent of class $c$, so that all terms of $c\_N$ are trivially zero for $N>c$). In the theory of identities one searches for a "basis" of identities. Yu P. Razmyslov found a finite basis f... | 1 | https://mathoverflow.net/users/32332 | 142735 | 77,671 |
https://mathoverflow.net/questions/142730 | 1 | Edit: Fix a lattice $N = \mathbb{Z}^n$ and let $N\_{\mathbb{R}} = N \otimes \mathbb{R}$.
Let $C(S)$ be a strongly convex rational cone in $N\_{\mathbb{R}}$ generated by a finite set $S \subset N$, with following two properties:
* No $v \in S$ is in the interior of $C(S)$.
* $S$ generates $N$ as a $\mathbb{Z}$-module.... | https://mathoverflow.net/users/40262 | A question about rational convex cone | Counterexample: lattice is $\mathbb Z^2$, $S = \{(1,0),(1,1),(1,3)\}$, $v=(1,2)$.
| 1 | https://mathoverflow.net/users/38468 | 142737 | 77,673 |
https://mathoverflow.net/questions/142619 | 3 | Let $M$ be a compact Fano Kähler–Einstein manifold, and $V$ a holomorphic $(1,0)$ vector field on $M$. The Fano conditions say that $V = \nabla^{1,0} f$ for some smooth complex-valued function. By Matsushima's theorem, the Kähler–Einstein condition implies that $\operatorname{div} V = \Delta f$ is an eigenfunction of t... | https://mathoverflow.net/users/40220 | Holomorphic vector field on Fano Kähler–Einstein manifold | actually, I just figure out an argument. Take any hol'c (1.0) vector field, say X, then Fano implies $X=\nabla^{1,0}f$, with complex valued f. And $div X=\Delta f$. Now let f=u+iv, then since div X is 1-eigenfunction of complex Laplacian, so is \Delta u and \Delta v, then by matsushima theorem, one knows that $\nabla^{... | 3 | https://mathoverflow.net/users/40220 | 142742 | 77,675 |
https://mathoverflow.net/questions/142743 | 7 | I asked this question on MSE but got no answer. I was advised to put it here. I am sorry if it is not suitable for MO.
What follows is a very nebulous question. I just seek for some help in understanding a "technique" which has proven itself very powerful.
I noticed that very often generating series appear in Algeb... | https://mathoverflow.net/users/30827 | The importance of generating series in Algebraic Geometry | Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the ... | 2 | https://mathoverflow.net/users/38783 | 142754 | 77,680 |
https://mathoverflow.net/questions/137782 | 9 | **BACKGROUND**
A *Salem number* is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their importance is derived for example from the fact that the minimal polynomial of a Salem number,
$$
P(x) = x^{10} + x^... | https://mathoverflow.net/users/35810 | Is the infimum of Salem numbers > 1? | I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the full Lehmer conjecture. It is not a coincidence that the smallest known Mahler measures are realized by Salem numbers, and... | 9 | https://mathoverflow.net/users/26522 | 142759 | 77,683 |
https://mathoverflow.net/questions/142760 | 7 | Does anyone know an example of a complete surface (certainly not necessarily embedded in $\Bbb R^3$) with negative curvature unbounded below? Any example I have computed so far is either not complete or has curvature bounded below. Any light that might be shed would be welcome. (Sadly, this corner of geometry is far fr... | https://mathoverflow.net/users/40245 | A complete surface with $K\to -\infty$ | Kazdan and Warner (MR0343206 (49 #7950) Reviewed
Kazdan, Jerry L.; Warner, F. W.
Curvature functions for open 2-manifolds.
Ann. of Math. (2) 99 (1974), 203–219.
53C20 (35J05 58G15) )
show that a smooth function on a $\mathbb{R}^2$ is the curvature of a complete metric if and only if $\lim\_{r\rightarrow \infty} \... | 13 | https://mathoverflow.net/users/11142 | 142762 | 77,684 |
https://mathoverflow.net/questions/142386 | 6 | Consider the following problem:
*Let $F \subseteq 2^{I}$ be a finite family of finite subsets of some index set $I.$*
*Let $F\_x$ be defined as the number of elements of $F$ that contains $x.$*
*Assume that for each $x \in I,$ $F\_x$ is an even number.
Under what conditions on $F$ can we find a subset $F'$ of $F$... | https://mathoverflow.net/users/1056 | Overlapping sets | Your question is equivalent to whether the combinatorial discrepancy of the dual hypergraph is zero. (In case you are unfamiliar with the notion, start here: <http://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs>) This problem is well-known to be NP-complete, you can easily reduce e.g., 1-in-3 SAT to it.
| 4 | https://mathoverflow.net/users/955 | 142781 | 77,693 |
https://mathoverflow.net/questions/142787 | 7 | **Definition (1):** An inner model of $ZFC$ is a tarnsitive proper class model of $ZFC$ which contains all ordinal numbers. Informally we denote the collection of all inner models of $ZFC$ by $Inn(ZFC)$.
Now if we consider the partial order (reflexive and transitive) $\... | https://mathoverflow.net/users/nan | Is there an inner model between two distinct inner models of ZFC? | Your question is related to the concept of *degrees of constructibility*, where we say that $c\equiv\_L d$ if and only if $L[c]=L[d]$. When $c$ is well-ordered in $L[c]$, then this will be an inner model of ZFC, and so the structure theory of the degrees of construtibility are a part of your informal treatment of inner... | 8 | https://mathoverflow.net/users/1946 | 142797 | 77,701 |
https://mathoverflow.net/questions/142791 | 2 | Let $q$ be odd, $G=PSU\_n(q)$ (Projective Special Unitary group) and $H=PSU\_{n-1}(q)$. Is it always true that $H$ is a subgroup of $G\ ?$
| https://mathoverflow.net/users/30252 | question about projective special unitary group | ${\rm PSU}\_5(3)$ (which is isomorphic to ${\rm SU}\_5(3)$) has a subgroup isomorphic to ${\rm SU}\_4(3)$ (which has centre of order $4$), but none isomorphic to ${\rm PSU}\_4(3)$.
In general ${\rm SU}\_n(q)$ contains ${\rm SU}\_{n-1}(q)$, but factoring out the centre of the first of these, which has order $(n,q+1)$,... | 10 | https://mathoverflow.net/users/35840 | 142799 | 77,703 |
https://mathoverflow.net/questions/142744 | 2 | Let $K$ be a field of characteristic zero but not algebraically closed. Let $C$ be a smooth projective curve over $K$. Let $r, d$ be two positive integers that are coprime. Consider the moduli space of stable vector bundles of degree $d$ and rank $r$ over $C$ with fixed determinantal line bundle. Is it fano? If so coul... | https://mathoverflow.net/users/38832 | Is the moduli space of stable vector bundles over a smooth projective curve fano? | Actually Drezet and Narasimhan prove that the canonical class is (-2n) times the positive generator of the Picard group, where n = g.c.d (r,d). Hence the moduli space (with fixed determinant) is Fano.
| 4 | https://mathoverflow.net/users/40297 | 142802 | 77,704 |
https://mathoverflow.net/questions/142775 | 11 | Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in [Roger Godement](http://godement.eu/site/)'s *Analyse mathématique*, IV, p.313 (perhaps the only treatise on Analysis which contains a statement of the Law in question).
>
> Legendre a deviné la formul... | https://mathoverflow.net/users/2821 | Le Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides | I disagree with Michael Grünewald's interpretation, which by the way doesn't answer the initial question: who Godement is he referring too? I think this is a joke made without acrimony. "Thought police", "innovation preventing", are much too strong phrases to translate Godement's light ironical quotation.
To a french... | 10 | https://mathoverflow.net/users/9317 | 142803 | 77,705 |
https://mathoverflow.net/questions/142766 | 2 | A step in the proof of Proposition 3.3 (p.6) in [this paper](http://arxiv.org/abs/1107.0606) argues the following:
Let $M^n$ be a topological sphere with a Riemannian metric $g$. Now assume further $(M^n,g)$ is locally symmetric, then it must be the standard round sphere by some results from [a paper by Borel](http:/... | https://mathoverflow.net/users/12904 | How to show 'symmetric topological sphere must be round'? | Perhaps the following is not the quickest way, but it works. A locally symmetric simply-connected manifold is symmetric, and hence homogeneous. Homogeneous simply-connected rational cohomology spheres are all classified. References are listed in Section 2.4 in the thesis (<http://opus.bibliothek.uni-wuerzburg.de/vollte... | 0 | https://mathoverflow.net/users/1573 | 142809 | 77,709 |
https://mathoverflow.net/questions/142646 | 2 | A square is bounded by the coordinates (0,0), (0,1), (1,0) and (1,1). Random x and y coordinates are chosen in the interval [0,1] for each of the n points. The n points are then randomly connected to form a cycle. What is the median area of this polygon?
| https://mathoverflow.net/users/40239 | What is the median area of a random n-gon inside a unit square? | Here is a technique which produces not just the median but the distribution of the area for $n=3$ in the square. See also [Johan Philip, "The area of a random triangle in a square"](http://www.diva-portal.org/smash/get/diva2:644463/FULLTEXT01.pdf), but it seems this method is simpler.
Consider the bounding rectangle ... | 6 | https://mathoverflow.net/users/2954 | 142844 | 77,723 |
https://mathoverflow.net/questions/142834 | 4 | Suppose $X\_{\max}$ is the maximum in a sequence $X\_1,X\_2,\ldots,X\_n$ where each $X\_i\sim\chi^2\_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom.
Since chi squared distribution has an exponential tail, for some fixed number of degrees of freedom $k$ we know that $\lim\_{n\rightarrow\infty}... | https://mathoverflow.net/users/18910 | Asymptotic behavior of max of chi-squared distribution | I am sure this must be written somewhere but since I don't know a reference, let me sketch the computation. I hope there is no error in computation below.
As I commented in your related post, what you are dealing with is the sum of $t=k/2$ exponentials (let's assume $k$ is even, the case of $k$ odd is not really that... | 5 | https://mathoverflow.net/users/35520 | 142846 | 77,724 |
https://mathoverflow.net/questions/142808 | 49 | Let us consider the following operation on positive integers: $$n=\prod\_{i=1}^{k}p\_i^{\alpha\_i} \qquad f(n):= \prod\_{i=1}^{k}\alpha\_ip\_i^{\alpha\_i-1}$$ (Is it true that if we apply this operation to any integer multilpe times, it will eventually get into a finite cycle?) Is there a constant $K$ such that any int... | https://mathoverflow.net/users/38267 | Strange (or stupid) arithmetic derivation | I will show that the answer to the second question is "no". Note that if the answer to the first question is "no", we are done. Hence assume that the answer is "yes" and for every number $n$, the chain eventually turns into a cycle. Take any number $n$ and consider a sequence of numbers $n, f(n)(n-1), f(f(n)(n-1))(n-2)... | 22 | https://mathoverflow.net/users/nan | 142851 | 77,726 |
https://mathoverflow.net/questions/142732 | 3 | Let $K$ be a cubic Galois extension of $\mathbb{Q}$.
I wonder if we can find a congruence for prime $p$ such that $p$ does not split completely in $K$. I know that we can do this for quadratic fields, but I am not sure for cubic fields.
Question: Does there exist a congruence for prime $p$, say $p\equiv a\textrm{... | https://mathoverflow.net/users/21090 | Prime splitting in cubic field, congruence | I don't see why the restriction to galois extensions is necessary. Consider, for example, the non-galois cubic field $K = \mathbb Q(\sqrt[3]{2})$. Then no prime congruent to 2 mod 3 splits completely in $K$. Indeed, if $p\equiv 2\pmod 3$, then 2 has a unique cube root mod p, and so the polynomial $x^3 - 2$ factors mod ... | 7 | https://mathoverflow.net/users/430 | 142852 | 77,727 |
https://mathoverflow.net/questions/142850 | 0 | Let $G=PSU\_3(q)$ and $q=p^n$, where $n$ is odd. Can we conclude that $PSU\_3(p)$ is a subgroup of $G$?
| https://mathoverflow.net/users/30252 | question about twisted group of Lie type A_n | Yes, because $SU\_3(q) \subset SU\_3(q^2)$ is the subgroup of elements Galois-conjugate to their inverse. $SU\_3(p)$ is the subgroup of elements defined over $p$ and Galois-conjugate to their inverse. Since the relevant Galois action is the same, they are the same. Then we mod out by the centers, which preserves inclus... | 3 | https://mathoverflow.net/users/18060 | 142853 | 77,728 |
https://mathoverflow.net/questions/130283 | 2 | As part of a different problem, I came across the following simplified question, for which I cannot exhibit a proof nor a counterexample. Note that the assumptions of smoothness and strict positivity of $f$ away from the $y$-axis are important.
Let $f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}$ be a smooth (${\cal C}... | https://mathoverflow.net/users/33889 | Lower bounds on derivative around zero set of a positive smooth function | Carlos pointed out that this works for analytic functions, leaving the general case open.
I believe, in the general case, this is false. I think something like $e^{-1/x^2} \left(\sin\left( y- \frac{1}{x} \right)+1\right)+g(x)$, for $g(x)$ a sufficiently small positive smooth function, should do. This is smooth, becau... | 1 | https://mathoverflow.net/users/18060 | 142856 | 77,729 |
https://mathoverflow.net/questions/142638 | 8 | If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO $P$ with a symbol $p(x,\xi)$ from some reasonable symbol class has a property that $Pf$ has compact $x$-support for any ... | https://mathoverflow.net/users/34984 | Pseudo-differential operators with compactly supported symbols | Yes. Denote by $S\_K$, $K$ compact, the closed subspace of Schwartz space $S$ consisting of all $f$ such that the support of $Pf$ is contained in $K$. The assumption and the Baire category theorem imply that $S=S\_K$ for some $K$. It follows that the $x$-support of the Schwartz kernel $k(x,y)$ of $P$ is contained in $K... | 13 | https://mathoverflow.net/users/nan | 142861 | 77,730 |
https://mathoverflow.net/questions/142858 | 4 | Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H\_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W\_{-k} H\_k(X) \to \mathrm{Gr}^W\_{-k} H\_k(Y) $$ is surjective.
**Question:** Does anyone know a reference for this fact?
I have a proof, but it's not as simple as it could be. And it ... | https://mathoverflow.net/users/1310 | Does anyone know this seemingly simple result in mixed Hodge theory? | See [Lewis's book](http://books.google.com/books?id=uNeu1hjOh7YC&lpg=PA284&ots=gPlXNOtiXu&dq=borel-moore%20weights%20mixed%20hodge%20structure&pg=PA285#v=onepage&q=borel-moore%20weights%20mixed%20hodge%20structure&f=false) (page 285) for the case of projective varieties
| 4 | https://mathoverflow.net/users/40331 | 142868 | 77,733 |
https://mathoverflow.net/questions/142699 | 11 | Let $F$ be a finitely generated free group.
>
> **Question:** Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be willing to write a brief summary answering the following questions.
>
>
> * Which analogues are thought most im... | https://mathoverflow.net/users/1463 | Analogues of the curve complex for Out(F) | I'll take a crack at this (this will be preliminary; I will correct if, as I expect, I forget things).
There is no survey that I know of. Knowledge is moving fast on this topic. It is perhaps too early to stretch too far at guessing which complexes will be important and which will not be, so I will stick to a more de... | 7 | https://mathoverflow.net/users/20787 | 142872 | 77,734 |
https://mathoverflow.net/questions/142871 | 1 | Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications.
People always said that most of the ideas in measurable dynamics system comes from the theory of ordinary differential equation. But when I began to find the topic... | https://mathoverflow.net/users/11966 | whether there are some books and original papers ergodic theory approach to ODE | The canonical reference for the relation between ergodic theory and the theory of ordinary differential equations (as it appears in classical mechanics) is:
V.I. Arnold, [Mathematical methods of classical mechanics](http://rads.stackoverflow.com/amzn/click/0387968903)
| 5 | https://mathoverflow.net/users/11260 | 142874 | 77,735 |
https://mathoverflow.net/questions/142875 | 4 | Assume that $ f : X \to Y $ is a morphism of schemes. Under what situations can we find some non empty open affine subscheme $ {\rm Spec} B \subseteq Y$ such that $ f^{-1}({\rm Spec}B)$ is affine? Here is what I have already thought about:
* If $ f $ is an affine morphism, we just take any open affine subscheme of $Y... | https://mathoverflow.net/users/4002 | Shrinking the target to make the domain affine | I don't know whether this is usefull: a necessary and sufficient condition, when $f$ is of **finite presentation**, is that $X\times\_Y \mathrm{Spec}(O\_{Y,y})$ is affine for some $y\in Y$.
Indeed, the condition is clearly necessary. Conversely, standard arguments show that $X\times\_Y \mathrm{Spec}(O\_{Y,y})$ exten... | 2 | https://mathoverflow.net/users/39387 | 142876 | 77,736 |
https://mathoverflow.net/questions/142873 | 3 | Let $G$ be a simple graph and the length of the longest odd cycle of $G$ is $2k+1$,then I guess the chromatic number of $G$ is no more than $2k+2$,is it right?
| https://mathoverflow.net/users/40096 | chromatic number of a simple graph whose length of the longest odd cycle is 2k+1 | Yes. If a graph $G$ is bipartite, by definition its chromatic number $\chi(G)$ is less than or equal to $2$. If $G$ contains an odd cycle, then $\chi(G) \leq l+1$, where $l$ is the length of a longest odd cycle. I found a couple papers that attribute this result to Erdős and Hajnal, but don't quote me on this. You can ... | 3 | https://mathoverflow.net/users/27829 | 142879 | 77,738 |
https://mathoverflow.net/questions/142870 | 9 | I asked this question [on M.SE](https://math.stackexchange.com/questions/483447/independent-families-versus-generators-in-boolean-algebras) a while ago and got no answers, so I'm asking it here.
Let $\kappa$ be an infinite cardinal. A family $\mathcal{A}\subseteq\mathcal{P}(\kappa)$ is *independent* if for any $A\_1,... | https://mathoverflow.net/users/11233 | Independent families versus generators | I think the answer is always no. Consider the algebra $\mathcal{B}$ generated by the independent family and the ideal $J$ of subsets of $\kappa$ of size $<\kappa$. Then look at $\mathcal{C} = \{ X \subseteq \kappa : \exists Y \in \mathcal{B}, X \triangle Y \in J \}$. It is easy to see that $\mathcal{C}$ is closed under... | 12 | https://mathoverflow.net/users/11145 | 142888 | 77,743 |
https://mathoverflow.net/questions/137593 | 8 | This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(\Omega)$. See the wikipedia page:
<http://en.wikipedia.org/wiki/H%C3%B6lder_condition>
More precisely, I want to know th... | https://mathoverflow.net/users/5656 | Is there a reference for compact imbedding theory of Hölder space? | For non-integer values of $\alpha$, the space $C^\alpha$ has a nice characterisation in terms of wavelet coefficients, see "[Wavelets and Operators](http://books.google.hu/books?id=y5L5HVlh3ngC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false)" by Yves Meyer. With that characterisation (essentially a weighted $\ell^\infty$ bou... | 2 | https://mathoverflow.net/users/38566 | 142904 | 77,749 |
https://mathoverflow.net/questions/142915 | 4 | I am struggling trying to understand an statement in a paper I am reading:
Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose real and imaginary parts are real analytic functions. Let $diag$($M$,$M$) be the diagonal of $M\times M$. Then there is a ho... | https://mathoverflow.net/users/40090 | Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$ | The statement you require (IIUC a simple consequence of analytic continuation) is e.g. in Bourbaki, [*Variétés différentielles et analytiques*, Fascicule de résultats, 5.14.7, page 60](http://books.google.com/books?id=bHIz7KNeuNIC&pg=PA60). It requires the manifold to be paracompact.
| 4 | https://mathoverflow.net/users/19276 | 142919 | 77,756 |
https://mathoverflow.net/questions/142913 | 15 | Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially an application of Stokes theorem to derive the following identity
$$\nabla^\*\_X = - \nabla\_X - div(X)$$
for the forma... | https://mathoverflow.net/users/17047 | Formal adjoint of the covariant derivative | **Ad 1:** Yes, there is. The formula is
$$\nabla^\*(X^\flat \otimes u) = - \nabla\_X u -\mathrm{div}(X) \cdot u,$$
as can easily seen by local computation. Here, $X$ is a vector field and $X^\flat$ is the dual one form w.r.t. the metric.
Note that unless you have a scalar product on the bundle $E$, too, the dual oper... | 13 | https://mathoverflow.net/users/16702 | 142933 | 77,759 |
https://mathoverflow.net/questions/142621 | 17 | If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by gluing the boundary components of $N^2\times [0,1]$ via $\phi$. In other words, $M^3$ is fibered over $S^1$, with fiber $... | https://mathoverflow.net/users/1446 | Hyperbolic manifolds which fiber over the circle | I make a remark here that is well-known to experts, that $\pi\_1(M)$ is an extension of $\pi\_1(N)$ by an infinite-order outer automorphism, explaining partly Misha's comment.
Consider the sequence $\pi\_1(N^{k-1}) \to \pi\_1(M^k) \overset{\varphi}{\to} \mathbb{Z} $. Choose $\alpha\in \pi\_1(M)$ such that $\varphi(\a... | 11 | https://mathoverflow.net/users/1345 | 142934 | 77,760 |
https://mathoverflow.net/questions/142927 | 4 | If I understood this correctly, the Gibbs Specification for the [Ising model](http://en.wikipedia.org/wiki/Ising_model#No_phase_transitions_in_finite_volume) on $ℤ^d$ dos not have a unique [Gibbs Measure](http://en.wikipedia.org/wiki/Gibbs_measure) for β above the critical level. But what about the Ising model [on a fi... | https://mathoverflow.net/users/39855 | Uniqueness of Gibbs Measure on Ising model | From the physics point of view, the answer to your question is an immediate "yes": a nonunique Gibbs measure arises if there is a phase transition into a phase with multiple ground states (say, a phase transition into a ferromagnetic state with all spins aligned either up or down). A finite system has no phase transiti... | 6 | https://mathoverflow.net/users/11260 | 142935 | 77,761 |
https://mathoverflow.net/questions/142912 | 3 | Let $X$ be a variety in $\mathbb{C}^{10}$ defined by the ideal
$I=\left<xz'-x'z, y'(u+z)-y(u'+z'), t'(u-z)-t(u'-z')+xy'-x'y\right>$ of $\mathbb{C}[x,y,z,u,t,x',y',z',u',t']$.
Note that $I+\left<u-z, u'-z'\right>$ gives a determinantal variety $D$ defined over the matrix
$ \left( \begin{array}{ccc} x & y & z \\ x' & y... | https://mathoverflow.net/users/39715 | rational singularities of a certain variety | Ok, $X$ is a complete intersection (dimension 7, cut out by three equations in $\mathbb{C}^{10}$).
You can then use the **hasRationalSing** function described [HERE](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6/share/doc/Macaulay2/Dmodules/html/_has__Rational__Sing_lp__List_rp.html) in the D-modules package ... | 5 | https://mathoverflow.net/users/3521 | 142944 | 77,763 |
https://mathoverflow.net/questions/142938 | 21 | I was asked two questions related to Diophantine equations.
1. Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all solutions but no points which do not satisfy the equation.
2. Is there an algorithm that will determine, given any qu... | https://mathoverflow.net/users/40368 | Is there an algorithm to solve quadratic Diophantine equations? | Sierpsinski proved that whenever this diophantine equation does not have a solution for $x$ then $x^2+(x+1)^2$ is prime. It is conjectured also by Sierpinski that there are infinitely many primes of the above form but this still remains open.
So, your question cannot be answered yet. For more details see <http://arxi... | 16 | https://mathoverflow.net/users/38851 | 142945 | 77,764 |
https://mathoverflow.net/questions/142937 | 18 | What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly elaborate on the motivation behind defining multiplier ideal sheaves.I think there are lots of experts here in mathoverflow who... | https://mathoverflow.net/users/30081 | motivation for multiplier ideal sheaves | Here is a sketch of Nadel's original motivation. Classical results of Aubin and Yau imply the existence of Kahler-Einstein metrics on manifolds with ample canonical bundle and and for all polarisations of Calabi-Yau manifolds. The method involved is a continuity method for the complex Monge-Ampère equation (see for exa... | 14 | https://mathoverflow.net/users/22294 | 142946 | 77,765 |
https://mathoverflow.net/questions/142947 | 3 | Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x\_1,\ldots,x\_n$.
---
**Construction 1:** it is well-known that the conformal class $[g]$ of the metric makes of $S$ a Riemann surface, denoted by $X$. Then to the pair $(S,g)$, one can associate... | https://mathoverflow.net/users/40381 | Euclidean surfaces with conical singularities and cusped hyperbolic surfaces | Your description is a little too abstract: it is easier to think of this geometrically [in a special case].
Fact (Rivin, 1991): Every complete finite area hyperbolic metric on a sphere with punctures can be realized in a unique way as the induced metric on a convex ideal polyhedron $P$ in $\mathbb{H}^3.$ Such a metri... | 3 | https://mathoverflow.net/users/11142 | 142956 | 77,767 |
https://mathoverflow.net/questions/142955 | 5 | All varieties are assumed to be projective over $\mathbb{C}$. Let $f\_1: Y \to X$ and $f\_2: Y' \to X$ be étale morphisms with same finite Galois groups (to be honest, I don't know what does Galois group really mean in this context), then can we conclude $Y,Y'$ are birational?
Even if the general situation might have... | https://mathoverflow.net/users/29730 | Étale covers and birationality of varieties | I think, for an elliptic curve $E/\mathbf{C}$, there are étale coverings $E\_i \to E$, $i = 1,2$ of degree a prime number (hence with the same Galois group; the étale fundamental group of an elliptic curve is Abelian), with $E\_1 \not\cong E\_2$.
| 3 | https://mathoverflow.net/users/nan | 142957 | 77,768 |
https://mathoverflow.net/questions/142954 | 5 | Particularly, how to characterize a set of chaotic nonlinear dynamical systems as a subset of nonlinear dynamical systems with respect to the set cardinality?
To explain the question more, a simple similar question would be "if I point to an arbitrary number on a real number line, what is the probability of this numb... | https://mathoverflow.net/users/40383 | What is the probability of an arbitrary nonlinear dynamical system to be chaotic? | This is an important and interesting question; unfortunately it's very hard to say anything in much generality.
First one needs to establish within which class of dynamical systems the question is being asked, and then to establish a notion of "largeness" within that class. Cardinality is not the most useful because ... | 8 | https://mathoverflow.net/users/5701 | 142961 | 77,770 |
https://mathoverflow.net/questions/142692 | 8 | Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.
>
> My Question:
> What is the geometric analogue of the restriction functor $Res^G\_L:Rep(G)\to Rep(L)$?
>
>
>
To be a little bit more precise:
Let $\check{G}$ be the dual group ... | https://mathoverflow.net/users/32972 | Restriction to Levi Subgroups and the Affine Grassmannian | On the level of sheaves, the construction is indeed a pull-push formula. I believe it was first worked out by Beilinson and Drinfeld in section 5.3 of their preprint "Quantization of Hitchin's Hamiltonians and Hecke eigen-sheaves". For published references, look at section 2.4 in Braverman and Gaitsgory's paper "Crysta... | 5 | https://mathoverflow.net/users/nan | 142966 | 77,771 |
https://mathoverflow.net/questions/142462 | 1 |
>
> Definition: A **$k$-factor** of a graph is a spanning $k$-regular
> subgraph.
>
>
> Definition: A **$k$-factorization** of a graph is a partition of the edge
> set into $k$-factors.
>
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Petersen's celebrated theorem says that a $2m$-regular graph has a $2$-factorization. A corollary is that a $2m$-regu... | https://mathoverflow.net/users/22051 | Converse of Petersen's 2-Factorization Theorem | If I understand your question correctly, the answer is yes and follows directly from Petersen.
Let $H\_1$ be your $2k$-factor, and $H\_2$ the $(2m-2k)$-factor that remains of $G$ after deleting the edges of $H\_1$. Now apply Petersen to each of $H\_1$ and $H\_2$ to get $2$-factorizations in each, which combine to a $... | 5 | https://mathoverflow.net/users/12487 | 142977 | 77,774 |
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