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https://mathoverflow.net/questions/309860 | 4 | The question is related to the question: [detecting weak equivalences in a simplicial model category](https://mathoverflow.net/questions/309538/detecting-weak-equivalences-in-a-simplicial-model-category/309546?noredirect=1#comment771217_309546)
Suppose that we have a simplicial model category $M$ and denote by $M^{f}... | https://mathoverflow.net/users/128371 | detecting weak equivalences in a simplicial model category II | I think the following gives a counterexample:
Take the category of morphisms in $\operatorname{SSet}$, i.e., the category $\operatorname{Fun}(C,\operatorname{SSet}),$ where $C$ is the category with two objects and one non-identity morphism. Equip this category with the projective model structure. $R$ will be the cate... | 3 | https://mathoverflow.net/users/51424 | 310365 | 135,106 |
https://mathoverflow.net/questions/310307 | 3 | Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(t)x$ faster than that of $\dot{x}=A(t)x$?
This question is initially asked on Mathematics:
<https://math.stackexchange.co... | https://mathoverflow.net/users/122100 | Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$? | A simple counterexample can be constructed as follows: take
$$ A(t) = \pmatrix{0&\tfrac{\pi}{2}\\-\tfrac{\pi}{2}&0} \qquad \text{for } t \in [2n, 2n+1) ,$$
and
$$ A(t) = \pmatrix{-\alpha&0\\0&-\beta} \qquad \text{for } t \in [2n+1, 2n+2) ,$$
where $0 < \alpha \leqslant \beta$.
The equation $x'(t) = c A(t) x(t)$ descr... | 5 | https://mathoverflow.net/users/108637 | 310367 | 135,108 |
https://mathoverflow.net/questions/310388 | 5 | Let $G=(V,E)$ be a finite, simple, undirected graph. A *matching* is a set $M\subseteq E$ of pairwise disjoint edges. A *vertex cover* is a set $C\subseteq V$ of vertices such that $C\cap e \neq \emptyset$ for all $e\in E$.
The *matching number* $\mu(G)$ of $G$ is the maximum size that a matching can have, and the *v... | https://mathoverflow.net/users/8628 | Vertex cover number vs matching number | Minimum $c$ equals 2 as is seen from the complete graph on a large number $n$ of vertices: $\tau$ equals $n-1$, $\mu$ equals $[n/2]$.
| 3 | https://mathoverflow.net/users/4312 | 310390 | 135,116 |
https://mathoverflow.net/questions/310369 | 2 | Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$.
Let $\{a\_{\lambda}\}\_{\lambda}$ be a family of ideals of $S$.
Is there some "natural" assumption on $\varphi$ to guaranty that
$$
\cap\_{\lambda} a\_{\lambda}^e = (\cap\_\lambda a\_\lambda)^e,
$$ where $\\_^e$ denotes the extension to $... | https://mathoverflow.net/users/80525 | Homomorphisms of ring extending nicely ideal intersections | [Note: I am assuming that you are considering commutative algebras. However, everything below also works in the non-commutative case if you use right (left) ideals, and assume that $R$ is finitely presented and flat as a *left* (*right*) $S$-module.]
Given an $S$-algebra $R$, the assumption that $R$ is **flat and fin... | 2 | https://mathoverflow.net/users/86006 | 310394 | 135,117 |
https://mathoverflow.net/questions/310315 | 12 | Let $k$ be a global field, $J$ its idele group, and $C = J/k^\times$ the idele class group. For any place $v$ of $k$, we have the familiar closed embedding $k\_v^\times \hookrightarrow J$. More generally, we can let $S$ be a finite set of places of $k$, consider $P = \prod\_{v \in S} k\_v^\times$ with the product topol... | https://mathoverflow.net/users/128759 | Inclusion of multiplicative group of one local field into the idele class group is a closed embedding, but inclusion of more than one isn't? | Let $J^1$ be the ideles of idelic norm one; the global points $k^\*$ is contained in $J^1$. In the number field case, we have a surjection $J/k^\* \rightarrow \mathbb R \_{>0}$ with kernel $J^1/k^\*$ (the latter is compact).
Now let $v$ be a place and $O\_v^\*$ the unit group of $k\_v^\*$ (if $v$ is complex, $O\_v^\... | 6 | https://mathoverflow.net/users/23291 | 310398 | 135,119 |
https://mathoverflow.net/questions/310306 | 9 |
>
> If $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group.
>
>
>
I hear that this is a classical result but I fail to see a clear proof in the literature. Dan Petersen says [here](https://mathoverflow.net/questions/254... | https://mathoverflow.net/users/118688 | Outline of the proof that Cech cohomology and singular cohomology coincide on any locally contractible space | I think looking at Bredon's Sheaf theory book (page $179$ chapter $3$) would be helpful.
It defines singular cohomology of a topological space $X$ with coefficients from **a sheaf of abelian groups** $\mathcal{A}$ by $H^k\_{Sing}(X,\mathcal{A})$ which boils down to (I did not check in detail but I am very much sure)... | 4 | https://mathoverflow.net/users/118688 | 310404 | 135,121 |
https://mathoverflow.net/questions/310372 | 5 | Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \quad d(f(x), x)=diam(M)$$ where $d$ is the metric arising from the Riemannian metric and diameter of $M$ with repect to this... | https://mathoverflow.net/users/36688 | A possible characterization of sphere or projective space | The article by X. Liu and Sh. Deng "The antipodal sets of compact symmetric spaces" gives many examples, e.g. $\mathrm{SU}(2n)$, $\mathrm{Spin}(5)$, $\mathrm{Spin}(7)$,.... All those spaces have unique antipodal points which implies that the antipodal map is a diffeomorphism, see below. Here the antipodal points at $p$... | 10 | https://mathoverflow.net/users/123897 | 310410 | 135,125 |
https://mathoverflow.net/questions/310391 | 13 | This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold:
<https://arxiv.org/abs/1709.10033>
What's the current consensus of the community? Is the proof considered to be correct? Does this imply that the Navier-Stokes equations ... | https://mathoverflow.net/users/88142 | Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid? | In regards to the question of the "consensus" or "correctness", I will only point out that Tristan Buckmaster has had a proven record of studying nonuniqueness problems for low-regularity solutions in incompressible fluids, and contributed significantly to the settling of Onsager's Conjecture on the nonuniqueness probl... | 31 | https://mathoverflow.net/users/3948 | 310411 | 135,126 |
https://mathoverflow.net/questions/310419 | 0 | I am reading an article. There is a step in which I suspect that they use a "result" that "Let $A$ be a local artinian $k$-algebra with residue field $k$. If $A$ is regular then $A$ is nothing but $k$."
I do not know whether it is true or not . If it is true, can anyone give me a source including it .
Thank you in... | https://mathoverflow.net/users/76288 | Regular local artinian k-algebra with residue field k is k | A local ring $(A,\mathfrak{m})$ is regular iff the minimal number of generators of $\mathfrak{m}$ equals the Krull dimension of $A$. But Artinian rings have Krull dimension $0$, i.e. $\mathfrak{m}=0$, hence $A=k$. The latter fact can for example be found in Atiyah and MacDonald's book on Commutative Algebra.
| 3 | https://mathoverflow.net/users/1849 | 310420 | 135,130 |
https://mathoverflow.net/questions/299452 | 8 | According to wiki: <https://en.wikipedia.org/wiki/Dedekind_eta_function>, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic format) on how to computing it. Where to find such one? Thanks!
| https://mathoverflow.net/users/103866 | How to compute Dedekind eta function efficiently? | Euler's formula
$$
\sum\limits\_{n \in \mathbb{Z}} {( - 1)^n q^{\frac{{(3n^2 - n)}}
{2}} } = \prod\limits\_{n = 1}^\infty {(1 - q^n ),}
$$
(which can be proven from Jacobi’s triple product identity by using the fact that $\prod\limits\_{n = 1}^\infty {(1 - q^{3n} )(1 - q^{3n - 2} )} (1 - q^{3n - 1} ) = \prod\limits... | 4 | https://mathoverflow.net/users/5690 | 310435 | 135,136 |
https://mathoverflow.net/questions/309916 | 3 | Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a modified homotopy equivalence, namely "partial homotopy in $X$" as following: Let $\gamma\_1,\gamma\_2: I \rightarrow \ma... | https://mathoverflow.net/users/97137 | Homotopy of paths at the boundary | If one doesn't impose any additional assumptions on $\Gamma$ or $\gamma$ then this doesn't hold.
**Counterexample.** Consider $\Gamma$ that is given by $xy(x+y)=0$, i.e. it is a union of three lines through $(0,0)$. Suppose that $\gamma\_1$ and $\gamma\_2$ are any two paths that join points $(1,0)$ and $(0,1)$ and su... | 0 | https://mathoverflow.net/users/943 | 310438 | 135,138 |
https://mathoverflow.net/questions/310434 | 2 | Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\geq 1$, let $M\_n:=\phi^{-1}(nN)$. Since $nN\cong N$, we have an exact sequence $0\to K\to M\_n \to N\to 0$. I was wonder... | https://mathoverflow.net/users/128812 | Extensions of lattices | It is what you think it is:
>
> **Lemma.** *The class of the short exact sequence $0 \to K \to M\_n \to N \to 0$ in $\operatorname{Ext}^1\_G(N,K)$ is $n$ times the class of the sequence $0 \to K \to M \to N \to 0$.*
>
>
>
*Proof.* By construction, the class in $\operatorname{Ext}^1\_R(C,A)$ of a short exact se... | 2 | https://mathoverflow.net/users/82179 | 310439 | 135,139 |
https://mathoverflow.net/questions/310409 | 1 | Recall that a Boolean space is a $\sigma$-space in case the closure of every open Borel set is open.
Let $\{B\_i\}$ be a denumerable family of open-closed sets in a $\sigma$-space $X$. Then $\bigcup\_i B\_i$ is open but not closed (hence, not open-closed). Since $X$ is a $\sigma$-space, however, the closure of $\bigc... | https://mathoverflow.net/users/126821 | Is the boundary of an open set in a $\sigma$-space empty? | In effect, you are asking if $\bigvee\limits\_{i \in \omega}B\_i = \bigcup\limits\_{i \in \omega}B\_i$, where the left hand side is the closure of the union, which is the join/supremum of the family $\{B\_i\}\_{i \in \omega}$ in the Boolean algebra of clopen sets of $X$, which we will write as $\mathrm{Clopen}(X)$. It ... | 0 | https://mathoverflow.net/users/61785 | 310441 | 135,140 |
https://mathoverflow.net/questions/310384 | 40 | I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.
I am quite new to the ideas of persistent homology and have not done a single computati... | https://mathoverflow.net/users/51223 | Reference on Persistent Homology | Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice [book](https://bookstore.ams.org/surv-209/) if the students had sufficient background, and ... | 36 | https://mathoverflow.net/users/18263 | 310442 | 135,141 |
https://mathoverflow.net/questions/310392 | 7 | Is there an infinite Hausdorff space $(X,\tau)$ with the following property?
>
> If $x\neq y \in X$ and $f:\{x,y\}\to X$ is a map, then there is exactly one continuous function $f': X\to X$ such that $f'|\_{\{x,y\}} = f$.
>
>
>
| https://mathoverflow.net/users/8628 | $2$-determined Hausdorff spaces | Using the technique of [van Mill](http://www.ams.org/journals/tran/1983-280-02/S0002-9947-1983-0716833-2/S0002-9947-1983-0716833-2.pdf) it is possible to prove the following
>
> **Theorem.** There exists a subset $Z$ in the complex plane $\mathbb C$ such that
>
>
> 1) for any complex numbers $x,a,b,c,d\in Z$ wi... | 5 | https://mathoverflow.net/users/61536 | 310448 | 135,144 |
https://mathoverflow.net/questions/310399 | 0 | Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 1.2:
>
> A line bundle $L$ on $X$ is said *numerically effective* (*nef*, for short) if for any $\epsilon>0$ there exists a smooth Hermitian ... | https://mathoverflow.net/users/57030 | Dual of a stable locally free subsheaf is a locally free quotient sheaf | If $F$ is a *subbundle* of $E$, then one has a natural surjection of vector bundles $E^\*\to F^\*\to 0$ induced by the restriction map. This is equivalent to saying the $F^\*$ is a quotient bundle of $E^\*$ (by the kernel of the map above, which is a vector subbundle of $E^\*$), or if you prefer, $\mathcal O\_X(F^\*)$ ... | 1 | https://mathoverflow.net/users/5659 | 310457 | 135,148 |
https://mathoverflow.net/questions/310462 | -2 | I am looking for an elegant proof of the fact that a countable metric space is complete iff its underlying topology is discrete.
It is easy to see that a discrete space is complete because its topology can be derived from the distance $d(x,y)=1$ iff $x$ and $y$ are distinct, so that every Cauchy sequence must be even... | https://mathoverflow.net/users/30395 | Must a countable Polish space be discrete? | This is not true. Take a countable successor ordinal with the order topology. This topology makes it compact hence Polish.
| 5 | https://mathoverflow.net/users/15129 | 310463 | 135,150 |
https://mathoverflow.net/questions/310461 | 6 | I wonder does the following statement holds: 1. For any sufficiently large $n$ the symmetric group $S\_n$ contains at least one 2-transitive subgroup other than $S\_n$ itself and $A\_n$?
2. Actually I'm interested in whether every symmetric group $S\_n$, for sufficiently large $n$, contains a transitive subgroup whos... | https://mathoverflow.net/users/128827 | 2-transitive subgroups of a symmetric group | It has been pointed out in comments that the answer to Question 1 is no. The answer to Question 2 is also no.
Let $p$ be a prime with $p \equiv 1 \bmod 72$, and let $n = 2p+1$. Then $n \equiv 3 \bmod 9$, so $n$ is not prime and is not a proper power. Also, $n \equiv 3 \bmod 8$, so $n+1$ is not a power of $2$.
I cla... | 14 | https://mathoverflow.net/users/35840 | 310479 | 135,156 |
https://mathoverflow.net/questions/310238 | 7 | The [Weyl character formula](https://en.wikipedia.org/wiki/Weyl_character_formula) and [the denominator identity](https://en.wikipedia.org/wiki/Weyl_character_formula#Weyl_denominator_formula) play important roles in the representation theory of classical simple Lie algebras and Kac-Moody Lie algebras over $\mathbb{C}$... | https://mathoverflow.net/users/33047 | Character formula for Lie superalgebras | I agree with the suggestion in the comments for searching the front of the math arXiv (as an entry point), because this is a quite broad and active topic (and i am not sure it can be fully covered within a couple of references).
However, i think it might be particularly useful to mention:
* The [Dictionary on Lie... | 4 | https://mathoverflow.net/users/85967 | 310489 | 135,158 |
https://mathoverflow.net/questions/310363 | 4 | Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the probability density function of $S(N,C)$?
| https://mathoverflow.net/users/114108 | The probability density function of the number of coins to first fill one box of $N$ | (I change notation from $N,C$ to $n,c$ since I use capitals throughout to denote rvs).
Let $X\_i$ be the random variable "number of the box the $i$-th coin", then $X\_1,X\_2,\ldots$ is an i.i.d. sequence of random variables
(each uniformly distributed on $\{1,\ldots,n\}$) and
you are asking for the distribution o... | 4 | https://mathoverflow.net/users/48831 | 310504 | 135,162 |
https://mathoverflow.net/questions/310502 | 3 | Suppose $a\_1,...,a\_n\geq0, \sum\_{i=1}^na\_i=1$ and $Z\_1,...,Z\_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order?
$$\mathbb{P}(\sum\_{i=1}^na\_iZ\_i^2\leq\delta)$$
How will the distribution of $a\_1,...,a\_n$ affect the upper bound?
| https://mathoverflow.net/users/123075 | Left tail of convex combinations of $\chi_1^2$ | $\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}$
One can use formula (1.14) in [Rozovsky](https://link.springer.com/article/10.1007/s10958-006-0251-2):
\begin{equation}
\P(\sum\_1^n X\_i\le r)\sim k\_n\prod\_1^n \P(X\_i\le r)
\end{equation}
for $r\downarrow0$, where th... | 2 | https://mathoverflow.net/users/36721 | 310511 | 135,167 |
https://mathoverflow.net/questions/310482 | 4 | Differential graded algebras *dga-s* are fundamental objects of study in homological algebra and category theory. On the **nlab** [webpage](https://ncatlab.org/nlab/show/differential+graded+algebra), they are defined as follows:
>
> a *dga* is a monoid in the symmetric monoidal category of (possibly unbounded) chai... | https://mathoverflow.net/users/125790 | DGA for a general abelian category | The concept you're asking about has been studied by Christensen and Hovey in [Quillen model structures for relative homological algebra](https://arxiv.org/abs/math/0011216). Specifically, see Example 3.1 on page 17 of the pdf. In this paper, $\mathscr{A}$ is only required to be closed monoidal (as well as bicomplete, t... | 3 | https://mathoverflow.net/users/11540 | 310517 | 135,169 |
https://mathoverflow.net/questions/308772 | 5 | I recently stumbled upon the [slides of a talk](http://kskedlaya.org/slides/nagoya2010.pdf) given by Kedlaya, in which the following appears:
>
> For $X$ of finite type over $F\_q$, a Weil cohomology theory, mapping $X$ to
> certain vector spaces $H^i(X)$ over a field of characteristic zero, provides a
> spectral... | https://mathoverflow.net/users/104436 | What is the spectral interpretation of the arithmetic zeta function? | First, it's worth mentioning that the above theorem refers to the (local) Hasse-Weil zeta function of a scheme $X$ of finite type over $\mathbb{F}\_q$ rather than the arithmetic zeta function of a scheme $X$ of finite type over $\mathbb{Z}$. Certainly these are related: for $X$ is a scheme of finite type over $\mathbb{... | 5 | https://mathoverflow.net/users/128825 | 310519 | 135,171 |
https://mathoverflow.net/questions/300256 | 5 | Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1\_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to the relations
$$ d(\lambda a + \mu b) = \lambda da + \mu db, ~~~~~~ d(ab) = a\,db + da \,b$$
for $a, b \in A$, $\lambda, ... | https://mathoverflow.net/users/16702 | Definition of Non-commutative de-Rham-Cohomology | I do not really know the reason, but if I would guess I think it might be $\Omega(A)$ is no longer an abelian category. The exact sequences you written down are not just exact sequences of abelian groups, but may be interpreted as exact sequences of algebras (note that we have $d(ab)=da\*b+a\*db=0\*b+a\*0=0$ for $a,b\i... | 2 | https://mathoverflow.net/users/18850 | 310520 | 135,172 |
https://mathoverflow.net/questions/310512 | 12 | It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}\_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.
More precisely, since the site has finite limits, the inverse image functors are induced by functors $\mathbf{CRing}^\mathrm{op}\_{\mathrm{fp}... | https://mathoverflow.net/users/nan | Points of the big Zariski site | Let's simplify and consider the presheaf topos.
I asked the same question [over at the nForum](https://nforum.ncatlab.org/discussion/7639/points-of-toposes-of-sheaves-on-largerthanusual-sites/) a while back. There Marc Hoyois reminded me of the following quite general fact: The category of topos-theoretic points of $... | 8 | https://mathoverflow.net/users/31233 | 310529 | 135,176 |
https://mathoverflow.net/questions/310515 | 2 | I am going through [Sims - Étale groupoids and their $C^\*$ algebras](https://arxiv.org/abs/1710.10897) and at Lemma 3.1.4. the author says that
$f^\*\*f\in C\_c(G^{(0)})$ is supported on $s(supp(f))$ and $(f^\*\*f)(s(\gamma))=|f(\gamma)|^2$ for all $\gamma\in supp(f)$. He says that it follows from the convolution form... | https://mathoverflow.net/users/nan | Convolution product in an étale groupoid | In the lemma he is supposing that $f$ is supported on a bisection $U$. Then by definition $f^\*$ is supported on $U^{-1}$. If $\gamma$ is in the support of $f^\*\*f$ then $\gamma=\alpha\beta$ with $\alpha\in U^{-1}$ and $\beta\in U$. Since $U$ is a bisection the source and range maps are injective on $U$ and you must h... | 1 | https://mathoverflow.net/users/15934 | 310535 | 135,178 |
https://mathoverflow.net/questions/310544 | 4 | We know that the norm of a unitary in a unital $C^\*$-algebra is one. Also, in a unital Banach algebra A, $u \in A$ is defined to be a unitary if $\|u\| = \|u^{-1}\| =1$. I tried to prove it for the unitaries in unital Banach $\*$-algebra, but not able to get an answer. Is it true that a unitary in a unital Banach$\*$-... | https://mathoverflow.net/users/91509 | Unitaries in Banach *-algebras | The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $\widetilde{K}:= \{\lambda Id + k: k\text{ is compact}\}$, given with natural involution and norm $\|\lambda Id + k\| := |\lambda| + \|k\|\_{op}$ (it's not equal to the operator norm!); it is a unital ... | 8 | https://mathoverflow.net/users/24953 | 310550 | 135,182 |
https://mathoverflow.net/questions/310542 | 8 | In the end of Section 9.2 of Bosch's book *[Lectures on Formal and Rigid Geometry](http://www.math.purdue.edu/~tongliu/seminar/rigid/Bosch.pdf)*, a rigid $S$-space $E\_Q$ is constructed, for a variable $Q$ replacing the classical parameter $q\in k$. (Here $k$ is a non-archimedean field and $S=\mathrm{Spf}\mathbb Z[[Q]]... | https://mathoverflow.net/users/69190 | Raynaud's universal Tate elliptic curves | **Q1:** Michel Raynaud, [Géométrie analytique rigide d’après Tate, Kiehl...](http://www.numdam.org/article/MSMF_1974__39-40__319_0.pdf), Bull. Soc. Math. France, Mémoire **39-40**, p. 319-327 (1974).
| 3 | https://mathoverflow.net/users/11260 | 310551 | 135,183 |
https://mathoverflow.net/questions/310526 | 19 | To a symmetric sequence $V\_\bullet$ of vector spaces, associate the generating function $F\_V(z) = \sum\_n \frac{\dim(V\_n)}{n!} z^n$. Then
$$F\_{Comm\_\ast}(z) = \exp(z)-1 \qquad F\_{Lie}(z) = \ln(1-z)$$
where $Comm\_\ast$ is the reduced commutative operad and $Lie$ is the Lie operad. Notice that
1. On the one ... | https://mathoverflow.net/users/2362 | Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$? | Let me flesh out the answer a little. The general statement is given by Theorem 7.5.1 in the book *Algebraic Operads* by Loday and Vallette.
First a definition. Let $P = P(E,R)$ be a quadratic operad, with generators $E$ (f.gen. s.t. $E(0) = 0$) and $R \subset E \circ E$ quadratic relations. Let $P^{(r)}(n)$ be the s... | 18 | https://mathoverflow.net/users/36146 | 310552 | 135,184 |
https://mathoverflow.net/questions/310533 | 11 | Let $G$ be a finite (or discrete) group, $M$ a $d$-dimensional manifold with smooth $G$-action (I am interested in the case where the action is not free, so $M/G$ is not a manifold). For an Abelian group $A$, let $\mathcal{C}^n(M,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-... | https://mathoverflow.net/users/81457 | Is there a kind of Poincare duality for Borel equivariant cohomology? | This kind of thing shows up quite naturally in parameterised stable homotopy theory. Let me translate an idea I know from there into the language in this question.
Cap product gives a map
$$C^{p}(M ; A) \otimes C\_q(M, \mathbb{Z}) \to C\_{q-p}(M;A),$$
which is $G$-equivariant and is a chain map when we sum over $(p,... | 7 | https://mathoverflow.net/users/318 | 310553 | 135,185 |
https://mathoverflow.net/questions/310539 | 16 | EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is available for readers who prefer being low-tech; just click on the abstract button and a download button for this will b... | https://mathoverflow.net/users/317 | Relationship between Smith's special homology groups and equivariant homology theory | Smith's special homology groups are special instances of Bredon homology. Here's roughly how it goes, taken from Peter May's ["A generalization of Smith theory"](http://www.ams.org/journals/proc/1987-101-04/S0002-9939-1987-0911041-3/home.html).
A Bredon coefficient system $M$ is some kind of functorial assignment of ... | 15 | https://mathoverflow.net/users/360 | 310559 | 135,186 |
https://mathoverflow.net/questions/310541 | 3 | The below is a simplification of part of a proof I'm working on, in numerical analysis. It is similar to a paper that I studied some months ago, for which I got [some advice here on MathOverflow](https://mathoverflow.net/questions/292107/resolving-an-inequality-in-the-final-step-of-derivation-of-an-a-priori-energy).
... | https://mathoverflow.net/users/15045 | How to estimate a recursive inequality with an upper bound | $\newcommand{\de}{\delta}$
The bound ($\star\star$) that you want is impossible in general. E.g., take $A\_n=0$, $C\_n=1$, $B\_n=n\Delta t$ for all $n$, with $\Delta t>0$. Then ($\star$) will hold, whereas ($\star\star$) will not hold for large enough $n$, for any choice of $f,g,h$.
**Added:** The bound ($\star\st... | 2 | https://mathoverflow.net/users/36721 | 310564 | 135,187 |
https://mathoverflow.net/questions/295739 | 1 | In the ring of shifted symmetric functions $\Lambda^\*$ there are many ways to generalize the symmetric power sums. First of all, we have the functions $$p^\*\_k=\sum\_{i=1} \left((x\_i-i+1/2)^k-(-i+1/2)^k\right)$$
(the factor 1/2 is convenient when the $x\_i$ are the parts of a partition, but can replaced by any const... | https://mathoverflow.net/users/115255 | On two types of shifted symmetric power sums | The answer to this question is precisely given by the Gromov-Witten/Hurwitz correspondence by Okounkov and Pandharipande (see <https://arxiv.org/abs/math/0204305>). Up to the constant $(1-2^{-k})\zeta(-k)$ the functions $p\_k^\*$ equal the renormalized shifted symmetric power sums $\mathbf{p}\_k$ which occur in express... | 0 | https://mathoverflow.net/users/115255 | 310566 | 135,188 |
https://mathoverflow.net/questions/310576 | 29 | let $M$ be a smooth $n$-manifold with boundary $\partial M$; I denote by $M^o$ the internal part of $M$, that is $M \smallsetminus \partial M$.
The question is the same as in the title: let $M$ and $N$ be two compact orientable topological manifolds such that $M^o$ is homeomorphic to $N^o$. Does this imply that $M$ is ... | https://mathoverflow.net/users/128408 | Does $M^o=N^o$ imply that $\partial M = \partial N$? | In general, this is wrong.
Take $M=L(7,1)\times S^{2n}\times[0,1]$ and $N=L(7,2)\times S^{2n}\times[0,1]$.
Their boundaries $L(7,1)\times S^{2n}$ and $L(7,2)\times S^{2n}$ are not homeomorphic but $h$-cobordant, as proven by Milnor in
>
> J. Milnor,
> Two complexes which are homeomorphic but combinatorially... | 32 | https://mathoverflow.net/users/84120 | 310579 | 135,192 |
https://mathoverflow.net/questions/310578 | 7 | The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternating projection".
| https://mathoverflow.net/users/11142 | Number of (distinct) knots with a bounded number of crossings | Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered).
Clearly from inclusion of sets $ak(n)\leq a(n)\leq l(... | 14 | https://mathoverflow.net/users/1345 | 310587 | 135,199 |
https://mathoverflow.net/questions/310595 | 11 | Let $G$ be a finitely presented group, $\widehat{G}$ be the profinite completion of $G$, and $f: G\rightarrow \widehat{G}$ be the natural map.
My question is:
Is there an example of $G$ for which $\text{Im} f$ is not finitely presented?
| https://mathoverflow.net/users/128887 | Profinite completion of finitely presented groups | Yes. Take the Baumslag-Solitar group
$$G=\mathrm{BS}(2,3)=\langle t,x\mid tx^2t^{-1}=x^3\rangle$$
Then $G$ is finitely presented; the image of $G$ in its profinite completion (i.e., the largest residually finite quotient of $G$) is $\mathbf{Z}[1/6]\rtimes\_{2/3}\mathbf{Z}$, which is not finitely presented. (Here $(2,3)... | 17 | https://mathoverflow.net/users/14094 | 310597 | 135,203 |
https://mathoverflow.net/questions/310373 | 5 | The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) seems to be that [every 4-*orientable* manifold X admits a $Spin^c$ structure.](https://mathoverflow.net/questions/266946/ev... | https://mathoverflow.net/users/27004 | Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure | $\newcommand{\RP}{\mathbb{RP}}\newcommand{\Z}{\mathbb Z}$
The conjecture is false: $\RP^4\amalg(\RP^2\times\RP^2)$ is an unorientable 4-manifold that has no
pin$c$, pin$\tilde c-$, or pin$\tilde c+$ structure.
It suffices to find three unorientable 4-manifolds $A$, $B$, and $C$, such that $A$ isn't pin$c$, $B$
isn't ... | 6 | https://mathoverflow.net/users/97265 | 310600 | 135,204 |
https://mathoverflow.net/questions/310572 | -1 | I am a new learner of optimization, and I am confused by the question below, (how to change a 0-norm constrain into binary and linear constrain ?)
---
Given a sparse data fitting problem:
$ minimize \quad \| Ax-b \|^{2}\_{2}$
$ s.t. \qquad \|x\|\_{0} \le K, $
$x \in R^{n}$
Suppose we are given a constant... | https://mathoverflow.net/users/128877 | sparse data fitting problem | This can be done using a Big M approach.
Introduce zero one binary variables $y\_i, i=1,..,n$
Replace the constraint $\qquad \|x\|\_{0} \le K, $ with the following constraints:
$$\Sigma\_{i=1}^n y\_i \le K$$
$$-x\_i \le My\_i, i=1,..,n$$
$$x\_i \le My\_i, i=1,..,n$$
As can be seen, when $y\_i = 0$, then $x\_i$ ... | 1 | https://mathoverflow.net/users/75420 | 310603 | 135,206 |
https://mathoverflow.net/questions/310565 | 6 | First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^\*O(1)$ for some (not-necessarily closed) immersion $j:X\rightarrow \mathbb{P}^n$.
Now, let $X$ be a variety and $L$ a line bundle on $X$... | https://mathoverflow.net/users/4181 | Does ampleness descend along finite maps? | There is a non-quasi-affine variety $X$ with quasi-affine normalization. See [Tag 0271](https://stacks.math.columbia.edu/tag/0271). Then $\mathcal{O}\_X$ is a counter example.
| 5 | https://mathoverflow.net/users/128893 | 310607 | 135,208 |
https://mathoverflow.net/questions/228166 | 8 | It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$.
A usual generalization of this structure to unorientable manifolds is to ask for a spin structure on $TX\oplus\mathcal{L}\oplus\mathcal{E}$ where $\mathcal{L}... | https://mathoverflow.net/users/81773 | Are all 4-manifolds $Pin^{\tilde{c}}$? | $\newcommand{\RP}{\mathbb{RP}}\newcommand{\Z}{\mathbb Z}$No, $\RP^2\times\RP^2$ isn't pin$\tilde c$. This came up while thinking about [another MathOverflow
question](https://mathoverflow.net/q/310373/97265), but I'll rewrite the argument here.
What you call a pin$\tilde c$-structure has been studied in physics, foll... | 8 | https://mathoverflow.net/users/97265 | 310608 | 135,209 |
https://mathoverflow.net/questions/310425 | 3 | Consider permutations $\pi$ of the set $\{1,\dots,n\}$ having the symmetry property $\pi \pi^\* \pi = \pi^\*$, where $\pi^\*$ is the "reflection" $k \mapsto n+1-k$. Are there references or other information about such a $\pi$?
| https://mathoverflow.net/users/127871 | A permutation with reflection property | The problem boils down to finding permutations $\pi$ such that $\pi(n+1-x)=y\iff\pi(n+1-y)=x$. Let $X\subset\{1,...,n\}$ be such that $n-\lvert X\rvert$ is even and let $P$ be a partition of $\{1,...,n\}\backslash X$ into pairs and denote, for every element $k$ of $\{1,...,n\}\backslash X$, its companion by $P(k)$. The... | 1 | https://mathoverflow.net/users/99279 | 310617 | 135,214 |
https://mathoverflow.net/questions/310620 | 9 | I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows:
As a set $D=E\oplus uE \oplus u^2 E$ where $u$ is an indeterminate. Addition is defined component wise while multiplication is defined as $e.u=u\sigma(e)$ where $\s... | https://mathoverflow.net/users/80991 | Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$ | There is a general strategy to tackle such question. Since the norm is multiplicative, it make sense to look for a prime $p$ not in the image. Then, we can ask about the number of times $p$ divides a general norm.
The norm map extends to prime ideals of the integer ring, and if the norm $N\_{E/F}(q)$ is $p^3$ for ever... | 8 | https://mathoverflow.net/users/115052 | 310621 | 135,215 |
https://mathoverflow.net/questions/309044 | 28 | Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture.
Goldbach's conjecture asserts that every even number greater than or equal to 4 can be written as the sum of two prime numbers. It's pretty straightforward how t... | https://mathoverflow.net/users/5736 | Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample? | Let's note that this is not a question of whether Collatz is undecidable.
The statement $\neg\mathrm{Con}(PA)$ is undecidable (by $PA$, assuming $PA$ is consistent) but nevertheless $\neg\mathrm{Con}(PA)$ is provably equivalent to a certain Turing machine halting (the one that searches for a proof of a contradiction in... | 22 | https://mathoverflow.net/users/4600 | 310622 | 135,216 |
https://mathoverflow.net/questions/310629 | 1 | Let $V$ be a finite dimensional real inner product space and $U$ a real inner product space of countable dimension. Why is the space of linear isometries from $V$ to $U$ paracompact?
| https://mathoverflow.net/users/128857 | Why is a certain space of linear isometries paracompact | Certainly if $U$ is a finite-dimensional inner product space, then $\text{Isom}(V, U)$ is paracompact (being a closed subspace of $\text{Lin}(V, U) \cong U^n$).
If $U$ has countably infinite dimension, then $U$ is a sequential colimit of finite-dimensional inner product subspaces $U\_n$ and inclusions between them, ... | 2 | https://mathoverflow.net/users/2926 | 310631 | 135,218 |
https://mathoverflow.net/questions/310638 | 2 | I am studying the quotient of
$$f(\varepsilon) = \sum\_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$
and $$g(\varepsilon) = \sum\_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$
for some $\varepsilon>0.$
Consider then the function $h(\varepsilon):=f(\varepsilon)/g(\varepsilon).$
Doing some numerics, I found tha... | https://mathoverflow.net/users/128387 | Quotient with positive second derivative in the limit? | I expect for small $\epsilon$ the approximation of the sum by an integral to be accurate. Here is a plot of $h(\epsilon)$ (blue curve) and
$$H(\epsilon)=\frac{\int\_0^\infty x^2 2^{-\epsilon x^2}\,dx}{\int\_0^\infty 2^{-\epsilon x^2}\,dx}=\frac{1}{\epsilon\log 4},$$
(gold curve).
$ is trivial.
For instance, this appears in Section 3.4 of Springer's Corvallis article. However, ... | https://mathoverflow.net/users/97316 | Anisotropic algebraic groups have no unipotent elements | What you have asked for is indeed a fact in characteristic zero. See Corollary 8.5 of Borel-Tits [(Publ. IHES at Numdam, unrestricted access)](http://www.numdam.org/item?id=PMIHES_1965__27__55_0).
However, it is false in positive characteristic. See also section 4 of the same article (where other fields are consider... | 19 | https://mathoverflow.net/users/23291 | 310660 | 135,230 |
https://mathoverflow.net/questions/310661 | 2 | Suppose $H=(V,E)$ is a $3$-uniform hypergraph with $n$ vertices $V$ and $O(n^{3/2})$ hyperedges, $E.$
My question is: How many vertices do I need to delete to make sure all hyperedges are destroyed?
I can solve this problem by Pigeonhole principle and greedy approach, but unfortunately, it does not give me a good... | https://mathoverflow.net/users/80245 | $3$-uniform hypergraph with $n$ vertices and $O(n^{3/2})$ hyperedges | It is more natural to search for a maximal independent set (which contains no hyperedges). Choose $K$ vertices at random. We have at most $C\cdot (K/n)^3\cdot n^{3/2}=CK^3/n^{3/2}$ edges on them. Assume that it is less than $K/2$, than removing a vertex from any edge we get an independent set of size $K/2$ (this trick ... | 1 | https://mathoverflow.net/users/4312 | 310664 | 135,231 |
https://mathoverflow.net/questions/310623 | 6 | Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct factors of $G$.
In general, $\mathcal{D}(G)$ is not a sublattice of $\mathcal{N}(G)$. For example, take $G=\mathbb{Z}/4\ma... | https://mathoverflow.net/users/6849 | Groups whose poset of direct factors are lattices | *(edit: Added Theorem 2 below, which gives half the case with abelian direct factors, and classifies the finite abelian $\mathcal{D}$-groups)*
>
> **Theorem 1**. Let $G$ be a non-trivial group satisfying both chain conditions on normal subgroups, and with no non-trivial abelian direct factors. Then $G$ is a $\mathc... | 3 | https://mathoverflow.net/users/41862 | 310667 | 135,232 |
https://mathoverflow.net/questions/310671 | 2 | Let $G=(V,E)$ be a simple, undirected graph. A *matching* is a set $M\subseteq E$ consisting of pairwise disjoint edges. We say $M$ is *maximal* if it is maximal amongst all matchings in $G$ with respect to $\subseteq$.
Given $n\in\mathbb{N}$ is there a graph $G$ such that $\chi(G)\geq n$ and every two maximal matchi... | https://mathoverflow.net/users/8628 | Graphs in which all maximal matchings intersect | Affirmative: take $G$ to be the disjoint union of $K\_n$ and $K\_2$, then the $K\_n$ forces $\chi(G)\geq n$, while any maximal matching contains the $K\_2$ edge.
OK, that was cheating. I suppose you want a connected example.
I don't know the answer there, but can give a negative answer if you replace the condition $\... | 4 | https://mathoverflow.net/users/75735 | 310672 | 135,235 |
https://mathoverflow.net/questions/310674 | 3 | Let $n\in\mathbb{N}$ be a positive integer. Is there a connected graph $G$ such that $G$ cannot be coloured with less than $n$ colours, and every two maximal [matchings](https://en.wikipedia.org/wiki/Matching_(graph_theory)) have non-empty intersection?
(I take maximality with respect to set inclusion.)
| https://mathoverflow.net/users/8628 | Maximal matchings in connected graphs | Another attempt. Take $K\_{n+1} $ and three paths $abc, ade, afg$, $a\in K\_{n+1} $, $b, c, d, e, f, g\notin K\_{n+1} $. Any maximal matching contains at most one of edges $ab, ad, af$, thus at least two from three edges $bc, de, fg$. Hence any two maximal matchings have a common edge.
| 4 | https://mathoverflow.net/users/4312 | 310677 | 135,236 |
https://mathoverflow.net/questions/153883 | 2 | Suppose $S\_3$ is the symmetric group of order 6. Which elements of the variety $Var(S\_3)$ are relatively free?
This question is related to my previous question
[Relatively free algebras in a variety generated by a single algebra](https://mathoverflow.net/questions/153743/relatively-free-algebras-in-a-variety-gener... | https://mathoverflow.net/users/44949 | relatively free groups in $Var(S_3)$ | Let me provide by hand free groups in the variety $V\_p$ generated by the dihedral group $D\_{2p}$ of order $2p$, $p$ odd prime.
First observe that $D\_{2p}$ satisfies the group identities $x^{2p}=1$, $[x^2,y^2]=1$.
In any group, let $G^2$ be the set of squares and $G\_p$ the set of elements of order dividing $p$. ... | 5 | https://mathoverflow.net/users/14094 | 310683 | 135,238 |
https://mathoverflow.net/questions/310652 | 2 | Let $\mathfrak f(\tau)=e^{-\pi i/24}\frac{\eta\left(\frac{\tau+1}{2}\right)}{\eta(\tau)}=q^{-1/48}\prod\_{n=1}^{\infty}\left(1+q^{n+1/2}\right)$ be the Weber modular function. The function $\mathfrak f$ has level $48$, it is nonzero on the upper half-plane, and it is positive on the imaginary axis.
Let $\mathfrak h=... | https://mathoverflow.net/users/122104 | Roots of modular functions | **EDIT**. Your function $\mathfrak{h}$ is not modular of any level. Indeed $\mathfrak{h}$ has Fourier coefficients in $\mathbf{Q}$. A standard argument shows that any modular unit $u$ with Fourier coefficients in some number field $K$ must have bounded denominators: take $n$ large enough such that $u \Delta^n$ is a cus... | 5 | https://mathoverflow.net/users/6506 | 310684 | 135,239 |
https://mathoverflow.net/questions/310682 | 1 | Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property:
$\forall x, \forall y \in X, \exists \{z\_n\}$ such that $\lim\_{n\to +\infty } d(x,z\_n)=+\infty$ and $\lim\_{n \to +\infty } \frac{d(x,z\_n)}{d(y,z\_n)}=1 $.
Note: I should say, I know this property is true for some ... | https://mathoverflow.net/users/117299 | A property on some unbounded metric spaces | Unboundedness guarantees that there is one sequence $(z\_n)\_n$ such that $d(x,z\_n)\to\infty$ for all $x$.
That sequence also satisfies the second requirement via the triangle inequality:
$$
\frac{d(x,z\_n)}{d(y,z\_n)}\le \frac{d(x,y)}{d(y,z\_n)}+1
$$
and symmetrically
$$
\frac{d(y,z\_n)}{d(x,z\_n)}\le \frac{d(y,x)}{d... | 2 | https://mathoverflow.net/users/5903 | 310685 | 135,240 |
https://mathoverflow.net/questions/309926 | 5 | Let $M$ be a submanifold of a Riemannian manifold $\widetilde{M}$. Let $A$ be the second fundamental form of $M$.
Suppose that, for all $p \in M$, the linear map $A(v, \cdot)\: \colon T\_{p}M \to N\_{p}M$ has rank $k$ for every nonzero $v \in T\_{p}M$.
Does this condition define some well-known class of submanifol... | https://mathoverflow.net/users/74033 | Submanifolds whose second fundamental form has constant rank in every direction | Sometimes, this is an open condition, so that it does not impose any differential equations on the submanifold. For example, consider the case of a surface $\Sigma^2$ in a $4$-manifold $M^4$. The condition that the rank of $A(v,\cdot):T\_p\Sigma\to N\_p\Sigma$ be equal to $2$ for every *nonzero* $v\in T\_p\Sigma$ is an... | 5 | https://mathoverflow.net/users/13972 | 310686 | 135,241 |
https://mathoverflow.net/questions/295679 | 0 | Well, one thing is special about it, but it takes a while to explain.
**Please let me know, whether this number occurs in other special occasions as well.**
The explanation: Let $p$ be a complex polynomial of degree $n$ with simple zeros $z\_j$ and simple critical points $z'\_k$ and consider the matrix
\begin{eq... | https://mathoverflow.net/users/37855 | What is special about 2 + $\sqrt{3}$? | The fun fact in a comment of @LucGuyot about the fundamental unit of $\mathbb{Q}[\sqrt{3}]$ is on target, and not only for $\mathbb{Q}[\sqrt{3}]$ or units. I hope the empirical facts I give here will be of use to people interested in quadratic number fields. The following relation between the geometric configuration an... | 1 | https://mathoverflow.net/users/37855 | 310692 | 135,243 |
https://mathoverflow.net/questions/310687 | 3 | Let $F\_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}\_4$, with Dynkin diagram
$$
\beta\_1-\beta\_2\Rightarrow\beta\_3-\beta\_4\,.
$$
Let $P\_{\{\beta\_2\}}$ be the minimal parabolic subgroup $\neq B$ such that its set of smiple roots is $\{\beta\_2\}$. Let
$$
\mu\,\... | https://mathoverflow.net/users/66288 | Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$ | The following paper says that your map $\mu$ is indeed surjective; the author attributes the result over $\mathbb C$ to Tits.
```
Demazure, M. Invent Math (1977) 39: 179. https://doi.org/10.1007/BF01390108
```
I also add the link
[Automorphism group of flag manifolds?](https://mathoverflow.net/questions/160292/... | 7 | https://mathoverflow.net/users/23291 | 310697 | 135,244 |
https://mathoverflow.net/questions/310694 | 4 | Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'\_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the topology of uniform convergence on compact subsets of $X$).
>
> **Question.** Is the space $X\_c'$ sequential (equivalen... | https://mathoverflow.net/users/61536 | Is the compact-open topology on the dual of a separable Frechet space sequential? | Yes. In the next paragraph I will show that if $X$ is a Fréchet space (without requiring separability) then $X'\_c$ with the compact-open topology is a $k$-space. As you note, this implies sequentiality if $X$ is separable.
The compact-open topology on $X'\_c$ is the same as the finest topology agreeing with $\sigma... | 4 | https://mathoverflow.net/users/61785 | 310698 | 135,245 |
https://mathoverflow.net/questions/310639 | 2 | Let $p\_{n,K}(x)$ be the polynomial of degree $n$ that minimizes $\epsilon\_{n,K} = ||p\_{n,K}(x) - sin(x)||\_\infty$ on the interval $[-K, K]$. Question: what is the asymptotic behavior of $\epsilon\_{n,K}$ where $n \to \infty$?
| https://mathoverflow.net/users/109563 | Minimax Approximation to Sine Function on interval [-K, K] | The rate of polynomial approximation
$$E\_{n}(f,\mathcal{K}):=\inf\{\max\_{z\in \mathcal{K}}|f(z)-P\_{n}(z)|,~\text{deg}~P\_{n}\leq n\}$$
to an entire function $f$ on a compact set $\mathcal{K}$ of the complex plane of positive capacity was derived by A. Batyrev in
>
> Batyrev, A. V., On the best approximation of ... | 4 | https://mathoverflow.net/users/89429 | 310700 | 135,247 |
https://mathoverflow.net/questions/310715 | 13 | Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1\_k,\mathbb{A}^1\_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{A}^1,\mathbb{A}^1)$ associates to a $k$-scheme $S$, the set of isomorphisms $\mathbb{A}^1\_S\to \mathbb{A}^1\_S$ of $... | https://mathoverflow.net/users/128936 | On non-representability of certain hom schemes | Welcome, new contributor. Let $k$ be a field. Let $Y$ be a finite type, separated $k$-scheme such that the $k$-algebra $\mathcal{O}\_Y(Y)$ is a $k$-vector space of infinite dimension. For instance, this holds for every finite type, affine $k$-scheme of positive dimension. Let $X$ be a finite type, separated $k$-scheme ... | 10 | https://mathoverflow.net/users/13265 | 310723 | 135,253 |
https://mathoverflow.net/questions/310640 | 2 | The question
============
Let $L$ be a [lattice (in the sense of combinatorics, not number theory)](https://en.wikipedia.org/wiki/Lattice_(order)).
An *$L$-bag* will mean a finite multiset of elements of $L$.
Given an $L$-bag $A$, we consider three possible operations that can be performed on $A$:
* *Raising* $... | https://mathoverflow.net/users/2530 | Gale order on multisets of elements of a lattice | The answer is "no" for the "only if" part of the question, even when the lattice is a Boolean lattice.
For a counterexample, let $L$ be the Boolean lattice of all subsets of $\left\{1,2,3\right\}$. Let $A$ be the $L$-bag whose elements are $\varnothing, \left\{2,3\right\}, \left\{3,1\right\}, \left\{1,2\right\}$. Let... | 2 | https://mathoverflow.net/users/2530 | 310724 | 135,254 |
https://mathoverflow.net/questions/310730 | 5 | I could swear I remember a result of the following form:
Suppose we have a pair of functors $$C\xrightarrow{F}D\xrightarrow{G} X,$$ with $X$ cocomplete.
then we obtain a functor $$D\to X$$ sending $$d\mapsto \operatorname{colim}\_{(F\downarrow d)}(G\circ \pi\_d)$$ where $\pi\_d:(F\downarrow d)\to D$ is the projecti... | https://mathoverflow.net/users/1353 | Factorization of colimits through slices? | The functor $\text{colim}\_{(F\downarrow -)}(G\circ\pi\_-)\colon D\to X$ is a left Kan extension of $G\circ F$ along $F$. The corresponding natural transformation $\ell\_F^{G\circ F}\colon G\circ F\to \text{colim}\_{(F\downarrow -)}(G\circ\pi\_-)\circ F$ is defined by
$$
\ell\_F^{G\circ F}(c)=\varphi^{F(c)}(id\_{F(c)})... | 6 | https://mathoverflow.net/users/35349 | 310739 | 135,255 |
https://mathoverflow.net/questions/310735 | 5 | The below identity I have found experimentally.
>
> **Question.** Is this true? If so, may you provide a "slick" (or any) proof.
> $$6\sum\_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum\_{k=1}^{\infty}\frac{kq^k}{1-q^k}\right)^2=\sum\_{k=1}^{\infty}\frac{(5k^3+k)q^k}{1-q^k}.$$
>
>
>
| https://mathoverflow.net/users/66131 | Searching for a proof for a series identity | Follow the comments of Lucia and note that
$$\sum\_{n\ge 1}\frac{n^2q^n}{(1-q^n)^2}=q\frac{\,d}{\,dq}\sum\_{n\ge 1}\frac{nq^n}{1-q^n}.$$
I believe the identity actually is the well known
$$q\frac{\,d}{\,dq}L=\frac{L^2-M}{12},$$
where
$$L=1-24\sum\_{n\ge 1}\frac{nq^n}{1-q^n}\;\mbox{
and}\;
M=1+240\sum\_{n\ge 1}\frac{n^... | 12 | https://mathoverflow.net/users/110368 | 310748 | 135,258 |
https://mathoverflow.net/questions/309223 | 2 | Let $M$ be a simply connected complex manifold (of dimension greater than one), $L$ a line bundle, and $\nabla$ a connection on $L$ with possibly singularities along a divisor $D$. We define the curvature as
$$
R\_{\nabla}(X,Y)=[\nabla\_X,\nabla\_Y]-\nabla\_{[X,Y]}
$$
Suppose that a second connection $\nabla'$ on $L$... | https://mathoverflow.net/users/48866 | Does the holomorphic curvature determine the connection? | The connection can be equivalently given by parallel transport along curves. So if two connections are equal (or isomorphic in appropriate sense) then they must have the same holonomy. Now by the Ambrose--Singer theorem there is a close relationship between holonomy and curvature. But full holonomy group is a global ob... | 0 | https://mathoverflow.net/users/6818 | 310755 | 135,261 |
https://mathoverflow.net/questions/310731 | 8 | In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such that any functor $C \to D$ that sends $S$ to isomorphisms factors uniquely through $L$.
Does a generalization of this no... | https://mathoverflow.net/users/101861 | Localization of $\infty$-categories | An *abstract* construction I find appealing is that a localization should satisfy a natural pullback square of spaces
$$ \require{AMScd} \begin{CD}
\hom(C[S^{-1}], X) @>>> \hom(C, X)
\\ @VVV @VVV
\\ \hom(S, \mathrm{Core}(X)) @>>> \hom(S, X) \end{CD}$$
expressing the universal property that a functor on $C[S^{-1}]$ ... | 8 | https://mathoverflow.net/users/nan | 310760 | 135,263 |
https://mathoverflow.net/questions/310718 | 4 | What is the current situation of the second part of the [Hilbert 16th problem](https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem#The_second_part_of_Hilbert's_16th_problem)? What are the most updated news on this problem?
| https://mathoverflow.net/users/36688 | Updated background on Hilbert 16th problem? | An update from April 2018 is given by [Patrick Speissegger](https://arxiv.org/abs/1804.03585).
The idea, going back to Poincaré, is to reduce the two-dimensional counting problem (counting limit cycles in the plane) to a one-dimensional counting problem (counting certain points on a line). Roussarie (1998) showed tha... | 8 | https://mathoverflow.net/users/11260 | 310763 | 135,264 |
https://mathoverflow.net/questions/310751 | 1 | Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P\_x:=\langle \bullet, x \rangle x$ and similarly $P\_y.$
Assume then that we know that $\left\lVert x-y \right\rVert\le \alpha.$
What is the optimal estimate on $\left\lVert P\_x-P\_y \right\rVert\_1$ we can get, whe... | https://mathoverflow.net/users/128387 | Optimal estimate in trace norm | $\newcommand{\R}{\mathbb{R}}
\newcommand{\ep}{\epsilon}$
Without loss of generality (wlog), the space is the two dimensional space spanned by $x,y$. So, wlog the space is $\R^2$, with $x=[1\ 0]^T$ and $y=[a\ b]^T$ for some real $a,b$ such that $a^2+b^2=1$. Then $$\ep:=\|x-y\|=\sqrt{2-2a}$$ and $a=1-\ep^2/2$. Also, id... | 2 | https://mathoverflow.net/users/36721 | 310769 | 135,265 |
https://mathoverflow.net/questions/310770 | 1 | Let $(M,g)$ be a globally hyperbolic Lorentzian manifold with Lorentzian volume density $dV$ and $\Sigma$ a Cauchy hypersurface, i.e. each inextendible causal curve hits $\Sigma$ exactly once. Furthermore let $P: C^\infty(M) \rightarrow C^\infty(M)$ be a formally self-adjoint wave operator, that is a $2^{nd}$-order dif... | https://mathoverflow.net/users/128963 | Existence of bisolutions for wave operators on globally hyperbolic Lorentzian manifolds | I think this just follows from linearity.
The condition that $u(\sigma, q) = u(q,\sigma)$ implies that $P u(\sigma,\cdot) = 0$. In fact, you have that $q\mapsto u(\sigma,q)$ is the unique solution to the initial value problem for $Pu(\sigma,\cdot) = 0$ with data prescribed on $\{\sigma\} \times \Sigma$ under your hy... | 0 | https://mathoverflow.net/users/3948 | 310776 | 135,268 |
https://mathoverflow.net/questions/297924 | 2 |
>
> Every totally disconnected separable metric space of dimension $n$ homeomorphically embeds into $C\times \mathbb R ^n$.
>
>
>
Is something like this known? $X$ is *totally disconnected* means that every point in $X$ is equal to the intersection of all clopen sets containing the point. $C$ is the Cantor set. ... | https://mathoverflow.net/users/95718 | Embedding into $C\times [0,1]$ | The answer to this question is negative even in dimension $n=1$.
A suitable counterexample can be constructed as follows.
By a [result of Dranishnikov](http://iopscience.iop.org/article/10.1070/SM1987v057n01ABEH003059), there exists a self-map $f:M\_1\to M\_1$ of the Menger cube $M\_1$ such that for any $y\in M\_1$... | 4 | https://mathoverflow.net/users/61536 | 310778 | 135,269 |
https://mathoverflow.net/questions/310773 | 4 | When dealing with a top degree differential form $\mu$ in a manifold $M$, a way of "computing" its cohomology class is integrating it through the whole manifold. For instance, if the integral $ \int\_M \mu=0 $ then the form is exact.
I've been trying to find a similar condition (computing an integral or something lik... | https://mathoverflow.net/users/108886 | Computing relative cohomology class of differential form | Assume that the pair $Z,M$ is oriented. There are two cases: If $Z$ represent a non-vanishing $n-1$-cycle, then the map
$H^{n-1}(M)\to H^{n-1}(Z)\stackrel{\int\_Z}{\to} \mathbb{R}$ is onto by Poincare duality (the restriction of a cocycle to $Z$ and integrating is like cupping with the poincare dual of $Z$ which can n... | 2 | https://mathoverflow.net/users/115052 | 310791 | 135,275 |
https://mathoverflow.net/questions/310725 | 1 | For some numerical analysis of a fluid, I am wondering if there is **any** inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In particular, I am seeking one that is useful for the form:
$$(D(x),D(v))\_{L^2} + (\nabla\cdot{x}, \nabla\cdot{v})\_{L^2... | https://mathoverflow.net/users/15045 | Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative? | I don't think what you want is possible.
In phase space $x$ and $v$ are basically quantities that can be independently prescribed. The first terms of your left hand side is linear in $v$. So fixing $x$, $v$ arbitrary, for $\lambda$ of the correct sign and $|\lambda|$ sufficiently small, you will have that the left h... | 3 | https://mathoverflow.net/users/3948 | 310798 | 135,278 |
https://mathoverflow.net/questions/310777 | 4 | Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle\_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there any sufficient conditions on $x, y$, and/or $M$ that ensure that $\langle{x}, {y}\rangle\_M > 0$ (other than the obvious $x... | https://mathoverflow.net/users/128729 | Which inner products preserve positive correlation? | Since $M$ is PSD, it can be decomposed as $M=A^T A$, where $A$ is another matrix; see
<https://math.stackexchange.com/questions/1801403/decomposition-of-a-positive-semidefinite-matrix>
Now $\langle x,y\rangle=x^T M y=x^T A^T Ay=\langle Ax,Ay\rangle$ and so your question is equivalent to: Which matrices $A$ preserve p... | 9 | https://mathoverflow.net/users/12518 | 310799 | 135,279 |
https://mathoverflow.net/questions/310803 | 11 | The Galois cohomology group $H^1(\mathbb{Q}, \mathbb{Z}/3\mathbb{Z})$ classifies cyclic cubic extensions $K/\mathbb{Q}$ (specifically: the non-trivial elements correspond to Galois cubic field extensions $K/\mathbb{Q}$ together with a choice of isomorphism $\mathrm{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$).
L... | https://mathoverflow.net/users/5101 | Cyclic cubic extensions and Kummer theory | It's just the map
$$x \mapsto y = \frac{x}{x^{\sigma}},$$
where the corresponding degree three extension of $\mathbb{Q}$ is the degree three subfield of $k(y^{1/3})$. The point is that it is obvious from the restriction map that
$$H^1(\mathbb{Q},\mathbb{Z}/3 \mathbb{Z})
= (k^{\times}/k^{\times 3})^{G = {\chi}},$... | 17 | https://mathoverflow.net/users/126439 | 310805 | 135,280 |
https://mathoverflow.net/questions/310808 | 16 | A group G is called ‘good’ if the canonical map $G\to\hat{G}$ to the profinite completion induces isomorphisms $H^i(\hat{G},M)\to H^i(G,M)$ for any finite $G$-module $M$. I’ve had multiple academics in my department claim to me that the orientation-preserving mapping class group of an orientable surface of genus $g$ wi... | https://mathoverflow.net/users/119286 | Are mapping class groups of orientable surfaces good in the sense of Serre? | This is an open and probably very difficult question. There have been purported proofs (for instance, [this one](https://arxiv.org/abs/math/0306379)), but they have all had fatal flaws.
The mapping class group is definitely not virtually abelian (or virtually solvable; for other readers, the relevant fact [alluded to... | 18 | https://mathoverflow.net/users/317 | 310811 | 135,282 |
https://mathoverflow.net/questions/310821 | 23 | The starting point of this question is the following:
>
> If $G$ is a group such that all elements have order at most $2$, then $G$ is commutative.
>
>
>
If $G$ is any group, let $G\_{>2}$ denote the set of elements $g\in G$ such that $g^2 \neq 1\_G$ where $1\_G$ denotes the neutral element of the group.
**Q... | https://mathoverflow.net/users/8628 | Are infinite groups in which most elements have order $\leq 2$ commutative? | Let $G$ be an infinite group in which the set of elements of order $\neq 2$ has cardinal $<|G|$. Then $G$ is 2-elementary abelian.
Indeed, by contradiction, let $g\in G$ be of order $3\le d\le \infty$. So the conjugacy class of $g$ has cardinal $<|G|$, and hence the centralizer $C\_g$ of $g$ has order $|G|$, and in t... | 40 | https://mathoverflow.net/users/14094 | 310830 | 135,290 |
https://mathoverflow.net/questions/310717 | 5 | Let M be a closed oriented manifold which has the structure of a "semi-bundle" (See Section 1.2. of Hatcher's notes on three-manifolds) over an interval I. Assume that M is Seifert fibered over a base B and that the Euler number of the Seifert fibering is zero. Suppose finally that M has a unique Seifert fibering, so B... | https://mathoverflow.net/users/120035 | Seifert fiberings of zero euler number which are semi-bundles | Let $\Sigma$ be the orientable generic fiber of the semi-bundle structure on $M$. I only consider the case $\chi(\Sigma) < 0$ for simplicity: the cases $\chi(\Sigma) \geq 0$ should be worked out by hand; in many cases the fibration is not unique there.
In that case the answer is yes, because of a stronger fact:
> ... | 4 | https://mathoverflow.net/users/6205 | 310837 | 135,293 |
https://mathoverflow.net/questions/310838 | 3 | **Short version:** Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional?
**Longer version:** Let $R$ be a commutative ring. Several slightly different definitions of the notion of *catenarity* of $R$ are in us... | https://mathoverflow.net/users/11025 | Local ring of infinite dimension | Let $k$ be a field. Let $x\_{i, j}$, $1 \leq j \leq i \in \mathbf{N}$ be variables. Consider the ring
$$
R = k[x\_{i, j}]/(x\_{i, j} x\_{i', j'}, i \not = i')
$$
Let $\mathfrak m$ be the maximal ideal generated by all $x\_{i, j}$. For any prime ideal $\mathfrak p \not = \mathfrak m$ there exists a unique $i$ such that ... | 7 | https://mathoverflow.net/users/128993 | 310843 | 135,296 |
https://mathoverflow.net/questions/310531 | 7 | Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p\_{f,A,n}(x)$ of degree $n$ such that $f(a)=p\_{f,A,n}(a),\forall a\in A$. Such a polynomial is called the interpolating polynomial of $f$ with nodes $A$. Due... | https://mathoverflow.net/users/127118 | Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise | The result that you mention in the first part of your question is a classical result by Faber
>
> G. Faber, Uber die interpolatorsche Darstellung stetiger Funktionen, Jahresber. der deutschen Math. Verein. 23 (1914), 190-210.
>
>
>
Of course, this result does not exclude pointwise convergence. This question wa... | 6 | https://mathoverflow.net/users/89429 | 310853 | 135,299 |
https://mathoverflow.net/questions/310874 | 6 | For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal H})$.
I have heard that this extends to the unbounded case, but I can't find a precise statement. So if $D$ is a dens... | https://mathoverflow.net/users/128876 | Unbounded version of continuous functional calculus | You find the precise abstract statment in the Internet Seminar lecture notes of Markus Haase, especially in Chapter 4.
See
<https://www.math.uni-kiel.de/isem21/en/course/phase1>
By the way, the answer is yes.
| 6 | https://mathoverflow.net/users/12898 | 310875 | 135,303 |
https://mathoverflow.net/questions/310835 | 5 | Consider two undirected graphs $G$ and $H$ of the same order (same number of vertices).
If $G$ and $H$ have a *common equitable partition*, then it is known (see e.g., Chapter 6 in [1](https://www.ams.jhu.edu/ers/wp-content/uploads/sites/2/2015/12/fgt.pdf)) that these graphs must have the same number of walks of any... | https://mathoverflow.net/users/9839 | Two graphs with the same number of walks but without a common equitable partition | It appears that one way to obtain a desired example pair $G$ and $H$ is to let
$G$ be the disjoint union $G\_1\cup G\_2$ and $H$ be the disjoint union of
$H\_1\cup H\_2$, such that
* $G\_1$ and $H\_1$ have a common equitable partition, yet are not co-spectral; and
* $G\_2$ and $H\_2$ have the same number of walks o... | 2 | https://mathoverflow.net/users/9839 | 310876 | 135,304 |
https://mathoverflow.net/questions/310673 | 19 | I am trying to properly write a proof of Minkowski's theorem in a self-contained way and understandable by (good) undergraduates.
>
> **Theorem (Minkowski)**
>
>
> Let $L$ be a lattice of $\mathbb{R}^n$ and $C$ a convex body, symmetric relatively to the origin and with $\mathrm{vol}{C} > 2^n \det(L)$. Then there ... | https://mathoverflow.net/users/128718 | Proof of Minkowski theorem using harmonic analysis | $$
\newcommand{\Vol}{\mathrm{Vol}\,}
\newcommand{\supp}{\mathrm{supp}\,}
\newcommand{\eps}{\varepsilon}
\renewcommand{\phi}{\varphi}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\<}{\langle}
\newcommand{\>}{\rangle}
$$
This is in fact fairly standard, but filling in the technical details can ... | 12 | https://mathoverflow.net/users/9924 | 310877 | 135,305 |
https://mathoverflow.net/questions/310704 | 6 | The Mulitiset monad, aka the free commutative monoid monad or "Bag" monad, takes a set to the set of all Multisets for that set. A Multiset is like a set, but can have duplicates. It is used in computer science as the Bag monad, where the Bag is a data structure that can take in symbols or readings from a set and pop t... | https://mathoverflow.net/users/10007 | Map from the Multiset Monad to the Giry Monad: From Data to Probabilities | **The intuition described in the question can be made precise as a natural map from the finite-multiset monad to a certain monad of (non-normalised) *measures*. However, this map doesn’t factor through the discrete Giry monad; in fact, there is no monad morphism from either the finite-multiset monad or the non-empty-fi... | 7 | https://mathoverflow.net/users/2273 | 310888 | 135,308 |
https://mathoverflow.net/questions/310869 | 4 | Let
$$
\beta\_g:=\inf\{\frac14\int\_\Sigma H^2 d\mu \hspace{0.2cm} | \hspace{0.2cm} \Sigma\subset \mathbb
R^{3}, \operatorname{genus}(\Sigma)=g \}
$$
be the infimum of the Willmore energy of embedded genus-$g$ surfaces. A standard result by Willmore says that $\beta\_0=4\pi$ and Marquez-Neves showed that $\beta\_1=2\p... | https://mathoverflow.net/users/127306 | Monotonicity of infimum of the Willmore energy with prescribed genus | As pointed out in the paper you reference, this question was posed on p. 446 of this paper
*Kühnel, Wolfgang; Pinkall, Ulrich*, [**On total mean curvatures**](http://dx.doi.org/10.1093/qmath/37.4.437), Q. J. Math., Oxf. II. Ser. 37, 437-447 (1986). [ZBL0627.53044](https://zbmath.org/?q=an:0627.53044).
I found 19 p... | 4 | https://mathoverflow.net/users/1345 | 310910 | 135,314 |
https://mathoverflow.net/questions/310898 | 16 | If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of limit theories $\mathcal{T}$ such that the category of models of $\mathcal{T}$ is a topos? The category of models of a $\mat... | https://mathoverflow.net/users/61823 | When is the category of models of a limit theory a topos? | My collaborator *Julia Ramos González* and I are working on this question precisely in these days.
A part of the answer is already cointained in a paper by *Carboni, Pedicchio and Rosický*: **[Syntactic characterizations of various classes of locally presentable categories](https://doi.org/10.1016/S0022-4049(01)0001... | 15 | https://mathoverflow.net/users/104432 | 310915 | 135,316 |
https://mathoverflow.net/questions/310818 | 16 | Is it true that the Baumslag-Solitar groups, say, $BS(1,n)$, $|n|\ge 2$, are finitely presented groups with largest Dehn functions (namely, exponential growth) known to be inside finitely presented groups with quadratic Dehn functions?
| https://mathoverflow.net/users/nan | What is the largest known Dehn function of f.p. subgroup of a f.p. group with quadratic Dehn function? | Yes, the highest known Dehn function of a subgroup of a finitely presented group with quadratic Dehn function is exponential. The example was found by Yves Cornulier and Romain Tessera in
Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups.
Confluentes Math. 2 (2010), no. 4, 431–443.
| 7 | https://mathoverflow.net/users/nan | 310916 | 135,317 |
https://mathoverflow.net/questions/310753 | 7 | Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$.
For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the Fourier transform of $f$.
**Question:** If $f\_n\stackrel{n\rightarrow\infty}{\longrightarrow}f$ in $\mathcal{E}'(\mathbb{... | https://mathoverflow.net/users/128952 | On the Fourier-Laplace transform of compactly supported distributions | Let me expand Nate Eldredge's comments. A more elementary proof than the one provided by Jochen Wengenroth, still requiring no computations, is as follows:
$\mathcal{E}'$ is the dual space of $C^\infty$, which is endowed with the Fréchet space topology given e.g. by the seminorms $\|f\|\_k:=\|f\|\_{C^k(B\_k(0))}$. Th... | 2 | https://mathoverflow.net/users/36952 | 310940 | 135,321 |
https://mathoverflow.net/questions/310610 | 7 | For $n\in\Bbb{N}$, define three matrices $A\_n(x,y), B\_n$ and $M\_n$ as follows:
(a) the $n\times n$ tridiagonal matrix $A\_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal all $-x$'s. For example,
$$ A\_4(x,y)=\begin{pmatrix} y&x&0&0\\-x&y&x&0\\0&-x&y&x
\\0&0&-x&y\end{pmatrix}. $$
(b)... | https://mathoverflow.net/users/66131 | Block matrices and their determinants | Flip the order of the Kronecker products to get $M'=A\_n(I\_n,I\_n)+B\_n\otimes T\_n$, where $T\_n=A\_n(1,0)$. Note that $\det M=\det M'$. Since all blocks are polynomial in $A$, they commute, and therefore the determinant of $M'$ is $\det(f(T\_n))$, where $f(x)=\det(A\_n(1,1)+xB\_n)$. That is, $f(x)=\det(B\_n)\det(x+A... | 6 | https://mathoverflow.net/users/112641 | 310947 | 135,325 |
https://mathoverflow.net/questions/310932 | 1 | I have the following linear matrix inequality:
$$F^T P + PF < 0,$$
where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension.
Let $Q$ be a positive definite matrix that obeys $Q \leq P$. Is this equivalence valid?
$$F^TP+PF< 0 \iff F^TQ+QF< 0$$
| https://mathoverflow.net/users/128605 | Linear matrix inequality | Here is a counterexample to what I think is being asked.
In this counterexample, $P$, $Q$, and $P - Q$ are all symmetric positive definite, $F^TP+PF$ is negative definite, and $F^TQ+QF$ is indefinite, as seen below.
```
>> disp(P)
30 15
15 50
>> disp(Q)
3 -2
-2 7
>> disp(F)
-2 ... | 1 | https://mathoverflow.net/users/75420 | 310953 | 135,328 |
https://mathoverflow.net/questions/310943 | 6 | Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int\_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$
is not always an element of $W^{1,1}(0,\tau;X)$. That seems odd to me. Can anyone think of an example?
| https://mathoverflow.net/users/113264 | Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$? | In other words, if $A$ is the infinitesimal generator of $T$, the mild solution of the abstract inhomogeneous Cauchy problem
$$\begin{cases}\dot v =A v +f\\v(0)=0\end{cases}$$
needs not to be $W^{1,1}\_{loc}(\mathbb{R}\_+, X)$.
For instance an $f\in C^0(\mathbb{R}\_+,X)$ of the form $f(t):=T(t)x$ for some $x=x(\theta... | 8 | https://mathoverflow.net/users/6101 | 310955 | 135,329 |
https://mathoverflow.net/questions/310886 | 8 | Suppose we have $n^2$ red points and $n(n-1)$ blue points in the plane in general position. Is it possible to find a subset $S$ of red points such that the convex hull of $S$ does not contain any blue points, where $|S|>n^{\epsilon}$?
| https://mathoverflow.net/users/80245 | monochromatic subset | By coloring the Horton set with two colors, periodically mod 3 according to the $x$-coordinate, [Devillers et al.](https://doi.org/10.1016/S0925-7721(03)00013-0) obtained arbitrarily large bicolored point sets with no monochromatic empty convex $5$-gon (that is, monochromatic $5$-hole).
Using the fact that every set ... | 7 | https://mathoverflow.net/users/24076 | 310961 | 135,330 |
https://mathoverflow.net/questions/310846 | 2 | I would like your help with the following tail condition, which arises in the theory of large deviations.
Let $P(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$, $ G:P(\mathbb{R}^{d})\longrightarrow \mathbb{R}$ continuous and bounded,
$W:P(\mathbb{R}^{d})\longrightarrow [0,\infty]$, ${Q}$ prob... | https://mathoverflow.net/users/128523 | Tail condition (Varadhan's lemma) | Since $G$ is bounded and $\sigma W\ge0$, for all large enough real $M$ we have $A=\emptyset$, $\int\_A\cdots=0$, and $\log\int\_A\cdots=-\infty$, which yields what you wanted.
| 1 | https://mathoverflow.net/users/36721 | 310969 | 135,333 |
https://mathoverflow.net/questions/310956 | 3 | If $X\subset \mathbb{P}^n$ is a smooth hypersurface (more generally a complete intersection) of dimension at least 2, and if $K\_X+\mathscr{O}\_X(-1)\geq 0$, why is it true that $H^0(T\_X(1))=0$?
(Source: *On a conjecture of Clemens on rational curves on hypersurfaces*, paragraph before Proposition 1.1)
| https://mathoverflow.net/users/37650 | Sections of tangent bundle on hypersurface | This has little to do with hypersurfaces. If $\dim(X)=d$, the only condition you need on $X$ is $H^0(X,\Omega ^{d-1}\_X)=0$.
The wedge product gives a non-degenerate pairing $\Omega ^1\_X\otimes \Omega ^{d-1}\_X\rightarrow K\_X$, hence an isomorphism $T\_X\cong \Omega ^{d-1}\_X\otimes K\_X^{-1}$; therefore $T\_X(1)\c... | 5 | https://mathoverflow.net/users/40297 | 310975 | 135,338 |
https://mathoverflow.net/questions/310926 | 6 | Does the concept of differential manifold with negative dimension make sense, in differential geometry?
If yes, how is it defined? Do you have any reference to recommend?
My problem was born in Einstein warped-product manifolds, i.e. to admit a type of metric, the fiber-manifold must have a negative dimension, but... | https://mathoverflow.net/users/90594 | Manifolds with negative dimension – Definition, References | Smooth manifolds of negative dimension are defined in derived geometry.
Recall that if A→M and B→M are two transversal submanifolds
of codimension a and b respectively,
then their intersection C is again a submanifold, of codimension a+b.
Derived geometry explains how to remove the transversality condition
and make... | 5 | https://mathoverflow.net/users/402 | 310976 | 135,339 |
https://mathoverflow.net/questions/310575 | 12 | This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< e\_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). Then I conjecture that there exists some **distinct** $P\_i$'s in $... | https://mathoverflow.net/users/108278 | Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$ | If we restrict to the case when $e\_1=e\_2=\cdots=e\_n=2$ the right condition is $\sum\_{i=1}^n e\_i=n+kp$ or $n+kp+1$ for some $k\geq 1$. This provides counterexamples to the stated conjecture (see below) and it shows that the right condition is more complicated than just one inequality.
Here is a proof of my claim:... | 8 | https://mathoverflow.net/users/2384 | 310980 | 135,340 |
https://mathoverflow.net/questions/310951 | 6 | Let $X$ be a toric variety containing the $n$-torus $T\overset{i}{\hookrightarrow} X$. The action of $T$ extends naturally to an action on the sheaf $i\_\*\mathcal{O}\_T$ by
$$(\alpha\cdot f)(x):=f(\alpha^{-1}x) \; \; \forall \, f\in (i\_\*\mathcal{O}\_T)(U),\forall \alpha\in T$$
on each invariant open set $U\subset... | https://mathoverflow.net/users/117675 | Isomorphic equivariant sheaves are equivariantly isomorphic on a toric variety | **Edit.** The short answer is that this fails for every toric variety except when $T$ equals all of $X$. I edited the original answer to prove this, and also to prove the positive result below by @S. carmeli.
Denote by $X^{(1)}$ the finite set of irreducible components of the closed subset $X\setminus T$ of $X$.
... | 7 | https://mathoverflow.net/users/13265 | 310984 | 135,342 |
https://mathoverflow.net/questions/310966 | 3 | For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $M$ have the same sign and are bigger than $\varepsilon$ in absolute value.
A theorem by Eschenburg states that if $M$ is ... | https://mathoverflow.net/users/85934 | Special spheres: principal curvatures with different signs | Most of Cartan's isoparametric hypersurfaces in $S^4$ have your desired properties: They have constant principal curvatures (in fact, they are homogeneous), nearly all of them have all three principal curvatures nonzero, and the ones near the minimal one (i.e., the one of maximal volume) have principal curvatures of tw... | 5 | https://mathoverflow.net/users/13972 | 310995 | 135,349 |
https://mathoverflow.net/questions/310924 | 2 | I need some philosophical explanation for JSJ decomposition theorem. It says that closed orientable irreducible 3-manifold can be cut along set of incompressible tori onto pieces which are:
* atoroidal or Seifert-fibered
* hyperbolic or Seifert-fibered
* hyperbolic or spherical or Seifert-fibered with infinite fundam... | https://mathoverflow.net/users/nan | JSJ decomposition and classification of 3-manifolds | Nobody knows how to classify the family zero manifolds. The Seifert ones are pretty well-known and were classified by Seifert himself: each such may be represented as a string like
$$(S, (p\_1,q\_1), \ldots, (p\_k,q\_k)$$
where $S$ is a surface and $(p\_i,q\_i)$ are coprime numbers, and there is a theorem that says ve... | 7 | https://mathoverflow.net/users/6205 | 311001 | 135,351 |
https://mathoverflow.net/questions/309502 | 2 | Is there an infinite, simple, undirected graph $G=(V,E)$ such that there is $n\in\mathbb{N}$ with the following properties?
1. $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$, but $K\_{n+1}$ is not a minor of $G$, and
2. if $F$ is a finite subgraph of $G$, then $K\_n$ is not a minor of $F$.
(... | https://mathoverflow.net/users/8628 | Compactness of Hadwiger number | Assume $K\_n$ is a minor of $G$. Each vertex of $K\_n$ corresponds to a connected subset of vertices of $G$ as it can only have been obtained by contracting edges. Each edge of $K\_n$ can only have been derived from a path through $G$, by contracting edges.
Now create $F$ as follows: collect the edges of the $n(n-1)/2... | 1 | https://mathoverflow.net/users/5903 | 311005 | 135,352 |
https://mathoverflow.net/questions/311004 | 1 | Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be *regular open* if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A standard exercise exercise shows that $(\text{RO}(X,\tau),\subseteq)$ is not only a lattice, but even a [Boolean algebra](... | https://mathoverflow.net/users/8628 | Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$? | No, $\mathrm{RO}(X)$ is complete; $\mathcal{P}(\omega)/\mathit{fin}$ is not (no strictly increasing sequence has a supremum).
| 6 | https://mathoverflow.net/users/5903 | 311007 | 135,354 |
https://mathoverflow.net/questions/310999 | 2 | I would like an explanation for the fact stated in the title. To repeat:
>
> **Question**: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of degree $n$?
>
>
>
Note that the statement might as well be false (though I would not bet on that). [EDIT: I th... | https://mathoverflow.net/users/47722 | Having a separable extension of degree $n$ implies having a Galois extension of degree $n$? | I am posting my comment as an answer (I had no intention of causing any dispute).
That is not true. Let $K$ be the maximal solvable extension of $\mathbb{Q}$ (or $\mathbb{C}(t)$, etc.). There are irreducible quintic polynomials over $K$ (otherwise every quintic over $\mathbb{Q}$ would have a solvable Galois group). T... | 10 | https://mathoverflow.net/users/13265 | 311012 | 135,355 |
https://mathoverflow.net/questions/310939 | 6 | Recall the [integer partition function](https://oeis.org/A000041) $P(n)$ with generating function
$$\sum\_{n\geq0}P(n)x^n=\prod\_{k=1}^{\infty}\frac1{1-x^k}.$$
Let $[n]\_q=\frac{1-q^n}{1-q}$ denote the $q$-analogue of the integer $n$ and let $\lambda=(\lambda\_1,\lambda\_2,\dots)\vdash n$ be a partition of $n$. Now, de... | https://mathoverflow.net/users/66131 | Partitions, $q$-polynomials and generating functions | A generating function of sorts is given by
$$ \sum\_{n\geq 1}\Psi\_q(n)x^n = P(x)\sum\_{m\geq 0}q^m\sum\_{k\geq m+1}
\frac{x^k}{1-x^k}, $$
where $P(x)=\prod\_{i\geq 1}(1-x^i)^{-1}$.
| 3 | https://mathoverflow.net/users/2807 | 311014 | 135,356 |
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