parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/285895 | 11 | Does there exist a Banach space $X$ such that $X^{\*\*}$ is separable but $X^{\*\*\*}$ is non-separable?
More generally, for every natural $n$ can someone construct an example of Banach space $X$ such that $X^{n}$ is separable but not $X^{n+1}$?
| https://mathoverflow.net/users/76412 | Separable bidual but nonseparable third dual | Yes to both. Lindenstrauss extended James' construction to show that for any separable $X$ there is a separable $Y$ s.t. $Y^{\*\*}/Y$ is isometrically isomorphic to $X$. Induct on that. Spaces built that way are called James-Lindenstrauss spaces. Another proof is contained in my "Factoring Weakly Compact Operator" pape... | 10 | https://mathoverflow.net/users/2554 | 285897 | 126,268 |
https://mathoverflow.net/questions/285800 | 6 | Let $f : X \to Y$ be a finite surjective morphism of smooth affine algebraic varieties over the complex numbers. Is it true that a function on $Y$ whose pullback via $f$ is an analytic function on $X$, is itself analytic?
I ask because I am interested in knowing that, for a reductive complex algebraic group $G$, an a... | https://mathoverflow.net/users/2095 | "Descent" of analytic functions along a finite morphism | Let $h:Y\to \mathbb{C}$ be such a function. If $U\subset Y$ is an open subset (for the complex topology) and $s:U\to X$ is an (analytic) local section of $f$ on $U$, then $h=h\circ f\circ s$ on $U$, hence $h\_{\vert U}$ is analytic since $h\circ f$ is.
So, $h$ is analytic on $Y\smallsetminus B$ where $B\subset Y$ is... | 9 | https://mathoverflow.net/users/7666 | 285900 | 126,269 |
https://mathoverflow.net/questions/278489 | 13 | It is well known that for a finite group $G$, the associator of the fusion category of $G$-graded $k$-vector spaces is given by an element of $H^3(G,k^\*)$, up to equivalence of categories. ($k^\*$ is the multiplicative group of units in $k$.)
A crucial step when showing this is the fact that in $G$-graded vector spa... | https://mathoverflow.net/users/13767 | A cohomology theory for fusion categories | There is no such cohomology theory known (in particular, this is not related to Davydov-Yetter cohomology which is about deformations and vanishes for finite groups). In my mind this is a very important open problem in the field which could have some major applications to classification of Izumi categories. One can spe... | 10 | https://mathoverflow.net/users/22 | 285907 | 126,272 |
https://mathoverflow.net/questions/285911 | 3 | Let $\mathcal{G}$ be a affine algebraic group scheme(may not be reductive) over a scheme $S$. How to define a rational representaion of $\mathcal{G}$ (over $S$)? Is there always a faithful representation?
Please provide references related to these questions.
| https://mathoverflow.net/users/nan | Representation of a group scheme | I'm not sure what your current sources are, but the definitions are laid out clearly in SGA3 (by Demazure and Grothendieck) and similarly in the book by Demazure and Gabriel, *Groupes algebriques* (North-Holland, 1970) which was later published in an English translation. (Their designation of this book as "Tome I" is o... | 6 | https://mathoverflow.net/users/4231 | 285912 | 126,275 |
https://mathoverflow.net/questions/285528 | 7 | Let $(E, \langle\cdot\;, \;\cdot\rangle)$ be a complex Hilbert space. Let $T\in\mathcal{L}(E)$ and $M\in \mathcal{L}(E)^+$.
Assume that $T(ker(M))\nsubseteq ker(M)$. We define the following subset:
\begin{eqnarray\*}
S\_M(T)
&=&\{\lambda\in \mathbb{C}\,;\;\; \exists\,(\alpha\_n,\beta\_n)\in ker(M)\times \overline{I... | https://mathoverflow.net/users/116483 | Is the following set convex or not? | Suppose $M\geq 0$ and $T(\ker M) \nsubseteq \ker M$. So there exists an $\alpha \in \ker M$ such that $T\alpha \notin \ker M$ which gives that $ MT\alpha \neq 0$ and since $M$ is positive then $M^{1/2}MT\alpha\neq 0$ as well.
Let $\beta = \frac{MT\alpha}{\|M^{3/2}T\alpha\|} \in {\textrm Im}(M)$ then $\|M^{1/2}\beta\|... | 4 | https://mathoverflow.net/users/76593 | 285913 | 126,276 |
https://mathoverflow.net/questions/285908 | 2 |
>
> Let $f : U \to \Bbb C$ be a holomorphic function in a neighborhood of 0 (or a
> polynomial). Is it true that for any integer $k \geq 1$,
> $$ |f^{(k)}(0)| \leq \|D^k\_0 |f|\| \left( = \max\_{|u\_1|=\dotsb = |u\_k|=1} \| D^k\_0 |f| (u\_1,\dotsc,u\_k)\| \right)? $$ where $D^k\_0 |f|$ is the
> $k$th derivative at... | https://mathoverflow.net/users/19205 | For a holomorphic function $f$, is $|f^{(k)}(0)| \leq \| D_0^k |f| \|$? | It is true for $k=1,2$ but not for $k\ge 3$.
Write $f=g^2$ near $0$ and let $g(z)=\sum\_{m\ge 0}a\_m z^m$. Then $|f|(z)=\overline{g(z)}g(z)$, so we want (up to $k!$ on both sides) that
$$
\left|\sum\_{0\le m\le k/2}a\_ma\_{k-m}\right|\le \max\_u\left|\Re\sum\_{0\le m\le k/2}\bar a\_ma\_{k-m}P\_{m}(u)\right|
$$
where... | 5 | https://mathoverflow.net/users/1131 | 285921 | 126,281 |
https://mathoverflow.net/questions/285651 | 8 | This is motivated by [a previous question of mine](https://mathoverflow.net/questions/283373/undetermined-copy-diagonalize-games-without-ch), but I think it is ultimately more interesting (and hopefully easier to answer in the positive). In that question, a class of games *(on $\omega$, of length $\omega$)* is consider... | https://mathoverflow.net/users/8133 | Undetermined games of "overdetermined" type | The club game is overdetermined by a projective equivalence relation, so stationary co-stationary subsets of $\omega\_1$ will give the counterexamples to overdeterminacy that you're looking for. The equivalence relation is actually $\Delta^1\_2$, and if you're careful you can probably show it's low in the difference hi... | 4 | https://mathoverflow.net/users/102684 | 285922 | 126,282 |
https://mathoverflow.net/questions/285575 | 10 | Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having
$$
f(0) = \widehat{f}(0) = 1
$$
and
$$
f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1],
$$
the following ridiculous argument based on the prime number theorem yields the strict upper bound
$$
\int f(t) \log{\frac{1}{|t|}} \, d... | https://mathoverflow.net/users/26522 | The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non-negative outside of $[-1,1]$ | One can take the continuum limit of your proof as $X \to \infty$, again using the prime number theorem, to obtain a proof that does not involve primes at all:
$$ \int f(t) \log \frac{1}{|t|}\ dt = \gamma - \sum\_{\sigma = \pm 1} \int\_0^\infty f(\sigma t) (\log t + \gamma)\ dt $$
$$ = \gamma - \lim\_{\varepsilon \to ... | 8 | https://mathoverflow.net/users/766 | 285929 | 126,283 |
https://mathoverflow.net/questions/285924 | 1 | Let $(X,d)$ be a metric space. $X$ is said to be a Busemann $G$-space provided it satisfies the following axioms:
(1) Menger Convexity: Given distinct points $x,y\in X$, there is a point $z\in X-\{x,y\}$, so that $d(x,z)+d(z,y)=d(x,y)$.
(2) Finite Compactness: Every $d$-bounded infinite set has at least one accumul... | https://mathoverflow.net/users/114032 | Small codimension 1 ball on the boundary of metric ball in Busemann G-spaces | The ball with *induced* metric generally will not be a G-space. Take $R^2$ with the Euclidean metric, and consider the unit ball $S^1$ with the induced metric. Let $x=(0,1)$ and $y=(1,0)$; then $d(x,y)=\sqrt{2}$ in that metric, but there is no $z$ in $S^1$ with $d(x,z)+d(z,y)=d(x,y)$. So Menger convexity fails and that... | 1 | https://mathoverflow.net/users/nan | 285930 | 126,284 |
https://mathoverflow.net/questions/285934 | 10 | I am looking for references where the following (or similar questions) have been studied:
Let $K$ be a number field or a function field in one variable over a finite field and let $E$ be an elliptic curve (or more generally, an abelian variety) over $K$. If $x \in E(K)$ is a point of infinite order then the order of ... | https://mathoverflow.net/users/519 | Orders of reductions of rational points on elliptic curves |
>
> For example, is it known that there is an infinite sequence of rational primes $p\_i$ and primes $P\_i$ of (the ring of integers of) $K$ such that $p\_i$ divides the order of the reduction of $x$ modulo $P\_i$?
>
>
>
Yes, a weak and ineffective form of this at least follows from Siegel's integrality finitene... | 4 | https://mathoverflow.net/users/26522 | 285940 | 126,286 |
https://mathoverflow.net/questions/285938 | 3 | Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a *cover* if $\bigcup {\cal C} = X$. A subset $D\subseteq X$ is a *choice set* for ${\cal C}$ if $|D\cap c| = 1$ for all $c\in C$.
[Ramiro de la Vega](https://mathoverflow.net/users/17836/ramiro-de-la-vega) showed in his ... | https://mathoverflow.net/users/8628 | Choice sets and the axiom of choice | Let $P\subseteq X\times Y$ be sets such that $\forall x\in X\exists y\in Y[(x,y)\in P]$. We wish to derive from (S) the existence of an $f\subseteq P$ such that $\forall x\in X\exists! y\in Y[(x,y)\in f]$.
Consider the following cover of $P$. Let $
\cal{C}$ $:= \{\{(x,y)|y\in Y, (x,y)\in P\}|x\in X\}$. Notice that an... | 6 | https://mathoverflow.net/users/101577 | 285941 | 126,287 |
https://mathoverflow.net/questions/285925 | 1 | Original question here:
[Global orthogonal coordinates on the open unit ball](https://mathoverflow.net/questions/285759/global-orthogonal-coordinates-on-the-open-unit-ball)
So let $v\_i:\mathbb{R}\to\text{Im }v\_i\subset\mathbb{R}$ be diffeomorphisms, $1\leq i\leq n$, $\mathbb{R}^n=\mathbb{R}^{n\_1}\times\cdots\times... | https://mathoverflow.net/users/117091 | Global orthogonal coordinates on the open unit ball II | The answer is already 'no' when $n=2$, which implies that the answer is 'no' for all $n\ge 2$.
To see why, note that the local orthogonal diffeomorphisms that you describe depend essentially only on arbitrary functions of one variable: The reparametrizations $\nu\_i$ and conformal diffeomorphisms of domains in $\math... | 3 | https://mathoverflow.net/users/13972 | 285954 | 126,290 |
https://mathoverflow.net/questions/285955 | 21 | I'm looking for an explanation or a reference to why there is this equivelence of triangulated categories: $${D}^b(\mathrm {Coh}(\Bbb P^1))\simeq {D}^b(\mathrm {Rep}(\bullet\rightrightarrows \bullet))$$
It is my understanding that the only reason why $\Bbb P^1$ appears at all is because it is used to index the regular ... | https://mathoverflow.net/users/54401 | Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecker quiver? | Let $\mathcal O$ be the structure sheaf of $\mathbb P^1$. Then $\mathcal O \oplus \mathcal O(1)$ is rigid and generates the derived category of coherent sheaves on $\mathbb P^1$. Thus, it is a tilting object, and so the derived category is equivalent to the category of modules over its endomorphism ring, which is the p... | 22 | https://mathoverflow.net/users/468 | 285961 | 126,295 |
https://mathoverflow.net/questions/285873 | 7 | The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id\_A$.
Question: Suppose that $X\_1$ and $X\_2$ are two polyhedra. If $A$ is homotopy dominated by $X\_1\vee X\_2$, then is $A$ of the form $A\_1 \vee A... | https://mathoverflow.net/users/114476 | Homotopy domination of a wedge of two polyhedra | The answer here is certainly yes under many sets of mild side hypotheses.
For example, once upon a time, I wrote a paper with Frank Adams (!) that seems of some relevance: [J.F.Adams and N.J.Kuhn, *Atomic spaces and spectra*, Proc. Edin. Math. Soc **32** (1989), 473-481]. We show that if $X$ is a space or spectrum th... | 6 | https://mathoverflow.net/users/102519 | 285973 | 126,299 |
https://mathoverflow.net/questions/285353 | 48 | Let $(p\_1, p\_2)$ be a twin prime pair, where we include $(2, 3)$. If $p\_1 \equiv 1$ mod $4$ then we let $t\_{(p\_1, p\_2)} := p\_1 ^ 2 / p\_2 ^ 2$ otherwise, we let $t\_{(p\_1, p\_2)} := p\_2 ^ 2 / p\_1 ^ 2$.
I conjecture that the product
$$
\prod\_{(p\_1, p\_2): \text{twin primes}}t\_{(p\_1, p\_2)}
=\tfrac{3 ^ 2... | https://mathoverflow.net/users/116870 | Twin primes conjecture and extrapolation method | From the product
$$
\prod\_{(p\_1, p\_2): \text{twin primes}}t\_{(p\_1, p\_2)}
=\tfrac{3 ^ 2}{2 ^ 2} \cdot \tfrac{5 ^ 2}{3 ^ 2}\cdot
\tfrac{5 ^ 2}{7 ^ 2}\cdot\tfrac{13 ^ 2}{11 ^ 2} \cdot\tfrac{17 ^ 2}{19 ^ 2} \cdot\tfrac{29 ^ 2}{31 ^ 2} \cdot\tfrac{41 ^ 2}{43 ^ 2} \cdot \tfrac{61 ^ 2 }{59 ^ 2} \cdot \tfrac{73 ^ 2}{ 7... | 3 | https://mathoverflow.net/users/116870 | 286006 | 126,311 |
https://mathoverflow.net/questions/285966 | 10 | Suppose I am given a set of $n$ intervals, each having length $\ell\_i$. Is there a bound on the number of possible orderings of their left and right endpoints? For example, if each interval is represented by $[x\_i,y\_i]$, with $y\_i-x\_i=\ell\_i$, then one possible ordering would be $x\_1\leq x\_3\leq y\_1\leq x\_2 \... | https://mathoverflow.net/users/70190 | Enumerating all arrangements of intervals with given lengths | If all the interval lengths are the same, then the number of ways is
$n!C\_n$, where $C\_n$ is a Catalan number. If we are interested only in
the number of ways we can specify whether $y\_i<x\_j$ or $x\_j<y\_i$ for
all $i,j$, then the number of ways is the number of regions of the
hyperplane arrangement $\mathcal{A}(\e... | 10 | https://mathoverflow.net/users/2807 | 286007 | 126,312 |
https://mathoverflow.net/questions/285995 | 2 | Let $\mathfrak g$ be a real simple **split** Lie algebra. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the Cartan decomposition. Let $\mathfrak a\subseteq \mathfrak p$ be a maximal abelian subalgebra. Let $\alpha \in \mathfrak a^\*$ be a (restricted) root and let $\mathfrak g\_{\alpha}$ be its root space.
>... | https://mathoverflow.net/users/23500 | Dimension of restricted root spaces of split Lie algebras | Yes. See Helgason (Prop. 6.3, [p. 430](https://books.google.com/books?id=a9KFAwAAQBAJ&pg=PA430); 6(a), [p. 531](https://books.google.com/books?id=a9KFAwAAQBAJ&pg=PA531)) or Onishchik–Vinberg [1990](https://mathscinet.ams.org/mathscinet-getitem?mr=1064110) (23–25, [p. 274](https://books.google.com/books?id=TV7sCAAAQBAJ&... | 6 | https://mathoverflow.net/users/19276 | 286009 | 126,313 |
https://mathoverflow.net/questions/286033 | 2 | Is there a connected $T\_2$ space $(X,\tau)$ with more than one point, such that the singletons and $X$ are the only connected subspaces of $X$?
| https://mathoverflow.net/users/8628 | Connected $T_2$ space with essentially no connected subspaces | There is no such space. For if $x$ is any point and $X\setminus \{x\}=U|V$ then $\{x\}\cup U$ and $\{x\}\cup V$ are connected sets, each with more than one point and different from $X$.
The closest thing you can get is a connected set whose connected subsets are cofinite. The axiom CH implies there is a countable con... | 6 | https://mathoverflow.net/users/95718 | 286034 | 126,319 |
https://mathoverflow.net/questions/286020 | 4 | It seems rather surprising that, given the Diophantine equation,
$$a^3+b^3+c^3 = n^3\tag1$$
then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this [MSE post](https://math.stackexchange.com/questions/2514643/statistics-for-n-in-sum-of-cubes-a3b3c3-n3).... | https://mathoverflow.net/users/12905 | What is so special about $a^3+b^3+c^3 = (13m)^3$? | A solution of (1) must contain $0, 2$ or $4$ terms divisible by $13$. Essentially, this is because the only cubic residues mod $13$ are $0, 1, 5, 8, 12$, and there is no combination (with or without repetition) of three of the non-zero residues with a sum $s$ such that $s \equiv 0 \pmod{13}$. A proof is in (A).
Altho... | 6 | https://mathoverflow.net/users/117220 | 286044 | 126,321 |
https://mathoverflow.net/questions/286024 | 9 | I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that classifying lens spaces was I problem that motivated Reidemeister to introduce torsion.
All of the definitions of torsion tha... | https://mathoverflow.net/users/99414 | Intuition for torsion of a chain complex and application to lens spaces | Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$
$$0\to U\_0 \stackrel{\pa}{\to} U\_1\to 0.$$
(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)
This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U\... | 15 | https://mathoverflow.net/users/20302 | 286045 | 126,322 |
https://mathoverflow.net/questions/286012 | 3 | I am looking for a space as in the title, i.e.,
>
> **Is there a metacompact, normal, CCC space which is not Lindelof?**
>
>
>
A space is ccc iff any family of pairwise disjoint open sets is at most countable.
A space $X$ is metacompact iff for any open cover $\mathcal U$ of $X$ there is
a point finite refin... | https://mathoverflow.net/users/39873 | Is there a metacompact, normal, CCC space which is not Lindelof | Yes, such an example can be obtained using the Pixley-Roy hyperspace construction.
Given a topological space $X$, the *Pixley-Roy hyperspace on $X$* ($PR(X)$) is defined as the space of all non-empty finite subsets of $X$ with the topology generated by sets of the form $[F,U]:=\{G \in PR(X): F \subset G \subset U \}$... | 5 | https://mathoverflow.net/users/11647 | 286050 | 126,324 |
https://mathoverflow.net/questions/286004 | 6 | A spherical variety is a normal variety $X$ together with an action of a connected reductive affine algebraic group $G$, a Borel subgroup $B\subset G$, and a base point $x\_0\in X$ such that the $B$-orbit of $x\_0$ in $X$ is a dense open subset of $X$.
A wonderful variety is a smooth complete variety $X$ with the act... | https://mathoverflow.net/users/nan | Spherical and Wonderful varieties | The only groups which act on only one wonderful variety are tori (with $X$ being a point). All other admit at least $X=G/B$ and $X=G/G$.
If one fixes the open $G$-orbit then there is at most one wonderful completion (Luna-Vust, Luna).
It is known that the number of wonderful varieties for $G$ is finite (work of Ale... | 7 | https://mathoverflow.net/users/89948 | 286059 | 126,326 |
https://mathoverflow.net/questions/286035 | -1 | It is a classical result in harmonic analysis that
$$ \|\|P\_kf\|\_{\ell^2\_k}\|\_{L^p\_x}\approx\|f\|\_{L^p} $$
for $p\in(1,\infty)$, where $P\_k$ is the Littlewood-Paley decomposition onto frquency $\approx 2^k$.
What if I replace the $\ell^2$ norm in $k$ by the $\ell^q$ norm? Is it possible that in addition to... | https://mathoverflow.net/users/37103 | $\ell^q$ analog of square function | The following counterexamples are taken from examples 6.1.10 and 6.1.11 in the textbook L. Grafakos, *Classical Fourier Analysis (Third Edition)*.
>
> **Claim 1.** Fix $1<p<\infty$ and $q<2$. Then the inequality
> $$\| (\sum\_{j\in\mathbb{Z}} |P\_{j}(f)|^{q})^{1/q}\|\_{L^{p}} \lesssim\_{p,q} \|f\|\_{L^{p}}$$
> ca... | 4 | https://mathoverflow.net/users/54316 | 286078 | 126,331 |
https://mathoverflow.net/questions/286073 | 2 | If we have two independent brownian motion in $x$ and $y$ direction. At time zero we sit at $(a,b)$ with $a>0, b>0$.
What is the probability that we will hit positive $x$ axis before hitting the negative $x$ axis?
I tried to look at some posts but no clue yet...
[2d-brownian-motion-hitting-a-point](https://math.s... | https://mathoverflow.net/users/104856 | 2 dimensional brownian motion hitting time | The probability in question is $1-p$, where $p$ is the probability that we will hit the negative $x$-semiaxis before hitting the positive $x$-semiaxis. Next, $p$ is the probability that (we will hit the positive $y$-semiaxis before hitting the positive $x$-semiaxis, and then we will hit the negative $x$-semiaxis before... | 4 | https://mathoverflow.net/users/36721 | 286081 | 126,332 |
https://mathoverflow.net/questions/286077 | 0 | In this question - [On a Hirzebruch surface.](https://mathoverflow.net/questions/122952/on-a-hirzebruch-surface) , the Hirzebruch surface is shown to be isomorphic to a hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$.
My question is, does such an isomorphism exist for all toric varieties (or at least simplicial on... | https://mathoverflow.net/users/99595 | Can any simplicial toric variety be embedded in a product of projective spaces? | There are well known examples of smooth (hence simplicial) complete toric varieties which are not projective. See for example p. 71 of Fulton's book *Introduction to Toric Varieties*. Any such variety gives a counterexample.
| 6 | https://mathoverflow.net/users/115593 | 286091 | 126,337 |
https://mathoverflow.net/questions/286097 | 1 | My coauthors and I are writing a paper based on MO questions and answers:
[Friedrich Knop's answer](https://mathoverflow.net/a/237585/4149),
[my answer 1](https://mathoverflow.net/a/239129/4149)
and
[my answer 2](https://mathoverflow.net/a/239450/4149).
For a linear algebraic group $G$ over a perfect field $k$, I cons... | https://mathoverflow.net/users/4149 | Notation for the restriction map in Galois cohomology | I've seen both $\text{Res}\_k^K$ and $\text{Res}\_{K/k}$, and similarly for the inflation map if $K/k$ is Galois. More generally, if $\Gamma$ is a group acting on a group $G$ and if $\Lambda\subseteq\Gamma$ is a subgroup, the restriction map $H^1(\Gamma,G)\to H^1(\Lambda,G)$ is commonly written as $\text{Res}^\Gamma\_\... | 3 | https://mathoverflow.net/users/11926 | 286099 | 126,339 |
https://mathoverflow.net/questions/285981 | 8 | Differentials of the second kind
================================
Gross and Rohrlich in the paper *On the periods of abelian integrals and a formula of Chowla and Selberg* state the claim below without citation (pg. 198), giving an explicit determination of the cohomology classes of the Fermat curve $X\_d := \{x^d + ... | https://mathoverflow.net/users/45609 | How to compute cohomology using differentials of the second kind on a Fermat curve? | If I recall correctly, you can find a proof of the claim in Lang's book ["Introduction to Algebraic and Abelian Functions"](https://books.google.co.cr/books?id=p4aA6DKzoeIC). He has a chapter on the Fermat curve and in fact (after looking at the google preview of the book) I think that the claim is essentially Theorem ... | 2 | https://mathoverflow.net/users/4170 | 286107 | 126,342 |
https://mathoverflow.net/questions/286105 | 3 | The conjectural density of twin primes is $\frac {c\cdot n}{(\log n)^2}$ at a $c>0$.
Consider integers of form $p,p+1=2^tq,p+2=r$ where $p,q,r$ are primes and $t\geq1$ holds.
1. Is there any reason to believe there are infinite of them at a given $t\geq1$? Is there a conjectural density for such triples at a given... | https://mathoverflow.net/users/10035 | Density of triple primes | Your questions (more precisely their affirmative answers) are special cases of the generalized Hardy-Littlewood conjecture. You can read about this conjecture in [Linear equations in primes](https://arxiv.org/abs/math/0606088). See especially Conjecture 1.4 on Page 5 and the subsequent remarks on Page 6.
| 4 | https://mathoverflow.net/users/11919 | 286109 | 126,343 |
https://mathoverflow.net/questions/286054 | 3 | Let $E$ be a complex Hilbert space. Let $T\_1,T\_2\in \mathcal{L}(E)$. Let
\begin{align}
W\_{\max}(T\_1,T\_2)
=\big\{ (\lambda\_1,\lambda\_2)\in \mathbb{C}^2; & \;\exists\,(x\_n)\_n;\;\|x\_n\|=1,\;(\langle T\_1 x\_n\; ,\;x\_n\rangle,\,\langle T\_2 x\_n\; ,\;x\_n\rangle)\to (\lambda\_1,\lambda\_2),\\
& \text{ and }\disp... | https://mathoverflow.net/users/113054 | Solving this problem of convexity | Consider the matrices
$$
T\_1 = \left[\begin{array}{ccc} 1&0&0 \\ 0& 0&0 \\0&1&0 \end{array}\right] \ \ \textrm{and} \ \ T\_2 = \left[\begin{array}{ccc} 0&0&0 \\ 1& 0&0 \\0&1&0 \end{array}\right].
$$
Note that because we are working in finite-dimensional Hilbert space we do not need the limits in the definition of $W\_... | 4 | https://mathoverflow.net/users/76593 | 286110 | 126,344 |
https://mathoverflow.net/questions/154885 | 7 | **Question:** Given a generic finite abelian group $G=\mathbb{Z}\_{N^{(1)}} \times \cdots \times \mathbb{Z}\_{N^{(k)}}$.
**(1) What is the explicit forms of its [cohomology group (see my definition)](http://ncatlab.org/nlab/show/Dijkgraaf-Witten+theory) in a generic $n$:**
$$
H^n(G,R/\mathbb{Z})=H^n(G,U(1)) =?
$$
... | https://mathoverflow.net/users/27004 | n-cocycles of finite abelian groups from cohomology group | 3 years later.
My recent paper (<https://arxiv.org/abs/1703.03266>) answers this question.
More precisely,
let $\mathbb{k}$ be an algebraically closed field of characteristic zero. By $\mathbb{k}^\*$ we denote the multiplicative group $\mathbb{k}-\{0\}$. Let $G=\mathbb{Z}\_{m\_{1}}\times\cdots \times\mathbb{Z}\_{m\... | 7 | https://mathoverflow.net/users/102515 | 286115 | 126,346 |
https://mathoverflow.net/questions/286108 | 6 | Let $X,Y$ be two centered Gaussian random variables each with variance at most $1$. Note that we do not assume independence. I would like to minimize
$$\mathbb{P}(|X|\leq 1, |Y|\leq 1).$$
Is it true that the latter quantity is minimized when $X,Y$ are independent and both have variance $1$?
| https://mathoverflow.net/users/24494 | Minimum probability that two Gaussian random variables are small | The minimum value is simply $2\alpha-1 = 0.365379$ where $\alpha = \Phi(1)-\Phi(-1) = P(|X|<1)$ where $X \sim N(0,1)$. This can be achieved by translating the percentile of $X$ (considering the percentile $\mod 1$) to produce the percentile of $Y$. For example, let $T=\Phi(X)+\alpha \mod 1$ and then $Y=\Phi^{-1}(T)$. O... | 5 | https://mathoverflow.net/users/2954 | 286118 | 126,348 |
https://mathoverflow.net/questions/286076 | 2 | I have a big problem to solve this system
$\Delta f-hf^2=0$
$|\nabla f|^2+hf^3=0$
where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f\_{xx}+f\_{yy}$) and $\nabla f$ is the gradient of $f$.
ADD
In first case $f$ is defined on $R^2$
and in second case $f... | https://mathoverflow.net/users/111304 | Pde system problem | I assume that, in the surface case, the OP wants to interpret $S$ as a surface endowed with a Riemannian metric and wants to understand the solutions to the equations $\Delta f - hf^2 = 0$ and $|\nabla f|^2 + hf^3 = 0$ for a given constant $h$.
Clearly, if $h=0$, the only solutions are to have $f$ be constant, so one... | 12 | https://mathoverflow.net/users/13972 | 286140 | 126,351 |
https://mathoverflow.net/questions/286135 | 43 | I was just watching Andrej Bauer's lecture [Five Stages of Accepting Constructive Mathematics](https://youtu.be/zmhd8clDd_Y), and he mentioned that in the constructive setting we cannot guarantee that every ideal is contained in a maximal ideal---since that obviously requires Zorn's Lemma (or is equivalent to Zorn's Le... | https://mathoverflow.net/users/56938 | Constructive algebraic geometry | Let me wrote a quick introduction to this idea:
**1) Locales**
I do not know if you are already familiar with the notion of locale that Andrej is referring to in his talk: They are a small variation on the idea of a topological space, where instead of defining a space by giving a set of points together with a colle... | 52 | https://mathoverflow.net/users/22131 | 286143 | 126,353 |
https://mathoverflow.net/questions/286106 | 2 | $\omega^{CK}\_1$ is the supremum of all the [recursive ordinals](https://en.wikipedia.org/wiki/Recursive_ordinal), where an ordinal $\alpha$ is recursive if there is a computable ordering of a subset of the naturals with order type $\alpha$.
For a [Turing degree](http://if%20there%20is%20a%20computable%20ordering%20o... | https://mathoverflow.net/users/65915 | Connection between countable ordinals and Turing degrees | The ordinals of the form $\omega\_D^{CK}$, as you denote it, are exactly the [countable admissible ordinals](https://en.wikipedia.org/wiki/Admissible_ordinal), and these ordinals are intensely studied in the context of admissible set theory and fine structure theory.
| 5 | https://mathoverflow.net/users/1946 | 286145 | 126,354 |
https://mathoverflow.net/questions/286072 | 11 | The following question came up while trying to determine whether the extension problems in a spectral sequence are trivial.
Given a noetherian ring $R$ and a finitely generated $R$-module $M$ with a filtration $M=F\_0 \supset F\_1 \supset \ldots \supset F\_n \supset F\_{n+1}=0$ such that $M \cong \bigoplus\_{i=0}^n F... | https://mathoverflow.net/users/117231 | Given a filtration of a finitely generated module over a noetherian ring that "looks" split, is it split? | Take the direct sum of the short exact sequences
$$0\to F\_{i+1}\to F\_i\to F\_i/F\_{i+1}\to0$$
for $0\leq i\leq n$.
This has the form
$$0\to \bigoplus\_{i=1}^n F\_i\to \bigoplus\_{i=0}^n F\_i\to F\_0\to 0$$
and so splits by the linked answer of Steven Landsburg.
A direct summand of a split short exact sequence is... | 7 | https://mathoverflow.net/users/22989 | 286147 | 126,355 |
https://mathoverflow.net/questions/286149 | 2 | Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $j \ge 0$ let $\mathcal{H}^j:=R^{j}f\_!(k\_X)$, i.e. the shefification of the presheaf on $Y$ given by:
$$U \mapsto H\... | https://mathoverflow.net/users/22810 | Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system? | First of all, in case $f$ is not proper, the sheaf $\mathcal{H}^j = R^j f\_!(k\_X)$ is not defined as a sheaf associated to a presheaf $$U\mapsto H^j\_c(f^{-1}(U),k),$$ since that rule is not a presheaf. Compactly supported cohomology is **covariant** for open inclusions, it is not contravariant (presheaves are contrav... | 5 | https://mathoverflow.net/users/13265 | 286154 | 126,357 |
https://mathoverflow.net/questions/286150 | 1 | Let $V$ be a general smooth projective cubic hypersurface. Doing literally as in case of cubic curves we define a relation on $V\times V\times V$: $(x,y,z)$ satisfy it iff $x+y+z$ is an intersection of $V$ with a line.
Contrary to the one-dimensional case this relation is not a graph of a binary operation ($x^2$ is no... | https://mathoverflow.net/users/13842 | Multiplication on cubic hypersurfaces and partially defined groups | Here is an explicit example over the rationals.
Consider the [diagonal Clebsch cubic surface](https://en.wikipedia.org/wiki/Clebsch_surface) given by $\sum\_{i=0}^4 X\_i = 0$ and $\sum\_{i=0}^4 X\_i^3 = 0$. Let me take the point $u := (0:0:0:1:-1)$ so that $x \mapsto u\circ x$ takes $(X\_0:X\_1:X\_2:X\_3:X\_4)$ to $(... | 2 | https://mathoverflow.net/users/17064 | 286161 | 126,359 |
https://mathoverflow.net/questions/286163 | 4 | Let $X$ be a non-reflexive Banach space. It is supposed to compare two locally convex topologies on $B(X)$:
Let $w$ be the topology on $B(X)$ implemented by all seminorms given by
$$B(X)\to [0,\infty) : T\to |\langle T^\*x^\*,x\rangle|$$
where $x\in X$ and $x^\*\in X^\*$.
We also denote $w^\*$ by the topology imp... | https://mathoverflow.net/users/84390 | Two locally convex topologies on $B(X)$. | Fix $x \in X$ and let $(f\_\alpha)$ be a net in $X^\*$. For each $\alpha$ let $T\_\alpha$ be the rank-one operator $y \mapsto f\_\alpha(y) x$. Then you want to compare the seminorms
$$ |\langle T\_\alpha^\*(x^\*), x \rangle| = |\langle x^\*, x\rangle| |\langle f\_\alpha, x\rangle| $$
against
$$ |\langle T\_\alpha^\*(x^... | 7 | https://mathoverflow.net/users/406 | 286165 | 126,360 |
https://mathoverflow.net/questions/286164 | 17 | The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1\_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi\_1^{et}(\mathbb{P}^1\_K) = 1$; equivalently, if $\phi\colon C\to \mathbb{P}^1\_{K}$ is finite etale, then $\deg\phi=1$).
For $K=\mathbb{C}$, this follows from the conne... | https://mathoverflow.net/users/2042 | A short proof for simple connectedness of the projective line | You can deduce this from the classification of vector bundles on $\mathbf{P}^1$. Say $f:C \to \mathbf{P}^1$ is a connected finite etale Galois cover of degree $n$. We must show $n=1$.
The sheaf $E := f\_\* \mathcal{O}\_C$ is a rank $n$ vector bundle on $\mathbf{P}^1$, so we can write it as $E \simeq \oplus\_{i=1}^n \... | 18 | https://mathoverflow.net/users/117273 | 286169 | 126,363 |
https://mathoverflow.net/questions/286166 | 7 | What goes wrong if you try to define the Kontsevich space $\overline{\mathscr{M}\_{0,n}}(\mathbb{P}^r,e)$ is positive characteristic?
It is a naive question, but I couldn't find much with a google search. I thought I found something saying they weren't DM stacks because you have inseparable maps, but I didn't clearl... | https://mathoverflow.net/users/16356 | Kontsevich space in positive characteristic | I am just writing my comments as an answer. The main computations have to do with the **cotangent complex** of a stable map. I will work with unpointed stable maps for simplicity (the associated cotangent complex is a bit more complicated in the pointed case).
**Notation and Hypotheses.** Let $k$ be an algebraically... | 10 | https://mathoverflow.net/users/13265 | 286170 | 126,364 |
https://mathoverflow.net/questions/286061 | 17 | I have asked this question exactly [here](https://math.stackexchange.com/questions/2450607/on-siegel-mass-formula). The question is as follows:
I am interested deeply in the following problem:
Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be an arbitrary natural number; then find a closed for... | https://mathoverflow.net/users/68462 | On Siegel mass formula | There are *many, many* references on quadratic forms. This is a huge area, depending on which way you want to go. One of the main approaches is to construct a theta series associated to your quadratic form whose Fourier coefficients give you the representation numbers you want. These are modular forms, and this is trea... | 6 | https://mathoverflow.net/users/6518 | 286176 | 126,366 |
https://mathoverflow.net/questions/256641 | 4 | Let $c\_1, \ldots, c\_k \in \mathbf N^+$ and $x\_1,\ldots,x\_k \in \mathbf Z \setminus \{0\}$. It is possible to prove by elementary means that $(\omega(c\_1 x\_1^n+\cdots+c\_kx\_k^n))\_{n\ge 1}$ is a bounded sequence only if $|x\_1|=\cdots=|x\_k|$. (As usual, $\omega(x)$ is, for every non-zero $x \in \mathbf Z$, the *... | https://mathoverflow.net/users/16537 | Proving that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is bounded only if $|x_1|=\cdots=|x_k|$ by the Subspace Theorem | Just in order to mark this question as answered: The answer is yes. Some details follow.
---
The basic idea (for some more general conclusion) was generously provided by the anonymous referee of a short note (joint work with Paolo Leonetti) that has been only recently accepted for publication in JNT (\*). The key... | 2 | https://mathoverflow.net/users/16537 | 286181 | 126,368 |
https://mathoverflow.net/questions/271718 | 2 | *All sets and groups in the question are finite.*
In order to understand equivariant sheaves better **I'm trying to prove some basic facts from Mackey theory using equivariant sheaves.** The main obstacle i've been faced with is the difficulty of keeping track of all the different equivalences.
Let $G$ be a group ... | https://mathoverflow.net/users/22810 | Orbit decomposition of the restriction of an equivariant sheaf? | In my opinion the Mackey formula is most easily seen if you think in terms of groupoids/stacks. This idea makes sense in many situations, but for simplicity let's restrict to representations of discrete groups.
The basic idea is:
1) There is a Cartesian diagram of groupoids:
$$
\require{AMScd}
\begin{CD}
[K\backsla... | 3 | https://mathoverflow.net/users/7762 | 286184 | 126,369 |
https://mathoverflow.net/questions/286189 | 9 | Let $K$ be a field and $C$ a smooth and projective curve over $K$. Then the kernel $Pic^0(C)$ of the degree map injects into $H^0(K,Pic^0\_C)$, where $Pic\_C^0$ is the connected component of the Picard variety.
I am wondering if there are examples where this is not an isomorphism for $K$ a global field. I am especia... | https://mathoverflow.net/users/26735 | Pic^0 and H^0(K,Pic^0) | By the long exact sequence of low degree terms for the Leray spectral sequence computing $H^r\_{\text{et}}(C,\mathbb{G}\_m)$ via $H^p\_{\text{et}}(\text{Spec}\ K,R^q f\_\*\mathbb{G}\_m)$, the cokernel of the map $$\text{Pic}(f):\text{Pic}(C) \to H^0\_{\text{et}}(\text{Spec}\ K,\text{Pic}\_{C/K})$$ equals the kernel of ... | 12 | https://mathoverflow.net/users/13265 | 286195 | 126,374 |
https://mathoverflow.net/questions/286123 | 16 | Consider the finite field ${\bf F}\_p$ and its cubic extension ${\bf F}\_{p^3}$. The multiplicative group ${\bf G}\_m({\bf F}\_{p^3})$ contains the multiplicative group ${\bf G}\_m({\bf F}\_p) \cong {\bf Z}/(p-1){\bf Z}$ as a subgroup. The quotient $A\_p = {\bf G}\_m({\bf F}\_{p^3})/{\bf G}\_m({\bf F}\_p)$ is an abelia... | https://mathoverflow.net/users/117251 | How do I see the equality $57 = 3 \times 19$ geometrically? | I'm not sure if this does what you want, but the subgroup of order $3$ in the additive group $(\mathbb{Z}\_{57},+)$ is $\{0,19,38\}.$
To expand on that, one construction for the plane $\mathbb{P}\_{7}$ of order $7$ is to take as points the elements of $\mathbb{Z}\_{57}$ with lines $$\ell\_k=[k,k+1,k+3,k+13,k+32,k+36,... | 3 | https://mathoverflow.net/users/8008 | 286201 | 126,375 |
https://mathoverflow.net/questions/286187 | 9 | I am curious if (any of) the various inequivalent constructions of the real line in constructive mathematics can be used to build a model of Kock and Lawvere's synthetic differential geometry? In other words, do any of the constructions of the real line (in say HoTT) satisfy the Kock-Lawvere axiom for a class of functi... | https://mathoverflow.net/users/56938 | Constructive analysis and synthetic differential geometry | In the smooth-topos models of SDG, the situation is generally something like this. The internally-definable Cauchy real numbers $\mathbf{R}\_c$ are the sheaf of locally constant $\mathbb{R}$-valued functions, while the internally-definable Dedekind real numbers $\mathbf{R}\_d$ are the sheaf of continuous $\mathbb{R}$-v... | 6 | https://mathoverflow.net/users/49 | 286207 | 126,377 |
https://mathoverflow.net/questions/286204 | 18 | What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.
A good example is Lovász's proof of [Kneser's conjecture](https://en.wikipedia.org/wiki/Kneser_graph).
| https://mathoverflow.net/users/10035 | Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications? | The question asks for *concepts*, not applications, so in a sense the example given in the OP isn't one.
Here are five quick examples:
(0) One could argue that **girth** is a transferral of the concept **systole** from metric-topology, though this is an ahistorical argumentation: the two concepts arose independen... | 13 | https://mathoverflow.net/users/108556 | 286211 | 126,378 |
https://mathoverflow.net/questions/286208 | 13 | I know from Wikipedia that in NBG, the surreal numbers are the largest possible ordered field (if a proper class is allowed to be a field). But then, it is written: "in theories without the axiom of global choice [...] *it is not necessarily true* that the surreals are the largest ordered field".
How would such a fi... | https://mathoverflow.net/users/114143 | Largest ordered "field" in NBG without axiom of global choice | There is no problem defining the surreal field without global choice.
One can define it in ZFC and considerably weaker theories, for
example with the hereditary birthday construction of left-sets and
right-sets, and also in other ways.
With global choice, the surreal field No is *largest* in the sense
of model-theore... | 13 | https://mathoverflow.net/users/1946 | 286227 | 126,383 |
https://mathoverflow.net/questions/286197 | 25 | Articles from the *Proceedings of the International Congress of Mathematicians*, Seoul, 2014 don't appear to be on Mathscinet. Why is this?
(Someone pointed this out to me recently, and I was reminded of it today when I tried to cite a lecture.)
| https://mathoverflow.net/users/919 | Why aren't proceedings from ICM 2014 on mathscinet? | We have had difficulty obtaining the requisite permissions from the publisher. The ICM2014 website has the Legal Disclaimer: "The Seoul ICM Organizing Committee, the legal copyright owner of the articles in the proceedings, hearby grants unlimited noncommercial download and use of the articles." This is not sufficient ... | 50 | https://mathoverflow.net/users/49409 | 286237 | 126,386 |
https://mathoverflow.net/questions/286230 | 2 | If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is an *edge cover* if $C\cap e \neq \emptyset$ for all $e\in E$.
The "best" covers in some sense are subsets $C\subseteq V$ that meet every edge in exactly one point - but in many graphs, such a nice cover does not exist; there are often "bad" edges $e$ ... | https://mathoverflow.net/users/8628 | Edge covers in infinite graphs | No.
The proof requires a lemma: *If $C$ is an edge cover with $|\mathrm{Good}(C)| < |E|$, then it is not minimal (I can remove a point without changing the fact that it's an edge cover).*
To prove the lemma, note that every "bad" edge has both its endpoints in $C$, which means that the number of bad edges is at mos... | 3 | https://mathoverflow.net/users/70618 | 286240 | 126,388 |
https://mathoverflow.net/questions/269595 | 19 | $\DeclareMathOperator{\rk}{rk}$
The question below is implicit in [this MO post](https://mathoverflow.net/questions/265468/pointwise-hadamard-matrix-product-and-the-rank), but I believe it deserves to be asked explicitly, particularly now that I have some more numerical evidence.
>
> Suppose that $A$ is a real, s... | https://mathoverflow.net/users/9924 | The rank of a perturbed triangular matrix | $\DeclareMathOperator{\rk}{rk}$It is possible to construct a matrix with $\rk(A)\leq 2\sqrt{n}$. Assuming that $n=r^2$ with an integer $r$, introduce two matrices $B$ and $C$, whose rows and columns are indexed by elements of $\{1,2,\dotsc,r\}^2$, and whose entries are defined by
$$
B\_{(x,y),(x',y')}=\begin{cases}1&\t... | 3 | https://mathoverflow.net/users/806 | 286254 | 126,394 |
https://mathoverflow.net/questions/286198 | 3 | Let's suppose I have an [LFSR](https://en.wikipedia.org/wiki/Linear-feedback_shift_register) that generates an m-sequence $y\_1[k]$ --- in other words, the LFSR has $N$ bits and $y\_1[k]$ has period $m=2^N - 1$.
Now suppose I know someone has decimated this and taken every $j$th element, so $y\_2[k] = y\_1[jk+b]$. An... | https://mathoverflow.net/users/1305 | Computing the decimation ratio between two m-sequences | In general, $p\_2$ has $x^j$ as a zero in $F\_1$. In other words, $p\_1(x)$ divides $p\_2(x^j)$ over $\mathrm{GF}(2)$.
To find $j$ from the given $p\_1$ and $p\_2$, one can factor $p\_2(y)$ in $F\_1[y]$, and for every zero $y\_0\in F\_1$ of $p\_2(y)$, find the discrete log of $y\_0$ base $x$ in $F\_1$.
Here is a s... | 2 | https://mathoverflow.net/users/7076 | 286263 | 126,398 |
https://mathoverflow.net/questions/286266 | 3 | I am reading a book by Billingsley (convergence of probability measures) and he makes a footnote on page 27 which I am struggling to understand. I'll explain the setup below.
Suppose $(X\_n,Y\_n)$ are random elements of $S\times S$, where $S$ is a metric space. Then since the projections $(x,y)\mapsto x$ and $(x,y)\m... | https://mathoverflow.net/users/117335 | Measurable functions in product space | The problem is that the Borel $\sigma$-algebra of the product space $S\times S$ need not be the same as the product $\sigma$-algebra $B(S)\otimes B(S)$ unless $S$ is separable (or more generally, a second-countable topological space). To make your argument work, you want to deal with $B(S)\otimes B(S)$ but measurabilit... | 2 | https://mathoverflow.net/users/15129 | 286267 | 126,401 |
https://mathoverflow.net/questions/286272 | 1 | Suppose we have a graph $G$ with $n$ vertices. If we color the edges of $G$ by two colors, then we can conclude by Konig's Theorem that there exists a monochromatic subtree $T$ with at least $n/{\alpha}$ vertices, $\alpha$ is the size of maximum independent set in the graph.
I was wondering if there is any theorem t... | https://mathoverflow.net/users/80245 | Induced monochromatic subtree in a graph which is colored by two colors | Every connected $n$-vertex graph with $O(n)$ edges has an induced tree of size $2\log\log n+O(\log\log\log n)$ (I guess the constants in the $O$'s depend on each other); see
P. Erdős, M. Saks, and V. T. Sós. Maximum induced trees in graphs. J. Combinatorial Theory, Series B, 41(1):61 – 79, 1986. doi:10.1016/0095-8956... | 2 | https://mathoverflow.net/users/440 | 286276 | 126,404 |
https://mathoverflow.net/questions/286285 | 3 | Let $a\neq 1$ be a positive constant and let $d(n)$ denote the number of divisors of $n.$
Can one obtain upper and lower bounds on
$S\_{a}(x)=\sum\_{n\leq x} d(n)^a$?
I am particularly interested in estimates for $a\in(0,1)$ and $a=2$. A weak upper bound on the latter is
$$S\_2(x) \leq S\_1(x)^2,$$
and can be use... | https://mathoverflow.net/users/17773 | Estimates for $\sum_{n\leq x} d(n)^a$ | One has $S\_a(x) \sim C(a) x (\log x)^{2^a -1}$ where
$$
C(a) = \Gamma(2^a)^{-1} \prod\_p \left( 1 - \frac{1}{p} \right)^{2^a} \left( \sum\_{k \geq 0} \frac{(k+1)^a}{p^k}\right).
$$
This follows for example from standard tauberian theorems and from the fact that $\sum\_{n \geq 1} d(n)^a n^{-s} = \zeta(s)^{2^a} F(s)$ wh... | 9 | https://mathoverflow.net/users/21724 | 286290 | 126,408 |
https://mathoverflow.net/questions/286205 | 7 | Consider a random walk $S\_t = \sum\_{i=1}^{t} X\_i$, with $X\_i$ i.i.d.. Assume that $X\_i \in [0,1]$. Define $\tau(y) := \inf\{t: S\_t\geq y\}$, i.e., $\tau(y)$ is the hitting time of $[y,\infty)$. Is this possible to show that $\mathbb{E}[\tau]$ is a Lipschitz function, under some "natural" condition?
One conditi... | https://mathoverflow.net/users/100482 | One dimension random walk. Is hitting time Lipschitz with respect to target? | *could you elaborate more on your high tech solution?*
OK, but it is just a standard boring exercise giving one no intellectual pleasure whatsoever. Write $X=t+Y$ where $EY=0$. Note that $t\ge\frac 1{2M}$. Let $f=f\_1$ be the pdf of $Y$. Then the pdf $f\_n$ of $Y+\dots+Y$ ($n$ times) is $f\*\dots\*f$, so $\widehat {f... | 5 | https://mathoverflow.net/users/1131 | 286301 | 126,411 |
https://mathoverflow.net/questions/286287 | 2 | Is the following true ? If so, is there a quick proof of it ? (Perhaps using the uniqueness of the graded object associated to a Jordan-Holder filtration or maybe otherwise)
Suppose $E$ is an $\omega$-semistable bundle with slope $\mu$ over a compact Kahler manifold $(X,\omega)$. There are only finitely many (upto is... | https://mathoverflow.net/users/3709 | Number of semistable subbundles of a semistable bundle | That is not true. The issue has to do with nontrivial extensions between semistable bundles of the same slope. If you have a compact Kähler manifold where every semistable sheaf of slope $0$ has vanishing $H^1$, then I suspect that it is true that there are only finitely many isomorphism classes occuring for semistable... | 9 | https://mathoverflow.net/users/13265 | 286307 | 126,414 |
https://mathoverflow.net/questions/286261 | 5 | Let $X, V\in\mathbb{R}^{n\times r}$ such that $X^\top V$ is symmetric. The central quantity I care about is
\begin{equation}
\|XV^\top\|\_{F}^2+\|X^\top V\|\_{F}^2 +[\text{Tr}(X^\top V)]^2.
\end{equation}
An easy lower bound for this quantity is given by $2\sigma\_{r}(X)^2\|V\|\_{F}^2$, where $\sigma\_{r}(X)$ is the sm... | https://mathoverflow.net/users/90066 | Nontrivial lower bound on the sum of matrix norms | No. With $n = r = 2$, set $$X = \bigg(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \bigg) \, , \quad V = \bigg( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \bigg) \, .$$
In particular, $X^T V = V^T X = 0$, the zero matrix.
If you restrict to invertible square matrices, the statement is still false. Set
$$X = \bigg(\... | 6 | https://mathoverflow.net/users/40264 | 286314 | 126,420 |
https://mathoverflow.net/questions/286316 | 7 | In GAP (<https://www.gap-system.org>), there is a function **IsSymmetricGroup**, which tells you whether a subgroup of $S\_n$ generated by given permutations is all of the $S\_n$. It looks like it takes virtually no time, even in large examples I tried ($n=50$). What is the method behind this function? Is it so easy to... | https://mathoverflow.net/users/1306 | recognition of symmetric groups in GAP | The method (I assume) uses Jordan's theorem, which says that an primitive subgroup of $S\_n$ with a cycle of prime order (at most $n-2,$ if memory serves) is either $A\_n$ or $S\_n.$ You rule out $A\_n$ by looking at the generators, you show transitivity by randomly generating an $n$-cycle (of which there are a lot, so... | 12 | https://mathoverflow.net/users/11142 | 286317 | 126,421 |
https://mathoverflow.net/questions/16960 | 7 | Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as in the affine case the theorem is still true so this seems like a reasonably naive question. Although there is no discus... | https://mathoverflow.net/users/6986 | Hochschild Kostant Rosenberg theorem for varieties in positive characteristic? | See [this](https://arxiv.org/abs/1710.06039) paper of mine and Gabriele Vezzosi. We prove that HKR holds in particular for smooth proper schemes $X$ of dimension at most $p$, the characteristic prime. In particular, it holds for smooth proper surfaces in characteristic $2$.
| 7 | https://mathoverflow.net/users/100 | 286321 | 126,423 |
https://mathoverflow.net/questions/286320 | 1 | The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":
>
> Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ be the function field in one variable, and let $L := K[y]/(y^{p}-y-x)$ be the Artin-Schreier extension of $K$ associ... | https://mathoverflow.net/users/112809 | Norms of elements in Artin-Schreier extensions | Let's define the leading coefficient of a rational function to be the leading coefficient of the numerator over the leading coefficient of the denominator. We will show that the leading term of any norm is a $p$th power. It will follow that any norm that is a constant is a $p$th power.
Observe that $L$ is isomorphic ... | 4 | https://mathoverflow.net/users/18060 | 286322 | 126,424 |
https://mathoverflow.net/questions/286278 | 1 | Let
* $R = \mbox{diag} (r\_1,\dots,r\_n)$, where $r\_1, \dots, r\_n > 0$, be a (positive) diagonal matrix.
* $1\_n \in \mathbb{R}^n $ denote the $n$-dimensional vector of all ones.
* $S$ be a matrix defined by $[S]\_{jk}=\sin (\theta\_{j}-\theta\_k +\delta)$, where $[\theta\_1, \dots, \theta\_n]^\mathrm{T} \in \math... | https://mathoverflow.net/users/102447 | Is this expression always non-negative? | Consider $R = {\rm diag}(r, 1/r)$ and $\delta = \pi/4, \theta\_1 = \pi/4, \theta\_2 = 0$. Then $$
S = \left[\begin{array}{cc} \sin(\theta\_1 - \theta\_1 + \delta) & \sin(\theta\_1 - \theta\_2 + \delta) \\ \sin(\theta\_2 - \theta\_1 + \delta) & \sin(\theta\_2 - \theta\_2 + \delta) \end{array}\right] =
\left[ \begin{arr... | 3 | https://mathoverflow.net/users/76593 | 286325 | 126,425 |
https://mathoverflow.net/questions/286329 | 7 | Let $C=(\mathbb{Z}/2\mathbb{Z})^N$ be the Hamming cube with its usual graph structure, and assume each edge $e=(x,x+\epsilon)$ (where $x\in C$ and $\epsilon$ has one $1$ and $N-1$ zeros) is given a length $\ell(x,\epsilon)$ satisfying the constraints
$$ \sum\_\epsilon \ell(x,\epsilon) \le 1 \qquad \forall x\in C.$$
Den... | https://mathoverflow.net/users/4961 | Diameter of a weighted Hamming cube | It looks like even the sharp upper estimate 1 may be obtained. We use the following
**Lemma.** If $q\_0,\dots,q\_{N-1}$ are non-negative real numbers such that $q\_i-q\_{i+1}+q\_{i+2}-\dots+q\_{i+2s}\geqslant 0$ for all $0\leqslant i\leqslant i+2s\leqslant N-1$, then there exist non-negative numbers $p\_0,\dots,p\_{N... | 5 | https://mathoverflow.net/users/4312 | 286346 | 126,432 |
https://mathoverflow.net/questions/286330 | 4 | Consider a Toeplitz matrix $T$, indexed by $\mathbb{N}\_0 \times \mathbb{N}\_0$. given by the sequence $t\_k,k \in \mathbb{Z}$ where $t\_k \geq 0,\sum\_{k=-\infty}^\infty t\_k=1$. By this I mean that $T\_{i,i+k}=t\_k$ for all $i \in \mathbb{N}\_0$ and $k \in \{ -i,-(i-1),\dots,0,1,\dots \}$.
It is easy to see that $T... | https://mathoverflow.net/users/92082 | When does iteration of an infinite Toeplitz matrix converge? | OK. The case when one $t\_k=1$ and the rest are $0$ is easy to figure out (left shifts are good, the rest are bad).
Assume that all $t\_k<1$. The $\ell\_1$ problem is essentially equivalent to the question when the corresponding random walk on $\mathbb Z$ has positive chance to stay above $0$ forever (this is not obvio... | 1 | https://mathoverflow.net/users/1131 | 286348 | 126,433 |
https://mathoverflow.net/questions/286337 | 6 | Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u\_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\in [0,\infty)$. Is there a way to generalize to higher order derivatives. That is, if $u''(t) \leq \alpha(t)u'(t) + \bet... | https://mathoverflow.net/users/110094 | Gronwall's inequality for higher order derivatives | No, comparison of this type only works for first order. Consider for example
$$
u'' \le -u, \quad u(0)=u'(0)=0 .
$$
The statement you were hoping for would here say that such a $u$ satisfies $u(x)\le 0$ for all $x\ge 0$, but this can easily be outmaneuvered. Start out by making $u$ negative; obviously there are no prob... | 5 | https://mathoverflow.net/users/48839 | 286350 | 126,434 |
https://mathoverflow.net/questions/286139 | 3 | If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow\_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow\_{\leq} x) = \{p\in P: p\leq x\}$.
Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The *Rudin-Keisler preorder* on... | https://mathoverflow.net/users/8628 | The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder | As far as I can see, the Wallman topology as defined here is the same as the topology that $NPU(\omega)$ gets as a subspace of the Stone-Cech compactification of the discrete space $\omega$. Specifically, for any $A\subseteq\omega$, any ultrafilter on $\omega$ contains either $A$ or $\omega-A$ but not both, so the subb... | 2 | https://mathoverflow.net/users/6794 | 286355 | 126,436 |
https://mathoverflow.net/questions/286199 | 4 | Let $f\colon \mathbb{R}^2 \to \mathbb{R}^2$ be a $C^2$ uniformly expanding diffeomorphism that fixes the origin: that is, $f(0)=0$ and there is $\lambda>1$ such that $d(f(x),f(y)) \geq \lambda d(x,y)$ for all $x,y\in \mathbb{R}^2$. *[The original question just asked for a locally expanding map; I've clarified that it s... | https://mathoverflow.net/users/5701 | Making images arbitrarily dense under an expanding map | You are looking at things from a totally wrong perspective, i.e., you try to construct a complicated mapping for a simple curve while it is much easier to construct complicated curves for simple mappings. Also expansion is something that grows and gets more complicated with every step and you do not want to fight monst... | 3 | https://mathoverflow.net/users/1131 | 286363 | 126,442 |
https://mathoverflow.net/questions/286334 | 6 | Let $l^{\infty}$ (respectively, $l^{1}$) be the space of bounded
(respectively, absolutely summable) real sequences. I need to find out if
$l^{\infty}$ equipped with the Mackey topology $\tau(l^{\infty},l^{1})$, i.e.
the finest locally convex topology that leads to the topological dual $l^{1}$,
is strongly/hereditarily... | https://mathoverflow.net/users/117369 | Is the Mackey topology $\tau(l^{\infty},l^{1})$ strongly Lindelöf? | This is true and follows from the fact that in this case the Mackey topology agrees with the weak $\ast$ topology on balls.
| 4 | https://mathoverflow.net/users/117380 | 286364 | 126,443 |
https://mathoverflow.net/questions/286286 | 2 | Is there an infinite connected simple undirected graph $G=(V, E)$ such that the identity map $\text{id}\_V: V\to V$ is the only graph self-homomorphism from $G$ to itself?
(A graph self-homomorphism is a map $f: V\to V$ such that for all $e\in E$ with $e = \{v, w\}$ we have $\{f(v), f(w)\} \in E$.)
| https://mathoverflow.net/users/8628 | Infinite strongly rigid graphs | Unfortunately, I don't have enough reputation to comment, but there seems to be a problem with both solutions suggested so far: The graphs are bipartite, meaning that they allow a homomorphism to a single edge, which of course is a non-trivial homomorphism from the graph to itself (note that homomorphisms are not assum... | 1 | https://mathoverflow.net/users/97426 | 286382 | 126,448 |
https://mathoverflow.net/questions/286075 | 1 | I am currently trying to read Colmez' "Série principale unitaire pour $Gl\_2(\mathbb{Q}\_p)$ et représentations triangulines de dimension 2", that you can find here
<https://webusers.imj-prg.fr/~pierre.colmez/triangulines> . In the proof of Lemma 4.1. at the very end I can not follow anymore.
The statement is the fol... | https://mathoverflow.net/users/104544 | Definition and properties of $\mathcal{B}^\dagger$ | The statement "if $b \in \mathcal{E}^\dagger$ then there exists $c \in \mathbf{B}^\dagger$ such that $\varphi(c)=bc$" is clearly incorrect (just take $b=p$).
The paper that you are reading is an abandoned preliminary version of other papers that were subsequently published. If you look in Colmez' corresponding publi... | 2 | https://mathoverflow.net/users/5743 | 286383 | 126,449 |
https://mathoverflow.net/questions/286368 | 5 | Let $f:X\longrightarrow Y$ be a map between CW-complexes $X$ and $Y$. By the Whitehead Theorems, if one of the conditions:
1- (homotopy version) $\pi\_n (f):\pi\_n (X)\longrightarrow \pi\_n (Y)$ is an isomorphism for all $n\geq 1$,
or
2- (homology version) $\pi\_1 (f):\pi\_1 (X)\longrightarrow \pi\_1 (Y)$ and $... | https://mathoverflow.net/users/114476 | A weak version of the Whitehead Theorems | Suppose you have a map $f\colon X\to Y$ of finite CW complexes such that $K(p,n)\_\*(f)$ is injective for all primes $p$ and integers $n\geq 0$ (where $K(p,n)$ is Morava $K$-theory). Then the Nilpotence Theorem of Hopkins, Devinatz and Smith implies that the map $\Sigma^kf^{(m)}\colon \Sigma^kX^{(m)}\to\Sigma^kY^{(m)}$... | 5 | https://mathoverflow.net/users/10366 | 286386 | 126,450 |
https://mathoverflow.net/questions/286384 | 5 | Suppose $C$ is a small category with a monoidal structure. Then by the special case of the Day convolution theorem for presheaves, $\operatorname{Psh}(C)$ is equipped with a corresponding biclosed monoidal structure. If $C$ is equipped with a Grothendieck topology, is there any useful condition for when the biclosed mo... | https://mathoverflow.net/users/1353 | Are there any useful conditions for a biclosed monoidal structure on presheaves to descend to a biclosed monoidal structure on sheaves? | There is a more general form of Day's theorem that does pretty much that, at least for sub-canonical topologies:
**Theorem (Day):** Let $C$ be a complete and co-complete Category, and $D \subset C$ a full subcategory of $C$ endowed with monoidal structure which contains a full subcategory dense in $C$. Then if there ... | 5 | https://mathoverflow.net/users/22131 | 286387 | 126,451 |
https://mathoverflow.net/questions/286389 | 8 | Is there a real number $A$ such that $$\left \lfloor n^{A} \right \rfloor$$ is a prime number (for all natural numbers $n$)? It is obvious that $A>1+\epsilon$ from the prime number theorem.
| https://mathoverflow.net/users/nan | A more dense analog of the Mills' constant | No, such an $A$ does not exist. First, $A$ cannot be an integer because then $\lfloor n^A\rfloor $ is never a prime for $n\geq 2$. So, assume that $A$ is not an integer. Then by Weyl's equidistribution theorem the fractional parts of $n^{A}/2$ are equidistributed modulo $1$. In particular $\{n^A/2\}\in [0,1/2)$ infinit... | 21 | https://mathoverflow.net/users/806 | 286391 | 126,452 |
https://mathoverflow.net/questions/286392 | 6 | The following is said without further explanation in Folland's *Real Analysis*:
>
> Some authors prefer to take the domains of measures to be $\sigma$-rings rather
> than $\sigma$-algebras. The reason is that in dealing with "very large"
> spaces one can avoid certain **pathologies** by not attempting to measure ... | https://mathoverflow.net/users/nan | Why are $\sigma$-algebras preferable to $\sigma$-rings? | Reference: Measure Theory by Halmos.
An example of a pathology: If measures are not $\sigma$-finite, you have to be careful how to formulate Fubini's theorem. The old way of doing this was to have the measure defined on the $\sigma$-ring of those sets on which the measure is σ-finite. The current way of doing this is... | 7 | https://mathoverflow.net/users/95282 | 286394 | 126,453 |
https://mathoverflow.net/questions/286390 | 19 | What is the simplest example (or perhaps best reference) for the fact that there are genus $1$ curves (over a field of your choice --- or if you wish, over $\mathbb{Q}$, to make it more exciting) with no points of degree less than $n$? Brian Conrad gave a slick answer here: <https://math216.wordpress.com/2011/04/22/fou... | https://mathoverflow.net/users/299 | For each $n$: show there is a genus $1$ curve over some field $k$ with no points of degree less than $n$, (simple argument / best reference)? | How about a universal example?
Let $E$ be an elliptic curve over any field $k\_0$ and let $L$ be a degree $n$ line bundle on $E$. Then the actions of $n$-torsion points of $E$ by translation preserves $L$, and hence these automrophisms act by projective linear automorphisms of $H^0(E,L)$, giving a map $E[n] \to PGL\_... | 14 | https://mathoverflow.net/users/18060 | 286396 | 126,455 |
https://mathoverflow.net/questions/286393 | 15 | I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$.
* **Vague question.** Recall that if $G$ acts on a finite set $X$, we get a permutation representation $$G \to GL\_{\lvert X \vert}(\mathbb C).$$ (Unless $X$ is very small,) this repre... | https://mathoverflow.net/users/105957 | How do I know if an irreducible representation is a permutation representation? | First of all, note that $S\_6$ has a doubly transitive action on 10 points, obtained from the action of $PSL\_2(9)=A\_6$ on the projective line over $\mathbb{F}\_9$ by adding the Galois automorphism of $\mathbb{F}\_9$, and this is the only faithful permutation action of $S\_6$ of degree 10. $S\_6$ has two irreducible c... | 6 | https://mathoverflow.net/users/11100 | 286404 | 126,457 |
https://mathoverflow.net/questions/286257 | 6 | If $G$ is a semisimple algebraic group over a local field with finite residue field $K$ and $x$ a point in the Bruhat-Tits building $B(G, K)$ then the parahoric group scheme $P\_x$ is a group scheme $P$ whose $O\_K$ points are the connected component of the stabilizer of $x$.
If $G$ is not semisimple there is a const... | https://mathoverflow.net/users/6084 | Parahorics in nonsemisimple reductive algebraic groups | I don't know how Bruhat-Tits theory is used in representation theory, but I think there is a little confusion of notions in your question.
For a connected reductive algebraic group $G$, given a point $x$ in $B(G,K)$, Bruhat and Tits define $4$ integral models denoted $\mathfrak{G}\_x^0$, $\mathfrak{G}\_x$, $\hat{\ma... | 3 | https://mathoverflow.net/users/47722 | 286405 | 126,458 |
https://mathoverflow.net/questions/286369 | 10 | Here I consider cuspidal automorphic representations $\pi$ over the similitude group $\mathrm{GSp}(4,\mathbb{A}\_\mathbb{Q})$. Let $f$ be a non-zero vector in the representation $\pi$. I want to know if there is any reference/work on relating special values of the complex adjoint $L$-function $L(s,\pi,\mathrm{Ad})$ of ... | https://mathoverflow.net/users/117171 | Special values of adjoint $L$-functions of automorphic representations of $\mathrm{GSp}(4)$ as Petersson norms | For $\mathrm{GL}\_2$, the relationship between the Petersson norm of a newform $f$ and its adjoint $L$-function is roughly a statement of the form
\[\frac{|a\_f(1)|^2}{\langle f, f\rangle} = \frac{c\_f}{\Lambda(1, \operatorname{ad} f)},\]
where $a\_f(n)$ denotes the first Fourier coefficient of $f$, $\Lambda(s,\pi)$ de... | 12 | https://mathoverflow.net/users/3803 | 286406 | 126,459 |
https://mathoverflow.net/questions/286401 | 1 | We know that, given an n-dimensional Euclidean simplex, for all $1\leq i,j,k,l\leq n+1$, we have(law of sines)$$\frac{A\_i A\_j}{A\_k A\_l}=\frac{c\_{ij}}{c\_{kl}}$$(from *Elementary Formulas for a Hyperbolic Tetrahedron*)
And in <http://www.emis.de/journals/JIPAM/images/106_03_JIPAM/106_03.pdf> it mentioned some inequ... | https://mathoverflow.net/users/111290 | Is there a law of cosine for n-dimensional hyperbolic simplex | Yes, something like that is proved in the paper by Simon Kokkendorff:
*Kokkendorff, Simon L.*, [**Polar duality and the generalized law of sines**](http://dx.doi.org/10.1007/s00022-006-1858-7), J. Geom. 86, No. 1-2, 140-149 (2006). [ZBL1115.51010](https://zbmath.org/?q=an:1115.51010).
It would be too cumbersome to ... | 5 | https://mathoverflow.net/users/11142 | 286416 | 126,460 |
https://mathoverflow.net/questions/286299 | 0 | I've been reading about Weil pairing from [Pairings for Beginners](http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf) and in section 5.1 an example is given. I took a look on the magma code of that example (see [here](http://www.craigcostello.com.au/pairings/beginners/5-1-1-WeilPairing1.txt)) and it wor... | https://mathoverflow.net/users/117113 | Weil Pairing Example fails | It seems that the author made a mistake in code.
In fADD function the vertical line equation should be computed as
`v:=F!(x-(lambda^2-P[1]-Q[1]));` instead of `v:=F!(x-(lambda^2-P[1]-P[2]));`
And now works with any points $P$ and $Q$ as expected.
| 1 | https://mathoverflow.net/users/117113 | 286418 | 126,461 |
https://mathoverflow.net/questions/286411 | 1 | Let $X$ be a Hausdorff space such that the irrationals $\mathbb P$ (in their usual topology) form a dense subspace of $X$.
Let $C$ be the Cantor set. The set of "non-endpoints" of $C$ is homeomorphic to $\mathbb P$.
**Question.** If $f:C\to X$ is a continuous surjection such that $f\restriction \mathbb P$ is the id... | https://mathoverflow.net/users/91061 | Cantor set onto connected set? | To rephrase and explicitate YCor's example in the comments, consider the [devil's staircase function](https://en.wikipedia.org/wiki/Cantor_function): it maps the non-endpoints of the standard Cantor set $C$ to the non-dyadic reals in $[0,1]$. Now compose (on the left) with the inverse of the [question mark function](ht... | 3 | https://mathoverflow.net/users/17064 | 286425 | 126,463 |
https://mathoverflow.net/questions/286409 | 1 | Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can find a norm $\|\cdot\|$ on the space of measurable functions (or at least a subspace of "sufficiently nice" functions) sati... | https://mathoverflow.net/users/117403 | Existence of a certain norm on space of measurable functions | What you are asking about is known as a **symmetric** (or rearrangement invariant) Banach space of measurable functions (not to be confused with Riemannian symmetric spaces) and the associated **fundamental function**, see the recent book [Foundations of symmetric spaces of measurable functions](https://mathscinet.ams.... | 1 | https://mathoverflow.net/users/8588 | 286431 | 126,465 |
https://mathoverflow.net/questions/286046 | 18 | Fix a prime $p\geq 5$ and an integer $n>0$. All spaces in this question are implicitly $p$-localized. Consider the spaces $X=J\_{p^n-1}S^2$ (the $p^n-1$'th stage in the James construction $JS^2\simeq\Omega S^3$) and $Y=\Omega X$. These appear naturally in a number of applications. The loop sum operation makes $Y$ into ... | https://mathoverflow.net/users/10366 | Is $\Omega J_{p^n-1}S^2$ commutative up to homotopy? | This was answered in the affirmative by Brayton Gray in his paper *Homotopy Commutativity and the EHP Sequence*. Specifically he shows that for all $n$ the space $\Omega J\_{p^s-1} S^{2n}$ is homotopy commutative for $s\geq 1$ when localised at any prime $p\geq 3$. Moreover he claims to be able to show that $\Omega J\_... | 5 | https://mathoverflow.net/users/54788 | 286434 | 126,466 |
https://mathoverflow.net/questions/265727 | 8 | For $n\ge 1$, let $f(n)$ be the number of rooted complete (unordered) binary trees with $n$ leaves labeled from $1$ to $n$ ("complete binary" means that every vertex has either $0$ or $2$ children and "unordered" means that the we do not specify which child is the left child or the right child). Then it is well known (... | https://mathoverflow.net/users/3106 | Bijective proof of formula for rooted binary forests | I think the following might do the trick:
*Erdös, Péter L.*, [**A new bijection on rooted forests**](http://dx.doi.org/10.1016/0012-365X(93)90154-L), Discrete Math. 111, No.1-3, 179-188 (1993). [ZBL0785.05049](https://zbmath.org/?q=an:0785.05049).
| 6 | https://mathoverflow.net/users/3032 | 286442 | 126,468 |
https://mathoverflow.net/questions/255516 | 14 | My original goal was to read the PTVV paper *Shifted Symplectic Structures* <https://arxiv.org/pdf/1111.3209v4.pdf>. I was quickly humbled!
Being told the theory ought to generalize symplectic structures on algebraic varieties and schemes I was unable to find a clear reference for these structures. I could get a hold... | https://mathoverflow.net/users/nan | Reference for symplectic structures on schemes? | Dear past life Jacob,
Find a specific geometric problem that this stuff solves, which you think is interesting, and then you will find yourself magically learning it. For you, this problem was extending Donaldson-Thomas theory to Calabi-Yau 4-folds. Even in the Calabi-Yau 3-fold case, shifted symplectic structures s... | 18 | https://mathoverflow.net/users/nan | 286445 | 126,469 |
https://mathoverflow.net/questions/286438 | 4 | A well-known problem is to classify all covering spaces of a topological space $X$. For example, if $X$ is a semi-locally simply connected space, then each equivalent class of a covering space of $X$ is corresponding to conjugacy class of a subgroup of $\pi\_1 (X)$. Now my question is that:
Is there any classificati... | https://mathoverflow.net/users/114476 | The Classification of all spaces for which $X$ is a covering space | In general, I would expect this to be a quite intractable problem. For instance, let's assume we are only interested in the category of manifolds, and we ask the question which $3$-manifolds are covered by $\mathbb{R}^{3}.$ Here, by the solution to the geometrization conjecture for $3$-manifolds, every closed, orientab... | 10 | https://mathoverflow.net/users/49247 | 286447 | 126,470 |
https://mathoverflow.net/questions/286443 | 8 | In Donaldson-Kronhiemer Section 4.2.5. (local models of the moduli space of YM instantons) they first get local models of the moduli space $M$ inside the space of all connections modulo gauge $\mathcal{B}$ by taking $(F^+)^{-1}(0)/\Gamma\_A$, where ($\Gamma\_A$ is the isotropy group of the connection).
Now here comes... | https://mathoverflow.net/users/nan | Deformation-Obstruction Theory of YM Instantons | $\newcommand{\A}{\mathscr{A}}$ $\newcommand{\G}{\mathscr{G}}$ Denote by $\A$ the space of connections, by $\A\_-$ the space of ASD connections and by $\G$ the gauge group. For ssimplicity I will not keep track of various Sobolev decorations. The moduli space $\newcommand{\M}{\mathscr{M}}$ $\M$ is defined as a set by th... | 5 | https://mathoverflow.net/users/20302 | 286449 | 126,471 |
https://mathoverflow.net/questions/286415 | 7 | I am reading Ben Andrews book about Ricci flow and at the start of the chapter about Perelman's gradient flow formulation for Ricci flow he says Robert Bryant exposed that there are no functionals defined on the $L^2$-space of Riemannian metrics that promotes Ricci flow as a gradient flow.
Does anyone know the name o... | https://mathoverflow.net/users/94097 | Ricci flow is not a gradient flow for $L^2$-space of metrics | If there were such a functional $\mathcal{F}$, observe that
1. Under Ricci flow the functional would have to decrease. That is, if $\partial\_t g(t) = -2 Rc[g]$ then $\partial\_t \mathcal{F}(g(t)) \leq 0$, with strict inequality of $-2 Rc[g] \neq 0$.
2. The functional would have to be invariant under diffeomorphisms... | 7 | https://mathoverflow.net/users/46591 | 286451 | 126,472 |
https://mathoverflow.net/questions/286444 | 5 | Let $X$ be a Banach space and consider $B(X)$, the set of all bounded linear maps on $X$. By the W-topology on $B(X)$ we mean the topology induced by the semi-norms
$$B(X)\to [0,\infty): T\to |\langle Tx,x^\*\rangle|$$
where $x\in X$ and $x^\*\in X^\*$.
The algebraic tensor product $X\otimes X^\*$ may be considered... | https://mathoverflow.net/users/84390 | A dense subset in $B(X)$ under the weak operator topology | I think yes, for the following general reason: If $E$ is a real vector space and $F$ a linear space of linear forms of $E$, then a linear subspace $V$ of $E$ is $\sigma(E,F)$-dense in $E$ if and only if no $f\in F\setminus\{0\}$ vanishes identically on $V$. (Equivalently, any proper $\sigma(E,F)$-closed linear subspace... | 4 | https://mathoverflow.net/users/6101 | 286467 | 126,474 |
https://mathoverflow.net/questions/286472 | 10 | In his paper "The pair correlation of zeros and the zeta function",
Montgomery defines a function
$$F(\alpha,T) = \left(\frac{T}{2 \pi} \log T\right)^{-1} \sum\_{0 < \gamma, \gamma' < T} T^{i \alpha (\gamma'-\gamma)} w(\gamma'-\gamma)$$
where $w(u)=\frac{4}{4+u^2}$, and the sum is over pairs of imaginary parts $\gamma,... | https://mathoverflow.net/users/9317 | Statement of the pair correlation conjecture | For any $\epsilon > 0$ and for any finite interval $I \subset [1, \infty)$, there is a $T\_{0}$ such that for all $T > T\_{0}$ and all $\alpha \in I$ we have $|F(\alpha,T) - 1| \leq \varepsilon$. The meaning of the conjecture is that the Fourier transform $F(\alpha, T)$ of the pair correlation of zeros up to height $T$... | 11 | https://mathoverflow.net/users/117430 | 286476 | 126,478 |
https://mathoverflow.net/questions/286478 | 6 | Is there an infinite, countable connected $T\_2$-space $(X,\tau)$ such that $(X,\tau)$ has the fixed point property? (This means that for every continuous map $f:X\to X$ there is $x\in X$ such that $f(x) = x$.)
| https://mathoverflow.net/users/8628 | Countably infinite connected Hausdorff space with the fixed point property | Yes, there is such an example.
This is nearly Problem 10705 in **The American Mathematical Monthly**, proposed by D. W. Brown in [**106** #1 (January 1999), p. 67](http://www.jstor.org/stable/2589591), where the problem asks for a countably infinite $T\_{2}$ example. In an editorial comment following John Cobb's solu... | 5 | https://mathoverflow.net/users/15780 | 286479 | 126,479 |
https://mathoverflow.net/questions/286428 | 0 | Any map $f \colon \mathbb{R} \to \mathbb{R}$ induces a "composition map"
$$f^\circ\colon \mathbb{R} \times \mathbb{N} \to \mathbb{R},$$
where
$$f^{\circ n}(x) = \underbrace{f \circ \dotsb \circ f}\_{n \textrm{ times}} (x).$$
If $f$ happens to have an inverse, the domain of $f^\circ$ can be extended to $\mathbb{R} ... | https://mathoverflow.net/users/92270 | Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter | Notice that $e^x$ does not have an inverse on the whole real line.
Extension of iterates is possible if $f$ has a fixed point $x\_0$. Suppose for example, that this fixed point is repelling that is $f(x\_0)=x\_0$ and $\lambda=f'(x\_0)>1.$
I assume that $f$ is analytic, strictly increasing on $R$ and maps $R$ onto
i... | 5 | https://mathoverflow.net/users/25510 | 286480 | 126,480 |
https://mathoverflow.net/questions/286481 | 3 | Consider a rooted tree of height $h$, such that all the leaves are at last layer. We perform the following random process: each edge is deleted with probability $0.5$, and otherwise it is retained. We are interested in the probability that after the process ends, there remains a path from the root to one of the leaves.... | https://mathoverflow.net/users/24226 | Percolation on finite irregular trees | You still need more information on the structure of the tree. The 1-3-tree (Example 1.2 in Lyons & Peres: [Probability on trees and networks](http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html)) shows that the probability can go to 0, even if the maximum degree is bounded.
Here is a quick proof sketch: It is not har... | 4 | https://mathoverflow.net/users/97426 | 286489 | 126,484 |
https://mathoverflow.net/questions/286491 | 17 | I was reading the paper [Towards Constructive Homological Algebra in
Type Theory](https://link.springer.com/chapter/10.1007%2F978-3-540-73086-6_4) by Thierry Coquand and Arnaud Spiwack, and they state that constructively, the category of abelian groups fails to be abelian, because we cannot verify that every monic and ... | https://mathoverflow.net/users/56938 | In constructive mathematics, why does the category of abelian groups fail to be abelian? | There are many different flavors of constructive mathematics. The theory that was used in this paper is weak, it lacks some useful constructions from the usual set theory such as quotient sets. Another problem is that it lacks [function extensionality](https://ncatlab.org/nlab/show/function+extensionality) (that is, if... | 22 | https://mathoverflow.net/users/62782 | 286498 | 126,487 |
https://mathoverflow.net/questions/286502 | -2 | If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is an *edge cover* if $C\cap e \neq \emptyset$ for all $e\in E$.
Let $G=(V,E) $ be a graph with infinite chromatic number. Is every edge cover $C\subseteq V$ infinite?
| https://mathoverflow.net/users/8628 | Edge covers of graphs with $\chi(G) \geq \aleph_0$ | Yes. If $C$ is edge cover of $G$ then if $|C| = n < \infty$ then we can color each vertex of $C$ by its own unique color and color everything else by color $n + 1$ and so chromatic number is finite.
| 3 | https://mathoverflow.net/users/104330 | 286508 | 126,488 |
https://mathoverflow.net/questions/286513 | 1 | Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a *cover* of $X$ if $\bigcup {\cal C} = X$. A subset $S\subseteq X$ is a *choice set* for ${\cal C}$ if $|S\cap K| = 1$ for all $K\in {\cal C}$.
Suppose $X$ is infinite and ${\cal C}$ is a cover of $X$ with the following ... | https://mathoverflow.net/users/8628 | Choice sets in covers with small intersections | Not always. Say, we may achieve that for any $x\in X$ there exists a set $K\in \mathcal{C}$ such that $K\subset \cup\_{L\in \mathcal{C};L\ni x} L\setminus\{x\}$. It means that $x$ can not belong to a choice set. This may be done as follows: let $X$ be a set of rationals, and each set in $\mathcal{C}$ be a convergent se... | 1 | https://mathoverflow.net/users/4312 | 286518 | 126,491 |
https://mathoverflow.net/questions/286505 | 10 | Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $Gr\_{i,n}$ denote the Grassmannian of $i$-dimensional linear subspaces in $F^n$.
>
> Can one describe explicitly the $GL(n,\mathcal{O})$-orbits on $Gr\_{i,n}\times Gr\_{i,n}$, i.e. on pairs of subspaces?
>
>
>
Remark. $G... | https://mathoverflow.net/users/16183 | Orbits of $GL(n, \mathcal{O})$ on pairs of linear subspaces over non-Archimedean local fields | The $\text{GL}(n,\mathcal{O})$-orbits refine the $\text{GL}(n,F)$-orbits, i.e., the Bruhat cells. Up to replacing $i$ by $n-i$, assume that $2i\leq n.$ For every integer $m$ with $0\leq m\leq i,$ the **Bruhat cell** $U\_m$ in $\text{Gr}\_{i,n}\times \text{Gr}\_{i,n}$ is the set of pairs $([V],[W])$ of $F$-vector subspa... | 4 | https://mathoverflow.net/users/13265 | 286519 | 126,492 |
https://mathoverflow.net/questions/286525 | 1 | For a compact Kähler manifold, we say that a form is **primitive** if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\omega$ is closed with respect to the $d$ de Rham exterior derivative if and only if $\omega$ is harmonic. I suspect... | https://mathoverflow.net/users/89074 | de Rham closed harmonic form on a Kähler manifold | If $\omega$ is harmonic, then it is $d$-closed. For the other direction, suppose $\omega$ is $d$-closed and primitive, i.e. $d\omega = 0$ and $\Lambda\omega = 0$. Then, by a Kähler identity,
$$i\bar{\partial}^\*\omega = [\Lambda, \partial]\omega = \Lambda(\partial\omega) - \partial(\Lambda\omega) = \Lambda(\partial\... | 3 | https://mathoverflow.net/users/21564 | 286529 | 126,495 |
https://mathoverflow.net/questions/286524 | 3 | Let $p$ be a (large) prime.
>
> Does there exist a function $f\colon\mathbb F\_p^\times\to\mathbb F\_p$ such that the three sets
> $$ \{f(z)-z\colon z\in\mathbb F\_p^\times\},\ \{f(z)\colon z\in\mathbb F\_p^\times\},\ \text{and}\ \{f(z)+z\colon z\in\mathbb F\_p^\times \} $$
> form a double-cover of $\mathbb F\_p$... | https://mathoverflow.net/users/9924 | Double-covering $\mathbb F_p$ with three sets | Start with $f(z)=3z$, then all non-zero elements are covered three times. Change $f(a)$ to 0 and $f(b)$ to $b$. Now 0 is covered two times and $2a,3a,4a$, $2b,3b,4b$ are covered two times also (assume that all these 6 numbers are different, this is possible for large $p$). Some other multiplicities could increase (and ... | 6 | https://mathoverflow.net/users/4312 | 286530 | 126,496 |
https://mathoverflow.net/questions/286497 | 2 | I have a question about the proof of proposition $3.3.6(3)$ in "Tensor Categories" by Etingof et al..
This part states that for $A$, transitive unital $\mathbb Z\_+$-ring, there is a unique character taking non-negative values on the basis elements.
The proof uses the fact that if $\chi$ is a character, and $f$ is... | https://mathoverflow.net/users/117445 | Uniqueness of character for Z_+-rings | Hm, I probably figured it out myself, but I won't delete the question since I think the formulation in the proof is slightly misguiding.
Use $b\_i$ to denote basis elements ($i \in I$), a take element $y = \sum y\_i b\_i$ and denote $\chi\_i = \chi(b\_i)$. We have $\chi(y) = \sum y\_i \chi\_i$. But if the multiplica... | 1 | https://mathoverflow.net/users/117445 | 286531 | 126,497 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.