parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/118913 | 4 | The motivation for this question is that I am working through an exercise to force the GCH (generalized continuum hypothesis) over a model of ZFC and obtain a model of ZFC where GCH holds.
The forcing is an ORD length iteration of $Add(\gamma^+, 1)$ at every cardinal $\gamma$ ($\gamma$ is a cardinal in $V^{P\_{\gamm... | https://mathoverflow.net/users/nan | The canonical forcing of the GCH and direct limits. | Let me take the case of $\aleph\_\omega$ as a specific example. Similar considerations apply at other singulars.
If you want $\sigma$-closure above $\omega\_1$, you need the full support at $\aleph\_\omega$; bounded support (i.e., direct limit) will not be closed enough.
Also: You need not worry about collapsing $... | 3 | https://mathoverflow.net/users/14915 | 118921 | 67,086 |
https://mathoverflow.net/questions/118849 | 15 | **Main question.** Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega\_{X\_{\mathbf Q}}$ of the generic fibre $X\_{\mathbf Q}$ is ample?
Replacing "relative dimension two" by "relative dimension one", the answer is negative by a theore... | https://mathoverflow.net/users/4333 | Is the set of surfaces over Spec Z with ample canonical sheaf empty | I don't think the answer to the first question is known.
Will has already pointed out the trivial answer to the second question. However this is not the right question. I mean this is kind of trivial. The interesting question is if you fix the genus and require that the curve over $K$ has good reduction everywhere (... | 9 | https://mathoverflow.net/users/10076 | 118924 | 67,089 |
https://mathoverflow.net/questions/118886 | 8 | We write $cl$ for the commutator length, i.e. the least number of commutators which multiply to a given element of a group.
Given an element $g$ in the commutator subgroup of the free group $G=F\_2$ on two generators, is it true that
$$cl\_G(g) = \displaystyle \max\_{\mbox{H < G finite index normal}} cl\_{G/H} (g \... | https://mathoverflow.net/users/30644 | Commutator length modulo finite index subgroups | I think the answer is no and that actually the "finite quotient" commutator length (defined by your formula) is bounded on $[F\_2,F\_2]$.
Indeed by Nikolov-Segal, in the profinite completion $P$ of $F\_2$, the derived subgroup $[P,P]$ is closed; since the set $C$ of commutators is compact, it follows by a Baire argum... | 11 | https://mathoverflow.net/users/14094 | 118926 | 67,091 |
https://mathoverflow.net/questions/72794 | 12 | Let $\mathfrak{A} = \langle A, \dots \rangle$ and $\mathfrak{B} = \langle B, \dots \rangle$ be structures for a signature $\mathscr{L}$. For each ordinal $\gamma$ we define a game of perfect information between two players, Spoiler and Duplicator:
>
> For each $\xi < \gamma$ Spoiler picks an element from $A$ or $B$... | https://mathoverflow.net/users/8547 | What are some other uses for Ehrenfeucht-Fraïssé games? | These games have many uses. I think they're a lot of fun, but proofs that Duplicator has a winning strategy tend to get tedious quickly. Several applications are given (either in the body of the text or as exercises) in *Elements of Finite Model Theory* by Libkin. One application of E-F games that I like: to show that ... | 5 | https://mathoverflow.net/users/5497 | 118936 | 67,096 |
https://mathoverflow.net/questions/118939 | 9 | I need a simple commutative algebra lemma for a paper, but can't find a reference. Maybe I don't know the right keywords. Here's the setup. $F:K$ is a field extension, $A$ is an algebra over $K$, and $M$ an $A$-module. If $M \otimes\_K F$ is a free $A \otimes\_K F$ module, must $M$ have been a free $A$-module?
I can ... | https://mathoverflow.net/users/13910 | If M is not a free A-module, can tensoring with a bigger field make it free? | Yes, there are such examples even with invertible $M$ and smooth $K$-algebras of dimension 1, with $K$ any field that is not algebraically closed.
Choose such a $K$, so we may and do also choose a nontrivial primitive finite extension $F$ of $K$. Consider the projective line over $K$ and remove a closed point $\xi$ ... | 12 | https://mathoverflow.net/users/30180 | 118942 | 67,098 |
https://mathoverflow.net/questions/118930 | 8 | Let $X$ be a smooth projective algebraic variety over an algebraically closed field $k$. If $k=\mathbb{C}$, we know by work of Kuranishi that the base of the versal deformation of $X$ is the germ at $0$ of the fiber over $0$ of a holomorphic map $K:H^1(X, T\_X)\to H^2(X, T\_X)$ (defined in the neighborhood of 0), calle... | https://mathoverflow.net/users/3847 | Algebraic definition of the Kuranishi map | You can look at Manetti's paper [Deformation theory via differential graded Lie algebras](http://arxiv.org/abs/math/0507284), arXiv:math/0507284.
As the title suggest, it follows the philosophy that every deformation problem is governed by a DGLA, via solution of Maurer-cartan equations (module gauge action).
One o... | 8 | https://mathoverflow.net/users/7460 | 118951 | 67,102 |
https://mathoverflow.net/questions/118960 | 5 | Let us consider the following configuration of hyperplanes in the real
vector space V with coordinates $z\_1,\ldots,z\_n$: the hyperplanes are
numbered by all the nonempty subsets $J\subset I=\{1,\ldots,n\}$, and the
hyperplane $H\_J$ is given by $\sum\_{i\in J}z\_i=0$.
Question: How many connected components does th... | https://mathoverflow.net/users/22180 | connected components of a real hyperplane arrangement | This is not known as far as I know, and seems to be a hard problem, see the references and comments in
[this MO question](https://mathoverflow.net/questions/62764/a-natural-refinement-of-the-a-n-arrangement-is-to-consider-all-2n-1-hyperpla).
| 6 | https://mathoverflow.net/users/730 | 118970 | 67,111 |
https://mathoverflow.net/questions/118941 | 0 | In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have problem but the question is with the ordered structure.
| https://mathoverflow.net/users/30267 | Direct limit of lattice-ordered groups | I assume "$\ell$-group" abbreviates "lattice-ordered group"; actually, all I need is that it means algebras in some variety. In any variety, every algebra is the direct limit of its finitely generated subalgebras.
| 6 | https://mathoverflow.net/users/6794 | 118973 | 67,112 |
https://mathoverflow.net/questions/118978 | 6 | It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences.
The counter-example usually given is the Folner sequence $F\_N=\{N^2,N^2+1...,N^2+N\}$, however I've only seen it referenced to the first such result by Akcoglu and del Junco in... | https://mathoverflow.net/users/18698 | Failure of the Pointwise Ergodic Theorem | Bellow, Alexandra(1-NW); Jones, Roger(1-DPL); Rosenblatt, Joseph(1-OHS)
Convergence for moving averages.
Ergodic Theory Dynam. Systems 10 (1990), no. 1, 43–62.
| 8 | https://mathoverflow.net/users/11054 | 118980 | 67,115 |
https://mathoverflow.net/questions/118988 | 9 | What are the automorphisms of $SL\_n$ as an algebraic variety?
In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL\_n$ regarded as an *algebraic variety* over $k$. Assume that $\tau$ takes the unit element $e$ of $G$ to itself. Is i... | https://mathoverflow.net/users/4149 | Automorphisms of $SL_n$ as a variety | The coordinate ring when $n=2$ is $A=k[a,b,c,d]/(ad-bc-1)$.
If $f\in k[b,c]$, there is an automorphism $\phi:A\to A$ such that $\phi(a)=a+bf$, $\phi(c)=c+df$, $\phi(b)=b$ and $\phi(d)=d$.
One could conjecture that the automorphism group in this case is generated by $SL\_2$, inversion and this sort of triangular a... | 9 | https://mathoverflow.net/users/1409 | 118992 | 67,121 |
https://mathoverflow.net/questions/118993 | 6 | It is well-known, that the moduli space $\mathcal M\_{1;1}$ of elliptic curves is isomorphic to an orbifold space $(S\_3\times S\_2) \backslash\backslash \mathcal M\_{0;4}$, where the first factor of the group acts by permutation of the first three distinguished points on a rational curve, and the second one acts trivi... | https://mathoverflow.net/users/13921 | Moduli space of genus 1 curves with two fixed points | Yes, it is.
See Leila Schneps, [*Special loci in moduli spaces of curves*](http://www.math.jussieu.fr/~leila/articles.html) (in Galois Groups and Fundamental Groups, MSRI series **41**, Cambridge University Press, 2003), pages 34-35.
| 6 | https://mathoverflow.net/users/7460 | 118997 | 67,124 |
https://mathoverflow.net/questions/118940 | 5 | I'm trying to understand some aspects of Oguiso's example of a Calabi-Yau threefold of Picard number 2, described in the paper ["Automorphism groups of Calabi-Yau manifolds of Picard number two"](http://arxiv.org/abs/1206.1649).
Take $X \subset \mathbb P^3 \times \mathbb P^3$ to be the intersection of general hypersu... | https://mathoverflow.net/users/30660 | Counting contracted curves | I set up the enumerative computation in a manner similar to Serge Lvovski, but I also get a different answer than the one in Oguiso's paper. Serge Lvovski's answer gives $4\times 4\times 6=96$ rather than $8^3$. My answer gives $120$, rather than $8^3$.
The discrepancy may be due to the same issue as in my previous com... | 4 | https://mathoverflow.net/users/13265 | 119001 | 67,127 |
https://mathoverflow.net/questions/118983 | 8 | Let $K$ be a local field with residue field of char $p$, denote $G$ its Galois group. Is it possible that we have two Abelian varieties $A\_1$ and $A\_2$, defined over $K$, such that they are not isogeny （over $K$ or $\bar{K}$ ）, but have isomorphic p-adic Galois representation of $G$?
| https://mathoverflow.net/users/3945 | Diferent abelian varieties over local field with the same p-adic representation? | Yes, this can happen. Here is a counterexample (which is probably not the simplest possible, but it's the one that first came to mind).
There are not very many 2-dimensional representations of the Galois group of $\mathbf{Q}\_p$ which are "crystalline" in Fontaine's sense. Fontaine's functor $\mathbf{D}\_{\operatorn... | 10 | https://mathoverflow.net/users/2481 | 119003 | 67,128 |
https://mathoverflow.net/questions/118989 | 4 | There are plenty of sources discussing dual abelian varieties, but I'm looking for a reference that discusses the construction and properties of the dual abelian scheme. I'm willing to accept general theory that the Picard scheme of an abelian scheme exists, but why is it an abelian scheme?
| https://mathoverflow.net/users/27736 | the dual abelian scheme | You can also look at chapter 6 of Mumford's "Geometric Invariant Theory", together with part of Kleiman's article in "FGA Explained" (section 9.6) for the construction of $Pic\_{X/S}^\tau$ from $Pic\_{X/S}$.
| 7 | https://mathoverflow.net/users/88 | 119007 | 67,131 |
https://mathoverflow.net/questions/118972 | 7 | If yes, how? Also, I know you can't do it for arbitrary statements about real numbers, but that's not what I'm asking, and by "real" numbers, I mean the numbers constructible from 1, -, /, and the operations mentioned in the title.
Also, I don't care about numbers that can't be constructed from said operations and co... | https://mathoverflow.net/users/25684 | Is equality of terms for "real" numbers with roots, logarithm, exponential, sin, cos, and other trigonometric operations decidable with a Turing-machine? | Assuming [Schanuel's conjecture](http://en.wikipedia.org/wiki/Schanuel%27s_conjecture), the answer seems to be yes, according to Daniel Richardson, "How to recognize zero" *J. Symbolic Comput* **24** (1997), 627–645 ([doi:10.1006/jsco.1997.0157](http://dx.doi.org/10.1006/jsco.1997.0157), available [here online](http://... | 13 | https://mathoverflow.net/users/17064 | 119013 | 67,134 |
https://mathoverflow.net/questions/118946 | 8 | In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his [*Morse Theory from an algebraic viewpoint*](http://www.maths.ed.ac.uk/%7Eaar/papers/skoldberg.pdf). I'm going through the paper and am having some difficulties. I'd be mos... | https://mathoverflow.net/users/11317 | Algebraic Morse theory | It's always nice to see people working on discrete Morse theory.
**Answer 1**
It is an "if and only if". Meaning: the partial order $\prec$ is *defined* by $\alpha \prec \gamma$ if and only if $\gamma$ precedes $\alpha$ in a path of the matching. The idea goes back to Forman's "[Morse theory for cell complexes](htt... | 5 | https://mathoverflow.net/users/18263 | 119014 | 67,135 |
https://mathoverflow.net/questions/119010 | 6 | Let us assume ZFC and let Q be the set of rational numbers ordered according to size. There is a
well known theorem which implies that if S is any totally ordered countable set containing a subset
ordinally similar to Q, then S contains well ordered subsets having arbitrarily large countable
ordinal numbers. Is there a... | https://mathoverflow.net/users/4423 | A question about well ordered subsets of totally ordered countable sets | If a linear order doesn't contain a copy of $\mathbb Q$, then, by a theorem of Hausdorff, it can be obtained by a transfinite sequence of steps, starting with singletons, and at each step forming well-ordered or reverse-well-ordered sums of previously constructed orderings. If the final result is to be a countable set,... | 12 | https://mathoverflow.net/users/6794 | 119016 | 67,136 |
https://mathoverflow.net/questions/118864 | 0 | How i can write general relativity Lorentz connection ( $A^{a} \_ {\ \ bc}$ ) in term of structure coefficients ( $f^a\_{\ \ bc}$ ) ? $\ \ $
I know the relation:
$A^a\_{\ \ bc}$=$1/2$( $f^{\ a} \_ {b\ c}$ +$f^{\ a} \_ {c\ b}$ -$f^{a}\_{\ \ bc} )$
But i couldn't derive it !
Any References ?
| https://mathoverflow.net/users/30480 | Lorentz connection | Use the Koszul formula in a orthonormal frame (see O'Neill B. "Semi-Riemannian geometry", AP, 1983, Chapter 3, Theorem 11).
| 1 | https://mathoverflow.net/users/30598 | 119022 | 67,141 |
https://mathoverflow.net/questions/69493 | 3 | Hi,
After the response to the following question, [Rational roots to quadratic forms in 4 variables](https://mathoverflow.net/questions/69335/rational-roots-to-quadratic-forms-in-4-variables), I am now considering the following question.
Let $d \geq 2$ be a positive integer, and suppose that $F(x\_1, x\_2, x\_3, x\... | https://mathoverflow.net/users/10898 | Integral roots to degree $d$-forms in four variables inside a box | There is a result of the form you want in the literature. Take a look at Theorem 4 in Heath-Brown's paper *The density of rational points on curves and surfaces* (Annals of Math., **155** (2002), 553–595). An easy generalisation of this gives the following. Let $G\in \mathbb{Z}[X\_0,\ldots, X\_n]$ be a primitive form o... | 4 | https://mathoverflow.net/users/30668 | 119026 | 67,142 |
https://mathoverflow.net/questions/118971 | 1 | Let $E$ be a globally generated vector bundle on a surface $S$ of rank $r\geq 2$. By standard facts about degeneracy loci, for a general $V\in G(r,H^0(E))$ one has:
(\*)the evaluation map $ev: V\otimes \mathcal{O}\_S\to E$ is injective and the cokernel is a line bundle supported on a smooth curve.
Now, let $E\_1$ ... | https://mathoverflow.net/users/33841 | Globally generated subvector bundles and evaluation maps. | No.
Suppose that $E$ has rank $2$, and write $V\_1=V\cap H^0(E\_1)$. Then you have an exact sequence
$$
0\to coker(V\_1\otimes \mathcal O\_S\to E\_1)\to coker(V\otimes \mathcal O\_S\to E)\to coker((V/V\_1)\otimes \mathcal O\_S\to E/E\_1)\to 0.
$$
If the line bundles $E\_1$ and $E/E\_1$ are such that every divisor co... | 0 | https://mathoverflow.net/users/6666 | 119028 | 67,143 |
https://mathoverflow.net/questions/118962 | 12 | A Riemannian manifold $(M,g)$ is said to have flat ends if the curvature tensor of $g$ vanishes outside a compact set $K$. I was wondering if such manifolds are of bounded geometry. Recall that a manifold is of bounded geometry if
1. The curvature tensor and all its covariant derivatives are uniformly bounded.
2. The... | https://mathoverflow.net/users/12156 | Is a manifold with flat ends of bounded geometry? | Here is a self-contained proof not using any classification. With some effort, it can be made to work under weaker assumptions: the curvature is nonnegative outside a compact set and is bounded from above. It is a variation of the proof of the Soul Theorem via Sharafutdinov's retraction.
Fix a reference point $o\in M... | 14 | https://mathoverflow.net/users/4354 | 119033 | 67,145 |
https://mathoverflow.net/questions/119032 | 9 | I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered papers talking about Wilson lines, boundary conditions, surface defects and so on. I looked up these terminology but I co... | https://mathoverflow.net/users/nan | Relation between TQFT and Wilson lines, boundary conditions, surface defects etc | Greg Moore recently gave the Felix Klein lectures and a draft of notes for his lectures is available at
<http://www.physics.rutgers.edu/~gmoore/FelixKleinLectureNotes.pdf>
You will find in the first few pages a discussion of (extended) TQFT, defects, Wilson lines and so on in a language which I imagine is more su... | 7 | https://mathoverflow.net/users/10475 | 119034 | 67,146 |
https://mathoverflow.net/questions/119030 | 5 | I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional time dependent ODE on the disk, forward in time. Let us assume that the ODE system (and hence the resulting diffeomorphi... | https://mathoverflow.net/users/30684 | A theory of bifurcation of braids ? | Here is a couple of references regarding the holomorphic world, although they do not represent an answer to your question per se (but I'm afraid this is too long a comment, and might anyhow interest people interested in your question).
First there is [a work](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&c... | 3 | https://mathoverflow.net/users/24309 | 119036 | 67,148 |
https://mathoverflow.net/questions/119031 | 6 | Let $G$ be a finite group. Then it is well known that a function $f\colon G\to \mathbb C$ is a linear combination of irreducible characters iff it is constant on conjugacy classes. What is the corresponding result for an arbitrary field $\Bbbk$, i.e., which $\Bbbk$-valued mappings on $G$ are $\Bbbk$-linear combinations... | https://mathoverflow.net/users/15934 | Which functions are linear combinations of irreducible characters for a given field $\Bbbk$? | I am interpreting your question as talking of $k$-linear combinations of traces of $k$-representations of $G.$ Note that such a function must not only be constant on conjugacy classes, but should also be constant on $p^{\prime}$-sections, where $k$ has characteristic
$p.$ Recall that every element of $G$ may be written... | 4 | https://mathoverflow.net/users/14450 | 119038 | 67,149 |
https://mathoverflow.net/questions/119037 | 1 | Is the statement below false?
"The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations."
---
Possible "counterexample":
Sp(2n,R) is a subgroup of O(4n,C) (or O(2n,2n) if you prefer).
So the Clifford algebraic Pin group will contain a double cover. The double cov... | https://mathoverflow.net/users/30685 | Is the metaplectic group not a matrix group - counterexample | Keep in mind that any finite-dimensional representation of a Lie group determines a finite-dimensional representation of its Lie algebra, and for a connected Lie group the induced Lie algebra representation determines the Lie group representation.
However, every finite-dimensional representation of $\operatorname{Lie... | 8 | https://mathoverflow.net/users/78 | 119040 | 67,151 |
https://mathoverflow.net/questions/110721 | 4 | Suppose that $A$ is an arbitrary fixed $n\times n$ matrix and $G$ a random $n\times n$ matrix with i.i.d. $N(0,1)$ entries. Is there a simple proof that $A+G$ is invertible with probability 1?
What if $G$ is a random Wigner matrix (symmetric, upper diagonal entries are i.i.d. $N(0,1)$)? Is $A+G$ still invertible with... | https://mathoverflow.net/users/14432 | invertibility of a matrix with a Gaussian perturbation | We use the idea suggested by [Alekk](https://mathoverflow.net/users/1590/alekk).
Let $A\_{i,j}$ the entries of $A$. Then in the first case, the entries of $M:=A+G$ are Gaussian independent random variables, that is, $M\_{i,j}\sim N(A\_{i,j},1)$. Denote $N$ the set of elements of $\Bbb R^{n^2}$ such that the matrix of... | 4 | https://mathoverflow.net/users/17118 | 119052 | 67,154 |
https://mathoverflow.net/questions/119050 | 8 | **Fact:** For any (continuous) $S^1$-action on the closed unit disk $\mathbb{D}^n$, there is a fixed point $x\_0\in\mathbb{D}^n$.
I have thought of a possible argument that re-proves this, but am not sure how to complete it:
Let $U\_p\subset S^1$ be the subgroup of $p^\text{th}$-roots of unity ($p$ prime). An $S^1... | https://mathoverflow.net/users/12310 | Fixed point of $S^1$-action using roots of unity | You might have a look at Chapter VI of Borel's "Seminar on transformation groups". This is the chapter on "Isotropy groups of toral actions" by E. E. Floyd.
In particular Theorem VI.1.2 seems to be saying that the fixed point sets eventually stabilize. (I can give more details if you don't have the reference to hand.... | 6 | https://mathoverflow.net/users/8103 | 119055 | 67,156 |
https://mathoverflow.net/questions/118314 | 12 | In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian manifold $M$, one has the formula
$$ \mathrm{Tr} P = \int\_{T^\*M} \sigma(P), $$
where $\sigma(P)$ is the full symbol of $P$... | https://mathoverflow.net/users/16702 | Trace formula for PSDOs | Consider the modified question: How does one define the operator $P=\mathrm{Op}(p)$ so that the trace formula holds with $\sigma(P)=p$ as the full symbol? The following quantization rule answers this with the help of the exponential map of the compact Riemannian manifold $M$, $n=\dim M$:
$$Pu(x)=(2\pi)^{-n}\int\_{T\_x^... | 8 | https://mathoverflow.net/users/nan | 119061 | 67,159 |
https://mathoverflow.net/questions/119059 | 14 | Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?
| https://mathoverflow.net/users/nan | What is the history of $\sqrt{}$ | We use "\sqrt" (in TeX) since Knuth chose that command for the square root symbol.
Square roots have been computed since the days of the Babyloneans, but they didn't use the symbol.
The symbol √ for the square root was first used in print in 1525 in Christoph Rudolff's Coss.
See <http://en.wikipedia.org/wiki/Squa... | 8 | https://mathoverflow.net/users/30364 | 119063 | 67,160 |
https://mathoverflow.net/questions/119066 | 17 | If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an invertible matrix A, is there an "explicit" way to recover/write down A?
If N=3 it is easy, as you can get the determi... | https://mathoverflow.net/users/24152 | 2x2 subdeterminants of a matrix | By "corresponding submatrices" I presume you mean those $2\times2$ minors obtained by deleting $n-2$ colums and $n-2$ rows, where these columns and rows have the same $n-2$ indices. Once you've calculated the determinants of these submatrices you recover the action of $A$ on the exterior square $\Lambda^2 V$.
Now the... | 21 | https://mathoverflow.net/users/801 | 119071 | 67,161 |
https://mathoverflow.net/questions/119067 | 3 | This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice of $C([0,1])$. Note that, by virtue of the formula $2(f\vee g) = f+g+|f-g|$, such a $V$ must contain an element $h$ such ... | https://mathoverflow.net/users/20300 | Banach lattice subspace of $C([0,1])$ not a sublattice | Yeah, it's easy, take $V$ to be the set of linear functions $ax + b$. Any $f$ and $g$ in $V$ have a least upper bound which is the line from $\max(f(0),g(0))$ to $\max(f(1),g(1))$. The sup norm of $f \vee -f$ equals the sup norm of $f$, so it's a Banach lattice.
| 6 | https://mathoverflow.net/users/23141 | 119074 | 67,162 |
https://mathoverflow.net/questions/119069 | 2 | Hi there!
Let $X$ be a left $G$-set, and $\Delta=${$x\_1,\ldots,x\_n$} a fundamental domain of $G$ in $X$. In other words, $G$ acts on $X$ from the left, and {$Gx\_1,\ldots,Gx\_n$} is the orbit space $X/G$.
Let us call $H\_i$ the stabilizer of $x\_i$, for all $i=1,2,\ldots, n$. Then the set $G/{H\_i}$ of left $H\_i... | https://mathoverflow.net/users/22606 | Is any $G$-set a coset geometry (in the sense of Tits-Buekenhout)? | Any $G$-set with $n$ orbits has a rank $n$ coset geometry structure that is unique up to relabeling. Indeed, there is computer algebra software (e.g., GAP, Magma) that will produce such a structure from a list of stabilizers of orbits.
Some Googling reveals that Tits came up with coset geometries, so it is likely tha... | 4 | https://mathoverflow.net/users/121 | 119076 | 67,163 |
https://mathoverflow.net/questions/118874 | 5 | To what extent is the structure theorem for finitely generated modules over principal ideal domains true over non-commutative domains? I'm in particular interested in non-commutative euclidean domains especially the twisted polynomial ring $K\langle X \rangle$ over a field $K$ (i.e. such that $Xa = \sigma(a)X$ for some... | https://mathoverflow.net/users/26756 | Structure of f.g. modules over a non-commutative ring | The question is thoroughly explored in Chapter 3 of Nathan Jacobson's *Theory of Rings*. I took a quick look, and it looks like the analogous results go through in the noncommutative case. For example, Theorem 19 in Chapter 3 states that a finitely-generated module over a noncommutative principal ideal domain is a dire... | 3 | https://mathoverflow.net/users/3711 | 119083 | 67,166 |
https://mathoverflow.net/questions/119041 | 12 | Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a\_1 \le a\_2 \le \ldots \le a\_k \le 1$ such that $\sum\_{i=1}^k (-1)^{k+1} f(a\_k) = f(a\_1) - f(a\_2) + f(a\_3) - \ldots = \frac{f(0)}{2}$?
When $k=1$ we have to find one point... | https://mathoverflow.net/users/17016 | A generalization of intermediate value theorem on R^k | The statement is true. It is almost precisely Lemma 2 in the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210. The proof is short but not easy to invent. The paper can be read on Google Books [here](http://books.google.com/books?id=8f8tky1MoisC&pg=PA207).
Notes on the text: the intervals in t... | 15 | https://mathoverflow.net/users/4354 | 119092 | 67,170 |
https://mathoverflow.net/questions/119095 | 13 | Let $A$ be a subset of natural numbers. Consider the following problem:
**Is there a group $G$ such that $\lbrace O(x) \; | \; x \in G \rbrace = A\cup\lbrace 1\rbrace$ ? (where $O(x)$ is the order of $x$)**
If you know any reference concerning this problem or any partial solution (containing a necessary or suffici... | https://mathoverflow.net/users/nan | The set of orders of elements in a group | Obviously not for every set $A \subset \mathbb{N}$ there is a group $G$ with $A$ as
set of orders of its elements (usually called 'spectrum') -- for example if $G$ has an element of order $n$, then $G$ also has an element of order $d$ for every divisor $d$ of $n$.
For a survey of what is known on this question, you m... | 13 | https://mathoverflow.net/users/28104 | 119097 | 67,173 |
https://mathoverflow.net/questions/119084 | 5 | Assume having a monoidal category $(\mathcal{A},I,\otimes)$ where each object has an inverse, i.e. for every object $A$ there is an object $\bar{A}$ with $A\otimes\bar{A}=\bar{A}\otimes A=I$. Then I wonder: If $f:X\rightarrow Y$ is an arrow, is $f$ invertible with respect to $\circ$ if and only if $f$ is invertible wit... | https://mathoverflow.net/users/27923 | Vertical and Horizontal Isomorphisms in 2-categories | Before answering, let me remark that it's generally considered a little weird (some would say, with tongue partly in cheek, ["evil"](http://ncatlab.org/nlab/show/principle+of+equivalence)) to posit categorical axioms which enforce equalities between objects, since models of such axioms will not be invariant with respec... | 10 | https://mathoverflow.net/users/2926 | 119100 | 67,175 |
https://mathoverflow.net/questions/119088 | 3 | Reduction of an ideal is an important method on commutative algebra. Let $R$ be a Noetherian ring, $J \subseteq I, J \neq I$ two ideals of $R$. Then $J$ is a reduction of $I$ is there exists $k$ such that $I^{k+1} = I^kJ$. Now I interested in a similar condition $J^{k+1} = J^kI$.
**Discussion:** if $\mathrm{ht}(J)>0$... | https://mathoverflow.net/users/17901 | reduction of an ideal | The idea you are flirting with is the ``Ratliff-Rush operation''. Given a *regular* ideal $J$ (which is a special case of an ideal with positive height), Ratliff and Rush define $\tilde{J} := \bigcup\_{n=0}^\infty (J^{n+1} : J^n)$.
Then they show that $\tilde{J}$ is the *largest* ideal among ideals $I$ such that $I^... | 2 | https://mathoverflow.net/users/19045 | 119108 | 67,177 |
https://mathoverflow.net/questions/119096 | 13 | It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ that divides the permanent of such a matrix.
Note that if $A$ is an odd $n\times n$ matrix, then $\det (A)\equiv 0 (\mod ... | https://mathoverflow.net/users/20748 | Permanent of a matrix of odd integers | Let us consider $s$ with $2^s\leq n<2^{s+1}$. First we prove the conjecture when all the entries of $A$ are $1$'s. Then $\mathrm{perm}(A)=n!$, hence by [Legendre's formula](http://www.cut-the-knot.org/wiki-math/index.php?n=Arithmetic.LegendresFormula) the exponent of $2$ in it equals $n-t$, where $t$ is the number of $... | 10 | https://mathoverflow.net/users/11919 | 119112 | 67,179 |
https://mathoverflow.net/questions/119118 | 13 | Suppose that $N$ is a totally geodesic submanifold of a complete Riemannian manifold $(M,g)$. Is it the case that a geodesic segment that minimizes length in the submanifold $N$ also minimizes length in the ambient manifold $M$?
| https://mathoverflow.net/users/14454 | Totally Geodesic Submanifolds | Let $M$ be the flat cylinder $\mathbb{R} \times S^1 \subset \mathbb{R} \times \mathbb{C}$ and $N = \{(t,e^{it}) | t \in \mathbb{R}\}$, which is a geodesic (hence a complete totally geodesic submanifold of $M$) minimizing between any two points of $N$ (among the geodesics of $N$). But the minimizing geodesic in $M$ betw... | 18 | https://mathoverflow.net/users/24152 | 119123 | 67,184 |
https://mathoverflow.net/questions/119121 | 1 | Hello,
My question is about the relation between representations of a Lie group and its quotient.
Let $G$ be a compact lie group and $H$ a central subgroup of $G$. What is the relation between representations of $G$ and those of $G/H$? Exactly, I want to know that, is any representation of $G/H$ correspond to a r... | https://mathoverflow.net/users/26148 | Representation of quotient group | Yes, if $H$ is closed (i.e. not some irrational-flow subgroup inside the closed subgroup $Z(G)$).
| 3 | https://mathoverflow.net/users/391 | 119125 | 67,186 |
https://mathoverflow.net/questions/119111 | 3 | In this question $R$ is a commutative noetherian local ring with unity.
One can construct examples of rings $R$ and *zerodivisors* $z$ such that $\dim R/(z)=\dim R-1$, e.g., $S\colon=k[a,b,c],\ \mathfrak{m}\colon=(a,b,c),\ R\colon=S\_\mathfrak{m}/(a^2,ab)S\_\mathfrak{m},\ z\colon=b^2$.
One can also construct examp... | https://mathoverflow.net/users/16046 | Can a zerodivisor reduce both the depth and the dimension? | $R = k[[a,b,c,d]]/(a,b,c)^2 \cap (c) \cap (c,d)^2 $, $\dim R = 3$ and $\mathrm{depth}R = 1$. We have $d$ is a zerodivisor. Because $R/d \cong k[[a,b,c]]/(a,b,c)^2 \cap (c)$.
So $\dim R/d = 2$ and $\mathrm{depth}R/d = 0$.
Edit: As Mahdi comment $\mathrm{depth}R/d = 2$. I repair as follows.
I need the following inter... | 8 | https://mathoverflow.net/users/17901 | 119135 | 67,189 |
https://mathoverflow.net/questions/119047 | 2 | Consider the property of a vertex $v$ of a planar graph $G$ that the [circular ordering](http://en.wikipedia.org/wiki/Rotation_system) of its edges is the same (upto orientation) for every [graph embedding](http://en.wikipedia.org/wiki/Graph_embedding) $\pi$ of $G$ into the plane $\mathbb{R}^2$.
>
> 1. Does this pr... | https://mathoverflow.net/users/2672 | Unique circular ordering of edges around a vertex | I'm assuming that the reverse of a cyclic ordering counts as the same ordering. I'm also assuming the graph is simple (otherwise subdivide edges with extra vertices to make it simple).
There is only one cyclic order for vertices of degree 1,2,3, so the interesting vertices have degree 4 or more.
3-connected graphs ... | 2 | https://mathoverflow.net/users/9025 | 119154 | 67,195 |
https://mathoverflow.net/questions/119070 | 4 | This question is in a way related to the one I posted on [math.se](https://math.stackexchange.com/questions/266444/generating-non-isomorphic-graph-by-adding-two-edges-to-a-fixed-graph). Since the question there did not produce any final answer I am trying my luck here!
I am given a fairly large graph $G$ and subsets ... | https://mathoverflow.net/users/1737 | Generating non-isomorphic graphs by adding edges to a given graph | Let me assume that Aut$(G)$ fixes $A$ and $B$ setwise, or at least the group you use to define equivalence is the setwise stabiliser of each of $A$ and $B$ (in which case use vertex colours to get this stabiliser).
Suppose recursively that you have all the non-equivalent graphs in which $k$ vertices of $A$ are match... | 3 | https://mathoverflow.net/users/9025 | 119162 | 67,200 |
https://mathoverflow.net/questions/119157 | 2 | Let $P(j,k,n)$ be the probability of getting $j$ uniform runs of length $k$ from $n$ fair coin flips. What's the best way to compute $P$? I have no idea how difficult it might be; if it's a very complicated combinatorial argument, I'm looking more to understand the mathematical tools than to actually be able to do this... | https://mathoverflow.net/users/30721 | Runs in coin flips | For large $n$, this sort of question can be tackled via concentration of measure. The idea is that the typical case should be very close to the average case, so that the probability is essentially either zero or one, depending on whether the property holds on average.
One useful tool is Talagrand's inequality. Let $\... | 4 | https://mathoverflow.net/users/25485 | 119165 | 67,202 |
https://mathoverflow.net/questions/118882 | 9 | This question is an outgrowth of this MathSE question: <https://math.stackexchange.com/questions/276068/members-of-lightface-borel-sets>.
A **Borel set** $X\subseteq 2^\omega$ is a member of the smallest collection of subsets of $2^\omega$ closed under complementation and countable union. As such, Borel sets come wi... | https://mathoverflow.net/users/8133 | Ensuring nonempty lightface Borel sets have elements via theories of second-order arithmetic | Given a tree $T \subseteq \omega^{\lt\omega}$, the statement "$B$ is an infinite path through $T$" is $\Pi^0\_2$. Therefore, if $\mathcal{M}$ is such that every nonempty $\Pi^0\_2$-class coded in $\mathcal{M}$ has a point in $\mathcal{M}$, then $\mathcal{M}$ is necessarily a $\beta$-model. So the models you want are pr... | 4 | https://mathoverflow.net/users/2000 | 119175 | 67,206 |
https://mathoverflow.net/questions/119113 | 14 | It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by an elementary argument as follows: using partitions of unity to construct a nowhere vanishing 1-form, integrate to obtain a... | https://mathoverflow.net/users/15819 | Essential uniqueness of the real-analytic structure on $\mathbb R$ | For Misha's comment and question 3 see the paper of David Minda Regular Analytic arcs and curves Colloq.Math 38(1977) no 1 73-82 .Regarding the vanishing theorem for real analytic manifolds this was proved by Henri Cartan Bulletin de la S.M.F tome 85 (1957) 77-99. Cartan assumed his manifolds were real analytically emb... | 7 | https://mathoverflow.net/users/4696 | 119201 | 67,215 |
https://mathoverflow.net/questions/119190 | 6 | Is there a known bijective proof of Ramanujan's congruence for the partition function modulo 5? E.g., is there a construction that for every $n$ congruent to 4 mod 5 gives a permutation of the partitions of $n$ that increases the Dyson rank of each partition by 1 mod 5?
| https://mathoverflow.net/users/3621 | Bijective proof of Ramanujan's congruence | It's hard to be absolutely certain of course, but I would say **no**.
The most recent reference I can find mentioning the absence of such a proof is [this article](http://www.iazd.uni-hannover.de/~bessen/bespak.ps) by Bessenrodt and Pak (2003). I also got some negative answers from modular form specialists to whom I... | 4 | https://mathoverflow.net/users/730 | 119204 | 67,217 |
https://mathoverflow.net/questions/119046 | 0 | Suppose a bipartite graph $G=(V\_1 \cup V\_2, E)$ is given, and one is interested in matching vertices $V\_1$ to vertices $V\_2$. Assume Hall's condition does not hold, so a perfect matching does not exist. What is a good lower bound on the amount of vertices in $V\_1$ which can be matched? I am not interested in algor... | https://mathoverflow.net/users/30689 | Good lower bound on matching in bipartite graph | Let $d$ denote the *deficiency* of the graph, i.e., the maximum difference between the a size of a subset of $V\_1$ and the number of its neighbors (#vertices $-$ #neighbors).
If $d \leq 0$ then the condition holds and the graph has a matching. Otherwise, there is a matching of size $|V\_1| - d$.
To see why, add to... | 4 | https://mathoverflow.net/users/27663 | 119206 | 67,219 |
https://mathoverflow.net/questions/119211 | 4 | Assume that $G$ is a group that fits into a short exact sequence
$$1 \longrightarrow F \longrightarrow G \longrightarrow \mathbb{Z}^n \longrightarrow 1$$
with $F$ a finite group. For $f \in F$ not the identity, can we find some finite-index subgroup $A\_f \subset G$ such that $f \notin A\_f$?
Actually, the answer to ... | https://mathoverflow.net/users/30735 | Separating elements in abelian by finite groups | It follows easily from the result that (for finitely generated $G$) $G'$ finite implies $|G:Z(G)|$ finite.
If that is still too sledgehammer-like, then by replacing $G$ by $C\_G(F)$ we can assume $F$ is central and hence $G$ is class 2 nilpotent. Let $|F|=k$, and let $x\_1,\ldots,x\_n$ generate $G$ modulo $F$. Since ... | 8 | https://mathoverflow.net/users/35840 | 119217 | 67,223 |
https://mathoverflow.net/questions/119150 | 8 | Let $X$ be a normal complex projective variety, let $V$ be a closed subset of $X$ (possibly reducible), and let $I\_V$ be its ideal sheaf (consider the reduced scheme structure for example).
If $A$ is an ample divisor on $X$, then $\mathcal{O}\_X(mA)\otimes I\_V$ is globally generated if $m \gg 0$ by one of the defin... | https://mathoverflow.net/users/6430 | Irreducible divisors containing an arbitrary closed set | EDIT : Olivier Wittenberg pointed out to me that a positive answer to the question follows from Theorem 1 of [Altman-Kleiman, Bertini theorems for hypersurface sections containing a subscheme]. I keep below my previous answer (whose argument is different, but more complicated).
---
The answer to your question is ... | 8 | https://mathoverflow.net/users/2868 | 119221 | 67,224 |
https://mathoverflow.net/questions/119189 | 13 | Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E\_6$, $E\_7$ and $E\_8$
1) $E\_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus \wedge^{6}V^{\star}\bigoplus \wedge^{3} V^{\star} \bigoplus \mathfrak{g}\mathfrak{l}(V)\bigoplus \wedge^{3} V\bigoplus \wedge^{6} V \bigoplus(... | https://mathoverflow.net/users/nan | decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ | The $E\_6$ and $E\_7$ decompositions you list are explained in Cartan's 1894 thesis (see pages 89–92 for these formulae). For $E\_8$, Cartan instead gives a decomposition (a $\mathbb{Z}\_3$-grading) of the form
$$
{\frak{e}}\_8 = \Lambda^3(W^\ast)\oplus {\frak{sl}}(W)\oplus \Lambda^3(W)
$$
for a $9$-dimensional space $... | 37 | https://mathoverflow.net/users/13972 | 119222 | 67,225 |
https://mathoverflow.net/questions/119213 | 5 | Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/non-zero entries. (I can expand on this if required.)
Anyway, I have a construction which appears to work (which I have... | https://mathoverflow.net/users/30734 | Systems of simultaneous real quadratic equations | Existence of real solutions of systems of algebraic equation is a difficult problem
and there is no general theory. You can get an impression of the modern state of it
from the nice book:
MR2830310
Sottile, Frank,
Real solutions to equations from geometry.
American Mathematical Society, Providence, RI, 2011.
Als... | 3 | https://mathoverflow.net/users/25510 | 119227 | 67,228 |
https://mathoverflow.net/questions/119220 | 9 | This is an extension of [this MSE](https://math.stackexchange.com/q/268091/12952) question, in which I asked whether there was a counterexample to the following statement,
>
> **Conjecture.** If a finite group $G$ contains a $\lbrace p,q \rbrace$-Hall subgroup for every pair of primes $p$ and $q$ dividing $|G|$, th... | https://mathoverflow.net/users/25494 | An extension of the converse to Hall's theorem. | For the record, I believe that P. Hall proved that if $|G|$ has $n$ prime divisors, then $G$ is solvable if and only if $G$ has $n$ Sylow subgroups $P\_{1},P\_{2}, \ldots ,P\_{n},$ one for each prime divisor, such that $P\_{i}P\_{j} = P\_{j}P\_{i}$ for $1 \leq i,j \leq n.$ You are asking whether the pairwise permutabil... | 7 | https://mathoverflow.net/users/14450 | 119245 | 67,233 |
https://mathoverflow.net/questions/119241 | 25 | While playing with something totally irrelevant I stumbled upon the recurrence:
$$a\_{n+1} = \frac{1}{a\_n} + a\_{n-1}$$
It turns out that given $a\_0 = 1, a\_1 = 1$,
$$lim \frac{a\_{2n}}{a\_{2n-1}} = \frac{\pi}{2}$$
I have a very crude idea (or rather a hint) on proving it (the iterations sort of unfold into a s... | https://mathoverflow.net/users/30747 | "Harmonacci" recurrence and identities for $\pi$ | The sequence $a\_n$ is closely related to the Wallis product
$$a'\_n = \prod\_{i = 1}^n \left(\frac{2i}{2i - 1} \frac{2i}{2i + 1}\right),$$
which converges to $\pi/2$ as $n$ goes to infinity. Namely, we have
$$a'\_n = a\_{n + 1} \cdot \frac{2n}{2n + 1}$$.
This could be proven by induction or maybe more easily by defini... | 23 | https://mathoverflow.net/users/30755 | 119251 | 67,236 |
https://mathoverflow.net/questions/119254 | 13 | Can anyone cite an example of a mathematics paper that has been retracted?
It is said that on the order of 100,000 new theorems enter the mathematics literature every year. For a number of reasons including hyper-specialization and demands on referee resources it is, in my view, unlikely that all their proofs are cor... | https://mathoverflow.net/users/4111 | Retracted Mathematics Papers | A general existence theorem is proved :
1933 : W. Grunwald, *Ein allgemeines Existenztheorem für algebraische Zahlkörper*, J. reine angew. Math. 169 (1933), 103–107.
and reproved :
1942: G. Whaples, *Non-analytic class field theory and Grünwald's theorem*.
Duke Math. J. 9, (1942). 455–473.
A counter-example i... | 49 | https://mathoverflow.net/users/2821 | 119257 | 67,238 |
https://mathoverflow.net/questions/119262 | 1 | In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A\_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such that $\newcommand{\cl}{\operatorname{cl}} \cl\_{X}{A} = \cl\_{X}({\bigcup\_{\alpha\in S}A\_{\alpha}})$.
Also, I defin... | https://mathoverflow.net/users/29349 | Example of a topological space | Not possible, in a completely regular space every open set is a union of cozero sets. Just collect the clopen sets that you use for these subordinate cozero sets.
| 4 | https://mathoverflow.net/users/23141 | 119265 | 67,243 |
https://mathoverflow.net/questions/119256 | 1 | Let $X$ be a smooth projective surface over $\mathbb{C}$ and $E$ be a vector bundle on it. Let $F\subset E$ be a normal subvector bundle of $E$, that is, the quotient $E/F$ is torsion free. Any section $s\in H^0(F)$ naturally defines a section of $E$, that I call $s\_E$. Does the zero locus of $s\_E$ always coincide wi... | https://mathoverflow.net/users/33841 | Sections of a normal subsheaf. | If you mean that you have a morphism of sheaves $\mathcal O\_X(F)\to\mathcal O\_X(E)$ such that the quotient sheaf is torsion free, then in general the answer is no.
Think at the example of a holomorphic foliation $\mathcal F$ by curves on a smooth projective surface $X$. It is given by definition by the datum of a ... | 5 | https://mathoverflow.net/users/9871 | 119268 | 67,244 |
https://mathoverflow.net/questions/119250 | 0 | My question relates to square roots of unity modulo N, ie $r^2 = 1 \mod N$.
I have an efficient algorithm for obtaining these for arbitrary $N$. But for a given $N$ what I really want is to obtain the roots for all $N\_f = \frac {N^2}{f^2}$ for all $f|N$.
My question is simply this - can these all be deduce... | https://mathoverflow.net/users/30445 | Square Roots of Unity modulo N^2 | Given the square roots of $1$ modulo $N$ you can deduce the square roots of $1$ modulo $N^2$ just by using Hensel's Lemma (without factoring). Specifically let $r$ be one of the square roots of $1$ modulo $N^2$. Then $r \equiv s \pmod{N}$ where $s$ is one of the square roots of $1$ modulo $N$. Now $r=s+\lambda N$ and y... | 5 | https://mathoverflow.net/users/4140 | 119275 | 67,245 |
https://mathoverflow.net/questions/119087 | 4 | The quotient of the affine space $\mathbb{A}^n$ by the symmetric group $Sym\_n$ is again an affine space of the same dimension, and invariants are given by elementary symmetric polynomials.
What about a $n$-cycle? More precisely, what are the generators of the invariants of the action of $\mathbb{Z}/n\mathbb{Z}$ on ... | https://mathoverflow.net/users/23758 | Quotient of affine space by cyclic permutation | Rita's answer is perfectly correct. However, Jérémy posed his question over a (not necessarily algebrically closed) characteristic $0$ field
$k=\mathbb{Q}$, so here is another approach. Consider the subring $S = k[s\_1,\dots,s\_n]$ generated by the elementary symmetric polynomials, i.e., the invariant ring under the en... | 5 | https://mathoverflow.net/users/13265 | 119284 | 67,250 |
https://mathoverflow.net/questions/119290 | 7 | I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior calculus. However, I do not know of how to reconcile the notion of a functional derivative with say an exterior derivative.... | https://mathoverflow.net/users/30745 | Formulating the calculus of varations with exterior calculus | There is a large literature on this, and the roots go back more than one hundred years. Some of the modern work along these lines can be found by looking for papers containing the term 'variational bicomplex'. For example, look at the papers and books by Ian Anderson and his group.
You can also look at papers and bo... | 11 | https://mathoverflow.net/users/13972 | 119292 | 67,254 |
https://mathoverflow.net/questions/119283 | 2 | Let $X = (X\_1,X\_2)$ and $\hat X = (\hat X\_1,\hat X\_2)$ be two random variables where $X\_i,\hat X\_i$ are taking values over the Polish space $E\_i$ endowed with their Borel $\sigma$-algebras, where $i=1,2$.
Let $X\_1$ has a distribution $\mu$ and $\hat X\_1$ has a distribution $\hat \mu$. Furthermore, suppose t... | https://mathoverflow.net/users/11768 | Coupling of vectors | 1. No, this is not true. For example, let $E\_1=E\_2=\{0,1\}$, let $P$ be the uniform distribution on $\{(0,0),(1,1)\}$ and let $\hat P$ be the uniform distribution on $\{(0,1),(1,0)\}$. By (2) the maximal coupling $\mathbb P$ is the product distribution $P\otimes\hat P$, so the pushforward measure on $E\_1\times E\_1$... | 2 | https://mathoverflow.net/users/408 | 119297 | 67,256 |
https://mathoverflow.net/questions/119294 | 12 | It is well known that the generating function of a regular language $L$, i.e. $\sum n\_kz^k$ where $n\_k$ is the number of words of length $k$ in $L$, is rational, i.e. a quotient of two polynomials $P(z)/Q(z)$. Suppose that $L$ is the language accepted by some finite automaton $\mathcal{A}$. How to find the polynomial... | https://mathoverflow.net/users/nan | Generating function of a regular language | Cannon does more than just refer to result that you ask for, he sketches the proof out. His proof is couched in the notation that he has set up for his application to hyperbolic groups. But it is easy enough to unravel the notation and express the proof in general.
Label the state set of the automaton as $0,\ldots,N$... | 8 | https://mathoverflow.net/users/20787 | 119302 | 67,259 |
https://mathoverflow.net/questions/119317 | 0 | I have the following questiom: let $X$ and $Y$ be two different points (represented by Riemann surfaces) in the Teichmuller space $T\_g$ of genus $g \geq 2$ Riemann surfaces. Then of course $X$ and $Y$ are homeomorphic and not bi-holomorphically equivalent. My question is, whether there exists a holomorphic covering fr... | https://mathoverflow.net/users/30776 | holomorphic covering between points in Teichmuller space | As $g\geq 2$, it follows by Riemann-Hurwitz that any topological covering $X\rightarrow Y$ is a homeomorphism, and any holomorphic homeomorphism is biholomorphic.
| 7 | https://mathoverflow.net/users/15819 | 119321 | 67,268 |
https://mathoverflow.net/questions/119266 | 4 | Let $G$ be an infinite group. It's (integral) group ring $\mathbb{Z}[G]$ has as its elements the *finite* formal linear combinations
$$
m\_1g\_1 + m\_2g\_2 + \cdots + m\_ng\_n,\qquad n\in\mathbb{N},\quad m\_i\in\mathbb{Z},\quad g\_i\in G,
$$
and these are added and multiplied in the obvious way such that the usual rin... | https://mathoverflow.net/users/8103 | A version of the group ring using direct product rather than direct sum? | OK, I will make my comment into an answer. The multiplication is not well defined, because of infinite sums :-)
| 2 | https://mathoverflow.net/users/6668 | 119327 | 67,271 |
https://mathoverflow.net/questions/119325 | 1 | What is the smallest simplest(non-trivial) $E\_8$ -module ?
| https://mathoverflow.net/users/nan | smallest simplest $E_8$ -module | Cartan showed that the lowest dimensional (nontrivial) $E\_8$-module is ${\frak{e}}\_8$ itself, i.e., the adjoint representation, which has dimension $248$. The next smallest nontrivial irreducible module is considerably larger dimension, $3875$, and I think that the next one after that has dimension $30380$.
At
<h... | 10 | https://mathoverflow.net/users/13972 | 119330 | 67,273 |
https://mathoverflow.net/questions/119319 | 4 | Let $X$ be space. A space $X$ is called right-separated if it can be well-ordered in such a way that every initial segment is open in $X$. [See the related link (left-separated).](https://mathoverflow.net/questions/118858/understanding-the-left-separated-spaces)
How could we show that hereditary lindelof number is th... | https://mathoverflow.net/users/18465 | A question on hereditary Lindelof number | In one direction, suppose $\kappa$ is a cardinal and $X$ has a subspace $Y$ with an open cover $\mathcal U$ that has no subcover of size $<\kappa$. Define in parallel a $\kappa$-sequence of points $y\_i\in Y$ and a $\kappa$-sequence of sets $U\_i\in\mathcal U$ by the following induction of length $\kappa$, in which one... | 3 | https://mathoverflow.net/users/6794 | 119336 | 67,275 |
https://mathoverflow.net/questions/111877 | 5 | The problem whether a real function $f$ has a root or not is undecidable, given that $f$ is from a class of functions including polynomials and the sine function (<http://dl.acm.org/citation.cfm?id=321856>). Usually, undecidability is proved by using a periodic function like sin to encode integer problems. Is there any... | https://mathoverflow.net/users/10072 | (Un)Decidability of the root existence problem for functions with bounded domain | Suppose there is an algorithm that decides whether a function $f\colon \Omega \to \mathbb{R}$ has a root. Then one can also compute a root of $f$ if $f$ has one.
One can see this using a standard bi-partition argument:
Cover $\Omega$ with finitely many balls of radius $1$. This is possible since the closure of $\Omeg... | 6 | https://mathoverflow.net/users/3365 | 119342 | 67,279 |
https://mathoverflow.net/questions/119335 | 0 | William Goldman used projective invariants in order to classify a triangle with three adjacent ones . Choi in " Geometric structures on low-dimensional manifolds "
wants to generalize Goldman's results to 3-dimensional manifolds , so he also used the projective invariants so that he could classify a tetrahedron with f... | https://mathoverflow.net/users/25609 | projective structure and holonomy | There are two kinds of [holonomy](http://en.wikipedia.org/wiki/Holonomy), which are well contrasted with each other in the opening paragraphs of the link given.
The first kind is exemplified by the holonomy of a Riemannian metric: parallel transport of tangent vectors around a closed loop. This kind of holonomy does... | 6 | https://mathoverflow.net/users/20787 | 119343 | 67,280 |
https://mathoverflow.net/questions/119329 | 26 | On page 263 of this [book review](http://www.ams.org/journals/bull/2004-41-02/S0273-0979-04-01007-9/S0273-0979-04-01007-9.pdf) appears the following:
>
> Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic number theory from the ... | https://mathoverflow.net/users/6269 | The Riemann Hypothesis and the Langlands program | One can use Langlands functoriality to eliminate the so-called Siegel zeros of an automorphic $L$-function. For example, Hoffstein-Ramakrishnan (IMRN 1995) proved that the $L$-function of a $GL(n)$ cusp form for $n>1$ has no Siegel zero if all $GL(m)\times GL(n)$ $L$-functions are $GL(mn)$ $L$-functions. There are seve... | 22 | https://mathoverflow.net/users/11919 | 119347 | 67,281 |
https://mathoverflow.net/questions/119326 | 24 | Apologies in advance if this is a bit too simple to ask here, but I think I'm probably more likely to get an answer here than at stackexchange.
I've been trying to learn the basics of the Langlands program over the last couple of months, and I've reached a funny point where I *think* I understand the rough idea of wh... | https://mathoverflow.net/users/29273 | Understanding the "idea" behind Langlands | I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).
I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphici... | 11 | https://mathoverflow.net/users/11919 | 119351 | 67,284 |
https://mathoverflow.net/questions/119353 | 9 | In the literature I can only find Chern-Simons terms for odd-dimensional manifolds. For example, for a $G$-bundle over a 3-dimensional manifold we have $A \wedge dA + 2/3 \* A \wedge A \wedge A$ with $A$ being a $\mathfrak{g}$-valued 1-form. Why can't I write such forms for even-dimensional manifolds?
| https://mathoverflow.net/users/25953 | Chern-Simons for 2n-dimensional manifolds | It has to do with the fact that the characteristic classes (over the reals) of a principal $G$-bundle have *even* degree. We can associate Chern-Simons-like theory to each characteristic class of degree $2k$ together with a $G$-bundle $P$ over a manifold of dimension $2k-1$.
To be a bit more technical a Chern-Simons-... | 16 | https://mathoverflow.net/users/20302 | 119359 | 67,287 |
https://mathoverflow.net/questions/119354 | 5 | I have two questions related to [Selinger: A survey of graphical languages for monoidal categories](http://www.mathstat.dal.ca/~selinger/papers/graphical.pdf).
1) On page 60 there is a summary of some monoidal categories with extra structure. Among them are the well known braided and symmetric versions. Due to the [p... | https://mathoverflow.net/users/27923 | String diagrams of special monoidal categories and higher categories | As you point out, in such a situation it may not be possible to compose horizontally, hence in full generality you cannot assume your underlying category to be monoidal. On the other hand, you still want to keep track somehow of the tensor product. I think it can be handled as follow:
The structure you are looking f... | 6 | https://mathoverflow.net/users/13552 | 119361 | 67,288 |
https://mathoverflow.net/questions/119362 | 5 | Hello,
We know that in Hilbert space it is, but what about these topology:
Let $\mathcal{T}\_{\infty}= \{ U \subset \mathbb{R}^{\infty}: \ U \cap \mathbb{R}^n \in \mathcal{T}\_n, \ for \ n=1,2,... \} $ which isn't metric space?
Of course $\mathcal{T}\_{\infty}$ is topology in $\mathbb{R}^{\infty}$. How to prove or di... | https://mathoverflow.net/users/30784 | Unit sphere in R^\infty is contractible? | The question doesn't seem to be very well expressed, but the intended question might be as follows. Take $\mathbb{R}^\infty$ to mean the vector space consisting of real tuples $(v\_1, v\_2, v\_3, \ldots)$ such that all but finitely many $v\_i$ are zero, equipped with the coherent topology (so that $U \subseteq \mathbb{... | 18 | https://mathoverflow.net/users/2926 | 119372 | 67,294 |
https://mathoverflow.net/questions/119385 | 0 | I'm trying to solve the following question:
Suppose $Y \subset R^n$ is a Euclidean neighborhood retract. I want to prove that if $Y$ is contractible, then it is a retract of $R^n$.
| https://mathoverflow.net/users/30791 | (Homotopy) Y ENR and contractible subset implies Y is a retract | Let me solve the case of compact $Y$. Thus $Y$ is a contractible ANR. Then Y is an AR.
**Indeed**, the equivalence "contractible ANR $\Leftrightarrow$ AR" is a well known theorem, certainly known in the past to the founder of the theory of ANRs, Karol Borsuk. To prove this equivalence, consider an arbitrary compact... | 4 | https://mathoverflow.net/users/8385 | 119388 | 67,302 |
https://mathoverflow.net/questions/119377 | 4 | If $G$ is a finite group, embedded as a transitive subgroup of $S\_n$ for some $n$, will every automorphism of $G$ extend to an inner automorphism of $S\_n$?
I'm trying to connect the language that's used in many different sources on Hurwitz spaces and Nielsen classes. On Michael Fried's website, he notes that "absol... | https://mathoverflow.net/users/15242 | Automorphism of finite groups and Hurwitz spaces | While the answers by Eric and ARupinski give negative examples for your question, here is the precise characterization for when the answer is yes: Let $\alpha$ be an automorphism of the transitive subgroup $G\le S\_n$, and $G\_1$ be the stabilizer of $1$ in $G$. Then $\alpha$ extends to an inner automorphism of $S\_n$ ... | 6 | https://mathoverflow.net/users/18739 | 119391 | 67,304 |
https://mathoverflow.net/questions/119392 | 5 | **Situation:** Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, *unramified* morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
**What I know:** For $\Bbbk=\mathbb C$, the second chern class of $X$ equals its topological Euler characteristic (i.... | https://mathoverflow.net/users/9947 | Top chern class under finite, unramified, dominant morphism | Yes, it does hold in positive characteristic. You can show that the degree of $c\_n(X)$ equals its Euler characteristic with respect to étale cohomology with coefficients in $\mathbb{Q}\_{\ell}$, where $\ell$ is a prime different from the characteristic ( [Top chern class in positive characteristic](https://mathoverflo... | 9 | https://mathoverflow.net/users/4790 | 119395 | 67,306 |
https://mathoverflow.net/questions/119397 | 2 | Suppose we have extended $ZF$ by adding to $ZF$ an unary function symbol $arb$ (an arbitrary element of a set) and a corresponding axiom "For every non-empty set $S$, $arb(S)$ is in $S$".
Will be the resulting theory a conservative extension of $ZF$?
| https://mathoverflow.net/users/5761 | Notation arb(x) | It depends how the remaining axioms of ZF are altered (or not). For example, do we broaden the scheme of replacement to apply to formulas of set theory which contain the symbol "$arb$"? If so, then as Zhen Lin says the resulting theory proves the axiom of choice: given a collection of sets $\mathcal{A}=\lbrace A\_i: i\... | 8 | https://mathoverflow.net/users/8133 | 119405 | 67,310 |
https://mathoverflow.net/questions/119406 | 1 | The typical characterization of points constructible by compass and straightedge is the following:
Let $S\subseteq\mathbb{C}$ with $0,1\in S$, $K\_0 = \mathbb{Q}(S\cup \bar{S})$ and $a\in\mathbb{C}$.
Then $a$ is constructible from $S$ by compass and straightedge if and only if there is a tower of quadratic field exte... | https://mathoverflow.net/users/40722 | Algebraic characterization of points constructible by compass and straightedge | Your question is about showing that the normal hull of $K\_n$ over $K\_0$ has $2$-power degree, if $[K\_i:K\_{i-1}]=2$ for all $i$. But that follows be induction: Let $L$ be the normal hull of $K\_{n-1}$ over $K\_0$, so $[L:K\_0]$ is a $2$-power.
The normal hull $N$ of $K\_n$ over $K\_0$ is the composite of the conju... | 2 | https://mathoverflow.net/users/18739 | 119411 | 67,312 |
https://mathoverflow.net/questions/119402 | 30 | About 20 years ago I read in textbook that
"all irreducible representations of compact groups are finite-dimensional", but
me and the proof of this fact never met each other :)
May I ask dear MO colleagues, is there (simple?) argument to prove it ?
As far as I heard this result can be generalized in the realm of... | https://mathoverflow.net/users/10446 | Why all irreducible representations of compact groups are finite-dimensional ? [EDIT: Subtleties: AC,etc] | [First proof] I will address only the first part. Suppose $G$ is a compact group and $\pi : G \rightarrow U(H)$ a unitary irreducible representation on a Hilbert space $H$. Suppose $\phi$ a continuous function on $G$. Then it is easy to show that $\pi (\phi)$ is a compact operator on $H$.
Since the OP has asked for c... | 19 | https://mathoverflow.net/users/23291 | 119415 | 67,314 |
https://mathoverflow.net/questions/119416 | 1 | A simple argument shows that if we choose $n$ positive integers at random, the probability of their greatest common divisor being 1 is $1/ \zeta (n)$ (in the sense that if we choose the numbers among $\lbrace 1, 2, 3, \dots , N \rbrace$ uniformly and independently, the above probability tends to some number $p(n)$ as $... | https://mathoverflow.net/users/26045 | Probability of all combinations of k numbers among n being coprime | For any prime $q$, define
$$
\ell(q,n,k) = q^{-n} \sum\_{j=0}^{n-k} \binom nj (q-1)^j,
$$
so that $\ell(q,n,k)$ is the probability, if a biased coin that comes up heads only $1/q$ of the time is tossed $n$ times, that at least $k$ heads are obtained. Equivalently, $\ell(q,n,k)$ is the probability, if $n$ numbers are ch... | 6 | https://mathoverflow.net/users/5091 | 119422 | 67,317 |
https://mathoverflow.net/questions/119401 | 6 | Suppose $(M, g)$ is an open complete nonnegatively curved Riemannian manifold with $d$ its distance.
A totally convex set $C\subset M$ has the property that for any two point $x, y \in C$ *any* geodesic (not only the minimal ones) joining them must lie in $C$.
If $C$ is totally convex, the fattened set $C^a=\{x \i... | https://mathoverflow.net/users/24152 | Fattening of totally convex sets | No. Consider a rotation-symmetric metric on $\mathbb R^2$ resembling a small spherical cap extented by a flat cone. A sufficiently short geodesic segment at the origin is totally convex in your sense. But a small neighborhood of such a segment is not convex due to positive curvature.
| 6 | https://mathoverflow.net/users/4354 | 119424 | 67,319 |
https://mathoverflow.net/questions/119426 | 7 | As the discussion here [Is singular cohomology representable by a (Voevodsky's) motivic complex?](https://mathoverflow.net/questions/42693/is-singular-cohomology-representable-by-a-voevodskys-motivic-complex)
shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex (and also by a... | https://mathoverflow.net/users/2191 | Ring structure for the motivic spectrum/complex that represents singular cohomology? | Singular cohomology is represented by an $E\_\infty$ motivic ring spectrum. That spectrum is $\mathbf{R}f(H\mathbb{Z})$ where $f$ is right adjoint to the stable topological realization functor and $H\mathbb{Z}$ is the topological Eilenberg-Mac Lane spectrum. Since this is a symmetric monoidal Quillen adjunction (see Th... | 8 | https://mathoverflow.net/users/20233 | 119434 | 67,322 |
https://mathoverflow.net/questions/119429 | 7 | Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l\_1\mathbb{Z}\oplus l\_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}$, are $\lambda\_{m\_1,m\_2}=(2\pi)^2(\frac{m\_1^2}{l\_1^2}+\frac{m\_2^2}{l\_2^2})$ with the associat... | https://mathoverflow.net/users/19325 | Eigenfunctions restricted on closed geodesics | The answer is 'no', as you can see by taking the case of $M$ being the unit $2$-sphere in $\mathbb{R}^3$ and the geodesic $\gamma$ being a great circle, say, the horizontal great circle given by $z=0$. If you consider the harmonic polynomials of degree $2$ in $x,y,z$ restricted to the $2$-sphere, these are eigenfunctio... | 12 | https://mathoverflow.net/users/13972 | 119437 | 67,324 |
https://mathoverflow.net/questions/119164 | 4 | Let's consider only global case.
Let $G\_n$ be classical algebraic group over global # field (eg, $GL(n),SO(n), U(n)$...) and let $\pi\_n$ be its irr. cusp. reps of $G\_n$.
Then we can define the period of two reps $\pi\_{n+1}$ & $\pi\_n$ as follows;
For $f\_{n+1}\in \pi\_{n+1}$ and $f\_{n}\in \pi\_{n}$ ,
defin... | https://mathoverflow.net/users/29334 | On the restriction of cusp. irr. representation and period. | I see now what Marc meant: this will look like an "unanswered" question. So here goes: both the functions $f\_1,f\_2$ are bounded since they are cuspidal (in fact they are of rapid decrease on Siegel sets). The space $[G\_n]$ has finite volume since you have quotiented out by the centre. Therefore, the product is integ... | 5 | https://mathoverflow.net/users/23291 | 119438 | 67,325 |
https://mathoverflow.net/questions/119364 | 3 | I think this question is famous, but I searched a lot in the internet with many keywords and unfortunately I did not find any related problem. I will appreciate any helpful answers and any guidance for similar questions and problems.
Suppose the integers $n$ and $k$ are given and the set $S$ shows all possible factor... | https://mathoverflow.net/users/19885 | Minimum sum among fixed length factors of a number | At the end are a few interesting numerical examples. Here is a general discussion:
This is not a very well specified question. The answer depends strongly on the prime factorization of $n$. Very easy cases include $n$ prime and $n$ a perfect $k$th power. $n$ a prime power is easy too. If the number of distinct prime ... | 2 | https://mathoverflow.net/users/8008 | 119446 | 67,328 |
https://mathoverflow.net/questions/119258 | 1 | Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a totally convex set $C\in M^n$, we mean for any two point $a, b\in C$ and any geodesic joining them must lie in $A$. In ord... | https://mathoverflow.net/users/30176 | Examples on small cut radius of totally convex set in non-negatively curved manifold | Let $M$ be the union of the unit northern hemisphere centered at the origin of $R^3$ with the cylinder $\{x^2+y^2=1\,,\, z\leq 0\}$ and $C$ be the geodesic segment $\{x^2+z^2=1\,,\, y=0\,,\, \vert x\vert\leq 1/2\}$. The point $P=(0,1,0)$ is at the same distance from every point of the segment $C$.
The metric here is... | 1 | https://mathoverflow.net/users/24152 | 119452 | 67,329 |
https://mathoverflow.net/questions/119454 | 11 | What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ?
| https://mathoverflow.net/users/nan | Category and the axiom of choice | Here's a somewhat trivial one, but it is one that category theorists use all the time:
>
> Let us say that a functor $F : \mathcal{C} \to \mathcal{D}$ is a **weak equivalence** if it is fully faithful and essentially surjective on objects, and that it is a **strong equivalence** if there exists a functor $G : \math... | 18 | https://mathoverflow.net/users/11640 | 119459 | 67,332 |
https://mathoverflow.net/questions/119463 | 0 | Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum\_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, where $C$ is much smaller than $n$?
I want to prove that this probability tends to zero as $n$ increases.
| https://mathoverflow.net/users/30821 | Probability of summing products of irreducible polynomials in a finite field to zero | Taking a view from old-fashioned algebra: your sum can be interpreted from a generating function of a product of three geometric series. That generating function has a partial fraction decomposition, with numerators say P, Q, and R. If these were rational functions in x alone the issue would be like
$Pf^m(x) + Qg^m(... | 0 | https://mathoverflow.net/users/6153 | 119468 | 67,336 |
https://mathoverflow.net/questions/119470 | 3 | Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map
$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)
$$
in group cohomology, where $M$ is any $G$-module (see Brown's "Cohomology of groups", Chapter III).
I think this construction should be natural, in the following sens... | https://mathoverflow.net/users/8103 | Naturality of the transfer in group cohomology | I don't believe this is true. Let $(G, H) = (\Sigma\_3, C\_3)$ and $f : C\_3 \to \Sigma\_3$. Then your square says that
$$H^1(C\_3;\mathbb{Z}/3) = \mathbb{Z}/3 \longrightarrow H^1(\Sigma\_3;\mathbb{Z}/3) = 0\longrightarrow H^1(C\_3;\mathbb{Z}/3) = \mathbb{Z}/3$$
is the identity, which is false.
| 6 | https://mathoverflow.net/users/318 | 119473 | 67,339 |
https://mathoverflow.net/questions/119474 | 1 | I was trying to solve one particular problem and to settle that problem I need to solve this problem, and it goes like this (sorry in advance for I still do not know LaTex symbolism so I will try to type it in the old-fashioned way):
If we have system of infinite number of equations that looks like this:
p(3)q(3) -... | https://mathoverflow.net/users/30730 | Does this system of equations admits a solution? | The $p\_k$ and $q\_k$ always appear together so let us for the moment write $a\_k= p\_kq\_k$ (I will come back to the condition later).
Then your system can be rewritten as $a\_{i+2} = 3a\_{i+1} - 2a\_i$ for $i$ a positive integer.
Now if you pick any two starting values $0 \lt a\_1 \lt a\_2$ you directly get from t... | 8 | https://mathoverflow.net/users/nan | 119476 | 67,340 |
https://mathoverflow.net/questions/119460 | 7 | The question [Are automorphism groups of hypersurfaces reduced ?](https://mathoverflow.net/questions/10743/are-automorphism-groups-of-hypersurfaces-reduced) reminded me of the following related question that I have not seen discussed.
If $X$ is a smooth projective variety over an algebraically closed field $k$ of cha... | https://mathoverflow.net/users/25792 | Automorphism groups of general type varieties | William Lang produced examples of surfaces of general type in positive characteristic with non-zero vector fields. Since surfaces of general type must have finite automorphism groups, this gives examples. See William Lang, "Examples of surfaces of general type with vector fields", Arithmetic and geometry, Vol. II, Prog... | 8 | https://mathoverflow.net/users/4790 | 119490 | 67,345 |
https://mathoverflow.net/questions/119479 | 4 | I'm reading Hatcher's book on algebraic topology. In Section 3.C, he proves as Theorem 3C.4 that if $A$ is a graded commutative associative Hopf algebra over a field of characteristic $0$ and $A^n$ is finitely generated for each $n$, then $A$ is isomorphic **as an algebra** to the tensor product of an exterior algebra ... | https://mathoverflow.net/users/30829 | Different Hopf algebra structures on same graded algebra | There are generalizations of the following construction, but here is a simple one. Let $\mathfrak n$ denote the Lie algebra of strictly-upper-triangular $n\times n$ matrices. It is $\mathbb Z\_{>0}$-graded, by declaring that for $1\leq k \leq n$, the degree-$k$ part of $\mathfrak n$ consists of those matrices supported... | 1 | https://mathoverflow.net/users/78 | 119492 | 67,346 |
https://mathoverflow.net/questions/119495 | 19 | I am trying to find proofs of the stability of an atom, says, for simplicity, the hydrogen atom. There are positive answers and negative answers in various atom models.
The naive "solar system" model of a negatively charged electron orbiting the positively charged nucleus is not stable, it radiates electro-magnetic e... | https://mathoverflow.net/users/30830 | Mathematical "proof" of the stability of atoms? | I think you can find more in Lieb and Seiringer's book "The Stability of Matter in Quantum Mechanics", or see also Freeman Dyson <http://www.webofstories.com/play/4415> and the book review <http://arxiv.org/abs/1111.0170>.
| 16 | https://mathoverflow.net/users/30364 | 119498 | 67,347 |
https://mathoverflow.net/questions/111980 | 5 | I am looking for an article or book where the theory of heights over function fields (in any characteristic) is treated. I am especially interested in Northcott-type statements. For instance, over a function field $K$ over $\bf Q$, say, a subvariety $X$ of ${\bf P}^n\_{K}$, which has a dense subset of $K$-points with b... | https://mathoverflow.net/users/17308 | Reference request for the theory of heights over function fields | This is discussed (using somewhat older language) in Lang's *Fundamentals of Diphantine Geometry*. See in particular Chapter 3, Section 3, which is called "Heights in Function Fields". There is a theorem of Neron (Theorem 3.6 on page 66) which says that bounded height implies that the associated map has bounded degree.... | 5 | https://mathoverflow.net/users/11926 | 119512 | 67,353 |
https://mathoverflow.net/questions/119516 | 5 | The question is as in the title:
>
> Is every finite subcomplex of a contractible simplicial complex $K$ contained in a finite contractible subcomplex of $K$? What if we are allowed to take subdivisions? What if the complex is locally compact and/or finitely dimensional?
>
>
>
I suppose the answer is no, but I... | https://mathoverflow.net/users/2578 | Is every finite subcomplex of a contractible simplicial complex contained in a finite contractible subcomplex? | Let $X$ be any finite complex such that any map $X\to X$ homotopic to the identity is surjective and for which there is a surjective but nullhomotopic PL map $f:X\to X$ (for instance, $X$ could be $S^n$ for any $n>0$). Let $K$ be the mapping telescope obtained by iterating $f$. Then $K$ is contractible, locally compact... | 13 | https://mathoverflow.net/users/75 | 119519 | 67,357 |
https://mathoverflow.net/questions/119475 | 8 | First of all, let me fix some notation.
1. Let $\mathcal D$ be a triangulated category in the sense of Verdier-Grothendieck (for example, the homotopy category $\mathbf{K}(k)$ of cochain complexes over a fixed commutative ring $k$). I call *cone* of a morphism $f : A \rightarrow B$ the object $C(f)$ (uniquely determ... | https://mathoverflow.net/users/20883 | Example: a pair of nonisomorphic parallel morphisms with isomorphic cones | Let $R$ be the ring $R=\mathbb C[x,y]$, and let $B$ be the $5$-dimensional $R$-module with shape like a 'W'. That is, basis elements are $a\_1,a\_2,a\_3,b\_1,b\_2$ and the module structure is given by $$y \cdot a\_1=b\_1,$$
$$x \cdot a\_2=b\_1,$$
$$y \cdot a\_2=b\_2,$$
$$x \cdot a\_3=b\_2,$$
and all other products of ... | 7 | https://mathoverflow.net/users/18756 | 119521 | 67,358 |
https://mathoverflow.net/questions/119482 | 4 | Say that a class $\mathcal{C}$ of countable first-order structures in some finite signature has the *effective substructure property* if $\mathcal{C}$ is closed under isomorphism and whenever $A\in \mathcal{C}$ has domain $\subseteq\omega$ and $B\in\mathcal{C}$ is isomorphic to a substructure of $A$, then $B$ has a pre... | https://mathoverflow.net/users/8133 | When do substructures have computable copies? | Two more non-sophisticated examples would be the class of all countable algebraically closed fields and the class of all countable vector spaces over any collection of (not necessarily computable) fields.
Perhaps it is worth making the trivial observation that any member $A$ of such a class $\mathcal{C}$ can only hav... | 3 | https://mathoverflow.net/users/18131 | 119525 | 67,360 |
https://mathoverflow.net/questions/119541 | 0 | Good moring,
Let $\Omega$ be a domain in $\mathbb{C}^n$ and $S\subset\Omega$ an analytic subset of codimension 1. *What can we say about the cohomology group $H^1(\Omega\backslash S, \mathbb{Z})$? E.g, when $\Omega$ is a ball?*
Any help is appreciated. Thanks in advance.
Duc Anh
| https://mathoverflow.net/users/11376 | Cohomology of Complements by an analytic subset? | I am not sure about cohomology with coefficients in $\mathbb Z$, but if we replace $\mathbb Z$ by $\mathbb Q$ and if $\Omega$ is a ball, then Poincar\'e duality implies that $H^1(\Omega\setminus S,\mathbb Q)$ is dual to $H^{2n-1}\_c(\Omega\setminus S,\mathbb Q)$ (cohomology with compact support). Now the exact sequence... | 1 | https://mathoverflow.net/users/29992 | 119545 | 67,368 |
https://mathoverflow.net/questions/118897 | 18 | Let $k$ be some field, algebraically closed and of characteristic $0$, if you like.
Let $U= \mathbb{A}^2\_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of $U\_1 = \mathrm{Spec} k[x^{\pm 1},y]$ and $U\_2 = \mathrm{Spec} k[x,y^{\pm 1}]$. Then deformations $U^\prime$ of $U$ to $... | https://mathoverflow.net/users/6074 | Deformations of the punctured affine plane | I think I can prove the following
>
> Theorem: Let $R$ be a $k$-algebra of finite type with a closed point $x\in \mathrm{Spec} R$ such that $\mathrm{Spec} R\setminus \{x\}\cong \mathbb{A}^2\_k\setminus \{(0,0)\}$. Let $\tilde{R}$ be a $t$-adically complete flat lift of $R$ to $k[[t]]$. Then $\mathrm{Spf} \tilde{R}\... | 12 | https://mathoverflow.net/users/6074 | 119555 | 67,372 |
https://mathoverflow.net/questions/119551 | 4 | Is it known who actually wrote Bourbaki's *Elements of the History of Mathematics*?
| https://mathoverflow.net/users/26039 | Elements of the history of mathematics | Different people wrote different parts. Weil indicates somewhere that he wrote the *Note historique* about infinitesimal calculus. A perusal of [*Archives de l'Association des Collaborateurs de Nicolas Bourbaki*](http://math-doc.ujf-grenoble.fr/archives-bourbaki/) might provide further hints.
Other possible sources o... | 5 | https://mathoverflow.net/users/2821 | 119556 | 67,373 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.