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https://mathoverflow.net/questions/443507 | 2 | I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant generalization of $\pi$). This led to the conclusion that $\zeta'(2)=\frac{\pi^2}{6}(\gamma + \ln (2\pi) -12\ln A)$.
I don't unde... | https://mathoverflow.net/users/109569 | Derivative of the Riemann zeta function at $z=-2$ | I just came across the article
* Marc-Antoine Coppo, *Generalized Glaisher-Kinkelin constants, Blagouchine’s integrals, and Ramanujan summation*, ([hal-03197403v20](https://hal.univ-cotedazur.fr/hal-03197403v20))
(see page 3) that talks about this topic. Indeed, there is a formula.
| 1 | https://mathoverflow.net/users/109569 | 443518 | 178,884 |
https://mathoverflow.net/questions/443538 | 2 | Let $f : \mathbb C \to \mathbb C$ be an entire function belonging to the Fock space $F\_\alpha^2$, that is,
$$
\int\_\mathbb{C} |f(z)|^2 e^{-\alpha|z|^2} \, dA(z)
$$
with $A$ the Euclidean are measure. To which space does the complex derivative of $f$ belong,
$$
f' \in F\_\beta^2
$$
$\beta > 0$. Can we choose $\beta = ... | https://mathoverflow.net/users/397017 | To which space does the derivative of a function in Fock space belong? | It is true for all $\beta>\alpha$. Use Cauchy estimate
$$|f'(z)|\leq \frac{1}{2\pi}\int\_{-\pi}^\pi |f(z+e^{it})|dt,$$
then Cauchy Schwarz,
$$|f'(z)|^2\leq \frac{1}{2\pi}\int\_{-\pi}^\pi|f(z+e^{it})|^2dt,$$
and substitute to your integral. You obtain
$$\int\_{\mathbf{C}}\int\_{-\pi}^\pi e^{-\beta|z|^2}|f(z+e^{it})|^2dt... | 4 | https://mathoverflow.net/users/25510 | 443542 | 178,892 |
https://mathoverflow.net/questions/443552 | 2 | I have to prove that for $0<\alpha<1$ and $\beta>0$,
\begin{equation}
\int\_{0}^{\infty} x^{-\alpha-1}\left(e^{-\beta x}-1\right)dx=\beta^\alpha\Gamma(-\alpha),
\end{equation}
and I have to show that the same equality is valid when $-\beta$ is replaced by any
complex number $w \neq 0$ with $Re(w)\leq 0$.
In the ... | https://mathoverflow.net/users/501039 | Integral calculus with Gamma function | You fix $\alpha$ and denote your integral to the left by $I(\beta )$. Then $I$ is convergent and analytic on the semi-plane $H=\{\beta\in{\mathbb C}\mid\Re (\beta )>0\}$. The right hand side too is analytic on $H$. Since the two analytic functions are the same on $(0,\infty )$, they coincide on the whole $H$.
| 6 | https://mathoverflow.net/users/140180 | 443555 | 178,896 |
https://mathoverflow.net/questions/443502 | 5 | Let $X$ be a condensed set, and let $G$ be a (discrete) group. Suppose we have an action $G$ on $X$, which is a group morphism $a:G \rightarrow \mathrm{Aut}(X)$, where $\mathrm{Aut}(X)$ is the group of condensed isomorphisms. In this case, what would be an appropriate definition of the orbit space of $a$?
We may proc... | https://mathoverflow.net/users/130868 | Group action on a condensed set and its orbit space | The answer is no (in general) for question 0 and no for question 2. In the edit of the question I have explained why $S \mapsto X(S)/G$ is not a condensed set in general.
In the following sense, the right generalization of the orbit space seems to be the categorical quotient (thanks to Echo's comment, I got to know t... | 1 | https://mathoverflow.net/users/130868 | 443556 | 178,897 |
https://mathoverflow.net/questions/443352 | 2 | As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself with understanding the case in which $I$ is the opposite category of a non-directed poset and $A$ is the category of abel... | https://mathoverflow.net/users/495743 | Non-cofiltered derived limits | Let $F:C\to Ab$ be the constant functor on an Abelian group $A$. Then $${\lim}^n F = H^n(BC, A),$$ where $BC$ is the classifying space or simplicial nerve of $C$. Take $C$ to be a finite poset with the homotopy type of $S^n$, e.g. the poset of faces of the usual CW structure with two cells in each dimension up to $n$. ... | 6 | https://mathoverflow.net/users/12166 | 443561 | 178,898 |
https://mathoverflow.net/questions/443551 | 9 | Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful?
If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full subcategory of $D(\operatorname{QCoh}(X))$ consisting of those complexes whose cohomology sheaves are coherent. The b... | https://mathoverflow.net/users/2191 | Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful? | No, not always.
In
*Positselski, Leonid; Schnürer, Olaf M.*, [**Unbounded derived categories of small and big modules: is the natural functor fully faithful?**](https://doi.org/10.1016/j.jpaa.2021.106722), J. Pure Appl. Algebra 225, No. 11, Article ID 106722, 23 p. (2021). [ZBL1464.18015](https://zbmath.org/?q=an:1... | 20 | https://mathoverflow.net/users/22989 | 443562 | 178,899 |
https://mathoverflow.net/questions/428010 | 9 | Recall that a graph is *triangle-free* if it does not contain a copy of $K\_3$. Also, for a graph $G$, $\alpha(G)$ shall denote its independence number. Lastly, we will write $o(1)$ to denote quantities that tend towards zero as $c \to 0$.
For $c \in [0,1/2]$, let $\mathcal{F}\_c$ denote the family of triangle-free g... | https://mathoverflow.net/users/130484 | Dense triangle-free graphs and their independent sets | The answer turns out to be a mix of options 2 and 3. As you have noticed, we always have that $\alpha(G)\geq d(G)$. We could ask for what values of $c$ could the equality hold, i.e. for any vertex $v$ in $G$, its neighbors always form a maximum independent set.
[Sidorenko](https://www.sciencedirect.com/science/articl... | 3 | https://mathoverflow.net/users/500054 | 443567 | 178,902 |
https://mathoverflow.net/questions/443564 | 4 | Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X\_P$ called [toric variety](https://en.wikipedia.org/wiki/Toric_variety). In general $X\_P$ may not be smooth, but the smoothness condition imposes ... | https://mathoverflow.net/users/16183 | Approximation of convex bodies by polytopes corresponding to smooth toric varieties | Yes. Let $\Sigma$ be the fan corresponding to $P$. Section 2.6 of Fulton's "Introduction to Toric Varieties" explains how to perform toric resolution of singularities on $\Sigma$ so as to produce a fan $\Sigma'$ whose toric variety is smooth and which refines $\Sigma$. This can be done in such a way that each step corr... | 5 | https://mathoverflow.net/users/468 | 443578 | 178,905 |
https://mathoverflow.net/questions/443568 | 3 | We are given a simple connected graph $G(V,E)$ with vertex and edge set $V$ and $E$ respectively. For any vertex $v\in V$, let $D\_T(v)$ the degree of $v$ in a uniformly generated random spanning tree $T$ of $G$.
---
**Question:** I am interested in bounding the maximum variance of the $D\_T(v)$ over all vertices... | https://mathoverflow.net/users/115803 | Random spanning trees probability problem | Here is a proof that the variance of $d\_T(v)$ does not exceed $\frac14(\deg v-1)$.
For every edge $e\in E$ take a variable $x\_e$ and consider the polynomial $$P:=\sum\_T \prod\_{e\in T} x\_e,$$
where the sum is taken over all spanning trees $T$ of $G$. It is known (see Proposition 1.1 [here](https://people.math.har... | 6 | https://mathoverflow.net/users/4312 | 443585 | 178,909 |
https://mathoverflow.net/questions/443421 | 8 | **Updated:** My first post had a mistake because I confused in my mind two different but related sets. Hopefully the description below is correct now.
Let $\Lambda$ be a finite set. Let $\Lambda^{(2)}$ be the set of unordered pairs $\{x,y\}\subset\Lambda$ with $x\neq y$. I will use ${\rm Part}(\Lambda)$ to denote the... | https://mathoverflow.net/users/7410 | Do this polyhedron and other set have names? | As I said in the comment, it seems to me that your two definitions are not equivalent. For example, the first definition yields a convex set, while the second one does not. I sort of hope and suspect that it is the second one that you want, because it is the more interesting space.
I believe that your second definiti... | 5 | https://mathoverflow.net/users/6668 | 443586 | 178,910 |
https://mathoverflow.net/questions/442993 | 4 | Err, not research but if anyone has read this part of Knapp's book recently, I'd be obliged if they could help me out. Also posted on [MSE](https://math.stackexchange.com/questions/4659572/step-in-the-bruhat-decompostion-for-reductive-lie-groups).
I'm stuck on a line in the proof of Theorem 7.40 in Knapp's 'Lie Group... | https://mathoverflow.net/users/105628 | Step in the Bruhat decomposition for reductive Lie groups | Er, the author kindly pointed out that this is a typo and that the conclusion (in the 8th line of the Proof of Existence in Theorem 7.40) should read
$$
Z \text{ is in } \mathfrak{a}\_0 \oplus \mathfrak{n}\_0.
$$
I guess one can see this by writing our element (in the language of the question above)
$$
Z:= \operatornam... | 3 | https://mathoverflow.net/users/105628 | 443601 | 178,913 |
https://mathoverflow.net/questions/443530 | 2 | I think proposition XIII.1.22 of Odifreddi is false but I wanted to check I wasn't being dumb. Here's the claim.
**Definition**: The set$^1$ $A$ is arithmetically-hyperimmune-free if every function $f$ arithmetic in $A$ is majorized by some arithmetic function (i.e. there is an arithmetic function $g$ s.t. $\forall x... | https://mathoverflow.net/users/23648 | Arithmetically-hyperimmune-free degrees are comeager | Absent any pushback and based on some further reading I'm going to say this claim is false and the correct claim is that the arithmetically-hyperimmune-free degrees have measure 0.
| 2 | https://mathoverflow.net/users/23648 | 443608 | 178,916 |
https://mathoverflow.net/questions/438085 | 4 | I'm now interested in classifying the indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ : the group ring of $\mathbb{Z}/p\mathbb{Z}$ over the ring $\mathbb{Z}/p^{2}\mathbb{Z}$.
I think that a piece of information such as an upper bound for the number of indecomposable modules whose $p$-... | https://mathoverflow.net/users/123226 | Classifying indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ | This was done by Szekeres, "Determination of a certain family of finite metabelian groups" Trans. Amer. Math. Soc. 66 (1949), 1–43.
See also Nazarova & Roiter, "Finitely generated modules over a dyad of two local Dedekind rings, and finite groups which possess an abelian normal divisor of index $p$" Math USSR Izvesti... | 7 | https://mathoverflow.net/users/460592 | 443611 | 178,917 |
https://mathoverflow.net/questions/443612 | 1 | Let $Y = (Y\_1, Y\_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y\_1, y\_2)$ and let
\begin{align}
F\_{\delta}(y) = \Pr[Y\_1 > y\_1 - \delta, Y\_2 > y\_2 - \delta].
\end{align}
Consider
\begin{align}
G\_{\delta}(y) = \frac{F\_{\delta}(y)
}{
F\_0(y)
},
\e... | https://mathoverflow.net/users/501711 | The monotonicity of the bivariate normal with non-isotropic covariance | For real $u,v$, let
\begin{equation\*}
Q(u,v):=P(Y\_1>u\sqrt2,Y\_2>v\sqrt2)=\int\_u^\infty dz\,\varphi(z)\Big(1-\Phi\Big(\frac{2v-z}{\sqrt3}\Big)\Big),
\end{equation\*}
where $\varphi$ and $\Phi$ are the standard normal pdf and cdf, respectively.
The question in the OP can be restated as follows: show that
\begin{eq... | 1 | https://mathoverflow.net/users/36721 | 443628 | 178,922 |
https://mathoverflow.net/questions/443626 | 4 | I am wondering if there is a known example of a pair of *non-isomorphic* graphs $G$ and $H$ that are both Cayley graphs for $\mathbb{Z}\_2^n$ (for some $n$) and are both distance regular and have the same intersection array. Or is it known that this is not possible?
| https://mathoverflow.net/users/18606 | Strongly/distance regular graphs over $\mathbb{Z}_2^n$ with the same parameters | The answer is **yes**.
There are many difference sets on the group $\mathbb Z\_2^6$, [which can be found in the La Jolla repository](https://ljcr.dmgordon.org/diffsets/show_ds.php?param_id=4237824).
Below is the SageMath code to generate two nonisomorphic $\mathbb Z\_2^6$-cayley graphs:
```
a1=[(0,0,0,0,0,0),(1,... | 3 | https://mathoverflow.net/users/125498 | 443648 | 178,926 |
https://mathoverflow.net/questions/443613 | 6 | I'm looking for a family of abelian varieties $A\rightarrow S$ over a base that is finite type over $\mathbb{Q}$ (or $\mathbb{Z}$) that is "comprehensive" in the following sense: for every characteristic zero field $K$ and every abelian variety $B/K$ (of specified dimension $g$ and polarization degree $d^2$), there exi... | https://mathoverflow.net/users/176407 | A "comprehensive" family of abelian varieties | Welcome new contributor. The idea of such "comprehensive" families goes back very far. These were studied by Amitsur under the name "generic splitting varieties", primarily in connection with problems in the theory of algebraic groups (with further connections to K-theory, Galois cohomology, etc.).
**Definition** (cf... | 10 | https://mathoverflow.net/users/13265 | 443660 | 178,931 |
https://mathoverflow.net/questions/443646 | 33 | The senses of vision and hearing are commonly recognized as being important for the study of mathematics, with fields like geometry and topology relying heavily on vision, while algebra and number theory can be communicated through hearing alone. There are many examples of mathematicians who have made significant contr... | https://mathoverflow.net/users/114032 | Do mathematicians rely on senses other than vision and hearing? | Anecdotally, it seems that many mathematicians use gestures to aid in understanding, not only when explaining but also when thinking privately. I believe that while this is connected to vision, it is something more general about spatial and temporal metaphor that can be associated with touch and the perception of one’s... | 44 | https://mathoverflow.net/users/11145 | 443661 | 178,932 |
https://mathoverflow.net/questions/443607 | 1 | I have recently been interested in some questions which stem from taking subshifts which converge to a limiting subshift in the Hausdorff metric.
More specifically, given an alphabet $\mathcal{A}$, I consider $\mathcal{A}^{\mathbb{Z}^d}$ as a dynamical system with a natural action from $\mathbb{Z}^d$. When I say *sub... | https://mathoverflow.net/users/143153 | Approximation of subshifts in Hausdorff distance | Let $X \subset A^{\mathbb{Z}^d}$ be a subshift. So $A$ a discrete finite set, $A^{\mathbb{Z}^d}$ carries the product topology; $X$ is topologically closed in this topology; for all $\vec v \in \mathbb{Z}^d$ we have $X = \sigma\_{\vec v}(X)$ where $\sigma\_{\vec v}(x)\_{\vec u} = x\_{\vec v + \vec u}$.
Given finite $N... | 4 | https://mathoverflow.net/users/123634 | 443666 | 178,933 |
https://mathoverflow.net/questions/443667 | 18 | On [Wikipedia](https://en.wikipedia.org/wiki/Minimal_volume), it is said that the minimal volume
$$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K\_{g}|\leq 1\}$$
is a topological invariant, introduced by [Gromov](https://en.wikipedia.org/wiki/Mikhael_Gromov_(m... | https://mathoverflow.net/users/153400 | Is the minimal volume a topological invariant? | Minimal volume is not a homeomorphism invariant. It is shown in [L. Bessières, [Un théorème de rigidité différentielle](https://dx.doi.org/10.5169/seals-55112), Comm. Math. Helv. **73** 443-479 (1998)] that the minimal volume of the connected sum of an exotic $7$-sphere and a closed hyperbolic manifold $M$ can be large... | 32 | https://mathoverflow.net/users/1573 | 443669 | 178,934 |
https://mathoverflow.net/questions/443672 | 2 | I am migrating this question from math stackexchange...
I have a semidirect product and I have shown that it is perfect. However, I would like to know whether every element is a simple commutator (rather than a product of commutators). This is closely related to Ore's Conjecture (which is actually a theorem now, cf. ... | https://mathoverflow.net/users/501771 | When are elements of a (perfect) semidirect product simple commutators? | Your general question seems too general. Here is a partial answer to your specific question.
Let $V = \mathbb{F}^{2m}$ and $G = \mathrm{Sp}\_{2m}(\mathbb{F})$. Consider some $vg \in VG$. We know that $g = [x,y]$ for some $x, y \in G$. *Assume $y$ can be chosen so that $y-1$ is invertible.* Now observe that
$$[wx,y] =... | 5 | https://mathoverflow.net/users/20598 | 443681 | 178,938 |
https://mathoverflow.net/questions/443682 | 8 | Suppose I have a commutative ring $R$. Given an element $(x\_1,x\_2)\in R^2$ there exists a homomorphism $\mathbb{Z} \to R\otimes R$ taking $1$ to $x\_1\otimes x\_2$, so there exists a map $f:S^0 \to HR \wedge HR$ giving this on $\pi\_0$. There is also a map $g:S^0 \to H\mathbb{Z}$ which is an isomorphism on $\pi\_0$. ... | https://mathoverflow.net/users/1378 | When can I extend a map of spectra? | You can consider $HR$ as a module over $H=H\mathbb{Z}$, so for any $u\colon S^0\to HR\wedge HR$ you can consider the composite
$$ u' = (H \xrightarrow{1\wedge u} H\wedge HR \wedge HR \xrightarrow{\mu\wedge 1} HR\wedge HR) $$
Then the composite $S^0\xrightarrow{\eta}H\xrightarrow{u'}HR\wedge HR$ is the same as $u$, so $... | 10 | https://mathoverflow.net/users/10366 | 443684 | 178,939 |
https://mathoverflow.net/questions/443201 | 13 | I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H\_2(\tilde{M},\mathbb{Z})=0$ and $H\_2(\tilde{M},\mathbb{Z}/v\mathbb{Z})=0$ for all integers $v\ge 2$.
Of course, when $H\_2$ is replaced with $H\_1... | https://mathoverflow.net/users/482183 | Compact closed aspherical manifolds with vanishing second homology in all the covering spaces | I think that the answer to this question is unknown in general. If one had a closed aspherical manifold with this property, then it could not contain a [Baumslag-Solitar subgroup](https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group?wprov=sfti1) since such a group has a subgroup with non-trivial $H\_2$ (possibl... | 4 | https://mathoverflow.net/users/1345 | 443690 | 178,941 |
https://mathoverflow.net/questions/443677 | 3 | I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, it even more fascinating to me because in my initial inequality the longer sum had a range $1\leq k\leq m+n$, however the experimen... | https://mathoverflow.net/users/66131 | Short sequence beats long sequence | $\newcommand\bi\binom\newcommand{\tr}{\tilde r}$In view of [Rob Pratt's comment](https://mathoverflow.net/questions/443677/short-sequence-beats-long-sequence#comment1145361_443677), it is enough to show that
\begin{equation\*}
L\_m:=A\_m-B\_m\overset{\text{(?)}}\ge R\_m \tag{10}\label{10}
\end{equation\*}
for integers... | 6 | https://mathoverflow.net/users/36721 | 443692 | 178,943 |
https://mathoverflow.net/questions/443697 | 1 | Let $M$ be a compact connected manifold, $X\subset M$ a closed subset, and $f:M \times [0;1] \to M$ an isotopy such that each $f\_t:M \to M$ is fixed on some open neighborhood $N\_t$ of $X$, but there are **no assumptions** on the "size" of $N\_t$ and "continuity" or "uniformity" of those neighbourhoods in $t$.
>
>... | https://mathoverflow.net/users/113688 | When a support of an isotopy is disjoint from a subset | Let $M=\mathbb{R}$ and let $X=\{0\}$ (edit: I see that you want the manifold to be compact, you can take $[-10,10]$ or $\frac{\mathbb{R}}{20\mathbb{\mathbb{Z}}}$ instead of $\mathbb{R}$ in the example below, in the end only what happens in a small nhood of $0$ matters).
Consider a smooth function $\varphi:\mathbb{R}^... | 1 | https://mathoverflow.net/users/172802 | 443701 | 178,945 |
https://mathoverflow.net/questions/443711 | 6 | I'm looking for Bruce W. Jordan's thesis: On the diophantine arithmetic of Shimura curves. Thesis, Harvard University, 1981.
I could not find the pdf at the following site.
<https://www.math.harvard.edu/graduate/dissertations/>
I could not find it in my university library either.
Would it be possible to read it online?... | https://mathoverflow.net/users/128235 | B. W. Jordan's thesis on arithmetic of Shimura curves | This is Bruce Jordan. Essentially all the relevant parts of the thesis are in print: there is a Crelle paper on global points on Shimura curves and a Math. Ann. paper with Livne on local points. The thesis works out the canonical principal polarization on a QM abelian surface -- but this was stated (without proof) by D... | 13 | https://mathoverflow.net/users/501807 | 443723 | 178,951 |
https://mathoverflow.net/questions/443713 | 3 | Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many equivalent ways, for example, decomposing $M-$ covariant derivative into tangential and normal components. This normal c... | https://mathoverflow.net/users/298774 | Definitions fundamental forms and their geometric Intuition | It is very unhelpful that you give the wrong title of the paper you are asking for. In reality, the paper appears to be
[Cauchy's problem for Bach's equations of general relativity.
R. Schimming](https://eudml.org/doc/265338)
Banach Center Publications (1984)
Volume: 12, Issue: 1, page 225-231
A MathSciNet sear... | 2 | https://mathoverflow.net/users/1306 | 443732 | 178,954 |
https://mathoverflow.net/questions/443624 | 17 | $\DeclareMathOperator\SL{SL}$ $\DeclareMathOperator\GL{GL}$The question is the one in the title: for a prime $p$, does the obvious surjection $\pi\colon \SL(n,\mathbb{Z}/p^2) \rightarrow \SL(n,\mathbb{Z}/p)$ split?
Actually, I know that the answer is "no" for $p \geq 5$. The point is that in that case, there is no wa... | https://mathoverflow.net/users/501730 | Does the map $\mathrm{SL}(n,\mathbb{Z}/p^2) \rightarrow \mathrm{SL}(n,\mathbb{Z}/p)$ split? | The map splits if and only if $(n,p)$ is either $(2,3)$ or $(3,2)$. As far as I know, this is a theorem of C.-H. Sah. See Theorem 7 in
*Sah, Chih-Han*, [**Cohomology of split group extensions**](https://doi.org/10.1016/0021-8693(74)90099-4), J. Algebra 29, 255-302 (1974). [ZBL0277.20071](https://zbmath.org/?q=an:0277... | 10 | https://mathoverflow.net/users/2381 | 443735 | 178,956 |
https://mathoverflow.net/questions/443319 | 2 | Is there a group $G$ which is at the same time a (compact-Hausdorff)-ly generated weakly Hausdorff space (or short CGWH space) such that inverse and product are continuous maps and the space is not Hausdorff?
Note that the group multiplication is a map from $G \times G$ to $G$. In the sense of this question, $G \time... | https://mathoverflow.net/users/501466 | Non-Hausdorff CGWH-group | To give this question the correct color, I give this proper answer. The mathematical part of this answer is purely based on [Tyrone](https://mathoverflow.net/users/54788/tyrone)'s comments.
Yes, there is a CGWH group which is not Hausdorff. This is shown in [Lamartin's article](https://www.google.com.hk/url?sa=t&sour... | 2 | https://mathoverflow.net/users/501466 | 443737 | 178,957 |
https://mathoverflow.net/questions/443746 | 1 | Let $f:\mathbb{R}\_+\rightarrow\mathbb{R}\_+$ be twice differentiable quasi-concave function satisfying $f(x)>0,\forall x \in \mathbb{R}\_+$. Let $g:\mathbb{R}\_+\rightarrow\mathbb{R}\_+$ be a positive, increasing and convex function (i.e. $g(x)>0,g'(x)>0,g''(x)>0$) satisfying
$$ g(x)\leq f(x), \forall x\in [a,b] \text... | https://mathoverflow.net/users/172374 | Establishing quasiconcavity | The answer is negative. Take $[a,b]=[1,3]$. Let
$f(x)=x+\epsilon(x-2)^2,$ where $\epsilon$ is small enough,
so that the function is increasing, thus quasiconcave.
Then take $g(x)=x+2\epsilon(x-2)^2-c$, where $c$ is such that
$g\leq f$ on $[a,b]$. Then the difference $f(x)-g(x)=-\epsilon(x-2)^2+c$ has a strict maximum a... | 2 | https://mathoverflow.net/users/25510 | 443750 | 178,960 |
https://mathoverflow.net/questions/442393 | -3 | **Hybrid Comprehension:** if $\phi,\varphi$ are formulas in which $x$ doesn't occur, and $\varphi$ is stratified; then: $$ \forall A \exists x \forall y \, (y \in x \leftrightarrow \varphi \land [wf(A) \land A \neq \emptyset \to y \in A \land\phi] )$$
Where $wf$ stands for being well founded, defined as:
$wf(A) \if... | https://mathoverflow.net/users/95347 | Can we have a hybrid comprehension between Z and NF? | NF+swf-Separation is inconsistent.
Let P1(X) be the set of one element subsets of X. Let z={Ø,{Ø},{{Ø}}}
Let S={{P1(A),z}| A is infinite}.
Let T be the set of subsets of S.
(1) Suppose A is infinite. Then {P1(A),z}∩P1(A)=Ø and therefore swf({P1(A),z}).
(2) Suppose A is infinite. Then {P1(A),z}∩S=Ø. and therefor... | 2 | https://mathoverflow.net/users/133981 | 443754 | 178,962 |
https://mathoverflow.net/questions/443674 | 2 | **Update**: I edited the question as I saw it was closed. Let's see if with some improvements it can be considered worth reopening... (I already accepted an answer, but I'd like to see something more specific, if possible).
---
**Premise**: I think that a reasonable definition of what a "correct proof" is goes li... | https://mathoverflow.net/users/167834 | What's a wrong proof? | You might be interested in the paper *[What do mathematicians mean by proof? A comparative judgement study of students’ and mathematicians’ views](https://doi.org/10.1016/j.jmathb.2020.100824)* by Davies, Jones and Alcock. (More generally, I consider the work of Lara Alcock to be interesting and insightful about a rang... | 3 | https://mathoverflow.net/users/10366 | 443760 | 178,965 |
https://mathoverflow.net/questions/443719 | 4 | I would like to be able to look at the ring $R=\mathbb{Z}[x\_1,x\_2,\ldots,x\_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. Is there a way to do so? Gorenstein rings seem complicated in the general setting, but may be there are some results f... | https://mathoverflow.net/users/157080 | Determining when quotient of a polynomial ring is a Gorenstein ring | Yes, this is possible. I think there are general criterion for checking whether such a toric ring is Gorenstein. Maybe check out the book of Cox-Little-Schenck. That's over a field, but I think many things will work over $\mathbb{Z}$ with minimal change (note those criteria are still going to be combinatorial in nature... | 6 | https://mathoverflow.net/users/3521 | 443763 | 178,967 |
https://mathoverflow.net/questions/443759 | 2 | I am reading the research article *"[The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets](https://doi.org/10.2307/121009)"* by Shishikura. In his article, he defined hyperbolic sets and hyperbolic dimensions for any rational map of $\overline{\mathbb{C}}$ onto itself, where $\overline{\mathbb{C}}$ ... | https://mathoverflow.net/users/499397 | How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself? | The hyperbolic dimension of $f$ is 1 and its maximal hyperbolic set is the unit circle $\mathbb{S}^1$.
First we show that a hyperbolic set for $f$ must be contained in the unit circle $\mathbb{S}^1$.
If $E$ is a closed, $f$-invariant set that is not contained in $\mathbb{S}^1$ then it must contain $0$ or $\infty$. Th... | 6 | https://mathoverflow.net/users/91134 | 443765 | 178,968 |
https://mathoverflow.net/questions/443775 | 5 | [ I asked the same question on stackexchange but attracted little attention. Besides, I made some progress after I posted it. So I decided to move it here. ]
---
Consider path-connected CW-complexes $A$, $B$, $C$, $X$.
I would like to show the equivalence of the following two conditions.
(i) They fit into fibr... | https://mathoverflow.net/users/472749 | Associativity of consecutive fibrations | You are right: they are not equivalent. For an example, choose a group $G$ that has a normal subgroup $H$ in which there is a subgroup $K$ that is normal in $H$ but not in $G$. Take $A$, $B$, and $C$ to be $BK$, $B(H/K)$, and $B(G/H)$. (So $Y\_2$ can be $BH$, but what is $Y\_1$ going to be? There is no $G/K$.)
| 8 | https://mathoverflow.net/users/6666 | 443776 | 178,974 |
https://mathoverflow.net/questions/443798 | 6 | I proved a variant of the Sauer-Shelah lemma and I was wondering if something like that is already known.
Let $S \subseteq \{0,1\}^n $. We say that a set of coordinates $K \subseteq [n]$ is *shattered* by $S$ if $S|\_K = \{0,1\}^K$. The [Sauer-Shelah lemma](https://en.wikipedia.org/wiki/Sauer%E2%80%93Shelah_lemma) sa... | https://mathoverflow.net/users/24226 | A Sauer-Shelah-like lermma for prefix tree | (Joint work with Ilkka Törmä.)
From winning set/order-shattering considerations, we get a tight upper bound for the size of $S$ not having a subtree of a similar kind as you describe. It's not exactly equal, but it seems we get stronger results in any case.
Say a set $S$ has property $P$ if it satisfies what you wr... | 6 | https://mathoverflow.net/users/123634 | 443812 | 178,989 |
https://mathoverflow.net/questions/443619 | 8 | Consider an $n\times n$ real matrix $A=(a\_{ij})$ with non negative entries. Assume that
- the sum of the three largest entries in each row is a constant $R$ (the same for all rows),
- the sum of the three largest entries in each column is a constant $C$ (the same for all columns).
Then $R$ and $C$ cannot be to... | https://mathoverflow.net/users/2480 | Big triples in a matrix | $3/4$ and $4/3$ are correct bounds. You can use the same linear algebra trick as for pairs but you should choose equations more carefully an there is some small casework.
Let $R=1$ (just not to write it as a factor every time).
In each row in which the maximal entry is $\ge 1/2$ mark its position red and the position... | 5 | https://mathoverflow.net/users/1131 | 443814 | 178,990 |
https://mathoverflow.net/questions/443807 | 6 | Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces.
Does there exist a family $m=(m\_x)\_{x\in X}$ of Radon probability measures on $Z$, such that
* the support of $m\_x$ is contained in the fibre $p^{-1}(\{ x\})$,
* for every $f\in C(Z)$ we have $f^m\in C(X)$, where $f^m(x)=\int\_Zf(z)\ d... | https://mathoverflow.net/users/473423 | Integration along fibres of continuous map on compact Hausdorff spaces | I think that such family might not exist even for rather well-behaved spaces. Consider $Z = [0,3]$ and $X = [0,2]$. Then, let $p \colon Z \to X$ be defined by:
$$
p(z) =
\begin{cases}
z, &\text{ if } z \in [0,1], \\
1, &\text{ if } z \in [1,2], \\
z-1, &\text{ if } z \in [2,3].
\end{cases}
$$
For $x \in X \setminus ... | 8 | https://mathoverflow.net/users/170491 | 443819 | 178,992 |
https://mathoverflow.net/questions/443454 | 11 | I'm currently going through a number of expository accounts of Huber's adic spaces in order to start understanding perfectoid spaces and I'd like to understand the motivation behind the definition of adic spaces.
Take $A$ a topological ring containing an open subring $A\_0$ with subspace topology the $I$-adic topolog... | https://mathoverflow.net/users/483597 | Why is the definition of the adic spectrum $\operatorname{Spa}\,(A,A^+)$ the "right" definition? | You aren't the first to ask these questions. You won't be the last.
First, to set the stage let me point out that definitions are neither right nor wrong. They are only right *for a purpose* or wrong *for a purpose*. Huber's papers "A generalization..." proposes to define adic spaces $\operatorname{Spa}(A,A^+)$ as a ... | 13 | https://mathoverflow.net/users/35330 | 443825 | 178,995 |
https://mathoverflow.net/questions/443781 | 6 | Given a (tame) knot $K \subset S^3$, let $t \in G = \pi\_1(S^3 - K)$ be any meridian. The Wirtinger presentation shows that $\langle \langle t \rangle \rangle = G$, where the notation indicates the normal closure of $t$ in $G$.
Let $I$ be an infinite subset of the natural numbers. A linking number argument shows that... | https://mathoverflow.net/users/202668 | Powers of meridians in knot groups | For hyperbolic knots, this holds for any infinite set $I\subset \mathbb{N}$ via hyperbolic Dehn filling. One may use Thurston’s original theorem, or the geometric group theory version of [Groves-Manning](https://mathscinet.ams.org/mathscinet-getitem?mr=2448064) and [Osin](https://mathscinet.ams.org/mathscinet-getitem?m... | 6 | https://mathoverflow.net/users/1345 | 443827 | 178,996 |
https://mathoverflow.net/questions/443769 | 11 | Suppose $X$ is a finite $n$-connected CW complex, then the function spectrum $F(\Sigma^\infty\_+ X,S^0)$ is an $E\_\infty$-ring spectrum, induced by the diagonal of $X$. I believe it is known that if we consider $F(\Sigma^\infty\_+ X,S^0)$ as an $E\_n$-ring spectrum, its Koszul dual is the $E\_n$-ring spectrum $\Sigma^... | https://mathoverflow.net/users/134512 | Koszul duals of n-fold loop spaces | I am not sure if this gives what you want, but maybe it is:
I went in the other direction in a paper *The McCord model for the tensor product of a space and a commutative ring spectrum,* in Progress in Math. **215** (2003).
Let $D(Y) = F(Y,S)$ denote the $S$-dual of a spectrum $Y$. Given a space $X$, $D(\Sigma^{\inft... | 8 | https://mathoverflow.net/users/102519 | 443836 | 178,999 |
https://mathoverflow.net/questions/443817 | 5 | I am hoping this question is alright for Math Overflow. I didn't get a definitive solution in [Math Stack Exchange](https://math.stackexchange.com/questions/4667105/if-a-homogeneous-polynomial-is-the-square-of-a-polynomial-modulo-x2-y2-1).
Let $f(x, y) \in \mathbb{R}[x, y]$ be a **homogeneous** polynomial with real c... | https://mathoverflow.net/users/136356 | Modulo $x^2 + y^2 - 1$, is every homogeneous polynomial that is a square of a polynomial, necessarily of sum of squares of homogeneous polynomials? | Yes. Will Sawin's answer uses a topological fact about the unit circle in $\mathbb{R}^2$. Here is an answer replacing $\mathbb{R}$ with any field of characteristic not $2$.
First, working modulo $x^2+y^2-1$, we can multiply terms of $g$ with powers of $x^2+y^2$, so we can assume $g = g\_{2k} + g\_{2k+1}$ for some $k$... | 3 | https://mathoverflow.net/users/88133 | 443840 | 179,001 |
https://mathoverflow.net/questions/443853 | 0 | Let $C\_n$ be the hypercube $[-1,1]^n$. For $a\_1,\cdots,a\_s \in C\_n$, define its dispersion $D(a\_1,\cdots,a\_s)$ as $\max\_{x \in C\_n}\min\_{i \in [s]} \|x-a\_i\|\_{2}$. Let $0< \lambda < 1$ be a constant. How small can $s(n)$ be so that
$$
\lim\_{n\rightarrow \infty}\{ \Pr\_{a\_1,\cdots,a\_s \sim C\_n}\left[ D(a\... | https://mathoverflow.net/users/316923 | How many samples do you need to get constant dispersion? | The packing number of the cube with L2 balls of radius $\lambda$ is at least $ \lambda^{-n} C^n (n/2)! \approx \lambda^{-n} C^n n^{n/2} $. By the argument you had before you get the upper bound in the question is tight.
| 1 | https://mathoverflow.net/users/116352 | 443858 | 179,007 |
https://mathoverflow.net/questions/443883 | 4 | I am interested in the following question. Let $q$ be a prime power and let $\mathbb{F}\_q$ be the finite field of cardinality $q$. Suppose $q>61$. Is it true that, for every $b\in \mathbb{F}\_q$ and for every $c\in \mathbb{F}\_q$ with $c\ne 0$, there exists $x,y,z\in\mathbb{F}\_q$ such that
\begin{align\*}
x^2+y^2+z^2... | https://mathoverflow.net/users/45242 | On a certain equation in finite fields | Generically the intersection of the surfaces described by these two equations is an algebraic curve of genus $4$. Once one has made sure that this curve is absolutely irreducible, by Weil there are $\mathbb F\_q$-points provided that $q$ is big enough. Weil assumes a smooth curve, so it can be a pain to handle singular... | 9 | https://mathoverflow.net/users/18739 | 443888 | 179,016 |
https://mathoverflow.net/questions/443845 | 4 | Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm wondering whether it's possible to do the same with the sphere spectrum $\mathbb{S}$, using the constructions given in [Lu... | https://mathoverflow.net/users/130058 | The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers) | Z.M.'s first comment is correct. If $S \subset \Bbb Z \cong \pi\_0(\Bbb S)$ is a multiplicatively closed subset, there is a localization $S^{-1} \Bbb S$ whose homotopy groups lift the algebraic localization:
$$
\pi\_\*(S^{-1} \Bbb S) \cong S^{-1} \pi\_\*(\Bbb S)
$$
This localization (a "smashing localization" of the sp... | 7 | https://mathoverflow.net/users/360 | 443897 | 179,019 |
https://mathoverflow.net/questions/443893 | 7 | Let Z be Zermelo's system without choice (C) and without axiom of foundation (AF), namely extensionnality, pair, union, power set, separation, empty set, infinity. My question is the following one :
Is Z equiconsistent with Z+C ?
[I would also be interested it knowing whether Z+AF is equiconsistent with Z+AF+C,
but... | https://mathoverflow.net/users/73529 | Are Z and ZC equiconsistent? | The **addendum** to my answer to [this related MO question](https://mathoverflow.net/questions/185338/is-there-an-l-like-inner-model-for-sf-z/185362) has a pointer to the work of Mathias (as pointed out by Avshalom's comment to my answer there) on a positive answer to the question, i.e., on establishing Con(Z + AC) ass... | 6 | https://mathoverflow.net/users/9269 | 443900 | 179,020 |
https://mathoverflow.net/questions/443886 | 2 | I am reading a [2007 article](https://www.sciencedirect.com/science/article/pii/S0001870807000497) of Bressler et al. on deformation quantization of gerbes. In the article, the authors state that a gerbe on a manifold is defined using certain two-cocycles $c\_{ijk}$ but how is this cocycle data used to write down a co... | https://mathoverflow.net/users/119114 | Connections on bundle gerbes from cocycle data | A gerbe on a manifold $M$ is a morphism of simplicial presheaves
$$\def\tB{{\sf B}}\def\U{{\rm U}}\def\cC{{\sf\check C}}\cC(U)→\tB^2\U(1),$$
where $\{U\_i\}\_{i∈I}$ is an open cover of $M$, $\cC(U)$ is the Čech nerve of $U$, $\U(1)$ is the representable presheaf of the Lie group $\U(1)$, and $\tB$ denotes the delooping... | 4 | https://mathoverflow.net/users/402 | 443907 | 179,021 |
https://mathoverflow.net/questions/443905 | 6 | This question might be more suitable for a law forum, but I thought somebody may be able to share their experience.
>
> Can the editors of a research journal unilaterally decide to move
> their journal to a different publisher, and keep the journal’s name?
>
>
>
I’m aware of cases where the editors resigned an... | https://mathoverflow.net/users/69681 | Can editors move a journal to a different publisher? | **Q:** Can editors move a journal to a different publisher?
**A:** Only if they represent the owner of the journal (a foundation or university). If the publisher owns the journal it is unlikely they will agree to a move, which will not prevent a fresh start under a (slightly) different name.
A few case studies are... | 9 | https://mathoverflow.net/users/11260 | 443911 | 179,023 |
https://mathoverflow.net/questions/443894 | 1 | $\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\Gamma}
$Let $F=\Q(\sqrt{7}\,)$, and consider the $\Gal(\Fbar/F)$-module $\mu\_8$.
The Galois group $\Gal(\Fbar/F)$ acts ... | https://mathoverflow.net/users/4149 | Decomposition groups for the Galois module $\mu_8$ | If I understand this, $E = \mathbf Q(\sqrt{7},\zeta\_8) = F(\zeta\_8)$. Since $\mathbf Q(\zeta\_8)/\mathbf Q$ ramifies only at $2$ (I am ignoring infinite places), $E/F$ can ramify only at a place over $2$, and there is just one such place in $F$ since $2$ is totally ramified in $F$: $(2) = \mathfrak p^2$ where $\mathf... | 4 | https://mathoverflow.net/users/3272 | 443915 | 179,026 |
https://mathoverflow.net/questions/438589 | 4 | Update:
1. Q1 is answered in the comments.
2. I think that the usual arguments show that every relatively almost compact set in a space is closed in the space.
**Original question:**
A set $K$ in a space $X$ is *almost compact* if every open cover of $K$ has a finite subset $\cal F$ with
$K\subseteq \bigcup\_{U\i... | https://mathoverflow.net/users/2415 | Almost compact sets | I don't specifically recall "almost compact" in the literature, but it's quite natural as it relates to "almost Menger" and "almost Lindelof".
In general, relative compactness is **not** equivalent to a set's closure being compact (sometimes called precompact). For example, in the [particular point topology on a coun... | 1 | https://mathoverflow.net/users/73785 | 443920 | 179,030 |
https://mathoverflow.net/questions/443606 | 5 | Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that $g \in G(\overline k)$ is such that the image of $g$ in $(G/H)(\overline k)$ is contained in $(G/H)(k)$. Is $g N\_{\over... | https://mathoverflow.net/users/2383 | Fields of definition of conjugates | The answer is yes, and affineness is unnecessary. We'll assume that $G$ is locally of finite type (as I imagine was intended), but this is not essential. In brief, the proof proceeds by using fppf descent in place of Galois descent; this will involve working over a base slightly more general than a field (i.e., a finit... | 3 | https://mathoverflow.net/users/108284 | 443928 | 179,033 |
https://mathoverflow.net/questions/443884 | 6 | Let $M$ be a closed subspace of $L\_p(0,1)$, $1<p<\infty$, $p\neq 2$.
Suppose that M contains copies of $\ell\_p^n$ uniformly.
Does $M$ contain a copy of $\ell\_p$?
The result is true for $p=1$, since subspaces of $L\_1(0,1)$ containing no copies of $\ell\_1$ are superreflexive.
| https://mathoverflow.net/users/39421 | Finite representability of $\ell_p$ in subspaces of $L_p(0,1)$ | Yes. There is probably a quick proof using ultraproducts but I am not sure. Here is a very rough sketch of a proof.
$1\le p<2$ case. Let $\varepsilon\_k\searrow 0$, and suppose $(x^n\_i)\_{i=1}^n\in X$ are $(1+\varepsilon\_k)$-equivalent to the unit vector basis of $\ell\_p^n$'s. We will use a theorem of Dor that fun... | 3 | https://mathoverflow.net/users/3675 | 443969 | 179,048 |
https://mathoverflow.net/questions/443867 | 14 | Let $A \subset \mathbb F\_p \setminus \{0\} $ and let $A+1/A = \{x+1/y:x,y \in A\}$.
**Question:** For fixed $c>0$, if $|A| \geq cp$, is $|A+1/A|$ at least $(1-o(1))p$ when $p \rightarrow \infty$?
**Known:**
This is false for $c<1/4$ according to Seva's comments.
Take $A$ to be $I \cap I^{-1}$ where $I=[1,m]$ and... | https://mathoverflow.net/users/125498 | A sum-product phenomenon on reciprocals | The claim is true for $c > 1/4$, although the proof I have either requires the machinery of the arithmetic regularity lemma, or the (morally equivalent) language of additive limits (as discussed in [this blog post](https://terrytao.wordpress.com/2014/10/12/additive-limits/) of mine, or in [this paper of Szegedy](https:... | 11 | https://mathoverflow.net/users/766 | 443991 | 179,057 |
https://mathoverflow.net/questions/443778 | 4 | In [How to glue perverse sheaves](https://people.math.harvard.edu/%7Egaitsgde/grad_2009/Beilinson%20--%20How%20to%20glue%20perverse%20sheaves.pdf) Beilinson claims that the category of perverse sheaves on the complex unit disk $D$ with the stratification with the closed strata $\{0\}\subset D$ is equivalent to the cate... | https://mathoverflow.net/users/131868 | Explicit description of perverse sheaves on a disk | Assume we have the data $(X,Y,\phi,u,v)$. Notice that $Y$ splits as a direct sum $(Y,\text{id}\_Y-\phi)^0\oplus \text{im}(\text{id}\_Y-\phi)^n$ for $n$ sufficiently large and denote this stable image $Y'$. Note that $\text{id}-\phi$ restricts to an automorphism $\psi$ on $Y'$. Let $\psi$ be this restriction.
Then we ... | 1 | https://mathoverflow.net/users/131868 | 444000 | 179,060 |
https://mathoverflow.net/questions/443954 | 10 | We say that a metric space $(X, d)$ is a *Banakh space* if for every $\rho \in \mathbb{R}\_{> 0}$ and every $x \in X$, there are
$a,b \in X$ such that $\{y \in X \, \vert \, d(x, y) = \rho\} = \{a, b\}$ and $d(a, b) = 2 \rho$.
**Question on Banakh spaces**. Let $(X, d)$ be a Banakh space. Is $(X, d)$ isometric to the... | https://mathoverflow.net/users/84349 | An incomplete characterisation of the Euclidean line? | Taras Banakh writes in the original question that *by transfinite induction of length $\mathfrak c$, one can construct a dense $\mathbb Q$-linear subspace $L$ of the Euclidean plane $\mathbb R^2$ such that $|\{x\in L:\|x\|=r\}|=2$ for every positive real number $r$*. This is indeed rather straightforward to do; it’s a ... | 7 | https://mathoverflow.net/users/12705 | 444003 | 179,063 |
https://mathoverflow.net/questions/443899 | 2 | It is well known that a smooth manifold $M$ is orientable if the first Stiefel-Whitney class of the tangent bundle vanishes. In particular, this implies that if $M$ is equipped with a pseudo-Riemannian metric of signature $(t,s)$, then the frame bundle $O(M)$ admits a reduction to a principal $SO(t,s)$ bundle under the... | https://mathoverflow.net/users/496509 | Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable | For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well.
The idea is as you suggested. Maximal-rank timelike (or spacelike) vector subspaces form a convex space. So you paste together local decompositions using a partition of unity to dec... | 1 | https://mathoverflow.net/users/1465 | 444019 | 179,069 |
https://mathoverflow.net/questions/437255 | 5 | 1. I was going through the paper of Brezis and Merle [1]. In theorem 3 step 2 it's written that the boundedness of $f\_n=V\_ne^{u\_n}$ in $L\_{loc}^p(\Omega)$ implies $\mu\in L^1(\Omega)\cap L\_{loc}^p(\Omega)$. Let $v\_n$ be a solution of $-\Delta v\_n=f\_n$ in $\Omega$ and $v\_n=0$ in $\partial\Omega$. Then $v\_n\to ... | https://mathoverflow.net/users/493046 | Two doubts in the paper of Brezis Merle in blow up analysis of the equation $-\Delta u=Ve^u$ | $\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$
I realize this is a bit late but I'd like to elaborate on Professor Metafune's very interesting and helpful comments. I'll prove how your claim 1. follows from the following:
**Lemma 1**
Let $\Omega$ be a $C^{1,1}$ bounded domain in $\mathbb{R}^d... | 2 | https://mathoverflow.net/users/479534 | 444029 | 179,075 |
https://mathoverflow.net/questions/444024 | 9 | Let $R = \Lambda \mathbb{C}^n$ be the exterior algebra on $\mathbb{C}^n$ for some positive integer $n$. It is an associative (graded-commutative) algebra of $\mathbb{C}$-dimension $2^n$.
Suppose we have an injective ring homomorphism $R\rightarrow M\_N(\mathbb{C})$ into the ring $M\_N(\mathbb{C})$ of $N\times N$ matr... | https://mathoverflow.net/users/12419 | Smallest faithful matrix representation of the exterior algebra | Let $M$ be a faithful module for $R$, and let $a\in R$ be the product of the exterior generators, so that $a$ generates the socle of $R$. Then there an element $m\in M$ such that $am\ne 0$. The map $r\mapsto rm$ is then an injective homomorphism from the regular representation of $R$ to $M$, so the dimension of $M$ is ... | 10 | https://mathoverflow.net/users/460592 | 444042 | 179,079 |
https://mathoverflow.net/questions/444044 | -1 | Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
**Is it true that $f$ is homotopic to 0?**
| https://mathoverflow.net/users/16183 | When morphism of complexes is homotopic to 0? | No. Take a diagram$$\require{AMScd}
\begin{CD}
0@>{}>>0 @>{}>> I^{1}@>{}>>0 @>{}>>\cdots\\
@VVV @VVV @VVV^{\!\!\!\operatorname{Id}}@VVV \\
0@>{}>>I^0 @>{d}>> I^1@>{}>>0@>{}>>\cdots
\end{CD}$$with $I^0,I^1$ injective and $d$ a surjective homomorphism which doesn't split. Then $H^0(A^{\bullet})=H^1(I^{\bullet})=0$, so $H... | 9 | https://mathoverflow.net/users/40297 | 444051 | 179,080 |
https://mathoverflow.net/questions/443934 | 1 | Background
----------
For a real-valued random variable $X$, define its entropy by $H(X) = E[\phi(X)] - \phi(E[X])$, where $\phi(u) = u \log u$.
It can be shown that, if the entropy satisfies the bound
$$
H(e^{\lambda X}) \leq \frac{\sigma^2 \lambda^2}{2} E[e^{\lambda X}]
$$
for all real $\lambda$, then $X$ must be a... | https://mathoverflow.net/users/488776 | Converse of the Herbst argument? | As in the comment, the converse is true up to a factor of 4. The proof is given below.
Suppose $X$ is $\sigma^2/4$-subgaussian, i.e. the following holds for any $\lambda \in \mathbb{R}$:
$$
\mathbb{E}[e^{\lambda X}] \leq \exp \left( \lambda \mathbb{E}[X] + \frac{\lambda^2 \sigma^2}{8} \right) \text{.}
$$
Let $Z := e^... | 2 | https://mathoverflow.net/users/488776 | 444055 | 179,081 |
https://mathoverflow.net/questions/444045 | 6 | Let $G$ be a group and $H$ a subgroup with finite index $d$. Then for any finite generating set $S$ of $G$, does $S^{\le k}$ contain a generating set of $H$ where $k$ is a constant depending only on $d$? When $S$ is symmetric, i.e. $S=S^{-1}$, the conclusion holds by [Shalen-Wagreich's Lemma 3.4](https://www.jstor.org/... | https://mathoverflow.net/users/397904 | Does the Shalen-Wagreich lemma holds for non-symmetric generating sets? | The answer is **yes**, with $k = 2d - 1$ as a sharp upper bound.
This follows from a classical combinatorial result of Otto Schreier known as [Schreier's lemma](https://en.wikipedia.org/wiki/Schreier%27s_lemma) [1, Proof of Proposition I.3.7].
These ideas of Otto Schreier go back to 1927 and already establish [2, Lem... | 4 | https://mathoverflow.net/users/84349 | 444060 | 179,083 |
https://mathoverflow.net/questions/443159 | 3 | Let $C\_p\equiv C\_p(\mathbb R\_+,\mathbb R\_+)$ be the space of right-continuous piecewise constant functions $f: \mathbb R\_+\to \mathbb R\_+$, i.e. $f\in C\_p$ iff
$$f(t)=\sum\_{k=1}^n {\mathbf 1}\_{[t\_{k-1},t\_k)}(t)x\_k,$$
where $0=t\_0<t\_1<\cdots<t\_{n-1} <t\_n=\infty$, $x\_1,\ldots, x\_n\in (0,\infty)$ and... | https://mathoverflow.net/users/493556 | An integral on the interval depending on the integrand | It takes some time to understand what is asked, but after that it becomes easy.
Suppose that $\newcommand{\wt}{\widetilde}$ $+\infty>\tau\ge \wt\tau\ge 0$ and $\int\_0^\tau F\le\int\_0^{\wt\tau}\wt F$ where $F=\max(F,e^{-2})$ (so my $F$ is your $f^2$). Write $F=e^{-2}+G$ and decompose $G=G\_+-G\_-$ where $G\_\pm=\max... | 2 | https://mathoverflow.net/users/1131 | 444061 | 179,084 |
https://mathoverflow.net/questions/444056 | 5 | Can someone give me a roadmap for learning Inverse Galois theory?
I am a PhD student in the representation theory of finite groups. I studied Galois theory when I was an undergraduate student. The Inverse Galois Problem is one of my favourite problems in mathematics. I want to read the book Inverse Galois Theory. (<h... | https://mathoverflow.net/users/138348 | Learning Inverse Galois Theory | I would like to suggest that a good place to start is with John Thompson's work on the subject. He initiated the modern approach using the notion of "rigid" tuples of conjugacy classes.
| 7 | https://mathoverflow.net/users/460592 | 444075 | 179,089 |
https://mathoverflow.net/questions/444040 | 13 | I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.
I am aware that for a topological space $X$ and a point $x \in X$ the fundamental group $\pi\_1(X,x)$ acts on all homotopy groups $\pi\_n(X,x)$. In particular, this means the $... | https://mathoverflow.net/users/502108 | Local systems arising from higher rational homotopy groups | Let's say we have rigorously defined the notion of a family of objects in an $\infty$-category $C$ parametrized by a space $X$. There are several ways to do this: if we model $\infty$-categories by quasicategories, we can simply consider $\mathrm{Fun}(X,C)$; and in other models it might be easier to consider the catego... | 6 | https://mathoverflow.net/users/134512 | 444085 | 179,091 |
https://mathoverflow.net/questions/427581 | 1 | Let $SM\_n(R)$ be the set of $n\times n$ symmetric matrices with entries in a ring $R$ and let $A\sim B$ for such matrices if $A=C^T\cdot B\cdot C$ for some $C\in SL(n,R).$ It is an equivalence relation (appearing for example in the theory of modular forms, at least in the context of positive matrices).
Each $M\in SM... | https://mathoverflow.net/users/23935 | Symmetric Integral Matrices | Partial answer:
Let $M$ be an $n \times n$ symmetric matrix over a PID $R$. Then using an algorithm resembling the one for computing Smith Normal Forms, we get that $M \sim E\oplus\bigoplus\_{i=1}^{\lfloor n/2\rfloor} D\_{i}$, where $E$ is either $1 \times 1$ or $0 \times 0$, and each $D\_i$ is $2 \times 2$ and symme... | 1 | https://mathoverflow.net/users/75761 | 444095 | 179,094 |
https://mathoverflow.net/questions/444079 | 5 | For integers $n$ and $k$, let $P(n,k)(x) = \sum\_{i=0}^k \binom ni x^i$ be the truncated binomial polynomial. There has been work on whether $P(n,k)$ is irreducible, but this is a different question. According to some Mathematica computations, if $n \le 200$ and $1 \le r < n/2$, then $P(n,2r)(x) > 0$ for all $x$. In ad... | https://mathoverflow.net/users/502148 | Are truncated even degree binomial polynomials psd? | We have $$P(n,k)'(x)=\sum\_{i=1}^k \binom ni ix^{i-1} \\
=n\sum\_{i=1}^k \binom{n-1}{i-1}x^{i-1}
=nP(n-1,k-1)(x)\tag{1}\label{1}$$
and
$$P(n,k)(x)-P(n-1,k)(x)
=\sum\_{i=1}^k \Big(\binom ni-\binom{n-1}i\Big)x^i \\
=\sum\_{i=1}^k \binom{n-1}{i-1} x^i
=xP(n-1,k-1)(x). \tag{2}\label{2}$$
Also, for $r=1,2,\dots$ and $n\ge2... | 5 | https://mathoverflow.net/users/36721 | 444096 | 179,095 |
https://mathoverflow.net/questions/441971 | 0 | **Background**:
The perfect numbers are the positive integers $n$ such that $$\sigma(n)=2n,$$ where $\sigma(n)$ is the sum of divisors function.
The function $\sigma(n)$ is multiplicative and satisfies $$\sigma(p^k)=\dfrac{p^{k+1}-1}{p-1}$$ for all primes $p$ and any positive integer $k$.
Every even perfect numbe... | https://mathoverflow.net/users/nan | Steuerwald's theorem | Here is a proof of this fact.
We start with a standard
**Lemma 1.** Any prime divisor $q$ of $1+x+x^2$ for an integer $x$ is either equal to 3 or is congruent to 1 modulo $3$.
**Proof.** If, on the contrary, that $q=3k+2$, then $x^3\equiv 1 \pmod q$ and also by Fermat's little theorem $x^{3k+1}\equiv 1 \pmod q$, ... | 2 | https://mathoverflow.net/users/4312 | 444097 | 179,096 |
https://mathoverflow.net/questions/444100 | 6 | Let us consider the following stronger version of the Axiom Schema of Replacement (let us call it the Axiom Schema of Replacement for Definable Relations):
Let $\varphi$ be any formula in the language of ZF whose free variables are among the symbols $x,y,A,w\_1,\dots,w\_n$.
Then:
$\forall w\_1\dots\forall w\_n \for... | https://mathoverflow.net/users/61536 | A strong form of the Axiom Schema of Replacement | Your axiom is known as [the axiom of collection](https://en.wikipedia.org/wiki/Axiom_schema_of_replacement#Collection).
There are a variety of contexts where it is known that replacement does not imply collection over what may seem reasonable theories.
For example, in set theory without the power set axiom, one can... | 7 | https://mathoverflow.net/users/1946 | 444107 | 179,098 |
https://mathoverflow.net/questions/444086 | 3 | Given two type I (von-Neumann algebra) factors $\mathcal M,\mathcal N$, is there a *smallest* type I factor containing both $\mathcal M,\mathcal N$?
**Notes:**
* $\mathcal M,\mathcal N$ are over the same Hilbert space, of course.
* Obviously, a type I factor containing both exists (namely the set of all bounded ope... | https://mathoverflow.net/users/101775 | Least upper bound of type I factors | No, there typically does not exist a smallest type I factor containing $\mathcal{M}$ and $\mathcal{N}$. Since the commutant of a type I factor is again a type I factor, by the bicommutant theorem, the question is equivalent with the existence of a maximal type I factor contained in $\mathcal{P} = \mathcal{M}' \cap \mat... | 3 | https://mathoverflow.net/users/159170 | 444116 | 179,101 |
https://mathoverflow.net/questions/443679 | 17 | The concept of [sparse polynomials](https://en.wikipedia.org/wiki/Sparse_polynomial) has its place, and solvable but irreducible quadrinomial examples such as,
$$x^7-7x^4-14x^3-7=0$$
$$x^8+x^7+29x^2+29=0$$
$$x^9-27x^4-9x^3-9^2=0$$
$$x^{12}-36x^5-12x^3-12^2=0$$
may be intriguing, especially the octic which needs the... | https://mathoverflow.net/users/12905 | On the solvable septic quadrinomial $x^7-7x^4-14x^3-7=0$? | It turns out ***there are*** infinitely many solvable septic quadrinomials after all (though with a caveat), and the answer was under my nose all along. For example, recall that,
$$e^{\pi\sqrt{163}} =640320^3+743.999999999999925\dots$$
Then the septic,
$$x^7+7\left(\tfrac{1-\sqrt{-7}}2\right)x^4+7\left(\tfrac{1+\sq... | 3 | https://mathoverflow.net/users/12905 | 444131 | 179,104 |
https://mathoverflow.net/questions/444142 | 1 | Let $L$ be a number field and let ${\mathcal{O}}\_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k\_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}\_L$ to $k\_B$ is surjective. What is the proof of this statement?
| https://mathoverflow.net/users/502220 | The map from the ring of integers to the residue field of a valuation subring is surjective | By Ostrowski's theorem, the valuation in question must be the $P$-adic one for some prime $P$ of $\mathcal{O}\_L$. This means that $k\_B=\mathcal{O}\_L/P$.
If you prefer to avoid using such a strong theorem, you can also use the characterization of valuation rings as maximal local subrings, since its maximal ideal ca... | 2 | https://mathoverflow.net/users/158123 | 444147 | 179,108 |
https://mathoverflow.net/questions/444110 | 2 | Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-torsors over $C$ is naturally equivalent to the groupoid of $C$-adelic $G$-automorphic forms. Notice that, once we pass to th... | https://mathoverflow.net/users/158123 | What is the sum operation on torsors induced by Weil uniformization? | As pointed out by Jesse Silliman in the comments, this question is ill-posed. In fact, adelic automorphic forms are global sections of the structure sheaf of the moduli of $G$-torsors, not elements of this stack.
| 1 | https://mathoverflow.net/users/158123 | 444148 | 179,109 |
https://mathoverflow.net/questions/444124 | 3 | Let $f: X \to Y$ a morphism between smooth varieties
over alg. closed field of characteristic zero. It is known that the deformation theory
in relative setting of $f$ is encoded in the cohomology of the trunicated cotangent
complex
$$ \mathbb{L}\_f = \tau\_{\ge -1}\left[f^\*\Omega\_Y \to \Omega\_X\right] $$
concent... | https://mathoverflow.net/users/501436 | Lower bound for the dimension of the space of deformations $\mathrm{Defor}(f : X \to Y)$ in relative setting | This is too long for a comment.
You need some sort of hypothesis to get the existence of a versal deformation space for morphisms $f$. The most common hypothesis is that $X$ is proper over your field $k$.
In that case, Schlessinger gives a versal formal deformation over Spf of a power series $k$-algebra, $k[[x\_1,\... | 2 | https://mathoverflow.net/users/13265 | 444151 | 179,111 |
https://mathoverflow.net/questions/444129 | 3 | How is the Petersson norm of a quaternionic modular form defined?
*Background:* In [Tamiozzo, On the Bloch-Kato conjecture for Hilbert modular forms](https://arxiv.org/pdf/1911.05019.pdf), section 3.3, it is written "We normalize $f\_B$ requiring its Petersson norm to be $1$ (the Petersson product being just a finite... | https://mathoverflow.net/users/471019 | Petersson norms of quaternionic modular forms | "Of Petersson norm one" in the paper you mention means "of norm one" in the sense of Theorem 7.1 of S. Zhang's paper "Gross-Zagier formula for GL(2) II".
With this normalisation of the Jacquet-Langlands transfer $f\_B$, as you write, $a(f)$ is not necessarily rational.
On the other hand, outside the statement of Theo... | 3 | https://mathoverflow.net/users/502223 | 444153 | 179,112 |
https://mathoverflow.net/questions/443786 | 2 | I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will implicitly assume that everything is finite-dimensional.
**Theorem.** If every abelian extension of the Lie algebra $\mathfrak g$ sp... | https://mathoverflow.net/users/147463 | Lie algebras for which all one-dimensional extensions split | Here is an example of a Lie algebra $\mathfrak g$ for which every one-dimensional extesion splits, but not every extension splits (which is based on @YCor's comment above). We can work over any field $k$ of characteristic zero. If $V$ is a $\mathfrak g$-module, then $H^2(\mathfrak g,V)=0$ if and only if the only extens... | 1 | https://mathoverflow.net/users/147463 | 444155 | 179,114 |
https://mathoverflow.net/questions/444149 | 1 | In this article [Interpolation inequalities with weights
Chang Shou Lin](https://www.tandfonline.com/doi/abs/10.1080/03605308608820473?journalCode=lpde20) the following lemma is stated and proved.
Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$ such that
$$|||x|^{\alpha}u||\_p ... | https://mathoverflow.net/users/123355 | Lemma about the weighted interpolation inequality | $\newcommand\R{\mathbb R}\newcommand{\al}{\alpha}$In view of conditions (0.1) and (0.4) in the linked paper, $p\in[1,\infty)$ and $u\in C\_0^\infty(\R^n)$. To prove the lemma in question (Lemma 2.2 in that paper), it suffices that $u\in C\_0^1(\R^n)$.
Indeed,
\begin{equation\*}
\|\,|x|^\al u\|\_p^p=I:=\int\_{\R^n}dx... | 1 | https://mathoverflow.net/users/36721 | 444163 | 179,117 |
https://mathoverflow.net/questions/444082 | 64 | I have been reading David Angell's lovely book, *Irrationality and Transcendence in Number Theory*, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of something that I've long been puzzled by, which the book hasn't cleared up for me. In the book's proof that $e^r$ is ... | https://mathoverflow.net/users/3106 | To prove irrationality, why integrate? | Here's an exposition of Niven's proof that makes the connection to orthogonal polynomials explicit. We start with an observation, easily proven by induction, that if $P\in \mathbb{Z}[x]$, then $\int\_0^\pi P(x)\sin x=Q(\pi)$, where $Q\in \mathbb{Z}[x]$, and $\deg Q\leq\deg P$. If we can find a sequence of polynomials $... | 22 | https://mathoverflow.net/users/56624 | 444167 | 179,119 |
https://mathoverflow.net/questions/444168 | 4 | Let $\mathbb{k}$ be an algebraically closed field of positive characteristic, $X$ an affine smooth variety over it. Then the ring of crystalline differential operators on $X$ is generated by $\mathcal{O}(X)$ and $\operatorname{Der}\_\mathbb{k} \mathcal{O}(X)$ with relations $f.\partial=f\partial$, $\partial.f-f.\partia... | https://mathoverflow.net/users/160378 | Reference request on rings of crystalline differential operators | Let $\pi: T^\ast X \to X$ be the projection of the cotangent bundle and $-'$ denote Frobenius twist.
Then $D(X)$ is an Azumaya algebra over its center $\pi\_\ast \mathcal O\_{T^\*X'}$. This appears in Roman Bezrukavnikov, Ivan Mirković, Dmitriy Rumynin: "Localization of modules for a semisimple Lie algebra in prime cha... | 2 | https://mathoverflow.net/users/125523 | 444170 | 179,120 |
https://mathoverflow.net/questions/444139 | 2 | I'm looking for triples $(K,z, |-|)$ where $K$ is a local field, $z \in K$, and $|-|$ is an absolute value on $K$, such that, for all $f(x),g(x),h(x) \in \mathbb N[x]$, the following inequality is satisfied:
$$|f(z) + g(z)| \leq |f(z) + h(z)| + |g(z) + zh(z)|$$
I know that the following triples work:
* $(\mathbb ... | https://mathoverflow.net/users/2362 | When can I satisfy the following twisted subadditivity inequality? | Surprisingly to me, these are the only possibilities! We will consider polynomials $f(z), g(z), h(z)$ as above. Set $a(z) = f(z)/h(z)$, and $b(z) = g(z)/h(z)$. So the desired inequality is
$$|a(z) + b(z)| \leq |a(z) + 1| + |b(z) + z| \qquad (\ast)$$
where $a(z),b(z)$ are rational functions of $z$ with natural numbe... | 1 | https://mathoverflow.net/users/2362 | 444182 | 179,122 |
https://mathoverflow.net/questions/444161 | 9 | *Note: Here $\mu$ denotes Lebesgue measure on $\mathbb R$.*
We say a function $f: \mathbb R \to \mathbb R$ is *uniformly Lebesgue differentiable* if there exists some measurable subset $E$ of $\mathbb R$ with $\mu(E^c) = 0$ such that the following holds:
>
> For all $\varepsilon > 0$, there exists some $\delta > ... | https://mathoverflow.net/users/173490 | Uniformly Lebesgue differentiable functions | Yes. Suppose $f$ is uniformly Lebesgue differentiable (ULD).
Let's first note that $f$ at least has a *continuous* representative, since the functions $f\_r(x) = \frac{1}{2r} \int\_{B\_r(x)} f(y)\,dy$ are each continuous (by dominated convergence if you like), and by the triangle inequality and the ULD condition, the... | 12 | https://mathoverflow.net/users/4832 | 444186 | 179,123 |
https://mathoverflow.net/questions/444118 | 1 | By the works of Michiel de Bondt and Arno van den Essen, and Ludwik Drużkowski, it is known that if $F=I+N$, where $I$ is the identity mapping and $N$ is cubic homogeneous polynomials in $n$ complex variables, is invertible, then the Jacobian conjecture is true for $n$ variables.
Consider $J(N)$, the jacobian of the ... | https://mathoverflow.net/users/172458 | Question about Jacobian conjecture on the reals | To (Q1): Yes, $F$ being invertible means in this context exactly that $F$ admits a polynomial *compositional* inverse. If you are working over a general commutative ring $k$, then invertibility of $J\_F$ is equivalent to invertibility of $\det J\_F$ in $k[x\_1,\dots,x\_n]$. In particular, if $k$ is not reduced, then th... | 5 | https://mathoverflow.net/users/1849 | 444188 | 179,124 |
https://mathoverflow.net/questions/444212 | 1 | Is it true that if I have $\alpha \in \mathbb{C}$, $q,w \in \mathbb{Z}$ such that for every automorphism $\sigma$ of $\mathbb{C}$, $$|\sigma(\alpha)|=q^{w/2}$$ then $\alpha$ must in fact be algebraic?
Note this is false without assuming some kind of axiom of choice, as else the identity and complex conjugation are th... | https://mathoverflow.net/users/143607 | Norm fixed under complex automorphisms implies algebraic | Assuming ZFC, as an abstract field $\mathbb C$ is an algebraic closure of a pure transcendental extension of $\mathbb Q$ on $2^{\aleph\_0}$ generators. The automorphism group, of order $2^{2^{\aleph\_0}}$, is transitive on transcendental elements. So the answer is yes.
| 3 | https://mathoverflow.net/users/460592 | 444213 | 179,126 |
https://mathoverflow.net/questions/444214 | 5 | what's the limit of
$\sqrt{1-t}\sum \_{n=0}^{\infty}t^{n^2}$ as $t$ goes to the left of $1$? i.e. $t\to 1^{-}$? I tried several times but failed. Here is my thought:
This is a $0\cdot\infty$ problem, so I just tried with $\frac{\sum \_{n=0}^\infty t^{n^2}}{\frac{1}{\sqrt{1-t}}}$. Then it becomes a $\frac{\infty}{\inf... | https://mathoverflow.net/users/169417 | Solving a limit about sum of series | The sum $\sum \_{n=0}^{\infty}t^{n^2}$ evaluates for $t<1$ to an elliptic theta function, and then taking the limit $t\rightarrow 1$ from below gives
$$\lim\_{t\nearrow 1}\sqrt{1-t}\sum \_{n=0}^{\infty}t^{n^2}=\tfrac{1}{2}\sqrt{\pi}.$$
Alternatively, I can write $t=1-\epsilon$, with $(1-\epsilon)^{n^2}\rightarrow e^{... | 9 | https://mathoverflow.net/users/11260 | 444216 | 179,127 |
https://mathoverflow.net/questions/444226 | 2 | Let $M$ be a compact Riemann manifold without boundary. Please is this true that each homotopy class of closed curves contains a geodesic?
| https://mathoverflow.net/users/486323 | geodesics on a compact manifold | Presumably this is a reference request (since the statement is sort of well-known).
See Theorem IV.5.1 in
*Chavel, Isaac*, Riemannian geometry. A modern introduction, Cambridge Studies in Advanced Mathematics 98. Cambridge: Cambridge University Press (ISBN 0-521-61954-8/pbk; 0-521-85368-0/hbk). xvi, 471 p. (2006). ... | 6 | https://mathoverflow.net/users/3948 | 444230 | 179,134 |
https://mathoverflow.net/questions/444210 | 1 | Bezout's Theorem concludes that if $f\_1,f\_2,\cdots,f\_n\in k[x\_1,x\_2,\cdots,x\_n]$ have finite intersection points, then they have at most $d\_1d\_2\cdots d\_n$ intersection points, where $d\_i$ is the degree of $f\_i$.
When $n=2$, if $f\_1$ and $f\_2$ don't have a common divisor of positive degree, then they hav... | https://mathoverflow.net/users/492228 | Sufficient conditions to guarantee finite intersection points in Bezout's Theorem | The condition you want is
$$
{\rm Res}(h\_1,\ldots,h\_n)\neq 0\ .
$$
Here $h\_i$ is the (leading/highest degree) homogeneous part of $f\_i$ of degree $d\_i$.
The expression on the left is the homogeneous resultant of the given collection of forms $h\_1,\ldots,h\_n$. It is a polynomial in the coefficients of these forms... | 1 | https://mathoverflow.net/users/7410 | 444238 | 179,136 |
https://mathoverflow.net/questions/444259 | 2 | The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following:
Do continuous families $t\mapsto G\_t$ of "games" (say each $G\_t$ is a stable finite game with unique Nash equilibrium) have that the Nash equilibrium varies continuously?; or a fixed point?; or... | https://mathoverflow.net/users/12310 | Continuity of Nash equilibrium for a family of games | Usually, one looks at the correspondence (set-valued map) that associates with each game its set of Nash equilibria. Under many conditions, this correspondence is [upper hemicontinuous](https://en.wikipedia.org/wiki/Hemicontinuity). For example, if one fixes finite action spaces for a fixed finite set of players, then ... | 4 | https://mathoverflow.net/users/35357 | 444267 | 179,141 |
https://mathoverflow.net/questions/444268 | 1 | As we know Bernstein's inequality for polynomials states that, if $P(z)$ is a polynomial of degree $n$ then
$$\max\_{|z|=1}|P'(z)|\leq n \max\_{|z|=1}|P(z)|. $$
There are results related to the reverse of the above inequality for some restricted class of polynomials. Are there any literature about the lower bound of $\... | https://mathoverflow.net/users/128472 | Lower bound for polynomials | For unrestricted polynomials of a given degree $n$, there is no lower bound. Indeed, consider
$$
P(z)=cz^n+1,
$$
with $|c|$ small. Then
$$
\frac{\|P'\|\_\infty}{\|P\|\_\infty}=\frac{|c|n}{1+|c|},
$$
which is arbitrarily small for $|c|$ small enough.
By the way, if the roots of $P$ lie in the closed disk, then there i... | 5 | https://mathoverflow.net/users/89429 | 444273 | 179,142 |
https://mathoverflow.net/questions/444248 | 2 | Let $S$ be a *uniruled* surface, ie admits a dominant map $ f:X \times \mathbb{P}^1$. Why then it's canonical divisor $\omega\_X$ cannot be trivial? Motivation: I want to understand why $K3$ surfaces cannot be uniruled?
| https://mathoverflow.net/users/108274 | $K3$ surfaces can't be uniruled | The comment by user @naf is completely correct: there is a simpler argument than the argument I sketched. However, the argument I sketched gives a stronger result, which Mumford conjectured is sharp.
Because of the semipositivity, for every smooth, projective variety $S$ over a characteristic zero field $k$, for a ge... | 2 | https://mathoverflow.net/users/13265 | 444286 | 179,146 |
https://mathoverflow.net/questions/444279 | 1 | In an article I read, I have the following inequality:
$\|A-B\|\_1 \geq \max \{ \|A 1\_m- B 1\_m \|\_1, \|A^T 1\_n - B^T1\_n\|\_1 \}$
Where $A, B \in \mathbb{R}\_+^{m\times n}$.
The $\|\cdot\|\_1$ refers either to the $l\_1$ vector norm, either to the matrix operator norm.
They use Jensen inequality for this result.
... | https://mathoverflow.net/users/145604 | Norm inequality | The inequality is true if $\|A - B\|\_1$ is interpreted as $\sum |A\_{ij} - B\_{ij}|$.
Just observe that
$$ \|A 1\_M - B 1\_M\|\_1 = \sum\_i | \sum\_j A\_{ij} - B\_{ij}| \leq \sum\_i \sum\_j |A\_{ij} - B\_{ij}| $$
by the triangle inequality.
| 1 | https://mathoverflow.net/users/3948 | 444288 | 179,147 |
https://mathoverflow.net/questions/444094 | 9 | **It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$.**
$F(m,N)=\frac{N^m}{2^m}\displaystyle \sum\_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\frac{\pi j}{N}\right)$
**How do I prove this?**
(Source: <https://twitter.com/SamuelGWalters/status/15132667048... | https://mathoverflow.net/users/501409 | How to prove this sum involving powers of cosec is an integer? | *It turns out rewriting the function in terms of Bernouilli numbers etc, while useful in generating specific values of $F(m,N)$, is not an easy path to proving they are integers. The following proof is reasonably elementary (knowledge of complex $n$th roots of $1$, no calculus) and is drawn from D. Zagier, Elementary A... | 1 | https://mathoverflow.net/users/501409 | 444292 | 179,149 |
https://mathoverflow.net/questions/444282 | 4 | The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum\_{\rho} \frac{x^{\rho}}{\rho}+O(1),
$$
where $\psi(x)=\sum\_{n \leq x} \Lambda(n)$ and the sum is over all non-trivial zeroes of $\zeta(s)$.
Let $L/K$ be a finite Galois extension of number fields and let $V$ be an irreducible... | https://mathoverflow.net/users/501378 | Explicit formula for Artin L-functions | One proves explicit formulae essentially by integrating the logarithmic derivative of the $L$-function. For simplicity, let $\psi$ be a Schwartz function with $\psi(1) = 1$, and let
$$ \widehat{\psi}(s) = \int\_0^\infty \psi(t) t^{s} \frac{dt}{t} $$
be its Mellin transform. Then one can get an explicit formula by consi... | 7 | https://mathoverflow.net/users/14508 | 444293 | 179,150 |
https://mathoverflow.net/questions/444297 | 0 | Let $\phi ：C\_1\to C\_2$ be morphism of projective singular curve. Let $\tilde{C}\_1$ and $\tilde{C}\_2$ be their smooth compactification.
Then $\phi$ extends to $\tilde{\phi} : \tilde{C}\_1\to \tilde{C}\_2$.
Let $\deg \phi$ be degree of $\phi$ and $e\_\phi (P)$ be ramification degree at $P \in C\_1$.
>
> Does $\de... | https://mathoverflow.net/users/144623 | Are degrees and ramification degrees preserved upon passing to the smooth compactification? | The first claim is true. The degree can be defined as the degree of the extension on function fields $k(C\_1)/k(C\_2)$, but $k(\tilde{C}\_1) = k(C\_1)$ and $k(\tilde{C}\_2)= k(C\_2)$ so $k(C\_1)/k(C\_2) = k(\tilde{C}\_1)/k(\tilde{C}\_2)$.
The second claim is true for smooth points lying over smooth points. This is be... | 4 | https://mathoverflow.net/users/18060 | 444303 | 179,153 |
https://mathoverflow.net/questions/444298 | 2 | Let $f : \mathbb C \to \mathbb C$ be an entire function with a separated zero set, i.e. there is a $\delta>0$ s.t. $|z-z'| > \delta$ for every distinct zeros of $f$. Further, suppose that all zeros of $f$ are simple and
$$
|f(z)| \geq C\_\varepsilon e^{a|z|^2}, \quad z \in Z\_\varepsilon
$$
where $Z\_\varepsilon$ denot... | https://mathoverflow.net/users/397017 | Lower bounding the derivative of a simple zero of an analytic function | Sure enough, the idea is that if $f(z\_0) = 0$ then $g(z) = \frac{f(z)-f(z\_0)}{z-z\_0}$ is an entire function which is non-vanishing and for which we know the lower bound on the circle $|z-z\_0| =\frac{\delta}{2} = r$. Moreover, $g(z\_0) = f'(z\_0)$. But then on this disk the function $u(z) = \log g(z)$ is harmonic, t... | 3 | https://mathoverflow.net/users/104330 | 444306 | 179,156 |
https://mathoverflow.net/questions/444299 | 20 | Is it true that, assuming the Axiom of Choice, every infinite-dimensional Banach space has an infinite-dimensional closed subspace with infinite codimension? Note that this is different from the indecomposability problem, which asks whether every infinite-dimensional Banach space has an infinite-dimensional closed subs... | https://mathoverflow.net/users/19444 | Closed subspaces of Banach spaces | Yes, I think this is true. Any infinite dimensional Banach space $V$ contains a [basic sequence](https://matthewdaws.github.io/files/notes/bases.pdf) $(x\_n)$. Then $\{x\_1, x\_3, x\_5, \ldots\}$ is linearly independent and therefore its closed span $V\_0$ is infinite dimensional. The codimension of $V\_0$ must be infi... | 8 | https://mathoverflow.net/users/23141 | 444320 | 179,157 |
https://mathoverflow.net/questions/444341 | 4 | Let $T\_t$ a Feller semigroup (see [this](https://en.wikipedia.org/wiki/Feller_process#Definitions)) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always true if $(A,D(A))$ is a diffusion on a compact manifold? I... | https://mathoverflow.net/users/498097 | approximation of a Feller semi-group with the infinitesimal generator | You look for the concept of analytic (or entire) vector. If you have a strongly continuous group, then they are dense, otherwise it might happen that there is only the 0 vector (for nilpotent semigroups). See e.g.
*Engel, Klaus-Jochen; Nagel, Rainer*, [**One-parameter semigroups for linear evolution equations**](http... | 5 | https://mathoverflow.net/users/12898 | 444342 | 179,163 |
https://mathoverflow.net/questions/444309 | 4 |
>
> **Question:** Given exponents $0<\alpha<\beta$ and an interval
> $[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
> $\lambda\_1,\lambda\_2\in\mathbb{R}$,
> $$\left|\int\_a^be(\lambda\_1x^\alpha+\lambda\_2x^\beta)dx\right|\leq C\max\left(\frac1{|\lambda\_1|^d},\frac1{|\lambda\_2|^d}\right)?$$... | https://mathoverflow.net/users/18698 | Generalization of van der Corput's estimate on oscillatory integrals | For convenience of notation write $\Phi(x) = \lambda\_1 x^\alpha + \lambda\_2 x^\beta$. The general argument is based on the integration by parts argument
$$ \int\_a^b e\circ\Phi = \int\_a^b \frac{1}{2\pi i \Phi'} \frac{d}{dx} e\circ \Phi = \frac{1}{2\pi i \Phi'(b)} e(\Phi(b)) - \frac{1}{2\pi i \Phi'(a)} e(\Phi(a)) + \... | 5 | https://mathoverflow.net/users/3948 | 444347 | 179,164 |
https://mathoverflow.net/questions/444310 | 3 | I work with i.i.d. variables $X\_1, \dots, X\_{N}$ such that $0 \le X\_i \le 1$, $E[X\_i] = \mu$, $\operatorname{Var}[X\_i] = \sigma^2$.
I am gradually sampling $X\_1, X\_2, \dots$ and want to ensure that the natural sample variance estimate stays within reasonable bounds. More formally, define $A\_n = \sum\_{i = 1}^... | https://mathoverflow.net/users/502360 | A maximal inequality for sample variance | $\newcommand{\si}{\sigma}$Note that
\begin{equation\*}
A\_n=B\_n-C\_n,
\end{equation\*}
where
\begin{equation\*}
B\_n:=\sum\_1^n Y\_i,\quad C\_n:=\frac1n\,\Big(\sum\_1^n Z\_i\Big)^2-\si^2,
\end{equation\*}
\begin{equation\*}
Z\_i:=X\_i-\mu,\quad Y\_i:=Z\_i^2-\si^2.
\end{equation\*}
Note also that, in view of the co... | 3 | https://mathoverflow.net/users/36721 | 444350 | 179,166 |
https://mathoverflow.net/questions/444345 | 6 | Let $(\mathcal C, \otimes, I)$ be a symmetric monoidal 2-category, and let $X \in \mathcal C$ be a dualizable object, with dual $X^\vee$, unit $coev: I \to X \otimes X^\vee$, and counit $ev : X^\vee \otimes X \to I$ satisfying the triangle identities. Recall that $X$ is [2-dualizable](https://ncatlab.org/nlab/show/full... | https://mathoverflow.net/users/2362 | Checking 2-dualizability | I will try to answer questions 1, 2, and 4. Question 3 is somewhat orthogonal to them, and I don't know the answer.
Clearly (1) implies both (2) and (3).
Suppose that (3) holds. Let me write $\sigma$ for the symmetry on $\mathcal{C}$, and consider the map $coev' := \sigma \circ ev^R : I \to X \otimes X^\vee$ and $e... | 5 | https://mathoverflow.net/users/78 | 444353 | 179,167 |
https://mathoverflow.net/questions/444352 | 0 | Let $ S\subset\mathbb{R}^n $ is of finite $ k $-dimensional Hausdorff and $ 0<\delta<1 $ is a constant. If for any $ x\in\mathbb{R}^n $ and $ r>0 $, we hae
$$
\mathcal{H}^k(S\cap B\_r(x))\leq A\omega\_kr^k.
$$
I want to ask if I can get that
$$
\mathcal{H}^k(S\cap B\_1(0))\leq A\omega\_k.
$$
Intuitively thinking it is ... | https://mathoverflow.net/users/241460 | If $ \mathcal{H}^k(B_1(0)\cap S)\leq A\omega_k $ when $ \mathcal{H}^k(B_r(x)\cap S)\leq A\omega_kr^k $ for all $ 0<r<\delta $, $ x\in\mathbb{R}^n $? | You cannot get this bound. My heuristic explanation for this failure would be that your bound 'does not see folds of $S$ at scales larger than $\delta$'. However, these folds may well make positive contributions to the $k$-dimensional area of $S$.
For a counterexample, you could for instance take any compact, smoothl... | 1 | https://mathoverflow.net/users/103792 | 444361 | 179,169 |
https://mathoverflow.net/questions/444364 | 1 | We add a bit to [Which polygons tessellate the hyperbolic plane?](https://mathoverflow.net/questions/398191/which-polygons-tessellate-the-hyperbolic-plane).
**Question:** Are there hyperbolic quadrilaterals with all angles different (not necessarily irrational fractions of π) that tile the hyperbolic plane? What abou... | https://mathoverflow.net/users/142600 | Tiling the hyperbolic plane by non-regular quadrilaterals | Here's a simple solution: start with the equilateral hyperbolic triangle with angles $2\pi/7$, which tiles the hyperbolic plane. Cut it into three congruent quadrilaterals with angles $(2\pi/7, \pi/2+\epsilon, 2\pi/3, \pi/2-\epsilon)$, meeting at the center of the triangle.
| 2 | https://mathoverflow.net/users/47484 | 444365 | 179,171 |
https://mathoverflow.net/questions/444308 | 3 | In complex analysis, by Poincare-Lelong theorem, we have
$$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T\_{z=0}
$$
as currents, where
$$
T\_{z=0}(\eta)=\int\_{z=0}\eta.
$$
Now suppose we have two variables $z\_1$, $z\_2$. We have $$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z\_1|^2)=T\_{z\_1=0}
$$
an... | https://mathoverflow.net/users/24965 | Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current? | Yes, since this corresponds to a proper intersection, this kind of product is very robust and can be defined for a couple of different reasons:
1. Since the unbounded loci of $\log |z\_1|^2$ and $\log |z\_2|^2$ are $\{ z\_1 = 0 \}$ and $\{ z\_2 = 0 \}$, and these intersect in a set of codimension $2$, the product can... | 2 | https://mathoverflow.net/users/49151 | 444366 | 179,172 |
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