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https://mathoverflow.net/questions/355122 | 0 | I trie to calculate the Hilbert class field of $\mathbb Q\left(\sqrt{-83}\right)$, whose class number is $3$, so I should find a cubic integral monic polynomial whose discriminant is $-83$, but I failed
| https://mathoverflow.net/users/147080 | Find a cubic monic integral coefficient whose discriminant is $-83$ | There's no guarantee that such a polynomial exists, because the corresponding cubic subfield of the Hilbert class field need not be monogenic. But it does actually exist in this case and one example is
$$x^3 - x^2 + x - 2$$
A really useful website for looking up polynomials whose corresponding number field has smal... | 10 | https://mathoverflow.net/users/153756 | 355132 | 149,951 |
https://mathoverflow.net/questions/355046 | 2 | Given a positive integer $n$, what is the minimum positive real number $b(n)$ such that for any $a\_1,\ldots,a\_n\in[0,1]$, some two subset sums differ by at most $b(n)$?
This is similar to [subset sum problems](https://cstheory.stackexchange.com/questions/17988/subset-sum-difference-problem), except it is a worst-ca... | https://mathoverflow.net/users/153694 | Minimum real number for subset sum difference | As I wrote in my comment, the trivial bounds are $1/2^{n-1}$ and $n/(2^n-1)$. Here is a proof of the estimate $b(n)<3\sqrt n/2^n$; maybe it can be improved further using the same idea.
Consider the random variable $X=c\_1a\_1+\dotsb+c\_na\_n$ where $c\_1,\dotsc,c\_n$ independently take values $0$ and $1$, with equal... | 2 | https://mathoverflow.net/users/9924 | 355133 | 149,952 |
https://mathoverflow.net/questions/355143 | 2 | Let $Z=[z\_1, \dots z\_n]$ be a $d \times n$ matrix, where the $z\_i$'s are iid random vactors with mean $\mu \in \mathbb{R}^d$ and $d \times d$ (population) covariance matrix $\Sigma$, but the entries $z\_{ij}'s $ are not necessarily iid. Consider the (unscaled) sample covariance matrix $C:= ZZ' \in \mathbb{R}^{d \tim... | https://mathoverflow.net/users/35936 | Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data | If the random vectors are *isotropic* (meaning $E[z\_i z\_i^T]=I$), you can use the lower bound derived by [Pavel Yaskov](https://projecteuclid.org/euclid.ecp/1465316785) (2014):
>
> **Abstract:** We provide tight lower bounds on the smallest eigenvalue of a sample
> covariance matrix of a centred isotropic random... | 1 | https://mathoverflow.net/users/11260 | 355145 | 149,955 |
https://mathoverflow.net/questions/355152 | 11 | In "Character Theory of Finite Groups" I.M. Isaacs mention the following conjecture:
It is only possible in a solvable group $G$ to have $\chi(1)^2=|G:Z(G)|$ with $\chi \in$ **Irr**$(G)$.
Is this problem still open? I tried to search for attempts to solve it but didn't find anything.
| https://mathoverflow.net/users/152333 | Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable? | Such groups are solvable. This has been solved by Howlett and Isaacs himself, in
*Howlett, Robert B.; Isaacs, I. Martin*, [**On groups of central type**](http://dx.doi.org/10.1007/BF01215066), Math. Z. 179, 555-569 (1982). [MR652860](https://mathscinet.ams.org/mathscinet-getitem?mr=652860) [ZBL0511.20002](https://zbm... | 13 | https://mathoverflow.net/users/10266 | 355154 | 149,958 |
https://mathoverflow.net/questions/355158 | 7 | Is there a refinement of Euler characteristic that distinguishes between the torus $S^1 \times S^1$ and the cylinder $S^1 \times [0,1]$?
(The intuition here is that $\chi$ is multiplicative, so that $\chi(S^1 \times S^1) = \chi(S^1) \times \chi(S^1) = 0 \times 0$, which “vanishes twice”.)
Relatedly, is there a sett... | https://mathoverflow.net/users/3621 | Refined Euler characteristic | Look up the [Poincare polynomial](https://topospaces.subwiki.org/wiki/Poincare_polynomial) $p\_X(t)$. It is still multiplicative, by the Kunneth formula. The Euler characteristic is $\chi\_X=p\_X(-1)$. We have $p\_{S^1\times S^1}(t)=(t+1)^2$ but $p\_{S^1\times [0,1]}(t)=(t+1)$, with the former having a double root at $... | 13 | https://mathoverflow.net/users/6107 | 355164 | 149,960 |
https://mathoverflow.net/questions/355171 | 0 | On pg 76 of Jech's Set Theory, he proves the existence of a nonprincipal ultrafilter on $\omega$ that is not a $p$-point.
>
> Given a partition $\{A\_n\}$ of $\omega$ into $\aleph\_0$ infinite pieces, let $F$ be the following filter on $\omega$.
>
> $X \in F$ if and only if except for finitely many $n$, $X \cap... | https://mathoverflow.net/users/153785 | A nonprincipal ultrafilter that is not a $p$-point | First of all, it is clear that the given Partition $\{A\_n\;|\;n\in\omega\}$ satisfies $A\_n\notin D$ for any n, since $\omega\smallsetminus A\_n\in F\subseteq D$ for any n (since $(\omega\smallsetminus A\_n)\cap A\_k=A\_k$ for all $k\neq n$).
Furthermore, assume $X$ has the property that $|X\cap A\_n|<\omega$ for an... | 3 | https://mathoverflow.net/users/138274 | 355173 | 149,962 |
https://mathoverflow.net/questions/354232 | 5 | Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these spaces interesting?
In particular, for what reasons would one like to consider spaces over
1. Complex-analytic space... | https://mathoverflow.net/users/130058 | Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc | This a small example and a couple of reflections.
Consider an *adic* map of formal schemes $f \colon \mathfrak{X} \to \mathfrak{Y}$. Its fiber are schemes. So, I would consider this a "relative scheme" over $\mathfrak{Y}$ where $\mathfrak{Y}$ might be some interesting completion of a scheme, perhaps of a scheme of ge... | 1 | https://mathoverflow.net/users/6348 | 355176 | 149,963 |
https://mathoverflow.net/questions/355174 | 14 | If $A,B\in{\bf M}\_n(k)$, then the following formula holds true:
$$\det(A+B)=\sum\_{r=0}^n\sum\_{|I|=|J|=r}\epsilon(I,I^c)\epsilon(J,J^c)A\binom IJ B\binom{I^c}{J^c}.$$
In this formula, $I$ and $J$ are ordered (increasingly) $r$-uplets in $[1,n]$, $A\binom IJ$ is the corresponding minor, $I^c$ is the ordered complement... | https://mathoverflow.net/users/8799 | Expansion of $\det(A+B)$ | The formula appears in
Marvin Marcus "determinants of sums". College mathematics journal, March 1990.
<https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinants-of-sums>
The author notes that it is hard to track it's origins, but mentions previous references in the text, in p... | 13 | https://mathoverflow.net/users/17980 | 355178 | 149,964 |
https://mathoverflow.net/questions/355192 | 0 | I have a bunch of ordered pairs x, y where 0 < x < y <= n (some given upper bound)
like S = [(1,2), (1,3), (1,4), (2,3), (3,4)]
I need to find the Length of the longest subset where all the numbers within the ordered pairs are unique.
Here that would be 2. eg: (1,4) (2,3)
I am writing code that deals with situa... | https://mathoverflow.net/users/153803 | longest possible chain from a collection of ordered pairs/ co-ordinates | This is an instance of the [*maximum independent set problem*](https://en.wikipedia.org/wiki/Independent_set_(graph_theory)#Maximum_independent_sets_and_maximum_cliques) (equivalently, maximum clique of the complement) in a graph with a node per ordered pair and an edge for each pair of ordered pairs that intersect. Yo... | 1 | https://mathoverflow.net/users/141766 | 355194 | 149,967 |
https://mathoverflow.net/questions/355053 | 4 | Let $1<p<\infty$ and $f\_n$ be a sequence in $L^p(S^1)$ that converges weakly to some $f$. Here $S^1$ is the circle so we are dealing with periodic functions.
Let us see if $| f\_n |^p - | f |^p - | f\_n-f |^p$ converges to zero in distribution sense. This means that for every smooth function $\phi$ on $S^1$, $\int\_... | https://mathoverflow.net/users/56524 | $| f_n |^p - | f |^p - | f_n-f |^p$ converges in distribution sense if $f_k$ converges almost everywhere and weakly to $f$? | Let $A\_{n,\epsilon}$ be the set where $|f|<\epsilon|f\_n|$. Now split the integral of $(|f\_n|^p-|f|^p-|f\_n-f|^p)\phi$ into an integral over $A\_{n,\epsilon}$ and an integral over its complement. Since the $L^p$ norm of $f\_n$ is bounded, the former part is bounded by a constant times $\epsilon$. The latter part conv... | 2 | https://mathoverflow.net/users/12120 | 355195 | 149,968 |
https://mathoverflow.net/questions/355111 | 3 | **Context:** In this interesting [blog post](https://golem.ph.utexas.edu/category/2009/10/generalized_operads_in_classic.html "Mike Shulman - Generalized Operads in Classical Algebraic Topology"), Mike Shulman indicates an approach for defining generalized types of operads. If I interpret the details correctly, (**edit... | https://mathoverflow.net/users/144100 | Semi-cocartesian operads | I'd like to emphasize that nowhere in the linked blog post did I talk about distributive laws. It's true that some people like to define generalized multicategories using distributive laws over $P$, but that's not my preferred framework. My preferred framework is the one I linked to in the post that Geoff Cruttwell and... | 4 | https://mathoverflow.net/users/49 | 355201 | 149,970 |
https://mathoverflow.net/questions/355220 | 1 | Is it possible to place the twelve pentominoes around a circle in such a way that if two of the pentominoes find themselves next to each other, it is because one of the two can be obtained from the other by cutting out one of its component squares (thus obtaining a tetramino) and glueing it elsewhere?
| https://mathoverflow.net/users/60732 | Generating all pentominoes by cutting and pasting | Yes. Ugly representation below (\* is the cell to be moved to . to obtain the next pentamino)
```
.
####* -> ###* -> ### -> ### -> ### -> *## -> ## -> ## -> ##. -> ### -> * -> ####. -> #####
. #. #* # . * #. ## ##... | 1 | https://mathoverflow.net/users/106512 | 355229 | 149,976 |
https://mathoverflow.net/questions/355235 | 5 | Let $A=KQ/I$ be a finite dimensional quiver algebra of finite global dimension.
Is it true that the dimension of $A/[A,A]$ is equal to the number of simples of $A$?
Here $[A,A]$ is the vector space generated by all elements of the form $ab-ba$.
Note that it is known that in general for such $A$ that the dimension of ... | https://mathoverflow.net/users/61949 | Commutator of finite global dimension algebras | Yes. See the result of Section 2.5 of a wonderful paper of Bernhard Keller :
<https://webusers.imj-prg.fr/~bernhard.keller/publ/ilc.pdf>
(and the references therein).
| 6 | https://mathoverflow.net/users/1306 | 355236 | 149,979 |
https://mathoverflow.net/questions/355208 | 1 | Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e\_i \otimes e\_j $ and a group of translations $T\_a$ defined by $T\_a e\_i \otimes e\_j = e\_{i+a} \otimes e\_{j+a}$.
Suppose that I have a bounded operator $A: l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z}) \to l^2(\math... | https://mathoverflow.net/users/143779 | Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ | Fine question. The first thing to notice is that your problem becomes simpler if you work in the $(1,1)$, $(0,1)$ basis of $\mathbb{Z}\times\mathbb{Z}$. I.e., we have an isomorphism from $\mathbb{Z}\times \mathbb{Z}$ to itself given by the matrix $ \left[\begin{matrix}1&0\cr
1&1
\end{matrix}\right]$, and conjugating by... | 1 | https://mathoverflow.net/users/23141 | 355246 | 149,981 |
https://mathoverflow.net/questions/355214 | 7 | Let $(X,d)$ be an arbitrary metric space and $E \subset X$ also arbitrary. Fix $s \in (0,\infty)$.
>
> Is it true that for any $ \delta > 0 $ and any collection of pairs $\{(A\_i,a\_i)\}\_{i \in \mathbb{N}}$ where $A\_i$ are subsets of $X$ and $a\_i \in [0,\infty]$, if
> $$
> \text{diam} \, A\_i \leq \delta \quad... | https://mathoverflow.net/users/91442 | Bounding an "integral" from below by the Hausdorff measure of the domain | It is true and deserves to be known better (it is morally equivalent to the statement that the Frostman lemma is just an exercise in linear programming duality though there are some pesky details I don't want to go into now).
I'll assume that all $A\_j$ are balls $B\_j$. It may change 5 to 10 but I don't think you ca... | 7 | https://mathoverflow.net/users/1131 | 355249 | 149,983 |
https://mathoverflow.net/questions/355226 | 4 | Let $X$ be a quasi-projective scheme, the followings are quite useful.
>
> 1. Every coherent sheaf is globally generated after tensoring with a suitable line bundle.
> 2. Every coherent sheaf has trivial higher cohomology groups after tensoring with a suitable line bundle.
> 3. Every coherent sheaf is a quotient of... | https://mathoverflow.net/users/110093 | Serre's theorem on global generations on stacks | About 1 & 2, I doubt there are sensible results without some hypothesis like the existence an ample family of line bundles that makes the stack actually a scheme, in fact a so-called divisorial scheme.
As for 3, there is a paper that settles the issue, namely a quasi-compact and quasi-separated algebraic stack has af... | 4 | https://mathoverflow.net/users/6348 | 355253 | 149,984 |
https://mathoverflow.net/questions/355198 | 2 | One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety
$X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.
In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "sp... | https://mathoverflow.net/users/108274 | Arc space & formal loops in motivic integration | I believe Nash originally chose the term "arc" to mean a short path, rather than a circle. The ring of functions on the completion of an algebraic curve over $k$ at a smooth $k$-point is isomorphic to $k[[t]]$, so maps from its spectrum to a variety $X$ can be viewed as infinitesimal pieces of a smooth curve in $X$. I ... | 4 | https://mathoverflow.net/users/121 | 355254 | 149,985 |
https://mathoverflow.net/questions/355242 | 11 | Let $L$ be the Lazard's universal ring, and $R=\mathbb{Z}[b\_1,b\_2,\cdots,b\_n,\cdots]$, regarded as a graded ring with the degree of $b\_i$ equal to $2i$. Let $\theta: L\rightarrow R$ be the homomorphism carrying the universal formal group law $\mu^L$ to the formal group law
$$\mu^R(x\_1,x\_2)=\exp(\log(x\_1)+\log(x\... | https://mathoverflow.net/users/100553 | Complex cobordism and Chern numbers | It is a key result that the composite
$$ MU\_\* \xrightarrow{h} H\_\*(MU;\mathbb Z) \xrightarrow[\sim]{\Phi^{\vee}}H\_\*(BU;\mathbb Z),$$
where $\Phi^{\vee}$ is the dual of the Thom isomorphism $\Phi$, agrees with evaluating on normal Chern numbers.
In other words, $\langle \Phi(c), h([M])\rangle = \bar c(M)$ for ... | 9 | https://mathoverflow.net/users/102519 | 355278 | 149,992 |
https://mathoverflow.net/questions/335924 | 3 | Let $X,Y$ be smooth complex varieties and let $G$ be an smooth affine algebraic group acting on $X$ and $Y$ such that $X,Y$ are $G$-homogeneous spaces (the $G$ action is transitive). We also let $f:Y \to X$ a smooth $G$-equivariant morphism.
Finally, consider $\mathcal{D}$ a sheaf of $G$-*homogeneous twisted differen... | https://mathoverflow.net/users/111845 | Pullback of homogeneous twisted differential operators | This is true.
One definition of htdo is analogues to that of equivariant sheaf: a htdo is a tdo $\mathcal D$ equipped with an isomorphism $act^\sharp \mathcal{D} \cong pr\_X^\sharp \mathcal{D}$ satisfying the cocycle condition, where $G \times X \xrightarrow{act} X$ and $G \times X \xrightarrow{pr\_X} X$ are the acti... | 1 | https://mathoverflow.net/users/99342 | 355284 | 149,994 |
https://mathoverflow.net/questions/354994 | 3 | Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
[Axiom of extension](https://en.wikipedia.org/wiki/Axiom_of_extensionality):
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \rightarrow x = y)
\end{equation}
[Axiom of heredity](https... | https://mathoverflow.net/users/74578 | Shortest axiom of infinity for foundationless set theory | **REMARK** I hope that *negation* is legal. Perhaps some $Sx$ statements can be harmlessly removed. Feel free to polish the statement below (I am not a logician).
**STATEMENT**
Set $b$ stands for *big* (infinite). Also, set $y$ strictly contains set $x$.
\begin{align}
\psi
&= \exists b (S b \land \exists a (a\in... | 2 | https://mathoverflow.net/users/110389 | 355285 | 149,995 |
https://mathoverflow.net/questions/355257 | 4 | Let $G/\mathbb{Q}$ be a connected reductive group, let $G^{\text{ad}}$ be the adjoint group, let $G^{\text{der}}$ be the derived group and let $\rho\colon G^{\text{sc}} \to G^{\text{der}}$ be the simply connected cover. Let $G^{\text{ad}}(\mathbb{R})^{0}$ be the identity component (in the real topology) of $G^{\text{ad... | https://mathoverflow.net/users/56856 | Quotienting $G(\mathbb{Q})_{+}$ by $G^{\text{sc}}(\mathbb{Q})$ and inner forms | The answer is **Yes**. We denote $K(G)=G({\mathbb Q})\_+/\rho G^{\rm sc}({\mathbb Q})$.
We compute $K(G)$; see the corollary below.
It is clear from the corollary that $K(G)$ is canonically isomorphic to $K(H)$.
We will use Section 3 of [M. Borovoi, Abelian Galois cohomology of reductive groups. Memoirs of the AMS 13... | 2 | https://mathoverflow.net/users/4149 | 355287 | 149,996 |
https://mathoverflow.net/questions/355286 | 8 | Consider a polynomial algebra $A=\mathbb{K}[x\_1,\ldots,x\_n]$ and its ideal $I$, such that $A/I=\mathbb{K}[y\_1,\ldots, y\_k]$. Is it true that there exist new polynomial generators $z\_1,\ldots,z\_n$ (in a sense that $A=\mathbb{K}[z\_1,\ldots,z\_n]$) such that $I=(z\_1,\ldots,z\_{n-k})$?
| https://mathoverflow.net/users/100359 | Polynomial algebra and its special ideals | This is called the Embedding Problem. I believe that it is false in positive characteristics and unknown in characteristic zero, except a few small cases. See [Kraft's review](https://mathscinet.ams.org/mathscinet-getitem?mr=1423629)
for the extent of my knowledge of the state of the problem. The counterexample in char... | 9 | https://mathoverflow.net/users/5301 | 355288 | 149,997 |
https://mathoverflow.net/questions/354548 | 2 | Let $G = (V,E)$ be a finite, connected, simple, undirected graph. By a *roundtrip of $G$* we mean a map $r:\{0,\ldots,n\} \to V$ for some $n\in\mathbb{N}$ with the following properties:
1. $r$ is surjective,
2. $\{r(k), r(k+1)\} \in E$ for all $k \in \{0, \ldots, n-1\}$, and
3. $r(0) = r(n)$.
An easy inductive argu... | https://mathoverflow.net/users/8628 | "Roundtrip"-chromatic number of (connected) graphs | There are graphs with $\chi(G) = 4$ and arbitrarily large $\chi\_r(G)$ (linear in the number of vertices) for a badly chosen roundtrip $r$.
A family of examples can be constructed as follows: For an even integer $n \geq 4$, consider the graph with vertex set $\{v\_i,v\_i',u\_i,u\_i'\mid 1 \leq i \leq n\}$ and edges
... | 3 | https://mathoverflow.net/users/97426 | 355299 | 149,999 |
https://mathoverflow.net/questions/354699 | 0 | This is a subquestion for an [older question](https://mathoverflow.net/questions/354548/roundtrip-chromatic-number-of-connected-graphs) about a certain kind of greedy coloring.
Let $G = (V,E)$ be a finite, connected, simple, undirected graph. By a *roundtrip of $G$* we mean a map $r:\{0,\ldots,n\} \to V$ for some $n\... | https://mathoverflow.net/users/8628 | Starting point of roundtrip coloring in connected graphs | My answer to your [other question](https://mathoverflow.net/questions/354699/starting-point-of-roundtrip-coloring-in-connected-graphs?noredirect=1&lq=1) can be modified to give a family of graphs and a roundtrip such that the difference $\chi(r,v)-\chi(r,v')$ is linear in the number of vertices. Let $G$ be a graph with... | 2 | https://mathoverflow.net/users/97426 | 355303 | 150,000 |
https://mathoverflow.net/questions/355309 | 7 | Observably, the number of primitive sorting networks on $n$ elements (or the number of commutation classes of reduced words of the longest element of $S\_n$) is even for $3\leq n\leq 15$. These are all the values for which it has been computed so far. Is there an abstract way to see that it is always even?
Sequence i... | https://mathoverflow.net/users/62135 | Is the number of commutation classes of reduced words of the longest element of $S_n$ even for $n\geq 3$? | On the linked OEIS entry I find the following:
>
> Also the number of mappings X:{{1..n} choose 3}->{+,-} such that for any four indices a < b < c < d, the sequence X(a,b,c), X(a,b,d), X(a,c,d), X(b,c,d) changes its sign at most once (see Felsner-Weil and Balko-Fulek-Kynčl reference). - Manfred Scheucher, Oct 20 20... | 7 | https://mathoverflow.net/users/1310 | 355319 | 150,004 |
https://mathoverflow.net/questions/355264 | 4 | Let $y^n = f(x)$ define a smooth projective curve $C$ over some field $k$ with $\deg f \geq n$ and odd and with $f(x)$ having no repeated roots. Let $J$ be the Jacobian of $C$ and $J[n]$ it's (geometric) n-torsion. Then is it true that the points $x= x\_i, y=0$ generate the group $J[n]$ for the roots $x\_i$ of $f(x)$?
... | https://mathoverflow.net/users/58001 | Torsion in the jacobian of a super elliptic curve | As mentioned by the previous answers, this cannot be true for $n \ge 3$ by size considerations. When you identify the points $(x\_i, 0)$ of $C$ inside its jacobian $J$, you are implicitly using some base-point. I will assume that $n$ and $\deg f$ are coprime, so that there is exactly one rational point at infinity (whi... | 9 | https://mathoverflow.net/users/153881 | 355320 | 150,005 |
https://mathoverflow.net/questions/355295 | 5 | I am interested in the topic of persistent homology (topological data analysis). According to what I read, there is some roadblock in the analysis of "big data" using persistent homology as it is considered rather computationally intensive. Computation of the simplicial complexes of a large point cloud often relies on ... | https://mathoverflow.net/users/83274 | Is there an upper bound on the number of points in point cloud for which we compute the persistent homology? | It depends very much on the type of simplicial complex you're using. If you have points in 3d then doing Cech/Delaunay complex is feasible with millions of points. If you have high dimensional data, complexes will generally blow up in size and millions of points will be too much. Finding ways to decrease the size of co... | 8 | https://mathoverflow.net/users/112954 | 355323 | 150,007 |
https://mathoverflow.net/questions/355228 | 5 | Let $U$ be an open subset of $\Bbb{R}^n$ and take $\omega$ to be a nowhere-vanishing smooth $1$-form on $U$. The [Frobenius Theorem](https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology)) implies that, near each point of $U$, $\omega$ may be written as $g\,{\rm{d}}f$ for suitable locally defined smoot... | https://mathoverflow.net/users/128556 | Obstruction to the existence of a globally defined integrating factor | To expand on [my comment](https://mathoverflow.net/questions/355228/obstruction-to-the-existence-of-a-globally-defined-integrating-factor?noredirect=1#comment891611_355228): suppose you have $U$ not simply connected, and $\omega$ closed but not exact, and suppose there exists a closed loop $\gamma: [0,1]\to U$ such tha... | 1 | https://mathoverflow.net/users/3948 | 355330 | 150,009 |
https://mathoverflow.net/questions/355294 | 1 | Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$ gives rise to a primitive L-function.
Is the converse true? That is, assuming Ramanujan conjectur... | https://mathoverflow.net/users/13625 | Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility? | Yes, and this is true without the Ramanujan conjecture (but see also Peter Humphries' comment below).
If $\pi$ is not irreducible, say $\pi=\pi\_1\oplus\pi\_2$, then $L(s,\pi)=L(s,\pi\_1)L(s,\pi\_2)$.
If $\pi$ is not cuspidal, then by Langlands' theory of Eisenstein series, there is a nontrivial partition $n=\sum... | 3 | https://mathoverflow.net/users/11919 | 355348 | 150,013 |
https://mathoverflow.net/questions/355354 | 7 | If $G$ is a group and $S\subseteq G$, let $\langle S \rangle$ be the intersection of all subgroups of $G$ containing $S$.
Let $S\_\omega$ denote the group of all bijections $f:\omega\to\omega$ with composition.
Is there $M\subseteq S\_\omega$ such that $\langle M \rangle =S\_\omega$, but for all $m\in M$ we have $\... | https://mathoverflow.net/users/8628 | Minimal generating set for $S_\omega$ | No.
Indeed, F. Galvin proved in 1995 that every countable subset of $S\_\omega$ is contained in a finitely generated subgroup (and also $S\_\kappa$ for every infinite $\kappa$). By contradiction suppose $M$ exists. Let $I$ be an infinite countable subset of $M$, so $I\subset \langle F\rangle$ for some finite $F$, and... | 14 | https://mathoverflow.net/users/14094 | 355356 | 150,017 |
https://mathoverflow.net/questions/355200 | 2 | Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric
$$
d(f,g):= \sum\_{n =0}^{\infty} \frac1{2^n} \sup\_{y \in K\_n} \min\left\{d\_X(f(y),g(y)),1\right\},
$$
where $\{K\_n\}\_{n \in... | https://mathoverflow.net/users/36886 | Metrizability of topology of compact convergence | According to Engelking (exercise 3.4E, which is based on a [paper](https://www.jstor.org/stable/1969087?seq=1) by Arens):
>
> If $C(X,\Bbb R)$ (with the compact-open topology and $X$ Tychonoff) is first countable, then $X$ is hemicompact.
>
>
>
A Hausdorff space $X$ is hemicompact if there is a countable famil... | 2 | https://mathoverflow.net/users/2060 | 355370 | 150,026 |
https://mathoverflow.net/questions/355379 | 7 | Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}\_\pi Y$ the projective tensor product.
**Question:**
If $X\hat{\otimes}\_\pi Y$ contains an isomorphic copy of $c\_0$, must then $X$ or $Y$ contain an isomorphic copy of $c\_0$ also?
| https://mathoverflow.net/users/46114 | Containment of $c_0$ in projective tensor products | The answer is no. Bourgain and Pisier have given a counterexample (A construction of $\mathcal{L}\_\infty$-spaces and related Banach spaces. Bol. Soc. Bras. Mat. 14, No. 2, 109-123 (1983). See Zbl 0586.46011 <https://zbmath.org/?q=an%3A0586.46011> ).
| 8 | https://mathoverflow.net/users/127871 | 355382 | 150,028 |
https://mathoverflow.net/questions/355385 | 2 | Suppose that $X\_{\theta}$ is an interpolation space between the Banach spaces $X\_0$ and $X\_1$. Let $\mathcal{B}$ be another Banach space.
**Is it true that $X\_{\theta}\times\mathcal{B}$ is an interpolation space between $X\_{0}\times\mathcal{B}$ and $X\_{1}\times\mathcal{B}$?**
Thank you for any suggestion.
| https://mathoverflow.net/users/54552 | Interpolation of product spaces | Yes, interpolation on product spaces works componentwise, so $$\Bigl(\prod\_{i=1}^n X\_i,\prod\_{i=1}^n Y\_i\Bigr) = \prod\_{i=1}^n (X\_i,Y\_i)$$ for any interpolation functor $(\cdot,\cdot)$ even with equal norms for a fixed choice of $\ell\_p$ norm on the product spaces. This follows from the restriction/corestrictio... | 6 | https://mathoverflow.net/users/85906 | 355389 | 150,031 |
https://mathoverflow.net/questions/212235 | 4 | Consider a (Hausdorff and complete) locally convex topological vector space $V$ and a countable subset $(v\_k)\_{k=1}^\infty \subset V$ of non-zero vectors.
>
> $(\*)$ Under what conditions on this subset are we guaranteed the existence of a sequence of positive real numbers $(\alpha\_k)\_{k=1}^\infty$, such that t... | https://mathoverflow.net/users/2622 | Existence of a countable linear combination with positive coefficients | As a partial answer, note the following necessary (and often sufficient) criteria.
Let $E$ be a (real or complex) topological vector space. Let us say that a sequence $\{x\_n\}\_{n=1}^\infty$ in $E$ *can be made summable/bounded/to converge to zero* if there is a sequence $\{\alpha\_n\}\_{n=1}^\infty$ of strictly pos... | 2 | https://mathoverflow.net/users/120251 | 355402 | 150,035 |
https://mathoverflow.net/questions/355398 | 0 | I have a problem with Proposition 7.2.10 in Bogachev's Measure Theory Volume II book on page 77 (I have link to my drive with that book <https://drive.google.com/file/d/1CgzgWhEiNPU1vy0YkyiVjGLH3Qiq1aCL/view>). Supposedly, from that proof one can deduced that $\text{supp}\,\mu$ is separable if $\mu$ is $\tau$-additive ... | https://mathoverflow.net/users/153916 | separable support of Borel measure, with tau-additive measure and full support | Following Bogachev's notation and terminology, let $S\_\mu$ be the intersection of all closed sets of full $\mu$-measure, and say $\mu$ "has support" if $S\_\mu$ itself has full measure. A useful fact is that $x \in S\_\mu$ if and only if every open neighborhood of $x$ has positive measure.
Now suppose $\mu$ is a fin... | 1 | https://mathoverflow.net/users/4832 | 355403 | 150,036 |
https://mathoverflow.net/questions/355397 | 16 | First of all I should apologies if this question does not count as a research level one. I asked the [same question](https://math.stackexchange.com/questions/3583228/understanding-sheaves-and-stacks) on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it here.
I am trying to understand... | https://mathoverflow.net/users/54507 | Understanding the definition of stacks | A canonical example of a sheaf of sets on a topological space $X$
is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$.
The gluing property then says that a continuous function on a union of open subsets $U\_i$ of $X$ is the same thing as a collection of continuous fun... | 20 | https://mathoverflow.net/users/402 | 355404 | 150,037 |
https://mathoverflow.net/questions/353288 | 3 | Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ implies $x=0$ (Similarly for $F$-non-degenerate). Equivalently, the two maps of $E$ to $F^{\*}$ and $F$ to $E^{\*}$ defi... | https://mathoverflow.net/users/150264 | Higher order functional derivatives |
>
> Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$
> \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}?
> $$
>
>
>
Yes, it is... | 2 | https://mathoverflow.net/users/113756 | 355408 | 150,039 |
https://mathoverflow.net/questions/355392 | 2 | I am looking for a good reference about the algebraic geometry of non-associative rings. I am in particular interested in derivation algebras.
Preferrably an online resource or a book that is available online, since every library near me is in lockdown right now.
| https://mathoverflow.net/users/70751 | Good reference on the algebraic geometry of non-associative rings | See "Seven Lectures on the Universal Algebraic Geometry" by
Boris Plotkin (Hebrew University). The text is in arXiv. Since the OP does not explain what "algebraic geometry" means, here are some explanations. The point is that there are several statements in the classical algebraic geometry which make sense and are eve... | 5 | https://mathoverflow.net/users/nan | 355418 | 150,042 |
https://mathoverflow.net/questions/355421 | 3 | I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}\_p^r$ for some integer $r\geq 0$ and let $V=G^\*={\rm Hom}(G,{\mathbb F}\_p)$. Then $(H(G,{\mathbb F}\_p),+,\cup )$ is a ring,
$$H(G,{\mathbb F}\_p)\cong\begin{cases}S(V)&p=2\\
\Lambda (V)\otimes S(V)&p>2\end{cases}.$$
Moreover, if $p=2... | https://mathoverflow.net/users/140180 | Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$ | They're essentially exercises, compute it for $r=1$ and then invoke Künneth, and I'd expect every book to include it: Adem--Milgram's classic book for example (Corollary II.4.3 and Theorem II.4.4).
You can look at the identical homology case in Brown's classic book (Theorem V.6.6) and in particular, his description i... | 9 | https://mathoverflow.net/users/12310 | 355423 | 150,044 |
https://mathoverflow.net/questions/353650 | 0 | **Question:** Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous?
For convexity to be well-defined, we need to assume that $X$ is a vector space, but not necessarily a normed space. The projection map is defined as
$$
P\_Sx
= \text{argmin}\_{y\in S} d(x,y... | https://mathoverflow.net/users/99132 | Example of a nonconvex Chebyshev set in a metric space with continuous projection? | Let $E$ be the (incomplete) subspace of sequences in $\ell^{2}(\mathbb{R})$ having at most finitely many nonzeros terms. In [1], a subset $S$ of $E$ is constructed (by a long induction argument), which has the following properties :
* $S$ is closed and nonconvex,
* each point in $E$ has a unique nearest point in $S$,... | 1 | https://mathoverflow.net/users/89429 | 355425 | 150,045 |
https://mathoverflow.net/questions/355338 | 7 | Let $P$ be a parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field. Let $M$ be a Levi subgroup of $P$, and let $N$ be the unipotent radical of $P$. If $(\pi,V)$ is a smooth, irreducible representation of $M$, extend $\pi$ to a representation of $P$ by making it trivial on $N$, and let $\sigma = \... | https://mathoverflow.net/users/38145 | Definition of functions in the induced space from parabolic induction | Let $H$ be any subgroup such that $H\backslash G$ is compact. Let $K$ be an open subgroup. Then there are $x\_1,...x\_n$ such that $G=Hx\_1 K \cup...\cup Hx\_nK.$ Suppose $f$ satisfies points 1 and 2. Replace $K$ with a smaller $K'$ so that $f(x\_i k)=f(x\_i)$ for all $i$ and for all $k \in K'.$ Given $a \in G,$ there ... | 2 | https://mathoverflow.net/users/64244 | 355436 | 150,051 |
https://mathoverflow.net/questions/354024 | 3 | **Problem:** Let $p$ be an odd prime number and consider the $n$-dimensional vector space over the field with $p$ elements. I want to prove that the number of $3$-term arithmetic progressions in a subset
$A\subseteq\mathbb{F}\_{p}^{n}$ is $cp^{2n}$ for every $n\geq N\_{0}$, for sufficiently large $N\_{0}$ and a constan... | https://mathoverflow.net/users/94132 | Asymptotic number of $3$-AP's in a set $A\subseteq\mathbb{F}_{p}^{n}$ of density $\epsilon$ | I was going to comment with a link to where this Varnavides idea is written up, but to my surprise I couldn't find one simply done in the case of $\mathbb{F}\_p^n$, so I thought I'd sketch the idea here. (Of course none of this is original to me, it's one of those proofs that is well-known in the field, and is a routin... | 3 | https://mathoverflow.net/users/385 | 355439 | 150,052 |
https://mathoverflow.net/questions/355447 | 1 | In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity:
$$f(x)=x^2\exp{(−x^8\sin^2{x})}.$$
Intuitively, it can be expected that $f(x)$ is square-integrable as its peaks become more and more nar... | https://mathoverflow.net/users/32389 | Square-integrable unbounded function | For $k=1,2,\dots$, let
$$I\_k:=\int\_{|x-k\pi|<1/k}f(x)^2\,dx
=\int\_{|x-k\pi|<1/k}x^4\exp(-2x^8\sin^2 x)\,dx.$$
Then, as $k\to\infty$,
$$I\_k\asymp k^4\int\_{|x-k\pi|<1/k}\exp\{-(2+o(1))(k\pi)^8\sin^2 x\}\,dx \\
=k^4\int\_{|x|<1/k}\exp\{-(2+o(1))(k\pi)^8\sin^2 x\}\,dx \\
=k^4\int\_{|x|<1/k}\exp\{-(2+o(1))(k\pi)^8 x^2... | 6 | https://mathoverflow.net/users/36721 | 355448 | 150,055 |
https://mathoverflow.net/questions/355449 | 2 | **Motivation.** I was wondering about the following when playing a card-shuffling game with my elder son.
If $\varphi: \omega \to \omega$ is a bijection, we define the *shuffling distance* of $\varphi$ by $$sh(\varphi) = \min\{|n-\varphi(n)| : n \in \omega\}.$$
Let $\varphi^{(1)}=\varphi$ and $\varphi^{(k+1)} = \varp... | https://mathoverflow.net/users/8628 | Increasing the "shuffling distance" by iterating a permutation $\varphi: \omega \to \omega$ | There's indeed an example of such a function.
Given a an infinite subset $I\subset\omega$, call indexation of $I$ the unique increasing bijection $\omega\to I$. If $(x\_n)$ is its indexation, define $f\_I$ the permutation of $I$, mapping $x\_1\mapsto x\_0$, $x\_{2n+1}\mapsto x\_{2n-1}$ ($n\ge 1$), $x\_{2n}\mapsto x\_... | 4 | https://mathoverflow.net/users/14094 | 355451 | 150,056 |
https://mathoverflow.net/questions/355459 | 1 | From Silverman's AEC page 332:
I need to understand why the determination of the following local kernel
$$
ker \Big( H^1(G\_v, E[\phi]) \rightarrow WC(E/K\_v)[\phi] \Big)
$$
is straightforward. The book says that it is the same as answering the questions whether a curve has a point over a complete local field (w... | https://mathoverflow.net/users/124242 | Clarification: Using Hensel's Lemma to determine $K_v$-rational points on a curve | you really shouldn't crosspost. Anyway, you've slightly misstated Hensel's lemma, you left out the assumption that $f(x)$ has a **simple** root in $R\_v/\mathcal{M}\_v$. That's where the $e$ is coming from. In general, if $f(x)$ has a root of higher multiplicity in $R\_v/\mathcal{M}\_v$, then you need to work in $R\_v/... | 2 | https://mathoverflow.net/users/11926 | 355467 | 150,057 |
https://mathoverflow.net/questions/355465 | 6 | It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act freely on $\mathbb{H}^n$ up to conjugation.
The correspondece is given by $\Gamma < Isom(\mathbb{H}^n)\mapsto \mathbb{H}^... | https://mathoverflow.net/users/99042 | Hyperbolic manifolds with infinite cyclic fundamental group | This consists in classifying non-elliptic elements of the Lie group $\mathrm{Isom}(\mathbf{H}^n)\simeq\mathrm{PO}(n,1)$ up to conjugacy and inversion.
One can do separately loxodromics and horocyclics ("parabolics").
*Loxodromics:* they have two invariants: the translation length (a positive real number), and the t... | 8 | https://mathoverflow.net/users/14094 | 355474 | 150,059 |
https://mathoverflow.net/questions/355453 | 7 | I found the bit count of Lucas sequence $U\_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason?
<https://oeis.org/A002487> : Stern's diatomic series
<https://oeis.org/A168081> : Lucas sequence $U\_n(x,1)$ over the field GF(2)
| https://mathoverflow.net/users/153948 | How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)? | I always work mod 2. Let $f(n)$ be the bit count (number of nonzero
coefficients) of $U\_n(x,1)$. $U\_n(x,1)$ satisfies
$$ U\_{2n}(x,1) = xU\_n(x,1)^2 $$
$$ U\_{2n+1}(x,1) = (U\_n(x,1)+U\_{n+1}(x,1))^2. $$
From the first equation $f(2n)=f(n)$, and only odd powers of $x$
can have nonzero coefficients. From the second e... | 7 | https://mathoverflow.net/users/2807 | 355475 | 150,060 |
https://mathoverflow.net/questions/355454 | 7 | Let $p$ be an odd prime. The $\mathbb F\_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C\_p, \mathbb F\_p) = \mathbb F\_p[\xi,x]$ where $\xi$ has degree 1, $x$ has degree 2, and the Koszul signs are imposed (so that in particular $\xi^2 = 0$). As a module over the Steenrod algebra, t... | https://mathoverflow.net/users/78 | How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$? | If $P$ is the group of order $p^3$ and exponent $p$, its mod $p$ cohomology ring is known to have its depth = its Krull dimension = rank of a maximal elementary abelian subgroup = 2. A theorem of Jon Carlson then implies that the product of restriction maps
$$ H^\*(P;\mathbb F\_p) \rightarrow \prod\_E H^\*(E;\mathbb F\... | 9 | https://mathoverflow.net/users/102519 | 355484 | 150,062 |
https://mathoverflow.net/questions/355472 | 4 | Let $V=k^n$ for an algebraically closed field $k$ of characteristic 0, and let $W \subseteq V$ a subspace. Let $G\_W\subseteq GL(V)$ be the set of invertible linear maps that preserve $W$, i.e.
$$
G\_W=\{x \in GL(V): x(W)=W\},
$$
and let
$$
\mathfrak{g}\_W=\{X \in \mathfrak{gl}(V) : X(W) \subseteq W\},
$$
where $\mathf... | https://mathoverflow.net/users/150898 | The Lie algebra of the subgroup of $GL(n)$ preserving a given variety | Doc, you are right but only amorally. You need to replace the tangent vectors with jets to capture the behavior of your cone.
Let $I(Y)$ be the ideal of zeroes of your $Y$. Then
$$
Lie (G\_Y) = \{ X \in {\mathfrak{gl}}(V) | X(I(Y))\subseteq I(Y)\}.
$$
Now you know that $I(Y)$ is homogeneous. Pick a finite set of its... | 3 | https://mathoverflow.net/users/5301 | 355493 | 150,064 |
https://mathoverflow.net/questions/355498 | 6 | I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p\_a(C)$ as the integer such that the Hilbert polynomial of $C$ (or of its ideal of polynomials vanishing on it) is
$$h\_C(X)=\deg(C) X+1-p\_a(C).$$
**I would then li... | https://mathoverflow.net/users/23758 | Simple proof that the arithmetic genus is non-negative | Mumford (*Complex Projective Varieties*, section 7) has the following, reasonably simple proof.
Let $d$ be the degree of $\mathrm{C}$, $m$ big enough such that $h\_{\mathrm{C}}(m)=\mathrm{dim}\_{\mathbf{C}} (\mathbf{C}[\mathrm{T}\_0,\ldots,\mathrm{T}\_n]/\mathrm{I}(\mathrm{C}))\_m$ and $md/2>p\_a$. Embed $\mathrm{C}$ i... | 2 | https://mathoverflow.net/users/104669 | 355506 | 150,069 |
https://mathoverflow.net/questions/355394 | 6 | In Hirsch's book "Differential Topology," he claims in Chapter 2, Theorem 1.4 that the set of $C^1$-embeddings is open in the strong Whitney topology $C^1(M, N)$ where $M$ and $N$ are $C^1$ manifolds. My question is: How do you see that the set $\mathcal{N}\_1$ is open? See below for the setup and description of $\math... | https://mathoverflow.net/users/4517 | The set of embeddings is open in the strong Whitney topology | A proof of this is also contained in Michor: Manifolds of mappings. There Proposition 5.3 establishes that the $C^r$-embeddings are $WO^1$ open in the space $C^r (M,N)$ (no restriction on the manifolds $M,N$). Thus if you set $r=1$ this implies that they are open with respect to the strong Whitney topology, as the $WO^... | 2 | https://mathoverflow.net/users/46510 | 355510 | 150,072 |
https://mathoverflow.net/questions/355516 | 5 | There is a list of open problems in my sub-field that was published in a journal some time ago and has had an impact on the area.
Many of the problems have been solved, some have partial solutions, and some are still unsolved.
I am considering trying to write a survey of the current status of this list. However, I ... | https://mathoverflow.net/users/153985 | Progress on a problem list | **Q3:** A classic of this type is [Erdős on Graphs : His Legacy of Unsolved Problems](https://rads.stackoverflow.com/amzn/click/com/156881111X)
>
> This book is a tribute to Paul Erdős, the wandering mathematician once
> described as the "prince of problem solvers and the absolute monarch
> of problem posers." It... | 0 | https://mathoverflow.net/users/11260 | 355517 | 150,074 |
https://mathoverflow.net/questions/355483 | 6 | We are in first order logic world.
Let $\sigma$ be a finite signature and $T$ a consistent theory of $\sigma$.
Due to Löwenheim–Skolem theorem, we can consider the $\underline{set}$ of all at most countable models of $T$ up to elementary equivalance(does not change what formulas hold).
Let $\mu$ be a nonprincipial ... | https://mathoverflow.net/users/127260 | The theory of a model of a theory that knows all formulas true in almost all its models | In this answer, I will basically repeat things that zeb said in their nice answer, but arranged differently (in a way that might or might not be more clear).
Let $\text{Mod}\_{\leq \aleph\_0}(T)$ be the "set" of all at most countable models of $T$. Your question is a bit ambiguous about what this means - I'll come b... | 7 | https://mathoverflow.net/users/2126 | 355522 | 150,075 |
https://mathoverflow.net/questions/351534 | 5 | It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}\_p$ would allow for a possible solution to the Weil conjectures.
It was also understood that such a cohomology theory could not take values in vector spaces over either the rational numbers $\mathbb... | https://mathoverflow.net/users/151698 | Motivating the coefficient field of $\ell$-adic cohomology | One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion). Furthermore, if s... | 6 | https://mathoverflow.net/users/121 | 355540 | 150,082 |
https://mathoverflow.net/questions/355505 | 1 |
>
> Consider the 2-dimensional optimal control problem of the LQR kind
> $$
> \min\_u \int\_0^\infty (x^T Q x + u^TRu) \, dt \quad\text{such that}\quad \begin{cases}\dot x(t) = Ax(t)+Bu(t) \\ x(0) = \begin{pmatrix}1\\-1\end{pmatrix}\end{cases}
> $$
> with $x=\begin{pmatrix}x\_1 \\ x\_2\end{pmatrix}$, $u=\begin{pma... | https://mathoverflow.net/users/136012 | Solve a 2-dimensional optimal control problem via Riccati nonlinear equation | The solution of a Riccati equation can be found by determining the eigenvectors of the Hamiltonian; see e.g. [here on Wikipedia](https://en.wikipedia.org/wiki/Algebraic_Riccati_equation#Solution). This is your best hope for a closed-form symbolic solution, in my view. In your case, the Hamiltonian is
[
\begin{bmatrix}
... | 1 | https://mathoverflow.net/users/1898 | 355543 | 150,083 |
https://mathoverflow.net/questions/355546 | 5 | Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system.
Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that $T^{R(y)}(y)\in Y$.
Then we can define a function $F:Y\to Y$ by $F=T^R$ and consider the induced system $(Y,\mathcal{F}\... | https://mathoverflow.net/users/153994 | Ergodicity of induced system | In general $F$ is not ergodic. A very simple example can be constructed as follows: let $X=\mathbb{Z}\_3=\{0,1,2\}$ and $\mu =1/3(\delta\_0+\delta\_1+\delta\_2)$ and $T(x):=x+1$. This is an ergodic system. Let us define $Y:=\{0,1\}$ and $R\equiv 3$. Since $F=T^3=id$, it is not ergodic.
| 4 | https://mathoverflow.net/users/889 | 355550 | 150,085 |
https://mathoverflow.net/questions/355476 | 1 | I'm considering the SDE, with $B$ the brownian motion and $\beta$ a scalar (it can be negative)
$$ X\_t = x\_0 + \int\_0^t (\beta + X\_s^2) ds + B\_t $$
and I would like to show that $X\_t$ almost surely diverges to infinity in finite time.
I really don't know how to do it since usually I have to control $X\_t$, ... | https://mathoverflow.net/users/153961 | Diverging solution to a SDE | Let us begin by showing that $X$ diverges to $+\infty$, possibly as time goes to infinity. Set
$$f : x\mapsto \int\_0^x\exp\left(-2\beta y-\frac 23y^3\right)\mathrm dy. $$
The point of $f$ is that $f(X)$ is a local martingale, possibly up to the explosion time $\tau$ of $X$ ($f$ is solution to $(\beta+x^2)\partial\_xf+... | 2 | https://mathoverflow.net/users/129074 | 355569 | 150,087 |
https://mathoverflow.net/questions/355548 | 2 | I know that Symplectic group has an action on Heisenberg group.
I am wondering how to extend this to non-trivial two fold metaplectic covering?
Thanks in advance!
| https://mathoverflow.net/users/29422 | Semidirect product of metaplectic group and Heisenberg group | I presume you are looking for a faithful action of $Mp\_{2n}$ on something related to the Heisenberg group $H\_{2n+1}$. This is well-known as Weil Representation.
In the modern language, consider $Mp\_{2n}$ acting on $H\_{2n+1}$ by automorphisms. This action has a kernel. Now consider the action on the category of u... | 3 | https://mathoverflow.net/users/5301 | 355577 | 150,090 |
https://mathoverflow.net/questions/355381 | 3 | In the paper [*Zum Beweise des Starkschen Satzes*](https://eudml.org/doc/141914) Siegel considers the function
$$L\_q(s)=\sum\_{n=1}^{\infty}\left(\frac{q}{n}\right)n^{-s},$$
where $q$ is a discriminant of a quadratic number field and the character is the Kronecker symbol. Then he writes that "according to Dirichle... | https://mathoverflow.net/users/122104 | Is this Siegel's formula correct? | Dirichlet proved his class number formula for quadratic forms; in particular he was working with class numbers $h^+$ in the strict sense, and his unit $\varepsilon$ was the fundamental solution of the Pell equation $t^2 - Du^2 = 1$, not the fundamental unit $\eta$ of the corresponding number field. The relation
$$ \eta... | 8 | https://mathoverflow.net/users/3503 | 355578 | 150,091 |
https://mathoverflow.net/questions/355574 | 2 | Is there an established name for cycles $C\subseteq G(V,E)$ with the property that
$$\lbrace u,v\rbrace\subseteq C\cap V\implies\mathrm{dist}\_{|C}(u,v)\le \mathrm{dist}\_{|G}(u,v)$$
I would be tempted to call them *facet*s because vertices and edges that constitute to the boundary of a facet of a polyhedron are ... | https://mathoverflow.net/users/31310 | Name for specific cycles in graphs | You are looking for the following:
>
> **Definition.** A subgraph $H\subseteq G$ is called *isometric* if $\mathrm{dist}\_H(u,v)=\mathrm{dist}\_G(u,v)$ for all $u,v\in V(H)$.
>
>
>
So your cycles could be called **isometric cycles**.
---
Note that not all facets of a polyhedron are induced in this sense.... | 5 | https://mathoverflow.net/users/108884 | 355590 | 150,097 |
https://mathoverflow.net/questions/355089 | 6 | I'm reading up on linear algebra over semirings, and I'm wondering why people seem to stop short of showing an equivalence between linear transformations between free modules and matrices.
It seems clear to me that over *commutative* semirings, the usual developments one does for commutative *rings* go through: a lin... | https://mathoverflow.net/users/30392 | Linear algebra over non-commutative semirings | What I said was mostly right; except that there's no need for the bimodule structure on the $R^n$:
Let $R$ be a semiring and, for $n,m\ge 0$, $R^n$ and $R^m$ the free bimodules over $R$. Say that a function $f:R^m\to R^n$ is *left-linear* if $f(x+y)=f(x)+f(y)$ and $f(ax)=af(x)$ for all $x,y\in R^m$ and all $a\in R$.
... | 2 | https://mathoverflow.net/users/30392 | 355591 | 150,098 |
https://mathoverflow.net/questions/355579 | 1 | An rr function (i.e. *rational rational function*) is a quotient
$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$
such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$
**QUESTION** Do there exist rr functions $\ \phi\ \psi\ $
such that set
$$ \{(\phi(x)\ \ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq
\ (\Bbb Q\... | https://mathoverflow.net/users/110389 | Rational Peano curves | Use the resultant to eliminate the variable $X$. Since the resultant is computed over the rationals, the resultant is a rational coefficient polynomial in the two variables of the plane, satisfied on the image of the parameterized curve. See my (undergraduate!) lecture notes [Concrete Algebra](https://github.com/Ben-Mc... | 3 | https://mathoverflow.net/users/13268 | 355604 | 150,099 |
https://mathoverflow.net/questions/355587 | 1 | I’m trying to resolve respect to $k$ the following inequality,
$$
k\left(\log k +\log \log k-\alpha+O\left(\frac{\log \log k}{\log k}\right)\right)\geq x,
$$
in order to obtain, under the condition $k\leq x$, that
$$
\ k\geq \dfrac{x}{\log x}\left(1+\dfrac{\alpha+o(1)}{\log x}\right)
$$
is this possible?
| https://mathoverflow.net/users/151852 | Resolution of an inequality on integers | $\newcommand\ka{\kappa}$
Let $a:=\alpha$. Let $f$ be a function such that
$$f(k)=k\Big(\ln k +\ln\ln k-a+O\Big(\frac{\ln\ln k}{\ln k}\Big)\Big)
=k\big(\ln k +\ln\ln k-a+o(1)\big)$$
as $k\to\infty$.
For any real $b$ and $x>0$, let
$$\ka:=\ka\_b(x):=\frac x{\ln x}\Big(1+\frac b{\ln x}\Big).$$
We have to show that
... | 3 | https://mathoverflow.net/users/36721 | 355609 | 150,102 |
https://mathoverflow.net/questions/320510 | 7 | Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if
for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$
$$
\left| \sum\_{x \in X, x \leq N} \exp(2\pi i \alpha x) \right| < C(\alpha).
$$
My conjecture is the following: If $X$ is super-equidistribu... | https://mathoverflow.net/users/133916 | Conjecture about an exponential sum | This was proven by Reynold Fregoli! <https://arxiv.org/abs/1912.08626>
| 4 | https://mathoverflow.net/users/133916 | 355610 | 150,103 |
https://mathoverflow.net/questions/355617 | 0 | I'm trying to become better with using proper terminologies and standard notation when taking notes, which lead me to think:
Similar to the indication of a completed proof by use of the **Q.E.D.** mark, or "∎", is there a standard method as to indicate the end of a simplified definition?
| https://mathoverflow.net/users/154034 | Is there a common notation to indicate the final form of a simplified definition? | Following Euclid, you could use [**QEF**](https://mathworld.wolfram.com/QEF.html) (*quod erat faciendum – which had to be done*). Euclid used the Greek version of this (ὅπερ ἔδει ποιῆσαι) to close propositions that were not proofs of theorems, but constructions of geometric objects.
| 1 | https://mathoverflow.net/users/11260 | 355620 | 150,105 |
https://mathoverflow.net/questions/355626 | 31 | If I had a vector space with a linear endomorphism $D$ satisfying $D^2 = 0$, I might call it a *differential* and study its *(co)homology* $\operatorname{ker}(D) / \operatorname{im}(D)$. I might say that $D$ is *exact* if this (co)homology vanishes. I would especially do this if $D$ increased by 1 some grading on my ve... | https://mathoverflow.net/users/78 | What should I call a "differential" which cubes, rather than squares, to zero? | Many years ago, when I was a graduate student, I remember seeing a couple of papers on the homologies of operators satisfying $\partial^p=0$, generalizing the case $p=2$. I seem to remember that they were by somebody like Steenrod, and it might evan have been in the Annals, sometime in the 40s or 50s.
Unfortunately,... | 19 | https://mathoverflow.net/users/13972 | 355633 | 150,109 |
https://mathoverflow.net/questions/355621 | 6 | Sard's famous theorem asserts that
**Theorem.** The set of critical values of a smooth function from a manifold to another has Lebesgue measure $0$.
I am asking for the curiosity that is it possible to find such a function whose set of critical values is
1. Cantor set or
2. Any other uncountably infinite set?
| https://mathoverflow.net/users/131172 | Sard's theorem and Cantor set | It is not hard to construct a smooth function $f$ on $\mathbb R$ such that $f \ge 0$ with $f(x) = 0$ if and only if $x$ is in the Cantor set $E$. If $F$ is an antiderivative of $f$, the critical values of $F$ will be an uncountably infinite perfect set.
| 14 | https://mathoverflow.net/users/13650 | 355634 | 150,110 |
https://mathoverflow.net/questions/354483 | 2 | I had a question that might be well-known but I'm not sure where to find it. Grayson defined a filtration on the algebraic $K$-theory of affine regular rings via commuting automorphisms which you can find [here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.154.9304&rep=rep1&type=pdf). You can define the sam... | https://mathoverflow.net/users/127776 | Grayson filtration and weight filtration | My impression is that Adams operations are "well known" to act coherently on all levels of the weight spectral sequence for K-theory (of smooth varieties); probably, this fact was established by Gillet and Soul´e. It easily follows that the filtration induced by this spectral sequence on rational K-theory is the Adams ... | 1 | https://mathoverflow.net/users/2191 | 355638 | 150,112 |
https://mathoverflow.net/questions/355629 | 6 | I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that:
>
> If $\lambda\gg 1$, then
> $$I(\lambda,x) = \int\_{\mathbb{S}^{n-1}}(x\cdot y)e^{i\lambda(x\cdot y)}\,d\sigma(y),$$
> is $\mathcal... | https://mathoverflow.net/users/32591 | Stationary phase in spherical integral | You have
$
I(\lambda, x)=x\cdot\int\_{\mathbb S^{n-1}} ye^{i \lambda x\cdot y} d\sigma(y)=x\cdot J(x,\lambda)
$
and you claim that for $\vert x\vert \lambda \ge 1$, you have
$$
J(x,\lambda)=O((\vert x\vert \lambda)^{-\frac{n-1}{2}}).
$$
Indeed, using coordinate charts and a finite partition of unity, you are reduced ... | 3 | https://mathoverflow.net/users/21907 | 355641 | 150,113 |
https://mathoverflow.net/questions/355640 | 7 | Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial\_x^2+c\_\*+\Phi$ repeatedly appear. Usually, on these contexts $\Phi$ is a smooth exponentially decaying function and $c\_\*\in\mathbb{R}$ is a positive constant.
I am very surpris... | https://mathoverflow.net/users/129131 | Spectrum of "classical" operators | Let $A$ be a self-adjoint operator with domain $D(A)\subset\mathcal H$ ($\mathcal H$ is some Hilbert space). An operator $C$ with $D(A)\subset D(C)$ is called relatively compact with respect to $C$ if $C(A-zI)^{-1}$ is compact for some (hence all) $z\notin\sigma(A)$. Paraphrasing Corollary 2, page 113 Section XIII.4, i... | 11 | https://mathoverflow.net/users/153800 | 355649 | 150,116 |
https://mathoverflow.net/questions/355630 | 1 | Let`s define ternary ECC as a code that its codewords can be defined by $ \{ xyz f(y,z) f(x,z) f(x,y) | x,y,z \in \{0,1\}^m \} $ for some function $f$. $f$ returns bitstring of constant length.
Are there any known good error correction codes that are ternary?
Such a family of LDPC codes would be best.
Is there ... | https://mathoverflow.net/users/142777 | Ternary error correction codes | Well, if the function $f$ has range $GF(2)^m$, represented by $GF(2^m)$ if convenient, it has rate 1/2. Such a function can really control symbol ($GF(2^m)$ ) not bit errors so it is a code over $GF(2^m)$ of length $n=6$ and rate 1/2 (dimension 3). If the code is MDS [best possible] it has symbol distance at most $n-k+... | 4 | https://mathoverflow.net/users/17773 | 355651 | 150,117 |
https://mathoverflow.net/questions/355588 | 3 | If $X$ is a topological space, we let $\text{End}(X)$ be the collection of continuous functions $f:X\to X$. We say that $f,g\in \text{End}(X)$ *meet* if there is $x\in X$ with $f(x) = g(x)$. We say that $D\subseteq \text{End}(X)$ is a *cover* for $\text{End}(X)$ if for every $f\in \text{End}(X)$ there is $g\in D$ such ... | https://mathoverflow.net/users/8628 | Continuous function covers in connected $T_2$-spaces | The answer is strong YES for connected spaces admitting a non-constant continuous function and NO in the opposite case.
1. If $X$ is a connected topological space that admits a non-constant continuous function
$\gamma:X\to\mathbb R$, then the family
$$\{f\in C(X):f(X)=\{q\}\subset\mathbb Q\}\cup\{q+\gamma:q\in\mathb... | 3 | https://mathoverflow.net/users/61536 | 355656 | 150,118 |
https://mathoverflow.net/questions/355466 | 7 | This is a follow-up to [this answer](https://mathoverflow.net/a/355402/120251).
If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x\_n\}\_{n=1}^\infty$ in $E$ *can be made to converge to zero* if there exists a sequence $\{\alpha\_n\}\_{n=1}^\infty$ of strictly positive real scalars su... | https://mathoverflow.net/users/120251 | Metrizability of a topological vector space where every sequence can be made to converge to zero | For the counterexample below we need a little lemma about barrelled spaces (i.e., every barrel = closed, absolutely convex, and absorbing set is a $0$-neighbourhood):
*If $(E,\mathcal T)$ is a barrelled locally convex space which has a finer metrizable vector space topology $\mathcal S$ then $(E,\mathcal T)$ is metri... | 3 | https://mathoverflow.net/users/21051 | 355661 | 150,119 |
https://mathoverflow.net/questions/355654 | 0 | I know that the Fenchel conjugate of a function is
$$f^\*(x^\*) = \sup\_x\{\langle x, x^\*\rangle - f(x)\}.$$
However, how do I find the Fenchel conjugate of the function
$$f(x) = \frac{1}{p}\sum\limits\_{i=1}^n |x\_i|^p$$ where $1 < p < \infty$.
I have tried differentiating the equation and taking it to be $= 0$ bu... | https://mathoverflow.net/users/154060 | Finding the conjugate of a function | It will be, somewhat expectedly, $$f^\*(x^\*) = \frac{\|x^\*\|\_q^q}{q}, \quad \frac1q + \frac1p = 1$$ as indicated in the comments to the OP. The paper linked there was already rightfully identified as sketchy, the derivation there is wrong on several levels, which is why I am including this answer.
I will use $q$ ... | 1 | https://mathoverflow.net/users/85906 | 355663 | 150,120 |
https://mathoverflow.net/questions/355669 | 2 | The locally convex space of *essentially compactly-supported $p$-integrable "functions"* $\operatorname{L}\_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined as the set
$$
\bigcup\_{n \in \mathbb{N}} \left\{
f \in L^p(\mathbb{R}^d,\mathbb{R}):\, \operatorname{ess-supp}(f)\subseteq [-n,n]^d
\right\},
$$
topologized... | https://mathoverflow.net/users/36886 | Sobolev topology on essentially compactly supported Sobolev-"functions" | To make my comment an answer: No. To see this, fix a cube $K$ and look at $L^p\_K=\{f\in L^p: \text{ ess-supp}(f) \subseteq K\}$. On this space, the $L^p\_{comp}$-topology coincides with the $L^p$-topology (this follows from the stricness of the inductive limit) and the $W^{k,p}$-topology is strictly finer on this subs... | 2 | https://mathoverflow.net/users/21051 | 355680 | 150,125 |
https://mathoverflow.net/questions/355687 | 2 | I couple of days ago, I asked extensively the same question on Stack-exchange (see <https://math.stackexchange.com/questions/3592151/riemann-hilbert-correspondence-versus-simpson-correspondence)and> go no answer.
Let us assume that X is a connected, smooth complex algebraic variety. Then the Riemann-Hilbert correspo... | https://mathoverflow.net/users/152554 | Riemann-Hilbert correspondence versus Simpson correspondence | From my point of view, Riemann-Hilbert and non-abelian Hodge are really two independent statements - though the statement of the latter may be wound up in the former in some sense.
There are three different types of objects at play:
1) Higgs bundles (Dolbeault),
2) flat connections (de Rham),
3) reps of $\p... | 9 | https://mathoverflow.net/users/7762 | 355692 | 150,128 |
https://mathoverflow.net/questions/355693 | 4 | I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter.
$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu\_E$ s.t. the Gauss Green theorem holds: that is for each compactly supported vector field $T$
$$ \int\_{E}div(T)=\int\_{\mathbb{R}^n}T\cdot ... | https://mathoverflow.net/users/127879 | Every convex set is of locally finite perimeter | First assume that $E$ is compact. Then, your inequality says that you can approximate it from above by a sequence $E\_n$ of convex polytopes with decreasing perimeters. Then, the sequence $\mu\_{E\_n}$ is weak\*-precompact by Banach$-$Alaoglu theorem, so we can assume by passing to a subsequence that $\mu\_{E\_n}\to \m... | 4 | https://mathoverflow.net/users/56624 | 355696 | 150,129 |
https://mathoverflow.net/questions/355684 | 8 | Some cobordism invariants are not cohomology classes. Such as the $\mathbb{Z}\_{16}$-valued eta invariant $\eta$ of $\Omega\_4^{Pin^+}$, the $\mathbb{Z}\_8$-valued Arf-Brown-Kervaire invariant ABK of $\Omega\_2^{Pin^-}$, and the $\mathbb{Z}\_4$-valued quadratic enhancement $q(a)$ of $\Omega\_2^{Pin^-}(B\mathbb{Z}\_2)$ ... | https://mathoverflow.net/users/102515 | Cobordism invariants: topological v.s. geometric | $\newcommand{\ko}{\mathit{ko}}
\newcommand{\MTSpin}{\mathit{MTSpin}}
\newcommand{\Z}{\mathbb Z}$
As Mike points out in his comment, it's not obvious what it means for a bordism invariant is
“topological” or “geometric.” The bordism invariants you mention can be described
topologically, and some bordism invariants which... | 8 | https://mathoverflow.net/users/97265 | 355702 | 150,131 |
https://mathoverflow.net/questions/355515 | 5 | Let $f(z)$ be a rational function of degree $d \geq 2$, with complex coefficients. I am interested in fully invariant measures for the dynamical system $(\mathbb C\_\infty,f)$, where $\mathbb C\_\infty$ is the Riemann sphere. By a fully invariant measure, I mean a probability measure $\mu$ such that $f^\ast \mu = d \mu... | https://mathoverflow.net/users/9317 | Fully invariant measures for rational functions | The unique measure of maximal entropy $\mu\_f$ supported on the Julia set of a rational map $f$ of degree $d \geq 2$ is indeed the unique balanced measure for $f$, i.e., the only probability measure $\mu$ not charging the exceptional set and satisfying $f^\*\mu =d \cdot \mu$. As you already noticed in the comments, uni... | 5 | https://mathoverflow.net/users/14493 | 355706 | 150,132 |
https://mathoverflow.net/questions/355652 | 6 | **EDIT:**
>
> Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that the system remains in a fixed ball all the time? (For $N=2$ this is impossible, and this is what I expect in gene... | https://mathoverflow.net/users/16183 | Movement of repelled particles in a ball | If all the particles remained in a bounded domain, the [virial theorem](https://en.m.wikipedia.org/wiki/Virial_theorem) would apply. In the case of a radial inverse square power law, it states that twice the asymptotic time average of kinetic energy of the system equals minus the asymptotic time average of its potentia... | 7 | https://mathoverflow.net/users/37059 | 355716 | 150,137 |
https://mathoverflow.net/questions/355708 | 4 | let $P(x)\in{\mathbb Q}[x]{}$ be a rational polynomial with $P(1) >1$ and $\zeta $ be the Riemann zeta function , I want to know if there exist a rational polynomial such that $P(\zeta(s))=\zeta(P(s))$ with $s>1$ ?
| https://mathoverflow.net/users/51189 | Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$? | Extending the argument by GH from MO, $\zeta(P(s))$ has a pole for any $s$ such that $P(s)=1$, while $P(\zeta(s))$ has unique pole for $s=1$. Therefore if $\zeta(P(s))=P(\zeta(s))$, then $P(s)=1$ has unique solution $s=1$ and $P(x)-1=c(x-1)^n$ for some complex $c$ and positive integer $n$, and we get $\zeta(1+c(s-1)^n)... | 15 | https://mathoverflow.net/users/4312 | 355719 | 150,139 |
https://mathoverflow.net/questions/355714 | 9 | Is there a non nuclear $C^\*$ algebra $A$ for which the minimum and maximum $C^\*$ norms on $A\otimes A$ coincide?
A somewhat similar question is [discussed here](https://mathoverflow.net/questions/127971/).
| https://mathoverflow.net/users/36688 | A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm | Pisier <https://arxiv.org/abs/1908.02705> very recently constructed a non-nuclear $C^\ast$-algebra $A$ with the weak expectation property (WEP) and the local lifting property (LLP). By a celebrated result of Kirchberg (see Corollary 13.2.5 in Brown and Ozawa's book) it follows that if $B$ has WEP and $C$ has LLP, then ... | 12 | https://mathoverflow.net/users/126109 | 355720 | 150,140 |
https://mathoverflow.net/questions/355127 | 16 | Let suppose that we are given a connected CW-complex $X$, such that we know
1. All its homology groups.
2. All its homotopy groups, in particular we know $\pi\_{1}(X)$.
As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I ... | https://mathoverflow.net/users/17895 | The homology of the universal covering space, why so difficult to compute | This is not a complete answer to your question, which I am not sure how to answer without having a precise meaning for what you mean by "computing" and what you mean by "knowing" $\pi\_1(X)$, as indicated in the comments above, and given that you have not included the data of any action of $\pi\_1(X)$.
However, I thi... | 6 | https://mathoverflow.net/users/5450 | 355723 | 150,142 |
https://mathoverflow.net/questions/355710 | 7 | This is a borderline question, but I'm going to risk posing it.
Cuthbert Edmund Cullis (1875?-1955?) was a somewhat obscure British mathematician whose opus magnum was a multi-volume treatise called *Matrices and Determinoids*. While his notation is somewhat idiosyncratic, it was a fairly systematic exposition of mos... | https://mathoverflow.net/users/2530 | Has vol. 3A of Cullis's "Matrices and Determinoids" been scanned and vol. 3B been archived? | **Q1:** Volume 3 part 1 (a.k.a. volume 3A) has been digitized and reissued as a paperback by Cambridge UP, see [Amazon](https://rads.stackoverflow.com/amzn/click/com/1107414261). The digital version is online in the [HathiTrust digital library](https://catalog.hathitrust.org/Record/000664163?type%5B%5D=author&lookfor%5... | 6 | https://mathoverflow.net/users/11260 | 355727 | 150,145 |
https://mathoverflow.net/questions/355740 | 6 | I've enocuntered the following question in my current research, and I'd appreciate any help you could give me. This is probably well known to experts on the subject.
Let $S = \langle K \rangle$ be a **finitely generated** inverse semigroup. Recall that the set $E$ of *idempotents* (i.e. elements $e \in S$ such that $... | https://mathoverflow.net/users/147609 | Ascending sequences of idempotents in inverse semigroups | Yes. Indeed, for $X$ a set, let $G\_X$ be the group of partial bijections of $X$, that are defined and identity outside a countable subset. I claim that, for $X$ uncountable, every countable subset of $G$ is contained in a (5-generator) finitely generated submonoid (and hence in a finitely generated inverse submonoid).... | 5 | https://mathoverflow.net/users/14094 | 355745 | 150,148 |
https://mathoverflow.net/questions/352723 | 7 | The following definitions are from [lecture notes](https://users.renyi.hu/~nemethi/JEGYZET.ps) of Némethi. A *surface singularity* $(X,0)$ is defined by $$(X,0) = (\{ f\_1 = \ldots = f\_m=0 \}) \subset \mathbb (\mathbb{C}^n,0),$$ where $f\_i : (\mathbb{C}^n ,0) \to (\mathbb{C},0)$ are germs of analytic functions with $... | https://mathoverflow.net/users/nan | Resolution graphs in the sense of Némethi | About question 2: I think that the software [Singular](http://www.singular.uni-kl.de/) has this feature; it's well-documented, and if you look for resolution graph you should find the reference.
About question 1: well, I must admit that that algorithm is not pleasant, and that it took me a while to work out the case ... | 3 | https://mathoverflow.net/users/13119 | 355748 | 150,150 |
https://mathoverflow.net/questions/355685 | 2 | Let $W$ be an irreducible affine Coxeter group (say of type $\widetilde{X}\_n$), and let $\Sigma$ be the associated Coxeter complex. Thus, $\Sigma$ is an $n$-dimensional Euclidean space tesselated by isometric copies of a given simplex $A$ (namely, by the alcoves of $\Sigma$). I would like to know what the isometry typ... | https://mathoverflow.net/users/154087 | Isometry type of alcoves in affine Coxeter complexes | Go to Bourbaki, Groupes et Algebres de Lie, Ch.4-6 where the fundamental weights $\varpi\_1, \ldots , \varpi\_n$ of $X\_n$ are listed. The vertices of the simplex are
$$0 \ \mbox{ and } \ x\_k \varpi\_k, \ k=1,\ldots,n$$
where $x\_k>0$ is such that $x\_k \varpi\_k$ lies on the fixed hyperplane of the last Coxeter gen... | 4 | https://mathoverflow.net/users/5301 | 355760 | 150,152 |
https://mathoverflow.net/questions/355759 | 6 | I was told that if we have an equivalence of categories $F : \mathcal{A} \rightarrow \mathcal{B}$ with $\mathcal{A}$ abelian, then it is not necessarily true that $\mathcal{B}$ is also abelian.
I would like to know if there are nice examples of an abelian category $\mathcal{A}$ which is equivalent to a non-abelian ca... | https://mathoverflow.net/users/nan | Abelian category equivalent to a non-abelian category | What you were told is wrong, for we have the following:
**Proposition.** *If two categories are equivalent and one of them is abelian, then so is the other.*
A proof (and some related results) can be found in Satz 16.2.4 in H. Schubert, *Kategorien II,* Springer, 1970 (likewise in the English version [https://www.a... | 26 | https://mathoverflow.net/users/11025 | 355765 | 150,154 |
https://mathoverflow.net/questions/307817 | 7 | Consider $A = \mathbb{R}[t\_1,\ldots,t\_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t\_1,\ldots,t\_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = t\_1 y + t\_2 y^2 + \cdots + t\_k y^k \in A\,.
$$
Let $I$ be the ideal in $A$ generated by $\{p(y)^2 \mid y \in \mat... | https://mathoverflow.net/users/78097 | A commutative variant of the exterior algebra | For $k=\infty$, this algebra appears when studying integrable representations of level 1 of the Lie algebra $\widehat{\mathfrak{sl}}\_2$, see, for example, discussion in Section 2 of
[A. V. Stoyanovsky, B. L. Feigin. Functional models for representations of current algebras and semi-infinite Schubert cells. Function... | 3 | https://mathoverflow.net/users/1306 | 355775 | 150,161 |
https://mathoverflow.net/questions/355512 | 9 | Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$ the $k$-algebra of regular functions on $Y$.
The $S$-unit equation for $k(X)$ is the equation $f+g =1$, with $f,g \i... | https://mathoverflow.net/users/6506 | The $S$-unit equation for functions on curves | The set of solutions to the $S$-unit equation for $k(X)$ is finite. Let me explain why. (You can "theoretically" find all solutions, as the finiteness eventually boils down to the "effective" finiteness result of de Franchis-Severi on maps of curves.)
Let $k$ be a number field, let $X$ be a smooth projective geometri... | 5 | https://mathoverflow.net/users/4333 | 355779 | 150,162 |
https://mathoverflow.net/questions/355762 | 5 | My question concerns the *Propsition 5.12 of the paper of Bhatt-Mathew on arc-topology* where they claim that the functor $X\mapsto D^b\_{\text{cons}}(X,\Lambda)^{[-n,n]}$ which assigns to a qcqs scheme $X$ the subcategory of the full derived category $D(X\_{ét},\Lambda)$ of étale sheaves of $\Lambda$-modules of amplit... | https://mathoverflow.net/users/39237 | descent implies hyperdescent | It is certainly true that descent implies hyperdescent whenever $\mathcal C$ is a $n$-category for some $n<\infty$ (it wasn't clear from your question whether you knew this or not). This is because, for any $\infty$-site $\mathcal A$:
1. A presheaf $F:\mathcal A^{op}\to\mathcal C$ is a sheaf or hypersheaf if and only... | 4 | https://mathoverflow.net/users/20233 | 355784 | 150,163 |
https://mathoverflow.net/questions/355795 | 6 | These days I found a mysterious [page on Google books](https://books.google.com.br/books/about/On_the_De_Rham_cohomology_of_schemes.html?id=plrvAAAAMAAJ&redir_esc=y) describing a book entitled *On the De Rham cohomology of schemes* by Grothendieck, Coates, and Jussila.
At once I thought this was an error and Google b... | https://mathoverflow.net/users/130058 | On a mysterious reference of Grothendieck | I think your hunch is correct that Google is in error, and that "Crystals and the de Rham cohomology of schemes" refers to the article in Dix Exposés. I doubt there is a book by the same name and author. There is also "On the de Rham cohomology of algebraic varieties" but that's different. The first is a long proposal ... | 8 | https://mathoverflow.net/users/4144 | 355798 | 150,164 |
https://mathoverflow.net/questions/355797 | 3 | In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to send three of the punctures at those location whilst the fourth puncture is free to move.
On the other hand, the moduli ... | https://mathoverflow.net/users/48526 | Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures | In your description of moduli space you say:
>
> I can use a Moebius transformation to send three of the punctures at
> those location whilst the fourth puncture is free to move.
>
>
>
That assumes that the punctures have names. Let's call them $a, b, c, d$ and we will agree to send them to $0, 1, \infty, z$... | 6 | https://mathoverflow.net/users/1650 | 355802 | 150,166 |
https://mathoverflow.net/questions/355808 | 41 | [Gromov's symplectic nonsqueezing theorem](https://en.wikipedia.org/wiki/Non-squeezing_theorem) asserts that in the symplectic space ${\bf R}^{2n}$ with canonical coordinates $p\_1,\dots,p\_n,q\_1,\dots,q\_n$, and two radii $0 < r < R$, it is not possible to symplectomorphically map the ball $B(0,R)$ into the cylinder ... | https://mathoverflow.net/users/766 | Is a symplectic camel actually prohibited from passing through the eye of a needle? | Eliashberg & Gromov sketched a proof in their paper [*"Convex symplectic manifolds"*](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/976.pdf) (Section 3.4). Written in the 4-dimensional case it says:
For $r>0$ define the subspace $X(r)\subset\mathbb{R}^4$ to be the union of the half-space $\lbrace q\_2<0\rbra... | 29 | https://mathoverflow.net/users/12310 | 355812 | 150,169 |
https://mathoverflow.net/questions/355806 | 2 | Consider a graph with vertices being people (in some region), and make an edge if one person pass another closer than say 1.5 meter during say one week.
(Such a graph might be thought a kind of useful for modelling epidemics on graphs ([papers](https://scholar.google.fr/scholar?hl=en&as_sdt=0%2C5&as_vis=1&q=epidemic%2... | https://mathoverflow.net/users/10446 | How many persons pass your 1.5 meter neighbourhood during 1 week ? If the distribution is power law what is the exponent? | • Concerning the first of the two questions in the title, *"How many persons pass your 1.5 meter neighbourhood during 1 week?"*
Here is a graph from [Mixing patterns between age groups in social networks](https://www.researchgate.net/publication/228649013_Mixing_patterns_between_age_groups_in_social_networks), showi... | 6 | https://mathoverflow.net/users/11260 | 355813 | 150,170 |
https://mathoverflow.net/questions/355787 | 3 | Let $S = \langle K\rangle$ be a finitely generated inverse semigroup, where $K \subset S$ is a fixed, finite and symmetric set of generators.
**Preliminaries:** Recall that we say that $s, t \in S$ are $\mathcal{L}$-related if $s^{-1}s = t^{-1}t$. Given an $\mathcal{L}$-class $L \subset S$, we may construct its' *Sch... | https://mathoverflow.net/users/147609 | Quasi-isometries and E-unitary inverse semigroups | The answer is no. Let $G$ be a free abelian group of rank 2 generated by $x,y$. Let $S$ be the Meakin-Margolis expansion of $G$. It consists of all pairs $(X,g)$ with $X$ a finite connected subgraph of the Cayley graph of $G$ containing the origin and $g$. The product is $(X,g)(Y,h)=(X\cup gY,gh)$. The projection to $G... | 3 | https://mathoverflow.net/users/15934 | 355816 | 150,172 |
https://mathoverflow.net/questions/353557 | 4 | A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a classification? In particular I am interested if the following VOAs are simple or not and under what conditions:
1. The rank $d... | https://mathoverflow.net/users/nan | Examples of simple vertex operator algebras (VOAs) | I expect there will never be a classification of simple VOAs, unless perhaps one is only sorting according to very rough criteria. This is because there are too many of them - even the rational case is wide open. For your examples, we have the following:
1. The irreducible modules of the rank $d$ free boson are natur... | 3 | https://mathoverflow.net/users/121 | 355834 | 150,178 |
https://mathoverflow.net/questions/355844 | 0 | I'm interested in knowing whether there exists any kind of theory for measures on groups without assuming that it's the Haar measure for a locally compact group topology.
| https://mathoverflow.net/users/15482 | measures on groups without assuming a locally compact group topology | Translastion-invariant, locally finite measure on a topological group.
As I recall, there is something like this in Hewitt & Ross Vol. I
>
> *Hewitt, E.; Ross, K. A.*, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Group representations, Die Grundlehren der mathematische... | 2 | https://mathoverflow.net/users/454 | 355846 | 150,180 |
https://mathoverflow.net/questions/352343 | 1 | Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties.
In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is well publicised, but not the exponent, as far as I can tell.)
Thanks!
| https://mathoverflow.net/users/152146 | Where can I find a table of the exponents of the sporadic groups? | From the comments,
This information can be calculated easily from the printed character tables in the ATLAS of Finite Groups (which include orders of elements in conjugacy classes) or, perhaps more conveniently, using the same information online via GAP or Magma. From there you can just load the character table from th... | 1 | https://mathoverflow.net/users/152146 | 355852 | 150,182 |
https://mathoverflow.net/questions/355789 | 1 | Let $A$ be a real/complex algebra (just a real/complex vector space with a multiplication; none PI's are required). Let $\mathcal{H}$ be a real/complex Hilbert space. Let $\operatorname{B}(\mathcal{H})$ be the algebra of bounded endomorphisms of $\mathcal{H}$ and let $\operatorname{Nil}(\mathcal{H})\subset\operatorname... | https://mathoverflow.net/users/153173 | Representation of algebras as bounded nilpotent operators | The problem is still ill-posed, because the set of nilpotent bounded operators is not a vector space. (The sum of two nilpotents need not be nilpotent.) Let's interpret the question as: which algebras admit faithful representations as algebras of nilpotent bounded operators?
Note that "nilpotent" usually means $T^n =... | 2 | https://mathoverflow.net/users/23141 | 355865 | 150,185 |
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