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https://mathoverflow.net/questions/97035 | 8 | What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that.
Namely, I'm interested in the homotopy groups of the free simplicial *commutative* ring on a simplicial set. Let $X\_\bullet$ be a simplici... | https://mathoverflow.net/users/344 | Derived functors of symmetric powers | The homology of all of the symmetric groups together is well understood, as Tyler says.
Taking mod $p$ coefficients, that is the special case when $X = S^0$ of the calculation
of $H\_\*(CX)$ as a functor of $H\_\*(X)$, where $C$ is the monad on based spaces associated
to any $E\_{\infty}$ operad of spaces. The calcula... | 7 | https://mathoverflow.net/users/14447 | 97225 | 56,772 |
https://mathoverflow.net/questions/97216 | 2 | Consider a linear system
$$Ax=b\qquad (\*)$$
and a sequence of perturbed linear systems $$(A+\delta A\_n)x=b+\delta b\_n. \qquad (n)$$
Suppose that all the linear systems are consistent (i.e., have solutions).
My question is: Let $\overline{x}$ be a solution to $(\*)$. Suppose that $\delta A\_n \rightarrow 0$ and... | https://mathoverflow.net/users/11870 | A question for solutions of perturbed linear systems | Consider $A= (1, 0 ; 1,0)$ and $b= (1 ; 1)$.
Let $x$ be any solution of this, so $x= (1; t)$ for some $t$.
Now, perturbe this to $(1, 0 ; 1 - 1/n,1/n^2)$ and $b$ unchanged, so pertruebed by $0$ (but one could also impose some nontrivial perturbation).
The perturbation(s) tends to zero. But the (only) solution is $(1... | 2 | https://mathoverflow.net/users/nan | 97228 | 56,774 |
https://mathoverflow.net/questions/97168 | 6 | Question 1: Does there exist an intrinsic characterization of groups $G$ isomorphic to some subgroup of some finitary symmetric group (i.e. all the permutations of a given set that fix
all but finitely many elements)?
Clearly every such $G$ enjoys local finiteness, but I see where (for a fixed $p$) the multiplicativ... | https://mathoverflow.net/users/10909 | Subgroups of finitary symmetric groups | The answer to Question 2 is "yes". Take the relatively free group with law $x^4=1$ and infinite set of generators. That group is locally finite (Sanov, I. N.
Solution of Burnside's problem for exponent 4.
Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 10, (1940). 166–170.).
It is not inside the group of fin... | 6 | https://mathoverflow.net/users/nan | 97229 | 56,775 |
https://mathoverflow.net/questions/97132 | 5 | Suppose that we have $q$ positive integers $a\_1, \ldots, a\_q$ satisfying $a\_1 \leq \cdots \leq a\_q \leq q$. I'm interested in the possible types of behaviors for the function given by $$f(x) = (a\_1^{x-1} + \cdots + a\_q^{x-1})^{1/x},$$ where $x \in [2,\infty)$. In particular, I'm interested in the behavior at inte... | https://mathoverflow.net/users/23578 | Sum of exponential functions | As @David points out, the function is log-convex -- this is a general fact about $L^p$ norms (as pointed out by @Tom, this is a special case of an $L^p$ norms where the measure of a point $x\_i$ is proportional to $1/a\_i.$) Not one, not two, but *three* proofs of this fact is given in [Terry Tao's excellent as usual b... | 1 | https://mathoverflow.net/users/11142 | 97240 | 56,780 |
https://mathoverflow.net/questions/96126 | 3 | Let $G= SL(2, F)$, given a torus $T$, the Weyl group with respect to $T$ is defined to be $W=N(T)/Z(T)$, the quotient of the normalizer $N(T)$ of the torus by the centralizer $Z(T)$ of the torus.
My question might be basic and elementary for some of you, but I would like to be in a complete understanding.
Why Weyl... | https://mathoverflow.net/users/9842 | The Weyl group of $SL(2, F)$ | Two people (so far) have tried to provide some sensible information, but I think the question itself is too loosely formulated to have a real answer. What you are looking at is the group of rational points of an algebraic group of rank 1 over a certain type of infinite field, which could be of characteristic 0 or not. ... | 8 | https://mathoverflow.net/users/4231 | 97265 | 56,793 |
https://mathoverflow.net/questions/97097 | 15 | In Mariusz Wodzicki's paper "Cyclic homology of differential operators," the following result is mentioned: for $D\_M$ the algebra of differential operators on a smooth manifold $M$ we have that $HH\_n(D\_M) \cong H\_{DR}^{2m-n}(M)$ where $m=\dim M$. I'm having trouble finding a reference for the Hochschild Cohomology ... | https://mathoverflow.net/users/112114 | Hochschild Cohomology of Differential Operators in characteristic 0 | Hi,
This is an instance of a more general fact concerning deformation quantization of symplectic varieties. The general theorem is:
`Let $X$ be an symplectic manifold, and let $A\_\hbar$ be any quantization of the Poisson algebra $(C^\infty(X), \{, \})$. Then we have $HH^\*(A\_\hbar)\cong H\_{DR}^\*(X)$.
See the ... | 14 | https://mathoverflow.net/users/1040 | 97267 | 56,794 |
https://mathoverflow.net/questions/97266 | 5 | A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a finite dimensional graded ring, can you find a domain of holomorphy having that ring as its cohomology ring?
| https://mathoverflow.net/users/1106 | How does pseudoconvexity restrict the topology? | A theorem of [Eliashberg](http://www.worldscinet.com/ijm/01/0101/S0129167X90000034.html) implies that an open subset of $\Bbb C^n$, $n \neq 2$, is isotopic to a Stein domain (hence to a domain of holomorphy) if and only if it admits a handlebody structure with all handles of index $\leq n$. For $n = 2$ the theorem stil... | 8 | https://mathoverflow.net/users/23193 | 97269 | 56,796 |
https://mathoverflow.net/questions/97207 | 9 | Given a strictly positive integer $A$, let $D(A)$ denote the set of all
real quadratic algebraic numbers with a continued fraction having almost all coefficients
$\leq A$.
Consider the field $Q\_A$ generated by all elements of $D(A)$. One has $Q\_1=\mathbb Q[\sqrt{5}]\subset Q\_2\subset Q\_3,\dots$.
The inclusion... | https://mathoverflow.net/users/4556 | Coefficients in the periodic part of continued fractions for real quadratic algebraic numbers | In [this paper of McMullen](http://dx.doi.org/10.1112/S0010437X09004102), he asks on p. 842 whether all real quadratic fields
contain infinitely many continued fractions with coefficients bounded by 2? In this case, one
would have $Q\_d=Q\_2=\mathbb{Q}(\sqrt{\mathbb{N}})$ for all $d\geq 2$.
See also [McMullen's slides]... | 11 | https://mathoverflow.net/users/1345 | 97270 | 56,797 |
https://mathoverflow.net/questions/97258 | 11 | I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell\_p$, $p\in (1,\infty)$ or $X=c\_0$.
Since the differences between *partiuclar* choices of ultrafilters are not clear to me enough, let me take *any* free ultrafilter $\... | https://mathoverflow.net/users/23779 | Do ultrapowers of classical Banach spaces have unconditional bases? | For $\ell\_1$ and $c\_0$, the answer to your (implicit) question is easy. If you take a free ultrafilter on the natural numbers (or most any ultrafilter used in Banach space theory) the ultrapower contains $L\_1$ (or $C[0,1]$) and hence does not have an unconditional basis.
For $\ell\_p$, $1<p\not= 2 < \infty$, the s... | 7 | https://mathoverflow.net/users/2554 | 97272 | 56,799 |
https://mathoverflow.net/questions/97274 | 6 | If $X$ is a noetherian scheme , then for any quasi-coherent sheaf $\mathscr{F}$ on $X$ that satisfies Serre's (S2) condition and an inclusion of an open subscheme $j:U\to X$ with complement of codimension at least 2 then we have $$j\_{\star}j^{\star}\mathscr{F}\cong\mathscr{F}.$$ This is like the Hartogs phenomenon for... | https://mathoverflow.net/users/16943 | When do we have derived "Hartogs" for quasi-coherent sheaves? | Unfortunately almost never (although it obviously holds if $j^\* \mathcal{F} = 0$). In particular, it virtually never holds for $\mathcal{F} = \mathcal{O}\_X$. I'm going to assume that $\mathcal{F}$ is coherent.
Let me give the quick answer first, then I'll explain it in more detail.
**Quick answer:** $R^i j\_\* j^... | 8 | https://mathoverflow.net/users/3521 | 97279 | 56,802 |
https://mathoverflow.net/questions/97289 | 0 | What is a locally cosmall category relative to a universe?
| https://mathoverflow.net/users/23785 | What is a locally cosmall category relative to a universe? | Unfortunately, there is a clash of terminology: Some people call a category locally small if the hom-classes are sets. These people call a category well-powered if the subobjects of any object form a set; dually, co-well-powered refers to quotients. For these people, there is no notion of locally cosmall. Other people ... | 3 | https://mathoverflow.net/users/2841 | 97300 | 56,806 |
https://mathoverflow.net/questions/97287 | 3 | Assume M affine algebraic manifold over C and we know that H^1(M,Z)=H^2(M,Z) = 0.
**Question** Does it imply Pic^algebraic(M) = 0 ?
Pic^algebraic means group of algebraic line bundles = H^1(M, O^\*) in Zariskky topology.
This is follow up to:
[Are there other ways to show Pic(G)is trivial when G is a simple-co... | https://mathoverflow.net/users/10446 | M - affine and H^1(M,Z)=H^2(M,Z) = 0 imply (?) Pic^algebraic(M) = 0. Note: in algebraic category NO exponential sequence | Let us assume that we work over an algebraically closed base field $k$, and fix a prime $\ell$ different from the characteristic of $k$.
First of all, the Kummer-sequence
$$0\rightarrow \mu\_{\ell^n}\rightarrow \mathbb{G}\_m\xrightarrow{\ell^n}\mathbb{G}\_m\rightarrow 0$$
for $\ell$ prime to the characteristic of $k$, ... | 9 | https://mathoverflow.net/users/259 | 97309 | 56,810 |
https://mathoverflow.net/questions/13831 | 1 | Sorry for the title, but I think it's funny. Can you write down a homomorphism (of additive groups)
$\mathbb{R}^\mathbb{N} \to \mathbb{R}$,
which is nontrivial and whose kernel contains the finite sequences? For example, on the subgroup of convergent sequences, we can take the limit. The question is not if such thi... | https://mathoverflow.net/users/2841 | Limit for divergent sequences | Since $\mathbb{R}^\mathbb{N}$ with the product topology is a polish group and the set $F$ of finite sequences is dense in it, it follows that the only Baire measurable homomorphism $\mathbb{R}^\mathbb{N} \to \mathbb{R}$ that contains $F$ in its kernel is the trivial one. So "writing down" a nontrivial one will be prett... | 2 | https://mathoverflow.net/users/17836 | 97311 | 56,812 |
https://mathoverflow.net/questions/97283 | 1 | Let $\pi: G \rightarrow S$ be a finite flat group scheme over a locally noetherian connected base scheme $S$.
Its degree is defined as the rank of the locally free $\mathcal O\_S$-module $\pi\_\* \mathcal O\_G$.
Let $H \subset G$ be a closed subgroup scheme of $G$ which is also finite flat over $S$.
I want to show th... | https://mathoverflow.net/users/18183 | Degree of finite group schemes | This can be seen from the existence of the quotient $G/H$ as a finite flat $S$-scheme (and an $H$-torsor). One shows that the natural map $G \rightarrow G/H$ is finite flat of order $[H : S]$; the conclusion then follows from the product formula $[G : S] = [G : G/H] [G/H : S]$. Let me give you a reference where all thi... | 3 | https://mathoverflow.net/users/5498 | 97330 | 56,820 |
https://mathoverflow.net/questions/97307 | 12 | I have two questions about the class of integer-coefficient polynomials all of whose roots are rational.
I asked [this at MSE](https://math.stackexchange.com/questions/146288/), but it attracted little interest (perhaps because it is not interesting!)
**Q1**. Is there some way to recognize such a polynomial from its ... | https://mathoverflow.net/users/6094 | Polynomials all of whose roots are rational | It seems to me that the obvious algorithm via the rational root theorem is somewhat inefficient in at least two cases: $a\_0$ or $a\_n$ is BIG (so that we might not even be able to factor it), or they have A LOT of prime factors.
Instead, I believe the following algorithm based on [Hensel's lifting lemma](http://en.w... | 10 | https://mathoverflow.net/users/23796 | 97331 | 56,821 |
https://mathoverflow.net/questions/97316 | 5 | The equation $c^2 = a^2 + b^2 + ab$ is the law of cosines for a triangle with integer sides $a$, $b$, and $c$, and a 120 degree angle opposite side $c$. By the substitution $x = (a-b)/2$, $y = (a+b)/2$ it can be transformed to $x^2 + 3y^2 = 4z^2$, which is a more familiar equation, whose solutions are given in parametr... | https://mathoverflow.net/users/12669 | c^2 = a^2 + b^2 + ab and its solutions | Here are some thoughts, too long for a comment.
You need to assume that $a$ and $b$ are coprime, otherwise there are infinitely many solutions with the same square-free part for $ab$ (scaling does not change it). Assuming $a$ and $b$ are coprime, bounding the square-free part of $ab$ is the same as requiring that $a... | 3 | https://mathoverflow.net/users/11919 | 97333 | 56,823 |
https://mathoverflow.net/questions/97304 | 9 | Let $G$ be a group acting on a locally finite tree $T$. Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric). However, even if $G$ is the full automorphism group of $T$, we can't get the full quasi-isometry group of $\partial T$, as the ... | https://mathoverflow.net/users/4053 | Recognising group actions on trees from the boundary | Besides our paper that Alessandro mentions, Sageev, Whyte and I also have a paper "Maximally Symmetric Trees" which comes pretty close to doing what you ask. For certain trees it gives nice descriptions of $Aut(T)$ in terms of the representation into the quasi-isometry group $QI(T)$ (which, by the paper "Un groupe hype... | 10 | https://mathoverflow.net/users/20787 | 97335 | 56,825 |
https://mathoverflow.net/questions/97329 | 6 | This is inspired by [this](https://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational) question. Let $f(x)=a\_nx^n+...+a\_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking all $p$ that divide $a\_0$ and all $q$ that divide ... | https://mathoverflow.net/users/nan | Complexity of finding a rational root of a polynomial | Didn't Lenstra, Lenstra and Lovász in their famous LLL paper prove that factorization of polynomials over $\mathbb Q$ can be done in polynomial time? You get a rational root if and only if there is a factor of degree 1, and the polynomial has only rational roots if and only if all factors have degree 1.
Lenstra, A.K.... | 10 | https://mathoverflow.net/users/21146 | 97338 | 56,827 |
https://mathoverflow.net/questions/97339 | 6 | Let $K$ be a field, and $F\_K$ be the fraction field of the polynomial ring $R\_K$ in $n^2$ indeterminates $X\_{11},X\_{12},...,X\_{nn}$ over $K$.
Now set $A = (X\_{ij})\_{i,j} \in M\_n (F\_K)$, and let $\chi\_A$ be the characteristic polynomial of $A$.
**Question** : Is it always true that $\chi\_A$ is irreducible ... | https://mathoverflow.net/users/21724 | Characteristic polynomial of a generic n*n matric | If it were reducible, all $n\times n$ matrices over any field extension of $K$ would have reducible characteristic polynomials. But consider the companion matrix of an irreducible polynomial over some not algebraically closed field extension of $K$.
| 9 | https://mathoverflow.net/users/15934 | 97342 | 56,830 |
https://mathoverflow.net/questions/97358 | 3 | This question is related to [this question](https://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational) of Joseph O'Rourke and [this question](https://mathoverflow.net/questions/97329/complexity-of-finding-a-rational-root-of-a-polynomial) of mine.
**Question.** Let $f$ be a polynomial w... | https://mathoverflow.net/users/nan | Solving polynomial equations in radicals provided all roots are rational | No.
Consider a generic polynomial $x^n+a\_1x^{n-1}+a\_2x^{n-2}+....+a\_n$ over $\mathbb Q(a\_1,...,a\_n)$. Adjoin $b\_1$, a root of $p\_1$ of degree $k\_1$, then adjoin $b\_2$, and so on. This is contained in some solvable galois extension of $\mathbb Q(a\_1,...,a\_n)$.
Let $q$ be the generic polynomial evaluated a... | 4 | https://mathoverflow.net/users/18060 | 97363 | 56,842 |
https://mathoverflow.net/questions/97364 | 7 | Let $f(x) \in \mathbb{Z}[x]$ be a monic irreducible polynomial of degree $n$, and let $\alpha\_1, \alpha\_2, ... , \alpha\_n \in \overline{\mathbb{Q}}$ be the $n$ distinct roots of $f(x)$.
Following Bewersdorff's "Galois Theory for Beginners" (and older sources?) I want to define the Galois group of $f(x)$ as follows... | https://mathoverflow.net/users/23806 | The Galois group and relations among the roots of a polynomial | The Galois ideal is one of the prime factors of the ideal generated by the trivial elements. Thus, an algorithm for primary decomposition in $\mathbb Q[x\_1,...x\_n]$, of which there are several, will do the trick.
Proof: The Galois ideal is prime, which is obvious from its definition. It contains the trivial ideal. ... | 6 | https://mathoverflow.net/users/18060 | 97366 | 56,843 |
https://mathoverflow.net/questions/97356 | 3 | Question 1. Let $X\_i$ be an infinite family of closed, orientable, smooth 4-manifolds with the following properties:
a) $\pi\_1(X\_i) = \mathbb{Z}\times \mathbb{Z\_{2}}$ for any $i = 1, 2, \cdots $
b) all the homology groups of $X\_i$ and $X\_j$ with integer coefficients are same
Is it true in this family there ... | https://mathoverflow.net/users/23802 | Homeomorphism classification of 4-manifolds | First, you also want to fix not just $H\_2$ but $H\_2(M, {\mathbb Z}[\pi\_1(M)])$ together with the intersection form on this group. With this in mind, if $M$ is a closed 4-manifold whose fundamental group is infinite cyclic, then Freedman-style classification is indeed available for $M$, but requires extra work which ... | 7 | https://mathoverflow.net/users/21684 | 97370 | 56,847 |
https://mathoverflow.net/questions/97357 | 2 | Hi, everyone, I want to ask following problem:
Is strict Henselian ring a excellent ring?
If not, could you give me a example?
| https://mathoverflow.net/users/5274 | Is strict Henselian ring a excellent ring? | Let $k$ be a field of characteristic $p>0$. Consider the field $k((t))$ of Laurent series, endowed with its $t$-adic valuation. Let $L$ be a subextension of $k((t))/k$ which is of finite type over $k$ and of positive transcendence degree over $k$. Consider the subring $A$ of $k[[t]]$ corresponding to $L$. Then the disc... | 11 | https://mathoverflow.net/users/1017 | 97382 | 56,853 |
https://mathoverflow.net/questions/97286 | 10 | Let $X$ be a scheme locally of finite type over a field $k$, and let $p \in X$ be a $k$-point with ${\rm dim}\; {\Omega\_{X}}\_{|p}={\rm dim}\; \mathfrak{m}\_{p}/\mathfrak{m}^{2}\_{p}=m$ (the 'embedding dimension' of the local ring at $p$).
Then a paper that I'm reading asserts that there is a Zariski open neighbour... | https://mathoverflow.net/users/6254 | Embedding dimension=minimum dimension of a local embedding? | One algebraic version of this statement is that if $A$ is a local ring with embedding dimension $m$ and $A$ is the quotient of a regular local ring, then $A$ is the quotient of a regular local ring of dimension $m$. Write $A = R/I$ where $R$ is a regular local ring and suppose that $R$ has dimension $n$. Then we have a... | 9 | https://mathoverflow.net/users/8914 | 97387 | 56,857 |
https://mathoverflow.net/questions/97298 | 3 | Oliver Heaviside, on page 387 of *Electrical Papers*, Vol. I, Macmillan and Co., 1892, available [here](http://archive.org/stream/electricalpapers01heavuoft#page/386/mode/2up), writes
$$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots = J\_0(nr)$$
and
**This function is us... | https://mathoverflow.net/users/22714 | Fourier and Bessel | There is a fundamental reference using Bessel functions in Fourier's works. This is ["Théorie analytique de la chaleur"](http://books.google.it/books?id=TDQJAAAAIAAJ&printsec=frontcover&hl=it#v=onepage&q&f=false) firstly published on 1822. You will find this series firstly given in chapter VI pag. 370. This chapter is ... | 4 | https://mathoverflow.net/users/19520 | 97389 | 56,859 |
https://mathoverflow.net/questions/97394 | 3 | Let $x\_1,x\_2,...,x\_k$ be irrational number,is it always true that:
$\liminf\_{n\rightarrow\infty} \sum\_{i=1}^k (nx\_i)=0$ (where $(x)$ denotes the fractional part of $x$)
If not,what are the necessary and sufficient conditions that {$x\_i$} must satisfy so that $\liminf\_{n\rightarrow\infty} \sum\_{i=1}^k (nx\_i)... | https://mathoverflow.net/users/22907 | sum of fractional parts (nx_i),x_i are irrational | Here is a counterexample: $x\_1 = \sqrt2, x\_2 = 1-\sqrt2$.
For $n>0$, $(nx\_1) + (n x\_2) =1$ so the limit infimum is $1$, not $0$.
I think the other answers assumed that you meant $x-[x]$, where $[x]$ is the nearest integer to $x$, instead of $(x)$.
If there is no rational dependency, then the multiples are dens... | 4 | https://mathoverflow.net/users/2954 | 97403 | 56,867 |
https://mathoverflow.net/questions/97412 | 9 | Suppose that $g$ is a $C^{1,1}$ (i.e., continuously differentiable with locally Lipschitz first derivative) Riemannian metric on a smooth manifold $M$. It seems to be known that locally the exponential map $\exp\_p$ corresponding to $g$ at some point $p$ of the manifold is still a local homeomorphism (and Lipschitz). I... | https://mathoverflow.net/users/23818 | Is the exponential map of a $C^{1,1}$ Riemannian metric a local homeomorphism? | Note that $C^{1,1}$ metric admits an approximation by $C^2$-metrics with uniformly bounded $C^2$-norm.
In particular the curvature is bounded, hence we get a bounds for the distortion of the exponential map depending on the size of the neighborhood of $0$.
It remains to pass to the limit.
| 6 | https://mathoverflow.net/users/1441 | 97417 | 56,878 |
https://mathoverflow.net/questions/97429 | 12 | we know that the maximal ideals of ${\mathbb Z}[x]$ are of the form $(p, f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in ${\mathbb Z}[x]$ which is irreducible modulo $p$.
Is it true that:
the maximal ideals of ${\mathbb Z}[x,y]$ are of the form $(p, f(x,y),g(x,y))$ where $p$ is a prime number and $... | https://mathoverflow.net/users/18970 | Maximal ideals of Z[x,y] | Better yet, you can replace $f(x,y)$ with $f(x)$. See the answer to [this question](https://mathoverflow.net/questions/26497/maximal-ideals-in-the-ring-kx1-xn).
**Edited to add:** At Martin Brandenburg's request, I'm expanding this to add the details I thought were too obvious to mention:
1) A maximal ideal $M$ of... | 17 | https://mathoverflow.net/users/10503 | 97431 | 56,885 |
https://mathoverflow.net/questions/97381 | 20 | I need this result for something else. It seems fairly hard, but I may be missing something obvious.
Just one non-trivial solution for any given $c$ would be fine (for my application).
| https://mathoverflow.net/users/10454 | Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rational c? | *[edited again mainly to add the Euler link, see last paragraph]*
Yes, and indeed there are infinitely many rational points:
the birationally equivalent Diophantine equation
given by J.Ramsden in his partial answer to his own question,
$$
X + Y = Z + T,
\phantom{and}
XYZT = c,
$$
was already studied by Euler (in the ... | 39 | https://mathoverflow.net/users/14830 | 97437 | 56,888 |
https://mathoverflow.net/questions/97439 | 8 | I have a question regarding a partial order $<$
on the set ${\rm Part}(n)$ of partitions of $n$.
Given $\lambda=(\lambda\_1,\lambda\_2,\ldots)\in{\rm Part}(n)$ with
$\sum\_{i\geq1} \lambda\_i=n$ and $\lambda\_1\geq\lambda\_2\geq\cdots\geq0$,
let $J\_\lambda$ denote the $n\times n$ block diagonal matrix
$\bigoplus\_{i\... | https://mathoverflow.net/users/23827 | Finite nilpotent orbits: GL(n,q)-conjugacy classes and a partial order on partitions | I think the right way is to factor $n\_\lambda(q)$ in general. In particular, it is obviously a quotient of the order of $GL\_n(q)$. The formula for the order of $GL\_n(q)$ does not have very many prime factors: just $q$ and the first $n$ cyclotomic polynomials.
One could consider an alternate question, the order of ... | 5 | https://mathoverflow.net/users/18060 | 97441 | 56,890 |
https://mathoverflow.net/questions/97432 | 3 | Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology sphere (I don't know how much this condition matters). It's not always the case that $G$ is solvable, since one can take $... | https://mathoverflow.net/users/3405 | When is a three-manifold deck transformation group solvable? | Cooper and Long showed [you can realize any finite group acting on a rational
homology sphere](http://dx.doi.org/10.1016/S0166-8641%2898%2900116-3).
There's no general sort of classification that I know of. Maybe what you're asking for is,
given $Y'$ a rational homology sphere, and a homomorphism $\varphi: \pi\_1(Y'... | 5 | https://mathoverflow.net/users/1345 | 97442 | 56,891 |
https://mathoverflow.net/questions/97449 | 6 | I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for this result, which uses the cellular decomposition of $\mathbb{C}\mathrm{P}^2$ to get $\chi(\mathbb{C}\mathrm{P}^2) = 3$,... | https://mathoverflow.net/users/23831 | Computing the Euler characteristic of the complex projective plane using differential topology | There is a canonical way to construct *holomorphic* vector fields on $\mathbb{C}P^2,$ and that way is [described in Zoladek's "Monodromy Group", page 335.](http://www.springer.com/us/book/9783764375355) If you read the description, it will be pretty clear what the index is (note that if the vector field is holomorphic,... | 6 | https://mathoverflow.net/users/11142 | 97456 | 56,899 |
https://mathoverflow.net/questions/97448 | 7 | Let $M$, $N$ be a symmetric matrix over a ring $R$.
$M$ and $N$ are said to be equivalent if there exist an invertible
matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U^T$ is the transpose of $U$).
A question is that what is the simple canonical form of $M$
under such an equivalent relation.
We know that ... | https://mathoverflow.net/users/17787 | Canonical form of symmetric integer matrix M | It's all in the correct reference. Cassels, *Rational Quadratic Forms*, chapter 9 "Integral Forms over the Rational Integers," pages 163-164, Examples 9-11. Example 11(i) says that, for "odd" matrices, we can cut down the dimension by 2 and write $y\_1^2 - y\_2^2 + g(z\_1, \ldots , z\_{n-2}).$ The determinant of $g$ is... | 6 | https://mathoverflow.net/users/3324 | 97458 | 56,901 |
https://mathoverflow.net/questions/97464 | 33 | The last remaining problem in this whole "everything is a sphere" business, is the *Smooth Poincare Conjecture* in dimension 4: If $X\simeq\_\text{homo.eq.} S^4$ then $X\approx\_\text{diffeo} S^4$. Freedman showed that this holds if we replace "diffeomorphism" by "homeomorphism", so another viewpoint would be that $S^4... | https://mathoverflow.net/users/12310 | How to Tackle the Smooth Poincare Conjecture | In principle the Ricci flow (with surgery) could also be used to prove the smooth Poincare' conjecture in dimension $4$.
There are some major problems to be overcome in this approach (problems which did not arise in dimension $3$, such as the absence of Hamilton-Ivey pinching estimates, and the new "hole-punch" sing... | 23 | https://mathoverflow.net/users/13168 | 97465 | 56,903 |
https://mathoverflow.net/questions/97469 | 0 | Fields are one of the following: scalars, vectors, spinors or some Lie algebra elements, right? And it's often said that scalars are spin-0 and vectors are spin-1. So, what's idea of correspondence between nature of field and it's spin value?
| https://mathoverflow.net/users/23834 | How the spin value is related to mathematical nature of the field? | A classical field is a section of a vector bundle on the space-time manifold $M$.
That vector bundle is typically obtained by using the [associated bundle](http://en.wikipedia.org/wiki/Associated_bundle) construction
applied to the frame bundle of $M$, and some irreducible representation of SO(d) (you've noted that I... | 8 | https://mathoverflow.net/users/5690 | 97473 | 56,906 |
https://mathoverflow.net/questions/97478 | 1 | I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell\_p$-sum of an arbitrary (possibly uncountable) family of Grothendieck spaces is Grothendieck.
Best wishes,
A.
| https://mathoverflow.net/users/23779 | Reference request for sums of Grothendieck spaces | For $1< p<\infty$ it is an exercise (which, in fact, some of my students did this past semester).
For $p=\infty$ it is false. The space $(\ell\_1^1\oplus \ell\_1^2 \oplus \ell\_1^3 \oplus \dots)\_\infty$ contains a norm one complemented subspace that is isometrically isomorphic to $\ell\_1$ (another exercise for my s... | 3 | https://mathoverflow.net/users/2554 | 97483 | 56,912 |
https://mathoverflow.net/questions/97477 | 19 | Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's
[article](http://projecteuclid.org/euclid.bams/1183528512)
in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}\_q$ for rank $q\ $ block bundles, which is a PL analog of $BO\_q$, and there are similar classifying spaces $B\... | https://mathoverflow.net/users/1573 | Characteristic classes for block bundles | I don't know where the results are written down in one place (perhaps in the book of Madsen and Milgram?), but see the the end of this post for a list of references.
In any case, here is a proof of the result for $q \ge 3$.
By a result of Haefliger, there is a homotopy cartesian square
$$
B\widetilde{PL}\_q \quad \... | 14 | https://mathoverflow.net/users/8032 | 97484 | 56,913 |
https://mathoverflow.net/questions/97486 | 2 | I'll be delighted to get some help in understanding the proof of the first theorem here:
<http://www.math.utah.edu/~malone/QI/notes.pdf>
"If G acts geometrically on X and Y (proper geodesic metric spaces)
then X and Y are quasi-isometric."
In his proof, he fixed $a,b \in X$ and took an arbitrary $q \in Y$ . He then p... | https://mathoverflow.net/users/20568 | Fundamental Lemma Of Geometric Group Theory | In the context of that proof, $p$ is a base point for $X$, $q$ is a base point for $Y$, and the function $f : X \to Y$ is defined on each $x \in X$ by first choosing $g \in G$ so that $d(x,gp) \le R$ and then defining $f(x) = gq$. The proof is showing that the function $f$ is a quasi-isometry.
| 2 | https://mathoverflow.net/users/20787 | 97487 | 56,915 |
https://mathoverflow.net/questions/97490 | 3 | I once read that the first eigenvalue of a Schrödinger operator always is simple, together with an easy proof of it. But I cannot remember where. Does anybody know a reference?
| https://mathoverflow.net/users/16702 | First eigenvalue of Schrödinger operator is simple | Say that $V$ be bounded by below. Up to the addition of a large enough constant, you may assume that $V\ge0$. Then argue as follow.
The operator $L=-\Delta+V$ (where $V(x)$ is the potential) satisfies the maximum principle: if $f\ge0$ and $f\not\equiv0$, then the solution $u$ of $-\Delta u+Vu=f$ exists, is unique and... | 8 | https://mathoverflow.net/users/8799 | 97491 | 56,916 |
https://mathoverflow.net/questions/97497 | 5 | What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int\_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but divergent in the lower.
Thank you for any ideas and/or references.
| https://mathoverflow.net/users/23841 | One-sided Cauchy principal value | Let me give an example: you want to define a distribution on $\mathbb R$ which coincides with $1/t$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$. Let us take
$$
T=\frac{d}{dt}(H(t)\ln t),\quad H=1\_{\mathbb R\_+},
$$
with a distributional derivative. You can easily generalize to a 1D situation with a finite number of... | 4 | https://mathoverflow.net/users/21907 | 97502 | 56,922 |
https://mathoverflow.net/questions/84345 | 11 | Again, there is a general and a concrete question:
**Concrete question.** Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it is known that $H$ is a Hopf algebra, thus has an antipode $S$. Let $E:H\to H$ be the $k$-linear map which sends eve... | https://mathoverflow.net/users/2530 | Connected graded involutive bialgebras (Hopf algebras) sought (for Dynkin idempotent checking) | The answers to both the Concrete Question and the analogous question for $\log\operatorname\*{id}$ are "No". By that I mean that now I am sufficiently convinced of the correctness of my [Sage 5.0 code](http://mit.edu/~darij/www/ordforst.py). I am still surprised that I had to go all the way up to a $5$-th graded compon... | 2 | https://mathoverflow.net/users/2530 | 97514 | 56,927 |
https://mathoverflow.net/questions/97495 | 52 | There are the following Nakayama style lemmata:
1. (the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m\_1, \ldots, m\_n$ generate $M$ modulo $I$, where $I \subset \mathrm{Jac}(R)$, then they generate $M$.
2. (the graded Nakayama lemma) See [How to memor... | https://mathoverflow.net/users/nan | a categorical Nakayama lemma? | Let me describe a common generalization of Nakayama's lemmas and Burnside's basis theorem which may shed some light here. Let $G$ be a group and $P$ a set of endomorphisms of $G$. A $P$-subgroup will be a subgroup of $G$ which is closed under acting by elements of $P$. We'll call $\mathbb{Fr}\_P(G)$ the "$P$-Frattini s... | 52 | https://mathoverflow.net/users/2384 | 97515 | 56,928 |
https://mathoverflow.net/questions/97506 | 8 | It is known that the boundary of a branched double cover of the four ball branched over a surface bounded by a knot in is a rational homology $3$-sphere. Two questions come to my mind simply out of curiosity:
1. Which rational homology $3$-spheres arise this way? By this I mean, is this set large or small (in the mos... | https://mathoverflow.net/users/9187 | Rational homology spheres and knots | For Question 1, I believe that the answer follows from:
Montesinos, José M. *Surgery on links and double branched covers of $S^3$.* Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 227–259. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., (1975; MR0380802).
Nam... | 8 | https://mathoverflow.net/users/2051 | 97519 | 56,930 |
https://mathoverflow.net/questions/97517 | 6 | We are given a matroid. Our goal is to find a set of elements of minimum size that has non-empty intersection with every base of the matroid. Is the problem studied before? Is it in P? For example, in a spanning tree matroid, the minimum hitting set should be a minimum cut. Thanks.
Crossposted at [CSTheory](https://c... | https://mathoverflow.net/users/11368 | A minimum set hitting every base of a matroid | The problem is hard in general. Note that a minimal set that intersects every base of a matroid $M$ is a dependent set in the dual matroid $M^{\*}$. Such sets are called *cocircuits*. So, you are looking for the shortest cocircuit of a matroid.
The shortest cocircuit problem (equivalently shortest circuit problem) i... | 8 | https://mathoverflow.net/users/2233 | 97521 | 56,932 |
https://mathoverflow.net/questions/97516 | 17 | So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X\_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X\_0$ at a prime $p$ is given by $Z\_p(T)=1-T$ (a polynomial of degree $1$). So the Hasse-Weil zeta function of $X\_0$ is given by
$$
L\_{X\_0}(s):=\prod\_{p}Z\_p(p^{-s})^{-1}=... | https://mathoverflow.net/users/11765 | On the Hasse-Weil L-function of $P^n$ | $\newcommand\GL{\mathrm{GL}}$
$\newcommand\SL{\mathrm{SL}}$
$\newcommand\R{\mathbf{R}}$
There is, of course, an autormorphic representation (Hecke character)
$\chi$ for $\GL(1)$ whose $p$-adic avatar is the cyclotomic character. From this, one thus has the isobaric sum (following Langlands and Jacquet-Shalika)
$$\pi... | 7 | https://mathoverflow.net/users/nan | 97536 | 56,939 |
https://mathoverflow.net/questions/97533 | 4 | Let $G$ be a group and let $H$ be a factor group of $G$. Is there any result that relates $\operatorname{Aut}(G)$ (the automorphism group of $G$) and $\operatorname{Aut}(H)$?
As a very special case of the question, let $F\_2$ be the free group with two generators $x$ and $y$. Let $G\_2$ be the factor group of $F\_2$ ... | https://mathoverflow.net/users/15770 | Automorphism group of factor groups | If $H=G/N$ and $N$ is a characteristic subgroup, then there is a natural homomorphism $\phi$ from $Aut(G)$ to $Aut(H)$. If $G=F\_n$, the image $\phi(Aut(F\_n))$ consists of *tame* automorphisms. In general $\phi(Aut(F\_n))$ may not coincide with $Aut(H)$ and there is a lot of work on this topic, especially for the cas... | 9 | https://mathoverflow.net/users/nan | 97537 | 56,940 |
https://mathoverflow.net/questions/97531 | 2 | Is there a characterization of measure zero subsets $A$ of $\mathbb R^n$, $n>1$ such that the set $A+A$ contains interior? Here $A+A$ is the set of points $\{ x+y \mid x, y\in A \}$.
Is it true that if the convex hull of the connected component of $A$ contains interior then so does $A+A$?
| https://mathoverflow.net/users/17822 | When a set of measure zero plus itself contains interior | Regarding Question #1.
There's one obvious dimensional obstruction (for example consider the Hausdorff dimension of A).
There is some research about it in the 1-dimensional case, such as the well-known theorem that K+K contains an interval, and related conjectures and works by Pallis, Furstenberg and Yoccoz. Even t... | 0 | https://mathoverflow.net/users/8857 | 97545 | 56,943 |
https://mathoverflow.net/questions/97540 | 1 | Given $x\_1,x\_2, \ldots, x\_n \ge 0, \alpha \ge 1$, show that
$\sum\_{i}\alpha x\_i(\sum\_{j \le i}{x\_j})^{\alpha-1} \ge (\sum\_{i}x\_i)^\alpha$
We're pretty sure the inequality holds for the given precondition. It can be validated using a small piece of matlab code. Now we post it for rigorous proof. Its continu... | https://mathoverflow.net/users/23849 | A polynomial inequality | Setting $s\_i = \sum\_{j \leq i} x\_j$, just write
$$ \sum\_{i \leq n} \alpha x\_i s\_i^{\alpha-1} = \sum\_{i \leq n} \int\_{s\_{i-1}}^{s\_i} \alpha s\_i^{\alpha-1} dt \geq \sum\_{i \leq n} \int\_{s\_{i-1}}^{s\_i} \alpha t^{\alpha-1} dt = \int\_{0}^{s\_n} \alpha t^{\alpha-1} dt = s\_n^{\alpha}
$$
| 5 | https://mathoverflow.net/users/21724 | 97546 | 56,944 |
https://mathoverflow.net/questions/97543 | 3 | Start by fixing invertible matrics $A\_1, \ldots, A\_m \in \mathbb{Z}^{n \times n}$.
For a sequence $i\_1, \ldots, i\_k$ we construct $A = A\_{i\_1} \cdots A\_{i\_k}$. We would like to know "Is 1 an eigenvalue of $A$?".
As we are doing this for a large number of sequences (the naive computations when $n \sim 6$, $... | https://mathoverflow.net/users/3121 | Does a product of matrices have eigenvalue 1 | You can use Sylvester's determinant theorem $\det(I+AB) = \det(I+BA)$ to reuse the results.
For example, $\det(I-A\_1 A\_2 A\_3 A\_4) = \det(I-A\_2 A\_3 A\_4 A\_1) = \det(I-A\_3 A\_4 A\_1 A\_2 )$
$ = \det(I-A\_4 A\_1 A\_2 A\_3 )$
| 10 | https://mathoverflow.net/users/23736 | 97552 | 56,947 |
https://mathoverflow.net/questions/97501 | 2 | The Banach integral is elegant in its definition, and I am intrigued as to why it is so rarely seen. Is it in practice difficult to calculate from the definition? And are there any other problems with it? I would also be interested to see examples of functions that are Banach-integrable, but not Lebesgue-integrable, an... | https://mathoverflow.net/users/7458 | Is there a good comparative study of the Banach integral? | One can find some information in the German language book *Reelle Zahlen* by Oliver Deiser. Banach apparently introduced his integral in the paper [Sur le problème de la mesure](http://matwbn.icm.edu.pl/ksiazki/fm/fm4/fm412.pdf), Fund. Math. 4, 1923. It was apparently introduced to show that a translation invariant and... | 3 | https://mathoverflow.net/users/35357 | 97554 | 56,949 |
https://mathoverflow.net/questions/97541 | 2 | Is there a characterization of graphs $G$ such that $\exists$ $\phi : G \rightarrow KG(n,k)$, where $KG(n,k)$ is the Kneser graph ($k \leq \lceil \frac{n}{2}\rceil $).
Any references on the subject will be appreciated.
| https://mathoverflow.net/users/23850 | Homomorphism into Kneser graphs $KG(n, k)$ | Homomorphisms into Kneser graphs are another way of describing fractional colourings; an introduction to how this all works is the topic of one of the chapters in my favourite book on Algebraic Graph Theory. There are other much more detailed references on fractional colourings, but not necessarily from the homomorphis... | 8 | https://mathoverflow.net/users/1492 | 97557 | 56,950 |
https://mathoverflow.net/questions/97564 | 4 | Recall that we call a category *rigid* if it contains no non-identity isomorphisms. Let $\mathbf{rig}$ denote the full 2-subcategory of $\mathbf{Cat}$ spanned by the small rigid categories. It is easy to see that a functor in $\mathbf{rig}$ is an equivalence of categories if and only if it is an isomorphism.
Then co... | https://mathoverflow.net/users/1353 | Equivalence of categories of abelian presheaves reflects isomorphisms of rigid categories? | No, this is false. Let $C$ be the monoid $\lbrace 1,\ldots, 2^n\rbrace$ with $\max$ as the operation and let $D$ be the power set of $\lbrace 1,\ldots, n\rbrace$ with $\cup$ as the operation. These are both join semilattices with identity of cardinality $2^n$. The integral monoid rings of two finite join semilattices w... | 6 | https://mathoverflow.net/users/15934 | 97565 | 56,954 |
https://mathoverflow.net/questions/97550 | 6 | I know pursuit-evasion has been studied in many contexts, including
on a manifold (e.g., Melikyan,
"[Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds](http://www.springerlink.com/content/k50v733542102p87/)"),
but I have not seen this version:
There is a pursuer, a point $p\_t$ at time $t$, while the
eva... | https://mathoverflow.net/users/6094 | Pursuit-Evasion on a Manifold | *My original answer was wrong, here I summarize the discussion in the comments, mostly by Barry Cipra .*
**Q1.** On the round sphere, 3 pursuers can catch the evader the following way.
If one pursuer starts from the pole then he can move staying on the same meridian,
as the evader and keeping him on a larger distance... | 6 | https://mathoverflow.net/users/1441 | 97584 | 56,958 |
https://mathoverflow.net/questions/97567 | 0 | Hi fellows,
Does anyone know the number of holes of a level 2 Menger Sponge ?
| https://mathoverflow.net/users/nan | Level 2 Menger Sponge | For $n=1$, $g=5$: you drill a vertical hole through the middle and four horizontal holes to meet that vertical hole.
For higher values of $n$, the right way to think of it is in terms of Euler characteristic.
For $n=2$ you start with $20$ copies of a small level $1$ Menger sponge, with Euler characteristic $-8$. Yo... | 6 | https://mathoverflow.net/users/18060 | 97586 | 56,959 |
https://mathoverflow.net/questions/97579 | 4 | Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?
| https://mathoverflow.net/users/22051 | upper bounds on a certain matrix norm | You can use the surprising identity $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, and take the norms of the three factors separately.
| 7 | https://mathoverflow.net/users/1898 | 97587 | 56,960 |
https://mathoverflow.net/questions/97585 | 1 | Hi all -- what would be good methods (and/or software packages) to try for solving a problem minimizing a quadratic function $f(x) = \sum\_{i=1}^N{(x\_i - y\_i)^2}$, where some constraints are non-linear (and non-differentiable), e.g. $0 \leq x\_i \leq 1$, and $\sum\_i x\_i \mathbf{1}\_{x\_i>a} < b$ ?
I am thinking ... | https://mathoverflow.net/users/10837 | Nonlinearly constrained optimization (quadratic) | The real issue here is the constraint
$\sum\_{i} x\_{i}1\_{x\_{i}>a} < b $
whose left hand side has horrible discontinuities.
Rather than using a solver designed for problems with continuous variables, you should formulate this as 0-1 mixed integer nonlinear programming problem, with binary decision variables $z... | 7 | https://mathoverflow.net/users/9022 | 97590 | 56,962 |
https://mathoverflow.net/questions/97589 | 10 | Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic question is more about references:
>
> 1) Are there textbook or other convenient sources summarizing properties of $G$ whic... | https://mathoverflow.net/users/4231 | Textbook source for finite group properties deducible from character table? | For nilpotency, you can deduce the character table of $G/Z$ from the character table of $G$. First, determine $Z$. Second, throw out all the representations where $Z$ is not in the kernel. Third, merge the conjugacy classes which have the same trace in every representation. (This works because the irreducible represent... | 8 | https://mathoverflow.net/users/18060 | 97594 | 56,965 |
https://mathoverflow.net/questions/97596 | 4 | Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of *non* invertible elements of this ring.
>
> For $k \geq 0$, what is the largest integer $n=n(k)$ such that $\mathbb{R}^... | https://mathoverflow.net/users/21724 | Ring structrures on R^n | For $k$ even, take the ring $\mathbb C[\epsilon]/\epsilon^{k/2+1}$. Non-invertible elements are multiples of $\epsilon$, which form a $k$-dimensional vector space. The ring has dimension $k+2$, so $n(k)=k+2$ for $k$ even.
For $k$ odd, take the ring $\mathbb R[\epsilon]/\epsilon^{k+1}$. By the same logic, this gives $... | 6 | https://mathoverflow.net/users/18060 | 97597 | 56,967 |
https://mathoverflow.net/questions/97576 | 3 | Is there any length function on additive group of $\mathbb{Q}$ such that $\mathbb{Q}$ is of polynomial growth WRT this length function? What about the multiplicative group of $\mathbb{Q}$ instead?
| https://mathoverflow.net/users/nan | growth of infinitely generated groups | Yes for $\mathbf{Q}$, no for $\mathbf{Q}^\*$.
For $\mathbf{Q}$, write it as the union of an increasing sequence $L\_n$ with $L\_1=\mathbf{Z}$ and $L\_n$ of finite index over $L\_1$. Pick a function $F$ with fast growth and define $l'(r)=|r|+F(\sup\{n:r\notin L\_n\})$.
For $\mathbf{Q}^\*$, it contains a subgroup iso... | 6 | https://mathoverflow.net/users/14094 | 97601 | 56,970 |
https://mathoverflow.net/questions/97581 | 3 | Do the additive group or the multiplicative group of $\mathbb{Q}$ have property (RD) (Rapid Decay)?
| https://mathoverflow.net/users/nan | Property (RD) for $\mathbb{Q}$ | Thanks to 'Yves Cornulier's answer to my other question about the growth of $\mathbb{Q}$, we now know (1) there is a length function on the additive group of $\mathbb{Q}$ which makes $\mathbb{Q}$ of polynomial growth. (2) there is no length function on $\mathbb{Q}^\times$ making it of polynomial growth.
We can modif... | 3 | https://mathoverflow.net/users/nan | 97622 | 56,980 |
https://mathoverflow.net/questions/97624 | 10 | At the DeKalb conference on [Hilbert's problems](http://books.google.co.in/books?id=4lT3M6F745sC&lpg=PA305&dq=hilbert%2520problems&pg=PA305#v=onepage&q&f=false), John Tate gave a masterly survey of Problem 9, the General Reciprocity Law. He ends with a discussion of the Langlands Programme, especially the case of odd A... | https://mathoverflow.net/users/2821 | The first odd degree-2 Artin representation for which the Artin conjecture was proved | $R$ is a 2-dimensional conductor 133 representation of the absolute Galois group of the rationals into $GL(2,\mathbf{C})$, whose associated representation to $PGL(2,\mathbf{C})$ cuts out the $A\_4$ extension of the rationals which is the splitting field of $x^4 + 3x^2 - 7x + 4$. To verify the existence of the weight 1 ... | 17 | https://mathoverflow.net/users/1384 | 97637 | 56,987 |
https://mathoverflow.net/questions/97636 | 3 | (May be a poor title, happy to update)
Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there is a sheaf $Hom\_{\mathcal{X}\_U}(x,y)$ on $Sch/U$. Call this a 'local hom-sheaf'.
Reading through Laumon & Moret-Ba... | https://mathoverflow.net/users/4177 | Representability of Hom-sheaves of various moduli spaces | For the examples you mention, this boils down to representability of Hom sheaves of flat finitely presented proper schemes, which is due to Grothendieck. These Hom sheaves are not of finite type, though, except for $\mathcal M\_g$ with $g ≥ 2$, in characteristic 0, in which all arrows are cartesian. And, of course, if ... | 4 | https://mathoverflow.net/users/4790 | 97639 | 56,989 |
https://mathoverflow.net/questions/97628 | 8 | I thought that the interesting question Gerry Myerson asked in the comments of [this question](https://mathoverflow.net/questions/97602/degrees-of-irreducible-characters-of-groups-of-order-48-closed) deserved to be asked in a non-closed mathoverflow question.
What can we say about groups of order $n$ with an irreduci... | https://mathoverflow.net/users/18060 | Groups with irreducible representations of the largest possible dimension | Just to give correct references. Let $d$ be the degree of an irreducible character of a finite group $G≠1$. Then $|G|=d(d+e)$ for some $e > 0 $ (that is because $d$ divides $|G|$ and $d^2 < |G|$). Therefore the condition $(d+1)^2 > |G|$ means $d(d+e)=|G|$ with $e=1$ or $2$. If $e=1$, then $G$ is a doubly transitive Fro... | 15 | https://mathoverflow.net/users/nan | 97641 | 56,991 |
https://mathoverflow.net/questions/97574 | 8 | **Van Den Berg-Kesten-Reimer inequality**
Let $n$ be a positive integer. For $i\in[n]$, let $\Omega\_i$ be a finite set and $\mu\_i$ a probability measure on it. Set $\Omega=\Omega\_1\!\times\!\ldots\!\times\!\Omega\_n$ and $\mu=\mu\_1\!\times\!\ldots\!\times\!\mu\_n$.
For $A\!\subset\!\Omega$ and $\sigma\!\subset\
In $\int\_{-\infty}^\infty e^{-st} f(t)\;dt$ write $z=e^{-s}$.
| 0 | https://mathoverflow.net/users/454 | 97669 | 57,003 |
https://mathoverflow.net/questions/97659 | 5 | Suppose I have e.g. the Witt algebra,
$\left[l\_n,l\_m \right] = -(n-m)l\_{n+m}$.
I want to realize the $l\_n$ as vector fields on some manifold. The classical example is when the $l\_n$ span the Lie algebra of diffeomorphisms of the circle, i.e.
$l\_n = -i e^{i n \phi} \partial\_\phi, \ \ \ 0 \leq \phi < 2\pi.... | https://mathoverflow.net/users/17660 | Representations of infinite dimensional Lie algebras as vector fields on manifolds | You could try É. Cartan's papers on infinite pseudogroups (mostly appearing 1904-05). In particular, see Paragraph 57 of *Sur la structure des groupes infinis de transformation (suite)*. There, for example, he proves that the (pseudo-)group of diffeomorphisms of the line has three distinct (i.e., nonequivalent) $2$-dim... | 7 | https://mathoverflow.net/users/13972 | 97670 | 57,004 |
https://mathoverflow.net/questions/97675 | 1 | Let $M = PD$, where $P$ is a permutation matrix and $D$ diagonal.
If $P$ is also symmetric, then does $M$ have all real eigenvalues?
| https://mathoverflow.net/users/23313 | Eigenvalues of monomial matrices | How about $M = \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}$?
| 6 | https://mathoverflow.net/users/19276 | 97676 | 57,007 |
https://mathoverflow.net/questions/97681 | 0 | $H$ is called a nearly normal subgroup of a group $G$ if it is of finite index in its normal closure, $H^G:=\cup\_{g\in G} gHg^{-1}$, in $G$. Clearly, every normal subgroup or subgroup of finite index of $G$ is nearly normal. I am not interested in finite subgroups either. I am looking for infinite nearly normal subgro... | https://mathoverflow.net/users/nan | Non-elementary examples of nearly normal subgroups | Take any infinite group with an infinite-index normal subgroup. Say, the free group on two generators with the commutator subgroup. Then take a finite index subgroup of the normal subgroup. This particular normal subgroup is the free group on countably infinite generators, and so has many finite-index subgroups. Most a... | 2 | https://mathoverflow.net/users/18060 | 97684 | 57,010 |
https://mathoverflow.net/questions/97682 | 12 | Let $M$ be a $\pi\_\*(MU)$-module. The Landweber exact functor theorem gives conditions for the functor that sends a space $X$ to $ MU(X) \otimes\_{\pi\_\*(MU)} M$ to define a homology theory on spaces, which thus comes from a spectrum.
It'd be nice, though, if one could construct the spectrum directly, instead of g... | https://mathoverflow.net/users/344 | A homotopyish Landweber exact functor theorem | Here are three methods that I know:
* In the case $M\_\*=(MU\_\*/I)[S^{-1}]$ (where $I$ is generated by a regular sequence) there is a more direct construction by reducing to the cases $M\_\*=MU\_\*/a$ and $M\_\*=MU\_\*[a^{-1}]$. My paper 'Products on MU-modules' is probably the sharpest version, but there are many e... | 16 | https://mathoverflow.net/users/10366 | 97695 | 57,015 |
https://mathoverflow.net/questions/97654 | 8 | Hi,
I have asked this question on [math.stackexchange](https://math.stackexchange.com/questions/146280/on-the-set-of-divergence-to-infinity-for-sequences-of-positive-continuous-functi) but it has not received much attention, so I ask it here.
This question is partly motivated by this [one](https://mathoverflow.net/... | https://mathoverflow.net/users/1162 | On the set of divergence to infinity for sequences of positive continuous functions | The condition that the set be an $F\_{\sigma\delta}$ is necessary and sufficient.
This is a result proved by Hahn in 1919. The reference is
H. Hahn, Ueber die Menge der Konvergenzpunkte einer Funktionenfolge,
Archiv. der Math. und Physik 28 (1919), 34-45.
| 8 | https://mathoverflow.net/users/12120 | 97697 | 57,016 |
https://mathoverflow.net/questions/97687 | 2 | Let $Q=[0,\infty)\times [0,\infty)\subset \mathbb C$ and $f: Q\times Q\to Q\times Q$ be a diffeomorphism.
such that $f$ is holomorphic in the interior of $Q\times Q$. Can we extend this map analytically across the boundary.
Motivation: We have following proposition:
Let $U$ and $V$ are open subsets of $\mathbb R^n\... | https://mathoverflow.net/users/16031 | Analytic extension across the boundary. | Not only $f$ admits an analytic continuation across boundary, in fact, $f$ is the restriction of a linear transformation. Indeed, the interior of $Q\times Q$ is the 2-dimensional polydisk (more precisely, it is biholomorphic to the standard polydisk by a product map). Biholomorhic automorphisms of polydisks are composi... | 3 | https://mathoverflow.net/users/21684 | 97717 | 57,023 |
https://mathoverflow.net/questions/97718 | 11 | In Keven Walker's answer to the question, [Cubical vs. simplicial singular homology](https://mathoverflow.net/questions/3656/cubical-vs-simplicial-singular-homology) it is written:
>
>
> >
> > Personally, I think it is more convenient to do singular homology with the larger collection of polyhedra which is close... | https://mathoverflow.net/users/14167 | What is the precise relationship between "prodsimplicial sets" and rooted trees? | There are a short list of operations described as generating the desired polyhedra:
* $ X : \mathrm{Prism} \vdash C X : \mathrm{Prism} $
* $ l : \mathrm{list}\ \mathrm{Prism} \vdash \Pi l : \mathrm{Prism} $
There are a short list of operations needed to generate the family of rooted trees:
* $ T : \mathrm{Tree}\... | 5 | https://mathoverflow.net/users/1631 | 97722 | 57,026 |
https://mathoverflow.net/questions/97721 | 3 | Let $f$ be a quadratic polynomial with leading coefficient $\alpha$, and suppose $\alpha$ is in a "minor arc" in the sense that $\alpha$ is not within $\frac{(\log N)^A}{q N^2}$ of any rational number $\frac{a}{q}$ with $q < (\log N)^A$. My advisor told me that the following holds:
$\sum\_{n=1}^N \Lambda(n) e(f(n)) \... | https://mathoverflow.net/users/23896 | Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes | This result (or at least the method) probably goes back to Vinogradov or Davenport. For an explicit statement/proof of this result you can see, for instance, Theorem 1 in:
[J. Liu, T. Zhan, Estimation of exponential sums over primes in short intervals. I. Monatsh. Math. 127 (1999), no. 1, 27–41.](http://www.springerl... | 0 | https://mathoverflow.net/users/630 | 97723 | 57,027 |
https://mathoverflow.net/questions/97710 | 1 | Hi all,
We consider the set
$$ S = \left\lbrace (F,h)\;\;\middle\vert\;\genfrac{}{}{0pt}{}{F\text{ is a decreasing function from }R^{+}\text{ to }R^{+}, h\in R}{0=1- \dfrac{\theta + 1}{\theta} \dfrac {\int^{h}\_{y=0} F(y) dy}{F(0)} \dfrac{F(0)-\frac{1}{2}F(h)}{F(0)-F(h)}} \right\rbrace $$
The function $L$ is def... | https://mathoverflow.net/users/23756 | A variational problem under a monotonicity constraint | It seems that for the maximization problem, as it is, we have $\sup\_{(F,h)\in S}=+\infty$, for all $\theta > 0$.
For $\lambda > 1$ consider the function $F\_\lambda$ such that $F\_\lambda(0)=\lambda$, and $F\_\lambda(x)=1$ for all $x > 0$. Then there exists exactly one $h=h\_\lambda > 0$ such that $(F\_\lambda,h\_\l... | 0 | https://mathoverflow.net/users/6101 | 97735 | 57,033 |
https://mathoverflow.net/questions/97698 | 3 | Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p\_{ij}=1/(2d(i))$ where $d(i)$ is the degree of node $i$. Note that $p\_{ii}=1/2$ for all $i$. Define $C(i)$ be the expected time until a walk starting from node... | https://mathoverflow.net/users/21162 | Cover time and intersection time of random walks | In Proposition 5 of [Chapter 14](http://www.stat.berkeley.edu/~aldous/RWG/Chap14.pdf) of the unpublished book on Markov chains by Aldous and Fill, they show that for continuous time reversible Markov chains,
`\[
I \le \max\{ \mathbb{E}_i T_j, i,j \in V\},
\]`
where $\mathbb{E}\_i T\_j$ is the expected time, starting f... | 5 | https://mathoverflow.net/users/3401 | 97744 | 57,037 |
https://mathoverflow.net/questions/97733 | 9 | What more can be said about the eigenvalues (especially the spectrum) of the $N \times N$ matrix ${\bf M} = {\bf A} + {\bf A}^T$ in terms of $\bf A$ if $\bf A$ is not symmetric and its eigenvalues are not necessarily positive ($\bf A$ is not necessarily positive semi-definite)?
1. The eigenvalues of $\bf M$ are real ... | https://mathoverflow.net/users/23900 | Eigenvalues of non-symmetric matrix and its transpose | You cannot do much, for the following reason. Take any real symmetric matrix $M$. It is a consequence of the Toepliz-Hausdorff Theorem about the Numerical Range that $M$ is unitarily similar to a (symmetric) matrix $N$ whose diagonal is constant:
$$n\_{jj}=\frac1n{\rm Tr} M.$$
Then $N=B+B^T$, where $B$ is upper triangu... | 19 | https://mathoverflow.net/users/8799 | 97746 | 57,038 |
https://mathoverflow.net/questions/97651 | 3 | A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G\_\delta$-subset of $X$ is open. (i.e $\tau$ is closed under countable intersection). Here we recall some special properties of $P$-spaces:
* Every countable subset of $X$ is obviously closed and discrete.
* Every countable subset of ... | https://mathoverflow.net/users/23317 | Lindelöf subsets of $P$-spaces | Every Lindelof subset of a $P$-space is closed, and the proof is almost the same as the proof of "*a compact subset of a Hausdorff space is closed*" (I´m assuming your space is Hausdorff since you wrote that every countable set is obviously closed).
I´m not so sure about the second question, but every $P$-space is an... | 4 | https://mathoverflow.net/users/17836 | 97752 | 57,040 |
https://mathoverflow.net/questions/97732 | 1 | Hello,
I just need some clarification (or a good reference) for the definition of the realization of a bisimplicial set, this is what i have when $X$ is a bisimplicial set its realization is
$\cup\_n X\_n \times \Delta[n]$ subject to the following equivalence relation
$(d\_ix,p) \sim (x,d\_ip), (x,p) \in X\_n \time... | https://mathoverflow.net/users/23754 | Realization of a bisimplicial set | I think the easiest way to see it is that :
$|X|\_k=\cup\_n (X\_{n,k} \times \Delta[n]\_k)/\sim$ subject to the following equivalence relation
$(d\_ix,p) \sim (x,d\_ip), (x,p) \in X\_{n,k} \times \Delta[n-1]\_k$
$(s\_ix,p) \sim (x,s\_ip), (x,p) \in X\_{n-1,k} \times \Delta[n]\_k$
| -4 | https://mathoverflow.net/users/23754 | 97763 | 57,046 |
https://mathoverflow.net/questions/97741 | 12 | I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask them in the one question).
In great generality for a Hopf algebroid $(A,\Gamma)$ we can define the cobar complex $C\... | https://mathoverflow.net/users/16785 | Some calculations with the Adams spectral sequence and the cobar complex | One quick answer to 1 is trial and error; 2 is a typo: you don't mean $u = h\_{11}^2$.
For 3, unhelpfully, you just have to do the computation from the definitions. But I'm
really answering the last question. I wish that people would use my original notations, and I wish that Doug had done so, since my notations are s... | 16 | https://mathoverflow.net/users/14447 | 97767 | 57,048 |
https://mathoverflow.net/questions/97766 | 2 | We know that any hyperelliptic curve over a field with characteristic not equal to $2$ has an affine model given by $y^2 = f(x)$, with $deg(f) = 2g+1$ or $2g+2$.
Can we always find a model such that $deg(f)=2g+1$?
| https://mathoverflow.net/users/7313 | Affine model of a hyperelliptic curve | Over an algebraically closed field, yes. Let the roots of $f$ in $\mathbb{P}^1$ be $r\_1$, $r\_2$, ..., $r\_{2g+2}$, and apply a Mobius transformation taking $r\_{2g+2}$ to $\infty$ as Qiaochu says.
Over a nonalgebraically closed field, not necessarily. Let $g \geq 2$ for simplicity, so the hyperelliptic involution $... | 5 | https://mathoverflow.net/users/297 | 97768 | 57,049 |
https://mathoverflow.net/questions/97776 | 2 | how can i classify strongly regular graph with parameter $(25,12,5,6)$?
just i know we have fifteen $SRG(25,12,5,6)$ that two come from latin square(5)
| https://mathoverflow.net/users/22967 | classify strongly regular graph with parameter (25,12,5,6) | What you're really looking for are conference graphs of order 25, which come from [symmetric conference matrices](http://en.wikipedia.org/wiki/Conference_matrix) of order 26. Your 15 known graphs are the Paulus graphs on 26 nodes, and the 10 are Paulus graphs on 26 nodes. That should be enough to find the answers you w... | 5 | https://mathoverflow.net/users/18086 | 97780 | 57,052 |
https://mathoverflow.net/questions/97648 | 10 | Every weighted limit can be constructed from conical limits and cotensors. However, yesterday, a friend of mine, asked a question that may be rephrased as follows.
What is the reason that in the world of $\mathbf{Set}$-enriched categories every weighted limit can be constructed from conical limits (and trivial cotens... | https://mathoverflow.net/users/13480 | Weighted limits and completeness | Yes, is directly related to that fact, as you surmise. The cotensor $X^K$, for $K\in \mathbb{V}$, preserves (co)limits in the variable $K$, that is we have
$$ X^{\mathrm{colim}\_i K\_i} \cong \lim\_i X^{K\_i}. $$
Even better, if $\lim\_i X^{K\_i}$ exists, then it automatically has the universal property to be $X^{\... | 12 | https://mathoverflow.net/users/49 | 97794 | 57,060 |
https://mathoverflow.net/questions/96960 | 5 | From Chang and Keisler's "Model Theory", section 7.2, we know that:
1. There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^+,\alpha)$ iff there exists a tree $T$ of height $\alpha^+$, with at most $\alpha$ elements at each level $\xi<\alpha^+... | https://mathoverflow.net/users/13694 | Is there a statement equivalent to a sentence admitting $(\alpha^{+n},\alpha)$? | Letting $\alpha$ be an infinite cardinal and $3\leq n\lt\omega$,
I think Peter Komjath's proposed statement $2^\alpha\geq\alpha^{+n}$ is the simplest and most natural equivalent of a first-order sentence admitting $(\alpha^{+n},\alpha)$: just let $\sigma$ say a binary relation is extensional with domain given by a pred... | 5 | https://mathoverflow.net/users/12106 | 97797 | 57,063 |
https://mathoverflow.net/questions/97793 | 0 | Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t\_0 < t\_1 < t\_2 < ... < t\_s=b$ of the interval $[a,b]$, and constants $m\_1,m\_2,...,m\_s$ with $v(t)=m\_i$ whenever $t \in I\_i := [t\_{i-1},t\_i)$ for $i=1,\dots,s$. Let $\mathcal{P}$ denote the collection of... | https://mathoverflow.net/users/21269 | Hölder continuity of uniform limit of piecewise constant functions | I can't think of a characterization which is not too close to a tautology; a sufficient condition is the following. Denote the modulus of the subdivision $\mathcal{P}$ by $\|\mathcal{P}\|:=\max \_ {1\le i\le s} (t \_ i-t \_ {i-1})$, and by $\mathcal{P}^M$ the set of mid-points of the intervals $I\in \mathcal{P}$.
Ass... | 2 | https://mathoverflow.net/users/6101 | 97798 | 57,064 |
https://mathoverflow.net/questions/97060 | 22 | Let $\Gamma$ be a finitely generated group of exponential growth and $gr(S)=\lim\_{k\rightarrow \infty} \sqrt[k]{|B\_k(S)|}$ be the growth rate of $\Gamma$ with respect to the generating set $S$. I am confused with the following question: Does there always exist a generating set $S'$ such that
$$\frac{|B\_k(S')|}{gr(S'... | https://mathoverflow.net/users/8699 | Asymptotics of the growth rate of a group | There is never a (finite) generating set with that property.
Consider a generating set $S=\{x\_1,\ldots,x\_{\ell}\}$ of cardinality $\ell$. Let $B\_k := B\_k(S)$, $S\_k := B\_k \setminus B\_{k-1}$, and $g := gr(S)$. Let $b\_k := |B\_k|$ and $s\_k: = |S\_k|$. Assume for simplicity that $L := \lim\_{k \to \infty} \frac... | 16 | https://mathoverflow.net/users/8410 | 97805 | 57,069 |
https://mathoverflow.net/questions/97807 | 6 | I have written a program which finds the roots of polynomial using Newton's Method. After finding the first root to within a tolerance (note that this also finds complex roots), I use synthetic division to remove that root from the original polynomial (f = f/(x-root))
My question is, how does this affect the error? I... | https://mathoverflow.net/users/23915 | Error in Polynomial Root Finding Algorithm with Synthetic Division | It is a terrible idea to divide out roots as they are found. There will be examples where the later roots are lost almost completely. See [this wikipedia article](http://en.wikipedia.org/wiki/Wilkinson%27s_polynomial) for a famous and remarkably simple example of a polynomial whose zeros are very sensitive to the coeff... | 5 | https://mathoverflow.net/users/9025 | 97810 | 57,072 |
https://mathoverflow.net/questions/97811 | 1 | Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or have tangency of order $k$ (for any $k$ between 3 and $d$) with $C$? Probably this can be done for small $d$ using the equ... | https://mathoverflow.net/users/23802 | Pencil of lines and degree $d$ curve in $\mathbb{CP}^2$ | You have some polynomial $f(x,y,z)$. A line through the point $(1:0:0)$ can be paramaterized by a map from $\mathbb P^1: (u:v) \to (u:av:bv)$ for some constant $a$ and $b$. $f$ restricts to a degree $d$ polynomial in $u$ and $v$. Since it has no roots where $v=0$, set $v=1$. You now have a univariate polynomial such th... | 3 | https://mathoverflow.net/users/18060 | 97815 | 57,074 |
https://mathoverflow.net/questions/97820 | 21 | I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.
In a similar vein to [this question](https://mathoverflow.net/questions/73075/a-recommended-roadmap-into-inner-models), I was hopi... | https://mathoverflow.net/users/22095 | A recommended roadmap to Fermat's Last Theorem | What about
* Cornell-Silverman-Stevens, Modular Forms and Fermat's Last Theorem
* Darmon-Diamond-Taylor, Fermat's Last Theorem, <http://modular.math.washington.edu/edu/2011/581g/misc/Darmon-Diamond-Taylor-Fermats_Last_Theorem.pdf>
* Diamond-Shurman, A First Course in Modular Forms
* some of Milne's course notes <htt... | 19 | https://mathoverflow.net/users/nan | 97823 | 57,079 |
https://mathoverflow.net/questions/97830 | 21 | [Homotopy groups of Lie groups](https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups/8996#8996)
I asked it also there, and I still don't know the answer, so I try again.
I would like to know a closed manifold (possibly of low dimension) such that $\pi\_2(\textrm{Diff}(M))\neq 0$.
| https://mathoverflow.net/users/5628 | nontrivial $\pi_2(\textrm{Diff}(M))$ | $\newcommand{\Diff}{\mathrm{Diff}}$Probably the simplest such manifold is $S^1 \times S^2$, whose diffeomorphism group has the homotopy type of $O(2) \times O(3) \times \Omega SO(3)$. This has $\pi\_2$ equal to $\pi\_2\Omega SO(3)=\pi\_3 SO(3) = {\mathbb Z}$. The $\Omega SO(3)$ term is realized by rotating the $S^2$ sl... | 43 | https://mathoverflow.net/users/23571 | 97832 | 57,082 |
https://mathoverflow.net/questions/97828 | 4 | I apologize if the question is a well-known theorem, but I'm just starting to learn about laminations, so I don't know much.
The question is roughly, if interval exchange maps have an underlying closed smooth surface, or if not, what is known about conditions on that.
Now I try to be more precise.
Usual interval ... | https://mathoverflow.net/users/5628 | interval exchange maps and surfaces | Your method certainly works, because you are just identifying boundary edges of the annulus $[0,1] \times S^1$ in pairs to form a surface. As usual, when one glues up edge pairs of a surface-with-boundary, the endpoints of the boundary edges form "vertex cycles" whose images are points of the quotient surface, and the ... | 8 | https://mathoverflow.net/users/20787 | 97836 | 57,086 |
https://mathoverflow.net/questions/97837 | 3 | I'm trying for some time now to prove or disprove the following conjecture to no avail:
>
> Let $S$ be a set and let $(\Sigma \_n)$
> be a sequence of countably generated
> $\sigma$-algebras on $S$ satisfying
> the following two conditions:
>
>
> 1. $\Sigma\_n\subseteq\Sigma\_{n+1}$ for all $n$.
> 2. If $A\in\... | https://mathoverflow.net/users/35357 | Atoms of a sequence of Sigma-algebras | Counterexample. First, let ${\cal B}$ be the Borel $\sigma$-algebra on ${\bf R}$ and let ${\cal B}'$ be the $\sigma$-algebra generated by ${\cal B}$ together with one non-Borel set $E$. Note that $E$ is a union of atoms of ${\cal B}$.
Now for each $n$ let $\Sigma\_n$ be the $\sigma$-algebra of subsets of ${\bf R} \ti... | 4 | https://mathoverflow.net/users/23141 | 97839 | 57,088 |
https://mathoverflow.net/questions/97160 | 3 | ### Question
The question asked is:
>
> On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast\_s$ the symplectic star, does $\ast=\ast\_s$ iff $(M,g,\omega)$ is Kähler?
>
>
>
### Answer: no
For the reason [posted below](https://mathov... | https://mathoverflow.net/users/11394 | Does equality of Hodge star and symplectic star imply Kähler structure? | Dear John,
The symplectic Hodge star and Riemannian Hodge star (associated to a Kahler metric) are never identical to each other.
In fact, it is proved in Brylinski's paper on symplectic Hodge theory that on a Kahler manifold, for a $(p,q)$ form $\alpha$ we have that
$$\* \alpha =(-1)^{p-q}\*\_s \alpha.$$
One ca... | 3 | https://mathoverflow.net/users/23926 | 97846 | 57,091 |
https://mathoverflow.net/questions/97674 | 3 | Let $G$ be a semi simple Lie group (or real reductive), $\mathfrak{g}$ its lie algebra and $B$ its killing form. We can defined the 3-form $k$ by
$$k(X,Y,Z)=B([X,Y],Z).$$
with $X,Y,Z\in \mathfrak{g}$.
In fact, $k$ is nothing but the structure constants.
It is easy to prove that $k$ is a closed forme on $G$. For examp... | https://mathoverflow.net/users/16326 | Topologic or geometric mean of the structure constants of a semi simple lie algebra | Of course, this has been well-studied. Cartan himself pointed out this form. You can read about it in his little 1936 book *La topologie des groupes de Lie*. There are, of course, more modern references, such as Adams' *Lectures on Lie groups*.
To answer your questions:
1. The form $k$ is always nonzero on Lie sub... | 3 | https://mathoverflow.net/users/13972 | 97848 | 57,093 |
https://mathoverflow.net/questions/97849 | 8 | Let $F$ be a field and $h \in F[x]$ be an irreducible, degree $n$ monic polynomial. Let $G$ denote the Galois group of $h$.
It is well known that $G \subset A\_n $ if and only if the discriminant of $h$, which we'll denote by $D(h)$, is a square in $F$. We could think of this as being a rationality condition: we are ... | https://mathoverflow.net/users/13828 | Rationality conditions for determining Galois groups | More or less, yes. Fix a transitive subgroup $H \subset S\_n$. Let $S\_n$ act in the usual way in the field $\mathbb{Q}(x\_1,\ldots,x\_n)$ where the $x\_i$ are algebraically independent. Then the fixed field $\mathbb{Q}(x\_1,\ldots,x\_n)^{S\_n} = \mathbb{Q}(a\_1,\ldots,a\_n)$ where $\prod(X-x\_i) = \sum a\_iX^{n-i}, a\... | 10 | https://mathoverflow.net/users/2290 | 97866 | 57,096 |
https://mathoverflow.net/questions/97855 | 6 | The $k^{\rm th}$ largest eigenvalue (arranged in decreasing order) of the sum of two $N \times N$ Hermitian (real symmetric) matrices $\bf{A}$ and $\bf{B}$ can be stated using the Weyl inequalities as
$L\_k \leq \lambda\_k({\bf A} + {\bf B}) \leq U\_k$
with the lower and upper bounds given by
$L\_k = {\rm max}\le... | https://mathoverflow.net/users/23900 | Weyl inequalities for largest eigenvalue of matrix sum | Weyl's inequalities are not the full story. The characterization of the possible spectra of $A+B$, given the spectra of $A$ and $B$ is the object of A. Horn's conjecture. This is now a theorem, after hard works by Fulton, Klyachko, Knutson, Terry Tao and others. The conjecture consists in linear inequalities (the simpl... | 8 | https://mathoverflow.net/users/8799 | 97870 | 57,098 |
https://mathoverflow.net/questions/97840 | 3 | If $f: X \to S$ is a projective smooth morphism between complex algebraic varieties. Does the $\pi\_1(S)$-representation corresponding to the local system $R^i f\_\* (C\_X)$ on $S$ maps $\pi\_1(S)$ onto a discrete subgroup of $GL(r, C)$?
| https://mathoverflow.net/users/11056 | Is the image of the representation of the fundamental group associated to a local system discrete? | Yes, because it lies in $GL(r, \mathbb{Z})$ (use universal coefficients: $R^if\_\*(\mathbb{C}\_X)= R^if\_\*(\mathbb{Z}\_X)\otimes \mathbb{C}$). A more interesting question -- which was open for a while -- was whether the monodromy group is always arithmetic. The answer turned out to be no. See Nori "A nonarithmetic mon... | 7 | https://mathoverflow.net/users/4144 | 97878 | 57,099 |
https://mathoverflow.net/questions/97872 | 2 | Let $\mathbb Z\_n$ denote the integers modulo $n$. Let $\mathbb Z\_n[i, j, k]$ be the quaternionic ring over $\mathbb Z\_n$, that is, the free module over $\mathbb Z\_n$ with basis $\{1, i, j, k\}$ and multiplication defined by
$$i^2=j^2=k^2=ijk=-1.$$
It is well-known that if $n=p$ where $p$ is a odd prime then $\ma... | https://mathoverflow.net/users/22475 | Quaternion ring | Yes, for odd $n$, the ring $(\mathbb Z/n)[i,j,k]$ is isomorphic to the ring of two-by-two matrices over $\mathbb Z/n$.
To explain this, it is better to write $\mathbb Z/n$ and $\mathbb Z/p$ rather than $\mathbb Z\_n$ and $\mathbb Z\_p$, because, in fact, the decisive statement concerns $p$-adic integers $\mathbb Z\_p... | 4 | https://mathoverflow.net/users/15629 | 97881 | 57,100 |
https://mathoverflow.net/questions/97865 | 7 | Let $X\subset\mathbb{P}^n$ be a (globally) complete intersection, let $(X\_t)\_{t\in\mathbb{C}^1}$ be a flat family, with $X\_1=X$. Which types of schemes can we get as $X\_0$?
Or, conversely, which (embedded, projective) schemes deform to complete intersections?
e.g. some non-ACM schemes (not arithmetically Cohen-... | https://mathoverflow.net/users/2900 | what can be reached by flat degeneration of (globally) complete intersection? | Anything with the same Hilbert polynomial as a globally complete intersection. This is true because two fibers of the same flat projective family have the same Hilbert polynomial, and because the Hilbert scheme is connected. EDIT: Since the base of the flat family is required to be irreducible, the projective scheme mu... | 5 | https://mathoverflow.net/users/18060 | 97885 | 57,104 |
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