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https://mathoverflow.net/questions/434962 | 3 | The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& & \ddots & \ddots & \ddots & \\
& & & 1& 0 & 1\\
& & & & 1 &0
\end{pmatrix}.$$
Is it possible to give analytic expressions for the eigenvalues and eigenvectors of $A$?
Wang et. al [[1](https://ww... | https://mathoverflow.net/users/495150 | Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix | Defining the $2 \times 2$ transfer matrix
\begin{align}\tag{1}
Q = \begin{pmatrix} -\lambda & 1 \\ -1 & 0 \end{pmatrix},
\end{align}
the characteristic polynomial (CP) of the $M \times M$ matrix $A\_M$ is given by
\begin{align}
P\_M(\lambda) &= \det(A\_M -\lambda \, I)\tag{2a}\\
&=\langle 1{-}\lambda, 2| \, Q^{M-1} \,|... | 3 | https://mathoverflow.net/users/90413 | 435026 | 175,885 |
https://mathoverflow.net/questions/434798 | 2 | Let $M^n$ be a complete simply connected Riemannian manifold with $\operatorname{sec}\_M \leq 0$ (i.e. a Hadamard manifold) and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}\_M \geq -a^2$. Do you know whether this implies an upper bound on the volume growth of area minimizing submanifolds i.e... | https://mathoverflow.net/users/313861 | Upper bound on volume growth of area minimizers | Consider the complex curve $w = z^k$ in $\mathbb{C}^2\simeq\mathbb{R}^4$, which is calibrated and therefore area-minimizing. The area of the part of this curve that lies inside the polydisk $\max\{|z|,|w|\}\le 1$ is $(k{+}1)\pi$. (The reason is that, because the standard Kähler form $\Omega = \tfrac{i}{2}(\mathrm{d}z\w... | 4 | https://mathoverflow.net/users/13972 | 435040 | 175,890 |
https://mathoverflow.net/questions/435025 | 4 | Let $f^{\*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive function on $\mathbb{R}\_{+}$. Is it true that $$\left(\int\_{0}^{\infty}(f^{\*}(s))^{p'}(g(s))^{-p'}ds\right)^{p/p'}\leq c\int\_... | https://mathoverflow.net/users/163368 | Inequality with decreasing rearrangement function | No (if $c$ cannot depend on $f^\*$ or $g$). Indeed, let $h:=f^\*/g$. Then $h$ can be any positive function and the inequality in question can be rewritten as
$$lhs:=\Big(\int\_0^\infty h(s)^{p'}ds\Big)^{p/p'}
\le rhs:=c\int\_0^\infty h(s)^p s^{p/p'-1}\,ds. \tag{1}\label{1}$$
Note that $p'>p>0$.
To obtain a contradict... | 6 | https://mathoverflow.net/users/36721 | 435042 | 175,891 |
https://mathoverflow.net/questions/435050 | 10 | $\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,
$$\pi\_1^{\orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$
$$\pi\_1^{\orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$
Want to know t... | https://mathoverflow.net/users/126243 | Centre of orbifold fundamental group of torus (Klein bottle) with one cone point | These groups have trivial centers. As one proof, they are both fuchsian and so embed in $\mathrm{PSL}(2, \mathbb{R})$ (well, the orientation preserving subgroups do). However, elements of $\mathrm{PSL}$ that commute must have common fixed points on the "circle at infinity" of the hyperbolic plane.
| 9 | https://mathoverflow.net/users/1650 | 435051 | 175,893 |
https://mathoverflow.net/questions/435070 | 3 | In IZF, we can easily prove there is a minimal cauchy complete field extending the rationals: the dedekind reals are cauchy complete, so just intersect all of its cauchy complete subfields.
CZF can still prove that there are dedekind reals, and that they are cauchy complete, but intersecting all of its cauchy complet... | https://mathoverflow.net/users/65915 | Does CZF prove there is a minimal cauchy completion of the rationals? | You can prove this from the regular extension axiom $\mathbf{REA}$ using the general theory of inductive definitions. See e.g. Theorem 5.11 in Aczel & Rathjen, [*Notes on Constructive Set Theory*](https://ncatlab.org/nlab/files/AczelRathjenCST.pdf). I suspect that $\mathbf{REA}$ is strictly necessary, although I don't ... | 3 | https://mathoverflow.net/users/30790 | 435072 | 175,903 |
https://mathoverflow.net/questions/435024 | 4 | Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^\*$-algebra together with a right action
$$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $\*$-homomorphism satisfying $(\beta \otimes \iota)\beta = (\iota \otimes \Delta)\beta$ and the Podles density condition. A right $B$-Hilbert module $X$ is ... | https://mathoverflow.net/users/470427 | Reference request: decomposability of $\mathbb{G}$-Hilbert modules | Let's assume that $G$ is a reduced compact quantum group, that is, the Haar state on $C(G)$ is faithful.
(1): A direct reference is [1, Lemma 4.2]. You can also get this from a careful study of [2, Théorème 3.2].
(2): This (for countably generated modules) follows from [2, Théorème 3.2] and [3, Proposition 4.6].
... | 3 | https://mathoverflow.net/users/9942 | 435074 | 175,904 |
https://mathoverflow.net/questions/435028 | 2 | ##### How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser and Schoenfeld [RS62, Corollary 3] prove:
>
> *For $λ≥20.5$, $\frac{3}{5}λ/\lnλ < π(2λ) - π(λ) < \frac{7}{5... | https://mathoverflow.net/users/16178 | Explicit bounds on number of primes of given size | Dusart proved in his [thesis](https://www.unilim.fr/laco/theses/1998/T1998_01.pdf) that
$$\frac{x}{\log x-1}<\pi(x)<\frac{x}{\log x-1.1},\qquad x\geq 60184.$$
It follows after a bit of calculation that
$$0.975\frac{x}{\log x}<\pi(2x)-\pi(x)<\frac{x}{\log x},\qquad x\geq 2^{31}.$$
Hence for $b\geq 31$, the value of $\al... | 8 | https://mathoverflow.net/users/11919 | 435077 | 175,906 |
https://mathoverflow.net/questions/435085 | 9 | Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the principal branch of the logarithm.)
It appears that $^9(-\sqrt2)$ is very close to $1$, but not exactly $1$ — so that the s... | https://mathoverflow.net/users/36721 | The $9$th tetration of $-\sqrt2$ | This is not a huge coincidence: the idea is that the sequence $a\_n={}^{n}(-\sqrt{2})$ has small norm until $n=6$, then it gets out of hand for $n=7$ ($a\_7\sim-33+29i$), so that $a\_8=e^{a\_7\ln(-\sqrt{2})}$ is almost $0$ and $a\_9$ is almost $1$.
In general when sequences of this type get out of hand (increase expo... | 22 | https://mathoverflow.net/users/172802 | 435088 | 175,908 |
https://mathoverflow.net/questions/435004 | 12 | Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ [formally smooth](https://en.wikipedia.org/wiki/Formally_smooth_map) over $\mathbb{R}$?
If it helps, feel free to assume that $M$ is compact.
---
(This is not a joke question. And yes, I know about $C^\infty$-rings and topologi... | https://mathoverflow.net/users/27013 | Are algebras of smooth functions formally smooth? | It seems $C^\infty(M;\mathbb{R})$ is not formally smooth for any positive-dimensional manifold. (The following argument came up in a discussion with Thomas Nikolaus, we later also found it [in this MO question](https://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials/9723#9723)):
Let's ... | 10 | https://mathoverflow.net/users/39747 | 435099 | 175,909 |
https://mathoverflow.net/questions/420365 | 4 | $\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in [this book](https://link.springer.com/book/10.1007/978-1-4612-4728-9) by Glimm and Jaffe.
It defines
\begin{equation}
\mathcal{A} = \left \{ A(\phi) = \sum\_{j = 1}^N c\_j \exp \left( \phi(f\_j) \right) \; \Big \vert \; c\_j \i... | https://mathoverflow.net/users/18936 | Understanding the Osterwalder-Schrader conditions as formulated by Glimm and Jaffe | Since this has confused me multiple times, I write this answer in the hope that it might help others.
First, recall that reflection positivity as formulated by [Osterwalder and Schrader](https://link.springer.com/article/10.1007/BF01645738) states that
\begin{equation}
\sum\_{n,m} G\_{n+m} \left( \theta f\_n^\* \otim... | 1 | https://mathoverflow.net/users/18936 | 435109 | 175,912 |
https://mathoverflow.net/questions/435110 | 41 | Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: <https://arxiv.org/abs/2210.11635>
In that preprint, Kirti Joshi claims that
* he agrees with Scholze and Stix that Mochizuki's proof of ABC is incomplete,
* Scholze and Stix's rigidity claim in Remark 9 of their paper... | https://mathoverflow.net/users/483446 | Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture | To give a simple answer: There would be no direct implications. The paper doesn't claim a proof of Corollary 3.12, the ABC conjecture, or any other Diophantine inequalities. I'm pretty sure that, if Joshi had a proof of one of these, he would say it.
| 34 | https://mathoverflow.net/users/18060 | 435117 | 175,914 |
https://mathoverflow.net/questions/435115 | 8 | Is there a predicate $P(x)$ such that $\mathrm{ZF} \vdash \exists! x. P(x)$, and $\mathrm{ZF} \vdash \forall x. P(x) \to (x \subseteq \mathbb R)$, but $\mathrm{ZF} \nvdash \forall x. P(x) \to \mathsf{Borel}(x)$?
Since ZF is consistent with the statement that every real subset is Borel, we have $\mathrm{ZF} \nvdash \f... | https://mathoverflow.net/users/136535 | Definable set in ZF that cannot be proved to be Borel | There is a common view amongst mathematicians that uses of the axiom of choice must somehow be inherently undefinable, and that if one sticks with definable sets only, then everything will be very nicely behaved.
The question here is an instance of this perspective, since it is asking whether every definable set of r... | 23 | https://mathoverflow.net/users/1946 | 435118 | 175,915 |
https://mathoverflow.net/questions/435128 | 2 | This question is a continuation of the question [here](https://mathoverflow.net/questions/435025/inequality-with-decreasing-rearrangement-function/435042?noredirect=1#comment1120917_435042).
Let $f^{\*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<... | https://mathoverflow.net/users/163368 | Inequality with decreasing rearrangement and non-decreasing function | $\newcommand\la\lambda\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}$The answer to this question is yes.
Indeed,
let $h:=f^\*/g$. Then $h$ is a nonincreasing function and the inequality in question can be rewritten as
\begin{equation\*}
\Big(\int\_0^\infty h(s)^{p'}ds\Big)^{p/p'}
\le c\int\_0^\infty h(s)^p s^{p/... | 2 | https://mathoverflow.net/users/36721 | 435130 | 175,917 |
https://mathoverflow.net/questions/360856 | 8 | In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact that the total curvature is $2|k|\pi$ with $k \not= 0$ the winding number.
Is there a known generalisation of this to... | https://mathoverflow.net/users/32210 | Length and curvature for closed curves in negatively curved spaces | The Reshetnyak majorization theorem (see [9.56](https://arxiv.org/pdf/1903.08539v5.pdf)) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be *majorized* by a convex plane figure $F$; that is, there is *short* (= 1-Lipschitz) map (= majorization) $m\colon F\to U$ such that $m|\_{\partia... | 4 | https://mathoverflow.net/users/1441 | 435142 | 175,924 |
https://mathoverflow.net/questions/434970 | 13 | As shown in Simpson's excellent [Subsystems of Second Order Arithmetic](https://www.personal.psu.edu/t20/sosoa), the ‘big five’ system ATR$\_0$ from second-order reverse mathematics is equivalent to the following principle:
*For arithmetical $\varphi$ such that $(\forall n)(\exists \text{ at most one } X)\varphi(X, n... | https://mathoverflow.net/users/33505 | "At most one" versus "at most finitely many" | The answer is positive, assuming extra induction, and a sketch is as follows.
Let $\varphi(X,n)$ be as in (\*).
1. Define an analytic code $A\_n$ as follows $X\in A\_n\leftrightarrow \varphi(X, n)$.
2. Use induction (say for $\Sigma\_2^1$-formulas) to show that $A\_n$ can be enumerated (as a finite sequence).
3. No... | 4 | https://mathoverflow.net/users/33505 | 435159 | 175,931 |
https://mathoverflow.net/questions/434423 | 8 | This is something which I'm sure is well known to experts which I would appreciate some information about. In his paper [1], Scholze proves (e.g. Theorem 8.4, Theorem 8.8) that on a proper adic space $X$ over $\textrm{Spa}(k,\mathcal{O}\_{k})$, with $k$ an appropriate non-archimedean field, the natural map $H^{i}(X\_{\... | https://mathoverflow.net/users/113933 | $p$-adic comparison of cohomology with coefficients in $\mathbb{Z}_{p}$ and $\mathbb{B}_{\textrm{dR}}$ on general smooth algebraic varieties | The result is false in the open case.
If true, a long exact sequence would show that also
$$H^i(X\_{\mathrm{proet}},\hat{\mathbb Z}\_p)\otimes k\to H^i(X\_{\mathrm{proet}},\hat{\mathcal O}\_X)$$
is an isomorphism, where I assume $k$ is algebraically closed (which I think is also implicit in the question, or other... | 5 | https://mathoverflow.net/users/6074 | 435163 | 175,933 |
https://mathoverflow.net/questions/434794 | -2 | This posting has been Edited. The edited material shall be noted.
[The projectively extended real line](https://en.wikipedia.org/wiki/Projectively_extended_real_line) $\hat {\mathbb R}= \mathbb R \cup \{\infty\}$ is one system which allows division by zero! Yet it has many [undefined](https://en.wikipedia.org/wiki/Pr... | https://mathoverflow.net/users/95347 | Is this extension of the projectively extended real line, consistent? | This is an account on the particulars of an interpretation of the original system [before the edit] presented in this question in set theory.
First we define an extended kind of rationals to suit adding a rational that is higher than all other rationals, the latter would correspond to $\infty$. The set of all those r... | 1 | https://mathoverflow.net/users/95347 | 435165 | 175,934 |
https://mathoverflow.net/questions/435167 | 1 | It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their relation?
>
> **Question:**
>
>
> can the *longest* side of a triangle in the Delaunay triangulation of a planar po... | https://mathoverflow.net/users/31310 | Relation of MSTs in the Euclidean plane to Delaunay triangulations | It cannot be for any planar triangulation. Say we have a triangle with vertices $x, y, z$ and $xy$ is the longest edge.
Consider a run of [Prim’s MST algorithm](https://en.m.wikipedia.org/wiki/Prim%27s_algorithm) which at each step adds an edge to a growing tree if it is minimum length among all those edges that have... | 2 | https://mathoverflow.net/users/97414 | 435168 | 175,935 |
https://mathoverflow.net/questions/435138 | 2 | I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e\_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x\_n)\_{n \in \mathbb{Z}} \in \mathcal{D}(J)$ by
$$(Jx)\_{n} =i((2n+1)x\_{n+1}-(2n-1)x\_{n-1}).$$
I know that $J$ is essent... | https://mathoverflow.net/users/142048 | Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$ | Under the Fourier series isomorphism $\ell^2(\mathbb{Z}) \cong L^2(-\pi,\pi)$, $u(t) = \sum\_{n\in\mathbb{Z}} x\_n e^{int}$, the operator becomes
$$\begin{aligned}
(Ju)(t) &= 4i\sin(t) u'(t) + 2i\cos(t) u(t) \\
&= \begin{cases}
+4i\left|\sin(t)\right|^{1/2} \partial\_t (\left|\sin(t)\right|^{1/2} u(t)) & t\in(0,\pi)... | 3 | https://mathoverflow.net/users/2622 | 435174 | 175,937 |
https://mathoverflow.net/questions/435187 | 3 | Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\in [0,2\pi)\}$ or $A×B,$ where $E,A,B$ are sets of positive Lebesgue measure in $\mathbb{R}?$
Note: I can show that the ... | https://mathoverflow.net/users/483450 | Property of sets of positive Lebesgue measure in $\mathbb{R}^2$ | Firstly, a set $P$ of positive measure need not contain anything of the form $A\times B$, for example consider for some $k\in\mathbb{R}\setminus\{0\}$ the set $P=\{(x,y)\in\mathbb{R}^2;x-ky\not\in\mathbb{Q}\}$. Then the complement of $P$ is a null set, and any set $A\times B$ where $A,B$ have positive measure contains ... | 2 | https://mathoverflow.net/users/172802 | 435188 | 175,943 |
https://mathoverflow.net/questions/435200 | 0 | The subject of this question are perfect matchings of a complete undirected graph $G(V,E), n:=\mathrm{card}(V)=2k$, without self-loops or parallel edges and $n=2k$ vertices.
The objective is to determine a perfect matching $\mathrm{M}\_{\text{opt}}$ of minimal weight.
Consider now the greedy heuristic that determin... | https://mathoverflow.net/users/31310 | Edge-length constraints from greedy matching | Let the weights of the edges in a 6-cycle in $K\_6$ be $1,2,5,6,5,2$ (in cyclic order), and let other weights be large. Then the optimal matching will be $2,2,6$, while the greedy one will consist of $1,5,5$.
A similar construction works with longer cycles.
| 2 | https://mathoverflow.net/users/17581 | 435201 | 175,945 |
https://mathoverflow.net/questions/435129 | 4 | Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ for each $y\in Y$, measurable and bounded by a $L^1$ function that does not depend on $x$). Let ${\Sigma}\_0$ be sub $\si... | https://mathoverflow.net/users/470906 | Is the conditional expectation of a Caratheodory function a Caratheodory function? | Here is a positive answer for the case that $\Sigma\_0$ is generated by a random variable with values in a Polish space, so that we can use regular conditional probabilities and for some kernel $\kappa:Y\to\Delta(Y)$, we can let
$$\mathbb{E}(f(\cdot,x)|\Sigma\_0)\_y=\int f(,\cdot,x)~\mathrm d\kappa\_y.$$
Then continuit... | 1 | https://mathoverflow.net/users/35357 | 435217 | 175,947 |
https://mathoverflow.net/questions/435232 | 4 | For a set $A\subseteq \omega$ we let the *upper density* of $A$ be defined as $d^+(A) := \lim\sup\_{n\to\infty}\frac{|A\cap(n+1)|}{n+1}$. Let $\text{FrU}(\omega)$ be the collection of [free ultrafilters](https://en.wikipedia.org/wiki/Ultrafilter) on $\omega$.
[Asaf Karagila](https://mathoverflow.net/users/7206/asaf-k... | https://mathoverflow.net/users/8628 | Supremum of infimum of measure of members of a free ultrafilter | The answer is: zero.
The reason is that every ultrafilter has zero as the infimum of the upper density of its members. To see this, observe that if a set $U$ is in the ultrafilter $\mathcal{U}$, with some positive upper density, then we can split $U$ in half $U=A\sqcup B$ each with half the upper density (just take e... | 9 | https://mathoverflow.net/users/1946 | 435233 | 175,950 |
https://mathoverflow.net/questions/435195 | 0 | I'm now solving an LP that has a few coupling rows (as in Dantzig-Wolfe decomposition) and a few coupling columns (as in Benders decomposition) simultaneously; other rows and columns are block-angular. Is there an algorithm that decomposes such LP? Thank you very much!
| https://mathoverflow.net/users/494373 | Combining Dantzig-Wolfe and Benders decomposition | You could apply Benders as the main algorithm and use Dantzig-Wolfe for the subproblem. Alternatively, you could apply Dantzig-Wolfe as the main algorithm and use Benders for the subproblem. For LP, Benders and Dantzig-Wolfe are equivalent if you take duals, so you could also either apply Benders as the main algorithm ... | 0 | https://mathoverflow.net/users/141766 | 435235 | 175,951 |
https://mathoverflow.net/questions/435241 | 34 | Does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that
$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$
and
$$\displaystyle \liminf\_{(x,y) \in \mathbb{R}^2} f(x,y) = -\infty?$$
In other words, does there exist a polynomial $f$ which takes on positive values at every integer point, but ... | https://mathoverflow.net/users/10898 | Ruling out the existence of a strange polynomial | The polynomial $f(x,y)=(x^2+1)(5y^2+5y+1)\in\mathbb{Z}[x,y]$ is an example. Note that $5y^2+5y+1>0$ for $y\in\mathbb{Z}$, but $5y^2+5y+1<0$ at $y=-\frac{1}{2}$.
| 52 | https://mathoverflow.net/users/95685 | 435244 | 175,953 |
https://mathoverflow.net/questions/435251 | 9 | This is a refinement of my [question asked earlier](https://mathoverflow.net/questions/435241/ruling-out-the-existence-of-a-strange-polynomial), which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same question, but with the extra assu... | https://mathoverflow.net/users/10898 | Ruling out the existence of a strange polynomial II | Just a slight modification of the previous [example](https://mathoverflow.net/a/435244/36721):
$$f(x,y):=\left(5 y^2+5 y+1\right) \left(x^2+y^2\right)+\left(10 y^2+10 y+1\right) y^2.$$
---
Your conditions $f(0,0) = 0$ and
$$f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$
contradict each other. So, I was assuming... | 20 | https://mathoverflow.net/users/36721 | 435252 | 175,955 |
https://mathoverflow.net/questions/433196 | 1 | The origin question: Let $\Omega \subset \mathbb{H}^2$ be a domain of the hyperbolic plane $\mathbb{H}^2$. Let $u: \Omega \to \mathbb{H}^2$ be injective and an isometry from $\Omega$ to its image. Does there exist a Mobius transformation $\gamma\in \text{PSL}(2,\mathbb{R})$ such that $u=\gamma\mid\_\Omega$?
The modif... | https://mathoverflow.net/users/161514 | Is a local isometry of the hyperbolic plane the restriction of a global isometry? | The hyperbolic plane has this property, as does the Euclidean plane.
If $E$ is any subset of $ \mathbb{H}^2$, and $u : K \to\mathbb{H}^2$ is an isometry, then there is an extension of $u$ which is an isometry of $\mathbb{H}^2$ onto itself. I use "isometry" in the sense: $d(u(x),u(y)) = d(x,y)$ for all $x,y \in E$, wh... | 3 | https://mathoverflow.net/users/454 | 435259 | 175,958 |
https://mathoverflow.net/questions/435231 | 5 | Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \quad (af)’’(x) \leq 0$$
for all $x \in I$?
| https://mathoverflow.net/users/50438 | On existence of a concave function | Such a function doesn't exist for some choices of $a$. For notational purposes I will change the unit interval by $[-2,2]$.
Consider a $C^\infty$ function $a:[-2,2]\to\mathbb{R}$ such that $a(0)=3$, $a(1)=1$ and $a(x)=a(-x)\forall x$.
Suppose that we have a convex function $f:[-2,2]\to\mathbb{R}$ such that $f''\leq... | 4 | https://mathoverflow.net/users/172802 | 435264 | 175,962 |
https://mathoverflow.net/questions/435268 | 2 | This is a variant of the Nash equilibrium. Let's say that there are 3 prizes: A Ferrari, a diamond watch, and a new boat. There are 6 players. 3 players with a motive while 3 players with another motive, but all 6 players are playing for themselves: 3 players, for example, are wearing red shirts. The people wearing red... | https://mathoverflow.net/users/495433 | Nash equilibrium at another level | With the given parameters, the Nash equilibria are exactly those situations where the three purple players pick each a different price. This ensures that the red players can't win anyway, so they'll do whatever.
To see this, we first observe that a purple player would never change to a prize another purple player alr... | 3 | https://mathoverflow.net/users/15002 | 435271 | 175,963 |
https://mathoverflow.net/questions/435272 | 1 | The general approach here is a follow up of the approach outlined in a prior posting on [extending the projectively extended real line](https://mathoverflow.net/questions/434794/is-this-extension-of-the-projectively-extended-real-line-consistent). In particular arithmetic operators break down to ternery relations that ... | https://mathoverflow.net/users/95347 | Is there an effective way to generalize this approach of affinely extending the number line? | First of all, mentioning set-theoretic background is really just making this more complicated than it needs to be, so I'll ignore it. Additionally, I'll use "$\leadsto$" in place of your "$\rightarrow$" since I also want to talk about conventional limits.
It sounds like you want $t\leadsto c$ whenever $t$ "has a form... | 3 | https://mathoverflow.net/users/8133 | 435273 | 175,964 |
https://mathoverflow.net/questions/435191 | 8 | This is about a rather concrete problem that occurs in the middle of a lecture by Scholze. First I'll refer to the lecture, but then I'll state the problem.
In <https://www.youtube.com/watch?v=q6Tv2vJJShg> , at around the 22 minute mark, Peter Scholze claims that if you have a simplicial hypercover of a profinite set... | https://mathoverflow.net/users/318125 | A hypercover of profinite sets as a limit of hypercovers of finite sets | I'm sorry for being cryptic.
The subtle point in the construction is that the maps $T\_n\to T\_{n,j}$ are not all surjective, i.e. one cannot construct this pro-system as a system of quotients.
By induction on $n$, one constructs a cofiltered system of $n$-skeletal simplicial hypercovers $T\_{\bullet,j}^{(n)}\to S\... | 9 | https://mathoverflow.net/users/6074 | 435277 | 175,966 |
https://mathoverflow.net/questions/435210 | 3 | **Question.** Let $u: B^3 \to \mathbf{R}$ be a harmonic function with $u(0) = 0$, $Du(0) = 0$, where its homogeneous harmonic blow-up is a polynomial $p = p(x,y)$ in two variables, so independent of $z$; in other words $p$ is a non-zero homogeneous harmonic polynomial so that
\begin{equation}
u(x,y,z) = p(x,y) + o( \lv... | https://mathoverflow.net/users/103792 | A harmonic function degenerate in one direction | The questions have been answered in the comments, I am just recording them here: Alexandre Eremenko pointed out that *no*, the function $u$ need not be translation-invariant, because the dependencies on $z$ could be 'hidden' inside a polynomial of higher degree, say
\begin{equation}
u = p(x,y) + q(x,y,z),
\end{equation... | 3 | https://mathoverflow.net/users/103792 | 435278 | 175,967 |
https://mathoverflow.net/questions/435248 | 0 | Consider a smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d$ over a nice field (such as $\mathbb{C}$), we know that the cone $C(\operatorname{id}:\mathcal{O}\_X\rightarrow\mathcal{O}\_X)=0$. Under the isomorphism
$$\hom(\mathcal{O}\_X,\mathcal{O}\_X)\cong\hom(\mathcal{O}\_X,\mathcal{O}\_X(d-n-2)[n])^\vee\con... | https://mathoverflow.net/users/nan | Compute the cone of $\mathcal{O}_X\rightarrow\mathcal{O}_X(d-n-2)[n]$ | The cohomology sheaves are easy to see from the long sequence of cohomology sheaves of the triangle
$$
\mathcal{O}\_X \to \mathcal{O}\_X(d-n-2)[n] \to E.
$$
If $n \ge 2$ it gives
$$
\mathcal{H}^i(E) =
\begin{cases}
\mathcal{O}\_X(d-n-2), & i = -n,\\
\mathcal{O}\_X, & i = -1,\\
0, & \text{otherwise.}
\end{cases}
$$
And... | 3 | https://mathoverflow.net/users/4428 | 435280 | 175,968 |
https://mathoverflow.net/questions/435283 | 3 | Consider the following fragment from the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset (p72, in section 2.5):
>
> $\ \ \ $ Assume $\mathscr{C}$ is a category having all the properties of a strict $\scr C^\*$-tensor category except existence of direct sums and subobjects. First... | https://mathoverflow.net/users/216007 | Adding finite direct sums to a C*-tensor category | If you care about completeness, you just want to observe that the norm defined this way restricts to the original norms on the direct summands $\operatorname{Mor}(U\_i, V\_j)$. Since there are only finitely many summands, a sequence in $\bigoplus\_{i,j} \operatorname{Mor}(U\_i, V\_j)$ is a Cauchy sequence if and only i... | 5 | https://mathoverflow.net/users/9942 | 435289 | 175,971 |
https://mathoverflow.net/questions/435216 | 8 | For the symmetric group on $n$ objects $S\_n$ the question of how to write its longest element $w\_0$ as a reduced decomposition is an important combinatorical problem. As example, in this [question](https://mathoverflow.net/questions/370333/number-of-reduced-decompositions-of-the-longest-element-of-the-weyl-group) the... | https://mathoverflow.net/users/491434 | One element commutation classes of reduced decompositions of the longest element of the Weyl group | The reduced words that are in their own commutation classes are:
* $s = [123\cdots(n-2)(n-1)(n-2)\cdots321][23\cdots(n-3)(n-2)(n-3)\cdots32][3\cdots(n-4)(n-3)(n-4)\cdots3]\cdots$
* the reversal of $s$ (writing the word in reverse order)
* the complement of $s$ (mapping each letter $i$ to $n-i$)
* the reverse compleme... | 12 | https://mathoverflow.net/users/495435 | 435293 | 175,973 |
https://mathoverflow.net/questions/433762 | 6 | I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.
From my understanding, to define the p-adic height (and more precisely the $p$-adic sigma function), you need an integral Weier... | https://mathoverflow.net/users/124772 | Does the $p$-adic regulator depend on Weierstrass model? | Here a long comment to settle this question.
This is really a bug in the implementation of $p$-adic heights in sagemath. I have announced it [as a bug here on the sage trac list](https://trac.sagemath.org/ticket/34790). I hope to add the code to fix it soon after this post.
The current implementation is based on [1... | 2 | https://mathoverflow.net/users/5015 | 435294 | 175,974 |
https://mathoverflow.net/questions/435205 | 4 | I suppose that what I look for is known, but I can't find it.
Let $\left\lbrace I\_n=[a\_n,b\_n)\right\rbrace$ and $\left\lbrace J\_n=[b\_n,c]\right\rbrace$ ($n\in\mathbb{N}$) be two countable families of intervals in the unit circle $S^1$. Notice that $I\_n$ and $J\_n$ are adjacent for every $n$, and that the extrem... | https://mathoverflow.net/users/167834 | First visit of intervals for an irrational rotation | No, it is not possible. In the following I will use $I\_n=(a\_n,b\_n)$ instead of $[a\_n,b\_n)$ (this is not a problem, you can just increase $a\_n$ a bit so that the statement with $I\_n=(a\_n,b\_n)$ is stronger).
For fixed $n$, we say that $x$ first visits $J\_n$ if $\min\left\lbrace k:R^k(x)\in J\_n\right\rbrace<\... | 2 | https://mathoverflow.net/users/172802 | 435298 | 175,975 |
https://mathoverflow.net/questions/16347 | 7 | $\newcommand\Box{\mathrm{Box}}\newcommand\Set{\mathrm{Set}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}$A cubical set $\Box\op \to \Set$ is a model for a homotopy type, via Grothendieck and Cisinski (here $\Box$ is the box category with objects the natural numbers and arrows generated by face and degeneracy ... | https://mathoverflow.net/users/4177 | Homotopy groups of cubical sets | I think a reference for this would be [Theorem 3.24](https://arxiv.org/pdf/2202.03511.pdf#page=26) of
>
> Homotopy groups of cubical sets, Daniel Carranza, Chris Kapulkin, 2022. arXiv:2202.03511, <https://doi.org/10.48550/arxiv.2202.03511>
>
>
>
| 4 | https://mathoverflow.net/users/130058 | 435306 | 175,978 |
https://mathoverflow.net/questions/435324 | 9 |
>
> Has anyone ever attempted to write down axioms capturing the behaviour of ${\bf Rel}$, the category of relations?
>
>
>
Lawvere's [ETCS](https://ncatlab.org/nlab/show/ETCS) attempts to axiomatize the behaviour of the subcategory ${\bf Set}$ of ${\bf Rel}$ and ends up with a theory equiconsistent with $ZF-{\s... | https://mathoverflow.net/users/92164 | Elementary theory of the category of relations | As Sam has pointed out in the comments, [SEAR](https://ncatlab.org/nlab/show/SEAR) is close in spirit to this sort of theory. The difference is that SEAR has "elements" as a basic notion in addition to sets and relations, whereas an axiomatization of Rel would contain only sets and relations. One could in theory transl... | 14 | https://mathoverflow.net/users/49 | 435330 | 175,985 |
https://mathoverflow.net/questions/435336 | 9 | In mathematical contexts the term *spline* essentially refers to interpolating or approximating piecewise functions with continuity constraints.
According to [the history of mathematical splines](https://en.wikipedia.org/wiki/Spline_(mathematics)#History)
>
> In the foreword to (Bartels et al., 1987), Robin Forre... | https://mathoverflow.net/users/31310 | True origin of the term "Spline" | The Oxford English Dictionary doesn't necessarily give the earliest uses of a word. But spline with the meaning "A long, narrow, and relatively thin piece or strip of wood, metal, etc.; a slat." is quoted from 1756, much earlier than "flexible strip of wood or hard rubber used by draftsmen in laying out broad sweeping ... | 11 | https://mathoverflow.net/users/9025 | 435337 | 175,988 |
https://mathoverflow.net/questions/435316 | 2 | Nancy Dykes says in the proof of Theorem 3.4 in her article [Generalizations of realcompact spaces](https://msp.org/pjm/1970/33-3/pjm-v33-n3-p05-s.pdf) that by a result of
John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative
upper semicontinuous function $f$ on $\beta X$ such that $f$ is positiv... | https://mathoverflow.net/users/86099 | A question about a realcompact space and upper semicontinuous function | The following characterisation is well known. It can be found in Engelking's book as Theorem 3.11.10.
>
> **Theorem**: A Tychonoff space $X$ is realcompact if and only if for each $p\in\beta X\setminus X$ there is a continuous function $\varphi:\beta X\rightarrow [0,\infty)$ such that $\varphi|\_X>0$ and $\varphi(p... | 2 | https://mathoverflow.net/users/54788 | 435339 | 175,990 |
https://mathoverflow.net/questions/435340 | 1 | Consider the partial differential equation
$$\psi\_t(t,x)=i\kappa \psi\_{xx}(t,x) ~\mbox{for}~ 0<(t,x)\in\mathbb{R}\times\mathbb{R}$$
with boundary conditions
$$\psi(0,x)=0 ~\mbox{for}~ x>0,$$
$$\psi(t,0)=\psi\_0(t) ~\mbox{for}~ t\ge0.$$
Are these equation uniquely solvable whenever $\psi\_0$ is sufficiently smooth?
... | https://mathoverflow.net/users/56920 | Schrödinger equation with nonstandard boundary conditions | The equation under consideration is uniquely solvable in $H^s(\mathbb{R}^+)$, as soon as $\psi\_0\in H^{(2s+1)/4}(\mathbb{R}^+)$, and there is a fairly explicit expression for the propagator. You can find these results, for example, in
J. L. Bona, S. Sun, B. Zhang,
Nonhomogeneous boundary-value problems for one-dimen... | 1 | https://mathoverflow.net/users/169603 | 435348 | 175,993 |
https://mathoverflow.net/questions/435344 | 2 | I need to find the following paper:
“K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, 15 (1967), p. 161-169.”
It is possible to find it on the internet?
Thanks
| https://mathoverflow.net/users/495513 | Reference request: “A random integral and Orlicz spaces” | I could not find it on the internet so I uploaded it here: <https://www.transfernow.net/dl/20221126nnUfCto7>
| 2 | https://mathoverflow.net/users/54263 | 435354 | 175,994 |
https://mathoverflow.net/questions/435368 | 3 | $\DeclareMathOperator\Var{Var}\DeclareMathOperator\Motives{Motives}$Let us assume for the moment that we have a "nice" category of motives, that is for fields $k$ we have a contravariant functor
$$\Var(k)\to \Motives(k).$$
Now for any field extension $l/k$, we have a natural forgetful functor
$$\Var(l)\to\Var(k).$$
Wou... | https://mathoverflow.net/users/152554 | Functor between categories of motives | This would be a motivic analogue of the induced representation functor, and should be adjoint to the restriction functor, at least for finite field extensions.
This is just because the cohomology of a variety obtained by the forgetful functor is the induced representation of the cohomology.
It should be possible to... | 6 | https://mathoverflow.net/users/18060 | 435369 | 175,999 |
https://mathoverflow.net/questions/435371 | 1 | Assume that for each $n\in\mathbb{N}$, there's a stochastic function $f\_n$ of type $\mathbb{R}^{m}\to\Delta\mathbb{R}^{m}$, and for each $x\in\mathbb{R}^{m}$, the distributions $\frac{f\_n(x)-x}{\frac{1}{n}}$ will weakly converge as $n$ limits to $\infty$, s.t. the n'th distribution is about $O(\frac{1}{n^2})$ away fr... | https://mathoverflow.net/users/148137 | How to rigorously prove that this sequence of stochastic processes converges to a deterministic process? | I am guessing in "The particular thing I'm trying to prove is that,..." you are talking about the convergence of discrete generator to continuous one. The natural topology for these questions is Skorokhod. See here for some ideas and references/keywords: <https://math.stackexchange.com/questions/4225750/continuous-limi... | 1 | https://mathoverflow.net/users/99863 | 435373 | 176,000 |
https://mathoverflow.net/questions/435377 | 2 | Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt points, like the Fermat cubic, which has $18$. Is there any such non-singular complex projective cubic surface where four, ... | https://mathoverflow.net/users/493889 | Is there a non-singular cubic surface that has a point where four lines intersect? | No, this is not possible. If p is a smooth point on any surface S, and is
contained in a line l on S, then l is contained in the tangent plane at p,
call it T\_p. Now if S is a cubic then it intersects T\_p in a cubic curve
(with some singularity at p, even though S is smooth at p); and a cubic curve
can contain at mos... | 12 | https://mathoverflow.net/users/14830 | 435379 | 176,002 |
https://mathoverflow.net/questions/428454 | 6 | A famous corollary of [Matiyasevich's theorem](https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich%27s_theorem) is that there exists a Diophantine equation such that it is undecidable (under some recursively axiomatizable theory $T$, such as ZFC) whether that equation has any natural-number solutions. I will som... | https://mathoverflow.net/users/95043 | How constructive is Matiyasevich's theorem? | Turning the proof that there exists a Diophantine equation encoding eg the consistency of $\mathrm{ZFC}$ into a program that actually computes the polynomial would be a bit tedious, but there should not be any significant obstacles. Whether running said program would actually yield an answer, some kind of overflow erro... | 2 | https://mathoverflow.net/users/15002 | 435391 | 176,006 |
https://mathoverflow.net/questions/435389 | 3 | In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E\_{1}$ and $E\_{2}$ with $N\_{G}(E\_{1})=E\_{1}.\operatorname{Sp}(4,2)$ and $N\_{G}(E\_{2})=E\_{2}.(2^{3}:S\_{3})$.
$N\_{G}(E\_{1})$ is a maximal subgroup of $G$ and $(2^{3}:S\_{3})$ is a point stabilizer of a non... | https://mathoverflow.net/users/488802 | Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$ | The difference between the examples arises principally because $5 \equiv 1 \bmod 4$ and $3 \equiv 3 \bmod 4$.
For $q \equiv 1 \bmod 4$, $G := {\rm GL}(4,q)$ has centre $Z$ divisible by $4$, and contains a group $S$ of symplectic type with $N\_G(S) = ZS.{\rm Sp}(4,2)$. The group $S$ maps onto your group $E\_1$ in ${\r... | 4 | https://mathoverflow.net/users/35840 | 435402 | 176,008 |
https://mathoverflow.net/questions/435401 | 5 | I am trying to prove the following. (I posted this on [math.se](https://math.stackexchange.com/questions/4568521/initially-horizontal-geodesic-is-always-horizontal) with no success)
>
> Let $E,B$ be Riemannian manifolds. Suppose
> $\pi: E\to B$ is a Riemannian submersion.
>
>
> For each $x\in E$, define $V\_x E =... | https://mathoverflow.net/users/220580 | Initially horizontal geodesic is always horizontal | After choosing local coordinates, by the implicit function theorem (I'm omitting a bunch of technical computations) there is a smooth function $\varphi:TE \to TE$ such that $\varphi(x,-): T\_x E \to T\_x E$ is the projection to $V\_x E$.
Let $I\subseteq [0,1]$ be the set on which $\gamma'$ is horizontal: this is the ... | 7 | https://mathoverflow.net/users/3948 | 435405 | 176,009 |
https://mathoverflow.net/questions/420558 | 1 | Both cohomology and homotopy groups capture global topological information of a manifold $X$. It is interesting to ask if they can be computed from local data. A triangulation $T$ is a natural presentation of a manifold.
The cohomology by definition can be computed from $T$. However, the problem seems much harder for... | https://mathoverflow.net/users/124549 | (Lower) homotopy groups from triangulations | Being a manifold or the dimension restriction $k\leq n$ doesn't matter, the following applies to finite simplicial complexes in general:
As others have explained, if the fundamental group is not finite, the higher homotopy groups might be infinitely generated, so in that case even the format of the answer is unclear.... | 3 | https://mathoverflow.net/users/39747 | 435424 | 176,012 |
https://mathoverflow.net/questions/435383 | 3 | Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum\_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$
I have checked this up to $n=100$, and I suspect there's some simple argument for why this should be true, probably using Eisenstein series identities but I'm not ... | https://mathoverflow.net/users/127690 | Divisibility relation with a specific sum of divisors | From the paper of [Touchard](https://oeis.org/A000385/a000385.pdf) that is linked in the question on we get the relation $$3nS\_0(n)-\frac{n(n-1)\sigma(n)}{6}=\frac{10}{n}S\_2(n)
....(1)$$
here $S\_i(n)=\sum\_{k=1}^{n-1}k^i\sigma(k)\sigma(n-k)$.
Now we can prove $n\nmid S\_2(n)$ in general when $n$ is even. For $n$... | 2 | https://mathoverflow.net/users/156029 | 435433 | 176,016 |
https://mathoverflow.net/questions/435411 | 3 | Nine different types of singularities are possible on a cubic surface, according to [Wikipedia.](https://en.wikipedia.org/wiki/Cubic_surface) How exactly is the "type" of singularity defined? I know that the number corresponding to the singularity is the number of degrees of freedom removed, but how can we say that the... | https://mathoverflow.net/users/493889 | How do we define the type of a singularity on a cubic surface? | All the singularities involved in this classification are Rational Double Points. These singularities are taut, in other words, their analytic type is uniquely determined by the configuration of curves in their minimal resolution.
Such a resolution is a finite set of (-2)-curves whose dual diagram is a Dynkin diagram... | 3 | https://mathoverflow.net/users/7460 | 435439 | 176,019 |
https://mathoverflow.net/questions/435438 | 0 | Can someone explain the Palm distribution? Or provide some information about Palm distribution. The
article called 《A tutorial on Palm distributions for spatial point processes》 is hard to understand.
| https://mathoverflow.net/users/495598 | About Palm distribution | C. Palm's theory of spatial point processes relies heavily on measure theory in an abstract setting. A more gentle introduction is given in the lecture notes [Conditioning in spatial point processes](https://data.math.au.dk/publications/csgb/2015/math-csgb-2015-14.pdf). Section 3, in particular, defines the Palm distri... | 1 | https://mathoverflow.net/users/11260 | 435442 | 176,021 |
https://mathoverflow.net/questions/433698 | 4 | A Riemannian manifold $(M, g)$ is said to be an almost Ricci soliton if there exists a complete vector field $X \in \Gamma(TM)$ and a smooth function $\lambda: M \to \mathbb{R}$ such that
$$\operatorname{Ric} + \frac{1}{2}\mathscr{L}\_{X} g = \lambda g$$ When this vector field is the gradient of a smooth function $f: M... | https://mathoverflow.net/users/119418 | What are some explicit examples of nontrivial gradient almost Ricci solitons with harmonic curvature? | **I'm revising my answer to shorten it, since there is a much simpler way to describe these solutions more fully.**
Let $(N^n,h)$ be a metric of constant sectional curvature $k$ and consider the quadratic form
$$
g = \frac{\mathrm{d}u^2}{k-a\,u^2+ b\,u^{1-n}} + u^2\,h
$$
on $M^{n+1} = \mathbb{R}^+\times N$, where $a$... | 4 | https://mathoverflow.net/users/13972 | 435448 | 176,023 |
https://mathoverflow.net/questions/435403 | 4 | Let $M$ be a Riemannian $n$-manifold and $x \in M$. For the metric tensor $g\_{ij}$ of geodesic normal coordinates at $x$, there is a formula $g\_{ij}(y) = \delta\_{ij} + \frac13 R\_{kijl} y^k y^l + O(\|y\|^3)$. Assume that the injectivity radius of $M$ is positive.
**Question.** Are there constants $c\_1, c\_2>0$ (t... | https://mathoverflow.net/users/156792 | Bounds for metric in normal coordinate | As mentioned by Deane Yang in the comments and his (deleted) answer, one can estimate the components of the metric in normal coordinates using a transport ODE (I know it from [Dolgov-Khriplovich (1983)](https://doi.org/10.1007%2fBF00760057) Eq.(34), and also in more implicit form from [Florides-Synge (1971)](https://do... | 7 | https://mathoverflow.net/users/2622 | 435449 | 176,024 |
https://mathoverflow.net/questions/434592 | 4 | Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x\_0 \in X$. Let us say that $f$ and $g$ are *tangent at* $x\_0$ if the following condition is satisfied: For every $\epsilon > 0$ there is some $\delta > 0$ such that for all $x \in X$ we have
$$d(x,x\_0) \leq \delta \implies d(f(x),g(x)) \leq \epsilon \... | https://mathoverflow.net/users/2841 | Reference request: "Tangent relation" in metric spaces | One reference:
>
> Elisabeth Burroni and Jacques Penon, [A metric tangential calculus](http://tac.mta.ca/tac/volumes/23/10/23-10abs.html). *Theory and Applications of Categories* 23 (2010), 199–220.
>
>
>
The first sentence of the paper says that a fuller account of their work can be found in a longer (99-page... | 2 | https://mathoverflow.net/users/586 | 435456 | 176,026 |
https://mathoverflow.net/questions/435428 | 7 | I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$.
Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group acting on $X$. Moreover, let $x\in X$ be a closed point and $G\_{x}$ its stabilizer under the $G$ action where $G\_{x}$ is... | https://mathoverflow.net/users/495308 | Intuition for Luna's Étale Slice Theorem | Luna Étale's Slice Theorem is probably the most powerful result we have to understand the local structure of a moduli space.
Moduli spaces (say of semi-stable vector bundles on a smooth projective curve) may be constructed by taking the geometric quotient of a Quot-scheme by a reductive group $G$. The stabilizer of a... | 3 | https://mathoverflow.net/users/37214 | 435458 | 176,027 |
https://mathoverflow.net/questions/435457 | 6 | This is a cross post from MSE of
<https://math.stackexchange.com/questions/4562196/normalizer-of-maximal-torus-is-maximal>
Let $ T $ be a maximal torus in a compact connected simple Lie group $ K $. For which groups $ K $ is the normalizer $ N(T) $ maximal among the proper closed subgroups of $ K $?
I know this i... | https://mathoverflow.net/users/387190 | When is the normalizer of the maximal torus maximal? | $\def\fg{\mathfrak{g}}\def\ft{\mathfrak{t}}\def\long{\text{long}}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SU{SU}$The point of this answer, as discussed in comments, is to note that this is NOT true in types BCFG. In comments, Mikhail Borovoi said he could prove the result is true in ty... | 10 | https://mathoverflow.net/users/297 | 435474 | 176,032 |
https://mathoverflow.net/questions/435469 | 1 | Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x\_n \in E$ with $n\in \mathbb N$,
$$
x\_n \to x \iff [f(x\_n) \to f(x) \quad \forall f \in \mathcal F]. \quad (\star)
$$
Let $\tau$ be the metric topology on $E$ induced by $d$ and $\tau'$ the initial... | https://mathoverflow.net/users/477203 | Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$ | If $\mathcal{F}$ is countable, $\tau'$ is metrizable with a compatible metric $\rho$ given by
$$\rho(x,y)=\sum\_n 2^{-n}~|f\_n(x)-f\_n(y)|\wedge1$$
for some enumeration of $\mathcal{F}$.
Since the topology of a metric space is determined by the convergent sequences, this is a sufficient condition.
| 2 | https://mathoverflow.net/users/35357 | 435477 | 176,033 |
https://mathoverflow.net/questions/435459 | 3 | Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}\_A$ the collection of radical morphisms in $\text{mod}\,A$. For a natural number $n$ we denote by $\text{rad}\_A^n$ the collection of $n$ compositions of morphisms in $\text{rad}... | https://mathoverflow.net/users/145920 | For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$? | This is proven in the paper (as the title suggests):
*Coelho, Flávio U.; Marcos, Eduardo N.; Merklen, Héctor A.; Skowroński, Andrzej*, [**Module categories with infinite radical square zero are of finite type**](https://doi.org/10.1080/00927879408825084), Commun. Algebra 22, No. 11, 4511-4517 (1994). [ZBL0812.16019](... | 5 | https://mathoverflow.net/users/18756 | 435484 | 176,035 |
https://mathoverflow.net/questions/433927 | 3 | According to [Wikipedia](https://en.wikipedia.org/wiki/Doubling_space),
>
> However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures.
>
>
>
My question is: what are some examples of these results, and where can a more thorou... | https://mathoverflow.net/users/408316 | Results in computational geometry utilizing doubling dimension of a metric space | The question is a bit unclear, so I'm not sure this is what you seek.
But here is a connection to computational geoemtry.
There is considerable literature on
[geometric spanners](https://en.wikipedia.org/wiki/Geometric_spanner) and [doubling metric spaces](https://en.wikipedia.org/wiki/Doubling_space).
Here is a samp... | 2 | https://mathoverflow.net/users/6094 | 435485 | 176,036 |
https://mathoverflow.net/questions/435394 | 3 | In the article titled Tilting Exercises (See <http://arXiv.org/abs/math/0301098v3>) the authors define a notion of tilting perverse sheaves on an algebraic vareity $X$ with respect to stratification $\lbrace X\_{\nu} \rbrace$. It appears that §1.3 onwards they assume that all the morphisms $i\_{\nu} : X\_{\nu} \hookrig... | https://mathoverflow.net/users/58056 | Perverse tilting sheaves | At @random123's request, I'm will try to argue that looking for tilting perverse sheaves takes one very close to a setting in which [BBM] work.
Suppose that we have a suitably stratified variety
\begin{equation}
X = \bigsqcup\_{\lambda \in \Lambda} X\_\lambda
\end{equation}
and we wish to understand $D = D^b\_\Lambda... | 4 | https://mathoverflow.net/users/919 | 435492 | 176,040 |
https://mathoverflow.net/questions/435483 | 6 | Let $K$ be a $p$-adic local field with uniformizer $\pi \in \mathcal{O}\_{K}$ and residue field $k = \mathcal{O}\_{K}/\pi$. Let $G$ be a Lubin-Tate formal $\mathcal{O}\_{K}$-module and $G\_{0}$ its reduction to $k$.
Let $G\_{0}[\pi]$ denote the $\pi$-torsion subgroup of $G\_{0}$, this is a finite group scheme over $k... | https://mathoverflow.net/users/16981 | Generating the coordinate ring of the Lubin-Tate formal group | I'll assume that $\pi=p$; I guess that the general case is the same but I have not checked. We now have $k=\mathbb{F}\_{p^n}$ and $\mathcal{O}\_K=W\mathbb{F}\_{p^n}$. The group of roots of unity in $W\mathbb{F}\_{p^n}$ is cyclic of order $p^n-1$, with generator $\omega$ say, and maps by an isomorphism to $\mathbb{F}\_{... | 2 | https://mathoverflow.net/users/10366 | 435512 | 176,043 |
https://mathoverflow.net/questions/435506 | 2 | If $X$ is any set, we let $[X]^2:=\big\{\{x,y\}:x\neq y\in X\big\}$. If $G=(V,E)$ is an undirected graph and $v\in V$, we define $N\_G(v) = \{w\in V:\{v,w\}\in E\}$.
For $i =1,2$, let $G\_i=(V\_i,E\_i)$ be non-empty, finite, simple, undirected graphs with $V\_1\cap V\_2 = \emptyset$. Suppose that $E \subseteq [V\_1\c... | https://mathoverflow.net/users/8628 | "Combined" chromatic number of $2$ graphs glued together with $2$ edges per vertex | This is not true even for $k=1$. Assume that $(V\_1,E\_1)$ is a complete $d$-partite graph with each part having $d$ vertices, and so is $(V\_2,E\_2)$. Take a bijection $f\colon V\_1\to V\_2$ and add edges between $v$ and $f(v)$ for all $d^2$ vertices $v\in V\_1$. $f$ is chosen so that for each part in $V\_1$, its vert... | 5 | https://mathoverflow.net/users/4312 | 435515 | 176,045 |
https://mathoverflow.net/questions/435496 | 1 | Suppose you're a shopkeeper in the business of selling Items. An "Item" is a thing whose only property is that the quantity that can be bought by a purchaser must be a positive integer; all Items are identical.
A book whose identity I do not recall (I'll post it here maybe tomorrow?) seems to hold that the way to mod... | https://mathoverflow.net/users/6316 | Why should the logarithmic series distribution model the number of "Items" bought? | $\newcommand\la\lambda\newcommand\La\Lambda\newcommand{\Ga}{\Gamma}\newcommand{\R}{\mathbb R}$**Some preliminaries:** To make it explicit that the distribution of $Y$ depends on the parameter $r$, let $Y\_r:=Y$, so that
\begin{equation\*}
\Pr(Y\_r=j) = \binom{-r}j (-q)^j p^r \tag{1}\label{1}
\end{equation\*}
for $j=0,1... | 4 | https://mathoverflow.net/users/36721 | 435530 | 176,049 |
https://mathoverflow.net/questions/428913 | 1 | I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function $\omega(N)$. Wikipedia has this section dedicated to near-square primes from the article [*Landau's problems.*](https... | https://mathoverflow.net/users/142929 | Number of distinct near-squares primes dividing an odd perfect number | In general, very few prime factors in an odd perfect number can be of the form $n^2+1$.
In particular, if N is an odd perfect number then $\frac{\sigma(N)}{N}=2$, and for any $m$ (perfect or not), $\frac{\sigma(n)}{n} \leq \prod\_{p|n}\frac{p}{p-1}$ with equality if and only if $n=1$.
Thus, for any perfect number, ... | 3 | https://mathoverflow.net/users/127690 | 435552 | 176,060 |
https://mathoverflow.net/questions/435500 | 1 | The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction map $H^2(X,\mathbb{G}\_m) \rightarrow H^2(X-Z,\mathbb{G}\_m)$ is an isomorphism. This means that the result cannot be ex... | https://mathoverflow.net/users/172132 | Counterexample to purity of Brauer group for curves | The results of [this answer](https://mathoverflow.net/a/380825/82179) carry over (mutatis mutandis) to curves over finite fields. This shows that if $C$ is a (smooth, geometrically integral) curve over $\mathbf F\_q$ with function field $K$, then there is an exact sequence
$$0 \to \operatorname{Br}(C) \to \operatorname... | 1 | https://mathoverflow.net/users/82179 | 435560 | 176,064 |
https://mathoverflow.net/questions/435556 | 1 | Recently, I came across the notion of [quasi-isometries](https://en.wikipedia.org/wiki/Quasi-isometry), while thinking of *"discrete* spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are preserved by quasi-isometries? Also, are there good references on the topic?
--... | https://mathoverflow.net/users/36886 | What properties are preserved by quasi-isometries | One can think of quasi-isometric spaces as spaces which look the same when seen from far away. Examples of properties preserved under quasi-isometries are for example Gromov-hyperbolicity (for geodesic metric spaces), growth types of Dehn functions and various notions of "rank". As a reference I would recommend Buyalo-... | 2 | https://mathoverflow.net/users/313861 | 435561 | 176,065 |
https://mathoverflow.net/questions/434711 | 1 | Given $X$=$l^1$ and its dual space $X^\*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^\*$. Then clearly $\|f\|\_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$.
Note: A subspace $Y$ of a normed linear space $X$ is called proximinal if for all $x \in X$, $P\_Y(x) \neq \emptyset$, whe... | https://mathoverflow.net/users/494605 | Finding the set of best approximation | Similar to $P\_Y(x)$, there is no such ready formula for evaluating $P\_{B\_Y}(x)$, when $Y=\ker (f)$, and so is for $d(x,B\_Y)$. In some cases, for instance when $d(x,Y)=d(x,B\_Y)$, it is easier to understand $P\_{B\_Y}(x)$. In fact in few cases one can write $P\_{B\_Y}(x)=B\_Y\cap P\_Y(x)$, when $d(x,Y)=d(x,B\_Y)$.
... | 0 | https://mathoverflow.net/users/76412 | 435574 | 176,068 |
https://mathoverflow.net/questions/435573 | 2 | Let $f \in W^{1,p}(U)$, then how to prove that $|f| \in W^{1,p}(U)$, where $W$ means the sobolev space over some open subset $U \in \mathbb{R}^n$.
In Lieb's Analysis he prove that Let $f$ be in $W^{1, p}(\Omega)$. Then the absolute value of $f$, denoted by $|f|$ and defined by $|f|(x)=|f(x)|$, is in $W^{1, p}(\Omega)... | https://mathoverflow.net/users/494763 | Derivative of the absolute value | Part 1 follows from the Cauchy-Schwartz inequality, applied to the two vectors $(R(x), I(x))$, $(\nabla R(x), \nabla I(x))$.
Part 2 follows from the simple inequality $|\partial\_j R(x)| \leq \sqrt {\sum\_i (\partial\_i R(x))^2 } = |\nabla R(x)|$ for all $j$ (and similarly for $I$).
I am not sure about part 3 mysel... | 2 | https://mathoverflow.net/users/173490 | 435580 | 176,071 |
https://mathoverflow.net/questions/311877 | 18 | I consider *convex polytopes* $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-edge pairs). Especially, this includes that the polytope is vertex- and edge-transitive. The *graph* of a polytope is the... | https://mathoverflow.net/users/108884 | Can the graph of a symmetric polytope have more symmetries than the polytope itself? | The answer is *No*, the graph of an arc-transitive polytope cannot have more symmetries than the polytope.
The polytope and its graph have the exact same symmetries!
---
In the article [*"Capturing Polytopal Symmetries by Coloring the Edge-Graph"*](https://arxiv.org/abs/2108.13483) I prove the following:
>
> ... | 5 | https://mathoverflow.net/users/108884 | 435585 | 176,073 |
https://mathoverflow.net/questions/435471 | 1 | Let $(E, d)$ be a Polish space and $\mathcal C\_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P\_p (E)$ the space of all Borel probability measures on $E$ with finite $p$-th moments. We define the [Wasserstein metric](https://en.wikipedia.org/wiki/Wasserst... | https://mathoverflow.net/users/99469 | Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$? | A topology generated by countably many point-separating real functions [is metrizable](https://mathoverflow.net/a/435477/35357). To apply this here, it suffices to show that there is a countable family $\mathcal{G}$ of bounded real functions on $E$ such that the topology of weak convergence of measures is generated by ... | 1 | https://mathoverflow.net/users/35357 | 435604 | 176,077 |
https://mathoverflow.net/questions/435609 | 3 | I have a math/stat problem where I need to show the convergence of the average of a sequence of experiments to an interval. The sequence of experiments is not i.i.d., hence the standard law of large number does not apply. However, the framework satisfies some assumptions which might facilitate the convergence proof. I ... | https://mathoverflow.net/users/42412 | A convergence problem | I assume that $(a\_n)\_{n \ge 1}$ are random variables taking values on a finite subset $B$, and that $\nu\_l(b) \le P[a\_n = b|a\_1,\ldots,a\_{n-1}] \le \nu\_u(B)$ almost surely for every $n \ge 1$ and $b \in B$.
If yes, then for each $b \in B$, the formula
$$M\_n(b) := \sum\_{k=1}^n\frac{1}{k}\big(1\_{[a\_k=b]}-P[a... | 4 | https://mathoverflow.net/users/169474 | 435611 | 176,079 |
https://mathoverflow.net/questions/435617 | 4 | Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ be the projective bundle associated to $\mathcal{E}$. Is it true that if the structural map $\mathbf{P}(\mathcal{E})\to ... | https://mathoverflow.net/users/14143 | Basic question on projective bundles | This can fail when $\mathscr E$ has torsion:
**Example.** Let $k$ be a field, let $R = k[t]$ with maximal ideal $\mathfrak m = (t)$, and let $S = \operatorname{Spec} R$. Take $\mathscr E = R\oplus R/\mathfrak m = Rx \oplus Ry/(ty)$. Then
$$\operatorname{Sym}^\*(\mathscr E) = R[x,y]/(ty),$$
so taking $\mathbf{Proj}$ g... | 8 | https://mathoverflow.net/users/82179 | 435619 | 176,080 |
https://mathoverflow.net/questions/435628 | 1 | Let $k \in \mathbb{Z}^+$.
Is it possible to prove that, for some given
$m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$,
there are only finitely many $k$ such that the closed interval $[k, k+99]$ contains (exactly) $m$ prime numbers?
Backstory: years ago, we wrote an informative article, i... | https://mathoverflow.net/users/481829 | Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers | It is compatible with current knowledge that all prime gaps are greater than 100\*, so it's not possible currently to show that for any of m in {2, ..., 23} that there are infinitely many such intervals (though it is surely true). Cases 0 and 1 follow from PNT (or even just Chebyshev's theorem on prime density) and Dir... | 3 | https://mathoverflow.net/users/6043 | 435632 | 176,086 |
https://mathoverflow.net/questions/435636 | 0 | Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert\_\infty)$. Let $\mathcal{B}\_1 = \mathcal{B}\_{\Vert\cdot\Vert\_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some norm $\Vert\cdot\Vert\_K$. Denote the $\varepsilon$-covering number of $\mathcal{B}\_1$ with respect to $\Vert\cdot\Ver... | https://mathoverflow.net/users/495129 | Right-continuity of covering number | Without additional assumptions on the metric space, it may appear that for every $\varepsilon>1$ the covering number equals 1, but for $\varepsilon=1$ it is infinite. For example, let positive integers be the points and the distance between $n$ and $m>n$ be equal $1+1/m$.
For compact metric space, it is right-continu... | 1 | https://mathoverflow.net/users/4312 | 435638 | 176,089 |
https://mathoverflow.net/questions/434685 | 2 | Let $a,b$ be two positive integers. Let the sequence $\{s\_n\}\_n$ be the set of all possible sums of squares $a^2+b^2$, such that they are in *ascending order*
\begin{align\*}
& n=1 & s\_1=1^2+1^2=2 \\
& n=2 & s\_2=1^2+2^2=5 \\
& n=3 & s\_3=2^2+1^2=5 \\
& n=4 & s\_4=2^2+2^2=8
\end{align\*}
etc. Here $n$ serves as a co... | https://mathoverflow.net/users/148012 | Asymptotic analysis of a peculiar sum of squares sequence | The answer has been provided by [Noam D. Elkies](https://mathoverflow.net/users/14830/noam-d-elkies), for which I'll copy here his comment.
Usually this is studied in the nearly equivalent form of asking for the number of $(x,y)$ such that $x^2+y^2\le N$.
The answer is asymptotic to the area of this circle, which i... | 0 | https://mathoverflow.net/users/148012 | 435643 | 176,091 |
https://mathoverflow.net/questions/435644 | 26 | Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized as being inefficient or roundabout (see, e.g., the question [“Why isn't integral defined as the area under the graph of f... | https://mathoverflow.net/users/17064 | What is the origin/history of the following very short definition of the Lebesgue integral? | This definition is due to Jan Mikusiński, see Mikusiński, Jan,
The Bochner integral. Basel, Stuttgart: Birkhauser, 1978.
Mikusiński has co-authored another book on integration with Hartman in 1961, where a standard exposition of Lebesgue integration is given. So we may infer that Mikusiński's definition was invented ... | 21 | https://mathoverflow.net/users/56624 | 435651 | 176,092 |
https://mathoverflow.net/questions/435479 | 5 | $\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I am looking to classify the homomorphisms from the group $\PSL(2,p)=\Aut(\mathbb{P}^1\mathbb{F}\_p)$ to $\PGL(n,2)=\GL(n,2)=\Aut(\mathbb{F}\_{2^n})$ when $p$ is a Mersenne prime, i.e. $2^n=p+1$.
There ... | https://mathoverflow.net/users/15133 | How to classify homomorphisms from $\operatorname{PSL}(2,p)$ to $\operatorname{PGL}(n,2)$ when $2^n=p+1$? | The map $T \colon \mathrm{PSL}\_2(p) \to \operatorname{Sym}(\mathbb{F}\_{2^n}) \colon f \mapsto T\_f$ does not have its image in $\mathrm{GL}\_n(2)$ for other Mersenne primes $p = 2^n - 1$, unlike the case $p = 7 = 2^3 - 1$.
For instance, let $p = 31$ and consider $\mathbb{F}\_{32} \cong \mathbb{F}\_2[x] / (g)$ with ... | 3 | https://mathoverflow.net/users/12858 | 435664 | 176,097 |
https://mathoverflow.net/questions/435662 | 7 | Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
* If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^1$. In particular $C(k)$ is infinite.
* If $g(C) = 1$ and $C(k) \neq \emptyset$, then $C(k)$ is a finitely generated ... | https://mathoverflow.net/users/5101 | Rational points on regular curves over global fields | *If the (geometrically integral, projective) curve $C$ over the global field $k$ is regular but not smooth over $k$, then $C(k)$ is finite*:
First of all $C$ is smooth over $k$ if and only if also the base change $C\_{\bar{k}}$ is regular.
This is equivalent to $C$ being conservative, i.e. $g(C)=g(C\_{\bar{k}})$, whe... | 7 | https://mathoverflow.net/users/50351 | 435667 | 176,098 |
https://mathoverflow.net/questions/375655 | -1 | Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$.
The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x\_u\oplus x\_s$ where $x\_u$ and $x\_s$ are unitary and unilateral shift respectively. Indeed $x\_u$ is the restriction of $x$ to $H\_u=\big... | https://mathoverflow.net/users/84390 | A commuting pair of isometries | Such a pair $(X,Y)$ is constructed as follows. Consider a Hilbert space $M$ with an orthonormal basis $\{e\_n:n\in\mathbb Z\}$ and the bilateral shift $U$ on $M$ such that $Ue\_n=e\_{n+1}$. Denote by $S$ the restriction of $U$ to the space $N$ generated by $\{e\_n:n\geq 0\}$, and construct the space $H=N\oplus M\oplus ... | 1 | https://mathoverflow.net/users/12465 | 435671 | 176,099 |
https://mathoverflow.net/questions/435598 | 1 | I have a cube $X\in \mathbb R^{N\times N\times N}$ such that no matter how the cube is rotated by $90^\circ$ along any of the axes, the result is unchanged. What is the maximum number of distinct entries in the cube if $N$ is odd?
Here is my answer. I am sure that it is less than or equal to the true answer.
Let $n... | https://mathoverflow.net/users/495707 | Number of distinct entries in a rotation invariant cube | Since rotations give only even permutations, the 3 distinct index triples $\{a,b,c\}$ with $a \neq b \neq c \neq a$ give twice as much distinct entries as you gave: entries at $(a,b,c)$ and $(b,a,c)$ may be distinct in this case. This adds $n$ choose $3$ distinct entries.
However, for entries $\{a,b,n\}$ a 90 degree ... | 1 | https://mathoverflow.net/users/490128 | 435680 | 176,101 |
https://mathoverflow.net/questions/335437 | 6 | A famous result of Sullivan (closely related to work of Wilkerson) says that the group of isotopy classes of diffeomorphisms of a simply-connected closed smooth manifold of dimension $\geq 5$ is commensurable with an arithmetic group. Sullivan gave an outline of its proof, and several later papers filled in more detail... | https://mathoverflow.net/users/798 | A result of Borel on extensions of arithmetic groups | A proof appears in Section 2.3 of my paper [Mapping class groups of manifolds with boundary are of finite type](https://arxiv.org/abs/2204.01945). As pointed out in the comments, it is crucial that $G$ is unipotent.
| 1 | https://mathoverflow.net/users/798 | 435683 | 176,102 |
https://mathoverflow.net/questions/376729 | 9 | Derived mapping spaces between little $d$-disks operads $E\_d$ play an important role in embedding calculus. For example, [Dwyer-Hess](https://projecteuclid.org/euclid.gt/1513732411) expresses the homotopy of framed long knots as loop spaces such mapping spaces, a result which was generalized by [Boavida de Brito-Weiss... | https://mathoverflow.net/users/798 | Maps between unitary little disks operads and non-unitary little disks operads | A positive answer is the main theorem of my paper with Krannich and Horel, [Two remarks on spaces of maps between operads of little cubes](https://arxiv.org/abs/2211.00908). The proof uses a result of Haugseng and Kock to reduce it to a theorem of Lurie.
| 3 | https://mathoverflow.net/users/798 | 435684 | 176,103 |
https://mathoverflow.net/questions/435647 | 2 | I have been looking into bootstrapping lately and although I believe to have understood the basic process somewhat, I am fuzzy on the mathematical details. I will begin with my understanding of what bootstrapping is and then my understanding of the mathematics going on in the background. I might very well be mistaken o... | https://mathoverflow.net/users/151546 | Bootstrapping and the central limit theorem | $\newcommand{\X}{\mathbf X}\newcommand{\x}{\mathbf x}\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Your understanding of the purposes of the bootstrap is largely incorrect.
Here is what the [foundational paper by Efron](https://projecteuclid.org/journals/annals-of-statistics/volume-7/issue-1/Bootstrap-Methods-An... | 1 | https://mathoverflow.net/users/36721 | 435687 | 176,105 |
https://mathoverflow.net/questions/435679 | 4 | The motivation for this question comes from Theorem 3.3 of the 1995 paper [*Tilings of Triangles*](https://core.ac.uk/download/pdf/82351337.pdf) by M. Laczkovich, which states:
>
> Let $x$ and $y$ be non-zero integers such that $x+2y\neq 0\neq y+2x$. Then there is a positive integer $k$ such that the equilateral tr... | https://mathoverflow.net/users/89672 | Squarefree parts of integers of the form $xy(x+2y)(y+2x)$ | If a given squarefree integer $s$ is given, determining whether there is a pair of integers $(x,y)$ for which $s(x,y) = s$ is a problem about rational points on elliptic curves. In particular, there is a bijection to solutions to
$$
xy(x+2y)(y+2x) = sz^{2}
$$
and rational points on the elliptic curve $E\_{s} : Y^{2} =... | 7 | https://mathoverflow.net/users/48142 | 435688 | 176,106 |
https://mathoverflow.net/questions/435589 | 1 | This is a follow-up to [this question of mine](https://mathoverflow.net/questions/428242/isometric-embeddings-of-c-0-into-metric-spaces).
It is well-known that the Banach space $\ell\_1$ does not contain any *isomorphic* copies of $c\_0$. One can even go further and show that $\ell\_1$ does not contain any *bilipschi... | https://mathoverflow.net/users/15860 | Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$ | Yes, this is known. Raynaud showed that $B\_{c\_0}$ does not uniformly (in particular, bilipschitz) embed into any stable Banach space. $\ell\_1$ is stable.
* [Yves Raynaud, Espaces de Banach superstables, distances stables et homeomorphismes uniformes](https://link.springer.com/article/10.1007/BF02763170)
**Added ... | 7 | https://mathoverflow.net/users/3675 | 435692 | 176,108 |
https://mathoverflow.net/questions/435690 | 2 | $\DeclareMathOperator\GL{GL}$Let $G$ be a $2$-generator pro-$p$-group of finite rank, i.e. it is isomorphic to a closed subgroup of $\GL\_d(\mathbb{Z}\_p)$ for some integer $d$. Assume that $G$ is torsion-free. Recall that the dimension $\dim(G)$ of $G$ as a $p$-adic analytic group can be described as $d(U)$ where $U$ ... | https://mathoverflow.net/users/492970 | The dimension of a torsion-free $p$-adic analytic group generated by two generators | A pro-$p$ group $G$ has $\textit{lower rank}$ $r$ if $r$ is minimal such that every open subgroup of $G$ contains an open subgroup generated by at most $r$ elements. Lubotzky and Mann showed that the lower rank of a $p$-adic analytic pro-$p$ group is the number of generators of its associated Lie algebra. On the other ... | 3 | https://mathoverflow.net/users/5034 | 435696 | 176,109 |
https://mathoverflow.net/questions/435705 | 1 | I am looking for a formula giving the asymptotic expansion of the renewal equation when there is exponential growth (for lack of better terms).
Consider the renewal equation, for an unknown $M(t)$:
\begin{equation}
M(t)=a\int\_0^\infty M(t-\tau)f(\tau)\mathrm{d}\tau
\end{equation}
where $f(\tau)$ is a probability den... | https://mathoverflow.net/users/420641 | Asymptotic expansion of the renewal function for an exponential growing population | For any $\theta>0$, if you substitute $M(t)=e^{\theta t}N(t),$ then the equation becomes
$$
N(t)=\int\_0^\infty N(t-\tau)ae^{-\theta\tau}f(\tau)d\tau.
$$
If $a>1$, then there is, by continuity, exactly one $\theta$ such that $ae^{-\theta \tau}f(\tau)$ is a probability density. For this $\theta$, the problem is reduced... | 0 | https://mathoverflow.net/users/56624 | 435712 | 176,112 |
https://mathoverflow.net/questions/424595 | 5 | I'm reading Casselman's notes "[Introduction to the theory of admissible representations of p-adic reductive groups](https://personal.math.ubc.ca/%7Ecass/research/pdf/p-adic-book.pdf)". In chapter 4 "The asymptotic behavior of matrix coefficients", the main result of this chapter states that for some torus elements $a$... | https://mathoverflow.net/users/123363 | Asymptotic behavior of matrix coefficients | For the $p$-adic case, the idea is as follows (thanks to [Elad](https://mathoverflow.net/users/103908/darkl) for pointing out the direction).
Recall (from the unpublished notes by Casselman, [*Introduction to the theory of admissible representations of p-adic reductive groups*](https://personal.math.ubc.ca/%7Ecass/rese... | 2 | https://mathoverflow.net/users/123363 | 435724 | 176,117 |
https://mathoverflow.net/questions/435723 | 3 | The [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) appears in the Sokhotsky formula,
$$\text{Im}\lim\_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$
to be understood in the integral sense
$$\text{Im}\lim\_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dy=\pi f(x),$$
for a real v... | https://mathoverflow.net/users/11260 | Representation of the Dirac delta function | As usual in such examples, there is no need to integrate against a test function. One can simply use the fact that if a sequence (or net) of distributions converges in the distributional sense, then so does the one obtained by differentiating term by term. In particular, this applies when the sequence consists of funct... | 12 | https://mathoverflow.net/users/495655 | 435730 | 176,120 |
https://mathoverflow.net/questions/435729 | 1 | The sum I am looking for is the following sum as $M \to \infty$:
$$ L(\omega) = \sum\_{m = 1}^{M} \frac{\sin\left( N \frac{\omega\_m - \omega}{2} \right)}{\sin\left( \frac{\omega\_m - \omega}{2} \right)} \cos\left(N \frac{\omega\_m - \omega}{2} + \beta\_m \right) $$
where
* $\omega\_m$ is a random number from a G... | https://mathoverflow.net/users/489481 | Is it possible to sum this analytically in any way? | Since the average over $\beta$ will give a vanishing expectation value of $L$, let me omit it for now and set $\beta=0$. I will also simplify the question by setting $\omega=\mu=0$ and $\sigma=1$. Then the expectation value of $L$ has a compact expression
$$\mathbb{E}[L(0)]=M\;\int\_{-\infty}^\infty \frac{dx}{\sqrt{2\p... | 2 | https://mathoverflow.net/users/11260 | 435734 | 176,123 |
https://mathoverflow.net/questions/435704 | 0 | Given the function
$$
E(M) = \sum\_{i=1}^N \sum\_{a=1}^K \left( M\_{ia} \cdot \left\lVert\sum\_{i=1}^N M\_{ia}\cdot x\_i\right\rVert\_2^2 \right)
$$
$x$ is a given constant matrix, $x\_i$ is a the $n\_\text{th}$ column of that.
$M\_{ia} \in \{0,1\}$ and $\sum\_{i=1}^N M\_{ia}=\frac{N}{K} $, and we also have $\sum\_{a=1... | https://mathoverflow.net/users/495775 | Can this function be simplified to use only quadratic, linear terms of M with given conditions? | Yes, $$E(M) = \frac{N}{K} \lVert X M \rVert\_F^2$$
writing $X$ for the matrix $\begin{bmatrix} x\_1 & \dots & x\_N \end{bmatrix}$.
Note $\sum\_{i=1}^N M\_{ia}x\_i = X M e\_a$ where $e\_a$ is the $a$th standard basis vector. The constraints imply $M$ is a binary $N \times K$ matrix with a single nonzero per row and each... | 0 | https://mathoverflow.net/users/70005 | 435740 | 176,128 |
https://mathoverflow.net/questions/414286 | 3 | I find the $d$-separation criterion (see, e.g., [Theorem 2 here](https://ftp.cs.ucla.edu/pub/stat_ser/r130-reprint.pdf); note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient criterion for conditional independence of random variables. However,... | https://mathoverflow.net/users/106046 | General version of $d$-separation | Yes, there are more general versions of the $d$-separation criterion, in particular also for standard Borel spaces (i.e. Polish spaces).
For completeness, the classical version of $d$-separation (i.e. on discrete random variables) has been generalized several times to the best of my knowledge.
### 1. $d$-separation... | 2 | https://mathoverflow.net/users/493730 | 435741 | 176,129 |
https://mathoverflow.net/questions/435685 | 5 | Let $A$ be a homotopy ring spectrum. Then the homology theory $A\_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi\_\ast A)$. If $A$ is homotopy commutative, then this functor $A\_\ast : Spectra \to GrMod(\pi\_\ast A)$ is lax monoidal. But it makes sense to ask for a lax monoidal structure on th... | https://mathoverflow.net/users/2362 | If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor? | As pointed out in the comments, the functor $A\_\*$ cannot in general be lax symmetric monoidal without making some alterations.
Here is an incomplete discussion of when $A\_\*$ can be lax monoidal.
The first observation is that, for any homotopy associative ring spectrum $A$, the functor $A\_\*$ naturally takes va... | 6 | https://mathoverflow.net/users/360 | 435747 | 176,131 |
https://mathoverflow.net/questions/435715 | 3 | Let us consider a Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $L\mathfrak{g} = C^\infty(S^1, \mathfrak{g})$ the Lie algebra of the loop group $LG$.
My question is about continuous Lie algebra 2-cocycles on $L\mathfrak{g}$.
It is well-known (see, e.g., Prop. 4.2.4 in Pressley-Segal "Loop groups") that if $G$... | https://mathoverflow.net/users/16702 | Non-invariant forms on loop Lie algebra of semisimple Lie group | Thanks to Yves Cornulier, for suggesting to look at the paper of Neeb and Wagemann. After reading Example 6.2 of that paper (arxiv version), I think the answer to my question is that in fact all 2-cycles have a representative of the form in my original post. In other words, the assumption of $G$-invariance in the propo... | 1 | https://mathoverflow.net/users/16702 | 435748 | 176,132 |
https://mathoverflow.net/questions/433646 | 3 | Consider a simple closed curve $\gamma$ in $\mathbb R^3$. Suppose that $\gamma$ has length $\ell$ and contains a line segment $s$ of length $k<\ell/2$. Let $\Sigma$ be a surface with boundary $\gamma$ of minimum area. I'm interested in knowing what such curve $\gamma$ causes the area of $\Sigma$ to be largest possible.... | https://mathoverflow.net/users/493914 | Isoperimetric inequality for minimal surfaces bounded by space curves containing a line segment | The area minimizing disc $\Sigma$ with boundary $\gamma$ has nonpositive curvature in the sense of Alexandrov.
Applying Reshetnyak majorization theorem (see [9.56](https://arxiv.org/pdf/1903.08539v5.pdf)), we get a convex plane figure $F$ and a length-nonincreasing map $m\colon F\to \Sigma$ such that the restriction $m... | 3 | https://mathoverflow.net/users/1441 | 435753 | 176,133 |
https://mathoverflow.net/questions/435728 | 4 | In a paper that I am reading the author quotes the following result about harmonic functions. According to him this should be "easy to show" but I don't seem to be able to do so.
Let $u:\overline{B^n}\to \mathbb{R}$ be harmonic, where $\overline{B^n}\subset\mathbb{R}^n$ is the closed unit ball. I would like to prove ... | https://mathoverflow.net/users/351083 | An inequality for harmonic functions | Consider first the $d=2$ case. Then, $u$ is a real part of an analytic function. We can write $$u(z)=\frac12\sum\_{n=0}^{\infty}(a\_nz^n+\overline{a}\_n\overline{z}^n)$$ and $$\partial\_\nu u(z)=\frac12\sum\_{n=0}^{\infty}(a\_nnz^n+\overline{a}\_nn\overline{z}^n)$$ for $|z|=1$. When multiplying them out and integrating... | 3 | https://mathoverflow.net/users/56624 | 435757 | 176,134 |
https://mathoverflow.net/questions/435642 | 15 | Lately I have become interested in solid $F$-modules where $F$ is some discrete field. Ideally, one would want a category that is as nicely behaved as solid abelian groups or solid $\mathbb{F\_p}$-modules.
I have managed to show that a discrete field is a solid module over itself, so we get that the usual $\prod\_I F... | https://mathoverflow.net/users/170467 | Is there a good theory of solid vector spaces? | I will prove that the result is true if $F$ is a finitely generated field, but fails if $F$ is countably generated field that is not finitely generated.
Let me first discuss the case $F=\mathbb Q$. For $F=\mathbb Q$, one has the idempotent solid $\mathbb Z$-algebra $\hat{\mathbb Z}=\mathrm{lim}\_n \mathbb Z/n\mathbb ... | 12 | https://mathoverflow.net/users/6074 | 435760 | 176,136 |
https://mathoverflow.net/questions/435774 | 3 | Recall a prior posting titled [Is there an effective way to generalize this approach of affinely extending the number line?](https://mathoverflow.net/questions/435272/is-there-an-effective-way-to-generalize-this-approach-of-affinely-extending-the), and especially the accepted [answer](https://mathoverflow.net/a/435273/... | https://mathoverflow.net/users/95347 | Can we always know if an algebraic rule over the reals is preserved over the extended reals or not? | Of course this depends on the signature (= choice of basic functions) used, but for a wide variety of such the answer will be **yes**. This is because $(\hat{\mathbb{R}};\overline{\hat{F}})$ is interpretable in $({\mathbb{R}};\overline{{F}})$ for any tuple of functions $\overline{F}$ containing at least $+$ and $\times... | 4 | https://mathoverflow.net/users/8133 | 435775 | 176,140 |
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