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https://mathoverflow.net/questions/374492 | 4 | Let $0<\beta<1$ and $ f \colon [0,1] \to [0,1]$ be $\beta$ Hölder continuous with constant $C$. Let $H$ be a Hilbert space and $A,B$ be self adjoint operators on $H$, such that $\sigma(A+B),\sigma(A) \subset [0,1]$. Then we can define $f(A+B)$ and $f(B)$ by the continuous functional calculus. Do we then have the estima... | https://mathoverflow.net/users/123409 | Hölder continuity of functional calculus | Such questions have been much studied, in particular by Aleksandrov and Peller. Probably the most relevant reference is the paper *Functions of operators under perturbations of class $S\_p$* by Aleksandrov and Peller,
J. Funct. Anal. 258 (2010). [Zbmath link](https://zbmath.org/?q=an%3A1196.47012) or [mathscinet link](... | 7 | https://mathoverflow.net/users/10265 | 374506 | 156,331 |
https://mathoverflow.net/questions/374484 | 4 | Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0.
The function can be defined in one of the following equivalent ways.
* The number $f(x,y,z)$.
* The number of all par... | https://mathoverflow.net/users/82465 | Combinatorial representation of function | I'll rename all three of your variables; you are asking for the number of partitions of $k$ that fit into an $m \times n$ box. This is famously known to be the coefficient of $q^k$ in the [$q$-binomial coefficient](https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient)
$${m+n \choose m}\_q = \frac{[m+n]\_q!}{[m... | 5 | https://mathoverflow.net/users/290 | 374513 | 156,333 |
https://mathoverflow.net/questions/374453 | 15 | Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets.
**Question:** Is there a nontrivial **signed** measure on $\mathfrak{L}(\mathbb{R})$ that is trivial on $\mathfrak{B}(\mathbb{R})$?
Obviously, any **positive** measure that is trivial on $\... | https://mathoverflow.net/users/166613 | Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets | So, promoting my answer to a comment, this is unprovable in ZFC (assuming ZFC is consistent). I claim that such a signed measure $\nu$ exists only if there exists a nontrivial, atomless, countably additive probability measure $\mu$ on the discrete $\sigma$-algebra of $\mathbb{R}$ (or equivalently $[0,1]$). As I underst... | 6 | https://mathoverflow.net/users/4832 | 374520 | 156,337 |
https://mathoverflow.net/questions/374521 | 3 | This problem is motivated by the problem of reconstructing a genome from the family of its short subwords.
Given a word $w$ and a positive integer $k$, let $M\_k(w)$ be the family of all subwords of length $k$ and $\mu\_{k,w}:M\_k(w)\to \omega$ be the function assigning to each subword $v\in M\_k(w)$ the number of su... | https://mathoverflow.net/users/61536 | The mean value of the reconstruction complexity of a random sequence | The expectation of the reconstruction complexity of a random word of length $n$ over the alphabet $A$ is $E[k(n)]=(2\pm o(1))\log\_{|A|}(n)$.
**Upper bound.** We show that $E[k(n)]\leq 2\log\_{|A|}(n)+2$, for any $n$ large enough. Given a word $w$, let $k'=\min\{l:\mu\_{l,w}\equiv 1\}$. Then, $k(n)\leq k'+1$. To see ... | 2 | https://mathoverflow.net/users/85550 | 374526 | 156,340 |
https://mathoverflow.net/questions/374543 | 14 | (I am most interested in the case $X=\mathbb R^2$, but of course one could ask the same question for manifolds, or metric spaces in general.)
Let $\text{Com}(\mathbb R^2)$ denote the space of nonempty compact subsets of the plane, equipped with the Hausdorff metric. Let $S\_\bullet:[0,1]\to\text{Com}(\mathbb R^2)$ be... | https://mathoverflow.net/users/57604 | Must a path of compact sets in $X$ descend to a path in $X$? | $\DeclareMathOperator{\R}{\mathbf{R}}\DeclareMathOperator{\Z}{\mathbf{Z}}$The answer is no, even in the circle (and hence in the plane).
As coordinates, write the circle as the 1-point compactification $\bar{\R}$ of $\R$.
For $t\in\mathopen]0,1]$, write $$X\_t=\{\infty\}\cup\big(t\Z+\sin(1/t)\big).$$
For $t\to 0$, ... | 13 | https://mathoverflow.net/users/14094 | 374548 | 156,345 |
https://mathoverflow.net/questions/374551 | 8 | $\newcommand{\Psh}{\operatorname{Psh}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\O}{{\mathcal{O}}}$
Let $X$ be a locale, $\O(X)$ the corresponding frame.
1. What's the localic reflection of $\Psh X$?
We know that
$$
\O(X) \cong \mathrm{Sub}\_{\Sh X}(1)
$$
Call $Y = \mathrm{Sub}\_{\Psh X}(1)$ the localic refl... | https://mathoverflow.net/users/50376 | What's the localic reflection of a presheaf topos? | I'm writing $\mathcal{O}(X)$ for the frame corresponding to $X$.
Opens of $Y$ are sieves on $X$, i.e. the collection of open subsets $V \subset \mathcal{O}(X)$ such that $v \in V$ and $u \leqslant v \Rightarrow u \in V$. Thus **$Y$ is the locale of downward closed subsets of $\mathcal O(X)$**.
More generally if $P$... | 11 | https://mathoverflow.net/users/22131 | 374555 | 156,347 |
https://mathoverflow.net/questions/374553 | 2 | My question is surely a classical one in the algebraic number theory, but I'm not working in it and I do not know the results and references. Please excuse me.
Let $k = \mathbf{F}\_q$ be a finite field and let $\ell$ be a prime number invertible in $k$. Let $\mathbf Z\_\ell(1) = \varprojlim\_{r\to\infty} \mu\_{\ell^r... | https://mathoverflow.net/users/38052 | Galois cohomology of $\mathbf{Z}_\ell(m)$ over finite fields | The Galois cohomology of finite fields is pretty straightforward: a $G\_k$-module $M$ is entirely determined by the action of Frobenius $\varphi$, and we have $H^0(k, M) = M^{\varphi = 1}$, $H^1(k, M) = M/(1 - \varphi)M$ (and it's zero in all other degrees).
On $\mathbf{Z}\_\ell(m)$, the Frobenius acts as multiplicat... | 5 | https://mathoverflow.net/users/2481 | 374556 | 156,348 |
https://mathoverflow.net/questions/374542 | 5 |
>
> Given a matroid $M$ with ground set $E$ of size $2n$, suppose there exists $A\subseteq E$ of size $n$ such that both $A$ and $E\setminus A$ are independent. What is the minimum number of $B\subseteq E$ such that both $B$ and $E\setminus B$ are independent?
>
>
>
With $n=2$, some casework shows that the answe... | https://mathoverflow.net/users/65718 | Minimum number of independent pairs in a matroid | As observed by Geva Yashfe, the answer is $2^n$. This can be achieved when each of $A$ and $\overline{A}:=E\setminus A$ are bases, with $A = \{a\_1,\ldots,a\_n\}$, $\overline{A} = \{b\_1,\ldots,b\_n\}$, and $a\_i$ parallel to $b\_i$ for all $i \in [n]$.
For the lowerbound, by truncation, we may assume that $A$ and $\... | 7 | https://mathoverflow.net/users/2233 | 374564 | 156,352 |
https://mathoverflow.net/questions/374566 | 15 | The following theorem is usually attributed to [Eduard Study](https://en.wikipedia.org/wiki/Eduard_Study):
>
> Let $f(x,y)$ and $g(x,y)$ be polynomials in two variables over a field, with $f$ irreducible. If $f\nmid g$ then the curves $C\_f:f=0$ and $C\_g:g=0$ have finitely many points of intersection. Consequently... | https://mathoverflow.net/users/33757 | History of Study's Lemma? | I found the lemma on page 63 of Study's [Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung](https://archive.org/details/einleitungindiet01studuoft/page/62/mode/2up) (1923).
The source cited for the proof is page ... | 17 | https://mathoverflow.net/users/11260 | 374568 | 156,353 |
https://mathoverflow.net/questions/374565 | 3 | Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum\_{i,j}{a\_{ij}}z\_i\frac{\partial}{\partial z\_j}$ be the linear term of $v$ and suppose that the matrix $a\_{ij}$ is invertible and all its eigenvalues have modulus different from $1$. Is it... | https://mathoverflow.net/users/13441 | Holomorphic vector fields with a non-degenerate isolated zero | A relevant result, but not a complete answer.
Vladimir Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, p. 181:
>
> A zero of a vector field is in the Poincare domain if the origin is not in the convex hull of the eigenvalues of the linearization.
>
>
>
>
> A resonance of a zero... | 4 | https://mathoverflow.net/users/13268 | 374572 | 156,355 |
https://mathoverflow.net/questions/374573 | 0 | The Sierpinski-Mazurkiewicz paradox yields a nonempty rigid-motion paradoxical subset $S$ of the Euclidean plane: $S$ is the disjoint union of $A$ and $B$, each of which is $G$-equidecomposable with $S$, for a group $G$ of rigid motions.
(Here sets $U$ and $V$ are $G$-equidecomposable for a group of $G$ acting on a s... | https://mathoverflow.net/users/26809 | Is there a $G$-paradoxical $G$-invariant subset of the plane for $G$ a group of rigid motions? | No, it's not possible because the isometry group of the Euclidean plane is amenable (as discrete group, since it's solvable), so every $G$-set admits an invariant mean defined on all subsets. If $S$ is $G$-invariant, this applies to $S$ as $G$-set and an immediate contradiction follows.
| 4 | https://mathoverflow.net/users/14094 | 374580 | 156,358 |
https://mathoverflow.net/questions/374547 | 2 | (By request from a comment: UF stands for [Univalent Foundations](https://en.wikipedia.org/wiki/Univalent_foundations))
Correct me if I'm wrong, but in a model $M$ of ZF each element $x$ of $M$ should produce a directed-graph-with-a-marked-sink $G\_x$ having $x$ as marked sink, as follows: to $\varnothing$, i. e. the... | https://mathoverflow.net/users/41291 | What do UF and ZF do to each other? | This is a standard way of building a model of "material" / membership-based set theory (such as ZFC) from a "structural" / categorical set theory (such as ETCS or the sets in HoTT/UF). In the context of comparing membership-based set theories to category theory and topos theory, it goes back at least to the work of Mit... | 8 | https://mathoverflow.net/users/49 | 374590 | 156,362 |
https://mathoverflow.net/questions/374417 | 1 | Let $H$ be a Hilbert space, $T\_+(H)$ the set of positive self-adjoint trace-class operators on $H$, and $f : T\_+(H) \to [0,m]$ a non-negative, bounded, convex functional. I don't necessarily know that it's continuous or semicontinuous.
Assume that a minimizer $x\_0 \in \operatorname{argmin}\_{x \in T\_+(H)} f(x)$ e... | https://mathoverflow.net/users/76565 | Linearity of the directional derivative of a convex functional at the minimum | A sufficient condition and a necessary and sufficient condition for the linearity of a [functional derivative](https://mathoverflow.net/questions/349057/question-about-functional-derivatives/349584#349584) were given by Mikhail Mordukhovich Vaĭnberg in his well known monograph [1], chapter 1 §3.2, pp. 37-40. Before bri... | 1 | https://mathoverflow.net/users/113756 | 374593 | 156,364 |
https://mathoverflow.net/questions/374597 | 1 | Let $I$ be a small category and $\mathcal{D}=D^b\_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. Define a bounded t-structure on $\mathcal{C}$ by lifting the one on $\mathcal{D}$, that is $\mathcal{C}^{\leq 0}=\math... | https://mathoverflow.net/users/138396 | Computing Ext groups in a functor stable $\infty$-category | $\newcommand{\Z}{\mathbb Z} \newcommand{\Ch}{\mathrm{Ch}} \newcommand{\Fun}{\mathrm{Fun}}$
Let $\Ch(\Z)$ be the projective model category of chain complexes. It is well known that it presents $D\_\infty(\Z)$.
Moreover, $\Fun(I,\Ch(\Z))$ with its projective model structure presents $\Fun(I,D\_\infty(\Z))$, and of cour... | 7 | https://mathoverflow.net/users/102343 | 374598 | 156,365 |
https://mathoverflow.net/questions/374595 | 6 | In Giraud's book "Cohomologie non-abelienne", the author repeatedly cites sources using something like [D blah]. E.g., Chapter 1, section 1, first line: "Nous renvoyons à [D 1] et à [SGA 1 VI]..."
The problem is there's nothing in the bibliography labelled [D]. The closest thing are two papers of Dedecker's. The firs... | https://mathoverflow.net/users/88840 | What are the references of the form [D blah] in Giraud's cohomologie nonabelienne? | In the bibliography you can find the following item:
>
> 11. Giraud,J.: Méthode de la descente. Mémoires Soc. Math. Fr. 2 (1964) (cité [D]).
>
>
>
My understanding then is that citationd like [D 1] refer to specific sections or results in the reference [D], which is above. Note something similar is done with r... | 13 | https://mathoverflow.net/users/30186 | 374601 | 156,366 |
https://mathoverflow.net/questions/374581 | 2 | Given real symmetric matrix $\mathbf{M}$ with eigenvalues $\lambda\_i$ and eigenvectors $\mathbf{v}\_i$, the derivative of an eigenvector is $$\dot{\mathbf{v}}\_i = \sum\_{j \ne i} \frac{\mathbf{v}\_j \mathbf{v}\_j^T}{\lambda\_i - \lambda\_j} \dot{\mathbf{M}} \mathbf{v}\_i$$
This is obviously not defined when $\lambd... | https://mathoverflow.net/users/167408 | Calculating second derivatives of eigenvectors of a matrix with some degenerate eigenvalues | If $\lambda\_i$ is simple, then the eigenvalue and eigenvector are as smooth as your matrix will allow. You start from $(M-\lambda\_i)v\_i=0$, $v\_i^Tv\_i=1$. Differentiating this once yields $(M-\lambda\_i)\dot v\_i+(\dot M-\dot\lambda\_i)v\_i=0$, $\dot v\_i^Tv\_i=0$. From this you calculate $\dot\lambda\_i=v\_i^T\dot... | 3 | https://mathoverflow.net/users/12120 | 374609 | 156,369 |
https://mathoverflow.net/questions/370698 | 8 | I have seen in some engineering departments that they manufacture models of periodic minimal forms (characterised by equal and opposite curvature at every points on the surface). In pure mathematics, they are known as triply periodic minimal surfaces.
If I understand rightly, these have been observed experimentally i... | https://mathoverflow.net/users/119114 | Work on triply periodic minimal surfaces | This is an active research topic. I'm currently working on the front line towards a classification of TPMSs of genus 3 (TPMSg3s). My collaborators include Weber and Traizet. I also know a Japanese team working on the moduli space.
Recent progress include surprising discoveries of new examples:
* <https://arxiv.org/... | 7 | https://mathoverflow.net/users/20595 | 374629 | 156,372 |
https://mathoverflow.net/questions/374631 | 5 | Let $I, J$ be two bases of a matroid. For every $x$ in $I$, there is some $y$ in $J$ such that, if we exchange $x$ with $y$, then both resulting sets ($I \setminus x \cup y$ and $J \setminus y \cup x$) are bases (this is the [strong basis exchange property](https://blog.zilin.one/2019/11/03/on-the-basis-exchange-proper... | https://mathoverflow.net/users/34461 | Exchanges between independent sets of a matroid | **No**, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K\_4$. The matroids that satisfy your property are called *base orderable* matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is... | 5 | https://mathoverflow.net/users/2233 | 374634 | 156,373 |
https://mathoverflow.net/questions/374636 | 9 | If $\{p\_i\}$ is the sequence of all primes, is it possible that there exist a non constant $P\in \mathbb{Z}[x\_1,\dots x\_n]$ such that $P(p\_i,p\_{i+1},\dots p\_{i+n-1})$ is bounded in $i$?
More precisely, can widely believed conjectures, or even heuristic arguments, help make such a claim (even more) unlikely.
| https://mathoverflow.net/users/2480 | Are polynomials bounded on the primes possible? | Here is a proof that such a polynomial does not exist assuming that every admissible [$n$-tuple](https://en.wikipedia.org/wiki/Prime_k-tuple) occurs infinitely often in the sequence of primes.
To see this let $a:=(0, a\_1, \dots, a\_{n-1})$ be an admissible $n$-tuple. Suppose $P \in \mathbb{Z}[x\_1, \dots, x\_n]$ is ... | 12 | https://mathoverflow.net/users/2233 | 374640 | 156,374 |
https://mathoverflow.net/questions/374637 | 2 | Let $V$ be a quasi-complete Hausdorff locally convex space. (By quasi-complete, one means that every bounded closed subset of $V$ is complete.) For a bounded closed absolutely convex subset $B$, denote by $V\_B$ the subspace of $V$ spanned by vectors in $B$ and define a norm function $q\_B$ on $V\_B$, namely, $$q\_B(v)... | https://mathoverflow.net/users/97981 | Subspaces of quasi-complete locally convex spaces | The answer to the first question is NO: For example, consider the inclusion of $\ell\_1$ into $\ell\_2$, then the unit ball $B$ of $\ell\_1$ is closed but $V\_B=\ell\_1$ is dense in $\ell\_2$.
The argument for the second question is sometimes attributed to Wendy Robertson although it is due to Grothendieck: Given a C... | 4 | https://mathoverflow.net/users/21051 | 374646 | 156,376 |
https://mathoverflow.net/questions/374610 | 0 | Assuming that $x$ is a real number, the function $f\_n(x)$ is defined as follows: the value of $f\_n(x)$ is equal to the number of bits before the first occurrence of $n$ consecutive zero bits in the binary representation of the fractional part of $x$. For example, $$\begin{array}{l}
f\_1(0.11100110001\ldots) = f\_2(0.... | https://mathoverflow.net/users/122796 | Explanation of unexpectedly large offset of the first occurrence of five consecutive zeroes in the sequence of second-to-last bits of primes | I don't think this is particularly surprising. In particular, it seems like it's a 1-in-400 event or so. Mathematics is big enough that there are [lots of rare events](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf)!
Let's first think about the simpler question of asking when we exp... | 3 | https://mathoverflow.net/users/8345 | 374659 | 156,377 |
https://mathoverflow.net/questions/374670 | 6 | It is a well known fact that if $\mathcal{F}$ is a non-principal ultrafilter on $\omega$, then the set $\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$ (conflating binary strings with subsets of $\omega$) is not a Borel subset of $2^\omega$ with its standard product topology.
The proof of this that I am familiar w... | https://mathoverflow.net/users/83901 | Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$ | A Borel ultrafilter would have the property of Baire. Therefore either $\mathcal F$ or its complement $2^\kappa\setminus\mathcal F$ is comeager relative to some basic open set. Since $\mathcal F$ is invariant under finite changes, this would mean $\mathcal F$ or its complement is comeager. Since $\mathcal F$ is the ima... | 4 | https://mathoverflow.net/users/164965 | 374679 | 156,384 |
https://mathoverflow.net/questions/374684 | 0 | Let $k\geq 2$, and let $P\_k$ be a sequence of polynomials, such that:
1. $P\_k=\sum\_{n=2}^{k+1}a\_{n,k}X^n \in \mathbb{Q}[X]$, $a\_{2,k}\neq 0$, $\deg P\_k \leq k+1$, and consider $P\_k :[0,1]\rightarrow \mathbb{R}$ as a real valued function.
2. $P\_k(1)=\frac{1}{k(k+1)}$ and $\mid a\_{n,k}\mid < \frac{2}{k}$, for ... | https://mathoverflow.net/users/163521 | Upper bound of a uniformly converging sequence of polynomials | Take
$$
P\_k(x)=\frac{x^2}{k(k+1)}+\frac{x^2(1-x)}k.
$$
| 3 | https://mathoverflow.net/users/17581 | 374686 | 156,386 |
https://mathoverflow.net/questions/374692 | 4 | I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
be a short exact sequence where $A$ maps to the center of $B$. Th... | https://mathoverflow.net/users/88840 | Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$ | See Section 5.7 of Serre's Galois Cohomology. (In general Chapter 5 of this book is a fairly definitive reference for non-abelian cohomology, and he works with an arbitrary profinite group $G$).
| 11 | https://mathoverflow.net/users/5101 | 374694 | 156,387 |
https://mathoverflow.net/questions/374690 | 3 | In Chapter 8.8 of Davis' "*The geometry and topology of Coxeter groups*" the smallest class $\mathcal{G}$ of Coxeter groups which contains all spherical Coxeter groups and which is closed under taking amalgamated free products of the form $W\_1 \ast \_{W\_0} W\_2 $ with $W\_1, W\_2 \in \mathcal{G}$ with common spherica... | https://mathoverflow.net/users/64444 | Decompositions of Coxeter groups into trees of groups | No. Indeed, every Coxeter group with no $\infty$ edge has Serre's Property FA and hence cannot be written as a nontrivial amalgam. Hence if it's not spherical or affine, it does not belong to the class you're defining. This notably applies to many Coxeter groups on 3 Coxeter generators.
(A group generated by a finite... | 4 | https://mathoverflow.net/users/14094 | 374697 | 156,388 |
https://mathoverflow.net/questions/374682 | 2 | Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the universal elliptic curve. For a given geometric point $b\to B$, the space $\Gamma(E\_b,\Omega^1\_{E\_b/b})$ is a 1 dimension vector s... | https://mathoverflow.net/users/44005 | Global section of vertical differential 1 forms on universal elliptic curve | What you're looking for is a section of the sheaf $\omega = \pi\_\* \Omega^1\_{E/B}$, where $\pi: E \to B$ is the structure map, which is a line bundle on $B$. This line bundle $\omega$ has a canonical extension to the compactification $\bar{B}$, and global sections of this line bundle over $\bar{B}$ are exactly weight... | 7 | https://mathoverflow.net/users/2481 | 374705 | 156,391 |
https://mathoverflow.net/questions/373067 | 9 | Let $S\_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:A \to S\_0$ be an abelian scheme of relative dimension $g$ such that the Newton polygon of $A[p^{\infty}]$ is constant.... | https://mathoverflow.net/users/56856 | A question about $p$-adic monodromy of abelian varieties | Let me briefly answer your question. There are two $p$-adic analogues of $R^1\pi\_\*\mathbb{Q}\_\ell/S\_0$: a (convergent) $F$-isocrystal $\mathcal E$ and an overconvergent $F$-isocrystal $\mathcal{E}^\dagger$. These two objects define algebraic monodromy groups $G\_F(\mathcal E)$ and $G\_F(\mathcal E^\dagger)$, where ... | 4 | https://mathoverflow.net/users/24479 | 374709 | 156,392 |
https://mathoverflow.net/questions/374718 | 1 | I was referred here from [this question](https://math.stackexchange.com/questions/3814733/curvature-waves-harmonic-curvature-and-curvature-flow/3875670#3875670) I asked on stackexchange. And now that I'm here, I see that [this other question about geometric wave equations](https://mathoverflow.net/questions/341325/geom... | https://mathoverflow.net/users/167518 | the curvature wave equation | A similar equation that's been used is the Penrose wave equation
\begin{equation}
\square R\_{a b c d} = 2 R\_{a e d f} R{\_b}{^e}{\_c}{^f} - 2 R\_{a e c f} R{\_b}{^e}{\_d}{^f} - R\_{a b e f} R{\_{c d}}{^{e f}} .
\end{equation}
This holds for a vacuum spacetime, i.e. $R\_{a b} = 0$.
I believe that it originates in
... | 3 | https://mathoverflow.net/users/39284 | 374722 | 156,394 |
https://mathoverflow.net/questions/374721 | 0 | Let $(X\_n)\_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma\_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}\_+^d,\,\sum\_{1\leq i\leq n} x\_i=1\big\rbrace$. Assuming that there exists $\alpha\in\Sigma\_d$ such that for $n\geq 1$, $\mathbb{E}[X\_n]=\alpha$, and that the sequence ... | https://mathoverflow.net/users/159940 | Weak convergence to a "multi-Bernoulli" distribution | $\newcommand\Ga\Gamma\newcommand\R{\mathbb R}$This answer is similar to the one linked by the OP.
Indeed, let $a:=\alpha$ and $(X\_{n,1},\dots,X\_{n,d})$.
We have $EX\_{n,1}=a\_1$ and $Var\,X\_{n,1}\to(1-a\_1)a\_1$, whence $EX\_{n,1}^2\to a\_1$ and $E(1-X\_{n,1})X\_{n,1}\to0$. So, for each $t\in(0,1)$,
$$P(t\le X\_... | 2 | https://mathoverflow.net/users/36721 | 374727 | 156,395 |
https://mathoverflow.net/questions/374617 | 3 | Sobczyk's theorem states that if a separable Banach space $X$ contains a subspace isometric to $c\_{0}$, then $X$ contains a subspace $Z$ which is isometric to $c\_{0}$ and is $2$-complemented in $X$. Since every projection from $c$ onto its subspace $c\_{0}$ has norm at least two, the projection constant in Sobczyk's ... | https://mathoverflow.net/users/41619 | An improvement of Sobczyk's Theorem | Narcisse and I gave a counterexample in
Johnson, William B.(1-TXAM); Randrianantoanina, Narcisse(1-MMOH)
On complemented versions of James's distortion theorems. (English summary)
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2751–2757.
You could have discovered this just by checking on MathSciNet reviews of papers tha... | 2 | https://mathoverflow.net/users/2554 | 374728 | 156,396 |
https://mathoverflow.net/questions/374710 | 1 | Let $x\_1,\dots,x\_n,y\_1,\dots,y\_n\in \mathbb{R}$ and such that $x\_i\neq x\_j$ and $y\_i\neq y\_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \sum\_{i=1}^n a\_i \delta\_{x\_i}$ and $\nu\triangleq \sum\_{i=1}^n b\_i \delta\_{y\_i}$. Are there known, (not too l... | https://mathoverflow.net/users/36886 | Closed-form upper-bounds for Wasserstein distance between finite measures | Let
$$F(x):=\mu((-\infty,x])=\sum\_i a\_i\,1(x\_i\le x)
=\sum\_{j=1}^n s\_j\,1(x\_{n:j}\le x<x\_{n:j+1}),$$
where $x\_{n:1}<\cdots<x\_{n:n}$ are the values $x\_1,\dots,x\_n$ put in the increasing order (with $x\_{n:n+1}:=\infty$),
$$s\_j:=\sum\_{i=1}^j a\_{n:i},$$
and $a\_{n:1},\dots,a\_{n:n}$ are the values $a\_1,\dot... | 3 | https://mathoverflow.net/users/36721 | 374736 | 156,398 |
https://mathoverflow.net/questions/374702 | 4 | Let $F$ be a free group on 2 generators and $G = \operatorname{SL}(d,\mathbb{C})$.
A word $w \in F$ induces the word map $\mathrm{ev}\_w: G \times G \to G$.
Can we find some (generic) conditions on $A,B \in G$ such that, for all $w \in F$,
the differential of $\mathrm{ev}\_w$ is surjective at $(A,B)$ ?
| https://mathoverflow.net/users/91134 | Local surjectivity of word maps | Since a theorem of Borel (which you quote in the comment) tells you that this regular map is dominant for every $w$, its differential is surjective on a Zariski-dense open subset $U\_w$.
Hence there intersection (over all $w\neq 1$) of these subsets $U\_w$ is a $\mathrm{G}\_\delta$-dense subset of full measure.
| 3 | https://mathoverflow.net/users/14094 | 374738 | 156,399 |
https://mathoverflow.net/questions/374696 | 4 | I want to know if the notion of completed tensor product in [Stacks Project Tag 0AMU](https://stacks.math.columbia.edu/tag/0AMU) is the one that yields
$$k[[x]] \widehat{\otimes} k[[y]]≅k[[x,y]].$$
Here I should be considering the inverse limit topology in the power series rings, and $R=k$ a field (with the trivial top... | https://mathoverflow.net/users/167503 | Completed tensor product and power series rings | $k[[x]]$ and $k[[y]]$ are topological $k$-algebras, and their completed tensor product is indeed isomorphic to $k[[x,y]]$ (with the topology defined by the maximal ideal $(x,y)$). This is because the completed tensor product is the projective limit of the algebras $k[x]/(x^m) \otimes k[y]/(y^n) = k[x,y]/(x^m,y^n)$ over... | 6 | https://mathoverflow.net/users/6506 | 374740 | 156,400 |
https://mathoverflow.net/questions/374423 | 6 | It is well-known that the Sobolev space $H^1(0,s)$ embeds continuously in the space of continuous functions $C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is $\sqrt{\coth(s)}$, with
$$\|\cosh\|\_\infty = \sqrt{\coth(s)} \|\cosh\|\_{H^1}.$$
Is the optimal embedding constant of $H^1\_0(0... | https://mathoverflow.net/users/26039 | Optimal constant in Sobolev embedding | I do not know a reference but the following argument gives the best constant. Consider the interval $[0,a]$ and $G(t,s)$ the Green function of $I-D^2$ with zero boundary conditions at $0,a$. If $u \in H^2 \cap H^1\_0$, then
$$u(t)=\int\_0^a G(t,s)\left (u(s)-u''(s)\right )ds=\int\_0^a \left(G(t,s)u(s)+G\_s(t,s)u'(s) \r... | 4 | https://mathoverflow.net/users/150653 | 374741 | 156,401 |
https://mathoverflow.net/questions/374732 | 42 | If $V \hookrightarrow W$ and $W \hookrightarrow V$ are injective linear maps, then is there an isomorphism $V \cong W$?
If we assume the axiom of choice, the answer is *yes*: use the fact that every linearly independent set can be extended to a basis and apply the usual [Schroeder-Bernstein theorem](https://en.wikipe... | https://mathoverflow.net/users/2362 | Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein? | Without the axiom of choice, it is possible that there is a vector space $U\neq 0$ over a field $k$ with no nonzero linear functionals.
Let $V$ be the direct sum of countably many copies of $U$, and $W=V\oplus k$.
Then each of $V$ and $W$ embeds in the other, but they are not isomorphic, since $V$ doesn’t have any ... | 45 | https://mathoverflow.net/users/22989 | 374744 | 156,403 |
https://mathoverflow.net/questions/374747 | 10 | For a logic $\mathcal{L}$, let the *compactness number* of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that there is no restriction here on the cardinality of the language of the theory in question.
For example, an uncountable ca... | https://mathoverflow.net/users/8133 | When do infinitary compactness numbers exist? | The compactness number for $\mathcal L\_{\kappa,\kappa}$ is equal to the least $(\kappa,\infty)$-strongly compact cardinal. A cardinal is $(\kappa,\infty)$-strongly compact if for every set $X$, there is a $j : V\to M$ such that $\text{crit}(j)\geq \kappa$, and $j[X]$ can be covered by and element of $M$ of $M$-cardina... | 11 | https://mathoverflow.net/users/102684 | 374752 | 156,406 |
https://mathoverflow.net/questions/350520 | 7 | [Sorry if the level here is wrong, I asked this on [math.SE](https://math.stackexchange.com/q/3498670/320311), but even with a bounty, it got no attention.]
I am currently reading [Hatcher's 3-Manifolds notes](https://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html), the part proving Alexander's theorem, which is a ... | https://mathoverflow.net/users/104963 | Generalized Schoenflies - formalizing step in proof? | For an interval $[a,b]\subset{\mathbb R}$ in which the height function $f:S\to {\mathbb R}$ has no critical values one obtains a product structure on $f^{-1}([a,b])$ by following flow lines of the gradient vector field of $f$. This vector field on $f^{-1}([a,b])$ can be extended to a vector field on ${\mathbb R}^2\time... | 10 | https://mathoverflow.net/users/23571 | 374755 | 156,407 |
https://mathoverflow.net/questions/374761 | 9 | It is probably an easy question, but somehow I am stuck.
**Question** Is the following statement true? If yes, how to prove it?
>
> Suppose that $f\in C^1(\mathbb{R}^n)$ is convex and
> $$
> \langle\nabla f(x)-\nabla f(y),x-y\rangle \leq L|x-y|^2
> $$
> for some $L>0$ and all $x,y\in\mathbb{R}^n$. Does it follow ... | https://mathoverflow.net/users/121665 | Convexity and Lipschitz continuity | That's a standard result in convex optimization. For example Theorem 2.1.5 in Nesterov's "Introductory Lectures on Convex Optimization" states that the following are equivalent:
* $f$ is $C^1$, convex and the gradient $\nabla f$ is $L$-Lipschitz
* for all $x,y$: $0\leq f(y) - f(x) - \langle\nabla f(x),y-x\rangle \leq... | 11 | https://mathoverflow.net/users/9652 | 374776 | 156,415 |
https://mathoverflow.net/questions/374695 | 3 | Let $\left\lbrace \mathsf{O}(n)\right\rbrace\_{n\in \mathbb{N}} $ be an operad in a symmetric monoidal category $(\mathsf{C},\otimes, \mathbf{1})$ which in addition has the structure of a model category (I think of topological spaces or chain complexes). In this case I think there exists the Boardman-Vogt construction ... | https://mathoverflow.net/users/130225 | Boardman-Vogt construction for PROP(erads) | Such a construction is described in the preprint *Boardman-Vogt resolutions of generalized Props* in [here](https://u.osu.edu/yau.22/main/). It uses the language of generalized props in the book [A Foundation for PROPs, Algebras, and Modules](https://bookstore.ams.org/surv-203/). Operads, properads, props, and their co... | 2 | https://mathoverflow.net/users/53034 | 374778 | 156,416 |
https://mathoverflow.net/questions/374766 | 3 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) that is a finite Fano plane, that is, $V$ is a finite set and $E$ has the following properties:
1. for $e\_1\neq e\_2\in E$ we have $|e\_1|=|e\_2|$, as well as $|e\_1\cap e\_2|=1$, and
2. for $v\neq w\in V$ there is a (unique) $e\in E$ with $\{... | https://mathoverflow.net/users/8628 | Injective choice function for finite Fano planes | Yes, there is always such a map. Let $k$ be the number of vertices in each edge of $H=(V,E)$. Consider an arbitrary vertex $v \in V$ and choose $e \in E$ such that $v \notin e$. For each $w \in e$ there is a unique edge $f\_w$ such that $\{v,w\} \subseteq f\_w$. Moreover, for distinct $w,w' \in e$, $f\_w \neq f\_{w'}$.... | 3 | https://mathoverflow.net/users/2233 | 374784 | 156,418 |
https://mathoverflow.net/questions/374771 | 3 | Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to FB \to FC$ is exact.
We know that for any right (resp. left) exact functor $F$, $L\_nF$ (resp. $R^nF$) are middle exa... | https://mathoverflow.net/users/94076 | Is every middle exact functor a derived functor? | It is not possible in general to infer $n$ and $G$ from a middle-exact functor $F$. Consider the module category of some self-injective, finite-dimensional algebra over a field, say a group algebra of a finite group. Then every left-derived functor vanishes on projectives and therefore factors through the stable module... | 3 | https://mathoverflow.net/users/3041 | 374786 | 156,420 |
https://mathoverflow.net/questions/374783 | 24 | What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far.
In [Lei2001], Leinster demonstrated 10 different definitions for an $n$-category, and made no comment on whether they are equivalent or not. In [BSP201... | https://mathoverflow.net/users/124549 | Definition of an n-category | First of all, there are important differences between the notions of strict $n$-category, weak $n$-category, and $(\infty,n)$-category. The easiest notion is that of a strict $n$-category, and [there's no doubt about the definition there](https://en.wikipedia.org/wiki/Higher_category_theory): a strict $0$-category is a... | 19 | https://mathoverflow.net/users/11540 | 374795 | 156,424 |
https://mathoverflow.net/questions/335645 | 6 | Let $P\subset\Bbb R^{24}$ be the *contact polytope* of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda\_{24}$.
>
> **Question:** What are the edges of $P$?
>
>
>
Let's say the norm of any vertex is $4\sqrt 2$.
I am pretty sure that any two vertices at distance $4\... | https://mathoverflow.net/users/108884 | Edges of the contact polytope of the Leech lattice | Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are:
* 1 vector $v$ with $\langle u, v \rangle = 4$ (namely $v = u$);
* 4600 vectors $v$ with $\langle u, v \rangle = 2$;
* 47104 vectors $v$ wi... | 5 | https://mathoverflow.net/users/39521 | 374818 | 156,430 |
https://mathoverflow.net/questions/374816 | 1 | Let us consider the equation:
$$
\dot{x}\_i = F\_i(x)
$$
with $x\in \mathbb{R}^n$ and $i=1\dots n$, and the equation for small displacements:
$$
\dot{\delta x} = \sum\_j \frac{\partial}{\partial x\_j} F\_i(x) \delta x\_j
$$
I often read (and checked in practice) the following: starting from a random initial $\delta x$,... | https://mathoverflow.net/users/138060 | Starting vector in Lyapunov exponents evaluation | To begin with, there is no reason whatsoever for the dynamical system determined by an **arbitrary** vector field on $\mathbb R^n$ to be Lyapunov regular.
If the system is Lyapunov regular, then the associated filtrations of the tangent space start from the bottom of the Lyapunov spectrum. If you are interested just ... | 1 | https://mathoverflow.net/users/8588 | 374821 | 156,432 |
https://mathoverflow.net/questions/374812 | 7 | I'm looking for references for the following closely related facts:
Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e\_b$ such that $e\_b e\_{b'} = e\_{b \cap b'}$ and $e\_b + e\_{b'} = e\_{b \cup b'}+ e\_{b \cap b'}$.
Then:
1. The $e\_b$ are the only idempotent of $\m... | https://mathoverflow.net/users/22131 | Functions on Stone spaces as "enveloping algebra" of Boolean algebra | Given a Boolean algebra, unital or non-unital, and a commutative ring $K$, the $K$-algebra $K[B]$ given by the generators and relations you give is isomorphic to the ring $C\_c(\widehat B,K)$ of locally constant $K$-valued functions on the Stone space $\widehat B$ with compact support.
The intuitive reason is that if... | 3 | https://mathoverflow.net/users/15934 | 374822 | 156,433 |
https://mathoverflow.net/questions/374824 | 4 | Let us define the following functions:
\begin{equation\*}
\small A(x)=\prod\_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
B(x)=\prod\_{\substack{p\leq x\\ p\equiv 1 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
C(x)=\prod\_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(... | https://mathoverflow.net/users/140356 | Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$ | I assume that you meant to write product and not sum. Defining
\begin{equation\*}
\small D(x)=\prod\_{\substack{p\leq x\\ p\equiv 1 \bmod 4}} \Big(1+\frac{1}{p}\Big) \mbox{ } \mbox{ }
\end{equation\*}
We see that
$$A(x)\cdot D(x) \sim L(1,\chi) = 1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$$
Where $\chi$ is... | 11 | https://mathoverflow.net/users/88679 | 374826 | 156,435 |
https://mathoverflow.net/questions/374834 | 9 | $\DeclareMathOperator{\Sp}{\mathrm{Sp}}$I am taking a special case $\Sp$ here, mainly because it has nice categorical properties.
Let $R$ be an $E\_\infty$-ring spectrum. In [Higher Algebra](http://people.math.harvard.edu/%7Elurie/papers/HA.pdf), Lurie proves we have a forgetful functor (part of monadic adjunction)
$... | https://mathoverflow.net/users/139900 | Is the forgetful functor $\mathrm{Mod}_R \mathrm{Sp} \rightarrow \mathrm{Sp}$ faithful? | $U\_R$ obviously preserves delooping, so if that were the case, because $\pi\_0 map(X,Y) = \pi\_1 map(X, \Sigma Y)$, you would also get an isomorphism on $\pi\_0$, so an equivalence of mapping spaces.
In other words, $U\_R$ is faithful if and only if it is fully faithful. But now for a map of ring spectra $R\to S$, t... | 13 | https://mathoverflow.net/users/102343 | 374839 | 156,437 |
https://mathoverflow.net/questions/374837 | 1 | Let $\phi\_1,...,\phi\_n,...$ be a sequence of real-valued functions so that $\phi\_j:[0,1)\to[0,1)$, $\phi\_j(0)=0$, and $\phi\_j(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\ge1$. Further suppose that $\sum\_{j=1}^\infty \phi\_j(\delta)$ **converges** and moreover, is strictly **smaller*... | https://mathoverflow.net/users/145053 | Convergence of the sum of a family of real-valued functions | Let $\phi\_i'(x)$ be defined as $1/2$ for $1/2^i<x<1/2^{i-1}$ and 0 elsewhere.
Clearly $\phi\_j':[0,1)\to[0,1)$, $\phi\_j'(0)=0$, and $\phi\_j'(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\ge1$. Also $\sum\_{j=1}^\infty \phi\_j'(x)$ converges and is equal to 0 if $x=0$ and $1/2$ elsewher... | 2 | https://mathoverflow.net/users/7113 | 374841 | 156,439 |
https://mathoverflow.net/questions/374846 | 6 | Say we toss $d$ pairwise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head?
If they had been fully independent then I believe the answer is $1/e$.
| https://mathoverflow.net/users/45564 | Lower bound for probability of getting exactly one head with pairwise independence | The highest lower bound is $1/d$.
Indeed, for each $j\in[d]:=\{1,\dots,d\}$, let $A\_j$ denote the event of the head on the $j$th coin and let $X\_j:=1\_{A\_j}$. Let $S:=X\_1+\dots+X\_d$. Then the event of getting exactly one head is $\{S=1\}$.
Note that $EX\_j=p$ and (by the pairwise independence) $EX\_jX\_k=p^2+p... | 9 | https://mathoverflow.net/users/36721 | 374852 | 156,443 |
https://mathoverflow.net/questions/374854 | 0 | Assuming we have a ordered list of 115 elements, if we select 60 elements from the list, prove that the interval for **at least** 2 selected elements are *exactly* 4 elements apart.
Example:
`A B C D E F G...`
Then A and F are 4 elements apart
I tried to find the probability that an element being selected, whic... | https://mathoverflow.net/users/167613 | Prove the the interval of selected elements in a list is exactly 4 | You can solve the problem via integer linear programming as follows. For $i\in\{1,\dots,115\}$, let binary decision variable $x\_i$ indicate whether $i$ is selected. For $i\in\{1,\dots,111\}$, let binary decision variable $y\_i$ represent $x\_i x\_{i+4}$. The problem is to minimize $\sum\_{i=1}^{111} y\_i$ subject to
\... | 2 | https://mathoverflow.net/users/141766 | 374855 | 156,444 |
https://mathoverflow.net/questions/374518 | 1 | In David Mumford's book Algebraic Geometry I, Complex Projective Varieties
treating mainly complex varieties as objects of interest on page
43 he defines what is a topologically unibranch variety $X$ at $x \in X$.
(3.9) Definition. Let $X$ be an affine variety over $\mathbb{C}$
and $x\in X$. Then $X$ is
topologically... | https://mathoverflow.net/users/108274 | Unibranch points (definition for varieties over arbitrary field) | For a scheme $X$, say that $X$ is *topologically* unibranch at $x$ if $\mathop{Spec} O\_{X,x}$ is geometrically unibranch (meaning that $O\_{X,y}$ is geometrically unibranch at all generisations $y$ of $x$).
Assume $X$ is irreducible for simplicity.
Then by <https://stacks.math.columbia.edu/tag/0BQ4>, $X$ is topologi... | 1 | https://mathoverflow.net/users/26737 | 374857 | 156,445 |
https://mathoverflow.net/questions/374859 | 0 | Let $\mu$ be a discrete, finitely supported probability measure in $\mathbb{R}^d$ and denote by $\phi$ be the characteristic function of $\mu$, i.e. $\phi(t)=\mathbb{E}e^{i<t,X>}$, where $X$ is a random variable with distribution $\mu$. Given a bounded open set $A\subset \mathbb{R}^d$, how does one express $\mu (A)$ in... | https://mathoverflow.net/users/24494 | Expressing the measure of a set in terms of the characteristic function of the measure | $\newcommand\R{\mathbb R}\newcommand\sn{\operatorname{sign}}$Let $n:=d$ and $f:=\phi$, so that $f(t)=Ee^{it\cdot X}$ for $t\in\R^n$. We have the following straightforward multivariate extension of (the special case, with $t=0$, of) formula (6) in [this arXiv paper](https://arxiv.org/pdf/1309.5928.pdf) or its [published... | 1 | https://mathoverflow.net/users/36721 | 374866 | 156,449 |
https://mathoverflow.net/questions/374872 | 4 | Let $X$ be a connected topological space with abelian fundamental group. Let $\mathcal{L}$ be a $\mathbb{Z}$-valued local system on $X$.
Suppose that I know the full homology $H\_\*(X;\mathbb{Z})$. Are there any tools which could allow me to compute (some part of) the local-coefficient homology $H\_\*(X; \mathcal{L})... | https://mathoverflow.net/users/155668 | Homology with local systems | One approach is to use mod $2$ homology. You know that
$H\_i(X;\mathbb Z/2)$ is isomorphic to both $ H\_i(X)\otimes \mathbb Z/2\oplus Tor(H\_i(X),\mathbb Z/2)$ and $H\_i(X;\mathcal L:)\otimes \mathbb Z/2\oplus Tor(H\_i(X;\mathcal L),\mathbb Z/2)$. If the integral homology groups are finitely generated, then this give... | 10 | https://mathoverflow.net/users/6666 | 374874 | 156,451 |
https://mathoverflow.net/questions/374850 | 4 | I found this topic in a book ['Metric Affine Geometry' by Ernst Snapper and Robert J. Troyer](https://books.google.ru/books/about/Metric_Affine_Geometry.html?id=PVbvAAAAMAAJ&redir_esc=y).
I call a field $k$ trigonometric iff there is a quadratic form $q$ over $k^2$ such that every two lines through the origin in $k^2$ ... | https://mathoverflow.net/users/91850 | Is there a trigonometric field which is different enough from real numbers? | A field $K$ is trigonometric iff the sum of 2 squares is a square and $-1$ is not a square (equivalently, the set of nonzero squares is stable under addition), in which case the standard scalar product on $K^2$ satisfies the required condition.
Indeed, suppose that $K$ is trigonometric, so there is a *nonzero* quadra... | 4 | https://mathoverflow.net/users/14094 | 374878 | 156,452 |
https://mathoverflow.net/questions/374893 | 6 | It seems to be the case that filtered colimits commute with finite limits in the category Set (for instance, this is shown in [Why do filtered colimits commute with finite limits?](https://mathoverflow.net/questions/57099/why-do-filtered-colimits-commute-with-finite-limits)), but does the same hold for the category of ... | https://mathoverflow.net/users/167636 | Do filtered colimits commute with finite limits in the category of pointed sets? | Yes, filtered colimits commute with finite limits in the category of pointed sets. This is because the forgetful functor from the category of pointed sets to the category of sets creates finite limits and filtered colimits -- in fact, it creates all limits and all connected colimits -- and so the category of pointed se... | 15 | https://mathoverflow.net/users/57405 | 374896 | 156,457 |
https://mathoverflow.net/questions/374683 | 2 | I am trying to figure out the structure of an M-matrix (<https://en.wikipedia.org/wiki/M-matrix>) whose inverse has a special form: Let $A$ be an inverse M-matrix (inverse M-matrices are those matrices whose inverse is an M-matrix, such that each row sum of the matrix is a fixed constant (greater than 1). Each diagonal... | https://mathoverflow.net/users/167252 | Inverse M-matrices structure | I got 2 counter-examples:
1. Let $A= \begin{pmatrix}1.1 & 0.89\\0.89 & 1.1 \end{pmatrix}$, then $A^{-1}=\begin{pmatrix} 2.6322 &-2.1297\\-2.1297 & 2.6322 \end{pmatrix}$.
Here $x=0.8, y=0.9$, so, $(0.8)(1.1)\leq 0.89\leq (0.9)(1.1)$.
2. Let $A= \begin{pmatrix}5.1 & 4.9\\4.9 & 5.1 \end{pmatrix}$, then
$A^{-1}=\b... | 0 | https://mathoverflow.net/users/167252 | 374898 | 156,458 |
https://mathoverflow.net/questions/374909 | 1 | If $A\subseteq\mathbb{N}$ is a subset of the positive integers, we let $$\mu^+(A) = \lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$ be the *upper density* of $A$.
For $n\in\mathbb{N}$ we let $\sigma(n)$ be the number of divisors of $n$, the numbers $1$ and $n$ included.
Do we have $\mu^+\big(\sigma^{-1}(\{k... | https://mathoverflow.net/users/8628 | Distribution of pre-images of the divisor function $\sigma$ | Notice that $\sigma(p^{k-1}) = k$ and so the image of $\sigma$ is all of $\mathbb{N}$.
By the way, $\sigma$ is usually used for the sum of divisors function, and it is more standard to use $d$ or $\tau$ for your function.
EDIT: I misread the question. I will use $\tau$ instead of $\sigma$.
I claim that $\mu ^ {+}... | 3 | https://mathoverflow.net/users/88679 | 374913 | 156,463 |
https://mathoverflow.net/questions/374921 | 8 | As I was reading Grothendieck's Tohoku paper(translated by M.L.Barr and M.Barr), I found that the definition of a generator in the category differs from that defined in wikipedia.
Let $\mathbf{C}$ be a category(It may be necessary that $\mathbf{C}$ is a locally small category), a family of generators {$U\_i$}$\_{i\i... | https://mathoverflow.net/users/167661 | Are generators defined in Tohoku paper equivalent to that defined in Wikipedia (Which I believe is a more widely used definition) | In this answer, let me use the terms **generator** and **extremal generator** for the Wikipedia and Tohoku definitions respectively.
In general, these two definitions are not equivalent, and neither implies the other.
**Example 1.** For an example of an extremal generator which is not a generator, consider a non-tr... | 18 | https://mathoverflow.net/users/57405 | 374923 | 156,466 |
https://mathoverflow.net/questions/374924 | 6 | I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special case of
1. A Riemannian manifold and its associated Dirac operator
$$
d+d^\*: \Omega^\bullet \to \Omega^\bullet,
$$
2. a Kaehler ma... | https://mathoverflow.net/users/167165 | Atiyah-Singer for Riemannian and Kaehler manifolds | I highly recommend the discussion in Shanahan's book, The Atiyah-Singer Index Theorem (An introduction), Lecture Notes in Math 638. In addition to a sketch of the proof, he gives a nice discussion of how the formidable general statement of the theorem gives the answers for your two examples, plus the (spin) Dirac opera... | 10 | https://mathoverflow.net/users/3460 | 374927 | 156,468 |
https://mathoverflow.net/questions/374773 | 7 | I found this intriguing remark at the end of Woodin's *[Supercompact cardinals, sets of reals, and weakly homogeneous trees](https://www.jstor.org/stable/32425)* (1988):
>
> The assertion that every set of reals, in $L(\mathbb{R})$, is the projection of a weakly homogeneous tree has consequences beyond the usual re... | https://mathoverflow.net/users/69827 | Weakly homogenously Souslin sets and the measurability of $\omega_1$ | This result is implied by a result in Kechris's paper ["Subsets of $\aleph\_1$ constructible from a real"](https://authors.library.caltech.edu/38904/). Kechris proves the following: If there is a measurable cardinal, then every subset of $\omega\_1$ is constructible from a real if and only if every subset of $\omega\_1... | 8 | https://mathoverflow.net/users/102684 | 374939 | 156,471 |
https://mathoverflow.net/questions/374772 | 1 | This is a [cross-post](https://math.stackexchange.com/questions/3873192/is-a-locally-invertible-weak-limit-of-injective-maps-injective-almost-everywhere).
Let $\Omega\_1,\Omega\_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f\_n:\bar\Omega\_1 \to \bar\Omega\_2$ be Lipsc... | https://mathoverflow.net/users/46290 | Is a locally invertible weak limit of injective maps injective almost everywhere? | Okay, let me try a writeup of the comment chain. For any reasonable subset $A\subset \Omega\_2$ and $B := f^{-1}(A)$ you get
$$\int\_A |f^{-1}(y)| dy = \int\_B \det df dx \leq \liminf\_{n\to\infty} \int\_B \det df\_n dx = \liminf\_{n\to\infty} \mathcal{H}^2(f\_n(B)). $$
Then if we know that $\mathcal{H}^2(f\_n(B)) \t... | 2 | https://mathoverflow.net/users/51695 | 374941 | 156,472 |
https://mathoverflow.net/questions/374933 | 3 | Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Box u+qu=0\,\quad &\text{on $(0,\infty)\times \Omega$},
\\
u=f\,\quad &\text{on $\Sigma=(0,\infty)\times \partial \Omega$,}\\
(u,\partial\_t u)\to 0 \,\quad &\text{ on $\Ome... | https://mathoverflow.net/users/50438 | wave equation with vanishing trace at infinity | If $q$ is not signed, then in general the solution need not be unique.
The question of uniqueness can be reduced to the case where $f \equiv 0$.
In this case, the constant $0$ function obviously solve the PDE. So you just need an example of a non-zero solution.
Let $v$ be a Dirichlet eigenfunction of the Laplacia... | 4 | https://mathoverflow.net/users/3948 | 374944 | 156,473 |
https://mathoverflow.net/questions/374911 | 6 | Let $f:\mathcal{X}\to \mathcal{Y}$ be a separated quasi-finite map of qcqs Deligne-Mumford stacks. Is there a version of Zariski's main theorem that makes sense in this context? Rydh proved a version of this in the case where the map $f$ is also assumed to be representable, in which case we recover a stacky version of ... | https://mathoverflow.net/users/1353 | Zariski's main theorem for non-representable morphisms? | You can take the relative coarse map to get a factorization of $f$ into $\mathcal{X} \to X \to \mathcal{Y}$ where $g : X \to \mathcal{Y}$ is representable and $\pi : \mathcal{X} \to X$ is proper + quasi-finite with $\mathcal{O}\_X \to \pi\_\*\mathcal{O}\_{\mathcal{X}}$ an isomorphism. Then you can apply the representab... | 5 | https://mathoverflow.net/users/12402 | 374947 | 156,476 |
https://mathoverflow.net/questions/374883 | 7 | The following problem arose in asymptotic analysis of difference equations.
---
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)\le2,\tag{1}$$
where
$$r(a,b,c):=\Big|\frac{a b + a c - b c}{a^2}\Big|$$
and $h(\cdot,\cdot,\cdot)$ is the har... | https://mathoverflow.net/users/36721 | An elementary inequality for three complex numbers | I will prove the original inequality.
First, performing the change of variables $x=1/a$, etc., and inverting the harmonic mean, we need
$$
\sum \left|\frac{yz}{x(y+z-x)}\right|\geq \frac32.
$$
Next, denoting $p=y+z-x$, etc., we transform the inequality to
$$
\sum\left|\frac{(p+q)(p+r)}{p(q+r)}\right|\geq 3,
$$
or
$... | 13 | https://mathoverflow.net/users/17581 | 374969 | 156,485 |
https://mathoverflow.net/questions/374979 | 6 | I have been wondering if there are many cases of an author having published two (or more?) papers in the same issue of the same journal. I vaguely recall having seen one or two cases like this, maybe be old papers, but cannot vividly remember. I have the impression such a situation would make sense should the two paper... | https://mathoverflow.net/users/15155 | Famous cases of multiple papers by the same author published in same issue of same journal | Roger Howe famously filled an entire issue of Pacific Journal of Mathematics ([volume 73, no.2](//doi.org/10.2140/pjm.1977.73-2), 1977) with 8 different papers. (Also, Euler...)
| 32 | https://mathoverflow.net/users/19276 | 374982 | 156,489 |
https://mathoverflow.net/questions/374936 | 4 | Let $E$ be a Banach space, $T:E\rightarrow E$ a continuous bounded **nonlinear** mapping., and $\{x\_n\}\_{n\in\mathbb N}$ such that $$x\_{n+1}=T(x\_n),\:\forall n\in \mathbb{N}:=\{0,1,\cdots\}.$$
Let $$X\_n=\overline{\text{Conv}}\{x\_n,x\_{n+1}\cdots\}.$$
I want to prove that $$T\big(\bigcap\_{n=0}^{+\infty}X\_n\big... | https://mathoverflow.net/users/102228 | Inclusion of infinite intersection | No, take $E:=\mathbb{R}$, $x\_0:=1$ and $T$ any continuous bounded function with $T(1)=-1$, $T(-1)=1$, $T(0)=2$.
| 5 | https://mathoverflow.net/users/6101 | 374985 | 156,492 |
https://mathoverflow.net/questions/374647 | 9 | I'm learning about tight vs. overtwisted contact structures in contact geometry. I understand that we care about the existence/nonexistence of overtwisted disks in a contact structure in part because the distinction has proved useful (e.g., in classification).
But since contact geometry has a lot of applications to p... | https://mathoverflow.net/users/146012 | Physical motivation for tight/overtwisted dichotomy | In the physics of fluids, a reason for caring about tightness of the contact structure is the idea/conjecture that overtwisted discs raise the energy of the fluid.
The velocity field of an inviscid, incompressible fluid flow on a Riemannian manifold corresponds to a contact 1-form in dimension three.$^\ast$ In this c... | 4 | https://mathoverflow.net/users/11260 | 375013 | 156,501 |
https://mathoverflow.net/questions/375007 | 5 | I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}\_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz + zx),$$
and I suspect the only point of equality is (1,1,1).
It feels like the sort of thing that ought to have a... | https://mathoverflow.net/users/49446 | Elementary inhomogeneous inequality for three non-negative reals | Denote $x^2=a^3,y^2=b^3,z^2=c^3$. By AM-GM we have $1+2xyz=1+(abc)^{3/2}+(abc)^{3/2}\geqslant 3\sqrt[3]{1\cdot (abc)^{3/2}\cdot (abc)^{3/2}}=3abc$, so LHS is not less then $$a^3+b^3+c^3+3abc\geqslant ab(a+b)+bc(b+c)+ac(a+c)\\ \geqslant 2(ab)^{3/2}+2(bc)^{3/2}+2(ca)^{3/2}=2(xy+yz+zx),$$
the first inequality is [Schur](h... | 8 | https://mathoverflow.net/users/4312 | 375036 | 156,508 |
https://mathoverflow.net/questions/375024 | 7 | [I originally posted this on stackexchange](https://math.stackexchange.com/questions/3877667/assigning-a-canonical-geometry-to-a-seifert-surface), but it hasn't gotten an answer. I hope it's not inappropriate for this forum.
Suppose I have a knot $K: S^1 \hookrightarrow S^3$ with minimal genus Seifert surface $S$. I ... | https://mathoverflow.net/users/150528 | Assigning a "canonical geometry" to a Seifert surface | If $K$ is a non-trivial knot, then $\chi(S)<0$, so $S$ admits a hyperbolic structure as a surface. But in general, that metric does not arise from the emdedding into $S^3\setminus K$.
If $S^3\setminus K$ is hyperbolic, and $S$ is a properly embedded $\pi\_1$-essential surface in $S^3\setminus K$, then $S$ is either v... | 4 | https://mathoverflow.net/users/126206 | 375050 | 156,511 |
https://mathoverflow.net/questions/374860 | 7 | Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}\_{\mathbb{Z}/n\mathbb{Z}}(X)$, I can write it using equivariant homotopy theory as
$\mathrm{Vect}^{1}\_{\mathbb{Z}/n... | https://mathoverflow.net/users/73712 | The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex | You can replace $GL\_1(\mathbb C)$ with its maximal compact subgroup, which is $S^1$. Since $S^1$ is an abelian compact Lie group, there is a natural $\mathbb Z/n$-equivariant equivalence
$$B\_{\mathbb Z/n}S^1\xrightarrow{\simeq} \mbox{Map}(E\mathbb Z/n, BS^1).$$
See, for example, [this](https://mathoverflow.net/ques... | 4 | https://mathoverflow.net/users/6668 | 375051 | 156,512 |
https://mathoverflow.net/questions/375046 | 2 | Let $E$ be a Banach space, $T:E\rightarrow E$ a continuous, norm-bounded, and **nonlinear** mapping., and $\{x\_n\}\_{n\in\mathbb N}$ such that $$x\_{n+1}=T(x\_n),\:\forall n\in \mathbb{N}:=\{0,1,\cdots\}.$$
Let $$X\_n=\overline{\text{Conv}}\{x\_n,x\_{n+1}\cdots\}.$$
Let $X\_{\infty}=\bigcap\_{n=0}^{+\infty}X\_n$. **... | https://mathoverflow.net/users/102228 | Compactness of a sequence | In general it may fail to be compact. Consider $E:=L\_2(\mathbb{R})$, and $x\_n:=\chi\_{[n,n+1]}$. Clearly, for any $n\in\mathbb{N}$, all functions in the set $\overline{\text{co}}\{x\_k:k\ge n\}$ have support in $[n,+\infty)$, and in the intersection we get a compact nonempty set, the singleton $X\_\infty=\{0\}$. Howe... | 1 | https://mathoverflow.net/users/6101 | 375054 | 156,514 |
https://mathoverflow.net/questions/375058 | 2 | I am trying to prove or disprove the following Lemma:
Let $S=[n]$ and $\mathcal{T}$ be the set of all $k$-subsets of $S$ that contain $t \in [n]$. Furthermore, let $\mathcal{R}$ be the set of all $k$-subsets of $S$ that do no contain $t$.
>
> Is it possible to choose $|\mathcal{R}|$ elements from $\mathcal{T}$ de... | https://mathoverflow.net/users/167752 | Recovering set of $k$-subsets without specific element $t$ by modifying subsets with element $t$ | I assume what is meant is whether it is always possible to choose a size $|\mathcal R|$ subcollection $\mathcal U$ of $\mathcal T$ and elements $e\_U \notin U$ for each $U \in \mathcal U$ such that $\{(U \setminus \{t\}) \cup \{e\_U\} \mid U \in \mathcal U\}$ is equal to $\mathcal R$. I claim that the answer is **yes**... | 3 | https://mathoverflow.net/users/2233 | 375067 | 156,518 |
https://mathoverflow.net/questions/375064 | 35 | In the [homepage](http://www.crm.umontreal.ca/2020/Nombres2020/index_e.php) for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-Kato conjecture for the symmetric square motive of an ellipti... | https://mathoverflow.net/users/85392 | The modularity theorem as a special case of the Bloch-Kato conjecture | That is not what the link says. To quote (emphasis mine):
>
> ... in which this conjecture was **reduced** to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic curve.
>
That means something quite different. You could equally say that Wiles "reduced" the proof to the... | 22 | https://mathoverflow.net/users/167758 | 375069 | 156,519 |
https://mathoverflow.net/questions/375074 | 6 | On the objects of the category of groups we define the mapping $G\mapsto \operatorname{Hol}(G)$, the holomorph $G\rtimes \operatorname{Aut}(G)$ of $G$. Can we extend this mapping to a functor on this category? (Via extension to morphisms)
| https://mathoverflow.net/users/36688 | Is $G\mapsto \operatorname{Hol}(G)$ the object component of any functor on the category of groups? | There is no such functor. Recall that a *split epimorphism* is a morphism $f : x \to y$ with a section (right inverse) $g : y \to x$, satisfying $fg = \text{id}\_y$. Split epimorphisms, as their name suggests, are epimorphisms, and moreover they are *absolute* epimorpisms in that they are preserved by any functor whats... | 9 | https://mathoverflow.net/users/290 | 375076 | 156,521 |
https://mathoverflow.net/questions/375063 | 2 | As the title of the question susggests, I would like to show that
The "trivial bound is that"
\begin{align\*}
\lim\_{Q\to\infty}\lim\_{N\to\infty}\frac{1}{N\pi(Q)}\sum\_{n<N}\left|\sum\_{\substack{p<Q\\p|n}}p-\pi(Q)\right|&\leq \lim\_{Q\to\infty}\lim\_{N\to\infty}\frac{1}{N\pi(Q)}\sum\_{n<N}\left(\sum\_{\substack{p... | https://mathoverflow.net/users/159298 | Showing that $\lim_{Q\to\infty}\lim_{N\to\infty}\frac{1}{N\pi(Q)}\sum_{n<N}\left|\sum_{\substack{p<Q\\p|n}}p-\sum_{p<Q}1\right|=0$ | It seems that the conjecture is **false**, if I did not miss some asymptotc issue The essence of what follows is that I show that the interior limit exceeds *any* function of $Q$ tending to $0$.
For any $t\geq 1$, denote by $p\_t(Q)$ the density of those $n$ divisible by at least one $p$ with $t\pi(Q)<p<Q$. We have
\... | 2 | https://mathoverflow.net/users/17581 | 375079 | 156,522 |
https://mathoverflow.net/questions/375071 | 8 | Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that makes it easier). I have a few questions regarding their interpolation theory:
* Is $C^1(M)$ an interpolation space for th... | https://mathoverflow.net/users/126651 | Interpolation theory and $C^k$-spaces | $C^1$ is not an interpolation space between $C$ and $C^2$. There is an example due to Mitjagin and Semenov of a sequence $T\_n$ of uniformly bounded operators both in $C,C^2$ whose norm blow up in $C^1$. You find this example in the booklet by A. Lunardi "Interpolation Theory" (SNS Pisa publisher). In the edition in my... | 8 | https://mathoverflow.net/users/150653 | 375081 | 156,524 |
https://mathoverflow.net/questions/374966 | 4 | I'm trying to understand convenient vector spaces, but I'm unsure about the definition of the topology on smooth maps.
A map $f : E \rightarrow F$ between locally convex vector spaces $E$ and $F$ is called smooth iff it maps smooth curves to smooth curves. Set of all smooth maps is denoted with $C^\infty(E,F)$.
I'm... | https://mathoverflow.net/users/167679 | Convenient vector space and its locally convex structure | None of your suggestions is correct. Your third description is the closest, but you should include also finitely many of the derivatives there to reproduce KM 3.11 correctly, in a different form. A description of a fundamental system of seminorms for $C^\infty(E,F)$ can be given as follows: Take all seminorms $q\_{n,C}... | 2 | https://mathoverflow.net/users/12643 | 375083 | 156,525 |
https://mathoverflow.net/questions/375066 | 6 | Can anyone suggest a way to numerically compute the following matrix vector product?
$$u=A^{-1}b=(AA\otimes BB + AB \otimes BA)^{-1}\operatorname{vec}(C)$$
Here $AA,BB,AB,BA$ and $C$ are $d\times d$ matrices with $d\approx 1000$. A naive method is to expand Kronecker products and use linear solver for an answer in ... | https://mathoverflow.net/users/7655 | Computing $(AA\otimes BB + AB \otimes BA)^{-1}$ | Your equation is equivalent to
$$
(B^{-1}A)X + X(B^{-1}A)^T = B^{-1}A^{-1}CB^{-T}B^{-T},
$$
where $X$ is the matricization of $u$. This is a [Sylvester equation](https://en.wikipedia.org/wiki/Sylvester_equation), which can be solved for $X$ in $O(d^3)$ time.
You'll get some numerical ugliness if $A$ or $B$ are close ... | 5 | https://mathoverflow.net/users/11236 | 375087 | 156,526 |
https://mathoverflow.net/questions/374828 | 1 | Computing the exact volume of a polytope given in half space representation seems to be NP-hard. One paper I found proved it is hard for rational coefficients. (However, the paper itself was behind a paywall, so I don't know what exactly they did.)
What are the weakest restrictions under which the problem is still di... | https://mathoverflow.net/users/167596 | Exact volume calculation of a polytope is NP hard under which restrictions? | Not sure answering ones own question is common practice here, but I did get the information I was looking for out of that paper:
*Dyer, M. E.; Frieze, A. M.*, [**On the complexity of computing the volume of a polyhedron**](http://dx.doi.org/10.1137/0217060), SIAM J. Comput. 17, No. 5, 967-974 (1988). [ZBL0668.68049](... | 1 | https://mathoverflow.net/users/167596 | 375089 | 156,527 |
https://mathoverflow.net/questions/375085 | 6 | Let $\Gamma\_1$ and $\Gamma\_2$ be two subgroups of the rank-$2$ free group $F\_2$. Can then one find a nontrivial lower bound on the growth exponent of their intersection $\Gamma\_1 \cap \Gamma\_2$, in terms of the growth exponents of the two subgroups?
Here by the *growth exponent* of a subgroup $\Gamma \subset F\_... | https://mathoverflow.net/users/39348 | Lower bound on growth for intersection of two subgroups of free group | OK, in fact I figured this out as soon as I properly understood how Stallings graphs work. Here is the analogous construction for groups.
Let $\Gamma\_0$ be the subgroup of the free group $F\_2 = \langle a, b \rangle$ generated by the set $\{a^{k+1} b^{-1} a^{-k} \;\mid\; k \geq 0\}$. I claim that elements of $\Gamma... | 4 | https://mathoverflow.net/users/39348 | 375095 | 156,528 |
https://mathoverflow.net/questions/375009 | 8 | Let $(M^n,g)$ be a complete Riemannian manifold with $|Rm| \le 1$. Can we find two positive constants $C$ and $\epsilon$, depending only on $n$, such that under the normal coordinates $(g\_{ij})$ with respect to any point $p \in M$, we have
$$
|\partial\_k g\_{ij}(x)| \le C
$$
for any $|x| \le \epsilon$?
As pointed o... | https://mathoverflow.net/users/105900 | First order estimates of geodesic normal coordinates | The answer is 'no' for $n=2$ (and hence for all higher $n$). Here is how one can see this.
First, when $n=2$, recall that, by the Gauss Lemma, a metric $g$ in geodesic normal coordinates $(x,y)$ centered on $p$ takes the form
$$
g = \mathrm{d}x^2 + \mathrm{d}y^2
+ h(x,y)\bigl(x\,\mathrm{d}y-y\,\mathrm{d}x)^2,
$$
wh... | 9 | https://mathoverflow.net/users/13972 | 375100 | 156,531 |
https://mathoverflow.net/questions/374980 | 4 | Consider the Lichnerowicz Laplacian arising in the study of the stability of Einstein metrics:
$\Delta\_L h\_{ij} := \nabla^\* \nabla h\_{ij} + 2 R\_{i p j q} h\_{pq}$.
I am interested to know, on $\mathbb {CP}^n$, as explicitly as possible, the first eigentensors for this operator on the space of traceless, diverg... | https://mathoverflow.net/users/40460 | Eigentensors for Lichnerowicz Laplacian on $\mathbb{CP}^n$ | The eigenvalues and the corresponding eigentensors of the Lichnerowicz Laplacian on the complex projective space are explicitly known. See here:
<https://www.sciencedirect.com/science/article/pii/S0393044010000926>
| 3 | https://mathoverflow.net/users/20823 | 375113 | 156,534 |
https://mathoverflow.net/questions/375110 | 22 | The characteristic polynomial of a real symmetric $n\times n$ matrix $H$ has $n$ real roots, counted with multiplicity.
Therefore the discriminant $D(H)$ of this polynomial is zero or positive.
It is zero if and only if there is a degenerate eigenvalue.
Thus $D(H)$ is a non-negative (homogeneous) polynomial in the $\... | https://mathoverflow.net/users/55893 | Discriminant of characteristic polynomial as sum of squares | The answer for a general $n$ is positive: the discriminant is a sum of squares of polynomials in the entries of $H$. The first formula was given by Ilyushechkin and involves $n!$ squares. This number was improved by Domokos into
$$\binom{2n-1}{n-1}-\binom{2n-3}{n-1}.$$
See Exercise #113 on my [page.](http://perso.ens-l... | 19 | https://mathoverflow.net/users/8799 | 375115 | 156,535 |
https://mathoverflow.net/questions/375106 | 5 | The Lawson minimal surfaces $\xi\_{1,g} \subset \mathbf{S}^3$ are minimal surfaces with genus $g$. In Lawson's original construction [[Law70]](https://www.jstor.org/stable/1970625?seq=1)
these were constructed from geodesic triangulations. An alternative construction was given by Kapouleas [[Kap10]](https://arxiv.org/a... | https://mathoverflow.net/users/103792 | Flapping wings: on a question of Kapouleas | My understanding of Kapouleas' heuristic is as follows.
In $\mathbb{R}^3$, a member Scherk family has four asymptotic half-planes. When you blow these down you get two intersecting planes (at any angle in $\theta\in (0, \pi/2]$) -- this union is the tangent cone at infinity. However, if you look at them in the origin... | 5 | https://mathoverflow.net/users/127803 | 375116 | 156,536 |
https://mathoverflow.net/questions/375119 | 11 | Suppose that we have two polynomials that split:
$$\begin{align\*}
f(x)=\sum\_{k=0}^d a\_{d-k}x^k&=\prod\_{i=1}^d (x-\lambda\_i),\\
g(x)=\sum\_{k=0}^e b\_{e-k}x^k&=\prod\_{j=1}^e (x-\mu\_j).\\
\end{align\*}$$
Then the following result is often attributed to [James Joseph Sylvester](https://en.wikipedia.org/wiki/James_J... | https://mathoverflow.net/users/33757 | History of Sylvester's resultant? | The [resultant](https://en.wikipedia.org/wiki/Resultant) of the [Sylvester matrix](https://en.wikipedia.org/wiki/Sylvester_matrix) first appeared in J. J. Sylvester, Philos. Magazine **16**, 132–135 (1840): [A method of determining by mere inspection the derivatives from two equations of any degree](https://zenodo.org/... | 19 | https://mathoverflow.net/users/11260 | 375122 | 156,539 |
https://mathoverflow.net/questions/375105 | 0 | On page 7 in the article referred to below an axiom $D9$ is stated as follows: $$A\to B\to.\lnot(A \& \lnot B)~\\ (\text{equivalently: } (A\to\lnot A)\to\lnot A)$$
How may one prove the alleged equivalence, using the rest of the axioms and the inference rules:
D1: $\vdash A\to A$
D2: $\vdash((A\to B)\wedge(B\to C... | https://mathoverflow.net/users/37385 | Counterexample equivalent in relevant logic DL | From D3 and D4,
1- $A\wedge\neg B\rightarrow A$.
2- $A\wedge\neg B\rightarrow \neg B$.
From 2, D8 and R1,
3- $B\rightarrow \neg(A\wedge\neg B)$.
From 1, 3 and R3 (taking $B=A$, $C=B$, $A=A\wedge\neg B$ and $D=\neg(A\wedge\neg B)$)
4- $(A\rightarrow B)\rightarrow (A\wedge\neg B\rightarrow \neg(A\wedge\neg B)... | 3 | https://mathoverflow.net/users/9825 | 375130 | 156,543 |
https://mathoverflow.net/questions/375092 | 13 | As everyone knows, every vector bundle on $\mathbf{P}^1$ splits as a direct sum of line bundles $\mathcal{O}(a\_1)\oplus\cdots\oplus\mathcal{O}(a\_n)$. This means that in the Weil-uniformisation description of vector bundles
$$\text{Bun}\_{\text{GL}\_n}(\mathbf{P}^1)\ =\ \text{GL}\_n(k(\mathbf{P}^1))\backslash \text{GL... | https://mathoverflow.net/users/119012 | Vector bundles on $\mathbf{P}^1$ and the Iwasawa decomposition | **Disclaimer:** I am not fully confident in my understanding of the terminology here. Corrections are welcome.
$\def\GL{\mathrm{GL}}\def\PP{\mathbb{P}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\Spec{\mathrm{Spec}\ }$My understanding is that this is Bruhat decomposition for the loop group of $\GL\_n$.
Rather than yo... | 4 | https://mathoverflow.net/users/297 | 375136 | 156,544 |
https://mathoverflow.net/questions/375132 | 13 | As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots?
Let's break it down to two questions:
1. Can you give an example of two non-isomorphic reductive groups $G\_1$ and $G\_2$ for which one gets the same ro... | https://mathoverflow.net/users/98901 | Why are coroots needed for the classification of reductive groups? | $\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Zent{Z}\newcommand\Q{\mathbb Q}\newcommand\Z{\mathbb Z}$The collections of roots and the coroots, as abstract root systems, provide the same information (each being recoverable as the dual of the other), which may be why it seems non-obvious that the co-roots matter. T... | 14 | https://mathoverflow.net/users/2383 | 375139 | 156,546 |
https://mathoverflow.net/questions/375077 | 0 | Suppose we have random variables Y, D and X, where Y is independent of D conditional on X (Y⊥D|X). If there is another variable Z=f(X), where f(.) is a measurable real function, my question is: (1) under what conditions can we have Y⊥D|Z ?; (2) do we need the sigma-algebra σ(Z) belongs to
σ(X), so σ(Z) is sub-σ-algebra... | https://mathoverflow.net/users/114831 | Relaxing conditional independent assumption | A sufficient condition to have $Y⊥D|Z$ is that $f$ is injective. The sharp condition (if $Y$ and $D$ are not specified) is
$(\*)$ $\sigma(X)$ should be contained in the completion of $\sigma(Z)$.
If $(\*)$ holds, then conditioning on $X$ is equivalent to conditioning on $Z$.
If (\*) does not hold, then there is an ... | 2 | https://mathoverflow.net/users/7691 | 375140 | 156,547 |
https://mathoverflow.net/questions/375125 | 0 | In a factor $M$, we know that for any two projections $P$ and $Q$ in $M$, either $P\preceq Q$ or $Q\preceq P$ holds true. Here $\preceq$ denotes the Murray-von Neumann subequivalence of two projections. Is there a von Neumann algebra which is not a factor, where such comparison holds true for any two projections? If ye... | https://mathoverflow.net/users/116379 | comparison of two projections in a non-factor von Neumann algebra | This can never happen. Let $M$ be a von Neumann algebra with a nontrivial center $Z(M)$. Take two nonzero mutually orthogonal projections $p,q \in Z(M)$. Suppose these projections are comparable. Then w.l.o.g. we have a partial isometry $v \in M$ so that $vv^\*=p$ and $v^\*v \leq q$ is a projection. Since $p$ is a cent... | 4 | https://mathoverflow.net/users/166500 | 375143 | 156,548 |
https://mathoverflow.net/questions/375142 | 3 | Let $K$ be a number field with ring of integers $O\_K$ is not PID. Can we estimate the cardinality of the following sets
$$\mathcal{A}= \{\mathcal{P}\subset O\_K \ |\ Nm(\mathcal{P})\leq x, \mathcal{P}\ \text{ is not principal}\},$$
$$\mathcal{B}= \{\mathcal{P}\subset O\_K \ |\ Nm(\mathcal{P})\leq x, \mathcal{P}\ \text... | https://mathoverflow.net/users/131448 | How many non principal prime ideals does a number field contain? | I assume that, in your question, $\mathcal{P}$ means a prime ideal of $\mathcal{O}\_K$.
It follows from the non-vanishing of Hecke $L$-functions $L(s,\chi)$ at $s=1$ (where $\chi$ is an unramified Hecke character of $K$) that $\#\mathcal{B}$ is asymptotically $\mathrm{li}(x)/h(K)$, and $\#\mathcal{A}$ is asymptotical... | 8 | https://mathoverflow.net/users/11919 | 375144 | 156,549 |
https://mathoverflow.net/questions/375094 | 6 | A metric space $(M,d)$ is *doubling* if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric generating sets $S$, is $\mathrm{Cay}(G, S)$ doubling under the path metric? Groups like this have polynomial growth, so th... | https://mathoverflow.net/users/123634 | Which groups are doubling? | I think this follows from a standard ball-packing argument.
Suppose that $G$ with the metric $\rho$ induced from the Cayley graph has growth $V(R)=|B\_R(1)| \sim R^d$, i.e. $\exists\ 0<c< C$ such that $cR^d\leq V(R)\leq CR^d$, where $B\_R(1)$ is the open ball of radius $R$ about the identity (indeed, this argument wo... | 4 | https://mathoverflow.net/users/1345 | 375146 | 156,551 |
https://mathoverflow.net/questions/375134 | 3 | I am actually interested in the same question for more general kinds of curves, but I will be specific.
Let $K$ be a field and $\overline{K}$ be an algebraic closure of $K$. Let's say that a "hyperelliptic curve" is a smooth projective $K$-curve $C$ of genus $\ge 2$ such that there is a degree $2$ morphism $C\_{\over... | https://mathoverflow.net/users/152899 | Is the set of hyperelliptic curves with a K-point closed? | In the "more sophisticated" direction, we can ask a similar question about the moduli *stack* $\mathscr{M}\_g$ of hyperelliptic curves of genus $g$. If $K$ is a topological field, there is a natural topology on the set $\vert\mathscr{M}\_g(K)\vert$ of isomorphism classes of of genus $g$ hyperelliptic curves over $K$: a... | 7 | https://mathoverflow.net/users/7666 | 375158 | 156,554 |
https://mathoverflow.net/questions/375161 | 4 | Is it true that a bounded real function $f:[0,1]\to[0,1]$ with only countably many discontinuities has to be of Baire class 1, that is pointwise limit of a sequence of continuous functions? Is there a counter-example?
This would be an easy consequence of a theorem stated here: <https://encyclopediaofmath.org/wiki/Bai... | https://mathoverflow.net/users/167834 | Baire class 1 and discontinuities | It sounds like you're looking for Baire's characterization theorem. The first reference Google gives me is [here](https://math.ucsd.edu/_files/undergraduate/honors-program/honors-program-presentations/2012-2013/Siuyung_Fung_Honors_Thesis.pdf).
| 4 | https://mathoverflow.net/users/23141 | 375166 | 156,556 |
https://mathoverflow.net/questions/375154 | 3 | So this is asking a basic and/or stupid question (my apology and appreciation) about Soergel modules that comes out of exercises by me who knows little about the subject.
Let $W$ be a finite Weyl group with standard representation $V$ ($=X^\*(T)\otimes\_{\mathbb{Z}}\mathbb{R}$). Let $R=\mathbb{R}[V]$ and $I\_W$ be th... | https://mathoverflow.net/users/31327 | Description of Soergel modules | The cyclicity of the Soergel modules (the ideals you write are principal) is equivalent to rational smoothness of the corresponding Schubert varieties. Or more generally, this can be written as the following condition on Kazhdan-Lusztig polynomials $P\_{xy}(q)=q^{l(x)-l(y)}$, which works e.g. for dihedral groups and an... | 4 | https://mathoverflow.net/users/120010 | 375169 | 156,557 |
https://mathoverflow.net/questions/374729 | 1 | In *Galois extensions of structured ring spectra*, Rognes introduces the notion of a faithful $G$-Galois extension of ring spectra. Let me recall what this means:
We have a commutative ring spectrum $R$ with an action of $G$, and a $G$-equivariant morphism of commutative ring spectra $S\to R$ where $S$ has the trivia... | https://mathoverflow.net/users/102343 | Galois extensions of ring spectra and subextensions | It turns out that I was misapplying Rognes' 5.6.3. : when $p\neq n$, the action of $\pi\_1(S^1)$ on $H\_\*(C\_n; \mathbb F\_p)$ need not be nilpotent.
In particular, the following is a counterexample : take any odd prime $p$, then the projection $S^1\to \mathbb RP^1$ is a mod $p$-equivalence, so that $\mathbb F\_p^{S... | 0 | https://mathoverflow.net/users/102343 | 375174 | 156,558 |
https://mathoverflow.net/questions/375096 | 2 | Consider a projective system $\dots X\_{n+1} \to X\_n \to \dots \to X\_1$ of completely regular Hausdorff spaces with projective limit $X$. Then the linking mappings $f\_n$ induce a projective system (in the category of sets) of spaces of probability measures $\dots P(X\_{n+1}) \to P(X\_n) \to \dots \to P(X\_1)$ with t... | https://mathoverflow.net/users/58682 | Projective limit of spaces of probability measures | Just some night thoughts on your question, but too long for a comment.
1. If all of your $X$’s are compact, then everythig is fine and the desired projective limit is just the family of probability measures on the (compact) projective limit of the $X$’s (I am assuming, by the way that the image of $X\_n$ is equal to ... | 2 | https://mathoverflow.net/users/131781 | 375181 | 156,561 |
https://mathoverflow.net/questions/375193 | 1 | I'm looking for an example of the following:
A hypothesis class $\mathcal{H}$ such that
* $\forall h \in \mathcal{H}$, the number of free parameters of $h$ is equal to $n \in \mathbb{N}$ (where $n < \infty$); and
* The VC dimension of $\mathcal{H}$ satisfy $\text{VC-dim}(\mathcal{H}) > n$.
I'm only familiar with ... | https://mathoverflow.net/users/150065 | Finite VC dimension > the number of free parameters | Here's a classic example. For $\alpha>0$, define $f\_\alpha(x)=\sin(\alpha x)$ and let $F$ be the collection of all functions $f\_\alpha$ thresholded at $0$ --- that is, every $h \in F$ is the sign function composed with some $f\_\alpha$. Then every member of $F$ is fully specified by a single parameter, $\alpha$, but ... | 1 | https://mathoverflow.net/users/12518 | 375195 | 156,563 |
https://mathoverflow.net/questions/375183 | 0 | Let $C\_1$ and $C\_2$ be two proper full dimensional closed convex cones in $\mathbb{R}^n$ that are pointed. Suppose that $C\_1\subseteq C\_2$ and that the boundary of $C\_1$ is contained in the boundary of $C\_2$. Then is $C\_1=C\_2$? Any references for a result of this form would be welcome. I suspect this to be true... | https://mathoverflow.net/users/125733 | If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$? | Assume that $q\in C\_2\setminus C\_1$. Let $p$ be an interior point of $C\_1$. Then the interval $(p,q)$ contains a boundary point of $C\_1$ but only interior points of $C\_2$. A contradiction.
| 4 | https://mathoverflow.net/users/4312 | 375196 | 156,564 |
https://mathoverflow.net/questions/374926 | 2 | Let $\mathcal{F}$ be the Grothendieck ring of an abelian fusion category. Let $(M\_i)$ be its fusion matrices and $(\mathrm{diag}(\lambda\_{i,j}))$ their simultaneous diagonalization. Take $M\_1=id$, so that $\lambda\_{1,j}=1$. The numbers $$c\_j:=\sum\_i \vert \lambda\_{i,j} \vert^2$$ are usually called the *formal co... | https://mathoverflow.net/users/34538 | Schur orthogonality relation on fusion categories | By Lemma 2.3 in [this paper](https://mathscinet.ams.org/mathscinet-getitem?mr=2576705) by V. Ostrik (which uses Proposition 19.2(b) in [this paper](https://mathscinet.ams.org/mathscinet-getitem?mr=1974442) by G. Lusztig): $$ \sum\_i \lambda\_{i,j} \overline{\lambda\_{i,j'}} = \delta\_{j,j'} c\_{j} $$
Let $U$ be the mat... | 1 | https://mathoverflow.net/users/34538 | 375201 | 156,566 |
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