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https://mathoverflow.net/questions/200243 | 7 | Not knowing elementary number theory well, I ask this one, which is not very clear to answer, rather I am looking for some results around this question or known theorems. The problem is the following:
The set of prime numbers $\mathbb{P}=2,3,5,7,11...$ generates $\mathbb{N}$ by multiplication. Now I am interested in ... | https://mathoverflow.net/users/69341 | Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers? | Let $T=\mathbb P\setminus S$. If $\sum\_{p\in T}1/p=\infty$, then $\prod\_{p\in T}(1-1/p)=0$ and for any $\epsilon>0$, there exist $p\_1,\ldots,p\_n\in T$ such that $\prod\_{i=1}^n(1-1/p\_i)<\epsilon$. Now modulo $P=p\_1\cdots p\_n$, the fraction of integers that have no factor of the form $p\_i$ with $i\le n$ is $\pro... | 12 | https://mathoverflow.net/users/11054 | 200254 | 96,718 |
https://mathoverflow.net/questions/200251 | 15 | Let $X$ be an affine scheme and $\mathcal{E}$ a finitely generated locally free sheaf on $X$. It is obvious that $\mathcal{E}$ is a projective object in the category Qcoh$(X)$ since we can pass to rings and modules.
My question is: is $\mathcal{E}$ still projective when we consider it in the larger category $\mathca... | https://mathoverflow.net/users/24965 | Is a locally free sheaf projective in the category of $\mathcal{O}_X$-modules when $X$ is an affine scheme? | This answer is inspired by the discussion [at this question](https://mathoverflow.net/questions/5378/when-are-there-enough-projective-sheaves-on-a-space-x). Let $X$ be an integral affine scheme admitting an open cover $X=U\cup V$ with $U$, $V$ and $X$ all distinct. I claim that $\mathscr O\_X$ is not projective in $\op... | 20 | https://mathoverflow.net/users/6856 | 200258 | 96,719 |
https://mathoverflow.net/questions/200255 | 5 | Is the ring $\mathbb{Z}\_p [[x]]\otimes\_{\mathbb{Z}\_p} \overline{\mathbb{Q}}\_p$ Noetherian?
| https://mathoverflow.net/users/3847 | Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian? | Yes. Consider any nonzero element of the ring. Using the norm, we can see that it divides some nonzero element of $\mathbb Z\_p [[x]] \otimes\_{\mathbb Z\_p} \mathbb Q\_p$. By multiplication by $p$, it divides some nonzero element of $\mathbb Z\_p[[x]]$. Applying Weierstrass preparation, we may factor it as a unit time... | 11 | https://mathoverflow.net/users/18060 | 200263 | 96,721 |
https://mathoverflow.net/questions/194830 | 5 | Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$.
It is well known that $x$ tends to $1$ as $n\to \infty$, even if $p\geq \frac {1+\epsilon}{n}$. However, I am looking for a more exact statement: Something of... | https://mathoverflow.net/users/31092 | Probability of a giant component existing in a $G(n,p)$ random graph with $p=\omega(\frac 1n)$ | In ["On tree census and the giant component in sparse random graphs](http://onlinelibrary.wiley.com/doi/10.1002/rsa.3240010306/abstract), B. Pittel proves the following (see Gap Theorem and Lemma 4): for $c>1,$ any $a>0$ and any $\omega \to \infty$ however slowly,
\begin{equation\*}
P(|\# \text{ of vertices in largest... | 4 | https://mathoverflow.net/users/36452 | 200275 | 96,724 |
https://mathoverflow.net/questions/200270 | 7 | This is the first time I ask a question on Mathoverflow, so I apologize in advance if it is not suitable/a duplicate/otherwise inappropriated.
I am thinking about Voevodsky's category of motives and I realized that in his presheaves with transfers formalism pullbacks for Chow groups are defined for arbitrary maps of ... | https://mathoverflow.net/users/43054 | When is the pullback in Chow groups defined? | You have to distinguish between pullbacks of cycles, pullbacks on Chow groups, and pullbacks of relative cycles.
* You cannot always pull back cycles. If $f: Y\to X$ is a morphism of (arbitrary) schemes and $Z\subset X$ is an elementary cycle, $f^\*(Z)$ is defined provided that $Z$ is ``in good position with respect ... | 10 | https://mathoverflow.net/users/20233 | 200279 | 96,727 |
https://mathoverflow.net/questions/199382 | 14 | Imagine you sample $n$ number with replacement uniformly from the integers $1,\dots, n$. Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but with a twist. All I know is that the samples are uniform and pairwise independent.
>
> Assuming $n$ is large, what bounds can one get for $\mathbb{... | https://mathoverflow.net/users/45564 | Expected value of the minimum with limited independence | Let $u\_k$ be the number of variables with value exactly $k$. If you pick a distribution of the $u\_k$ such that $E[u\_k]=1$, $E[u\_k^2] = 2-1/n$, $E[u\_ku\_l] = 1-1/n$ for $k \neq l$ and $\sum\_{k=1}^n u\_k$ is always $1$, then by choosing a random $u\_1, \dots ,u\_n$ according to the distribution and then choosing a ... | 8 | https://mathoverflow.net/users/18060 | 200280 | 96,728 |
https://mathoverflow.net/questions/200274 | 1 | Given a projective plane I'd like to form a latin rectangle from the lines. In particular, I'd like to take each line from the plane, order the elements in some way, and stick them into the matrix as a column.
I liked to know whether this is possible. Better yet, I'd love an algorithm for creating the matrix.
Thank... | https://mathoverflow.net/users/43928 | Creating a Latin rectangle from a projective plane | By a theorem of Singer, the automorphism group of $PG(2,q)$ contains a cyclic subgroup $\langle\sigma\rangle$ which acts regularly on points and regularly on lines. Fix a base block, $B$, and list its elements in the first column of your matrix in any order you choose. Then form the rest of the columns by applying $\si... | 5 | https://mathoverflow.net/users/27513 | 200291 | 96,730 |
https://mathoverflow.net/questions/200305 | 1 | Let $\mathcal{A}$ be a non-empty systems of non-empty sets such that there is an injective map $f:\bigcup \mathcal{A}\to \mathcal{A}$ such that $a\in f(a)$ for all $a\in\bigcup\mathcal{A}$. Assuming that $|\mathcal{A}| = |\bigcup \mathcal{A}|$, is there always a bijection $\varphi: \bigcup \mathcal{A}\to \mathcal{A}$ s... | https://mathoverflow.net/users/nan | Surjectivity from union of a set system to the set system | No - let $\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega\}$.
| 0 | https://mathoverflow.net/users/8628 | 200307 | 96,738 |
https://mathoverflow.net/questions/200310 | 6 | Let $E$ be a vector bundle of rank $r$ and degree $d$ over a smooth curve $X$. Is there any canonical exact sequence for $Sym^k(E)$? in particular what is the degree of $Sym^k(E)$?
Suppose $E$ is stable, is $Sym^k(E)$ is stable ?
Is there any interesting information about $Sym^k(E)$?
Thanks
| https://mathoverflow.net/users/66528 | Symmetric product of a vector bundle | There is no "canonical exact sequence", whatever that means. The determinant of $Sym^k(E)$ is $(\det E)^m$, with $m=\binom{r+k-1}{r}$; this follows from the analogous equality of $GL(V)$-modules $\det(Sym^k(V))=(\det V)^m$ for a vector space of dimension $r$ (which one gets easily by looking at the action of the scalar... | 5 | https://mathoverflow.net/users/40297 | 200319 | 96,744 |
https://mathoverflow.net/questions/200299 | 5 | Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B\_2(0) \to \mathbb{R}$, say $f \in L^2(B\_2(0))$ . Suppose that there exists $\alpha \in (0,1)$ and $c > 0$ such that the following is true: For every $y \in B\_1(0)$, there exists an ... | https://mathoverflow.net/users/4281 | Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}' | First you can rescale so $\|f\|\_{L^2(B\_2)} = 1$ and you see that you are basically talking about the boundedness of the first order Campanato norm of your function $f$. Your conclusion then is a classical theorem concerning the comparison of Campanato and Holder spaces.
See e.g. <http://link.springer.com/chapter/1... | 5 | https://mathoverflow.net/users/3948 | 200320 | 96,745 |
https://mathoverflow.net/questions/200317 | 0 | Let $X$ be the hyper-surface defined by
$$f:=\sum\_{i=1}^k x\_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x\_1^{n-1},\dots , x\_k^{n-1}) $$
What is the multiplicity of $Y$ in $X$? Is it $n(n-1)^{k-1}$?
thanks
| https://mathoverflow.net/users/48866 | Samuel multiplicity | This follows from the discussion in Fulton's book on intersection theory, Example 4.3.5. However, your terminology isn't quite right. What you want is the *multiplicity of $Y$ along $X$ at $0$*, see Fulton, Example 4.3.4. Fulton quotes a result of Samuel that one can reduce this multiplicity to the multiplicity for an ... | 1 | https://mathoverflow.net/users/69386 | 200325 | 96,746 |
https://mathoverflow.net/questions/200322 | 7 | Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$?
(We assume that $\{0,1\}$ is endowed with the discrete topology, and $\{0,1\}^\kappa$ has the product topology.)
| https://mathoverflow.net/users/8628 | Images of $\{0,1\}^\kappa$ | It is well known that the space $\{0,1\}^{\kappa}$ satisfies the countable chain condition. Recall that a topological space $X$ satisfies the countable chain condition if and only if every collection $\mathcal{A}$ of pairwise disjoint open sets is countable. However, it is easy to show that the surjective continuous im... | 19 | https://mathoverflow.net/users/22277 | 200333 | 96,749 |
https://mathoverflow.net/questions/200323 | 4 | Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$.
I am interested in hypersurfaces of $X$ which again satisfy this condition.
>
> Q: Do there always exist hypersurfaces $X'\subset X$ such that $X'$ is again normal and $Cl(X')=0$? How 'many'... | https://mathoverflow.net/users/69353 | Existence of Factor rings of UFDs which are UFDs | The answer is NO in small dimension. Take a general surface $\bar{X}$ of degree $d\gg 0$ in $\mathbb{P}^3$, and take for $X$ the complement of a general hyperplane section. Then $X$ is smooth, $\mathrm{Pic}(X)=\mathrm{Cl}(X)=0$, but $\bar{X}$ contains no rational curve, so any normal (= smooth) curve in $X$ has a large... | 2 | https://mathoverflow.net/users/40297 | 200337 | 96,750 |
https://mathoverflow.net/questions/200327 | 3 | For $A\subseteq {\mathbb F}\_2^n$ let
$$
Q(A)=\{\alpha+\beta\mid \alpha,\beta \in A,\ \alpha\neq\beta \}.
$$
I want to prove or disprove that if $|A|=2^k+1$ for some integer $k$, then
$$
|Q(A)|\ge2^{k+1}-1.
$$
I have checked using a computer that this is true when $n\le5$. Also, this is true when $k=n-1$. I run my b... | https://mathoverflow.net/users/47837 | On a problem about $GF(2)^n$ | It **is** true that $|Q(A)|\ge 2^{k+1}-1$; this can be proved using [Kneser's theorem](https://en.wikipedia.org/wiki/Kneser's_theorem_(combinatorics)) as follows.
Let $2A:=\{a'+a''\colon a',a''\in A\}$ be the *sumset* of $A$; we thus want to prove that if $A\subset{\mathbb F}\_2^n$ has size $|A|=2^k+1$, then $|2A|\ge... | 6 | https://mathoverflow.net/users/9924 | 200338 | 96,751 |
https://mathoverflow.net/questions/200331 | 2 | We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS):
$$i\partial\_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$
$$\phi (x,0) = \phi\_{0}(x)\in H^{s}(\mathbb R);$$
where $H^{s}(\mathbb R)$ is usual [Sobolev space](http://en.wikipedia.org/wiki/Sobolev_s... | https://mathoverflow.net/users/33018 | global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$ | For the NLS in 1D, quintic is $L^2$ critical. So you are quite comfortably in the subcritical regime. Indeed, you have the result of
[Tsutsumi, Yoshio; "$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups"](http://www.ams.org/mathscinet-getitem?mr=MR0915266)
which implies for $H^s$ (and hence ... | 3 | https://mathoverflow.net/users/3948 | 200354 | 96,756 |
https://mathoverflow.net/questions/200347 | 3 | Let us say have a sequence of $n$ 2-$D$ random variables $X\_i=(\varepsilon\_i/\sqrt{n},i\varepsilon\_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon\_{i}$ are independent random variables such that $\mathbb{P}(\varepsilon\_i =\pm 1)=1/2$. Denote by $S\_n$ their sum and take a $2$-dimensional Gaussian random variable $Z$ wit... | https://mathoverflow.net/users/24494 | Berry-Esseen bound in 2 dimensions for linear combinations | I'm fairly confident the answer is yes, but unfortunately I can't specify an exact reference just now.
If you just want to see that one can get a similar bound (but with worse dependence on $\gamma$), then it's not too hard to prove multidimensional Berry--Esseen theorems via the "Lindeberg replacement method"; see, ... | 4 | https://mathoverflow.net/users/658 | 200362 | 96,759 |
https://mathoverflow.net/questions/199243 | 1 | **Question**. Let $(R,m)$ be a Noetherian local ring and $M$ be a finite *faithful* $R$-module. Let $I$ be an ideal of $R$ such that $IM=mM$. Can we say that $I$ is a *reduction ideal* of $m$? Recall that $I$ is a reduction ideal of $m$ if $Im^n=m^{n+1}$ for some (or equivalently all) sufficiently large $n$.
If not ... | https://mathoverflow.net/users/47763 | $IM=mM$. can we say that $I$ is a reduction ideal of $m$? | The answer is always yes. It's a basic tool in studying the integral closures of ideals.
Let $x \in m$. Let $\phi: M \rightarrow M$ be the endomorphism of $M$ given by multiplication by $x$. Then since $\phi(M) \subseteq IM$, the Cayley-Hamilton theorem shows that there exist $a\_j \in I^j$ and $n \in \mathbb N$ such... | 2 | https://mathoverflow.net/users/19045 | 200363 | 96,760 |
https://mathoverflow.net/questions/200365 | 2 | I want to learn about multivariate orthogonal polynomials. Is there a good textbook/survey that you could suggest? I need to see common examples like Jack's polynomials etc .. and also general theorems.
Thanks in advance.
| https://mathoverflow.net/users/69320 | Reference for multivariate orthogonal polynomials | The only book fully dedicated to the topic seems to be "Orthogonal Polynomials of Several Variables", by Charles F. Dunkl and Yuan Xu.
Cambridge University Press, 2001
| 6 | https://mathoverflow.net/users/11100 | 200374 | 96,762 |
https://mathoverflow.net/questions/200336 | 6 | Suppose $K \subset \mathbb{C}$ is a Cantor set and let $u:\mathbb{C} \setminus K \to \mathbb{R}$ be the maximal smooth function such that the conformal metric $e^{2u}(\mathrm{d}x^2 + \mathrm{d}y^2)$ has constant curvature $-1$ on $\mathbb{C} \setminus K$.
Suppose we put the Hausdorff distance on the set of compact su... | https://mathoverflow.net/users/7631 | Is the Poincaré metric continuous with respect to the domain? | In "D.A. Hejhal, Universal covering maps for variable regions, Mathematische Zeitschrift 137 (1974), 7--20." it is shown that the universal covering map of a hyperbolic domain depends locally uniformly continuously on the domain. The convergence of the derivatives follows from Cauchy's integral formula.
Since the hy... | 8 | https://mathoverflow.net/users/38319 | 200387 | 96,769 |
https://mathoverflow.net/questions/200371 | 0 | I recently started the study of vector bundles on $\mathbb{P}^n$, and started to read Rao's article 'A family of vector bundles on $\mathbb{P}^3$'. There, there is a notion of spectrum of a vector bundle. I searched this in some other articles, even in the excellent book "Vector Bundles on Complex Projective Space" by ... | https://mathoverflow.net/users/43027 | Vector Bundles of small rank | It seems that the notion of spectrum of vector bundles used in Rao's article is defined in Section 3 of
* C. Okonek and H. Spindler. Reflexive Garben vom Rang $r>2$ auf $\mathbb{P}^n$. Crelle Journal Reine Angew. Math. 344 (1983), pp. 38-64.
If you are having trouble reading German, I could give the definition her... | 2 | https://mathoverflow.net/users/50846 | 200388 | 96,770 |
https://mathoverflow.net/questions/200393 | 9 | I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their connections?
| https://mathoverflow.net/users/37103 | Cricket and the Hardy-Littlewod maximal function | One hesitates to explain a joke, but this is quite a nice joke, and I cannot resist answering a cricket question (especially now). Suppose a batsman scores 20, 100, 30, 40, 70 and 0 in his last six innings (0 being the most recent). Being upset at scoring 0 in his last innings, he might say to himself -- at least I am ... | 36 | https://mathoverflow.net/users/38624 | 200395 | 96,774 |
https://mathoverflow.net/questions/200303 | 3 | Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific down-to-earth detail that is hard to track down, namely charts that are complements of points and which are 'unitary' in the... | https://mathoverflow.net/users/4177 | 'Unitary' charts on odd-dimensional spheres | **Part I: The original question:**
Now that the question has been clarified, I can answer it. The answer is 'no', there is no CR-isomorphism $\phi: \mathrm{Heis}\to U$ that is unitary on the holomorphic tangent bundles.
To see this, it's probably better to look at the dual $1$-forms. Let $\alpha:T S^{2n+1}\to\mathb... | 5 | https://mathoverflow.net/users/13972 | 200400 | 96,776 |
https://mathoverflow.net/questions/200401 | 3 | Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^\*(M,F)$ is known. We want to obtain $H^\*(M/G;F)$. Is there any method or procedure to follow? Can I quotient the action directly $H^\*(M/G;F)=H^\*(M;F)/G$?
| https://mathoverflow.net/users/41075 | cohomology of the orbit space of a group action | If $F$ is a field of characteristic $0$, then $H^k(M/G;F)$ equals the invariants of the action of $G$ on $H^k(M;F)$. For two different proofs of this, see Proposition III.2.4 of Bredon's "Introduction to compact transformation groups" and Proposition 1.1 of my note "The action on homology of finite groups of automorphi... | 8 | https://mathoverflow.net/users/317 | 200403 | 96,777 |
https://mathoverflow.net/questions/200413 | 7 | In [Fifteen problems about MCG](http://arxiv.org/abs/math/0608325) Ivanov stated the following metaconjecture:
**Every object naturally associated to a surface S and having
a sufficiently rich structure has $Mod(S)$ as its groups of automorphisms.
Moreover, this can be proved by a reduction to the theorem about the a... | https://mathoverflow.net/users/9485 | Ivanov's metaconjecture on surface homeomorphisms | EDIT: Brendle-Margalit have released their paper. See [here](https://arxiv.org/abs/1710.08929).
---
One should observe that these are *not* all examples of Ivanov's metaconjecture (for instance, the automorphism group of the disk complex is the handle body group, not the whole mapping class group). In any case, D... | 12 | https://mathoverflow.net/users/317 | 200414 | 96,780 |
https://mathoverflow.net/questions/200348 | 2 | Based on a couple of references, it seems that the answer is yes, see for example
[Boneta-Dierolf, 1992](http://www.sciencedirect.com/science/article/pii/S0304020808703207) and [Bierstedt-Bonet, 1989](http://dmle.cindoc.csic.es/revistas/detalle.php?numero=592).
However, from a comment to the answer of [this MO questi... | https://mathoverflow.net/users/47925 | Is a Fréchet Montel space distinguished? | Yes, they are. This should be in Grothendieck's *Sur les espaces (F) et (DF)*. A good reference is the book *Introduction to Functional Analysis* of Meise and Vogt. Corollary 25.14 there says that reflexive Frechet spaces are distinguished (and, of course, Frechet Montel spaces are reflexive).
| 4 | https://mathoverflow.net/users/21051 | 200424 | 96,784 |
https://mathoverflow.net/questions/200344 | 2 | For a problem I am working over, I would like to prove that numbers of the following type are not squares
$p(l^4+6l^2m^2-3m^4)$
where $p,l,m$ are integers an $p$ prime. I have already found various necessary condition but could not conclude what I want, so I won't include them here in order not to lead you astray. ... | https://mathoverflow.net/users/69391 | Can you find squares in this class? | Here's a bit more detail about my comment. We're searching for integer solutions to $y^{2} = p(l^{4} + 6l^{2}m^{2} - 3m^{4})$. Assume for simplicity that $\gcd(l,m) = 1$ and $p \geq 5$. If $p \equiv 2 \pmod{3}$ and $\gcd(l,3) = 1$, then the right hand side is $\equiv 2 \pmod{3}$, which is a contradiction, while if $l$ ... | 6 | https://mathoverflow.net/users/48142 | 200432 | 96,786 |
https://mathoverflow.net/questions/200422 | 5 | If $X$ and $Y$ are two spectra, I denote by $F(X,Y)$ their mapping spectrum. This is uniquely determined by the existence of a natural isomorphism $[X\wedge Y, Z]\cong [X,F(Y,Z)]$.
I denote by $H\_\*$ the rational homology functor. If $X$ is a finite spectrum, I have an isomorphism $H\_\*(F(X,Y))\cong Hom(H\_\*(X),H\... | https://mathoverflow.net/users/10707 | homology of a mapping spectrum | This was a comment, but at the OP's suggestion I have promoted it to an answer.
There is no Milnor exact sequence for homology groups of an inverse limit, only for the homotopy groups. If $I$ is the Brown-Comenetz dual of the sphere, then $H\_∗(I)=0$ but $F(I,I)$ is the profinite completion of S and so $H\_∗F(I,I)≠0$... | 6 | https://mathoverflow.net/users/10366 | 200434 | 96,787 |
https://mathoverflow.net/questions/200423 | 1 | Let $f:X \to Y$ be a proper surjective morphism of reduced connected noetherian schemes. Assume $Y$ is irreducible. Let $y \in Y$ be a closed point. Denote by $X\_y$ the fiber over $y$ to the morphism $f$. Suppose that $X\_y$ is a non-singular, irreducible variety and for every closed point $x \in X\_y$, the tangent sp... | https://mathoverflow.net/users/58203 | Relative tangent space to proper morphism and irreducibility of fibers | That is certainly not true. Let $Y$ be $\mathbb{A}^2$ with coordinates $(s,t)$. Denote by $[u\_0,u\_1]$ homogeneous coordinates on $\mathbb{P}^1$. Let $X$ be the hypersurface of $\mathbb{A}^2\times \mathbb{P}^1$ with homogeneous defining equation $stu\_1 = 0$. Let $f:X\to Y$ be the restriction to $X$ of the projection ... | 2 | https://mathoverflow.net/users/13265 | 200436 | 96,788 |
https://mathoverflow.net/questions/200425 | 4 | I am currently doing my masters studies in financial mathematics. However, I have had a good background in number theory and I don't feel like leaving it just like that. I am thus inquiring on any applications of algebraic number theory in financial mathematics. When I finish this degree, can I be allowed to enroll for... | https://mathoverflow.net/users/69424 | Algebraic Number Theory in Financial Mathematics | I do not know any applications of algebraic number theory in financial mathematics. However, there are attempts to use string theory inspired approaches to financial markets: <http://arxiv.org/abs/1109.0435> (The string prediction models as an invariants of time series in forex market, by R. Pincak and M. Repasan). On ... | 1 | https://mathoverflow.net/users/32389 | 200438 | 96,789 |
https://mathoverflow.net/questions/200442 | 1 | For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative rings). The arrow
$$
R^X\otimes\_R R^Y\rightarrow R^{X\times Y}
$$
is given by the product $f\otimes g\rightarrow ((x,y)\r... | https://mathoverflow.net/users/25256 | Commutation of tensor products with inverse limits in a specific case | The key point is to generalize the problem in order to make more effective the use of limits. Consider more generally for any $R$-module $R$ the natural map
$$f\_{M,Y}: M \otimes\_R \prod\_{y \in Y} R \rightarrow \prod\_{y \in Y} M$$
given by $m \otimes (r\_y) \mapsto (r\_y m)$. In the special case $M = R^X$ this recov... | 4 | https://mathoverflow.net/users/61939 | 200443 | 96,792 |
https://mathoverflow.net/questions/175812 | 14 | Shelah's Main Gap Theorem states that for all first-order, complete theories, T, in a countable language, we have that either $$I(T,\aleph\_\alpha)=2^{\aleph\_\alpha}$$ or $$I(T,\aleph\_\alpha)<\beth\_{\omega\_1}(\alpha)$$ This result, while beautiful, leaves some questions. Can analogues of this result be extended to ... | https://mathoverflow.net/users/51323 | Main Gap Phenomenon | Extending Shelah's main gap to non first-order or even to first-order theories in an uncountable language is a major hard open problem.
I am familar with couple of extensions to non f.o. (both papers are available from my web page):
1. Rami Grossberg and Bradd Hart. The classification theory of excellent classes, ... | 9 | https://mathoverflow.net/users/68958 | 200451 | 96,797 |
https://mathoverflow.net/questions/200470 | 19 | For subsets $A$ and $B$ of $[0,1)$, say $A\sim B$ iff $\lambda(A\Delta B)=0$ where $\lambda$ is Lebesgue measure.
**Question:** How many equivalence classes of subsets of $[0,1)$ are there given AC?
I would guess the answer is $2^c$ given AC, but I haven't got a proof.
What got me thinking about this was trying... | https://mathoverflow.net/users/26809 | How many subsets of $[0,1)$ are there modulo null sets? | Yes, there are $2^\mathfrak{c}$ equivalence classes. In fact, I claim that there is a collection $S$ of $\mathfrak{c}$ disjoint non-null subsets of $[0,1)$; taking all unions of subcollections of $S$ gives $2^\mathfrak{c}$ inequivalent subsets of $[0,1)$.
To construct this $S$, let $N$ be the set of all null Borel se... | 20 | https://mathoverflow.net/users/75 | 200473 | 96,801 |
https://mathoverflow.net/questions/200367 | 2 | Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ acts on $(M,g,\omega)$ with corresponding moment map $\mu\colon M\to\mathbb{R}^{\ast}$. The symplectic reduction manfiold... | https://mathoverflow.net/users/66688 | Symplectic reduction: from indefinite signature to Riemannian signature | Just take $R^4=C^2$ with complex
coordinates $z\_1=x\_1+ i y\_1, z\_2= x\_2 + i y\_2$ and the flat
metric $-dz\_1 d\bar z\_1 + dz\_2 d \bar z\_2$. It has signature (2,2). As the group of isometries take the group of $x\_1 $-translations, it preserves the canonic symplectic form $dx\_1\wedge dy\_1 + dx\_2 \wedge dy\... | 1 | https://mathoverflow.net/users/14515 | 200481 | 96,806 |
https://mathoverflow.net/questions/200454 | 2 | Let $\mathbf{Z,R}$ two Hermitian semidefinite positive matrices with all eigenvalues larger than one. Intuition drives me that
>
> $\mathbf{R}^{-1/2}\mathbf{Z} \left(\mathbf{R}^{-1/2}\right)^H - \mathbf{Z} \preceq \mathbf{0}$
>
>
>
Any idea of how to proceed with the inequality verification?
Thank you.
| https://mathoverflow.net/users/11825 | Matrix inequality | Let $A$ and $B$ be Hermitian positive definite matrices. Then, the following matrix
\begin{equation\*}
\begin{pmatrix}
A & X\\
X^\* & B
\end{pmatrix}
\end{equation\*}
is positive definite *if and only if* $X=A^{1/2}ZB^{1/2}$ for some $Z$ that satisfies $\|Z\| \le 1$ (thus, the range of $X$ is a subspace of the range of... | 2 | https://mathoverflow.net/users/8430 | 200484 | 96,807 |
https://mathoverflow.net/questions/200421 | 2 | Motivated by this post on [cubic graphs decompositions](https://mathoverflow.net/questions/156455/cubic-graphs-decompositions) and the connection to Barnette's conjecture, I am interested in decomposing a connected bridgeless cubic graph into edge-disjoint paths of length 3 (P4). My intuition is that it should be $NP$-... | https://mathoverflow.net/users/8784 | Connection between Barnette conjecture and hardness of cubic graph decomposition | It is easy to prove the following result:
Proposition. Every bridgeless cubic graph admits a decomposition into paths of length 3.
Petersen (1891) proved that every bridgeless bipartite graph contains a perfect matching (<http://en.wikipedia.org/wiki/Petersen%27s_theorem>).
Koztig (57) proved that a cubic graph ... | 7 | https://mathoverflow.net/users/69461 | 200485 | 96,808 |
https://mathoverflow.net/questions/200486 | 1 | This topic came out today during a discussion with a colleague. I realized that a counter-example to his claim could be constructed if there exits a subset $A \subset [0,1]$ such that $0 < \mu(A) < 1$ (where $\mu$ is Lebesgue measure), $A$ is dense in $[0,1]$ (with respect to the standard topology on $\mathbb{R}$), $\t... | https://mathoverflow.net/users/10898 | Are there dense sets of positive but not full measure? | By the Central Limit Theorem, something like this should have the
property that $0<\mu([a,b]\cap A)<b-a$ for all $0\leq a<b\leq 1$. Let
$0.a\_1a\_2\cdots$ be the binary expansion of $x\in[0,1]$. Let $A$
consist of all $x$ such that for all $n$ sufficiently large,
$$ -1 < \frac{\frac n2-(a\_1+a\_2+\cdots+a\_n)}{\sqrt{n... | 3 | https://mathoverflow.net/users/2807 | 200491 | 96,810 |
https://mathoverflow.net/questions/200462 | 12 | Say that $a\_1, \cdots, a\_{n-1}$ is an independent generating set for $S\_n$. Let $b$ be any element in $S\_n$. Is it true that $b$ can replace one of the generators, i.e. that there exists an index $i$, such that we have that $a\_1,\cdots, \hat{a\_i},\cdots, a\_{n-1}, b$ generate $S\_n$?
If $a\_1, \cdots, a\_{n-1}$... | https://mathoverflow.net/users/69443 | generating set for symmetric group $S_n$ | The answer is yes.
There is a paper by Cameron and Cara which describes all maximal generating sets of length $n-1$ of $S\_n$. They are not very hard to describe and basically are variants of the standard $n-1$ length generating sets.
<http://www.maths.qmul.ac.uk/~pjc/preprints/igsgsn.pdf>
Cameron and Cara say bu... | 10 | https://mathoverflow.net/users/69463 | 200492 | 96,811 |
https://mathoverflow.net/questions/200482 | 1 | Is there any (simple) approximation of this Hypergeometric function: $ \_2F\_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$.
Thanks!
| https://mathoverflow.net/users/51469 | Approximation of $ _2F_1((b-1)a,b;ba;x) $ | There are Pade approximations. See [these slides by Matala-aho.](http://cc.oulu.fi/~tma/TOKYOSLIDES.pdf) I believe the results in the classical case you are asking about are due to Grisha Chudnovsky (1979, cited in the slides).
| 0 | https://mathoverflow.net/users/11142 | 200494 | 96,812 |
https://mathoverflow.net/questions/200497 | 3 | In the paper *configuration spaces: applications to Gelfand-Fuks cohomology*, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use to obtain Theorem 1?
Theorem 1:
Let $M=R^n\times V$, $V$ connected, $M$ a manifold of dimension $m$. How to obtai... | https://mathoverflow.net/users/41075 | cohomology ring of configuration spaces | I believe the proof is [published in Geometric Applications of Homotopy Theory I.](http://www.dropbox.com/s/w5iqazs0if3300e/CohenTaylor.pdf?dl=0)
| 2 | https://mathoverflow.net/users/11142 | 200500 | 96,815 |
https://mathoverflow.net/questions/200507 | 2 | I am Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma \in (0,1)$, where $M$ is a compact manifold.
Any references are appreciated.
PS
I am also looking for a reference that gives interpolation inequalities that justify (for when $u\_\epsilon \in C(I, H^k) \ca... | https://mathoverflow.net/users/13904 | Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma\in (0,1)$ | Surely there is a typo for the first inclusion, since you have strictly less than a derivative in $L^2(M)$ on the left hand side and one full derivative in $L^{\infty}(M)$ on the right hand side.
Regarding the interpolation question, that's an application of the usual interpolation inequality between $H^k(M)$ and $H... | 3 | https://mathoverflow.net/users/62629 | 200509 | 96,817 |
https://mathoverflow.net/questions/194674 | 4 | Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int\_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth and rapidly decaying at infinity, for all derivatives). Then $T$ is Hilbert-Schmidt (in particular, compact) with Hilbert... | https://mathoverflow.net/users/23753 | Nuclearity properties of Integral operators with Schwartz type kernels | To answer my own question: If the kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ is of Schwartz type, then the associated operator $T$ is $p$-nuclear for all $p>0$, i.e. its singular values satisfy $\sum\_k\mu\_k^p<\infty$ for all $p>0$, i.e. they decay faster than any inverse power of $k$.
One way of seeing this is as ... | 1 | https://mathoverflow.net/users/23753 | 200523 | 96,821 |
https://mathoverflow.net/questions/200528 | 6 | I've been reading a little bit about the definition of symmetries on General Relativity, and they are related with the concept of *Killing vector*, i.e., vectors along which the Lie derivative of the metric vanishes $\mathcal{L}\_X g =0$.
However, afaik the most symmetric geometrical object is the Ricci tensor ([see ... | https://mathoverflow.net/users/25356 | Is there a way to define a Lie derivative of a connection? | Of course, yes. Lie derivative is defined for any geometric object (= when it is defined what happends when we change a coordinate system): take the flow $\phi\_t$ of the vector field, consider the pullback $\phi\_t^\*\Gamma$ of your geometric object $\Gamma$ and define Lie derivative as the $\tfrac{d}{dt}$-derviative ... | 14 | https://mathoverflow.net/users/14515 | 200531 | 96,823 |
https://mathoverflow.net/questions/200539 | 6 | A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in ZFGC$^{\text{−f}}$+BAFA, there are nontrivial automorphisms and elementary embeddings of the universe $V$ into itself.
Accordingly, Kunen inconsistency is circums... | https://mathoverflow.net/users/69488 | $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication | I'm glad to hear you're reading our paper, which can be found here: [The foundation axiom and elementary self-embeddings of the universe](http://jdh.hamkins.org/foundation-axiom-and-self-embeddings-of-the-universe/). Click through to the arxiv for a pdf — and I note that the title you mention is from an earlier draft o... | 5 | https://mathoverflow.net/users/1946 | 200544 | 96,825 |
https://mathoverflow.net/questions/200516 | 3 | is there a reference on the structure of the space of metrics on a compact manifold that induce a given measure $\mu $?
i have a given manifold $M$, a given measure $\mu$ with an everywhere positive $C^{\infty}$ density with respect to Lebesgue. i want to construct a metric in this class satisfying some additional pr... | https://mathoverflow.net/users/69474 | structure of metrics on a compact manifold | Have a look at
* Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. Journal of Differential Geometry 94, 2 (2013), 187-208. [(pdf)](http://www.mat.univie.ac.at/~michor/rie-met2.pdf).
| 0 | https://mathoverflow.net/users/26935 | 200549 | 96,827 |
https://mathoverflow.net/questions/200501 | 3 | Find all symmetric matrices $X=X^{T}$ such that
\begin{align}
XDX^{T}=-D \quad (1)
\end{align}
where $D\ne 0$ is a real diagonal matrix.
For example, $X=iI$ satisfies $(1)$. Can you get a necessary and sufficient characterization for such matrices $X$? Thanks a lot.
| https://mathoverflow.net/users/58802 | quadratic matrix equation | Since $XD^2=XD(-XDX)=-XDXDX=(-XDX)DX=D^2X$, the matrices $X$ and $D^2$ commute. W.L.O.G we can assume that entries of $D$ which have the same absolute value are consecutive (multiply with a permutation matrix). It follows that $X$ has block diagonal form where each absolute value corresponds to a block. The block corre... | 3 | https://mathoverflow.net/users/35593 | 200559 | 96,832 |
https://mathoverflow.net/questions/200235 | 0 | $n$ number of balls are thrown randomly to $m$ number of bins, standing in a row. The balls are labeled as $1,2,3,....n$ and bins are also labeled as $1,2,3,...,m$. The probability of $i\_{th}$ ball enters in the $j\_{th}$ bin is $p\_{ij}$ where $\sum\_{j=1}^{m}p\_{ij}=1$ for all $i$. What is the expected number of bal... | https://mathoverflow.net/users/64387 | Generalized expression for balls and bins problem | The odds of having all the balls in bins $j\_1, \cdots, j\_d$ is $\Pi\_{i\leq n}(p\_{ij\_1} + \cdots + p\_{ij\_d})$; call this number $q\_{j\_1, \cdots, j\_d}$
By inclusion/exclusion we get that the odds the that set of nonempty bins is exactly $J = \{j\_1, \cdots, j\_d\}$ are $\sum\_{S \subset J} (-1)^{d - |S|} q\_S... | 1 | https://mathoverflow.net/users/25229 | 200561 | 96,833 |
https://mathoverflow.net/questions/200540 | 3 | Let $(X,\mathcal{O}\_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}\_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is a two-side bounded complex of finitely generated locally free $\mathcal{O}\_X)$-modules.
Then we have the following ... | https://mathoverflow.net/users/24965 | An alternative definition of pseudo-coherent complex | **No**, this is true under noetherian hypothesis. See the relevant exposé by Illusie in SGA 6, It is related to the phenomenon that not every finitely presented module is coherent whenever the ring is not coherent itself.
| 5 | https://mathoverflow.net/users/6348 | 200567 | 96,836 |
https://mathoverflow.net/questions/185723 | 8 | Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?
What methods are available for proving such a property for some family of polytopes?
Remember, the Ehrhart polynomial $p(k)$ for a convex polytope $P$ with integer vertices is given by the ... | https://mathoverflow.net/users/1056 | Positivity of Ehrhart polynomial coefficients | I have converted a comment (slightly modified) to the following answer as requested by Per Alexandersson. If $Ω\_P(k)$ is the order polynomial of a poset $P$, then $Ω\_P(k+1)$ is the Ehrhart polynomial of the order polytope $\mathcal{O}(P)$. For any n≥1, the Ehrhart polynomial of the order polytope of the poset $P\_n$ ... | 10 | https://mathoverflow.net/users/2807 | 200574 | 96,838 |
https://mathoverflow.net/questions/200584 | 21 |
>
> ***Q***. When did the notion of *homeomorphism* reach its
> modern formulation as a bicontinuous bijection, i.e., a
> continuous bijection
> between topological spaces whose inverse is also continuous?
>
>
>
Was this present in Riemann's work (1826-1866)?
Or in the work of Möbius (1790-1868); or Jordan ... | https://mathoverflow.net/users/6094 | Homeomorphism historically: When did it reach its modern formulation? | Well, all I did was a search on "homeomorphism history", but... I tried to extract some points that are made in conjunction to your question (Riemann, Möbius, Jordan), though feel free to edit it down if it is too long (and apologies to those who think this should be remapped to a History of Math Q/A).
**The evolutio... | 42 | https://mathoverflow.net/users/69507 | 200585 | 96,841 |
https://mathoverflow.net/questions/200297 | 2 | The class $\textsf{Cs}\_{\omega}^{reg}\cap \textsf{Lf}\_{\omega}$ of locally finite and regular cylindric set algebras (of dimension $\omega$) can be seen as the algebraic counterpart of first-order models (details below).
I also know that we can associate, to each model, a representable cylindric algebra in $RCA\_{\... | https://mathoverflow.net/users/69371 | Representable cylindric algebras and correspondence with first-order models | $RCA$ is defined to be $SP(Cs)$, the subalgebras of products of cylindric set algebras. It is written $Gs$ in Henkin, Monk, and Tarski's book Cylindric Algebras Volumes I and II. (Or pedantically $IGs$, the isomorphic copies.) It turns out this is a variety, and that book HMT II gives equations characterizing it in sec... | 3 | https://mathoverflow.net/users/69470 | 200587 | 96,842 |
https://mathoverflow.net/questions/200505 | 3 | Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a smooth curve that is such that this results in a 2-dimensional manifold.
How can I tell from the properties of $M$ whether... | https://mathoverflow.net/users/42714 | Determining the Fourier transform | An interesting article may be *A sharp form of the Cramér-Wold theorem* by Cuesta-Albertos, Fraiman and Ransford, which can be found here: [klick](http://link.springer.com/article/10.1007/s10959-007-0060-7)
Their Theorem 3.1 states that under a moment condition a measures is well-defined by the values of the Fourier-... | 0 | https://mathoverflow.net/users/42537 | 200616 | 96,850 |
https://mathoverflow.net/questions/200597 | 6 | (Cross-posted from math-SE).
I am trying to estimate the values of the following integral for large $n$,
$$\frac{1}{n!}\intop\_{\Omega}\prod\_{1\leq i<j\leq n}(x\_{j}-x\_{i})^{2}\,\prod\_{j=1}^{n}e^{-x\_{j}^{2}}dx\_{j},$$
where $\Omega$ is one of these infinite rectangular regions in $\mathbb{R}^n$:
$$\Omega\_1^{(n... | https://mathoverflow.net/users/22773 | Upper bound for a Selberg-type integral over a rectangular region | Mathematica can do these integrals explicitly, for small $n$. One gets
$$
\frac1{n!}\int\_{\Omega\_1^{(n)}}\prod\_{1\leq i<j\leq n}(x\_i-x\_j)^2
\prod\_{i=1}^ne^{-x\_i^2}\,dx = c\_n\,\pi^{\frac{n-1}2}
e^{-\lambda^2}\lambda^{2n-3}\left(1+O\left(\frac1{\lambda^2}\right)\right) \tag{1}
$$
with $c\_2=\frac14$, $c\_3=\fr... | 7 | https://mathoverflow.net/users/69194 | 200622 | 96,853 |
https://mathoverflow.net/questions/200633 | 3 | Consider the following equivalence relation on $\{0,1\}^\omega$:
>
>
> >
> > $x\simeq y$ iff there is $n\in\omega$ such that $x(k)=y(k)$ for all $k\in\omega$ with $k\geq n$.
> >
> >
> >
>
>
>
It is easy to see that the following is a well-defined ordering relation on $\{0,1\}^\omega/\simeq$ :
>
>
> >... | https://mathoverflow.net/users/8628 | Is $\{0,1\}^\omega$ the order-preserving image of $\{0,1\}^\omega$ modulo some finiteness relation? | Yes, in fact, there is a surjective lattice-homomorphism. Your two posets are better known as the Boolean algebras $P(\omega)/fin$ and $P(\omega)$ ($fin$ being the ideal of finite sets). Let $\omega=\bigcup A\_n$ be a partition of $\omega$ into infinitely many infinite sets and let $U\_n$ be a nonprincipal ultrafilter ... | 7 | https://mathoverflow.net/users/75 | 200637 | 96,857 |
https://mathoverflow.net/questions/200027 | 1 | In a paper I am reading on the Hankel transform ([this paper](http://arxiv.org/abs/1205.2268) to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a convolution theorem for the Hankel transform in mind. The Hankel transform, as given in the ... | https://mathoverflow.net/users/20460 | Motivating the Bessel translation operator | After some thought, I decided to see what happens in the Fourier case. Since $j\_{\alpha}$ is an eigenfunction of $D\_{\alpha}$, it stood to reason that we should look to the following PDE in the Fourier case:
$$ \frac{\partial}{\partial x}u(x,y) = \frac{\partial}{\partial y}u(x,y) \tag{1}$$
since the exponential i... | 1 | https://mathoverflow.net/users/20460 | 200640 | 96,858 |
https://mathoverflow.net/questions/200639 | 4 | Consider the random variable $Y = Y\_1 + \dots + Y\_k$, where each $Y\_i$ is iid distributed as a geometric random variable with sucess probability $p$; here we should think of $p$ as being close to zero. The mean of $Y$ is $k (1-p) / p \approx k/p$. I would like to show that there is a large probability that $Y$ is ab... | https://mathoverflow.net/users/46681 | Anti-concentration for sums of geometric random variables | This is just the [negative binomial distribution](https://en.wikipedia.org/wiki/Negative_binomial_distribution#Sum_of_geometric_distributions).
So you can work with the corresponding cdf, [the regularized incomplete beta function](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function).
| 5 | https://mathoverflow.net/users/4600 | 200643 | 96,859 |
https://mathoverflow.net/questions/200613 | 5 | Let $S$ be a set and $\vartheta$ be an equivalence relation on $S$. We say that $\vartheta$ is *proper* if there are $x\neq y\in S$ with $(x,y)\in\vartheta$.
Is there an infinite Hausdorff space $(X,\tau)$ such that for every proper equivalence relation $\vartheta$ on $X$ we have $X\not\cong X/\vartheta$?
| https://mathoverflow.net/users/8628 | Infinite Hausdorff space that is not homeomorphic to any proper quotient | Let $X$ be a strongly rigid infinite Hausdorff space. Let $\vartheta$ be an equivalence relation on $X$. Let $q \colon X \to X/\vartheta$ be the quotient map. If there is a homeomorphism $f \colon X/\vartheta \to X$, then $(f \circ q)$ is a continuous map from $X$ to itself; hence constant or the identity. Since $X$ is... | 3 | https://mathoverflow.net/users/21815 | 200647 | 96,861 |
https://mathoverflow.net/questions/200635 | 3 | I would like to see what names that has been suggested for useful modal axioms. By name here I mean some abbreviation such as $T$, $K$, $4$, $.2$, $E$ and so on. In particular I am interested in suggestions that would amount to a good name for the useful modal axiom $\Box\Box\alpha\rightarrow\Box\alpha$ which correspon... | https://mathoverflow.net/users/37385 | Is there a good list of nomenclature for modal axioms? | One of the most complete online summaries of modal logic systems is still [Halleck's list](http://home.utah.edu/~nahaj/logic/structures/systems/). Even though the axiom corresponding to *density* is not explicitly identified in the latter list, it is a particular example of the Chellas/Lemmon/Geach **G**$(a,b,c,d)$-axi... | 5 | https://mathoverflow.net/users/34544 | 200651 | 96,864 |
https://mathoverflow.net/questions/200626 | 4 | Given a compact, simple Lie group $G$, and a compact, oriented three manifold $M$, we can consider the following smooth homotopy groups:
>
> * $(C^\infty(M;G)/{\sim},\cdot )$: Here, $\sim$ is smooth homotopy equivalence, and $(g\cdot h)(p):=g(p)\cdot h(p)\in G$.
> * $(C^\infty(M;G)/{\sim},\circ )$: Here, $\circ$ is... | https://mathoverflow.net/users/69531 | Computational trick used in QFT and the Jones Polynomial | I don't know what you mean by "composition as in $\pi\_3(G)$" unless either $M$ is a $3$-sphere or $G$ is, in addition, assumed to be simply connected. If the latter, then $G$ is $2$-connected, and so any map $M \to G$ factors up to homotopy through a map $M \to S^3$ inducing an isomorphism on $H^3$ (here I'm assuming ... | 4 | https://mathoverflow.net/users/290 | 200652 | 96,865 |
https://mathoverflow.net/questions/200572 | 3 | Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $P$ a submanifold of dimension $k.$ Let us define the tube of radius $r$ about $P$ by
$$T(P,r):=\{x\in M: d(x,P)\le r\}$$ and the tubular hypersurface at distance $t$ from $P$ by
$$P\_t=\{x\in T(P,r):d(x,P)=t\}.$$
Let $p\in P$ be a point of the submanifol... | https://mathoverflow.net/users/51420 | Conditions for tubular hypersurfaces to be a Riemannian product | Here is the counterexample.
Consider the metric on the tangent bundle $TS^2$ of the 2-sphere, given by the Riemannian submersion $S^3\times (R^2,g) \to S^3\times (R^2,g)/S^1$, where $S^3$ is the 3-sphere of unit quaternions, and $S^1$ acts on it (as usual) by multiplications by unit complex numbers (same action on $R^... | 1 | https://mathoverflow.net/users/1988 | 200663 | 96,870 |
https://mathoverflow.net/questions/200664 | 14 | In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated surgeries on knots.
What are the methods to show this? In the simplest intuitive level?
I understand this... | https://mathoverflow.net/users/44768 | Obtain any 3-manifold from repeating surgeries on knots in $S^3$ | This is a result of Lickorish, in his paper "A representation of orientable combinatorial 3-manifolds". The paper is only eleven pages, and is very readable. In his proof, Lickorish rediscovers some ideas first investigated by Max Dehn 40 years earlier.
Lickorish's theorem was unexpected at the time; I don't think t... | 21 | https://mathoverflow.net/users/1650 | 200667 | 96,871 |
https://mathoverflow.net/questions/200656 | 22 | In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$ with his pairing function. Is there a natural bijection bethween these sets?
Naturalness is of course not ... | https://mathoverflow.net/users/37385 | Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$? | There is a following result which is quite lovely, I think (I don't remember right away whose result this is):
Let us define a function $f\colon\mathbb{N}\to\mathbb{Q}^+$ as follows: $f(1)=1$, and also $f(2n)=f(n)+1$, $f(2n+1)=\frac{1}{f(n)+1}$. Then:
1. $f$ is a bijection (Sketch of a proof: A. Show using inductio... | 41 | https://mathoverflow.net/users/1306 | 200681 | 96,877 |
https://mathoverflow.net/questions/200630 | 2 | Here is my question : I have a harmonic function $h$ on the open unit disc in $D \subset \mathbb{C}$, such that $\iint\_D e^{2h} d\lambda(z) \leq A < \infty$ ($d\lambda$ is the Lebesgue measure on $\mathbb{C}$).
Can one have an upper bound of $e^h$ near the boundary of the disc ?
With elementary tools (mean value ... | https://mathoverflow.net/users/69533 | Growth of a harmonic function on the disc | Function $f(z)=(1-z)^\alpha$ with $\alpha=-1+\epsilon$ is zero-free in the unit disk, and
its coefficients satisfy $a\_n\sim cn^{-\alpha-1}=cn^{-\epsilon}$.
Setting $u=\log|f|$, we obtain, using Parseval,
$$\int\_{|z|<1}e^{2u}d\lambda=\int\_{|z|<1}|f|^2d\lambda=\sum\_n\frac{1}{2n+1}|a\_n|^2<\infty.$$
So your trivial es... | 2 | https://mathoverflow.net/users/25510 | 200682 | 96,878 |
https://mathoverflow.net/questions/200700 | 2 | Let $A\subseteq B$ be normal affine doamins over a field $k$ with same field of fractions. If the induced morphism of schemes $i^\*:Spec\ B \rightarrow Spec\ A$ is an open immersion, how to prove that the complement of $Spec\ B$ in $Spec\ A$ is a divisor?
| https://mathoverflow.net/users/66365 | complement of an open immersion | Let $Z$ be an irreducible component of the complement; its generic point corresponds to a prime ideal $\mathfrak{p}$ of $A$. Put $S:=\mathrm{Spec}(A\_{\mathfrak{p}})$ and $s:=\mathfrak{p}A\_{\mathfrak{p}}$. Consider the affine morphism $j: S\rightarrow \mathrm{Spec}(A)$. Then $S\smallsetminus \{s\} =j^{-1}(\mathrm{Spec... | 4 | https://mathoverflow.net/users/40297 | 200703 | 96,885 |
https://mathoverflow.net/questions/200699 | 9 | From Wikipedia.
In linear algebra, two n-by-n matrices A and B are called similar if
$$ B = P^{-1} A P$$
for some invertible n-by-n matrix $P$.
If $P$ is a permutation matrix, $A$ and $B$ are **permutation similar**.
Two graphs $G,H$ are isomorphic iff their adjacency matrices
$A\_G,A\_H$ are permutation simila... | https://mathoverflow.net/users/12481 | When are the adjacency matrices of non-isomorphic graphs similar? | There is no characterization known of when a graph is determined by its spectrum. The probability that a tree on $n$ is determined by its characteristic polynomial goes to zero as $n$ tends to infinity. It is *conjectured* that for graphs the probability goes to one.
| 9 | https://mathoverflow.net/users/1266 | 200709 | 96,887 |
https://mathoverflow.net/questions/200722 | 8 | It's well known that if $\alpha $ is a rational root to an integer coefficient polynomial, then its denominator divides the leading coefficient and its numerator divides the constant term. I'm asking if there is an analog if the coefficients of the polynomial live in a number field and I'm looking for solutions in that... | https://mathoverflow.net/users/37103 | Root criterion for polynomial over number fields | Let $f(x) = a\_nx^n+\cdots+a\_0$ be a polynomial with coefficients in the ring of integers $\mathcal{O}\_K$ of a number field $K$. Then for every nonzero root $\alpha$ of $f$ in $K$ one has $a\_n \alpha, \frac{a\_0}{\alpha} \in \mathcal{O}\_K$. This can be seen by the simple observation that $a\_n \alpha$ and $\frac{a\... | 8 | https://mathoverflow.net/users/51663 | 200726 | 96,893 |
https://mathoverflow.net/questions/200711 | 4 | Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?
| https://mathoverflow.net/users/38889 | Finite p-groups and their fibered products | No, you cannot obtain all finite $p$-groups in this way; you even miss some $p$-groups of nilpotency length 2 (although you have all abelian $p$-groups as I mentioned in a comment).
Lemma: Let $G$ be a fibered product of $G\_1$ and $G\_2$. Let $L$ be a subgroup of $G$ both of whose projection on $G\_1$ and $G\_2$ are... | 6 | https://mathoverflow.net/users/14094 | 200727 | 96,894 |
https://mathoverflow.net/questions/200357 | 14 | Imagine you sample $n$ numbers with replacement uniformly from the integers $1,\dots, n$ (we can assume $n$ is large). Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but with a twist. All I know is that the samples are uniform and $k$-wise independent for some $k$.
>
> What is the small... | https://mathoverflow.net/users/45564 | Smallest $k$ so that $k$-wise independence guarantees a constant expected minimum | I can do $k\geq 4$. This is done using a method similar to the upper bound from last time. Let $N\_m$ be the number of samples that are at most $m$. Then we wish to upper bound the probability that $N\_m=0$. We can do this using the fourth moment method, because the first four moments are the same as for a totally inde... | 9 | https://mathoverflow.net/users/18060 | 200730 | 96,895 |
https://mathoverflow.net/questions/200737 | 4 | *I asked this question over at [Math.StackExchange](https://math.stackexchange.com/questions/1182799/matrix-elements-of-real-represententations) and despite having had a bounty on it I did not receive an answer.*
Suppose that $G$ is a finite group and we have a unitary irreducible representation $\rho:G\rightarrow \h... | https://mathoverflow.net/users/35482 | Matrix Elements of Real Representations | The following is more of a long comment than a proper answer.
Let $\chi$ be the character of $\rho$. Look at the character $\overline\chi$ of $\overline\rho$. We have $\overline\chi(s)=\overline{\chi(s)}$ for $s\in G$. As
the trace is preserved under matrix conjugation, and as $\overline\chi(s)=Tr(\overline\rho(s))$... | 3 | https://mathoverflow.net/users/11100 | 200741 | 96,902 |
https://mathoverflow.net/questions/200658 | 5 | The group $\operatorname{SL}\_2(\mathbb Z)$ acts on polynomials in two variables $\mathbb C[x,y]$ via $A\cdot f(x,y)\mapsto f(A^{-1}.(x,y))$ where $(x,y)$ is regarded as a column vector. There are two standard matrices $S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ and $T=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ which generate... | https://mathoverflow.net/users/9417 | Dimensions of a vector space akin to modular symbols | Martin Kassabov sent me an elegant solution to this problem. Start with the observation that $\epsilon S$ and $ST$ generate a copy of the symmetric group $S\_3$. Now decompose $R$ into a direct sum of irreducible $S\_3$-modules. Modding out by $1+ST+(ST)^2$ and $1-\epsilon S$ kill both $1$ dimensional representations a... | 3 | https://mathoverflow.net/users/9417 | 200750 | 96,907 |
https://mathoverflow.net/questions/200582 | 3 | The Blanchfield pairing is usually defined on the homology of the infinite cyclic cover over the knot exterior.
In his article "cobordism of satellite knots", Litherland works with the $0$-framed surgery instead of the knot exterior.
What is the difference between the two resulting pairings?
| https://mathoverflow.net/users/36098 | Blanchfield pairing: knot exterior versus $0$-framed surgery | They are essentially identical.
Let $K$ be a knot with exterior $X$ and $M$ the closed 3-manifold
obtained by 0-framed (longitude) surgery on $K$,
and let $X\_\infty$ and $M\_\infty$ be the infinite cyclic covers.
Thus $M\_\infty$ is obtained from $X\_\infty$ by adjoining a 2-cell to kill the longitude and then a 3-ce... | 3 | https://mathoverflow.net/users/58488 | 200754 | 96,908 |
https://mathoverflow.net/questions/200746 | 4 | Does $\textsf{PA}$+Con($\textsf{PA}$) entail the existence of non-standard models of $\textsf{PA}$?
Is there a reasonable way in which to code, inside $\textsf{PA}$, the statement that $\textsf{PA}$ has non-standard models?
If so, suppose we start from $\textsf{PA}$ and iteratively add consistency statements, obtai... | https://mathoverflow.net/users/69371 | Does PA+Con(PA) entail the existence of non-standard models of PA? | $PA$ is a weird theory to work with, here: if you want to talk about models, then a two-sorted theory like $RCA\_0$ or $ACA\_0$ is probably better. $PA$ cannot directly talk about models, since it can't directly talk about sets.
However, $PA$ (in fact, much less than $PA$) proves that if $PA$ is consistent, then so i... | 12 | https://mathoverflow.net/users/8133 | 200756 | 96,909 |
https://mathoverflow.net/questions/200764 | 7 | Let $p\_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E\_n(\mathbb{R}^\infty)\to G\_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, *The Cohomology of BSO n and BO n with Integer Coefficients, Proceedings of the American Mathematical Society 1982 Vol 85-2, Edgar H.Brown JR.*,
$H^\*(G\_... | https://mathoverflow.net/users/65800 | rational cohomology of finite real grassmannian | In the monography "Algebraic models in geometry" by Félix, Oprea and Tanré look at section 1.12.
And also: Mimura, Mamoru and Toda, Hirosi (1991). Topology of Lie Groups. I, II, Volume 91 of Translations of Mathematical Mono- graphs. American Mathematical Society, Providence, RI.
| 6 | https://mathoverflow.net/users/27816 | 200765 | 96,910 |
https://mathoverflow.net/questions/200769 | 4 | We define an equivalence relation on $\mathcal{P}(\omega)$: for $x,y\in\mathcal{P}(\omega)$ we say $$x\simeq\_{fin} y \text{ iff there is } n \in \omega \text{ such that }
x\setminus \{0,\ldots,n\} = y \setminus \{0,\ldots,n\}.$$
The set $\mathcal{P}(\omega)/\simeq\_{fin}$ is usually written as $\mathcal{P}(\omega)/f... | https://mathoverflow.net/users/8628 | Antichain on $\mathcal{P}(\omega)/fin$ of cardinality $2^{\aleph_0}$? | Yes. This is an easy exercise:
For every $r\in\Bbb R$ fix some sequence of rational numbers $r\_n$ such that $\lim r\_n=r$. Now enumerate $\Bbb Q$ as $\{q\_n\mid n\in\Bbb N\}$ and consider $A\_r=\{k\mid\exists n:q\_k=r\_n\}$.
Then given $r\neq r'$, the sequences $r\_n$ and $r'\_n$ must be disjoint from some point o... | 12 | https://mathoverflow.net/users/7206 | 200770 | 96,913 |
https://mathoverflow.net/questions/200771 | 12 | I asked [this](https://math.stackexchange.com/questions/1190984/weak-equivalence-testable-on-invariant-open-covers) question on math.stackexchange, but did not get an answer.
Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in ... | https://mathoverflow.net/users/69525 | When are (weak) homotopy equivalence testable on open covers? | For weak homotopy equivalences this holds always (Theorem 6.7.9 in tom Dieck's *Algebraic Topology*).
For homotopy equivalences this holds provided the open covers are numerable (Theorem 4.2.7 loc. cit.)
| 14 | https://mathoverflow.net/users/12547 | 200785 | 96,923 |
https://mathoverflow.net/questions/200733 | 6 | In ([reference](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCUQFjAA&url=http%3A%2F%2Fwww.philosophy.ox.ac.uk%2F__data%2Fassets%2Fpdf_file%2F0015%2F1194%2FSufficient_conditions.pdf&ei=7DAPVczxCoXfaqD_glg&usg=AFQjCNFrPLZT7E00XZ5-GjdmzvJ2Lwoqzw))The following result is attributed t... | https://mathoverflow.net/users/65878 | Generalizing a result of Kreisel on $\omega$-consistency | The property does not hold in general.
First, I’ll recall some basic properties of $\omega$-consistency. Let $T\vdash\_1\phi$ denote the relation that $\phi$ is derivable from $T$ using rules of first-order logic, and *unnested* instances of the $\omega$-rule.
1. If $T\vdash\_1\phi$, then $\phi$ is derivable from $... | 11 | https://mathoverflow.net/users/12705 | 200795 | 96,926 |
https://mathoverflow.net/questions/200803 | 4 | I am only considering commutative rings with $1$. Dimension refers to Krull dimension.
In the paper "Products of commutative rings and zero-dimensionality", Gilmer and Heinzer give necessary and sufficient conditions for an arbitrary direct product of rings to be at least countably infinite-dimensional. As there are ... | https://mathoverflow.net/users/69591 | Uncountable chain of prime ideals in an arbitrary direct product of rings | Yes, whenever the product is not zero-dimensional there is an uncountable chain of primes (in fact, a chain of cardinality $\mathfrak{c}$ and uncountable cofinality). Let $(A\_\alpha)\_{\alpha\in I}$ be an infinite collection of rings such that $\prod A\_\alpha$ is not zero-dimensional. It follows from Gilmer and Heinz... | 5 | https://mathoverflow.net/users/75 | 200813 | 96,931 |
https://mathoverflow.net/questions/200784 | 2 | Is $\mathcal{P}(\omega)/fin$ with the [interval topology](https://mathoverflow.net/questions/194642/product-of-posets-with-hausdorff-interval-topology) a connected space? (You find the definition of $\mathcal{P}(\omega)/fin$ [here](https://mathoverflow.net/questions/200769/antichain-on-mathcalp-omega-fin-of-cardinality... | https://mathoverflow.net/users/8628 | Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space? | The answer is yes, because any two nonempty open sets have points
in common. And this also shows directly that the space is not
Hausdorff.
From what you describe in the other question, the topology is
generated by the complements of the upper cones $\uparrow
f=[f,1]=\{h\mid h\geq^\* f\}$ and the lower cones $\downarr... | 5 | https://mathoverflow.net/users/1946 | 200830 | 96,936 |
https://mathoverflow.net/questions/200673 | 3 | Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ($\boldsymbol{\Lambda}$ diagonal, $\boldsymbol{X}$ orthonormal matrix of eigenvectors). Assume that we reduce each eigenvalue $\lambda\_i$ by ... | https://mathoverflow.net/users/69546 | Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements? | I don't think you can say whether the ratios $r\_{ij}$ increase or decrease. Consider the following two extremes when $S$ is positive definite with smallest eigenvalue $\lambda\_{min}>0$.
(1) Pick every $\psi$ the same positive number, and smaller than $\lambda\_{min}$. Then $S^\psi=S-\lambda\_{min}I$ since $\lambda\... | 0 | https://mathoverflow.net/users/59011 | 200840 | 96,941 |
https://mathoverflow.net/questions/200820 | 9 | Is the following algorithmic problem known to be decidable/undecidable?
Input: a finite group presentation $P$.
Decide: is the commutator subgroup of the group presented by $P$ finitely generated?
| https://mathoverflow.net/users/61502 | Can you decide whether the commutator subgroup of a f.p. group is f.g? | It's undecidable.
Lemma: a group $G$ is nontrivial if and only if the free product $H=G\ast\mathbf{Z}$ has an infinitely generated derived subgroup.
Proof: assume $G$ finitely generated and nontrivial. The kernel $N$ of the canonical epimorphism $H\to\mathbf{Z}$ is isomorphic to $G^{\ast\mathbf{Z}}$ and hence is in... | 16 | https://mathoverflow.net/users/14094 | 200854 | 96,946 |
https://mathoverflow.net/questions/200836 | 12 | Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of $G$, and group automorphisms which preserve $S$.
In some simple cases, these are everything one has (up to composition) ... | https://mathoverflow.net/users/56465 | Isometries of some simple Cayley graphs | Question (b) (and, therefore, also Question (a)) was answered affirmatively (for all torsion-free nilpotent groups) in 1998 by R.G.Möller and N.Seifter [[Europe J. Comb. 19, 597-602]](http://www.sciencedirect.com/science/article/pii/S0195669898902104) (see Theorem 4.1(1)).
The answer to Question (a) was rediscovered ... | 10 | https://mathoverflow.net/users/68305 | 200862 | 96,950 |
https://mathoverflow.net/questions/200874 | 4 | Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation isomorphic to $\pi$ (there are $(\dim \pi)^2$ linearly independent such elements.) It should be some standard `canonical' way to ... | https://mathoverflow.net/users/4312 | how to find explicitly given component in a regular representation | The space that you seek is the two-sided ideal in $\mathbb C[G]$ generated by the character of $\pi$ (see details below).
This follows from the explicit Wedderburn decomposition of $\mathbb C[G]$. If we write
$$
\mathbb C[G] = \bigoplus\_{\pi \in \hat G} M\_{d\_\pi}(\mathbb C),
$$
then the identity element of the $\p... | 8 | https://mathoverflow.net/users/9672 | 200875 | 96,954 |
https://mathoverflow.net/questions/200886 | 8 | I'm looking for a tool which generates a BibTeX item for a given arxiv id. I only found <http://www.crcg.de/arXivToBibTeX/> using Google but this tool always tells me that the arxiv ids I enter don't exist which is of course not true.
Edit: For example <http://www.crcg.de/arXivToBibTeX/?q=1503.06747&format=bibtex> gi... | https://mathoverflow.net/users/69627 | Automatically generate BibTeX item from arxiv | If you omit the initial zero after the dot in the identifier, it [works](http://www.crcg.de/arXivToBibTeX/?q=1503.6747&format=bibtex).
I guess that the tool does some formatting mojo that hasn't been updated when arXiv [switched to 5-digit identifiers](http://arxiv.org/help/arxiv_identifier) last January.
| 15 | https://mathoverflow.net/users/1898 | 200890 | 96,960 |
https://mathoverflow.net/questions/200877 | 6 | Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
$$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\gamma\wedge\alpha,\mathrm{d}\gamma+\alpha\wedge\beta)$$
**Is it injective / surjective? Which is its kernel / cokernel? ... | https://mathoverflow.net/users/62367 | Strange problem about triplets of differential forms | Your map is *not* onto for $n>2$, even locally. If $(A,B,C)$ is a triple of $2$-forms on $\mathbb{R}^n$ that can be written in the form
$$
(A,B,C) = \bigl(\mathrm{d}\alpha + \beta\wedge\gamma,
\mathrm{d}\beta + \gamma\wedge\alpha, \mathrm{d}\gamma + \alpha\wedge\beta\bigr),
$$
then, taking the exterior derivative of th... | 6 | https://mathoverflow.net/users/13972 | 200891 | 96,961 |
https://mathoverflow.net/questions/200846 | 5 | Let $\overline{M}\_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation maps:
$$ev:=ev\_1\times ...\times ev\_n:\overline{M}\_{0,n}(\mathbb{P}^N,d)\rightarrow (\mathbb{P}^N)^{n},\quad [C,f,... | https://mathoverflow.net/users/nan | Evaluation maps for moduli of stable maps | For $N>1$ and large $n$ the map $ev$ will not be surjective for dimension reasons:
$$\text{dim} \overline{M}\_{0,n}(\mathbb{P}^N,d) = Nd + N + d + n-3$$
whereas
$$\text{dim} (\mathbb{P}^N)^{n} = N n.$$
Thus the pushforward of the structure sheaf cannot be the structure sheaf on the target.
On the other hand for $n=1$... | 4 | https://mathoverflow.net/users/69630 | 200895 | 96,964 |
https://mathoverflow.net/questions/74763 | 6 | In Kevin Buzzard's [recent question](https://mathoverflow.net/questions/74663/how-badly-can-strong-multiplicity-one-fail-in-the-theory-of-automorphic-represent), a warm up question was: if two automorphic representations are nearly equivalent, then are the central characters of their local components equal?
Working m... | https://mathoverflow.net/users/2024 | Finite field analogue of representations in same packet have equal central character | This is quite an old question but I believe the answer to your question is given in Lemma 2.2 of Malle's paper "Height 0 characters of finite groups of Lie type" (2007) which is freely available online [here](http://www.ams.org/journals/ert/2007-11-09/S1088-4165-07-00312-3/).
His lemma states that any two characters ... | 5 | https://mathoverflow.net/users/22846 | 200901 | 96,966 |
https://mathoverflow.net/questions/200876 | 33 | Is there a topological space $(C,\tau\_C)$ and two points $c\_0\neq c\_1\in C$ such that the following holds?
>
>
> >
> > A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c\_0) = x$ and $f(c\_1) = y$.
> >
> >
> >
>
>
>
Is there also a Haus... | https://mathoverflow.net/users/8628 | Connectedness in the language of path-connectedness | No such space $C$ can exist.
We will derive a contradiction from the assumption that $C,c\_0,c\_1$ as desired exists.
Let κ be any cardinal greater than $|C|$. View $\kappa$ as an ordinal. For each β in κ add a copy of the unit interval between β and β+1, and add a point ∞ at the end. The resulting "Very Long Lin... | 33 | https://mathoverflow.net/users/14915 | 200903 | 96,967 |
https://mathoverflow.net/questions/200910 | 3 | Let $\mu\_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu\_n=\ker[{\mathbb{C}}^\*\overset{n}{\longrightarrow}{\mathbb{C}}^\*]$.
We set
$$ \mu=\varinjlim\_n \mu\_n\subset {\mathbb{C}}^\*,\quad {\widehat{\mathbb{Z}}}(1)=\varprojlim\_n \mu\_n,\quad {\mathbb{A}}(1)={\widehat{\mathbb{Z}}}(1)\otime... | https://mathoverflow.net/users/4149 | Adeles and twisted adeles | Doesn't this boil down to:
$$ \mathbb{A}(1)/\hat{\mathbb{Z}}(1) \cong \mathbb{Q}/\mathbb{Z} \otimes \hat{\mathbb{Z}}(1) \cong (\mathrm{colim}\, \mathbb{Z}/n\mathbb{Z}) \otimes \hat{\mathbb{Z}}(1) \cong \mathrm{colim}\, (\mathbb{Z}/n\mathbb{Z} \otimes \hat{\mathbb{Z}}(1)) \cong \mathrm{colim}\, \mu\_{n} = \mu$$
Here... | 5 | https://mathoverflow.net/users/21815 | 200911 | 96,970 |
https://mathoverflow.net/questions/144902 | 7 | In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar results about other norms? Such as $l^1$-norm of the coefficients?
| https://mathoverflow.net/users/39756 | Norms of B-spline coefficients | This question is closely related to the so-called "condition number" of the B-spline basis. Basically, for a spline $f$ of some degree $p$ with a coefficient vector $c=(c\_i)$, you generally have for any $q \in [1,\infty]$ that
$$
A\_{p,q} \|c\|\_{\ell\_q} \le \| f \|\_{L\_q} \le B\_{p,q} \|c\|\_{\ell\_q},
$$
and the ... | 5 | https://mathoverflow.net/users/56150 | 200920 | 96,975 |
https://mathoverflow.net/questions/200919 | 8 | There is a famous theorem due to J.-M. Fontaine,
[Il n'y a pas de variété abélienne sur Z](http://link.springer.com/article/10.1007%2FBF01388584)
(and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether there is a function field analog of this result. More precisely: Le... | https://mathoverflow.net/users/69640 | Abelian varieties with good reduction everywhere over function fields | There are non-isotrivial families of supersingular abelian varieties of dimension $g$ over $\mathbb P^1\_{\overline {\mathbb F\_p}}$ if $g\geq 2$; see
Goren, E. Z.(3-MGL); Oort, F.
Stratifications of Hilbert modular varieties. (English summary)
J. Algebraic Geom. 9 (2000), no. 1, 111–154.
There are many other pap... | 12 | https://mathoverflow.net/users/4333 | 200921 | 96,976 |
https://mathoverflow.net/questions/200922 | 0 | [Editted: The assertion is wrong; see Jay's answer]
My apology if this question is too simple. I am reading Deligne-Lusztig *"Reductive groups over finite fields"* and at the beginning of Chap. 4, there is a statement saying something about Green functions which seems to claim the following: Let $G$ be a adjoint semi... | https://mathoverflow.net/users/31327 | Unipotent orbit in adjoint group over finite field | That's not what they're claiming and your statement is not true. Your claim is that every unipotent element is rational. However Lemma 5.6 of [this](http://www.sciencedirect.com/science/article/pii/S0021869303001741) article by Tiep and Zalesskii provides a counter example. Indeed, non-rational unipotent elements exist... | 7 | https://mathoverflow.net/users/22846 | 200932 | 96,979 |
https://mathoverflow.net/questions/200929 | 3 | A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group.
Is it true that every totally ordered group has a nontrivial normal subgroup (not necessarily convex) ?
| https://mathoverflow.net/users/69647 | Normal subgroup of a totally ordered group | No. Thompson's group F is bi-orderable (or "totally ordered", in your terminology), and its commutator subgroup [F,F] is simple. So [F,F] is an (infinite) bi-orderable simple group. See [this paper](http://arxiv.org/abs/0808.1688) of Navas and Rivas for a discussion of all the bi-invariant orderings of F, and a referen... | 9 | https://mathoverflow.net/users/68305 | 200936 | 96,980 |
https://mathoverflow.net/questions/200353 | 7 | Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.)
Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group $\mathrm{O}(V,\phi) \subset \mathrm{GL}(V)$ of linear automorphisms preserving $\phi$. The connected component of the identit... | https://mathoverflow.net/users/21815 | When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras? | Here's a proof assuming $n\ge 3,n\neq 8$ that the Lie algebras are isomorphic only when the quadratic forms are equivalent up to rescaling (I assume $K$ has characteristic zero and fix an algebraically closed extension $C$).
Let $f:\mathfrak{so}(\phi)\to \mathfrak{so}(\psi)$ be a $K$-defined isomorphism. We can assum... | 7 | https://mathoverflow.net/users/14094 | 200947 | 96,986 |
https://mathoverflow.net/questions/200870 | 6 | I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some other theorems may be. But this doesn't give me a geometric intuition so far.
| https://mathoverflow.net/users/69320 | reflexive banach space | There is a beautiful result of Odell and Schlumprecht that gives an answer to this question for separable Banach spaces.Odell, E.(1-TX); Schlumprecht, Th.(1-TXAM) Asymptotic properties of Banach spaces under renormings. (English summary) J. Amer. Math. Soc. 11 (1998), no. 1, 175–188.
A separable Banach space is refle... | 12 | https://mathoverflow.net/users/2554 | 200950 | 96,988 |
https://mathoverflow.net/questions/200953 | 2 | Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme.
Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially on $S$.
I was wondering what is $Pic(BG)$.
Is it true that $Pic(BG)= H^1(k,G)$ when $S=Speck$ the spectrum of a fie... | https://mathoverflow.net/users/69558 | Picard group of classifying stack | $\mathrm{Pic}([S/G])$ is the group of line bundles on $S$ together with a $G$-linearization. When $S=\mathrm{Spec}(k)$, any line bundle on $S$ is trivial, and a $G$-linearization is given by a character $G\rightarrow \mathbb{G}\_m$, so $\mathrm{Pic}[S/G]=\mathrm{Hom}(G,\mathbb{G}\_m)$. No relation with $H^1(k,G)$, whic... | 6 | https://mathoverflow.net/users/40297 | 200954 | 96,989 |
https://mathoverflow.net/questions/200960 | 0 | Let $G$ be an algebraic group, and $G\_{Id}$ the connected component of the identity. Then $G\_{Id}$ is a normal subgroup of $G$ and $G/G\_{Id}$ is the component group of $G$.
Let $G\_{c}\subset G$ be another connected component of $G$. Is is possible to define a group structure on $G\_c$?
Assume we know that for a... | https://mathoverflow.net/users/nan | Connected components of algebraic groups | Each connected component is (in a natural way) a torsor under the identity component. The choice of a rational point (if there is one) defines an isomorphism with the identity component, and makes the component into an algebraic group.
| 8 | https://mathoverflow.net/users/57398 | 200974 | 96,998 |
https://mathoverflow.net/questions/200965 | 3 | Suppose $G$ is a smooth and abelian $k$-group scheme, for $k$ a field.
Is it possible to get back galois cohomology groups $H^\*(k,G)$ studying the cohomology of the classifying stack $BG=[\*/G]$ ?
| https://mathoverflow.net/users/69558 | Galois cohomology out of the classifying stack | Hmm, a priori Galois cohomology is cohomology of $Spec(k)$ with values in $G$. For $H^1$, for instance, this amounts to considering the set (in fact group) of isomorphism classes of maps of stacks $Spec(k) \to BG$. Whereas the degree-one cohomology of $BG$ with values in some $A$ (some sort of abelian group scheme over... | 3 | https://mathoverflow.net/users/4177 | 200977 | 97,000 |
https://mathoverflow.net/questions/200506 | 1 | Let $\tau(n)$ be the divisor function. Let $a$ be either a constant, or a function of $X$ that is slowly varying with $X,$ say $X/\log(X)<a(X)<X \log(X),$ for example. I want to lower bound sums of the following form
$$
\sum\_{1\leq n\leq X} a^{1-\frac{\tau(n)}{D}},\quad(1)
$$
and
$$
\sum\_{1\leq n\leq X: n\in I} a^{1-... | https://mathoverflow.net/users/17773 | Lower Bound on "exponential" sum | I think I have an answer, whereby the same asymptotic order lower bound as in (1) is obtained for (2). *Please comment if there is an issue with the argument:*
Consider
$$
\sum\_{1\leq n\leq X: n\in I} a^{1-\frac{\tau(n)}{D}},\quad(2)
$$
where the question specified $I$ to be an index set of roughly $X/2$ integers. T... | 1 | https://mathoverflow.net/users/17773 | 200982 | 97,004 |
https://mathoverflow.net/questions/200990 | 4 | It is known, thanks to Gabber, that algebraic spaces are sheaves in the fpqc topology:
[Stacks project 03W8](http://stacks.math.columbia.edu/tag/03W8)
Is the analogous statement for algebraic (Artin) stacks true? If not, is it true under some reasonable hypotheses?
| https://mathoverflow.net/users/45657 | Do algebraic stacks satisfy fpqc descent? | It may be helpful to have a look at [these notes](http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept15-17%28stacks%29.pdf) by Anatoly Preygel (see also [MO/15910/2503](https://mathoverflow.net/a/15910/2503)). In particular, Proposition 3.3.6 says that an algebraic stack is an fpqc sheaf if the diagonal i... | 6 | https://mathoverflow.net/users/2503 | 200991 | 97,008 |
https://mathoverflow.net/questions/200992 | 0 | Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
>
>
> >
> > for $X, Y \in \text{Top}(\kappa)$ we set $X \to Y$ iff there is a continuous surjective map $f:X\to Y$.
> >
> >
> >
>
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We... | https://mathoverflow.net/users/8628 | Continuous image relation on topological spaces | A partial answer: It is not a lattice, at least if you allow non-Hausdorff spaces.
For any natural numbers $n$ and $k$, write $\bf n+k$ for the sum of an indiscrete space of size $n$ (i.e., a space that has no nontrivial open sets) and an indiscrete space of size $k$. Then both $\bf 4$ and $\bf 2+1$ are lower bounds... | 1 | https://mathoverflow.net/users/14915 | 201007 | 97,011 |
https://mathoverflow.net/questions/200723 | 22 | Let $\lambda = (\lambda\_1, \ldots, \lambda\_r)$ and $\mu = (\mu\_1, \ldots, \mu\_r)$ be partitions such that $\mu\_j = \lambda\_j +1$ for one index $j$ and $\mu\_i = \lambda\_i$ for all other $i$. Then there is a natural transformation $\alpha\_{\mu/\lambda}: \mathbb{S}\_{\lambda}(V) \otimes V \to \mathbb{S}\_{\mu}(V)... | https://mathoverflow.net/users/297 | Are there any natural differential operators besides $d$? | I think your question, the way it is stated, makes one want to classify unary and binary (depending how far you generalise the question as written) invariant differential operators on tensor fields. This has been done for unary operators by an awful lot of people, and the statement indeed is that $d$ is the only operat... | 18 | https://mathoverflow.net/users/1306 | 201011 | 97,013 |
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