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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/420538 | 3 | I'm trying to show that manifolds are affine, i.e. $\text{Man}(M,N)\cong \mathbb{R}\text{-Alg}(C^\infty(N),C^\infty(M)) $. If I could show this for $N=\mathbb{R}^n$, then I know how to do the rest using the strong Whitney's embedding theorem. But it seems hard to show the surjectivity of $ \text{Man}(M,\mathbb{R}^n)\ho... | https://mathoverflow.net/users/167862 | How to show that $\text{Man}(M,\mathbb{R}^n)\cong \mathbb{R}\text{-Alg}(C^\infty(\mathbb{R}^n),C^\infty(M))$? | The proof of this fact is available in modern textbooks.
For example, see Theorem 7.16 in Jet Nestruev's *Smooth Manifolds and Observables* (Second Edition, 2020).
In fact, the cited book contains a lot of material that explains how to pass between differential geometric objects and the corresponding algebraic obje... | 10 | https://mathoverflow.net/users/402 | 420552 | 171,130 |
https://mathoverflow.net/questions/420547 | 4 | I'm looking for the proof that $\mathtt{PSP}$, the statement that every uncountable subset of the the Baire space $\mathbb{N}^\mathbb{N}$ contains an homeomorphic copy of the Cantor space $2^\mathbb{N}$, implies the consistency of inaccessible cardinals.
I was able to track down Mycielski's paper *On the axiom of det... | https://mathoverflow.net/users/141146 | $\mathtt{PSP}$ implies the consistency of inaccessible cardinals | See Proposition 11.5 (and the discussion leading up to it) in Kanamori's book **The Higher Infinite**. Note that Kanamori states the result a bit more optimally: if $M\models\mathsf{ZF}$ + "$\omega\_1$ is regular" + $\mathsf{PCP}$ then $\omega\_1^M$ is inaccessible in $L^M$ (and in fact $M\models$ "$\omega\_1$ is inacc... | 4 | https://mathoverflow.net/users/8133 | 420553 | 171,131 |
https://mathoverflow.net/questions/420501 | 1 | As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between vectors a and b, b and c, c and a.
I am trying to bound the area of sphere in which the normal of hyperplane can resid... | https://mathoverflow.net/users/480694 | Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given | $\newcommand\al\alpha\newcommand\be\beta\newcommand\ga\gamma$It appears that the question is as follows: Given unit vectors $a,b,c$ with angles
$$\al:=\cos^{-1}(b\cdot c),\quad \be:=\cos^{-1}(a\cdot c),\quad \ga:=\cos^{-1}(b\cdot a)$$
(where $\cdot$ denotes the dot product), find the probability, say $p$, that the vect... | 5 | https://mathoverflow.net/users/36721 | 420559 | 171,133 |
https://mathoverflow.net/questions/403291 | 1 | Throughout, by "logic" I mean regular logic (in the sense of Ebbinghaus–Flum–Thomas) whose sentences are coded by elements of $\mathsf{HC}$. Say that $\mathcal{L}$ is **Barwise compact** iff whenever $\mathbb{A}$ is a countable admissible set and $X\subseteq\mathbb{A}$ is a $\mathbb{A}$-c.e. $\mathcal{L}$-theory each o... | https://mathoverflow.net/users/8133 | Natural strong logic with Barwise compactness property | A very nice family of examples is provided by Harrington's 1980 paper *[Extensions of countable infinitary logic which preserve most of its nice properties](https://link.springer.com/content/pdf/10.1007/BF02021129.pdf)*. Harrington shows that if we expand $\mathcal{L}\_{\omega\_1,\omega}$ by any infinitary propositiona... | 1 | https://mathoverflow.net/users/8133 | 420561 | 171,134 |
https://mathoverflow.net/questions/389952 | 2 | I'd like to know if there is a concentration inequality for the sample covariance matrix that don't assume the knowledge of the true mean.
---
**Background.**
Given a probability distribution $\mu$ on $\mathbb R^d$, the covariance matrix of $\mu$ is defined as follows:
$$\Sigma := \mathbb E [(x - \bar \mu)(x -\... | https://mathoverflow.net/users/156792 | Concentration inequality for the sample covariance matrix | A variant of this (taking $\frac1m$ instead of $\frac1{m-1}$ for the empirical covariance) is answered on Proposition 2.6, page 10 of <https://arxiv.org/pdf/2110.06357.pdf> .
| 2 | https://mathoverflow.net/users/156792 | 420569 | 171,137 |
https://mathoverflow.net/questions/229088 | 6 | There exists a Latin square of order $8$ which can be partitioned into $2 \times 4$ subrectangles:
$$
\begin{bmatrix}
\color{red} 1 & \color{red} 2 & \color{red} 3 & \color{red} 4 & \color{purple} 5 & \color{purple} 6 & \color{purple} 7 & \color{purple} 8 \\
\color{red} 2 & \color{red} 3 & \color{red} 4 & \color{red} 1... | https://mathoverflow.net/users/48278 | For which divisors $a$ and $b$ of $n$ does there exist a Latin square of order $n$ that can be partitioned into $a \times b$ subrectangles? | I eventually co-authored a paper which includes this topic. The non-trivial results are:
>
> * There exists a Latin square of order $n$ which decomposes into $2 \times (n/2)$ subrectangles for all even $n \not\in \{2,6\}$.
> * There exists a Latin square of order n which decomposes into $3 \times (n/3)$ subrectangl... | 5 | https://mathoverflow.net/users/48278 | 420575 | 171,138 |
https://mathoverflow.net/questions/420567 | 1 | Is there study on polynomial-like functions of the following kind?
$$f(z) = c\_0 + a\_1z+b\_1\bar{z} + a\_2z^2+b\_2\bar{z}^2 + ...+ a\_nz^n+b\_n\bar{z}^n$$
My reason for studying it is polynomials are analytic so that they can't approximate non-analytic complex functions with arbitrary precision. While the above func... | https://mathoverflow.net/users/151671 | Complex polynomial-like functions with conjugate terms | Yes, there are studies under the title harmonic maps of the plane. In general, a harmonic map is of the form $f+\overline{g}$ where $f,g$ are analytic. Most of this literature is about univalent harmonic maps (in a region) but there are some papers on the general case, including polynomials and entire functions in the ... | 3 | https://mathoverflow.net/users/25510 | 420588 | 171,140 |
https://mathoverflow.net/questions/417973 | 6 | It is known that if a finite group $G$ admits a faithful *topological* action on the 3-sphere $S^3$, then $G$ admits a faithful action on $S^3$ by *isometries*. (Pardon proved that a topological action implies a smooth action, and Dinkelbach & Leeb proved that a smooth action implies an isometric one.) I wonder if this... | https://mathoverflow.net/users/69681 | From topological actions on $\mathbb{R}^3$ to isometric actions | I believe (though have not checked carefully) that the argument in my paper proves:
>
> If $\Gamma$ (discrete) acts continuously and properly discontinuously on a smooth three-manifold $M$, then that action can be uniformly approximated by a smooth action.
>
>
>
The point is simply that each step in the argume... | 2 | https://mathoverflow.net/users/35353 | 420591 | 171,142 |
https://mathoverflow.net/questions/420102 | 6 | Let $G \neq 1$ be a finite perfect group which is not simple.
Is it true that $G$ necessarily has a maximal subgroup whose derived subgroup
has nontrivial core in $G$?
*Remark 1:* This holds for all such $G$ of order less than 100000.
*Remark 2:* In case the answer is negative, I would mainly be interested
in a cou... | https://mathoverflow.net/users/128342 | Does a non-simple perfect group always have a maximal subgroup whose derived subgroup has nontrivial core? | The answer to the title question is 'No.' Following YCor's comment, an example is furnished by $S=J\_1$, the smallest Janko sporadic group, and its complex irreducible character $\chi$ of degree 76 (in Atlas notation, 76a). I have checked by hand (hopefully correctly), using the Atlas, that $(\chi\downarrow M,1\_M)>0$ ... | 6 | https://mathoverflow.net/users/99221 | 420592 | 171,143 |
https://mathoverflow.net/questions/420583 | 2 | Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \mathrm{Der}(\frak{a})$. For the zero map $\frak{g} \to \mathrm{Der}(\frak{a})$ we get the usual direct product. But can there... | https://mathoverflow.net/users/176218 | Non-isomorphic direct products of a solvable and a semisimple Lie algebra | Let $K$ be the ground field of characteristic zero.
For $n\ge 1$, let $\mathfrak{v}\_n$ denote an $n$-dimensional irreducible representation of $\mathfrak{sl}\_2(K)$ over $K$ ($(n-1)$-th symmetric power of the standard 2-dimensional representation). Then $\mathfrak{v}\_3\rtimes\mathfrak{sl}\_2(K)$ and ($\mathfrak{v}\... | 2 | https://mathoverflow.net/users/14094 | 420595 | 171,145 |
https://mathoverflow.net/questions/420551 | 4 | A real matrix is called totally positive if all of its minors are positive. Of course, to check that a given matrix is totally positive it is not necessary to check all the minors: for example, it suffices to check that all initial minors are positive. A minor $A\begin{pmatrix}I\\J\end{pmatrix}$ of $A$, corresponding t... | https://mathoverflow.net/users/5018 | Total positivity tests: optimal in the number of minors vs. the computational cost | There are at least a few places looking at complexity for total positivity counting arithmetic operations. I don't know if people have looked at trying to find smaller minors. These use initial minors but try to compute smartly.
>
> Cryer, Colin W. Some properties of totally positive matrices. Linear Algebra Appl. ... | 5 | https://mathoverflow.net/users/51668 | 420597 | 171,146 |
https://mathoverflow.net/questions/420600 | 2 | Assume factorization of $N$ is unknown. What is the best complexity we know to find roots of the irreducible equation $$ax^2+bx+c\equiv0\bmod N?$$
Is this problem equivalent to any hardness results?
| https://mathoverflow.net/users/10035 | On roots of irreducible quadratics modulo composites | Ability to find all roots leads to finding factorization of $N$. For example, if $N=pq$ is a product of two odd primes, and we find a root $x'$ of $x^2-1\equiv 0\pmod N$ such that $x'\equiv 1\pmod p$ and $x'\equiv -1\pmod q$, then we can compute $p$ as $\gcd(N,x'-1)$.
| 6 | https://mathoverflow.net/users/7076 | 420604 | 171,148 |
https://mathoverflow.net/questions/420605 | 3 | I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest .
---
Consider a Heegaard splitting $M = V\cup\_F W$ of a $3$-manifold $M$ with splitting surface $F = \partial V = \partial W$. Suppos... | https://mathoverflow.net/users/136505 | Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$ | Yes, you are correct that the two handlebodies ($V$ cut along $D$ and $V$ plus a neighbourhood of $E$) are isotopic. This is discussed when defining *stabilisation* and the inverse operation *destabilisation* of Heegaard splittings. I don’t have a copy of Hempel’s book with me, but I am fairly sure it is discussed ther... | 5 | https://mathoverflow.net/users/1650 | 420606 | 171,149 |
https://mathoverflow.net/questions/420607 | 5 | Many connected [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) graphs $G=(V,E)$ have the property that some of their automorphisms other than the identity have fixed points. To point out two simple examples:
* If $G = K\_3$ then the automorphism swapping the points of an edge and leaving th... | https://mathoverflow.net/users/8628 | Connected vertex-transitive graph with the fixed-point property | Let me convert my comments into an answer.
There is no such **[EDIT:] finite** graph $G$. Indeed, something stronger can be said. Suppose that a group $\Gamma$ acts transitively by permutations on a finite set $X$ , with $\#X \geq 2$. Then there is some $\gamma \in \Gamma$ for which $\gamma\colon X \to X$ has no fixe... | 9 | https://mathoverflow.net/users/25028 | 420611 | 171,150 |
https://mathoverflow.net/questions/420264 | 2 | I got this question because of [this post](https://mathoverflow.net/questions/415806/limit-of-the-extremal-process-of-i-i-d-gaussians-see-from-the-tip), discussing the limiting distribution of extreme point process of i.i.d. Gaussians seen from the tip. For notations please refer that post.
At first, @Iosif Pinelis g... | https://mathoverflow.net/users/174600 | How to identify a point process as Poisson point process with (possibly) random intensity measure? | $\bullet$ The answer to the first question is affirmative, but the notion of "independence" needs to be modified if we allow for the point process to be directed by an external random variable.
A Poisson point process with a random intensity is known as a [Cox process.](https://en.wikipedia.org/wiki/Cox_process) The ... | 1 | https://mathoverflow.net/users/11260 | 420624 | 171,154 |
https://mathoverflow.net/questions/420510 | 6 | Propositional logic can be presented as in Mendelson’s book, with the sole inference rule of modus ponens, and with the following three axioms:
$$B \Rightarrow (C \Rightarrow B)$$
$$(B \Rightarrow (C \Rightarrow D)) \Rightarrow ((B \Rightarrow C) \Rightarrow (B \Rightarrow D))$$
$$((\neg C) \Rightarrow (\neg B)) \Right... | https://mathoverflow.net/users/419791 | Decidability of completeness in propositional logic | It is undecidable, because it is even undecidable to recognize whether a finite set of axioms together with the rule of modus ponens axiomatizes exactly classical propositional logic by the Post-Linial theorem. This was shown in 1948 by Linial and Post, see their [announcement (p. 50)](https://www.ams.org/journals/bull... | 6 | https://mathoverflow.net/users/58913 | 420630 | 171,157 |
https://mathoverflow.net/questions/420608 | 11 | Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we find an open cover $\{U\_i\}\_{i∈I}$ of $\R^m$
such that for any $i∈I$ the restriction of $f$ to $U\_i$
is a smooth map ... | https://mathoverflow.net/users/402 | Does every smooth map of rank at most d factor through a d-manifold? | There is a counterexample with $d=1$ and $m=n=2$. Here is one way to construct such an example: Let $g:\mathbb{R}\to\mathbb{R}$ be a smooth function such that $g'(t)>0$ for $t\not=0$ and $g^{(k)}(0) = 0$ for $k\ge 0$. Now, define a smooth mapping $f:\mathbb{R}^2\to\mathbb{R}^2$ by the rules
$$
f(x,y) = \begin{cases}\bi... | 10 | https://mathoverflow.net/users/13972 | 420646 | 171,161 |
https://mathoverflow.net/questions/420641 | 5 | Let $X$ be a compact Kähler manifold with Kähler form $\omega$, then from Kodaira & Spencer's paper on deformations [III](https://www.jstor.org/stable/1969879?casa_token=Lm4RJWHwjh4AAAAA%3Ahj6Mji5mA0uzR5pq4cNuPWCRZOUp24xKCZ50qXtMxR7_N3CaUW0ltkqwtMlvuoSb3oRx_D8wgolZsQPjd0UfEmFPzNk8jTXurtrtgmoObNCQLU-0MFokMg&seq=1), p.75... | https://mathoverflow.net/users/99826 | Does the Kähler form $\omega$ satisfy $d^*\omega=0$? | Recall that $d^\* = -\ast d\ast$ and $\ast\omega = \frac{1}{(n-1)!}\omega^{n-1}$, see Example 1.2.32 of Huybrechts' *Complex Geometry: An Introduction* for example. Therefore
$$d^\*\omega = -\ast d\ast\omega = -\ast d\left(\frac{1}{(n-1)!}\omega^{n-1}\right) = 0$$
as $\omega$ is closed.
| 13 | https://mathoverflow.net/users/21564 | 420658 | 171,165 |
https://mathoverflow.net/questions/420656 | 3 | A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F\_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have seen this definition is in:
*Balogh, Zoltán*, [**There is a $Q$-set space in ZFC**](http://dx.doi.org/10.2307/2048543), Pr... | https://mathoverflow.net/users/141146 | Co-analytic $Q$-sets | I do not know about full references but if you haven't seen it yet there is a nice discussion of this topic in Miller's [https://people.math.wisc.edu/~miller/res/dstfor.pdf](https://people.math.wisc.edu/%7Emiller/res/dstfor.pdf). (Incidentally this may be the most fun math book to read). See in particular sections 2-5.... | 7 | https://mathoverflow.net/users/114946 | 420660 | 171,166 |
https://mathoverflow.net/questions/420663 | 13 | Historically, the theory of ultracategories was invented [by Makkai](https://www.sciencedirect.com/science/article/pii/000187088790020X) to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such that the ultracategory of models of $T$ is equivalen... | https://mathoverflow.net/users/480841 | Ultracategories with one object | $\newcommand{\cat}{\mathrm}
\newcommand{\St}{\cat{Stone}^\cat{fr}}
\newcommand{\Cat}{\cat{Cat}}
\newcommand{\Cart}{\cat{Cart}}
\newcommand{\Fun}{\cat{Fun}}
\newcommand{\Mon}{\cat{Mon}}
\newcommand{\Set}{\cat{Set}}
\newcommand{\Po}{\cat{Poset}}$
EDIT : I misread the definition of ultracategory fibration, apparently only... | 11 | https://mathoverflow.net/users/102343 | 420671 | 171,168 |
https://mathoverflow.net/questions/420657 | 7 | Let $G$ be a finite group and let $M$ be a representation of $G$ over a field $k$. Suppose that, for every cyclic subgroup $C$ of $G$, we have $M|\_{C} \cong k[C]^{\oplus [G:C]}$. Can we conclude that $M \cong k[G]$?
In characteristic zero, this is immediately true by character theory, but I don't know what happens i... | https://mathoverflow.net/users/297 | Is a $k[G]$-module which is free on every cyclic subgroup free? | Suppose that $E = C\_p \times \dots \times C\_p$ be an elementary abelian $p$-group, say of rank $r$, and let $k$ be a field of characteristic $p$. Then the group algebra $kE$ is isomorphic to $k[x\_1, \dots, x\_r]/(x\_1^p, \dots, x\_r^p)$. If $v$ is in the vector space spanned by the $x\_i$, then $(1+v)^p=1$, so $1+v$... | 10 | https://mathoverflow.net/users/4194 | 420672 | 171,169 |
https://mathoverflow.net/questions/420678 | 8 | Let $\Lambda \subset\mathbb{R}^n$ be a lattice satisfying $\|x-y\|\_2^2 \in \mathbb{Z}$ for all $x,y\in\Lambda$. How small can $\text{vol}(\Lambda)=\det(\Lambda)$ be?
For example, in dimension $2$, the hexagonal lattice has smallest volume of any integral lattice, equal to $\frac{\sqrt{3}}{2}$. In higher dimensions, ... | https://mathoverflow.net/users/12176 | The smallest volume possible for a lattice with integer distances? | After scaling your lattice by $\sqrt{2}$, the Gram matrix has integral entries, so the absolute value of its determinant, being a nonzero integer, is at least $1$. This gives a lower bound of $2^{-n/2}$.
| 10 | https://mathoverflow.net/users/40821 | 420682 | 171,172 |
https://mathoverflow.net/questions/420679 | 4 | Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a bounded analytic function such that its derivative is also bounded. What kind of bound can we get on the higher order derivatives of $f$? Does it follow that they are bounded as well?
| https://mathoverflow.net/users/78173 | Bounded real analytic function with bounded derivative and its higher order derivatives | No, $f(x)=\int\_0^x \sin t^2\, dt$ is a counterexample.
| 10 | https://mathoverflow.net/users/48839 | 420686 | 171,175 |
https://mathoverflow.net/questions/420706 | 0 | Let $\left( S\_{n}^{1}\right) $ and $\left( S\_{n}^{2}\right) $ two sequences
of operators in $\mathcal{L}(E\_{1},F\_{1})$ and $\mathcal{L}(E\_{2},F\_{2})$
where $E\_{i},F\_{i},i=1,2$ are Hilbert spaces such that $\left\Vert
S\_{n}^{1}x-S^{1}x\right\Vert \_{F\_{1}}\underset{n\rightarrow \infty }{%
\rightarrow }0,$ $\fo... | https://mathoverflow.net/users/106804 | Kernels of sequences of operators | No, even if $\|S^2\_n x-S^2x\|\_{F\_2}\to0$ for some $S^2\in\mathcal L(E\_2,F\_2)$ and all $x\in E\_2$.
Indeed, let e.g. $E\_1=F\_1=E\_2=F\_2=H:=\ell^2$, $S^1\_n=I$, $S^1=I$, $S^2=0$, and $S^2\_n x=(e^{-|k-2n|}x\_k)\_{k=1}^\infty$ for $x=(x\_k)\_{k=1}^\infty\in\ell^2$. Then all the conditions hold. In particular,
$$\... | 2 | https://mathoverflow.net/users/36721 | 420707 | 171,181 |
https://mathoverflow.net/questions/308196 | 4 | Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a *triple threat*.
I'd like to be able to show that there are no triple threats. I've been able to prove the following weak... | https://mathoverflow.net/users/127690 | A specific Diophantine equation restricted to prime values of variables. | Pace Nielsen and Cody Hansen just [put this preprint on the Arxiv](https://arxiv.org/abs/2204.08971) which shows that no triple threats exist.
| 3 | https://mathoverflow.net/users/127690 | 420712 | 171,183 |
https://mathoverflow.net/questions/420711 | 2 | I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate.
I am reading a paper of Brezis and Oswald about existence and uniqueness of positive solutions to sublinear elliptic equations:
\begin{equation}
- \Delta u = f(x, u), \ u \geq 0,... | https://mathoverflow.net/users/113406 | Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$ | The fact that both $u\_j$ solve the PDE is not an issue. What matters is that both $u\_j$ are smooth over the closure $\overline\Omega$, positive in $\Omega$, vanish on $\partial\Omega$ and their normal derivatives don't vanish. This is where you may invoque L'Hopital's rule, if you like this tool.
What is more impor... | 3 | https://mathoverflow.net/users/8799 | 420720 | 171,184 |
https://mathoverflow.net/questions/420717 | 2 | It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum\_{n = 0}^\infty a\_n T\_n $$ which is absolutely convergent under the $\lVert \, \cdot \, \rVert\_\infty$ norm (see Approximation Theory and Approximation Practice by Lloyd Tref... | https://mathoverflow.net/users/47757 | Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$? | No, the space of Lipschitz functions on an infinite metric space is non-separable so it can't have a Schauder basis.
| 6 | https://mathoverflow.net/users/15129 | 420721 | 171,185 |
https://mathoverflow.net/questions/420585 | 1 | Let $X=(X\_n)\_{n\in\mathbb N}$ be a stochastic process in $\{0,1\}^{\mathbb N}$ with distribution $\mu$. I do not at first make any assumptions about $X$ being stationary or having any kind of correlations or associations. Let $Y=(Y\_n)\_{n\in\mathbb N}$ be an iid Bernoulli process with distribution product measure $\... | https://mathoverflow.net/users/68851 | Stochastic process on $\{0,1\}^{\mathbb N}$ domination of product measures, necessary and sufficient conditions | Ok, here is a counterexample with the index set $\{1,2\}$ (you can easily extend it to whole $\mathbb N$ if you wish).
Let $(Y\_1,Y\_2)$ be independent Bernoulli($1/2$) and set
$$
(X\_1,X\_2) = \begin{cases}
(Y\_1,Y\_2), & Y\_1 + Y\_2>0,\\
(0,1), & Y\_1 = Y\_2 = 0.
\end{cases}
$$
In this case $$
\mathrm P(X\_1 = 1\... | 2 | https://mathoverflow.net/users/8146 | 420728 | 171,186 |
https://mathoverflow.net/questions/420732 | 5 | I've copied over this question from [what I asked on Mathematics Stack Exchange](https://math.stackexchange.com/q/4422568/976076), in the hope that some experts here can help me find a way to check the residual finiteness of this group.
Consider the group $G=K\rtimes \mathbb{Z}$ defined as follows:
The subgroup $K$... | https://mathoverflow.net/users/479955 | Is this semi-direct product residually finite? | Yes, it's residually finite.
This group maps onto the residually finite wreath product $C\_2\wr\mathbf{Z}$ and the kernel is central, free over the $y\_k$, $k\ge 0$, as 2-elementary abelian group.
So one needs to show that for every non-empty subset $J$ of the set $J$ of positive integers, the element $y\_J=\prod\_... | 6 | https://mathoverflow.net/users/14094 | 420735 | 171,187 |
https://mathoverflow.net/questions/420695 | 6 | Let $M$ be a simply connected, (orientable), non-compact, 3-manifold without boundary. Must $M$ be homeomorphic with a topological subspace of $\mathbb{R}^3$?
| https://mathoverflow.net/users/69681 | Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$? | When the manifold is the universal cover of a compact $3$-manifold $M$ (to begin with, lets say without boundary) then you construct the embedding by hands, using geometrization. In your question let's replace $\Bbb R^3$ with $S^3$ and we will see the embedding into $\Bbb R^3$ comes for free in the situation you are in... | 4 | https://mathoverflow.net/users/1465 | 420738 | 171,188 |
https://mathoverflow.net/questions/420684 | 11 | Recall the notation $(z;q)\_n=(1-z)(1-zq)(1-zq^2)\cdots(1-zq^{n-1})$.
My earlier [MO question](https://mathoverflow.net/questions/420603/congruence-modulo-2-for-q-series) did not find enough interest or yield an answer. Perhaps the modulo $2$ part might have thrown people off. So, I now can write another question whi... | https://mathoverflow.net/users/66131 | Equality of two $q$-series. Proof? | I take it from looking at the previous problem that you are familiar with the [Dyson rank](https://en.wikipedia.org/wiki/Rank_of_a_partition) on partitions with distinct parts. Let's denote by $Q(r,n)$ the number of partitions of $n$ into distinct parts that have rank $r$. The following expression for the generating fu... | 18 | https://mathoverflow.net/users/2384 | 420742 | 171,189 |
https://mathoverflow.net/questions/419776 | 3 | Note: Here all processes take values in $[0, 1]$.
Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Let $X$ be the solution to the SDE
$$dX\_t = \sigma X\_t \, dW\_t$$
with $X\_0 = 1$ a.s.
For every $\varepsilon > 0$, let $A\_\varepsilon$ be the event $\{\text{max}\_{0 \leq t... | https://mathoverflow.net/users/173490 | Another large noise limit | $\newcommand{\si}{\sigma}\newcommand{\ep}{\varepsilon}\newcommand\num{\operatorname{num}}\newcommand\den{\operatorname{den}}\newcommand{\R}{\mathbb R}
\newcommand{\vpi}{\varphi}$The conjecture is not true in general.
The limit depends on $\si$. In particular, let us show that the limit in question is, not $0$, but $\... | 1 | https://mathoverflow.net/users/36721 | 420752 | 171,192 |
https://mathoverflow.net/questions/420734 | 3 | I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is somewhat vague since I don't know precisely what I'm looking for and I'm hoping someone can point me in the right direction. ... | https://mathoverflow.net/users/480683 | Conformal groupoid |
>
> That is, what algebraic structure captures this kind of groupoid-with-restriction and how do we describe its action on a given sheaf more precisely?
>
>
>
This structure is well known and has many equivalent incarnations:
[inductive groupoid](http://www-users.york.ac.uk/%7Evarg1/esnslidesf.pdf), [inverse sem... | 3 | https://mathoverflow.net/users/402 | 420755 | 171,194 |
https://mathoverflow.net/questions/420722 | 5 | **Question 1:** Suppose that two hyperbolic 3-manifolds $M\_1$ and $M\_2$ with finitely generated fundamental group satisfy the property that for every closed geodesic in $M\_1$, there is a closed geodesic in $M\_2$ with the same length. Are $M\_1$ and $M\_2$ isometric?
**Question 2:** The same as question one, but s... | https://mathoverflow.net/users/479962 | Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies? | I think your questions are answered by [this](https://arxiv.org/abs/math/0606343) paper of Leininger, McReynolds, Neumann and Reid. In their terminology, your first question is asking about manifolds with equal *length sets*, and your second question is asking about manifolds with equal *complex length sets*. Their The... | 5 | https://mathoverflow.net/users/1463 | 420758 | 171,195 |
https://mathoverflow.net/questions/420751 | 5 | Let $G$ be an elementary abelian group, so that $G = (\mathbb{Z}/p)^k$ for some $k$. We can then compute the Morava $K$-theory of $BG = (BZ/p)^k$ pretty easily: $K(n)^\*(BG) = K(n)^\*[[x]]/[p](x) (x)\_{K(n)^\*} ... (x)\_{K(n)^\*} K(n)^\*[[x]]/[p](x)$, the tensor being taken $k$ times. We know that $[p](x) = x^{p^n}$. W... | https://mathoverflow.net/users/480262 | Computation of cohomology of Morava $K$-Theory | Bjorn Schuster and I looked at this problem in our article `On the $GL(V)$-module structure of $K(n)^\*(BV)$' Math Proc Cambridge Phil Soc vol 122 (1997) pp73--89. In particular, we show that in at least some cases it is not a permutation module. If you don't have access to the journal, there is a preprint that started... | 6 | https://mathoverflow.net/users/124004 | 420761 | 171,197 |
https://mathoverflow.net/questions/420759 | 2 | I'm almost not at all knowledgable in either Freidlin-Wentzel theory or Kramers' escape problem as it is known in the physics community, so please excuse some of my naivety.
One can use Freidlin-Wentzel theory to study the problem of a particle with position given by the SDE
$$dX\_t = V(X\_t)\,dt + \sqrt{\epsilon}\... | https://mathoverflow.net/users/78650 | Kramers' escape problem: statistical physics vs. Large deviations | The Kramers theory is limited to reversible processes in equilibrium, while the Freidlin-Wentzel theory generalises this to irreversible processes out of equilibrium.
The distinction appears in the stochastic differential equation
$$dx\_t=f(x\_t)dt+ \sqrt{2\epsilon}dW\_t,$$
in the Kramers theory it is assumed that $f... | 2 | https://mathoverflow.net/users/11260 | 420762 | 171,198 |
https://mathoverflow.net/questions/420354 | 6 | In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}\_M$ can be identified with the colimit of a diagram of $\infty$-operads parametrized by $M$, each of which is equivalent to $\mathbb{E}\_k$". I do not see how this follows from the preceding discu... | https://mathoverflow.net/users/156537 | $\mathbb{E}_M$ as colimit of little cubes operads | I agree this is sort of scattered around as remarks (e.g. HA.2.3.3.4, say). Let's try to spell it out more cleanly.
1. I claim that, if $X$ is a groupoid, then the category of $X$-families of operads is the same as the category of functors $\mathsf{Fun}(X, \mathsf{Op})$. Indeed, $\mathsf{Op}$ is a (non-full) subcateg... | 2 | https://mathoverflow.net/users/6936 | 420770 | 171,202 |
https://mathoverflow.net/questions/420737 | 4 | Given a group $G$, the outer automorphism group $Out(G)$ acts on the center by $Z(G)$ by lifting an outer automorphism to an actual automorphism and evaluating this on elements of $Z(G)$. What is classified by the degree three group cohomology $H^3(Out(G),Z(G))$ ?
Given an algebra $A$, say finite-dimensional over a f... | https://mathoverflow.net/users/3473 | 3-cocycles on outer automorphism groups | Among other things, the third cohomology contains an invariant for the existence of group-graded algebras whose degree-1-piece is $A$ / group extension with $G$ as normal subgroup. This is a theorem of Schreier. If I'm not mistaken, the cohomology class $\xi$ that comes from crossed modules is precisely Schreier's inva... | 2 | https://mathoverflow.net/users/3041 | 420772 | 171,203 |
https://mathoverflow.net/questions/420765 | 11 | I am embarrassed to be stuck on this seemingly simple question.
Suppose that $X,Y$ are mean-zero real-valued random variables and $\tilde X,\tilde Y$ are their "independent copies": $\tilde X,\tilde Y$ are mutually independent and independent of $(X,Y)$, and $\tilde X$ (resp., $\tilde Y$) is distributed identically t... | https://mathoverflow.net/users/12518 | Decoupling inequality/counterexample | [**EDITED** *to ensure that the random variables have expectation 0*]
I think the answer is no.
Let $Z$ be a random variable taking the values $\pm 1$ with probability $p$ each and 0 with probability $1-2p$; and let $N$ be a standard normal independent of $Z$.
Set $X=ZM+(1-|Z|)N$ and $Y=-ZM+(1-|Z|)N$, where $M=\sqr... | 13 | https://mathoverflow.net/users/11054 | 420773 | 171,204 |
https://mathoverflow.net/questions/420776 | 10 | Let $A\in \mathbb{S}^{N\times N}$ be a symmetric, real and stable matrix, i.e., $\rho(A)<1$, where $\rho(A)$ stands for the spectral radius of $A$. Then, $$\sum\limits\_{i=0}^{\infty} A^{2i}=\left( I-A^2 \right)^{-1}.\tag{$\star$}\label{star}$$
Now, let $A$ be asymmetric. It is not clear whether the power series $$F(... | https://mathoverflow.net/users/138242 | (Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$ | Let me answer the first question:
**Q:** Does $F=\sum\_{i=0}^{\infty} A^i (A^i)^{\top}$ for non-symmetric $A$ have a closed-form solution analogous to the solution $\sum\_{i=0}^{\infty} A^{2i}=\left( I-A^2 \right)^{-1}$ for a symmetric $A$?
**A:** Yes, in terms of the [vectorization operation](https://en.wikipedia.... | 7 | https://mathoverflow.net/users/11260 | 420782 | 171,207 |
https://mathoverflow.net/questions/420744 | 14 | Let $\mathbb{S}=\{x\in\mathbb{R}^n|x\_1^2+\ldots +x\_n^2=1\}$ be the unit sphere in $\mathbb{R}^n$, $\mathbb{C}[x]=\mathbb{C}[x\_1,\ldots ,x\_n]$ the complex-valued polynomial functions on $\mathbb{R}^n$, and $\Delta=\partial\_1^2+\ldots +\partial\_n^2$ the Laplacian. Let $H\_n=\ker(\Delta)$ be the space of harmonic po... | https://mathoverflow.net/users/4721 | Harmonic polynomials on the sphere | I view this as a concatenation of two facts:
---
**Fact 1:** Let $k$ be a field, let $I$ be an ideal of $k[x\_1, \ldots, x\_n]$ and let $J$ be associated graded ideal of $I$, meaning that a degree $d$ homogenous polynomial $g(x)$ is in $J$ if and only if there is an element of $I$ of the form $g(x) + (\text{terms... | 6 | https://mathoverflow.net/users/297 | 420784 | 171,208 |
https://mathoverflow.net/questions/420781 | 2 | Let $(X,\mu)$ be a measure space and let $1<p<\infty$.
>
> **Question.** Is the space $L^p(X,\ell^p)$,
> $$
> \lVert f\rVert\_p=\Bigl(\int\_X\sum\_{i=1}^\infty \lvert f\_i\rvert^p\, dx\Bigr)^{1/p},
> \qquad
> f=(f\_i)\_{i=1}^\infty,
> $$
> uniformly convex?
>
>
>
If the answer if "yes" I would appreciate a ref... | https://mathoverflow.net/users/121665 | Is $L^p(X,\ell^p)$, $1<p<\infty$, uniformly convex? | By Proposition 1.2.24 in T. Hytonen, J. Van Neerven, M. Veraar and L. Weis, Analyis in Banach spaces Vol I, Springer, the spaces $L^p(X; \ell^p)$ and $L^p(X\times \mathbb N)$, $X \times \mathbb N$ with the product measure, are isometric (at least when $\mu$ is $\sigma$-finite). Then the uniform convexity of $L^p(X; \el... | 8 | https://mathoverflow.net/users/150653 | 420791 | 171,211 |
https://mathoverflow.net/questions/420785 | 5 | Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $p \mid 2n$?
| https://mathoverflow.net/users/477248 | Dihedral extension unramified at primes dividing order of group? | $\DeclareMathOperator\Gal{Gal}$The answer is "yes", and this is an easy exercise in class field theory: if, for example, $q$ is a prime number that is $1\pmod n$, and $F$ is a quadratic number field in which $q$ splits, then there is a quotient of the ray class group of $F$ with modulus $q$ that has order $n$ and on wh... | 4 | https://mathoverflow.net/users/35416 | 420793 | 171,212 |
https://mathoverflow.net/questions/231770 | 6 | The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$
In [Mazur-Swinnerton-Dyer](https://eudml.org/doc/142281) and [Zagier](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084592/fulltext.pdf)'s construction, a theta function associ... | https://mathoverflow.net/users/18286 | Integral quaternary forms and theta functions | The answer is true, using the following construction.
Let $B$ be the quaternion algebra of discriminant $p$ and let $O$ be a maximal order with an element $x$ satisfying $x^2 = -p$. The reduced norm is a quadratic form on $O$, with positive definite Gram matrix $A$ of determinant $p^2$. The matrix $A^{-1}$ then represe... | 4 | https://mathoverflow.net/users/74819 | 420800 | 171,216 |
https://mathoverflow.net/questions/420799 | 3 | Let $Pic\_n^0$ denote the even part of the $K(n)$-local Picard group, and let $Pic\_n^\*$ denote $Hom(Pic\_n^0, W(\mathbb{F}\_{p^n})^x)$. Denote by $L$ the profinite group ring $\mathbb{Z}\_p[[Pic\_n^\*]] $. Let $\lambda$ be an element of $Pic\_n$; this gives a map $Pic\_n^\* \to W(\mathbb{F}\_{p^n})^x$, given by evalu... | https://mathoverflow.net/users/480262 | Studying a connection between Iwasawa theory and the $K(n)$-local Picard group by some geometry on the Lubin-Tate stack | A long time ago there were various attempts to make this sort of approach work, but they were not very successful. If you want to try again then you should make sure that you are familiar with the phenomena observed by computation in the height two case (which are quite complex). The computations are due to Shimomura, ... | 6 | https://mathoverflow.net/users/10366 | 420804 | 171,219 |
https://mathoverflow.net/questions/420816 | 8 | Usually, having more comprehension axiom means the more you can prove. We wonder if the converse can be the case.
Is there a natural problem $\mathsf{P}$ so that $\mathsf{P}+\neg(\Gamma-\mathsf{ComprehensionAxiom})$ implies
$\mathsf{Q}$, but $\mathsf{P}$ does not implies $\mathsf{Q}$.
Also I'd like to exclude some ... | https://mathoverflow.net/users/74918 | Comprehension axiom that helps in the opposite direction | David Belanger's work is relevant.
The principle $\mathsf{WKL\_0}$ (= "Every infinite subtree of $2^{<\omega}$ has an infinite path") is not literally a comprehension principle, so its negation can be used as a choice of $\mathsf{Q}$. Belanger [showed](https://www.jstor.org/stable/43303769) that over $\mathsf{RCA\_0}... | 10 | https://mathoverflow.net/users/8133 | 420818 | 171,223 |
https://mathoverflow.net/questions/420756 | 6 | I was wondering if it is well understood under what circumstances say three univariate polynomials $f(x),g(x),h(x)$ have a common root.
In this situation, I can see that the resultant of each pair must vanish but that only ensures that each pair has a common root. Is there a way to generate a finite set of polynomials ... | https://mathoverflow.net/users/164946 | When do multiple polynomials have a common root? | Assume that $f\_n$ is monic. Then for indeterminates $u\_1,\ldots,u\_{n-1}$, all coefficients of the polynomial $Res\_x(f\_n,u\_1f\_1+\ldots+u\_{n-1}f\_{n-1})\in k[u\_1,\ldots,u\_{n-1}]$ vanish if and only if $f\_1,\ldots,f\_n$ have a common root [expanding out this resultant then gives the list of polynomials].
This... | 5 | https://mathoverflow.net/users/3404 | 420821 | 171,224 |
https://mathoverflow.net/questions/420825 | 7 | I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos which admits a cover by the étale spectra of $E\_{\infty}$-rings. The definition of a nonconnective spectral *scheme*, on... | https://mathoverflow.net/users/158123 | If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get? | Yes, they are more general. This is in fact already the case with ordinary rings. Let's call a classically-ringed $\infty$-topos which is locally the Zariski $\infty$-topos of an affine scheme an *$\infty$-scheme*. A classical scheme is then the same as a $0$-localic $\infty$-scheme. To construct an $\infty$-scheme whi... | 13 | https://mathoverflow.net/users/20233 | 420829 | 171,225 |
https://mathoverflow.net/questions/419672 | 3 | I am having trouble finding any results concerning real interpolation of vector-valued Sobolev spaces. Namely, I would like to know if a continuous embedding of the type,
$$
L^p(0,T;X\_1)\cap W^{1,p}(0,T;X\_0) \hookrightarrow \mathcal{C}([0,T]; (X\_0, X\_1)\_{\theta, p})
$$
can hold for some $\theta \in (0,1)$, whe... | https://mathoverflow.net/users/146986 | Real interpolation for vector-valued Sobolev spaces | The desired embedding is indeed correct for $\theta = 1-1/p$. This is a classical result in interpolation theory and the theory of evolution equations. See for example the book of Amann [2], Theorem III.4.10.2.$^1$
In fact, the real interpolation space $$X:= \bigl(X\_0,X\_1\bigr)\_{1-1/p,p}$$ can be *defined* (equiva... | 3 | https://mathoverflow.net/users/85906 | 420840 | 171,228 |
https://mathoverflow.net/questions/420837 | 2 | Suppose I have two random variables $X\_1$ and $X\_2$. $X\_1,X\_2$ are both sums of random variables, and I can find Chernoff bounds for both variables independently. That is to say, I have
$$Pr[\mid X\_i - \mathbb{E}X\_i \mid \geq \delta \mathbb{E}X\_i] \leq 2\exp(-\frac{\delta^2 \mathbb{E}X\_i}{3}) $$ for both $i=1,2... | https://mathoverflow.net/users/480998 | Chernoff style concentration bound for ratio of variables | $\newcommand{\de}{\delta}$You have the bounds
\begin{equation}
P(|X\_i-m\_i|\ge \de\_i m\_i]\le2e^{-\de\_i^2 m\_i/3}
\end{equation}
for $i=1,2$ and $\de\_i>0$, where
\begin{equation}
m\_i:=EX\_i.
\end{equation}
Clearly, these bounds are useful only if $m\_i>0$ for $i=1,2$, which will be henceforth assumed. Assume al... | 1 | https://mathoverflow.net/users/36721 | 420851 | 171,230 |
https://mathoverflow.net/questions/420857 | 1 | Let $P\in\mathbb{F}\_p[T]$ (not supposed irreducible). All roots $\xi$ of $P$ have a certain order $k$ such that $\xi^k=1$.
**Question:** is it possible to know the order of the roots of the given polynomial $P$, or at least a upper bound of the order?
It is clear that if $k$ is an order of (a root of) $P$ then by ... | https://mathoverflow.net/users/34066 | Order of roots for a polynomial $P\in\mathbb{F}_p[T]$ | Let $d = \deg P$. The list of possible orders is simply the list of divisors of $p^d-1$.
Proof that every such order appears: Let $n$ divide $p^d-1$. Let $\alpha$ be an element of $\mathbb F\_{p^d}$ of order $n$, which exists since the multiplicative group of $\mathbb F\_{p^d}$ is cyclic. Then $\alpha$ generates a su... | 3 | https://mathoverflow.net/users/18060 | 420859 | 171,233 |
https://mathoverflow.net/questions/420839 | 6 | I was wandering if there was a book, thesis or some notes where Shelah's argument for
1. $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{ZFC} + \exists \kappa$ inaccessible
2. $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals have the Baire property" is equiconsistent ... | https://mathoverflow.net/users/141146 | Shelah's "Can you take Solovay's inaccessible away?" | Chapter 9.5 of the book by Bartoszynski-Judah presents Raisonnier's proof (answering your question 1):
Assume that $\aleph\_1$ is not inaccessible in $L$, hence a successor in $L$. So there is a real $x$ which knows that the $L$-predecessor is countable, hence $\aleph\_1^{L[x]}=\aleph\_1$, so $X:=\mathbb R \cap L[x]$... | 9 | https://mathoverflow.net/users/14915 | 420867 | 171,235 |
https://mathoverflow.net/questions/420870 | 4 | I am reading a very nice paper of Newton and Thorne, [*Symmetric power functoriality for holomorphic modular forms*](https://doi.org/10.1007/s10240-021-00127-3), and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic Galois representations that I do not fully understand. In case it is he... | https://mathoverflow.net/users/165625 | Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero | Let $T\_H$ be a maximal torus in $H$.
Suppose that $H^\circ$ is not a torus. Then $T\_H$ has non-trivial roots on $\operatorname{Lie}(H)$, and the subgroup of $H$ generated by opposite root groups is $\operatorname{SL}\_2$.
Thus, if $H$ does not contain $\operatorname{SL}\_2$, then its identity component equals $T\... | 6 | https://mathoverflow.net/users/2383 | 420875 | 171,237 |
https://mathoverflow.net/questions/420874 | 13 | Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a **spectral gap** for such a $\Delta$, that is, finding an interval $(0,c)$ which does not contain any eigenvalues of $\Delta$. Why are people interested in producing such spectral gaps? What interesting informa... | https://mathoverflow.net/users/438034 | Why are we interested in spectral gaps for Laplacian operators | A spectral gap gives information on geometry of the manifold via Cheeger's inequality, <https://en.wikipedia.org/wiki/Cheeger_constant> See also Buser's inequality discussed there. More directly, a spectral gap for the Laplacian yields exponential decay for the heat kernel and determines the asymptotic rate of decay. T... | 7 | https://mathoverflow.net/users/7691 | 420879 | 171,239 |
https://mathoverflow.net/questions/420894 | 6 | [Bhargava 2021](https://arxiv.org/abs/2111.06507) proves van der Waerden's conjecture about Galois groups of random integer polynomials: over all $x^n + a\_{n-1} x^{n-1} + \cdots + a\_0 = 0$ with $a\_k \in \{-H, \ldots, H\}$, the number of polynomials $E\_n(H)$ with Galois group not equal to the full symmetric group $S... | https://mathoverflow.net/users/22930 | Is there a conjectured dependence on $n$ in van der Waerden's conjecture? | There are two different questions you could ask here.
One is the dependence on $n$ in the conjectured asymptotic best constant, i.e. the $c\_n$ such that we have $E\_n (H) < c\_n H^{n-1} + o(H^{n-1})$ with the little $o$ depending on $n$. This one has a precise answer, because we do not need to worry about all the gr... | 10 | https://mathoverflow.net/users/18060 | 420895 | 171,242 |
https://mathoverflow.net/questions/420906 | 0 | Given two primes $p,q\in[T,2T]$, how many integers $m$ of size $O(T^{3/2+\epsilon})$ are there such that the residues $m\bmod p$ and $m\bmod q$ are both $O(polylog(T))$? **Looking for an answer**
Is it possible to construct any such in $polylog(T)$ time without integer programming? *Answered below*
| https://mathoverflow.net/users/10035 | Constructing an integer with small residues for two distinct primes in polynomial time | If such $m$ exists, then $m=up+a=vq+b$ for some $u,v\in O(T^{1/2+\epsilon})$ and $a,b\in O(\mathrm{polylog}(T))$. Then $up-vq=b-a$ and thus $\frac{p}q - \frac{v}u=\frac{b-a}{uq}$, implying that $\frac{v}u$ represents a rational approximation to $\frac{p}q$, which so good that it must be a convergent. So, it enough to c... | 3 | https://mathoverflow.net/users/7076 | 420907 | 171,245 |
https://mathoverflow.net/questions/420900 | 0 | Find the first derivative of a SUM of all elements of an INVERSE of a square matrix (whose elements are functions of $z$) at $z=1$ knowing that all of the matrix' elements evaluate to $1$ at $z=1$.
This is needed in Statistics to find the expected number of trials to generate one of several preselected patterns (such... | https://mathoverflow.net/users/141969 | Derivative involving a singular matrix | I first rephrase the question using formulas and then will provide a proof.
Start from the $n\times n$ matrix $A(z)$, such that $[A(1)]\_{ij}=1$ for all $i,j\in\{1,2,\ldots n\}$. The matrix $B(z)$ is the elementwise derivative, $B\_{ij}=dA\_{ij}/dz$.
Now the question which the OP asks is whether
$$\lim\_{z\rightar... | 2 | https://mathoverflow.net/users/11260 | 420909 | 171,246 |
https://mathoverflow.net/questions/420896 | 31 | Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we know (Hasse principle, quadratic reciprocity, Vieta jumping, search for large solutions, etc.) to try to decide the solvabi... | https://mathoverflow.net/users/89064 | Is equation $xy(x+y)=7z^2+1$ solvable in integers? | There is **no solution**.
It is clear that at least one of $x$ and $y$ is positive and that neither is divisible by 7. We can assume that $a := x > 0$. The equation implies that there are integers $X$, $Y$ such that
$$ X^2 - 7 a Y^2 = a (4 + a^3) $$
(with $X = a (a + 2y)$ and $Y = 2z$).
First consider the case that... | 36 | https://mathoverflow.net/users/21146 | 420918 | 171,248 |
https://mathoverflow.net/questions/420899 | 6 | Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$
always separable? An example of non separable polynomial comes
from Puiseux series: the polynomial $y^2-x$ has a double root
$y=\sqrt{x}$. So I wonder what happen if we restrict to Laurent serie... | https://mathoverflow.net/users/122378 | Is the minimal polynomial of an algebraic formal Laurent series always separable? | Yes. Let $p=char(K)$ and $\alpha\in \overline{K(x)}\cap K((x))$ assumed to be inseparable over $K(x)$.
* Let $L= K^{1/p^\infty}$ which is perfect. If $\alpha$ is inseparable over $L(x)$ then $\alpha$'s monic $L(x)$-minimal polynomial is $$f(y)=g(y^p)=h(y)^p$$ with $h(y)\in L(x)^{1/p}[y]=L(x^{1/p})[y]$ so $$[L(x^{1/p}... | 4 | https://mathoverflow.net/users/84768 | 420919 | 171,249 |
https://mathoverflow.net/questions/420887 | 5 | Given a multi-variable polynomial $F$, denote the number of monomials by $N(F)$. Take for instance, \begin{align\*}N(x(x+y)+(x+y)^2-(x-y)^2)=N(x^2+5xy)&=2 \qquad \text{and} \\
N((x+z)(x+y)^2)=N(x^3 + 2x^2y + x^2z + xy^2 + 2xyz + y^2z)&=6.\end{align\*}
Denote the symmetric group on $n$ letters $\{1,2,\dots,n\}$ by $\mat... | https://mathoverflow.net/users/66131 | Counting monomials and the Catalan numbers | Any monomial $P:=\prod x\_i^{c\_i}$ of degree $\sum c\_i=n$ maps to a non-zero constant after symmetrization
$$
P\to \Phi(P):=G\_{\mathfrak{S}\_{n+1}}\frac{P}{(x\_1-x\_2)(x\_2-x\_3)\ldots (x\_n-x\_{n+1})}.
$$
Indeed, $\Phi(P)$ is a constant by a degree consideration, to prove that this constant is non-zero you may use,... | 9 | https://mathoverflow.net/users/4312 | 420923 | 171,252 |
https://mathoverflow.net/questions/420700 | 3 | Crosspost from Math.SE as I did not receive an answer there:
In Lang's book *Elliptic Functions*, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $k$ using the $j$-invariant of an elliptic curve $A/\mathbb{C}$ with $\mathrm{End}(A)\cong\frak{o}\_\text{$k$}$ and t... | https://mathoverflow.net/users/480786 | Lang's proof concerning ray class fields of imaginary quadratic number fields | I have found out what's going on here and it is so trivial that I wonder why this did not occur to me earlier: The conclusion of the proof is that $K$ is the ray class field of $k$ modulo $N$ - but this means that $K$ is abelian over $k$, so the intermediate field $k(j\_A, h(A\_N))$ must be Galois over $k$ because all ... | 6 | https://mathoverflow.net/users/480786 | 420937 | 171,258 |
https://mathoverflow.net/questions/420913 | 2 | Let $p$ be large prime in $[T,2T]$ where $T$ is a parameter.
1. Can we have an integer pair $a,b$ satisfying $ab\equiv c\bmod p$ such that $|c|$ is of size $O(\operatorname{polylog}(T))$ and $a,b$ are of size $O(T^{1/2 +\varepsilon})$?
2. If $1/2$ is impossible what is the best rational in exponent we can get?
3. Giv... | https://mathoverflow.net/users/10035 | Small near-reciprocals | There is a nice article of Heath-Brown in the Mathematical Intelligencer, called Arithmetic applications of Kloosterman sums, where this problem is discussed under the heading, "An elementary problem." He derives exponent $3/4$ and remarks that it is open to improve on it. I am not aware of any subsequent improvements.... | 4 | https://mathoverflow.net/users/2627 | 420941 | 171,260 |
https://mathoverflow.net/questions/420081 | 5 | Let $G$ be a group and $F$ a field. I am particularly interested in the case where $G$ is a uniform lattice in a Lie group and $F=\mathbb{F}\_2$, or in finite groups $G$ where
$\operatorname{char} F$ divides
$|G|$, but the discussion applies to general $G$ and $F$.
Let $\rho:G\to \mathrm{GL}(V)$ be a representation o... | https://mathoverflow.net/users/86006 | Examples of a group $G$ and an $F$-representation $V$ where $\cup:H^1(G,F)\otimes H^1(G,V)\to H^2(G,V)$ is injective | $\newcommand{\bZ}{\mathbb{Z}}$Let $G$ be a finite group of order divisible by $p:=\mathrm{char}\, F$ such that $\dim\_F \mathrm{Hom}(G,F)\geq 2$ (e.g. $G=\bZ/p\times\bZ/p$). Take $V$ to be the kernel of the augmentation map $e(\sum a\_g\cdot g)=\sum a\_g$ from $F[G]$ to $F$. Since $F[G]$ is an injective $G$-module, the... | 3 | https://mathoverflow.net/users/39304 | 420945 | 171,262 |
https://mathoverflow.net/questions/420927 | 3 | Let $X\_0, X\_1, X\_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of empirical CDFs $F\_n(x) = \frac{1}{n} \sum\_{i = 1}^n \mathbf{1}\_{\{X\_i \leq x\}}(x)$ ($\mathbf{1}\_A$ is an indicator f... | https://mathoverflow.net/users/113992 | For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable? | $\newcommand\Om\Omega\newcommand\om\omega\newcommand\R{\mathbb R}$This is indeed straightforward. Spelling out
$$\sup\_{x\in\R}|F\_n(x)-F(x)|\to0 \text{ a.s.}, $$
we see that for some subset $N$ of $\Om$ of outer probability $0$ and all $\om\in\Om\_0:=\Om\setminus N$ we have
$$\sup\_{x\in\R}\Big|\frac1n\sum\_{j=1}^n 1(... | 2 | https://mathoverflow.net/users/36721 | 420951 | 171,264 |
https://mathoverflow.net/questions/420947 | 1 | I am wondering can we say something about the cover time $T$ for a box, eg. $[-n,n]^2\cap \mathbb{Z}^d$, by the simple symmetric random walk on $\mathbb{Z}^2$ starting from zero?
For example, the expected time, the variance, or some scale $a\_n$ such that $T\leq a\_n$ with high probability?
(I tried search on web b... | https://mathoverflow.net/users/174600 | Cover time of a box by SRW in $\mathbb{Z}^2$? | "Cover times for Brownian motion
and random walks in two dimensions": Annals Math 160 (2004), 433-464,
By Dembo, Peres, Rosen, Zeitouni.
See Theorem 1.4
| 5 | https://mathoverflow.net/users/35520 | 420954 | 171,265 |
https://mathoverflow.net/questions/420939 | 0 | how to show that non-decision problem is NP-hard?
So far I could find out that problems which are NP-hard do not have to be decision problems.
But how to show a non-decision problem is NP-hard?
Is it possible to show problem L is NP-hard by showing that another NP-hard problem B is reducible to L?
Does B ha... | https://mathoverflow.net/users/480697 | NP-hardness of non-decision problems | A definition of NP-hard is: if the problem can be solved in polynomial time, then every problem in NP can be solved in polynomial time. This definition works for function problems.
Example: Consider the optimization problem "find the length of the shortest path in an instance of the traveling salesman problem." This ... | 3 | https://mathoverflow.net/users/29697 | 420957 | 171,266 |
https://mathoverflow.net/questions/420949 | 6 | Context:
--------
In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists [1]:
>
> My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with ... | https://mathoverflow.net/users/56328 | Freeman Dyson's approach to string theory | Dyson's [A walk through Ramanujan's garden](https://www.semanticscholar.org/paper/A-walk-through-Ramanujan%E2%80%99s-garden-Dyson/eb6d4ec50b6666a07521157c22c14c81380ce424) gives the background of this comment: He explains that the "seeds from Ramanujan's garden have been
blowing on the wind and have been sprouting all ... | 22 | https://mathoverflow.net/users/11260 | 420958 | 171,267 |
https://mathoverflow.net/questions/420959 | 7 | This question was asked at the french ENS oral examination. I do not really know how to approach it. I think the answers no.
What I've gathered so far :
Lets call $T$ the subset of $\mathbb{R}^2$ in the title (for obvious reasons). If the union exists, by Baire's theorem it must be uncountable. Let $E$ be the set o... | https://mathoverflow.net/users/466576 | Can $\mathbb{R}^2$ be covered by disjoint sets homeomorphic to the union of the segments $[(0,0), (0,1)], [(0,0), (1,1)], [(0,0), (1,0)]$? | Such sets are called *triods*. R. L. Moore (Concerning triods in the plane and the junction points of plane continua, Proceedings of the National Academy of Sciences USA, vol. 14, 1928, pp. 85-88) proved that every set of pairwise disjoint triods in the plane is countable.
<https://www.pnas.org/doi/abs/10.1073/pnas.1... | 7 | https://mathoverflow.net/users/43266 | 420965 | 171,269 |
https://mathoverflow.net/questions/420914 | 5 | Taken from Math Stack Exchange.
>
> Let $\mathcal{F}$ be a set of $\mathcal{L}\_\in$-formulae, $\kappa$ be a cardinal and $A \subset \textrm{Ord}$. Then, $\kappa$ is called $\mathcal{F}$-Mahlo if $A \cap \kappa$ intersects every club definable in $H\_\kappa$ by a formula $\varphi \in \mathcal{F}$. $\kappa$ is $\mat... | https://mathoverflow.net/users/473200 | Equivalences of $\mathcal{F}$-Mahloness | The paper ["Small Definably-large Cardinals" by Roger Bosch](https://doi.org/10.1007/3-7643-7692-9_3) proves that an inaccessible cardinal is $\Sigma\_{n+1}$-Mahlo if and only if it is $\Pi\_n$-Mahlo except for $n=1$ (I'm referring to the boldface hierarchy, not the lightface hierarchy which is less relevant to your qu... | 6 | https://mathoverflow.net/users/352898 | 420971 | 171,270 |
https://mathoverflow.net/questions/420822 | 4 | A lower bound of the number of conjugacy classes in the automorphism group of a simple Lie algebra $\mathfrak{s}$, of finite dimension over an arbitrary field $\mathbb{F}$, can be the size of the image of the automorphism group by the trace.
I think this map $\operatorname{tr}:\operatorname{Aut}(\mathfrak{s})\longrig... | https://mathoverflow.net/users/235856 | Conjugacy classes in the automorphism group of a simple Lie algebra | 1. If $\mathfrak{s}$ is $K$-anisotropic, where $K$ is a real or $p$-adic field (this is equivalent to $\mathfrak{s}$ not containing $\mathfrak{sl}\_2$, and also to the corresponding group be compact), then the trace is bounded, hence non-surjective.
2. Suppose that $\mathrm{Aut}(\mathfrak{s})^0$ is Zariski-dense in the... | 4 | https://mathoverflow.net/users/14094 | 420976 | 171,272 |
https://mathoverflow.net/questions/420993 | 2 | Let us consider the space $M\_n(\mathbb{C})$. By a unitary matrix $U=(u\_{ij})$ we mean that $U^{-1}=(\overline{u\_{ji}})$.
>
> Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ satisfying the following conditions: (1) $D$ is a diagonal matrix in $M\_n(\mathbb{C})$, (2) $UAU^{-1}=D$, (3... | https://mathoverflow.net/users/84390 | On the eigen vectors of a diagonalizable matrix | I [mentioned](https://mathoverflow.net/questions/420993/on-the-eigen-vectors-of-a-diagonalizable-matrix#comment1081449_420993) the silly example $U = A = D$ offhand, but, in fact, it's essentially the only example. In general, note that the columns of $U^{-1}$ are the eigenvectors of $A$. So we're asking for, at least,... | 2 | https://mathoverflow.net/users/2383 | 420994 | 171,277 |
https://mathoverflow.net/questions/421012 | -2 | Which extension of $\sf ZFC$ prove that
$$ {\sf ZFC} \not \vdash \exists x \, ( \operatorname {CH}(x) \land x \neq \emptyset \land x \neq 1)$$
Where $\operatorname {CH}(x) \iff \neg \exists \kappa \, (|x| < \kappa < |P(x)|) $
In English: $\sf ZFC$ doesn't prove the continuum hypothesis of any set other than the emp... | https://mathoverflow.net/users/95347 | Which extension of ZFC proves that ZFC can only prove CH satisfied by the first two sets? | The statement you propose is equivalent to consistency of GCH failing for all (infinite) cardinals. This problem is well-studied. Of course the theory ZFC+Con(ZFC+"GCH fails everywhere") gives an answer to your problem, but this is completely tautological. It is standard in such situations to calibrate the consistency ... | 5 | https://mathoverflow.net/users/30186 | 421038 | 171,288 |
https://mathoverflow.net/questions/420999 | 6 | The simplest case of the Fundamental Theorem of Projective Geometry states that, if $f: \mathbb{R}^2\to\mathbb{R}^2$ is a bijection that preserves lines – in the sense that if $L\subseteq\mathbb{R}^2$ is a line then so is $f[L]$ – then $f$ is an affine transformation. [1]
The condition that $f$ be a bijection is stro... | https://mathoverflow.net/users/8217 | Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines | Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function that maps lines to lines, and suppose that there are three non-collinear points in the image of $f$.
**Lemma 1:** If $f[\ell\_1]$ and $f[\ell\_2]$ are distinct parallel lines, then $\ell\_1$ and $\ell\_2$ are distinct parallel lines.
**Proof:** Suppose o... | 3 | https://mathoverflow.net/users/39521 | 421042 | 171,291 |
https://mathoverflow.net/questions/421006 | 2 | One of the most intriguing things I've read about over the last few years is [Diaconescu's theorem](https://en.wikipedia.org/wiki/Diaconescu%27s_theorem), which says that, in some forms of constructivist/intuitionistic set theory, even if the law of the excluded middle is not presupposed in the background logic, the LE... | https://mathoverflow.net/users/147890 | The LNC as a mathematical theorem | Non-contradiction can only be a mathematical theorem, rather than a logical assumption, against a background of some logic which does not already contain it. One of the few well-motivated systems with that property is relevant logic.
So the best answer to your question would be some mathematical theory, like a releva... | 5 | https://mathoverflow.net/users/nan | 421048 | 171,293 |
https://mathoverflow.net/questions/421005 | 4 | Suppose $M$ is a cusped finite-volume hyperbolic $3$-manifold, say with a single cusp for simplicity. Following [NZ, Section 4] we can parametrize deformations of the hyperbolic structure with a complex paramter $u$, chosen so that $u = 0$ gives the complete hyperbolic structure on $M$. We can identify $u$ with the der... | https://mathoverflow.net/users/113402 | Complex length of geodesic added in hyperbolic Dehn surgery | After more thought, I realize this can't possibly be true, by an easy extension of the methods in [NZ]. We can re-write our two equations $pu + qv = 2\pi i$ and $\operatorname{L}\_{\mathbb C} (\gamma) = -(ru + sv)$ as the matrix equation
$$
\begin{pmatrix}
p & q \\
r & s
\end{pmatrix}
\begin{pmatrix}
\Re u & \Im ... | 2 | https://mathoverflow.net/users/113402 | 421065 | 171,302 |
https://mathoverflow.net/questions/421068 | 6 | The singular cohomology with integer coefficients of a projective variety is isomorphic to the Čech cohomology of the constant sheaf of integers on this variety.
If the above statement is correct then consider the following example:
Look at $\mathbb{P}^1$ and the standard open cover $U\_0,U\_1$. Then the Čech complex f... | https://mathoverflow.net/users/481166 | Čech cohomology is isomorphic to singular cohomology | The Cech complex only computes cohomology if the subspaces have vanishing cohomology themselves: you need
$$
H^i(U\_0; \mathcal{F}) \cong
H^i(U\_1; \mathcal{F}) \cong
H^i(U\_0 \cap U\_1; \mathcal{F}) \cong 0
$$
for all $i > 0$. In the example you've written, this is not true for $U\_0 \cap U\_1$. In general there is ac... | 23 | https://mathoverflow.net/users/360 | 421069 | 171,303 |
https://mathoverflow.net/questions/421062 | 3 | **Motivation.** [Swiss license plates](https://en.wikipedia.org/wiki/Vehicle_registration_plates_of_Switzerland) consist of $2$ letters indicating the region, followed by a number, such that the pairing (region, number) is unique by car. In the small town where I live, I saw two cars today, both from my region, having ... | https://mathoverflow.net/users/8628 | Probability of picking neighbors in $\{1,\ldots, n\}$ | **Claim** The number of sets $S$ of cardinality $m$ with no neighbors is precisely $\binom{n+1-m}{m}$.
**Proof** Encode a subset $S$ of $[n]$ as a binary string $x\_0 x\_1 \ldots x\_n$ where $x\_0=0$ and $x\_j = 1$ if and only if $j \in S$. Then the sets we want correspond to strings with $m$ ones and no consecutive ... | 7 | https://mathoverflow.net/users/297 | 421070 | 171,304 |
https://mathoverflow.net/questions/421059 | 6 | Let $G$ be a compact Lie group with a compatible biinvariant metric $d$. The hyperspace $K(G)$ of nonempty compact *subsets* of $G$ is a compact metric space with the Hausdorff metric, and it is easy to check that *subgroups* of $G$ form a closed subspace in $K(G)$, hence we may talk about the (compact) **space of clos... | https://mathoverflow.net/users/479121 | Hausdorff distance in compact Lie groups | Assume a sequence of subgroups $G\_n$ converges to $G$. Up to extraction, we can assume that $\mathrm{Lie}(G\_n)$ converges to a Lie subalgebra $\mathrm{Lie}(H)$. Since the adjoint action of $G\_n$ preserves $\mathrm{Lie}(G)$, by passing to the limit we get that $\mathrm{Lie}(H)$ is an ideal of $\mathrm{Lie}(G)$.
Now... | 2 | https://mathoverflow.net/users/173096 | 421076 | 171,306 |
https://mathoverflow.net/questions/421028 | 3 | Galois revealed that an algebraic equation $f(x)=0$ with coefficients in a field $K$ of zero characteristic is solvable by radicals if and only if the Galois group of $f(x)$ over $K$ is solvable. However, many mathematicions actually expect that one could give a criterion for solvability by radicals simply by coefficie... | https://mathoverflow.net/users/124654 | A criterion for the equation $ax^n+bx+c=0$ not solvable by radicals via $a,b,c$ and $n$ | No, the conjecture is false at least for $n = 5$. The irreducible quintic trinomial $f(x) = 85x^{5} - 4x + 1$ satisfies $\gcd(b,5ac) = \gcd(-4,5 \cdot 85 \cdot 1) = 1$. However, the Galois group of $f(x)$ is solvable.
This can be found as follows. The family of solvable quintic trinomials is
$$
f(x) = (4u^{2} + 16)x... | 13 | https://mathoverflow.net/users/48142 | 421081 | 171,308 |
https://mathoverflow.net/questions/420539 | 3 | **Definition** Let $E\_{1} \xrightarrow{\pi\_{1}} M\_{1}$, $E\_{2} \xrightarrow{\pi\_{2}} M\_{2}$ be two vector bundles over $M\_{1}$ and $M\_{2}$ with fibers $V\_{1}$, $V\_{2}$ respectively. The exterior tensor product $E\_{1} \boxtimes E\_{2}$ is defined as the vector bundle over $M\_{1} \times M\_{2}$, whose fiber o... | https://mathoverflow.net/users/138482 | Fundamental result on the projective tensor product of sections of a vector bundle | The tensor product (either one) is symmetric $A \otimes B \cong B \otimes A$, associative $(A\otimes B) \otimes C \cong A \otimes (B \otimes C)$ and distributive over direct sums $A \otimes (B \oplus C) \cong (A\otimes B) \oplus (A\otimes C)$. Spaces of sections preserve direct sum decompositions $\Gamma(M, E\_1 \oplus... | 1 | https://mathoverflow.net/users/2622 | 421098 | 171,313 |
https://mathoverflow.net/questions/421106 | 3 | Let $M$ be an $n$-dimensional complete Riemannian manifold and $r$ is the distance function to a fixed point.
The Hessian comparison theorem says that if the sectional curvature of $M$ is bounded (precisely $k\le \operatorname{sec}\le K$), then the Hessian of $r$ is bounded by the Hessians of the distance function fo... | https://mathoverflow.net/users/102515 | Hessian of the distance function--comparison with the space form with constant sectional curvature 0 | $dr$ is defined regardless of the metric. If $g$ and $g\_{0}$ are two metrics, and $\nabla^{g}$ and $\nabla^{g\_{0}}$ are their corresponding Levi-Civita connections, then the connection difference yields a tensorial operation $D:T^\*M\rightarrow T^\*{}M\otimes T^{\*}M$ such that $\mathrm{Hess}\,r-\mathrm{Hess}\_{0}\,r... | 5 | https://mathoverflow.net/users/144247 | 421118 | 171,318 |
https://mathoverflow.net/questions/421104 | 3 | If $K/\mathbb{Q}$ is an infinite algebraic extension, define as usual the class group $Cl\_K$ by the direct limit via the natural (conorm) map $Cl\_K := \lim\limits\_{\rightarrow} Cl\_F$,
where $F$ runs over all finite extensions of $\mathbb{Q}$. (I think this definition coincides with the Picard group of the ring of i... | https://mathoverflow.net/users/95241 | What's the class group of $\mathbb{Q}^{\mathrm{ab}}$? | From Armand Brumer, [*The class group of all cyclotomic integers*](https://www.sciencedirect.com/science/article/pii/0022404981900864):
>
> As an abelian group, $\mathrm{Pic}(O\_\infty)$ is isomorphic to a countable direct sum of copies of $\mathbb Q/\mathbb Z$.
>
>
>
Here $O\_\infty$ is the ring of cyclotomic... | 4 | https://mathoverflow.net/users/30186 | 421121 | 171,320 |
https://mathoverflow.net/questions/421115 | 3 | I am an amateur in $K$-theory, I have just started reading from "The K-book" by Charles Weibel. I have only read the definition of $K\_1$ which is stated as a quotient of $GL(R)$. The union of the sequence $R^{ \times} = GL\_1(R) \subset GL\_2(R) \subset ...\subset GL\_n(R) \subset GL\_{n+1}(R) \subset..$ is called $GL... | https://mathoverflow.net/users/443060 | $K_1(\mathbb{Z}_4)$ and $K_1(\mathbb{Z_4}[t])$ | As mentioned in Example 1.3.5, the group $SK\_1$ vanishes over euclidean domains. The reason is that then you can transform any invertible matrix into diagonal form using elementary row and column operations, by the Smith normal form algorithm. (Note that we are using more than just the abstract elementary divisor theo... | 5 | https://mathoverflow.net/users/39747 | 421122 | 171,321 |
https://mathoverflow.net/questions/421123 | 2 | Let $f:X\rightarrow Y$ be a proper and surjective morphism of projective $k$-varieties, where $k$ is a field of characteristic $0$. Let $L$ be a line bundle over $X$. Assume that $L$ is nef and generically ample ( here generically ample means $L$ is ample on the generic fiber of $f$). Does there exists an line bundle $... | https://mathoverflow.net/users/297229 | Nef and generically ample line bundles | This holds for *any* ample line bundle $A$ on $Y$. Indeed, then $f^\*A$ is nef [Laz, Ex. 1.4.4(i)], hence so is $D= L + f^\*A$. Setting $n = \dim X$ and $m = \dim Y$, we see that $D^n \geq 0$ [Laz, Thm. 1.4.9] and $D$ is big if and only if $D^n > 0$ [Laz, Thm. 2.2.16].
For positivity of $D^n$, we may replace $(L,A)$ ... | 4 | https://mathoverflow.net/users/82179 | 421129 | 171,323 |
https://mathoverflow.net/questions/420984 | 0 | Causality seems to play an important role in physics. There also seems to be a close parallel between "$P$ causes $Q$" and "if $P$ then $Q$." Mathematical logic studies logical inference; has there been any formal mathematical study of causal inference?
| https://mathoverflow.net/users/14024 | Has there been any mathematical study of causality? | I am converting my comments into an answer.
Setting aside the alleged parallel between causation and inference for a moment, there has indeed been some mathematical investigation of cause and effect. The Stanford Encyclopedia of Philosophy article on [causal models](https://plato.stanford.edu/entries/causal-models/) ... | 9 | https://mathoverflow.net/users/3106 | 421141 | 171,325 |
https://mathoverflow.net/questions/421113 | 1 | Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =x) = \begin{cases} &1-\alpha, &x=0 \\ &\alpha, &x=1 \end{cases}$.
Now my problem is given a random vector $\boldsymbol... | https://mathoverflow.net/users/472652 | The maximum trace of a covariance can be achieved by a discrete random vector? | The answer is yes. Indeed, let $X:=\boldsymbol{X}$, $H:=\boldsymbol{H}$, and $a:=\boldsymbol{\alpha}$. By approximation and compactness, without loss of generality (wlog), the matrix $H$ is nonsingular, so that the trace in question is
$$\sum\_{i=1}^n Var\,l\_i (X),$$
where the $l\_i$'s are linearly independent linear ... | 1 | https://mathoverflow.net/users/36721 | 421143 | 171,326 |
https://mathoverflow.net/questions/421130 | 5 | The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L\_1=\frac{1}{3}p\_1,$$
$$L\_2=\frac{1}{45}(7p\_2-p\_1^2),$$
$$L\_3=\frac{1}{945}(62p\_3-13p\_1p\_2+2p\_1^3)$$
Let $a\_k$ denote the coefficient of $p\_k$ in $L\_k$. Is there any estimate of $a\_k$? Is it tru... | https://mathoverflow.net/users/40517 | Coefficient of the top Pontryagin class in $L$-genus | The coefficient of $p\_k$ is given by
$$2^{2n}(2^{2n-1}-1)\frac{B\_n}{(2n)!} = \zeta(2n)\frac{2^{2n}-2}{\pi^{2n}},$$
see e.g. Appendix A of this older version of [Weiss](https://arxiv.org/pdf/1507.00153v3.pdf) (warning: for Weiss, the convention is that the Bernoulli numbers $B\_n$ are positive for all positive $n$... | 11 | https://mathoverflow.net/users/798 | 421144 | 171,327 |
https://mathoverflow.net/questions/419301 | 15 |
>
> Are the weak topologies of $\ell\_1$ and $L\_1$ homeomorphic?
>
>
>
Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. And indeed, no linear homomorphism between weak topologies of non-isomorphic spaces exists as prevented... | https://mathoverflow.net/users/15129 | Distinguishing topologically weak topologies of Banach spaces | The weak topologies of the spaces $\ell\_1$ and $L\_1$ are not homeomorphic because of the following
**Theorem.** *Assume that $X,Y$ are two Banach spaces whose weak topologies are homeomorphic. If $X$ has Shur's property, then $Y$ has Shur's property, too.*
Proof. For a topological space $T$ let $T\_s$ be the sequ... | 3 | https://mathoverflow.net/users/61536 | 421149 | 171,329 |
https://mathoverflow.net/questions/421154 | 1 |
>
> Let $A \in M(n)$, let $\lambda \in \mathbb{R}$, let $V\_{\lambda} := \ker(\lambda I - A)$ and let $x:\mathbb{R} \to \mathbb{R}^{n}$ be a solution of $\dot x= Ax$ such that $x(t\_0) \in V\_{\lambda}$ for some $t\_0 \in \mathbb{R}$, then $x(t) \in V\_{\lambda}$ for every real $t \in \mathbb{R}$.
>
>
>
---
... | https://mathoverflow.net/users/481556 | $V_{\lambda}$ is invariant under $A$ | Define $y(t) = \exp(\lambda(t - t\_0)) x(t\_0)$. Verify that $y' = Ay$ and that $y(t) \in V\_\lambda, \forall t \in \mathbb R$. Uniqueness of solutions gives $y = x$ and, therefore, the desired result.
| 2 | https://mathoverflow.net/users/115302 | 421156 | 171,332 |
https://mathoverflow.net/questions/421138 | 6 | Homomorphisms $B\_n \to B\_{2n}$ and $B\_n \to S\_{2n}$ have been classified in [Chen–Kordek–Margalit - Homomorphisms between braid groups](https://people.math.gatech.edu/%7Edmargalit7/papers/bn2bn.pdf) and [Lin - Braids and permutations](https://arxiv.org/abs/math/0404528) respectively. I am interested in the correspo... | https://mathoverflow.net/users/153494 | Is there a classification of homomorphisms $S_n \to S_{n+k}$ for small $k$? | As spin observed in a comment below, if you know all (conjugacy classes of) subgroups of $S\_n$ of index up to $m$, then you can determine the equivalence classes of homomorphisms $\phi:S\_n \to S\_m$, because the subgroups tell you the possible actions on the orbits of the image of $\phi$. The subgroups of index up to... | 7 | https://mathoverflow.net/users/35840 | 421166 | 171,336 |
https://mathoverflow.net/questions/421164 | 1 | Let $A$ and $B$ be two, possibly dependent, random variables, and let $X$ be a random variable independent of $(A,B)$. For simplicity, let's concern ourselves with discrete random variables. Is the following inequality always true?
$$I(A+X : B+X) \geq I(A:B) \label{eq:conj} \tag{\*}$$
This is clearly true when $A$ ... | https://mathoverflow.net/users/436290 | A question about mutual information | This is not true in general. E.g., let each of the random variables $A,B,X$ take values in the set $\{1,2\}$. Let
the matrix $(p\_{a,b}\colon a=1,2,\,b=1,2)$ of the probabilities $p\_{a,b}:=P(A=a,B=b)$ be the following matrix:
$$\frac1{10^4}\left(
\begin{array}{cc}
1456 & 3987 \\
4533 & 24 \\
\end{array}
\right);$$
i... | 3 | https://mathoverflow.net/users/36721 | 421170 | 171,337 |
https://mathoverflow.net/questions/421163 | 1 | Suppose $d\in \{3,4,\dotsc\}$ and $A\subseteq \mathbb{R}^d$ is non-empty, open and connected with its complement $A^c$ connected too and $\text{int}(A^c)\neq \emptyset$. Its boundary $S:=\partial A$ is then [connected too](https://math.stackexchange.com/questions/170337/connectedness-of-the-boundary?noredirect=1&lq=1).... | https://mathoverflow.net/users/33927 | Can one explore a surface along ‘piecewise planar’ curves? | Consider the set
$S=\{\,(t,t^2,t^3)\in\mathbb R^3\mid\,t\in\mathbb R\}$.
Clearly $S$ is connected, and so is its complement $A=\mathbb R^3\setminus S$.
Note that each plane has at most 3 points of intersection with $S$.
It follows that $S\_x=\{x\}$ for any $x\in S$.
The only problem is the set $A^c=S=\partial A$ ha... | 2 | https://mathoverflow.net/users/1441 | 421178 | 171,341 |
https://mathoverflow.net/questions/421034 | 1 | I am reading one lecture note [Dynamics for Spherical Models of Spin-Glass and Aging](https://doi.org/10.1007/978-3-540-40908-3_5) by Alice Guionnet. On page 124, it says that
for some $\alpha>0$,
$$
e^L=P\left(\exp(\alpha\sup\_{|s-t|\le\delta}\frac{|B\_s-B\_t|^2}{|s-t|})<\infty\right)
$$
Moreover, how to show that
$$
... | https://mathoverflow.net/users/168083 | For some $\alpha>0$, $ e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\frac{|B_s-B_t|^2}{|s-t|})<\infty\right) $? | First, there is a typo there - should be E instead of P, as in the second display in your question. Second, the argument as written is not quite right, but it can be rescued. Indeed, $\sup\_{t<\delta} (B\_t/\sqrt{t})=\infty$, and therefore that expectation is $\infty$. The solution is not to divide by $|t-s|$ but rathe... | 2 | https://mathoverflow.net/users/35520 | 421179 | 171,342 |
https://mathoverflow.net/questions/421066 | 5 | I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly can be answered on MO, so I am now posting it here.
---
A number of conditions that are equivalent to the stateme... | https://mathoverflow.net/users/165007 | Radon-Nikodym property in Diestel & Uhl: a definition clarification | Diestel & Uhl can only mean the first interpretation you gave for two reasons:
1. For the second interpretation, the term “off a fixed set of measure zero” makes no sense.
2. Even in case $X=\mathbb R$ (which certainly has the RNP) the condition is not satisfied if interpreted in the second way: Cantor's stairway fun... | 6 | https://mathoverflow.net/users/165275 | 421185 | 171,345 |
https://mathoverflow.net/questions/421114 | 1 | In this question, the term “word” implies a binary word, i.e. a sequence of bits.
Let $W(w)$ denote the number of non-zero bits in a word $w$.
Assuming that $l \geq 2$ is even, an $l$-bit word $w$ is *balanced* if and only if $W(w) = l/2.$
Assuming that $w$ is an $l$-bit word and $0 \le k < l$, let $R(w, k)$ deno... | https://mathoverflow.net/users/122796 | Is there an efficient generalized algorithm to generate a set of binary words satisfying a particular cross-correlation property? | If $m = 2^a b$ where $b$ is odd, the words $(0^{2^k} 1^{2^k})^{2^{a-k-1}b}$ for $k < a$ have cross-correlation $2^{a-1}b$. This appears to give a (potentially one of many) set of maximum size which achieves the upper bound on cross-correlation (since the average Hamming distance over all rotations of two balanced words... | 1 | https://mathoverflow.net/users/46140 | 421189 | 171,346 |
https://mathoverflow.net/questions/419705 | 3 | This question is a cross post from the Math StackExchange since it got no attention at all there: <https://math.stackexchange.com/questions/4414601/maps-that-preserve-winding-numbers>
I am looking for a characterisation of the continuous maps on some subset of $A\subseteq \mathbb{C}$ that preserve the winding numbers... | https://mathoverflow.net/users/32355 | Maps that preserve winding numbers | I found an answer to this question: the question of how a continuous function changes the winding number of a closed curve can be studied quite generally. The important concept here is the [degree of the function](https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping) (also known as the Brouwer degree).
If a c... | 0 | https://mathoverflow.net/users/32355 | 421194 | 171,347 |
https://mathoverflow.net/questions/132878 | 16 | I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.
Here a finite tensor category is, according to [Etingof-Ostrik](http://arxiv.org/abs/math/0301027), a rigid monoidal category whose underlying category is equivalent to the category of finite dimensional modu... | https://mathoverflow.net/users/22 | Are there small examples of non-pivotal finite tensor categories? | Monoidal categories $\mathcal{C}$ which admit a monoidal natural isomorphism $\Phi\_X:X^{\*\*} \rightarrow \beta\otimes X\otimes \beta^\*$ are called $\textit{quasi-pivotal}$ . For $\mathcal{C}=$Rep$(H)$ with $H$ a finite dimensional Hopf algebra, quasi-pivotal structures on $\mathcal{C}$ are in bijection with pairs $(... | 3 | https://mathoverflow.net/users/58202 | 421197 | 171,349 |
https://mathoverflow.net/questions/404064 | 3 | While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's *User's guide*: the problem to determine whether we have $T\in W^{1,1}$ for the transport map $T$ is still open?
More precisely, assume $\Omega\_1,\Omega\_2\subset \mathb... | https://mathoverflow.net/users/137743 | Open problem 1.28: $W^{1,1}$ regularity for optimal transport map | The problem has indeed been solved by De Philippis-Figalli. The reference is:
<https://link.springer.com/article/10.1007/s00222-012-0405-4>
the paper linked above is a subsequent, related, development of the theory and concerns stability of optimal maps.
| 2 | https://mathoverflow.net/users/58975 | 421200 | 171,350 |
https://mathoverflow.net/questions/419722 | 4 | This question is inspired by the [recent answer](https://mathoverflow.net/q/419324), where @RobPratt proposed to use integer linear programming (ILP) for solving a number-theoretic optimization problem. I will consider a very similar problem described by sequence [OEIS A061057](https://oeis.org/A061057), which for a gi... | https://mathoverflow.net/users/7076 | Reliability of ILP approach to number-theoretic optimization | Well, the question is broad, but let us address at least some of it.
>
> Q1. What is the cause of failure for ILP solvers in this problem?
>
>
>
Let's concentrate in the GLPK failure with $n=27$. [Here is a SageMath cell](https://sagecell.sagemath.org/?q=lfikax) for demonstrating what happens, based on your or... | 3 | https://mathoverflow.net/users/171662 | 421204 | 171,353 |
https://mathoverflow.net/questions/421107 | 6 | $\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}\newcommand{\ab}{\mathrm{ab}}$Let $G(\mathbb Q) = \Gal(\overline{\mathbb Q} / \mathbb Q)$ be the absolute Galois group. It's well-known that the abelianization $G(\mathbb Q)^{\ab}$ of $G(\mathbb Q)$ is isomorphic to $\Aut(\mathbb Q / \mathbb Z) = \widehat {\mat... | https://mathoverflow.net/users/2362 | What is $\mathbb Q^{\mathrm{hypoab}}$? | My [comment](https://mathoverflow.net/questions/421107/what-is-mathbb-q-mathrmhypoab?noredirect=1#comment1081832_421107), Wojowu's [answer](https://mathoverflow.net/a/421125/82179), YCor's [comment](https://mathoverflow.net/questions/421107/what-is-mathbb-q-mathrmhypoab?noredirect=1#comment1081902_421125), and Z. M's [... | 8 | https://mathoverflow.net/users/82179 | 421232 | 171,358 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.