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/398083 | 3 | I would like to estimate from above the following sum
$$
\sum\_{1 \leq x\_1 < X} .. . \sum\_{1 \leq x\_n < X} \frac{\prod\_{1 \leq i \leq n } \phi(x\_i)}{\mathrm{lcm}(x\_1, .., x\_n)^a}.
$$
$\phi$ is the Euler totient function and $a$ is a positive integer less than $2n$. A trivial estimate would be $\ll X^{2n - a}$... | https://mathoverflow.net/users/84272 | Estimating $\sum_{x_i < X} \prod_i \phi(x_i)/ \mathrm{lcm}(x_i)^a$ | One can improve on $X^{2n-a}$ as long as $(a,n)\neq(1,1)$ (for $a=n=1$, the sum grows like $X/\zeta(2)$ so there's no room for improvement). Let us introduce $$f\_n(m) := \# \{ (x\_1,\ldots,x\_n) : \mathrm{lcm}(x\_1,\ldots,x\_n) = m\},$$ which satisfies $f\_n(m) \le \tau(m)^n \ll\_{n,\varepsilon} m^{\varepsilon}$ where... | 1 | https://mathoverflow.net/users/31469 | 398110 | 164,306 |
https://mathoverflow.net/questions/398109 | 2 | Let $f\in L^p(\mathbb{R})$ and define $f\_\theta(x)=f(x-\theta)$. I would like to compute (or at least lower bound) the following:
$$
\inf\_{\theta\ne\theta'}\frac{\Vert f\_\theta - f\_{\theta'}\Vert\_p}{|\theta-\theta'|}.
$$
In particular, I want to understand how this depends on $f$, and would like a bound that depen... | https://mathoverflow.net/users/323995 | Lower bounds on translates of a function | For any real $p\ge1$ and any real $t\ne0$,
$$\frac{\|f\_t-f\_0\|\_p}{|t-0|}
\le\frac{\|f\_t\|\_p+\|f\_0\|\_p}{|t|}
=\frac{2\|f\|\_p}{|t|}\to0$$
as $|t|\to\infty$.
So, the least lower bound in question is always $0$.
| 3 | https://mathoverflow.net/users/36721 | 398111 | 164,307 |
https://mathoverflow.net/questions/395996 | 8 | Consider the following statement:
***If a vector space has a basis then its dual vector space also has a basis.***
It is not an axiom of ZF. It clearly follows from the Axiom of Choice. But it is also strictly weaker than the Axiom of Choice.
I have convinced myself that this statement holds in a variety of model... | https://mathoverflow.net/users/13480 | If a vector space has a basis then its dual vector space has a basis | Here are two statements you might find interesting:
1. With base field $k\subseteq\mathbb C$ your axiom implies that any free $\mathbb Z$-action has a choice of representatives.
2. For each prime $p,$ there is an $\omega$-categorical theory such that the associated permutation model contains a set $X$ with no basis f... | 1 | https://mathoverflow.net/users/164965 | 398116 | 164,310 |
https://mathoverflow.net/questions/398106 | 2 | Let $Y$ be a compact Riemann surface and $B$ a finite subset of $Y$. It is a standard fact that isomorphism classes of holomorphic ramified covers $f:X\rightarrow Y$ of degree $d$ with branch points in $B$ are in a correspondence with homomorphisms $\rho:\pi\_1(Y-B)\rightarrow S\_d$ with transitive image modulo conjuga... | https://mathoverflow.net/users/128556 | Recovering a family of rational functions from branch points | The reason for this phenomenon is that you are afflicted with a serious mathematical condition, that being:
Your monodromy has monodromy.
---
To be less cryptic, the key thing is that the fundamental group is not actually generated by small loops around these four points, except if by "small loop around $x$" yo... | 5 | https://mathoverflow.net/users/18060 | 398125 | 164,316 |
https://mathoverflow.net/questions/398132 | 3 | **Problem set up:**
Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space.
We say that a measure preserving transformation $T$ on $\mathbf X$ is *$\varepsilon$-almost weakly mixing* if for every $\delta > \varepsilon$, and every pair of non-null measurable sets $A, B \in \mathcal A$, there exists ... | https://mathoverflow.net/users/173490 | Does an “almost weakly mixing” transformation admit a non-null ergodic component? | Your question is equivalent to asking whether $T$ being $\epsilon$-almost weak mixing implies that the invariant $\sigma$-algebra, $\mathcal I$ contains atoms. The answer is yes.
I will prove the contrapositive. I claim if $\mathcal I$ contains no atoms, then $T$ is not $\epsilon$-almost weak mixing for any $\epsilon... | 5 | https://mathoverflow.net/users/11054 | 398133 | 164,318 |
https://mathoverflow.net/questions/398138 | 2 | In Qing Liu's Algebraic Geometry and Arithmetic Curve, we have the following proposition(6.3.13):
Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be smooth schemes over $S$. Then any immersion $f:X\rightarrow Y$ of $S$-schemes is a regular immersion, and we have a canonical exact sequence on $X$:
$0\rightarrow... | https://mathoverflow.net/users/nan | Construction of Jacobian Ideal | I think this is true: since $Y\rightarrow X$ is birational, the homomorphism $\mathcal{C}\_{Y/W}\rightarrow i^\*\Omega\_{W/X}$ is bijective on a dense open subset of $Y$. Thus its kernel is torsion, hence zero since $\mathcal{C}\_{Y/W}$ is locally free.
| 3 | https://mathoverflow.net/users/40297 | 398140 | 164,320 |
https://mathoverflow.net/questions/398033 | 2 | Let FSym$(\mathbb{N})$ denote the finitary symmetric group on the natural numbers. Are all Sylow p-subgroups of FSym$(\mathbb{N})$ isomorphic (maybe to the iterated wreath product $C\_p\wr C\_p\wr\dots$ of cyclic $p$-group $C\_p$)?
It is well-known that the Sylow $p$-subgroups of $S\_{p^k}$, for the positive integer ... | https://mathoverflow.net/users/98061 | Isomorphic Sylow p-subgroups of FSym$(\mathbb{N})$ | $\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}\newcommand\N{\mathbf{N}}$The answer is no, as already mentioned in the comments: a transitive Sylow subgroup, or more generally any subgroup without finite orbit, has a trivial center (clear, since for a nontrivial element, its centralizer preserves its supp... | 3 | https://mathoverflow.net/users/14094 | 398147 | 164,323 |
https://mathoverflow.net/questions/398143 | 4 | There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/R$.
I would like to replace here $\operatorname{Commutative rings}$ with some ($\infty$-?)category $\bf X$ of symmetr... | https://mathoverflow.net/users/41291 | Is there essentially unique notion of module over monoidal stable $\infty$-categories? | The ∞-categorical analog of the fact you mention can be found in *Higher Algebra*, corollary 7.3.4.14:
>
> Let $\operatorname{CAlg}$ be the category of $E\_\infty$-rings and $A\in \operatorname{CAlg}$. Then there is a canonical equivalence
>
>
> $$\operatorname{Sp}(\operatorname{CAlg}\_{/A})\simeq \operatorname{M... | 11 | https://mathoverflow.net/users/43054 | 398148 | 164,324 |
https://mathoverflow.net/questions/398008 | 5 | This question is a repost of the following: <https://math.stackexchange.com/questions/4195805/sections-of-a-polar-action-are-totally-geodesic>. I've decided it to post it here because it didn't seem to get much traction there.
Suppose $G\curvearrowright M$ is an isometric action of a Lie group on a complete Riemannia... | https://mathoverflow.net/users/175146 | Sections of a polar action are totally geodesic | I will try to answer your question posed in (4). From now on, $G$ is a Lie group acting properly and isometrically on a connected Riemannian manifold $M$. First, we need a little preparation.
**Fact.** Let $Q \subseteq M$ be an orbit of $G$. There exists $\varepsilon > 0$ such that the exponential map is defined on $... | 2 | https://mathoverflow.net/users/132126 | 398154 | 164,325 |
https://mathoverflow.net/questions/398129 | 1 | Denote by $\mathcal L$ the set of continuously differentiable real valued functions on $[0, 1]$ with Lipschitz continuous derivative. Does there exist a Borel measurable function $ f: [0, 1] \times \mathbb R \to \mathbb [0, \infty) $ such that
$$\inf\_{g \in \mathcal L} \int\_{0}^{1} f(t, g(t)) \ dt < \inf\_{h \in C^... | https://mathoverflow.net/users/173490 | Lavrentiev phenomenon between $C^1$ + Lipschitz derivative and $C^2$ | Edited.
Let $g\_0^\prime(x)=|x-1/2|,\; g\_0(x)=\int\_0^xg\_1(t)dt,\; 0\leq x\leq1$.
Then $g\_0$ has continuus derivative, namely $g\_0^\prime$, which is Lipschitz,
but the second derivative is discontinuous. Let $L=\{(x,y):y=g\_0(x)\}$
be the graph of $g\_0$.
Now define $f(x,y)=0$ when $0\leq x\leq 1, y=g\_0(x)$ an... | 1 | https://mathoverflow.net/users/25510 | 398163 | 164,329 |
https://mathoverflow.net/questions/398156 | 3 | Automorphisms of connected, reductive groups are well understood: the outer automorphism group is an essentially combinatorial object associated to the root datum. I am trying to understand automorphisms of possibly *disconnected* reductive groups, by which I just mean groups whose identity components are reductive.
... | https://mathoverflow.net/users/2383 | Automorphisms of étale-by-torus groups | For any group $\Gamma$ that acts on $G$, $\Gamma$ acts on $G^{\circ}$ and on $G/G^{\circ}$. Since $G/G^{\circ}$ is finite and $G^{\circ}$ is a torus, both their automorphism groups are discrete groups, thus because $\Gamma$ is connected, the action of $\Gamma$ on them is trivial.
Automorphisms of $G$ fixing $G^{\circ... | 3 | https://mathoverflow.net/users/18060 | 398166 | 164,330 |
https://mathoverflow.net/questions/398001 | 3 | In the book of *S.Osher & R.Fedkiw - Level set methods and dynamic implicit surfaces*, at page 15, is stated without proof, a formula like that:
$$\text{Per}\_{\Omega}(\omega)=\lim\_{\varepsilon\to 0} \int\limits\_{\Omega}\delta\_{\varepsilon}(\phi(x))\ |\nabla\phi(x)|\ dx,$$
where
* $\Omega\subset\mathbb{R}^2$ is ... | https://mathoverflow.net/users/61629 | Perimeter continuity of $BV$ sets on any sequence from $W^{1,1}$ | This seems too strong, regardless of which set $\omega \subset \Omega$ one works with. We suppose that $\omega$ has bounded perimeter, so that $\chi\_\omega \in BV(\Omega)$.
Let $(f\_n \mid n \in \mathbf{N})$ be a sequence of functions in $W^{1,1}(\Omega)$ with
\begin{equation}
\lvert f\_n \rvert\_{L^1} \to 0
\text{ ... | 1 | https://mathoverflow.net/users/103792 | 398171 | 164,333 |
https://mathoverflow.net/questions/395216 | 4 | Let's say we have a principal bundle $(P,B,\pi;G)$ and associated bundle $E=P \times\_{(G,\rho)}V$and $Ad(P)=P\times\_{(G,Ad)} \mathfrak{g}$ the adjoint bundle. The Yang-Mills-Higgs action (without potential) is
\begin{equation}
\int\_M(- \frac{1}{2}\langle F^A, F^A \rangle\_{\operatorname{Ad}(P)} +\langle d\_A \phi,... | https://mathoverflow.net/users/209074 | The Yang-Mills Higgs Lagrangian | 1. The first question has been already answered in a comment by NicAG, but let us repeat the 3-line long argument here for completeness:
>
> For the adjoint bundle , ρ∗=ad(⋅)(⋅)=[⋅,⋅]. Hence $⟨j,α⟩=−⟨d\_Aϕ,[α,ϕ]⟩$. Because of ad-invariance, the commutator can be 'moved': $⟨j,α⟩=⟨[d\_Aϕ,ϕ],α⟩$. The rest its then jus... | 4 | https://mathoverflow.net/users/108862 | 398175 | 164,335 |
https://mathoverflow.net/questions/398180 | 4 | $\newcommand\P{\mathcal P}$A "partition" $P$ (of the interval $[0,1]$) is a finite sequence $(t\_0,\dots,t\_n)$ such that $0=t\_0<\cdots<t\_n=1$; then the mesh of $P$ is $\|P\|:=\max\_{1\le j\le n}(t\_j-t\_{j-1})$.
Fix any sequence $\P:=(P\_k)$ of "partitions" $P\_k=(t\_{k,0},\dots,t\_{k,n\_k})$ such that $\|P\_k\|\t... | https://mathoverflow.net/users/36721 | A narrower dichotomy for the quadratic variation of differentiable functions? | Using your previous example of $f(x) = x^2 \cos(x^{-4})$. Note that on any interval $[\epsilon,1]$ the function is continuously differentiable, and hence has quadratic variation 0.
Construct $P\_k$ so that the following points are contained in the partition:
1. 0
2. $\frac{1}{\sqrt[4]{\pi}} \ell^{-1/4}$ for natural... | 5 | https://mathoverflow.net/users/3948 | 398183 | 164,336 |
https://mathoverflow.net/questions/398128 | 12 | Atiyah duality is the equivalence $M/\partial M \simeq (M^{-T(M)})^\vee$, i.e. the Spanier-Whitehead dual of the space $M/\partial M$ is the Thom complex of the stable normal bundle of $M$. The theorem is proven by taking an appropriate embedding $i$ of $M$ into a sphere $S^d$, identifying the complement with the Thom ... | https://mathoverflow.net/users/134512 | Atiyah duality without reference to an embedding | Here is another short construction which is much simpler and just takes a few lines.
1. Let $M$ be a closed $n$-manifold. Consider the diagonal $M \to M \times M$. It is an embedding. Take its Pontryagin-Thom construction to get a map
$$
M\_+ \wedge M\_+ \to M^\tau
$$
(we have identified a tubular neighborhood of the... | 9 | https://mathoverflow.net/users/8032 | 398188 | 164,337 |
https://mathoverflow.net/questions/398153 | 4 | The set theory with atoms ([ZFA](https://ncatlab.org/nlab/show/ZFA)), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements.
I have thusfar found two applications of ZFA-Set Theory. Firstly, ZFA can be used... | https://mathoverflow.net/users/324481 | Applications of ZFA-Set Theory | Although you prefer models other than permutation models, let me point out an appearance of permutation models, particularly the basic Fraenkel model, at the border between computer science and logic. The topic concerns operators that bind variables, like $\forall$ and $\exists$ in logic, the $\lambda$ in lambda-calcul... | 12 | https://mathoverflow.net/users/6794 | 398193 | 164,340 |
https://mathoverflow.net/questions/398187 | 11 | It is well-known that when passing to $\infty$-categories the notion of commutativity gets replaced by an infinite array of notions of commutativity: $\mathbb{E}\_{1}$, $\mathbb{E}\_{2}$, ..., $\mathbb{E}\_{\infty}$. This is already apparent when passing from sets to categories and $2$-categories:
* For sets, we have... | https://mathoverflow.net/users/130058 | Intermediate notions of bilinearity in higher algebra | Let me clarify a bit what I meant in [my comment](https://mathoverflow.net/questions/398187/intermediate-notions-of-bilinearity-in-higher-algebra#comment1020095_398187) on how the notion of bilinearity will depends on "how commutative" are $A$, $B$ and $C$, and this is one way to define a hierarchy of notion of bilinea... | 9 | https://mathoverflow.net/users/22131 | 398194 | 164,341 |
https://mathoverflow.net/questions/397624 | 1 | Consider any 3D body with an axis of rotational symmetry (e.g. cone, cylinder...) and packing the 3d space efficiently with infinitely many copies of this body. Is the following claim valid?
**Claim:** The densest packing with any such body is necessarily such that all units are aligned along or opposite to the same ... | https://mathoverflow.net/users/142600 | On packing axisymmetric bodies in 3D | Your claim is false for axially-symmetric ellipsoids: when restricted to all have their axis of symmetry in the same direction, they cannot pack more densely than spheres (in terms of volume fraction). However, they can in fact pack more densely than spheres, as discussed in [Donev et al](https://dx.doi.org/10.1103/Phy... | 6 | https://mathoverflow.net/users/20186 | 398206 | 164,345 |
https://mathoverflow.net/questions/395592 | 9 | Fix $(R,m)$ a complete DVR of mixed characteristic $(0,p)$ with perfect residue field, and consider finite flat commutative group schemes $G = Spec(A)$ over $R$. One can associate a differential invariant to $G$, an integer $d \geq 0$, in two ways:
1. the "absolute different", such that the dual $A^\*$ of $A$ under t... | https://mathoverflow.net/users/367 | Cotangent spaces of finite flat group schemes in short exact sequences | By Proposition 5.1(i) in [Mazur and Roberts, Local Euler characteristics, *Invent. Math.* **9** (1970), 201-234](https://doi.org/10.1007/BF01404325), $G$ fits in an exact sequence $0 \to G \to A \to B \to 0$, where $A$ and $B$ are smooth commutative affine group schemes over $R$ of the same relative dimension, say $n$.... | 4 | https://mathoverflow.net/users/2757 | 398211 | 164,347 |
https://mathoverflow.net/questions/398088 | 2 | When we talk about the theory of variation of Hodge structures, we always assume that the central fiber is a Kähler manifold $X$, then consider a family of deformations $\pi:\mathcal X\to B$ and the period map $\mathcal P:B\to Grass(b^{p,k},H^k(X,\mathbb C))$, $b^{p,k}=dim F^pH^k(X,\mathbb C)$.
What if we replace the... | https://mathoverflow.net/users/99826 | Period map for $\partial\bar\partial$-manifolds | Let me start with a disclaimer that I think the following facts are true, but I'm doing this over coffee and I haven't checked the details carefully. First, I'll redefine $F^pH^k(X,\mathbb{C})$ to be the space of de Rham classes represented by the sum of $(p', p'-k)$ forms with $p'\ge p$, or equivalently as
$$F^pH^k(X,... | 1 | https://mathoverflow.net/users/4144 | 398228 | 164,349 |
https://mathoverflow.net/questions/398210 | 1 | I'm encountering a lot of problems when dealing with the root of unity sheaf $\mu\_\infty := \mathrm{colim}\_n\mu\_n$.
Let $X$ be a smooth geometrically integral variety over a number field $k$. Although we have the canonical inclusion $\mu\_\infty \subset \mathbb{G}\_m$, the cohomology groups with coefficients in th... | https://mathoverflow.net/users/172132 | Cohomology with coefficients in $\mu_\infty$ | **Question 2.** Does $H^2(\bar{X},\mu\_\infty)$ have trivial Galois action? If so, is it true for all $H^i(\bar{X}, \mu\_\infty)$?
The answers are "often not" and "almost never".
The exact sequence $1 \to \mu\_{\infty} \to \mathbb G\_m \to \mathbb G\_m \otimes \mathbb Q \to 1$ you mention gives a map on cohomology ... | 5 | https://mathoverflow.net/users/18060 | 398232 | 164,351 |
https://mathoverflow.net/questions/398220 | 1 | Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated.
It is used a few times in Bosch, [Lectures on Formal and Rigid Geometry](https://www.math.purdue.edu/%7Etongliu/seminar/rigid/Bosch.pdf) e.g. first lines of pg. 164, Cor. 5 and Cor. 6 (their condition (V) is what I stated a... | https://mathoverflow.net/users/97321 | Flatness criterion for $I$-adic ring: $I$-torsion free | In Section 7.3 it is assumed that $I$ is the ideal of definition or $R$. It follows that the $I$-adic topology is separated, so (because $R$ is a valuation ring) every nonzero ideal of $R$ contains some power of $I$, so everything that is $R$-torsion is $I$-torsion.
| 2 | https://mathoverflow.net/users/18060 | 398240 | 164,354 |
https://mathoverflow.net/questions/398243 | 5 | Given a group $G$, find subsets $A,B$ such that $G=A\sqcup B$ and $A$ and $B$ are closed under multiplication: $x,y\in A$ (corr. $x,y\in B$) implies $xy\in A$ (corr. $xy\in B$).
For example, if $G$ is finite then all splitting are trivial: if $1\in A$ then $A=G$ and $B=\emptyset$.
If $G=\mathbb{Z}^d$ then any split... | https://mathoverflow.net/users/90980 | Splitting a group into two subsets closed under multiplication | This is a way to rediscover a quite well-studied class of groups:
**Proposition** *Let $G$ be a group. Equivalent statements: (a) $G$ admits a partition $G=A\sqcup B$ with $1\in A$, $B$ nonempty, and both $A,B$ subsemigroups. (b) $G$ admits a nontrivial order-preserving action on some totally ordered set [which can b... | 10 | https://mathoverflow.net/users/14094 | 398250 | 164,357 |
https://mathoverflow.net/questions/398241 | 5 | I am reading the book *Applications of Diophantine Approximation to Integral Points and Transcendence* by Zannier and Corvaja and, after their proof of the Chevalley-Weil theorem, in Example 3.8 they suggest that it is possible to prove the weak Mordell-Weil theorem using Chevalley-Weil. Honestly, I can't see that, but... | https://mathoverflow.net/users/167909 | Weak Mordell-Weil for EC using Chevalley-Weil theorem | The multiplication-by-$m$ map $[m]:E\to E$ is unramified, so there exists a finite set of primes $S$, depending only on $E$ and $m$, so that for every $P\in E(K)$, the field generated by the coordinates of the points in $[m]^{-1}(P)$ is unramified outside of $S$. (This is where we use Chevelley-Weil.) The degree of tha... | 12 | https://mathoverflow.net/users/11926 | 398252 | 164,358 |
https://mathoverflow.net/questions/398167 | 5 | Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that:
$\bullet$ $\phi^{-1}(0)\neq\emptyset$;
$\bullet$ $\nabla\phi(x)\neq 0$ on a neighborhood $W\subset\Omega$ of the curve $\phi^{-1}(0)$.
WLOG we can assume that $W=\{x\in\mathbb{R}... | https://mathoverflow.net/users/61629 | Continuity of Hausdorff measure on level sets | As Leo Moos suggested in the comments, in any dimension $d$, this is a simple consequence of the implicit function theorem.
The implicit function theorem implies that every point $x\in\phi^{-1}(0)$ has a neighborhood $\Omega\_x$ that is a diffeomorphic image $\varphi\_x(Q\_{\delta\_x})$ of a box $Q\_{\delta\_x}=\{|y\... | 6 | https://mathoverflow.net/users/56624 | 398254 | 164,359 |
https://mathoverflow.net/questions/398238 | 1 | It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page [Binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution)), where in the limit $n \to \infty$ the entropy of a binomial approaches the Gaussian one. The... | https://mathoverflow.net/users/nan | Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution | The confusion about $1/\sqrt n$ versus $1/n$ corrections is addressed below. First, for reference, let me quote the relevant results, all following from this [source](https://core.ac.uk/download/pdf/82776425.pdf).
The Gaussian entropy is $S\_0=\tfrac{1}{2}\ln(2\pi \sigma^2)+\tfrac{1}{2}$, the leading correction $\del... | 0 | https://mathoverflow.net/users/11260 | 398260 | 164,362 |
https://mathoverflow.net/questions/398261 | 1 | Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different.
I am interested in finding the distribution of $X/\|X\|\_2$.
As a start let $X$ be isotropic. Then $\|X\|\_2^2$ wil... | https://mathoverflow.net/users/325572 | Norm contrained Gaussian distribution | In two dimensions, with
$$\mu=\begin{pmatrix} m \\ n \end{pmatrix},\ \
\Sigma=\begin{pmatrix} v & 0 \\ 0 & w \end{pmatrix},$$
an integration over all possible radii gives the distribution of $X/\|X\|\_2$ as
$$f(\cos t,\sin t)=\frac{
1+\sqrt{\pi}u
\exp(u^2)
(1+\text{erf}(u))
}{
\exp(a)c\pi \sqrt{v w}
}
$$
where
$$a=\fr... | 3 | https://mathoverflow.net/users/nan | 398272 | 164,365 |
https://mathoverflow.net/questions/398251 | 20 | In Zhu's [Coherent sheaves on the stack of Langlands parameters](https://arxiv.org/abs/2008.02998) theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global Langlands parameters:
>
> Let $H\_{I,V}^i=H^iC\_c\left(\mathrm{Sht}\_{\Delta(\bar\eta),K... | https://mathoverflow.net/users/85392 | Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters | The anticipated analogue is as follows:
There is a map $f: \mathcal X \to \prod\_{v \in S} \mathcal X\_v,$
where $\mathcal X$ is the stack of $p$-adic representations of $G\_{E,S}$ into ${}^LG$ (the $L$-group over $E$ of some group $G$ that is part of a Shimura datum, with reflex field $E$) unramified outside $S$, an... | 13 | https://mathoverflow.net/users/169863 | 398274 | 164,367 |
https://mathoverflow.net/questions/398282 | 5 | I am struggling to understand what an invariant section with respect to a linearization of a line sheaf is. In Geometric Invariant Theory, given a $k$-scheme $X$ (being $k$ an algebraically closed field of characteristic zero) where a reductive algebraic group $G$ acts on by $\sigma$, and a line bundle $L\rightarrow X$... | https://mathoverflow.net/users/140062 | Invariant section of a linearized sheaf | **Question:** "What is a dual action of the group?"
**Answer:** For simplicity, if $X:=Spec(A)$ and $G:=Spec(R)$ is a linear algebraic group over a field $k$ acting on $X$ via $\sigma^\*: G\times\_k X \rightarrow X$ you get a "dual action"
$$\sigma: A \rightarrow R\otimes\_k A.$$
If $L\in Pic(A)$ is an invertible... | 6 | https://mathoverflow.net/users/nan | 398295 | 164,371 |
https://mathoverflow.net/questions/398012 | 0 |
>
> Donsker's invariance principle:
> Let $X\_1,X\_2,...$ be i.i.d. real-valued random variables with mean 0 and variance 1. We define $S\_0=0$ and $S\_n= X\_1+ ... + X\_n$ for $n \geq 1$. To get a process in continuous time, we interpolate linearly and define for all $t \geq 0$
> $$
> S\_t = S\_{[t]}+ (t-[t])(S\_{[t... | https://mathoverflow.net/users/168083 | How to prove the coupling version of the Donsker's Invariance Principle? | Yes, this can be done in any dimension but the two dimensional case is especially simple. Rotating a simple RW in two dimensions by 45 degrees yields a random walk where the $x$ and $y$ coordinates are independent. Couple each coordinate to a BM using Skorokhod embedding to obtain the desired coupling.
| 0 | https://mathoverflow.net/users/7691 | 398314 | 164,376 |
https://mathoverflow.net/questions/398306 | -1 | I would say no. Because a constant curvature manifold is symmetric. Any 2 dimensional subspace should be like a hyperboloid or a sphere. These objects do not have torsion.
| https://mathoverflow.net/users/105352 | Do exist constant curvature manifolds (hyperbolic or elliptic) with torsion? | You should consider the following example:
Let $M^3=\mathrm{SU}(2)\simeq S^3$ endowed with its biïnvariant metric $g$ (unique up to a constant multiple, let's fix this by requiring that the $g$ has constant sectional curvature equal to $1$). Now consider the unique connection $\nabla$ for which the left-invariant vec... | 11 | https://mathoverflow.net/users/13972 | 398323 | 164,380 |
https://mathoverflow.net/questions/398316 | 6 | This question was [posted](https://math.stackexchange.com/posts/4199651/edit) on Math Stack Exchange, but did not attract an answer. Here is the question:
Informal Description
====================
Let me start with an example. Let $X$ be the set $\{a, b, c, d, e\}$ and $E$ be the set $\{a, b, c\}$. Let $f$ be a fun... | https://mathoverflow.net/users/324481 | What do you call the generalisation of the direct image? | This idea is commonly used in set theory with atoms. I'm not sure whether it has a standard name, but I would be inclined to call it the *natural extension* of $f$ to sets.
There is no need to stop the iteration at $\omega$ as you do, for one can continue the cumulative hierarchy through the ordinals. If $A$ is any c... | 4 | https://mathoverflow.net/users/1946 | 398329 | 164,385 |
https://mathoverflow.net/questions/398317 | 3 |
>
> **QUESTION.** Let $x>0$ be a real number or an indeterminate. Is this true?
> $$\sum\_{n=0}^{\infty}\frac{\binom{2n+1}{n+1}}{2^{2n+1}\,(n+x+1)}=\frac{2^{2x}}{x\,\binom{2x}x}-\frac1x.$$
>
>
>
**POSTSCRIPT.** I like to record this presentable form by Alexander Burstein:
$$\sum\_{n=0}^{\infty}\frac{\binom{2n}n}... | https://mathoverflow.net/users/66131 | Proving a binomial sum identity | \begin{align}\sum\_{n=0}^{\infty}\frac{\binom{2n+1}{n+1}}{2^{2n+1}\,(n+x+1)}&= \int\_0^1\sum\_{n=0}^{\infty}\frac{\binom{2n+2}{n+1}y^{n+x}}{2^{2n+2}}\,{\rm d}y\\&=\int\_0^1 y^{x-1}\big((1-y)^{-1/2}-1\big){\rm d}y\\&=B\left(x,\frac12\right)-\frac1x\end{align}
and the rest follows from the properties of [beta function](h... | 9 | https://mathoverflow.net/users/7076 | 398338 | 164,391 |
https://mathoverflow.net/questions/398267 | 3 | Let $\omega\_{\text{WP}}$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $f : X \to Y$ be a surjective holomorphic map with connected fibres such that, over the regular locus, the fibres $X\_y$ of $f^{\circ} : X^{\circ} \to Y^{\circ}$ are smooth compact Kähler manifolds wi... | https://mathoverflow.net/users/105103 | Ricci curvature of the Weil-Petersson metric? | A positive answer to this question is given in part (2) of Theorem 3.1. of C.-L. Wang's paper [Curvature properties of the Calabi-Yau moduli](https://www.math.uni-bielefeld.de/documenta/vol-08/18.pdf).
The specific reference is:
Wang, C.-L., *Curvature properties of the Calabi-Yau moduli*, Documenta Math., **8** (200... | 0 | https://mathoverflow.net/users/105103 | 398340 | 164,393 |
https://mathoverflow.net/questions/398268 | 18 | The recent [article on Quanta](https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/)
(by Natalie Wolchover)
concerning $\aleph\_1$ vs. $\aleph\_2$ suggests that there is
excitement within that community:
>
> Juliette Kennedy: "It’s one of the most intellectua... | https://mathoverflow.net/users/6094 | Heat map of current mathematics | <https://paperscape.org/> is a 'heat map' of the arxiv if you color the graph by age. Unfortunately, its ability to detect links between mathematics papers is a bit lacking compared to physics papers for some reason, but it still gives a very interesting view of the subject.
| 23 | https://mathoverflow.net/users/152678 | 398347 | 164,395 |
https://mathoverflow.net/questions/398343 | 1 | Let $W$ be a standard Brownian motion on a probability space $(X, \mathcal F, \mathbb P)$ let and $\mathcal F\_t$ its natural filtration.
For $\varepsilon > 0, T \in [0, \infty)$ let $A\_{\varepsilon, t}$ be the event $\{\sup\_{s \in [0, T]} |W\_s|\ < \varepsilon\}$.
Let $X\_t$ be the solution to the SDE
$$dX\_t ... | https://mathoverflow.net/users/173490 | Is the integral against a Brownian motion conditioned to stay bounded a local martingale? | In case $\sigma = 1$ the claim is that the conditioned process itself is a martingale, however, as it paths are the paths of a brownian motion, it will have quadratic variation t, and therefore, it is an ordinary brownian motion.
| 2 | https://mathoverflow.net/users/143907 | 398348 | 164,396 |
https://mathoverflow.net/questions/398288 | 7 | Assume $X$ is a Tychonoff space. Then $A(X)$ is the free topological abelian group over $X$. I know that $A(X)$ is Hausdorff and the canonical embedding from $X$ to $A(X)$ is a topological embedding.
Now consider the subgroup $N:=2A(X)=\left\{g+g\ \colon\ g\in A(X)\right\}$.
The quotient $B(X):=A(X)/N$ is the free to... | https://mathoverflow.net/users/326011 | Why are free Boolean topological groups Hausdorff? | You can use the universality property with the following Boolean group as codomain: $B$ is the measure algebra over the unit interval (the quotient of the $\sigma$-algebra of Lebesgue measurable sets by the ideal of sets of measure zero), with symmetric difference as operation this is a Boolean group and $d(A,B)=\lambd... | 8 | https://mathoverflow.net/users/5903 | 398353 | 164,398 |
https://mathoverflow.net/questions/398375 | 2 | Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A\_X$ on $X$, the higher direct images $R^n f\_\* A\_X$ are locally constant, with stalk $H^n(F, A)$. This can be seen from the description of said higher direct images as the sheaf... | https://mathoverflow.net/users/173545 | Higher direct image with compact support of a constant sheaf | With the information added in the last update, the result is already proved once you realize that the unit of the adjunction
$$ id \to a\_{B,\*} a\_B^\* $$
is actually an isomorphism on the category of abelian groups if the space $B$ is connected.
| 1 | https://mathoverflow.net/users/173545 | 398379 | 164,405 |
https://mathoverflow.net/questions/398380 | 0 | For $s\in (0,1)$, is there are an explicit expression for
$$(-\Delta)^s \left(\frac{1}{\left(1+|x|^2\right)^s}\right)?$$
Edit: My goal is to show that for the function $u(x)=\frac{1}{\left(1+|x|^2\right)^s}$ defined on the ball $B(0,R)$ where $R>1$ we have $(-\Delta)^s u(x)\geq c(n,s) u^2(x)$ where $c(n,s)>0$ is cons... | https://mathoverflow.net/users/68232 | Explicit expression for the fractional Laplacian of $1/(1+|x|^2)^s$ | Unless I am making a typo, the result is:
$$ 2^{2s} \frac{\Gamma(\tfrac n2+s) \Gamma(2s)}{\Gamma(\tfrac n2)^2} {\_2F\_1}(\tfrac n2+s, 2s, \tfrac n2, -|x|^2), $$
where $n$ is the dimension and ${\_2F\_1}$ is the Gauss's hypergeometric function. See Table 1 on page 168 in my survey [1], or Corollary 2 in the original pap... | 4 | https://mathoverflow.net/users/108637 | 398382 | 164,406 |
https://mathoverflow.net/questions/398278 | 10 | It is well-known that the free symmetric monoidal category on one object is the [category $\mathbb{F}$ of finite sets and bijections](https://ncatlab.org/nlab/show/FinSet). This is supposed to be the categorification of the monoid of natural numbers, and its algebraic $K$-theory is given by the sphere spectrum $\mathbb... | https://mathoverflow.net/users/130058 | What is the free symmetric monoidal $\infty$-category on one object? | Yes, it is the same as $\mathbb{F}$.
As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints between groupoids and categories).
Symmetric monoidal $\infty$-groupoids are the same as $E\_\infty$-spaces (careful... | 17 | https://mathoverflow.net/users/184 | 398384 | 164,407 |
https://mathoverflow.net/questions/398265 | 5 | Picard $n$-groupoids are expected to model stable homotopy $n$-types. So far this has been proved for $n=1$ in
>
> Niles Johnson, Angélica M. Osorno, *Modeling stable one-types*. Theory Appl. Categ. 26 (2012), No. 20, 520–537; [arXiv:1201.2686](https://arxiv.org/abs/1201.2686).
>
>
>
and for $n=2$ in
>
> N... | https://mathoverflow.net/users/130058 | Categorical models for truncations of the sphere spectrum | I don't understand what you mean about the "directed sphere" so will focus on the other questions.
The free Picard $n$-category on one object has a description as a bordism $n$-category. Specifically it has:
* Objects are stably framed 0-manifolds;
* 1-Morphisms are stably framed 1-dimensional bordisms;
* 2-morphis... | 7 | https://mathoverflow.net/users/184 | 398385 | 164,408 |
https://mathoverflow.net/questions/397900 | 1 | Let $\log(1),\log(2),\log(3),\log(4)...\log(n)$ be approximated by fractions generated by the truncated sums:
$k=0$
$c=1$
$$\text{log1}=\sum\_{n=0}^k \frac{0}{(1 n+1)^c}=0$$
$$\text{log2}=\sum \_{n=0}^k \left(\frac{1}{(2 n+1)^c}-\frac{1}{(2 n+2)^c}\right)=\frac{1}{2}$$
$$\text{log3}=\sum \_{n=0}^k \left(\frac{1... | https://mathoverflow.net/users/25104 | Are the Riemann zeta zeros of the form $-\text{integer } i \pi +\log \left(\text{polynomial root}\right)$? | The actual question asked here is a *Mathematica* question, so this is not really the right site for it, but here goes. Suppressing the extraneous notation in $c$ and $k$, the expression in $s$ is
$$
-e^{-13 s/12}+e^{-5 s/6}-e^{-s/2}+1
$$
which is a polynomial of degree 13 in $x=e^{-s/12}$:
$$
- x^{13} + x^{10} - x^6+1... | 6 | https://mathoverflow.net/users/6756 | 398389 | 164,409 |
https://mathoverflow.net/questions/398388 | 1 | The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of applications of simple *modules* over some ring/algebra $A$, but I can barely know an application of them for finite simple g... | https://mathoverflow.net/users/146933 | Why are finite simple groups useful? | There's an entire book on this subject, "Applying the Classification of Finite Simple Groups: A User’s Guide" by Stephen D. Smith, published through the AMS, though you can find a draft version [here](http://homepages.math.uic.edu/%7Esmiths/book.pdf).
The applications are not as simple as they are for modules, but ma... | 9 | https://mathoverflow.net/users/3711 | 398393 | 164,411 |
https://mathoverflow.net/questions/398263 | 5 | Definition
==========
Call a [partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) $\lambda$ of an even integer $2n$ "black-white balanced" if the following equivalent conditions are satisfied:
* In the usual (Ferrers-)[Young diagram](https://en.wikipedia.org/wiki/Young_tableau#Diagrams) of $\lambda$... | https://mathoverflow.net/users/1079 | Bijection from "black-white balanced" partitions to pairs of partitions | [Richard Stanley](https://mathoverflow.net/users/2807/richard-stanley) and [Sam Hopkins](https://mathoverflow.net/users/25028/sam-hopkins) have answered this question in the comments. (Thanks!)
Professor Stanley mentions that this is a special case of exercise 7.59(e) in *Enumerative Combinatorics*, volume 2:
>
>... | 3 | https://mathoverflow.net/users/1079 | 398399 | 164,417 |
https://mathoverflow.net/questions/398386 | 3 | Let $A$ be a pre-$C^\*$-algebra, i.e. $A$ satisfies all axioms for a $C^\*$-algebra except completeness. In other words, $A$ is an involutive algebra with a $C^\*$-norm.
We say that $x \in A$ is positive (notation: $x \ge 0$) if there exists $a\in A$ with $x = a^\*a$ and on self-adjoint elements we define the usual r... | https://mathoverflow.net/users/216007 | The inequality $a^*ca \le \|c\| a^*a$ in a pre-$C^*$-algebra | Given your definitions, the first one is an easy no. If $a = 1$ then it says $c \leq \|c\|$, which fails when $A$ is the polynomials in $C[0,1]$: let $c$ be the polynomial $x$, then $\|c\|= 1$ and $1-x$ is not of the form $p\bar{p}$ for any polynomial $p$.
For the second question, take $a = x^2 + y^2$ and $b = z^2$ i... | 9 | https://mathoverflow.net/users/23141 | 398410 | 164,426 |
https://mathoverflow.net/questions/398360 | 6 | I'm interested in finding solutions a fourth order version of the standard wave equation in $d$ dimensional Minkowski spacetime $\mathcal{M}^d$. Defining $\Box := \partial\_0^2 - \sum\_{i = 1}^{d-1} \partial\_{i}^2$, I want to find solutions to $$\Box^2\Phi(x) = 0,$$ which are not also solutions to $\Box \Phi(x) = 0$, ... | https://mathoverflow.net/users/171026 | Space of solutions to a fourth order wave equation | You talk about the non-separability of the $\Box^2 \phi = 0$ equation, which I don't understand. Each plane wave $e^{ik\cdot x} = \prod\_{j=0}^{d-1} e^{i k\_j x^j}$ is already in separated form with the components of the null vector $k=(k\_j)$ playing the role of the separation constants.
But rather than get into tec... | 3 | https://mathoverflow.net/users/2622 | 398411 | 164,427 |
https://mathoverflow.net/questions/398436 | 2 | The intersection of (countably many) 'spheres' in a Hilbert space can be non-empty. If we make this situation moving real analytically, the mid points and the radii, can it happen that the intersection becomes empty while it is not empty on an open set of the parameter space?
Precisely:
Let $H$ be a (separable) com... | https://mathoverflow.net/users/109905 | Intersection of 'spheres' in Hilbert space with respect to real analytically moving mid points | **This is my *new* answer for the edited question**
Here is a counterexample. Let $H = \mathbb{C}^2$ and let $z\_1(t) = (1,t)$ and $z\_i(t) = (2,2)$ for all $i > 1$ and for all $t$. Also let $r\_i(t) = 1$ for all $i$ and $t$.
Then $M\_t$ is the set of points $(z,w) \in \mathbb{C}^2$ such that $z + tw = 1$ and $2z +... | 1 | https://mathoverflow.net/users/175976 | 398439 | 164,438 |
https://mathoverflow.net/questions/386213 | 13 |
>
> It should be the case that, in some appropriate sense
> $$\pi (x)\sim \operatorname{Ri}(x)-\sum\_{\rho}\operatorname{Ri}(x^{\rho}) \tag\*{(4)}$$
> with $\operatorname{Ri}$ denoting the *Riemann function* defined:
> $$\operatorname{Ri}(x)=\sum\_{m=1}^\infty \frac{\mu (m)}{m}\operatorname{li}\left(x^{\frac{1}{m}}\r... | https://mathoverflow.net/users/175751 | Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all? | *Preliminaries.* We will denote the slight modifications of the prime counting function and the prime power counting function with $\pi^\*$ and $J^\*$ respectively, which assume a halfway step at discontinuities. Furthermore we will denote the non-trivial zeros of the zeta function with $\rho$ and all zeros i.e. the tr... | 11 | https://mathoverflow.net/users/175826 | 398456 | 164,444 |
https://mathoverflow.net/questions/398461 | 0 | Let $M$ be a monoid, and let $x\in M$. One says that $x$ is *periodic* if
$$x^{i+j}=x^j$$
for some integers $i\geq 1$ and $j\geq 0$.
An easy division algorithm argument shows that if $m$ is the smallest value of $i$ where this happens (for some $j$), and similarly $n$ is the smallest value of $j$ where this happens (... | https://mathoverflow.net/users/3199 | More vocabulary for periodic elements in monoids | Question 1: See Clifford and Preston, volume 1.
Question 2: $m$ is the period, $n$ is called the index of the element. See this [Wikipedia text](https://en.wikipedia.org/wiki/Monogenic_semigroup).
Question 3: If $n=1$, the element is called a group element of finite order. If $n=0$, it is called a unit of finite or... | 2 | https://mathoverflow.net/users/157261 | 398462 | 164,446 |
https://mathoverflow.net/questions/398363 | 10 | I want to prove that weak descent of a $1$-category implies the classical Giraud axioms.
More precisely, let $\mathsf{C}$ be a cocomplete, finitely complete $1$-category. We say that $\mathsf{C}$ satisfies weak descent if the following conditions are satisfied:
* $(\mathbf{D1}a)$-(Universal coproducts): Given a col... | https://mathoverflow.net/users/124010 | Weak descent and effective equivalence relations | I find it a bit surprising, but I think you are correct. The proof I have is maybe a little too long for MO, so I'm only sketching it, but I'll be happy to provide more details if needed.
First one observe a form of "weak descent" for coequalizer diagram:
**Lemma:** Assume that the category $C$ satistifes all four ... | 5 | https://mathoverflow.net/users/22131 | 398463 | 164,447 |
https://mathoverflow.net/questions/397925 | 8 | I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42].
>
> It is a well-known fact that a weakly converging sequence of stationary integral currents may have a limit which is not a stationary current.
>
>
>
**Question.** How should I interpret this quote? What does L... | https://mathoverflow.net/users/103792 | How to interpret this quote of Lin? | I believe the following sequence demonstrates the failure of flat limits to be stationary. This would be consistent with the natural interpretation of the quote, meaning: a current $T$ is called *stationary* if the varifold $\lvert T \rvert$ is.
(A quick side remark before the construction: on second thought whether ... | 2 | https://mathoverflow.net/users/103792 | 398464 | 164,448 |
https://mathoverflow.net/questions/398449 | 1 | I am trying to study the asymptotic behavior of $k^{th}$ order statistic of $n$ i.i.d chi-square distribution. Let $X\_1, \cdots , X\_n$ be i.i.d $\chi^2\_1$ random variables and $X\_{(k:n)}$ be the $k^{th}$ order statistic of these random variables. I do know that,
$$
\frac{X\_{(n:n)}}{\log n} \overset{p}{\to} 2\; \te... | https://mathoverflow.net/users/151115 | $k^{\text{th}}$ maxima of $n$ i.i.d chi-square random variables | $\newcommand{\ep}{\varepsilon}
\newcommand{\pp}{\overset p\to}$Let $Y\_k:=X\_{(k:n)}$, where $n-1\ge k\sim n$. The correct asymptotics for $Y\_k$ is as follows:
\begin{equation\*}
Y\_k/l\_{n,k}\pp2,\tag{1}
\end{equation\*}
where
\begin{equation\*}
l\_{n,k}:=\ln\frac n{n-k},
\end{equation\*}
so that $l\_{n,k}\to\inft... | 1 | https://mathoverflow.net/users/36721 | 398466 | 164,449 |
https://mathoverflow.net/questions/398457 | 0 | Suppose that I have 2 CW-complexes $A\subset B $ such chat thé inclusions is a retract.
Let $H\_{\ast}$ be the ordinary homology with integral coefficients.
Let
$$\mu: B\times B \rightarrow B $$
be continiuous map such that:
* the restriction map $\mu:A\times A\rightarrow A$ is well defined.
* There exists an eleme... | https://mathoverflow.net/users/141114 | Retractions, homology and multiplication | The role of $C$ in the problem is irrelevant.
Let $r : C \to B$ be a retraction, and set
$$m := r\circ \mu : B \times B \to B .
$$ Set $\ast := b$ and think of it as the basepoint of $B$. Then the restriction of $m$ to the wedge $B\vee\_\ast B$ is the fold map $B \vee\_\ast B\to B$ (so $B$ is an $H$-space).
Since $... | 2 | https://mathoverflow.net/users/8032 | 398470 | 164,451 |
https://mathoverflow.net/questions/398020 | 7 | I'm looking at chapter 4 of Waterhouse's *"Abelian varieties over finite fields"*; and Theorems 4.1 and 4.2 seem to use the following fact:
>
> Suppose that $E/\mathbb{F}\_q$ is an elliptic curve over a finite field with $q=p^n$ elements and let $\pi\_E$ denote the $q$th-power Frobenius map acting on $E$. Suppose t... | https://mathoverflow.net/users/103423 | Supersingular curves over $\mathbb{F}_q$ and the splitting of $p$ | Here is a solution that avoids explicit use of $\mathbf{Q}\_p$ and in particular does not require knowing that $(\operatorname{End} E) \otimes \mathbf{Q}\_p$ is a division ring. The key is to use *inseparable degree of endomorphisms*.
Identify $\alpha$ with $\pi\_E$. Let $a = \operatorname{tr} \alpha \in \mathbf{Z}$.... | 5 | https://mathoverflow.net/users/2757 | 398482 | 164,459 |
https://mathoverflow.net/questions/398505 | 2 | Let us recall this fact. Let $G$ be a semisimple algebraic group over $\mathbb C$ and let $V,V'$ be two irreducible $G$-representations. We denote by $X,X'$ the unique closed $G$-orbits contained in $\mathbb P V, \mathbb P V'$ respectively. We know that if
$$
\mathbb P V \supset X \cong X' \subset \mathbb P V'
$$
as pr... | https://mathoverflow.net/users/147236 | Do representations of same dimension implies isomorphic closed orbits? | No, this is not true. For instance the symplectic group
$$
G = \mathrm{Sp}(V),
$$
where $V$ is a symplectic vector space of dimension 6
has two irreducible representations of dimension 14:
$$
V\_1 = \wedge^2V^\vee / \langle \omega \rangle,
\qquad\text{and}\qquad
V\_2 = \wedge^3V^\vee / (V^\vee \wedge \omega),
$$
where... | 4 | https://mathoverflow.net/users/4428 | 398506 | 164,464 |
https://mathoverflow.net/questions/398481 | 3 | Let $\mathfrak{A}$ be a C\*-algebra, and let $\phi \in \mathfrak{A}^\*$ be a self-adjoint bounded linear functional on $\mathfrak{A}$. Then there exists a unique pair $\phi^+, \phi^-$ of positive bounded linear functionals on $\mathfrak{A}$ such that $\phi = \phi^+ - \phi^-$ and $\| \phi \| = \left\| \phi^+ \right\| - ... | https://mathoverflow.net/users/62469 | Jordan decomposition of tracial functionals on a C*-algebra | Assuming $\mathfrak{A}$ is unital, every algebra element is a linear combination of unitaries, so a linear functional being tracial is equivalent to $\phi = \phi \circ \operatorname{Ad} u$ for every unitary $u$. In this case, for every unitary $u$ we have
$$
\phi = \phi \circ \operatorname{Ad} u = \phi^+ \circ \opera... | 4 | https://mathoverflow.net/users/2085 | 398507 | 164,465 |
https://mathoverflow.net/questions/398519 | 0 | For any set $X$ we let $[X]^2 = \big\{\{x, y\}: x\neq y \in X\big\}$. Consider the following statement:
>
> (S) : If $G =(V,E)$ is a simple, undirected graph such that $E \neq [X]^2$, then there is $e^\* \in [X]^2 \setminus E$ such that $G \not \cong (V, E\cup\{e^\*\})$.
>
>
>
For finite graphs, (S) is true si... | https://mathoverflow.net/users/8628 | Edge-adding conjecture for graphs | No - for a countexample, take the disjoint union of all possible finite graphs, each repeated countably many times.
There is a similar counterexample even if we ask that the graph be connected - just add all possible edges between vertices from different 'copies'.
| 5 | https://mathoverflow.net/users/385 | 398521 | 164,468 |
https://mathoverflow.net/questions/398526 | 2 | In a [posting](https://math.stackexchange.com/questions/4206809/is-countability-absolute-over-supertransitive-models-of-set-theory?noredirect=1#comment8734837_4206809) to mathstackexchange I've alluded to the concept of supertransitive model. Now $M$ is a supertransitive model of a set $Q$ of first order sentences, den... | https://mathoverflow.net/users/95347 | Is there a lower bound on the size of a supertransitive model of ZFC? | If $M$ is supertransitive and satisfies $\sf ZFC$, then $\omega\in M$, and more importantly, $V\_\omega\in M$.
Now by recursion, if $\alpha$ is an ordinal in $M$, then $V\_\alpha\in M$ as well.
Therefore $M$ must agree on the $V\_\alpha$ hierarchy, and therefore it must have the form $V\_\kappa$ for a worldly cardi... | 6 | https://mathoverflow.net/users/7206 | 398527 | 164,470 |
https://mathoverflow.net/questions/398533 | 2 | Let $R$ be a PID with field of fraction $K$.
Let $L$ be a lattice with non-degenerate quadratic form $q:L\times L \to R$.
Let
$$
L^\* = \{x \in L\otimes K \text{ s.t. } q(x,l) \in R \text{ for all } l \in L \}.
$$
By integrality of $q$, we have $L \subseteq L^\*$.
I heard the following
**Claim.** The unique decomposi... | https://mathoverflow.net/users/148575 | Reference request: Given a non-degenerate integral quadratic lattice $L,q$ over a PID, the quotient $L^*/L$ is given by SNF of $q$ | The following more general statement is easier to prove:
Let $L\_1, L\_2$ be lattices with a nondegenerate bilinear form $b: L\_1 \times L\_2 \to R$. Let $$L\_1^\* = \{ x \in L\_2 \otimes K \textrm{ s.t. } b(l,x) \in R \textrm{ for all } l \in L\_1 \}$$
Claim: The unique decomposition of the quotient $L\_1^\*/L\_2$... | 3 | https://mathoverflow.net/users/18060 | 398536 | 164,474 |
https://mathoverflow.net/questions/398525 | 5 | Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X\_A := f^{-1}(A)$.
Let $D\subset X$ be a divisor such that $D\_{|X\_A}$ (the restriction of $D$ to $X\_A$) is big for $A\in H^0(Y,\mathca... | https://mathoverflow.net/users/nan | Divisors whose restriction is big | Consider a product $X = \mathbb{P}^n\times\mathbb{P}^1$, with projections $g:X\rightarrow\mathbb{P}^n$ and $f:X\rightarrow\mathbb{P}^1$.
Set $H\_1:= g^{\*}\mathcal{O}\_{\mathbb{P}^n}(1)$ and $H\_2:= f^{\*}\mathcal{O}\_{\mathbb{P}^1}(1)$. The effective cone of $X$ is closed and generated by $H\_1,H\_2$.
Now, take a ... | 4 | https://mathoverflow.net/users/14514 | 398537 | 164,475 |
https://mathoverflow.net/questions/398414 | 24 | For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is **$\mathcal{X}$-detectable** iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary predicate symbol $U$ such that for all $A\subseteq X$, the expansion of $\mathcal{X}$ gotten by interpreting $U$ as $A\sub... | https://mathoverflow.net/users/8133 | Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$ | For the first question (distinct regular cardinals $>\aleph\_1$): Force ZFC + MA + $2^{\aleph\_0}=\aleph\_3$ over $L$ in the usual way (see Jech, Theorem 16.13; note the forcing is ccc and it forces MA + $2^{\aleph\_0}=\aleph\_3$, which is all we need here). Then in $L[G]$, $\aleph\_2$ and $\aleph\_3$ are both $\mathca... | 13 | https://mathoverflow.net/users/160347 | 398547 | 164,476 |
https://mathoverflow.net/questions/398371 | 6 | **Notation.**
1. $\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints.
2. $\mathsf{Topoi}\_{\text{ess}}$ is the bicategory of topoi, essential geometric morphisms and natural transformations between left adjoints.
3. $\mathsf{Presh}$ is the full subcategor... | https://mathoverflow.net/users/104432 | Stability properties of essential geometric morphisms | This is only a partial answer. With '(co)limit' I will always mean pseudo(co)limit.
1. If $\mathcal{C}$ and $\mathcal{D}$ are Cauchy-complete, then the category of essential geometric morphisms $\mathbf{PSh}(\mathcal{C}) \to \mathbf{PSh}(\mathcal{D})$ (and geometric transformations between them) is equivalent to the ... | 5 | https://mathoverflow.net/users/37368 | 398558 | 164,480 |
https://mathoverflow.net/questions/398176 | 3 | What is the generating function of $f\_{m,n}$?
$ f\_{m,n} = \begin{cases} 0 , & \text{if $m<0 $ or $ n<0$ }; \\
f\_{n,m} , & \text{ if $n<m$}; \\
1, & \text{ if $0=m$ and $ n\in\{0,1\} $}; \\
f\_{0 ,n-1}+ f\_{1,n-1}, & \text{ if $0=m$ and $ n>1 $}; \\ f\_{m-1 ,n}+ f\_{m ,n-1}+ f\_{m-1,n-1}, & \text{ if $0<m\in \{n... | https://mathoverflow.net/users/168671 | A generating function related to the Delannoy numbers | Consider the following generating function:
$$F(x,y) := \sum\_{n=0}^\infty \sum\_{m=0}^n f\_{m,n} x^n y^m$$
and its diagonal
$$D(z) := \sum\_{n=0}^\infty f\_{n,n} z^n.$$
Then the recurrence implies the following functional equation:
$$2(1-x-y-xy)F(x,y) - 2x\frac{F(x,y)-F(x,0)}y = 1 + (1-xy-2y)D(xy).$$
I'm not yet sure ... | 2 | https://mathoverflow.net/users/7076 | 400574 | 164,485 |
https://mathoverflow.net/questions/398572 | 4 | In the prominent examples of weak double categories (or bicategories), the weak composition is typically defined by a universal property (specifically, it's usually the tensor product of some kind of bimodules), and as such it's not a uniquely designated arrow.
However, the currently accepted definitions also requir... | https://mathoverflow.net/users/44339 | Proper weakness condition of double categories | There's a general theory of this sort of "replacing algebraic structure by universal properties", called [generalized multicategories](https://ncatlab.org/nlab/show/generalized+multicategory). The simplest case is for an ordinary monoidal category, in which case the corresponding "virtual" structure is a [multicategory... | 6 | https://mathoverflow.net/users/49 | 400585 | 164,488 |
https://mathoverflow.net/questions/398275 | 1 | Let $R\_{\theta}$ be the rotation by an angle $\theta$.
Is it then true that for multi-indices $\alpha$ of fixed order $j$ and any smooth function $f$ we have
$$\sum\_{\vert \alpha \vert=j}(R\_{\theta}z)^{\alpha} \partial^{\alpha}f(x) = \sum\_{\vert \alpha \vert=j}(z)^{\alpha} (R\_{-\theta}\partial)^{\alpha}f(x)$$
... | https://mathoverflow.net/users/325814 | Generalized adjoint operation valid? | If I am understanding your notation correctly (which admittedly, I may not be), I believe this is already false for $j = 2$. Let me write:
$$ R\_{\theta} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \qquad f = f(x,y) \qquad z = \begin{bmatrix} u \\ v \end{bmatrix} $$
Then for $j =... | 0 | https://mathoverflow.net/users/61479 | 400589 | 164,490 |
https://mathoverflow.net/questions/400586 | 4 | Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers. We can consider different moduli spaces related to those. On one hand we have:
-$M\_{Dol}$ the moduli space of stable Higgs field of rank $n$ and degree $d$
-$M\_B$ moduli space of (twisted) representations of fundamental group of $\S... | https://mathoverflow.net/users/146464 | Explicit example de Rham moduli space of connections | Yes, there is a description of the rank 1 de Rham moduli space, and you can find it for example in Goldman's notes: Higgs Bundles and Geometric Structures on Surfaces.
The deRham moduli space is given by an affine holomorphic bundle over the Jacobian with underlying holomorphic bundle the cotangent bundle, where the ... | 6 | https://mathoverflow.net/users/4572 | 400590 | 164,491 |
https://mathoverflow.net/questions/398058 | 0 | [The collecting numbers](https://cses.fi/problemset/task/2216) problem in the CSES problem set has a [greedy solution](https://www.youtube.com/watch?v=lhhHCZYoJvs) where we compare the position of a number x with the position of x-1. If pos(x) < pos(x-1) then we increment rounds because this means that x can't be in th... | https://mathoverflow.net/users/323644 | How does the greedy algorithm for CSES problem collecting numbers work? | I was actually confused about the problem itself. The problem stated that we had to collect numbers in an increasing order in each round. I assumed that they didn't have to be consecutive. For ex: in one round, I could have [2, 4]
But this is not the case. In the above example, I will have to pick 3 only after 2 in t... | 0 | https://mathoverflow.net/users/323644 | 400601 | 164,495 |
https://mathoverflow.net/questions/400604 | 11 | Let $M$ be a closed connected smooth manifold. We define the degree of symmetry of $M$ by $N(M):=\sup\_\limits{g}\mathrm{dim}\,\mathrm{Isom}(M,g)$, where $g$ is a smooth Riemannian metric on $M$ and $\mathrm{Isom}$ is the isometry group of the Riemannian manifold $(M,g)$.
The torus $T^n$ does not admit a Riemannian m... | https://mathoverflow.net/users/90512 | Scalar curvature and the degree of symmetry | It seems that there are examples. By a theorem of Gromov and Lawson every simply connected manifold of dimension $n \geq 5$ which is not spin admits a metric of positive scalar curvature.
There are many examples of simply connected, non-spin, closed $6$-manifolds which cannot admit a smooth circle action, constructed... | 18 | https://mathoverflow.net/users/99732 | 400605 | 164,496 |
https://mathoverflow.net/questions/398541 | 13 | Thirty or so years ago, someone showed me a clever proof of the Fundamental Theorem of Arithmetic that did not make use of the lemma "If $p\mid ab$ then $p\mid a$ or $p\mid b$". I'm unable to reconstruct the argument; all I remember is that it used induction and that it didn't generalize to other number rings. Can anyo... | https://mathoverflow.net/users/3621 | Nonstandard proofs of the fundamental theorem of arithmetic | To summarize the comments, this is also known as Zermelo's proof. A version can be found on [wikipedia](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic#Uniqueness_without_Euclid%27s_lemma). I will give the proof here to avoid link rot.
The proof is by contradiction. If FTA did not hold, then use the w... | 21 | https://mathoverflow.net/users/1106 | 400607 | 164,497 |
https://mathoverflow.net/questions/400606 | 7 | Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\sigma$-algebra of $X$ is said to be ergodic for $\varphi$ if it is $\varphi$-invariant, i.e.,
$$\int\_X f \, d\mu = \... | https://mathoverflow.net/users/169719 | Count of non-trivial ergodic measures of a topological dynamical system | Suppose $X$ is the unit circle and $\varphi$ is the doubling map (multiplicatively, $X = \{ z\in \mathbb{C} : |z| = 1\}$ and $\varphi(z) = z^2$, or additively, $X = \mathbb{R}/\mathbb{Z}$ and $\varphi(x) = 2x \mod \mathbb{Z}$). Then the set of ergodic measures is uncountable and in fact is path-connected and dense in t... | 8 | https://mathoverflow.net/users/5701 | 400609 | 164,499 |
https://mathoverflow.net/questions/400613 | 4 | I was interested in the question of figuring out the rational homotopy type of mapping spaces (regular or rational) between two algebraic varieties over $\mathbb{C}$. I encountered the following paper ["Contractibility of the space of rational maps"](http://people.math.harvard.edu/%7Egaitsgde/GL/Contractibility.pdf). T... | https://mathoverflow.net/users/127776 | Rational homotopy type of rational mapping spaces | In Gaitsgory's setup, $Rat(X,Y)$ is not really a topological space (its an algebraic prestack). If you want to think about it topologically, then it is probably best to think of its associated homotopy type.
By definition, this homotopy type obtained as a homotopy colimit over the category $fset^{op}$ of finite sets ... | 3 | https://mathoverflow.net/users/131945 | 400616 | 164,500 |
https://mathoverflow.net/questions/400621 | 0 | Given 2 finite sets $S$ and $M$, with $\operatorname{card}(S) \geq \operatorname{card}(M)$, and an item $z \not\in M$. There is an unknown function $f: S \to M \cup \{z\}$, which is known to be one-to-one for all $s \in S$ for which $f(s) \in M$ (i.e. for which $f(s) \neq z$). The goal is to find $f$. To this end, I ca... | https://mathoverflow.net/users/332154 | Mapping problem reminiscent of Mastermind | Since $S$ is finite, we can label its elements $s\_1, s\_2, \ldots, s\_n$. Then take as query sets $S\_1 = \{ s\_i \mid i \,\&\, 1 = 1 \}$ where $\&$ represents bitwise conjunction; $S\_2 = \{ s\_i \mid i \,\&\, 2 = 2 \}$, $S\_k = \{ s\_i \mid i \,\&\, 2^{k-1} = 2^{k-1} \}$. This gives $\lceil \lg \operatorname{card}(S... | 1 | https://mathoverflow.net/users/46140 | 400629 | 164,504 |
https://mathoverflow.net/questions/398257 | 10 | Let $G=F\_2$ be the free group of rank $2$. Is there a constant $c>0$ such that the word length $|[u,v]|$ of every commutator $[u,v]=uvu^{-1}v^{-1}$ where $u,v\in G$, $|u|,|v|>0$ is at least $c(|u|+|v|)$ unless $[u,v]=[u\_1,v\_1]$ for some $u\_1,v\_1$ with $|u\_1|+|v\_1|<|u|+|v|$?
| https://mathoverflow.net/users/157261 | Length of commutators in the free group | Victor Guba sent me a proof. The proof is based on the old result by Wicks, Wicks, N. J. Commutators in free products. J. London Math. Soc. 37 (1962), 433–444., which describes all words which are commutators in a free group (Lemma 5 in the paper). By that result, a word is a commutator iff it is a conjugate of a reduc... | 7 | https://mathoverflow.net/users/157261 | 400630 | 164,505 |
https://mathoverflow.net/questions/400634 | 15 | Green and Tao's version of Freiman's theorem over finite fields ([doi:10.1017/S0963548309009821](https://www.doi.org/10.1017/S0963548309009821)) is as follows:
If $A$ is a set in $\mathbb{F}\_2^n$ for which $|A+A| \leqslant K|A|$, then $A$ is contained in a subspace of size $2^{2K+O(\sqrt{K}\log(K))}|A|$.
Does anyb... | https://mathoverflow.net/users/332227 | Explicit constant in Green/Tao's version of Freiman's Theorem? | Even-Zohar (On sums of generating sets in $\mathbb{Z}\_2^n$, Combin. Probab. Comput. 21 (2012), no. 6, 916–941, available at <https://arxiv.org/abs/1108.4902>) has proved a completely explicit and sharp version of Frieman's theorem in $\mathbb{F}\_2^n$ - see Theorem 2 of that paper.
As a corollary, one gets that if $... | 16 | https://mathoverflow.net/users/385 | 400640 | 164,509 |
https://mathoverflow.net/questions/400622 | 4 | Suppose I have a topological space $X$ with a collection of closed subsets $X\_\tau$ for $\tau \in P$ where I think of $P$ as a poset with $\tau \leq \lambda \iff X\_\tau \subset X\_\lambda$. Is there some nice (algorithmic?) way to get information about the (compactly supported) cohomology of $U\_{\tau} = X\_\tau - \b... | https://mathoverflow.net/users/58001 | Can I reconstruct the cohomology from a collection of open sets? | I wrote a paper on precisely this question:
[A spectral sequence for stratified spaces and configuration spaces of points.](https://msp.org/gt/2017/21-4/p16.xhtml)
Geom. Topol. 21 (2017), no. 4, 2527–2555.
| 7 | https://mathoverflow.net/users/1310 | 400645 | 164,511 |
https://mathoverflow.net/questions/400644 | 2 | **Motivation.** I was trying to prove that whenever $G$ is a simple, undirected graph and $\kappa< \chi(G)$ is a cardinal, then there is an induced subgraph with chromatic number exactly $\kappa$. This is easy to do when $\chi(G)$ is finite. For $\chi(G)$ infinite, my strategy is to consider the collection of induced s... | https://mathoverflow.net/users/8628 | Is the union of a chain of $\kappa$-colorable subgraphs $\kappa$-colorable? | For a counterexample, let $G$ be the complete graph on the ordinal $\omega\_1$, let the $W$'s be the countable ordinals, and let $\kappa=\aleph\_0$.
| 9 | https://mathoverflow.net/users/75735 | 400647 | 164,512 |
https://mathoverflow.net/questions/400648 | 2 | Over projective space, it is well-known that given a graded $S^\bullet$-module $M\_\bullet$, where $S^\bullet = k[x\_0, \dots, x\_N]$, there is a unique minimal free resolution
$$
\cdots \to \bigoplus\_q S^\bullet(-q) \otimes B\_{p, q} \to \cdots \to \bigoplus\_q S^\bullet(-q) \otimes B\_{1, q} \to \bigoplus\_q S^\bul... | https://mathoverflow.net/users/129738 | Minimal free resolution over arbitrary varieties | This is not a good definition, because the complex $\mathcal{E}^\bullet \otimes\_{\mathcal{O}\_{X,x}} \kappa(x)$ computes $\mathrm{Tor}\_i^{\mathcal{O}\_{X,x}}(\mathcal{F}, \kappa(x))$, so if this complex has zero differential and at least two terms, it follows that
$$
\mathrm{Tor}\_1^{\mathcal{O}\_{X,x}}(\mathcal{F}, ... | 4 | https://mathoverflow.net/users/4428 | 400650 | 164,513 |
https://mathoverflow.net/questions/398476 | 9 | Let $M$ be an open orientable three-manifold such that $\pi\_1 (M)$ is isomorphic to the fundamental group of a closed orientable surface $S\ncong \mathbb{S}^2$. Furthermore, suppose that $\tilde{M} \cong \mathbb{R}^3$. Is it true that $M \cong S \times \mathbb{R}$?
Without making any assumptions about $\tilde{M}$, t... | https://mathoverflow.net/users/148805 | Noncompact three-manifold with fundamental group isomorphic to a surface group | I'll restate the question for the convenience of the reader.$\newcommand{\RR}{\mathbb{R}}$
>
> Suppose that $M$ is a non-compact, connected, oriented three-manifold without boundary, with universal cover homeomorphic to $\RR^3$. Suppose that $F$ is a compact, connected, oriented surface with genus at least one. Sup... | 8 | https://mathoverflow.net/users/1650 | 400657 | 164,515 |
https://mathoverflow.net/questions/398528 | 4 | Let $M$ and $N$ be von Neumann algebras, and $\mathcal{H}$ a cyclic $M-N$ correspondence with unit cyclic vector $\xi$. For which $\eta\in \mathcal{H}$ is the bimodule map extending $\xi\mapsto \eta$ a closable operator on $\mathcal{H}$? Is it closable for every $\eta$?
| https://mathoverflow.net/users/6269 | Closability of a natural bimodule map between cyclic correspondences of von Neumann algebras | Here is a down-to-Earth counter-example. Set $N=\mathbb C$ and $M=\mathcal B(K)$ with $H=K\otimes K$ and $M$ acting on the first tensor factor (should perhaps be $K\otimes\overline K$ but this will not affect the argument). Let $K$ have orthonormal basis $(e\_n)$ and take for example $\xi = \sum\_n n^{-1} e\_n\otimes e... | 1 | https://mathoverflow.net/users/406 | 400659 | 164,516 |
https://mathoverflow.net/questions/400667 | 2 | Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure.
Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\overline{S}$ at $p$. Is it possible to chose the point $p$ in such a way that $\overline{X}$ is the algebraic closure... | https://mathoverflow.net/users/nan | Blow-ups of surfaces over a field | The answer is yes for some $S$ and no for other $S$. I think the answer is no for most $S$ for a reasonable definition of "most".
For a positive example, take $S$ to be $\mathbb P^2$ blown up at a point, of Picard rank $2$. Then $\overline{X}$ will be $\mathbb P^2$ blown up at two points, which is the base change to ... | 2 | https://mathoverflow.net/users/18060 | 400671 | 164,518 |
https://mathoverflow.net/questions/400678 | 1 | If $B\in\mathbb{R}^{n\times e}$ is the incidence matrix corresponding to a graph with $n$ vertices and $e$ edges, we know that $BB^T\in\mathbb{R}^{n\times n}$ is the graph Laplacian matrix.
I am curious that if there is any special meaning of the matrix $B^TB\in\mathbb{R}^{e\times e}$.
| https://mathoverflow.net/users/178204 | Incidence matrix in a graph, meaning of $B^TB$ | Well, "special" is maybe an overstatement, because the meaning is really boring: If we consider edges of an undirected graph as two-element subsets of its vertex set, then
$$(B^T B)\_{e,e'} = \sum\_v B\_{v,e} B\_{v,e'} = \#\{v | v \text{ is incident to $e$ and $e'$}\}=|e\cap e'|$$
which is $2\cdot 1\_{n\times n}+$ the ... | 4 | https://mathoverflow.net/users/3041 | 400679 | 164,521 |
https://mathoverflow.net/questions/400649 | 8 | Let $G = (V,E)$ be a simple, undirected graph with $\chi(G)$ infinite. Given a cardinal $\kappa$ with $0 < \kappa < \chi(G)$, is there an induced subgraph $S$ of $G$ with $\chi(S) = \kappa$?
**What I tried:** Let ${\cal S}$ be the collection of all subgraphs of $G$ colorable with $\kappa$ colors. I hoped to find a ma... | https://mathoverflow.net/users/8628 | Induced subgraphs of any given smaller chromatic number | Not necessarily. Komjáth showed that it is consistent that there is a graph of chromatic number $\aleph\_2$ which does not have a subgraph (not just induced) of chromatic number $\aleph\_1$. See P. Komjáth, Consistency results on infinite graphs, Israel J. Math., 61 (1988), pp. 285-294.
| 15 | https://mathoverflow.net/users/332589 | 400684 | 164,522 |
https://mathoverflow.net/questions/398430 | 1 | According to [Riemann surfaces, dynamics and
geometry](http://people.math.harvard.edu/%7Ectm/home/text/class/harvard/275/09/html/base/rs/rs.pdf) by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by
$$
\|\phi\|\_p = \left(\int\_X \rho^{2-2p} |\phi|^p\righ... | https://mathoverflow.net/users/121404 | Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials | The powers of $\rho$ are necessary to make the integrals well-defined. In probably excessive detail:
1. **Because $X$ is a Riemann surface, the integrands here need to be $(1,1)$-forms.** This is just the familiar fact (change of variables theorem) that integrals over (oriented) manifolds are only well-defined on obj... | 3 | https://mathoverflow.net/users/174177 | 400692 | 164,525 |
https://mathoverflow.net/questions/400680 | 12 | Suppose that $\{I\_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, we have
\begin{equation}
\Big\| \Big( \sum\_{m} |(\hat{f}\chi\_{I\_m})^{\vee}|^2 \Big)^{1/2} \Big\|\_{L^p}\lesssim \|f\|\... | https://mathoverflow.net/users/69086 | A generalization of Rubio de Francia's inequality | The answer is "no" for any $p<2$ (obviously the inequality holds for $p=2$), but the construction I have is rather indirect (analogous to how the Hardy-Littlewood majorant conjecture is disproved, see e.g., [this paper](https://arxiv.org/abs/1203.2378)). A shame, because restriction theory (and other related areas of h... | 13 | https://mathoverflow.net/users/766 | 400694 | 164,526 |
https://mathoverflow.net/questions/400668 | 4 | The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.
In the plane let $v\_i$ be $n$ unit vectors no two of them are colinear. Take $P\_0$ any point in the plane and construct a successive set of points $P\_i$ such that for every $i$, $1\le i\le... | https://mathoverflow.net/users/121643 | Computational (conjecture) choices for a path | Let $S=\Sigma v\_i$. If $S=0$, sort the vectors according to their angle along the unit circle. Then the corresponding closed path traces the boundary of a convex polygon.
In fact, the vectors $v\_i$ can be of arbitrary length.
If $S\neq 0$, then add an auxiliary vector $v\_{n+1}=-S$ and proceed as in the first cas... | 11 | https://mathoverflow.net/users/24076 | 400697 | 164,528 |
https://mathoverflow.net/questions/400663 | 4 | I'm looking for a finite-dimensional Hopf algebra (over any field) that is unimodular, has unimodular dual, but is not involutory. Is there such a thing?
Here's what I know:
* By Radford's formula, the antipode $S$ must have order 4.
* Suzuki [has constructed](https://www.jstor.org/stable/43685968) unimodular Hopf ... | https://mathoverflow.net/users/27013 | Hopf algebra that is unimodular and counimodular but not involutory | It turns out that an explicit example of such a Hopf algebra already appears at the end of Radford's seminal paper [The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite](https://www.jstor.org/stable/2373888) as Example 2.
The example is a bit too complicated to reproduce here, so let me just note ... | 1 | https://mathoverflow.net/users/27013 | 400705 | 164,532 |
https://mathoverflow.net/questions/400615 | 4 | Let $A$ be an (abstract) affine plane. We call $A$ a translation plane if the group of translations acts transitively on the set of points (Axiom 4a in Artin's book "Geometric algebra").
Desargues' "little" theorem characterizes translation planes by the following geometric assertion:
Let $l\_1, l\_2, l\_3$ be distinct... | https://mathoverflow.net/users/332108 | How does the affine Desargues theorem imply the little Desargues theorem? | I just found a marvelous proof in the style I was looking for in Bennett's book "Affine and projective geometry" (see the corollary on page 60).
| 3 | https://mathoverflow.net/users/332108 | 400719 | 164,534 |
https://mathoverflow.net/questions/400732 | 10 | Vopenka's principle implies the existence of weakly compact cardinals (a proper class of them, I believe). My question is whether Vopenka's principle is consistent with the assertion that the universe itself is weakly compact. Alternatively, can a Vopenka cardinal be weakly compact?
There are several versions of this... | https://mathoverflow.net/users/2362 | Is Vopenka's Principle + "ORD has the tree property" consistent? | Yes, a Vopenka cardinal can be weakly compact, at least assuming the consistency of a huge cardinal (though this is certainly a bit of an overkill). A huge cardinal is a weakly compact (in fact, measurable) Vopenka cardinal.
EDIT: Actually almost huge cardinals suffice to get measurable Vopenka cardinals. Theorem 24.... | 11 | https://mathoverflow.net/users/26319 | 400739 | 164,536 |
https://mathoverflow.net/questions/400743 | 7 | As Hilbert spaces, $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I wonder if there is an integral transform
$$
f(x,y) \mapsto (K f)(z)=\int dx\, dy K(x,y,z) f(x,y)
$$
with a nice explicit ... | https://mathoverflow.net/users/38654 | Explicit isomorphism between $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$? | The following result was obtained by an "explicit" construction in [1]. It is related to the comment of Terry Tao. A modification of the argument allows one to replace the cube by the whole space.
>
> **Theorem.** If $k\geq n$ and $1\leq p\leq \infty$, then there is an isometric isomorphism $\Phi: L^p([0,1]^k)\to L... | 4 | https://mathoverflow.net/users/121665 | 400748 | 164,539 |
https://mathoverflow.net/questions/400726 | 4 | For a $k$-Hopf algebra $H$ and element $h \in H$ is called **grouplike** is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1\_k$ ($\epsilon$ is the counit). The identity $1\_H$ is clearly grouplike, but in general non-trivial grouplike elements may fail to exist. See this question for Is there a name for a Hopf algebra who... | https://mathoverflow.net/users/326091 | Name for a Hopf algebra whose only grouplike element is the identity? | There is a one-to-one correspondence between the grouplike elements and the simple, $1$-dim subcoalgebras. So if the only grouplike element is $1\_H$, then there is a unique $1$-dim simple subcoalgebra (which is $k\cdot 1\_H$). In that case, the HA is - by definition- called: *connected HA*.
Also, notice that a conn... | 3 | https://mathoverflow.net/users/85967 | 400761 | 164,545 |
https://mathoverflow.net/questions/400763 | 10 | **Observation:** Every $\aleph\_1$-directed colimit $\varinjlim\_i X\_i$ of finite sets is finite.
**Proof:**
Because the $X\_i$'s are finite, the Mittag-Leffler condition holds, so by passing to the diagram of essential images, we may assume that the transition maps are injective. Therefore the cardinalities of the ... | https://mathoverflow.net/users/2362 | Which abelian groups are $\aleph_1$-filtered colimits of finitely-generated abelian groups? | Yes. This can be rephrased as: let $G$ be an abelian group [resp. group] with a chain of f.g. subgroups $G\_\alpha$ for $\alpha<\omega\_1$. Is $G$ f.g.?
The answer is yes for abelian groups:
The answer is clearly yes if $G\_\alpha=G$ for large $\alpha$. Otherwise, up to extract we can suppose that all inclusions ar... | 8 | https://mathoverflow.net/users/14094 | 400768 | 164,547 |
https://mathoverflow.net/questions/398025 | 3 | Let $(X\_t)\_{0\le t\le 1}$ be a continuous Markov martingale (with respect to its natural filtration) s.t. $X\_0=0$ and $X\_1\in\{-1,1\}$. Can we prove the existence of some measurable function $\sigma: [0,1]\times \mathbb R\to\mathbb R\_+$ s.t.
$$X\_t=\int\_0^t\sigma(s,X\_s)dW\_s,\quad \forall 0\le t\le 1?$$
Here... | https://mathoverflow.net/users/261243 | Question on the martingale representation theorem | No. There are at least two reasons for that.
First, the underlying probability space $(\Omega, \mathcal F)$ and the natural filtration can be too small for the Brownian motion to exist. Indeed: consider the case when $X\_t$ is constant for $t \geqslant \tfrac12$, and $\Omega$ is just the space of paths of $X\_t$. If ... | 5 | https://mathoverflow.net/users/108637 | 400788 | 164,552 |
https://mathoverflow.net/questions/400779 | 8 | In [page 6, RH Equivalence 5.3](https://web.archive.org/web/20120731034246/http://aimath.org/pl/rhequivalences). An equivalence of the Riemann Hypothesis says that
$$\sum\_{\rho} \frac{1}{|\rho|^2} =\sum\_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$
where $\rho$ is over nontrivial zeros of the Riemann z... | https://mathoverflow.net/users/159935 | A question on an equivalence of RH | Note that if $1/2< \sigma <1, t \in \mathbb R$ one has $\frac{2\sigma-1}{\sigma^2+t^2} < \frac{2\sigma-1}{(1-\sigma)^2+t^2}$.
By a little manipulation, one gets:
$\frac{2\sigma}{\sigma^2+t^2} + \frac{2(1-\sigma)}{(1-\sigma)^2+t^2} < \frac{1}{\sigma^2+t^2} + \frac{1}{(1-\sigma)^2+t^2} $
But if RH is false and ther... | 12 | https://mathoverflow.net/users/133811 | 400789 | 164,553 |
https://mathoverflow.net/questions/400795 | 7 | Selmer's curve is the equation $3x^3 +4y^3 +5z^3=0$. This equation is famous for having non-trivial solutions in every completion of $\mathbb{Q}$ but only having the trivial solution in the rationals. This curve has been discussed on Mathoverflow before such as [here](https://mathoverflow.net/questions/47442/diophantin... | https://mathoverflow.net/users/127690 | A family of Diophantine equations with no integer solutions but solutions modulo every integer | The answer is no. Heath-Brown has shown every cubic form over the integers in at least 14 variables represents zero nontrivially. The Wikipedia article on [Hasse principle](https://en.wikipedia.org/wiki/Hasse_principle#Cubic_forms) contains references as well as related results for forms in higher odd degrees.
| 18 | https://mathoverflow.net/users/30186 | 400797 | 164,557 |
https://mathoverflow.net/questions/400801 | 1 | Suppose that ${\bf x} \in\mathbb C^n$ is a complex random vector, we know the mean vector and covariance matrix of $\bf x$ are defined as follows:
$${\bf m}\_{\bf x} = \mathbb{E} ({\bf x}) \\
{\bf C}\_{\bf x x} = \mathbb{E} (({\bf x}-{\bf m}\_{\bf x})({\bf x}-{\bf m}\_{\bf x})^H)$$
How is 4th order cumulant of a ${\bf ... | https://mathoverflow.net/users/164342 | How is 4th order cumulant of a complex random vector defined? | $\newcommand{\C}{\mathbb C}\newcommand{\ip}[2]{\langle #1,#2\rangle}$First of all, cumulants are defined, rather than derived.
Now, let $X:=\mathbf x$. Suppose $E\|X\|^m<\infty$ for some natural $m$. Let $Y:=X-EX$. Then for the respective characteristic functions $f$ an $g$ of $X$ and $Y$ we have
\begin{equation}
f(... | 0 | https://mathoverflow.net/users/36721 | 400806 | 164,562 |
https://mathoverflow.net/questions/400810 | 3 | Let $K$ be a field which is a (transcendental) extension of $\mathbb{C}$. Let $L\_1, L\_2$ and $M\_1, M\_2$ be two field extensions of $K$ (not necessarily algebraic) such that $$L\_1 \otimes\_K L\_2 \cong M\_1 \otimes\_K M\_2$$
My question is: Is one of $M\_1$ or $M\_2$ a field extension (not necessarily finite) of $L... | https://mathoverflow.net/users/38832 | On tensor product of field extensions | Not necessarily.
Let $p\_1,p\_2,p\_3,p\_4$ be primes, and let $A\_1,A\_2,A\_3,A\_4$ be extensions of $K$ of degrees $p\_1,p\_2,p\_3,p\_4$ respectively.
Let $$L\_1 = A\_1 \otimes\_K A\_2$$ $$L\_2 = A\_3 \otimes\_K A\_4$$ $$M\_1 = A\_1 \otimes\_K A\_3$$ $$M\_2= A\_2 \otimes\_K A\_4.$$
Then neither $M\_1$ nor $M\_2$... | 6 | https://mathoverflow.net/users/18060 | 400811 | 164,563 |
https://mathoverflow.net/questions/400793 | 10 | Let $C\_n=\frac1{n+1}\binom{2n}n$ be the all-familiar Catalan numbers. Then, the following identity has received enough attention in the literature (for example, [Lagrange Inversion: When and How](https://doi.org/10.1007/s10440-006-9077-7)):
\begin{equation}
\label1
\sum\_{k=0}^n\binom{2n-2k}{n-k}C\_k=\binom{2n+1}n \qq... | https://mathoverflow.net/users/66131 | In search of a $q$-analogue of a Catalan identity | This identity is known as Jonah's formula (special case with $n\rightarrow 2n$ and $r\rightarrow n$, see "Catalan Numbers with Applications" by Thomas Koshy, pg. 325-326 for a combinatorial proof)
$$\sum\_{k=0}^r\binom{n-2k}{r-k}C\_k=\binom{n+1}r$$
and a $q$-analogue was obtained by Andrews in "$q$-Catalan identities... | 9 | https://mathoverflow.net/users/302667 | 400812 | 164,564 |
https://mathoverflow.net/questions/398094 | 3 | Let $f\_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f\_i\|\_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \mathbb R$ by
$$F(h) = \limsup\_{i \to \infty} \|f\_i - h\|\_{L^1}.$$
>
> **Question:** Does this functional admit a mi... | https://mathoverflow.net/users/173490 | Minimiser of a certain functional | As it has been already noted in the comments, the minimizer doesn't need to be unique. However, it always exists. It is not *terribly* hard to show but it is not trivial either, so I wonder why the question attracted so few votes.
The proof consists of two independent parts. The first one is that the limit of every m... | 3 | https://mathoverflow.net/users/1131 | 400816 | 164,565 |
https://mathoverflow.net/questions/400814 | 3 | Let $R$ be a reduced Noetherian ring. Assume $R$ is quasi-excellent and Cohen-Macaulay.
>
> Is $R$ the quotient of a Gorenstein ring?
>
>
>
If the answer is yes, then $R$ has a dualizing complex. The question can, therefore, be rephrased into:
"is there an example of a reduced quasi-excellent Cohen-Macaulay ri... | https://mathoverflow.net/users/nan | Quotients of Gorenstein rings | No, $R$ is not necessarily a quotient of a Gorenstein ring, because of the following:
**Example** [[Nishimura 2012](https://doi.org/10.1215/21562261-1503754), Example 6.1]**.** *There exists a two-dimensional Cohen–Macaulay factorial excellent local domain with a Gorenstein module, which has no dualizing (i.e., canon... | 7 | https://mathoverflow.net/users/33088 | 400822 | 164,567 |
https://mathoverflow.net/questions/400819 | 8 | Can you prove or disprove the following claim:
>
> **Claim:**
> $$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum\_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
>
>
>
The SageMath cell that demonstrates this claim can be found [here](https://sagecell.sagemath.org/?z=eJwrTbI1MgACa65i2-LSXI08WwOd0iQdQ30NDTOtPG1DTS0wbaqpqWnNV... | https://mathoverflow.net/users/88804 | An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$ | Here is an elementary proof. We rewrite the series as
$$\frac{1}{4}\int\_0^1\frac{1-x^4}{1-x^6}\,dx=\frac{1}{8}\int\_0^1\frac{dx}{1-x+x^2}+\frac{1}{8}\int\_0^1\frac{dx}{1+x+x^2}.$$
It is straightforward to show that
\begin{align\*}
\int\_0^1\frac{dx}{1-x+x^2}&=\frac{2\pi}{3\sqrt{3}},\\
\int\_0^1\frac{dx}{1+x+x^2}&=\fra... | 27 | https://mathoverflow.net/users/11919 | 400824 | 164,569 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.