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https://mathoverflow.net/questions/410044 | 8 | I am wondering if there is any literature on general formula of the Moebius function of subgroup lattices of any finite abelian group $G$? What I know is
When $G$ is cyclic, the Moebius function is simply the classical number theoretic one.
When $G=(\mathbb Z/p\mathbb Z)^r$, the formula involves the number of $k$-d... | https://mathoverflow.net/users/175127 | Moebius function of finite abelian groups | Since every interval of the subgroup lattice $\mathcal{L}(G)$ of a
finite abelian group $G$ is isomorphic to the subgroup lattice of some
finite abelian group, we can restrict ourselves to
$\mu(\hat{0},\hat{1})$, where $\hat{0}$ is the bottom element (the
trivial subgroup of $G$) and $\hat{1}$ is the top element (the g... | 20 | https://mathoverflow.net/users/2807 | 410062 | 167,799 |
https://mathoverflow.net/questions/409898 | 6 | Apologies if this is not quite at the level of MathOverflow, but it has already been asked at [MSE](https://math.stackexchange.com/questions/1400501/question-about-hartshornes-proof-of-halphens-theorem) and gone unresolved for several years despite a bounty.
Hartshorne states the theorem as follows:
>
> **Proposi... | https://mathoverflow.net/users/169571 | Hartshorne's proof of Halphen's theorem | I think this Hartshorne's explanation is very rough. By using Picard variety, this step can be proved more clearly as follows.
Write $\mathrm{Pic}^d(X)$ for the scheme which parametrized all line bundles of degree $d$ on $X$, $\mathrm{Div}^d(X)$ for the scheme which parametrized all effective divisors of degree $d$ o... | 3 | https://mathoverflow.net/users/442153 | 410071 | 167,800 |
https://mathoverflow.net/questions/410032 | 4 | Who was the first to consider that categories were semi-simplicial sets (and in particular groupoids were simplicial sets)?
I think there was a concept of nerve of a covering in algebraic topology before (maybe Alexandroff).
| https://mathoverflow.net/users/429204 | Who introduced nerves in category theory? | In Peter Johnstone's 1977 "Topos theory" (p.48) the simplicial description of categories is attributed to Grothendieck and he cites the "Technique de la descente"-series of Bourbaki seminars 1959-62 for it. I guess what he has in mind is in particular prop.4.1 on page 108 of the third installment [Préschémas quotients]... | 11 | https://mathoverflow.net/users/470544 | 410076 | 167,803 |
https://mathoverflow.net/questions/409912 | 15 | Let $S$ be a closed orientable surface of genus at least $2$. I'm interested in the connected components of $\operatorname{Hom}(\pi\_1(S),\operatorname{SL}\_n(\mathbb{R}))$ for $n$ at least $3$.
I know that for $n$ odd there is only $3$ connected components. My first question is:
1. For $n$ even, do we still have $3$... | https://mathoverflow.net/users/197544 | What do the components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ look like? | $\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\R{\mathbb{R}}$Just to set the notation, as in the OP's question, let $S$ be a closed orientable surface of genus $g$ at least 2, a... | 5 | https://mathoverflow.net/users/12218 | 410077 | 167,804 |
https://mathoverflow.net/questions/353599 | 7 | Recently, I proved the following Lipschitz-continuity like result for convex polytopes:
>
> Let $A\in\mathbb R^{m\times n}$ and $b,b'\in\mathbb R^m$ be given such that $\{x\,:\,Ax\leq 0\}=\{0\}$ (which is equivalent to boundedness of all induced polytopes) and that $\{x\in\mathbb R^n\,:\,Ax\leq b\},\{x\in\mathbb R^... | https://mathoverflow.net/users/116991 | Lipschitz-continuity of convex polytopes under the Hausdorff metric | This is a classic question in the literature on linear programming, since it is related to the stability of the feasible set (and hence the solutions) under perturbation of the parameters.
The classic work in this field is:
* A. J. Hoffman, Approximate solutions of systems of linear inequalities, *J. Res. Nat. Bur.... | 3 | https://mathoverflow.net/users/76565 | 410080 | 167,805 |
https://mathoverflow.net/questions/410115 | 5 | **Problem**: Given three positive integers $0 < n\_1 < n\_2 < n\_3$. Is there always a real number $x$ such that
$$\cos n\_1 x + \cos n\_2 x + \cos n\_3 x < -2?$$
| https://mathoverflow.net/users/141801 | Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$? | The answer is negative. If $n\_3=n\_1+n\_2$, then
$$\cos n\_1 x + \cos n\_2 x + \cos n\_3 x\geq -2.$$
Indeed, the left-hand side equals
$$(1+\cos n\_1 x)(1+\cos n\_2 x)-1-\sin n\_1 x\sin n\_2 x,$$
where $1+\cos n\_j x\geq 0$, and $\sin n\_1 x\sin n\_2 x\leq 1$.
Furthermore, if $n\_2=2 n\_1$ and $n\_3=3 n\_1$, then
$$\m... | 17 | https://mathoverflow.net/users/11919 | 410117 | 167,819 |
https://mathoverflow.net/questions/410123 | 11 | By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.
Is this result known to fail for nonseparable spaces? That is, is there a known example of two (necessarily nonseparable) Banach spaces $X,Y$ such that $X$ em... | https://mathoverflow.net/users/470606 | What is an example of two Banach spaces $X,Y$ such that $X$ embeds isometrically but not linearly into $Y$? | Yes, if $H$ is a nonseparable Hilbert space then it embeds isometrically into the Arens-Eells space ${\rm AE}(H)$, but not linearly isometrically, or even linearly homeomorphically. See Theorem 5.21 of my book *Lipschitz Algebras* (second edition).
As I explain in the notes to that chapter, a more general version of ... | 13 | https://mathoverflow.net/users/23141 | 410137 | 167,823 |
https://mathoverflow.net/questions/410135 | 0 | This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with boudary condition $\left.u\right|\_{\partial U}=g$ where $g$ is smooth
under the assumption that $\frac{\partial f... | https://mathoverflow.net/users/469129 | A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III | It is much simpler than Arzela-Ascoli.
Since you have the uniform bound
$$ \sup\_U |u\_j| \leq \sup\_U 2|\Phi| $$
For simplicity I will assume your $K\_j = j$. Take $J = \sup\_U 2|\Phi|$, then for every $j,k \geq J$ you have that
$$ f\_j(x,u\_j) = f\_k(x,u\_j) = f(x,u\_j).$$
This tells you that for every $j,k \geq J$... | 4 | https://mathoverflow.net/users/3948 | 410144 | 167,826 |
https://mathoverflow.net/questions/410089 | 5 | I have been reading ['Improved bounds for the sunflower lemma'](https://annals.math.princeton.edu/2021/194-3/p05) by Alweiss, Lovett, Wu and Zhang (Ann. of Math., Vol. 194(3), 2021), and have some gaps in my understanding of the paper. They are as follows:
1. In Lemma: 2.8 (pg. 802), the authors bound the number of b... | https://mathoverflow.net/users/nan | Questions on 'Improved bounds for the sunflower lemma' | Let me try to answer your questions.
1. We can recover the pair $(W, S\_i)$ from the four quantities you mentioned. So the number of their combinations upper bounds the number of pairs $(W, S\_{i})$, which is why we multiply the number of options for each one.
2. Indeed, we switch from a set system with not necessari... | 6 | https://mathoverflow.net/users/20709 | 410160 | 167,833 |
https://mathoverflow.net/questions/410125 | 2 | Say we have the line segment $L(t) = [0,t]$, and randomly remove open intervals of length $1$ from $L(t)$ until no more open intervals of length $1$ remain. Define $u(t)$ as the expected measure of what remains.
Note $u(t) = t$ for $0 \le t < 1$ and $u(t) = t-1$ for $1 \le t < 2$. In general, we have the integral equ... | https://mathoverflow.net/users/795 | Asymptotics of a delay differential equation | Let us show that
\begin{equation\*}
u(t)\ge c(1+t)\text{ for some real $c>1/4$ and all $t\ge4$. }\tag{1}
\end{equation\*}
Let
\begin{equation\*}
u\_n(z):=u(n+z),\quad w\_n(z):=u\_n(z)-\tfrac14\,(1+n+z);
\end{equation\*}
here and in what follows, $z\in[0,1]$ and $n=0,1,\dots$.
Then for $n\ge1$
\begin{equation\*}
w\_... | 2 | https://mathoverflow.net/users/36721 | 410162 | 167,834 |
https://mathoverflow.net/questions/410168 | 2 | Lets consider two views of zeta functions of curves.
For the following, let $\mathbb{F}\_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}\_p}$ be the algebraic closure of $\mathbb{F}\_p$. Let $X$ be a smooth curve over $\text{Spec}(\mathbb{F}\_p)$.
**View 1:** Using a Weil cohomol... | https://mathoverflow.net/users/30211 | Connecting two pictures of the Zeta function | The quickest answer I can give is that the Lefschetz formula gives the identity $$\prod\_{i }\det (1 - ft, H^i(X))^{ (-1)^{i+1}} = \prod\_{v \in |X| } \frac{1}{1 - t^{\deg v}} $$
and then the product on the right side can be expanded out into a sum over the divisors $D$ on $X$ of the function $t^{\deg D}$ (i.e. a chara... | 4 | https://mathoverflow.net/users/18060 | 410172 | 167,836 |
https://mathoverflow.net/questions/410167 | 1 | Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z\_{ij} \sim \text{Bernoulli}(p) \implies X\_{ij} = \frac{Z\_{ij}}{\sum\_{k=1}^m Z\_{ik}}.$$
Is there a name for the distribution of $X$? Is there a closed-form for $E[XX^\top]$? I am struggling to find any inf... | https://mathoverflow.net/users/128729 | Moments of rescaled Bernoulli random matrix | It is apparently assumed that the $Z\_{ij}$'s are independent, as we will do here -- since otherwise hardly anything can be said. Suppose also that $m\ge2$ and $0<p<1$.
The $ab$-entry of the matrix $Y:=XX^\top$ is
\begin{equation}
Y\_{ab}=\sum\_{r\in[m]}X\_{ar}X\_{br}
=\sum\_{r\in[m]}\frac{Z\_{ar}Z\_{br}}{\sum\_{k\... | 2 | https://mathoverflow.net/users/36721 | 410178 | 167,838 |
https://mathoverflow.net/questions/368988 | 3 | Let $X\_m$ denote a set of $m\geq 3$ lines in $\mathbb{R}^2$ that are not all parallel. Consider the problem of determining a closed path of $kn$ points in $X\_m$ $k, n \in \mathbb{Z}^+$, such that the "forward orbit" of the path has angles of incidence with the lines in $X\_m$ which are strictly contained in a given s... | https://mathoverflow.net/users/129192 | Finding particular closed paths in geometric plane regions | The answer, given in this paper: <https://arxiv.org/abs/2112.02207>, turns out to be "yes", there always exist such closed curves. The theorem stating the answer to this question is given in the introduction to the linked paper, and I restate it here for reference:
***Theorem:*** *For any space $X\_m$ with labeled li... | 0 | https://mathoverflow.net/users/129192 | 410189 | 167,842 |
https://mathoverflow.net/questions/410140 | 2 | If $H= (V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph), a *matching* is a set $M\subseteq E$ such that $e\_1\cap e\_2 = \emptyset$ whenever $e\_1\neq e\_2 \in M$. The *matching number* $\mu(H)$ of a hypergraph $H=(V,E)$ with $V$ finite is the maximum number of elements a matching can have.
For infi... | https://mathoverflow.net/users/8628 | Matching number in infinite hypergraphs | Let $(P,\leq)$ be a poset, let $H$ be the set of maximal chains in $P$ —- so $H$ will be the set of vertices. For $a\in P$, let $E\_a$ denote the set of all chains in $H$ containing $a$; these sets are the edges.
Now, $E\_a$ and $E\_b$ are disjoint iff $a$ and $b$ are incomparable. So the edges are pairwise disjoint ... | 3 | https://mathoverflow.net/users/17581 | 410197 | 167,844 |
https://mathoverflow.net/questions/410122 | 5 | In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a subspace $V$ of $T\_{p}Q$ such that $[u, [v, w]] \in V$ for all $u,v,w \in V$; see for example Theorem 7.2 in Helgason, *D... | https://mathoverflow.net/users/74033 | Proof of equivalence between Lie triple systems and totally geodesic submanifolds | Try pages 71 and 72 of Cartan's "La géométrie des groupes de transformations" <https://eudml.org/doc/235668>
| 1 | https://mathoverflow.net/users/21123 | 410199 | 167,845 |
https://mathoverflow.net/questions/410186 | 14 | I apologize in advance if this is well-known, but I can't seem to find the answer in the literature. Let me be precise about my question. I am looking for concrete examples of locally compact Hausdorff groups $G$ such that that there exists $a, b \in G$ with $a \ne b$, but for any continuous representation $\pi : G \to... | https://mathoverflow.net/users/128540 | Examples of locally compact groups that do not admit enough finite dimensional representations | There is an example which satisfies something much stronger: there exist nontrivial groups $G$ such that *any* homomorphism (not even necessarily continuous) $\pi:G\to GL\_n(k)$ for *any* field $k$ (and, in fact, any commutative integral domain) is trivial, so in particular for *any* $a,b\in G$ we have $\pi(a)=\pi(b)$.... | 12 | https://mathoverflow.net/users/30186 | 410205 | 167,847 |
https://mathoverflow.net/questions/410204 | 5 | I have a stupid question about a topology on $C\_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps.
**Definition (topology on $C\_c(X)$):** For each compact $K \subset X$, $C\_K(X)$ is the set of functions in $C\_c(X)$ with support in $K$. $C\_K(X)$ is given a Banach space structur... | https://mathoverflow.net/users/105628 | Product of inductive limit topologies on $C_c(X)\times C_c(X)$ | The topology $\tau$ you describe makes $(C\_c(X),\tau)$ the *colimit (or inductive or direct limit) in the category LCS of locally convex spaces* of the system $C\_K(X)$ with inclusions $i\_{K,L}:C\_K(X) \hookrightarrow C\_L(X)$ for compact subsets $K\subseteq L$ of $X$, i.e., the inclusions $i\_K:C\_K(X)\to C\_c(X)$ h... | 6 | https://mathoverflow.net/users/21051 | 410207 | 167,848 |
https://mathoverflow.net/questions/410176 | 5 | *Previously [asked and bountied](https://math.stackexchange.com/questions/4318032/does-the-absolute-fragment-of-second-order-logic-satisfy-a-strong-lowenheim-skol) at MSE:*
Let $\mathsf{SOL\_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such tha... | https://mathoverflow.net/users/8133 | Can the forcing-absolute fragment of SOL have a strong Lowenheim-Skolem property? | Yes, and a proper class of Woodin cardinals suffices. For $n<\omega$ and $X$ a set of ordinals, $M\_n(X)$ denotes the minimal iterable proper class model $M$ of ZFC with $n$ Woodin cardinals above $\mathrm{rank}(X)$ and $X\in M$.
Because we have a proper class of Woodin cardinals, $M\_n(X)$ exists for every set $X$ of ... | 8 | https://mathoverflow.net/users/160347 | 410213 | 167,850 |
https://mathoverflow.net/questions/408965 | 1 | Let $Q$ be a quiver of type $ADE$, $I$ is the set of vertices of $Q$. Let $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ be a Nakajima quiver variety for such quiver (here ${\mathbf{v}}=(v\_i)\_{i \in I}$ is the dimension vector and ${\mathbf{w}}=(w\_i)\_{i \in I}$ is framing). We can associate to $\mathfrak{M}({\mathbf{v}}... | https://mathoverflow.net/users/144206 | Nakajima quiver varieties for ADE quiver with one dimensional framing | We, at least, have the explicit answer for their Betti numbers. For an example, let $Q$ be type $E\_8$, and $k$ be the triplet vertex. Take $\mu = 0$. The normalized Poincare polynomial of $\mathfrak M(\mathbf v,\mathbf w)$ was computed in <https://arxiv.org/pdf/math/0606637.pdf> as $$1357104 + 2232771t^2 + 2002423t^4 ... | 4 | https://mathoverflow.net/users/3837 | 410215 | 167,851 |
https://mathoverflow.net/questions/410047 | 3 | In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous arithmetic function of the logarithm function which is of the form $$\beta(n)=\sum\_{p\_{i}^{\alpha\_{i}}\mid n}\alpha\_{i}\cd... | https://mathoverflow.net/users/164499 | Literature on analogous arithmetic function of logarithm function | Your function $\beta$ is a completely additive function, in the sense that $\beta(nm)=\beta(n)+\beta(m)$ for all $n,m$.
There is a vast literature on the statistical behavior of additive functions, e.g. Erdös and Kac's 1940 paper and various textbooks: Kubilius' book "Probabilistic methods in the theory of numbers", ... | 3 | https://mathoverflow.net/users/31469 | 410221 | 167,853 |
https://mathoverflow.net/questions/410209 | 20 |
>
> **Problem**: Given three positive integers $0 < n\_1 < n\_2 < n\_3$ such that
> $$n\_1 + n\_2 \ne n\_3, \quad n\_2 \ne 2n\_1, \quad n\_3 \ne 2n\_1, \quad n\_3 \ne 2n\_2,$$
> is there always a real number $x$ such that
> $$\cos n\_1 x + \cos n\_2 x + \cos n\_3 x < -2?$$
>
>
>
This is a follow-up of the questi... | https://mathoverflow.net/users/141801 | (update) Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$? | In principle this problem can be resolved numerically in finite time, by exploiting the dichotomy between structure (small linear relations between the frequencies $n\_1,n\_2,n\_3$) and randomness (equidistribution), though I do not know if the approach below can actually be implemented in a feasible amount of time. (O... | 26 | https://mathoverflow.net/users/766 | 410227 | 167,854 |
https://mathoverflow.net/questions/410222 | 8 | The classification of oriented compact smooth manifolds up to oriented cobordism is one
of the landmarks of 20th century topology. The techniques used there form the part of the foundations of differential topology and stable homotopy theory.
It is a popular knowledge to find the oriented bordism groups $\Omega\_d^{S... | https://mathoverflow.net/users/27004 | Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$) | Most of the main results needed for this calculation can be found in Wall's paper "[Determination of the oriented cobordism ring](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/cobord.pdf)", but [this note](http://math.uchicago.edu/%7Emay/REU2016/REUPapers/Gwynne.pdf) by Gwynne might be helpful to express this in more m... | 15 | https://mathoverflow.net/users/360 | 410228 | 167,855 |
https://mathoverflow.net/questions/410225 | 1 | I have heard about the Schottky problem and the related Novikov's conjecture about the characterization of matrices in the Siegel upper half-space which are indeed the Riemann matrix of a compact Riemann surface.
Instead of such a global statement, I was wondering about the following: given $\Omega$ a Riemann matrix ... | https://mathoverflow.net/users/201087 | operations on matrices preserving the property of being the Riemann matrix of a surface | No, this is not true. If $C$ is a general curve of genus $\geq 4$ with period matrix $\Omega$ and $p$ is a prime, $p\Omega$ is not the period matrix of a curve. This is proved (in an equivalent, more geometric, form) by Donagi and Livné, Ann. Sc. Norm. Sup. Pisa 28, no. 2 (1999), p. 323-339 — see §7.
| 1 | https://mathoverflow.net/users/40297 | 410231 | 167,856 |
https://mathoverflow.net/questions/410223 | 4 | Let $m$ be a positive integer divisible by $6$, and let $q$ be one of $8,9$,
or a prime $\gt 3$.
**Question :** Is there always an $x\in [1,m]$, coprime to $m$, such
that $x\not\equiv\pm 1 \ \mod{q}$ ?
The main difficulty in that problem I think is that it combines two very different requirements, the congruence (a... | https://mathoverflow.net/users/10341 | System of congruences with bound condition | Yes. Denote $m=2^ab$ where $b$ is odd. One of numbers $b\pm 2$ is not congruent to $\pm 1$ modulo $q$.
| 5 | https://mathoverflow.net/users/4312 | 410232 | 167,857 |
https://mathoverflow.net/questions/409470 | 2 | In Aczel's Constructive Set Theory (CZF), no non-degenerate complete lattice can be proved to be a set. There are hallmark examples of complete lattices that are proper classes in CZF, including the Dedekind–MacNeille completion of a lattice/poset.
Is there a way to talk about the collection of Complete Lattices in C... | https://mathoverflow.net/users/312621 | Collection of proper classes with in CZF | Unfortunately the only answer I can give is that there is not a good way to do this in general, and since no one else has answered yet, that is probably the only real answer.
Having said that, there are a few techniques that work in some special situations, and can still be useful.
The most relevant one here is tha... | 4 | https://mathoverflow.net/users/30790 | 410233 | 167,858 |
https://mathoverflow.net/questions/410226 | 4 | $X = (x\_1,...x\_n) \in \mathbb{R}^n, X \sim \mathcal{N}(O, \Sigma\_X)$ and $Y = (x\_1,...x\_n) \in \mathbb{R}^n, Y \sim \mathcal{N}(O, \Sigma\_Y)$ are two independent gaussian vectors.
If $\Sigma\_Y - \Sigma\_X$ is positive semidefinite, then $\forall \alpha \in \mathbb{R}^n$, $\alpha\Sigma\_Y\alpha^T \ge \alpha\Sig... | https://mathoverflow.net/users/470763 | L_infinity norm of two gaussian vector | $\newcommand\R{\mathbb R}$Let $\|\cdot\|$ be any norm on $\R^n$. Take any real $t$. Let $Z$ be a random vector in $\R^n$ such that (i) $Z$ is independent of $X$ and (ii) $Z\sim N(0,\Sigma\_Y-\Sigma\_X)$. Then $X+Z$ equals $Y$ in distribution.
So, it suffices to show that
\begin{equation\*}
P(\|X\|\le t)\ge P(\|X+Z\|... | 2 | https://mathoverflow.net/users/36721 | 410242 | 167,861 |
https://mathoverflow.net/questions/410230 | 5 | Let $S\_p(X)$ be the $p$-th singular chain group and $\mathcal S(X)$ be the singular chain complex of a topological space $X$. There is a barycentric subdivision operator (which is also a chain map) $\operatorname{sd}: S\_p(X) \to S\_p(X)$ as is defined in standard topology textbooks such as Munkres. There is a chain h... | https://mathoverflow.net/users/166613 | Homology of singular chain complex modulo subdivision | I believe $\mathcal D$ is not acyclic, unless $X$ is a discrete space, and therefore $S(X)/\mathcal D$ is not quasi-isomorphic to $S(X)$.
In fact, assuming that I did not make a mistake in the claim below, we have the following description of $\mathcal D$: it is zero in degree zero, and is isomorphic to $S(X)$ in pos... | 7 | https://mathoverflow.net/users/6668 | 410248 | 167,863 |
https://mathoverflow.net/questions/409871 | 6 | This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can't find a solution anywhere online. Here is the problem:
Let $E \subset [0,1]$ be Lebesgue measurable with measure $\lv... | https://mathoverflow.net/users/19673 | A problem concerning a divergent function on $[0, 1]$ | I asked a colleague in Hungary, and he found the solution here (on page 170):
<http://real-j.mtak.hu/9393/1/MTA_MatematikaiLapok_1992.pdf>
It is in Hungarian, but with some effort and Google translator, finally I understood.
EDIT This is the translation of the argument above. I keep the same notation but write $E... | 8 | https://mathoverflow.net/users/150653 | 410251 | 167,864 |
https://mathoverflow.net/questions/410255 | 5 | I am interested in a morphism of $S$-schemes $f : X \to Y$ such that $X$ and $Y$ are proper over $S$ and there is some $s\_0 \in S$ such that $f : X\_{s\_0} \to Y\_{s\_0}$ is a closed immersion. Is it true that there is an open neighborhood $U \subset S$ of $s\_0$ so that $X\_U \to Y\_U$ is a closed immersion?
Becaus... | https://mathoverflow.net/users/154157 | Proper morphisms that are closed immersion on a fiber | This is proved in [EGA III, tome 1, Proposition 4.6.7(i)](http://www.numdam.org/item/PMIHES_1961__11__5_0/).
| 9 | https://mathoverflow.net/users/70322 | 410259 | 167,866 |
https://mathoverflow.net/questions/382285 | 5 | I posted this question on [Math Stack Exchange](https://math.stackexchange.com/questions/3988744/find-the-maximum-trigonometric-polynomial-coefficient-a-k) but did not get any answer. I am trying my luck here.
>
> Let $n,k$ be given positive integers and $n>k$. If for all real numbers $x$ we have $$A\_{1}\cos{x}+A\... | https://mathoverflow.net/users/38620 | Find the maximum trigonometric polynomial coefficient $A_{k}$ | It is a known result:
$$\max A\_k = 2 \cos \frac{\pi}{\lfloor \frac{n}{k} \rfloor + 2}, \quad 1\le k \le n.$$
See:
1. Theorem 6 and the references therein, "Extremal Positive Trigonometric Polynomials", [https://www.dcce.ibilce.unesp.br/~dimitrov/papers/main.pdf](https://www.dcce.ibilce.unesp.br/%7Edimitrov/papers/... | 0 | https://mathoverflow.net/users/141801 | 410270 | 167,870 |
https://mathoverflow.net/questions/410269 | 5 | Let $(X\_1,X\_2,\ldots)$ be a stationary ergodic process with each $X\_n$ a real random variable taking values in $[-1,+1]$. Suppose that $\mathbb{E}[X\_n]=0$. Let $S\_n = \sum\_{k=1}^n X\_k$. Is the process $(S\_1,S\_2,\ldots)$ necessarily recurrent, in the sense that there exists some $M$ such that almost surely $|S\... | https://mathoverflow.net/users/23661 | Recurrence of ergodic processes | Yes. I will use some different notation, but the idea is the same.
Let $(\Omega,\mu)$ be a probability space, and let $\sigma \colon \Omega \to \Omega$ be an ergodic measure preserving transformation. Let $f$ be a measurable function taking values in $[-1,1]$ such that $\int f\,d\mu=0$. Write $S\_nf(\omega)=f(\omega)+\... | 4 | https://mathoverflow.net/users/11054 | 410276 | 167,873 |
https://mathoverflow.net/questions/410191 | 0 | This is a variant on the question posed [here](https://mathoverflow.net/questions/409899/is-every-matrix-involution-over-a-ufd-diagonalisable), in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}\_n(A)$, where $A$ is a $k$-algebra for some field $k$ of characteristic $\ne... | https://mathoverflow.net/users/175051 | Order 2 matrices with entries in the polynomial ring over a field are diagonalisable | As abx pointed out in the comments, I misread the fact that this is an equivalence, so it is unlikely that there is a more elementary proof.
| 1 | https://mathoverflow.net/users/175051 | 410280 | 167,875 |
https://mathoverflow.net/questions/410096 | 9 | This identity came up in my research:
$$
\sum\_{m=1}^n m^2 \frac{(\frac{xy}n + m-1)\_{2m-1} (n+m-1)\_{2m-1}}{(x+m)\_{2m+1} (y+m)\_{2m+1}} = \frac{n^2}{(x^2-n^2) (y^2 - n^2)}.
$$
Here $n$ is a fixed positive integer and $x,y$ are variables. So for each $n$ this is an identity of rational functions in $x,y$. I denote by ... | https://mathoverflow.net/users/89514 | Identity involving a quadratic term inside the Pochhammer symbol | Your identity is not just a curiosity. It is a special case of a result that has been used to obtain many quadratic and cubic identities for hypergeometric series. Perhaps the most general formulation is given by Warnaar (Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002... | 6 | https://mathoverflow.net/users/10846 | 410284 | 167,876 |
https://mathoverflow.net/questions/410234 | 3 | I am reading K.D.Bierstedt's paper [Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I. Journal für die reine und angewandte Mathematik 259 (1973): 186-210](https://www.degruyter.com/document/doi/10.1515/crll.1973.259.186/html). It is written in German, and, perhaps, because of that ... | https://mathoverflow.net/users/18943 | $\varepsilon$-product in Bierstedt's paper | I don't have access to Bierstedt's article right now. Nevertheless, let me try to answer your question.
0. Yes, precompact sets are also called totally bounded.
1. The lemma is true and probably this is what Bierstedt says.
2. Yes, $X\varepsilon Y$ is complete whenever so are $X$ and $Y$. The proof should be quite st... | 3 | https://mathoverflow.net/users/21051 | 410289 | 167,878 |
https://mathoverflow.net/questions/410212 | 4 | Let $X$ be a manifold and $E\to X$ a complex vector bundle and let's work in $H^\bullet(X,\mathbb{Z})$. Given the total Chern class of $E$, $c(E)=1+c\_1(E)+\cdots+c\_n(E)$, we can define the total Segre class of $E$ to be $s(E)=1+s\_1(E)+\cdots+s\_n(E)$ to be the inverse of the total Chern class.
Equivalently, recall... | https://mathoverflow.net/users/148223 | Real analogue of Segre classes | Unfortunately $\int\_Xp\_n(V)\neq\chi(V) \operatorname{mod} 2$ in general. In particular $\int\_Xp\_n(TX)=0$ and it can be shown as follows.
By the multiplicativity of the Stiefel-Whitney classes, given two bundles $V, W$ such that $V\oplus W=\mathbb{R}^d$, then $p\_k(V)=w\_k(W)$. It is now easy to show that $p\_n(TX... | 2 | https://mathoverflow.net/users/148223 | 410294 | 167,879 |
https://mathoverflow.net/questions/410293 | 4 | In Girard's $\Pi^1\_2$-logic, a *dilator* $D$ is a endofunctor which commutes with pull-back and direct limit on $\mathrm{ON}$, the category whose objects are ordinals and morphisms are strictly increasing functions. For dilator $D\_0,D\_1$, an *embedding* from $D\_0$ to $D\_1$ is a natural transformation from $D\_0$ t... | https://mathoverflow.net/users/149565 | Are bi-embeddable dilators equal? | No. For example consider the dilator $D$ that maps a well-order $A$ to the well-order consisting of denotations
1. $(2n;x,y)$, where $y<\_Ax$ are elements of $A$;
2. $(2n+1;x)$, where $x\in A$.
The denotations are compared by lexicographical order. For a morphism $f
\colon A\to B$ we as usual put $D(f)((m;x\_1,\ldo... | 5 | https://mathoverflow.net/users/36385 | 410296 | 167,880 |
https://mathoverflow.net/questions/410295 | 2 | For any real random variable $X$, define
$$\|X\|\_{2,1}=\int\_0^\infty \sqrt{\Pr(|X|>t)}dt.$$
This quantity (it is not a norm) appears in various problems, e.g. the multiplier central limit theorem (see, e.g., Section 2.9 in [this book](https://link.springer.com/book/10.1007/978-1-4757-2545-2)) or in L-statistics (see,... | https://mathoverflow.net/users/174236 | Comparison between $\|X\|_2$ and $\|X\|_{2,1}$ | $$\begin{aligned}
\|X\|\_{2,1}^2&=\int\_0^\infty\int\_0^\infty ds\,dt\,\sqrt{P(|X|>s)}\sqrt{P(|X|>t)} \\
&\ge\int\_0^\infty\int\_0^\infty ds\,dt\,P(|X|>\max(s,t)) \\
&=E\int\_0^\infty\int\_0^\infty ds\,dt\,1(|X|>\max(s,t)) \\
&=E|X|^2=EX^2.
\end{aligned}$$
So, we have an improvement of your bound. Moreover, the lowe... | 7 | https://mathoverflow.net/users/36721 | 410303 | 167,882 |
https://mathoverflow.net/questions/410302 | 2 | If $G=(V,E)$ is a simple, undirected graph, $C\subseteq V$ is said to be a *vertex cover* if for every $e\in E$ we have $C\cap e \neq \emptyset.$
If $G=(V,E) $ is infinite, is there necessarily a vertex cover $C\_0\subseteq V$ of $G$ such that for every $v\in C\_0$ we have that $C\_0\setminus\{v\}$ is no longer a ver... | https://mathoverflow.net/users/8628 | Minimal vertex-covering set | Yes: take an inclusion-maximal independent set (exists by Zorn lemma) and pass to a complement.
| 1 | https://mathoverflow.net/users/4312 | 410306 | 167,883 |
https://mathoverflow.net/questions/409941 | 2 | Let $ M $ be a smooth manifold.
Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $.
Recall that a manifold $ M $ is Riemannian homogeneous if it admits a metric with respect to which the isometry group is transitive, moreover this metric can always be chosen ... | https://mathoverflow.net/users/387190 | Riemannian homogeneous equivalent to linear group orbit | $ \textbf{(1)} \implies \textbf{(2)} $
Up to diffeomorphism a Riemannian homogeneous space is just a trivial vector bundles over a compact Riemannian homogeneous space ([noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous](https://mathoverflow.net/questions/410334/riemannian-homogeneou... | 0 | https://mathoverflow.net/users/387190 | 410309 | 167,884 |
https://mathoverflow.net/questions/410275 | 3 | Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively?
My intuition is that yes we can because $ M $ is the total space of a vector bundle over a compact Riemannian homogenous space (... | https://mathoverflow.net/users/387190 | Is the Moebius strip Riemannian homogeneous? | Suppose by contradiction it is. Write it as $G/K$ where $G$ is the identity component of the isometry group and $K$ is compact, and $G$ acts faithfully on $G/K$. Since $G$ is connected, maximal compact subgroups are connected. Since $G/K$ is not contractible, $K$ is not maximal compact and it follows that maximal compa... | 2 | https://mathoverflow.net/users/14094 | 410312 | 167,885 |
https://mathoverflow.net/questions/410315 | 1 | Let $(X, d)$ be a uniformly discrete metric space of bounded geometry, that is, $\sup\_{x \in X} |B\_r(x)| < \infty$ for every $r \geq 0$ and there is a uniform $\delta > 0$ such that $d(x, y) \geq \delta$ for all $x \neq y \in X$.
Let $G$ be the so-called *wobbling group of $X$*, that is, the set of all bijections $... | https://mathoverflow.net/users/147609 | Countability of the wobbling group of a bounded geometry metric space | Say that a metric space $X$ is *hyperdiscrete* if the distance map $X\times X\to\mathbf{R}\_{\ge 0}$ is a proper map, i.e. for every $R$, the set of $(x,y)$ such that $d(x,y)\le R$ is finite. For instance, with the Euclidean distance, $\mathbf{N}$ is not hyperdiscrete, but the set $\{0,1,4,\dots\}$ of squares in $\math... | 5 | https://mathoverflow.net/users/14094 | 410319 | 167,887 |
https://mathoverflow.net/questions/410305 | 5 | The basic question is:
Does vanishing of homology with trivial coefficients imply triviality of an infinite-dimensional Lie algebra?
My question is motivated by acylic groups in group theory. In particular, there is a an acylic group $\textrm{Aut}\_f(\mathbb{R})$ of autohomeomorphisms of $\mathbb{R}$ with finite su... | https://mathoverflow.net/users/143549 | Infinite dimensional Lie algebras with trivial homology | There exist acyclic infinite-dimensional Lie algebras, i.e., for which the trivial homology vanishes in all nonzero degree (recall that $H\_0$ of every Lie algebra is always 1-dimensional over the ground field $K$).
>
> **Lemma.** Let $\mathfrak{g}$ be the increasing union of Lie subalgebras $\mathfrak{g}\_n$. Supp... | 8 | https://mathoverflow.net/users/14094 | 410338 | 167,892 |
https://mathoverflow.net/questions/410339 | 0 | Let $d$ be a positive integer, and suppose $c$ is an integer such that $\gcd(c,d) = 1$. Then the following identity holds:
$$\displaystyle \left \lvert \{b \pmod{d} : b^2 \equiv c \pmod{d} \}\right \rvert = \sum\_{\substack{\chi \pmod{d} \\ \chi^2 = \chi\_0}} \chi(c).$$
What is the correct analogue for the right ha... | https://mathoverflow.net/users/10898 | Analogues of an identity involving quadratic characters | If $\gcd(c,d) = 1$ we have
$$\displaystyle \left \lvert \{b \pmod{d} : b^k \equiv c \pmod{d} \}\right \rvert = \sum\_{\substack{\chi \pmod{d} \\ \chi^k = \chi\_0}} \chi(c).$$
This is not really anything to do with numbers - it works in any finite abelian group, and here we are applying it to the multiplicative grou... | 7 | https://mathoverflow.net/users/18060 | 410341 | 167,893 |
https://mathoverflow.net/questions/410323 | 4 | I have the following exact sum for the expectation of an event
$$\sum\_{m=0}^{nk} \sum\_{j=1}^n (-1)^{j-1}\binom{n}{j} \binom{(n-j)k}{m} / \binom{nk}{m}$$
which is exactly correct but I want to give an upper bounding approximation that is easier to interpret. In particular, I want to see the expectations dependence... | https://mathoverflow.net/users/214997 | Approximating binomial coefficient sum | Actually we may simply compute this sum (and I guess that your expectation may be computed differently to give the answer in the below simplified form).
We start with $1/{nk\choose m}=(nk+1)\int\_0^1 x^m(1-x)^{nk-m}dx$ by the Beta function $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ formula. Next, we sum up over $m$ with f... | 8 | https://mathoverflow.net/users/4312 | 410344 | 167,896 |
https://mathoverflow.net/questions/410334 | 2 | Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive **($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant vector bundle** where $ K $ is a compact group acting transitively on the base $ B $ of the bundle?
If the vector b... | https://mathoverflow.net/users/387190 | noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous | The only if direction fails. That is, there are $K$-equivariant vector bundles which are not homogeneous. For example, the Mobius band has the form $O(2)\times\_{O(1)} \mathbb{R}$, and is not Riemannian homogeneous as you mention.
On the other hand, it seems the if direction is true. In fact, I think I can prove that... | 1 | https://mathoverflow.net/users/1708 | 410347 | 167,898 |
https://mathoverflow.net/questions/410333 | 10 | It is known that a random series
$$
\sum\_{n\geq 1} X\_n
$$
whose terms $X\_n$ are independent converges a.s. if and only if it converges in probability.
Is it true that a martingale $(Y\_n)$ converges a.s. if and only if it converges in probability? If not, are there any counter-examples? Thanks.
| https://mathoverflow.net/users/20302 | Martingales converging in probability but not a.s | $\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\F}{\mathcal{F}}\newcommand{\Om}{\Omega}\newcommand{\Z}{\mathbb{Z}}$A counterexample can be obtained as follows. Let $T\_1,T\_2,\dots$ be independent (say) geometrically distributed random variables with fast growing means, say with $ET\_i=2^i$.
Let $Y\_n... | 11 | https://mathoverflow.net/users/36721 | 410350 | 167,899 |
https://mathoverflow.net/questions/410346 | 13 | It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as
$$\mathcal{M}=P\_{1}\#\dots\# P\_{n}$$
where $P\_{i}$ are prime manifolds, i.e. manifolds which can not be written as a non-trivial connected sum of two 3-manifolds.
>
> **My main question is the ... | https://mathoverflow.net/users/259525 | Classification of 3-dimensional manifolds with boundary | With regard to the first question: There are a couple of versions of unique decompositions for 3-manifolds with boundary, with respect to boundary connected sum. See Gross, Jonathan L. [The decomposition of 3-manifolds with several boundary components](https://www.ams.org/journals/tran/1970-147-02/S0002-9947-1970-02580... | 9 | https://mathoverflow.net/users/3460 | 410361 | 167,900 |
https://mathoverflow.net/questions/410357 | 7 | A divisor $d$ of a natural integer $N$ defines a permutation
of $\{0,\ldots,N-1\}$ by considering
$$x\longmapsto \pi\_{d\vert N}(x)=\left\lfloor \frac{x}{d}\right\rfloor+\frac{N}{d}
\left( x\pmod d\right)$$
for $x \pmod d$ in $\{0,\ldots,d-1\}$.
One can show that the group $A(N)$ generated by all permutations $\pi\_{... | https://mathoverflow.net/users/4556 | An abelian group associated to divisors of an integer $N$ | All the $\tau$'s fix $0$ so we can look at how they behave on the set $\{1,2,\dots,N-1\}$. We have
$$\tau\_{d|N}(kd+r)=k+\frac{rN}{d}=\left[\frac{N}{d}(kd+r)\right]\_{\pmod{N-1}}$$
where $[a]\_{\pmod{N-1}}$ is the unique integer in $\{1,2,\dots, N-1\}$ congruent to $a\pmod{N-1}$. So $\tau\_{d|N}$ is the same as multipl... | 3 | https://mathoverflow.net/users/2384 | 410363 | 167,901 |
https://mathoverflow.net/questions/410365 | 7 | Let $f : X\rightarrow S$ be a flat finite type morphism of schemes with $S$ integral and Noetherian. Let $\eta\in S$ be the generic point.
Let $\{\sigma\_i\}$ be a collection of sections of $f$ (possibly infinite), which are Zariski dense in $X\_\eta$. I'm interested in additional conditions on $f,\{\sigma\_i\}$ unde... | https://mathoverflow.net/users/88840 | When must a set of sections which is Zariski dense in the generic fiber also be dense in some special fiber? | Building on Yosemite Stan's example, for any scheme over any number ring, even if you take all the sections, it will still not be Zariski dense in any special fiber, because all sections go through the rational points of the special fiber, which are not dense as the base field is finite. Of course, there are many examp... | 7 | https://mathoverflow.net/users/18060 | 410369 | 167,902 |
https://mathoverflow.net/questions/403877 | 1 | I have been looking at constructions satisfying the Johnson-Lindenstrauss Lemma (e.g., projections onto random subspaces, random Gaussian matrices, random Rademacher matrices, etc.). It seems that other than the "random subspace" construction, none of the other projections are true projections in the sense that the vec... | https://mathoverflow.net/users/128729 | Johnson-Lindenstrauss with Orthogonalization | Suppose $v$ has unit norm and $\|Xv\|\in[1-\epsilon,1+\epsilon]$. Then
$$\|X^\top(XX^\top)^{-1}Xv\|=\|(XX^\top)^{-1/2}Xv\|\in\bigg[\frac{1-\epsilon}{\sigma\_\max(X)},\frac{1+\epsilon}{\sigma\_\min(X)}\bigg].$$
If $X$ has subgaussian entries, then one may control its extreme singular values with standard techniques. (Se... | 1 | https://mathoverflow.net/users/29873 | 410371 | 167,903 |
https://mathoverflow.net/questions/410049 | 7 | I was thinking of a way to prove [this](https://math.stackexchange.com/questions/586862/non-vanishing-vector-fields-on-non-compact-manifolds) and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is compact or $\ma... | https://mathoverflow.net/users/172802 | Is the union of a compact and the relatively compact components of its complementary in a manifold compact? | Actually, there is an elementary proof of this. I will imitate the one given in O.Forster Lectures on Riemann Surfaces. We assume $M$ is connected. Let $Y$ be equal to the union of $X$ with all the relatively compact components of $M \setminus X$
Let $U$ be a relatively compact, open subset containing $X$ and let $C\... | 3 | https://mathoverflow.net/users/176397 | 410372 | 167,904 |
https://mathoverflow.net/questions/410377 | 0 | I'm a newbie in the field of mathematical research and I'm not able to find the following paper:
>
> " M. Radić, A definition of the determinant of a rectangular matrix,
> (Serbo-Croatian summary) Glasnik Mat. Ser. III 1(21) (1966), 17–22 "
>
>
>
Please could you provide me a link to this paper.
| https://mathoverflow.net/users/470930 | Where can I find the following math paper? | You can find the paper at this [link](https://books.google.it/books?id=5z0GkLQ6LfgC&pg=PA17&redir_esc=y&hl=it#v=onepage&q&f=false). The table of contents is at the [journal site](https://web.math.pmf.unizg.hr/glasnik/).
| 4 | https://mathoverflow.net/users/19520 | 410381 | 167,905 |
https://mathoverflow.net/questions/410330 | 7 | Consider $f\_n(x) = \min\_{|z|=x} \Re \sum\_{j=1}^{n} \frac{z^j}{j}$, a real function of positive variable $x>0$.
I am interested in lower bounds on $f\_n(x)$. Specifically, I ask: what lower bounds can be given on $f\_n(x)$ in the regime where $x=1+O(1/n)$ and $n$ tends to $\infty$?
Trivially, $f\_n(x) \ge - \sum\... | https://mathoverflow.net/users/31469 | Asymptotics of truncated logarithm on a cricle | So, we have a function $u\_n(z) = \Re \sum\_{j = 1}^n \frac{z^j}{j}$. As a real part of an analytic function, it is a harmonic function. We are interested in its behaviour on the circle $|z| = x = 1 + \frac{c}{n}$, where $c > 0$ is some constant and we want to estimate the minimal value of $u\_n$ on it. Let me start wi... | 2 | https://mathoverflow.net/users/104330 | 410384 | 167,908 |
https://mathoverflow.net/questions/410337 | 5 | I’ve considered the diffusion equation $$\frac{\partial f(x,t)}{\partial t}=\frac12 \frac{\partial^2 f(x,t)}{\partial x^2}$$ with the conditions $f(x,0)=\delta(x)$ and $f(-1,t)=f(1,t)=0\ \forall t>0$ and I’ve found the solution $$f(x,t)=\sum\_{n=0}^\infty \cos\left[ \left( n+\frac12 \right)\pi x \right] e^{-\frac12 (n+... | https://mathoverflow.net/users/470349 | Approximation for a series involving the derivative of a Jacobi theta function | Write your $\Lambda$ as
$$\Lambda(t)=\frac{1}{2}\sum\_{-\infty}^\infty f(n),$$
where
$$f(x)=\pi(x+1/2)\sin\pi(x+1/2)\exp\left(-\pi^2(x+1/2)^2t/2\right)=y\sin y\,e^{-ty^2/2},$$
where $y=\pi(x+1/2),$ and $t>0$.
Then use [Poisson's summation formula](https://en.wikipedia.org/wiki/Poisson_summation_formula)
$$\sum\_{-\inft... | 5 | https://mathoverflow.net/users/25510 | 410411 | 167,919 |
https://mathoverflow.net/questions/410164 | 7 | Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of [Hawking, King, and McCarthy](https://aip.scitation.org/doi/abs/10.1063/1.522874) (see Thm 5) says that $M$ is determined up to conformal isomorphism by its class of null geodesic curves (where the parameterization of the null geodes... | https://mathoverflow.net/users/2362 | In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure? | Trying to recover as much of the topology/geometry from the causal order as possible has been studied quit a bit since the early paper of Hawking et al that you cite. A quick summary of my understanding of the situation is that there are some purely order-theoretic topologies (like Alexander or Scott topologies) that r... | 3 | https://mathoverflow.net/users/2622 | 410412 | 167,920 |
https://mathoverflow.net/questions/410410 | 1 | A rooted tree is a tree with a distinguished root node. When a rooted tree is embedded in a plane, a cyclic ordering is induced on the subtrees of the root. Such trees are called rooted plane trees.
Given a tree $T$, is there an algorithm that can output a rooted plane tree $T\_r$ isormorphic to $T$ such that every n... | https://mathoverflow.net/users/148974 | Construct a rooted plane tree with nodes labelled | Easy answer: Denote the root by $(0)$. Denote its $a\geq 0$ children by $(0,1),(0,2),\ldots,(0,a)$.
Denote recursively the $b$ children of a vertex $(0,a\_1,\ldots,a\_{k-1})$ at height $k-1$ by $(0,a\_1,\ldots,a\_{k-1},1),(0,a\_1,\ldots,a\_{k-1},2),\ldots,(0,a\_1,\ldots,a\_{k-1},b)$.
Draw the vertex $(0,a\_1,\ldots,a... | 1 | https://mathoverflow.net/users/4556 | 410417 | 167,923 |
https://mathoverflow.net/questions/410421 | 3 | Let $\mathbb{G}$ be a $C^\*$-algebraic compact quantum group. Consider the associated dense Hopf$^\*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ denote its antipode. In the book "Compact quantum groups and their representation categories", it is shown that $S$ i... | https://mathoverflow.net/users/216007 | Norm antipode on a Kac-type compact quantum group | $S$ is an anti-$\*$-homomorphism, and extends by continuity to $A = C(\mathbb G)$ (the closure of $\mathcal O(\mathbb G)$ acting on the GNS space for the Haar state). Let $A^{\operatorname{op}}$ be the opposite $C^\*$-algebra to $A$. Then we can consider $S$ as a map $A\rightarrow A^{\operatorname{op}}$, which is now a... | 3 | https://mathoverflow.net/users/406 | 410426 | 167,926 |
https://mathoverflow.net/questions/410387 | 7 | Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following [this article](https://www.ams.org/journals/tran/1970-147-02/S0002-9947-1970-0258047-X/S0002-9947-1970-0258047-X.pdf), it is stated that $\mathcal{M}$ can be written as
$$\mathcal{M... | https://mathoverflow.net/users/259525 | Decomposition of manifolds with toroidal boundary | The situation is similar to the one of closed manifolds. One defines "boundary-prime" manifolds as those that cannot be decomposed nontrivially in a boundary-connected sum.
Note that if $M$ is connected, has nonempty boundary and is not prime, then $M$ is never boundary-prime. Namely, take a 2-sphere $S\subset M$ sep... | 5 | https://mathoverflow.net/users/39654 | 410427 | 167,927 |
https://mathoverflow.net/questions/410358 | 3 |
>
> Theorem(Guillemin Sternberg Marle) Let $(M, \omega, \mu) $ be a symplectic manifold together with a Hamiltonian group action. Let $p$ be a point in $M$ such that $O\_p $ is contained in the zero level set of the moment map. Denote $G\_p$ the stabilizer and $O\_p$ the orbit of $p$. There is a $G$-equivariant sympl... | https://mathoverflow.net/users/172459 | Proof of the Hamiltonian slice theorem | The following textbooks contain a proof of the symplectic slice theorem:
* [Juan-Pablo Ortega and Tudor S. Ratiu: Momentum Maps and Hamiltonian Reduction](https://link.springer.com/book/10.1007/978-1-4757-3811-7)
* [Gerd Rudolph and Matthias Schmidt: Differential Geometry and Mathematical Physics](https://link.spring... | 5 | https://mathoverflow.net/users/17047 | 410429 | 167,928 |
https://mathoverflow.net/questions/410438 | 12 | For any $a = [a\_0; \dots; a\_n]\in \mathbb{P}^n(\mathbb{Q})$, the corresponding Galois group $G\_a$ of $f(X) = a\_n X^n + \cdots + a\_1 X + a\_0\in \mathbb{Q}[X]$ is a subgroup of $S\_n$. (I'm interested primarily in the case where $f$ is irreducible here, but we can take $G\_a = \text{Aut}(k/\mathbb{Q})$ for the spli... | https://mathoverflow.net/users/61829 | Structure of coefficients of polynomials giving a specified Galois group | Every Galois group is Zariski dense, so that's not a very exciting measurement. In fact, for any degree n field, minimal polynomials of generators of that field are Zariski dense. This follows from the fact that the map taking an element to its minimal polynomial is finite-to-one, thus sends Zariski dense sets to Zaris... | 16 | https://mathoverflow.net/users/18060 | 410439 | 167,931 |
https://mathoverflow.net/questions/410436 | 2 | In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}\_p[[T]]$.
>
> If $M$ is a finitely generated torsion $\Lambda$-module, then there is a pseudo-isomorphism
> $$M \to \left( \bigoplus\_{i=1}^s \Lambda/(p^{n\_i}) \ri... | https://mathoverflow.net/users/394740 | Structure theorem for finitely generated $\Lambda$-modules - uniqueness part | By Nakayam's lemma, since $M$ is finitely generated, $x\_1,\dots, x\_n$ generate $M$ if and only if they generate $M/ (p, T)M$ (where $(p,T)$ is the maximal ideal of the local ring $\mathbb Z\_p[[T]]$.)
So $x\_1,\dots, x\_n$ are a minimal generating set if and only if they are a basis of $M/ (p,T)M$, and thus in this... | 1 | https://mathoverflow.net/users/18060 | 410446 | 167,934 |
https://mathoverflow.net/questions/410393 | 9 | Let $A$ be a $C^\*$ algebra of operators acting on some Hilbert space $H$, and $A\_0$ is a norm dense $\*$-subalgebra of $A$. Suppose there exists some unit vector $\xi \in H$, such that (i) $A\_0 \xi$ is dense in $H$; (ii) $a\xi = 0$ if and only if $a=0$ for all $a \in A\_0$, i.e. $\xi$ is separating for $A\_0$.
**Q... | https://mathoverflow.net/users/128540 | Can one detect a cyclic and separating vector for a concrete $C^*$-algebra using a dense subalgebra? | Here's a counter-example. Take $A\_0:=\mathbb{C}[F(s,t)]\subset A:=\mathrm{C}^\*\_{\mathrm{r}}(F(s,t))$, where $F(s,t)$ is the free group on $\{s,t\}$, $E\colon A\to \mathrm{C}^\*\_{\mathrm{r}}(F(s))$ the canonical conditional expectation, and $\psi\colon \mathrm{C}^\*\_{\mathrm{r}}(F(s)) \cong C(\mathbb{R}/\mathbb{Z})... | 14 | https://mathoverflow.net/users/7591 | 410452 | 167,936 |
https://mathoverflow.net/questions/410285 | 9 | Consider matrices $M$ of size $L\times L$ over a finite field $\mathbb{Z}\_p$, for simplicity focus on $p$ prime. The size $L$ is even. We want to find the order of a specific class of matrices, namely we want to find the smallest non-zero integer $n$ such that $$M^n=1,$$ where 1 is here the identity matrix of size $L\... | https://mathoverflow.net/users/150048 | Find the order of a class of finite matrices over finite fields | Big primes enter the picture as follows: Compute the characteristic polynomial $P\_L$ of your square matrix $M$ of size $L$.
Factor $P\_L$ over your working prime $p$. This gives you the eigenvalues of the involved Jordan-blocs over $\mathbb F\_p$. These eigenvalues are elements of field extensions of degree at most $L... | 4 | https://mathoverflow.net/users/4556 | 410455 | 167,938 |
https://mathoverflow.net/questions/410444 | 4 | We denote by $C[0, 1]$ the space of continuous functions on $[0, 1]$ under the supremum norm, equipped with the Borel sigma algebra.
A *covering* of $C[0, 1]$ is a (possibly countably infinite) collection of Borel sets $E\_i$ such that $\bigcup\_i E\_i = C[0, 1]$.
A covering is said to have *finite eccentricity* if... | https://mathoverflow.net/users/173490 | Does $C[0, 1]$ admit a covering by sets of arbitrarily small eccentricity? | To give a positive answer to the question it is enough to, for a fixed $\varepsilon$, give a collection of disjoint balls in $C[0,1]$ of radius $\varepsilon$ which is dense in $C[0,1]$. Indeed, then for $\delta$ as small as you want you can take the sets $E\_i$ to be the balls and adjoin every point $f$ outside the bal... | 5 | https://mathoverflow.net/users/172802 | 410458 | 167,940 |
https://mathoverflow.net/questions/410462 | 30 | Consider the set of ultrafilters $\beta(\mathbb N)$ on $\mathbb N$.
Any function $f\colon\mathbb N\to\mathbb N$ extends to a function $\beta f\colon \beta \mathbb N \to \beta\mathbb N$. We say that two ultrafilters $\mathcal U$ and $\mathcal V$ are *isomorphic* if there is some bijection $f$ with $f(\mathcal U) = f(\ma... | https://mathoverflow.net/users/470870 | Are all free ultrafilters 'the same' in some sense? | Certain important properties are shared by all free ultrafilters. In many applications of ultrafilters, especially more elementary applications, only these properties are used. In such a situation, it does not matter which free ultrafilter is chosen -- any one will do.
But there are some proofs that require (or seem ... | 38 | https://mathoverflow.net/users/70618 | 410463 | 167,941 |
https://mathoverflow.net/questions/410375 | 7 | I have two discrete groups $G\_1$ and $G\_2$ sitting in the following exact sequences:
$1\to H\_1\to G\_1\to K\_1\to 1$ and $1\to H\_2\to G\_2\to K\_2\to 1$.
$H\_1$, $K\_1$, $H\_2$ and $K\_2$ are all non-abelian free groups of ranks $k+n$, $k$, $k+l+n$ and $k+l$ respectively. Also $k,l>1$ and $n\geq 1$. Somehow I f... | https://mathoverflow.net/users/126243 | Distinguishing poly-free groups | Euler characteristic is multiplicative in the setting of your exact sequences
$1\to H\_i\to G\_i\to K\_i\to 1$,
i.e. $\chi(G\_i)=\chi(H\_i)\chi(K\_i)$.
(You can see this directly by building a model for each $G\_i$ as a graph of graphs, or by more sophisticated arguments.)
In your case, this gives
$\chi(G\_1)... | 5 | https://mathoverflow.net/users/1463 | 410465 | 167,942 |
https://mathoverflow.net/questions/410443 | 7 | Let $ G $ be the real points of a linear algebraic group and $ G' $ a Zariski closed subgroup. **Then is $ G/G' $ a Cartesian product
$$
(K/K') \times F
$$
where $ F $ is contractible?** Here $ K,K' $ are maximal compacts of $ G,G' $.
Some relevant information:
Let $ G,G' $ be Lie groups with finitely many connected ... | https://mathoverflow.net/users/387190 | Is a quotient of real linear algebraic groups always a Cartesian product of compact and contractible factors? | The answer is no.
At least when $G$ and $H$ are semisimple, the quotient $G/H$ is diffeomorphic to the normal bundle of $K\_G/K\_H$ inside $G/H$ (where $K\_G$ and $K\_H$ denote respectively maximal compact subgroups of $G/H$), but this normal bundle might not be trivial.
For a concrete example take $G= SO(n+1,\math... | 7 | https://mathoverflow.net/users/173096 | 410467 | 167,944 |
https://mathoverflow.net/questions/410099 | 3 | Preliminaries
=============
Let $ n $ be an integer such that $ n \geq3 $. Denote $ \left[ n \right] \equiv \{1,2, \ldots ,n \} $. Let $ P $ be a non-empty subset of $ \left[ n \right] $ such that $ \left|P \right| \geq 3 $. Denote $ X\_{P} \equiv \left( x\_{i} \colon i \in P \right) $ as an ordered alphabet. Let $ \... | https://mathoverflow.net/users/91155 | Polynomial function defined recursively by a resultant - is it well defined? | Notice that
$$
\let\eps\varepsilon
q(x\_1,x\_2,x\_3)
=-\bigl(\sqrt{x\_1}+\sqrt{x\_2}+\sqrt{x\_3}\bigr) \bigl(-\sqrt{x\_1}+\sqrt{x\_2}+\sqrt{x\_3}\bigr) \bigl(\sqrt{x\_1}-\sqrt{x\_2}+\sqrt{x\_3}\bigr) \bigl(-\sqrt{x\_1}-\sqrt{x\_2}+ \sqrt{x\_3}\bigr)\\
=-\prod\_{\eps\_1,\eps\_2\in\{\pm1\}}\left(
\sqrt x\_3+\sum\_{I... | 2 | https://mathoverflow.net/users/17581 | 410470 | 167,946 |
https://mathoverflow.net/questions/410437 | 3 | In *"On numbers and games"*, Conway writes that the surreal Numbers form a *universally embedding totally ordered Field.* Later Jacob Lurie proved that (the equivalence classes of) the partizan games form a *universally embedding partially ordered abelian group.*
As far as I can tell from Lurie's paper, the latter me... | https://mathoverflow.net/users/470978 | What does it mean for the surreal numbers/partizan games to be "universally embedding"? | Besides the two types of structures being considered, there are two types of universality here. They are indeed related, but the stronger one (a la Lurie and I believe also Conway) is in my opinion the "right" one.
The story I like to imagine is this. We have some putatively-"universal" object $\mathcal{U}$ that we u... | 5 | https://mathoverflow.net/users/8133 | 410471 | 167,947 |
https://mathoverflow.net/questions/410459 | 9 | Let $f \in S\_2(\Gamma\_0(N))$ be a newform with trivial character. I want to compute the Petersson norm $\lVert f\rVert^2$ of $f$, *not* normalized by $1/[\operatorname{SL}\_2(\mathbf{Z}):\Gamma\_0(N)]$, as in Gross–Zagier.
From [Numerical evaluation of the Petersson product of elliptic modular forms](https://mathov... | https://mathoverflow.net/users/471019 | Computing the Petersson norm of newforms of weight 2 from the symmetric square $L$-function | My guess is that the formula you are trying to use is only valid for $N=1$, and thus needs correction in general.
Maybe Shimura's paper can help sort this out. <https://doi.org/10.1002/cpa.3160290618>
In (2.1), which Shimura writes for $\Gamma\_1(N)$ but that doesn't matter when the character is trivial, his defini... | 2 | https://mathoverflow.net/users/334725 | 410483 | 167,949 |
https://mathoverflow.net/questions/410489 | 5 | The May Recognition Theorem establishes an equivalence between the $\infty$-categories
* The $\infty$-category of grouplike $E\_n$ monoids
* The $\infty$-category of pointed $(n-1)$-connected spaces
There is also an equivalence between the $\infty$-categories
* The $\infty$-category of $E\_1$ monoids
* The $\inft... | https://mathoverflow.net/users/445753 | Are $E_k$ monoids higher categories? | This is closely related to the Baez–Dolan [stabilization hypothesis](https://ncatlab.org/nlab/show/stabilization+hypothesis).
There are numerous proofs of this statement.
One line of reasoning is to establish
a general 1-category statement first: given a symmetric monoidal presentable (∞,1)-category $C$,
the (∞,1)-ca... | 6 | https://mathoverflow.net/users/402 | 410493 | 167,953 |
https://mathoverflow.net/questions/410450 | 4 | $\DeclareMathOperator\Bl{Bl}\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Sym{Sym}\newcommand\Sch{\mathit{Sch}}\newcommand\Sets{\mathit{Sets}}$Let $X$ be a smooth variety. The Hilbert scheme of two points $X^{[2]}$ on $X$ can be obtained by blowup the diagonal of the symmetric product $\Sym^2X:=(X\times X)/\mathb... | https://mathoverflow.net/users/74322 | $\text{Bl}_{\Delta}(X\times X)$ as the "Hilbert scheme" of ordered two points on $X$ | Let $X$ be a smooth variety, and $X[n]$ the variety obtained from $X^n = X\times\dots \times X$ by blowing-up the diagonals in order of increasing dimension.
The variety $X[n]$ is a wonderful compactification (the added boundary divisor is simple normal crossing) of the configuration space of $n$ ordered points on $X... | 4 | https://mathoverflow.net/users/14514 | 410500 | 167,955 |
https://mathoverflow.net/questions/410507 | 1 | I'm trying to show the curvature of a one dimensional vector bundle with a Riemannian metric vanishes, no matter what the connection is. I found this can be done for orientable bundles, because an orientable Riemannian line bundle is trivial.
My questions are,
1. Are Riemannian line bundles trivial?
2. If not, how ... | https://mathoverflow.net/users/471097 | Riemannian vector bundle | Given a connection $A\_{i\alpha}^\beta$ the curvature is
$$F\_{ij\alpha}^\beta=\frac{\partial A\_{j\alpha}^\beta}{\partial x^i}-\frac{\partial A\_{i\alpha}^\beta}{\partial x^j}+A\_{i\gamma}^\beta A\_{j\alpha}^\gamma-A\_{j\gamma}^\beta A\_{i\alpha}^\gamma.$$
In the case of a line bundle one can only have $\alpha=\beta=1... | 2 | https://mathoverflow.net/users/156492 | 410513 | 167,959 |
https://mathoverflow.net/questions/410505 | 2 | Suppose $K$ is a number field and $E$ is an elliptic curve defined over $K$. *My question is:* how do you compute the local cohomology group $H^1(K\_v, \, E[p^{\infty}])$?
As to why I'm asking this, it comes up in [Iwasawa theory for elliptic curves](https://arxiv.org/abs/math/9809206) by Greenberg in page 74. In the... | https://mathoverflow.net/users/394740 | Calculating the Galois cohomology group $H^1(K_v, \, E[p^{\infty}])$ | The corollary 3.4 of chapter 1 Arithmetic Duality theorems by Milne says that:
If $K$ has char 0 local field, there is a canonical pairing $$H^r(K,A^t)\times H^{1-r}(K,A)\to \mathbb{Q}/{\mathbb{Z}}$$ so because eliptic curves are self-dual for computing $H^1(K,E)$ it is enough to compute $E(K)$, It also implies $H^2(K,... | 3 | https://mathoverflow.net/users/65846 | 410525 | 167,962 |
https://mathoverflow.net/questions/410535 | 1 | Suppose that $f(t)$ is a square-integrable, band-limited function, i.e. the Fourier transform $\hat f$ has compact support.
**Problem:** Under which assumptions on a function $g(t)$ is the map $h(t) := e^{ig(t)}f(t)$ band-limited, i.e. $\hat h$ has compact support.
My idea would be to use the Paley-Wiener theorem w... | https://mathoverflow.net/users/170539 | Under which assumptions is $e^{ig(t)}f(t)$ of exponential type if $f$ is of exponential type? | A function is entire and of exponential type if and only if its Fourier transform has bounded support (This is Paley-Wiener theorem). Since your
$f$ belongs to this class, then, assuming that $g$ is also entire, $h=e^gf$ will belong to this class if and only if $g$ is linear.
If you do not want to assume that $g$ is ... | 2 | https://mathoverflow.net/users/25510 | 410539 | 167,967 |
https://mathoverflow.net/questions/410512 | 3 | Let $\mathcal I$, $\mathcal C$ be $2$-categories (or $(\infty, 2)$-categories, I'm interested in both cases) and assume that $\mathcal I$ is small, $\mathcal C$ has enough weighted (co)limits as you need. We can define the $2$-category of (co)lax functors $\operatorname{Fun}^\text{(co)lax}(\mathcal I, \mathcal C)$. (**... | https://mathoverflow.net/users/109318 | (Co)limits in lax functor categories | It depends on what kind of morphisms between your lax functors you're interested in. (As Tim says, lax functors are not the objects of the Gray internal hom.)
For any 2-category $I$, there is a 2-category $I'$ such that lax functors out of $I$ are the same as strict functors out of $I'$. Thus, the 2-category of lax f... | 5 | https://mathoverflow.net/users/49 | 410544 | 167,970 |
https://mathoverflow.net/questions/410360 | 1 | Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma\_d$ be a psd matrix such that
* $\mbox{trace}(\Sigma\_d) = 1$.
* $\|\Sigma\_d\|\_{op} = \mathcal O(1/d)$.
* The empirical eigenvalue distribution of $d \cdot \Sigma\_d$ converges weakly to some distribution ... | https://mathoverflow.net/users/78539 | Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$ | The distribution of $v^TB^{-1}AB^{-1}v$ is the same for every vector $v$ in the unit sphere either deterministic or independent of $W$. Once this is established, you are allowed to take $v=z/\|z\|$ independent of $W$; you can then apply the argument given at the end of the question together with concentration of the qu... | 1 | https://mathoverflow.net/users/141760 | 410551 | 167,971 |
https://mathoverflow.net/questions/410519 | 3 | Let $E\rightarrow M$ be a vector bundle.
Kirill Mackenzie in the book General theory of Lie groupoids and Lie algebroids associates a Lie algebroid to $E\rightarrow M$ in the following steps:
1. talk about zero-th and first order differential operators on $E\rightarrow M$, which are some nice maps of sections $\Gam... | https://mathoverflow.net/users/118688 | Lie algebroid associated to a vector bundle | **Question 1.** The two procedures indeed give the same Lie algebroid. One possible way of seeing this is by considering the flows of vector fields: a section of $T(GL(E))/GL(n)$ is a vector field on the frame bundle that is $GL(n)$-invariant, and this means that its flow is by $GL(n)$-equivariant diffeomorphisms. But ... | 2 | https://mathoverflow.net/users/69713 | 410554 | 167,974 |
https://mathoverflow.net/questions/410336 | 0 | In Kanamori's [Bernays and Set Theory](https://www.researchgate.net/publication/220366286_Bernays_and_Set_Theory) pages 20-21, a first order reflection principle due to Bernays is mentioned, that of:
>
> $$\sf \varphi \to \exists y \, (\text {Trans}(y) \land \varphi^y)$$ for formulas $\varphi$ without $\sf y$ or an... | https://mathoverflow.net/users/95347 | What is the proof of replacement from Bernays first order reflection? | The above reflection scheme does not imply the existence of an empty set, nor does it imply the existence of more than one element. In the domain with one element which is an element of itself, the reflection scheme holds.
Let T0 be the theory whose axioms are extensionality, foundation and the above reflection schem... | 1 | https://mathoverflow.net/users/133981 | 410568 | 167,978 |
https://mathoverflow.net/questions/410565 | 3 | Suppose that $ M $ is non-orientable with transitive action by a Lie group $ G $. **Does that imply that some Lie group $ G' $ acts transitively on the orientable double cover $M'$?**
This is true for compact dimension 2: the Klein bottle is an $ \operatorname{SE}\_2 $ manifold and so is the torus. The projective pla... | https://mathoverflow.net/users/387190 | Transitive action on non-orientable $ M $ lifts to orientable double cover | There is a general theory for lifting Lie group actions to covering spaces, see Bredon's monograph "Introduction to compact transformation groups", chapter 1, section 9. In the case of orientation covers one gets that any (effective, continuous) action of a Lie group $G$ on a non-orientable manifold lifts to a $G^\prim... | 7 | https://mathoverflow.net/users/1573 | 410571 | 167,981 |
https://mathoverflow.net/questions/409916 | 6 | Working in *mono-sorted* first order logic with equality $``="$ and membership $``\in"$:
* *Define:* $set(x) \equiv\_{df} \exists y \, (x \in y)$
* Axiomatize:
1. **Extensionality:** $( a \subseteq b \land b \subseteq a \to a=b)$
2. **Separation:** $(set(a) \to \exists \ set \ x : \forall y \, (y \in x \leftrightar... | https://mathoverflow.net/users/95347 | What's the consistency strength of this form of reflection? | This theory is inconsistent.
We note that by 1 and 2 that if set(x) and y⊆x, then set(y).
(a) There is a v such that ∀x(set(x)-->x∈v).
Proof:Suppose not. Then ∀v∃s∃t(s∈t∧s∉v). By 3 there is transitive x such that
```
set(x) and ∀v(v⊆x-->∃s∃t(s⊆x∧t⊆x∧s∈t∧s∉v). In particular
(x⊆x-->∃s∃t(s⊆x∧t⊆x∧s∈t∧s∉x). ... | 1 | https://mathoverflow.net/users/133981 | 410575 | 167,983 |
https://mathoverflow.net/questions/410580 | 2 | A Banach space $X$ has the *weak Phillips property* if the canonical projection $X^{\*\*\*}\to X^{\*}$ is sequentially weak$^{\*}$-weak continuous [[FreedmanÜlger2000](https://doi.org/10.1090/S0002-9939-00-05703-8), [Ülger2001](https://doi.org/10.4064/cm87-2-1)].
Let $1<p<\infty$ and $E = (\oplus\_{n=1}^{\infty}\ell^... | https://mathoverflow.net/users/164350 | Does $K( (\oplus_{n=1}^{\infty}\ell^1_n)_{\ell^p})$ have the weak Phillips property? | Yes, it does. This is essentially an unpublished result due to Hermann Pfitzner, see III.3.6 and III.3.7 in *$M$-ideals in Banach spaces and Banach algebras* by P. Harmand, W. Werner and myself ([Zbl 0789.46011](https://zbmath.org/?q=an%3A0789.46011)) along with the fact that $K(E)$ is an $M$-ideal in its bidual $L(E)$... | 4 | https://mathoverflow.net/users/127871 | 410588 | 167,986 |
https://mathoverflow.net/questions/410601 | 3 | Let ${\cal U}$ be a free ultrafilter on $\omega$. Is the linearly ordered set $(\omega+1)^\omega/{\cal U}$ complete?
| https://mathoverflow.net/users/8628 | Is $(\omega+1)^\omega/{\cal U}$ complete for ${\cal U}$ free ultrafilter? | No. Every ultraproduct by a free ultrafilter on $\omega$ is $\aleph\_1$-saturated. And infinite $\aleph\_1$-saturated linear orders cannot be complete!
*Proof:* Let $L$ be an infinite $\aleph\_1$-saturated linear order. Since $L$ is infinite, it contains an infinite increasing sequence or an infinite decreasing seque... | 6 | https://mathoverflow.net/users/2126 | 410606 | 167,991 |
https://mathoverflow.net/questions/385019 | 4 | This is a question in stable homotopy theory which I will boil down to a pure combinatorics question. If you're not interested in the homotopy theory, feel free to skip to the end for the combinatorial formulation.
**Homotopy theory:**
The question is basically whether the stable version of Serre's method of killin... | https://mathoverflow.net/users/2362 | Is there an elementary subexponential upper bound on the size of the stable stems? | There is indeed a subexponential bound on the size of the stable stems, which can be seen already at the $E\_2$ page of the May spectral sequence. See [John Palmieri's comment](https://mathoverflow.net/questions/385019/is-there-an-elementary-subexponential-upper-bound-on-the-size-of-the-stable-stem#comment981054_385019... | 0 | https://mathoverflow.net/users/2362 | 410608 | 167,992 |
https://mathoverflow.net/questions/410498 | 2 | I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.
That is, has anyone proved the following?
>
> Let $\{x\_{k,\alpha}\}\_{k\in \mathbb N,\alpha \in \mathcal A}\subseteq \mathbb R\_+$ with
> $\mathcal A$ a directed set. Then
>
>
> $$\sum\_{k=1}^{\infty} \sup\_{\bar \alp... | https://mathoverflow.net/users/470906 | Fatou's lemma and dominated convergence for nets and the counting measure | If $S \mathrel{:=} \sup\_{\overline\alpha} \inf\_{\alpha \ge \overline\alpha} \sum\_{k = 1}^\infty x\_{k, \alpha}$ were strictly less than $\sum\_{k = 1}^\infty \sup\_{\overline\alpha} \inf\_{\alpha \ge \overline\alpha} x\_{k, \alpha}$, then there would be some $N$ such that $S$ was strictly less than $\sum\_{k = 1}^N ... | 2 | https://mathoverflow.net/users/2383 | 410609 | 167,993 |
https://mathoverflow.net/questions/410617 | 6 | I'm dealing with the expression $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$. What is this approximately, if one is explicitly writing y in terms of x? There's no general formula for sixth powers unfortunately.
Also can one given an approximation of this so that the difference between the true y and the approximation g... | https://mathoverflow.net/users/265714 | Solution to sixth order equation | For $x,y>0$ there is a unique solution $y(x)$ to $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$ given by
$$y=\tfrac{1}{2} 3^{-1/3} \sqrt{\frac{\left(\sqrt{11664 x^2-3}+108 x\right)^{2/3}+3^{1/3}}{\bigl(\sqrt{11664 x^2-3}+108 x\bigr)^{1/3}}}- \tfrac{1}{2}.$$
Here is a plot of $y$ versus $x$.
$ have idempotents distinct from $0$ and $1$?
| https://mathoverflow.net/users/104638 | Idempotents in group rings of finite cyclic groups | More generally, when $G$ is a finite group and $\mathbb{K}$ is a field, the group algebra $\mathbb{K}G$ has no non-trivial idempotent if and only if the regular $\mathbb{K}G$-module is indecomposable. If $G$ is non-trivial and the characteristic of $\mathbb{K}$ is either zero or is coprime to $|G|$, then this is never ... | 5 | https://mathoverflow.net/users/14450 | 410670 | 168,010 |
https://mathoverflow.net/questions/410581 | 0 | This posting comes as a possible salvage to [this](https://mathoverflow.net/questions/409916/whats-the-consistency-strength-of-this-form-of-reflection) earlier presented reflective theory which was [proved inconsistent](https://mathoverflow.net/questions/409916/whats-the-consistency-strength-of-this-form-of-reflection/... | https://mathoverflow.net/users/95347 | Is this reflection schema equivalent to second order Bernays reflection? | This theory is inconsistent.
We note that by 1 and 2 that if set(x) and y⊆x, then set(y).
(a) There is a v such that ∀x(set(x)-->x∈v).
Proof:Suppose not. Then ∀v∃s∃t(s∈t∧s∉v). By 3 there is transitive x such that
```
set(x) and ∀v(v⊆x-->∃s∃t(s∈x∧t∈x∧s∈t∧s∉v). In particular
(x⊆x-->∃s∃t(s∈x∧t∈x∧s∈t∧s∉x). ... | 2 | https://mathoverflow.net/users/133981 | 410682 | 168,016 |
https://mathoverflow.net/questions/410577 | 17 | Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold which is of type $n$ after $p$-localization?
When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R \mathbb P ^2$.
Other than that, I'm not sure.
**Notes:**
* Recall that a finite CW complex $X$ is said to be of... | https://mathoverflow.net/users/2362 | For which $n$ does there exist a closed manifold of (chromatic) type $n$? | After discussing this with Tim we came up with the following answer:
The first steifel whiteny class $\omega\_1$ of $M$ can be written as the following composition:
$$M \to BO(n) \to BO \to BAut(\mathbb{S}) \to BAut(\mathbb{Z}) \simeq B\mathbb{Z}/2$$
But if $M$ is of type $\ge 2$ then $[M,BO]\simeq [\Sigma^\infty... | 10 | https://mathoverflow.net/users/22810 | 410684 | 168,017 |
https://mathoverflow.net/questions/410655 | 3 | Let $G$ be a reductive group over a number field $\mathbb{Q}$ and $K$ be a maximal compact subgroup of $G$.
Let $\Gamma$ be an arithmetic subgroup of $G(\mathbb{Q})$.
Let $\mathfrak{g}$ be the complexfied Lie alogebra of $G$ and $Z(\mathfrak{g})$ the center of the universal enveloping algebra $U(\mathfrak{g})$ of $\m... | https://mathoverflow.net/users/35898 | Why differential operator preserves $K$-finiteness of automorphic form? | Consider the map $\mathfrak{g}\otimes C^\infty(\Gamma\backslash G)\to C^\infty(\Gamma\backslash G)$, $X\otimes f\mapsto Xf$.
The group $K$ acts on both sides and the map is equivariant. If $f$ lies in a finite-dimensional $K$-module $M$, then $Xf$ lies in the finite-dimensional $K$-module that is the image of $\mathfra... | 3 | https://mathoverflow.net/users/nan | 410685 | 168,018 |
https://mathoverflow.net/questions/410640 | 3 | Consider the following nonlocal ODEs on $[1,\infty)$.
#1)
$$\begin{align}
r^2 f''(r) + 2rf'(r)-l(l+1) f(r) &= -\frac{(f'(1) + f(1))}{r^2}\\
f(1) &= \alpha \\
\lim\_{r\to \infty} f(r) &= 0
\end{align}$$
#2
$$\begin{align}
r^2 f''(r) + 2rf'(r)-l(l+1) f(r) &= -\frac{(f'(1) + f(1))}{r^2}\\
f'(1) &= \beta \\
\lim\_{r\to \... | https://mathoverflow.net/users/138705 | Dirichlet to Neumann operator for a nonlocal ODE | You have a linear ODE with explicit coefficients: the solutions can actually be written down explicitly via [variation of constants](https://en.wikipedia.org/wiki/Variation_of_parameters). To summarize the result1 when $\ell > 1$, let $g\_{a,b}(r)$ be given by
$$ g\_{a,b}(r) = \frac{a}{r^{\ell+1}} + \frac{b}{r^2} $$
... | 3 | https://mathoverflow.net/users/3948 | 410686 | 168,019 |
https://mathoverflow.net/questions/410678 | 2 | Given two SDEs $X^1$, $X^2$ :
$$X^i\_t=1+t+\int\_0^t\sigma\_i(s)dW\_s,\quad \forall t\ge 0,$$
where $\sigma\_i:\mathbb R\_+\to [1/2,1]$ are non-decreasing s.t. $\sigma\_1(t)\le \sigma\_2(t)$ for all $t\ge 0$. Can we prove $\mathbb P[\inf\_{0\le s\le t}X^1\_s>0]\ge \mathbb P[\inf\_{0\le s\le t}X^2\_s>0]$ for all $t\... | https://mathoverflow.net/users/261243 | Does higher volatility of SDE imply lower probability of staying positive? | Yes. This follows by time change: The process $(X^i\_t)$ equals the process $(1+t+W\_{\tau\_i(t)})$ in distribution, where
$$\tau\_i(t):=\int\_0^t\sigma\_i(s)^2\,ds,$$
so that $\tau\_2\ge\tau\_1$ and hence for the corresponding inverse functions we have $\tau\_2^{-1}\le\tau\_1^{-1}$.
It follows that
$$\begin{aligned}... | 3 | https://mathoverflow.net/users/36721 | 410689 | 168,021 |
https://mathoverflow.net/questions/410693 | 3 | Let $R$ be a ring and $I$ an ideal. I am interested under which conditions the following holds:
>
> **Claim.** Suppose that any two elements in $I$ have a non-trivial $\operatorname{gcd}$. Then $I$ is contained in some non-trivial principal ideal.
>
>
>
For simplicity you may assume that $R$ is a UFD so that $... | https://mathoverflow.net/users/127269 | A criterion for whether an ideal is contained in a principal ideal | This is true in UFDs. Let $f$ be any element of $I$ and let $\pi\_1,\dots, \pi\_n$ be its irreducible factors (choosing one from each class of irreducible factors that differ by a unit). If, for some $i$, $I \subseteq (\pi\_i)$ then we are done. Otherwise, for each $i$ let $g\_i$ be an element of $I$ that is nonzero mo... | 2 | https://mathoverflow.net/users/18060 | 410694 | 168,023 |
https://mathoverflow.net/questions/410677 | 8 | What is the étale fundamental group of the circle $X({\bf R})$, where
$$
X(k) = \{(x,y) \in k^2 \mid x^2+y^2 = 1\}?
$$
I know that there is a sequence
$$
1 \rightarrow \pi\_1^{et}(X({\bf C})) \rightarrow \pi\_1^{et}(X({\bf R})) \rightarrow Gal({\bf C}/{\bf R}) \rightarrow 1
$$
with $Gal({\bf C}/{\bf R})= \{z\mapsto z, ... | https://mathoverflow.net/users/6129 | Etale fundamental group of the circle | As Donu explained, the sequence splits by choosing an $\mathbf R$-point of $X$. So the only question remaining is what the $\operatorname{Gal}(\mathbf C/\mathbf R)$-action on $\pi\_1^{\text{ét}}(X\_{\mathbf C})$ is. I claim that the action is trivial, because the two points at infinity $V(x^2+y^2)$ are not defined over... | 10 | https://mathoverflow.net/users/82179 | 410695 | 168,024 |
https://mathoverflow.net/questions/410578 | 1 | This question is an example in the book Introduction to Probability Models 11th edition (Sheldon M.Ross). 3.6.2 A random graph:
A graph has $V$ nodes and a set $A$ of pairs of nodes in $V$ called arcs.
$V = \{1,2,...n\}$ and $A = \{(i,X(i), i=1,...,n\}$. The probability that node $i$ is connected to node $j$ is:
$P... | https://mathoverflow.net/users/471176 | how to compute the probability that a random graph has two components? | There are several alternative derivations. It seems that in the derivation you are following, the first component is defined as the component of the node 1, so if this component has size $k$, then only $k-1$ additional nodes from $\{2,\ldots,n\}$ must be chosen for this component.
| 1 | https://mathoverflow.net/users/7691 | 410696 | 168,025 |
https://mathoverflow.net/questions/410664 | 1 | Actually, I have asked this question in <https://math.stackexchange.com/questions/4330127/orthogonal-transformation-of-multivariate-bernoulli-gaussian-distribution>, but I think mathoverflow might be more appropriate for it.
Recently, I studied multivariate Bernoulli-Gaussian distribution which is very useful for spa... | https://mathoverflow.net/users/471275 | Orthogonal transformation of multivariate Bernoulli-Gaussian distribution | Each $X\_j$ has the Bernoulli-Gaussian distribution,
$$P(x\_j)=(1-p)\delta(x\_j)+pN(x\_j;0,\sigma^2).$$
To characterize the distribution of the variables $Y\_i=\sum\_{j}A\_{ij}X\_j$, for an orthogonal $n\times n$ matrix $A$, I calculate the moment generating function:
$$F(z\_1,z\_2,\ldots z\_n)=\mathbb{E}\left[e^{\... | 1 | https://mathoverflow.net/users/11260 | 410697 | 168,026 |
https://mathoverflow.net/questions/410533 | 7 | $\DeclareMathOperator\PSL{PSL}$ Let $ \mathbb{Z} $ be the ring of integers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma\_g $ be a surface of genus $ g \geq 2 $. Let $ \pi\_1(\Sigma\_g) $ be the fundamental group of the surface. There are many way to embed $ \pi\_1(\Sigma\_g) $ into $\PSL\_2(\mathbb{R}) $.... | https://mathoverflow.net/users/387190 | Do surface groups embed into PSL_2 over a real quadratic integer ring? | For the new question an answer is given by arithmetic Fuchsian groups. For example it is well-known that the reflection group associated with the regular right-angled pentagon in $\mathbb H^2$ contains every surface group as it contains the fundamental group of the non-orientable surface of Euler characteristic -1 as a... | 4 | https://mathoverflow.net/users/32210 | 410715 | 168,030 |
https://mathoverflow.net/questions/410716 | 0 | I'm stuck on a passage I do not understand, which reads:
>
> $$\int\_{r<|y|<1} \bigg| \frac{1}{(|y|^2 - r^2)^s |y|^n} - \frac{1}{|y|^{n+2s}}\bigg|\ \text{d}y$$
> $$\int\_1^r \bigg| \frac{1}{(t^2 - r^2)^s} - \frac{1}{t^{2s}}\bigg|\ \frac{\text{d}t}{t}$$
> $$r^{s}\int\_1^{1/r} \bigg| \frac{1}{(\tau^2 - 1)^s} - \frac{... | https://mathoverflow.net/users/471329 | Understanding the performed change of variable in this integration | Note that the integrand only depends on the absolute value of $y$, and not on the direction of the vector.
That means it is a good idea to substitute $t = |y|$.
Let's denote $$f(t) = \bigg| \frac{1}{(t^2 - r^2)^s t^n} - \frac{1}{t^{n+2s}}\bigg|$$ for positive real $t$.
Denote $t\mathbb S^n$ to be the sphere of radius $... | 2 | https://mathoverflow.net/users/470870 | 410720 | 168,032 |
https://mathoverflow.net/questions/410735 | 1 | A finite, simple, undirected graph $G=(V,E)$ is said to be *(vertex)-critical* if removing any vertex decreases its [chromatic number](https://en.wikipedia.org/wiki/Graph_coloring). By $\delta(G)$ we denote the *minimum degree* of $G$. The *degeneracy* of $G$ is defined by $$\text{degen}(G) = \max\{\delta(H): H \text{ ... | https://mathoverflow.net/users/8628 | Degeneracy vs minimal degree in critical graphs | There is such a graph, provided by the graph6 string
```
U???????wE?kAgWoAdGpQQ`GdA^?ldgsRW[C~w??
```
and is [uploaded to the House of Graphs](https://hog.grinvin.org/ViewGraphInfo.action?id=48171).
The graph is critical, as verified by the SageMath code:
```
a=Graph('U???????wE?kAgWoAdGpQQ`GdA^?ldgsRW[C~w??'... | 3 | https://mathoverflow.net/users/125498 | 410739 | 168,036 |
https://mathoverflow.net/questions/410368 | 10 | I am looking for a simple reference to the following fact:
>
> If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\varepsilon$ we can find a $C^\infty$ smooth function $f\_\varepsilon:\Omega\to\mathbb{R}$ such that $|f(x)-f\_\varepsilo... | https://mathoverflow.net/users/121665 | Density of smooth function in Hilbert spaces | I think the paper of Bonic and Frampton is a good reference, but it is true that for a specific result like the one you are after, there is some amount of lingo that one can bypass. Here is my take on their proof, very heavily inspired by their Theorem 1. I must add that there might very well be a more readable proof o... | 2 | https://mathoverflow.net/users/129074 | 410740 | 168,037 |
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