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https://mathoverflow.net/questions/366846 | 0 | Let $G$ be a regular simple graph with degree $\Delta=n-k-1$ and order $m$. Let $C\_k$ be the regular graph which is formed by removing a $k$-factor from the complete graph $K\_{n}$. I think we could always find a proper induced subgraph of $C\_k$ with maximum degree at least $\ge\frac{\Delta}{2}$ as a subgraph of the ... | https://mathoverflow.net/users/100231 | Decomposition of regular graphs | The answer is **no**.
Let $G$ be the Hoffman-Singleton graph (hence n=50, k=42). Let $C\_k$ be the disjoint union of 5 $K\_8$s and a $K\_{10}-C$ ($K\_{10}$ with a 10-cycle removed). Any proper induced subgraph of $C\_k$ with maximum degree at least 4 will contain a $C\_3$ or $C\_4$, which $G$ does not contain.
| 3 | https://mathoverflow.net/users/125498 | 366869 | 154,042 |
https://mathoverflow.net/questions/366858 | 7 | This is a [cross-post](https://math.stackexchange.com/questions/3763305/is-a-function-of-several-variables-convex-near-a-local-minimum-when-the-derivati).
Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a **strict local minimum** point of $f$.
Let $d... | https://mathoverflow.net/users/46290 | Is a function of several variables convex near a local minimum when the derivatives are non-degenerate? | Let
$$\begin{aligned} f(x,y) & = x^4 - x^2 y^2 + y^4 \\ & = \tfrac{1}{2} x^4 + \tfrac{1}{2} y^4 + \tfrac{1}{2} (x^2 - y^2)^2 . \end{aligned}$$
Then $f$ is a strictly positive (except at the origin, of course) homogeneous polynomial of degree $4$, and hence $d^j f(\vec 0) = 0$ for $j < 4$ and $d^4 f(\vec 0) > 0$ (indeed... | 12 | https://mathoverflow.net/users/108637 | 366873 | 154,044 |
https://mathoverflow.net/questions/366675 | 20 | I have some questions on derivators and $(\infty,1)$-categories,
I would be grateful if someone could help me.
* Is there some problems that $(\infty,1)$-categories/derivators can resolve but derivators/$(\infty,1)$-categories cannot resolve?
* Why do so many people prefer $(\infty,1)$-categories than Grothendieck de... | https://mathoverflow.net/users/155635 | Grothendieck derivators vs $\infty$-categories | The short answer is that $(\infty,1)$-categories are the "real" object of interest. Derivators are a tool for working with them that is sometimes (for some people) easier to use, but doesn't remember all the information and hence is not always applicable.
When you work with $(\infty,1)$-categories, you have to deal e... | 24 | https://mathoverflow.net/users/49 | 366887 | 154,047 |
https://mathoverflow.net/questions/366627 | 0 | I'm [crossposting](https://math.stackexchange.com/questions/3759310/functional-equation-for-etas-following-riemanns-2nd-method).
Being
\begin{equation\*}
\eta(s)=\sum\_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots
\end{equation\*}
and following Riemann'... | https://mathoverflow.net/users/6842 | Functional equation for $\eta(s)$ following Riemann's $2^{nd}$ method | Ignoring technicalities of convergence, in Riemann's second proof, you start with the Poisson summation formula $\sum\_{n\in\mathbb Z} f(n / x) = x \sum\_{n\in\mathbb Z} \hat f (n x)$, take the Mellin transform of both sides, and use the self-dual function $f(x)=e^{-x^2}$.
To get the alternating sum you want, you cou... | 2 | https://mathoverflow.net/users/9849 | 366892 | 154,048 |
https://mathoverflow.net/questions/366881 | 4 | $\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper [Čadek, Crabb, and Vanžura - *Obstruction theory on 8-manifolds*](https://arxiv.org/abs/0710.0734), the authors discussed the "$\spin^c$-index" for a $\spin^c$ manifold $M$ (display (3.1) of the paper):
$$y\in K^0(M)\m... | https://mathoverflow.net/users/100553 | The $\operatorname{spin}^c$ index for manifolds | I suggest consulting section 26 of Hirzebruch's *Topological Methods in Algebraic Geometry* and the references therein. In particular, it contains the following statement:
>
> Theorem 26.1.1. Let $d$ be an element of $H^2(X, \mathbb{Z})$ whose reduction mod 2 is the Whitney class $w\_2(X)$, and $\eta$ a continuous ... | 4 | https://mathoverflow.net/users/21564 | 366899 | 154,052 |
https://mathoverflow.net/questions/364349 | 6 | In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that [the directed graph minor theorem](https://web.math.princeton.edu/%7Epds/papers/GM20/GM20.pdf) is true, using the definition
>
> A directed graph is a minor of another if the first can be obtained from a subgraph of t... | https://mathoverflow.net/users/45118 | Directed graph minor theorems | So directed graphs are not well-quasi-ordered by butterfly minors; see the intro of [[BPP]](https://arxiv.org/abs/1707.03563). Furthermore, there are reasons to think that many of the FPT results for graph minors may not hold in the directed setting (ie [[PW]](https://arxiv.org/abs/1507.02178)).
Yet, perhaps surprisi... | 5 | https://mathoverflow.net/users/161944 | 366903 | 154,053 |
https://mathoverflow.net/questions/366035 | 10 | Given are a positive integer $n$ and positive real numbers $a\_1,\dots,a\_n,b\_1,\dots,b\_n$. A subset $S\subseteq N=\{1,\dots,n\}$ is called *$a$-good* if $$\sum\_{i\in S}a\_i\geq \frac{1}{2}\left(\sum\_{i\in N\backslash S}a\_i-\min\_{i\in N\backslash S}a\_i\right),$$
and $b$-good if $$\sum\_{i\in S}b\_i\geq 2\left(\s... | https://mathoverflow.net/users/83212 | Disjoint sets with twice ratio | An affirmative answer to your question would follow from a famous conjecture about envy-free allocations. It is also known that if you replace the '2' in $b$-goodness with '$\frac 12$', then it is true.
A good paper to read (first) on the topic is [Almost Envy-Freeness with General Valuations](https://arxiv.org/abs/1... | 4 | https://mathoverflow.net/users/955 | 366904 | 154,054 |
https://mathoverflow.net/questions/348137 | 8 | Given a projective variety $X$ (over $\mathbb{C}$, say) with an affine paving $X=\sqcup\_i C\_i$, one can construct a poset $P\_X$ on the set of cells $\{C\_i\}$ by saying $C\_i \leq C\_j$ whenever $C\_i \subseteq \overline{C\_j}$. For example, doing this for Schubert cells in the flag variety gives the Bruhat order.
... | https://mathoverflow.net/users/33089 | A "polar dual" for projective varieties? | The answer to your question is no. If you have an irrdeucible projective variety $X$ of dimension $d$ with an affine paving, then for all $i < d/2$, the number of $i$-cells is less than or equal to the number of $(d-i)$-cells. This is the main idea of this paper by Bjorner and Ekedahl:
<https://arxiv.org/pdf/math/050... | 9 | https://mathoverflow.net/users/10273 | 366905 | 154,055 |
https://mathoverflow.net/questions/366884 | 6 | Consider the graph $G$ of order $n$ consisting of two disjoint cliques of even order $\frac{n}{2}=p+1$ (where $p$ is odd prime) joined by a bipartite graph (that is, deleting the edges of the two disjoint cliques from $G$ leaves a bipartite graph) of maximum degree $p$. Then, does the graph have list chromatic index $\... | https://mathoverflow.net/users/100231 | List chromatic index of a particular graph | Greedy coloring works here to show $2p$-choosability, I believe, and the hypothesis that $p$ is prime doesn't appear to be necessary. Write the cliques as $A = \{a\_1, \ldots, a\_{p+1}\}$ and $B = \{b\_1, \ldots, b\_{p+1}\}$, taking the notation so that $a\_i$ has exactly $i-1$ neighbors in $B$ and vice versa.
First ... | 5 | https://mathoverflow.net/users/6322 | 366906 | 154,056 |
https://mathoverflow.net/questions/365364 | 5 | For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following [majorization inequality](https://en.wikipedia.org/wiki/Majorization) holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \prec\_w \sigma^{\downarrow}(\mathbf{A})\sigma^{\downarrow}(\mathbf{B}),
$$
... | https://mathoverflow.net/users/75323 | Proving a majorization inequality for the singular value of the product of two matrices without using tensor product | We prove that
$$\sum\_{i=1}^k \sigma^\downarrow\_i(AB)
= \sup\_{U}|\mathrm{Tr}(UAB)|
\le \sup\_{U,V}|\mathrm{Tr}(UAV^\*B)|
=\sum\_{i=1}^k \sigma^\downarrow\_i(A)\sigma^\downarrow\_i(B),$$
where $U$ and $V$ run over all partial isometries (or contractions) of rank (at most) $k$.
The only nontrivial is $\le$ part of t... | 3 | https://mathoverflow.net/users/7591 | 366908 | 154,057 |
https://mathoverflow.net/questions/366396 | 6 | Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|\_p = \| |A|^p + |B|^p\|^{1/p}.
$$
Q: Is this a norm?
Consider the matrices $C = \left[\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right]$ and $D = \left[\begin{matrix}0 & ... | https://mathoverflow.net/users/76593 | Potential p-norm on tuples of operators | No, the expression
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|\_p = \| |A|^p + |B|^p\|^{1/p}.
$$
is not a norm for any $2<p<\infty$ (so it is a norm if and only if $p=2,\infty$).
I will justify that by proving that the triangle inequality for this expression would imply that the map $t\mapsto t^{p... | 7 | https://mathoverflow.net/users/10265 | 366924 | 154,060 |
https://mathoverflow.net/questions/366927 | 4 | For finite dimensional manifolds, there is a lot of theory about when the number of intersections (modulo $2$) of certain objects are preserved under homotopy. I'll give two quick examples:
Let $f:X \to Y$ be a smooth map from a compact manifold $X$ to a connected manifold $Y$ of the same dimension. Then if both $x,y... | https://mathoverflow.net/users/161947 | Intersection modulo 2 theory for infinite dimensional manifolds? | One can speak of transersality of intersections in an infinite dimensional context.
If one goes beyond Hilbert manifolds (e.g. Banach, Frechet) one needs to be a bit careful with the definition of transverse, because one needs to impose splitting conditions. For a submanifold one typically demands that the tangent spac... | 6 | https://mathoverflow.net/users/12156 | 366931 | 154,062 |
https://mathoverflow.net/questions/366912 | 3 | I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something along the lines of
$$-\Delta \phi(x)+ \phi(x) + 2 \int\_0^1 \frac{ \phi(tx)}{t} dt = f(x) \ge 0 \qquad B\_1$$ with $ \phi=0... | https://mathoverflow.net/users/66623 | Maximum principle for an elliptic like operator | *(For an actual answer, see the edit below.)*
Let $\phi$ be smooth near zero and non-negative. Suppose that the Taylor expansion of $\phi$ at zero is non-trivial, and let $P(x)$ be the leading term. Then $P(x)$ is a non-negative homogeneous polynomial of degree $2 k \geqslant 2$. Then $-\Delta P$ is a homogeneous pol... | 5 | https://mathoverflow.net/users/108637 | 366940 | 154,065 |
https://mathoverflow.net/questions/366935 | 12 | If $E$ is a supersingular elliptic curve over $\mathbb{F}\_{p^m}$ with $m\geq 2$ its endomorphism ring is a maximal order in a quaternion algebra ramified at $p$ and $\infty$ so there can't be a Weil cohomology with coefficients in $\mathbb{Q}\_p$ or $\mathbb{R}$.
For varieties over $\mathbb{F}\_p$ there is a $\mathb... | https://mathoverflow.net/users/nan | Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$? | The answer is no. If $A$ is a simple abelian variety over $\mathbb{F}\_p$, then $End(A)\otimes\mathbb{R}$ cannot act on a real vector space of dimension $2dim(A)$ if the center of the endomorphism algebra of $A$ has a real embedding. Let $E=End(A)\otimes\mathbb{Q}$. A theorem of Tate shows that $2dim(A)=[E:F]^{1/2}[F:\... | 12 | https://mathoverflow.net/users/161984 | 366951 | 154,067 |
https://mathoverflow.net/questions/366945 | 2 | Given a random variable $X$ that is uniformly distributed on $[-b,b]$ and $Y=g(X)$ with
$$g(x) = \begin{cases} 0, ~~~ x\in [-c,c] \\ x, ~~~ \text{else}\end{cases}$$
Now I want to compute the information dimension $d(X), d(Y)$ and the conditional information dimension $d(X|Y)$ and show that $d(X) = d(X|Y) + d(Y)$ in t... | https://mathoverflow.net/users/161976 | Conditional entropy - solve example | From the context, it appears that $b\in(0,\infty)$ and $c\in[0,\infty)$ (and, likely, $c\le b$). Anyway, let $c\_1:=\min(b,c)$. Note that
* With probability $1$, either $Y=0$ or $c\_1<|Y|\le b$;
* The conditional distribution of $X$ given $Y=0$ is the uniform distribution on the interval $[-c\_1,c\_1]$ and hence $d(X... | 2 | https://mathoverflow.net/users/36721 | 366956 | 154,069 |
https://mathoverflow.net/questions/366953 | 3 | Let $g: [0, 1] \to \mathbb R$ be a Lebesgue-measurable function (in the classical sense: the inverse images of Borel sets are Lebesgue-measurable). It is a classical fact in analysis that $f \circ g$ is Lebesgue-measurable as soon as $f$ is continuous, for instance, or Borel-measurable (the inverse images of Borel sets... | https://mathoverflow.net/users/161983 | Sharp assumption for preserving Lebesgue measurability by left composition | The answer is: $\cal F$ is the family of universally measurable functions.
---
For simplicity, let us consider functions on $[0,1]$ rather than on $\mathbb R$. Let $\cal B$ be the family of Borel sets, $\cal B^\star$ the family of universally measurable sets, and $\cal L$ the family of Lebesgue sets.
---
Cl... | 4 | https://mathoverflow.net/users/108637 | 366962 | 154,072 |
https://mathoverflow.net/questions/366966 | 19 | Two pointed, connected CW complexes with the same homotopy groups need not be homotopy equivalent ([Are there two non-homotopy equivalent spaces with equal homotopy groups?](https://mathoverflow.net/q/3540/39910)). Moreover, having the same homotopy and homology groups is also not enough ([Spaces with same homotopy and... | https://mathoverflow.net/users/39910 | Homotopy equivalent Postnikov sections but not homotopy equivalent | This is a pretty well-known phenomenon, linked with phantom maps.
One of the first existence results was Brayton Gray's paper
>
> *Spaces of the same $n$-type, for all $n$*, Topology
> **5** (1966) 241--243
>
>
>
Clarence Wilkerson classified the spaces of the same $n$-type for all $n$ in
>
> *Classifica... | 24 | https://mathoverflow.net/users/3634 | 366968 | 154,075 |
https://mathoverflow.net/questions/366920 | 3 | This question was posted at [MSE](https://math.stackexchange.com/questions/3764904/containment-of-bruhat-cells-on-flag-variety), but it did not receive any answer there.
Let $G$ be a connected semisimple algebraic group over $\mathbb C$, $X$ the flag variety of $G$, $B\_0$ a Borel subgroup, $\mathbb O$ a $B\_0$-orbit... | https://mathoverflow.net/users/99342 | Containment of Bruhat cells on flag variety | I'd say that the relevant fact here is as follows. For two Borels $B\_1$ and $B\_2$ with a common maximal torus $T$ let $x\_1$ be the unique $T$-fixed point in the open $B\_1$-orbit. Then the $B\_2$-orbit $B\_2x\_1$ lies in the open $B\_1$-orbit. For instance, you can see this by proving that
(a) for every $x\in B\_2x\... | 3 | https://mathoverflow.net/users/19864 | 367987 | 154,082 |
https://mathoverflow.net/questions/366893 | 5 | An $n\times n$ matrix $A$ with nonegative real entries $a\_{ij}$ is said to be *doubly stochastic* if $\sum\_{i=1}^na\_{ij} = 1$,
for all $j$, and $\sum\_{j=1}^na\_{ij}=1$, for all $i$.
Much is known [1] about the algebraic structure of the semigroup $\Omega \_n$ formed by all doubly stochastic $n\times n$ matrices. ... | https://mathoverflow.net/users/97532 | Doubly-stochastic partial-isometric matrices | The following is an attempt to validate the conclusion proposed by
@vidyarthi.
>
> Theorem: Every
> doubly-stochastic partial-isometric matrix is the product of a
> permutation matrix and a doubly-stochastic projection.
>
>
>
Proof:
Given a doubly-stochastic partial-isometric matrix $A$, one has that $A^tA$ an... | 0 | https://mathoverflow.net/users/97532 | 367993 | 154,085 |
https://mathoverflow.net/questions/367998 | 6 | I am following Schroeder's work on pursuit-evasion games on graphs (often called "cops and robbers"). In his 2001 publication ("The copnumber of a graph is bounded by $\lfloor 3/2 {\ \rm genus}(G)+3\rfloor$". In: Categorical perspectives (Kent, OH, 1998). Trends in Mathematics, pp. 243-263. Birkhäuser, Boston 2001) he ... | https://mathoverflow.net/users/163004 | pursuit-evasion based on Schroeder's upper bound for graphs of genus $g$ | What you conjecture has been conjectured (more or less explicitly) a few times before. In the paper by Bonato and Mohar that you reference, it is dubbed the Andreae-Schroeder conjecture.
I recently proved that it is true, i.e. the cop-number of toroidal graphs is at most 3, see [this ArXiv preprint](https://arxiv.org... | 7 | https://mathoverflow.net/users/97426 | 367999 | 154,086 |
https://mathoverflow.net/questions/368006 | 0 | Let $\{\omega\_i\}\_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C\_{\omega\_i}(\mathbb{R}^n,\mathbb{R}^d):=
\left\{
f \in (\mathbb{R}^n,\mathbb{R}^d):\,
\|f\|\_{\omega\_i,\infty}<\infty
\right\} \mbox{ where }
\|... | https://mathoverflow.net/users/36886 | Ultrabornological representation for the space of uniformly continuous functions? | Maybe, I miss something, but the answer seems to be easy: If $f:\mathbb R^n\to\mathbb R^d$ is continuous with $f(0)=0$ you can define the weight function $$\omega(r)=\sup\{\|f(x)\|: \|x\|\le r\}$$ which is obviously increasing with $\omega(0)=0$. Moreover, it is continuous at $r\ge 0$ because of the uniform continuity ... | 1 | https://mathoverflow.net/users/21051 | 368008 | 154,089 |
https://mathoverflow.net/questions/367989 | 8 | Consider the propositional modal language in one propositional letter, $p$.
Recall that a pointed Kripke frame is a Kripke frame $(W,R)$ with a designated world $w\_0\in W$, and a sentence is valid in a pointed Kripke frame iff it is true at $w\_0$ for every interpretation of the propositional letters as subsets of $... | https://mathoverflow.net/users/78564 | Interpretations of modal logic where $\Box$ means "valid" | $\def\R{\mathrel R}$No, this is not possible.
Recall that the *depth* of a point $x$ in a transitive frame $(W,R)$ is the maximal length $d$ of a strictly increasing chain starting at $x$, i.e., $x\_1,\dots,x\_d$ such that $x\_d=x$ and $x\_{i+1}\R x\_i$, $x\_i\not\R x\_{i+1}$.
There are formulas in one variable tha... | 10 | https://mathoverflow.net/users/12705 | 368010 | 154,090 |
https://mathoverflow.net/questions/368000 | 1 | Let $f,g$ are the functions of $S^{1}$. Are there $f, g$ such that
$$\int\_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int\_{S^{1}}f'g<0,$$
where $f'=\frac{\partial f}{\partial \theta}$?
| https://mathoverflow.net/users/147073 | Are there $f, g$ such that $\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$ where $f'=\frac{\partial f}{\partial \theta}$ | No, there are no such functions. Indeed, if $h$ is periodic and has mean zero, then $\|h\|\_2 \le \|h'\|\_2$. Take $f,g$ as above and write $g=g\_1+c$ with $g\_1$ having mean zero. Then
$$
\|f'\|\_2^2+\|g'\|\_2^2<2\int\_{S^1}f'g=2 \int\_{S^1} f'g\_1 \le 2\|f'\|\_2\|g\_1'\|\_2=2\|f'\|\_2\|g'\|\_2 \le \|f'\|\_2^2+\|g'\|\... | 5 | https://mathoverflow.net/users/150653 | 368013 | 154,091 |
https://mathoverflow.net/questions/368011 | 4 | Let $\mathbb{S}\_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}\_m$, and consider paths in the Bruhat order like this: $1\lessdot v\_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the covering relation in the (strong) Bruhat order. Let $N\_v$ be the number of such paths.
It is intuitively clear that $... | https://mathoverflow.net/users/66288 | Number of paths in the Bruhat order in the symmetric group | $\ell(v)!$ is of course even if $\ell(v)>1$, so the statement is really that $N\_v$ is even for $\ell(v)>1$. We find a fixed-point free involution on the set of such Bruhat paths. Suppose that $v\_2,v\_3,\ldots$ are fixed. By the diamond property of Bruhat order there are exactly two possibilities for $v\_1$. This give... | 8 | https://mathoverflow.net/users/1310 | 368014 | 154,092 |
https://mathoverflow.net/questions/368017 | 9 | In their 2009 paper (“On a graph property generalizing planarity
and flatness”. In: Combinatorica 29.3 (May 2009), pp. 337–361. issn: 1439-6912.
doi: 10.1007/s00493-009-2219-6.), van der Holst and Pendavingh defined a new minor monotone
graph invariant $\sigma(G)$ for a graph $G$: the minimal integer $k$ such that ever... | https://mathoverflow.net/users/163004 | Conjecture of van der Holst and Pendavingh related to bound for Colin de Verdière invariant | Kaluza and Tancer have actually proved $\mu(G)\leq\sigma(G)$ in 2019: See their proof in the preprint "Even maps, the Colin de Verdière number, and representations of
graphs" on arxiv. Here is the link <https://arxiv.org/pdf/1907.05055.pdf>
You are right, the invariant $\sigma(G)$ of Holst and Pendavingh does not see... | 9 | https://mathoverflow.net/users/156936 | 368020 | 154,094 |
https://mathoverflow.net/questions/368005 | 1 | We have a matrix valued function $A:\mathbb{R}\_+\to \mathbb{R}^{m\times m}$. It is known that $A(\lambda)$ is a positive definite matrix for all $\lambda\in\mathbb{R}\_+$ Denoting $\rho\_i(A(\lambda))$ the $i^{th}$ eigenvalue of the matrix $A(\lambda)$, we know that asymptotically each eigenvalue of the matrix $A(\lam... | https://mathoverflow.net/users/14414 | Asymptotic behavior of a matrix equation and its eigenvalues | Of course, without assumptions on the behavior of the eigenvectors, your desired conclusion will not hold. E.g., for $t:=\lambda$, let
$$A(t):=\left(
\begin{array}{cc}
2+\cos t & \sin t \\
\sin t & 2-\cos t \\
\end{array}
\right),\quad L:=\left(
\begin{array}{c}
1 \\
0 \\
\end{array}
\right).$$
The eigenvalues of $... | 2 | https://mathoverflow.net/users/36721 | 368026 | 154,097 |
https://mathoverflow.net/questions/368024 | 7 | Can you prove or disprove the following claim:
>
> Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime.
>
>
>
You can run this test [here](https://sagecell.sagemath.org/?z=eJzzszW35krLL9JItDXSyc_XSbLNyUwr0fDNT9FI1PHTjNPQ8N... | https://mathoverflow.net/users/88804 | Primality test for $N=2^a3^b+1$ | Yes. Obviously this $c$ and $N$ are coprime. We get $c^{(N-1)/2}+1=(c^{(N-1)/6}+1)(c^{(N-1)/3}-c^{(N-1)/6}+1)$ is divisible by $N$. Therefore $c^{N-1}-1$ is divisible by $N$, and $N-1$ is divisible by $k:={\rm {ord}}(c)$, where ${\rm ord}(x)$ denotes the multiplicative order of $x$ modulo $N$. But $(N-1)/2$ is not divi... | 15 | https://mathoverflow.net/users/4312 | 368036 | 154,102 |
https://mathoverflow.net/questions/368003 | 3 | Recall that: let $0<r<s<2$, then $\ell\_r$ uniformly contains a subspace isomorphic to $\ell\_s^m$, $m\ge 1$ (see [JS]).
I am wondering whether are any result for the case when $r>s>2$?
[Johnson, William B.; Schechtman, Gideon Embedding $l\_p^m$ into $l\_1^m$, Acta Math. 149 (1982), 71--85.][JS]
| https://mathoverflow.net/users/91769 | Banach embedding of finite dimensional spaces | For $2<r<\infty$, if $\ell\_s^n$ embeds uniformly into $\ell\_r$ for all $n$, then either $s=r$ or $s=2$. This is basically the localization to finite dimensions of the classical dichotomy theorem of Kadec and Pelczynski. The book of Albiac and Kalton is a good source for this.
| 3 | https://mathoverflow.net/users/2554 | 368039 | 154,103 |
https://mathoverflow.net/questions/368041 | 5 | If $G$ is a finite abelian group, then we have a decomposition
$$G\cong \prod\_{p} G(p)$$
where $G(p)$ is the $p$-Sylow subgroup of $G$. This product makes sense as for all but finitely many primes $p$, we have $G\_p=\{0\}$. This is proven by showing that the cardinality of $G$ and $\prod\_{p} G(p)$ agree. If we now as... | https://mathoverflow.net/users/152554 | Sylow subgroups of abelian profinite groups | This is Proposition 2.3.8 of [Ribes and Zaleskii - Profinite groups (second edition)](https://doi.org/10.1007/978-3-642-01642-4). (I originally gave references specifically for the finer structure of profinite Abelian groups, but assuming finite generation, in Section 4.3 of the same book.)
| 10 | https://mathoverflow.net/users/2383 | 368042 | 154,104 |
https://mathoverflow.net/questions/366983 | 1 | After reading [this question](https://math.stackexchange.com/questions/3546800/countable-dense-subset-of-c-c-infty-mathbb-rn), which asked for some examples of commonly used (proper) dense subsets of $C\_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" examples of countable subsets of ... | https://mathoverflow.net/users/36886 | Known dense subset of Schwartz-like space and $C_c^{\infty}$? | I don't know about "well known" or canonical answers to this question, but it is easy to construct an $X$ that works as follows.
Using the definition of Hermite polynomials given by
$$
H\_n(x)=(-1)^n e^{x^2}\left(\frac{d}{dx}\right)^n e^{-x^2}\ ,
$$
we define the one-dimensional Hermite functions
by
$$
h\_n(x)=\pi^{-... | 4 | https://mathoverflow.net/users/7410 | 368043 | 154,105 |
https://mathoverflow.net/questions/368015 | 4 | Let $\{\alpha\_i\}\_{i=1}^n$ be complex numbers such that $|\alpha\_i|<1$, and consider the following $n\times n$ structured matrix
$$
X=\left[\frac{1}{1-\bar\alpha\_i \alpha\_j}\right]\_{ij}.
$$
Such matrix arises in the solution of particular Stein matrix equations (e.g., see p. 11 of Bhatia, "Positive definite matri... | https://mathoverflow.net/users/62673 | Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$ | This matrix is also related to the [Nevanlinna-Pick Theorem](https://en.wikipedia.org/wiki/Nevanlinna%E2%80%93Pick_interpolation). Namely, if $z\_i, \lambda\_i \in \mathbb D, 1\leq i\leq n$ then $$\left[\begin{matrix} \frac{1- \overline{z\_j}z\_i}{1-\overline{\lambda\_j}\lambda\_i}\end{matrix}\right]\_{i,j=1}^n \geq 0$... | 6 | https://mathoverflow.net/users/76593 | 368044 | 154,106 |
https://mathoverflow.net/questions/368040 | 4 | I expected the following formula to hold:
$\int^{2n\pi}\_0\cos(\sin t+t/n)dt=0$,
for ${}^\forall n\in\mathbb{N},\ n\geq2$
But I can't prove it.
Could you please tell me.
| https://mathoverflow.net/users/152099 | An integral of composite function of triangle functions | You can rewrite the integral as
$$
\int\_0^{2\pi} \left(\sum\_{j=0}^{n-1}\cos\Big(\sin t+\tfrac tn+2\pi \tfrac jn\Big)\right)\,dt.
$$
But $\sum\_{j=0}^{n-1}\cos\big(a+2\pi \tfrac jn\big)=0$ for all $a$.
In particular, the equality holds if $\sin t$ is replaced by *any* $2\pi$-periodic function.
| 4 | https://mathoverflow.net/users/11054 | 368050 | 154,109 |
https://mathoverflow.net/questions/368053 | 11 | Is there a generalization of the Cauchy-Schwarz inequality along the following lines? Let $V$ be an inner product space (for simplicity of notation, let us work over the real numbers). Let $v\_1, \ldots, v\_n$ be in $V$. Let $G$ denote the Gram matrix of the $v\_i$, namely, $G$ consists of all possible $(v\_i, v\_j)$, ... | https://mathoverflow.net/users/81645 | Higher order generalization of Cauchy-Schwarz? | Yes, because the OP stated that the ground field is $\mathbb{R}$, one can simply take the octic polynomial
$$
Q(v\_1,v\_2,\ldots,v\_n) = \sum\_{1\le i < j\le n} \bigl((v\_i,v\_i)(v\_j,v\_j)-(v\_i,v\_j)^2\bigr)^2,
$$
which will do the trick.
| 16 | https://mathoverflow.net/users/13972 | 368058 | 154,112 |
https://mathoverflow.net/questions/366828 | 1 | In my question here I suggest a possibility for generalization of Furstenberg recurrence theorem needing some hypothesis for that generalization to be hold in the side of convergence of the below average (Multiple recurrence theorem) and existence of positive integer $n$ such that we have positive measure (Furstenberg ... | https://mathoverflow.net/users/51189 | What are the hypotheses we should add for the generalizations of Furstenberg recurrence theorem? | The extension you want was proved in the 1990s by Bergelson and Leibman. See [1] and also further developments in [2].
[1] Bergelson, Vitaly, and Alexander Leibman. "Polynomial extensions of van der Waerden’s and Szemerédi’s theorems." Journal of the American Mathematical Society 9, no. 3 (1996): 725-753.
[2] Berge... | 3 | https://mathoverflow.net/users/7691 | 368065 | 154,116 |
https://mathoverflow.net/questions/368070 | 3 | The Weisfeiler-Lehman test for graph isomorphism is based on iterative graph recoloring and works for almost all graphs, in the probabilistic sense. If we extend the domain to general hypergraphs, does there exist an analogous test for hypergraph isomorphism?
| https://mathoverflow.net/users/122916 | Weisfeiler-Lehman test for hypergraphs | You can represent a hypergraph by its vertex-edge incidence graph and apply W-L to that.
| 5 | https://mathoverflow.net/users/9025 | 368076 | 154,119 |
https://mathoverflow.net/questions/368067 | 3 | Let $V$ denote the von Neumann universe and $L$ Gödel's constructible universe. For any set $X$, let $P(X)$ denote the power set of $X$.
Assume that $0^\sharp$ exists (and ZFC).
What is the smallest ordinal $\alpha$ such that $L \cap P(L\_{\alpha})$ is uncountable? (If $V = L$, then $\alpha = \omega$, but if $0^\sh... | https://mathoverflow.net/users/17218 | smallest ordinal $\alpha$ such that $L \cap P(L_\alpha)$ is uncountable | *Really, this was answered in the comments; I'm putting this answer down to move this off the unanswered queue. I've made this CW and will delete it if one of the original commenters adds their own answer.*
---
We have in $L$, for each (infinite) $\alpha$, the following bijections:
* $f\_\alpha:\alpha\rightarro... | 4 | https://mathoverflow.net/users/8133 | 368094 | 154,125 |
https://mathoverflow.net/questions/368106 | 0 | Edit: According to the comment by LSpice we come back to the initial version of this question
Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the derivative $A=Df(x)$ satisfies the condition $a\_{ii}=1$ and $a\_{ij}=k(x),\; \forall i\neq j$ where $k(x)$ depends only o... | https://mathoverflow.net/users/36688 | Is it a sufficient condition for linearity? | I answer the version where you assume that the diagonal entries of $Df$ are constant (all $1$, in your problem), and the off-diagonal entries of each $Df(x)$ are all the same value $k(x)$.
I assume that, by linear, you mean affine linear. Then this is clear if $n = 1$. Otherwise, fix $i$, let $f\_i$ be the appropriat... | 3 | https://mathoverflow.net/users/2383 | 368109 | 154,130 |
https://mathoverflow.net/questions/368082 | 9 | Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$, $t>0$. I would like to ask for some literature on the consequences of the condition
\begin{equation}
e^{-t(A+B)}-e^{... | https://mathoverflow.net/users/102949 | Literature request: Schatten class difference of semigroups | The problem is discussed in a more general setting (operator ideals in Banach spaces) for the so-called analytic semigroups (parabolic problems) in
*Blunck, S.; Weis, L.*, [**Operator theoretic properties of differences of semigroups in terms of their generators**](http://dx.doi.org/10.1007/s00013-002-8292-3), Arch. ... | 8 | https://mathoverflow.net/users/12898 | 368126 | 154,132 |
https://mathoverflow.net/questions/368127 | 6 | Let $R$ be a commutative ring with identity. It is not always true that ideal $J$ and annihilator of ideal $ann(J)$ are co maximal (ex: integral domain)
>
> Is there a sufficient (necessary) condition( or ring) under which this happen $J$ and $ann(J)$ are comaximal?
>
>
>
| https://mathoverflow.net/users/163091 | When annihilator of ideal and ideal is co maximal | The necessary and sufficient condition is that $J$ be generated by an idempotent.
A. Assume $J=(e)$ is generated by an idempotent $e$ ($e^2=e$). Then the annihilator of $J$ contains $1-e$, so the sum of $J$ and its annihilator contains $e + (1-e)=1$. This shows that $J$ and its annihilator are comaximal.
B. Now sup... | 11 | https://mathoverflow.net/users/75735 | 368131 | 154,134 |
https://mathoverflow.net/questions/366871 | 10 | It is well-known that $\mathbf{PRA}$ plus $\epsilon\_0$-induction on bounded formulas cannot prove all $\mathbf{PA}$ theorems ([essentially because $I\Sigma\_1$ plus $\epsilon\_0$-induction on bounded formulas is finitely axiomatizable while the latter isn't](https://math.stackexchange.com/questions/3130538/how-do-we-k... | https://mathoverflow.net/users/151697 | Concrete examples of statements not provable in PRA + $\epsilon_0$-induction that are provable in PA? | One example is $I\Sigma\_2$ (or rather, the conjunction of its finite axiomatization).
Notice that the theory $I\Sigma\_1+\epsilon\_0$-*induction for bounded formulas* is axiomatized by $\Pi\_3$ sentences: in particular, the $\epsilon\_0$-induction schema can be written in prenex form as
$$\forall u\,\forall x\,\exis... | 8 | https://mathoverflow.net/users/12705 | 368134 | 154,135 |
https://mathoverflow.net/questions/366977 | 3 | Let $X$ be a proper geometrically integral $\mathbb{F}\_p$-scheme.
Assume that $X$ is the special fiber of a proper flat $\mathbb{Z}\_p$-scheme with a smooth generic fiber and that for each point $x\in X$ we have $\dim\_{\kappa (x)}(\Omega \_{X/\mathbb{F}\_p, x}\otimes\_{\mathcal{O}\_{X, x}} \kappa (x))\leq 1+\dim(X)... | https://mathoverflow.net/users/nan | Smoothable $\mathbb{F}_p$-variety embeds in a regular scheme | **Edit**. The original post is likely wrong: it would be better to work with a $\mathfrak{g}^r\_d$ that gives a closed immersion of $C$, but which is a singular point of $\mathcal{G}^r\_d(C)$. The following example is easier conceptually to understand.
Let $k$ be a field, i.e., $\mathbb{F}\_p$. Let $C$ be a $k$-curve... | 0 | https://mathoverflow.net/users/13265 | 368140 | 154,138 |
https://mathoverflow.net/questions/368047 | 7 | It is almost two decades since the now classical books by McConnell and Robinson's
* **[** *Noncommutative Noetherian rings*. With the cooperation of L. W. Small. Revised edition. Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001 **]**,
and Krause and Lenagan's
* **[** *Grow... | https://mathoverflow.net/users/160378 | Survey of recent developments of the Gelfand-Kirillov dimension | This list is certainly far from being complete, but it contains some important results obtained in the last 20 years.
The following thesis discusses some recent results obtained by Bell (see Section 5):
*Michelle Roshan Marie Ashburner (2008). A Survey of the Classification of Division Algebras over Fields. Master ... | 4 | https://mathoverflow.net/users/160051 | 368147 | 154,140 |
https://mathoverflow.net/questions/368150 | 12 | Suppose that we take constructive set theory and add the axiom $\forall x \forall y\ (x \in y) \lor \lnot (x \in y)$. Does this imply excluded middle, or are there still some formulas $\varphi$ for which $\varphi \lor \lnot \varphi$ isn't provable using this new axiom?
| https://mathoverflow.net/users/163109 | Does $\forall x \forall y\ (x \in y) \lor \lnot (x \in y)$ imply excluded middle? | It depends how much separation is available. If you can construct the set $\{ z \in \{ \emptyset \} \;|\; \varphi \}$ then you can show $\varphi \vee \neg \varphi$. So for theories with full separation, like IZF, you can derive excluded middle, whereas for CZF where you only have separation for bounded formulas, you ca... | 19 | https://mathoverflow.net/users/30790 | 368154 | 154,143 |
https://mathoverflow.net/questions/368130 | 4 | This question is related to my [previous question.](https://mathoverflow.net/q/368024/88804)
Can you prove or disprove the following claim:
>
> Let $N=2^mp^n+1$ , $m>0 , n>0$ and $p$ is an odd prime . If there exists an integer $a$ such that $$\displaystyle\sum\_{i=0}^{p-1} (-1)^i \cdot a^{i \cdot(N-1)/2p} \equiv... | https://mathoverflow.net/users/88804 | Primality test for $N=2^mp^n +1$ | I think this is true. Let $b = a^{\frac{N-1}{2p}} = a^{2^{m-1}p^{n-1}}$, and note that we
have $\frac{b^{p}+1}{b+1} \equiv 0$ (mod $N$).
Now $a$ and $N$ must be coprime, so that $b$ and $N$ are coprime. We have $b^{2p} \equiv 1$ (mod $N$).
Now $b^{p}-1$ and $b^{p} +1$ have gcd dividing $2$. However $\frac{b^{p}+1}{... | 8 | https://mathoverflow.net/users/14450 | 368161 | 154,146 |
https://mathoverflow.net/questions/368163 | 11 | I have seen it conjectured several times that Artin groups are torsion-free. This is a very basic question one could ask about these groups. Intuitively, to me it seems like it must be true however it seems impossible to prove. I am curious if anyone is actually working on this / what kind of methods people may have tr... | https://mathoverflow.net/users/149915 | Torsion & Artin groups | This is a consequence of the [$K(\pi,1)$ conjecture](https://link.springer.com/chapter/10.1007%2F978-81-322-1814-2_13), stating that there is an explicit $K(\pi,1)$ for Artin groups which is the complement of a complex hyperplane arrangement whose real locus are the hyperplanes of the reflections in the associated Coxe... | 10 | https://mathoverflow.net/users/1345 | 368170 | 154,148 |
https://mathoverflow.net/questions/368125 | 1 | Fix $p \in [1,\infty)$. Let $f\_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^\*$, suppose that the sequence of functions $$\{f\_{n}\psi\_m(f\_n)\}\_{n \in \mathbb N}$$ has a strongly convergent subsequence in $L^p([a,b])$. Here $\psi\_m$ is a smooth cut-of... | https://mathoverflow.net/users/157076 | $L^p$ compactness for a sequence of functions from compactness of cut-off | Let $$g^m\_n := f\_n \psi\_m(f\_n).$$ The assumptions mean that $(f\_n)\_n$ is a bounded sequence in $L^p(a,b)$ and that $(g\_n^m)\_n$ is relatively compact in $L^p(a,b)$ for each $m$. We use the [Frechet-Kolmogorov theorem](https://en.wikipedia.org/wiki/Fr%C3%A9chet%E2%80%93Kolmogorov_theorem) characterizing compactne... | 2 | https://mathoverflow.net/users/85906 | 368198 | 154,158 |
https://mathoverflow.net/questions/368093 | 6 | Let $G$ be a group and
$$0\rightarrow K\rightarrow M\rightarrow N\rightarrow 0$$
a short exact sequence of groups. Now these are abelian groups, if I want to show that $\text{Hom}(G,M)\rightarrow \text{Hom}(G,N)$ is surjective, I would show that $\text{Ext}^1(G,K)=0$. However, if I'm studying the same question for non-... | https://mathoverflow.net/users/152554 | Non-abelian Ext functor and non-abelian $H^2$ | EDITED, taking into account the [comments](https://mathoverflow.net/questions/368093/non-abelian-ext-functor#comment928981_368204) of Donu Arapura.
As JLA wrote, a homomorphism $f\colon G\to N$ gives an extension
\begin{equation}\label{e:E}
1\to K\to E\to G\to 1.\tag{E}
\end{equation}
This extension defines a homomor... | 6 | https://mathoverflow.net/users/4149 | 368204 | 154,161 |
https://mathoverflow.net/questions/368189 | 6 | I'm looking for a good quotation and comprehensive explaination of the theorem of Chow-Rashewski.
I'm writing my thesis on sub-Riemannian Geometry and a special control problem. Therefore I want to state the theorem of Chow–Rashewski in its sub-Riemannian version and prove it:
Let $M$ be a connected manifold and $\... | https://mathoverflow.net/users/163130 | Proof of Rashevskii-Chow theorem | As a reference, in addition to the classical ones cited above, I can recommend the following:
*Agrachev, Andrei; Barilari, Davide; Boscain, Ugo*, [**A comprehensive introduction to sub-Riemannian geometry.**](http://dx.doi.org/10.1017/9781108677325), [ZBL07073879](https://zbmath.org/?q=an:07073879).
The proof of th... | 5 | https://mathoverflow.net/users/13915 | 368215 | 154,164 |
https://mathoverflow.net/questions/368213 | 65 | The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary manipulations show that if you zoom in by a factor of $\lambda$, then you expect viscosity to scale as $\lambda^{\f... | https://mathoverflow.net/users/161947 | Should water at the scale of a cell feel more like tar? | There is a beautiful article (a write-up of a talk, actually), by E.M. Purcell, [Life at low Reynolds number](https://science.curie.fr/wp-content/uploads/2016/04/Purcell_life_at_low_reynolds_number_1977.pdf), that explains how bacteria swim.
Low Reynolds number is the technical way to phrase the statement in the OP t... | 102 | https://mathoverflow.net/users/11260 | 368216 | 154,165 |
https://mathoverflow.net/questions/368185 | 3 | Consider a fiber square
$\require{AMScd}$
\begin{CD}
X' @>i'>> Y'\\
@V g V V @VV f V\\
X @>>i> Y,
\end{CD}
where $i$ and $i'$ are regular immersions, and consider the *excess normal bundle* defined by the exact sequence
$$ 0 \to N\_{X'/Y'} \to N\_{X/Y} \to E \to 0, $$
which measures the failure of $f$ to be transver... | https://mathoverflow.net/users/16914 | Reference request: excess normal bundle and derived pullback | See Lemma 3.2 in the following paper: R. W. Thomason, *Les K-groupes d'un schéma éclaté et une formule d'intersection excédentaire*, Invent. Math. **112**, 195--215 (1993), [DOI](https://doi.org/10.1007/BF01232430).
| 4 | https://mathoverflow.net/users/nan | 368233 | 154,167 |
https://mathoverflow.net/questions/368225 | 1 | Is
$$U\_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=v\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$
identical with the set $V\_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?
| https://mathoverflow.net/users/37385 | Adjunction, infinity and hereditarily finite sets | I assume that the definition of $U\_\omega$ has a typo: adjunction operation usually means the following binary operation:
$$u;v:=u\cup\{v\}$$
If you just assume $z$ in the definition of $U\_\omega$ is closed under successor operator $u\mapsto u\cup\{u\}$, then $U\_\omega$ would be $\omega$.
---
It suffices to ... | 5 | https://mathoverflow.net/users/48041 | 368235 | 154,168 |
https://mathoverflow.net/questions/366739 | 2 | The [Fubini-Study metric](https://en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric#In_local_affine_coordinates) $g:=g\_{FS}$ is the unique $U(n+1)$-invariant
Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space
which by $U(n+1)$-invariance can be wlog definined on
tangent bu... | https://mathoverflow.net/users/108274 | Fubini-Study metric induced by submersion | Let me start from scratch. Note that everything below uses only the definition of complex projective space and the natural Hermitian inner product on $\mathbb{C}^{n+1}$. Also, the construction is coordinate-independent in the sense that everything below can be done with an abstract complex vector space with a Hermitian... | 2 | https://mathoverflow.net/users/613 | 368247 | 154,171 |
https://mathoverflow.net/questions/368241 | 5 | This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding.
For instance on the unit ball, nonnegative radially symmetric functions which are nondecreaing in the radial direction s... | https://mathoverflow.net/users/66623 | improved Sobolev embedding | The answer is indeed "no." To see this, instead of considering translations of radial bump functions, one needs to use bump functions whose level sets are deformed e.g. to ellipsoids. More precisely, denote $x \in \mathbb{R}^n$ by $(x\_1,\,x')$ and let $h$ be any decreasing function on $\mathbb{R}$. Then for
$$H(x) := ... | 3 | https://mathoverflow.net/users/16659 | 368258 | 154,176 |
https://mathoverflow.net/questions/368244 | 2 | In the developments I've seen of primitive recursive and computable functions, the functions always have codomain $\mathbb{N}$, but are allowed to have domain $\mathbb{N}^{m}$ for any natural number $m$. This seems odd to me---treating the domains and codomains as fundamentally different.
One solution would be to all... | https://mathoverflow.net/users/3199 | Computable functions with limited domains | Two papers of Julia Robinson seem to do the sort of thing you're looking for. Here are the MathSciNet data.
*Robinson, Julia*, [**General recursive functions**](http://dx.doi.org/10.2307/2031973), Proc. Amer. Math. Soc. 1, 703-718 (1950). [ZBL0041.15101](https://zbmath.org/?q=an:0041.15101) [MR0038912](https://mathsc... | 4 | https://mathoverflow.net/users/6794 | 368262 | 154,177 |
https://mathoverflow.net/questions/368250 | 6 | Le $\theta$ be irrational. One can define the noncommutative torus $A\_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an abstract defnition: however one can show that this algebra is simple and can be concretely represented as a $C^\*$-subal... | https://mathoverflow.net/users/24078 | Noncommutative torus as a von Neumann algebra | No. It's irreducible. The element $U$ generates the maximal abelian subalgebra $L^\infty({\mathbb T})$ and hence one computes the commutant:
$$\{U,V\}'=\{U\}'\cap\{V\}'=L^\infty({\mathbb T})\cap\{V\}'={\mathbb C}1.$$
By the way, the invariant subspace problem for the Bishop operator $f(x)\mapsto xf(x+\theta)$ is still ... | 8 | https://mathoverflow.net/users/7591 | 368263 | 154,178 |
https://mathoverflow.net/questions/368228 | 1 | Let $u$ an harmonique function on $\Omega=(a,b)\times (0,+\infty)$ and boundary conditions :
$\displaystyle u(a,y)=u(b,y)=0,\quad\forall y\geq 0$
$\displaystyle u(x,0)=0,\,\lim\_{y\to +\infty} u(x,y)=0 \quad \forall x\in (a,b)$
Can we conclude that $\quad u=0$ on $\Omega$ ?
My adempt
Let $$\Omega\_{R}=(a,b)\t... | https://mathoverflow.net/users/126827 | A question of uniqueness | If you want the positive answer you should state your last condition more carefully.
For example, add that $u$ is bounded, or that $u(x+iy)$ tends to $0$ as $y\to\infty$
UNIFORMLY with respect to $x$.
As you presently stated, the answer is negative. I sketch the construction of a counterexample.
1. There exists a n... | 3 | https://mathoverflow.net/users/25510 | 368268 | 154,180 |
https://mathoverflow.net/questions/368271 | 10 | Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete entropy
\begin{equation}
H(\lfloor X \rfloor) = - \sum\_{n\in\mathbb{Z}} \mathbb{P}( \lfloor X \rfloor = n ) \log( \mathbb{P}... | https://mathoverflow.net/users/39261 | Discrete entropy of the integer part of a random variable | Since $\lfloor X\rfloor$ has finite entropy if and only if $|\lfloor X\rfloor|$ has finite entropy, it suffices to consider random variables taking values in the natural numbers. Write $p\_n$ for $\mathbb P(X=n)$ (so that $\sum\_n p\_n=1$). We have $X\in L^q$ if and only if $\sum p\_n n^q<\infty$.
Suppose $X\in L^q$ ... | 7 | https://mathoverflow.net/users/11054 | 368274 | 154,182 |
https://mathoverflow.net/questions/368195 | 6 | I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow.
---
Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\neq 2$, and $Q$ a non-degenerate quadratic form on $V$. The spinor norm is a homomophism
$$sn: O(V,Q) \rightar... | https://mathoverflow.net/users/84165 | Explicit computation of spinor norm | Posted from the comments ([1](https://mathoverflow.net/questions/368195/explicit-computation-of-spinor-norm#comment929028_368195) [2](https://mathoverflow.net/questions/368195/explicit-computation-of-spinor-norm#comment929029_368195) [3](https://mathoverflow.net/questions/368195/explicit-computation-of-spinor-norm#comm... | 4 | https://mathoverflow.net/users/2383 | 368294 | 154,187 |
https://mathoverflow.net/questions/368234 | 2 | How do I show the [Clausen identity](https://en.wikipedia.org/wiki/Clausen%27s_formula)
$$
{}\_2F\_1\left(a, b; a+b+\frac{1}{2}; z\right)^2 = {}\_3F\_2\left(2a, 2b, a+b; a+b+\frac{1}{2}, 2a+2b, z\right)?
$$
I saw this on [MathWorld](https://mathworld.wolfram.com/ClausensProductIdentity.html) but am unsure how to progre... | https://mathoverflow.net/users/nan | Proving Clausen hypergeometric identity | Let
$$y\_1(z):={}\_2F\_1\left(a, b; a+b+\frac{1}{2}; z\right)^2$$
and
$$ y\_2(z):= {}\_3F\_2\left(2a, 2b, a+b; a+b+\frac{1}{2}, 2a+2b, z\right) $$
You can verify that $y\_1$ and $y\_2$ both satisfy the following differential equation:
$$ z^2(z-1)y''' -3z \left (a+b+\frac{1}{2} - (a+b+1)z \right) y'' + ((2(a^2+b^2+4ab)+... | 5 | https://mathoverflow.net/users/160051 | 368323 | 154,197 |
https://mathoverflow.net/questions/368317 | 6 | I am a bit confused with the relations among **Gelfand pairs**, **weakly symmetric pairs**, and **spherical pairs** defined in the book "**Harmonic analysis on commutative spaces**" written by professor Joseph A. Wolf.
For convenience, let me recall the definitions in this book, and just consider connected groups $G$... | https://mathoverflow.net/users/56989 | Gelfand pair, weakly symmetric pair, and spherical pair | I like this question! I originally thought it was much less subtle, and so posted some ill informed guesses in the comments.
Although the notions of "weakly symmetric" and "Gelfand pair" differ in general ([Lauret - Commutative spaces which are not weakly symmetric](https://doi.org/10.1112/S0024609397003925)), they c... | 4 | https://mathoverflow.net/users/2383 | 368329 | 154,199 |
https://mathoverflow.net/questions/366944 | 2 | Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by [Golomb (1991)](https://oeis.org/A005185/a005185_1.pdf) and [Tanny (1992)](https://www.sciencedirect.com/science/article/pii/0012365X92901456).
And in literature, there are very few studies that have different approa... | https://mathoverflow.net/users/161528 | Why are attempts to define chaos with discrete states so scarce? | The biggest problem I see with discrete-state chaos is that you would want it to capture some characteristic features of normal chaos (i.e., in the sense of chaos theory), namely:
* Chaos is sensitive to initial conditions, i.e., arbitrarily small changes to the initial condition blow up.
* Chaotic dynamics are aperi... | 1 | https://mathoverflow.net/users/38453 | 368335 | 154,202 |
https://mathoverflow.net/questions/368338 | -3 | $\DeclareMathOperator\CM{CM}$
I recently came across [Okhezin - Study of families of monotone continuous functions on Tychonoff spaces](https://link.springer.com/content/pdf/10.1007/s10958-007-0259-2.pdf) describing monotone functions on general topological spaces and I wonder the following more basic question. Let $\C... | https://mathoverflow.net/users/36886 | Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ | $\newcommand\R{\mathbb R}$$\newcommand\LBV{\mathrm{LBV}}$As was noted in comments, the set of all monotone functions (or your version of it,
$\mathrm{CM}^+(\R)$) is not a linear space.
An appropriate linear space here is $\LBV$, the space of all (say) right-continuous functions from $\R$ to $\R$ of locally bounded va... | 4 | https://mathoverflow.net/users/36721 | 368344 | 154,205 |
https://mathoverflow.net/questions/368325 | 15 | In Wall's paper [Unknotting tori in codimension one and spheres in codimension two](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/wallunknot.pdf), he states the following conjecture:
>
> Any $h$-cobordism of $S^3 \times S^1$ to itself is diffeomorphic to $S^3 \times S^1 \times I$.
>
>
>
What is the status of t... | https://mathoverflow.net/users/157284 | Status of a conjecture of C.T.C. Wall? | The conjecture has been solved. This is Theorem 16.1 in *C.T.C. Wall. (1999). Surgery on Compact Manifolds, Second edition. Mathematical Surveys and Monographs,
Vol. 69*. Here, two proofs of this theorem are given: the first one is an application of a more general method presented in the book, while the second one reli... | 9 | https://mathoverflow.net/users/160051 | 368371 | 154,209 |
https://mathoverflow.net/questions/368373 | 16 | I have a matrix
$$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$
As you can see, the matrix is always self-adjoint for any $a, b, c, d \in \mathbb C$.
But it has a funny property (that I found by playing with some numbers):
If $... | https://mathoverflow.net/users/119875 | Spectral symmetry of a certain structured matrix | For real $a,b,c$ and imaginary $d$ the matrix $A$ has *chiral symmetry*, meaning it anticommutes with a matrix $X$ that squares to the identity:
$$X=\left(
\begin{array}{cccc}
0 & 0 & 0 & -i \\
0 & 0 & i & 0 \\
0 & -i & 0 & 0 \\
i & 0 & 0 & 0 \\
\end{array}
\right),\;\;XA+AX=0,\;\;X^2=I.$$
Hence the spectrum of $A$... | 38 | https://mathoverflow.net/users/11260 | 368379 | 154,211 |
https://mathoverflow.net/questions/368206 | 5 | I am trying to understand the relationship between rigid monoidal categories and closed monoidal
categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ given by $X^\* \otimes -$.
Let $\mathcal{C}$ be a closed monoidal category (i.e., with internal homs), such that for ... | https://mathoverflow.net/users/153228 | Rigid monoidal and closed monoidal categories | Let $1$ be the unit of $C$. For every $X$, we define $X^\* = Hom(X,1)$. I will assume $C$ is strict closed symmetric monoidal. Further assuming the condition the OP specified, we can show $C$ is rigid.
Let's unpack the additional condition the OP wants to assume. For every $X$, the functors $F(-) = X\otimes -$ and $G... | 3 | https://mathoverflow.net/users/11540 | 368382 | 154,213 |
https://mathoverflow.net/questions/366682 | 1 | The setup of my question is the following: Suppose that we have a measurable space $(\Omega,\mathcal{F})$ and a filtration $\mathbf{F} = (\mathcal{F}\_t)\_{t \geq 0}$ on it. Let $\mathcal{P}(\mathbf{F})$ be the predictable $\sigma$-algebra, that is, the $\sigma$-algebra generated by all real-valued left-continuous proc... | https://mathoverflow.net/users/157982 | When does the predictable $\sigma$-algebra $\mathcal{P}$ coincide with the optional $\sigma$-algebra $\mathcal{O}$? | (Answering your comment)
Off the top of my head, I'd look in vol.2 of *Probabilités et Potentiel* (Dellacherie & Meyer) or in *Limit Theorems for Stochastic Processes* (Jacod & Shiryaev). Another convenient resource is the blog <https://almostsure.wordpress.com> of Geo. Lowther.
The key is that for a bounded rc mar... | 1 | https://mathoverflow.net/users/42851 | 368393 | 154,216 |
https://mathoverflow.net/questions/368385 | 1 | I consider the curve $c(t)=(x(t),y(t))$ in $\mathbb{R}^2$ such that
$\frac{d^2x(t)}{dt^2}=-(a\sin t+b)\frac{dy(t)}{dt}$
$\frac{d^2y(t)}{dt^2}=(a\sin t+b)\frac{dx(t)}{dt}$
$a,b\in\mathbb{R}$
Is the orbit curve of solution of above equations known?
| https://mathoverflow.net/users/152099 | Is this curve well known? | It is not clear what you mean by "known" but this system can be solved explicitly, in quadratures of elementary functions. Set $x'=u,\; y'=v,\; g(t)=a\sin t+b$. Then your system becomes
$$u'=-gv,\quad v'=gu.$$
Multiplying the first equation on $u$ and second on $v$ and adding, we obtain
$u'u+v'v=0,$ therefore $u^2+v^2=... | 8 | https://mathoverflow.net/users/25510 | 368400 | 154,219 |
https://mathoverflow.net/questions/365129 | 4 | In many books about conformal field theory, when we talk about a coset $\mathfrak{g}\_k/\mathfrak{h}\_{k'}$, we would talk about how the modules of $\mathfrak{g}\_k$ are decomposed into those of $\mathfrak{h}\_{k'}$ tensoring those of $\mathfrak{g}\_k/\mathfrak{h}\_{k'}$, for example, for the vacuum module
$$
\mathcal{... | https://mathoverflow.net/users/15884 | coset of affine Lie algebra | This is typically given by the commutant, or coset construction. You take the vector subspace of $\mathcal{R}\_\text{vac}[\mathfrak{g}\_k]$ spanned by vectors $v$ satisfying $Y(u,z)v \in \mathcal{R}\_\text{vac}[\mathfrak{g}\_k][[z]]$ for all $u \in \mathcal{R}\_\text{vac}[\mathfrak{h}\_{k'}]$. Equivalently, you take fi... | 5 | https://mathoverflow.net/users/121 | 368402 | 154,220 |
https://mathoverflow.net/questions/368389 | 0 | I was reading [this](https://link.springer.com/book/10.1007/BFb0084913) book by Coorneart, Delzant, and Papadopoulos. I am stuck with this proposition in Chapter 1 (Proposition 1.5)
>
> If $Y$ is a bounded $\delta$-hyperbolic subset of $X$, then $X$ is $\delta'$-hyperbolic with $\delta' = \delta + 6 \eta$ where $\e... | https://mathoverflow.net/users/163326 | Bounded subsets of $\delta$-hyperbolic metric spaces | Let $x,y,z,b \in X$ be given and assume that $\eta = \sup\_{x\in X} d(x,Y) < \infty$. Fix $x' , y' , z' , b' \in Y$ such that $|x-x'|, |y-y'|$, etc. are all $\leq \eta$. (For a 'properly done' proof, use $|x-x'|<\eta + \varepsilon$ for some $\varepsilon>0$ and take $\varepsilon \to 0$ at the end).
Using the triangle ... | 0 | https://mathoverflow.net/users/160011 | 368406 | 154,221 |
https://mathoverflow.net/questions/368410 | 4 | My vague intuition is that not only it is common for a simple arithmetic proposition $p$ to be independent of ZFC, but it is common for the statement "$p$ is independent of ZFC" to be independent, and so on. If we let $I(p)$ be the statement that $p$ is independent of ZFC, then this "iterative independence" property is... | https://mathoverflow.net/users/22930 | Arithmetic statement which is independent, and whose independence is independent, and so on? | If we fix things to avoid Will Sawin's observation, then the answer is **yes** under any reasonable interpretation I can think of.
For example, consider the following: let $J(p)$ be the sentence "If $\mathsf{ZFC}$ is consistent then $p$ is independent over $\mathsf{ZFC}$." We can indeed prove in $\mathsf{ZFC}$ many i... | 9 | https://mathoverflow.net/users/8133 | 368415 | 154,223 |
https://mathoverflow.net/questions/365447 | 11 | I read that one of the main goals of utilization simplicial methods is to prove that a space is a loop space. On the other hand where lies the main importance to recognize topological spaces as loop spaces? Surely, if a space is a loop space then its connected components obtain a magma structure via concatenation becau... | https://mathoverflow.net/users/108274 | Loop spaces motivation | There are several useful points in the comments, but I want to go beyond them and try to give a more comprehensive answer, so this question doesn't linger unanswered. Some great sources are May's [Geometry of Iterated Loop Spaces](https://www.math.uchicago.edu/%7Emay/BOOKS/gils.pdf) (GILS) and [A Concise Course in Alge... | 7 | https://mathoverflow.net/users/11540 | 368416 | 154,224 |
https://mathoverflow.net/questions/368407 | 12 | One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: <https://www.gap-system.org/Manuals/doc/ref/chap62.html> .
When $K$ is a finite field, this is a finite dimensional $K$-algebra and thus has finitely many elements. Let $A\_q$ denote the octonio... | https://mathoverflow.net/users/61949 | Unit group of octonions over finite fields | This is all worked out in the article ["A class of simple Moufang loops"](https://www.ams.org/journals/proc/1956-007-03/S0002-9939-1956-0079596-1/) by L.J. Paige. The short answer is that the loop of units has size $q^3(q^4-1)(q-1)$, and is not associative for any $q$. The example given by Paige (lemma 3.5) is given in... | 14 | https://mathoverflow.net/users/2384 | 368417 | 154,225 |
https://mathoverflow.net/questions/366560 | 3 | There are some properties that are easily studied for random d-regular graphs, but that are very hard to extend to random graphs with a given degree sequence (e.g. whether a graph is w.h.p. hamiltonian).
Does anyone have any other examples of papers/problems that successfully (and **not** trivially) extend properties... | https://mathoverflow.net/users/134361 | Reference request - random regular graphs vs random graphs w/ degree sequence | Simple random walk (and non-backtracking walk) on random regular graphs exhibit the cutoff phenomenon [1]. The extension to graphs with degree sequences came later; see [2] for nonbacktracking walks and [3] for simple random walk where backtrackings cause additional difficulties.
In another direction, component struc... | 1 | https://mathoverflow.net/users/7691 | 368419 | 154,227 |
https://mathoverflow.net/questions/368343 | 3 | I'm reading "UNSTABLE MOTIVIC HOMOTOPY THEORY" by Kirsten Wickelgren and Ben Williams (<https://arxiv.org/pdf/1902.08857.pdf>). There they have a version of Whitehead's Theorem, namely Prop 2.3, which says that $f:\mathcal{X}\rightarrow \mathcal{Y}$ is a weak equivalence if and only if the morphisms on homotopy sheaves... | https://mathoverflow.net/users/152554 | Whitehead Theorem in $\mathbb{A}^1$-homotopy theory | The condition you've stated implies that the homotopy sheaves are equivalent, and it is implied by the map being a weak equivalence, so they are equivalent. You're nullifying $\mathbb{A}^1$ in the $\infty$-category of Nisnevich sheaves of spaces on the category of (affine) schemes (smooth over the base). This ultimatel... | 1 | https://mathoverflow.net/users/1353 | 368421 | 154,228 |
https://mathoverflow.net/questions/368424 | 5 | The real number given by the absolutely convergent series
$$\displaystyle A = \sum\_{k=1}^\infty \frac{|\mu(k)|}{k \phi(k)}$$
is known as Landau's Totient Constant. It can be explicitly evaluated to be $\frac{\zeta(2)\zeta(3)}{\zeta(6)}$. Indeed, we see that $A$ can be expanded into an Euler product
$$\displaysty... | https://mathoverflow.net/users/10898 | A number similar to Landau's Totient constant | This constant is known as [Artin's Constant.](https://mathworld.wolfram.com/ArtinsConstant.html) The book *Finch S. R. Mathematical constants* (section 2.4) gives the following information.
>
> A rapidly convergent expression for Artin's constant is as follows [12-18]. Define Lucas' sequence as $$
> l\_{0}=2, \quad... | 8 | https://mathoverflow.net/users/5712 | 368434 | 154,229 |
https://mathoverflow.net/questions/354769 | 19 |
>
> Is [Thompson's group $F$](https://en.wikipedia.org/wiki/Thompson_groups) definably left-orderable? definably bi-orderable?
>
>
>
*Orderability definitions:* Recall that a group $G$ is left-orderable (resp. bi-orderable) if it admits a left-invariant (resp. bi-invariant) total order. If $S$ is a submonoid of ... | https://mathoverflow.net/users/14094 | Is Thompson's group definably orderable? | Yes, Thompson's group $F$ is definably bi-orderable.
Let $a$ be some element of $F$ with the support of $a$ equal to $(0,1/2)$. Let $b$ be some element of $F$ with the support of $b$ equal to $(1/2,1)$.
We will rely upon the following facts
1. If $g$ and $h$ are in $F$ then $[a^g,b^h] = 1\_F$ if and only if $(1/2... | 8 | https://mathoverflow.net/users/125391 | 368438 | 154,231 |
https://mathoverflow.net/questions/368443 | 1 | The computation below (part 1) shows that if two finite groups of order at most $100$ have the same (ordered) list of conjugacy class sizes, then they also have the same (ordered) list of (irreducible) character degrees.
**Question**: Is it true in general?
If so, is there an explicit way to determine the characte... | https://mathoverflow.net/users/34538 | Are the character degrees determined by the conjugacy class sizes? | SmallGroup(128,227) and SmallGroup(128,731)) are counterexamples.
```
gap> S:=List([227,731],n->SmallGroup(128,n));;
gap> for g in S do L:=List(ConjugacyClasses(g),c->Size(c));; Sort(L);; Print(L); od;
[ 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8 ]
[ 1, 1, 1, 1, 1,... | 10 | https://mathoverflow.net/users/22989 | 368449 | 154,232 |
https://mathoverflow.net/questions/365901 | 10 | What makes an infinite loop space an interesting object of study for homotopy theorists? The reason I ask this question is that I found a lot of results treating the question of whether a given space is an infinite loop space. So it seems that the property of a space being homotopy equivalent to an infinite loop space ... | https://mathoverflow.net/users/108274 | Why study infinite loop spaces? | I just wrote [an answer to the other thread](https://mathoverflow.net/a/368416/11540), and can expand it into an answer here, about infinite loop spaces instead of just loop spaces.
As mentioned there, spaces of the form $\Omega^\infty \Sigma^\infty X$ contain a great deal of information that helps when computing the... | 4 | https://mathoverflow.net/users/11540 | 368457 | 154,234 |
https://mathoverflow.net/questions/368404 | 4 | During my studies I faced a function $f:\mathbb{R} \to \mathbb{R}^+ $ with the property: for all $x \in \mathbb{R} $ and all $y$ in open interval $(x-\frac{1}{f(x)} ,x+\frac{1}{f(x)}) $ we have $f(x) \leq f(y)$.
At first I guessed maybe it is necessarily constant function but I could not show this. I tried to generate ... | https://mathoverflow.net/users/117299 | If all points of a real function with positive values would be local minimum, can one say it is constant function? | My answer is: Such a function must be constant. Suppose (for puoposes of contradiction) $f$ is a nonconstant function $f : \mathbb R \to (0,+\infty)$ such that
$$
\forall x\in\left(a-\frac{1}{f(a)},a+\frac{1}{f(a)}\right),\quad
f(x) \ge f(a) .
$$
For $m > 0$, define $U\_m := \{x : f(x)>m\}$.
**Lemma 1:** For all $m... | 6 | https://mathoverflow.net/users/454 | 368458 | 154,235 |
https://mathoverflow.net/questions/368461 | 1 | Let $G=(V,E)$ be a finite simple graph. We say a map $p:V\to [n]:=\{1,\ldots,n\}$ is a *pseudo-coloring* if for all $a\neq b\in[n]$ there is $v\in\psi^{-1}(\{a\})$ and $w\in\psi^{-1}(\{b\})$ such that $\{v,w\}\in E$. We denote the maximal number $m$ such that there is a pseudo-coloring $p:V\to [m]$ by $\psi(V)$.
An e... | https://mathoverflow.net/users/8628 | Hedetniemi for pseudo-chromatic number $\psi(G)$ | It is not true. Let $G$ and $H$ be graphs, and let $p\_{max}$ be maximal pseudo-coloring of the graph $H$. Show that the map $p((x,y))=p\_{max}(y)$ is pseudo-coloring of graph $G\times H$. Fix some $\{u,v\}\in E(G)$. For arbitrary distinct colors $a,b$ there exist $\{k,l\}\in E(H)$ such that $p\_{max}(k)=a$ and $p\_{ma... | 4 | https://mathoverflow.net/users/144883 | 368477 | 154,244 |
https://mathoverflow.net/questions/368133 | 0 | Suppose
$$X\_{ij} = \mu\_j + \varepsilon\_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N\_j$$
ANOVA can allow us to test whether $\mu\_1 = \cdots = \mu\_J$.
In traditional ANOVA, however, the number of groups $J$ is fixed. But I want to know the proof technique in the case that $J$ goes to infinity.
Does anyo... | https://mathoverflow.net/users/153595 | Asymptotic properties of ANOVA when the number of groups goes to infinity | The following article is what you are looking for:
>
> Akritas, M., Arnold, S. (2000). Asymptotics for Analysis of Variance When the Number of Levels is Large, Journal of the American Statistical Association, 95:449, 212-226.
>
>
>
| 1 | https://mathoverflow.net/users/99279 | 368481 | 154,245 |
https://mathoverflow.net/questions/368286 | 1 | Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found [Johanis - Smooth partitions of unity on Banach spaces](https://www.sciencedirect.com/science/article/abs/pii/S0022123617301398), which provides several classes of Banach spaces with thi... | https://mathoverflow.net/users/56931 | Does the Skorokhod space with the uniform topology admit a smooth partition of unity? | The Skorohod space $D$ has $C([0,1])$ as a Banach subspace that does not have smooth bump functions by results of Bonic and Frampton from 1965. If $D$ had smooth partitions of unity, it would posses smooth bump functions, hence also $C([0,1])$ which is false. So the answer is **no**.
See Section 14 (pp. 152−158, in p... | 2 | https://mathoverflow.net/users/12643 | 368499 | 154,252 |
https://mathoverflow.net/questions/261467 | 18 | The original paper on Steenrod squares, [Steenrod's "Products of cocycles and extensions of mappings", 1947](https://www.jstor.org/stable/1969172), uses an explicit combinatorial formula for the squares in terms of simplicial cochains: given a simplicial cochain $\alpha$ on some simplicial set, Steenrod defines a cocha... | https://mathoverflow.net/users/78 | A cochain-level Adem relation? | Explicit homotopies for a cochain-level Adem relation were first worked out in:
[Greg Brumfiel, Anibal M. Medina-Mardones, John Morgan. A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra. 2020](https://arxiv.org/abs/2006.09354)
The odd-prime case is still open.
| 5 | https://mathoverflow.net/users/78 | 368502 | 154,253 |
https://mathoverflow.net/questions/368439 | 5 | What is the consistency strength of the following situation?
1. $j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$.
2. $\mathbb P$ is a forcing that collapses all ordinals between $\kappa$ and $j(\kappa)$.
3. $j$ can be lifted through $\mathbb P$.
One can deduce ... | https://mathoverflow.net/users/11145 | Consistency strength of lifting through a lot of collapsing | $\text{AD}^{L(\mathbb R)}$ suffices. The situation actually holds in the model $H = \text{HOD}^{L(\mathbb R)}$. We will have $\kappa = \omega\_1$ and $j : H\to \text{Ult}(H,U)$ equal to the ultrapower of $H$ by the club measure $U$ over $\omega\_1$ as computed in $L(\mathbb R)$ (using all functions in $L(\mathbb R)$).
... | 5 | https://mathoverflow.net/users/102684 | 368506 | 154,256 |
https://mathoverflow.net/questions/368489 | 10 | It is well-known that the Mertens function $M(n)=\sum\_{k=1}^n\mu(k)$ changes sign infinitely many times when $n\rightarrow +\infty$. Let $f(n)=\sum\_{k=1}^n\frac{\mu(k)}{k}$, then $\lim\limits\_{n\rightarrow +\infty}f(n)= 0$.
**Question:** Does the function $f(n)=\sum\_{k=1}^n\frac{\mu(k)}{k}$ also change sign infin... | https://mathoverflow.net/users/160959 | Sign of the function $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$ | Yes it does. To see this, note that by partial summation,
$$\frac{1}{\zeta(s + 1)} = s\int\_{1}^{\infty}\sum\_{n \leq x} \frac{\mu(n)}{n} x^{-s} \, \frac{dx}{x}$$
for all $\Re(s) > 0$. Now let $\Theta$ denote the supremum of the real part of the zeroes of $\zeta(s)$, and suppose in order to obtain a contradiction that ... | 18 | https://mathoverflow.net/users/3803 | 368511 | 154,257 |
https://mathoverflow.net/questions/368513 | 31 | I'm refereeing a Banach spaces paper and it looks pretty good. I'm about ready to recommend it for publication.
However, its main result depends crucially on some other results that are in preprints on the arxiv. Is it the referee's responsibility to verify those results too? Or should I just alert the editor that we... | https://mathoverflow.net/users/73784 | Is it the referee's responsibility to verify results from arXiv preprints used in the refereed paper? | I'm going to use the word "I" in this answer since there is no universally agreed-upon standard for what a referee should do.
I feel that the referee's only job is to make an informed recommendation to an editor as to whether or not a paper should be accepted. The extent to which that includes verifying that a paper ... | 54 | https://mathoverflow.net/users/317 | 368516 | 154,260 |
https://mathoverflow.net/questions/368515 | 10 | I am currently reading Kervaire-Milnor's paper "Groups of Homotopy Spheres I", *Annals of Mathematics*, and I am trying to prove (or disprove) the following result. The more elementary the proof, the better.
>
> If two smooth manifolds are homeomorphic, then their stable tangent
> bundles (i.e. the Whitney sum of t... | https://mathoverflow.net/users/152049 | If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic | The result you are hoping for is in fact false.
In section 9 of *[Microbundles: Part I](http://dx.doi.org/10.1016/0040-9383(64)90005-9)*, Milnor constructs an open set $U \subset \mathbb{R}^m$. With its standard smooth structure, the (stable) tangent bundle of $U\times\mathbb{R}^k \subset \mathbb{R}^{m+k}$ is trivial... | 31 | https://mathoverflow.net/users/21564 | 368517 | 154,261 |
https://mathoverflow.net/questions/368508 | 12 | Fix a cardinal $\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}$. Consider the equivalence relation on $\beta\lambda$ given by $\cU\sim \cV$ when for all first-order structures $M$ we have $M^{\cU}\cong M^{\cV}$.
By considering the structure $(\lambda,A)\_{A\subseteq \lambda}$, one can show that the... | https://mathoverflow.net/users/54415 | When do two ultrafilters yield isomorphic ultrapowers? | If one allows an arbitrary signature, the answer to this question is fairly well-known. Consider the structure $(\lambda,A)\_{A\subseteq \lambda}$. Let $(M,R\_A)\_{A\subseteq \lambda}$ be the common ultrapower by $\mathcal U$ and $\mathcal V$. Note that $A\in \mathcal U$ if and only if $[\text{id}]\_U \in R\_A$ by the ... | 9 | https://mathoverflow.net/users/102684 | 368522 | 154,263 |
https://mathoverflow.net/questions/368483 | 17 | The Riemann curvature tensor ${R^a}\_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.
**Question:** Is there a similarly direct geometric interpretation of the [Weyl conformal tensor](https://en.wikipedia.org/wiki/Weyl_tensor) ${C^a}\_{bcd}$?
**Background:** My... | https://mathoverflow.net/users/2362 | Geometric interpretation of the Weyl tensor? | There is such an interpretation, with a few caveats. Essentially, there is a canonical connection on a certain vector bundle for which the "principal part" of the curvature is the Weyl tensor in dimensions $n\geq4$, and the Cotton tensor when $n=3$. I will describe this from the point of view of the tractor calculus, b... | 17 | https://mathoverflow.net/users/121820 | 368525 | 154,265 |
https://mathoverflow.net/questions/368531 | 15 | I am a PhD student from India working on representations of quantum groups. I want to organize a workshop on Hopf Algebra and Quantum groups but there are only 2 or 3 specialists in India currently working on it. So I googled some professors outside India who are currently working on this field. Now the problem is how ... | https://mathoverflow.net/users/163401 | Tips to organize a successful math workshop | The [Lorentz Center](https://www.lorentzcenter.nl/uploadedfiles/4-lorentzcenterapplicationinstructions.pdf) has some advice that you might find useful, I have organized several workshops there and followed a route similar to the one you describe.
Tentative answers to your specific questions:
1. Since you can only i... | 17 | https://mathoverflow.net/users/11260 | 368539 | 154,269 |
https://mathoverflow.net/questions/368387 | 22 | It's known that we can have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms.
My question is whether we can have global failures of GCH where there is a weakly inaccessible cardinal between $\lambda$ and $2^\lambda$ for each $\lambda$. Simi... | https://mathoverflow.net/users/17968 | How badly can the GCH fail globally? | In the Foreman-Woodin model [The generalized continuum hypothesis can fail everywhere.](https://www.jstor.org/stable/2944324?origin=crossref&seq=1) for each infinite cardinal $\kappa, 2^\kappa$ is weakly inaccessible.
This answers your last question. The answer to the first two questions can be yes as well. In the ca... | 14 | https://mathoverflow.net/users/11115 | 368546 | 154,271 |
https://mathoverflow.net/questions/368552 | 6 | Let $G$ be a compact connected Lie group and $T\subset G$ a maximal torus. Let $V$ be a representation of $G$ and $U=\{v\in V: tv=v\textrm{ for all }t\in T\}$. For any $g\in N(T)$ we have for all $t\in T$ and $v\in U$ that $g^{-1}tgv=v \Rightarrow t(gv)=gv$. This shows that for all $v\in U$ we have $gv\in U$ as well. F... | https://mathoverflow.net/users/36563 | Fixed space of maximal torus and Weyl group | This paper of Humphreys addresses your second question (the first is answered in the comments - the $W$-module structure is independent of the choice of torus): [https://people.math.umass.edu/~jeh/pub/zero.pdf](https://people.math.umass.edu/%7Ejeh/pub/zero.pdf)
Here is a quote from the paper (section 1.4):
>
> In... | 7 | https://mathoverflow.net/users/7762 | 368557 | 154,274 |
https://mathoverflow.net/questions/368540 | 4 | A matrix valued function is of the form $\psi:\mathbb{R}\_+\to\mathbb{R}^{n\times n}$ and it is known that $\psi(\lambda)$ is always a **positive definite** matrix. The asymptotic exapnsion of $\psi(\lambda)$ is given as $$\psi(\lambda) = A + \frac{B\_1}{\lambda}+ \frac{B\_2}{\lambda^2}+ \frac{B\_3}{\lambda^3}+...$$ wh... | https://mathoverflow.net/users/14414 | Asymptotic expansion of the inverse of a matrix valued function | If we disregard the positivity constraint, this is not true in general, the leading order term can be of order $n-1$ rather than of order 1.
The problem is treated in [Laurent expansion of the inverse of perturbed, singular matrices](https://e-archivo.uc3m.es/bitstream/handle/10016/23560/laurent_JCP_2015_ps.pdf). The... | 1 | https://mathoverflow.net/users/11260 | 368570 | 154,279 |
https://mathoverflow.net/questions/363602 | 5 | I am recently studying ergodic actions of Lie groups acting on Riemannian symmetric spaces. Since I am also interested in operator algebras, it makes me wonder if there are some very natural noncommutative analogues of Riemannian symmetric spaces and measure preserving ergodic group actions by classical Lie groups?
| https://mathoverflow.net/users/136860 | Noncommutative symmetric spaces | You might be interested in the work of Lezter and others on quantum symmetric spaces. See for example this paper
<https://arxiv.org/pdf/math/0406193.pdf>
These objects seem to be closely connected to the representation theory of quantum groups. People have also considered $C^\*$-completions, but I'm not sure about ... | 1 | https://mathoverflow.net/users/153228 | 368575 | 154,281 |
https://mathoverflow.net/questions/368478 | 5 | I am analyzing Hadwiger's original article (Hadwiger, Hugo (1943), "Uber eine Klassifikation der Streckenkomplexe", Vierteljschr. Naturforsch. Ges. Zurich, 88: 133–143) for my work related to the Hadwiger Conjecture in graph theory.
This article is in German, and Hadwiger's terminology from 1943 is very different fro... | https://mathoverflow.net/users/163385 | Hadwiger number of a graph: Question about the original article from 1943 | Let me give it a try. As a disclaimer, English is not my mother tongue, so my translation might have linguistic flaws.
First of all, I would say the sentence is hard to translate and it is a bit informal, i.e. it is not a rigorous mathematical statement. In my view, this sentence gives an informal motivation why it i... | 4 | https://mathoverflow.net/users/156936 | 368587 | 154,285 |
https://mathoverflow.net/questions/368556 | 5 | I wrote a program to calculate the minimal primitive root modulo $p^a$ where $p > 2$ is a prime, by enumerating $g$ from $2$ and checking whether it's a primitive root, but I forgot to check $\gcd(g, p) = 1$. However, it still worked in all the test cases.
So is it true that the smallest primitive root modulo $p^a$ i... | https://mathoverflow.net/users/158466 | The smallest primitive root modulo powers of prime | This is known, see
<https://arxiv.org/abs/1908.11497>
where it is show for squares of primes. Higher powers then follow from other elementary arguments
| 5 | https://mathoverflow.net/users/nan | 368588 | 154,286 |
https://mathoverflow.net/questions/368586 | 6 | All I can remember is that it was very **high-level / abstact and kind of philosophical**, explaining **(the discovery or interdependence of) small world networks**. I assume that it was **+50 years old** and 'might' be an iconic paper, but maybe not - surely it was by far not as popular as the, already mentioned, pape... | https://mathoverflow.net/users/163441 | Searching for an early, highly theoretical, even philosophical, math paper on models or small-world networks | Stanley Milgram, [The Small World Problem](http://snap.stanford.edu/class/cs224w-readings/milgram67smallworld.pdf), Psychology Today **2**, 60 (1967)
seems to fit the bill: +50 years old, "kind of philosophical", and yes, iconic -- cited more than 9,000 times. There are a few related papers in that time frame, listed... | 10 | https://mathoverflow.net/users/11260 | 368594 | 154,289 |
https://mathoverflow.net/questions/368593 | 1 | If $(X,T)$ is a minimal system uniquely ergodic with $\mu$, is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$?
| https://mathoverflow.net/users/163445 | Balls in minimal systems | Let $T$ be an irrational rotation of the circle. We modify the metric on the circle as follows, letting $d(\cdot,\cdot)$ be the standard metric on the circle; and for $C$ a non-empty closed subset of the reals, let $D(x,C)$ denote the distance from $x$ to $C$. Let $C$ be a Cantor set of positive measure contained in $[... | 2 | https://mathoverflow.net/users/11054 | 368597 | 154,290 |
https://mathoverflow.net/questions/368520 | 0 | I have the following question:
Does there exist a non-negative function $g$ on $(0,1)$ such that
$$1\leq F(x):=\dfrac{\displaystyle\sum\_{k=0}^{\infty}a\_{k}\,(k+1)^{2}\,x^{k}}{\displaystyle\sum\_{k=0}^{\infty}(k+1)\,x^{k}}\leq 2,\;\forall\;x\in (0,1),$$ and $$\displaystyle\lim\_{x\rightarrow 1^{-}}F(x)\;\text{ does ... | https://mathoverflow.net/users/120300 | Construct a function with certain growth property | Such a function $g$ exists.
Indeed, we have
\begin{equation\*}
\sum\_{k=0}^\infty(k+1)\,x^k=\frac1{(1-x)^2}
\end{equation\*}
and
\begin{align\*}
\sum\_{k=0}^\infty a\_k(k+1)^2\,x^k&=
\int\_0^1 du\,g(u)\,\sum\_{k=0}^\infty x^ku^{2k+1} \\
&= \int\_0^1 du\,g(u)u\,\frac{1+xu^2}{(1-xu^2)^3} \\
&=\frac12\int\_0^1 dv... | 1 | https://mathoverflow.net/users/36721 | 368599 | 154,292 |
https://mathoverflow.net/questions/368596 | 12 | I think I have a pretty good *intuitive* understanding of most types of [fibrations of quasicategories](https://ncatlab.org/nlab/show/fibrations+of+quasi-categories):
* a (trivial) Kan fibration is a bundle of (contractible) spaces with equivalent fibers,
* a left/right fibration is a bundle of spaces with covariant/... | https://mathoverflow.net/users/56392 | Intuition for categorical fibrations? | Categorical fibrations are not particularly meaningful in their own right. Luckily, there is a characterization in the most interesting case, of categorical fibrations $p:Q\to R$ between quasicategories. Namely such a map $p$ is nothing more than an inner fibration and an *isofibration*, that is, it is weakly orthogona... | 10 | https://mathoverflow.net/users/43000 | 368602 | 154,293 |
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