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https://mathoverflow.net/questions/180573 | 6 | In a problem I am trying to solve, the following situation occurs. $X$ is a smooth variety and $G$ is a reductive group acting transitively on $X$. We have the stack $X/G$ and a morphism $\pi : X \to X / G$. Fix a basepoint $\* \in X$ and let $H$ be the stabilizer of $\*$. I don't know much about stacks but I was wonde... | https://mathoverflow.net/users/4002 | Pulling back quasi-coherent sheaves from a quotient stack | For the first question, there is an equivalence of stacks between $X/G$ and $\*/H$, and Theorem 4.46 of [Vistoli's notes](http://arxiv.org/abs/math/0412512) gives an equivalence between $H$-equivariant quasicoherent sheaves on a point (i.e., representations of $H$) and quasicoherent sheaves on $\*/H$.
Your conjecture... | 5 | https://mathoverflow.net/users/121 | 180599 | 90,639 |
https://mathoverflow.net/questions/180571 | 2 | It's well-known that on a Riemannian manifold $(M,g)$ with dimension larger than 2, the dimension of its conformal group $\text{conf}(M,g)$ is bounded above by ${n+2\choose 2}$. A Riemannian manifold whose conformal group has the maximum possible dimension is said to have a "large" conformal group. Some important examp... | https://mathoverflow.net/users/41626 | Large and Small Conformal Groups | By the so-called conformal Lichnerowicz conjecture (proved by Alekseevsky, Ferrand, Schoen) a manifold has either big conformal group or there exists a metric in the conformal class such that the conformal group acts by isometries of this metric. Thus, the answer to your question is the same as for the isometry group
... | 3 | https://mathoverflow.net/users/14515 | 180608 | 90,644 |
https://mathoverflow.net/questions/180618 | 1 | Let $E\subset{\mathbb R}^n$ be a set of the type $I\_1\times \dots \times I\_n$, where $I\_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is there a quick way for checking whether the intersection between $E$ and the space generated by the columns of $X$ is empty... | https://mathoverflow.net/users/58068 | Checking the intersection of two sets | [Emil Jeřábek posted a similar comment while I was writing this…]
Probably linear programming is the simplest way:
Let's say that $I\_i = [l\_i,u\_i]$. Now plug the following linear program into any linear programming solver:
$$
\min\_{x,y} 1\quad\text{such that}\quad l\_i\leq x\_i \leq u\_i,\quad \begin{bmatrix} X... | 3 | https://mathoverflow.net/users/9652 | 180622 | 90,646 |
https://mathoverflow.net/questions/180614 | 9 | Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people working with valuations (like Ostrowski, Krull, Dedekind, Hensel et al) would have started this idea. Just an idea.. the... | https://mathoverflow.net/users/1245 | How did height in algeb. number theory/elliptic curves started? | I think that Keith Conrad is correct and heights via maxs of valuations are in Weil's "Mordell-Weil" paper. But note that Weil's paper generalized Mordell's work in two ways. First, he extended from $\mathbb{Q}$ to number fields, and second from elliptic curves to abelian varieties (or at least Jacobians). So he would ... | 16 | https://mathoverflow.net/users/11926 | 180631 | 90,649 |
https://mathoverflow.net/questions/180624 | 9 | Equip $\mathbb S^n$ with the standard round metric. Let $f : \mathbb S^n \to \mathbb S^n$ be a continous map satisfying $\vert d(f(x),f(y)) - d(x,y)\vert \leq \epsilon$.
Is $f$ is surjective for all $0 \leq \epsilon < \epsilon\_0$ for some positive $\epsilon\_0$?
My guess would be that the answer is yes and maybe... | https://mathoverflow.net/users/29319 | When is a continous $\epsilon$-isometry of the sphere surjective? | Your guess seems to be true. If a map $f$ is not surjective then $f$ can be considered as a continuous map from $S^n$ to $R^n$. Hence there exist two opposite points on $S^n$ which maps to the same point by Borsuk-Ulam theorem.
| 15 | https://mathoverflow.net/users/2823 | 180635 | 90,651 |
https://mathoverflow.net/questions/180632 | 0 | In [this](http://arxiv.org/pdf/0809.1958v3.pdf) article by Iyama and Wemyss there is the following formula:
Let $R$ be a Cohen-Macaulay ring with canonical module $\omega$, let $X$ be a finitely generated $R$-module. Then
$$\mbox{depth}(X)=\dim{R}-\sup\{i\ge 0\ |\ \mbox{Ext}^i\_R(X,\omega)\not= 0\}.$$
I am not very fam... | https://mathoverflow.net/users/58072 | Depth formula in CM-ring involving canonical module | Put $d:=\dim R$. Grothendieck duality identifies the local cohomology group $H^i\_{\mathfrak{m}}(X)$ with the Matlis dual of
$\mathrm{Ext}^{d-i}\_R(X, \omega )$ (the Matlis dual of a $R$-module $M$ is $\mathrm{Hom}\_R(M,I)$, where $I$ is an injective hull of the residual field). This implies your equality because by d... | 1 | https://mathoverflow.net/users/40297 | 180639 | 90,652 |
https://mathoverflow.net/questions/179627 | 1 | Is there a structure theorem for such varieties?
If X is a smooth and proper/projective variety whose canonical bundle $\omega\_X$ has finite order in the Picard group, do we know anything about X?
EDIT: As was pointed out in the comments, if $\omega\_X^n = O\_X$ then one can find a cycling covering of order n, $Y ... | https://mathoverflow.net/users/46690 | varieties whose canonical bundle has finite order in Pic? | (Comment copied here at user125763's request.)
There seem to be two parts to the problem.
The first part is the Bogomolov–Beauville theorem that any such variety $X$ has a finite étale cover $Y \rightarrow X$ such that $Y$ is a product of 1) "strict" Calabi–Yaus, 2) abelian varieties, and 3) hyperkähler varieties.
... | 1 | https://mathoverflow.net/users/nan | 180642 | 90,653 |
https://mathoverflow.net/questions/180652 | 10 | I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there in the world there exists a division algebra $D$ with the following properties:
1. The characteristic of $D$ (as a norm... | https://mathoverflow.net/users/66 | Can a division algebra have degree divisible by its characteristic? | There are counterexamples for each $p$. The easiest maybe is the following: Let $F$ be the field of order $p^p$, and $\sigma$ be an automorphism of $F$ of order $p$. Let $D=F((t))$ be the set of Laurent series of the form $\sum a\_it^i$ with the usual addition. Define multiplication by $ta=a^\sigma t$. Then $D$ is a di... | 15 | https://mathoverflow.net/users/18739 | 180658 | 90,659 |
https://mathoverflow.net/questions/180620 | 8 | In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra with an action of an $A\_\infty$-operad (if I understand correctly) and commutative rings can be identified with spectra w... | https://mathoverflow.net/users/11546 | Higher coherent multiplicative structures on S-algebras | You can pick a model for the operad $E\_n$ which receives a map from the associative operad. For instance, the Boardman-Vogt tensor product of the associative operad with $E\_{n-1}$ has this property. Then, if you have an algebra $A$ over that operad, it is in particular an associative algebra and you can put a model s... | 10 | https://mathoverflow.net/users/10707 | 180661 | 90,661 |
https://mathoverflow.net/questions/180486 | 1 | Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult\_xD < 1$ then the pair $(X,D)$ is klt.
I am wondering if it possible to check that a pair is klt by compunting the discrepancies o... | https://mathoverflow.net/users/nan | A question on klt pairs | If I interpreted correctly your question I guess the answer is positive. Here is the statement:
Let $f:Y\rightarrow X$ be a proper birational morphism between normal $\mathbb{Q}$-factorial varieties, and let $D$ be an effective $\mathbb{Q}$-divisor in $X$. Let us write
$$K\_Y = f^{\*}(K\_X+D)+\sum\_{i}a\_iE\_i-\widet... | 0 | https://mathoverflow.net/users/14514 | 180664 | 90,662 |
https://mathoverflow.net/questions/180681 | 3 | It is easy to verify that
$$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2}
\quad \quad 0<t\leq1$$
I want to ask if there exist a real polynomial $h(t)$ such that$$ \frac{t}{2}\leq h(t^2)\leq \frac{t}{1-(1-t^2)^n}-\frac{t}{2} \quad \quad 0<t\leq1$$
when the positive integer $n\geq3?... | https://mathoverflow.net/users/58096 | the existence of a real polynomial satisfying the following property | Basically this is asking for an even polynomial $f=2h$ on $[-1,1]$
such that $f(t) \geq |t|$ but $f(t)-t \ll (1-t)^n$ as $t \rightarrow 1$
from below. There's a standard construction of a uniform approximation to
$|t|$ on $[-1,1]$: truncate the Taylor expansion
$$
(1-x)^{1/2} =
1 - \frac{x}{2} - \frac{x^2}{8} - \frac{... | 14 | https://mathoverflow.net/users/14830 | 180684 | 90,668 |
https://mathoverflow.net/questions/180424 | 2 | Background: I have a function $g(\omega)\in C^{\infty}(\mathbb{R})$, which vanishes like $O(|\omega|^{-\beta})$ at infinity for some $\beta>0$.
[This answer](https://mathoverflow.net/a/8102/57993) states that functions that decays "too slowly" failes to be in $\mathcal{F}L^1(\mathbb{R})$, and exercise VI.1.7 of Kazne... | https://mathoverflow.net/users/57993 | Is a polynomial decay sufficient for a smooth function to be in $\mathcal{F}(L^1)$? | This does not follow. There are compactly supported, finite, purely singular measures $\mu$ whose Fourier transform has power decay: $|\widehat{\mu}(x)|\lesssim (1+|x|)^{-\beta}$; for more background, you can search for *Fourier dimension*, for example on this site.
This function $g=\widehat{\mu}$ satisfies all your ... | 7 | https://mathoverflow.net/users/48839 | 180685 | 90,669 |
https://mathoverflow.net/questions/180651 | 3 | I asked this question on <http://math.stackexchange.com> but no unswers!
I have this paragraph from K.C. Chang [Infinite dimensional Morse theory](http://books.google.sk/books?id=wPo_AQAAIAAJ)
>
> In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in esti... | https://mathoverflow.net/users/49045 | Morse theory Vs degree theory | To see why Morse theory gives more info in the variational case consider system of equations on a torus $T^n$:
$$f\_1(\theta\_1,\dotsc, \theta\_n)=\cdots=f\_n(\theta\_1,\dotsc, \theta\_n)=0.$$
where $f\_,\dotsc, f\_n$ are smooth functions. Degree theory cannot say much about this system. However if
$$ (f\_1,\do... | 5 | https://mathoverflow.net/users/20302 | 180691 | 90,672 |
https://mathoverflow.net/questions/180706 | 0 | Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1\_n\\1\_n&0\end{array}\right]$. Then $X^t\Omega X$ is symmetric with $0$ diagonal, for each $2n\times2n$-matrix $X$.
Question: can we express each symmetric matrix with zero diag... | https://mathoverflow.net/users/20764 | Symmetric Zero-Diagonal Matrices | In fact, the symmetric matrix with zero diagonal over $F$ with $\mathop{\rm char} F=2$ is skew symmetric. It is a standard fact that every skew symmetric (bilinear) form in some basis has matrix $\Omega$ surrounded by zeroes. Each such matrix can be easily obtained from $\Omega$ by an appropriate $X$.
| 2 | https://mathoverflow.net/users/17581 | 180711 | 90,677 |
https://mathoverflow.net/questions/179901 | 7 | **Formulation of the Conjecture**
Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum\_{k\in S \,j\in S'} \sin(kx)\left( a\_{kj}\sin(jt)+b\_{kj}\cos(jt)\right),$$ where $\int\_\Omega \psi^2 = 1$ and $S,S'\subset \mathbb N$ are finite subsets of $\math... | https://mathoverflow.net/users/9144 | A conjecture about the measure estimates of a trigonometric polynomial | This solution is based on the suggestion of [Ian](https://mathoverflow.net/users/1840/ian-morris). First we need an extension of the Nazarov-Turán Lema in infinite dimentions. It can be found [here](http://www.sciencedirect.com/science/article/pii/S0021904505002340).
The formulation of the Nazarov-Turán Lema in high... | 1 | https://mathoverflow.net/users/9144 | 180717 | 90,681 |
https://mathoverflow.net/questions/180710 | 5 | A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):
For every $\mu\_1,...,\mu\_n$ discrete (i.e., purely atomic) probability measures on $\mathbb{R}^n$, there is a hyperplane $H$ defined by $\sum\_{i=1}^n a\_i x\_i =b$ such that f... | https://mathoverflow.net/users/3461 | Ham sandwich theorem for discrete measures - reference request | You could not find a reference because the statement is not true. Suppose that we are in two dimensions but both are measures are concentrated on a line. The first measure is uniform on $1,\ldots,k$ while the second measure is uniform on $-1,\ldots,-k$. In this case any open halfplane has measure $0$ or $1$ with respec... | 5 | https://mathoverflow.net/users/955 | 180728 | 90,684 |
https://mathoverflow.net/questions/180725 | 2 | This question was also asked on MSE.
Does there exist an asymptotic estimate for the following sum over primes
$$
\sum\_{p\leq x} \frac{\tau(p-1)}{p}\;,
$$
where $\tau(n)=\sum\_{d|n}1$ is the divisor function?
| https://mathoverflow.net/users/50610 | sum over primes involving divisor function (variation of the Titchmarsh divisor problem) | For this kind of things it's always a good idea to check in the two volumes of the Handbook of Number Theory of J. Sandor, D. S. Mitrinovic and B. Crstici.
Here I found the formula:
$$(\star)\quad \sum\_{p \leq x} \tau(p - 1) = \frac{315 \,\zeta(3)}{2 \pi^4} \cdot x + O\!\left(\frac{x}{(\log x)^\alpha}\right), $$
as ... | 9 | https://mathoverflow.net/users/nan | 180729 | 90,685 |
https://mathoverflow.net/questions/179930 | 3 | Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a spherical fusion category to a ribbon fusion category. Interesting means, that they should differ from those functors obtained... | https://mathoverflow.net/users/13767 | Pivotal functors of that are substantially different from finite group homomorphisms | I've been thinking about this since longer already and just realised a really easy example. I was a bit of a blockhead in thinking that for any inclusion functor $F$, we must have that $F\Omega\_\mathcal{C}$ is a subobject of $\Omega\_\mathcal{D}$, which is not true.
Consider the category of $U\_qSU(2)$-tilting modul... | 1 | https://mathoverflow.net/users/13767 | 180731 | 90,687 |
https://mathoverflow.net/questions/180727 | 9 | I consider definability to mean one of either cases:
1. Definability without parameters (in the language of set theory), or
2. Definability from ordinals and a real (in the same language).
So my question is:
Is there a model $M$ of ZFC (or at least of ZF) such that every definable family of sets (not necessarily of... | https://mathoverflow.net/users/38200 | Is it consistent with ZFC (or ZF) that every definable family of sets has at least one definable member? | The following theorem seems to express how the various
definability witness properties are connected with each other and
with $V=\text{HOD}$.
**Theorem.** The following are equivalent in any model $M$ of ZF:
1. $M$ is a model of $\text{ZFC}+\text{V}=\text{HOD}$.
2. $M$ has a definable well-ordering of the universe.... | 16 | https://mathoverflow.net/users/1946 | 180734 | 90,690 |
https://mathoverflow.net/questions/180753 | -2 | I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics law by which the quasicrystals are produced or formed.
What are the formula of representation of quasicrystals and the law... | https://mathoverflow.net/users/14024 | What are the formula of representation of quasicrystals and the law or mechanism of the formation | You might look at work of Charles Radin: [listed here](http://www.ma.utexas.edu/users/radin/tiling.html), especially the survey article #8 and the book review #11.
| 2 | https://mathoverflow.net/users/13650 | 180754 | 90,697 |
https://mathoverflow.net/questions/180742 | 3 | Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's [Propriétés galoisiennes...](http://www.college-de-france.fr/media/jean-pierre-serre/UPL5874918517843398173_Serre_proprie_te_s_galoisiennes_des_courbes_elliptiques.pdf%22Propri%C3%A9t%C3%A9s%20g... | https://mathoverflow.net/users/58124 | Theorem 7b of Serre's "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques" | This is ultimately an application of Lang's vanishing theorem for degree-1 Galois cohomology of connected algebraic groups over finite fields (applied to tori).
What follows may look complicated if you haven't worked much with tori, but to Serre in those days this sort of thing was bread and butter (and it is all "sta... | 6 | https://mathoverflow.net/users/52824 | 180772 | 90,703 |
https://mathoverflow.net/questions/180769 | 26 | I am interested in collecting a list of research papers with a mainly mathematical focus that appeared in high-reputation general science journals without a dedicated mathematics section. This would include things like Nature or Science, but exlcude, for example, PNAS.
By a "papers with a mainly mathematical focus" I... | https://mathoverflow.net/users/30264 | Mathematical research papers in general science journals | I guess you have already tried this, but just in case, you can search the publishers' websites. For example, here's all research and review papers under the "Mathematics and Computing" category in Nature Publishing Group's journals:
<http://www.nature.com/subjects/mathematics-and-computing#research-and-reviews>
You... | 9 | https://mathoverflow.net/users/27829 | 180773 | 90,704 |
https://mathoverflow.net/questions/180762 | 11 | Let $F$ be a (finitely generated) free group, $H \leq F$ of infinite index. Is it possible that $$ \bigcup\_{g \in F} gHg^{-1} = F?$$
| https://mathoverflow.net/users/38889 | Union of conjugates in free groups | The problem of characterizing groups that are union of conjugates of a proper subgroup was considered in some papers by Wiegold and others.
In particular, you can look at
[*Transitive groups with fixed point free permutations*](http://link.springer.com/article/10.1007%2FBF01224701), Archiv der Mathematik **27** (197... | 11 | https://mathoverflow.net/users/7460 | 180775 | 90,705 |
https://mathoverflow.net/questions/180779 | 5 | What's the current state of knowledge regarding packings of spheres in $n$-space that minimize the supremum of the sizes of the holes? This notion of tightness is more rigid than asymptotic density. I would expect tightest packings to coincide with densest periodic packings in low dimensions but not when $n$ is suffici... | https://mathoverflow.net/users/3621 | Minimizing deep holes in sphere packings | The problem you are asking about is sometimes known as the packing-covering problem, since it asks for a configuration with a fixed packing radius that minimizes the covering radius, irrespective of mean density. The lattice version of the problem is solved in some dimensions (see Table 3 of [arXiv:math/0412320](http:/... | 5 | https://mathoverflow.net/users/20186 | 180794 | 90,706 |
https://mathoverflow.net/questions/180767 | 1 | Let $ (A,G,\alpha) $ be a $ C^{\*} $-dynamical system, i.e., $ A $ is a $ C^{\*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by $ \* $-automorphisms. Equip $ {C\_{c}}(G,A) $, the linear space of continuous $ A $-valued functions on $ G $ with co... | https://mathoverflow.net/users/32467 | An unconventional definition of the $ C^{*} $-algebraic reduced crossed product | It seems that I have answered my own question. For the benefit of anyone who might have an interest in this sort of thing, I have decided to post my answer.
My idea is to find a unitary mapping
$$
U: {L^{2}}(G,\mathcal{H}) \to {L^{2}}(G,\mathcal{H})
$$
that intertwines $ (\tilde{\pi} \rtimes\_{\alpha} \lambda)(f) $ a... | 3 | https://mathoverflow.net/users/32467 | 180797 | 90,707 |
https://mathoverflow.net/questions/180803 | 4 | I need to answer the following question, hopefully in the negative.
>
> **Question:** Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk $\{0<|z|<r\}$, such that $f$ extends to a continuous map $\{1\leq|z|<R\}\rightarrow\{|z|<r\}$ sending the inner circle $\{|z|=... | https://mathoverflow.net/users/17294 | Non-bijective conformal maps between annuli | If a bounded holomorphic function $f$ on the unit disk has boundary value zero on a positive measure subset, then $f\equiv 0$ (this is a well known fact from the theory of Hardy spaces, and it holds more generally).
This rules out the existence of functions such as the ones you describe above (by conformal mapping of... | 9 | https://mathoverflow.net/users/48839 | 180804 | 90,710 |
https://mathoverflow.net/questions/180777 | 2 | Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements.
We call a *quasi-partition* or *q-p* of $A$ a subset $W \subset \mathcal P(A)$ such that we have:
* $|A\_i|=t$ for every $A\_i\in W$;
* $|A\_i\cap A\_j|\leq 1$ for every $A\_i, A\_j\in W$ with $i\neq j$;
* $W$ i... | https://mathoverflow.net/users/45664 | On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections | Lucia's answer given in the linked question from the comment gives an upper bound (i.e., no pair should appear twice as a subset of $A\_i$). But of course, the real question starts from here:
*When can we attain the upper bound? When we can't, what's the largest cardinality of $W$?*
The question as stated (*without... | 7 | https://mathoverflow.net/users/27829 | 180813 | 90,715 |
https://mathoverflow.net/questions/180672 | 2 | Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or countable?
| https://mathoverflow.net/users/nan | Question on the number of equilibria | Building upon what others have already said, the number of hyperbolic fixed points is indeed countable, but need not be finite.
First, it follows from the definition of a hyperbolic fixed point $x$ that they are isolated: its total derivative $Df(x)$ does not have any eigenvalues on the unit circle, which precludes a... | 5 | https://mathoverflow.net/users/3928 | 180830 | 90,722 |
https://mathoverflow.net/questions/179568 | 5 | Given an operator $A \in \mathcal L(B)$, $B$ being a Banach space, I came across the following question: assume $\mathrm{dom}(A)=\mathrm{range}(A)$, $\mathrm{dom}(A)$ dense in $B$.
Under which conditions is it possible to obtain the boundedness of $A$ from the boundedness of $A^2$. It is clear that "in most" cases th... | https://mathoverflow.net/users/10893 | Can the boundedness of $A^2$ imply the boundedness of $A$? | Your first query has been answered, but not the second. We make the simple remark that it is the case when $A$ is a self-adjoint (even normal) operator on Hilbert space by the spectral theorem. Presumably this can be extended to operators on general Banach spaces with good spectral properties (spectral operators, opera... | 4 | https://mathoverflow.net/users/58171 | 180845 | 90,729 |
https://mathoverflow.net/questions/180810 | 6 | This question arises from an issue arising in user38200's recent question concerning models of set theory in which [every definable set has a definable element](https://mathoverflow.net/a/180734/1946). In my answer to that question, with François's help, it turned out that $V=\text{HOD}$ is equivalent to the assertion ... | https://mathoverflow.net/users/1946 | Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable element? | **Update.** (June, 2017) François Dorais and I have completed a paper growing out of this answer and our others on related posts.
>
> F. G. Dorais and J. D. Hamkins, [When does every definable nonempty set have a definable element?](http://jdh.hamkins.org/when-does-every-definable-nonempty-set-have-a-definable-ele... | 8 | https://mathoverflow.net/users/1946 | 180850 | 90,731 |
https://mathoverflow.net/questions/180846 | 66 | Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
* A positive polynomial not reaching its infimum. Impossible in dimension $1$ and possible in dimension $2$ or more. See more details [here](http://www.mathcounterexamples.... | https://mathoverflow.net/users/41060 | Results true in a dimension and false for higher dimensions | An n-dimensional brownian motion visits every neighborhood of $\mathbb{R}^n$ infinitely often with probability 1 iff $n \leq 2$
| 55 | https://mathoverflow.net/users/8737 | 180853 | 90,734 |
https://mathoverflow.net/questions/180869 | 5 | The normal gradient descent is additive: $w\_{t+1}=w\_t-\lambda\_t\nabla f(w\_t)$, but is there a multiplicative gradient descent that looks something like $w\_{t+1}=w\_t[-\lambda\_t\nabla f(w\_t)]$?
I know there is a well-known exponentiated gradient descent (EG) algorithm, which gives $w\_{t+1}\propto w\_t\exp[-\la... | https://mathoverflow.net/users/58180 | Multiplicative gradient descent? | The most general form of such algorithms are named Mirror-Descent. This algorithm is an extension of gradient descent for non-Euclidean geometries.
For a formal explanation on how multiplicative weights (or exponentiated gradient descent) is a particular setup for Mirror-Descent see Appendix A.2 from <http://arxiv.or... | 8 | https://mathoverflow.net/users/39129 | 180872 | 90,744 |
https://mathoverflow.net/questions/180883 | 2 | What is the best known growth bound of $r\_k(n)$, where $$r\_k(n)=\#\{(a\_1,\dots,a\_k\in\mathbb{Z}^k:\sum\_{i=1}^ka\_i^2=n\}?$$ Please provide some reference if known. Thanks.
| https://mathoverflow.net/users/36735 | Growth of $r_k(n)$ | As Igor Rivin said, the question was answered [here](https://math.stackexchange.com/questions/72378/representing-a-number-as-a-sum-of-at-most-k-squares) for $k\geq 5$ by Greg Martin. For $n$ not divisible by $8$, the asymptotic formula described there remains valid for $k=3$ and $k=4$ as well, but the proof techniques ... | 3 | https://mathoverflow.net/users/11919 | 180887 | 90,749 |
https://mathoverflow.net/questions/180886 | 0 | For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}\_{\geq 0}^n$ let $x^\alpha=x\_1^{\alpha\_1} \cdots x\_n^{\alpha\_n}$. Let $$\ell^2=\{z=(z\_\alpha)\_{\alpha \in \mathbb{Z}\_{\geq 0}^n}:\, z\_{\alpha} \in \mathbb{R}, \,\, \|z\|^2=\sum\_{\alpha \in \mathbb{Z}\_{\geq 0}^n } z\_{\alpha}^2 < \infty\}.$$
Let $1 \geq... | https://mathoverflow.net/users/36563 | Is the span of those vectors dense in $\ell_2$? | I'm puzzled by the notation $l^2(\mathbb{R})$. (Also you surely mean $\|z\|^2$, not $\|z\|$, in the definition.)
One way to show that the span of a set is dense in a Hilbert space is by showing that the only vector orthogonal to the set is the zero vector. So let $(a\_\alpha)\_{\alpha \in \mathbb{Z}^n\_{\geq 0}}$ be ... | 5 | https://mathoverflow.net/users/23141 | 180889 | 90,751 |
https://mathoverflow.net/questions/180898 | 0 | I'm having trouble finding a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$
1. $(t,x)\mapsto \partial\_x^2 u(t,x)$ is $C^{0,\alpha}$;
2. $(t,x)\mapsto \partial\_x u(t,x)$ is not $C^{0,\alpha}$.
3. $t\mapsto u(t,x)$ is $C^{1,\alpha}$.
All the statements must be true in a neig... | https://mathoverflow.net/users/58196 | A function with one partial derivative Hölder continuos is Hölder continuos? | From conditions 1) and 3) it follows that $u(x,t)$ belongs to $C^{2+\alpha,1+\alpha}(\bar Q)$ for some cube $Q$. This space is a special case of anisotropic Besov spaces $B^{\mathbf s}\_{\mathbf p\mathbf q}$. From the embedding theorem it follows (if I evaluated the exponents correctly) that $\partial\_x u\in C^{1+\alp... | 1 | https://mathoverflow.net/users/14551 | 180903 | 90,753 |
https://mathoverflow.net/questions/180819 | 4 | Let $(\lambda\_n)\_{n\geq0}$ be a sequence of positive numbers such that $\lambda\_n\rightarrow \lambda$ as $n\rightarrow +\infty$. These $\lambda\_n$ are the parameters of a sequence of Poisson Processes $N(\lambda\_n)$. Let $(S\_i)\_{i\geq0}$ be a sequence of reasonably smooth, positive i.i.d. random variables (say t... | https://mathoverflow.net/users/56384 | Continuity of the stationary distribution of $M/G/1$ queue w.r.t. the input rate | Yes. To be precise about this you might have to specify more about what space you are working on, etc. For example, the number of customers in the queue is not Markov for a general service time distribution, so you have to be careful what you mean by "stationary distribution". You could look at the total amount of work... | 2 | https://mathoverflow.net/users/5784 | 180904 | 90,754 |
https://mathoverflow.net/questions/180912 | 9 | There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but in most publications on hyperbolic geometry, it is almost given that the tracioid (tractrix rotated about its asymptope) ... | https://mathoverflow.net/users/38835 | Besides the tracioid are there other surfaces of revolution that have a constant negative curvature? | There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called *parametrization by Chebyshev nets.* Have a look at the paper by Robert McLachlan
[*A gallery of constant-negative-curvature surfaces*](http://link.springer.com/article/10.1007%2FBF0302... | 14 | https://mathoverflow.net/users/7460 | 180915 | 90,757 |
https://mathoverflow.net/questions/180909 | 3 | Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space,
and let $X:\Omega\to\mathbb R$ be a random variable.
Then,
one can generate a random variable $Y$ from the probability space $\big([0,1],\mathscr B,\lambda\big)$ (where $\mathscr B$ denotes the Borel algebra and $\lambda$ denotes the Lebesgue measure) to $\... | https://mathoverflow.net/users/nan | Is it possible to construct any random variable on the Euclidean Probability space? | Your question is a little bit unprecise, because of the fuzziness of the word "construct". Since all Borel $\sigma$-algebras of complete separable metric spaces are equivalent, your desired $Y$ exists (and in fact, you can even take $Y$ from $[0,1]$ to $\mathbb{R}^n$, for that matter). I guess you would like to have sp... | 4 | https://mathoverflow.net/users/4961 | 180920 | 90,759 |
https://mathoverflow.net/questions/180913 | 1 | Let $X$ be a smooth projective variety, $A$ a complete discrete valuation ring, $Y=\mbox{Spec} A$ and $f:X \to Y$ a smooth, projective, surjective morphism. Denote by $y$ the closed point of $Y$. Let $\mathcal{L}$ and $\mathcal{M}$ be two line bundles on $X$ such that its restrictions to the fiber $X\_y$ over $y$ are i... | https://mathoverflow.net/users/58203 | An application of the Grauert's upper semi-continuity theorem | No. First of all note that your line bundle $\mathcal{N}$ on $Y$ is trivial, so your assertion is $\mathcal{L}\cong\mathcal{M}$.
Take a smooth projective curve $C$ of genus $\geq 1$ (say over $\mathbb{C}$), with a closed point $p$. Consider the first projection $f:C\times C\rightarrow C$, and take for $\mathcal{L}$,... | 5 | https://mathoverflow.net/users/40297 | 180925 | 90,762 |
https://mathoverflow.net/questions/180917 | 3 | Let $K$ be a simplicial complex with $n$ vertices and $n-t$ facets, where $t \geq 3$.
Is it true that the $(n-3-j)$th reduced homology (with coefficients in a field) of $K$ vanishes for $0 \leq j \leq t-2\,$?
| https://mathoverflow.net/users/58206 | Vanishing homology of simplicial complexes with few facets | By the nerve lemma, your complex is homotopy equivalent to a complex with $n-t$ vertices and therefore has trivial homology in degrees greater than $n-t-2$
| 7 | https://mathoverflow.net/users/36466 | 180943 | 90,772 |
https://mathoverflow.net/questions/180948 | 11 | I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z\_0$ of $z'=Az$ (here $z(t)\in\mathbb{R}^n$, $A\in{\mathbb R}^{n\times n}$) into formulas for nonlinear problems. In particular, to solve $y'=Ay+G[y]$ for some ... | https://mathoverflow.net/users/25311 | Generalizing "variation of parameters" | Yes, this is called the nonlinear variation of constants formula due to Alekseev: “An estimate for the perturbations of the solutions of ordinary differential equations”, in: Vestnik Moskov. Univ. Ser. I Mat. Meh. 2 (1961), pp. 28–36. I don't think that that article is available in English.
It can also be found in th... | 14 | https://mathoverflow.net/users/3928 | 180951 | 90,775 |
https://mathoverflow.net/questions/180897 | 7 | **Definition.**
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous pregeometry if the following holds:
**(a)** $(\mathcal{A}, cl)$ is a pregeometry,
**(b)** $dim(\mathcal{A})$ is... | https://mathoverflow.net/users/38966 | Groups and pregeometries | This question has been around for some time.
Connections of homogeneous pregeomtries, quasiminimal structures and regular types have been studied in a recent [article](http://www1.maths.leeds.ac.uk/~pillay/regular.7.pdf) of Pilay and Tanovic. They show that the generic type of a homogeneous pregeometry is strongly r... | 6 | https://mathoverflow.net/users/57712 | 180955 | 90,777 |
https://mathoverflow.net/questions/179964 | 3 | Let $Q=[-1,1]^2$ denote the unit square and let $f:Q\to Q$ be a Lipschitz function such that for any ball $B(a,r)\subset Q$ with radius $r$, the width of the image $f(B(a,r))$ is at least $cr$ for some absolute constant $c>0$.
(The width of a compact set is defined (as in [here](http://en.wikipedia.org/wiki/Curve_of_co... | https://mathoverflow.net/users/18698 | Lipschitz function with somewhere dense image | This answer builds on Bill Johnson's comment.
This is not a full answer since a step is missing, but too long for a comment.
A Lipschitz function is almost everywhere differentiable, so let $a\in Q$ be a point where $f$ is differentiable.
For simplicity, we can take $a$ to be an interior point.
There is a $2\times2$ ... | 1 | https://mathoverflow.net/users/55893 | 180961 | 90,779 |
https://mathoverflow.net/questions/180950 | 2 | Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ the corresponding induced representation of $G$. When does there exist a (continuous in a proper sense) involutive homomo... | https://mathoverflow.net/users/18943 | When is the induced representation factored through the initial one? | This is false in many cases where $G$ is finite. Let $\rho\circ \iota$ and $\pi$ denote the corresponding maps of group algebras $\mathbb{C}[G]$. The equation above can only hold if any element killed by $\pi$ must also be killed by $\rho\circ \iota$, that is $\mathrm{ker}(\pi)\subset \mathrm{ker}(\rho\circ \iota)$. If... | 5 | https://mathoverflow.net/users/66 | 180964 | 90,781 |
https://mathoverflow.net/questions/180946 | 3 | As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see [this question](https://mathoverflow.net/questions/159987/characterizations-of-the-goe-gue-family-of-distributions) for more background). The question is: how quickly is it k... | https://mathoverflow.net/users/11142 | GOE convergence | [On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles](http://www-brs.ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/SchubertKristinaBeatrice/diss.pdf), K.B. Schubert (2012).
>
> In this thesis we consider the empirical distribution of the spacings
> of adjac... | 6 | https://mathoverflow.net/users/11260 | 180966 | 90,782 |
https://mathoverflow.net/questions/180987 | 9 | Next day: apparently my original question is harder, by far, than the other bits. So: it is a finite check, I was able to confirm by computer that, if the polynomial below satisfies $$ f(a,b,c,d) \equiv 0 \pmod {27}, \;\; \mbox{THEN} \; \; a,b,c,d \equiv 0 \pmod 3, $$ and if
$$ f(a,b,c,d) \equiv 0 \pmod {125}, \;\; \mb... | https://mathoverflow.net/users/3324 | Go I Know Not Whither and Fetch I Know Not What | Yes, this is a field norm; it is the norm of $a + b \sqrt{3} + c \sqrt{5} + d \sqrt{15}$, from $K = \mathbb{Q}(\sqrt{3}, \sqrt{5})$ down to $\mathbb{Q}$. Note that $a+b \sqrt{3} + c \sqrt{5} + d \sqrt{15}$ acts on the basis $(1, \sqrt{3}, \sqrt{5}, \sqrt{15})$ by
$$a \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 ... | 12 | https://mathoverflow.net/users/297 | 180988 | 90,792 |
https://mathoverflow.net/questions/180914 | 5 | I would like to ask the broad community what is known about the solutions of diophantine equation $$\frac{u}{v} +\frac{v}{w} +\frac{w}{u} =t$$ where $t,v,u,w\in \mathbb{N}.$
I read a book of W. Sierpinski, $\text{ 250 Problems in Elementary Number Theory}$ and there is that this is still an open problem. Is this true... | https://mathoverflow.net/users/54245 | Diophantine equation | The values of $t$ for which a solution exists are tabulated at the [Online Encyclopedia of Integer Sequences](https://oeis.org/A072716). There are also references there and links to further information. In particular, there is a reference to the paper, Andrew Bremner and Richard K. Guy, Two more representation problems... | 9 | https://mathoverflow.net/users/3684 | 180990 | 90,794 |
https://mathoverflow.net/questions/181000 | 1 | Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic graphs with circulant graphs on $p$ vertices.
Which circulant graphs over prime number of vertices greater than or equal to 11 are determined by the spectrum?
| https://mathoverflow.net/users/31179 | Non-DS circulant graphs | No, and you don't need $p\ge 11$ either.
B Elspas, J Turner
Graphs with circulant adjacency matrices,
J. Combinatorial Theory, 9 (1970), pp. 297–307.
This paper shows that no two non-isomorphic circulant graphs on a prime number of vertices have the same spectrum. It doesn't say anything about circulant graphs cosp... | 2 | https://mathoverflow.net/users/9025 | 181001 | 90,798 |
https://mathoverflow.net/questions/180977 | 6 | In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical points and gluing/compactifying appropriately). Has something similar been proven for Discrete Morse Theory?
| https://mathoverflow.net/users/58228 | Using Discrete Morse Theory to represent hom classes | In section 11 of [this paper](http://www3.nd.edu/~lnicolae/tameflow.pdf) I show that a discrete Morse function on a simplicial complex leads to a dynamical description of Forman's theory. More precisely there is a canonical flow associated to the function such that the (open) faces of the barycentric subdivision are in... | 7 | https://mathoverflow.net/users/20302 | 181004 | 90,800 |
https://mathoverflow.net/questions/180997 | 3 | The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation
$[X]=[Y] + [X\setminus Y]$
for any $Y \subset X$ a closed immersion and with the product structure given by
$[X\times Y]= [X]\cdot[Y... | https://mathoverflow.net/users/58240 | Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes? | If I understand the setup of the question correctly, the answer is yes. Using non separated schemes makes no difference.
This is explained [here](http://arxiv.org/pdf/1002.4372v1.pdf).
In brief, whether you do the Grothendieck ring of "integral, finite type, separated" or "finite type" or even "finite type algebrai... | 3 | https://mathoverflow.net/users/46690 | 181010 | 90,803 |
https://mathoverflow.net/questions/181007 | 1 | Given the Sturm-Liouville type (time independent Schroedinger) equation
\begin{equation}
\frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R}
\end{equation}
where $V(x)$ is symmetric and exponentially decreasing as $|x| \to \infty$, choose the two linearly independent solutions $y\_\p... | https://mathoverflow.net/users/58247 | Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue | If $\mu$ is real, and $\sqrt{\mu+\lambda}$ is not pure imaginary, the zeros are real, because
they are eigenvalues of one of the problems:
$y(-\infty)=0,\; y(x^\*)=0$ or $y(+\infty)=0,\; y(x^\*)=0$, and these problems are self-adjoint.
| 1 | https://mathoverflow.net/users/25510 | 181011 | 90,804 |
https://mathoverflow.net/questions/181003 | 0 | Let $a$ be an arbitrary sequence and denote by $\mbox{gap}\_k(a) = a\_{(k)} - a\_{(k+1)}$, where $a\_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of $a$. Let $b$ be an arbitrary sequence with the same length as $a$. I would like to prove or disprove
$$\mathbb{E} \mb... | https://mathoverflow.net/users/8369 | Monotonicity of the gap of permutated sequence | It is not necessarily true. Set $a=(0,3,4)\;$ and $b=(0,0,2).$ Then possible sequences of the form $\sigma(a)+b\;$ (reordered) are $(0,3,6),$ $(0,4,5)\;$ and $(2,3,4),\;$ so
$$
\mathop{\mathbb E}\mathop{\rm gap}\nolimits\_2(\sigma(a)+b)=\frac83
<3=\mathop{\rm gap}\nolimits\_2(a).
$$
| 1 | https://mathoverflow.net/users/17581 | 181012 | 90,805 |
https://mathoverflow.net/questions/181017 | 2 | Let $X = (x\_1,\ldots,x\_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod\_{i=1}^n x\_i$
Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that *on average* considers only a subsample of $X$?
Weak conditions may be imposed on $F$, for instance we may assume it has finite mean and vari... | https://mathoverflow.net/users/8737 | Unbiased sample from a product | The solution is the Poisson estimator. Let $a\_i$ = $\log x\_i$. We would like to approximate $\exp (\sum\_{i=1}^n a\_i) = \exp (n \hat{a})$
Draw $\kappa \sim \textrm{Poisson}(\lambda)$, then draw with replacement $a\_{i\_1}, \ldots, a\_{i\_\kappa}$ and compute $y = e^{\lambda}\prod\_{j=1}^{\kappa} n a\_{i\_j}$
$$E... | 1 | https://mathoverflow.net/users/8737 | 181024 | 90,810 |
https://mathoverflow.net/questions/181026 | 3 | Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c\_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, basically meaning that $B$ is a fat matrix. Is there any result (or any suggestion on how to approach the problem) that state... | https://mathoverflow.net/users/44722 | Strictly positive solutions of a random linear system | Let $1\leq i\leq m$. The probability that $B$ has no column with a unique one in the $i$th position is $(1-2^{-m})^n$. Thus the probability that $B$ has each such column is at least $1-m(1-2^{-m})^n$. If $B$ has all such columns (let their numbers be $j\_1,\dots,j\_m$ respectively) then one may assign very small values... | 2 | https://mathoverflow.net/users/17581 | 181031 | 90,812 |
https://mathoverflow.net/questions/181042 | 10 | We say that a compact subset $E$ of the Riemann sphere $\mathbb{C}\_\infty$ is *(conformally) removable* if every homeomorphism of $\mathbb{C}\_\infty$ conformal outside $E$ is actually conformal everywhere, i.e. is a Mobius transformation.
My question is the following :
Suppose $\Gamma$ is a *non-removable* Jordan... | https://mathoverflow.net/users/1162 | On the conformal removability of Jordan curves | In 1994 this was open, as evidenced by [this paper by Chris Bishop](http://www.acadsci.fi/mathematica/Vol19/bishop.pdf) (which has a nice survey). (Some homeomorphisms of the sphere conformal off a curve).
| 2 | https://mathoverflow.net/users/11142 | 181047 | 90,818 |
https://mathoverflow.net/questions/180708 | 3 | I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not understand goes as follows. Let $M$ be a (not necessarily continuous) local martingale, $a \in \mathbb{R}$ and define ... | https://mathoverflow.net/users/58112 | An identity for the exponential of a martingale | I think your last equality is true: in your notation,
$[M]^c=[M-a[M]]^c=[f(X)]^c=[f´(X\_{-})\cdot X]^c=[(X\_{-})^{-1} \cdot X]^c=(X\_{-})^{-2} \cdot [X]^c$ using your Ito expansion of $f(X)$. Hope it helps.
| 2 | https://mathoverflow.net/users/58271 | 181051 | 90,819 |
https://mathoverflow.net/questions/181063 | 23 | This is a chaser for the [examples of using physical intuition to solve math problems](https://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems) question.
Physical intuition seems to be used relatively frequently for solving math problems as well as stating new interesting... | https://mathoverflow.net/users/38448 | Examples of intuition from fields other than Physics to solve math problems | Douglas Zare's comment mentioning linguistics brings to mind the important example of the Chomsky hierarchy. In the 50s, in the field of linguistics, Noam Chomsky introduced the notion of formal grammars and identified certain levels of complexity of formal grammars (regular, context-free, etc, these forming the above-... | 31 | https://mathoverflow.net/users/29697 | 181074 | 90,827 |
https://mathoverflow.net/questions/181059 | 7 | Let $C$ be a fusion category with simple objects $X\_1,...,X\_n $, and let $Y\_1,...,Y\_n$ be objects with each $Y\_i$ isomorphic to $X\_i$. Is there a monoidal auto-equivalence $F:C \rightarrow C $ which takes each $X\_i$ to $Y\_i$, and such that $F$ is naturally isomorphic (as a monoidal functor) to the trivial auto-... | https://mathoverflow.net/users/58281 | Does an equivalence of fusion categories depend on choice of simple objects within isomorphism classes? | Let's forget about the monoidal structure for a moment. $C$ is semisimple, and semisimplicity means that to give a functor out of $C$ is the same thing as specifying where it sends simple objects. So that means that saying $\mathcal{F}(X\_i)=Y\_i$ determines a unique functor $\mathcal{F}$. Now, what does it mean for th... | 7 | https://mathoverflow.net/users/22 | 181075 | 90,828 |
https://mathoverflow.net/questions/180884 | 22 | For positive integers $n\_1, \ldots, n\_k$, let $H(n\_1, \ldots, n\_k)$ denote $1/n\_1 + \ldots + 1/n\_k$. Let $V(N)$ be the largest possible value of $H(n\_1, \ldots, n\_k)$ that is less than 1, subject to the condition that $n\_1 + \ldots +n\_k \le N$. So $V(5) = 5/6$, realized as $1/2 + 1/3$. My question is, how doe... | https://mathoverflow.net/users/8252 | Representing a number close to 1 with a sum of reciprocals of natural numbers | $K(N) := 1 / (1 - V(N))$ grows faster than any power of $N$.
This can be seen by finding for each $k$ an identity
$$
\sum\_{i=1}^m \frac{A\_i x + B\_i}{C\_i x + D\_i}
= 1 - c x^{-(2k+1)} + O(x^{-(2k+2)})
$$
where the coefficients $A\_i,B\_i,C\_i,D\_i$ are integers
with $A\_i, C\_i > 0$ and $c$ is a positive rational... | 19 | https://mathoverflow.net/users/14830 | 181081 | 90,830 |
https://mathoverflow.net/questions/181087 | 5 | In the first-order context, "reflection" of a formula $\varphi(x)$ below $\kappa$ refers to the the following situation:
>
> There are many ordinals $\alpha<\kappa$ such that for all $a \in V\_\alpha$, $V\_\alpha \models \varphi(a)$ iff $V\_\kappa \models \varphi(a)$.
>
>
>
If $\kappa$ is inaccessible, then t... | https://mathoverflow.net/users/11145 | higher-order reflection | It depends on how we define elementarity for the $(n+1)^{th}$-order language of set theory, $\mathcal L^{n}\_\in$. For $X\subseteq V\_\kappa$, two salient definitions are:
$V\_\alpha \prec^n\_X V\_\kappa$ iff $\forall \vec{x}\in V\_\alpha(V\_\alpha\vDash \phi(X\cap V\_\alpha,\vec{x}) \leftrightarrow V\_\kappa\vDash \... | 7 | https://mathoverflow.net/users/17968 | 181108 | 90,839 |
https://mathoverflow.net/questions/181006 | 6 | The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ergodic (in fact mixing). On an hyperbolic surface with infinite volume however, I am not aware of any result about dyn... | https://mathoverflow.net/users/32210 | Geodesic flow on infinite surfaces | Dynamics of the geodesic flow for infinite volume manifolds has been studied a lot, but with respect to the most relevant measure, that is the Bowen-Margulis-Patterson-Sullivan measure. A complete reference could be Roblin, Mémoires SMF.
But I guess that you are interested only in the Lebesgue/Liouville measure ?
... | 6 | https://mathoverflow.net/users/30691 | 181110 | 90,840 |
https://mathoverflow.net/questions/181123 | 9 | In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm of a Hecke Mass newform. As they indicate themselves, their method also works for classical holomorphic newforms of arbi... | https://mathoverflow.net/users/11920 | Holomorphic Hoffstein-Lockhart | You can find a detailed treatment for all cuspidal representations of $GL(2)$ over a number field in Péter Maga's [thesis](http://www.renyi.hu/~magap/publications/dissertations/phd.pdf), see his Proposition 3.2 on Page 20. Note that this is really what you need, because the residue appearing in the proposition equals, ... | 10 | https://mathoverflow.net/users/11919 | 181126 | 90,846 |
https://mathoverflow.net/questions/181111 | 7 | Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an estimate of the form
$$ ||f(x)||\_k \leq C ||x||\_{k+r}$$
for some $C$ and $r$ (and all $k$), where $|| \cdot ||\_k$ denotes ... | https://mathoverflow.net/users/17047 | Inverse of partial differential operator as a smooth tame map | In fact, it's hard to find an example of a PDO which has a right inverse that is *not* smooth tame. It's certainly true for the standard types: elliptic, hyperbolic, and parabolic.
On the other hand, why do you need this for hyperbolic operators? Local solvability of a quasilinear hyperbolic PDE can be proved using t... | 8 | https://mathoverflow.net/users/613 | 181132 | 90,848 |
https://mathoverflow.net/questions/181101 | 1 | For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as
$$(X(B\_1),\cdots,X(B\_n))\sim Dir(\alpha H(B\_1),\cdots,\alpha H(B\_n))$$
where$\{B\_i\}\_{i=1}^n$is a partition of the measurable space.
While the Dirichlet distribution is like that
$$f(x\_1,x\_2, \cdots ... | https://mathoverflow.net/users/58306 | How to extend Dirichlet distribution to Dirichlet process | Here $\alpha$ is a number, and $H$ is a measure.
I believe that you should take a finite space, say, $\{1,...,K\}$. Then for any partition of $\{1,...,K\}$,
your first formula will hold. $H$ measures the asymmetry of your Dirichlet process, as the parameters $\alpha\_1,...,\alpha\_K$ do in the second formula.
| 0 | https://mathoverflow.net/users/979 | 181133 | 90,849 |
https://mathoverflow.net/questions/181136 | 3 | Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V\_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of $V\_y$ is trivial for all $y\in S-p$, does $V\_p$ have trivial determinant? If not, under what hypotheses would it have?
... | https://mathoverflow.net/users/4096 | deformations of vector bundles on curves | Yes if $X$ is proper. Note that $\det(V)$ is itself a line bundle on $X\times S$, so the question is: given a line bundle $L$ on $X\times S$, with $X$ proper, is the locus of $s\in S$ such that $L\_s$ is trivial a closed subset of $S$? In fact, there is a natural maximal closed *subscheme* of $S$ over which $L$ is fibe... | 3 | https://mathoverflow.net/users/3847 | 181137 | 90,850 |
https://mathoverflow.net/questions/180663 | 21 | Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary version we also drop the (Cantor's) axiom of continuity, Greenberg calls such geometries Archimedean H-planes in his [s... | https://mathoverflow.net/users/51484 | Is every elementary absolute geometry Euclidean or hyperbolic? | Your answer is correct, except that in the example you gave, one has to adjoin far more numbers than you described in order to get to that ordered Pythagorean not-Euclidean field *K*.
That example is described on p.594 of the fourth edition of my book *Euclidean and Non-Euclidean Geometries: Development and History* ... | 11 | https://mathoverflow.net/users/58328 | 181144 | 90,853 |
https://mathoverflow.net/questions/181143 | 0 | How can we prove that a coin graph is 4-colorable???Also, can we find any example of an non-3-colorable coin graph.
| https://mathoverflow.net/users/58327 | Coin graph is 4-colorable | The phrase "coin graph" sometimes requires equal coins and sometimes allows non-equal coins. I am assuming the former meaning as indicated in the comments: equal coins.
[This paper](http://infoscience.epfl.ch/record/129193/files/coincikk.ps) says that the 4-colourability can be proved using a "simple induction". I'll... | 7 | https://mathoverflow.net/users/9025 | 181149 | 90,856 |
https://mathoverflow.net/questions/181095 | 5 | Let $\Gamma\_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x\_1,y\_1, \dots,x\_g,y\_g) \in \Gamma\_g^{2g}$ will be called a ***Surface Basis*** if we have the presentation $$\Gamma\_g = \langle x\_1, y\_1, \dots, x\_g, y\_g \vert \prod\_{i = 1}^{g}[x\_i,y\_i] = 1\rangle$$ Take some $1 \leq k \leq g$ and $H ... | https://mathoverflow.net/users/38889 | Bases of surface groups | There is indeed a surface basis for $H$ containing $x\_1,\ldots,x\_k$. I'll give a topological proof, basically the same as the proof suggested in the comment of @HJRW. First, I'll give a topological re-interpretation of your problem, by formulating a topological property which is equivalent to the property "there is a... | 3 | https://mathoverflow.net/users/20787 | 181161 | 90,858 |
https://mathoverflow.net/questions/181163 | 26 | Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
* How are individual eigenvectors distributed (probably zero-mean multi-variate Normal, but what is the covariance)?
* If $u\_i$ and $u\_j$ are eigenvect... | https://mathoverflow.net/users/58340 | What is known about the distribution of eigenvectors of random matrices? | If you choose the matrix elements of $A$ independently from a Gaussian distribution you have the socalled *Ginibre ensemble* of random-matrix theory. The statistics of the eigenvalues is known, see for example [Eigenvalue statistics of the real Ginibre ensemble](http://arxiv.org/abs/0706.2020). The statistics of the ei... | 21 | https://mathoverflow.net/users/11260 | 181174 | 90,862 |
https://mathoverflow.net/questions/181178 | 11 | Assuming ZFC. We can make $(\mathbb{R},+)$ into a nontrivial (scalar multiplication is not identically zero) $\mathbb{C}$-module.
Now my questions are?
### 0.Is it consistent with $ZF$ that $\mathbb{R}$ is not a $\mathbb{C}$-module?
### 1.Does $AD$ (Axiom of determinacy) implies that $\mathbb{R}$ is not a $\mathbb{... | https://mathoverflow.net/users/38866 | Is $\mathbb{R}$ a $\mathbb{C}$-module without AC? | If all sets of reals have the Baire Property (as holds under $\sf AD$ and in Solovay's model, and in Shelah's model whose consistency strength does not require any large cardinals), then the answer is negative. Let us denote by $\sf BP$ this principle.
To see that, first we use the fact that under $\sf BP$ if $\varph... | 13 | https://mathoverflow.net/users/7206 | 181179 | 90,864 |
https://mathoverflow.net/questions/180978 | 1 | Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of degree $r$ each of whose terms involves at most $d−1$ variables?
The linear polynomials are truly linear functions: e.g... | https://mathoverflow.net/users/2586 | Decomposition of symmetric homogeneous polynomials | This is true if $r$ is odd or $r<2d$, and false otherwise. I assume, when you say a polynomial is symmetric you mean in fact it is homogeneous.
**1.** Let $r=2d$ and take $P(x)=-x\_1^2x\_2^2\dots x\_d^2$. The term $x\_1^2\dots x\_d^2$ appears in every $r$th power of a linear form with a nonnegative coefficient, so th... | 4 | https://mathoverflow.net/users/17581 | 181181 | 90,865 |
https://mathoverflow.net/questions/181175 | 4 | Let $\Gamma\_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma\_g = \langle x\_1, y\_1, \dots, x\_g, y\_g \vert \prod\_{i = 1}^{g}[x\_i,y\_i] = 1\rangle$$
Is there a nontrivial, finitely generated $N \lhd \Gamma\_g$ of infinite index?
More generally, Can there be a finitely genera... | https://mathoverflow.net/users/38889 | A Karrass-Solitar theorem for surface groups | The answer to both questions is 'no'. This was proved by Greenberg for Fuchsian groups. One outline of the proof is as follows.
1. Any finitely generated subgroup $H$ of a surface group $\Gamma$ is quasiconvex. So the map $H\to\Gamma$ induces an injection of Gromov boundaries $\partial H\to\partial\Gamma\cong S^1$.
2... | 7 | https://mathoverflow.net/users/1463 | 181186 | 90,868 |
https://mathoverflow.net/questions/181142 | 2 | Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int\_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A quick application of Sobolev embedding gives $f\leq C |x|^{2-n/p}$, which then immediately shows $$\int\_{B(0,r)} (f|x|^{-2+\ep... | https://mathoverflow.net/users/40746 | Hardy-type inequality for point boundary | If $p > n/2$ and if $f \in C^2\_c (\mathbb{R}^n \setminus \{0\})$ (twice continuously differentiable functions whose support is compact in $\mathbb{R}^n \setminus \{0\}$), then the weighted Hardy inequality to $f$ says that
$$
\int\_{\mathbb{R}^n} \frac{| f (x) |^p}{| x |^{2 p}} \,dx
\le \Bigl(\frac{p}{2 p - n}\Bigr)... | 2 | https://mathoverflow.net/users/42047 | 181201 | 90,874 |
https://mathoverflow.net/questions/180787 | 1 | It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I am curious about what kinds of additional assumptions one would need to make in order for something like the converse to hold. That is:
I'd like to know when, given... | https://mathoverflow.net/users/24525 | Nonpositive curvature of Stein manifolds | Complete, simply connected manifolds of non-positive sectional curvature are diffeomorphic to $R^n$, by Cartan-Hadamard theorem. Conversely, if it's $R^n$, you can put the metric of negative curvature on it.
| 3 | https://mathoverflow.net/users/3377 | 181204 | 90,876 |
https://mathoverflow.net/questions/181219 | 0 | Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. Call a relation $U \to V$ a (linear) Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus V$, where $\overline U$ is the conjugate symplectic vector space $(U,-\omega)$.
These linear Lagrangian relations have ... | https://mathoverflow.net/users/6345 | If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation? | Counterexample: set $U$ to be anything, $V := \text{pt}$, and $\Lambda \subset \overline{U} \oplus \text{pt}$ to be a non-Lagrangian subspace.
(But maybe true with some hypotheses, e.g. $\Lambda$ induces an injection on Lagrangian subspaces? Hmm, a linear symplectic analogue of Orlov's theorem...)
| 2 | https://mathoverflow.net/users/20391 | 181229 | 90,882 |
https://mathoverflow.net/questions/181225 | 0 | Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{minimize}\_{x\_{ij}}\sum\_{i,j}u\_{ij}x\_{ij}s.t.$$
subject to the constraints that
$$\sum\_{j}w\_{ij}x\_{ij}\leq c\_{i}\text{ ... | https://mathoverflow.net/users/58367 | Generalized assignment problem with no integrality gap | This is trivial. A solution to the problem is precisely a solution to the linear programming relaxation that happens to have all entries 0 or 1. If there was an easy (i.e. polynomial-time) way to find the "nice solution" when it exists, this would lead to a polynomial-time decision procedure: try the easy way to find t... | 0 | https://mathoverflow.net/users/13650 | 181233 | 90,884 |
https://mathoverflow.net/questions/181213 | 5 | (1) Can anybody suggest a readable reference for Schneider's theorem that the number
$$
\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$ is transcendental for $a, b \in \mathbb{Q}$ such that none of $a, b, a+b$ is an integer?
(2) Fix some integer $n \geq 3$ Is the degree of transcendence of the field generated ... | https://mathoverflow.net/users/58364 | transcendence of beta values | Scheider's original paper is available [online](http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0183&DMDID=dmdlog12). The theorem you quote is proved at the end of Section 1, on Page 114. I don't know the answer to your second question, but my guess is "no".
| 2 | https://mathoverflow.net/users/11919 | 181237 | 90,886 |
https://mathoverflow.net/questions/181241 | 7 | [Helge von Koch](http://link.springer.com/article/10.1007%2FBF02403071) proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
>
> Q1: Is von Koch's result (and proof) also valid for the general... | https://mathoverflow.net/users/45947 | Is there a von Koch-type theorem for the generalized Riemann hypothesis? | As GH from MO and Felipe Voloch have already indicated it is standard to show that $\psi(x;q,a) = x/\phi(q) +O(x^{\frac 12+\epsilon})$ for all reduced residue classes $a\pmod q$ is equivalent to GRH for the characters $\pmod q$. I want to make the following small (but amusing) refinement: it is enough to know that $\ps... | 14 | https://mathoverflow.net/users/38624 | 181248 | 90,891 |
https://mathoverflow.net/questions/181226 | 54 | If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
UPDATE1. It is a duplicate of the question here
<https://math.stackexchange.com/questions/94280/if-every-open-set-is-a-countable-union-of-balls-is-the-space-separable/94301#94301>
UPDATE2. Let me summari... | https://mathoverflow.net/users/4312 | If any open set is a countable union of balls, does it imply separability? | Towards a contradiction, let us assume that we have a metric space $X = \{x\_i : i < \omega\_1\}$ in which any two points are at least unit distance apart and every subset of $X$ is the union of a countable family of open balls. Let $r\_i$ be the supremum of all $r > 0$ such that $B(x\_i, r)$ is countable. Construct $\... | 37 | https://mathoverflow.net/users/2689 | 181249 | 90,892 |
https://mathoverflow.net/questions/181239 | 3 | Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j\_\*\mathbb G\_m$ ?
It should vanish, shouldn't it? In this case, it would be enough to show that its stalks vanish. Then, if $p ... | https://mathoverflow.net/users/58372 | Compute higher direct image for Gm under open embedding | As $R^1 j\_\*\mathbb{G}\_m$ is the sheafification of $(V\to \mathbb{P}^1 \text{ etale})\mapsto H^1(V\_U, \mathbb{G}\_m)$ (here $V\_U = V\times\_{\mathbb{P}^1} U$), it suffices to prove that given an etale $V\to \mathbb{P}^1$, an element $\zeta\in H^1(V\_U, \mathbb{G}\_m)$, and a geometric point $\bar x$ of $V$, there e... | 1 | https://mathoverflow.net/users/3847 | 181252 | 90,893 |
https://mathoverflow.net/questions/181257 | 3 | It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be proved to see this is that the sheaves of isomorphisms between two elliptic curves over a base is representable (I'm si... | https://mathoverflow.net/users/4177 | Sheaf of isogenies representable? | In general you have a Hom scheme $\mathrm{Hom}\_S(X,Y)$ for $X$ and $Y$ two schemes over $S$ whenever $S$ is noetherian, $X$ flat and projective, $Y$ quasi-projective. It decomposes into connected components depending on the Hilbert polynomial of the graph of $f$ in $X \times\_S Y$. If $X$ and $Y$ are curves, this give... | 7 | https://mathoverflow.net/users/1310 | 181262 | 90,895 |
https://mathoverflow.net/questions/181260 | 12 | Let $M$ be a Riemannian manifold and $G$ a closed connected subgroup of isometries of $M$. Call the pair $(M,G)$ an *isotropic pair* if $G$ acts transitively on the sphere bundle $SM$. As an example, the pair $(S^6,SO(7))$ is isotropic, but also $(S^6,G\_2)$.
I am looking for a reference for the classification of all... | https://mathoverflow.net/users/57961 | Isotropic Riemannian manifolds | The reason there are no 'negatively curved' analogs of $(S^6,\mathrm{G}\_2)$ or $\bigl(S^7,\mathrm{Spin}(7)\bigr)$ is that, in each of these cases of homogeneous Riemannian manifolds $G/H$, the corresponding $H$-structure ($H=\mathrm{SU}(3)$ in the first case, $H=\mathrm{G}\_2$ in the second) is not torsion-free.
In... | 5 | https://mathoverflow.net/users/13972 | 181267 | 90,897 |
https://mathoverflow.net/questions/181266 | 2 | My problem is the inconsistency between the definition and the computation of the Approximate entropy (`ApEn`).
Suppose $u = (u\_i:1\leq i \leq N)$ is a sequence of positive real numbers and $x = (x\_i:1\leq i\leq N - m + 1)$ the sequence of components of $u$ of length $m$ (i.e. $x\_i = (u\_i,\ldots,x\_{i + m -1})$)... | https://mathoverflow.net/users/13938 | Which is the right way to compute the Approximate Entropy (ApEn)? | From a computational perspective it is more advantageous to take the logarithm of the average (definition B) rather than the average of the logarithm (definition A). In particular if your time series is long, taking the logarithm just once at the end (definition B) rather than for each data point (definition A) can spe... | 1 | https://mathoverflow.net/users/11260 | 181274 | 90,898 |
https://mathoverflow.net/questions/181258 | 6 | Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators.
Is there a p-adic analogue of the story, that is a p-adic L-function $L\_p(C, s)$ and a conjecture about the special values?
I have se... | https://mathoverflow.net/users/58364 | p-adic L-function of curves | To complete Chris's answer in comments, yes, we do expect such a $p$-adic $L$-function to exist, but we are far from being able to prove it. There are two problems: first we very likely need to show that the curve $C$, or equivalently its Jacobian, or equivalently its $L$-function $L(C,s)$ is automorphic, because all $... | 5 | https://mathoverflow.net/users/9317 | 181281 | 90,901 |
https://mathoverflow.net/questions/179221 | 3 | I have a $K$ equations of the form $x\_1^{a\_{i1}} \cdots x\_n^{a\_{in}}=c\_i$ where $a\_{ij}$ are non-negative integer constants and $c\_i$ are real constants -- i.e. each equation is a monomial in $n$ variables and I have $k$ equations. I wish to find all real solutions for $x\_1 \cdots x\_n$. It is assumed that ther... | https://mathoverflow.net/users/19899 | Real solutions for systems of monomial equations | As @Oleg Eroshkin has already pointed out in the comments, this is closely related to solving a linear algebraic system. You could take absolute values and then logs to obtain a linear system of the form $Ax=b$ where the entries of $x$ are absolute values of your original variables. Once you solve that, you're left wit... | 2 | https://mathoverflow.net/users/20507 | 181288 | 90,902 |
https://mathoverflow.net/questions/181279 | 4 | Let $\mathbb{Q}\_p$ be the field of $p$-adic numbers. Consider an unramified representation $\rho : Gal(\bar{\mathbb{Q}}\_p / \mathbb{Q}\_p) \to \mathbb{F}\_p^{\times}$ which sends the arithmetic Frobenius to an element $\mu \in \mathbb{F}\_p^{\times}$.
I know that the corresponding $(\varphi, \Gamma)$-module is of r... | https://mathoverflow.net/users/58392 | Rank one (phi,Gamma)-modules | Nice question! I remember doing this exercise myself once. This can be extracted from Fontaine's article in the Grothendieck Festschrift, but it takes a little bit of work. The key observation is that since your representation is unramified, it factors through $\operatorname{Gal}(\overline{\mathbb{F}}\_p / \mathbb{F}\_... | 4 | https://mathoverflow.net/users/2481 | 181297 | 90,906 |
https://mathoverflow.net/questions/181286 | 6 | Consider the stochastic differential equation on $\mathbb R$
$$
dx\_t = f(x\_t) dt + g(\omega t)\, dW\_t
$$
with $W\_t$ a standard Brownian motion, $f:\mathbb R \to \mathbb R$ a smooth function, and $g:\mathbb R\to \mathbb R$ a 1-periodic function. Let $c^2 := \int\_0^1 g^2(s)ds$ be the square of the average of $g$... | https://mathoverflow.net/users/29661 | Ito diffusion with highly oscillatory diffusion coefficient | This is true, but the limiting process is driven by a Brownian motion $B$ which is different from $W$. To prove this, use first the Dambis-Dubins-Schwarz representation of a continuous martingale (see the book by Revuz & Yor for example) to see that $A\_\omega(\cdot) = \int\_0^\cdot g(\omega t)\,dW$ is a time-change of... | 9 | https://mathoverflow.net/users/38566 | 181322 | 90,916 |
https://mathoverflow.net/questions/181218 | 3 | a) How to construct a conformal mapping $f$ of the unit disk $D$ onto a Jordan domain with $C^1$ boundary such that $$\int\_D|f''(z)|^2 dxdy =\infty.$$ (This is done in two different ways in the sequel)
b) It follows by Kellogg theorem that if $\partial D\in C^{1,\alpha}$ with $\alpha>1/2$, then $\int\_D|f''(z)|^2 dx... | https://mathoverflow.net/users/57714 | Integrability of second derivative of conformal mappings | Let $g$ be analytic in $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$ such that $\int \int\_{\mathbb{D}} |g'|^2=\infty$. It is well-known that such functions exist (the disc algebra is not contained in the Dirichlet space).
We can assume that $\operatorname{Re} g>0$ and $ | \operatorname{Im} g |<\pi/2$ on $\math... | 1 | https://mathoverflow.net/users/1162 | 181325 | 90,918 |
https://mathoverflow.net/questions/181284 | 5 | The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV equations) (cf. OEIS-[A074060](https://oeis.org/A074060)) is the compositional inverse of the generating function for the B... | https://mathoverflow.net/users/12178 | Compositional inversion and generating functions in algebraic geometry | I think you would enjoy reading Curt McMullen's paper "Moduli spaces in genus zero and inversion of power series". In some sense there is nothing there that isn't already in Getzler's paper, but everything is stated in a down-to-earth and combinatorial fashion.
Let me summarize the story, first for the spaces $\overl... | 6 | https://mathoverflow.net/users/1310 | 181327 | 90,920 |
https://mathoverflow.net/questions/181330 | 1 | Does anyone know of a convergence test for a complex series of the form
$$\sum\_n a\_n \cdot \exp(i \cdot b\_n)$$
?
The particular series I need to understand has a\_n going to zero as n goes to
infinity, but it fails the absolute convergence test. However numerically
I do find it to converge. It should have so... | https://mathoverflow.net/users/40588 | Convergence of complex series that are not absolutely convergent? | If your $a\_n$s are nonnegative, decrease monotonically, and approach zero, and the $b\_n$s are such that $\sum e^{i x b\_n}$ remains bounded, then Dirichlet's test would apply here and give you convergence. If the $b\_n$'s are an arithmetic sequence, as in the case of Fourier series, then you have the second of these ... | 4 | https://mathoverflow.net/users/20186 | 181332 | 90,923 |
https://mathoverflow.net/questions/181323 | 4 | In Siegel's 1969 paper, *Abschätzung von Einheiten*, on page 73, he states the inequality
$$\log\sqrt d\le n-1+{n\over 2}\log\pi+r\_2\log 2\qquad (\*)$$
and compares with the bound due to Minkowski that
$$n-{1\over 12n}-\log\sqrt{2\pi n}-r\_2\log\left({4\over\pi}\right)\le \log\sqrt{d}$$
where $n=[\mathbf{Z}:\B... | https://mathoverflow.net/users/4701 | Inequality due to Siegel (assumptions) and upper bounds on number field discriminants | I don't have access to Siegel's paper at the moment, but $(\*)$ is clearly false in general. For example, for $n=2$ it would mean that there are only finitely many quadratic number fields. In fact $(\*)$ is *equivalent* to the bound $\log a\leq n-1$ that you mention under (3).
| 2 | https://mathoverflow.net/users/11919 | 181333 | 90,924 |
https://mathoverflow.net/questions/181328 | 1 | Let's define , $$R\_{p^m,k}(n)=\#\{(a\_1,\dots,a\_k)\in\mathbb{Z}^k:\sum\_{i=1}^ka\_i^2\le n \ \text{and} \ p^m|\sum\_{i=1}^ka\_i^2\}$$
what will be growth bound of $R\_{p^m,k}(n)$? This can be thought as a extended version of Gauss's Circle problem.
I am interested only in the case of $k=4$ but would be happy to kno... | https://mathoverflow.net/users/36735 | Expression and growth bound for $r_{p^m,k}(n)$ | By a simple volume argument (resembling Gauss's original argument in the circle problem) it is easy to see that
$$R\_{p^m,k}(n)\sim \frac{S\_{p^m,k}}{p^{km}}\cdot\frac{(\pi n)^{k/2}}{\Gamma(k/2+1)},$$ where $S\_{p^m,k}$ is the number of solutions of the congruence
$$\sum\_{i=1}^k x\_i^2\equiv 0\pmod{p^m}.$$
The quant... | 4 | https://mathoverflow.net/users/11919 | 181339 | 90,926 |
https://mathoverflow.net/questions/181222 | 9 | I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$\*-algebra and let $p,q\in A^{\*\*}$ be two commuting projections. Then there exist self-adjoint nets $(x\_i)\_i$ and $(y\_j)\_j$ in $A$ such that $x\_i\to p$ and $y\_j\to q$ in the weak$^\*$-t... | https://mathoverflow.net/users/58366 | Commuting nets for commuting projections | I think the following provides a counterexample, though the bidual of a $C^\*$-algebra always makes me nervous.
Let $A=M\_2\otimes C[0,1]$. Any bounded, Borel measurable, $M\_2$ valued function on $[0,1]$ will give an element of $A^{\*\*}$; for the projection $p\in A^{\*\*}$ we take the function
\begin{equation}
p(t)... | 5 | https://mathoverflow.net/users/13360 | 181341 | 90,927 |
https://mathoverflow.net/questions/181122 | 1 | In Federer's Theorem, $ \mathcal{H}^{n-1} (\partial ^{m}E \setminus \partial ^{\*}E)=0 $, where $E$ is a set of finite perimeter in $ \mathbb R^n $, $\partial ^{e}E$ is the essential boundary of E, and $\partial ^{\*}E$ is the reduced boundary of E.
From Maggi's book Prop. 12.19, We know that $\operatorname{spt}(\mu... | https://mathoverflow.net/users/51546 | Is it true that $ \mathcal{H}^{n-1} (\operatorname{spt} \mu _E \setminus \partial ^{*}E)=0$? | In Maggi's book, Example 12.25, there you have an open set of finite perimeter $E$ in $\mathbb{R}^2$ with $|\text{spt}\, \mu\_E|>0$. Since $H^{n-1}(\partial^\* E)$ is finite, you have $|\partial^\* E|=0$. Thus $|(\text{spt}\, \mu\_E)\setminus \partial^\*E|>0$, so $H^{n-1}((\text{spt}\, \mu\_E)\setminus \partial^\*E)= +... | 1 | https://mathoverflow.net/users/58420 | 181343 | 90,928 |
https://mathoverflow.net/questions/181344 | 0 | Let G be a graph which has the following properties:
1) For every $e\_1,e\_2 \notin E(G)$, $G \cup e\_1 \cong G \cup e\_2$
2) For every $e\_1,e\_2 \in E(G)$, $G\setminus e\_1 \cong G\setminus e\_2$
i.e. adding one more edge anywhere gives rise to the same graph and deleting one edge also gives rise to the same g... | https://mathoverflow.net/users/43701 | edge transitivity and edge deletion | There are no such graphs.
1. Consider the effect that edge addition and edge deletion have on the sum of the squares of the degrees. It shows that there are two constants $A,B$ such that for any two vertices $u,v$ we have $d\_u+d\_v=A$ if $uv\in E(G)$ and $d\_u+d\_v=B$ otherwise (where $d\_x$ is the degree of vertex ... | 2 | https://mathoverflow.net/users/9025 | 181346 | 90,929 |
https://mathoverflow.net/questions/176489 | 4 | I would like to numerically solve a hyperbolic PDE of the form
$\frac{\partial\theta\_t}{\partial t}(x,y)+\frac{\partial\left[\theta\_t \gamma\_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta\_t \gamma\_t^y\right]}{\partial y}(x,y)=0,$
which is very similar to the 2D advection equation, except that the part... | https://mathoverflow.net/users/56169 | Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions | The reason you don't get conservation is that you've **used the product rule before discretizing**, so conservation would require an exact cancellation of truncation errors in the different product terms (which generally won't happen).
Instead, you should **directly discretize the conservative form of the equation**... | 1 | https://mathoverflow.net/users/20507 | 181356 | 90,931 |
https://mathoverflow.net/questions/181347 | 3 | This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed non-simple geodesic. $\gamma$ is not homotopic to a point, a puncture or a boundary. Let $p$ be a self intersection point of... | https://mathoverflow.net/users/9485 | Intersection of closed geodesics in hyperbolic surface | The answer to (1) is yes.
Take $P$ a hyperbolic surface with one geodesic boundary, called $\delta$, and two punctures. Form $S$, a sphere with four punctures, by doubling $P$ across $\delta$. Note that $S$ has a reflection symmetry $f$ that fixes $\delta$ pointwise. Let $\gamma$ be a figure of eight curve, about tw... | 6 | https://mathoverflow.net/users/1650 | 181364 | 90,935 |
https://mathoverflow.net/questions/181289 | 2 | Let $N(v)$ be the (open) neighbourhood set of a vertex $v$, and let $N[v]$ be the closed neighbourhood set of $v$.
A graph $G$ is called *4-chordal* if $G$ has no induced cycle with five or more vertices.
**Question:** Does every 4-chordal graph $G$ have two vertices $x$ and $y$ such that
$N(x)$ is a subset of $N(y... | https://mathoverflow.net/users/58396 | Existence of neighborhood inclusion for 4-chordal graphs | The answer is **no.** A counterexample is the [triangular prism](http://en.wikipedia.org/wiki/Triangular_prism) graph. Up to symmetry, there is a unique 5-cycle and a unique 6-cycle in the triangular prism and both these cycles have chords. Hence the triangular prism is 4-chordal. On the other hand, it is also easy to ... | 3 | https://mathoverflow.net/users/2233 | 181372 | 90,939 |
https://mathoverflow.net/questions/181154 | 5 | Let $G$ be a real algebraic group, and let $X$ be a real affine $G$-variety. I am looking for conditions on $G$ and $X$ for which the $G$-orbits are known to be locally closed in the Zariski topology on $X$. Please feel free to offer any conditions you desire. References would also be much appreciated. Of particular in... | https://mathoverflow.net/users/25358 | Locally Closed Orbits in Real Algebraic Geometry | Let $x\_0\in X(\mathbb{R})$, and consider the orbit map $f:g\mapsto g.x\_0$, as a morphism of $\mathbb{R}$-schemes. Denote by $S\subset G$ the stabilizer of $x\_0$. Chevalley's therorem asserts that $f$ factors as
$$G\longmapsto G/S \xrightarrow{\sim} Y \hookrightarrow X$$
where the first map is the (faithfully flat) c... | 7 | https://mathoverflow.net/users/7666 | 181375 | 90,941 |
https://mathoverflow.net/questions/181385 | 1 | I'd like to ask the following easy question, since I can't find a reference.
Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in terms of $L(H^i(X),s)$. Do they differ by only a shift in $s$?
| https://mathoverflow.net/users/25854 | L-function of twist | The answer is yes. On page 12 of [1], line 7 from the top, it reads
$$ L(M (n), s) = L(M, s + n). $$
The article deals with compatible systems of cohomological realisations (one way to think and work with motives). Every variety gives rise to such a system, and every such system gives you an $L$-function.
Actually,... | 3 | https://mathoverflow.net/users/21815 | 181387 | 90,946 |
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