parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/301882 | 9 | If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U\_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
Suppose $(C, I, \otimes)$, $(D, J, \boxtimes)$ are (strict) monoidal categories, and call $Mon(C)$, $Mon(D)$ the correspondi... | https://mathoverflow.net/users/111265 | On functors preserving monoid objects | Here is a simple counterexample: take $C=D=\mathrm{Set}$, and let $\tilde{G}$ be the abelianization of a monoid. Then $\tilde{G}$ cannot arise from any $G:\mathrm{Set}\to\mathrm{Set}$, because the cardinality of the underlying set of $\tilde{G}(M)$ depends on the monoid structure of $M$, not just on its cardinality. Fo... | 9 | https://mathoverflow.net/users/49 | 301885 | 132,260 |
https://mathoverflow.net/questions/301881 | 0 | For every positive integer $n>1$ , let $f(n)$ denote the largest prime factor of $n$. How fast does $f(1+n^k)$ grow with respect to $k$ ? Is it true that $f(1+n^k) > 2k, \forall n >2, \forall k >1$ ?
| https://mathoverflow.net/users/nan | On the largest prime factor of $1+n^k$ | I do not attempt to answer the first question. The last question is closely related to Zsygmondy's Theorem: Given your hypotheses, and Zygmondy's theorem, there will be a prime $p$ which divides $n^{2k}-1,$ but does not divide $n^{j}-1$ for any $j < 2k.$ Note then that $p$ must be a prime divisor of $n^{k}+1.$
But als... | 5 | https://mathoverflow.net/users/14450 | 301889 | 132,262 |
https://mathoverflow.net/questions/301892 | 2 | Let $\mathbb{R}^\omega$ be endowed with the product topology. Is there a nonempty open set $U\subseteq \mathbb{R}^\omega$ such that $\mathbb{R}^\omega\cong \mathbb{R}^\omega\setminus \text{cl}(U)$?
(By $\text{cl}(\cdot)$ we denote the topological closure.)
**Edit.** Apologies for omitting the word "open" in the qu... | https://mathoverflow.net/users/8628 | Punching a hole into $\mathbb{R}^\omega$ | If $U= (-\infty,0) \times \mathbb R \times \mathbb R \times \dots$, then
$\mathrm{cl}(U) = (-\infty,0] \times \mathbb R \times \mathbb R \times \dots$, and the complement of this is $(0,+\infty) \times \mathbb R \times \mathbb R \times \dots$, which is homeomorphic to the whole space.
| 11 | https://mathoverflow.net/users/454 | 301893 | 132,264 |
https://mathoverflow.net/questions/301838 | 0 | Let $X \sim \text{Binom}(n, p)$ a binomial random variable. I want to show that : $$\forall 0 < t < 0.9, \quad \exists C, \quad \forall n >1, \quad \mathbb P\bigg(|X-np| \leq C\sqrt{n}\bigg) \geq t$$
I want to prove the result for all $n$, so pure asymptotic is not enough.
---
**1-** Using the normal approximat... | https://mathoverflow.net/users/47823 | Show that interval of maximum probability grows no faster than $\sqrt{n}$ for binomial distribution | $\newcommand{\de}{\delta}
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https://mathoverflow.net/questions/301903 | 8 | I encountered the following double sum which requires an evaluation.
>
> Is there a closed form for this?
> $$\sum\_{n=0}^{\infty}\frac{\sum\_{k=0}^n\binom{n}k^{-1}}{(n+1)(n+2)}.$$
> Incidentally, it seems that the following gives (empirically) the same value. Does it?
> $$\sum\_{n=1}^{\infty}\frac1{2^{n-1}(n+1)... | https://mathoverflow.net/users/66131 | What is the value of this double sum in closed form? | Consider the integral $$I=\int\_0^1\int\_0^1\frac{zdzdt}{(1-zt)(1-z(1-t))}=\sum\_{k,j\geqslant 0} \int\_0^1\int\_0^1 z^{k+j+1}t^k(1-t)^jdzdt=\\
\sum\_{k,j\geqslant 0} \frac1{k+j+2}\cdot \frac{k!j!}{(k+j+1)!},$$
that is your sum (denote $n=k+j$). For evaluating the integral, we first integrate by $t$, get $-2\log(1-z)/(... | 27 | https://mathoverflow.net/users/4312 | 301912 | 132,267 |
https://mathoverflow.net/questions/301134 | 2 | Let $X$ be a sooth algebraic variety over $\mathbb{C}$.
Let us assume that there exists the commutative diagram
$\require{AMScd}$
\begin{CD}
U @>{i}>> \hat{X}\\
@| @VV{\phi}V\\
U @>{j}>> X
\end{CD}
where $\phi : \hat{X} \rightarrow X$ is the blowing-up, and assume that the inclusion maps $i : U\hookrightarrow \hat{X}... | https://mathoverflow.net/users/124883 | The commutativity of minimal extension and direct image by blowing-down | No. Here's an example. Let $X=\Bbb C^2$, $\hat{X}$ the blow-up of $X$ at the origin, $U=(\Bbb C^\*)^2$, and $M=\mathcal{O}\_U$. Then $j\_{!\*}M=\mathcal{O}\_X$ and $i\_{!\*}M=\mathcal{O}\_{\hat{X}}$.
>
> **Claim:** $\phi\_+\mathcal{O}\_{\hat{X}}\ncong \mathcal{O}\_X$.
> *Proof.* Let $E=\phi^{-1}(0)$, and let $i\_E\... | 0 | https://mathoverflow.net/users/36720 | 301914 | 132,268 |
https://mathoverflow.net/questions/293171 | 9 | Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|\_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as an operator on $\mathbb{C}^n$ by $\|A\|\_p$, too. Moreover, let $|A|$ denote the matrix whose entries are the absolute ... | https://mathoverflow.net/users/102946 | Regular $p$-norm of a matrix | It is trivially $n^{1/p}$ for $p>2$ (and, therefore, $n^{1-\frac 1p}$ for $1<p<2$ because the norm of the adjoint operator in the dual space is the same as the norm of the operator in the space itself). To see it, choose the vector $x$ of $\ell^p$ norm $1$ so that
$$
\sum\_i\left|\sum\_j |a\_{ij}|x\_j\right|^p=\||A|\|^... | 6 | https://mathoverflow.net/users/1131 | 301917 | 132,269 |
https://mathoverflow.net/questions/301755 | 5 | Let $Q$ be a quaternion $k$-algebra (namely, a dimension 4 $k$-central simple algebra).
Then it is possible to (canonically) attach a smooth projective conic $C\_Q\subseteq \mathbf{P}\_k^2$ to $Q$: if $\mathrm{char}\,k \neq 2$, then $Q\simeq \langle x,y : x^2 = a, y^2 = b, xy+yx=0\rangle$ for certain $a\in k,b\in k^... | https://mathoverflow.net/users/83833 | A generalization of Witt's theorem for quaternion algebra isomorphism | Norms of quaternion algebras are particular cases of n-fold Pfister forms for n=2 (in any characteristic), and the result holds true for any Pfister forms. See, for example, Elman, Karpenko, Merkurjev "The algebraic and geometric theory of quadratic forms": <https://sites.ualberta.ca/~karpenko/publ/Kniga.pdf>, Corollar... | 3 | https://mathoverflow.net/users/5107 | 301919 | 132,271 |
https://mathoverflow.net/questions/301901 | 6 | I'm trying to learn about Embedded Contact Homology. To understand the proof of $d^2=0$, I started by watching Hutchings' lectures on Obstruction Bundle Gluing on YouTube ([1](https://www.youtube.com/watch?v=_4rJnq9Fn1Y), [2](https://www.youtube.com/watch?v=FcP8vJSJj3k), [3](https://www.youtube.com/watch?v=0EuXhQpezeE)... | https://mathoverflow.net/users/110236 | Question about Obstruction Bundle Gluing paper of Hutchings-Taubes | I was there during this IHES conference to scribe the lectures and also give a discussion session on it. So in case it helps, my notes from both of these are available [here](http://www.math.ias.edu/~joelfish/Hutchings_full.pdf).
Your thought on the reason for (i) and (ii) occurring is correct, in lieu of the fact th... | 6 | https://mathoverflow.net/users/12310 | 301920 | 132,272 |
https://mathoverflow.net/questions/301831 | 11 | In some circumstances I've been using a form of choice over the first uncountable ordinal knowing a priori that only a countable number of choices were going to be made (without any a priori upper bound). I would like to know whether I was using the classical dependent choice or something stronger and in case how much ... | https://mathoverflow.net/users/58975 | Countable (?) dependent choice | There has been some recent interest in the reverse mathematics of Ekeland's theorem. At the latest meeting of the Association of Symbolic Logic, Paul Shafer presented some of his work with David Fernández-Duque, Henry Towsner and Keita Yokoyama the reverse analysis of Ekeland's theorem and Caristi's theorem. Hopefully ... | 7 | https://mathoverflow.net/users/2000 | 301921 | 132,273 |
https://mathoverflow.net/questions/301930 | 2 | I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log\_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is convex (see [this](https://en.wikipedia.org/wiki/Quasiconvex_function)) but apparently it is not that easy.
| https://mathoverflow.net/users/123067 | Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$ | We have
\begin{equation}
f\_2(x):=f''(x)\Big/\frac{2^{x-3}}{125 \left(2^x+1\right)^2}=
-x \ln2-2 \left(2^x+1-500 \ln2\right)
\end{equation}
and
\begin{equation}
f''\_2(x)=-2^{1 + x} \ln^2 2<0,
\end{equation}
so that $f\_2$ is concave. Also, $f\_2(0)>0$ and $f'\_2(0)<0$. So, $f\_2$ decreases on $[0,1000]$ from $f\_... | 4 | https://mathoverflow.net/users/36721 | 301932 | 132,277 |
https://mathoverflow.net/questions/301925 | 4 | While thinking of whether any web (spatial trivalent graph) without an embedded bridge can be realized as a branching locus of a finite branched cover over $S^3$, I realized that this problem is closely related to whether fundamental groups of web complements are residually finite or not. I heard that knot groups are r... | https://mathoverflow.net/users/45553 | Are fundamental groups of web complements residually finite? | That the fundamental groups of compact $3$-manifolds are residually finite was proved by Hempel in
Hempel, John,
Residual finiteness for 3-manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379–396,
Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987.
Hempel's proof required ge... | 5 | https://mathoverflow.net/users/317 | 301933 | 132,278 |
https://mathoverflow.net/questions/301934 | 4 | Let $a\_i, i=1, \ldots, n$ be real numbers. Let $\epsilon\_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r\_i, i=1, \ldots, n$ be random variables that take values $\alpha$ and $-\alpha$ with equal probability such that $\sum\_{i=1}^nr\_i=K\in R$. Note, $r\_i$ and $\ep... | https://mathoverflow.net/users/122182 | Bound for a conditional expectation | $\newcommand{\al}{\alpha}
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\newco... | 5 | https://mathoverflow.net/users/36721 | 301935 | 132,279 |
https://mathoverflow.net/questions/301868 | 18 | The Brown-Comenetz dualizing spectrum $I\_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}\_2, X(n)$ for $n\geq 2$, and even $I\_{\mathbf{Q/Z}}$-acyclic. However, if $X$ is any nontrivial finite spectrum, then $X\wedge I\_{\mathbf{Q/Z}}$ is not contractible. This motivates a natur... | https://mathoverflow.net/users/102390 | Detecting the Brown-Comenetz dualizing spectrum | Your question appears to be equivalent to the 'dichotomy conjecture' of Hovey, which I believe is still open.
First, note that any finite spectrum has a type, and all finite spectrum of type $n$ have the same Bousfield class, usually denoted $F(n)$. In Hovey and Strickland's memoir (Appendix B) they conjecture that ... | 12 | https://mathoverflow.net/users/16785 | 301948 | 132,281 |
https://mathoverflow.net/questions/300531 | 8 | Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts):
1. $h^\*$: Scalar extension by means of $h$, i.e. $h^\*(M)=M\otimes\_RS$;
2. $h\_\*$: Scalar restriction by means of $h$;
3. $\widetilde{h}$: Scalar c... | https://mathoverflow.net/users/11025 | Adjoints of scalar extension and scalar coextension | If $X$ is an $R$-module, there is a natural map $M\otimes\_RX\to\text{Hom}\_R\left(\text{Hom}\_R(X,R),M\right)$ given by $m\otimes x\mapsto[\varphi\mapsto m\varphi(x)]$ that is easily checked to be an isomorphism when $X=R$, and hence (by additivity) when $X$ is a finitely generated projective.
So assuming (i)-(iii),... | 8 | https://mathoverflow.net/users/22989 | 301950 | 132,282 |
https://mathoverflow.net/questions/301927 | 3 | Let $:f:X\to Y$ be a projective surjective morphism between two normal varieties over $\mathbb{C}$. Assume that $f$ has only $1$-dimensional fibers. Let $D$ be a multi-section of $f$, i.e., $D$ is a prime Weil divisor on $X$ such that $f|\_D: D\to Y$ is a generically finite surjective morphism. Let $D\to Y''\to Y $ be ... | https://mathoverflow.net/users/80473 | How to split a Multi-section into finitely many Sections via base-change? | First, you have a multisection $D\_1 = D \times\_Y Y'$ for the family $X' \to Y'$, which is still generically finite of the same degree over $Y'$.
On the other hand, let $D' = D \times\_{Y''} Y'$. Then the maps $D' \to D \to X$ and $D' \to Y'$ induce a map $D' \to X'$ which is generically finite over $Y'$ with conne... | 5 | https://mathoverflow.net/users/4428 | 301955 | 132,284 |
https://mathoverflow.net/questions/301942 | 0 | Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x\_i - y\_i|=1$ and $ x\_k = y\_k$ for all $k\in \omega\setminus\{i\}$.
$K\_\omega$, the complete graph on $\omega$ points, is a [minor](https://dominiczypen.wordpress.com/2018/06/02/a-definition-of-mino... | https://mathoverflow.net/users/8628 | Large complete minors of $\mathbb{Z}^\omega$ | If I understand the definition correctly, the answer is no.
If $K\_\lambda$ is a minor of $G$, then there is a 1-1 map from $K\_\lambda$
into the family of (nonempty) connected subsets of $G$, such that any two images are in the same connected component of $G$, but disjoint. So there must be a component of size $\ge... | 6 | https://mathoverflow.net/users/14915 | 301973 | 132,292 |
https://mathoverflow.net/questions/301977 | 5 | A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that **a zero dimensional local ring is a chain ring whenever its maximal ideal is principal**?
| https://mathoverflow.net/users/125235 | When is a zero dimensional local ring a chain ring? | A famous theorem by Kaplansky says that a commutative ring is a principal ideal ring iff all of its prime ideals are principal. By using a zero-dimensional local ring with a principal maximal ideal, you are in that situation.
A commutative, local principal ideal ring is well-known to be a chain ring (a.k.a. uniseria... | 4 | https://mathoverflow.net/users/19965 | 301981 | 132,295 |
https://mathoverflow.net/questions/302002 | 12 | Background:
===========
$FGL(R)$ will be the category of formal group laws over the ring $R$. If $(E,x\_E)$ is an oriented spectrum (i.e. $x\_E \in \tilde{E}^2(\mathbb{P}^{\infty})$ is sent to the unit $1 \in \tilde{E}^2(\mathbb{P}^1)$ by the pullback of the inclusion $i: \mathbb{P}^1 \hookrightarrow \mathbb{P}^\inft... | https://mathoverflow.net/users/125244 | Morphism between formal groups and cohomology theories | Yes, if $E$ and $E'$ are Landweber exact, then ring maps $E\to E'$ biject with isomorphisms of the associated formal groups, suitably interpreted. One version of this (for the context where $E$ and $E'$ are $2$-periodic) is explained in Proposition 8.43 of [this Memoir](https://arxiv.org/abs/math/0011121). There is als... | 12 | https://mathoverflow.net/users/10366 | 302005 | 132,301 |
https://mathoverflow.net/questions/302007 | 7 | It is known that if $R$ is a DVR with fraction field $K,$ then the $R$-submodules of $K$ are $0,K,x^nR,$ with $n$ any integer and $x$ a generator of the maximal ideal of $R.$ I was wondering if there is a simple structure theorem for the $R$-submodules of $K^n,$ the $n$-dimensional vector space over $K$ (similar to tha... | https://mathoverflow.net/users/8027 | $R$ a DVR with fraction field $K.$ What are the $R$-submodules of $K^n?$ | Indeed, it seems that the situation gets nicer, but certainly not as nice as what I depicted in my first and **very flawed** answer. (See the remains below and the enlightening counter-example of Wilberd van der Kallen.)
Still, the situation is as tamed as it can be in the case of a **complete** discrete valuation ri... | 8 | https://mathoverflow.net/users/84349 | 302011 | 132,302 |
https://mathoverflow.net/questions/301965 | 8 | Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear).
Of course, if $Y$ is complemented, then the quotient always has a Lipschitz (bounded linear) right inverse.
The only examples I know of that do **... | https://mathoverflow.net/users/12248 | Lipschitz right inverses of Banach space quotients | The answer to (2) is yes and is contained in your reference [2]. That was how Aharoni and Lindenstrauss were able to construct two Lipschitz equivalent Banach spaces that are not isomorphic. Note that their example is non separable. Whether there are two Lipschitz equivalent non isomorphic separable Banach spaces is a ... | 7 | https://mathoverflow.net/users/2554 | 302012 | 132,303 |
https://mathoverflow.net/questions/302010 | 1 | Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with
1. $E'|\_B = E$
2. For $x\in B'\setminus B$ each fibre is only one point.
3. $E'$ is a compact space.
4. Each section $\om... | https://mathoverflow.net/users/124042 | Extend a bundle "trivially" | You should be able to construct this 'directly' in a similar way to the construction of the one-point compactification. Explicitly, you put $E' = E \cup (B'\setminus B)$, and take the open sets as follows: for each open set $U$ of $B'$, and each open set $V$ of $E$, take $V \cup \pi^{-1}(U \cap B) \cup (U \setminus B)$... | 2 | https://mathoverflow.net/users/5279 | 302013 | 132,304 |
https://mathoverflow.net/questions/301454 | 3 | I read the following statement in these [slides](http://shelah.logic.at/ecm/shelah-6ecm.pdf) of Saharon Shelah:
"$K$ is stable iff for every $M \in K$ there are only "few" complete types
over $M$." About the notation: here $K$ consists of all structures $N$ with the same *theory* of $M$, that is, the same set of first... | https://mathoverflow.net/users/12884 | Stability and complete types (in Model Theory) | My previous answer addressed the question "How can I make sense of the counting types definition of stability outside of a first-order model theory context?"
After discussion in the comments, I realized that you really wanted to know what stability means in the context of an (abstract) projective plane. So I'll try t... | 3 | https://mathoverflow.net/users/2126 | 302015 | 132,305 |
https://mathoverflow.net/questions/302023 | 5 | This [MO question](https://mathoverflow.net/questions/301903/what-is-the-value-of-this-double-sum-in-closed-form) prompted me to ask:
>
> What is the second order asymptotic growth/decay rate for the sum
> $$\sum\_{k=0}^n\frac1{\binom{n}k}$$
> as $n\rightarrow\infty$?
>
>
>
| https://mathoverflow.net/users/66131 | Asymptotic rate for $\sum\binom{n}k^{-1}$ | This sum can be written in the form, see [2-adic Logarithm and Resistance of n-dimensional Cube](https://mathoverflow.net/questions/146285/2-adic-logarithm-and-resistance-of-n-dimensional-cube)
$$S\_{n+1}=\frac{n+1}{2^{n+1}}\sum\_{k=1}^{n+1}\frac{2^k}{k}.$$
The last term in the sum gives the estimate $S\_n>1.$ You can ... | 6 | https://mathoverflow.net/users/5712 | 302025 | 132,310 |
https://mathoverflow.net/questions/119683 | 17 | It is conjectured that the automorphism group of the Turing degrees, $Aut(\mathcal{D})$, is trivial. However, to the best of my knowledge, the current state-of-the-art is that $Aut(\mathcal{D})$ is countable.
My question is twofold: first, is my understanding correct? Is there any countable group $G$ which we know c... | https://mathoverflow.net/users/8133 | Automorphism group of the Turing degrees | Let $p\_i$ denote the $i$th prime number, and let $\oplus$ be the recursive join on $\omega$. Let $\mathcal O$ be Kleene's $\Pi^1\_1$-complete set and $\mathcal O'$ its Turing jump.
For any $B$, let $G\_B$ be the direct sum of $\mathbb Z/p\_i\mathbb Z$ over all $i\in B\oplus\overline B$. So $G\_B$ is a countably infi... | 4 | https://mathoverflow.net/users/4600 | 302029 | 132,311 |
https://mathoverflow.net/questions/302030 | -3 | Let $\sigma\_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma\_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots by $\lambda\_1$, $\lambda\_2$ ...$\lambda\_n$.
May I ask how to get $\lambda\_i \in \sigma\_A(x)$ from $0=p(\lambd... | https://mathoverflow.net/users/113410 | How to show $\lambda_i \in \sigma_A(x)$? | If $p(\lambda) = \phi((\lambda e - x)^n) = 0$, that says $0 \in \sigma\_A((\lambda e - x)^n)$. By the Spectral Mapping Theorem, $\sigma\_A((\lambda e - x)^n)$ is the image of $\sigma\_A(x)$ under the map $t \to (\lambda - t)^n$, i.e. there is some
$t \in \sigma\_A(x)$ such that $(\lambda - t)^n = 0$, but that says $t =... | 1 | https://mathoverflow.net/users/13650 | 302032 | 132,312 |
https://mathoverflow.net/questions/302054 | 1 | This problem actually arose as a question in the real world (see the paragraph "Origin of the problem" below).
Let $\mathbb{N}$ denote the set of the positive integers and let $n\in\mathbb{N}$. If $w:\{1,\ldots,n\}\to \mathbb{N}$ is a function (the letter $w$ stands for "weight function") and $S\subseteq \{1,\ldots,... | https://mathoverflow.net/users/8628 | Achieving every possible ranking by rearranging weights | The following should give an example where you can't achieve every ranking.
Let $n=4$ and let $\mathcal{A} = \{ (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\} = [4]^{(2)}$.
I claim that we can't achieve any ranking which starts $(1,2) > (3,4) > \ldots$.
Indeed, since $sc\_w(1,2) > sc\_w(3,4)$ if follows that $\max \{ w(1)... | 2 | https://mathoverflow.net/users/35545 | 302058 | 132,318 |
https://mathoverflow.net/questions/302061 | 3 | I have just read this question [Short proof of $\frak p=t$](https://mathoverflow.net/questions/168125/short-proof-of-frak-p-t).
The link present in the answer about the proof given by Steprans doesn't work anymore.
Since I don't have enough reputation neither to talk in the chat nor to comment directly there to ask to ... | https://mathoverflow.net/users/121875 | Short proof of $\mathfrak{p}=\mathfrak{t}$ by Juris Steprans | Have a look at the Fremlin's [webpage](https://www1.essex.ac.uk/maths/people/fremlin/), in the section "[Miscellaneous research notes](https://www1.essex.ac.uk/maths/people/fremlin/preprints.htm)".
| 0 | https://mathoverflow.net/users/7460 | 302063 | 132,321 |
https://mathoverflow.net/questions/301970 | 8 | I have read this article
<https://arxiv.org/abs/1307.5708>
about vertix-frequency analysis on graph.
**David IShuman**
in this article claims that,"we generalize one of the most important signal processing tools – windowed Fourier analysis – to the graph setting and When we apply this transform to a signal with freq... | https://mathoverflow.net/users/124827 | graph signal processing |
>
> "I am looking for some simple concrete examples of the ways in which
> real problems go through graph signal processing and how graph Fourier
> transforms are obtained."
>
>
>
• A concrete example of a graph Fourier transform, to the Minnesota road network, is presented in [Fourier Analysis on Graphs](http... | 8 | https://mathoverflow.net/users/11260 | 302068 | 132,324 |
https://mathoverflow.net/questions/302072 | 5 | An abelian surface $A$ is called *singular* if it has maximal Picard number $\rho(A) = 4$.
By work of Shioda-Mitani, any singular abelian surface $A$ is the product $A = E\_1 \times E\_2$ of two isogenous elliptic curves with complex multiplication. If both $E\_1$ and $E\_2$ are defined over $\mathbb Q$, then $A$ is... | https://mathoverflow.net/users/43951 | Singular abelian surfaces that can be defined over $\mathbb Q$ | This is a complement to Joe Silverman's answer and is a bit long for a comment. One seeks to find a $\mathbb{Q}$-curve $E$ so that $\mathbb{Q}(j(E))$ is a quadratic extension of $\mathbb{Q}$. One such $\mathbb{Q}$-curve is $E : y^{2} + \sqrt{2} xy + y = x^{3} + x^{2} + (-2\sqrt{2} - 3) x + \sqrt{2} + 1$, which has CM b... | 6 | https://mathoverflow.net/users/48142 | 302080 | 132,329 |
https://mathoverflow.net/questions/302051 | 4 | *Question*: Is there a decidable theory sufficient to formulate and prove (many) theorems of classical analysis?
What I have in mind is a theory with two kinds of objects, reals (which are introduced as a Dedekind complete ordered field) and subsets of ${\mathbb R}^n$.
For each ${\mathbb R}^n$ there are axioms of Ext... | https://mathoverflow.net/users/9833 | A weak fragment of analysis? | I say this is impossible: any expansion of the theory of reals by anything vaguely resembling sets is doomed to undecidability.
>
> **Theorem:** Let $T$ be a two-sorted theory with one sort for reals, and the second sort for sets of reals, which includes the theory of ordered rings (on the first sort). Assume that ... | 5 | https://mathoverflow.net/users/12705 | 302093 | 132,332 |
https://mathoverflow.net/questions/301042 | 3 | Let $A$ be a $g$-dimensional, complex abelian variety, let $H$ be an ample divisor, let $D\in Pic^0(A)$, and let $0\leq k\leq g$.
**Question 1:** Does $D^k\neq 0\in CH^k(A,\mathbb{Q})$ imply that $H^{g-k}\cdot D^k\neq 0\in CH^g(A,\mathbb{Q})$?
(As Jason points out, it is necessary to work with $\mathbb{Q}$-coeffici... | https://mathoverflow.net/users/124840 | Intersections with a Power of an Ample Divisor on an Abelian Variety | Let me just explain the statement of the result which alluded to by abx and proved by Beauville. Let $A$ be an abelian variety over $\mathbb{C}$, and let $\operatorname{CH}^\*(A)$ be the Chow ring of cycles with rational coefficients modulo rational equivalence.
Beauville considers eigenspaces for the action of pulli... | 1 | https://mathoverflow.net/users/17630 | 302128 | 132,346 |
https://mathoverflow.net/questions/302125 | 1 | My friends are preparing project about a Solow model. The asked me to calculate such integral:
$ s(1-a)\int e^{(1-a)(b+c)t} \cdot (d-ge^{ft})^{1-a}dt$
where: $b,c,s,d,g>0$ and $a∈(0,1)$. $[a,b,c,d,g,f$-constans] They recived a hint to use hypergeometric series.
Unfortunately, I do not know anything about hypergeometric... | https://mathoverflow.net/users/117789 | Difficul integral- Solow model | It is long continuation of a comment above, only on the rights of comment (Thanks to user @alpoge, please check and type your own full answer):
If I did not mistake somewhere in arithmetic after change of variable $t=\ln z$
we have $$s(1-a)\int e^{(1-a)(b+c)t}(d-e^{ft})^{(1-a)}dt =s(1-a)\int z^{(1-a)(b+c)-1}(d-gz^f... | 2 | https://mathoverflow.net/users/73577 | 302135 | 132,348 |
https://mathoverflow.net/questions/301745 | 3 | Let $A/L$ be an elliptic curve, with complex multiplication by a quadratic imaginary field $K$.
A theorem by Deuring ([13, paragraph 4], Theorem 12 on page 182 of Elliptic Functions by Serge Lang) states that if $\mathfrak{P}$ is a prime of $L$ lying over $p$ at which $A$ has good reduction, then $A$ is supersingular... | https://mathoverflow.net/users/105764 | Confusion on supersingular reduction of elliptic curves with complex multiplication | It seems like Silverman's exercise is wrong: [see this errata](https://www.math.brown.edu/~jhs/ATAEC/ATAECErrata.pdf).
| 4 | https://mathoverflow.net/users/105764 | 302166 | 132,358 |
https://mathoverflow.net/questions/302050 | 2 | Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^\*~ \textrm{is separable} $$
The converse is clearly true, since the closed unit ball is relatively weak star second countable.
**Def.** A topological space $X$ is hereditarily Lindelöf if e... | https://mathoverflow.net/users/84390 | Relation between the weak star topology and hereditary Lindelöfness | Under CH there exists an example of a non-metrizable compact scattered Hausdorff space $K$ such that the Banach space $X=C(K)$ endowed with the weak topology is hereditarily Lindelof. The non-metrizability of $K$ implies that the Banach space $X=C(K)$ is not separable and then the dual $X^\*$ is not separable as well. ... | 4 | https://mathoverflow.net/users/61536 | 302168 | 132,359 |
https://mathoverflow.net/questions/301638 | 3 | I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group.
Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}\_0$. By an isotropy group of $\mathcal{G}$ we mean the collection of all arrows from $x$ to itself. Some people write $\m... | https://mathoverflow.net/users/118688 | Isotropy group of a Lie groupoid is a Lie group | There is also a rundown in the more modern book of Mackenzie: General Theory of Lie groupoids and Lie algebroids (p.26 Corollary 1.4.11)
Let me summarise the idea: Fix a Lie groupoid $G \rightrightarrows M$ with source map $s$ and target map $t$.
**Idea**:
1. Every fibre $s^{-1} (x)$ is a closed embedded subman... | 3 | https://mathoverflow.net/users/46510 | 302171 | 132,360 |
https://mathoverflow.net/questions/300846 | 7 | [Rough paths](http://www.hairer.org/notes/RoughPaths.pdf) can be thought of as taking values in a Lie group embedded in the tensor algebra of $\Bbb R^d$. See page 17/section 2.3. Lie groups represent the continuous symmetries of some object. That is, elements of the Lie group act on some other object in a way that pres... | https://mathoverflow.net/users/nan | What does the group action of a rough path in a Lie group look like? | Though I like the Arnoldian spirit of the question ("a group is not some set with a forgettable system of axioms but something which acts on a space"), I think the comments given above are already spot on:
Rough paths are paths of a certain regularity (often one takes finite $p$-Variation or equivalently $1/p$-Hölder ... | 13 | https://mathoverflow.net/users/46510 | 302173 | 132,361 |
https://mathoverflow.net/questions/302165 | 2 | The sum of uniform i.i.d. random variables follows the Irwin-Hall distribution. Through observation it seems that the convergence is faster in comparison to the sum of uniform independent but not identically distributed random variables.
Is there any result that proves this conjecture?
| https://mathoverflow.net/users/117503 | Fastest convergence of sum of uniform independent distributions to a Gaussian | Let $U\_1,U\_2,\dots$ be iid rv's uniformly distributed on $[-1,1]$. If a natural number $n$ and real $a\_1,\dots,a\_n$ vary so that
\begin{equation}
\sum\_1^n a\_i^2=3\quad \text{and}\quad \max\_1^n|a\_i|\to0\tag{1}
\end{equation}
(whence $n\to\infty$), then (say) by the Berry--Esseen inequality,
\begin{equation\*}... | 7 | https://mathoverflow.net/users/36721 | 302174 | 132,362 |
https://mathoverflow.net/questions/302170 | 0 | Question is as in the title.
Why study orbifolds?
I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. Groupoid) $\mathcal{G}$ and a homeomorphism $|\mathcal{G}|\rightarrow X$, where $|\mathcal{G}|$ is orbit space of the gr... | https://mathoverflow.net/users/118688 | Why study orbifolds? | I can not say why one studies orbifolds (or e.g. why one studies math at all). However, I can try the approach which might convince your funding agency: There are tons of interesting examples of how orbifolds arise in "applications" (= mathematics):
1. The quotient spaces appearing in symplectic reduction are not alw... | 5 | https://mathoverflow.net/users/46510 | 302176 | 132,364 |
https://mathoverflow.net/questions/302148 | 6 | Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ which is an analytic cycle on $T^\*X$ whose components are conic Lagrangian subvarieties and their multiplicities are positi... | https://mathoverflow.net/users/16183 | Additivity of characteristic cycle of holonomic D-module | Isn’t this Björk’s [**3.1.4** (p. 130)](https://books.google.com/books?id=iLnnCAAAQBAJ&pg=PA130)?
| 4 | https://mathoverflow.net/users/19276 | 302187 | 132,368 |
https://mathoverflow.net/questions/302192 | 3 | If $a^2+b^2+c^2+d^2=1$ in which $a,b,c,d>0$, prove or disprove
\begin{equation\*}
\begin{aligned}
(a+b+c+d)^8&\geq 2^{12}abcd;\\
a+b+c+d+\frac{1}{2(abcd)^{1/4}}&\geq 3.
\end{aligned}
\end{equation\*}
Can you tell any general algorithm for this type of problems? Thanks.
| https://mathoverflow.net/users/115637 | Algebraic inequalities on different means | Put
\begin{align\*}
f(a,b,c,d) &= \frac{(a+b+c+d)^8}{2^{12}abcd} \\
g(a,b,c,d) &= \frac{a+b+c+d}{3} + \frac{1}{6(abcd^{1/4})}
\end{align\*}
so the conjecture is that $f,g\geq 1$. I used Maple to search randomly for places where $f$ is as small as possible, then used Maple's fsolve() to find a local minimimum of $f$ ... | 3 | https://mathoverflow.net/users/10366 | 302197 | 132,372 |
https://mathoverflow.net/questions/302074 | 3 | Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$.
When $V$ is a free module, $H$ will be given by a matrix from $M\_n(K\times K)\cong M\_n(K)\times M\_n(K)$ where $n$ is the rank of $V$. Further ... | https://mathoverflow.net/users/69977 | Hermitian forms over $K\times K$ | **The situation is quite the same when $V$ is not free over $R$.** You only need an extra parameter, namely the dimension $n\_1$ of $(1, 0)V$ over $K$. The equivalence classes of Hermitian forms over $V$ are in one-to-one correspondance with the Smith normal forms of $n\_1$-by-$n\_2$ matrices $A$ over $K$ where $n\_2 =... | 2 | https://mathoverflow.net/users/84349 | 302199 | 132,374 |
https://mathoverflow.net/questions/302196 | 4 | Let $k$ be an algebraically closed field (of characteristic zero, if it helps).
Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the cardinality of $k$.
For every $i$ in $I$, let $S\_i$ be a proper closed subset of $X$.
>
>
> >
> > Is is true t... | https://mathoverflow.net/users/125327 | Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$ | If $dim X=1$ then it follows from cardinalities, and if $dimX>1$ then apply induction on dimension to $X \cap H, S\_i \cap H$ where $H$ is a hyperplane such that dimensions of $X \cap H, S\_i \cap H$ all drop by 1 (such $H$ exists by cardinality). (Embed an affine open of $X$ in projective space in general.)
| 5 | https://mathoverflow.net/users/59248 | 302200 | 132,375 |
https://mathoverflow.net/questions/302188 | 1 | Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it".
But in practice if we consider some widely used regularization methods: analytic continuation, Cesaro, Abel, Borel, Ramanujan etc we always get the same resul... | https://mathoverflow.net/users/10059 | Why we cannot speak about the main or natural regularization? | **Preliminary comment:** I second the idea that, for certain integrals, there should be a "well-behaved" class of regularization methods which all give the same value - of course, the important thing then is to give a good description of such a class of regularizations.
**Answer to your question:** Yet, your claim t... | 8 | https://mathoverflow.net/users/102946 | 302205 | 132,379 |
https://mathoverflow.net/questions/302194 | 6 | Let $G$ be a connected compact separable Hausdorff metric group, which is monothetic, i.e., has a dense subgroup generated by a single element. Such a group is necessarily Abelian.
**Question:**
Can the cardinality of the set of all closed subgroups of $G$ be uncountable?
Thank you.
| https://mathoverflow.net/users/89313 | The number of distinct closed subgroups of a compact monothetic group | Yes. Take $G=\prod\_{i<\omega}S^1$. It admits a metric which is compatible with the group action. An element $x=(e^{i\pi a\_0},e^{i\pi a\_1},\ldots,e^{i\pi a\_n},\ldots)$ generates a dense subgroup if the set $\{1,a\_0,a\_1,\ldots,a\_n,\ldots\}$ is $\mathbb{Q}$-linearly independent.
For every subset $I\subseteq\omega... | 7 | https://mathoverflow.net/users/16678 | 302207 | 132,380 |
https://mathoverflow.net/questions/302113 | 11 | Let $p$ be a real polynomial and $N$ be a positive integer. Suppose I tell you that $|p(\frac{1}{k})| \le 1$ for all $k\in\{1,\ldots,N\}$, and also that $p(\frac{1}{N})\le -\frac{1}{2}$ while $p(\frac{2}{N})\ge \frac{1}{2}$. What bounds can you give me on the minimal possible degree $\deg(p)$?
An upper bound of $O(\s... | https://mathoverflow.net/users/2575 | Real polynomial bounded at inverse-integer points | **Part I: $CN^{1/3}$ is enough.**
Start with $P(x)=\prod\_{k\le N^{1/3}}(1-k^2x^2)$. Notice that it vanishes at $1/k$ with $k\le N^{1/3}$, is bounded by $1$ on $[0,N^{-1/3}]$, the degree of $P$ is $2N^{1/3}$ and on the interval $[0,2N^{-1}]$ we have $\log P\ge -2\sum\_{k\le N^{1/3}}\frac{4k^2}{N^2}=o(1)$. Now just ta... | 20 | https://mathoverflow.net/users/1131 | 302208 | 132,381 |
https://mathoverflow.net/questions/302216 | 7 | $\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}$
I know from Bott-periodicity that $K\_0(C\_0(\mathbb{C}))\simeq \Z$, is there any easy way to compute an explicit generator of $K\_0(C\_0(\mathbb{C}))$? By this I mean a projection $q$ in $M\_k(C\_0(\mathbb{C})^+)$ such that $[q]-[p\_n]$ generates $K\_0(C\_0(\mat... | https://mathoverflow.net/users/125055 | Generator of $K_0(C_0(\mathbb{C}))$ | The group $K\_0(C\_0(\mathbb{C}))$ is generated by by the class $[p\_{Bott}] - [1]$ where $p\_{Bott} \in M\_2(C\_0(\mathbb{C})^\sim)$ is the so-called "Bott projection" given by
$$
p\_{Bott}(z) = \frac{1}{1+|z|^2} \begin{pmatrix} |z|^2 & z \\ \overline{z} & 1 \end{pmatrix}.
$$
This class comes from the tautological lin... | 11 | https://mathoverflow.net/users/61935 | 302224 | 132,387 |
https://mathoverflow.net/questions/302231 | 5 | Let $\Sigma\_{g}$ be a closed oriented surface of genus $g$, Goldman defined a Lie algebra structure on the free module generated by the free homotopy
classes of loops on $\Sigma\_{g}$. Roughly speaking, the Lie bracket is defined via intersection and concatenation of loops. In detail, see his paper [*Invariant functio... | https://mathoverflow.net/users/91245 | Formula for Goldman Lie bracket of surface | I would recommend looking at the work of [Moira Chas](http://www.math.stonybrook.edu/~moira/) to start.
Here are two interesting papers of hers to read:
*[The Goldman bracket and the intersection of curves on surfaces.](http://charvarworkshop.wdfiles.com/local--files/references/Chas-Almora.pdf)*
*[Combinatorial ... | 10 | https://mathoverflow.net/users/12218 | 302232 | 132,390 |
https://mathoverflow.net/questions/302242 | 3 | Proving $\lnot\lnot(A\lor\lnot A)$ in intuitionistic sequent calculus with cut seems to be easy:
We use cut to prove $\lnot(A\lor\lnot A)\vdash \bot$ from $\lnot(A\lor\lnot A)\vdash \lnot A \land\lnot\lnot A$ and $\lnot A \land\lnot\lnot A\vdash \bot$.
But how can one ever prove $\lnot(A\lor\lnot A)\vdash \bot$ witho... | https://mathoverflow.net/users/20781 | Does cut elimination fail here? | $$
\dfrac{\dfrac{\dfrac{\dfrac{}{A\vdash A}}{A\vdash A\lor(A\to C)}\qquad\dfrac{}{C\vdash C}}{\dfrac{\dfrac{(A\lor(A\to C))\to C,A\vdash C}{(A\lor(A\to C))\to C\vdash A\to C}}{(A\lor(A\to C))\to C\vdash A\lor(A\to C)}}\qquad\lower3em\hbox{$\dfrac{}{C\vdash C}$}}{\dfrac{(A\lor(A\to C))\to C\vdash C}{\vdash((A\lor(A\to C... | 11 | https://mathoverflow.net/users/12705 | 302245 | 132,393 |
https://mathoverflow.net/questions/302250 | 2 | In the answer to [MO question 132247](https://mathoverflow.net/questions/132247/quotient-groups-of-the-lower-central-series-of-a-free-group/132253), it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
>
> **Q.** What are the quotient grou... | https://mathoverflow.net/users/7460 | Quotient groups of the lower central series of a surface group | This paper seems to be relevant:
* S. Papadima and S. Yuzvinsky. On rational $K[π,1]$ spaces and Koszul algebras. J. Pure Appl. Algebra 144 (1999), no. 2, 157–167. [(link to paper on ScienceDirect)](https://www.sciencedirect.com/science/article/pii/S0022404998000589)
At least it contains the formula for dimenions ... | 3 | https://mathoverflow.net/users/50846 | 302253 | 132,397 |
https://mathoverflow.net/questions/302249 | 7 | I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute
$$
Z\_N[J]=\int dM e^{{\rm Tr} (J^T M)},
$$
where $dM$ is the $SO(N)$ Haar measure, and $J$ is an arbitrary $N\times N$ matrix. From this generating function, I can generate all correlations $\langle M\_{ij}M\_{kl}\cdots\ran... | https://mathoverflow.net/users/125359 | Generating function of $SO(N)$ random matrix | **Singular value decomposition:** The real $N\times N$ matrix $J$ has singular values $\sigma\_1,\sigma\_2\ldots\sigma\_N\geq 0$ in the decomposition $J=U\,{\rm diag}\,(\sigma\_1,\sigma\_2\ldots\sigma\_N)V$ with $U,V\in{\rm O}(N)$. We need to restrict $U,V$ to ${\rm SO}(N)$, which we can do by allowing for one of the s... | 4 | https://mathoverflow.net/users/11260 | 302271 | 132,405 |
https://mathoverflow.net/questions/302288 | 5 | Edit 2: Please note the new, more specific version of the question: [Does there always exist an irreducible representation occurring with multiplicity one when inducing from $M=Z\_K(A)$ to $K$?](https://mathoverflow.net/questions/302308/does-there-always-exist-an-irreducible-representation-occurring-with-multiplicit)
... | https://mathoverflow.net/users/58125 | Does there always exist an irreducible representation occurring with multiplicity one when inducing from a closed subgroup to a compact Lie group? | Let $K=SU(2)$, $M=Z(SU(2)) = \{\pm I\}$, and $\tau$ the sign representation of $M$. Then given an irreducible representation $\sigma$ of $SU(2)$, $\sigma|\_M$ is necessarily a character with multiplicity $dim(V)$. Moreover, the sign character appears exactly for the even dimensional representations (i.e. those of odd h... | 7 | https://mathoverflow.net/users/7762 | 302300 | 132,415 |
https://mathoverflow.net/questions/302261 | 2 | Suppose that $(X, \|\cdot\|\_X)$, $(Y, \|\cdot\|\_Y)$ are two Banach spaces such that $X\subset Y$ and $\|x\|\_Y\leq \|x\|\_X$ for all $x\in X$ and $X$ is dense in $(Y, \|\cdot\|\_Y)$.
Every functional $y^\*\in Y^\*$ also yields a functional $y^\*|\_X\in X^\*$ with $\|y^\*|\_X\|\_{X^\*}\leq \|y^\*\|\_{Y^\*}$. Suppos... | https://mathoverflow.net/users/125366 | Norming functionals for vectors in intersections | Let $X=\ell^1$, $Y=\ell^2$. Take $y\_n^\*=(1,1/n,1/n,\dots,1/n,0,0,\dots)$ with $\frac 1n$ repeated $n^3$ times. Looks like we are fried, or am I missing something in the setup?
| 1 | https://mathoverflow.net/users/1131 | 302307 | 132,418 |
https://mathoverflow.net/questions/302147 | 7 | Denote the [unsigned Stirling numbers of the first kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind) by $s(n,j)$.
>
> **Question.** Is it true that the polynomials
> $$P\_n(x)=\sum\_{j\geq0}s(n,j)\binom{x}j$$
> have only real roots?
>
>
>
**Note.** Obviously, the roots of $F\_n(x)=\sum\_{... | https://mathoverflow.net/users/66131 | Real-rootedness of some polynomials | Fix $n$ and consider $P\_k(x)=(1+\alpha)(1+2\alpha)\cdots(1+(k-1)\alpha)\cdot{x\choose n}$, where $\alpha$ is the operator sending ${x\choose k}$ to ${x\choose k-1}$. That is, $\alpha$ sends $f(x)$ to $f(x+1)-f(x)$. We'll show by induction that each $P\_k(x)$ has all real roots and that the distance between consecutive... | 8 | https://mathoverflow.net/users/112641 | 302310 | 132,420 |
https://mathoverflow.net/questions/302279 | 10 | Is $\mathbb{R}^\omega \cong (\mathbb{R}^\omega \setminus \{x\})$, where $\mathbb{R}^\omega$ is given the product topology, and $x\in\mathbb{R}^\omega $?
| https://mathoverflow.net/users/8628 | Is $\mathbb{R}^\omega \cong (\mathbb{R}^\omega \setminus \{x\})$? | The answer is yes. Any two homotopy equivalent [Hilbert manifolds](https://en.wikipedia.org/wiki/Hilbert_manifold) are homeomorphic. The countable product of lines is homeomorphic to the separable Hilbert space, see p.174 of [Bessaga C., Pelczynski A., Selected Topics in Infinite-Dimensional topology] where much more i... | 9 | https://mathoverflow.net/users/1573 | 302314 | 132,421 |
https://mathoverflow.net/questions/132795 | 19 | I have downloaded from [this link](http://www.math.jussieu.fr/~leila/grothendieckcircle/crystals.pdf) a quite poor quality scan of the letter dating May 1966 that Grothendieck sent to Tate mentioning his ideas about generalizing Monsky-Washnitzer cohomology. I am trying to put it in LaTeX, at least for my own referen... | https://mathoverflow.net/users/18238 | Letter from Grothendieck to Tate on "crystals" | In the light of past events ("[Les Archives Grothendieck](https://grothendieck.umontpellier.fr/archives-grothendieck/)"), we now have an "ameliorated" version of the letter:
[**Cote n° 6**](https://grothendieck.umontpellier.fr/archives-grothendieck/). *Lettre Tate (mai 66) (Cristaux) : lettre (1966), tapuscrit, notes... | 23 | https://mathoverflow.net/users/nan | 302330 | 132,426 |
https://mathoverflow.net/questions/302332 | 7 | Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.
Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum.
>
> **Question**. Can we conclude the norm topology on $X$ is separable?
>
>
>
**Remark.** It was proved by [T... | https://mathoverflow.net/users/84390 | Does separability of the strong operator topology imply separability of the underlying space? | Let $y$ be a norm-one vector in $X$. Consider the evaluation map $T\mapsto Ty$, which is a (linear) surjection from $B(X)$ onto $X$. This map is SOT-norm continuous. Indeed, suppose that $T\_n\to T$ in SOT. In particular, $\|T\_ny - Ty\|\to 0$ as $n\to \infty$.
A continuous image of a separable space is separable, he... | 10 | https://mathoverflow.net/users/15129 | 302336 | 132,428 |
https://mathoverflow.net/questions/302255 | 7 | Define the sum of the non-negative numbers $\{r\_s \mid s \in S\}$ $S$ uncountable to be
$$\sup \_{D \subseteq S} \sum \_{d \in D} r\_d$$
($D$ being finite), which exists if this supremum is finite.
Define a point function to be a function from $[0, 1]$ to $\mathbb R$ that is $0$ everywhere except for a single po... | https://mathoverflow.net/users/125362 | A conjecture on writing a function as a sum of uncountably many points | OK, here goes as promised.
Fix $\delta>0$ and consider the set $A\_\delta$ of all points $a$ such that $S[0,a]$ is continuous and $\max f\_a>\delta$ (this maximum is just that exceptional positive point value in your case but we can do arbitrary non-negative not identically $0$ functions). If we could find $a\_n,a\i... | 7 | https://mathoverflow.net/users/1131 | 302346 | 132,430 |
https://mathoverflow.net/questions/302355 | 3 | If $Y\subset X^\*$ is a closed subspace (where $X$ is a separable Banach space), the preannihilator of $Y$ in $X$ is $Y\_{\perp}:=\{x\in X : y^\*(x)=0, \forall y^\*\in Y \}$. If $Y$ is a proper subspace of $X^\*$, and $X$ is reflexive then it can be proved that $Y\_{\perp}\neq\{0\}$. On the other hand if $X=l\_1$ and $... | https://mathoverflow.net/users/69275 | Preannihilators of subspaces of separable duals | For (1), take $X = c\_0$ so that $X^\* = \ell^1$, which is separable. Take $Y = \{y \in \ell^1 : \sum\_i y(i) = 0\}$ which is a proper closed subspace. Since $Y$ contains all the elements of the form $e\_i - e\_{i+1}$, where $e\_i$ are the usual basis functions, you can see that any $x \in Y\_\perp$ satisfies $x(1) = x... | 3 | https://mathoverflow.net/users/4832 | 302357 | 132,432 |
https://mathoverflow.net/questions/280789 | 14 | Consider a Riemannian manifold $(M,g)$.
How much regularity is required of $g$ so that for any $x\in M$ and $v\in T\_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to M$ so that $\gamma(0)=x$ and $\dot\gamma(0)=v$?
All regularity is considered with respect to a fixed smooth (or somewh... | https://mathoverflow.net/users/55893 | Existence and uniqueness of geodesics in low regularity | There is a classical example by Hartman that shows failure of uniqueness for $C^{1,\alpha}$ metrics. (P. Hartman, On the local uniqueness of geodesics, Amer. J. Math. 1950).
You could lower the regularity if the metric is smooth off some hypersurface but globally only $C^{0,1}$, see for example our recent review: [On... | 7 | https://mathoverflow.net/users/47189 | 302366 | 132,435 |
https://mathoverflow.net/questions/302367 | 5 | A question I was asked recently lead me to the following question. For every closed first order formula $\theta$ in the group signature consider the set $N\_\theta$ of natural numbers $n$ such that the symmetric group $S\_n$ satisfies $\theta$.
**Question.** Can $N\_\theta$ always be defined by a first order formula... | https://mathoverflow.net/users/nan | Translating first order statements about symmetric groups into the language of numbers and back | If by "signature of natural numbers" you mean the usual signature of arithmetic $\{+,\times\}$ or similar, the answer is **yes**. This is because using this language we can talk in a first-order way about finite objects: in the same manner as Godel describes strings of symbols, we can explicitly write something like "F... | 8 | https://mathoverflow.net/users/8133 | 302368 | 132,436 |
https://mathoverflow.net/questions/302351 | 2 | For any integer $m>2$, let $P\_m$ be the set of primes less than $m$, and let
$$
f(m) = \sum\limits\_{p \in P\_m} \frac{1}{m-p}.
$$
For example, $f(3)=\frac{1}{3-2}=1$, $f(4)=\frac{1}{4-2}+\frac{1}{4-3}=\frac{3}{2}$, and so on.
The question is to estimate $I=\inf\limits\_{m>2} f(m)$.
A simple Mathematica calculatio... | https://mathoverflow.net/users/31472 | Sum of reciprocals of integers minus primes | We have $f(m)>0.53899$ for $m$ sufficiently large. Under the Riemann hypothesis, we even have $f(m)>0.69314$ for $m$ sufficiently large, which would also imply that $f(m)$ attains a minimum.
If $m-1$ is a prime, then clearly $f(m)\geq 1$. Otherwise we have
$$f(m)=\int\_0^{m-1}\frac{d(\pi(x)-\pi(m))}{m-x}=\frac{\pi(m)... | 4 | https://mathoverflow.net/users/11919 | 302369 | 132,437 |
https://mathoverflow.net/questions/302270 | 6 | Let $G$ be a locally compact group and let $H$ be an open subgroup in $G$.
Then the full group $C^\*$-algebra of $H$, $C^\*(H)$, is a subalgebra of $C^\*(G)$ and there is a conditional expectation $$E\colon C^\*(G)\to C^\*(H),$$
which is induced by restriction $f\in L^1(G) \mapsto f\_{|H}\in L^1(H)$ of functions which ... | https://mathoverflow.net/users/75338 | Is the conditional expectation faithful? | As $G$ and $H$ are amenable, we have that $C^\*(G) = C^\*\_r(G) \subseteq VN(G)$ the group von Neumann algebra. So let's prove the stronger result that $E:VN(G)\rightarrow VN(H)$ is faithful.
As $H$ is open, it is also closed (write $H$ as the complement of the union of cosets of $H$). It follows that $G/H$ is discre... | 3 | https://mathoverflow.net/users/406 | 302377 | 132,440 |
https://mathoverflow.net/questions/302365 | 3 | Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^\*$ and an N$\times$N covariance matrix $R\_s$ with its eigendecomposition $R\_s = U\_s\Lambda\_sU\_s^\*$. We also have the eigendecomposition of $HR\_sH^\*$ as $U\_A\Lambda\_AU\_A^\*$. In my research problem setting, I ... | https://mathoverflow.net/users/125418 | Connections between eigenvectors after matrix multiplication | $U\_s$ is not recoverable from $H$ and $U\_A$.
Consider the following examples, one each for $M < N, M > N, M = N$. In each example, there are 2 different $R$'s, $R1$ and $R2$, having different $U\_s$'s while having the same $U\_A$.
MATLAB output for $M=2, N=3$ example:
```
>> disp(H)
1 0 0
0 ... | 2 | https://mathoverflow.net/users/75420 | 302378 | 132,441 |
https://mathoverflow.net/questions/302313 | 4 | Let $d(n)$ denote the number of positive divisors of $n$. Is it known how to evaluate the sum
$$\displaystyle \sum\_{1 \leq m < n \leq X} d(m) d(n) d(n-m)?$$
A slightly more difficult question is if we change the height condition in the summation, to obtain the sum
$$\displaystyle \sum\_{\substack{1 \leq mn(n-m) ... | https://mathoverflow.net/users/10898 | A sum of divisor functions | The first problem is completely solved in the paper:
Tim Browning - The divisor problem for binary cubic forms.
J. Théorie Nombres Bordeaux 23 (2011), 579-602.
The method is to change the order of summation to reduce to a lattice point counting problem. One controls the error term using a variant of Dirichlet's hyp... | 8 | https://mathoverflow.net/users/5101 | 302383 | 132,442 |
https://mathoverflow.net/questions/302387 | 7 | $\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr\_M(r)=\Tr\_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the ... | https://mathoverflow.net/users/119736 | Is there a converse to the Brauer–Nesbitt theorem? | Not always — e.g. $g(1)$ should be an integer. The desired description is given in
*Helling, H.*, [**Eine Kennzeichnung von Charakteren auf Gruppen und assoziativen Algebren**](http://dx.doi.org/10.1080/00927877408548718), Commun. Algebra 1, 491-501 (1974). [ZBL0288.16019](https://zbmath.org/?q=an:0288.16019).
-... | 8 | https://mathoverflow.net/users/19276 | 302394 | 132,445 |
https://mathoverflow.net/questions/302348 | 8 | Consider the spherical shell (annulus)
$$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq |
x|\leq R+r \}.$$ Think of the limit $R \to \infty$.
Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are interested in the distribution of lattice points in $A(R,r)$.
From results on the Gauss circle problem in three dimensio... | https://mathoverflow.net/users/125281 | Are lattice points in thin spherical shells uniformly distributed? | Yes, they are equidistributed as long as $\delta<11/16$ and $r=R^{-\delta}$ and $R\to\infty$. Without loss of generality, we shall assume that $\delta>-1$ (i.e. $r<R$).
To see this, let $\mathcal{F}\subset S^2$ be a fixed convex region with piecewise smooth boundary on the unit sphere. Let $\mu(\mathcal{F})$ be the n... | 10 | https://mathoverflow.net/users/11919 | 302398 | 132,446 |
https://mathoverflow.net/questions/302385 | 9 | In their [paper](https://arxiv.org/abs/1508.07205), Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my limited understanding, $J^{\#}$ is a sort of a TQFT (with $\mathbb{F}\_2$ coefficient),... | https://mathoverflow.net/users/45553 | Is the instanton homology for webs and foams a categorified Chern-Simons? | It’s not clear what you mean by categorify here: the J# invariant doesn’t have a grading in their definition. Just taking dimension to get a number then doesn’t seem to yield a polynomial invariant.
However, they have a variation of this invariant for SU(3) representations whose dimension yields the number of Tait co... | 9 | https://mathoverflow.net/users/1345 | 302404 | 132,449 |
https://mathoverflow.net/questions/302409 | 14 | A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ordered field.
Is there a similar characterization of $\mathbf{R}$ as an ordered group?
Is any linearly ordered group th... | https://mathoverflow.net/users/6129 | Characterizing $\mathbf{R}$ as an ordered group | The linearly ordered group $(\mathbb{Z},+,\le)$ is a counterexample, but that is probably not what the OP had in mind. To give a detailed description of the situation, let us use the following notation:
* By a *linearly bi-ordered group* we mean a tuple $(G,\cdot,\le)$ where $(G,\cdot)$ is a group and $\le$ is a line... | 29 | https://mathoverflow.net/users/102946 | 302413 | 132,453 |
https://mathoverflow.net/questions/302286 | 8 | Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?
| https://mathoverflow.net/users/84390 | If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable? | An example of a non-separable Banach space $X$ with $|B(X)|=\mathfrak c$ is any non-separable Banach space $X$ whose dual $X^\*$ is $w^\*$-separable and has cardinality $|X^\*|=\mathfrak c$.
This follows from the observation that the map $B(X)\to B(X^\*)$, $T\mapsto T^\*$, is injective and hence for a countable $w^\... | 6 | https://mathoverflow.net/users/61536 | 302414 | 132,454 |
https://mathoverflow.net/questions/302390 | 9 | A little bird told me that Dirichlet/Chebotarev's Theorem is true not only for Dirichlet density, but also for analytic/natural density. I'm having trouble finding a citation for this. Does anyone know of one?
| https://mathoverflow.net/users/56362 | Dirichlet/Chebotarev's Theorem for natural/analytic density | First, here is some perspective on why most accounts only treat Dirichlet density: the difference between proving a theorem about primes using Dirichlet density or natural density is essentially about the difference between proving $L$-functions are nonzero at $s = 1$ or on the whole line ${\rm Re}(s) = 1$. Dirichlet d... | 14 | https://mathoverflow.net/users/3272 | 302419 | 132,456 |
https://mathoverflow.net/questions/302420 | 0 | If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a [partially ordered set](https://en.wikipedia.org/wiki/Partially_ordered_set) we let $\text{Max}(P)$ and the set of maximal elements.
Let $\frak{S}$ be the group of all bijections $f:\ome... | https://mathoverflow.net/users/8628 | $\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$ | As was written by Andreas Thom in his comment, Google gives an extensive literature on this subject. For example, look at the papers
<https://link.springer.com/chapter/10.1007/978-94-011-2080-7_18>
or <https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s2-42.1.85>
of Macpherson and his coauthors.
The qu... | 6 | https://mathoverflow.net/users/61536 | 302425 | 132,457 |
https://mathoverflow.net/questions/302382 | 2 | Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm looking for an elementary proof that there are infinitely many distinct (unitary) characters $\chi$ of $E^\times$ such that $... | https://mathoverflow.net/users/74112 | Characters of the kernel of the norm map of an extension of local fields | The sequences
$ 1 \rightarrow K \rightarrow E^\times \rightarrow \operatorname{Im} N\_{E/F} \rightarrow 1$
and
$ 1 \rightarrow \operatorname{Im} N\_{E/F} \rightarrow F^\times \rightarrow G \rightarrow 1$
are exact (the second by Local Class Field Theory). Taking Pontryagin duals we obtain exact sequences:
$ 1... | 0 | https://mathoverflow.net/users/74112 | 302435 | 132,460 |
https://mathoverflow.net/questions/302453 | 3 | Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover.
Now consider the following situation:
Everything I say in the following is with respect to the standard Borel sigma algebra with Lebesgue measure.
Let $M$ be a bounded measurable subset of $\mathbb R.$ For every $x... | https://mathoverflow.net/users/nan | How "compact" are sets of finite measure? | Vitali's covering theorem says if you take a sequence of balls $M\_{x\_i}^{\epsilon\_i}$ such that every $x \in M$ is covered by balls of arbitrarily small diameter, then there is a disjoint subsequence whose union contains $M$ except for a set of measure $0$.
| 4 | https://mathoverflow.net/users/13650 | 302454 | 132,466 |
https://mathoverflow.net/questions/301824 | 9 | Let $ f:A\to B$ be an injective, local homorphism between two Noetherian local rings. Consider the completions $\hat A$ and $\hat B$ with respect the maximal ideals. We have an induced homomorphism $\hat f: \hat A \to\hat B$. What assumptions do we need in order to ensure that also $\hat f$ is injective?
My main inte... | https://mathoverflow.net/users/65980 | When does completion preserve injectivity? | This may be useful:
>
> **Proposition** (Zariski) (see EGA I, (3.9.8) in Springer edition)
>
>
> *Let $f: (A,\mathfrak{m})\to (B,\mathfrak{n})$ be a local homomorphism of noetherian local rings. Assume that:*
>
>
> * *$f$ is injective.*
> * *$\hat{A}$ is a domain.*
> * *$f$ is essentially of finite type.*
>
> ... | 4 | https://mathoverflow.net/users/7666 | 302460 | 132,467 |
https://mathoverflow.net/questions/302467 | 5 | Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some **complex** $B \in \text{End}(\mathbb{C}^d)$.
>
> Does there exist $M \in \text{End}(\mathbb{R}^d)$ such that $A=\bigwedge^k M$?
>
>
>
In other words: Does every **real image** of the **comp... | https://mathoverflow.net/users/46290 | If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source? | Counterexample: $d=3$, $k=2$, $A = -I$, $B = iI$. There is no real $M$
because $\det \bigl(\bigwedge^2 M \bigr) = \det^2 M$ and $\det(A) = -1$.
| 9 | https://mathoverflow.net/users/14830 | 302469 | 132,472 |
https://mathoverflow.net/questions/302468 | 5 | Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ?
It is easy to see that the answer is yes when $R$ is integral domain by considering an ideal $r\mathfrak m$ for $r\in \mathfrak m $ and n... | https://mathoverflow.net/users/nan | local ring all whose non-maximal ideals are finitely generated | There exists no such non-noetherian local ring.
Below I assume by contradiction that we have such a ring.
(a) The first observation is a particular case Proposition 1.2(a) in your reference to Armendariz: for every $r\in R$, we have $r\mathfrak{m}\in\{\mathfrak{m},\{0\}\}$. Indeed, $r\mathfrak{m}$ is a quotient of ... | 12 | https://mathoverflow.net/users/14094 | 302474 | 132,474 |
https://mathoverflow.net/questions/302354 | 3 | Let be $S$ a separable(non compact) metric space and $X=C\_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive measures endowed with the variation norm. Denote by $\mathscr{P}(S)$ the subset of $rba(S)$ of all additive probability mea... | https://mathoverflow.net/users/98969 | Existence of a separating affine functional | Here is a characterization of those $\mu$'s which have the desired property:
**Theorem.** Let $S$ be an arbitrary topoligcal space, let $C\_b(S)$ denote the space of bounded real-valued continuous functios on $S$ (endowed with the supremum norm) and let $\mathscr{P}(S)$ denote the subset of the dual space $C\_b(S)'$ ... | 3 | https://mathoverflow.net/users/102946 | 302479 | 132,478 |
https://mathoverflow.net/questions/302480 | 3 | Let $H$ be a separable infinite-dimensional real Hilbert space. We consider operators in $H.$
Nuclear norm of a nuclear operator is the sum of its singular values.
A nuclear, positive and self-adjoint operator is called S-operator.
Does the following criterion hold true?
A sequence $A\_n$ of S-operators converge... | https://mathoverflow.net/users/125472 | Convergence of nuclear operators | Looks correct. Split $H=H\_n\oplus H^n$ where $H\_n=span(e\_1,\dots,e\_n)$. Let $P\_n$ and $P^n$ be the corresponding projections. The conditions are $P\_nA\_kP\_n\to P\_nAP\_n$, $\operatorname{Tr}(P^nA\_kP^n)\le\varepsilon\_n$ with $\varepsilon\_n\to 0$ as $n\to\infty$.
Now just observe that $A\_k-A+\delta n^{-1} P\... | 3 | https://mathoverflow.net/users/1131 | 302484 | 132,480 |
https://mathoverflow.net/questions/302486 | 18 | I am looking for a book or other reference which develops category theory 'from the ground up' assuming a healthy background in set and model theory, not one in homological algebra or Galois theory etc.
Up until now I have been satisfied by the heuristic that most of what takes place in category theory can be transla... | https://mathoverflow.net/users/92164 | Category theory for a set/model theorist | See [here](https://mathoverflow.net/questions/903/resources-for-learning-practical-category-theory) for a more comprehensive list with a slightly different audience in mind.
* Books aimed at various sorts of students:
[Awodey](http://angg.twu.net/MINICATS/awodey__category_theory.pdf), [Leinster](https://arxiv.org/a... | 20 | https://mathoverflow.net/users/2362 | 302489 | 132,482 |
https://mathoverflow.net/questions/302411 | 3 | In Milnor/Stasheff characteristic classes there is the proof that $H^\*(BO(n);\mathbb{Z}\_2)$ is the polynomial ring on the first n Stiefel-Whitney classes. I understand the part that the latter ring is contained in $H^\*(BO(n);\mathbb{Z}\_2)$, but for the other inclusion he writes:
>
> Let $C^i(BO(n))$ represent t... | https://mathoverflow.net/users/125439 | Milnor's proof of cohomology of BO(n) | One approach is as follows. All cohomology groups will be taken with coefficients $\mathbb{Z}/2$.
1. Define $w\_1(L)\in H^1(X)$ for real line bundles $L$ over $X$, and prove various properties. (There are several ways to do this, which I will not summarise here.)
2. Prove a real projective bundle theorem. In more det... | 4 | https://mathoverflow.net/users/10366 | 302493 | 132,483 |
https://mathoverflow.net/questions/302399 | 7 | Consider the class of functions
$$X:=\{f\in \mathcal{C}\_0^{\infty}(\mathbb{R})\;s.t.\;f\equiv 1 \mbox{ in a neighbourhood of}\;\;x=0\}$$
**Is it true that, for every $\varepsilon > 0$, I can find $f\in X$ such that $\|f\|\_{H^{1/2}(\mathbb{R})}<\varepsilon$?**
For $s\in[0,1/2)$, it is easy to show that the analo... | https://mathoverflow.net/users/54552 | Smallness of cut-off functions at critical Sobolev regularity | It is well known and easy to verify that (Exercise 14 p. 309 in [1])
$$
\log\Big|\log\sqrt{x^2+y^2}\Big|\in H^1(B^2(0,e^{-1}))
$$
so the trace of this function on the $x$-axis belongs to the trace space
$$
f(x)=\log\Big|\log|x|\Big|\in H^{1/2}((-e^{-1},e^{-1})).
$$
Let
$$
f\_t(x)=\begin{cases} 0 & \text{if } f(x)\leq t... | 4 | https://mathoverflow.net/users/121665 | 302496 | 132,485 |
https://mathoverflow.net/questions/302361 | 5 |
>
> Given a mapping in the Sobolev space $f\in W^{2,n}\_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the
> Sobolev regularity of the Jacobian $J\_f=\operatorname{det} Df$.
>
>
>
It is well known and easy to prove that if $u,v\in W^{1,p}\cap L^\infty(\mathbb{R}^n)$, then $uv\in W^{1,p}\cap ... | https://mathoverflow.net/users/121665 | Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping | Here is an answer from
[**Andrea Cianchi**](https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=260742):
A form of Hölder's inequality in Orlicz spaces asserts that, if $f\_1\in L^{A\_1},\ldots,f\_n\in L^{A\_n}$, and $B$ is such that
$$
A\_1^{-1}(t)\cdots A\_n^{-1}(t)\leq cB^{-1}(t)
\quad
\text{for $t\g... | 2 | https://mathoverflow.net/users/121665 | 302499 | 132,486 |
https://mathoverflow.net/questions/302498 | 6 | Here $Sp(2n,\mathbb{F}\_2)$ means the group of matrices preserving the form $\Omega = \left( \begin{array}{cc} 0&I \\ -I&0& \end{array} \right)$, i.e. the symplectic group over an even characteristic ($-1=1$). It is well known that, $Sp(2n,\mathbb{F}\_2) \leq SL(2n,\mathbb{F}\_2)$. Additionally both groups are generate... | https://mathoverflow.net/users/125476 | How much do we need to add to the generating set of the symplectic group to get $SL(2n,2)$? | For general $q$, it's not quite maximal. For $n>1$, the maximal $M$ subgroup of $G={\rm SL}(2n,q)$ containing $H={\rm Sp}(2n,q)$ contains $H$ as a subgroup of index $\gcd(q-1,n)$. So, as Noam D. Elkies points out in a comment, in the case $q=2$ it is indeed maximal for all $n>1$.
We have $M = \langle H, X \rangle$, w... | 8 | https://mathoverflow.net/users/35840 | 302514 | 132,489 |
https://mathoverflow.net/questions/302071 | 12 | Let $V$ a **real** vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor:
$$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, \,\psi(A)=\bigwedge^k A$$
>
> Is the image of $\psi$ closed in the standard topology on the $\text{Hom}$-space?
>
> ... | https://mathoverflow.net/users/46290 | Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$? | The answer is **negative**: In general $\psi({\rm End}(V))$ is not closed. Here is a proof when $d\ge4$ is *even* and $k=d-1$. Notice that $\Lambda^{d-1}V$ can be identified with $V$, so that $\psi(A)$ is just the cofactor matrix $\widehat{A}$.
${\rm End(V)}$ is the disjoint union of ${\rm GL}(V)$ and $\Delta$, the s... | 13 | https://mathoverflow.net/users/8799 | 302524 | 132,492 |
https://mathoverflow.net/questions/302026 | 17 | Let $\pi\_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}\_{n\_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\pi\_1$ and $\pi\_2$:
* $\pi\_1 \times \pi\_2$ (Rankin–Selberg product?)
* $\pi\_1 \boxplus \pi\_2$ (isobaric sum)
* $\p... | https://mathoverflow.net/users/38145 | Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$ | The right person to answer this question is probably Dinakar Ramakrishnan (a good working reference is his paper "[Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}(2)$](https://doi.org/10.2307/2661379)"), but since he doesn't seem to use MathOverflow, here is my understanding of this. ... | 15 | https://mathoverflow.net/users/3803 | 302527 | 132,493 |
https://mathoverflow.net/questions/302536 | 8 | Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional.
A well known theorem independently discovered by Eilenberg and Watts states that every $k$-linear, right exact, cocontinuous functor $F: A\mathsf{-Mod}\to B\mathsf{-Mod}$ is of the form $M\otimes... | https://mathoverflow.net/users/3041 | Eilenberg-Watts theorem for the derived category | Your first question seems to be a famous open question due to Rickard, see <https://academic.oup.com/jlms/article-abstract/s2-43/1/37/888935?redirectedFrom=PDF> .
It is known in several special cases, such as for hereditary algebras, see <http://home.ustc.edu.cn/~xwchen/Personal%20Papers/A%20note%20on%20standard%20e... | 4 | https://mathoverflow.net/users/61949 | 302537 | 132,496 |
https://mathoverflow.net/questions/302424 | 15 | Let $A = [a\_{ij}]\_{n\times n}$ be a Hermitian matrix, such that $|a\_{ij}| =1$ for $i \neq j$, and $a\_{ii} = 0$ for each $i$.
I am interested in a tight lower bound of $\|A\|\_\*:=\sum\_{i=1}^n |\lambda\_i(A)|$, where $\lambda\_i(A)$'s are eigenvalues of $A$.
Note that,
by minimizing $\sum\_{i=1}^n |\lambda\_i|$ ... | https://mathoverflow.net/users/53059 | Is this lower bound for a norm of some complex matrices true? | No, it is not; in fact, $2(n-1)$ is a local *maximum*.
Let $B$ be a Hermitian matrix such that $|B\_{ij}|=1$ and $B\_{ii}=1$. We denote its eigenvalues by $\mu$ (not to confuse them with eigenvalues of $A$). It is easy to see that always $\mu\le n$: if $(x\_1,\dots,x\_n)$ is an eigenvector and $|x\_i|=\max\_j|x\_j|$ ... | 5 | https://mathoverflow.net/users/9833 | 302540 | 132,497 |
https://mathoverflow.net/questions/302543 | 7 | In algebraic topology, we have the map lifting lemma which says that given a covering space $p:(\tilde{X},\tilde{x})\rightarrow (X,x)$ and a map $f:(Y,y)\rightarrow (X,x)$ with $Y$ connected and locally path-connected, a lift $\tilde{f}:(Y,y)\rightarrow(\tilde{X},\tilde{x})$ of $f$ exists if and only if $f\_\*(\pi\_1(Y... | https://mathoverflow.net/users/120280 | Map Lifting lemma and Etale fundamental group | The answer is yes, at least for finite covers (or even pro-finite covers). The key is to carefully translate everything into the language of finite $\pi\_1$-sets using the Galois correspondence. The canonical reference is of course SGA 1, Expose V - in particular $\S5,6$ and 7 once you're familiar with the basics.
By... | 8 | https://mathoverflow.net/users/15242 | 302548 | 132,498 |
https://mathoverflow.net/questions/302556 | 6 | There has been much research related to [web graphs](https://en.wikipedia.org/wiki/Webgraph) and [social graphs](https://en.wikipedia.org/wiki/Social_graph).
They can be thought of as a kind of [random graphs](https://en.wikipedia.org/wiki/Random_graph), but the point is that
they are different from the well-known [Erd... | https://mathoverflow.net/users/10446 | Citations graphs: what is known? | A great deal is known about citation graphs. For example, an [analysis](http://www.ist.drexel.edu/faculty/yan/publications/i2cs2002.pdf) of the citation graph for the computer science literature answers the three questions in the OP as:
• the connected component has a diameter of 18;
• the sparsity is characteris... | 6 | https://mathoverflow.net/users/11260 | 302559 | 132,500 |
https://mathoverflow.net/questions/302501 | 14 | There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M\_1, \omega\_1)$ and $(M\_2, \omega\_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M\_1, \omega\_1)\simeq \mathrm{Ham}(M\_2, \omega\_2)$ then there exists a diffeomorphism $f:M\_1\rightarrow M\_2$ such that ... | https://mathoverflow.net/users/nan | Recovering topological invariants of symplectic manifold from the group of Hamiltonian diffeomorphisms? | Let $M$ be compact and connected. For every closed $1$-form $\alpha$ on $M$ consider the Roger $2$-cocyle $\Psi\_\alpha$ on the Lie algebra $\operatorname{ham}(M, \omega)$ defined by
$$
\Psi\_\alpha(X\_f, X\_g) = \int\_M f \, \alpha(X\_g) \, \frac{\omega^n}{n!}.
$$
It was stated by Roger and proven by [Janssens & Vizma... | 5 | https://mathoverflow.net/users/17047 | 302560 | 132,501 |
https://mathoverflow.net/questions/302554 | 4 | For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):
finite groups $\leftrightarrow$ finite groups
discrete groups $\leftrightarrow$ compact groups
discrete torsion groups $\leftrightarrow$ profinite ... | https://mathoverflow.net/users/117693 | What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group? | Totally disconnected LCA groups are (profinite)-by-(discrete) LCA groups. Hence their Pontryagin dual are (compact)-by-(discrete torsion) groups. These are precisely **locally elliptic** LCA groups.
(I'm using the kernel-by-quotient convention.)
A locally compact group is called locally elliptic if each of its comp... | 8 | https://mathoverflow.net/users/14094 | 302563 | 132,503 |
https://mathoverflow.net/questions/302550 | 4 | Let $\{X\_t\}\_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let the support of $X\_1$ equal $[-1,1]$. Can the support of $(X\_1,X\_2)$ equal the unit disc centered at the origin?
(Under the stronger condition that the sequence is i.i.d., the support of the joint... | https://mathoverflow.net/users/125519 | Support of bivariate joint distribution of stationary and ergodic sequence | Sure!
Let $(U,V)$ be a pair of random variables with values from $[-1,1]$ such that
1. The joint distribution of $(U,V)$ is supported on the unit disk,
2. $U$ and $V$ have the same individual distributions.
For instance, you can choose $(U,V)$ uniformly at random from the unit disk.
Now construct a Markov proce... | 1 | https://mathoverflow.net/users/23297 | 302565 | 132,504 |
https://mathoverflow.net/questions/302539 | 8 | For theories with well known proof-theoretic-ordinals, (what) is there a correspondence between their proof-theoretic-ordinal and (ordinal indexed families of?) fast growing functions provable total in a given theory ?
| https://mathoverflow.net/users/8631 | Correspondence between proof-theoretic ordinals and fast growing functions? | Yes: for many theories, $\alpha$ is the proof-theoretic ordinal of T exactly when T proves that $f\_\beta$ is total for all $\beta<\alpha$, but does not prove $f\_\alpha$ is total. (Where $f\_\alpha$ is the $\alpha$-th function in the fast-growing hierarchy.)
[Avigad](http://www.andrew.cmu.edu/user/avigad/Papers/ordi... | 11 | https://mathoverflow.net/users/8991 | 302566 | 132,505 |
https://mathoverflow.net/questions/302497 | 3 | Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}\_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the entropy of $\pi$:
$$
\mathbb{E}\_\pi \left[ \ln \pi \right] \geq C
$$
Is there an analytical solution to $\min\_\pi F(... | https://mathoverflow.net/users/17596 | Minimizing the expectation of a functional of probability distribution subject to an entropy constraint | Here is the lower bound of $0.49$ for all $\alpha\ge 1$. Note that $\min\_\pi F(\pi)$ is a non-decreasing function of $\alpha$, so it is enough to consider $\alpha=1$. Also, the truth is about $0.55$ for $\alpha=1$ and $0.62$ for $\alpha=2$, so, if you care about entropy, the uniform distribution that gives $\frac 23$ ... | 2 | https://mathoverflow.net/users/1131 | 302577 | 132,509 |
https://mathoverflow.net/questions/302578 | 0 | Is there an example of a $n$ dimensional manifold $M$ and a natural number $k<n$ with a Lie subalgebra $L$ of $\chi^{\infty}(M)$ with the following property:
>
> For every $x\in M$ the space $\{V\_x \in T\_x M\mid V\in L\} $ is a $k$ dimensional vector space $D\_x$ but the distribution $D$ consisting of all $D\_x,\... | https://mathoverflow.net/users/36688 | A non integrable distribution arising from a Lie algebra of vector fields | No, this a consequence of the Frobenius theorem. Let $x\in M$, you can find $X\_1,..,X\_k\in L$ which is a basis of $D\_x$, for $Y,Z$ tangent to $D$ in a neighbourood $U$ of $x$, you can write $Y=f\_1X\_1+...+f\_kX\_k$ and $Y=g\_1X\_1+...+g\_kX\_k$, $[X,Y](y)\in D\_y, y\in U$. You can apply Frobenius.
Let $f,g$ be fu... | 4 | https://mathoverflow.net/users/80891 | 302579 | 132,510 |
https://mathoverflow.net/questions/302580 | 1 | Denote the [elementary symmetric functions](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial) in $n$ variables by $e\_k(x\_1, x\_2,\dots, x\_n)$. In the special case $x\_j=j$, simply write $e\_k(n)$ for $e\_k(1, 2, \dots, n)$. Next, define the sequence
$$a\_{+}(n)=\sum\_{k\geq0}(-1)^ke\_{2k}(n).$$
I a... | https://mathoverflow.net/users/66131 | Modular arithmetic and elementary symmetric functions | It follows that $e\_k(n) = \begin{bmatrix} n+1\\ n-k+1\end{bmatrix}$, i.e., unsigned Stirling numbers of first kind.
Now, let
$$f\_n(x) := \sum\_{k\geq 0} e\_k(n) x^k = (1+x)(1+2x)\cdots (1+nx).$$
Then
$$\sum\_{k\geq 0} e\_{2k}(n) x^{2k} = \frac{1}{2}(f\_n(x)+f\_n(-x))$$
and
$$a\_+(n) = \frac{1}{2}(f\_n(I)+f\_n(-I)) ... | 8 | https://mathoverflow.net/users/7076 | 302585 | 132,511 |
https://mathoverflow.net/questions/302445 | 9 | We know that, under MA, every linear order $(X,\le)$ with $|X|<\mathfrak c$ embedds in $\wp(\omega)/Fin$. Does this hold for linear orders with cardinality $\mathfrak c$?
| https://mathoverflow.net/users/122189 | Embeddings of linear orders in $\wp(\omega)/Fin$ under Martin's axiom | This is actually independent of $MA+\neg CH$.
For example, under $MA+OCA$ there are no gaps in $\mathcal{P}(\omega)/fin$ of type $(\mathfrak{c}, \mathfrak{c}^\ast)$, which excludes the possibility that every linear order of size $\mathfrak{c}$ embeds. This is because assuming $MA$, the linear order $L=(2^{<\mathfrak{... | 9 | https://mathoverflow.net/users/8843 | 302590 | 132,514 |
https://mathoverflow.net/questions/302593 | 0 | I feel that there is a good chance to apply certain integrals of
$\ \exp(-(\sum\_{k=1}^n x\_k))\ $ over corners (see below) to the analytic number theory. I have obtained two formulas to start with but am asking about their joint generalization. If all this is well known then let me be in the known too (and I apologiz... | https://mathoverflow.net/users/110389 | Corner integrals of $\exp$ | By a substitution $x\_k = A\_k y\_k$,
$$\begin{aligned}
I & := \idotsint\limits\_{\;\;\Delta(\mathbf{A})} \exp\left(-\sum\_{k=1}^n x\_k\right) dx\_1 \ldots dx\_n \\
& = \idotsint\limits\_{\;\;\Delta(\mathbf{1})} \exp\left(-\sum\_{k=1}^n A\_k y\_k\right) A\_1 \ldots A\_n dy\_1 \ldots dy\_n \\
& = \idotsint\limits\_{\;\;... | 3 | https://mathoverflow.net/users/108637 | 302602 | 132,518 |
https://mathoverflow.net/questions/302619 | 4 | A topological space $X$ is called a [Fréchet–Urysohn space](https://en.wikipedia.org/wiki/Sequential_space#Fr%C3%A9chet%E2%80%93Urysohn_space) if for each subset $A\subseteq X$ and for each point in its closure, $x\in\overline{A}$, there is a sequence (not just a net, but a sequence) $\{a\_n\}\subseteq A$ that converge... | https://mathoverflow.net/users/18943 | Fréchet–Urysohn subspaces in $[0,1]^{[0,1]}$ | An affirmative answer to this problem is given by a Theorem on page 216 of [this book](https://books.google.com.ua/books/about/Sequences_and_series_in_Banach_spaces.html?id=UVmqAAAAIAAJ&redir_esc=y) of Diestel. This theorem says that the space $B\_1[0,1]$ of functions of the first Baire class is angelic in the topology... | 3 | https://mathoverflow.net/users/61536 | 302621 | 132,522 |
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