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https://mathoverflow.net/questions/378069 | 2 | Given a basis for a full-rank lattice $\mathcal{L} \subset \mathbb{R}^n$ I want to find a vector with totally positive entries, in other words an element belonging to $\mathcal{L} \cap Q$ where $Q$ is the "quadrant" $\{(x\_1, ..., x\_n) \in \mathbb{R}^n | x\_i \gt 0, 1 \leq i \leq n\}$.
The only construction I can co... | https://mathoverflow.net/users/106850 | How can I construct an element in a particular "quadrant" of a lattice (preferably short)? | Since I haven't had a response I'll add my temporary solution. You can use Babai round off with suitably short vectors in the positive orthant to find vectors in the lattice that may be in that orthant. Its not optimal but it is fast.
| 2 | https://mathoverflow.net/users/106850 | 382612 | 159,211 |
https://mathoverflow.net/questions/382604 | 11 | What is the rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$? Here $\Omega$ denotes based loop space.
| https://mathoverflow.net/users/91826 | Rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$ | Let us first consider the homotopy fiber of the map $f\colon\mathbb CP^n\vee S^d \to \mathbb CP^\infty$ which is the inclusion on $\mathbb CP^n$ and is trivial on $S^d$. The homotopy fiber of the map $\mathbb CP^n\to \mathbb CP^\infty$ is $S^{2n+1}$. The homotopy fiber of the map $\*\to \mathbb CP^\infty$ is $S^1$, and... | 19 | https://mathoverflow.net/users/6668 | 382616 | 159,212 |
https://mathoverflow.net/questions/382418 | 3 | In this [paper by Jon Chaika and Howard Masur](https://arxiv.org/pdf/1410.1576.pdf) it is remarked at the end of page 1 that for an interval exchange transformation $T$ with $n$-intervals, one can bound the number of invariant measures generic with respect to $T$ and $T^{-1}$ by $n$. Generic in this situation means tha... | https://mathoverflow.net/users/56183 | Bounding the number of generic measures on an interval exchange transformation | I ended up asking Jon Chaika this question, and he gave a rough sketch, in which I filled in the details, so if there are any mistakes, they are probably mine.
Suppose that there were $n+1$ measures $\{\mu\_1, \ldots, \mu\_{n+1}\}$ that were generic with respect to $T$ and $T^{-1}$, and let $\{x\_1, \ldots, x\_{n+1}\... | 0 | https://mathoverflow.net/users/56183 | 382619 | 159,213 |
https://mathoverflow.net/questions/382582 | 3 | It is well-known that in the finite-dimensional case one can use the notion of Fréchet differentiability and Carathéodory differentiability interchangeably. See for example the 194 AMM article *Frechet vs. Carathéodory* by Acosta and Delgado (doi:[10.2307/2975625](https://doi.org/10.2307/2975625)). I am trying to obtai... | https://mathoverflow.net/users/98139 | Fréchet vs. Carathéodory differentiability on Banach spaces | We replace coordinates in $\mathbb{R}^n$ with the Hahn-Banach theorem in infinite-dimensional spaces. Suppose that $f$ is Fréchet-differentiable at $x\_0 \in X$ with derivative $f'(x\_0) \in \mathcal{L}(X,Y)$, so $$\lim\_{x\to x\_0} \frac{\|f(x)-f(x\_0) - f'(x\_0)(x-x\_0)\|\_Y}{\|x-x\_0\|\_X} = 0.$$
Set first $\varph... | 6 | https://mathoverflow.net/users/85906 | 382625 | 159,217 |
https://mathoverflow.net/questions/382593 | 12 | The *ordered Bell numbers* (also known as *Fubini* numbers, sequence A000670 in OEIS) count the number of ordered partitions of an n-element set. Experimentally I have found the following expression for the n-th ordered Bell number $a\_n$:
$$a\_n = \sum\_{\sigma \in S\_n}\prod\_{i=1}^n \binom{i}{\sigma(i)-1}$$ where ... | https://mathoverflow.net/users/21946 | Ordered Bell numbers | I would accept Sam's and lambda's comments as the answer. For the record, I'll just flesh it out a bit for the first formula.
In terms of compositions of $n$, the following is all but self-evident
$$a\_n = \sum\_{n\_1+n\_2+\ldots+n\_k=n} \binom{n}{n\_k}\binom{n-n\_k}{n\_{k-1}}\binom{n-n\_k-n\_{k-1}}{n\_{k-2}}\ldots... | 4 | https://mathoverflow.net/users/21946 | 382632 | 159,219 |
https://mathoverflow.net/questions/382630 | 6 | Let $X$ be a differentiable manifold and $G$ a finite group acting
differentiably on $X$. The following formula for the Euler number $\text{e}(X/G)$ of the orbit space $X/G$ appears to be well-known:
\begin{equation\*}\tag{1}
\text{e}(X/G) = \dfrac{1}{\lvert G \rvert} \sum\_{g \in G} \text{e}(X^g)
\end{equation\*}
whe... | https://mathoverflow.net/users/161310 | On the Euler number of an orbit space | I like to think of it as a (kind of) categorification of Burnside's formula. If $X$ is a finite $G$-set, then Burnside's formula says that
$$
|X/G|=\frac{1}{G} \sum\_{g\in G} |X^g|.
$$
If $X$ is (homeomorphic to the geometric realization of) a finite $G$-simplicial set, then you can apply Burnside's formula degree-wise... | 14 | https://mathoverflow.net/users/6668 | 382635 | 159,220 |
https://mathoverflow.net/questions/382502 | 1 | I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper,
<https://arxiv.org/abs/math/0411613>.
However I cannot purpose analogously on $D^b(P^1 \times P^2)$.
How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?
| https://mathoverflow.net/users/173120 | (Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$? | I assume you are looking for an analogous result of [your previous question](https://mathoverflow.net/questions/382331/bridgeland-stability-conditions-how-can-i-get-the-heart-of-a-bounded-t-structu).
In this case I think you could use Corollary 2.7 of Orlov's paper ["PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND ... | 2 | https://mathoverflow.net/users/44499 | 382636 | 159,221 |
https://mathoverflow.net/questions/382618 | 2 | Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w\_1,\ldots,w\_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I\_d)$, and define the $k \times k$ matrix $C(W)$ by setting $C(W)\_{i,j} := 2(w\_i^\top w\_j)^2 + \|w\_i\|^2\|w\_j\|^2$.
>
> **Question.** I... | https://mathoverflow.net/users/78539 | Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$ | We have
$$ C(W) = 2 A \circ A + v v^\top$$
where $v$ is the vector with entries $\|w\_i\|^2$, $A$ is the Wishart matrix with entries $w\_i^\top w\_j$, and $\circ$ is the Hadamard product. From the [Schur product theorem](https://en.wikipedia.org/wiki/Schur_product_theorem) (and the fact that adding a positive semi-defi... | 10 | https://mathoverflow.net/users/766 | 382655 | 159,227 |
https://mathoverflow.net/questions/382666 | 3 | Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and minimal parabolic in $G(\mathbb A)$, for a reference see for example pg. 37 of Arthur's [introduction to the trace formula](... | https://mathoverflow.net/users/38145 | Finiteness of the volume of $G(F) \backslash G(\mathbb A)$ | $\DeclareMathOperator\intr{int}\DeclareMathOperator\meas{meas}$This is one—probably not the best—way of thinking about it. (I am used to the $p$-adic world, where one can think about (real-valued) measures much less subtly than in the real world.) I hope someone will come along and write something more elegant.
Since... | 4 | https://mathoverflow.net/users/2383 | 382669 | 159,232 |
https://mathoverflow.net/questions/382325 | 1 | For sets $A, B$ we write $B^A$ for the set of all functions $f:A\to B$.
Let $H = (V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $V,E\neq\varnothing$ and $|e| \geq 2$ for all $e\in E$. Let $\kappa>1$ be a cardinal, finite or infinite, and let $c:V\to \kappa$ be a map. By $\text{Mono}(H,c)... | https://mathoverflow.net/users/8628 | Minimizing the set of monochromatic edges | Let $H=(V,E)$ be a hypergraph, $\kappa$ a cardinal.
**Observation 1.** $\operatorname{Min}(\mathcal M(H,\kappa))\ne\emptyset$ if and only if $H$ has a maximal $\kappa$-colorable subhypergraph.
**Proof.** If $F\subseteq E$, then $(V,F)$ is a maximal $\kappa$-colorable subhypergraph of $H$ if and only if $E\setminus ... | 1 | https://mathoverflow.net/users/43266 | 382678 | 159,236 |
https://mathoverflow.net/questions/382677 | 0 | Let $M = \mathbb{R}^3 \setminus B\_1$ where $B\_1$ is unit ball. I am trying to solve the following PDE for $f$:
$$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|\_{\partial M}}{r^2} = 0, \qquad \text{on} \, M$$
$$f + a \partial\_r f = h, \qquad \text{on} \, \partial M$$
$$\lim\_{|x|\to \infty} f= 0 $$
where $a>... | https://mathoverflow.net/users/138705 | Is this PDE solvable? | One can try the Kelvin transforn and expansion of solutions into series of spherical harmonics in $B\_1$. Since the equation is rotationally inveriant it should lead to simple enough equations for the harmonics' coefficients.
| 3 | https://mathoverflow.net/users/14551 | 382679 | 159,237 |
https://mathoverflow.net/questions/382370 | 3 | We'll consider $(N, g)$ a Riamannian Manifold and $\overline{g} = e^{2f}g$ a conformal metric. Let M be a hypersurface in N, $\overline{H}\_M$ and $H\_M$ the mean curvature of M with respect
to the metrics $\overline{g}$ and g, respectively. I would like some help to prove that
$$ \overline{H}\_M = e^{-f}( H\_M -2g( ... | https://mathoverflow.net/users/173003 | Relation between mean curvature and conformal metric | Let me use the transformation $\overline g = e^{2f}g$ to simplify some notations (and I guess your formula also use this convention). Near a point $p\in N,$ let $\{e\_i\}$ be an orthonormal frame with respect to $g,$ and $\eta$ be a normal. Then with respect to $\overline g,$ we have $\overline e\_i = e^{-f}e\_i$ form ... | 3 | https://mathoverflow.net/users/105980 | 382683 | 159,239 |
https://mathoverflow.net/questions/382680 | 4 | Let $\{U\_i\}\_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup\_{i =1 }^{ I} \overline{U\_i}$. Under what condition does there exist a continuous function $f:[0,1]^n\rightarrow [0,1]^I$ such that
$$
x\in U\_i \Le... | https://mathoverflow.net/users/170917 | Condition for existence of a continuous function realizing a partition | For $x \in U\_i$ let $f(x) = r e\_i$ where $r = \min\_{y \notin U\_i} |x - y|$, i.e. the radius of the largest open ball centered on $x$ and contained in $U\_i$. If $x$ is not in any $U\_i$ let $f(x) = 0$. Note that $|f(x) - f(y)| \leq |x - y|$
| 2 | https://mathoverflow.net/users/120134 | 382689 | 159,242 |
https://mathoverflow.net/questions/382688 | 8 | In Bousfield and Kan's book"Homotopy Limits, Completions and Localizations",they define homotopy direct limit for system of pointed simplicial sets(Ch XII S2 2.1 P327), while they define homotopy inverse limit for system of simplicial sets without pointed condition(P 295), why do they insist this condition for direct l... | https://mathoverflow.net/users/173314 | About definition of homotopy colimit of Kan and Bousfield | You can define homotopy limits and colimits in pointed as well as unpointed spaces. It so happens that the two notions of homotopy limit coincide, basically because the forgetful functor from pointed to unpointed spaces is a right Quillen adjoint. That is, the homotopy limit of a diagram of pointed spaces is the same w... | 18 | https://mathoverflow.net/users/6668 | 382691 | 159,243 |
https://mathoverflow.net/questions/381270 | 6 | Let $k$ be a nonnegative integer and let $m,n$ be coprime positive integers. Let $\phi\_k$ be the number of lattice paths from $(0,0)$ to $(km,kn)$ with steps $(0,1)$ and $(1,0)$ that are never rising above the line $my=nx$. A path having this property will be called a $\phi$-path. Then, $\phi\_k$ satisfies the recurre... | https://mathoverflow.net/users/165719 | What is the direct proof of the recurrence relation about lattice path enumeration given by Bizley? | The trick is to add $\phi\_k$ to both sides of the equation, and interpret the left hand side as counting paths with a prepended horizontal step, and one of its steps marked. Then, make the marked step the first step of a path from $(-1, 0)$ to $(km, kn)$. Let $j$ be minimal such that this path hits $(jm, jn)$ and stay... | 3 | https://mathoverflow.net/users/3032 | 382703 | 159,247 |
https://mathoverflow.net/questions/382699 | 1 | Suppose we draw a independent random vector $X'$ uniformly from a unit hypercube, $[0, 1]^d$. Given similarly drawn vectors $X\_1 \dots X\_n$ we can define the following quantity
$\rho\_{\infty}(d, n):= \mathbb{E}\_{X', \mathbf{X}} \left[ min\_{i \in [1, n]}|X' - X\_i|\_{\infty} \right]$
Here $l\_{\infty}$ norm is ... | https://mathoverflow.net/users/56778 | Lower bound on mean minimum distance($l_{\infty}$) between a test random vector $X'$ and vectors $X_1, \dots X_N$ | Let $\rho:=\rho\_{\infty}(d,n)$ and $m:=\min\_{1\le i\le n}\|X'-X\_i\|\_\infty$. We want to show that
$$\rho\overset{\text{(?)}}\ge\frac d{2(d+1)}\,n^{-1/d}.\tag1$$
For $I:=[0,1]$, we have
$$\rho=\int\_I dt\,P(m>t)$$
and, for $t\in I$,
\begin{align}P(m>t)&=\int\_{I^d}P(X'\in dx)P(\min\_{1\le i\le n}\|x-X\_i\|\_\infty... | 2 | https://mathoverflow.net/users/36721 | 382705 | 159,248 |
https://mathoverflow.net/questions/382690 | 0 | This question concerns distributions $\mu$ over the naturals $\mathbb{N}=\{1,2,\ldots\}$. For $q\ge1$, let us define the $q$th moment of entropy:
$$
H\_q(\mu)=\sum\_{i=1}^\infty \mu(i)|\log\mu(i)|^q,
$$
so $H\_1(\mu)$ is just the usual entropy.
I am interested in a sequence of distributions $\mu\_n$ satisfying the fo... | https://mathoverflow.net/users/12518 | Existence of sequence of distributions | Suppose that $\mu\_n(1)=1-t\_n$, $\mu\_n(2)=\cdots=\mu\_n(n+1)=t\_n/n$, and $\mu\_n(n+2)=\mu\_n(n+3)=\cdots=0$, where $t\_n:=1/\ln n$ and $n\ge3$. Then $\mu\_n(1)\to1$ and $H\_1(\mu\_n)\to1$. So, 1' and 2' hold; one may say 2' holds with an infinitely slow rate. It is easy to modify this example to have 2' hold with an... | 2 | https://mathoverflow.net/users/36721 | 382707 | 159,249 |
https://mathoverflow.net/questions/382712 | 13 | Let $t\in\Bbb{N}$ and consider the sequences $p\_t(n)$ defined by
$$\sum\_{n\geq0}p\_t(n)x^n=\prod\_{i\geq1}\frac1{(1-x^i)^t}=(x;x)\_{\infty}^{-t}.$$
The numbers $p\_t(n)$ can be regarded as enumerating *partitions of $n$ into parts of $t$ colors*.
Furthermore, $p\_t(n)=\sum\_{\lambda\vdash n}\prod\_{j\geq1}\binom{k\_j... | https://mathoverflow.net/users/66131 | Congruences for "colored partitions" a la Ramanujan | Yes, these are all true and they are all in the literature. The first two congruences are part of infinite families of the form
$$\begin{cases}
p\_{\ell-1}(\ell n+a)\equiv 0\mod \ell, \\ p\_{\ell-3}(\ell n+b)\equiv0\mod \ell,
\end{cases}$$
where $\ell \geq 5$ is a prime and $a,b$ are such that: $24a+1$ is a quadratic ... | 15 | https://mathoverflow.net/users/2384 | 382714 | 159,250 |
https://mathoverflow.net/questions/382480 | 6 | I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations
$$\partial\_{z\_2}f\_1(z\_1,z\_2) + \partial\_{z\_1} f\_2(z\_1,z\_2)=0 \text{ and }$$
$$\partial\_{\bar z\_1}f\_1(z\_1,z\_2) - \partial\_{\bar z\_2} f\_2(z\_1,z\_2)=0$$
and in addition, is doubly-periodic in both it... | https://mathoverflow.net/users/119875 | Complex-doubly periodic function in two variables? | The answer is that the only solutions have the form
$$
f = (f\_1,f\_2) = \bigl(c, h(\,\overline{z}\_1, z\_2)\bigr)
$$
where $h:\mathbb{C}^2\to\mathbb{C}$ is holomorphic and $c$ is a constant,
which must equal zero unless $k\_1$ and $k\_2$ are integers.
The argument is as follows: The first equation implies that there... | 4 | https://mathoverflow.net/users/13972 | 382716 | 159,252 |
https://mathoverflow.net/questions/159881 | 14 | In [Higher Topos Theory](http://arxiv.org/pdf/math/0608040v4), Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of $n$-category, as well necessary and sufficient conditions for a given quasicatogory... | https://mathoverflow.net/users/11546 | Are n-truncated quasicategories a model for n-categories? | Let $C$ be an $\infty$-category, and $n\geq -1$. The following are equivalent:
1. $C$ is $n$-truncated.
2. The $\infty$-groupoids $\def\Map{\operatorname{Map}}\Map(\Delta^0,C)$ and $\Map(\Delta^1,C)$ are $n$-truncated. (Remember that $\Map(B,C)$ is the maximal Kan complex inside $\operatorname{Fun}(B,C)$.)
3. ($n\geq... | 17 | https://mathoverflow.net/users/437 | 382721 | 159,254 |
https://mathoverflow.net/questions/382708 | 4 | I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal slope) but it does not admit subbundles with greater slope. This is the simplest example I have in mind in order to explai... | https://mathoverflow.net/users/129919 | Push-out in the category of coherent sheaves over the complex projective plane | This example is useful not to understand that stability should be checked on subsheaves (instead of checking only subbundles), but that it should be checked for **all quotient sheaves** (and not only quotient bundles). Indeed, let us show that $\mathcal{E}$ has no quotient bundle of rank $1$ with slope $0$. Assume, by ... | 3 | https://mathoverflow.net/users/37214 | 382722 | 159,255 |
https://mathoverflow.net/questions/382525 | 2 | Given any triangular region and two integers $n$ and $p$ which can be large and $p > 4$. It is needed to cut the triangle into $n$ $p$-gons (e.g., cut a triangle into 10 heptagons). Among the $p$-gons, we need the maximum possible number to be convex. No other requirements on the pieces.
To our knowledge, for any hig... | https://mathoverflow.net/users/142600 | To cut a triangle into $n$ $p$-sided polygonal regions | This is nothing like a complete answer to the question, but it takes care of a few cases. It summarizes the string of comments I posted.
Quadrilaterals:
Any triangle can be cut up into $n$ quadrilaterals, all of them convex, for any $n\ge3$, as follows:
Take a point in the interior of the triangle, and for each s... | 5 | https://mathoverflow.net/users/158000 | 382727 | 159,258 |
https://mathoverflow.net/questions/382720 | 3 | When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral
$$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$
where $P$ is a linear partial differential operator and $\phi$ is a compactly support smooth function. In order for $t$ to be a root of the... | https://mathoverflow.net/users/64302 | Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$? | **Edit notice:** The answer is completely rewritten due to user2520938's comment. My original answer was that the linear operator $P$ depends on $t$, and we have $(1)$ as long as $P(t) = 0$. But as user2520938 pointed out, we always have $P(t) \ne 0$, since otherwise it contradicts the minimality of the Bernstein-Sato ... | 1 | https://mathoverflow.net/users/14037 | 382734 | 159,260 |
https://mathoverflow.net/questions/382650 | 4 | Let $B$ be the Auslander algebra of a representation-finite algebra $A$.
>
> Question: When do we have $Ext\_B^1(D(B),B)=0$? Can this be expressed in terms of nice properties of $A$?
>
>
>
This is for example true when $A$ is a hereditary Nakayama algebra, but other than for hereditary $A$ I do not know any ex... | https://mathoverflow.net/users/61949 | Which Auslander algebras satisfy $Ext_B^1(D(B),B)=0$? | I believe this is answered by Theorem 1.20 in the following article:
<https://arxiv.org/pdf/0809.4897.pdf>
| 1 | https://mathoverflow.net/users/115085 | 382739 | 159,262 |
https://mathoverflow.net/questions/382751 | 3 | I am a beginner in the subject, and at the moment I am trying to understand basic properties of the main objects of the M. Saito's theory of the mixed Hodge modules in general.The question in the title might be trivially wrong or correct.
| https://mathoverflow.net/users/16183 | Is the abelian category of pure Hodge modules semi-simple? | The category of polarizable pure Hodge modules (of a given weight) on an algebraic variety $X$ is semi-simple. This is Theorem 14.37 in the book "Mixed Hodge Structures" by Peters and Steenbrink.
If you don't insist on polarizability the category is not semisimple even for $X=pt$ (where we just get pure Hodge structu... | 5 | https://mathoverflow.net/users/7762 | 382769 | 159,269 |
https://mathoverflow.net/questions/382771 | 0 | Fix $x \in \mathbb{R}$ and let $I\_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g\_n$ to $I\_{[x]}$ such that
* $g\_n$ converge uniformly to $I\_{[x]}$ on $\mathbb{R}$,
* $|g\_n(y)-I\_{[x,x+n^{-1})}(y)|\in (\frac1{2n},\frac1{n}]$?
Is this possible?
| https://mathoverflow.net/users/172598 | Uniform approximation of indicator function of a point | Such a sequence $(g\_n)$ does not exist -- if by $[x]$ you mean $\{x\}$ and if you want $|g\_n(y)-I\_{[x,x+n^{-1})}(y)|\in (\frac1{2n},\frac1{n}]$ to hold for all real $y$.
Indeed, then $|g\_n(x+1/(2n))-1|\in (\frac1{2n},\frac1{n}]$, so that $g\_n(x+1/(2n))\to1$ (as $n\to\infty$) and hence
$$\liminf\_n\sup\_{t\in\mat... | 1 | https://mathoverflow.net/users/36721 | 382776 | 159,271 |
https://mathoverflow.net/questions/382749 | 1 | Let $X$ be a compact metric space, $\{\delta\_n\}\_{n=1}^{\infty}$ be a strictly monotonically decreasing sequence in $[0,1]$ converging to $0$, and $\{h\_n\}\_{n=1}^{\infty}$ be a uniformly convergence sequence of continuous functions on $X$ converging to $h:X\rightarrow [0,1]$. Distinguish a non-empty compact subset ... | https://mathoverflow.net/users/172598 | Conditions for pointwise convergence of indicators precomposed with uniformly continuous sequence | $\newcommand\de\delta\newcommand\N{\mathbb N}\newcommand\R{\mathbb R}$The answer is yes. Indeed, fix any $x\in X$. We need to show that
$$l\_n:=I\{0\le h\_n(x)<\de\_n\}\to r:=I\{h(x)=0\}\tag1$$
as $n\to\infty$.
Let
$$N:=\{n\in\N\colon0\le h\_n(x)<\de\_n\}.$$
If $N\ni n\to\infty$, then $l\_n=1$, $h\_n(x)\to0$, and hen... | 1 | https://mathoverflow.net/users/36721 | 382778 | 159,272 |
https://mathoverflow.net/questions/382255 | 7 | Recall that a group $G$ is *pseudofinite* if every first-order sentence $\varphi$ (in the language of groups) satisfied in $G$ is also satisfied in some finite group. Also recall that an instance of the *order property* in $G$ is a pair of sequences of finite tuples of elements of $G$, $\{\bar{a}\_i\}\_{i<\omega}$ and ... | https://mathoverflow.net/users/83901 | Is there a pseudofinite group with a quantifier-free instance of the order property? | **Short answer:** Given a prime $p>2$, an infinite extra-special $p$ group is pseudofinite, and the quantifier-free formula $xy=yx$ witnesses the independence property (and so witnesses the order property too).
**Details:** I am basically just quoting from the Appendix in *[Definable envelopes in groups having a simp... | 5 | https://mathoverflow.net/users/38253 | 382786 | 159,274 |
https://mathoverflow.net/questions/382752 | 4 | Let $\Gamma$ be a simple, locally finite, acyclic graph. Let $v\_0$ be some vertex in $\Gamma$.
We let $X\_n$ denote the simple random walk on $\Gamma$ where $X\_0 = v\_0$. If we almost surely have $\limsup\_{n\to\infty} \frac{d(X\_0,X\_n)}{n}=0$ (using the path metric), does it follow that almost surely $X\_t = X\_0... | https://mathoverflow.net/users/130484 | Does there exist a non-recurrent acyclic graph with sublinear expansion? | For a spherically symmetric tree (where all vertices at the same distance from the root have the same degree) it is well known that transience is equivalent to $ \sum\_n |T\_n|^{-1} <\infty$, (where $T\_n$ is level $n$ of the tree) since that means the resistance from the root to infinity is finite. On the other hand, ... | 5 | https://mathoverflow.net/users/7691 | 382789 | 159,276 |
https://mathoverflow.net/questions/382784 | 10 | Let $X \subseteq \mathbb{R}$. Let $A$ and $B$ be *finite* subsets of $X$. The statement $$\sum\_{a \in A} 2^a = \sum\_{b \in B}2^b \iff A = B $$ is true if $X = \mathbb{N}$ or $X = \mathbb{Z}$; this follows from the uniqueness of finite binary represntation (for naturals and dyadic rationals). However, the statement is... | https://mathoverflow.net/users/170682 | Representing finite sums of rational powers of 2 | If $A$ and $B$ are distinct finite subsets of $\mathbb Q$ which are not both subsets of $\mathbb Z$, let $d$ be the least common denominator and $m$ the minimum of $A \cup B$. Thus $\sum\_{a \in A} 2^a - \sum\_{b \in B} 2^b = 2^m P(2^{1/d})$ where
$P$ is a polynomial with coefficients in $\{-1,0,1\}$. Now for this to b... | 15 | https://mathoverflow.net/users/13650 | 382790 | 159,277 |
https://mathoverflow.net/questions/382628 | 4 | Let $n \ge 2$ be a positive integer. Do there exist $n$ non-zero distinct integers such that the sum of their square is a perfect square and their product is a *n*th power?
For $n=2$ the answer is no, by infinite descent. Is this true for all $n$? What happens for other values of $n$?
| https://mathoverflow.net/users/70464 | $n$ variables Diophantine | There are such solutions for each $n>2$.
We seek distinct nonzero integers $x\_1,\ldots,x\_n$ such that
$\sum\_{i=1}^n x\_i^2 = y^2$ and $\prod\_{i=1}^n x\_i = z^n$.
These equations are homogeneous, so it is enough to consider
the distinct nonzero rationals $r\_i := x\_i / z$, which satisfy
$\sum\_{i=1}^n r\_i^2 = (y... | 15 | https://mathoverflow.net/users/14830 | 382801 | 159,281 |
https://mathoverflow.net/questions/382795 | 27 | The convention that $\sin^2 x = (\sin x)^2$, while in general $f^2(x) = f(f(x))$, is often called illogical, but it does not lead to conflicts because nobody uses $\sin(\sin x)$.
But is this really true? Or is there a real-world application in which $\sin(\sin x)$ occurs? Or maybe something a bit more general, like $... | https://mathoverflow.net/users/nan | Is there any use for $\sin(\sin x)$? | The intensity of light diffracted at a slit as a function of the angle actually involves a term $\sin\left(\frac{\alpha\beta}{2}\sin(\theta)\right)$, see
<https://en.wikipedia.org/wiki/Fraunhofer_diffraction>
(I'm no physicist at all, but this has been stuck in my head since high school just because it is such an u... | 55 | https://mathoverflow.net/users/39747 | 382803 | 159,282 |
https://mathoverflow.net/questions/382802 | 3 | This is a second part of my [previous question](https://mathoverflow.net/questions/382463/rigorous-euler-lagrange-equations-for-fields). I'm trying to figure it out by myself how to deduce Hamilton's equations in classical field theory as it is usually obtained in physics books.
**Notation:** If ${\bf{x}} = (x\_{1},.... | https://mathoverflow.net/users/152094 | Hamilton equations for Classical Field Theory | There is a fundamental misunderstanding in your translation of Hamilton's formalism to classical field theory, which pertains to the proper identification of dynamical variables.
In classical mechanics, the position variables are dynamical variables, whereas time is the external parameter in terms of which we registe... | 8 | https://mathoverflow.net/users/11211 | 382821 | 159,290 |
https://mathoverflow.net/questions/382809 | 9 | $\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^\*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators $\mathcal{B}$ on some Hilbert space $H$.
A state is defined as a positive linear functional $\varphi: A\rightarrow... | https://mathoverflow.net/users/98901 | Why does Riesz's Representation Theorem apply in quantum mechanics? | Okay, there is a lot of confusion in this question.
First, I'm not sure why you say ``it is common to begin the discussion with embedding $A$'' into $B(H)$. The point of the C${}^\*$-algebra approach to quantum mechanics is doing things in a representation-independent manner, so I would say it's unusual to begin the ... | 14 | https://mathoverflow.net/users/23141 | 382826 | 159,291 |
https://mathoverflow.net/questions/382810 | 0 | **Question:**
Equip $\{0,1\}$ with the [Sierpiński topology $\{\{1\},\{0,1\},\emptyset\}$](https://ncatlab.org/nlab/show/Sierpinski+space), let $X$ be a compact metric space, and equip $C(X,\{0,1\})$ with the compact-open topology. Let $\{B\_n\}\_{n=1}^{\infty}$ be a sequence of open subsets of $X$ and $B$ also be open... | https://mathoverflow.net/users/36886 | Convergence in compact-open topology on the Sierpiński space | Yes, the answer is not correct. Let Y the space $\{0,1\}$ with the Sierpinsky-topology. The compact-open topology is generated by the set of $T(K,U) := \{f \in C(X,Y) \colon f(K) \subset U\}$ with $K \in \mathcal{K}(X)$ (compact subsets) and $U$ open in $Y$. W.l.o.g. $U = \{1\}$. Since $C(X,Y) = \{1\_B \colon B \text{ ... | 1 | https://mathoverflow.net/users/100904 | 382837 | 159,294 |
https://mathoverflow.net/questions/381642 | 10 | Let $\mathbb{M}$ be the monster group, i.e. the largest finite
simple sporadic group.
**Question**: Are the conjugation classes of pairs of
involutions in $\mathbb{M}$ known?
**What I have found so far**:
According to the ATLAS of finite groups, $\mathbb{M}$ has two classes
of involutions 2A and 2B. The 9 classes... | https://mathoverflow.net/users/105705 | Conjugation classes of pairs of involutions in the monster group | You could use the character table of the Monster $M$ and its maximal subgroups to find more information. For the pairs from (2a, 2a) and (2a, 2b) the orbits are in bijection with the conjugacy classes containing their products. For pairs from (2b, 2b) this is not correct, but at least you can split the problem into sma... | 6 | https://mathoverflow.net/users/61095 | 382840 | 159,295 |
https://mathoverflow.net/questions/382808 | 5 | I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means.
My first instinct was to do something like "the center of a random tiling of a large square". More formally, consider the following property for a random distrib... | https://mathoverflow.net/users/89672 | Random domino tilings: Is this distribution well-defined, and how can it be sampled from? | This distribution is the maximal entropy Gibbs measure for domino tilings of the plane. Burton and Pemantle (<https://projecteuclid.org/euclid.aop/1176989121>) proved many important facts about this distribution, including some remarkable formulas for specific probabilities. (But beware of typos: if I remember right, a... | 8 | https://mathoverflow.net/users/4720 | 382844 | 159,296 |
https://mathoverflow.net/questions/382799 | 5 | Let $X\_1, \ldots, X\_n$ be independent symmetric variables. Now I would like to know whether there exists a constant $C$, such that
$$
\Pr[\max\_{k \in [n]} |\sum\_{i \in [n] \setminus \{k\}} X\_i| \ge t] \le C\Pr[|\sum\_{i \in [n]} X\_i| \ge t]
$$
Now similar inequalities are Levy's inequalities
\begin{align}
\labe... | https://mathoverflow.net/users/152785 | Does there exist a constant $C$, such that, $\Pr[\max_k|\sum_{i\neq k}X_i|\ge t]\le C\Pr[|\sum_i X_i|\ge t]$ for all independent symmetric variables? | Alas, no such inequality can hold. Suppose that the symmetric $X\_i$ take values $\pm 1$ and $t=n-1$. Then
$$
\Pr[\max\_{k \in [n]} |\sum\_{i \in [n] \setminus \{k\}} X\_i| \ge t] =(n+1)2^{1-n}
$$
but
$$\Pr[|\sum\_{i \in [n]} X\_i| \ge t]=2^{1-n} \,.
$$
| 6 | https://mathoverflow.net/users/7691 | 382850 | 159,297 |
https://mathoverflow.net/questions/382843 | 26 | It is conjectured (see [[1](https://www.ams.org/journals/mcom/1992-59-200/S0025-5718-1992-1146835-5/home.html)]) that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Numerical investigations of this conjecture show that for some $k$ solutions are easily fou... | https://mathoverflow.net/users/101078 | The "stubborn" solutions to sums of three cubes | This lim sup indeed goes to $\infty$. We can prove this using exactly the strategy Lucia suggested.
We will count the number of $x,y,z$ in a box with $x^3+y^3+z^3$ not a cubic residue modulo $p$ for a large finite list of primes $p$, all congruent to $1$ mod $3$. We can similarly count the number of $n$ not a cubic r... | 30 | https://mathoverflow.net/users/18060 | 382852 | 159,298 |
https://mathoverflow.net/questions/382856 | 1 | In this experiment, I have checked how many times different gapped primes occur out of the first 10000, 100000, 1000000 first primes.
Please view the following as ($X$:$Y$) where $X$ represents the gap and $Y$ represents how many times it occurs.
Out of first 1000000 primes:
**2:** 40405
**4:** 40233
**6:** 68311... | https://mathoverflow.net/users/173298 | Comparing densities of different gapped primes (twin, cousin, sexy...) | Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118 suggest that somewhere around $x=1.7427\times10^{35}$, the most common gap between consecutive primes less than $x$ switches from $6$ to $30$.
See The Most Common Prime Gaps, posted by John Baez, at <http... | 2 | https://mathoverflow.net/users/158000 | 382867 | 159,301 |
https://mathoverflow.net/questions/382792 | 5 | It seems to me that by the algebraic Riemann Hilbert functor(which factors through the analytification map) and also the analytic Riemann Hilbert functor that the (derived) category of (algebraic) regular holonomic D-modules embeds into the (derived) category of analytic regular holonomic D-modules. Is this correct? I ... | https://mathoverflow.net/users/136287 | Is analytification of regular holonomic D modules a fully faithful functor? | Yes, for a smooth algebraic variety $X$, the analytification functor
$D^b(\mathcal D\_X)\_{rh} \to D^b(\mathcal D\_{X^{an}})\_{rh}$
is fully faithful.
As you note, it is a consequence of the usual algebraic and analytic versions of the Riemann-Hilbert correspondence for $D$-modules: analytic $D$-modules correspon... | 5 | https://mathoverflow.net/users/7762 | 382894 | 159,313 |
https://mathoverflow.net/questions/382907 | 2 | 1. Let $X$ be a smooth affine algebraic variety. Does there necessarily exist an embedding into some affine space $A^n$ of codimension $1$?
I guess so. Next one I'm less sure.
2. Let $X$ be a complete intersection inside an affine space $A^m$. Does there exist a different embedding into another $A^n$ of codimension... | https://mathoverflow.net/users/173476 | Embedding varieties as divisors | Not every smooth affine curve can be embedded into the affine plane so the answer is no.
| 6 | https://mathoverflow.net/users/173477 | 382909 | 159,317 |
https://mathoverflow.net/questions/382904 | 7 | In the textbook Homotopy Type Theory: Univalent Foundations of Mathematics, the authors give a predicative constructive construction of the initial Cauchy complete reals $\mathbb{R}\_C$ in terms of a higher inductive-inductive type ${\sim}\_C$ defined at the same time as $\mathbb{R}\_C$. Is there a similar construction... | https://mathoverflow.net/users/nan | Construction of Dedekind reals using higher inductive-inductive types | The Dedekind reals are constructed as Dedekind cuts in section 11.2 of the HoTT book. A Dedekind cut is a pair $(L,U)$ of subsets of rational numbers $\mathbb{Q}$ satisfying the conditions listed at the beginning of the section. The Dedekind reals as a type are therefore *not* a higher inductive-inductive type, but are... | 6 | https://mathoverflow.net/users/1176 | 382910 | 159,318 |
https://mathoverflow.net/questions/382912 | 2 | Let $G$ be a symmetric group on a finite set acting on another finite set $X$ through a natural action $\alpha:G \times X \to X$, $\alpha(g,x)=gx$. Let $x \in X$ and consider the orbit $G \cdot x := \{gx: g \in G\}$. Assuming that $|G\cdot x| <|G|$, can we always find a subgroup $H$ of $G$ such that (i) $|H|=|G\cdot x|... | https://mathoverflow.net/users/132350 | Group action: 'Minimal' subgroup generating an orbit | No. Take $G=S\_5$. It has a subgroup $H$ of order $4$ and index $30$. Then the group $G$ acts transitively on $G/H$ but $G$ does not have a subgroup of order $30$.
| 10 | https://mathoverflow.net/users/157261 | 382915 | 159,320 |
https://mathoverflow.net/questions/382924 | 19 | I am a retired mathematics professor and AMS member continuing to do research and publish papers.
Unfortunately, my former university (39 years) allows library access only to Emeritus Professors so I have no access to JSTOR or MathSciNet, putting me at somewhat of a disadvantage.
Needless to say, living on a retire... | https://mathoverflow.net/users/74683 | Is there any limited access to MathSciNet for retired mathematics faculty? | [Zbmath](https://www.zbmath.org/) is now completely open, and hence it is a free alternative to Mathscinet.
| 25 | https://mathoverflow.net/users/1898 | 382927 | 159,323 |
https://mathoverflow.net/questions/382891 | 1 | Let $M$ be a locally compact (Hausdorff) space, and $g:M\to M$ an isomorphism (think of an action of a finite cyclic group).
By some generalities one can show that the "obvious" map $(M^g)^+\to(M^+)^g$ is continuous. Here $M^+$ is the one-point-compactification, and $M^g$ are the fixed points with the subspace topology... | https://mathoverflow.net/users/168301 | Fixed points of one-point-compactification | $M^+$ is still Hausdorff, so also $(M^+)^g$ is. Now we observe that the "obvious" map is a continuous bijection from a compact to a Hausdorff space, thus by standard textbook contents, an isomorphism.
| 2 | https://mathoverflow.net/users/168301 | 382933 | 159,325 |
https://mathoverflow.net/questions/382930 | 2 | In the second paragraph of [Soulé - Perfect forms and the Vandiver conjecture](https://arxiv.org/abs/math/9812171), it is written that:
*For any natural integer $i \le p − 2$, let $C^{(i)}$ be the subgroup of $C$, where the Galois group of $\Bbb Q(\zeta\_p)$ over $\Bbb Q$ acts by the $i$-th power of the Teichmuller cha... | https://mathoverflow.net/users/166540 | Galois group acts by the $i$-th power of the Teichmuller character on $H$ | $\DeclareMathOperator\Gal{Gal}$So as you defined it, the Teichmüller character operates on $\mathbb{Z}\_p^\times$, and is basically a homomorphism
$$
\mathbb{Z}\_p^\times\rightarrow \mu\_{p-1}.
$$
Under the isomorphism $\mathbb{Z}\_p^\times \cong \Gal(\mathbb{Q}(\zeta\_{p^\infty})/\mathbb{Q})$, it can be thought of as ... | 3 | https://mathoverflow.net/users/5513 | 382942 | 159,327 |
https://mathoverflow.net/questions/382947 | 5 | Suppose $\Omega$ is a bounded domain in $\mathbb R^3$ with Lipchitz boundary $\partial\Omega$, and $u\in H\_0^1(\Omega)\cap C(\Omega)$. Is $u$ continuous to the boundary i.e. do we have $u \in C( \overline{\Omega})$?
In other words, is is true that $H\_0^1 (\Omega)\cap C(\Omega)\subset C(\overline \Omega)$?
Dependi... | https://mathoverflow.net/users/137640 | Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior? | Not necessarily- let $\Omega = B\_1 \cap \{x\_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x\_3}{|x|}$ is in $H^1\_0(\Omega) \cap C^{\infty}(\Omega),$
but $u$ is discontinuous at the origin.
| 11 | https://mathoverflow.net/users/16659 | 382950 | 159,329 |
https://mathoverflow.net/questions/382948 | 1 | I am reading the proof of Theorem 1(a) in the [paper](http://www.math.sjtu.edu.cn/faculty/weidongl/Publication/04.pdf) that proposed the CLIME method for estimating precision matrix. I am puzzled by an inequality on Page 605 three lines above formula (29). I isolate the specific question as follows for your convenience... | https://mathoverflow.net/users/172610 | A result about sub-exponential random variables | The best I can prove has an extra $K$, but I think it does not matter too much for the proof of the original paper.
\begin{align}
n t^2 \mathbb{E}\left(Y\_{kij}^2 e^{t|Y\_{kij}|} \right) &= n (\eta\sqrt{\log p /n})^2 \mathbb{E}\left(Y\_{kij}^2 e^{\eta\sqrt{\log p /n}|Y\_{kij}|} \right) \\
&= \eta^2\log p \mathbb{E}\... | 1 | https://mathoverflow.net/users/163923 | 382954 | 159,331 |
https://mathoverflow.net/questions/382906 | 3 | Let $\mathcal{M}$ be the vector space of Borel finite signed measures on $\mathbb{R}^d$. On $\mathcal{M}$ we can consider the weak topology $\tau$: the coarsest topology on $\mathcal{M}$ s.t. all the maps $\mu \mapsto \int \varphi d\mu$ are continuous on varying of $\varphi \in C\_b(\mathbb{R}^d)$, the continuous and b... | https://mathoverflow.net/users/142961 | Borel sigma algebra on measures generated by distance inducing weak convergence and the one generated by weak topology | The Borel $\sigma$-algebras generated by these two topologies seem to be equal.
The idea of the proof is as follows. Let $\mathcal M\_+$ be the subspace of $\mathcal M$ consisting of measures. It is known that the weak topology on $\mathcal M\_+$ is metrizable and the space $\mathcal M\_+$ is Polish. Consider the sub... | 2 | https://mathoverflow.net/users/61536 | 382958 | 159,332 |
https://mathoverflow.net/questions/382957 | 1 | Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F\_2^{n\times n}$ where $A\in\mathbb F\_2$ and $D\in\mathbb F\_2^{(n-1)\times(n-1)}$ are square.
Through the determinant result on Schur complement
$$\det(M)=\det(D-CB)$$
if $A=1$ holds.
It suggests an algorithm for $\det(M)$. Without loss of ge... | https://mathoverflow.net/users/10035 | Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$ | As the comments note, what you are doing is Gaussian elimination with complete pivoting: permute rows and column to bring an 1 to the top-left corner, make one step of Gaussian elimination, repeat.
Note that in this algorithm we can choose to apply all permutations directly before step 1, and the result won't change ... | 1 | https://mathoverflow.net/users/1898 | 382959 | 159,333 |
https://mathoverflow.net/questions/382941 | 8 | The basic idea of this question is to see if there is any other derivations than 'formal derivations'.
Let $\mathbb{K}$ be a field. Given a commutative $\mathbb{K}$-algebra $A$, a derivation of $A$ is a $\mathbb{K}$-linear map $D:A\rightarrow A$ satisfying $D(ab)=D(a)b+ aD(b)$. Consider the case when $\mathbb{K}=\mat... | https://mathoverflow.net/users/65841 | Derivation of formal power series | Every $\mathbb{K}$-derivation $D$ of $\mathbb{K}[[x\_1,\dots,x\_n]]$ has the standard form $$D(f)=\sum\_{i=1}^n p\_i \frac{\partial f}{\partial x\_i},$$ where of course $p\_i=D(x\_i)$.
Indeed, let $\mathfrak{m}$ be the maximal ideal of $\mathbb{K}[[x\_1,\dots,x\_n]]$: then clearly $D(\mathfrak{m}^{N+1})\subset \math... | 6 | https://mathoverflow.net/users/7666 | 382965 | 159,336 |
https://mathoverflow.net/questions/382938 | 2 | Consider the Laplace equation in $\mathcal{R}^3$
\begin{equation}
\Delta u = f, ~~~\lim\_{x\to \infty} u(x) = 0.
\end{equation}
Here we assume $f$ is a smooth, compactly supported function. Of course, $u$ can be explicitly solved with the Green function. I am considering if we can use a stochastic process (like Brownia... | https://mathoverflow.net/users/114951 | Use stochastic process to express solution to Laplace equation in the whole space | If $f(x) / (1 + |x|)$ is integrable, then the solution $u$ is equal to the Newtonian potential of $f$:
$$ -u(x) = \frac{1}{4\pi} \int\_{\mathbb R^3} \frac{f(y)}{|x - y|} \, dy . $$
And the Newtonian potential kernel is the occupation density of the Brownian motion:
$$ \frac{1}{4\pi} \, \frac{1}{|x|} = \int\_0^\infty \f... | 6 | https://mathoverflow.net/users/108637 | 382966 | 159,337 |
https://mathoverflow.net/questions/382963 | 2 | Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again locally free since $dim(X)=1$. Let $E,F$ be the corresponding vector bundles associated with $\mathcal{E},\mathcal{F}$ res... | https://mathoverflow.net/users/129919 | Locally free sheaves and vector bundles over smooth connected projective curve | Welcome to MathOverflow!
**Question 1**: this can be checked locally, on affine opens or local rings, and then becomes an exercise: if $0 \to M' \to M \overset{\pi}{\to} M'' \to 0$ is an exact sequence of $R$-modules, then the submodule of $M$ generated by $M'$ and $\pi$-preimage of torsion in $M''$ has torsion free ... | 4 | https://mathoverflow.net/users/111491 | 382970 | 159,338 |
https://mathoverflow.net/questions/382983 | 1 | Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
---
**Question**: What ***computationally efficient*** strategy can we use to sample a point $\mathbf{p}$ from $T$ with probability linearly proportional to $\|\mathbf{p}\|\_2$?
| https://mathoverflow.net/users/115803 | Geometric sampling problem in the Euclidean space in high dimensions | It doesn't make any difference that $d\gg 1$. If $0<a<b<1$, $Q(a,b)=\|a\mathbf x+(b-a)\mathbf y+(1-b)\mathbf z\|\_2^2=a^2\|\mathbf x\|^2+(b-a)^2\|\mathbb y\|^2+(1-b)^2\|\mathbf z\|^2+2a(b-a)\langle\mathbf x,\mathbf y\rangle+2(b-a)(1-b)\langle\mathbf x,\mathbf z\rangle$. That is, you should pick the point in the 2d simp... | 4 | https://mathoverflow.net/users/11054 | 382987 | 159,342 |
https://mathoverflow.net/questions/382984 | 1 | So the title is quite self-explanatory,
suppose we have a stochastic process $(X\_t: t\in[0,T])$ where for a fixed $t$, $X\_t$ is a $\mathbb R$-valued square integrable random variable, we could even assume it's in the Gaussian Sobolev space $\mathbb D^{1,2}$.
Then in which cases does the following hold?
$$D\_u \in... | https://mathoverflow.net/users/132216 | Under which conditions does Malliavin derivative and Lebesgue integral commute? | The hypotheses are sometimes stated differently, depending on the kind of processes you have in mind. The following is taken from Pratelli's [lecture notes](https://people.dm.unipi.it/pratelli/Didattica/Appunti-Malliavin.pdf), Theorem 3.2.1, but he gives no further reference.
**Theorem.** Let $(X\_t)\_{t \in [0,T]}$ ... | 1 | https://mathoverflow.net/users/nan | 383008 | 159,348 |
https://mathoverflow.net/questions/382995 | 2 | Let T be a compactly generated triangulated category and let T' be a localizing subcategory. Is it automatic that T' is comapctly generated by $T^c \cap T'$, where $T^c$ is compact objects of $T$?
Edit: I would be interested if there is a useful sufficient criteria (that takes advantage of the compact generation of T... | https://mathoverflow.net/users/173527 | Subcategory of compactly generated triangulated category | Let $T' \subset D(\Bbb Z)$ be the collection of complexes whose homology is uniquely divisible; i.e. $T'$ is the essential image of $D(\Bbb Q)$. Then $T'$ is a localizing subcategory. However, compact objects of $D(\Bbb Z)$ have finitely generated homology groups, and so the only compact objects in $T'$ are zero object... | 5 | https://mathoverflow.net/users/360 | 383011 | 159,349 |
https://mathoverflow.net/questions/382973 | 11 | If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$.
Now if I take a deformation $A\_{\epsilon}/k[\epsilon]$, does there still exist a canonical lift of this deformation to a deformat... | https://mathoverflow.net/users/161405 | Canonical lift of the deformation of an ordinary abelian variety | No. The picture over a general base is this: let $A\_0\to S\_0$ be an ordinary abelian variety over a characteristic $p$ scheme $S\_0$, and let $S\_n$ ($n\geq 0$) be compatible flat liftings of $S\_0$ over $\mathbf{Z}/p^{n+1}$. Let $F\colon S\_0\to S\_0$ be the Frobenius. Then the pull-back $(F^n)^\* A\_0$ has a canoni... | 14 | https://mathoverflow.net/users/3847 | 383013 | 159,350 |
https://mathoverflow.net/questions/382986 | 0 | I do have a real positive random vairable, which distribtuion I only known through some truncature of the orthogonal projection of it's density in a Laguerre basis, and I want to find the best way of simulating from this random variable.
Denote $\phi\_k(x) = \sqrt{2} e^{-x}\sum\limits\_{\ell \le k} \binom{k}{\ell} \f... | https://mathoverflow.net/users/143783 | Simulate from signed mixtures of erlangs? | You have $f=\sum\_{k=0}^m c\_k f\_k$ for some real $c\_k$. (So, your $f$ may take negative values and/or not integrate to $1$ on $[0,\infty)$, and thus fail to be a pdf.) However, you can write
$$0\le f^+\le h:=\sum\_{k=0}^m c\_k^+ f\_k,$$
where $u^+:=\max(0,u)$. So,
$$0\le f^+\le cg,\tag1$$
where
$$c:=\int h=\sum\_{k=... | 1 | https://mathoverflow.net/users/36721 | 383014 | 159,351 |
https://mathoverflow.net/questions/382889 | 0 | Given $\{v^i\}\_{i \in \mathbb{N}} \subseteq \mathbb{N}^n$, and $\cup\_{k=1, \ldots, m} C\_j = \mathbb{N}^n$ for some $m$, where each $C\_k$ is a cone generated by rational vectors. My question is: does there exist $i,j \in \mathbb{N}$ and $k=1, \ldots, m$, such that $v^i, v^j, v^j - v^i \in C\_k?$
For example, when ... | https://mathoverflow.net/users/129960 | Generalization of Dickson's Lemma | Yes.
The condition that the union of the $C\_k$ is $\mathbb N^n$ is not strictly necessary - all you need is that each $v^i$ is in some $C\_k$.
Indeed, there must be some $k$ such that infinitely many $v^i$ lie in $C\_k$.
By Gordan's lemma, $C\_k$ is generated (as a semigroup) by finitely many vectors in $C\_k$. ... | 2 | https://mathoverflow.net/users/18060 | 383015 | 159,352 |
https://mathoverflow.net/questions/383007 | 0 | In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave.
My questions:
1. What can we say about this in the discrete setting?. For ex: Is the CDF of a Poisson random variable log-concave?.
My 2nd question could be somewhat relevant to above.
2. If $X \s... | https://mathoverflow.net/users/165072 | CDF of a log-concave discrete random variable | Indeed, if the probability mass function of an integer-valued random variable is log concave as a function on $\mathbb Z$, then the corresponding cdf is also log concave as a function on $\mathbb Z$.
This is a special case of [Theorem 2, p. 152](https://www.google.com/books/edition/Advances_in_Stochastic_Inequalities... | 1 | https://mathoverflow.net/users/36721 | 383016 | 159,353 |
https://mathoverflow.net/questions/382971 | 1 | Let $X$ be a Banach space with an unconditional basis $(x\_{n})\_{n}$.
Question. If $X$ contains a subspace isomorphic to $l\_{1}$, does $(x\_{n})\_{n}$ admit a block basic sequence equivalent to the unit vector basis of $l\_{1}$ ?
I do not know whether the question has already existed as a known result. But a self... | https://mathoverflow.net/users/41619 | $l_{1}$-block basic sequences in Banach spaces with an unconditional basis | Yes. This result of R. C. James can be found in standard references. See, for example, Theorem 3.3.1 in the book by Albiac and Kalton.
| 1 | https://mathoverflow.net/users/2554 | 383018 | 159,354 |
https://mathoverflow.net/questions/382774 | 5 | Let $(L\_k)\_{k\geq 0}$ be the Laguerre polynomials. These polynmials are orthogonal with respect to the inner product: $$\langle f,g\rangle = \int\_0^\infty f(x)g(x)\mathrm e^{-x}\,\mathrm dx.$$
Hence, the functions $\psi\_k(x) = \sqrt{2} L\_k(2x) \mathrm{e}^{-x}$ form a basis of $\mathrm L^2(\mathbb R\_+)$ called t... | https://mathoverflow.net/users/173383 | Proving that the primitives of the Laguerre functions are uniformly bounded | OK, here is the argument. We want to show that the partial integrals of $u(x)=L\_n(x)e^{-x/2}$ are not (much) larger in the absolute value than the full integral.
We'll just use the differential equation
$$
xL\_n''+(1-x)L\_n+nL\_n=0\,.
$$
Plugging $L\_n=e^{x/2}u$, we get
$$
x(u''+u'+\tfrac 14 u)+(1-x)(u'+\tfrac 12u)+... | 3 | https://mathoverflow.net/users/1131 | 383022 | 159,356 |
https://mathoverflow.net/questions/382961 | 8 | I was reading the paper `actions of discrete groups on nonpositively curved spaces' written by Kapovich and Leeb.
In this paper, they proved that generic mapping class groups are not Hadamard groups, i.e. no discrete actions on CAT(0) spaces by semi-simple isometries.
In their proof, they said that since the unit t... | https://mathoverflow.net/users/173504 | Why does not a closed 3-manifold modelled on SL(2,R) admit a metric of nonpositive curvature? | If you read our paper a bit further, you will find that on page 348 we mention that this result is due to Eberlein and give a reference to his 1982 paper.
More precisely, he proves a more general theorem that a nonpositively curved compact Riemannian manifold whose fundamental group has nontrivial center has a finite-s... | 10 | https://mathoverflow.net/users/21684 | 383029 | 159,359 |
https://mathoverflow.net/questions/382998 | 3 | Consider the moduli of smooth curves $M\_{g,n}$ (genus $g$, $n$ marked points) and its Deligne-Mumford compactification $\overline{M}\_{g,n}$ of *stable* nodal curves (genus $g$, $n$ marked points). This is usually only defined for $2g-2+n>0$. It seems the point of this condition is that $M\_{g,n}$ has no infinite auto... | https://mathoverflow.net/users/173528 | Infinite automorphisms in the moduli of curves | A keyword to google for is *prestable curves*. They are defined just as stable curves except the conditions that ensure finite automorphism groups are removed. There is a stack $\mathfrak M\_{g,n}$ of $n$-pointed prestable curves of genus $g$. Stabilization defines a map $\mathfrak M\_{g,n} \to \overline M\_{g,n}$ when... | 3 | https://mathoverflow.net/users/1310 | 383033 | 159,361 |
https://mathoverflow.net/questions/382999 | 2 | Let $ S $ be a finite group. Denote by $\mathcal{B}\_0(S)$ the set of the subgroups $H$ of $S$ satisfying $|H:H'| > |K:K'|$ for every proper subgroup $K$ of $H$ ($H'$ denotes the drived subgroup of $H$), and let $\mathcal{K}(S)$ be the subgroup generated by the minimal elements of $\mathcal{B}\_0(S)$ (the latter being ... | https://mathoverflow.net/users/131634 | Status of a conjecture of Thompson | The conjecture is still open. The most recent paper where this Thompson's paper was mentioned was published in 2016: Rowley, Peter; Taylor, Paul An algorithm for the Thompson subgroup of a p-group. J. Algebra 461 (2016), 375–389.
There is also a more recent paper in the arXive: An extension of the
Glauberman ZJ-Theor... | 3 | https://mathoverflow.net/users/157261 | 383040 | 159,363 |
https://mathoverflow.net/questions/382815 | 7 | Motivated by a similar question [Complex-doubly periodic function in two variables?](https://mathoverflow.net/questions/382480/complex-doubly-periodic-function-in-two-variables), I would like to ask if there exists a non-zero function $(z\_1,z\_2) \mapsto f(z\_1,z\_2)$, where $z\_1,z\_2 \in \mathbb C$ are two complex v... | https://mathoverflow.net/users/150564 | Existence of complex function? | The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $\Sigma\subset\mathbb{C}^2$, and here is how one can understand them:
First, it helps to change coordinates, though, perhaps, a little more subtly than Fedor Petrov suggested: Let
$$
y\_1 = \tfrac i2({\overl... | 4 | https://mathoverflow.net/users/13972 | 383049 | 159,367 |
https://mathoverflow.net/questions/382501 | 27 | This title probably seems strange, so let me explain.
Out of the several different ways of modeling $(\infty, n)$-categories, [complicial
sets](https://arxiv.org/abs/1610.06801) and [comical sets](https://arxiv.org/abs/2005.07603) allow $n = \infty$,
providing mathematical definitions of $(\infty, \infty)$-categories... | https://mathoverflow.net/users/97265 | "Non-categorical" examples of $(\infty, \infty)$-categories | As mentioned by Simon Henry: **The $(\infty,\infty)$-category of cobordisms.**
(Not constructed, but if you did it you could presumably have any of the usual bells and whistles you might want.)
To clarify Simon Henry's comment: The statement is that that $(\infty,\infty)$-category of cobordisms in the coinductive s... | 11 | https://mathoverflow.net/users/2362 | 383060 | 159,369 |
https://mathoverflow.net/questions/383058 | 9 | $\DeclareMathOperator\GL{GL}$Let $A\in \GL\_d(\mathbb{Z})$ have finite order $n.$ Suppose that $k\in \mathbb{Z}$ is relatively prime to $n.$ Is $A^k$ conjugate to $A$ in $\GL\_d(\mathbb{Z})$?
For $d\leq 4$ the answer is yes. Indeed the papers *"On the finite subgroups of $\GL(3,\mathbb{Z})$"* by K. Tahara, 1971 and *... | https://mathoverflow.net/users/34640 | Finite order elements of $\mathrm{GL}_d(\mathbb{Z})$ that are conjugate to powers of themselves | The answer is "no" in general. There may be an elementary way of seeing this, but I will frame this in representation theoretic terms and will describe a general construction.
The question is equivalent to the following question: let $C\_n$ be a cyclic group of order $n$, let $g$ be a generator, and let $\rho\colon C... | 14 | https://mathoverflow.net/users/35416 | 383064 | 159,370 |
https://mathoverflow.net/questions/81975 | 15 | I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant derivatives (specifically regarding maps of type $ \phi\colon M\to S $ between Riemannian manifolds $ \left(M,g\right) $ and $ \l... | https://mathoverflow.net/users/19516 | Good reference for globally formulated calculus of variations on Riemannian manifolds? | I had forgotten about this question I had asked until I stumbled upon it again today. The work I mentioned is here -- <https://arxiv.org/abs/1212.2376> -- for anyone interested.
The punchline of the paper is the covariant Euler-Lagrange equation for maps between Riemannian manifolds, using a "strongly typed" global t... | 2 | https://mathoverflow.net/users/19516 | 383071 | 159,372 |
https://mathoverflow.net/questions/383055 | 4 | Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T\_xX$, of $X$ at $x\in X$, intersects $X$ in $x\in X$ with multiplicity at least $s$.
For instance, any point of $X$ in an inflection point of order $s = 2$, and the ... | https://mathoverflow.net/users/nan | Higher order inflection points | (The statements you quote are only true if you are working over a field of characteristic zero or $p > d$. I will continue to make that assumption)
The formula is not $I(3)=3d(d-2)$ but rather $\sum\_{s>2} (s-2)I(s) = 3d(d-2)$ (changing notation slightly so $I(s)$ counts the points with contact exactly $s$ instead of... | 11 | https://mathoverflow.net/users/2290 | 383072 | 159,373 |
https://mathoverflow.net/questions/383068 | 3 | All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a **semisimple** complex group. I am interested in whether the same results also apply to arbitrary **reductive** complex $G$? In particular, that one can make d... | https://mathoverflow.net/users/114985 | Schubert cells in G/P for reductive G | You already answered your question: the center of any reductive group lies in any parabolic, so if $G$ is reductive, and $G\_{\operatorname{ad}}$ its adjoint quotient (which is, of course, semi-simple), then $G/P\cong G\_{\operatorname{ad}}/P'$ (where $P'$ is the image of $P$ in $G\_{\operatorname{ad}}$).
| 8 | https://mathoverflow.net/users/66 | 383075 | 159,375 |
https://mathoverflow.net/questions/383065 | 8 | In 2009, Jochen Koenigsmann [showed](https://annals.math.princeton.edu/wp-content/uploads/Koenigsmann.pdf) that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park [proved a result](https://arxiv.org/abs/1202.6371) which implies that $\mathbb{Z}$ is $\exists\forall$-definable in ... | https://mathoverflow.net/users/5017 | What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$ | There is an existential definition of $\mathbb{Z}$ in the rational function field $\mathbb{R}(t)$ by a beautiful result of Denef using elliptic curves (Proposition 2 of *The diophantine problem for polynomial rings and fields of rational functions*, Trans. Amer. Math. Soc. **242** (1978), 391-399 doi:[10.1090/S0002-994... | 10 | https://mathoverflow.net/users/50351 | 383087 | 159,380 |
https://mathoverflow.net/questions/383037 | 2 | Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there are distinct lines $L\_1,\dots,L\_7$ such that $p\_{i,j} = L\_i\cap L\_j \in C$ for all $i,j = 1,\dots,7$.
Does anyone k... | https://mathoverflow.net/users/nan | Configuration of points on a plane curve | A set of points in the plane is called a star configuration of type $\ell$ if it is the set of pairwise intersections of some $\ell$ lines, no three concurrent. If the lines are defined by $L\_1,\dotsc,L\_\ell$, it's clear that each product $\hat{L}\_j = L\_1 \dotsm \widehat{L\_j} \dotsm L\_\ell = (\prod L\_i)/L\_j$ va... | 4 | https://mathoverflow.net/users/88133 | 383093 | 159,383 |
https://mathoverflow.net/questions/382711 | 7 | $\DeclareMathOperator{\Sym}{Sym}$For $N>0$, consider the $O\_N$-representations $V = \mathbb R^N$ and $M\_n = \ker (\Sym^n{V}\otimes\Sym^2 V\to \Sym^{n+1} V\otimes V)$ (the irreducible $GL\_n$-representation corresponding to the partition $(n+2) = n + 2$). There is an equivariant map $\rho\_n:M\_{n}\to M\_{n-1}\otimes ... | https://mathoverflow.net/users/35687 | de Rham-invariants of a Riemannian metric | $\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\Map}{Map}$After Robert Bryant's helpful comment, I was able to find a positive answer to Question\* as Theorem 1.2 in the article [Gilkey, Peter B.
Local invariants of an embedded Riemannian manifold. (English)
Ann. Math. (2) 102, 187-203 (1975).](https://www.jstor.... | 2 | https://mathoverflow.net/users/35687 | 383102 | 159,387 |
https://mathoverflow.net/questions/383105 | 2 | Given a set $S\_n$ of $n$ points $\mathbf{x}\_1, \mathbf{x}\_2, \ldots, \mathbf{x}\_n\in\mathbb{R}^d$, such that every $(d+1)$-tuple in $S\_n$ is affinely independent, and let $C(S\_n)$ be the convex hull of the points of $S$. Let now $T(S\_n)$ be the set of all triplets $\{\mathbf{x}\_i, \mathbf{x}\_j, \mathbf{x}\_k\}... | https://mathoverflow.net/users/115803 | Triangles and convex hulls in high dimensions | A $3$-[neighborly polytope](https://en.wikipedia.org/wiki/Neighborly_polytope)
is one in which every triple of vertices forms a face.
Such $k$-neighborly polytopes exist and achieve the
maximum number of $k$-faces,
by the [upper bound theorem of McMullen](https://en.wikipedia.org/wiki/Upper_bound_theorem).
So your ma... | 5 | https://mathoverflow.net/users/6094 | 383113 | 159,389 |
https://mathoverflow.net/questions/383104 | 9 | According to [this Quanta article](https://www.quantamagazine.org/a-path-less-taken-to-the-peak-of-the-math-world-20170627/) about June Huh, there exists a memoir by Heisuke Hironaka called The Joy of Learning.
It seems to be this short article:
* Heisuke Hironaka, *The joy of learning*, SEIBUTSU BUTSURI KAGAKU **4... | https://mathoverflow.net/users/173582 | Translation of "The joy of learning" by Hironaka | It seems like that there is no such translation. Moreover, the essay you linked and the book you linked are *different* material, although they have the same title.
I searched the book whose author is Heisuke Hironaka, but I cannot find any essays written by Heisuke Hironaka, except for those in Japanese. Does it mea... | 13 | https://mathoverflow.net/users/48041 | 383114 | 159,390 |
https://mathoverflow.net/questions/383107 | 5 | $\DeclareMathOperator\Mod{Mod}$Let $S$ be a closed surface and $\Mod(S)$ be its mapping class group.
It is a well known fact, proved in the Primer on Mapping class groups for example, that the subgroup of $\Mod(S)$ generated by two Dehn twists $T\_a$ and $T\_b$ depends only on the geometric intersection number $i(a,b... | https://mathoverflow.net/users/150711 | Subgroups of Mod(S) generated by Dehn twists depend only on intersection numbers? | This is not true and I don't see the "right" side-conditions to make it true. Here is an example.
Suppose that $a\_1, a\_2, a\_3$ all lie in a single handle (surface of genus one, with one boundary component). and all meet exactly once, pairwise. Then the twist about $a\_3$ lies in the group generated by the others, ... | 6 | https://mathoverflow.net/users/1650 | 383118 | 159,392 |
https://mathoverflow.net/questions/383035 | 8 | Among the families of sequences studied by Nicolaas de Bruijn (*Asymptotic Methods in Analysis*, 1958), let's focus on the (modified)
$$\hat{S}(4,n)=\frac1{n+1}\sum\_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^4.$$
An all-familiar fact states: *the Catalan number $C\_n=\frac1{n+1}\binom{2n}n$ is odd iff $n=2^m-1$*. In the same tra... | https://mathoverflow.net/users/66131 | De Bruijn's sequence is odd iff $n=2^m-1$: Part I | That's true. This follows easily from the formula
$$
\hat{S}(4,n)=\frac1{n+1}\binom{2n}n\sum\_{k=0}^n (-1)^k\binom{2n+k}k^2\binom{2n}{n+k}.\quad\quad\quad\quad\quad(\*)
$$
First of all, I prove your parity claim using $(\*)$, and next prove $(\*)$.
If $n+1$ is not a power of 2, then $C\_n=\frac1{n+1}\binom{2n}n$ is e... | 9 | https://mathoverflow.net/users/4312 | 383119 | 159,393 |
https://mathoverflow.net/questions/383126 | 2 | Does there exist a monotone function $f: [0, 1] \to \mathbb R$ that is differentiable everywhere, but its derivative is discontinuous a.e.?
| https://mathoverflow.net/users/173490 | Monotone differentiable function whose derivative is discontinuous on a full measure set | I think you want the [Pompeiu function](https://en.wikipedia.org/wiki/Pompeiu_derivative) (Feel free to add questions on the details of the construction).
| 3 | https://mathoverflow.net/users/6101 | 383131 | 159,398 |
https://mathoverflow.net/questions/383112 | 2 | Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $GL\_n(\mathcal{O})$ denote the group of invertible $n\times n$ matrices with entries in $\mathcal{O}$ with the inverse having entries in $\mathcal{O}$.
Consider the natural (unitary) representation of $GL\_2(\mathcal{O})$ in fun... | https://mathoverflow.net/users/16183 | Representation of $GL_2(\mathcal{O})$ in space of functions on projective line | Let $F \mathbb{P}^{n - 1}$ denote $(n - 1)$-dimensional $F$-projective space, which we may identify with $\mathrm{P}\_{(n - 1,1)}(\mathcal{O}) \backslash \mathrm{GL}\_n(\mathcal{O})$, where
$$\mathrm{P}\_{(n - 1,1)}(\mathcal{O}) :=\left\{\begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in \mathrm{GL}\_n(\mathcal{O}) : a \... | 2 | https://mathoverflow.net/users/3803 | 383146 | 159,403 |
https://mathoverflow.net/questions/383142 | 6 | Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p\_1}{p\_2}$ such that $p\_1,p\_2$ are positive co-prime integers minimizing $p\_2 \cdot |p\_2\alpha - p\_1|$? Clearly, there are going to be some irrational numbers for which this makes ... | https://mathoverflow.net/users/140356 | Algebraic and rational parts of a real number | Let $\alpha$ be an irrational. We shall consider its continued fraction $[a\_0;a\_1,a\_2,\dots]$. Recall some basic results about convergents of continued fractions (see e.g. [here](https://en.wikipedia.org/wiki/Continued_fraction#Some_useful_theorems)): letting $p\_n,q\_n$ be the sequence of numerators and denominator... | 10 | https://mathoverflow.net/users/30186 | 383147 | 159,404 |
https://mathoverflow.net/questions/383115 | 18 |
>
> Does there exist a finite group $G$ of order greater than two containing a unique element $g$ such that
> $$
> g\notin\langle x\rangle
> \hbox{ for all $x\in G\setminus\{g\}$ ?}
> $$
>
>
>
Or we have another [fantastic property of the order-two group](https://mathoverflow.net/q/148925/24165)?
Clearly, suc... | https://mathoverflow.net/users/24165 | Groups with a unique lonely element | I think that the nontrivial semidirect product of a cyclic group of order 4 $\langle x\rangle$ acting on another cyclic group of order 4 $\langle y\rangle$ is an example of such a group. The center of this group is $\langle x^2,y^2\rangle$ and $x^2y^2$ is not a square.
I came up with this trying to prove that no such... | 23 | https://mathoverflow.net/users/173071 | 383148 | 159,405 |
https://mathoverflow.net/questions/383080 | 2 | Let $X\_1, \cdots, X\_n \sim \mathrm{Unif}[0,1]$ be $n$ random variables, each with marginal distribution being a standard uniform distribution. I want to characterize the set of covariance matrices (or correlation matrix if it is easier) that can be attained by these $n$ variables. Is there a simple characterization? ... | https://mathoverflow.net/users/97310 | Covariance/Correlation matrix of $n$ random variables with uniform marginal distributions | Interesting question. I am not an expert but quoting
Admissible Bernoulli correlations, by Huber and Marić
*Journal of Statistical Distributions and Applications* (2016):
<https://jsdajournal.springeropen.com/articles/10.1186/s40488-019-0091-5>
>
> Correlation matrices are symmetric positive semi-definite and h... | 1 | https://mathoverflow.net/users/17773 | 383153 | 159,407 |
https://mathoverflow.net/questions/383121 | 7 | Does there exist a finitely presented (preferably $\text{FP}\_{\infty}$) group $\Gamma$ and an element $\alpha \in \text{H}^{\ast>0}(B\Gamma;\mathbf{Q})$ that is not nilpotent?
If non-discrete groups were allowed, the Euler class $e \in \text{H}^2(BS^1;\mathbf{Q})$ would do the trick, and there are corresponding clas... | https://mathoverflow.net/users/14233 | A finitely presented group whose rational cohomology is not nilpotent | Let me compile the comments into an official answer: yes, such a group exists. As @dodd predicted in a comment, Thompson's group $F$ does the trick. Brown's computation of the cohomology ring ([http://pi.math.cornell.edu/~kbrown/papers/homology.pdf](http://pi.math.cornell.edu/%7Ekbrown/papers/homology.pdf)) reveals non... | 7 | https://mathoverflow.net/users/164670 | 383160 | 159,410 |
https://mathoverflow.net/questions/383178 | 8 | I am trying to understand constructions of exceptional groups of type $G\_2$ (over rings). In this post, by a model (of type $G\_2$) I mean an affine smooth group scheme over $\mathbb{Z}$ such that the fibres are connected simple algebraic groups of type $G\_2$.
In Gross' paper *[Groups over Z](https://link.springer.... | https://mathoverflow.net/users/56217 | Does $G_2(\mathbb{Z})$ depend on the choice of an integral model? | The short answer is no, because a Chevalley group has infinitely many $\mathbb{Z}$-points (even Zariski dense by the Borel density theorem).
For the long answer, let me first completely describe all $\mathbb{Q}$-models and $\mathbb{Z}$-models of groups of type $G\_2$. Let $G\_0/\mathbb{Q}$ be the split reductive grou... | 12 | https://mathoverflow.net/users/110362 | 383185 | 159,414 |
https://mathoverflow.net/questions/383167 | 12 | One possible approach to constructive field theory is to define it on a lattice and take the scaling limit, and there are famous results stating that in $d\geq4$ this cannot lead to a non-trivial theory.
What is the status of approaches using the Gaussian free field?
What I mean is this: let $\Omega\subseteq\mathbb... | https://mathoverflow.net/users/68927 | Mathematical construction of $\phi^4$ Euclidean field theory | When $d=2$, this works fine and this is precisely how Nelson originally constructed the $\Phi^4$ measure (in finite volume). Already for $d = 3$, the $\Phi^4$ measure is singular with respect to the free field, even in finite volume, so this approach is bound to fail. The reason why it is singular is subtle, but you ca... | 11 | https://mathoverflow.net/users/38566 | 383187 | 159,415 |
https://mathoverflow.net/questions/382902 | 8 | This is a cross post from [MSE](https://math.stackexchange.com/questions/4004859/characterization-of-pretty-compact-spaces).
I believe that the following problem have already been considered by some sophisticated topologist.
**Definition 1.** A non-compact Hausdorff topological space $X$ is called *almost compact* ... | https://mathoverflow.net/users/19593 | Characterization of pretty compact spaces | A partial answer: other examples of pretty compact spaces are uncountable powers of $\{0,1\}$ and $[0,1]$, and in general products of uncountably many non-trivial compact Hausdorff spaces. See Problem 3.12.24(c) in Engelking's *General Topology*, or [Glicksberg, *Stone-Čech compactifications of products*](https://doi.o... | 10 | https://mathoverflow.net/users/5903 | 383199 | 159,419 |
https://mathoverflow.net/questions/383186 | 8 | I was looking at [this question](https://mathoverflow.net/questions/57099/why-do-filtered-colimits-commute-with-finite-limits) about a "soft proof" of the fact that finite limits (shape $I$) commute with filtered colimits (shape $J$) in **Set**, using only the fact that the diagonal $J \to J^I$ is final.
If we consid... | https://mathoverflow.net/users/4080 | Commutation of limits and colimits: Is there a choice diagram? | This isn't true in general. Take $I = BG$ and $J = BH$ to be one-object groupoids, so that $A(i, j)$ becomes a set $A$ with commuting actions of $G$ and $H$. The left hand side is obtained by taking the $H$-orbits of $A$, and then the $G$-fixed points of the result. The right hand side isn't as easy to describe precise... | 5 | https://mathoverflow.net/users/126667 | 383200 | 159,420 |
https://mathoverflow.net/questions/383170 | 3 | Let $X\_n, X$ be $[0, 1]$-valued random variables whose laws are absolutely continuous with respect to Lebesgue measure. Suppose $X\_n \to X$ a.s. Does this imply that the pdfs of $X\_n$ converge to that of $X$ in some suitable sense?
For concreteness, the three I have in mind are convergence a.e., convergence in $L^... | https://mathoverflow.net/users/173490 | Almost sure convergence vs convergence of probability density functions | The answer is no. Indeed, for natural $n$ let
$$p\_n(x):=1+\tfrac12\,\text{sign}\,\sin(2\pi nx)$$
if $x\in[0,1]$, with $p\_n(x):=0$ if $x\notin[0,1]$. Then $p\_n$ is a pdf, with the corresponding cdf $F\_n$, so that
$$F\_n(x)=\int\_{-\infty}^x p\_n(t)\,dt$$
for all real $x$. The cdf $F\_n$ is continuously and strictly ... | 6 | https://mathoverflow.net/users/36721 | 383202 | 159,421 |
https://mathoverflow.net/questions/383165 | 2 | The essential spectrum of a bounded linear operator $A$ on a separable Hilbert space $\mathcal{H}$ is defined as $$ \sigma\_{\mathrm{ess}}(A) \equiv\left\{z\in\mathbb{C}\left.\right|A-z\mathbb{1}\text{ is not a Fredholm operator}\right\} \tag{1}\,. $$
This way to define the essential spectrum immediately implies that... | https://mathoverflow.net/users/68927 | Characterization of absolutely-continuous spectrum | You wrote "normal" earlier, but I assume (from your use and description of $A\_{ac}$) that you want to consider self-adjoint $A$. Then your set (denote it by $T$) is simply $T=\sigma\_{ess}(A)$ again.
If $A-z$ isn't invertible modulo compacts, then it won't be invertible modulo trace class, so $\sigma\_{ess}(A)\subse... | 2 | https://mathoverflow.net/users/48839 | 383215 | 159,428 |
https://mathoverflow.net/questions/383217 | 3 | A finite lattice is [geometric](https://en.wikipedia.org/wiki/Geometric_lattice) if it is [semimodular](https://en.wikipedia.org/wiki/Semimodular_lattice) and [atomistic](https://en.wikipedia.org/wiki/Atom_(order_theory)). Geometric lattices can have arbitrarily high rank $r$, as evidenced by the Boolean lattice $B\_r$... | https://mathoverflow.net/users/171662 | Does every geometric lattice of rank $r$ contain the Boolean $B_r$ as a sublattice? | According to [Wikipedia](https://en.wikipedia.org/wiki/Geometric_lattice), "geometric lattice" is equivalent to "lattice of flats of a finite matroid". If this is true, the answer is yes.
Let $M$ be a matroid of rank $r$, and choose a basis $(e\_1, e\_2, \ldots, e\_r)$. Then the flats spanned by the subsets of this b... | 4 | https://mathoverflow.net/users/297 | 383218 | 159,429 |
https://mathoverflow.net/questions/382807 | 4 | In the work of [Connes and Marcolli](http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2006/1.pdf), on page 20, it state that:
>
> Just as in the classical case of a (commutative) manifold, what ensures that the derivations
> considered are enough to span the whole tangent space is the condition of ellipticit... | https://mathoverflow.net/users/172458 | Why Der($A_{\theta}$) is spanned by two elements only? | [Bratteli–Elliott–Jorgensen](https://eudml.org/doc/152592) prove a range of classification results for unbounded derivations on a totally irrational noncommutative torus $C(\mathbb{T}^n\_\theta)$, which basically say that any reasonable $\ast$-derivation will be the sum of a $\mathbb{R}$-linear combination of the infin... | 3 | https://mathoverflow.net/users/6999 | 383224 | 159,431 |
https://mathoverflow.net/questions/383192 | 21 | What sorts of mathematical statements are predicted by the AdS/CFT correspondence?
My "understanding" (term used very loosely) is that this correspondence isn't a mathematically rigorous statement, but I can still imagine that certain baby cases might spit out interesting well-defined identities or relationships. I'm... | https://mathoverflow.net/users/76409 | Mathematical predictions of AdS/CFT | Although it might seem futile, given how far most of the activity on AdS/CFT is from rigorous mathematics, I think this a good question, provided one is happy, for now, with (very) baby versions of this correspondence.
An example of nontrivial mathematical prediction, using a baby version of AdS/CFT (the [Caffarelli-... | 14 | https://mathoverflow.net/users/7410 | 383226 | 159,432 |
https://mathoverflow.net/questions/383172 | 2 | To edify my understanding of fiber bundles with structure groups, I was currently trying to reconcile two classifications (in a particular case). For simplicity, I'm taking the base to be $S^1$ and the group $G$ to be discrete. Initially, I was getting a different answer from each. **Update:** The mistake there has bee... | https://mathoverflow.net/users/147463 | Principal G-bundles over the circle | Okay, I figured out my mistake! I was trying to use left-actions everywhere, but we need some left and some right. This comes up when constructed associated bundles: if you have a left $G$-space $F$, then you put it together with a right principal $G$-bundle $E\rightarrow X$ to get the total space $E\times\_G F$ of the... | 3 | https://mathoverflow.net/users/147463 | 383236 | 159,436 |
https://mathoverflow.net/questions/383247 | 0 | How to verify if a linear system of symmetrical matrix blocks has solution?
I have the matrix:
* $\left[M\right]\_{p \times p}$, symmetrical
* $\left[G\right]\_{p \times q}$
and then, I would like to solve the following linear system:
$$
\underbrace{\begin{bmatrix}
\left[M\right] & \left[G\right] \\
\left[G^T\rig... | https://mathoverflow.net/users/173662 | Conditions to solve linear system with matrix blocks | You find various conditions in Section 3 of the [classical review paper](http://page.math.tu-berlin.de/%7Eliesen/Publicat/BenGolLie05.pdf) by Benzi-Golub-Liesen on this kind of problems, which are known as *saddle-point problems*.
| 3 | https://mathoverflow.net/users/1898 | 383249 | 159,441 |
https://mathoverflow.net/questions/383244 | 1 | $\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M, \N$ be smooth two-dimensional Riemannian manifolds.
>
> Are there any local obstructions for the existence of a smooth map $f:\M \to \N$ with constant **distinct** singular values?
>
>
>
(The singular values of $df$ can be defined to be t... | https://mathoverflow.net/users/46290 | Local obstructions for maps with constant singular values | There are no local obstructions for constant distinct singular values when $M$ and $N$ have dimension $2$. This is locally a determined symmetric hyperbolic system of two equations for two unknowns, so it's always locally solvable.
**Note 1: Local character of the equations**
Giving a general analysis in all dimens... | 5 | https://mathoverflow.net/users/13972 | 383251 | 159,442 |
https://mathoverflow.net/questions/383252 | 2 | Suppose $A=(a\_{jk})\_{j,k=1}^n$ is a symmetric complex valued matrix, that is to say, $a\_{jk}=a\_{kj}$ for all $j,k=1,\dotsc,n$. Suppose that given any two **linearly independent** vectors $\alpha=(\alpha^j)\_{j=1}^n, \beta=(\beta^j)\_{j=1}^n \in \mathbb C^n$ that satisfy $$\sum\_{j=1}^n(\alpha^j)^2=\sum\_{j=1}^n(\be... | https://mathoverflow.net/users/50438 | A matrix identity | For $n = 1$, the conclusion that $A = 0$ trivially fails, as noted in the [comments](https://mathoverflow.net/questions/383252/a-matrix-identity#comment974839_383252).
For $n = 2$, the conclusion that $A = 0$ seems also to fail. We have that any vector $(v\_1, v\_2)$ of ‘norm’ $v\_1^2 + v\_2^2 = 0$ is a multiple of $... | 1 | https://mathoverflow.net/users/2383 | 383253 | 159,443 |
https://mathoverflow.net/questions/383174 | 24 | $\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Abs[1]{\left\lvert#1\right\rvert}$**Question**: Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Does the following inequality hold? $$\int\_{0}^{1}\int\_{0}^{1}\dotsi\int\_0^1\int\_0^1\abs{f(x\_{1})+f(x\_{2})+\dotsb+f(x\_{n})}dx\_1 \; dx\_{... | https://mathoverflow.net/users/38620 | Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$? | Here is the proof using the alternative route.
Let $X$, $Y$ be two independent real-valued random variables such that $EX,EY\ge 0$ and $\min(E|X|,E|Y|)=I$. We want to prove that $E|X+Y|\ge I$. Again, as in both the OP and Iosif's post, we can consider only the case when $X$ is $A$ with probability $P$ and $-B$ with p... | 20 | https://mathoverflow.net/users/1131 | 383254 | 159,444 |
https://mathoverflow.net/questions/383273 | 5 | How to prove the following continued fraction of $e^{x/y}$
$${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}}$$
Since $a\_i \geq b\_i$ for all $i \geq 1$. By the condition of irrationality of generalized continued fraction, ... | https://mathoverflow.net/users/172447 | Irrationality of $e^{x/y}$ | I think this might be a solution.
The Continued Fraction Expansion of the hyperbolic tanh function discovered by Gauss is
$$\tanh z = \frac{z}{1 + \frac{z^2}{3 + \frac{z^2}{5 + \frac{z^2}{...}}}} \\\\$$
We also know that the hyperbolic tanh function is related to the exponential function with the following formul... | 7 | https://mathoverflow.net/users/172447 | 383280 | 159,450 |
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