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https://mathoverflow.net/questions/151542 | 1 | My question is quite simple: Let $X$ be an irreducible algebraic variety over a field $\Bbbk$. Is there a name for such varieties with perfect function field $\Bbbk(X)$? Is this very rare? Is there literature that deals with this kind of question and object?
| https://mathoverflow.net/users/9947 | Variety with perfect function field? | This is a pretty easy question, but I don't see why not to record an answer. (The answer was clearly known to user ACL...as it would be to almost any arithmetic geometer of his pay grade.)
Case 1: $k$ characteristic $0$. Then $k(V)$ is always perfect, of course.
Case 2: $k$ has characteristic $p > 0$ but is perfec... | 2 | https://mathoverflow.net/users/1149 | 151583 | 80,864 |
https://mathoverflow.net/questions/151553 | 2 | We know that the full automorphism group of the $\pi\_q = PG(2,q)$ acts *imprimitively* on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full correlation group (semi-direct product of the automorphism group with cyclic group of order 2). I have been able to ... | https://mathoverflow.net/users/34180 | Flag primitivity of the correlation group of classical projective planes. | Let $G=GL(3,q)$ act on $\mathbb F\_q^3$ from the right. Set $e\_1=(1,0,0)$ and $e\_2=(0,1,0)$, and $P=\langle e\_1\rangle$, $L=\langle e\_1,e\_2\rangle$. Then the stabilizer of the flag $F=(P,L)$ is the group $B$ of lower triangular matrices.
There are precisely two groups $B\_1$ and $B\_2$ properly between $B$ and $... | 2 | https://mathoverflow.net/users/18739 | 151599 | 80,874 |
https://mathoverflow.net/questions/151595 | 2 | It is a long standing problem to investigate whether irreducible integral polynomials not divisible by a fixed square integer assumes square-free values infinitely often. The result is known conditioned on the $abc$-conjecture (due to Granville), and the best unconditional results are for $k$-free values of polynomials... | https://mathoverflow.net/users/10898 | Polynomials with few prime factors | 1) The polynomial in two variables $f(x,y)=2(x^2-2y^2)$ will take the prime
value $p=2$ infinitely often (Pell's equation).
2) Nontrivial results in the direction of your question
are due to Gihan Marasingha:
a) Almost primes represented by binary forms. J. Lond. Math. Soc. (2) 82 (2010), no. 2, 295–316.
b) On th... | 5 | https://mathoverflow.net/users/36707 | 151607 | 80,877 |
https://mathoverflow.net/questions/140899 | 1 | I would like to estimate the following sum
$\sum\_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N).
The first idea may be to apply Weyl-Van der Corput inequality several times($[l]$ times) then invoke the theory of exponentia... | https://mathoverflow.net/users/4486 | Exponential sums | There is the article [Primes in Special Intervals and Additive Problems with Such Numbers](http://link.springer.com/article/10.1023/A:1023270112701) of Maris Changa. He considered exponential sum with $n^l$ $(l>1)$ and estimated them using double sums
$$n^l\to n^l\left(1+\frac{xy}n\right)^l$$
and Vinogradov's mean valu... | 1 | https://mathoverflow.net/users/5712 | 151613 | 80,879 |
https://mathoverflow.net/questions/151609 | 1 | Let E be an elliptic curve over $\mathbb{Q}$. Is there an efficient algorithm which can solve an elliptic curve discrete logarithm in E?
| https://mathoverflow.net/users/41032 | Is there an efficient algorithm to solve ECDLP over global field? | As Joro says, you can use the height pairing. And it's worth pointing out that it is generally possible to compute canonical heights even when the coefficients of $E$ are so large that it's infeasible to factor the discriminant.
* Computing canonical heights with little (or no) factorization, *Math. Comp.* **66** (1... | 3 | https://mathoverflow.net/users/11926 | 151627 | 80,881 |
https://mathoverflow.net/questions/151612 | 5 | The question is kind of self contained, but let me develop a bit further.
Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is a set $\Phi$ consisting of $g$ complex embeddings of $K$ such that $\mathrm{Hom}(K, \mathbb{C})=\Phi \cup \overline{\P... | https://mathoverflow.net/users/44067 | can all CM types be realized by Jacobians? | ``Given $(K,\Phi)$ , there always exists a $g$ -dimensional abelian variety $A$ such that $End(A)\otimes Q=K$ and that $K$ acts on $H^0(A,\Omega^1)$ through $\Phi$. One easily constructs as a complex torus, starting from the embedding $K\subset C^g$ given by $\Phi$."
Actually, this is not always the case. For example... | 6 | https://mathoverflow.net/users/9658 | 151628 | 80,882 |
https://mathoverflow.net/questions/151594 | 9 | Let $-CP^{2}$ denote the complex projective surface $CP^{2}$ with the reverse orientation. I have seen some results about the existence of symplectic structures on the connected sums $\#\_{l}CP^{2}\#\_{k}(-CP^{2})$ for **some** positive integers l,k.
My question is whether there is a complete result which can decrib... | https://mathoverflow.net/users/44052 | Does the smooth manifold $\#_{l}CP^{2}\#_{k}(-CP^{2})$ admit a symplectic structure? | For $l=1$, these are blowups of the projective plane, which are all Kaehler and hence symplectic. For $l>1$, these do not have symplectic structures. For if $l$ is even, then they don't even have almost complex structures [(cf. this MO thread)](https://mathoverflow.net/questions/59820/a-question-on-classification-of-al... | 12 | https://mathoverflow.net/users/3460 | 151638 | 80,886 |
https://mathoverflow.net/questions/151637 | 1 | Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with
no restriction on the degree.
Write a point $P = (X , Y , Z)$ with the smallest coprime integers
$X,Y,Z$.
Is it true that for every fixed $ a > 0$
$$
\log \max(|X|,|Y|,|Z|)- \log \min(|X|,|Y|,|Z|) > a
$$
finitely often?
I believe it is... | https://mathoverflow.net/users/12481 | The relative sizes of coordinates of a point on projective genus 1 curve | I don't think so. Suppose that $T=(0,0)$ is a rational point on your curve $C$, and suppose that the rational points on $C$ lie dense around $T$ in the real topology. (It is easy to find such a $C$.) Your assertion would preclude $x=X/Z$ and $y=Y/Z$ from getting arbitrarily close to $(0,0)$, but this is exactly what ha... | 2 | https://mathoverflow.net/users/44022 | 151640 | 80,887 |
https://mathoverflow.net/questions/151431 | 2 | Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of $\phi$ on the set $I$. For each orbit $J$ consider the longest element $s\_J$ of the parabolic subgroup $W\_J$. Let $W\_\... | https://mathoverflow.net/users/35957 | Which subgroups of a finite reflection group have distingushed coset representatives? | The comments suggest that your notational choices may be obscuring the question,
which I and others have found difficult to untangle. If stated more precisely, the question might answer itself.
There is some variation of notation and terminology in the liteature, since twisted groups of Lie type are treated a little ... | 1 | https://mathoverflow.net/users/4231 | 151641 | 80,888 |
https://mathoverflow.net/questions/151540 | 3 | Good day. The question is on proving the following relation ($\|\cdot\|$ here and on denotes $\ell\_2$ norm):
$$
\frac{dJ(f)}{df} = -\mathrm{div}\left(\frac{\nabla f}{\|\nabla f\|}\right) \qquad =: DJ,
$$
where $J(f)$ is the total variation norm:
$$
J(f) = \int \|\nabla f\| \, dx
$$
So what I was trying to do was an ... | https://mathoverflow.net/users/44025 | Fréchet derivative of the Total Variation norm | Not near zero, which is an issue (!), instead of using an inequality, write
\begin{eqnarray\*}
\| \nabla f+ \nabla\delta f \| - \|\nabla f\| &=& \frac{\| \nabla f+ \nabla\delta f \|^2 - \|\nabla f\|^2}{\| \nabla f+ \nabla\delta f \| +\|\nabla f\|}\\
&=& \langle \nabla\delta f , \frac{ 2\nabla f}{\| \nabla f+ \nabla \de... | 1 | https://mathoverflow.net/users/40120 | 151656 | 80,893 |
https://mathoverflow.net/questions/151654 | 6 | Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}\_K(E)\cong \mathcal{O}=\mathcal{O}\_K$. Let $\mathfrak{p}$ be a prime of $K$ such that the map $\mathcal{O}^\times \to(\mathcal{O}/\mathfrak{p})^\times$ is not surjective. With this situation, I tried to p... | https://mathoverflow.net/users/44006 | Rational points and torsion points of CM elliptic curve | See Cor. 5.18 of Rubin "Elliptic curves with complex multiplication" in LNM 1716 for a proof (that uses the main theorem of complex multiplication).
| 4 | https://mathoverflow.net/users/5498 | 151664 | 80,897 |
https://mathoverflow.net/questions/151635 | 6 | Is it known if the direct limit of hyperbolic groups can have finite commutator width? Every hyperbolic group has infinite verbal width for any word $w$, so in particular for the commutator word $w=x^{-1}y^{-1}xy$ (<http://arxiv.org/abs/1107.3719>)
Especially I would like to know if the examples of infinite torsion ... | https://mathoverflow.net/users/23232 | Commutator Width of a direct limit of hyperbolic groups | Assuming that you refer to Ivanov's construction from Ol'shanskii's book of a $2$-generated infinite group $G$ of exponent $p$, for a large prime $p$, having exactly $p$ conjugacy classes, then the commutator width in this group is bounded above by $p-1$. The reason for this is that for any non-trivial commutator $w$ i... | 7 | https://mathoverflow.net/users/7644 | 151668 | 80,898 |
https://mathoverflow.net/questions/49472 | 9 | There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of $S$-algebras in EKMM (Rings, Modules, and Algebras in stable homotopy theory by Elmendorf-Mandell-May-Kriz) is one such mod... | https://mathoverflow.net/users/3901 | Technology for various models of spectra | I don't think I noticed this question before. One point is that now that we
have multiplicatively well-behaved Quillen equivalences between all reasonable
models for the stable category, hence between reasonable models for categories
of ring and module spectra, it is formal to transport constructions like spectral
se... | 5 | https://mathoverflow.net/users/14447 | 151675 | 80,902 |
https://mathoverflow.net/questions/151617 | 1 | How does one prove that the space $B(\mathbb H)$ of bounded operators on a infinite dimensional (separable) Hilbert space is **not** reflexive?
I guess this should go along the lines of the non-reflexivity of $l\_\infty$, but I was unable to find this written down somewhere.
| https://mathoverflow.net/users/44073 | Non-reflexivity of $B(\mathbb H)$ | It seems to be a good idea to extend my comment to an answer.
First of all, pick any orthonormal basis of $\mathbb{H}$ so that $B(\mathbb{H})$ can be identified with $B(\ell\_2)$. The subspace of diagonal operators is clearly isometric to $\ell\_{\infty}$. It is a general truth that if $Y$ is a closed subspace of a r... | 5 | https://mathoverflow.net/users/24953 | 151678 | 80,903 |
https://mathoverflow.net/questions/151631 | 1 | Let $R\rightarrow S$ be a morphism of rings in characteristic $p$ which is formally smooth.
Is it true that $R$ is Frobenius splitting if and only if $S$ is Frobenius splitting?
One direction seems to be easy but I am not sure if both implications are true.
| https://mathoverflow.net/users/44081 | Frobenius splitting and smoothness | Consider the following. I'm going to assume that all rings are Noetherian (some things probably generalize, but I want to be careful). First a definition,
**Definition:** An extension of rings $A \subseteq B$ is *pure* if $M \otimes\_A A \to M \otimes\_A B$ is injective for every $A$-module $M$. A ring $A$ is called... | 2 | https://mathoverflow.net/users/3521 | 151681 | 80,906 |
https://mathoverflow.net/questions/151669 | 1 | This question was on math.stackexchange but got no answer. The link is <https://math.stackexchange.com/questions/598811/kolmogorov-backward-equation-for-ito-diffusion>
I want to know the answer too, so here it is
Let $(X\_t)\_{t\ge 0}$ be the solution of the SDE
$$
X\_t = X\_0 + \int\_0^t \mu(s,X\_s) \,ds + \int\_0... | https://mathoverflow.net/users/32325 | Kolmogorov backward equation question | First note the following fact: if $f : [0,T] \to \mathbb{R}$ is an integrable function and $\int\_0^t f(s)\,ds = 0$ for all $0 \le t \le T$, then $f = 0$ almost everywhere. (It's immediate that $\int\_a^b f(s)\,ds = 0$ for all $a,b$. Now use a monotone class argument to show that $\int\_A f(s)\,ds = 0$ for every measur... | 2 | https://mathoverflow.net/users/4832 | 151685 | 80,908 |
https://mathoverflow.net/questions/151662 | 5 | Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded?
By small knots, I'm referring to things like torus knots, 2-bridge knots and the Rolfsen knot table.
For 2-sheeted coverings the Bonahon-Siebenman paper is a... | https://mathoverflow.net/users/1465 | The cyclic branched covers of "simple" knots in $S^3$ | Such a classification follows from the [orbifold theorem](https://en.wikipedia.org/wiki/William_Thurston#Orbifold_theorem). The point is that the $n$-fold cyclic branched cover over a knot is an $n$-fold orbifold cover over the orbifold whose singular locus is the knot with a cone angle of $2\pi/n$ (in fact, one only n... | 8 | https://mathoverflow.net/users/1345 | 151687 | 80,909 |
https://mathoverflow.net/questions/151682 | 2 | Does there exist a strictly increasing and continuous function of ordinal numbers whose
smallest critical number (i.e. fixed point) is the smallest non-constructive ordinal
number (in the sense of Church and Kleene)? If so, does there exist such a function all
of whose critical numbers (i.e. fixed points) are non-const... | https://mathoverflow.net/users/4423 | Some questions about functions of Ordinal Numbers | Any admissible ordinal is the limit of indecomposable ordinals. Pick an $\omega$-sequence of these ordinals, $\alpha\_0<\alpha\_1<\dots$ with limit $\omega\_1^{CK}$, so the order type of $[\alpha\_i,\alpha\_{i+1}]$ is $\alpha\_{i+1}$. Now define $f$ on $[0,\alpha\_0]$ by $f(\beta)=\alpha\_0+1+\beta$, on $(\alpha\_0,\al... | 4 | https://mathoverflow.net/users/6085 | 151689 | 80,910 |
https://mathoverflow.net/questions/151657 | 6 | Let $X, X\_1, X\_2, \ldots, X\_N$ be a sequence of identically distributed random variables with $0 \leq X \leq 1$.
We do **not** assume the sequence is iid, but rather allow the random variables to be dependent on their neighbors: we suppose that there exists $k\geq 1$ such that for every $I,J \subseteq \{ 1,\ldots... | https://mathoverflow.net/users/5678 | Is there a McDiarmid-type inequality for sequences with a finite range of dependence? | There are several versions of this type:
1) K. Marton has results for dependent variables. Maybe closest to what you ask (for convex functions $f$) is the paper of Samson:
[Samson paper](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aop/1019160125)
2) For a martingale di... | 8 | https://mathoverflow.net/users/35520 | 151690 | 80,911 |
https://mathoverflow.net/questions/151676 | 4 | Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is an example of a manifold which is not parallelizable, but $T^{n}(M)$ is parallelizable for some n?
| https://mathoverflow.net/users/36688 | Totally non parallelizable manifold | If $\xi$ is a vector bundle over a manifold $B$ with total space $E(\xi)$, then the restriction of $TE(\xi)$ to the the zero section of $\xi$ is isomorphic to $\xi\oplus TB$. It follows that $T^k M$ restricted to $T^{k-1}M$ is isomorphic to $T^{k-1}M\oplus T^{k-1}M$, and iterating we conclude that $T^k M$ restricted to... | 9 | https://mathoverflow.net/users/1573 | 151695 | 80,912 |
https://mathoverflow.net/questions/151688 | 5 | Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to Pic^0(A)$. Let $P$ be the Poincare sheaf on $A \times \hat{A}$. It is known that $\Phi\_{A \to \hat{A}}^{P}$ (i.e.the Four... | https://mathoverflow.net/users/29730 | Fourier-Mukai transform for abelian varieties | **No.** As Will Sawin indicates, every finite subgroup $H$ of $A$ is contained in $K(L)$ for some very ample line bundle on $A$: indeed, let $L\_1$ be your favorite very ample line bundle on $A$, and let $n = \# H$; then $H \subset A[n] \subset K(nL\_1)$. Thus you are asking whether every abelian variety $B$ which is i... | 8 | https://mathoverflow.net/users/1149 | 151703 | 80,914 |
https://mathoverflow.net/questions/151426 | 3 | At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the origin and a point in $P$, i.e. $r = \max\_{x\in P} \|x\|$. Let $S$ denote a sector of a circle with radius $r$ that is... | https://mathoverflow.net/users/43968 | Map from a convex polygon that increases distance | Begin with a lemma: Let $\ T=\Delta oab$ be a triangle contained in a circle centered at the origin $o$ and let the arc $\alpha$ of the circle be such that the circle's sector $S$ based on $\alpha$ has the same area as $T$. Let the map $f\_T$ from $T$ onto $S$ be defined so that $f\_T(o)=o$, $f\_T$ maps the segment $ab... | 3 | https://mathoverflow.net/users/36904 | 151712 | 80,918 |
https://mathoverflow.net/questions/151718 | 7 | I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what background I should have before starting to read the relevant papers. If you could offer me some books/papers/articles, I w... | https://mathoverflow.net/users/33900 | Mazur's torsion theorem on elliptic curves and its generalisations | Andrew Snowden has just finished teaching a course on Mazur's torsion theorem--video-taped lectures and extensive notes may be found [here](http://asnowden.com/679/). I've watched several of the lectures; they are excellent.
| 14 | https://mathoverflow.net/users/6950 | 151720 | 80,922 |
https://mathoverflow.net/questions/151709 | 1 | Let $M$ be a variety and $Y\subset M$ a subscheme. Is the scheme-theoretic complement $M\setminus Y$ of $Y$ in $M$ equal to the scheme-theoretic complement $M\setminus Y\_{\text{red}}$, where $Y\_{\text{red}}$ denotes the underlying reduced scheme upon which $Y$ is supported? I'm thinking the answer should be "yes", si... | https://mathoverflow.net/users/24132 | Complement of a subscheme | If $Y\subseteq X$ is a closed subscheme, then the topological space underlying $Y$ is a closed subset of $X$. Its complement $X\setminus Y$ is an open subset of $X$ and the restriction of $\mathcal{O}\_X$ to it makes it into a scheme. It has the universal property Allen Knutson described in his comment. It is clear fro... | 6 | https://mathoverflow.net/users/3847 | 151727 | 80,926 |
https://mathoverflow.net/questions/151644 | 4 | Suppose $M$ is a $n$-dimensional closed Riemannian manifold. Let the packing number $N(t)$ be the maximum number of balls with radius $t$ in $M$ that are disjoint. I am wondering whether the following limit has some geometric meaning:
$$
\lim\_{t\to 0}\frac{N(t)t^n}{Vol(M)}
$$
Does the limit depend only on dimension $n... | https://mathoverflow.net/users/3922 | Normalized packing number | If I am not mistaking, your number is the same for all manifolds, does not depend on the metric and on the manifold and coincides with the packing number of the standard ball in the euclidean $R^n$.
Indeed, two metrics $g$ and $g'$ on $M$
that are $\epsilon$ close one to another
(in the sence that for any $i,j$ the ... | 3 | https://mathoverflow.net/users/14515 | 151730 | 80,927 |
https://mathoverflow.net/questions/151739 | 4 | How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$
| https://mathoverflow.net/users/33900 | $\lambda$-invariant is constant for isogenous elliptic curves | Given the brevity of the question, I am not sure about the precise setup, so let me assume that you are looking at some $\mathbf{Z}\_p$-extension and that you assume (or know in the situation at hand) that the appearing Iwasawa modules are torsion, so that the $\lambda$-invariant is defined in the first place. For an I... | 13 | https://mathoverflow.net/users/5498 | 151740 | 80,930 |
https://mathoverflow.net/questions/151649 | 5 | Is it correct to say that the Zariski closure of any set of $k$-rational points in affine space over $\overline{k}$ is always defined over $k$?
| https://mathoverflow.net/users/15482 | closure of a set of k-rational points always defined over k? | While the question can be answered briefly and narrowly in an *ad hoc* style (as Peter Mueller has done), it may be useful to add a reference to Borel's textbook while putting the question in context.
The language of $k$-closed sets and $k$-varieties (varieties defined over a field $k$) was developed during the evol... | 4 | https://mathoverflow.net/users/4231 | 151749 | 80,935 |
https://mathoverflow.net/questions/17578 | 49 | I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's book on Riemann surfaces and Moise's book "Geometric topology in dimensions 2 and 3". Both of these strike me as being a bi... | https://mathoverflow.net/users/317 | Triangulating surfaces | [Three years later …]
All the published proofs of triangulability of surfaces that I am aware of use the Schoenflies theorem, which is not exactly an easy thing to prove. There is however another line of proof which avoids the Schoenflies theorem and instead uses the Kirby torus trick that underlies Kirby-Siebenmann ... | 46 | https://mathoverflow.net/users/23571 | 151760 | 80,938 |
https://mathoverflow.net/questions/151752 | 1 | Hopefully this is better than what I asked yesterday and Milton solved.
Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with
no restriction on the degree.
Write a point $P = (X , Y , Z)$ with the smallest coprime integers
$X,Y,Z$.
Is it true that for every fixed $ a > 1$
$$
\frac{\log \m... | https://mathoverflow.net/users/12481 | The relative sizes of coordinates of a point on projective genus 1 curve (second try) | Let $f$ be a nonconstant rational function on your curve $C$. For any point $P\in E(\mathbb{Q})$, write
$$
f(P) = \frac{a\_f(P)}{b\_f(P)} \in \mathbb{Q}
$$
in lowest terms. Then Siegel's theorem implies that
$$
\lim\_{P\in E(\mathbb{Q}), h(P)\to\infty}
\frac{\log|a\_f(P)|}{\log|b\_f(P)|} = 1.
$$
This looks pretty c... | 4 | https://mathoverflow.net/users/11926 | 151762 | 80,939 |
https://mathoverflow.net/questions/151731 | 19 | Following [Thurston](http://library.msri.org/books/gt3m/), an **orbifold** is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable manifolds, the finite groups being part of the structure.
A fundamental exam... | https://mathoverflow.net/users/6205 | Is there a good notion of morphism between orbifolds? | I believe this is worked out very nicely in "Geometrization of Three-Dimensional Orbifolds via Ricci Flow" by Bruce Kleiner, John Lott (<http://arxiv.org/abs/1101.3733>).
An *atlas* for an $n$-orbifold $\mathcal O$ consists of a Hausdorff paracompact topological space $|\mathcal O|$ together with an open covering $\... | 10 | https://mathoverflow.net/users/15155 | 151764 | 80,941 |
https://mathoverflow.net/questions/151700 | 2 | I posted a similar question on math stack exchange with the same title, but I didn't get a helpful response. I am trying to develop a logical language where one can express variable numerical quantifiers. In first order logic, one can express statements like "there exist at most 10 x such that Px" and "there exist exac... | https://mathoverflow.net/users/43439 | Variable numerical quantifiers | There is an old notion of a generalized quantifier introduced by Andrzej Mostowski in late '50s:
A. Mostowski, "On a generalization of quantifiers", Fundamenta Mathematicae 44, 1957.
The paper is actually well-written, but you may also enjoy reading the following introduction:
J. A. Väänänen, "Generalized Quantif... | 1 | https://mathoverflow.net/users/13480 | 151768 | 80,943 |
https://mathoverflow.net/questions/151758 | 4 | I am interested in knowing whether the space of trace class operators is (crudely) finitely representable in an $L^1$-space. I suspect that the answer is negative but I am unable to find any argument confirming my intuition.
As for motivation, I am working on matrix-valued versions of some inequalities coming from ha... | https://mathoverflow.net/users/24953 | Is the space of trace class operators finitely representable in an $L^1$-space? | You can deduce that $S\_1$ is not finitely crudely representable in an $L\_1$ space from the paper
Pisier, Gilles Some results on Banach spaces without local unconditional structure. Compositio Math. 37 (1978), no. 1, 3–19.
However, I think that the result you want might have been known earlier. Maybe it follows f... | 4 | https://mathoverflow.net/users/2554 | 151771 | 80,944 |
https://mathoverflow.net/questions/151778 | 1 | Call two sets, $A$ and $B$ close iff there exists a finite $k$ such that there are infinitely many pairs of elements $(a,b)$ with $a\in A$ and $b\in B$ where $|a-b|\le k$. If two sets are not close, call them far. If such a $k$ exists, call the smallest such one the radius of closeness of those sets.
Call two numbers... | https://mathoverflow.net/users/44115 | Bounded differences in exponential sequences | For any fixed integers $a$ and $b$ that are multiplicatively independent and for any integer $k$, there are only finitely many pairs of positive integers $m$ and $n$ such that
$$ a^n - b^m = k. $$
To see this, write $n=3N+i$ and $m=3M+j$ with $i,j$ between $0$ and $2$. Then $(a^N,b^M)$ is a solution to the Thue equatio... | 3 | https://mathoverflow.net/users/11926 | 151789 | 80,951 |
https://mathoverflow.net/questions/84279 | 9 | Is there any elementary argument showing that there exist uncountably many distinct quasi-isometry classes of elementary amenable groups? How about solvable groups?
For amenable groups it follows from the result of Grigorchuk (proved in the 80's) stating that there are uncountably many groups of intermediate growth ... | https://mathoverflow.net/users/10251 | Quasi-isometry classes of elementary amenable groups | The affirmative answer in given by Yves de Cornulier and Romain Tessera in arXiv:1203.4696.
| 4 | https://mathoverflow.net/users/10251 | 151791 | 80,953 |
https://mathoverflow.net/questions/151792 | 2 | Recall continued fractions: <http://en.wikipedia.org/wiki/Continued_fraction>
Now take a look at this question: <https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-exp-1-exp-2-exp-3-ldots-exp/601890#601890>
You cannot help but notice that there is some resemblance between recur... | https://mathoverflow.net/users/38448 | Expressions in "continued" monotone functions | In other words you want to write $f$ as a limit $g\_1\circ g\_2\circ g\_3\circ\ldots$, where
$g\_n(x)=a\_n+\exp(-x)$. Putting $h\_n(t)=\exp(- g\_n(-\log t))$, we obtain
$$h(t)=\exp f(\log t)=h\_1\circ h\_2\circ\ldots,$$
now $h\_n(t)=\lambda\_n e^{-t}, \lambda\_n=\exp(- a\_n)$. Infinite compositions of such functions an... | 2 | https://mathoverflow.net/users/25510 | 151795 | 80,955 |
https://mathoverflow.net/questions/151784 | 13 | As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like **$0^{\sharp}$ exists** such that all large cardinal axioms which are equivalent or stronger than this axiom are contradictory with $V=L$ and all large cardinal assumptions below it are consis... | https://mathoverflow.net/users/nan | V=HOD & The Height of the Large Cardinal Tree | $\newcommand\HOD{\text{HOD}}$
There is no such border, because almost all the large cardinal properties, including the very strongest large cardinal axioms, are relatively consistent with $V=\HOD$. For the larger large cardinals, this is generally proved by forcing, and there are several natural ways to force $V=\HOD... | 16 | https://mathoverflow.net/users/1946 | 151805 | 80,960 |
https://mathoverflow.net/questions/151750 | 4 | I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve:
$(a,b) \* (c,d) = (a+c,b+d+ac^q)$.
Over $GF(q^2)$ the group has $q^3$ points, the identity is $(0,0)$ and
the inverse of $(a,b)$ is $(-a,b^q)... | https://mathoverflow.net/users/44138 | Non-abelian group from affine hermitian curve | Yes, that group law is known. It is a disguised form of a Sylow $p$-subgroup of the automorphism group of the Hermitian curve (namely $\text{PGU}\_3(q^2)$), where $p$ is the characteristic of $\mathbf{F}\_q$. This group is well understood, for instance it's an extraspecial $p$-group.
For any $\mathbf{F}\_{q^2}$-ratio... | 3 | https://mathoverflow.net/users/30412 | 151806 | 80,961 |
https://mathoverflow.net/questions/151815 | 5 | Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$
**Question.** Let $\mathbb{P} \in \{Add(\omega, \kappa), Col(\omega\_1, \kappa) \}.$
(1) Is there a subforcing of $\... | https://mathoverflow.net/users/11115 | Bad subforcings of nice forcing notions | No. A subforcing of a c.c.c. forcing is c.c.c. A subforcing of a countably closed forcing is countably-strategically-closed, which implies proper. (This is easy to see via countable elementary submodels. Use the strategy to construct a generic condition.)
Furthermore, every subforcing of a proper forcing is proper. P... | 8 | https://mathoverflow.net/users/11145 | 151818 | 80,965 |
https://mathoverflow.net/questions/151802 | 3 | Let $G$ be a $2$-connected $3$-regular graph. Can $V(G)$ be partitioned into $V\_1$ and $V\_2$ where
$G[V\_1]$(the induced subgraph on $V\_1$) is a cycle of $G$ and $G[V\_2]$ is a forest (Acyclic subgraph) of $G$?
**Edit:**
Since the counterexamples presented so far are $2$-connected, what happen if the connectivi... | https://mathoverflow.net/users/23850 | Can the Vertices of cubic graph be partitioned into and induced cycle and a forest? | I believe this is false.
**EDIT**
The previous counterexample was wrong, let me try again.
A program found counterexample on $10$ vertices and exhaustive search
confirmed it.
The edges are:
```
[(0, 3), (0, 5), (0, 7), (1, 4), (1, 6), (1, 9), (2, 6), (2, 7), (2, 8), (3, 5), (3, 7), (4, 8), (4, 9), (5, 8), (6... | 3 | https://mathoverflow.net/users/12481 | 151819 | 80,966 |
https://mathoverflow.net/questions/151793 | 6 | It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example <http://mathworld.wolfram.com/LatticePolygon.html>.) The same is true for lattices in $\mathbb{R}^n$, simply because any such polygon would lie in a two-dimensional sublattice. If we relax the requirement that the penta... | https://mathoverflow.net/users/29328 | Nonplanar equilateral lattice "pentagons" | Here are the vertices of an equilateral pentagon of side $\sqrt{2}$ in $\mathbb{Q}^3$ in order:
$(0,0,0), (1,1,0), (1,2,1), (0,1,1), (-\frac{1}{3}, -\frac{1}{3}, \frac{4}{3}).$
The first $4$ points form a rhombus. The last point satisfies $z=1-y, x^2+y^2+(1-y)^2=2.$ This has the rational solution $(1,1,0)$ and lin... | 5 | https://mathoverflow.net/users/2954 | 151821 | 80,967 |
https://mathoverflow.net/questions/151840 | 3 | Let $ML$ be a modal logic which contains the Reflection Rule (from $\vdash\Box F$ infer $\vdash F$). For a modal formula $F$, let $H(F)=\{\ \Box G\rightarrow G~|~\Box G$ is a subformula of $F\}$. A Kripke model for $ML$ is called $F$-sound if the root of the model satisfies $H(F)$.
**Soundness**. If $ML\vdash F$, the... | https://mathoverflow.net/users/44170 | Soundness of modal logics which contain the reflection rule | I don’t think your proof strategy will work as is. Here is one way to prove the result. Assume $\vdash\_{ML}A$, and let $M$ be a model with a root $r$ such that $r\models\bigwedge H(A)$. Let $M^\circ$ be the model which differs from $M$ only in that $r$ is made reflexive (accessible from itself). Prove by induction on ... | 1 | https://mathoverflow.net/users/12705 | 151849 | 80,977 |
https://mathoverflow.net/questions/151608 | 2 | Start with a closed Riemann surface with $g$ handles $\Sigma\_g.$
I'm interested in the cohomology of its Jacobian $Jac(\Sigma\_g)=T^{2g},$
in particular how the $SU(2)$ or $SL(2,\mathbb{R})$ Lefschetz decomposition acts on it.
The picture I have in mind is, following Gopakumar-Vafa ideas
(see <http://arxiv.org/abs/h... | https://mathoverflow.net/users/40154 | SU(2) Lefschetz decomposition for cohomology of Riemann surface Jacobian | Your description of the states for g=1 is correct and $dz |0>$ , $\overline{dz} |0>$
are spin 0 representations whereas $|0>$ and $dz \wedge \overline{dz} |0>$ gives a
spin 1/2 representation : the creation operator is the cup-product by the class
$dz \wedge \overline{dz}$.
The "attempt of answer" is correct : I\_g ... | 2 | https://mathoverflow.net/users/25309 | 151855 | 80,981 |
https://mathoverflow.net/questions/151824 | 2 | I have some problems with homotopies.
The situation is this:
Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open Annulus with a finitely many punctures).
$f: X \rightarrow X$ is a homeomorphism, which is isotopic to the identity.
... | https://mathoverflow.net/users/44066 | Homotopy versus path-homotopy on punctured surface | The special feature of $X$, a sphere with three or more punctures, that is being used here is that the space $E(X)$ of all homotopy equivalences $X\to X$ has $\pi\_1 E(X)=0$. (Here we take the identity map of $X$ as the basepoint of $E(X)$ for computing $\pi\_1 E(X)$.) The corresponding statement when $X$ is an annulus... | 11 | https://mathoverflow.net/users/23571 | 151859 | 80,984 |
https://mathoverflow.net/questions/151863 | 1 | Let $c\_i,d\_j <n$, be a set of integers and define
$$ M=\prod Gr(c\_i,n),\quad N=\prod Gr(d\_j,n),$$
where $Gr(k,n)$ is the grassmannian of k-planes in $\mathbb{C}^n$.
Let $E\_M=\oplus E\_i^\*$, where $E\_i$ is the pullback of tautological bundle to $M$ from the i-th component.
Similarly, let $E\_R=\oplus E\_j^\*$, w... | https://mathoverflow.net/users/5259 | How can we show that a transverse section exist | If a vector bundle on a smooth variety is globally generated then its generic section is transversal to zero. This is Bertini Theorem, so of course you need the base field to be infinite. In your special case note that the dual tautological bundles are globally generated and that pullbacks, tensor products, and direct ... | 2 | https://mathoverflow.net/users/4428 | 151876 | 80,990 |
https://mathoverflow.net/questions/151868 | 3 | In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true:
>
> Let $G$ be a group, $g\in G$, and $\rho:G \to \operatorname{End}\_k(V)$ a $G$-module over some commutative ring $k$. Let $H < G$ denote the subgroup centralizing $g$ (i.e. $... | https://mathoverflow.net/users/78 | For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element? | Here is a counterexample. Let $G = \langle g, k : g^2 = x^5 = 1, x^g = x^{-1} \rangle$ be the dihedral group of order $10$. Take $k = \mathbb{C}$ and let $U$ and $U'$ be the two distinct $2$-dimensional irreducible representations of $G$. Let $\rho : G \rightarrow \mathrm{End}(U \oplus U')$ be the corresponding represe... | 2 | https://mathoverflow.net/users/7709 | 151879 | 80,992 |
https://mathoverflow.net/questions/151884 | 1 | Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain.
Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$
where $w'$ means the weak derivative that satisfies
$$\int\_0^T w(t)\phi'(t) = -\int\_0^T w'(t)\phi(t)$$
for all $\phi \in C\_c^\infty(0,T).$
Let $u \... | https://mathoverflow.net/users/43959 | If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$? | Really a comment, unfortunately too long for the comment box:
Your terminology is incorrect: the theory of distributions always requires the "domains" to be open. So here your $\Omega$ is an open set, fixed once and for all as the "space domain" of your given distribution $u\in W^1$, and it really makes no sense to c... | 2 | https://mathoverflow.net/users/33741 | 151888 | 80,995 |
https://mathoverflow.net/questions/151873 | 3 |
>
> Does there exist a group with a normal countable-index abelian subgroup but without characteristic countable-index abelian subgroups?
>
>
>
It is well known that *any finite-index subgroup contains a normal (in the whole group) finite-index subgroup*. A lesser-known proposition states that
($\*$) *any grou... | https://mathoverflow.net/users/24165 | Large abelian characteristic subgroups in abelian-by-countable groups | Here is an example (the group $H\_K^\mathbf{Q}$ defined below, which is actually a split extension abelian-by-countable).
Let $(Q,\le)$ be a total ordering. Let $K$ be a countable (possibly finite) field. Let $V^Q\_K=K^{(Q)}$ be the free $K$-module with basis $(e\_q)\_{q\in Q}$. Let $G^Q\_K$ be the group of $K$-modul... | 3 | https://mathoverflow.net/users/14094 | 151889 | 80,996 |
https://mathoverflow.net/questions/151729 | 2 | Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some minimality/universality property? I.e.: does $X'\to X$ always factor through $S\to X$?
| https://mathoverflow.net/users/4096 | minimality/universality of the Springer resolution of a determinantal variety | I am not sure what you call a "Springer resolution", because I guess most determinantal varieties do not have any Springer resolution.
But anyway, say that we look at $X = \{ A \in End(V)\, \text{such that}\, A^2 = 0\, \text{and}\, \operatorname{rk}(A) \leq 1 \}$, then it has two Springer resolutions which are given... | 2 | https://mathoverflow.net/users/37214 | 151915 | 81,007 |
https://mathoverflow.net/questions/151905 | 2 | It is well-known that the function $f(z)=\sum\_{n=0}^\infty z^{n!}$ is analytic in the open unit disk and it can not be extended analytically to any proper open superset of the unit disk, i.e., the unit circle is the natural boundary of $f$.
Is there an example of a function, analytical in the unit disk, with "natura... | https://mathoverflow.net/users/43681 | Can the natural boundary be part of the unit circle? | Every closed set $F$ on the unit circle is the set of singularities of some analytic function.
Take a countable dense subset $z\_k$ of $F$ and then choose positive $a\_k$ so small
that the series
$$f(z)=\sum\_k\frac{a\_k}{z-z\_k}$$
converges uniformly on compact subsets of $C\backslash F$.
| 8 | https://mathoverflow.net/users/25510 | 151918 | 81,009 |
https://mathoverflow.net/questions/151914 | 4 | While perusing Kluener's *Database of Number Fields*, I noticed that a lot of [the discriminants of 7T5](http://reh.math.uni-duesseldorf.de/cgi-klueners/groups3.pl?deg=7&t=5&zer=&bet=&teil=¬eil=&maxteil=&numprime=) came in pairs. After some doodling, I found four families. The first two are,
$$x^7 - x^6 + x^5 + (n... | https://mathoverflow.net/users/12905 | Parametric septic fields $L(7) = L(3,2)$ with the same discriminant | *Every* septic number field $F$ with this Galois group
has a twin septic field $F'$ with the
same Galois closure $K$, the same Dedekind zeta functions
$\zeta\_F$ and $\zeta\_{F'}$, and the same discriminant.
These twins arise in the same way as twin sextic extensions with
Galois groups $S\_6$ and $A\_6$: each of tho... | 7 | https://mathoverflow.net/users/14830 | 151920 | 81,010 |
https://mathoverflow.net/questions/151892 | 8 | Suppose you have the mapping torus $M\_\phi$ of some pseudo-Anosov map $\phi.$ Is there some sufficient or necessary condition on $\phi$ to assure that $M\_\phi$ has large injectivity radius? I am aware of Jeff Brock et al's results on volume bounds, but this is not quite the same...
| https://mathoverflow.net/users/11142 | Hyperbolic 3-manifolds fibering over the circle | A related question is studied by Minsky in the paper "Bounded geometry for Kleinian groups": <http://arxiv.org/abs/arXiv:math/0105078>
The main theorem gives a necessary and sufficient condition for the infinite cyclic cover $\tilde{M}\_{\phi}$ of a pseudo-Anosov mapping torus to have large injectivity radius, in ter... | 8 | https://mathoverflow.net/users/37118 | 151923 | 81,011 |
https://mathoverflow.net/questions/151906 | 3 | A while back I remember reading that F. Jaeger proved that Tutte's $5$-flow conjecture is equivalent to a statement about the co-planarity of a certain set of points in some euclidean space. But I cannot remember the exact statement and I cannot find it anywhere.
Does anyone out there remember the equivalent statemen... | https://mathoverflow.net/users/23850 | About an equivalent to Tutte's 5-flow Conjecture | You are probably thinking of the paper:
F. Jaeger, [Geometrical aspects of Tutte’s 5-flow conjecture](http://www.ams.org/mathscinet-getitem?mr=737025).
Graphs and other combinatorial topics, Teubner-Texte Math, 1983.
The following is taken from the MathSciNet review, which in turn is taken from the author's summary... | 3 | https://mathoverflow.net/users/630 | 151924 | 81,012 |
https://mathoverflow.net/questions/151846 | 6 | Every automorphism of an algebraic number field $F$ extends to an automorphism of $\mathbb{\overline{Q}}$, but an order 2 automorphism of $F$ need not extend to one of order 2 on $\mathbb{\overline{Q}}$,
Is there an algebraic criterion, just looking at the action on $F$, to tell whether a given involution on $F$ wil... | https://mathoverflow.net/users/38783 | Can you identify complex conjugations in a number field? | An involution $\sigma$ is a complex conjugation if not every element of $F^\sigma$ is a sum of squares of elements of $F^\sigma$ and, if $\sigma$ is nontrivial, some element of $F^\sigma$ which is not a sum of squares of elements of $F^\sigma$ is a sum of squares of elements of $F$.
Proof that every complex conjugati... | 6 | https://mathoverflow.net/users/18060 | 151928 | 81,014 |
https://mathoverflow.net/questions/151801 | 2 | Let $G$ be a $2$-connected $3$-regular graph. Is it true that $E(G) = E\_1 \cup E\_2$ where
$G[E\_1]$(the induced subgraph on $E\_1$) is a cycle of $G$ and $G[E\_2]$ is a forest (Acyclic subgraph) of $G$?
**Edit:**
Since the counterexamples presented so far are $2$-connected, what happen if the connectivity of the... | https://mathoverflow.net/users/23850 | Is the set of edge of a cubic graph the union of a cycle and and an Acyclic graph? | Let $G$ be the cubic graph on $30$ vertices obtained by replacing each vertex of the [Petersen graph](http://en.wikipedia.org/wiki/Petersen_graph) with a triangle. Then $G$ is $3$-connected because the Petersen graph is $3$-connected, and there is no cycle in $G$ that meets all ten triangles because there is no Hamilto... | 2 | https://mathoverflow.net/users/43266 | 151931 | 81,015 |
https://mathoverflow.net/questions/151869 | 12 | Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as
$$
N= \prod\_{j=1}^{n} j^{x\_j}
$$
where $x\_1$, $\ldots$, $x\_n$ are non-negative rational numbers with $\sum\_{j}x\_j \in {\Bbb N}$. Does there necessarily exist a representation ... | https://mathoverflow.net/users/38624 | An integrality question about expressing an integer as a product of numbers below $n$ | It seems that here is an example for rational exponents.
Let $n=2209=47^2>3^7>2^{11}$, and $N=4\,385\,664=2^7\cdot 3^6\cdot 47=2048^{7/11}\cdot 2187^{6/7}\cdot 2209^{1/2}\cdot 1^{1/154}$ with $7/11+6/7+1/2+1/154=2$. If $N=ab$ with $a,b\leq 47^2$, then 47 divides one of $a$ and $b$ (say, $a=47k$); since $a,b\leq 47^2$... | 18 | https://mathoverflow.net/users/17581 | 151943 | 81,021 |
https://mathoverflow.net/questions/151922 | 2 | We have a smooth space $X$ with an action from $G\_1 \times G\_2$ on it; we also have a differential operator $P \in \mathscr{D}(X)$. If $P$ takes $G\_1 \times G\_2$-invariant functions to $G\_1 \times G\_2$-invariant functions and also is $G\_1$-invariant, is it necessarily true that $P$ takes $G\_2$-invariant functio... | https://mathoverflow.net/users/44191 | Differential operators and commuting actions | Let $X$ be ${\mathbb R}\times{\mathbb R}$ and let $G\_1\cong G\_2$ be the group of reals, $G\_1$ acting by translation on the first argument, $G\_2$ on the second.
Let $f$ be any smooth function on $\mathbb R$ and let $D=f(y)\frac{\partial}{\partial x}$, where the coordinates on $X$ are $(x,y)$. Then $D$ ist $G\_1$-inv... | 3 | https://mathoverflow.net/users/nan | 151945 | 81,023 |
https://mathoverflow.net/questions/151942 | 8 | I expect that the holonomy group of an Enriques surface $S$ is $SU(2)\times C\_2$. I think this can be proven by the fact that its double cover, which is a K3 surface, has the full $SU(2)$ holonomy, but I failed proving it. The holonomy group should be either $SU(2)$ or its $\pi\_1(S)=C\_2$-extension $SU(2)\times C\_2$... | https://mathoverflow.net/users/44202 | Holonomy group of Enriques surface | Assume that you have endowed an Enriques surface $S$ with a Ricci-flat Kähler metric $g$. The holonomy $H$ of $g$ cannot be contained in $\mathrm{SU}(2)$ because the canonical bundle of $S$ is not trivial (though its square is trivial). Meanwhile, the identity component of $H$ has to be equal to $\mathrm{SU}(2)$ becaus... | 14 | https://mathoverflow.net/users/13972 | 151949 | 81,024 |
https://mathoverflow.net/questions/151957 | 2 | Let $\mathcal{H}$ be Hilbert space and $\mathfrak{B(}\mathcal{H}\mathcal{)}$ of all bounded linear operators on $\mathcal{H}$. Let $\mathcal{A}$ be a maximal commutative sub-algebra of $\mathfrak{B(}\mathcal{H)}$.
1. Is there an explicit formula a conditional expectation (a norm one projection) $\pi:\mathfrak{B(}\mat... | https://mathoverflow.net/users/40214 | A norm one projection | frege, maximal abelian subalgebras of von Neumann algebras are von Neumann algebras, hence are of the form $L\_\infty(\mu)$ for some measure $\mu$. See [this post](https://mathoverflow.net/questions/110461/direct-proof-of-injectivity-of-l-infty) of Bill Johnson to see a neat way to produce a projection. The problem is ... | 1 | https://mathoverflow.net/users/15129 | 151958 | 81,029 |
https://mathoverflow.net/questions/137111 | 6 | Let S be a propositional modal logic system (extension of K, or even E) with a single unary modal operator and defined by a single non-iterative axiom (i.e. of modal degree 1).
Is it true that for such a system S, every theorem that is a non-iterative formula has a proof consisting only of non-iterative formulas?
... | https://mathoverflow.net/users/37336 | Non-iterative modal logics | $\let\ET\bigwedge\let\LOR\bigvee\let\EQ\Leftrightarrow$The property is true for extensions of K (i.e., normal modal logics). You didn’t really describe the proof system you are interested in, but based on the discussion in the question, I will assume it is a Hilbert-style proof system with the rules of modus ponens and... | 4 | https://mathoverflow.net/users/12705 | 151961 | 81,030 |
https://mathoverflow.net/questions/147068 | 4 | I would like to diagonalize a very large matrix that has the same property as quaternion matrices (in a sense that the matrix can be written as a linear combination of quaternions):
$$
H = \left(\begin{array}{cc} H\_{11} & H\_{12} \\ -H^\ast\_{12} & H^\ast\_{11} \end{array}\right),
\quad U^\dagger HU = \epsilon
$$
wher... | https://mathoverflow.net/users/42395 | Diagonalization of quaternion matrices | I am rather certain that you cannot modify a standard diagonalization to the structured diagonalization that you want. I have seen such a method used, but it is expected to fail occasionally, especially for a large matrix. What you expect are the eigenvalues to all have even multiplicity, and then you modify the basis ... | 3 | https://mathoverflow.net/users/6133 | 151971 | 81,033 |
https://mathoverflow.net/questions/151978 | 4 | Let $P\_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d\_{n,m}(x) = |\frac{d^m}{dt^m}P\_n(t)|\_{t=x}$ assuming that $|x| \le 1$? Clearly
$$d\_{n,m}(x) \le d\_{n,m}(1) = \frac{(m+n)!}{2^m m! (n-m)!}$$
works, but this is pessimistic unless is $x$ is very close to 1. For example, if $n = 10... | https://mathoverflow.net/users/4854 | Accurate bounds for derivatives of Legendre polynomials | A cheap way to improve the trivial bound is to use Bernstein (if you are far away from the endpoints ) or Markov (if you are close to the endpoints) inequalities for the derivatives of polynomials. In case $m=1,$ Berstein inequality gives
$$\|P'\_n\|\le \frac{\|P\_n\|}{\sqrt{1-x^2}},$$
while Markov leads to
$$\|P'\_... | 3 | https://mathoverflow.net/users/17503 | 151983 | 81,037 |
https://mathoverflow.net/questions/151976 | 0 | so: I have a M/G/1-queue with Poisson arrivals with rate lambda=1 and the service time being the sum of two exp-distributed variables vith rates u1=1 and u2=2.
If we let Wq be the time an average customer spends in the queue, what is the probability that Wq equals 0, i.e. P(Wq=0)=?.
Formulas for M/M/1-queues can't ... | https://mathoverflow.net/users/44203 | M/G/1 queue - probability that waiting time is zero | Did you look at the Pollaczek–Khinchine transform?
The distribution is hypo-exponential, which is simple in the bi-variate case,
$$2(\exp(-x)-\exp(-2x)).$$
The Laplace transform of this is simple to find which should enable the Pollaczek–Khinchine transform.
| 0 | https://mathoverflow.net/users/42619 | 151986 | 81,038 |
https://mathoverflow.net/questions/151991 | 1 | Given a finite Abelian group: $G=\mathbb{Z}\_{n\_1} \times \mathbb{Z}\_{n\_2} \times \mathbb{Z}\_{n\_3}$, where ${n\_1},{n\_2},{n\_3}$ are arbitrary positive integers. ${n\_1},{n\_2},{n\_3}$ may have or may not have common divisors.
**Question**: What is **the smallest number of $N$** such that the following Lie grou... | https://mathoverflow.net/users/27004 | The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$ | As soon as a compact connected Lie group has rank at least 3, i.e. it contains a 3-dimensional torus, it will contain your group $G$ (for all values of the parameters). So for example $SU(4), SO(6), SU(3)\times SU(3),SO(4)\times SO(4)$ do the job. Of course you must see whether these values are optimal...
Where does th... | 6 | https://mathoverflow.net/users/14497 | 151993 | 81,041 |
https://mathoverflow.net/questions/151702 | 3 | In 1978 [Doyen, Hubaut and Vandensavel](http://link.springer.com/article/10.1007/BF01174898#page-1) proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is
$$
Rk\_{2}(N)=v-(d\_{p}+1),
$$
where $d\_{p}$ is the *projective dimension* of $S$.
I would like to know if ... | https://mathoverflow.net/users/22051 | Ranks of higher incidence matrices of designs | For the generalization in the first direction, the $p$-rank of the incidence matrix $N$ of an $S(2,k,v)$ is lower bounded by the dimension of the Steinberg module:
$$\operatorname{rank}\_2(N)(\operatorname{rank}\_2(N)-1) \geq \frac{(v-1)(v-k)}{k} \quad \text{if} \frac{v-k}{k-1} \text{is even}$$
and if further $k$ is ... | 3 | https://mathoverflow.net/users/27829 | 151995 | 81,042 |
https://mathoverflow.net/questions/151968 | 14 | This entire question takes place in the $HF\_p$-local category of $p$-local spectra, i.e. the essential image of $HF\_p$-localization on the stable homotopy category. $HF\_p$ itself is in there, and of course so is the $HF\_p$-local sphere $L\_{HF\_p}S^0$. Let $loc(X)$ denote the smallest localizing subcategory contain... | https://mathoverflow.net/users/14220 | localizing subcategories of $HF_p$-local spectra | First note that it is equivalent to ask whether the mod p Moore spectrum $M(p)$ is in the localizing subcategory (in the local sense) generated by $HF\_p$. Indeed, the fiber $C\_0 S$ of the map $S \xrightarrow{} H\mathbb{Q}$ is in the localizing subcategory generated by $M(p)$ (in the usual sense), and $L\_{HF\_p}C\_0 ... | 14 | https://mathoverflow.net/users/1698 | 152002 | 81,046 |
https://mathoverflow.net/questions/152004 | 7 | Let $A$ be an abelian variety of dimension $n$. Over $\mathbb{C}$, at least, it is known that the Picard number (that is, the rank of the Neron-Severi group of $A$) is less than or equal to $n^2$, with equality if and only if $A$ is isogenous to the self product of an elliptic curve with complex multiplication.
Is t... | https://mathoverflow.net/users/14143 | Picard number of principally polarized abelian varieties | A tight bound for simple $A/\mathbb{C}$ is $\rho(A) \leq 3n/2$. This follows from Proposition 5.5.7 in Birkenhake-Lange. If $A$ does not have indefinite quaternionic multiplication, the stronger bound $\rho(A) \leq n$ holds.
| 7 | https://mathoverflow.net/users/949 | 152017 | 81,050 |
https://mathoverflow.net/questions/152015 | 4 | Is the packing of the plane by disks of radius 1/2 centered at the points of ${\bf Z} \times {\bf Z}$ "locally rigid" in the sense that no finite subcollection of the disks admits any joint infinitesimal deformations subject to the locations of the other disks?
Certainly it is clear that if you fix the locations of a... | https://mathoverflow.net/users/3621 | Local rigidity of square disk packing | Yes, every finite subset of a square lattice packing is locked in place if you hold the rest fixed. See [Finite and Uniform Stability of Sphere Packings](http://dx.doi.org/10.1007/PL00009374) by [A. Bezdek](http://www.auburn.edu/cosam/faculty/math_stats/bezdek/index.htm), [K. Bezdek](http://math.ucalgary.ca/profiles/ka... | 10 | https://mathoverflow.net/users/4720 | 152018 | 81,051 |
https://mathoverflow.net/questions/151390 | 9 | In one variant of the classic counterfeit coins problem you are given a bag of $n$ numbered but otherwise identical looking coins and a scale and your job is to find which coins are counterfeit. Counterfeit coins all have one weight and the other coins all have another weight. The scale can only tell you if two sets of... | https://mathoverflow.net/users/nan | How to find counterfeit coins by weighing | This problem was solved (up to a small multiplicative factor) by Erdos and Renyi:
<http://www.renyi.hu/~p_erdos/1963-12.pdf>
Ps. Wow, Douglas Zare has a good intuition!
| 8 | https://mathoverflow.net/users/955 | 152023 | 81,055 |
https://mathoverflow.net/questions/152016 | 4 | In this [Math Stack Exchange post](https://math.stackexchange.com/questions/606172/if-lim-n-to-infty-2-x-n-1-x-n-x-then-is-it-true-that-li/608550#608550), I proved the following result.
>
> **Theorem:** Let $ X $ be a locally convex topological vector space. Let $ x \in X $ and suppose that $ (x\_{n})\_{n \in \math... | https://mathoverflow.net/users/nan | Extending a Certain Result from Locally Convex Topological Vector Spaces to General Topological Vector Spaces | I think that you can construct a counterexample if $X$ is the space of measurable functions on a nice probability space (like the unit interval with the Lebesgue measure) endowed with stochastic convergence where $y\_n \to 0$ if and only if $P(|y\_n| >\varepsilon) \to 0$ for all $\varepsilon >0$. Take a suitable sequen... | 7 | https://mathoverflow.net/users/21051 | 152026 | 81,056 |
https://mathoverflow.net/questions/152042 | 6 | Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec }K\times\_{\text{Spec }k}V$ is also regular?
| https://mathoverflow.net/users/40504 | Does regular field extension preserve regularity? | Yes, and it is only necessary to assume $K$ is separable over $k$ (i.e., not necessary to assume in addition that $k$ is algebraically closed in $K$). The idea is to use Serre's regularity criterion to reduce to the case when $K/k$ is finitely generated, and then use a separating transcendence basis in such cases to co... | 9 | https://mathoverflow.net/users/43107 | 152054 | 81,065 |
https://mathoverflow.net/questions/152027 | 2 | Let $X$ denote a complex $C^\*$-algebra and $\{Z(t)\}\_{t\geq 0}$ is a $C\_0$-semigroup of operators on $X$.
Let $x\in X$ satisfy have $x=x^\*$ (x is self-adjoint), such that its spectrum satisfies $\sigma(x)\subset [0,\infty)$.
Then under what conditions does it follow that $[Z^\*(t)]=[Z(t)]$, and $\sigma(Z(t)x)... | https://mathoverflow.net/users/43418 | One-parameter semigroup of operators of a $C^*$-algebra applied to positive self-adjoint element | Apart from the fact that I do not understand some parts of your question, self-adjoint elements with positive spectrum define a positive cone in your $C^\ast$ algebra. Positivity-preserving semigroups in $C^\ast$ and von Neumann algebras were extensively studies, you should consult the chapter written by Ulrich Groh in... | 4 | https://mathoverflow.net/users/12898 | 152057 | 81,068 |
https://mathoverflow.net/questions/152073 | 6 | 1. Does the minimal standard model of ZF contain all recursive ordinals or is it limited (probably by the proof theoretic ordinal of ZF as I suspect but cannot prove)?
2. Paul J. Cohen's definition of the minimal standard model for ZF.(<http://www.ams.org/journals/bull/1963-69-04/S0002-9904-1963-10989-1/S0002-9904-1963... | https://mathoverflow.net/users/16554 | Recursive ordinals and the minimal standard model of ZF | The minimal transitive (or synonymously, standard) model of ZF is $L\_\alpha$, where the ordinal $\alpha$ is chosen to be smallest such that $L\_\alpha\models\text{ZF}$. Since ZF proves that $\omega\_1^{CK}$ exist, and the recursive ordinals are absolute between $V$ and $L\_\alpha$, it follows that $\omega\_1^{CK}<\alp... | 6 | https://mathoverflow.net/users/1946 | 152075 | 81,074 |
https://mathoverflow.net/questions/151090 | 5 | it is known that tropicalization of a variety(irreducible and subvariety of some torus.) is a support of a polyhedral complex. I wonder which kinds of polyhedra can occur in this polyhedral complex. In other words, if P is any polyhedron of dimension d, then does some variety X always exist such that P is in a polyhedr... | https://mathoverflow.net/users/nan | Polyhedra from a tropical variety | Provided that the facets of $P$ have integral normal vectors, this is always the case.
To construct such a tropical variety, first let $\Sigma$ be the normal fan to $P$.
This comes along with a piecewise linear function $\psi$ defined by
$\psi(n)=-inf\{ \langle n, m \rangle\,|\, m\in P\}$. Now choose any lattice polyt... | 4 | https://mathoverflow.net/users/23917 | 152091 | 81,080 |
https://mathoverflow.net/questions/152081 | 16 | Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes\_K A$ be the algebra given by extension of scalars, and let $M' = L \otimes\_K M$ and $N' = L \otimes\_K N$ be the A'-modules given by extension of scalars.
Does $M' \cong N'$... | https://mathoverflow.net/users/22 | Does base extension reflect the property of being isomorphic? | I hope I'm not misunderstanding the question. Here goes:
We'll show that if $M,N$ are finite-dimensional over $K$,
then they are isomorphic over $K$.
Think of the linear space $X=\mathrm{Hom}\_{A}(M,N)$
as a variety over $K$. Inside $X$ look at
the $K$-subvariety $X'$ of maps that are not isomorphisms $M \righta... | 10 | https://mathoverflow.net/users/42059 | 152093 | 81,082 |
https://mathoverflow.net/questions/151926 | 1 | There is a matrix A where each entry is either 0 or 1. Each column has exactly a 1's and each row has at most b 1's. What's the upper bound of abs(|A|)?
The condition is stronger than the Hadamard's maximal determinant problem. Is there any known result?
| https://mathoverflow.net/users/44193 | Estimate the determinant of sparse 0-1 matrix | It looks like a previous answer disappeared, so I will expand on my comments.
Using that a determinant corresponds to the volume of a parallelipiped, an immediate
upper bound (using columns) of $a^{n/2}$ results. If you know the distribution across the
rows, use the product of their lengths (don't forget the square r... | 1 | https://mathoverflow.net/users/35626 | 152097 | 81,083 |
https://mathoverflow.net/questions/152096 | 2 | It is a theorem of Selberg that a lattice $\Gamma$ in a linear group has a torsion-free subgroup of finite index. Page 64 in 'Introduction to Arithmetic Groups' by Dave Morris asserts these can be realized as follows. Since $\Gamma$ is finitely generated, $\Gamma \subseteq \mathrm{SL}\_k(R)$ where $R = \mathbb{Z}[a\_1,... | https://mathoverflow.net/users/30721 | Finite-index free subgroups in lattices and matrix rings | A co-compact lattice in $SL\_2({\mathbb C})$ cannot contain a finite index free subgroup because of cohomological reasons. The cohomological dimension of the finite index torsion free subgroup is 3. \vskip 5mm
A non-cocompact lattice $\Gamma $ in $SL\_2({\mathbb C})$, after a conjugation, intersects the upper triangu... | 3 | https://mathoverflow.net/users/23291 | 152105 | 81,084 |
https://mathoverflow.net/questions/152099 | 6 | I have asked this question [here](https://math.stackexchange.com/questions/387033/constructing-a-odd-homeomorphism-between-a-and-sn) seven months ago and until now I got no answer.
Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is homeomorphic to some sp... | https://mathoverflow.net/users/39678 | Constructing a odd homeomorphism between $A$ and $S^n$ | The answer is "NO".
It follows from existence of exotic smooth involutions of sphere.
Say, [this movie](http://homepage.ruhr-uni-bochum.de/Thomas.Puettmann/XInvolution.html) explains a construction of an involution $\iota$ of $\mathbb S^5$ such that
the quotient $\Pi=\mathbb S^5/\iota$ is homotopy equivalent, but no... | 7 | https://mathoverflow.net/users/1441 | 152106 | 81,085 |
https://mathoverflow.net/questions/152094 | 2 | Let $\lambda(G)$ denote the edge-connectivity of $G$.
Consider the following parameter:
$\rho(G) = \max\_{X \subset V(G)} \min(\lambda(G[X]), \lambda(G[V(G) - X]))$
Has this parameter been studied? Are there any known bounds on it in terms of $\lambda(G)$?
| https://mathoverflow.net/users/23850 | Is this Graph parameter known? | **Note**: The question can be rephrased as follows: Two players $A$ and $B$ play a game on a graph. $A$ cuts the graph into two connected components, and $B$ chooses one component of the resulting graph. $A$ wants to maximize and $B$ wants to minimize the edge connectivity of the result. The bounds are the question: Wh... | 4 | https://mathoverflow.net/users/38267 | 152110 | 81,088 |
https://mathoverflow.net/questions/151952 | 3 | Let $S$ be a generic cubic surface and let $C$ be its intersection with a generic quadric surface.
In the linear system of hyperplane sections of $S$, how many points represent the planes $H$ tangent to $C$ at two points, such that the curve $H \cap S$ is singular?
| https://mathoverflow.net/users/27125 | Points of a linear system on a cubic surface | OK, I'll try to answer your corrected question : you are looking at planes which are bitangent to $C$, and tangent to $S$ somewhere. Now these are two independent properties, so in the dual $(\mathbb{P}^3)^\*$ your planes are the intersection points of the dual surface $S^\*$ and the curve of bitangent planes, say $\Ga... | 3 | https://mathoverflow.net/users/40297 | 152113 | 81,089 |
https://mathoverflow.net/questions/152090 | 9 | I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to H^n(BG;\mathbb{R})$ certainly kills all torsion, but can it have non-torsion elements in the kernel? For instance, if $G=SU\_n$, then th... | https://mathoverflow.net/users/5937 | Integral versus real (universal) characteristic classes | If $X$ is a space of finite type (meaning that the homology groups $H\_i(X)$ are all finitely generated, a condition which applies in particular to $X=BG$ for $G$ a compact Lie group) then for each $n$ the map $H^n(X)\to H^n(X;\mathbb{R})$ is injective if and only if $H\_{n-1}(X)$ is torsion free. Here and below intege... | 11 | https://mathoverflow.net/users/8103 | 152117 | 81,090 |
https://mathoverflow.net/questions/152131 | 10 | Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms?
I am not quite sure how smoothing works for trivial $G$, but maybe this can be generalized to the situation described above?
| https://mathoverflow.net/users/37353 | Equivariant smoothing of PL structures on $S^3$ | I think the answer to your question is "always". You may want to look at the following paper:
Kwasik, Sławomir; Lee, Kyung Bai. Locally linear actions on $3$-manifolds. Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 2, 253--260. MR0948910.
In particular, Corollary 2.2: *A topological action of a finite group ... | 9 | https://mathoverflow.net/users/1944 | 152138 | 81,096 |
https://mathoverflow.net/questions/151551 | 9 | Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all Pontryagin numbers of the manifold? I would be grateful for some nice reference on this topic.
For four-manifolds such a formu... | https://mathoverflow.net/users/13441 | Pontryagin numbers on manifolds with an $S^1$-action | One can find the Pontryagin numbers. It is an application of a more general version of the G-signature theorem, when one considers the signature operator on M twisted by a vector bundle. It is Theorem 8.11 respectively formula 8.12 in the paper which proves the G-signature theorem:
Atiyah, M. F.; Singer, I. M. The in... | 6 | https://mathoverflow.net/users/36799 | 152142 | 81,099 |
https://mathoverflow.net/questions/152135 | 6 | Let $B\_t$ be a Brownian motion with variance 1. We know that $\int\_0^1 B(t) \mathrm{d} t \sim \mathcal{N}(0,1/3)$. I am interested to know what we can say about the law of the two random variables
$X = \int\_{0}^1 B(t)^2 \mathrm{d}t = \langle B,B\rangle\_{L^2([0,1])}$ and $Y = \int\_{0}^{1} \left( B(t) - \int\_0^1 B... | https://mathoverflow.net/users/39261 | Law of the $L^2$ norm of a Brownian motion and related | [Aspects of Brownian Motion](http://www.springer.com/mathematics/probability/book/978-3-540-22347-4) (Mansuy & Yor) give an expression for the joint Laplace transform of $B\_t$ and $\int\_0^t B\_s^2 ds$ (section 2.1). For $\delta$-dimensional Brownian motion,
$$ \mathbb{E}\left[\exp\left(-\alpha|B\_t|-\frac{b^2}{2}\int... | 8 | https://mathoverflow.net/users/32723 | 152149 | 81,101 |
https://mathoverflow.net/questions/152051 | 3 | For vector basis $b\_1,..,b\_n$ on a finite extension $F$ of $\mathbb{Q}$, where $-1$ is not a sum of squares, each linear order on $F$ is determined by an order on the basis. This uses information about where the $b\_i$ sit among the rationals, and that information can be gained from Sturm's algorithm applied to any p... | https://mathoverflow.net/users/38783 | Definability of orderings on a formally real number field | (Originally a comment, reposted as an answer:)
Choose a primitive element $\alpha$ of F (i.e. such that $F=\mathbf{Q}(\alpha)$). Let $f$ be its minimal polynomial. Then the data of a field ordering of $F$ is equivalent to the data of an interval in $\mathbf{Q}$ which contains a unique root of $f$. Given any such inte... | 3 | https://mathoverflow.net/users/2481 | 152150 | 81,102 |
https://mathoverflow.net/questions/152145 | 3 | The value of $k$ can be very large indeed (up to $10^{12}$). Is there an efficient way to calculate the output?
Edit : 'm' is a prime number.
| https://mathoverflow.net/users/44317 | Calculating (n ^ fibonacci(k)) MOD m for a large value of k | Assuming $n$ and $m$ are coprime and $m$ is factored, first compute $a=\text{fibonacci}(k) \mod \varphi(m)$. Computing linear recurrence efficiently modulo $n$ is possible, e.g. via matrix exponentiation.
Then your expression is equal to $n^a \mod m$ which is easy to compute efficiently
working $\mod m$ and fast expo... | 7 | https://mathoverflow.net/users/12481 | 152151 | 81,103 |
https://mathoverflow.net/questions/151871 | 0 | Any [algebraically closed field](http://en.wikipedia.org/wiki/Algebraically_closed_field) (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order [Peano arithmetic](http://en.wikipedia.org/wiki/Peano_axioms#First-order_theory_of_arithmetic) (PA) except $\forall x(Sx \neq 0)$ is replaced wit... | https://mathoverflow.net/users/26766 | Recursive Non-standard Models of Modular Arithmetic? | The only part of the question that looks research-level is the last sentence, provided it is interpreted as “are there recursive models of *the theory* of $\mathbb{Z}^\* /n^\* \mathbb{Z}^\*$”, so let me address that. (Models do not have models, theories have models. The preceding discussion is quite nonsensical, as one... | 6 | https://mathoverflow.net/users/12705 | 152155 | 81,105 |
https://mathoverflow.net/questions/152107 | 4 | Assume you have a source of random binary information that has a bias but no correlation between consecutive bits. John von Neumann describes an algorithm to debias the random source and output a perfectly unbiased sequence of 1s and 0s as follows:
1. Extract two bits from the source
2. If the two bits are the same, ... | https://mathoverflow.net/users/44295 | Proof of Von Neumann's debiasing algorithm | The original article by von Neumann
(<https://dornsifecms.usc.edu/assets/sites/520/docs/VonNeumann-ams12p36-38.pdf>)
does not bother to prove this.
Most likely because if the probability of "1" is $p$ and that of "0" is $q$ in a Bernoulli sequence, it was evident for him that both "10" ad "01" have probability $pq$.
| 2 | https://mathoverflow.net/users/20804 | 152156 | 81,106 |
https://mathoverflow.net/questions/152139 | 4 | So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y$, a continuous map $f:X\rightarrow Y$
and a sheaf of abelian groups $\mathcal{F}$ on $X$ such that the presheaf $f\_\*^... | https://mathoverflow.net/users/11765 | Explicit examples presheaves associated to higher direct images which fail to be sheaves | For (i), take $X=S^3$, $Y= S^2$, $f$ the Hopf fibration. There is a local $H^1$ everywhere, coming from the fact that the fibers are circles. This local section is defined consistently everywhere (because there is no monodromy since $Y$ is simply-connected), but it does not glue to a global section because $H^1(X,\math... | 11 | https://mathoverflow.net/users/18060 | 152159 | 81,107 |
https://mathoverflow.net/questions/152160 | 3 | I want a reference that treated the proof of this proposition :
>
> **Proposition :** Suppose that $X$ is a finitely generated, torsion $\Lambda-$module. Then there are uniquely determined $\Lambda-$submodules $Z$ and $Y$ of $X$ with the following properties :
>
>
> 1. $Z$ is finite and $X/Z$ has no nonzero, fini... | https://mathoverflow.net/users/44319 | Iwasawa's invariants | I'm assuming $\Lambda=\mathbf{Z}\_p[[\mathbf{Z}\_p]]\cong\mathbf{Z}\_p[[T]]$ (with a topological generator $\gamma$ of $\Gamma=\mathbf{Z}\_p$ going to $1+T$).
Let $Z$ be the $\mathbf{Z}\_p$-torsion submodule of the $\Lambda$-submodule $X^\prime=\bigcup\_{n\geq 0}X^{\Gamma^{p^n}}$. Since $X^\prime$ is finitely generat... | 3 | https://mathoverflow.net/users/4351 | 152169 | 81,112 |
https://mathoverflow.net/questions/151907 | 11 | The Sato-Tate conjecture for elliptic curves $E$ predicts the distribution of the eigenvalues of Frobenius at $p$ on the Tate module of $E$ as $p$ varies in terms of the distribution of the eigenvalues of a random element of an associated compact group. There are conjectural generalizations for curves of higher genus a... | https://mathoverflow.net/users/290 | The Sato-Tate conjecture for hypersurfaces? | The conjecture is that associated to any cohomology group of an algebraic variety, or set of cohomology group on the algebraic variety, there should be a complex Lie group (with representations), which can be described in a number of ways:
The Zariski closure of the image of the Galois group inside $GL\_n(\mathbb Q\_... | 6 | https://mathoverflow.net/users/18060 | 152176 | 81,116 |
https://mathoverflow.net/questions/152182 | 3 | This question is about the formal smoothness property for schemes. A morphism $X\to S$ is formally smooth if for every affine $S$-scheme $Y$ and every subscheme $Y\_0\subset Y$ cut out by a nilpotent ideal, morphisms $Y\_0\to X$ always extend to $Y\to X$. (Here all morphisms are $S$-morphisms.)
Let's suppose $f\colon... | https://mathoverflow.net/users/271 | Does formal smoothness work compatibly across morphisms? | Certainly not for a fixed morphism $Y \to X$. Perhaps if you choose the morphism $Y \to X$ it might work. Counterexample:
$Y\_0 =Y\_0'= X'= \operatorname{Spec} k$. $Y= Y' = \operatorname{Spec} k [x]/x^2$. $X = \operatorname{Spec} k[x]$. The first two commutative diagrams are the obvious ones. Then the obvious morphis... | 1 | https://mathoverflow.net/users/18060 | 152183 | 81,118 |
https://mathoverflow.net/questions/150613 | 10 | For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The [Brun-Titchmarsh theorem](https://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem) states that $$\pi(x,q,a)\leq \frac{(2+o(1))x}{\phi(q)\log(x/q)}$$ for all $q<x$. Letting $\theta=\fra... | https://mathoverflow.net/users/12176 | Can the Brun-Titchmarsh theorem be improved when the modulus is smooth? | (Credit goes also to James Maynard for this answer.) If you assume that $x\ge q^{12/5+\epsilon}$ for some fixed $\epsilon>0$ and you take $q$ to be $x^\theta$-smooth, then you do get that
$$
\pi(x;q,a)\le (2+\delta) \frac{x}{\phi(q)\log x},
$$
with $\delta$ tending to 0 as $\theta\to0$ and $x\to\infty$. This follows b... | 8 | https://mathoverflow.net/users/4003 | 152189 | 81,120 |
https://mathoverflow.net/questions/136887 | 11 | The [Bombieri-Vinogradov Theorem](http://en.wikipedia.org/wiki/Bombieri%E2%80%93Vinogradov_theorem) states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum\_{q\leq Q}\max\_{y\leq x}\max\_{\begin{array}{c}
a\text{ mod q}\\
(a,q)=1
\end{array}}\left|\psi(y;q,a)-\frac... | https://mathoverflow.net/users/12176 | The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$ | [Elliott has a result](http://www.ams.org/mathscinet-getitem?mr=2281164) in this direction. Let $A>0$ and $a \geq 2$ a fixed integer. Furthermore, let $q \leq x^{1/3-\epsilon} $ be a large power of $a$. One then has that
$$ \sum\_{\substack{d \leq q^{-1}x^{1/2} \log^{-A-6}(x) \\ (d,q)=1 }} \max\_{(r,qd)=1} \max\_{y\leq... | 5 | https://mathoverflow.net/users/630 | 152200 | 81,123 |
https://mathoverflow.net/questions/152199 | 3 | Assume that $M$ is the long line. Is $TM$, the tangent bundle, isomorphic to $T^{\*}(M)$, the cotangent bundle?
| https://mathoverflow.net/users/36688 | A question on long line | No (for any differentiable structure on the long line, there are many). Since $TM$ is a line bundle, an isomorphism $TM\cong T^\*M$ is necessarily symmetric, i.e. given by a global section of $\mathrm{Sym}^2T^\*M$, locally of the form $f(t)dt^2$. Since $M$ is connected the form is either positive or negative, thus prov... | 8 | https://mathoverflow.net/users/40297 | 152201 | 81,124 |
https://mathoverflow.net/questions/152147 | 5 | This is a question about the base-rings appearing in the the theory of $(\varphi, \Gamma)$-modules in $p$-adic Hodge theory.
Let $p$ be prime, $n \ge 1$, and let
$$ \mathbf{A}\_{\mathbf{Q}\_p}^{\dagger, n} = \left\{ \sum\_{k \in \mathbf{Z}} a\_k T^k : a\_k \in \mathbf{Z}\_p, v\_p(a\_k) + \frac{k}{(p-1)p^{n-1}} \ge 0\... | https://mathoverflow.net/users/2481 | Psi operator on Phi-Gamma modules | It seems to me that the statement follows from some formulas appearing in other papers of Colmez. Lemma I.9 (page 224) of "La série principale unitaire de $GL\_2(Q\_p)$" has some very precise estimates for the coefficients of $\psi(T^k)$ with $k<0$. It says that if $k<0$, then $\psi(T^k)= \sum\_{i=k}^{\lfloor k/p \rflo... | 5 | https://mathoverflow.net/users/5743 | 152214 | 81,129 |
https://mathoverflow.net/questions/152166 | 6 | This question stems from [this question on Mathematics stack exchange](https://math.stackexchange.com/questions/592373/how-to-identify-surfaces-of-revolution). The top answer there provides a geometric solution; I'm curious though about an algebro-geometric solutions.
Let me rephrase that question a bit: Let $V$ be a... | https://mathoverflow.net/users/38448 | Varieties invariant under affine transformations | Probably, the easiest method is this (at least in characteristic zero, which I will assume henceforth): Suppose that $V\subset \mathbb{A}^n\_k$ is the set of zeros of a polynomial ideal $I\subset k[x^1,\ldots,x^n]$, say, generated by some finite set $\{f\_1,\ldots,f\_m\}\subset k[x^1,\ldots,x^n]$. Let $\frak{a}$ be the... | 4 | https://mathoverflow.net/users/13972 | 152222 | 81,130 |
https://mathoverflow.net/questions/152221 | 3 | I'm trying to calculate the limit for the sum of binomial coefficients:
$$S\_{n}=\sum\_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum\_{j=0}^i {i \choose j}^{n+1} \right).$$
Numerically it seems to converge rapidly to zero and I have asked at <https://math.stackexchange.com/questions/608296/limit-of-sum-i-1n-left-f... | https://mathoverflow.net/users/44284 | Limit of sum of binomials | By Stirling approximation for the central binomial coefficient, we have
$$\sum\_{j\le i}\binom ij^{n+1}\le(i+1)2^{i(n+1)}\left(\frac2{i\pi}\right)^{(n+1)/2},$$
hence
$$S\_n\le\frac n{2^{n-1}}+2\left(\frac2\pi\right)^{(n+1)/2}\sum\_{i=2}^n\binom ni\frac{2^i}{i^{(n-1)/2}}.$$
Put
$$c\_i=\binom ni\frac{2^i}{i^{(n-1)/2}}.$$... | 4 | https://mathoverflow.net/users/12705 | 152230 | 81,134 |
https://mathoverflow.net/questions/151606 | 7 | Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For Zariski-almost all $x\in G$, there is a unique decomposition $x=lu$ with $l$ lower triangular and $u$ upper unipotent triangular.... | https://mathoverflow.net/users/9947 | Open cell decomposition after applying a Weyl group element | In lieu of any other answers till now: the question reminded me somewhat vaguely of "cell multiplication" in a Bruhat decomposition, namely, in a Bruhat decomposition $G=\bigsqcup\_{w\in W} BwB$ of $G=GL\_n(k)$ with $B$ a minimal parabolie (Borel), and $W$ (representatives for) the (spherical) Weyl group, the "cells" $... | 2 | https://mathoverflow.net/users/15629 | 152231 | 81,135 |
https://mathoverflow.net/questions/152235 | 5 | Let $S$ be a K3 surface over the complex numbers $\mathbb{C}$. If $C\subset S$ is a smooth rational curve, the normal bundle $N\_{C/S}$ is isomorphic to $\mathbb{O}(-2)$ and thus $C$ is rigid. What about a curve $C'\subset S$ whose normalization is a rational curve? I think it may admit deformation in higher genus fami... | https://mathoverflow.net/users/44357 | Are singular rational curves on K3 surfaces rigid? | To develop what Jason says: if your curve deforms in a family of rational curves, it means that you can find a dominant rational map from a ruled surface onto your K3. This is forbidden (over $\mathbb{C}$): e.g. because the nonzero 2-form of the K3 would lift to a nonzero 2-form on the ruled surface.
| 11 | https://mathoverflow.net/users/40297 | 152245 | 81,139 |
https://mathoverflow.net/questions/152254 | 7 | A Banach space $H$ is said to have Schur's property if weak convergence of a sequence implies converge in norm. The most famous example of such a space is $\ell^1(\mathbb N)$, while $L^1[0,1]$ does not have this property.
My question is the following:
Is there a characterization of such spaces?
Is there a list of... | https://mathoverflow.net/users/43681 | Characterization of Schur's property | Rosenthal's $\ell\_1$ theorem (Google) says that every bounded sequence in a Banach space contains a subsequence that is either weakly Cauchy or is equivalent to the unit vector basis of $\ell\_1$. From this you get that a Banach space has the Schur property iff for every $\epsilon > 0$, every $\epsilon$ separated boun... | 14 | https://mathoverflow.net/users/2554 | 152256 | 81,144 |
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