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https://mathoverflow.net/questions/108792 | 3 | It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the fundamental weights are irreducible if the group is either $SL\_n$ or $SO\_n$ (Wong 71). For the symplectic group, this is no... | https://mathoverflow.net/users/26032 | Irreducibility of fundamental Weyl modules | It is true in types $G\_2$, $F\_4$ and $E\_6$, but in types $E\_7$ and $E\_8$ one has to exclude $p=7$, $p=13$ and $p=19$ as well (I think $p=19$ occurs only in type $E\_8$). More on this can be found in Jantzen's paper "First cohomology groups for classical Lie algebras" published in Progress in Math., vol. 95, 1991 (... | 8 | https://mathoverflow.net/users/24386 | 108811 | 62,647 |
https://mathoverflow.net/questions/108809 | 7 | In the related literature one often sees the phrase "The missing Moore graph" which (to me) tacitly implies that the missing Moore graph (if exists) is unique.
Is there a result of this type or is or is this just a limitation of words that do not express the fact that there could be nonisomorphic graphs of diameter 2... | https://mathoverflow.net/users/1737 | (The) missing Moore graph(s) - uniqueness | The uniqueness of a Moore graph of degree 57 and diameter 2 is not known. See, for instance, this paper -
<http://www.sciencedirect.com/science/article/pii/S0024379509003735>
- where they refer to `the missing Moore graph(s)' to indicate this fact.
Other discussion can be found here:
<http://symomega.wordpress.com/20... | 14 | https://mathoverflow.net/users/801 | 108812 | 62,648 |
https://mathoverflow.net/questions/108770 | 9 | Let $M$ be a 3-smooth manifold and $g\_{1}$ and $g\_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K\_{1}$ and $K\_{2}$ be the Gaussian curvatures of $S$ with respect to the induced ambient metrics $g\_{1}$ and $g\_{2}$.
Question : Has anyone already studied the problem of finding e... | https://mathoverflow.net/users/27001 | Surfaces in a 3-manifold with the same Gaussian curvature with respect to two ambient conformal metrics | I don't know the answers to your questions, i.e., I don't know whether there has been any work already done on this problem, nor do I know whether there is any kind of 'heat flow' argument for constructing solutions.
However, I suspect that the latter, if possible, is not going to be completely straightforward. Afte... | 12 | https://mathoverflow.net/users/13972 | 108813 | 62,649 |
https://mathoverflow.net/questions/108783 | 8 | Let $G$ be a group (if it helps, assume that $G$ is a Lie group or finite). Is a pair of elements $(g, h) \in G \times G$ determined up to simultaneous conjugacy by the conjugacy class of every element $w(g, h) \in G$, where $w$ runs over all words in the free group on two generators?
If $G$ is finite, can we bound ... | https://mathoverflow.net/users/290 | Are two elements of a group determined up to simultaneous conjugacy by the conjugacy classes of all of their products? | For a concrete example consider the symmetric group on 6 symbols. The pairs ((1,2)(3,4),(1,3)(2,4)) and ((1,2)(3,4),(3,4)(5,6)) both generate Klein 4-groups.
Words in them are either trivial of conjugate to (1,2)(3,4).
The generated subgroups (and thus the pairs) are not conjugate as one fixes two points.
| 12 | https://mathoverflow.net/users/22250 | 108817 | 62,651 |
https://mathoverflow.net/questions/108818 | 3 | Let $Q$ follow subspace topology from $R$
Then I think it is true that $Q^n$ and $Q^m$ (with product topology) are not homeomorphic.I also think it will be possible to define "rational" homotopy groups by considering $[0,1]\cap Q$ etc. But I can't find any reference regarding topology of product spaces of rationals in ... | https://mathoverflow.net/users/21488 | product spaces of rationals | **Edit:** The following is a second attempt to repair problems in earlier proposed solutions, as pointed out by Gerald Edgar in comments. Hopefully this time I've gotten it right this time.
It's classical (by a back-and-forth argument; see for instance A Shorter Model Theory by Hodges) that any two countable dense u... | 7 | https://mathoverflow.net/users/2926 | 108821 | 62,652 |
https://mathoverflow.net/questions/86334 | 5 | Let $T(R)$ denote the space of tempered functions on the line,
i.e. the smooth functions that give Schwartz function after a
multiplication by any Schwartz function, equipped with the natural
nuclear topology. e.g. the topology induced from the strong
(convergence on bounded sets) topology on the endomorphi... | https://mathoverflow.net/users/4690 | Schwartz Kernel theorem for tempred functions | This is proved in 4.1 of:
Michel Dubois-Violette, Andreas Kriegl, Yoshiaki Maeda, Peter W. Michor: Smooth \*-algebras. Progress of Theoretical Physics Supplement 144 (2001), 54-78. arXiv:math.QA/0106150.
[pdf](http://www.mat.univie.ac.at/~michor/ncdg-smo.pdf)
According to L. Schwartz, there are two kind of tempered ... | 4 | https://mathoverflow.net/users/26935 | 108826 | 62,654 |
https://mathoverflow.net/questions/108831 | 3 | Let $K$ be a number field, and let $X\_K$ be a $K$-variety.
The etale fundamental group of $X\_K$, as defined in SGA1, classifies the automorphisms of finite etale covers of $X\_K$. Some of these etale covers are not geometric. For example, if $L$ is a finite field extension of $K$, then $X\_L\rightarrow X\_K$ is a f... | https://mathoverflow.net/users/5756 | Why is there no "regular etale fundamental group"? | $K = \mathbb{Q}$, $X\_K = \mathrm{Spec} \mathbb{Q}[t, t^{-1}]$. Let $Y$ and $Z$ be the covers of $X\_K$ corresponding to adjoining $\sqrt{t}$ and $\sqrt{-t}$ respectively. They are both regular covers, but their composite contains $\sqrt{-1}$ and thus is not.
| 11 | https://mathoverflow.net/users/297 | 108836 | 62,659 |
https://mathoverflow.net/questions/106926 | 10 | I have been able to find a lot of information on the **category of contexts** -- for example, the page on [syntactic categories](http://ncatlab.org/nlab/show/syntactic+category) at the nLab is a good starting point. However, when I try to find similar information on the **category of judgements**, I find a whole lot le... | https://mathoverflow.net/users/3993 | Category of Judgements? | One way to set up a category is to use contexts as objects, and declare that a morphism from the context $\Gamma = x\_1 : A\_1, \ldots, x\_m : A\_m$ to the context $\Delta = y\_1 : B\_1, \ldots, y\_n : B\_n$ is an $n$-tuple $(t\_1, \ldots, t\_n)$ where $$\Gamma \vdash t\_i : B\_i.$$
Composition is given by substitution... | 8 | https://mathoverflow.net/users/1176 | 108841 | 62,661 |
https://mathoverflow.net/questions/108825 | 1 | Hello,
This problem looks very simple and I conjecture it's true but I have a hard time proving it. It'd be very useful for my work (I'm doing a PhD) and I'll be glad to cite you in a future article if you help me.
Let $P$ be a random permutation of $\mathbb{Z}/N\mathbb{Z}$ with the condition that $P$ verifies $q$ ... | https://mathoverflow.net/users/22976 | Conditional probability with permutations | My second comment indicates that I think you need to amend your conjecture, or else I don't understand.
Allow me to sketch some relevant ideas for getting non-trivial lower bounds. If my sketch does not suffice, I'll try to flesh it out when I have more time.
One should think of permutations here in terms of their... | 2 | https://mathoverflow.net/users/10909 | 108843 | 62,662 |
https://mathoverflow.net/questions/108844 | 0 | Following question: Let's assume that W is a wellfounded set, i.e. it has a partial order and every nonempty subset of W has minimal elements with respect to the order.
Now we can easily define a binary relation 'preceeds' with the definition
a.preceeds(b) = b.is\_minimal({x: a < x})
I am not able to prove that t... | https://mathoverflow.net/users/27009 | wellfounded sets and predecessors | Sure. Just take the natural numbers with the usual ordering, and slap on a new maximum element. This is well-founded (it is the ordinal $\omega + 1$), but the maximum element has no predecessor, despite not being minimal either.
| 2 | https://mathoverflow.net/users/3902 | 108845 | 62,663 |
https://mathoverflow.net/questions/108819 | 2 | For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E\_{1}^{p,q}=\oplus\_{i\_{0}< \cdots < i\_{p}} H\_{ Y\_{i\_{0} \cdots i\_{p} }}^{q} (X, F) \Longrightarrow H\_{Y}^{q-p}(X, F).$
Here, $X$ is a scheme, $Y\to X$ is a closed subscheme, an... | https://mathoverflow.net/users/39742 | Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings? | I believe that cycle groups do not behave nicely for singular varieties. On the other hand, one could define certain reasonable motivic cohomology via Voevodsky's motives. In any case, look at (section 2 of) the paper <http://www.math.uiuc.edu/K-theory/0529/glr.pdf>
| 2 | https://mathoverflow.net/users/2191 | 108847 | 62,664 |
https://mathoverflow.net/questions/108823 | 0 | For an *alternating* knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1
kinks exist (K has minimal crossing number =C). Call black-white areas D and the writhe W.
Then for the Thurston-Bennequin numbers of K ... | https://mathoverflow.net/users/11504 | Thurston-Bennequin number vs. checkerboard coloring difference | The first identity you conjecture is true only for alternating knots.
The identity for alternating knots is mentioned in one of [Lenny Ng's papers](http://arxiv.org/pdf/math/0612356.pdf) (at the end of page 3), where he also gives references and a sketch of the proof.
In the same paper, he reproves a result of Mats... | 4 | https://mathoverflow.net/users/13119 | 108850 | 62,665 |
https://mathoverflow.net/questions/108829 | 0 | Consider 2 related aspects of a process for prices in a financial market:
* time &
* return.
**Time**
Say I've identified a distribution that reasonably models the occurrence of the lengths of price trends (how long in time the price moves up or down). Given the length of a given trend up to some point in time (i... | https://mathoverflow.net/users/25999 | Can one combine (join) probabilities from 2 aspects of a related process? | I think I've found a simple solution to what I wanted to do. Others here may (likely) suggest a way to streamline the approach. Any vetting of the idea much appreciated.
So, I generated equal length sets of random variates from each of the distributions mentioned above and did a linear model fit of the 2 data sets us... | 0 | https://mathoverflow.net/users/25999 | 108865 | 62,670 |
https://mathoverflow.net/questions/108842 | 9 | On pg. 1 of the [slides of a talk](http://www.math.mcgill.ca/darmon/slides/harvard08/slides.pdf), Henri Darmon wrote:
>
> **Question:** What is an interesting Diophantine equation?
>
>
> **A “working definition”**. A Diophantine equation is interesting
> if it reveals or suggests a rich
> underlying mathemati... | https://mathoverflow.net/users/683 | The significance of modularity for all Galois representations | Your question reminds me of a current strain of research whose starting point is Serre's conjecture, now the Khare-Wintenberger Theorem:
>
> any continuous odd irreducible two-dimensional Galois representation over a finite field arises from a modular form
>
>
>
The question one might ask is then "Where are th... | 10 | https://mathoverflow.net/users/3384 | 108867 | 62,672 |
https://mathoverflow.net/questions/108861 | 5 | Are there any examples of $H < G$ such that for any pmp ergodic action of the group $G$ on a standard proba space $(X,\mu)$ there exists a set $A$ of $\mu(A)>C$ such that the action of the subgroup $H$ on $A$ is ergodic? It seems that this can happen only in finite index case...
**Conclusion:**
The combination of the... | https://mathoverflow.net/users/8699 | Ergodic action of a subgroup | We seem to be talking past each other in the limited space provided by the comments, so maybe I can express myself better in the room provided by the answer box. You indicated that you were focused on countable discrete groups. For countable discrete $G$ and $H < G$ the following are equivalent.
1. There exists a con... | 6 | https://mathoverflow.net/users/14913 | 108873 | 62,677 |
https://mathoverflow.net/questions/108855 | 1 | I'm looking for an example of the following situation, related to Max Noether's AF+BG Theorem (see Bill Fulton's book on algebraic curves, page 61, at <http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf>).
Fulton motivates the AF+BG Theorem by saying that if $F, G, H$ are curves in the projective plane with no com... | https://mathoverflow.net/users/27011 | Max Noether's AF+BG theorem | Define $F$ as $x^2-y^2=0$, $G$ as $x^2+y^2=0$ and $H$ as $xy=0$. (here $(x:y:z)$ are homogeneous coordinates in $\mathbb P^2$.)
It is clear that $H$ can not be expressed as $AF+BG$. At the same time if we take $B=0$, then
$H\cdot F=B\cdot F+G\cdot F=4\cdot (0:0:1)$.
| 2 | https://mathoverflow.net/users/943 | 108874 | 62,678 |
https://mathoverflow.net/questions/108877 | 3 | if E is a splitting field for a division algebra D, is it always true that E can be embedded in D? Jacobson (BA II, Th. 4.8) only states that E can be embedded into Mat(r,D) for some r.
| https://mathoverflow.net/users/27018 | splitting field for a division algebra | It depends on what you mean by "splitting field". What is true is that for a division algebra $D$ (over $F$) of dimension $n^2$, if $E$ is a field extension of degree $n$ over $F$ that splits $D$, then $E$ is isomorphic to a subfield of $D$. See the Theorem on page 16 of Paul Garrett's [notes](http://math.umn.edu/~garr... | 6 | https://mathoverflow.net/users/6753 | 108878 | 62,679 |
https://mathoverflow.net/questions/108846 | 11 | In [another question](https://mathoverflow.net/questions/107500/counting-higher-dimensional-abelian-varieties-of-a-given-conductor) I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, [one can give a proof using Baker's ... | https://mathoverflow.net/users/683 | Modularity of higher dimensional abelian varieties | There has been recent work in some concrete cases. There's a paper by [Poor and Yuen](http://arxiv.org/abs/0912.0049) that gives computational evidence for a special case of the so-called Paramodular Conjecture. This Conjecture is described as "a precise and testable modularity conjecture for rational abelian surfaces ... | 7 | https://mathoverflow.net/users/16419 | 108880 | 62,681 |
https://mathoverflow.net/questions/108871 | 6 | I tend to ask questions on mathstackexchange, because I don't feel adequate yet for mathoverflow. I had previously asked this question there (I have now deleted it), where it was quite popular, but it didn't seem like anybody knew the answer. I thought that perhaps this question should graduate to mathoverflow. Here it... | https://mathoverflow.net/users/27015 | Lifting the Frobenius to the absolute Galois group of the $p$-adics | As others have mentioned, the exact sequence is continuously split. Furthermore, any lift of Frobenius will act as you expect on the subfield $\mathbb{Q}\_p^{nr}$, i.e., the extension you get by adjoining all prime-to-$p$ roots of unity. Indeed, you can form a $\mathbb{Q}\_p$-basis of $\mathbb{Q}\_p^{nr}$ using roots o... | 4 | https://mathoverflow.net/users/121 | 108885 | 62,685 |
https://mathoverflow.net/questions/108860 | 24 | I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.
| https://mathoverflow.net/users/24541 | Anabelian geometry study materials? | There is this very beautiful survey
Nakamura, Hiroaki; Tamagawa, Akio; Mochizuki, Shinichi
The Grothendieck conjecture on the fundamental groups of algebraic curves
<http://www.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf>
You could also have a look at
Szamuely, Tamás
Heidelberg Lectures on Funda... | 15 | https://mathoverflow.net/users/11682 | 108897 | 62,691 |
https://mathoverflow.net/questions/108820 | 2 | The action of $GL\_6$ on $P(\wedge^3 \mathbb{C}^6)=P^{19}$ has 4 orbits (of dim 19, 18, 14, 9). Can you describe how the springer resolution applies to each of these orbits? It should have positive dimensonal fibers over the 14 and 9 dimensional orbits (probably some flag variety?).
| https://mathoverflow.net/users/4096 | springer resolution over $\wedge^3 \mathbb{C}^6$ | A representative of the quartic orbit is $e\_{156} + e\_{246} + e\_{345}$ (I write $e\_{ijk}$ for $e\_i\wedge e\_j\wedge e\_k$). A resolution of this orbit is given by the projective closure of the tangent bundle to the Grassmannian. Indeed, if $V = \mathbb{C}^6$ and $U$ is the tautological bundle on $X = Gr(3,V)$ then... | 3 | https://mathoverflow.net/users/4428 | 108902 | 62,695 |
https://mathoverflow.net/questions/108886 | 16 | I am currently using Fulton and Harris for a course on representation theory, and I have noticed that there are a few errors throughout the book. A search on google with the keywords "Errata for Fulton and Harris" doesn't come up with anything.
I believe the book is widely used in many representation theory courses,... | https://mathoverflow.net/users/21278 | Erratum for Fulton and Harris | Being somewhat error-prone myself, I'm well aware of the need to collect errata in some systematic way. In the Internet age this has often been done in ad hoc ways on individual homepages, though some publishers (like AMS) are trying to establish durable book pages at their site with updates and errata posted by the au... | 12 | https://mathoverflow.net/users/4231 | 108908 | 62,696 |
https://mathoverflow.net/questions/108816 | 0 | Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let
$$
\Phi(p,i) := \frac{1}{2^p+1}
+ \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right),
$$
where $\lg x$ is the binary logarithm. With the help of a Computer Algebra System, it *seems* that
* If $0 \leqslant p \leqslant 3$, then $... | https://mathoverflow.net/users/26078 | Root and sign of a complicated bivariate function | Suppose $p\ge3$, so the cubic $q(i)$ has $3$ real roots. As the product of the roots is $-1$, at most two of them are positive. From $q(0)=1>0$, $q(1)<0$, $q(+\infty)>0$ we see that there are exactly two positive roots, one in $]0,1[$, the other one in $]1,\infty[$. Now $\Phi(0+)=-\infty$, $\Phi(1)>0$, $\Phi(2^p)<0$, a... | 0 | https://mathoverflow.net/users/18739 | 108911 | 62,699 |
https://mathoverflow.net/questions/108913 | 1 | In the preprint arXiv:math/0505432v1
by Batyrev and Kreuser I have found (on pages 2 and 10) the claim that
"by a recent result of Kresch and Vistoli [arXiv:math/0301249]"
the (usual) Brauer group of a Calabi-Yau threefold is isomorphic
to its cohomological Brauer group. However, in that preprint of
Kresch and Visto... | https://mathoverflow.net/users/9833 | Usual vs. cohomological Brauer groups of Calabi-Yau threefolds | It was proved by Gabber and De Jong that the cohomological Brauer group of a quasi-compact scheme with an ample invertible sheaf equals its Brauer group. Our preprint does not contain this result.
| 4 | https://mathoverflow.net/users/4790 | 108915 | 62,701 |
https://mathoverflow.net/questions/108907 | 2 | Let D be a division algebra over F, E its maximal field. Is it true that:
1) all such fields are equivalent over F?
2) all such fields are conjugate by inner automorphisms of D?
| https://mathoverflow.net/users/27018 | equivalence of maximal fields in division algebras | No, this is very untrue. Let $F=\mathbb{Q}$ and let $D$ be the quaternions with $\mathbb{Q}$ coefficients. For any nonzero $\alpha = bi+cj+dk$ in $D$, $\mathbb{Q}(\alpha)$ is a maximal subfield of $D$. Since $\alpha^2 = -b^2-c^2-d^2$, we can achieve the field $\mathbb{Q}(\sqrt{-D})$ for any $D$ not of the form $4^n (8m... | 10 | https://mathoverflow.net/users/297 | 108917 | 62,703 |
https://mathoverflow.net/questions/108896 | 3 | 1. Put $B(n,k)=k! S(n,k)$, here $S(n,k)$ be the Srirling numbers of second kind.
Prove that
$$
\sum\_{j=i+1}^n \frac{(-1)^{j+1-i}}{j-i} B(n,j)= n B(n-1,i),i=0,1,\ldots,n-1
$$
1. Put $T(n,i)=\displaystyle \sum\_{j=i}^n (-1)^{j-i} 2^{n-j} B(n,j) {j-1 \choose i-1}$.
Prove that
$$
\sum\_{j=i+1}^n \frac{1-(-1)^{j-... | https://mathoverflow.net/users/9645 | How to prove recursion formulas for Stirling numbers? | Here is a proof of the first formula. We use the exponential generating function
$\sum\_{n=0}^\infty S(n,k) x^n/n! = (e^x-1)^k/k!$.
Multiply the left side by $x^n/n!$ and sum on $n$ from 1 to $\infty$. We obtain
\begin{align\*}
\sum\_{j=i+1}^\infty\frac{(-1)^{j+1-i}}{j-i} (e^x-1)^j
&=\sum\_{k=1}^\infty \frac{(-1)^{k... | 6 | https://mathoverflow.net/users/10744 | 108929 | 62,709 |
https://mathoverflow.net/questions/108912 | 55 | Let $A\_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A\_n|$? Will it be $o(n^2)$?
| https://mathoverflow.net/users/19078 | Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$ | This question is known as the multiplication table problem, and was originally posed by Erdős in 1955. Erdős proved that $|A\_n|=o(n^2)$, and this was sharpened by Tenenbaum in 1984. In 2008, Ford gave the exact magnitude and proved that $$\left|\lbrace a\cdot b:\ a,b\leq N\rbrace\right|\asymp \frac{N^2}{(\log N)^c(\lo... | 73 | https://mathoverflow.net/users/12176 | 108939 | 62,711 |
https://mathoverflow.net/questions/108938 | 2 | Hello,
I would like to know how to currify a partial mapping $A \times B \to\_p C$?
Here, by "partial mapping from $X$ to $Y$", I mean a function from a subset of $X$ to $Y$, as explained here
[Partial Function](http://en.wikipedia.org/wiki/Partial_function).
It is known that $(A \times B) \to C$ is isomorphic t... | https://mathoverflow.net/users/22707 | Currying a " partial mapping A*B -->_p C | For plain vanilla sets and functions, a partial function $f$ from $A$ to $B$ can be viewed as essentially the same thing as a total function $g: A \to B + 1$, where the codomain is the disjoint union of $B$ with a 1-element set. The idea is that whenever $f(a)$ is undefined, we define $g(a)$ to be the element of $1$.
... | 3 | https://mathoverflow.net/users/2926 | 108941 | 62,712 |
https://mathoverflow.net/questions/108945 | 4 | I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts on fields on definition this would help me a lot (because these aren't the only questions I have concerning fields of d... | https://mathoverflow.net/users/22189 | Defining isogenies over smaller fields | No: Consider the elliptic curve $E: y^2 = x^3 + x$ defined over $\mathbb{Q}$. Then the isogeny $y \mapsto iy$ and $x \mapsto -x$ is defined over $\mathbb{Q}(i)$ but obviously not over $\mathbb{Q}$.
In general if $K \subset L$ is Galois, then a morphism defined over $L$ comes from one over $K$ if and only if it is inv... | 11 | https://mathoverflow.net/users/5101 | 108946 | 62,715 |
https://mathoverflow.net/questions/108848 | 7 | The following should be true: every normal subgroup of a non-Abelian Right Angled Artin Group should contain a free group on two generators. Is there a standard reference one can cite for this?
| https://mathoverflow.net/users/11142 | Right Angled Artin Group Reference request | Igor, I assume the question is about centerless RAAGs. Then it follows from the classical result of A. Baudisch (see MR0634562): every 2-generated subgroup of a RAAG is abelian or $F\_2$.
Indeed if $N$ is a non-trivial normal subgroup in a RAAG $G$, take any $x\in N$. Since $G$ is centerless there exists $g\in G$ th... | 13 | https://mathoverflow.net/users/10251 | 108947 | 62,716 |
https://mathoverflow.net/questions/107637 | 9 | Can you name a ccc forcing with the following properties?
1) Atomless and separative
2) The least size of a dense set is large, say at least $\aleph\_3$, hopefully as big as you like.
3) Existence is consistent with CH.
4) Preserves CH.
If you can think of any, please list as many as you can.
| https://mathoverflow.net/users/11145 | large ccc forcing that preserves CH | Thanks to Michael Blackmon for this idea:
If P has the ccc and preserves CH, then there is a contiuum sized regular suborder R that adds all the reals P will add. The factor forcing P/R is ccc and $\omega$-distributive, a Suslin algebra. A theorem (of Jech?) says that any Suslin algebra has size at most $2^{\omega\_1... | 6 | https://mathoverflow.net/users/11145 | 108952 | 62,719 |
https://mathoverflow.net/questions/108797 | 33 | One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G\_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. It has also been shown by Scott that $\mathbb{C}[G\_{k,n}]$ carries a cluster algebra structure for any $k$.
When I'm... | https://mathoverflow.net/users/21483 | What do cluster algebras tell us about Grassmannians? | I'm afraid that as far as I know, the answer is no. That is, the cluster structure hasn't (yet) told us anything new. There are two reasons why we might have expected that, though.
Firstly, the Grassmannians have been studied over a long period of time in many different settings, in which much of the time this same ... | 10 | https://mathoverflow.net/users/13215 | 108956 | 62,721 |
https://mathoverflow.net/questions/108968 | 6 | Floer's paper, *An Instanton-Invariant for 3-Manifolds*, makes reference to Casson's construction of a topological invariant for homology 3-spheres. He literally references it by placing the bibliographic-tag [C] in one of his sentences... but he doesn't actually list it in the bibliography!
I subsequently search ar... | https://mathoverflow.net/users/12310 | Seeking a seemingly missing reference of Casson | According to [the following paper](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.jdg/1214510050&page=record), this invariant's introduction is sourced as: A. Casson, Lecture notes, MSRI Lectures, Berkeley, 1985.
The first published discussion appears to be found in: S. Akbulut & J. ... | 9 | https://mathoverflow.net/users/22971 | 108969 | 62,724 |
https://mathoverflow.net/questions/108971 | 0 | I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V\_\alpha \setminus X$ where $V\_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My understanding is that ZFC doesn't support ordinal-indexed sequences, so my question is, what is a good set theory in whi... | https://mathoverflow.net/users/26080 | Good set theory in which to study ordinal-indexed sequences? | Note that the class-length sequence $V\_\alpha$ is definable, so given a set $X$, so is the sequence $V\_\alpha\setminus X$ going to be definable (via the parameter $X$).
If you intend to use more, perhaps a theory like ZFC+Global choice; or NBG which is more suitable for handling proper classes.
If you are going t... | 2 | https://mathoverflow.net/users/7206 | 108972 | 62,726 |
https://mathoverflow.net/questions/108890 | 14 | Consider $V\_{(n-1, 1)}$, the $n-1$ dimensional irreducible representation of $S\_n$, i.e. the "standard" or "defining" representation. Is there a nice formula for how the $k$-th tensor power of $V\_{(n-1, 1)}$ decomposes into irreps?
| https://mathoverflow.net/users/6045 | Tensor powers of the standard representation | For convenience consider the representation $Y=V\_n\oplus V\_{n-1,1}$ instead of $V\_{n-1,1}$. Then the multiplicity of the representation of $S\_n$ indexed by the partition $\lambda$ of $n$ in the $k$th tensor power of $Y$ equals the scalar product of the symmetric function $s\_1^k$ (where $s\_1=x\_1+x\_2+\cdots$ deno... | 16 | https://mathoverflow.net/users/2807 | 108974 | 62,727 |
https://mathoverflow.net/questions/108926 | 15 | I have two questions about *q-continued fractions*, but a little intro first. Given [*Ramanujan's theta function*](https://mathworld.wolfram.com/RamanujanThetaFunctions.html),
$$f(a,b) = \sum\_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}$$
then the following,
$$A(q) = q^{1/8} \frac{f(-q,-q^3)}{f(-q^2,-q^2)}$$
$$B... | https://mathoverflow.net/users/12905 | The complete list of continued fractions like the Rogers-Ramanujan? | [**Edited** *again to give a second identity relating $E$ to eta products*]
Continued fraction or not, an expression
$q^{\frac{(r-s)^2}{8(r+s)}} f(\pm q^r, \pm q^s)$
is a modular form of weight $1/2$ for all integers $r,s$ with $r+s>0$,
because it is a sum $\sum\_{n=-\infty}^\infty \pm q^{(cn+d)^2}$
with rational $c,... | 20 | https://mathoverflow.net/users/14830 | 108980 | 62,728 |
https://mathoverflow.net/questions/108979 | 1 | For space pairs $(X\_1,A\_1)$ and $(X\_2,A\_2)$ ,is there a groups M,such that $\pi\_{k}(X\_1\vee X\_2,A\_1\vee A\_2)\cong \pi\_k(X\_1,A\_1)\oplus M$?
| https://mathoverflow.net/users/21719 | is $\pi_k(X_1,A_1)$ a direct summand of $\pi_{k}(X_1\vee X_2,A_1\vee A_2)$ | The pair $(X\_1,A\_1)$ is a retract of the pair $(X\_1\vee X\_2,A\_1\vee A\_2)$, so the map $\pi\_k(X\_1,A\_1)\to \pi\_k(X\_1\vee X\_2,A\_1\vee A\_2)$ is a split injection for all $k$. If $k\ge 3$ then the groups are abelian, and your result follows with $M$ the cokernel.
| 6 | https://mathoverflow.net/users/8103 | 108986 | 62,731 |
https://mathoverflow.net/questions/108955 | 14 | Wilkie's well known question asks whether $I\Delta\_{0}$ proves the unboundedness of primes. We know that by adding a sentence to $I\Delta\_{0}$ which says "the exponential function is total", it is possible to prove the unboundedness of primes. This sentence is $\Pi\_{2}$. Suppose $\Pi\_{1}\text{-Th}(\mathbb{N})$ deno... | https://mathoverflow.net/users/27034 | Unboundedness of primes in bounded arithmetic | Yes, because $\: I\Delta\_0 + \text{WPHP}\left(\Delta\_0\right) \:$ proves the unboundedness of primes (see [this answer](https://mathoverflow.net/questions/75995/formalizing-euclids-proof-of-the-infinitude-of-primes/76058#76058)),
since the assetion that a $\Delta\_0$-defined relation is an injection from $\:[0\hs... | 12 | https://mathoverflow.net/users/nan | 108988 | 62,732 |
https://mathoverflow.net/questions/108975 | 5 | I ran into this problem when studying Morse theory. My professor referred to the torus and riemann surface of genus 2 as an example. And after some manipulating on the long exact sequence of cohomology groups he came to the conclusion that attaching an n-cell(passing a critical point of index n) would either kill a cla... | https://mathoverflow.net/users/27040 | How do you intepret "kill a cohomology class" intuitively for attaching an n-cell? | First of all, by Morse theory you know that $\mathcal{M}^+$ is obtained by attaching an $m$-cell to $\mathcal{M}^-$, i.e. $\mathcal{M}^+=\mathcal{M}^-\cup\_{\phi}e$, where $e=D^m$ is an $m$-cell and $\phi:\partial(e)=S^{m-1}\to \mathcal{M}^-$ is some attaching map. Therefore you can compute $H^{\\*}(\mathcal{M}^+,\math... | 4 | https://mathoverflow.net/users/3465 | 108989 | 62,733 |
https://mathoverflow.net/questions/108930 | 2 | By well-pointed i mean that the inclusion of the base point is a h-cofibration, weak equivalences are the usual weak homotopy equivalences between spaces.
this is claimed as part of theorem 6.9 (i) in [model categories of diagram spectra](http://www.math.uni-bonn.de/people/schwede/MMSS.pdf)
but as far as i can see with... | https://mathoverflow.net/users/27028 | Smashing with a cw-complex preserves weak equivalences between well-pointed spaces | I would like to just comment but don't see how. Christian, here's an answer to your last question.
For based spaces, you can just do what you want by hand, as Fernando suggested, taking care to use
disjoint basepoints to make your attaching maps of $A$ based. You are right to complain that the
interplay of $h$ and $q$-... | 5 | https://mathoverflow.net/users/14447 | 108996 | 62,735 |
https://mathoverflow.net/questions/109009 | 2 | The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$.
A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$.
The four vertices of the tetrahedron may be parametrized by four complex numbers $z\_1, z\_2, z\_3, z\_4$.
What is the surface... | https://mathoverflow.net/users/27048 | Surface of a Ideal Tetrahedron in Hyperbolic Space H3 | The answer is $4\pi.$
| 10 | https://mathoverflow.net/users/11142 | 109012 | 62,740 |
https://mathoverflow.net/questions/109005 | 1 | is the question whether a polynomial is non-negative on some semi-algebraic set (equivalently, is it in the cone denerated by some polynomials in the field of rational functions) known to be decidable?
| https://mathoverflow.net/users/27047 | Decidability of the generated order | Assuming you're working over the real numbers (or at least over a real-closed field), the answer is yes. The question lies within the first-order theory of real-closed fields, and that theory is decidable by an old theorem of Tarski.
| 1 | https://mathoverflow.net/users/6794 | 109013 | 62,741 |
https://mathoverflow.net/questions/109015 | 3 | The Bloch-Wigner function $D(z)$ gives the volume of an ideal tetrahedron in the hyperbolic space $\mathbb{H}^3$. Here $z$ is the cross-ratio $(z\_1,z\_2,z\_3,z\_4)$ parametrizing the tetrahedron in $\mathbb{C}P^1$.
Put $\tilde D(z\_1,z\_2,z\_3,z\_4) = D(z)$.
The five-term relation for the dilogarithm could be int... | https://mathoverflow.net/users/27048 | Dilogarithm, tetrahedrons, and hyperbolic space | The five term relation comes from the fact that the sum of the volumes of tetrahedra $ABCD$ and $ABCE$ equals the sum of the volumes of the three tetrahedra $ABDE, ACDE, BCDE.$ One can think of $ABCDE$ as a degenerate four-dimensional simplex.
| 12 | https://mathoverflow.net/users/11142 | 109016 | 62,743 |
https://mathoverflow.net/questions/108655 | 0 | Consider two hypergraphs $H\_1 = (V, \mathscr{E}\_1), H\_2 = (V, \mathscr{E}\_2)$ over the same vertex set $V$. am interested in what could be called a "cartesian join" operation building a new hypergraph $H\_3=H\_1 \sqcup H\_2 = (V, \mathscr{E}\_1 \times \mathscr{E}\_2)$ where "$\times$" is the cartesian join over edg... | https://mathoverflow.net/users/20793 | Hypergraph cartesian join operation (over same vertex set) | You may want look at the so called "fractional Cartesian products" studied extensively by Blei in connection with some extremal problems in Harmonic Analysis. You may have to translate some of his formalism to fit your needs that are stated in graph-theoretic language...
| 3 | https://mathoverflow.net/users/27052 | 109019 | 62,745 |
https://mathoverflow.net/questions/99588 | 3 | Let $X$ be a complex algebraic variety and consider the category $P(X)$ of perverse sheaves of complex vector spaces.
Let $f:X\rightarrow \mathbb C$ be a regular function, $Z$ its zero set and $U$ its complement.
A glueing data consists of a tuple $({\cal F}\_U,{\cal F}\_Z, u,v)$, where ${\cal F}\_U$ and ${\cal F}\... | https://mathoverflow.net/users/2837 | How to glue perverse sheaves of abelian groups? | Everything I know about gluing perverse sheaves is written in my paper [Notes on Beilinson's "How to glue perverse sheaves"](http://arxiv.org/abs/1002.1686), which deconstructs the construction sufficiently that it is possible to identify the following minimal axioms for making gluing work:
1. The nearby cycles funct... | 2 | https://mathoverflow.net/users/6545 | 109030 | 62,747 |
https://mathoverflow.net/questions/95701 | 33 | I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology.
From <http://en.wikipedia.org/wiki/Mandelbrot_set> I see that conjecture has implications concerning the structure of th... | https://mathoverflow.net/users/10909 | The deep significance of the question of the Mandelbrot set's local connectedness? | If a connected compact $K \subset C$ is locally connected then the Riemann map
$h\colon C \setminus \Delta \to C \setminus K$ extends continuously to $\partial \Delta$. For each $z \in \partial K$, the boundary of the convex hull of $h^{-1}(\{z\})$ is the union of a set $\Lambda\_z$ of chords; the union of these $\Lam... | 29 | https://mathoverflow.net/users/8252 | 109036 | 62,750 |
https://mathoverflow.net/questions/109040 | 2 | Let $A$ be Noetherian ring that is an integral domain and let $\frak a$ be a proper ideal.
I would like to know if it can happen that $\cap\_{1}^{\infty}{\frak a}^n\ne 0$ and at the same time the sequence of ideals ${\frak a}^n$ does not stabilise? What would be a "natural example"?
(this question is motivated by try... | https://mathoverflow.net/users/13441 | Infinite power of in ideal in a Noetherian ring | If $P$ is a prime ideal containing $a$ then $\cap a^n$ is contained in $\cap P^n$, whose image in $A\_P$ vanishes, so $\cap a^n$ has vanishing image in $A\_P$ for every point $P$ of Spec($A/a$). Such $P$ exist as long as $a$ isn't the unit ideal, so if $A$ is an integral domain then $\cap a^n = 0$ in $A$ since $A \righ... | 10 | https://mathoverflow.net/users/27056 | 109041 | 62,754 |
https://mathoverflow.net/questions/109022 | 3 | In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as $(M, \omega)$ where $\omega$ is a sympathetic form on $M$ corresponding to the Poisson bracket. Given a Hamiltonian $H$... | https://mathoverflow.net/users/27053 | Infinite dimensional manifold | Have a look at section 48 of:
Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997,
[(pdf)](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf).
There is a mistake there that I did not yet correct on... | 3 | https://mathoverflow.net/users/26935 | 109054 | 62,758 |
https://mathoverflow.net/questions/109048 | 10 | Let $F$ be a finite degree extension over $\mathbf{Q}\_p$ and consider the locally profinite group $G:=GL\_2(\mathbf{Q}\_p)$.
**P1**: Give an interesting example (non-artificial one, i.e., one that arises in real life for a representation theorist) of a non-smooth representation $\rho$ of $G$ on a topological $\math... | https://mathoverflow.net/users/11765 | non-artificial examples of non-smooth and non-admissible representations of GL_2 | The "smoothness" prevents taking Hilbert-space completions in general, for example. That is, for example, with $G=GL\_n(F)$ for a $p$-adic field $F$ and $n\ge 1$, the Hilbert space $V=L^2(G)$ with right translation by $G$ *is* a continuous representation in the strong topology but is not smooth. In fact, this would be ... | 11 | https://mathoverflow.net/users/15629 | 109059 | 62,762 |
https://mathoverflow.net/questions/109070 | 6 | The Hodge-de Rham Laplacian $L=(d+d^\*)^2$, where $d$ is the boundary operator of the de Rham complex, is well-known in the math community. Recently, I tried very hard to search for examples of its use in physics and engineering applications (for giving talks or future teaching). More specifically, I want to find appli... | https://mathoverflow.net/users/16464 | Applications of Hodge-de Rham Laplacian on p-forms ($p\neq 0,n$) in physics or engineering | Witten wrote a classic paper in which he uses Morse theory to study supersymmetric quantum mechanics. Harmonic forms play a central role. Here's the link: <http://intlpress.com/JDG/archive/1982/17-4-661.pdf>
| 2 | https://mathoverflow.net/users/24525 | 109073 | 62,768 |
https://mathoverflow.net/questions/37540 | 2 | Let $X$ a dual Banach space (there exists a Banach space $Y$ such that $X=Y'$).
A weak\* semigroup on $X$ is a semigroup $(T\_t)\_{\geq 0}$ on $X$ such that, for all $x\in X$, we have $T\_tx\to x$ in the weak\* topology when $t\to 0^+$.
I know a lot of books about $C\_0$-semigroups but not about weak\* semigroups.
... | https://mathoverflow.net/users/5210 | Reference for weak*-semigroup | Echoing the remark of @Bill Johnson, one possibility is [van Neerven's book](http://books.google.hu/books/about/The_adjoint_of_a_semigroup_of_linear_ope.html?id=9jbvAAAAMAAJ&redir_esc=y) on adjoint semigroups.
| 4 | https://mathoverflow.net/users/12898 | 109075 | 62,770 |
https://mathoverflow.net/questions/109044 | 11 | Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d\_1,\dots,d\_k)$.
1. Is there some nice (i.e. "explicit") parameter space for them?
(even if all the $d\_i$'s are distinct one cannot take just the product of linear systems :)
Or, does the corresponding loc... | https://mathoverflow.net/users/2900 | Parameter space for complete intersections and their discriminant | The description of the Hilbert scheme of complete intersections (obtained by taking in an iterative way open subsets of grassmannian bundles, as explained in the answer above) may be found in part 2.2 of my thesis <http://www.math.ens.fr/~obenoist/articles/Thesefinale.pdf>. If the distinct degrees are $\delta\_1<\dots<... | 10 | https://mathoverflow.net/users/2868 | 109077 | 62,771 |
https://mathoverflow.net/questions/109071 | 8 | Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is there an algorithm for reducing this word to my chosen decomposition using the Coxeter relations? That is to say, an algor... | https://mathoverflow.net/users/21037 | Algorithm for reducing words in a Coxeter group | Yes, at least if your chosen element is the lexicographically first reduced expression for your permutation. This is how people usually prove that any two reduced expressions for a given permutation are connected by a series of long and short braid moves -- by reducing both to the lexicographically first reduced expres... | 10 | https://mathoverflow.net/users/23408 | 109079 | 62,773 |
https://mathoverflow.net/questions/109008 | 2 | Let $\ell\_p^n$ be the $n$-dimensional real or complex $\ell\_p$-space and let $\mathscr{B}(\ell\_p^n)$ be the space of matrices on $\ell\_p^n$ endowed with the operator norm. I am looking for any references which give (reasonable) estimates for the Banach-Mazur distance between $\mathscr{B}(\ell\_p^n)$ and $\ell\_p^{n... | https://mathoverflow.net/users/15129 | BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$ | For the first question, $n^{1/2}$ is the right order for $p=1,2,\infty$. The case $p=\infty$ is arguably the easiest, because $B(\ell\_\infty^n)=\ell\_\infty^n(\ell\_1^n)$ isometrically and the Banach-Mazur distance between $\ell\_\infty^n$ and $\ell\_1^n$ is of order $n^{1/2}$. That gives the upper bound, and the lowe... | 3 | https://mathoverflow.net/users/2554 | 109080 | 62,774 |
https://mathoverflow.net/questions/93584 | 7 | Let $G$ be a simple algebraic group over an algebraically closed
field $k$ of characteristic $p>0$. For $x\in G$, let $C\_{G}(x)$
denote the centraliser, considered as a group scheme over $k$. If
$p$ is a very good prime for $G$ it is known that $C\_{G}(x)$ is
smooth (over $ $$k$). Assume that $p$ is not very good for ... | https://mathoverflow.net/users/2381 | Lie algebras and non-smoothness of centralisers in bad characteristic | For all groups of exceptional types the answer can be deduced from the paper "Jordan block sizes of unipotent elements in exceptional algebraic groups" by Ross Lawther, published in
Comm. Algebra, 23, Issue 11, 1995, 4125-4156. In order to determine the Jordan block sizes of ${\rm Ad}\ u$, where $u\in G$ is unipotent, ... | 5 | https://mathoverflow.net/users/24386 | 109083 | 62,777 |
https://mathoverflow.net/questions/109068 | 2 | Maskit's combination theorem says: if $M=M\_1\cup\_\Sigma M\_2$ is a union of hyperbolic 3-manifolds $M\_1=\Gamma\_1\backslash H^3, M\_2=\Gamma\_2\backslash H^3$ along a surface $\Sigma$, and if the limit set of $H:=\pi\_1\Sigma$ is a codimension 1 submanifold of $\partial H^3$ dividing $\partial H^3$ into domains $\Om... | https://mathoverflow.net/users/39082 | Combination theorems for discrete subgroups of isometry groups | There's a [combination theorem of Bestvina-Feighn](https://mathscinet.ams.org/mathscinet-getitem?mr=1152226) for hyperbolic groups.
There are gluing theorems for $CAT(\kappa)$ spaces in [Chapter 11 of
Bridson-Haefliger](https://web.archive.org/web/20130423104708/http://www.math.psu.edu/petrunin/papers/scans/books/bri... | 6 | https://mathoverflow.net/users/1345 | 109086 | 62,779 |
https://mathoverflow.net/questions/108936 | 6 | This question is a generalization of the question posed in [this page](https://mathoverflow.net/questions/100032/existence-of-an-arbitrary-small-positive-continuous-real-valued-function) to lower semi continuous functions. so let me describe the Question in the following way.
---
**Def**: Let $(X,\tau)$ be a Tych... | https://mathoverflow.net/users/23317 | Arbitrary small positive lower semi continuous functions | Since we are dealing with upper and lower semicontinuous functions instead of continuous functions, it is fruitful to consider all topological spaces instead of just completely regular spaces. The following theorem characterizes all such spaces, and $2\rightarrow 1$ incorporates François Dorais's idea in his answer to ... | 4 | https://mathoverflow.net/users/22277 | 109109 | 62,792 |
https://mathoverflow.net/questions/109118 | 7 | Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees $>n$, or of any other version of this statement)? I have found some proofs; all of them are step by step, and it is not ... | https://mathoverflow.net/users/2191 | The main idea in the proof of Artin's vanishing | I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the Artin-Grothendieck theorem). The corresponding statement for smooth varieties over the complex numbers and singular cohomology (theorem of Andreotti-Frankel) follows from the fact that Morse theory shows t... | 15 | https://mathoverflow.net/users/3847 | 109123 | 62,800 |
https://mathoverflow.net/questions/108614 | 1 | It is well known, that the intermediate extension functor $j\_{!\*}$ preserves injections and surjections. However it seems that it is not exact in general!
1) What would be an example which shows that $j\_{!\*}$ is not exact?
2) It is easy to see, that $j\_{!\*}$ is exact if $j$ is the inclusion of a cell into a c... | https://mathoverflow.net/users/2837 | Intermediate extension functor exact? | I think that the answer for 2) is no.
Consider the affine Grassmannian for $SL\_2$ and let $X$ denote the Schubert variety corresponding to the three dimensional representation of $PSL\_2$. Then $X$ is stratified by Iwahori orbits into three affine strata of dimensions 2, 1 and 0. Let $U$ denote the open stratum.
C... | 3 | https://mathoverflow.net/users/919 | 109124 | 62,801 |
https://mathoverflow.net/questions/109125 | 7 | This is related to [another question](https://mathoverflow.net/questions/77022/diophantine-xpyqxyr)
I am interested in the non-trivial integer solutions of
$$ x^n + y^n = z^{n-1} $$
for $n \ge 4$. A solution is trivial if $xyz=0$ or $x = \pm y$.
There are infinitely many rational solutions to $x^n + y^n = (x+y)^{n-... | https://mathoverflow.net/users/12481 | Integer solutions of x^n + y^n = z^{n-1} | Take any $a,b$ and set $c=a^n+b^n$. Then the triple $(ac^{n-2},bc^{n-2},c^{n-1})$ is a solution of your equation.
Conversely, if $(x,y,z)$ is a solution and $d$ is their gcd, so $(x,y,z)=(ad,bd,cd)$, then you get $d(a^n+b^n)=c^{n-1}$. One of the solutions is presented above (with $d=c^{n-2}$). But there also exist sm... | 33 | https://mathoverflow.net/users/17581 | 109128 | 62,802 |
https://mathoverflow.net/questions/109130 | 1 | Is it true that if M is existentially closed in N then N can be embedded in an ultraproduct of M ?
Best regards.
| https://mathoverflow.net/users/27076 | Existentially closed substructure and ultraproducts | Yes; isn't this in standard textbooks like Chang-Keisler? Anyway, here's a construction. Adjoin, to the vocabulary (= signature = language) of $M$ and $N$, new constant symbols $\dot n$ for all the elements of $N$, and let $D$ be the set of all atomic sentences and negations of atomic sentences that are true in $N$ (wi... | 4 | https://mathoverflow.net/users/6794 | 109136 | 62,803 |
https://mathoverflow.net/questions/109116 | 3 | Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected subset.
The answer is negative if the space is assumed to be connected and locally path-connected. Since every compon... | https://mathoverflow.net/users/26608 | A closed connected component in a topological space does not contain any path-connected subset? | What you are asking for is a connected and totally path-disconnected space. Apparently there is such a beast on page 145 of "[Counter-examples in Topology](http://www.amazon.co.uk/Counterexamples-Topology-Dover-Books-Mathematics/dp/048668735X#reader_048668735X)" by Steen and Seebach (I don't have a copy of the book, an... | 5 | https://mathoverflow.net/users/8103 | 109140 | 62,804 |
https://mathoverflow.net/questions/109134 | 1 | Let $A \subseteq V(G)$ be a set of vertices in a graph $G$ and let $v \in V(G)$ be some vertex. Define $d\_{A}(v)$ as the number of neighbours of $v$ inside $A$.
Now suppose you have a graph whose vertex set is partitioned into $A,B,U$ and define for every vertex $u \in U$ the ***"AB-degree"*** of $u$ as $\Delta\_{u}... | https://mathoverflow.net/users/22051 | Have you come across this kind of "degree" concept? | This should perhaps be a comment, but I don't have enough reputation.
What sort of bound are you hoping for? The trivial bound $d(|A|+|B|)$ appears to be tight: take $G=K\_{d,d}$ with $A$ one of the parts and $B$ empty. (Two copies of this will give you an example with $A$ and $B$ the same size.)
| 2 | https://mathoverflow.net/users/25485 | 109141 | 62,805 |
https://mathoverflow.net/questions/108933 | 6 | Consider two Erdos-Renyi random graphs $G\_1,G\_2$ on $n$ nodes, with the edges in each graph generated independently at random with probability $1/2$. My question is about the cut-distance between these two graphs. Recall that the cut-distance is the maximum over all cuts of the difference between the cut values of th... | https://mathoverflow.net/users/27029 | Cut-distance between two Erdos-Renyi random graphs | (To answer Anthony, I'm taking the two graphs to be on the same vertex set.)
This is an argument, without details, that the answer is $\Omega(n^{3/2})$.
Generate the graphs two vertices at a time. As each pair of vertices is generated and their adjacencies with the previous vertices are chosen randomly, put one ver... | 5 | https://mathoverflow.net/users/9025 | 109144 | 62,807 |
https://mathoverflow.net/questions/109149 | 24 | Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:
(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?
(anot... | https://mathoverflow.net/users/8799 | Cyclotomic polynomials with coefficients $0,\pm1$ | There are other families, but there is by no means a complete characterization known.
Even for products of three primes there is no complete answer known. A relevant keyword is flat cyclotomic polynomial. Some results to give a flavor of the problem.
The follwing is due to N. Kaplan (from some years ago, Journal of... | 23 | https://mathoverflow.net/users/nan | 109155 | 62,813 |
https://mathoverflow.net/questions/109151 | 1 | The question is in the title! I know that a globally rigid graph is 3-connected and redundantly rigid, so my question could be rephrased as: "does there exist a graph which is 3-connected and chordal but not redundantly rigid?"
It seems fairly intuitive to me that there does not, but my intuition about graphs has a f... | https://mathoverflow.net/users/2189 | Does there exist a 3-connected, chordal graph which is not globally rigid? | Every such graph is generically globally rigid in $E^2$. A 3-connected chordal graph can be built by starting with a triangle and then sequentially attaching new vertices to at least 3 previous ones. See [this paper](http://arxiv.org/abs/1205.3990) for some explicit statements. This idea generalizes to any dimension.
... | 5 | https://mathoverflow.net/users/14832 | 109157 | 62,814 |
https://mathoverflow.net/questions/109174 | 6 | I would like to find a source giving the exact formula for the product of two Hecke operators $T\_{\kappa}(n^2)$ and $T\_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - \mathbb{Z}$. I am searching for a formula which is similar to
$$
T\_k(m)T\_k(n) = \sum\_{d|(m,n)} d^{k-1}T\_k\left(\fr... | https://mathoverflow.net/users/16389 | Half integral weight Hecke operators | I think you can find the answer [here](https://arxiv.org/abs/1208.4326).
| 4 | https://mathoverflow.net/users/11919 | 109178 | 62,823 |
https://mathoverflow.net/questions/109160 | 28 | The function $F(x) = \sum\_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I know of no name for it, nor any use for it aside from pedagogical, so this is a pure curiosity question which I hope i... | https://mathoverflow.net/users/26327 | The function $\sum_{0}^{\infty} x^n/n^n$ | [**Edited** *to outline the end of the argument that $f(-M) \rightarrow 0$
(and to correct a few typos etc. while I'm at it)*]
Yes, $F(x) \rightarrow 0$ from below as $x \rightarrow -\infty$.
The convergence is slow, and precise asymptotic analysis seems to be
somewhat annoying because it involves the lower branch o... | 37 | https://mathoverflow.net/users/14830 | 109185 | 62,828 |
https://mathoverflow.net/questions/108923 | 3 | I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between $L$ and $K$ (the Euler characteristic) such that in the case $Y = \emptyset$
$$ \chi([V\_0, \dots, V\_n]) = \sum\_{k=0}^n ... | https://mathoverflow.net/users/16702 | Euler characteristics and the difference bundle construction | You missed that the sequence is not exact at $K(X,Y)$ (neither is it exact at $K(Y)$, but that does not matter here). There is an ambiguity coming from $K^{-1} (Y)$, i.e. automorphisms of bundles. If $Y$ is a point, your construction works, and the purpose of the arguments in Atiyah-Bott-Shapiro is to extend it to $Y \... | 4 | https://mathoverflow.net/users/9928 | 109188 | 62,829 |
https://mathoverflow.net/questions/109198 | 0 | Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the conjugate transpose.
Let $f$ be a real-analytic function that converges in a neighbourhood of the origin in ${\mathbb{... | https://mathoverflow.net/users/nan | Positive definite Hermitian matrices of countable rank | I'm puzzled as to why you would expect this ... no, let $(c\_{ij})$ be the identity matrix ($c\_{ii} = 1$, $c\_{ij} = 0$ for $i \neq j$). That is positive semidefinite but it has infinite rank.
There's a well-developed theory of positive semidefiniteness for operators on $l^2$. If the operator is bounded, then $\lang... | 2 | https://mathoverflow.net/users/23141 | 109201 | 62,834 |
https://mathoverflow.net/questions/109138 | 3 | Let $X$ be a smooth degree $d$ $(d \ge 5)$ surface in $\mathbb{P}^3$. Let $D$ be an effective Cartier divisor (hence locally of complete intersection) on $X$ and $D\_{red}$ the associated reduced scheme which is also a Cartier divisor on $X$. Then, we have a natural inclusion of ideal sheaves,
$$0 \to \mathcal{O}\_X(-... | https://mathoverflow.net/users/9164 | linear system of non-reduced divisor and associated reduced divisors | The second short exact sequence is wrong. You should recognize this without knowing where the mistake is: $D\_{\mathrm{red}}\leq D$, so $\mathscr{O}\_X(D\_{\mathrm{red}}) \subseteq \mathscr{O}\_X(D)$ and so the map you have is surjective if and only if $D\_{\mathrm{red}}= D$. In fact that map is always injective as you... | 9 | https://mathoverflow.net/users/10076 | 109203 | 62,836 |
https://mathoverflow.net/questions/109190 | 2 | Hi,
I have two commutative finite-dimensional $k$-Algebras $A$ and $B$ ($k$ is a field).
I wonder, whether there is a way to get the finite-dimensional indecomposable modules of $A \otimes\_k B$,
if I know the finite-dimensional indecomposable modules of $A$ and $B$.
For example, is there a way to do this, if ... | https://mathoverflow.net/users/12826 | Indecomposable modules of a tensor product of two algebras | No, in general there isn't. In your particular case by accident for $n=1$ this is possible. But even for $n=2$, there are only $2$ indecomposable $A$-modules up to isomorphism. But Kronecker already clasified the $A\otimes\_k B$-modules and there are infinitely many.
For $n>2$ there are still only finitely many indecom... | 2 | https://mathoverflow.net/users/15887 | 109208 | 62,838 |
https://mathoverflow.net/questions/109205 | 4 | I looked on the web to search for weakly compact cardinals.
The web sources indicate many properties of weakly compact cardinals and say that their existence is not entailed by the axioms of ZFC. However it is not indicated whether their existence is {\it consistent} with the axioms of ZFC.
So my question is: Is it ... | https://mathoverflow.net/users/nan | Existence of weakly compact cardinals | As weakly compact cardinals are in particular [(strongly) inaccessible](http://en.wikipedia.org/wiki/Inaccessible_cardinal), it follows from Gödel's Second Incompleteness Theorem that ZFC cannot prove the implication "Con(ZFC) implies Con (ZFC + $\exists$weakly-compact )." (Unless, of course, if ZFC is itself inconsist... | 5 | https://mathoverflow.net/users/13653 | 109212 | 62,841 |
https://mathoverflow.net/questions/108983 | 1 | I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. This complex structure $J$ is unique due to the Bruhat and Whitney complexification method. With this complex structure... | https://mathoverflow.net/users/27043 | Unique symplectic form in an adapted complex structure | This is a rather tough question to answer.
Check out: D. Burns & R. Hind, Symplectic geometry and the uniqueness of Grauert tubes, Geom. Func. Anal. Vol. 11 (2001), 1-10.
In this paper, they prove that, if two Grauert tubes of $M$ in $TM$ with respect to two Riemannian metrics $g\_1,\ g\_2$ are biholomorphic, then ... | 1 | https://mathoverflow.net/users/9102 | 109216 | 62,842 |
https://mathoverflow.net/questions/109211 | 20 | As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex variety of dimension n vanishes in degrees >n). My question is: are there any ideas how to extend any of the consequences o... | https://mathoverflow.net/users/2191 | Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic? | A Morse function is a map of a manifold to the real line locally equivalent to:
$$f(x\_1,\ldots, x\_n)=-x\_1^2\ldots -x\_k^2+ x\_{k+1}^2+\ldots+x\_n^2$$
for some $k$. In other words,
for which the singularities are as simple as possible.
While a Lefschetz pencil is a map of a smooth projective variety to the projecti... | 20 | https://mathoverflow.net/users/4144 | 109219 | 62,843 |
https://mathoverflow.net/questions/109206 | 7 | There should be a category $3\text{CobTang}$ whose
* objects are some kind of surfaces with a finite set of marked points
* morphisms $M : S \to T$ are some kind of $3$-dimensional cobordisms $M$ between surfaces together with a choice of (framed) tangle in $M$ beginning at the marked points in $S$ and ending at the... | https://mathoverflow.net/users/290 | Reference request: the "Kauffman bracket skein category"? | I'm not sure where in the literature this exact construction occurs, but it is certainly well-known. I think some skein-module-centric papers consider non-empty boundary conditions on the skein modules (which would be equivalent to what you describe), so you might try looking for that. You could also have a look at the... | 10 | https://mathoverflow.net/users/284 | 109220 | 62,844 |
https://mathoverflow.net/questions/109213 | 10 | In classical complex analysis it is easy to prove that a meromorphic function has at most one analytic continuation (on an open connected subset of $\mathbb C$, say).
The problem of non-uniqueness of analytic continuation is one of the reasons why it is not possible (if one wants a good theory) to translate the compl... | https://mathoverflow.net/users/27110 | Uniqueness of analytic continuation in rigid analytic geometry | No: let $X$ be the union of the coordinate axes in the affine plane. As over $\mathbf{C}$, the answer is affirmative on a connected *normal* analytic space. Hint: prove in any rigid-analytic space that connected components are witnessed via finite linked chains of connected affinoid opens (and after thereby reducing to... | 5 | https://mathoverflow.net/users/27056 | 109221 | 62,845 |
https://mathoverflow.net/questions/109226 | 8 | Hello to everyone,
My problem is the following: I have this version of the Hahn-Banach theorem:
Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any
convex function. Let $W$ be a vector subspace of $V$ and let $L:W\rightarrow \mathbb{R}$ be a linear functional dominated by $p$ on $W$. Then there is a (n... | https://mathoverflow.net/users/27116 | Hahn-Banach theorem with real extended valued function | The Hahn-Banach theorem is wrong for extended real $p$: For $V=\mathbb R^2$ let
$p$ be the Minkowski functional of $A= \mathbb R \times (0,\infty)$
(so that $p(x,y)=0$ if $y>0$ and $p(x,y)=\infty$ if $y\le 0$),
$W= \mathbb R \times \lbrace 0 \rbrace$, and $L: W\to\mathbb R$ defined by $L(x,0)=x$.
Then $L$ is $p$-domi... | 11 | https://mathoverflow.net/users/21051 | 109231 | 62,849 |
https://mathoverflow.net/questions/109218 | 3 | It is well known that if $c(K)=2n+1$, then $u(K)$ is less than $n+1$. It can not be sharper because of the trefoil knot. On the other hand, if $c(K)=2n$, then similarly we have $u(K)$ is less than $n+1$. I think $u(K)=n$ is impossible in this case, i.e. there does not exist a knot $K$ with $c(K)=2n$ and $u(K)=n$. Maybe... | https://mathoverflow.net/users/27111 | Unknotting number and crossing number | You can see the answer in Proposition 2.1 of [link text](http://arxiv.org/abs/0705.3337)
| 2 | https://mathoverflow.net/users/25322 | 109232 | 62,850 |
https://mathoverflow.net/questions/109227 | 0 | For any compact abelian group $K$.$$K\cong H\_0\times \mathrm{U}(1)^k,$$where $H\_0$ is a finite group.
| https://mathoverflow.net/users/27115 | How to prove this equation | Any topological group of the form $K\cong H\_0\times U(1)^k$ (with $H\_0$ finite and $k$ a positive integer) is a closed subgroup of $U(1)^h$ (for some positive integer $h\geq k$). Furthermore, all the closed subgroups of $U(1)^h$ are of this form.
The proof is an easy application of the Pontryagin-Van Kampen duality... | 3 | https://mathoverflow.net/users/24891 | 109236 | 62,852 |
https://mathoverflow.net/questions/109238 | 7 | Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior.
Can you help me to find a proof of the following theorem?
"Arbitrary union of meagre open subsets of $X$ is meagre."
The cas... | https://mathoverflow.net/users/27120 | Arbitrary union of meagre open sets | First, consider Gerhard's easier special case, where the open sets are
disjoint.
**Claim.** The union of an arbitrary family of pairwise
disjoint open meager sets is meager.
Proof. Suppose that $U\_i$ are pairwise disjoint and meager, so
that $U\_i\subset\bigcup\_n C\_n^i$, where each $C\_n^i$ is closed and
nowhere... | 11 | https://mathoverflow.net/users/1946 | 109241 | 62,854 |
https://mathoverflow.net/questions/109243 | 3 | Is it possible to find a genus two curve $C$ (over the field of complex numbers) with an endomorphism $\phi: C \to C$, such that $\phi$ has no fixed points and $\phi$ does not take any point to its hyperelliptic involution?
| https://mathoverflow.net/users/17673 | genus two curve with special automorphisms | No, $\phi$ must come from an automorphism of $\mathbb{P}^1$ and that has fixed points.
| 11 | https://mathoverflow.net/users/2290 | 109246 | 62,856 |
https://mathoverflow.net/questions/109250 | 3 | I was reading the following question: [About isogeny theorem for elliptic curves](https://mathoverflow.net/questions/41931/about-isogeny-theorem-for-elliptic-curves) and was interested in the following statement at the end of Torsten Ekedahl's answer:
"Note also that the situation is similar (not by chance) to the ca... | https://mathoverflow.net/users/16858 | Another question related to the isogeny theorem for elliptic curves | I think that maybe what is meant is that there is no *functorial* way to define the bijection in the category of algebraic geometry. So suppose that we let $\hbox{Ell}(R)$ denote the set of elliptic curves with $\hbox{End}(E)\cong R$, where for simplicity $R$ is the maximal order of an imaginary quadratic field. (The i... | 7 | https://mathoverflow.net/users/11926 | 109256 | 62,862 |
https://mathoverflow.net/questions/109242 | 4 | Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by
$$x^4-y^4-z^4+w^4 = 0$$
and consider the involution
$$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$
The surface $X$ can be seen as a narural elliptic fibration over $\mathbb{P}^{1}$ as explained here
[construct the elliptic fibration of ellipti... | https://mathoverflow.net/users/14514 | Involution of the Fermat quartic | Using the notation of the question [construct the elliptic fibration of elliptic k3 surface](https://mathoverflow.net/questions/87633/construct-the-elliptic-fibration-of-elliptic-k3-surface), one sees that the elliptic curve $C\_{[\lambda:\mu]}$ is sent to the curve $C\_{[-\lambda: \mu]}$ by the involution $i$. So the ... | 4 | https://mathoverflow.net/users/7460 | 109262 | 62,863 |
https://mathoverflow.net/questions/109222 | 2 | Hi,
I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject.
Thanks in advance
| https://mathoverflow.net/users/27114 | Jacobi method on first order partial differential equations | Chapter VII of É. Goursat's book, [Leçons sur l'intégration des équations aux dérivées partielles du premier ordre](http://name.umdl.umich.edu/ACR1803.0001.001), exposes the method and ends with 14 examples of applying it (pp. 168-169).
| 3 | https://mathoverflow.net/users/19276 | 109268 | 62,867 |
https://mathoverflow.net/questions/100701 | 15 | A few months ago, I have asked a [question](https://mathoverflow.net/questions/81079/primes-of-the-form-x2ny2mz2-and-congruences) about primes represented by ternary quadratic forms. I got two wonderful answers, which showed me how the theory was way richer and more complex that I naively expected from the case of bina... | https://mathoverflow.net/users/9317 | Primes and $x^2+2y^2+4z^2$ | Here's a simpler argument. We may assume p is 7 mod 8. Let N be the number of triples of squares (r,s,u) with r+2s+4u=p. We will show that N is odd if p is 7 mod 16 and even if p is 15 mod 16. Let M be (1/64)(the number of representations of p by xx+yy+zz+tt). Jacobi's 4 square theorem (which has elementary proofs usin... | 5 | https://mathoverflow.net/users/6214 | 109275 | 62,870 |
https://mathoverflow.net/questions/109266 | 1 | Let $x\_1^4+x\_2^4+x\_3^4+x\_4^4=0 \subset \mathbb{P}^4$ be the Fermat K3 surface. Is it possible to start from some involution on $\mathbb{P}^3$, do blow-ups to get rid of fixed points and then quotient with respect to resulting involution to get an Enrique surface inside the quotient of some blow-up of $\mathbb{P}^3$... | https://mathoverflow.net/users/5259 | How to construct Enriques surface from Fermat K3 | Let me show that the answer to your last question is **yes**, by explaining how to construct an Enriques surface inside a simply connected threefold.
Let $\psi \colon V \to \mathbb{P}^3$ be a double covering branched on a smooth quartic surface and let $\tau \in \textrm{Aut}(V)$ be an involution fixing eight points.... | 2 | https://mathoverflow.net/users/7460 | 109277 | 62,872 |
https://mathoverflow.net/questions/109254 | 3 | I am trying to completely characterize the conditions on $f : \mathbb{R}^n \to \mathbb{R}$ under which $\{x | f(x) \le 0 \}$ is path-connected. There are many obvious conditions that are sufficient (e.g. $f$ concave), but is there any suite of conditions that is necessary and sufficient?
We can assume $f$ is continuo... | https://mathoverflow.net/users/21816 | When is a sublevel set path-connected? | If $f$ is $C^1$ and coercive, the existence of a non-path connected sub-level set implies the existence of a critical point, e.g. by the [mountain pass theorem](http://en.wikipedia.org/wiki/Mountain_pass_theorem). If $f$ is $C^2$, the Hessian of this critical point has a "strong" Morse index $m \_ \* $ (number of negat... | 2 | https://mathoverflow.net/users/6101 | 109281 | 62,874 |
https://mathoverflow.net/questions/109264 | 7 | Consider a symmetric N-player game in which all players partition one total unit of
energy among individual games. The probability of winning each game is simply proportional to the spent energy (player #1 wins with probability $\frac{E\_1}{E\_1+E\_2+...+E\_N}$).
The winner is the first player to win G games.
Before ... | https://mathoverflow.net/users/20757 | Optimum Tournament Strategy | I will address the case $N=2$.
The best strategy is the boring one of distributing your energy evenly. This can be proved inductively in the number of games left. If the players need $a$ and $b$ games to win, respectively, then we say the number of games left is $a+b-1$.
It is trivial for $1$ game left, and a simp... | 4 | https://mathoverflow.net/users/2954 | 109286 | 62,877 |
https://mathoverflow.net/questions/109288 | 5 | I came across this rather week small cancellation condition $C'\left(\frac{5}{11}\right)$ of a group $G$. It has been proved that $C'\left(\frac16\right)$ is enough for $G$ to contain free subgroups. I was therefore wondering if $\frac{5}{11}$ is maybe enough to still have exponential growth.
Does anyone know of any... | https://mathoverflow.net/users/23232 | Does $C'\left(\frac{5}{11}\right)$ imply exponential growth? | Every finitely presented group has presentation satisfying $C'(1/5)$. Note that $1/5 < 5/11$. See the book by Olshanskii's book "Geometry of defining relations of groups".
| 11 | https://mathoverflow.net/users/nan | 109294 | 62,881 |
https://mathoverflow.net/questions/109258 | 0 | Suppose $L/K$ and $M/K$ are algebraic extensions of a field $K$, such that $L\cap M=K$, and $L/K$ is a normal extension. It is well-known that, with these conditions, we have:
$$\text{Gal}(L/K)\cong \text{Gal}(LM/M).$$
However, suppose we now complicate matters by specifying that $M/K$ (and hence also $LM/L$) is no... | https://mathoverflow.net/users/2189 | Does this isomorphism between Galois groups hold for transcendental extensions? | The answer is yes, no matter what the characteristic is. Since $L/K$ is not assumed to be separable, I believe that $\text{Gal}(L/K)$ denotes the group of $K$-automorphisms of $L$. Let $K'$ be the fixed field of this group. By the usual properties of linearly disjoint extensions, we know that the fields $L$ and $K'M$ a... | 1 | https://mathoverflow.net/users/18739 | 109299 | 62,884 |
https://mathoverflow.net/questions/109297 | 1 | Let $A$ be a semistable real matrix (*i.e.* the real parts of all the eigenvalues of $A$ are nonnegative). Let $P$ be a positive definite matrix.
**Is it always true that $\operatorname{trace}{A^{T}P+PA} \leq 0$?**
P.S.
In fact, slightly more is known about $A$: that $A+A^{T}$ is negative semidefinite.
| https://mathoverflow.net/users/22051 | Is the trace of a Lyapunov transform of a semistable matrix always nonpositive? | Here is a counterexample. Notice that $\mathrm{Tr}(A^T P)=\mathrm{Tr}(PA)$, so it suffices to calculate $\mathrm{Tr}(PA)$. Suppose $n=2$ and $A$ triangular:
$$P=\begin{pmatrix} x & y \\ y & z \end{pmatrix},\qquad A=\begin{pmatrix} a & b \\ 0 & c \end{pmatrix},$$
where $x,z,xz-y^2$ positive, and $a,c\le0$. We have $\mat... | 2 | https://mathoverflow.net/users/8799 | 109300 | 62,885 |
https://mathoverflow.net/questions/109261 | 7 | It seems like there is an algorithm to find the Heegard diagram of a 3 manifold obtained by surgery on a link. Also someone told me I can find it in the Gompf and Stipciz's book. But I could not find it. Can anyone help?
| https://mathoverflow.net/users/27129 | heegard diagram | (1) Choose a planar presentation of your link, approximately in the plane of the blackboard. Let $N$ be a tubular neighborhood of the link in this position.
(2) For each crossing $c$ of the presentation, add a 1-handle $T\_c$ to $N$ which is perpendicular to the blackboard and connects the upper and lower parts of th... | 14 | https://mathoverflow.net/users/284 | 109307 | 62,891 |
https://mathoverflow.net/questions/109310 | 18 | I have been thinking about quotients lately and pondered the following:
Let $G$ be a connected linear algebraic group and $X$ a $G$-variety where the action is the morphism $\sigma:G\times X\rightarrow X$. Let $p:L\rightarrow X$ be a line bundle on $X$.
A $G$-linearisation of $L$ is an action of $G$ on $L$ such th... | https://mathoverflow.net/users/9970 | A line bundle that does not admit a G-linearisation | Let me give an example showing that the normality hypothesis is necessary.
Let $Y=\mathbb{P}^1$ with natural $G=\mathbb{G}\_m$-action. Let $X$ be the $G$-variety obtained by glueing transversally the two fixed points $0$ and $\infty$. Consider the line bundle $\mathcal{O}(l)$ with $l\neq 0$ on $Y$ and glue the fibers... | 18 | https://mathoverflow.net/users/2868 | 109312 | 62,894 |
https://mathoverflow.net/questions/108646 | 2 | I want to numerically integrate the equation $\partial\_t u= a(t) \partial\_xu+b\partial\_{xxx}u+c$ to get $u(t)$. Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives? Thanks. I am using a Semi-Discretization method called Method of Lines (vertical) to treat this problem.
| https://mathoverflow.net/users/19294 | Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives to solve this IVP problem ? | You could use the method of lines to solve this PDE. If you use an explicit finite difference method, you will need to take a rather small time step (${\mathcal O(\Delta x^3))}$ due to the $u\_{xxx}$ term.
Given that you don't specify any boundary conditions, I will assume that you are solving the Cauchy problem. In ... | 2 | https://mathoverflow.net/users/20507 | 109314 | 62,895 |
https://mathoverflow.net/questions/109319 | 10 | Let $H$ be a Hilbert space,
and let $A\_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R\_{\ge 0}$. Consider the ordinary differential equation
$$
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\frac{d}{dt} E\_t = -A\_tE\_t
\quad\qquad\qquad\qquad\qquad\qquad\qquad(1)
$$
t... | https://mathoverflow.net/users/5690 | ordered exponential of unbounded operators | This is usually called the magnus expansion method and has a nice literature in numerical analysis. Kato also used this method to show the existence of solutions in the hyperbolic case.
I would say that strong resolvent continuity and a sufficiently big common domain is sufficient in your case. See Section 5.3 in [P... | 1 | https://mathoverflow.net/users/12898 | 109324 | 62,899 |
https://mathoverflow.net/questions/109318 | 5 | Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random variables. I am working on a model from mathematical physics involving such measures, and I would like to express them in... | https://mathoverflow.net/users/238 | Stationary, ergodic measures from the structuralist point of view | I'm not familiar enough with the notion of localisable measurable space in Pavlov's sense to say anything too authoritative, but I can make the following comments which apply to at least the case $d=1$ (as a dynamicist working mostly with group actions generated by a single continuous map $f\colon X\to X$ this is the n... | 4 | https://mathoverflow.net/users/5701 | 109327 | 62,902 |
https://mathoverflow.net/questions/109306 | 6 | In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}\_2$
$lim\_{t \to \infty}\begin{pmatrix} it & b \\\ b & \tau\end{pmatrix} = \tau.$
Since we have that $M\_2$ is an open of $A\_2$, it is natural to look for a family of genus 2 curves ... | https://mathoverflow.net/users/27125 | Genus 2 curves vs Abelian surfaces | The analytic solution is easy enough to describe: Compute the gradients of the Theta function at the six odd 2-torsion points, and projectivize these gradients. You now have six points on the projective line. This are the 6 Weierstrass points of the curve, alternatively there is the "Rosenhaon normal form", which expre... | 3 | https://mathoverflow.net/users/404 | 109331 | 62,904 |
https://mathoverflow.net/questions/109334 | 13 | Let $R$ be the set of homogeneous polynomials of degree $n$ in $d$ variables over $\mathbb{C}$. When $n>2$, the set of elements of $R$ that split into a product of linear factors forms a proper subset $S$ of $R$.
1. Is $S$ an algebraic variety, or something almost as nice?
2. If so, how can $S$ be described implicit... | https://mathoverflow.net/users/9611 | which homogeneous polynomials split into linear factors? | 1) This is the Chow variety of degree $n$ zero cycles in $\mathbb{P}^{d-1}$.
2) Yes, this collection of polynomials can be bundled together into the Brill form or covariant.
3) Rather explicit descriptions of the Brill equations can be found in the book
by Gelfand, Kapranov and Zelevinsky on resultants. There is al... | 16 | https://mathoverflow.net/users/7410 | 109340 | 62,910 |
https://mathoverflow.net/questions/109298 | 6 | I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model theoretically nice e.g. stable).
So I have some idea as to why these open image theorems should hold from a model the... | https://mathoverflow.net/users/16858 | Example of a diophantine application of an open image theorem | Here's an application to independence of Heegner points. (But if you search on MathSciNet for papers that reference Serre's two results, I expect you'll find a very large number of applications.)
Let $E/\mathbb{Q}$ be an elliptic curve with no CM, and let $\Phi:X\_0(N)\to E$ be a modular parametrization. (Wiles et.al... | 5 | https://mathoverflow.net/users/11926 | 109348 | 62,914 |
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