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https://mathoverflow.net/questions/262825 | 4 | In order to finish a paper on 'metric space magnitude' I need to prove that a certain distribution on $\mathbb{R}^{2p+1}$ is in Mark Meckes' weighting space (see
Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces). My question requires no knowledge of that background, however: for what I want to do, it s... | https://mathoverflow.net/users/458 | Is the distribution $f\mapsto \int_{S} \frac{\partial^i }{\partial \nu^i}f\,\mathrm{dvol}$ in a Bessel potential space? | Yes, this is true. Your distribution is defined as an integral over the sphere of a derivative of order i. By using the divergence theorem, you can convert this to an integral over the ball of a derivative of order i+1. Hence the distribution is an element of $H^{-(i+1)}$.
| 2 | https://mathoverflow.net/users/12120 | 262834 | 118,291 |
https://mathoverflow.net/questions/262839 | 0 | Let us call $f,g:\omega\to \omega$ *almost totally distinct* if $$|\{n\in \omega: f(n) = g(n)\}| < \aleph\_0.\;\;\;\; (\star)$$
It is known that there are uncountable collections of almost totally distinct functions.
**Question.** Is the above statement still true if we replace $\aleph\_0$ by $2$ in $(\star)$?
| https://mathoverflow.net/users/8628 | Almost totally distinct functions | No. By pigeonhole principle any uncountable family of functions $f:\omega\to \omega$ contains two functions with the same pair $(f(1),f(2))$.
| 7 | https://mathoverflow.net/users/4312 | 262840 | 118,292 |
https://mathoverflow.net/questions/262788 | 6 | This is a rather technical question. I cannot find my mistake in a proof of the (obviously wrong) following sentence: Every countable ordinal is $\Sigma\_2$-definable in $J\_{\omega\_1 + 1}$ by a formula with no parameter.
I first recall the definition of $J\_\alpha$, closely following the notations and ideas of "Fin... | https://mathoverflow.net/users/14490 | Problem with definability in the constructible hierarchy | I may have found where is the trick hidden: I mean two different things by "being $\Sigma\_2$-definable". When I write "the smallest ordinal which is not $\Sigma\_2$-definable" I mean being definable by a formula of the form
$\exists \forall \psi$ where $\psi$ only have bounded quantifiers.
But when I define this s... | 3 | https://mathoverflow.net/users/14490 | 262875 | 118,305 |
https://mathoverflow.net/questions/262865 | 3 | Let $\zeta$ be a primitive $n$-th root of unity, and for each function $f : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ define its Fourier transform $\widehat{f} : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ by
$$\widehat{f}(a) = \sum\_{x \in \mathbf{Z}/n\mathbf{Z}} f(x) \zeta^{ax} .$$
Further, for any $A \subseteq \mathbf{Z}/n\... | https://mathoverflow.net/users/100299 | Fourier transform of subgroups of $(\mathbf{Z}/n\mathbf{Z})^*$ | The dimension is $\phi(n)/|G|$.
Denote the $\mathbb{Q}$-span of $\{\hat{G}(a)\mid a\in\mathbb{Z}/n\mathbb{Z}\}$ in $\mathbb{C}$ by $V$.
Observe that $V<\mathbb{Q}(\zeta)$, which is a Galois extension of $\mathbb{Q}$ with Galois group $\Gamma\simeq (\mathbb{Z}/n\mathbb{Z})^\*$.
Identifying $G$ with a subgroup of $\Gam... | 1 | https://mathoverflow.net/users/89334 | 262876 | 118,306 |
https://mathoverflow.net/questions/262058 | 21 | I posted this question [in MSE](https://math.stackexchange.com/questions/2125473/a-differentiable-isometry-is-smooth?noredirect=1#comment4390675_2125473) but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differen... | https://mathoverflow.net/users/46290 | A differentiable isometry is smooth? | By the nontrivial fact that $f$ is a local homeomorphism, we can assume without loss of generality that $f$ is a bijective homeomorphism. The usual textbook proof of the formula for the differential of the inverse map works here, so that $f^{-1}$ is again a differentiable isometry in the sense of the OP.
So **it suff... | 6 | https://mathoverflow.net/users/36952 | 262880 | 118,308 |
https://mathoverflow.net/questions/262886 | 0 | Recall that Cantor set can be defined as the set of numbers in $[0,1]$ that don't contain $1$ when written in ternary number system.
Alternatively if we consider the map $\varphi: [0,1]\to [0,1]$, $x\to (3x\mod 1)$, then Cantor set consists of points whose orbits does not intersect the interval $(\frac{1}{3}, \frac{2}... | https://mathoverflow.net/users/13441 | A modified Cantor and its measure | This is called a *cookie cutter*. If by smooth, you mean $f$ is $C^{1+\epsilon}$ or smoother, then it's known that $f$ preserves a fully supported absolutely continuous invariant measure on $[0,1)$. In particular, almost every point enters the middle interval (there's no reason this should be $(\frac 13,\frac 23)$). So... | 4 | https://mathoverflow.net/users/11054 | 262888 | 118,311 |
https://mathoverflow.net/questions/258981 | 7 | Let $G$ be a reductive algebraic group, and let $M$ be a Levi subgroup of $G$. In Urban's paper [*Eigenvarieties for Reductive Groups*](http://annals.math.princeton.edu/2011/174-3/p07), it seems to be assumed that if $G(\mathbb{R})$ and $M(\mathbb{R})$ both have discrete series, then the region of convergence of Eisens... | https://mathoverflow.net/users/93798 | Region of convergence of Eisenstein series is a union of Weyl chambers when groups have discrete series? | The assumption in Urban's paper is incorrect: the region of convergence need
not be a union of Weyl chambers. Hence the character distribution of the
Eistenstein series coming from a single cusp generally does not have a
unique $p$-adic interpolation. However, the sum of all of the character
distributions will still ha... | 2 | https://mathoverflow.net/users/93798 | 262896 | 118,314 |
https://mathoverflow.net/questions/242587 | 13 | Motivated by a [geometric proof of the Fundamental Theorem of Algebra](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Geometric_proofs) we ask:
Is there a geometric proof for the [Gauss-Lucas theorem](https://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem)? Since we are working on a half plane, can o... | https://mathoverflow.net/users/36688 | A geometric proof of the Gauss-Lucas theorem | Good question! I think the following article may qualify as a "yes":
Arnaud Ch´eritat, Yan Gao, Yafei Ou, Lei Tan. "A refinement of the Gauss-Lucas theorem (after
W. P. Thurston)". 2015.
<https://hal.archives-ouvertes.fr/hal-01157602/document>
(An essential observation here is that you **can** make the Gauss-Lucas... | 13 | https://mathoverflow.net/users/81295 | 262908 | 118,317 |
https://mathoverflow.net/questions/262920 | 2 | This must be very classical, but I can't find a reference.
>
> Is there an explicit description of the (generic?) fibers of the Prym map $\mathcal{R}\_3 \to \mathcal{A}\_2$?
>
>
>
By this I mean the map that to any double etale cover of a genus 3 curve associates the corresponding Prym variety.
| https://mathoverflow.net/users/4096 | Fiber of the Prym map in dim 2 | The fibres of the extended Prym map $\overline{P} \colon \overline{\mathcal{R}}\_3 \to \mathcal{A}\_2$ are studied in detail in the paper
Verra, Alessandro: *[The Fibre of the Prym Map in Genus Three](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002329107)*, Mathematische Annalen **276** (1986), 433-448.
... | 3 | https://mathoverflow.net/users/7460 | 262923 | 118,321 |
https://mathoverflow.net/questions/262910 | 1 | Consider a countable-state Markov chain; for the sake of concreteness identify the set of states with the set of nonnegative integers {0, 1, 2, ... }. Suppose the transition probabilities $P\_{ij} = \mathbb{P}(j \to i)$ are such that the chain has a stationary probability distribution $(\pi\_j)$: that is, $\pi\_j \geq ... | https://mathoverflow.net/users/5701 | Return time estimates in countable state Markov chains | Assume $p\_{i(i+1)}>0$ for all $i$. For any fixed $k$, you have $\mathbb{P}(\tau\_1>n)\geq p\_{12}p\_{23}\dots p\_{(k-1)k}p\_{kk}^n$ for all $n$. So if $\sup\_k p\_{kk}=1$, then the exponential tail bound that you want for the return time can't hold.
But you can still get exponential decay of the stationary probabil... | 3 | https://mathoverflow.net/users/5784 | 262926 | 118,322 |
https://mathoverflow.net/questions/262905 | 6 | Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories. Let $\tau : \mathcal{A} \rightarrow \mathcal{B}$ be a left-exact functor. Let $\mathcal{C}$ be the mapping cylinder category of $\tau$. Then, there are various functors $j\_\*, i\_\*$ ... defined on page 524 of Mazur's expository article in 1973, in Ann. Sc... | https://mathoverflow.net/users/46108 | Mazur's article, Notes on etale cohomology of number fields | Form an injective resolution
$$
0 \to M \to I^0 \to I^1 \to I^2 \to \dots
$$
and apply $j\_\*$. Then $j\_\* I^k = (\tau I^k, I^k, id\_{\tau I^k})$ in the mapping cylinder category for all $k$, and so to calculate the derived functors you calculate the cohomology of the sequence
$$
0 \to (\tau I^0, I^0, id) \to (\tau I^... | 8 | https://mathoverflow.net/users/360 | 262934 | 118,324 |
https://mathoverflow.net/questions/262863 | 10 | The standard Hilbert matrix $H$ is given by
$$H\_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example [in this MO question](https://mathoverflow.net/questions/47561/deriving-inverse-of-hilbert-matrix).
Now I have encountered a matrix $M$ of similar form, namely,
$$M\_{ij}=\frac{1+(-1)^{i+j}}{i+j-1}.$$
... | https://mathoverflow.net/users/105214 | Find the inverse of a matrix that is very similar to the Hilbert matrix | We present a generalization and also give an explicit solution.
If $M$ is the $n\times n$ matrix
$$M=\left[\frac{1+(-1)^{i+j}}{x\_i-y\_j}\right]\_{i,j=1}^n$$
then the inverse matrix $K:=M^{-1}$ has entries given by
\begin{align}
K\_{a,b}=\begin{cases}
2\frac{\prod\_{2j-1\neq b}x\_{2j-1}-y\_a}{\prod\_{2j-1\neq a}y\_a-... | 6 | https://mathoverflow.net/users/66131 | 262951 | 118,326 |
https://mathoverflow.net/questions/262935 | 3 | I asked the same question on math.stackexchange recently (<https://math.stackexchange.com/questions/2134978/is-it-possible-to-orbifold-torus-td-into-a-sphere-sd-using-mathbbz-2>), but it didn't receive much attention there, so I decided to move the question to this forum.
Consider a torus $T^d$ constructed as a hyper... | https://mathoverflow.net/users/103233 | Does the torus $T^d$ 2-fold cover an orbifold $Q^d$ with underlying space $S^d$? | Given the specific quotient described in the question, the answer is no. Under this quotient there is a fixed point at the origin and the neighborhood of this point quotients to a cone over $RP^{d-1}$. For $d=2$, $RP^1=S^1$ so the cone has underlying space a disk, however for $d>2$, the underlying cone will not be home... | 6 | https://mathoverflow.net/users/27453 | 262963 | 118,328 |
https://mathoverflow.net/questions/262950 | 3 | Let $K$ be a finite field of characteristic $p$, $G/K$ be a connected, reductive, split algebraic group. Fix some maximal split torus $T$ and a system of positive roots $\Phi^+ \subset \Phi (G,T)$.
Consider a $p$-restricted weight $\lambda \in X\_\*(T)$: this is a (dominant) weight such that $0 \le \langle \lambda, \... | https://mathoverflow.net/users/101091 | Absolute irreducibility of p-restricted highest weight modules | Yes, it's absolutely irreducible. The standard reference is R. Steinberg's 1963 paper [*here*](http://www.ams.org/mathscinet-getitem?mr=0155937) (which is freely available online). His 1967-68 Yale lectures (see $\S13$), now published by AMS in a LaTeX version, may be a good alternative source.
It should be emphasiz... | 3 | https://mathoverflow.net/users/4231 | 262966 | 118,329 |
https://mathoverflow.net/questions/262965 | 6 | Suppose that I'm given a set of rational primes $S$ with positive Dirichlet density, and a finite set of primes $R$, disjoint from $S$.
Does there exist a number field $K$ that is;
* unramified outside $R$;
* splits only at primes in some subset $S'\subseteq S$?
In my situation, I have a semisimple representation ... | https://mathoverflow.net/users/54339 | Extension of $\mathbb Q$ which splits only at primes in $S$ | For many choices of $R$ and $S$ the answer is obviously no. For example, if $R$ is empty, then the answer is no, because there are no unramified extensions of $\mathbb{Q}$.
For a more interesting example, let $S$ be the set of primes $p \equiv 3 \pmod{4}$ and $R$ be any finite subset of $\{ 2, 5, 13, 17, 29, \ldots \... | 12 | https://mathoverflow.net/users/48142 | 262970 | 118,331 |
https://mathoverflow.net/questions/262944 | 4 | A real sequence $\mathbf{x}=(x\_n)\_{n=1}^\infty\subseteq[0,1]$ is *equidistributed* if, for all $0\leq a< b\leq 1$,
$$
\lim\_{n\to\infty}\frac{|\{1\leq k\leq n:x\_k\in (a,b)\}|}{n}=b-a
$$
This can be restated as follows. Given $\mathbf{x}$, $a$, $b$ as above, let
$$
A(\mathbf{x},a,b)=\{n\in\mathbb{N}:x\_n\in (a,b)... | https://mathoverflow.net/users/38253 | Equidistributed sequences and lower banach density | As it is stated, question **1** has a negative answer: take any equidistributed sequence, and insert a sequence of $n$ $0$'s between $x\_{2^n}$ and $x\_{2^n+1}$. This will change the natural density of no set $A=A({\bf x},a,b)$, although it will make vanish the limit $\lim\_{d\to\infty }\min\_n{|A\cap\{n+1,\dots,n+d\}|... | 7 | https://mathoverflow.net/users/6101 | 262974 | 118,334 |
https://mathoverflow.net/questions/262074 | 5 | Suppose that
* $X$ is the $n \times n$ matrix of all ones
* $Y$ is an arbitrary $n \times n$ matrix with zeroes on the diagonal and all other entries equal to $0$ or $1$
* $0 < \delta < 1$
Let $Z = -X - \delta Y$. If $Y$ has any ones, then does $Z$ have an eigenvalue with positive real part?
This question is bas... | https://mathoverflow.net/users/104830 | Proving that a certain non-symmetric matrix has an eigenvalue with positive real part | Suppose that $Y\_{ij} = 1$ for some $i, j$. Construct a vector $x$ with $n$ coordinates such that $x\_t = 1+\frac{\delta}{n}$ if $t = i$ and $x\_t = 1$ otherwise.
Note that $\frac{(-Zx)\_t}{x\_t} > n$ for each $1 \leq t \leq n$. By the Collatz-Weilandt formula, the Perron-Frobenius eigenvalue of $-Z$ exceeds $n$.
... | 1 | https://mathoverflow.net/users/104830 | 262977 | 118,336 |
https://mathoverflow.net/questions/262989 | 2 | What is an example of a Hamiltonian graph $G=(V,E)$ such that there is one path visiting all vertices that is *not* chromatic (definition see below)?
---
Let $G= (V,E)$ be a simple undirected graph on $n\geq 1$ vertices, and let $b:[n]\to V$ be a bijection. We assign to $b$ the greedy coloring $c\_b$ constructed ... | https://mathoverflow.net/users/8628 | Non-chromatic paths in Hamiltonian graphs | **Counterexample.** Let $G$ be the graph with vertices $v\_1,v\_2,v\_3,v\_4,v\_5,v\_6$ and edges $v\_1v\_2,v\_2v\_3,v\_3v\_4,v\_4v\_5,v\_5v\_6,v\_6v\_1,v\_1v\_5,v\_4v\_6.$
The graph $G$ is Hamiltonian, since $v\_1,v\_2,v\_3,v\_4,v\_5,v\_6,v\_1$ is a Hamiltonian cycle.
The graph $G$ is $3$-chromatic; for a proper co... | 4 | https://mathoverflow.net/users/43266 | 262997 | 118,340 |
https://mathoverflow.net/questions/262986 | 2 | What are some examples of a (connected) compact complex manifold with a non-constant global complex valued function, $f$, such that
$\partial {\bar{\partial}} f = 0$.
In other words, what are examples of a connected compact complex manifold with
Aeppli cohomology, $H^{0,0}\_A > 1$ ?
It is clear such a manifold must ... | https://mathoverflow.net/users/30172 | Are there compact complex manifolds with non-constant pluriclosed functions? | A function $f : M \to \mathbb{C}$ satisfying $\partial\bar{\partial}f = 0$ is called *pluriharmonic*.
In any holomorphic coordinates $(z^1, \dots, z^n)$, a pluriharmonic function satisfies $\partial\_{z\_i}\partial\_{\bar{z}\_j}f = 0$ for all $1 \leq i, j \leq n$. In particular,
$$0 = \sum\_{j=1}^n\partial\_{z\_j... | 7 | https://mathoverflow.net/users/21564 | 263003 | 118,343 |
https://mathoverflow.net/questions/263005 | 3 | Let $G=GL\_n(K)$ where $K$ is an algebraically closed field of characteristic zero. Let $V$ be a finite dimensional rational representation of $V$.
Assume that $v\in V$ has a reductive stabilizer $H\subseteq G$.
I would like to ask for a reference for the following fact:
>
> There is a rational finite dimensional... | https://mathoverflow.net/users/41644 | Closed orbits for the action of general linear groups | I don't know a reference to the question as asked, but below I give a brief argument.
Let $G$ be an algebraic group and $H$ a reductive subgroup.
By Matsushima's theorem, $G/H$ is affine. Theorem 1.12 in Borel's "Linear algebraic groups" tells that there any $G$ affine action could be $G$-equivariantly closedly embed... | 3 | https://mathoverflow.net/users/89334 | 263016 | 118,347 |
https://mathoverflow.net/questions/262794 | 4 | (If anyone has a better title please change it!)
Given two finite words $v,w$ in the alphabet $\{a,b\}$, define the $v$-proportion of $w$ to be the largest number of letters in $w$ which can be covered by (not necessarily disjoint) copies of the word $v$ divided by the length of $w$. Denote this quantity by $Pr(w;v)$... | https://mathoverflow.net/users/35269 | Covering sequences of words | Try $w\_0 = b$, $w\_1 = bab$, $w\_2 = baba^2bab$, $w\_3 = baba^2baba^3baba^2bab$, and recursively $w\_{i+1} = w\_ia^{i+1}w\_i$.
Then $\displaystyle \limsup\_{i \rightarrow \infty} Pr(w\_i,a^n) = \lim\_{i \rightarrow \infty} \frac{2^{i + 2 - n} +n - i -3}{3\cdot 2^i - i - 2} = \frac{2^{2-n}}{3} \rightarrow\_n 0$.
-D... | 3 | https://mathoverflow.net/users/99278 | 263031 | 118,351 |
https://mathoverflow.net/questions/262911 | 3 | I [learned](https://mathoverflow.net/questions/252698/does-bf-prof-admit-all-pseudolimits) that the bicategory $\bf Prof$, while having lots of interesting properties, does not admit inverters, so it does not admit arbitrary pseudolimits.
Does this imply that it is impossible to define some useful constructions, like... | https://mathoverflow.net/users/7952 | Comma objects in the bicategory of profunctors | I don't know about comma objects, and I don't think there is a known characterization of which limits and colimits exist in Prof.
But cotensors do exist and are given as you say. More generally, lax limits of lax functors exist and coincide with lax colimits, the projections being the right adjoints of the coprojecti... | 3 | https://mathoverflow.net/users/49 | 263036 | 118,352 |
https://mathoverflow.net/questions/262985 | 2 | $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
| https://mathoverflow.net/users/102865 | Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative | If a nonzero $x\geq 0$ is a solution to $Ax=0$ then $\sum\_j x\_j=a>0$, and $x$ is also a solution to the system
$$A x=0,\quad 0\leq x,\quad e^\top x\leq a,\qquad\qquad\qquad\qquad(\*) $$
where $e$ denotes the all-1 vector.
It is a standard application of Farkas lemma in an appropriate form (one might want to use inst... | 0 | https://mathoverflow.net/users/11100 | 263045 | 118,356 |
https://mathoverflow.net/questions/263023 | 6 | Let $\mathcal{W}$ be a Weyl group, acting variously on a root system $\Phi$, the real vector space $V = \langle\Phi\rangle\_{\mathbb{R}}$ in which they live, or the associated weight lattice $P \subset V$ (vectors on which the coroots $\Phi^\vee\subset V^\vee$ take integral values). There are two obvious "rings of inva... | https://mathoverflow.net/users/17064 | Relation between linear and multiplicative invariants of Weyl groups | You can describe both constructions directly in terms of the weight lattice $P$. The first one (up to taking a dual) is the $W$-invariants of the symmetric algebra $\text{Sym}(P \otimes \mathbb{C})$; the second one is the $W$-invariants of the group algebra $\mathbb{C}[P]$, or maybe more evocatively, $\mathbb{C}[e^P]$,... | 5 | https://mathoverflow.net/users/290 | 263051 | 118,358 |
https://mathoverflow.net/questions/263020 | 12 | A student asked me this, and I can't believe I never knew the answer to this.
Let $R$ be a commutative ring, and $M$ be an $R$-module.
1. If $M$ has a set of $n$ linearly independent vector for each $n\in\mathbb{N}$, does that necessarily imply that $M$ has an infinite set of linearly independent vectors?
2. More g... | https://mathoverflow.net/users/105303 | Maximum cardinal of a set of linearly independent vectors in a module | Question 1. has a negative answer.
Denote $A := \{(i,j) \in \mathbb{N}^2, j\leq i\}$ and $S := \{$finite subsets of A with at least 2 different first coordinates$\}$. Define the ring $$R := \mathbb{F}\_2[t\_s: s \in S]\,/\,\big(t\_{s\_1}t\_{s\_2}: s\_1,s\_2 \in S\big),$$ and the $R$-module
$$M := \bigoplus\_{(i,j) \... | 8 | https://mathoverflow.net/users/59248 | 263053 | 118,360 |
https://mathoverflow.net/questions/263060 | 17 | What are known examples of two smooth, closed, oriented Manifolds $M,N$ of the same dimension that are simple homotopy equivalent, but not homeomorphic ?
It is well-known that the homotopy type of a given such $2$-manifold is the same as its homeomorphism type, so there won't be any such easy examples. Moreover (and... | https://mathoverflow.net/users/78554 | Simple homotopy equivalent, non-homeomorphic manifolds | The manifold $\*\mathbb{C}P^2$ (or the Chern manifold) is homotopy equivalent to $\mathbb{C}P^2$, but it is not homeomorphic to $\mathbb{C}P^2$, since its Kirby-Siebenmann invariant is non-trivial.
| 12 | https://mathoverflow.net/users/66131 | 263065 | 118,364 |
https://mathoverflow.net/questions/263013 | 2 | I would like to know a reference for Grothendieck duality in a resolution of singularities. More precisely, let $Y$ be a normal, Gorenstein variety with finite quotient singularities, and suppose that $f\colon X \to Y$ is a crepant resolution of singularities, meaning that $f^\*\omega\_Y \cong \omega\_X$. In particular... | https://mathoverflow.net/users/45285 | Grothendieck duality for resolution of singularities | In general, $f^! = R Hom\_X(L f^\*R Hom\_Y(\\_\_, \omega\_Y^\bullet), \omega\_X^\bullet)$, so given that both $X$ and $Y$ are Gorenstein, this simplifies quite a bit. In particular, if $G$ is a locally free sheaf, then
$f^!G \simeq f^\*G$.
| 1 | https://mathoverflow.net/users/10076 | 263069 | 118,366 |
https://mathoverflow.net/questions/262709 | 6 | Let $G$ be a simply-connected compact topological group (you can think of $SU(n)$ if you like it more concrete), and let $X$ be a finite-dimensional simply-connected $G$-CW-complex. If we know that all the isotropy groups are finite, does this imply they are trivial? And if not, are they at least bounded in size (in so... | https://mathoverflow.net/users/14233 | Almost free actions on simply-connected spaces | Since it was requested, here is the answer (from the comment) again.
The statement is wrong. For each $k$, one can start with a unique $0$-cell $G/(\mathbb Z/k)$ and attach a $2$-cell $G \times D^2$ by a map $G \times S^1 \to G/(\mathbb Z/k)$ that equivariantly extends a generator $S^1 \to G/(\mathbb Z/k)$ of the fun... | 2 | https://mathoverflow.net/users/14233 | 263074 | 118,369 |
https://mathoverflow.net/questions/263076 | 9 | It seems that there are different conventions in the literature as to what is a locally compact space (when the space is not supposed Hausdorff).
The two main non equivalent definitions I've seen are :
* (LC1) every point has a compact neighborhood
* (LC2) every neighborhood of any point contains a compact neighb... | https://mathoverflow.net/users/100552 | On the definition of locally compact for non-Hausdorff spaces | To me, the second definition of local compactness is much to be preferred for the simple reason that such locally compact spaces $X$ are exponentiable in $Top$, meaning that $X \times -: Top \to Top$ has a right adjoint $(-)^X: Top \to Top$ (even without the Hausdorff condition), and all this implies (such as $X \times... | 15 | https://mathoverflow.net/users/2926 | 263078 | 118,371 |
https://mathoverflow.net/questions/263028 | 8 | Let $V$ be a $n$-dimensional real vector space with standard inner product $(\cdot,\cdot)$. For any $\alpha \neq 0 \in V$, set $\alpha^\vee := \frac{2}{(\alpha,\alpha)}\alpha$. For $\alpha \neq 0,\beta \in V$ set $n\_{\alpha}(\beta) := (\beta,\alpha^\vee)$ and $s\_\alpha(\beta) := \beta - n\_{\alpha}(\beta)\cdot\alpha$... | https://mathoverflow.net/users/25028 | Non-reduced, non-crystallographic root systems | Let's stick to the OP's definition of a root system.
Let $\Phi\_0$ be the set of normalized roots $\frac{\alpha}{||\alpha||}$, $\alpha\in\Phi$. This is a root system satisfying 1, 2 and 3. Thus it is in the list of not necessarily crystallographic root systems. To reconstruct $\Phi$ we need for every $\alpha\_0\in\Ph... | 8 | https://mathoverflow.net/users/89948 | 263094 | 118,375 |
https://mathoverflow.net/questions/263049 | 2 | For vector fields in $R^3$ one knows that there exists a unique decomposition of vector fields in to solenoidal (divergence free) and potential parts. A generalization of this Theorem for symmetric tensors of arbitrary order is known for compact Riemannian manifolds with boundary , e.g [Theorem 3.3.2], Sharafutdinov, *... | https://mathoverflow.net/users/21422 | Tensor Field Decomposition in Space time | The class of Lorentzian manifolds that is best suited for this kind of question is that of [*globally hyperbolic*](https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold) ones. Analyzing the space of solutions of a hyperbolic PDE that does not satisfy an analog of the globally hyperbolic condition is rather messy b... | 4 | https://mathoverflow.net/users/2622 | 263096 | 118,376 |
https://mathoverflow.net/questions/263091 | 4 | A total recursive function $f(x)$ is provably total in $PA$ if there's some formula $\phi(x,y)$ such that
1. $f(x)=y \iff PA\vdash \phi(x,y)$ and
2. $PA\vdash \forall x \exists y \phi(x,y)$
I know (not in much detail) that a total recursive function is not provably total if it grows as fast as/faster than $f\_{\eps... | https://mathoverflow.net/users/75935 | $f_{\epsilon_0}$ and provably total functions in $PA$ | I think usually one adds the condition that $\phi$ be a $\Delta\_1$ (i.e. computable) formula.
As Gro-Tsen has pointed out, the answer is no: there are lots of functions which are provably total, dominated by $f\_{\epsilon\_0}$, but not provably total in PA. First of all, you're not stating the sharpest version of th... | 7 | https://mathoverflow.net/users/8991 | 263100 | 118,378 |
https://mathoverflow.net/questions/263088 | 1 | On page 530 in his paper,
```
Notes on etale cohomology of number fields, Ann. Scient. ENS (1973),
```
Mazur insisted that
$$
\text{Ext}^q\_{G\_K} (M,~\bar{K}^\*) \simeq H^q(G\_K,~\widehat{M}),
$$
where $G\_K=Gal(\bar{K}/K)$ for a local field $K$, $\widehat{M}=\text{Hom}\_{\mathbb{Z}}(M,~\bar{K}^\*)$, and $... | https://mathoverflow.net/users/46108 | Spectral sequence of Galois cohomology over local fields | This works with $K$ any field with $\#M \in K^{\times}$; local fields (of characteristic 0) play no special role. The key points are (i) to re-interpret certain Hom-constructions as "sheaf Hom" in a way that is not noticed for finite $M$ but makes a big difference for general discrete $G\_K$-modules, (ii) "derive" in t... | 2 | https://mathoverflow.net/users/81332 | 263107 | 118,381 |
https://mathoverflow.net/questions/262987 | 2 | Let $\mathfrak{g}$ be a semisimple Lie algebra (over $\mathbb{C}$), $\mathfrak{b}$ a Borel subalgebra of $\mathfrak{g}$, and $L$ a semisimple element contained in $\mathfrak{b}$.
I know that $L$ is contained in a Cartan subalgebra (CSA) of $\mathfrak{g}$. Is it true that $L$ is contained in a CSA of $\mathfrak{b}$? ... | https://mathoverflow.net/users/105284 | Is every semisimple element of a Borel subalgebra contained in a Cartan subalebra of the Borel subalgebra? | Here is another answer, using only Lie algebra theory. After applying ${\rm ad}\_{\mathfrak{g}}$ we can assume that $\mathfrak{g}$ is a semisimple subalgebra of some $\mathfrak{gl}\_k(V)$. As such, $\mathfrak{g}$ is almost algebraic, as defined in Jacobson's book, page 98 (called "decomposable" in the English translati... | 5 | https://mathoverflow.net/users/97435 | 263111 | 118,382 |
https://mathoverflow.net/questions/262983 | 2 |
>
> Given a field $\mathbb{F}$ and a consistent underdetermined system $Ax=b$ over $\mathbb{F},$ $A\in \mathbb{F}^{m \times N}$ and $b \in \mathbb{F}^m,$ finding a vector $z \in \mathbb{F}^N$ such that $Az=b$ and $\|z\|\_0 \leq \|x\|\_0$ for all $x \in \mathbb{F}^N$ such that $Ax=b$ is NP-hard.
>
>
>
I have bee... | https://mathoverflow.net/users/85776 | NP-Hardness of finding minimal-support solutions of underdetermined systems over any field | This proof looks correct to me. You don't have to worry about cancellations since the only way $z\in\mathbb{F}^N$ can have $|z|\_0=m/3$ and $Az={\bf 1}$ is if $z$ is the indicator function for an exact 3-cover. To see this, let $Z=\{j\in[m]:j\in C\_i\text{ for some }i\in[N]\text{ st }z\_i\neq0\}$. Note that $|Z|\leq3|z... | 2 | https://mathoverflow.net/users/90531 | 263122 | 118,390 |
https://mathoverflow.net/questions/263109 | 1 | In *$D$-Modules, Perverse Sheaves and Representation Theory* from R. Hotta, K. Takeuchi and T. Tanisaki, I found the following statement (in section 8.2, the lines before Definition 8.2.2):
Setting: Let $X$ be an analytic space, $U\subset X$ Zariski-open, $j\colon U\hookrightarrow X$ the open embedding, $D^b\_c(U)$ t... | https://mathoverflow.net/users/103697 | condition for constructibility of direct images of constructible sheaves under open embedding | For a discussion of Whitney stratifications, see for example *Stratified Morse theory* Goresky and Macpherson, *Notes on topological stability* by Mather, or *Stratifications de Whitney et théorème de Bertini-Sard* by Verdier. The point is that the conditions imply that everything is topologically locally trivial along... | 3 | https://mathoverflow.net/users/4144 | 263124 | 118,391 |
https://mathoverflow.net/questions/263123 | 6 | What are the best references for finding explicit formulas for Belyi maps for rational dessin d'enfants?
I am most interested in a formula for the Belyi map that corresponds to a specific rational dessin d'enfant: it is clean, with four black vertices, each of valency 3, and with ramification indices above the infin... | https://mathoverflow.net/users/105345 | Explicit formulas Belyi maps for a rational dessin d'enfants | Every elliptic surface on $\mathbb P^1$ with four semistable fibers and no other singular fibers has a $j$ invariant with ramification of order $3$ around $j=0$ and ramification of order $2$ around $j=1728$. By Riemann-Roch, one can check that this is the only ramification, so this means the $j$ invariant is a Belyi ma... | 9 | https://mathoverflow.net/users/18060 | 263126 | 118,392 |
https://mathoverflow.net/questions/263103 | 11 | Let $f(z) = \sum\_{n=1}^{\infty} a(n)n^{(k-1)/2} q^n$ be a cusp form, I am interested to know what is currently the best bound on the growth of $\sum\_{n\le X}a(n)$ in the following two case :
1. When $f$ is a cusp form of integral weight $k$ on $\Gamma\_0(N)$ with non-trival character $\chi.$
2. When $f$ is a cusp f... | https://mathoverflow.net/users/101794 | The best bound on the growth of $\sum_{n\le X}a(n)$ | For integral weight $k$ and level $N=1$, I believe that the best result is due to Rankin (1990):
$$ \sum\_{n\le X}a(n)\ll\_\epsilon x^{1/3}(\log x)^{-\delta+\epsilon},
\qquad \delta:=\frac{8-3\sqrt{6}}{10}\approx 0.065.$$
On the other hand, Jutila (1987) proved that in square mean the sum is of size $\asymp x^{1/4}$, s... | 12 | https://mathoverflow.net/users/11919 | 263129 | 118,393 |
https://mathoverflow.net/questions/237859 | 0 | [section 6 of the link](http://projecteuclid.org/download/pdf_1/euclid.jca/1292249710)
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration $\{m^rJ^s\}$ where J is the Jacobian ideal of the defining polyn... | https://mathoverflow.net/users/9485 | Milnor numbers and mixed multiplicities | Let $f(x,y,z)=0$ be the equation of a surface with isolated silgularity at the origin. Assume that the coordinates are chosen generically. Then in addition to the Milnor number the mixed multiplicities are the multiplicity of the ideal (x,y,$\partial f/\partial z$) and $(x,\partial f/\partial y,\partial f/\partial z)$.... | 3 | https://mathoverflow.net/users/105360 | 263158 | 118,403 |
https://mathoverflow.net/questions/263151 | 3 | An edge clique cover of an undirected graph $G$ is a set of cliques such that every edge of $G$ belongs to some clique in the set. The edge clique cover number $\theta(G)$ is the minimum size of edge clique cover of $G$.
Let's define $k$-restricted edge clique cover of $G$ as a set of cliques such that every edge of ... | https://mathoverflow.net/users/105356 | Edge clique cover of a graph with restriction on how many times an edge can be covered | In general there can be a rather large difference in these quantities. If you take $k = 1$ you are considering the clique partition vs. clique covering problem. This is studied in [Clique partitions and clique coverings](http://www.sciencedirect.com/science/article/pii/0012365X88901975) by Erdős, Faudree, and Ordman. I... | 4 | https://mathoverflow.net/users/51668 | 263170 | 118,406 |
https://mathoverflow.net/questions/263165 | 1 | Let $V$ be a vector space with a basis $v\_1, v\_2, \ldots, v\_n$. Let $T(V)$ be the tensor algebra of $V$. Let $S(Lie(V))$ be the symmetric algebra of the free Lie algebra of $V$. I think that $T(V)$ is isomorphic to $S(Lie(V))$ as a graded vector space. For example, let $n=2$. We have the degree $2$ component of $T(V... | https://mathoverflow.net/users/11877 | How to write down the map $T(V)_n \to S(Lie(V))_n$ explicitly? | The natural map is rather from $TV \to U(FreeLie(V))$: consider the forgetful functors $Assoc \to Lie \to Vec$ and compose their left adjoints to get the left adjoint $T$ of the composite. Then, as Alex Suciu comments, your question is a special case of the comparison of $U\mathfrak g$ and $S\mathfrak g$, accomplished ... | 7 | https://mathoverflow.net/users/391 | 263171 | 118,407 |
https://mathoverflow.net/questions/263169 | 5 | Let $\cal{P}$ be a $k$-linear semisimple abelian rigid monoidal category with finite dimensional (over $k$) Hom-spaces (for a field $k$).
By a tensored $\cal{P}$-category we mean a $\cal{P}$-category which admits tensors with objects in $\cal{P}$, i.e. for every two objects $X,Y$ in the category and $P$ in $\cal{P}$,... | https://mathoverflow.net/users/51663 | Does every enriched functor preserve tensors? | No. The functor "take a vector space to its double dual" is linear but does not preserve tensors with infinite-dimensional vector spaces.
Enriched functors are automatically "lax tensored". An enriched functor $F$ provides a natural map $[X,Y] \overset F\to [FX,FY]$ for any $X,Y$. Consider setting $Y=P\otimes X$, and... | 6 | https://mathoverflow.net/users/78 | 263172 | 118,408 |
https://mathoverflow.net/questions/263142 | 1 | Let $[n]=\{1,2,\dots,n\}$ and fix $r\in[n]$. Define the set $\mathcal{B}\_{n,r}$ as the set of all **set partitions** of $[n]$ into disjoint non-empty blocks such that each block $B$ satisfies:
$\,\,\,\,\,\,\,\,\,\,$ either $\vert B\vert=1$, or $r\in B$, or there exist $i,j\in B$ such that $i<r<j$.
For example, the... | https://mathoverflow.net/users/66131 | Is this variant on set partition explored? | Here is an alternative approach to what is proposed by Brendan.
The key idea is to notice that the only forbidden parts in the set partitions are non-singleton subsets of $\{1,2,\dots,r-1\}$ and $\{r+1,r+2,\dots,n\}$, which enables one to employ the inclusion-exclusion principle to enumerate the set partitions witho... | 3 | https://mathoverflow.net/users/7076 | 263174 | 118,409 |
https://mathoverflow.net/questions/263178 | 3 | I am working over the paper: [Target Enumeration via Euler Characteristic Integrals](http://epubs.siam.org/doi/abs/10.1137/070687293) and in order to follow a proof I need to prove:
>
> If $A$ is compact nonempty subset of $\mathbb{R}^2$, then the singular homology
> groups of $A$, $H\_k(A)$ vanish for $k\geq 2$.
... | https://mathoverflow.net/users/73539 | Homology groups of compact subset of $\mathbb{R}^2$ | This is answered in the paper
>
> *The singular homology group of planar sets do not behave anomalously*
> by Andreas Zastrow
>
>
>
This appears to be a link to the paper: <http://at.yorku.ca/i/d/e/b/11.htm>
| 4 | https://mathoverflow.net/users/3634 | 263180 | 118,411 |
https://mathoverflow.net/questions/263153 | 3 | Here's a question that (I hope) may seem very trivial for you, and I hope one of you may provide me with a reference answering it (unless it's a trivial colloquial knowledge).
Let $f$ be an indefinite ternary quadratic form that is anisotropic (does not represent 0).
By Dickson's theorem a universal (representing... | https://mathoverflow.net/users/39331 | Indefinite quadratic form universal over negative integers | Two parts. If the form is anisotropic, there is a specific prime (actually an even number of primes, by the product formula for the Hilbert Norm Residue symbol) $p$ for which the form is anisotropic. The problem is that the form does not integrally represent anything in the $p$-adic squareclass of the discriminant of t... | 3 | https://mathoverflow.net/users/3324 | 263183 | 118,413 |
https://mathoverflow.net/questions/263164 | 5 | An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square\_n$ denote all such paths. Define the **area of a path** $P$ to be the number of full squares below the path and above the diagonal. Then one va... | https://mathoverflow.net/users/66131 | generating $q$-Catalan numbers | The functions
$$
C\_n(q)=\sum\_{P\in\square\_n}q^{area(P)}
$$
satisfy the following recurrence relation
$$
C\_n(q)=\sum\_{k=1}^nq^{k-1}C\_{k-1}(q)C\_{n-k}(q).\tag{1}
$$
*Proof.*
(taken from the book ["The q, t-Catalan Numbers and the Space of Diagonal Harmonics" by J. Haglund](https://www.math.upenn.edu/~jhaglund/books... | 10 | https://mathoverflow.net/users/82588 | 263184 | 118,414 |
https://mathoverflow.net/questions/263191 | 12 | I am reading Lewis' paper "Is there a convenient category of spectra?". To prove the main result on the non-existence of such a nice category, he shows that otherwise the unit component of $QS^0= \varinjlim \Omega^n S^n$ would have to be weakly equivalent a product of Eilenberg-Mac Lane spaces. So far so good, but it i... | https://mathoverflow.net/users/39713 | $QS^0$ isn't a product of Eilenberg-Mac Lane | If $X$ is a product of Eilenberg-MacLane spaces then the map
$$ \eta^\*\colon \pi\_2(X) = [S^2,X]\to[S^3,X]=\pi\_3(X) $$ is easily seen to be zero (where $\eta\colon S^3\to S^2$ is the Hopf map). However, standard calculations give:
\begin{align\*}
\pi\_2(QS^0) & =\pi\_2^S(S^0)=\mathbb{Z}/2.\eta^2\\
\pi\_3(QS^0) &=... | 22 | https://mathoverflow.net/users/10366 | 263197 | 118,417 |
https://mathoverflow.net/questions/263185 | 6 | Let $\mathcal C$ be a finite tensor category, and $\mathcal M$ a finite left $\mathcal C$-module category. By a result of P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik (<http://www-math.mit.edu/~etingof/tenscat1.pdf> , Thm. 2.11.6), there is an algebra object $A \in \mathcal C$ and an equivalence of left $\mathcal ... | https://mathoverflow.net/users/105173 | Bimodule categories realized as internal bimodules | The equivalence is unfortunately not correct: take for example the category $\mathcal{C}=Vec\_G$ for some finite group $G$. Take $\mathcal{M}=Vec$ be the category of vector spaces. Then the forgetful functor from $\mathcal{C}$ to $Vec$ enriches $\mathcal{M}$ with the structure of a $\mathcal{C-C}$ bimodule category.
I... | 2 | https://mathoverflow.net/users/41644 | 263199 | 118,418 |
https://mathoverflow.net/questions/262060 | 2 | Given a homomorphism of a group $G$ to the mapping class group of a manifold $M$, is there any condition that guarantees that it is defined by an action of $G$ on $M$? Thank you very much.
| https://mathoverflow.net/users/19838 | Lifting from mapping class group to groups of homeo/diffeomorphisms | I'm not sure if this is exactly what you were asking about, but one way of parsing your question is to ask when a homomorphism from $G$ to $\operatorname{MCG}(M)$ is induced from a homomorphism $G \to \operatorname{Diff}(M)$ (or replace Diff with your favorite alternative). This is a very general (and very hard!) quest... | 6 | https://mathoverflow.net/users/960 | 263203 | 118,419 |
https://mathoverflow.net/questions/263214 | 1 | Let $\mathcal{V}$ be a closed monoidal category, and $\mathscr{C}$ be a category enriched over $\mathcal{V}$. One says that the *power* or *cotensor* of an objec $A \in \mathscr{C}$ by an object $U \in \mathcal{V}$ is an object $A^U$ of $\mathscr{C}$ with a natural isomorphism $$\mathscr{C}(B, A^U) \cong \mathcal{V}(U,... | https://mathoverflow.net/users/90058 | Morphisms of cotensors | It's pretty simple, really: using the universal property of $A^U$, one may define maps $E^i: E^V \to E^U$ and $p^V: E^V \to B^V$, and similarly maps $B^i: B^V \to B^U$ and $p^U: E^U \to B^U$.
For example, if $I$ is the monoidal unit of $\mathcal{V}$, one has a canonical arrow $\theta\_{E, V}: I \to \mathcal{V}(V, \m... | 1 | https://mathoverflow.net/users/2926 | 263215 | 118,427 |
https://mathoverflow.net/questions/263054 | 1 | Is there a general (integral) solution to $du(t)= -a(t)u(t)\,dt +\sigma(t)\,dz$? Is the following $u(t)=e^{-\int\_{t\_0}^{t} \alpha(s) \, ds}u(t\_0)+\int\_{t\_0}^t \sigma(v) e^{-\int\_v^t a(s) \, ds} \, dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wiener process. Is there a g... | https://mathoverflow.net/users/23935 | A solution to stochastic PDE $du(t)= a(t)u(t)\,dt +s(t)\,dz$ | About the differentiation formula of $u(t)$: Since
\begin{align}
u(t)&=e^{-\int\_{t\_0}^ta(s)\mathrm{d}s}u(t\_0)+\int\_{t\_0}^t\sigma(v)e^{-\int\_{v}^ta(s)\mathrm{d}s}\mathrm{d}Z(v)\\
&=e^{-\int\_{t\_0}^ta(s)\mathrm{d}s}\Bigl[u(t\_0)+\int\_{t\_0}^t\sigma(v)e^{\int\_{t\_0}^va(s)\mathrm{d}s}\mathrm{d}Z(v)\Big]\stackrel{... | 1 | https://mathoverflow.net/users/103256 | 263217 | 118,428 |
https://mathoverflow.net/questions/263223 | 26 | Is there any analogue of Morse theory in Number theory? Naive idea arising in my head is that defining a Morse function on scheme and find etale cohomology using that function. Since I'm not an expert about algebraic geometry and Morse theory, I can't advance my thoughts.
| https://mathoverflow.net/users/95471 | Arithmetic Morse theory? | One usually considers the analogue of Morse theory in algebraic geometry to be the theory of vanishing cycles and Lefschetz pencils.
Because of the nature of algebraic functions, Morse theory must be a little more complicated. A Morse function on a compact manifold lets us build the manifold up step by step, starting... | 48 | https://mathoverflow.net/users/18060 | 263224 | 118,430 |
https://mathoverflow.net/questions/263231 | 8 | It is (I believe) a very easy exercise to prove that the general recursive functions over the natural number object $N$ form a category. But what sort of category is it? From the fact that one can prove the Recursion Theorem one might surmise that the category of general recusive functions over $N$ is cartesian closed,... | https://mathoverflow.net/users/20597 | Recursion theory from the standoint of category theory | I am not sure this is what you are looking for but I think that the following paper may provide an answer to your question 1:
J. Robin B. Cockett, Pieter J. W. Hofstra:
Introduction to Turing categories. Ann. Pure Appl. Logic 156(2-3): 183-209 (2008)
It gives a categorical axiomatization of computability in arbitra... | 8 | https://mathoverflow.net/users/45027 | 263238 | 118,432 |
https://mathoverflow.net/questions/263241 | 9 | I am working on Tate's thesis, and I have some problems with computations, yet the result seems to be a good natural motivation for introducing the arithmetic conductor of a character.
Let $F$ be a non-archimedean local field, and $\psi$ a non-trivial additive character. Define the $\psi$-Fourier transform for a loca... | https://mathoverflow.net/users/43737 | Conductor as volume of the integers ring | Apply the Fourier Inversion Formula to the characteristic function $\Phi(x) = \chi\_\mathcal{O}(x)$ of the ring $\mathcal{O}$ of integers in $F$. The Fourier transform is the integral $\widehat{\Phi }(x)=\int\_\mathcal{O} \psi (xy)dy$. This integral is zero if and only if $y\mapsto \psi (xy)$ is a non-trivial character... | 15 | https://mathoverflow.net/users/23291 | 263243 | 118,434 |
https://mathoverflow.net/questions/263233 | 1 | The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))\_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$. This is equivalent to say that
the family $(\widehat f(k) e\_k)\_{k\in \mathbb{Z}^n}$ (where $(e\_k)$ is the Fourie... | https://mathoverflow.net/users/100552 | On a weaker condition of summability for Fourier series | If a series $\sum c(k) e\_k$ is summable in $C({\mathbb T}^n)$, in particular, the series $\sum c(k) e\_k(0)=\sum c(k)$ is summable (in $\mathbb{C}$), thus $\sum |c(k)|<\infty$.
| 1 | https://mathoverflow.net/users/4312 | 263244 | 118,435 |
https://mathoverflow.net/questions/262096 | 1 | I am reading section 7.B on Clifford theory in
[this paper](https://arxiv.org/abs/1511.04714) and hope someone can help me understand some of the arguments there.
Let me shortly explain the part of the setup which I need for my question:
$X$ is a normal subgroup of $Y$, $M$ is a $kX$-module ($k$ is a field) which ... | https://mathoverflow.net/users/68519 | Splitting of certain short exact sequences in context of Clifford theory | The idea is to consider your extension as a factor set $$f:Y/X\times Y/X\to 1+J(A)$$ and show that it must be equivalent to a trivial one. For this you first consider the image of $f$ in $(1+J(A))/(1+J(A)^2)\cong J(A)/J(A)^2$.
The last group is abelian, so since $char(k)\nmid |Y/X|$ the resulting sequence $$1\to (1+J(... | 1 | https://mathoverflow.net/users/41644 | 263253 | 118,440 |
https://mathoverflow.net/questions/263228 | 5 | Let $G$ be a directed graph without loop and suppose that $M$ is its endomorphism monoid.
First how we can create a simple graph $\Gamma$ (using $G$), with the same endomorphism monoid? and then how we can create it with the smallest possible number of vertices?
Thanks for your help
| https://mathoverflow.net/users/97333 | The smallest number of vertices for a graph with the same endomorphism monoid | These kinds of results are considered in the paper <http://onlinelibrary.wiley.com/doi/10.1002/jgt.20396/epdf>
| 3 | https://mathoverflow.net/users/15934 | 263256 | 118,442 |
https://mathoverflow.net/questions/263260 | 4 | Let $K,T\subset\mathbb{R}^{n}$ be convex bodies (i.e. nonempty, compact and convex) containing the origin, and let $\lambda\in(0,1)$.
Fix $\langle \cdot ,\cdot \rangle$ to be the standard dot product in $\mathbb{R}^n$ (though probably here any dot product would do).
Let us denote by $K^\circ$ the dual to $K$ with respe... | https://mathoverflow.net/users/103184 | Is duality of convex bodies in itself a convex function? | Does this work?
First prove it when $n=1$. Now reduce the general case to this one as follows:
If $L$ is a line through the origin in $\mathbb R^n$ and $K$ is convex in $\mathbb R^n$ then the intersection of the dual of $K$ with $L$ is the dual (with respect to $L$) of the image of $K$ in $L$ under orthogonal pro... | 3 | https://mathoverflow.net/users/6666 | 263263 | 118,444 |
https://mathoverflow.net/questions/263273 | 2 | In the mathematics literature, for example in Lee's Introduction to Smooth Manifolds, the singular homology groups of a smooth manifold $M$ are defined using singular $p$-chains, which are formal superpositions of singular $p$-simplices. For the purposes of this question I will define singular $p$-simplices as smooth m... | https://mathoverflow.net/users/98045 | Singular Homology Groups from Compact Oriented Submanifolds? | Expanded version of my comment:
Please google "the steenrod realization problem". This is a foundational problem in algebraic topology. It has a nice answer -- Glen Bredon's book "Topology and Geometry" covers it pretty well.
Bredon's book is not the full story but it gets you quite close to it.
For example if y... | 4 | https://mathoverflow.net/users/1465 | 263276 | 118,450 |
https://mathoverflow.net/questions/263265 | 4 | I hope it is okay that I re-post [my question](https://math.stackexchange.com/questions/2156119) from Math.SE. I know it is very specific. I would be grateful for thoughts on how to tackle this problem. Any idea is welcome!
For $n \in \mathbb{N}$, let the positive real numbers $x\_1, \dots, x\_n, y\_1, \dots, y\_n$ s... | https://mathoverflow.net/users/75070 | Inequality connected to Lagrange's interpolation formula | Write $x\_i$ for $x\_i^2$ and $y\_i$ for $y\_i^2$, our sum is
$$
a\_{ij}=\sum\_{k=1}^n \frac{f(x\_k)}{\prod\_{l\ne k} (x\_k-x\_l)},\,f(t)=-t^{-1/2}\prod\_{l\ne i,j} (t-y\_j).
$$
By Lagrange interpolation, it is a coefficient of $t^{n-1}$ in the polynomial $g(t)$ of degree at most $n-1$, which interpolates the functi... | 4 | https://mathoverflow.net/users/4312 | 263277 | 118,451 |
https://mathoverflow.net/questions/263279 | 4 | I posted this question on MSE earlier, however could not elicit a reply. I am not sure if this belongs here, if not, please flag it. Moreover, I am not sure what tags to put on it, so if this question remains, please add appropriate tags.
I was reading Bruhat Tits' "Groupes réductifs sur un corps local : I. Données r... | https://mathoverflow.net/users/36035 | Why is the image of $G$ under a $B-$adapted homomorphism normal? (Question from Bruhat Tits paper 1) | Bruhat-Tits are not saying that $\phi(G)$ is normal "since we have the Bruhat decomposition", they are saying it is normal *since the conjugates of $B$ generate $G$*.
(*Proof.* Let $a\in G$ and $g\in G'$. To see that $g\phi(a)g^{-1}\in\phi(G)$, pick $(b\_i ,k\_i)\in B\times G$ such that $a=\prod\_i k\_ib\_ik\_i^{-1}$... | 7 | https://mathoverflow.net/users/19276 | 263284 | 118,454 |
https://mathoverflow.net/questions/263290 | 2 | There is a huge amount of research dealing with analysis of representation of integers by a quadratic polynomials only with terms of degree 2 (here is for example review of some known methods by J.Hanke: <http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf>). There is a theory where we can expess corresponding theta ... | https://mathoverflow.net/users/61438 | Representation of integers by positive definite ternary quadratic polynomials with linear terms | Yes, it is possible to extend these methods, although the picture is somewhat
less clear than in the quadratic form case. When one discusses representations by a quadratic polynomial, this is equivalent to asking for representations by a *coset* of a lattice.
The theta series of a lattice coset is still a modular f... | 5 | https://mathoverflow.net/users/48142 | 263296 | 118,458 |
https://mathoverflow.net/questions/263294 | 3 | I'm trying to figure out second moment of the following quantity
$$y = \frac{\langle x\_1, x\_2 \rangle}{\left\|x\_1\right\|\left\|x\_2\right\|}$$
Where $x\_1$, $x\_2$ are sampled independently from $\mathcal{N}(0, \Sigma)$
This can be solved exactly in 2-dimensions using algebraic manipulation: suppose eigenvalu... | https://mathoverflow.net/users/7655 | Second moment of cos(x,y) for Normal x,y? | This $y = \frac{<x\_1, x\_2>}{\|x\_1\|\|x\_2\|}$ is the distribution of the $cos \theta$ where $\theta$ is known as the canonical angle/principal angle of two random vectors $x\_1,x\_2$. $cos\theta$ is known as the canonical correlation between $X\_1,X\_2$ since we know their joint distribution $(X\_1,X\_2)$ from indep... | 1 | https://mathoverflow.net/users/25437 | 263298 | 118,459 |
https://mathoverflow.net/questions/263318 | 1 | If ${\cal C}$ is a collection of subsets of a set $X$, we associate to ${\cal C}$ a graph $G\_{\cal C} = (V,E)$ where $V = {\cal C}$ and $$E = \big\{\{A,B\}: A\neq B\in {\cal C} \land A\cap B \neq \emptyset\big\}.$$
If $\kappa$ is a cardinal and ${\cal C}$ is a collection of subsets of $\kappa$, it is easy to see tha... | https://mathoverflow.net/users/8628 | Representability of "large" graphs by "small" intersection graphs | Assuming that $\kappa$ is an **infinite** cardinal, I answer your question with a **characterization.**
For $\mathcal C\subseteq\mathcal P(\kappa),$ the intersection graph $G\_\mathcal C=(V,E)$ has the following properties:
(1) $|V|\le2^\kappa;$
(2) there is a collection $\mathcal K$ of cliques, with $|\mathcal... | 3 | https://mathoverflow.net/users/43266 | 263335 | 118,469 |
https://mathoverflow.net/questions/263323 | 12 | Let $R$ be a topological integral domain. Let $K=\mathrm{Frac} R$. Is there any "natural" topology on $K$? Actually, since $K$ can be regarded as a quotient of $R\times R$ quotient some equivalence relation, so maybe we can equip $R\times R$ with the product topology, and that induces topology on $K$?
In particular, ... | https://mathoverflow.net/users/100985 | Any "natural" topology on fractional field of a topological ring? | As said Fred Rohrer, exercise 27 exists in the French edition and seems to answer the question of the MO. Let
$$
s\_{frac}\ :\ R\times R'\to Frac(R)
$$
($R'=R\setminus \{0\}$) be the canonical surjection. The "quotient field equivalence" $\equiv\_{frac}$ on $R\times R'$ is that given by $s\_{frac}$ i.e.
$$
(a,p)\equ... | 13 | https://mathoverflow.net/users/25256 | 263336 | 118,470 |
https://mathoverflow.net/questions/263359 | 3 | Let $H$ be a monoid, and denote by $H^\times$ and $\mathcal A(H)$, respectively, the *set of units* (or *invertible elements*) and the *set of atoms* (or *irreducible elements*) of $H$ (an element $a \in H$ is an atom if $a \notin H^\times$ and $a = xy$ for some $x, y \in H$ implies $x \in H^\times$ or $y \in H^\times$... | https://mathoverflow.net/users/16537 | A non-reduced, commutative BF-monoid s.t. $au = u$ for all $a \in \mathcal A(H)$ and $u \in H^\times$ | The answer seems yes if I understood the question. Take $G$ to be any commutative group and $N$ be the free semigroup on one-generator $x$ (so isomorphic to the positive natural numbers under $+$, but I want to use multiplicative notation). Let $H=G\cup N$ where the operation on $G$ and $N$ are their original operation... | 3 | https://mathoverflow.net/users/15934 | 263362 | 118,479 |
https://mathoverflow.net/questions/260525 | 5 | Given $n$ and $B\_n= \lceil H\_n \rceil$, where the latter is the $n$th harmonic number $\sum^n\_{i=1} 1/i$, for most $n$ it is easy to pack the first $n$ terms of the harmonic series into $B\_n$ many unit bins. An interesting question is for which $n$ one cannot perform such a packing. My guess is there are no such $n... | https://mathoverflow.net/users/3402 | Exact bin packing the harmonic series: references? | The next $n$ for which there is an exact packing are $24 \leq n \leq 30$. This is because
$$
1 = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{1}{4} + \frac{1}{5} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{15}+ \frac{1}{18} + \frac{1}{20} + \frac{1}{24}.
$$
The sum of the remaining fractions is $< 1$ for $... | 5 | https://mathoverflow.net/users/48142 | 263363 | 118,480 |
https://mathoverflow.net/questions/263351 | 2 | This question is a follow-up on [another MO query here](https://mathoverflow.net/questions/263265/inequality-connected-to-lagranges-interpolation-formula).
>
> **Question.** For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P\_r(x\_1,\dots,x\_n)$ with **positive coefficients** s... | https://mathoverflow.net/users/66131 | Lagrange interpolation vs homogeneous symmetric polynomials? | Denote by $\delta\_n$ the partition $(n-1,n-2,\dots,1)$ of $\binom{n}2$, and the associated schur function is
$$s\_{\delta\_n}(x\_1,\dots,x\_n)=\prod\_{i<j}(x\_i+x\_j).$$
Consider now the ratio of determinants
$$\begin{vmatrix}
x\_1^{2r-1} & x\_2^{2r-1} &\cdots & x\_n^{2r-1} \\
x\_1^{2n-4} & x\_2^{2n-4} &\cdots & x\_n^... | 6 | https://mathoverflow.net/users/2384 | 263372 | 118,484 |
https://mathoverflow.net/questions/262741 | 7 | Let $\lambda$ be a partition. Suppose that $\lambda$ is both $2$- and $3$-decomposable, in the sense that $\lambda$ admits a total decomposition by both $2$-rim hooks (aka dominos) and $3$-rim hooks. Equivalently, assume that the $2$-core and $3$-core of $\lambda$ is zero. Then what can be said about the $6$-core of $\... | https://mathoverflow.net/users/105141 | What can be said of a $6$-core Young diagram whose $2$-and $3$-cores are empty | Let me comment a bit further on the fact that there are other 6-cores with trivial 2 and 3-core, in fact infinitely many of them. The fundamental paper of Garvan, Stanton, Kim, "Cranks and T-cores", gives a bijection between t-cores and integer tuples $(n\_0,n\_1,\dots,n\_{t-1})$ which satisfy $\sum\_{i=0}^{t-1}n\_i=0$... | 3 | https://mathoverflow.net/users/2384 | 263375 | 118,487 |
https://mathoverflow.net/questions/257084 | 2 | Let say I have a hyperbolic system of conservation law. How do I show there is a blow up in finite time? For a single conservation law, I think, I could just show that there is collision of characteristics given a certain velocity for example $a(u)=u$ in Burgers' equation.
I would also like to know good books or any ... | https://mathoverflow.net/users/99453 | blow up in finite time of hyperbolic system of conservation law | For general hyperbolic systems in one space and on time dimension, the result is treated in
John, F.
Formation of singularities in one-dimensional nonlinear wave propagation
Comm. Pure Appl. Math., 1974, 27, 377-405
The higher dimensional case is not completely understood at present. There's a lot of work by... | 1 | https://mathoverflow.net/users/3948 | 263379 | 118,488 |
https://mathoverflow.net/questions/263360 | 5 | Let $P$ denote the set of positive primes and let $p$ be a fixed prime. Then define $q\_{p}:=\min{q\in P:p<q,(q/p)=1}$ where $(⋅/p)$ is the Legendre symbol. So for instance $q\_{3}=7$, $q\_{3}=7$, and $q\_{5} = 11$. Is there anything known about $q\_{p}$ and specifically are there known bounds on $q\_{p}$ as a function... | https://mathoverflow.net/users/90403 | Smallest prime that is a quadratic residue modulo a fixed prime | Elaborating on Gerhard Paseman's idea, under a generalized Elliott-Halberstam conjecture the [Polymath8b paper](https://arxiv.org/abs/1407.4897) shows that there are infinitely many $n$'s such that at least two elements of $\{n,n+36,n+100\}$ are primes. In other words, assuming this conjecture, there are infinitely man... | 8 | https://mathoverflow.net/users/11919 | 263380 | 118,489 |
https://mathoverflow.net/questions/263345 | 15 | I am looking for an extension $F/\mathbb{Q}$ with the following properties:
1. $F/\mathbb{Q}$ is Galois with $\mathrm{Gal}(F/\mathbb{Q}) \simeq A\_4$.
2. $F$ is totally real.
3. The prime $2$ has full decomposition group.
4. For every odd prime $p$ that ramifies in $F/\mathbb{Q}$, the primes above $p$ all have decomp... | https://mathoverflow.net/users/7443 | Search for $A_4$-extension of $\mathbb{Q}$ with particular ramification properties | There is no such field. In fact, let $F$ be an $A\_4$-extension of the rationals,
and let $K$ denote the cyclic cubic subfield of $F$. Then $F = K(\sqrt{\alpha},\sqrt{\alpha'}\,)$ for some $\alpha \in K$ such that $\alpha\alpha'\alpha''$ is a square in ${\mathbb Q}$. Here $\alpha'$ is the conjugate of $\alpha$ etc.
E... | 14 | https://mathoverflow.net/users/3503 | 263383 | 118,491 |
https://mathoverflow.net/questions/263175 | 2 |
---
**Setup:**
Let $C\_n$ be a closed $n$-simplex in $\mathbb{R}^n$ and let $r \in (0,R)$ where $R$ is the distance any one of the vertices $\{v\_1,\cdots , v\_{n+1}\}$ of $C\_n$ to the centroid $\frac{v\_1+ \cdots v\_{n+1}}{n+1}\in C\_n$.
---
**Question:**
Is there a way or removing a connected open set... | https://mathoverflow.net/users/36886 | Isometry between punctured sphere and punctured triangle? | There is no such $A$. Let $c$ be a point from the boundary of $C\_n$ that belongs to an intersection of at least two distinct facets $F\_1, F\_2$ (so $c$ is in a face of dimension at most $n-2$). In the plane, $c$ is a vertex of the triangle $C\_2$.
Since the simplex is regular, the angle between any two of its facet... | 2 | https://mathoverflow.net/users/24076 | 263387 | 118,493 |
https://mathoverflow.net/questions/263407 | 5 | In the paper, P Erdos, R Graham, I Ruzsa, E Straus, *On the prime factors of $\binom{2n}n$*, Math. Comp., 29:83–92, 1975, it was conjectured that the central binomials are never square-free for $n>4$. The proof was given in A Granville, O Ramare, *Explicit bounds on exponential sums and the scarcity
of squarefree binom... | https://mathoverflow.net/users/66131 | Central binomial coefficients deprived of $2$'s: not radicals? | The real thrust of the work of Granville and Ramare was to get explicit bounds, so that one could get a complete resolution of the Erdos problem. Earlier work of [Sarkozy](http://www.sciencedirect.com/science/article/pii/0022314X85900174) already gave asymptotic results that are quite a bit sharper. Thus, Sarkozy showe... | 14 | https://mathoverflow.net/users/38624 | 263414 | 118,503 |
https://mathoverflow.net/questions/259691 | 3 | Let $G$ be a locally compact topological group with closed subgroups $H, N$ and $H$ normalizing $N$. Then $H$ acts continuously on $N$ by conjugation. If it will help, assume that $N$ is nilpotent, so that Haar measures on $N$ and all its closed subgroups are unimodular.
If $n \in N$, let $O = O(n)$ be the orbit of $... | https://mathoverflow.net/users/38145 | Measure on orbits of $N$ under conjugation by $H$ | One should take care here.
While when considering algebraic group actions typically orbits are locally closed, in general this is not the case.
Consider for example the multiplication action of $H=\mathbb{Q}^\*$ on $N=\mathbb{R}$ and form the corresponding semidirect product $G=H\ltimes N$.
Topologize it by taking t... | 1 | https://mathoverflow.net/users/89334 | 263416 | 118,504 |
https://mathoverflow.net/questions/262300 | 10 | Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called *tamed* if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $v$.
**Question.** Is there an example of an almost complex structure on $\mathbb CP^2$ such that any $C^{\infty}$ smal... | https://mathoverflow.net/users/13441 | Almost complex structures on $\mathbb CP^2$ that are not tamed | Summarising the discussion above and Daniel Ruberman's helpful clarifications below.
Any symplectic structure on $\mathbb{C}P^2$ is standard by a result due to Gromov and Taubes. By Siebert-Tian every symplectic surface in $\mathbb{C}P^2$ of degree at most 17 is smoothly isotopic to an algebraic surface. In particula... | 5 | https://mathoverflow.net/users/48067 | 263418 | 118,505 |
https://mathoverflow.net/questions/263395 | 1 | Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi\_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ which are not in the inner set, given an outer automorphism $\phi(x)$, we can create a new group which has the presentat... | https://mathoverflow.net/users/105528 | What happens when you internalize outer automorphisms? | Your group is just the semidirect product $\Gamma=G\rtimes\_\psi\mathbb{Z}$. It's easy to check that the isomorphism class of $\Gamma$ only depends on the conjugacy class of $\psi$ in $\mathrm{Out}(G)$; in particular, if $\psi$ were inner this would be the direct product.
One trivial corollary is that such $\Gamma$ i... | 3 | https://mathoverflow.net/users/1463 | 263421 | 118,507 |
https://mathoverflow.net/questions/263420 | 4 | Let $G= (V,E)$ be a simple undirected graph on $\kappa$ vertices, where $\kappa$ is a finite or infinite cardinal. Let $b:\kappa\to V$ be a bijection. We assign to $b$ the greedy coloring $c\_b$ constructed by traversing the graph in the order $b$. Formally, with recursive definition of $c\_b:\kappa \to V$:
* $c\_b(0... | https://mathoverflow.net/users/8628 | Greedy coloring for infinite graphs | Yes, the same holds for infinite graphs. I'll provide here a probably overlong proof of this.
Let $G = (V,E)$ be an infinite graph of size $\kappa$, and let $\chi(G) = \lambda$. If $\lambda = \kappa$, then any bijection is chromatic, so we may assume that $\lambda < \kappa$.
Let $c:V \rightarrow \lambda$ be a chrom... | 6 | https://mathoverflow.net/users/26002 | 263434 | 118,513 |
https://mathoverflow.net/questions/263436 | 1 | I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank constraint). In particular I'm searching for a penalization (soft constraint) or hard constraint in the form
$$f(x) <= n\_{max... | https://mathoverflow.net/users/105484 | minimize number of unique elements in a vector | While $f(x)=\sum\_i \sum\_j (x\_i-x\_j)^2$ is not quite what you might want, as it would favour vectors with small discrepancy in the entries, it's at least a very nice convex function to minimise, and could be very efficient for certain sets of feasible data.
On the other hand if the entries of your $x$ are only, say,... | 0 | https://mathoverflow.net/users/11100 | 263437 | 118,514 |
https://mathoverflow.net/questions/263397 | -7 | Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
>
> **Question.** Is this true? If so, is there an underlying transformation or just a proof?
> $$\int\_0^{\infty}x^2e^{-x^2}\frac{{dx}}{\cosh\sqrt{\pi}x}
> =\frac14\int\_0^{\infty}e^{-x^2}\frac{dx}{\... | https://mathoverflow.net/users/66131 | Is there a transformation or a proof for these integrals? | This is a generous explanation of Lucia's comment above.
The functions
$$\mathcal H\_n(x)=\frac{2^{1/4}}{(2^n n!)^{1/2}}H\_n(\sqrt{2\pi}\; x) e^{-\pi x^2}$$
form an orthonormal system in $L^2(\textbf{R})$. Here $H\_n(x)$ are the usual Hermite polynomials
defined by
$$e^{2xz-z^2}=\sum\_{n=0}^\infty \frac{H\_n(x)}{n!}... | 9 | https://mathoverflow.net/users/7402 | 263471 | 118,524 |
https://mathoverflow.net/questions/263262 | 9 | This question was [originally asked and bountied at MSE](https://math.stackexchange.com/questions/2035333/dodgy-turing-degrees), but received no answer there, so I'm asking it again here.
*Below, I'm specifically interested in **weak truth table** (wtt) reducibility, but other reducibilities between truth table and T... | https://mathoverflow.net/users/8133 | Dodgy Turing degrees | Every sufficiently large degree is dodgy. Assuming $X \geq\_T \emptyset'$, we can define $F(Y)$ as
follows. First compute a set $Z$ from $X$ and $Y$: Given $n
= \langle i,j \rangle$, ask $X$ whether $\Phi\_i(n)$ converges. If not
then $Z(n)=0$. If yes then ask $X$ whether $\Phi\_j^{Y \upharpoonright
\Phi\_i(n)}(n)$ con... | 6 | https://mathoverflow.net/users/47312 | 263475 | 118,526 |
https://mathoverflow.net/questions/263435 | 1 | My question could be resume in the following way :
>
> Let $\mathfrak{t} \to \mathrm{End}(V)$ a representation of an *abelian* Lie algebra into an *infinite* dimensional vector space.
>
>
> What can we say about this representation ? In particular, can we decompose it into a *product* of representation of dimens... | https://mathoverflow.net/users/83320 | link between completion of the universal enveloping algebra and an endomorphism of functor | It sounds like you want to prove that $\text{End}(F)$ is the profinite completion of the universal enveloping algebra $U(\mathfrak{g})$. I don't understand your strategy for proving this (in particular, the answer to your first question as stated is clearly "no," consider an infinite direct sum), but the following stat... | 4 | https://mathoverflow.net/users/290 | 263476 | 118,527 |
https://mathoverflow.net/questions/262629 | 1 | It is known that for a completely bounded map $\psi:A\to B(H)$ there exist completely positive maps $\phi\_1,\phi\_2:A\to B(H)$ such that
$$\Vert \phi\_i\Vert\_{cb}=\Vert \psi\Vert\_{cb},$$
and the map $\Phi:M\_2(A)\to B(H\oplus H)$, given by
$$\Phi\left(\left[\begin{array}{ll}a&b\\c&d\end{array} \right]\right)=\left[... | https://mathoverflow.net/users/104535 | Extensions of completely positive maps | It's always worth exploring results and proofs to see if you can get more from them. By pushing the result you claim a bit further, we arrive at the standard representation theorem for cb maps (produced using Stinespring on the $2\times 2$ matrix map; see Effros + Ruan section 5.3 or Paulsen's book, for example). Namel... | 4 | https://mathoverflow.net/users/406 | 263485 | 118,529 |
https://mathoverflow.net/questions/263356 | 6 | Let $P$ be an infinite extra special $p$-group for some prime $p$, namely, $Z(P)=P'=\Phi(P)$ and $P/Z(P)$ is infinite elementary abelian.
Let $C$ be a Prufer $q$-group for some prime $q\neq p$.
**Question**
Is it possible to find an action of $C$ on $P$ such that $C$ acts irreducibly on $P/Z(P)$? (i.e. $P/Z(P)$ doe... | https://mathoverflow.net/users/45296 | Extra special p-groups | My construction does not work for all $p$ and $q$. I need to assume that $p$ and $q$ are both odd, that $P$ is of exponent $p$, and that the multiplicative order of $p$ modulo $q$ is even. I don't know whether such an action exists for other $p,q$ - I would guess not.
Let us assume that $P$ is the central product of ... | 2 | https://mathoverflow.net/users/35840 | 263487 | 118,530 |
https://mathoverflow.net/questions/263482 | 7 | Let $a\_1,\dots,a\_n, b$ be positive real numbers.
>
> \*Question.\*\* Is this true?
> $$\int\_{-\infty}^{\infty}\frac{\sin(bx+a\_1x+\cdots+a\_nx)}{x}\prod\_{j=1}^n\frac{\sin(a\_jx)}{a\_jx}\,\,dx=\pi.$$
> My most immediate quest is "why is it independent of $b$, in particular?"
>
>
>
| https://mathoverflow.net/users/66131 | "sinc-ing" integral | Let $s(x) = \frac{\sin(\pi x)}{\pi x}$ be the normalized sinc function, $a\_j' = \frac{a\_j}{\pi}$ and $c = \frac{b+\sum\_j a\_j}{\pi}$. Then you want to compute $\pi\int\_{-\infty}^\infty c s(cx)\prod\_j s(a\_j' x)dx$. This is equal to the value of the Fourier transform of the integrand at zero. The Fourier transform ... | 10 | https://mathoverflow.net/users/5963 | 263489 | 118,531 |
https://mathoverflow.net/questions/263274 | 3 | Considering the following equation,
$$
u\_t + \operatorname{div} \, (u \, \mathbf{b}(\mathbf{x},t)) = 0
$$
in a cylinder $K = \{(\mathbf{x},t) \in \Omega \times (0,T) \}$ where $\Omega \subset \mathbb{R}^d$ is regular (whatever we need), with initial condition
$$u |\_{t=0} = u\_0$$
and
$$\mathbf{b} \cdot \mathbf{n} =... | https://mathoverflow.net/users/97620 | Uniqueness conditions for linear transport equation with nonconstant velocity | Well, the answer comes from a quite simple energy argument.
After multiplying the equation by $u$, integrating by parts and using
$$
\pmb{b} \cdot \nabla u = \operatorname{div} ( \pmb{b} u ) - \left( \operatorname{div} \pmb{b} \right) u,
$$
one can get
$$
\frac{1}{2} \frac{d}{dt} \int\_{\Omega} u^2 d\Omega = - \int\_{... | 1 | https://mathoverflow.net/users/97620 | 263494 | 118,532 |
https://mathoverflow.net/questions/263478 | 1 | Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section.
What can we say about the singularities of $X$?
| https://mathoverflow.net/users/nan | Divisor class group of quartic surfaces | Here are a few comments-too long to be in a comment.
First, it is clear that a singular point can only have multiplicity at most 4 and if it is of multiplicity 4, then it is a cone over a smooth quartic plane curve, and then your hypothesis will be violated.
If the multiplicity is 3, again, by projectiing, you see ... | 3 | https://mathoverflow.net/users/9502 | 263497 | 118,533 |
https://mathoverflow.net/questions/263133 | 1 | Let $(M, g)$ be a compact, connected Riemann without boundary and let $f: M\rightarrow M$ be Anosov. The nonwandering set of $f$, $\Omega (f)$, has a decomposition into finitely many closed, invariant, "basic" sets
$\Omega (f) =B\_1 \;\cup \;... \;\cup \;B\_n$
So that $f|\_{B\_i}$ is topologically transitive. We a... | https://mathoverflow.net/users/105318 | Is there some point of intersection between the global stable and unstable manifolds of basic sets of an Anosov diffeomorphism? | Not a priori. This would immediately imply transitivity which is an open problem
| 1 | https://mathoverflow.net/users/105588 | 263515 | 118,539 |
https://mathoverflow.net/questions/263505 | 5 | Given a topological group $G$, a $G$-space is a topological space $X$ equipped with an action of $G$, such that the map $(g,x) \mapsto g.x$ is continuous. The action is distal if no non-diagonal orbit of $G$ on $X \times X$ has an accumulation point on the diagonal, and minimal if every orbit is dense.
A theorem of F... | https://mathoverflow.net/users/4053 | Furstenberg decomposition for non-compact spaces | Here is a counter example:
take $G=\text{SL}\_2(\mathbb{R})$ and $X=\mathbb{R}^2-\{0,0\}$.
It is easy to check that the action is distal, and that there is no non-trivial factor carrying an invariant metric (the only proper factor is $\mathbb{P}^1(\mathbb{R})$).
---
More generally (I think), taking any simple Lie... | 4 | https://mathoverflow.net/users/89334 | 263522 | 118,541 |
https://mathoverflow.net/questions/263488 | 3 | I expect this to be true and proven, but I can't find any proofs of this. So anyone can confirm or deny this?
Let $R$ be a commutative ring, and let $M$ be a $kn\times kn$ matrix, which can be split into $n^{2}$ block of dimension $k\times k$. Assuming the block matrices form a solvable Lie algebra. Then the determin... | https://mathoverflow.net/users/105303 | Determinant of block matrix | It is true when $R$ is reduced, without $\mathbb{Z}$-torsion. If your blocks are $(M\_{i,j})\_{1 \leq i,j \leq n}$ and if
$$N = \sum\_{\sigma \in \mathfrak{S}\_n} \epsilon(\sigma) M\_{1,\sigma(1)} \dots M\_{n,\sigma(n)},$$
then $\mathrm{det}(M) = \mathrm{det}(N)$.
Indeed, if $R$ is reduced without $\mathbb{Z}$-torsi... | 1 | https://mathoverflow.net/users/21724 | 263526 | 118,543 |
https://mathoverflow.net/questions/263543 | 1 | May be this question turns out trivial, but I can't figure out. I asked on StackExchange, but no one answers.
Let $S$ be a set, or equivalently the topological space with discrete topology and $2$ two point set with discrete tiopology. The $βS$ be the Stone–Čech compactification of $S$. By Tychonoff theorem the topol... | https://mathoverflow.net/users/73577 | Clopen subsets of $P(S)\times {\beta S}$ | Your $X = \{(A,\Sigma) : A\in\Sigma\} \subseteq P(S) \times \beta S$ is never open.
Indeed, let $\Sigma\_0$ be a nonprincipal ultrafilter on $S$. If $X$ were open, then in particular $\{A : A\in\Sigma\_0\}$ would be open in $P(S)$ (as the inverse image of $X$ under the continuous map $P(S) \to P(S) \times \beta S$ gi... | 3 | https://mathoverflow.net/users/17064 | 263545 | 118,548 |
https://mathoverflow.net/questions/263426 | 4 | Let $u(t, x)$ be the unique solution of the heat equation on the unit interval with Dirichlet boundary conditions and initial data $u\_0$:
$$
\left\{
\begin{array}{l}
\partial\_t u(t, x) = \partial\_x^2 u(t, x), \quad t > 0,\ x \in [0, 1] \\
u(t, 0) = u(t, 1) = 0, \quad t > 0 \\
u(0, x) = u\_0(x), \quad x \in [0, 1]
\... | https://mathoverflow.net/users/nan | Heat equation close to the steady state | Let $v:=\partial\_xu$. It still satisfies the heat equation $\partial\_tv=\partial\_x^2v$, but with the **Neumann boundary condition**:
$$\partial\_xv(t,0)=\partial^2u(t,0)=\partial\_tu(t,0)=\partial\_t0=0,$$
and the same at $x=1$. Therefore you may apply the maximum principle:
$$\sup\_x|v(t,x)|\le\sup\_x|v(0,x)|=\sup\... | 2 | https://mathoverflow.net/users/8799 | 263553 | 118,551 |
https://mathoverflow.net/questions/263538 | 2 | I am a beginner in graph theory and I am interested in finding an upper bound for the chromatic number of the following class of graphs:
>
> 1. If two vertices $a$ and $b$ are adjacent in $G$, then there exist vertex $c$ such that $abc$ is a triangle graph. In other words, there exist vertex $c$ such that $c$, $a$ ... | https://mathoverflow.net/users/90655 | Upper bound on chromatic number for some graphs | For any $H$ (in particular for $K\_4$) there exist a constant $c(H)$ such that the chromatic number of any $H$-free graph $G$ does not exceed $c(H)\frac{d\log\log d}{\log d}$, $d=\Delta(G)$ (provided that $d>0$). It is [(conjecture 3.1.)](https://www.tau.ac.il/~nogaa/PDFS/logf4.pdf) conjectured (or already proved? oir ... | 2 | https://mathoverflow.net/users/4312 | 263559 | 118,554 |
https://mathoverflow.net/questions/263182 | 6 | I have a polynomial of degree 8 in 6 variables given explicitly by
$$ (\sqrt{1+(x\_1+x\_2+x\_3)^2+(y\_1+y\_2+y\_3)^2}+\sqrt{1+x\_1^2+y\_1^2}+\sqrt{1+x\_2^2+y\_2^2}+\sqrt{1+x\_3^2+y\_3^2})\times\text{the other seven of its Galois conjugates}. $$
I fed it into Maple and it shows it's irreducible. But being unsure of ... | https://mathoverflow.net/users/37103 | Check irreducibility of an explicit polynomial, without computer | For new variables $x$ and $y$ set $x\_1=x\_2=x\_3=x$ and $y\_1=y$, $y\_2=2y$, $y\_3=3y$. Then
$$
\prod\left(\sqrt{1+9x^2+36y^2}\pm\sqrt{1+x^2+y^2}\pm\sqrt{1+x^2+4y^2}\pm\sqrt{1+x^2+9y^2}\right)=-1024\left((3y^2 + 4)x^6 + (23y^4 + 47y^2)x^4 + (45y^6 + 180y^4)x^2 + 225y^6\right).
$$
This polynomial has degree $8$, and if... | 8 | https://mathoverflow.net/users/18739 | 263564 | 118,555 |
https://mathoverflow.net/questions/263562 | 1 | **Question.** Is there a finite, simple undirected graph $G=(V,E)$ with more than $1$ vertex such that there is only $1$ coloring bijection (defined below) for $G$?
---
We denote by $\mathbb{N}$ the set of positive integers and set $[n] = \{1,\ldots,n\}$ for $n\in\mathbb{N}$.
Let $G= (V,E)$ be a simple undirect... | https://mathoverflow.net/users/8628 | Graph with only one coloring bijection | For every graph, every optimal coloring of the graph, and every ordering of the colors of the graph, there is another optimal coloring (possibly the same) in which each color class is maximal among the remaining vertices not included in earlier color classes. For this maximal coloring, any ordering of the vertices cons... | 2 | https://mathoverflow.net/users/440 | 263565 | 118,556 |
https://mathoverflow.net/questions/263563 | 9 | L Moser and M Wyman, *On solutions of $x^d = 1$ in symmetric groups*, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations:
$$f\_d(n):=\#\{\pi\in\mathfrak{S}\_n:\, \pi^d=1\}.$$
In particular, $f\_2(n)$ counts the number of *involutions* in the symmetric group $\... | https://mathoverflow.net/users/66131 | Solutions of $x^d=1$ in the symmetric group | Q2. Of course, this is a general thing for exponential generating functions. Assume that $a(n)$ is the number of ways to make lunch from $n$ distinct ingredients, $f(z)=\sum \frac{a(n)}{n!} z^n$ is an exponential generating function. Then, say, $f^2$ is an exponential generating function for making two enumerated lunch... | 8 | https://mathoverflow.net/users/4312 | 263566 | 118,557 |
https://mathoverflow.net/questions/263509 | 5 | Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I may be missing something rather obvious here, but unlike for Hilbert-Schmidt norms, very little useful methods seem to b... | https://mathoverflow.net/users/105584 | Trace-norm of integral operator | This is typically not a trace class operator. The problem is that the kernel
$$
K(t,u) = \int \overline{f(s+t\_1,t\_2)}f(s+u\_1,u\_2)\, ds
$$
of $T^\*T$ depends on the first coordinates only through the difference $t\_1-u\_1$, so has no uniform decay in these directions. It follows that $\int\!\!\int |K|^2\, dt\,du =\i... | 3 | https://mathoverflow.net/users/48839 | 263568 | 118,558 |
https://mathoverflow.net/questions/263585 | 2 | For which odd values of $n\in \mathbb{N}$, $n \neq 7$, does the volume form $\alpha$ on $S^n$ admit a wedge product decomposition $\alpha = \beta \wedge \gamma$ such that neither $\beta$ nor $\gamma$ is a $1$-form?
For which even values of $n\in \mathbb{N}$ does the volume form $\alpha$ on $S^n$ admit a wedge product... | https://mathoverflow.net/users/36688 | Decomposition of the volume form on the sphere | For all odd $n=2k{+}1>3$ one can write the volume form $\alpha$ on $S^n$ as a wedge product $\alpha=\beta\wedge\gamma$ with $\mathrm{deg}(\beta)$ and $\mathrm{deg}(\gamma)$ both greater than $1$. Just note that there is a $1$-form $\theta$ such that $\alpha = \theta\wedge(\mathrm{d}\theta)^k$ where $n=2k{+}1$, and set ... | 9 | https://mathoverflow.net/users/13972 | 263591 | 118,565 |
https://mathoverflow.net/questions/263391 | 2 | Let $M$ be a compact connected 3-manifold and let $S$ be a closed connected surface in $\partial M$. Let $G$ be the image of the map $\pi\_1(S) \to \pi\_1(M)$ induced by inclusion. I was reading the first chapter of Jaco's "Lectures on 3-Manifolds" and as a corollary to the Loop theorem, he states that $G \cong F \ast ... | https://mathoverflow.net/users/99414 | Images of boundary surfaces in 3-manifold groups | Here's the reference I mentioned at tea today:
[MR0732345 (85j:57011)
Bonahon, Francis
Cobordism of automorphisms of surfaces.
Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 237–270.](http://www.numdam.org/item?id=ASENS_1983_4_16_2_237_0)
See section 2, which shows that given any irreducible manifold
$M$, t... | 2 | https://mathoverflow.net/users/1345 | 263608 | 118,570 |
https://mathoverflow.net/questions/263221 | 2 | Let $R$ be a complete DVR of mixed characteristic $(0, p)$, let $K$ be its fraction field, and assume that the absolute ramification index $e$ of $R$ satisfies $e < p - 1$ and that the residue field of $R$ is perfect. Let $G$ be a commutative finite $K$-group scheme of $p$-power order.
It is a classical result of Ray... | https://mathoverflow.net/users/63877 | Uniqueness of finite flat models over bases of low ramification via Breuil-Kisin modules | Yes, I think you can extract Raynaud's result straightforwardly from the Breuil--Kisin theory. Write $k$ for the residue field of $R$. Let's recall how the Breuil--Kisin theory works. Let $\phi : k[[u]] \to k[[u]]$ be the $p$th power map. Then there's an anti-equivalence between the category of $p$-torsion finite flat ... | 3 | https://mathoverflow.net/users/379 | 263610 | 118,571 |
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