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https://mathoverflow.net/questions/442004 | 1 | Let $A$ be a normalized $N\times N$ GOE matrix. Let $\sigma^1<\sigma^2<\dots <\sigma^N$. We know that the largest eigenvalue converges $\sigma^N$ to 2 almost surely. Assume that $X\_1,\dots, X\_N$ are iid random variables with law $\mu$ so that $E\_{X\sim \mu}[e^{\alpha X}]<\infty$ for some $\alpha>0$.
Now, I try to ... | https://mathoverflow.net/users/168083 | How to bound $ P(\frac{1}{N}\sum_{i=1}^N \sigma^i X_i^2\ge ax)$ for eigenvalues of a normalized $N\times N$ GOE matrix? | Since we have the joint law [GOE eigenvalues](https://www.lpthe.jussieu.fr/%7Eleticia/TEACHING/Master2019/GOE-cuentas.pdf) density $\rho\_{N}$
$$P[\sigma\_{i}\in A\_{i}]=\iint\_{A\_{i}}\frac{1}{Z\_{N}}e^{-\frac{1}{4}\sum\_{i=1}x\_{i}^{2}}\prod\_{k>m}|x\_{k}-x\_{m}|\prod dx\_{i}$$
and so
$$P(\frac{1}{N}\sum\_{i=1}... | 1 | https://mathoverflow.net/users/99863 | 452064 | 181,729 |
https://mathoverflow.net/questions/451961 | 3 | I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I\_s(U)=\int\_U\int\_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take for simplicity $U$ to be a ball I am not sure how to evaluate the integration. If I shift the integral to origin i.e... | https://mathoverflow.net/users/493060 | Calculation of Riesz energy for balls | This is just an extended comment to [Iosif Pinelis's answer](https://mathoverflow.net/a/452021/108637) above, which provides an answer in terms of an unknown constant $C\_{d,s}$. Here we evaluate this constant.
Let $B$ be the unit ball. If $f(x)=(1-|x|^2)^p$ if $x\in B$ and $f(x)=0$ otherwise, then
$$(-\Delta)^{\alph... | 4 | https://mathoverflow.net/users/108637 | 452065 | 181,730 |
https://mathoverflow.net/questions/452073 | 1 | This is a follow up to [this question](https://math.stackexchange.com/q/4747105/1176963), where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.
(In this case, I want to find $\lim\_{t\to\infty} \lambda(A\cap [-t,t]... | https://mathoverflow.net/users/87856 | What is the measure of two sets which partition the reals into subsets of positive measure? | We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described [here](https://twitter.com/JDHamkins/status/1683929594175401984)), an... | 3 | https://mathoverflow.net/users/1946 | 452078 | 181,733 |
https://mathoverflow.net/questions/452072 | 0 | Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I haven't seen a way to do it with a strictly increasing sequence.
| https://mathoverflow.net/users/509865 | Finding a strictly increasing Collatz sequence of arbitrary length | Start with a number equal to $-1$ modulo $2^n$. Then, after one step, the number is $\frac{3(-1)+1}{2}=-1$ modulo $\frac{2^n}{2}=2^{n-1}$, so inductively it will increase $n$ steps.
| 3 | https://mathoverflow.net/users/35593 | 452080 | 181,734 |
https://mathoverflow.net/questions/452026 | 4 | I'm making a research on Galois theory, and found something interesting regarding the ring of symmetric polynomials:
At least up to 5 variables, we can rewrite the elementary symmetric polynomials with powers of symmetric polynomials of degree 1.
For the formulation of the question, i will introduce the index set $I\_q... | https://mathoverflow.net/users/504252 | Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1? | Yes.
For variables $x\_1,\dots, x\_k$, we have
$$ \sum\_{J \subseteq \{1,\dots, k \} } (-1)^{ k- |J|} \left(\sum\_{j \in J} x\_j\right)^k = k! x\_1 \dots x\_k $$
so that
$$ e\_k = \sum\_{(i\_1; \dots; i\_k )\in S} x\_{i\_1} \dots x\_{s\_k} = \frac{1}{k!} \sum\_{(i\_1; \dots; i\_k )\in S} \sum\_{J \subseteq \{i\_1... | 9 | https://mathoverflow.net/users/18060 | 452081 | 181,735 |
https://mathoverflow.net/questions/451272 | 4 | One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that
cannot decrease below zero and cannot be explicitly tested for zero.
An OCN $A$ over alphabet $\sum$ accepts a word $w \in \sum^\*$ from initial counter value $c\in\mathbb{N}$ if there is a run of $A$ on $w$ from an initial state ... | https://mathoverflow.net/users/377873 | Equivalence between deterministic and non-deterministic counter net | The short answer is that as far as I'm aware, this question is open.
It is however very close to ones that are settled. I provide some more detail below.
As you've correctly pointed out, the decidability status of decision problems relating to counter automata
often depends on the presence of non-determinism, but als... | 4 | https://mathoverflow.net/users/509800 | 452092 | 181,739 |
https://mathoverflow.net/questions/452037 | 5 | Let us say that a partial order is "countably closed with infima" if every descending $\omega$-sequence has an infimum.
Suppose $P$ and $Q$ are posets that are countably closed with infima, and for some dense $D \subseteq Q$, there is a projection $\pi : D \to P$, i.e. $\pi$ is an order-preserving map such that whene... | https://mathoverflow.net/users/11145 | Countable closure of quotient forcing | This does (unfortunately?) not hold in general.
Consider the following (trivial) forcings $P$ and $Q$:
The only thing $Q$ does is generically pick out some $N\leq\omega$, it does this in the following way. $Q$ has two types of conditions:
* atoms deciding $N$, one atom $a\_m$ representing "$N=m$" for $m\leq\omega... | 6 | https://mathoverflow.net/users/125703 | 452097 | 181,740 |
https://mathoverflow.net/questions/260561 | 5 | To have the composition of two monads be a monad itself, we need a
distributive law natural transformation satisfying certain coherence
laws.
I'm interested in the strict 2-monad case, i.e. a strict 2-functor
equipped with unit and counit natural transformations that satisfy the
zig-zag equations on the nose.
I pre... | https://mathoverflow.net/users/756 | Coherence laws when composing 2-monads | These are known as [pseudo-distributive laws](https://ncatlab.org/nlab/show/pseudo-distributive+law) and are the most common notion of distributive law of 2-dimensional monads, even when both 2-dimensional monads in question are strict (i.e. 2-monads rather than pseudomonads). These have been well studied, and there ar... | 1 | https://mathoverflow.net/users/152679 | 452100 | 181,742 |
https://mathoverflow.net/questions/452095 | 9 | The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}\_\*$ to the stable ring of homotopy groups of spheres $\pi\_\*(\mathbb{S})$. I have a few questions about geometric interpretations of algebraic facts under this correspondence.
1. Every framed manif... | https://mathoverflow.net/users/484277 | Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map | 1. I don't know how satisfied you would be by the paper:
"AN ELEMENTARY GEOMETRIC PROOF OF TWO THEOREMS OF THOM"
SANDRO BUONCRISTIANO and DEREK HACON
4. I think this happens already with the framed circle. One framing is nullbordant, but the other one is not (so k=1 or k=2). But maybe I don't understand the questio... | 9 | https://mathoverflow.net/users/12156 | 452107 | 181,744 |
https://mathoverflow.net/questions/450520 | 4 | This is inspired by the [problem of the Hoffman-Singleton Decomposition of](https://mathoverflow.net/a/450325) [$K\_{50}$](https://mathoverflow.net/a/450325). I wanted to look at smaller variants of this kind of problem, and so naturally I started wondering:
>
> Can the (edges of the) complete graph $K\_{16}$ be pa... | https://mathoverflow.net/users/29783 | Are there decompositions of $K_{16}$ by certain 3-regular graphs? | Such a decomposition exists. In particular a decomposition of 5 copies of $2Q\_3$ exists.
We can label each vertex with a binary number from 0000 to 1111 (or more mathematically with elements of $C\_2^4$). Note that each three linearly independent numbers $a, b, c$ correspond to a copy of $2Q\_3$ by connecting $u$ an... | 1 | https://mathoverflow.net/users/502833 | 452109 | 181,745 |
https://mathoverflow.net/questions/452105 | 4 | $\DeclareMathOperator\Hom{Hom}$Let $A$ be a finite abelian group and $\text{Sym}(A)$ the (abelian) group of symmetric bilinear forms over $A$ valued in $\mathbb{R}/\mathbb{Z}$.
A quadratic function on $A$ is $q:A\rightarrow \mathbb{R}/\mathbb{Z}$ such that $q(-a)=q(a)$ and
$$
\chi\_q(a,b)=q(a+b)-q(a)-q(b)
$$
is bilin... | https://mathoverflow.net/users/495347 | Quadratic refinements of a bilinear form on finite abelian groups | 1. By the classification of finite abelian groups we have $A \cong C\_1 \oplus \cdots \oplus C\_k$ for some cyclic groups $C\_i = \langle g\_i \rangle$ of prime power order $r\_i$. Let $\chi: A \times A \to \mathbb R / \mathbb Z$ be a symmetric bilinear map. Observe that $\chi(g\_i, g\_j)$ has order dividing $\gcd(r\_i... | 7 | https://mathoverflow.net/users/20598 | 452113 | 181,747 |
https://mathoverflow.net/questions/452068 | 3 | Let $U\subset\mathbf{P}^n\_{\mathbf{Z}}$ be an open subscheme such that the smooth morphism $U\to\text{Spec}(\mathbf{Z})$ is surjective. Suppose $U(\mathbf{Q})\neq\varnothing$ and $U(\mathbf{Z}\_p)\neq\varnothing$ for all primes $p$.
>
> Do we have $U(\mathbf{Z})\neq\varnothing$?
>
>
>
The answer is probably "... | https://mathoverflow.net/users/509694 | $\mathbf{Z}$-points of quasi-projective schemes | As Jason Starr points out, $U = \mathbf P^1{\setminus}\{[1{:}0],[1{:}4],[0{:}1]\}$ is a counterexample to the first question:
* It has a $\mathbf Z[1/3]$-point $[1{:}1]$ and a $\mathbf Z[1/2]$-point $[1{:}2]$, so $U(\mathbf Q) \neq \varnothing$ and $U(\mathbf Z\_p)\neq\varnothing$ for all primes $p$;
* But it does no... | 3 | https://mathoverflow.net/users/82179 | 452122 | 181,750 |
https://mathoverflow.net/questions/452121 | 0 | Suppose the following situation: we have to buy $n$ goods $g\_1,\,\dots,\,g\_n$ starting at day 1 and we can't buy more than one good per day.
On day $d$ the prices are $p\_1^d,\,\dots,\,p\_n^d;\quad p\_i^d\lt p\_i^{d+1}$, i.e. the prices for the goods keep rising.
>
> **Question:**
>
> what is the optimal bu... | https://mathoverflow.net/users/31310 | Optimality of a "shopping" heuristic | With the information given, for any strategy you might pick, there exists an instance of the game that makes this strategy awful. For example, suppose $n=2$. Let $i$ be the index of the good you buy at day $1$, and suppose that $p\_i^3=p\_i^2+\epsilon$ and that for all $j\neq i$, $p\_j^3=p\_j^2+N$ for small $\epsilon$ ... | 3 | https://mathoverflow.net/users/158721 | 452123 | 181,751 |
https://mathoverflow.net/questions/452116 | 5 | In quantum mechanics we have position and momentum operators $P$ and $Q$ acting on $L^2(\mathbb{R})$ in the usual way. I'm wondering what the von Neumann algebra generated by the bounded functions of $P$ and $Q$ is. I.e., what is
$$
\{f(Q), f(P)\,\vert\,f\in L^\infty(\mathbb{R})\}''?
$$
I have a hunch that the answer m... | https://mathoverflow.net/users/480683 | von Neumann algebra of canonical commutation relations | The C$^\*$-algebra generated by the exponentials $e^{isQ}$ and $e^{itP}$, $s,t \in \mathbb{R}$, is the CCR algebra. The von Neumann algebra they generate in this representation is all of $B(L^2(\mathbb{R}))$. That is because $e^{isQ}$ is multiplication by $e^{isx}$ and $e^{itP}$ is translation by $t$. Any operator that... | 8 | https://mathoverflow.net/users/23141 | 452125 | 181,752 |
https://mathoverflow.net/questions/452118 | 4 | I believe there is no good notion of homotopy groups for an arbitrary simplicial set $S$. However, when $S$ is *fibrant* - meaning that $S\to \*$ is a fibration - there is a definition. The singular realization of a topological space is fibrant, and one recovers the homotopy groups this way. Given a small category $\ma... | https://mathoverflow.net/users/66686 | Can one bypass the geometric realization in the definition of algebraic $K$-theory? |
>
> I believe there is no good notion of homotopy groups for an arbitrary simplicial set S.
>
>
>
It depends on what “good” means.
Kan's original definition works for arbitrary pointed simplicial sets: $$\def\Exi{\mathop{\sf Ex^∞}}π\_k(S,\*):=[S^k,\Exi S],$$
where $\Exi$ was defined by Kan in 1950s and $S^k$ can... | 6 | https://mathoverflow.net/users/402 | 452130 | 181,753 |
https://mathoverflow.net/questions/452127 | 0 | I would like to pose a question regarding the distance between a geodesic $\gamma(t)$ and a perturbed geodesic $\gamma\_{\epsilon}(t)$ on a Riemannian manifold. Specifically, is the distance controlled by a bound on the associated Jacobi field?
Following @Deane and @Otis (With heartfelt thanks), I'd like to attempt t... | https://mathoverflow.net/users/127966 | Compute distance between geodesics and perturbed geodesics on a Riemannian manifold via Jacobi field $\vert J \vert$ | You should formulate your question in more detail and rigor. Let me try to do this here and elaborate on Otis's comments.
Let $M$ be a Riemannian manifold and $\gamma: [0,1] \rightarrow M$ be a constant speed geodesic. You say that $\gamma\_\epsilon: [0,1] \rightarrow M$ is a perturbed geodesic but do not say what th... | 3 | https://mathoverflow.net/users/613 | 452147 | 181,755 |
https://mathoverflow.net/questions/452161 | 4 | Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong C\_{n\_1,...,n\_k}:=\mathbb{Z}\_{n\_1}\times ...\times \mathbb{Z}\_{n\_k}
$$
Hence I was wandering if there is any kn... | https://mathoverflow.net/users/495347 | Projective representations of a finite abelian group | The answer is $\ \displaystyle\bigoplus\_{i<j}\ \mathbb{Z}/\!\gcd(n\_i,n\_j)$. The reason is that for $G$ finite, $H^2(G,U(1))$ is the dual abelian group of $H\_2(G,\mathbb{Z})$. Now use the Künneth formula.
| 5 | https://mathoverflow.net/users/460592 | 452166 | 181,759 |
https://mathoverflow.net/questions/452168 | 7 | $\require{AMScd}$Given an endofunctor $F : C\to C$, its category of algebras is the inserter of $F$ and the identity functor. This means that there is a square
$$\begin{CD}
Alg(F) @>j>> C \\
@VjVV \Rightarrow@| \\
C @>>F> C
\end{CD}$$
filled by a 2-cell $Fj\Rightarrow j$ and 1- and 2-terminal among all such.
Given ... | https://mathoverflow.net/users/7952 | Eilenberg-Moore category as a 2-dimensional limit | Yes, the Eilenberg–Moore object for a monad $T$ can be presented in terms of two [equifiers](https://ncatlab.org/nlab/show/equifier) of the inserter $\mathbf{Ins}(T, 1)$. Denoting by $\phi \colon TU \Rightarrow U$, we equify $1\_U$ and $\phi \cdot \eta U \colon U \Rightarrow U$, as well as $\phi \cdot T \phi$ and $\phi... | 10 | https://mathoverflow.net/users/152679 | 452172 | 181,760 |
https://mathoverflow.net/questions/452171 | 8 | From the answer to another question I asked ([Projective representations of a finite abelian group](https://mathoverflow.net/questions/452161/projective-representations-of-a-finite-abelian-group/452166#452166)) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian group with... | https://mathoverflow.net/users/495347 | Trivial group cohomology induces trivial cohomology of subgroups | For any abelian group $A$ we have a canonical isomorphism $\bigwedge^2A\to H\_2(A,\mathbb{Z})$, given by the (anti-symmetric) Pontrjagin product $H\_1(A,\mathbb{Z})\times H\_1(A,\mathbb{Z}) \to H\_2(A,\mathbb{Z})$, see for example Section 6 of Breen, "[On the functorial homology of abelian groups](https://doi.org/10.10... | 11 | https://mathoverflow.net/users/460592 | 452176 | 181,762 |
https://mathoverflow.net/questions/452160 | 2 | As usual, for an $r$-uniform hypergraph $G$, denote by $ex\_r(n,G)$ the maximum number of edges an $r$-uniform, $G$-free hypergraph on $n$ vertices can have, and let $\lim \frac{ex\_r(n,G)}{\binom nr}\stackrel{n\to\infty}\longrightarrow\pi\_r(G)$.
Define the maximum of this quantity over graphs with $m$ edges as $\p... | https://mathoverflow.net/users/955 | Turán density of hypergraphs with very few edges | Seems, the answer is negative for $m=3$: consider $H = \{AB, BC, AC\}$ for disjoint $k$-element sets $A,B,C$. Then for $r=2k$ the is an $r$-graph $F$ which density is close to 0.5: consider only $r$-tuples that have an odd intersection with the first half-part $V\_1$ of vertex set. Suppose there is a copy of $H$. Then ... | 5 | https://mathoverflow.net/users/479618 | 452179 | 181,764 |
https://mathoverflow.net/questions/452030 | 7 | This is a less ambitious version of [Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive?](https://mathoverflow.net/questions/451975/is-the-lebesgue-measure-of-the-x-so-that-this-exponential-sum-is-o-sqrtn) .
Consider $$S\_N(x):=\sum\_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\al... | https://mathoverflow.net/users/479223 | Does there exist some irrational $x,\alpha$ so that this Weyl sum is $o(\sqrt N)$? | The estimate
$$ S\_N(x) = S\_N(x,\alpha) = o(N^{1/2}) \tag {1}$$
cannot hold when $x$ is irrational.
Heuristically, the reason comes from the (approximate) modularity properties of $S\_N(x,\alpha)$; this expression is periodic mod $1$ in both $x$ and $\alpha$, and from the [van der Corput B-process](https://en.wikipe... | 9 | https://mathoverflow.net/users/766 | 452184 | 181,767 |
https://mathoverflow.net/questions/452185 | 10 |
>
> **Problem.** Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is infinite and the intersection $X\cap\overline{ab}\cap\overline{cd}$ is not empty.
>
>
>
Here $\overline {uv... | https://mathoverflow.net/users/61536 | A projective plane in the Euclidean plane | Let $\ P^2(\mathbb Q)\ $ and $\ P^2(\mathbb R)\ $ be the projective planes over rationals and reals. Let $\,\ L\subseteq P^2(\mathbb R)\,\ $ be a straight line in the real plane such that
$$ L\cap P^2(\mathbb Q)\,\ =\,\ \emptyset, $$
and let projective map $\,\ F: P^2(\mathbb R)\to P^2(\mathbb R)\,\ $ map $\ L\ $ o... | 16 | https://mathoverflow.net/users/110389 | 452190 | 181,768 |
https://mathoverflow.net/questions/442196 | 1 | Let $W$ be a standard one dimensional Brownian motion, and $X$ a continuous process adapted to $W$ such that $\int\_0^T X^2 \, ds < \infty$ almost surely for some $T > 0$.
Define for any sequence of partitions $\mathcal P\_n = \{t\_1^n, \dotsc t\_{k\_n} ^n\}$ of $[0, T]$, the elementary integral process $Y^n$ associa... | https://mathoverflow.net/users/173490 | Convergence in sup norm of elementary integrals to the Itô integral process | Let
\begin{equation\*}
X^{(n)}\_t=\sum\_{i=1}^{k\_n-1} X\_{t\_i^n}1\_{(t\_i^n,t\_{i+1}^n]}(t).
\end{equation\*}
Since $X=\{X\_t,0\le t\le T\}$ is a continuous adapted process, then
\begin{gather\*}
\lim\_{n\to\infty} \sup\_{0\le t\le T}|X^{(n)}\_t - X\_t |=0, \qquad \text{a.s.}, \\
\lim\_{n\to\infty} \int\_{0}^{T} (... | 2 | https://mathoverflow.net/users/103256 | 452204 | 181,773 |
https://mathoverflow.net/questions/452211 | 7 | It’s well known that the heart of a t-structure is an abelian category. My question is that can we find some structure on a triangulated category which can “produce” an exact category in analogy with the t-structure? I would be appreciated if someone can answer this question or give me some related references.
| https://mathoverflow.net/users/510027 | structure in triangulated category similar to t-structure | In
*Jørgensen, Peter*, [**Abelian subcategories of triangulated categories induced by simple minded systems**](https://doi.org/10.1007/s00209-021-02913-5), Math. Z. 301, No. 1, 565-592 (2022). [ZBL1503.16015](https://zbmath.org/?q=an:1503.16015).
The following theorem due to Matthew Dyer is provided:
>
> Let C ... | 10 | https://mathoverflow.net/users/44499 | 452215 | 181,774 |
https://mathoverflow.net/questions/452213 | 0 | How to use the contraction mapping theorem to prove the following result: Let $X$ and $Y$ be Banach spaces, let $a>0$, and let $$B\_a=B\_a\left(z\_0\right)=\left\{z \in X:\left\|z-z\_0\right\| \leq a\right\}.$$ Suppose that $F$ is a $C^1$ map of $B\_a$ into $Y$, with $F^{\prime}\left(z\_0\right)$ invertible, and satisf... | https://mathoverflow.net/users/499114 | A contraction mapping theorem | With $G(z)=F'(z\_0)^{-1}(F(z))$ you reduce your problem to the following assertion:
>
> $\|G'(z)-id\|\le \theta$ for $\|z-z\_0\|\le a$ and $\|G(z\_0)\|\le (1-\theta)a$ imply that $G$ has a zero in $B\_a(z\_0)$.
>
>
>
Indeed, we are looking for a fixed point of $H(z)=z-G(z)$ in $B\_a(z\_0)$. From $\|H'(z)\|=\|i... | 3 | https://mathoverflow.net/users/21051 | 452216 | 181,775 |
https://mathoverflow.net/questions/452159 | 13 | I want to check if $$\left\lfloor \left( \sum\_{k=n}^{2n}{\frac{1}{F\_{2k}}} \right)^{-1} \right\rfloor =F\_{2n-1}~~(n\ge 3) \tag{$\*$}$$ where $\lfloor x \rfloor$ is th floor function.
The Fibonacci sequence is defined by $F\_1=1$, $F\_2=1$, $F\_{n+1}=F\_n+F\_{n-1}~(n\ge 2)$. Then we can get $$F\_n=\dfrac{\alpha^n-\... | https://mathoverflow.net/users/494116 | On the finite sum of reciprocal Fibonacci sequences | We need to prove, equivalently
$$\frac1{F\_{2n-1}+1}<\sum\_{k=n}^{2n}\frac1{F\_{2k}}\le \frac1{F\_{2n-1}}, $$
that is, by the above expression for $F\_{k}$, since $\beta=-\alpha^{-1}$, we need to check the double inequality
$$\frac1{\alpha^{2n-1}+\alpha^{-2n+1}+\sqrt5}<\sum\_{k=n}^{2n}\frac1{\alpha^{2k}-\alpha^{-2k}... | 9 | https://mathoverflow.net/users/6101 | 452222 | 181,776 |
https://mathoverflow.net/questions/452221 | 26 | This problem has been originally posted at [math.stackexchange](https://math.stackexchange.com/q/4746506/573047).
Since there are no answers and no comments there yet, I am crossposting it here to know if it is already known and tractable.
Here is the problem, slightly restricted with respect to the original:
sta... | https://mathoverflow.net/users/136218 | "Make all numbers equal" game | I claim that it is always possible to make all of the integers equal to each other. To see this, consider the allowable transformations
1. $(a,b,c,d,e,f)\mapsto(a+b,a+b,c,d,e,f)$,
2. $(a\_1,\dots,a\_6)\mapsto(a\_{\sigma(1)},\dots,a\_{\sigma(6)})$, and
3. $(a,b,c,d,e,f)\mapsto(a/n,b/n,c/n,d/n,e/n,f/n)$ whenever $a/n,b... | 26 | https://mathoverflow.net/users/22277 | 452229 | 181,778 |
https://mathoverflow.net/questions/452189 | 16 | In FGA 3.V, there is a citation for
>
> Mumford D. and Tate J., Séminaire de géométrie algébrique, Harvard University, Spring term 1962 (à paraître).
>
>
>
This seems to be the same seminar mentioned by Grothendieck in a letter to M. Artin on the 23rd of February, 1963 (see footnote 2 of letter number 19, page... | https://mathoverflow.net/users/73622 | Mumford–Tate 1962 "Algebraic geometry seminar" citation | I asked Steve Lichtenbaum, and he wrote "I started to write up the notes for the Grothendieck-Mumford seminar but then Grothendieck left and I never finished. The notes do not exist. They probably should not have been cited."
| 21 | https://mathoverflow.net/users/11926 | 452230 | 181,779 |
https://mathoverflow.net/questions/452228 | -2 | The context of this question is related to proving the first incompleteness by alternative ways related to Rosser's trick. So, for a proof by negation, we assume that $T$ is complete, and fulfills Gödel's criteria of effectively capturing all computable functions, etc..
Can $T$ decide on sentence $\sigma$ defined bel... | https://mathoverflow.net/users/95347 | Can this Rosser-like trick also work as a proof of the first incompleteness theorem? | I see two major issues with the proposal.
First, if $\sigma$ is refutable, then in the standard model the biconditional will be vacuously true, since there will be no $k$ for which $\text{Proof}\_T(k,\ulcorner\sigma\urcorner)$. This seems to trivialize the idea.
But second, even if one were to fix that somehow, the... | 2 | https://mathoverflow.net/users/1946 | 452234 | 181,780 |
https://mathoverflow.net/questions/452027 | 5 | Let $\{x\_i\}\_{i\geq 1}$ be iid random elements of the sequence space $\ell^2(\mathbb{N})$;
assume that $\|x\_i\|\_2 \leq 1$ almost surely. Let $\Sigma = \mathbb{E}[x\_1 \otimes x\_1]$.
Define
$$
\Sigma\_n = \frac{1}{n} \sum\_{i=1}^n x\_i \otimes x\_i.
$$
I am interested in upper bounding the following lower tail
$$... | https://mathoverflow.net/users/121486 | Lower tail of random rank one sums? | Warning: This is not a proper answer, just a dump of the thoughts I have had about this problem so far. Also, I'm not an expert in random matrix theory, so some bounds I'll be using may cry for improvement and if someone can do any of them better, I would appreciate both a helping hand and a criticism. We'll still get ... | 7 | https://mathoverflow.net/users/1131 | 452247 | 181,781 |
https://mathoverflow.net/questions/452151 | 2 | If $X$ and $Y$ are random variables, then a **maximal coupling** of $X$ and $Y$ is a coupling $\left(X', Y'\right)$ such that $\mathbf{P}\left(X'=Y'\right)$ is maximal (that is, the probability that the coupled variables coincide is optimal). By $``$coupling$"$, I'm referring to a random vector $\left(X',Y'\right)$ suc... | https://mathoverflow.net/users/95756 | Measures of dependence in a maximal coupling | $\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$Let $\mu$ and $\nu$ stand for the distributions of real-valued random variables (r.v.'s) $X$ and $Y$, respectively. Let $\Pi(\mu,\nu)$ denote the set of all probability distributions over $\R^2$ with marginals $\mu$ and $\nu$.
By Strassen's [Theorem 11](https://pro... | 2 | https://mathoverflow.net/users/36721 | 452250 | 181,783 |
https://mathoverflow.net/questions/452237 | 1 | This is a natural follow-up to my previous question, here: [A question regarding equational bases of the theory of the commutative and associative properties](https://mathoverflow.net/questions/409080/a-question-regarding-equational-bases-of-the-theory-of-the-commutative-and-assoc). As before, suppose we are working in... | https://mathoverflow.net/users/43439 | Follow up to a question on equational bases of the theory of the commutative and associative properties | In more simple terms, you want to show that if $S$ is any set of equations equivalent to $\def\ac{\mathrm{AC}}\ac=\{x+y=y+x,x+(y+z)=(x+y)+z\}$, then there exists $T\subseteq S$ of size $2$ such that $T$ is already equivalent to $\ac$.
This follows from the answer to the linked question: since $\ac$ is finite, $S$ is ... | 1 | https://mathoverflow.net/users/12705 | 452263 | 181,785 |
https://mathoverflow.net/questions/452264 | 2 | I would like to know some global geometry of the blowup of projective spaces.
Question: Let $Z\subseteq \mathbb{P}^n$ be a subvariety. How can I embed $BL\_Z\mathbb{P}^n$ in a projective space or product of projective spaces?
Examples:
1. Let $Z\subseteq \mathbb{P}^4$ be a union of two disjoint lines, then $BL\_Z... | https://mathoverflow.net/users/192152 | Blowing up the projective space along a subvariety | Let $I\_Z$ be the ideal sheaf of $Z$. If the sheaf $I\_Z(k)$ is globally generated (i.e., if hypersurfaces of degree $k$ cut out $Z$ as a scheme) and $m := h^0(I\_Z(k)) - 1$, then
$$
\mathrm{Bl}\_Z(\mathbb{P}^n) \subset \mathbb{P}^n \times \mathbb{P}^m.
$$
| 6 | https://mathoverflow.net/users/4428 | 452267 | 181,786 |
https://mathoverflow.net/questions/452055 | 4 | Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, where $U$ ranges over the open subsets of $X$.
Question: Is the set $\mathcal{C}(X)$ of all *clopen* subsets of $X$ a ... | https://mathoverflow.net/users/16107 | Is the set of clopen subsets Borel in the Effros Borel space? | Here is a negative answer for $\mathbb{N}^\mathbb{N}$.
Given a countably-branching tree $T$, we built a new countably-branching tree $T'$ in two steps. First, for any $\sigma \in T$ we place $(\sigma + 2)$ into $T'$, where $\sigma + 2$ is just obtained by adding $2$ to any number appearing in $\sigma$. In the second ... | 1 | https://mathoverflow.net/users/15002 | 452272 | 181,789 |
https://mathoverflow.net/questions/452142 | 6 | Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup\_{i=1}^r Z\_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for simplicity that $Z$ is connected and choose a point $x\in X\setminus Z$. As $Z$ has real codimension $2$, it is known that the ... | https://mathoverflow.net/users/476832 | Presentation of the fundamental group of a complex variety | I found an affirmative answer to this question on [Complex reflection groups, braid groups, and Hecke algebras](https://www.degruyter.com/document/doi/10.1515/crll.1998.064/html?lang=en). The proof is in Appendix A1, specifically on page 181.
| 5 | https://mathoverflow.net/users/476832 | 452273 | 181,790 |
https://mathoverflow.net/questions/452269 | 7 | This is a slight variation of [this recommended blog post](https://karagila.org/2022/dual-df-maps/) by [Asaf Karagila](https://mathoverflow.net/users/7206/asaf-karagila). Let $A$ be a set. Then:
1. $A$ is said to be *Dedekind-finite* if every injective map $f:A\to A$ is also surjective.
2. $A$ is said to be *dually D... | https://mathoverflow.net/users/8628 | Dedekind-finite-to-one vs Dedekind-finite | The answer is Yes.
For the first question, suppose that $A$ is Dedekind infinite and let $f$ be an injection from $\omega$ into $A$. Then the function that maps $f(2n)$ to $f(n)$ for each $n\in\omega$, maps $f(2n+1)$ to $f(0)$ for each $n\in\omega$, and leaves all other elements of $A$ fixed, is a surjection from $A$... | 7 | https://mathoverflow.net/users/101817 | 452287 | 181,792 |
https://mathoverflow.net/questions/452143 | 4 | Let $A\_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A\_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A\_n$ correspond to the ways of inserting one nontrivial parentheses into the expression $a\_1 a\_2 \cdots a\_n$, and the $(n-3)$-dimens... | https://mathoverflow.net/users/509958 | Orienting the dual of the associahedron | There are many combinatorial models for the associahedron -- parenthesizations of $n$ variables, triangulations of the $(n+1)$-gon, planar binary trees with $n$-leaves. I'll follow the OP's lead and use parenthesizations of $n$-variables.
First, let's remember how we label the vertices of the dual associahedron. We'l... | 2 | https://mathoverflow.net/users/297 | 452294 | 181,793 |
https://mathoverflow.net/questions/447472 | 11 | The following problem was considered in
Cohen and Kontorovich, "Local Glivenko-Cantelli",
<https://arxiv.org/abs/2209.04054>,
to appear in COLT 2023 (henceforth, CK'23).
Let $Y\_j$, $j\in\mathbb{N}$
be a sequence of independent
$\operatorname{Binom}(n,p\_j)$
random variables,
where $n\in\mathbb{N}$
and $1/2\ge p\_j\d... | https://mathoverflow.net/users/12518 | Open problem: $\log n$ factor in Binomial empirical process | See the preprint by
Moïse Blanchard and Václav Voráček, titled "Tight Bounds for Local Glivenko-Cantelli", available here:
<https://arxiv.org/abs/2308.01896>
.
It clearly explains their construction.
| 1 | https://mathoverflow.net/users/12518 | 452296 | 181,794 |
https://mathoverflow.net/questions/451222 | 3 | We say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ has *property ${\bf B}$* if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $S\cap e \neq \emptyset \neq e \setminus S$.
Let $(P,\leq)$ be a [partially-ordered set (poset)](https://en.wikipedia.org/wiki/Partially_or... | https://mathoverflow.net/users/8628 | Posets such that the collection of principal down-sets does not have property ${\bf B}$ | Let $M$ be the ordered Mostowski model (T. Jech, *The Axiom of Choice*, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subset of $A$; then $S$ is a union of finitely many intervals. So if $S$ has a lower bound then $S$ is disjoint from (many)... | 4 | https://mathoverflow.net/users/5903 | 452297 | 181,795 |
https://mathoverflow.net/questions/452286 | 2 | Recall that a domain $D \subseteq \mathbb C$ is called *regular* if for each point $x \in \partial D$, we have $\mathbf P\_x\lbrack \tau\_D = 0\rbrack = 1$, where $\tau\_D = \inf\{t > 0 : B\_t \notin D\}$, and $(B\_t)\_{t \ge 0}$ is a Brownian motion.
1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
2. Is every... | https://mathoverflow.net/users/510096 | Is every simply connected domain regular? | Yes to both. Both were already answered here <https://math.stackexchange.com/questions/4389689/is-every-simply-connected-set-in-the-plane-regular-for-brownian-motion>
So for Q1 we can use the simple-arc criterion.
>
> The regularity of every boundary point of an open set in $\mathbb R^2$ is in fact strongly relat... | 2 | https://mathoverflow.net/users/99863 | 452303 | 181,797 |
https://mathoverflow.net/questions/452276 | 2 | Let $f$ be a :
1. $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$,
2. for all $x> 0,~f(x)>0$,
3. for all $x< 0,~f(x)<0$,
I am struggling to find a bound for the distance between the root of $f$ and the root of its Gaussian convolution. Formaly,
>
> Let $f$ satisfies the previous properties and define $g$ as:
> ... | https://mathoverflow.net/users/510079 | Distance between root of $f$ and its Gaussian convolution | $\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}\newcommand{\si}{\sigma}$After some simple rewriting, we see that $\la\_\si$ is the root $x=x\_\si$ of the equation $G(\si,x)=0$, where
\begin{equation\*}
G(\si,x):=Ef(x+\si Z)[=\si\sqrt{2\pi}\,g\_\si(x)\text{ if }\si>0]
\end{equation\*}
and $Z$ is a standard normal... | 3 | https://mathoverflow.net/users/36721 | 452305 | 181,798 |
https://mathoverflow.net/questions/451627 | 1 | On page 268 of Prof. John M Lee's book "Introduction to Smooth Manifolds" (second edition), it says if $E$, $M$ and $F$ are smooth manifolds with or without boundary, $\pi:E\to M$ is a smooth map and all the local trivializations $\Phi:\pi^{-1}(U)\to U\times F$ are diffeomorphisms, where $U$ is an open subset of $M$, t... | https://mathoverflow.net/users/41686 | On the definition of smooth fiber bundle and smooth manifolds with boundary | Allowing boundaries makes no big difference. Figure out, for example, the simple case of the first projection $\pi:D^2\times I\to D^2$ where $F=I$ is the compact interval and $M=D^2$ is the compact $2$-disk; the general case is not much more complicated.
First, there is nothing to change in the definition of a fibre ... | 1 | https://mathoverflow.net/users/105095 | 452306 | 181,799 |
https://mathoverflow.net/questions/452304 | 15 | I am interested in the amenability properties of infinite products of solvable groups. The following facts are well-known:
1. Any solvable group is amenable.
2. The class of solvable groups is closed under finite products and quotients (but not infinite products).
3. Infinite products of solvable groups of a fixed de... | https://mathoverflow.net/users/117822 | Is the infinite product of solvable groups amenable? | The free group $F\_2$ is residually nilpotent, meaning that the intersection of its lower central series is trivial, because the length of an element in the $k$th term of the lower central series is bounded below by $k$. It follows that $F\_2$ embeds in a direct product of nilpotent groups. So there is an infinite prod... | 21 | https://mathoverflow.net/users/460592 | 452307 | 181,800 |
https://mathoverflow.net/questions/452013 | 14 | This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help.
Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{F}\_{q}$. Is there a clean reference in the literature to the fact that there must be at least one regular semisimple ... | https://mathoverflow.net/users/155467 | Existence of a regular semisimple element over $\mathbb{F}_{q}$ | See Proposition 7.1.4 in Dat, Orlik, and Rapoport, *[Period domains over finite and $p$-adic fields](https://doi.org/10.1017/CBO9780511762482)*, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements i... | 7 | https://mathoverflow.net/users/4494 | 452314 | 181,804 |
https://mathoverflow.net/questions/452331 | -4 | Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a unit vector $v \in \mathbb{R}^2$ and a continuous function $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $S^2$ and
$$F(x)\neq f(x)v \ \ \ \ \forall x\in S^2?$$
| https://mathoverflow.net/users/115905 | Uncountable Cantor's diagonal argument on $S^2$ | Sure. Simply choose $f(x)=\|F(x)\|+1$, and pick an arbitrary unit vector $v\in\mathbb{R}^2$.
| 1 | https://mathoverflow.net/users/11919 | 452332 | 181,809 |
https://mathoverflow.net/questions/452337 | 1 | The following lemma is from the book *Discrete groups* by Ohshika.
>
> If a language $L$ is accepted by a non-deterministic automaton, then $L$ is regular, i.e., there exists a finite state automaton $M$ such that $L = L(M)$.
>
>
>
Proof. Let $M = (\Sigma,A,\mu,F,\Sigma\_0)$ be a non-deterministic automaton ac... | https://mathoverflow.net/users/323920 | If a language $L$ is accepted by a non-deterministic automation, then $L$ is regular | [Wikipedia](https://en.wikipedia.org/wiki/Powerset_construction) and [Hopcroft and Ullman](https://en.wikipedia.org/wiki/Introduction_to_Automata_Theory,_Languages,_and_Computation) require a unique start state for their NFA's. You can transform an NFA with many start states into an NFA with just one by adding a new st... | 3 | https://mathoverflow.net/users/1650 | 452338 | 181,811 |
https://mathoverflow.net/questions/452295 | 5 | Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x = f(0)\ne f(1)$?
That is, does then there always exist a nontrvial continuous curve starting at $x$ and staying out... | https://mathoverflow.net/users/36721 | On the boundary of a simply connected set | Here's a variant of Fernando Muro's construction.
Let $X$ be the closure in $\mathbb R^2$of the graph of the function
$g(x):=x\sin\big(1/\sin(1/x)\big)$ defined on $[0,+\infty)\setminus\{1/n\pi:n\in \mathbb Z\_+\}$. Let $U:=\mathbb R^2\setminus X$, again a simply connected open set. The function $g$ has a set of esse... | 5 | https://mathoverflow.net/users/6101 | 452358 | 181,817 |
https://mathoverflow.net/questions/452355 | 0 | Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere that the class of a vector bundle of odd degree on $\mathbb P^1$ is not equal to the class of a sheaf equivariant under... | https://mathoverflow.net/users/12395 | Equivariant sheaves on $\mathbb P^1$ | Let me explain why the line bundle $\mathcal{O}(1)$ does not admit a $\mathrm{PGL}(2)$-equivariant structure. Indeed, if it does, then the vector space
$$
\mathrm{Hom}(\mathcal{O}, \mathcal{O}(1))
$$
would have a structure of a $\mathrm{PGL}(2)$-representation, compatible with the standard $\mathrm{SL}(2)$-representati... | 3 | https://mathoverflow.net/users/4428 | 452361 | 181,818 |
https://mathoverflow.net/questions/452369 | 2 | Is there a tabulation somewhere of the isomorphism classes of distributive lattices $L$ with $|L|=n$ for small $n$? Google has not found me one.
| https://mathoverflow.net/users/10366 | Classification of small finite distributive lattices | I have the same problem to collect posets with certain properties and for distributive lattices I have the following solution which was good enough for my purposes.
Here is a sage program to obtain all distributive lattices (in form of their leq matrices) on $n$ points which works for $n \leq 9$ in the sage online cell... | 2 | https://mathoverflow.net/users/61949 | 452371 | 181,822 |
https://mathoverflow.net/questions/452353 | 7 | Let $G$ be an infinite group. Let $N\_0$ be the set of all $x\in G$ for which the conjugacy class $\{y^{-1}xy: y\in G\}$ is a finite set. Clearly $N\_0$ is a normal subgroup. Iteratively, form an ascending transfinite sequence by
* for $n$ a non-limit ordinal, let $N\_n\subseteq G$ be the set of all $x\in G$ for whic... | https://mathoverflow.net/users/164350 | Finite conjugacy classes | **Q0:** For references try starting with Robinson's *A course in the theory of groups*, starting around 14.5.5. There the FC center is defined and some characterizations are given for FC groups (groups with finite conjugacy classes). From there it is not a great leap to define the second FC center and so on, but I am n... | 8 | https://mathoverflow.net/users/20598 | 452372 | 181,823 |
https://mathoverflow.net/questions/452366 | 3 | The question concerns a very general setting and a very general inequality about KL divergence. I'm writing this thread to verify whether my intuition is correct.
Let $E\_1, E\_2$ be two measurable spaces, $f, g: E\_1 \rightarrow E\_2$ two measurable functions, and $A, B$ two random variables taking values in $E\_1$.... | https://mathoverflow.net/users/510181 | A general inequality for KL divergence of functions of variables | $\newcommand{\si}{\sigma}\newcommand{\F}{\mathcal F}\newcommand{\G}{\mathcal G}\newcommand{\pa}{\parallel}$The first inequality is obviously false in general, e.g. when $A=B$ but $f(A)$ differs from $g(B)=g(A)$ in distribution.
The second inequality is true. A bit more generally, let $\mu$ and $\nu$ be probability me... | 2 | https://mathoverflow.net/users/36721 | 452373 | 181,824 |
https://mathoverflow.net/questions/452359 | 3 | $\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) constant speed, first horizontally right to $(2,0)$, then horizontally left to $(1,0)$, and finally goes counterclockwi... | https://mathoverflow.net/users/36721 | Can such a set be simply connected? | Let $I\_0$ be the interval where $f$ is a horizontal motion, and let $I\_1$ the interval where it is a circular motion. So up to reparametrisation (and adopting complex notation) $I\_0=[-3,0]$ and $I\_1=[0,4\pi]$, and $f(t)=\min(t+3,1-t)$ in $I\_0$ and $f(t)=e^{it}$ in $I\_1$.
Since $|g(3\pi/2)+i|<1/16$ and $|g(5\pi/... | 4 | https://mathoverflow.net/users/6101 | 452377 | 181,826 |
https://mathoverflow.net/questions/451805 | 6 | Suppose we have a compact three-manifold $M$, a codimension-one foliation $F$ of $M$, and a one-form on $\alpha$, chosen so that $F$ is tangent to $\alpha = 0$. We deduce that $\alpha \wedge d\alpha = 0$. Also, there is some one-form $\beta$ so that $d\alpha = \alpha \wedge \beta$.
The *Godbillon-Vey invariant* of $F... | https://mathoverflow.net/users/36688 | The current situation of the Godbillon-Vey invariant conjecture | Check out:
*Hilsum, Michel*, [**Functions with bounded variation and the class of Godbillon-Vey**](https://doi.org/10.1093/qmath/hav013), Q. J. Math. 66, No. 2, 547-562 (2015). [ZBL1401.57040](https://zbmath.org/?q=an:1401.57040).
| 4 | https://mathoverflow.net/users/1345 | 452384 | 181,829 |
https://mathoverflow.net/questions/452386 | 2 | Motivation:
-----------
I'm studying certain properties of conjugation in $SL(n,q)$. There's a nice number, a bit like a covering number, that one can associate with an arbitrary group. In writing a programme in GAP to compute this number, a shortcut comes to mind - I'm not going to share what exactly - that involves... | https://mathoverflow.net/users/42153 | Is there a maximum length of the chief series of $SL(n,q)$? | Well, $PSL(n,q)$ is usually simple, so apart from $SL(2,3)$ you're looking at two plus the number of prime factors, counting multiplicity, of the greatest common divisor of $n$ and $q-1$. Of course, this is unbounded.
| 8 | https://mathoverflow.net/users/460592 | 452387 | 181,830 |
https://mathoverflow.net/questions/452385 | 1 | Let $H\_n$ be the $n$th probabilistic Hermite polynomial of degree n and $\eta = \exp(-x^2/2)/\sqrt(2 \pi)$ be the standard Gaussian density.
I would like to compute the integral $f\_n(x) = \int H\_n(x - z) \eta(z) dz$. Any hope to get a closed form expression?
Some ideas:
1. The $n$th Hermite polynomial $H\_n$ c... | https://mathoverflow.net/users/510195 | Convolution of a Hermite polynomial with Gaussian kernel | Using the definition of the Hermite polynomials and then integrating by parts $n$ times, we get $f\_n(x)=x^n$.
Details:
\begin{equation}
\begin{aligned}
f\_n(x)&=\int(-1)^n\eta^{(n)}(x-z)\frac{\eta(z)}{\eta(x-z)}\,dz \\
&=(-1)^ne^{x^2/2}\int\eta^{(n)}(x-z)\ e^{-xz}\,dz \\
&=(-1)^{n-1}e^{x^2/2}x\int\eta^{(n-1)}(x... | 1 | https://mathoverflow.net/users/36721 | 452388 | 181,831 |
https://mathoverflow.net/questions/452365 | 4 | A factor $R$ is called stable if $M\_n(R)\cong R$ for all $n>0$. For the sake of this question, we call a factor *backwards stable* if $R\cong M\_n(S)$ implies $S\cong R$ where $S$ is allowed to be any other factor.
If $R$ is stable then it is backwards stable if $M\_n(R)\cong M\_n(S) \implies R\cong S$.
My question ... | https://mathoverflow.net/users/485160 | Backwards stable factors | For $\textrm{II}\_1$ factors, your definitions of backwards stable and stable are the same. The point here is that for a $\textrm{II}\_1$ factor $R$ you can talk about $M\_n(R)$ for all positive number $n$ instead of just positive integers. This is called amplification and is usually written as $R^n$. This is done by c... | 5 | https://mathoverflow.net/users/504602 | 452401 | 181,836 |
https://mathoverflow.net/questions/452421 | 1 | Let $S$ be the set of integers with largest prime factor bounded by a given positive integer $k$. Is there a formula for the asymptotic density of such a set $S$?
| https://mathoverflow.net/users/18659 | Prime factors bounded by $k$ | If $k$ is fixed, then the simple bound $|S\cap[1,x]|\leq(\log\_2 x)^{\pi(k)}$ shows that the aymptotic density of $S$ is zero. For stronger bounds, see Chapter III.5 in Tenenbaum: Introduction to analytic and probabilistic number theory.
| 8 | https://mathoverflow.net/users/11919 | 452422 | 181,840 |
https://mathoverflow.net/questions/452425 | 6 | I am not familiar with the definition of total positivity. I am not sure about the link between *log-concavity* and *total positivity*.
* In a paper [On Variation-Diminishing Integral Operators of the Convolution Type](https://www.jstor.org/stable/87970) of Schoenberg, he defines *Pólya frequency functions* as a *tot... | https://mathoverflow.net/users/510079 | Total positivity, log-concavity and Pólya frequency | $\newcommand{\R}{\mathbb R}$For a positive integer $r$, a measurable function $f\colon\R\to\R$ is called a Pólya frequency function of order $r$ (abbreviated as PF$\_r$) if the matrix $(f(x\_i-y\_j))\_{i,j,=1}^r$ is totally positive for all real $x\_i$ and $y\_j$ such that $x\_1\le\dots\le x\_r$ and $y\_1\le\dots\le y\... | 8 | https://mathoverflow.net/users/36721 | 452439 | 181,846 |
https://mathoverflow.net/questions/452442 | 1 | Let $d \in \mathbb N^\*,p \in [1, \infty]$ and $T>0$. Let
$$
F :[0, T] \to L^p (\mathbb R^d; \mathbb R\_{\ge 0}), t \mapsto F\_t
$$
be measurable. I would like to ask if there is a measurable function $G:[0, T] \times \mathbb R^d \to \mathbb R\_{\ge 0}$ such that
* $G(t, \cdot) \in L^p (\mathbb R^d; \mathbb R\_{\ge 0... | https://mathoverflow.net/users/99469 | Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version? | $\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\LL}{\mathcal L}\newcommand{\si}{\sigma}$The answer is yes, at least for $p\in[1,\infty)$.
Indeed, $L^p(\R^d)$ is a separable metric space. So, for each real $\ep$ there is a countable measurable partition $(B\_{\ep,j})$ of $L^p(\R^d)$ such that for ... | 2 | https://mathoverflow.net/users/36721 | 452445 | 181,848 |
https://mathoverflow.net/questions/452398 | 2 | Prove that:
$$ f(x) = \log\big(
{}\_2F\_1(a,\,b\,;\,c\,;\,x^{-1})\big),\;\;a,b,c>0
$$
is convex (and decreasing) on $(1,\infty)$.
It actually seems that the stronger result that $f\big((x+1)^{\beta}\big)$, $\beta>0$, is completely monotonic, is true. I saw a post on here proving a similar result using continued ... | https://mathoverflow.net/users/510206 | Log convexity of hypergeometric function for $a,b,c>0$ | Using the series representation of the hypergeometric function, we see that $\_2F\_1(a,b\,;c\,;(1+x)^{-1})$, $x>0$, is the pointwise convergent limit of positive sums of completely monotonic functions for $a,b,c>0$, and is thus completely monotonic.
Since completely monotonic functions are log convex, it follows that... | 0 | https://mathoverflow.net/users/510206 | 452453 | 181,850 |
https://mathoverflow.net/questions/449715 | 6 | Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations
$$G(M) := \text{Fun}(M,G)$$
(viewing $M$ as a discrete groupoid). One of the multiple ways of improving this prestack to a stack is to ins... | https://mathoverflow.net/users/139911 | Anafunctors vs the plus construction | The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$
Suppose for simplicity, we are in a site $(S,J)$ equipped with a subcanonical singleton pretopology $J$. This means ... | 4 | https://mathoverflow.net/users/4177 | 452456 | 181,852 |
https://mathoverflow.net/questions/452423 | 1 | The following equation may be meaningful, but how can we make it well-defined
$$\delta(x-a)\cdot\delta(x-b)=0$$
**Question**: How do we defined this equation? Or more broadly define product between generalized functions with certain restrictions. Whether this definition satisfies the product rule [$D(fg)=Df\cdot g+f\cd... | https://mathoverflow.net/users/510003 | Product of Dirac delta function | You can, of course, if you wish, consult the highly sophisticated treatise of Hörmander to verify that your result holds (for distinct values of $a$ and $b$—there is no serious text which claims this for the case $a=b$). Or you can refer to work which precedes this by decades and is completely elementary, in order to v... | 4 | https://mathoverflow.net/users/510259 | 452459 | 181,854 |
https://mathoverflow.net/questions/452405 | 6 | Let $A$ be a C\*-algebra, and consider the infinite tensor power $A^{\otimes {J}}$, where $J$ is infinite (we consider the minimal or maximal tensor product). To any finite permutation, which is a bijection $\sigma\colon J\to J$ fixing all but finitely many elements of $J$, we can associate a \*-homomorphism $\hat{\sig... | https://mathoverflow.net/users/510216 | Is there an element in an infinite tensor power of a C*-algebra that is invariant under finite permutations? | Thanks to the comments of @DiegoMartinez and @CalebEckhardt, I can answer my question.
Briefly, the answer is no: any $x$ satisfying the hypothesis is of the form $\lambda 1$.
Let us consider $x \in A^{\otimes J}$, where $J$ is infinite, such that $\hat{\sigma}(x) = x$ for every finite permutation $\sigma$. As sugg... | 3 | https://mathoverflow.net/users/510216 | 452472 | 181,858 |
https://mathoverflow.net/questions/452474 | 0 | Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|\_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of Lebesgue measurable functions $f:\mathbb R^d \to \mathbb R$ such that
$$
\|f\|\_{\tilde L^p} := \sup\_{x \in \mathbb R^d}... | https://mathoverflow.net/users/99469 | Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$? | $\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}\newcommand\J{\mathcal J}$No. Consider first the case $d=1$. Let $\J$ denote the set of all subsets of $\Z$.
For $J\in\J$, let
$$f\_J:=\sum\_{j\in J}1\_{[j,j+1)},$$
so that $f\_J\in\tilde L^p(\R^d)$.
For any two distinct $J$ and $K$ in $\J$, we have $\|f\_J-f\_K\|\_{\til... | 3 | https://mathoverflow.net/users/36721 | 452475 | 181,859 |
https://mathoverflow.net/questions/452469 | 5 | In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.
If I am not wrong, over a prime there is a similar theorem for $\mathbb{F}\_p$ cohomology, that is Mandell's Theorem: under ad... | https://mathoverflow.net/users/140013 | Analogues of Sullivan Theory at a prime for coformality | Mandell shows that, under some hypotheses, the $\mathbb{F}\_p$-cochains detect the $\mathbb{F}\_p$-homotopy type in the sense that there is such an equivalence $X \simeq Y$, if and only if, there is an equivalence of $E\_\infty$-algebras $C\_\*(X; \mathbb{F}\_p) \simeq C\_\*(X;\mathbb{F}\_p)$. This is different than sa... | 7 | https://mathoverflow.net/users/134512 | 452480 | 181,860 |
https://mathoverflow.net/questions/452464 | 1 | Maple seems to suggest the following formula for $n>0$, $p \le q$:
\begin{align}
\frac{d^n}{d x^n} & {}\_p F\_q (a\_1,\ldots,a\_p;b\_1,\ldots,b\_q;1/x) \\[8pt]
= {} & (-1)^n \hspace{1pt} n!\hspace{2pt} \frac{\prod\_j a\_j}{\prod\_k b\_k}\hspace{2pt} x^{-n-1} {}\_{p+1} F\_{q+1} (n+1, a\_1+1, \ldots, a\_p+1; 2, b\_1+1,... | https://mathoverflow.net/users/510206 | $n$th Derivative of $_p F_q(a_1,...,a_p; b_1,...,b_q;x^{-m})$, $p \le q$ | The first displayed identity can be verified in a straightforward manner, by differentiating the power series for ${}\_{p} F\_{q} (a\_1,...,a\_p;b\_1,...,b\_q;1/x)$ (in the powers of $1/x$) term-wise $n$ times in $x$, and then comparing the coefficients of the resulting power series with the coefficients of the power s... | 2 | https://mathoverflow.net/users/36721 | 452490 | 181,862 |
https://mathoverflow.net/questions/452256 | 4 | $\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$A subset $A$ of a topological space $X$ is called regular closed if $A=\cl
\_{X}\int\_{X}A$.
The family of all regular closed sets of a topological space is denoted by $%
\mathcal{R}\left( X\right) $.
An ultrafilter $\mathcal{U}$ on $\mathcal{R}\left( X\righ... | https://mathoverflow.net/users/86099 | Stone–Čech compactification and an ultrafilter of regular closed sets | The family $\mathcal{F}=\{ F\in \mathcal{R}( \beta X) :p\in \operatorname{int}\_{\beta X}F\}$ is indeed a filterbase; it is a base for the neighbourhood filter at $p$. As [noted](https://mathoverflow.net/questions/452256/stone-%c4%8cech-compactification-and-an-ultrafilter-of-regular-closed-sets#comment1169517_452256) i... | 3 | https://mathoverflow.net/users/5903 | 452494 | 181,863 |
https://mathoverflow.net/questions/447782 | 10 | This is a cross-post of [this question from MSE](https://math.stackexchange.com/questions/4617947/serre-fibrations-between-spaces-of-embeddings-reference-request).
---
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ cons... | https://mathoverflow.net/users/144250 | Reference request - Fibrations between spaces of embeddings | It is a Serre fibration, and this is one of the rare cases where you can give an elementary argument and don't need something hard like [Edwards-Kirby](https://www.jstor.org/stable/1970753?origin=crossref), [Lees](https://www.projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-3/Imm... | 4 | https://mathoverflow.net/users/798 | 452495 | 181,864 |
https://mathoverflow.net/questions/452341 | 3 | Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ satisfies something". In my case, I have something close to being the identity, yet I don't know the best approach to show it is an e... | https://mathoverflow.net/users/140013 | A fiber-like method to show equivalence of infinity categories | An obvious necessary condition for $f$ to be a categorical equivalence is that $f$ is weakly equivalent to a (co)cartesian fibration of quasicategories, i.e., $f$ is an analogue of a [Street fibration](https://ncatlab.org/nlab/show/Street+fibration) of quasicategories.
Since $f$ is already a Joyal fibration, a plausi... | 3 | https://mathoverflow.net/users/402 | 452499 | 181,868 |
https://mathoverflow.net/questions/452509 | 20 | Say we have DC-λ where λ is some inaccessible cardinal. Is that enough to develop all of ordinary mathematics? If not, is there a strengthening that is but that nevertheless does not assume full choice high up in the universe of sets?
| https://mathoverflow.net/users/168572 | How much of the axiom of choice do you need in mathematics? | Your hypothesis is in a sense stronger than just assuming ZFC outright.
Namely, if we have $\lambda$-DC for some inaccessible cardinal $\lambda$, and ZF in the background, then in particular, we will have the full axiom of choice inside the universe $V\_\lambda$, consisting of all sets of rank less than $\lambda$, si... | 32 | https://mathoverflow.net/users/1946 | 452512 | 181,871 |
https://mathoverflow.net/questions/452360 | 2 | Let's take a $G\_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+\*\_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\in\Omega^3(M),\Psi\in\Omega^4(M\times\mathbb{R})$ determining the corresponding $G\_2$ and Spin$(7)$ structures. Now t... | https://mathoverflow.net/users/131004 | Decomposition of forms on a Spin$(7)$ manifold | Observe that, in a $\mathrm{Spin}(7)$-manifold, since $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, the $\mathrm{Spin}(7)$ decomposition $\Lambda^4 = \Lambda^4\_1\oplus \Lambda^4\_7\oplus\Lambda^4\_{27}\oplus\Lambda^4\_{35}$ must refine the splitting $\Lambda^4=\Lambda^4\_{+}\oplus\Lambda^4\_{-}$ into self-dual and anti-sel... | 2 | https://mathoverflow.net/users/13972 | 452513 | 181,872 |
https://mathoverflow.net/questions/452412 | 3 | Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$.
In one of the talks, speaker mentions that this is easier to believe if one think of this in terms of differential graded manifolds.
Think of $\mathfrak{g}$ as a ... | https://mathoverflow.net/users/118688 | Lie's third theorem via graded geometry | This is the Duistermaat–Kolk construction of a simply connected Lie group that integrates the given Lie algebra $\def\g{{\frak g}}\g$.
The starting observation is that for any simply connected Lie group $G$ the canonical morphism of group objects in diffeological spaces (or smooth sets) $$\def\Hom{\mathop{\rm Hom}}\H... | 2 | https://mathoverflow.net/users/402 | 452516 | 181,874 |
https://mathoverflow.net/questions/452447 | 4 | Can one get cancellation in exponential sums such as, say,
$$
\sum\_{n\sim N} e(\lfloor n^\theta\rfloor^\beta),
$$
for fixed positive $\theta,\beta\not\in\mathbb Z$? When $\theta < 1$, it seems possible without issue by grouping together values taken on by $\lfloor n^\theta\rfloor$ and then using exponent pairs, but fo... | https://mathoverflow.net/users/40983 | Exponential sum involving floor function | Let me try to provide a partial answer in the case when $0<\beta<2$ much in line with what Terry suggested in the comments. Perhaps it is possible to extend this method to all $\beta>2$ but the computations get more complicated.
Using Taylor expansion and $\lfloor n^\theta\rfloor=n^\theta-\{n^\theta\}$ we obtain
$$
\... | 4 | https://mathoverflow.net/users/510318 | 452535 | 181,880 |
https://mathoverflow.net/questions/452533 | 0 | Related to another question I asked, some questions came up, the most important is the following:
>
> Are there any 4-regular planar graphs without 2-cycles + 3-cycles?
>
>
>
Could someone draw an example if there is one? I couldn't find any paper that answers this question.
| https://mathoverflow.net/users/135618 | Even regular planar graphs without 2-cycles | In a connected planar 4-regular graph without cycles of length less than 4 we have $E=2V$ and $2E\geqslant 4F$ (since every face has at least 4 edges and every edge belongs to at most 2 faces). Thus $2E\geqslant 2V+2F$, contradiction to Euler formula $E+2=V+F$.
| 1 | https://mathoverflow.net/users/4312 | 452537 | 181,881 |
https://mathoverflow.net/questions/452476 | 6 | Let $X={\rm Gr}\_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}\_{n,k,{\Bbb C}}$.
We consider the twisted $\Bbb R$-forms $\_c X$ of $X$ where $c\in {\rm Aut\,} X\_{\Bbb C}$ is a $1$-cocycle... | https://mathoverflow.net/users/4149 | Twisted forms with real points of a real Grassmannian | **Updated**
Let $G:={\rm Aut}(X)^0$. Then it is well-known that
$$G\_{\mathbb C}= ({\rm Aut}(X)^0)\_{\mathbb C}= ({\rm Aut}(X)\_{\mathbb C})^0= {\rm Aut}(X\_{\mathbb C})^0\cong {\rm PGL}(n,\mathbb C).$$ So $G$ is a real form of ${\rm PGL}(n,\mathbb C)$ which means that $G$ is isomorphic to ${\rm PGL}(n,\mathbb R)$, $... | 4 | https://mathoverflow.net/users/89948 | 452543 | 181,883 |
https://mathoverflow.net/questions/452481 | 4 | One may see the modular interpretation of (points of) modular curves in the very first course on modular forms and modular curves. I am wondering if it is well-known that modular interpretation of the stalks of the structure sheaf at points of modular curves, i.e. if $Y$ is an open modular curve parameterising elliptic... | https://mathoverflow.net/users/44005 | Modular interpretation of the stalks of modular curves | I guess I'll make my comment into an answer so this question is no longer marked unanswered.
If the modular curve corresponds to a torsion-free subgroup, then the completion of the etale local ring at $x$ is the universal deformation ring of the object corresponding to $x$. In general, the completion of the etale loc... | 3 | https://mathoverflow.net/users/15242 | 452551 | 181,884 |
https://mathoverflow.net/questions/452539 | 1 | This is a repost from [MSE](https://math.stackexchange.com/questions/4744769/supremum-or-upper-bound-of-bivariate-function-involving-logarithms-and-combinato) because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get ... | https://mathoverflow.net/users/158098 | Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers | Note that
\begin{equation\*}
\frac{\ln\frac{\binom{n}{d}}{2^{d}4}}{d\ln\frac{n}{d}}
\le r(n,d):=\frac{\ln\frac{\binom{n}{d}}{2^{d}}}{d\ln\frac{n}{d}};
\end{equation\*}
here and in what follows, $n$ and $d$ are integers such that $1\le d\le n-1$, as in the OP. So, it is enough to bound $r(n,d)$ from above by a univ... | 2 | https://mathoverflow.net/users/36721 | 452556 | 181,887 |
https://mathoverflow.net/questions/452547 | 5 | **Problem:**
Given a random isotropic unit vector in $\mathbb{R}^p$ for $p\ge2$, we are trying to compute (preferably exactly, otherwise to upper bound):
$$\mathbb{E}\_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\left[{w\_1}^{\!4}\,{w\_2}^{\!4}\right]\,.$$
Any help would be greatly appreciated!
---
**A related expect... | https://mathoverflow.net/users/100796 | Expectation of a function of two entries of an isotropic unit vector $\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\![{w_{1}}^{\!4}\,{w_{2}}^{\!4}]$ | As noted in the [previous answer](https://mathoverflow.net/a/449401/36721),
the joint distribution of $w\_1^2$ and $w\_2^2$ is the [Dirichlet distribution](https://en.wikipedia.org/wiki/Dirichlet_distribution#Definitions) with parameters $1/2,1/2,p/2-1$. [Therefore](https://en.wikipedia.org/wiki/Dirichlet_distribution#... | 3 | https://mathoverflow.net/users/36721 | 452559 | 181,889 |
https://mathoverflow.net/questions/452570 | 9 | Today, We call the Kronecker's Jugendtraum Hilbert's 12th problem. But, Hilbert's interpretation of the "Jugendtraum" was not that intended by Kronecker.
And Weber missed his chance to disprove Hilbert's claim in the third volume of his *Lehrbuch der Algebra*.
I read Schappacher's "[On the History of Hilbert's Twel... | https://mathoverflow.net/users/167507 | What is Weber's mistake about Hilbert's 12th problem? | It would help to be clearer about where in Schappacher's paper this issue is brought up. I think you are asking about pages 256-258, yes?
The end of the 2nd paragraph on p. 257 says "Translating back to the characterization by ray class groups, Weber overlooked precisely the possibility of choosing different signs in... | 17 | https://mathoverflow.net/users/3272 | 452581 | 181,898 |
https://mathoverflow.net/questions/452584 | 4 | I’ve been looking for a quantitative notion of the Borel-Cantelli lemma along the following lines:
Let $(X,\Omega,p)$ be a probability space, and let $(A\_n)\_n\subseteq \Omega$ be a sequence of measurable sets s.t $p(A\_n)\geq 1-\delta$ for all $n$ (for some small $\delta>0$).
Then of course, the Borel-Cantelli le... | https://mathoverflow.net/users/70853 | Quantitative Borel-Cantelli | Imagine that for all $k=1,2,\ldots$ all events $A\_n$ for $n\in [k!,(k+1)!-1)$ are the same event $C\_k$. Then if $x$ belongs to all $A\_i$ along a subsequence of positive density yields that it belongs to all but finitely many $C\_k$. It may well appear that there is no such $x$.
| 4 | https://mathoverflow.net/users/4312 | 452587 | 181,900 |
https://mathoverflow.net/questions/452612 | 4 | Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
Does there exist any such counterexample where the restricted Lie algebra comes from an associative algebra?
I recall t... | https://mathoverflow.net/users/17582 | Derived subalgebra of a restricted Lie algebra | There is no such a counterexample, as the derived subalgebra of the restricted Lie algebra associated to an associative algebra over a field of positive characteristic is always restricted.
For instance, this follows from Lemma 4.5 in Chapter 2 of the book "D. Passmann: The algebraic structure of group rings" ([MathS... | 5 | https://mathoverflow.net/users/14653 | 452614 | 181,904 |
https://mathoverflow.net/questions/452600 | 3 | Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x\_1,...,x\_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its [**matroid base polytope**](https://en.wikipedia.org/wiki/Matroid_polytope) (i.e. the convex hull of the characteristic vectors of the bases $B\in\mathcal B$).
The *circumcenter*... | https://mathoverflow.net/users/108884 | Does a matroid base polytope contain its circumcenter? | I do not think so. Consider a uniform matroid on 3 elements $a, b, c$ of rank 2, and take 100 copies of $a$ (so, totally we have 102 elements). Then the matroid base polytope has full dimension 101, thus the circumcentre has all coordinates $1/51$. But each base has the sum of coordinates corresponding to $b$ and $c$ n... | 3 | https://mathoverflow.net/users/4312 | 452616 | 181,905 |
https://mathoverflow.net/questions/452623 | 0 | Start by introducing the finite sums
$$A\_n:=\sum\_{m=1}^nq^m\prod\_{j=1}^{m-1}(1-q^j) \qquad \text{and} \qquad
B\_n:=\sum\_{m=1}^nq^m\prod\_{j=m+1}^n(1-q^j).$$
An algebraic proof is *facile*: Clearly, $A\_1=B\_1=q$. Note
$A\_{n+1}=A\_n+q^{n+1}\prod\_{j=1}^n(1-q^j)$ while $B\_{n+1}=B\_n+q^{n+1}-q^{n+1}B\_n$. By inducti... | https://mathoverflow.net/users/66131 | A combinatorial proof: where art thou? | Consider all partitions $\lambda$ with distinct parts not exceeding $n$ and sum up $(-1)^{r(\lambda)+1}q^{|\lambda|}$, where $r$ goes for the number of parts. You may count the sum by fixing the smallest part, or by fixing the largest part, getting representations $B\_n, A\_n$ respectively.
| 7 | https://mathoverflow.net/users/4312 | 452625 | 181,909 |
https://mathoverflow.net/questions/452621 | 2 | Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale Cohomology (Example 12.4, page 81) is claimed that the stalk of derived direct image sheaf of sheaf $\mathcal{F}$ on etale site... | https://mathoverflow.net/users/108274 | Calculate stalk of etale derived pushforward sheaf (Milne's LEC) | This is because $U$ is again normal [Tag [033C](https://stacks.math.columbia.edu/tag/033C)], so it is irreducible since it is connected [Tag [033M](https://stacks.math.columbia.edu/tag/033M)]. Clearly the generic point of $U$ maps to the generic point of $X$, and conversely any point mapping to the generic point of $X$... | 2 | https://mathoverflow.net/users/82179 | 452629 | 181,911 |
https://mathoverflow.net/questions/451824 | 6 | Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. In the paper TOWARDS THE KAZHDAN-LUSZTIG CONJECTURE by Gabber and Joseph, they claim that there is a formula, due to D ... | https://mathoverflow.net/users/466793 | An alternative form of the Kazhdan-Lusztig conjecture | Let $G$ be the simply-connected Lie group with Lie algebra $\mathfrak g$, let $B\subset G$ be the Borel subgroup, and let $X=G/B$ be the flag variety. There are equivalences $$\mathrm{Mod}\_f(\mathfrak g,B,\chi\_\lambda)\xrightarrow{-\otimes\_{U(\mathfrak g)}D\_X}\mathrm{Mod}\_c(D\_X,B)\xrightarrow{DR\_X}\mathrm{Perv}(... | 3 | https://mathoverflow.net/users/123673 | 452633 | 181,913 |
https://mathoverflow.net/questions/452632 | 4 | Show that $f(x)\ge 0$ for $0\le x \le 1$, where:
$$f(x) = \arccos(x)^2 -8x(5x^2-2) \sqrt{1-x^2}\arccos(x)+36 x^8-112 x^6+93 x^4-17 x^2$$
The endpoints are $f(0)=\pi^2/4$ and $f(1)=0$. Plotting verifies that the function is positive, but it is not monotonic, or concave. The maximum is the root of a complicated funct... | https://mathoverflow.net/users/510206 | Non-negativity of a complicated function | We have to show that
\begin{equation\*}
g\overset{\text{(?)}}\le0 \text{ on }[1/2,1].
\end{equation\*}
Note that $p(x):=x^4-\frac{19}{20}x^2+\frac{7}{80}$ is of the same sign as $x-x\_\*$ for $x\in[1/2,1]$, where
$x\_\*:=\frac{1}{2} \sqrt{\frac{1}{10} \left(19+\sqrt{221}\right)}=0.920\dots$.
For $x\in[1/2,1]\setmin... | 3 | https://mathoverflow.net/users/36721 | 452636 | 181,915 |
https://mathoverflow.net/questions/452074 | 6 | Let $A$ be an analytic subset of a complex manifold $M$ and $O\_{M}$ be the sheaf of complex analytic functions on $M$. The sheaf of ideals $\mathcal{J}\_{A}$ is defined as the subsheaf of $O\_{M}$ whoes stalk at each point $x\in M$ consists of germs of analytic functions vanishing on $A$. If $x\notin A$, then $\mathca... | https://mathoverflow.net/users/480953 | A question on Demailly's proof of coherence of ideal sheaf | We need the concept of multi-symmetric polynomial which is a straight forward generalization of symmetric polynomial. The definitions and relevant results concerning it can be found in [this paper](https://%20arxiv.org/pdf/math/0205233.pdf).
Indeed, the coefficients $\delta b\_{j}$ of $B\_{k}$ can be expressed as a m... | 1 | https://mathoverflow.net/users/480953 | 452641 | 181,916 |
https://mathoverflow.net/questions/452607 | 2 | A smooth $m$-dimensional submanifold of $\mathbb{R}^{d}$ is said to be $k$-*ruled* if it is foliated by $k$-dimensional planes, called *rulings*.
Let $M$ be a $k$-ruled submanifold. Then $M$ can be parametrized (locally) by a smooth map $\sigma \colon (U\subset \mathbb{R}^{m-k}) \times \mathbb{R}^{k} \to \mathbb{R}^{... | https://mathoverflow.net/users/74033 | Parametrization of $k$-ruled submanifold: can we choose the base to be orthogonal to the rulings? | The answer is already 'no' in the first nontrivial case: A 3-manifold in $\mathbb{R}^d$ (where $d>3$) that is ruled by lines (i.e., $k=1$).
One can see this as follows: As the OP notes, one can write $\sigma$ as above in the form
$$
\sigma(u\_1,u\_2,v\_1) = \xi(u\_1,u\_2) + v\_1 X(u\_1,u\_2)
$$
where, without loss of... | 1 | https://mathoverflow.net/users/13972 | 452647 | 181,918 |
https://mathoverflow.net/questions/452644 | 8 | In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$?
Also, more generally, does this also apply for $Lu=a^{ij}u\_{ij}+b^iu\_i+cu$?
I tried to prove it by using variational way through considering $\frac{\int|Du|^2}{\int... | https://mathoverflow.net/users/348579 | Are all positive eigenfunctions principal eigenfunctions? | **Answer.** Yes, for appropriate boundary conditions (e.g., Dirichlet or Neumann) the Laplace operator on bounded domains with sufficiently smooth boundary has no positive eigenfuntions, except for those that belong to the leading eigenvalue.
Here are the details:
**Part 0.** Notation.
Consider an $L^p$-space ove... | 12 | https://mathoverflow.net/users/102946 | 452650 | 181,920 |
https://mathoverflow.net/questions/452519 | 1 | Define the family of densities:
$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta)\big)\Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}, \quad 0 \le \phi,\theta\le \pi/2$$
where $f(x)=g(x^2)$ with $g$ non-negative, increasing, and convex or concave, on $[0,... | https://mathoverflow.net/users/510206 | Monotone likelihood ratio of a family of densities with compact support | $\newcommand{\ep}{\varepsilon}$This conjecture is not true in general.
Indeed, suppose the "convex" part of your conjecture is true. Then (letting $x:=\phi$, $t:=\theta\_1$, and $\theta\_2\downarrow\theta\_1=t$) we see that for any strictly increasing convex smooth function $g$ and all $x$ and $t$ in $(0,\pi/2)$ we w... | 1 | https://mathoverflow.net/users/36721 | 452659 | 181,922 |
https://mathoverflow.net/questions/452657 | -3 | I was wondering whether it would be possible to do propositional logic without any rules of inference and assumptions (except modus ponens).
I have the following axioms:
1. $ p \to (q \to p) $
2. $ (p \to (q \to r)) \to ((p \to q) \to (p \to r)) $
3. $ (\lnot p \to \lnot q) \to (q \to p) $
4. MP: From $ p $ and $ p... | https://mathoverflow.net/users/510414 | Propositional logic without rules of inference and assumptions (except MP) | I'll do one of them, $p \to (p \vee q)$.
**Define** $a \vee b$ as $\lnot b \to a$.
We want to prove
$$
p \to (p \vee q) .
$$
This is now merely alternate notation for
$$
p \to (\lnot q \to p) .
$$
But this is an instance of Axiom 1.
| 1 | https://mathoverflow.net/users/454 | 452661 | 181,923 |
https://mathoverflow.net/questions/452656 | 4 | Let $f = (f\_0,f\_1,\ldots,f\_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum\_{n\geq 0} z^n f\_n$$ for its probability generating function. It has been shown [in this post](https://math.stackexchange.com/questions/220705/how-to-obtain-probability-fun... | https://mathoverflow.net/users/163454 | Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$ | $\newcommand\R{\mathbb R}$Assume that the integral
$$I\_n:=\oint\_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\,\frac{d\xi}{\xi^{n+1}}
=\int\_0^{2\pi} \frac{1-\hat{f}(e^{it})}{1-e^{it}}\,\frac{e^{it}\,i\,dt}{e^{i(n+1))t}} $$
exists in the Lebesgue sense -- that is,
$$\int\_0^{2\pi} \Big|\frac{1-\hat{f}(e^{it})}{1-e^{it}... | 5 | https://mathoverflow.net/users/36721 | 452662 | 181,924 |
https://mathoverflow.net/questions/452322 | 14 | 1. $\DeclareMathOperator\pcf{pcf}$For simplicity say $\aleph\_\omega$ is a strong limit. Let $A=\pcf\{\aleph\_n:n\in\omega\}$. Then it follows from basic properties of pcf operation that $X\subseteq A\rightarrow\pcf(X)\subseteq A$, and that pcf is a Kuratowski closure operator on $A$, which gives $A$ a topology. Is $A$... | https://mathoverflow.net/users/499983 | Topology and pcf theory | For (1) and (2), you may want to look at Burke and Magidor's [exposition](https://core.ac.uk/download/pdf/82500424.pdf) of pcf theory, as they adopt a topological stance at a couple of places. For my money, the part of the ``$\aleph\_{\omega\_4}$ Theorem'' where the topological flavor is strongest is when you prove the... | 7 | https://mathoverflow.net/users/18128 | 452682 | 181,931 |
https://mathoverflow.net/questions/452691 | 2 | Does the Riemann hypothesis predict an upper bound for
$$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$
where
$$f(x)=\sum\limits\_{n=2}^x \frac{\Lambda(n)}{\log^2(n)}\tag{2}$$
and $\Lambda(n)$ is the [von Mangoldt function](https://en.wikipedia.org/wiki/Von_Mangoldt_... | https://mathoverflow.net/users/110710 | Does the Riemann hypothesis predict a bound for this prime-counting function? | The Riemann hypothesis is *equivalent* to the following statement:
$$f(x)=\mathrm{li(x)}-\frac{x}{\log x}+O(\sqrt{x}),\qquad x\geq 2.$$
Note that
$$\mathrm{li(x)}=\mathrm{li(2)}+\frac{x}{\log x}-\frac{2}{\log 2}+\int\_2^x\frac{dt}{\log^2 t},$$
hence the claim is that the Riemann hypothesis is equivalent to
$$f(x)=\int\... | 9 | https://mathoverflow.net/users/11919 | 452694 | 181,936 |
https://mathoverflow.net/questions/446460 | 2 | **EDIT 1:** All topological groups in this question are assumed to be second countable. In particular, this forces every group to be metrizable and every Lie group to have at most countably many components. In particular, all discrete groups are countable. The general case (where a Lie group is just assumed to be local... | https://mathoverflow.net/users/153400 | Does every locally compact group G contain a maximal open subgroup P which is a pro-Lie group? | After thinking about the problem for some time, I came up with a counter-example, so as YCor wrote in his comment, it is indeed true that there is not always a maximal open pro-Lie subgroup.
Let $A$ be a nontrivial compact group, for example $\mathbb Z/2\mathbb Z$ or the circle group $\mathbb R /\mathbb Z$. Then the ... | 3 | https://mathoverflow.net/users/153400 | 452706 | 181,938 |
https://mathoverflow.net/questions/452681 | 13 | Let $A$ and $G$ be two groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ be a group homomorphism and $\beta: A\rightarrow\operatorname{Bij}(G)$ an anti-homomorphism satisfying some conditions given in [Wikipedia](https://en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p_product#External_Zappa%E2%80%93Sz%C3%A9p_product... | https://mathoverflow.net/users/95592 | Is it possible for a direct product to be isomorphic to the Zappa–Szép product? | Let $S$ be a nonabelian group with a fixed-point-free automorphism $\alpha$. (Such groups for which $\alpha$ has prime order are necessarily nilpotent. I think the smallest example is the nonabelian group of order $7^3$ and exponent $7$, which has a fixed-point-free automorphism of order $3$.)
Now let $G = S \times S... | 18 | https://mathoverflow.net/users/35840 | 452712 | 181,942 |
https://mathoverflow.net/questions/452654 | 2 | Given an odd natural integer $2a-1$ with $a\geq 1$, associate to it recursively the composition $\psi(1)=\emptyset$ and $\psi(2^{-n}a)+(n+\delta\_{>1}(m))$ if $a=2^n m$ with $m$ odd where $\delta\_{>1}(1)=0$ and $\delta\_{>1}(m)=1$ otherwise.
For $2a-1=37$ we get for example
\begin{align\*}
\psi(2\cdot 19-1)&=\psi(2\cd... | https://mathoverflow.net/users/4556 | A bijection between odd natural integers and compositions | More or less a comment. I'd say the closest known bijection is just given by the binary representation, namely, if I understand it correctly, for e.g. $37$ we do $3$ times the last power of $2$ less than $37$, here $3\cdot32=96$, and the binary representation of the difference to $37$, here $96-37=59=2^5+2^4+2^3+2^1+2^... | 3 | https://mathoverflow.net/users/6101 | 452714 | 181,943 |
https://mathoverflow.net/questions/452707 | 2 | Consider over $\mathbb{C}$. Let $(X,\mathcal{O}(1))$ be a smooth projective scheme with an ample polarisation. Let $P(t):=\chi(X,\mathcal{O}(t))$ denote the Hilbert polynomial of $\mathcal{O}\_X$. Choose a decomposition $P(t)=I(t)+Q(t)$ such that $I(t),Q(t)$ are also Hilbert polinomials. If a sheaf of ideals $\mathcal{... | https://mathoverflow.net/users/105537 | When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism? | This works for any smooth projective variety $X$ under the assumption
$$
\mathrm{Pic}^0(X) = 0
$$
and any $Z$ of codimension at least 2. For the proof see Lemma B.5.6 in Kuznetsov, Alexander G.; Prokhorov, Yuri G.; Shramov, Constantin A. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Jp... | 1 | https://mathoverflow.net/users/4428 | 452720 | 181,945 |
https://mathoverflow.net/questions/451365 | 1 | This is a cross-post from [this other question](https://math.stackexchange.com/q/4726158/610053) that I asked ~1 month ago in the mathematics forum, with no reaction. I am still stuck on this, looking for references or approaches to proofs. I hope I have a bit more luck in this forum! let me know if any clarifications ... | https://mathoverflow.net/users/172514 | Expected value of the projective metric between random orthogonal Stiefel matrices in $\mathbb{R}^{N \times k}$ equals $1 - \frac{k}{N}$ | Fortunately, there's nothing deep going on here.
We'll use slightly different notation. Let $\mathbf{Q} \in \mathbb{R}^{N \times k}$ be a random matrix drawn uniformly from the Stiefel manifold of $N \times k$ orthonormal frames. In particular,
* The columns of $\mathbf{Q}$ are orthonormal.
* The marginal distribut... | 1 | https://mathoverflow.net/users/510467 | 452727 | 181,947 |
https://mathoverflow.net/questions/452723 | 3 | For any finite, simple, undirected graphs $G, H$ we denote by $G\times H$ their [categorical product](https://en.wikipedia.org/wiki/Tensor_product_of_graphs). For any graph $G$ we let $G^1 = G$ and for $n\geq 1$ we let $G^{n+1} = G \times G^n$.
It is easy to see that the sequence of the power chromatic numbers $(\chi... | https://mathoverflow.net/users/8628 | The sequence of the power chromatic numbers $(\chi(G^n))_{n\in\mathbb{N}}$ | Yes.
First note that $\chi(G^k) \leq \chi(G)$. We can colour $(x\_1, \ldots, x\_k)$ with $c(x\_1)$ where $c$ is a colouring of $G$.
Vice versa $\chi(G^k) \geq \chi(H) = \chi(G)$, where $H$ is the induced subgraph of $G^k$ on the vertices $\{(v, v, \ldots, v) \mid v \in V(G)\}$.
Thus $\chi(G^k) = \chi(G)$, which i... | 7 | https://mathoverflow.net/users/502833 | 452738 | 181,950 |
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