parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/149363 | 5 | For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x\_\alpha(t))=x\_{\sigma\alpha}(t')$. While conjugating by certain elements of the torus normalizer realizes the action of the Weyl group of $\Phi$, the automorphisms corresponding ... | https://mathoverflow.net/users/5018 | Root system automorphisms as inner automorphisms of extended Chevalley group | Let me rephrase your question. Fix a representation $\rho: G \to GL(V)$ of the Chevalley group $G$. You want a criterion for whether there is an element $x \in GL(V)$ so that $\rho(\sigma(g)) = x\rho(g)x^{-1}$ for all $g \in G$.
In your example of $G = SO\_{2n}$ and $V$ the natural representation, you can take $\sigm... | 7 | https://mathoverflow.net/users/6486 | 149547 | 80,033 |
https://mathoverflow.net/questions/149448 | 4 | We are looking at directed graphs with no loops or parallel edges, but given two vertices $x$ and $y$, we allow the presence of both the edge $(x, y)$ and $(y, x)$. Thus, if $G$ is a directed graph on $n$ vertices, $|E(G)| \le n(n-1)$.
A tournament is a directed complete graph where for any pair of vertices $x$ and $... | https://mathoverflow.net/users/20940 | Extremal functions for tournaments | Short answer: Yes, the bound you are suggesting is correct.
The problem was solved by Brown and Harary in ``Extremal digraphs" (1970). I could not find the complete text of the paper, so let me try to give a (possibly different) proof here. (*Disclosure*: The proof has been substantially modified after Paul found an ... | 6 | https://mathoverflow.net/users/8733 | 149550 | 80,034 |
https://mathoverflow.net/questions/149548 | 5 | Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d\_{L^{\infty}}$. Each element $f$ in $C[0,1]$ corresponds uniquely to a compact subset of $\mathbb{R}^2$, by taking the graph of $f$. Thus, we can consider t... | https://mathoverflow.net/users/23434 | Hausdorff metric on C[0,1] | Yes, this is true.
You need to show that for any $\varepsilon>0$ and any function $f$ in $C[0,1]$
there is $\delta>0$ such that $\varepsilon$-neighborhood in one metric contains $\delta$-neighborhood in the other metric and the other way around.
One way you know already; it follows since $\rho\le d\_{L^\infty}$.
... | 1 | https://mathoverflow.net/users/1441 | 149552 | 80,035 |
https://mathoverflow.net/questions/149536 | 1 | The spectral $l^2$ norm of a complex matrix is given by:
$\|A\|= \left( \sum\_{k=0}^{N-1} s\_k(A)^2 \right)^{1/2}$ where $s\_k(A)$ are the singular values of $A$ ordered so as to be non decreasing in $k$.
The weighted $l^2$ norm of real vector $x \in \mathbb{R}^N$ is given by: $\|x\|= \left( \sum\_{k=0}^{N-1} w\_k ... | https://mathoverflow.net/users/41654 | Weighted Spectral l-2 norms arising from matrix inner products | I don't think it is even a norm. If you take $w\_0 = 1$ and $w\_k = 0$ for all $k\geq 1$ then you get $\|A\| = |s\_{\min}(A)|$ and it does not satisfy the triangular inequality:
$\|e\_1e\_1^T\| =0$, $\|e\_2e\_2^T\| =0$, but $\|e\_1e\_1^T+e\_2e\_2^T\| \neq 0$.
Am I missing something?
| 3 | https://mathoverflow.net/users/4878 | 149557 | 80,038 |
https://mathoverflow.net/questions/149567 | 5 | Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of finite index according to [this MO question](https://mathoverflow.net/questions/133072/on-the-f-rational-points-of-the-de... | https://mathoverflow.net/users/24114 | When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$ | Yes, since in characteristic 0 all unipotent groups are connected and split. More generally, for any field $k \ne \mathbf{F}\_2$ and any connected reductive $k$-group $G$ and split unipotent smooth connected closed $k$-subgroup $U \subset G$, necessarily $U(k) \subset (G(k), G(k))$. (This is false for ${\rm{SL}}\_2(\ma... | 7 | https://mathoverflow.net/users/39487 | 149573 | 80,044 |
https://mathoverflow.net/questions/149520 | 15 | A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise polarized abelian varieties (which sounds like a hard geometric problem) by parametrising their H^1's with the extra structure ... | https://mathoverflow.net/users/43076 | Moduli space of motives vs moduli space of varieties | Let me expand my (and ulrich's) comment slightly concerning your last question. Let $D$ be the period domain of all Hodge structures with fixed Hodge numbers and polarization. For the sake of simplicity, let's say the weight is $2$. Suppose that $H\in D$ is a summand of $H^2$ of some smooth projective variety, then by ... | 11 | https://mathoverflow.net/users/4144 | 149576 | 80,046 |
https://mathoverflow.net/questions/149549 | 6 | Are there any verbal subgroups in a rank 2 free group $F(a,b)$ arising as normal closures $\langle\langle r \rangle\rangle$ of a (nontrivial) element $r \in F(a,b)$ other than the commutator subgroup $[F,F]=\langle\langle [a,b] \rangle\rangle$?
More generally, are there any subgroups of the given type $\langle\langle... | https://mathoverflow.net/users/25643 | A question on verbal subgroups of free groups | 1) The answer is no: namely, suppose that $n\ge 2$, $r\in F\_n$ and $N=\langle\langle r\rangle\rangle$ is verbal, then either $r=1$, or $n=2$ and $r$ is conjugate to $[x\_1,x\_2]$. Indeed $F\_n/N$ is a 1-relator group; it was established; if $r\neq 1$ it satisfies a law, and Magnus [a,b] proved that this only happens w... | 8 | https://mathoverflow.net/users/14094 | 149583 | 80,048 |
https://mathoverflow.net/questions/149559 | 3 | Consider the measurable space $2^{\mathbb R}$, equipped with the tensor-product $\sigma$-algebra. Famously, this space has a measurable structure which is not generated by a topology (see [this answer](https://mathoverflow.net/a/87888/238)).
Can you provide an example of a non-trivial probability measure on $2^{\math... | https://mathoverflow.net/users/238 | A non-trivial probability measure on $2^{\mathbb R}$ | $2^{\mathbb{R}}$, being a product of compact Hausdorff groups, is a compact Hausdorff group, so it has a normalized Haar measure ("flipping uncountably many coins").
| 8 | https://mathoverflow.net/users/290 | 149587 | 80,051 |
https://mathoverflow.net/questions/149588 | 19 | Let $z \in \overline{\bf Q}$ be an algebraic number. Define the "denominator" of z to be the least natural number $n$ such that $nz$ is an algebraic integer.
By a rather *ad hoc* argument (playing with the minimal polynomial of $z$ to compute high powers of $z$ in terms of low powers of $z$, and measuring the coeffic... | https://mathoverflow.net/users/766 | Growth of the "denominator" of powers of an algebraic number | There is some number field $K$ containing $z$, and there we have a prime factorization
$$
(z)=\frac{\mathfrak{p}\_1\ldots\mathfrak{p}\_a}{\mathfrak{q}\_1\ldots\mathfrak{q\_b}}
$$
of fractional ideals, where no $\mathfrak{p}\_i$ is equal to a $\mathfrak{q}\_j$. If $nz^m$ is an algebraic integer, then $(\mathfrak{q}\_1\l... | 24 | https://mathoverflow.net/users/5263 | 149590 | 80,052 |
https://mathoverflow.net/questions/149591 | 1 | Let $f:X \to \mbox{Spec } R$ be a projective morphism between irreducible Noetherian schemes. Assume that $R$ is a discrete valuation ring and its residue field is algebraically closed. Suppose now that $f$ restricted to the special fiber is smooth. Assume further that the special fiber is irreducible and smooth (over ... | https://mathoverflow.net/users/9164 | Does flatness/smoothness over special fiber imply flatness/smoothness globally? | No, it does not. As a counterexample, let $f$ be the inclusion of the closed point $Spec\ k \hookrightarrow Spec\ R$. Equally well, one could also take other irreducible projective smooth $k$-varieties for $X$.
| 8 | https://mathoverflow.net/users/5498 | 149593 | 80,054 |
https://mathoverflow.net/questions/149574 | 6 | I am currently reading the paper
D.A. Goldston, J. Pintz, C.Y. Yildirim, $\textit{Primes in tuples I}$, Annals of Mathematics $\textbf{170}$ (2009), 819-862
and in particular I found equation (8.16), which records the identity
$$\displaystyle \frac{1}{u!} \sum\_{i=0}^u (-1)^i \binom{u}{i} \frac{d(d+1)\cdots(d+i-1... | https://mathoverflow.net/users/10898 | The combinatorial interpretation of an identity found in "Primes in tuples I" | Since the terms aren't integers we can't find a combinatorial interpretation directly.
If we multiply both sides by $(u+v+d)!$ and rearrange, we can rewrite the identity as
$$
\sum\_{i=0}^u (-1)^i \binom{u+v+d}{u-i}\binom{d+i-1}{i} =\binom{u+v}{u},
$$
which we can interpret combinatorially. Let $U$, $V$, and $D$ be d... | 13 | https://mathoverflow.net/users/10744 | 149602 | 80,057 |
https://mathoverflow.net/questions/149619 | 2 |
>
> I'm interested in *triangulations* with few vertices of a given orientable compact surface $S$.
>
>
>
By triangulation, I don't mean a "*simplicial triangulation*" but a "*decomposition of $S$ by triangles*", these being topological triangles glued by identifying edges. The only condition required is that a... | https://mathoverflow.net/users/36575 | Non-simplicial triangulations of compact surfaces with few vertices | This is an incomplete answer, but maybe the pointers can be of some use to you.
Question 1: It looks to me that the triangulations you describe are essentially triangulated multigraphs embedded on a surface. It is a heavily studied topic in topological graph theory, about which Graphs on Surfaces by Mohar and Thomass... | 3 | https://mathoverflow.net/users/6325 | 149629 | 80,066 |
https://mathoverflow.net/questions/149342 | 1 | Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. Suppose there exists a divisor in $X$ which is flat over $\mathbb{P}^n$. Under what condition on $f$ or $X$, does this i... | https://mathoverflow.net/users/32151 | Existence of rational section to a flat projective morphism | This is a follow-up to Mike Roth's important, correct comment (now disappeared). The following is one of a series of examples that I learned of from Tom Graber, but which I guess goes back to the work on "normic forms". Assume that the characteristic is not $3$ (there are similar examples in every characteristic). Let ... | 2 | https://mathoverflow.net/users/13265 | 149636 | 80,069 |
https://mathoverflow.net/questions/149643 | 1 | N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) [polygonal chain](http://en.wikipedia.org/wiki/Polygonal_chain) connecting all the points. It can be proved that $L(N)\sim c\sqrt{AN}$, for some constant $c$, where A is the area of... | https://mathoverflow.net/users/43124 | Expected length of the shortest polygonal chain connecting N random points in the unit square | Some partial answers:
1) The fact that the length is asymptotic to $c\sqrt{N}$ follows from sub-additivity.
2) This is the ``random travelling salesman'' problem. Joe Yukich wrote several papers on it and its variants.
3) See also the wikipedia page
<http://en.wikipedia.org/wiki/Travelling_salesman_problem#TSP_p... | 6 | https://mathoverflow.net/users/35520 | 149647 | 80,075 |
https://mathoverflow.net/questions/149600 | 9 | Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality
$$|\alpha-\frac{p}{q}|<\frac{1}{q^{2+\epsilon}}$$
has only finitely many solutions in integers $(p,q)$.
I seek confirmation (or not) fo... | https://mathoverflow.net/users/10909 | History question: Roth's theorem on approximating algebraic numbers...before Roth | I believe that both statements are correct. The ineffective results of Thue, Siegel, Gel'fond and Dyson do not lead to irrationality measures arbitrarily close to $2$, while earlier effective results (Thue, Siegel, etc) for classes of algebraic numbers give measures as close to $2$ as you like, but not of the $2 + \eps... | 7 | https://mathoverflow.net/users/7302 | 149653 | 80,077 |
https://mathoverflow.net/questions/149654 | 1 | Suppose there are $m$ balls to be randomly thrown into $n$ bins ($m>n$). Let $X\_i$ be the number of balls ending up in bin $i$.
Let $X\_{max}$ be the heaviest bin and $X\_{min}$ be the lightest bin. In [Raab and Steger's paper](https://doi.org/10.1007/3-540-49543-6_13), the authors state that
$$
Pr[X\_{max}≥k]≤o(1).... | https://mathoverflow.net/users/43113 | Balls and bins: Exact probability | You can find the answer in [this paper by Reviriego, Holst and Maestro](http://www.nebrija.es/~jmaestro/esa/papers/JDA2011.pdf).
Especially interesting should be formula 23 giving the exact probability distribution.
Of course it can be easily seen that your term “heaviest bin” is equivalent to the term “Longest Lengt... | 1 | https://mathoverflow.net/users/34050 | 149671 | 80,084 |
https://mathoverflow.net/questions/149678 | 4 | I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$$ 1 \rightarrow \pi\_1 (\bar{X},\bar{x\_0})\rightarrow \pi\_1 (X,x\_0)\rightarrow Gal(k)\rightarrow 1 $$
is exact.
where $\bar{X}= X \times\_k k\_s $ and $X$ is a scheme of finite type over the field $k$ and $k... | https://mathoverflow.net/users/42021 | homotopy exact sequence for the étale fundamental group | Your second question has a negative answer because you've got some variances backwards. When $f:G \rightarrow G'$ is a map of groups, by composition with $f$ one gets a functor from the category of $G'$-sets to the category of $G$-sets. As Sawin notes, there is a functor from finite etale covers of $X$ to those of $Y$,... | 6 | https://mathoverflow.net/users/43107 | 149685 | 80,087 |
https://mathoverflow.net/questions/149661 | 4 | In a nutshell, if $u$ is a solution to
$$
\partial\_r^2 u(r)+ \frac{1}{r} \partial\_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r\_0>0\\
\lim\_{r \to \infty} u(r) = 0,
\quad \text{and} \quad
u(r\_0) = u\_0 > 0 .
$$
I'd like to know the behavior of $u(r)$ as $r \to \infty$.
**Motivation:** $u$ is the steady-... | https://mathoverflow.net/users/32723 | Asymptotic behavior for the solution of a nonlinear ODE | I do not have a complete solution, just a comment. As $u(r)\to 0$, one may try to neglect
the quadratic term. Then the equation becomes linear, and its solution tending to $0$
is called the modified Bessel function $K\_0$. $K\_0(x)=Y\_0(ix)$, where $Y\_0$ is the Weber
function ("second" solution of the Bessel equation)... | 5 | https://mathoverflow.net/users/25510 | 149689 | 80,089 |
https://mathoverflow.net/questions/149656 | 22 | I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$. Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type.
I already "understand" the pointed loop space of $S^n$ using the James reduced product. (See Hatcher, *Algebraic Topology*, Sect... | https://mathoverflow.net/users/6514 | Is there a good way to understand the free loop space of a sphere? | Stably the free loop space of the suspension of a connected space splits up, just as the based loop space does. Just as $\Omega\Sigma X$ is stably the wedge of the smash products $X^{\wedge n}$, $L\Sigma X$ is stably the wedge of $S^1\_+\wedge\_{C\_n}X^{\wedge n}$. Here $C\_n$ is a cyclic group of order $n$ acting free... | 23 | https://mathoverflow.net/users/6666 | 149694 | 80,093 |
https://mathoverflow.net/questions/149700 | 9 | The twin prime conjecture says there are infinitely many pairs $p,p+2$ that are both prime, and although we still don't know whether it's true there's been a lot of progress recently showing that there are infinitely many pairs $p,p+k$ that are both prime, for $k$ bounded by a constant.
It's also an old conjecture th... | https://mathoverflow.net/users/440 | Primes $p$ for which $pk+1$ is prime for small $k$ (or approximating Sophie Germain) | [This paper](http://www.ams.org/journals/mcom/2005-74-252/S0025-5718-05-01749-7/S0025-5718-05-01749-7.pdf#page=2) uses "$\hspace{.01 in}P(n)$ to denote the greatest prime factor of an integer $n\hspace{.01 in}$",
and obtains the following result as Theorem 2 on page 2.
>
> Let $a\in \mathbb{Z},\theta < 1-\frac1... | 5 | https://mathoverflow.net/users/nan | 149705 | 80,095 |
https://mathoverflow.net/questions/149670 | 13 | Given a prime $p$, define $f(p)$ to be the smallest prime congruent to $1$ modulo $p$. For example, $f(7)=29$. It has been conjectured that $f(p)<p^2$ always: by Schinzel in his "Hypothesis H" paper with Schinzel (Acta Arith. 1958), and also by Kanold (Arch. Math. 1963). Of course this is also related to Linnik's theor... | https://mathoverflow.net/users/5091 | Reference for a conjecture on the first primes congruent to 1 modulo other primes | There are even stronger conjectures in the literature:
1) D.R. Heath-Brown, Almost-primes in arithmetic progressions and
short intervals. Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3,
357–375.
<http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2079092>
quoting from the first few lines:... | 8 | https://mathoverflow.net/users/36707 | 149706 | 80,096 |
https://mathoverflow.net/questions/149702 | 35 | I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual [$j$-invariant](http://en.wikipedia.org/wiki/J-invariant), and $\alpha = (-1+i\sqrt{163})/2$, then
\begin{align\*}
\frac{j(i)}{1728} &= 1 \\
\frac{-j(\alpha)}{1728} &= 151931373056000 = 2^{12}5^323^329^3 \\
\frac{j(i)-j(\alpha)}{1728... | https://mathoverflow.net/users/5091 | Difference of j-invariant values and the abc conjecture | There is a beautiful theory by Gross and Zagier (On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220) that explains completely the factorizations of
$$
j(\tau\_1)-j(\tau\_2)
$$
for $\tau\_1,\tau\_2$ lying in (possibly two different) imaginary quadratic fields. There are recent extensions by [Kristin Lauter ... | 34 | https://mathoverflow.net/users/3132 | 149708 | 80,098 |
https://mathoverflow.net/questions/149376 | 7 | The starting point of this question is the (presumably) well-known theorem (the proof I know is from *Abelian $\ell$-adic representations and elliptic curves* from J-P.Serre in which it is a lemma for $n=2$ and an exercise for $n>2$; which suggests that the result was already classical in the 60s).
**Theorem I:** If ... | https://mathoverflow.net/users/2284 | Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$ | For the first question, at least for $p=5, n=2$ there is a counterexample.:
The group $\text{SL}\_2({\mathbb F}\_5)$ has a 2-dimensional symplectic representation with character in ${\mathbb Q}(\sqrt 5)$ and Schur index 2, so it is realizable over $A={\mathbb Z}\_5(\zeta\_5)$. So $\text{SL}\_2({\mathbb F}\_5)$ inject... | 4 | https://mathoverflow.net/users/3132 | 149709 | 80,099 |
https://mathoverflow.net/questions/149726 | 4 | Given n independent random variables $x\_1,x\_2,...,x\_n$, they have standard uniform distributions over [0,1]. Then what's the probability that there is at least one $|x\_i-x\_j| >= d$ for any different $i,j$ and $0<=d<=1$?
The discrete form of this problem is as follow:
Given n independent random variables $x\_1,... | https://mathoverflow.net/users/42931 | What's the probability of differences among n independent uniform distribution variables? | Quite generally, if $P(x)$ is the probability density of $n$ independent random variables and $F(x)=\int\_{-\infty}^{x}P(x')dx'$ is their cumulative distribution function, then the joint distribution of the smallest and largest variables $x\_{\rm min}<x\_{\rm max}$ is given by
$$P(x\_{\rm min},x\_{\rm max})=n(n-1)P(x... | 2 | https://mathoverflow.net/users/11260 | 149733 | 80,108 |
https://mathoverflow.net/questions/149675 | 3 | The real Eisenstein series
$G\_s^\* = \frac{\Gamma(s)}{\pi^s} \sum'\_{m,n}\frac{Im(\tau)}{|m+n \tau|^{2s}}$
admits the following integral representation (their Mellin transform):
$G\_s^\* = \frac{1}{2} \int\_0^\infty (\Theta\_\tau(t)-1) t^{s-1} dt$
where $\Theta\_\tau(t)$ is the associated theta series $\Thet... | https://mathoverflow.net/users/41940 | Real modular form, inverse transform | Yes, this is essentially the $L^2$ spectral decomposition of the orthogonal complement to cuspforms (waveforms). It is spanned (for example) by pseudo-Eisenstein series $E\_\phi$, formed by winding up functions of the form $\Phi(x+iy)=\phi(y)$ with $\phi\in C^o\_c(0,\infty)$. The more mundane spectral decomposition of ... | 3 | https://mathoverflow.net/users/15629 | 149741 | 80,112 |
https://mathoverflow.net/questions/131051 | 9 | I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int\_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, and notions of entropy from statistical mechanics. Would you please explain it in details?
| https://mathoverflow.net/users/32817 | The relations between the Perelman's entropy functional and notions of entropy from statistical mechanics | For metrics on $S^{2}$ with positive curvature, Hamilton introduced the
entropy $N\left( g\right) =-\int\ln(R\operatorname{Area})Rd\mu.$ If the
initial metric has $R>0,$ he proved that this is nondecreasing under the Ricci
flow on surfaces; note that $Rd\mu$ satisfies $(\frac{\partial}{\partial t}-\Delta
)(Rd\mu)=0.$ L... | 10 | https://mathoverflow.net/users/nan | 149743 | 80,114 |
https://mathoverflow.net/questions/149744 | 2 | I sometimes come across this notion called "unitary automorphic representation". But I have never seen the precise definition. When they say $(\pi, V)$ is a unitary automorphic representation of a group $G(\mathbb{A})$, does that mean that $\pi$ is unitary as an abstract representation of $G(\mathbb{A})$ (assuming $\pi... | https://mathoverflow.net/users/32746 | On a unitary automorphic representation | As you suspect, there are many implicit assumptions and abuses of language in this terminology.
First, it is *not* safe to assume that an "automorphic" repn of $G(\mathbb A)$ is literally a repn of that topological group. Sometimes, only the finite-prime groups are allowed to act, and at archimedean places one does n... | 4 | https://mathoverflow.net/users/15629 | 149747 | 80,115 |
https://mathoverflow.net/questions/149696 | 2 | If we have $R^{4}$ with basis $\{e\_{1},e\_{2},e\_{3}=Je\_{1},e\_{4}=Je\_{2}\}$, then we know that $\{e\_{1},e\_{2}\}$ is totally real minimal submanifold of $R^{4}$. Is there a nontrivial example of totally real minimal submanifold of $R^{4}$.
Also, the above totally real minimal immersion is an inclusion map. How w... | https://mathoverflow.net/users/43143 | Examples of totally real minimal submanifolds in complex space forms | Yes, there are many such examples.
A totally real submanifold in this case is simply a surface $S\subset\mathbb{R}^4$ such that, for each $p\in S$, the space $J(T\_pS)$ is orthogonal to $T\_pS$. In other words, if $x^i$ are the coordinates dual to your basis $e\_i$, then the $2$-form $\Upsilon\_1 = dx^1\wedge dx^3 +... | 2 | https://mathoverflow.net/users/13972 | 149748 | 80,116 |
https://mathoverflow.net/questions/149746 | 3 | Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus\_i M\_{n\_i}(\mathbb{C})$$
>
> **Question** : Is it singly generated (as von Neumann algebra)? how ?
>
>
>
**Allowed operations** : $() \mapsto I$ , $(A) \mapsto \lambda A$ or $\mathbf{A^\*}$ and $(A,B) \mapst... | https://mathoverflow.net/users/34538 | Are the finite dimensional von Neumann algebras, singly generated? | Pick distinct complex numbers $\lambda\_1,\ldots,\lambda\_k$ and consider the element
$$
X:=\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda\_1&1&0&0&0\\
0&\lambda\_1&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda\_1&1\\
0&0&0&0&\lambda\_1\\
\end{smallmatrix}}\_{n\_1}\,\Bigg)
\oplus
\Bigg(\,\underbrace{\begin{smallmatrix}
... | 12 | https://mathoverflow.net/users/5690 | 149752 | 80,118 |
https://mathoverflow.net/questions/149737 | 30 | Suppose $m$ is a positive integer. A quantity of interest is
$$
H\_m = \liminf\_{n\to\infty} \left(p\_{n+m} - p\_n \right)
$$
The twin prime conjecture, is, of course $H\_1 = 2$, the the prime k-tuples conjecture of Hardy and Littlewood asserts that $H\_2 = 6$, $H\_3 = 8$ and so on. Goldston-Pintz-Yildirim showed t... | https://mathoverflow.net/users/37327 | What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$ | The major difference is the choice of the Selberg Sieve weights. I strongly recommend reading [Maynard's paper](http://arxiv.org/abs/1311.4600). It is well written, and the core ideas are nicely explained.
---
In what follows, I'll give a brief explanation of what Selberg Sieve weights are, why they appear, and ... | 40 | https://mathoverflow.net/users/12176 | 149753 | 80,119 |
https://mathoverflow.net/questions/149755 | 2 | **Definition 1:**
Let $(G,\leq)$ be a nonzero partially ordered Abelian group with order unit $u$. (Recall that $u\in G$ is a order unit if, for every $g\in G$, there exists $N\in\mathbb N$ such that $-Nu\leq g\leq Nu$.) For any $g\in G$, define the quantities (as it is done in proposition 4.7 of the book [Partially Or... | https://mathoverflow.net/users/nan | Example involving partially ordered Abelian groups | Let $G=\mathbb{R}^2$ with order defined by
$(a\_1,a\_2)\leq (b\_1,b\_2) \iff a\_1\leq b\_1$ and $a\_2\leq b\_2$.
Let $u=(1,1)$ and let $g\_0=(1,2)$.
Then $p(g\_0)=1$ and $r(g\_0)=2$.
| 1 | https://mathoverflow.net/users/5513 | 149757 | 80,120 |
https://mathoverflow.net/questions/129000 | 2 | In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then $\frac{d}{dt}\mathcal{F}(g,f)=\int \langle-Ric-Hess(f),\dot{g}\rangle e^{-f}dV$. So by definition, gradient of $\mathcal{F}$ is given by $\nabla \ma... | https://mathoverflow.net/users/32817 | Ricci flow as a gradient flow and its Lyapunov function | If $\frac{\partial}{\partial s}g=v$, then $\frac{\partial R}{\partial
s}=-\Delta V+\operatorname{div}^{2}v-\left\langle v,\operatorname{Ric}
\right\rangle $ and $\frac{\partial}{\partial s}d\mu=\frac{1}{2}Vd\mu$, where
$V=\operatorname{tr}\_{g}v$. So
$$
\frac{\partial}{\partial s}(Rd\mu)=(-\Delta V+\operatorname{div}^{... | 6 | https://mathoverflow.net/users/nan | 149759 | 80,122 |
https://mathoverflow.net/questions/149761 | 3 | Let $X$ be a scheme and $ \mathscr{L}$ be a line bundle on $X$. In a few proofs I have seen the scheme
$$ L = \mathscr{S}{\rm pec} \oplus\_{n \in \mathbb{Z}} \mathscr{L}^{\otimes n} \to X$$
pop up. Does the scheme $ L $ have a name? How should I be thinking about $L$?
| https://mathoverflow.net/users/4002 | How should I think about this scheme constructed from a line bundle | Denote by $\mathcal{A}$ the $\mathbb{Z}$-graded sheaf of $\mathcal{O}\_X$-algebras, $$ \mathcal{A} = \oplus\_{n\in \mathbb{Z}} \mathcal{L}^{\otimes n}. $$
There is a natural isomorphism of invertible $\mathcal{A}$-modules,
$$\phi: \mathcal{A}\otimes\_{\mathcal{O}\_X}\mathcal{L} \to \mathcal{A}.$$
As a map of graded mod... | 6 | https://mathoverflow.net/users/13265 | 149764 | 80,123 |
https://mathoverflow.net/questions/149754 | 4 |
>
> Is there a ([$\mathrm{T}\_0$](http://en.wikipedia.org/wiki/Kolmogorov_space)) [topological group](http://en.wikipedia.org/wiki/Topological_group) that is connected and locally connected but is not path-connected?
>
>
>
This is a cross-post from MSE, since [my question there](https://math.stackexchange.com/qu... | https://mathoverflow.net/users/nan | topological group that is connected and locally connected but not path-connected | A similar question was answered here: <https://mathoverflow.net/a/119962/2926>. The idea is to start with your favorite non-trivial abelian group $A$, say $A = \mathbb{Z}/(2)$ (viewed as a group object in $\mathbf{Set}$) and apply to it a sequence of product-preserving functors
$$\mathbf{Set} \stackrel{K}{\to} \math... | 7 | https://mathoverflow.net/users/2926 | 149766 | 80,124 |
https://mathoverflow.net/questions/149765 | 0 | Who can give me a table of supersingualr elliptic curves over F\_p? At lsist for small p. If I make such a table using magma on my laptop, how long (as a function of p) will I use.
| https://mathoverflow.net/users/42690 | list of supersingular elliptic curves | For $p \leq 307$ you can use [this table](http://wstein.org/Tables/antwerp/table6) (from Antwerp IV = **LNM** 476 (1975)).
| 3 | https://mathoverflow.net/users/14830 | 149776 | 80,126 |
https://mathoverflow.net/questions/149735 | 3 | Elkies proved *The existence of infinitely many supersingular primes for every elliptic curve over Q*. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how about the others?
| https://mathoverflow.net/users/42690 | The existence of infinitely many supersingular primes for every elliptic curve over Q | It's the *auxiliary prime* $l$ that must be $3 \bmod 4$;
the *supersingular* prime $p$ is not guaranteed to be congruent to $3 \bmod 4$,
and indeed the residue of $p \bmod 4$ is unpredictable
(unless the curve has CM by an order in ${\bf Q}(i)$).
Several examples on page 565 have $p \equiv 1 \bmod 4$.
For example, the ... | 11 | https://mathoverflow.net/users/14830 | 149779 | 80,129 |
https://mathoverflow.net/questions/149781 | 3 | For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?
| https://mathoverflow.net/users/42690 | The existence of elliptic curves with prescribed supersingular primes | No. In fact the set can be arbitrarily sparse, i.e. the $n$-th prime
in the set can be chosen to exceed $a\_n$ for any sequence $\{a\_n\}$.
This is because the rationals are countable. Fix an enumeration
$j\_1,j\_2,j\_3,\ldots$ of $\bf Q$. For each $n$ let $p\_n$ be the smallest
prime such that $p\_n > a\_n$ and $j\_n$... | 7 | https://mathoverflow.net/users/14830 | 149782 | 80,131 |
https://mathoverflow.net/questions/149692 | 23 | The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int\_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu + 1}\|\mathbf a\|^{-\nu}J\_\nu(\|\mathbf a\|),
\qquad\nu=\frac n2 -1,
\tag1
$$
found e.g. in [1, p. 198] or [2, p. 154... | https://mathoverflow.net/users/19276 | Fourier transform of the unit sphere | At the risk of answering my own question, here is what I have since found:
1. For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, [*Summation of multiple Fourier series by spherical means*](http://www.ams.org/mathscinet-getitem?mr=1501870), Trans. AMS **40** (1936) 175-207. Bochner exposes it a... | 18 | https://mathoverflow.net/users/19276 | 149787 | 80,133 |
https://mathoverflow.net/questions/149790 | 7 | I know that if $x$ is a rational multiple of $\pi$, then $tan(x)$ is [algebraic](http://divisbyzero.com/2010/10/28/trigonometric-functions-and-rational-multiples-of-pi/).
Is there a fairly simple way to express $x$ as $\pi\ m/n$, if $tan(x)$ is given as a square root of a rational?
| https://mathoverflow.net/users/43192 | arctan of a square root as a rational multiple of pi | An angle $x$ with $\tan^2 x$ rational has been called "geodetic" by Conway, Radin, and Sadun, [On Angles Whose Squared Trigonometric Functions are Rational](http://arxiv.org/abs/math--ph/9812019). Geodetic angles have a simple representation, see their Theorem 2, but they are not in general rational multiples of $\pi$.... | 11 | https://mathoverflow.net/users/11260 | 149795 | 80,137 |
https://mathoverflow.net/questions/149810 | 2 | Given a coherent sheaf $\mathcal{F}$ we denote by $\Gamma\_\*(\mathcal{F})=\oplus H^0(\mathcal{F}(d))$. Suppose, $\mathcal{F}\_1$ and $\mathcal{F}\_2$ are two coherent sheaves on $\mathbb{P}^n$. Denote by $M\_1$ (resp. $M\_2$) the modules $\Gamma\_\*(\mathcal{F}\_1)$ (resp. $\Gamma\_\*(\mathcal{F}\_2)$). Is it true tha... | https://mathoverflow.net/users/43198 | Does $\Gamma_*$ commute with tensor product? | No, this is not true. For simplicity, let's work relative to an affine base scheme $\text{Spec} R$.
Denote $\Gamma\_\*(\mathcal{O}\_{\mathbb{P}^n})$ by $S$. Then $S$ is (naturally) a graded $R$-algebra that is isomorphic to $R[x\_0,\dots,x\_n]$.
Take $\mathcal{F}\_1$ to be $\mathcal{O}\_{\mathbb{P}^n}(1)$, and take ... | 2 | https://mathoverflow.net/users/13265 | 149814 | 80,145 |
https://mathoverflow.net/questions/149816 | 0 | Let us work in ZFC set theory, let A and B be two sets and C be the set of functions with domain A and range B.
Question: what can be said about c=rank(C), knowing a=rank(A) and b=rank(B) ?
Gérard Lang
| https://mathoverflow.net/users/30395 | About the rank of sets | Note that every function $f$ is a subset of $A\times B$, and so the rank of $C$ is at most $\newcommand{\rank}{\operatorname{rank}}\rank(\mathcal P(A\times B))$.
It is possible that $\rank(A\times B)=\rank(A)=\rank(B)$. For example in the case $A=B=\omega$. But it is possible that $\rank(A\times B)>\rank(A),\rank(B)$... | 2 | https://mathoverflow.net/users/7206 | 149817 | 80,146 |
https://mathoverflow.net/questions/149811 | 4 | Let us consider $n$-dimensional simplex $K$ in $n$-dimensional Euclidean space. Let $r\_0$ be the radius of the inscribed sphere of $K$, and let be $r\_1, r\_2, \cdots, r\_{n+1}$ be each radius of the [exsphere](http://en.wikipedia.org/wiki/Exsphere_%28polyhedra%29) of $K$.
Then, here is my question.
>
> **Quest... | https://mathoverflow.net/users/34490 | About the inscribed sphere and the exspheres of a $n$-dimensional simplex | Let $S\_1,\dots,S\_{n+1}$ be the $(n-1)$-dimensional volumes of the corresponding faces, and $V$ be the $n$-dimensional volume of the simplex. Then $\displaystyle V=\frac{r\_0S}n=\frac{r\_i(S-2S\_i)}n$, where $S=\sum\_i S\_i$. Thus
$$
\frac {n-1}{r\_0}=\frac {(n-1)S}{nV}=\frac{\sum\_i (S-2S\_i)}{nV}
=\sum\_i\frac{1}... | 8 | https://mathoverflow.net/users/17581 | 149820 | 80,149 |
https://mathoverflow.net/questions/149826 | 6 | I have proved that every Paley graph $P(p^{2})$ over $p^{2}$ vertices, where $p\geq 5$ is a prime number has a cospectral mate, i.e. for every prime number $p\geq 5$ there exists a graph $\Gamma\_{p}$ such that $P(p^{2})$ and $\Gamma\_{p}$ are cospectral but non-isomorphic. Is it well-known? If so, Could one please giv... | https://mathoverflow.net/users/31179 | Paley graphs over $p^{2}$ vertices | Choose a projective plane of order $p$, where $p$ is odd. Choose a line and view it as a line at infinity. Choose a partition of the points on the line into two classes $C\_0$ and $C\_1$ of size $(p+1)/2$. Now construct a graph on the affine points, the $p^2$ points not on the line, where two affine points are adjacenc... | 4 | https://mathoverflow.net/users/1266 | 149831 | 80,153 |
https://mathoverflow.net/questions/149830 | 1 | Is there a finite $p$-group $G$ such that :
(a) $G= \langle A,x,y \rangle$, with $G/Z(G)$ has exponent $p$, $A$ is a maximal abelian normal subgroup of $G$, and $G/A$ has order $p^2$ (thus it is elementary abelian of rank $2$);
(b) the class of $G$ is equal to $p$;
(c) the subgroup $\langle B,y \rangle$ has class... | https://mathoverflow.net/users/31883 | A finite $p$-group with certain properties | I think the same example works that I gave to your question <https://math.stackexchange.com/questions/571949>.
Let $H = C\_p \wr C\_p$, and $G = H\_1 \times H\_2$ with $H\_1 \cong H\_2 \cong H$. So $|H|=p^{p+1}$, $|G|=p^{2(p+1)}$.
The maximal abelian normal subgroup $A$ is the direct product of the base groups of $... | 6 | https://mathoverflow.net/users/35840 | 149832 | 80,154 |
https://mathoverflow.net/questions/149823 | 2 | Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ellipsoid containing $K$ that minimizes the sum of squared axis lengths (the square root of this sum is what I called the Hil... | https://mathoverflow.net/users/35733 | Circumscribed ellipsoid of minimum Hilbert-Schmidt norm | One reference that I could locate is the following [Minimum norm ellipsoids as a measure in high-cycle fatigue criteria](http://www.engopt.org/wcsmo6/papers/4721.pdf) by *Nestor Zouain*, presented at a conference in 2005 (see $\S5$ of that pdf). However, I believe this question must have been considered earlier---if I ... | 2 | https://mathoverflow.net/users/8430 | 149833 | 80,155 |
https://mathoverflow.net/questions/149844 | 9 | The Weyl algebra (say, over $\mathbb{C}$) is an universal unital algebra with two generators $x,y$ subject to the relation $xy-yx=1$. This algebra can be constructed in the following way: take two dimensional vector space with basis $\{x,y\}$ and construct the tensor algebra $T(V)$. Then take an ideal $I=(xy-yx-1)$ gen... | https://mathoverflow.net/users/24078 | Weyl algebra and its nontriviality | Generally the strategy for showing that some syntactic construction is nontrivial is to find a semantic model of it. E.g. the strategy for showing that some relations don't force a group to be a trivial group is to find, say, some matrices satisfying the relations, and the strategy for showing that some axioms in some ... | 6 | https://mathoverflow.net/users/290 | 149849 | 80,160 |
https://mathoverflow.net/questions/149695 | 0 | Recently I came up with a type of variational problem in stochastic process.
It can be stated in the following way:
Given $a$ and $b$ positive, and an increasing function $f$ on $(0,1)$ (may be not strictly, but $f$ is possibly unbounded), which satisfies the following equation:
$$ \int\_{0}^{1}H\_{1}(f(\alpha))d\alp... | https://mathoverflow.net/users/11966 | variational problem related to an integral | You question is too vague. If anything is allowed for $H$, of course, then the minimum is $-\infty$: take
$$
H\_n = -n \mbox{ on } f((0,1)) \mbox{ and } 0 \mbox{ otherwise.}
$$
If you decide that you just want to avoid that, then you can that $H$ is bounded below, and then on whichever space you decide $H$ should be i... | 1 | https://mathoverflow.net/users/40120 | 149862 | 80,165 |
https://mathoverflow.net/questions/149839 | 4 | Let $R$ be the hyperfinite $II\_1$ factor and let $X$ be any $R$-$R$-bimodule.
>
> **Question**: Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules)?
>
>
>
**Example**: If $(N \subset M)$ is an irreducible, finite depth and finite index (hyperfinite $II\_1$) subfactor, then $... | https://mathoverflow.net/users/34538 | Are all the R-R-bimodules completely reducible? | Let $\_RM\_R:={}\_R(L^2R\otimes\_{\mathbb C} L^2R)\_R$, where the first $R$ acts on the first $L^2R$ and the second $R$ acts on the second $L^2R$.
Its algebra of $R$-$R$-bimodule endomorphisms is $R^{\mathrm{op}}\,\bar\otimes\, R$.
Using (misleading!) intuition from the representation theory of separable $C^\*$-algeb... | 3 | https://mathoverflow.net/users/5690 | 149872 | 80,168 |
https://mathoverflow.net/questions/149875 | 0 | Let $X$ be a K3 surface (algebraic, complex). An involution $\sigma:X\rightarrow X$ is called non-symplectic if it acts trivially on $H^{2,0}(X)=\Bbb{C}\omega\_X$ $\ $ (where $\omega\_X$ is any nowhere vanishing 2-form), i.e. if $\sigma^\ast\omega\_X=-\omega\_X$.
Now, the Picard group can be identified with the latti... | https://mathoverflow.net/users/40038 | K3 surface with a non-symplectic involution: a basic question | A typical example is given by K3 surfaces of genus 2, that is, double coverings of $\Bbb{P}^2$ branched along a sextic plane curve $C$. Here $S(\sigma )$ is 1-dimensional (it is the pull back of $\mathrm{Pic}(\Bbb{P}^2)$), while $\mathrm{rank}(S\_X)$ can take all values from 1 to 20 -- for instance it is 20 when $C$ is... | 4 | https://mathoverflow.net/users/40297 | 149877 | 80,171 |
https://mathoverflow.net/questions/149895 | -2 | I am going through a sketch of the proof of Dixon's Theorem (the probability that two randomly chosen elements of A\_n generate A\_n -> 1 as n -> infinity) due to M. Liebeck and its underlying idea is that if two even permutations fail to generate A\_n, then they must both be contained in some maximal subgroup of A\_n.... | https://mathoverflow.net/users/42751 | Dixon's Theorem | If you have two elements of ${\rm A}\_n$ which do not lie both in any maximal subgroup of
${\rm A}\_n$, then they in particular do not lie both in *any* proper subgroup of ${\rm A}\_n$.
This in turn means that the only subgroup of ${\rm A}\_n$ which contains your elements is
${\rm A}\_n$ itself. Hence your two elements... | 2 | https://mathoverflow.net/users/28104 | 149896 | 80,178 |
https://mathoverflow.net/questions/149863 | 6 | I have a Heegaard diagram which produces a non-orientable 3-manifold. I want to know any 3-manifold invariant which can be calculated from Heegaard diagrams for non-orientable 3-manifold. (As far as I know, Roklin invariant or Heegaard Floer homologies are defined on oriented 3-manifolds, right?)
| https://mathoverflow.net/users/36445 | Is there "nonorientable Heegaard Floer homology"? | In general, any invariant of oriented manifolds is an invariant of non-orientable ones, since one can pass to the oriented double cover. Note that this cover admits an orientation-reversing diffeomorphism, so in some sense it shouldn't matter which orientation you choose. (This point should be treated with great care, ... | 7 | https://mathoverflow.net/users/3460 | 149899 | 80,179 |
https://mathoverflow.net/questions/149885 | 4 | Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category $\mathcal{E}^I$, the projective one (fibrations and weak equivalences are pointwise) and the injective one (cofibrations ... | https://mathoverflow.net/users/36625 | Factorization of morphisms in a diagram category | **It is not possible in the case $\mathcal{E} = \mathbf{sSets}$, $\mathcal{I} = (\cdot \to \cdot)$.**
To see this, note that since $\mathcal{I}$ is both an inverse and a direct category, the projective cofibrations (resp. injective fibrations) are exactly the Reedy cofibrations (resp. fibrations).
So your condition... | 5 | https://mathoverflow.net/users/2273 | 149909 | 80,184 |
https://mathoverflow.net/questions/149898 | 1 | I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper *[Partial flag varieties and preprojective algebras](http://arxiv.org/abs/math/0609138)* by Geiss, Leclerc and Schröer. One way of writing down such a cluster structure is to give an initial seed, and ind... | https://mathoverflow.net/users/21483 | Finding particular reduced words for Weyl group elements | I think you can use minimal coset representatives for this.
Each $w \in W(\Delta)$ can be uniquely written as $w\_K w^K$ where $w\_K \in W(\Delta\_K)$ and $l(w\_K) + l(w^K) = l(w)$. The element $w^K$ is called minimal left coset representative (and all of this works also for the opposite side) and the set of these i... | 6 | https://mathoverflow.net/users/6818 | 149916 | 80,185 |
https://mathoverflow.net/questions/149919 | 3 | **Definition 1:** A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions $\langle\mathbb{P}, <\_{\mathbb{P}}\rangle\in M,\langle\mathbb{Q}, <\_{\mathbb{Q}}\rangle\in N$ and $\mathbb{P}$-generic f... | https://mathoverflow.net/users/nan | Joint Forcing Extension Property | First, I note that the strong JFEP is the same as the medium JFEP, since two transitive sets such as $M[G]$ and $N[H]$ are isomorphic if and only if they are equal.
Next, note that the strong property is just too much.
**Theorem.** No consistent extension of ZF, which allows a model to be a forcing extension by a... | 5 | https://mathoverflow.net/users/1946 | 149920 | 80,186 |
https://mathoverflow.net/questions/149936 | 4 | Let $(\Omega,\Sigma,\mu)$ be a probability space. A $\mu$-atom is an $A\in\Sigma$ such that $\mu(A)>0$ and for all $B\in\Sigma$ such that $B\subseteq A$, either $\mu(B)=\mu(A)$ or $\mu(B)=0$ holds.
Now let $(X,\mathcal{X})$ be a measurable space with the property that for all $x,y\in X$ there is $B\in\mathcal{X}$ suc... | https://mathoverflow.net/users/35357 | Are measurable functions almost surely constant on atoms? | No.
Let $\Omega$ be an uncountable set, and $\Sigma$ the $\sigma$-algebra consisting of all countable and co-countable sets. Let $\mu$ be the probability measure assigning measure 0 to all countable sets, and measure 1 to all co-countable sets. Note every co-countable set is an atom. Let $(X, \mathcal{X}) = (\Omega, ... | 6 | https://mathoverflow.net/users/4832 | 149944 | 80,194 |
https://mathoverflow.net/questions/149946 | 4 | Discontinuous linear representations of $GL(n,\mathbb{C})$ can be obtained from the so-called "wild" (field) automorphisms of $\mathbb{C}$; but these wild automorphisms in turn require some choice to construct. Is it consistent with ZF that all linear representations of $GL(n,\mathbb{C})$ are continuous?
| https://mathoverflow.net/users/25028 | Discontinuous representations of GL(n,C) in ZF | It is consistent with $\sf ZF+DC$ that every homomorphism between Polish groups is continuous.
This holds in Solovay's model, whose existence requires an inaccessible cardinal, but also in Shelah's model where every set of real numbers have the Baire property -- something which does not require any additional consist... | 11 | https://mathoverflow.net/users/7206 | 149947 | 80,195 |
https://mathoverflow.net/questions/149937 | 0 | In T. Taos *Analysis 1* book, on page 26, we have a proposition that tells us that recursive definitions are actually well-defined.
>
> Proposition 2.1.16: *Suppose for each naturla number $n$, wh have some function $f\_n:\mathbb{N}\rightarrow\mathbb{N}$. Let $c$ be a natural number. Then we can assign a unique nat... | https://mathoverflow.net/users/43263 | An informal version of the recursion theorem | The proof of 2.1.16 uses an informal notion of induction. The induction axiom 2.5 says that
* if P(n) is a property, such that P(0) holds etc etc… then P(n) holds for all n.
Now the "proof" of 2.1.16 does not define any property P(n), so it does not use induction in a formal way.
In Exercise 3.5.12, on the othe... | 5 | https://mathoverflow.net/users/14915 | 149948 | 80,196 |
https://mathoverflow.net/questions/149959 | 4 | Let $\kappa$ be a (finite or infinite) cardinal. Assuming consistency of $\text{ZFC}$ (and probabely some additional assumptions) is the following consistent with $\text{ZFC}$?
There is a countable transitive model of $\text{ZFC}$ like $M$ and a partial order $\mathbb{P}$ in $M$ and $\mathbb{P}$-generic filter $G$ ov... | https://mathoverflow.net/users/nan | On the number of models between a ground model and its forcing extension | Let me answer the question you have asked, but then make a remark that there is another related question that perhaps you had intended to ask, which might be more interesting.
First, let me note that no uncountable cardinal $\kappa$ (uncountable in $V$) can arise in the way you have stated, where $M$ and $M[G]$ are ... | 5 | https://mathoverflow.net/users/1946 | 149963 | 80,199 |
https://mathoverflow.net/questions/149965 | 5 | Let $G$ be a connected graph on $n$ vertices and $\mathcal{T}$ be the set of all spanning trees of $G$.
Consider the graph whose vertices are the elements of $\mathcal{T}$
and
$T, T' \in \mathcal{T}$ are connected by an edge if $|E(T)\cap E(T')| = n - 2 $
Does this graph have a name?
One can see that the grap... | https://mathoverflow.net/users/23850 | Does this graph have a name? | I have seen this called the *Tree Graph* of $G$. It has been investigated by a lot of authors. For example Cummings proved in "Hamilton circuits in tree graphs" IEEE Trans. Circuit Theory,13(1966), pp.82-90, that this graph is Hamiltonian. Holzmann and Harary generalized this to tree graphs of matroids in ["On the Tree... | 7 | https://mathoverflow.net/users/2384 | 149968 | 80,203 |
https://mathoverflow.net/questions/149969 | 2 | I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability measures, such that the latter fails to be a perfectly normal topological space.
Almost all results start with $\Omega$ sep... | https://mathoverflow.net/users/43277 | When is a space of probability measures not perfectly normal? | One of the most natural topologies on spaces of probability measures on a compact space is the weak star topology. Suppose that $X$ is a compact Hausdorff space. Then a set of the form $f^{-1}[\{0\}]$ for some continuous $f:X\rightarrow\mathbb{R}$ is said to be a zero set. The Baire $\sigma$-algebra $\mathcal{M}$ on $X... | 2 | https://mathoverflow.net/users/22277 | 149979 | 80,207 |
https://mathoverflow.net/questions/149952 | 4 | If a PDE has a unique classical solution, must it have a unique viscosity solution?
The particular problem I am interested in is parabolic, but I would be interested in the general case.
A short answer would be good. An answer with references would be great!
| https://mathoverflow.net/users/32325 | If a PDE has a unique classical solution, must it have a unique viscosity solution? | The answer in general is no, because viscosity solutions are not suitable for every problem.
The viscosity solution idea based on an idea of solution comparison. If you look at the user's guide, Crandall-Ishii-Lions 1992, you see that for a problem of the form
$$
F(x,u,Du,D^2u)=0
$$
what they call the fundamental ass... | 8 | https://mathoverflow.net/users/40120 | 149984 | 80,208 |
https://mathoverflow.net/questions/149950 | 7 | For a seminar I am working on a Moreau-Yosida regularization in Banach spaces.
The regularization is defined by
$$f\_\lambda(x) := \inf \left \{ \frac{\|x-y\|^2}{2\lambda} +f(y) : y \in X \right \}, ~ x \in X$$
where $X$ is a reflexive and strictly convex Banach space, $f: X \rightarrow \mathbb{R} \cup \{+\infty\... | https://mathoverflow.net/users/43270 | Moreau-Yosida regularization in Banach spaces | I deleted a previous wrong and misleading answer.
The Moreau-Yoshida envelope is a special case of th infimal convolution of two convex functions $f$ and $g$ which is defined as
$$
f\Box g(x) = \inf\_{y\in Y} f(y) + g(x-y).
$$
In other words: The infimal convolution of $f$ and $g$ is the largest (extended) real value... | 7 | https://mathoverflow.net/users/9652 | 149989 | 80,210 |
https://mathoverflow.net/questions/149954 | 15 | It is a well-known fact that the generalized von Mangoldt function, defined by
$$\displaystyle \Lambda\_k(n) = \sum\_{d | n} \mu(d) \left(\log \frac{n}{d}\right)^k$$
vanishes whenever $n$ has more than $k$ distinct prime factors. I was able to prove this fact through a relatively lengthy and cumbersome combinatoria... | https://mathoverflow.net/users/10898 | On the vanishing of the generalized von Mangoldt function $\Lambda_k(n)$ when $n$ has more than $k$ prime factors | Write the Riemann zeta function as a product of its Euler factors
$\zeta (s)=\prod\_i E\_{i}(s)$. Repeated application of the Leibniz rule shows
$$\frac{\zeta^{(k)}(s)}{\zeta (s)}=\sum\_{i\_1+\cdots+i\_k=k}\sum\_{t\_1,\dots,t\_k}\frac{E\_{t\_1}^{(i\_1)}(s)}{E\_{t\_1}(s)}\cdots \frac{E\_{t\_k}^{(i\_k)}(s)}{E\_{t\_k}(s)}... | 19 | https://mathoverflow.net/users/2384 | 149990 | 80,211 |
https://mathoverflow.net/questions/150000 | 0 | I would like to know if there exists a relation between $ \mathrm{Hdg}\_k(X \bigcup Y ) $, $ \mathrm{Hdg}\_k(X) $, $ \mathrm{Hdg}\_k( Y) $ and $ \mathrm{Hdg}\_k(X \bigcap Y )$ ( short exact sequence, or direct sum or something like that ), such that $ \mathrm{Hdg}\_k ( X ) = H^{k,k} ( X ) \bigcap H^{2k} ( X , \mathbb{Q... | https://mathoverflow.net/users/43299 | Group of Hodge classes | I'll assume that $X$ and $Y$ are *open* subvarieties of $M$, so that this looks like a Mayer-Vietoris sequence; I think this is the only sensible interpretation of the question. Since $X$ and $Y$ are not going to be projective I also interepret $\newcommand{\Hdg}{\operatorname{Hdg}}\Hdg(-)$ in the sense of *mixed* Hodg... | 3 | https://mathoverflow.net/users/1310 | 150011 | 80,214 |
https://mathoverflow.net/questions/150009 | 12 | Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d\_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d\_p(x)$=the distance between $p$ and $x$). Let $\epsilon>0$ be an arbitrary positive number. Is there a smooth function $\tilde{d}\_p(x)$ on $M$, such that
$$ | d\_p(x)-\tilde... | https://mathoverflow.net/users/37354 | Smoothing of the distance function on a Riemannian manifold | You function $d\_p$ is a Lipschitz function (w.r.t. to the Riemannian distance) with the Lipschitz constant $1$ and it is possibly, for every $\varepsilon>0$, to $\varepsilon$-approximate it by a smooth function with the length of the gradient $\le 1 + \varepsilon$.
The proof of this statement, even if we replace 'R... | 10 | https://mathoverflow.net/users/14515 | 150030 | 80,219 |
https://mathoverflow.net/questions/149900 | 1 | I was wondering, have you ever seen a formula in the Riemannian (more specially Kahlerian but not essential) setting for the derivative $X \cdot |Ric|^2 = 2 g(\nabla\_X Ric, Ric)$ for a vector field $X$?
Or more in general $\nabla\_X Ric$.
Thank you
David
| https://mathoverflow.net/users/19545 | Derivative of (the length of) the Ricci tensor | The comments section was getting unwieldy, so I'll answer here. Hopefully this is helpful.
---
What I was trying to say is as follows: suppose that $(M,g,X)$ is a steady gradient soliton, i.e.
$$
\mathcal{L}\_X(g) = 2 Ric\_g
$$
for $X=\nabla f$ for some function $f$. Then, let $\Phi\_t$ denote the flow of $-X$. Y... | 2 | https://mathoverflow.net/users/1540 | 150035 | 80,220 |
https://mathoverflow.net/questions/150036 | 7 | Shoenfield's absoluteness states that if $M \subseteq N$ are models of $ZF$ and $M \supseteq \omega\_1^N$, then every $\Sigma^1\_2$ formula with parameters in $M$ is absolute between $M$ and $N$. In particular, $\Sigma^1\_2$ properties are preserved under generic extensions of the universe.
What I'm looking for are ... | https://mathoverflow.net/users/43322 | Failure of Shoenfield's Absoluteness | I'm turning my comment into an answer. With $\Sigma^1\_2$ statements we can discuss well-foundedness: A real codes a well-founded model of enough set theory iff it codes a model (which is an arithmetic statement) and the model is well-founded (this you can express by saying that no sequence through the ordinals of the ... | 10 | https://mathoverflow.net/users/6085 | 150041 | 80,221 |
https://mathoverflow.net/questions/150039 | 2 | I am wondering whether the region $H:=\{(t,x):x^2−t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g\_H$ of the standard Minkowski metric $g=−\mathrm{d} t\otimes \mathrm{d}t + \mathrm{d}x \otimes \mathrm{d} x$, is globally conformally equivalent to the vertical strip $S:=\{(t,x):|x|<1... | https://mathoverflow.net/users/43324 | Global conformal equivalence of two regions of Minkowski spacetime | In $H$ you have two lightlike geodesics such that every lightlike geodesic intersects one of these two, namely the geodesics $\{x=y\}$ and $\{x=-y\}$. In $S$ you do not have such two geodesics. Hence, the regions are not conformally equivalent
| 3 | https://mathoverflow.net/users/14515 | 150043 | 80,223 |
https://mathoverflow.net/questions/148832 | 1 | I'm looking for a method to efficiently compute a numerical approximation of
$$F^n\_D(x\_1,\ldots,x\_n) = \sum\_{m=0}^{\infty} \sum\_{i\_1 +\ldots+i\_n=m}\frac{(a)\_{m}(b\_1)\_{i\_1}\ldots (b\_n)\_{i\_n}}{(c)\_{m}i\_1!\ldots i\_n!}x\_1^{i\_1}\ldots x\_n^{i\_n}$$
I'm restricted to the cases where
$$\left\{\begin{arr... | https://mathoverflow.net/users/8737 | Fast numerical approximation of Lauricella series of the fourth kind for real variables and real parameters | I'm not an expert on this, but did you try the software and article here:?
It uses up to 2nd-order Laplace's approximation for the integrals.
<http://faculty.smu.edu/rbutler/Lauricella.zip>
Best, Alex
| 1 | https://mathoverflow.net/users/40437 | 150055 | 80,226 |
https://mathoverflow.net/questions/150051 | 5 | Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X\_1, \ldots, X\_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality $k$, and take $A\_{i,j} = 1$ if $j \in X\_i$, $0$ otherwise. What can be said about the probability $P(k,m,n)$ that $A... | https://mathoverflow.net/users/13650 | Full-rank rectangular matrices over GF(2) | Actually, something a bit stronger is true: For fixed $k\geq 3$ the threshold is sharp in the sense that above $t(k)$ the probability that $A$ is of full rank tends to $1$. In computer science this question has received a fair amount of study under the name of "random XOR-SAT". The existence of a threshold was apparent... | 9 | https://mathoverflow.net/users/405 | 150058 | 80,228 |
https://mathoverflow.net/questions/150049 | 7 | Let $G\_1,G\_2$ be two Gromov-hyperbolic groups. If $G\_1\times \Bbb{Z}$ is quasi-isometric to $G\_2\times \Bbb{Z}$, must it be true that $G\_1$ is quasi-isometric to $G\_2$?
This is similar to the classic topology problem where crossing two non-homeomorphic spaces with $\Bbb{R}$ can create two homeomorphic spaces wh... | https://mathoverflow.net/users/43328 | Quasi-isometric rigidity of certain products of groups | See Kapovich and Leeb *[On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds](http://www.math.ucdavis.edu/%7Ekapovich/EPR/KL_1995.pdf)*, or Kapovich, Kleiner and Leeb
*[Quasi-isometries and the de Rham decomposition](http://www.math.ucdavis.edu/%7Ekapovich/EPR/kkl.pdf)*. This result can b... | 9 | https://mathoverflow.net/users/21684 | 150062 | 80,229 |
https://mathoverflow.net/questions/150052 | 12 | Let $X$ be a collection of cardinals such that if $\kappa,\lambda\in X$ and $\kappa<\lambda$, then there is a non-trivial elementary embedding $j:V\_{\kappa+1} \to V\_{\lambda+1}$ with $crit(j) = \kappa$.
What is the consistency strength of the claim that there are such $X$ for various sizes of $X$? In particular, w... | https://mathoverflow.net/users/17968 | What is the strength of chains of 1-extendibles? | First, let me point out that having a chain of $3$ such cardinals, with elementary embeddings $$V\_{\kappa+1}\to V\_{\lambda+1}\to V\_{\eta+1},$$ already implies that $\kappa$ is $1$-inaccessible with target $\lambda$ in $V\_{\eta+1}$, and so by elementarity $\kappa$ will be $1$-extendible with arbitrarily large target... | 11 | https://mathoverflow.net/users/1946 | 150064 | 80,230 |
https://mathoverflow.net/questions/150014 | 7 | The classical PL [Reidemeister Theorem](http://mathworld.wolfram.com/ReidemeistersTheorem.html) reads:
>
> **Reidemeister Theorem**: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by [Reidemeister moves](http://en.wikipedia.org/wiki/Reidem... | https://mathoverflow.net/users/2051 | Is there a combinatorial version of PL ambient isotopy in dimension $>3$? | Unfortunately, this is not quite an answer. So for knotted surfaces in 4-space, there is Roseman's Theorem. I think that the context of Dennis's proof is in the smooth category. Or certainly the proof that I know depends on a smooth structure.
I would imagine that the proof could be cranked up to PL-locally flat, but ... | 3 | https://mathoverflow.net/users/36108 | 150074 | 80,232 |
https://mathoverflow.net/questions/150067 | 5 | In Mark Hovey's article *Model category structures on chain complexes of sheaves* ([arXiv:math/9909024](http://arxiv.org/abs/math/9909024)) a model structure on the category $Ch(A)$ of unbounded chain complexes for a Grothendieck abelian category $A$ is constructed. This is called the *injective structure* and has the ... | https://mathoverflow.net/users/43339 | Is the injective structure on unbounded chain complexes simplicial? | It is instructive to consider a simpler case, namely that of chain complexes over a ring. Hovey deals with this example in great detail in his book *Model Categories.* In particular, on page 114 Hovey states that $Ch(R)$ is not a simplicial model category. In algebraic situations like this, it's not really the right qu... | 7 | https://mathoverflow.net/users/11540 | 150077 | 80,235 |
https://mathoverflow.net/questions/150073 | 11 | Over Q, the definite quaternion algebras with a unique conjugacy class of maximal orders, i.e. "with class number one", are those with discriminant 2,3,5,7, and 13.
Three questions:
1. What is a reference for this result?
2. What are some examples of definite quaternions of class number one, over other totally real... | https://mathoverflow.net/users/29980 | Which quaternion algebras have class number one? | For the first question, the statement goes back to Brzezinski (<http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1995__7_1/JTNB_1995__7_1_93_0/JTNB_1995__7_1_93_0.pdf>), who treats all definite quaternion orders over $\mathbb{Z}$, not just maximal ones. For maximal orders, this is an almost-immediate consequence of Eichler'... | 15 | https://mathoverflow.net/users/4433 | 150086 | 80,240 |
https://mathoverflow.net/questions/150088 | 4 | The concept of elementary embedding is very important in the definition of several large cardinals ideas. The usual model-theoretic definition can not be expressed for some of these ideas within $\text{ZFC}$. There are various tricks to give satisfactory formalization of these ideas. My question is what are the usual w... | https://mathoverflow.net/users/43354 | Formalizing Elementary Embeddings and Substructures in ZFC | Proposition 5.1 in Kanamori's Higher Infinite provides the answer to one major part of your question:
If $j:M\_1\to M\_2$ is a $\Sigma\_1$-elementary embedding between inner models,
then it is an elementary embedding.
Hence you can define in ZFC what elementary embeddings are, at least between inner models,
but tha... | 6 | https://mathoverflow.net/users/7743 | 150091 | 80,242 |
https://mathoverflow.net/questions/150026 | 1 | Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR transformation is then
$$
\mathfrak{cr}(D, J) = \{X \in \Gamma(T M): [X, D] \subseteq D, L\_X J = 0 \}
$$
where the Lie... | https://mathoverflow.net/users/19545 | A subspace of the algebra of infinitesimal CR automorphisms | An even stronger statement is true: If $[X,D]\subset D$ and $X$ belongs to $\Gamma(D)$, then $X = 0 $. The reason is that, because $D$ is a contact $2n$-plane field (since your CR structure is strictly pseudoconvex), if $X$ were nonzero, there would have to be another vector field $Y\in\Gamma(D)$ such that $[X,Y]$ does... | 2 | https://mathoverflow.net/users/13972 | 150092 | 80,243 |
https://mathoverflow.net/questions/150008 | 7 | Throughout this question, the following notation holds: Let $q$ be a power of a prime $p$, and let $d>4$ be a positive integer. Let $G$ be a finite group with a normal subgroup $E$ which is an elementary-abelian $p$-group, and such that $G/E\cong\mathrm{SL}\_d(q)$.
I am interested in the situation where this extensi... | https://mathoverflow.net/users/801 | On non-split extensions of $\mathrm{SL}_d(q)$ | I will try and answer Question 2, although I only have a superficial knowledge of the representation theory involved. I claim that, when $|E|=q^d$, the induced module action of ${\rm SL}\_d(q)$ on $E$ must either be trivial, or it must be quasi-equivalent to the action on the natural module. (Quasi-equivalent means equ... | 7 | https://mathoverflow.net/users/35840 | 150094 | 80,244 |
https://mathoverflow.net/questions/149987 | 5 | The following notion is introduced by Assaf Rinot:
**Definition.** A singular cardinal $\kappa$ is a prevalent singular
cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with $|\mathbb{A}| = \kappa$
and $sup\{|A| : A\in \mathbb{A}\} < \kappa$ such that any $B\subset \kappa$ with
$|B| < cf(\kappa)$ is c... | https://mathoverflow.net/users/11115 | Prevalent singular cardinals hypothesis | A singular cardinal $\lambda$ is prevalent iff there exists some cardinal $\mu<\lambda$ such that $Cov(\lambda,\mu,cf(\lambda),2)=\lambda$. In his solution of the pcf conjecture, Gitik has constructed a model where there exists a singular cardinal $\lambda$ of cofinality $\aleph\_1$ such that $pp(\mu)>\lambda$ for cofi... | 5 | https://mathoverflow.net/users/20033 | 150106 | 80,248 |
https://mathoverflow.net/questions/150104 | 5 | If $G$ is a finite simple group then is it true that an abelian subgroup $H$ of $G$ of maximal order has order $|H| < |G|^{\frac{1}{3}}$? If so, could you please point me to a reference for this, or where a proof is given? Secondly, if $G$ is now a finite nilpotent group of class $2$ and $H$ is again an abelian subgrou... | https://mathoverflow.net/users/15825 | How large can abelian subgroups of class 2 nilpotent groups or simple groups be? | The answer to both questions is *no*:
Counterexample to first assertion: $G = {\rm A}\_5$, $H = \langle (1,2,3,4,5) \rangle$.
Counterexample to second assertion: $G = \langle (1,2,3,4), (1,3), (5,6) \rangle$, $H = \langle (1,2,3,4), (5,6) \rangle$.
| 6 | https://mathoverflow.net/users/28104 | 150107 | 80,249 |
https://mathoverflow.net/questions/149762 | 5 | I've recently encountered the following cobordism theory modulated by a class $\sigma \in H^{d+1}(B^2\mathbb{Z}/2,U(1))$.
My objects are $d$-dimensional spin manifolds with chosen spin structure. Note that a spin structure, thought of as a null-homotopy of $w\_2$, gives a trivialization of $w\_2^\*\sigma$.
My morp... | https://mathoverflow.net/users/36591 | Cobordism modulated by a cohomology operation | To bring this into a classical setting, it seems to me that one can use the bordism groups $\Omega\_d^{\sigma}$ where both d-manifolds and (d+1)-bordisms are not necessarily spin, but have only a trivialization of $w\_2^\*\sigma$.
Then there is a natural forgetful map $\Omega\_d^{Spin}\to \Omega\_d^{\sigma}$ with image... | 5 | https://mathoverflow.net/users/36799 | 150108 | 80,250 |
https://mathoverflow.net/questions/150109 | 1 | I'm interested in a concrete example of an infinite metabelian quotient of the free product $C\_2 \* C\_2 \* C\_2$, where $C\_2$ is the cyclic group of order $2$. In particular, I would be interested in a homomorphism onto a wreath product of abelian groups. How would this look like on the standard generating set $\lef... | https://mathoverflow.net/users/23232 | Infinite metabelian/nilpotent quotients of $C_2 * C_2 * C_2$ | Perhaps the easiest metabalian quotient of $C\_2\ast C\_2\ast C\_2$ is $C\_2\ast C\_2 \cong \mathbb{Z}\rtimes C\_2$ the infinite dihedral group.
| 5 | https://mathoverflow.net/users/3041 | 150112 | 80,251 |
https://mathoverflow.net/questions/150099 | 0 | Let $G=GL(n,q)$ be a generalized linear group whose center is a cyclic group of prime order $p$. Does there always exists an element $x\in G$ such that $C\_G(x)$ is a group of exponent $p$?
| https://mathoverflow.net/users/40723 | Centralizers of p-elements in general linear groups | As I mentioned in comments, either $q=3$ or $q$ is even.
**Suppose that $q>n$**. Then $G$ contains a regular semisimple element $x$ with eigenvalues in $\mathbb{F}\_q$, then its centralizer is a maximal split torus of $G$, which is elementary abelian of order $(q-1)^n$. Thus the answer is **YES** in this case.
**S... | 2 | https://mathoverflow.net/users/801 | 150116 | 80,254 |
https://mathoverflow.net/questions/150133 | 11 | Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to $H$? I am only interested in the case when $H$ is a continuous subgroup, and "locally isomorphic" means has the same Li... | https://mathoverflow.net/users/38654 | Realizing a subgroup of a Lie group as a stabilizer subgroup | You need $H$ to be closed.
The Mostow-Palais theorem (1,2,3) then gives what you want -- with "equal" in place of "locally isomorphic", but with a possibly reducible representation. I'm not aware of conditions ensuring that the representation can be chosen irreducible.
(1): <http://en.wikipedia.org/wiki/Mostow-Palais... | 9 | https://mathoverflow.net/users/19276 | 150139 | 80,262 |
https://mathoverflow.net/questions/149411 | 1 | Let $\mathcal{M}\_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}\_g$ whose fiber over a curve $C\in\mathcal{M}\_g$ is the space of holomorphic $k$-differentials $H^0(C,K\_C^k)$. This is often called the Hodge bundle.
My question is, **is there a nat... | https://mathoverflow.net/users/17294 | Connections on the Hodge bundle? | The fourth power of the Hodge bundle is isomorphic to the $E\_8$ conformal block bundle at level one, and for that bundle (or rather its projectivization) you have the Hitchin/KZ/WZW connection, which is projectively flat.
| 1 | https://mathoverflow.net/users/940 | 150140 | 80,263 |
https://mathoverflow.net/questions/150142 | 11 | Let $A$ a set with $|A|=n$ that contains only perfect squares of integers.
What lower bounds can we give for $|A+A|$?
I think the lower bound $\gg \frac{n^2}{\sqrt{log \,n}}$ holds (this would be the best bound possible, with "equality" for $A=\{1^2,2^2,...n^2\}$). However, this estimate seems really hard.
I could... | https://mathoverflow.net/users/43383 | Lower bounds for $|A+A|$ if $A$ contains only perfect squares | This is a well-known (and difficult) problem.
The current record is $|A+A| \geq \log(|A|)^{c\log\log(|A|)}|A|$ [due to Schoen](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1304429491) (in 2011), using his near optimal form of Freiman's theorem. Note this is just shy of a power ... | 17 | https://mathoverflow.net/users/630 | 150147 | 80,266 |
https://mathoverflow.net/questions/150157 | 3 | If $X$ and $Y$ are smooth projective varieties, $p: X \times Y \to X$ and $q: X \times Y \to Y$ are the projections, and $\mathcal{P}$ is an object in $D^b(X \times Y)$, then the Fourier--Mukai transform associated to $\mathcal{P}$ is a functor $\Phi\_\mathcal{P}: D^b(X) \to D^b(Y)$ that sends $\mathcal{E}^\bullet$ to ... | https://mathoverflow.net/users/20391 | Fourier--Mukai transforms and adjunction | A pair of functors $(F,G)$ which are both left and right adjoints of each other is called a *Frobenius pair*. If you google that you'll find plenty of literature. The canonical example is that the induction and restriction functors in the representation theory of finite groups form a Frobenius pair. This also explains ... | 5 | https://mathoverflow.net/users/1310 | 150160 | 80,270 |
https://mathoverflow.net/questions/150167 | 2 | I am unable to find the MO comments about the first use of the phrase "fat slags" in an article. [On page 26 of this](http://arxiv.org/abs/math/0104196) we find "these correspond to thickenings of the
corresponding special Lagrangian (giving fat SLags, as they are known in Britain,
or multiply-wrapped cycles in physics... | https://mathoverflow.net/users/3324 | Special Lagrangians and fat | This answer and its comments, is the reference you are searching for:
<https://mathoverflow.net/a/22384/2051>
The paper in which the term appears appears is <http://arxiv.org/abs/math.DG/0104196>
| 3 | https://mathoverflow.net/users/2051 | 150172 | 80,273 |
https://mathoverflow.net/questions/149998 | 21 | Let $p$ be a prime number and $P=\{1,2,...,p-1\}$
In how many ways we can sum *all* the elements of $P$ in such a way that we will reach a multiple of $p$
only when we sum the last summand?
For example let $p=7$ .
Clearly, $1+2+3+4+5+6$ is such a sum (In fact there exist $408$ such sums)
but $2+3+5+4+1+... | https://mathoverflow.net/users/38851 | Avoiding multiples of $p$ | As I mentioned in the comments above, the number of permutations of elements $1,2,\dots,p$ (i.e., including $p$) is just by factor of $p-2$ larger than the amount in question (in fact, this is true for any odd $p$). Such permutations are now counted in <http://oeis.org/A232663>
Here is an explicit formula for the num... | 3 | https://mathoverflow.net/users/7076 | 150193 | 80,283 |
https://mathoverflow.net/questions/142548 | 18 | Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int\_{R} \frac{{\xi}^\prime(z)}{\xi (z)} dz$ over appropriate rectangle(s) in the critical strip. This still seems difficult, however, without a computer.
| https://mathoverflow.net/users/8435 | How did Riemann calculate the first few non-trivial zeros of the zeta-function? | In searching through the Riemann Nachlass in Gottingen (including those
folders not listed as connected with $\zeta(s)) $ there is no
evidence -- at least that has been saved -- that Riemann computed
anything more than the first few zeros (I think up to ordinate about 80).
The method he used was the expansion that i... | 13 | https://mathoverflow.net/users/43406 | 150196 | 80,285 |
https://mathoverflow.net/questions/150198 | 1 | Theorem (Rellich). Let $\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$ be a Hermitian matrix function that depends
on $t$ analytically.
**(i)** The $n$ roots of the characteristic polynomial of $\boldsymbol{A}(t)$ can be arranged so that each root $\lambda\_j(t)$ for $j = 1,\cdots,n$ is an analytic... | https://mathoverflow.net/users/38974 | Is Rellich's function valued theorem valid for a rank defficient function valued matrix? | Yes, Rellich's theorem does not require the eigenvalues to be distinct. See e.g. Reed and Simon, "Methods of modern mathematical physics vol. 4: Analysis of Operators", Chapter XII (in particular Problems XII.16 and XII.17).
| 1 | https://mathoverflow.net/users/13650 | 150201 | 80,286 |
https://mathoverflow.net/questions/150224 | -1 | Could you tell me an example to an $(X,\varrho)$ metric-space with balls $B(x\_1,r\_1)$ and $B(x\_2,r\_2)$ where $r\_1<r\_2$ but also $B(x\_2,r\_2)\subset B(x\_1,r\_1)$?
| https://mathoverflow.net/users/43421 | Metric-space with a ball inside a smaller ball | A slightly simpler, 1-dimensional version of Alexandre's example: On the closed half-line $[0,\infty)$, the ball of radius 3 around 0 (i.e., the interval $[0,3)$) equals the ball of radius 2 around the point 1. For a proper inclusion of balls, shrink 3 or enlarge 2 slightly.
| 3 | https://mathoverflow.net/users/6794 | 150231 | 80,296 |
https://mathoverflow.net/questions/150222 | 5 | According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L\_{AV}\subset X$ is defined by the equations
$\sum\_{i=1}^{k+3}l\_i^1|X\_i|^2=c\_1$, $\sum\_{i=1}^{k+3}l\_i^2|X\_i|^2=c\_2$, $\sum\_{i=1}^{k+3}\phi\_i=c\_3$
... | https://mathoverflow.net/users/13244 | Question about the Aganagic-Vafa A-brane | I believe that the special Lagrangians that Aganagic-Vafa want to consider are contained in special fibres of the Harvey-Lawson fibration. First, we had better
take $k=0$ in your equations, since otherwise the three equations will give a subspace which is of too high dimension to be a Lagrangian. (But these equations d... | 6 | https://mathoverflow.net/users/23917 | 150239 | 80,300 |
https://mathoverflow.net/questions/150215 | 5 | I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also discovered several multiply-perfect numbers ($n$ such that $\sigma(n)=kn$ for some integer $k\geq 2$). No odd multiply-perfect ... | https://mathoverflow.net/users/40984 | Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even? | The earliest reference seems to be from 1966: E.A. Bugulov, *On the question of the existence of odd multiperfect numbers*, Kabardino-Balkarskaya State University Učen. Zap. **30** (1966) 9-19. [I could not find this article online.]
Bugulov showed that an odd multiperfect number must have at least 11 distinct prime... | 5 | https://mathoverflow.net/users/11260 | 150240 | 80,301 |
https://mathoverflow.net/questions/150207 | 8 | I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine algebraic group over $k$ and let $X$ be an affine $k$-variety. Then it's a basic fact that orbits $Gx$ are locally closed su... | https://mathoverflow.net/users/43412 | Orbits of group scheme action | Presumably you meant to assume the schemes are finite type over $k$. To work naturally with orbit questions for such schemes one just has to bring in appropriate use of flatness to adapt intuition and experience from the traditional smooth setting over algebraically closed fields (e.g., one uses the robust theory of qu... | 13 | https://mathoverflow.net/users/43107 | 150244 | 80,302 |
https://mathoverflow.net/questions/150168 | 25 | This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent sheaves be constructed categorically from the category of vector bundles? I am thinking of Coh being some kind of 'abelian... | https://mathoverflow.net/users/36922 | is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles? | Here are a few comments that might be useful. I don't think there is a chance that this can work unless the scheme in question has the resolution property (meaning every coherent sheaf is a quotient of a locally free sheaf of finite rank). Otherwise the category of locally free sheaves does not even form a generator of... | 17 | https://mathoverflow.net/users/1649 | 150248 | 80,303 |
https://mathoverflow.net/questions/150249 | 4 | If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n variables out of a linear combination of conjugacy classes of maximal orders in A.
When A has class number one, there is a sin... | https://mathoverflow.net/users/29980 | What does the Jacquet-Langlands correspondence say about quaternion algebras of class number one? | The Jacquet-Langlands correspondence in the case of a totally definite quaternion algebras over a strict class number one field, parallel weight $2$ gives a Hecke equivariant map from the space of (set theoretic) maps from the set of left ideal classes of an Eichler order $\mathcal{O}$ to $\mathbb{C}$ that are *orthogo... | 6 | https://mathoverflow.net/users/40821 | 150255 | 80,305 |
https://mathoverflow.net/questions/150258 | 2 | Let $R$ be a commutative ring and $M$ a not necessary finitely presented $R$-module. I am looking for a prove or a counterexample to the following statement: $M$ is flat as an $R$-module if and only if $\mathrm{Tor}^R\_1(R/I,M)=0$ for all radical ideals $I$.
| https://mathoverflow.net/users/41973 | Checking flatness using radical ideals | If you're not assuming $R$ is Noetherian, this is false.
Namely, let $R$ be a valuation domain with value group $\mathbb Q$, let $m$ be its maximal ideal, and let $M=k=R/m$. Then the only radical ideals of $R$ are $0$ and $m$, and one can show that $\operatorname{Tor}\_1^R(R/I,k) = 0$ for $I=0, m$, but it is also cle... | 3 | https://mathoverflow.net/users/19045 | 150261 | 80,307 |
https://mathoverflow.net/questions/150253 | 13 | The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. But the K-theoretic proof has the advantage that it proves Atiyah-Singer not only for Dirac-type operators, but much more ... | https://mathoverflow.net/users/13356 | Atiyah-Singer for pseudodifferential operators via heat kernel? | The answer is no. Even for the differentil elliptic operator, the heat kernel method can not give the result.
In the heat method proof, we use McKean-Singer formula,
$$\mathrm{Ind} D= \mathrm{Tr} \left[e^{-tD^\*D}-e^{-tDD^\*}\right]=\int\_M \mathrm{Tr} [p\_t(x,x)-q\_t(x,x)]dx,$$
where $p\_t(x,y), q\_t(x,y)$ are the h... | 16 | https://mathoverflow.net/users/16326 | 150265 | 80,308 |
https://mathoverflow.net/questions/147275 | 0 | Convergence of a sequence of sections of a bundle is defined as follows:
**Definition:** Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on $TM$. Let $Ω ⊂ M$ be an open set with compact closure $\bar{Ω}$ in $M$, and let $(ξ\_k)$ be a
sequence of sections of ... | https://mathoverflow.net/users/32817 | On the definition of convergence of a sequence of sections of a bundle | This is just to supply some details to what Rafe Mazzeo wrote.
Let $g\_{i}$ be metrics and $^{\left( i\right) }\nabla$ be connections on $E$
and on $M$ for $i=1,2$. Since $\bar{\Omega}$ is compact,
the uniform equivalence of norms reduces to a local coordinate chart
$(U,\{x^{i}\})$ over which the bundle $E$ is triviali... | 1 | https://mathoverflow.net/users/nan | 150268 | 80,309 |
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