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/369281 | 6 | I was playing around with the chromatic polynomial (denoted here by $\chi\_G(x)$) and I have made the following conjecture.
Let $(G\_n)\_{n \ge 1}$ be a sequence of graphs with $v(G\_n) \to \infty$ ($v(G\_n)$ denotes the number of vertices of $G\_n$) and $e(G\_n) \to \infty$ ($e(G\_n)$ denotes the number of edges of ... | https://mathoverflow.net/users/161863 | On a limit involving a transform of the chromatic polynomial | Here is a heuristic argument which perhaps someone can make
rigorous. I write $v\_n=v(G\_n)$ and $e\_n=e(G\_n)$. Let
$$ \chi\_{G\_n}(x) =
x^{v\_n}-c\_{n,v\_n-1} x^{v\_n-1}+c\_{n,v\_n-2}x^{v\_n-2}-\cdots. $$
I claim that for fixed $k\geq 0$,
$$ \lim\_{n\to\infty} \frac{c\_{n,v\_n-k}}{e\_n^k} = \frac{1}{k!}. $$
One can ... | 6 | https://mathoverflow.net/users/2807 | 369327 | 154,570 |
https://mathoverflow.net/questions/369336 | 0 | Let $x\_1, \ldots, x\_n$ be **possibly dependent** random variables, each taking values $x\_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. Now for each $i \in \{1, \ldots, n\}$ define
$$
f\_i = \begin{cases}
\Pr[x\_i = 2 \mid x\_i \geq 1] & \text{if }... | https://mathoverflow.net/users/153090 | Bounds on variance of sum of dependent random variables | $Var\,f$ can be on the order of $n$ (but not more than that).
Indeed, let $U$ and $N$ be independent random variables such that $P(U=1)=:p=1-P(U=0)=:q$ and $P(N=i)=1/n$ for all $i\in[n]:=\{1,\dots,n\}$. Let
$$x\_i:=1(U=1,N\ne i)+2\times1(N=i).
$$
Then with $p=1/n$
$$Var\,f\sim n/4\tag{1}$$
(as $n\to\infty$).
On the... | 4 | https://mathoverflow.net/users/36721 | 369341 | 154,575 |
https://mathoverflow.net/questions/369328 | 0 | Assume $f\in k[x\_1,\ldots, x\_n]$ is irreducible. Let for $g\in k[x\_1,\ldots, x\_n]$, $\partial(g)$ is divisible by $f$ for each derivation $\partial$ with $f\in\ker\partial$. Is it true that $g-t$ is divisible by $f$ for some $t\in k$?
| https://mathoverflow.net/users/100359 | Is it true that $g-t$ is divisible by $f$? | Consider the map $g:X\to\mathbb{A}^1$, where $X$ is defined by $f=0$ in $\mathbb{A}^n$. Your condition implies $dg=0$ and thus this map must be constant. This is what you wanted to prove.
| 4 | https://mathoverflow.net/users/9502 | 369342 | 154,576 |
https://mathoverflow.net/questions/369304 | 4 | I've seen a couple of similar questions asking to verify computations of Bredon cohomology [here](https://mathoverflow.net/q/298217) and [here](https://mathoverflow.net/q/300432), so I will ask one such question myself.
Let $\mathbb{Z}/2$ act on $S^3\subset \mathbb{C}^2$ by restriction of a permutation action on $\ma... | https://mathoverflow.net/users/143549 | Bredon cohomology of a permutation action on $S^3$ | Your final answer is correct, but the cell structure you're using isn't a $G$-CW structure: $T\times T$ can't be used as a cell in this way.
I would approach it like this: The action of $G = {\mathbb Z}/2$ on $\mathbb{C}\times\mathbb{C}$ can be written as the representation $\mathbb{C}\oplus\mathbb{C}^\sigma$, where ... | 3 | https://mathoverflow.net/users/58888 | 369345 | 154,577 |
https://mathoverflow.net/questions/369284 | 2 | The integral is of the form $\int\_{-\infty}^\infty \sigma(x)\mu(x)\,\mathrm{d}x$.
Where the Fourier transform of the $\sigma$ function is $\tilde \sigma(p)= e^{-iap}\frac{1}{1+e^{-c|p|}}$ and the function $\mu(x)$ is given by $\mu(x)=-2 \tan ^{-1}\left(\frac{2 x-2}{c}\right)$.
The Fourier transform of $\mu(x)$ can b... | https://mathoverflow.net/users/163076 | When is it possible to use the Parseval-Plancherel identity to solve an integral? | Recall the identity that Fourier transform of $K(x)=\text{sech}(x)$ is $\tilde K(p)=\pi \text{sech}\left(\frac{\pi p}{2}\right)$.
Using this identity the Fourier transform of $\frac{\text{sech} {x}}{x}$ can be easily computed
\begin{equation}
\int\_{-\infty}^{-\infty} e^{-i x p} \frac{\text{sech}{x}}{x} \, \mathrm... | 2 | https://mathoverflow.net/users/163821 | 369354 | 154,579 |
https://mathoverflow.net/questions/369333 | 8 | I'm wondering if it's possible, given a prime p and an infinite list of primes $q\_1$, $q\_2$, ... to find an integer d which (1) is *not* a square mod p, but (2) *is* a square mod $q\_i$ for all i. Always, sometimes, never? Probably sometimes --- what are some conditions? In the application I have in mind, the $q\_i$ ... | https://mathoverflow.net/users/163849 | Is it possible to find a (nonsquare) integer which is a quadratic residues modulo a given infinite list of primes? | It depends on the given list of primes. A simpler but necessary condition is that there be a $d$ so that all the primes of the list (greater than $d$) are concentrated in a few congruence classes $\bmod 4d.$ We can stick to odd prime divisors since everything is a quadratic residue $\bmod 2.$
If the list is all prime... | 5 | https://mathoverflow.net/users/8008 | 369358 | 154,581 |
https://mathoverflow.net/questions/369347 | 0 | A simple, undirected graph is said to be $1$-*factorizable* if there is a partition of the edge set $E$ such that every member of the partition is a perfect matching of $G$. Let us call $G$ *weakly $1$-factorizable* if there is a partition of $E$ into maximal (but not necessarily perfect) matchings.
$K\_3$ is weakly ... | https://mathoverflow.net/users/8628 | Weak $1$-factorizability | The [Petersen graph](https://en.wikipedia.org/wiki/Petersen_graph) $G$ is $3$-regular and has a perfect matching but is not $1$-factorizable. To see that $G$ is weakly $1$-factorizable, regard it as the complement of the line graph of $K\_5$. For each vertex $v$ of $K\_5$, let $M\_v$ be the set of all pairs $\{e,f\}$ w... | 2 | https://mathoverflow.net/users/43266 | 369359 | 154,582 |
https://mathoverflow.net/questions/369334 | 11 | Let $B\to X$ be a surjective submersion over the smooth integral scheme $X$ over $\mathbb{C}$. Associated to this we have in the $C^\infty$ world the notion of the $k$ jet-bundles $J\_k(B)$, which are affine bundles over $X$. I wonder what the best way to define this notion in the setting of algebraic geometry is. If $... | https://mathoverflow.net/users/64302 | How to define jet bundles algebraically? | Let $S$ be a scheme, e.g., $\text{Spec}\ \mathbb{C}$. Let $f:X\to S$ be a morphism that is separated and smooth. Denote the associated relative diagonal morphism by $$\Delta\_{X/S}:X \to X\times\_S X.$$ This is a closed immersion whose ideal sheaf $\mathcal{I}$ is everywhere locally generated by a regular sequence. For... | 11 | https://mathoverflow.net/users/13265 | 369366 | 154,583 |
https://mathoverflow.net/questions/369295 | 2 | It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a projective bundle over an open subset of $Y$. We can even assume $Y$ smooth, even if I don't think it is necessary. Is it still... | https://mathoverflow.net/users/4096 | Decomposition of a morphism with positive dimensional fibers | I am posting my comment as an answer. This already fails for relative dimension $1$ when the base scheme has dimension $n$ at least $3$.
Let $k$ be a field. Let $n\geq 3$ be an integer. Denote $\text{Proj}\ k[x\_0,x\_1,x\_2, \dots,x\_n]$ by $\mathbb{P}^n\_k$. Denote $\text{Proj}\ k[y\_0,y\_1,y\_2]$ by $\mathbb{P}^2\_... | 1 | https://mathoverflow.net/users/13265 | 369368 | 154,584 |
https://mathoverflow.net/questions/369367 | 4 | I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim:
Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function.
Let $\lambda\_n \in [0,1],a\_n\le c\_0<b\_n \in [0,\infty)$ satisfy
$$ \lambda\_n a\_n +(1-\lambda\_n)b\_n=c\_n $$ and suppose that $c\... | https://mathoverflow.net/users/46290 | Does strict convexity plus asymptotic affinity imply bounded mean? | Yes, $b\_n$ must be bounded. Assume the contrary. Passing to a subsequence we may suppose that $a\_n\to a$, $b\_n\to \infty$. We have $$\lambda\_n=\frac{b\_n-c\_n}{b\_n-a\_n}\to 1;\, 1-\lambda\_n=\frac{c\_n-a\_n}{b\_n-a\_n}\sim (c-a)b\_n^{-1},$$
and using $F(b\_n)\geqslant F(c\_n)+(b\_n-c\_n)F'(c\_n)$ we get
$$
D\_n+F(... | 6 | https://mathoverflow.net/users/4312 | 369371 | 154,587 |
https://mathoverflow.net/questions/369343 | 3 | Let $M$ be a connected smooth manifold and $f \in C^\infty(M)$ such that $0$ is a regular value of $f$. Moreover, suppose that $f^{-1}(0)$ is connected. Is it true that $M \setminus f^{-1}(0)$ has exactly two connected components?
As $0$ is a regular value of $f$, we know that $f^{-1}(-\infty,0]$ and $f^{-1}[0,+\inft... | https://mathoverflow.net/users/98139 | Connected manifold without connected regular level set admits exactly two connected components | Since $M$ is connected and a manifold, it is path-connected. Thus, any two points $x,y \in M$ such that $f(x), f(y) > 0$ can be joined by a path $\gamma$. Suppose $\gamma$ does not lie entirely in $f^{-1}(0,\infty)$.
Then let $p = \text{inf}\{ t \in [0,1] : f(\gamma(t)) \leq 0\}$ and $q = \text{sup}\{ t \in [0,1] : f... | 2 | https://mathoverflow.net/users/14233 | 369374 | 154,588 |
https://mathoverflow.net/questions/369377 | 1 | Let $X,Y$ be compact, connected, simply-connected, and separable, metric spaces each with at-least $2$-points, and let $f,g:X\rightarrow Y$ be continuous functions. Does there always exist a homeomorphism $\Phi:X\times Y \rightarrow X\times Y$ such that
$$
g(x) =\pi\_Y\circ \Phi(x,f(x))
$$
for all $x \in X$, where $\pi... | https://mathoverflow.net/users/36886 | Projecting Graph of a Function acted on by a homeomorphism | No. Take $I=[0,1]$, $Q=[0,1]\times[0,1]$ and $J=\{0\}\times[1,2]$. Let $X=I$ and $Y=Q\cup J$. Let $f:X\to Y$ be the constant function $f(x)=(0,\frac32)$ and let $g$ be the constant function $g(x)=(\frac12,\frac12)$. Then any homeomorphism $\Phi$ of $X\times Y$ will preserve the square $I\times J$, so $\pi\_Y(\Phi(x,f(x... | 1 | https://mathoverflow.net/users/16447 | 369381 | 154,591 |
https://mathoverflow.net/questions/369355 | 0 | Let $x\_1, \ldots, x\_n$ be **possibly dependent** random variables, each taking values $x\_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. Now for each $i \in \{1, \ldots, n\}$ define
$$
f\_i = \begin{cases}
\Pr[x\_i = 2 \mid x\_i \geq 1] & \text{if }... | https://mathoverflow.net/users/153090 | Independent sampling of dependent random variables | The answer is "no" (if i understand the question correctly).
Consider the following exchangeble joint distribution of the $x\_i$s. In event $A$, which occur with probabiluty $1/\sqrt n$, all the $x\_i$s are 1, exept for one 2. In the complement event $B$, all the $x\_i$s are 0 exept for one 2.
Under this distributi... | 2 | https://mathoverflow.net/users/85550 | 369389 | 154,595 |
https://mathoverflow.net/questions/368633 | 6 | Let $M$ be a compact path metric space in $\mathbb{R}^d$, and for $\sigma>0$,
$$
M\_\sigma:=\{y\in\mathbb{R}^d:\min\_{x\in M}\|x-y\|\leq\sigma\}
$$
the $\sigma$-tube around $X$ in $\mathbb{R}^d$. I consider both $M$ and $M\_\sigma$ metric spaces with respect to the shortest path metric (geodesic, not necessarily Euclid... | https://mathoverflow.net/users/159398 | Gromov Hausdorff distance to tubular neighborhood | I think I have figured this out. More specifically, it should hold that
$$
d\_{GH}(M, M\_\sigma) \leq \max\left\{2\sigma, \left(\frac{\epsilon}{s-2\sigma}-1\right)(\mathrm{diam}(M)+2\sigma)+\epsilon\right\},
$$
whenever $\sigma < s/2$.
Sketch of the proof:
Define the correspondence $C$ as
$$
(x,y)\in C\leftrightarr... | 1 | https://mathoverflow.net/users/159398 | 369393 | 154,597 |
https://mathoverflow.net/questions/369379 | 4 | In the book of Kra and Farkas on Riemann surfaces the following (slightly unusual) definition is given:
**Definition IV.3.2** (*Section IV.3*). Let $M$ be a Riemann surface. We will call $M$ *elliptic* if and only if $M$ is compact. We will call $M$ *parabolic* if and only if $M$ is not compact and $M$ doesn't carry ... | https://mathoverflow.net/users/13441 | Elliptic, parabolic and hyperbolic Riemann surfaces: classification? | This is somewhat unusual terminology, but it is common in the theory of classification of open Riemann surfaces. The more standard notation
is $P\_G$ for "parabolic", and $O\_G$ for "hyperbolic".
The surface $M\backslash\{ x\_1,\ldots,x\_n\}$ is parabolic in this sense,
by the "removable singularity theorem" (a subha... | 2 | https://mathoverflow.net/users/25510 | 369397 | 154,599 |
https://mathoverflow.net/questions/369375 | 5 | Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected.
Given $n$ pairs of points $(x\_i, y\_i)$ where $x\_i\in X$ and $y\_i\in Y$ for $1\leq i\leq n$ and a continuous map $f:X\to Y$ can we find a continuous map $g:X\to Y$ homotopic to $f... | https://mathoverflow.net/users/nan | Any continuous map is homotopic to one assuming fixed values at finitely many points | Let $X$ be the real line with a doubled origin and $Y$ be $\Bbb R$, and let $f$ be the projection map that collapses the two origins $0^+$ and $0^-$ to $0$. Then any map $g: X \to Y$ satisfies $g(0^+) = g(0^-)$ because $\Bbb R$ is Hausdorff. Therefore, $f$ is not homotopic to any map that sends these two points to dist... | 13 | https://mathoverflow.net/users/360 | 369398 | 154,600 |
https://mathoverflow.net/questions/369400 | 0 | My universe has M different items. I run m=10 independent samplings over M. In each sampling, n elements are picked without replacement (n<<M). What is the expected number of pair duplicates we shall get across the m independent samplings? I know that the universe has M(M-1)/2 distinct pairs, and in each sampling one c... | https://mathoverflow.net/users/148279 | Number of duplicate pairs in multiple samplings | The answer is
$$\binom M2(1-m p q^{m-1}-q^m),$$
where
$$p:=\frac{n(n-1)}{M(M-1)},\quad q:=1-p.\tag{0}$$
---
*Details:* Fix any "pair" -- that is, any subset $a\subseteq[M]:=\{1,\dots,M\}$ of cardinality $2$. For each $i\in[m]$, let $S\_i$ denote the $i$th random sampling, that is, the $i$th random set of size $|S... | 2 | https://mathoverflow.net/users/36721 | 369413 | 154,603 |
https://mathoverflow.net/questions/369414 | 7 | $\DeclareMathOperator\Spec{Spec}$Let $A$ be a finite dimensional $\*$-algebra over $\mathbb C$.
(Namely, an associate algebra equipped with an involution $\*:A\to A$ satisfying $(ab)^\*=b^\*a^\*$ and $(\lambda a)^\*=\bar\lambda a^\*$.)
>
> Assume that for $\forall a\in A$ we have $\Spec(a^\*a)\subset\mathbb R\_+... | https://mathoverflow.net/users/5690 | Characterisation of finite dimensional C*-algebras? | Let $V$ be a complex vector space equipped with an involutive anti-linear star operation (e.g. a C\*-algebra whose multiplication has been forgotten). Equip $V$ with the identically zero multiplication, namely
$xy=0$
for all $x$ and $y$ in $V$. Then the unitization of $V$ is a counter-example. In fact, every element $a... | 7 | https://mathoverflow.net/users/97532 | 369417 | 154,605 |
https://mathoverflow.net/questions/369416 | 17 | This question concerns some counterintuitive results (to me at least) regarding the number of points on a projective curve over a finite field. Namely, if one fixes the degree of the curve, but *increases* the dimension of the ambient projective space, one can get tighter bounds on the number of $\mathbb{F}\_q$ points ... | https://mathoverflow.net/users/106264 | Why should the number of $\mathbb{F}_q$ points on degree $d$ curves $C\subset \mathbb{P}_{\mathbb{F}_q}^n$ decrease as $n$ increases? | One way to get some intuition comes from looking at the (weaker) combinatorial bound. Suppose you had a nondegenerate curve $C$ in some projective space $\mathbb P^n$. Suppose that that $L$ is a subspace of codimension $2$ in $\mathbb P$ and that $|C\cap L|=m$. The higher the dimension $n$ gets, the higher value we are... | 8 | https://mathoverflow.net/users/2384 | 369429 | 154,610 |
https://mathoverflow.net/questions/369430 | 2 | To begin with, i would like to apologize if my question is not up to the level of this forum.
I have tried asking a variant of the following question on math.stackexchange.com and my question generated some comments (even one upvote) but no answers, so i decided to give it a shot over here.
My original question was:
... | https://mathoverflow.net/users/163824 | Fermat's little theorem, Poulet numbers, Carmichael numbers, and primes | The answer is No. E.g., see [OEIS A153580](https://oeis.org/A153580). for smaller examples.
| 4 | https://mathoverflow.net/users/7076 | 369458 | 154,614 |
https://mathoverflow.net/questions/369459 | 3 | If $G=(V,E)$ is a simple, undirected graph, is there a regular graph $G\_R$ such that $G$ is isomorphic to an induced subgraph of $G\_R$ and $\chi(G) = \chi(G\_R)$?
| https://mathoverflow.net/users/8628 | Embedding any graph in a regular graph with the same chromatic number | Yes, you can find such a $G\_R$ of any degree greater than or equal to the maximum degree of $G$. This is the main theorem of the paper ["On regular bipartite-preserving supergraphs"](https://link.springer.com/article/10.1007%2FBF01834121) by G. Chartrand and C. E. Wall .
| 7 | https://mathoverflow.net/users/2384 | 369462 | 154,615 |
https://mathoverflow.net/questions/369433 | 9 | I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the basics of elliptic curves, Mordell's Theorem etc (though I'm not up to the point where I can understand the proof).
What ... | https://mathoverflow.net/users/38744 | Reference request: Diophantine equations | This may be a good choice for someone who (like yourself) is already superficially acquainted with some of the definitions and methods of Diophantine geometry:
* Marc Hindry, Joseph H. Silverman -- *Diophantine Geometry: An Introduction*, Graduate Texts in Mathematics **201**, Springer (2000), <https://doi.org/10.100... | 8 | https://mathoverflow.net/users/17907 | 369473 | 154,618 |
https://mathoverflow.net/questions/369480 | 6 | Let
$$
P\_m(x):=\begin{cases}4x+1\quad&\ \text{if}\ m=1,\\
0\quad&\ \text{if}\ m=2,\\
8x^m+(x+1)^{m-3}(2x+1)^3\quad&\ \text{if}\ m\geq3.\end{cases}
$$
How to prove that for any positive odd integer $n$, there exist integers $a\_1^{(n)},a\_2^{(n)},\ldots,a\_n^{(n)}$ such that
$$
(4x+1)^n=\sum\_{k=1}^n a\_k^{(n)}P\_k(x).... | https://mathoverflow.net/users/111873 | Write $(4x+1)^n$ as the linear combination of certain polynomials | I enjoyed very much this question! My solution contains two ideas, each of which addresses one of two distinct subproblems:
1. show that the coefficients $a\_m^{(n)}$ are **integers**;
2. show that the coefficients $a\_m^{(n)}$ **exist**.
The subproblem (1) is not completely obvious because the polynomials $P\_m$ a... | 14 | https://mathoverflow.net/users/58242 | 369498 | 154,630 |
https://mathoverflow.net/questions/369441 | 2 | (This post is an offshoot of this [MSE question](https://math.stackexchange.com/q/3629818).)
Let $\sigma(x)$ denote the sum of divisors of $x$. (<https://oeis.org/A000203>)
**QUESTION**
>
> Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$ equal to zero?... | https://mathoverflow.net/users/10365 | Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$ equal to zero? | I think the density does go to zero, but quite slowly. If $p \equiv 1 \bmod 6$ is prime then there are two solutions $0<r<s<p-1$ of $$x^2+x+1=0 \bmod p$$
If $p\parallel n$ then, with probability $1,$ there are two distinct primes $x $ and $ y,$ each congruent to $r \bmod p,$ with $x \parallel n$ and $y \parallel n.$ ... | 2 | https://mathoverflow.net/users/8008 | 369521 | 154,635 |
https://mathoverflow.net/questions/369511 | 3 | This is a follow-up question to [this](https://mathoverflow.net/questions/369421/is-it-always-possible-to-partition-a-b-timesc-d-into-disjoint-blocks-d-i).
Since it is not always possible to construct such partition, I would like to know if there are additional restrictions which we could impose so that the wanted pa... | https://mathoverflow.net/users/nan | Under which conditions the domain of the surjective function $f:[a,b]\times[c,d]\to[0,1]^{2}$ can be split s.t. the restrictions are bijective? | The trivial answer to this question is tautological: such a partition exists if and only if it exists.
---
A more informative answer is: almost never (unless such a partition obviously exists). To be more specific, consider the following counterexample, when such a partition does **not** exist. Define the functio... | 5 | https://mathoverflow.net/users/36721 | 369526 | 154,636 |
https://mathoverflow.net/questions/369524 | 7 | Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}\_2 (X), W\_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. It is known that discrete measures are dense inside $(\mathcal{P}\_2 (X), W\_2)$ - namely, given any $\mu \in \mathcal{... | https://mathoverflow.net/users/100163 | Stability of displacement interpolation in optimal transport | The displacement interpolation $\mu\_t$ should not be fixed *a priori*, due to nonuniqueness of Wasserstein Geodesics. Thus, the correct question should be: fix the approximating sequences $(\mu\_{0,n}),(\mu\_{1,n})$ and $W\_2$ geodesics $\mu\_{t,n}$, and ask if there exists **one** $\mu\_t$ close to $\mu\_{t,n}$ for $... | 3 | https://mathoverflow.net/users/150328 | 369538 | 154,638 |
https://mathoverflow.net/questions/369453 | 3 | I'm reading Kauffman's 1990 paper "An Invariant of Regular Isotopy" about knots that are isotopic through only Reidemeister Type II and III moves, which is known as a regular isotopy. His paper claims there is a relationship between regular isotopy and embedded bands ($S^1 \times [0,1]$) in $S^3$. He refers to Burde's ... | https://mathoverflow.net/users/118997 | Embedded ribbons and regular isotopy | From any knot diagram, one can obtain a framed knot by taking the "blackboard framing." The point of regular isotopy of knot diagrams is that it preserves this blackboard framing. Since framed knots and embedded bands are the same thing, regular isotopy will also preserve the embedded band corresponding to the blackboa... | 2 | https://mathoverflow.net/users/113402 | 369563 | 154,643 |
https://mathoverflow.net/questions/369568 | 4 | Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies
$$ \int\_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$
Does it follow that $f\geq 0$ on $I$?
| https://mathoverflow.net/users/50438 | A functional integral inequality | The answer is no. E.g., take
$$f(t):=1-\frac{3}{2} \max \left(0,1-2 \left| t-\frac{1}{2}\right| \right).$$
Then $f(1/2)=-1/2<0$, but
$$ \int\_0^1 f(t) e^{at}\,dt=
\frac{2 e^{a/2}}{a^2}\,\left(a \sinh\frac{a}{2}-3 \left(\cosh
\frac{a}{2}-1\right)\right)\ge0$$
for all real $a$. (The inequality here follows from the ineq... | 5 | https://mathoverflow.net/users/36721 | 369572 | 154,644 |
https://mathoverflow.net/questions/369465 | 1 | A **digraph** is a directed graph.
A **directed cycle** or **simple directed circuit** is a directed circuit in which the only repeated vertices are the first and last vertices.
A **digraph is primitive** if its adjacency matrix is primitive.
A square non-negative **matrix** $A$ is said to be **primitive** if the... | https://mathoverflow.net/users/163905 | Does every primitive digraph have a directed cycle? | Yes.
For all $i,j$, $(A^k)\_{ij}>0$, so there is at least one walk of length $k$ from $v\_1$ to $v\_2$ and there is at least one walk of length $k$ from $v\_2$ to $v\_1$. This closed directed walk which goes from $v\_1$ to $v\_2$ and then back to $v\_1$ must contain a non-trivial directed cycle (i.e. a cycle of length ... | 0 | https://mathoverflow.net/users/17798 | 369578 | 154,648 |
https://mathoverflow.net/questions/369435 | 0 | I wonder the difference between $L^1(\mu\times\nu)$ and $L^1(\mu;L^1(\nu))$, as if partial derivatives can be exchanged with integration in the second spaces in many articles. In Folland's real analysis, Fubini-Tonelli theorem can't be used without the assumptions $L^1(\mu\times\nu)$ or $L^+(X\times Y)$.
Precisely, d... | https://mathoverflow.net/users/163895 | a question about vector valued Banach spaces | There are two parts to your questions and the second hasn’t been touched on so far. Before bringing some suggestions which I hope will be useful, let me add to the information alrady given in the comments in the first part.
The fact that a function on a product $S\times T$ can be regarded as a function on $S$ with va... | 1 | https://mathoverflow.net/users/131781 | 369583 | 154,651 |
https://mathoverflow.net/questions/368800 | 5 | Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ admits local cross-sections. More precisely, for every $z\in X/G$ there are an open set $U$ in $X/G$ containing $z$, an... | https://mathoverflow.net/users/29566 | Local cross-sections for free actions of finite groups | Let $X=[-1,1]^\infty\setminus\{0\}$, which is a metrizable, locally compact space. Consider the two-element group $G$, and the free $G$-action on $X$ given by $(x\_j)\_{j=1}^\infty\mapsto (-x\_j)\_{j=1}^\infty$. We show that the fibration $X\to X/G$ has infinite Schwarz genus.
Consider the $n$-sphere $S^n$ with the a... | 4 | https://mathoverflow.net/users/24916 | 369584 | 154,652 |
https://mathoverflow.net/questions/369585 | 4 | Let $X$ be a smooth, projective ireducible scheme over an algebraically closed field $k$. I'm trying to understand when there exists an abelian variety $A$ such that $X$ is isomorphic to a prime divisor on $A$.
There are some simple cases, of course. If $X$ is zero-dimensional, i.e. a point, then it is isomorphic to ... | https://mathoverflow.net/users/152554 | Which schemes are divisors of an abelian variety? | Any curve of genus greater than two, whose Jacobian $J$ is simple, will do. If it were a divisor on an abelian surface $S$, then there would be a surjection $J\to S$ with positive dimensional kernel, contradicting the simplicity of $J$. Most curves of genus larger than two have this property; a randomly chosen example ... | 11 | https://mathoverflow.net/users/949 | 369589 | 154,653 |
https://mathoverflow.net/questions/369598 | 4 | A lot is known about the Fermat numbers $2^{2^k}+1$. For example, the first few
$$
2^1+1=3,\;2^2+1=5,\;2^4+1=17,\;2^8+1=257,\;2^{16}+1=65537
$$
are known to be prime, and Euler showed that the next ($2^{32}+1$) is not prime, being divisible by 641.
But what about the subset of special Fermat numbers formed by "tetrat... | https://mathoverflow.net/users/12965 | Primality of Fermat numbers associated with "tetration" | It's composite. It's divisible by 825753601.
Edit: It's also divisible by 188981757975021318420037633.
| 6 | https://mathoverflow.net/users/95685 | 369599 | 154,658 |
https://mathoverflow.net/questions/369106 | 14 | Is there a finitely generated computably presentable group $G$ on generator set $A$ and a computable function $f$ from first-order formulas to words on $A$ such that $\mathsf{ZFC}\vdash\sigma\leftrightarrow\tau$ iff $f(\sigma)$ and $f(\tau)$ represent the same element in $G$?
| https://mathoverflow.net/users/49223 | Is there a finitely generated group with the same structure as ZFC? | The relation $\text{ZFC}\vdash\varphi\leftrightarrow \psi$ is a $\Sigma\_1^0$-definable equivalence relation on the set $\mathcal L$ of formulas in the language of set theory. It is a corollary of Theorem 3.2 of Neis-Sorbi's ["Calibrating word problems of groups via the complexity of equivalence relations"](https://arx... | 18 | https://mathoverflow.net/users/102684 | 369602 | 154,659 |
https://mathoverflow.net/questions/369603 | 1 | I'm interested to see a result where for large degree of freedom $m,$ the [chi distribution](https://en.wikipedia.org/wiki/Chi_distribution) $\chi\_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $m.$ The motivation comes from the fact that for large $m, \chi^2\_m \... | https://mathoverflow.net/users/35936 | Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$ | Such a result can be obtained by the so-called [delta method](https://en.wikipedia.org/wiki/Delta_method), which yields, in particular, the following: if $X\_m\sim\chi^2\_m$, then the distribution of $\sqrt X\_m$ is approximately $N(\sqrt m,1/\sqrt2)$ (for large $m$).
*Details:* By the central limit theorem, $\overli... | 2 | https://mathoverflow.net/users/36721 | 369604 | 154,660 |
https://mathoverflow.net/questions/369605 | 3 | We say that a Hermitian symmetric (i.e., $f\_{-n} = f\_n^\*$ for any $n \in \mathbb{Z})$ sequence $(f\_n)\_{n\in \mathbb{Z}}$ is positive-definite if, for any $N \geq 0$ and any $z\_0 , \ldots, z\_N \in \mathbb{C}$,
\begin{equation}
\sum\_{n,m =0}^N f\_{n-m} z\_n z\_m^\* \geq 0. \tag{1}
\end{equation}
According to th... | https://mathoverflow.net/users/39261 | Existence of probability measure on the circle with given Fourier coefficients | Yes, this works. Condition (1) says that $\int |p(e^{ix})|^2\, d\mu(x)\ge 0$ for every polynomial $p(z)=\sum\_{n=0}^N p\_n z^n$. By the [Fejer-Riesz theorem](https://encyclopediaofmath.org/wiki/Fej%C3%A9r-Riesz_theorem), these squares $|p|^2$ range exactly over the trigonometric polynomials $f=\sum\_{|n|\le N} f\_n z^n... | 5 | https://mathoverflow.net/users/48839 | 369616 | 154,661 |
https://mathoverflow.net/questions/368923 | 4 |
>
> *Condition* : Given a $4 \times 5$ matrix, where each element is denoted by $p/q$, we have $|p|<10$ and $1\leq |q|<10$.
>
>
>
Example: $A$ given by
```
-9/7 -5/8 -1 4/5 4
-1 9/7 -6/5 -7 2/9
-1/3 -2 5/7 -2/9 -7
3 0 -5/3 8/9 -2/5
```
I have found that when I calculate the row reduced echelon form (RREF) o... | https://mathoverflow.net/users/163626 | Maximum length of numerator/denominator in calculating RREF | Surprisingly the question hasn't been closed yet, so I'm posting the solution, as promised. As I said, bringing the matrix to the row echelon form is equivalent to the left multiplication by the inverse to the $4\times 4$ sub-matrix (if the rank is full) or smaller size sub-matrix if the rank is smaller than $4$. I'll ... | 5 | https://mathoverflow.net/users/1131 | 369620 | 154,662 |
https://mathoverflow.net/questions/369624 | 7 | Inspired by this question [Is there a known asymptotic for $A(x):=\sum\_{1\leq i,j \leq X} \frac{1}{\text{lcm}(i,j)}$?](https://mathoverflow.net/questions/368957/is-there-a-known-asymptotic-for-ax-sum-1-leq-i-j-leq-x-frac1-mathr) I tried to find the asymptotic of the following function.
$$
\Lambda(x)=\sum\_{\substack{ ... | https://mathoverflow.net/users/156029 | Finding the asymptotic of the function $\Lambda(x):=\sum_{1 \leq m,n \leq x \,\land \,\gcd(m,n)=1} \frac{1}{mn}$ | We have, for $x\geq 2$,
\begin{align\*}
\sum\_{\substack{ 1 \leq m,n \leq x \\ \mathrm{gcd}(m,n)=1}} \frac{1}{mn}
&=\sum\_{1 \leq m,n \leq x}\frac{1}{mn}\sum\_{k\mid\mathrm{gcd}(m,n)}\mu(k)\\
&=\sum\_{1\leq k\leq x}\mu(k)\sum\_{\substack{ 1 \leq m,n \leq x \\ k\mid\mathrm{gcd}(m,n)}} \frac{1}{mn}\\
&=\sum\_{1\leq k\leq... | 15 | https://mathoverflow.net/users/11919 | 369625 | 154,664 |
https://mathoverflow.net/questions/369399 | 9 | Let $k$ be an algebraically closed field. Let $V$ be a smooth projective variety over $k$. For a map $\phi:V\to V$, do the coefficients of the characteristic polynomial of $\phi^\*:H^ i\_{dR}(V/k)\to H^ i\_{dR}(V/k)$ lie in the prime subfield of $k$?
| https://mathoverflow.net/users/nan | Coefficients of the characteristic polynomial of the map on algebraic de Rham cohomology | Suppose that $\mathrm{char}\, k=p>0$. It is easy to give an example of a stack $V$ with an endomorphism that violates this property. Take $V=B\alpha\_p$, the scaling action of $\mathbb{G}\_m$ on $\alpha\_p\subset \mathbb{G}\_a$ induces an action on $B\alpha\_p$ so, in particular, the group $k^{\times}$ acts on $V$. By ... | 5 | https://mathoverflow.net/users/39304 | 369626 | 154,665 |
https://mathoverflow.net/questions/369576 | 3 | I want to sample a signal whose derivative I know to be bounded by physical constraints. The sampling is disturbed by gaussian noise, hence I need to filter the sample with a lowpass filter.
Since I know precisely the bound on the derivative magnitude, I was wondering if there is a way to translate this bound in a fr... | https://mathoverflow.net/users/163970 | Relation between signal derivative and frequency spectrum | A lot depends on how you want to formalize your question. Here is one possible approach. Let's say that the signal can be any function on $\mathbb Z$ with the derivative bounded by $1$ and the noise has the standard deviation $\sigma$ and its values at different samples are independent. You apply a linear filter and yo... | 4 | https://mathoverflow.net/users/1131 | 369631 | 154,667 |
https://mathoverflow.net/questions/360941 | 4 | At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist approach to mathematics, formed by Bourbaki. Intuitively (and on many examples) the notion of a mathematical structure is c... | https://mathoverflow.net/users/61536 | What is a good definition of a mathematical structure? | I doubt that there is any generally accepted definition of "structured set" in mathematics that includes a notion of morphism and does not already use the technology of category theory. (For a "behavioral" definition that does use category theory, see for instance [here](https://ncatlab.org/nlab/show/stuff,+structure,+... | 11 | https://mathoverflow.net/users/49 | 369637 | 154,669 |
https://mathoverflow.net/questions/369632 | 5 | Let $f:X\to Y$ be a surjective morphism of smooth irreducible varieties over $\mathbb{C}$. Assume further that $Y$ is complete and that every fiber $f^{-1}(y)$ for $y\in Y$ is complete and irreducible. Does it necessarily follow that $X$ is complete as well? If no, what additional assumptions can we put so that this fo... | https://mathoverflow.net/users/36563 | Complete target and complete fibers imply complete source? | Let $k$ be a field. Let $Y$ be a separated, finite type $k$-scheme that is geometrically connected and normal. Let $f:X\to Y$ be a separated, finite type morphism from a geometrically connected and reduced $k$-scheme to $Y$ such that the fiber over every geometric point of $Y$ is connected and proper.
**Proposition.*... | 8 | https://mathoverflow.net/users/13265 | 369645 | 154,670 |
https://mathoverflow.net/questions/369640 | 8 | Is there a general formula for the number of unramified quadratic extensions of a number field $K$?
When $K$ is quadratic, this is known (by genus theory) to be $2^{\omega(\Delta\_K)-1}$, where $\omega(n)$ denotes the number of distinct prime factors of $n$ and $\Delta\_K$ is the discriminant of $K$. I'm interested i... | https://mathoverflow.net/users/145167 | Number of unramified quadratic extensions of a number field | The answer seems to be no.
1. The number of unramified quadratic extensions of $K$ is equal to the number of index-two subgroups of the ideal class group $\text{Cl}\_K$ by class field theory.
2. The index-two subgroups of $\text{Cl}\_K$ correspond to the non-zero elements of $\text{Hom}(\text{Cl}\_K, \mathbb{Z}/2\mat... | 8 | https://mathoverflow.net/users/145167 | 369649 | 154,671 |
https://mathoverflow.net/questions/369614 | 8 | Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a **t**-structure? If a **t**-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?
My question is motivated by the ongoing search for an abe... | https://mathoverflow.net/users/nan | When does a triangulated category have a heart? | A silly remark is that "trivial" $t$-structures always exist. You should probably say that you want a bounded or a non-degenerate $t$-structure. As far as I remember, non-zero negative $K$-groups of $T$ should give an obstruction for the former condition if you believe that the heart is noetherian or something like thi... | 7 | https://mathoverflow.net/users/2191 | 369654 | 154,672 |
https://mathoverflow.net/questions/369613 | 0 | Let $G=(\mathcal{V}\_G,\mathcal{A}\_G)$ be an oriented acyclic graph. Assume that $G$ has a unique source $s\in \mathcal{V}\_G$ and a unique sink $t\in \mathcal{V}\_G$. Now, fix $u,v\in \mathcal{V}\_G$ such that $(v,u)\in \mathcal{A}\_G$. Is it true that there exists an oriented path in $G$ of type: $v\_0=s, v\_1, \cdo... | https://mathoverflow.net/users/137269 | Oriented path in a graph | With the acyclic condition, the answer is yes. Starting at $v$ and repeatedly following any edge exiting the current vertex, you will eventually end up at $t$, by acyclicity and uniqueness of the sink. Thus there exists a path from $v$ to $t$ and, similarly, there exists a path from $s$ to $u$, which shows what you wan... | 1 | https://mathoverflow.net/users/160416 | 369671 | 154,677 |
https://mathoverflow.net/questions/244051 | 7 | Back when I was first learning about forcing and trying to understand the need to consider *generic* filters, I came up with the following question. Suppose we have a countable transitive model $M$. Let's say that "$p$ pseudoforces $\phi$" if for every filter (not necessarily generic) $G\in P$, $p\in G$ implies that $\... | https://mathoverflow.net/users/3106 | Dropping "generic" from the definition of forcing | In a way what I am going to say here echoes what Joel has already mentioned in his comments above (\*I suppose in that case one would want M[F] to be some kind of reduced power, analogous to what you get with ultrapowers by a filter in place of an ultrafilter. Thus, one might take M[F] naturally as a B/F-valued model),... | 2 | https://mathoverflow.net/users/15293 | 369682 | 154,684 |
https://mathoverflow.net/questions/369690 | 11 | The $n$-th taxicab number, denoted $\text{Ta}(n)$, is the smallest integer that can be expressed as a sum of two positive integer cubes in $n$ different distinct ways.
$\text{Ta}(1) = 2 = 1^3 + 1^3$ is trivial, and the infamous $\text{Ta}(2) = 1729$ was known as early as the 17th century, much before the well-known H... | https://mathoverflow.net/users/156061 | What is the roadblock in the discovery of new taxicab numbers? | There are a few issues here.
(1) It is relatively easy to show that Ta($n$) exists, for example by using a point of infinite order on an elliptic curve $x^3+y^3=mz^3$ to show that there is at least one number with $n$ distinct representations. However, the number tends to be divisible by a large cube, or alternativel... | 21 | https://mathoverflow.net/users/11926 | 369700 | 154,690 |
https://mathoverflow.net/questions/369681 | 2 | For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\Delta(f)(a,b) = f(ab)$ is well-defined.) What is the corresponding construction for a non-unital algebra. The coproduct p... | https://mathoverflow.net/users/121660 | Non-counital coalgebras | The finite dual of a non-unital algebra has been introduced in [Semiperfect and coreflexive coalgebras, S. Dăscălescu, M. C. Iovanov, Forum Math. 27 (2015), No. 5, 2587--2608](https://www.degruyter.com/view/journals/form/27/5/form.27.issue-5.xml). See also: [arXiv:1512.09344 [math.RT]](https://arxiv.org/abs/1512.09344)... | 2 | https://mathoverflow.net/users/85967 | 369705 | 154,693 |
https://mathoverflow.net/questions/369710 | 72 | Let me begin by formulating a concrete (if not 100% precise) question, and then I'll explain what my real agenda is.
Two key facts about forcing are (1) the definability of forcing; i.e., the existence of a notion $\Vdash^\star$ (to use Kunen's notation) such that $p\Vdash \phi$ if and only if $(p \Vdash^\star \phi)^... | https://mathoverflow.net/users/3106 | A better way to explain forcing? | I have proposed such an axiomatization. It is published in Comptes Rendus: Mathématique, which has returned to the Académie des Sciences in 2020 and is now completely open access. Here is a link:
<https://doi.org/10.5802/crmath.97>
The axiomatization I have proposed is as follows:
Let $(M, \mathbb P, R, \left\{\V... | 33 | https://mathoverflow.net/users/9825 | 369718 | 154,700 |
https://mathoverflow.net/questions/369307 | 3 | Let $X\_1$ and $X\_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$.
Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X\_1$ mod $2$. Let
$$f\_1:X\_1\to Z,\qquad f\_2:X\_2\to Z,\qquad F:W\to Z$$
be smooth maps such that $F|\_{X\_1}=f\_1$ and $F|\_{X\_2}=f\_2$. We ca... | https://mathoverflow.net/users/78729 | Does the Gysin map in $K$-theory respect bordism? | Let $N^n=\partial M^{n+1}$, $E\in K^\bullet(M)$ and $f:M\to X$
Choose a smooth embedding $i:X\to \mathbb{R}^N,N>>1$,
denote by $\chi$ the normal bundle of $X$ and by $\mu$ the normal bundle of $M$ after suitable small deformation of $i\circ f$.
Let $\nu=\mu|\_N$ and $\eta$ be the normal bundle of $N\subset M$ (whic... | 2 | https://mathoverflow.net/users/8906 | 369730 | 154,707 |
https://mathoverflow.net/questions/369579 | 8 | I recently learned about automorphic spectral decomposition from the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger. (Let me call it M-W)
I have a question about the characterization of discrete spectra.
Let me explain the basic notation as in M-W.
Let $G$ be a connected reductive ... | https://mathoverflow.net/users/163485 | Characterization of automorphic discrete spectra | It is true by Harish-Chandra's admissibility theorem.(c.f. 1.7 of the article by Borel-Jacquet in the Corvallis book.
| 0 | https://mathoverflow.net/users/163485 | 369733 | 154,710 |
https://mathoverflow.net/questions/369655 | 4 | Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}\_C$ be the set of all $d$-dimensional shapes such that their minimum bounding box is a $d$-dimensional cube $C$. I am interested in ... | https://mathoverflow.net/users/115803 | Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box | This is a bit too long for the comment box, so I'm posting it as an answer.
The worst case scenario is when $X$ is the intersection of a ball of radius $r\ge 1$ with the cube $C=[-1,1]^d$. Indeed, if we take the difference body $\frac{X-X}{2}$ of any body $X$ contained in the cube and of diameter $\ell=2r$, we'll get... | 4 | https://mathoverflow.net/users/1131 | 369739 | 154,713 |
https://mathoverflow.net/questions/369648 | 3 | Since the paper [Smooth and proper noncommutative schemes and gluing of DG categories](https://arxiv.org/abs/1402.7364) by Orlov, dg categories are considered the non-commutative counterpart of algebraic geometry. More specifically, we call a dg category a non-commutative scheme if it is an admissible dg subcategory of... | https://mathoverflow.net/users/91572 | Is there a notion of projective dg category? | If $X$ is a smooth projective threefold with a flopping curve $C$ then typically the variety $Y$ obtained from $X$ by a flop in $C$ is not projective, but smooth, proper, and derived equivalent to $X$. This shows that projectivity is not invariant under derived equivalence, hence does not correspond to a property of th... | 9 | https://mathoverflow.net/users/4428 | 369756 | 154,720 |
https://mathoverflow.net/questions/369659 | 5 | Defining a quasi-coherent module $\mathcal{M}$ on a scheme $X$ to be a compatible family of modules $(\mathcal{M}(x))\_{x \in X(A), A \in \textbf{Rings}}$ (as in [here](https://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheavesII)), is there a straightforward way to show the existence of (finite) limits (and that it f... | https://mathoverflow.net/users/116837 | Existence of finite limits of quasi-coherent modules on a scheme | Here is the precise statement alluded to in the comments:
>
> Let $C = \lim\_i C\_i$ be a limit of categories
> with projections $\pi\_i : C \to C\_i$. Let $\{X\_j\}\_j$ be a
> diagram in $C$. If for every $i$ the induced diagram
> $\{\pi\_i(X\_j)\}\_j$ in $C\_i$ has a limit $X\_i$, and the
> transition functors $C... | 5 | https://mathoverflow.net/users/nan | 369763 | 154,725 |
https://mathoverflow.net/questions/369689 | 6 | I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a quasi-triangular Hopf algebra lurking in the background. In what follows $V$ will always be a vector space of dimension $n$.... | https://mathoverflow.net/users/155559 | Confusion around the reflection equation algebra | 1. The only reasonable definition of the REA associated with a quasi-triangular Hopf algebra is 1). This is, of course, a somewhat abstract definition but provides a solution of the RE which is universal in a precise sense.
2. is reminiscent of the so-called Faddeev-Reshetikhin-Takhtajan (usually abbreviated as FRT) co... | 4 | https://mathoverflow.net/users/13552 | 369767 | 154,727 |
https://mathoverflow.net/questions/369734 | 9 | **EDIT**: I've made a mistake with the matrices. Now it is corrected.
A couple of days ago I asked [this question](https://mathoverflow.net/questions/369214/deciding-if-mathbbz-ltimes-a-mathbbz5-and-mathbbz-ltimes-b-mathbb). There, answerers gave me excellent hints to solve that case and others too. But I've found tw... | https://mathoverflow.net/users/150901 | A "subtle" isomorphism testing problem: $\mathbb{Z}\ltimes_{A} \mathbb{Z}^5\cong \mathbb{Z}\ltimes_{B}\mathbb{Z}^5$ or not? | **Claim.** The groups $G\_A$ and $G\_B$ are not isomorphic.
We will use the following lemma.
**Lemma.** Let $\Gamma\_A = G\_A/Z(G\_A) = C\_6 \ltimes\_{A'} \mathbb{Z}^4$ and $\Gamma\_B = G\_B/Z(G\_B) = C\_6 \ltimes\_{B'} \mathbb{Z}^4$, where $C\_6 = \langle \alpha \rangle$ is the cyclic group of order $6$ and $A'$ a... | 7 | https://mathoverflow.net/users/84349 | 369770 | 154,729 |
https://mathoverflow.net/questions/369775 | 4 | For a prime (or prime power) $p$ and some absolute constant $C$ (say $C$ = 100), consider the set $A$ of all $1 \leq a \leq p/C$ such that $1 \leq a^2 \leq p/C$ modulo $p$. Is it known that $|A| = \Omega(p)$?
| https://mathoverflow.net/users/141963 | Distribution of quadratic residues in an interval | Yes. The points $(\frac{a}p,\frac{a^2\pmod p}p)$ are asymptotically equidistributed in $[0,1]^2$ by Weyl's criterion.
| 7 | https://mathoverflow.net/users/4312 | 369777 | 154,732 |
https://mathoverflow.net/questions/369778 | 4 | Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real number line $\mathbb{R}\cup\{\pm\infty\}$ such that the preimage of $\mathbb{R}$ is dense in $X$. By the Ogasawara theorem,... | https://mathoverflow.net/users/58366 | Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit? | It is not true. Let $X = \beta \mathbb{N}$, so that $C(X) \cong l^\infty$. For each $i, k\in \mathbb{N}$ let $f\_{i,k}$ be the function which is constantly $1$ on $\{1, \ldots, i\}$ and constantly $\frac{1}{k}$ on the rest of $X$. Also let $g\_i$ be the function which is constantly $0$ on $\{1, \ldots,i\}$ and constant... | 2 | https://mathoverflow.net/users/23141 | 369781 | 154,733 |
https://mathoverflow.net/questions/369779 | 1 | Let $(X, \mu)$ be a probability space, and let $p \in (1, 2)$ be arbitrary. It is known from Corollary 2.4 of this [paper](https://www.math.uwo.ca/faculty/sinnamon/pdf/holdmink.pdf) by G. Sinnamon that for any measurable $f : X \to [0, +\infty],$ we have
$$0 \leq \left( \int\_X f^p \ d\mu\right) - \left(\int\_X f \ d... | https://mathoverflow.net/users/94701 | Mean deviation in $p$-norm for $1 < p < 2$ | Let us rewrite the inequality in question as
$$E|f-1|^p\le C\_p(Ef^p-1),$$
where $E$ is the expectation with respect to $\mu$ -- assuming, without loss of generality, that $Ef=1$.
This inequality does not hold in general for any $p\in(1,2)$ and any real constant $C\_p$. Indeed, suppose e.g. that $f=1+tR$, where $P(R=... | 1 | https://mathoverflow.net/users/36721 | 369786 | 154,737 |
https://mathoverflow.net/questions/369787 | 1 | I asked this [question](https://math.stackexchange.com/questions/3788580/the-rank-of-a-linear-combination-of-operators) a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow.
Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now let $a,b\in\mathbb{C}... | https://mathoverflow.net/users/50468 | Rank of a linear combination of linear operators | I think one can cook a similar counter-example to that you mention. Consider the matrices: $X = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}$ and $\ Y = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$.
Then we have : $X... | 1 | https://mathoverflow.net/users/37214 | 369801 | 154,743 |
https://mathoverflow.net/questions/369780 | 5 | Let $G$ be the absolute Galois group of some number field. Can there be a semisimple continuous representation $G\to GL\_n(\overline{\mathbb{F}\_p})$ (the latter has Zariski topology) with infinite image?
| https://mathoverflow.net/users/nan | $\mathrm{mod}\:p$ Galois representation with respect to Zariski topology | No. A quick proof uses the existence of Haar measure on compact topological groups like the Galois group.
The kernel would be a closed subgroup of the Galois group with infinite index, and thus would have Haa measure $0$. However, because $GL\_n (\overline{\mathbb F\_p})$ is countable, countably many translates cover... | 7 | https://mathoverflow.net/users/18060 | 369807 | 154,746 |
https://mathoverflow.net/questions/369797 | 7 | This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I gave up trying to decipher conflicting notations and conventions).
Let $\lambda$ be an integer partition of $n$. A You... | https://mathoverflow.net/users/7410 | A basic question about Young symmetrizers | For Q1 the answer in general is no. Young symmetrizers can be used to give a decomposition of $\mathbb C[S\_n]$ into a direct sum of minimal left ideals but in general they are not pairwise orthogonal. One can actually characterize precisely when $Y(T)Y(T')\neq 0$ holds: (i) the underlying shape of $T$ and $T'$ needs t... | 10 | https://mathoverflow.net/users/2384 | 369808 | 154,747 |
https://mathoverflow.net/questions/369646 | 1 | I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across [this post](https://math.stackexchange.com/questions/2463116/frechet-derivative-of-evaluation-function) I could not help but wonder the following.
Let $H$ be the reproducing-kernel Hil... | https://mathoverflow.net/users/36886 | Fréchet derivative of evaluation-like functional (multivariate) | The general procedure for the identification of a Fréchet derivative is the following
1. Calculate the [functional derivative](https://mathoverflow.net/questions/349057/question-about-functional-derivatives) of the given functional, then
2. verify its linearity and
3. verify its continuity respect to the topology tha... | 1 | https://mathoverflow.net/users/113756 | 369809 | 154,748 |
https://mathoverflow.net/questions/369783 | 6 | Given algebraic spaces $X$, $Y$, $Z$ with a finite morphism $Y \rightarrow X$ and a closed immersion $Y \hookrightarrow Z$, the pushout $P \cong X \amalg\_Y Z$ exists as an algebraic space (cf. [Temkin and Tyomkin - Ferrand pushouts for algebraic spaces](https://arxiv.org/abs/1305.6014), Theorem 6.2.1 (ii),(b)).
Does... | https://mathoverflow.net/users/164072 | Ferrand pushouts for algebraic stacks | Yes, this is exactly Theorem A.4 in my old preprint [*Compactification of tame Deligne–Mumford stacks*](https://people.kth.se/%7Edary/tamecompactification20110517.pdf) which is long overdue to appear on the arXiv. The proof is rather terse but fairly standard (compare with Appendix A of Jack Hall's [*Openness of versal... | 10 | https://mathoverflow.net/users/40 | 369812 | 154,751 |
https://mathoverflow.net/questions/369750 | 5 | Suppose I have a reduced l.c.i. scheme with two irreducible components: $X = Y \cup Z$. I want to say that if $Y$ is Cohen-Macaulay then $Z$ is as well.
I think this follows from Eisenbund Theorem 21.23 (which has a typo: the first $J = (0:\_A I)$ should be deleted). Or from Peskine and Szpiro, "Liaison des variétés ... | https://mathoverflow.net/users/16914 | Linkage and Cohen-Macaulay-ness | The question is local. So, let $R$ be a local ring which is Gorenstein. $I,J\subset R$ define $Y,Z$ as in your question. Then you have an exact sequence $0\to I\to R\to R/I\to 0$ and we are assuming that $R/I$ is Cohen-Macaulay. Notice that all $R,R/I,R/J$ have the same dimension $d$. Dualizing, one gets $0\to\omega\_{... | 4 | https://mathoverflow.net/users/9502 | 369814 | 154,752 |
https://mathoverflow.net/questions/369618 | 1 | For one of the near universal hash functions $f(x) = ax \bmod p \bmod m$ where $p$ is prime and $m < p, m>1$ and $x$ ranges over $1 \dots p-1$ , what is the probability that given $x\_r \in \{ x | x \bmod p \bmod m = x \bmod m = r\}$, $f(x\_r) = s$? That is, find $Pr\_{x\_r}(f(x\_r)=s)$. The probability is the fraction... | https://mathoverflow.net/users/149410 | Probability a near universal hash function $ax \bmod p \bmod m$ produces an output from inputs equal modulo $m$ | Denote $\mathbb Z\_p^\*:=\{1,2,\dots,p-1\}$ and $\mathbb Z\_m:=\{0,1,2,\dots,m-1\}$.
I assume $p\nmid a$. Then $f(x) = g(h(x))$, where $h:\mathbb Z\_p^\*\to \mathbb Z\_p^\*$ is a bijection defined by $h(x):=ax\bmod p$, and $g:\mathbb Z\_p^\*\to \mathbb Z\_m$ is defined by $g(x):=x\bmod m$.
Let $b:=(p-1)\bmod m$ and... | 2 | https://mathoverflow.net/users/7076 | 369817 | 154,753 |
https://mathoverflow.net/questions/369815 | 4 | Does anyone have a reference for the universal deformation space of a cuspidal plane cubic curve? Specifically, a reference that discusses its discriminant locus -- Apparently it has a cuspidal discriminant.
| https://mathoverflow.net/users/105103 | Universal deformation space of a cuspidal plane cubic curve | In *Moduli of Curves* by Harris-Morrison, page 97, it says
>
> The space of first-order deformation of a singular point $p$ of a plane curve $C\subseteq \mathbb A^2$ given by $f(x,y)=0$ is the local ring of $C$ at $p$ modulo the Jacobian ideal $\mathcal{J}$ generated by the partial derivatives $\partial f/\partial ... | 5 | https://mathoverflow.net/users/74322 | 369822 | 154,757 |
https://mathoverflow.net/questions/369832 | 1 | Let $M(x)$ be an $m$ by $n$ matrix with entries in $\mathbb{C}[x]$. Suppose that for all $x\in \mathbb{C}$ the rank of $M(x)$ is constant and equal to $r<n$. Therefore, for any $x\_0\in \mathbb{C}$ we can find a full-rank $N\in \mathbb{C}^{n,n-r}$ such that
$$
M(x\_0)N=0.
$$
Question: is it possible to find an $n$ by $... | https://mathoverflow.net/users/32985 | Global polynomial basis for the kernel of a matrix polynomial | Yes, and there is a constructive algorithm. Put $M$ into Smith normal form:
$ PMQ = D$ for invertible $P$ and $Q$ and diagonal $D$. Since $M(x\_0)$ is full rank for all $x\_0$, the same is true for $D$, and thus $D$ is of the form
$$ D = \begin{bmatrix} c\_1 & 0 & \cdots & 0 & 0 & \cdots &0\\
0 & c\_2 & \cdots & 0 & 0 ... | 3 | https://mathoverflow.net/users/125523 | 369843 | 154,764 |
https://mathoverflow.net/questions/369477 | 2 | In the last semester I learned homological algebra and higher category theory/homotopy theory.
But I am kind of confused when I try to really understand the link between the two subjects (this is really not my comfort zone ...)
Therefore I try to write (a kind of self-exercise) a text on homological algebra and hom... | https://mathoverflow.net/users/155635 | On the link between homology and homotopy | I'd encourage the OP to read the writings of others on this topic, before trying to write something from scratch. I attended lectures at OSU where Aaron Mazel-Gee motivated $\infty$-categories very much as the OP suggests in Question 1. It appears some of the ideas from those lectures have now appeared [here](https://e... | 1 | https://mathoverflow.net/users/11540 | 369844 | 154,765 |
https://mathoverflow.net/questions/369846 | 14 | This should be well-known, but I can't find a reference (or a proof, or a counter-example...). Let $d$ be a positive square-free integer. Suppose that there is no element in the ring of integers of $\mathbb{Q}(\sqrt{d})$ with norm $-1$. Then I believe that no element of $\mathbb{Q}(\sqrt{d})$ has norm $-1\ $
(in fancy ... | https://mathoverflow.net/users/40297 | Norms in quadratic fields | This is false. The smallest counterexample is $d = 34$. Let $K = \mathbb{Q}(\sqrt{34})$. The fundamental unit in $\mathcal{O}\_{K} = \mathbb{Z}[\sqrt{34}]$ is $35 + 6 \sqrt{34}$, which has norm $1$, and therefore, there is no element in $\mathcal{O}\_{K}$ with norm $-1$.
However, $\frac{3}{5} + \frac{1}{5} \sqrt{34}$... | 24 | https://mathoverflow.net/users/48142 | 369857 | 154,769 |
https://mathoverflow.net/questions/369722 | 8 | Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of trace-class operators, one can show using standard result of Banach space theory that, any bounded linear functional $\... | https://mathoverflow.net/users/116379 | Are (completely) positive maps approximated by normal (completely) positive maps? | Also the answer to the second question is **yes**, and the approximation may be chosen to converge in the point-ultrastrong$^\*$ topology.
First, by choosing a net of finite rank orthogonal projections $p\_i \in B(\mathcal{H})$ such that $p\_i \rightarrow 1$ strongly, the completely positive maps $\Phi\_i : M \righta... | 8 | https://mathoverflow.net/users/159170 | 369860 | 154,771 |
https://mathoverflow.net/questions/369840 | 3 | I'm not sure this question is research level question. Sorry in advance.
**Hypothesis**
1. $k$ is a commutative ring.
2. $A$ is an augmented $k$-algebra.
3. $A^e$ is defined as the $k$-algebra $A\otimes\_{k}A^{op}$. It is naturally augmented $k$-algebra.
**assumptions**
1. $k$ (as left $A$-module) is quasi-isom... | https://mathoverflow.net/users/128371 | Smallness condition for augmented algebras | No.
Let $k$ be a field, and let $A$ be the algebra of upper triangular $2\times 2$ matrices over $k$, with augmentation map $\pmatrix{a&b\\0&c}\mapsto a$.
$A$ and $A^e$ have finite global dimension, so all modules have finite projective dimension, and are therefore quasi-isomorphic to perfect complexes.
Let $\mat... | 3 | https://mathoverflow.net/users/22989 | 369867 | 154,773 |
https://mathoverflow.net/questions/369871 | 4 | In 1976 Tijdeman proved that the Catalan equation
$$
x^{p}-y^{q}=1
$$
has finitely many solutions in integers $x,y,p,q>1$ in his paper
* R. Tijdeman, *On the equation of Catalan*, Acta Arith. **29** (1976) pp 197–209 ([EuDML](https://eudml.org/doc/205418))
He just found the following upper bound for $p$ and $q$ usi... | https://mathoverflow.net/users/158974 | How to prove that a diophantine equation has only finitely many solutions in integers? | By the initial remarks in the paper, one can restrict to $p,q\geq 5$. By Theorem A in the paper (which is a result of Baker's from 1969), $x$ and $y$ can be effectively bounded in terms of $p$ and $q$:
$$\max(|x|,|y|)<\exp\exp(5^{10}p^{10} q^{10q^3}).$$
Hence it suffices to bound $p$ and $q$.
| 4 | https://mathoverflow.net/users/11919 | 369872 | 154,777 |
https://mathoverflow.net/questions/369880 | 3 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $V\neq \varnothing$ and $\varnothing \notin E$. A *matching* is a subset $M\subseteq E$ such that $m\_1\neq m\_2 \in M$ implies $m\_1\cap m\_2 = \varnothing$, and $M$ is said to be *perfect* if $\bigcup M = V$. We say that $H$ is *$1$-f... | https://mathoverflow.net/users/8628 | $1$-factorizability for "complete" finite hypergraphs | This is [Baranyai's theorem](https://en.wikipedia.org/wiki/Baranyai%27s_theorem). Other than in Baranyai's original paper you can also find a cool proof in the article "Uniform hypergraphs" by Brouwer and Schrijver which uses max-flow min-cut.
| 6 | https://mathoverflow.net/users/2384 | 369883 | 154,782 |
https://mathoverflow.net/questions/369882 | 13 | Let $Y\_d$ be a Fano threefold of Picard rank $1$ and index $2$ (eg cubic 3fold). There is a natural anticanonical map $Y\_d\to \mathbb{P}^{d+1}$. Smooth sections of the anticanonical bundle are $K3$ surfaces, so we can ask the following
>
> **Question:** does such a general $K3$ surface contain lines in $\mathbb{P... | https://mathoverflow.net/users/48616 | Lines on an anticanonical K3 on a Fano 3-fold | If $Y$ is a Fano threefold with Picard rank $1$, then its general anticanonical element $X \in |-K\_Y|$ is a $K3$ surface with Picard rank $1$. In particular, $X$ contains no lines.
This is explained (for instance) at p. 797 of
C. F. Doran, A. Harder, A. Y. Novoseltsev, A. Thompson: [Calabi–Yau threefolds fibred by... | 15 | https://mathoverflow.net/users/7460 | 369886 | 154,783 |
https://mathoverflow.net/questions/369890 | 2 | I have a question about a specific proof that all finite group schemes in characteristic 0 are etale. The proof is [here](http://www-personal.umich.edu/%7Easnowden/teaching/2013/679/L05.html), Proposition 8 in lecture notes by Andrew Snowden.
In his notation, let $A = k\oplus I$ be a (finite) local group scheme over ... | https://mathoverflow.net/users/58001 | Proving that finite, connected group schemes in characteristic 0 are trivial | I don't understand that claim either.
It seems to me that if you follow the chain rule you get $$D\_i (\varphi(f)) = \sum\_{j=1}^n D\_i(x\_j) \varphi \left( \frac{\partial f}{\partial x\_j } \right). $$ You have $D\_i(x\_j) \equiv \delta\_{ij} \mod I$ so the matrix with entries $D\_i(x\_j)$ is invertible which means ... | 6 | https://mathoverflow.net/users/18060 | 369898 | 154,786 |
https://mathoverflow.net/questions/369874 | 5 | Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only **continuous, complex-valued** functions on $X$.
>
> Let $\varepsilon >0$. Is there $\delta > 0$ such that for any given functions $f, g$ on $X$ that are **nowhere zero** and any function $d$ with $\|d\|\_\infty... | https://mathoverflow.net/users/15129 | A functional equation in two complex variables | $Hello$, Tomasz! (for some reason the MO prohibits saying "Hi" or "Hello" in the normal text mode). Nice to see you back. Apparently you are still asking the same question whether a function $H$ close to the product $fg$ can be represented as a product $FG$ where $F$ is close to $f$ and $G$ is close to $g$ but now just... | 8 | https://mathoverflow.net/users/1131 | 369899 | 154,787 |
https://mathoverflow.net/questions/369908 | 3 | I'm studying relationships between trace entropy functionals and combinatorics and I'm faced with the following problem. Lets $\mathcal {D}$ be the following differential operator $1 -x\cdot \cfrac{d}{dx}$ i.e. $\mathcal {D} g = g - x\cdot g'$.
For $m\ge 0$ integer, if $\Phi\_m(x) := x\cdot \log(x)^m$ then $\mathcal ... | https://mathoverflow.net/users/150973 | Trace entropies | For any $c\in(0,\log2]$, the function $g$ defined by the formula $g(x)=cx$ for $x\in[0,1]$ satisfies your conditions I)–V), but it is not the Boltzman–Gibbs–Shannon entropy trace.
---
There are many more functions $g$ satisfying your conditions I)–V) that are not the Boltzman–Gibbs–Shannon entropy trace. In parti... | 1 | https://mathoverflow.net/users/36721 | 369925 | 154,794 |
https://mathoverflow.net/questions/369924 | 1 | I have defined $S\_4$ (Symmetric group of order 4), and with the base field $Z\_5$, groupring $Z\_5S\_4$ is constructed. Then I have taken two elements of this group ring and I want to multiply them to get the simplest result.
```
gap> f := FreeGroup( "a", "b","c" );;
gap> G := f / [ f.1^2, f.2^3,f.3^4, f.1*f.2*f.3 ... | https://mathoverflow.net/users/160231 | Problem while multiplying under a set of relators | Algorithms for finitely presented groups are hard -- generically problems, such as testing whether a word represents the identity (or finding a shortest word expression) do not have (they cannot exist as they are equivalent to the Halteproblem for Turing machines) general algorithmic solutions.
Therefore GAP will by de... | 3 | https://mathoverflow.net/users/59303 | 369937 | 154,798 |
https://mathoverflow.net/questions/369931 | 10 | The question was [asked on Mathematics Stackexchange](https://math.stackexchange.com/q/3792791/660)
but has remained unanswered so far.
A self-map is a map $f:X\to X$ from a set $X$ to itself. There is an obvious notion of *morphism*, and thus of *isomorphism* and *automorphism*, of self-maps. [A morphism from $f:X\t... | https://mathoverflow.net/users/461 | Does every set have a rigid self-map? | Yes. Actually, this was part of [my first answer](https://mathoverflow.net/a/358067/14094) to [this question](https://mathoverflow.net/q/358057/14094), but this was a digression there (and I also posted there another answer to the same question which addressed it and was accepted). So I'm copying this digression here a... | 11 | https://mathoverflow.net/users/14094 | 369938 | 154,799 |
https://mathoverflow.net/questions/369130 | 9 | Let $G$ be a Fuchsian group of first kind contained in $\text{PSL}\_2(\mathbb{R})$. A result of Eichler says, there exists a finite set $S\subset G$ such that any $\gamma$ in $G$ can be written as a product $\prod\_{i=1}^{k} \gamma\_i,$ where each $\gamma\_i$ are either in $S,$ or power of some parabolic element coming... | https://mathoverflow.net/users/100578 | Fuchsian groups and Eichler's result | This follows from Theorem 2(i) and Theorem 4 in [*The structure of words in discrete subgroups of $\mathrm{SL}(2,\mathbb{C})$*](https://doi.org/10.1112/jlms/s2-10.2.201), by Beardon.
---
Since it isn't explicitly stated, I will roughly summarize/explain how you get the result.
Lets $D$ be a convex fundamental p... | 9 | https://mathoverflow.net/users/nan | 369947 | 154,800 |
https://mathoverflow.net/questions/272640 | 7 | In (function space) interpolation theory, a Banach space $E$ is of *class $J(\theta)$* (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an interpolation couple, and there exists a constant $C>0$ such that
$$\|x\|\_E \leq C \|x\|\_X^{1-\theta} \|x\|\_Y^\theta \... | https://mathoverflow.net/users/85906 | "Reversion" of class $J(\theta)$ interpolation property for Besov spaces | Here is a negative answer for a certain range of $p$ and $\theta$. It shows that one can have convergence to zero in a $\gamma$-Besov norm, where $\gamma$ may be larger than $\theta$. Hope I didn't mess up the parameters.
>
> **Claim.** For any $p\in (1,\infty)$ and $\theta, \gamma \in (0, 1 - 1/p)$ there are $f\_\... | 2 | https://mathoverflow.net/users/136913 | 369950 | 154,801 |
https://mathoverflow.net/questions/366563 | 3 | Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}\_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber over a closed point has a trivial étale fundamental group. Let $s, s'\in S$ be two closed points.
For $l\neq p$ is there an equivalence between the localized-at-$l$ étale hom... | https://mathoverflow.net/users/nan | Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes | By finding a path between $s$ and $s'$, it suffices to consider the case $S$ local strictly henselian, $s$ the closed (geom.) point, $s'\to S$ some other geometric point. Cutting down with curves, we can even assume $S$ is the spectrum of a strictly henselian discrete valuation ring, and that $s'$ is the geometric gene... | 4 | https://mathoverflow.net/users/3847 | 369953 | 154,804 |
https://mathoverflow.net/questions/369891 | 3 | Let $\Gamma$ and $\Delta$ be theories in the language of set theory (LST), and let $M = \{x: \phi(x)\}$ be a class, where $\phi(x)$ is some formula of LST. Let us say that $M$ is a (standard) class model of $\Gamma$ in $\Delta$ if and only if $\Gamma \vdash \psi$ implies $\Delta \vdash \psi^M$ for all sentences $\psi$ ... | https://mathoverflow.net/users/17218 | What are some further examples of proper class models of ZF that are contained in their own "self-relativization"? | Let $AC$ stand for the axiom of choice, let $L$ denote the constructible universe and let $L^\*$ the universe of constructible sets transitively containing $\emptyset$ as an element.
Although $L^\*$ is a proper subclass of $L$, it collapses to $L$, so it is isomorphic to $L$.
Let $\phi(x)$ be the formula
* $(AC\rig... | 2 | https://mathoverflow.net/users/9825 | 369960 | 154,807 |
https://mathoverflow.net/questions/369941 | 3 | Let $n$ be a natural number and $D\_n$ be the set of divisors.
We can make this set to a ring by observing that each divisor $d$ has
$$0 \le v\_p(d) \le v\_p(n)$$
Hence we can add two divisors $d,e$ by setting:
$$d \oplus e := \prod\_{p | n} p^{v\_p(d)+v\_p(e) \mod (v\_p(n)+1)}$$
and similarily we can multiply ... | https://mathoverflow.net/users/nan | Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix? | In both cases you are really only using the additive structure of your rings, so this is really a question about abelian *groups*.
Assuming $n = p\_1^{a\_1} \cdots p\_r^{a\_r}$, when studying $A\_n$ we are working with the abelian group $$G=\mathbb{Z}/(a\_1+1)\mathbb Z \times \cdots \times \mathbb{Z}/(a\_r+1)\mathbb ... | 3 | https://mathoverflow.net/users/2384 | 369962 | 154,808 |
https://mathoverflow.net/questions/369983 | 1 | Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi\_1,\phi\_2$ on $\mathbb{R}^n$ and on $\mathbb{R}^m$, respectively, such that
$$
\sup\_{x \in K}\left\|
\phi(x,y) -(\phi\_1(x),\phi\_2(y))
\right\|<\e... | https://mathoverflow.net/users/36886 | Is every homeomorphism approximately a product of homeomorphisms? | Take $m=n=1$, $K=[0, 1]$ and $\phi$ to be rotation by 90 degrees.
Then the $x$-coordinate of $(\phi\_1(x), \phi\_2(y))$ is bounded for $x\in K$ and any $\phi\_1, \phi\_2$ while there are points with arbitrarily large $y$-coordinate (which becomes arbitrarily large $x$-coordinate upon applying $\phi$).
| 2 | https://mathoverflow.net/users/nan | 369985 | 154,810 |
https://mathoverflow.net/questions/369790 | 7 | In *Birational Geometry of Algebraic Varieties*, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line bundle $L$ on a projective variety $X$ of dimension $d$ is *big* if
$$ \limsup\_{n \to \infty } \dfrac{H^0(X,L^n)}{n^d} \neq 0.$$... | https://mathoverflow.net/users/158012 | Multiplication maps for big line bundles | If $R(L)=\oplus H^0(mL)$ is not finitely generated, the above surjectivity will fail, however it will hold "asymptotically" for any big line bundle $L$. In fact, by Fujita's approximation of big classes (see eg. Lazarsfeld's Positivity book Theorem 11.4.4), for any $\epsilon >0$ there is a birational modification $f:X'... | 2 | https://mathoverflow.net/users/19369 | 370009 | 154,816 |
https://mathoverflow.net/questions/369930 | 34 | Below, we compute with exact real numbers using a realistic / conservative model of computability like [Type Two Effectivity](https://link.springer.com/book/10.1007/978-3-642-56999-9).
Assume that there is an algorithm that, given a symmetric real matrix $M$, finds an eigenvector $v$ of $M$ of unit length.
Let
$$... | https://mathoverflow.net/users/75761 | Why is uncomputability of the spectral decomposition not a problem? | The singular value decomposition, when applied to a real symmetric matrix $A = \sum\_i \lambda\_i(A) u\_i(A) u\_i(A)^T$, computes a stable mathematical object (spectral measure $\mu\_A = \sum\_i \delta\_{\lambda\_i(A)} u\_i(A) u\_i(A)^T$, which is a [projection-valued measure](https://en.wikipedia.org/wiki/Projection-v... | 77 | https://mathoverflow.net/users/766 | 370014 | 154,819 |
https://mathoverflow.net/questions/369995 | 8 | I am studying the proof that if $A$ is a $C^\*$-algebra such that $A^{\*\*}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the unital case, but I am stuck in the non-unital setting: The authors (N. Brown and N. Ozawa) suggest that one should prove... | https://mathoverflow.net/users/164203 | The double dual of the unitization of a $C^*$-algebra | Believe it or not, these are $\*$-isomorphisms as C${}^\*$-algebras. If $J$ is a closed two-sided ideal of $B$ then $J^{\*\*}$ is a weak\* closed two-sided ideal of $B^{\*\*}$, and every weak\*-closed two-sided ideal of a von Neumann algebra is a direct summand. I suppose these are good exercises. The supremum in $B^{\... | 11 | https://mathoverflow.net/users/23141 | 370015 | 154,820 |
https://mathoverflow.net/questions/369875 | 0 | Let $X=(X\_1,\ldots,X\_n)$, where $X\_i \sim P\_{p\_i}(0,\frac{1}{\lambda})$ are iid, $P\_{p\_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm looking for a result on the concentration of $\|X\|\_2^2$ something of the form $E \|X\|\_2^2 \leq c$ with $P(\|X\|... | https://mathoverflow.net/users/121695 | Concentration of $\ell_2$ norm of a vector sampled from a distribution | WLOG, let $\lambda = 1$ (rescale your problem appropriately, if necessary). Then, it is well-known consequence of [Bernstein's inequality](https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory)) (e.g see theorem 3.1.1 of "High-dimensional Probability" book by R. Vershynin) that
$$
\mathbb P\left(\l... | 3 | https://mathoverflow.net/users/78539 | 370020 | 154,821 |
https://mathoverflow.net/questions/369403 | 6 | Let $S$ be a regular Noetherian scheme, and let $U\subset S$ be the complement of a divisor. If $A\to B$ is an isogeny of abelian schemes over $U$, and $A$ extends to a semi-abelian scheme over $S$, does $B$ also extend to a semi-abelian scheme over $S$?
I'm happy to assume that the degree of the isogeny is invertibl... | https://mathoverflow.net/users/163886 | Extensions of (semi-)abelian schemes | Under the hypothesis that the degree of the isogeny is invertible in $S$, the answer is yes; I attempted to sketch a proof below under slightly more general assumptions on the base. [On the other hand, if the degree of the isogeny is not invertible in $S$, the answer is no. There are counterexamples over regular rings ... | 2 | https://mathoverflow.net/users/117533 | 370025 | 154,823 |
https://mathoverflow.net/questions/370028 | 18 | I stumbled upon these very nice looking [notes](https://math.uchicago.edu/%7Ebrianrl/notes/fibonacci.pdf) by Brian Lawrence on the period of the Fibonacci numbers over finite fields. In them, he shows that the period of the Fibonacci sequence over $\mathbb{F}\_p$ divides $p$ or $p-1$ or $p+1$.
I am wondering if there... | https://mathoverflow.net/users/92401 | The period of Fibonacci numbers over finite fields | This maybe too elementary for this site, so if your question is closed, you might try asking on MathStackExchange. Many questions about the period can be answered by using the formula
$$ F\_n = (A^n-B^n)/(A-B), $$
where $A$ and $B$ are the roots of $T^2-T-1$. So if $\sqrt5$ is in your finite field, then so are $A$ and ... | 36 | https://mathoverflow.net/users/11926 | 370030 | 154,824 |
https://mathoverflow.net/questions/370044 | 6 | A local fusion category ${\cal R}$ is a unitary fusion category equipped with a top-faithful surjective monoidal functor to the fusion category of vector spaces: $\beta: {\cal R} \to {\cal V}ec$. Here, top-faithful means that the functor $\beta$ is injective when acting on the morphisms.
What are those local fusion c... | https://mathoverflow.net/users/17787 | Local fusion categories | Let $\mathcal{R}$ be a fusion category and $\beta : \mathcal{R} \to \mathrm{Vec}$ an additive monoidal functor.
I first claim that $\beta$ is automatically faithful. (I know why you use "top faithful" — in higher categories, you want faithfulness just on top-morphisms — but here in 1-category land "top faithful" is j... | 11 | https://mathoverflow.net/users/78 | 370049 | 154,829 |
https://mathoverflow.net/questions/370024 | 6 | If $G=(V,E)$ is a simple, undirected graph, is there a [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) graph $G\_v$ such that $\chi(G) = \chi(G\_v)$ and $G$ is isomorphic to an induced subgraph of $G\_v$?
| https://mathoverflow.net/users/8628 | Embedding any graph into a vertex-transitive graph of the same chromatic number | For $k\in\mathbb N$ the random $k$-chromatic countably infinite graph is vertex transitive and contains an isomorphic copy of every $k$-colorable countable graph as an induced subgraph. I suppose this can be generalized somehow to uncountable graphs and infinite chromatic numbers, but I don't think anyone is interested... | 5 | https://mathoverflow.net/users/43266 | 370052 | 154,832 |
https://mathoverflow.net/questions/370034 | 7 | Let's say I'm thinking about an unsolved mathematical problem for a hobby and I draw some conclusions of my own. I'd like to make these ideas public, allow anyone to use them absolutely freely (even without mentioning me) and maintain these ideas on my blog/notes website.
Now let's say a reader leaves a comment and c... | https://mathoverflow.net/users/164232 | Public personal ideas: using contributions from other people | When it comes to legal rights, I think a disclaimer might suffice, though IANAL and you maybe should consult one.
When it comes to giving proper attribution and credit for ideas, I think this is not (or not only) something that an author owes to a specific originator of the ideas, but rather (or also) something an au... | 6 | https://mathoverflow.net/users/3077 | 370054 | 154,833 |
https://mathoverflow.net/questions/370043 | 0 | I am struggling to solve an optimization problem of the following form:
$$\begin{array}{ll} \underset{A}{\text{maximize}} & \log \det (A) \\ \text{subject to} & a^T A^{-1} a \le b\end{array}$$
Is there any solution for it?
| https://mathoverflow.net/users/164237 | Optimization problem involving matrix | I will presume you want $A$ to be constrained to be symmetric (hermitian) psd. In that case, this is a convex optimization problem which is a Linear Semidefinite Programming problem (SDP) a.k.a. Linear Matrix Inequality (LMI).
A convex optimization modeling tool, such as CVX, can formulate this as a standard Linear S... | 2 | https://mathoverflow.net/users/75420 | 370057 | 154,834 |
https://mathoverflow.net/questions/184954 | 15 | Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed Hodge structure, by Deligne.
Is there an example where this mixed Hodge structure does not split? In other words, where... | https://mathoverflow.net/users/1310 | Mixed Hodge structure on configuration spaces | In fact the conclusion in Gorinov's paper seems to be false, see
>
> E. Looijenga, "Torelli group action on the configuration space of a surface", [arXiv:2008.10556](https://arxiv.org/abs/2008.10556)
>
>
>
| 7 | https://mathoverflow.net/users/318 | 370060 | 154,835 |
https://mathoverflow.net/questions/370053 | 1 | I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked [this](https://math.stackexchange.com/questions/3799694/infinite-pro-p-group-of-finite-solvable-length-and-finite-coclass) question in math.stackExchange before ... | https://mathoverflow.net/users/89515 | Infinite pro-$p$ group of finite solvable length and finite coclass | The answer is yes, $S$ is an inverse limit of its lower central quotients. As these have bounded derived length, the same goes for the Cartesian product of these groups.
By the way, all pro-$p$ groups of finite coclass are solvable, that's Theorem D of the coclass theory.
| 3 | https://mathoverflow.net/users/6339 | 370062 | 154,836 |
https://mathoverflow.net/questions/369520 | 6 | Recently, I've been studying what the definable subsets of the countable ordinals "look like" from the perspective of bare-bones first order logic (not set theory) equipped with various ways to "access" the structure of the ordinals.
For example, we may have a signature consisting only of a 2-arity relational symbol ... | https://mathoverflow.net/users/115247 | Definability of ordinals in various signatures | I do not have enough reputation to add a comment. The following paper may be useful for you. It contains results extending the work of Tarski, Mostowski, and Doner, as well as some very nice historical overview and references.
[Buchi, Siefkes - The Complete Extensions of the Monadic Second Order Theory of Countable O... | 6 | https://mathoverflow.net/users/164256 | 370063 | 154,837 |
https://mathoverflow.net/questions/370026 | 19 | In this [interview by Eric Weinstein to Roger Penrose, Timestamp 1:24:05.](https://youtu.be/mg93Dm-vYc8?t=5045), what result is the host talking about?
Transcription of the relevant part:
>
> "If you have two sets of symmetries, known as Lie groups, that act transitively on the same sphere in usual position, then... | https://mathoverflow.net/users/4721 | A result on Lie group actions on 15-dimensional spheres? | My guess is that Weinstein was thinking of this fact, but didn't get it out correctly:
For every $n\not=15$, there is a compact Lie group $H\_n\subseteq\mathrm{SO}(n{+}1)$ that acts transitively on the $n$-sphere such that any Lie group $G$ that acts transitively and effectively on the $n$-sphere contains a subgroup ... | 35 | https://mathoverflow.net/users/13972 | 370065 | 154,838 |
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