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https://mathoverflow.net/questions/368600 | 3 | Let $M$ be a finite-dimensional compact smooth manifold and
$$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$
**Q1-a:** What metrics $g$ are very close to the given metric $g\_0$? I.e. Is it possible $g\in B\_\varepsilon(g\_0,M)$ and $g$ has completely different curvature for sufficiently small ... | https://mathoverflow.net/users/90655 | Open neighbourhood of a point of space of Riemannian metrics | $\mathcal{M}et(M)$ carries many natural (= invariant under the action of the group of diffeomorphisms of $M$) Riemannian metrics. See the following papers (and references therein):
* Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. Journal of Differential Geomet... | 8 | https://mathoverflow.net/users/26935 | 368603 | 154,294 |
https://mathoverflow.net/questions/368559 | 5 | I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products:
>
> If $\{ X\_i \}\_{i \in I}$ is any collection of nonempty sets indexed by an index set $I$, then $\prod\_{i \in I} X\_i$ is nonempty.
>
>
>
To me, ... | https://mathoverflow.net/users/155425 | Foundational results dependent on/equivalent to the continuum hypothesis or its negation? | Concerning your first question, there is a simple, if not "self-evident", order-theoretic statement equivalent to $CH$ admitting a generalization equivalent to $GCH$:
* If $L$ is a linear ordering of size $2^{\omega}$, then $L$ embeds every cardinal less than $2^{\omega}$ or $L^\*$ ($L$ reversed) embeds every cardina... | 5 | https://mathoverflow.net/users/9825 | 368610 | 154,297 |
https://mathoverflow.net/questions/368624 | 0 | Let $ f: X\to (-\infty,+\infty]$ that $ X$ is **an infinite dimensional space**.
What are the conditions for $f$ and space $X$ to have the following equality correct?
$$\partial f(x)=\{\nabla f(x)\}$$ for all $x\in X$
I know when space X is **finite dimensional** and $f$ is proper convex function then
$$\partial ... | https://mathoverflow.net/users/147309 | the subdifferential at points of differentiability in infinite dimensional space | That depends on what you mean by $\nabla f$. If $X$ is a normed vector space and $f:X\to \mathbb{R}$ is Gâteaux differentiable, then $\partial f(x) = \{DF(x)\}$ for the Gâteaux derivative $DF(x)\in X^\*$ of $F$ at $x$; the proof is virtually identical to the finite-dimensional one using the definition of the Gâteaux de... | 1 | https://mathoverflow.net/users/30516 | 368627 | 154,303 |
https://mathoverflow.net/questions/368566 | 1 | Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $F$ and $SS(F)$ be its singular support (I follow the terminology and notation of the book “Sheaves on manifolds” by Kash... | https://mathoverflow.net/users/16183 | Relation between characteristic cycle and singular support of constructible sheaf | No.
Consider $M = \mathbb R$, $F$ the direct sums of the constant sheaves on the positive real numbers, negative real numbers, and $0$, extended by zero to the whole space.
Then $F$ is the associated graded of a filtration on the constant sheaf, hence has the same characteristic cycle as the constant sheaf, which d... | 2 | https://mathoverflow.net/users/18060 | 368641 | 154,309 |
https://mathoverflow.net/questions/368658 | 5 | Is arc connected-ness well-behaved with respect to products?
That is -
>
> $\prod X\_\alpha$ is arc connected iff $X\_\alpha$ is arc connected $\forall \alpha$
>
>
>
In [this question](https://math.stackexchange.com/q/3779642) on MathStackexchange, an answer is provided only for the reverse implication, that... | https://mathoverflow.net/users/140681 | Arc connectedness of product spaces | This is false when the spaces are not Hausdorff. Let $X$ be the line with two origins $\{O\_1,O\_2\}$ and $Y$ be the usual line. Then $X\times Y$ is arc connected because you can pick an arc that starts at $(O\_1,y\_1)$ travels outside $\{O\_1,O\_2\}\times Y$ and then comes back to $(O\_2,y\_2)$, but $X$ itself is not ... | 7 | https://mathoverflow.net/users/2384 | 368663 | 154,315 |
https://mathoverflow.net/questions/364393 | 4 | Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$?
By "measurable total ordering" we mean that any initial segment $I\_y:=\{x: x<y\}$ is Borel-measurable.
Edit: We know that separability is sufficient for a meas... | https://mathoverflow.net/users/12518 | Measurable total order | As mentioned in the OP, Vladimir Pestov has answered the question affirmatively. See Appendix D here:
<https://arxiv.org/pdf/1906.09855.pdf>
| 1 | https://mathoverflow.net/users/12518 | 368664 | 154,316 |
https://mathoverflow.net/questions/368660 | 1 | Consider an equation like
$$-\Delta u = |u|^p u $$ in $\Omega$ with $u=0$ on $ \partial \Omega$ where $\Omega$ a domain in $ R^N$ and $ u:\Omega \rightarrow R^N$. Here $p$ is arbitrary or maybe $p=2$. Or consider Neumann problems like this with a zero order term $u$ added to the left.
Do these equations have a name... | https://mathoverflow.net/users/66623 | name of elliptic pde with a power law nonlinearity | The case $p=2$ is the [nonlinear Schrödinger equation,](https://en.wikipedia.org/wiki/Nonlinear_Schr%C3%B6dinger_equation) more generally written as
$$Eu=-\Delta u+\kappa|u|^2 u,$$
with coefficients $E,\kappa\in\mathbb R$. It describes the propagation of light in nonlinear optical fibers and is also a model for a super... | 2 | https://mathoverflow.net/users/11260 | 368665 | 154,317 |
https://mathoverflow.net/questions/368581 | 2 | Define
\begin{equation}
F(\sigma) = \Re \sum\_{h=1}^{\infty} \sum\_{n=1}^{\infty} \frac{b\_{n, h}}{n^{2\sigma}}(1+h/n)^{-\sigma}\Bigg( \frac{e^{i\log(1+h/n)}-1}{i\log(1+h/n)} \Bigg)
\end{equation} where $\sigma \in \mathbb{R}$ and $|b\_{n, h}|\ll \log n$. *What is the minimal real number $c$ such that $F(\sigma)$ has a... | https://mathoverflow.net/users/480516 | Does this function have a holomorphic continuation in $\sigma > \frac{1}{2}$? | Observe first that
$$\lim\_{x\to 0}\frac{e^{i\log(1+x)}-1}{i\log(1+x)}=1,$$
hence there exists an absolute constant $C>0$ such that
$$\Re\frac{e^{i\log(1+h/n)}-1}{i\log(1+h/n)}>\frac{1}{2}\qquad\text{for}\qquad n>Ch.$$
Now assume that $b\_{n,h}$ is the indicator function of $n\in(Ch,2Ch)$. Then for $\sigma\in[1,2]$ we ... | 2 | https://mathoverflow.net/users/11919 | 368671 | 154,320 |
https://mathoverflow.net/questions/368684 | 4 | Let $x, y, z$ be pairwise coprime positive integers. Does one have $x^5 + y^5 = z^p$ for any prime $p \geq 2$ ?
| https://mathoverflow.net/users/480516 | On the Diophantine equation $x^{5} + y^5 = z^p$ | To the best of my knowledge, this is open for general $p$.
As mentioned by Alapan Das, Bjorn Poonen has solved the case
$p = 2$ and also $p = 3$ [B. Poonen, Some diophantine equations
of the form $x^n + y^n = z^m$, Acta Arith. 86 (1998), 193-205].
The case $p = 5$ is part of FLT. Sander Dahmen and Samir Siksek
[Perfect... | 10 | https://mathoverflow.net/users/21146 | 368689 | 154,326 |
https://mathoverflow.net/questions/368090 | 11 | This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, **axiomatically**, a quantum group?
There are many classes of noncommutative algebras that everybody agrees is a quantum group (or quantum algebra): quantizati... | https://mathoverflow.net/users/160378 | Axiomatic definition of quantum groups | I would say that if you are looking for a concrete definition then it's better to adopt the [Tannakian point of view](https://ncatlab.org/nlab/show/Tannaka+duality) and to focus on the category of representations of the quantum group rather than on the algebra itself. So take as your fundamental object a tensor categor... | 6 | https://mathoverflow.net/users/3072 | 368699 | 154,331 |
https://mathoverflow.net/questions/368696 | 6 | Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the category of simplicial objects $s\mathcal{C}$ (which is an $\infty$-category), take its stabilization $\text{Stab}(s\mathcal{C})$ and take the homotopy category $\text{Ho}(s\mathcal{C})$ of the simplicial catego... | https://mathoverflow.net/users/152554 | "Universal" triangulated category | I will give a partial answer. I note that the OP has asked a LOT of questions recently (I count 12 so far in the first 9 days of August), and many of them are good questions on which much research has already been done. I would encourage the OP to slow the rate of question-asking, to spend more time reading the referen... | 14 | https://mathoverflow.net/users/11540 | 368711 | 154,337 |
https://mathoverflow.net/questions/368518 | 4 | In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in the category of $\mathbb{A}^1$-homotopy, i.e. do we have
$$H^n\_{ét}(X,F)\cong H^{n+a+b}\_{ét}\left( (\mathbb{A}^1/\{0,1\... | https://mathoverflow.net/users/152554 | Suspension Theorem in $\mathbb{A}^1$-homotopy | This question has already been satisfactorily answered in the comments. To avoid it lingering around as "unanswered," I am providing a CW answer with a reference for the observations made by Denis-Charles Cisinski.
A complete treatment of etale cohomology operations is provided in [Operations in étale and motivic coh... | 2 | https://mathoverflow.net/users/11540 | 368714 | 154,338 |
https://mathoverflow.net/questions/368718 | 1 | Let $n\_1>n\_2\geq 1$ be integers. Are there a known algorithms for generating $n\_2\times n\_1$-dimensional random matrices $A$ such that
$$
\|Ax - Ay\|<\|x-y\| \mbox{ if $x\neq y$}?
$$
| https://mathoverflow.net/users/36886 | Reference Request: Randomly Generated Contraction | [On some properties of contracting matrices:](https://arxiv.org/abs/math/0604457)
For $n\_1=n\_2$ and if the norm is the $\|\cdots\|\_\infty$ norm, then the contractive property (with $\leq$ instead of $<$) is satisfied if the matrix is a [Markov matrix](https://en.wikipedia.org/wiki/Stochastic_matrix) (nonnegative ... | 1 | https://mathoverflow.net/users/11260 | 368719 | 154,340 |
https://mathoverflow.net/questions/368721 | 4 | It is [known](https://mathworld.wolfram.com/LogisticMap.html) that for $r=-2,2,4$ the logistic map $x\_{n+1}=r x\_n (1-x\_n)$ has exact solutions of the form
$$
x\_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x\_0)\right)\right\} \qquad \qquad{(\*)}
$$
for suitable functions $f$. The same source further claims, with refer... | https://mathoverflow.net/users/163525 | Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$? | Explicit solutions for arbitrary $r$ exist in various forms:
* [Logistic map: an
analytical solution](http://havlin.biu.ac.il/PS/rbbsh260.pdf) (1995) represents the solution as a power of
a transfer matrix.
* [An explicit solution
for the logistic map](http://havlin.biu.ac.il/PS/rmh365.pdf) (1999) gives a functional ... | 5 | https://mathoverflow.net/users/11260 | 368724 | 154,341 |
https://mathoverflow.net/questions/368716 | 15 | Let $Q\in \mathbb{Z}[x]$ be a polynomial defining an injective function $\mathbb{Z}\to\mathbb{Z}$. Does it define an injective function $\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ for some prime $p$?
| https://mathoverflow.net/users/nan | Injective integer polynomial is injective modulo some prime | Consider $Q(x)=x(2x-1)(3x-1)$. This gives an injective map $\mathbb Z\to \mathbb Z$, because $n<m \implies Q(n)<Q(m)$. However, this $Q$ is not injective over $\mathbb Z/p\mathbb Z$ for any $p$ because $Q(x)=0$ has three solutions when $p\geq 5$ and two solutions when $p\in \{2,3\}$.
| 32 | https://mathoverflow.net/users/2384 | 368728 | 154,343 |
https://mathoverflow.net/questions/368733 | 1 | Edit: Version 2:
Suppose that $A,B,C$ are chain complexes and $f: A \rightarrow B$ is a chain map. Suppose that there is a homotopy equivalence
$$ \text{Cone}(f: A \rightarrow B) \simeq C.$$
The chain map $u: \text{Cone}(f) \rightarrow C$ provided by the homotopy in particular means that I am provided with a chai... | https://mathoverflow.net/users/163478 | Homotopy equivalences and Mapping Cones | It is true. It is not trivial (this is an opinion), however it is standard. In any triangulated category, two objects and a map determine the third, up to (usually non unique) isomorphism. And the category of Chain complexes with maps up to homotopy is a classical example of triangulated category. A reference for those... | 3 | https://mathoverflow.net/users/98863 | 368737 | 154,345 |
https://mathoverflow.net/questions/368639 | 1 | Let $X\subset \mathbb{C}^{n}$ be a domain. You can assume that it is nice (e.g. bounded convex balanced ). Let $\{x\_n\}$ be a sequence of points that does not have a limit point in $X$.
Let $D$ be the (unit) disc on the plane.
>
> Is there a holomorphic $\varphi:D\to X$ such that $\varphi(D)$ contains an infinit... | https://mathoverflow.net/users/53155 | How many points of a sequence can we catch with an analytic disc? | The answer to your quesiton is yes.
In particular in [Discs in Stein manifolds containing given discrete sets](https://www.fmf.uni-lj.si/%7Edrinovec/research/SteinMZ.pdf) B. Drinovec Drnovšek proves that given a discrete subset $S$ of a connected Stein manifold $M$ (every domain of holomorphy in $\mathbb{C}^m$ is if ... | 1 | https://mathoverflow.net/users/47862 | 368739 | 154,346 |
https://mathoverflow.net/questions/368744 | 0 | Let $(X, \mathcal X)$ be a measurable space. Say that a net $(\mu\_\alpha)$ of finitely additive probability measures converges to a finitely additive probability measure $\mu$ if and only if $\mu\_\alpha(A) \to \mu(A)$ for all $A \in \mathcal X$.
If $f$ is an extended-real-valued simple $\mathcal X$-measurable funct... | https://mathoverflow.net/users/96899 | Does the finitely additive integral preserve convergence for non-negative measurable functions? | No, not even for sequences of countably additive measures.
Take $X = \mathbb{N} = \{0,1,2,\dots\}$ with its discrete $\sigma$-algebra, and let $\mu\_n$ put mass $1/n$ at the point $n$ and mass $1-1/n$ at $0$. Let $\mu$ put mass $1$ at $0$. Then it is clear that $\mu\_n(A) \to \mu(A)$ for every set $A$ (consider the c... | 5 | https://mathoverflow.net/users/4832 | 368746 | 154,347 |
https://mathoverflow.net/questions/368709 | 1 | An algebraic stack or Artin stack is a stack in
groupoids $\mathcal{X}$ over the étale site such that the diagonal
map of $\mathcal{X}$ is representable and there exists a smooth
surjection from (the stack associated to) a scheme to $\mathcal{X}$.
In [Wikipedia's article on stacks](https://en.wikipedia.org/wiki/Stack... | https://mathoverflow.net/users/108274 | Stabilizer $G_x$ of a $k$-valued point of an algebraic Stack | This was getting a little bit long for a comment, so I'll just write it here:
Let $X\simeq S//R$ be an algebraic stack presented by a smooth surjective map $S\to X$ with $S$ a scheme, then $R=S\times\_X S$, and the pair of maps $R\rightrightarrows S$ has the canonical structure of a groupoid in algebraic spaces (with... | 2 | https://mathoverflow.net/users/1353 | 368752 | 154,350 |
https://mathoverflow.net/questions/368292 | 5 | This question is two-fold.
The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of examples coming from Borromean rings. By performing $-1/m$ and $-1/n$ surgery on two components of the Borromean rings, we g... | https://mathoverflow.net/users/45553 | Negative surgeries on negative knots | $(-7)$-surgery on the left-handed trefoil yields the lens space $L(7,2)$ which is defined to be the $(-7/2)$-surgery along the unknot.
Similarly one can get more examples along negative torus knots producing lens spaces. Moser classified all surgeries along torus knots in [L. Moser, Elementary surgery along a torus k... | 5 | https://mathoverflow.net/users/84120 | 368753 | 154,351 |
https://mathoverflow.net/questions/368760 | 1 | First of all, I am sorry for the ''not clear title' for this question but I cannot find a better way to describe this seemingly very simple and standard inequality,
So.. I am reading a paper '**Two-dimensional Navier-Stokes Equation Driven by a space time white noise**' by Daprato and Debussche. And I came across an ... | https://mathoverflow.net/users/127918 | Inequality regarding a probability measure | It looks fine to me.
If we let $I\_k = [k t^\ast\_M, (k+1) t^\ast\_M]$ be the relevant subintervals of $[0,T]$, then the supremum of $|u\_n|$ over $[0,T]$ must be almost attained along some sequence of points, and by pigeonhole infinitely many of them must be in one of the $I\_k$, call it $I\_{k\_0}$, so that $\sup\_... | 1 | https://mathoverflow.net/users/4832 | 368762 | 154,353 |
https://mathoverflow.net/questions/368761 | 2 | I am interested in finding bounds on cumulants in terms of moments.
For example, [this paper](https://link.springer.com/content/pdf/10.1007%2FBF01043479.pdf) alludes to the bound
\begin{align}
|\kappa\_n|\le n^n E[|X-E[X]|^n]
\end{align}
where $\kappa\_n$ is the $n$-th cumulant.
However, due to the language barrier, tr... | https://mathoverflow.net/users/69661 | Bounds on cumulants in terms of moments | Let $k\_n:=\kappa\_n$, $a\_n:=E(X-EX)^n$, $b\_n:=E|X-EX|^n$, so that $|a\_n|\le b\_n$. We have to show that
$$|k\_n|\le n^n b\_n$$
for natural $n$. For $n=1,2$ this is obvious. The key is the recursion
$$k\_n=a\_n-\sum\_{m=1}^{n-1}\binom{n-1}{m-1}k\_m a\_{n-m}$$
at the end of [this section](https://en.wikipedia.org/wik... | 2 | https://mathoverflow.net/users/36721 | 368764 | 154,354 |
https://mathoverflow.net/questions/368659 | 9 | Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", not assuming smoothness.
Two groups can have the same Lie algebras without being equal. For example, if $k$ has characteristic $2$, then ev... | https://mathoverflow.net/users/2383 | Showing subgroups with equal Lie algebras are equal | $\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Cent{C}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Lie{Lie}$The key point is not, as I expected, whether $\Cent\_G(T)^\circ$ is a torus, but whether it equals $\Cent\_G(\Lie(T))^\circ$. Certainly it is contained in the latter group, so this is the same as asking whet... | 5 | https://mathoverflow.net/users/2383 | 368772 | 154,356 |
https://mathoverflow.net/questions/368792 | 1 | I've come up with an idea of an integer sequence. It can be formulated (perhaps a bit loosely) as follows: **For *n* points *N(n)* is the number of configurations where each point either lies on some circle or is a center of some circle. Each point lying on a circle can belong to only 1 circle and each center point can... | https://mathoverflow.net/users/163415 | What OEIS sequence is this? | Your sequence is the same as the linked OEIS sequence. This is the
>
> Number of partitions of $n$ into parts of two kinds.
>
>
>
In your case, the two kinds are circles for which the centre is occupied and circles for which the centre is not occupied. See the "example" section in the OEIS entry where you can ... | 8 | https://mathoverflow.net/users/17647 | 368802 | 154,365 |
https://mathoverflow.net/questions/368803 | 4 | On the Wolfram Research Reference page for the cotangent function (<https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/>), I saw the following partial sum formula
$$\sum\_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n.$$
I was unable to find a reference for it but eventually proved it as described bel... | https://mathoverflow.net/users/149093 | The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$ | The expression under the limit sign in question is just a Riemann sum for
$$\int\_0^{\pi/2} \frac12\,\Big(\frac1x-\cot x\Big)\,dx=\frac12\,\ln\frac\pi2,$$
which therefore is the value of the limit.
| 4 | https://mathoverflow.net/users/36721 | 368805 | 154,367 |
https://mathoverflow.net/questions/368568 | 6 | I admit I am not a differential geometer (a probabilist actually). However recently I get interested and I would like to have more intuitions and insight of what is the Riemann curvature.
This is the way I see it so far (please correct me if I am wrong):
* We start from a connection $\nabla$.
* This defines a paral... | https://mathoverflow.net/users/99045 | Is it possible to calculate the parallel transport on a loop from the Riemann curvature? | In the case that the Riemannian manifold $M$ in question has dimension $2$ and is oriented and $\gamma([0,1])\subset M$ is the piecewise-$C^1$ oriented boundary of a compact domain $S\subset M$, we have the famous Gauss-Bonnet Theorem, which asserts that the holonomy around $\gamma$ is equal to rotation by the angle
$$... | 11 | https://mathoverflow.net/users/13972 | 368808 | 154,369 |
https://mathoverflow.net/questions/368422 | 6 | The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align\*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \\ \dot{(\rho u)}+\nabla \cdot (\rho u u) + \nabla p=0 \;&\text{(momentum conservation)} \\ \dot{(\rho e)} +\nabla \cdot (\... | https://mathoverflow.net/users/161947 | Explanation for why an ideal fluid doesn't have increasing entropy? | This is a very important issue, to which an answer must be made in mathematical terms, rather than by waving hands.
Yes, the Euler system (conservation of mass, momentum and energy) is time-reversible. So where is the error when we say *Entropy is non-decreasing, but the system is time-reversible, therefore the entro... | 9 | https://mathoverflow.net/users/8799 | 368812 | 154,370 |
https://mathoverflow.net/questions/368730 | 3 | Let $\mathcal{B}$ be a braided ($n-1$)-category. I will assume that $\mathcal{B}$ is a fully-dualizable object in some $n+1$-category of braided ($n-1$)-categories. Hence, from $\mathcal{B}$, using the cobordism hypothesis, one gets a TQFT $\int\_{\Box}\mathcal{B}$.
I have read recently that $$\int\_{S^{n-1}\_b}\math... | https://mathoverflow.net/users/105094 | Factorization homology of a braided (n-1)-category on an (n-1)-sphere | I think this is an error in my paper. Thank you for finding it. The overall result is correct, but the proof is wrong as written. To correct it, I need to replace "TQFT" with "relative TQFT", and replace $S^{n-1}\_b$ with the pair $(D^n, S^{n-1}\_b)$, and the rest is correct. In detail:
$\newcommand\cB{\mathcal{B}}\new... | 1 | https://mathoverflow.net/users/78 | 368813 | 154,371 |
https://mathoverflow.net/questions/368811 | 3 | Consider, just as an example, an action of $\mathbb{C}^\*$ on $\mathbb{P}^2$ of the form
$$t\cdot p=[p\_0:tp\_1:t^2p\_2]$$
There are $3$ fixed points, namely $e\_1,e\_2,e\_3$. If I consider a $\mathbb{C}^\*$-linearizable line bundle -like $L=\mathcal{O}(1)$-, then I have an induced action
$$\phi:\mathbb{C}^\*\times... | https://mathoverflow.net/users/124705 | Weights on the linearization | If your $\mathbb{P}^2$ is $\mathbb{P}(\mathbb{C}^3)$, you can identify the complement of the zero section in $L^{-1}$ with $\mathbb{C}^3\smallsetminus 0$, viewed as a bundle over $\mathbb{P}^2$ via the projection $p:\mathbb{C}^3\smallsetminus 0\rightarrow \mathbb{P}^2$. One possible way to extend your action is to have... | 4 | https://mathoverflow.net/users/40297 | 368818 | 154,374 |
https://mathoverflow.net/questions/368537 | 4 | For any integer $N \geq 2$, we have the identity:
$$\frac{\ \prod \_{n=1}^{N-1}\ \left(2+2\sum \_{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)\ }{\prod \_{n=1}^{N-1}\ \left(1+2\sum \_{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)}=N$$
So how to prove it? Any help and suggestion will be appreciated, thank you!
| https://mathoverflow.net/users/163164 | The complex trigonometric function degenerates to the positive integer | Following Johann Cigler's [suggestion](https://mathoverflow.net/a/368807), set $q=e^{\frac{i\pi}{N}}$. We will need the two evaluations
$$\prod\_{n=1}^{2N-1}(1-q^n)=\left.\frac{x^{2N}-1}{x-1}\right|\_{x=1}=2N \tag{1}$$
$$\prod\_{n=1}^{N-1}(1-q^{2n})=\left.\frac{x^{N}-1}{x-1}\right|\_{x=1}=N \tag{2}$$
In your expression... | 8 | https://mathoverflow.net/users/2384 | 368821 | 154,375 |
https://mathoverflow.net/questions/368799 | 0 | Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H\_1 + \epsilon^2 H\_2 + \epsilon^3 H\_3 + ... $$ All matrices are symmetric. Assume $\tilde{A}$ to be positive definite. Let $E = [e\_0,e\_1,e\_2,...e\_{n-1}]$ be the eigenvectors matrix of $A$ and $\t... | https://mathoverflow.net/users/14414 | Convergence of the eigenvector matrix for an analytic perturbation of a singular matrix | Because of the degeneracy in the eigenvalues of $A$, "the eigenvectors matrix of $A$" is far from well-defined. Rather, there is a one-dimensional eigenspace of $A$ for eigenvalue $n$ (spanned by $(1,\ldots,1)^T$), and its orthogonal complement is the eigenspace for eigenvalue $0$. For any $\eta > 0$, there is $\delta ... | 2 | https://mathoverflow.net/users/13650 | 368823 | 154,376 |
https://mathoverflow.net/questions/368828 | 2 | Let $p:C\to\mathbb{P}^1$ be a degree $k$ morphism from a smooth projective curve $C$ to the projective line and $L$ a very ample line bundle on $C$. We know that $p\_\*\mathcal{O}\_C(L)$ is a rank $k$ locally free sheaf on $\mathbb{P}^1$ and hence is in the form $\mathcal{O}(e\_1)\oplus\cdots\oplus\mathcal{O}(e\_k)$ by... | https://mathoverflow.net/users/148748 | Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$ | No, not in general. Take $C=\mathbb{P}^1$, $L=\mathcal{O}(1)$, $p$ to be map $x\mapsto x^2$ in affine coordinates. Then $p\_\*L$ has rank $2$, but
$$2=h^0(L)=h^0(p\_\*L)=h^0(\mathcal{O}(e\_1))+h^0(\mathcal{O}(e\_2))$$
If $e\_1$ and $e\_2$ were both positive, then term on the right would be at least $4$. So this is i... | 8 | https://mathoverflow.net/users/4144 | 368830 | 154,378 |
https://mathoverflow.net/questions/368833 | 2 | Let $n$ be a positive integer, and
$$2 = p\_1 < p\_2 < \dots < p\_m \le n$$
be the sequence of all primes less than or equal to $n$.
For each index $j$ let $p\_j^{e\_j}$ be the largest power of $p\_j$ still less than or equal to $n$.
Define
$$S\_n = p\_1^{e\_1} + p\_2^{e\_2} + \dots + p\_{m}^{e\_m} $$
to be the s... | https://mathoverflow.net/users/138628 | What is the growth rate of the sum of powers of distinct primes closest to a given a integer? | A better lower bound is $S(n)$, the sum of all primes below $n$, and this lower bound makes a good asymptotic value. One can tweak this by observing that for every term corresponding to a prime less than $\sqrt{n}$ that term is at least $n^{2/3}$, so a tighter lower bound like $S(n) - S(\sqrt{n}) + n^{7/6}/\log n$ is a... | 2 | https://mathoverflow.net/users/3402 | 368838 | 154,381 |
https://mathoverflow.net/questions/368755 | 4 | Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G\_\delta$-subset of $\mathbb Q ^\omega$?
**Remark 1**. Suppose that $f:\mathbb P\to \mathbb Q ^\omega$ is a continuous bijection ... | https://mathoverflow.net/users/95718 | Mapping $\mathbb P$ onto $\mathbb Q ^\omega$ | The "canonical" continuous bijection works. We start by observing that $\mathbb{P}$ is homeomorphic to $\mathbb{N}^\omega$. We pick some bijection $\tau : \mathbb{N} \to \mathbb{Q}$, which is trivially continuous, and has a Baire class 1 inverse. We can then lift $\tau$ to obtain a continuous bijection $\tau^\omega : \... | 4 | https://mathoverflow.net/users/15002 | 368841 | 154,383 |
https://mathoverflow.net/questions/368801 | 6 | This question was posted to MSE but didn't get any answers, so I am posting it here. [Original post](https://math.stackexchange.com/questions/3784248/applications-of-the-infinitesimal-lifting-property)
Hartshorne in his book gives the 'Infinitesimal Lifting Property' as an exercise in chapter 2, section 8 and mention... | https://mathoverflow.net/users/152391 | Applications of the Infinitesimal Lifting Property | This is known as the formal criterion for "formal smoothness." In [this stacks project entry](https://stacks.math.columbia.edu/tag/02GZ) they prove that a morphism of schemes (in your case $\text{Spec }A \to \text{Spec }k$) is smooth if and only if it's formally smooth and locally finite presentation.
Aside from phil... | 7 | https://mathoverflow.net/users/86614 | 368852 | 154,387 |
https://mathoverflow.net/questions/321724 | 6 | Is it possible to show (the trivial statement)
$\sum \_{n\leq x}1=x+\mathcal O\left (1\right )$
using Perron's formula?
For $c$ a little bigger than $1$ and $1>c'>0$, a quantitative form of Perron's formula and then the Residue Theorem implies (with some parameter $T>0$)
$\begin {eqnarray\*}
\sum \_{n\leq x}1&=... | https://mathoverflow.net/users/110603 | Perron's formula | One could try to not even bother with the cut-offs and just integrate over the complete vertical line:
$$\sum\_{n \leq x}' 1 = \frac 1{2\pi i} \int\_{(c)} \zeta(2s) x^{2s} \frac{ds}s$$ for some $\frac 12 < c < \frac 34$.
Next, we shift the contour to some negative number, $-(c-\frac 12)$ say, and apply the function... | 8 | https://mathoverflow.net/users/157181 | 368868 | 154,391 |
https://mathoverflow.net/questions/368592 | 6 | $\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{D}$ be the inclusion of a full subcategory. Consider a functor
$$F:\mathcal{C}^{op}\rightarrow \Ab.$$
I've often seen e... | https://mathoverflow.net/users/152554 | (Pro-)representable functors and full subcategories in homotopy theory | This is a partial answer. Broadly speaking, representability theorems break down into two types. In both cases, the functor $F$ has to satisfy some exactness condition. For Freyd type theorems, $F$ must satisfy some set-theoretic condition such as accessibility or a solution set condition. For Brown type theorems, the ... | 2 | https://mathoverflow.net/users/11540 | 368881 | 154,395 |
https://mathoverflow.net/questions/368776 | 12 | Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$.
Now, one can define *integral* cohomology classes as those cohomology classes $a$ so that $\int\_S a \in \mathbb{Z}$. On the othe... | https://mathoverflow.net/users/143492 | Different definitions for integral de Rham cohomology classes | $\def\RR{\mathbb{R}}\def\ZZ{\mathbb{Z}}$I've considered assigning this when I've taught sheaf cohomology but it always seemed a little too hard. Let's see if I can do it. I'll be a little more general while I am at it and do the case of a smooth compact oriented $n$-fold. Choose a triangulation $S$ of the $n$-fold; let... | 12 | https://mathoverflow.net/users/297 | 368882 | 154,396 |
https://mathoverflow.net/questions/368880 | 4 | This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations:
**(1)** $\Omega\_{0} := \{-1,1\}$ and $\mathcal{F}\_{0} := 2^{\Omega\_{0}}$ are, respectivelly, the single particle configuration space and its associated $\sigma$-algebra.
**(2)** If $\Lamb... | https://mathoverflow.net/users/152094 | What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure? | One way to construct the thermodynamic limit of the states $\mu\_{\Lambda,\beta,h}^\varnothing$ is to observe that, for any local function $f$ and any increasing sequence of sets $\Lambda\_n\uparrow\mathbb{Z}^d$, the support of $f$ will be included inside $\Lambda\_n$ for all large enough $n$. In particular, for any lo... | 4 | https://mathoverflow.net/users/5709 | 368887 | 154,398 |
https://mathoverflow.net/questions/368877 | 2 | Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
>
> Is there an open neighborhood $U$ of $x$ such that $f(y)\ne 0$, for every $f\in B$ and $y\in U$?
>
>
>
| https://mathoverflow.net/users/53155 | Equicontinuity-like property of a convex compact set | Since $C(X)$ is not complete one cannot take the closed convex hull of the example in the comment. But what about this:
Let $g\_n=1-f\_n$ with $f\_n$ as in my comment.
Since the $g\_n$ are bounded by one, the linear map $T:\ell^1\to C(X)$, $\lambda\mapsto \sum\limits\_{n=1}^\infty \lambda\_ng\_n$ is well defined. We ... | 1 | https://mathoverflow.net/users/21051 | 368890 | 154,399 |
https://mathoverflow.net/questions/368896 | 0 | We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$
$$r^2=|4pq|$$
holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable conditions on $p,q,r$.
Are there other non trivial sum of squares form with higher powers being cancelled off and leaving ... | https://mathoverflow.net/users/136553 | On a sum of squares representation | Notice that for a form to be expressible as a sum of squares it must be nonnegative everywhere. In particular setting $a=0$ or $b=0$ tells you $p,q\geq 0$ and setting $a=\pm \sqrt{q}, b=\sqrt{p}$ tells you $|r|\le 2\sqrt{pq}$. Therefore the only regimes of $p,q,r$ where your form is a sum of squares are the ones you al... | 3 | https://mathoverflow.net/users/2384 | 368899 | 154,400 |
https://mathoverflow.net/questions/368840 | 10 | I am wondering if there is some example of a famous or well-known mathematician who often had trouble with peer review, or who often had to publish in obscure journals because referees didn't 'get' what they were saying (there could be any reasons for these troubles).
Now, you might say this is paradoxical (a well-kn... | https://mathoverflow.net/users/119114 | Example of a mathematician who had problems with peer review system? | There does not seem to be a direct mathematical analogue of Alfven. Nobody who has won a Fields Medal or an Abel Prize has made well-publicized complaints about how they have had an unduly difficult time with the peer-review system.
Some partial analogues have been mentioned in the comments. Louis de Branges had trou... | 22 | https://mathoverflow.net/users/3106 | 368903 | 154,402 |
https://mathoverflow.net/questions/368901 | 3 | I am looking for infinite set of Diophantine solutions.
1. Suppose we require
$$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$
$$a,b,c,d\in\mathbb Z$$
then can we still find solutions to
$$ab-cd=1?$$
2. Is it possible to do this if only
$$a,b,c,d\in\mathbb Z$$
$$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b... | https://mathoverflow.net/users/136553 | Close integer solutions to $ab-cd=1$? | It is possible with any constant $\lambda>1$ on the place of $\sqrt{2}$. Take $a=t^2$, $d=t^2+t-1$, $b=t^2+2t+1$, $c=t^2+t+1$ for large $t$.
| 6 | https://mathoverflow.net/users/4312 | 368911 | 154,404 |
https://mathoverflow.net/questions/368910 | 6 | Consider the $p$-adic exponential defined over $\mathbb C\_p$. One knows $\exp$ is analytic in the domain $\mathcal D=\{z\in\mathbb C\_P\mid v\_p(z)>\frac1{p-1}\}$. Does it exist an element $z\_0\in\mathcal D$ such that $\exp(z\_0)=0$?
Thanks in advance.
| https://mathoverflow.net/users/33128 | Zero of the exponential p-adic | The exponential function satisfies $\exp \left({x + y}\right) = \exp\left(x \right) \exp\left(y \right)$ for $x, y$ in the convergence domain. It also satisfies $\exp \left( 0 \right) = 1$. So if $\exp \left( z\_0 \right) = 0$, then $0 = \exp \left( z\_0 \right) \cdot \exp\left(-z\_0 \right) = \exp \left( z\_0 + (-z\_0... | 12 | https://mathoverflow.net/users/103908 | 368912 | 154,405 |
https://mathoverflow.net/questions/368913 | 4 | I've been reading "Structured Brown representability via concordance" by D.Pavlov (<https://dmitripavlov.org/concordance.pdf>) and I'm strugeling with a point and was wondering if someone could help me with my confusion. In the text there is a criterion which says that if a simplicial presheaf
$$F:Man^{op}\rightarrow \... | https://mathoverflow.net/users/152554 | Homotopy descent and cohomology |
>
> if a simplicial presheaf F:Man^op→sSet satisfies homotopy descent, where Man is the category of smooth manifolds, then there exists a K such that F≅[−,K].
>
>
>
Here one must also mention that F is required to be concordance-invariant (alias **R**-local), i.e., the map F(X)→F(**R**⨯X) must be a weak equivale... | 6 | https://mathoverflow.net/users/402 | 368917 | 154,407 |
https://mathoverflow.net/questions/368768 | 3 | If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $V\neq\varnothing\neq E$ and $|e| > 1$ for all $e\in E$, and $\kappa\neq\varnothing$ is a cardinal, we say that a map $c:V\to\kappa$ is a *coloring* if the restriction $c\restriction\_e: e\to \kappa$ is non-constant for each $e\in E$. We... | https://mathoverflow.net/users/8628 | Does every maximal almost disjoint family have the same chromatic number? | A negative answer to the question follows by the proof of Theorem 1.1 in the following paper of Erdős and Shelah, where for every $n<\omega$ they construct a mad family that is $(n+1)$-colorable but not $n$-colorable:
*Erdős, Paul; Shelah, Saharon*, [**Separability properties of almost-disjoint families of sets**](ht... | 4 | https://mathoverflow.net/users/37613 | 368928 | 154,411 |
https://mathoverflow.net/questions/368904 | 5 | The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary", by A. Fraser and M. Li:
>
> Let $M^n$ be a compact $n$-dimensional Riemannian manifold with nonempty boundary $\pa... | https://mathoverflow.net/users/85934 | Manifolds with boundary admitting no closed embedded minimal hypersurface | A solid torus should work. Choose cylindrical coordinates $(r,\theta, \lambda)$, $0\leq r \leq r\_0 < \pi/2, 0\leq \theta \leq 2\pi, 0\leq \lambda \leq l$, where we equate $(r,\theta,0)\sim (r,\theta, l)$ and $(0,\theta, \lambda)\sim (0,0,\lambda)$. Put a Riemannian metric on this solid torus of the form $dr^2+ f(r)^2 ... | 7 | https://mathoverflow.net/users/1345 | 368937 | 154,416 |
https://mathoverflow.net/questions/368938 | 3 | We say a matrix $(a\_{ij})$ is 0-1 matrix if $a\_{ij}\in \{0,1\}$ for all $i,j$. We say a matrix $(a\_{ij})$ is *monochromatic* if for some $a$, $a\_{ij} = a$ for all $i,j$.
>
> Question: Let $c\geq 1/2$ be a constant and $n$ be very large. Given a $n\times n$ 0-1 matrix $M$, must there be a $c\log\_2 n\times c\log... | https://mathoverflow.net/users/74918 | The size of monochromatic submatrix | Even $c=1-\varepsilon$ works when $n$ is large enough. This question is about bounding the diagonal bipartite Ramsey, see the recent achievement of David Conlon here
[http://people.maths.ox.ac.uk/~conlond/Bipartite.pdf](http://people.maths.ox.ac.uk/%7Econlond/Bipartite.pdf)
| 2 | https://mathoverflow.net/users/4312 | 368941 | 154,417 |
https://mathoverflow.net/questions/368870 | 3 | As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L\_{-2}^{k\_1} \cdots L\_{-n}^{k\_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we define
$$C\_2(V) = \{\psi^i\_{-h\_i-1}v | v \in V\}$$
where $i$ labels all the operators strongly generating the VO... | https://mathoverflow.net/users/50447 | Zhu's $V/C_2(V)$ algebra | You have underdefined the $C\_2(V)$ subspace.
It is as you wrote but with $\psi^i$ being *any* element of $V$. You can intuitively think of this as being the subspace of iterated normally ordered products of vertex operators for which at least one is a derivative.
Thus in your Virasoro example, $L\_{-10}\Omega\in C... | 3 | https://mathoverflow.net/users/17647 | 368946 | 154,422 |
https://mathoverflow.net/questions/368963 | 59 | Recently I saw an MO post [Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?](https://mathoverflow.net/q/368129/161328) that got me interested. It is about a graph parameter that is derived from the Laplacian of a graph. Its or... | https://mathoverflow.net/users/161328 | Intuitively, what does a graph Laplacian represent? | **How to understand the Graph Laplacian (3-steps recipe for the impatients)**
1. read the answer [here](https://www.quora.com/Whats-the-intuition-behind-a-Laplacian-matrix-Im-not-so-much-interested-in-mathematical-details-or-technical-applications-Im-trying-to-grasp-what-a-laplacian-matrix-actually-represents-and-wha... | 40 | https://mathoverflow.net/users/15293 | 368970 | 154,430 |
https://mathoverflow.net/questions/368885 | 1 | Let $L$ be a number field and let $\zeta\_L(s)$ be its associated Dedekind zeta function. It is known that $\zeta\_L(s)$ has at most one zero in the region
$$1 - \frac1{4 \log d\_L} \leq \sigma \leq 1, \qquad |t| \leq \frac1{4\log d\_L},$$
(as usual, $s=\sigma + it$) where $d\_L$ is the discriminant of $L / \mathbb{Q}$... | https://mathoverflow.net/users/163613 | For which number fields we know the nonexistence of Stark zeros? | In [Real zeros of real odd Dirichlet $L$-functions](https://www.ams.org/journals/mcom/2004-73-245/S0025-5718-03-01537-0/S0025-5718-03-01537-0.pdf), Mark Watkins showed in 2003 that $L(s,\chi\_d)$ (as in the title) has no positive real zeros for $d<300,000,000$. (I think this is still state of the art.). This answers th... | 1 | https://mathoverflow.net/users/6756 | 368971 | 154,431 |
https://mathoverflow.net/questions/368932 | 3 | I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X\_i$,
$$\frac{X\_1+\cdots+X\_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\gamma,\delta)$$
in distribution sense, whenever $X\_i$ belongs to the domain of attraction of its corresponding limit.... | https://mathoverflow.net/users/115114 | Reference for multivariate generalised CLT | $\newcommand\al\alpha\newcommand\R{\mathbb R}$The domains of attraction to multidimensional stable distributions are characterized by Rvačeva's [Theorems 4.1 (p. 194, for $\al=2$) and 4.2 (p. 196, for $\al<2$)](https://books.google.com/books?id=5d__hmE9q34C&pg=PA183&lpg=PA183&dq=%22On%20domains%20of%20attraction%20of%2... | 3 | https://mathoverflow.net/users/36721 | 368974 | 154,432 |
https://mathoverflow.net/questions/368979 | 1 | [Here](https://mathoverflow.net/questions/326033/prove-that-a-sub-gaussian-random-vector-over-a-finite-set-s-subset-mathbb-rn) the original question was asked and answered. However I have a question to the solution. If I get it right they try to show $\frac 12 I\_n \leq \mathbf{E} YY^T \leq I\_n$ by proving
$$ \mathbf{... | https://mathoverflow.net/users/163533 | Proof of "Prove that a sub-gaussian and isotropic random vector over a finite set T implies that the set is exponentially large" | In the linked answer, the inequality sign $\le$ in $\frac12\,I\_n\le EYY^T\le I\_n$ is not meant in the sense of the entrywise comparison. Rather, it is meant in this sense: for any two symmetric matrices $A$ and $B$, we write $A\le B$ if $B-A$ is positive semidefinite.
In this case, we have $Y=X1\_{\|X\|\le4C\sqrt{n... | 1 | https://mathoverflow.net/users/36721 | 368985 | 154,438 |
https://mathoverflow.net/questions/368955 | 6 | Let $K\_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$. Let $V\_{p,q}$ and $V$ denote the Seifert matrices of $K\_{p,q}$ and $K$, respectively.
Is it possible to obtain a closed formula for the matrix $V\_{p,q}$ in terms of $V$?
| https://mathoverflow.net/users/nan | Seifert matrices of cable knots | It also appeared in the [article](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/seifert2.pdf) of H. Seifert as Theorem II:
>
> Seifert, H. (1950). On the homology invariants of knots. The Quarterly Journal of Mathematics, 1(1), 23-32.
>
>
>
| 4 | https://mathoverflow.net/users/131172 | 368991 | 154,439 |
https://mathoverflow.net/questions/368957 | 11 | My guess is that there exists a constant $C$ such that $A(X) \sim C (\log X)^2$.
| https://mathoverflow.net/users/163643 | Is there a known asymptotic for $A(X):= \sum_{1 \leq i,j \leq X} \frac{1}{\mathrm{lcm}(i,j)}$? | $$ \sum\_{1 \leq i,j \leq X} \frac{1}{\mathrm{lcm}(i,j)} = \sum\_{1 \leq i,j \leq X} \frac{\mathrm{gcd}(i,j)}{ij} $$
$$ = \sum\_{1 \leq i,j \leq X} \frac{\sum\_{d|i,j} \phi(d)}{ij}$$
$$ = \sum\_{d \leq X} \phi(d) \sum\_{1 \leq i,j \leq X: d|i,j} \frac{1}{ij}$$
$$ = \sum\_{d \leq X} \frac{\phi(d)}{d^2} \sum\_{1 \leq i',... | 28 | https://mathoverflow.net/users/766 | 368996 | 154,441 |
https://mathoverflow.net/questions/368873 | 4 | In a paper on Hadwiger's conjecture, [https://web.math.princeton.edu/~pds/papers/hadwiger/paper.pdf](https://web.math.princeton.edu/%7Epds/papers/hadwiger/paper.pdf), Seymour explains various results on excluding the complete graph as a minor.
In particular, there is a nice bound on the number of edges, due to Mader,... | https://mathoverflow.net/users/62562 | Are K_t-minor free graphs on small vertex sets understood? | There is no known straightforward answer, but pseudorandom graphs must come into the answer. See the paper by Myers and Thomason.
[In response to the comment below] Look at recent papers by Postle--Norine--Song, plus earlier work by Reed--Kawarabayashi, all on Hadwiger's Conjecture. You will see that the difficulty w... | 3 | https://mathoverflow.net/users/25980 | 369004 | 154,446 |
https://mathoverflow.net/questions/368796 | 4 | I was looking at a [paper by Takao Suyama on GT theory](https://arxiv.org/abs/1008.3950), and I couldn't figure out how he derived his formula (3.59):
$$\frac{1}{\pi}\int\_a^bdx\frac{1}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{|(x-a)(x-b)|}}\frac{\log(e^{-t\_1}x)}{2}=\frac{1}{2}\log\left[\frac{e^{-t\_1}}{2\sqrt{ab}+a+b}\left... | https://mathoverflow.net/users/133758 | Deriving integral in Gaiotto-Tommasiello theory | Okay, so I think I may have found the answer myself. So, really, the absolute value symbol is a trick. You can get rid of it by pulling out an $i$, and then you have
$$\mathcal{I}:=\frac{1}{2\pi i}\int\_a^b dx \frac{\log(x e^{-t\_1})}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{(x-a)(x-b)}}$$
What you have to do is take a dumbb... | 2 | https://mathoverflow.net/users/133758 | 369008 | 154,448 |
https://mathoverflow.net/questions/368948 | 2 | Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions from $X$ to $\mathbb{R}$) such that for every pair of points $x \neq y$ there is a function $f \in D$ satisfying $f(x) \... | https://mathoverflow.net/users/14870 | Are there minimal topological conditions on a space for it to have a countable separating set? | With the help of the comments by erz, I will prove the following fact:
>
> $(X,\tau)$ admits a countable separating function set if and only if there exists a weaker topology $\tau^\*\subset\tau$ such that $(X,\tau^\*)$ is Hausdorff regular (i.e. $T\_3$) and second countable.
>
>
>
### Comments
Let me first ... | 3 | https://mathoverflow.net/users/129074 | 369009 | 154,449 |
https://mathoverflow.net/questions/365399 | 1 | The Besicovitch class of $B^p$ almost-periodic functions is defined as the closure of the set of trigonometric polynomials (of the form $t \mapsto \sum\_{n=1}^N a\_n e^{i \lambda\_n t}$ with $\lambda\_1, \dots, \lambda\_n \in \mathbb R$) under the semi-norm $$||f||\_{B^p} := \left(\limsup\_{X \to +\infty} \frac{1}{X} \... | https://mathoverflow.net/users/133679 | On $B^1$ and $B^2$ almost-periodic functions | **The answer depends on what exactly you mean by the question.** The subtle thing about $B^p$ is that it represents not functions by classes, and the classes depend on $p$.
This is a problem since if you replace $F$ by another representative $F'$ in its class, you could have $F \in B^2$ but $F' \notin B^2$.
The cor... | 2 | https://mathoverflow.net/users/11552 | 369015 | 154,453 |
https://mathoverflow.net/questions/368889 | 5 | Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int\_0^T u(t)\phi'(t) = -\int\_0^T g(t)\phi(t) \qquad\forall \phi \in C\_c^\infty(0,T).$$
Suppose that $u$ also has a weak derivative $h \in L^1(0,T;Z)$ where $Y \subset Z$.
In Boyer and Fabrie... | https://mathoverflow.net/users/137958 | Why is density and separability needed for uniqueness of weak (time) derivatives? | $\def\bbR{\mathbb R}\def\inc{\subseteq}$The requirements on density or separability are superfluous because of the following
**Lemma.** *Let $J$ be a real open interval, and let $E$ be any real or complex Banach space. Let the function $f$ in $L^1(J,E)$ be such that $\int\_J(\varphi\,f)=0\_E$ holds for all compactly ... | 4 | https://mathoverflow.net/users/12643 | 369025 | 154,456 |
https://mathoverflow.net/questions/368129 | 25 | 30 years ago, Yves Colin de Verdière introduced the algebraic graph invariant $\mu(G)$ for any undirected graph $G$, see [1]. It was motivated by the study of the second eigenvalue of certain Schrödinger operators [2,3]. It is defined in purely algebraic terms as the *maximum corank* in a set of generalized Laplacian m... | https://mathoverflow.net/users/156936 | Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? | Embeddability in any surface but the sphere (or plane) can probably not be characterized via the Colin de Verdière number.
Suppose that $K\_n$ is the largest complete graph that embedds into a surface $S$.
This shows that the best we can hope for is "$G$ embedds in $S$ $\Leftrightarrow$ $\mu(G)\le\mu(K\_n)= n-1$".
... | 6 | https://mathoverflow.net/users/108884 | 369032 | 154,459 |
https://mathoverflow.net/questions/369005 | 9 | Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL\_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \mathbb C$ or in finite characteristic, on the etale cohomology group $H^1(X(N),\mathbb Z\_\ell)\otimes\_{\mathbb Z\_\ell}\ov... | https://mathoverflow.net/users/58001 | The cohomology of modular curves as a module over the Galois group | Jared Weinstein's PhD thesis (<http://math.bu.edu/people/jsweinst/jswthesis.pdf>) is an excellent reference for this kind of thing. See section 3.4 in particular, where he computes the space $S\_k(\Gamma(N), \mathbb{C})$ as a $\mathbb{C}[\mathrm{SL}\_2(\mathbb{Z}/N)]$-module using an equivariant version of the Riemann-... | 6 | https://mathoverflow.net/users/2481 | 369045 | 154,463 |
https://mathoverflow.net/questions/369035 | 6 | I find this is the best site to post this question, even though I considered [cs](https://cs.stackexchange.com/).
It is a Monte Carlo experiment over the set of 10.000 n×n matrices.
If a single matrix eigenvalue is complex then python numpy package will return all the eigenvalues as `numpy.complex128` type, else it w... | https://mathoverflow.net/users/161386 | Probability of complex eigenvalues | The probability that a $n\times n$ real matrix (with elements that are independent random variables with standard normal distributions) has only real eigenvalues is given by
$$ 2^{-n(n-1)/4}$$
Reference: A. Edelman, The Probability that a Random Real Gaussian Matrix has $k$ Real Eigenvalues, Related Distributions, an... | 10 | https://mathoverflow.net/users/78061 | 369057 | 154,466 |
https://mathoverflow.net/questions/368998 | 7 | A combinatorial game I am studying has given rise to the following question. Consider the group $\Bbb Z/n\Bbb Z$. What is the largest $m$ such that there exist $k$ sets of $m$ residues such that the intersection of a translation of each of these sets has at most 1 element? That is, if the sets are $A\_1, \ldots A\_k$, ... | https://mathoverflow.net/users/163672 | Sets of residues with only a single intersection under translation | It is necessary and sufficient that for any nonzero $d\in\Bbb Z/n\Bbb Z$, there exists $i\in\{1,2,\dots,k\}$ such that $d\notin (A\_i-A\_i)$. In other words,
$$\bigcap\_{i=1}^k (A\_i-A\_i) = \{0\}.$$
This holds even for sets of varying sizes.
---
Since $| A\_i-A\_i|\geq |A\_i| = m$, we get a necessary condition: ... | 1 | https://mathoverflow.net/users/7076 | 369061 | 154,467 |
https://mathoverflow.net/questions/368886 | 7 | Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of [Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple Lie Algebra in prime characteristic](https://arxiv.org/abs/math/0205144), we have the following sentence:
>
> The sh... | https://mathoverflow.net/users/153148 | Weyl algebra as an Azumaya algebra over its centre |
>
> But my question remains the same: is there a direct way to prove the
> claim of the paper in the case when is the affine -space (or even
> the affine line) over .
>
>
>
Yes, read the proof of Proposition 1 in "[The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture](https://arxiv.org/abs/math... | 3 | https://mathoverflow.net/users/2275 | 369063 | 154,468 |
https://mathoverflow.net/questions/368956 | 5 | In [this](https://arxiv.org/abs/1705.07442 "A type theory for synthetic ∞-categories - Emily Riehl, Michael Shulman") article, Emily Riehl and Michael Shulman describe a type theory in which one can do $\infty$-category theory synthetically. Their framework allows them to define simplices $\Delta^n$, and a *morphism* i... | https://mathoverflow.net/users/144100 | Defining (infinity,1)-categories in HoTT using only an interval type | The shape/tope type theory is indeed just a "convenience". When I [first suggested](https://golem.ph.utexas.edu/category/2012/06/directed_homotopy_type_theory.html) this approach to synthetic $(\infty,1)$-categories, I took the approach you describe with a simple axiomatic interval. But the coherence paths very quickly... | 7 | https://mathoverflow.net/users/49 | 369070 | 154,473 |
https://mathoverflow.net/questions/369083 | 3 | I am sorry if this is a basic question, but I don't think in MSE I will receive any answers.
Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral currents supported on the compact, connected, oriented embedded surfaces $\Sigma\_1$ and $\Sigma\_2$ with multiplicitie... | https://mathoverflow.net/users/85934 | Behaviour of mass for currents with disjoint supports | Yes, this is true, from the fact that the supports of these currents are disjoint. In fact, I think the same argument should work for two general currents, as long as their supports have disjoint neighborhoods.
Write $M(\alpha)$ and $M(\alpha + \beta)$ for the masses of $\alpha$ and $\alpha + \beta$, respectively.
... | 3 | https://mathoverflow.net/users/43158 | 369085 | 154,478 |
https://mathoverflow.net/questions/369001 | 5 | I am a CS person so please excuse the hand-waving.
Given a set of machine-represented proofs, each different (but not necessarily proving a different thing), what sort of information-theoretic statements could we make about these?
I would define the information "density" of a proof by simply losslessly compressing ... | https://mathoverflow.net/users/163669 | Information density of proofs? | First, beware: this can only be done meaningfully within a fixed "language of proof". If you try to compare across different systems, you can get wildly different results. There is a whole domain of [Proof Complexity](https://en.wikipedia.org/wiki/Proof_complexity) which has lots of results about "expressivity". Adding... | 3 | https://mathoverflow.net/users/3993 | 369087 | 154,479 |
https://mathoverflow.net/questions/369044 | 5 | 1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group?
2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metric. Can $G$ be embbeded isometrically in a locally connected Lie group with its left invariant metric?
**Remark**: We emp... | https://mathoverflow.net/users/36688 | Is every countable discrete group a subgroup of a non discrete Lie group? | Malcev proved that every finitely-generated matrix group $\Gamma$ (over any field) is [residually finite](https://en.wikipedia.org/wiki/Residually_finite_group), i.e. the intersection of all finite-index subgroups of $\Gamma$ is $\{1\}$. [Baumslag-Solitar groups](https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_g... | 6 | https://mathoverflow.net/users/39654 | 369091 | 154,482 |
https://mathoverflow.net/questions/369082 | 7 | In Lawvere and Rosebrugh's *Sets for Mathematics*, they write
>
> It is a theorem [MM92] that a topos is well-pointed if and only if it is Boolean, two-valued, and supports split.
>
>
>
[MM92] is a reference to Mac Lane and Moerdijk's *Sheaves in Geometry and Logic*. I have found the proof that a well-pointed ... | https://mathoverflow.net/users/92424 | Prove that a Boolean two-valued topos in which supports split is well-pointed | Given a Boolean, two-valued topos in which supports split, we want to prove it's well-pointed, meaning that, if $f,g:A\to B$ are distinct, then there is a point $p:1\to A$ such that $fp\neq gp$. Since $f$ and $g$ are distinct, their equalizer $e:E\to A$ is a proper subobject of $A$. (I'm ignoring the distinction betwee... | 6 | https://mathoverflow.net/users/6794 | 369093 | 154,483 |
https://mathoverflow.net/questions/369092 | 10 | Suppose $S$ is a subset of the primes with natural density $0 < \alpha < 1$ within the primes. If
$$D(X) := \{n \leq X \mid p \not \mid n \text{ for all } p \in S \}$$
(so $D(X)$ is numbers at most $X$ not divisible by any $p \in S$),
then is there a nice form for the asymptotic value of $D(X)$ as a function of $X$?
... | https://mathoverflow.net/users/145167 | Natural density of set of numbers not divisible by any prime in an infinite subset | The recent paper
*Matomäki, Kaisa; Shao, Xuancheng*, [**When the sieve works. II**](http://dx.doi.org/10.1515/crelle-2018-0034), [ZBL07207214](https://zbmath.org/?q=an:07207214).
gives a fairly satisfactory answer to this question in the setting of arbitrary $S$. General sieve theory gives the upper bound
$$ |D(X... | 12 | https://mathoverflow.net/users/766 | 369102 | 154,489 |
https://mathoverflow.net/questions/369100 | 0 |
>
> **Problem**:
> Can an $f$ function be created where:$$f\colon\mathbb Q\_{+}^{\*}\to \mathbb Q\_{+}^{\*}$$
> The function is defined on the set of fully positive rational numbers and is achieved:
> $\forall(x,y)\in \mathbb Q\_{+}^{\*}\times\mathbb Q\_{+}^{\*},f(xf(y))=\frac{f(f(x))}{y}$
>
>
>
This question is... | https://mathoverflow.net/users/163245 | can there be a function $f:\mathbb Q_{+}^{*}\longmapsto\mathbb Q_{+}^{*}$ such that $f(xf(y))=\frac{f(f(x))}{y}$? | There is no function $f\colon Q\to Q$ such that
$$f(xf(y))=\frac{f(f(x))}y \tag{1}$$
for all $x$ and $y$ (in $Q$), where $Q:=\mathbb Q\_{+}^{\*}$.
Indeed, for $x=1$ equality (1) is
$$f(f(y))=\frac{f(b)}y,$$
where $b:=f(1)$. Replacing here $y$ by $x$, from (1) we get
$$f(xf(y))=\frac{f(b)}{xy}.$$
This with $y=1$ yield... | 7 | https://mathoverflow.net/users/36721 | 369107 | 154,491 |
https://mathoverflow.net/questions/369012 | 0 | I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. **My question is:** What is the LDP of
$$\lambda^{-1} \int\_0^\lambda f(B\_s) ds, \quad\lambda \to \infty \
$$
Here, $B\_s, s \in... | https://mathoverflow.net/users/125415 | Large deviation for Brownian occupation time | This is precisely the Donsker-Varadhan LDP, coupled with an application of the contraction principle. Namely, the rate function is
$$I(x)=\inf\{ J(\mu): \int f d\mu =x\}$$
where $J$ is the Donsker-Varadhan rate function. Look at the series of Donsker-Varadhan papers from 1975 (#I is the one you need) and any text on la... | 2 | https://mathoverflow.net/users/35520 | 369108 | 154,492 |
https://mathoverflow.net/questions/369088 | 3 | In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z\_{p^{2}}}\rtimes \mathbb{Z\_{p^{}}}) \rtimes\_{\phi}\mathbb{Z\_{p^{}}} $, have presentation
$$<a,b,c : a^{p^{2}}=b^p=c^p=e, ab=ba^{1+p},ac=cab,bc=cb>$$
I was trying... | https://mathoverflow.net/users/160231 | Permutation representation of a finite $p$-group | Using the relations, you can represent every element uniquely as $c^k b^l a^m$ with $0\leq k < p$, $0\leq l < p$, $0\leq m< p^2$. Now you can work out how left multiplication with $a, b, c$ acts on the set of such representatives by again using the relations, this is easy to do explicitly. The corresponding action on t... | 1 | https://mathoverflow.net/users/39747 | 369115 | 154,493 |
https://mathoverflow.net/questions/369120 | 8 | Consider a number field $K/\mathbb{Q}$ and the embedding of $K^\* \hookrightarrow GL\_n(\mathbb{Q})$. This is the set of rational points of a $\mathbb{Q}$-algebraic group $G \subseteq GL\_n(\mathbb{C})$. Then is it true that any $\mathbb{Q}$-characters of $G$ will look like $g \mapsto \det(g)^k$ for some $k \in \mathbb... | https://mathoverflow.net/users/94546 | Rational characters of a number field are powers of norm | If $H$ denotes the multiplicative group defined over $K$, then $G=\mathrm{Res}\_{K/\mathbb{Q}}H$. By Section 2.61 of Milne's "Algebraic Groups - The Theory of Group Schemes of Finite Type over a Field", the group $G\_{\overline{\mathbb{Q}}}$ obtained from $G$ by extension of scalars is isomorphic to the product of $H\_... | 8 | https://mathoverflow.net/users/11919 | 369146 | 154,501 |
https://mathoverflow.net/questions/369143 | 11 | Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively.
1. Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ and $\widetilde A$?
2. Also, do $A$ and $\widetilde A$ have the same set of eigenvectors?
Thank you.
| https://mathoverflow.net/users/33047 | Eigenvalues of the complement of a graph | **Edit** (bis). There are two answers, depending on whether loops about vertices are allowed or not. In addition, the case of regular graphs is completely described.
1. If loops are allowed
The relation between matrices is
$$A+{\widetilde A}=J$$
where $J={\bf1}{\bf1}^T$ is the all-ones matrix. The first consequence... | 15 | https://mathoverflow.net/users/8799 | 369147 | 154,502 |
https://mathoverflow.net/questions/369131 | 5 | Let $X^n$ be a compact Kähler manifold with $K\_X$ semi-ample, i.e., a sufficiently high power of $K\_X$ is basepoint free. The associated pluricanonical system $| K\_X^{\ell} |$ furnishes a birational map $$f : X \dashrightarrow \mathbb{P}^{\dim H^0(X, K\_X^{\ell})-1}$$ onto some normal projective variety $Y \subset \... | https://mathoverflow.net/users/105103 | Fibrations in complex geometry | I would say that the answer is in general *no*.
Think of an elliptic surface $X$ with Kodaira dimension $1$ and whose elliptic fibration contains a cuspidal curve. Then the general fibre is not homotopically equivalent to the special one (the former is homeomorphic to $S^1 \times S^1$, the latter to $S^1$, in particu... | 6 | https://mathoverflow.net/users/7460 | 369148 | 154,503 |
https://mathoverflow.net/questions/369135 | 3 | I have a question about the proof of Lemma 78.12.1 from [Stacks Project](https://stacks.math.columbia.edu/tag/076L). The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an isomorphism. By [Lemma 7.11.2](https://stacks.math.columbia.edu/tag/00WN) our job i... | https://mathoverflow.net/users/108274 | Algebraic spaces in the étale topology (proof from Stacks project) | Recall from the beginning of the [proof of the lemma](https://stacks.math.columbia.edu/tag/076L) that $R$ is defined to be $U \times\_F U$, so the surjections $U \to F \to U/R$ induce a canonical étale sheaf map $U \times\_{U/R} U \to U \times\_F U$ that was shown to be an isomorphism.
Let's see what this means for a... | 3 | https://mathoverflow.net/users/121 | 369150 | 154,504 |
https://mathoverflow.net/questions/369144 | 6 | Can you prove or disprove the following claim:
>
> Let $N=4p+1$ where $p$ is an odd prime number , let $T\_n(x)$ be the nth Chebyshev polynomial of the first kind and let $F\_n(x)$ denote an irreducible factor of degree $\varphi(n)$ of $T\_n(x)$ . If there exists an integer $a$ such that $F\_{p}(a) \equiv 0 \pmod{N... | https://mathoverflow.net/users/88804 | Primality test for $N=4p+1$ | The claim is true, and it holds more generally for every odd integer $p\geq 3$; the assumption that $p$ is prime is not needed. By the known [factorization of Chebyshev polynomials](http://icm.mcs.kent.edu/reports/1998/ICM-199802-0001.pdf),
$$F\_p(x/2)=\prod\_{\substack{1\leq m\leq 2p-1\\(m,4p)=1}}(x-\zeta^m-\zeta^{-m}... | 8 | https://mathoverflow.net/users/11919 | 369151 | 154,505 |
https://mathoverflow.net/questions/369142 | 4 |
>
> Let $C$ be an abelian category. Suppose that $(N\_i)\_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}\_C^1(M,N\_i)=0\quad\forall i\in I\tag{1}$$ imply $$\operatorname{Ext}\_C^1\left(M,\lim\limits\_\longleftarrow N\_i\right)=0?\tag{2}$$
>
>
>... | https://mathoverflow.net/users/163737 | When does $\operatorname{Ext}_C^1(M,N_i)=0$ imply $\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0$? | The standard result in this direction is the dual Eklof lemma (for your first problem) or the Eklof lemma (for your dual problem). Any version of the Eklof lemma presumes that your direct/inverse system is indexed by a well-ordered set. For an inverse system, it should be a smooth chain of epimorphisms with the kernels... | 3 | https://mathoverflow.net/users/2106 | 369160 | 154,507 |
https://mathoverflow.net/questions/365227 | 3 | The Kochen-Stone theorem says that if $A\_n$ is sequence of events with $\sum\_{i=1}^{\infty} P(A\_i) = \infty$, then:
$$
P(A\_n \mbox{ i.o.}) \ge \limsup\_{n \to \infty} \frac{(\sum\_{i=1}^nP(A\_i))^2}{\sum\_{i, j= 1}^nP(A\_i \cap A\_j)}
$$
I am interested in cases where the $A\_n$ are not mutually independent, but ... | https://mathoverflow.net/users/8187 | When is the Kochen-Stone inequality an equality? | In case it's of interest to others, I can now show that if $A\_n$ is any increasing sequence of events then the Kochen-Stone inequality is an equality. This was enough for my application (I am looking at issues of computability and wanted to arrange for equality when the probabilities $P(A\_n)$ form a Specker sequence)... | 0 | https://mathoverflow.net/users/8187 | 369166 | 154,509 |
https://mathoverflow.net/questions/369154 | 7 | Suppose that $F$ is a subfield of a field $E$ and, for
$n\times n$ matrices $A\_1,\dots,A\_m, B\_1,\dots,B\_m$
over $F$, there exists a matrix $T\in{\rm GL}\_n(E)$
such that $T^{-1}A\_iT=B\_i$ for all $i$.
>
> Does this imply that such a matrix $T$ can be chosen from ${\rm GL}\_n(F)$?
>
>
>
It is easy to se... | https://mathoverflow.net/users/24165 | Is simultaneous similarity of matrices independent from the base field? | This question is [answered in comments](https://mathoverflow.net/questions/369154/is-simultaneous-similarity-of-matrices-independent-from-the-base-field?__=1674402206#comment931814_369154):
>
> "As everyone is saying, this follows from Noether-Deuring.
> See [mathoverflow.net/questions/28469/hilbert-90-for-algebras... | 5 | https://mathoverflow.net/users/24165 | 369176 | 154,511 |
https://mathoverflow.net/questions/369175 | 3 | I have to show that a random vector $X$ who ist uniformly distributed on the Ball with Radius $\sqrt{n}$ is sub-gaussian with
$$\lVert X \rVert\_{\psi\_2}\leq C$$
I already know that the same result does hold for a random vector on the sphere with radius $\sqrt{n}$ (1).
I tried to show that $r Y$ is uniformly distribut... | https://mathoverflow.net/users/163533 | Uniform distribution in Ball with radius $\sqrt{n}$ is sub-gaussian | Let $X$ be uniformly distributed on the ball $B\_{\sqrt n}$ of radius $\sqrt n$ in $\mathbb R^n$. Then
$$X=RY,$$
where $R:=|X|/\sqrt n$ and $Y:=\sqrt n\,X/|X|$ is uniformly distributed on the sphere $S\_{\sqrt n}$ of radius $\sqrt n$ in $\mathbb R^n$.
Note that $0\le R\le1$ and hence $E\exp\{c(X\cdot t)^2\}=E\exp\{cR... | 4 | https://mathoverflow.net/users/36721 | 369178 | 154,512 |
https://mathoverflow.net/questions/369133 | 6 | A Banach space $X$ is said to be Grothendieck if the weak and the weak\* convergence of sequences in $X^{\*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck if and only if every weak\*-Cauchy sequence in $X^{\*}$ is weakly Cauchy?
Question 2. If $(x^{\*}\_{n})\_{n}$ is ... | https://mathoverflow.net/users/41619 | A question on Grothendieck space | I find the following criterion useful: A sequence $(x\_n)$ is Cauchy iff for all subsequences $(x\_{n\_{k+1}}-x\_{n\_k})$ tends to $0$. This works for the norm topology, the weak topology and the weak$^\*$ topology. This answers Q1 in the positive.
As for Q2, if $(x\_n^\*)$ is weakly Cauchy and weak$^\*$ null, it has... | 5 | https://mathoverflow.net/users/127871 | 369182 | 154,513 |
https://mathoverflow.net/questions/369084 | 5 | I'm finding myself a little confused about Koszul-Malgrange holomorphic structures in a certain context.
Suppose $M$ is a complex manifold, $N$ is a smooth manifold with a smooth complex vector bundle $V$ and bundle connection $\nabla$, and $f:M\to N$ is a smooth map. We form the pullback bundle $f^\*V$ over $M$ and ... | https://mathoverflow.net/users/158773 | Koszul-Malgrange Holomorphic structure on a pullback bundle | Okay I thought about it a bit more, and here is what I can say in general. We can find a neighbourhood $U\subset N$ such that $V|\_U$ splits diffeomorphically as $U\times \mathbb{C}^n$ and the section $s$ takes the form $s(z) = (f(z), A(z)\circ h(z))$, where $A(z)$ is a smoothly varying family of invertible $n\times n$... | 1 | https://mathoverflow.net/users/158773 | 369184 | 154,514 |
https://mathoverflow.net/questions/369185 | 15 | I apologize for a question that is not about mathematics, but I believe it is of interest to research mathematicians, and I believe there may be people on MathOverflow who can answer it objectively. If it is deemed unacceptable, I can survive.
For many years, I (and many others I know) have used a ranking of mathemat... | https://mathoverflow.net/users/163772 | Australian Mathematical Society journal rankings | There are several sources online that [rank math journals by impact factor](http://myweb.ecu.edu/katsoulise/ranking.html) (and, this comes from a professor's webpage) or [journal citation reports](https://clarivate.com/webofsciencegroup/solutions/journal-citation-reports/?subsector=18127). However, it is important to r... | 11 | https://mathoverflow.net/users/11540 | 369189 | 154,517 |
https://mathoverflow.net/questions/369202 | 2 | We say that a simple, undirected graph $G=(V,E)$ is *$1$-factorizable* if there is a partition of $E$ such that every member of the partition is a [perfect matching](https://en.wikipedia.org/wiki/Perfect_matching) of $G$. It is easy to see that any $1$-factorizable graph is regular (every vertex has the same degree).
... | https://mathoverflow.net/users/8628 | Are countable graphs with infinite minimal degree $1$-factorizable? | Yes, any $\aleph\_0$-regular graph $G=(V,E)$ is $1$-factorizable. (By "graph" I mean "simple graph" as in the question. Actually a loopless multigraph is OK provided no two vertices are joined by an infinite number of edges.)
Let $C$ be a set of colors, $|C|=\aleph\_0$. We will color the edges of $G$ with colors from... | 5 | https://mathoverflow.net/users/43266 | 369210 | 154,526 |
https://mathoverflow.net/questions/368975 | 17 | The Graph Minor Theorem of Robertson and Seymour asserts
that any minor-closed graph property is determined by a finite set
of forbidden graph minors. It is a broad generalization e.g. of the Kuratowski-Wagner theorem, which characterizes planarity in terms of two forbidden minors: the complete graph $K\_5$ and the com... | https://mathoverflow.net/users/156936 | Graph embeddings in the projective plane: for the 35 forbidden minors, do we know their Colin de Verdière numbers? | Here's a table containing the Colin de Verdière numbers:
```
Name Graph6 μ Reason
K33 + K33 4 (components linklessly embeddable)
K5 + K33 4 (components linklessly embeddable)
K5 + K5 4 (components linklessly embeddable)
K33 . K33 4 (ape... | 8 | https://mathoverflow.net/users/125498 | 369221 | 154,530 |
https://mathoverflow.net/questions/368892 | 5 | I am reading Hitchin's Self-Duality [paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.557.2243&rep=rep1&type=pdf). In section 5 (page 85), he is trying to prove that $Dim H^1=12(g-1)$. In doing so, he defines an operator $d^\*\_2+d\_1$, where $d^\*\_2$ and $d\_1$ are given by
$d\_1\dot{\psi}=(d\_{A}\do... | https://mathoverflow.net/users/155740 | A question on moduli space of Hitchin's equations | The operator
$$ d\_2^\*+d\_1\colon \Omega^0(M,ad P\otimes\mathbb C)\oplus\Omega^0(M,ad P\otimes \mathbb C)\to\Omega^{0,1}(M,ad P\otimes \mathbb C)\oplus\Omega^{1,0}(M,ad P\otimes \mathbb C)$$
is just the sum of $d\_1$ and $d\_2^\*,$ and it suffices to describe these two operators using the identification $$\Omega^{0,1}... | 2 | https://mathoverflow.net/users/4572 | 369232 | 154,532 |
https://mathoverflow.net/questions/369227 | 3 | This question is related to my [previous question](https://mathoverflow.net/q/369144/88804).
Can you prove or disprove the following claim:
>
> Let $N=2n+1$ where $n$ is an odd natural number greater than one , let $L\_m(x)$ be the mth Lucas polynomial and let $F\_m(x)$ denote an irreducible factor of degree $\va... | https://mathoverflow.net/users/88804 | Primality test for specific class of natural numbers using factors of Lucas polynomials | This claim can be proved in essentially the same way as the [previous one](https://mathoverflow.net/q/369144/88804). We have
$$F\_n(x)=\prod\_{\substack{|m|<n/2\\(m,n)=1}}(x+\zeta^m-\zeta^{-m}),$$
where $\zeta\in\mathbb{C}$ is a primitive $2n$-th root of unity. The splitting field of $F\_n(x)$ is the $n$-th cyclotomic ... | 2 | https://mathoverflow.net/users/11919 | 369237 | 154,534 |
https://mathoverflow.net/questions/369073 | 8 | Take a compact homogeneous space $G/K$, and a left $G$-invariant differential $k$-form $\omega \in \Omega^k(G/K)$. Will $\omega$ necessarily be closed? Might it even be harmonic when $G/K$ is endowed with a Riemannian metric?
| https://mathoverflow.net/users/160055 | Are invariant forms on homogeneous spaces necessarily closed? | Note that the answer depends on the pair $(G,K)$.
For example, if $K=\{e\}$, then one is asking whether the ring of left-invariant forms on $G$ consists only of closed forms. This only happens when $G$ is abelian.
On the other hand, if $M=G/K$ is a compact Riemannian symmetric space and $G$ is the identity componen... | 8 | https://mathoverflow.net/users/13972 | 369243 | 154,536 |
https://mathoverflow.net/questions/369214 | 14 | I asked this in [this MSE question](https://math.stackexchange.com/questions/3789119/deciding-if-mathbbz-ltimes-a-mathbbz5-and-mathbbz-ltimes-b-mathbb) but I didn't get answers. I think maybe here someone can help me.
I have the two following groups
$G\_A=\mathbb{Z}\ltimes\_A \mathbb{Z}^5$, where $A=\begin{pmatrix}... | https://mathoverflow.net/users/150901 | Deciding if $\mathbb{Z}\ltimes_A \mathbb{Z}^5$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^5$ are isomorphic or not |
>
> **Claim.** The groups $G\_A$ and $G\_B$ are **not** isomorphic.
>
>
>
We will use the following lemmas.
>
> **Lemma 1.** Let $A \in \text{GL}\_n(\mathbb{Z})$ and let $G\_A \Doteq \mathbb{Z} \ltimes\_A \mathbb{Z}^n$. Then the following hold:
>
>
>
>
> * The center $Z(G\_A)$ of $G\_A$ is generated b... | 20 | https://mathoverflow.net/users/84349 | 369248 | 154,539 |
https://mathoverflow.net/questions/369255 | 5 | Let $G$ be a semi simple algebraic group, $B \subset G$ is a Borel subgroup and $U \subset B$ is the unipotent radical of $B$. We can consider the variety $G/U$. Let us also denote $\overline{G/U}:=\operatorname{Spec}(\mathbb{C}[G/U])$. It is known that the natural morphism $G/U \rightarrow \overline{G/U}$ is an open e... | https://mathoverflow.net/users/98560 | Ideal of the boundary of $G/U \subset \overline{G/U}$ | Here is one way to see it, via classifying $G$-invariant radical ideals. (This has the bonus that it implicitly describes the boundary.)
**Lemma:** $G$-invariant ideals $I$ of $\mathbb{C}[G/U]$ are in bijection with sets of weights $S$ so that for $\lambda\in S$ and $\mu > \lambda$, $\mu\in S$. Such an ideal is radic... | 8 | https://mathoverflow.net/users/51424 | 369259 | 154,543 |
https://mathoverflow.net/questions/369226 | 6 | This is a question which I asked on StackExchange first, but might be more suited here.
I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the statement of said theorem in any other source, so I am completely at a loss how to proceed.
The result is called the E... | https://mathoverflow.net/users/131868 | Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5 | You have constructed, up to this point, a monomorphism $H\_n C / \bigcap F\_p H\_n C \to \lim H\_n(C/F\_p C)$, and this is compatible with the map from $H\_n C$. This allows you to construct the right-hand square, and then assemble all of, the following map of exact sequences:
$$
\require{AMScd}
\begin{CD}
0
@>>>
\bigc... | 5 | https://mathoverflow.net/users/360 | 369267 | 154,547 |
https://mathoverflow.net/questions/369257 | 8 | Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$.
$\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that:
1. $\frac{F(x)}{x}\in L^1(0,1)$,
2. $\frac{F'(x)}{x}\in L^1(0,1)$,
3. $\frac{F(x)}{x^2}\notin L^1(0,1)$?
I'm guessing that the answer ... | https://mathoverflow.net/users/157356 | Example of a function with a curious property | There is no such function. First of all, $|F(a)-F(b)|\leqslant \int\_a^b |F'(x)|dx\to 0$ when $a,b\to 0$. So $F$ has a limit $c$ at point 0. If $c\ne 0$, then 1) fails. So $\lim\_{x\to 0} F(x)=0$.
Next, $$|F(a)|\leqslant \int\_{0}^a|F'(x)|dx\leqslant a\int\_{0}^a\frac{|F'(x)|}x dx=o(a),\quad\text{when}\quad a\to 0.\q... | 11 | https://mathoverflow.net/users/4312 | 369277 | 154,552 |
https://mathoverflow.net/questions/336231 | 1 | I apologise if this is off topic.
Consider the quantity
$$
F(m,n,k)=\frac{(m)\_k}{k!n^{k-1} }
$$
where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation
$$
\sum\_{k=1}^{K} k F(n,n,k) \approx en,
$$
where $K$ is the largest integer $k$ for which $k!\leq n,$
is excellent with the quality o... | https://mathoverflow.net/users/17773 | A (surprising?) expression for $e$ | Just typing in the answer from the comments so the question does not stay "unanswered".
The expression on the left can be rewritten
$$
\sum\_{k} k \binom{n}{k} n^{-k}=\left(1+\frac{1}{n}\right)^{n-1},
$$ and the contribution to the sum comes from the very few first terms, those with $k!\leq n.$
| 1 | https://mathoverflow.net/users/17773 | 369286 | 154,557 |
https://mathoverflow.net/questions/369239 | 3 | This question is a duplicate of
[that 2010 MO question](https://mathoverflow.net/questions/27129/involutions-in-gl-nz/27145#27145).
I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C\_2$ of order $2$.
Clearly, any integral representation of $C\_2$ is ... | https://mathoverflow.net/users/4149 | Indecomposable integral representations of a group of order 2 "by hand" | In [Computing with real tori](https://www.math.ubc.ca/%7Ecass/research/pdf/realtori.pdf), Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C\_2$, explic... | 2 | https://mathoverflow.net/users/2383 | 369306 | 154,560 |
https://mathoverflow.net/questions/369272 | 1 | It is well-known that compact Hausdorff topological *unital* rings are profinite. The proof generalises to (left or right) s-unital rings (i.e. rings such that for all $r\in R$ we have $r\in Rr$ or for all $r\in R$ we have $r\in rR$).
Is there a reference for this more general fact? Is there a further generalisation ... | https://mathoverflow.net/users/54415 | S-unital compact rings are profinite | This is essentially answered in one of the answers to [Is every compact topological ring a profinite ring?](https://mathoverflow.net/questions/48718/is-every-compact-topological-ring-a-profinite-ring).
If a compact ring $R$ either admits no element $r\neq 0$ with $rR=0$ or the left-right dual condition then it is pro... | 1 | https://mathoverflow.net/users/15934 | 369309 | 154,561 |
https://mathoverflow.net/questions/369253 | 8 | As I said before, I'm not a QFT expert but I'm trying to understand the basics of its rigorous formulation.
Let's take [Dimock's book](https://www.amazon.com.br/Quantum-Mechanics-Field-Theory-Mathematical/dp/1107005094), where the foundation of QM and QFT is discussed. If we consider, say, two particles, one living i... | https://mathoverflow.net/users/150264 | Creation and annihilation operators in QFT | The connection can be seen by taking $H = L^2(\mathbb{R}^3)$ in the first explanation. This is the Hilbert space of a nonrelativistic, spinless, three-dimensional particle. By direct summing the symmetric (antisymmetric) tensor powers of $H$ we get the Hilbert space of an ensemble of noninteracting Bosonic (Fermionic) ... | 7 | https://mathoverflow.net/users/23141 | 369320 | 154,566 |
https://mathoverflow.net/questions/369258 | 6 | Let $G$ be a lattice in $SL(2,\mathbb{R})$. Is it always true that there exists a finite index subgroup $F$ of $G$ such that the quotient surface $\mathbb{H}/F$ has positive genus? Is the statement true under some general enough set of assumptions? Please can you add a reference?
| https://mathoverflow.net/users/163814 | Positive genus Fuchsian groups | Yes, this is true, but proving this is easier than finding a reference.
1. Every finitely-generated matrix group (e.g. a lattice in $PSL(2, {\mathbb R})$ contains a torsion-free subgroup. The general result is due to Selberg, but for *discrete* subgroups of $PSL(2, {\mathbb R})$ it was surely known earlier.
2. In vie... | 3 | https://mathoverflow.net/users/39654 | 369323 | 154,568 |
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