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https://mathoverflow.net/questions/420757 | 2 | Some time ago I had a chat with a friend (and colleague) about some statement I wanted to prove. I was (and am) sure the statement is true, but couldn't prove it. I described some of my attempts and explained my difficulties.
After a month or so, he came to me and said he thought about it, tried different ideas but n... | https://mathoverflow.net/users/167834 | Is time spent without a result enough for authorship, in some cases? | For question 2, consider the following scenario.
There are two mathematicians. Alice chooses a problem and comes up with $N$ possible approaches to solve it. Bob tries $N-1$ of the approaches and can't make them work, and reports this to Alice, who tries the $N$th approach, and succeeds. I think it's clear that for $... | 8 | https://mathoverflow.net/users/18060 | 421234 | 171,360 |
https://mathoverflow.net/questions/421193 | 6 | I asked this question ten days ago on MathStackexchange (see [here](https://math.stackexchange.com/questions/4429556/linear-logic-and-linearly-distributive-categories)). Despite having placed a bounty on the question, I have not received any answers or comments until now. Following [Nick Champion's advice](https://meta... | https://mathoverflow.net/users/160778 | Linear logic and linearly distributive categories | Yes, Cockett and Seely's comment about proof theory is a reference to the theory/category adjunction. Each kind of theory corresponds to a kind of category, for instance:
| Theory | Category |
| --- | --- |
| simply typed lambda-calculus | cartesian closed category |
| intuitionistic multiplicative linear logic | ... | 6 | https://mathoverflow.net/users/49 | 421237 | 171,362 |
https://mathoverflow.net/questions/421210 | 4 | Let $\lambda=(\lambda\_1\geq\lambda\_2\geq\cdots\geq\lambda\_{\ell(\lambda)}>0)$ be an integer partition of $n\in\mathbb{N}$; i.e., $\lambda\_1+\cdots+\lambda\_{\ell(\lambda)}=n$.
One may now associate $f\_{\lambda}=\dim(\lambda)=\#SYT(\lambda)$ which is computed by $f\_{\lambda}=\frac{n!}{H\_{\lambda}}$ where $H\_{\... | https://mathoverflow.net/users/66131 | The fraction $\frac{g_{\mu}}{f_{\lambda}}$ is an integer | With computer search one finds that the premise is first violated at $n = 19$. The obstruction is as follows: consider $\mu\_1 = 1^{19}$, $\mu\_2 = 1^{17}2$, $\mu\_3 = 1^{16} 3$. We have $g\_{\mu\_1} = 1$, $g\_{\mu\_2} = {19 \choose 2} = 9 \cdot 19$, $g\_{\mu\_3} = 2{19 \choose 3} = 2 \cdot 3 \cdot 17 \cdot 19$. There ... | 10 | https://mathoverflow.net/users/106512 | 421248 | 171,367 |
https://mathoverflow.net/questions/421231 | 28 | (*Note:* This has been [asked on Math SE](https://math.stackexchange.com/q/3783159/42969), but without an answer after almost two years and one offered bounty.)
For an entire function $f$ let $M(r,f)=\max\_{|z|=r}|f(z)|$ be its maximum modulus function. $M(r, f)$ does not change
* if $f$ is replaced by $e^{i\varphi... | https://mathoverflow.net/users/116247 | Are entire functions “essentially” determined by their maximum modulus function? | This is a classical problem, but only partial results are available:
MR3155684
Hayman, W. K.; Tyler, T. F.; White, D. J.
The Blumenthal conjecture, in [*Complex Analysis and Dynamical Systems V*](https://web.archive.org/web/20220428174502/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.364.3299&rep=rep1&type... | 30 | https://mathoverflow.net/users/25510 | 421249 | 171,368 |
https://mathoverflow.net/questions/421265 | 4 | **Update:** In light of Fedor Petrov's answer, I added an additional requirement that all strings in $A$ and $B$ have Hamming weight *exactly* $n/2$, which hopefully makes the question more interesting.
Without this additional requirement on the Hamming weight, as Fedor Petrov pointed out, we can let $A$ contain all ... | https://mathoverflow.net/users/481320 | Bipartite version of Hamming bound (two families of codewords with large Hamming distance)? | If the Hamming distance between $A$ and $B$ is at least $d$, it yields that $B$ is disjoint from the $(d-1)$-neighborhood of $A$. By isoperimetric inequality for a Boolean cube (Harper's theorem), the smallest $d$-neighborhood of $A$ for a given size $|A|$ is attained when $A$ consists of all points with sum of coordin... | 11 | https://mathoverflow.net/users/4312 | 421266 | 171,371 |
https://mathoverflow.net/questions/421263 | 0 | I have already posted [this on stackexchange](https://math.stackexchange.com/questions/4437252/semi-simplicity-over-commutative-algebras-over-non-algebraically-closed-fields)
I have a question:
If k is an arbitrary field then is it true that if $M$ a finite dimensional $k[x, y]$ is semisimple as a $k[x]$ module and... | https://mathoverflow.net/users/92129 | Semi simplicity over commutative algebras over non-algebraically closed fields | There is an inseparable field extension $L/k$ with $L = k(a)$, such that the ring $L\otimes\_k L$ has nilpotents, so is not semisimple. See for example
<https://math.stackexchange.com/questions/345497/tensor-product-of-inseparable-field-extensions>
Now there is a surjective ring homomorphism $k[x,y]\to L\otimes\_k ... | 6 | https://mathoverflow.net/users/425351 | 421276 | 171,373 |
https://mathoverflow.net/questions/421267 | 1 | Consider the partial Loewner order $\le\_L$ for symmetric matrices: let $A,B$ be symmetric matrices of the same dimension, we say $A\le\_L B$ if $B-A$ is positive definite. Now let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a Lipschitz function, i.e., there exists $L\ge 0$ such that $ |f(x)-f(y)|\le L|x-y|$. I was wondering w... | https://mathoverflow.net/users/91196 | Lipschitz continuity and quadratic growth in Loewner order | $\newcommand\R{\mathbb R}\newcommand\ep{\varepsilon}$Such a statement does not hold for any $n\ge2$. Indeed, in view of the natural embedding of $\R^2$ into $\R^n$ for $n\ge2$, without loss of generality $n=2$.
The inequality in question is
$$2u^\top f(x)u^\top x\le\ep|u|^2+\frac C\ep\,(u^\top x)^2 \tag{1}\label{1}$$... | 2 | https://mathoverflow.net/users/36721 | 421277 | 171,374 |
https://mathoverflow.net/questions/421209 | 2 | Consider $f \in \mathbb{C}\{x\_1,\dots,x\_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x\_1,\dots,x\_n,t\}$ be a deformation of $f$ such that there exists some integer $m$ such that $(\partial\_t(F))^m \in \langle \partial\_1(F), \dots, \partial\_n(F) \rangle$.
Can we conclude that ... | https://mathoverflow.net/users/113200 | Deformation of isolated singularities and non zero divisors | I am just posting my comment as one answer. No, you cannot conclude that the image of $t$ is a nonzerodivisor modulo the ideal $I$.
Let $\ell$ and $k$ be positive integers $\geq 2$ with $\ell \geq 2k$. Consider the following polynomial,
$$ F=\ell(x\_1^{\ell+k}+x\_2^{\ell+k})-(\ell+k)t^2x\_1^\ell x\_2^\ell. $$ The par... | 4 | https://mathoverflow.net/users/13265 | 421282 | 171,376 |
https://mathoverflow.net/questions/421207 | 4 | In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\begin{cases}
\partial\_t \rho =\Delta\rho +\operatorname{div}(\rho\nabla V) & \mbox{for }t>0,x\in\Omega\\
(\nabla\rho +... | https://mathoverflow.net/users/33741 | Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity? | One has the ordering of the three $\lambda$'s, i.e.
$$
\lambda\_{\text{convex}} \leq \lambda\_{\text{LSI}} \leq \lambda\_{\text{SG}},
$$
where $\lambda\_{\text{convex}}$ is the one from convexity, $\lambda\_{\text{LSI}}$ is the one from the inverse Log-Sobolev constant (both as in your question) and $\lambda\_{\text{S... | 7 | https://mathoverflow.net/users/13400 | 421283 | 171,377 |
https://mathoverflow.net/questions/421290 | 3 | Let $\mathcal C$ and $\mathcal D$ be pre-additive (enriched over the abelian groups) categories and $F : \mathcal C \Rightarrow \mathcal D$ a functor which is left-exact in the sense that it preserves kernels (or, equivalently, equalizers). Can we prove that $F$ is semi-additive in the sense that $F(f + g) = F(f) + F(g... | https://mathoverflow.net/users/124454 | Does left-exactness imply semi-additivity? | So I don't know if assuming biproducts exists is enough or not but preserving kernel alone is not enough. Here is a counter-exemple. Let $C$ be the pre-additive category with only one object $\*$ and such that $Hom(\*,\*) = \mathbb{Z}$.
If we want, we can add a zero object to $C$ if we want $C$ to have all kernel.
... | 4 | https://mathoverflow.net/users/22131 | 421294 | 171,380 |
https://mathoverflow.net/questions/421002 | 2 | Consider a smooth surface of the following form
$$
S = \{f(x,y,t) = p\_0(t)x^2+p\_1(t)xy+p\_2(t)x+p\_3(t)y^2+p\_4(t)y+p\_5(t) = 0\}\subset\mathbb{A}^3
$$
over $\mathbb{Q}$, and set
$$
U\_S = \{t' \in \mathbb{Q} : |\: f(x,y,t') = 0 \text{ for some } (x,y)\in\mathbb{Q}^2\}\subset\mathbb{Q}.
$$
Is there any example of suc... | https://mathoverflow.net/users/14514 | Rational points on a special class of surfaces | I am just posting my comments as an answer. Without a hypothesis that the geometric generic fiber of $\pi:S\to \mathbb{A}^1\_t$ is irreducible, the result is false. For a smooth compactification of $S$ on which $\pi$ extends to a morphism to $\mathbb{P}^1\_t$, the finite part of the Stein factorization of $\pi$ is eith... | 5 | https://mathoverflow.net/users/13265 | 421297 | 171,382 |
https://mathoverflow.net/questions/421269 | 8 | Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P\_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n\_x(y)$ is the probability that a simple random walk started at $x$ ends at $y$ after $n$ steps).
Let further
$$
H\_n :=\displaystyle \sup\_{x \... | https://mathoverflow.net/users/18974 | Does entropy of the random walk control the return probability | This is a very partial answer.
For simple random walks on Cayley graphs, linear growth of $H\_n$ implies that
for any $\epsilon>0$, the inequality $r\_n \le \exp((\epsilon-1/2)n)$
must hold for infinitely many $n$, see Theorem 1.1 in [1].
For infinite connected graphs of bounded degree, it is possible for $H\_n$ to g... | 5 | https://mathoverflow.net/users/7691 | 421300 | 171,383 |
https://mathoverflow.net/questions/421299 | 1 | $\DeclareMathOperator\iCard{iCard}$In a prior posting [If we limit matters what ZFC can prove, would that be consistent?](https://mathoverflow.net/questions/420979/if-we-limit-matters-what-zfc-can-prove-would-that-be-consistent) to MO, I tried to capture the informal principle of *whatever ZFC proves, it is*, which I'v... | https://mathoverflow.net/users/95347 | What's the consistency status/strength of this limitation principle? | This principle is inconsistent: consider the formula $\theta(x)$ = "$x^+$ is the smallest infinite cardinal at which $\mathsf{CH}$ fails." The formula $\theta$ cannot hold on more than one infinite cardinal, let alone on all infinite cardinals, yet your principle (applied to $\phi:=\neg\theta$) would require this.
*(... | 4 | https://mathoverflow.net/users/8133 | 421301 | 171,384 |
https://mathoverflow.net/questions/421321 | 16 | A prime $p$ is called a [Sophie Germain prime](https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes) if $2p+1$ is also prime:
[OEIS A005384](https://oeis.org/A005384).
Whether there are an infinite number of such primes is unsolved.
My question is:
>
> If there are an infinite number of Germain primes,
> is... | https://mathoverflow.net/users/6094 | Sum of reciprocals of Sophie Germain primes | Here is a general result. For a sequence of nonnegative numbers $\{a\_n\}$, let $A(x) = \sum\_{n \leq x} a\_n$. For example, if $S \subset \mathbf Z^+$ and we set $a\_n = 1$ when $n\in S$ and $a\_n = 0$ when $n \not\in S$, then $A(x)$ is the number of elements of $S$ that are $\leq x$.
Exercise: If $A(x) = O(x/(\log ... | 23 | https://mathoverflow.net/users/3272 | 421324 | 171,389 |
https://mathoverflow.net/questions/421316 | 1 | Let $X\_{i}$ be a collection of iid random variables of cardinality $n$, and let $S\_{n}=\frac{1}{\sqrt{n}}\sum\_{i=1}^{n}X\_{i}.$
Let $|| X||:=\inf\_{B}\{E[\exp(X/B)-1]\leq 1\}$. This is the so-called sub-exponential norm.
What can be said about $|| S\_{n}||$ in terms of $||X\_{i}||$?
In the $L^{2}$ norm, we hav... | https://mathoverflow.net/users/116781 | Independent Sums and Orlicz Norms | 1. For the equality $\|S\_n\|\_{L^2}=\|X\_i\|\_{L^2}$ you need the zero-mean condition -- that $EX\_i=0$.
2. Let $X,X\_1,\dots,X\_n$ be any random variables with the same norm: $\|X\|=\|X\_1\|=\cdots=\|X\_n\|$.
Then, by the norm inequality for the sub-exponential norm $\|\cdot\|$,
\begin{equation\*}
\|S\_n\|\le\fra... | 1 | https://mathoverflow.net/users/36721 | 421325 | 171,390 |
https://mathoverflow.net/questions/421320 | 2 | In [Show that $\sum\limits\_pa\_p$ converges iff $\sum\limits\_{n}\frac{a\_n}{\log n}$ converges](https://math.stackexchange.com/questions/1458492/show-that-sum-limits-pa-p-converges-iff-sum-limits-n-fraca-n-log-n-c) it is said that this sequence of partial sums converges
$$
\begin{split}
\sum\_{1<n\leq N}\frac{a\_{n}}... | https://mathoverflow.net/users/481355 | One series converges iff the other converges | Since $a\_n$ is nonincreasing and nonnegative, $a\_n$ converges to some real $a\ge0$. If $a>0$, then neither one of the two series converges. It remains to consider the case $a=0$. Then
\begin{equation\*}
a\_n=\sum\_{j\ge n}b\_j \tag{1}\label{1}
\end{equation\*}
for some nonnegative $b\_j$'s.
By the prime number the... | 2 | https://mathoverflow.net/users/36721 | 421326 | 171,391 |
https://mathoverflow.net/questions/421191 | 7 | **Question:** *Is every (finitely generated) virtually free group residually finite?*
A well-known question asks whether every hyperbolic group is residually finite (Mladen Bestvina. Questions in geometric group theory. <http://www.math.utah.edu/> ~bestvina/eprints/questions- updated.pdf.). My question is a very spec... | https://mathoverflow.net/users/69681 | Is every virtually free group residually finite? | As already mentioned, it is an easy exercise to show that a group containing a residually finite subgroup of finite index is itself residually finite. But proving that free groups are residually finite is (standard but) not so easy. There exist several possible arguments. Here are two of my favourites.
**First proof.... | 6 | https://mathoverflow.net/users/122026 | 421331 | 171,394 |
https://mathoverflow.net/questions/421117 | 6 | I must first preface that while this is indeed a question on an exercise, I believe this is advanced enough for MathOverflow.
Let $\kappa$ be a regular uncountable cardinal. Recall that the notion of a stationary subset makes sense for subsets of limit ordinals of uncountable cofinality. In Jech's *Set Theory* (Third... | https://mathoverflow.net/users/146831 | Properties of Jech's hierarchy of stationary sets (Exercise 8.13, 8.14 of Jech) | After some googling I found [the notes by J. D. Monk](http://euclid.colorado.edu/%7Emonkd/jech.pdf), which have answered the questions in the span of Theorem 2.68 to 2.91.
| 2 | https://mathoverflow.net/users/146831 | 421344 | 171,397 |
https://mathoverflow.net/questions/421372 | 1 | Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}\_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}\_\ell^\times$ to construct the "Lang sheaf" $\mathscr{L}\_\varphi$ [$\S$1.4, SGA 4 $\frac{1}{2}$, Sommes trig.]. This is a rank... | https://mathoverflow.net/users/131975 | Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules? | Question 1: Yes. See Lemma 2.14 of my paper [On the Ramanujan conjecture for automorphic forms over function fields](https://arxiv.org/abs/1805.12231) with Nicolas Templier, although this simple argument is surely not original to us.
Question 2: Any rank one connection has the form $\partial-f$ for a function $f$ on ... | 5 | https://mathoverflow.net/users/18060 | 421374 | 171,406 |
https://mathoverflow.net/questions/421371 | 7 | **Statement:** Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T\_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $\sum\_{n=0}^{+\infty} \dim(\mathcal{S}^n V)^K\,t^n$ (encoding the dimension of the invariants on the $n$-th symmetr... | https://mathoverflow.net/users/17064 | Invariants for the isotropy representation of a Riemannian symmetric space | One reference is in Helgason's 1984 book *Groups and Geometric Analysis*. The result you want appears there as Corollary 5.12.
The notation he uses is $X=G/K$ is a symmetric space where $G$ is connected and semisimple and $K$ is a maximal compact. As usual, write ${\frak{g}} = {\frak{k}} + {\frak{p}}$ as the Cartan d... | 6 | https://mathoverflow.net/users/13972 | 421382 | 171,410 |
https://mathoverflow.net/questions/421309 | 3 | I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ matrix, $I$ is the $m\times m$ identity matrix, $\Lambda$ is an $m\times m$ diagonal matrix, and $X$ is an $n \times m$ mat... | https://mathoverflow.net/users/140356 | Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration | 1. As Iosif Pinelis pointed out, you presumably meant
$$ \lambda\_i^{-1} = \frac{e\_i^T M e\_i}{\lVert M e\_i \rVert\_2^2}, $$
where $e\_i$ is the $i$th standard basis vector,
which is indeed a reasonable choice, and can be motivated as being the unique minimizer of the Frobenius norm $$ \lVert I - \Lambda^{-1} M \rVer... | 2 | https://mathoverflow.net/users/70005 | 421392 | 171,414 |
https://mathoverflow.net/questions/421383 | 2 | For $t\in(-1,1)$, let
$$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$
and
$$g(t):=\frac1{f(t)}.$$
Note that the functions $f$ and $g$ are even.
**Question 1:** Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$?
**... | https://mathoverflow.net/users/36721 | Coefficients of certain Taylor series | Note that
$$\ln(2g(t))=\frac{1}{2} \,\ln \left(1-t^2\right)+ t \tanh ^{-1}(t)
=\sum\_{k=1}^\infty\frac{t^{2k}}{2k(2k-1)}.$$
This immediately yields the positive answer to Question 2.
| 2 | https://mathoverflow.net/users/36721 | 421393 | 171,415 |
https://mathoverflow.net/questions/421397 | 3 | Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
| https://mathoverflow.net/users/85664 | How close are two Gaussian random variables? | As the measure of the closeness of two distributions $p\_A$ and $p\_B$ You could use the [Bhattacharyya coefficient](https://en.wikipedia.org/wiki/Bhattacharyya_distance)
$$w=\int \sqrt{p\_A(x)p\_B(x)}\,dx\in[0,1],$$
which for two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$) is given by $w=e^{-d}$ with
... | 7 | https://mathoverflow.net/users/11260 | 421398 | 171,416 |
https://mathoverflow.net/questions/421218 | 4 | A crucial aspect of the Bruhat–Tits theory of affine buildings is the [Bruhat–Tits fixed-point theorem](https://en.wikipedia.org/wiki/Bruhat%E2%80%93Tits_fixed_point_theorem), which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an affine building and $S$ is a closed, bounded, convex... | https://mathoverflow.net/users/2383 | Fixed points on spherical buildings | Since spherical buildings are CAT(1), we get a fixed point if $$\mathop{\rm rad}S<\tfrac \pi 2.$$
| 5 | https://mathoverflow.net/users/1441 | 421410 | 171,420 |
https://mathoverflow.net/questions/421405 | 3 | I am currently helping teach a course about foundations of mathematics, which has thus far focused mostly on propositional and first-order logic. As part of the course, the students are each required to present a lecture about a specific piece of material. We recently had a student "prove" quantifier-elimination in alg... | https://mathoverflow.net/users/175051 | Tarski's original proof of quantifier elimination in algebraically closed fields | I doubt you’ll find a shorter *proof* than Swan’s which is equally elementary. In particular:
* For algebraically closed fields, you can stop in the middle of page 10 of the document, which should make it less overwhelming.
* Tarski’s published papers on this are longer and more difficult to read (or were for me), an... | 5 | https://mathoverflow.net/users/nan | 421411 | 171,421 |
https://mathoverflow.net/questions/421391 | 2 | Assume that two smooth quasi-projective vareities $X,Y$ have the same class in the Grothendieck ring of varieties.
>
> Do the symmetry products $X^{(n)}$ and $Y^{(n)}$ also have the same class?
>
>
>
Moreover, if there are some feasible criteria to judge if two varieties have the same class?
Edit: I leave th... | https://mathoverflow.net/users/nan | Symmetry product in Grothendieck ring of varieties | This is just a quick explanation. Let $k$ be an algebraically closed field. Let $\mathcal{Var}\_k$ denote the set of (embedded) quasi-projective $k$-varieties. The Grothendieck group of varieties, $K\_0(\mathcal{Var}\_k)$, is the quotient of the free Abelian group on $\mathcal{Var}\_k$ by all relations, $$[X] = [U]+[C]... | 7 | https://mathoverflow.net/users/13265 | 421426 | 171,427 |
https://mathoverflow.net/questions/420955 | 5 | In Pressley and Segal's book *Loop Groups*, they define a "basic inner product" $\langle-,-\rangle$ on a simple Lie algebra to be (minus) the Killing form scaled so that $\langle h\_\alpha,h\_\alpha\rangle=2$ where $h\_\alpha$ is the coroot associated to a long root.
[**Aside** I believe there is some terminological ... | https://mathoverflow.net/users/4177 | What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C? | I found a reference that gave the correct inner products explicitly, without requiring the reader to assemble the relevant facts: Chapter II, section 1.2 (bottom of page 583) of:
* McKenzie Y. Wang, Wolfgang Ziller, *On normal homogeneous Einstein manifolds*, Annales scientifiques de l'École Normale Supérieure, Série... | 1 | https://mathoverflow.net/users/4177 | 421475 | 171,443 |
https://mathoverflow.net/questions/421445 | 4 | I have $$X\_i \sim N(0,1), \quad S\_n=X\_1+\cdots+X\_n,$$
$$ \mathscr{S}\_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum\_{i=1}^{n} \left[ S\_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X\_{i}(\omega) \right] \textbf{1}\_{ \left( \frac{i-1}{n}, \frac{i}{n} \right] } (t) $$ and let $$d(f,g) := \sup\_{x \in [0,1]}| ... | https://mathoverflow.net/users/481401 | Distance between trunctated random walk and its normal form | You have
$$
\begin{aligned}
&\sup\_{n\ge1}\sqrt{E[d\left( \mathscr{S}\_n^{(v) }, \mathscr{S}\_n \right)^2]} \\
&=\sup\_{n\ge1}\|d\left( \mathscr{S}\_n^{(v) }, \mathscr{S}\_n \right) \|\_2 \leq 2 \sqrt{E \left[ X\_1^2; |X\_1| > v \right] } \text{ for all } v>0.
\end{aligned}
$$
So, by the Chebyshev/Markov inequality,... | 2 | https://mathoverflow.net/users/36721 | 421479 | 171,444 |
https://mathoverflow.net/questions/421480 | 2 | Let $k$ be a field with $\operatorname{char} k = 0$. Let $L$ be a Lie $k$-algebra. Then the universal envelope $U(L)$ is a PI-algebra iff $L$ is abelian.
Remark:
PI-algebra means polynomial identity algebra, an algebra that satisfies a nonzero polynomial.
Obviously, if $L$ is abelian, then $U(L)$ is commutative. Bu... | https://mathoverflow.net/users/476040 | The universal envelope U(L) is a PI-algebra iff L is abelian | The article is available [here](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=smj&paperid=4949&option_lang=eng), it seems that it hasn't been translated. The result you're interested in is Теорема 2; the proof is to notice that a non-abelian Lie algebra in characteristic zero contains as a subalgebra either... | 1 | https://mathoverflow.net/users/43309 | 421482 | 171,445 |
https://mathoverflow.net/questions/421465 | 8 | Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}\_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is *harmonic* if
$\varphi(\emptyset)=1$ and
\begin{equation}
\varphi(u)=\sum\_{\stackrel{\scriptstyle u \, \triangleleft \, v}{|v| \, = \, |u| + ... | https://mathoverflow.net/users/70119 | Harmonic flow on the Young lattice | Let us identify $s\_\lambda$ with the character of the irreducible representation $S^\lambda$ of the symmetric group $S\_n$ indexed by the partition $\lambda$. Then
$$
s\_\Box^{n-k}({\bf x})
\sum\_{|\mu| = k} s\_\mu({\bf x}) \dim(\mu,\lambda)
$$
defines the character of a certain representation of $S\_n$. We can explic... | 5 | https://mathoverflow.net/users/159272 | 421484 | 171,447 |
https://mathoverflow.net/questions/421493 | 3 | Let $H$ be a monoid (written multiplicatively) with the property that $H = H^\times A H^\times$ for some finite $A \subseteq H$ (shortly, an f.g.u. monoid), where $H^\times$ is the group of units of $H$.
>
> **Question.** Is it true that, if $H$ is duo (i.e., $aH = Ha$ for every $a \in H$), then it's also unit-duo ... | https://mathoverflow.net/users/16537 | An f.g.u. duo monoid is unit-duo: True or false? | This is false. Let $G$ be a group with a non-normal subgroup $A$ with finitely many double cosets (eg $G=S\_3$ and $A$ generated by a transposition). Consider $M=A'\cup G$ where $A'$ is a group isomorphic to $A$ via $a'\mapsto a$. Here $G$ is a two-sided ideal of $M$, $A'$ is the group of units, $G$ and $A'$ multiply a... | 4 | https://mathoverflow.net/users/15934 | 421497 | 171,452 |
https://mathoverflow.net/questions/421500 | 1 | I thought of the following large cardinal axiom, extending the notion of $\theta$-upliftingness:
>
> Let $\eta$ be be an ordinal, and $X$ be a class of ordinals. $\kappa$ is called $\eta$-iteratively uplifting onto $X$ iff, for every ordinal $\theta$, there is a monotonically increasing sequence $(\gamma\_i)\_{0 \l... | https://mathoverflow.net/users/473200 | How do chains of elementary extensions compare to shrewdness? | As [Cantor's Attic](http://cantorsattic.info/Uplifting#Consistency_strength_of_uplifting_cardinals) explains, if $\kappa$ is 0-uplifting, that is, there is a cardinal $\lambda \gt \kappa$ such that $V\_\kappa \prec V\_\lambda$ and $\lambda$ is inaccesible, $\lambda$ has a club subset $C$ of cardinals such that $V\_\kap... | 2 | https://mathoverflow.net/users/352898 | 421507 | 171,456 |
https://mathoverflow.net/questions/421211 | 4 | Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of [Tits building of a linear algebraic group](https://mathoverflow.net/questions/299422/tits-building-of-a-linear-algebraic-group).
I reference the papers [Ti: Tits -... | https://mathoverflow.net/users/2383 | Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group? | [@UriyaFirst](https://mathoverflow.net/questions/421211/does-the-building-of-parabolics-of-a-semisimple-group-have-a-simplex-correspon#comment1082314_421211) points out that every abstract simplicial complex must have an empty face of dimension $-1$, and it would make sense for this to correspond to the entire group. (... | 2 | https://mathoverflow.net/users/2383 | 421512 | 171,458 |
https://mathoverflow.net/questions/421389 | 11 | **Question:** I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the image of the Bockstein homomorphism $H^{2}(X,\mathbb{Z}\_2) \rightarrow H^{3}(X,\mathbb{Z})$ maps the second Stiefel... | https://mathoverflow.net/users/99732 | Examples of 6-manifolds without an almost complex structure | Turning comments into answer: An example of a closed 6-manifold not admitting an almost complex structure is $S^1 \times (SU(3)/SO(3))$. From the obstruction theory for lifting the map $M \to BSO(6)$ classifying the tangent bundle of an oriented 6-manifold through $BU(3) \to BSO(6)$, one sees that the unique obstructio... | 10 | https://mathoverflow.net/users/104342 | 421515 | 171,459 |
https://mathoverflow.net/questions/421513 | 0 | Suppose $x>0$ and let $f(x)=\sum\_{k\le x}\frac{1}{\varphi(k)}$, where $\varphi(k)$ is the Euler totient function. It is well known that $\sum\_{k\le x}\frac{1}{k}\sim\log x$. What is the asymptotic behavior of the sum $f(x)=\sum\_{k\le x}\frac{1}{\varphi(k)}$?
| https://mathoverflow.net/users/160959 | Asymptotic behavior of the sum $\sum_{k\le x}\frac{1}{\varphi(k)}$ | Here is a question that addresses the mentioned asymptotics - <https://math.stackexchange.com/questions/2683190/showing-sum-n-leq-x-frac1-phi-n-c-log-x-o1?noredirect=1&lq=1>
Roughly, you can proceed by the standard convolution method via approximating $1/\varphi(n)$ as $1/n$.
| 2 | https://mathoverflow.net/users/95838 | 421517 | 171,461 |
https://mathoverflow.net/questions/421492 | 1 | Let $a\_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int\_{\mathbb{R}} \int\_{\mathbb{R}^n} \alpha w(t) e(\alpha (a\_1t\_1 + \dotsb + a\_n t\_n)) dt\,d \alpha.
$$
If the factor $\alpha$ were not present ... | https://mathoverflow.net/users/84272 | Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$? | The OP asks whether the integral is 0, the answer is no in general. For example, let me take $n=2$, $a\_1,a\_2>0$, $w(t\_1,t\_2)=\theta(t\_1)\theta(1-t\_1)\theta(1-|t\_2|)$, with $\theta(x)$ the unit step function. Then
$$I=\begin{cases}
\frac{i}{2\pi a\_1a\_2}&\text{if}\; a\_1>a\_2\\
0&\text{if}\; a\_2>a\_1.
\end{case... | 1 | https://mathoverflow.net/users/11260 | 421518 | 171,462 |
https://mathoverflow.net/questions/420250 | 6 | Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla\_A\phi\rvert^2\geq \lvert D\_A\phi\rvert^2$ for some $C>0$. My question is what's the best value of $C$ one can hope for? Ideally this should depend on the... | https://mathoverflow.net/users/131004 | Weitzenböck formula and comparison of norms | I am not an expert in all the delicate points of clifford algebras and spin structures, but I think the following shows the constant can't be 1 in general: let $(M,g)$ be a Riemannian manifold of dimension $n$. Take the usual Dirac operator $d+\delta:\Omega^{\*}(M)\rightarrow\Omega^{\*}(M)$ and the covariant derivative... | 3 | https://mathoverflow.net/users/144247 | 421523 | 171,463 |
https://mathoverflow.net/questions/421467 | 2 | I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int\_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$How can I compute probability densities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgu... | https://mathoverflow.net/users/102071 | Probability density of a hyperplane for a Gaussian distribution | $\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}$First, one should not denote a random vector in $\R^n$ (which is not actually a vector in $\R^n$ but a function with values in $\R^n$) and a true, non-random vector in $\R^n$ by the same symbol, such as $\mathbf x$.
Accordingly, let $X\sim N(0,S)$, where $S:=H^{-1}$... | 3 | https://mathoverflow.net/users/36721 | 421527 | 171,464 |
https://mathoverflow.net/questions/421529 | 2 | For real numbers $t>0$ and $x$, let $f(x)=\sum\_{k=1}^Ne^{ikx}$ and $g(t)=\int\_{-t}^{t}\lvert f(x)\rvert^2dx$. Then $g(\pi)=\int\_{-\pi}^{\pi}\lvert f(x)\rvert^2dx=2\pi N$.
I want to know is there any results about the value of $g(t)$ for small $t$ relevant to $N$. In particular, what is the asymptotic behavior (or ... | https://mathoverflow.net/users/160959 | For estimation on the integral $g(t)=\int_{-t}^{t}\left\vert\sum_{k=1}^Ne^{ikx}\right\rvert^2dx$ for small $t>0$ | Using the formula for the sum of the first $n$ terms of a geometric series, we have
$$|f(x)|^2=\frac{\sin^2(Nx/2)}{\sin^2(x/2)}$$
and hence for $t\downarrow 0$
$$g(t)=2\int\_0^t |f(x)|^2\,dx
=2\int\_0^t \frac{\sin^2(Nx/2)}{\sin^2(x/2)}\,dx \\
\sim8\int\_0^t \frac{\sin^2(Nx/2)}{x^2}\,dx
=4N\,\int\_0^{Nt/2} \frac{\si... | 5 | https://mathoverflow.net/users/36721 | 421530 | 171,465 |
https://mathoverflow.net/questions/421539 | 4 | One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi\_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\times \mathbb{R})$ and a form $\omega\in \Omega^2(M,M\times \mathbb{R})$ which is closed and non degenerate, i.e. $d\ome... | https://mathoverflow.net/users/171666 | Existence of non-trivial "line-symplectic" manifolds | Let's assume that your connection is flat. This is a reasonable assumption, since more generally, you would at least need that your isomorphisms preserve the connection 2-form, which is a pretty stringent (non-topological) requirement in dimensions $\geq 4$, so unless you have a good reason to prescribe a specific curv... | 3 | https://mathoverflow.net/users/66405 | 421554 | 171,470 |
https://mathoverflow.net/questions/421538 | 9 | For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta\_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta\_K(t)$ is the Alexander-Conway polynomial.
One explanation is that it’s the unique nontrivial order $2$ finite-type invariant, so it’s go... | https://mathoverflow.net/users/113402 | Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$? | Given a knot in $S^3$, think of it as an embedding
$$f : S^1 \to S^3.$$
The configuration space of $5$ distinct points in $S^3$ is denoted $C\_5(S^3)$, this is a $15$-dimensional manifold and it consists of all $5$-tuples of distinct points in $S^3$.
Similarly, we can talk about $C\_5(S^1)$, but since $S^1$ has a... | 6 | https://mathoverflow.net/users/1465 | 421555 | 171,471 |
https://mathoverflow.net/questions/421560 | 1 | Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\mathbb C^g\to GL(r,\mathbb C)$ such that $J(\gamma'\gamma,x)=J(\gamma',\gamma x)J(\gamma,x)$. If $f:M\to M$ is a holomorp... | https://mathoverflow.net/users/356774 | Pull-back of factor of automorphy | I think every holomorphic map $f:\mathbb{C}^g/\Gamma\to \mathbb{C}^g/\Gamma$ lifts to a $\Gamma$-equivariant holomorphic map $\tilde{f}:\mathbb{C}^g\to \mathbb{C}^g$ (indeed, every holomorphic map is a composition of a homomorphism, that lifts, with a translation, that lifts too).
Hence a factor of automorphy giving ... | 3 | https://mathoverflow.net/users/7031 | 421563 | 171,474 |
https://mathoverflow.net/questions/421551 | 2 | I have looked through books such as Matrix Analysis by R.A. Horn and C.R. Johnson and would not find an answer to the following question:
Given $V^TV \in S^{n}$, where $V$ is an invertible matrix with each column of $V$ of unit length. Can the norm of $(V^TV)^{-1}$ be bounded above by a constant that does not depend ... | https://mathoverflow.net/users/155703 | An upper bound on an invertible matrix | As noted in the comments, the quantity you want is $\sigma\_{\min}(V)^{-2}$, the inverse square of the minimum singular value of $V$.
Unfortunately you can't get any meaningful bound from below for matrices with unit column norms, as they can still be arbitrarily close to singular. For instance,
$$
\begin{bmatrix}
1 ... | 0 | https://mathoverflow.net/users/1898 | 421565 | 171,476 |
https://mathoverflow.net/questions/421470 | 9 | Let $A=KQ$ be a path algebra over a field $K$ with finite connected quiver $Q$.
A slope function $\mu$ is a function of the form $\mu=\sigma/dim$ defined on the Grothendieck group $K\_0(A) \setminus 0$ (without the zero module), where $\sigma$ is linear and dim is just the sum of entries or equivalently the dimension o... | https://mathoverflow.net/users/61949 | Is this a counterexample to Reineke's conjecture on total stability conditions for Dynkin type quivers? | Your argument looks correct to me. Note that the corresponding question for the derived category has been answered positively, and a parameterisation of total stability conditions given by [QiuYu and ZhangXiaoting](https://arxiv.org/abs/2202.00092). In the case of $\mathsf{E}\_7$, this space is $7$-dimensional (over $\... | 4 | https://mathoverflow.net/users/21483 | 421577 | 171,478 |
https://mathoverflow.net/questions/421582 | 6 | **Motivation:** Take an algebraic number $\lambda$. In my research, I've stumbled upon the question in which cases the expression $\sum\_{\sigma \in S} \sigma(\lambda)$, where $S$ is a subset of the field embeddings of $K=\mathbb{Q}(\lambda)$, can be 'less irrational' then $\lambda$ itself.
**Question:** Take a finit... | https://mathoverflow.net/users/481532 | Cancellation of irreducibility for Galois conjugates | No.
Let $a, b \in \mathbb Q(i)$. Let $\alpha\_1$ be a root of $x^3 + ax + b$. Let $L$ be the field generated by $i, \alpha\_1, \overline{\alpha}\_1$. Assume that $a,b$ are sufficiently general that $L$ has Galois group $S\_3 \wr \mathbb Z/2$, i.e. $(S\_3 \times S\_3 ) \rtimes \mathbb Z/2$.
Let $K$ be the subfield g... | 8 | https://mathoverflow.net/users/18060 | 421585 | 171,481 |
https://mathoverflow.net/questions/421574 | 3 | Let $p$ be a prime number and let $q = p^2$. Let $C$ be a separated scheme of finite type over $\mathbb F\_q$ of dimension $1$.
If we know that for every $\alpha \in \mathbb Z\_{>0}$, "the number of $Spec(\mathbb F\_{q^{\alpha}})$-points on $C$" $= C\_1 q^{\alpha} + C\_2p^{\alpha}$ for some $C\_1, C\_2 \in \mathbb Z$... | https://mathoverflow.net/users/481539 | If we have a nice formula for number of points on a curve over finite fields, can we get some geometric information of the curve from the formula? | This formula implies that the zeta function of $C$ is given by the formula
$$\zeta\_C( u) = e^{ \sum\_{\alpha=1}^{\infty} | C(\mathbb F\_{q^\alpha})| u^\alpha / \alpha } = e^{ \sum\_{\alpha=1}^{\infty} (C\_1 q^{\alpha} + C\_2 p^\alpha) u^\alpha / \alpha } = \frac{1}{ (1 - q u)^{C\_1} (1 - p u)^{C\_2}} $$
Now, the z... | 11 | https://mathoverflow.net/users/18060 | 421587 | 171,483 |
https://mathoverflow.net/questions/421573 | 3 | If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$.
But if $g$ is a little bit more than $L^s$, say $L^s \log L$ , can we say that $fg$ is a little bit more than $L^1$ ? For instance $L^1 \log L^1$ ?
| https://mathoverflow.net/users/481540 | Hölder inequality between different Orlicz spaces | Yes, we can say so. Indeed, let us show that the conditions $f\in L^r$ and $g\in L^s\ln L$ imply $fg\in L\ln^t L$ for $t:=1/s$. Moreover, we shall show that the value $t=1/s$ here is optimal, as it cannot be replaced by any greater value. Of course, by $h\in L\ln^t L$ we mean $\int |h|\ln^t(|h|+1)<\infty$.
let $\psi\... | 2 | https://mathoverflow.net/users/36721 | 421588 | 171,484 |
https://mathoverflow.net/questions/421581 | 3 | According to [MacMahon formula](https://en.wikipedia.org/wiki/Plane_partition) the total number $P\_3(r, s, t)$ of plane partitions that fit in the $r \times s \times t$ box $\mathcal{B}(r,s,t)$ is equal to the following product formula:
$$
P\_3(r,s,t)=\prod\_{(i,j,k)\in \mathcal{B}(r,s,t)}\frac{i+j+k-1}{i+j+k-2}=\pr... | https://mathoverflow.net/users/136218 | Total number of plane partitions for $4$ or more dimensions | My comments above were a bit condensed, so let me spell things out in a little more detail here.
First of all, there is a question of what "dimension" one should consider a plane partition to be. When we say a plane partition "fits inside an $r\times s \times t$ box" we are treating it as a 3-dimensional object. But ... | 6 | https://mathoverflow.net/users/25028 | 421590 | 171,485 |
https://mathoverflow.net/questions/421571 | 4 | Any number with of a form $\frac{1}{n}$ has a decimal with a repetend of finite length that is never longer than $n$ (provable by Dirichlet principle). (Example: $\frac{92}{99}=0.929292\ldots$ in which case it is 92 that is repeating and the length of the series is 2.) Is there a way to find ALL numbers of the form $\f... | https://mathoverflow.net/users/481506 | Is there a way to specify a special kind of reciprocals of natural numbers? | This is a textbook example of a question for which one should turn to the [OEIS](https://oeis.org/) for assistance. The first few elements of this set are
$$ 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, \dotsc$$
and the OEIS then tells you that these are the [full r... | 20 | https://mathoverflow.net/users/766 | 421598 | 171,488 |
https://mathoverflow.net/questions/421596 | 5 | Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]\_0$. We work with the category of $\Delta$-generated spaces equipped with the mixed model structure. Suppose that the injective model structure exists (the paper <http://dx.doi.org/10.4... | https://mathoverflow.net/users/24563 | Fibrant replacement of an injective model category of enriched diagrams | Section 8 of my paper [All (∞,1)-toposes have strict univalent universes](https://arxiv.org/abs/1904.07004) shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of Corollary 8.16). I think this will apply to your situation if the hom-objects of $\... | 5 | https://mathoverflow.net/users/49 | 421602 | 171,490 |
https://mathoverflow.net/questions/421611 | 24 | Loosely inspired by the game [Abalone](https://en.wikipedia.org/wiki/Abalone_(board_game)), I've encountered the following simple problem I cannot solve.
Suppose that we are given a finite set of marbles on an infinite chessboard.
One move consists of one marble jumping over another to an empty space.
For exam... | https://mathoverflow.net/users/955 | Can an odd number of marbles jump to infinity? | With $5$ you can using the following moves:
```
..... ..... ..... ..... ..... ..... ..... ..oo. ...oo
..... ..o.. ..o.. ..o.. ..oo. ...oo ..ooo ..ooo ..ooo
.oo.. .oo.. .oo.. ..oo. ..oo. ..oo. ..oo. ..... .....
ooo.. oo... .oo.. ..oo. ..o.. ..o.. ..... ..... .....
```
So the only... | 34 | https://mathoverflow.net/users/172802 | 421612 | 171,495 |
https://mathoverflow.net/questions/421616 | 7 | $\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and geometric morphisms, where a geometric morphism points in the direction of its *inverse* image functor. Then $\Logos$ is a non-... | https://mathoverflow.net/users/2362 | Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic? | Regarding *monadicity* (rather than comonadicity), the (2-categorical variant of the) question is answered in Bunge–Carboni's [The symmetric topos](https://doi.org/10.1016/0022-4049(94)00157-X). In their paper, $\mathbf A$ denotes the 2-category of locally presentable categories and cocontinuous functors (i.e. left adj... | 7 | https://mathoverflow.net/users/152679 | 421620 | 171,496 |
https://mathoverflow.net/questions/421607 | 6 | Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set
$$
D\_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0
$$
Show that $D\_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to 0$.
**I'm trying to use the fact that $C\_{c}^{1}(\mathbb{R}$) is dense in $W^{1,p}(\mathbb{R}$)**
| https://mathoverflow.net/users/481556 | Exercise 8.13 - Brezis | The proof is not short, because it is done from first principles, without using any theorems about Sobolev space except its definition.
By the definition of $W^{1,p}$, there exist $v\_n \in C\_{c}^{1}(\mathbb{R})$
and $w \in L^p(\mathbb{R})$ such that $v\_n \to u$ in $L^p(\mathbb{R})$ and $v\_n' \to w$ in $L^p(\mathb... | 18 | https://mathoverflow.net/users/7691 | 421625 | 171,499 |
https://mathoverflow.net/questions/421622 | 1 | Let $\{x\_{n}\}\_{n=0}^{\infty}$ be decreasing sequence of non-negative reals. Suppose that there exist constants $a, s>0$ and $b>1$ such that $$x\_{n+1}\leq ab^{n}x\_{n}^{1+s}$$ and $$x\_{0}\leq a^{-1/s}b^{-1/s^{2}}.$$ Is it true that then $$\lim\_{n\to \infty}x\_{n}=0?$$
Any help is appreciated!
| https://mathoverflow.net/users/163368 | Sequence of reals such that $x_{n+1}\leq ab^{n}x_{n}^{1+s}$ converges to $0$? | By induction on $n$, we check that
$$x\_n\le a^{-1/s}b^{-1/s^2-n/s}$$
for all integers $n\ge0$.
Now the desired result immediately follows.
---
The condition that $x\_n$ is decreasing in $n$ was not needed or used.
| 4 | https://mathoverflow.net/users/36721 | 421635 | 171,501 |
https://mathoverflow.net/questions/421576 | 1 | The infinite sums involving mobius function and a multiplicative function has got quite interest in past. In particular, sums of the form $$\sum\_{d=1}^{\infty}\frac{\mu(d)}{f(d)}$$ for mobius function $\mu$ and multiplicative function $f$ have been investigated for various $f.$ I am interested in knowing about any arg... | https://mathoverflow.net/users/160943 | Non-negativity of an infinite absolutely convergent sum | In general, it is better to approach such a question numerically, since your sum is absolutely convergent. However, in your particular case, it is possible to compute this explicitly without any numerical calculations. Notice that non-zero summands that appear in your sum correspond to squarefree $d$ (otherwise $\mu(d)... | 5 | https://mathoverflow.net/users/101078 | 421645 | 171,503 |
https://mathoverflow.net/questions/421648 | 3 | Consider the sentence $\mathtt{PSP}\_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set".
A priori this sentence is weaker than the usual $\mathtt{PSP}$, since $\mathtt{PSP}\_\mathfrak{c}$ requires the set not only to be uncountable, but to be of the size of th... | https://mathoverflow.net/users/141146 | $\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$ | Here are the answers you're looking for:
1. No.
2. No.
3. Yes, no large cardinals needed! (Which explains the previous two answers.)
Look no further than John Truss' paper:
>
> *Truss, John*, [**Models of set theory containing many perfect sets**](http://dx.doi.org/10.1016/0003-4843(74)90015-1), Ann. Math. Logi... | 5 | https://mathoverflow.net/users/7206 | 421651 | 171,505 |
https://mathoverflow.net/questions/421644 | 2 | I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf\_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup\_{r \in [0,u]}|X\_r|=\frac{\pi}{2\sqrt{2}}\text{ a.s.}$$
| https://mathoverflow.net/users/138491 | Chung's law of the iterated logarithm for Brownian motion | A detailed proof with weakened conditions is given by Pakshirajan in [in a 1959 paper.](https://doi.org/10.1137%2F1104036)
>
> In the present work the results of K. L. Chung (1948) concerning the
> maximum partial sums of sequences of independent random variables are
> obtained for a weaker condition. The method em... | 2 | https://mathoverflow.net/users/11260 | 421653 | 171,506 |
https://mathoverflow.net/questions/421646 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). A set $M\subseteq E$ consisting of mutually disjoint members of $E$ is said to be a *matching*. We say $S\subseteq V$ is *matchable* if there is a matching $M$ such that $\bigcup M = S$.
One might think that[Zorn's Lemma](https://en.wikipedia.... | https://mathoverflow.net/users/8628 | Maximal matchable set in hypergraph with finite edges | Let $n = 2$, and $H$ be the complete bipartite graph with halves $V\_1$ a copy of $\mathbb{N}$ and $V\_2$ a copy of $\mathbb{R}$. Let $r(M)$ be an element of $V\_2$ not covered by a matching $M$. If $M$ doesn't cover an $x \in V\_1$, extend $M$ with $(x, r(M))$, otherwise increment every $V\_1$-endpoint in $M$, and add... | 2 | https://mathoverflow.net/users/106512 | 421672 | 171,512 |
https://mathoverflow.net/questions/421659 | 4 | A Banach space $X$ has *property (V)* whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to Y$ is an isomorphism on a subspace of $X$ isomorphic to $c\_0$.
The space $X$ has the *Grothendieck property* whenev... | https://mathoverflow.net/users/39421 | Does property (V) imply the Grothendieck property for dual Banach spaces? | I need a few preliminaries:
A Banach space $X$ is a Grothendieck space if and only if every bounded linear $T:X\to c\_0$ is weakly compact.
A bounded linear operator $T:X\to Y$, between two Banach spaces $X$ and $Y$, is either unconditionally converging or fixes a copy of $c\_0$.
If $V$ is a subspace of $c\_0$ th... | 6 | https://mathoverflow.net/users/164350 | 421674 | 171,513 |
https://mathoverflow.net/questions/421599 | 2 | More specifically, let $B$ be a open ball and $C, D$ be open disjoint sets in $\mathbb{R}^n$, $n>1$. Suppose that $B\cap C\neq\emptyset$ and $B\cap D\neq\emptyset$, furthermore, $B\subset \bar{C}\cup\bar{D}$. Is there **at least** one path in $B\cap\partial C$?
Edit: for what i need, the statement actually can be a l... | https://mathoverflow.net/users/481551 | Is there at least one path in the common boundary of two open sets? | The answer to the second question is yes: there is an arc containing uncountable points of $B\cap\partial C$. It is enough to prove it in the case $n=2$.
Applying an affine transformation if necessary, we can suppose that $[0,1]^2\subseteq B$, $[0,1]\times\{0\}\subseteq C$ and $[0,1]\times\{1\}\subseteq D$. This impl... | 1 | https://mathoverflow.net/users/172802 | 421677 | 171,514 |
https://mathoverflow.net/questions/421675 | 2 | Let $T$ be a single directed tree, by parameters $(\kappa, \lambda, \zeta)$ of $T$ we mean: the number of root nodes in $T$, the strict upper bound on the number of children nodes per a node in $T$, the strict upper bound on the level of a node in $T$ respectively.
>
> Would ZFC be interpreted in a Graph theory tha... | https://mathoverflow.net/users/95347 | Can ZFC sets be interpreted as single rooted trees with accessible degree and countable height? | The details of your graph theory will matter - how, for example, are you going to experess "$(1,\mathsf{icc}, \omega)$" in your setting? - but certainly some version of this will work: in a (well-founded) model $M$ of $\mathsf{ZFC}$ there's a natural way to code sets by well-founded trees, and so a "rich enough" theory... | 3 | https://mathoverflow.net/users/8133 | 421679 | 171,515 |
https://mathoverflow.net/questions/421663 | 2 | $\newcommand\Psh{\mathit{Psh}}\newcommand\Pr{\mathit{Pr}}$Let $\Psh$ be the category of presheaf categories and cocontinuous functors which preserve tiny objects. There is a functor $(-)^\ast : \Psh \to \Psh$ sending $\Psh(C) \mapsto \Psh(C^\text{op})$. This functor is an involution in the sense that $(\Psh(C)^\ast)^\a... | https://mathoverflow.net/users/2362 | Is there a "duality involution" on presentable categories? | The answer is no, even if you restrict to the full subcategory of $Pr^L$ spanned by the $Psh(C)$'s. I'll answer in the $1$-categorical case but : a- the $\infty$-categorical case follows because presentable $1$-categories are presentable $\infty$-categories and b- even if it didn't strictly follow, one easily convinces... | 3 | https://mathoverflow.net/users/102343 | 421684 | 171,517 |
https://mathoverflow.net/questions/421660 | 3 | Fix an integer $n \ge 5$. Let $\mathcal{V}$ be a *countable* collection of closed subvarieties of $\mathbb{P}^n\_{\mathbb{C}}$ of codimension at least $2$. Choose a point $p \in \mathbb{P}^n$. Does there exist a curve $C$ (affine or projective) containing the point $p$ and not intersecting any subvariety $V \in \mathca... | https://mathoverflow.net/users/45397 | Moving lemma for countable collection of subvarieties | Consider the case when $C$ is a line through $p$. Lines through $p$ correspond to points of $\mathbb P^{n-1}$, and this gives a projection map $\mathbb P^n \setminus p \to \mathbb P^{n-1}$ Then $C$ intersects $V$ if and only if the image of $V \setminus p$ under the projection map doesn't contain the point correspondin... | 2 | https://mathoverflow.net/users/18060 | 421696 | 171,519 |
https://mathoverflow.net/questions/421700 | 4 | Let $p, q\geq 2$, $s\geq p$ and $f,g$ be non-negative smooth enough functions. Then why does the following inequality hold: $$-f^{q-2}g^{s}|\nabla f|^{p}+f^{q-1}g^{s-1}|\nabla f|^{p-1}|\nabla g|\leq C(s, q)(-|\nabla (f^{\frac{p+q-2}{p}})|^{p}g^{s}+|\nabla g|^{p}g^{s-p}f^{p+q-2}),$$ for some constant depending $C(s, q)$... | https://mathoverflow.net/users/163368 | Using Young's inequality to show elementary inequality? | This inequality cannot hold in general. Indeed,
\begin{equation\*}
|\nabla(f^{\frac{p+q-2}{p}})|^p=k^pf^{q-2}|\nabla f|^p,
\end{equation\*}
where
\begin{equation\*}
k:=\frac{p+q-2}p=1+\frac{q-2}p\ge1.
\end{equation\*}
So,
at all points where $g>0$, $|\nabla g|>0$, and $|\nabla f|>0$, we can rewrite the inequality in... | 6 | https://mathoverflow.net/users/36721 | 421701 | 171,520 |
https://mathoverflow.net/questions/421633 | 3 | For a given $a\in \mathbb{Z}$, define $P(a)$ to be the set of all prime numbers dividing $a$. Also define $\mathcal{P}$ to be the set of all prime numbers. Let $a,b,c\in \mathbb{Z}\setminus \{0\}$ be such that neither $\frac{b}{c}$ nor $\frac{c}{b}$ is a power of $a$. Then is it true that $$\mathcal{P}\setminus\cup\_{n... | https://mathoverflow.net/users/481562 | Question about iterations not divisible by infinitely many prime numbers | Yes. This follows from a result of Corrales-Rodrigáñez and Schoof (see the paper [here](https://www.sciencedirect.com/science/article/pii/S0022314X97921144?via%3Dihub)) solving the support problem of Erdős.
In particular, suppose that there are only finitely many primes $p$ that do not divide $ca^{n} - b$ for any $n$... | 7 | https://mathoverflow.net/users/48142 | 421703 | 171,521 |
https://mathoverflow.net/questions/421686 | 2 | Let's say we have two matrices $M$ and $G$ with $G, M \in \{0, 1\}^{n, n}$, we denote by $m\_{i, j}$ the element of $M$ in the $i^\text{th}$ row and $j^\text{th}$ column, same for $G\_{i, j}$.
Let's define $K$ the matrix resulting from the matrix operation $G \oplus M$ as follows:
$$\forall i, j \in [1\ldots n] \ \... | https://mathoverflow.net/users/481615 | What is the name of a matrix operation using the OR operator instead of addition? | Your operation is known as Boolean matrix multiplication. There is a considerable literature on efficient algorithms; see for example [An improved combinatorial algorithm for Boolean matrix multiplication](https://doi.org/10.1016/j.ic.2018.02.006) by Huacheng Yu.
| 3 | https://mathoverflow.net/users/3106 | 421713 | 171,525 |
https://mathoverflow.net/questions/421711 | 6 | Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is *computably saturated* (or *recursively saturated*) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{L}$-formulas in the variables $\bar{x}y$ and any $\bar{a} \in M^{\bar{x}}$, if $\Sigma(\bar{a},y)$ is finitely satis... | https://mathoverflow.net/users/83901 | How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated? |
>
> The answer to Question 1 is positive (thus the answer to Question 2 is also positive). More explicitly, the positive answer to Question 1 follows from the following well-known facts:
>
>
>
**Lemma 1.** $(M,\mathbb{P},G)$ *is parametrically definable in* $M[G]$.
**Lemma 2.** $M[G]$ *is recursively saturated... | 8 | https://mathoverflow.net/users/9269 | 421715 | 171,526 |
https://mathoverflow.net/questions/352974 | 5 | Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is positive.
Now consider also the *left* Perron eigenvector $u^T A=\lambda u^T$. Another result of Perron-Frobenius theory i... | https://mathoverflow.net/users/108314 | Significance of the length of the Perron eigenvector | That quantity $s = \frac{|u^Tv|}{\|u\|\|v\|}$ is the inverse of the eigenvalue condition number. The smaller it is, the more sensitive to perturbation the Perron value is.
More precisely, any perturbed matrix $A+E$ with $\|E\| \leq \varepsilon$ has a Perron value $\tilde{\lambda}$ that satisfies $|\tilde{\lambda}-\la... | 3 | https://mathoverflow.net/users/1898 | 421727 | 171,528 |
https://mathoverflow.net/questions/421305 | 1 | I'm reading [Tawfik - The Yamabe problem](https://www.math.mcgill.ca/gantumur/math580f12/Yamabe.pdf): the PDE is
$$
\Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1}
$$
**Theorem (Yamabe)**. For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\infty}$ strictly positive $\varphi\_{q}$ satisfying \eqref{1... | https://mathoverflow.net/users/469129 | A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used | ### Application of Holder's inequality
Notice that the estimate on the Green's function means that
$$ \int |G(P,Q)|^\alpha ~dQ $$
is bounded whenever $\alpha < \frac{n}{n-2}$ (and the bound can be taken to be uniform; that is independent of $P$).
By Holder's inequality, we have
$$ \int G(P,Q) F(Q) ~dQ \leq \|G(P,-)\|... | 2 | https://mathoverflow.net/users/3948 | 421743 | 171,532 |
https://mathoverflow.net/questions/421748 | -3 | In Voisin's book *Hodge theory and complex algebraic geometry, I* Section 9.1.2, p.223, the author writes:
>
> Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi\_\*$ is a morphism of holomorphic vector bundles $T\_{\mathcal X}\to \phi^\*T\_B$.
>
>
>
If my understanding is righ... | https://mathoverflow.net/users/99826 | Pull back a vector field | You are **not** pulling back vector fields.
You are pulling back the vector bundle $T\_B$ to be a bundle over $\mathcal{X}$. (See, e.g. <https://en.wikipedia.org/wiki/Pullback_bundle> for a description.)
Notice that in general, the nomenclature "pushforward of a vector field" is imprecise. When the mapping is not b... | 4 | https://mathoverflow.net/users/3948 | 421750 | 171,536 |
https://mathoverflow.net/questions/421737 | 3 | Let $A$ be a unital C\*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}\_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$
Suppose that $A$ satisfies the following property:
For every $S\subseteq A$ there is a projection $q\in A$ such that
$\operatorname{Ann}\_r(S)=Aq$.
>
> Q. Is $A$ necessarily a von Neumann... | https://mathoverflow.net/users/84390 | Impact of annihilators in C*-algebras | An AW${}^\*$-algebra is a C${}^\*$-algebra which satisfies this condition for both right and left annihilators. So every AW${}^\*$-algebra has your property, and any C${}^\*$ algebra that is isomorphic to its opposite algebra has your property iff it is AW${}^\*$.
There are lots of AW${}^\*$-algebras that aren't von ... | 11 | https://mathoverflow.net/users/23141 | 421753 | 171,538 |
https://mathoverflow.net/questions/421754 | 2 | As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series
$$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$
Thus, to compute $\log n$ with $n>1$, we may employ the series
$$\log n=-\log\left(1-\frac{n-1}n\right)=\sum\_{k=1}^\infty\frac{1}k\left(\frac{n-1}n\right)^k,$$
which converges at geometr... | https://mathoverflow.net/users/124654 | What's the fastest way to compute $\log n$ for $n>1$? | Theorem 9.1 by [Brent](https://arxiv.org/abs/1004.3412) states the following:
>
> If $x>0$ is a precision $n$ number, then $\log(x)$ may be evaluated to precision $n$ in time
> $\sim13M(n) \log\_2 n$ as $n\to\infty$ [assuming $\pi$ and $\log(2)$ precomputed to precision $n+O(n/ \log(n))$].
>
>
>
Here
$$M(n)=O(... | 10 | https://mathoverflow.net/users/36721 | 421757 | 171,539 |
https://mathoverflow.net/questions/421759 | 0 | Does the the equivalence of Total variation distance formulas presented here([https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf](https://ece.iisc.ac.in/%7Eparimal/2019/statphy/lecture-14.pdf)) assumes that the two distributions are symmetrical ?
| https://mathoverflow.net/users/481678 | Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical? | All the expressions for the total variation distance given in Definition 1.1 and Propositions 1.2, 1.4, 1.7 in the linked lecture notes hold in general, without any symmetry assumptions.
Indeed, the proofs there do not use any symmetry assumptions.
Moreover, all these expressions hold when $\mathscr X^N$ is replace... | 1 | https://mathoverflow.net/users/36721 | 421762 | 171,541 |
https://mathoverflow.net/questions/421733 | -5 | If by a size preserving model we mean any bijection between any two elements of it is an element of it. Then:
>
> is it a thoerem of $\sf ZFC$ that for any theory $T$ any two equinumerous size preserving models of $T$ are isomorphic? That is, there is a bijection between the domains of those models that preserves t... | https://mathoverflow.net/users/95347 | Are equinumerous size preserving models of a theory isomorphic? | There is no version of this question I can think of which has an affirmative answer. Let $\alpha,\beta$ be distinct countable ordinals such that $L\_\alpha\equiv L\_\beta\equiv L\_{\omega\_1^L}$ (which exist by downward Lowenheim-Skolem + condensation). Then $L\_\alpha\not\cong L\_\beta$ (since distinct levels of $L$ a... | 5 | https://mathoverflow.net/users/8133 | 421784 | 171,547 |
https://mathoverflow.net/questions/421768 | 1 | Let $g \in R^{d}$ have $iid$ Gaussian components. Let $a \in R^{d}$, and let $b \in R^{d}$. be arbitrary vectors.
Consider the random variable $Y\_{g,g}:= \frac{1}{n}\langle g,a \rangle \langle g, b \rangle$. What can be said about the tails of the random variable $Y$?
If $g$ were replaced with $\bar{g}$ a gaussian... | https://mathoverflow.net/users/116781 | Non-independent Sub-gaussian variables and concentration | Let $X:=a\cdot g$ and $Y:=b\cdot g$. We want to bound the tails of the random variable (r.v.) $XY$ (the factor $\frac1n$ is clearly inessential). The r.v.'s $X$ and $Y$ are zero-mean jointly normal, with $Var\,X=|a|^2$, $Var\,Y=|b|^2$, and $Cov\,(X,Y)=a\cdot b$, where $|\cdot|$ is the Euclidean norm. By further rescali... | 2 | https://mathoverflow.net/users/36721 | 421785 | 171,548 |
https://mathoverflow.net/questions/421778 | 2 | $\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree of a dominant morphism $C\to\mathbb P^1$.
I am interested in curves $C$ such that there is only one linear system $g\_d... | https://mathoverflow.net/users/119184 | Curves having only one linear system realizing its gonality | A generic $d$-gonal curve of genus $g$ satisfies this property unless $g \leq 2d-2$. So the only possible restriction for curves with this property is $g \geq 2d-1$, which I believe follows from Brill-Noether-Petri theory for $d>2$.
Indeed, let $C$ be a generic such curve and $\pi : C \to \mathbb P^1$ the projection ... | 5 | https://mathoverflow.net/users/18060 | 421786 | 171,549 |
https://mathoverflow.net/questions/421669 | 15 | We work in ZFC throughout. The following question was posed to me by a friend:
>
> Can there exist cardinals $\kappa,\lambda$ such that $\lambda<\mathrm{cof}(\kappa)$ and $2^\lambda<\kappa<\kappa^\lambda$?
>
>
>
Originally I thought this should be easy, after all many minor variants are almost immediate: if we... | https://mathoverflow.net/users/30186 | Can $\kappa^\lambda$ be large if $2^\lambda$ is small and $\lambda<\mathrm{cof}(\kappa)$? | It is consistent that such a pair exists, see my paper [Singular cofinality conjecture and a question of Gorelic](https://arxiv.org/abs/1506.07634).
To show that some large cardinals are needed, suppose for example $\lambda=\aleph\_0 < \aleph\_1=cf(\kappa)$ and $\kappa^\omega > \kappa > 2^\omega.$ Then for some $\mu ... | 15 | https://mathoverflow.net/users/11115 | 421797 | 171,553 |
https://mathoverflow.net/questions/421800 | 2 | I am confused in finding the right bound for the following oscillatory integral
$$I = \int\_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth cutt-off function supported on the annulus
$A:= \{ 2^{k-1} \leq | \xi| \leq 2^{k+1} \}$, $y \in \mathbb{R}$, $t >0$ ... | https://mathoverflow.net/users/471464 | Estimate for an oscillatory integral of the first kind | Write $s=\eta t$ and note that $I(s,y)$ solves the 1D Schrödinger equation $iI\_s-3I\_{yy}=0$. Thus it satisfies the sharp estimate $|I(s,y)|\le c\_0s^{-1/2}$ where $c\_0$ is a multiple of $\int|I(0,y)|dy$. Now, $I(0,y)$ is the Fourier transform of $\chi(2^{-k}\xi)$ where $\chi=\psi^2$, that is to say $I(0,y)=2^k\wideh... | 2 | https://mathoverflow.net/users/7294 | 421803 | 171,555 |
https://mathoverflow.net/questions/421804 | 0 | Let us consider the spaces $C\_\infty(E)$, $C\_c(E)$, $C\_b(E)$, where $E$ is locally compact, $C\_\infty(E）$ is all continuous functions vanishing at the ends of $E$, $C\_c(E)$ is all the continuous functions with compact support and $C\_b(E)$ is all the bounded continuous functions.
How do we find the dual spaces $C^... | https://mathoverflow.net/users/147009 | How to calculate the dual spaces of the following spaces? | There is a concrete but disappointing answer to your question. Let me discard $C\_c(E)$ as it is not a Banach space and let us focus on what you call $C\_\infty$ but it is more commonly denoted by $C\_0(E)$. In this case the dual space is the space of all Borel measures on $E$ with the total variation norm, see also [A... | 5 | https://mathoverflow.net/users/15129 | 421807 | 171,557 |
https://mathoverflow.net/questions/421812 | 2 | $\DeclareMathOperator\PSL{PSL}$(Classical, finitely generated) Schottky groups are groups generated by finitely many hyperbolic elements of $A\_i\in \PSL(2,\mathbb{C}), $ $i<n$ such that the isometric circles of $\{A\_i,A\_i^{-1}\}\_{i<n}$ are pairwise disjoint.
Quasi-Fuchsian groups are discrete subgroups $\Gamma$ o... | https://mathoverflow.net/users/62647 | Is every finitely generated classical Schottky group quasifuchsian? | Yes - every Schottky group is quasi-fuchsian. See Lemma 1 of Chuckrow's paper "On Schottky Groups with Applications to Kleinian Groups" published in Annals of Mathematics, 1968.
The argument there is nice. Start with a different, classical, Schottky group $\Gamma'$ where all of the circles are perpendicular to one, g... | 3 | https://mathoverflow.net/users/1650 | 421813 | 171,559 |
https://mathoverflow.net/questions/421806 | 1 | I’m considering a $H^1$ function u on a open domain D. Is the integral:
$$ \int\_{\partial B\_r(x)} u \hspace{2pt}dH^{n-1}$$
continuous with respect to x?
I tried to prove that it’s differential by showing that the derivative can be written as the integration:
$$ \int\_{\partial B\_r(x)} Du \hspace{2pt}dH^{n-1}... | https://mathoverflow.net/users/348579 | About the continuity of the integral on the boundary of a ball | I would say so. Denote your integral by $b\_u(x)=\int\_{|x-y|=r}u(y)dH^{n-1}$.
Approximate $u$ in $H^1$ with test functions $u\_j$. The property is certainly true for $u\_j$ thus it is enough to prove that $b\_{u\_j}\to b\_u$ uniformly. You can estimate $|b\_{u\_j}(x)-b\_u(x)|$ with the $L^2$ norm of the trace of $u-u\... | 1 | https://mathoverflow.net/users/7294 | 421818 | 171,560 |
https://mathoverflow.net/questions/421773 | 0 | We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected *uniformly at random* from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\in\mathbb{R}^d$, and $H:=\{\mathbf{x}\in\mathbb{R}^d : \langle\mathbf{x},\mathbf{v}\rangle=0\}$ be the *random* hyperpl... | https://mathoverflow.net/users/115803 | Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions | Given $v\_1$ and $v\_2$ in $\mathbb{R}^d$ linearly independent, their Gram matrix $G$ is defined to be
$$ G = V^T V, $$
where $V = (v\_1 v\_2)$ is the $d \times 2$ matrix having $v\_1$ as first column and $v\_2$ as second column. More explicitly, we have
$$ G = \begin{pmatrix} (v\_1, v\_1) & (v\_1, v\_2) \\ (v\_2, v\_1... | 2 | https://mathoverflow.net/users/81645 | 421819 | 171,561 |
https://mathoverflow.net/questions/421627 | 4 | Say I have two integrable codistributions
$$ U = \langle du^1, \ldots, du^m \rangle, \qquad Z = \langle dz^1, \ldots, dz^N \rangle $$
on a manifold $M$, with $N >> m$. Suppose that the intersection $U \cap Z$ is nontrivial of rank $m'$ (with $0 < m' < m$) and completely nonintegrable, (i.e., $(U\cap Z)^{(\infty)} = \la... | https://mathoverflow.net/users/18048 | Question about differential operators in a completely non-integrable distribution | Consider the following example: On $\mathbb{R}^4$ with coordinates $u^1,u^2,u^3, z^1$, define $z^2 = u^2 - z^1 u^1$ and $z^3 = u^3 - z^1u^2$.
We have that $U\cap Z$ is spanned by $\mathrm{d}u^2-z^1\,\mathrm{d}u^1$ and $\mathrm{d}u^3-z^1\,\mathrm{d}u^2$, and its last derived system is zero. Following the OP's descript... | 3 | https://mathoverflow.net/users/13972 | 421847 | 171,570 |
https://mathoverflow.net/questions/421859 | 16 | Below, I mean smooth oriented closed connected manifolds and smooth maps (but am happy to hear about the topological category, or unoriented manifolds, etc instead).
Say that $X^n$ has the Hopf property if two maps $f\_0,f\_1 : M^n\to X^n$ are homotopic if and only if they have the same degree.
Say that $X$ has the... | https://mathoverflow.net/users/1540 | Converse to Hopf degree theorem | *See the second half of the answer for a complete characterisation of closed orientable manifolds with the Hopf property.*
---
Note that $X$ having the Hopf property is equivalent to the injectivity of $\deg : [M, X] \to \mathbb{Z}$ for every $M$; here $[M, X]$ denotes the free homotopy classes of maps $M \to X$.... | 19 | https://mathoverflow.net/users/21564 | 421867 | 171,574 |
https://mathoverflow.net/questions/421870 | 8 | There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 stable $\infty$-categories are known to be equivalent:
1. The $\infty$-category $Fun(\mathbb N, \mathcal C)$ of filtered objects in $... | https://mathoverflow.net/users/2362 | Is there a Dold-Kan theorem for circle actions? | No, they are not equivalent, even for $C = Sp$.
Indeed, the category of spectra with $S^1$-action is also the category of $\mathbb S[S^1]$-modules, and is compactly generated by a single object.
On the other hand, compact objects of $Fun(\mathbb Z, Sp)$ are retracts of finite colimits of representables, and any fin... | 9 | https://mathoverflow.net/users/102343 | 421872 | 171,576 |
https://mathoverflow.net/questions/421871 | 6 | Let $S$ be a unit sphere in the [Urysohn space](https://en.wikipedia.org/wiki/Urysohn_universal_space) $\mathbb{U}$.
Is it true that any isometry $S\to S$ can be extended to an isometry $\mathbb{U}\to \mathbb{U}$?
| https://mathoverflow.net/users/1441 | Sphere in Urysohn space | This is not true, no.
There is a proof in Section 4.4 of this [old paper of mine](http://math.univ-lyon1.fr/%7Emelleray/NoteBeerSheva3.pdf) ; the key fact is that if $B$ is an open unit ball in the Urysohn space $\mathbb U$, then $\mathbb U$ is isometric to $\mathbb U \setminus B$.
(the proof of the fact about extens... | 9 | https://mathoverflow.net/users/8923 | 421875 | 171,579 |
https://mathoverflow.net/questions/421879 | 5 | After my [earlier question](https://mathoverflow.net/questions/421582/cancellation-of-irreducibility-for-galois-conjugates) question turned out to have a negative answer (Thank you to all respondents!), here is a more modest one. Both a positive answer and a counterexample would help my work. If some context is of inte... | https://mathoverflow.net/users/481532 | Rationality of field embeddings | The answer is yes. Suppose $S$ is nonempty. Write $K=\mathbb Q(\alpha)$ (using the [primitive element theorem](https://en.wikipedia.org/wiki/Primitive_element_theorem)). Applying your assumption to $\alpha^n$ for all $n\in\mathbb N$ we get that all sums $\sum\_{\sigma\in S}(\sigma(\alpha))^n$ are rational. Using [Girar... | 12 | https://mathoverflow.net/users/30186 | 421880 | 171,580 |
https://mathoverflow.net/questions/421889 | 7 | In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small ball by a bigger one. That is, for the elliptic equation $ \operatorname{div}(A(x)\nabla u)=0 $, we have estimates like $ \le... | https://mathoverflow.net/users/241460 | Why don't we study hyperbolic equations as elliptic and parabolic equations? |
>
> Why we do not study such estimates for hyperbolic equations?
>
>
>
Because they are false.
---
Now: you may ask "why are they false?" This is a fairly deep question, and answers often involve discussion of propagation of singularities and characteristics. Quite a few chapters in Hörmander's *Analysis o... | 28 | https://mathoverflow.net/users/3948 | 421891 | 171,582 |
https://mathoverflow.net/questions/420654 | 1 | Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $ and $ T>0 $ be a positive number. Consider the wave equation in the domain $ \Omega\times(0,T) $
\begin{align}
\left\{\begin{matrix}
\partial\_t^2u-\Delta u=F&\text{ in }&\Omega\times(0,T),\\
u=0&\text{ on }&\partial\Omega\times(0,T),\\
u(x,0)=f(x),\partial\... | https://mathoverflow.net/users/241460 | Wave equation in $ \Omega\times(0,T) $ | Strichartz estimates on domains is a difficult problem!
First: on bounded domains you cannot have any *global in time* Strichartz estimates. This is because of the presence of standing waves. (Set initial data to be an eigenfunction of the Laplacian.)
On the other hand, there is still the possibility of *local in t... | 4 | https://mathoverflow.net/users/3948 | 421893 | 171,584 |
https://mathoverflow.net/questions/421749 | 4 | An **inseparable minimal pair** is a pair of sets $A, B \subseteq \mathbb{N}$ which are
* inseparable: there is no computable $C \subseteq \mathbb{N}$ such that $A \subseteq C$ and $B \subseteq \mathbb{N} \setminus C$, and
* minimal pair: if $C \leq\_T A$ and $C \leq\_T B$ then $C$ is computable.
(I do not care whe... | https://mathoverflow.net/users/1176 | Existence of an inseparable minimal pair | Here's a construction of a computably inseparable minimal pair. I believe that it is not hard to modify this construction to give c.e. sets by using a priority construction. However, I have not checked this fact carefully and to keep things as simple as possible I will not do so here (i.e. I will not prove the c.e. ver... | 3 | https://mathoverflow.net/users/147530 | 421895 | 171,585 |
https://mathoverflow.net/questions/414709 | 0 | Fix a language $\mathcal{L}$ of first-order set theory. For this question, we can assume that $\mathcal{L}$ is the language described in Chapter 1 of “An introduction to set theory” [William A. R. Weiss | October 2, 2008].
Assuming that the complexity of a formula is not restricted and every formula has a finite leng... | https://mathoverflow.net/users/122796 | How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST? | Your ordinal $\beta\_\mathcal{L}$ is perfectly well-defined: in my opinion it's more easily thought of as $$\sup\{\alpha: \forall \beta<\alpha(V\_\beta\not\equiv V\_\alpha)\},$$ and this definition should be clearly unproblematic (note that [Tarski](https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem) notwi... | 3 | https://mathoverflow.net/users/8133 | 421898 | 171,587 |
https://mathoverflow.net/questions/421890 | 1 | Strassen demonstrated a seven multiplication algorithm for $2\times 2$ matrix multiplication and Winograd showed its optimality.
Let $A$ be $2\times k$ and $B$ be $k\times 2$.
What is the minimum number of multiplications needed for the product $AB$ at any fixed $k\geq3$? Is there a reference?
| https://mathoverflow.net/users/10035 | What is the minimum number of multiplications for $2\times 3$ and $3\times 2$ multiplication? | A recent paper with relevant results is *New lower bounds for matrix multiplication and the 3x3 determinant* by Austin Conner, Alicia Harper, J.M. Landsberg, <https://arxiv.org/abs/1911.07981>. A fairly comprehensive book is *Tensors: Geometry and Applications* by J.M. Landsberg, <https://bookstore.ams.org/gsm-128/>. T... | 6 | https://mathoverflow.net/users/88133 | 421901 | 171,588 |
https://mathoverflow.net/questions/421816 | 7 | I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes\_\pi W\to Y \otimes\_\pi V$ need not be injective. Here $\otimes\_\pi$ stands for the projective tensor product of Banach spaces.
1. Can this happe... | https://mathoverflow.net/users/15129 | Projective tensor product of injective operators | $\require{AMScd}\newcommand{\id}{\operatorname{id}}$I use a common characterisation of the [approximation property](https://en.wikipedia.org/wiki/Approximation_property) as found in e.g. Ryan's book [Zbl 1090.46001](https://www.zbmath.org/?q=an%3A1090.46001).
>
> A Banach space $X$ has the approximation property if... | 5 | https://mathoverflow.net/users/406 | 421902 | 171,589 |
https://mathoverflow.net/questions/421900 | 2 | Let $X$ be a smooth projective surface and $D$ be an effective Cartier divisor (not necessarily ample) on $X$. Is there a connection between these two conditions?
$(i)$ for a large enough $n$, the linear system $|nD|$ is base point free (semiample divisor)
$(ii)$ $h^1(\mathcal O\_X(D)^{\otimes t})=0$ for all $t >0$... | https://mathoverflow.net/users/133832 | Two conditions on divisors on surfaces | Neither condition implies the other.
$(i)\, \not\!\Rightarrow\, (ii)$: Take for $X$ a surface with $K$ ample, but $h^1(K)=h^1(\mathscr{O}\_X)>0$ (e.g. a product of 2 curves of genus $>1$). Then take $D=K$.
$(ii)\, \not\!\Rightarrow\, (i)$: Consider a smooth cubic curve $C\subset \mathbb{P}^2$, take $9$ general poin... | 6 | https://mathoverflow.net/users/40297 | 421904 | 171,590 |
https://mathoverflow.net/questions/421865 | 10 | It is known that genus one fibred knots are two trefoils and the figure-eight knot. Is there any characterization of the knot $5\_2$? Specifically, is there any other genus one knot that shares the same Alexander polynomial $2t^2-3t+2$ with $5\_2$?
| https://mathoverflow.net/users/169890 | Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$? | Ian Agol, in the comments says:
>
> Yes, there should be plenty. Think of the Seifert surface for the 5\_2
> knot as a disk with two strips (1-handles) attached. By tying knots
> into the strips (with zero framing so as not to change the linking
> form), you can obtain many knots with a genus 1 Sefert surface with
... | 6 | https://mathoverflow.net/users/1650 | 421909 | 171,592 |
https://mathoverflow.net/questions/421844 | 5 | If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $G$ by the conjugation action on itself and the quotient $T/W$.
This result has several generalisations. In particular, in [Orbit... | https://mathoverflow.net/users/143492 | Is there a Chevalley map for spherical varieties? | **Edit:** The answer to question 1 is yes if $G/H$ is a symmetric variety as the OP pointed out.
For arbitrary spherical varieties the answer is no in general. If my memory serves me right, the spherical variety $Sp(4,\mathbb C)/(\mathbb C^\*\times SL(2,\mathbb C))$ is a counterexample. As far as I know, the $H$-orbi... | 6 | https://mathoverflow.net/users/89948 | 421920 | 171,596 |
https://mathoverflow.net/questions/421916 | 1 | This is a soft question.
I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.
Given a measure $\mu$ on metric space $(X,d)$, if there is a function $F$ on $X$ for which
$$\lim\_{\varepsilon\t... | https://mathoverflow.net/users/479223 | When is the mode of a stochastic process a better statistic than the mean? | A "mode" of the Onsager-Machlup action functional identifies a *locally* most probable transition pathway between metastable states. If there is a single minimizer then mode and mean will be equally informative, but there may well be multiple local minima of the action functional, and then the mean does not tell you wh... | 1 | https://mathoverflow.net/users/11260 | 421923 | 171,599 |
https://mathoverflow.net/questions/420090 | 1 | Let $f \in S\_2(\Gamma\_0(N))$ be a newform with associated residual Galois representation $\rho: \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \operatorname{GL}\_2(\mathbf{F})$, $\mathbf{F}$ a residue field of the coefficient ring of $f$ of characteristic $p > 0$.
Is there an explicit bound $M$ such that ... | https://mathoverflow.net/users/471019 | Explicit Chebotarev density theorem for Galois representations associated to newforms | An explicit bound on $M$ can be proved. It is not clear to me if the modularity of $\rho$ would help improve such bounds. One can use the best available numerical bounds on the least norm of an unramified prime ideal with a given Artin symbol. For the Galois extension inherent in your setting, the best such uncondition... | 1 | https://mathoverflow.net/users/111215 | 421927 | 171,601 |
https://mathoverflow.net/questions/421928 | 4 | Let $A$ be a unital $C^\*$-algebra and let $K$ be an inner product space (not necessarily complete!). Let $\pi: A \to \operatorname{End}\_{\mathbb{C}}(K)$ be a unital algebra homomorphism such that
$$\langle \pi(a)\xi, \eta\rangle = \langle \xi, \pi(a^\*)\eta\rangle$$
for all $a \in A$ (i.e. the adjoint of $\pi(a)$ exi... | https://mathoverflow.net/users/216007 | Is a unital $*$-morphism from a unital $C^*$-algebra $A$ to $\operatorname{End}_{\mathbb{C}}(K)$ automatically contractive? | $\newcommand{\End}{\operatorname{End}}$For $T\in\End\_{\mathbb C}(H)$ I write $T^\*$ if the adjoint exists. Given the hypotheses in the question, if $u\in A$ is an isometry, then $1 = \pi(1) = \pi(u^\*u) = \pi(u)^\*\pi(u)$. Thus, for $\xi\in H$,
$$ \|\xi\|^2 = (\xi|\xi) = (\pi(u)^\*\pi(u)\xi|\xi) = (\pi(u)\xi|\pi(u)\xi... | 5 | https://mathoverflow.net/users/406 | 421934 | 171,605 |
https://mathoverflow.net/questions/421952 | 5 | Let $p\_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p\_n-p\_{n-1}\le C(\sqrt p\_n \log p\_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ that works? In Cramer's original paper ([On the order of magnitude of the difference of consecutive prime numbers](... | https://mathoverflow.net/users/130113 | A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis | [On the Riemann hypothesis and the difference between primes](https://doi.org/10.1142/S1793042115500426) by Adrian W. Dudek states the result (Theorem 3, at least in [the arXiv version](https://arxiv.org/abs/1402.6417)) that any $C>1$ works (for $n$ sufficiently large).
| 15 | https://mathoverflow.net/users/18060 | 421953 | 171,612 |
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