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https://mathoverflow.net/questions/402873 | 0 | Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded lipschitz domain $\Omega\subset\mathbb{R}^3$).
I want to determine the essential spectrum of $M$ as a multiplication operator... | https://mathoverflow.net/users/152870 | Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces | $H^3$ is $\mathbb{C}^3 \otimes H$ and the action of $M$ on it is just the usual action on the first tensor factor. It is a completely straighforward exercise to prove that the $\lambda$-eigenspace of $M\otimes\operatorname{id}\_H$ is exactly $\operatorname{Eig}\_\lambda(M) \otimes H$ and similarly for generalized eigen... | 0 | https://mathoverflow.net/users/3041 | 402888 | 165,306 |
https://mathoverflow.net/questions/402808 | 3 | Let $\Omega$ be an algebraically closed field of characteristic $0$, $k$ a subfield such that $\mathrm{tr.deg}(\Omega/k)=\infty$. Let $u\_1,\dots,u\_n,u\_{n+1}\in \Omega$ be algebraically independent over $k$, $P$ be a prime(not maximal) ideal of $k[X\_1,\dots,X\_n]$. Does there exists $x\_1,\dots,x\_n\in \Omega$ such ... | https://mathoverflow.net/users/313627 | Existence of generic zeros | I found the answer from Serge Lang’s book. As I mentioned in the description of the question, take arbitrary $x\_1,\dots,x\_n\in \Omega$ such that $P$ is the kernel of $k(u\_1,\dots,u\_n)[X]\rightarrow \Omega, X\_i\mapsto x\_i$. Let $u\_{n+1}’:=u\_1x\_1+\cdots+u\_nx\_n$, then $u\_{n+1}’$ is algebraically independent ov... | 0 | https://mathoverflow.net/users/313627 | 402892 | 165,309 |
https://mathoverflow.net/questions/402910 | 6 | Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or disprove it. I suspect that this is true. Do you have a reference for this?
| https://mathoverflow.net/users/94414 | Absolute values of two functions and absolute values of their Fourier transform coincides | This is true. This was proved by Hardy and Littlewood and the proof is reproduced in Zygmund's Trigonometric Series (which I don't have access to at the moment).
(Contradicting my prior "answer")
The answer to this question is negative. Such counterexamples are known as "Pauli partners" and are studied in, among ot... | 9 | https://mathoverflow.net/users/630 | 402912 | 165,317 |
https://mathoverflow.net/questions/402919 | 4 | The question is in the title. It was asked by Paul Erdős, e.g. as part of Section 9 in this [paper](https://www.ime.usp.br/%7Eyoshi/resenhas/abstracts/Erdos.pdf).
| https://mathoverflow.net/users/174530 | Does the equation $\varphi(n)=\sigma(m)$ have infinitely many solutions? | Yes, this was proved by [Ford, Luca and Pomerance](https://faculty.math.illinois.edu/%7Eford/wwwpapers/phi=sig.pdf) in 2010 (paper in Bulletin of the London Math. Soc.).
| 19 | https://mathoverflow.net/users/38624 | 402920 | 165,318 |
https://mathoverflow.net/questions/402905 | 0 | I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the [*Ramanujan Prime*](https://mathworld.wolfram.com/RamanujanPrime.html) from the online encyclopedia Wolfram MathWorld. This week I wondered what about experimental mathematics concerning this sequence of... | https://mathoverflow.net/users/142929 | New experiments involving Ramanujan primes: Benford's law | If I am following what is being asked, the answer is no.
Set $R$ to be the set of Ramanujan primes. Let $R\_d$ be the set of Ramanujan primes with lead digit $d$. For a set of positive integers integers $S$, we'll write $S(x)$ to be the number of elements in S which are at most $x$. Then you are asking whether for an... | 4 | https://mathoverflow.net/users/127690 | 402935 | 165,324 |
https://mathoverflow.net/questions/402939 | 5 | It is known that with $M(x) = \sum\_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way to make this result effective in the following sense: show that for large $X$, there exists some $x\in [X, f(X)]$ s.t. $... | https://mathoverflow.net/users/40983 | Frequency of large values of the Mertens function | It was proved by Kaczorowski and Pintz (Acta Math. Hungar. 48 (1986), 173-185, doi: [10.1007/BF01949062](http://dx.doi.org/10.1007/BF01949062)) that there exists $x\in[X,X^{1+o(1)}]$ such that $M(x)\geq x^{\sigma-\varepsilon}$, and there also exists $x\in[X,X^{1+o(1)}]$ such that $M(x)\leq -x^{\sigma-\varepsilon}$. See... | 3 | https://mathoverflow.net/users/11919 | 402971 | 165,333 |
https://mathoverflow.net/questions/402929 | 24 | To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F\_E(x,y)\in E\_\*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and suppose I have an isomorphism of coefficient rings $\phi:E\_\*\rightarrow F\_\*$ that carries $F\_E(x,y)$ to $F\_F(x,y)... | https://mathoverflow.net/users/163893 | Are complex-oriented ring spectra determined by their formal group law? | The following is a communal answer from the algebraic topology Discord [[1](http://nodorek.net)], primarily put forward by Irakli Patchkoria (correcting previous half-answers by Tyler Lawson and me).
Kiran suggested it be recorded here to ease future reference.
The idea is to produce two topological realizations $M$,... | 20 | https://mathoverflow.net/users/1094 | 402987 | 165,338 |
https://mathoverflow.net/questions/402986 | 2 | I recently came across the paper *Les variétés de dimension 4 à signature non nulle dont
la courbure est harmonique sont d’Einstein* by Jean Pierre Bourguignon. What he shows in §8 is that the Weitzenböck curvature operator $\mathfrak{Ric}\_\text{R}$ on $p$-forms is given by $$\mathfrak{Ric}\_\text{R}(\omega)(X\_1,\dot... | https://mathoverflow.net/users/142232 | Invariant description of the Weitzenböck curvature operator by Bourguignon | I think the idea is to think of $\hat{R}\_p$ as a mapping from $\Lambda^pM$ to itself, and $\hat{\omega}$ as a mapping from $\Lambda^pM$ to $E$ (the vector bundle in which $\omega$ takes values), and then just compose these two operators (to get something $E$ valued in the end).
In the case where $E$ is the trivial (... | 1 | https://mathoverflow.net/users/3948 | 402989 | 165,339 |
https://mathoverflow.net/questions/402762 | 18 | Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{(x,y) \in \mathfrak{g} \times \mathfrak{g} \ | \ xy=yx\} .$$
The group $G$ acts on both of them by conjugation: I'd l... | https://mathoverflow.net/users/146464 | Variety of commuting matrices | There's been a good deal of work since the papers Mark Grant cited.
The rational cohomology of $\mathrm{Hom}(\mathbb{Z}^n,K)//K$ for $K$ a compact connected Lie group was computed by Stafa (<https://arxiv.org/abs/1705.01443>). It's a theorem of Florentino and Lawton that if $G$ is a linearly reductive Lie group with ... | 8 | https://mathoverflow.net/users/4042 | 402991 | 165,340 |
https://mathoverflow.net/questions/402992 | 1 | I have question about a statement from *Etale Cohomology and the Weil Conjecture* by Freitag, Kiehl
at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15
and is seemingly a consequence of following two facts.
Let $A$ be a *strict Henselian* ring (i.e. Henselian + residue c... | https://mathoverflow.net/users/108274 | Exactness of functor $ Et(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $ (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl ) | I think they meant to say that the functor $\mathcal F \mapsto \mathcal F(A)$ from sheaves on $A$ to abelian groups is exact.
The proof is:
If $\mathcal F\_2\to \mathcal F\_3 \to 0$ is an exact sequence, we need to show $\mathcal F\_2(A) \to \mathcal F\_3(A)$ is surjective.
Let $s \in \mathcal F\_3(A)$ be a secti... | 4 | https://mathoverflow.net/users/18060 | 402995 | 165,341 |
https://mathoverflow.net/questions/402994 | 11 | Often times, we consult resources, like Abramowitz and Stegun's Handbook of Mathematical Functions [https://www.math.ubc.ca/~cbm/aands/](https://www.math.ubc.ca/%7Ecbm/aands/), NIST's database on special functions <https://www.nist.gov/programs-projects/special-functions>, or Mathematica to find identities which aid us... | https://mathoverflow.net/users/115325 | How to determine if you've discovered a new identity for a special function | ***Q:*** Are there journals which would publish identities of classical functions?
***A:*** Elsevier's *Applied Mathematics and Computation* has published quite a number of papers in that category, see this [search listing](https://www.sciencedirect.com/search?qs=identities&pub=Applied%20Mathematics%20and%20Computati... | 11 | https://mathoverflow.net/users/11260 | 402998 | 165,343 |
https://mathoverflow.net/questions/402950 | 2 | Recall that a ternary $C^\*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle variable, $\|[a,a,a]\|= \|a\|^3$ and $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$
>
> Does there exist a unique $C^\*$-algeb... | https://mathoverflow.net/users/129638 | Trying to understand construction of $C^*$-algebra corresponding to a ternary $C^*$-ring from a paper | Yes, Proposition 3.2 from the article [MR0700979] that you link to gives you the answer for ternary $\rm C^\*$-rings. The reason for taking the opposite algebra is that the algebra $V$ acts on the right of $X$ rather than on the left. A right action is the same thing as a left action by the opposite algebra. You can se... | 1 | https://mathoverflow.net/users/351 | 403000 | 165,345 |
https://mathoverflow.net/questions/402993 | 9 | The permanent $\mathrm{per}(A)$ of a matrix $A$ of size $n\times n$ is defined to be:
$$\mathrm{per}(A)=\sum\_{\tau\in S\_n}\prod\_{j=1}^na\_{j,\tau(j)}.$$
Let
$$A=\left[\tan\pi\frac{j+k}n\right]\_{1\le j,k\le n-1},$$
$$B=\left[\sin\pi\frac{j+k}n\right]\_{1\le j,k\le n-1},$$
$$C=\left[\cos\pi\frac{j+k}n\right]\_{1\... | https://mathoverflow.net/users/287674 | Permanent identities | Let $\zeta$ be a primitive $n$-th root of unity. Then
$$\prod\_{j=1}^{n-1}(x-\zeta^j)=\frac{x^n-1}{x-1}=1+x+\cdots+x^{n-1}$$
and hence
$$\sigma\_k=\sum\_{1\le i\_1<\cdots<i\_k\le n-1}\zeta^{i\_1+\cdots+i\_k}=(-1)^k$$
for all $k=1,\ldots,n-1$.
Observe that
\begin{align\*}\mathrm{per}[1-\zeta^{j+k}]\_{1\le j,k\le n-1}=... | 10 | https://mathoverflow.net/users/124654 | 403004 | 165,348 |
https://mathoverflow.net/questions/403011 | 12 | I am looking for a proof of the following claim:
First define the function $\chi(n)$ as follows:
$$\chi(n)=\begin{cases}1, & \text{if }n \equiv \pm 1 \pmod{10} \\
-1, & \text{if }n \equiv \pm 3 \pmod{10} \\ 0, & \text{if otherwise }
\end{cases}$$
Then,
$$\frac{\pi^2}{5\sqrt{5}}=\displaystyle\sum\_{n=1}^{\infty}\fra... | https://mathoverflow.net/users/88804 | A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$ | More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger,
$$S\_{N,k} := \sum\_{n=0}^\infty\biggl( \frac{1}{(N n+N-k)^2} + \frac{1}{(N n+k)^2}
\biggr) = \frac{\pi^2}{N^2\sin^2(\pi k/N)}.$$
Your sum is $S\_{10,1}-S\_{10,3}$.
| 24 | https://mathoverflow.net/users/9025 | 403016 | 165,349 |
https://mathoverflow.net/questions/397768 | 3 | Suppose $x$ is a non-zero vector in a Banach space, and $T$ is a fixed operator. Is the following true:
For any $\varepsilon, \delta$, there exists $S$ in the commutant of $T$ such that $1\leq\|S\|<1+\delta$ and $\|Sx\|<\varepsilon$.
This is true for some $T$ (for example the identity), but is it true for *all* $T$... | https://mathoverflow.net/users/69275 | Operator in the commutant which is small on a given vector | Not even true for $2\times 2$ matrices. Let $T$ be the nilpotent $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ and $x$ be the vector $(0,1)$. Then anything in the commutant of $T$ has form $\lambda+\mu T$. So if $S$ is in the commutant of $T$, then $\|Sx\| = \sqrt{\lambda^2 + \mu^2}$, so if $\|Sx\| < \varepsilon$, then $\max(... | 5 | https://mathoverflow.net/users/356618 | 403031 | 165,355 |
https://mathoverflow.net/questions/403020 | 14 | Let $\mathcal{P}$ be a set of prime numbers of relative density $\kappa \in (0, 1)$, which means that
$$\#\left(\mathcal{P} \cap [1, x]\right) = \kappa \,\pi(x) + E(x) \quad (x \to \infty)$$
for a "suitable" error term $E(x)$ (of course, $E(x) = o(\pi(x))$).
Let $\mathcal{S}$ be the set of natural numbers having no p... | https://mathoverflow.net/users/342938 | Numbers without prime factors in a set of positive relative density | You basically ask about the sum
$$ \sum\_{n \le x} \alpha(n)$$
where $\alpha$ is a completely multiplicative function with $\alpha(p) = \mathbf{1}\_{p \notin \mathcal{P}}$.
This is addressed by Wirsing in his famous paper ``Das asymptotische Verhalten von Summen über multiplikative Funktionen'' (Math. Ann. 143 (1961)... | 20 | https://mathoverflow.net/users/31469 | 403033 | 165,356 |
https://mathoverflow.net/questions/403036 | 25 | According to Claude Shannon, von Neumann gave him very useful advice on what to call his measure of information content [1]:
>
> My greatest concern was what to call it. I thought of calling it 'information,' but the word was overly used, so I decided to call it 'uncertainty.' When I discussed it with John von Neum... | https://mathoverflow.net/users/56328 | John von Neumann's remark on entropy | An [alternative version](https://en.wikipedia.org/wiki/Talk%3AHistory_of_entropy) of Von Neumann's quote says *"no one understands entropy very well"*. At the intuitive level, this makes sense, it is much harder to explain the concept of entropy to a novice than it is to explain energy.
One debate that existed at the... | 18 | https://mathoverflow.net/users/11260 | 403040 | 165,359 |
https://mathoverflow.net/questions/403008 | 10 | Is there an increasing function on
$[a, b]$ which is differentiable,
but not absolutely continuous？
| https://mathoverflow.net/users/345221 | Is there an increasing function on $[a, b]$ which is differentiable, but not absolutely continuous？ | An elementary non-existence proof may be of interest.
Let $f:[a,b]\to\mathbb{R}$ an increasing, continuous, and not absolutely continuous function: I claim there exists a point $c\in[a,b]$ where the Dini derivative $D^\*f(c):=\limsup\_{x\to c}\frac{f(x)-f(c)}{x-c}$ is infinite. For the proof, we may assume w.l.o.g. t... | 13 | https://mathoverflow.net/users/6101 | 403041 | 165,360 |
https://mathoverflow.net/questions/403037 | 1 | $\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess that it can only be a constant map. Is it correct ?
| https://mathoverflow.net/users/356774 | Holomorphic map to Möbius group | Condition $U\subseteq \mathbb{C}^2$ can be replaced by $U\subseteq \mathbb{C}^1$ (just restrict your map
on a little disk in a complex line in $U$. Moreover, the restricted map lifts to a map into $SU(2)$. Then existence of a map to $SU(2)$
essentially means that you have two analytic functions satisfying
$|f|^2+|g|^2=... | 2 | https://mathoverflow.net/users/25510 | 403042 | 165,361 |
https://mathoverflow.net/questions/403039 | 2 | Consider the (complex) geometric realization of the motivic cohomology theory on simplicial presheaves over complex smooth schemes, which is a functorial homomorphism of $R$-modules, where $R$ is the coefficient ring:
\begin{equation}\label{eq}
\varphi: H^{s,t}\_{mot}(X;R)\to H^s(X(\mathbb{C});R).
\end{equation}
Whe... | https://mathoverflow.net/users/100553 | The multiplicativity of the (complex) geometric realization of motivic cohomology | Complex realization sends the motivic Eilenberg-MacLane spectrum to the classical Eilenberg-MacLane spectrum [Theorem 5.5 in Marc Levine's "A comparison of motivic and classical stable homotopy theories]. It is also a symmetric monoidal functor, hence preserves ring spectra. The result follows.
| 4 | https://mathoverflow.net/users/356850 | 403045 | 165,362 |
https://mathoverflow.net/questions/403043 | 3 | I am currently reading through differential geometry as a mathematics graduate.
Can somebody give me a brief explainer on the purpose of connections?
I could also use explainers on differential forms. Lie derivatives, and the tangent bundle generally.
I understand that you can have these structures, I just don't ... | https://mathoverflow.net/users/356829 | The purpose of connections in differential geometry | If you are interested in local-to-global results, i.e., collecting local info about the manifold and then patch it together to get a global info then you need tools for the patching part of the process. These often come under the guise of some form of integration.
Here is a simple example. If a compact surface admits... | 7 | https://mathoverflow.net/users/20302 | 403057 | 165,370 |
https://mathoverflow.net/questions/397198 | 13 | Let $G$ be a group acting highly transitively (and faithfully) on a set $S$. Suppose that $G$ is finitely presented, and that every stabilizer in $G$ of a finite subset of $S$ is finitely generated. I think I can prove that $G$ embeds in a finitely presented simple group, which in particular implies $G$ has decidable w... | https://mathoverflow.net/users/164670 | Decidability of word problem for group admitting certain action | Yes. There is such a reason.
I will write a subset of $G$ is RE if the set of those words over the generators for $G$ which represent elements of the subset is recursively enumerable.
As IJL argued, since $G$ is finitely presented the subset $\{1\}$ of $G$ containing only the identity is RE. It remains to show that... | 7 | https://mathoverflow.net/users/125391 | 403058 | 165,371 |
https://mathoverflow.net/questions/403053 | 1 | I am working on a research paper where I need to investigate conditions for the existence of probability distributions satisfying certain characteristics. I have already asked a related question ([here](https://mathoverflow.net/questions/401905/conditions-for-existence-of-a-distribution-with-full-support)), whose answe... | https://mathoverflow.net/users/42412 | From probability distribution in $\mathbb{R}^3$ to probability distribution in $\mathbb{R}^4$ | I think the answer to both questions is negative.
**Question A.**
You ask whether for all $G\in\mathcal{G}$, there exist a random vector $\eta=(\eta\_i)\_{1\le i\le 6}$ such that they satisfy Condition 1 together.
If $G$ were not assumed to be fully supported, the answer would be easily seen to be negative: taking ... | 2 | https://mathoverflow.net/users/4961 | 403059 | 165,372 |
https://mathoverflow.net/questions/403029 | 14 | In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With complex coefficients, a simple argument for this is to compute the cohomology in terms of the cohomology of the de Rham co... | https://mathoverflow.net/users/6074 | Artin vanishing for Stein manifolds and restriction maps | The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of
Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse
Theory.The title of the paper is "A Topological property of Runge pairs"
The paper by Coltoiu Mihalache titled On the Homology Groups of... | 13 | https://mathoverflow.net/users/4696 | 403065 | 165,375 |
https://mathoverflow.net/questions/403047 | 7 | Is there a homeomorphism $f:CP^2\to CP^2$ that has finitely many fixed points and acts by -1 on $H^2$?
| https://mathoverflow.net/users/356859 | Homeomorphism with finitely many fixed points acting by -1 on $H^2(CP^2)$ | Let $a\_1$, $a\_2$ and $a\_3$ be three distinct positive real numbers. Take the map
$$[z\_1 : z\_2 : z\_3] \mapsto [a\_1 \overline{z\_1}: a\_2 \overline{z\_2} : a\_3 \overline{z\_3}]$$
where the bar is complex conjugation. I claim that the only fixed points are $[1:0:0]$, $[0:1:0]$ and $[0:0:1]$.
Suppose that to the ... | 12 | https://mathoverflow.net/users/297 | 403066 | 165,376 |
https://mathoverflow.net/questions/403074 | 2 | Let $H$ and $H'$ be two Hopf algebras, and let $\phi:H \to H'$ be an bialgebra map. Then is $\phi$ automatically a Hopf algebra map?
| https://mathoverflow.net/users/352001 | Bialgebra maps and Hopf algebra maps | Yes it is.
It is actually a standard result, for any hopf algebra, that under the circumstances you are describing any bialgebra map $\phi$ commutes with the antipode, i.e. we have: $$S\_{H'}\circ \phi=\phi\circ S\_H.$$
(The simplest way to show this is to consider $\operatorname{Hom}(H,H')$ as an algebra —under the... | 3 | https://mathoverflow.net/users/85967 | 403075 | 165,378 |
https://mathoverflow.net/questions/402681 | 0 | I am studying properties of the **two-parameter** Mittag-Leffler function.
$$ E\_{\alpha,\beta}(z)=\sum\_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and relations, such as the duplication formulas.
However, I am seeking some relation in which **a product of two two-pa... | https://mathoverflow.net/users/172600 | When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function? | The product of two MLF is another MLF, only when $\alpha = \beta =1.$ In the other cases we obtain a similar formula but have to introduce generalized binomial coefficients. To show it, it is enough to use the rule used for multiplication of power series. The coefficients of the product are the discrete convolution of ... | 2 | https://mathoverflow.net/users/345734 | 403079 | 165,379 |
https://mathoverflow.net/questions/402842 | 24 | Let $P$ be a (convex, bounded) polytope with the following property: for every vertex $v$, there are exactly two facets which do not contain $v$. Does it follow that $P$ is (combinatorially) a Cartesian product of two simplices?
Some remarks
* by "facet" I mean "face of codimension $1$"
* a Cartesian product of two... | https://mathoverflow.net/users/908 | Polytope where each vertex belongs to all but two facets | There are other polytopes with this property that can be obtained via the *free join* construction.
Given two polytopes $P\_1\subset\Bbb R^{d\_1}$ and $P\_2\subset\Bbb R^{d\_2}$, the free join $P\_1\bowtie P\_2$ is obtained by embedding $P\_1$ and $P\_2$ into skew affine subspaces of $\Bbb R^{d\_1+d\_2+1}$ and taking... | 18 | https://mathoverflow.net/users/108884 | 403080 | 165,380 |
https://mathoverflow.net/questions/403023 | 7 | Setting
-------
Consider two independent orthogonal matrices, which are decomposed into 4 blocks:
\begin{align}
Q\_{1}
=
\left[\begin{array}{cc}
A\_{1} & B\_{1}\\
C\_{1} & D\_{1}
\end{array}\right]
,
\,Q\_{2}=\left[\begin{array}{cc}
A\_{2} & B\_{2}\\
C\_{2} & D\_{2}
\end{array}\right]\in \mathbb{R}^{d\times d}
\end... | https://mathoverflow.net/users/100796 | Proving a lemma for a decomposition of orthogonal matrices | Using the direction Federico Poloni gave, I was finally able to prove it.
Notice that the matrices are real, so I use $A^\top$ and $A^H$ interchangeably.
---
Let $v\in\mathbb{\mathbb{C}}^{d-r}$ be a normalized eigenvector as required.
* **Lemma:** The norm of $v$ is preserved under the following transformatio... | 3 | https://mathoverflow.net/users/100796 | 403096 | 165,385 |
https://mathoverflow.net/questions/403099 | 10 | In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell\_\infty$ are of the from $(a\_i) \mapsto (\varepsilon\_i a\_{\pi(i)})$ where $\pi$ is a permutation of $\mathbb{N}$ and $\varepsilon\_i \in\{\pm 1\}$. They mention that the proof is a consequence of t... | https://mathoverflow.net/users/15388 | Surjective linear isometries on $\ell_\infty(\mathbb{N})$ | I don't know a reference (other users will probably provide one). Here's a (quite immediate) proof anyway. I'll replace $\mathbf{N}$ with an arbitrary set $X$ since the integers play no role.
**Fact** The extremal points of the closed 1-ball of $\ell^\infty(X)$ precisely consists of the maps $X\to\{-1,1\}$.
Proof: ... | 14 | https://mathoverflow.net/users/14094 | 403100 | 165,386 |
https://mathoverflow.net/questions/403102 | 3 | Let $M^3$ be a compact $3$-manifold such that $\pi\_1(M)$ contains a normal subgroup isomorphic to $\mathbb Z$.
Can we show either $\pi\_1(M)$ is torsion-free or $\pi\_1(M)=\mathbb Z \oplus \mathbb Z\_2$ or $\mathbb Z\_2 \* \mathbb Z\_2$?
| https://mathoverflow.net/users/280895 | $\pi_1(M^3)$ containing a normal infinite cyclic subgroup | The answer is "no" (see below for an example) but it is almost "yes". If $M$ does not have any real projective plane boundary components then this follows from Theorem 7 of the paper [A survey on Seifert fibre space conjecture](https://arxiv.org/abs/1202.4142) by Jean-Philippe Préaux.
In general, one has to understan... | 6 | https://mathoverflow.net/users/1650 | 403105 | 165,388 |
https://mathoverflow.net/questions/403073 | 5 | I'm looking for the following:
(1) an example of a $\Pi^1\_1$ class-theoretic sentence that has no known equivalence to a $\Sigma^1\_1$ sentence, even if large cardinal hypotheses or reflection principles are assumed.
(2) an example of a $\Pi^1\_2$ class-theoretic sentence that has no known equivalence to a $\Sigma... | https://mathoverflow.net/users/170446 | Class-theoretic sentences that are $\Pi^1_1$ or $\Pi^1_2$ | Much as described by @ElliotGlazer above, let's consider the question for models of the form $(V\_\kappa,V\_{\kappa+1})$, assuming ZFC in the background, considering all such where $\kappa$ is inaccessible in the background.
Then *with respect to this particular class of models*, a $\Pi^1\_2$, non-$\Sigma^1\_2$ state... | 4 | https://mathoverflow.net/users/160347 | 403108 | 165,389 |
https://mathoverflow.net/questions/403110 | 0 | Is there a lower bound on the slope (i.e. ratio of degree to rank) of normal bundles of smooth projective curves embedded in smooth projective varieties?
| https://mathoverflow.net/users/357747 | Are the slopes of normal bundles of curves bounded from below? | No. For instance the Hirzebruch surface $F\_n$ contains a smooth rational curve with normal bundle $\mathcal{O}(-n)$ of slope $-n$.
| 3 | https://mathoverflow.net/users/4428 | 403113 | 165,392 |
https://mathoverflow.net/questions/403118 | 26 | **Question 1:** Let $C$ be a small category. Does there exist a poset $P$, a set of $W$ of morphisms in $P$, and an equivalence $P[W^{-1}] \simeq C$?
Here $P[W^{-1}]$ is the universal category receiving a functor from $P$ which carries each morphism of $W$ to an isomorphism.
I'm hoping for an affirmative answer. I'... | https://mathoverflow.net/users/2362 | Is every category a localization of a poset? | Yes, this is true. It follows form the work of Barwick and Kan on relative categories as a model for $\infty$-categories.
The idea is similar to how Thomason's work shows that every homotopy type can be modeled by a poset (by taking subdivisions of the category of simplicies of the simplicial set).
Specifically in ... | 23 | https://mathoverflow.net/users/184 | 403122 | 165,394 |
https://mathoverflow.net/questions/342846 | 1 |
>
> I [know](https://mathoverflow.net/a/136067/143814) we can't formalize the Kunen inconsistency as an assertion in the first-order language of set theory. But
> [the wikipedia say](https://en.wikipedia.org/wiki/New_Foundations#Strong_axioms_of_infinity) "**Con([MK](https://en.wikipedia.org/wiki/Morse%E2%80%93Kelle... | https://mathoverflow.net/users/143814 | Can we use Kunen's inconsistency theorem in NFU+AC? | In general, NFU + AC is going to have exactly the same possible large cardinals as ordinary set theory. NFU + AC has the same stratified mathematics as ordinary set theory. There will be no interesting results of this kind, unless they involve unstratified assertions in the language of NFU.
I should be slightly more ... | 0 | https://mathoverflow.net/users/345616 | 403142 | 165,400 |
https://mathoverflow.net/questions/342915 | 0 | J. D. Hamkins proved in ["The foundation axiom and elementary self-embeddings of the universe"](http://jdh.hamkins.org/foundation-axiom-and-self-embeddings-of-the-universe/) that, working in $ZFGC^− +BAFA$, there are nontrivial automorphisms and elementary embeddings of the universe **V** into itself.
But, he say "so... | https://mathoverflow.net/users/143814 | Is there no anti-foundational theory exists Reinhardt and hold Global Choice? | In general, anti-foundation axioms have nothing to do with issues of large cardinal consistency strength. You can take your universe of set theory with foundation and convert it into a universe with your favorite antifoundation axiom and no large cardinals will be created or destroyed. This process is reversible: large... | 3 | https://mathoverflow.net/users/345616 | 403143 | 165,401 |
https://mathoverflow.net/questions/403139 | 12 | In algebraic number theory, we constantly make use of the nine-term *Poitou-Tate* sequence: Let $K$ be a number field and $M$ a finite $K$-Galois module. Then we have the nine-term exact sequence
$$
H^0(K, M) \to \prod' H^0(K\_v,M) \to H^2(K, M^\vee)^\vee \mathop{\to}\limits^\delta H^1(K, M) \mathop{\to}^{\operatorname... | https://mathoverflow.net/users/105625 | Finite Galois module whose Ш¹ is nonzero? | Wang's conterexample to Grunwald's theorem: $K=\mathbb{Q}(\sqrt{7})$ and $M=\mu\_8$. Then $H^1(K,M) \cong K^\times/(K^\times)^8$. Now $16$ is not an $8$-th power in this field but locally an $8$-th power everywhere. Your group is cyclic of order $2$. See [wikipedia](https://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_t... | 12 | https://mathoverflow.net/users/5015 | 403145 | 165,402 |
https://mathoverflow.net/questions/403146 | 7 | There is a classification of simple Lie algebras in $\text{Vec}\_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones.
**Question**: How about simple Lie algebras in the bigger category $\text{sVec}\_{\mathbb{C}}$ of super vector spaces? Is there a known c... | https://mathoverflow.net/users/41644 | Semisimple super Lie algebras | Yes there is a complete classification of finite dimensional, simple Lie superalgebras (over $\mathbb{C}$), which -up to a certain extent- goes very much in parallel with the corresponding case of Lie algebras and incorporates the later as a special case. There are significant conceptual differences though (as to the r... | 12 | https://mathoverflow.net/users/85967 | 403150 | 165,403 |
https://mathoverflow.net/questions/403140 | 0 | If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kappa \neq \emptyset$ is a cardinal, then a map $c:V\to\kappa$ is called a *coloring* if the restriction $c\restriction\_e$ is non-constant for all $e\in E$ with $|e|\geq 2$. The smallest cardinal $\kappa$ for which a coloring $c:V\to \kappa... | https://mathoverflow.net/users/8628 | Discrepancy of chromatic number and independent covering number for $k$-regular hypergraphs | No, this is false already for $k=3$. Let $A$ and $B$ be disjoint sets of size $n$, and let $H$ be the hypergraph with vertex set $A \cup B$, whose hyperedges are all $3$-subsets $e$ of $A \cup B$ such that $|e \cap A| \in \{1, 2\}$. Note that $\chi(H)=2$, since we can colour all vertices in $A$ red and all vertices in ... | 2 | https://mathoverflow.net/users/2233 | 403155 | 165,404 |
https://mathoverflow.net/questions/403101 | 3 | A $3$-manifold $M$ is called $P^2$-irreducible if it is irreducible and there is no $2$-sided $P^2$ contained in $M$.
Can we show $M$ is $P^2$-irreducible iff $\pi\_2(M)=0$?
Notice that one direction follows directly from the Sphere theorem.
| https://mathoverflow.net/users/280895 | $P^2$-irreducibility of a $3$-manifold | Yes.
One direction is immediate by the Sphere theorem (projective plane theorem) as pointed out by the OP.
Assume $M$ satisfies $\pi\_2(M)=0$. Note that this condition with the Poincare conjecture means any sphere in $M$ bounds a $3$-ball (see [here](https://math.stackexchange.com/questions/1494524/homotopically-tr... | 3 | https://mathoverflow.net/users/16323 | 403165 | 165,406 |
https://mathoverflow.net/questions/403175 | 5 | A "randomly chosen" 2-generated dense subgroup
$$
G \ = \ \langle a, b \rangle \ < \
{\rm A}\_5 \times {\rm A}\_6 \times {\rm A}\_7 \times \dots
$$
of the cartesian direct product of the finite simple alternating groups
is "typically" free of rank 2. The "interesting" cases are those where
such group $G$ is *not* fre... | https://mathoverflow.net/users/28104 | Quotients of a 2-generated dense subgroup of a Cartesian product of infinitely many finite alternating groups | A very close variant of this group (the one in the question) was introduced and studied by B.H. Neumann (Neumann, B. H. Some remarks on infinite groups. J. Lond. Math. Soc. 12, 120-127 (1937).).
Namely, it's the same group, but restricting to the product of $A\_n$ for odd; this shouldn't make much difference.
This ... | 6 | https://mathoverflow.net/users/14094 | 403176 | 165,408 |
https://mathoverflow.net/questions/402985 | 20 | I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral over $(2n+1)$ variables $u\_1 \cdots u\_{2n+1}$
\begin{equation}
\mathcal{I}\_{2n} = \int\_0^1 du\_1 \cdots \int\_0^1 du... | https://mathoverflow.net/users/106463 | A multiple integral that seems related to the $\zeta$ function at even integers | We have $$I\_{2n}=\frac{(2n)!!}{(2n+1)!!}\cdot \frac1{2n+2}\cdot \pi^{2n}.$$
To see this, we follow the suggestion by Terry Tao in the comments and apply the diagonalization of the integral operator with the kernel $1/(x+y)$ on $[0,1]$. Change the variable to $1/x\in [1,\infty)$ and use (1.18) [here](https://www.ams.or... | 12 | https://mathoverflow.net/users/4312 | 403179 | 165,411 |
https://mathoverflow.net/questions/401964 | 6 | First, consider group extensions with non-abelian kernel
$$1\to K\to G \to Q \to 1$$
It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $\omega: Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}... | https://mathoverflow.net/users/3041 | Connection between the classifications of group extensions and group-graded algebras in terms of non-abelian cohomology | In the meantime I found the right nlab pages to read... The answer seems to be [2-groups](https://ncatlab.org/nlab/show/2-group)! Specifically [automorphism 2-groups](https://ncatlab.org/nlab/show/2-group#automorphism_2groups). I will write up what I have come to understand so far (though I have not checked every last ... | 1 | https://mathoverflow.net/users/3041 | 403180 | 165,412 |
https://mathoverflow.net/questions/403181 | 1 | Let $\Omega\_1$, $\Omega\_2$ be bounded,convex, open domains with smooth boundary in $\mathbb{R}^2$ and $\overline\Omega\_1\subset\Omega\_2$. Suppose we are given a $C^1$ function $f:\overline\Omega\_1\cup(\mathbb{R}^2\setminus\Omega\_2)\rightarrow\mathbb{R}$ satisfies the following properties:
(1)$f\equiv 1$ on $\pa... | https://mathoverflow.net/users/157939 | Looking for a reference for an extension problem of function | The boundary of a convex domain is a Jordan curve (this is the only property of
the boundaries that will be used). Then $\Omega\_2\backslash\Omega\_1$ is a topological ring, and by the well known theorem, there is a conformal map
$\phi:\Omega\_2\backslash\Omega\_1\to A$, where $A=\{ z:r<|z|<1\}$, for some $r\in(0,1)$. ... | 1 | https://mathoverflow.net/users/25510 | 403182 | 165,413 |
https://mathoverflow.net/questions/403184 | 63 | A (non-mathematical) friend recently asked me the following question:
>
> Does the golden ratio play any role in contemporary mathematics?
>
>
>
I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for almost every other mathemat... | https://mathoverflow.net/users/352001 | Golden ratio in contemporary mathematics | The "Cleary group" $F\_\tau$ is a version of Thompson's group $F$, introduced by Sean Cleary, that is defined using the golden ratio, and it's definitely of interest in the world of Thompson's groups. See *[An Irrational-slope Thompson's Group](https://arxiv.org/abs/1806.00108)* ( Publ. Mat. 65(2): 809-839 (2021). DOI:... | 47 | https://mathoverflow.net/users/164670 | 403186 | 165,415 |
https://mathoverflow.net/questions/403169 | 2 | Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? Or under what condition this can be true? I... | https://mathoverflow.net/users/165074 | Rank of sumsets in matroids | Without any condition on the matroid structure, there is really no reason for your inequality to hold.
For example, take $X=\{a\},Y=\{b\}$ so that $X+Y=\{a+b\}$ where $a,b$ are any nonzero elements of $G$.
Then take any matroid structure such that $a+b$ is a loop but $a,b$ and $0$ are not. We have $0=r(X+Y)<r(X)+r(Y)... | 2 | https://mathoverflow.net/users/160416 | 403192 | 165,417 |
https://mathoverflow.net/questions/403172 | 6 | $\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
[Taras Banakh](https://mathoverflow.net/users/61536/taras-banakh) and me proceed a long quest answering a [question](https://math.stackexchange.com/questions/1203722/approximate-vanishing-in-pontryagin... | https://mathoverflow.net/users/43954 | A property of rapid sequences of natural numbers | A sequence is called *lacunary* if, in your terminology, its minimum growth rate is strictly greater than $1$. The following articles prove that every lacunary sequence is remote. If I understand your question correctly, this means that the (only) maximal $M$ you seek is the interval $(1,\infty)$.
*Pollington, Andrew... | 4 | https://mathoverflow.net/users/10457 | 403197 | 165,419 |
https://mathoverflow.net/questions/403187 | 3 | Define a structure made of objects $A, B, C, \dots$ and morphisms $f, g, \dots$. Each morphism has *a collection of* domain objects and codomain objects. For simplicity we consider the domains and codomains as unordered sets. We require that
* For each $A$ there exists an $f : A \to A$.
* For every $f:B\_1, B\_2, \do... | https://mathoverflow.net/users/136535 | What's the terminology for a sequent-like variant of category? | It sounds like you want the notion of a [polycategory](https://ncatlab.org/nlab/show/polycategory).
| 6 | https://mathoverflow.net/users/126667 | 403204 | 165,422 |
https://mathoverflow.net/questions/403166 | 5 |
>
> $\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\
> \begin{eqnarray}
> \sqrt{2}\left\|\triangledown u\right\|\_{L^2(\Omega)}\leq \left\|\triangledown u+(\triangledown u)^T\right\|\_{L^2(\Omega)}
> \end{eqnarray}
> for an... | https://mathoverflow.net/users/241460 | How to prove the second Korn inequality? | You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work... | 5 | https://mathoverflow.net/users/113756 | 403211 | 165,425 |
https://mathoverflow.net/questions/403218 | 8 | Is there a known explicit description of the [abelian $2$-group](https://en.wikipedia.org/wiki/Abelian_2-group) $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi\_{\leq1}(QS^0)$?
| https://mathoverflow.net/users/130058 | What is the homotopy category of the sphere spectrum? | This is the groupoid given by the 1-truncation $\tau\_{\leq 1}(QS^0)$. This groupoid has $\mathbb Z$-many objects (since $\pi\_0^s = \mathbb Z$), and each one has automorphism group $C\_2$ (since $\pi\_1^s = C\_2$). The tensor product on objects is given by addition in $\mathbb Z$, and on morphisms by addition in $C\_2... | 12 | https://mathoverflow.net/users/2362 | 403221 | 165,430 |
https://mathoverflow.net/questions/403222 | -3 | I was looking for a natural power of 3 that could be written like
>
> Binary format:
>
>
> 11..(N times)..11011..(M times)..11
>
>
> Example: 1111110111111111111111 (...isn't a power of 3)
>
>
>
Or could also be written like
>
> 3^x = 2^a - 2^b - 1
>
>
> (x is arbitrary, "a" and "b" are natural numbe... | https://mathoverflow.net/users/359116 | Prove that the equation $2^a - 2^b - 1=3^c$ has no integral solution with $a,b\geq 3$ | The equation $2^a-2^b-1=3^c$ has no integral solution with $a,b\geq 3$. Indeed, in this case the left-hand side is $\equiv 7\pmod{8}$, while the right-hand side is either $\equiv 1\pmod{8}$ or $\equiv 3\pmod{8}$.
| 3 | https://mathoverflow.net/users/11919 | 403226 | 165,433 |
https://mathoverflow.net/questions/402936 | 2 | I wish to have a proof for the following result:
Let $U\_n$ be an $n\times n$ upper [shift matrix](https://en.wikipedia.org/wiki/Shift_matrix), and $L\_n = U\_n^T$ be a lower shift matrix. For example,
$$
U\_5 = \begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 ... | https://mathoverflow.net/users/113258 | Prove: Lie algebra generated by two $n\times n$ shift matrices is $\mathfrak{so}(n,\mathbb{C})$ ($n$ odd) or $\mathfrak{sp}(n,\mathbb{C})$ ($n$ even) | Let ${\mathfrak g} $ be the Lie algebra generated by $U\_n$ and $L\_n$. It's easy to check that $U\_n$ and $L\_n$ preserve the bilinear form determined by the matrix with $1,-1,1,\ldots$ down the antidiagonal and zero elsewhere. So ${\mathfrak g}$ is contained in (a Lie algebra isomorphic to) $\mathfrak{so} \_n$ for od... | 4 | https://mathoverflow.net/users/26635 | 403240 | 165,443 |
https://mathoverflow.net/questions/403242 | 4 | Let $G$ be a group quasi-isometric to the fundamental group of a genus 2 surface group $H$. It is well known that $G$ is quasi-isometrically rigid, i.e. $G$ and $H$ are virtually isomorphic. Does the stronger property, that $G$ and $H$ are commensurable, also hold?
If so is there a reference for this?
| https://mathoverflow.net/users/121307 | Quasi-isometric rigidity of surface groups and commensurability | Yes (I assume that by "virtually isomorphic" you mean commensurable modulo finite kernels, which is a nonstandard misleading use of "virtually"). This is because surface groups have Serre's property D$\_2$ meaning that each 2-cohomology class (in a finite abelian group with trivial action) it trivial on some finite ind... | 8 | https://mathoverflow.net/users/14094 | 403243 | 165,444 |
https://mathoverflow.net/questions/379869 | 4 | $\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$. I'm wondering whether this generalizes to the case where $\GL(V)$ is replaced by an arbitrary reductive group?
More ... | https://mathoverflow.net/users/69713 | Centralizers of semisimple subgroups | In characteristic zero, the connected closed subgroups of $G$ are in 1-1 correspondence with the Lie subalgebras of ${\rm Lie}(G)={\mathfrak g}$, and the Killing form a suitably chosen symmetric bilinear $G$-equivariant form on ${\mathfrak g}$ is non-degenerate. *In fact we can choose this form to be the trace form for... | 3 | https://mathoverflow.net/users/26635 | 403246 | 165,447 |
https://mathoverflow.net/questions/402257 | 7 | Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta\_+$, the augmented simplex category. Because Day convolution allows us to lift monoidal structures on a category $\mathcal{C}$ to its category of presheaves $\mathrm{Sets}^{\Delta^\circ}$, it is th... | https://mathoverflow.net/users/130058 | Is $\oplus$ the only monoidal structure on the simplex category? | Here is half of a classification. Let $\otimes$ be a monoidal structure on $\Delta\_+$. As I mentioned in a [comment](https://mathoverflow.net/questions/402257/is-oplus-the-only-monoidal-structure-on-the-simplex-category#comment1032524_402257), the monoidal unit must be $[-1]$ or $[0]$ because these are the only object... | 6 | https://mathoverflow.net/users/2362 | 403250 | 165,448 |
https://mathoverflow.net/questions/403217 | 8 | $\mathbb{Z}$-graded rings play an important role in algebra and algebraic geometry, so when moving to derived algebra and spectral algebraic geometry, it's natural to ask about ring spectra graded in the sphere spectrum $\mathbb{S}$.
As discussed in Section 2 of [Bunke–Nikolaus's *Twisted differential cohomology*](ht... | https://mathoverflow.net/users/130058 | Grading ring spectra over the sphere spectrum | One of the default examples of ordinary graded commutative rings is the polynomial ring $\mathbf Z[t]$. Let us first examine the analogue of that, and then see where else that leads!
**1. $S$-grading on $S\{t\}$**
For the sake of clarity, allow me to denote the underlying infinite loop space (equivalently: grouplik... | 10 | https://mathoverflow.net/users/39713 | 403254 | 165,450 |
https://mathoverflow.net/questions/403235 | 3 |
>
> Is there any integral expression for $\log (X + Y) - \log (X)$ if $X$ and $Y$ are positive definite matrices?
>
>
>
Could anyone give some suggestion as to how to find such an integral expression if there is any?
Thanks a bunch.
| https://mathoverflow.net/users/352167 | Question on integral expression of positive definite matrices | The [formula](https://math.stackexchange.com/a/2085547/87355)
$$\frac{d}{ds}\log Z(s) = \int\_0^1 [(1-t)I+tZ(s)]^{-1}Z'(s) [(1-t)I+tZ(s)]^{-1}\, dt,$$
with $Z(s)=X+sY$, gives upon integration of
$$\int\_0^1 \frac{d}{ds}\log Z(s)\,ds=\log Z(1)-\log Z(0)$$
an integral expression for
$$\log(X+Y)-\log X=\int\_0^1\int\_0^1 ... | 3 | https://mathoverflow.net/users/11260 | 403258 | 165,452 |
https://mathoverflow.net/questions/403272 | 7 | Given a ring spectrum $R$ and an $R$-module $E$, we have the [spectral symmetric algebra](https://ncatlab.org/nlab/show/spectral+symmetric+algebra) $\mathrm{Sym}\_R(E)$ of $E$ over $R$, defined by
$$
\begin{align\*}
\mathrm{Sym}\_R(E) &\overset{\mathrm{def}}{=} \mathrm{colim}\_{\mathbb{F}}(\Delta\_{E})\\
&\cong \bigop... | https://mathoverflow.net/users/130058 | Is there a "spectral exterior algebra" construction in higher algebra? | Interesting question! I can't give a real answer, but here are some idle musings:
Note that one way to encode exterior powers is as $\Lambda^i\_R(E) = \Sigma^{-i}(\mathrm{Sym}^i\_R(\Sigma(E)))$ (since placing the generators in degree one switches, by virtue of the Koszul sign rule from the usual $\Sigma\_n$-action to... | 6 | https://mathoverflow.net/users/39713 | 403274 | 165,460 |
https://mathoverflow.net/questions/403275 | 29 | Let $\mathrm{SO}(3)$ be the group of rotations of $\mathbb{R}^3$ and let $S\_\infty$ be the group of all permutations of $\mathbb{N}$. Is $\mathrm{SO}(3)$ isomorphic to a subgroup of $S\_\infty$?
This question is due to Ulam. It is discussed in V.2 of Ulam's "A collection of mathematical problems". It is also discuss... | https://mathoverflow.net/users/152899 | Does $\mathrm{SO}(3)$ act faithfully on a countable set? | From MathSciNet:
Thomas, Simon Infinite products of finite simple groups. II.
J. Group Theory 2 (1999), no. 4, 401–434.
Summary: "(...) In the course of our classification proof, we also show that if $K$ is a field of cardinality $2^\omega$ and $G$ is a non-trivial linear group over $K$, then there exists a subgrou... | 16 | https://mathoverflow.net/users/14094 | 403299 | 165,469 |
https://mathoverflow.net/questions/403294 | 4 | $\DeclareMathOperator\Gr{Gr}$Let $E$ be a real, finite dimensional vector space of dimension $n$. Let $\Gr(k)$ be the set of linear subspaces of dimension $k$ of $E$. I am wondering what structures the manifold $\Gr(k)$ inherits from $E$, beyond differentiability. When $E$ is Euclidean, $\Gr(k)$ gets a Riemannian metri... | https://mathoverflow.net/users/176470 | Grassmannians on a vector space without metric | $\DeclareMathOperator\Gr{Gr}$I discussed this in my thesis.
**Lemma:** Every tangent space of the Grassmannian is a tensor product $T\_P \Gr(k)=P^\* \otimes (E/P)$; these isomorphisms are invariant under the general linear group $\operatorname{GL}\_E$.
**Proof:** Take a smooth path $P(t)\in \Gr(k)$. Write each $P(t... | 14 | https://mathoverflow.net/users/13268 | 403301 | 165,471 |
https://mathoverflow.net/questions/403304 | 2 | Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1\_m 1\_m^\top + b XX^\top + c I\_m$.
>
> **Question 1.** In the limit $m,d \to \infty$ with $m/d \to \rho \in (0,\infty)$, what does $d^{-1}\mbox{tr... | https://mathoverflow.net/users/78539 | Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix | Let me try to answer the updated question. If $A$ and $B$ are Hermitian $m\times m$ matrices with $B$ of rank $r$ then the two sequences of eigenvalues $\lambda\_k(A+B)$ of $A+B$ and $\lambda\_k(A)$ of $A$, each sorted in ascending order, are related by
$$\lambda\_k(A)\leq\lambda\_{k+r}(A+B)\leq\lambda\_{k+2r}(A),\;\;1... | 2 | https://mathoverflow.net/users/11260 | 403312 | 165,475 |
https://mathoverflow.net/questions/403313 | 3 | In an exercise of Voisin book, says:
Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set
$H=ker(j\_\*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$.
We also write $A\subset J(C)$ for the Abelian subvariety corresponding to the Hodge substructure $H$.
I do not un... | https://mathoverflow.net/users/150116 | Abelian varieties corresponding to Hodge substructures | Hodge substructures of $H^1(C,\mathbb Z)$ and Abelian subvarieties of $J(C)$ ar essentially the same thing:
In general, let $V,\omega$ be a free $\mathbb Z$-module of even rank with an integral symplectic form. A *polarized Hodge structure of weight 1* on $V$ is the data of a decomposition
$V\_{\mathbb C} := V\otimes... | 5 | https://mathoverflow.net/users/173096 | 403317 | 165,477 |
https://mathoverflow.net/questions/402940 | 3 | The Auslander correspondence said there exists a bijection between the set of Morita-equivalence classes of representation-finite finite-dimensional algebras $\Lambda$ and that of finite-dimensional algebras $\Gamma$ with gl.dim $\Gamma \leq 2 \leq$ dom.dim $\Gamma$. It is given by $\Lambda \mapsto \Gamma:=\operatornam... | https://mathoverflow.net/users/118028 | Why is Auslander correspondence a bijection between the set of Morita-equivalence classes? | Let me answer your second question about the importance of Morita-equivalence classes first. Two rings $R$ and $S$ are called Morita equivalent if $\operatorname{Mod} R$ and $\operatorname{Mod} S$ are equivalent as categories. If one considers representation theory as the study modules over a ring, then in some sense $... | 2 | https://mathoverflow.net/users/15887 | 403320 | 165,479 |
https://mathoverflow.net/questions/403310 | 9 | In the context of [another MO question](https://mathoverflow.net/q/400714), the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial equation?
Failing that, has anyone written down a detailed description of such an... | https://mathoverflow.net/users/3106 | Software for detecting Brauer-Manin obstructions? | I strongly disagree with the assertion that "the language of modern algebraic geometry [...] is unfamiliar to many people who might otherwise have the right skills to write the software". You are denying the existence of a flourishing research field, computational arithmetic geometry! See e.g. this paper:
*Bright, M.... | 11 | https://mathoverflow.net/users/2481 | 403323 | 165,480 |
https://mathoverflow.net/questions/204790 | 17 | So this question probably shows my inner model theoretic ignorance, but:
In "Two remarks on elementary embeddings of the universe" (<http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567>), Jech defines - for $j: V\rightarrow M$ a definable-with-parameters elementary embedding - an inner model $L(j)$, and pro... | https://mathoverflow.net/users/8133 | Whatever happened to $L(j)$? | I recently noticed that Mitchell returned to $L[j]$ in ["Applications of the covering lemma for sequences of measures,"](https://www.ams.org/journals/tran/1987-299-01/S0002-9947-1987-0869398-2/S0002-9947-1987-0869398-2.pdf) where he shows that the model is quite a bit larger than one might expect. The thing is, $L[j]$ ... | 6 | https://mathoverflow.net/users/102684 | 403329 | 165,481 |
https://mathoverflow.net/questions/403321 | 1 | Let $$r\_k(x)=\prod\_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$
Computations suggest that $$r\_{2k}(x)=\sum\_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r\_{2k+1}(x)=\sum\_{j=0}^{k^2-k}a(2k+1,j)\binom{k^2+k+x-j}{k^2+k},$$ where the coefficients $a(k,j)$ are positive, palindromic and gamma positive.
Is there an... | https://mathoverflow.net/users/5585 | Worpitzky-like identities? | Upgrading my comment to an answer. It is not hard to see that $r\_k(x)$ is the same as MacMahon's famous formula for the number of [plane partitions](https://en.wikipedia.org/wiki/Plane_partition) in a $\lfloor k/2 \rfloor \times \lceil k/2 \rceil \times x$ box. Then the same argument as in [my previous answer](https:/... | 3 | https://mathoverflow.net/users/25028 | 403330 | 165,482 |
https://mathoverflow.net/questions/300151 | 5 | **Problem.** Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane?
**Remark.** The results of [this paper](https://arxiv.org/pdf/1805.01997) imply the affirmative answer to this problem for compact ... | https://mathoverflow.net/users/61536 | A compact set in the plane with small sum-set and large projections | The answer is negative. The key point is the following
**Lemma:** For every $L>0$ and $x\in\mathbb R^2, x\ne 0$, there exist $\ell>0$ and a closed origin-symmetric set $F\subset\mathbb R^2$ such that every interval $I$ of length $L$ on the plane contains a subinterval $J\subset I\cap F$ of length $\ell$ while $F\cap(... | 3 | https://mathoverflow.net/users/1131 | 403358 | 165,488 |
https://mathoverflow.net/questions/403348 | 18 | I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples.
Is the underlying condensed abelian group of a finitely generated module (with the unique induced topology) over a Banach ring solid? Do yo... | https://mathoverflow.net/users/118220 | Examples of solid abelian groups | Here's a rule of thumb: As long as the construction is nonarchimedean and does not involve noncompleted tensor products, it's solid.
More precisely, anything you can build from discrete abelian groups by repeatedly forming limits (in particular, kernels), colimits (in particular, cokernels), extensions, and internal ... | 31 | https://mathoverflow.net/users/6074 | 403371 | 165,491 |
https://mathoverflow.net/questions/313956 | 5 | In two important papers of Wirsing, namely ["Das asymptotische Verhalten von Summen über multiplikative Funktionen"](https://link.springer.com/article/10.1007/BF01351892) (1961) and its [follow up](https://akademiai.com/doi/abs/10.1007/BF02280301?pubCode=mobile&journalCode=10473) (1967), several results on mean values ... | https://mathoverflow.net/users/31469 | Uniformity in Wirsing's Mean Value Theorems | Effective versions of Wirsing's theorems are worked out in a recent paper of G. Tenebaum: ``Moyennes effectives de fonctions multiplicatives complexes'', published in Ramanujan J. 44 (2017), no. 3, 641–701. Questions 1 and 2 are resolved by his Théorème 1.3. He shows that error term is uniform in the sense that it depe... | 0 | https://mathoverflow.net/users/31469 | 403388 | 165,499 |
https://mathoverflow.net/questions/403389 | 9 | Formula (12) in the paper
* Bauer, M., Chetrite, R., Ebrahimi-Fard, K., & Patras, F. (2013).
Time-ordering and a generalized Magnus expansion. Letters in
Mathematical Physics, 103(3), 331-350.
expresses the derivative of an exponential of a parameter-dependent Lie algebra element. The authors call it 'Duhamel's for... | https://mathoverflow.net/users/56920 | Duhamel's formula | Duhamel's formula for the derivative of the exponent of a matrix refers to [Jean-Marie-Constant Duhamel](https://en.wikipedia.org/wiki/Jean-Marie_Duhamel), who described it in [Eléments de calcul infinitésimal](http://catalogue.bnf.fr/ark:/12148/cb3037024) (volume 2, 1856; page 36) as a method to obtain solutions to in... | 11 | https://mathoverflow.net/users/11260 | 403404 | 165,504 |
https://mathoverflow.net/questions/403381 | 4 | Let $C$ be a connected simplicial 2-complex, and $f: C \to \mathbb{S}^3$ an embedding in the 3-sphere. Assume that each of the link graphs of $C$ is connected, and that $f$ nice, e.g. locally flat or piecewise linear.
Is it true that there is a *simply connected* 2-complex $C'$, containing $C$ as a topological subspa... | https://mathoverflow.net/users/69681 | Extending a 2-complex embedded in $\mathbb{S}^3$ into a simply connected one | Yes. [The tameness of $f$ makes our life *much* easier.]
It will be convenient for the proof to assume that $C$ is *pure*: that is, every vertex and edge of $C$ lies in some triangle of $C$. If this is not the case, then glue triangles on until it is the case.
Take $D = f(C)$; since $f$ is piecewise linear $D$ is t... | 5 | https://mathoverflow.net/users/1650 | 403405 | 165,505 |
https://mathoverflow.net/questions/403345 | 3 | Fix a field $k$ of characteristic zero, and let $G$ be a connected reductive algebraic $k$-group of isotropic rank $\ge 1$. Fix a maximal $k$-split torus $S$, and let $\Phi\_k$ be the relative root system of $G$ with respect to $S$. Assume that $\Phi\_k$ is reduced and irreducible.
By Theorem 2 of Petrov and Stavrova... | https://mathoverflow.net/users/361094 | Conjugation of root subgroups by the Weyl group | Let me use $a$ and $b$ for relative roots, so that I can later switch to $\alpha$ and $\beta$ for absolute roots.
If $b$ is a non-multipliable root, then, as you have [said](https://mathoverflow.net/questions/403345/conjugation-of-root-subgroups-by-the-weyl-group#comment1032860_403345), $V\_b$ is the $b$-root space, ... | 1 | https://mathoverflow.net/users/2383 | 403409 | 165,506 |
https://mathoverflow.net/questions/192252 | 4 | I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``*$B$ structures on $G$-bundles and local triviality*" in which they showed that under some nice conditions, every $G$-bundle on a cur... | https://mathoverflow.net/users/24706 | A step in the proof of the Drinfeld-Simpson theorem | Fix two principal $G$-bundles $P\_G$ and $P'\_G$ on $X$ along with an identification $\beta$ away from a divisor $D$. Also fix a simple root $\alpha\_i$.
Let $P\_B$ and $P\_B'$ be arbitrary $B$-reductions (of $P\_G$ and $P\_G'$ respectively) compatible with $\beta$. We write
$$
L = P\_B \overset{B}{\times} \alpha\_i... | 1 | https://mathoverflow.net/users/362094 | 403411 | 165,507 |
https://mathoverflow.net/questions/403133 | 18 | **Question:** Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
>
> 1. all roots of $f$ are distinct integers; and
> 2. all roots of $f'$ are distinct integers?
>
>
>
**Background:**
I asked a related question about four years ago on MSE ([**2312516**](https://math.stackexchange.com... | https://mathoverflow.net/users/22971 | Distinct integer roots for a degree 7+ polynomial and its derivative | Suppose one wishes to solve a system $P\_1(n\_1,\dots,n\_k) = \dots = P\_m(n\_1,\dots,n\_k)=0$ of diophantine equations involving polynomials $P\_1,\dots,P\_m$ of various degrees. We have the following probabilistic heuristic (discussed for instance in [this blog post of mine](https://terrytao.wordpress.com/2012/09/18/... | 18 | https://mathoverflow.net/users/766 | 403425 | 165,510 |
https://mathoverflow.net/questions/352051 | 10 | Every monoidal category $(\mathcal{C},\otimes)$ can be seen as a one-object bicategory: the morphisms are the objects of $\mathcal{C}$, and the $2$-morphisms are the morphisms of $\mathcal{C}$. In every bicategory, we can speak of left/right Kan extensions. Specifically, if $P$ is an object of $\mathcal{C}$ (the one "a... | https://mathoverflow.net/users/2841 | Kan extensions inside a monoidal category | It is certainly the case that the duals of internal homs have appeared significantly less in the categorical literature. I've included a few more references below, but I am not sure this is a satisfying answer; I have not found any reference that lists many naturally-occurring examples.
In the cocartesian monoidal se... | 6 | https://mathoverflow.net/users/152679 | 403432 | 165,511 |
https://mathoverflow.net/questions/403415 | 6 | I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with or without boundary) admits a triangulation, i.e. its homeomorphic to (the geometric realization of) an abstract simplici... | https://mathoverflow.net/users/259525 | Properties a triangulation must have in order to describe a manifold | From the comments:
1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)
2. There are triangulations of topological manifolds (... | 6 | https://mathoverflow.net/users/1650 | 403435 | 165,512 |
https://mathoverflow.net/questions/403429 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). For $v\in V$ we let $v^\* = \{e\in E:v\in e\}$. We define the *dual* of $H$ by $H^\*= (E, V^\*)$ where $V^\* = \{v^\*: v\in V\}$. We say that a $H$ is *self-dual* if $H \cong H^\*$.
Is there a self-dual hypergraph $H = (\omega, E)$ with the fo... | https://mathoverflow.net/users/8628 | Self-dual hypergraph on $\omega$ | Expanding my comments into an answer: by rephrasing this as a question of constructing a symmetric $\omega\times\omega$ matrix with the same properties, we can give an explicit construction. For convenience's sake I'm going to start numbering at 0. Let $e\_0 = \{v\_0, v\_1\}$ and for $n\geq 1$ let $e\_n = \{v\_i\} \cup... | 2 | https://mathoverflow.net/users/7092 | 403436 | 165,513 |
https://mathoverflow.net/questions/403441 | 7 | $\DeclareMathOperator\Diff{Diff}$Suppose for simplicity that $X$ is affine, it is then possible to define $\Diff(X)$ — the ring of Grothendieck differential operators. When $X$ is smooth, then
>
> **Definition.** the category of $D$-modules on $X$ is defined to be modules over $\Diff(X)$. (Category **1**)
>
>
>
... | https://mathoverflow.net/users/111070 | Why is the ring of Grothendieck differential operators bad when $X$ is singular? | One thing that's wrong with the Grothendieck definition on singular varieties is the same thing that's wrong with defining the tangent space at a singular point rather than the tangent complex - it's not sufficiently derived (ie it's a strange truncated notion of the "true" derived object), which is why several differe... | 9 | https://mathoverflow.net/users/582 | 403445 | 165,516 |
https://mathoverflow.net/questions/403418 | 1 | I am working on a paper of Elie Aidekon : [*‘Speed of the biased random walk on a Galton–Watson tree’*](https://doi.org/10.1007/s00440-013-0515-y) and have a question about one transformation in a proof:
\begin{align}
& 1+\frac{1}{1-\lambda}+\mathbb{E} \biggl[ \frac{\beta\_n(I) \mathbf{1}\_{\{ \beta(j)=0 \ \forall j\ne... | https://mathoverflow.net/users/362153 | Random walks on Galton–Watson trees | It seems no one checked this calculation prior to the publication of this paper. The factor $\frac{\lambda}{1-\lambda}$ in (1) seems mistaken, it should be $\frac{2-\lambda}{1-\lambda}$. This has no effect on the finiteness claimed in the lemma. The denominator in (1) indeed takes the given form because all the other s... | 1 | https://mathoverflow.net/users/7691 | 403455 | 165,520 |
https://mathoverflow.net/questions/403356 | 14 | $\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M\_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices. Let these matrix variables be $X\_1,\ldots,X\_n$. For an $d\times d$ matrix $A$, let $c\_k(A)$ be the coefficient of ... | https://mathoverflow.net/users/88840 | Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings | It is true. The standard reference is the Book by Jantzen, Representations of Algebraic Groups, Second edition. In particular we need the Appendix `Chapter B', and the base change Proposition in part I, 4.18. By Donkin the $G\_{\mathbb Z}$ module $M={\mathbb Z}[n]$ has good filtration. So $H^1(G\_{\mathbb Z},M)$ vanish... | 13 | https://mathoverflow.net/users/4794 | 403460 | 165,521 |
https://mathoverflow.net/questions/403473 | 7 | I came to know that the statement below could be proved using *Stallings' binding tie* argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me telling something related to this? Another question, except this, does there exist any other application of binding ti... | https://mathoverflow.net/users/363264 | Stallings' binding tie | Several of the papers citing Stallings paper *A topological proof of Grushko's theorem on free products* are using the binding tie argument. Here are the ones I found using a Google search. More can probably be found using MathSciNet.
1. Jaco [1968] *Constructing 3-manifolds from group homomorphisms*
2. Heil [1972] *... | 6 | https://mathoverflow.net/users/1650 | 403481 | 165,529 |
https://mathoverflow.net/questions/403048 | 2 | Once-punctured torus bundles are a well-studied class of hyperbolic 3-manifolds, but unfortunately I have been unable to find out whether they always have integral traces (in the sense of [1], Def. 5.2.1). Is this already known?
In several dozens of examples that I have calculated with the help of Snappy or the appro... | https://mathoverflow.net/users/129446 | Do once-punctured torus bundles have integral traces? | Floyd and Hatcher give a classification of essential surfaces in once-punctured torus bundles. In particular, there are no closed incompressible surfaces (other than the boundary torus), if there were they would persist in high parameter Dehn fillings, contradicting Corollary 1.2:
*Floyd, W.; Hatcher, Allen E.*, [**I... | 2 | https://mathoverflow.net/users/27453 | 403488 | 165,531 |
https://mathoverflow.net/questions/403469 | 3 | An interesting representation of the Euler-Mascheroni constant
$$ \gamma~=~ \lim \limits\_{n\to \infty} \sum \limits\_{k,s=1}^n \frac{s-k}{k\left( s\,n +k\right)},\label{1}\tag{$\*$}$$
can be proved using the relatively recent identity of Macys (Mat. Zametki, 94/5, 695–701, 2013), but it also can be reformulated to
alm... | https://mathoverflow.net/users/20804 | An Euler-Mascheroni double sum | $\newcommand\ga\gamma$As usual, let $H\_n$ denote the $n$th harmonic number. Note that
$$\frac{s-k}{k (s n+k)}=\frac1n\frac1k-\frac{n+1}n\frac1{n s+k}$$
and hence
$$
\begin{aligned}
&\sum\_{k,s=1}^n\frac{s-k}{k (s n+k)} \\
&=\frac1n\,\sum\_{k,s=1}^n\frac1k-\frac{n+1}n\,\sum\_{k,s=1}^n\frac1{n s+k} \\
&=H\_n-\frac{n+1}... | 2 | https://mathoverflow.net/users/36721 | 403495 | 165,533 |
https://mathoverflow.net/questions/403489 | 4 | Let $(C, J)$ be a small site and let $\mathsf{Sh}\_{(2, 1)}(C, J)$ be the $(2, 1)$-sheaf topos of sheaves of (small) groupoids on $(C, J)$. Let $G$ be a sheaf of groups on $(C, J)$, and let $\mathbf{Bun}\_G$ be the hom-stack $[-, \mathbf{B}G]$, which is typically known as the moduli stack of principal $G$-bundles on $(... | https://mathoverflow.net/users/143390 | Internal principal $G$-bundles | The easiest way to see local trivializations is to compute the homotopy pullback using the local projective model structure.
For differential geometry, we can $C$ to be the category of cartesian spaces and smooth maps, whereas $J$ is the usual topology of open covers.
(Other sites work in the same manner.)
Then map... | 5 | https://mathoverflow.net/users/402 | 403499 | 165,536 |
https://mathoverflow.net/questions/403414 | 45 | The following simple-looking inequality for complex numbers in the unit disk generalizes [Problem B5 on the Putnam contest 2020](https://www.maa.org/sites/default/files/pdf/Putnam/2020/2020%20Putnam%20Session%20B%20Solutions.pdf):
>
> **Theorem 1.** Let $z\_1, z\_2, \ldots, z\_n$ be $n$ complex numbers such that $\... | https://mathoverflow.net/users/2530 | Putnam 2020 inequality for complex numbers in the unit circle | Here is a detailed and self-contained proof for general $n$, which also covers the "equality" case. It is based on Fedja's post, but it only uses (a variant of) the Gauss-Lucas theorem once.
Let $a\_0,\dotsc,a\_{n-1}\in\mathbb{C}$ be coefficients such that
$$p(z):=(z-z\_1)\dotsb(z-z\_n)=(z-1)^n+\sum\_{k=0}^{n-1}a\_k ... | 24 | https://mathoverflow.net/users/11919 | 403506 | 165,538 |
https://mathoverflow.net/questions/403520 | 2 | I suspect that
$$ \lim\_{n\rightarrow\infty}\frac{1}{n}\sum\_{i=1}^n\sum\_{j=1}^n \frac{1}{\sqrt{i^2+j^2}} =a\approx 1.76$$
$$ \lim\_{n\rightarrow\infty}\frac{1}{n}\sum\_{i=1}^n\sum\_{j=1}^n \frac{1}{\sqrt{n^2+(i-j)^2}} =b\approx 0.934$$
Is there an analytic expression for $a$ or $b$?
| https://mathoverflow.net/users/94200 | Limiting behavior of lattice sums | You seem to have a typo in your value of $a$ - evaluating the sum as is yields something more like 1.76. Converting to an integral,
$$
\lim\_{n\rightarrow \infty } \frac{1}{n^2 } \sum\_{i=1}^{n} \sum\_{j=1}^{n} \frac{1}{\sqrt{(i/n)^2 + (j/n)^2 } } =\\ \int\_{0}^{1} dx \int\_{0}^{1} dy\, \frac{1}{\sqrt{x^2 +y^2 } } = 2\... | 9 | https://mathoverflow.net/users/134299 | 403522 | 165,541 |
https://mathoverflow.net/questions/403547 | 4 | Consider the Hirzebruch surface $\mathbb{F}\_n = \mathbb{P}(\mathcal{O}\_{\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}\_n$ is generated by the class of the negative section $\Gamma$, such that $\Gamma^2 = -n$, and the class of a fiber $F$ of the morphism o... | https://mathoverflow.net/users/nan | Cohomology of divisors on Hirzebruch surfaces | Sure. To compute $H^0$ one can first pushforward to the base $\mathbb{P}^1$. If $a \ge 0$ one obtains
$$
p\_\*\mathcal{O}(a\Gamma + bF) \cong
p\_\*\mathcal{O}(a\Gamma) \otimes \mathcal{O}(b) \cong
S^a(\mathcal{O} \oplus \mathcal{O}(-n)) \otimes \mathcal{O}(b) \cong
\bigoplus\_{i=0}^a \mathcal{O}(b-in),
$$
and if $a < 0... | 6 | https://mathoverflow.net/users/4428 | 403548 | 165,547 |
https://mathoverflow.net/questions/403546 | 10 | In [1] (section 3), C. Scott introduces the following concept of **regular atlas** for closed $C^\infty$-smooth Riemannian manifolds. He says:
>
> When referring to a coordinate system $(U,\phi)$ as **regular**, we shall mean that there is another system $(V,\psi)$ with $\overline{U}$ compact, $\overline{U} \subset... | https://mathoverflow.net/users/364344 | Sobolev spaces of differential forms and regular atlases | Ok this is already quite a mouth full, so let me try to give answers to some of your questions:
The main issue is that Sobolev mappings are defined via a boundedness concept (you ask for $L^p$-integrability conditions) and boundedness is not an intrinsic concept on a manifold. This means that definitions in charts usua... | 11 | https://mathoverflow.net/users/46510 | 403551 | 165,548 |
https://mathoverflow.net/questions/403545 | 4 | Let $R$ be a (commutative, otherwise the answer is easy, see the comment below) ring and let $M$ be a finitely generated $R$-module. Is it possible that $M$ admits an infinite linearly independent set? [Cardinality of maximal linearly independent subset](https://mathoverflow.net/questions/30066/cardinality-of-maximal-l... | https://mathoverflow.net/users/7845 | Infinite linearly independent set in finitely generated module | In a commutative ring $R$ this does not exist. Better for any $n\ge 0$, if $M$ is an $R$-module generated by $n$ elements, then $R^{n+1}$ doesn't embed into $M$.
Indeed, lifting if necessary, we can suppose $M=R^n$. So we get an $n\times (n+1)$ matrix $u$ over $R$ defining an injective operator $R^{n+1}\to R^n$. Let ... | 7 | https://mathoverflow.net/users/14094 | 403552 | 165,549 |
https://mathoverflow.net/questions/403554 | 3 | Lately I have been studying reflection groups, and there is a particular example of a complex reflection group that has been very good for guiding my intuition. I would like to know if there is an analogue over the real numbers. To keep this post self-contained, let me start by stating my definition of a reflection gro... | https://mathoverflow.net/users/175051 | Are there infinite abelian real reflection groups? | I think that you can prove by induction on the dimension of $V$ that there is no infinite Abelian (finite-dimensional) real reflection group $G$. Suppose that $G \subseteq {\rm GL}(V)$ is such an Abelian infinite real reflection group with ${\rm dim}\_{\mathbb{R}}(V)$ minimal. Then $V$ is certainly not $1$-dimensional.... | 3 | https://mathoverflow.net/users/14450 | 403557 | 165,551 |
https://mathoverflow.net/questions/403533 | 2 | From the standard literature it is well known that for sequences of random variables $X\_{1, n} \stackrel{P}{\rightarrow} X\_1$ and $X\_{2, n} \stackrel{P}{\rightarrow} X\_2$ as $n \rightarrow \infty$ it holds that $(X\_{1, n}, X\_{2, n}) \stackrel{P}{\rightarrow} (X\_1, X\_2)$ for $n \rightarrow \infty$. Using a conti... | https://mathoverflow.net/users/302666 | Convergence in probability of series of random variables | $\newcommand\ep\varepsilon\newcommand\de\delta\newcommand{\P}[1]{\overset P{\underset{#1}\longrightarrow}}$What you need is the uniform summability (in probability).
Here are details: Let $Y\_{l,n}:=X\_{l,n}-X\_l$, so that $$Y\_{l,n}\P{n\to\infty}0 \tag{0}$$
for each $l$. We want to have
$$S\_{n,n}\P{n\to\infty}0,\ta... | 3 | https://mathoverflow.net/users/36721 | 403561 | 165,553 |
https://mathoverflow.net/questions/403517 | 20 | Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
It is a nice exercise with rational generating functions (or equivalently, linear recurrence relations) to show that for a random domino tiling of a $2\times n$ rectangle, with $n$ large, we ca... | https://mathoverflow.net/users/25028 | Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical? | Here is at least a heuristic argument for why we should expect fewer vertical dominoes than horizontal dominoes. As we increase the length of the strip (to the right, say), let us think about how the rightmost four squares are covered. There are three cases: (1) two horizontal dominoes; (2) two vertical dominoes; (3) t... | 16 | https://mathoverflow.net/users/3106 | 403563 | 165,554 |
https://mathoverflow.net/questions/403536 | 3 | I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1.
In the paper they used the Hopf lemma to show that $u\_\nu>c>0$, but, as the boundary regularity is just $C^{1, \alpha}$, I don’t think that we can directly use Hopf lemma.
I tri... | https://mathoverflow.net/users/348579 | About the proof of higher regularity boundary Harnack inequality | *Edit:* The result is fine: Hopf's lemma was proved in
* G. Giraud, *Problèmes de valeurs à la frontière relatifs à certaines donn ás discontinues*, Bull. de la Soc. Math. de France, 61 (1933), 1–54
Below is my incorrect answer (which I keep for reference), where I mistakenly assumed that $C^{1,Dini}$ is stronger t... | 3 | https://mathoverflow.net/users/108637 | 403564 | 165,555 |
https://mathoverflow.net/questions/403560 | 3 | Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half sum of all positive roots with respect to $B$. For a character $\lambda \in X^\*(T)$ we define a line bundle $L\_\lambda$... | https://mathoverflow.net/users/135674 | Character which defines canonical bundle on flag variety | I think the most direct way comes from the following fact. There is a natural $G$-equivariant isomorphism $$ T^\*(G/B) \cong G \times\_{B} \mathfrak{b}^{\bot}$$ where $\mathfrak{b}^{\bot}$ is the sub-Lie algebra of $\mathfrak{g}^\*$ given by the annihilator of $\mathfrak{b}$. This can be found in " Representation theor... | 2 | https://mathoverflow.net/users/146464 | 403574 | 165,557 |
https://mathoverflow.net/questions/403512 | 1 | Consider the definition of "lost melody" given by Merlin Carl in his arXiv preprint, "[The Lost Melody Phenomenon](https://arxiv.org/abs/1407.3624v5)" (arXiv: 1407.3624v5 [math.LO] 16 Mar 2015):
>
> A lost melody is a real number $x \subseteq \omega$ which is recognizable, i.e., for some ITTM program $P$, the the c... | https://mathoverflow.net/users/20597 | Can $\{x \mathrel| \text{$\varphi_{x}$ total}\}$ be deemed a "lost melody" relative to classical recursion theory? | If we disregard the model of computation we want to use, a *lost melody* is a decidable singleton $\{x\}$ such that the point $x$ is non-computable. Classical computability theory cannot admit any lost melodies in $2^\omega$ because there are no decidable singletons in the first place, and it cannot admit lost melodies... | 2 | https://mathoverflow.net/users/15002 | 403578 | 165,560 |
https://mathoverflow.net/questions/403572 | 1 | In the question [here](https://mathoverflow.net/questions/403523/explicit-eigenvalues-of-matrix)
the author asks for the eigenvalues of an operator
$$A = \begin{pmatrix} x & -\partial\_x \\ \partial\_x & -x \end{pmatrix}.$$
Here I would like to ask if one can extend this idea to the operator
$$A = \begin{pmatri... | https://mathoverflow.net/users/150549 | Eigenvalues of operator | The extended operator can be treated along similar lines as the $c=0$ case. One merely has to modify the algebra a little. Again, denote the standard harmonic oscillator eigenfunctions (i.e., the eigenfunctions of $-\partial\_{x}^{2} +x^2 $ with eigenvalues $\lambda\_{n} =2n+1$) as $\psi\_{n} (x)$. Introduce also the s... | 3 | https://mathoverflow.net/users/134299 | 403584 | 165,562 |
https://mathoverflow.net/questions/403587 | 4 | In his pioneering paper *An extension of Schwarz's lemma*, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows:
Let $W$ be a Riemann surface. For an arbitrary point $\mathfrak{m}$ of $W$, let $\rho(\mathfrak{m})$ denote the radius of the ... | https://mathoverflow.net/users/172177 | Ahlfors' proof of Bloch's theorem | He explains his choice in lines 8-9 on p. 364 of the paper:
"This metric has the curvature $-4$ for it is obtained from the hyperbolic metric by the transformation $w'=w^{1/2}$ ."
Remarks. By more sophisticated choices of metrics later authors,
M. Bonk (MR0979048) and Chen and Gauthier (MR1428103) were able to impr... | 8 | https://mathoverflow.net/users/25510 | 403589 | 165,565 |
https://mathoverflow.net/questions/403562 | 3 | Let $X\subset \mathbb{P}\_{\mathbb{C}}^3$ be a projective singular cubic surface with two singular points. Is the rationalcohomology of such objects known? As an example of the type of surfaces I'd be interested in one can take the hypersurface defined by the homogenous cubic $$x^2w+y^2w+z^2w+xyz-zw^2-w^3=0 .$$
In th... | https://mathoverflow.net/users/146464 | Cohomology of singular projective cubic surface | By the classification theorem of cubic surfaces (p.6 in [this paper](https://arxiv.org/pdf/1305.0178.pdf)), a cubic surface belongs to the following classes
1. Has at worst ADE singularities.
2. Has an elliptic singularity, i.e., the surface is cone over a smooth cubic curve.
3. Non-normal or non-integral, and singul... | 6 | https://mathoverflow.net/users/74322 | 403592 | 165,567 |
https://mathoverflow.net/questions/403437 | 1 | Let $\overline{\widehat{Z}\_i} = \frac{E\_i\left[ \int\_{t\_i}^{t\_{i+1}}\widehat{Z}\_sds\right] }{\Delta t\_i}$ with $\widehat{Z}$ a square integrable process, $\Delta t\_i := t\_{i+1} - t\_i$, and $E\_i$ denotes the conditional expectation w.r.t. $F\_{t\_i}$, with standard probability space/filtration.
Why is then ... | https://mathoverflow.net/users/358044 | Stochastic Integral + conditional expectation | Breaking the integral into two terms, the first term is simply $E\_i\left[ \int\_{t\_i}^{t\_{i+1}}\widehat{Z}\_sds\right]$.
The second term is $E\_i \left[\int\_{t\_i}^{t\_{i+1}} \overline{\widehat{Z}\_i} ds\right]$. The term in the expectation is $\mathcal F\_{t\_i}$ measurable, and so the second term is just $\int\... | 1 | https://mathoverflow.net/users/173490 | 403609 | 165,571 |
https://mathoverflow.net/questions/403604 | 2 | Let $\sigma: \mathbb R \times \mathbb R \to \mathbb R$ be a Lipschitz continuous function bounded below by some $M > 0$.
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the SDE
$$dX\_t = \sigma(t, X\_t) dW\_t$$
with $X\_0 = 0$.
**Question:** Fix $T > 0$. Does there exist, for every $\varep... | https://mathoverflow.net/users/173490 | A bound for the occupation time of a diffusion | I think so. The martingale $X\_t$ is a time-changed Brownian motion: $X\_t = B\_{A\_t}$, where $$A\_T = \int\_0^T \sigma^2(t, X\_t) dt$$ and $B\_t$ is some Brownian motion. Now write
$$ \int\_0^T \mathbb 1\_{[-\delta,\delta]}(X\_t) dt = \int\_0^{A\_T} \mathbb 1\_{[-\delta,\delta]}(B\_s) dA^{-1}\_s = \int\_0^{A\_T} \fra... | 3 | https://mathoverflow.net/users/108637 | 403610 | 165,572 |
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