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https://mathoverflow.net/questions/242250 | 3 | I am interested in a particular group $G$, where
$$ (A\_4\times C\_\ell) \lhd G \lhd S\_4 \times D\_\ell$$
Here, $C\_\ell$ is cyclic, $D\_\ell$ is dihedral of order $2\ell$, and the two inclusions both have index $2$. Since
$$ (S\_4\times D\_\ell)/(A\_4\times C\_\ell) \cong C\_2\times C\_2,$$
my description of $G$ coul... | https://mathoverflow.net/users/801 | Do the irreducible modules of this finite group preserve a tensor product structure? | This is all about Clifford's theorem in the complex case, which I treat in detail. The Klein $4$ subgroup that is acting on $A\_{4}\times C\_{\ell}$ contains three involutions: one inverts the $C\_{\ell}$ but centralizes the $A\_{4}$, one induces the outer automorphism of $A\_{4}$, but centralizes the $C\_{\ell}$, and ... | 4 | https://mathoverflow.net/users/14450 | 242254 | 111,119 |
https://mathoverflow.net/questions/242263 | 1 | Let $\Sigma\_k$ be the symmetric group on $k$-letters. Let $W$ be a **contractible** topological space with a **free** $\Sigma\_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the Cartesian product of $k$-copies of $X$. Let $\Sigma\_k$ act on $X^k$ from the right by permuting the order of coordina... | https://mathoverflow.net/users/41075 | free group actions on a contractible topological space | A couple of scattered things to address the "how to deal with" part of the question.
---
If $X$ is connected, then probably so is $X^n$, and so you are looking at a fibration
$$ X^n \to X^n // \Sigma\_n \to \mathbf{B}\Sigma\_n $$
for which the Leray-Serre spectral sequences are usually called Cartan-Leray, and th... | 1 | https://mathoverflow.net/users/35529 | 242274 | 111,125 |
https://mathoverflow.net/questions/242271 | 4 | Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a [Gelfand pair](https://en.wikipedia.org/wiki/Gelfand_pair) if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative.
Let $H$ be a subgroup of $Aut(G)$. We will consider $H$ and $G$ as subgroups of $G\rtimes H$.
*Lemma... | https://mathoverflow.net/users/34538 | Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian? | No, let $G$ be an arbitrary group and take $H=G$ acting on itself by conjugation. Then $H$ becomes the diagonal in $G\rtimes H\cong G\times G$. This is well known to be a Gelfand pair.
PS: The Hecke algebra of the pair $(G\rtimes H,H)$ is in fact equal to the fixed point algebra $\mathbb C[G]^H$. This explains your o... | 10 | https://mathoverflow.net/users/89948 | 242278 | 111,127 |
https://mathoverflow.net/questions/242277 | 6 | Let $G$ be a connected semisimple Lie group with trivial centre and $\mathfrak{g}$ its Lie algebra. The adjoint representation of $G$ defines an isomorphism of $G$ onto the connected component of the identity of $Aut(\mathfrak{g})$. The latter is a real algebraic group, but its identity component (in the Lie group topo... | https://mathoverflow.net/users/93839 | Centreless semisimple Lie group that is not real algebraic | You basically answered your own question: The connected component $G$ of $SO(2,1)$ is not real algebraic. As such, $G$ should be isomorphic to (the real points of) $PSL(2,\mathbb R)$. But it isn't. As algebraic groups $PSL(2,\mathbb R)$ and $PGL(2,\mathbb R)$ are isomorphic (in fact, the natural homorphism between them... | 7 | https://mathoverflow.net/users/89948 | 242281 | 111,128 |
https://mathoverflow.net/questions/242253 | 5 | Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a complete projective resolution, i.e. that are expressible as the image of the map $P\_{-1}\to P\_0$ in some sequence
$$... | https://mathoverflow.net/users/21483 | When is the category of Gorenstein projective $R$-modules Frobenius? | It's always a Frobenius category and its projective-injective objects are the projective modules. Your analysis is essentially right. In fact, this holds very generally. I worked out the following when I read a bit in Enochs-Yenda, but I suppose it's well-known:
Given an exact category $(\mathcal{A}, \mathcal{E})$, l... | 4 | https://mathoverflow.net/users/93935 | 242282 | 111,129 |
https://mathoverflow.net/questions/242283 | 2 | Where can I find details about the irreducible unitary representations of SO(1,4) and SO(2,3)?
| https://mathoverflow.net/users/56920 | Unitary representations of SO(1,4) and SO(2,3) | There is Volume 2, Chapter 9 of the monumental (4-volume!) work of N. Ya. Vilenkin and A. U. Klimyk, *Representations of Lie Groups and Special Functions* (Kluwer, 1993).
| 3 | https://mathoverflow.net/users/11211 | 242293 | 111,132 |
https://mathoverflow.net/questions/242292 | 2 | Let $(M,g)$ be a compact Riemannian three-fold such that $H\_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual to the surface $S$, meaning for all closed 2-form $\alpha$ we have :
\begin{equation}
\int\_M \eta \wedge \alpha = \i... | https://mathoverflow.net/users/nan | harmonic differential form integer class | The norms in cohomology are isometric to suitable norms in homology which can be expressed using geometric quantities, but this is far from trivial. The question is studied in
Bangert, Victor; Katz, Mikhail.
Stable systolic inequalities and cohomology products.
Dedicated to the memory of Jürgen K. Moser.
Comm. Pure... | 2 | https://mathoverflow.net/users/28128 | 242296 | 111,133 |
https://mathoverflow.net/questions/242295 | 2 | Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following inequality true?
$$\inf\_{\phi \in A} \int\_K (\phi - u) dx = 0?$$
It seems true, if $d = 1$. Indeed, one can simply ta... | https://mathoverflow.net/users/5656 | Continuous upper envelope of upper semicontinuous function | The equality is true, independently of $d$.
By Theorem 2.1.3 of Ransford's book "Potential theory in the complex plane", since $u$ is bounded above on the compact set $K$, there is a decreasing sequence of continuous functions $f\_n$ that converge pointwise to $u$.
Then, by the monotone convergence theorem,
$$\lim... | 2 | https://mathoverflow.net/users/89429 | 242297 | 111,134 |
https://mathoverflow.net/questions/242302 | 9 | Consider the following statement:
*If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$
the "Peano phenomenon" can arise only at those values of $\bar x$ for which $f(\bar x) = 0$.*
(The "Peano phenomenon" = Cauchy problems associated to the above equation c... | https://mathoverflow.net/users/14101 | The "Peano phenomenon" for differential equations | I'm not sure this is "well known" (it wasn't to me). We can do this with a [Lyapunov function](https://en.wikipedia.org/wiki/Lyapunov_function) $V(x,y)=g(y)-g(x)$, with $g(x)=\int\_a^x (1/f(u))\, du$. This is well defined near an $a$ with $f(a)\not= 0$.
Now if $x(t),y(t)$ both solve $x'=f(x)$, $x(0)=a$, then $W(t)= V... | 6 | https://mathoverflow.net/users/48839 | 242313 | 111,138 |
https://mathoverflow.net/questions/242276 | 5 | Johansson's theorem states the following:
Given $f:M\_1\rightarrow M\_2$ (not a pair map) an homotopy equivalence between 3-manifolds with incompressible boundary.
Let $V\_i$ be the components of the characteristic submanifolds meeting the boundary then we can homotope $f\simeq g$ so that $g$ is an homotopy equiva... | https://mathoverflow.net/users/93931 | On Johansson's Theorem on homotopy equivalences of 3-manifolds | Yes, this is true, with the appropriate assumption that $M\_1$ is irreducible (this is needed for Johansson and Waldhausen's statements),
and let's say orientable.
One may reduce to Waldhausen's theorem if we know that $f\_{|\partial M\_1}$ is homotopic to a homeomorphism $f'\_{|\partial M\_1}:\partial M\_1\to \part... | 6 | https://mathoverflow.net/users/1345 | 242314 | 111,139 |
https://mathoverflow.net/questions/242280 | 7 | In [the paper](http://people.maths.ox.ac.uk/~laugwitz/2010-08%20-%20Yetter-Drinfeld%20and%20Hopf%20Modules%20-%20Bachelorarbeit.pdf), page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is
$$
h\_{(1)} v\_{(-1)} \otimes h\_{(2)}.v\_{(0)} = (h\_{(1)}.v)\_{(-1)}h\_{(2)} \otimes (h... | https://mathoverflow.net/users/11877 | Compatibility conditions for Yetter-Drinfeld modules | Assume that the second equation holds. Then
\begin{align\*}
\delta(h\_{1}.v)=(h\_{1}. v)\_{-1}\otimes(h\_{1}.v)\_{0}=h\_{1,1}v\_{-1}Sh\_{1,3}\otimes h\_{1,2}. v\_{0}
\end{align\*}
and therefore
$$
(h\_{1}.v)\_{-1}h\_{2}\otimes(h\_{1}.v)\_{0}=h\_{1,1}v\_{-1}Sh\_{1,3}h\_{2}\otimes h\_{1,2}.v\_{0}=h\_{1}v\_{-1}\otimes h\... | 3 | https://mathoverflow.net/users/17845 | 242315 | 111,140 |
https://mathoverflow.net/questions/242307 | 4 | In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C\* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any explanation or definition. A google search brought no results, so what is a generalized limit?
To be more precise, he ... | https://mathoverflow.net/users/nan | What is a generalized limit? | I tried to search for [renault ideal "generalized limit"](https://www.google.com/search?q=renault+ideal+%22generalized%20limit%22) to see whether I will find some related works where the definition of this notion is included.
---
I found this thesis: Groupoid Crossed Products by Geoff Goehle, <https://arxiv.org/a... | 4 | https://mathoverflow.net/users/8250 | 242318 | 111,141 |
https://mathoverflow.net/questions/242290 | 1 | Let $M$,$N$ be two modules over commutative ring $R$.
Can we say that $pd (M \otimes N )= $ $pd( M) + pd( N)$?
Thanks!
| https://mathoverflow.net/users/92487 | Projective dimension and tensorial product of two modules | If $M$ and $N$ are Tor-independent (i.e., Tor$\_i^R(M,N)=0$ for $i>0$), then you have some positive results. For instance if $R$ is local complete intersection, the three modules are of finite projective dimension (and finite) then you have an affirmative answer as a consequence of Auslander-Buchschaum formula and Hune... | 2 | https://mathoverflow.net/users/92322 | 242329 | 111,142 |
https://mathoverflow.net/questions/242325 | 2 | Let $\mathcal F$ be an ample subsheaf of $T\_{\mathbb P^n}$. Is it actually locally free? If not, is there a counterexample?
| https://mathoverflow.net/users/62798 | ample subsheaf contained in the tangent bundle of projective space | Yes, $\mathcal{F}$ is locally free. See e.g. Corollary A.11 in [this paper](http://arxiv.org/pdf/0705.4602v2.pdf) of Aprodu, Kebekus, and Peternell.
| 3 | https://mathoverflow.net/users/6950 | 242330 | 111,143 |
https://mathoverflow.net/questions/242328 | 2 | Let $\ p\ $ be an arbitrary prime. Then an integer $\ s\ $ is called $p$-simple $\ \Leftarrow:\Rightarrow\ s\ $ is not divisible by any prime $\ q<p.\ $
Could you prove my conjecture (or is it known one way or another?):
>
> For every prime $\ p\ $ and for every every integer $\ n\ $ there exists a $p$-simple int... | https://mathoverflow.net/users/8385 | $p$-simple integers from between $n$ and $n+p-1$ | The conjecture is false. Rankin (1938) proved that there exists a constant $c>0$ such that for all $x>20$, there exist at least $$ c x\frac{(\log x)(\log\log\log x)}{(\log\log x)^2} $$
consecutive integers, each of which are divisible by some prime less than $x$. Note that the fraction here tends to infinity as $x\to\i... | 8 | https://mathoverflow.net/users/11919 | 242331 | 111,144 |
https://mathoverflow.net/questions/242327 | 10 | Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A\_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic?
This question is motivated by the answer by Kasper Andersen to [my question](https://mathoverflow.net/q/241876/4149). Namely, a desired example (if exi... | https://mathoverflow.net/users/4149 | A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups | Yes. Note that Daniel Loughran's comment to David Speyer's answer to [this question](https://mathoverflow.net/questions/240439/hasse-principle-for-rational-times-square) states that for any solvable group $G$, there is a Galois extension $L/\mathbb{Q}$ with all decomposition groups cyclic.
It's not hard to find a con... | 15 | https://mathoverflow.net/users/48142 | 242336 | 111,146 |
https://mathoverflow.net/questions/242342 | 1 | Let $ X\rightarrow Spec(\mathbb{C}[t])=\mathbb{C}$ be a projective variety over $\mathbb{C}[t]$ (a flat family of projective varieties $X\_{t}$, $t\in \mathbb{C}$), and $X\_{\eta}$ be the base change via $\mathbb{C}[t] \rightarrow \mathbb{C}((t))$ by $t\mapsto t$. ($X\_{\eta}$ is the formal neighborhood of $X\_{0}$ in ... | https://mathoverflow.net/users/93958 | Possible `surgery' via formal neighborhoods | That's fpqc-descent. For this to work you need additionally ample line bundles on both $X\_{\ne0}$ and $Y$ and a compatible extension of the isomorphism $X\_\eta\to Y\_\eta$ to these line bundles. See SGA 1,Proposition 7.8.
| 3 | https://mathoverflow.net/users/89948 | 242344 | 111,149 |
https://mathoverflow.net/questions/242312 | 9 | Consider a real $m\times n$ matrix $A$ and the $p$-norms in $\mathbb{C}^n$ and $\mathbb{C}^m: \|x\|=\left(\sum|x\_i|^p\right)^{1/p}$.
One defines the real $p$-norm of $A$ as $\|A\|=\sup\frac{\|Ax\|}{\|x\|}$, where the $\sup$ ranges over $0\neq x\in\mathbb{R}^n$.
Similarly, one can define the $p$-norm of the complex... | https://mathoverflow.net/users/1172 | Equivalence between complex and real operator norms | Since you seem really interested in the inequality $\rho(A)\le\|A\|$ ($\rho$ the spectral radius), here is a simple and elegant proof. In the 2nd edition of my book *Matrices* (Springer Verlag GTM**216**), it is Proposition 7.6.
Take any operator norm $N$ on ${\bf M}\_n({\mathbb C})$ ; you may take $N=\|\cdot\|\_c$, ... | 5 | https://mathoverflow.net/users/8799 | 242346 | 111,150 |
https://mathoverflow.net/questions/242334 | 3 | Given a quasi-smooth toric variety $X$ in the sense of Gelfand-Kapranov-Zelevinsky,
i.e. a (not necessarily normal) toric variety $X$ whose normalization is a $\mathbb{Q}$-factorial toric variety and such that the normalization morphism is bijective. It is known that such non-normal toric varieties share nice propertie... | https://mathoverflow.net/users/37338 | Cohen-Macaulay non-normal toric variety | Counterexample to CM: Let $M$ be the monoid $\{x^iy^j\mid i,j\ge0\}\setminus\{x,y\}$ and $X=\text{Spec}\ \mathbb C[M]$ the corresponding toric variety. Then $x^2,y^2$ is a system of parameters of $X$. But $\mathbb C[M]$ is not a free $\mathbb C[x^2,y^2]$-module since the fiber $\mathbb C[M]/(x^2,y^2)=\langle 1,xy,x^3,x... | 3 | https://mathoverflow.net/users/89948 | 242352 | 111,151 |
https://mathoverflow.net/questions/75784 | 21 | The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then it could be rather hard; I'm asking this question here because MO is lucky enough to have some of the foremost experts o... | https://mathoverflow.net/users/1463 | Is there a non-Hopfian lacunary hyperbolic group? | Contrary to I think the expectations of the earlier answers, in [a recent paper](http://arxiv.org/abs/1606.00679), Coulon--Guirardel showed that the answer to the title question is 'no'!
>
> **Theorem (Coulon--Guirardel):** Every lacunary hyperbolic group is Hopfian.
>
>
>
The proof is remarkably short, and al... | 6 | https://mathoverflow.net/users/1463 | 242364 | 111,155 |
https://mathoverflow.net/questions/242379 | 42 | In his [2014 book](http://www.ams.org/mathscinet-getitem?mr=3342785), Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7:
>
> Capitolo I
>
>
> Esempi e metodi dimostrativi
>
>
> 1. Introduzione
>
>
> In The Calculus as Algebraic Analysis, Craig Fraser, riferendosi
> all'opera di Eulero ... | https://mathoverflow.net/users/28128 | Did Euler prove theorems by example? | There's some evidence that precisely the *opposite* can be said: that Euler is aware of the fallacies of proving theorems by example (of course, this does not necessarily mean he has never used it). One memorable instance is his *Exemplum Memorabile Inductionis Fallacis*, where he described how he was almost led to con... | 37 | https://mathoverflow.net/users/3948 | 242385 | 111,159 |
https://mathoverflow.net/questions/242332 | 2 | I am trying to use the following result
>
> **Theorem:**
> A pde of the form
> $$\frac{\partial w}{\partial t} = F\{x, \frac{\partial w}{\partial x},\frac{\partial^2 w}{\partial^2 x}\}$$
> has an Additive separable solution
> $$w\left(x, t\right) = M t + N + \phi(x),$$
> where $M$,$N$ are arbitrary constants,... | https://mathoverflow.net/users/18929 | IBVP with transformed boundary conditions | You can't use your theorem. Your boundaries are not straight. Furthermore, your boundary conditions are independent of $t$, which requires $M = 0$, and this puts strong requirements on the initial data.
Basically:
1. Your theorem shows that there exists some family of solutions to a PDE of the form as written. Em... | 2 | https://mathoverflow.net/users/3948 | 242389 | 111,160 |
https://mathoverflow.net/questions/242369 | 0 | Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and a homomorphism of sheaves in groups $f:H\rightarrow G$ the fiber product $H\times\_{f, G,e}S$ is representable by an al... | https://mathoverflow.net/users/92355 | representing base changes of the unit section | No. For one thing, it would imply that every group subsheaf $K$ of an (abelian) algebraic space in groups $H$ is an algebraic space (just take $G=H/K$). Now let me give a "concrete" example.
For any field $k$, take $S=\mathrm{Spec}(k[t])$ and $H=(\mathbb{Z}/2\mathbb{Z})\_S$. Put $U=\mathrm{Spec}(k[t,t^{-1}])$, and l... | 1 | https://mathoverflow.net/users/7666 | 242400 | 111,165 |
https://mathoverflow.net/questions/242410 | 1 | Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \cdot)$ are equivalent, i.e. for any $A \in \mathcal B(E)$ and $x \in E$, $P(x,A) > 0 \Leftrightarrow Q(x,A) > 0$.
Supp... | https://mathoverflow.net/users/22157 | Uniqueness of invariant measure for equivalent transition probabilities | Since you don't want to exclude the case of infinite stationary measures, it is very easy to produce a counterexample. The simple random walk on $\mathbb Z$ has a unique stationary measure (which coincides with the counting one), whereas for $p\neq q$ the random walk with the transition probabilities $p(n,n+1)=p$ and $... | 2 | https://mathoverflow.net/users/8588 | 242414 | 111,168 |
https://mathoverflow.net/questions/242365 | 17 | I have already asked this question on MathStackExchange ([here](https://math.stackexchange.com/questions/1810985/why-is-12-the-smallest-weight-for-which-a-cusp-forms-exists)) but did not receive an answer, therefore I am trying it here:
On wikipedia ([here](https://en.wikipedia.org/wiki/12_%28number%29 "here")) I hav... | https://mathoverflow.net/users/50081 | Why is 12 the smallest weight for which a cusp forms exists | These facts are related via the geometric interpretation of cusp forms. Essentially, identifying cusp forms as differential forms on modular curves allows to establish the relation to the ${\rm SL}\_2\mathbb{Z}$-action on the upper half-plane. The following is very sketchy, but I hope it properly outlines the main poin... | 8 | https://mathoverflow.net/users/50846 | 242415 | 111,169 |
https://mathoverflow.net/questions/242416 | 4 | Let $G$ be a group. Let $\Bbbk$ be a field of char. $0$. We denote with $C^{n}(G, \Bbbk)$ the set of maps $f\: : \: G^{n}\to \Bbbk$ and with $\partial\_{G}\: : \: C^{n-1}(G, \Bbbk)\to C^{n}(G, \Bbbk)$ the differential
$$\partial\_{G}f(g\_{1},\ldots,g\_n)=f(g\_{2},\ldots,g\_n)+\sum\_{1\leq j\leq n-1}(-1)^{j-1}f(g\_1,\l... | https://mathoverflow.net/users/41970 | Coboundary for the Cohomology of free groups | There is a projective resolution $P\_\bullet$ of $\mathbb Z$ as an $F\_n$-module of the form $$0\to\mathbb ZF\_n\otimes V\to\mathbb ZF\_n\to\mathbb Z\to0$$ in which $V$ is the free $\mathbb Z$-module spanned by the generators $x\_1,\dots,x\_n$ of $F\_n$, and the differential is the $\mathbb ZF\_n$-linear map such that ... | 5 | https://mathoverflow.net/users/1409 | 242420 | 111,172 |
https://mathoverflow.net/questions/242412 | 21 | A *homology sphere* is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to [a previous question of mine](https://mathoverflow.net/q/236029/21564) shows that the smoothness hypothesis is not necessary except in dimension four where it is not yet known whether ever... | https://mathoverflow.net/users/21564 | Are homology spheres stably parallelisable? | Yes, they have stably trivial tangent bundles. A remark to this effect can be found on page 70 of
M. Kervaire "[Smooth Homology Spheres and their Fundamental Groups](http://www.ams.org/journals/tran/1969-144-00/S0002-9947-1969-0253347-3/S0002-9947-1969-0253347-3.pdf)"
but it is a little terse. It is essentially the... | 23 | https://mathoverflow.net/users/318 | 242423 | 111,173 |
https://mathoverflow.net/questions/242440 | 2 | Let $a,b,c>0$ with $a>b$ and let $X\_1,\dots,X\_n$ be a collection of independent uniform samples from the unit interval. Assume that we re-order the samples so that $X\_1\leq X\_2 \leq \cdots \leq X\_n$, and then let $Z\_i := X\_{i+1}-X\_i$ denote the differences between neighbors. My question is, is there any way to ... | https://mathoverflow.net/users/70190 | Expectation of a function of sorted uniform random variables | Let's throw in $X\_0=0$ and $X\_{n+1}=1$. Then there are $n+1$ variables $(Z\_i)$ lying on the simplex $\{z\in\mathbb R^{n+1}\_+\colon z\_1+\ldots+z\_{n+1}=1\}$. The distribution is uniform. Each $Z\_i$ has the same marginal distribution with density $f\_Z(z)=(n+1)(1-z)^n$. So your expectation is just $(n+1)\int\_0^1 (... | 2 | https://mathoverflow.net/users/11054 | 242447 | 111,181 |
https://mathoverflow.net/questions/242441 | 10 | It is well known that Kurt Gödel had doubts concerning the US constitution and believed that it somehow was inconsistent and opened up for a dictatorial grab.
What was he thinking?
| https://mathoverflow.net/users/37385 | What was Gödel's Constitutional Problem? | The history of how Gödel studied the US constitution in preparation for the citizenship exam is well documented by [Jeffrey Kegler](http://morgenstern.jeffreykegler.com/), as pointed out in the comments. Perhaps more relevant for MO (as opposed to HSM) is what the flaw in mathematical logic might have been that Gödel h... | 12 | https://mathoverflow.net/users/11260 | 242452 | 111,184 |
https://mathoverflow.net/questions/242443 | 4 | What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?
In a similar way let us consider a stationary vector valued Gaussian process indexed by an infinite discrete abelian group with mean zero. What is the condition of ergodicity of such a pro... | https://mathoverflow.net/users/651 | On the spectrum of stationary Gaussian process | If $\mu$ is the spectral measure of the Gaussian process, then the maximal spectral type of the shift map on the distribution of the process is $\exp(\mu):=\sum\_{k=1}^n\mu^{\ast k}/k!$ where $\mu^{\ast k}$ is the $k$ fold convolution of $\mu$ (this comes from the Fock spaces decomposition). Using this characterisation... | 5 | https://mathoverflow.net/users/78465 | 242456 | 111,185 |
https://mathoverflow.net/questions/242362 | 7 | For Banach spaces $X,Y$ and an open subset $U$ of $X$ a function $f:U\to Y$ is $C^1$ if $U\to L(X,Y)$, $x\to f'(x)$ is continuous where, by definition, the derivative $f'(x)$ is a continuous linear operator from $X$ to $Y$ satisfying $$\|f(x+h)-f(x)-f'(x)(h)\|\_Y = o(\|h\|\_X)$$ and $L(X,Y)$ is the Banach space of cont... | https://mathoverflow.net/users/21051 | $C^1$-functions on Banach spaces | In Banach spaces, continuity of $Df:U\times X \to Y$ and linearity in the second variable implies that $Df: U\to L(X,Y)$ is continuous for the topology of uniform convergence on compact subsets of $X$, and not the operator norm in general.
If $f$ is $C^2\_c$ then $Df: U\to L(X,Y)$ is actually continuous into the operat... | 5 | https://mathoverflow.net/users/26935 | 242463 | 111,187 |
https://mathoverflow.net/questions/242448 | 7 | As Theorem 8.1 in "*Lectures on the h-cobordism theorem* (written by J.Milnor)" show, we can choose a handle decomposition of cobordism (satisfying some connectivity and dimensional assumptions) with no 0,1-handles, which sometimes we call this techniques *handle trading* (we actually did trading all 1-handles with the... | https://mathoverflow.net/users/85988 | Generalizations of the handle trading techniques | You might find the paper by C.T.C Wall: Geometrical connectivity I, J. London Math. Soc. 3 (1971), p. 597-604, interesting.
What Wall proves, entirely by handle trading, is that if $W:M\_0 \to M\_1$ is an $n$-dimensional cobordism and the inclusion $M\_0\to W$ is r-connected, then you can built W from M\_0 using only ... | 6 | https://mathoverflow.net/users/9928 | 242469 | 111,190 |
https://mathoverflow.net/questions/242472 | 9 | I would like to know whether there is an example of a reductive algebraic group $G$ (say, over the complex numbers $\mathbb{C}$) and a finite dimensional representation $V$ of $G$ such that dim$(V//G)$ is different from dim$(V^\*//G)$. Here $V//G=Spec\ \mathbb{C}[V]^G$ is the categorical quotient and $V^\*$ is the dual... | https://mathoverflow.net/users/23236 | Is the dimension of $V//G$ always the same as the dimension of $V^*//G$? | If $G$ is connected and $T\subseteq G$ is a maximal torus then there is an involution $\theta:G\to G$ with $\theta(t)=t^{-1}$ for all $t\in T$. It has the property that as a representation $V^\*$ is isomorphic to the $\theta$-twisted representation $V$, i.e., to $G\overset\theta\to G\to GL(V)$. To see this just look at... | 15 | https://mathoverflow.net/users/89948 | 242475 | 111,191 |
https://mathoverflow.net/questions/241920 | 2 | Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where $E$ is a supersingular elliptic curve) iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b\_2(... | https://mathoverflow.net/users/69852 | How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$? | Now let $k$ be a finite field. If $E$ is an elliptic curve over $k$ then $End(E)\otimes\mathbf{Q}$ is either an imaginary quadratic field or a definite quaternion algebra over $\mathbf{Q}$.
Warning: $E$ may be supersingular even if $End(E)\otimes\mathbf{Q}$ is an imaginary quadratic field; this means that not all en... | 3 | https://mathoverflow.net/users/9658 | 242480 | 111,194 |
https://mathoverflow.net/questions/242266 | 0 | In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). Now let $\xi: H \to \mathbb R$ a continuous, linear functional with $\xi(1) = 1$ (where the first $1$ denotes the constan... | https://mathoverflow.net/users/93924 | Reproducing Kernel Hilbert Spaces with positive kernels | The answer is no, and a simple counterexample can be obtained by taking, say, $E = \{1,\dots,n\}$, where $n \ge 3$, and $K(x,y) = C (\delta\_{xy} - \frac{1}{n})$, where $C > n$. In fact, what we will show is that no rank one perturbation of $1 + K$ is positive pointwise.
Normalizing your $\xi$ to have norm one (which... | 1 | https://mathoverflow.net/users/22758 | 242482 | 111,195 |
https://mathoverflow.net/questions/215102 | 53 | In a recent talk [Finite groups, yesterday and today](https://youtu.be/MZ6_JKYdKog?t=54m) Serre made some comments about proofs that rely on the [classification of finite simple groups](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups) (CFSG) and on the [ATLAS of Finite Groups](http://brauer.maths.q... | https://mathoverflow.net/users/4177 | How much of the ATLAS of finite groups is independently checked and/or computer verified? | It may also worth to look at the paper
* T. Breuer, G. Malle, and E. A. O'Brien, *Reliability and reproducibility of Atlas information*, Contemporary Mathematics **694** (2017) pp 21–31, doi:[10.1090/conm/694/13960](https://doi.org/10.1090/conm/694/13960), arXiv:[1603.08650](https://arxiv.org/abs/1603.08650)
in whi... | 17 | https://mathoverflow.net/users/13525 | 242488 | 111,198 |
https://mathoverflow.net/questions/242508 | 10 | Here are four possible definitions for an etale, finite, surjective map $X\rightarrow Y$ between integral schemes to be considered Galois:
1. There exists a finite group $G$, and an action $\varphi: G\times X\rightarrow X$, so that the induced map $\varphi\times p\_2: G\times X\rightarrow X\times\_Y X$ is an isomorph... | https://mathoverflow.net/users/73798 | Which of these 4 definitions of Galois coverings of integral schemes are equivalent? | I can prove $(3) \Rightarrow (1)$ when $X$ and $Y$ are irreducible (**Lemma 1** below; no integrality assumptions needed. Irreducible is needed for the question to make sense), and I can prove $(4) \Rightarrow (1)$ when $X$ and $Y$ are normal, connected, and Noetherian (in particular integral, see [Tag 033M](http://sta... | 14 | https://mathoverflow.net/users/82179 | 242510 | 111,208 |
https://mathoverflow.net/questions/242424 | 2 | **Definition.** An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a *covering morphism* if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i.e such that the square below is a pullback.
$$\require{AMScd} \begin{CD}
E\times\_BA @>{\eta\_{E\times\_BA}}>> H... | https://mathoverflow.net/users/69037 | For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected? | Here is a general claim: in an extensive category $\mathbf{C}$, if $p: E \to B$ is an epimorphism and $E$ is connected (i.e., $\hom(E, -): \mathbf{C} \to \text{Set}$ preserves finite coproducts), then also $B$ is connected.
Lemma: If a coproduct inclusion $i\_U: U \to U + V$ is epic, then $V$ is initial. Proof: let ... | 2 | https://mathoverflow.net/users/2926 | 242527 | 111,212 |
https://mathoverflow.net/questions/242522 | 5 | It is known that a 3 by 3 real symmetric matrix $A$ has an eigendecomposition
$$ A = Q E Q^T $$
where $Q$ is an orthogonal matrix and $E$ is a diagonal matrix whose elements, $E\_{11}$, $E\_{22}$ and $E\_{33}$, are the eigenvalues of $A$.
Moreover, if those eigenvalues are non-negative then $A$ is positive-semide... | https://mathoverflow.net/users/24224 | Structure of a real 3x3 positive-semidefinite matrix whose eigenvalues verify the triangle inequalities | Condition
$$E\_{11}+E\_{22}\ge E\_{33}$$
is equivalent to
$$E\_{11}+E\_{22}+E\_{33}\ge 2 E\_{33}$$
or
$$E\_{33} \le \frac12 tr\ A$$
since the sum of eigenvalues is equal to the trace. Combining with the other two conditions gives
$$\lambda\_{max}(A) \le \frac12 tr\ A$$
which is semidefinite representable as
$$A \pr... | 8 | https://mathoverflow.net/users/1184 | 242528 | 111,213 |
https://mathoverflow.net/questions/242470 | 6 | Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer:
$$
S\_{p,k} := \lim\_{r\to+\infty} \sum\_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k
$$
Convergence is easy to see. This can also be written as
$$
S\_{p,k} = \frac{1}{2} \sum\_{0\neq x \in B\_p} x^{-k}
$$
where
$$
\begin{aligned... | https://mathoverflow.net/users/17064 | A $p$-adic sum of reciprocals of powers | As far as I know this quantity does not have a name. There is a mention of a general family of $p$-adic limits, of which $S\_{k,p}$ is a special case, on p. 31 of [1].
The quantity $S\_{k,p}$ can be expressed as an infinite series involving the Kubota-Leopoldt $p$-adic zeta function $\zeta\_p(n):=L\_p(n,\omega\_p^{1-... | 3 | https://mathoverflow.net/users/5263 | 242532 | 111,214 |
https://mathoverflow.net/questions/242494 | 1 | Given real numbers $x\_1$, $\dots$, $x\_n$ and $r\_1$, $\dots$, $r\_n$ (with reasonable restrictions), is there a trigonometric series $T(x)=\sum\_k a\_k \cos(kx)$ with $a\_k\ge 0$ such that
$$
|T(x\_i)-r\_i|<\epsilon
$$
for all $i$?
| https://mathoverflow.net/users/94031 | trigonometric series with nonnegative coefficients and prescribed values | **[EDITED]** $\def\conv{\mathop{\rm conv}\nolimits}$The answer is **yes** if $x\_1,\dots,x\_n\in(0,\pi]$ are pairwise distinct (are these `reasonable restrictions'?).
Consider a set $C=\{x(t)=(\cos tx\_1,\dots,\cos tx\_n)\colon t\in\mathbb Z\}\subset \mathbb R^n$. Let $D$ be the closure of the convex hull of $C$.
... | 2 | https://mathoverflow.net/users/17581 | 242534 | 111,216 |
https://mathoverflow.net/questions/148691 | 22 | I previously asked a version of this question [on Math.SE](https://math.stackexchange.com/q/554841/822), but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to assume that $X$ is separable.) In this question, "subspace" means a linear... | https://mathoverflow.net/users/4832 | Meager subspaces of a Banach space and weak-* convergence | The answer to Q2 is **No**.
I am grateful to Damian Sobota for drawing my attention to the following paper:
>
> Darst, R. B.
> On a theorem of Nikodym with applications to weak convergence and von Neumann algebras.
> *Pacific J. Math.* **23** (1967) 473–477. MR0238084
>
>
>
(Damian mentioned it in the ques... | 1 | https://mathoverflow.net/users/4832 | 242545 | 111,219 |
https://mathoverflow.net/questions/242381 | 2 | I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version
This question is motivated by the following fact in complex variable:(I learned this fact from the book of Ahlfors, Complex Analysis)
**Fact: If all roots of a complex polynomial $p(z)$ lie in a half plane then all roots of its ... | https://mathoverflow.net/users/36688 | Half spaces free of roots of a given polynomial | The property that the zeros of the derivative of a polynomial $P$ lie in the convex hull of the zeros of $P$ is usually called the Gauss-Lucas theorem.
About question 2), the algebra of entire functions of order less than 1 satisfy Property P : it is closed under derivation and satisfies the property concerning the z... | 6 | https://mathoverflow.net/users/89429 | 242546 | 111,220 |
https://mathoverflow.net/questions/242559 | 4 | Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=C(X)$. Further we can make these into a group (the Picard group) by taking the group operation $\otimes\_A$. However af... | https://mathoverflow.net/users/29625 | Does a nonabelian Picard group exist? | By the Eilenberg-Watts theorem, the 2-group of invertible $(A, A)$-bimodules is naturally equivalent to the 2-group of automorphisms of $\text{Mod}(A)$. There is a natural map
$$\text{Aut}(A) \to \pi\_0 \text{Aut}(\text{Mod}(A))$$
whose kernel is the subgroup of inner automorphisms. So it suffices to find $A$ whose... | 8 | https://mathoverflow.net/users/290 | 242560 | 111,226 |
https://mathoverflow.net/questions/200437 | 3 | Let us consider an invertible matrix $\mathbf{A}\in GL\_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}\_{ii}=-1 \; \forall \, i$.
My question is the following:
Does it always exists a diagonal matrix $\mathbf{\Lambda}\in \mathbb{R}^d) $ such that
$\sigma(\mathbf{\Lambda A}) \subset \{ z\in\mathbb{C}... | https://mathoverflow.net/users/69428 | Stability of a linear system and spectrum of the product of two matrices | The answer is no, in general. Of course it is true for $d=1,2$. But already in the case $d=3$ there are counterexamples. One such counterexample is the matrix
$$
\mathbf{A} = \begin{bmatrix} -1 & 1 & 1\\ 1 & -1 & 1 \\ 1 & 1 & -1\end{bmatrix}
$$
All the 2x2 principal minors of this matrix are zero, so that the coeff... | 3 | https://mathoverflow.net/users/85570 | 242564 | 111,227 |
https://mathoverflow.net/questions/242562 | 6 | I read that one of the current challenging problems in mathematics is [constructing a minimal graph with a specified number of spanning trees](https://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees) (say, $k$).
* However, is there a quick way to create some graph $G$ (*not*... | https://mathoverflow.net/users/86412 | Create a graph with a specified number of spanning trees | A $k$-cycle works if $k>2$. For $k=1$ any tree works. I don't think $k=2$ is possible unless you allow double edges: if the graph is not a tree then it has an $m$-cycle $C$ for some $m>2$; remove edges off $C$ until the graph is connected but has no cycle other than $C$, and then removing an arbitrary edge of $C$ yield... | 13 | https://mathoverflow.net/users/14830 | 242565 | 111,228 |
https://mathoverflow.net/questions/242572 | 4 | Could someone be so kind to point me in the direction of a citeable proof of the following version of the Closed Graph Theorem? (i.e. assuming this is true, could someone give me a literature reference, or if it is false, a counter-example.))
Suppose $A$ and $B$ are topological spaces. For the correspondence $f : A \... | https://mathoverflow.net/users/94073 | Reference or counter-example for Closed Graph Theorem for multivalued maps in general topological spaces | Since this result is mentioned in [Wikipedia article on hemicontinuity](https://en.wikipedia.org/wiki/Hemicontinuity#Closed_Graph_Theorem), the [references](https://en.wikipedia.org/wiki/Hemicontinuity#References) mentioned there might be a reasonable place to look for this result.
If you simply try searching for ["u... | 5 | https://mathoverflow.net/users/8250 | 242576 | 111,232 |
https://mathoverflow.net/questions/242577 | 12 | The following observation makes me quite confused when I am trying to count the number of solutions of the equation:
$$\sum\_{k=0}^{M}{M \choose k}^2x^k=0$$
on finite fields $\mathbb{F}\_p$ with the prime number $p>3$ and $M=(p-1)/2$.
**Observation:** I tried the count the number of solutions with Mathematica. To... | https://mathoverflow.net/users/18286 | Congruence equation and quadratic residue | When $p \equiv 1 \bmod 4$, both expressions are $0$ and I understand why. For $p \equiv 1 \bmod 4$, the number $k$ is a QR iff $p-k$ is. Pairing off $k$ and $p-k$, the OEIS expression is
$$\frac{1}{p} \left( \frac{p\cdot \# \mbox{QRs}}{2} - \frac{p \cdot \# \mbox{non-QRs}}{2} \right) = 0.$$
Meanwhile, values of $x$ i... | 10 | https://mathoverflow.net/users/297 | 242600 | 111,236 |
https://mathoverflow.net/questions/242608 | 15 | There is a foreword, written by professor Snow, to the book [A mathematician's apology](https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology).
In the foreword, it is written some thing like the following:
"Hardy was opposed to a certain mathematical competition in the UK because he believed that such competiti... | https://mathoverflow.net/users/36688 | A certain mathematical competition in the UK | Hardy's opposition was to the Mathematical Tripos (the Cambridge undergraduate mathematics degree), as it was prior to its reform in 1909, which Hardy did much to bring about.
The text of "A Mathematician's Apology", with Snow's preface, is [here](https://archive.org/stream/AMathematiciansApology/Hardy-AMathematician... | 40 | https://mathoverflow.net/users/2481 | 242610 | 111,239 |
https://mathoverflow.net/questions/242544 | 2 | Let $C\subset\mathbb{P}^n$ be a smooth curve, and let $Y\subseteq\mathbb{P}^n$ be its tangent developable.
Given two general points $y\_1,y\_2\in Y$ does there exist a smooth curve $\Gamma\subset Y$ joining $y\_1$ and $y\_2$ ?
Assume that $C$ is rational. In this case does there exist a smooth rational curve in $Y... | https://mathoverflow.net/users/nan | Smooth curves in Tangent Developables | **Definition of the Tangent Developable Surface.** Let $k$ be a field. For every $k$-scheme $X$ and for every quasi-coherent $\mathcal{O}\_X$-module $\mathcal{F}$, denote by $\mathcal{P}^1\_X(\mathcal{F})$ the bundle of principal parts $\text{pr}\_{2,\*}(\text{pr}\_1^\*\mathcal{F}\otimes \mathcal{O}/\mathcal{I}^2)$ wit... | 1 | https://mathoverflow.net/users/13265 | 242614 | 111,241 |
https://mathoverflow.net/questions/242597 | 1 | Let $F:\mathbb{S}^{2}\times\lbrack0,1]\rightarrow\mathbb{R}$ be a smooth
($C^{\infty}$) function and $f\_{t}(x)=F(x,t)$. Suppose that $f\_{0}=f\_{1\text{
}}$is the projection over $z$-axis, so point $P=(0,0,1)$ is an absolute
maximum of both $f\_{0}$ and $f\_{1}$. Let $A\_{t}$ be the critical points set of
$f\_{t}$ and... | https://mathoverflow.net/users/94090 | Stability of the critical points set | If $f\_t$ is generic then $A$ is formed by a collection of curves in $S^2\times [0,1]$.
So, $(P,0)$ is connected to either
1. $(P,1)$ --- this happens for the constant $f\_t$.
2. $(-P,1)$ --- this happens for the projection of a generic sphere eversion, otherwise the orientation would not change.
3. $(-P,0)$ --- It ... | 1 | https://mathoverflow.net/users/1441 | 242618 | 111,244 |
https://mathoverflow.net/questions/242570 | 6 | I'm trying to understand a proof on "Sheaves of Continuous Definable Functions" (Pillay, Anand. "Sheaves of continuous definable functions." The Journal of symbolic logic 53.04 (1988): 1165-1169.)
Let $\mathcal{N} = (N,<,\ldots)$ be an o-minimal structure and let $X \subset N^m$ be a definable set, the o-minimal spe... | https://mathoverflow.net/users/58963 | O-minimal spectrum is a spectral space | In this answer, I'm going to differ from Pillay's terminology by writing "compact" instead of "quasicompact" and "[sober](https://en.wikipedia.org/wiki/Sober_space)" for the condition that every irreducible closed set is the closure of a unique point (called the generic point).
In the statement of Lemma 1.1, Pillay ... | 5 | https://mathoverflow.net/users/2126 | 242623 | 111,245 |
https://mathoverflow.net/questions/242588 | 4 | I have a kind of vague question.
Two non-degenerate symplectic vector space of same dimension (say $\mathbb{R}^{2n}$) are isomorphic. Then why all noncommutative tori of same dimension aren't isomorphic?
| https://mathoverflow.net/users/92111 | isomorphism of noncommutative tori | If you view a noncommutative $2n$-torus not as a strict deformation quantisation along a $\mathbb{R}^{2n}$-action (that happens to be periodic) but as a strict deformation quantisation along a $\mathbb{T}^{2n}$-action, then what actually matters isn't the non-degenerate symplectic form $\theta : \mathbb{R}^{2n} \times ... | 7 | https://mathoverflow.net/users/6999 | 242626 | 111,247 |
https://mathoverflow.net/questions/242641 | 17 |
>
> Do there exist commutative rings $A$ and $B$ and multiplicative subsets $S\subseteq A$, $T\subseteq B$ such that $A\not\simeq B$ but $S^{-1}A \simeq B$ and $T^{-1} B\simeq A$?
>
>
>
This question comes from a deleted claim in an [answer](https://mathoverflow.net/posts/239420/revisions) of Qiaochu Yuan. The o... | https://mathoverflow.net/users/35484 | Non-isomorphic rings that are localizations of each other | **Example.** Let $k$ be a field, and let $K = k(x\_1,x\_2,\ldots)$ be the fraction field of $k[x\_1,x\_2,\ldots]$. Let
$$A = K[y\_1,y\_2,\ldots],$$
and
$$B = A[y\_1^{-1}].$$
Then $B$ is a localisation of $A$. If we further localise at the multiplicative set $S = K[y\_1]\setminus\{0\} \subseteq B$, we get a ring that is... | 37 | https://mathoverflow.net/users/82179 | 242643 | 111,249 |
https://mathoverflow.net/questions/242661 | 8 | Assume that $X$ is a metric space, and $\sim$ is an equivalence relation on $X$.
Furthermore we assume that the number of elements in each equivalence class
is bounded by a positive constant.
Does the quotient topology on $X/{\sim}$ and the topology induced by the Hausdorff-metric on $X/{\sim}$ coincide?
| https://mathoverflow.net/users/78413 | Does the topology induced by the Hausdorff-metric and the quotient topology coincide? | Let $X=[0, \infty)$ with the metric $d(x,y)=|x-y|$.
The equivalence relation is $x\sim y$ iff $x=y$ or $xy=1$.
In the Hausdorff metric on $X/{\sim}$, the open ball with radius $1/2$ around the equivalence class $\{0\}$ contains only $\{0\}$.
However $\{0\}$ is not open in $X$, so $\{\{0\}\}$ is not open in the quoti... | 12 | https://mathoverflow.net/users/35484 | 242668 | 111,254 |
https://mathoverflow.net/questions/242659 | 4 | Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's equation $\Delta u^\phi=0$ on $\Omega$ with Dirichlet boundary conditions $u^\phi|\_{\partial\Omega}=\phi$. Is it true that... | https://mathoverflow.net/users/35593 | $H^1$-continuity of Laplace's equation with respect to boundary data | It is not true. An $H^1$ function has a boundary trace in $H^{1/2}$, and you cannot bound the $H^{1/2}$ norm by the $L^\infty$ norm.
| 3 | https://mathoverflow.net/users/12120 | 242669 | 111,255 |
https://mathoverflow.net/questions/210799 | 6 | Let $K$ be a field complete for a discrete valuation. Assume that the residue field has characteristic $p > 0$. Let $A$ be an abelian variety over $K$ having the property that (for some prime $\ell \neq p$) the action of the absolute Galois group of $K$ on the $\ell$-adic Tate module $T\_\ell(A)$ is tamely ramified.
... | https://mathoverflow.net/users/61815 | Log smooth models for abelian varieties | This question has been answered in the following preprint:
<http://arxiv.org/abs/1512.02464>
| 5 | https://mathoverflow.net/users/61815 | 242675 | 111,257 |
https://mathoverflow.net/questions/242604 | 2 | Let a $\Pi\_1^0$ sentence be a sentence asserting that some given Turing machine never halts at the empty input tape. Let Q1 be a (potentially *consistently lying*) oracle for deciding $\Pi\_1^0$ sentences, and let a Q1-TM be a Turing machine with access to Q1. Let TT be a given first order theory whose axioms are enum... | https://mathoverflow.net/users/20781 | Model existence and consistency conditions for $\Pi_1^0$ oracles | There's a natural way to define "consistently lying" (let's say *plausible*, to include the true $\Pi^0\_1$ theory):
>
> A set $A$ of (indices for) $\Pi^0\_1$ sentences is *plausible* if $PA\cup A\cup\{\neg \varphi: \varphi\in\Pi^0\_1\setminus A\}$ is consistent.
>
>
>
Note that this definition requires us to ... | 4 | https://mathoverflow.net/users/8133 | 242679 | 111,258 |
https://mathoverflow.net/questions/242654 | 3 | Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum\_{k\in\mathbb Z}\hat u\_k\,\mathrm{e}^{ikx},$ such that
$$
\lvert u\rvert\_s = \left(\sum\_{k\in\mathbb Z}
\lvert k\rvert^{2s}\lvert \hat u\_k\rvert^2\right)^{\!1/2}<\infty
$$
Then, for $\,u\in \ove... | https://mathoverflow.net/users/43681 | Optimal constant for a Sobolev-type inequality | Taking the Fourier transform and using $L^2$ orthogonality you are equivalently trying to estimate
$$ \sum\_{i + j +k = 0} \hat{u}\_i \hat{u}\_j \hat{u}\_{k} |i+j|^{n+1}|k|^{n} $$
Now from the equality $i + j +k = 0$ we have that either
1. $|k| >\max(|i|,|j|)$ in which case $i$ and $j$ have the same sign,
2. $|i... | 4 | https://mathoverflow.net/users/3948 | 242681 | 111,259 |
https://mathoverflow.net/questions/242634 | 10 | Given $n$, is it possible to upper bound the smallest $x > 1$ that satisfies the congruence $x^2 \equiv x\pmod{n}$? Obviously when $n$ is a prime power $x = n$, and we are in the worst situation. However for other $n$ we can make $x \leq \frac{n}{2}$. Perhaps better bounds can be obtained if we know that $n$ has more p... | https://mathoverflow.net/users/90626 | Smallest solution to $x^2 \equiv x\pmod{n}$ | We may really get better estimates if $n$ have at least three prime divisors, like, say $0<x\leqslant n/3$.
If $n=\prod q\_i$ for prime powers $q\_i$, denote by $u\_i$ a solution which is 1 modulo $q\_i$ and 0 modulo all $q\_j,j\ne i$. Assume that they all belong to $(n/3,2n/3)$ (if $2n/3<u\_i<n$, then $n/3>n-1-u\_i... | 6 | https://mathoverflow.net/users/4312 | 242683 | 111,261 |
https://mathoverflow.net/questions/242647 | 2 | Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by $[\Lambda(f)(b)](a) = f(a,b) \in C$.
Let $X, Y$ be topological spaces; we ... | https://mathoverflow.net/users/8628 | Spaces $Y$ such that $C(-, Y)$ is always acceptable | Following up on Simon's comments on 2): in the literature, one of the usual terms for this is that $X$ is *exponentiable*. There is in fact quite a lot of literature on this. Categorically, one is asking that there be a right adjoint $C(X, -)$ to $X \times -: \text{Top} \to \text{Top}$. As it turns out, this is equival... | 7 | https://mathoverflow.net/users/2926 | 242692 | 111,264 |
https://mathoverflow.net/questions/242685 | 3 | I wonder about the notion of a spin structure for varieties over any field and results in this direction. For example, I wonder if there is something like a spin-bundle for the sphere $x^2+y^2+z^2=R^2$ and the projective space qith $q^2+q+1$ elements for the finite fields $\mathbb{F}\_q$ depending on $q$
Thanx in ad... | https://mathoverflow.net/users/22709 | Spin structure for varieties, especially finite field | Let me work with arbitrary affine schemes $X = \text{Spec } k$. I believe there is a reasonable notion of a spin structure on a quadratic module $(V, q)$ over $k$, by which I mean a pair consisting of a finitely generated projective $k$-module (vector bundle over $X$) and a map $q : V \to k$ such that
1. $q(\lambda ... | 5 | https://mathoverflow.net/users/290 | 242693 | 111,265 |
https://mathoverflow.net/questions/143320 | 3 | The holonomy provides a bijection from
* the space of flat $G$-connections (modulo gauge equivalence) on a trivial $G$-bundle over $M$
to
* a connected component of the representation variety $Hom(\pi\_1M,G)/G$.
Is this a homeomorphism for the $C^\infty$-topology on the space of connections?
If not, what ca... | https://mathoverflow.net/users/39082 | Representation variety vs. space of flat connections | The answer to your question is yes, the bijection between the Betti moduli space and the de Rham moduli space is a homeomorphism.
See [here](http://arxiv.org/pdf/1404.5025v1.pdf) for a nice exposition on this topic with references for further reading.
| 4 | https://mathoverflow.net/users/12218 | 242700 | 111,269 |
https://mathoverflow.net/questions/242453 | 3 | Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a kernel means an equalizer of a morphism and a zero morphism, and cokernel dually). Then my question is:
>
> Can we m... | https://mathoverflow.net/users/69184 | Does a binormal category always admit an additive structure? | No. Let $C$ be a category constructed as follows. Let $G$ be a group and let $BG$ be the one-object category with automorphisms $G$. Then adjoin to $BG$ a zero object. This means that $C$ has two objects, $0$ (the zero object) and another object we'll call $c$. There are zero morphisms $0 \to 0, 0 \to c, c \to 0, c \to... | 6 | https://mathoverflow.net/users/290 | 242713 | 111,272 |
https://mathoverflow.net/questions/242652 | 0 | Let $(e\_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle\_{\mathcal{H}})$ and $(\mathcal{S\_H}, \langle\cdot, \cdot\rangle\_{\mathcal{S\_H}})$ be the separable Hilbert space of Hilbert Schmidt operators. Then, e.g. $(\phi\_{ij})$ is a ONB of $\mathcal{S\_H}$ with ... | https://mathoverflow.net/users/66236 | Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators | Thanks to "Jan-Christoph Schlage-Puchta", who edited my question before, for giving me a hint to write it down correctly, hopefully ;).
If $(E\_j)$ is a ONB of $\mathcal{H}^p,$ then I can choose a subsequence $(j\_k)\_k \subseteq \mathbb{N}$ such that $(E^{(1)}\_{j\_k})\_k$ is maximal linear independent. Then the ort... | 0 | https://mathoverflow.net/users/66236 | 242716 | 111,273 |
https://mathoverflow.net/questions/242671 | 7 | Let $X$ be a smooth, projective variety over $\mathbb{C}$ for which $\chi(X) = 0$. Here by $\chi$, I mean the topological Euler characteristic of $X(\mathbb{C})$; this number can also be computed as the degree of the top Chern class of the tangent bundle of $X$, and there are various other equivalent definitions.
I w... | https://mathoverflow.net/users/61815 | Hyperplane sections with chi non-zero | There is no easy way to express $\chi (Y)$ in terms of $\chi (X)$. Using $\chi (X)= c\_n(T\_X)$ (with $n:=\dim X$ and the identification $H^{2n}(X,\mathbb{Z})=\mathbb{Z}$), and the
exact sequence $0\rightarrow T\_Y\rightarrow T\_{X|Y}\rightarrow N\_{Y/X}\rightarrow 0\ $ you get $\ \ \ \chi (Y)=c\_{n-1}\cdot h-c\_{n-2}\... | 6 | https://mathoverflow.net/users/40297 | 242717 | 111,274 |
https://mathoverflow.net/questions/242707 | 3 | Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \mathfrak{l}$, and $\mathfrak{n}$ denote the corresponding Lie algebras. Recall that $N$ is normal in $P$; therefore, $P$ and ... | https://mathoverflow.net/users/41301 | Invariant theory for parabolics | The map $\mathbb C[\mathfrak g]^G\to\mathbb C[\mathfrak g]^P$ is an isomorphism for trivial reasons: In any *quasi-affine* $G$-variety, $P$ and $G$ have the same fixed points. Just look at the orbit map of a $P$-fixed point which factors through the complete variety $G/P$. Applied to representations, this means $V^G=V^... | 11 | https://mathoverflow.net/users/89948 | 242719 | 111,276 |
https://mathoverflow.net/questions/242691 | 15 | I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to <https://arxiv.org/abs/1408.2760>, $ SO(2n+1)/U(n) $ is not a symmetric space because it does not have the right Cartan decomposition of the Lie al... | https://mathoverflow.net/users/54448 | Is SO(2n+1)/U(n) a symmetric space? | Let me complement Claudio's answer. There is indeed a definition of symmetric space which works for any Riemannian manifold $M$: For any point $p\in M$ there is an involutive isometry $\iota\_p$ of $M$ such that $p$ is an isolated fixed point. This involution is easy to describe: Take any geodesic $\gamma(t)$ with $\ga... | 12 | https://mathoverflow.net/users/89948 | 242725 | 111,277 |
https://mathoverflow.net/questions/242709 | 7 | The various large cardinal axioms are usually described in terms of some roughly linear hierarchy of varying consistency strengths. However, some cardinal axioms potentially contradict one another. As an example, asserting the existence of unboundedly many strong limit cardinals implies AC. However, asserting the exist... | https://mathoverflow.net/users/94150 | Are There Mutually Exclusive Large Cardinal Axioms in ZFC? | The answer to your question depends on what counts as a large
cardinal axiom, and there is no agreed-upon official definition
for that term.
On the one hand, it is easy to formulate incompatible theories
involving large cardinals. If these theories themselves count as
large cardinal axioms, then they provide the answ... | 6 | https://mathoverflow.net/users/1946 | 242727 | 111,278 |
https://mathoverflow.net/questions/242715 | 1 | There are four versions of compatibility conditions of Yetter-Drinfeld modules (left-left, left-right, right-left, right-right) in the [article](https://ncatlab.org/nlab/show/Yetter-Drinfeld+module). Are there some references which derive these compatibility conditions? Thank you very much.
| https://mathoverflow.net/users/11877 | Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules | All of these conditions can be derived from Majid's construction of the dual of a monoidal category $\mathcal{C}$ [Maj1, Def. 3.2, 3.3] (in the reference, we want to consider the category denoted by $(\mathcal{C},F)^\circ$; see especially Expl. 3.4 which gives the monoidal center $\mathcal{Z}(\mathcal{C})$ as a special... | 4 | https://mathoverflow.net/users/33854 | 242738 | 111,283 |
https://mathoverflow.net/questions/242708 | 3 | Let $H$ be a Hopf algebra and $V$ a right $H$-module and right $H$-comodule. The module $V$ is a Yetter-Drinfeld module over $H$ if and only if
\begin{align}
( v \triangleleft h\_{(2)} )\_{(0)} \otimes h\_{(1)} ( v \triangleleft h\_{(2)} )\_{(1)} = ( v\_{(0)} \triangleleft h\_{(1)} ) \otimes v\_{(1)} h\_{(2)}. \quad (1... | https://mathoverflow.net/users/11877 | How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding | The YD-condition is not equivalent to $\Psi$ being a braiding. The condition (1) is related to $\Psi$ being a morphism of $H$-modules (see the more detailed answer to your other question [Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules](https://mathoverflow.net/questions/242715/r... | 3 | https://mathoverflow.net/users/33854 | 242740 | 111,284 |
https://mathoverflow.net/questions/242678 | 7 | I've been searching around looking for the (maths component) of the scholarship papers to Trinity College (Cambridge) from around 1890. Can anyone provide a link to a pdf scan of these papers?
Was the scholarship paper some kind of combined paper or did it cover maths only? Anyway I'd like to have a look at the bread... | https://mathoverflow.net/users/94129 | Trinity College, Cambridge, circa 1896 maths scholarship papers | I'll answer my own question. There are many questions that were included in the college entrance and scholarship papers to be found in for example Routh's Dynamics of a Particle and various other text books of the time.
Still I'd love to see a scan of a few complete papers from this period circa 1890.
Cheers,
Aeler... | 0 | https://mathoverflow.net/users/94129 | 242741 | 111,285 |
https://mathoverflow.net/questions/242726 | 3 | Planning of the question:
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle
The isotropic almost complex structures $J\_{\delta , \sigma}$ were introduced by Aguilar on the tangent bundle of a Riemannian manifold $(M,g)$ in [this paper](http://link.springer.com/article/10.1007%2FBF02568316#page-1)... | https://mathoverflow.net/users/86401 | An answer to this system of PDE's | Unless I am somehow misunderstanding the notation, we can forget about the x dependence due to your last condition. So let $\alpha$ be a function of y. Your third equation can be written as
$${\partial^2 \over\partial y\_i\partial y\_j}(\alpha^{(n+1)/2})=0.$$
So it follows that
$$\alpha^{(n+1)/2}=\sum\_i f\_i(y\_i).$$
... | 3 | https://mathoverflow.net/users/12120 | 242743 | 111,286 |
https://mathoverflow.net/questions/242748 | 6 | It is written on wikipedia article (<https://en.wikipedia.org/wiki/Analytic_torsion>) that the Reidemeister torsion is the first invariant that could distinguish between spaces which are homotopy equivalent but not homeomorphic. Can any one give me examples of other invariants which do this?
| https://mathoverflow.net/users/89956 | List of invariants that distinguish homotopy equivalent non-homeomorphic spaces | Usual homotopy-type invariants of the configuration spaces associated to a given space may be able to distinguish between spaces which are homotopy equivalent but not homeomorphic. For instance, Riccardo Longoni and Paolo Salvatore [*Configuration spaces are not homotopy invariant*, Topology **44** (2005), no. 2, 375–3... | 14 | https://mathoverflow.net/users/17846 | 242750 | 111,290 |
https://mathoverflow.net/questions/242749 | -2 | In the well-known book "*THE PRINCETON COMPANION TO MATHEMATICS*" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document [Spectral Geometry in Non-standard
Domains](https://www.fields.utoronto.ca/programs/scientific/14-15/summer-research14/mini-conf/Spectra... | https://mathoverflow.net/users/91822 | The spherical harmonics are the EIGENVECTORS of Beltrami operator | The article linked by the OP proves that every degree $k$ spherical harmonic is a $k(k+1)$-eigenfunction of the Laplacian on $S^2$. The OP asks: is every eigenfunction of the Laplacian a spherical harmonic?
The answer is yes, though it seems that neither the linked article nor the Wikipedia page on spherical harmonic... | 8 | https://mathoverflow.net/users/4362 | 242755 | 111,292 |
https://mathoverflow.net/questions/242756 | 16 | Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}\_G(H)=H$. Then by results of [Brion and Pauer](http://link.springer.com/article/10.1007%2FBF02564447)
and [Knop](http://www.ams.org/journals/jams/1996-9-01/S0894-03... | https://mathoverflow.net/users/4149 | Is the wonderful compactification of a spherical homogeneous variety always projective? | The wonderful compactification is always projective. One way to see is to use a theorem of Sumihiro which says that a normal $G$-variety is covered by $G$-invariant quasiprojective open subsets. Since a wonderful variety $X$ has only one closed orbit $Y$ the only $G$-invariant open subset meeting $Y$ is $X$ itself.
| 19 | https://mathoverflow.net/users/89948 | 242758 | 111,294 |
https://mathoverflow.net/questions/242633 | 4 | Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl\_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l\_{2k}\equiv Cl\_{2k}\otimes\_{\mathbb{R}}\mathbb{C}.$$
This has both a $\mathbb Z\_2$-grading and a $\mathbb Z$-grading, which we distinguish by the formulae
$$\ma... | https://mathoverflow.net/users/69531 | Why does the Bogolyubov transformation work? - In language of Clifford Algebras? | There is an elegant formulation of the Bogolyubov transformation in terms of Clifford algebras. Note that a quadratic Hamiltonian (noted by a hat), is a hermitian element of the representation of a Clifford algebra on the Hilbert space,
$$\mathbb{C} l\_{2k}\xrightarrow{~~Q~~} \text{End}(\Lambda(\mathbb{C}^{k})),~~~~\ha... | 3 | https://mathoverflow.net/users/69531 | 242761 | 111,295 |
https://mathoverflow.net/questions/242760 | 3 | The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f\_2} & R\_2 \\
\downarrow{f\_1} & & \downarrow{j\_2} \\
R\_1 & \xrightarrow{j\_1} & R\_3
\end{array}
$$
such that $j\_1$ or $j\_2$ is surjective, a pair of projective m... | https://mathoverflow.net/users/58651 | Milnor patching for general modules | Patching tends not to work well at the level of modules, especially if you want to preserve properties like finiteness of projective dimension. For most purposes, it's far better to work not with modules but with perfect complexes of modules (that is, complexes that are quasi-isomorphic to complexes of projectives).
... | 7 | https://mathoverflow.net/users/10503 | 242764 | 111,296 |
https://mathoverflow.net/questions/242703 | 3 | Let $A$ be a ring and $f \in A$ an element. If $M$ is an $A$-module on which multiplication by $f$ is an isomorphism, then $M$ is in fact an $A\_f$-module.
Now suppose that $C \in D(A)$ is a complex in the derived category of $A$-modules and that the multiplication by $f$ map $C \xrightarrow{f} C$ is a (quasi-)isomo... | https://mathoverflow.net/users/63877 | Characterizing the image of $D(A_f) \rightarrow D(A)$ | The two sides of your "that is" are different questions:
1) Does $C$ necessarily come from $D(A\_f)$ ?
2) Is there some complex $C'$ of $A\_f$ modules and a quasi-isomorphism $C'\rightarrow C$ of complexes of $A$-modules?
Regarding 1): By assumption, $f$ induces an isomorphism on homology. Therefore, by your firs... | 3 | https://mathoverflow.net/users/10503 | 242768 | 111,298 |
https://mathoverflow.net/questions/242680 | 15 | Recall the Chevalley‒Warning theorem:
>
> **Theorem.** Let $f\_1, \ldots, f\_r \in \mathbb F\_q[x\_1,\ldots,x\_n]$ be polynomials of degrees $d\_1, \ldots, d\_r$. If
> $$d\_1 + \ldots + d\_r < n,$$
> then the number of common zeroes of $f\_1, \ldots, f\_r$ is $0$ modulo $p$.
>
>
>
In particular, if there exi... | https://mathoverflow.net/users/82179 | Can you use Chevalley‒Warning to prove existence of a solution? | Let $f\_i \in \mathbb{F}\_q[X\_1,\dots,X\_n]$ have degree at most $d$ for each $i \in [|1,r|]$, and assume that the affine scheme
$$
X = \operatorname{Spec} \mathbb F\_q [x\_1,\ldots,x\_n]/(f\_1, \ldots, f\_r)
$$
is nonempty. Take a prime number $\ell \neq p$. Consider the Frobenius $F$ as an endomorphism of the vector... | 14 | https://mathoverflow.net/users/21724 | 242794 | 111,306 |
https://mathoverflow.net/questions/242664 | 5 | In a new survey on $E\_8$, namely
Skip Garibaldi - E8 the most exceptional group
, the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-Index. This construction was actually invented by Tits and bears his name..
From the inclusion of $F\_4 \subset E\_... | https://mathoverflow.net/users/51251 | Constructing groups of Type E7 with certain Tits Index | This might shed some light on relationship between anisotropic quadratic forms in 10 variables and the desired forms of $E\_7$, though it uses results more recent than Tits, and doesn't quite answer your questions as stated. Bruce Allison worked out the following results in his paper "Structurable division algebras and... | 5 | https://mathoverflow.net/users/3545 | 242800 | 111,308 |
https://mathoverflow.net/questions/242781 | 1 | A topological space $(X,\tau)$ is connected if and only if the only continuous maps $f:X\to\{0,1\}$ (where $\{0,1\}$ carries the discrete topology) are the constant maps.
Are there other examples of topological properties that can be discribed via topological properties of continuous maps? (I apologize for the somewh... | https://mathoverflow.net/users/8628 | Topological properties via properties continuous maps | There are numerous examples and it's not too difficult to come up with a few on your own, but here is a few examples:
A topological space $X$ is said to be functionally Hausdorff if and only if each pair of distinct points of $X$ can be separated by a continuous function $f:X\rightarrow [0,1].$
Another would be to ... | 1 | https://mathoverflow.net/users/92399 | 242809 | 111,311 |
https://mathoverflow.net/questions/242762 | 8 | In [FSIT] and [OIMT] it is claimed that there is a surjection from $P(\kappa)\cap J^{\vec E}\_{\nu(E\_\alpha)}\times[\nu(E\_\alpha)]^{<\omega}$ onto $\alpha$, and that this surjection lies in $J\_{\alpha+1}^{\vec E}$. It is stated as a rather trivial fact, but I'm having trouble with seeing how this map should look lik... | https://mathoverflow.net/users/38602 | Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequence $\vec E$ | We know that $\alpha = (\nu^{+})^{Ult(J^{\vec{E}}\_{\alpha}, E\_{\alpha})}$ and that $i\_{E\_{\alpha}} (\kappa) > \nu$, where $i\_{E\_{\alpha}}$ denotes the ultrapower embedding. Thus working in $Ult(J^{E\_{\alpha}}\_{\alpha}, E\_{\alpha})$ any $\beta <\nu^{+}=\alpha$ can be represented in the ultrapower $Ult(J^{\vec{E... | 5 | https://mathoverflow.net/users/4753 | 242812 | 111,313 |
https://mathoverflow.net/questions/242651 | 2 | For a Noetherian local ring $(R,m)$, and a finite $R$-module $M$ with $\operatorname{depth} M=t,$ *type* of $M$ is defined to be $r(M):=dim\_{R/m}Ext^t \ (R/m, M).$
>
> Is there a characterization of $r(M)$ by *local cohomology* instead of $Ext$?
>
>
>
If so, can you give a reference?
Thank you.
| https://mathoverflow.net/users/47763 | Is there a characterization of r(M) by local cohomology instead of Ext | I think (it is well-known that) $r(M) = \dim \mathrm{Soc}(H^t\_{\mathfrak{m}}(M)) = \ell (\mathrm{Hom}(R/\mathfrak{m}, H^t\_{\mathfrak{m}}(M)))$.
Indeed, choose an $M$-regular sequence $x\_1, \ldots, x\_t$ of $M$. We have
$$\mathrm{Ext}^t\_R(R/\mathfrak{m}, M) \cong \mathrm{Hom}(R/\mathfrak{m}, M/(x\_1, \ldots, x\_t... | 2 | https://mathoverflow.net/users/17901 | 242813 | 111,314 |
https://mathoverflow.net/questions/242808 | 1 | Specifically, I want to compute the set of values of $|G:\text{ker}(\chi)|/\chi(1)$ for all the characters of a p-group, for a lot of p-groups. I don't know how to use either program, so before I invest in learning it, I'm wondering about the pros and cons of GAP and MAGMA (when compared to each other).
| https://mathoverflow.net/users/94212 | Which is better for creating tables of group theory info, GAP or MAGMA? | The corresponding Magma function (using the same name as Stefan) is
```
CharacterKernelIndicesDividedByValueAtOne := function ( G )
return [Index(G,Kernel(c))/Degree(c) : c in CharacterTable(G) ];
end function;
```
Both the GAP and Magma versions returned the answer almost instantly on SmallGroup(3^6,300). The M... | 4 | https://mathoverflow.net/users/35840 | 242824 | 111,320 |
https://mathoverflow.net/questions/242457 | 4 | Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by $[\Lambda(f)(b)](a) = f(a,b) \in C$.
Let $X, Y$ be topological spaces; we ... | https://mathoverflow.net/users/8628 | Admissible and proper topologies on $C(X,Y)$ | First, another name for a proper topology on a function space is *splitting topology*, and other names for an admissible topology are *approximating topology* and *conjoining topology*. This may be of aid in further Google searches.
Here are some basic facts:
* Any topology that is coarser (smaller) than a proper... | 3 | https://mathoverflow.net/users/2926 | 242831 | 111,325 |
https://mathoverflow.net/questions/242789 | 2 | I'm reading the Bruzzo and Graña Otero's paper *Semistable and Numerically Effective Principal (Higgs) Bundles*; here: $X$ is a smooth, complex, projective variety; $G$ is a connected, complex, reductive, (affine) algebraic group; $E$ is a principal $G$-bundle on $X$.
A *Higgs field* $\varphi$ on $E$ is a global sect... | https://mathoverflow.net/users/57030 | Scheme of Higgs reductions | Generalising t3suji's comment, let us consider $GL(n)$ Higgs bundles, i.e., rank $n$ Higgs vector bundles $\mathcal E$. The Higgs Grassmannian of rank $m$ quotients is a closed subscheme of the usual Grassmannian bundle; in general it is a proper subset, as there are quotients of $\mathcal E$ that are not Higgs quotien... | 4 | https://mathoverflow.net/users/94220 | 242832 | 111,326 |
https://mathoverflow.net/questions/193663 | 8 | Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the closed unit disc? I can only find papers which discuss analytic complex valued functions, which is not what I need.
UPDATE: ... | https://mathoverflow.net/users/10366 | Approximation theory on the disc | If the main interest lies in numerical methods designed to compute specifically with functions defined on a disk, the following recent manuscript may be useful:
<https://arxiv.org/pdf/1604.03061.pdf>
A few basic (theoretical) differences between Chebyshev polynomial approximations on an interval and on a disk are a... | 1 | https://mathoverflow.net/users/89429 | 242838 | 111,328 |
https://mathoverflow.net/questions/242835 | 13 | In [this paper](http://www.sciencedirect.com/science/article/pii/S0020019013002214) (Construction 2.6 p860) the authors have built examples of
connected $k$-regular graph without Hamiltonian path, but with a cut-vertex (i.e. it is not $2$-connected).
*Question:* Is there a $2$-connected $k$-regular graph without Ha... | https://mathoverflow.net/users/34538 | Is there a 2-connected k-regular graph without Hamiltonian path? | Take $k>3$ long paths between two vertices $a,b$. This graph is already 2-connected. Now add some edges to make it $k$-regular so that every edge joins only interior vertices of the same path (it is not hard). Then $G\setminus\{a,b\}$ has more connected components than any graph having Hamiltonian path with two vertice... | 14 | https://mathoverflow.net/users/4312 | 242841 | 111,330 |
https://mathoverflow.net/questions/242460 | 30 | Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has **algebro-geometric application** in mind **they seem more attractive then stable $(\infty,1)-$categories** which seem to carry data in a sli... | https://mathoverflow.net/users/22810 | DG categories in algebraic geometry - guide to the literature? | Let me try to address the bulleted questions and simultaneously advertise the G-R book everyone has mentioned. Since the main question was about literature, I could also mention Drinfeld's article "DG quotients of DG categories," which nicely summarizes the state of the general theory before $\infty$-categories shook e... | 23 | https://mathoverflow.net/users/3544 | 242842 | 111,331 |
https://mathoverflow.net/questions/240213 | 7 | Suppose we have a symmetric polynomial $P$ in $n$ variables.
We can partition this alphabet into sets with one or two letters, e.g. ${ {x\_1}, {x\_2, x\_3}}.
We can thus see $P$ as an element in $Q[x\_1][x\_2,x\_3]$,
and expand it in the Schur basis in each sub-set of variables.
E.g, $P=5 s\_{2}(x\_1)s\_{32}(x\_2,x\... | https://mathoverflow.net/users/1056 | Schur positivity on 2 letter alphabets implies Schur-positivity on n letters? | My intuition is that if $P$ is in three variables then your requirements only force $P$ to be unimodal in each pair of variables, while being Schur positive is much more restrictive. For example, take
$$P(x\_1,x\_2,x\_3)=s\_3(x\_1,x\_2,x\_3)+s\_{21}(x\_1,x\_2,x\_3)-s\_{111}(x\_1,x\_2,x\_3).$$
Equivalently,
$$P(x\_1,x\... | 3 | https://mathoverflow.net/users/54838 | 242844 | 111,333 |
https://mathoverflow.net/questions/242847 | 3 | Let us parametrize the set of lattices inside $\mathbb{C}^g$ with the open dense subset $U = \text{GL}\_{2g}(\mathbb{R})$ of $\mathbb{R}^{4g^2}$. Does there exist a countable family $(Z\_n)\_{n \in \mathbb{N}}$ of algebraic real hypersurfaces of $\mathbb{R}^{4g^2}$ such that for every matrix $M \in U \setminus \bigcup\... | https://mathoverflow.net/users/83593 | A very general complex torus is simple | I think yes. Let $J$ be the complex structure on $\mathbb{R}^{2g}$. Let $L$ be the lattice. Then there is a complex subtorus if and only if $L$ intersects some complex-linear subspace in a full sub-lattice. Thus, there are no complex subtori if and only if for all $x\_1,\dots,x\_k\in L$ there do not exist $y\_1,\dots,y... | 5 | https://mathoverflow.net/users/8003 | 242850 | 111,336 |
https://mathoverflow.net/questions/241679 | 5 | A solution to the **$l$-isoperimetric problem** on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant:
$$h(\gamma) = \frac{l}{\text{vol}(M\_\gamma)}.$$
Where $M\setminus \gamma$ denotes the submanifold of minimal volume of $M$ with boundary ... | https://mathoverflow.net/users/14549 | The radially symmetric isoperimetric problem | **Foreword**. Given my unfamiliarity with the subject, and the amount of literature on the isoperimetric problem, it is highly likely that the answer can be shortened or improved. Moreover I will take some days to verify the argument before accepting the answer. I would be happy for any comment or additional contributi... | 1 | https://mathoverflow.net/users/14549 | 242871 | 111,340 |
https://mathoverflow.net/questions/242877 | 3 | There seem to be two definitions of what a saturated class should be:
1. A class of morphisms closed under retracts, pushouts and transfinite composition.
2. A class of monomorphisms containing all isomomorphisms, closed under retracts, pushouts, arbitrary coproducts and countable composition.
My question is, does ... | https://mathoverflow.net/users/20356 | Saturated classes and cofibrantly generated model structures | It doesn't make a difference as long as we restrict attention to compactly generated saturated classes, i.e. cofibrantly generated ones where generators can be chosen to have $\aleph\_0$-small domains. This is the case in all your examples. (In the case of topological spaces, compact spaces are not $\aleph\_0$-small wi... | 3 | https://mathoverflow.net/users/12547 | 242881 | 111,346 |
https://mathoverflow.net/questions/242776 | 16 | Let $X$ be a separable complete metrizable space. Does there exist a complete metric $d$ and a Borel measure $\mu$ such that
(a)
$\mu(B\_r(x))<\infty$ for every open ball $B\_r(x)$ of radius $r>0$ around $x\in X$,
(b) for each $r>0$ the map
$
x\mapsto \mu(B\_r(x))
$
is continuous on $X$,
(c) $\mathrm{supp}(\mu)... | https://mathoverflow.net/users/nan | Does there exist a ``continuous measure'' on a metric space? | By continuing the line of thought from Nate Eldredge's comment you can handle finitely many separated parts of the space by a distance having values at most one and exactly one between points in different connected components. However, a possible counter-example (I did not yet check the details) should follow by contin... | 6 | https://mathoverflow.net/users/11716 | 242888 | 111,350 |
https://mathoverflow.net/questions/242891 | 5 | I have a question concerning cardinal arithmetic in $\sf{ZF}$.
Write $ X \cong Y$ if there is a bijection between $X$ and $Y$.
Let $M$ be a set such that $2M\cong M$.
Can one show, in $\sf{ZF}$, that for any infinite subset $X$ of $M$, we also have $2X \cong X$ ?
Thank you :)
| https://mathoverflow.net/users/74128 | Subsets of $M$ such that $2M \cong M$ | No.
Simply because if $A$ is a set of any cardinality, there is $B$ such that:
1. $|A|\leq|B$, and
2. $|B|+|B|=|B|$, and in fact we can require $|B\times B|=|B|$.
Just take $B=A^\omega$, all the functions from $\omega$ to $A$. Then we have that $|B|=|A|^{\aleph\_0}$ and therefore $|B\times B|=|A|^{\aleph\_0+\alep... | 12 | https://mathoverflow.net/users/7206 | 242893 | 111,351 |
https://mathoverflow.net/questions/242899 | 1 | Let $(X,d)$ be a non-empty complete metric space, let
M be the set of all non-empty compact subsets equipped
with the Hausdorff metric, and $N$ be a positive integer.
Is
$$
\{A\subset X : 1\le \# A \le N \}
$$
a closed subset of $M$?
| https://mathoverflow.net/users/78413 | Is the following set closed with respect to the Hausdorff metric? | We need to prove that $\{A\subseteq X:\#A>N\}$ is open. Let $d\_H$ denote the Hausdorff metric.
We need to prove that for any nonempty compact $A$ with $\#A>N$, there exists $\epsilon>0$ such that any nonempty compact $B$ with $d\_H(A, B)<\epsilon$, we have $\#B>N$.
Assume for the sake of contrary that there exists... | 4 | https://mathoverflow.net/users/35484 | 242902 | 111,352 |
https://mathoverflow.net/questions/242863 | 6 | Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?
| https://mathoverflow.net/users/23500 | Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group? | No, there are cocompact lattices for which no such homomorphism exists. This is Warning 16.4.3 on page 330 of my book [Introduction to Arithmetic Groups](https://arxiv.org/src/math/0106063v6/anc/IntroArithGrps-FINAL.pdf).
Assume $\Gamma$ is irreducible, $G$ has no compact factors, and $\mathrm{rank}\_{\mathbb{R}} G \... | 14 | https://mathoverflow.net/users/68305 | 242912 | 111,356 |
https://mathoverflow.net/questions/242892 | 7 | In a 1993 letter, Deligne posed the following (paraphrased from a [paper of Gerstenhaber and Voronov's](http://www.math.umn.edu/~voronov/del.pdf)):
>
>
> >
> > **Conjecture (Deligne).** The Hochschild cochain complex $CC^\*(A)$ of an associative algebra $A$ has a natural structure of an algebra over a chain opera... | https://mathoverflow.net/users/20391 | Can the Hochschild cochain complex be given the structure of a "homotopy BV algebra"? | Yes, this is the "cyclic Deligne conjecture", which also has several proofs by now. I believe the first one was Kaufmann's, using the cacti operad.
| 5 | https://mathoverflow.net/users/1310 | 242914 | 111,358 |
https://mathoverflow.net/questions/242925 | 5 | Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.
Can the same be done for a family of projections? That is, given finitely many projections $P\_1$,...,$P\_n$ in $B(X)$ is there an equivalent norm under which all projections become contractive?
Are ther... | https://mathoverflow.net/users/94259 | Renorming a Banach space to make projections contractive | Let $X$ be the Euclidean plane, and consider the two projections
$$
P\_1=\begin{pmatrix}1&A\\0&0\end{pmatrix}\quad\text{and}\quad
P\_2=\begin{pmatrix}0&0\\A&1\end{pmatrix},
$$
where $A$ is a big constant (actually any $A>1$ will do). If a norm on $X$ gave both of them norms $\leq1$, then the same would be true for thei... | 16 | https://mathoverflow.net/users/6794 | 242927 | 111,360 |
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