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https://mathoverflow.net/questions/412680 | 5 | Ross Street's 1991 paper [Parity Complexes](http://www.numdam.org/item/CTGDC_1991__32_4_315_0/) (apologies; I don't know how to find DOI links for *Cahiers* papers) develops some very useful tools for working with free strict $\omega$-categories. There is a [corrigenda](http://www.numdam.org/item/CTGDC_1994__35_4_359_0... | https://mathoverflow.net/users/2362 | What is the correct statement of Theorem 4.2 in Street's "Parity Complexes"? | I hope I can offer some quick answers to your questions without errors.
Let's tackle the breaking down:
>
> 1. As indicated in the paper, the notion of "free generation" comes from Street's earlier [The Algebra of Oriented
> Simplices](https://doi.org/10.1016/0022-4049(87)90137-X). I believe
> this notion is to b... | 4 | https://mathoverflow.net/users/110515 | 413042 | 168,529 |
https://mathoverflow.net/questions/412388 | 2 | It is an obvious fact that the sum $\sum\_{n\geq 0} \binom{2n}{n} x^n$ defines an algebraic function. I am interested in the variation of this sum, namely
$$A(x)=\sum\_{n\geq 0} \binom{2n}{n}^2 x^n$$
which is not an algebraic function due to the growth of the coefficients (see Enumerative Combinatorics, Vol. 2 from... | https://mathoverflow.net/users/46573 | Algebraicity of a generating function and binomial numbers | It is a straightforward computation to express $A(x) = \sum\_{n=0}^{\infty} \binom{2n}{n}^2x^n$ in terms of hypergeometric functions, namely
$$A(x) = {}\_2F\_1 (\tfrac12,\tfrac12;1;16x).$$
Then in turn one can express this function in terms of elliptic integrals,
$${}\_2F\_1 (\tfrac12,\tfrac12;1;16x) = \frac{1}{\pi} \i... | 5 | https://mathoverflow.net/users/7263 | 413050 | 168,533 |
https://mathoverflow.net/questions/413027 | 1 | Let $G$ be a compact connected Lie group and $T$ is a maximal torus of $G$. Let $K$ be a non trivial connected Lie subgroup of $G$.
We say that $r \in \mathfrak{g}$ is a regular element of the Lie algebra $\mathfrak{g}$ if the stabilizer subgroup $G\_r$ of the adjoint action of $G$ on $\mathfrak{g}$ is a maximal toru... | https://mathoverflow.net/users/172459 | Question about regular elements in a Lie subalgebra | It is possible to have a non-trivial, closed, connected Lie subgroup $K$ of a compact, connected Lie group $G$ such that no element of $\mathfrak k$ is regular in $\mathfrak g$. For example, consider $K = \operatorname{SU}\_2$ embedded in $G = \operatorname{SU}\_4$ as $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \maps... | 3 | https://mathoverflow.net/users/2383 | 413061 | 168,540 |
https://mathoverflow.net/questions/413070 | 1 | [This paper by Maslov et al.](https://www.nature.com/articles/s41567-021-01271-7) uses that the probability of two $n$-bit Boolean functions $l(x)$ and $g(x)$ being equal is bound in terms of $\hat{g}\_\text{max}$, the largest Fourier coefficient of $g(x)$, in the following way (between Eq. (4) and (5) of the *Methods*... | https://mathoverflow.net/users/474032 | Probability of two Boolean functions being equal expressed in terms of the maximum Fourier coefficient | As the RHS is independent of $l(x)$, the statement should only be true if $l(x)$ lies in a restricted subset of Boolean functions. I cannot read the paper linked, but I suspect it is intended to restrict $l(x)$ to linear functions, i.e. $l(x)=l\_y(x)=x\cdot y$.
In that case it is easy to show
$$Pr\left[l\_y(x)=g(x)\r... | 2 | https://mathoverflow.net/users/474034 | 413072 | 168,543 |
https://mathoverflow.net/questions/413078 | 1 | Let $(X\_t)\_{t\ge0}$ be a càdlàg Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F\_t)\_{t\ge0},\operatorname P)$ and $B\in\mathcal B([0,\infty)\times\mathbb R)$.
>
> How can we show that $$\pi:=\sum\_{\substack{s\:\ge\:0\\\Delta X\_s\:\ne\:0}}1\_B(s,\Delta X\_s)$$ is $\mathcal A$-measur... | https://mathoverflow.net/users/91890 | Is $\sum_{\substack{s\:\ge\:0\\\Delta X_s\:\ne\:0}}1_B(s,\Delta X_s)$ measurable for fixed $B\in\mathcal B([0,\infty)\times\mathbb R)$? | This is routine (and I am quite sure covered by standard textbooks), although somewhat tedious. First, for a compactly supported, non-negative and continuous $f$, one writes
$$ \tag{1} S\_t[f] := \sum\_{s \leqslant t} f(s, X\_{s-}, X\_s) = \lim\_{n \to \infty} \sum\_{i = 0}^{\lfloor n t\rfloor} f(\tfrac in, X\_{(i-1)/n... | 2 | https://mathoverflow.net/users/108637 | 413081 | 168,544 |
https://mathoverflow.net/questions/413089 | 3 | Consider the following statement in $\sf ZF$:
>
> (I) Whenever $X$ is a set with more than $1$ element, there is an injective map $\iota: X\to X$ such that $\iota(x) \neq x$ for all $x\in X$.
>
>
>
The [Axiom of Choice (AC)](https://en.wikipedia.org/wiki/Axiom_of_choice) implies (I) -- but does (I) imply (AC)?... | https://mathoverflow.net/users/8628 | Injections without fixed-points and the Axiom of Choice | It is shown in
Tachtsis, E.
On the existence of permutations of infinite sets without fixed points in set theory without choice.
Acta Math. Hungar. 157 (2019), no. 2, 281-300.
that ZF+(every infinite set supports a permutation with no fixed points)
does not imply AC. It is easy to see in ZF that a finite set sup... | 2 | https://mathoverflow.net/users/75735 | 413095 | 168,545 |
https://mathoverflow.net/questions/413087 | 9 | In this [previous post](https://mathoverflow.net/q/412923/7113) I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the question I gave the example of three rational functions that generate $\mathbb... | https://mathoverflow.net/users/7113 | Is $\mathbb{Q}$ the orbit of a rational function under iteration? | As was mentioned in the [comments](https://mathoverflow.net/questions/413087/is-mathbbq-the-orbit-of-a-rational-function-under-iteration#comment1058750_413087) by pregunton, it is possible to do using two rational functions. I claim it is not possible using just one. As Fedor Petrov suggests in [another comment](https:... | 19 | https://mathoverflow.net/users/30186 | 413097 | 168,547 |
https://mathoverflow.net/questions/413101 | 5 | I have a random walk $$R(t)= \sum\_{n<t} X\_n,$$ with $X\_n \sim U(-\tfrac{1}{n^\alpha}, \tfrac{1}{n^\alpha}),$ where $X\_n$ are independant and $\alpha >0$.
I think that someone must have studied this before. I am interested in understanding the behavior of $R(t)$ for large $t$.
For example can we estimate the pro... | https://mathoverflow.net/users/422944 | Random walk with decreasing steps | Let $a:=\alpha$. Note that
\begin{equation}
Ee^{zX\_n}=\frac{\sinh(z/n^a)}{z/n^a}
\end{equation}
for real $z>0$. Using the inequality $\dfrac{\sinh u}u<e^{u^2/6}$ for real $u\ne0$ (see e.g. [this MathSE answer](https://math.stackexchange.com/a/1759418/96609)) and the independence of the $X\_n$'s, we get
\begin{equatio... | 5 | https://mathoverflow.net/users/36721 | 413118 | 168,550 |
https://mathoverflow.net/questions/413112 | 2 | In a physics paper (pubs.acs.org/doi/10.1021/j100210a011), I see the following transformation:
$$\sum\_q \frac{2[1-\cos(\textbf{q} \cdot \textbf{r})]}{q^2} =\frac{1}{\pi} \int\_0^{+\infty}[1-J\_0(qr)]\frac{dq}{q}$$
in which $\textbf{q}$ is a wave vector (spatial frequencies in 2D), $\textbf{r}$ is a 2D position vec... | https://mathoverflow.net/users/474097 | From a sum of cosines to an integral of Bessel function | So this is a bit of physics notation. The sum over wave vectors is short hand for an integral over $n$-dimensional reciprocal space,
$$\sum\_{\mathbf{q}}\mapsto \int\frac{d^n \mathbf{q}}{(2\pi)^n}.$$
Then the integral follows for $n=2$, in polar coordinates,
$$(2\pi)^{-2}\int\_0^\infty qdq\int\_0^{2\pi}d\phi \, \frac{2... | 3 | https://mathoverflow.net/users/11260 | 413123 | 168,552 |
https://mathoverflow.net/questions/413121 | 0 | I'm currently struggling with concluding a proof and need a hint. So the first part of the exercise
was that given an open subset $\Omega \subset \mathbb{R}^2$ and a harmonic function
$f: \Omega \to \mathbb{R}$ with a (local) maximum or minimum in $\Omega$, then $f$ is constant. This part is done.
Now I have to show ... | https://mathoverflow.net/users/407441 | A minimal surface with a local extremum in normal direction is a plane | I don't completely understand what you're asking (what is point in the normal direction?) but minimal surfaces must have non-positive curvature, because $H = k\_1 + k\_2 = 0 \implies K = k\_1k\_2 \leq 0$. Local maximal and minima would necessarily be in the interior of the domain, as the domain is open, and at such a p... | 0 | https://mathoverflow.net/users/104812 | 413126 | 168,554 |
https://mathoverflow.net/questions/413037 | 2 | This is from the paper [Representations up to homotopy of Lie algebroids](https://doi.org/10.1515/CRELLE.2011.095) by Camilo Arias Abad and Marius Crainic
Let $M$ be a smooth manifold.
Let $E\rightarrow M$ be a vector bundle. A connection on the vector bundle $E\rightarrow M$ is a map
$$\nabla:\Gamma(M,TM)\times \G... | https://mathoverflow.net/users/118688 | Connection on the complex of vector bundles | The concept of a linear $A$-connection on a vector bundle predates the paper of Arias Abad and Crainic. It goes back at least to
*Sam Evens, Jiang-Hua Lu and Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Q. J. Math., Oxf. II. Ser. 50, No. 200, 417-436 (1999)*.
[Ar... | 3 | https://mathoverflow.net/users/104042 | 413132 | 168,556 |
https://mathoverflow.net/questions/413075 | 1 | Consider any probability density function $f(x)$ that has mean zero variance one and say all finite moments. You may assume standard normal density if you like.
Given $a\_1,a\_2>0$, I consider two copies of independent Random Walk bridges $(S\_0^{(i)}=0,S\_1^{(i)},S\_2^{(i)},\ldots,S\_n^{(i)}=a\_p\sqrt{n})\_{p=1,2}$ ... | https://mathoverflow.net/users/62327 | Existence of a process on $\mathbb{R}^2$ that looks like two 'independent' brownian bridges $B_1(x)$ and $B_2(x)$ conditioned on $B_1(x)+B_2(x) > 0$ | There is a paper by Durrett, Iglehart and Miller, which also sounds related to what you want for the sum,
[Weak convergence to Brownian meander and Brownian excursion](https://projecteuclid.org/journals/annals-of-probability/volume-5/issue-1/Weak-Convergence-to-Brownian-Meander-and-Brownian-Excursion/10.1214/aop/117699... | 1 | https://mathoverflow.net/users/471081 | 413134 | 168,557 |
https://mathoverflow.net/questions/413107 | 4 | I am trying to understand an argument in Guillemin and Sternberg's paper *Geometric Quantization and Multiplicities of Group Representations* (Inventiones, 1982). The argument (Proof of Theorem 3.2) seems to be based on the following fact:
**Lemma (I think).** *Let $G$ be a compact connected Lie group acting smoothly... | https://mathoverflow.net/users/409915 | Quotient of line bundle by compact Lie group (after Guillemin-Sternberg) | Yes, the claim is true. It's a special case of a more general fact, that, in quite some generality, equivariant vector bundles are equivariantly locally trivial.
In your case, given $p$, there is a "slice" through $p$, a set $S\subset M$ containing $p$ such that $G\times S \to M$ is one-to-one and a homeomorphism ont... | 4 | https://mathoverflow.net/users/58888 | 413138 | 168,560 |
https://mathoverflow.net/questions/413130 | 4 | The function $F(x) = \exp(x) + \exp(\exp(x))x$ plays a role in the formulation of the Lagarias inequality:
$$\sigma(n) \le H\_n + \exp(H\_n) \log(H\_n)$$
If we put $x = \log(H\_n)$, then this inequality is equivalent to :
$$\sigma(n) \le F(\log(H\_n))$$
I wanted to look at some properties of this function and f... | https://mathoverflow.net/users/165920 | Is the function $F(x) = \exp(x) + \exp(\exp(x))x$ a hypertranscendental function? | Yes, $F(x)=e^x + x e^{e^x}$ satisfies an algebraic differential equation, and we can find it explicitly.
Looking at the expressions for $F$ and $F'$, we find that
$$x(F'-e^x)=(1+xe^x)(F-e^x).$$
We can rewrite this equation and its derivative as
\begin{align}xe^{2x}+\quad\quad\quad\ (1-x-xF)e^x &= F-xF'\\
(-1-2x)e^{... | 10 | https://mathoverflow.net/users/nan | 413140 | 168,561 |
https://mathoverflow.net/questions/412422 | 7 | I've been trying to find the following article: "S. Shelah, *Remarks on cardinal invariants in topology*, General topology Appl. **7**(3) (1977), 251-259". I tried to go directly to the journal page, but it turns out that Issue 3 isn't registered there (see: [General Topology and its Applications](https://www.sciencedi... | https://mathoverflow.net/users/146942 | Where can I find the following S. Shelah's paper? | please get the pdf [here](https://drive.google.com/file/d/19tNBdYlUMe-KLWUHo04u8H3pC3Xb2hbF/view?usp=drivesdk).
note that the text itself starts on page 5.
| 9 | https://mathoverflow.net/users/11100 | 413144 | 168,564 |
https://mathoverflow.net/questions/413079 | 0 | The $n$th **cumulant** $\kappa\_n$ of a probability distribution for $n\ge2$ is functional that is a polynomial in the first $n$ **moments** of the distribution, that has the properties of $(1)$ homogeneity, $(2)$ translation invariance, and $(3)$ additivity.
If $X\_1,\ldots,X\_m$ are independent random variables and... | https://mathoverflow.net/users/6316 | Was this proposition on cumulants of compound Poisson distributions known before I put it into a Wikipedia article? | This is a simple fact, below is a short proof. It is certainly very-well known for quite some time, a sample reference is formula (6.6) in
* Cacoullos T. (1989) *Generating Functions. Characteristic Functions*. In: *Exercises in Probability*. Problem Books in Mathematics. Springer, New York, NY. [DOI:10.1007/978-1-46... | 6 | https://mathoverflow.net/users/108637 | 413147 | 168,565 |
https://mathoverflow.net/questions/413153 | 3 | My questions come from the proof of Theorem 5.14 in section 5.7 of [Boucheron, Lugosi, and Massart - Concentration inequalities](https://www.hse.ru/data/2016/11/24/1113029206/Concentration%20inequalities.pdf). My first question can be stated as follows:
---
Suppose for positive numbers $V,y,\Delta,x,\delta >0$ we... | https://mathoverflow.net/users/163454 | Deriving inequalities from other inequalities | If $y^2$ is defined by (1) and
$$\epsilon = \frac{168}{97},x= \frac{1168561}{2916},z= \frac{121}{8281},\Delta = \frac{1}{1000},K=
\frac{10201}{9604},$$
then the difference between the right-hand side of inequality (2) and its left-hand side is $-2.5248\ldots<0$, so that (2) fails to hold.
(What might be interesting,... | 2 | https://mathoverflow.net/users/36721 | 413163 | 168,571 |
https://mathoverflow.net/questions/313245 | 7 | I am having some issues computing Poincare duality for the quantum homology $QH(M)$ when $(M,\omega)=(S^2 \times S^2,\omega\_{FS}\oplus \omega\_{FS})$. I am using the simple novikov ring $\Lambda$ consisting of polynomials in variables $t$ and $t^{-1}$. I know that a basis for $QH(M)$ (over $\Lambda$) is given by the e... | https://mathoverflow.net/users/47228 | Quantum homology of $(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$ and Poincare duality | A bit late for this one, but I'll still post the answer for future visitors.
Poincaré duality on the quantum homology is just the same as Poincaré duality on normal homology, see for example the famous PSS paper [1, Section 2]. This means that for an element $\alpha= \sum\_{A\in \Gamma} \alpha\_A e^A$ with $\alpha\_A... | 3 | https://mathoverflow.net/users/168284 | 413174 | 168,575 |
https://mathoverflow.net/questions/413179 | 2 | Let $T\_1$ and $T\_2$ be algebraic tori over a field of characteristic 0. Let $T$ be an extension of $T\_1$ by $T\_2$, namely
$$
1\longrightarrow T\_1\longrightarrow T\longrightarrow T\_2\longrightarrow 1.
$$ Is $T$ necessarily an algebraic torus? If so, is there any simple proof?
| https://mathoverflow.net/users/32746 | An extension of algebraic torus | @MartinSkilleter has posted an [answer](https://mathoverflow.net/questions/413179/an-extension-of-algebraic-torus#comment1059067_413179) in the comments. I'll summarise here an elementary proof; it is almost the same as in @MartinSkilleter's link [Extensions of tori by tori are tori](https://blog.jpolak.org/?p=1125), j... | 2 | https://mathoverflow.net/users/2383 | 413181 | 168,579 |
https://mathoverflow.net/questions/396640 | 6 | Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n\_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). Suppose that $T$ extends along codimension $1$ points of $\mathbb{A}^n\_k$, i.e. the torsor class $[T\_{k(\mathbb{A}^n)}] ... | https://mathoverflow.net/users/110362 | Extending $G$-torsors on open subsets of affine space | In case this might be useful to anyone, it turns out that results of Colliot--Thelene can be used to resolve a closely related question: see Theorem 6.1 in the following preprint (apologies for the self-promotion):
[https://www.dpmms.cam.ac.uk/~jcsl5/ADEpaper.pdf](https://www.dpmms.cam.ac.uk/%7Ejcsl5/ADEpaper.pdf)
(v... | 1 | https://mathoverflow.net/users/110362 | 413188 | 168,583 |
https://mathoverflow.net/questions/413109 | 22 | [Edited due to [YCor](https://mathoverflow.net/users/14094/ycor)'s [comment](https://mathoverflow.net/questions/413109/when-does-g-times-g-times-g-admit-a-faithful-group-action-on-a-set-of-size-g#comment1058839_413109):]
Given a finite group $G$, under what conditions does $G\times G\times G$ (the direct product of thr... | https://mathoverflow.net/users/17496 | When does $G\times G\times G$ admit a faithful group action on a set of size $|G|$? | I think it is possible to give a fairly precise description of the groups $G$ which fail to satisfy the condition given by @YCor in comments, with the exception of the case that $G$ is a $2$-groups. I will say a few words later about the case that $G$ is a $2$-group, and I think understanding the exceptional $2$-groups... | 20 | https://mathoverflow.net/users/14450 | 413193 | 168,585 |
https://mathoverflow.net/questions/413171 | 4 | $\DeclareMathOperator\Mp{Mp}$Let $F$ be a number field and $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{PGL}\_2(\mathbb{A}\_F)$.
Then we can think a submodule $L\_{\pi}^2$ of $L\_{\mathrm{disc}}^2(\Mp\_2)$, the discrete spectrum of automorphic functions on $\Mp\_2(F) \backslash \Mp\_2... | https://mathoverflow.net/users/35898 | Global Waldspurger packet is finite or infinite? | **Revised.** The global Waldspurger packet of $\pi$ is indeed finite, as you say in your comment. It's elements are metaplectic representations which are in bijection with the Vogan packet of $\pi$, i.e., me the set of all cuspidal representations $\pi\_D$ of a quaternion algebra $D/F$ such that $\pi\_D$ corresponds to... | 5 | https://mathoverflow.net/users/6518 | 413194 | 168,586 |
https://mathoverflow.net/questions/413192 | 4 | If we have $n-1$ quadratic forms for $n$ variables $x\_i$,
$$p\_i(x) = M^{(i)}\_{jk} x\_j x\_k$$
for $1\leq i \leq n-1$ and $1 \leq j,k \leq n$ then the zeros of all $p\_i(x)$,
$$p\_i(x) = 0$$
is generically $0$ dimensional in projective space, i.e. points. I guess generically there are $2^{n-1}$ of such points... | https://mathoverflow.net/users/41312 | $n-1$ quadratic forms for $n$ variables | To reach a satisfactory understanding of the problem at hand, I think you need to learn about **multidimensional resultants** (see below for where to get started).
Working over the field $\mathbb{C}$, let $F\_1(x),\ldots, F\_n(x)$ be $n$ homogeneous polynomials of respective degrees $d\_1,\ldots,d\_n$. Then there is ... | 7 | https://mathoverflow.net/users/7410 | 413198 | 168,589 |
https://mathoverflow.net/questions/413197 | 14 | If someone can prove Goldbach conjecture assuming the continuum hypothesis, do we consider the Goldbach conjecture proved?
If ZFC+CH implies Goldbach, and if the Goldbach turn out to be false, then it would mean that ZFC+CH is not consistent, but we know that ZFC+CH is consistent assuming that ZFC is consistent...
What... | https://mathoverflow.net/users/122378 | If someone can prove Goldbach conjecture assuming the continuum hypothesis, do we consider the conjecture proved? | Because the Goldbach conjecture is an arithmetic statement, it is absolute between any two models which agree on the natural numbers.
Now, given any model of $\sf ZFC$, $M$, there is a forcing extension $M[G]$ with the same ordinals (and in particular, the same natural numbers, which are the just the finite ordinals)... | 33 | https://mathoverflow.net/users/7206 | 413199 | 168,590 |
https://mathoverflow.net/questions/413139 | 3 | Given a geodesically complete manifold M, can we define a global identification of tangent spaces by starting from a base point, and parallel transporting along smooth geodesics? For this to be consistent, we need the parallel transport along every geodesic loop to leave the tangent space invariant. Is there a simple c... | https://mathoverflow.net/users/138208 | Identification of tangent spaces by parallel transport along geodesics | Ok, given the comments, what you are really asking for, is for a class of connected Riemannian manifolds for which the following construction (or the map $\Phi$) is a (smooth) trivialization of the tangent bundle of a Riemannian manifold $M$:
Fix a $p\in M$. For each $q\in M$ let $\gamma\_{qp}$ denote a unit speed ge... | 3 | https://mathoverflow.net/users/39654 | 413216 | 168,595 |
https://mathoverflow.net/questions/413205 | 1 | Recall that an integral domain $R$ with quotient field $K$ is
an almost valuation domain if for every $0 \not= x \in K$, there is a positive integer
$n$ (depending on $x$) such that $x^n \in R$ or $x^{−n} \in R$. Now for a prime number $p$ let $R := \mathbb{Z}p +XF[[X]]$ and $F := \overline{\mathbb{Z}p}$ be the
algebra... | https://mathoverflow.net/users/338309 | why is $R := \mathbb{Z}p +XF[[X]]$ an almost valuation domain? | I assume that $\mathbb Zp$ means the field with $p$ elements, and denote it by $\mathbb F\_p$.
Since $\lambda= (\lambda X)/X$, the fraction field of $R$ is equal to the fraction field of $F[[X]]$, which is the ring of Laurent power series $F((X))$.
Let $f=\sum\_{n=m}^{\infty} a\_n X^n \in F((X))$ with $a\_m\neq 0$.... | 3 | https://mathoverflow.net/users/459579 | 413223 | 168,597 |
https://mathoverflow.net/questions/413058 | 0 | Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ q(x) \neq 0 $ for all non zero $ x \in U $. Let us consider two quadratic forms $ q\_{1}$, $q\_{2} $, defined by
$$ q\_{1}(a... | https://mathoverflow.net/users/215016 | Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $ | Dimension 4 is clearly impossible since your quadratic forms are isotropic, but dimension 3 is possible.
The following SageMath code generate random 3-dimensional subspaces and check whether they are simultaneously isotropic for both forms. It finds quite a lot of such subspaces, as an example you can take the subspa... | 2 | https://mathoverflow.net/users/160416 | 413232 | 168,603 |
https://mathoverflow.net/questions/413165 | 72 | I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(x) := \lim\_{n \to \infty} d^n f(x)$$ converges. A simple example for such an $f$ would be $ce^x + h(x)$ for any consta... | https://mathoverflow.net/users/143629 | Does iterating the derivative infinitely many times give a smooth function whenever it converges? | I was able to adapt [the accepted answer](https://mathoverflow.net/questions/34059/if-f-is-infinitely-differentiable-then-f-coincides-with-a-polynomial/34067#34067) to [this MathOverflow post](https://mathoverflow.net/questions/34059/if-f-is-infinitely-differentiable-then-f-coincides-with-a-polynomial) to positively an... | 61 | https://mathoverflow.net/users/766 | 413247 | 168,606 |
https://mathoverflow.net/questions/413245 | 5 | In G. M. L. Powell's note 'Steenrod operations in motivic cohomology', he stated that if $\mathrm{char}(k)=0$,
$$H^{\*,\*}(k,\mathbb{Z}/2)=K\_\*^M(k)/2[\tau]$$
where $\tau\in H^{0,1}$ is the unique nonzero element.
I wonder whether this result holds when $char(k)>0$?
| https://mathoverflow.net/users/149491 | Motivic cohomology with $\mathbb{Z}/2$ coefficients in positive characteristic | This holds if the characteristic of $k$ is not 2, and it follows from the Milnor conjecture proved by Voevodsky.
Voevodsky ultimately proved the following (Theorem 6.17 in <https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-s.pdf>):
If $m>0$ and $X$ is smooth over a field $k$ of characteristic pr... | 5 | https://mathoverflow.net/users/20233 | 413256 | 168,608 |
https://mathoverflow.net/questions/413250 | 6 | I am looking how to prove the following fact:
If $ X \subseteq A^\mathbb{Z}$ is an infinite minimal subshift, then for any $N\ge 1$, $X$ is conjugate to a minimal subshift $Y\subseteq B^\mathbb{Z}$ such that for any $y\in Y$, $y(i)\neq y(j)$ if $|i-j|\le N , i\neq j$.
(I have encountered this in a paper in which th... | https://mathoverflow.net/users/7307 | Subshifts with special property | Define $X(m)$ as the image of $X$ in $(A^m)^\mathbf{Z}$, mapping $(a\_n)\_{n\in\mathbf{Z}}$ to $((a\_{n+k})\_{0\le k<m})\_{n\in\mathbf{Z}}$. This is an equivariant embedding.
Fix $N$. We claim that if $X$ has no periodic element then for $m$ large enough, $X(m)$ has the required property: for $0<|i-j|\le N$ and $y\in... | 8 | https://mathoverflow.net/users/14094 | 413258 | 168,609 |
https://mathoverflow.net/questions/413209 | 3 | A triangulation of a topological manifold $\mathcal{M}$ possibly with boundary is an abstract simplicial complex $\Delta$ together with a homeomorphism $\varphi:\vert\Delta\vert\to\mathcal{M}$, where $\vert\Delta\vert$ denotes the geometric realization and where an abstract simplicial complex is a collection of simplic... | https://mathoverflow.net/users/199422 | Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected? | Suppose that $M$ is a connected $d$-dimensional topological manifold without boundary. (We make the last assumption to simplify matters.) Let $\Delta$ be the given pseudo-triangulation. So the realisation $|\Delta|$ has the same local homology groups as $M$. These are $H\_k(M, M - x) \cong \mathbb{Z}$ if $k = d$ and ar... | 7 | https://mathoverflow.net/users/1650 | 413262 | 168,610 |
https://mathoverflow.net/questions/413211 | 3 | Consider two compact, oriented and connected manifolds $\mathcal{M},\mathcal{N}$ with possibly non-empty connected boundaries $\partial\mathcal{M}$ and $\partial\mathcal{N}$. Now, in some project, I encounted the following manifold:
$$\mathcal{Q}:=(\mathcal{M}\# B^{d})\#\_{\partial}\mathcal{N}$$
Let me briefly expl... | https://mathoverflow.net/users/199422 | Kind of "associativity" of certain connected sum involving both manifolds with and without boundary | This is true in the piecewise linear category. As you note, the boundary connect sum of $B$ and $N$ is homeomorphic to $N$. Now apply a result of Gugenheim [1953]: if $C$ and $D$ are $n$-balls embedded in the interior of a manifold, then there is an isotopy taking $C$ to $D$. This obtains the middle homeomorphism in yo... | 1 | https://mathoverflow.net/users/1650 | 413264 | 168,611 |
https://mathoverflow.net/questions/413282 | 1 | This is concerning Eq. (3.7) of C R Rao's 1945 paper (see p.81 of [this article](https://www.ias.ac.in/article/fulltext/reso/020/01/0076-0090)). Can someone help me in figuring out the second equality in Eq. (3.7)?
His claim is (since $\phi(x,\theta) = \Phi(T,\theta) \psi(x\_1,\dots,x\_n)$ from Eq. (3.6)) can be writ... | https://mathoverflow.net/users/7699 | Multiple integral and integral with respect to a function of variables | $\newcommand\th\theta$$T$ is a sufficient statistic and thus a random variable. So, the integral $\int f(T)\Phi(T,\theta)dT$ cannot possibly have a meaning.
The conclusions that Rao is trying to reach here are true, though:
(i) "there exists a function $f(T)$ of $T$, independent of $\theta$ and is an unbiased estim... | 2 | https://mathoverflow.net/users/36721 | 413285 | 168,615 |
https://mathoverflow.net/questions/413246 | 2 | [Modes, Medians and Means: A Unifying Perspective](http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medians-and-means-an-unifying-perspective/) defines the following centers based on the $L\_p$ norms:
$$
\begin{aligned}
\text{mode of x} = \arg \min\_s \sum\_i \lvert x\_i - s \rvert^0 \\
\text{median of x} = \a... | https://mathoverflow.net/users/474258 | Can anything be said about the roots of the L4 center? | The function $\sum\_i(x\_i-s)^3$ is strictly increasing in $s$, from $-\infty$ to $\infty$. It is also continuous, so there is a unique real root.
| 2 | https://mathoverflow.net/users/470870 | 413287 | 168,616 |
https://mathoverflow.net/questions/373511 | 23 | Let $X$ be a connected CW-complex with $\pi\_k(X)$ trivial for $k >2$. Is it known under which circumstances $X$ is an $H$-group?
I have been able only to deduce the necessary condition that $\pi\_1(X)$ has to be abelian and act trivially on $\pi\_2(X)$. Furthermore, if the necessary condition holds, vanishing of the... | https://mathoverflow.net/users/94647 | Which homotopy 2-types are H-spaces? | The necessary condition is a "additivity"-type condition on the $k$-invariant.
Suppose $\pi\_1 X = G$ and $\pi\_2 X = A$. As you correctly point out, $G$ must be abelian and act trivially on $A$. Under these circumstances, the $k$-invariant is expressible as a natural map of pointed spaces
$$
\beta: K(G,1) \to K(A,3)... | 13 | https://mathoverflow.net/users/360 | 413299 | 168,620 |
https://mathoverflow.net/questions/413281 | 0 | The joint distributions of the brownian and both the minimum and the maximum respectively are known. What could be said about the joint distribution of the maximum and the minimum of a Brownian process?
| https://mathoverflow.net/users/3898 | The joint distribution of the min and max of a Brownian | I agree with @MattF. This seems like a standard question. Here is a reference by Biane, Pitmann and Yor
<https://arxiv.org/pdf/math/9912170.pdf>
Look at page 16, equation (56). It is expressed as a series similar to the Elliptic theta series that one gets for Brownian motion on a circle or using the reflection method f... | 2 | https://mathoverflow.net/users/471081 | 413300 | 168,621 |
https://mathoverflow.net/questions/413162 | 6 | Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int\_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
Since the integration is taking over the sphere, we have rotation invariance and the value of the integration only depen... | https://mathoverflow.net/users/75323 | Monotonic dependence on an angle of an integral over the $n$-sphere | That is technically a 2D question. We can assume that $v=e^{-it}, w=e^{it}\in\mathbb R^2$ ($0<t<\pi/4$). Then in the polar coordinates the integral becomes
$$
\int\_0^1 \varphi(r)dr\int\_0^{2\pi}|\cos(s-t)\cos(s+t)|\,ds
$$
where $\varphi(r)$ is some non-negative function which I leave to you to compute (and whose exac... | 5 | https://mathoverflow.net/users/1131 | 413307 | 168,624 |
https://mathoverflow.net/questions/413284 | 1 | I'm trying to follow the proof of [Lemma 4 of "Strong NP-Hardness of the Quantum Separability Problem", by S. Gharibian, 2010](https://arxiv.org/abs/0810.4507) [1], which, roughly, states that there is a many-one reduction from the problem of Robust Semidefinite Feasability (RSDF) and the problem of Weak Optimization (... | https://mathoverflow.net/users/474351 | Understanding statement about bounds of vector in the context of a RSDF ≤ₘ WOPT proof | $\def\Tr{\mathop{\text{Tr}}}$ Let $\langle \cdot, \cdot \rangle\_F$ be the [Frobenius inner product](https://en.wikipedia.org/wiki/Frobenius_inner_product).
$C$ is symmetric and real, so $C^\dagger \equiv C$, then writing out $\lVert \hat c \rVert\_2$, and with $(\Tr C\sigma\_i)^2 \equiv \lvert \Tr C \sigma\_i\rvert^... | 0 | https://mathoverflow.net/users/474351 | 413309 | 168,625 |
https://mathoverflow.net/questions/413084 | 4 | Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ admits a symplectic structure, i.e. $X$ carries a closed nondegenerate 2-form $\omega$, then is $X$
Kähler?
| https://mathoverflow.net/users/99826 | Fujiki class $\mathcal C$ with a symplectic structure | If $X'$ is a Mukai flop of a compact hyper-Kähler manifold $X$, then $X'$ is in Fujiki class $\mathcal{C}$ and carries a holomorphic symplectic form $\sigma$. Taking the real part or the imaginary part of $\sigma$ gives a symplectic form on $X'$.
There exist however Mukai flops which are not Kähler, see e.g. [this pa... | 6 | https://mathoverflow.net/users/14037 | 413327 | 168,636 |
https://mathoverflow.net/questions/413111 | 5 | I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with the theory of orthogonal polynomials. I would like to understand it, but the sources I found were hard to follow.
It w... | https://mathoverflow.net/users/78061 | Riemann-Hilbert approach to Selberg integral | Let me formulate the problem in a slightly more general way: We seek to evaluate the large-$N$ limit of the matrix integral
$$\int e^{-\beta\,{\rm Tr}\,V(X)}|\Delta(X)|^\beta dX\equiv e^{-\beta N^2 F},$$
integrated over $N\times N$ Hermitian matrices $X$. In the OP the index $\beta=2$ and $V(X)=(a/2)\sum\_m m^{-1}X^m$,... | 5 | https://mathoverflow.net/users/11260 | 413330 | 168,637 |
https://mathoverflow.net/questions/413332 | 20 | In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c\_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c\_2\Big(\frac{n}{\log n}\Big)^{1/2}$$
for all sufficiently large $n$, where $c\_1, c\_2$ are positive constants. Then he said that *"ther... | https://mathoverflow.net/users/474442 | Distinct exponents in the factorization of the factorial, a problem of Erdős | The primes $p \leq \sqrt{n}$ can be ignored as the number of them is $$\approx \sqrt{n} / \log (\sqrt{n} )= 2 \sqrt{n}/\log n = o (1) \cdot \sqrt { n/\log n}$$
For primes $p> \sqrt{n}$, the exponent of $p$ is simply $\lfloor n/p \rfloor$. So an equivalent problem is to count the number of $1 \leq k \leq \lfloor \sqrt... | 15 | https://mathoverflow.net/users/18060 | 413344 | 168,639 |
https://mathoverflow.net/questions/413340 | 2 | Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$,
$$
f\_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\},
$$
is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?
| https://mathoverflow.net/users/36886 | Smoothness of Minkowski functional is equivalent to smoothness of boundary | Note that since you are on a finite dimensional space, the Minkowski functional yields a norm. For this norm the boundary of the convex body is the unit sphere. Now the differentiability condition for the boundary is equivalent to asking the unit sphere to be a $C^1$-submanifold.
Now on a Banach space a norm is (off ... | 3 | https://mathoverflow.net/users/46510 | 413350 | 168,640 |
https://mathoverflow.net/questions/413346 | 3 | My question is: if $E$ is an elliptic curve over $\mathbf{Q}$, and $p$ is a prime number such that $E[p]$ is *irreducible* as a Galois module, how does one go about bounding the $p$-primary Selmer group of $E$? Is this known to be a difficult / intractable problem, or are there known techniques to attack it?
If $E[p]... | https://mathoverflow.net/users/394740 | Bounds on $p$-primary Selmer groups when $E[p]$ is irreducible | The question is a bit too broad. I will make a few comments that may answer some of the questions that seem to be behind it.
For a given, fixed elliptic curve $E$ over a number field $K$, the Selmer group for $E[p]$ is *in principle* computable. However in practice this is very difficult and not really feasible for a... | 5 | https://mathoverflow.net/users/5015 | 413355 | 168,641 |
https://mathoverflow.net/questions/413358 | 10 | I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent\* with Euclidean geometry. I would like to make an end-of-term project for them to write about an alternate route to the hyperbolic plane via Riemannian geometry, but every res... | https://mathoverflow.net/users/1231 | Reference for shortest educational path to (Riemannian) hyperbolic plane | Try sections 1-15 of this paper:
*Cannon, James W.; Floyd, William J.; Kenyon, Richard; Parry, Walter R.*, [**Hyperbolic geometry**](http://www.msri.org/communications/books/Book31/), Levy, Silvio (ed.), Flavors of geometry. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 31, 59-115 (1997). [ZBL089... | 8 | https://mathoverflow.net/users/39654 | 413363 | 168,643 |
https://mathoverflow.net/questions/413338 | 3 | Let $A$ be an abelian variety over a field $K$. [It is shown](https://www.math.ru.nl/personal/bmoonen/BookAV/TateBT.pdf) that its $p$-adic Tate module $T\_p(A)= \varprojlim\_{n} A[p^n](\overline{K}) \cong Hom(\mathbb{Q}\_p/\mathbb{Z}\_p,A(\overline{K})= \varprojlim Hom (\mathbb{Z}/p^n \mathbb{Z}, A(\overline{K}))$.
N... | https://mathoverflow.net/users/146212 | Universal covering of abelian variety | I think the group $(\mathbf{Q}\_p/\mathbf{Z}\_p)\oplus\mathbf{Z}[\tfrac1p]$ does the job, i.e.
$$
\mathrm{Hom}((\mathbf{Q}\_p/\mathbf{Z}\_p)\oplus\mathbf{Z}[\tfrac1p],A(\bar K))\cong B(A).
$$
Indeed, let $A(\bar K)\_{\mathrm{tor}}$ be the torsion subgroup of $A(\bar K)$. We have a short exact sequence
$$
0\rightarrow A... | 2 | https://mathoverflow.net/users/85592 | 413367 | 168,644 |
https://mathoverflow.net/questions/413354 | 5 | Let $F$ be a finite-rank free group, $g$ an element of $F$ and $\Phi\colon F \to F$ an automorphism. Consider the dynamical system $\psi\_g\colon F \to F$ defined by $x \mapsto g\Phi(x)$. Say that $g$ is *principal of period $k$ for $\Phi$* if the identity of $F$ is periodic with period $k$ under the dynamical system $... | https://mathoverflow.net/users/135175 | Bound on the period of the identity (in a free group) for an automorphism followed by left-multiplication | Ahhhh, suppose $\psi\_g^k(1) = 1$, i.e. that $g\Phi(g)\cdots\Phi^{k-1}(g) = 1$. Then $g = \psi^{k+1}\_g(g) = g\Phi(g)\cdots \Phi^{k-1}(g)\Phi^k(g) = \Phi^k(g)$, so $g$ is $\Phi$-periodic with period dividing $k$. Therefore if there is a uniform bound to the period of $\Phi$-periodic elements, then there is a uniform bo... | 5 | https://mathoverflow.net/users/135175 | 413373 | 168,648 |
https://mathoverflow.net/questions/413339 | 7 | There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in great detail, it feels like the relations between the latter should be well-understood as well. But since I was unable t... | https://mathoverflow.net/users/129445 | How do the various homotopy 2-categories compare? | 1. **The simplicial sets $h\_2(N^\Delta(\mathcal{C}))$ and $N^D(H\_2(\mathcal{C}))$ are isomorphic**. To prove this, observe that the universal property of $h\_2(N^\Delta(\mathcal{C}))$ applied to the image under $N^{\Delta}$ of the quotient simplicial functor $\mathcal{C} \to H\_2(\mathcal{C})$ yields a map of simplic... | 8 | https://mathoverflow.net/users/57405 | 413383 | 168,653 |
https://mathoverflow.net/questions/413382 | 3 | I am looking for closed forms, or at least a good approximation for
$$f(n) = \sum\_{k=1}^{k=n} \genfrac\{\}{0pt}{}{n}{k}(n)\_kk$$
I know that
$$\sum\_{k=1}^{k=n} \genfrac\{\}{0pt}{}{n}{k}(n)\_k = n^n$$
I have the intuition that $f(n)$ is bounded above by $n^{n+1}$ and approaches $n^{n+1}$ for large $n$ but I am... | https://mathoverflow.net/users/474500 | Sum of the Stirling numbers of the second kind multiplied by $k$ and falling factorials | We have that $(x)\_k - (x-1)\_k = k (x-1)\_{k-1}$. So applying the linear operator $f \mapsto xf(x) - xf(x-1)$, to the identity $$ \sum\_{k=1}^{n} \genfrac\{\}{0pt}{}{n}{k}(x)\_k = x^n $$ we get that $$\sum\_{k = 1}^n \genfrac\{\}{0pt}{}{n}{k} k (x)\_k = x^{n+1} - x(x-1)^n.$$
**Edit:** In retrospect, there is also a ... | 7 | https://mathoverflow.net/users/382874 | 413385 | 168,654 |
https://mathoverflow.net/questions/413381 | 4 | $\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\mathbb{R})$.
One can see Theorem 2.6 of Gelbart's book *Automorphic Forms on Adele Groups*.
$L^2\_{\text{cusp}}(\SL(2,\math... | https://mathoverflow.net/users/169800 | $\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$ | You are asking what is known about the dimension of weight $k$ holomorphic cusp forms for $\mathrm{SL}\_2(\mathbb{Z})$, and the multiplicities of Laplace eigenvalues of weight $0$ and weight $1$ Maass forms for $\mathrm{SL}\_2(\mathbb{Z})$. This question is very open ended, similar to asking what is known about the dis... | 4 | https://mathoverflow.net/users/11919 | 413395 | 168,656 |
https://mathoverflow.net/questions/413361 | 7 | $\DeclareMathOperator\TAut{TAut}\DeclareMathOperator\Homeo{Homeo}$Let $G$ be a topological group, and let $\TAut(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the group of maps $f \colon G \to G$ that are simultaneously group automorphisms and self-homeomorphisms).
We wish to give $... | https://mathoverflow.net/users/474271 | Is there a natural topology on the automorphism group of a topological group? | $\DeclareMathOperator\Aut{Aut}$There is a recent paper [Uniformly locally bounded spaces and the group of automorphisms of a topological group](https://arxiv.org/abs/2001.00548) by Maxime Gheysens where he among other nice things systematically investigates the topologies on $\Aut(G)$ for any topological group $G$. On ... | 8 | https://mathoverflow.net/users/1275 | 413400 | 168,659 |
https://mathoverflow.net/questions/413386 | 2 | Let $E$ be a separable Banach space with symmetric basis $\{e\_i\}$ (it is also called a symmetric sequence space).
Let $\{x\_i\}$ be a normalized disjoint sequence in $E$, i.e.,
$\lVert x\_i\rVert\_E=1$ and $x\_i =\sum\_{n=n\_i}^{n\_{i+1}-1}a\_n e\_n $ for some strictly increasing sequence $\{n\_i\}$.
Assume that th... | https://mathoverflow.net/users/91769 | On the symmetric basic sequence of a symmetric sequence space | The question is not well formulated but i will answer the way I understood it. I think you are asking if there is any space with a symmetric basis other than $c\_0, \ell\_p$ which contains symmetric basic sequences with sup norm tends to zero and equivalent to the basis. The answer is yes. For instance, a minimal Orlic... | 3 | https://mathoverflow.net/users/3675 | 413417 | 168,665 |
https://mathoverflow.net/questions/413418 | 17 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$One can compute the (group cohomological) Euler characteristic of $\SL\_2(\mathbb{Z})$ via
$$ \chi(\SL\_2(\mathbb{Z})) = \chi(\mathbb{Z}/2) \cdot \chi(\PSL\_2(\mathbb{Z})) = \frac{1}{2}\cdot \left(\frac{1}{2} + \frac{1}{3} - 1\right) = -\frac{1}{12} = \zeta(-1) ... | https://mathoverflow.net/users/474608 | Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function | (Expanding my comment into an answer)
It is not a coincidence. Relating the Euler characteristic of certain arithmetic groups to the Zeta function is a theorem due to Harder [1] from 1971. It is expanded on in Brown's "Cohomology of Groups", Chapter IX.8.
Taken from Gruenberg's [AMS review](https://www.ams.org/jour... | 14 | https://mathoverflow.net/users/120914 | 413421 | 168,666 |
https://mathoverflow.net/questions/413422 | 0 | The Question
------------
Suppose that $X\_n$ are independent random variables, $|X\_n| \leq 1$, and $\mathbb{E}[X\_n] = 0$. Let $S\_n = \sum\_{i=1}^n X\_i$ and let $s\_n = \sqrt{\sum\_k \sigma^2(X\_k)}$ be the standard deviation of $S\_n$. Does $P(1 + |S\_n| <\epsilon s\_n)$ go to zero uniformly in $n$, as $\epsilon... | https://mathoverflow.net/users/173134 | Is a sum of a bounded random variables the same order as its standard deviation? | $\newcommand\ep\epsilon$Let $Z$ denote a standard normal random variable. The condition $|X\_i|\le1$ implies that $A\_3:=\sum\_{i=1}^n E|X\_i|^3\le\sum\_{i=1}^n E|X\_i|^2 =s\_n^2$. Also note that $P(1+|S\_n|\le\epsilon s\_n)=0$ unless $1\le\ep s\_n$.
Therefore and in view of the [Berry--Esseen inequality with Shevtso... | 2 | https://mathoverflow.net/users/36721 | 413426 | 168,667 |
https://mathoverflow.net/questions/413409 | 2 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f\_n $ be an orientation reversing isometry of the round sphere $ S^n $. Let $ M\_n $ be the mapping torus of $ f\_n $. What can we say about $ M\_n $?
Here are the things I ... | https://mathoverflow.net/users/387190 | Mapping torus of orientation reversing isometry of the sphere | For $n$ even, $M$ admits such an action. Indeed, the antipodal map of the even-dimensional sphere is orientation reversing, so you can realize $M$ as the quotient
$$\langle \gamma \rangle \backslash \left(S^n\times \mathbb R\right)$$
where $\gamma = (-\mathrm{Id}, 1)\in O\_{n+1} \times \mathbb R$.
Since $\langle \gamma... | 2 | https://mathoverflow.net/users/173096 | 413430 | 168,669 |
https://mathoverflow.net/questions/413424 | 2 | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin\_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible representations of its Lie algebra $\mathfrak{so}\_n$. $\Spin\_n$ is the universal cover of $\SO\_n$, another compact simple Lie ... | https://mathoverflow.net/users/378228 | Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$ | Your answer is correct for $n$ odd. For $n$ even you instead need the sum of the coefficients of the last two weights $a\_{l-1} + a\_{l}$ to be even. Either way you are effectively asking for the non-trivial element of the centre of $\mathrm{Spin}(n)$ to act trivially. Since it acts as $-1$ on the spin representation (... | 7 | https://mathoverflow.net/users/163024 | 413431 | 168,670 |
https://mathoverflow.net/questions/413321 | 1 | Let $T\_n = \frac{1}{6}n(n+1)(n+2)$ denote the $n$th Tetrahedral number. The first several solutions to squares as sums of two Tetrahedral numbers are {T\_n,T\_m,a^2}
1 5 6\
1 8 11\
1 22 45\
1 24 51\
1 63 209\
2 9 13\
2 23 48\
2 94 378\
2 96 390\
8 12 22\
8 17 33\
8 38 100\
8 111 484\
9 12 23\
9 21 44\
9 28 65\
10 15 3... | https://mathoverflow.net/users/265714 | Prove there are infinitely many squares which are the sum of two tetrahedral numbers | There are infinitely many integer solutions of equation $(1).$
$$y^2=\frac{n(n+1)(n+2)}{6}+\frac{m(m+1)(m+2)}{6}\tag{1}$$
Substitute $n = s - m$ to equation $(1)$ then we get
$$6y^2 = (3s+6)m^2+(-3s^2-6s)m+s^3+2s+3s^2$$
Let $s = 2t$ and $x = m - t$ then we get
$$3y^2 = (3t+3)x^2+t(t+2)(t+1)$$
Let $t = u^2-1$... | 6 | https://mathoverflow.net/users/150249 | 413440 | 168,674 |
https://mathoverflow.net/questions/413425 | 2 | We can put a metric on the set of isomorphism classes of connected, locally finite rooted graphs as follows:
Let $G$, $H$ be locally finite graphs, and let $u \in V(G)$ be a vertex of $G$, $v \in V(H)$ be a vertex of $H$. Define $R((G,u),(H,v))$ to be the largest number such that $(B\_G(u,R),u)$ is isomorphic to $(B\... | https://mathoverflow.net/users/169294 | Which graphs produce a compact set of rooted graphs when we vary the basepoint over all vertices? | The answer is **no**.
The reason is that in such a graph, every vertex $v$ is a limit point in the local topology, and [a nonempty closed set in which every point is a limit](https://en.wikipedia.org/wiki/Perfect_set) is uncountable. As the latter claim is [standard](https://math.stackexchange.com/questions/2604777/e... | 3 | https://mathoverflow.net/users/125498 | 413445 | 168,675 |
https://mathoverflow.net/questions/413441 | 0 | Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X\_n\subseteq X$ with,
1. For every $n\in \mathbb{N}$, the topology $\tau$ on $X\_n$ is second countable and metrizable space.
2. $X\_n\subseteq X\_{n+1}$ and $X=\bigcup X\_n$.
Q. Is the Borel sigma algebra coming from the w... | https://mathoverflow.net/users/84390 | Borel sigma algebra coming from the weak topology on TVS | There are separable metric TVS $X$ whose topological dual is trivial (one example is $L\_p([0,1])$ for $0\le p<1$). Such $X$ satisfies 1 and 2 (w.r.to the sequence $X\_n:=X$), and the two sigma algebras are of course not the same.
| 1 | https://mathoverflow.net/users/6101 | 413446 | 168,676 |
https://mathoverflow.net/questions/413448 | 1 | Let $K$ be a finite extension of $\mathbb{Q}\_p$ and $L/K$ a finite unramified extension.
Let $M$ be a $(\phi, \Gamma\_L)$-module over the Robba ring of $L$ (with coefficients in some other $p$-adic field $E$).
>
> If $M$ is trianguline then is $\mathrm{Ind}^L\_K(M)$ also trianguline?
>
>
>
Since $L/K$ is unra... | https://mathoverflow.net/users/519 | Triangularizability of induced $(\phi, \Gamma)$-modules | The answer is NO in general. Laurent Berger studies in this paper:
<http://perso.ens-lyon.fr/laurent.berger/articles/article18.pdf>
inductions of 1-dimensional representations of the absolute Galois group of $\mathbb Q\_{p^2}$ (these are always trianguline) to the Galois group of $\mathbb Q\_p$. The resulting induc... | 3 | https://mathoverflow.net/users/459579 | 413455 | 168,678 |
https://mathoverflow.net/questions/413452 | 3 | Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity
$$
D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \},
$$
$\| x \|$ denotes the distance to the closest integer. Dirichlet's approximation theorem implies
$D(\alpha, N) \geq 1$ for all $N$. What I am interested knowing is d... | https://mathoverflow.net/users/84272 | number of integers $n$ with $\|n \alpha \|$ small? | Yes. Assume that $\alpha$ is irrational, and its continued fraction digits do not exceed $K$. Then, for any positive integer $q$, we have
$$q\|q\alpha\|>1/(K+2).$$
In particular, for $q\leq N$, we have $\|q\alpha\|>1/(N(K+2))$. This implies that $D(\alpha,N)\leq 2K+4$, because one cannot accommodate $2K+5$ real numbers... | 11 | https://mathoverflow.net/users/11919 | 413459 | 168,680 |
https://mathoverflow.net/questions/413466 | 2 | The 3x6 matrix G is as follows,
$\text{G} = [\text{V}\_\times| I\_{3\times3}]$
$\text{V}$ is a skew matrix of a vector with 3 elements about a 3D point. The dimension of $\text{V}$ is 3x3.
$I$ is the 3x3 identity matrix.
I think the vertical line between $\text{V}$ and $I$ is used to concatenate these two 3x... | https://mathoverflow.net/users/474715 | What does the subscript 'x' of a matrix mean? | It is the *skew-symmetric form* defined [here](https://en.wikipedia.org/wiki/Skew-symmetric_matrix#Cross_product).
| 4 | https://mathoverflow.net/users/141766 | 413469 | 168,681 |
https://mathoverflow.net/questions/413349 | 0 | I am reading *Fluctuations of Levy Processes with Applications* by A.E. Kyprianou and I am having struggles understanding a part in the proof of theorem 5.6. Let $Y$ be a subordinator and $\mathbf{e}$ an independent exponential random variable with parameter $\eta$. Let $X$ be the killed subordinator associated with $Y... | https://mathoverflow.net/users/474494 | Expectation of killed subordinator at first-passage time | Because $f(0)=g(0)=0$, the passage over $x$ is being counted in $\Bbb E[f(X\_{\tau\_x^+}-x)g(x-X\_{\tau\_x^+-})]$ iff $X$ jumps at the crossing time. And
$$
\eqalign{
1\_{\{\Delta X\_{\tau\_x^+}>0\}}f(X\_{\tau\_x^+}-x)g(x-X\_{\tau\_x^+-})
&=\sum\_{t\in J}1\_{\{t<\mathbf{e}\}}1\_{\{X\_{t-}\le x, X\_t>x\}}f(X\_{t}-x)g(x-... | 1 | https://mathoverflow.net/users/42851 | 413471 | 168,683 |
https://mathoverflow.net/questions/413372 | 10 | Given a strictly positive integer $m$ let $\alpha(m)=\mathrm{rad}(m\phi(m))$
be the radical (product of all distinct prime divisors) of the product of $m$ and of Euler's totient function $\phi(m)=m\prod\_{p\vert m}
\left(1-\frac{1}{p}\right)$
(where the product is over all prime-divisors of $m$).
(Equivalently, $\alpha... | https://mathoverflow.net/users/4556 | Fixpoints of $m\longmapsto \mathrm{rad}(\phi(m^2))$ under iteration | Your observation on towers of primes is spot on. To study the iterations of $\alpha$ at $m$, for each prime $p$ dividing $m$, one builds a tree whose descending branches are the primes that divide $p-1$, and repeats iteratively. These are known as *Pratt trees*; the primes that appear in (the union of) the tree for $m$... | 6 | https://mathoverflow.net/users/nan | 413472 | 168,684 |
https://mathoverflow.net/questions/413475 | 2 | Let $A$ be a local ring, which we can assume is reduced. Let $k$ be the residue field of $A$.
In the Stacks project (<https://stacks.math.columbia.edu/tag/06DT>), I have learned some notion of the number of "geometric branches" of $A$ as being the number of minimal primes of the strict henselianization of $A$. Equiva... | https://mathoverflow.net/users/165625 | Strict henselianization and branches of explicit curve at singularity | Let us assume that $k$ is separably closed. In the case of local rings of finite type $k$-schemes such as here, it is usually easier to look at the completion with respect to the maximal ideal. Note that such rings are excellent, so the number of geometric branches of $\mathcal{O}\_{X, (0,0)}$ will coincide with the nu... | 2 | https://mathoverflow.net/users/339730 | 413486 | 168,691 |
https://mathoverflow.net/questions/413464 | 2 |
>
> Let $E$ be a subset of all $4\times 4$ real skew symmetric matrices with the property that for any $A,B \in E,\ \operatorname{rank}(A-B)\leq 2$, then what can be said about the maximal dimension of $\newcommand{\span}{\operatorname{span}}\span E$?
>
>
>
For less calculations, I considered $\Lambda^2(\mathbb ... | https://mathoverflow.net/users/112546 | Maximal dimension of linear span of a subset of all $4\times 4$ real skew symmetric matrices | Yes, such a subset $E$ must be contained in a $4$-dimensional subspace of
$W := \Lambda^2({\bf R}^4)$.
The key fact here is that the 6-dimensional space $W$
carries a symmetric bilinear pairing $(\cdot,\cdot) : W \times W \to {\bf R}$
such that $(\omega,\omega) = 0$ if and only if $\omega$ has rank at most $2$.
Indee... | 2 | https://mathoverflow.net/users/14830 | 413488 | 168,693 |
https://mathoverflow.net/questions/413491 | 6 | While reading the paper *Seifert Fibred Homology 3-Spheres and the Yang-Mills Equations on Riemann Surfaces with Marked points* by M. Furuta and B. Steer, I stumbled upon the following statement:
>
> Any compact orbifold Riemann surface, with $n\geq 3$ singular points or $n=2$ and $\alpha\_1=\alpha\_2$ if the genus... | https://mathoverflow.net/users/123694 | When is a compact orbifold Riemann surface a global quotient of a Riemann surface | Let $M$ be a your compact orbifold Riemann surface and let $S\subset M$ be the finite set of orbifold singularities.
**Theorem:**
The following are equivalent:
* $M$ is the quotient of a closed Riemann surface by a finite group
* $M$ admits a conformal metric of constant curvature (with the correct cone angles at t... | 8 | https://mathoverflow.net/users/173096 | 413493 | 168,695 |
https://mathoverflow.net/questions/413460 | 1 | Let $f$ be a balanced Boolean function.
>
> Are there $g$ linear functions, with $$\frac1{2^n}\mathrm{card} \big(\big\{\mathrm{sign} (g (x)) = 2f (x) -1, x \in \{0,1\}^n\big\}\big) > 0.55\quad ?$$
>
>
>
$g (x) = a\_1 (2x\_1-1) + ... + a\_n (2x\_n-1)$ and the $a\_i$ reals.
Ps : if the answer is yes, then NP=P... | https://mathoverflow.net/users/110301 | Boolean function : approximation by a linear function | The answer is no. In fact, noise sensitive functions are characterized by being asymptotically uncorrelated with all weighted majority functions.
See Theorem 1.7 in [1]. A simple example of a noise sensitive function is the xor of all the Boolean variables. A more interesting example is percolation, see section 4 of [1... | 4 | https://mathoverflow.net/users/7691 | 413498 | 168,696 |
https://mathoverflow.net/questions/413438 | 4 | The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit.
We begin with a Hilbert space $\mathcal{H}$, and let $\mathcal{L}(\mathcal{H})$ be the set of all of its projection operators to (closed, I guess? I wil... | https://mathoverflow.net/users/473051 | In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability? | No direct answer. Only "hidden variables in quantum logic" (NO physics needed).
Quite long; subdivisions hopefully help.
---
Disclaimer
This expands what I said about hidden variables in (more general) lectures in Bologna, [2010](http://nohay.homepc.it/mat/sid/Bologna.08-09-10/)
My study of this subject goe... | 5 | https://mathoverflow.net/users/474159 | 413515 | 168,700 |
https://mathoverflow.net/questions/413432 | 3 | For $G$ any finite group and $V$ any irreducible complex representation of $G$ with character $\chi\_V$, is it always true that
$$ \frac{1}{\left| G \right|} \displaystyle\sum\_{g\in G} \chi\_V(g)\chi\_V(g^{-1}h) = \frac{\chi\_V(h)}{\dim(V)}?$$
I see already that if $V$ is one-dimensional, then $\chi\_V$ is a group... | https://mathoverflow.net/users/91316 | A "shifted" orthogonality relation for characters of irreducible representations of finite groups? | Consider the linear transformation $\rho(e\_V)$ to which $e\_V$ maps in the representation $\rho$ on $V$. The element $e\_V$ is in the center of the group algebra. So its image commutes with $\rho(g)$ for all $g\in G.$ But since $V$ is irreducible, by Schur's lemma, $\rho(e\_V)=\lambda \textit{Id}$ for some constant $\... | 1 | https://mathoverflow.net/users/91316 | 413516 | 168,701 |
https://mathoverflow.net/questions/413494 | 3 | I'd like to ask two questions about congruences: one about modular forms and one about elliptic curves.
1. Suppose we are given a cusp form $f$ of weight $2$ and level $\Gamma\_0(N)$. Given a good ordinary prime $p$ for $f$, does there always exist a Hida family passing through $f$? (I've heard that such a Hida famil... | https://mathoverflow.net/users/394740 | Existence of congruences between modular forms / elliptic curves | 1. Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This is for instance Theorem I of *Galois representations into $\operatorname{GL}\_{2}(\mathbb Z\_{p}[[X]])$ attached to or... | 5 | https://mathoverflow.net/users/2284 | 413524 | 168,704 |
https://mathoverflow.net/questions/413513 | 7 | Consider an entire function $f:\mathbb C \to \mathbb C$ that is real on the real line and even.
This function has a Taylor series of the form
$$f(z) = \sum\_{i=0}^{\infty} a\_i z^{2i} \text{ with } a\_i \in \mathbb R.$$
I have an estimate of the form
$$\vert e^{z^2 \mu} f(z) \vert \le e^{\vert z \vert^4 \nu}$$
fo... | https://mathoverflow.net/users/457901 | Improving Cauchy estimates? | Let $z=re^{i\theta}$. Then your original inequality implies
$$|f(re^{i\theta})|\leq \exp(\nu r^4-\mu r^2\cos2\theta).$$
Then instead of Cauchy estimate you can use the exact formula for the coefficient:
$$|a\_{2n}|=\left|\frac{1}{2\pi i}\int\_{|z|=r}\frac{f(z)}{z^{2n+1}}dz\right|\leq\frac{1}{2\pi}
\int\_0^{2\pi}|f(re^{... | 8 | https://mathoverflow.net/users/25510 | 413525 | 168,705 |
https://mathoverflow.net/questions/413252 | 0 | Is the chromatic number of a Cayley graph on $p$-groups with any generating set bounded by the chromatic number of the maximal induced circulant subgraph?
I think yes. Because for one, the main obstruction to the chromatic number, the clique size directly corresponds to a circulant subgraph. For another probable reas... | https://mathoverflow.net/users/100231 | Bound on chromatic number of graphs on any finite $p$-group | First, if you mean the largest induced circulant subgraph, you should call it a *maximum* induced circulant subgraph, not *maximal*. (That is quite standard in this kind of area, where *maximal* would mean "not contained in a larger one".)
Second, "the" maximal (or maximum) induced circulant subgraph is generally **n... | 1 | https://mathoverflow.net/users/22377 | 413533 | 168,708 |
https://mathoverflow.net/questions/413451 | 12 | **Background:**
It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ equal to the closed unit ball in the dual $X^\*$, and embed $X$ in $C((X^\*)\_1)$ using the duality mapping $x \map... | https://mathoverflow.net/users/1193 | Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space? | In [Ray - On Isometric Embedding $\ell\_p^m \to S\_\infty^n$ and Unique operator space structure](https://arxiv.org/abs/1911.00241) it is shown that for $p\in (2,\infty)\cup \{1\}$ and $n\ge 2$, $\ell\_p^n$ does not isometrically embed into the space of compact operators on $\ell\_2$.
(This was previously in the comm... | 6 | https://mathoverflow.net/users/3675 | 413534 | 168,709 |
https://mathoverflow.net/questions/413530 | 4 | In laying down the equality rules in Martin-Löf type theory, e.g., for the type $\mathsf{N}$ of natural numbers, there seems to be an implicit assumption that any natural number is either $0$ or $S(a)$ for some $a\in\mathsf{N}$; in other words, only the so-called 'canonical' elements of the type $\mathsf{N}$ are given ... | https://mathoverflow.net/users/163629 | Why in Martin-Löf type theory any natural number is assumed to be either $0$ or $S(a)$ for some $a\in\mathsf{N}$? | There is no such implicit assumption at all. You are missing one rule for natural numbers, namely the induction principle. The rules for natural numbers are:
$$\frac{}{\vdash \mathbb{N} \; \mathsf{type}} \qquad
\frac{}{\vdash 0 : \mathbb{N}}
\qquad
\frac{\vdash t : \mathbb{N}}{\vdash \mathsf{S}(t) : \mathbb{N}}
\\[4ex]... | 8 | https://mathoverflow.net/users/1176 | 413546 | 168,713 |
https://mathoverflow.net/questions/413547 | 5 | $\DeclareMathOperator\THH{THH}\DeclareMathOperator\HH{HH}$A version of the strict graded commutativity (i.e. graded commutativity & $x^2=0$ for every homogeneous element $x$ of odd degree) of $\pi\_\*(\THH(A))$ seems to be used in Construction 6.8 of [Bhatt–Morrow–Scholze, *Topological Hochschild homology and integral ... | https://mathoverflow.net/users/176381 | Strict graded commutativity of $\pi_*(\operatorname{THH}(A))$? | To my knowledge, there is no such result for THH of a commutative ring. (Could be?)
However, we need less for this result; we only need degree 1 elements to square to zero. Every element in $\pi\_1 THH(A)$ is a finite linear combination of elements $a \cdot \sigma(b)$ for $a, b$ in $A$, where $\sigma$ is the circle a... | 3 | https://mathoverflow.net/users/360 | 413553 | 168,717 |
https://mathoverflow.net/questions/413558 | 1 | $\DeclareMathOperator\Idl{Idl}$Let $P$ be a finite, connected poset with at least two elements, and let $\Idl\_{\neq \emptyset, P}(P)$ be the set of downward closed sets $S \subset P$ such that $S \neq \emptyset$ and $S \neq P$, ordered under inclusion.
**Question:** Is $\Idl\_{\neq \emptyset, P}(P)$ weakly contracti... | https://mathoverflow.net/users/2362 | Is the poset $\mathrm{Idl}_{\neq \emptyset, P}(P)$ of nonempty, proper ideals in a finite connected poset $P$ (empty or) weakly contractible? | Let me be a little more historical. As mentioned, it is well known that (open intervals of) distributive lattices are shellable. This is mentioned in Corollary 3.2 of Björner's classic paper "Shellable and Cohen-Macaulay partially ordered sets" (<https://doi.org/10.1090/S0002-9947-1980-0570784-2>) but is an older resul... | 3 | https://mathoverflow.net/users/25028 | 413559 | 168,718 |
https://mathoverflow.net/questions/413510 | 0 | Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}\_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $L^2(\mathbb{R}^d)$ denote the Bochner space of all Borel functions $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$ satisfying $... | https://mathoverflow.net/users/36886 | Building the Wasserstein space by pushforwards | Here is an alternative answer, which is weaker than @Benoît Kloeckner's strong form and only asserts density.
Yes, your set is dense. Indeed, $\mathcal X$ contains finitely atomic measures, which are obviously weakly-$\ast$ dense in the Wasserstein space and therefore also dense in distance (it is well known that the... | 1 | https://mathoverflow.net/users/33741 | 413585 | 168,726 |
https://mathoverflow.net/questions/413602 | 8 | I came across the following cute fact about partitions:
\begin{align}
& |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt]
& {} - |\{ \lambda \vdash n \text{ with an odd number of even parts}\}| \\[8pt]
= {} & |\{ \lambda \vdash n \text{ which are self-conjugate } (\text{i.e. } \lambda = \lambda... | https://mathoverflow.net/users/39120 | Bijective proof for a partition identity | **Lemma.** For $n>1$, the number of partitions of $n$ onto an even number of powers of 2 (here powers of 2 are 1,2,4,...) and the number of partitions of $n$ onto an odd number of powers of 2 are equal.
**Proof.** If $n$ is odd, we must have a part equal to 1, so subtract it and work with $n-1$ instead. If $n=2k$, th... | 14 | https://mathoverflow.net/users/4312 | 413605 | 168,732 |
https://mathoverflow.net/questions/413609 | 4 | Consider full second-order Heyting arithmetic, axiomatized in two-sorted first-order intuitionistic logic (with “number” and “class” variables) by the usual Peano axioms (with induction being stated quantified over classes) and a class-forming notation which, for every formula $\varphi(n)$ with a free number variable $... | https://mathoverflow.net/users/17064 | Does second-order Heyting arithmetic have the disjunction and existence properties? | Yes to all. For example see Chapter IX, Section 2 of Beeson, Foundations of Constructive Mathematics.
| 6 | https://mathoverflow.net/users/30790 | 413611 | 168,734 |
https://mathoverflow.net/questions/413577 | 3 | Let $X$ be a Banach space and let $(x\_{n})\_{n=1}^\infty$ be a (Schauder) basis for $X$. Let $(x^{\*}\_{n})\_{n=1}^{\infty}$ be the biorthogonal functionals associated to the basis $(x\_{n})\_{n=1}^\infty$. We shall use the notation $\|x^{\*}\|\_{n}:=\|x^{\*}|\_{[x\_{i}\colon i>n]}\|, \quad (x^{\*}\in X^{\*}, n\in \ma... | https://mathoverflow.net/users/41619 | Quantifying shrinking bases | The answer to your first question is "yes". Here is a sketch of an argument that may be clumsier than needed. If $(e\_n)$ is not shrinking, take a norm one linear functional $x^\*$ that has distance $d$ arbitrarily close to one from the span of $(e\_n^\*)$ and take a norm one $x^{\*\*}$ with $\langle x^{\*\*}, x^{\*}\r... | 3 | https://mathoverflow.net/users/2554 | 413612 | 168,735 |
https://mathoverflow.net/questions/413614 | 2 | My university has a subscription at IOPSciences
I am interesting in this article: Galochkin A.I. (1984): On estimates, unimprovable with respect to height, of some linear forms. Mat. Sb. 124 (166), 416-430. English transl.: Math. USSR, Sb. 52, 407-419.
So I went to IOPsciences site to download it (the english version... | https://mathoverflow.net/users/33128 | IOPsciences and a missing reference | I'm pretty sure a machine translation of the [paper](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=2059&option_lang=eng) from russian into english will be sufficient. For starters, I fed two paragraphs on pages 417-418 to <https://deepl.com>, with the following output (unedited, the only changes ... | 4 | https://mathoverflow.net/users/11260 | 413616 | 168,736 |
https://mathoverflow.net/questions/413576 | 5 | Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ lies in the center $\mathcal Z(R)$ of $R$; and condition (C') if there exists an integer $n \ge 1$ such that $a^n \in \ma... | https://mathoverflow.net/users/16537 | Rings s.t. each element has a power lying in the center (and their completely prime ideals) | Assume (C), so that for each $a\in R$, there exists some integer $n\geq 1$ (possibly depending on $a$) such that $a^n\in Z(R)$. The equivalence of conditions (1) and (3) in Theorem 12.11 from Lam's “[A First Course in Noncommutative Rings](https://doi.org/10.1007/978-1-4419-8616-0)” tells us that $R/J(R)$ is commutativ... | 3 | https://mathoverflow.net/users/3199 | 413618 | 168,737 |
https://mathoverflow.net/questions/403583 | 7 | Let $X$ be a scheme or fs log scheme over a finite field. There seem to be several slightly different definitions of the (log) crystalline site of $X/S$ available in the literature, depending on whether you take the objects of $\mathrm{Cris}(X/S)$ to be:
1. ($X$ a scheme) diagrams $X \leftarrow U\to T$ where $U\to T$... | https://mathoverflow.net/users/126183 | Choice of topology in the (log) crystalline site | It seems that the categories of log-crystals in the strict etale and Kummer etale topologies are not actually equivalent to one another, contrary to what I expected.
Here's a sketch of a proof. Firstly, we'll restrict attention to the case that $X=S$ is log-smooth, so that the category of log-crystals on $X/S$ in the... | 1 | https://mathoverflow.net/users/126183 | 413620 | 168,739 |
https://mathoverflow.net/questions/412451 | 3 | If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation
$$
yz=x^2
$$
are given by $x=uvw$, $y=uv^2$, $z=uw^2$ for some integers $u,v,w$. More formally, a subset $S \subset {\mathbb ... | https://mathoverflow.net/users/89064 | Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$ | For $x^2+x+1=yz$ we may factorize LHS in the unique factorization domain $\mathbb{Z}[\omega]$, where $\omega=e^{2\pi i/3}$: $$(x-\omega)(x-\omega^2)=yz.$$
Denote by $A$ the greatest common divisor of $x-\omega$ and $y$, say $x-\omega=AB$, $y=AC$. Then $B$ and $C$ are coprime, and $B(x-\omega^2)=Cz$. Thus $C$ divides $x... | 2 | https://mathoverflow.net/users/4312 | 413625 | 168,742 |
https://mathoverflow.net/questions/413622 | 2 | I understand that a combinatorial class $\mathcal{A}$ is a set of objects, with a function of size $\lvert\cdot\rvert\_{\mathcal{A}}:\mathcal{A}\to \mathbb{N}$. With objects of size $n$: $\mathcal{A}\_n=\{\alpha\in\mathcal{A}\;:\; \lvert\alpha\rvert\_{\mathcal{A}}=n\}$, and a condition of finitude $\#\mathcal{A}\_n<\in... | https://mathoverflow.net/users/474924 | Confusion in definition of class of structures and combinatorial class | Part of the problem is that you have stated the definition of "class of structures" incorrectly.
(Wagner states it correctly.) The first sentence should be: "A class $\mathcal{A}$ of structures associates to every finite set $X$ another finite set $\mathcal{A}\_X$, in such a way that the following two conditions are sa... | 3 | https://mathoverflow.net/users/10744 | 413628 | 168,743 |
https://mathoverflow.net/questions/413626 | 1 | I found this exercise while reading some notes on Large Deviation Principle. This exercise is at the end of the very first chapter, including Cramer's Theorem and essentially nothing more (no Sanov Theorem). Maybe it's because I'm a bit new to this subject, but I actually can't find any plausible solution. Let $X\_1,\l... | https://mathoverflow.net/users/70112 | Large deviation for empirical median | $\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\la}{\lambda}\newcommand{\be}{\beta}\newcommand{\Y}[1]{\hat Y\_{2n+1}^{#1}}$Since $\Y t$ is the sum of $2n+1$ independent copies of $Y:=Y\_1^t$, by [Cramér's theorem](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_theorem_(large_deviations)),
\begin{equat... | 1 | https://mathoverflow.net/users/36721 | 413636 | 168,745 |
https://mathoverflow.net/questions/413608 | 2 | I have an integral to minimize that writes like $$F: \mathbb R^d \to \mathbb R: \theta \mapsto \int\_{[0,1]^d} f(\langle x,\theta\rangle) dx$$.
The function $f$ is a convex function, which makes $F$ a convex function.
Q : Let $x \in [0,1]^d$. Is $\frac{df(\langle x,\theta\rangle)}{d\theta}$a subgradient of $F$ at $... | https://mathoverflow.net/users/143783 | Subgradient of a convex integral | For $d=1$ and $f(z)=z^2 $, $dF/d\theta = 2\theta /3$, but $df(x\theta )/d\theta = 2\theta x^2 $. Not a subderivative of $F$ for almost all $x$.
| 0 | https://mathoverflow.net/users/134299 | 413641 | 168,746 |
https://mathoverflow.net/questions/413632 | 3 | I am trying to prove this theorem. I have not found anything similar to it on the internet.
**Special version of Tonelli’s theorem**
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{R}$, $ g(x, \xi): [a,b] \times \mathbb{R} \to \mathbb{R}$ are continuous, $f$ is bounded below, $g$ is convex in $... | https://mathoverflow.net/users/471464 | Special version of Tonelli’s theorem | $\renewcommand\bar\overline$Indeed, it is not obvious why "$u'\_{n\_k} \to \overline{u}'$ in the sense of $L^r[a,b]$".
Look at this example: $[a,b]=[0,2\pi]$, $u\_n(x)=\dfrac{\sin nx}n$, $\bar u=0$. Then $u\_n\to\bar u$ uniformly, but $u\_{n\_k}'\not\to\bar u'$ in $L^r$ for any increasing sequence $(n\_k)$ of natural... | 4 | https://mathoverflow.net/users/36721 | 413645 | 168,748 |
https://mathoverflow.net/questions/413610 | 1 | [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula) expresses a sum over some finite number of naturals to the $m^{th}$ power in terms of the [Bernoulli numbers](https://en.wikipedia.org/wiki/Faulhaber%27s_formula#Summae_Potestatum) $B\_{j}$ (using the $B\_{1} = 1/2$ convention) or [polynomials](... | https://mathoverflow.net/users/118385 | Simplifying a rational function in terms of Bernoulli numbers and polynomials | For the special case $n\_1 = n\_2 = n$ and $1 \le m \le 9$ it turns out that the double sum yields a polynomial of the form $P(n)(2n+1)(n+1)n^2$ where $P(n)$ is irreducible of degree $2(m-1)$. Obviously the ratio of two irreducible polynomials of different degrees cannot be a polynomial. It may be that for higher value... | 1 | https://mathoverflow.net/users/46140 | 413664 | 168,754 |
https://mathoverflow.net/questions/413675 | 2 | What is the geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group? Could Springer theory of Weyl group representations be used to obtain such a geometric meaning?
| https://mathoverflow.net/users/198061 | Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group | Thanks to colleagues, it turns out that the Springer correspondence is a functor which associates representations of the Weyl group to sheaves on the nilpotent cone, and this functor maps induction from a parabolic subgroup $W\_L$ to the Weyl group $W\_G$ to parabolic induction of sheaves on the nilpotent cone defined ... | 3 | https://mathoverflow.net/users/198061 | 413685 | 168,756 |
https://mathoverflow.net/questions/412690 | 4 | Fix $k > 0$ for $\lceil \frac{2k+1}{3} \rceil \le j \le k$,
$$ \sum\_{i = 0}^{2j-k-1} \binom{j}{i} + \sum\_{b = \lceil \frac{k+1}{2} \rceil}^{\lfloor \frac{2k-j}{2} \rfloor} \left( \sum\_{l = 0}^{2b-k-1} \binom{b}{l} \right) \binom{j-b-1}{2j-(2k+1)+b}$$
I think that this expression (mod 2) should be 1 when $j=k$ an... | https://mathoverflow.net/users/473047 | Does anyone have ideas about how to simplify this combinatorial expression (mod 2)? | First we may notice that
$$\sum\_{i=0}^m \binom{n}{i} = [x^m]\ \frac{(1+x)^n}{1-x} \equiv\_2 [x^m]\ \frac{(1+x)^n}{1+x} = [x^m]\ (1+x)^{n-1} = \binom{n-1}m,$$
which was already pointed out by @FedorPetrov in the comments.
Then modulo 2 the given expression is congruent to
$$(\star)\qquad \binom{j-1}{2j-k-1} + \sum\_b... | 3 | https://mathoverflow.net/users/7076 | 413706 | 168,762 |
https://mathoverflow.net/questions/413702 | 6 | Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers.
\begin{equation}
(\mathcal{T} f)(z\_1) = \mathrm{p.v.} \int\_0^1 \frac{i}{z\_1-z\_2}f(z\_2) dz\_2
\end{equation}
where $\mathrm{p.v.}$ means the Cauchy principal value. This is an analogue of the Hilbert Transfor... | https://mathoverflow.net/users/106463 | Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$ | This was carried out in
*Koppelman, W.; Pincus, J. D.*, [**Spectral representations for finite Hilbert transformations**](http://dx.doi.org/10.1007/BF01181411), Math. Z. 71, 399-407 (1959). [ZBL0085.31701](https://zbmath.org/?q=an:0085.31701).
The spectrum is purely continuous on $[-\pi,\pi]$ (with the OP's normali... | 9 | https://mathoverflow.net/users/766 | 413713 | 168,765 |
https://mathoverflow.net/questions/413719 | 3 | I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case.
I admit things like: "no analog to cauchy hypersuface can be defined in arbitrary pseudoriemanniann manifolds, just in lorentzian ones". I ... | https://mathoverflow.net/users/148711 | Properties that only Lorentzian manifolds have | A pseudo-Riemannian manifold of signature $(p,q)$ with $p, q\geq 2$ certainly does not admit Cauchy hypersurfaces, since spacelike submanifolds have dimension at most $p$. However, you could define a notion of "Cauchy $p$-surface" as a spacelike submanifold of dimension $p$ intersecting any inextensible timelike $q$-di... | 9 | https://mathoverflow.net/users/173096 | 413723 | 168,769 |
https://mathoverflow.net/questions/413671 | 11 | The theory of classifying topoi due to Makkai, Reyes, Hakim, and Grothendieck supplies a bijection between geometric theories (up to Morita equivalence) and Grothendieck topoi, by assigning to each geometric theory $\mathbb T$ the unique Grothendieck topos $\mathcal E$ such that
$$\hom(\mathcal F, \mathcal E)\cong \tex... | https://mathoverflow.net/users/474987 | Compactness theorem and topos theory | The following is an analogy with the compactness theorem. "Compactness" in logic refers to the fact that the first-order theory you're working with is finitary. In topos theory, I believe the closest approximation to this idea is restricting to coherent theories (Johnstone in "Topos Theory" calls them finitary theories... | 9 | https://mathoverflow.net/users/37368 | 413732 | 168,770 |
https://mathoverflow.net/questions/413727 | 6 | Let $f$ be a bounded and continuous function, $0<a < 1$. $U(x,r)$ is the neighborhood of $x$ with diameter $r$. Can we prove the following equation of two limits
$$ \lim\_{r\rightarrow 0} \sup\_{y,z \in U(x,r)} \frac{|f(y)-f(z)|}{|y-z|^a} = \lim\_{r\rightarrow 0} \sup\_{y,z \in U(x,r)} \frac{|f(y)-f(z)|}{r^a}$$
wheneve... | https://mathoverflow.net/users/152618 | A limit problem | This equality is not true. E.g., if $f(u)=u$, $x=0$, and $a=1$, then the left-hand side of the equality is $1$, whereas its right-hand side is $2$.
---
The OP has now switched the meaning of $r$ from radius to diameter. The equality is still not true. E.g., if $f(u)=|u|$, $x=0$, and $a=1$, then the left-hand side... | 12 | https://mathoverflow.net/users/36721 | 413741 | 168,772 |
https://mathoverflow.net/questions/413737 | 1 | **Informal version**
What is the maximum number of distinct $n$-runs that a $\{0,1\}$-sequence of length $2^n$ can have?
**Formal version**
If $A, B$ are sets, we denote by $B^A$ the set of all functions $f:A \to B$. For any positive integer $n\in\mathbb{N}$ we let $[n]:=\{0,\ldots,n-1\}$.
**"Map of runs":** Le... | https://mathoverflow.net/users/8628 | Maximum number of distinct $n$-runs that binary sequence of length $2^n$ can have | I understand that $n$-run here refers just to a substring of length $n$ (sometimes called $n$-mer).
The answer to this question is given by any [de Bruijn sequence](https://en.wikipedia.org/wiki/De_Bruijn_sequence) $B(2,n)$, where all $2^n-n+1$ (or $2^n$ if the sequence is viewed as cyclic) $n$-mers are distinct.
| 3 | https://mathoverflow.net/users/7076 | 413749 | 168,775 |
https://mathoverflow.net/questions/413748 | 9 | Is it possible to get an upper bound better than $\ll\_\sigma T^{3/2-\sigma}$ for $$\int\_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
| https://mathoverflow.net/users/155294 | Moments of the Riemann zeta function | This answer is based on Lucia's remark, and is included for completeness.
By (8.111) in Ivić's book "The theory of the Riemann zeta function with applications", we have
$$\int\_T^{2T}|\zeta(\sigma+it)|\,dt\asymp\_\sigma T,\qquad T\geq 1,\quad 1/2<\sigma<1.$$
Hence, by the functional equation for $\zeta(s)$ and Stirli... | 13 | https://mathoverflow.net/users/11919 | 413754 | 168,777 |
https://mathoverflow.net/questions/413540 | 7 | Let $A$ be an infinite dimensional Banach algebra. Even if separable the primitive ideal space of $A$ need not be second-countable when endowed with the hull-kernel topology. Can we at least find an infinite-dimensional subalgebra with this property?
For C\*-algebras, separability implies second-countable primitive i... | https://mathoverflow.net/users/15129 | Does every infinite-dimensional Banach algebra contain an infinite-dimensional subalgebra with second-countable primitive ideal space? | The answer is no.
Let $A=A(D)$ be the [disk algebra](https://en.wikipedia.org/wiki/Disk_algebra). Every character on $A$ is a point evaluation at some $x\in D\_1$, the closed unit disk of $\mathbb{C}$, i.e. the spectrum $\sigma(A)$ of $A$ is $D\_1$. Let $D\_r = \{z\in\mathbb{C}: |z|\leq r\}$ for $r<1$. Then, the restri... | 3 | https://mathoverflow.net/users/164350 | 413755 | 168,778 |
https://mathoverflow.net/questions/413730 | 2 | Consider a random walk on an infinite connected vertex-transitive graph. Let $f(t)=P\_{o,o}^{2t}$ be the probability that the random walk is at its origin at time $2t$. What can be said about the asymptotics of $f(t)$? Specifically, I would like to know whether $\lim\_{t\to\infty}\frac {\log f(t)}{\log t}$ always exist... | https://mathoverflow.net/users/85550 | Asymptotics of the return probabilities of a random walk on a transitive graph | the limit $L=\lim\_{t\to\infty}\frac {\log f(t)}{\log t}$ always exists, in the wide sense. If a transitive graph does not have polynomial growth, then the limit is $-\infty$, while if it has polynomial growth, the exponent is necessarily an integer $d$ by Trofimov's [1] extension of Gromov's Theorem, and in that case ... | 5 | https://mathoverflow.net/users/7691 | 413759 | 168,780 |
https://mathoverflow.net/questions/413753 | 2 | There are different models of quantum computing like quantum circuits, adiabatic or annealing. Another thing to mention is the complexity class BQP. It is pretty much a given that the different models are equivalent as computational models and in turn they are equivalent to deterministic turing machines. However, like ... | https://mathoverflow.net/users/467143 | Different quantum computation models equivalence | Quantum Turing machines, quantum circuits, and quantum adiabatic algorithms are polynomially equivalent, in complexity class BQP [1,2]. Concerning quantum annealers, it is unknown whether they offer any speedup relative to classical annealers.
To find a computational model that is in a different complexity class, one... | 5 | https://mathoverflow.net/users/11260 | 413765 | 168,782 |
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