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/327835 | 7 | According to the Wikipedia article about [monodromy](https://en.wikipedia.org/wiki/Monodromy#Definition_via_Galois_theory), the monodromy group can be defined in terms of Galois theory in following way:
>
> Let $F(x)$ denote the field of the rational functions in the variable $x$ over the field $F$, which is the fi... | https://mathoverflow.net/users/108274 | Monodromy groups from Galois's viewpoint | All of the following can be found in the third chapter of Szamuley's "Galois Groups and Fundamental Groups" but I will try to sum it up a bit:
I think the starting point of making this precise is the fact that the function field $\mathbb{C}(t)$ is isomorphic to the field of meromorphic functions on the Riemann sphere... | 8 | https://mathoverflow.net/users/138270 | 327854 | 141,017 |
https://mathoverflow.net/questions/327783 | 2 | Can a sequence of degree one maps on, say, the unit circle, converges to a constant map in $W^{1,2}$ norm?
If the answer is yes, would you please provide an explicit example?
| https://mathoverflow.net/users/74664 | Can a sequence of degree one maps converges to a constant map in $W^{1,2}$ norm? | Lift your map to the universal cover to obtain a function $f:R\to R$ satisfying
$$f(x+1)=f(x)+1,$$
if the degree is $1$.
If the Sobolev norm of $f-c$ is $<\epsilon$ then by Schwarz inequality
$$(f(1)-f(0))^2\leq\int\_0^1 (f'(t))^2dt<\epsilon,$$
which contradicts the first inequality if $\epsilon<1$.
| 1 | https://mathoverflow.net/users/25510 | 327862 | 141,021 |
https://mathoverflow.net/questions/327871 | 6 | Let $\mathcal{O}$ be an operad in the monoidal category $M$. Then $\mathcal{O}(1)$ together with the morphisms
$$\mathcal{O}(1)\otimes \mathcal{O}(1)\to \mathcal{O}(1)$$
and the unit $\eta:1\to \mathcal{O}(1)$ is a monoid object. Moreover, a morphism $\varphi:\mathcal{O}\to \mathcal{O}'$ of operads induces a morphism $... | https://mathoverflow.net/users/124042 | Free operad over a monoid object | Let me mention that this is related to [this earlier question of mine](https://mathoverflow.net/q/254714/36146) (which is unanswered :-( ) and more generally to semi-direct products of operads by bialgebras. You construction is the semi-direct product $\mathtt{Com} \rtimes T$ of the commutative operad $\mathtt{Com}$ by... | 9 | https://mathoverflow.net/users/36146 | 327874 | 141,024 |
https://mathoverflow.net/questions/327851 | 1 | Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $f\_{n,j}$ be a doubly indexed sequence of positive functions in $L^p(\Omega),$ $1<p<\infty.$ Suppose $f\_{n,j}$ converges pointwise a.e. to $f\_n\in L^p$ and also in $\|.\|\_p.$ Denote $f\_{n}^\*(x)=\sup\_{j\geq 1} f\_{n,j}(x)$ and $f^\*(x)=\sup\_{n\geq 1}f\_n(... | https://mathoverflow.net/users/136860 | Limit of doubly indexed functions | The answer is no. E.g., suppose that $\Omega=\mathbb R$ with the Lebesgue measure $\mu$ over $\mathbb R$ and $f\_{n,j}=f\_n=1\_{[n,n+1)}$ for natural $n,j$. Then $f\_n^\*=f\_n=1\_{[n,n+1)}$ and $f^\*=1\_{[1,\infty)}$. So,
$$\|f^\*\|\_p=\infty>1=\sup\_n\|f\_n^\*\|\_p.
$$
| 1 | https://mathoverflow.net/users/36721 | 327875 | 141,025 |
https://mathoverflow.net/questions/327869 | 1 | I am not expertise in graph theory. So have to ask this question here. The term "good" means that the algorithms should be efficient for general undirected simple connected graphs with a higher success probability.
I also read the question [Efficient Hamiltonian cycle algorithms for graph classes](https://mathoverfl... | https://mathoverflow.net/users/103866 | Which are good algorithms for finding Hamiltonian path (not necessarily a circle) up to now? | The Hamiltonian path problem is NP-complete in general, so only heuristics could exist if P≠NP.
The rotation-extension heuristic may be the simplest heuristic:
```
Input: undirected graph G with size n
Let P be a path
Let e be a random edge of G
P:=[e]
Loop:
If Extension(G,P)≠∅:
P:=Extension(G,P)
... | 2 | https://mathoverflow.net/users/125498 | 327893 | 141,030 |
https://mathoverflow.net/questions/313922 | 9 | In page 15 of the article [New Applications of Min-max Theory](https://arxiv.org/pdf/1409.7537.pdf), Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi\_{1,2}$ in the 3-sphere to have a Morse index 9, but there is no real evidence.
My question is, what suggests the index to be 9? ... | https://mathoverflow.net/users/74664 | Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9? | Due to a recent preprint by Kapouleas and Wiygul (arXiv:1904.05812), the index of the Lawson $\xi\_{1,g}$ surface is $2g+3$. I have not looked at the paper in detail so far, but let me cite from the authors introduction:
"The ideas of our proof originate with work of NK on the approximate kernel for Scherk surfaces. ... | 6 | https://mathoverflow.net/users/4572 | 327903 | 141,034 |
https://mathoverflow.net/questions/327863 | 2 | Consider the continuity equation
$$\partial\_t u(t,x) + \mathrm{div}(a(t,x)u(t,x)) = 0,$$
where $u: [0,T]\times \mathbb{R}^N \to \mathbb{R}$ is the solution and $a:[0,T]\times \mathbb{R}^N \to \mathbb{R}^N$ is a vector field.
I've head many times that assumptions on $\mathrm{div}\,a$, for example that it is bounde... | https://mathoverflow.net/users/122620 | Role of the divergence of the vector field in transport equations: mass concentration? | It is instructive to think about 1 dimensional case. Take $a(x)=b-\alpha x$, ($\alpha\geq 0$) then the divergence is simply $-\alpha$.
Case 1: $b=0,\alpha\geq 0$: A trajectory starting at $x(0)=x\_0$ on the real line will follow the dynamics $x(t)=x\_0e^{-\alpha t}$. Hence, each point exponentially attracted to the o... | 1 | https://mathoverflow.net/users/30684 | 327905 | 141,036 |
https://mathoverflow.net/questions/327907 | 3 | Consider the integrals
$$I\_n(\zeta,\epsilon)=\int\_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$
I would like to know the asymptotic behavior of $I\_n(\zeta,\epsilon)$ for each fixed $\zeta>0$ as $\epsilon$ approaches zero, and hope that this will be independent of $\zeta$. For example, w... | https://mathoverflow.net/users/50438 | Oscillatory integrals | $\newcommand{\de}{\delta}
\newcommand{\ga}{\gamma}
\newcommand{\si}{\sigma}
\newcommand{\ep}{\epsilon}$
Take any real $n>0$ and any $z:=\zeta\in(0,\infty)$. Let $\ep\downarrow0$.
For real $t\ne0$, let $u:=\arctan(\ep/t)$, so that $t=\ep\cot u$.
Then for $t>0$
\begin{equation\*}
\left|(t-i\ep)^{-n}-(t+i\ep)^{-n}\righ... | 2 | https://mathoverflow.net/users/36721 | 327916 | 141,038 |
https://mathoverflow.net/questions/327908 | 2 | * Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.
* Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables.
* Furthermore, let
$$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$
be a $\mathcal{B}(\mathbb{R}^2)/\mathcal{B}(\mathbb{R})$-measurable function
such that, for all $y \in \mathbb{R}$, ... | https://mathoverflow.net/users/124116 | Substitute Concrete Value in Conditional Expectation | $\newcommand{\R}{\mathbb{R}}
\newcommand{\vpi}{\varphi}$
The answer is: the condition
>
> $E(f(X,Y)|Y)=\vpi(Y)$ for all $f$ such that $f(X,Y)$ has a finite expectation
>
>
>
holds iff $X$ and $Y$ are independent.
Indeed, if $X$ and $Y$ are independent, then for each Borel subset $B$ of $\R$
\begin{align\*... | 1 | https://mathoverflow.net/users/36721 | 327925 | 141,040 |
https://mathoverflow.net/questions/327891 | 2 | Let V be a vector space over an algebraically closed field.
Let $\{H\_i\}\_{i \in I}$ be a collection of closed connected subgroups of $\operatorname{GL}(V)$ (wrt. Zariski topology). It is a basic result in algebraic groups that the subgroup $G$ generated by the $H\_i$ is a closed connected subgroup of $\operatornam... | https://mathoverflow.net/users/38068 | Lie algebra of an algebraic group generated by connected subgroups | Let $p=\mathrm{char}(k)>0$. The vector group $G=\mathbf{G}\_a^2$ is generated by $H\_1=\{(t,0)|t\in k\}$ and $H\_2=\{(t,t^p)\mid t\in k\}$ but $\mathrm{Lie}H\_1=\mathrm{Lie}H\_2=k\oplus 0$.
| 4 | https://mathoverflow.net/users/89948 | 327933 | 141,042 |
https://mathoverflow.net/questions/327860 | 7 | Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$ is a sum of finitely many squares of linear forms with integer coefficients
$$
Q(x)=\sum\_{i=1}^N (\ell\_i(x))^2\quad ... | https://mathoverflow.net/users/4312 | Integer positive definite quadratic form as a sum of squares | This is a well-known problem, called the Waring's problem of integral quadratic forms. Every semi-positive definite quadratic form in $n \leq 5$ variables is a sum of $n + 3$ squares of linear forms. This was proved by Chao Ko, but this can be explained by the fact that the quadratic form of sum of $n + 3$ squares has ... | 11 | https://mathoverflow.net/users/29241 | 327945 | 141,048 |
https://mathoverflow.net/questions/327960 | 1 | How to prove following for $n\geq0$ ?
$$\int\_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$
Where, at any point $\vec{r}$, the $\vec{\Omega}$ can be described by the polar angle $\theta$ measured with respect to the *z* axis and an azimuthal angle $\phi$ m... | https://mathoverflow.net/users/138233 | Identity involving dot product of solid angle and gradient | Because of isotropy, the differential operator
$$J\_n=\int\_{|\vec{\Omega}|=1} d\vec{\Omega}\,(\vec{\Omega}\cdot\vec{\nabla})^{2n}=C\_n\Delta^n$$
with $\Delta$ the Laplacian. To find the coefficient $C\_n$, let $J\_n$ act on $z^{2n}$,
$$J\_n z^{2n}=(2n)!\int\_{|\vec{\Omega}|=1} d\vec{\Omega}\,\Omega\_z^{2n}=2\pi(2n)!\... | 2 | https://mathoverflow.net/users/11260 | 327963 | 141,056 |
https://mathoverflow.net/questions/327961 | 8 | The category of coherent sheaves on a locally Noetherian scheme is abelian. Are there some geometric conditions on the scheme that imply that the category of coherent sheaves is semisimple?
Edited in response to posic's comments.
| https://mathoverflow.net/users/nan | Semisimplicity of the category of coherent sheaves? | Let $X$ be a locally Noetherian scheme. Then the abelian category of coherent sheaves on $X$ is semisimple if and only if $X$ is the disjoint union of finitely many reduced points.
The if direction is clear: the category of coherent sheaves on a finite union of reduced points is a direct sum of categories of finite d... | 12 | https://mathoverflow.net/users/25309 | 327976 | 141,060 |
https://mathoverflow.net/questions/327980 | 3 | How to calculate:
$$\sum \_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}.$$
| https://mathoverflow.net/users/106253 | How to calculate: $\sum\limits_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}$ | Notice that $\frac{1}{n-k} = \int\_0^1 x^{n-k-1} dx$. Hence,
$$\sum\_{k=0}^{n-m} \frac{1}{n-k}\binom{n-m}k = \int\_0^1 dx \sum\_{k=0}^{n-m} x^{n-k-1}\binom{n-m}k = \int\_0^1 x^{m-1}(1+x)^{n-m}dx = (-1)^m B\_{-1}(m,n-m+1),$$
where $B\_{\cdot}(\cdot,\cdot)$ is [incomplete beta function](https://en.wikipedia.org/wiki/Beta... | 8 | https://mathoverflow.net/users/7076 | 327982 | 141,062 |
https://mathoverflow.net/questions/327338 | 7 | Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [[1](https://books.google.com/books?id=Jyj7CAAAQBAJ&pg=PA28&lpg=PA28&dq=milnor+immersion+circle+disks&source=bl&ots=BsDZoA7Rkx&sig=ACfU3U2MG6TtFsOccPuXInjgi5ytk098CQ&hl=en&sa=X&ved=2ahUKEwjR1MLC_bvhAhWKTd... | https://mathoverflow.net/users/12310 | Milnor immersion of circle, disks, and a ball | If one assumes a couple of simple conditions on $f$, namely that the restriction of $z$ on $f(S^2)$ is a Morse function with two critical points and that projections of two halves of $f(S^2)$ ($z>0$, $z<0$) onto the plane $z=0$ realize two original different immersions then $f$ can not be extended to an immersion of th... | 4 | https://mathoverflow.net/users/943 | 327988 | 141,065 |
https://mathoverflow.net/questions/327981 | 4 | Are there infinitely many squares in the set $$\{\sum\_{j=1}^m j^2: m\in\mathbb{N}\} ?$$
| https://mathoverflow.net/users/8628 | Squares in the set $\{\sum_{j=1}^m j^2: m\in\mathbb{N}\}$ | We have $\sum\_{j=1}^m j^2 = \frac{m(m+1)(2m+1)}6$. Hence, the question reduces to finding integral points on the elliptic curve:
$$y^2 = \frac{x(x+1)(2x+1)}6.$$
Turning it to Weierstrass equation with integer coefficients, we get:
$$(72y)^2 = (12x)^3 + 18 (12x)^2 + 72 (12x).$$
For many curves, integral points can be... | 14 | https://mathoverflow.net/users/7076 | 327996 | 141,067 |
https://mathoverflow.net/questions/327706 | 2 | Who first discovered the concept corresponding to the symbol of class comprehension
$\{x/\varphi\}$ used today in set theory ?
| https://mathoverflow.net/users/30395 | Who first discovered the concept corresponding to the symbol of class comprehension? | The concept is found with different notation in **Peano** ([1894](//zbmath.org/?q=an:25.0103.03), p. 20) (translated):
>
> Let $p\_x$ be a proposition containing a variable letter $x$, that is, a condition on $x$. By the notation $\overline{x\ \varepsilon}\,p\_x$ we shall indicate the class of those $x$ that satisf... | 8 | https://mathoverflow.net/users/19276 | 327999 | 141,068 |
https://mathoverflow.net/questions/327926 | 1 | Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.
| https://mathoverflow.net/users/36688 | A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf | Such a bounded leaf can not exist even if we assume that the vector field is analytic and the leaf is merely non-constant.
Indeed, suppose by contradiction that $i:\cal F\to \mathbb C^2$ is a leaf of an analytic vector field $v$ that is bounded subset of $\mathbb C^2$. Since $i(\cal F)$ is bounded, the vector field i... | 4 | https://mathoverflow.net/users/943 | 328000 | 141,069 |
https://mathoverflow.net/questions/328007 | 5 | An Osgood modulus of continuity is an increasing function $\omega:(0,1]\to(0,1]$ such that $\int\_0^1\frac{dt}{\omega(t)}=\infty$.
We say a vector field $X$ satisfies Osgood condition with modulus $\omega$ if locally there is a and $|X(x)-X(y)|\le\omega(|x-y|)$ (here I don't want a constant).
When $X$ satisfies Osg... | https://mathoverflow.net/users/118469 | Modulus of continuity of flow for non-Lipschitz vector fields satisfies Osgood condition | Let $X:\mathbb{R}^n\to\mathbb{R}^n$ a vector field with modulus of continuity $\omega:[0,\infty)\to[0,\infty)$ (we have no reason here to bound the domain of $\omega$ to $(0,1]$) that satisfies the Osgood condition $\int\_0^1{ds\over\omega(s)}=+\infty$. Being uniformly continuous on $\mathbb{R}^n$, the vector field $X$... | 4 | https://mathoverflow.net/users/6101 | 328030 | 141,075 |
https://mathoverflow.net/questions/328032 | 0 | Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle\_t$. I know that for two independent Brownian motions, that this quadratic covariation is zero, but does this also hold when we cannot say whether or not t... | https://mathoverflow.net/users/138346 | Quadratic covariation of two not independent Brownian motions | If the pair $(X,Y)$ is a local martingale wrt. some filtration, and since the quadratic variations are $[X,X]\_t=t$, $[Y,Y]\_t=t$ and $[X,Y]\_t=0$, [Lévy's characterisation](https://en.wikipedia.org/wiki/Brownian_motion#L%C3%A9vy_characterisation) of Brownian motion gives that $(X,Y)$ is a two dimensional Brownian moti... | 1 | https://mathoverflow.net/users/78493 | 328036 | 141,076 |
https://mathoverflow.net/questions/328049 | 1 | Let $Z$ be a random variable with finite moment-generating function $M\_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define
$C\_Z^\delta := \inf\_{\theta>0}\theta\log M\_Z(\theta) - \theta\log(\delta)$,
and $SVP\_Z^\delta:=E[Z] + \sqrt{(1/\delta)Var[Z]}$. By the Chernoff... | https://mathoverflow.net/users/78539 | Bounds on difference between "logsumexp" and variance? | By [Theorems 3.3 and 3.4](https://www.mdpi.com/2227-9091/2/3/349),
$$
C\_Z^\delta\ge C\_{Z;\alpha}^\delta:=\inf\_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)\_+\|\_\alpha}{\delta^{1/\alpha}}\Big)
$$
for any $\alpha\in(0,\infty)$, where $\|X\|\_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C\_Z^\delta$ ... | 1 | https://mathoverflow.net/users/36721 | 328056 | 141,084 |
https://mathoverflow.net/questions/328020 | 5 | It is known that any holomorphic bundle of any rank over a noncompact Riemann surface is trivial. A proof can be found in Forster's "Lectures on Riemann surfaces", section 30.
Let $E$ be a holomorphic vector bundle over a compact Riemann surface $X$ with gauge group $G$. A consequence of the above theorem is the rest... | https://mathoverflow.net/users/131977 | Any holomorphic vector bundle over a compact Riemann surface can be defined by only one transition function? |
>
> It is known that any holomorphic bundle of any rank over a noncompact
> Riemann surface is trivial.
>
>
>
The idea of proof is the following: non-compact curve is actually affine manyfold. On affine manyfolds cohomologies of coherent sheaves vanishes (Serre). Extensions of one bundle by another is controll... | 5 | https://mathoverflow.net/users/10446 | 328063 | 141,087 |
https://mathoverflow.net/questions/326971 | 3 | We call a topological space $X$ extremally disconnected if the closures of its open sets remain open. Obviously, Hausdorff extremally disconnected spaces are totally disconnected in the sense that their connected components are singletons.
Suppose $\mathbb{Z}\_p = \varprojlim\_{n \ge 1} \mathbb{Z}/p^n\mathbb{Z}$ is t... | https://mathoverflow.net/users/128540 | Is the ring of $p$-adic integers extremally disconnected? | Since questions recirculate to the front page forever if left unanswered, I will amalgamate the comments into an answer, which I've made community wiki.
Firstly, by the characterization of the Cantor space as the only metrizable Stone space with no isolated points (up to homeomorphism), $\newcommand{\Z}{\mathbb{Z}}\Z... | 8 | https://mathoverflow.net/users/61785 | 328070 | 141,092 |
https://mathoverflow.net/questions/328078 | 5 | Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following regularity estimates hold for $p=1$?
$$||u||\_{W^{k+1,p}(\Omega)}\leq C(||f||\_{W^{k,p}(\Omega)}+||u||\_{L^p(\Omega)}),$$
for s... | https://mathoverflow.net/users/42326 | $W^{k,1}$ regularity for elliptic equations | In the case of the Laplace operator and $k=0$, see Lemma 14 in <https://arxiv.org/pdf/0809.2172.pdf>. The argument should work for more general divergence elliptic equation in the divergence form and higher order derivatives. See also <https://mathoverflow.net/a/298962/121665>
| 2 | https://mathoverflow.net/users/121665 | 328084 | 141,096 |
https://mathoverflow.net/questions/328102 | 13 | Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent with a measurable cardinal $\kappa$ satisfying $o(\kappa)=\kappa^{++}$.
But this failure is at $\aleph\_\omega$. Suppos... | https://mathoverflow.net/users/7206 | Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality? | Suppose $\kappa$ is a singular cardinal and there are $cf(\kappa)$-many measurable cardinals $\lambda < \kappa$ with $o(\lambda)=\lambda^{++}$ cofinal in $\kappa.$ Then you can perform a Prikry type iteration and get the failure of $SCH$ at cofinally many singular cardinals below $\kappa.$
Now suppose we also want fo... | 10 | https://mathoverflow.net/users/11115 | 328106 | 141,104 |
https://mathoverflow.net/questions/328110 | 1 | Suppose that $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
>
> How does one prove that
> $$\tilde \Phi(x,y,t) = \left(\Phi(x,t), \frac{\Phi(x + r y,t) - \Phi(x,t)}{r} \right)$$
> is the flow of the ODE with
> ... | https://mathoverflow.net/users/122620 | Difference quotient for solutions of ODE and Liouville equation | *Question 1:* Denoting $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ the components of $\tilde \Phi(x,y,t)$ , we have $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} .$$
So
$$\partial\_t\tilde \Phi(x,y,t)= $$$$... | 2 | https://mathoverflow.net/users/6101 | 328113 | 141,107 |
https://mathoverflow.net/questions/328099 | 3 | Consider the von Neumann subalgebra of $M\otimes M$ by $ B= \mathrm{vN} \{T\otimes T: T\in M\}$. What is the commutant of B?
| https://mathoverflow.net/users/136400 | Commutant of subalgebra of tensor product | We need $M \subseteq B(H)$ in order for the commutant to make sense. So $B \subseteq B(H\otimes H)$. The commutant of $B$ is the von Neumann algebra $C$ generated by $M' \otimes M'$ and the flip unitary $u$ acting on $H \otimes H$. It's clear that $B$ is contained in $C'$; conversely, if $x \in C'$ then $\phi(x) = x$ w... | 2 | https://mathoverflow.net/users/23141 | 328114 | 141,108 |
https://mathoverflow.net/questions/328055 | 2 | Let $X$, $Y$ be connected smooth schemes of finite type over an algebraically closed field of characteristic $0$. Let $f:X\rightarrow Y$ be a non-birational morphism surjective on the underlying topological spaces. Can the non-flat locus of $f$ be non-empty of codimension$\geq 2$ in $X$? For birational morphism, I beli... | https://mathoverflow.net/users/nan | Non-flat locus for smooth schemes | Suppose that $f \colon X \to Y$ is generically finite. Then the locus in $X$ where $f$ is not flat is the locus where the fiber is positive dimensional.
Now, let $Z$ be a projective threefold with a unique singular point $p \in Z$, admitting a small resolution $X \to Z$ (that is, a proper birational map that is an is... | 5 | https://mathoverflow.net/users/4790 | 328141 | 141,114 |
https://mathoverflow.net/questions/328081 | 2 | Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)?
Any reference would be highly appreciated.
| https://mathoverflow.net/users/137675 | Reference request: Schauder estimates for parabolic equations | I found the answer in the book "Elliptic and Parabolic Equations" of Z. Wu, J. Yin and C. Wang, Theorem 7.2.24.
| 1 | https://mathoverflow.net/users/137675 | 328142 | 141,115 |
https://mathoverflow.net/questions/328149 | 4 | Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ N \neq \emptyset$ and $\bullet M \approx \bullet N$, where $\bullet M$ denotes the interior of $M$ and $\approx$ denote... | https://mathoverflow.net/users/32487 | Is a manifold-with-boundary with given interior and non-empty boundary essentially unique? | No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $n\ge 6$) for every element $\tau$ of the Whitehead group of $\pi\_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $\tau$ is the Whitehead torsion of the pair $(M,P)$. The interi... | 13 | https://mathoverflow.net/users/6666 | 328150 | 141,118 |
https://mathoverflow.net/questions/327897 | 8 | Sorry for perhaps naive questions, I am not at all a specialist in the subject
but I need it for my research.
I know that there are close relations between Riemann-Hilbert problems and
orthogonal polynomials. My precise questions are as follows:
1) There are several well-known methods for the numerical computation of... | https://mathoverflow.net/users/81776 | Riemann-Hilbert and orthogonal polynomials | Let $P\_{n}(z)=\gamma\_{n}z^{n}+\cdots$ be a sequence of orthonormal polynomials with respect to some weight $w$ on $\mathbb{R}$. Given an $n\geq0$, consider the following Riemann-Hilbert problem for the $2\times2$ matrix-valued function $Y\_{n}(z)$ of the complex variable $z$,
1) $z\to Y\_{n}(z)$ is analytic on $\ma... | 7 | https://mathoverflow.net/users/89429 | 328153 | 141,119 |
https://mathoverflow.net/questions/328133 | 1 | What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
| https://mathoverflow.net/users/136553 | Projections of particular simplex yielding boundary of a regular polygon? | I think you should be able to do this for arbitrary $m$ by taking the points $0$, $e\_1$,
$$\cos\left(\frac{2\pi}{m}\right) e\_1 + \sin\left(\frac{2\pi}{m}\right) e\_2,$$
and
$$\cos\left(\frac{2\pi}{m} k\right) e\_1 + \sin\left(\frac{2\pi}{m} k\right) e\_2 + e\_{k+1}$$
for $2 \leq k \leq m -1$.
Subtracting 0 ... | 1 | https://mathoverflow.net/users/6427 | 328154 | 141,120 |
https://mathoverflow.net/questions/328093 | 5 | Kleber's [Best card trick](http://www.apprendre-en-ligne.net/crypto/magie/card.pdf) proceeds as follows: The mark (audience member) freely selects five playing cards from a standard deck of $52$ and passes these five to the magician's assistant. The assistant studies those cards, returns one *mystery card* to the mark,... | https://mathoverflow.net/users/89654 | Questions about "The best card trick" | The answer to **Question 1** is $52!/48! - 4 \binom{13}{2} \binom{50}{2}= 6115200$.
That is, I claim that the number of $4$-tuples that *cannot* occur is $4 \binom{13}{2} \binom{50}{2}$. To see this, note that a $4$-tuple $(a,b,c,d)$ cannot occur if and only if the fifth card $e$ that $(a,b,c,d)$ signifies is among ... | 6 | https://mathoverflow.net/users/2233 | 328158 | 141,121 |
https://mathoverflow.net/questions/328155 | 12 | The Segal conjecture describes the Spanier-Whitehead dual $D \Sigma^\infty\_+ BG$ for certain $G$. Is there a similar description of $D\Sigma^\infty\_+ K(G,n)$ when $n \geq 2$ when $G$ is finite (and abelian)?
Notes:
* I'd be happy to understand the case of cyclic groups $G = C\_p$.
* $K(G,n)$ can be modeled by an ... | https://mathoverflow.net/users/2362 | Is there a "higher Segal conjecture"? | In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(\mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea: view $K(A,n+1)$ as the bar construction on $K(A,n)$.)
(No time right now to write more ... but maybe this is enough.)
| 17 | https://mathoverflow.net/users/102519 | 328159 | 141,122 |
https://mathoverflow.net/questions/328134 | 4 | Let $I\subseteq k[x\_0,\ldots,x\_n]$ be an ideal, generated by some polynomials $F\_1,\ldots,F\_r$, all homogeneous and of the same degree. Suppose $r$ is the smallest number of generators that will suffice to generate the ideal.
>
> Can one choose a monomial order or change coordinates to ensure that this generati... | https://mathoverflow.net/users/83755 | Can a minimal generating set for an ideal always be made into a Groebner basis? | No. Let $k$ not have characteristic $2$, let $I = \langle x^2, xy, y^2 \rangle$ and consider the generating set $x^2$, $(x+y)^2$, $y^2$. After a linear change of coordinates, these are $(a\_1 x + b\_1 y)^2$, $(a\_2 x + b\_2 y)^2$ and $(a\_3 x + b\_3 y)^2$ for some $a\_j$, $b\_j$. But the leading term of $(ax+by)^2$ wil... | 3 | https://mathoverflow.net/users/297 | 328163 | 141,123 |
https://mathoverflow.net/questions/328182 | 3 | Let $\Phi$ be the flow (defined as in page 6 of [this paper](http://www.numdam.org/article/JEDP_2004____A1_0.pdf)) of the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}.
\end{cases}$$
>
> Is it true that if $f$ is monotone in the first variable then $\P... | https://mathoverflow.net/users/nan | Flow of ODE with monotone source | Suppose that $f$ is decreasing in $x$. Let $x(t)$, $y(t)$ be two solutions of the ode. Then
$$
\dot{x}-\dot{y}= f(x,t)-f(y,t).
$$
Multiplying both sides by $x-y$ we deduce
$$
(\dot{x}-\dot{y})(x-y) =\big(f(x,t)-f(y,t)\big)(x-y)\leq 0,
$$
where the last equality holds because $f$ is decreasing.
Hence
$$
\frac{1}{... | 7 | https://mathoverflow.net/users/20302 | 328185 | 141,129 |
https://mathoverflow.net/questions/328138 | 6 | For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
More specifically, can somebody point me to a table detailing, for each series, those dominant weights for faithfulness fa... | https://mathoverflow.net/users/125941 | Non-faithful irreducible representations of simple Lie groups | Let $G\_{sc}$ be as in the answer by Victor Protsak and let $\varpi\_1$, $\ldots$, $\varpi\_l$ be the fundamental dominant weights.
Let $\lambda$ be a dominant weight and write $\lambda = \sum\_{i = 1}^l a\_i \varpi\_i$ for $a\_i \in \mathbb{Z}\_{\geq 0}$.
Then the irreducible representation of $G\_{sc}$ with high... | 10 | https://mathoverflow.net/users/38068 | 328201 | 141,134 |
https://mathoverflow.net/questions/310445 | 28 | I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on [Math SE](https://math.stackexchange.com/q/2888221/268333), I found that a few experts seem to disagree on the answer, and I didn't get enough responses to indicate a statistically significant co... | https://mathoverflow.net/users/95043 | Does the Gauss-Bonnet theorem apply to non-orientable surfaces? | The answer is already given in the comments (by Ryan Budney and Mizar). But I think it makes sense to clear this confusing point. The classical Gauss-Bonnet formula is [e.g. <https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem> ]
$$\int\_M K dA+\int\_{\partial M} k\_g ds=2\pi \chi(M).$$
In this formula nothing ... | 16 | https://mathoverflow.net/users/9833 | 328211 | 141,139 |
https://mathoverflow.net/questions/316160 | 2 | Let $G$ be a a finite group and $S$ a subnormal subgroup of $G$. The lenght of a fastest chain of subgroups $(U\_i)\_{1\le i\le n}$ such that $U\_1=S$, $U\_i$ normal in $U\_{i+1}$ and $U\_n=G$ is called the defect of subnormality of $S$. Examples in $S\_4$ Show that the series of repeated normalizers can be stable and ... | https://mathoverflow.net/users/57804 | Defect of subnormality and repeated normalizer series | Let H be a cyclic group of order 2 and let K be the dihedral group of order 8 with y a non-central element of order 2. Then let G be the wreath product of H by K with base group B.
Then $J = \langle H,Hy\rangle$ is normal in B which is normal in G. But the normalizer of J in G is $B\cdot \langle y \rangle$, which is no... | 1 | https://mathoverflow.net/users/57804 | 328219 | 141,144 |
https://mathoverflow.net/questions/328190 | 1 | Is there a generalization to Bolzano theorem when $f: \mathbb{R}^n \to \mathbb{R}^n$
| https://mathoverflow.net/users/137903 | Generalization of Bolzano theorem | There can be various generalizations. My favorite one is this.
Let $f\_1,\ldots,f\_n$ be continuous functions on the unit cube.
And suppose that each of them takes values of opposite sign on the opposite facets
(each on its own pair of the opposite facets). Then all $f\_j$ have a common
zero inside the cube.
| 3 | https://mathoverflow.net/users/25510 | 328220 | 141,145 |
https://mathoverflow.net/questions/327873 | 13 | Let us say a cardinal $\kappa$ *end-extending* if there is a function $F : V\_\kappa^{<\omega} \to V\_\kappa$ such that:
(a) If $M \subseteq V\_\kappa$ is closed under $F$, then $M \prec V\_\kappa$.
(b) If $M$ is closed under $F$ and of size $<\kappa$, then there is $N \supseteq M$ closed under $F$ such that $N \cap \s... | https://mathoverflow.net/users/11145 | End-extending cardinals | Suppose $\kappa$ carries an $\omega\_1$-saturated $\kappa$-complete ideal $I$, given $M\prec (V\_{\kappa+2},\in , <)$ ($<$ well orders $V\_{\kappa+2}$) of size $<\kappa$ containing $I$, we show how to find an end-extension $N\prec V\_{\kappa+2}$ of $M$ (i.e. $N\cap \sup(M\cap \kappa)=M\cap \kappa$).
Let $G\subset P(... | 6 | https://mathoverflow.net/users/23835 | 328221 | 141,146 |
https://mathoverflow.net/questions/328204 | 1 | I want a proof or a reference for the identity
$$
\int\_0^\infty \frac{s^{n-1}}{\Gamma(n)} p\_\beta(s,x)\,ds =\frac{x^{n\beta-1}}{\Gamma(\beta n)},\quad x>0, \,n\in\mathbb N,
$$
where $x\mapsto p\_\beta(s,x)$, $x>0$ is the density of the $\beta$-stable Lévy subordinator at time $s>0$, $\beta\in(0,1)$.
A possible pro... | https://mathoverflow.net/users/120741 | Identity for stable Lévy subordinator | It is of course possible to show that the left-hand side is continuous (e.g. using bounds for the derivative of $p\_\beta(s, x)$), but there is a more direct way. Observe that $$p\_\beta(s, x) = s^{-1/\beta} p\_\beta(1, s^{-1/\beta} x).$$ Thus,
$$
\int\_0^\infty s^{n - 1} p\_\beta(s, x) ds = \int\_0^\infty s^{n - 1/\b... | 1 | https://mathoverflow.net/users/108637 | 328225 | 141,150 |
https://mathoverflow.net/questions/328230 | 0 | Consider a right semicircular cone with height $h$ and radius $r$ given by $\mathcal{C}=\left\{\left(\frac{h-u}{h}r\cos(\theta),\frac{h-u}{h}r\sin(\theta),u\right)\,:\,u\in[0,h],\,\theta\in[0,\pi]\right\}$.
Letting $A=(r,0,0)$ and $B=(-r,0,0)$, find the path $\gamma$ whose initial and final points are $A$ and $B$ and... | https://mathoverflow.net/users/136352 | Geodesic in half cone | To elaborate on the comment: Let $A$ be the point at $(+r,0,0)$, $B$ be the point at $(-r,0,0)$, $O$ be the vertex of the cone; and $P$ be the point at $(0,r,0)$. For a right circular cone, $OA = OP = OB = r$ (and we know $AB = 2r$).
In a plane, draw the circular wedge diagram $OAPB$, with angle $AOB =\frac\pi 2$, wh... | 0 | https://mathoverflow.net/users/82067 | 328236 | 141,154 |
https://mathoverflow.net/questions/328234 | 1 | I am a theoretical physicist and
I need help in proving the alternate Whittaker formula
$W \_ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M \_ { k , m } ( z ) + \frac { \Gamma ( 2 m ) } { \Gamma \left( \frac { 1 } { 2 } + m - k \right) } M \_ { k , - m } ( z )$
i... | https://mathoverflow.net/users/83754 | How to prove the following Whittaker formula | If you start from the expression for $W\_{k,m}(z)$ given [here](https://en.wikipedia.org/wiki/Whittaker_function),
$$W\_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}U\left(m-k+\frac{1}{2}, 1+2m, z\right)$$
and express the Tricomi function $U\left(m-k+\frac{1}{2}, 1+2m, z\right)$ in terms of the Kummer f... | 2 | https://mathoverflow.net/users/6101 | 328237 | 141,155 |
https://mathoverflow.net/questions/328247 | 1 | given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \alpha\circ \sigma^3 \cdots : \Omega \to \Omega$ are all Independent identical distribution :
Let $(\Omega, \mu) = ([... | https://mathoverflow.net/users/124254 | Independent identical distribution sequence | The easiest way to do this is to take $\Omega=\{[0,1)\}^{\mathbb N\_0}$., $\sigma$ the shift map and $\alpha(\omega)=\omega\_0$.
If you don’t like the infinite-dimensional $\Omega$, you can build an example with $\Omega=[0,1)$ at the cost of making the transformation uglier (there is an measure space isomorphism mapp... | 2 | https://mathoverflow.net/users/11054 | 328250 | 141,158 |
https://mathoverflow.net/questions/328254 | 6 | This is purely exploratory and inspired by curiosity.
**Setup:** For an integer $k>0$, let $k=\sum\_{j\geq0}k\_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum\_jk\_j$. Further, introduce a *binary factorial*
$$[n]!\_b:=\eta(1)\eta(2)\cdots\eta(n).$$
Given an [integer partition](https... | https://mathoverflow.net/users/66131 | A binary hook-length formula? | I am afraid they are not always integers. Take large $p$ and $n=2^{2p}-1$. Then $[n]!\_b$ is divisible by $p^N$ for $N={2p\choose p}+1$. And $[2n+1]!\_b$ by $p^K$ for $K={2p+1\choose p}+2p+1<2N$. Then for $2\times N$ diagram we get $p$ in the denominator.
| 11 | https://mathoverflow.net/users/4312 | 328260 | 141,162 |
https://mathoverflow.net/questions/328152 | 5 | Let $X$ be a Banach space and $T \in \mathcal L(X)$.
The authors Engel and Nagel introduce in their book ["One-Parameter Semigroups for Linear Evolution Equations"](https://www.math.uni-tuebingen.de/de/forschung/agfa/members/engel-nagel_one-parameter-semigroups.pdf) on p. 248 the concept of the **Fredholm domain** of... | https://mathoverflow.net/users/91126 | Unbounded Component of the Fredholm Domain | A maybe different proof can be obtained using techniques related with the single-valued extension property (SVEP for short) of an operator. A good reference for this topic is the book [[Aiena]:](https://www.researchgate.net/profile/Pietro_Aiena/publication/31607282_Fredholm_and_Local_Spectral_Theory_with_Applications_t... | 3 | https://mathoverflow.net/users/39421 | 328261 | 141,163 |
https://mathoverflow.net/questions/326539 | 6 | Let $C$ be a hyperelliptic curve $y^2 = f(x)
$ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian.
I know Zarhin's result [Hyperelliptic Jacobians without complex multiplication], which states that if the Galois group $G= Gal(f)$ of $f$ is ei... | https://mathoverflow.net/users/106190 | Hyperelliptic Jacobians with (or without) CM | The curve $$y^2 = x^5 + 2x^4 + 2x^3 +2x^2 + 2x +1$$ has $G= C\_4$ and its Jacobian $J$ has endomorphism algebra ${\rm End}^0(J) \cong \mathbb{Q}$.
For the curve $$y^2 = x^5 + x^4 + x^3 + x^2 + x$$ we have $G= C\_4$ and its Jacobian $J$ has endomorphism algebra ${\rm End}^0(J) \cong \mathbb{Q} \times \mathbb{Q}$.
As... | 2 | https://mathoverflow.net/users/138474 | 328268 | 141,164 |
https://mathoverflow.net/questions/327631 | 5 | [This question](https://mathoverflow.net/questions/327167/lower-bound-for-integral-of-a-negativ-part-of-a-brownian-motion-with-respect-to) was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading [this paper]... | https://mathoverflow.net/users/36721 | Density near at $0$ for the integral of the positive part of the Brownian motion | Thank you for your work on this question! I am one of the authors of the cited paper and, yes, we stated a wrong claim (1) in the appendix of this paper (let me remark that the results in the main part remain unaffected as they do not rely on the exact bound). I am sorry for the confusion.
I want to add here how a tr... | 5 | https://mathoverflow.net/users/138483 | 328285 | 141,170 |
https://mathoverflow.net/questions/328263 | 0 | Let $M$ be a vN algebra (represented GNS space with respect to state) in standard form. Under which condition we can say a subalgebra $B$ of $M$ is also in standard form? If there exist $\varphi$ preserving conditional expectation $E\_{\varphi}: M\rightarrow B$, will $B$ be in standard form (on GNS space of M)?
| https://mathoverflow.net/users/136400 | A question on standard form in von Neumann algebra | I'm not an expert on von Neumann algebras either, but here are some easy observations.
1. Having a conditional expectation is not sufficient. Any state on $M$ can be regarded as a conditional expectation onto the one dimensional subalgebra $B$ consisting of the scalar multiples of the identity. If ${\rm dim}(M) > 1$ t... | 2 | https://mathoverflow.net/users/23141 | 328287 | 141,171 |
https://mathoverflow.net/questions/328289 | 10 | Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ has no a holomorphic extension to any open neighborhood of $\bar D$?
| https://mathoverflow.net/users/36688 | Is there a holomorphic function on open unit disc with this property? | Yes, for example
$$f(z)=\sum\_{n=1}^\infty e^{-\log^2n}z^{n^2}.$$
The radius of convergence is $1$ by Hadamard's test. The function has gaps (the density of the sequence $n^2$ among the integers is zero) which are sufficient to
apply Fabry's gap theorem (or Polya's gap theorem) to ensure that this $f$ does not have any... | 16 | https://mathoverflow.net/users/25510 | 328294 | 141,175 |
https://mathoverflow.net/questions/328077 | 6 | Let's say $p$ is a prime and $t\neq 0$ is a trace of Frobenius that occurs over $\mathbb{F}\_p$. The discriminant of the Frobenius polynomial is
$\Delta:=t^2-4p.$
So we obtain $4p=t^2-\Delta.$ If $E$ is an elliptic curve over $\mathbb{F}\_p$ with trace of Frobenius $t$, then the Frobenius, call it $\sigma$, generates a... | https://mathoverflow.net/users/138366 | Endomorphism rings of ordinary elliptic curves | The paper "Abelian varieties over finite fields" of Waterhouse (Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 2 (1969) no. 4, p. 521-560), available at
<http://www.numdam.org/item/?id=ASENS_1969_4_2_4_521_0>
has, I think, relevant information (see Section 4). There might be more recent works ... | 2 | https://mathoverflow.net/users/138103 | 328299 | 141,178 |
https://mathoverflow.net/questions/328297 | -2 | Can someone give a reasonably explicit example of an irreducible one-dimensional scheme with no closed points?
| https://mathoverflow.net/users/nan | One-dimensional scheme with no closed points | If $X$ is a 1-dimensional scheme with no closed points, let $x\in X$ be a point. Since $x$ is not closed, then there is some point $y\neq x$ in its closure. Since $y$ is not closed, there is a point $z\neq y$ in its closure. Now we have a chain $$\overline{\{z\}}\subset\overline{\{y\}}\subset\overline{\{x\}}$$ of irred... | 8 | https://mathoverflow.net/users/75 | 328303 | 141,179 |
https://mathoverflow.net/questions/328306 | 18 | The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H\_n(M;\mathbb{Z}\_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=\sum v\_i\in H^\*(M;\mathbb{Z}\_2)$ is the unique cohomology class such that
$$\langle v\cup x,z\rangle=\langle Sq(x... | https://mathoverflow.net/users/102515 | Wu formula for manifolds with boundary | A relative Wu formula for manifolds with boundary is discussed in Section 7 of
*Kervaire, Michel A.*, [**Relative characteristic classes**](http://dx.doi.org/10.2307/2372561), Am. J. Math. 79, 517-558 (1957). [ZBL0173.51201](https://zbmath.org/?q=an:0173.51201).
In particular, there are *relative Wu classes* $U^q\... | 18 | https://mathoverflow.net/users/8103 | 328316 | 141,182 |
https://mathoverflow.net/questions/328262 | 2 | Let $G$ be a reductive group over $\mathbb{Q}$ and let $P\_0$ minimal parabolic subgroup.
>
> If $P\_1=M\_{1}N\_{1} \subset P\_2=M\_2N\_2$ are standard parabolic subgroups of $G$, then can we decompose $P\_1=(P\_1 \cap M\_2 )N\_2$?
>
>
>
It seems it does hold but I don't know why it holds.
Any comments are we... | https://mathoverflow.net/users/29422 | Decomposition of parabolic subgroup in reductive group | Over an algebraically closed field (of any characteristic), it is fairly obvious using the omitted definition of "standard parabolic" that the assertion here for a pair of included standard parabolics is true: one just wants to trace the pairs of positive and negative roots involved. However, the question is too loosel... | 2 | https://mathoverflow.net/users/4231 | 328324 | 141,183 |
https://mathoverflow.net/questions/328300 | 5 | Let $\{v\_n\}\_{n \in \mathbb{N}}$ be a Schauder basis of $V$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $V\_m = \overline{\operatorname{span}} \{v\_n\}\_{n \geq m}$
Let $\{u\_n\}\_{n \in \mathbb{N}}$ be a Schauder basis of $U$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \... | https://mathoverflow.net/users/108867 | Intersection of spaces with Schauder basis | In general it is not true: $V\_m$ and $U\_m$ could even be transverse for all $m$, giving
$$
\bigcap\_{m=1}^\infty
\left(
V\_m + U\_m
\right)
=
\ell\_2.
$$
Let $\kappa:\mathbb{N}\to\mathbb{N}$ be a map such that every $p\in\mathbb{N}$ has fiber $\kappa^{-1}(p)$ of infinite cardinality.
Let $\{v\_n\}\_{n\ge1}$ be t... | 6 | https://mathoverflow.net/users/6101 | 328327 | 141,185 |
https://mathoverflow.net/questions/328270 | 3 | **Question.** Let $K$ be a field (assume $K=\mathbb{C}$ if this simplifies the problem). What is the right global dimension of the $K[x,y]$-algebra:
$$
A=\left[\begin{array}{cc}
K[x,y] & xK[x,y] \\
K[x,y] & K[x,y]
\end{array}\right] .
$$
I am not an expert in homological algebra so I would also appreciate advice on how... | https://mathoverflow.net/users/86006 | Global dimension of a certain $K[x,y]$-algebra | Have a look at the paper:
Kirkman, Ellen; Kuzmanovich, James; Matrix subrings having finite global dimension.
J. Algebra 109 (1987), no. 1, 74–92.
Using Theorem 1.6 one has:
$\operatorname{rgldim}(A) = \max\{\operatorname{rgldim}(K[x,y]), \operatorname{rgldim}(K[x,y]/(xK[x,y])+1\} = 2$
if I am not misunderstandi... | 5 | https://mathoverflow.net/users/130741 | 328329 | 141,186 |
https://mathoverflow.net/questions/328322 | 4 | Dear Colleagues and Friends,
Here's a question that I hope some of you, more experienced in programming, can answer.
Once SnapPy is used to compute the symmetry group of a hyperbolic manifold by way of, say,
>
> K = Manifold('4\_1');
>
>
> S = K.symmetry\_group();
>
>
>
it outputs the action of the isome... | https://mathoverflow.net/users/39331 | SnapPy isometry routine | As Ryan says, "isomorphisms of triangulations" are computed under the hood in snappy (and that is how it computes symmetry groups). However, this functionality is not shown to the user.
If you have a convincing use case, you could contact Nathan Dunfield and Marc Culler with your request.
| 4 | https://mathoverflow.net/users/1650 | 328339 | 141,190 |
https://mathoverflow.net/questions/328347 | 5 | I am reading [this](https://arxiv.org/pdf/1303.5664.pdf) paper and I came across the following sentence:
>
> Throughout the paper we silently assume [...] that ***the density character (i.e. the minimum cardinality of a
> dense subset) of every metric space is an Ulam number***. This guarantees that every finite ... | https://mathoverflow.net/users/100976 | Density character of a metric space is an Ulam number | Two mutually equivalent definitions of an Ulam number are given in Section [Weak inaccessibility of real-valued measurable cardinals](https://en.wikipedia.org/wiki/Measurable_cardinal#Weak_inaccessibility_of_real-valued_measurable_cardinals).
**Added in response to a comment by the OP:** First here, the first of the... | 5 | https://mathoverflow.net/users/36721 | 328352 | 141,193 |
https://mathoverflow.net/questions/328344 | 2 | Is there a locally countable and weakly Lindelöf space which is not ccc?
A space $X$ is locally countable if for each point $x\in X$ there is an open neighbourhood $O\_x$ of $x$ such that $|O\_x| < \omega\_1$;
Here ccc denotes the countable chain condition;
A space $X$ is called weakly Lindeöf if for any open c... | https://mathoverflow.net/users/138518 | Is there a locally countable and weakly Lindelöf space which is not ccc | No, because "locally countable + weakly Lindelöf" $\Rightarrow$ separable $\Rightarrow$ ccc.
For the first implication, use local countability to choose for each $x \in X$ some countable open neighborhood $O\_x \ni x$. Then $\mathcal U = \{O\_x \,:\, x \in X\}$ is an open cover of $X$, and we may use the weakly Linde... | 4 | https://mathoverflow.net/users/70618 | 328358 | 141,195 |
https://mathoverflow.net/questions/328281 | 2 | Does the problem
$$(-\Delta)^s u=\lambda u \text{ in } \mathbb R^N $$ admit a non-trivial solution when $s\in (0, 1)$ and $\lambda>0.$
| https://mathoverflow.net/users/132031 | Eigenvalue problem involving fractional laplacian | This is too long for a comment, and actually there is a number of questions that appeared in the comments, so let me summarize my points here.
1. The answer to the original question: is there a solution to $(-\Delta)^s u = \lambda u$, is positive: $u(x) = \cos(\lambda^{1/(2s)} x\_1)$ is an example.
More generally, ... | 2 | https://mathoverflow.net/users/108637 | 328365 | 141,199 |
https://mathoverflow.net/questions/328400 | 4 | My question might be very naive.
Let $X$ and $Y$ be reduced schemes of finite type over an algebraically closed field $k$ and $f:X\longrightarrow Y$ a morphism of $k$-schemes. Let us assume that $Y$ is normal, connected and that $f$ induces a bijection on closed points. Can we conclude from these assumptions that $f$... | https://mathoverflow.net/users/83945 | Morphism which is a bijection on closed points | If $k$ is algebraically closed of non-zero characteristic $p$, then bijectivity on closed points is not enough (take the morphism $\mathrm{Spec}\,k[x]\rightarrow \mathrm{Spec}\,k[x]$ corresponding to $f(T)\rightarrow f(T^p)$).
If $k$ is algebraically closed of characteristic 0, then bijectivity on closed points [is](... | 5 | https://mathoverflow.net/users/nan | 328407 | 141,209 |
https://mathoverflow.net/questions/328208 | 1 | Consider the following ODE initial value problem
\begin{align\*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \mathbb{R}^N.
\end{align\*}
We say that $\Phi: [0,T] \times \mathbb{R}^N \to \mathbb{R}^N$ is the flow of the ODE (as in [this paper]... | https://mathoverflow.net/users/122620 | Growth assumption and example of finite (arbitrarily small) time blow up for ODE | The definition of $\Phi$ you are considering is different from the one in the paper, which is Definition 13:
>
> Given a certain class $\mathcal L\_b$ of measure-valued solutions of the continuity equation (*see the paper for the details*), $\Phi(t,x)$ is a $\mathcal L\_b$-lagrangian flow of $b$ starting from a mea... | 2 | https://mathoverflow.net/users/44463 | 328408 | 141,210 |
https://mathoverflow.net/questions/328269 | 3 | A left shelf $(S, \rhd)$ is a magma with the left self-distributive law:
$$
\forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z).
$$
Shelves are generalization of [racks and quandles](https://en.wikipedia.org/wiki/Racks_and_quandles) from the knot theory.
I am looking for examples of shelves with the... | https://mathoverflow.net/users/22795 | Shelves with "trichotomy" | Linearly ordered sets when considered as lattices satisfy this property. Suppose that $(X,\wedge)$ is a meet-semilattice with corresponding partial ordering $\leq$. Then $\wedge$ is a self-distributive operation. Furthermore, $x\wedge y=y$ if and only if $y\leq x$. Therefore, the meet-semilattice $(X,\wedge)$ satisfies... | 2 | https://mathoverflow.net/users/22277 | 328409 | 141,211 |
https://mathoverflow.net/questions/328403 | 2 | Let $P\_1$ and $P\_2$ be complex polynomials with complex coefficients and $c > 0$. Can we find polynomial $P\_3$ and $c’>0$ such that
>
> $\{z \in \mathbb C : |P\_1(z)| \geq c\} \cap \{ z \in \mathbb C : |P\_2(z)| \geq c\}= \{ z \in \mathbb C : |P\_3(z)| \geq c’\}$
>
>
>
holds?
It seems like this is either ... | https://mathoverflow.net/users/130742 | Intersection of superlevel set of polynomials | You are right: this is trivially false.
Let $P\_1(z)=z^2,\; P\_2(z)=(z-a)^2$, and $c=1$. Then the boundaries of the first two level sets are circles of radius $1$,
and choosing an appropriate $a$ you can make them cross at any given angle.
On the other hand the boundary of the set in the RHS is a polynomial lemnisc... | 3 | https://mathoverflow.net/users/25510 | 328415 | 141,214 |
https://mathoverflow.net/questions/328399 | 2 | I need a help about the following:
Maple gave that the following equality is true for n =1,2,3,4,5, $$ \sum\_{h=0}^{\infty}\binom{n+h}{n}{\_3}F\_2\left( \substack{-h,n+1,n+1\\ 1,1}; x\right)= \frac{1}{x^{n+1}}.$$
Note that ${\_3}F\_2\left( \substack{-h,n+1,n+1\\ 1,1}; x\right)$ is a polynomial of degree $h$.
I woul... | https://mathoverflow.net/users/136807 | question about equality series containing hypergeometric term and a simple term | By equation (1.22) in
*Nørlund, Niels Erik*, [**Hypergeometric functions**](http://dx.doi.org/10.1007/BF02392494), Acta Math. 94, 289-349 (1955). [ZBL0067.29402](https://zbmath.org/?q=an:0067.29402).
the sum in question equals
$$\lim\_{z\to 1} \frac{1}{(1-z)^{n+1}}\ {\_3}F\_2\left({n+1,n+1,n+1\atop 1,1}; \frac{xz}... | 1 | https://mathoverflow.net/users/7076 | 328419 | 141,216 |
https://mathoverflow.net/questions/328396 | 12 | Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra.
Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$.
**Question**: Is there any criterion for when $A = k[H]$ for some subgroup $H$? Also, in that case, how do we read of the generating subgroup $... | https://mathoverflow.net/users/124549 | Subalgebra of a group algebra | The characteristic of the field is important here, when considering Hopf sub-algebras. The Cartier-Kostant-Milnor-Moore theorem says that a cocommutative Hopf algebra $H$ over an algebraically closed field $k$ of characteristic 0, is a semidirect product of a group algebra and an enveloping algebra of a Lie algebra. In... | 10 | https://mathoverflow.net/users/130741 | 328429 | 141,217 |
https://mathoverflow.net/questions/328437 | 5 | It is a well-known fact that $S^2 \times S^1$ can be obtained by $0$-surgery on unknot.
What about the $(-1)$-surgery on $S^2 \times S^1$? It seems the resulting manifold, say $W$, bounds contractible manifold.
But I cannot prove it yet or refutes my argument. Any help will be appreciated.
| https://mathoverflow.net/users/138574 | Integral surgery on $S^2 \times S^1$ | This is true, with some points to clarify. First, you are presumably talking about surgery along a knot that generates the first homology (and hence fundamental group) of $S^1\times S^2$. Then the result of adding the corresponding 2-handle to $S^1\times B^3$ is contractible. The construction (called a `Mazur manifold'... | 8 | https://mathoverflow.net/users/3460 | 328441 | 141,219 |
https://mathoverflow.net/questions/328426 | 4 | In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$
with all objects cofibrant and weak equivalences stable under filtered colimits, the category $\mathbf{Cat\_S}$ is a left proper combinatorial model ctegory, ... | https://mathoverflow.net/users/123746 | Why is the category of all small $\mathbf{S}$-enriched categories locally presentable? | This is proved in the paper
>
> Kelly, G. M.; Lack, Stephen. $\scr V$-Cat is locally presentable or locally bounded if $\scr V$ is so. *Theory Appl. Categ.* 8 (2001), 555--575. <http://tac.mta.ca/tac/volumes/8/n23/8-23abs.html>
>
>
>
as an instance of the fact (proved by Gabriel and Ulmer) that the category of... | 6 | https://mathoverflow.net/users/57405 | 328447 | 141,220 |
https://mathoverflow.net/questions/328449 | 18 | [Lagrange's four-squares theorem](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem) states that every natural number can be represented as the sum of four integer squares. [Rabin and Shallit](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem#Algorithms) gave a randomised algorithm that finds ... | https://mathoverflow.net/users/138575 | Lagrange four-squares theorem --- deterministic complexity | As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper [Finding the four squares in Lagrange's theorem](http://campus.lakeforest.edu/trevino/finding4squares.pdf) by Pollack and Treviño. They mention that there is a deterministic polynomial-time algorithm when $n$ is a prime via q... | 19 | https://mathoverflow.net/users/2233 | 328455 | 141,222 |
https://mathoverflow.net/questions/328454 | 6 | I am trying to read Mumford's *Geometric Invariant Theory*, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's *Algebraic Geometry*. Mumford cites a lot of results from EGA and SGA, but I cannot read French. Therefore, I want to ask:
>
> **What should I r... | https://mathoverflow.net/users/138583 | Preparation for GIT (Geometric Invariant Theory) | Before I started reading/referencing *Geometric Invariant Theory* by Mumford, Fogarty, Kirwan I read Dolgachev's book *Lectures on Invariant Theory*. I also like Peter Newstead's book *Introduction to Moduli Problems and Orbit Spaces*. Given your stated interest, these treatments might be sufficient. Even if they are n... | 9 | https://mathoverflow.net/users/12218 | 328459 | 141,225 |
https://mathoverflow.net/questions/328461 | 5 | Denote by $\pi(x)$ the number of primes $\leq x$. I'm interested in knowing who came up with the **first elementary proof** that $\pi(x)=o(x)$.
I know that Chebyshev demonstrated elementarily before Hadamard and de la Vallee-Poussin the slightly stronger result that $\pi(x)=O(x/\log x)$.
| https://mathoverflow.net/users/480516 | What was the first elementary proof that $\pi(x)=o(x)$? | Leonhard Euler knew that the infinite product:
$$ \prod\_{p \textrm{ prime}} \left(1 - \frac{1}{p} \right)^{-1} = \sum\_{n=1}^{\infty} \frac{1}{n} $$
is divergent (and used this to prove the infinitude of primes), so would have also known that the product:
$$ \prod\_{p \textrm{ prime}} \left(1 - \frac{1}{p} \righ... | 4 | https://mathoverflow.net/users/39521 | 328464 | 141,227 |
https://mathoverflow.net/questions/328378 | 7 | Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$.
Is it true that $\operatorname{dim} H^1(A)=2g$? I think this will follow from some standard conjectures, but do we know this unc... | https://mathoverflow.net/users/102104 | Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology? | Yes, it's the same for all Weil cohomology theories --- see 2A.8 of Kleiman, S. L. Algebraic cycles and the Weil conjectures. Dix exposés sur la cohomologie des schémas, 359--386, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968.
| 6 | https://mathoverflow.net/users/137572 | 328471 | 141,229 |
https://mathoverflow.net/questions/328100 | 4 | A regular topological space $X$ is called
$\bullet$ *analytic* if $X$ is a continuous image of a Polish space;
$\bullet$ *$K$-analytic* if $X$ is the image of a Polish space $P$ under an upper semicontinuous compact-valued map $\Phi:P\multimap X$.
It is well-known that an analytic space $X$ is countable if and o... | https://mathoverflow.net/users/61536 | K-analytic spaces whose any compact subset is countable | I looked at the paper of Fremlin and have seen that a minor modification of his example yields the following theorem showing that my question is independent of ZFC.
>
> **Theorem.** The following statements are equivalent:
>
>
> 1) $\omega\_1<\mathfrak b$;
>
>
> 2) A K-analytic space $X$ is analytic if and only... | 1 | https://mathoverflow.net/users/61536 | 328485 | 141,233 |
https://mathoverflow.net/questions/328423 | 3 | (This is a follow-up question; [the original question](https://mathoverflow.net/questions/328269) was about *shelves*.)
A [rack](https://ncatlab.org/nlab/show/rack) $(R, \rhd, \lhd)$ is a set $R$ with two binary operations $\rhd$ and $\lhd$ such that for all $x, y, z \in R$:
1. $x \rhd (y \rhd z) = (x \rhd y) \rhd ... | https://mathoverflow.net/users/22795 | Racks with "trichotomy" | There are no such racks. We shall first show that quandles with multiple elements cannot satisfy the Trichotomy property. We shall then show that every rack that satisfy the Trichotomy property must be a quandle.
Suppose that $(Q,\*,\*^{-1})$ is a quandle ($\*$ is left-distributive i.e. $x\*(y\*z)=(x\*y)\*(x\*z)$). T... | 3 | https://mathoverflow.net/users/22277 | 328487 | 141,234 |
https://mathoverflow.net/questions/328495 | 9 | Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect\_k$ for a field $k$ (of characteristic $0$) there exists a reductive algebraic group $G \cong Aut(F)$ such that $\mathcal{C} \cong Rep(G)$. This means that any su... | https://mathoverflow.net/users/101861 | Tannaka duality for semisimple groups | Another criterion is that there should be only finitely many objects of bounded dimension. This condition might be easy to check in practice from abstract finiteness theorems. The proof is that, if the group is not semi simple, you can take any 1-dimensional character of the identity component and induce up to the main... | 8 | https://mathoverflow.net/users/18060 | 328500 | 141,237 |
https://mathoverflow.net/questions/328438 | 1 | Let $R$ be a ring and let $f:P\rightarrow P'$ be a surjective morphism of smooth $R$-algebras. Let $J$ be the kernel of this map. If $R$ is Noetherian, one can show that $J$ is locally generated by a regular sequence. I am looking for a reference where the non-noetherian case is studied: I don't expect the existence of... | https://mathoverflow.net/users/62127 | Koszul -regular sequences (reference request) | In the stacks project this can be found as [Tag 067U](https://stacks.math.columbia.edu/tag/067U) stated in the language of schemes.
| 1 | https://mathoverflow.net/users/138600 | 328502 | 141,239 |
https://mathoverflow.net/questions/328504 | 4 | What is an example of a $p$-adic representation of the absolute Galois group of a $p$-adic field that has finite image on the inertia subgroup, but is not crystalline?
| https://mathoverflow.net/users/138598 | Finite image but not crystalline | If a crystalline representation has finite image when restricted to inertia, then this restriction has to be trivial.
Indeed, suppose that $K$ is a discretely valued extension of $\mathbb{Q}\_p$ and $\rho:G\_K\to GL(V)$ is a crystalline representation which has finite image when restricted to the inertia subgroup. R... | 4 | https://mathoverflow.net/users/39304 | 328508 | 141,241 |
https://mathoverflow.net/questions/328467 | 2 | Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the [weak $L^1$ space](https://en.wikipedia.org/wiki/Lp_space#Weak_Lp) such that
$$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$
holds for a.e. $x,y$?
---
This question is motivated by [Convergence of the diff... | https://mathoverflow.net/users/122620 | Weak Lebesgue spaces and an estimate for BV functions | A result of this kind can be found in [1] (Lemma A.3):
>
> If $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ such that
> $$
> |u(x) - u(y)| \le c\_N |x-y| (M\_R Du(x) + M\_R Du(y))
> $$
> for $x,y\in \mathbb R^N \setminus F$ with $|x-y|\le R$.
>
>
>
Here
$$
M\_R Du... | 1 | https://mathoverflow.net/users/44463 | 328526 | 141,244 |
https://mathoverflow.net/questions/328372 | 3 | I am writing a paper on K-analytic spaces and need the following known characterization.
>
> **Theorem.** For a regular topological space $X$ the following conditions are equivalent:
>
>
> (1) $X$ is a continuous image of a Lindelof Cech-complete space;
>
>
> (2) $X$ is the image of a Polish space under an uppe... | https://mathoverflow.net/users/61536 | A reference for a (folklore?) characterization of K-analytic spaces | See "Descriptive Topology" by R.Hansell in "Recent Progress in General Topology" (1992), p. 281-282
| 1 | https://mathoverflow.net/users/112448 | 328528 | 141,245 |
https://mathoverflow.net/questions/328469 | 16 | The Eulerian number $A(n,m)$ is defined as the number of permutations $\sigma \in S\_n$ having precisely $m$ descents, i.e. indices $i$ such that $\sigma(i)>\sigma(i+1)$.
The wikipedia entry on these numbers mentions one identity for which I cannot locate a proof, and I would be grateful for a reference. Namely
$$\s... | https://mathoverflow.net/users/10901 | Eulerian number identity | Here's a proof of the identity. Unfortunately, it doesn't show how anyone would come up with this formula. Like the proofs referred to in the comments, this proof is based on the beta integral
$$\int\_0^1 u^m (1-u)^{n-m}\, du = \frac{m!\,(n-m)!}{(n+1)!} = \frac{1}{(n+1)\binom{n}{m}}.$$
By the generating function for... | 21 | https://mathoverflow.net/users/10744 | 328534 | 141,249 |
https://mathoverflow.net/questions/328532 | 0 | Given some non-empty set $X$, does the set of positive definite kernels on $K\_X$ form a ring with pointwise addition and multiplication. I am convinced it does not as surely if $k \in K\_X$ then we would require that $-k \in K\_X$ also, which I don't think is true?
I ask because Hagan (the user) seems to think it d... | https://mathoverflow.net/users/138625 | Does the set of positive definite kernels on some set X form a ring? | To make it answered (cw):
>
> It's not stable under taking $x\mapsto -x$ so is not a ring, as it's not a group under addition.
>
>
>
| 0 | https://mathoverflow.net/users/14094 | 328547 | 141,252 |
https://mathoverflow.net/questions/328552 | 13 | Denote by $\mu$ the Mobius function. It is known that for every integer $k>1$, the number $\sum\_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen integer is $k$-free.
Letting $k\rightarrow 1^+$, why shouldn't this entail the Prime Number Theorem in the form
$$\sum\_{n... | https://mathoverflow.net/users/480516 | Why shouldn't this prove the Prime Number Theorem? | You ask:
>
> Denote by $\mu$ the Mobius function. It is known that for every integer $k>1$, the number $\sum\_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen integer is $k$-free.
>
>
> Letting $k\rightarrow 1^+$, why shouldn't this entail the Prime Number Theorem i... | 20 | https://mathoverflow.net/users/17773 | 328557 | 141,256 |
https://mathoverflow.net/questions/328524 | 5 | Suppose that $T$ is a bounded set in $\mathbb{R}^n$ and $f,g$ are two nonnegative functions such that $0\leq f(x)\leq g(x)$ for all $x\geq 0$.
Let $\epsilon\_1,\epsilon\_2,\dots,$ be a Rademacher sequence. Does it hold that
$$
\mathbb{E} \sup\_{t\in T} \left| \sum\_i \epsilon\_i f(|t\_i|) \right| \leq C\cdot \mathbb{... | https://mathoverflow.net/users/48609 | Comparison of Rademacher processes | $\newcommand{\ep}{\varepsilon}$
Letting $a\_{it}:=f(|t\_i|)$ and $b\_{it}:=g(|t\_i|)$, rewrite your inequality ($\ast$) as
\begin{equation}
E\sup\_{t\in T}\Big|\sum\_{i=1}^N\ep\_i a\_{it}\Big|\le C\,E\sup\_{t\in T}\Big|\sum\_{i=1}^N\ep\_i b\_{it}\Big|
\end{equation}
for any natural $N$ and any set $T$,
with the cond... | 3 | https://mathoverflow.net/users/36721 | 328559 | 141,257 |
https://mathoverflow.net/questions/328570 | 5 | I am looking for a reference, preferably as elementary as possible, for the following statement.
Let $X\_{m,n}$ be a bi-simplicial object in an **additive** category $\mathcal{A}$. Then the complex $|X\_{m,n}|$ obtained from iterative realization (i.e. the total complex of the associated bi-complex) is homotopy equi... | https://mathoverflow.net/users/115052 | Simplicial Objects in Additive Categories | For abelian categories this is known as the Eilenberg–Zilber theorem, see, for instance, Theorem 8.5.1 in Weibel's book. One can write down explicit comparison maps in both directions (namely, the Alexander–Whitney and Eilenberg–Zilber maps) and also write down an explicit chain homotopy that shows these maps to be cha... | 2 | https://mathoverflow.net/users/402 | 328580 | 141,261 |
https://mathoverflow.net/questions/328541 | 16 | The Kakeya conjecture posits that any [Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture) in $\mathbb{R}^n$ has dimension $n$.
A discrete (finitized?) version of this problem is the [Finite Field Kakeya conjecture](https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-... | https://mathoverflow.net/users/138628 | Examples of problems where considering "discrete analogues" has provided insight or led to a solution of the original problem | As a non-expert, I will tell a story (since I am not qualified to do more):
Once upon a time (1859) there was conjecture about the zeros of a complex function known as the [Riemann Hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis) (RH).
Years later Weil formulated a "discrete version" (in terms of fini... | 5 | https://mathoverflow.net/users/12218 | 328582 | 141,262 |
https://mathoverflow.net/questions/328583 | 1 | It is well known that there can be at most 20 consecutive integers (in base 10) that are divisible by their digital sum, so called Harshad or Niven numbers.
How long can a run of consecutive integers be in which each number is not relatively prime to its digital sum?
| https://mathoverflow.net/users/60732 | Runs of consecutive numbers that are not relatively prime to their digital sum | As long as you wish. Let $s(n)$ denote the sum of decimal digits of $n$.
**Lemma.** For any positive integer $k$ there exist distinct prime numbers
$p\_1<p\_2<\ldots<p\_k$ such that $p\_1>5$ and $10^{P}-1$ is coprime to $P:=p\_1p\_2\ldots p\_k$.
**Proof.** Induction in $k$. For $k=1$ choose $p\_1=7$. If $k$ primes ... | 4 | https://mathoverflow.net/users/4312 | 328589 | 141,264 |
https://mathoverflow.net/questions/328592 | 0 | Let $f:X\rightarrow Y$ be a morphism of schemes. Can the image of $f$ endowed with the subspace topology not be sober?
| https://mathoverflow.net/users/nan | Morphism of schemes with non-sober image | Let $X$ be a disjoint union of $\operatorname{Spec}\mathbb F\_p$ over primes $p$ and consider the obvious map $X\to\operatorname{Spec}\mathbb Z$. The image $\operatorname{Spec}\mathbb Z\setminus\{(0)\}$ is not sober.
| 3 | https://mathoverflow.net/users/30186 | 328593 | 141,265 |
https://mathoverflow.net/questions/309342 | 14 | *Weak Vopěnka's principle* says that
>
> the opposite of the category of ordinals cannot be fully embedded in any locally presentable category.
>
>
>
Recall that one form of Vopěnka's principle says that the category of ordinals cannot be fully embedded in any locally presentable category.
Adámek and Rosický... | https://mathoverflow.net/users/2362 | What is the consistency strength of weak Vopenka's principle? | The weak Vopěnka principle (WVP) and semi-weak Vopěnka principle (SWVP) are both equivalent to the large cardinal principle "Ord is Woodin". These results now appear in a [paper on the arXiv](https://arxiv.org/pdf/1907.00284.pdf). I kept the proof that Ord is Woodin implies SWVP as part of this MathOverflow answer (see... | 14 | https://mathoverflow.net/users/1682 | 328605 | 141,269 |
https://mathoverflow.net/questions/254714 | 3 | Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}\_A$ by $\mathtt{P}\_A(r) = A^{\otimes r}$ (so that $\mathtt{P}\_A(0) = 1$ is the unit of $\otimes$), and the composition is given by the following formula:
$$\gamma(a\_1 \otimes \dots \oti... | https://mathoverflow.net/users/36146 | An interpretation of this construction giving an operad from a bialgebra? | It might be worth to first consider the particular case of the symmetric monoidal category $({\rm Set},\times)$ of sets and Cartesian products. Let us mildly extend the setting to include possibly multi-colored operads. Then your construction becomes a particular case of the construction which associates to a category ... | 3 | https://mathoverflow.net/users/51164 | 328606 | 141,270 |
https://mathoverflow.net/questions/328585 | 8 | Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite sequence $v\_1, v\_2, \ldots.$ Is it the case that, almost surely, the closed span of $v\_1, v\_2, \ldots$ is all of $\mathc... | https://mathoverflow.net/users/32470 | Does a random sequence of vectors span a Hilbert space? | (This may turn out to be a simplified version of J. E. Pascoe's answer).
The *support* of (the distribution of) $v$, that we denote by $\operatorname{supp} v$, is the set of vectors $h \in \mathcal{H}$ such that $P(v \in B(h, \varepsilon)) > 0$ for every $\varepsilon > 0$. We list some properties of this set.
1. Th... | 6 | https://mathoverflow.net/users/108637 | 328612 | 141,273 |
https://mathoverflow.net/questions/328607 | 8 | I am looking for a reference for the following inequality: if $\Omega \subset [0,1]^d$ satisfies $\mbox{vol}(\Omega) \leq 1/2$, then
$$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq c\_d \mbox{vol}(\Omega)^{\frac{d-1}{d}},$$
where $\mathcal{H}^{d-1}$ is the $(d-1)-$dimensional Hausdorff measure and $c... | https://mathoverflow.net/users/138664 | An isoperimetric-type inequality inside a cube | This result is known as the *relative isoperimetric inequality*, see e.g. *Functions of Bounded Variation* by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).
It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $\chi\_\Omega$ (the indicator of the set $\Omega$). Indeed, by P... | 7 | https://mathoverflow.net/users/44463 | 328617 | 141,275 |
https://mathoverflow.net/questions/328619 | 11 | This is not a research problem, but challenging enough that I've decided to post it in here:
>
> Determine all triples $(a,b,c)$ of non-negative integers, satisfying
> $$
> 1+3^a = 3^b+5^c.
> $$
>
>
>
| https://mathoverflow.net/users/127150 | Diophantine equation $3^a+1=3^b+5^c$ | I can't resist this: The young [Chris Skinner](https://www.sciencedirect.com/science/article/pii/0022314X90901125) showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^w
$$
has a bounded number of solutions in $(x,y,... | 29 | https://mathoverflow.net/users/38624 | 328621 | 141,277 |
https://mathoverflow.net/questions/328624 | 10 | Say we have some symmetric positive definite $n\times n$ matrix $M$ with $n$ distinct eigenvalues $\{\lambda\_1,...,\lambda\_n\}$. Is there a general formula for the maximum angle $\theta$ for which $M$ can rotate some vector, in terms of matrix invariants?
I worked out the $2\times 2$ case and the answer is
$$\thet... | https://mathoverflow.net/users/138671 | Maximum rotation made by a symmetric positive definite matrix? | As explained in these [notes](https://calculus7.org/2013/05/04/condition-number-and-maximal-rotation-angle/), the maximum rotation angle $\theta$ of a symmetric positive definite matrix $M$ is related to the condition number $K=\mu\_{\rm max}/\mu\_{\rm min}$ of the matrix (the ratio of largest and smallest eigenvalue) ... | 20 | https://mathoverflow.net/users/11260 | 328635 | 141,283 |
https://mathoverflow.net/questions/328349 | 0 | We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum\limits\_{k=1}^{m} p\_k (f\circ f\_k)(x):=\mathbb E( f(X\_{n+1}|X\_n=x)$ for a system $X\_{n+1}=f\_{\omega\_n}(X\_n), n=0,1,2\dots.$ and $\omega\_n$ are i.i.d discrete r.v over $\{1,2,\dots,m\}$, $p\_k=\text{ Prob} (\omega\_i=k)$, $f\_k$ are bounded Lipschitz.
Could anyo... | https://mathoverflow.net/users/93713 | Expectation of a linear operator | By the tower law, $$ \mathbb{E} [f(X\_n) | X\_0=x] = \mathbb{E} [\mathbb{E} \{f(X\_n) | X\_{n-1}\} | X\_0=x]. $$
Observe that $\mathbb{E} \{f(X\_n) | X\_{n-1}\}=Tf(X\_{n-1})$ (because $(X\_n)\_{n\in \mathbb{N}}$ forms a time-homogeneous Markov chain) . By induction it follows that $E[f(X\_n) | X\_0=x]=T^n f(x)$ and you... | 2 | https://mathoverflow.net/users/138576 | 328644 | 141,285 |
https://mathoverflow.net/questions/328615 | 11 | Do you know a place where the integral cohomology of $G\_2$-homogeneous spaces is computed?
Great computational efforts using representation theory in order to determine the characteristic classes of homogeneous spaces were done by Borel and Hirzebruch in a series of papers:
1. Characteristic classes and homogeneo... | https://mathoverflow.net/users/21985 | Reference requests: Integral cohomology of $G_2$-homogeneous spaces | I don't know how "systematic" the answer you are looking has to be, but for the quotients of the form $G\_2/T$ and $G\_2/P$ (P: Parabolic subgroup) you can find the results in [Schubert presentation of the integral cohomology ring of the flag manifolds G/T, by Haibao Duan and Xuezhi Zhao](https://arxiv.org/pdf/0801.244... | 4 | https://mathoverflow.net/users/43326 | 328646 | 141,286 |
https://mathoverflow.net/questions/328618 | 6 | Two eulerian trails of $K\_{2n+1}$ are defined to be equivalent if the orientations obtained by orienting the edges as traversed by the trails are isomorphic as digraph. How many non-equivalent trails are there?
| https://mathoverflow.net/users/23850 | Non-equivalent eulerian trails in $K_{2n+1}$ | If I understand your question, you are asking for the number of isomorphism classes of regular tournaments.
There are no exact formulas. See <http://oeis.org/A096368> for counts up to 15 vertices.
The asymptotic number is known. It is $RT(2n+1)/(2n+1)!$, where $RT(2n+1)$ is given in the abstract of [this paper](htt... | 4 | https://mathoverflow.net/users/9025 | 328648 | 141,287 |
https://mathoverflow.net/questions/328657 | 23 | Inspired by the well known $$\int\_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given [here](https://math.stackexchange.com/questions/908108/how-to-find-large-int-01-frac-ln31x-ln-xx-mathrm-dx) (writing $\zeta\_r:=\zeta(r)$ for easier reading)$$\int\_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=12\zeta(5)-\pi... | https://mathoverflow.net/users/29783 | Why these surprising proportionalities of integrals involving odd zeta values? | For $n\geq 1$ and $m\geq 0$, an application of integration by parts ($u=\log^n(1-x)$, $dv=\log^m(x)\,dx/x$) followed by the substitution $x\mapsto 1-x$ shows that
$$
\frac{I\_{n,m}}{I\_{m+1,n-1}}=\frac{n}{m+1}.
$$
All of your examples are special cases of this identity.
| 39 | https://mathoverflow.net/users/5263 | 328661 | 141,289 |
https://mathoverflow.net/questions/328655 | 1 | Let $n>1$ be an integer and let $[n]=\{1,\ldots,n\}$. An *intersecting family* on $[n]$ is a set $E\subseteq {\cal P}([n])$ such that for all $a,b\in E$ we have $a\cap b\neq\emptyset$. It is easy to see that an intersecting family on $[n]$ can have size $2^{n-1}$: fix $j\in[n]$ and let $E = \{s\subseteq [n]:j\in s\}$. ... | https://mathoverflow.net/users/8628 | Size of intersecting families on $\{1,\ldots,n\}$ | For $n > 2$ the answer to question 1 is no. If $n$ is odd take $E$ to be all subsets of size greater than $n/2$, and if $n$ is even take all subsets of size greater than $n/2$ along with all sets of size $n/2$ that *exclude* a certain element.
| 3 | https://mathoverflow.net/users/112113 | 328664 | 141,290 |
https://mathoverflow.net/questions/328654 | 2 | Consider the following ODE:
$$f'(r) = f^2(r) + O \left( \frac{1}{r^4} \right)$$
as $r$ goes to infinity. The initial conditions are $f(1) = C <0$.
What is the behaviour of a solution $f$ at infinity? (not only the leading term).
This ODE is equivalent to:
$$w'' + O \left( \frac{1}{r^4} \right) w = 0$$
where $f... | https://mathoverflow.net/users/138705 | Asymptotic Behaviour of Solutions to a Riccati-type ODE with Small Forcing Term | In your example you obtained two linearly independent solutions with different behavior: $\cot(1/r)\sim r,\; r\to\infty$, so your second solution is $O(r^{-2})$.
This is the general pattern if you assume that your perturbation is analytic at $\infty$. Write your equation as
$$f'=f^2+q(r),\quad q(r)=r^{-4}\sum\_{0}^\... | 4 | https://mathoverflow.net/users/25510 | 328673 | 141,295 |
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