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https://mathoverflow.net/questions/198241 | 1 | I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example.
This question is inspired by my earlier question
[when-is-the-topology-generated-by-countable-subsets](https://mathoverfl... | https://mathoverflow.net/users/58628 | A countable tight topological group where every countable subset is metrizable | Actually, Santi Spadaro almost gave the answer in a comment to your previous question: Let $X$ be the $\Sigma$-product of $2^{\omega\_1}$ (where $2=\{0,1\}$), that is, the subset of $2^{\omega\_1}$ of all points with at most countably many coordinates different from $0$. Of course $X$ is a subgroup of $2^{\omega\_1}$.
... | 4 | https://mathoverflow.net/users/29491 | 198301 | 95,946 |
https://mathoverflow.net/questions/198272 | 0 | We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see [here](https://mathoverflow.net/a/167461/34538)).
Now about infinite single chain lattice:
*Question*: Is there an irreducible s... | https://mathoverflow.net/users/34538 | Is there an irreducible subfactor with an infinite homogeneous single chain lattice? | Yes, for example $(R \subset R \rtimes \mathbb{Z}(p^{\infty}))$ the [Prüfer $p$-group](http://en.wikipedia.org/wiki/Pr%C3%BCfer_group) subfactor.
More precisely:
Let $p$ be a prime number and let the chain of embeddings $\mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathbb{Z}/p^2\mathbb{Z} \hookrightarrow \mathbb{Z}/p^3... | 0 | https://mathoverflow.net/users/34538 | 198303 | 95,947 |
https://mathoverflow.net/questions/198205 | 6 | Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes\_{BV}$, the category $\mathbf{Op}$ is symmetric monoidal.
**Questions:**
1) is $(\mathbf{Op}, \otimes\_{BV})$ a symmetric monoidal model categ... | https://mathoverflow.net/users/61328 | Boardman-Vogt tensor product | The answer to 2) is yes for cofibrant operads, see <http://arxiv.org/abs/1102.1311> by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with cofibrations.
EDIT: As an answer to Chris' comment, here is a counterexample to 2) in the setting of colored operads, althou... | 10 | https://mathoverflow.net/users/36302 | 198310 | 95,948 |
https://mathoverflow.net/questions/198306 | 17 | Suppose $G$ is a finite group and that $E$ is an equation of the form $x\_1 x\_2 ... x\_n = e$, where each $x\_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.
>
> Is it always true that the number of ordered pairs $(g, h)$ of elements of $G$ satisfying $E$ is a multiple of $|G|$?
>
>
>
I know that it i... | https://mathoverflow.net/users/68449 | Number of solutions to equations in finite groups | Yes. This is a special case of results in <http://arxiv.org/pdf/1205.2824.pdf>
| 18 | https://mathoverflow.net/users/15934 | 198313 | 95,950 |
https://mathoverflow.net/questions/198143 | 7 | I'm interested in an asymptotic expansion of the following Riemann zeta-type function
$$
\begin{align}
\displaystyle \zeta(s \mid a,b) := \sum\_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)},
\quad \Re a >-1, \, \Re b >-1, \, s>0, \tag1
\end{align}
$$ as $s \to 0^+$.
The case $a=b$ in $(1)$ leads to the Riemann Hurwitz ... | https://mathoverflow.net/users/68382 | Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$ | For $s > 0$ we have
$$
\sum\_{n=1}^{\infty} \frac{1}{(n+a)^s(n+b)} - \sum\_{n=1}^{\infty} \frac{1}{(n+a)^{s+1}} = \sum\_{n=1}^{\infty} \frac{a-b}{(n+a)^{s+1}(n+b)} =: g(s).
$$
The series on the right-hand side converges and is analytic on $\operatorname{Re} s > -1$, so the difference on the left-hand side can be an... | 8 | https://mathoverflow.net/users/21917 | 198315 | 95,951 |
https://mathoverflow.net/questions/196312 | 5 | Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (**note carefully**).
>
> Suppose $g$ is holomorphic on $\mathbb{C}^n$ and satisfies $\int |g|^p\,d\mu < \infty$. Do there exist holomorphic polynomials $g\_n$ such that $\int |g-g\_n|^p... | https://mathoverflow.net/users/4832 | Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$? | Yes, they are dense.
The key result comes from a paper of R. Wallstén [1], in which Theorem 3.1 implies that the set $\mathcal{E}$ of functions of the form
$$f(z) = \sum\_{j=1}^m a\_j e^{\langle z, w\_j \rangle}, \qquad a\_j \in \mathbb{C}, \, w\_j \in \mathbb{C}^n$$
is dense in $\mathcal{H} L^p(\mu)$ for any $0 < p ... | 3 | https://mathoverflow.net/users/4832 | 198324 | 95,952 |
https://mathoverflow.net/questions/198320 | 6 | Good evening, I am currently taking a class which has combinatorial designs as the first topic, we are using Peter Cameron's book [Designs, Graphs, Codes and their Links](http://books.google.com.mx/books/about/Designs_Graphs_Codes_and_Their_Links.html?id=j1CZeuHI7q0C&redir_esc=y) which I am finding extremely interestin... | https://mathoverflow.net/users/24478 | Combinatorial designs textbook recommendation | There is a list of references in
<http://www.maths.qmul.ac.uk/~pjc/design/resources.html#books>
and
<http://en.wikipedia.org/wiki/Block_design#References>
From the latter, I know books by Beth et al (a long and detailed) and by Hughes and Piper (shorter and more readable).
All of it is outdated in several import... | 7 | https://mathoverflow.net/users/11100 | 198329 | 95,954 |
https://mathoverflow.net/questions/198265 | 10 | Let $$f(n,k) = \sum\limits\_{j = - k}^k {{{( - 1)}^{k - j}}}
\binom{n-j}{k-j}\binom{n+j}{k+j}.$$
Then $f(n,k)=\binom{n}{k}$
because it satisfies $f(n,k)=f(n-1,k)+f(n-1,k-1)$ and the obvious boundary values.
Let $ {n\brack {k}}$ be a $q-$binomial coefficient. I want to know if there is a similar proof for the ident... | https://mathoverflow.net/users/5585 | Quest for a human proof of a $q-$binomial identity | At first, we use a formula $\binom{u}{m}=(-1)^m\binom{m-u-1}{m}$. Thus
$$\binom{n\pm j}{k\pm j}=(-1)^{k-j}\binom{k-n-1}{k\pm j}.$$
Denote $k-n-1=x$, and $k-j$ by $s\in \{0,\dots,2k\}$, we have to prove that
$$
\sum\_{s=0}^{2k} (-1)^s \binom{x}{s}\binom{x}{2k-s}=(-1)^k \binom{x}{k}.
$$
LHS counts the coefficient of $t^... | 9 | https://mathoverflow.net/users/4312 | 198331 | 95,955 |
https://mathoverflow.net/questions/198302 | 3 | So recently I heard someone claiming that if $X\rightarrow S$ is a smooth curve (not necessarily proper?) and $S$ is an arbitrary scheme over $\text{Spec }R$ (for $R$ sufficiently nice), then there is an fpqc(fppf? can we go as far as etale?) cover $T\rightarrow S$ and a scheme $S'$ of finite type over $R$ factoring $S... | https://mathoverflow.net/users/15242 | reference for "curves over S are locally the base change of a curve over S' which is finite type over R" | If $S$ is quasicompact and quasiseparated you can even take $T=S$. By Thomason's approximation theorem (in The Grothendieck Festschrift, vol. III), $S$ is a projective limit of $R$-schemes $(S\_\lambda)$ of finite presentation, with affine transition maps. By general results from EGA IV.8, $X$ can be obtained by base c... | 7 | https://mathoverflow.net/users/7666 | 198332 | 95,956 |
https://mathoverflow.net/questions/165138 | 7 | I have trouble working out a proof in the second part of
>
> Jean-Pierre Ressayre and Alex Wilkie. **Modèles non standard en arithmétique et théorie des ensembles**. Publications Mathématiques de l'Université Paris VII, 1987.
>
>
>
On page 140, Ressayre writes:
>
> **4.6 Théorème** […] – b – En revanche, l... | https://mathoverflow.net/users/38196 | Foundation scheme for $\Sigma_{n+1}$-formulas | The point is that the $\omega$ in an $\omega$-model satisfies full induction, even if the whole model does not.
Take a nonstandard $\omega$-model $M\models\rm \Pi\_n\text{-}collection+V\,{=}\,L$. Let $I$ be that given by Corollary 4.5. Suppose $\varphi(v,x)$ is a $\Pi\_n$-formula for which
$$A=\{x\in\mathrm L\_I^M:\... | 1 | https://mathoverflow.net/users/38196 | 198334 | 95,957 |
https://mathoverflow.net/questions/198330 | 10 | Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
This question was first posted [here](https://math.stackexchange.com/questions/1160453/a-group-whose-automorphism-group-is-cyclic?noredirect=1#comment2370809_1160453).
| https://mathoverflow.net/users/66046 | A group whose automorphism group is cyclic | There's a construction of a rank two (and therefore not locally cyclic) abelian group with endomorphism ring $\mathbb{Z}$, and therefore automorphism group cyclic of order 2, in "On the cancellation of modules in direct sums over Dedekind domains" by L. Fuchs and F. Loonstra,
Indagationes Mathematicae, Volume 74, (1971... | 17 | https://mathoverflow.net/users/22989 | 198339 | 95,958 |
https://mathoverflow.net/questions/198336 | 3 | I am interested in infinite loop structures on the infinite dimensional projective space $\mathbb{R} P^\infty$. Is it unique? I think this has to be known in work of May, and If so, then I presume its proof should boil down to some simple fact. Can anyone please give me some advise on this, and the simple fact that one... | https://mathoverflow.net/users/51223 | Loop space structures on $RP^\infty$ | Essentially by definition, the category of infinite loop spaces is equivalent to the category of $(-1)$-connected spectra. Thus, you are asking about $(-1)$-connected spectra $X$ such that there exists an equivalence $\Omega^\infty X\simeq\mathbb{R}P^\infty$ of spaces (ignoring any infinite loop structures). From such ... | 10 | https://mathoverflow.net/users/10366 | 198340 | 95,959 |
https://mathoverflow.net/questions/198295 | 1 | Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In particular, is it true that if $Eff(X)$ has infinitely many extremal rays then $Eff(Y)$ has infinitely many extremal rays as well... | https://mathoverflow.net/users/nan | A question on the effective cone | Here is an example with $Pic(X)$ finitely generated. Take $X = \overline{M}\_{1,n}$ the moduli space of genus one curves with $n$ marked points. Then $Pic(\overline{M}\_{1,n})$ is finitely generated.
Now, by Theorem $1.1$ of this paper:
<https://www2.bc.edu/dawei-chen/Extremal.pdf>
we have that $Eff(\overline{M}\... | 0 | https://mathoverflow.net/users/14514 | 198355 | 95,963 |
https://mathoverflow.net/questions/198284 | 4 | Call a Borel set $A \subseteq [0,1]$ *good* if $$0 < \dim(A) \leq \overline{\dim\_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim\_\text{M}}(A)$ is the upper Minkowski dimension of $A$. (Note that the second inequality in the string of three inequalities holds automatic... | https://mathoverflow.net/users/19012 | Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension | A counterexample:
Let $E = \{0,1\}$ and let $\pi:E^{\mathbb N}\to [0,1]$ be the coding map for the Cantor set. Now fix $N\in\mathbb N$, let
$$
A = \{n\in\mathbb N : \lfloor \log\_N(n)\rfloor \text{ is even}\},
$$
let $S = \{\omega\in E^{\mathbb N} : \omega\_n = 0 \; \forall n\in A\}$, and let $X = \pi(S)$. A standard... | 4 | https://mathoverflow.net/users/54267 | 198358 | 95,964 |
https://mathoverflow.net/questions/198051 | 12 | For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y\_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.
I denote by $\Delta$ the category of [simplices](http://ncatlab.org/nlab/show/simplex+category), by $\Omega$ the categ... | https://mathoverflow.net/users/42137 | Tensor product of dendroidal sets: counter-examples | Associativity of the tensor product in fact does not hold even for representable presheaves, which addresses your Questions 2 and 3. Proposition 3.6.9 and its proof (in HHM, as you cite) give a typical counterexample, worked out explicitly there.
As to Question 1: consider the associative operad $\mathbf{A}$. Tensori... | 5 | https://mathoverflow.net/users/36302 | 198361 | 95,966 |
https://mathoverflow.net/questions/198364 | 3 | **Question:** Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}\_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and whose morphisms are chain maps. Is it true that every object in $\mathrm{Ch}\_{\mathbb{F}}$ is injective? If so, does... | https://mathoverflow.net/users/22763 | Are chain complexes over a field always injective? | No.
Let $X$ be the chain complex with $\mathbb{k}$ in degree $n$ (and all differentials zero) and let $Y$ be the chain complex with $\mathbb{k}$ in degrees $n$ and $n+1$ with the identity as differential $Y\_{n+1}\to Y\_n$ (I choose for my convention that the differential lowers degree).
Then $X$ is not injective b... | 11 | https://mathoverflow.net/users/3075 | 198370 | 95,969 |
https://mathoverflow.net/questions/198349 | 4 | I have been stuck for some time, thinking about the following question.
Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension $-dim(G)$.
Can something similar be said for the based loop group $\Omega G$ of pointed maps from $S^1$ to $G$ ?
I mea... | https://mathoverflow.net/users/68473 | Based loop groups as stacks? | $\Omega G$ won't be a differentiable stack unless you are willing to go to infinite dimensions. Provided you are considering $S^1$ as a smooth manifold, Nerses answer above is correct- it gives you a nice sheaf on the site of manifolds (it's even concrete- a so-called diffeological space). It's also representable by an... | 10 | https://mathoverflow.net/users/4528 | 198372 | 95,970 |
https://mathoverflow.net/questions/198177 | 4 | Given an ideal $I$ of $\mathbb{R}[X\_1,X\_2,X\_3,X\_4,X\_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p\_1 \in I$ and $p\_2 \in I$ such that
1. $p\_1$ is of degree 2 and up to a multiplicative constant the polynomial of smallest degree
2. $p\_2$ is of degree 3 and up to a linear combin... | https://mathoverflow.net/users/19874 | Generators vs minimal degree polynomials of ideals | As stated, I believe the answer is no. Set $q\_1=x\_1^2$, $q\_2=x\_2^4$, and consider $I=(q\_1,q\_2)$. Let $p\_1=q\_1$, and let $p\_2=x\_1p\_1$. Then $p\_1$ is (up to a constant) the only degree 2 polynomial in $I$, and $p\_2$ is (up to a linear combination of multiples of $p\_1$) the only polynomial of degree 3 in $I$... | 5 | https://mathoverflow.net/users/3199 | 198376 | 95,972 |
https://mathoverflow.net/questions/196764 | 3 | Let $G\subseteq GL(n)$ be a linear algebraic group, and let $G({\Bbb Q}\_p)\subseteq GL(V)$ act on a ${\Bbb Q}\_p$-vector space V of finite dimension.
Consider the action of $G$ on abelian subgroups $L\subseteq V$ such that $ L\otimes {\Bbb Q}\_p=V$.
What is known about the orbits of this action ? Where can I read ab... | https://mathoverflow.net/users/4745 | orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ | Probably the easiest case to understand is when $\mathbf{G}$ is an inner form that is adjoint. In this case, there will always be infinitely many orbits, unless $\mathbf{G}$ manages to be simply connected (even though it is adjoint), which means that every simple factor of $\mathbf{G}$ is of type $E\_8$, $F\_4$, or $G\... | 5 | https://mathoverflow.net/users/68305 | 198389 | 95,974 |
https://mathoverflow.net/questions/198386 | 3 | Let $f: \mathbb{R}\rightarrow \mathbb{C}$ be a measurable function, $H$ some Hilbert space and
$$ f\_E := \int\_{\mathbb{R}} f dE$$ for some spectral measure $E$.
Now, my question is: When do we have $f\_E\circ g\_E = (fg)\_E$ for two measurable functions $f$ and $g$?
My idea was that $g$ should be bounded.
Th... | https://mathoverflow.net/users/68375 | Composition of spectral measures | The best reference I know on these questions is Fell & Doran, [Representations of ${}^\*$-algebras, locally compact groups, and Banach ${}^\*$-algebraic bundles. Vol. 1](http://www.ams.org/mathscinet-getitem?mr=936628) (1988).
First they show your equality for $f$ and $g$ "$E$-measurable and $E$-essentially bounded" ... | 3 | https://mathoverflow.net/users/19276 | 198391 | 95,975 |
https://mathoverflow.net/questions/198383 | 5 | I know the general theorem of Kadeishvili which says that, for a DGA $C$, when $H^{i}(C)$, $i\geq 0$, is free, $H(C)$ can be made into an $A\_{\infty}$ algebra. If my understanding is correct, the proof essentially uses the freeness of $H^{i}(C)$ to produce a section
$$s:H(C)\to C$$
of the projection $p:C\to H(C)$. The... | https://mathoverflow.net/users/43687 | Massey products and $A_{\infty}$ structures | Let $C = \Bbb Z[x,y] \otimes \Lambda[u,v]$, with $x$ and $y$ in (homological) degree 2 and $u$ and $v$ in degree 3, with $dx = dy = 0$ and $du = 2x$, $dv = 2y$. Then $H\_5(C)$ is $\Bbb Z/2$, generated by the Massey product $x v - u y = \langle x,2,y\rangle$ (with no indeterminacy in this case).
This structure does no... | 7 | https://mathoverflow.net/users/360 | 198392 | 95,976 |
https://mathoverflow.net/questions/198371 | 2 | If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\sum\_{x\in C}x)\_C$ as a $k$-basis, where $C$ runs over conjugacy classes, then if each $x$ is conjugated to its inverse ... | https://mathoverflow.net/users/68479 | Involution on the components of a group algebra | In the book "Quadratic and hermitian forms" (by Winfried Scharlau), there is an entire section (in the Chapter 8) on the subject. The involutions induced by $g\mapsto g^{-1}$ are called the canonical involutions of $k[G]$.
For the case where the base field $k$ is a real closed field, for instance $k=\mathbb{R}$, ever... | 4 | https://mathoverflow.net/users/30062 | 198411 | 95,980 |
https://mathoverflow.net/questions/198404 | 6 | Terence Tao has shown [see his blog post](https://terrytao.wordpress.com/2009/08/30/an-elementary-inequality-involving-the-mobius-function/) that
$$\left| \sum\_{n\leq x} \frac{\mu(n)}{n} \right|\leq 1,$$
for $x$ a positive real number, where $\mu(n)$ is the Möbius function. Let $\lambda(n)$ denote Liouville's lamb... | https://mathoverflow.net/users/17773 | Bound on a scaled sum of the Liouville function | Yes, and this can be derived from your first inequality in an elementary way. Indeed, the formal Dirichlet series identity
$$ \sum\_{n=1}^\infty\frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)} $$
is equivalent to the convolution identity $\lambda=1\_{\square}\ast\mu$, i.e.
$$ \lambda(n)=\sum\_{d^2\mid n}\mu\left(\fr... | 12 | https://mathoverflow.net/users/11919 | 198413 | 95,981 |
https://mathoverflow.net/questions/198410 | 0 | Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}\_n$ by
1. $V(\text{HN}\_n) = \mathbb{R}^n$;
2. $E(\text{HN}\_n) = \{\{v\_1, v\_2\}: v\_1, v\_2 \in \mathbb{R}^n \text{ and } |v\_1-v\_2| = 1\}$.
Is it true that the clique number $\omega(\text{HN}\_n)$ equals $n+1$ for all $n\... | https://mathoverflow.net/users/8628 | Hadwiger-Nelson problem in higher dimensions | Yes. The moral reason is that cliques are 1-skeletons of regular simplices. If you want convincing quickly then it's straightforward to show inductively that there is exactly one clique in $\mathbb R^n$ of order $n+1$ up to the obvious symmetries, because when you go up a dimension you have exactly two choices for wher... | 3 | https://mathoverflow.net/users/25485 | 198414 | 95,982 |
https://mathoverflow.net/questions/198387 | 11 | The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
arbitrarily long sequences of elements of $S$ in arithmetic progression?
To make this more precise (following a comment by R... | https://mathoverflow.net/users/6094 | Most dense subset of numbers that avoids arbitrarily long arithmetic progressions | You are essentially asking for quantitative estimates on Szemerédi's theorem, which states that the largest subset of $[1,n]$ without a k-term arithmetic progression has size $o(n)$. To be precise, let us define $r\_k(n)$ to be the largest subset of [1,n] with no k-term arithmetic progression. Then a construction due t... | 16 | https://mathoverflow.net/users/66275 | 198421 | 95,984 |
https://mathoverflow.net/questions/193848 | 7 | I am interested in the distribution of the $\text{argmax}\_{t \in [0,1]} \{B(t) + f(t)\}$, where $B$ is a Brownian motion (or Brownian bridge) and $f:[0,1] \to \mathbb{R}$ is a continuous function. There are good results when the function $f$ is constant, linear or parabolic. However, I am particular interested in the ... | https://mathoverflow.net/users/41379 | Location of maximum of Brownian motion with rough drift | They are almost surely singular to each other.
First of all, here is a basic observation from measure theory that is useful in the proof. Consider two random measures, $M\_1$ and $M\_2$ on the same space - say, $[0,1]$, depending on some randomness $\omega \in \Omega$. With these measures one associates measures $\ma... | 2 | https://mathoverflow.net/users/22758 | 198431 | 95,986 |
https://mathoverflow.net/questions/198430 | 16 | The factorial function is primitive recursive, and therefore definable by a $\Sigma\_1$ formula.
Is it also definable by a $\Delta\_0$ formula (i.e. bounded quantifiers)?
If not, why?
| https://mathoverflow.net/users/68515 | Is factorial definable using a $\Delta_0$ formula? | (Sorry for the earlier confusion.)
Yes, the graph of factorial is $\Delta\_0$.
First, the problem can be restated in terms of computational complexity. By a characterization going back to Bennett, $\Delta\_0$-definable predicates are exactly those computable in the linear-time hierarchy. A convenient sufficient con... | 20 | https://mathoverflow.net/users/12705 | 198434 | 95,988 |
https://mathoverflow.net/questions/198249 | 4 | I have the following system of first order quasi-linear pde:
$$ -(\Delta+1) a^{\alpha\beta} [b\_{\beta\rho} I\_{\alpha;\sigma}+b\_{\beta\sigma} I\_{\alpha;\rho}]
+ a^{\alpha\beta} [(\Delta+1) b\_{\beta\rho;\sigma}+b\_{\beta\rho}\Delta\_{;\sigma}+b\_{\beta\sigma}\Delta\_{;\rho}]I\_\alpha
+a^{\alpha\beta} b\_{\rho\si... | https://mathoverflow.net/users/25516 | Existence and uniqueness of a quasi-linear pde system on a surface | This is not a full solution, but it indicates the main points. (A full solution would involve an EDS analysis that would be too long to include here; see below.)
First, setting $I\_\alpha = (\Delta + 1)a\_{\alpha\beta} J^\beta$ and noting that $\Delta+1$ is nonvanishing, the above equation becomes
$$
b\_{\beta\rho} ... | 4 | https://mathoverflow.net/users/13972 | 198435 | 95,989 |
https://mathoverflow.net/questions/198422 | 13 | Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ also the Hodge conjecture.
Are there analogues in the general case (when cohomology is no longer pure)?
Of course t... | https://mathoverflow.net/users/21815 | Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)? | (**Edited**) The short answer is yes. There are analogues of these conjectures, starting with work of Beilinson, Jannsen and perhaps others in the 1980's. Basically, they would say that cycle maps from motivic cohomology$^1$ $\otimes \mathbb{Q}$ (resp. $\mathbb{Q}\_\ell$) to $Hom\_{MHS}(\mathbb{Q}(-p), H^i(X))$ (resp. ... | 10 | https://mathoverflow.net/users/4144 | 198438 | 95,990 |
https://mathoverflow.net/questions/198380 | 2 | I am familiar with the recent Keevash paper [here](http://arxiv.org/pdf/1401.3665v1.pdf) which proves that given some $t,n,k,\lambda$ then provided standard divisibility conditions hold, and $n$ is suitably large, there exists a $t-(n,k,\lambda)$ design.
My question is in a similar vein, given some $n,k,t$ with $n\ge... | https://mathoverflow.net/users/68491 | Existence of Steiner system designs given $n,k,t$ | The best place to look for specific quadruples is the Handbook of Combinatorial Designs. If you want general statements, then one of Teirlinck's work (for large multiplicities), Kuperberg-Lovett-Peled (for large but usually more reasonable multiplicities) and Keevash's work are more or less all you can look for.
On t... | 1 | https://mathoverflow.net/users/59289 | 198439 | 95,991 |
https://mathoverflow.net/questions/198432 | 5 | Consider a domain $\Omega\_0 \subset \mathbb{R}^n$, and deformations of $\Omega\_0$, called $\Omega\_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is known that the Dirichlet and Neumann eigenvalues of the Laplacian on $\Omega\_t$ vary real analytically with respect to... | https://mathoverflow.net/users/68517 | Analytic perturbation of eigenfunctions | See this [recent paper](http://www.mat.univie.ac.at/~michor/DC-perturb.pdf) for answers to this and similar questions. You have to translate the parameter dependence of the domain to the parameter dependence of the operator first.
| 2 | https://mathoverflow.net/users/26935 | 198444 | 95,992 |
https://mathoverflow.net/questions/195668 | 2 | Assume that $B$ is a $C^{\*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any hereditary subalgebra of $A$.
Example: For every $C^{\*}$ algebra $A$, $c(A)$ is a totally non hereditary subalgebra of $\ell^{\in... | https://mathoverflow.net/users/36688 | Totally non hereditary $C^{*}$-subalgebras | So you want to consider a C\*-algebra $A$ with the following property: Every sub-C\*-algebra of $A$ is isomorphic to a hereditary sub-C\*-algebra of $A$.
We can distinguish two cases:
1. If $A$ is finite-dimensional, then $A$ has to be commutative. Indeed, assume $A\cong M\_{k\_1}\oplus\ldots\oplus M\_{k\_l}\oplus\... | 2 | https://mathoverflow.net/users/24916 | 198456 | 95,997 |
https://mathoverflow.net/questions/198453 | 8 | First of all, I need to declare my extreme ignorance on the topic of modular forms, so, please, does not assume that I know Deligne's construction in details.
In [Motives for modular forms](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/mf.pdf), Scholl constructs motives associated to modular forms . Deligne's also c... | https://mathoverflow.net/users/40883 | Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations? |
>
> More explicitly, I would like to know if from these motives $M\_{f}$ I can create an $\ell$-adic representation with values in some object of cohomological nature arising from $M\_{f}$ (like motivic cohomology) such that this representation is the one constructed by Deligne.
>
>
>
The answer is: of course, a... | 7 | https://mathoverflow.net/users/2284 | 198459 | 95,999 |
https://mathoverflow.net/questions/198403 | 10 | My question is a reference request for the following fact: if $k$ is a field and $X$ a proper smooth surface over $k$, then $X \rightarrow \mathrm{Spec}\, k$ is projective. Where is this well-known fact proved (in the stated generality)?
| https://mathoverflow.net/users/63877 | A proper smooth surface is projective | Quoting from a very nice paper by Stefan Schroeer we have: "The criterion of Zariski [3, Cor. 4, p. 328] tells us that a normal surface $Z$ is projective if and only if the set of points $z \in Z$ whose local ring $\mathcal{O}\_{Z,z}$ is not $\mathbf{Q}$-factorial allows an affine open neighborhood." The reference [3] ... | 7 | https://mathoverflow.net/users/68366 | 198487 | 96,007 |
https://mathoverflow.net/questions/198437 | 3 | For any graph $G=(V,E)$, the coloring number $\text{Col}(G)$ is defined to be the smallest cardinal $\kappa$ such that there is a well-ordering $\leq$ on $V$ such that for every vertex $v\in V$ we have $$|N(v) \cap \{w\in V: w \leq v\}| \leq \kappa,$$ where $N(v)=\{x\in V:\{x,v\}\in E(G)\}$.
It is known that $\chi(G)... | https://mathoverflow.net/users/8628 | Hadwiger's conjecture for coloring number instead of chromatic number | Take the complete 3-partite graph $K\_{n,n,n}$. Coloring number $2n+1$ (with the correct +1 definition). Any $K\_{2n+1}$ minor would have to have at least $n+1$ singleton branch sets, i.e. there would be a $K\_{n+1}$ subgraph (which is not there for $n\ge 3$).
| 4 | https://mathoverflow.net/users/12487 | 198490 | 96,010 |
https://mathoverflow.net/questions/198448 | 5 | In Walker and Wang's [article](http://arxiv.org/abs/1104.2632) about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory $\mathcal{S\_C}$ consisting of transparent objects. That quotient is promised to be mod... | https://mathoverflow.net/users/13767 | Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories? | Short answer: Look for papers on "deequivariantization". (I think the original references are by Müger and Brugières, but I am not sure whether they used the term "deequivariantization".)
Longer answer:
The procedure mentioned in the paper is actually a sort of predual to the deequivariantization procedure found i... | 6 | https://mathoverflow.net/users/284 | 198500 | 96,014 |
https://mathoverflow.net/questions/198501 | 1 | Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In <http://arxiv.org/abs/1502.04258>, the cohomology ring
$$
H^\*(F(\mathbb{R}P^n,k);R)$$
is obtained for any commutative ring $R$ with unit and $2$ invertible.
I want to find $$
H^\*(F(\mathbb{R}P^n,2);\mathbb{Z}).$$
When I use ... | https://mathoverflow.net/users/41075 | configuration spaces of real projective space | $F(\mathbb RP^n,2)$ is the space of ordered pairs of distinct points in real projective space. If you pass to the cover $S^n$ that means you're studying ordered pairs of distinct 1-dimensional subspaces of $\mathbb R^{n+1}$.
So there is a fibre bundle $F(\mathbb RP^n) \to Gr\_{n+1, 2}$ (the Grassmannian) and the fib... | 4 | https://mathoverflow.net/users/1465 | 198503 | 96,015 |
https://mathoverflow.net/questions/198479 | 2 | I've been trying to understand what happens to the cohomology ring (say with coefficients in $\mathbb{R}$) of a smooth complex projective manifold after blowing up along a smooth complex submanifold. I would first just like to double check my facts, and then I have some other general, perhaps naive, questions about blo... | https://mathoverflow.net/users/66536 | What happens to the cohomology ring after a "flip-flop"? | There are many questions here. Let me focus on **Question 4**.
The answer is *all smooth rational varieties.* This is a consequence of the following
>
> **Weak Factorization Theorem.** A birational map between complete nonsingular varieties over an algebraically closed field $\mathbb{K}$ of characteristic zero is... | 4 | https://mathoverflow.net/users/7460 | 198516 | 96,020 |
https://mathoverflow.net/questions/198515 | 1 | In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I know (of a non-cyclic group) with cyclic automorphism group lead me to $\mathbb{Z}\_2$.
Is it known if such examples d... | https://mathoverflow.net/users/66046 | An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$ | "Groups with a small number of automorphisms" by H. de Vries, A. B. de Miranda (Math Zeitschrift (1957/58) Volume 68, Issue 1, pp 450-464) [link](http://link.springer.com/content/pdf/10.1007/BF01160361.pdf)
gives examples with cyclic automorphism groups of order 4 and 6 (and it looks as though 2,4 and 6 may be the only... | 4 | https://mathoverflow.net/users/22989 | 198518 | 96,021 |
https://mathoverflow.net/questions/198440 | 2 | I know that [if $R$ is Noetherian local with a finite module of finite injective dimension, then *$R$ is Cohen-Macaulay*](https://math.stackexchange.com/questions/710644/does-operatornameid-m-dim-r-hold-for-finite-modules-of-finite-injective-d).
>
> Can one add assumptions on $M$, so that $R$ be *Gorenstein* or *C... | https://mathoverflow.net/users/47763 | if $R$ is Noetherian local with a finite module of finite injective dimension and if "?" , then $R$ is "Gorenstein" | If you admit $M$ cyclic as additional assumtion, then $R$ is Gorenstein by a theorem in Peskine-Szpiro paper "Dimension projective finie et cohomologie locale", Theorem II.5.5.
| 4 | https://mathoverflow.net/users/36672 | 198532 | 96,026 |
https://mathoverflow.net/questions/198524 | 6 | A mob is a word used for a topological semigroup which is a Hausdorff space. A clan is a compact connected mob with a two-sided identity element.
Who used these words with these meanings first and when? Why were these words chosen? I'm guessing it's a play on "group", but is it really?
| https://mathoverflow.net/users/20803 | Who coined "mob" and "clan" and why these words? | The mob/clan terminology goes back to [Alexander Wallace](http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Wallace_Alexander.html):
* [A note on
mobs](http://www.ams.org/mathscinet-getitem?mr=0053120) (1952).
* [The
structure of topological semigroups](http://www.ams.org/journals/bull/1955-61-02/S0002-9904... | 8 | https://mathoverflow.net/users/11260 | 198534 | 96,027 |
https://mathoverflow.net/questions/198531 | 3 | During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ should be interpreted as the [Moore-Penrose pseudoinverse](http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinve... | https://mathoverflow.net/users/68559 | Upper bounds on elements of a matrix | $C$ is an idempotent matrix, so its eigenvalues are either 1 or zero. $C$ is also Hermitian so Schur's Theorem says the sum of the $k$ largest eigenvalues is greater then the sum of the $k$ largest diagonal elements of $C$ for $k=1,2,...,m$. No diagonal element of $C$ can therefore exceed unity.
| 5 | https://mathoverflow.net/users/68572 | 198547 | 96,030 |
https://mathoverflow.net/questions/198400 | 6 | I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows:
Pick $k$ distinct numbers out of numbers $[1,n]$ uniformly at random, order them increasingly by size and denote them with $\alpha\_1, \alpha\_2, \dots , \alpha\_k$.... | https://mathoverflow.net/users/37757 | Expected value of a function over random sets | I assume you mean picking a random $k$-subset from {1,...,n}, i.e. drawing without
replacement? In this case one finds for $X\_j=\alpha\_j-j$ that
$$\mathbb{E}(t^{X\_j})= {k!\,(n-k)! \over n!} [x^{n-k}]{1 \over (1-tx)^j(1-x)^{k+1-j}}$$
and $$\mu(t):=\mathbb{E} \sum\_{j=1}^k t^{X\_j}={n+1 \over n+1-k}{1-t^{n+1-k} \o... | 1 | https://mathoverflow.net/users/48831 | 198551 | 96,032 |
https://mathoverflow.net/questions/198546 | 2 | I have been trying to build the function field of the jacobian of a genus 2 smooth curve over a finite field, but I am having problems making it explicit, I need to work with another curve with points in that field.
Let $H$ be a smooth curve of genus 2 over $\mathbb{F}\_q$ defined by the equation $y^2 = f(x)$ where $... | https://mathoverflow.net/users/91023 | Function field of the Jacobian of genus 2 curve over $\mathbb{F}_q$ | You get an open subset of the Jacobian by looking at points "in general
position", i.e., points represented by divisors of the form $(P)+(P')-2(\infty)$,
where $\infty$ denotes the point at infinity, such that $P, P' \neq \infty$
and $P$ and $P'$ are not images of each other under the hyperelliptic involution.
Such po... | 4 | https://mathoverflow.net/users/21146 | 198553 | 96,033 |
https://mathoverflow.net/questions/198525 | 12 | One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference is given to Milnor's paper, [*Differential Topology Forty-six Years Later*](http://www.ams.org/notices/201106/rtx1106008... | https://mathoverflow.net/users/10696 | Piecewise linear (PL) structures on $\mathbf R^4$ | There are three facts:
1. existence of uncountably many non-diffeomorphic exotic $\mathbf R^4$'s.
2. any smooth manifold has a PL structure.
3. Any PL manifold of dimension $<7$ has a smooth structure which is unique up to diffeomorphism.
For the latter two facts see 1.5 and 1.8 in [this survey](http://arxiv.org/p... | 13 | https://mathoverflow.net/users/1573 | 198558 | 96,036 |
https://mathoverflow.net/questions/198567 | 6 | If I imagine that (the self-adjoint part of) a C\*-algebra $A$ represents the algebra of observables of some quantum system, then [certain](http://arxiv.org/abs/1412.2177) [perspectives](http://dx.doi.org/10.1007/s00220-009-0865-6) on algebraic quantum theory would ask me to imagine that each (maximal) commutative C\*-... | https://mathoverflow.net/users/778 | Comparing cardinalities of the spectrum of two masas in $B(H)$ | **Yes**. The spectra of $\ell\_\infty$ and $L\_\infty$ have the same cardinality, namely $2^{\mathfrak{c}}$.
Indeed, every infinite, compact $F$-space space (in particular, an extremely disconnected compact space such as the spectrum of $L\_\infty$) contains a copy of $\beta \mathbb{N}$ (which happens to be the spect... | 5 | https://mathoverflow.net/users/15129 | 198572 | 96,039 |
https://mathoverflow.net/questions/198578 | 3 | Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L\_1, L\_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n\_1:= \langle c\_1(L\_1), [D]\rangle , ~~n\_2 := \langle c\_1(L\_2), [D]\rangle , ~~n\_3:= \langle c\_1(E), [D]\rangle . $$... | https://mathoverflow.net/users/4463 | Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle? | It looks like $n\_3-n\_1-n\_2$, but double check the computation. Tensor everything by $L\_1^{-1}$ to make $L\_1$ trivial and recompute the classes to get $0$, $n\_2-n\_1$, and $n\_3-2n\_1$. Then project a section of (trivial now) $L\_1$ to $E/L\_2$: you are interested in the zeroes of this projection, which are counte... | 5 | https://mathoverflow.net/users/44953 | 198581 | 96,043 |
https://mathoverflow.net/questions/198570 | 2 | Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other words (equivalently), let $\ C\ $ be a complement in $\ H\ $ of a $\sigma$-compact set (where a $\sigma$-compact set is... | https://mathoverflow.net/users/4423 | A question about open subsets of Hilbert space whose complements are compact sets | General fact: if we remove a countable collection $K\_1,K\_2,\ldots$ of compact sets from an infinite-dimensional Banach space $X$, the remaining set $V$ is locally and globally path-connected (actually any open ball is path-connected).
Proof. If $K$ is a compact subset of $X$, and $x\in X$ is a point, then the set u... | 10 | https://mathoverflow.net/users/4312 | 198585 | 96,044 |
https://mathoverflow.net/questions/198552 | 10 | Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike submanifolds?
Geroch showed that global hyperbolicity is equivalent to admitting a foliation by Cauchy hypersurfaces, ... | https://mathoverflow.net/users/17913 | Foliations of Lorentzian manifolds by Spacelike Hypersurfaces | In the case of a globally hyperbolic spacetime, what you want is a smooth *Cauchy temporal function* (the gradient is everywhere timelike, not just causal, and each level set is a Cauchy surface that is necessarily spacelike). That global hyperbolicity is also sufficient the the existence of a smooth temporal function ... | 8 | https://mathoverflow.net/users/2622 | 198588 | 96,045 |
https://mathoverflow.net/questions/198461 | 8 | Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of $\mathbb{C}$-points. Moreover, assume that for every point $x$ of $X$ the differential
$$df(x)\colon T\_xX \to T\_{f(y)}Y... | https://mathoverflow.net/users/48866 | Implicit Function Theorem on Singular Varieties | In Joe Harris' book, Algebraic Geometry, this is theorem 14.9.
| 3 | https://mathoverflow.net/users/9449 | 198592 | 96,047 |
https://mathoverflow.net/questions/198590 | 13 | Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C\* algebra structure, just consider $C(X)$ as a ring) and its (purely algebraic) spectrum $\text{Spec}(C(X))$. There is a injective continuous map $X \rightarrow \text{Spec... | https://mathoverflow.net/users/14233 | When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism? | The map $i:X\to\operatorname{Spec}(C(X))$ is a homeomorphism onto its image iff $X$ is completely regular; this is essentially the definition of complete regularity. However, it is very rarely surjective.
Indeed, suppose $X$ is completely regular and $i:X\to\operatorname{Spec}(C(X))$ is surjective and hence a homeomo... | 16 | https://mathoverflow.net/users/75 | 198595 | 96,049 |
https://mathoverflow.net/questions/198612 | 2 | Let $(P,\leq)$ be a partially ordered set. A *down-set* is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x'\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ordered down-set (tods).
Let $d\_1, d\_2$ be tods. We say that they are *incompatible* if neither $d\_1\subseteq d\_2$ ... | https://mathoverflow.net/users/nan | Complete sets of incompatible totally ordered down-set in a partially ordered set | The answer is No.
We define two sets of elements of $\{0,1,2\}^\omega$ in the following way:
1. For $n\in\omega$ let $u\_n$ be defined by $u\_n(k)=1$ for $k\leq n$
and $u\_n(k)=0$ for $k>n$;
2. For $n\in\omega$ let $t\_n$ be defined by $t\_n(k)=1$ for $k\leq n$
and $t\_n(n+1) = 2$ and $t\_n(k)=0$ for $k>n+1$;
N... | 2 | https://mathoverflow.net/users/8628 | 198614 | 96,054 |
https://mathoverflow.net/questions/198615 | 3 | Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?
Let $n=p\_1^{a\_1}\cdots p\_t^{a\_t}$ be the canonical prime factorization of $n$. Combining these bounds we get the well-known inequality, $\dfrac{\phi(n)\sigma(n)}{n^2}<1$, but fr... | https://mathoverflow.net/users/40984 | Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions? | In Tenenbaum's *Introduction à la théorie analytique et probabiliste des nombres*, chapter I.5, the following is proved.
A maximal order for $\sigma\_k(n)$ is
* $\exp((1+o(1))\log 2 \log n / \log \log n)$, if $k=0$.
* $$ n^k \exp \left ( (1+o(1)) \frac{(\log n)^{1-k}}{(1-k)\log \log n} \right ) $$ if $0<k<1$.
* $\... | 5 | https://mathoverflow.net/users/nan | 198623 | 96,057 |
https://mathoverflow.net/questions/182994 | 7 | Suppose $f:\mathbb{R} \to \mathbb{R}$ has the property that for every fixed $t\in\mathbb{R}$ the function
$$
g\_t : x \mapsto f(x) - f(x-t)
$$
is $C^\infty(\mathbb{R})$. Does it follow that $f$ is smooth?
Edit: The answer is no in generality (see answer below), but what if we impose the condition that $f$ is measur... | https://mathoverflow.net/users/14566 | Non-smooth function with all differences of translates smooth? | Here's a solution for $f\in L^1\_{\mathrm{loc}}$, or even for $f$ a distribution, which avoids Fourier analysis by mollifying.
Let $\phi \in C^\infty\_c(\mathbf{R})$ be a smooth bump function of total mass $1$. Define $\tilde{f} := \phi \* f$. Then $f\in C^\infty(\mathbb{R})$. Moreover,
$$
(f-\tilde{f})(x) = \int (... | 0 | https://mathoverflow.net/users/14566 | 198629 | 96,059 |
https://mathoverflow.net/questions/198602 | 3 | Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. Must there be $h \in H$, and $a \in G$ such that $\langle h,a \rangle = G$ ?
**NOTE:** $H\langle g \rangle = \{hg^n | h \... | https://mathoverflow.net/users/38889 | If d("G/H") < d(G) = 2, must H contain a primitive element? | I think I can modify Peter Mueller's example to make it a counterexample to the question that was asked. We still let $H$ be elementary abelian of order $4$, but now we take $V = C\_{105} \cong C\_3 \times C\_5 \times C\_7$. Again we let the three involutions of $H$ act as diagonal matrices on the three prime order dir... | 2 | https://mathoverflow.net/users/35840 | 198638 | 96,060 |
https://mathoverflow.net/questions/198628 | 4 | My curiosity was raised by the [following question](https://mathoverflow.net/questions/104400/when-is-an-hnn-extension-finitely-presented)
and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting.
Let $G$ be a finitely presented group which admits ... | https://mathoverflow.net/users/35269 | Finiteness properties for graph of groups decompositions | Does it follow that each vertex stabiliser is finitely presented: No.
Just notice that there exist infinitely presented groups $H$ with an automorphism such that the corresponding semidirect product $G=H\rtimes\mathbf{Z}$ is finitely presented. This is a non-trivial graph of groups decomposition, namely an HNN-decomp... | 4 | https://mathoverflow.net/users/14094 | 198641 | 96,061 |
https://mathoverflow.net/questions/198645 | 7 | Suppose I have a polynomial $p(x) = a\_n x^n + ... + a\_0$ where $a\_n, \dots, a\_0$ are integers. I would like to show that any root of this polynomial is either an integer or is far from an integer. That is, if $r$ denotes the fractional part of a root $x$, I want to show that either $r = 0$ or $r > w$ for some real ... | https://mathoverflow.net/users/9896 | How close to an integer can a polynomial root be? | A bound follows from the general root separation theory. See [Schonhage's 2006 paper](http://ac.els-cdn.com/S0747717106000472/1-s2.0-S0747717106000472-main.pdf?_tid=3e55d828-be9e-11e4-8390-00000aab0f26&acdnat=1425054938_40447d42a9f0f57fb2aec11709601c4c) (Journal of Symbolic Computation)> (see inequality (3) and the dis... | 1 | https://mathoverflow.net/users/11142 | 198646 | 96,063 |
https://mathoverflow.net/questions/198648 | -2 | I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). There is a natural extension to a map $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$, considering $\mathbb{S}^2=\mathbb... | https://mathoverflow.net/users/62367 | Degree of a rational function | The fundamental theorem of algebra tells us that the number of solutions to $f(z) = a$ is the maximum of the degrees of the numerator and denominator.
| 2 | https://mathoverflow.net/users/11142 | 198652 | 96,065 |
https://mathoverflow.net/questions/198643 | 5 | I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which MAXIMIZES the Frobenius norm of the commutator, i.e.
$$
R = \arg\max\_{R' \in O(N)}||[R',S]||\_F^2 = \arg\max\_{R' \in O(N... | https://mathoverflow.net/users/4047 | Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem) | Unless I'm mistaken, the following argument provides a solution.
Since the Frobenius norm is orthogonally invariant we can assume without loss of generality that $S$ is diagonal. I'll write $Q$ instead of $R'$ to avoid confusion with matrix transposition.
\begin{equation\*}
\|QS-SQ\|\_F^2 = \|QS\|^2 + \|SQ\|^2 - 2... | 7 | https://mathoverflow.net/users/8430 | 198659 | 96,067 |
https://mathoverflow.net/questions/196625 | 11 | Let $h:\pi\_{2n-1}(S^n) \rightarrow \mathbb{Z}$ be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if $x \in \pi\_{2n-1}(S^n)$ satisfies $h(x)=0$, then $x$ is in the image of the suspension map $\pi\_{2n-2}(S^{n-1}) \rightarrow \pi\_{2n-1}(S^n)$. Observe ... | https://mathoverflow.net/users/67144 | Maps with Hopf invariant zero are suspensions | In my notes "Homotopy groups of spheres and low-dimensional topology" (available on my page of notes [here](http://www.nd.edu/~andyp/notes/)), I have written up a modern account of Pontryagin's approach to calculating the homotopy groups of spheres. In particular, Section 9 contains a detailed account of Pontryagin's p... | 7 | https://mathoverflow.net/users/317 | 198662 | 96,068 |
https://mathoverflow.net/questions/198502 | 7 | Given two graphs $G=(V\_1,E\_1)$ and $H=(V\_2,E\_2)$, the tensor product of $G$ and $H$ is the graph $G \times H = (V,E)$, where $V=V\_1 \times V\_2$ is the Cartesian product of the $V\_i$ and
$ (u,v) \ E \ (u',v') \Leftrightarrow u E\_1 u' \wedge v E\_2 v'$.
Is anyone aware of a characterization of which $G,H$ give... | https://mathoverflow.net/users/68544 | When is the tensor product of two graphs planar? | Yuichiro Fujiwara's comments seem to answer the question in full so I am making it an answer. I quote below from [Kronecker products and local joins of graphs](http://cms.math.ca/cjm/v29/cjm1977v29.0255-0269.pdf) by M. Farzan and D. A. Waller, *Can. J. Math.* **29** (1977), 255–269.
>
> By a *1-contraction* of $G$ ... | 6 | https://mathoverflow.net/users/3106 | 198668 | 96,071 |
https://mathoverflow.net/questions/198300 | 6 | Let $T$ be an stable theory.
Further we work in the monster model of $T^{eq}$.
We say that a chain of types of the form
$$tp(a\_1/A\_1)\subset tp(a\_2/A\_2) ... \subset tp(a\_n/A\_n)$$
is a *forking chain* if for every $1< i \le n$
the type $tp(a\_i/A\_i)$ forks over $A\_{i-1}$.
What can we say about the length of ... | https://mathoverflow.net/users/47687 | Chains of forking extension in stable theories | The answer is no to all three questions.
First notice that if there exists a maximal finite forking chain,
then its length is the SU-rank
(or U-rank in the stable context) of $tp(a\_1/A\_1)$.
Since there are theories of Morley-rank $>1$ and U-rank $=1$,
this gives negative answer to 1 and 2.
Then note that i... | 3 | https://mathoverflow.net/users/47687 | 198671 | 96,073 |
https://mathoverflow.net/questions/198625 | 3 | Let $G$ be a finite group. When does there exist a finite group $H$ such that every $h\in H$ is in the kernel of some epimorphism $H\to G$?
This is well-known to be true for $G$ abelian, for example $H=G\times G$ works. I would like very much to know such an $H$ for non-abelian $G$, e.g. for symmetric groups $G$.
N... | https://mathoverflow.net/users/10481 | Covering finite groups by kernels | If $G$ is nonabelian simple, then $H$ cannot exist.
Lemma : Let $\cal X$ be a collection of normal subgroups of a group $H$ such that $\bigcap \cal X = 1$, and assume that $H/X$ is nonabelian simple for all $X \in \cal X$. Then $H$ is isomorphic to a direct product of nonabelian simple groups.
Proof: Let $M \triang... | 5 | https://mathoverflow.net/users/9694 | 198672 | 96,074 |
https://mathoverflow.net/questions/198605 | 7 | Let $k > 1$ and $n$ be positive integers. Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Let $D$ be a digraph which has exactly $k$ vertices $v\_0$, $v\_1$, ..., $v\_{k-1}$ and exactly $k$ arcs $v\_0 \to v\_1$, $v\_1 \to v\_2$, ..., $v\_{k-2} \to v\_{k-1}$, $v\_{k-1} \to v\_0$. (That is, $D$ is a directed cycle on $k$... | https://mathoverflow.net/users/2530 | Flooding a cycle digraph via chip-firing: $n^{k-1} + n^{k-2} + \cdots + 1$ bound (a Norway 1998-99 problem generalized) | It now looks to me that conjecture 2 is only easier.
We induct on $|J|$. Base $|J|=1$, say, $J=\{k\}$. If $f(k)>0$ there is nothing to prove, so assume that $f(k)=0$. We have $n^{k-1}$ coins in vertices $1,\dots,k-1$ (there may be more coins, but we use only $n^{k-1}$) and perform operations until $f(k)$ becomes posi... | 4 | https://mathoverflow.net/users/4312 | 198679 | 96,078 |
https://mathoverflow.net/questions/198667 | 5 | By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on Sato-Tate theorem calculated the symmetry type of many families of automorphic L-functions of varying weight or level, with th... | https://mathoverflow.net/users/31814 | Symmetry type of non-cohomological automorphic forms | You should read the following preprint of Sarnak, Shin, and Templier:
<http://arxiv.org/abs/1401.5507>
In particular, they study "nice" families $\mathfrak{F}$ of automorphic representations. They assume RH and that there exists $A < \infty$, $\delta < 1$, such that uniformly in $n \geq 1$,
\[\sum\_{\pi \in \mathfr... | 2 | https://mathoverflow.net/users/3803 | 198683 | 96,080 |
https://mathoverflow.net/questions/198596 | 7 | Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, $\operatorname{Ext}$, local cohomology, etc) group vanishes/does not vanish".
Note that this is a local question, sinc... | https://mathoverflow.net/users/45505 | Homological criterion for $A(B \cap C) = AB \cap AC$? | Ah, here it is: we have $A(B\cap C) = AB \cap AC$ if and only if the natural map
$$\operatorname{Tor}\_1^R(R/A,B) \oplus \operatorname{Tor}\_1^R(R/A,C) \to \operatorname{Tor}\_1^R(R/A,B+C)$$
is surjective.
This is not very pretty, but it's still entirely homological (for example, it can be computed in the comple... | 5 | https://mathoverflow.net/users/45505 | 198686 | 96,081 |
https://mathoverflow.net/questions/198235 | 18 | Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi\_k(x)=\#\{n\leq x: \omega(n)=k;\mu(n)\neq0 \}$$
and consider the generating functions
\begin{eqnarray}f(z,x)&=&\sum\_{k=0}^{m(x)}\pi\_k(x) z^k\\
&=&\sum\_{n\leq x}|\mu(n)|z^{\omega(n)}.
\end{eqnarray}
... | https://mathoverflow.net/users/10980 | Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts? | In any bounded region, for large $x$, the polynomial can only take on zeros very near the negative real axis (and indeed near the non-positive integers). This follows from the work of Selberg (Note on a paper by L.G. Sathe, see Theorem 2 there) which shows that, for bounded $z$ and large $x$,
$$
f(z,x) = x C(z) \frac... | 8 | https://mathoverflow.net/users/38624 | 198688 | 96,082 |
https://mathoverflow.net/questions/198685 | 8 | Does there exist a meager set of reals M such that every meager set can be covered by countably many translates of M? This is the category analogue of the [following](https://mathoverflow.net/questions/198122/translates-of-null-sets).
| https://mathoverflow.net/users/2689 | Translates of meager sets | No, there is no such set.
The situation for meager sets is dual to that described by Pietro Majer in a comment on [Translates of null sets](https://mathoverflow.net/questions/198122/translates-of-null-sets),
>
> "I was vaguely thinking to Hausdorff measures w.r.to gauge functions. One needs to know that, given $N$,... | 6 | https://mathoverflow.net/users/4600 | 198689 | 96,083 |
https://mathoverflow.net/questions/194818 | 6 | Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M\_A$ be a maximal ideal of $A$, and $M\_B$ be a maximal ideal of $B$ that lies above $M\_A$ (that is, $M\_B\cap A = M\_A$). Denote by $F\_A$ the field $A/M\_A... | https://mathoverflow.net/users/62826 | Divisibility of the degree of an extension by the degree of its residual field | Let $k$ be the field with $2$ elements. Let $R = k[t\_i, x\_i; i \in \mathbf{N}]$. Let $\sigma : R \to R$ be the order $2$ automorphism sending $t\_i$ to $t\_i$ and $x\_i$ to $x\_i + t\_i$. Let $S \subset R$ be the fixed elements under the action. Then $R$ and $S$ are normal domains and $R$ is integral over $S$. In par... | 5 | https://mathoverflow.net/users/68628 | 198690 | 96,084 |
https://mathoverflow.net/questions/198452 | 9 | Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL\_n}$ for $n>2$ and $F =\mathbb{Q}$.
Then for an automorphic form,
$\varphi : \mathrm{G}(F)\backslash\mathrm{G}(\mathbb{A}) \to \mathbb{C}$ is said to be cuspidal if
for each parabolic subgroup... | https://mathoverflow.net/users/62154 | Geometric interpretation of Cusps for general groups? | It's actually a lot easier to write a short comment than to say something more precise. In any case, I can only say something in the classical language of locally symmetric spaces.
The shortest answer is that the geometric understanding of the cusps is revealed in the various compactifications of locally symmetric s... | 2 | https://mathoverflow.net/users/50846 | 198703 | 96,091 |
https://mathoverflow.net/questions/198466 | 16 | Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^\* \to B(X)^\*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^\*, a\in B(X))
$$
Put differently, we consider the pairing between $X\hat{\otimes} X^\*$ (the projective tensor product of ... | https://mathoverflow.net/users/24916 | Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric? | The answer is no.
Let $X$ be a separable Pisier counterexample [P] to Grothendieck’s problem. That is, both $X$ and $X^\*$ have cotype 2 and every operator from $X$ or $X^\*$ into a Hilbert space is 2-absolutely summing. As Pisier points out, there is a constant $C$ so that if $T$ is a finite rank operator from $X$ ... | 14 | https://mathoverflow.net/users/2554 | 198710 | 96,094 |
https://mathoverflow.net/questions/198708 | 0 | A set $\sum$ of formulas in propositional logic is complete if for each propositional formula $\phi$ either $\sum \vdash \phi$ or $\sum \vdash \neg \phi$. Clearly every inconsistent set of formulas is complete because of the following lemma
>
> Lemma: Let $\sum$ be an inconsistent set, then for every propositional ... | https://mathoverflow.net/users/67097 | Completeness of a set of propositional formulas | Given a finite propositional theory, one can decide completeness by checking the truth table. As Emil mentions, in general completeness for a finite theory will be NP-complete.
But your examples are infinite. In this case, one needs to take more care with the precise formulation of the question. For the decidability ... | 3 | https://mathoverflow.net/users/1946 | 198711 | 96,095 |
https://mathoverflow.net/questions/198634 | 2 | Let $C$ be a pointed convex cone in a vector space $V$. This means that $C$ satisfies the three following axioms:
* $C + C \subset C$,
* $\mathbb{R}\_+ \cdot C \subset C$, and
* $C \cap (-C) = \{ 0 \}$.
Say that $K \subset C$ is a *base* of $C$ if, for every $x \in C \setminus \{ 0 \}$, there is a unique $\lambda ... | https://mathoverflow.net/users/58226 | Base of a cone in a vector space: can one always choose a convex base? | This is an extended version of my observations in the comments. The upshot is that there exist pointed convex cones without a convex base, but every cone has a base. Hence what the OP is trying to do is bound not to work.
**(1)** *There are pointed convex cones that do not have a convex base*. To see this,
take $V=\... | 9 | https://mathoverflow.net/users/27013 | 198712 | 96,096 |
https://mathoverflow.net/questions/198713 | -2 | Any combinatorical meaning or interpretation of
$$1^{\alpha\_1}2^{\alpha\_2}3^{\alpha\_3}...s^{\alpha\_s}\alpha\_1!\alpha\_2!...\alpha\_s!$$
for partition $(1^{\alpha\_1},2^{\alpha\_2},3^{\alpha\_3},...,s^{\alpha\_s})\vdash{n}$.
In addition, this expression is diviser of $n!$.
| https://mathoverflow.net/users/41522 | Combinatorical meaning of such expression | Given a permutation $(x\_1,\dots,x\_n)$ of numbers from 1 to $n$, we get a new permutation: $x\_1$, $\dots$, $x\_{\alpha\_1}$ are fixed points, $(x\_{\alpha\_1+1},x\_{\alpha\_1+2})$ form a 2-cycle, and so on. Thus we get a permutation with $\alpha\_i$ cycles of length $i$ and each of them is calculated as many times as... | 2 | https://mathoverflow.net/users/4312 | 198715 | 96,098 |
https://mathoverflow.net/questions/198705 | 7 | Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by
$$\times\_d(x) = dx \mod 1$$
where the degree $d$ is any integer greater than $1$.
There are a wealth of ergodic invariant measures; I know at least of uniform measures on periodic orbits, the Lebesgue measure, Gibbs measur... | https://mathoverflow.net/users/4961 | Classification of ergodic measures for circle expanding maps | You're right. The set of measures for these maps is a zoo! There is the obvious map (base $d$ expansion) $\pi$ from $\{0,1\ldots,d-1\}^{\mathbb N}$ to $[0,1)$ which is a bijection off a countable set. $\pi$ is then a factor map from the full one-sided $d$-shift to $\times\_d$.
There are only two ergodic invariant meas... | 3 | https://mathoverflow.net/users/11054 | 198720 | 96,100 |
https://mathoverflow.net/questions/188695 | 4 | I'd ideally like a categorical definition of differentiability that can then be trivially translated into locales. Barring this, I'm still interested in whether the notion make sense for locales.
| https://mathoverflow.net/users/62519 | Is there a straightforward way to define a differentiable structure on a localic manifold? | This is exactly what I've been working on. I've tried a few approaches.
Grothendieck gives a definition of formal smoothness which can be translated into just about any category. The problem is that, for the definition to be interesting, it relies on the fact that commutative rings sometimes have non-trivial nilpote... | 1 | https://mathoverflow.net/users/2884 | 198725 | 96,103 |
https://mathoverflow.net/questions/198727 | 2 | Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}\_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$.
Let $C$ be the Casimir of $\mathfrak{g}$. It is in the center of the universal enveloping algebra $U(\mathfrak{g})... | https://mathoverflow.net/users/16326 | Unitary representation with fixed Casimir | No, for higher-rank groups, such as $SL\_n(\mathbb R)$ with $n\ge 3$, it is straightforward to compute that the eigenvalue of Casimir on a unitary principal series is a quadratic polynomial in the parameters for the character, so there is a continuum of unitary principal series (generically irreducible) with the same e... | 5 | https://mathoverflow.net/users/15629 | 198733 | 96,104 |
https://mathoverflow.net/questions/198732 | 11 | I asked this question about two weeks ago on [MSE](https://math.stackexchange.com/questions/1149637/non-forcing-and-independence) and haven't gotten an answer, so I thought I would post the question here.
Do there exists sentences which are independent of **ZFC**, cannot be shown to be independent through some method... | https://mathoverflow.net/users/51323 | Non-Forcing and Independence | The [Gödel-Rosser sentence](http://en.wikipedia.org/wiki/Rosser%27s_trick) $R$ for $\text{ZFC}$ is an arithmetic assertion, such that $\text{ZFC}$ is equiconsistent with $\text{ZFC}+R$ and with $\text{ZFC}+\neg R$. So the Rosser sentence does not increase consistency strength. Since arithmetic assertions are preserved ... | 13 | https://mathoverflow.net/users/1946 | 198734 | 96,105 |
https://mathoverflow.net/questions/198704 | 1 | **Edit:** According to the comment of Pietro Majer, I revise the question
Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?:
"Constant maps and the identity are the only maps with fixed point"
| https://mathoverflow.net/users/36688 | Totally non fixed point property | The paper <http://matwbn.icm.edu.pl/ksiazki/fm/fm60/fm60123.pdf> contains an example of a compact continuum with the property the only continuous mappings from $X$ to $X$ are the identity mappings and the constant mappings. See also the answer [Strongly rigid Hausdorff spaces](https://mathoverflow.net/q/188729/22277).
... | 7 | https://mathoverflow.net/users/22277 | 198736 | 96,106 |
https://mathoverflow.net/questions/198726 | 0 |
>
> **Definition(Integral closure)**: Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisfies a monic equation
> $x^n + i\_1x^{n−1} + ··· + i\_n = 0$ such that $i\_j ∈ I^j$ .
>
>
>
Let $ R $ be a ring and $ I $ ideals of $ R $ and $ I $ be a finitely generated.
... | https://mathoverflow.net/users/68302 | Properties of Integral Closure | 1) is obvious : a radical ideal is integrally closed, so $I\subset \mathrm{rad}(I)$ gives $\bar{I}\subset \mathrm{rad}(I)$, hence $\mathrm{rad}(\bar{I})=\mathrm{rad}(I)$.
2) is more subtle. This is Corollary 5.2.3 in "Integral Closure of Ideals, Rings, and Modules" by Swanson and Huneke, London Mathematical Society ... | 1 | https://mathoverflow.net/users/40297 | 198762 | 96,112 |
https://mathoverflow.net/questions/163810 | 2 | For a formalisation of the Giry monad in a theorem prover, I think I require some notion of measurability of “curried” functions. I.e. I have measure spaces $A$, $B$, and $C$ and a function $f: A \rightarrow (B \rightarrow C)$ and want it to be “measurable”, but for that I would, of course, require some way of construc... | https://mathoverflow.net/users/32355 | Measurability of functions with multiple parameters | My advisor, Johannes Hölzl, apparently found a solution to this (and proved it formally, in Isabelle/HOL): there is no solution, since measurable spaces are not a cartesian-closed category, as, e.g.:
Let $\mathcal M\_1 = \mathfrak P(\mathbb R)$, $\mathcal M\_2 = \{\{x\}\,|\,x\in\mathbb R\}$, and $\mathcal N = \mathfr... | 0 | https://mathoverflow.net/users/32355 | 198771 | 96,113 |
https://mathoverflow.net/questions/198766 | 3 | Let $G\_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi\_i$ be a monomorphism from $G\_i$ to $G\_{i+1}$.
Let $\Sigma$ be the direcet limits of $G\_i$ under the embeddings of $\phi\_i$.
Let $\varphi\_i$ be another monomorphism from $G\_i$ to $G\_{i+1}$ s.t.
$$\phi\_i(G\_i)=\varphi\_i(G\_i)$$
i.e the... | https://mathoverflow.net/users/47344 | About direct limit of groups | Here is an example where they are not isomorphic (where the $G\_i$ are countable abelian groups).
Write $C\_k$ for the cyclic group of order $k$ and $C\_k^{(I)}$ the group of finitely supported functions $I\to C\_k$.
Let $I,J$ be two disjoint infinite countable sets, and define $G=C\_2^{(I)}\oplus C\_4^{(J)}$, her... | 5 | https://mathoverflow.net/users/14094 | 198772 | 96,114 |
https://mathoverflow.net/questions/198545 | 6 | I asked this question a day ago on [math.stackoverflow](https://math.stackexchange.com/q/1165234/172301) but figured it could have an interest here.
I'm interested in the set $\mathcal{P}\_N$ of boolean functions of boolean variables $p\_1, p\_2, \ldots, p\_N$ that can be written as products of operators of 2 variabl... | https://mathoverflow.net/users/68554 | Product of binary Boolean operators | $\let\ET\bigwedge$I’ll summarize basic facts about the first question already mentioned in the comments, and add some bounds.
First, if we can write $\phi$ as $\ET\_{i<j}\phi\_{i,j}(x\_i,x\_j)$, can can expand each $\phi\_{i,j}$ as a conjunction of 2-clauses (i.e., disjunctions of two literals $A,B$, which are variab... | 3 | https://mathoverflow.net/users/12705 | 198777 | 96,116 |
https://mathoverflow.net/questions/117517 | 32 | Universes seem to first enter Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is credited to Grothendieck and Verdier. The appendix on them there is credited to N Bourbaki.
Is there any known evidence of w... | https://mathoverflow.net/users/38783 | Authorship of Grothendieck universes | Pierre Cartier has told me everyone at the time (i.e. everyone in those circles) knew Pierre Samuel wrote the appendix.
Incidentally this makes a third person breaking the general rule that all writings signed N Bourbaki were collective. Weil and Dieudonné wrote historical/philosophic pieces signed Bourbaki, and Samu... | 24 | https://mathoverflow.net/users/38783 | 198778 | 96,117 |
https://mathoverflow.net/questions/198740 | 5 | Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is trivial. Is the ring of invariants $R^G$ necessarily Noetherian?
| https://mathoverflow.net/users/63877 | Is the ring of invariants Noetherian? | Yes, and regularity isn't needed (assuming noetherian). By the Eakin-Nagata Theorem (3.7, Matsumura CRT), it is enough that $R$ is $R^G$-finite. For the Cohen ring $W$ of the perfect residue field, the unique local map $W\rightarrow R$ lifting the identity on residue fields is $G$-invariant. Pick a surjection $W[\![x\_... | 8 | https://mathoverflow.net/users/61939 | 198780 | 96,118 |
https://mathoverflow.net/questions/198784 | 6 | Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where:
1) $s\in 2^{<\omega}$,
2) $N\in \mathbb{N}$,
3) (by identifying $s$ with a subset of $lh(s)$) $s$ contains no arithmetic progressions of length $3$, and $\Sigma\_{n\in s}1/n \geq N$.
The ordering is defined in the natu... | https://mathoverflow.net/users/11115 | Adding sets not containing arithmetic progressions of length three by forcing | A "yes" answer to your question is equivalent to the statement "there exists a large set of natural numbers that admits no arithmetic progression of length three." I'm submitting the proof of this equivalence as an answer since I don't expect to see an actual answer unless it shows up in Annals too :)
So, to the proo... | 10 | https://mathoverflow.net/users/11233 | 198789 | 96,123 |
https://mathoverflow.net/questions/198793 | 2 | Let $K$ be the set of all total recursive functions of non-negative integers having only non-negative integers as values. Let $L$ be any well-ordered subset of $K$ in which the ordering $<$ is defined as follows. If $f(n),g(n)$ are elements of $L$, then $f(n)< g(n)$ just in case there exists a non-negative integer $h$ ... | https://mathoverflow.net/users/4423 | Questions about a possible way of representing construcive ordinal numbers | The answer to the first question is No and the second question Yes, because in fact every countable (successor) ordinal arises that way. (The successor part is only because you insisted that $L$ has a maximal element; otherwise we could say that every countable ordinal arises this way.)
The reason is that $K$ contain... | 5 | https://mathoverflow.net/users/1946 | 198794 | 96,126 |
https://mathoverflow.net/questions/198785 | 3 | Which asymptotic bounds (upper and lower) are known for $s\_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
| https://mathoverflow.net/users/38889 | What is the growth of the rank of a power of a finite simple group? | One has
$$1 \leq s\_n - \frac{\log(n)}{\log|S|} \leq 2r$$
based on an elementary argument in Remark 1.1 in [Moshe Jarden and Alexander Lubotzky, *Random normal subgroups of free pro-finite groups*, J. Group Theory 2 (1999) 213-224], where $r$ denotes the minimal number of generators of $S$. By the classification of fin... | 8 | https://mathoverflow.net/users/8176 | 198801 | 96,129 |
https://mathoverflow.net/questions/198820 | 1 | This is a follow-up question to [Complete sets of incompatible totally ordered down-set in a partially ordered set](https://mathoverflow.net/questions/198612/complete-sets-of-incompatible-totally-ordered-down-set-in-a-partially-ordered-se).
Let $(P,\leq)$ be a partially ordered set such that for every $p\in P$ the se... | https://mathoverflow.net/users/8628 | Completion of a single totally ordered down-set | I think that in this case there will always be a complete bunch $B$.
* For every maximal chain $m$ with $m \cap t = \varnothing$, take the minimal element of $m$, and add it as tods to $B$.
* If $m \cap t \ne \varnothing$, and $t \not\subset m$, then take the minimal element $x$ of $m \setminus t$, and add $\{x\} \cu... | 3 | https://mathoverflow.net/users/21815 | 198822 | 96,132 |
https://mathoverflow.net/questions/198821 | 0 | Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$
We set $E = \{\{f,g\}: f,g \in V\text{ and } d(f,g) = 1\}$. Setting $G:=(V,E)$ it is easy to see that $G$ has a countable clique, but do we also have $\chi... | https://mathoverflow.net/users/8628 | Graph of bounded continous functions with distance 1 | It is a separable space and so may be covered by countably many sets of diameter less then 1.
| 2 | https://mathoverflow.net/users/4312 | 198823 | 96,133 |
https://mathoverflow.net/questions/198760 | 0 | In my research the following equation appeared:
$$\frac{1}{4\pi}\int\_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int\_0^{\infty} f(a) Q^{i\sqrt{a}}\_{s-1}(2\rho-1) da,$$
where $\rho,s>1$, $Q^{\mu}\_{\nu}$ is the associeted Legendre function of second kind and $f(a)$ is to be found.
All my attempts to solve t... | https://mathoverflow.net/users/68283 | Integral Transform with associated Legendre Function of second kind as kernel | [Disclaimer: the below text is not mine, but by Vladimir Petrov, who does not have MO account]
Consider generalized Mehler-Fock transform:
$$
\begin{cases}
F(\xi,\,\mu)&=\intop\_1^\infty f(y)P^{-\mu}\_{-1/2+i\xi}(y)\,dy,\ \ \ 0\le\Re\mu<1;\\
f(x)&={1\over\pi}\intop\_0^\infty \xi\sinh\pi\xi\,\Gamma(\mu+1/2+i\xi)\Gamma... | 1 | https://mathoverflow.net/users/4312 | 198827 | 96,135 |
https://mathoverflow.net/questions/195950 | 3 | Assume that $A$ is a unital $C^{\*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such codimensions. For example $n(A)=1$ if $A$ is commutative. Or $n(M\_{n}(\mathbb{C}))=n$.
For a commutative $A$, is it true to sa... | https://mathoverflow.net/users/36688 | Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements | Let $A$ be a unital C\*-algebra.
As was already noted in the comments, we have $n(A)=1$ if and only if $A$ has a character.
Let $M\_n=M\_n(\mathbb{C})$.
It is easy to see that $n(M\_n)\leq n$. Conversely, a linear subspace of $M\_n$ of codimension less than $n$ contains an invertible matrix, as shown by Dieudonné, [1... | 6 | https://mathoverflow.net/users/24916 | 198831 | 96,136 |
https://mathoverflow.net/questions/197815 | 5 | I am reading Burago, Burago and Ivanov's book *A course in metric geometry*. In chapter 10 the mention that Alexandrov spaces of curvature bounded below have a stratification into topological manifolds. On further reading, I found that on Perelman's article *Elements of Morse theory on Alexandrov spaces* that the strat... | https://mathoverflow.net/users/52863 | Codimension of the set of topologically singular points of an Alexandrov space. | Your intuition is correct. Since the Alexandrov space is locally homeomorphic to the cone on the space of directions, your induction hypothesis implies that the statement is true locally, and hence globally.
| 5 | https://mathoverflow.net/users/68708 | 198832 | 96,137 |
https://mathoverflow.net/questions/198828 | 0 | **Context:**
Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}\_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F \rightarrow G^F$ is surjective, where $F$ denotes the Frobenius morphisms inducing the $\mathbb{F}\_q$-rational structures... | https://mathoverflow.net/users/68519 | A bijection between Lusztig series induced by inflation | Your approach is correct and is proven in the book by Digne-Michel (in fact a more general statement is proven there). Indeed, by Proposition 13.22 in Digne-Michel we know that
$$R\_{T\subseteq B}^G(\theta)\circ \pi = R\_{\widehat{T}\subseteq \widehat{B}}^{\widehat{G}}(\theta\circ\pi)$$
where $\widehat{T} = \pi^{-1... | 3 | https://mathoverflow.net/users/22846 | 198835 | 96,139 |
https://mathoverflow.net/questions/198838 | 12 | The coherence theorem for bicategories, as usually stated, reads
>
> Any bicategory $B$ is biequivalent to a (strict) 2-category.
>
>
>
It is possible to give an explicit construction of the strictification as the full image of its Yoneda embedding $y:B\rightarrow [B,\text{Cat}]$, see for instance [this refere... | https://mathoverflow.net/users/27870 | On the coherence theorem for bicategories | Probably you had some trouble finding this because the search term $2$-$\text{Cat}$ is not accurate enough; you want not the cartesian monoidal product on $2$-$\text{Cat}$, but rather what is called the Gray monoidal product; the tricategory you want then is denoted $\text{Gray}$, the tricategory of strict 2-categories... | 12 | https://mathoverflow.net/users/2926 | 198845 | 96,142 |
https://mathoverflow.net/questions/198840 | 2 | Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl\_V(G)$ should mean.
Of course for affine space $\mathbb{A}^n$, the blow-up at the origin is a subset of $\mathbb{A}^n\times \mathbb{P}^{n-1}$ defined by $B=\{(x... | https://mathoverflow.net/users/41901 | Blowing-up the Grassmannian at a point | Let me discuss a more general question of blowing up a subscheme $Z$ in $X$. Imagine that there is a resolution
$$
F \stackrel{s}\to E \to I\_Z \to 0
$$
of the ideal of $Z$ by vector bundles $E$ and $F$. Then there is an embedding
$$
Bl\_Z(X) \to P\_X(E^\*)
$$
(it is induced by a surjection of graded algebras $\oplus S... | 8 | https://mathoverflow.net/users/4428 | 198850 | 96,144 |
https://mathoverflow.net/questions/198855 | 0 | Let $G$ be a semisimple and simply connected linear algebraic group over $\mathbb{C}$.
Let $H$ be a connected, Zariski closed and semisimple linear algebraic $\mathbb{Q}$-subgroup of $G$.
Is $H$ a simply connected linear algebraic group?
Here "simply connected" means every central isogeny to $G$ is an isomorphism. ... | https://mathoverflow.net/users/68719 | Are Zariski connected and closed semisimple subgroups of semisimple and simply connected algebraic groups again simply connected? | As Venkataramana says, the answer is no. In fact, every linear algebraic $\mathbb{Q}$-group $H$, whether simply connected or not, is (isomorphic to) a Zariski closed $\mathbb{Q}$-subgroup of some $SL\_n$ (which is semisimple and simply connected). (By definition of being an algebraic group, $H$ is a Zariski closed $\ma... | 3 | https://mathoverflow.net/users/68305 | 198862 | 96,146 |
https://mathoverflow.net/questions/198852 | 4 | This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
* $\mathbf{N}$ is the following matrix:
\begin{equation}
\mathbf{N}=\begin{bmatrix} \mathbf{I}\_n & \mathbf{0}\_n \\... | https://mathoverflow.net/users/54797 | Why are 1 and -1 eigenvalues of this matrix? | First, one should conjugate all matrices by
$$
\begin{pmatrix}
\operatorname{diag}(\omega\_1,\dots,\omega\_n) & 0 \\ 0 & 1
\end{pmatrix}
$$
as this converts $S(t)$ to a rotation matrix while leaving the reflection $N$ unchanged.
The matrix $P^{-1} \operatorname{diag}(1,\dots,1,-1) P$ has a line as its -1 eigenspace ... | 13 | https://mathoverflow.net/users/766 | 198863 | 96,147 |
https://mathoverflow.net/questions/198868 | 4 | Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ring over a commutative polynomial ring) is often referred to as the "Jordan plane" in the literature.
It can be see ... | https://mathoverflow.net/users/68726 | The Jordan Plane and Enveloping Algebras | Original answer without the finite-dimensionality requirement:
Yes, trivially so. For *every* algebra $A$, one can consider the underlying Lie-algebra $\mathfrak{a}$ of $A$ and gets a surjection $U(\mathfrak{a}) \twoheadrightarrow A$ because of the universal mapping property of $U(\mathfrak{a})$.
---
New answer... | 5 | https://mathoverflow.net/users/3041 | 198869 | 96,149 |
https://mathoverflow.net/questions/198872 | 15 | $V=\mathbb C^n$ is a $\mathbb CS\_n$-module, where $S\_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^{\otimes m}$ is therefore also a $\mathbb CS\_n$-module via the action $\sigma(v\_1\otimes\cdots\otimes v\_m) = \sigma... | https://mathoverflow.net/users/15934 | Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$ | By Schur–Weyl duality there is an isomorphism of $\mathrm{GL}(V) \times S\_m$-representations
$$V^{\otimes m} \cong \bigoplus\_\lambda \Delta^\lambda(V) \boxtimes S^\lambda$$
where the sum is over all partitions $\lambda$ of $m$ with at most $n$ parts, $\Delta^\lambda$ is the Schur functor for $\lambda$, $S^\lambda... | 26 | https://mathoverflow.net/users/7709 | 198874 | 96,151 |
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