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https://mathoverflow.net/questions/440311 | 10 | I know that the topic is classical and even "folklore", but many treatments make use of local coordinates and such treatments are rather messy. Could somewhere maybe provide some reference(s) to a modern treatment, using the language of modern differential geometry (with complex line bundles, connections and so on), of... | https://mathoverflow.net/users/81645 | Modern treatment of Dirac monopoles and related topics | It is from 1993 but I find it definitely worth mentioning for this question.
*Brylinski, Jean-Luc*, [**Loop spaces, characteristic classes and geometric quantization**](https://dx.doi.org/10.1007/978-0-8176-4731-5), Modern Birkhäuser Classics. Basel: Birkhäuser. xvi, 300 p. (2008).
Chapter VII (pages 257-277) is ca... | 2 | https://mathoverflow.net/users/41291 | 442758 | 178,640 |
https://mathoverflow.net/questions/442720 | 2 | Landau in the first equation of [Über die Gitterpunkte in einem Kreise](https://doi.org/10.1007/BF01203524) uses the following formula for the Bessel function of the first kind:
$$\frac{1}{2\pi i } \int\_{1-\infty}^{1+i\infty} \frac{\mathrm e^{As-B/s}}{s^4} \, \mathrm d s= (A/B)^{3/2} J\_3(2 \sqrt{AB}))
$$ valid for al... | https://mathoverflow.net/users/9232 | infinite integral defining Bessel functions | First, change the variable in the integral $s=kz$, where $k=\sqrt{B/A}$.
You obtain
$$\left(A/B\right)^{3/2}
\int \frac{\exp{\sqrt{AB}\left(z-1/z\right)}}{z^4}dz.$$
Then deform the contour of integration from the vertical line
to $|z|=1$, this is possible since the real part of
$(z-1/z)$ is bounded in a simply connecte... | 2 | https://mathoverflow.net/users/25510 | 442769 | 178,644 |
https://mathoverflow.net/questions/442469 | 7 | Given a covering space $p \colon X \to Y$, we get an injection $p^\* \colon \pi\_1(X) \to \pi\_1(Y)$, and we know that the image $p^\*(\pi\_1(X))$ is normal in $\pi\_1(Y)$ if an only if $p$ is *regular*, that is if the deck group of the covering $p$ acts transitively on preimages of points in $Y$. This is a satisfyingl... | https://mathoverflow.net/users/14257 | When do covering spaces correspond to characteristic subgroups? | Suppose $X$ is a $K(\pi,1)$. Then by definition every endomorphism, hence every automorphism $f\_\star\colon \pi\_1(X,x) \to \pi\_1(X,x)$ may be realized by a continuous map $f\colon (X,x) \to (X,x)$.
If $p\colon (\tilde X,\tilde x) \to (X,x)$ is a covering map such that $p\_\star(\pi\_1(\tilde X, \tilde x))$ is char... | 6 | https://mathoverflow.net/users/135175 | 442771 | 178,645 |
https://mathoverflow.net/questions/442762 | 3 | Given an $m\times n$ chess board, we place $p$ $2\times 1$ dominoes on the board so that they don't overlap. How many ways can we place them?
When each square of the board is covered by a domino this is the well-known tiling problem. Is there any research on the case where $p$ is small such that we have to leave some... | https://mathoverflow.net/users/115114 | An "incomplete" tiling? | If we sum over all $p$, then we obtain the number of monomer-dimer tilings of an $m\times n$ chessboard. This number is known to be #P-complete and hence very likely computationally intractable. For a reference, see citation [7] of the paper by Yong Kung [here](https://arxiv.org/abs/cond-mat/0610690).
| 9 | https://mathoverflow.net/users/2807 | 442778 | 178,647 |
https://mathoverflow.net/questions/442788 | 1 | Suppose that $X,Y$ are independent random $d$-dimensional vectors each uniformly distributed on the unit sphere, and let $Z=Y\cdot\sqrt{1-\alpha}$ be a uniformly selected vector on a slightly smaller sphere.
I'm interesting in calculating (or lower-bounding) the probability that $X$ and $Z$ are far from each other, n... | https://mathoverflow.net/users/501044 | Let $\alpha\in(0,1),d\in\mathbb N^+$ and $X,Y\in\mathbb S^d$ be uniform, what is $\Pr[\lVert X-Y\cdot\sqrt{1-\alpha} \rVert^2\le \alpha]$? | $\newcommand\al\alpha$By spherical symmetry, the conditional distribution of $\|X-Y\,\sqrt{1-\al}\|$ given $Y$ does not depend on $Y$. So, letting $e\_1:=(1,0,\dots,0)$ and writing $X=(X\_1,\dots,X\_d)$, we have
$$\begin{aligned}
&P(\|X-Y\,\sqrt{1-\al}\|^2\le\al) \\ &=P(\|X-e\_1\,\sqrt{1-\al}\|^2\le\al) \\
&=P((X\_1-\... | 1 | https://mathoverflow.net/users/36721 | 442790 | 178,651 |
https://mathoverflow.net/questions/442776 | 1 | I have two questions.
Let $(X\_t)\_{t\geq 0}$ be a Lévy process with Lévy measure $\nu$. The jump process $\Delta X=\left(\Delta X\_t\right)\_{t\geq 0}$ is defined by
$\Delta X\_t=X\_t-X\_{t-}$, for every $t\geq0$, with $X\_{t-}$ left limit in $t$.
For every $0\leq t <\infty$ e $A \in \mathcal{B}(\mathcal{R}-\{0\... | https://mathoverflow.net/users/501039 | The Lévy process jumps | Your questions are covered by, say, Theorem 3 in Section 4 of [Lalley's notes](http://galton.uchicago.edu/%7Elalley/Courses/385/LevyProcesses10-23-2017.pdf). The proof (given there) is somewhat involved.
| 0 | https://mathoverflow.net/users/36721 | 442800 | 178,654 |
https://mathoverflow.net/questions/442791 | 2 | Let $A$ be a closed (densely defined) operator on a Hilbert space $H$.
We define for a natural number $k$, the operator $A^k$ with its natural domain.
Is $A^k$ closed?
| https://mathoverflow.net/users/216007 | If $A$ is a closed operator, is $A^k$ closed? | Here's a counterexample (subject perhaps to what you consider "natural").
Take a separable Hilbert space with orthonormal basis $\{u\_n : n = 1, 2, \ldots\}$ and the operator $A$ defined by
$$ A u\_n = \cases{ n u\_{n+1} & if $n$ is odd\cr 0 & if $n$ is even}$$
with domain $D = \{x = \sum\_n c\_n x\_n : \sum\_n |c\_n|^... | 7 | https://mathoverflow.net/users/13650 | 442808 | 178,656 |
https://mathoverflow.net/questions/442816 | 19 | I’m currently writing my first paper, a little paper, and I’d like to have some nice graphics in it. Much of the proof work is tedious analysis and I’d like to give any potential reader visual references to help.
To give an idea of what I’m looking for in particular, I’d like to have some pictures of the plane with s... | https://mathoverflow.net/users/138669 | Where can I create nice looking graphics for a paper? | Ipe is a fantastic tool. There's also a bit of a learning curve, but it's not too bad. I've seen people who are proficient at it generate amazing figures in a matter of minutes.
<https://ipe.otfried.org/>
Edit: I wish I had known about it a few years ago, it would've saved me many hours of TikZ coding.
| 19 | https://mathoverflow.net/users/103164 | 442822 | 178,661 |
https://mathoverflow.net/questions/442818 | 7 | $\DeclareMathOperator\SL{SL}$By the celebrated results of Culler and Shalen, a closed $3$-manifold contains an incompressible surface if its $\SL\_2(\mathbb{C})$ character variety is infinite.
Now, for manifolds with positive $b\_1,$ there is a much easier argument for existence of incompressible surfaces. In that re... | https://mathoverflow.net/users/62201 | Hyperbolic homology spheres with infinite $\mathrm{SL}_2(\mathbb{C})$ character variety | Take two knot complements (say of $K,K'$) and glue them together, interchanging meridians and longitudes. This is called splicing and produces a homology sphere $S(K,K')$; if both knots are non-trivial then the boundary torus where you glued is incompressible. It's standard that this produces a higher dimensional compo... | 8 | https://mathoverflow.net/users/3460 | 442823 | 178,662 |
https://mathoverflow.net/questions/442055 | 16 | $\DeclareMathOperator\Ind{Ind}$Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $\Ind(C^I) \to \Ind(C)^I$ is an equivalence of categories? where $\Ind$ denotes the category of ind-objects (so the free completion under filtered colimits) and the exponentials are for categorie... | https://mathoverflow.net/users/22131 | $\operatorname{Ind}(C^I) = \operatorname{Ind}(C)^I$? | So, the question appeared more subtle than I initially thoughts so I have written a short paper with more references and the details of what I'm going to say below. It is available [here](https://arxiv.org/abs/2307.06664).
Here is the summary:
First, Makkai's theorem cited by Ivan is indeed false. Building on Ben W... | 17 | https://mathoverflow.net/users/22131 | 442827 | 178,663 |
https://mathoverflow.net/questions/405276 | 2 | When the tensor algebra is presented, it is usually constructed as the direct sum of all tensor powers of the space. By this construction, the graded structure of the tensor algebra is easy to prove. However, I have not been able to find any proofs of the graded structure of the tensor algebra using just the universal ... | https://mathoverflow.net/users/394901 | Proving the graded structure of the tensor algebra from only the universal property | After all of this time I found a simple and direct solution. I can't believe I didn't think of this before:
Let $V$ be an $R$-Module and let $T(V)$ be its tensor algebra. Define $T^n(V)$ to be the span of all products of $n$ vectors in $T(V)$, and consider the (for now possibly distinct) space $\bigoplus\_{n = 0}^\in... | 0 | https://mathoverflow.net/users/394901 | 442828 | 178,664 |
https://mathoverflow.net/questions/442856 | 10 | $\DeclareMathOperator{\Hom}{Hom}$
Dear all,
The question is for teaching purposes and rather basic, so I hope that it also allows (relatively) easy answer.
By abstract homotopy theory we know that if the object $A$ in a model category $\mathcal{C}$ is cofibrant, then for any object $X$ the left homotopy relation on... | https://mathoverflow.net/users/123432 | Example of non-transitive homotopy relation | This is not possible in the Quillen model structure on topological spaces. (Here I am interpreting in the weak sense: "$f$ and $g$ are left homotopic if there exists *some* cylinder object $A \amalg A \to C \xrightarrow{\sim} A$ and a map $C \to Y$ restricting to $f$ and $g$". If you restrict to using one fixed cylinde... | 11 | https://mathoverflow.net/users/360 | 442868 | 178,673 |
https://mathoverflow.net/questions/442861 | 5 | Let $E$ be a Banach space. A set $C \subseteq E$ is called *ideally convex* if for every bounded sequence $(x\_n)$ in $C$ and for every sequence $(\lambda\_n)$ in $[0,1]$ that sums up to $1$ the vector $\sum\_n \lambda\_n x\_n$ is also in $C$.
(So a bounded set is ideally convex if and only if it is *$\sigma$-convex*... | https://mathoverflow.net/users/102946 | Large ideally convex sets | Just take your favorite decreasing sequence $\lambda\_k$ of positive numbers with sum $1$ and inductively construct the vectors $x\_k\in C$ such that $\left \|\sum\_{k=1}^n\lambda\_k x\_k-x\right\|\le\lambda\_{n+1}$ (then automatically $\|x\_n\|\le \frac{\lambda\_{n}+\lambda\_{n+1}}{\lambda\_n}\le 2$ for $n\ge 2$) to g... | 7 | https://mathoverflow.net/users/1131 | 442870 | 178,674 |
https://mathoverflow.net/questions/442772 | 47 | Is the following metric characterization of the real line true (and known)?
>
> A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real number $r$ there exist two points $a,b\in X$ such that $d(a,b)=2r$ and $\{a,b\}=\{x\in X:d(x,c)=r\}$.
>
>
>
... | https://mathoverflow.net/users/61536 | A metric characterization of the real line | Yes: in the new version of the question, with the word "complete" added, this is indeed a characterization of the real line.
In order to generate as much confusion as possible, but also for convenience, let's give the name "Banakh space" to any metric space satisfying the condition in your question with the word "com... | 27 | https://mathoverflow.net/users/70618 | 442872 | 178,675 |
https://mathoverflow.net/questions/442850 | 0 | Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$.
I am trying to determine the asymptotic behaviour of
$$F(a,b):=\int\_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\frac{dx}{|x-a|^{\alpha}|\sqrt{1-x^2}-b|^{\beta}}$$
as $a^2+b^2\to 1$ and $a\rightarrow 0$ simultaneously.
Notation: Given $A,B>0$, as cu... | https://mathoverflow.net/users/116555 | The asymptotic behaviour of a singular integral | $\newcommand{\al}{\alpha}\newcommand{\be}{\beta} $As discussed in comments, given
\begin{equation\*}
a/2<x<2a \tag{00}\label{00}
\end{equation\*}
and $a\to0$ (so that $b\to1$), the restriction $b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b$ will automatically hold (eventually). So,
\begin{equation\*}
F(a,b)=\int\_{a/2}^{2a}\frac{d... | 3 | https://mathoverflow.net/users/36721 | 442892 | 178,681 |
https://mathoverflow.net/questions/442886 | 2 | Let $X$ be a set and $\widehat{F}(X)$ the restricted free profinite group on $X$. To get $\widehat{F}(X)$ we define a profinite topology on $F$ (the free abstract group on $X$) and take $\widehat{F}(X)$ as the completion relative to this topology. If $X$ is a finite set, then $\widehat{F}(X)$ is the profinite completio... | https://mathoverflow.net/users/123172 | Is the free profinite group (or pro-$p$) torsion-free? | Both the free profinite and free pro-$p$ groups are torsion-free (also free pro-solvable groups or free pro-$\mathcal C$ groups for any collection $\mathcal C$ of groups closed under extensions, subgroups and homomorphic images). The reason is any closed subgroup of such a group is a projective profinite group; this ca... | 7 | https://mathoverflow.net/users/15934 | 442893 | 178,682 |
https://mathoverflow.net/questions/442902 | 2 | Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I need this for the final step for my proof that I've been working on for 4 years straight; I think that it might intuitiv... | https://mathoverflow.net/users/113020 | Question on density of certain set of matrices | It suffices to check whether $B^{-1}S=:S'$ has measure zero in $B^{-1}Q=:Q'$. We have
$$Q'={\bf Sym}\_n(\mathbb R),\qquad S'=\{{\bf Sym}\_n(\mathbb R)|B(\Sigma+\Sigma^3)\in{\bf Sym}\_n(\mathbb R)\}.$$
The subset $S'$ is algebraic in $Q'$. Either it has measure zero, or $S'=Q'$.
Let us consider the latter case: then... | 4 | https://mathoverflow.net/users/8799 | 442904 | 178,685 |
https://mathoverflow.net/questions/442890 | 0 | Let $X,Y$ be two independent random variables.
The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal.
Are there analogues of this theorem for non-normal, continuous distributions? For instance, what functions $f\_i(X,Y)$ whose mutual independences guarantee that $X,Y$ are ex... | https://mathoverflow.net/users/113397 | Analogues of Kac-Bernstein characterisation theorem for non-normal distributions | Such problems were systematically considered in the book *Characterization Problems in Mathematical Statistics* by Kagan, Linnik, and Rao; see e.g. [this review](https://projecteuclid.org/journals/annals-of-statistics/volume-5/issue-3/Review--A-M-Kagan-Yu-V-Linnik-C-Radhakrishna/10.1214/aos/1176343861.full). In particu... | 1 | https://mathoverflow.net/users/36721 | 442920 | 178,690 |
https://mathoverflow.net/questions/442927 | 1 | Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$.
For $\epsilon \to 0$ (say, $\epsilon= 1/n$ with $n \in \mathbb{N}$ so that $1/\epsilon$ is always an integer) we should have that, at th... | https://mathoverflow.net/users/109382 | Quantitative version of Lebesgue points theorem | $\newcommand\ep\epsilon\newcommand\de\delta$Let $\ep=1/m$ for a natural $m$, so that
$$N\_\ep:=\ep^{-n},$$
the number of the $\ep$-cubes, is an integer.
Take any $c\in(0,1]$ and let
$$N\_{A,\ep}(c):=\#\{i\in[N\_\ep]\colon f\_{A,\ep}(i)\ge c\},$$
the number of the $\ep$-cubes with relative $A$-content $\ge c$; as usua... | 1 | https://mathoverflow.net/users/36721 | 442931 | 178,693 |
https://mathoverflow.net/questions/442928 | 2 | Let $C$ be the square $[-1,1]^2$. Let $a\_1,\dots,a\_m$ be points chosen independently and uniformly at random from $C$. Let $d\_m$ (dispersion) be the random variable $\max\_{x \in C}{\min\_{j \in [m]}{\|x-a\_j\|\_2}}$. I would like to find a function $\epsilon(m)$ such that $\Pr[d\_m \leq \epsilon(m)] \geq $ constant... | https://mathoverflow.net/users/316923 | Dispersion of a "random" subset of $[-1,1]^2$ | For every constant dimension $d$, (I.e. when $C = [0,1]^d$), the answer is asymptotically $\varepsilon\_d(m) = \Theta\left(\frac{\log m}{m}\right)^{1/d}$. In the $\Theta$ notation I’m hiding constant factors that depend on $d$. This follows from the [coupon collectors problem](https://en.wikipedia.org/wiki/Coupon_colle... | 3 | https://mathoverflow.net/users/468679 | 442938 | 178,695 |
https://mathoverflow.net/questions/442921 | 0 | I am going through Terence Tao's "*Nonlinear Dispersive Equations (Local & Global Analysis)*" and trying to work through some of his exercises. However, I find myself being stumped by Exercise A.21. The question goes as follows:
>
> Let $0<\alpha<1$ and $1\le p\le\infty$. If $f\in S\_x(\mathbb{R}^d)$, we define the... | https://mathoverflow.net/users/501150 | Littlewood-Paley characterisation of Hölder regularity | You need to remember that $N$ is a **dyadic number**, so that $\sum\_{N \geq c} N^{-\alpha}$ converges for positive $\alpha$.
So you have
$$ \sum\_{N \geq |h|^{-1}} \frac{\|P\_Nf^h-P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}}{|h|^\alpha} = \sum\_{N \geq |h|^{-1}} N^\alpha\|P\_Nf^h-P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}\cdot \frac{1}{... | 3 | https://mathoverflow.net/users/3948 | 442939 | 178,696 |
https://mathoverflow.net/questions/442733 | 2 | Are there incompatible Turing degrees $a,b$ s.t any degree computable in $a$ either computes $a$ or is computed by $b$?
Obviously, if $a$ was above $b$ then $a$ would be a strong minimal cover of $b$. But do incomparable such degrees exist and is there a name for them?
Or am I missing some obvious fact that implies... | https://mathoverflow.net/users/23648 | Incompatible degrees $a,b$ s.t. $x < a$ implies $x \leq b$ | This type of algebraic property is observed in the local structure of the enumeration degrees: there we call them Ahmad pairs. Ahmad (a student of Lachlan) showed that there are incomparable $\Sigma^0\_2$ enumeration degrees $a$ and $b$ such that for all enumeration degrees $x$ if $x<a$ then $x<b$. She also showed that... | 6 | https://mathoverflow.net/users/501163 | 442941 | 178,698 |
https://mathoverflow.net/questions/442926 | 1 | Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that
\begin{equation}\tag{1}\label{1}
\int\_a^b \varphi' v o ~\mathrm{d}x \leq 0 \quad \forall \varphi \in C\_0^\infty([a, b], [0, \infty))=:U, \quad \frac{o(b)}{o(a)} \leq v... | https://mathoverflow.net/users/500621 | How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$ | Let's upgrade Iosif's [comment](https://mathoverflow.net/questions/442926/how-much-room-in-inequality-displaystyle-int-ab-varphi-ov-mathrmdx#comment1143194_442926) to an answer.
Let $\chi$ be a smooth bump function supported in $[-\epsilon,\epsilon]$. For any $\varphi\in U$ with support within $[a+\epsilon,a-\epsilon... | 3 | https://mathoverflow.net/users/3948 | 442942 | 178,699 |
https://mathoverflow.net/questions/442953 | 5 | I have been using the notion of "semisimple linear category" for a while now, but I never bothered to write down a definition. I've always just waved my hands and said "Schur's lemma holds and every object is the direct sum of simple objects". I find myself writing some notes on the subject, and it occurred to me that ... | https://mathoverflow.net/users/159298 | What is the correct definition of semisimple linear category? | If I were to try to define "semisimple linear category", and this is indeed something I do try to define, I would say: it is a category, linear over your base commutative ring, which has finite direct sums and splittings of idempotents, and such that for every object $X$, the ring $\mathrm{End}(X)$ is semisimple.
I w... | 8 | https://mathoverflow.net/users/78 | 442957 | 178,704 |
https://mathoverflow.net/questions/442981 | 3 | Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$ is a connected component of $A$.
ii) $0\in \overline{A\setminus\{0\}}$ and
iii) each connected component $C\ne... | https://mathoverflow.net/users/66777 | Closed subset of unit ball with peculiar connected components | No.
Note that for any integer $n>0$ there is a component $C\_n\subset A$ that contains two points $x\_n$ and $y\_n$ such that $|x\_n|=\tfrac1n$ and $|y\_n|=1$.
Recall that $C\_n$ is a closed set.
Pass to a subsequence of $C\_n$ that converges in the sense of Hausdorff; denote its limit by $C\_\infty$.
We may assume... | 8 | https://mathoverflow.net/users/1441 | 442983 | 178,711 |
https://mathoverflow.net/questions/442511 | 5 | Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see short discussion at [Galois Group as a Sheaf](https://mathoverflow.net/questions/30030/galois-group-as-a-sheaf)). For now... | https://mathoverflow.net/users/351164 | “Sheaf cohomology” of Galois groups | The usual geometric point of view on the Galois group of $F$ is that the Galois group is the etale fundamental group of $\operatorname{Spec} F$. It's possible that you're already aware of this point of view and are looking for something different, but let me say a few words about it anyway.
This analogy comes not fro... | 3 | https://mathoverflow.net/users/501202 | 442992 | 178,714 |
https://mathoverflow.net/questions/442851 | 1 | Let $f:[0, \infty)\to [0, \infty)$ be non-increasing (and not necessarily differentiable nor continuous) and satisfy $$f(t)\leq f(0)-C\int\_{0}^{t}f(s)^{1/2}ds,$$ where $C>0$. How can one show that then $$f(t)\leq g(t)\quad \text{for all}~t\leq t\_{\*},$$ where $t\_{\*}>0$ and $g$ is a differentiable function on the in... | https://mathoverflow.net/users/163368 | Integral inequality implies majorization by solution of ODE | The problem with this question, compared to [this one of yours](https://math.stackexchange.com/questions/4658708/integral-inequality-implies-majorization-by-solution-of-ode), is that the vector field on the right hand side of the ODE is not a non-decreasing function of $g$. If you try to make the example of @fedja rigo... | 2 | https://mathoverflow.net/users/272040 | 442995 | 178,715 |
https://mathoverflow.net/questions/442994 | 15 | In [his 1926 paper](https://ipparco.roma1.infn.it/pagine/deposito/2011/TF.pdf) Fermi states without further explanation that it follows from the Thomas-Fermi equation
$$\frac{d^2\psi(x)}{dx^2}=\frac{\psi(x)^{3/2}}{\sqrt{x}},\label{1} \tag{1}$$
and boundary conditions
$$\psi(0)=1,\quad\psi(\infty)=0\label{2}\tag{2},$$
t... | https://mathoverflow.net/users/32389 | How did Fermi calculate this integral? | The way this is taught in text books,$^\ast$ which is likely the way Fermi reasoned, is to compare two alternative integrations by parts. One the one hand,
$$\int\_0^\infty \frac{\psi^{5/2}(x)}{\sqrt{x}}\,dx=-5\int\_0^\infty \psi'(x)\psi^{3/2}(x)\sqrt{x}\,dx$$
$$\qquad\qquad=-\frac{5}{2}\int\_0^\infty x\frac{d}{dx}[\ps... | 19 | https://mathoverflow.net/users/11260 | 443000 | 178,716 |
https://mathoverflow.net/questions/442765 | 1 | Let $W$ be a standard one dimensional Brownian motion, $\mathcal F\_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$.
Given a $C[0, 1] $ valued random variable $F$, define the translation map $T\_F: \Omega \to \Omega$ by $T\_F (\omega) = \omega + F(\omega)$, and denote t... | https://mathoverflow.net/users/173490 | Converse Cameron-Martin theorem for shifts by adapted processes | If $F\_t(\omega)=\int\_0^t h\_s(\omega)\,ds$, with $h$ progressive and such that $\int\_0^1 h\_s^2(\omega)\,ds\le C$ for some finite constant $C$, then indeed $T^\*\_F\Bbb P$ is equivalent to $\Bbb P$.
Let $X\_t$ denote the coordinate process on $\Omega$, a Brownian motion under $\Bbb P$. Now define the martingale $M... | 2 | https://mathoverflow.net/users/42851 | 443006 | 178,719 |
https://mathoverflow.net/questions/442445 | 2 | Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book.
Suppose that $g: \Omega \to \mathbb{C}$ is Borel-measurable and that $\xi \in \mathscr{D}\left(\int\_\Omega g dE\right)=: \mathsc... | https://mathoverflow.net/users/216007 | Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$ | If $g=\sum\_k\beta\_k\,1\_{X\_k}$ is simple,
$$
\Big\langle\Big(\int\_\Omega g\,dE\Big)\xi,\xi\Big\rangle=\int\_\Omega g\,dE\_{\xi,\xi}=\sum\_k\beta\_k\,E\_{\xi,\xi}(X\_k)=\Big\langle\sum\_k\beta\_k\,E(X\_k)\xi,\xi\Big\rangle.
$$
Since $g$ is measurable, then there exist simple functions $g\_n$ such that $g\_n\to g$... | 1 | https://mathoverflow.net/users/3698 | 443020 | 178,720 |
https://mathoverflow.net/questions/443026 | 0 | I'd like to know the estimate of the following sum
$$\sum\_{n\leq x}\sum\_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$
where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. Do you know how to compute this?
| https://mathoverflow.net/users/159935 | Estimating a sum involving the von Mangoldt function | Following Joshua Stucky's remark, the sum can be rewritten as the following sum over prime numbers:
$$\sum\_{p\leq x}(\log p)\left(1-\frac{1}{p}\right)\sum\_{k=1}^\infty\left\lfloor\frac{x}{p^k}\right\rfloor.$$
Hence the sum is upper bounded by
$$\sum\_{p\leq x}(\log p)\left(1-\frac{1}{p}\right)\frac{x}{p-1}=x\sum\_{p\... | 5 | https://mathoverflow.net/users/11919 | 443029 | 178,722 |
https://mathoverflow.net/questions/442888 | 3 | I have non-negative $d\times d$ matrices $A$, $B$ and need a tractable way to compute the sum of all entries of $\exp(-t(A-B))$ where $A$ is diagonal and $B$ symmetric rank-$1$. IE
$$f(t)=\langle\exp(-t(A-B))\rangle$$
Where $t>0$ and $\langle M\rangle$ represents sum of all entries of $M$.
**What expansion would ... | https://mathoverflow.net/users/7655 | Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$? | This Asymptote code seems to work perfectly and for any $t$ in your range the estimate uses $Cd$ operations and is a guaranteed upper bound though I am not sure whether $C$ is small enough for you (I hope it is).
If you try to run it online on <http://asymptote.ualberta.ca/> , the final pause() command should be remo... | 5 | https://mathoverflow.net/users/1131 | 443030 | 178,723 |
https://mathoverflow.net/questions/443032 | 6 | I am looking for deformations of the 4-sphere with 8-dimensional isometry group, like a 4-dimensional Berger sphere.
| https://mathoverflow.net/users/142151 | Deformations of the 4-sphere with 8-dimensional isometry groups | There cannot be an 8-dimensional group $G$ acting effectively on $S^4$ by Riemannian isometries. The following argument may not be the best, but it explains why this is true. (I will assume that $G$ acts effectively, since, otherwise, we can quotient by the closed subgroup $K\subset G$ that acts trivially, and work wit... | 11 | https://mathoverflow.net/users/13972 | 443036 | 178,724 |
https://mathoverflow.net/questions/442859 | 3 | Consider a smooth map $f:M\rightarrow N$ between smooth manifolds.
Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that nearby fibers are homeomorphic. If we weaken the condition of properness, we can still say something of the sort near a c... | https://mathoverflow.net/users/147463 | When is compactness of fiber components an open condition? | Let $f:X\rightarrow Y$ be a continuous map between locally compact Hausdorff spaces. For $x\in X$ denote by $A\_x:=f^{-1}(f(x))$ the fiber over $f(x)$ and by $C\_x$ the connected component of $A\_x$ containing $x$.
Assume that $C\_{x}$ is compact for some given $x\in X$.
**Claim**: there is an open neighborhood $V\... | 3 | https://mathoverflow.net/users/66777 | 443037 | 178,725 |
https://mathoverflow.net/questions/443013 | 2 | Suppose $X \in \mathbb{R}^{n \times d}$ is a random matrix where $n > d$. Given a matrix $A \in \mathbb{R}^{n \times n}$ such that $AX$ is a zero matrix in expectation, i.e., $\mathbb{E}\_{X}[AX] = 0$. Let $\sigma^2$ be the variance of the norm of $AX$, i.e., $\sigma^2:=\mathbb{V}[\lVert AX \rVert^2\_F]$.
Now I would... | https://mathoverflow.net/users/478836 | expectation and variance of the norm of a random matrix | $\newcommand\si\sigma\newcommand\bm[1]{\begin{bmatrix}#1\end{bmatrix}}$No and no: In general, (i) $EB\ne0$ and (ii) we cannot bound $Var\,\|B\|\_F^2$ by $\si^2$.
E.g., let $n=3$, $d=1$,
$$y\_1:=\bm{1\\ -1\\ -1},\quad y\_2:=\bm{-1\\ 1\\ -1},\quad
y\_3:=\bm{-1\\ -1\\ 1},\quad y\_4:=\bm{1\\ 1\\ 1},
$$
$$A:=\bm{1&-1&0\\... | 1 | https://mathoverflow.net/users/36721 | 443045 | 178,727 |
https://mathoverflow.net/questions/443047 | 2 | Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.
Let $\Delta\_X$ be the closed subscheme of $X\times X$ that is the scheme-theoretic image of the diagonal morphism $X\to X\times X$, and $\Delta\_Y$ the scheme-theoretic image... | https://mathoverflow.net/users/497064 | Finite flat pullback of the diagonal | If $f$ is finite flat of degree $d$, then $f \times f \colon X \times X \to Y \times Y$ has degree $d^2$, but $\Delta\_f \colon \Delta\_X \to \Delta\_Y$ has degree $d$. So equality cannot hold scheme-theoretically unless $f$ is an isomorphism.
A fairly explicit case is the finite étale Galois case, where $(f \times f... | 5 | https://mathoverflow.net/users/82179 | 443049 | 178,728 |
https://mathoverflow.net/questions/443016 | 6 | I'm looking to compute normalizers of finite subgroups of $\mathrm{GL}(n, \mathbb{Z})$ and its possible that they are infinite but they are always finitely presented. For $\mathrm{GL}(n, \mathbb{Z})$ the solution is actually described here:
<https://www.ams.org/journals/mcom/1973-27-121/S0025-5718-1973-0333025-7/S002... | https://mathoverflow.net/users/173855 | Is there a general method for computing finitely generated normalizers? | Here’s a strategy that sometimes works but is likely hopeless in general. Suppose $G$ acts on a complex $K$ with the property that each finite subgroup of $G$ fixes a point. For example, if the complex $K$ happens to enjoy the CAT(0) property, this will be satisfied.
In this situation, suppose $H$ is a finite subgrou... | 3 | https://mathoverflow.net/users/135175 | 443053 | 178,730 |
https://mathoverflow.net/questions/443024 | 5 |
>
> Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
> $$\hat f(k):=\int\_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
> Do we have $\|f\|\_{L^\infty}\leq \frac CN \|f'\|\_{L^1}$ for some $C>0$ indpendent of $f$ and $N$?
>
>
>
We have $\|f\... | https://mathoverflow.net/users/141451 | If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$? | There is no chance. If you could do it, the linear functional $g\mapsto (\int g)(0)$ would have small norm on the corresponding subspace of $L^1$ and, thus, extend to a functional of small norm in the whole $L^1$. Thus, we would have a small in $L^\infty$ function $g$ whose Fourier coefficients are $\frac 1n$ for $|n|>... | 10 | https://mathoverflow.net/users/1131 | 443057 | 178,732 |
https://mathoverflow.net/questions/443060 | 2 | Let $A\subset B$ be an inclusion of $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domains. Is the integral closure of $A$ in $B$ also $\mathbb{Z}/n\mathbb{Z}$-graded?
This is true for the $G$-graded case with $G$ a nice torsion-free integral commutative monoid, see Theorem 2.3.2 of [Huneke-Swanson, Integral Closure of I... | https://mathoverflow.net/users/3848 | Is the integral closure of a $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domain in a bigger $\mathbb{Z}/n\mathbb{Z}$-graded domain also graded? | Example 2.3.3 in Huneke and Swanson answers this in the negative.
| 2 | https://mathoverflow.net/users/460592 | 443066 | 178,736 |
https://mathoverflow.net/questions/443052 | 10 | Let
$r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$
Then it is a known theorem of Landau that
$$
r(B) \sim C \frac{B}{\sqrt{\log B}}
$$
for some $C > 0$. I was wondering if there is a generalization of this result for
slightly more general function $ax^2 +... | https://mathoverflow.net/users/84272 | A generalisation of theorem of Landau on sum of two squares? | The general result you seek is due to Paul Bernays, who studied under E. Landau in Göttingen and proved in his dissertation (1912) that $$\tag{$\star$}\sum\_{n \le x} b\_Q(n) \sim C\_Q \frac{x}{\sqrt{\log x}}$$
as $x \to \infty$ where $Q(x,y)=ax^2+bxy+cy^2$ is a primitive positive-definite binary quadratic form ($b^2−4... | 25 | https://mathoverflow.net/users/31469 | 443080 | 178,740 |
https://mathoverflow.net/questions/443046 | 6 | $\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top}
\DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such that the singular simplicial complex functor $\Sing\_\bullet:\Top \to \sSets$ is fully faithful ?
>
> Is $\Hom\_{\To... | https://mathoverflow.net/users/494312 | Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces? | Here is a simple counterexample with $X = Y = \mathbb{R}$:
Send a simplex $\sigma : |\Delta^n| \to \mathbb{R}$ to the affine function $F(\sigma) : |\Delta^n| \to \mathbb{R}$ with the same values at the vertices of $|\Delta^n|$. This is the identity on 0-simplices, so it could only come from the identity map of $\mathbb... | 15 | https://mathoverflow.net/users/126667 | 443092 | 178,744 |
https://mathoverflow.net/questions/443093 | 7 | I do not know whether this question (in history of math) is proper for MathOverflow, but I know no other places where it can be asked with a hope to obtain an answer.
Reading the biography of Henry Maurice Sheffer (a famous logician whose name is attributed to the "Sheffer stroke", one of two binary operations that g... | https://mathoverflow.net/users/61536 | The place and year of birth of Henry Maurice Sheffer | The confusion on the year of birth can be traced back to official records. The 1918 draft registration card of Henry Maurice Sheffer states September 1, 1883 as date of birth. However, the 1904 naturalization record states September 1, 1882. In one more document, a passport application from 1910, Henry Sheffer "solemnl... | 11 | https://mathoverflow.net/users/11260 | 443095 | 178,745 |
https://mathoverflow.net/questions/442789 | 5 | Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V\_\mathbb{Q}$. Let $x \in V\_\mathbb{Q}$ be a non-zero rational vector. Consider the stabilizer group $G\_{x}$ of $x$. This is a closed subgroup in $G$.
Consider a compactly supported continuou... | https://mathoverflow.net/users/94546 | Integrating on orbits of algebraic groups | One condition that I came across is that it is sufficient to have $G(\mathbb{C}) \cdot x $ a closed subvariety of $V\_\mathbb{C}$. Then $G(\mathbb{C}) \cdot x$ is a closed affine variety. This guarantees in particular that $G\_x(\mathbb{C})$ must be reductive from Matsushima's criterion which says if $G$ is a connected... | 1 | https://mathoverflow.net/users/94546 | 443099 | 178,747 |
https://mathoverflow.net/questions/443097 | 6 | Let $X$ be a compact Hausdorff space and $p\in X$ be a non-isolated point. Is it always possible to find a net $(x\_\alpha)\_{\alpha\in (I,\leq)}$ in $X\setminus\{p\}$ converging to $p$ such that $(I,\leq)$ is *totally ordered* (or a regular cardinal, which gives you the same thing)?
I would expect this to be false, ... | https://mathoverflow.net/users/153400 | When can we find a net, defined on a totally ordered index set, converging to a non-isolated point in a compact Hausdorff space? | Yes. For every non-isolated $p\in X$, there is a well-ordered net in $X\setminus\{p\}$ converging to $p$ as long as $X$ is compact and Hausdorff. This result was observed in the 1992 paper Convergent free sequences in compact spaces by I. Juhász and Z. Szentmiklóssy, so let me paraphrase their argument. Suppose that $\... | 4 | https://mathoverflow.net/users/22277 | 443108 | 178,751 |
https://mathoverflow.net/questions/442370 | 3 | The following post builds on [this post](https://mathoverflow.net/posts/442315/edit); I'll begin by quoting the setting.
---
**Background from Previous Question:**
$\newcommand\SS{P}\newcommand\TT{Q}$Call a Gaussian probability measure $\SS$ on $\mathbb{R}^d$ *isotropic* if its covariance matrix is diagonal wit... | https://mathoverflow.net/users/491352 | Conditions for: (local) lipschitz stability of I-projection | The answer is still no -- for any, however small $R>0$.
Indeed, let $d=1$. Let $g$ be the pdf of $\nu$; that is, $g$ is the standard normal pdf. Let $Q\_h$ be the probability measure with pdf $(1-p)g+pq\_h$, where $q\_h$ is as in [the previous answer](https://mathoverflow.net/a/442343/36721) and $p\in(0,1)$ is a fixe... | 2 | https://mathoverflow.net/users/36721 | 443118 | 178,753 |
https://mathoverflow.net/questions/442998 | 16 | I am interested in the low-degree stable homotopy group $\pi\_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H\_1(X,\mathbb{Z}/2)\to \pi\_2^{s}(X)\to H\_2(X,\mathbb{Z})\to 0$.
**Question:** Does it always split?
---
**Edit:** I've b... | https://mathoverflow.net/users/472749 | The second stable homotopy group | I love this question! I've enjoyed thinking of it. Below, I show why the sequence splits always.
Let $X\to Y=K(H\_1(X,\mathbb{Z}/2),1)$ be the map inducing the identity in $H\_1(-,\mathbb{Z}/2)$. By naturality, we have have a commutative diagram
$$
\begin{array}{cccccccc}
0&\to& H\_1(X,\mathbb{Z}/2)&\to& \pi\_2^{st}(... | 11 | https://mathoverflow.net/users/12166 | 443120 | 178,754 |
https://mathoverflow.net/questions/443127 | 5 | Let $\mathcal{F}$ be a saturated coherent subsheaf of $T\mathbb{CP}^n$. In particular, $\mathcal{F} \subset T\mathbb{CP}^n$ is a holomorphic vector subbundle outside a subset $Z \subset \mathbb{CP}^n$ of complex codimension $2$. Assume that $c\_1(\mathcal{F}) = 0$.
**Question:** Does there exists a non-zero holomorph... | https://mathoverflow.net/users/195890 | Vector fields tangent to distributions with zero first Chern class | Not in general.
Let $F \in H^0(\mathbb P^3, \mathcal O\_{\mathbb P^3}(3))$ be a general cubic form and let $H \in H^0(\mathbb P^3, \mathcal O\_{\mathbb P^3}(1))$ be a linear form. The kernel of
$\omega = 3FdH - HdF \in H^0(\mathbb P^3, \Omega^1\_{\mathbb P^3}(4))$ is a saturated subsheaf $T\_{\mathcal F}$ of $T\_{\ma... | 4 | https://mathoverflow.net/users/605 | 443129 | 178,756 |
https://mathoverflow.net/questions/442777 | 0 | I am looking for existing constructions of [vertex-symmetric graphs](https://en.wikipedia.org/wiki/Vertex-transitive_graph) on $n$ nodes that have a girth at least $g$ and are dense, i.e., have at least $n^{1 + \epsilon}$ edges, where $\epsilon>0$ may depend on $g$: in particular, if $g = O(1)$, then $\epsilon$ must be... | https://mathoverflow.net/users/501040 | Dense vertex-symmetric graphs with high girth | If you want an explicit construction (that does not involve randomness), you can use classical constructions of Ramanujan graphs, such as the one from [this paper of Morgenstern](https://doi.org/10.1006/jctb.1994.1054).
Given an odd prime power $q$ and an even integer $d$, Theorem 4.13 in that paper gives you a Cayle... | 1 | https://mathoverflow.net/users/45855 | 443130 | 178,757 |
https://mathoverflow.net/questions/443117 | 21 | *(I asked this question on [Math.SE earlier](https://math.stackexchange.com/questions/4660421/why-do-we-need-canonical-well-orders) but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-theorist, but for an outsider it see... | https://mathoverflow.net/users/328267 | Why do we need "canonical" well orders? | This isn't about just any choice for a 'canonical' well-order, but the von Neumann ordinals in particular have nice properties that you don't get just from well-orders. They admit a logically simple definition which enables certain arguments about complexity of definitions to go through.
A relation $R$ being well-fou... | 25 | https://mathoverflow.net/users/64676 | 443135 | 178,759 |
https://mathoverflow.net/questions/443133 | 2 | For the proof of a certain combinatorial statement on subsets of ${\mathbb F}\_2^n$
in a paper I and several other people are working on, the following statement was helpful.
**Proposition.**
Let $V$ be a vector space over a field $F$, let $I$ be a finite set, and let
$U\_i \subseteq V$ for $i \in I$ be linear subspa... | https://mathoverflow.net/users/21146 | Linearly independent vectors from a family of subspaces | This is exactly Rado theorem on independent transversal (for the corresponding linear matroid).
Strictly speaking, to use it you should deal with finite sets, but a dimension of a vector subspace always equals to a rank of its appropriate finite subset, so there is no problem with it.
| 2 | https://mathoverflow.net/users/4312 | 443138 | 178,761 |
https://mathoverflow.net/questions/443144 | 9 | In a now [classical paper](https://zbmath.org/?q=an:0438.10030), Iwaniec proved the following theorem.
**Theorem.** Let $T \geq 2$, $T^{1/2} < T\_0 \leq T$ and $T \leq t\_1 < t\_2 < \cdots < t\_R \leq 2T$, $t\_{r+1} - t\_r \geq T\_0$. Then
$$
\tag{1}
\sum\_{r=1}^R \int\_{t\_r}^{t\_r+T\_0} \left|\zeta\left(\tfrac{1}{2... | https://mathoverflow.net/users/307675 | Large values of $\zeta(1/2+it)$ from sums of short moments | I wrestled with this question for an annoyingly long time. Now understanding things much better, I'm a bit embarrassed that this took me so long to figure out (such is the process of learning though). I was quite frustrated that I couldn't find the details of this anywhere, aside from some helpful comments in [this pap... | 10 | https://mathoverflow.net/users/307675 | 443145 | 178,763 |
https://mathoverflow.net/questions/443124 | 2 | I apologize in advance for how vague this request is.
A few weeks ago, I came upon a paper that (if I recall correctly) proves that **the hull of a cut-and-project tiling is a fiber bundle over a torus**. This is *not* the paper by Sadun cited below, but rather it specifically deals with cut-and-project tilings. If I... | https://mathoverflow.net/users/165348 | Reference request: Cut-and-project method gives rise to a fiber bundle over the torus | I have found the sought-after paper. It was *Duneau, M., and Christophe O.* "Displacive transformations and quasicrystalline symmetries." Journal de Physique 51.1 (1990): 5-19.
In section 3, they show that a cut and project pattern can be mapped onto a lattice by what they call (but never define) a "modulation".
| 3 | https://mathoverflow.net/users/165348 | 443147 | 178,765 |
https://mathoverflow.net/questions/442538 | 5 | I was asked to post a different question following a wording error on [my previous question](https://mathoverflow.net/questions/442533/jigsaw-puzzle-on-set-family), so here it is.
>
> Given a set family $\mathcal{F}$ of $[n]$ (with certain additional properties), such that every element of $[n]$ lies in exactly $K$... | https://mathoverflow.net/users/475875 | "JigSaw Puzzle" on Set Family II | This problem is NP-complete, a nice reference is this answer here by András Salamon:
<https://cstheory.stackexchange.com/a/356/419>
If you are interested in results about the complete family, see [Baranyai's theorem](https://en.wikipedia.org/wiki/Baranyai%27s_theorem).
| 2 | https://mathoverflow.net/users/955 | 443163 | 178,769 |
https://mathoverflow.net/questions/443078 | 2 | Suppose $Y \to X$ is a finite morphism of varieties over $\mathbb C$, with $Y$ and $X$ both smooth. Is $Y$ always birational to a smooth hypersurface $Y' \subset X \times \mathbb A^1$, such that the projection $Y' \to X$ induces $\mathbb C(X) \subset \mathbb C(Y)$?
| https://mathoverflow.net/users/5279 | Does every finite map of smooth varieties birationally embed as a smooth hypersurface of 1d trivial bundle over the base? | This is an answer to the question in the original post. The answer is positive in many cases, in particular whenever $X$ is affine. There is nothing to prove if the degree $n$ of the finite morphism equals $1$. Thus, assume that the degree $n$ is at least $2$.
By the Primitive Element Theorem, $Y$ is birational over ... | 4 | https://mathoverflow.net/users/13265 | 443181 | 178,774 |
https://mathoverflow.net/questions/443112 | 3 | Let $R$ be a regular local ring of dimension $n$. Let $i:\text{Flat}\to\text{Fin}$ be the fully faithful inclusion of the category of flat finitely generated type $R$-modules into all finitely generated modules.
* If $n=1$ then $M$ is flat iff $M$ is torsionfree, and the functor $M\mapsto M/(torsion)$ is left adjoint... | https://mathoverflow.net/users/17988 | Flatness over regular local rings of dimension 3 | Fix $R$ (not necessarily local) and $M$. Let us call a map $u:M\to \overline{M}$ a *free hull* if $\overline{M}$ is finite free and for every finite free $F$ the induced map $\mathrm{Hom}(\overline{M},F)\to \mathrm{Hom}({M},F)$ is bijective.
>
> **Claim**: $M$ has a free hull if and only if $M^\vee$ is free, and in... | 1 | https://mathoverflow.net/users/7666 | 443182 | 178,775 |
https://mathoverflow.net/questions/182440 | 13 | The aperiodic monotile problem asks whether there exists a single tile that every tiling of the plane made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? how could it (not) be constructed?
EDIT: I'm thinking about the most restrictive case where the single ... | https://mathoverflow.net/users/49443 | (non-)existence of the aperiodic monotile | This recent preprint claims to find such a tile.
[David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, “An aperiodic monotile”,](https://arxiv.org/abs/2303.10798) (2023-03-20) arXiv:2303.10798
>
> A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape... | 22 | https://mathoverflow.net/users/5340 | 443193 | 178,778 |
https://mathoverflow.net/questions/443169 | 28 | I would like to ask the following question:
>
> Is it possible to find two non-constant polynomials $p(x), q(x)$ with integer
> coefficients, such that $\gcd(p(n), q(m))=1$ for every $(n, m)\in
> \mathbb{N}^2$?
>
>
>
If such $p(x), q(x)$ exist, we will call them "completely" coprime, since all of their values... | https://mathoverflow.net/users/38851 | Two polynomials which are "completely" coprime | Gro-Tsen's solution is elegant. Here's a more elementary solution that doesn't directly use any algebraic number theory. I'll change the polynomials to $f(x)$ and $g(x)$. We'll assume that
$$\gcd(f(n),g(m))=1 \quad\hbox{for all $m,n\in\mathbb N$ } \qquad(\*)
$$
and derive a contradiction. We may assume, WLOG, that $f(x... | 21 | https://mathoverflow.net/users/11926 | 443203 | 178,781 |
https://mathoverflow.net/questions/442937 | 3 | Weighted $K\_5$ have the unique property that their edge set can be interpreted as the disjoint union of their shortest and their longest Hamilton cycle.
That makes $K\_5$ attractive for designing new TSP heuristics; especially extremal $K\_5$ in TSP instances seem interesting for generating starting tours (e.g. the ... | https://mathoverflow.net/users/31310 | Fastest algorithm for calculating optimal tours in weighted $K_5$ | OK, I guess I have exhausted my ideas, so I'll make a claim on 14 additions and 8 (actually 7 and a half in a certain sense) comparisons.
Let A,B,C,D,E be the vertices of the pentagon. Consider all cycles that contain the edge AB (there are 6 of them and the other 6 are their complements, so if I find *both* the shor... | 3 | https://mathoverflow.net/users/1131 | 443209 | 178,784 |
https://mathoverflow.net/questions/443191 | 2 | $\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let
* $(\Omega, \mathcal F, \mathbb P)$ be a probability space.
* $B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions.
* $X=(X\_0^1, \ldots, X\_0^N)$ independent real-valued random variables.
* $X$ independent of $B$
* $(P\_t, t\ge0)$ a Marko... | https://mathoverflow.net/users/99469 | Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions? | Let $\{\nu\_x\}\_{x \in \mathbb R^n}$ be the regular conditional probability measures on $\Omega$ associated with $X$, and $\mu\_X$ the law of $X$ on $\mathbb R^n$.
Denote by $E$ the event
$$\left \{ \nu\_X ( \bigcap\_i \, \{X^i \in A\_i\} ) =
\prod\_i \nu\_X (X^i \in A\_i) \, , \, \forall A\_i \in \mathcal B(C[0, T]... | 1 | https://mathoverflow.net/users/173490 | 443213 | 178,785 |
https://mathoverflow.net/questions/442626 | 7 | I am struggling to understand lemma 7.20 of the paper *Stack Semantics and the Comparison of Material and Structural Set Theories* by Mike Shulman ([arXiv:1004.3802](https://arxiv.org/abs/1004.3802)). It contains formal sequents of the form $$U \Vdash \ulcorner V\Vdash \phi\urcorner$$ and I do not understand how I can ... | https://mathoverflow.net/users/219922 | The idempotence of Mike Shulman's "Stack semantics" | The stack semantics as described in that paper doesn't involve any type theory, only the first-order language of categories. Writing $\ulcorner V \Vdash \phi \urcorner$ means to write out the definition of $\Vdash$, which is by induction over the structure of $\phi$. In your example, to say that
$$V \Vdash \forall Z.... | 4 | https://mathoverflow.net/users/49 | 443217 | 178,787 |
https://mathoverflow.net/questions/443224 | 2 | Let $C$ be an algebraic curve over $\mathbb{C}$ and $\omega\_C$ be its canonical bundle. We may assume that $C$ has genus $g\geq2$. Let $x\in C$ be an arbitrary point.
>
> **Question:** What is the image of $H^0(C,\omega\_C-x)$ in the Grassmannian $G(g-1,\, H^0(C,\omega\_C))$ as $x$ varies along $C$?
>
>
>
| https://mathoverflow.net/users/148748 | Image of $H^0(C,\omega_C-x)$ in $G(g-1,H^0(C,\omega_C))$ | First, note that projectification provides a natural isomorphism $$G(g-1, \, H^0(C, \omega\_C)) \simeq \mathbb{G}(g-2, \, \mathbb{P}H^0(C, \omega\_C))=\mathbb{P}H^0(C, \omega\_C)^\*.$$
Next, observe that the canonical map $$\varphi \colon C \to \mathbb{P}H^0(C, \omega\_C)^\*$$ associates to $x \in C$ the hyperplane $... | 2 | https://mathoverflow.net/users/7460 | 443226 | 178,788 |
https://mathoverflow.net/questions/443212 | 16 | $\DeclareMathOperator{\ap}{ap}$ $\DeclareMathOperator{\rad}{rad}$ The average power of an integer $m$ is given by
$$
\ap(m):=\log\_{\rad(m)}(m)=\frac{\log(m)}{\log(\rad(m))},
$$ where $\rad(m)=\prod\_{p|m}p$.
$\ap(m) \ge 2$ if and only if $\rad(m)^2 \le m$ , which is obviously satisfied by the set of powerful inte... | https://mathoverflow.net/users/112259 | Does the sum of reciprocal of integers with average power at least two converge? | $\DeclareMathOperator{\ap}{ap}$ $\DeclareMathOperator{\rad}{rad}$
I think so. Fix $\rad(m)=p\_1\ldots p\_k=:P$. Denote by $\Omega$ the set of positive integers with all prime divisors in $\{p\_1,\ldots,p\_k\}$. Then $m=PQ$ where $Q\in \Omega$ and $Q\geqslant P$.
For $s\in (0,1)$ we have
$$
\prod\_{p|m}(1-p^{s-1})^{-1... | 19 | https://mathoverflow.net/users/4312 | 443228 | 178,790 |
https://mathoverflow.net/questions/442971 | 3 | Does positive set theory [$\sf GPK^+\_\infty$](https://en.wikipedia.org/wiki/Positive_set_theory#Axioms) prove the existence of a set $K= \{x \mid x \text { is von Neumann ordinal } \lor x=K\}$
| https://mathoverflow.net/users/95347 | Does positive set theory prove the existence of a set of all ordinals and itself? | I came to know that this question had been answered to the positive by the founder of $\sf GPK^+\_\infty$ himself at an article of him titled: [Inconsistency of GPK + AFA](https://onlinelibrary.wiley.com/doi/10.1002/malq.19960420109). At Journal Math.Log.Quart 1996: vol. 42, issue 1, page 107
| 3 | https://mathoverflow.net/users/95347 | 443234 | 178,791 |
https://mathoverflow.net/questions/443031 | 14 | Question is as mentioned in the title:
Are there any introductory notes on deformation theory that are easier to read for differential geometers?
I am learning about differential graded Lie algebras (and $L\_\infty$-algebras). I am aware of Marco Manetti's notes [Deformation theory via differential graded Lie algeb... | https://mathoverflow.net/users/118688 | (An introduction to) deformation theory (written) for differential geometers | A classical subject in deformation theory with a differential-geometric flavour is that of deformation quantization. There are several good introductory references I can recommend, all of which introduce the Maurer–Cartan formalism for DG Lie algebras in deformation theory:
* *Cattaneo, Alberto S.*, Formality and sta... | 6 | https://mathoverflow.net/users/163420 | 443237 | 178,792 |
https://mathoverflow.net/questions/443231 | 7 | Let $G$ be a finitely presented simple group. By Kuznetsov (1958), $G$ has decidable word problem. However, by Scott [1], $G$ may have undecidable conjugacy problem. Is anything known about other decision problems for $G$, in generality? I am particularly interested in the question of the subgroup membership problem, i... | https://mathoverflow.net/users/120914 | Subgroup membership problem in simple groups | As another example, the problem of computing the order of an element of the finitely presented simple Brin–Thompson group $2V$ is undecidable by
*Belk, James; Bleak, Collin*, [**Some undecidability results for asynchronous transducers and the Brin-Thompson group (2V)**](https://doi.org/10.1090/tran/6963), Trans. Am. ... | 9 | https://mathoverflow.net/users/24447 | 443241 | 178,794 |
https://mathoverflow.net/questions/443235 | 2 | Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int\_{-\infty}^\infty e^{-tx}\,d\mu(x) = \int\_{-\infty}^\infty e^{-tx}\,d\nu(x) <\infty
$$
for all $t\in(a,b)$.
Does this imply that $\mu = \nu$?
| https://mathoverflow.net/users/485160 | Injectivity of two sided Laplace transform | I suppose that $\mu$ and $\nu$ are real measures and "exist"
means absolute convergence. Then both transforms can be extended to bounded analytic functions in the strip
$\{ t+i\sigma:a<t<b\}$ and coincide in this trip by uniqueness theorem for analytic functions. But then the restrictions on the line $t=t\_0$ for some ... | 2 | https://mathoverflow.net/users/25510 | 443243 | 178,795 |
https://mathoverflow.net/questions/443247 | -1 | In a German science journal I read that PH Schoute asked this in the Educational Times around 1900:
What is the sum of
$$S = \frac{1}{1}+\frac{1}{2+3}+\frac{1}{4+5+6}+\frac{1}{7+8+9+10} +\cdots\text{?}$$
I don't know if an answer has been found. Anyone?
| https://mathoverflow.net/users/501420 | What's this infinite sum? | You can have a very good approximation of the partial sums writing first
$$\frac2{n(n^2+1)}=\frac 2{n(n+i)(n-i)}=\frac{2}{n}-\frac{1}{n-i}-\frac{1}{n+i}.$$
Using generalized harmonic numbers
$$S\_p=\sum\_{n=1}^p\frac2{n(n^2+1)}=2 H\_p-H\_{p-i}-H\_{p+i}+H\_i+H\_{-i}.$$
Using asymptotics
$$S\_p=\left(H\_{-i}+H\_i\right)-... | 1 | https://mathoverflow.net/users/42185 | 443249 | 178,797 |
https://mathoverflow.net/questions/443245 | 7 | On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum\_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it amounts to the evaluation of the moments of a certain distribution". Does someone know what exactly Gould is referring to? I am ... | https://mathoverflow.net/users/159851 | A reference for a sum found in Gould's Combinatorial Identities book | $\renewcommand{\b}{\binom}\renewcommand{\B}{\text{B}}$As noted in [Carlo Beenakker's comment](https://mathoverflow.net/questions/443245/a-reference-for-a-sum-found-in-goulds-combinatorial-identities-book#comment1144176_443245),
\begin{equation\*}
s\_n:=\sum\_{k=0}^{n-1}(-1)^k\b nk\b{2n}{2k}^{-1}=\frac{2 n+1-(-1)^n}... | 9 | https://mathoverflow.net/users/36721 | 443255 | 178,799 |
https://mathoverflow.net/questions/443262 | 3 | I don't know how to solve part (a) of exercise 1.5.8 in Robin Hartshorne's book Deformation Theory 1.5.8 (page 42):
>
> Consider the Hilbert scheme of zero-dimensional closed subschemes
> of $\mathbb{P}^4\_k$ of length $8$, the field $k$ is algebraically closed. There is one component of dimension $32$
> that has a... | https://mathoverflow.net/users/501436 | Exercise 1.5.8 from Robin Hartshorne's Deformation Theory | As the comment also indicates, consider the Grassmannian $ Gr\_3(\operatorname{Sym}^2 \Omega\_{\mathbb{P}^4} ) $ where the vector bundle $ \operatorname{Sym}^2 \Omega\_{\mathbb{P}^4} $ has fiber $ m\_p^2/m\_p^3 $ over a point $ p \in \mathbb{P}^4 $. Then there is a morphism $ Gr\_3(\operatorname{Sym}^2 \Omega\_{\mathbb... | 2 | https://mathoverflow.net/users/152391 | 443270 | 178,803 |
https://mathoverflow.net/questions/443264 | 2 | As a term of a Serre spectral sequence, I would like to compute the cohomology group with compact support $H^\*\_c(\mathcal{M}\_{1,[2]},\mathbb{V}\_1)$ of the moduli space of genus $1$ curves with $2$ unordered marked points, with values in $\mathbb{V}\_1$, the standard representation of $SL\_2(\mathbb{Z})$, coming fro... | https://mathoverflow.net/users/477884 | Computation of $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$ | Your approach is good but you must have messed something up with the Hochschild-Serre spectral sequence.
I prefer the sheafy approach over thinking about the Hochschild-Serre spectral sequence, so let me explain it in these terms. So we have the forgetful map $\pi : M\_{1,2} \to M\_{1,1}$, and we may write
$$ H^\bull... | 7 | https://mathoverflow.net/users/1310 | 443274 | 178,804 |
https://mathoverflow.net/questions/443280 | 1 | [Can move to cs or tcs stackexchange if thats a better home]
I remember back around 2016 DIMACS used to host a list of problem instances of various famous problems in the NP-complete class and harder classes. Usually "problem instance data", "solution" and sorted by difficulty for testing new algorithms/heuristics ag... | https://mathoverflow.net/users/46536 | Where to find hard instances of subsetsum and other famous np-complete problems for testing heuristics against? | Here's the DIMACS Implementation Challenges page:
<http://dimacs.rutgers.edu/programs/challenge/>
| 3 | https://mathoverflow.net/users/141766 | 443281 | 178,806 |
https://mathoverflow.net/questions/443251 | 4 | To be more precise, suppose that $M$ is a model of ZF, for simplicity (or tactility) a set in some larger model $V$ of ZFC+Con(ZF), and suppose that $M\vDash``\alpha$ is an ordinal$"$. Must there be a model $N$ of ZFC (appearing in $V$) such that $\alpha\in N$ and an isomorphism of the partial structures given by the t... | https://mathoverflow.net/users/478588 | Does every ordinal appearing in a model of ZF appear in a model of ZFC? | As Noah mentioned in the comments, the answer to the first question is "yes".
The answer to the second question is consistently "no", assuming a little consistency: if there is a transitive model of ZF in $V$, then there will be transitive models $M$ of ZF in $V$ and ordinals $\alpha$ of $M$ for which the extra requi... | 8 | https://mathoverflow.net/users/160347 | 443286 | 178,808 |
https://mathoverflow.net/questions/443268 | 3 | Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it.
QUESTION:
Let $P\_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ parts bounded by $N.$ Prove that, if $n$ and $N$ are sufficiently large, $$P\_{n,N}(k)\sim N^{n-1}\sqrt{\dfrac{6}{\... | https://mathoverflow.net/users/165036 | Proof of an asymptotic formula by Tricomi | $\newcommand{\si}{\sigma}
\newcommand{\Z}{\mathbb Z}$As noted in [Pietro Majer's](https://mathoverflow.net/questions/443268/proof-of-an-asymptotic-formula-by-tricomi#comment1144269_443268) comment, the meaning of $\sim$ in the claim that, if $n$ and $N$ are sufficiently large, then
\begin{equation\*}
P\_{n,N}(k)\sim N... | 5 | https://mathoverflow.net/users/36721 | 443290 | 178,809 |
https://mathoverflow.net/questions/443272 | 5 | Suppose $B$ and $C$ are commutative unital $C^{\ast}$-algebras with $B \subseteq C$ (unital). Let $c$ be an element of $C$ such that $c \ast 1 = 1 \ast c$ in the pushout (in the category of commutative unital $C^{\ast}$-algebras) $C \ast\_B C$. Is it valid to conclude that $c$ belongs to $B$?
| https://mathoverflow.net/users/130868 | Elements that commute with $1$ in the pushout of a $C^{\ast}$-algebra | The answer is yes: $1 \ast c = c \ast 1$ implies $c \in B$.
---
To see this, let me first translate the statement into fully categorical language. If we restrict to $\|c\| \le 1$ wlog, then the elements $c \in C$ are in bijection with unital $\*$-homomorphisms $\mathscr{C}(D) \to C$, where $D \subseteq \mathbb{C}... | 3 | https://mathoverflow.net/users/27013 | 443302 | 178,814 |
https://mathoverflow.net/questions/443310 | 2 | Suppose $x$ is a **unknown** sequence of non-negative integers with $n$ elements and $\sum\_{i=1}^n x\_i = c$ for a constant $c$.
What is the minimum possible value of $\sum\_{i=1}^n x\_i(x\_i-1)(x\_i-2)$?
I know the answer when we want a lower bound for $\sum\_{i=1}^n x\_i(x\_i-1)$. We can expand it and use the *C... | https://mathoverflow.net/users/501463 | Minimum possible value of $\sum_{i=1}^n x_i(x_i-1)(x_i-2)$ for fixed sum | (Updated as per suggestions in the comments.)
I will use
$$
S[x]\equiv \sum\_{i=1}^n x\_i(x\_i-1)(x\_i-2).
$$
Obviously, $S[x]\geq 0$.
First, let $n=2$, and set $x\_1=c/2+y$ and $x\_2=c/2-y$. Then
$$
\sum\_{i=1}^n x\_i(x\_i-1)(x\_i-2)=\frac{(c-2)(12y^2+c(c-4))}{4}.
$$
This means that for $c>2$ the minimum is achiev... | 6 | https://mathoverflow.net/users/32985 | 443313 | 178,818 |
https://mathoverflow.net/questions/443306 | 2 | Consider $L^p[ 0,1]$ for $1\leq p < \infty$ or, if you prefer, $L^p(\mu)$ where $\mu$ is a finite Borel measure with compact support. Let $(\phi)\_{i\in I}$ be a subset of measurable functions that is contained in $L^p$ for every $1\leq p < \infty$ and assume that it is total for $L^{p\_0}$ for some $1\leq p\_0< \infty... | https://mathoverflow.net/users/145367 | Total sets for $L^p$ for every $1\leq p < \infty$ | No, this is not true. The set $S$ of all simple function orthogonal to $x^{-\frac 13}$ is not dense $L^2(0,1)$. To see that it is dense in $L^1(0,1)$, take $s$ simple, let $A=\int\_0^1 s(x)x^{-\frac 13}$ and choose $t=\frac 23 A \delta^{-\frac 23}\chi\_{(0, \delta)}$. Then $s-t \in S$ and $\|t\|\_1 \leq C \delta^{\frac... | 4 | https://mathoverflow.net/users/150653 | 443317 | 178,819 |
https://mathoverflow.net/questions/443305 | 1 | I want a reference of the following statement which I think is true. Let $X$ and $Y$ be Banach spaces with $X$ finite dimensional. Then $(X\otimes\_\epsilon Y)^\*$ is isometrically isomorphic to $(X^\*\otimes\_\pi Y^\*)$ and $(X\otimes\_\pi Y)^\*$ is isometrically isomorphic to $(X\otimes\_\epsilon Y)^\*$ where $\otime... | https://mathoverflow.net/users/136860 | Duality of projective and injective tensor product | Since Ryan's book is mentioned in the original question, here's some pointers for an answer based on this source:
* See Section 3.4 for what the dual space of $X\otimes\_\epsilon Y$ is. If you combine Proposition 3.14 (and the comments after) with Proposition 3.22 (see the comments after it) we get $(X\otimes\_\epsil... | 2 | https://mathoverflow.net/users/406 | 443321 | 178,820 |
https://mathoverflow.net/questions/443323 | 2 | Let $p$ be an odd prime, and consider the group
$$\{U\in \operatorname{SL}\_n(\mathbb{Z}/p\mathbb{Z}) : U^{t}U=I \bmod p \}\subseteq \operatorname{SL}\_n(\mathbb{Z}/p\mathbb{Z}).$$
I wonder what is the structure of this subgroup? Can the generators of this subgroup be written or calculated ?
We know that if $U^{t}U... | https://mathoverflow.net/users/482299 | A subgroup of $\mathrm{SL}_n(\mathbb{Z}/p\mathbb{Z})$ | This is the [special orthogonal group](https://groupprops.subwiki.org/w/index.php?title=Orthogonal_group_for_the_standard_dot_product) over the field $\mathbb{Z}/p \mathbb{Z}$. Or, rather, one of the "special orthogonal groups". For any invertible symmetric matrix $Q$, one can consider the group of matrices $U$ in $\te... | 7 | https://mathoverflow.net/users/297 | 443328 | 178,824 |
https://mathoverflow.net/questions/443312 | 1 | Let $\mathbb{F}$ be a field of characteristic $2$ and define $S$ to be the set of all triples $(i,j,k)\in\lbrace 1,\dotsc,n\rbrace^3$ with $\left|i-j\right|=1$, $\left|i-k\right|>1$, and $\left|j-k\right|>1$. Define $V=\bigoplus\limits\_{(i,j)\in\lbrace 1,\dots,n\rbrace^2}\mathbb{F}\sigma\_{i,j}$ to be the vector space... | https://mathoverflow.net/users/482329 | Counting the number of summands in a vector space over characteristic $2$ to get a direct sum | The dimension is $n(n-3)/2$. The argument is as follows. The set of pairs $(i,j)$ with $i$ and $j$ between $1$ and $n$, and $j-i \geqslant 2$ has cardinality $\binom{n-2}{2}=\frac{(n-1)(n-2)}{2}$. The $\sigma\_{i,j}$ with these indices are the only ones involved. So let $W$ be the vector space spanned by these. For eac... | 1 | https://mathoverflow.net/users/460592 | 443351 | 178,833 |
https://mathoverflow.net/questions/443340 | 1 | I am trying to recover the result given by equation 10 in the article [here](http://proceedings.mlr.press/v119/lu20b/lu20b.pdf). I am unable to get rid of the integral, any help would be much appreciated. To keep the description as self contained as possible, I will describe the relevant notations etc., a more detailed... | https://mathoverflow.net/users/21422 | Adjoint sensitivity analysis for a cost functional under an ODE constraint | Ah, that is just about the meaning of the expression $\frac{\partial E(x,\rho)}{\partial\rho}$. Since $\rho$ is *a function* of $t$, it really means "a function $D(t)$ such that
$$
E(x,\rho+\Delta\rho)-E(x,\rho)\approx \int\_0^1 D(t)\Delta\rho(t)\,dt
$$
for all small perturbations $\Delta\rho(t)$".
What you did was t... | 3 | https://mathoverflow.net/users/1131 | 443356 | 178,834 |
https://mathoverflow.net/questions/443367 | 4 | Inspired by [The set of all limits of sub-series of an absolute convergent series](https://math.stackexchange.com/questions/2062357/the-set-of-all-limits-of-sub-series-of-an-absolute-convergent-series) is the following true?:
>
> Let $a\_n$ be a strictly decreasing sequence and $\sum\_1^\infty a\_n=\ell<\infty$ is ... | https://mathoverflow.net/users/36688 | The set of all possible values of subseries of a convergent positive term series | For convenience define $S\_n = \sum\_{j\le n} a\_j$ and $T\_n = \sum\_{j > n} a\_j$.
Suppose there is $n$ such that $a\_n > T\_n$. Then for $S\_{n-1} + T\_n < x < S\_n$,
$x$ is not the sum of a subseries.
On the other hand, if $a\_n \le T\_n$ for all $n$, then every $x \in [0,\ell]$ is the sum of a subseries. This ... | 9 | https://mathoverflow.net/users/13650 | 443370 | 178,838 |
https://mathoverflow.net/questions/443353 | 1 | Let $C$ be a connected curve of arithmetic genus $g$
over algebraically closed field $k$ of characteristic zero having only nodes
as singularities together with finite morphism
$f: C \to \mathbb{P}^1$.
In [3264 and All That](https://www.cambridge.org/core/books/3264-and-all-that/DC062983CC5F8B7CDD37CFEBCCA5FEA4) by E... | https://mathoverflow.net/users/501436 | Finiteness of automorphism group of finite map $f: C \to \mathbb{P}^1$ | Q1: since maps of curves are determined by their action on the function field, we get an injection $\mathrm{Aut}(C/\mathbb{P}^1) \to \mathrm{Aut}(k(C)/k(t))$ but the latter is just a Galois group (or automorphism group of a field extension if you don't like this terminology for a non-Galois extension) of a finite exten... | 3 | https://mathoverflow.net/users/154157 | 443375 | 178,840 |
https://mathoverflow.net/questions/443364 | 2 | I read the following on Wikipedia's page on [Monadic Second-Order Logic of Two Successors (MS2S)](https://en.wikipedia.org/wiki/S2S_(mathematics)):
>
> Weak S2S (WS2S) requires all sets to be finite (note that finiteness
> is expressible in S2S using Kőnig's lemma).
>
>
>
Is this statement an error? I would th... | https://mathoverflow.net/users/38049 | How can Kőnig's Lemma be expressed in Monadic Second-Order Logic of 2 Successors? |
>
> I would think that only Weak Kőnig's Lemma(WKL) would be expressible in MS2S
>
>
>
You're slightly misreading the passage - the point is that Konig's Lemma can be used to show that finiteness is definable in MS2S. (That said, you are right that even WKL would be enough.)
Here's the idea. First, note that i... | 3 | https://mathoverflow.net/users/8133 | 443380 | 178,842 |
https://mathoverflow.net/questions/440126 | 9 | First I would like to apologize if this post breaks any rule regarding career advice or opinion-based questions. Given that construct QFT (CQFT) is a rather small community, I found this is the only site where some current/past practitioners can chime in.
For an idea of my mathematical level, in terms of analysis I h... | https://mathoverflow.net/users/498931 | Approach to learning constructive QFT | CQFT is very much still an open research subject. I don't think it is known what the best approach is. So all I can do is share my own opinion. (And a warning: I'm just an interested observer!)
First: **Learn you some classical electrodynamics!** Quantum field theory is after all *a theory of fields*, and classical e... | 11 | https://mathoverflow.net/users/35508 | 443387 | 178,843 |
https://mathoverflow.net/questions/443297 | 3 | I'm interested in computing eta invariants of Dirac operators (on spinor bundles tensored with some vector bundles) on the total space of $S^{2n}$ bundles over odd-dimensional manifolds. I found the papers of [Bismut-Cheeger](https://www.jstor.org/stable/1990912) and [Dai](https://www.jstor.org/stable/2939276) where a ... | https://mathoverflow.net/users/5420 | Explicit computations of Bismut-Cheeger eta form for $S^{2n}$ bundles | If you consider a fibre bundle with compact structure group that acts by isometries on the typical fibre and preserves a given Dirac operator, then the $\eta$-form can be read of from an infinitesimally equivariant $\eta$-invariant on the typical fibre, the fibre bundle curvature, and its action on the typical fibre. T... | 2 | https://mathoverflow.net/users/70808 | 443391 | 178,845 |
https://mathoverflow.net/questions/443376 | 3 | I apologize if this question is basic in some sense. I was looking for an example of a non-proper HNN-extension and I found [this](https://mathoverflow.net/questions/405536/examples-of-non-proper-profinite-hnn-extensions).
In the comments, markvs mentioned the Baumslag-Solitar group $B(2,3)$. We can show that this is... | https://mathoverflow.net/users/123172 | Profinite completion of Baumslag-Solitar group as a profinite HNN-extension | The largest residually finite quotient of $\mathrm{BS}(2,3)$ is the semidirect product $\mathbf{Z}\ltimes\_{2/3}\mathbf{Z}[1/6]$.
So, if I'm correct, its profinite completion is
$$\widehat{\mathbf{Z}}\ltimes\_{2/3}\widehat{\mathbf{Z}[1/6]}.$$
Each time, the $2/3$ means that the generator $1$ acts by multiplication ... | 2 | https://mathoverflow.net/users/14094 | 443400 | 178,846 |
https://mathoverflow.net/questions/443198 | 2 | Given a word $w \in X^{\pm 1}$ representing an element of the free group $F(X)$ there is a (usually non-unique) sequence $w=w\_0 \to w\_1 \to \cdots \to w\_r$ with $|w\_i|>|w\_{i+1}|$ where $w\_r$ is the unique reduced form of $w$, i.e. there are no subwords of the form $xx^{-1}, x \in X^{\pm 1}$. I know of this fact, ... | https://mathoverflow.net/users/38698 | Passing to normal forms in graphs of groups | In comments, the OP indicates that what they really want is a *uniqueness* result for reduced words in arbitrary graphs of groups. (Indeed, what the question actually asks for, that any word can be transformed into a reduced word by successively cancelling, is obvious by induction on length.)
The desired uniqueness r... | 6 | https://mathoverflow.net/users/1463 | 443408 | 178,847 |
https://mathoverflow.net/questions/443068 | 1 | In [Juven Wang, Zheng-Cheng Gu, and Xiao-Gang Wen - Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond](https://arxiv.org/abs/1405.7689), the authors calculated many group cocycles in explicit form. See [n-cocycles of finite abelian groups from cohomology... | https://mathoverflow.net/users/500508 | Additivity of group cocycles? | There are plenty of cocycles which are not additive in any variable, and plenty of cohomology classes that do not admit additive representatives. For example, if $G$ is finite, then there are no [nontrivial] additive functions $G \to \mathbb{Z}$, and so no [nontrivial] $\mathbb{Z}$-valued cohomology class has an additi... | 3 | https://mathoverflow.net/users/78 | 443409 | 178,848 |
https://mathoverflow.net/questions/443406 | 5 | Let f be a polynomial with integer coefficients.
Let B(f) be the set of all values of f on positive integers.
B(f) = {f(n)| n is a positive integer} = {f(1), f(2), ...}
A positive integer k is called "good" if it is a sum of distinct members of B(f).
Otherwise we say k is "bad".
For instance, if f(x)=x^3, then it i... | https://mathoverflow.net/users/30650 | Covering all but finitely many integers via some given polynomials | With the right adaptations, the answer should be yes. In particular, the trivially necessary assumption is that the gcd of the values (not just of the coefficients) is 1. Then it is shown in
K. F. Roth, G. Szekeres, Some asymptotic formulae in the theory of partitions (1954) that every sufficiently large integer is a s... | 9 | https://mathoverflow.net/users/127660 | 443410 | 178,849 |
https://mathoverflow.net/questions/427785 | 3 | Let $X$ be an affine variety and $G$ an affine algebraic group (for example $\operatorname{PGL}\_n$). How do I compute the Selmer set
$$ \operatorname{Sel}\_\zeta(\mathbb{Q},G) = \{\tau \in H^1(\mathbb{Q},G) \ | \ \tau\_\nu \in \zeta(X(\mathbb{Q}\_\nu)) \ \text{for all places} \ \nu\} $$
where $(\zeta \mapsto \zeta(x))... | https://mathoverflow.net/users/489009 | How to compute Selmer set? | Assume that $X(\mathbb{Q}\_\nu)$ is nonempty for every place $\nu$. Since $X$ is affine, $H^1(X,G)$ is trivial. This means that $\zeta(x) = e$ for every $x \in X(\mathbb{Q}\_\nu)$, which means that $e \in \operatorname{Sel}\_\zeta(\mathbb{Q},G)$. So the Selmer set is nonempty.
| -1 | https://mathoverflow.net/users/489009 | 443420 | 178,851 |
https://mathoverflow.net/questions/443392 | 3 | Suppose $\mathcal{M} = (M, +, 0)$ is a cancelable commutative monoid. Let $G$ be the maximal subgroup of $M$, i.e.
$$G = \{ a \in M \colon (\exists b \in M)\, a + b = 0 \}.$$
For $a, b \in M$ say $a \preceq b$ if $(\exists c \in M)\, a + c = b$ and say $a \equiv b$ if $a \preceq b$ and $b \preceq a$. Note for all $... | https://mathoverflow.net/users/8106 | Cancelable commutative monoids with finite maximal subgroups | Here is a general construction that encompasses @R. van Dobben de Bruyn's example but the idea is taken from his answer. I'll write commutative monoids additively. I'll use $K(M)$ for the Grothendieck group of a commutative monoid $M$ and $M^\times$ for the group of units.
Let $0\to A\to B\xrightarrow{f} C\to 0$ be a... | 4 | https://mathoverflow.net/users/15934 | 443433 | 178,856 |
https://mathoverflow.net/questions/443432 | 2 | Suppose $(Z\_1, Z\_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want to know a valuable tighy lower bound of the probability
$$\mathrm{P} \left( Z\_1 > z\_1, Z\_2 < z\_2\right)$$
where $z\_1, z\_2$ a... | https://mathoverflow.net/users/153595 | The lower bound of bivariate normal distribution | Let $h:=z\_1$, $k:=z\_2$, and $r:=\rho\ge0$. We want to lower-bound $P(Z\_1>h,Z\_2<k)$. Formula (2.11) in the [paper by Willink](https://www.tandfonline.com/doi/abs/10.1081/STA-200031505) (cited the book you linked) gives the following upper bound on $P(Z\_1>h,Z\_2>k)$:
$$P(Z\_1>h,Z\_2>k) \\
\le\Phi(-h)\Big[\Phi\Bi... | 2 | https://mathoverflow.net/users/36721 | 443435 | 178,857 |
https://mathoverflow.net/questions/443444 | 11 | Let $k\in\mathbb{Z}\_{>0}$, and $s\in\mathbb{N}$, and for $m\_1,\ldots,m\_k$ some nonnegative integers, consider the problem of maximizing the product
$$
(1+m\_1)(1+m\_2)\cdots(1+m\_k)
$$
under the constraint $m\_1+\cdots+m\_k=s$.
I would like to know:
1. The exact formula $M(k,s)$ for the maximal value of the prod... | https://mathoverflow.net/users/7410 | A discrete optimization problem related to the AM-GM inequality | This was essentially answered by Nate in the comments, but here are some more details. As Nate argues, $|m\_i - m\_j| \leq 1$ for all distinct $i,j$. Thus, if $s=ak+r$, where $a,r \in \mathbb{N}$ and $r < k$, then there is a unique choice (up to permuting variables) which maximizes the product. Namely, set $r$ of the v... | 10 | https://mathoverflow.net/users/2233 | 443448 | 178,861 |
https://mathoverflow.net/questions/443438 | 1 | Consider the (inhomogeneous) minimal surface equation for functions $u,f:D\to \mathbb{R}$ for some smooth domain $D\subset \mathbb{R}^n$
$$Lu:=\operatorname{div} \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=f.$$
Then is it true that $u\_1\leq u\_2$ on $\mathbb{R}^n\setminus D$ and $Lu\_1\geq Lu\_2$ implies $u\_1\leq u\_2... | https://mathoverflow.net/users/68232 | comparison principle for the minimal surface equation | For the case that $D$ is bounded, as Leo Moos has noted, more details can be found in a textbook of minimal surface, cf. Lemma 1.26 in T. Colding, W. Minicozzi, *A Course in Minimal Surfaces*.
That is, let $F(X)=\frac{X}{\sqrt{1+|X|^2}},$ then
\begin{equation}
F(\nabla u\_1)-F(\nabla u\_2)=\left(\int\_0^1dF\big(\nabl... | 3 | https://mathoverflow.net/users/166368 | 443452 | 178,863 |
https://mathoverflow.net/questions/443450 | -1 | [This answer](https://mathoverflow.net/a/295742/501568) says that if $X$ is a random variable and $X\_+ = \mathrm{max}(0, X)$, then $X\_+ = \int\_0^\infty I\_{\{X > x\}}\mathrm{d}x$. I'd like to know how to derive this starting with $A \in \mathcal{S} \implies \int\_S 1\_A\mathrm{d}\mu = \mu(A)$ (from ["Desired Propert... | https://mathoverflow.net/users/501568 | Random variable as an integral of an indicator function | The layer cake representation of a non-negative measurable function, $X$, is applied in the proof of proposition 2.1 [here](https://stat.uiowa.edu/sites/stat.uiowa.edu/files/cae/Lo_Expectation.pdf).
| 0 | https://mathoverflow.net/users/501568 | 443470 | 178,869 |
https://mathoverflow.net/questions/442086 | 7 | In Yau and Schoen's differential geometry,in Ch5 before Thm 3.5,the author says
When $R$(scalar curvature of a manifold M)$>0$,there exists a unique Green's function $G$ to the operator $L=-\Delta+aR$ and $LG\_{P}=\delta\_{P}$.Here $P$ is an arbitrary point of $M$.
In the normal coordinate of $P$,we have
$$
G\_P(x)... | https://mathoverflow.net/users/148247 | Existence and estimates of Green's function on Riemannian manifold | For further details on the existence argument, see Chapter 4 of Aubin's book, "Nonlinear Analysis on Manifolds: Monge-Ampère Equations" [1] for the harmonic case, and [2] for the metaharmonic case. The asymptotic expansion formula near $P$ is derived from the first term of the Green function, which is obtained by multi... | 3 | https://mathoverflow.net/users/166368 | 443480 | 178,876 |
https://mathoverflow.net/questions/443488 | 7 | The Lie operad $\text{Lie}$ is generated by a binary operator $[\ ,\ ]$, modulo a degree two relation (skew commutativity $[x,y]=-[y,x]$) and a degree three relation (Jacobi $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]$). See e.g. the paper ["The operad Lie is free"](https://arxiv.org/pdf/0802.3010.pdf).
**Question:** do "non(skew... | https://mathoverflow.net/users/119012 | Non(skew)commutative Lie algebras? | Such objects are known as *Leibniz algebras*.
A Leibniz algebra is a module $M$ together with a bilinear pairing $$[-,-]\colon M⊗M→M$$ that satisfies the Leibniz identity: $$[a,[b,c]]=[[a,b],c]+[b,[a,c]].$$
Leibniz algebras for which $[a,b]=-[b,a]$ are precisely Lie algebras.
The concept was introduced and studied ... | 9 | https://mathoverflow.net/users/402 | 443494 | 178,879 |
https://mathoverflow.net/questions/443239 | 6 | I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case.
The integral over $G(\mathbb{A})^1\_\gamma \backslash G(\mathbb{A})^1$ may diverge for a given conjugacy class $\gamma$. Say $\gamma$ is the matrix of the translation by $1$. Then $G(\ma... | https://mathoverflow.net/users/128718 | Divergence of integrals in the trace formula | I take it that the question is "Why does this iterated integral diverge?," but correct me if that is not the question.
The issue here is the outer integral: when $\gamma$ is as you describe, then if $P$ is the group of upper triangular matrices, the quotient $G\_\gamma\backslash P$ is a rank $1$ split torus, the adel... | 4 | https://mathoverflow.net/users/62154 | 443508 | 178,881 |
https://mathoverflow.net/questions/443443 | 6 | In [Generalisation of the quantum exterior algebra](https://mathoverflow.net/questions/258512/generalisation-of-the-quantum-exterior-algebra) the quantum exterior algebra is discussed:
$$
K\langle x\_1,\dotsc x\_n\rangle/(x\_i^2,x\_i x\_j + q\_{i,j}x\_j x\_i),
$$
with nonzero field elements $q\_{i,j}$ for $i<j$.
Whe... | https://mathoverflow.net/users/499575 | Quantum exterior algebra | Quantum affine space has been studied since the nineties. The coordinate ring of quantum affine space is a quantum polynomial algebra, whose definition I think you can imagine, and the Koszul dual of a quantum polynomial algebra is a quantum exterior algebra. There are conditions on the $q\_{i,j}$ in order for the alge... | 7 | https://mathoverflow.net/users/460592 | 443512 | 178,882 |
https://mathoverflow.net/questions/443491 | 1 | Let $c>0$ be a very small constant and $N \in \mathbb N$ very large.
Assume we have a function $f(x)$ for $x \in S^1$ defined as
$$
f(x) = \sum\_{k=\lfloor N/(1+c) \rfloor}^{N} c\_k \sin(kx+b\_k)
$$
for some coefficients $b\_k, c\_k \in \mathbb R$, that is all the sines appearing have frequency similar to $N$.
I know... | https://mathoverflow.net/users/173610 | Inequality for sums of sines with similar frequency | In other words, in particular, you want to create an entire function $f$ of exponential type $a=\pi/4$ bounded on the real axis by $C$ and satisfying $f(n)=(-1)^n$ near $N$ on an interval of length $cN$. Note that by symmetrizing ($F(z)=\frac 12(f(z)+\bar f(\bar z))$), you can make it real-valued on $\mathbb R$, and th... | 2 | https://mathoverflow.net/users/1131 | 443514 | 178,883 |
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