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https://mathoverflow.net/questions/89232 | 1 | I'm not even sure if what I want to do is a good idea but I figure I'll experiment and see.
I have two predicted ratings in the range of 1-5 based on two different algorithms for predicting movie ratings. I also have two global averages for the given movie and given user.
e.g.
predicted rating 1 = 4.5
predicted r... | https://mathoverflow.net/users/21642 | weighted to centre mean | Possibly what you're looking for is an [empirical Bayes method](http://en.wikipedia.org/wiki/Empirical_Bayes_method).
Here's a bit more on empirical Bayes methods: <http://dspace.mit.edu/bitstream/handle/1721.1/45587/18-441Spring-2002/NR/rdonlyres/Mathematics/18-441Statistical-InferenceSpring2002/0636681D-CFBE-4A76-B... | 2 | https://mathoverflow.net/users/6316 | 89236 | 52,769 |
https://mathoverflow.net/questions/89245 | 0 | Given two locally compact topological spaces $X$ and $Y$, and a local homeomorphism $f : X \to Y$, Gelfand duality gives us a homomorhism $Cf : C\_0(Y) \to C\_0(X)$. How does the fact that $f$ is a local homeomorphism manifest itself in the homorphism $Cf$ between $C^\*$-algebras?
| https://mathoverflow.net/users/21647 | Dual notion of a local homeomorphism between topological spaces for C*-algebras | Your first claim seems to need a small correction. In general, the continuous maps $X\to Y$ that induce homomorphisms $C\_0(Y)\to C\_0(X)$ are those which are proper, i.e. where preimages of compact sets are compact. (Just take $X={\mathbb R}$ and $Y={\mathbb R}/{\mathbb Z}\cong {\mathbb T}$ to see what can go wrong ot... | 3 | https://mathoverflow.net/users/763 | 89246 | 52,773 |
https://mathoverflow.net/questions/89125 | 3 | For scalar variables $x$, we have a simple solution for the following problem.
\begin{eqnarray}
\min\_x&&\alpha(x-a)^2+\beta(x-b)^2 \\\
\mathrm{s.t. }&&x\leq a\\\
&&x\leq b
\end{eqnarray}
where $\alpha,\beta>0$.
The optimal solution $x=\min(a,b)$ is straightforward and independent of $\alpha,\beta$.
However, in th... | https://mathoverflow.net/users/19399 | What is the minimum of the Frobenius norm in the intersection of positive semidefinite cones? | I looked at the problem again and saw that it can be simplified nicely.
In summary:
1. If $\alpha, \beta > 0$, the solution is independent of $\alpha$ and $\beta$
2. The solution can be easily computed (one eigenvector decomposition is needed)
**Details**
Introduce the variables $X=B+C$ and $Z=A-B$. We optimize... | 6 | https://mathoverflow.net/users/8430 | 89247 | 52,774 |
https://mathoverflow.net/questions/89229 | 11 | I apologize if this is a trivial question. If $X$ is a smooth irreducible codimension two subvariety of projective space $\mathbb P^n$, then does there always exist a smooth irreducible codimension one subvariety $Y \subset \mathbb P^n$ with $X \subset Y$ ?
| https://mathoverflow.net/users/11661 | Smooth variety contained in another smooth variety | For $n = 3$ this is always true. It is also true when $X$ is a complete intersection.
Suppose $n ≥ 4$, and suppose that $Y$ exists; then by a well known result of Lefschetz $X$ is a hyperplane section of $Y$, and so $X$ is a complete intersection. Now, for $m = 4$ there are subvarieties of $\mathbb P^4$ that are not ... | 16 | https://mathoverflow.net/users/4790 | 89251 | 52,776 |
https://mathoverflow.net/questions/89184 | 5 | The cuspidal representations of $\operatorname{GL}\_n(F)$ a non archimedean field $F$ with ring of integers $o$ can be classified by inducing irreducible representation from $Z\operatorname{GL}\_n(o)$.
The general question:
>
> How does the representation theory of $\operatorname{GL}\_2(F)$ and the representation... | https://mathoverflow.net/users/10400 | Representations of GL(2, Q_p) and GL(2, Z_p) | Abbreviate $G={\rm GL}(n,F)$ and $K={\rm GL}(n,{\mathfrak o})$.
The general philosophy of "type theory" is the following.
When you restrict an irreducible representation of $G$ to $K$, you get a semisimple representation whose irreducible components are of two sorts:
-- irreducible representations of $K$ occuring... | 6 | https://mathoverflow.net/users/4767 | 89254 | 52,777 |
https://mathoverflow.net/questions/89267 | 3 | Let $(R, m, k)$ be a Noetherian local ring. Let $E\_R(k)$ be the injective hull of $k$ as an $R$-module. It is well known that $E\_R(k)$ is Artinian and is an $(R, \hat{R})$-bimodule (see, Brodmann-Sharp: local cohomology, 1998), where $\hat R$ is the completion of $R$. My question is
**Question**: Is $E\_R(k)$ injec... | https://mathoverflow.net/users/17901 | Is $E_R(k) = E_{\hat{R}}(k)$? | You can see Theorem 18.6 in Commutative ring theory by H.Matsumura.
| 3 | https://mathoverflow.net/users/18970 | 89276 | 52,784 |
https://mathoverflow.net/questions/89274 | 24 | It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of Radon measures with bounded variation.
However, according to my knowledge, there are few books which discuss the case when... | https://mathoverflow.net/users/11966 | the dual space of C(X) (X is noncompact metric space) | What you state in the first paragraph is the Riesz Representation Theorem (see [wikipedia](https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem)). This is valid for all locally compact Hausdorff spaces; so in particular for $\mathbb R$ (ah, I guess, if you look at $C\_0(\mathbb R)^... | 34 | https://mathoverflow.net/users/406 | 89278 | 52,786 |
https://mathoverflow.net/questions/88121 | 7 | Below I describe an infinite (but locally finite) quiver with relations. My question is whether anyone recognizes it and can provide appropriate pointers to the literature. I'm mainly interested in computing its Hochschild homology. The path category of the quiver (with relations) is related to the Temperley-Lieb categ... | https://mathoverflow.net/users/284 | Does anyone recognize this quiver-with-relations? | The Hochschild homology (for the finite truncations) is computed in:
MR2248284 (2007j:16014) de la Peña, José A. ; Xi, Changchang .
Hochschild cohomology of algebras with homological ideals.
Tsukuba J. Math. 30 (2006), no. 1, 61--79.
| 6 | https://mathoverflow.net/users/3992 | 89282 | 52,787 |
https://mathoverflow.net/questions/89279 | 4 | This is a follow up question to my previous one, thanks to Karl and Angelo for their answers, and to the commenters: [Smooth variety contained in another smooth variety](https://mathoverflow.net/questions/89229/smooth-variety-contained-in-another-smooth-variety)
If $X$ is a local complete intersection (irreducible) s... | https://mathoverflow.net/users/11661 | Codim 2 subvariety Cartier on a hypersurface containing it | The Grothendieck-Lefschetz theorem implies that if $Y$ is complete intersection in $\mathbb P^n$ of dimension at least 3, any Cartier divisor on $Y$ is obtained by intersecting with a hypersurface ($Y$ does not have to be smooth). So if $Y$ exists $X$ is a complete intersection, and the previous discussion applies.
| 4 | https://mathoverflow.net/users/4790 | 89297 | 52,794 |
https://mathoverflow.net/questions/89157 | 1 | I'm interested in the theory of additive monads, but I can't find much, so does exist a source?
My interest came in the sense of n-lab, that is a monad over an additive category such its endofunctor is an additive functor. I'm interested in the study of additive categories, particurarly, small abelian and Grothendiec... | https://mathoverflow.net/users/8648 | Is there a survey of additive monads? | First, a small remark: if $C, D$ are additive categories, then an additive functor $C \to D$ is the same as an $Ab$-enriched functor $C \to D$. So for search terms, one might consider "monads enriched in $Ab$" or the like.
I am not aware of any comprehensive survey article for $Ab$-enriched monads, although I can th... | 4 | https://mathoverflow.net/users/2926 | 89300 | 52,797 |
https://mathoverflow.net/questions/89270 | 3 | Dear mathematicians,
**The title says it all**. I would be grateful if you answer the following questions:
* I know that RH is mainly studied under Analytic Number Theory. But again I see Algebraic Number Theory books discussing L-functions. What specific branch of Number Theory studies for instance the generalized... | https://mathoverflow.net/users/21656 | Roadmap to understand the statement and current status of the most general statement of the Riemann Hypothesis | It is generally believed that every $L$-function in arithmetic can be built up from principal $L$-functions associated with cuspidal irreducible representations of $\mathrm{GL}\_n$ over $\mathbb{Q}$. Langlands formulated precise conjectures to support this belief. When properly normalized, principal $L$-functions have ... | 14 | https://mathoverflow.net/users/11919 | 89301 | 52,798 |
https://mathoverflow.net/questions/89304 | 5 | How large is the largest transitive subgroup of $S\_n$ other than itself and $A\_n$? In particular, does its size grow at least exponentially in $n$?
| https://mathoverflow.net/users/20598 | Faithful transitive actions by large groups on small sets | Re-edited to include Derek Holt's observation: This question seems to depend on the smallest prime divisor $p$ of $n.$ I am not sure how to proceed if $n =p.$ For example, if $n =p$ and (apart from $S\_{p}$ and $A\_{p}$) every doubly transitive permutation group of degree $p$ is solvable, then $S\_p$ has no transitive ... | 6 | https://mathoverflow.net/users/14450 | 89309 | 52,801 |
https://mathoverflow.net/questions/89303 | 2 | Let $G$ be a semisimple algebraic group, and let $\Gamma$ be a finitely generated subgroup. Given the generators of $\Gamma$, is there a good way to determine if $\Gamma$ is Zariski-dense in $G$?
For example, let $\Gamma \subseteq \mathrm{PSL}\_2$ where $ \displaystyle
\Gamma = \left\langle \gamma\_1 , \gamma\_2 \rig... | https://mathoverflow.net/users/21662 | Is there a good way to show that a subgroup is Zariski-dense? | 1. What do you mean by "good"? In practice, a theorem of either Lubotzky or Weigel (depending on who you ask) states that if a congruence quotient is surjective for SOME prime (bigger than 3, say), then the subgroup is Zariski-dense, so generally checking for a couple of primes gives a certificate.
2. In $PSL\_2,$ Zari... | 6 | https://mathoverflow.net/users/11142 | 89311 | 52,802 |
https://mathoverflow.net/questions/89205 | 3 | Let $f\_k$ be a sequence of *non-negative* functions from $L\_2(\Omega)$, where $\Omega$ is a bounded open set. Assume that $f\_k\to f$ weakly in $L\_2$ and strongly in $L\_p$, $\forall p<2$. Assume also that $f\_k^2\to F$ weakly in $L\_1$. Does it imply that $F=f^2$?
| https://mathoverflow.net/users/21629 | Weak L_1-convergence of squares | The answer is "yes", even for non non-negative $f\_n$. First assume $f=0$. Since $f\_n\to 0$ in $L\_1$, WLOG by passing to a subsequence $f\_n\to 0$ a.e. and hence $f\_n^2\to 0$ a.e. But $f\_n^2$ converges weakly in $L\_1$ hence is uniformly integrable, whence $\|f\_n^2\|\_1 \to 0$.
The general case follows from the ... | 4 | https://mathoverflow.net/users/2554 | 89314 | 52,805 |
https://mathoverflow.net/questions/89306 | 9 | I am searching for a constructive proof of the following fact: If $X$ is an infinite set, there exists an uncountable family $(X\_\alpha)\_{\alpha \in A}$ of infinite subsets of $X$ such that $X\_\alpha \cap X\_\beta$ is finite whenever $\alpha \neq \beta$. The way I know how to prove this statement is as follows.
Fi... | https://mathoverflow.net/users/703 | Uncountable family of infinite subsets with pairwise finite intersections | This all hinges on what you mean by constructive. An easy way to get such a family is to proceed as follows:
Put your countable set $X$ in bijective correspondence with the collection of finite sequences of 0s and 1s.
For every every subset $A$ of the natural numbers, let $\chi\_A:\mathbb{N}\rightarrow\{0,1\}$ be t... | 9 | https://mathoverflow.net/users/18128 | 89317 | 52,807 |
https://mathoverflow.net/questions/89287 | 0 | Let $p : W \rightarrow S$ be a smooth morphism of algebraic varieties over $\mathbb{C}$. Is it always true that:
$$p^\*H^{p,n-p}(S)=p^\*H^n(S,\mathbb{C})\cap H^{p,n-p}(W),$$
whenever $n\geq p$? If not, are there any more restrictive hypothesis under which it gets true?
| https://mathoverflow.net/users/4096 | pullback of certain forms | The answer is yes, and the morphism doesn't have to be smooth.
A fancy way to explain this
is that if $f:W\to S$ is a morphism of complex algebraic varieties, $f^\*:H^n(S)\to H^n(W)$
is compatible with mixed Hodge structures (by Deligne), and the functor
$H\mapsto H^{pq}:= Gr^p\_FGr\_{p+q}^WH$ form MHS to vector spa... | 4 | https://mathoverflow.net/users/4144 | 89320 | 52,809 |
https://mathoverflow.net/questions/89315 | 5 | A surprising (to me) consequence of Hahn-Banach is that when a sequence converges weakly then there is another sequence made of (finite) convex combination which converges in norm (to the same element).
In particular, this has a consequence in the context of $\ell^p$ spaces (the space of $p$-summable sequences, with ... | https://mathoverflow.net/users/18974 | Weak convergence implying norm convergence | No. In $\ell\_r \oplus\_r \ell\_r$ let $y\_n = (z\_n, x\_n)$ where $x\_n$ are disjoint, $\|x\_n\|\_r=4^n$, $\|x\_n\|\_p \to 0$, $\|z\_n\|\_r = 1$, and $\|z\_n-y\|\_r \to 0$ for some non zero $y$.
| 7 | https://mathoverflow.net/users/2554 | 89321 | 52,810 |
https://mathoverflow.net/questions/89295 | 4 | I'm reading Siu's peper"Fujita conjecture and the extension theorem of Ohsawa-Takegoshi".He refered to a "well-known techniques of using Rieamann-Roch to produce singular metrics".
The situation is:
$L$ is a positive line bundle over an $n$-dimensional compact complex manifold M,$\epsilon$ is a sufficiently small p... | https://mathoverflow.net/users/15882 | How to use Hirzebruch-Riemann-Roch to produce sections of a positive line bundle? | I would suggest you two references; the first one is Demailly's survey on Hodge theory ([B3] here: <http://www-fourier.ujf-grenoble.fr/~demailly/books.html> ); exercise 15.11 explains exactly how to construct sections with desired vanishing order at some point.
This is a consequence of Nadel's theorem applied to sing... | 6 | https://mathoverflow.net/users/5659 | 89326 | 52,812 |
https://mathoverflow.net/questions/89302 | 3 | I have several *positive* random variables $x\_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these variables as many times as needed; all variables are sampled simultaneously.
From each sample my software selects the m... | https://mathoverflow.net/users/21661 | Scale random variables in a way they have equal probabilities of being minimal | Such simple adjustment is not possible. First, take the logarithm: consider new random variables $y\_i=\log x\_i$. Their ordering is the same, but correction is now additive rather than multiplicative: you change $y\_i$ to $y\_i+c\_i$ where $c\_i=\log k\_i$.
Assume that the distribution of each $y\_i$ is normal. The ... | 6 | https://mathoverflow.net/users/4354 | 89332 | 52,814 |
https://mathoverflow.net/questions/89330 | 2 | Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle \geq \langle K(n) \rangle$?
It seems to me that the first relation is accurate if we somehow restricted ourselves to fini... | https://mathoverflow.net/users/11546 | Relationship of Bousfield Classes of Morava K-theories | It is standard that $K(n)\wedge K(m)=0$ for $n\neq m$. One way to think about this is as follows: if $E$ and $F$ are complex oriented ring spectra then the corresponding formal group laws become isomorphic over $\pi\_\*(E\wedge F)$, but it is easy to see that formal group laws of different heights can only become isomo... | 15 | https://mathoverflow.net/users/10366 | 89333 | 52,815 |
https://mathoverflow.net/questions/89324 | 54 | The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does.
Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then:
$\Gamma(s)-\Gamma(1-s)$ yields zeros at:
$\frac12 \pm 2.70269111740240387016556585336 i$
$\frac12 \pm 5.05334476784919736779735104686 i$
$\frac12 \pm 6.821... | https://mathoverflow.net/users/12489 | Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ? | **In the first part, we show that there are no zeros for $z = s + i t$
with $|t| \ge 4$** .
Let $\psi(z):= \Gamma'(z)/\Gamma(z)$ be the digamma function.
If $z = s + i t$, then
$$\frac{d}{ds} |\Gamma(z)|^2 = \frac{d}{ds} \Gamma(z) \Gamma(\overline{z})
= |\Gamma(z)|^2 \left(\psi(z) + \psi(\overline{z})\right).$$
(Bot... | 23 | https://mathoverflow.net/users/nan | 89349 | 52,824 |
https://mathoverflow.net/questions/89340 | 2 | Let $\;\;\; \big\langle V,+,\cdot, \;\; ||\cdot|| \;\; \big\rangle \;\;\;$ be a [Banach space](http://en.wikipedia.org/wiki/Banach_space) over the field $\mathbb{K}$, which is either $\mathbb{R}$ or $\mathbb{C}$.
Suppose there exists a subset $B\:$ of $V\:$ such that:
For all functions $\: f : B\to \mathbb{... | https://mathoverflow.net/users/nan | Uniqueness of dimension in Banach spaces | Well, Kofi is right. Your set $B$ is an unconditional basis in the extended sense. I think Singer treats this kind of basis in one of his volumes on basis theory. Anyway, standard techniques show that the (obviously well defined) biorthogonal functionals are continuous so that $B$ is a Markuschevich basis. It is essent... | 2 | https://mathoverflow.net/users/2554 | 89351 | 52,826 |
https://mathoverflow.net/questions/89345 | 75 | Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth manifold which is not a homogeneous space of any Lie group?
The only candidates for examples I can come up with are tw... | https://mathoverflow.net/users/703 | Example of a manifold which is not a homogeneous space of any Lie group | $\pi\_2$ of a Lie group is trivial, so $\pi\_2(G/H)$ is isomorphic to a subgroup of $\pi\_1(H)$, which is finitely generated (isomorphic to $\pi\_1$ of a maximal compact subgroup of the identity component of $H$). But $\pi\_2$ of a closed manifold is often not finitely generated. For example, the connected sum of two c... | 72 | https://mathoverflow.net/users/6666 | 89352 | 52,827 |
https://mathoverflow.net/questions/89240 | 2 | Hello All,is This conclusion true?
>
> If $(R,m)$ is a local ring and $ Min Ass R=Ass R$ then can we conclude that $Min Ass \hat{R}=Ass \hat{R}$? ($\hat{R}$ is $m$-adic completion of $R$)
>
>
>
$MinAss$ means minimal primes in $Ass(R)$. "$Min Ass R = Ass R$" means that $R$ has no embedded prime ideals. In fact... | https://mathoverflow.net/users/18970 | Minimal prime divisors (MinAss R) | The answer is *no* in general. In the paper *[Fibres formelles d'un anneau local noethérien](http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1970_4_3_3/ASENS_1970_4_3_3_295_0/ASENS_1970_4_3_3_295_0.pdf)* D. Ferrand and M. Raynaud give an example of a two-dimensional local domain whose $\mathfrak{m}$-adic completion has e... | 4 | https://mathoverflow.net/users/16046 | 89354 | 52,829 |
https://mathoverflow.net/questions/89264 | 0 |
>
> **Possible Duplicate:**
>
> [minimal prime devisor(MinAss R)](https://mathoverflow.net/questions/89240/minimal-prime-devisorminass-r)
>
>
>
Hello All,is This conclusion true?
$(R,m)$ be a local ring.if every associated prime ideal of $R$ be minimal then every associated prime ideal of $\hat{R}$ be minim... | https://mathoverflow.net/users/18970 | associated prime ideal | I posted a response to this question in your other post. Please see [the other post](https://mathoverflow.net/questions/89240/associated-prime-ideal)!
| 1 | https://mathoverflow.net/users/16046 | 89355 | 52,830 |
https://mathoverflow.net/questions/89258 | 8 | Let $G$ be a free group. Then $G/G^{(n)}$ ($G^{(n)}$ is the $n$th derived subgroup.) acts on $G^{(n)}/G^{(n+1)}$ by conjugation, which makes $G^{(n)}/G^{(n+1)}$ a $\mathbb{Z}[G/G^{(n)}]$-module. What can I say about this module? Namely, I wonder whether $G^{(n)}/G^{(n+1)}$ is a free $\mathbb{Z}[G/G^{(n)}]$-module. If I... | https://mathoverflow.net/users/6569 | question about derived subgroup | $G/G^{(n)}$ is the free solvable group of class $n$. The module mentioned in your question is projective (a submodule of a free module of rank = rank of $G$).
You can read about free solvable groups in Hanna Neumann's "Varieties of groups", and in Karras-Magnus-Solitar "Combinatorial group theory". Among newer paper... | 2 | https://mathoverflow.net/users/nan | 89360 | 52,835 |
https://mathoverflow.net/questions/89337 | 11 | The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x\_1,\ldots,x\_n]^{S\_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary symmetric polynomials $e\_i(\bar{x})$, for $i$ between $1$ and $n$. I'm looking for a reference in the literature for... | https://mathoverflow.net/users/1474 | Generalizing the Fundamental Theorem of Symmetric Polynomials | I heard of three relatively recent works in that direction, taking different routes and arriving to interesting information about diagonal invariants.
First, there is the paper of Vaccarino that Darij mentioned in his comment (<http://arxiv.org/abs/math/0205233>).
Second, there are results of Domokos (<http://arxi... | 7 | https://mathoverflow.net/users/1306 | 89385 | 52,843 |
https://mathoverflow.net/questions/89374 | 2 | Dear All,
I am dealing with a problem relating to monolithic group.
Let $L$ be a monolithic group with socle $N = S^r$, where S is a nonabelian simple group. Consider the projection $p:N \to S$. A maximal subgroup $H$ of $L$ is of product type if $HN=L$ and $ 1< p( H\cap N ) < S $.
I am considering four simple gr... | https://mathoverflow.net/users/18653 | Monolithic groups with all maximal subgroups of product type | If I am understanding your question correctly, then the answer must be no. You do not say why you are interested in those four simple groups in particular, but for any finite nonabelian simple group and any such $L$ with unique minimal normal subgroup $N=S^r$ with $r>1$, there will be maximal subgroups of $L$ containin... | 2 | https://mathoverflow.net/users/35840 | 89386 | 52,844 |
https://mathoverflow.net/questions/89375 | 6 | I have a subset $B\subset\mathbb{R}^n\times\mathbb{R}^m$ that I want to show has measure zero. I know that the sections $B^x = \{y : (x,y)\in B\}$ all have measure zero. I do not know if $B$ is measurable. Is this enough to conclude that $B$ is a measurable set with measure zero?
EDIT : I would like to say that when ... | https://mathoverflow.net/users/21586 | Sections measure zero imply set is measure zero? | Since the Sierpinski article is in French an uses slightly old-fashioned notation, let me sketch a proof of the result.
Theorem. There is a function $f:\mathbb R\to\mathbb R$ whose graph is not a measurable subset of $\mathbb R^2$.
Proof. We first show that a set $A\subseteq\mathbb R$ of size $<2^{\aleph\_0}$ canno... | 13 | https://mathoverflow.net/users/7743 | 89394 | 52,848 |
https://mathoverflow.net/questions/89327 | 12 | Let $\mathcal{A}(1)$ denote the subalgebra of the $\mathrm{mod}\ 2$ Steenrod algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$.
The cohomology with $\mathbf{F}\_2$ coefficients of the semidihedral group
$$SD\_{16} = \langle g, h \mid g^8 = h^2 = 1, hgh = g^3\rangle$$
of order $16$ is isomorphic to $\mathrm{Ext}^... | https://mathoverflow.net/users/6023 | The semidihedral group of order 16 and ko | Not a 'topological' explanation, but A(1) is the 8 dimensional member of the family of semi-dihedral algebras, whose members of dimension $2^n$ for $n > 3$ are the mod 2 group rings of the semidihedral group of that order. Their cohomology ring is insensitive to the differences; that difference is reflected in the orde... | 11 | https://mathoverflow.net/users/6872 | 89403 | 52,852 |
https://mathoverflow.net/questions/89399 | 3 | Consider linear $N$-dimensional space $F\_2^N$.
Consider its $K$ dimensional subspace $V \subset F\_2^N$.
Let us calculate $w(k,V,N)$ number of vectors in $V$ of Hamming weight $k$ in $V$.
Since there is finite number of subspaces we can calculate average:
$\sum\_{V} w(k,V,N)$.
**Question** is there something known... | https://mathoverflow.net/users/10446 | How many vectors of Hamming weight L in "random" K dimensional subspace of F_2^N ? Or how good/bad is random linear block code ? | You can compute the average very explicitly. For $v\in F\_2^N$, let $||v||$ denote the Hamming weight of $v$. Changing the order of summation, your sum can be re-written as
$$ \sum\_{v\colon\ ||v||=k} T(v), $$
where $T(v)$ is the number of $K$-dimensional subspaces of $F\_2^N$, containing $v$. Now, if $k>0$, then $v\... | 5 | https://mathoverflow.net/users/9924 | 89408 | 52,856 |
https://mathoverflow.net/questions/89414 | -2 | If the stated question is true then what are the consequencies to mathematical physics as an aspect of Hilbert's 6th Problem.
| https://mathoverflow.net/users/21268 | Does Godel's Incompletenss Theorem mean there is no solution to Hilbert's 6th Problem? | No. The incompleteness theorem does not say anything about whether a particular informal theory is capable of being formalized. The incompleteness theorem only applies to formal theories, so it doesn't tell us anything about a theory that has not yet been formalized. You might want to read a book by Franzen called Gode... | 8 | https://mathoverflow.net/users/nan | 89417 | 52,861 |
https://mathoverflow.net/questions/89416 | 4 | A finite group $G$ is an $n$-transposition group if there exists a union $D\subset G$ of conjugacy classes of involutions such that $\langle D \rangle = G$ and for all $a,b\in D$, the product $ab$ is of order at most $n$.
The almost simple $3$-transposition groups were classified by Bernd Fischer. Among the groups cl... | https://mathoverflow.net/users/3516 | How do 3-transposition groups generalise? | Aschbacher and Hall classified groups generated by a class elements of elements of order 3. I do not know what kind of generalizations are you looking for, but these references could be useful:
Aschbacher, Michael; Hall, Marshall, Jr. Groups generated by a class of elements of order $3$. Finite groups '72 (Proc. Gain... | 2 | https://mathoverflow.net/users/17845 | 89418 | 52,862 |
https://mathoverflow.net/questions/88741 | 8 | Given a hyperbolic PDE, the *domain of influence* of a spacetime point $x$, say $I\_x$ though $x$ could be replaced by any set, can be defined in two ways. Lets call one of them *geometric* ($I\_x^G$) and the other *analytical* ($I\_x^A$). In Lorentzian geometry, the geometric domain of influence consists of the interi... | https://mathoverflow.net/users/2622 | Methods for determining domains of influence | I highly doubt the result you actually asked for is true.
Consider the **linear wave equation** on $(1+3)$ Minkowski space. The *analytic domain of influence* of a point $x$ as Lax defined it, which morally says that $y$ is in the analytic domain only if one can find perturbations in arbitrary small neighborhoods of... | 3 | https://mathoverflow.net/users/3948 | 89419 | 52,863 |
https://mathoverflow.net/questions/89423 | 3 | I'm reading [Pierre Cartier's *A primer of Hopf algebras*](http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf) to educate myself. In its subsection 3.3 (which doesn't need any Hopf algebra theory), he sketches a proof why compact Lie groups are algebraic. One step in this proof is the Peter-Weyl t... | https://mathoverflow.net/users/2530 | Peter-Weyl theorem as proven in Cartier's Primer | (Same as pm's answer, with details.)
Well, if $$ R\_f(\varphi)(h) = \int\_G \varphi(g) f(g^{-1}h) \ dg $$
then the adjoint satisfies
\begin{align\*}
(R\_f^\*(\varphi)|\psi) &= (\varphi|R\_f(\psi))
= \int\_{G\times G} \varphi(h) \overline{ \psi(g) f(g^{-1}h) } \ dg \ dh \\
&= \int\_{G\times G} \varphi(h) \tilde f(h^{-... | 4 | https://mathoverflow.net/users/406 | 89428 | 52,870 |
https://mathoverflow.net/questions/89436 | 2 | Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K\_X)^n$ is at most $C^n (n+1)^n$?
This would follow from the weakening of Ehrhart's conjecture that I proposed in [Reference re... | https://mathoverflow.net/users/21123 | Bound on the (anticanonical) degree of toric Fano varieties | I think the answer to this is no. In fact according to [1] there is no universal polynomial bound on the $n$-th root of $c\_1(X)^n$ as X runs over all toric Fanos of dimension n (this is referenced to Debarre but I am afraid I do not have this source with me at the moment).
By contrast the purpose of [1] is to prove ... | 3 | https://mathoverflow.net/users/9202 | 89440 | 52,877 |
https://mathoverflow.net/questions/87354 | 11 | Let $N\_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures.
I'm looking at ideal triangulations of the surface, namely maximal collection of pairwise disjoint ideal arcs (joining punctures) on the surface.
Given a triangulation and an arc ... | https://mathoverflow.net/users/21084 | Flips of triangulations on non-orientable surfaces | My article ["Tiling the measured foliation space of a punctured surface", Trans. Math. 306 no. 1 (1988)](http://www.jstor.org/pss/2000830) contains a proof of this fact in the case of oriented surfaces. It is essentially the same as Hatcher's proof of contractibility, but focussing solely on the issue of connectivity, ... | 11 | https://mathoverflow.net/users/20787 | 89449 | 52,883 |
https://mathoverflow.net/questions/89439 | 21 | By Malcev's theorem, every finitely generated linear group is residually finite (RF).
On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to impose some restrictions on linear groups to avoid this example. Discreteness sounds like a reasonable hypothesis. It is ... | https://mathoverflow.net/users/21684 | Non-residually finite matrix groups | The answer to Question 1 is yes.
For each prime $p>2$, take the [von Dyck group](http://en.wikipedia.org/wiki/Von_Dyck_group#von_Dyck_groups) $D(p,p,\infty)$, generated by two rotations of order $p$ whose product is a parabolic $q\_p$. These groups lie in $PSL(2,\mathbb{R})< PSL(2,\mathbb{C})$. We may normalize the ... | 14 | https://mathoverflow.net/users/1345 | 89451 | 52,884 |
https://mathoverflow.net/questions/89459 | 6 | I'm teaching an undergraduate combinatorics class, using Harris et al.'s book ``Combinatorics and Graph Theory''. In Section 1.6 there is an exercise asking to show that for the complement of a bipartite graph, the chromatic number equals the clique number. I assigned the problem to my students, without thinking much a... | https://mathoverflow.net/users/21690 | Proving that the complement of a bipartite graph has chromatic number equal to clique number | According to [this Wikipedia entry](https://en.wikipedia.org/wiki/K%C5%91nig%27s_theorem_%28graph_theory%29#Connections_with_perfect_graphs)
the statement that $\chi(\overline{G}) = \omega(\overline{G})$ for all bipartite $G$ is actually equivalent to König's Theorem.
| 3 | https://mathoverflow.net/users/19729 | 89463 | 52,888 |
https://mathoverflow.net/questions/89458 | 7 |
>
> the conformal structure does not see the conical singularities of a polyhedral surface.
>
>
>
This is a quote from the Preface of *[Quantum Triangulations](http://www.springer.com/physics/book/978-3-642-24439-1)* (eds.: Carfora, Marzuoli).
The sentiment is attributed to Marc Troyanov, in a 1991 *Trans. AMS* ... | https://mathoverflow.net/users/6094 | Conformal structure does not see conical singularities | Consider the following pair of surfaces:
1. $P\;$ is the plane with the origin removed.
2. $C$ is the cone $z = \sqrt{x^2+y^2}$ in $\mathbb{R}^3$, with the origin removed.
There are several possible structures we could put on these surfaces. For example, both of the surfaces support a differentiable structure. As d... | 12 | https://mathoverflow.net/users/6514 | 89465 | 52,890 |
https://mathoverflow.net/questions/89473 | 8 | Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\mathcal{X}$. Is $H^i(X,\mathcal{F})$ finitely generated over $\Gamma(O\_X)$ if $\mathcal{F}$ is coherent ? This statement... | https://mathoverflow.net/users/6986 | Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent? | This is false even for $\mathrm H^0$: take $X$ to be $\mathbb A^2 \smallsetminus \{0\}$, and as $F$ the structure sheaf of $L \smallsetminus \{0\}$, where $L$ is a line through $0$.
| 17 | https://mathoverflow.net/users/4790 | 89484 | 52,900 |
https://mathoverflow.net/questions/89472 | 0 | Consider a random vector $\mathbf{Z}$ of length $N$ whose elements are i.i.d. and zero-mean. We construct two new scalar random variable $X$ and $Y$ from $\mathbf{Z}$ as follows:
$X = \langle\mathbf{a}, \mathbf{Z}\rangle = \mathbf{a}^T\mathbf{Z}$ and $Y = \langle\mathbf{b}, \mathbf{Z}\rangle = \mathbf{b}^T\mathbf{Z}$... | https://mathoverflow.net/users/15385 | Again a question related to uncorrelatedness and independence. | OK, I respond because I need some reputation points...
Counterexample: $N = 2$, $a = (1,1)$, $b = (1,-1)$. Entries of $Z$ are iid uniform on {$-1,1$}. Then $X = 2$ implies $Y = 0$, such that the variables are not independent.
| 5 | https://mathoverflow.net/users/18032 | 89488 | 52,902 |
https://mathoverflow.net/questions/89487 | 10 | A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).
Boolean Algebras that are complete as well as atomic (also called CABAs) are of course precisely those that are isomorphic to some power set (equipped with the... | https://mathoverflow.net/users/8590 | Examples for "nice" Boolean algebras that are not complete or not atomic | First of all, let me point out an error in the question. It is not true that "the Boolean algebras that are not atomic or not complete are precisely those that are carried to non-discrete Stone spaces via the Stone Duality." If $X$ is an infinite set, then, even though the power set algebra $P(X)$ is atomic and complet... | 13 | https://mathoverflow.net/users/6794 | 89491 | 52,904 |
https://mathoverflow.net/questions/86792 | 66 | The two standard approaches to the quantization of [Chern-Simons theory](http://www.ams.org/mathscinet-getitem?mr=990772) are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first published in 1991 (the geometric quantization picture [here](http://www.... | https://mathoverflow.net/users/35353 | Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent? | The equivalence of these two construction is actually known now. It follows by combining the main result of:
>
> Yves Laszlo, *Hitchin's and WZW connections are the same*., J. Differential Geom. **49** (1998), no. 3, 547–576, doi:[10.4310/jdg/1214461110](http://doi.org/10.4310/jdg/1214461110)
>
>
>
with my jo... | 43 | https://mathoverflow.net/users/21082 | 89498 | 52,908 |
https://mathoverflow.net/questions/89441 | 0 | Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$.
Let $Y$ be another Banach space embedded in $X$. We consider$A\_Y$, the part of $A$ in $Y$, defined as the operator with domain
$$D(A\_Y) := \{ y \in D(A) \cap Y: Ay \in Y \}$$
and
$$A\_Y \ y := Ay$$
Then it seems to me that $A\... | https://mathoverflow.net/users/17035 | The part of an operator as an analytic generator | This is wrong. Let us assume that $Y$ is a closed subspace of $X$ to clearify the problem. As Matthew Daws already said, you have to assume that the semigroup $(T(t))$ generated by $A$ leaves $Y$ invariant: suppose that $A\_Y$ indeed generates a (strongly continuous) semigroup $(S(t))$ on $Y$. Then for example the Yosi... | 1 | https://mathoverflow.net/users/21704 | 89499 | 52,909 |
https://mathoverflow.net/questions/89483 | 9 | Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a\_i>0$, $i=\overline{1,n}$, and there exist $j$ and $k$, $j\neq k$ such that $a\_j=a\_k\leq 1$. $a\_j$ and $a\_k$ are the smallest diagonal entries.
I'm interested... | https://mathoverflow.net/users/21702 | 0 eigenvalue for a symmetric tridiagonal matrix | To simplify things a little, I describe conditions under which the smallest eigenvalue is strictly positive. These can be adjusted to get equality to zero.
**Necessary and sufficient** conditions for positive definiteness of the tridiagonal matrix in question are described below.
**Definition (Chain Sequence).** A ... | 10 | https://mathoverflow.net/users/8430 | 89500 | 52,910 |
https://mathoverflow.net/questions/89494 | 1 | This is coming out of Mumford's GIT, section 7.2, page 131.
$A/S$ an abelian scheme of dimension $g$ with polarization $\bar{\omega}$ of degree $d^2$. Then $\pi\_\*(L^\Delta(\bar{\omega})^3)$ is locally free on $S$ of rank $6^gd$ which defines the closed immersion $\varphi\_3 : A \rightarrow \mathbb{P}(\pi\_{\*}(L^\D... | https://mathoverflow.net/users/18403 | Hilbert polynomial of an abelian scheme | Let us look at a single geometric fiber $X$. Let $L^\Delta(\bar\omega)|\_X = \mathcal{O}\_X(D)$. The Riemann-Roch theorem for abelian varieties (Mumford "Abelian Varieties", Chap. 3 Section 16) states that
$$ \chi(\mathcal{O}\_X(D)) = D^g/g!$$
and moreover that $\chi(\mathcal{O}\_X(D))^2 = \deg \phi$, where $\phi$ is t... | 2 | https://mathoverflow.net/users/3847 | 89502 | 52,911 |
https://mathoverflow.net/questions/89503 | 4 | Let $G$ and $H$ be two finitely generated groups, where $H$ is abelian. I'm curious in which cases $Hom(G,H)$ turns out to be cyclic or virtually cyclic.
| https://mathoverflow.net/users/12996 | When is Hom(G,H) cyclic? | If $H$ is abelian, any homomorphism $G \to H$ factors through the abelianization $G/[G, G] \to H$, so we may assume WLOG that $G$ is also abelian, so we can apply the structure theorem to both $G$ and $H$. Then $\text{Hom}(G, H)$ is virtually cyclic if and only if it has rank at most $1$, hence if and only if both $G$ ... | 7 | https://mathoverflow.net/users/290 | 89506 | 52,912 |
https://mathoverflow.net/questions/89512 | 4 | In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures.
Closed forms/exact forms
real parts of analytic functions/harmonic functions
Analytic functions/analytic functions that have holomorphic antiderivatives.
One can see that for open conne... | https://mathoverflow.net/users/21708 | Notions related to De Rham Cohomology | It's a somewhat broad question, but yes there are connections between various things on
your list under quite general conditions. Since it's a big topic, I'll mostly be
content to list references since you asked for them. If your manifold is simply connected then
closed $1$-forms are exact as you surmised.
So the firs... | 3 | https://mathoverflow.net/users/4144 | 89519 | 52,916 |
https://mathoverflow.net/questions/89504 | 6 | I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan - Introduction to Topological Groups" and is very tedious.
I thought so: $\mathbb{T}^n$ is compact then $\widehat{\ma... | https://mathoverflow.net/users/21706 | Quick computation of the Pontryagin dual group of torus | The key case is the 1-dimensional torus. We want to show every continuous homomorphism
$\chi \colon {\mathbf R}/{\mathbf Z} \rightarrow {\mathbf T}$ has the form $x \bmod {\mathbf Z} \mapsto e^{2\pi inx}$ for some integer $n$.
Any character of ${\mathbf R}/{\mathbf Z}$ can be pulled back to a character of ${\mathbf... | 14 | https://mathoverflow.net/users/3272 | 89520 | 52,917 |
https://mathoverflow.net/questions/89514 | 2 | What is the form of zeta function of an elliptic curve over $\mathbb{F}\_p(t)$? Does it satisfy a Riemann hypothesis?
| https://mathoverflow.net/users/nan | Zeta function of an elliptic curve over $\mathbb{F}_p(t)$ | It is a polynomial in $p^{-s}$ and yes, it satisfies the Riemann hypothesis. See for example Theorem 2.2.1 of Lecture 4 of Douglas Ulmer's *Elliptic curves over function fields* available on the arXiv: [1101.1939v1](http://arxiv.org/abs/1101.1939v1).
| 5 | https://mathoverflow.net/users/1021 | 89521 | 52,918 |
https://mathoverflow.net/questions/89518 | 26 | The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
[Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?](https://mathoverflow.net/questions/89324/are-all-zeros-of-gammas-pm-gamma1-s-on-a-line-with-real-part... | https://mathoverflow.net/users/12489 | Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line? | If $\zeta(s)$ is nonzero, but $\zeta(s)\pm\zeta(1-s)=0$, then by the functional equation of the Riemann zeta function we have
$$ \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\pm \pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)=0.$$
That is, your question is just the Riemann Hypothesis plus a more elementary one ... | 21 | https://mathoverflow.net/users/11919 | 89523 | 52,919 |
https://mathoverflow.net/questions/89522 | 4 | I was wondering if anyone could give me tips on the following question:
Suppose $\alpha \in\text{GL}\_2^{+}(\mathbb{Q})$ has integral entries and is such that det$(\alpha) = D > 0$.
If $\Gamma$ is a congruence subgroup of level $N$ then $\alpha^{-1}\Gamma\alpha$ contains a congruence subgroup of level $ND$.
I am ... | https://mathoverflow.net/users/21698 | Conjugate of congruence subgroup of level N contains congruence subgroup of level ND | Let $A\equiv 1\mod ND$, then $$\alpha (A-1) \alpha^{-1}=\frac{1}{D}\alpha (A-1) \mathrm{adj}(\alpha)\equiv 0 \mod N.$$ Hence $\alpha A\alpha^{-1}\equiv 1 \mod N$. Here $\mathrm{adj}(\alpha)$ is the adjoint matrix (the matrix of co-factors); we use the fact that all entries of $\mathrm{adj}(\alpha)$ are integers, and al... | 7 | https://mathoverflow.net/users/nan | 89526 | 52,921 |
https://mathoverflow.net/questions/89533 | 4 | Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order,
Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?
| https://mathoverflow.net/users/5259 | When a quotient singularity is toric? | Yes, it is true. Let $G \cong \mathbb{Z}/m$ act by $\mathrm{diag}(\zeta^{a\_1}, \zeta^{a\_2}, \ldots, \zeta^{a\_n})$, where $\zeta$ is a primitive $n$-th root of unity. Let $S$ be the semigroup $\{ (b\_1, \ldots, b\_n) \in \mathbb{Z}\_{\geq 0}^n : \sum a\_i b\_i \equiv 0 \mod m \}$. Then the quotient is Spec of the sem... | 6 | https://mathoverflow.net/users/297 | 89534 | 52,923 |
https://mathoverflow.net/questions/89528 | 4 | The setting is measure on $2^\omega$. That product (independent) measures obey a 0/1 law, i.e, that measurable tail sets all have measure 0 or 1, is well known. I've made some progress extending this to measures that satisfy a weak symmetry property that's a little complicated to state, but roughly is that the limit of... | https://mathoverflow.net/users/20809 | Measures that satisfy a 0/1 law | These measures are well studied in ergodic theory. They are measures with the $K$ (for Kolmogorov) property. It's known that they are a bit of a zoo from an ergodic point of view: Ornstein's celebrated theorem for Bernoulli shifts says that two Bernoulli shifts are isomorphic as measure-preserving systems if and only i... | 2 | https://mathoverflow.net/users/11054 | 89538 | 52,924 |
https://mathoverflow.net/questions/89540 | 9 | If $j : V \rightarrow M$ is an elementary embedding, what can we learn in $M$ from $j''ORD$? That is, what is $M[j''ORD]$?
In particular,
> Is it $M[j''ORD]$ equal to all of $V$?
If not, do we get a model intermediate between $M$ and $V$? If $\kappa$ is the critical point of $j$, is $\kappa$ still a large card... | https://mathoverflow.net/users/10671 | What can we learn about an elementary embedding from the image of the ordinals? | Nice question, Jonas!
Yes, in the case that $V$ satisfies ZFC and $M\subset V$, then indeed $M[j''\text{Ord}]=V$. To see this, consider first the case of a set of ordinals $A\subset\theta$ in $V$. Notice that from $j''\theta$ we may reconstruct $j\upharpoonright\theta$. Further, $j(A)$ is in $M$, and from $j(A)$ and ... | 8 | https://mathoverflow.net/users/1946 | 89545 | 52,926 |
https://mathoverflow.net/questions/89542 | 5 | [This question](https://mathoverflow.net/questions/89345/example-of-a-manifold-which-is-not-a-homogeneous-space-of-any-lie-group) asks for an example of a manifold which is not a homogeneous space of any Lie Group, and many examples are given in the answers. However: is there a an example known with a metric of positiv... | https://mathoverflow.net/users/11142 | A followup on non-homogeneous spaces. | Yes, [Eschenburg constructed an infinite family of simply connected 7-dimensional examples](https://doi.org/10.1007/BF01389224 "Eschenburg, Jost-Hinrich. New examples of manifolds with strictly positive curvature. Invent. Math. 66, 469–480 (1982), EuDML:142893. zbMATH review at https://zbmath.org/0484.53031") and prove... | 10 | https://mathoverflow.net/users/18050 | 89550 | 52,929 |
https://mathoverflow.net/questions/89527 | 4 | Search problem $MIN^P$ is, given a polynomial-time computable predicate that is a partial order, to find its minimum (any will do).
Search problem $MIN^L$ is, given a polynomial-time computable predicate that is a linear order, to find its minimum.
Search problems can be interpreted as computational models (complex... | https://mathoverflow.net/users/21163 | Is $MIN^P$ search problem (partial order) reducible to $MIN^L$ (linear order) search problem? | Because of the issue I mention in my comment, it seems that the question admits an unsatisfactory affirmative answer.
Namely, as I expect you know, it is a standard fact that every partial order is contained, as a set of relation pairs, within a linear order. In other words, for every partial order on a set there is... | 3 | https://mathoverflow.net/users/1946 | 89552 | 52,931 |
https://mathoverflow.net/questions/89537 | 4 | Let $G=GL(n,F)$, where $F$ is a non-archimedean local field. If we consider a smooth representation $\pi$ of $G$ such that every irreducible generic representation of $G$ embeds in $\pi$, is it true that the representation $Ind\_U^G\chi$ embeds in $\pi$, where $U$ is the standard unipotent subgroup of $G$ and $\chi$ is... | https://mathoverflow.net/users/6849 | Generic representations of $GL(n,F)$ | No. At least I think not.
I assume that $Ind\_U^G \chi$ means the space of all functions $f:G\to \mathbb C$ which satisfy (1) $f(ug) = \chi(u) f(g)$ and (2) there is an open compact subgroup $K$ of $G$ such that $f(gk) = f(g)$ for all $g \in G$ and $k \in K.$
What I want to do is construct a subrepresentation of th... | 5 | https://mathoverflow.net/users/21252 | 89554 | 52,932 |
https://mathoverflow.net/questions/89546 | 0 | Is there a way to determine if all points of a bezier curve are visible from an endpoint? For instance, if you're given a cubic bezier curve in the plane: $\textbf C(t) = \sum\_{i=0}^3 B\_i^3(t) \textbf P^i$ would any of the rays from $\textbf P^0$ to $\textbf C(t)$ intersect the curve at any other point?
So I want ... | https://mathoverflow.net/users/21715 | None or infinite #solns for Bezier Curve Problem? | WLOG take ${\bf P}^0 = (0,0)$ and ${\bf P}^3 = (1,0)$, and let ${\bf P}^1 = (x\_1, y\_1)$ and ${\bf P}^2 = (x\_2, y\_2)$. For $t \ne 0$ the tangent to the curve at ${\bf C}(t)$ is parallel to ${\bf C}(t)-{\bf P}^0$ iff $$ \left( -3x\_{{2}}y\_{{1}}+3x\_{{1}}y\_{{2}}-y\_{{2}}+2y\_{{1}}
\right) {t}^{2}-\left( -6x\_{{2}}y... | 0 | https://mathoverflow.net/users/13650 | 89559 | 52,936 |
https://mathoverflow.net/questions/89431 | 1 | Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system.
My apology to Yemon Choi, Will Jagi , Theo Johnson-Freyd and all other readers. My question was formulated extremely short wi... | https://mathoverflow.net/users/21665 | Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal ? | I'm no expert, and I haven't asked Professors Gell-Mann or Ne'eman, but with their choice of matrices, L\_3 measures the familiar Heisenberg iso-spin quantum number, while L\_8 measures the then-novel hypercharge. Mixing the operators would mix the quantum numbers.
| 2 | https://mathoverflow.net/users/36655 | 89561 | 52,937 |
https://mathoverflow.net/questions/89525 | 0 | This precise question grew out from the question whether a smooth commutative $k$-algebra (char($k$)=$0$) is always cofibrant as a non-positively graded commutative differential graded co-chain $k$-algebra. I think the answer is no (while the converse is true). For this I'd need at least one example of the following si... | https://mathoverflow.net/users/21710 | Example of a non-liftable morphism from a smooth algebra | Let me try to give a quick answer (but double check it).
Take $R=k[x,y]\_{xy}$, $S=k[x,y]$ and $I=(xy-1)$, with the obvious map $R\rightarrow S/I$. It's easy to prove that no lifting exists (any map $R\rightarrow S$ factors through $k$, so it cannot be a lifting of the given map). One more remark: $R$ discrete and cofi... | 1 | https://mathoverflow.net/users/16882 | 89583 | 52,947 |
https://mathoverflow.net/questions/89581 | 3 | Let $X$ be a holomorphic vector bundle over $Y$ (where $Y$ is an arbitrary complex manifold, not necessary projective). Does there exist an analytic subset $Z$ of $Y$ such that the restriction of $X$ to $Y \setminus Z$ is a trivial vector bundle?
| https://mathoverflow.net/users/2234 | non-trivial locus of a holomorphic vector bundle | If your manifold is complex projective, then the answer is yes. Otherwise it is no. You can take a $K3$ surface without complex curves and just consider its tangent bundle. Of curse it will stay *holomorphically* non-trivial, if you throw away finite number of points from $K3$, since any holomorphic vector field on a $... | 5 | https://mathoverflow.net/users/943 | 89585 | 52,948 |
https://mathoverflow.net/questions/89584 | 4 | Hello,
I would like to know clear references about the following facts:
Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I don't assume that $G$ is affine, but if there is a nice simple reference with $G$ affine, then I'll like it too.
>
> I... | https://mathoverflow.net/users/2095 | About $G$-modules versus $Lie(G)$-modules for algebraic groups | I'm sure that there are a number of ways of answering this in the affine case; here's one. Let's assume $G$ affine and let's say that $k$ is our algebraically closed field. The following argument may not be the slickest one, but it has the benefit of working in arbitrary characteristic with the enveloping algebra repla... | 8 | https://mathoverflow.net/users/1528 | 89589 | 52,952 |
https://mathoverflow.net/questions/89590 | 5 | In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S\_{\lambda})\_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective limit of this system.
The first version of my question I think assumes that you are somewhat familiar with this part of... | https://mathoverflow.net/users/17907 | Spreading out flat morphisms of schemes | I think what you are looking for is in the book, only later: see EGA IV, (11.2.6).
| 5 | https://mathoverflow.net/users/7666 | 89593 | 52,955 |
https://mathoverflow.net/questions/88659 | 17 | Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move $n$, one must jump $a\_n$ steps in one of the directions north,south,east or west, and mark the ending square as visited?
If $a\_n=n$ or if $a\_n=n^2$?
Allowing diagonal moves as well, is there a general alg... | https://mathoverflow.net/users/21452 | Traversing the infinite square grid | It's possible for $a\_n=n$ and probably most stepsizes without modular or growth obstructions.
We have covered some subset of an mxm square, are situated at the boundary, and want to visit a cell (x,y) in our square. Choose one of the x,y axes and move far away along it, (but not upon it), until stepsize s>>m and dis... | 6 | https://mathoverflow.net/users/21452 | 89595 | 52,956 |
https://mathoverflow.net/questions/89565 | 7 | Hi,
I'm looking for a reference for the full isometry groups of the
**(i)** complex Stiefel manifolds $U(m)/U(m-l)$, either for the Euclidean metric (i.e. identifying it with orthonormal $m \times l$-matrices and using the inner product $\operatorname{tr}(X^\ast Y)$) or for the induced quotient metric (where we use... | https://mathoverflow.net/users/13356 | Full isometry groups of Stiefel and Grassmann manifolds | As long as *connected* groups of isometries are concerned, Grassmann manifolds are symmetric spaces,
so the identity component of its isometry group
is $G$ in its symmetric presentation $G/H$ ($G$ connected) as a homogeneous space, namely, $SO(n)$ for $n$ odd and $SO(n)/\mathbf Z\_2$ for $n$ even in the real case,
a... | 4 | https://mathoverflow.net/users/15155 | 89601 | 52,958 |
https://mathoverflow.net/questions/89608 | 3 | Let's consider the moduli space $M\_g$ of curves of genus $g$ over $\mathbf{C}$.
Not every curve of genus $g$ is a Galois cover (of the projective line) if $g\geq 3$.
How big is the locus of Galois covers in $M\_g$? It contains the locus of cyclic covers. So there is a lower bound on the dimension ($2g-2$ if I'm no... | https://mathoverflow.net/users/21730 | How big is the locus of Galois covers in the moduli space of curves | The bigger the Galois group, the smaller the dimension. So, the biggest component is the hyperelliptic locus. There are only finitely many possibilities for the Galois group. For each fixed group, you get a quasi-projective variety, it may be affine but is definitely not projective. You can count parameters by noticing... | 9 | https://mathoverflow.net/users/2290 | 89619 | 52,968 |
https://mathoverflow.net/questions/89626 | 5 | I have heard that the rational homology of a covering space is easy to compute, compared with the ordinary homology. However, I don't know any details about that. Can anyone help me? Any reference will be greatly appreciated.
| https://mathoverflow.net/users/15770 | Rational Homology of a Covering Space | There are several reasons why this is true. Here's one: For a finite cover $p:\tilde X\to X$, there is a *transfer map*
$t:H\_i(X)\to H\_i(\tilde X)$ which, on the chain level, takes a chain $\sum a\_i \sigma\_i$ to $\sum a\_i \sum g\sigma\_i$, where the inner sum is over all lifts of $\sigma\_i$. This holds with any c... | 13 | https://mathoverflow.net/users/3874 | 89628 | 52,972 |
https://mathoverflow.net/questions/89600 | 15 | For a given real number $x$, let $R\_x$ be the set of real numbers $r$ such that the inequality
$$\displaystyle \left| x - \frac{p}{q} \right| < \frac{1}{q^r}$$
has at most finitely many solutions with integers $p,q$. Define the irrationality measure of $x$, say $\mu(x)$, to be the infimum of $R\_x$.
It is known t... | https://mathoverflow.net/users/10898 | Numbers with known irrationality measures? | If the elements $a\_n$ of the simple continued fraction of the irrational number $x$ satisfy $a\_n < c n + d$ for some positive constants $c$ and $d$, then $\mu(x) = 2$. Besides $e^{2/k}$ for positive integers $k$,
interesting examples of such numbers include $\tanh(1/k)$, $\tan(1/k)$, and $I\_0(1)/I\_1(1)$ where $I\_... | 22 | https://mathoverflow.net/users/13650 | 89637 | 52,977 |
https://mathoverflow.net/questions/89598 | 11 | I am a number theory graduate student learning a bit of homological algebra, and I am curious about higher complexes in abelian categories. I apologize if my post is slightly vague as I am not an expert in this area. I will use capital roman letters to denote objects or complexes, but the usage is clearly stated.
Mot... | https://mathoverflow.net/users/1437 | Higher "Cartan-Eilenberg" Resolutions | The process can certainly be iterated as explained by Marc (see also Weibel, Homological Algebra, 1.2.5. Moreover cf. 1.2.3, 2.2.2 for the fact that the category of chain complexes over an abelian category with enough projectives is again an abelian category with enough projectives).
However, it seems to me that it ... | 3 | https://mathoverflow.net/users/10194 | 89641 | 52,978 |
https://mathoverflow.net/questions/89635 | 5 | Let $\mathcal{C}$ be a small category and $P$ a presheaf on $\mathcal{C}$. Then there is an equivalence $$\widehat{\mathcal{C}} / P \cong \widehat{\int\_{\mathcal{C}}} P$$ Now there are (at least) two ways to see this (from what I've managed to ascertain.) You might derive the equivalence as it were 2-categorically usi... | https://mathoverflow.net/users/13753 | How can I see that the slice of a presheaf category is equivalent to the presheaf category of the category of elements? | You can verify this equivalence elementarily (without the language of fibrations etc.):
Assume first that $C$ is the category with only one morphism (i.e. the terminal category), so that a presheaf on it is just a set. Then the statement is as follows: If $P$ is a set, then a set $F$ together with a map $F \to P$ is ... | 7 | https://mathoverflow.net/users/2841 | 89648 | 52,980 |
https://mathoverflow.net/questions/89651 | 2 | I'm looking for a proof that the Pontryagin dual $G^\*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^\*$ is a group, my troubles are in proving that the map $G^\* \times G^\* \to G^\* : (f,g) \mapsto fg^{-1}$ is continuous and so $G^\*$ is topological.
I read in "Rudin - Fouri... | https://mathoverflow.net/users/21706 | Proof that the Pontryagin dual of a topological group is a topological group | I don't think this is a research level question, but here is an argument.
The topology of $G^\*$ is given by uniform convergence on compact subsets of $G$. Let $K\subset G$ be compact, then we need to show that if $f\_n\to f$ and $g\_n\to g$ uniformly on $K$, then $f\_ng\_n^{-1}\to fg^{-1}$ uniformly on $K$. This is ... | 4 | https://mathoverflow.net/users/11919 | 89653 | 52,982 |
https://mathoverflow.net/questions/89657 | 6 | In linear algebra one learns a lot of normal forms (Which I want to think of as a classification of the orbits of a group action on some set). For example if $V$ is a $k$-vectorspace $GL(V)$ acts on $V$ with exactly two orbits - the orbit of $0$ and the other orbit.
Now what happens if we let $GL(V)$ act diagonally o... | https://mathoverflow.net/users/3969 | Orbits of exterior products | To the best of my knowledge, already for $n=3$ case (which probably is close, if not identical, to $n=\dim(V)-3$ because of the duality argument) the situation is far from clear for $\dim(V)$ starting from $10$ or so (the complexity somewhat depends on the ground field). Some hints on that are in papers like [[1](http:... | 5 | https://mathoverflow.net/users/1306 | 89660 | 52,985 |
https://mathoverflow.net/questions/89578 | -1 | The following is a well known fact and due to the functorial properties of the jet functor:
Suppose you have two smooth manifolds $M$ and $N$ and maps $f:M \rightarrow N$ as well as
$g: M \rightarrow N$, then if there is a $m \in M$ and an open neighbourhood $U$ of $m$ in $M$ such that the equation
$$f(x) = g(x)$$
... | https://mathoverflow.net/users/21302 | Inverse Problem for jet equations | If you assume $U$ is a co-ordinate chart, the answer is yes. Simply choose local co-ordinates everywhere and assume each of your functions is an affine function of the co-ordinates. In that case, each function is equal to its own 1-jet map. So the desired equations involving the composition of functions follow directly... | 2 | https://mathoverflow.net/users/613 | 89663 | 52,988 |
https://mathoverflow.net/questions/89644 | 9 | A week or so ago, I was saddened to read Jim Stasheff's post on the AlgTop mailing list, announcing the passing of Colin Day, after a long bout with cancer.
I was thinking of reading Colin Day's PhD thesis in his memory, and trying to understand it (the title certainly sounds interesting); but I found it to be diffic... | https://mathoverflow.net/users/2051 | How to find Colin Day's PhD Thesis | I appreciate the interest shown. It apparently is on microfiche.
I am in contact with the UNC math library
and hope to have access I can share.
**Update**: It is now available at
<http://hans.math.upenn.edu/~jds/>
scroll down to Colin...
Please let me know if you access it or if you have trouble accessing it.
... | 15 | https://mathoverflow.net/users/36067 | 89665 | 52,989 |
https://mathoverflow.net/questions/89669 | 2 | If we fix a locally profinite group $G$ , we note $R(G)$ the category of smooth representations of $G$, $\mathcal{E}$ the set of equivalence classes of $R(G)$, and finaly $Irr(G)$ the set of irreducible equivalence calasses. I recall that the theorem of Cantor-Bernstein says : If $E$ and $F$ two sets. If there is an in... | https://mathoverflow.net/users/6849 | Theorem of Cantor-Bernstein in the category of smooth representation of $G$ | Here are some observations, too long for a comment:
1) Note that cuspidal irreducible representation are compactly induced
$\sigma = c-ind\_K^G \tau = Ind\_K^G \tau$
2) You have the second adjointness theorem (proposition on pg. 20 Bushnell-Henniart "Local Langlands for GL(2)".)
$Hom\_G( c-ind\_K^G \tau, \pi) =... | 1 | https://mathoverflow.net/users/10400 | 89674 | 52,992 |
https://mathoverflow.net/questions/89664 | 4 | **Update July 29, 2013**.
I have still not found a good **textbook** for this topic, if you point one to me I will be **grateful** :) I have accepted BS's answer anyway, since their explanation was useful to me and gave me a good starting point for further research. I ended up finding a very helpful resource for thi... | https://mathoverflow.net/users/12793 | Finite, abelian, yet "fugitive" orthogonal subgroups | These results belong to what is called *duality in finite abelian groups*, a theory that has been generalized by [Pontryagin](http://en.wikipedia.org/wiki/Pontryagin_duality) and others in the 30's to locally compact abelian groups.
Another keyword here is "Discrete Fourier Transform", although it is mainly used for... | 9 | https://mathoverflow.net/users/6451 | 89676 | 52,994 |
https://mathoverflow.net/questions/89687 | 7 | If $G$ is a finitely generated free group, then its classifying space $B G$ can be presented as a finite CW complex (a finite bouquet of circles), and therefore is Spanier-Whitehead dualizable. Are there any other discrete groups $G$ with the property that $B G$ is Spanier-Whitehead dualizable?
| https://mathoverflow.net/users/49 | Dualizable classifying spaces | Any finitely presented group of type FL admits a finite classifying space. (A group $G$ is of type FL if $\mathbb{Z}$ admits a finite length resolution by finitely generated, free $\mathbb{Z}G$-modules.) This is Theorem VIII.7.1 in Brown's book "Cohomology of groups".
More examples are given in section VIII.9 of Brow... | 8 | https://mathoverflow.net/users/8103 | 89698 | 53,000 |
https://mathoverflow.net/questions/89682 | 2 | Let us consider an embedding of smooth manifolds $i: (Y, \partial Y) \rightarrow (X, \partial X)$. It is neat (see Hirsh, "Differential topology") if $i(\partial Y) = i(Y) \cap \partial X$ and, for every $y \in \partial Y$, there exists a boundary chart $(U, \varphi)$ of $X$ in $i(y)$ such that $U \cap i(Y)$ is the cou... | https://mathoverflow.net/users/10758 | Neat maps between manifolds with boundary | Regarding your question 2, yes all maps of pairs $(Y,\partial Y) \to (X, \partial X)$ are homotopic to neat maps. There are many ways to prove it but it boils down to a collar construction. I would be surprised if this (or something very similar to it) isn't in Hirsch, since he proves many similar things in that text. ... | 7 | https://mathoverflow.net/users/1465 | 89702 | 53,004 |
https://mathoverflow.net/questions/89697 | 2 | Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open sets is composed of sets of the form $D(I)=\{p~\mathrm{prime~in~A}:I\nsubseteq p\}$.
So, given a prime ideal, it is n... | https://mathoverflow.net/users/11546 | Topology on Set of Prime Filters of a Distributive Lattice | If you just take the basis of sets $D(I)$ that you gave for the space of prime ideals and transport it via the bijection you gave, you obviously get a basis for the space of prime filters. It consists of the sets $M(I)=\{p \text{ prime filter}:p\cap I\neq\varnothing\}$. Clearly, this $M(I)$ is the union, over all $a\in... | 3 | https://mathoverflow.net/users/6794 | 89703 | 53,005 |
https://mathoverflow.net/questions/89606 | 3 | We know that a finitely generated $R$-module $M$ satisfies the $(S\_n)$ condition if $$\operatorname{depth} M\_p \geq \min(n,\dim M\_p)$$ for every $p\in \operatorname{Spec}R$.
It's well known that Cohen-Macaulay rings satisfy $(S\_n)$ for all $n \geq 0$. Now is the following conclusion true:
>
> If $A$ is a quoti... | https://mathoverflow.net/users/18970 | Serre condition $(S_n)$ | This is too long for a comment, so I am writing it here. It reduces the problem to the case where $A$ can be assumed to be Cohen-Macaulay. But there is still an exercise remaining for you to do! All theorem and page numbers refer to Matsumura's *Commutative ring theory*.
Here are steps to do exercise 23.2 (p. 185). I... | 4 | https://mathoverflow.net/users/16046 | 89705 | 53,007 |
https://mathoverflow.net/questions/89707 | 15 | One can cut a manifold up along the critical levels of a Morse function and deduce something about the topology. In particular the critical points (and the connecting gradient flowlines) define a chain complex which can be used to compute homology.
A minimal Morse function on a compact manifold is a Morse function wh... | https://mathoverflow.net/users/10839 | Nonisotopic homotopy equivalent Morse functions | I no longer think 3-dimensional lens spaces are a productive strategy. What you need is to have a manifold $M$ as a level-set of the Morse function and you need a non-trivial diffeomorphism of $M$ to be pseudo-isotopic to the identity.
The idea is that roughly, between any two consecutive critical levels of your Mor... | 8 | https://mathoverflow.net/users/1465 | 89710 | 53,009 |
https://mathoverflow.net/questions/89693 | 2 | Let $k$ be a perfect field of characteristic $p>0$. For any positive integers $n$, let $W\_n(k)$ be the truncated Witt vectors of length $n$ with coefficients in $k$. For any positive integers $a,b$, is $\textrm{Ext}(W\_a(k), W\_b(k))$ as an abelian group well understood? Are there any references? Thank you in advance!... | https://mathoverflow.net/users/4191 | Extensions of truncated Witt vectors | Answer reposted from comment:
The short exact sequence $0\to W\to W\to W\_a\to 0$, where the map $W\to W$ is multiplication by $p^a$, is a projective resolution of $W\_a$ over $W$. Applying $\mathrm{Hom}\_W(-, W\_b)$ to this resolution, we get that $\mathrm{Ext}\_W(W\_a, W\_b)$ is just $W\_b/p^a W\_b = W\_{min(a, b)}... | 3 | https://mathoverflow.net/users/3847 | 89714 | 53,010 |
https://mathoverflow.net/questions/89670 | 5 | I am interested in constructing the following "counter-example" to the Banach's fixed point theorem.
Let $K=$ {$ g\in L\_1: \|g\|=1, g(\cdot)\ge0 $}.
Clearly, $K$ is not a compact and $K$ is ~~not~~ closed.
~~My~~ A question is: is it possible to construct a [*edit*] *nonexpansive* mapping $f: K\to K$ with no fi... | https://mathoverflow.net/users/7646 | Contraction mapping with no fixed point | There need not be a fixed point. First note that by composing with a conditional expectation onto the closed span of indicator functions of disjoint sets it is sufficient to build an example on $W:=\{x\in \ell\_1 : x\_i \ge 0, \sum x\_i =1\}$. Given $x\in W$, define $y=Tx \in W$ by $y\_1=0$, $y\_2 = x\_1/2$, and, for $... | 4 | https://mathoverflow.net/users/2554 | 89716 | 53,011 |
https://mathoverflow.net/questions/89725 | 5 | Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^\*(O(1))$ is very ample. But can one say something if $\phi$
is birational (but not isomorphism) to its image? Is it ample?
| https://mathoverflow.net/users/4690 | What can be said about a pullback of a very ample line bundle w.r.t birational maps? | Suppose $\phi$ is a morphism (i.e., defined everywhere) which is birational, but not an embedding. Then there are two cases:
1. $\phi$ is finite. In this case $\phi^\*\mathscr L$ is ample for any ample $\mathscr L$ on the target. An example (pretty much the only one) when this happens is if $\phi$ is the normalizati... | 11 | https://mathoverflow.net/users/10076 | 89727 | 53,015 |
https://mathoverflow.net/questions/89726 | 1 | Let $X$ be a smooth projective variety. Let $E$ be a line bundle (or, more generally,
a vetor bundle) on $X$. Are there any nice criteria for acyclicity of $E$ (that is,
for the property $H^i(X,E)=0$ for all $i>0$)? Here $E$ is considered as a coherent sheaf.
In my case the structure sheaf $O\_X$ is acyclic and $E$ i... | https://mathoverflow.net/users/4690 | Criteria for acyclicity | Search for the topic *vanishing theorems*. There are many such criteria, and perhaps this question should be community wiki?
Anyways, I'll highlight the one of the most common situations, for adjoint line bundles. Set $\omega\_X = \Omega\_X^{\dim X}$. Suppose that:
$E = \omega\_X \otimes \text{ample}$
In this ca... | 6 | https://mathoverflow.net/users/3521 | 89729 | 53,016 |
https://mathoverflow.net/questions/89735 | 0 | Does anyone know an example of a rational ruled surface $X=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-e))$ for $e\ge 0$ which admits a transitive algebraic group action? except the trivial case $\mathbb{P}^1\times\mathbb{P}^1$.
| https://mathoverflow.net/users/14854 | Homogeneous rational ruled surface | A rational ruled surface with $e>0$ has a unique irreducible curve with negative self-intersection, so any automorphism has to fix that. Therefore it cannot have a transitive automorphism group. (Actually it also has to fix the ruling, because it has to fix the cone of curves and the negative curve and the fiber of the... | 5 | https://mathoverflow.net/users/10076 | 89739 | 53,020 |
https://mathoverflow.net/questions/89718 | 4 | In what classes of fields does CSB hold? That is to say, in what classes of fields is it true that if there exist embeddings $F\to K$ and $K\to F$ then $F$ and $K$ must be isomorphic?
I know this holds for algebraically closed fields, but all of the counter-examples I've seen are variations on the same idea ($F=\over... | https://mathoverflow.net/users/21765 | Classes of fields and Cantor-Schröder-Bernstein | A trivial example of a family that satisfy CSB is the set of completions $\{\mathbb{Q}\_{v}\}\_{v}$ of $\mathbb{Q}$. They satisfy the condition that you require about not containing any ACF, however their transcendence degree is big. This is a trivial example in the sense that even if one of such fields is embedding in... | 0 | https://mathoverflow.net/users/2089 | 89757 | 53,028 |
https://mathoverflow.net/questions/89752 | 2 | Consider system of linear equations Ax=0 over $F\_2$ (field with two elements {0,1}).
Where number of variables is bigger than equations - so we have many solutions $x$.
**Question** How to estimate minimal [Hamming weight](http://en.wikipedia.org/wiki/Hamming_weight) of $x$ ($x\ne 0$) ? (I.e. minimal number of $1$ i... | https://mathoverflow.net/users/10446 | Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ? | It is known that to determine the minimal weight of a nonzero code word (i.e., the minimum distance of the code) is a hard problem.
Here is a part of the abstract of a paper by Vardy (The intractability of computing the minimum distance of a code, IEEE Information Theory, 1997):
>
> It is shown that the problem ... | 1 | https://mathoverflow.net/users/nan | 89758 | 53,029 |
https://mathoverflow.net/questions/89759 | 6 | For a commutative square of spaces (of manifolds, or of simplicial sets):
$$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & D\end{array}\right)$$ I am trying to understand when the relative homotopy of $(B,A)$ is isomorphic up to a certain degree to the one of $(C,D)$. Acco... | https://mathoverflow.net/users/2191 | (Co)homological characterization of homotopy pullbacks | I think that part of what you are looking for is from work of Eilenberg-Moore.
Suppose that all four spaces are simply-connected. On taking cohomology, you get a commutative diagram of graded-commutative rings
$$
\begin{array}{ccc}
H^\*(A)&\leftarrow &H^\*(B)\\
\uparrow & & \uparrow\\
H^\*(C)&\leftarrow& H^\*(D).
\e... | 4 | https://mathoverflow.net/users/360 | 89764 | 53,031 |
https://mathoverflow.net/questions/89765 | 6 | Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation).
Let $|M|$ be the matrix obtained by taking the absolute value of each entry of $M$. Clearly $\operatorname{rk} |M|$ can be much smaller than $r$ --- take for instance Hadamard matrices.
Howeve... | https://mathoverflow.net/users/1898 | Rank of the absolute-value matrix $|M|$ vs. rank of $M$ | I do not think much/anything can be done.
Let us leave the simple special cases of rank $M$ equal $0$ or $1$ aside.
So, an example of a $n$ times $n$ rank two matrix $M$ such that the rank of $|M|$ is full:
Take the two vectors $e=(1, \dots, 1)$ and $u = (0, -1, -2, \dots, -(n-1))$.
Consider the matrix $M$ form... | 6 | https://mathoverflow.net/users/nan | 89768 | 53,034 |
https://mathoverflow.net/questions/89381 | 11 | Let $s\_1, s\_2 \in (1/2,1\rbrack$. I would like to bound the product
$$A=\prod\_p \left(1 + \frac{p^{-s\_1} p^{-s\_2}}{(1-p^{-s\_1}+p^{-1}) (1-p^{-s\_2}+p^{-1})}\right)$$
Now, I am almost positive that $$A\leq \frac{\zeta(s\_1+s\_2) \zeta(2 s\_1+ s\_2) \zeta(s\_1+2s\_2)}{\zeta(s\_1+2) \zeta(s\_2+2) \zeta(4)}$$ Is... | https://mathoverflow.net/users/398 | Bounding Euler products (or almost) by products of zeta functions | Following up on Boris's suggestion, let me tell of my mostly happy experience with QEPCAD.
First of all - QEPCAD seems to crash on three variables (at least for the slightly hairy expressions we are dealing with here). So we have to start by reducing our problem to a two-variable problem by means of human.
The ineq... | 6 | https://mathoverflow.net/users/398 | 89772 | 53,037 |
https://mathoverflow.net/questions/89761 | 1 | Given a locally compact group $G$ with a compact subgroup $K$.
Assume we are given two irreducible, infinite dimensional, admissible representations $\pi$ and $\pi'$ of $G$.
>
> What are examples, where $Res\_K \pi$ and $Res\_K \pi'$ are not isomorphic $K$ representation except for a finite dimensional represent... | https://mathoverflow.net/users/10400 | Restriction of irreducible representations | If Archimedean local fields are ok, then the simplest example probably occurs with $G=GL(2, \mathbb R)$ and $K=SO(2, \mathbb R).$ The irreducible representations of $K$ are in bijection with the integers. One can construct representations of $GL(2, \mathbb R)$ such that the set of $K$-types is the set of odd integers, ... | 5 | https://mathoverflow.net/users/21252 | 89776 | 53,040 |
https://mathoverflow.net/questions/89785 | 4 | I am asking about good references (both books and papers) for the well-known Borel–Weil theorem. Thank you very much!
| https://mathoverflow.net/users/20783 | Borel–Weil theorem - reference request | J.P. Serre: "Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)", Séminaire Bourbaki (Paris: Soc. Math. France) 2 (100), 1995, 447–454.
J. Tits: "Sur certaines classes d'espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. Mém. Coll. 2... | 4 | https://mathoverflow.net/users/14653 | 89789 | 53,045 |
https://mathoverflow.net/questions/89778 | 1 | Background/Motivation
---------------------
I'm currently interested in the duality theorem for projective varieties and more specifically in properties of the conormal variety over the dual variety.
Let $V$ be a $k$-vector space over a field of characteristic 0. Denote by $V^\ast$ its dual space. Note that a point... | https://mathoverflow.net/users/21778 | In what sense is a generically submersive morphism of varieties subermersive over singular points? | Negative answer to first question: let $C \subset \mathbb{A}^2$ be the plane curve (in characteristic $\neq 2,\,3$) with equation $y^2=x^3$, with singular point $Q=(0,0)$, and let $p:\mathbb{A}^1\to C$ be the normalization morphism $t\mapsto (t^2,t^3)$.
Now take $X=\mathbb{A}^2$, $Y=C\times \mathbb{A}^1$, and $f: X\to... | 2 | https://mathoverflow.net/users/7666 | 89793 | 53,048 |
https://mathoverflow.net/questions/89802 | 1 | I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron.
Anyone has any clue how to approach this problem?
| https://mathoverflow.net/users/21785 | Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron | What do you mean by "corresponding sides"? If what you mean that you have a gluing diagram which is consistent, just take your triangles $ABC, ABD, ACD, BCD$ in such a way that the angles at $A$ in all three triangles sharing that vertex is $5\pi/6,$ and otherwise the three triangles with vertex at $A$ are isosceles (s... | 1 | https://mathoverflow.net/users/11142 | 89803 | 53,051 |
https://mathoverflow.net/questions/89805 | 7 | Suppose I have a quasiprojective variety $X$ and a finite surjective map
$$f: X \rightarrow Y$$
to a scheme $Y$. Is it true that $Y$ is quasiprojective as well? It seems like the answer could be no, but I don't know enough examples of non-projective schemes.
| https://mathoverflow.net/users/21787 | Finite map from quasi-projective variety | This isn't true in general.
For example, see Section 6 of *Conducteur, Descente et Pincement* by D. Ferrand. There Ferrand gives an example of a non-normal proper variety $Y$ whose normalization is projective.
If I recall correctly, many examples of proper non-projective schemes have finite maps from projective o... | 12 | https://mathoverflow.net/users/3521 | 89808 | 53,053 |
https://mathoverflow.net/questions/89774 | 5 | Hey all,
I try to understand an argument in Lücks "A basic introduction to surgery theory" on page 51 which goes as follows:
Let $\mathbb{Z} \pi$ be the group ring where $\pi$ denotes the fundamental group of a finite connected CW-Komplex. Furthermore we regard the zellular $\mathbb{Z}\pi$-chain complex $C\_\*(\wid... | https://mathoverflow.net/users/21779 | Chain Homotopy classes as n-homology of a double complex | Part of the problem is that you're probably only used to seeing chain homotopies of degree zero chain maps. In this case $n=1$ (because the homotopy itself raises degree by 1 and so is an element of $Hom^1$) and the formula reduces to the standard formula for a chain homotopy that you'd find, for example, in Hatcher or... | 3 | https://mathoverflow.net/users/6646 | 89809 | 53,054 |
https://mathoverflow.net/questions/89753 | 10 | Let $A$ be a commutative $\mathbb{C}$-algebra, and consider $C\_{\bullet}(A,A)$ the simplicial Hochschild homology module of $A$ with respect to itself (i.e. $C\_{n}(A,A)=A^{\otimes (n+1)}$). This is a simplicial commutative $\mathbb{C}$-algebra and we can take its $Spec$ levelwise, to get a cosimplicial $\mathbb{C}$-s... | https://mathoverflow.net/users/21710 | Geometric realization of Hochschild complex | Nice question. I did this computation a while ago, but I guess what you get, by taking the levelwise analytification of $X\_A$, is the canonical cosimplicial model for the (topological) free loop space of $(SpecA)^{top}$ (i.e. its totalization is homeomorphic to the free loop space $L((SpecA)^{top})$).
| 4 | https://mathoverflow.net/users/16882 | 89814 | 53,056 |
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