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https://mathoverflow.net/questions/427057 | 1 | Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that
\begin{align}
u \notin C^{as} \text{ and }(-u) & \in \text{Int}\big(C^{as}\big).
\end{align}
For $x\in C$, let $p(x)$ be the pr... | https://mathoverflow.net/users/159940 | Lipschitz aspect of a projection on the boundary of a convex | $\newcommand{\ep}{\varepsilon}\newcommand{\R}{\mathbb R}\newcommand{\epi}{\operatorname{epi}}$Yes, $p$ is Lipschitz.
Indeed,
\begin{equation\*}
p(x)=x+f(x)u
\end{equation\*}
for $x\in C$, where
\begin{equation\*}
f(x):=\inf E\_x,\quad E\_x:=\{t\ge0\colon x+tu\notin C\}.
\end{equation\*}
So, it suffices to show t... | 2 | https://mathoverflow.net/users/36721 | 427913 | 173,530 |
https://mathoverflow.net/questions/427916 | 0 | Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?
Definition of homologically locally connected:
"A space $X$ is said to be homologically locally connected in dimension $n$ if for every $x \in X$ and neigh... | https://mathoverflow.net/users/489082 | Does contractible imply homologically locally connected? | Recall that $E$, the *earring space*, is the union in the plane of a countable collection of shrinking circles, all tangent to the $y$ axis at the origin. To obtain the desired counterexample, form the cone on $E$.
---
**Edit:** My example just above is very similar to that of Henrik Rüping (in the comments) and ... | 2 | https://mathoverflow.net/users/1650 | 427926 | 173,535 |
https://mathoverflow.net/questions/427899 | 1 | Let $\{ X\_n(\omega,x)\}\_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ergodic measure which is full supported.
I would like to estimate $\mu(C)$, where $C \subset \mathbb{S}^1$, is a particul... | https://mathoverflow.net/users/480325 | Connection between invariant measure and positive recurrence for continuum state space markov chain | Yes, let's assume that sup (which I think should be in the denominator in the last expression) is finite, as otherwise it is always true. Suppose it is bounded by A, the each return time is also quite finite, as by chebyshev inequality $P(T > 2A) < \frac 1 2$, and using the markov property this makes each return time s... | 0 | https://mathoverflow.net/users/143907 | 427929 | 173,536 |
https://mathoverflow.net/questions/427924 | 1 | It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows
$$E\_{s}(\tau) = \sum\limits\_{\gamma\in\Gamma\_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^{s}\right\vert\_{\gamma},$$
where $s\in\mathbb{C}$ and $\Gamma\_{\infty}$ is the stabilizer of $\text{SL}(2,\mathbb{... | https://mathoverflow.net/users/99716 | Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups | One natural generalization of $E\_s$ to $\Gamma\_0(N) \backslash \mathbb{H}$ is the family of Eisenstein series, $$E\_{s; \chi\_1, \chi\_2}(\tau) = \sum\_{\gamma \in \Gamma\_{\infty} \backslash \mathrm{SL}\_2(\mathbb{Z})} \chi\_1(c) \chi\_2(d) (q\_2 y)^s \Big| \gamma, \quad \gamma = \begin{pmatrix} a & b \\ c & d \end{... | 4 | https://mathoverflow.net/users/489142 | 427930 | 173,537 |
https://mathoverflow.net/questions/427934 | 4 | Below is a simple determinant. I need to show that it is not 0, so that the corresponding matrix is invertible.
$$
D = \begin{vmatrix}
0! & 1! & 2! & \ldots & x!\\
1! & 2! & 3! & \ldots & (x+1)! \\
2! & 3! & 4! & \ldots & (x+2)! \\
\vdots & \vdots & \vdots & \ldots & \vdots \\
y! & (y+1)! & (y+2)! & \ldots & (x+... | https://mathoverflow.net/users/85939 | Determinant with factorials is not 0? | This is the [Hankel determinant](https://en.wikipedia.org/wiki/Hankel_matrix) associated to the sequence $m\_n = \mathbb{E}(X^n) = n!$ of moments of an exponential distribution with mean $1$. Some general results can be used to show that the sequence of Hankel determinants associated to the moments of a random variable... | 10 | https://mathoverflow.net/users/290 | 427939 | 173,541 |
https://mathoverflow.net/questions/427561 | 3 | Say that a logic $\mathcal{L}$ is **nowhere-negative** iff for every $\mathcal{L}$-theory $T$ there is a structure $\mathfrak{A}$ such that $$\mathit{Th}\_\mathcal{L}(\mathfrak{A})=\mathit{Ded}\_\mathcal{L}(T),$$ where $\mathit{Th}\_\mathcal{L}(\mathfrak{A})=\{\varphi\in\mathcal{L}:\mathfrak{A}\models\_\mathcal{L}\varp... | https://mathoverflow.net/users/8133 | Is there a maximal fragment of FOL with "no negation at all?" | There is a simple characterization that $\def\LL{\mathcal L}\LL$ is nowhere negative iff all $\LL$-theories have the *disjunction property*:
>
> **Proposition.** For any $\LL\subseteq\mathrm{FO}$, the following are equivalent:
>
>
> 1. $\LL$ is nowhere negative.
> 2. For every $\LL$-theory $T$ and every finite se... | 6 | https://mathoverflow.net/users/12705 | 427946 | 173,543 |
https://mathoverflow.net/questions/427965 | 2 | Let $M$ a non-well founded model of Finite $\sf ZF$, which is $\sf ZF$ with axiom of infinity replaced by the axiom stating that all sets are finite. So there must be a set $\zeta$ that $M$ thinks it's a natural yet it possess's an infinite descending membership chain as seen from the outside of $M$, i.e. we have $\{ \... | https://mathoverflow.net/users/95347 | About external automorphism on non-well founded model of Finite ZF? | The answer is no, there cannot be even a single instance of this for an automorphism $j:M\to M$. The reason is that if $j(V\_{n+1})=V\_n$ for some (possibly nonstandard $n$), then we would have $j(n+1)=n$, and so $j$ does not respect that one of these numbers is even and the other odd. So the map couldn't be truth-pres... | 3 | https://mathoverflow.net/users/1946 | 427967 | 173,551 |
https://mathoverflow.net/questions/427970 | 1 | Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}\_q$. There exists a finite field extension $k/\mathbf{F}\_q$ of degree $d$ and an alteration $X'\to X\_k$ defined over $k$, where $X'$ is smooth and projective over $k$.
For $\ell$ a prime not dividing $q$, we have an inject... | https://mathoverflow.net/users/nan | Purity for proper varieties | The claim that alteration induces an injection on cohomology is wrong, as the example of the resolution of a nodal cubic curve by $\mathbb P^1$ shows (resolutions being a special case of alterations).
The induced map from $H^1$ of the nodal cubic to $H^1$ of $\mathbb P^1$ is not an injection, because $H^1$ of $\mathb... | 3 | https://mathoverflow.net/users/18060 | 427971 | 173,552 |
https://mathoverflow.net/questions/427942 | 15 | For an integer $n$, let $\ell(n)$ denote the maximal number of consecutive $1$s in the binary expansion of $n$. For instance,
$$ \ell(71\_{10}) = \ell(1000111\_2) = 3. $$
Consider the set $E$ of all integers $n \in \mathbb{N}$ such that $\ell(n)$ is even.
It seems intuitively obvious that $E$ should have natural dens... | https://mathoverflow.net/users/14988 | The parity of the maximal number of consecutive 1s in the binary expansion of an integer | Perhaps surprisingly, the random variable $\ell(n)$ (with $n$ drawn uniformly from $[0,N)$) concentrates too much around $\log\_2\log\_2 N$ (where $\log\_2$ denotes the logarithm to base $2$) to have a limiting parity probability - the variance stays bounded as $N \to \infty$, as opposed to growing to infinity. One onl... | 25 | https://mathoverflow.net/users/766 | 427983 | 173,556 |
https://mathoverflow.net/questions/427984 | 9 | Classically, given a compact Lie group $G$, there is a topological space $BG$ which classifies principal $G$-bundles. This means that there is an equality of sets {principal $G$-bundles up to isomorphism} = {maps $X \to BG$ up to homotopy}.
**Question:** is there a "infinity categorical" refinement of this statement?... | https://mathoverflow.net/users/396191 | Homotopy coherent generalization of classifying space theory | $\newcommand{\cS}{\mathcal{S}}\newcommand{\Fun}{\mathrm{Fun}}\newcommand{\LMod}{\mathrm{LMod}}\newcommand{\Sp}{\mathrm{Sp}}$Hey Laurent :) Let $X$ be a space (which I'll view as a Kan complex), and let $\cS$ denote the $\infty$-category of spaces. You could think of the homotopy-coherent categorification of "maps $X \t... | 10 | https://mathoverflow.net/users/102390 | 427985 | 173,557 |
https://mathoverflow.net/questions/427849 | 2 | The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1
**1.1. Lemma.**
If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ is any symmetric two tensor, then
$$
\delta\left[\operatorname{DEin}\left({ }^{(4)} g\right) \cdot ^{4} h\right... | https://mathoverflow.net/users/298774 | Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper | I presume the formula you are asking about is the long one highlighed by $\*$ $\*$ in your question, while the standard "contracted Bianchi identity" $\delta \operatorname{Ein}\left({ }^{(4)} g\right)=0$ poses no mystery to you. The Longer equation is simply obtained by using the Leibniz rule while applying the functio... | 1 | https://mathoverflow.net/users/2622 | 427991 | 173,560 |
https://mathoverflow.net/questions/427989 | 7 | Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible 4-manifold. Is there other explicit examples or is there an invariant of homology sphere characterizing this property(wh... | https://mathoverflow.net/users/1190 | Examples of homology sphere that bound a nonsmoothable contractible 4-manifold | There are certainly lots of obstructions coming from gauge theory, for instance Frøyshov's $h$-invariant or Ozsváth and Szabó's $d$-invariant. If an integer homology sphere $P$ has $d(P) \neq 0$, then $P$ does not even bound a *rational* homology ball. There are $\mathbb{Z}/2\mathbb{Z}$ refinements (more precisely, obs... | 5 | https://mathoverflow.net/users/13119 | 428014 | 173,566 |
https://mathoverflow.net/questions/428020 | -3 | Let be a positive integer and
= () be the digit sum of such that
+ 1 ≡ 0 (mod 2).
Is it that if is prime then is also prime?
e.g. =47(prime)-> =4+7=11 (prime)
| https://mathoverflow.net/users/489235 | Digit sum of a prime number | Not necessarily, but it takes a while to see this, because
you're not allowing $q$ to be a multiple of $2$, and it cannot be
a multiple of $3$ (other than $3$ itself) because then the same
would be true of $p$. So the smallest candidate for $q$ is $25$,
which happens for the first time at $p=997$.
| 13 | https://mathoverflow.net/users/14830 | 428021 | 173,568 |
https://mathoverflow.net/questions/427997 | 5 | Let $d \geq 2$, and let $f \in W^{1, 1} (\mathbb R^d)$ be a Sobolev function.
**Question:** For any $a, b \in \mathbb R$ such that $\text{essinf } f \leq a < b \leq \text{esssup } f$, is it true that $\mu\left(f^{-1} ([a, b])\right) > 0$?
*Note:* Here $\mu$ denotes the Lebesgue measure.
| https://mathoverflow.net/users/173490 | Intermediate value property for Sobolev functions | the one dimensional case is clear since $W^{1,1}$ functions have representatives that are absolutely continuous, see [1] Sec. 4.9.1. The general case can be reduced to that case:
Let $a<a\_1<b\_1<b$. Since $A:=\{x \in \mathbb R^d: f(x)<a\_1\}$ has $\mu(A)>0$,
it has a point of density; WLOG that point is the origin. ... | 5 | https://mathoverflow.net/users/7691 | 428035 | 173,572 |
https://mathoverflow.net/questions/428004 | -1 | In the 3-dimensional hyperbolic space there are given a plane $\mathcal{P}$ and four distinct lines $a\_1, a\_2, r\_1, r\_2$ in such positions that $a\_1$ and $a\_2$ are perpendicular to $\mathcal{P}$, $r\_1$ is coplanar with $a\_1, r\_2$ is coplanar with $a\_2$, finally $r\_1$ and $r\_2$ intersect $\mathcal{P}$ at the... | https://mathoverflow.net/users/489216 | 3-dimensional hyperbolic space | Let $L$ be the geodesic segment in $P$ with one endpoint at $a\_1 \cap P$ and the other endpoint at $a\_2 \cap P$. Let $b$ be the midpoint of $L$. We define a plane $Q$ by requiring it to contain $b$ and by requiring it to be perpendicular to $L$ (and thus to $P$). Thus reflection in $Q$ exchanges $a\_1$ and $a\_2$. It... | 0 | https://mathoverflow.net/users/1650 | 428038 | 173,573 |
https://mathoverflow.net/questions/428041 | 3 | The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$.
It is also true that such a $f$ cannot be injective unless $X=[0,1]$ (since then $f$ would be a homeomorphism). I was wondering... | https://mathoverflow.net/users/151406 | Space filling curves | I would call such $f\_\epsilon$ *coarsely $\epsilon$-dense*, or more simply *$\epsilon$-dense*. These exist in the compact case as follows. Tile $M$ by $n$-balls (these may overlap, but only finitely). Each ball is homeomorphic to $[0, 1]^n$, so admits an $\epsilon/2$-dense smooth Hilbert arc. A small perturbation of t... | 4 | https://mathoverflow.net/users/1650 | 428042 | 173,575 |
https://mathoverflow.net/questions/428043 | 2 | The last problem in 2022 IMC Day 1 strongly correlates with graph theory. In its official solution, the fundamental approach can be rephrased as follows.
>
> Give a digraph $G=(V,E)$. We call a subset of $E$ *admissible* such that it doesn't contain any directed paths of length $2$. Let $b(G)$ denote the minimal nu... | https://mathoverflow.net/users/151440 | Minimal digraph covering with no 2-path edge sets is of size $\left( 1 + o \left( 1 \right) \right) \log_2 \chi(G)$ | This is beautiful application of Sperner theorem on antichains. Consider a bijection between the vertices and the family $\binom{[n]}{n/2}$, which realizes the maximum in the mentioned theorem.
Element $i$ from 1 to $n$ are the palette. One may color a vertex in any color containing in the corresponding set. The antich... | 2 | https://mathoverflow.net/users/479618 | 428050 | 173,577 |
https://mathoverflow.net/questions/428045 | 2 | Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a *chain* if for all $A, B\in {\cal C}$ we have $A\subseteq B$ or $B \subseteq A$.
Interestingly, [chains in ${\cal P}(\omega)$ can have length $2^{\aleph\_0}$](https:... | https://mathoverflow.net/users/8628 | Maximal uncountable chains in ${\cal P}(\omega)$ | Any maximal chain in $\mathcal{P}(\omega)$ is determined by (that is, is the unique maximal chain containing) a *countable* subchain; this is basically just a restatement of the fact that $\omega\_1$ doesn't embed into $\mathcal{P}(\omega)$. This gives an upper bound of $2^{\aleph\_0}$ for the number of distinct maxima... | 7 | https://mathoverflow.net/users/8133 | 428051 | 173,578 |
https://mathoverflow.net/questions/427972 | 4 | $\newcommand\norm[1]{\lVert#1\rVert}$For any $p \in [1,2]$, $r \ge 0$, and integer $d \ge 1$, define a mixed-norm $\eta:\mathbb R^d \to \mathbb R$ by $\eta(x) := \norm x\_2 + r\norm x\_p$, for any $x \in \mathbb R^d$.
Do there exist scalars $a = a(d,p,r) \ge 0$ and $q=q(p,r) \in [1,\infty]$ such that $c\_1 \le \dfrac... | https://mathoverflow.net/users/78539 | Is a mixture of $\ell_p$-norms $\eta(x):=\lVert x\rVert_2 + r\lVert x\rVert_p$ always dimensionlessly equivalent to some $\ell_q$-norm? | This is impossible unless the following holds:
>
> $q=2$ and either $p=2$ or $r=0$.
>
>
>
The above exception is due to the fact that it implies $\frac{1}{1+r}\eta(\cdot)=\|\cdot\|\_q=\|\cdot\|\_2$, for which the desired inequality trivially exists.
We argue ad absurdum; hence, suppose the desired inequality... | 2 | https://mathoverflow.net/users/166628 | 428054 | 173,579 |
https://mathoverflow.net/questions/427950 | 1 | I am trying to use the coarea formula to get estimates on the measure of an epsilon-neighbourhood of a set. Specificly, given a compact 'nice' set $A\subseteq \mathbb{R^d}$, possibly with more than one connected components which are not convex, I'm hoping to get an upper estimate on $$\lambda(A^\epsilon)-\lambda(A),$$
... | https://mathoverflow.net/users/143153 | Coarea formula for measure of epsilon neighbourhood | Your formula is true (up to appropriate constants) if you take $\nu$ to be the Minkowski content, assuming that the volume of the tubular neighborhood is locally Lipschitz as a function of the tube radius. This is because the Minkowski content is defined as the derivative of the volume of the tubular neighborhood.
On... | 2 | https://mathoverflow.net/users/112954 | 428061 | 173,582 |
https://mathoverflow.net/questions/428029 | 2 | I asked this question on [Mathematics Stackexchange](https://math.stackexchange.com/questions/4505957/characterization-of-extendible-distributions), but got no answer.
I found the following [question](http://www.cmls.polytechnique.fr/perso/golse/MAT431-10/pc10.pdf) which characterize the extension of a distribution i... | https://mathoverflow.net/users/115618 | Characterization of extendible distributions | $\newcommand{\R}{\mathbb R}$Without loss of generality, $k$ is a positive integer.
First, the "if" part. For $t\in(0,1]$, let
\begin{equation\*}
(Jf)(t):=\int\_t^1 dx\,f(x).
\end{equation\*}
Here it is assumed that
\begin{equation\*}
(Jf)(t)=O(t^{-k})
\end{equation\*}
for $t\in(0,1]$.
As suggested in the comment ... | 2 | https://mathoverflow.net/users/36721 | 428065 | 173,583 |
https://mathoverflow.net/questions/427918 | 13 | It's well known that there are a shocking number of identities for the usual Jacobi theta function $$ \theta\_3(x) = \sum\_{n=-\infty}^{\infty} x^{n^2}. $$
So I wanted to turn my attention to slowly decreasing exponents. If I make my $n$ decay too slow such as $$f(x) = \sum\_{n=1}^{\infty} x^{\log n}, $$
we basical... | https://mathoverflow.net/users/46536 | Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $? | The series
$$\tag{0}\label{0}
G\_a(x)=\sum\_{n=0}^\infty x^{\sqrt[a]{n}}
$$
can be calculated for $0<x<1$ and $a\geq1$ using Cauchy's residue theorem. We use the fact that $\pi \cot(\pi n)$ has residues 1 at $n\in \mathbb Z$, and deform the integration contour to the imaginary axis in the usual way. After the substitut... | 10 | https://mathoverflow.net/users/90413 | 428077 | 173,587 |
https://mathoverflow.net/questions/428026 | 17 | While playing around with the MO question [Determinant with factorials is not 0?](https://mathoverflow.net/questions/427934/determinant-with-factorials-is-not-0) about a determinant of the Hankel matrix of entries $(i+j-2)!$, having the value $\prod\_{k=0}^{n-1}k!^2$, I stumbled on the following.
A permutation $\pi\i... | https://mathoverflow.net/users/66131 | A coincidence or a fact: determinants of two matrices | Orangeskid's guess is correct: a more general fact holds that the binomial transform preserves Hankel determinants.
For a matrix $(a\_{ij})$ (it is convenient to enumerate rows and columns from 0,not from 1) denote $$b\_{ij}=\sum\_{k, s}{i\choose k}{j\choose s}a\_{ks}.$$
This matrix transform corresponds to a left an... | 21 | https://mathoverflow.net/users/4312 | 428093 | 173,592 |
https://mathoverflow.net/questions/428072 | 6 | Let $G$ be a group of order $2^n$. Does $G$ have a normal abelian subgroup of order at least $2^{n/2}$?
(This is true, via computations in GAP, for $n \le 8$.
The question is similar to one posed here: <https://math.stackexchange.com/questions/44275/abelian-subgroups-of-p-groups/44283#44283>
However, that question, a... | https://mathoverflow.net/users/11124 | Is the largest normal abelian subgroup of a finite 2-group $P$ of order at least the square root of the order of $P$? | In
*Alperin, J. L.*, [**Large abelian subgroups of p-groups**](http://dx.doi.org/10.2307/1994193), Trans. Am. Math. Soc. 117, 10-20 (1965). [ZBL0132.27204](https://zbmath.org/?q=an:0132.27204),
the second part of Theorem 1 gives a group of order $2^{50}$ with no abelian subgroups of order greater than $2^{24}$.
| 14 | https://mathoverflow.net/users/22989 | 428096 | 173,593 |
https://mathoverflow.net/questions/427969 | 6 | By a egg-box **diagram** I will simply mean a (possibly infinite) rectangular array of holes, with some of the holes containing an egg (denoted by a filled-in circle) and the rest of the holes are empty (denoted with an empty circle). For instance
$$
\begin{array}{|c|c|c|c|}
\hline
\bullet & \circ & \circ &\circ \\ \hl... | https://mathoverflow.net/users/3199 | Automorphisms of special egg-box diagrams | It turns out (surprisingly) that the answer to my question is yes, and there are even finite examples. I came up with the following diagram
$$
\begin{array}{|c|c|c|c|c|c|}
\hline
\circ & \circ & \bullet & \circ & \bullet & \bullet\\\hline
\circ & \bullet & \circ & \bullet & \circ & \bullet\\\hline
\bullet & \circ & \ci... | 1 | https://mathoverflow.net/users/3199 | 428112 | 173,597 |
https://mathoverflow.net/questions/428081 | 6 | Let $L$ be a non-splittable link in $S^3$. Non-splittable means that there is no smooth embedding $s:S^2\to S^3\setminus L$ which splits $L$, i. e. such that both connected components of $S^3\setminus\operatorname{im}s$ intersect $L$.
The [Hopf link](https://en.wikipedia.org/wiki/Hopf_link) is an example.
>
> Pro... | https://mathoverflow.net/users/485324 | Can a non-splittable link be split by a wild sphere? | Let $L\_1 \cup L\_2$ be a nonsplittable link in $S^3$, and let $\phi : S^2 \to S^3 \setminus (L\_1 \cup L\_2)$ be an embedding of a $2$-sphere. We want to show that the $S^2$ does not separate $L\_1$ from $L\_2$.
Choose a points $p\_1$ and $p\_2$ on $L\_1$ and $L\_2$, so $S^3 \setminus \{ p\_1, p\_2 \} \cong S^2 \tim... | 2 | https://mathoverflow.net/users/297 | 428114 | 173,598 |
https://mathoverflow.net/questions/428018 | 4 | This is a second part of my [previous question](https://mathoverflow.net/questions/428017/what-is-a-large-field-problem), which I decided to split into two parts not to mix up different topics at one giant question.
Again, according to V. Rivasseau (section 1.5 of [Constructive Renormalization Theory](https://arxiv.o... | https://mathoverflow.net/users/152094 | Cluster expansion, Mayer expansion and perturbative renormalization group | I think there is a misunderstanding here on what is the expansion parameter.
Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.
Constructive renormalization, instead, does not expand in power... | 2 | https://mathoverflow.net/users/11260 | 428127 | 173,602 |
https://mathoverflow.net/questions/428134 | 5 | Here is what I observed :
Let $8p+1 = 256a^2+(2b-1)^2$ with $a$ and $b$ be a positive integers, $p$ and $8p+1$ both prime numbers.
Then $8p+1$ divides $(2^p+1)/3$ only if you can write $8p+1$ as $256a^2+(2b-1)^2$.
For example :
* $1049 = 256 \cdot 2^2+(2 \cdot 3 - 1)^2$ and $1049 = 131 \cdot 8+1$ so $1049$ divi... | https://mathoverflow.net/users/264367 | Condition for $8p+1$ divides $(2^p+1)/3$? | This follows easily from the octic reciprocity law for $2$. More precisely, it states that if $q$ is a prime of the form $q=8n+1$ and $n$ is odd, then $q=a^2+256b^2$ for some integers $a,b$ (note that $a$ is necessarily odd) if and only if $2$ is a biquadratic residue but not an octic residue, i.e. there is $x \in \mat... | 9 | https://mathoverflow.net/users/101078 | 428139 | 173,605 |
https://mathoverflow.net/questions/428084 | 3 | This question might be below the level of MO, so apologies in advance. I posted the [same question](https://math.stackexchange.com/questions/4503962/does-the-set-of-ideals-whose-jacobson-radical-nilradicals-coincide-form-a-la) in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring and $... | https://mathoverflow.net/users/164350 | Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice? | Let's call an ideal $I\lhd R$ *Jacobson* if $J(I)=\sqrt{I}$.
I will answer the question in the title by constructing, in stages, an example of a unital ring $R$ where the set of Jacobson ideals
is not a sublattice of the ideal lattice of $R$. It will turn out that more is true about this example: its ordered set of Jac... | 2 | https://mathoverflow.net/users/75735 | 428144 | 173,608 |
https://mathoverflow.net/questions/427811 | 4 | A metric space $M$ is called *two-point homogeneous* if for any pair of points $(p,q)$ in $M$ any distance preserving map $f\colon\{p,q\}\to M$ can be extended to an isometry $\bar f\colon M\to M$.
The following statement seems to be an easy corollary of [Gleason--Yamabe theorem](https://terrytao.wordpress.com/2011/0... | https://mathoverflow.net/users/1441 | Locally compact + two-point homogeneous => Riemannian | Tits proves in [Tits, J. Sur certaines classes d'espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8$^\circ$ 29 (1955), no. 3, 268 pp. MR0076286], page 220, the following.
*If $M$ is a locally compact and connected metric space which is 2-point homogeneous (in your sense) then $M$ is isometri... | 5 | https://mathoverflow.net/users/37911 | 428148 | 173,609 |
https://mathoverflow.net/questions/428149 | 0 | Let be given positive integers $m,n,r$, with $r \leq \min(m, n)$, and a finite field of $q$ elements $\mathbb{F}\_q$.
I'm looking for an efficient algorithm to enumerate (i.e., generate one by one) all the $m \times n$ matrices over $\mathbb{F}\_q$ that have rank equal to $r$.
One obvious solution is to enumerate t... | https://mathoverflow.net/users/489345 | Enumerating (i.e. generating one by one) matrices of given rank over a finite field | You may fix an ordering of $\mathbb{F}\_q^n$ (like lexicographic or which you prefer), then start the following: choose the first row, then choose the second etc. Denote by $r(i)$ the dimension of the span of the first chosen $i$ rows; also let $B(i)$ be (some) set of linearly independent rows between the first $i$ row... | 1 | https://mathoverflow.net/users/4312 | 428150 | 173,610 |
https://mathoverflow.net/questions/428137 | 3 | Let $\Gamma \subset \mathrm{SL}(2,\mathbb{R})$ be a lattice. If $N\_1, N\_2$ are a pair of independent parabolic subgroups contained in $\Gamma$, why must $\Gamma$ contain a hyperbolic element? By parabolic subgroup, I mean "subgroup containing only parabolic elements, other than the identity." This is used in the proo... | https://mathoverflow.net/users/123002 | Parabolic elements and hyperbolic elements in SL(2,R) | This very special case follows from "general principles" (namely a version of the ping-pong lemma) but it is also possible to give a direct proof, as follows.
---
Suppose that $a$ and $b$ are the distinct points at infinity fixed by the two parabolic subgroups $A$ and $B$ (note change of notation). Since $\mathrm... | 4 | https://mathoverflow.net/users/1650 | 428154 | 173,611 |
https://mathoverflow.net/questions/427956 | 2 | Let $T\_{p,q}$ be line joining $(0,0)$ and $(p,q).$ Now let us define the set
$$L= \bigcup\_{p\in[0,1]\cap \mathbb{Q}}T\_{p,1} \bigcup\_{q\in[0,1]\cap \mathbb{Q}}T\_{1,q} $$
and consider $P=[0,1]\times[0,1]\setminus L.$ $P$ should be a set of positive Lebesgue measure.
Question: Does there exist set of positive Lebes... | https://mathoverflow.net/users/483450 | Problem regarding set of positive Lebesgue measure in $\mathbb{R}^2$ | There are no such $A,B$. If $A\subset[0,1]$ is a Lebesgue measurable set of positive measure, the set $\mathbb Q\_+ A:=\{qa: q\in\mathbb Q\_+,\, a\in A\}$ has full measure in $\mathbb R\_+$, for it has at least one point of density $1$ and it is invariant by multiplication by positive rationals. Therefore if $B\subset[... | 2 | https://mathoverflow.net/users/6101 | 428156 | 173,613 |
https://mathoverflow.net/questions/427995 | 2 | Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of compact Hausdorff spaces?
Note added in the light of @Tyrone's comment. The answer to this question is NO
as can be seen b... | https://mathoverflow.net/users/124943 | Possible characterisation of compactly generated weakly Hausdorff spaces | Peter Scholze's [comment](https://mathoverflow.net/questions/427995/possible-characterisation-of-compactly-generated-weakly-hausdorff-spaces#comment1101190_427995)
>
> The correct characterization is that they are the topological spaces that can be written as filtered colimits of compact Hausdorff spaces along inje... | 0 | https://mathoverflow.net/users/124943 | 428161 | 173,614 |
https://mathoverflow.net/questions/428172 | 3 | Montesinos proved that the double branched cover $\Sigma(T)$ of an algebraic tangle $T$ in a $3$-ball is a graph manifold. I wonder if the converse true: Is $T$ algebraic if $\Sigma(T)$ is a graph manifold?
If unknown, how about this simpler question:
Is $T$ algebraic if $\Sigma(T)$ is a Seifert manifold?
| https://mathoverflow.net/users/23935 | Characterizing algebraic tangle by their double branched covers | Yes, this is true. Suppose that $T$ is the given tangle in the three-ball, and the branched double cover is a graph manifold. By the uniqueness of the [JSJ decomposition](https://en.wikipedia.org/wiki/JSJ_decomposition), the decomposition of the graph manifold into Seifert manifolds descends to give a tangle decomposit... | 5 | https://mathoverflow.net/users/1650 | 428178 | 173,618 |
https://mathoverflow.net/questions/428145 | 2 | Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$ for a positive function $\Phi$ "slowly going to zero" as $t\to \infty$, in the sense that the limit of the fraction of... | https://mathoverflow.net/users/111052 | 'Almost all' zeros of the Dirichlet L function lies 'near' the critical line? | One way to address this question is via a zero density result, of which there are a great variety. One can find a nice survey of this technology in Chapter 10 of Iwaniec and Kowalski. For instance, their Theorem 10.4 (which they call the Grand Density Theorem) gives a pretty general result that certainly works for an i... | 6 | https://mathoverflow.net/users/2627 | 428182 | 173,621 |
https://mathoverflow.net/questions/428183 | 5 | Let $p \equiv 1 \pmod 4$ be a prime and $E\_n$ denote the $n$-th Euler number. While investigating $E\_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$)
\begin{align\*}
\sum\_{k =1}^{\frac{p-3}{2}} \frac{a\_k}{2k+1} \pmod p.
\end{align\*}
where $a\_k = \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k}$.
... | https://mathoverflow.net/users/171396 | Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$? | It is known that $$\sum\_{k=0}^\infty\frac{\binom{2k}k}{(2k+1)4^k}=\frac{\pi}2.$$ Motivated by this, I proved in my paper [*On congruences related to central binomial coefficients*](https://doi.org/10.1016/j.jnt.2011.04.004) [J. Number Theory 131(2011), 2219-2238] the following result (as part (i) of Theorem 1.1 in the... | 8 | https://mathoverflow.net/users/124654 | 428190 | 173,623 |
https://mathoverflow.net/questions/423336 | 3 | Let $V$ be a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear operators from $V$ to $V$. An operator $\mathcal{E}:L(V)\rightarrow L(V)$ is said to be positive if whenever $A\geq 0$, we have $\mathcal{E}(A)\geq 0$ as well. We say that $\mathcal{E}$ is completely positive if $\math... | https://mathoverflow.net/users/22277 | Approximations of the spectral radii of completely positive superoperators | Yes. We can characterize $\rho\_{2,d}(\mathcal{E})$ whenever $\mathcal{E}$ is completely positive without needing to first decompose $\mathcal{E}$ as $\Phi(A)$ or $\Phi(A\_1,\dots,A\_r).$ As a consequence, we can define $\rho\_{2,d}(\mathcal{E})$ for all linear operators $\mathcal{E}:L(V)\rightarrow L(V)$, but if $\mat... | 1 | https://mathoverflow.net/users/22277 | 428197 | 173,625 |
https://mathoverflow.net/questions/428217 | 11 | Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X\_0 ⊆ X\_1 ⊆ X\_2 ⊆ \cdots ⊆ X\_k = \mathbb{P}^n$ whose dimension increases by one at every step by writing $X = \mathcal{V}(f\_1, \dots, f\_n)$ and dropping some of the $f\_i$ until the... | https://mathoverflow.net/users/123448 | Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension? | Suppose that $\operatorname{dim}(X)>1$ and that such a chain exists. Since $\operatorname{Pic}(\mathbf{P}^n)\simeq \mathbf{Z}$, the variety $X\_{k-1}$ is an ample divisor in $\mathbf{P}^n$, and hence by the Lefschetz hyperplane theorem we have $\operatorname{Pic}(X\_{k-1})\simeq \mathbf{Z}$. So $X\_{k-2}$ is an ample d... | 18 | https://mathoverflow.net/users/3847 | 428220 | 173,632 |
https://mathoverflow.net/questions/428214 | 5 | I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(<https://link.springer.com/article/10.1007/s00208-018-1742-4>) and I have a related question, which, despite some search in the literature, remains puzzling to me (I should say that I am not quite familiar with this f... | https://mathoverflow.net/users/478677 | modularity lifting theorems for non-compact unitary groups | You might like to read the introduction of Harris' 2013 Crelle paper "The Taylor-Wiles method for coherent cohomology" (see [link](https://webusers.imj-prg.fr/%7Emichael.harris/coherentTW.pdf)). Here is an excerpt:
>
> In practice, all the higher-dimensional results, with the exception of
> [GT] and [Pi], have been... | 8 | https://mathoverflow.net/users/2481 | 428233 | 173,634 |
https://mathoverflow.net/questions/428254 | 0 | Working in a suitable extension of $\sf ZF-Reg.$, can we have a transitive model $M$ of $\sf ZF-Reg.$ such that for every set in $M$ there is a partition on it in $M$ all compartments of which are non-singleton finite sets. And such that any subset $X$ of $M$ that is a family of pairwise disjoint larger than singleton ... | https://mathoverflow.net/users/95347 | Can we have all classes that can be partitioned into non-singleton finite sets, be sets? | No, this is not possible: the ordinals of $M$ will always provide a counterexample.
Even if regularity fails, the class $\mathsf{Ord}$ of ordinals cannot be a set. But we can partition $\mathsf{Ord}$ into (say) two-element sets via the equivalence relation $$\alpha\sim\beta\quad\iff\quad \sup\{\lambda: 2\cdot\lambda<... | 7 | https://mathoverflow.net/users/8133 | 428257 | 173,639 |
https://mathoverflow.net/questions/428231 | 5 | Let $F$ be a finite field extension of the $p$-adic numbers $\mathbb{Q}\_p$, whose residue field has $q$ elements. Let $\mathfrak{p}$ be the prime ideal of $F$. Given a finite field extension $K/F$, write $\operatorname{Disc}\_\mathfrak{p}(K)$ for $q^{v\_\mathfrak{p}(\operatorname{Disc}(K/F))}$. Then **Serre's mass for... | https://mathoverflow.net/users/169035 | Looking for proof of Serre's mass formula | The article is not only available in Serre's Collected Papers: it first appeared in a journal, after all. Here's a scan from the Comptes Rendus archives: <https://gallica.bnf.fr/ark:/12148/bpt6k6234149b/f323.item>.
| 7 | https://mathoverflow.net/users/3272 | 428260 | 173,640 |
https://mathoverflow.net/questions/428232 | 5 | In the page 482 of his [article](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.449.3854&rep=rep1&type=pdf), Fickle wrote the following argument:
Let $Y$ be a homology $3$-sphere. Next
* Add a $2$-handle to $Y \times [0,1]$ and produce a $4$-manifold with
boundary $S^1 \times S^2$,
* Cap off the boundary... | https://mathoverflow.net/users/475366 | Fickle's argument for Mazur manifolds | Call the manifold built in this way $W$. If you turn the handlebody for $W$ relative to $Y$ upside down, it becomes a handlebody with a single handle in indices $0$, $1$, and $2$. This is almost the same as saying it's a Mazur manifold; you need to check that it's contractible. Because $\pi\_1(W)$ is cyclic (it's got j... | 7 | https://mathoverflow.net/users/3460 | 428261 | 173,641 |
https://mathoverflow.net/questions/428153 | 2 | Given positive integer $n$, we are looking for a set
of $n$ positive integers $a\_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
* $0 \le x\_i \le \frac{n}{2}$
* $\sum x\_i = n$
* $\sum a\_i x\_i = \sum a\_i$
One exponential example is to take $a\_i=C^i$.
We ex... | https://mathoverflow.net/users/12481 | Only trivial solution to a pair of constrained linear diophantine equations | The answer to the first question is negative.
Let $A$ denote the set of weights $\{a\_i\}$. Strengthen the first constraint to $0 \le x\_i \le 2$.
If we have two different subsets $S\_1, S\_2 \subset A$ of the same cardinality, their sums must be different, since otherwise we can assign $$x\_i = \begin{cases} 2 & \... | 1 | https://mathoverflow.net/users/46140 | 428266 | 173,642 |
https://mathoverflow.net/questions/426170 | 6 | Suppose $A,B \in M\_{n}(\Bbb{R})$ such that $A = \left[C\_{1}\middle|\frac{I}{0\dots0}\right], B= \left[C\_{2}\middle|\frac{I}{0\dots0}\right]$ , where $A$ and $B$ have different first columns (represented as $C\_{1}, C\_{2}$).
Thus we have $B = A+ \xi e\_{1}^T$, where $\xi$ is a $n \times 1$ column vector and $e\_{1... | https://mathoverflow.net/users/105018 | Is it impossible for determinants of these matrices to both be negative? | This is a partial answer, now with added material.
Assume that $n$ is a multiple of 3, the other two cases should be similar. I'll denote $C\_1\mapsto a$ and $C\_2\mapsto b$, such that for, e.g., $n=6$,
$$\tag{1}\label{1}
A = \begin{pmatrix}
a\_1 & 1 & 0 & 0 & 0 & 0 \\
a\_2 & 0 & 1 & 0 & 0 & 0 \\
a\_3 & 0 & 0 & 1 & 0... | 1 | https://mathoverflow.net/users/90413 | 428268 | 173,643 |
https://mathoverflow.net/questions/428128 | 4 | Consider two measurable spaces $X\_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu\_1)$ and $X\_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu\_2)$ and the product spaces
$$X\_1^{q} = (\times\_{i=1}^q\mathbb{R}^m,\otimes\_{i=1}^q\mathcal{B}(\mathbb{R}^{m}),\mu\_1^{\otimes q})\ \ \ \text{ and }\ \ \ X\_2^{q} = (\times\... | https://mathoverflow.net/users/62673 | Bounds on discrepancy metric of product measures | Analogous to the TV metric, the requested upper bound holds for the discrepancy metric with $k=q$. The result given below can also be easily extended to general product probability measures $\mu=\otimes\_{i=1}^q \mu\_{i}$ and $\nu=\otimes\_{i=1}^q \nu\_i$ to obtain $$
D(\mu,\nu)\le \sum\_{i=1}^q D(\mu\_i, \nu\_i)\;.
$$... | 2 | https://mathoverflow.net/users/64449 | 428269 | 173,644 |
https://mathoverflow.net/questions/428064 | 3 | Let $S$ be a scheme and $X\to Y$ be a morphism over $S$. Then we have an induced homomorphism of sheaves $h\_X=\operatorname{Hom}\_S({-}, X)\to h\_Y=\operatorname{Hom}\_S({-}, Y)$ over the small étale site $S\_\text{étale}$.
Question: When is $h\_X\to h\_Y$ a surjective homomorphism between sheaves of sets over $S\_\... | https://mathoverflow.net/users/153842 | Surjective sheaf homomorphisms induced by morphisms of schemes | **Proposition.** Let $f\colon X\to Y$ be a surjective morphism such that $\forall y\in Y$ there exists a point $x\in f^{-1}(y)$ where $f$ is smooth (i.e. every fiber has a non-reduced point). Then $h\_X\to h\_Y$ is surjective on the (small or big) étale site.
**Proof:** Take $f\in h\_Y(S)$. The property of having a s... | 1 | https://mathoverflow.net/users/88385 | 428270 | 173,645 |
https://mathoverflow.net/questions/428259 | 11 | Let $H\_1,\ldots,H\_n$ be hyperplanes in $\Bbb R^d$. Denote $\mathcal{H} :=\{H\_1,\ldots,H\_n\}$ and let $c(\mathcal{H})$ be the number of regions in the complement: $\Bbb R^d\setminus \bigcup H\_i$.
**Question:** What is the complexity of computing $c(\mathcal{H})$?
Here we are assuming that $H\_i$ are defined exp... | https://mathoverflow.net/users/4040 | Complexity of counting regions in hyperplane arrangements | The problem is $\#\mathsf{P}$-complete. As you already noted, the problem is $\#\mathsf{P}$-hard even when we restrict to graphical arrangements, so it remains to show that the problem is in $\#\mathsf{P}$. Directly from the definition of $\#\mathsf{P}$, we see that it suffices to show that we can give a short descript... | 9 | https://mathoverflow.net/users/3106 | 428272 | 173,647 |
https://mathoverflow.net/questions/428224 | 2 | Let $\alpha>1$ be a constant and define $B\_n$ as the number of (labeled) *balanced* graphs with $n$ vertices and $\left\lceil \alpha n\right\rceil $ edges. The paper [Strongly Balanced Graphs
and Random Graphs](https://people.clas.ufl.edu/avince/files/StronglyBalanced.pdf "Ruciński, A. and Vince, A. (1986). J. Graph T... | https://mathoverflow.net/users/163685 | Lower bound on the number of balanced graphs | The bound of Ruciński and Vince is for *strongly balanced*, which is a more strict condition. If only *balanced* is required, the example of connected regular graphs provides a bound much greater than $n^{\Omega(n)}$.
The total number of regular graphs with $n$ vertices is
$$\alpha(n) \frac{2^{n^2/2}\sqrt{2e}}{\pi^{n... | 2 | https://mathoverflow.net/users/9025 | 428273 | 173,648 |
https://mathoverflow.net/questions/384146 | 4 | The question is inspired by G. Rzadkowski and M. Urlinska's examples in their paper [A Generalization of the Eulerian Numbers](https://arxiv.org/abs/1612.06635). They refer to the discussion
[Expressions involving Eulerian numbers of the second kind](http://mathoverflow.net/questions/45756/), with Pietro Majer's
[answe... | https://mathoverflow.net/users/174295 | How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order? | The identity is valid. This is a corollary to a proof of Amy M. Fu, [Some Identities Related to the Second-Order Eulerian Numbers](https://arxiv.org/abs/2104.09316).
A second proof follows from recent work of Cormac O'Sullivan, [Stirling's approximation and a hidden link between two of Ramanujan's approximations](htt... | 0 | https://mathoverflow.net/users/174295 | 428286 | 173,652 |
https://mathoverflow.net/questions/428288 | 3 | Serre's criterion says that for a scheme to be normal is equivalent to it being $R\_1$ (i.e. regular in codimension $1$) and $S\_2$ (i.e. regular functions on $X-Y$ extend to $Y$ if $Y$ has codimension at least $2$).
What would be examples of:
1. a scheme which is $R\_1$, but not $S\_2$ (i.e. not normal)?
2. a sche... | https://mathoverflow.net/users/198139 | Normal schemes and Serre's criterion | 1. Glue two planes at a point, i.e., take take $\mathrm{Spec}(A)$, where
$$
A = \{(f,g) \in k[x\_1,x\_2] \oplus k[y\_1,y\_2] \mid f(0,0) = g(0,0) \}.
$$
2. Take any singular curve, e.g., $\mathrm{Spec}(k[x,y]/xy)$.
| 7 | https://mathoverflow.net/users/4428 | 428291 | 173,654 |
https://mathoverflow.net/questions/428256 | 3 | Let $\Sigma\_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma\_g \times \Sigma\_g$. Then can there exist holomorphic embeddings of $\Sigma\_l$ into $X$ for $l < g$?
What about in the symplectic category i.e
if $\omega$ denotes the area 1 form on $\Sigma\_g$ and we equip $X = \Sigma\_g \times... | https://mathoverflow.net/users/92483 | Holomorphic/Symplectic embedding of Riemann surfaces | I am just posting my comment as one answer.
**Lemma.** For integers $\ell < g$, for every continuous map from $\Sigma\_\ell$ to $\Sigma\_g$, the pullback map on $H^2$ is the zero map.
**Proof.** The pullback map on $H^1$ has rank no greater than $2\ell$, since that is the rank of $H^1(\Sigma\_\ell)$. Since $H^1(\Si... | 3 | https://mathoverflow.net/users/13265 | 428294 | 173,656 |
https://mathoverflow.net/questions/428278 | 4 | Let $G$ be a group. I have two questions about the homology of $G$:
1. Consider a finite exact sequence
$$0 \rightarrow M\_1 \rightarrow \cdots \rightarrow M\_m \rightarrow 0$$
of $G$-modules. How are the homology groups $H\_k(G;M\_i)$ related?
2. Consider a filtration of $G$-modules
$$0 = N\_0 \subset N\_1 \subset \... | https://mathoverflow.net/users/489452 | Groups homology with coefficients fitting into filtration or exact sequence | 1. Think of your exact sequence as a resolution of $M\_m$. It's not necessarily a resolution by free $G$-modules, or by projective $G$-modules; it's just a resolution by $G$-modules. You get a spectral sequence whose $E\_1$-term is the direct sum of the $H\_\*(G; M\_i)$ for all $i<m$, and which converges to $H\_\*(G; M... | 2 | https://mathoverflow.net/users/nan | 428309 | 173,660 |
https://mathoverflow.net/questions/428311 | 2 | I am wondering what would be the value of $C(f)$ for the following inequality to hold? E.g., $C(f)$ could be some quantity related to the Lipschitz constant or the size of the domain.
$$\left(\int f(x,x) dx\right)^2 \leq C(f)\cdot \iint f(x,y)^2 dxdy$$
| https://mathoverflow.net/users/118287 | For which value of $C(f)$ would the following inequality hold? | The question is not well posed, as it is not quite clear in what terms you want $C(f)$ to be expressed.
If one only uses the terms you did mention -- the Lipschitz constant and the size of the domain, then no finite $C(f)$ exists. Indeed, suppose that the domain, of "size" $a\in(0,\infty)$, is $[0,a]^2$. Take any rea... | 1 | https://mathoverflow.net/users/36721 | 428313 | 173,661 |
https://mathoverflow.net/questions/428315 | 3 | Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$
\begin{align\*}
C = A^TXA + B^TXB.
\end{align\*}
In my case, the coefficient matrices are such that the equation has a unique real solution which is ... | https://mathoverflow.net/users/168454 | Solution to a Sylvester equation with positive definite coefficients | This is not true. For example, if
$$
A = \begin{bmatrix}
1 & 0 \\ 0 & 4
\end{bmatrix}, B = \begin{bmatrix}
4 & 0 \\ 0 & 1
\end{bmatrix}, C = \begin{bmatrix}
17 & 16 \\ 16 & 17
\end{bmatrix}
$$
then each of $A$, $B$, and $C$ is symmetric and positive definite. However, it is straightforward to check that the unique solu... | 7 | https://mathoverflow.net/users/11236 | 428317 | 173,664 |
https://mathoverflow.net/questions/428240 | 3 | I'm currently learning about sheaf theory with Angelo Vistoli’s 2007 [Notes on Grothendieck topologies,
fibered categories and descent theory](http://homepage.sns.it/vistoli/descent.pdf). And in page 35, there is the following definition of a refinement and a subordinate grothendieck topology:
**Refinement:**
Let $C$... | https://mathoverflow.net/users/485069 | Proof without sieves: Equivalent grothendieck topologies have the same sheaves | Let me break down the statement you are trying to prove into two independent facts. This answer is not really in the spirit of the question since I will make maximal use of sieves, but for such foundational matters I think it is much more efficient to embrace sieves rather than to avoid them (so my subjective answer to... | 5 | https://mathoverflow.net/users/20233 | 428343 | 173,673 |
https://mathoverflow.net/questions/428346 | 7 | Consider a set $X\subseteq \mathbb{R}$ such that
1. $X$ is *not* separable wrt its subspace topology
2. For all $r\in\mathbb{R}$ there exists a sequence $(x\_n)\_{n\in\omega} \subset X$ converging to $r$
In a model containing such a set $\text{AC}\_\omega(X)$ (choice for countable families of non-empty subsets of $... | https://mathoverflow.net/users/141146 | Consistency of a strange (choice-wise) set of reals | The existence of such a set follows from $``\mathbb{R}$ is a countable union of countable sets.$"$ Let $\mathbb{R} = \bigcup\_{n<\omega} S\_n,$ each $S\_n$ countable. Let $T\_n = \{x \in \mathbb{R}: \exists m \le n \exists y \in S\_m (x \le\_T y)\}$ and $X\_n = (2^{-n-1}, 2^{-n}) \setminus T\_n.$
We will show $X = \b... | 13 | https://mathoverflow.net/users/109573 | 428354 | 173,674 |
https://mathoverflow.net/questions/428308 | 2 | I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0\\
x\_{1}^{2} & Nx\_{1} & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & c & 0 & 0 & 0 & 0 & 0 ... | https://mathoverflow.net/users/119725 | Full-rank matrix | OK, let's call the block matrix above $M$. First eliminate $N$ by a substitution $c\mapsto d N$. Then substitute $z\_i \mapsto d y\_i$ to eliminate $d$. Then you can construct the Schur complement w.r.t. the first and last rows/columns of $M$ to get $\det(M) = -c^2 N x\_1^2 x\_2^2 x\_3^2 \det P$, with
$$
P =
\begin{pm... | 5 | https://mathoverflow.net/users/90413 | 428358 | 173,675 |
https://mathoverflow.net/questions/427876 | 5 | Given any topological space $(X,\tau)$, it seems to me that the uniformity generated by all continuous pseudometrics $d:(X,\tau)\times (X,\tau)\to [0,\infty)$ is the *fine uniformity* of the associated completely regular topology of $\tau$. This would yield an easy desciption of the left adjoint of the functor assignin... | https://mathoverflow.net/users/21051 | Is the fine uniformity generated by all continuous pseudometrics? | It is true, and I learned of many of the details in the somewhat obscure paper
>
> D. Thampuran, [*On Completely Regular Spaces*](https://purl.pt/2750/1/j-5293-b-vol33-fasc4-art4_PDF/j-5293-b-vol33-fasc4-art4_PDF_01-B-R0300/j-5293-b-vol33-fasc4-art4_0000_capa1-208_t01-B-R0300.pdf), Portugaliae Mathematica \**33*, (... | 3 | https://mathoverflow.net/users/54788 | 428368 | 173,677 |
https://mathoverflow.net/questions/428375 | 5 | This line
>
> The symbol may naturally be thought of as an element in the K-theory
> of X
>
>
>
appears in the [nLab page on principal symbols](https://ncatlab.org/nlab/show/symbol+of+a+differential+operator) for differential operators. What does this mean? Are they talking about K-theory or K-homology? How do... | https://mathoverflow.net/users/153228 | The principal symbol as an element in the K-theory | It's a bit easier to see this using a slightly non-standard definition of topological K-theory. Given a locally compact Hausdorff space $X$, let $\bf{E}$ be a complex of vector bundles, i.e. a sequence
$$0 \to E\_0 \xrightarrow{\alpha\_0} E\_1 \xrightarrow{\alpha\_1} \ldots \xrightarrow{\alpha\_{n-1}} E\_n \to 0$$
... | 14 | https://mathoverflow.net/users/4362 | 428382 | 173,682 |
https://mathoverflow.net/questions/427601 | 7 | Let $\epsilon <1/2$. Let $X$ be a random variable in $\mathbb Z$ such that $\mathbb P (X=x)\le \epsilon $ for any $x\in \mathbb Z$ (you may add any moment or regularity conditions on $X$ if needed). Let $S\_n$ be a sum of $n$ independent copies of $X$. Show that for any $x\in \mathbb Z$
$$\mathbb P (S\_n=x) \le C\cdot ... | https://mathoverflow.net/users/161778 | Local probabilities for lattice random walk | For the one dimensional case, a quite nice bound is in Theorem 4.2 of [1]. See also [2]. The dependence on $\epsilon$ that you seek was first shown by Kesten[3].
The combinatorial approach was revived in [4]. The sharpest result is quite recent, see [5], which also identifies the worst case. Look there first.
[1] E... | 4 | https://mathoverflow.net/users/7691 | 428389 | 173,685 |
https://mathoverflow.net/questions/427528 | -1 | This posting is a follow up of [this](https://mathoverflow.net/questions/427351/can-we-write-tangled-type-theory-without-reference-to-type-sequences)
Language multi-sorted FOL, with sorts (types) indexed by the naturals, equality symbol restricted to same type, while membership symbol restricted from lower to higher ... | https://mathoverflow.net/users/95347 | Is there an obvious inconsistency with this extension of Tangled Type Theory? | I do not know whether acyclicity avoids paradox here.
| 2 | https://mathoverflow.net/users/485780 | 428390 | 173,686 |
https://mathoverflow.net/questions/428384 | 9 | This is a cross-post! For the original post on SE (9 upvotes, no answer) see:
<https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-points-alrea>
Let $X$ be a complex algebraic set, i.e. the (not necessarily irreducible) vanishing set of some polynomia... | https://mathoverflow.net/users/489043 | Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers? | More is true. Let $K/F$ be a field extension. Let $X$ be the vanishing set of some polynomials in $K[X\_1,\dots, X\_n]$. If $X$ contains a Zariski dense set of points with coordinates in $F$, then $X$ is the vanishing set of some polynomials in $F$.
Proof: Choose a basis $\alpha\_i$ for $K/F$. Every polynomial in $K[... | 14 | https://mathoverflow.net/users/18060 | 428394 | 173,688 |
https://mathoverflow.net/questions/428360 | 4 | There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E\_7$ and $\mathbb{F}\_2^6 \setminus \{0\}$ (where $\mathbb{F}\_2$ is the field with two elements. Btw, this also preserves orthogonality. There is also a relation between ... | https://mathoverflow.net/users/4096 | Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$ | Let $L$ denote the root lattice of $E\_8$. The group $W(E\_8)$ acts (linearly) on $L/2L \cong \mathbb{F}\_2^8$, and hence as a permutation of $2^8=256$ elements. This permutation action has the following three orbits:
1. $\{0\}$, an orbit of size 1.
2. The classes represented by roots, an orbit of size 120. There are... | 8 | https://mathoverflow.net/users/78 | 428401 | 173,690 |
https://mathoverflow.net/questions/428406 | 4 | An element $s$ of a group $G$ is a *logical generator* of $G$ iff every element of $G$ can be defined in the first order language of groups with $s$ as a parameter. In this case we may call $G$ a *logically cyclic* group. This is equivalent to say that for any elementary extension $G^{\ast}$ of $G$ and any automorphism... | https://mathoverflow.net/users/44949 | Logical generators of groups and $\mathrm{Aut}$-bases | $\{1\}$ is an Aut-basis of the group of profinite integers $\hat{\mathbb Z}$, but since this group has cardinality $2^\omega$, it has no logical generator. However, every element of $\hat{\mathbb Z}$ is *type-definable* from $1$ (i.e., any pair of distict elements are distinguishable by a first-order formula with param... | 6 | https://mathoverflow.net/users/12705 | 428413 | 173,692 |
https://mathoverflow.net/questions/428396 | 3 | **1. Question**
How to get from the formulas
$$ \left| \frac{\log 2}{\log 3} - \frac{p}{q} \right| < c\_a\frac{1}{q^{B\_a}} \ \ \ \ \ \ \ \ \ \ \ (1.1)$$
and / or
$$ \left| \frac{\log 2}{\log 3} - \frac{p}{q} \right| \geq c\_b\frac{1}{q^{B\_b}} \ \ \ \ \ \ \ \ \ \ \ (1.2)$$
to the formula
$$ \left| 3^p - 2^... | https://mathoverflow.net/users/163727 | Simple estimation of difference of powers of 2 and powers of 3? | This is about bounding $e^x$ such as $e^x \geq 1 + x$.
For example, when $2^q>3^p$ (ie. $\log\_3 2>\frac{p}q$), from (1.2) we have
$$q\log 2 \geq c\_b \frac{\log 3}{q^{B\_b-1}} + p\log 3,$$
which under exponentiation translates to
$$2^q \geq e^{c\_b \frac{\log 3}{q^{B\_b-1}}}3^p.$$
Then
$$2^q - 3^p \geq \big(e^{c\_b ... | 2 | https://mathoverflow.net/users/7076 | 428425 | 173,693 |
https://mathoverflow.net/questions/258394 | 12 | Is it possible to unite two $n$-vertex trees such that the resulting graph has bounded tree-width?
Formally, does there exists a constant $k$ such that given two $n$-vertex trees $T\_1$ and $T\_2$ there exist labeled trees $T\_1'=([n], E\_1)$ and $T\_2' =([n],E\_2)$ with $T\_1' \simeq T\_1$, $T\_2' \simeq T\_2$ and
$... | https://mathoverflow.net/users/83519 | Tree-width of a union of two trees | It is shown in [Bogdan Alecu, Vadim Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev: "The treewidth and pathwidth of graph unions"](https://arxiv.org/abs/2202.07752) that the general question has a negative answer.
| 2 | https://mathoverflow.net/users/151145 | 428426 | 173,694 |
https://mathoverflow.net/questions/428408 | 4 | Consider a norm on $\mathbb C^2$ as $\|(z\_1,z\_2)\|:=\max\{|z\_1|,|z\_2|,\frac{1}{\sqrt{2}}|z\_1+iz\_2|\}.$
*Question.* Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|\_{\infty})$ where $\|(z\_1,z\_2)\|\_\infty:=\max\{|z\_1|,|z\_2|\}?$
| https://mathoverflow.net/users/136860 | A space isometric to $\ell_\infty^2$ | There is no such map $f$. Let's try to map from the second space (with the funny norm, which I'll denote simply by $\|\cdot\|$) back to $(\mathbb C^2, \|\cdot \|\_{\infty})$. Let $u=f(e\_1)$, $v=f(e\_2)$, so $\|u\|\_{\infty}=\|v\|\_{\infty}=1$. Since $|1\pm i|^2=2$, so $\|(1,\pm 1)\|=1$, we also have $\|u\pm v\|\_{\inf... | 3 | https://mathoverflow.net/users/48839 | 428432 | 173,698 |
https://mathoverflow.net/questions/428438 | 0 | I'm looking at the <https://arxiv.org/abs/1904.09193> paper (version 2, from 2021) and think it has a few errors. I think I found three small places where the paper needs to be corrected (in the sense that the corrected version is valid - none of the key conclusions are undermined by this).
1. On page 5 it says "Lemm... | https://mathoverflow.net/users/489586 | Did I find a few (small) errors in the Pradic and Brown 2021 paper that Schroeder-Bernstein implies excluded middle? | It may interest you that Martín Escardó also formalized the proofs, see [`Cantor-Schröder-Bernstein-gives-EM`](https://www.cs.bham.ac.uk/%7Emhe/TypeTopology/CantorSchroederBernstein.CSB.html#6327) in his [TypeTopology](https://www.cs.bham.ac.uk/%7Emhe/TypeTopology/index.html) library.
Regarding contacting the authors... | 1 | https://mathoverflow.net/users/1176 | 428445 | 173,700 |
https://mathoverflow.net/questions/428447 | 10 | I've seen the following sentence come up a few times in papers:
>
> Let $E$ be the universal elliptic curve over the modular curve $Y\_1(N)$. Then the localization of $E$ at any choice of cusp is isomorphic to the Tate curve with some suitable level structure.
>
>
>
Could somebody explain what exactly this sen... | https://mathoverflow.net/users/394740 | Universal elliptic curve and the Tate curve | Let $E\_1(N)$ denote the universal elliptic curve you are referring to. Note that we must assume $N\ge 4$, or else there is no such universal elliptic curve. Localizing $E\_1(N)$ at a cusp means, roughly, to look at how it behaves near a cusp. As you rightly point out, $Y\_1(N)$ has no cusps. In this case, the analytic... | 7 | https://mathoverflow.net/users/15242 | 428459 | 173,703 |
https://mathoverflow.net/questions/428355 | 1 | Suppose that $V\_1,V\_2$ are two commutative von Neumann algebras and $V\_1 \subset V\_2$. Being in particular commutative $C^\*$-algebras we have that $V\_1 \cong C(X\_1), V\_2 \cong C(X\_2)$ for some topological compact spaces $X\_1,X\_2$. Our inclusion $V\_1 \subset V\_2$ yields a continuous surjection $\pi:X\_2 \to... | https://mathoverflow.net/users/24078 | Continuous surjection between spectra of commutative von Neumann algebras |
>
> Is it true that π maps clopen sets into clopen sets?
>
>
>
This is true if and only if the inclusion $V\_1→V\_2$ is a morphism of von Neumann algebras, i.e., its image is closed in the ultraweak topology.
See Proposition 2.48 in [arXiv:2005.05284](https://arxiv.org/abs/2005.05284).
| 2 | https://mathoverflow.net/users/402 | 428477 | 173,707 |
https://mathoverflow.net/questions/428352 | 6 | I have posted a few questions on MSE, ~~most notably this one~~, which revolve around the same issue and have received no answers, so I decided to ask the same here.
In the following, $K(A, n)$ is the minimal Eilenberg-MacLane Kan complex given by $$K(A, n)\_q=\{\text{normalized $n$-cocycles } \Delta^q \to A\}.$$ Thi... | https://mathoverflow.net/users/482564 | Is every simplicial map $\Phi:K(A, n) \to K(A', n)$ a simplicial homomorphism of groups? | Denote by $$\def\U{{\sf U}}\def\sSet{{\sf sSet}}\def\sAb{{\sf sAb}}\def\Ch{{\sf Ch}}\def\N{{\sf N}}\U\colon \sAb→\sSet$$ the forgetful functor,
which is the right adjoint of $$\def\Z{{\bf Z}}\Z\colon\sSet→\sAb$$
and by $$Γ\colon \Ch→\sAb$$ the Dold–Kan functor given by the right adjoint of the normalized chains functor... | 3 | https://mathoverflow.net/users/402 | 428479 | 173,708 |
https://mathoverflow.net/questions/428472 | 3 | I am trying to show that
$$\lim\_{n\to\infty}\frac{1}{n}\sum\_{i=1}^{n}\prod\_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)^{k+1}}{(k+1)k^{k}}$$
for all $k\in\mathbb{N}$, $k\geq 4$.
I could verify the statement with Mathematica, but I could not find a self-standing proof. The product is telescopic, but I could onl... | https://mathoverflow.net/users/489614 | Limit of the average of telescopic products | Start by re-expressing the product term
\begin{align}\frac{(kn-k)(kn-k-1)\cdots(kn-k-i)}{(kn)(kn-1)\cdots(kn-i)}
&=\frac{(kn-i-1)\cdots(kn-i-k)}{(kn)\cdots(kn-k+1)}=\frac{\binom{kn-1-i}k}{\binom{kn}k}.
\end{align}
So, the given sum $S\_n(k):=\sum\_{i=1}^n\prod\_{j=0}^i\frac{kn-j-k}{kn-j}$ reads
\begin{align}
S\_n(k)&=\... | 8 | https://mathoverflow.net/users/66131 | 428486 | 173,709 |
https://mathoverflow.net/questions/428501 | 12 | A *corner* of an integer partition is a location at where a box can be added to its Ferrers diagram to give a new partition.
E.g. the partition $\{1,1,1\}$ has two corners, and $\{1,2\}$ has three corners.
Let $p(n,r)$ denote the number of integer partitions of $n$ with $r$ corners.
Let $P(x,t) = \sum\_{n,r\geq0}... | https://mathoverflow.net/users/489627 | Generating function for counting partitions with corners | Maybe this is not the kind of answer you are looking for, but it is easy to see that
$$ P(x,t) = \prod\_{k=1}^{\infty} \frac{1-x^{k-1}(1-t)}{1-x^k}$$
because the number of corners of a partition is one plus its number of distinct parts (you always have a corner at the "top" of a series of repeated parts, plus one at th... | 13 | https://mathoverflow.net/users/25028 | 428502 | 173,713 |
https://mathoverflow.net/questions/428505 | 0 | Let M be the generator matrix of a $N\times N$ lattice, and $\tilde{N}$ the set of Voronoi relevant vectors. The Voronoi cell for the origin can be written as $\text{Vor}\_{\bf 0}(M)=\left\{{\bf x}: |{\bf x}|\leq|{\bf x-c}|\text{ for } \forall {\bf c}\in \tilde{N}\right\}$. My question is given $\tilde{N}$, how to find... | https://mathoverflow.net/users/476103 | How to find the closest point given the Voronoi relevant vectors? | I believe state of the art for this is [Short Paths on the Voronoi Graph and Closest Vector
Problem with Preprocessing](https://arxiv.org/abs/1412.6168) by Bonifas and Dadush.
For instance, see the following quote from the abstract
>
> Our main technical contribution is a randomized procedure that given the Voronoi... | 0 | https://mathoverflow.net/users/101207 | 428506 | 173,714 |
https://mathoverflow.net/questions/428410 | 8 | Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert x+y\rVert^2=4$, does it follow that $X$ is isometric to a Hilbert space?
I suspect the answer is "no", but I cannot c... | https://mathoverflow.net/users/69275 | Parallelogram law for vectors of equal length | I will give it a try, based on Day's idea. Let $X$ be a two-dimensional Banach space with the given property and denote by $B\_X$ its unit ball. Consider the ellipsoid of maximal volume (denoted by $B\_2$) contained in $B\_X$ (the John's ellipsoid) and denote by $\|\cdot\|\_2$ the induced Euclidean norm. The goal is to... | 4 | https://mathoverflow.net/users/69275 | 428510 | 173,717 |
https://mathoverflow.net/questions/428519 | 11 | Consider, for instance, the categories of $C^k$-manifolds, where $k=0,1,2,...,\infty,\omega$. ($C^\omega$ means real analytic.) Are these categories pairwise non-equivalent?
Of course, the obviuos forgetful functor is not an equivalence, because it is not full: for $k<l$, there are $C^k$ functions which are not $C^l$... | https://mathoverflow.net/users/485324 | Are different categories of manifolds non-equivalent (as abstract categories)? | Your method of looking for automorphisms of objects works for $C^k$ manifolds as well, $0 < k \le \infty$. This follows from [a result of Filipkiewicz](https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/isomorphisms-between-diffeomorphism-groups/BE5563E03AD7DA6A94D08F92CFA4AF90) (which... | 17 | https://mathoverflow.net/users/40804 | 428521 | 173,721 |
https://mathoverflow.net/questions/428531 | 2 | Let $X, Y, Z$ be discrete random variables with $X$ and $Y$ independent of $Z$, while $X$ and $Y$ can be dependent. For the mutual information, we have $I(X; Y,Z) = I(X;Y)$. Now consider $I(X; f(Y,Z))$ for some deterministic function $f$. Does $I(X; f(Y,Z))$ depend on $Z$? If not, is there a way to express $I(X; f(Y,Z)... | https://mathoverflow.net/users/101100 | Mutual information and bivariate function of independent variables | $I(X;f(Y, Z))$ can depend on the $Z$ (or more specifically the distribution of $Z$). Consider the following example, $Z \sim B(p)$, $X \sim B(0.5)$, $Y=X$ satisfying $X,Y \perp \!\!\! \perp Z$. Let $F=\max(Y, Z)$ be a variable from the output of a deterministic function of $Y, Z$. $X, F$ have the following joint distri... | 1 | https://mathoverflow.net/users/478836 | 428538 | 173,728 |
https://mathoverflow.net/questions/426330 | 2 | Let $X$, $Y$, $Z$ be discrete random variables, with $Y$ and $Z$ independent. Does the following equality hold?
$$
\max\_{f\_{Y,Z}} \big\{ \ I(X; f\_{Y,Z}(Y,Z)) \ \big\} \le \max\_{f\_X, f\_Y} \big \{ \ I(X; f\_Y(Y), f\_Z(Z)) \ \big \}
$$
where the maximization is taken over all non-injective deterministic functions.
... | https://mathoverflow.net/users/101100 | Maximization of information over set of non-injective functions | It seems the inequality should be held in the opposite way due to the fact that the function family on the left that you are taking maximum over is a superset of the function family on the right. More specifically, let $F\_1:=\{(y,z) \to f\_1(y,z)\}$ and $F\_2:= \{(y,z) \to (f\_y(y),f\_z(z))\}$ where $f\_1,f\_y,f\_z$ a... | 0 | https://mathoverflow.net/users/478836 | 428550 | 173,731 |
https://mathoverflow.net/questions/428493 | 1 | Let $X\neq \emptyset$ be a set. We say $E\subseteq {\cal P}(X)$ is a *linear set system* if for all $a\neq b\in X$ there is exactly one $e\in E$ with $\{a,b\}\subseteq e$.
Is there an infinite cardinal $\kappa$ and a linear set system $E\subseteq {\cal P}(\kappa)$ with $\kappa \notin E$ such that for all $x\in \kappa... | https://mathoverflow.net/users/8628 | Flat linear set systems | Denote $\deg x =|\{e\in E\colon x\in e\}|$.
Take any $e\in E$; choose $x\notin e$. Then $e$ meets any $e'\ni x$ by at most one element, whence $|e|\leq \deg x<\kappa$.
Now, any $x$ is contained in $\deg x<\kappa$ sets of cardinality $<\kappa$ each; moreover (if this is needed), any of them except one has cardinalit... | 2 | https://mathoverflow.net/users/17581 | 428561 | 173,736 |
https://mathoverflow.net/questions/428530 | 1 | This post records a little bit more on this question: [Partitioning convex polygons into triangles of equal area and perimeter](https://mathoverflow.net/questions/428243/partitioning-convex-polygons-into-triangles-of-equal-area-and-perimeter).
The basic question of the above linked post was about this claim: ""For an... | https://mathoverflow.net/users/142600 | Partitioning convex polygons into quadrilaterals of equal area and perimeter | Convex quadrilateral decompositions of equal area are always possible. Given a convex polygon $P$ with at least $5$ sides, draw a line from one of the vertices of $P$ to a point $Q$ on one of the edges furthest from $P$ (in the graph sense). As we vary the location of $Q$ continuously, we'll find some decomposition int... | 2 | https://mathoverflow.net/users/89672 | 428563 | 173,737 |
https://mathoverflow.net/questions/428433 | 2 | My apologies if this question is below the level of MO. I posted the [same question](https://math.stackexchange.com/questions/4507690/dcc-on-the-powers-of-ideals) in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring. $R$ is called *strongly $\pi$-regular* if it satisfies the DCC on th... | https://mathoverflow.net/users/164350 | DCC on the powers of ideals | I am not aware of a term for rings with this property, and I have rarely seen it in the literature. There is a small result about such rings in Proposition 3.22 of the following paper:
*Lam, T. Y.; Reyes, Manuel L.*, [**A prime ideal principle in commutative algebra**](http://dx.doi.org/10.1016/j.jalgebra.2007.07.016... | 3 | https://mathoverflow.net/users/778 | 428569 | 173,741 |
https://mathoverflow.net/questions/428562 | 2 | Given a complete smooth Toric surface (over $\mathbb C$), is its intersection form well-known? Or is there an algorithm to calculate it? Thanks in advance.
| https://mathoverflow.net/users/88180 | Intersecton form of complete smooth Toric surface | The Picard group of a toric surface is generated by the simple toric divisors $D\_1,D\_2,\dots,D\_n$, and if $v\_1,v\_2,\dots,v\_n$ are the generators of the rays of the corresponding fan, there are relations
$$
\sum f(v\_i)D\_i = 0
$$
for all linear functions $f$. The intersection product is determined by these linear... | 6 | https://mathoverflow.net/users/4428 | 428583 | 173,746 |
https://mathoverflow.net/questions/428577 | 5 | So I was interested in formally assigning values to the completely divergent series $G(x) = \sum\_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to tackle this but feel free to continue reading for a strategy i think MIGHT work.
We start by consider a different totally ... | https://mathoverflow.net/users/46536 | Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives | $$
G(x) = \sum\_{n=0}^\infty n!x^n
\tag1$$
Another approach is to observe that the series $G(x)$ formally satisfies the differential equation
$$
x^2 G'(x) + (x-1) G(x) + 1 = 0 .
\tag2$$
The unique solution of $(2)$ with $\lim\_{x\to 0}G(x) = 1$ is
$$
\widetilde{G}(x) = -\frac{1}{x}\;e^{-1/x}\;\operatorname{Ei}\_1\left(... | 8 | https://mathoverflow.net/users/454 | 428595 | 173,752 |
https://mathoverflow.net/questions/428223 | 4 | $\DeclareMathOperator\BS{BS}$The linearity of the Baumslag-Solitar groups $\BS(m, n)=\langle a, t\mid t^{-1}a^mt=a^n\rangle$ is completely understood, and it may be phrased as: $\BS(m, n)$ is linear if and only if it is residually finite\*.
A *generalised Baumslag-Solitar (GBS) group* is a group which may be realised... | https://mathoverflow.net/users/6503 | When is a generalised Baumslag-Solitar group linear? | (1) I've been looking a little more. My belief is now:
**Conjecture.** *Let $G$ be a generalized Baumslag-Solitar group (i.e. the Bass-Serre fundamental group of a nonempty finite graph of groups in which every vertex and edge group is infinite cyclic). Let $\langle x\rangle$ be one vertex group. Then we have one of ... | 2 | https://mathoverflow.net/users/14094 | 428598 | 173,754 |
https://mathoverflow.net/questions/428446 | 6 | When I started studying the basics of $\phi^{4}\_{d}$, I looked for papers or lecture notes which would give me some general ideas about the topic and which would construct and/or prove the basic results of the theory. One of the main targets of this theory is, for example, the rigorous construction of the formal objec... | https://mathoverflow.net/users/152094 | Reference request for $\phi^{4}_{d}$ theory - where to begin? | For an introduction to the basics of quantum field theory you could look into [Introduction to Quantum Field Theory for Mathematicians](https://souravchatterjee.su.domains//qft-lectures-combined.pdf). Lectures 13 and 18-22 introduce the $\phi^4$ model in 3+1 dimensions and the perturbative calculation of transition pro... | 6 | https://mathoverflow.net/users/11260 | 428604 | 173,755 |
https://mathoverflow.net/questions/428581 | 0 | I have an integral of the form
$$
I = \int\limits^{1}\_{0} \exp\left(\dfrac{vt}{(v+1)^2 + v^2} - vt\right) dv
$$
and I want to prove that $I\leq c t^{-1}$ for the sufficiently large $t$, where $c$ is a constant independent of $t$.
Can anyone give me some hints or references to prove this expansion?
| https://mathoverflow.net/users/489455 | Asymptotic estimation of an integral | (This question should be on math.stackexchage.com.)
Substitute $v=t^{-1/2}u$, then it becomes
$$ t^{-1/2} \int\_0^{t^{1/2}} e^{-2u^2}\bigl(1 + O(u^3/t^{1/2})\bigr)\,du
= \sqrt{\frac{\pi}{8t}} + O(t^{-1}).$$
| 2 | https://mathoverflow.net/users/9025 | 428605 | 173,756 |
https://mathoverflow.net/questions/428555 | 24 |
>
> **Question:** Can I have an arbitrarily large finite family of lines $\ell\_1,\dotsc,\ell\_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$?
>
>
>
We can assume that all lines pass through the origin, and by angle I mean the smaller one of the two angles defined by two ... | https://mathoverflow.net/users/108884 | Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$? | Start with $n$ times each coordinate axis, so $3n$ lines in total. The average angle between non necessarily distinct lines is $1/3\*0 + 2/3\*\pi/2 = \pi/3$. So the average angle between "distinct" lines (here, distinct lines may coincide geometrically due to multiplicities) is greater than $\pi/3$.
You can then pert... | 15 | https://mathoverflow.net/users/112954 | 428607 | 173,757 |
https://mathoverflow.net/questions/428336 | 6 | Let $a,b\in\mathbb C$ be suc that $\max\{|a+b|,|a-b|\}\leq 1$ but $|a|+|b|>1.$ According to this paper by Arias, Figiel, Johnson and Schechtman <https://www.jstor.org/stable/2155206?origin=crossref#metadata_info_tab_contents>, the two dimensional complex subspace spanned by $(1,0,a)$ and $(0,1,b)$ in complex $\ell\_\in... | https://mathoverflow.net/users/136860 | Subspaces of $\ell_\infty^3$ | A funny isometry invariant to distinguish these normed spaces is: *The space of spheres of radius $2$ in the unit sphere of $(X,\|\cdot\|\_X)$ which are maximal by inclusion*, as described below. It turns out that for $\ell\_\infty^2(\mathbb C)$ it is a torus $\mathbb S^1\times \mathbb S^1$, and for the space $Y:=\text... | 5 | https://mathoverflow.net/users/6101 | 428609 | 173,758 |
https://mathoverflow.net/questions/428135 | 6 | I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came from Morse theory in differential topology. But I am not sure how the definition of "Morse index" in [2] can be related to the defini... | https://mathoverflow.net/users/151368 | Morse index in PDEs | In finite dimensional Morse theory, you study a function $f:M\to\mathbb{R}$ and look for it's critical points, i.e., where $(df)\_p = 0$. Then, Morse theory says that a count of these critical points is related to the underlying topology of $M$, as long as you count correctly. To do so, you should define the **Hessian*... | 8 | https://mathoverflow.net/users/1540 | 428613 | 173,759 |
https://mathoverflow.net/questions/428616 | 3 | I've edited this post two years ago on Mathematics Stack Exchange, with identifier **3590406** and same title [*On conjectures about the arithmetic function that counts the number of Sophie Germain primes*](https://math.stackexchange.com/questions/3590406/on-conjectures-about-the-arithmetic-function-that-counts-the-num... | https://mathoverflow.net/users/142929 | On conjectures about the arithmetic function that counts the number of Sophie Germain primes | A reasonable conjecture is that
$$ \operatorname{Germain}(x) = 2 C \int\_{2^x} \frac{dx}{\log^2 x} + O( x^{1-\delta} )$$ for some $\delta >0$. This is the Hardy-Littlewood conjecture with power-saving error term.
A function-field analogue of this conjecture follows from the methods of my paper [On the Chowla and tw... | 6 | https://mathoverflow.net/users/18060 | 428621 | 173,760 |
https://mathoverflow.net/questions/428542 | 5 | Let $G$ be a connected reductive group defined over a finite field $F$ of characteristic $> 3$. Is it true that the commutator group of $G(F)$ is perfect? This is true if $G$ is assumed to be semisimple.
| https://mathoverflow.net/users/4690 | Is the derived group of the G(F) perfect | $\newcommand{\ssc}{\text{sc}}
\newcommand{\der}{\text{der}}
\DeclareMathOperator{\im}{im}$**Yes,** this is true for a reductive $F$-group $G$, if
for the *simply connected semisimple* $F$-group $G^\ssc$, see below,
the group of $F$-points $G^\ssc(F)$ is perfect,
that is, the derived group $G^\ssc(F)^\der$ coincides wit... | 6 | https://mathoverflow.net/users/4149 | 428626 | 173,763 |
https://mathoverflow.net/questions/428631 | 2 | For any set $X$ and cardinal $\kappa$, we denote by
$\ [X]^\kappa :=\ \binom X\kappa\ $ the collection of subsets of $X$ having cardinality $\kappa$.
If $X$ is a set, we call a set system $E\subseteq {\cal P}(X)$ *linear* if for all $a\neq b\in X$ there is exactly one $e\in E$ with $\{a,b\}\subseteq e$. For example, ... | https://mathoverflow.net/users/8628 | $k$-regular linear set systems | I think they exist for all $k \geq 2$. Here is a proof. Fix an enumeration of $[\omega]^2$. For the first step in the process add an arbitrary $k$-set $q\_1=q(p\_1)$ containing the first pair $p\_1$ of the enumeration. At step $i$, if $\ p\_i\subseteq q(p\_j)\ $ for some $\ j<i\ $ then let $\ q(p\_i):=q(p\_j).\ $ Other... | 4 | https://mathoverflow.net/users/2233 | 428639 | 173,768 |
https://mathoverflow.net/questions/427603 | 6 | In a compact connected Lie group $G$, each element is conjugate to an element of a maximal torus $T$. For a classical group, one can pick a basis of the tautological representation such that $T$ is represented by diagonal matrices. The conjugacy classes in $G$ are in bijective correspondence with the orbits of the Weyl... | https://mathoverflow.net/users/5018 | How to distinguish conjugacy classes in SO(2n) efficiently? | One method, I don't know whether it is the most efficient, is to compute the following polynomial quantity $Q(g) = \mathrm{Pf}(g-g^t)$ for $g\in\mathrm{SO}(2n)$, where $g^t$ means the transpose of $g$ and $\mathrm{Pf}$, known as the *Pfaffian*, is a polynomial of degree $n$ on the skew-symmetric $2n$-by-$2n$ matrices t... | 7 | https://mathoverflow.net/users/13972 | 428641 | 173,770 |
https://mathoverflow.net/questions/428610 | 0 | A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e. for each $\lambda\in \Omega$ the map $f\mapsto f(\lambda)$ is a continuous linear functional on $\mathscr H$. The Ries... | https://mathoverflow.net/users/113054 | A reproducing kernel Hilbert space | A useful test case for RKHS (which is not like the interesting examples, but does satisfy the definitions) is $\Omega={\mathbb N}$ and $H=\ell^2({\mathbb N})$. Note that $\hat{k\_n}$ is just the usual unit basis vector that is $1$ in position $n$ and $0$ everywhere else.
Viewing $T$ as an ${\mathbb N}\times {\mathbb ... | 2 | https://mathoverflow.net/users/763 | 428643 | 173,771 |
https://mathoverflow.net/questions/428638 | 1 | Let $N$ be a Riemannian manifold, and $\widetilde{N}\subset N$ a closed submanifold. If we look at the total spaces of the tangent bundles, we get that $T\widetilde{N}$ is a submanifold of $TN$.
If we now use the metric to identify $\widetilde{M}=T^\*\widetilde{N}$ with $T\widetilde{N}$, and $M=T^\*N$ with $TN$, this... | https://mathoverflow.net/users/940 | Cotangent bundles to Riemannian manifolds and submanifolds | May I please call the submanifold $M$ instead of $\tilde N$?
For a smooth manifold $N$, $T^\ast N$ does not just have a canonical symplectic form $\omega$. It also has a canonical $1$-form $\alpha$ such that $d\alpha=\omega$. For $w\in T^\ast N$, the element $\alpha\_w\in T^\ast\_w(T^\ast N)$ can be defined in a coor... | 3 | https://mathoverflow.net/users/6666 | 428652 | 173,776 |
https://mathoverflow.net/questions/428651 | 7 | Jech exercise 13.3 states:
>
> If $M$ is closed under Gödel operations and extensional, and $\pi$ is the transitive collapse of $M$, then $\pi(G\_i(X,Y))=G\_i(\pi X,\pi Y)$ for all $i=1,\ldots,10$ and all $X$, $Y\in M$.
>
>
>
I'm able to show this for the first 5 Gödel operations ($\{X,Y\}$, $X\times Y$, $\var... | https://mathoverflow.net/users/4133 | Why do $\pi$ and $\bigcup$ commute for Gödel-closed extensional classes? | Here is what I wrote when I solved this problem a few years ago:
As $M$ is extensional, the transitive collapse is an isomorphism. The statement $C=G\_i(A,B)$ can be expressed by a $\Delta\_0$ formula $\phi\_i(A,B,C)$. If we assume that these operations are absolute for extensional classes, then $Z=G\_i(X,Y)$ if and ... | 7 | https://mathoverflow.net/users/3199 | 428656 | 173,777 |
https://mathoverflow.net/questions/428653 | 0 | Consider a random scalar variable $X$ with arbitrary measure.
I'm after a basis of polynomial functions $\{p\_k\}\_{k=0}^\infty$ which are orthonormal with respect to $X$ in the sense that
\begin{equation}
\mathbb{E}\_X [p\_k(X)p\_{k'}(X)] = \delta\_{kk'}.
\end{equation}
When discussing orthogonal polynomial bases,... | https://mathoverflow.net/users/60511 | Orthogonal polynomials w.r.t. an arbitrary measure | For example, you can write orthogonal polynomials as determinants
$p\_n(x) = c\_n \, \det \begin{bmatrix}
m\_0 & m\_1 & m\_2 &\cdots & m\_n \\
m\_1 & m\_2 & m\_3 &\cdots & m\_{n+1} \\
\vdots&\vdots&\vdots&\ddots& \vdots \\
m\_{n-1} &m\_n& m\_{n+1} &\cdots &m\_{2n-1}\\
1 & x & x^2 & \cdots & x^n
\end{bmatrix}$,
wher... | 1 | https://mathoverflow.net/users/483414 | 428666 | 173,778 |
https://mathoverflow.net/questions/428669 | 11 | The character table of a finite group will be called *integral* if all its entries are integers. There are $11$ such groups up to order $16$, namely $C\_1$, $C\_2$, $C\_2^2$, $S\_3$, $D\_8$, $Q\_8$, $C\_2^3$, $D\_{12}$, $C\_2 \times D\_8$, $C\_2 \times Q\_8$ and $C\_2^3$. There are $76$ such groups up to order $120$ (s... | https://mathoverflow.net/users/34538 | Finite groups with integral character table | There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their structure: [Structure and Representations of $\mathbb{Q}$-Groups](https://link.springer.com/book/10.1007%2FBFb0103426) by D.... | 16 | https://mathoverflow.net/users/35416 | 428671 | 173,780 |
https://mathoverflow.net/questions/428685 | 0 | We call an $n\times n$-matrix ${\bf A}\in \text{Mat}(n\times n, \mathbb{R})$ a *metric matrix* if
1. ${\bf A}\_{ii} = 0$ for all $i\in \{1,\ldots,n\}$,
2. ${\bf A}\_{ij} = {\bf A}\_{ji}$ for all $i,j \in \{1,\ldots,n\}$ (that is, ${\bf A}$ is symmetric), and
3. ${\bf A}\_{ik} \leq {\bf A}\_{ij} + {\bf A}\_{jk}$ for a... | https://mathoverflow.net/users/8628 | Realizability of metric matrices | In other words, you ask whether every finite metric space may be isometrically embedded to Euclidean space $\mathbb{R}^n$. Not every. A necessary and sufficient condition is given by the non-negativity of the so called [Cayley--Menger determinants](https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant).
| 5 | https://mathoverflow.net/users/4312 | 428692 | 173,783 |
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