parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/266013 | 5 | Given a smooth manifold $M$ and a Lie algebroid $E\rightarrow M$ we can consider the $E$-polydifferential operators $D\_E$ and the $E$-polyvectorfields
$T\_E$ as
$$D\_E:=\bigoplus\_{k=-1}^\infty\mathcal{U}(E)^{\otimes k+1}\hspace{0.3cm}\mbox{and}\hspace{0.3cm}T\_E=\bigoplus\_{k=-1}^\infty\Gamma\left(M;\bigwedge^{k+1}... | https://mathoverflow.net/users/27224 | Functoriality of the formality quasi-isomorphism of E-polydifferential operators | The construction is functorial with respect to **algebraic morphisms** of Lie algebroids (as opposed to geometric ones): see for instance my paper with Van de Bergh <https://arxiv.org/pdf/0708.2725.pdf> (it encompasses the globalization techniques that are used in the paper you cite).
As Stefan pointed out, the choi... | 3 | https://mathoverflow.net/users/7031 | 273405 | 122,298 |
https://mathoverflow.net/questions/273399 | 13 | In process of doing some computations on Hilbert schemes, I stumbled across the following identity, for $k \ge 2$:
$$
k^{k-3} = \frac{1}{2} \sum\_{i=1}^{k-1} \binom{k-2}{i-1} i^{i-2} (k-i)^{k-i-2}
$$
which can be verified computationally for small values of $k$ (I did it for the first 500).
This is actually a specia... | https://mathoverflow.net/users/19088 | Identity with binomial coefficients and k^k | Let's denote $T(x)=\sum\_{n\geq 1}\frac{n^{n-1}x^n}{n!}$ the exponential generating function of labeled rooted trees, and $U(x)=\sum\_{n\geq 1}\frac{n^{n-2}x^n}{n!}$ the corresponding function for unrooted trees. We will use the fact that $xU'=T$ and $xe^T=T$, both of which can be derived algebraically or combinatorial... | 20 | https://mathoverflow.net/users/2384 | 273411 | 122,301 |
https://mathoverflow.net/questions/273401 | 3 | The Narayana polynomials (OEIS [A001263](https://oeis.org/A001263)) are the h-polynomials of the associahedra (the Stasheff polytopes) and their dual simplicial polytopes (cf. the Fomin and Reading ref in the OEIS entry).
Are these polynomials related to the [Ehrhart series](https://en.m.wikipedia.org/wiki/Ehrhart_p... | https://mathoverflow.net/users/12178 | Narayana polynomials as numerators of Ehrhart series rational functions? | The $n$th Narayana polynomial is the numerator of the generating function for the Ehrhart polynomial of the order polytope and the chain polytope of the product of a 2-element chain and an $n$-element chain.
| 7 | https://mathoverflow.net/users/2807 | 273434 | 122,309 |
https://mathoverflow.net/questions/273436 | 6 | Let $K\_0(Var\_k)$ be the grothendieck group of the category of $k$-varieties, and call its elements virtual motives. $\mathbb{L}:=[\mathbb{A}^1\_k]$ is called the Lefschetz motive. I think that if a virtual motive is divisible by arbitrarily high powers of $\mathbb{L}$, then it must be $0$. Is this true? If so, is the... | https://mathoverflow.net/users/110215 | Virtual Motives Infinitely Divisible by Lefschetz Motive | This is an open problem. It's probably quite hard.
| 6 | https://mathoverflow.net/users/1310 | 273439 | 122,311 |
https://mathoverflow.net/questions/238186 | 2 | The OEIS entries [A019538](http://oeis.org/A019538), [A049019](http://oeis.org/A049019), and [A133314](http://oeis.org/A133314), relate a refinement of the face polynomials of the [permutohedra](http://en.wikipedia.org/wiki/Permutohedron) (A049019) to partition polynomials (A133314) defined by multiplicative inversion ... | https://mathoverflow.net/users/12178 | Relating face polytopes of permutohedra to integer partitions | Rodica Simion in "Convex polytopes and enumeration" affirms the bijection on pages 162 and 163.
| 2 | https://mathoverflow.net/users/12178 | 273448 | 122,315 |
https://mathoverflow.net/questions/273400 | 3 | Let $G=K\_{3,7}$ be a complete bipartite graph with partitions $X=\{x\_1,x\_2,x\_3\}$ and $Y=\{y\_1,y\_2,y\_3,y\_4,y\_5,y\_6,y\_7\}$. We know that the genus of graph $G$ is $2$. If we delete three edges $\{x\_1y\_1\}$, $\{x\_2y\_2\}$ and $\{x\_3y\_3\}$ of graph $G$ and add three edges $\{x\_1x\_2\}$, $\{x\_2x\_3\}$ and... | https://mathoverflow.net/users/105561 | Genus of The Complete Bipartite Graph | [SageMath](http://doc.sagemath.org/html/en/reference/graphs/sage/graphs/generic_graph.html) math says the genus is $1$.
```
sage: G = Graph([[0,1],[0,2],[1,2],[0,4],[0,5],[0,6],[0,7],[0,8],[0,9],[1,3],[1,5],[1,6],[1,7],[1,8],[1,9],[2,3],[2,4],[2,6],[2,7],[2,8],[2,9]])
sage: G.genus()
1
sage: G.get_embedding()
{0: [... | 4 | https://mathoverflow.net/users/51668 | 273451 | 122,316 |
https://mathoverflow.net/questions/273452 | 1 | In another question on mathoverflow ([What is the best currently proven bounds on prime gaps?](https://mathoverflow.net/questions/239987/what-is-the-best-currently-proven-bounds-on-prime-gaps?rq=1)) the following bound on the prime gap was quoted:
$G(X)\ll \frac{X^{0.525}}{\log X}$
How do you prove this, and what paper... | https://mathoverflow.net/users/111800 | Proof of prime gap bound? | The result is slightly misquoted, it should read $G(X)\leq X^{0.525}$ for $X\geq X\_0$. That is, there is no $\log X$ denominator, but $\ll$ can be improved to $\leq$ (when $X$ is sufficiently large). The result appeared in Baker-Harman-Pintz - The difference between consecutive primes, II (Proc. Lond. Math. Soc. 83 (2... | 3 | https://mathoverflow.net/users/11919 | 273454 | 122,317 |
https://mathoverflow.net/questions/273435 | 5 | Assume that a Heegaard diagram $(\Sigma\_g,\{\alpha\_1,\ldots,\alpha\_g\},\{\beta\_1,\ldots,\beta\_g\})$, defining a Heegaard splitting of a closed 3-manifold, is given. So, by attaching 3-dimensional 2-handles to $\Sigma\_g$ along $\alpha\_i$ and $\beta\_i$, one gets a closed 3-manifold with the Heegaard surface $\Sig... | https://mathoverflow.net/users/88357 | Irreducibility of a Heegaard splitting from its Heegaard diagram | There is a criterion which is usually used to show that a Heegaard splitting is irreducible, which in fact shows that the Heegaard splitting is strongly irreducible (meaning that any pair of meridian disks for $H\_1$ and $H\_2$ have intersecting boundaries). The original criterion was given by Casson and Gordon in an u... | 6 | https://mathoverflow.net/users/1345 | 273466 | 122,323 |
https://mathoverflow.net/questions/273463 | 7 | For common theories that talk about something more general than first-order arithmetic (e.g. set theories and subsystems of second-order arithmetic), are there nice axiomatizations of their arithmetic parts? I know that the arithmetic part of $ACA\_0$ is $PA$ and the arithmetic parts of $RCA\_0$ and $WKL\_0$ are both $... | https://mathoverflow.net/users/83073 | Axiomatizations of arithmetical parts of theories | A somewhat general method has been explained by Azriel Lévy
[[*Axiomatization of induced theories*](https://doi.org/10.1090/S0002-9939-1961-0122702-2), Proc. Am. Math. Soc. 12, 251-253 (1961); [ZBL0178.31603](https://zbmath.org/?q=an:0178.31603); [MR0122702](http://www.ams.org/mathscinet-getitem?mr=122702)] The method ... | 7 | https://mathoverflow.net/users/2000 | 273471 | 122,326 |
https://mathoverflow.net/questions/273202 | 5 | Let $K$ be a primitive (i.e. not biquadratic) quartic CM-field. That is we have $[K:\mathbb{Q}]=4$ and let $K\_0=\mathbb{Q}(\sqrt{d})$ be the totally real quadratic subfield, here $d> 1$, that is we have $[K:K\_0]=[K\_0:\mathbb{Q}]=2$.
Denote by $O\_K^{\times}$ the set of units in $K$ and by $(O\_{K\_0}^{\times})^{+}... | https://mathoverflow.net/users/70593 | Are there primitive quartic CM fields whose norms of units give all totally positive units of the real quadratic subfield? | Except for the finitely many fields where nontrivial roots of unity are present, the fundamental unit $\varepsilon$ of the quadratic subfield $k$ is the norm of a unit from $K$ if and only if $K = k(\sqrt{\pm \varepsilon})$. If $N(\varepsilon) = +1$, then $K$ is biquadratic, and if $N(\varepsilon) = -1$, then $K$ is no... | 2 | https://mathoverflow.net/users/3503 | 273476 | 122,331 |
https://mathoverflow.net/questions/268544 | 20 | Here's an interesting inequality involving binomial coefficient and Stirling numbers of the second kind that I believe holds for all $n,k$:
$$ k^n {n \choose k} \leq n^k {n \brace k} $$
On the left-hand side we choose a $k$-element subset of the set $[n]=\{1,\ldots,n\}$, and then choose a function from $[n]$ into the c... | https://mathoverflow.net/users/1716 | Are surjections $[n]\to [k]$ more common than injections $[k]\to [n]$? | Let $Y = \{1,\ldots, n\}$. Throughout $r \in \{1,\ldots, k\}$ will be the size of the range of a function.
The left-hand side $k^n \binom{n}{k}$ is the size of the union of the sets $\mathcal{L}\_r$ of all pairs $(Z, g)$ where $Z$ is an $k$-subset of $Y$ and $g : \{1,\ldots, n\} \rightarrow Z$ has range of size $r$. ... | 9 | https://mathoverflow.net/users/7709 | 273478 | 122,332 |
https://mathoverflow.net/questions/271295 | 7 | I am studying the basics of constructible and lisse sheaves, and am trying to understand [SGA 4](http://fabrice.orgogozo.perso.math.cnrs.fr/SGA4/details/index.html), IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with *torsion* sheaves, and there are several "moral" explanat... | https://mathoverflow.net/users/18238 | What is the need for torsion in the definition of lisse sheaves? | I think you are misinterpreting things slightly. Lemma 2.1 says nothing about abelian groups. Lemma 2.1(i) is about sets, and Lemma 2.1(ii) is about A-modules. For $A = \mathbb Z$, $\mathbb Z$-modules are equivalent to abelian groups and so Lemma 2.1(ii) applies.
It is only 2.1(i) that is used in Remark 2.3(i). That ... | 2 | https://mathoverflow.net/users/18060 | 273485 | 122,334 |
https://mathoverflow.net/questions/273438 | 3 | I am looking for a good set of sufficient conditions to put on a category $C$ such that the following becomes true:
Given any $\infty$-category $S$ and a functor $f\colon N(C)\to S$ which is essentially constant (i.e. it maps all edges to equivalences), a colimit $\overline f\colon N(C)^\triangleright\to S$ of $f$ ex... | https://mathoverflow.net/users/88153 | Conditions on a diagram category such that constant limits always exist and is constant | I think that Proposition 4.3.1.12. in Higher Topos Theory (Lurie) is exactly what I was looking for:
Given: a coCartesian fibration $p\colon X\to S$ of $\infty$-categories and a diagram $\overline q\colon K^\triangleright\to X$ such that $K$ is weakly contractible and such that $\overline q$ carries each edge of $K$ ... | 1 | https://mathoverflow.net/users/88153 | 273490 | 122,335 |
https://mathoverflow.net/questions/273495 | 4 | How to prove that
>
> in $A\_n$ (Alternating group), the subgroup of second smallest index has index $n \choose 2$ if $n\ge 9$ ?
>
>
>
I know how to prove it for the smallest index, but for the second smallest I don't know how to prove it.
| https://mathoverflow.net/users/nan | How to prove that index of second smallest index subgroup in $A_n$ is $n \choose 2$? | Although some fiddling will be necessary to get a precise result, one can understand problems of this type without using powerful modern results in group theory.
For large enough $n$, there are a lot of primes strictly between $n/2$ and $n-2$. Any subgroup whose index is at most $n^2$ will have order divisible by at ... | 7 | https://mathoverflow.net/users/36466 | 273509 | 122,341 |
https://mathoverflow.net/questions/273259 | 27 | In 1994, Mike Hopkins wrote a paper called [Topological Modular Forms, the Witten Genus, and the Theorem of the Cube](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf). As usual, the introduction was fantastic, explaining the power of various cobordism invariants and connections between homotopy ... | https://mathoverflow.net/users/11540 | Atiyah-Singer style index theorem for elliptic cohomology? | The status of this question is *OPEN*.
This theory has *NOT* been developed yet.
That being said, the evidence is as compelling as ever, I don't know of any obstructions to making this work, and I'm convinced that there's an awesome theory out there, waiting to be discovered.
Roughly 8 years ago, I wrote an unsuc... | 21 | https://mathoverflow.net/users/5690 | 273518 | 122,344 |
https://mathoverflow.net/questions/272786 | 2 | This question follows [this one](https://mathoverflow.net/questions/271936/closed-form-for-int-0t-e-x-fraci-n-alpha-xxdx/271955), where the general problem has apparently no simpler form than the integral one. I focus now on the limit case:
\begin{align}
\int\_0^T e^{-x}\frac{nI\_n(x)}{x}dx=\int\_0^T e^{-x}\frac{I\_{n-... | https://mathoverflow.net/users/111000 | Integral with Bessel function and hypergeometric function ${}_2F_2$: explicit expression for these polynomials? | From [DLMF](http://dlmf.nist.gov/10.43.E6), one has, for $n=0,1,2...$
$$\int\_{0}^{x}e^{-t}I\_{n}\left(t\right)\mathrm{d}t=xe^{-x}(I\_{0}\left(x\right)+I%
\_{1}\left(x\right))+n(e^{-x}I\_{0}\left(x\right)-1)+2e^{-x}\sum\_{k=1}^{n-1}(n-k%
)I\_{k}\left(x\right)$$
which may be obtained by recurrence, I guess. Then
\begin{... | 1 | https://mathoverflow.net/users/46744 | 273521 | 122,345 |
https://mathoverflow.net/questions/273522 | 7 | Let $(X,\omega)$ be an $n$-dimensional complete Kähler manifold. Then when it is isometric to complex Euclidean space $\mathbb C^n$
| https://mathoverflow.net/users/83444 | When a Kähler manifold is isometric to $\mathbb C^n$ | The story of characterization of the isometry class of $\mathbb C^n$ equipped with the flat metric comes back to around 80 of the nice work of Burns in Annals of Mathematics and also others
I know the following theorem, it may help in the foliation language.
**Theorem**: An $n$-dimensional Kähler manifold $(X,\omeg... | 12 | https://mathoverflow.net/users/nan | 273525 | 122,346 |
https://mathoverflow.net/questions/273457 | 5 | Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and let $|\mathbf{x}|$ denote the L2 norm. There is a fairly standard argument involving successive minima to obtain the estimate on $N(R)$ which is the number of points $\mathbf{x}$ on $\Lambda$ satisfying $|\mathbf{x}| \leq R$.
Let $R\_1, ..., R\_n$ be positive real num... | https://mathoverflow.net/users/84272 | Counting points on lattices in inside a box- Geometry of numebrs | Let $\mathcal{S}\subset\mathbb{R}^n$ be a convex compact set lying in the closed ball of radius $R$ centered at the origin. Then
$$\left|\#(\mathcal{S}\cap\Lambda)-\frac{\mathrm{vol}(\mathcal{S})}{\det(\Lambda)}\right|\ll\_n 1+\max\_{1\leq m\leq n-1}\frac{R^m}{\lambda\_1\dots\lambda\_m},$$
where $\lambda\_1\leq\dots\le... | 2 | https://mathoverflow.net/users/11919 | 273534 | 122,349 |
https://mathoverflow.net/questions/273535 | 1 | To what extent have all unital $C^\*$ algebras $A$ with the following property been classified? Is there a simple $C^\*$ algebra with this property? Does $C(K)$ satisfy this property, where $K$ is an arbitrary compact extremally disconnected space?
>
> The scalar elements, are the only elements with connected spect... | https://mathoverflow.net/users/36688 | Classification of $C^*$ algebras whose all non scalar elements have disconnected spectrum | In the commutative case $A=C(K)$, this is equivalent to asking for which compact Hausdorff spaces there are no continuous surjections $K\to[0,1]$, since the spectrum of $f\in C(K)$ is its image. A compact Hausdorff space $K$ has this property iff $K$ is *scattered*: that is, iff every nonempty subset of $K$ has an isol... | 6 | https://mathoverflow.net/users/75 | 273539 | 122,351 |
https://mathoverflow.net/questions/273532 | 2 | I'm looking specifically at the optimization problem
$$
\begin{align\*}
\text{maximize: }& n - \frac{\|\lambda\|\_1^2}{\|\lambda\|\_2^2}\\
\text{subj. to: }& \lambda \succeq \epsilon\mathbf{1}
\end{align\*}
$$
over $\mathbb{R}^n$ where $\epsilon\geq 0$ is some constant. This is equivalent to the problem
$$
\begin{align... | https://mathoverflow.net/users/108236 | Tighest Gap $\|x\|_1/\|x\|_2$ between $\ell^1$ and $\ell^2$ norms | (EDIT: changing this to at least solve the stated problem...)
We first look for the optimal vector for fixed $\|\lambda\|\_1=M$. Hence the problem is: maximize $\|\lambda\|\_2^2$ s.t. $\|\lambda\|\_1=M$, $\lambda\geq \varepsilon\mathbf{1}$.
The standard trick to solve this kind of inequalities is called *unsmoothin... | 3 | https://mathoverflow.net/users/1898 | 273542 | 122,353 |
https://mathoverflow.net/questions/273520 | 2 | Let $P$ be a point in $\mathbb{P}^n$. Then we have a regular map $\mathbb{P}^n\setminus P\rightarrow \mathbb{P}^{n-1}$, which is not defined only at $P$. Then We know that the graph closure of this map is the Blow Up of $\mathbb{P}^n$ at $P$. $Blow\_P \mathbb{P}^n\hookrightarrow \mathbb{P}^n\times \mathbb{P}^{n-1}$. No... | https://mathoverflow.net/users/nan | Ample line bundle of Blow up | You can undertstand this by thinking about the simplest case $n=2$, the case $n \geq 3$ being analogous.
The blow-up of $\mathbb{P}^2$ at a point $P$ is the Hirzebruch surface $\mathbb{F}\_1$, and the two projections $$p\_1 \colon \mathbb{F}\_1 \longrightarrow \mathbb{P}^2, \quad p\_2 \colon \mathbb{F}\_1 \longrighta... | 2 | https://mathoverflow.net/users/7460 | 273543 | 122,354 |
https://mathoverflow.net/questions/273519 | 4 | Let $\Sigma$ be a 2-sphere with a (smooth) Riemannian metric $g$ of nonnegative curvature. Let $\mathcal{C}$ be a simple closed smooth curve on $\Sigma$. Then $\mathcal{C}$ splits $\Sigma$ into two components $\Sigma^+$, $\Sigma^-$. Set $S(t):=\{p\in\Sigma^+\mid d(\mathcal{C},p)=t\}$ where $d$ is the distance function ... | https://mathoverflow.net/users/111820 | Bishop-Gromov type inequality for Jordan curve on 2-sphere | Cut out a small arc $AB$ of the equatorial circle of the round $S^2$ and replace it by minimizing geodesics connecting $A$ and $B$ to the north pole. Denote the resulting curve $\mathcal C$. Then the curves $S\_t$ south of $\mathcal C$ vary in length wildly for small $t$, violating your bound.
| 5 | https://mathoverflow.net/users/28128 | 273544 | 122,355 |
https://mathoverflow.net/questions/271256 | 3 | First, define a sequence $F\_0,F\_1,\dots$ of functions by
$$F\_0(x,z) = z,$$
$$F\_k(x,z)=x\exp\left(F\_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$
So, for example,
$$F\_1(x,z) = x e^z, \quad F\_2(x,z)=xe^{xe^z},\dots$$
etc. Also, set $F\_{-1}(x,z)=0$. Now, let
$$G(x,z) = \sum\_{k=0}^\infty \left(F\_k(x,z) - F\_{k-1}... | https://mathoverflow.net/users/39146 | Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property | The asymptotic distribution of the number of nodes at maximal height in a random tree is known.
The following was known as "Wilf's conjecture" (H. S. Wilf stated it in 1991, evidently unaware of the fact that it had been popularized among Russian combinatorialists
by V. F. Kolchin earlier):
Choose a labeled tree of... | 1 | https://mathoverflow.net/users/48831 | 273578 | 122,366 |
https://mathoverflow.net/questions/273575 | 4 | Let $k$ be an arbitrary field (in my case $k = \mathbb Q\_p$), and $G\subset \mathrm{GL}(n)\_{/k}$ a reductive group. Let $G^0$ be its identity connected component.
Suppose that $G^0(k)$ contains an element with pairwise distinct eigenvalues (in the natural representation $G(k)\subset \mathrm{GL}(n, k)$), and hence a... | https://mathoverflow.net/users/54339 | Distribution of regular elements in a disconnected algebraic group | No. Consider the group $H$ of $2\times 2$ matrices of the form $d\_t=\begin{pmatrix}t & 0 \\ 0 & t^{-1}\end{pmatrix}$ or $c\_t=\begin{pmatrix} 0 & u^{-1} \\ u & 0\end{pmatrix}$ with $t\in k^\*$. Let $G$ be the group of $4\times 4$ matrices of the form either $D\_{t,u}=\begin{pmatrix}d\_t & 0 \\ 0 & d\_u\end{pmatrix}$ o... | 3 | https://mathoverflow.net/users/14094 | 273582 | 122,367 |
https://mathoverflow.net/questions/273562 | 3 | Let $A$ be a noetherian ring, and let $\{I\_i \mid i \in J\}$ be a collection of K-injective complexes over $A$.
Is the direct sum
$$
\bigoplus\_{i \in J} I\_i
$$
also a K-injective complex over $A$?
Recall that a complex $I$ is called K-injective if for any acyclic complex $M$, the complex $\operatorname{Hom}\_A(M... | https://mathoverflow.net/users/111849 | Direct sum of K-injectives over a noetherian ring | No.
The $K$-injective complexes form a triangulated subcategory of the homotopy category of complexes, and so if they were also closed under coproducts, the homotopy colimit of a sequence of $K$-injective complexes would also be $K$-injective.
Let $A=\mathbb{C}[x]/(x^2)$ and let $X$ be the complex
$$\dots\stackrel{... | 3 | https://mathoverflow.net/users/22989 | 273592 | 122,370 |
https://mathoverflow.net/questions/273607 | 4 |
>
> Could You give your ideas, your comment, or a referen for a conjecture as follows:
>
>
>
Consider $A, B, C$ be three positive integers numbers. By [Fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) we write:
$A=a\_1^{x\_1}a\_2^{x\_2}...a\_n^{x\_n}$,
$B=... | https://mathoverflow.net/users/88319 | Another conjecture on sum $A+B=C$ | This conjecture is false, though only barely.
It should be true for $N\_0 \geq 4$ by the ABC conjecture.
But for $N\_0 = 3$ there are infinitely many counterexamples such as
$$
191114642^3 + 4309182809^3 = 7^4 \, 321817873^3.
$$
I obtained this by starting from the point
$P\_1 = (2:1:1)$ on the elliptic curve $x^3+... | 20 | https://mathoverflow.net/users/14830 | 273608 | 122,374 |
https://mathoverflow.net/questions/273602 | 2 | Let $f:X\rightarrow Y$ be a proper surjective morphism. Suppose X have only normal crossing singularity. Let H and E are two irreducible components of the singular locus of X i.e, $Sing(X)=H\cup E$. So clearly H and E are codimension 1 sub varieties. Suppose $f|\_{X\setminus E}$ is an isomorphism i.e f is isomorphism o... | https://mathoverflow.net/users/nan | Blowing up a singular variety | Yes, it is possible.
Well, it is kind of cheap, but if $f$ is not projective, then there is no chance of it being a blow-up, so you might as well say that $f$ is projective. Also, I assume you mean $f(E)$ with the reduced subscheme structure.
Let $Y$ be the quintessential example of a non-Cohen-Macaulay surface: T... | 3 | https://mathoverflow.net/users/10076 | 273610 | 122,375 |
https://mathoverflow.net/questions/273614 | 7 | This is a generalization of [an older question](https://mathoverflow.net/questions/273611/about-the-existence-of-a-convergent-sequence).
Let $(X,d)$ be a metric space and let $(A\_n)\_{n\in\mathbb{N}}\subseteq X$ be a sequence of non-empty closed subsets such that for all $\varepsilon > 0$ there is $N\in\mathbb{N}$ s... | https://mathoverflow.net/users/8628 | Hausdorff distance and Cauchy sequences | Yes. Choosing a subsequence $n\_1<n\_2<\dots$ such that $d\_H(A\_{n\_i},A\_{n\_i+1})\leqslant 2^{-i}$. Then inductively choose points $x\_{n\_i}\in A\_{n\_i}$ so that $d(x\_{n\_i},x\_{n\_{i+1}})\leqslant 2^{-i}$. It is Cauchy sequence by triangle inequality. For any index $m\notin \{n\_1,n\_2,\dots\}$ choose any $i$ su... | 6 | https://mathoverflow.net/users/4312 | 273629 | 122,384 |
https://mathoverflow.net/questions/273632 | 85 | In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example,
**Set theory**
*An injective map between two finite sets with the same cardinality is surjective.*
**Linear algebra**
*An injective linear map between two finite dimensional vector spaces of the same dimensio... | https://mathoverflow.net/users/6129 | Injectivity implies surjectivity | A famous result in this spirit is the [Ax-Grothendieck theorem](https://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem), whose statement is the following:
>
> **Theorem.** If $f \colon \mathbb{C}^n \to \mathbb{C}^n$ is an injective polynomial function then $f$ is bijective.
>
>
>
| 62 | https://mathoverflow.net/users/7460 | 273633 | 122,385 |
https://mathoverflow.net/questions/273580 | 9 | In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ [can be written as](https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry#Gradient.2C_divergence.2C_Laplace.E2.80.93Beltrami_operator):
$$
\Delta\_g = g^{ij}\partial\_{ij} - g^{jk}\Gamma^\ell\_{jk}\partial\_\ell
$$
The second... | https://mathoverflow.net/users/102881 | Intuition for the Drift Term of the Laplace-Beltrami Operator | You really should think of $\Delta$ as an $L^2$ self-adjoint, elliptic operator in it's own right irrespective of coordinates. $-\Delta$ has positive spectrum, with countable eigenvalues accumulating at $\infty$ etc. It's the average of the Hessian $\nabla^2 f$ in the sense that contracting the $(0, 2)$ tensor $\nabla^... | 3 | https://mathoverflow.net/users/78645 | 273644 | 122,390 |
https://mathoverflow.net/questions/273630 | 5 | I am trying to understand the non-commutative analysis for nilpotent Lie groups, so I've been reading Corwin's and Greenleaf's book on the representation theory of nilpotent groups and going through examples of the orbit method.
My problem is that I would like to apply the Fourier transform to the Lie algebra generat... | https://mathoverflow.net/users/97557 | Harmonic analysis on nilpotent Lie groups and the Campbell-Hausdorff formula | One friendly class comes from parabolic induction. So in case your nilpotent algebra is nilradical of a parabolic subgroup, there is a general formula that is straightforward to apply. See Theorem 1.3 of [Algebraic analysis on scalar generalized Verma modules of Heisenberg parabolic type I.: $A\_n$-series](https://arxi... | 2 | https://mathoverflow.net/users/6818 | 273646 | 122,391 |
https://mathoverflow.net/questions/273628 | 1 | if assume that $X$ is normed spaces, a closed subspace $Y\subset X$ is called $M-$Ideal, If there is a linear projection $P^\*:X^\*\to X^\*$ such that $ker(P)=Y^{\perp}$ and $$||x^\*||=||P(x^\*)||+||x^\*-P(X^\*)||$$
I'm looking an example of mentioned definition but I couldn't give an example in euclidean spaces with... | https://mathoverflow.net/users/31058 | Looking an example for following definition of M-ideal on normed spaces | I'm not sure what you mean by "euclidean spaces with different norm". Do you just mean a finite-dimensional normed space?
If you just want examples of M-ideals, then the motivating example is: all closed ideals in $C(X)$, where $X$ is compact Hausdorff, are M-ideals; and every M-ideal in $C(X)$ is a closed ideal in t... | 4 | https://mathoverflow.net/users/763 | 273663 | 122,397 |
https://mathoverflow.net/questions/273669 | 17 | In [this](https://groups.google.com/d/topic/comp.soft-sys.matlab/mwk1yXTuFWc/discussion) thread on MATLAB Central, I found a discussion on finding the nearest matrix with real eigenvalues. The first hypothesis was to simply truncate the complex part of the eigenvalues. So, given matrix $A$, the closest matrix to $A$ in... | https://mathoverflow.net/users/68336 | Finding the nearest matrix with real eigenvalues | I have no idea what is going on, but your conjecture is not correct. This is more transparent perhaps in the complex version. Consider
$$
A = \begin{pmatrix} i & a \\ 0&-i \end{pmatrix} , \quad a>0.
$$
This matrix is already in (complex) Schur form, and the obvious procedure to make the eigenvalues real and similar in ... | 12 | https://mathoverflow.net/users/48839 | 273677 | 122,401 |
https://mathoverflow.net/questions/273635 | 44 | Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say that a smooth $1$-form $\theta$ on $M$ is *adapted* to this flow if
1. $\theta(X)$ is everywhere positive; and
2. The Li... | https://mathoverflow.net/users/766 | Finding a 1-form adapted to a smooth flow | If I understand correctly, there is already a counterexample on the torus:
On the $xy$-plane $\mathbb{R}^2$, let $X$ be the vector field
$$
X = \sin x\,\frac{\partial\ }{\partial x} + \cos x\,\frac{\partial\ }{\partial y}.
$$
Now let $T^2=\mathbb{R}^2/\Lambda$ where $\Lambda$ is the lattice generated by $(2\pi,0)$ an... | 48 | https://mathoverflow.net/users/13972 | 273689 | 122,408 |
https://mathoverflow.net/questions/273670 | 37 | I don't know the following is a known result, but it would be very useful to me in my research if it were true.
**Conjecture:** Let $G$ be a planar graph. The sum
$$
\sum\_{\{x,y\} \in E(G)}{\min(\deg(x),\deg(y))}
$$
is at most linear in the number of vertices.
**What I know about this problem:**
* This conjectur... | https://mathoverflow.net/users/17599 | A conjecture on planar graphs | Let $L(G)=\sum\_{xy\in E(G)} \min\lbrace\deg(x),\deg(y)\rbrace$.
THM. For a simple planar graph with $n$ vertices, $L(G)\le 18n-36$ for $n\ge 3$.
PROOF. Recall that a simple planar graph with $k\ge 3$ vertices cannot have more than $3k-6$ edges, achieved by a triangulation. Let the degrees of the vertices be $d\_1\... | 43 | https://mathoverflow.net/users/9025 | 273694 | 122,410 |
https://mathoverflow.net/questions/273707 | 29 | We have the known [BBP(Bailey–Borwein–Plouffe)-type formulas](https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula),
$$\ln3 = \sum\_{n=0}^\infty\frac{1}{2^{2n}}\left(\frac{1}{2n+1}\right)$$
$$\ln5 = \frac{1}{2^2}\sum\_{n=0}^\infty\frac{1}{2^{4n}}\left(\frac{2^2}{4n+1}+\frac{2^2}{4n+2}+\frac{... | https://mathoverflow.net/users/12905 | Is this BBP-type formula for $\ln 257$ and $\ln 65537$ true? | Yes, this actually holds for all Fermat numbers! Let's start with the identity
$$\log 2=\sum\_{k=1}^{\infty}\frac{1}{k2^k}$$
and try to work out an expression for
$$\log (2^{2^s}+1)=2^{s}\log 2+\log\left(1-\frac{1}{2^{2^{s+1}}}\right)-\log\left(1-\frac{1}{2^{2^s}}\right)$$
$$=2^s\left(\sum\_{k=1}^{\infty}\frac{1}{k2^k... | 37 | https://mathoverflow.net/users/2384 | 273714 | 122,417 |
https://mathoverflow.net/questions/273718 | 0 | What is the solution for the following sum?
$\sum\_{i=1}^n i^{-\alpha}$ for $-2\leq \alpha\leq 0$.
For example if we have $\alpha =-1 $ we know it is the harmonic series thus $=\log n$ and for $\alpha=-2$ it is $\Theta(1)$. What if $\alpha$ ranges? Is there any more general formula?
How about if $\alpha>1$? Can we... | https://mathoverflow.net/users/148158 | Sum of series $\sum_{i=1}^n i^{-\alpha}$ | $$\sum\_{i=1}^n i^{-\alpha}=H\_{n,\alpha}$$
the [generalized Harmonic number.](https://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers) For $\alpha>1$ one has the limit
$$\lim\_{n\rightarrow\infty}H\_{n,\alpha}=\zeta(\alpha),$$
the [Riemann zeta function.](https://en.wikipedia.org/wiki/Riemann_zeta_f... | 1 | https://mathoverflow.net/users/11260 | 273719 | 122,419 |
https://mathoverflow.net/questions/273207 | 1 | Let $R$ be a commutative Noetherian ring and $M$ an $R$-module (not finitely generated). Are $\dim M$ and $\dim Supp(M)$ same?
| https://mathoverflow.net/users/111680 | On Krull dimension \dim M and \dim Supp(M) | Let $R$ be a ring and let $M$ be an $R$-module. We consider the topological space ${\rm Spec}(R)$. For a subset $Z\subseteq{\rm Spec}(R)$ we denote by $\dim(Z)$ its Krull dimension. In particular, we may consider the Krull dimensions of ${\rm Var}(0:\_RM)$ and of ${\rm Supp}\_R(M)$.
**Fact 1:** ${\rm Supp}\_R(M)\subs... | 2 | https://mathoverflow.net/users/11025 | 273727 | 122,422 |
https://mathoverflow.net/questions/273732 | -4 | Has nontrivial solution in positive integers of a [diophantine equation](https://en.wikipedia.org/wiki/Diophantine_equation) as follows ?
$$x\_1^2+x\_2^2+x\_3^2+x\_4^2=y\_1^2+y\_2^2+y\_3^2+y\_4^2$$
Where trivial solutions are $x\_i=y\_j$.
Can you send me any reference for this question?
| https://mathoverflow.net/users/88319 | Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$ | There are many non-trivial solutions. By [Lagrange's four-square theorem](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem), *every* natural number can be represented as a sum of $4$ squares (with $0^2$ allowed), and [Jacobi's four-square theorem](https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theor... | 5 | https://mathoverflow.net/users/2233 | 273734 | 122,424 |
https://mathoverflow.net/questions/273730 | 7 | Let $X$ and $Y$ be two algebraic varieties over $\mathbb{C}$ and $f\colon X\rightarrow Y$ be a proper map. Assume that $Y$ is smooth.
I am interested in sufficient and necessary conditions for $f$ to be topologically locally trivial on the target, i.e., whether for each $y\in Y$ there exists an open neighborhood $V$ ... | https://mathoverflow.net/users/85640 | Ehresmann's theorem for singular varieties | It seems that you are looking for a good notion of "equisingularity." About this I know quite little and I hope someone else will give a nice answer, but my guess is that a sufficient and necessary condition would be the existence of a horizontal Whitney stratification, at least locally on the source. Perhaps the book ... | 4 | https://mathoverflow.net/users/3847 | 273736 | 122,426 |
https://mathoverflow.net/questions/273699 | 3 | This is not a research level question, but I have not received any feedback on the other side, so I thought I'd try here. My apologies if the answer (counterexample) is a triviality: Suppose $(X,\mathscr A)$ is a measurable space, $Y$ is Polish and $f\_n:X\to Y$ is a sequence of measurable functions. Then the set $L\su... | https://mathoverflow.net/users/49155 | When is the set of points for which limit of $f_n $ exists measurable? | This may be the trivial counterexample you are looking for:
Let $A\subseteq \mathbb R$ be a nonmeasurable set.
Let $X=\mathbb R$, and let $Y = (\mathbb R \times \{1,1/2,1/3,\ldots\} )\cup (A \times \{0\})$ with the Euclidean metric.
Let $f\_n(x) = (x,1/n)$ for all $n\ge 1$. These functions are continuous.
The... | 7 | https://mathoverflow.net/users/14915 | 273742 | 122,427 |
https://mathoverflow.net/questions/273449 | 0 | Let's consider the propagator corresponding to the one-dimensional equation
$$
u\_t=\Lambda^\alpha u\_x,\; u(x,0)=f(x)
$$
where
$$
\widehat{\Lambda^\alpha u}=|\xi|^\alpha\hat{u}(\xi),
$$
and $-1< \alpha\leq 1$.
QUESTION 1) Do you know any good reference in the dispersive estimates for this operator? I mean inequali... | https://mathoverflow.net/users/33135 | Dispersive estimate for linear semigroup | First note that when $\alpha = 0$ you have the linear advection equation with no decay whatsoever, in either case, as $L^\infty$ is conserved. So we focus on the situation where $\alpha \neq 0$.
### Question 1
Since you have scaling homogeneity, it suffices to localize in frequency. Let $\chi\_\pm$ be bump functio... | 1 | https://mathoverflow.net/users/3948 | 273752 | 122,432 |
https://mathoverflow.net/questions/273737 | 4 | Given (not-necessarily linear) fibre bundles $E \to M$ and $F \to M$ over a smooth manifold $M$ with structure sheaf denoted $\mathcal{O}\_M$, a (not-necessarily-linear) differential operator taking sections of $E$ to sections of $F$ can be defined as a bundle map $D: J^\infty E \to F$ covering the identity on $M$, whe... | https://mathoverflow.net/users/56938 | Alternative definitions of infinite-order differential operators | I think your answer to your first question is correct. If I remember correctly, the definition of the sheaf of differential operators $\mathrm{Diff}(E,F)$ is exactly the one usually used in the theory of D-modules and I think that the only operators of infinite order arise from global sections.
To your second questi... | 2 | https://mathoverflow.net/users/6818 | 273766 | 122,439 |
https://mathoverflow.net/questions/222732 | 9 | Is there any approach for the following conjecture?
Let $X$ be a projective Fano manifold and let $Pic(X) \cong \mathbb Z$, then $(X, −K\_X) $ is $K$-semistable.
| https://mathoverflow.net/users/nan | Let $X$ be a projective variety and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable | The answer correspond to weighted stability of Fedor Bogomolov and result of Gang Tian. We explain it here as an additional comment to previous answer
Tian introduced a stronger notion of strong stability around 90 as follows and proved the following theorem albeit strong stability does not satisfied for any Fano man... | 9 | https://mathoverflow.net/users/nan | 273774 | 122,445 |
https://mathoverflow.net/questions/273040 | 4 | Let $G$ be a finite group and let $C$ be a cyclic subgroup such that $N\_G(C) = C$. Such a subgroup is in particular a Carter subgroup and Vdovin (\*) has shown that any two such subgroups are conjugate in an arbitrary finite group. Moreover, Zhang (\*\*) showed that a group with cyclic self-normalizing subgroup cannot... | https://mathoverflow.net/users/15302 | Finite groups with cyclic self-normalizing subgroup | $\mathrm{P\Sigma L}(2,8)$ is a counterexample. All its cyclic groups of order $6$ are self-normalising. (They fall into two conjugacy classes.)
(This the automorphism group of $\mathrm{PSL}(2,8)$, it is $\mathrm{PSL}(2,8)$ extended by a field automorphism.)
EDIT: Examples do not seem to be plentiful. The above is t... | 5 | https://mathoverflow.net/users/22377 | 273777 | 122,448 |
https://mathoverflow.net/questions/215998 | 7 | In Toen's *Affine Stacks*, he computes that $\mathcal{O}(B\mathbb{G}\_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we choose dg algebras as a model for derived rings). To prove this, he explicitly constructs a cosimplicial ring whose $\te... | https://mathoverflow.net/users/6059 | Derived global functions on (derived) stacks $BG$ and $G/G$ | For a smooth groupoid $G\_1\rightrightarrows G\_0$ , here is a way to compute derived global sections $\mathcal O(X)$ of $\mathcal O\_X$, where $X=[G\_0/G\_1]$.
Observe that $X$ is the homotopy colimit of the nerve $G\_\bullet$ of the groupoid. As you said then $\mathcal O(X)$ is the homotopy limit of $\mathcal O(G\... | 4 | https://mathoverflow.net/users/7031 | 273778 | 122,449 |
https://mathoverflow.net/questions/273798 | 1 | This is a follow-up to [an older question](https://mathoverflow.net/questions/273614/hausdorff-distance-and-cauchy-sequences).
Let $(X,d)$ be a metric space and let $(A\_n)\_{n\in\mathbb{N}}\subseteq X$ be a sequence of [bounded](https://en.wikipedia.org/wiki/Bounded_set#Metric_space) non-empty subsets of $X$ such th... | https://mathoverflow.net/users/8628 | Cauchy subsequences in "Hausdorff Cauchy sets" | No. The unit sphere in any infinite dimensional normed space contains a sequence of elements that all have distance $1/2$ between them by [Riesz's lemma](https://en.wikipedia.org/wiki/Riesz%27s_lemma). So this fails even if the sequence $\langle A\_n\rangle$ is constant.
| 2 | https://mathoverflow.net/users/35357 | 273799 | 122,457 |
https://mathoverflow.net/questions/273788 | 1 | On $L^p(\mathbb{R}^+)$, we consider the following operator:
$$
Af:= f'',\qquad D(A):=\{u\in W^{2,p}(\mathbb{R}^+),u'(0)=0\}
$$
Now I want to know if this operator is a generator of a positive analytic semigroup of contractions. Can anybody give me a hint or references about this question.
| https://mathoverflow.net/users/103083 | Positive analytic semigroup | You can check directly that it $A$ dispersive and that the numerical range is on a half-line, i.e.,
$$\langle Af,f^+\rangle \leq 0$$
holds for all $f\in D(A)$. You have to use integration by part and the fact that $W^{1,2}$ is a lattice. See for example Chapter 11.3 in
*Bátkai, András; Kramar Fijavž, Marjeta; Rhandi... | 2 | https://mathoverflow.net/users/12898 | 273800 | 122,458 |
https://mathoverflow.net/questions/273805 | 2 | Let X be an integral quasi projective variety. Let Y be the normalisation. Given a line bundle on Y, does some power of it descends to X?
| https://mathoverflow.net/users/nan | Descent of a line bundle from normalisation | Posting the comment above as an answer:
The answer is yes for curves and no in higher dimension.
For curves, we have a short exact sequence $$0\to \mathcal{O}\_X^\*\to \pi\_\*\mathcal{O}\_Y^\*\to Q\to 0 $$
where $Q$ is supported in finitely many points. In particular, $H^1(X, Q)=0$. Moreover, $R^1\pi\_\* \mathcal{O... | 5 | https://mathoverflow.net/users/3847 | 273811 | 122,460 |
https://mathoverflow.net/questions/273802 | 3 | For a local field $E$, denote by $U(E)$ the units of the corresponding valuation ring $\mathcal{O}\_E$, and denote by $U\_n(E)$ the prinicipal $n$-units, i.e. $U\_n(E)=1+M\_E^n$ where $M\_E$ is the maximal ideal of $\mathcal{O}\_E$.
Now let $L/K$ be a finite unramified extension of local fields, of characteristic $0$... | https://mathoverflow.net/users/100534 | Inverse image of norm map on principal units for an unramified extension | Let $\pi$ be a uniformizer for $K$ and $L$.
For sufficiently large $m$, the $\log$ and $\exp$ maps give isomorphism between $U\_m(L)$ and $\pi^m \mathcal{O}\_L$. Same for $K$. This $m$ should be thought as a fixed "small" integer.
These maps commute with Galois action, i.e. for any $\sigma \in Gal(\overline{K}/K)$,... | 2 | https://mathoverflow.net/users/76332 | 273814 | 122,461 |
https://mathoverflow.net/questions/273810 | 41 | Given a group $G$, its centre $Z(G)$ is a normal subgroup of $G$ and one can consider the quotient $G/Z(G)$.
Which groups $H$ can occur as $G/Z(G)$ for some group $G$?
For example, it is easily proved that $G/Z(G)$ cannot be cyclic unless it is trivial.
A certain amount of general literature survey and google s... | https://mathoverflow.net/users/50650 | Which groups can occur as the quotient of a group by its centre? | In the literature the following terminology is common. If $H$ is a group such that $H \cong G/Z(G)$ for some $G$, we say that $H$ is *capable*. As you mention, non-trivial cyclic groups are not capable. Another example is the quaternion group of order $8$, which is not capable.
The earliest paper on this seems to be ... | 52 | https://mathoverflow.net/users/38068 | 273818 | 122,462 |
https://mathoverflow.net/questions/273274 | 3 | Recall Lemma 0.5.1 from the Helemskii's monograph "The homology of Banach and Topological Algebras":
$\textbf{Lemma}$ Let $\phi\colon X\to Y$ be an injective map between Banach spaces with dense range. If the dual map $\phi^\*$ is surjective then so is $\phi$ itself.
My question is: what if $X,Y$ are complete DF-sp... | https://mathoverflow.net/users/75127 | DF-algebras and DF-modules | Here is a counterexample: For every countable inductive limit $E=\lim E\_n$
of Banach (or locally convex) spaces one has (either by definition or a simple construction) an algebraically exact sequence $0\to \bigoplus E\_n \to \bigoplus E\_n \to E\to 0$ where the map $\phi$ on the direct sum is
defined by $(e\_n)\_{n\in... | 2 | https://mathoverflow.net/users/21051 | 273822 | 122,465 |
https://mathoverflow.net/questions/256815 | 5 | Let $(\Omega, \mathcal{A},P)$ be a probability space, and let $(\mathcal{F}\_k)\_{k \geq 1}$ be a filtration which converges to $\mathcal{A}$. I suppose it is true that
$$
E \left( \big(E \left( X | \mathcal{F}\_k \right) \big)^2 \right) \to E \left(X^2 \right).
$$
How to prove this? I guess one will need Jensen's ine... | https://mathoverflow.net/users/46852 | Convergence of conditional second moments | Let us state Corollary 2.1 of [these notes](http://www.statslab.cam.ac.uk/~ps422/mynotes.pdf).
>
> Let $p>1$, $X\in\mathbb L^p$ and let $\left(\mathcal F\_n\right)\_{n\geqslant 1}$ be a filtration. Denote by $\mathcal F$ the $\sigma$-algebra generated by $\bigcup\_{n\geqslant 1}\mathcal F\_n$. Then the convergence ... | 5 | https://mathoverflow.net/users/17118 | 273840 | 122,471 |
https://mathoverflow.net/questions/273678 | 4 | This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum\_{|\alpha|\le m} a\_{\alpha}(x)D^{\alpha}$ has by definition the symbol $P\_m(x,\xi)=\sum\_{|\alpha|=m} a\_{\alpha}(x)\xi^... | https://mathoverflow.net/users/57155 | Symbol of the Laplace-Beltrami on $\mathbb{S}^2$ | Your expression above is correct. To see this, note that every Riemannian metric $g=g\_{ij} \, dx^i \, dx^j$ in any coordinate system on any manifold $M$ has "dual metric" a function on the cotangent bundle, given as $g^\*(\xi) = g^{ij} \xi\_i \xi\_j$ for any $\xi=\xi\_i dx^i \in T^\* M$, where $(g^{ij})$ is the invers... | 6 | https://mathoverflow.net/users/13268 | 273841 | 122,472 |
https://mathoverflow.net/questions/273831 | -1 | The distribution $g(x)$ has the following properties:
$$\int\_{-\infty}^{\infty}g'(x)f(x)dx=f(\pi g(0)-1/2)-f(\pi g(0)+1/2)$$
for any analytic $f(x)$.
How can I find $g(x)$?
| https://mathoverflow.net/users/10059 | Are there solutions for this functional equation? | If your equation extends to smooth test functions, this is just $g'=\delta\_{\pi g(0)-1/2}-\delta\_{\pi g(0)+1/2}$, which is equivalent to $g=c+\mathbb1\_{[\pi g(0)-1/2\ ,\ \pi g(0)-1/2]}$ for some $c$ . If you chose $c$ such that $|c|>\frac1{2\pi}$, then $g=c+\mathbb1\_{[c\pi-1/2\ ,\ c\pi-1/2]}$ is a solution.
| 1 | https://mathoverflow.net/users/75422 | 273842 | 122,473 |
https://mathoverflow.net/questions/273298 | 6 | I'm in the following situation:
Consider a centred Gaussian measure $\mu\_0$ on a separable Hilbert space $X$ with covariance operator $Q \in \mathcal{L}(X)$ (positive definite, self-adjoint, trace class). Denote the Cameron-Martin space of $\mu\_0$ by $E := Q^\frac12(X)$ with $\|\cdot\|\_E := \|Q^{-\frac12}\cdot\|$... | https://mathoverflow.net/users/111726 | Probabilities of small balls with convergent center points under Gaussian measure | Masoumeh Dashti kindly provided the following proof, which I reproduce here in my words and notation:
Let $u \mapsto W\_u$ denote the unique linear extension of $E \to L^2(X,\mu\_0)$, $u \mapsto \widetilde{W}\_u = (Q^{-\frac12}u,\cdot)\_X$, to $X$.
We first note that $Z := Q(X)$ is dense in $E = Q^{\frac12}(X)$, and ... | 2 | https://mathoverflow.net/users/111726 | 273847 | 122,475 |
https://mathoverflow.net/questions/273835 | 3 | According to the papers [The absolutely continuous spectrum of Jacobi matrices](https://arxiv.org/abs/0706.1101) and [these lecture notes](https://nms.kcl.ac.uk/alexander.pushnitski/ebd65/simon.pdf):
* periodicity ~ potential well or lattice (order)
* lack of absolutely continued spectrum ~ Anderson localization(diso... | https://mathoverflow.net/users/14024 | what is about the corresponding power series? | Quasiperiodic potentials, such as the [Fibonacci chain](https://en.wikipedia.org/wiki/Fibonacci_quasicrystal), have a spectrum that is called [singular continuous](https://en.wikipedia.org/wiki/Singular_distribution), with a fractal structure, see
* [Fractal energy spectrum of a polariton gas in a Fibonacci quasi-per... | 3 | https://mathoverflow.net/users/11260 | 273857 | 122,478 |
https://mathoverflow.net/questions/272239 | 8 | Let $R$ be a Cohen-Macaulay noetherian local ring. Let $\Lambda$ be a noetherian $R$-algebra which is maximal Cohen-Macaulay as an $R$-module, where for every nonmaximal prime $\mathfrak{p}$, $\Lambda\_{\mathfrak{p}}$ has finite global dimension. How can I prove that every $\Lambda$-module which is maximal Cohen-macaul... | https://mathoverflow.net/users/111161 | For isolated singularity algebra, is every maximal Cohen-Macaulay module locally projective? | The answer is no. Because locally, you can have a module over $\Lambda$ which is mCM over $R$ but not free. There are examples of singular $R$ with a module $M$ such that $\Lambda= End\_R(M)$ has finite global dimension. But $M$ itself is a $\Lambda$-module and if it is free, then $R$ is Morita equivalent to $\Lambda$,... | 5 | https://mathoverflow.net/users/2083 | 273864 | 122,481 |
https://mathoverflow.net/questions/273745 | 8 | I was trying to calculate the number of points of the curve $E:y^2 = x^3 + 4x^2 + 2x$ over $\mathbb{F}\_p$ for $p\equiv 1\bmod 8$ (In order to have $\sqrt{-2}\in\mathbb{F}\_p$) but I did not succeed. This curve is mentioned in Silverman's Advanced Topics in the Arithmetic of Elliptic curves (Proposition 2.3.1) to have ... | https://mathoverflow.net/users/91023 | Number of points of elliptic curve over $\mathbb{F}_p$ with CM by $\sqrt{-2}$ when $p\equiv 1\bmod 8$ | Sorry for self-advertisement: the question of the "sign" that one must choose
(equivalently which $\pi$ or $\overline{\pi}$) is entirely answered in my
book GTM239, Section 8.5.2, some of the results being due to Mark Watkins
(unpublished I believe). The main idea is to show that any CM curve is
equivalent in an eviden... | 7 | https://mathoverflow.net/users/81776 | 273867 | 122,482 |
https://mathoverflow.net/questions/273823 | 3 | Consider a real vector space $V$ with dimension $n$ (say $V=\mathbb{R}^n$). The construction I'm describing is similar to the construction of the projective space. Instead of the space of lines, it is the set of (affine) halfspaces, or equivalently the space of "oriented" (affine) hyperplanes.
Define $A$ the space of... | https://mathoverflow.net/users/111959 | Space of halfspaces | The space is just $\mathbb{R} \times S^{n-1}$: each oriented hyperplane $H$ is identified by the normal unit vector $v$ together with the inner product of $v$ with an arbitrary $w \in H$.
| 6 | https://mathoverflow.net/users/39521 | 273874 | 122,486 |
https://mathoverflow.net/questions/273862 | 7 | Let $\Omega\_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega\_n^O\rvert$. What are the asymptotics of $d(n)$?
It's possible to express $\log\_2 d(n)$ as a modified partition function: by work of Thom, $\Omega\_n^O$ is an $\mathbb F\_2$... | https://mathoverflow.net/users/97265 | What is the growth rate of the number of unoriented cobordism classes? | If you denote by $P'(n)$ the number of partitions of $n$ with parts not of the form $2^k-1$, it holds that
$$\log P'(n)\sim \pi \sqrt{\frac{2n}{3}}$$
just like for the usual partition function. This will be true whenever you allow the parts to come from a set of integers of asymptotic density 1. See for example Nathans... | 10 | https://mathoverflow.net/users/2384 | 273876 | 122,487 |
https://mathoverflow.net/questions/272264 | 10 | Let $T$ be a first order theory, $M$ its elementary class and consider its forgetfull functor $$M \stackrel{U}{\to} \text{Set}. $$
I would like to characterize those theories such that $U$ has a left adjoint. In fact this can be done.
---
>
> If $M$ is cocomplete then $U$ has a left adjoint iff $T$ is equival... | https://mathoverflow.net/users/104432 | Free objects in first order theories | Here's a second attempt at answering the "homomorphism" version of Q3.
**Proposition:** Suppose that $U: \mathrm{Hom}(T) \to \mathsf{Set}$ has a left adjoint $F$. Expand $T$ by those definable functions in $T$ which are preserved by all homomorphisms between models of $T$ to obtain a theory $T'$. Let $V(T')$ be the v... | 1 | https://mathoverflow.net/users/2362 | 273878 | 122,489 |
https://mathoverflow.net/questions/273762 | 13 | Let $k$ be a field of positive characteristic and let $G$ be a connected semisimple algebraic group over $k$ with fundamental group $\mu$. Note that $\mu$ can be non-smooth. It is stated in [Sansuc's 1981 Crelle paper *Groupe de Brauer et arithmétique des groupes algébriques...*](https://www.digizeitschriften.de/dms/im... | https://mathoverflow.net/users/5641 | The Picard group of a semisimple algebraic group in positive characteristic | I'm not sure if there is really a literature gap, or maybe it is addressed in some reference that just hasn't come to mind. Anyway, let $G$ be a perfect smooth connected affine group over a field $k$, so $G\_K$ has no nontrivial $K$-homomorphism to ${\rm{GL}}\_1$ for any extension field $K/k$, and hence $K[G]^{\times} ... | 10 | https://mathoverflow.net/users/81332 | 273883 | 122,493 |
https://mathoverflow.net/questions/273789 | 6 | I have been struggling with following Heat equation IBVP,
\begin{equation}
\frac{\partial v\left(x, t\right)}{\partial t} = \alpha \frac{\partial^2 v\left(x, t\right)}{\partial x^2}, \quad t \in \left(0, T\right], x \in \left(- \infty, 1\right]. \label{heatEqn}
\end{equation}
with the following initial and boundary con... | https://mathoverflow.net/users/18929 | Heat Equation with an integral boundary condition | Here is the answer to your modified question via potential approach but simpler than proposed in the comments.
Denote $Z(x,t)=(4\alpha\pi t)^{-1/2} e^{-x^2/(4\alpha t)}$ the fundamental solution for the heat equation. I'll write $t$ instead of $T$ in the question. Subtracting $Z(x,t)$ from $v$ reduces the initial con... | 2 | https://mathoverflow.net/users/14551 | 273903 | 122,498 |
https://mathoverflow.net/questions/273888 | 6 | I'm writing a paper about a math puzzle and the thing I'm studying ends up equivalent to finding the following parameter of a bipartite graph G with parts X and Y:
>
> The largest $k$ such that any $k$ vertices in $X$ all have a common neighbor.
>
>
>
I'm trying to figure out if this parameter has a name and i... | https://mathoverflow.net/users/57139 | Looking for the name or reference regarding a bipartite graph parameter | The problem is equivalent to the [set cover problem](https://en.wikipedia.org/wiki/Set_cover_problem) and thus NP-hard.
The set formulation you gave is:
>
> Given a finite set $\mathcal{X}$, a family $\mathcal{F} \subseteq \mathcal{P}(\mathcal{X})$, what is the largest $k$ such that the intersection of every $k$-... | 3 | https://mathoverflow.net/users/48859 | 273909 | 122,500 |
https://mathoverflow.net/questions/273919 | 13 | I am porting this question across from StackExchange, since it has received no answers and perhaps is sufficiently deep to fit here.
I am considering the set of upper triangular matrices
$$D\_N=\left\{\begin{pmatrix}a&b\\0&d\end{pmatrix}:a,b,d\in\mathbb{Z}, ad=N, \gcd(a,b,d)=1, 0\leq b<d\right\}.$$
Standard facts te... | https://mathoverflow.net/users/112012 | Action of SL(2,Z) on upper triangular primitive integer matrices of determinant N, from the right. Is it transitive? | Yes, the action is transitive. It is easier not to choose a set of coset representatives for the left action, so the question is whether $SL\_2(\mathbb{Z}) \times SL\_2(\mathbb{Z})$ acts transitively on $M\_N$ (under left and right action). We will want the following notations: Let $\tilde{M}\_N$ be **all** integer mat... | 13 | https://mathoverflow.net/users/297 | 273920 | 122,504 |
https://mathoverflow.net/questions/273892 | 3 | I have a set of nodes in a very large graph which I call Cluster Points. I also have for each point in the graph, the distance from each point in the Cluster point set.
For example: `node_id: 2493` is 5 hops (distance) away from the first cluster point, 2 hops from the second and 8 hops from the third.
What I woul... | https://mathoverflow.net/users/111701 | Finding the farthest point from a set of other points | The problem of choosing a sequence of nodes in a graph in which each is approximately as far as possible from the previously chosen ones was considered in my preprint with Har-Peled and Sidiropoulos, [arXiv:1507.01555](https://arxiv.org/abs/1507.01555). We provided a randomized $(1+\epsilon)$-approximation (that is, ea... | 3 | https://mathoverflow.net/users/440 | 273921 | 122,505 |
https://mathoverflow.net/questions/273894 | 3 | In [this post](https://mathoverflow.net/questions/273707/is-this-bbp-type-formula-for-ln-257-and-ln-65537-true), a binary BBP-type formula for ***Fermat numbers*** $F\_m$ was discussed as (with a small tweak),
>
> $$\ln(2^b+1) = \frac{b}{2^{a-1}}\sum\_{n=0}^\infty\frac{1}{(2^a)^n}\left(\sum\_{j=1}^{a-1}\frac{2^{a-1... | https://mathoverflow.net/users/12905 | Is this BBP-type formula for $\ln 31$, $\ln 127$, and other Mersenne numbers also true? | I'll give a proof of the corrected version of your conjectured series expansion and then comment at the end about the limits of this method for finding base 2 BBP formulas for $\log n$.
Let's denote $\alpha\_b=2^b-2-b\lfloor\frac{2^b-2}{b}\rfloor$, notice that $\alpha\_b=0$ for all primes $b$ but it can be nonzero fo... | 7 | https://mathoverflow.net/users/2384 | 273928 | 122,509 |
https://mathoverflow.net/questions/273905 | 5 | Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, is $\gamma$ a proper map?
The only counter-example I can think of is a curve looping back to itself such that $\lim\_{t... | https://mathoverflow.net/users/111997 | Flow of a nowhere vanishing complete vector field | I assume that your vector field is at least continuous.
I will answer your second question. Such a situation cannot arise.
Assume that $\gamma(t)$ converges to some point $p\in M$. Then, $X(\gamma(t))$ converges to $X(p)$, so $\gamma'(t)$ converges to some vector $a$. Using local coordinates, you can assume that $M=\ma... | 4 | https://mathoverflow.net/users/111917 | 273941 | 122,512 |
https://mathoverflow.net/questions/273940 | 4 | Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an integrable system result in quasi-periodic (and, in particular, not periodic) orbits
But fixed points of the diffeo... | https://mathoverflow.net/users/86065 | Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus | 1) is correct: by saying “hamiltonian diffeomorphism of” you imply that the torus has a symplectic structure $\omega$ and the diffeo is (something like) time 1 flow of a hamiltonian vector field $X$: $\omega(X,\cdot)=-dH$. As the torus is compact, $H$ has extrema, so $X$ has zeros, which the flow fixes.
2) is not: th... | 5 | https://mathoverflow.net/users/19276 | 273950 | 122,516 |
https://mathoverflow.net/questions/273953 | 0 | Up to how many subsets of $\{1,2,\dots,2n\}$ of size $n$ can we choose so that each pair has an intersection of size at least two?
The original Erdős-Ko-Rado [paper](https://www.renyi.hu/~p_erdos/1961-07.pdf) shows that taking all subsets that contain two fixed elements (thus yielding $\binom{2n-2}{n-2}$ subsets) is ... | https://mathoverflow.net/users/83212 | Erdős-Ko-Rado with intersections of size at least two | The family sets such that $\lvert A\cap B\rvert\geq t$ for every pair of distinct sets $A,B$ is called *$t$-intersecting*. The harder problem of finding the largest $t$-intersecting $k$-uniform family was solved by [Ahlswede and Khachatrian](http://www.ams.org/mathscinet-getitem?mr=1429238). The problem that you are in... | 4 | https://mathoverflow.net/users/806 | 273955 | 122,519 |
https://mathoverflow.net/questions/273583 | 5 | The conjecture 51 in Levine's Mixed Motives in Handbook of K-theory
<http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf>
states that the functor induced by $hs:\text{ECM} \rightarrow \text{MHS}$, denoted by,
\begin{equation}
\mathfrak{hs}:\text{NMM}(k)\_{\mathbb{Q}} \rightarrow \text{MHS}\_{\mathbb{Q}}
\e... | https://mathoverflow.net/users/87910 | Nori's Mixed Motives and Realisation Functors | Did you read <http://home.mathematik.uni-freiburg.de/arithgeom/preprints/buch/buch-v1.pdf>?
About 1: since the forgetful functor $MHS\_{\mathbb{Q}}\to \mathbb{Q}-Vect$ is exact and faithful, it suffices to verify that the singular cohomology functor from Nori motives is exact. The latter statement should be an immedi... | 3 | https://mathoverflow.net/users/2191 | 273961 | 122,521 |
https://mathoverflow.net/questions/273967 | 7 | 1. When does $\det(A+B)=\det(A)+\det(B)$ hold?
2. I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
| https://mathoverflow.net/users/10035 | When does the determinant distribute over addition? | let me assume $A$ is invertible, then you ask when
$$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$
so if $X$ has eigenvalues $x\_i$, $i=1,2,\ldots n$, you would need
$$\prod\_{i}(1+x\_i)=1+\prod\_i x\_i$$
basically you can take arbitrary values for $x\_1,x\_2,\ldots x\_{n-1}$ and then the only requirement is that
$$x\_n=\frac{1... | 15 | https://mathoverflow.net/users/11260 | 273972 | 122,522 |
https://mathoverflow.net/questions/273970 | 4 | What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C\_1,C\_2$ of the system surrounds an annular region $R$ such that $R$ does not contain any singular point and the flow orientation of $C\_1$ is opposite to the flow-orientation of $C\_2$.
Th... | https://mathoverflow.net/users/36688 | A cubic system with two nested limit cycles with opposite orientations | It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take
$$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\end{align}
$$
The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field ... | 10 | https://mathoverflow.net/users/13972 | 274981 | 122,529 |
https://mathoverflow.net/questions/273907 | 4 | I have a question about Dynkin **Hunt** formula.
Last day, I found a formula in this paper [enter link description here](https://pdfs.semanticscholar.org/06bb/40bf01bee9cecaea7aa2e9e7cda7cb5c35e9.pdf).
The formula is the equation (2.5) in this paper, which is called Dynkin **Hunt** formula. I know Dynkin formula. Dyn... | https://mathoverflow.net/users/68463 | Dynkin Hunt formula | The authors of the paper you cite mis-state the matter slightly. Dynkin's formula is indeed a direct consequence of the strong Markov property. Your Laplace inversion argument (with no assumption of continuity) shows that for each fixed $x$,
$$
p(t,x,y)=p^{U}(t,x,y)+E\_{x}[1\_{\{\tau\_{U} \le t\}}p(t-\tau\_{U},x,y)]
$$... | 2 | https://mathoverflow.net/users/42851 | 274982 | 122,530 |
https://mathoverflow.net/questions/274985 | 1 | In his Functional Analysis book (2nd ed) Rudin asserts on page 203 that
$$
(1+|y|)^{2r} < (2n+2)^r (1 + y\_1^{2r}+...+y\_n^{2r})
$$
where $r$ is a positive integer and $|y|$ is the Euclidean norm of the real vector $y = (y\_1,...,y\_n)$. Can anyone see how to prove this?
| https://mathoverflow.net/users/112053 | simple inequality in Rudin | Recall the power mean inequality, which, in particular, tells you that
$$ \frac{\sum\_{i=1}^{m} x\_i}{m} \le \sqrt[s]{\frac{\sum\_{i=1}^{m} x\_i^s}{m}},$$
for any $s \ge 1$ and any non-negative $x\_i$. Rudin's inequality is a double application of this. First apply it with $m=2, s=2, x\_1=1, x\_2 =|y|$:
$$(1+|y|)... | 8 | https://mathoverflow.net/users/31469 | 274986 | 122,531 |
https://mathoverflow.net/questions/274994 | 10 | For $X/{\sim}$ a quotient space,
$$
Map(X/{\sim},Y)\subset Map(X,Y).
$$
But is this inclusion always a homeomorphism on its image? (Assuming compact-open topology on the mapping spaces.) If not what would be the most general setting to make it true? We can also assume that $X$ and $Y$ are compactly generated.
A rela... | https://mathoverflow.net/users/9800 | Mapping space from a quotient space | The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the ... | 6 | https://mathoverflow.net/users/61536 | 275007 | 122,539 |
https://mathoverflow.net/questions/273828 | 50 | Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available [here](http://www.ams.org/samplings/math-history/hmath1-gorenstein33.pdf)).
>
> During that year in Harvard, Thompson began his monumental classification of the minimal simple groups. He soon... | https://mathoverflow.net/users/38068 | What is Atiyah's topological formulation of the odd order theorem? | In an email correspondence with Atiyah, I brought this up. The comments are meant to provide background for Thompson's remark.
*"When I first heard of FT I thought there should be a simpler proof using fixed point theorems and K-theory and I propagated the idea. The problem was that fixed point theorems could only d... | 30 | https://mathoverflow.net/users/12310 | 275009 | 122,540 |
https://mathoverflow.net/questions/274984 | 1 | Let $M $ be a manifold, and $\sf X, \sf Y$ vecor fields with the same set of zeros: i.e., for every $x\in M$, $\mathsf{X}(x)=0$ iff $\mathsf{Y}(x)=0$. Denote $V=\{ x\in M \mid \mathsf{X}(x)=0 \}$.
>
> Is there a diffeomorphism $\varphi:M\to M$ such that $\varphi\_{\*}\sf X = \sf Y$?
> What if $\varphi$ is require... | https://mathoverflow.net/users/4721 | Is the action of diffeomorphisms transitive on the set of vector fields with prescribed zero locus? | Let me summarize the conclusions in the comments. This is rarely true:
There are local obstructions around fixed points: For example the order of vanishing (Andreas Cap), or the Hopf index of an isolated equilibrium (more generally the Conley index) must be preserved by the diffeomorphism.
There are also global ob... | 6 | https://mathoverflow.net/users/12156 | 275023 | 122,546 |
https://mathoverflow.net/questions/274999 | 8 | I am looking for modern textbooks, expository papers and journal articles regarding the use of Lie algebras in combinatorics. In particular, I am interested to know the extent to which people still care about this.
For example, the representation theory of $\mathfrak{sl}\_2(\mathbf{C})$ can be used to show that certa... | https://mathoverflow.net/users/111804 | Modern sources on Lie algebraic methods in combinatorics | * A lot of the literature on crystals (in the modern sense of algebraic combinatorics) uses the theory of crystal bases (which is part of quantum group theory) as a black box. For one example of such a use, see [Mark Shimozono, *Crystals for dummies*](http://www.aimath.org/WWN/kostka/crysdumb.pdf), which proves various... | 6 | https://mathoverflow.net/users/2530 | 275028 | 122,549 |
https://mathoverflow.net/questions/275032 | 4 | There is possibly a huge literature on the subject but I am a newcomer on analytic representations and my need is rather specific. I simplify it below.
Let $A,B$ be two symbols (standing for annihilation and creation) and set
$$
H\_\mathbb{C}:=\frac{\mathbb{C}<A,B>}{<[A,B]-1>}=<A,B;[A,B]=1>\_{\mathbb{C}-AAU}
$$
wh... | https://mathoverflow.net/users/25256 | Representation of Heisenberg-Weyl elements and their exponentials | The *Heisenberg-Weyl algebra* or the *Weyl algebra* or *the algebra of the Canonical Commutation Relations (CCR)* is generated by the $p,q$ generators subject to the relation
$$
[q, p] = i \hbar I \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
$$
This is isomporphic to the algebra of the OP under the "cha... | 6 | https://mathoverflow.net/users/85967 | 275048 | 122,558 |
https://mathoverflow.net/questions/275049 | 3 | Let $[n] = \{1, \ldots, n \}$, and let $S(n,k)$ be the family of $k$-element subsets of $[n]$.
A cyclic word $w$ over alphabet $[n]$ is called a *universal cycle for* $S(n,k)$ if each element of $S(n,k)$ appears in $w$ as a factor (continuous subword) **exatly once**.
---
**Examlpe**
$n=8$, $k=3$
$w = 02456... | https://mathoverflow.net/users/83519 | Universal cycles for $k$-sets of an $n$-set | I believe the conjecture is still open. See the comment on Problem 476 and references therein for partial results in the paper below.
>
> Jackson, Brad; Stevens, Brett; Hurlbert, Glenn. [Research problems on Gray codes and universal cycles.](https://pdfs.semanticscholar.org/6844/94b7d88c34caf8c9e611cb3f73c1992b3543... | 2 | https://mathoverflow.net/users/51668 | 275051 | 122,560 |
https://mathoverflow.net/questions/275016 | 5 | Work over a model of $\sf GCH$. Suppose that $\kappa$ is some regular cardinal, and $U$ is a uniform ultrafilter on $\kappa^+$.
Does $U$ have some canonical extension after forcing with $\operatorname{Add}(\kappa,\kappa)$?
(Alternatively, replace $\kappa^+$ by an arbitrary regular $\lambda>\kappa$.)
| https://mathoverflow.net/users/7206 | Extending ground model ultrafilters | At Asaf's suggestion, I'm posting my earlier comment as an answer; I'll also append something about Asaf's subsequent question.
I can't imagine any of the extensions deserving the name "canonical". Each of the subsets $A$ of $\kappa$ that you've added produces a subset of $\kappa^+$, by chopping the ordinal $\kappa^+... | 4 | https://mathoverflow.net/users/6794 | 275060 | 122,565 |
https://mathoverflow.net/questions/273551 | 10 | I'm looking for a text that treats category theory from the perspective of MK class theory.
MK is already very well-designed and equipped for the type of abstraction that occurs in category theory, and I suspect that this together with its ability to discuss proper classes with relative ease would make a treatment in... | https://mathoverflow.net/users/92164 | Category theory from MK class theory perspective? | Morse-Kelley set theory doesn't seem adequate for all the things one would like to do in category theory. It provides a nice treatment of proper classes, so it can deal with large categories like the category of sets or the category of groups. It can deal with a functor between two such categories, like the forgetful f... | 19 | https://mathoverflow.net/users/6794 | 275062 | 122,567 |
https://mathoverflow.net/questions/273511 | 6 | Let $\mathcal L$ be a bounded modular lattice. Suppose also that $\mathcal L$ is complete and upper-continuous (i.e. for any directed subset $\{x\_i:i\in I\}$ of $\mathcal L$ and any $x\in \mathcal L$,
$x\wedge\bigvee\_{i\in I}x\_i=\bigvee\_{i\in I}(x\wedge x\_i)$).
Is there any known sufficient condition for $\math... | https://mathoverflow.net/users/24891 | Functions from a lattice to $[0,1]$ | Mathematical economists have been interested in special cases of the following general question, which they call "existence of utility representations."
$\quad$ Given $(X, \precsim)$ an ordered set, when does there exist a strictly increasing function $f: X \to \mathbb R$?
A classical result is that for $\precsim$ ... | 4 | https://mathoverflow.net/users/76065 | 275065 | 122,568 |
https://mathoverflow.net/questions/275013 | 5 | **Part I**
Let $f: X\to Y$ be an arbitrary map between topological spaces/simplicial sets $X$ and $Y$. Then we can associate with $f$ a tower of fibrations
$$
\dots \to F\_i\to\dots \to F\_1\to F\_0=Y
$$
constructed in a following way:
0.let $F\_0 = Y$ and we take $C\_0$ as homotopy cofiber of $f$
1.define $F\... | https://mathoverflow.net/users/106301 | Postnikov-type tower for a map between spaces | Here is an answer to your "Part I."
The construction you outlined (and its dual) was considered in detail in the papers of Ganea. In particular:
Ganea, T.
Induced fibrations and cofibrations.
Trans. Amer. Math. Soc. 127 1967 442–459.
(See Section 3 of the above for your case.)
For the dual question see my ans... | 3 | https://mathoverflow.net/users/8032 | 275069 | 122,570 |
https://mathoverflow.net/questions/275057 | 5 | I post this naive question on Math stackexchange, but have got no reply, so I decide to bother the mathoverflow community.
<https://math.stackexchange.com/questions/2350436/rational-structures-of-cokernel-of-linear-maps>
In Nekovar's paper on Beilinson Conjecture,
<http://math.stanford.edu/~conrad/BSDseminar/ref... | https://mathoverflow.net/users/87910 | A Naive Question about Nekovar's Paper on Beilinson's Conjecture | For every short exact sequence of $k$-vector spaces
\begin{equation\*}
0 \to V\_1 \xrightarrow{\alpha} V \xrightarrow{\beta} V\_2 \to 0
\end{equation\*}
there is an isomorphism $\det(V) \cong \det(V\_1) \otimes \det(V\_2)$. Looking at the special case $V\_2=0$, it is natural to define $\det(0)=k$. Now if $k\_0$ is a su... | 4 | https://mathoverflow.net/users/6506 | 275081 | 122,572 |
https://mathoverflow.net/questions/275084 | 0 | The famous plot of $\zeta(1/2+it)$ for real $t$ seems to show that this function gets a non zero real value exactly once between two consecutive Riemann zeros. Moreover, letting $\rho\_{i}$ the $i$-th critical zero of positive imaginary part, it seems that such a non zero real value is attained at around $1/2(\rho\_{i}... | https://mathoverflow.net/users/13625 | Is the number of real values of Zeta on the critical line up to some ordinate known? | $\zeta(1/2+i g\_n)$ is nonzero real iff $g\_n$ if [Gram point](https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function#Gram_points), which is efficienly computable and you can use binary search to count up to given $t$.
Your claim "his function gets a non zero real value exactly once between two consecuti... | 8 | https://mathoverflow.net/users/12481 | 275087 | 122,575 |
https://mathoverflow.net/questions/275092 | 12 | When proving that conditions $A$ and $B$ are equivalent, it is often an arbitrary choice whether to first prove $A\implies B$ or $B\implies A$. Are there examples where the second implication uses the first in a nontrivial way, and becomes significantly more difficult without invoking the former?
| https://mathoverflow.net/users/12518 | Asymmetric $A \iff B$ proofs | The proof of the Shephard-Todd theorem proceeds exactly that way. The theorem can be stated as:
For a finite subgroup $G\subset GL(n,\mathbb C)$ the following are equivalent:
(A) $G$ is generated by complex reflections (i.e., elements $g$ with $\text{rank}(g-\mathbf1\_n)=1$).
(B) The ring of invariants $\mathbb C... | 8 | https://mathoverflow.net/users/89948 | 275122 | 122,586 |
https://mathoverflow.net/questions/275072 | 11 | I'm no expert on these things (and this may not be cutting edge research level; it's really motivated by this [MSE question](https://math.stackexchange.com/questions/2332292/do-flat-functions-with-flat-fourier-transform-exist/2353043#2353043)), but it seems that there are non-zero measures (and also functions (?), I wo... | https://mathoverflow.net/users/48839 | Function and Fourier transform vanish on an interval | Over at MSE (see linked question), user1952009 (= reuns on MO) has given a very elegant construction of a Schwartz function with this property. I am reproducing his/her answer here.
Let $g,h\in C\_0^{\infty}$ with support contained in $[1/3,1/2]$, and write $\widehat{h}\_n=\int\_0^1 h(x)e^{-2\pi inx}\, dx$ for the di... | 8 | https://mathoverflow.net/users/48839 | 275125 | 122,587 |
https://mathoverflow.net/questions/273918 | 3 | My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint:
$$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum\_{j=1}^n x\_j = M-1 \}$$
Is there a formula for this index? That is, I would like an expression for $i\_S(\mathbf{x})$ explicitly ... | https://mathoverflow.net/users/74350 | Source coding lexicographic index of finite alphabet sequence with weight (partitions) | I was able to put the information from the two propositions in the paper together to get an explicit expression. The index should be:
$$i\_S (\mathbf {x})=\sum\_{k=1}^n \sum\_{j=1}^{x\_k-1} n\_S (x\_1, \ldots,x\_{k-1},j)$$
And further:
$$n\_S (x\_1, \ldots,x\_{k-1},j)={{n-\sum\_{l=1}^{k-1} x\_l -j +m -k-1} \choose {m-k... | 0 | https://mathoverflow.net/users/74350 | 275128 | 122,589 |
https://mathoverflow.net/questions/275133 | 2 | If I have a function $f \in H^s(\mathbb R^n)$, what can I say about $f^2$? In particular, I'm interested in the case when $s < 0$ and would like to conclude that, for example, $f \in H^{2s}(\mathbb R^n)$. Is something like this true?
| https://mathoverflow.net/users/90100 | If a function is in a Sobolev space, what can we say about its square? | No, something like this is not true. In any negative Sobolev space, you can find a function f with a point singularity such that $f^2$ is not integrable. Hence $f^2$ is not even a distribution, and certainly not in $H^{2s}$.
| 1 | https://mathoverflow.net/users/12120 | 275136 | 122,591 |
https://mathoverflow.net/questions/275135 | 7 | The forgetful functor $\mathbf{CompHaus} \rightarrow \mathbf{Top}$ from compact Hausdorff spaces to topological spaces famously has a left-adjoint, the Stone-Cech compactification.
>
> **Question.** Does the forgetful functor $$\mathbf{Comp} \rightarrow \mathbf{Top}$$ from compact spaces to topological spaces have ... | https://mathoverflow.net/users/26080 | Does the forgetful functor $\mathbf{Comp} \rightarrow \mathbf{Top}$ have a left-adjoint? | No, it does not. If it did, then $\mathbf{Comp}$ would be a reflective subcategory of the [total category](https://ncatlab.org/nlab/show/total+category) $\mathbf{Top}$, and hence would be total itself. Now, total categories admit all small limits, and there is the rub: $\mathbf{Comp}$ does not have all small limits. In... | 14 | https://mathoverflow.net/users/2926 | 275145 | 122,594 |
https://mathoverflow.net/questions/275103 | 3 | I'm trying to find a reference or a proof to the following statement used by [Matringe, N., 2012. Cuspidal representations of GL (n, F) distinguished by a maximal Levi subgroup, with F a non-archimedean local field. Comptes Rendus Mathematique, 350(17-18), pp.797-800.](https://arxiv.org/abs/1207.3925) (used in the proo... | https://mathoverflow.net/users/103908 | Distributions on the mirabolic subgroup which are left invariant to the unipotent radical | For a totally disconnected group $G$ and closed subgroup $H$, the averaging map $\alpha:C^\infty\_c(G)\to C^\infty\_c(H\backslash G)$ is surjective, certainly for scalar-valued functions, and I think for $V$-valued functions where $V$ is quasi-complete, locally convex, so that Gelfand-Pettis (or some equivalent kind of... | 3 | https://mathoverflow.net/users/15629 | 275147 | 122,595 |
https://mathoverflow.net/questions/275143 | 0 | For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ and ideally a large set of value for $q$ (although $q \in (0;1]$ probably suffices).
There are two approaches that I ... | https://mathoverflow.net/users/32355 | Analytic extension of the Hurwitz ζ function | Summing by parts, using the binomial series and inverting the double sum works fine, obtaining
$$\zeta(s,a)-s \zeta(s,N)
= \sum\_{n=1}^{N-1} n ((n+a-1)^{-s}-(n+a)^{-s})\\+ \sum\_{k=0}^\infty {-s \choose k+2} ((a-1)^{k+2} -a^{k+2}) \zeta(s+k+1,N)$$
Taking $a=1$ shows $(s-1)\zeta(s,N)$ is entire, from which you obtai... | 4 | https://mathoverflow.net/users/84768 | 275149 | 122,596 |
https://mathoverflow.net/questions/275142 | 16 | I've never had a problem with the axiom of choice, but it has often confused me how many authors find full choice so much different from finite choice. In my head they seem quite similar. We are picking elements in either case--why treat the infinite case as **fundamentally** different from the finite one? [Note: This ... | https://mathoverflow.net/users/3199 | The axiom of choice as a consequence of a stronger semantics? | I like this question very much.
I find the difference between the two readings of $\forall x\exists y\ P(x,y)$ to be similar to the issues often brought up by questions of *uniformity* in mathematical existence assertions. So this may be a good keyword leading to further discussion. For example, the discussion of un... | 16 | https://mathoverflow.net/users/1946 | 275150 | 122,597 |
https://mathoverflow.net/questions/275123 | 7 | Let $S(x) = \sum\_{n=0}^{N-1} a\_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that
$$ \sum\_{r=1}^R |S(x\_r)|^2 \leq (N + \delta^{-1}-1) \int\_{0}^{1} |S(x)|^2 dx$$
where $x\_1,x\_2,\ldots,x\_R$ are $\delta$ separated points... | https://mathoverflow.net/users/630 | Examples of the large sieve inequality where a constant larger than 1 is needed | Some comments:
1) Montgomery's survey paper
<https://projecteuclid.org/euclid.bams/1183540922>
page 559, Theorem 3 and equation (19).
It essentially says that Selberg gave an example,
with the comment that this is the only situation where this is sharp.
See also page 563 for lower bounds on $\Delta$, and also up... | 3 | https://mathoverflow.net/users/36707 | 275155 | 122,599 |
https://mathoverflow.net/questions/273897 | 4 | Let $X$ be a compact zero-dimensional space, let $S \subseteq \mathrm{Homeo}(X)$ and let $U$ be a compact open subset of $X$. Suppose that $s^{-1} \in S$ for all $s \in S$, and that $S$ restricts to an equicontinuous family of functions from $U$ to $X$. Is the set $\{sU \mid s \in S\}$ finite?
My thinking is that $S$... | https://mathoverflow.net/users/4053 | Equicontinuity and orbits of compact open sets | I couldn't quite answer the question as posed, but thanks to John Griesmer's hint I managed to get a positive answer with slightly different (and perhaps more 'natural') hypotheses, so I will put something here so that the question doesn't remain unanswered.
Claim: Let $X$ be a compact zero-dimensional space, let $S ... | 1 | https://mathoverflow.net/users/4053 | 275174 | 122,605 |
https://mathoverflow.net/questions/275153 | 7 | Consider a graph with the vertices being all subsets of size $n$ of a set of size $2n$. Two vertices are connected if their overlap has size at most one. What is the chromatic number of this graph?
If we have two vertices connected when the overlap is empty, then this corresponds to the [Kneser graph](https://en.wiki... | https://mathoverflow.net/users/83212 | Kneser graph with overlap | The graph you are interested in is the generalized Kneser graph $KG(2n,n,1)$. The *generalized Kneser graph* $KG(n,r,s)$ has the same vertex set as a Kneser graph $KG(n,r)$ but with two vertices joined by an edge if their intersection has no more than $s$ elements.
A web search for chromatic numbers of such graphs g... | 2 | https://mathoverflow.net/users/94968 | 275181 | 122,609 |
https://mathoverflow.net/questions/275146 | 1 | Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex as an object $T$ such that $\mathrm{Hom}\_{K^b(\mathrm{proj}\, A)}(T,T[n])=0$ for all $n \neq 0$ and such that the full ... | https://mathoverflow.net/users/95742 | Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory | It means that $K^b(\text{proj }A)$ is the smallest full triangulated subcategory containing $\text{add}(T)$. This is spelled out more explicitly in Section 5 of the earlier paper "Morita theory for derived categories", where the definition of a tilting complex first appears.
More concretely, starting with the objects... | 4 | https://mathoverflow.net/users/22989 | 275194 | 122,613 |
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