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/384912 | 2 | Let $X$, $Y\_1$ and $Y\_2$ be a compact Hausdorff spaces and let $\varphi\_i:X\to Y\_i$ be a continuous surjection (and so a quotient map).
Let $\sim$ be the minimal closed equivalence relation on $X$ that includes $(x,y)$ with $\varphi\_1(x)=\varphi\_1(y)$ and $\varphi\_2(x)=\varphi\_2(y)$ (recall that an equivalenc... | https://mathoverflow.net/users/53155 | Is the union of good equivalence relations on a compact space good? | It seems that the quostion about the skeletal property of $\psi$ has negative answer.
Let us recall that a map $f:X\to Y$ between topological spaces is *skeletal* if for any nonempty open set $U\subseteq X$ the set $\overline{f[U]}$ has non-empty interior in $Y$.
A corresponding counterexample looks as follows. Fix... | 2 | https://mathoverflow.net/users/61536 | 384920 | 160,041 |
https://mathoverflow.net/questions/384911 | 0 | I approximate a probability distribution $P\_x(x)$ with a $P\_x^{app}(x)$,
such that $P\_x(x)-P\_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic observable $Y = Y(x)$; $Y$ is a generic function of the random variable $x$, and so it is also a random variab... | https://mathoverflow.net/users/174176 | Perturbative approach starting from a probability distribution approximated form | $\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Translated into a standard language of probability theory, your question appears to be as follows:
>
> Let $p$ and $q$ be two probability density functions (say on $\R$) such that
> $$|p-q|\le\ep\tag{0}$$ for some real $\ep>0$. Let $Y$ be a Lebesgue-measurable functio... | 1 | https://mathoverflow.net/users/36721 | 384927 | 160,044 |
https://mathoverflow.net/questions/384919 | 20 | Given a linear system $Ax=b$, the pseudoinverse of $A$ is found as the matrix $A^+$ such that $x=A^+ b$ where $x$ solves the least squares problem $\min \| Ax - b \|^2 $ and $x \perp \mathcal{N}(A)$. That is, $x$ is the shortest vector in the solution space.
That is, find $x$ :
\begin{eqnarray}
\min \| x \| \text{ su... | https://mathoverflow.net/users/80663 | Is the pseudoinverse the same as least squares with regularization? | Here's one way to see this directly. I will assume that $A$ is $m \times n$. Let
$$ A = U \Sigma V^T$$
be the SVD for A. Recall that the regularized solution to the least squares problem $Ax = b$ is given by
$$\hat{x} = (A^TA + \lambda I)^{-1}A^Tb.$$
Now substitute the SVD of $A$ in place of $A$. Simplifying algebra, w... | 9 | https://mathoverflow.net/users/21278 | 384930 | 160,047 |
https://mathoverflow.net/questions/384692 | 6 | Let $SU\_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called **the quantum special unitary group**. For details see
<https://en.wikipedia.org/wiki/Compact_quantum_group>
(Note that on the Wikipedia page it is the $C^\*$-algebra that is discussed, but this question is about the dense Hopf algebra ... | https://mathoverflow.net/users/153228 | Invertible elements of the Hopf algebra quantum $SU(2)$ | Yes, the nonzero multiples of the identity $1$ are the only invertible elements in this algebra. I am sure that someone with more expertise in Hopf algebras than I, can provide a 'high level' proof of the result. The following is a 'low level' direct argument.
For the proof, I am using some notations and basic result... | 5 | https://mathoverflow.net/users/159170 | 384944 | 160,052 |
https://mathoverflow.net/questions/384934 | 7 | In the [lecture notes on condensed mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) the solidification of the free condensed abelian group $\mathbb{Z}[S]$ on a profinite set $S$ is defined as the inverse limit $\lim\_{\leftarrow} \mathbb{Z}[S\_i]$, but it can also alternatively be described as
$$... | https://mathoverflow.net/users/91925 | Solidification of free abelian group on compact Hausdorff space | Good question! You essentially already give the answer, but let me spell it out.
First, in the solid case, as you say one can compute the derived solidification $\mathbb Z[S]^\blacksquare$ for any compact Hausdorff $S$ as
$$
\mathbb Z[S]^\blacksquare = R\underline{\mathrm{Hom}}(R\Gamma(S,\mathbb Z),\mathbb Z),
$$
whe... | 10 | https://mathoverflow.net/users/6074 | 384945 | 160,053 |
https://mathoverflow.net/questions/384683 | 8 | A permutation avoiding a consecutive pattern $\underline{123}$ is permutation
$\pi = \pi\_1 \pi\_2 \ldots \pi\_n$ with the property that there does not exists $i \in [1, n-2]$
such that $\pi\_i < \pi\_{i+1} < \pi\_{i+2}$.
Example: $53241$ is $\underline{123}$-avoiding; while $314562$ is not $\underline{123}$ avoiding, ... | https://mathoverflow.net/users/31830 | Random permutations without double rises (avoiding consecutive pattern $\underline{123}$) | I have a sampling algorithm for you written in sage / python.
Instead of $\underline{123}$ avoiding permutations of length $n$ I consider words $w\_1 w\_2 \dots w\_n$ satisfying the following properties:
* $0\le w\_k \le n-k$
* the word is weakly $\underline{123}$ avoiding, which means there is no index $i\in [1,n-... | 3 | https://mathoverflow.net/users/174867 | 384947 | 160,054 |
https://mathoverflow.net/questions/384935 | 4 | We are given a $d$-dimensional hypercube $H$, where each vertex is labeled with an integer $\ell\in\{1, 2, \ldots, 2^d\}$. Let $L$ be this labelling.
---
**Question:** How many labelling permutations $L'$ of $L$ satisfy the property that *each* vertex $v$ of $H$ is labeled in $L'$ with one of the $d$-many labels ... | https://mathoverflow.net/users/115803 | Number of permutations with combinatorial geometric constraints | As indicated in the comments, the number in question is given by <https://oeis.org/A233001>, the square of the number of perfect matchings of the hypercube.
To see this, note that the hypercube $H\_n$ is a bipartite graph. Let $H\_n^b$ be a proper $2$-coloring of the vertices of $H\_n$ with colours black and white, a... | 3 | https://mathoverflow.net/users/3032 | 384951 | 160,056 |
https://mathoverflow.net/questions/384961 | 10 | I'm looking for a list of all (non-isomorphic) posets on 9 points. I know there are 183231 of them ([OEIS A000112](https://oeis.org/A000112)), but in order to progress with a problem I'm working on, I'd need the posets themselves, not only their number.
Background: A poset is a set $S$ together with a relation $\le$ ... | https://mathoverflow.net/users/30392 | Listing all posets on 9 points? | See the "Partially-ordered sets (posets)" section [here](http://users.cecs.anu.edu.au/%7Ebdm/data/digraphs.html), where Brendan McKay has provided the posets up to 10 points.
| 19 | https://mathoverflow.net/users/141766 | 384963 | 160,059 |
https://mathoverflow.net/questions/383189 | 4 | Given a ring $A$ and an ideal $I$, consider the completion $\hat{A}$. What does usually mean by a vector bundle on $\hat{A}$? One way is to consider projective $\hat{A}$-modules. Another one is a system of compatible vector bundles on $A/I^n$. Compatible mean there are isomorphisms of vector bundles when you pullback t... | https://mathoverflow.net/users/127776 | Vector bundles on complete rings | I will assume that $A$ is Noetherian (if you wish to work in the non-Noetherian setting, I'll see what I can do to modify my answer).
Without loss of generality, we may assume that $A$ is in fact $I$-adically complete. We know that
$$ \operatorname{Coh}( A) \stackrel{\sim}{\to} \varprojlim \operatorname{Coh}(A/I^{n+1... | 6 | https://mathoverflow.net/users/21278 | 384972 | 160,064 |
https://mathoverflow.net/questions/384971 | 4 | Given a smooth curve $C$, denote by $\text{Sym}^d(C)$ its $d$-th symmetric power. Let $\Delta$ be the diagonal subvariety which is defined as the codimension $1$ subvariety that at least two of the points coincide. Let $\text{Sym}^{d-1}(C)$ be the closed variety that its embedding is given by adding some extra fixed po... | https://mathoverflow.net/users/127776 | Symmetric powers of curves and completion along the diagonal | $\operatorname{Sym}^{d-1}(C) $ is ample, and $\Delta $ is not unless $C\cong \mathbb{P}^1$. To see this, consider the finite map $\pi :C^d\rightarrow \operatorname{Sym}^{d}(C) $, and the projections $\pi\_i:C^d\rightarrow C$. Your divisor $Z$ is ample if and only if $\pi ^\*Z$ is ample. Now $\pi ^\*\operatorname{Sym}^{... | 9 | https://mathoverflow.net/users/40297 | 384975 | 160,065 |
https://mathoverflow.net/questions/365336 | 11 | Koszul duality for operads allows for straightforward generalizations of $A$-infinity algebras and $A$-infinity morphisms for the so called Koszul operads $\mathcal{O}$, among which we find the associative operad. Good accounts of this can be found in standard references. However, I've been unable to find a generalizat... | https://mathoverflow.net/users/12166 | Infinity-homotopies | I don't know if you found an answer since you posted the question, but I will write this just in case: there is a "cute" (easy) definition in case of nonsymmetric operads which generalises the A-infinity story rather trivially (derivation homotopy), while for symmetric operads the definition is more involved. There is ... | 5 | https://mathoverflow.net/users/1306 | 384977 | 160,066 |
https://mathoverflow.net/questions/384957 | 3 | I'm interested in computing the fundamental group of the twisted loop space $$\Omega\_f(M)=\{ \gamma \in C^{\infty}(\Bbb R,M) \mid \gamma(s+1)=f\gamma(s)\}$$
where $f \in \text{Aut}(M,x\_0)$, for example a diffeomorphism with a fixed point $x\_0$.
The twisted loop space is part of a fibration $$\Omega\_{x\_0}M \to \O... | https://mathoverflow.net/users/93538 | Fundamental group of twisted loop space | **Edit: The following is incorrect, see below.** It might be more useful to think of $\Omega\_f(M)$ as sitting in a homotopy pullback
$\require{AMScd}$
\begin{CD}
\Omega\_f(M) @>>> M\\
@V V V @VVfV\\
M @>>\operatorname{Id}> M
\end{CD}
Then you can make use of the "Mayer-Vietoris sequence" of homotopy groups
$$\cd... | 4 | https://mathoverflow.net/users/8103 | 384981 | 160,067 |
https://mathoverflow.net/questions/384962 | 6 | Let $M$ be a factor (von Neumann algebra with trivial center), and let $L^1M:=M\_\*$ be its predual.
Let $\omega:M\to\mathbb C$ be a faithful normal state.
The Hilbert space $L^2M:=L^2(M,\omega)$ admits a distinguished vector $\omega^{1/2}$, defined as the image of $1\in M$ under the inclusion $M\hookrightarrow L^2M$.
... | https://mathoverflow.net/users/5690 | Image of $L^2M$ inside $L^1M$, for $M$ a von Neumann algebra | No, such a $\xi \in L^2(M)$ need not exist.
Denote by $L$ the space of maps from $\Omega$ to $L^1(M)$ that are continuous on the closed strip $\Omega$, holomorphic on the interior of $\Omega$ and satisfy $f(s+t) = \delta^{it}(f(s))$ for all $s \in \Omega$ and $t \in \mathbb{R}$. One can identify $L$ with the subspace... | 6 | https://mathoverflow.net/users/159170 | 384996 | 160,071 |
https://mathoverflow.net/questions/384873 | 3 | $\DeclareMathOperator\ss{ss}$Let $G$ be a reductive group over a field $F$ of characteristic 0. (Here not necessarily $F=\overline{F}$.) Consider the square map
$$
G(F)\longrightarrow G(F), \quad g\mapsto g^2.
$$
I know this map is not surjective in general even when $F=\overline{F}$. But how big is the image? Is the... | https://mathoverflow.net/users/32746 | Image of square map on reductive group | The answers to both questions are Yes.
Let $F$ be a field of characteristic 0 and ${\overline F}$ be a fixed algebraic closure of $F$.
Let $G$ be a connected reductive $F$-group.
By abuse of notation we identify $G$ with the set of ${\overline F}$-points $G({\overline F})$.
Let $G\_s$ denote the subset of semisimple ... | 2 | https://mathoverflow.net/users/4149 | 384999 | 160,073 |
https://mathoverflow.net/questions/385005 | 1 | I'm reading Beauville's book, Complex Algebraic Surfaces, and I'm trying to understand an affirmation in a proposition that characterizes the Picard group of a geometrically ruled surface.
First, let $S= \mathbb{P}\_C(E)$ be a geometrically ruled surface over a curve $C$, $p: S \rightarrow C$ the structure map. The t... | https://mathoverflow.net/users/174474 | Intersection of the tautological bundle with a fiber of a geometrically ruled surface | By [**1**, Chapter II, Proposition 7.12], the surjective map of sheaves $p^\*E \to \mathcal{O}\_S(1)$ gives a morphism of schemes
$C \to \mathbb{P}(E)$ over $C$, namely a section of $p \colon \mathbb{P}(E) \to C$.
Moreover, the image of this section corresponds to a divisor in $|\mathcal{O}\_S(1)|$, see [$1$, Chapter... | 1 | https://mathoverflow.net/users/7460 | 385010 | 160,076 |
https://mathoverflow.net/questions/385011 | 14 | In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be ****stably diffeomorphic**** if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some natural number $r$.
**Question**: What is known about the stable diffeomorphism classification of exotic $2k$... | https://mathoverflow.net/users/184 | What is known about exotic spheres up to stable diffeomorphism? | The inertia group $I\_M$ of a closed oriented $d$-manifold $M$ is the subgroup of $\theta\_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$.
[Wall](https://www.ams.org/journals/proc/1962-013-06/S0002-9939-1962-0143223-8/S0002-9939-1962-0143223-8.pdf) and [Kosinski... | 18 | https://mathoverflow.net/users/14233 | 385015 | 160,078 |
https://mathoverflow.net/questions/384492 | 4 | Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $2$-sheaves. According to <https://mathoverflow.net/q/307366>, this is an epimorphism if and only if it is *almost surjective*, that is to say if for all $U$, and any $g\in G(U)$, there exists an open cover $\{U\_i\to U\}$ such that there exis... | https://mathoverflow.net/users/152554 | Do stalks see epimorphism of stacks? | By definition (e.g. Remark 6.5.4.7 of *Higher topos theory*), an $n$-topos $\mathcal{E}$ has enough points if for every morphism $f:X\to Y$ in $\mathcal{E}$, whenever $p^\*(f)$ is an equivalence for all points $p:\mathcal{S}\_n \to \mathcal{E}$ (where $\mathcal{S}\_n$ is the $n$-topos of $(n-1)$-groupoids), then $f$ is... | 6 | https://mathoverflow.net/users/49 | 385022 | 160,080 |
https://mathoverflow.net/questions/308645 | 9 | Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can form a new left module structure on $M\otimes N$ via the structure map $$H\otimes M\otimes N\overset{\Delta\otimes 1\otim... | https://mathoverflow.net/users/11546 | Monoidal structures on modules over derived coalgebras | I couldn't make the above answer work, so here's an approach explained to me by Rune Haugseng (of course any errors are entirely my own). Let $C$ be symmetric monoidal and $p\colon C^\otimes\to Fin\_\ast$ be the cocartesian fibration witnessing this. First notice that $CoAlg(C)^{op}\simeq Alg(C^{op})$ has a "pointwise"... | 2 | https://mathoverflow.net/users/11546 | 385023 | 160,081 |
https://mathoverflow.net/questions/384526 | 6 | The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set, and there's also a generalization by Moore and Kline:
If $M \subset \mathbb{R}^2$ is compact, then it's contained in ... | https://mathoverflow.net/users/110965 | Proof of Denjoy-Riesz Theorem and Moore's Generalization? | I ended up just proving it myself here: <https://math.stackexchange.com/questions/4027493/accessible-proof-of-denjoy-riesz-theorem>
How to extend these methods to the Moore-Kline Theorem is still unclear, please take a look if you think that you might be able to help!
| 0 | https://mathoverflow.net/users/110965 | 385031 | 160,085 |
https://mathoverflow.net/questions/385036 | 2 | Denote the upper half space by $\mathcal{H}\_{3}=\Bbb{C}\times (0,\infty)$. A point $P \in \mathcal{H}\_{3}$ is given as, $P=(z, t)=(x, y, t)=z+t j$ where $z=x+i y$ and $j=(0,0,1) .$ The group $P S L\_{2}(\mathbb{C})$ has a natural action on $\mathcal{H}\_{3} .$ Let $M=\left(\begin{array}{c}\alpha & \beta \\ \gamma & \... | https://mathoverflow.net/users/174946 | Classification of isometries of hyperbolic 3-space | Let $M\in\mathrm{PSL}\_2(\mathbb{C})$ be distinct from $\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)$. If $M$ has distinct eigenvalues, then it is conjugate within $\mathrm{PSL}\_2(\mathbb{C})$ to a diagonal matrix. If $M$ has equal eigenvalues, then it is conjugate within $\mathrm{PSL}\_2(\mathbb{C})$ to ... | 6 | https://mathoverflow.net/users/11919 | 385037 | 160,087 |
https://mathoverflow.net/questions/385038 | 4 | By a general theory $X\_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$.
And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X\_1(13)$.
However, according to MAGMA, $J$ is bad at $2$.
>
> What is wrong with my argument?
>
>
>
Here is m... | https://mathoverflow.net/users/128235 | An explicit equation for $X_1(13)$ and a computation using MAGMA | To get a model with good reduction at $2$, take $y = 2Y + x^3 + x^2 + 1$,
subtract $(x^3+x^2+1)^2$ from both sides, and divide by $4$ to get
$$ Y^2 + (x^3+x^2+1) \, Y = -x^5-x^3+x^2-x. $$
(A similar tactic of un-completing the square
is well-known for elliptic curves.)
| 14 | https://mathoverflow.net/users/14830 | 385042 | 160,088 |
https://mathoverflow.net/questions/385045 | 1 | Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G\_{\pi,\pi'}$ be union of $G$ and $G\_{\pi,\pi'}$ where $G\_{\pi,\pi'}$ is $G$ but having vertices of permuted by permutation $\pi,\pi'$ and an edge is in the union iff it is in either $G$ ... | https://mathoverflow.net/users/10035 | Unique bipartite perfect matchings and cycles? | Counterexample: Let $G$ be a path on 10 vertices $y\_1,y\_2, \ldots y\_{10}$. This has a unique matching and this matching includes $e=y\_5y\_6$. Then $G\setminus \{e\}$ is 2 paths w $5$ vertices each; $y\_1y\_2y\_3y\_4y\_5$ and $y\_6y\_7y\_8y\_9y\_{10}$. So let $\pi\_1$ be the permutation on $\{y\_1,y\_3, y\_5,y\_7,y\... | 1 | https://mathoverflow.net/users/122188 | 385056 | 160,093 |
https://mathoverflow.net/questions/385061 | 1 | Let $C$ be a smooth geometrically integral affine curve. This question concerns the smooth completion $C\_1$ of $C$, both defined over a number field $k$.
We know that given any smooth projective geometrically integral curve $X$, and for a finite number of points $\{p\_1,...,p\_r\} \subset X$, the curve $X\backslash ... | https://mathoverflow.net/users/172132 | The smooth completion of a curve | Yes, in fact whenever $C$ is a non-empty Zariski open inside a smooth connected curve $C\_1$, then the complement $C\_1\setminus C$ is finite. This is easy to see in the case that $C$ is affine, since in that case $C\_1=D(f)$ is the locus of non-vanishing of a non-zero algebraic function $f$ on $C\_1$. So the complemen... | 2 | https://mathoverflow.net/users/126183 | 385065 | 160,095 |
https://mathoverflow.net/questions/385064 | 1 | In *"Knots and Links"* by Rolfsen, he mentioned words like \*"the collar of a boundary", "bicollared boundary", "a bicollar on the boundary". I just wonder what the definition of "collar" is. Also, what a flat ball is?
| https://mathoverflow.net/users/174967 | Questions about a few terminologies in "Knots and Links" by Rolfsen | A *collar* is a small product neighbourhood of the boundary.
That is: Suppose that $M$ is a compact manifold. Let $\partial M$ be its boundary. A *collar* of $\partial M$ in $M$ is a submanifold $N \subset M$ homeomorphic to $\partial M \times [0, 1]$ so that $\partial M$ is the zeroth slice.
As for a flat ball, co... | 2 | https://mathoverflow.net/users/1650 | 385068 | 160,097 |
https://mathoverflow.net/questions/385071 | 3 | I asked this question on math.stackexchange:
[Does this integral converge when $\frac{1}{p}+\frac{1}{q}\ge1$?](https://math.stackexchange.com/questions/4041263/does-this-integral-converge-when-frac1p-frac1q-geq1)
No answers or very useful comments there.
May be it is more appropraite for mathoverflow.
Fix a small... | https://mathoverflow.net/users/116555 | A question on a simple integral with a singular kernel? | It converges whenever $p$ and $q$ are both greater than 1. For fixed $x$, the integral against $y$ of $|y-x|^{-1/p}$ is uniformly bounded. Then, $\int\_{0}^2 |1-x|^{-1/q}dx$ converges.
| 5 | https://mathoverflow.net/users/4312 | 385076 | 160,099 |
https://mathoverflow.net/questions/385074 | 0 | Let $\Omega\subseteq \mathbb{R}^N$ be an open, bounded and connected set (it can be assumed with smooth boundary if necessary).
Consider $\phi:\Omega\to\mathbb{R}$, $\phi\in C^1(\overline{\Omega})$ (the Banach space of continuous functions on the closure of $\Omega$ having continuous partial derivatives on the closur... | https://mathoverflow.net/users/61629 | Lebesgue measure of sets in $\mathbb{R}^N$ | The answer is no even for $N = 1$ and $\Omega = (-1,1)$, thus $\bar \Omega = [-1,1]$. Let $\phi\_1 \colon [-1,1] \to [-1,1]$ be any increasing function (which may even be in $C^\infty$) with $\phi\_1(0) = 0$ and $\phi\_1'(x) > 0$ for all $x \in \bar \Omega$. Let $\phi\_t := t \cdot \phi\_1$. Then given $\epsilon > 0$ a... | 2 | https://mathoverflow.net/users/100904 | 385078 | 160,100 |
https://mathoverflow.net/questions/385084 | 0 | First, we note that there is a natural bijection ${\cal P}(\omega) \to \{0,1\}^\omega$ and endow the latter with the product topology (where $\{0,1\}$ carries the discrete topology). So we get a compact space that [can be endowed with a Haar measure](https://en.wikipedia.org/wiki/Haar_measure).
Is there a concrete de... | https://mathoverflow.net/users/8628 | Haar measure on ${\cal P}(\omega)$ | As Arno said, for "Haar measure" you need a group operation.
On $\{0,1\}$, we can define a group using "addition modulo 2". Normalized Haar measure for that group has each point of measure $1/2$.
Then on $\{0,1\}^\omega$ we use the product group. In that case, Haar measure is the product measure. $A\_0$ depends on... | 5 | https://mathoverflow.net/users/454 | 385085 | 160,103 |
https://mathoverflow.net/questions/385097 | 4 | The following is a question asked to me these days by Gülin Ercan.
Let $G = L(q^f)$ be a finite simple group of Lie type,
and let $L(q) \cong H \le G$ be the group of fixed points
of the automorphisms of $G$ induced by field automorphisms.
In case $H$ is nonsolvable, can there exist a nontrivial
subgroup $N$ of $G$ w... | https://mathoverflow.net/users/28104 | Subgroups of finite simple groups $L(q^f)$ of Lie type normalized by $L(q)$ | The answer is no. By a theorem of Burgoyne, Griess, and me ([Maximal subgroups and automorphisms of Chevalley groups](http://projecteuclid.org/euclid.pjm/1102811433), Pacific J. Math. 71 (1977), 365-403, Theorem 1), there would exist an integer $f\_0$ such that $G\_0\mathrel{:=}L(q^{f\_0})\le HN\le G\_0^\*$, where $G\_... | 5 | https://mathoverflow.net/users/99221 | 385110 | 160,109 |
https://mathoverflow.net/questions/385101 | 2 | Let $X$ be a smooth variety. Because $\mathcal{O}\_X$ admits a canonical connection $\mathrm{d} : \mathcal{O}\_X \to \Omega\_X$ the sequence,
$$ 0 \to \Omega\_X \to J^1(\mathcal{O}\_X) \to \mathcal{O}\_X \to 0$$
splits canonically.
I say that $X$ has the jet splitting property at level $n$ if the sequence,
$$ 0 \to \... | https://mathoverflow.net/users/154157 | Splitting of higher order jet sequence | Look at the paper P. Jahnke and I. Radloff, [Splitting jet sequences](https://arxiv.org/abs/math/0210454). They classify such splittings on compact Kaehler manifolds. Those which admit a vector bundle with splitting jet sequence are precisely projective spaces, compact complex manifolds covered by a complex torus, and ... | 3 | https://mathoverflow.net/users/13268 | 385112 | 160,110 |
https://mathoverflow.net/questions/384932 | 7 | I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to mention the classical Ramanujan-Petersson conjecture. From my point of view what's really interesting is its generalization ... | https://mathoverflow.net/users/174855 | Statement of classical Ramanujan-Petersson conjecture | For classical modular forms and Maass forms, you can find the definition of the associated $L$-function in Section 5.11 of the book *Analytic number theory* by Iwaniec-Kowalski. The statement of the Ramanujan-Petersson conjecture is given at the bottom of page 95 (a definition of general $L$-functions is also given on ... | 5 | https://mathoverflow.net/users/6506 | 385114 | 160,112 |
https://mathoverflow.net/questions/385122 | 4 | A Richaud-Degert type real quadratic field is a number field of the form $K = \mathbb{Q}(\sqrt{d})$ where $d = {(an)}^2 + ka > 0$ for positive integers $a, n$ and $k \in \{ \pm 1, \pm 2, \pm 4 \}$, $-n < k \leq n$, $d \neq 5$ and $d$ square free.
The reason to define such strange $d$ is that it turns out that the fun... | https://mathoverflow.net/users/167999 | Richaud-Degert type quadratic extensions | Watkins has recently (2019) improved Goldfeld's lower bound for $L\_\chi(1)$ by a constant, by using elliptic curves of rank 5 and computing $L'''\_E(1)$ to 1000 digits.
This allows the explicit solution for class number problems up to (say) 5 for various families with small fundamental unit.
Edit: he has a newer v... | 4 | https://mathoverflow.net/users/174996 | 385123 | 160,115 |
https://mathoverflow.net/questions/385115 | 2 | Let $A: D(A) \subset H \rightarrow H$ generate a strongly continuous semigroup $T(t)$ on a Hilbert space $H$ and $B\in \mathcal{B}(H)$. Consider the two control systems:
$$(1)\; x'(t)=Ax(t)+ Bu(t) \qquad \text{ and } \qquad (2)\; x'(t)=R(\lambda\_0,A)x(t)+Bu(t),$$
where $\lambda\_0 \in \rho(A): \Re \lambda\_0 \ge\omega... | https://mathoverflow.net/users/151918 | Link between exact null controllability of two systems | No. Many PDE systems, e.g. the heat and wave equation, allow exact null controllability for controls restricted to a proper subset of the physical domain. On the other hand, resolvent operators are smoothing, so if the initial condition for (2) has a singularity outside the controlled region, there is no way for the co... | 3 | https://mathoverflow.net/users/12120 | 385130 | 160,119 |
https://mathoverflow.net/questions/385137 | 1 | Suppose we have a homomorphism $f : A^{\bullet} \longrightarrow B^{\bullet}$ of differential graded algebras over a field $k$, and consider the filtration
\begin{align\*}
A^{\bullet} \supseteq F^0A^{\bullet} \supseteq F^1A^{\bullet} \supseteq F^2A^{\bullet} \supseteq \cdots \hspace{0.5cm} (\*)
\end{align\*}
of $A$.... | https://mathoverflow.net/users/153921 | Properties of filtrations preserved by a DG-algebra homomorphism | 1. Yes, if $A = F^0A$ then $1 \in F^0A$ so $f(1) = 1 \in F^0 B$ and so the ideal $F^0 B$ contains all of $B$. If you do not require unital dgas/maps, then the answer is no, just take the zero map and any exhaustive filtration at the source.
2. No, consider $A = k[x]$, $F^n A = (x^n)$, $B = k$, and $f : A \to B$, $f(x) ... | 2 | https://mathoverflow.net/users/36146 | 385143 | 160,123 |
https://mathoverflow.net/questions/385140 | 1 | This question is related to my post [Interpretation of some maps involving cohomology groups](https://mathoverflow.net/questions/385059/interpretation-of-some-maps-involving-cohomology-groups?noredirect=1#comment980961_385059).
$C$ is a smooth geometrically integral affine curve over a number field $k$, and $C\_1$ is... | https://mathoverflow.net/users/172132 | The groups $H^i(k,\mathbb{Z})$ for $i=1,2$ | **Question 1**: No. By the exact sequence $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}/n\rightarrow 0$, the $n$-torsion of $H^2(k,\mathbb{Z})$ is $H^1(k, \mathbb{Z}/n)\cong \operatorname{Hom}(\Gamma \_k,\mathbb{Z}/n) $, a huge group.
**Question 2**: I think so. Indeed the $\Gamma \_k$-module $... | 5 | https://mathoverflow.net/users/40297 | 385147 | 160,124 |
https://mathoverflow.net/questions/385159 | 2 | In the question "[Derivative of eigenvectors of a matrix with respect to its components](https://mathoverflow.net/questions/229425)", [Liviu Nicolaescu](https://mathoverflow.net/a/229467/175010) has provided an answer valid for a real matrix. As outlined in the following, the same proof applies to Hermitian matrices, b... | https://mathoverflow.net/users/175010 | Derivative of eigenvectors of an Hermitian matrix | This is first order perturbation theory, see [Wikipedia](https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#First_order_corrections). As explained there (search for "*Since the overall phase is not determined in quantum mechanics*"), you need to use the freedom you have to multiply the eigenstate by ... | 4 | https://mathoverflow.net/users/11260 | 385164 | 160,130 |
https://mathoverflow.net/questions/385099 | 10 | Let $X$ be a smooth projective variety over $\mathbb{C}$.
I know that a vector bundle $\mathcal{E}$ on $X$ admits a holomorphic/algebraic connection iff its Atiyah class vanishes, $A(\mathcal{E}) = 0$. If we choose a Hermitian structure on $\mathcal{E}$ giving a Chern connection $\nabla$ then $A(\mathcal{E}) = [\omeg... | https://mathoverflow.net/users/154157 | When do flat holomorphic connections exist? | In the affine algebraic case there is always an algebraic connection: Let $A$ be a commutative unital ring and let $E$ be a finite rank projective $A$-module. There is the Atiyah sequence
$$ 0 \rightarrow \Omega^1\_A \otimes E \rightarrow J^1(E) \rightarrow^{\pi} E \rightarrow 0$$
and since $E$ is projective and $\... | 4 | https://mathoverflow.net/users/nan | 385170 | 160,133 |
https://mathoverflow.net/questions/385124 | 1 | My question is related to the uniformization theorem. But not being an expert on this, I am not sure if my question is covered by this result. Before going deeply into the question, I would like to know if one of you already has an answer to my question.
Here comes the question.
Let $\varphi \in C^\infty\_c(\mathbb... | https://mathoverflow.net/users/41568 | Global isothermal coordinates for particular surfaces of revolution | This is a consequence of the Uniformization Theorem (UT) and Riemann's theorem on removable singularities:
1. By the $UT$, $S$ is conformal either to the open unit disk $\Delta$ or to the complex plane.
2. Suppose that $S$ is conformal to $\Delta$. Observe that the complement to a large disk in $S$ is contained in th... | 1 | https://mathoverflow.net/users/39654 | 385171 | 160,134 |
https://mathoverflow.net/questions/385167 | 12 | If $G=(V,E)$ is a finite, simple, undirected graph, and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$, and $\text{deg}(v)= |N(v)|$. We say a vertex $v\in V$ is a *king* if $\text{deg}(v) > \text{deg}(w)$ for all $w\in N(v)$.
In the graph $G=(\{0,1,2\}, \big\{\{0,1\}, \{1,2\}\big\})$, one of the $3$ vertices is a ... | https://mathoverflow.net/users/8628 | Are at most $1/3$ vertices "kings"? | For this discussion I am assuming we do not consider isolated vertices to be "Kings", even though technically your definition considers them to be so in a vacuous sense (I guess this convention goes [back to Shakespeare](https://www.enotes.com/shakespeare-quotes/king-infinite-space)). Otherwise of course one can make e... | 35 | https://mathoverflow.net/users/766 | 385179 | 160,137 |
https://mathoverflow.net/questions/384991 | 2 | Help me please.
Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}\_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ over the rational function field $k(t)$. Suppose that $C$ is absolutely irreducible, i.e., irreducible over $\overli... | https://mathoverflow.net/users/69852 | Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases? | This is an answer to the further question in the comments. The original question is answered in the comments (as was the further question while I was typing this :-)).
The vector of coefficients of a curve of degree $d$ gives a point in $\mathbb{P}^{N\_d}, N\_d = (d+1)(d+2)/2-1$. Multiplication of equations defines a... | 2 | https://mathoverflow.net/users/2290 | 385182 | 160,139 |
https://mathoverflow.net/questions/385052 | 7 | $\newcommand\tf{\text{tf}}\newcommand\tor{\text{tor}}$Let $G$ be a finitely generated group. Let $\gamma\_k(G)$ denote the $k$th term in the lower central series for $G$, so $\gamma\_1(G) = G$ and $\gamma\_{k+1}(G) = [\gamma\_k(G),G]$ for all $k \geq 1$.
There is also a natural central series that it makes sense to c... | https://mathoverflow.net/users/174957 | Lower central series vs torsion-free lower central series | You can prove this by repeatedly applying the following lemma:
**Lemma**:
Let $G$ be a group and let $A,B \lhd G$ be finitely generated normal subgroups. Assume that $A$ and
$B$ are commensurable. Then $[G,A]$ and $[G,B]$ are commensurable and
$$\left(A / [G,A]\right) \otimes \mathbb{Q} \cong \left(B / [G,B]\right) \... | 5 | https://mathoverflow.net/users/317 | 385205 | 160,145 |
https://mathoverflow.net/questions/385050 | 6 | Let $G$ be a finitely generated nilpotent group and let $A\le G$ be a finite-index subgroup. I have two questions about $A$:
1. Is it true that the inclusion map $A \rightarrow G$ induces isomorphisms $H\_k(A;\mathbb{Q}) \rightarrow H\_k(G;\mathbb{Q})$ for all $k$, and thus that the Betti numbers of $A$ and $G$ are t... | https://mathoverflow.net/users/174957 | Betti numbers and lower central series quotients of finite-index subgroups of nilpotent groups | Both statements are true. To prove them, we will need the following lemma:
**Lemma**: Let $G$ be a finitely generated nilpotent group of class $k$ and let $H$ be a subgroup of $G$.
Then $\gamma\_k(H)$ is a finite-index subgroup of $\gamma\_k(G)$.
**Proof**: Iterated $k$-fold commutators induce a surjective homomorp... | 5 | https://mathoverflow.net/users/317 | 385207 | 160,146 |
https://mathoverflow.net/questions/385204 | 1 | Is the group of automorphisms of the ring $\mathbb{F}[t,t^{-1}]$ of Laurent polynomials known? Here, $\mathbb{F}$ is an algebraically closed field of characteristic $0$ and I am considering not necessarily unital automorphisms. Thanks in advance.
| https://mathoverflow.net/users/137269 | Automorphisms of the ring of Laurent polynomials | First of all I think the unitality is automatic, since there is the unique element which multiplies trivially with any other (and the image of 1 satisfies this).
All invertible elements in $\mathbb F[t,t^{-1}]$ are of the form $\alpha\cdot t^n$ where $\alpha\in \mathbb F^\times $ and $k\in \mathbb Z$. Since the image... | 2 | https://mathoverflow.net/users/42606 | 385210 | 160,148 |
https://mathoverflow.net/questions/383259 | 27 | We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider the following theory. We take as axioms the axioms for ordered fields, and then add an axiom schema that states that every... | https://mathoverflow.net/users/43439 | Is this theory the complete theory of the real ordered field? | It is not. Using set forcing, we can add 'undefinable' reals in a controlled manner, while keeping complexity of parameter-free definable sets low.
Specifically, let $M$ be a countable $ω$-model of ZFC\P, real $r$ be Cohen generic over $M$, and $ℝ\_M(r)$ be the minimal field of reals containing $r$ and all reals in $... | 15 | https://mathoverflow.net/users/113213 | 385212 | 160,149 |
https://mathoverflow.net/questions/384672 | 3 | Let $k$ be a perfect field of characteristic $p \neq 2,3$ such that $\omega := \sqrt[3]{1} \in k$, where $\omega \neq 1$. Consider an absolutely irreducible (not necessarily homogenous) quadratic polynomial $Q \in k[s\_1, s\_2]$ in two variables $s\_1, s\_2$. Under what conditions is the polynomial $R(t\_1,t\_2) := Q(t... | https://mathoverflow.net/users/69852 | Under what conditions is the polynomial of degree $6$ irreducible? | The following argument might help reduce the problem to one in elimination of variables, which can be solved using a computer algebra system.
Write $S = k[s\_1,s\_2]$, $T = k[t\_1, t\_2]$ and $\phi : S \rightarrow T$,
$s\_i \mapsto t\_i^3$.
The ramification locus of $\phi$ is defined by $t\_1t\_2$.
Since $Q(s\_1, s\_... | 1 | https://mathoverflow.net/users/14895 | 385220 | 160,151 |
https://mathoverflow.net/questions/385102 | 3 | Working in a suitable extension of $\sf Z$ like $\sf ZfC + wholeness \ axiom$, or $\sf ZFj + Reinhardt \ axiom$.
Can we have a sequence $(j\_n)\_{n \in \mathbb N} $ of nontrivial elementary embeddings from $V$ to $V$, such that for some $V\_\lambda$ we have:
$$\exists (\alpha\_n)\_{n \in \mathbb N} : \forall n \in ... | https://mathoverflow.net/users/95347 | Is this sequence of embeddings possible? | Work in ZF+AC$\_\omega$. Suppose $V\_\theta$ is inaccessible. Suppose $j,k:V\_{\theta+1}\to V\_{\theta+1}$ are elementary and $\mathrm{crit}(j)=\mathrm{crit}(k)=\kappa$ and $\kappa\_\omega(j)<k(\kappa)$. Here $\kappa\_n(j)$ for $n\leq\omega$ is the critical sequence of $j$; that is, $\kappa\_0(j)=\mathrm{crit}(j)$ and ... | 4 | https://mathoverflow.net/users/160347 | 385223 | 160,153 |
https://mathoverflow.net/questions/384611 | 17 | The integer $(1^2+1)(2^3+1)(3^4+1)(4^5+1)$ is a square, namely $2^23^25^241^2$.
>
> **Question.** What will be the next occurrence, or is there an occurrence of $$\prod\_{i=1}^n (i^{i+1}+1)=k^2?$$
>
>
>
| https://mathoverflow.net/users/174610 | Diophantine equation: $\prod_{i=1}^n (i^{i+1}+1)=k^2$ beyond $(n,k) = (4,1230)$? | The following criterion will most likely cover all large $n$, but actually proving this is out of reach of current technology.
**Proposition**. Let $p$ be a [Sophie Germain prime](https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes) (so that $q := 2p+1$ is also prime) with $p = 11 \hbox{ mod } 12$, such that... | 22 | https://mathoverflow.net/users/766 | 385227 | 160,154 |
https://mathoverflow.net/questions/385228 | 2 | In the iterative methods for solving a system of linear equations, a term called relaxation method is often appears along with Jacobi and Gauss Seidel methods. As per the Earliest Known Uses website,
>
> RELAXATION, as a term in numerical analysis for a particular method of successive approximation, derives from So... | https://mathoverflow.net/users/142414 | Origin of the term relaxation method in numerical analysis for iteratively solving linear equations | **Q:** what is being "relaxed" in the relaxation method?
**A:** The relaxation method is an iterative approach to solve the set of linear equations $\sum\_{j}A\_{ij}u\_j-b\_i=0$ by *relaxing* the requirement that the right-hand-side should vanish. Residuals $Z\_i$ allow the right-hand-side to be nonzero, $\sum\_{j}A\... | 2 | https://mathoverflow.net/users/11260 | 385236 | 160,157 |
https://mathoverflow.net/questions/385221 | 3 | Suppose that $(x\_{n})\_{n}$ is a sequence in a Banach space $X$. We let $\textrm{clust}\_{X^{\*\*}}((x\_{n})\_{n})$ be collection of all the weak\*-limit points of $(x\_{n})\_{n}$ in $X^{\*\*}$.
Let $(e\_{n})\_{n}$ be the unit vector basis of $c\_{0}$. Let $s\_{n}=\sum\limits\_{i=1}^{n}e\_{i}(n=1,2,\cdots)$. It is e... | https://mathoverflow.net/users/41619 | The weak*-convergence of the summing basis of $c_{0}$ | For a Banach space $E$, let $\kappa = \kappa\_E:E\rightarrow E^{\*\*}$ be the canonical inclusion. Consider
$$ E^\perp = \{ M\in E^{\*\*\*} : M(\kappa\_E(x))=0 \ (x\in E) \} = \ker\kappa\_E^\*. $$
A simple calculation shows that $\kappa\_E^\* \circ \kappa\_{E^\*} = 1\_{E^\*}$ and so $\kappa\_{E^\*}\circ\kappa\_E^\*$ is... | 1 | https://mathoverflow.net/users/406 | 385238 | 160,158 |
https://mathoverflow.net/questions/385243 | 2 | The paper [Stable maps and tautological classes](https://arxiv.org/pdf/math/0304485.pdf) of Faber-Pandharipande shows that the Gromov-Witten classes on spaces of relative stable maps of the projective line push forward to tautological classes on moduli spaces of stable curves.
It is claimed on p. 6 that $\overline{M}... | https://mathoverflow.net/users/119354 | admissible covers vs. stable maps to P^1 | The claim is that these moduli spaces coincide under the numerical condition that
$$ 2g−2 + 2d=\sum\_{i=1}^m (d−\ell(\mu\_i)).$$
In this case there will be no ramification points outside the fibers $f^{-1}(q\_i)$ and no contracted components on the domain curve. Indeed any contracted component must have positive genus ... | 5 | https://mathoverflow.net/users/1310 | 385246 | 160,160 |
https://mathoverflow.net/questions/382642 | 3 | **Motivation.** As I was playing the pairs-matching game "[Memory](https://en.wikipedia.org/wiki/Concentration_(card_game))" (known as "Concentration" in some parts of the world) with my children, I was surprised that even thorough shuffling could not prevent quite a few pairs of cards lying next to each other. This in... | https://mathoverflow.net/users/8628 | Memory game inspired problem | First, a simple answer (evocated in the comments) : since expectation is linear (even without independence), when we express $\text{Adj}$ as the sum of each index being adjacent, we can sum their expectation. Each index $i \in \{1, \dots, 2n-1\}$ is connected to exactly one of the other indexes in $\{1, \dots, 2n\} \se... | 1 | https://mathoverflow.net/users/174620 | 385257 | 160,163 |
https://mathoverflow.net/questions/385055 | 1 | There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space):
$$\mu(0) = \{y\in{}^{\*}X: \forall x\in X ~~ \langle y|x\rangle\simeq0\}.$$
That is collection of vectors (possibly non standard and not near-standard) that have infinites... | https://mathoverflow.net/users/123836 | What's the size of non standard monad for weak topology? | Let's try this, as a negative answer. It has been a long time since I seriously worked on non-standard analysis—so criticism is welcome. I follow mostly the terminology of Robinson's book *Non-Standard Analysis*.
We will assume a [$2^{\aleph\_0}$ saturated model](https://en.wikipedia.org/wiki/Nonstandard_analysis#%CE... | 2 | https://mathoverflow.net/users/454 | 385267 | 160,167 |
https://mathoverflow.net/questions/372398 | 9 | Assume everything is finite.
Let $G$ be a primitive permutation group with point stabiliser $G\_\alpha$ for some $\alpha$. For $\beta\ne\alpha$, by an arc stabiliser we mean $G\_{\alpha\beta}=G\_\alpha\cap G\_\beta$ and an edge stabiliser we mean $G\_{\{\alpha,\beta\}}$, the stabiliser of the set $\{\alpha,\beta\}$. ... | https://mathoverflow.net/users/131819 | Can $1\ne H\cap H^g\lhd H$ happen if $G$ is a primitive permutation group with stabiliser $H$? | An example was constructed by Pablo Spiga:
<https://arxiv.org/abs/2102.13614>
"A generalization of Sims conjecture for finite primitive groups and two point stabilizers in primitive groups"
| 6 | https://mathoverflow.net/users/22377 | 385276 | 160,169 |
https://mathoverflow.net/questions/385070 | 3 | Lets define function T as
$$T(0) = \aleph\_0$$
$$T(1) = \aleph\_{\aleph\_0}$$
$$T(2) = \aleph\_{\aleph\_{\aleph\_0}}$$
etc
No finite tower of alephs can reach the first inaccessible cardinal
My questions are:
1. Can we 'feed' infinite ordinal numbers as a parameter to function T? I read somewhere - it was about t... | https://mathoverflow.net/users/174974 | Cardinality of infinite towers of Alephs - can tower be more than countable? | "Transfinite towers" often run into the problem of **termination**: when $F$ is a "nice" function on ordinals (= monotonic, nondecreasing, and continuous), we get $$F(\sup\{F^n(0): n\in\omega\}=\sup\{F^n(0):n\in\omega\}.$$ That is, the "iterating $F$" tower stops at level $\omega$. Moreover, if the sequence $(F^n(0))\_... | 4 | https://mathoverflow.net/users/8133 | 385279 | 160,170 |
https://mathoverflow.net/questions/385274 | 2 | Let $f $ be a periodic function and denote by $c\_n$, for $n \in \mathbb{N}$, its Fourier coefficients, i.e.
$$
c\_n := \frac{1}{2\pi}\int\_{-\pi}^{\pi}f(x)e^{inx}\ dx.
$$
It is well known that [Bochner's theorem](https://en.wikipedia.org/wiki/Bochner%27s_theorem) states that the Fourier transform of a positive definit... | https://mathoverflow.net/users/160454 | Does Bochner's Theorem apply to Fourier coefficients? | Following from the comment made by @abx, and some further research. The answer is yes, and this statement is often called Herglotz's Theorem. I found a useful resource regarding this [here](https://www.orsj.or.jp/%7Earchive/pdf/e_mag/Vol.60_02_122.pdf).
| 3 | https://mathoverflow.net/users/160454 | 385282 | 160,171 |
https://mathoverflow.net/questions/385195 | 1 | This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) + k(t,x)u(t,x) = g(t,x) &\textit{in}\quad\left[0,T\right]\times\mathbb{R}\\
u(T,x)=\phi(x)&\textit{in}\quad\mathbb{R}... | https://mathoverflow.net/users/134368 | Forwards Feynman–Kac formula | Given $t \in (0,T)$, define $\tilde{X}^{(t),x}$ to be the solution of the SDE
\begin{equation\*}
d\tilde{X}^{(t),x}\_{s} = \mu(t + s,\tilde{X}^{(t),x}\_{s}) \, ds + \sigma(t + s, \tilde{X}^{(t),x}\_{s}) \, d B\_{s}, \quad \tilde{X}^{(t),x}\_{0} = x.
\end{equation\*}
Notice that $\tilde{X}^{(t),x}\_{\cdot} = X^{t,x}\_{\... | 0 | https://mathoverflow.net/users/143683 | 385283 | 160,172 |
https://mathoverflow.net/questions/385230 | 3 | Let $k$ be a field, and $A$ an associative $k$-algebra with an identity element. Say that $A$ is *quadratic* if any subalgebra of $A$ generated by a single element has dimension at most two.
I am looking for a reference for the following (or any similar) result:
>
> Assume that ${\rm char}(k)\ne2$, and let $A$ be... | https://mathoverflow.net/users/136180 | Charaterisation of quaternion algebras | Yes, this is a great and well-studied question with a super nice answer! See Theorem 3.5.1 of my book (<http://quatalg.org>). You can also say something in characteristic 2 (see Theorem 6.2.8) if you refine "quadratic" = "degree 2" to "nonidentity standard involution".
| 2 | https://mathoverflow.net/users/4433 | 385287 | 160,173 |
https://mathoverflow.net/questions/385286 | 0 | I'm trying to figure out the connections between two contructions of Gaussian measure.
Let $(U, \langle\cdot,\cdot\rangle\_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-algebra.
---
[Definition 2.1, page 10](http://www.math.chalmers.se/%7Estig/underv/doktorandkurs/SPDE0708/spdenotes.p... | https://mathoverflow.net/users/170508 | Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure | If $U$ is a Banach space, then the natural thing to do is replacing the $\langle v,\cdot\rangle\_U$ by any continuous form $\phi\in U^\*$. Then the definition of a Gaussian measure is a measure $\mu$ such that the pushforward $\phi^\*\mu$ is Gaussian for all $\phi\in U^\*$ (note that this works in finite dimension too)... | 2 | https://mathoverflow.net/users/129074 | 385294 | 160,176 |
https://mathoverflow.net/questions/247672 | 6 | Suppose I have a CohFT, which is basically a vector space $V$ and a collection of maps $V^{\otimes n} \rightarrow H^\*(\overline{M\_{g,n}})$ satisfying certain properties under pull-backs by the gluing maps. Under which assumptions is it completely determined by its high genus values? High means with respect to the coh... | https://mathoverflow.net/users/89514 | Is a CohFT completely determined by its high genus values? | Teleman's trick works as follows. Suppose I want to deduce $\Omega\_{g,n}$ from $\Omega\_{g+1,n}$.
Note that $\overline{\mathcal{M}}\_{g+1,n}$ contains a copy of $\overline{\mathcal{M}}\_{g,n}$. It is constructed as follows. Take any fixed elliptic curve $E$ with two marked points. Now consider the locus of curves ob... | 4 | https://mathoverflow.net/users/175087 | 385300 | 160,180 |
https://mathoverflow.net/questions/385303 | 51 | The number $f(n)$ of graphs on the vertex set $\{1,\dots,n\}$,
allowing loops but not multiple edges, is $2^{{n+1\choose
2}}$, with exponential generating function $F(x)=\sum\_{n\geq 0}
2^{{n+1\choose 2}}\frac{x^n}{n!}$. Consider
$$ \sqrt{F(x)} = 1+x+3\frac{x^2}{2!}+23\frac{x^3}{3!}
+393\frac{x^4}{4!}+13729\frac{x^5}{... | https://mathoverflow.net/users/2807 | The "square root" of a graph? | There is a fixed-point-free involution on these graphs which I will call *loop-switching*, given by adding a loop to every vertex that doesn't have one while simultaneously deleting the loops from all vertices that do. Then $\sqrt{F(x)}$ counts equivalence classes of graphs, where two graphs are in the same class if on... | 55 | https://mathoverflow.net/users/112113 | 385306 | 160,182 |
https://mathoverflow.net/questions/385312 | 10 | Where can I find the latest revision of *A term of Commutative Algebra* by Allen B. ALTMAN and Steven L. KLEIMAN? Is my 2013 version ok?
It is hard to locate the latest one; many old revisions and pointers to them are randomly scattered across the web. (Details: The first page of a web search showed me all 4 versions... | https://mathoverflow.net/users/175094 | Latest "A Term of Commutative Algebra" by Altman and Kleiman? | You can get the latest from these sites:
* [ResearchGate](https://www.researchgate.net/publication/325591008_A_term_of_Commutative_Algebra)
* [Worldwide Center of Mathematics](https://the-center-of-math-store.constantcontactsites.com/textbooks/post/p_2398391)
* (Obsolete) [DSpace @ MIT](http://dspace.mit.edu/handle/1... | 17 | https://mathoverflow.net/users/175094 | 385313 | 160,183 |
https://mathoverflow.net/questions/385309 | 5 | *This was [asked and bountied at MSE](https://math.stackexchange.com/questions/4036036/are-there-nonstandard-mathsfpa-models-without-delta1-1-cuts) with no response:*
My question is the following:
>
> Is there a nonstandard model $\mathcal{M}\models\mathsf{PA}$ such that $\mathcal{M}$ has no $\Delta^1\_1$-with-pa... | https://mathoverflow.net/users/8133 | Can a nonstandard model of $\mathsf{PA}$ be "$\Delta^1_1$-well-ordered?" | I believe you don't need this, but assume that there is a strongly inaccessible cardinal $\kappa$. Fix a first-order completion $T$ of $\mathsf{PA}$ and let $\mathcal{M}$ be a saturated model of $T$ of cardinality $\kappa$. I will show that $\mathcal{M}$ has no $\Delta^1\_1$-with-parameters-definable cuts.
*Claim.* F... | 5 | https://mathoverflow.net/users/83901 | 385315 | 160,185 |
https://mathoverflow.net/questions/385120 | 1 | This concerns difference/limit ratio results for special restricted partitions.
Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $b$ positive integers and each of them is at most $a$; if $r = 0$ and $a,b >0$, then $p(0,a,b)$ is taken to be one. Th... | https://mathoverflow.net/users/42278 | Ratio limit results for restricted partition functions | This is exactly Lemma 1 in the recent [paper](https://link.springer.com/article/10.1007/s10958-019-04370-2) by Vershik and Malyutin. By the way, motivated by random walks on Heisenberg group.
| 2 | https://mathoverflow.net/users/4312 | 385333 | 160,190 |
https://mathoverflow.net/questions/385338 | 1 | Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup\_{x \in X} d\_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in Wasserstein distance; i.e.:
$$
W\_1\left(f\_{\#}\mathbb{P},g\_{\#}\mathbb{P}\right)<\delta(\epsilon)
,
$$
for every $\mathbb{P... | https://mathoverflow.net/users/36886 | Continuity of pushforward operation | Yes. If the uniform distance of $f$ and $g$ is less than $\epsilon$, simply take your coupling to be the push-forward of the function $x\mapsto\big(f(x),g(x)\big)$. The resulting coupling verifies that the Wasserstein-$1$-distance is less than $\epsilon$.
| 2 | https://mathoverflow.net/users/35357 | 385340 | 160,192 |
https://mathoverflow.net/questions/385280 | 5 | A Künneth formula by [Grothendieck/Schwartz](http://www.numdam.org/item/?id=SLS_1953-1954__1__A25_0) states the following:
>
> Let $A, B$ be chain complexes of nuclear Fréchet spaces. If the differentials $d\_A, d\_B$ are topological homomorphisms (meaning in this setting: if they have closed ranges), then we have ... | https://mathoverflow.net/users/126256 | The tensor product of two topological complexes with closed range | The discussion in the comments catapulted me onto the right track! It seems the solution is exactly to note that the isomorphism $H(A \, \hat \otimes \, B ) \cong H(A) \, \hat \otimes \, H(B)$ is not only an isomorphism of abstract vector spaces, but indeed an isomorphism of topological vector spaces, where all homolog... | 2 | https://mathoverflow.net/users/126256 | 385341 | 160,193 |
https://mathoverflow.net/questions/385345 | 0 | Let $I\_{n}:=]p\_{n},p\_{n+1}[$ be the open interval between the $n$-th and $(n+1)$-th prime. Under Goldbach's conjecture, denote by $r\_{0}(m)$ the smallest positive integer $r$ such that both $m-r$ and $m+r$ are prime for any large enough composite integer $m$ and by $k\_{0}(m)$ the quantity $\pi(m+r\_{0}(m))-\pi(m-r... | https://mathoverflow.net/users/13625 | Upper bound for the number of $k$-central numbers in a prime gap | $N\_{I\_n}(k)\leq k$ indeed always holds. Let $a\_1<\dots<a\_m$ be $m$ elements in $I\_n$ such that $k\_0(a\_i)=k$ for each $k$. Let $b\_i=a\_i-r\_0(a\_i),c\_i=a\_i+r\_0(a\_i)$ so that $b\_i,c\_i$ are primes and there are exactly $k-1$ primes strictly between $b\_i$ and $c\_i$ for each $i$.
I claim $b\_1<\dots<b\_m$.... | 3 | https://mathoverflow.net/users/30186 | 385353 | 160,195 |
https://mathoverflow.net/questions/385339 | 5 | I've been reading Tao's *An introduction to measure theory*, a draft can be found [here](https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/). An exercise from it is
Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely integrable function, and let ${... | https://mathoverflow.net/users/175105 | Question on an exercise from Terry Tao's blog | If $f=1\_{[0,1]}$, then for real $x$ we have
$$f^\*(x)=\frac1{1-x}1(x<0)+1(0\le x<1).$$
So, for $\lambda=1/2$ the left-hand side is
$$\tfrac12\,m(\{x\colon f^\*(x)>\tfrac12\})=\tfrac12\,m((-1,1))=1,$$
which is the same as the right-hand side.
| 11 | https://mathoverflow.net/users/36721 | 385355 | 160,197 |
https://mathoverflow.net/questions/385360 | 3 | Let $C$ be an algebraic curve (one dimensional projective regular connected scheme of finite type) of genus $g$ over an algebraically closed field $k$ with structure morphism $\pi$. By Riemann-Roch, the global sections of the sheaf of differentials $\Omega$ are a $g$-dimensional vectorspace over $k$ and every element $... | https://mathoverflow.net/users/164782 | Topological properties of differentials with prescribed zeroes on an algebraic curve | This probably belongs to MSE. In any case, it works for any line bundle $L$, there is nothing special about $\omega$ for this problem. Let $n$ be the degree of $L$ and choose a partition of $n$ with $d$ parts, a point $p$ of $C^d$ gives a set $Z\_p\subset C$ of zeroes with multiplicities. You can construct a line bundl... | 3 | https://mathoverflow.net/users/45660 | 385364 | 160,200 |
https://mathoverflow.net/questions/385363 | 10 | Let $F: \mathcal A^\to\_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the [category of algebras](https://ncatlab.org/nlab/show/Eilenberg-Moore+category) for the monad $UF$. The [Beck monadicity the... | https://mathoverflow.net/users/2362 | Characterization of functors whose right adjoint is monadic? | Let $F: C \to D$ be a left adjoint functor. I hope I'm not saying anything stupid, but I think you can just rephrase the two conditions of Beck Monadicity theorem in terms of the left adjoint:
The condition that $U$ is conservative translate as:
* The $Hom(F(x),\\_)$ are jointly conservatives.
It can also be repl... | 11 | https://mathoverflow.net/users/22131 | 385368 | 160,202 |
https://mathoverflow.net/questions/385285 | 2 | Let $\sigma\_D(x)=\sup \{ \left< x, y \right> : y\in D \}$ for a closed convex $D\subseteq \mathbb R^n$. Then $\sigma\_D$ is convex and lower semicontinuous (it's the supremum of linear functions). Let the effective domain of $f$ be the set of points where $f$ is finite. Then $\sigma\_D$ restricted to its effective dom... | https://mathoverflow.net/users/26809 | Is the support function continuous on its effective domain? | The answer is negative as can be seen by putting together these two facts:
1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is [here](https://math.stackexchange.com/questions/2487705/is-a-convex-and-lower-semicontinu... | 1 | https://mathoverflow.net/users/26809 | 385369 | 160,203 |
https://mathoverflow.net/questions/385308 | 0 | Define the frequency variance as:
$$ \sigma^2 = \int^\infty\_{-\infty}\omega^2 P(\omega) d\omega$$
Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore,
$$ \sigma^2 = \frac{\int^\infty\_{-\infty}\omega^2 X(\omega)\bar{X}(\omega) d\omega}{\int^\infty\_{-\infty} X(\omega)\b... | https://mathoverflow.net/users/173974 | Variance of spectral density is related to the gradient of signal? | It turns out that the relationship is obvious when I use Parseval's Theorem. First I am rewriting my $v$ (variance term) here:
$$ v = \int^{\infty}\_{-\infty}|i \omega X(\omega))|^2 d\omega$$
$ i \omega X(\omega)$ is Fourier transform of $\frac{dx(t)}{dt}$. Using Perseval's Theorem,
$$\int^{\infty}\_{-\infty}|i \... | 0 | https://mathoverflow.net/users/173974 | 385372 | 160,204 |
https://mathoverflow.net/questions/385377 | 2 | In his book *Algebraic Geometry and Arithmetic Curves* Qing Liu claims in Exercise 3.4, page 56, the following for a scheme $X$ and a global function $f\in \mathcal O\_X(X)$:
"The map $U\mapsto f\vert \_U\mathcal O\_X(U)$ for every affine open subset $U$ defines a sheaf of ideals on $X$."
The presheaf thus define... | https://mathoverflow.net/users/157954 | On the definition of a principal ideal sheaf | Note that we only need consider the case (of your setup) where $U \cup V$ is affine. If the function obtained by gluing $fs$ and $ft$ is not a multiple of $f$, then it is a nonzero element in $ \mathcal O (U \cup V) /f$, hence a nonzero function on $\operatorname{Spec} ( \mathcal O (U \cup V)/f)$. But $\operatorname{Sp... | 5 | https://mathoverflow.net/users/18060 | 385378 | 160,207 |
https://mathoverflow.net/questions/385225 | 5 | This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:
$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 \cdot \sin \left(\frac{2\pi}{x} \right) \cdot \left(3+2 \cos \left(\frac{2\pi... | https://mathoverflow.net/users/174962 | Is the solution to this trig function known to be algebraic or transcendental? | It should be possible to show that $x$ is irrational using Theorem 7 of [Trigonometric diophantine equations (On vanishing sums of roots of unity)](http://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3033.pdf) by J. H. Conway and A. J. Jones, *Acta Arithmetica* **30** (1976), 229–240, although I have not carried out the full ca... | 4 | https://mathoverflow.net/users/3106 | 385386 | 160,209 |
https://mathoverflow.net/questions/385272 | 6 | Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2).
In particular this is true for $\mathbb{Z}\_p[G]$-modules, and I was wondering if this version of the result can be generaliz... | https://mathoverflow.net/users/145915 | Cohomology of finite $p$-groups over integers in local fields | If you take $G=\mathbb Z/p^a\mathbb Z$, then the cohomology of any $M$ are computed by the Tate complex $M\xrightarrow{\sigma-\mathrm{id}} M \xrightarrow{1+\sigma+\ldots+ \sigma^{p^a-1} }M\xrightarrow{\sigma-\mathrm{id}} M \xrightarrow{1+\sigma+\ldots+ \sigma^{p^a-1} }M\xrightarrow{\sigma-\mathrm{id}}\ldots$. In partic... | 2 | https://mathoverflow.net/users/42606 | 385389 | 160,210 |
https://mathoverflow.net/questions/385387 | 0 | Do there exist a (non-trivial) globally Lipschitz continuous function $g:\mathbf{R}\to\mathbf{R}$ and a non-decreasing function $f:\mathbf{R}\_+\to\mathbf{R}\_+$ such that the identity
\begin{equation}
g(\alpha f(\alpha)s) = f(\alpha) g(s)
\end{equation}
holds for all $\alpha>0$ and $s\in\mathbf{R}$?
Remark:
I am pre... | https://mathoverflow.net/users/115381 | Existence of functions satisfying a homogeneity condition | Since $f$ is non-decreasing, the function $\alpha\mapsto \alpha f(\alpha)$ is strictly increasing and hence invertible with strictly increasing inverse. Therefore there is a non-decreasing function $k$ such that $k(\alpha f(\alpha)) = f(\alpha)$.
---
*Edit*: As Iosif pointed out in a comment, the notion of the in... | 1 | https://mathoverflow.net/users/3948 | 385395 | 160,212 |
https://mathoverflow.net/questions/385307 | 6 | This is a quick follow up on [R. Stanley's interesting post on MO](https://mathoverflow.net/questions/385303/the-square-root-of-a-graph) in a different direction, which might be easier.
For positive integers $a$, define the family of functions (infinite series) given by
$$F\_a(x)=\sum\_{n\geq0}a^{\binom{n+1}2}\frac{x... | https://mathoverflow.net/users/66131 | $a^{th}$-root of exponential generating functions | In fact, we can let $a$ be an indeterminate, and the coefficient of $x^n/n!$ in $\sqrt[a]{F\_a(x)}$ will be a polynomial with integer coefficients. This is because $a^{{n+1\choose 2}}$ counts graphs with edges colored with $a$ colors (including the color 0, which denotes no edge). Thus the number $c\_a(n)$ of connected... | 7 | https://mathoverflow.net/users/2807 | 385402 | 160,215 |
https://mathoverflow.net/questions/385405 | 6 | The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$.
* $\Sigma$ is *regular* if it acts transitively and freely on $\Omega$, i.e., for any two $i,j\in \Omega$ there is a unique $\sigma\in\Sigma$ with $\sigma(i)... | https://mathoverflow.net/users/108884 | Is there a known classification of regular multiplicity-free permutation groups? | These are the abelian regular permutation groups. The permutation character in this case is the character of the regular representation and in the regular representation a character appears with multiplicity equal to the dimension of the irreducible representation. So it can be multiplicity free iff all irreducibles ar... | 9 | https://mathoverflow.net/users/15934 | 385406 | 160,217 |
https://mathoverflow.net/questions/385413 | 14 | *This is a modified version of a question which was [asked and bountied at MSE](https://math.stackexchange.com/questions/3593969/is-there-a-natural-intermediate-version-of-pa) without success.*
---
Below, "$\mathsf{PA}$" refers to *first-order* Peano arithmetic.
There are various ["schematic" theories](https://... | https://mathoverflow.net/users/8133 | How special is first-order $\mathsf{PA}$? | This argument has a couple of iffy points, but I believe it does work.
In [this](https://www.sciencedirect.com/science/article/pii/0003484378900098) paper, Shelah introduced a logic $\mathcal{L}(Q\_{\mathrm{Brch}})$ which is fully compact, has the property that any countable theory with infinite models has models of ... | 10 | https://mathoverflow.net/users/83901 | 385418 | 160,219 |
https://mathoverflow.net/questions/322487 | 5 | Let $X$ be a complex algebraic variety. The numerical invariants associated with the Mixed Hodge Structure of $X$ can be encoded in a polynomial in three variables called the mixed Hodge polynomial $H(X, x,y,t)$. There is also a version for compactly supported cohomology $H\_c(X, x,y,t)$. The polynomial $E(X,x,y)=H\_c(... | https://mathoverflow.net/users/41301 | Mixed Hodge Polynomial for Algebraic Stacks | The answer is yes, these constructions make sense also for stacks, although you typically obtain a power series rather than a polynomial. Funnily enough, the best reference I know is Deligne's Hodge III, even though it doesn't talk at all about algebraic stacks.
Before asking how to put a mixed Hodge structure on the... | 5 | https://mathoverflow.net/users/1310 | 385424 | 160,220 |
https://mathoverflow.net/questions/379705 | 1 | This question has been partly answered in MSE, see [here](https://math.stackexchange.com/questions/3959775/bound-for-order-of-a-group-depending-on-number-of-elements-of-maximal-order).
In a paper [On the Number of Elements of maximal order in a Group](https://doi.org/10.1080/00029890.2019.1528826), it is proven that ... | https://mathoverflow.net/users/170420 | Bound for order of a group depending on number of elements of maximal order | My advisor found a way to prove it. I will add a full proof, if I find time!
Step 1: Prove that $m$ ist not divisible by $8$ or a square of an odd prime.
Step 2: We see that $N=\langle x \rangle \times M$ with $M$ being an elementary abelian $2$-group. If $4\not|\;\; m$, then $B=1$ and we are finished. Else we show... | 2 | https://mathoverflow.net/users/170420 | 385430 | 160,223 |
https://mathoverflow.net/questions/385440 | 1 | What does it mean that the function space $L^q\_tL^p\_x$ is invariant under the (3D) Navier-Stokes scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$ if $2/q + 3/p = 1$?
Does it mean that you compute the integral $$\left(\int\_0^t\left( \left(\int\_{\mathbb R^3} (\lambda u(\lambda x, \lambda^2 t))^pdx\right)^... | https://mathoverflow.net/users/175203 | What does the "scaling invariant" Serrin condition mean? | Yes.
---
(Body of answer must be 30 characters, and I only entered four. So here are a bit more.)
---
*The Computation*:
On $\mathbb{R}^3$, set $y = \lambda x$ and so $dx = \lambda^{-3} dy$. Set $s = \lambda^2 t$ so that $dt = \lambda^{-2} ds$. Plug into you integral expression you have
$$ \left(\int\_0... | 2 | https://mathoverflow.net/users/3948 | 385443 | 160,225 |
https://mathoverflow.net/questions/385422 | 3 | Say that a [preorder](https://en.wikipedia.org/wiki/Preorder) (i.e., a reflexive and transitive binary relation) $\preceq$ on a set $X$ is
* *artinian* if there is no sequence $(x\_n)\_{n \ge 1}$ of elements of $X$ with $x\_{n+1} \prec x\_n$ for each $n$, where $u \prec v$ means as usual that $u \preceq v$ and $v \no... | https://mathoverflow.net/users/16537 | Well-foundedness of divisibility vs well-foundedness of right- and left-divisibility | I will give a semigroup example. You can adjoin an identity to get a monoid example.
I think your question (and also what Green had in mind, which is something different) is answered by Baer-Levi semigroups. Let $X$ be a countably infinite set and let $S$ be the semigroup of all one-to-one maps $f\colon X\to X$ with ... | 4 | https://mathoverflow.net/users/15934 | 385447 | 160,227 |
https://mathoverflow.net/questions/385429 | 3 | Let $G=(V,E)$ be a complete bipartite graph with $2n$ vertices and $M \subset E$ some unknown perfect matching of $G$.
The goal is to determine $M$ by repeatedly choosing some perfect matching $M\_i \subset E$ and asking for $|M \cap M\_i|=m\_i$.
**Question**: What would be a good strategy of choosing the matchings... | https://mathoverflow.net/users/151546 | Determining a specific perfect matching $M$ by repeatedly asking for $|M \cap M_i|$ for other perfect matchings $M_i$ | This is a variant of the Mastermind game, a classical problem of black-box optimization. This problem (reformulated with permutations instead of perfect matchings, but it's exactly the same) is addressed here : <https://www.sciencedirect.com/science/article/pii/0196677486900131>
They give a polynomial-time algorithm ... | 1 | https://mathoverflow.net/users/174620 | 385448 | 160,228 |
https://mathoverflow.net/questions/385410 | 4 | I know that mean curvature and diffusion-type flows are common in manifold learning because of their smoothing effects. I haven't seen Ricci flow used as much. Given that Ricci and diffusion-type flows behave quite similarly, what is the reason for this (e.g. computational)?
I was thinking that a discrete Ricci flow ... | https://mathoverflow.net/users/174866 | Ricci flow for manifold learning | My limited exposure to manifold learning suggests that it is often searching for a submanifold in a high dimensional Euclidean space. There is one property that makes Ricci flow difficult for deforming submanifolds, which is that it is an intrinsic flow, and does not depend on the embedding. On the other hands, somethi... | 4 | https://mathoverflow.net/users/125275 | 385450 | 160,229 |
https://mathoverflow.net/questions/385449 | 1 | I need to find "Algorithms for Polynomials Over a Real Algebraic Number Field
Ph.D. thesis, University of Wisconsin, Madison (1974) by Rubald". However I cannot find it online nor in my university library, even though it seems to be quite a popular reference in the field (eg. Collins and McCallum often cite it).
>
... | https://mathoverflow.net/users/146338 | Algorithms for Polynomials Over a Real Algebraic Number Field, a reference | The thesis can be found [here](https://ftp.cs.wisc.edu/pub/techreports/1974/TR206.pdf).
| 2 | https://mathoverflow.net/users/120914 | 385458 | 160,231 |
https://mathoverflow.net/questions/385361 | 6 | Suppose we are given a finite family of points $p\_1,...,p\_n\in \Bbb R^d$, so that any two points have a rational distance square, that is,
$$\|p\_i-p\_j\|^2\in\Bbb Q,\quad\text{for all $i,j\in\{1,...,n\}$}.$$
Is it know whether these points can be isometrically embedded into $\Bbb Q^{D}$ for some sufficiently lar... | https://mathoverflow.net/users/108884 | Can every set of points with rational distance squares be isometrically embedded in $\Bbb Q^d$? | The following argument gives a bound of $D =4d$, based on a suggestion of LSpice in the comments.
Set $p\_1=0$. Using the distances and the fact that $p\_1=0$, we can find the dot products $p\_i \cdot p\_j$, which are all rational.
Assume we've embedded $p\_1,\dots,p\_k$ in $\mathbb Q^m$ and let's embed $p\_{k+1}$.... | 6 | https://mathoverflow.net/users/18060 | 385470 | 160,237 |
https://mathoverflow.net/questions/385371 | 11 | This is a re-post of a question I asked a month ago on MSE, but unfortunately didn't receive any answers. I'm hoping someone could help me with it. Here it goes:
Recently I've been self-studying the theory of covering spaces from "Introduction to Topological Manifolds", by John M. Lee. At the end of Chapter 11, there... | https://mathoverflow.net/users/175146 | Construction of the universal covering space via compact-open topology | Here is the key step you need to finish the proof: We are supposing $X$ is locally path-connected and semilocally simply connected, $\pi:P(X,x\_0)\to \widetilde{X}$ is the quotient map identifying path-homotopy classes of paths, and $p:\widetilde{X}\to X$ is the induced surjection. Also, $ev:P(X,x\_0)\to X$, $ev(\alpha... | 8 | https://mathoverflow.net/users/5801 | 385474 | 160,238 |
https://mathoverflow.net/questions/385472 | 2 | It is stated on the [nlab](https://ncatlab.org/nlab/show/skeleton#equivalents_of_choice) that the axiom of choice is equivalent to the statement that all small categories have a weak skeleton, meaning a skeletal category which is equivalent to them.
>
> Is the [axiom of global choice](https://ncatlab.org/nlab/show/... | https://mathoverflow.net/users/92164 | Global choice and skeletons of large categories | The answer is yes, by the same argument as for small categories and regular choice. First, assuming global choice, you can form skeleta by picking out one element from each isomorphism class of objects.
EDIT: As Sergei Akbarov points out in the comment, the argument is not quite right - to apply global choice we need... | 1 | https://mathoverflow.net/users/30186 | 385477 | 160,241 |
https://mathoverflow.net/questions/385478 | 4 | This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it.
Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite group $G$ is acting on $X$ and its quotient is an $S$-scheme $X//G$.
Is the quotient projective or at least proper? (I ... | https://mathoverflow.net/users/127776 | Is quotient of projective scheme over arbitrary base by a finite group also projective | I think the reference you want is:
Seshadri, C. S., **Geometric reductivity over arbitrary base**. *Advances in Math*. 26 (1977), no. 3, 225–274.
In particular, I think you will find conditions for a positive answer in Theorem 4 and Remark 10.
Here is part of Remark 10:
"We have been so far principally working ... | 4 | https://mathoverflow.net/users/12218 | 385480 | 160,242 |
https://mathoverflow.net/questions/385482 | 4 | If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is said to be a *vertex cover* if $C\cap e\neq \varnothing$ for all $e\in E$.
Is there an infinite graph $G=(V,E)$ such that for any vertex cover $C$ there is a vertex cover $C'\subseteq C$ with $C'\neq C$?
| https://mathoverflow.net/users/8628 | Infinite graph with no minimal vertex cover | No, by Zorn's Lemma!
It suffices to check that the intersection of a chain of vertex covers is a vertex cover. If the intersection $C$ fails to be a vertex cover, then there is some edge $(v,w)$ such that neither $v$ nor $w$ is in $C$. But then both $v$ and $w$ are excluded at some point in the chain, so not every se... | 10 | https://mathoverflow.net/users/2126 | 385485 | 160,244 |
https://mathoverflow.net/questions/385397 | 9 | I have heard that one application of $\infty$-categories is that they allow us to formulate a meaningful theory of descent for derived categories (say of sheaves on a scheme). While I'm sure the details are somewhere in Lurie's exposition of stable $\infty$-categories, I was hoping that someone familiar with the proces... | https://mathoverflow.net/users/126543 | How do $\infty$-categories allow us to do descent on the derived level? | Let $X$ be a topological space covered by open sets $U$ and $V$.
Let $\mathscr{F}$ and $\mathscr{G}$ be complexes of sheaves defined on $U$ and $V$, respectively. Suppose you are given an isomorphism $\alpha: \mathscr{F}|\_{ U \cap V} \rightarrow \mathscr{G}|\_{ U \cap V}$ in the derived category of the intersection $U... | 23 | https://mathoverflow.net/users/7721 | 385492 | 160,247 |
https://mathoverflow.net/questions/385493 | 2 | While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose an element $x$ of $X$' or something like this.
That is, while I understand that 'every surjection in ${\bf Set}$ split... | https://mathoverflow.net/users/92164 | A simple form of choice | Nothing is wrong with this version of choice. In $\sf ZF$, and the theories extending it, it is indeed equivalent to Global Choice, exactly by using Scott's trick. You just smooth it out by putting this into the language.
The difference will come if you start allowing proper class of atoms (non-sets). In that case, i... | 4 | https://mathoverflow.net/users/7206 | 385496 | 160,248 |
https://mathoverflow.net/questions/385296 | 2 | Let $E \to B$ be a Hermitian vector bundle. If $E$ has a projectively flat connection, then its total Chern character has the form $\mbox{ch}(E) = \mbox{rank} \cdot \exp(\mbox{slope})$. Is the converse true? In other words, can I deduce that a vector bundle has a projectively flat connection just by looking at its Cher... | https://mathoverflow.net/users/175087 | Projectively flat connection | Let $E$ is a Hermitian vector bundle with vanishing Chern classes.
**Proposition:** If $E$ admits a projectively flat Hermitian connection, then it admits a flat Hermitian connection.
**Proof:** Let $\nabla$ be a projectively flat Hermitian connection. Then its curvature $F\_\nabla$ has the form
$$F\_\nabla = i\ome... | 1 | https://mathoverflow.net/users/173096 | 385500 | 160,249 |
https://mathoverflow.net/questions/385451 | 16 | I'm trying to think of an example of a group $G$ with non-trivial center such that there exist subgroups $H\_1,H\_2\le G$ both isomorphic to $G$ and satisfying $H\_1\cap H\_2=\{1\}$. Does such a group exist? Preferably an easy-to-state example that I'm just not thinking of? (Ideally finitely generated, but I won't insi... | https://mathoverflow.net/users/164670 | Group with non-trivial center containing trivially-intersecting copies of itself | Here is a construction for a group similar to the braided Thompson group $BV$ that ought to have this property. Define the $n$th ***ribbon group*** to be the semidirect product
$$
R\_n = \mathbb{Z}^n \rtimes B\_n
$$
where the braid group $B\_n$ acts on $\mathbb{Z}^n$ by permuting the $n$ factors. Elements of $R\_n$ can... | 14 | https://mathoverflow.net/users/6514 | 385505 | 160,253 |
https://mathoverflow.net/questions/385513 | 21 | The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see [this comment](https://mathoverflow.net/questions/192230/is-there-a-manifold-with-fundamental-group-mathbbq#comment479763_192230) or [this answer](https://math.stackexchange.com/a/... | https://mathoverflow.net/users/21564 | Does every group arise as the fundamental group of a complete Kähler manifold? | Any Stein manifold admits a complete Kähler metric. Start with a connected real analytic manifold with the given fundamental group. A suitable tubular neighbourhood of the complexification will be Stein. This Stein manifold will have the same fundamental group as the original manifold. For details see page 383 of *Nara... | 26 | https://mathoverflow.net/users/4696 | 385514 | 160,259 |
https://mathoverflow.net/questions/385543 | 3 | Anyone know of an English translation of the oft-cited paper "Funktionaloperationen und Gleichungen" by Salvatore Pincherle, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Vol. 2 (Analysis), Part I.2, pp. 761--817 (B.G. Teubner, Leipzig, 1904-1016).
This is Pincherle of the Pincherle... | https://mathoverflow.net/users/12178 | Translation of a paper by Salvatore Pincherle | You can find an OCR-ed text at [https://gdz.sub.uni-goettingen.de/id/PPN360506208?tify={%22pages%22:[77],%22view%22:%22fulltext%22}](https://gdz.sub.uni-goettingen.de/id/PPN360506208?tify=%7B%22pages%22:%5B77%5D,%22view%22:%22fulltext%22%7D). As suggested by Carlo Beenakker in comments, running this through Google tran... | 2 | https://mathoverflow.net/users/45250 | 385547 | 160,271 |
https://mathoverflow.net/questions/385545 | 3 | Let $X$ and $Y$ be standard Borel spaces, $Y$ uncountable, and $f : X \to Y$ a surjective Borel map. Is it possible that there is a countable ordinal $\alpha$ such that for each Borel set $B \subseteq Y$, the Borel rank of the set $f^{-1}(B) \subseteq X$ is at most $\alpha$?
| https://mathoverflow.net/users/175248 | Can there be an upper bound on the Borel rank of the preimages of Borel sets under a surjective Borel map? | How about this: Construct a Cantor set $E \subseteq X$ on which $f$ is bijective. Then $f$ is a homeomorpism of $E$ onto $f(E)$, and $f(E)$ has Borel subsets of arbitrarily high rank.
| 2 | https://mathoverflow.net/users/454 | 385551 | 160,273 |
https://mathoverflow.net/questions/385540 | 9 | There is a very natural way to define generators of $\pi\_{4n-1}(SO(4n))\cong \mathbb{Z}\oplus \mathbb{Z}$ in terms of quaternions when $n=1$ and octonions when $n=2$ (see for example Tamura, *On Pontrjagin classes and homotopy types of manifolds*, 1957). Since there are no normed division algebras in higher dimensions... | https://mathoverflow.net/users/147200 | Homotopy groups $\pi_{4n-1}(SO(4n))$ | One way to think about this would be to consider the (injective!) map $\pi\_{4n-1} SO(4n) = \pi\_{4n} BSO(4n) \to \mathbf Z^2$ that sends the classifying map of a $4n$-dimensional oriented real vector bundle $\xi\colon E \to S^{4n}$ to the pair $(e,p\_n)$ consisting of its Euler and $n$th Pontryagin class evaluted agai... | 14 | https://mathoverflow.net/users/14233 | 385554 | 160,274 |
https://mathoverflow.net/questions/385553 | 4 | Let us consider the Minkowski space $(\mathbb{R}^{4},\eta)$ and the mass shell $H\_{m}$, $m\ge 0$, given by:
\begin{eqnarray}
H\_{m}:=\{x=(x\_{0},x\_{1},x\_{2},x\_{3}) \in \mathbb{R}^{4}: \hspace{0.1cm} x\cdot \tilde{x} = m^{2}, x\_{0}>0\} \tag{1}\label{1}
\end{eqnarray}
where $\tilde{x}$ is defined by $x=(x\_{0},x\_{1... | https://mathoverflow.net/users/150264 | Topology on Minkowski space $\mathbb{R}^{4}$ and Lorentz invariant measure | **Q1** The topology on $\mathbb{R}^4$ is the usual one. This is the general case for Lorentzian geometry: the topology is the one defined by the charts in your atlas.
**Q2** Given a fixed Lorentz transformation $\Lambda$, it sends $H\_m\to H\_m$ as you observed. It is a linear transformation of $\mathbb{R}^4$, and he... | 5 | https://mathoverflow.net/users/3948 | 385556 | 160,275 |
https://mathoverflow.net/questions/372226 | 2 | Suppose that $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$. Given a two plane $\Pi=\textrm{Span}\{X,Y\}$ with $X,Y \in T\_pM$, we say that $\Pi$ is non-degenerate if
$$ g(X,X)g(Y,Y)-g(X,Y)^2 \neq 0.$$
Moreover, given a non-degenerate two-plane we say that it is timelike or spacelike if the above quanti... | https://mathoverflow.net/users/50438 | stability of two-sided sectional curvature bounds in Lorentzian geometry | To add on to [Ettore Minguzzi's answer](https://mathoverflow.net/a/385541/3948):
* If $g(R(X,Y)X,Y) < 0$ for all pairs $X,Y$ as he defined, then since this is an open condition stability is automatic for small smooth compactly supported perturbations.
* If $g(R(X,Y)X,Y) = 0$ for some pair, then stability can be viola... | 4 | https://mathoverflow.net/users/3948 | 385561 | 160,278 |
https://mathoverflow.net/questions/385565 | 13 | I have two questions about the definition of a Hopf algebra in Hatcher's book on algebraic topology. He defines it as follows (see Section 3.C, page 283):
**Definition**: A *Hopf algebra* is a graded algebra $A = \oplus\_{n \geq 0} A^n$ over a commutative base ring $R$ satisfying the following two conditions:
1. Th... | https://mathoverflow.net/users/174957 | Question about definition of Hopf algebra in Hatcher | You have not transcribed Hatcher's definition correctly. He has a sum $\sum\_i\alpha'\_i\otimes\alpha''\_i$ over some unspecified index set, not over $\{1,\dotsc,n-1\}$, and he requires that the terms $\alpha'\_i,\alpha''\_i$ are homogeneous of arbitrary positive degree (although the requirement that $\Delta$ preserves... | 8 | https://mathoverflow.net/users/10366 | 385569 | 160,280 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.