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https://mathoverflow.net/questions/118609 | 4 | Hallo,
Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{\*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure $J$ and a Kähler form $-i\partial \overline{\partial} \phi$ such that $M$ is a Lagrangian manifold and $\phi = d\phi =0... | https://mathoverflow.net/users/22073 | Uniqueness of Kähler form with same volume | Your question doesn't really refer to the metric $g$ as part of the data, so I'll ignore it. What you really seem to be asking is this: Let $X$ be a complex $n$-manifold and let $M\subset X$ be a compact, analytic, totally real $n$-dimensional submanifold of $X$. Suppose you have two Kähler forms $\Omega\_1$ and $\Omeg... | 3 | https://mathoverflow.net/users/13972 | 119559 | 67,375 |
https://mathoverflow.net/questions/119548 | 3 | Given a $K\times M$ matrix $X$, where $M\gg K$, comprising independent complex Gaussian random variables, each one with mean
$$E[X\_{k,m}]=B\_{k,m}$$
and variance
$$Var[X\_{k,m}]=\Sigma\_{k,m}$$
define the random matrix $R(X)$ as
$$R(X)=I +XX^{H}.$$
My problem is now to compute the expected value of $R^{-1}(X)$, i.e... | https://mathoverflow.net/users/30847 | Expectation of random matrix inverse | I'm not sure that I understand well the notations, but:
(1) If $H$ is the transpose, then $XX^H$ is a Wishart matrix, its asymptotic law with $M,N\to\infty$ and $M/N\simeq t>0$ fixed is the Marchenko-Pastur law.
(2) If in addition $R^{-1}(X)=R(X)^{-1}$, then you can recover the law of $R^{-1}(X)$ from that of $XX^H... | 1 | https://mathoverflow.net/users/29333 | 119566 | 67,378 |
https://mathoverflow.net/questions/119537 | 3 | Lets say 2 players A and B try to have the most money at the end after playing a casino game in which they have a $49\%$ chance to double a wager.
Here are the rules to the bet between A and B:
* Both start with $100
* Player A goes first. Player A plays the casino game as many times as he wants then decides to sto... | https://mathoverflow.net/users/30842 | game theory - coin flipping question | I'll ignore the granularity of money. With arbitrarily small bets allowed, player B only has to reach the same amount as A to win with probability arbitrarily close to $1$. So, let's assume that player B wins if the totals are equal.
The simple strategy for player A of betting everything once is pretty good, but not... | 4 | https://mathoverflow.net/users/2954 | 119568 | 67,380 |
https://mathoverflow.net/questions/119561 | 5 | Clearly Very important results in Math require the Axiom of choice, for example "any vector space has a base". But in the absence of AC (i.e., only in ZF) it is possible that a vector space has no basis.
In another direction **append the negation of AC to ZF**. What happens to algebra or analysis now ?
If you know ... | https://mathoverflow.net/users/nan | Mathematics with the negation of AC | The negation of the axiom of choice only allows us to prove that there is *some* set which cannot be well-ordered. There is *some* family of non-empty sets whose product is empty. There is *some* partially ordered set in which every chain is bounded, but there is no maximal element. And so on.
Of course, from a famil... | 14 | https://mathoverflow.net/users/7206 | 119569 | 67,381 |
https://mathoverflow.net/questions/119494 | 7 | Next week I am going to teach two lessons on induction to very motivated students from high schools. At some point I would like to talk about ordered sets, well-ordered sets, and mention the fact that induction works on well-ordered sets. More precisely, I will prove the following formulation of induction: let $S$ be a... | https://mathoverflow.net/users/6206 | Examples of "exotic" induction | Goodstein's Theorem
<http://en.wikipedia.org/wiki/Goodstein%27s_theorem>
is proved using an induction of length $\epsilon\_0$.
| 8 | https://mathoverflow.net/users/15819 | 119579 | 67,384 |
https://mathoverflow.net/questions/119578 | 4 | I'm wondering why (and therefore also if) the notions of "a projective variety/submanifold of projective space is a complete intersection" as used in algebraic geometry and the theory of, say, Riemann surfaces agree.
The following are the precise versions of these notions I refer to:
In algebraic geometry:
An ... | https://mathoverflow.net/users/27272 | Complete intersections in complex and algebraic geometry | $\star$ By Thm. I.5.1, page 32 of Hartshorne the Jacobian condition is equivalent to that every local ring of $A$ is a regular local ring, i.e., that $A$ is a non-singular variety.
From AG to CG:
This is (probably) the easy direction: Let the coordinate functions of $f$ be the generators of the ideal $I(A)$. By $\... | 8 | https://mathoverflow.net/users/10076 | 119583 | 67,386 |
https://mathoverflow.net/questions/119543 | 9 | Let $F\_m$ be the free group with $m$ generators $S:=\{x\_1,\dots, x\_m\}$. I am interested in the following quantity
$$
F(n):=\frac{|\{w\in [F\_m,F\_m]: \|w\|\_S\leq n\}|}{|B\_S(n)|}
$$
where $B\_S(n):=\{w\in F\_m: \|w\|\_S\leq n\}$ and $\|w\|\_S$ is the word metric of $w$, and $[F\_m,F\_m]$ is the commutator subgrou... | https://mathoverflow.net/users/8419 | Commutators in free groups | The question is indeed very close to the one about the return probabilities on the abelian group $\mathbb Z^m$. Still, as Benjamin has noticed, there is some difference, since here one has to consider the non-backtracking paths only.
There is a standard way to deal with it which is based on the so-called **random wa... | 11 | https://mathoverflow.net/users/8588 | 119589 | 67,387 |
https://mathoverflow.net/questions/119593 | 3 | It is known that the Paley graph $P(q)$ for $q = 5, 9, 13$ or $17$ vertices are the only strongly regular graph with the parameters as $P(q)$.
If $q \geq 25$, is the following assertion true:
There are strongly regular graphs $G$ with the same parameters as $P(q)$ such that $G\not\cong P(q)$.
| https://mathoverflow.net/users/19075 | Strongly regular graphs with the same parameters as Paley graph | If $p$ is a prime congruent to 3 (mod 4) and $q$ is an even power of $q$, there are the
Peisert graphs which are arc-transitive, self-complementary conference graphs, not isomorphic to Paley graphs.
More generally, start with the affine plane over $GF(q)$ where $q$ is odd, with point set
$GF(q)\times GF(q)$. Choose a... | 3 | https://mathoverflow.net/users/1266 | 119595 | 67,388 |
https://mathoverflow.net/questions/119594 | 2 | It is well-known that the Paley graph $P(q)$ is a strongly regular graph with parameters
$(4t+1,2t,t-1,t)$. Suppose that $v$ is a vertex in the Paley graph $P(q)$. Suppoe that $C$ is the set of all neighbours of $v$ and $D$ be the complement of $C$.
If one applies Godsil-Mckay switching, then under what conditions
t... | https://mathoverflow.net/users/19075 | Godsil-Mckay switching applied on the Paley graph | If you apply what I call local switching to $C$, then the graph that results can be obtained by Seidel switching as follows. Let $Y$ be the Paley graph with an extra isolated vertex.
Apply Seidel switching to set $C$, to get a new graph $Y'$. Then $Y'$ still has an isolated vertex, the remaining vertices induce a graph... | 3 | https://mathoverflow.net/users/1266 | 119597 | 67,389 |
https://mathoverflow.net/questions/119608 | 3 | Cover a table with a tablecloth, crumple it up in the middle (while still leaving the edges hanging over the edge of the table), then stab the folds with a pin. You will almost surely poke an odd number of holes in the tablecloth, because one end of the pin is above the tablecloth, the other is below, and each hole ind... | https://mathoverflow.net/users/21816 | Does this result exist in the literature? | Of course there are results implying such things in degree theory and as @unknown (google) mentions.
Our "teaching" proof of Sperner's Lemma here <http://link.springer.com/article/10.1007%2Fs00199-007-0257-0> develops this intuition for the purpose of pedagogy. So the connection between what you are talking about and... | 15 | https://mathoverflow.net/users/26674 | 119612 | 67,393 |
https://mathoverflow.net/questions/119481 | 20 | This question was originally posted on [math.SE](https://math.stackexchange.com/questions/136821/can-the-n-string-sphere-braid-group-embed-in-to-the-n1-string-sphere-braid-gr) by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was m... | https://mathoverflow.net/users/21271 | Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group? | **Revision:** For $n>6$, there is no embedding of $\mathcal{S}\_n \hookrightarrow \mathcal{S}\_{n+1}$.
First, recall that there is an extension $\mathbb{Z}/2\mathbb{Z} \to \mathcal{S}\_n \to Mod(S\_{0,n})$, where $Mod(S\_{0,n})$ is the (orientation preserving) mapping class group of the $n$-punctured sphere.
Insid... | 18 | https://mathoverflow.net/users/1345 | 119616 | 67,395 |
https://mathoverflow.net/questions/119604 | 15 | $\newcommand\Sq{\mathrm{Sq}}$I am trying to compute $ [\mathbb{HZ}/4,\mathbb{HZ}/4 ]$ the mod 4 Steenrod algebra. For some reason, I need to work it out till dimension 6 or so. My approach is to use the cofiber sequence
$\mathbb{HZ}/4 \to \mathbb{HZ}/2 \xrightarrow{\Sq^{1}}\Sigma \mathbb{HZ}/2 $
twice.
I did the ... | https://mathoverflow.net/users/19186 | Computation of mod 4 Steenrod algebra [ HZ/4, HZ/4] | Let me write $H$ for $H\mathbb{Z}/2$ and $X$ for $H\mathbb{Z}/4$. In these terms we can compute $H^\*(X) = [H\mathbb{Z}/4,H\mathbb{Z}/2]$; it is isomorphic to $A/Sq^1 \oplus \Sigma A/Sq^1$, where $A$ is the Steenrod algebra.
Let's take the sledgehammer to the walnut and hit this with the Adams spectral sequence, and ... | 16 | https://mathoverflow.net/users/360 | 119624 | 67,399 |
https://mathoverflow.net/questions/119627 | 6 | **Background**: A Noetherian ring is said to be regular if its localizations at all prime (or maximal) ideals are regular local rings. Without this assumption, there are counter-examples.
Thanks.
| https://mathoverflow.net/users/370 | Does every regular Noetherian domain have finite Krull dimension? | No. An example is given in K. Fujita, [*Infinite dimensional Noetherian Hilbert domains,*](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.hmj/1206136627&page=record) Hiroshima Math. J. 5 (1975), 181-185.
| 7 | https://mathoverflow.net/users/11025 | 119629 | 67,403 |
https://mathoverflow.net/questions/118151 | 1 | I conjecture the following.
Let $\Omega=\mathbb{R}^3-\overline{B\_1(0)}$. Define
$$E\_{\Omega}(u)=\frac{1}{2}\int\_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int\_{\Omega}|u|^6dx.$$
$E\_{\mathbb{R}^3}$ is defined similarly:
$$E\_{\mathbb{R}^3}(u)=\frac{1}{2}\int\_{\\mathbb{R}^3}|\nabla u|^2dx-\frac{1}{6}\int\_{\mathbb{R}^3}|... | https://mathoverflow.net/users/30263 | A 'conjecture' on critical elliptic pde | Try taking a Kelvin transform of the PDE. It will send it to a new PDE on $|x| \le 1$. IN this case, since hte PDE is critical, you should get the same PDE. Some care will need to be taken at the origin. If you are looking for a "fast decay" solution of the exterior problem, ie. one for which $ |x| |u(x)| $ is bounded ... | 3 | https://mathoverflow.net/users/29444 | 119637 | 67,406 |
https://mathoverflow.net/questions/119639 | 0 | Hello
For some part of my research I want to know that PSL(3,q).2, the extension of PSL(3,q) by the graph automorphism, has the same maximal abelian subgroups as PSL(3,q)?
Also if f is a field automorphism of GF(p^3) where q=p^3, then the maximal abelian subgroups of PSL(3,q).f are the same as PSL(3,q)?
Thank you v... | https://mathoverflow.net/users/30252 | The maximal Abelian subgroups of PSL(3,q).2 the extension of PSL(3,q) by the graph automorphismAutomorphism | It's not possible for extensions of ${\rm PSL}(3,q)$ to have the same maximal abelian subgroups as ${\rm PSL}(3,q)$ since these don't include the abelian subgroups not wholly contained in ${\rm PSL(3,q)}$.
However we can still try and classify the maximal abelian subgroups of extensions of ${\rm PSL}(3,q).$
Here's... | 4 | https://mathoverflow.net/users/801 | 119644 | 67,407 |
https://mathoverflow.net/questions/119646 | 13 | This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the problem is as follows:
We say that $A$ has full outer measure in the unit interval $[0,1]$ if
$$m^\ast(A) = m^\ast([... | https://mathoverflow.net/users/26815 | Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure? | In [this article](http://cs.pwr.edu.pl/cichon/Math/Bernstein.pdf), J. Cichon shows that the real line (and hence the unit interval) can be partitioned into continuum many Bernstein subsets. Of course, any Bernstein set has full outer measure.
| 18 | https://mathoverflow.net/users/17836 | 119652 | 67,411 |
https://mathoverflow.net/questions/119640 | 3 | Clifford theory relates the representation theory of a group to that of a normal subgroup. A good reference for this is Curtis and Reiner's "Methods in Representation theory II", Theorem 11.1.
>
> **Theorem (Clifford theory)**
>
>
> Let $N$ be a normal subgroup of a finite group $G$. Let $M$ be a simple $kG$-modu... | https://mathoverflow.net/users/19113 | Reference for Clifford theory of algebras | I would suggest [Reiten, Riedtmann: Skew group algebras in the representation theory of Artin algebras, Journal of Algebra 92, 1985] if you have that $|G|$ is invertible in $A$ and $A$ is an Artin algebra.
If not, there is also a recent preprint (which I haven't read yet): [Liping Li: Representations of Modular skew... | 4 | https://mathoverflow.net/users/15887 | 119656 | 67,414 |
https://mathoverflow.net/questions/119375 | 29 | Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
([<http://en.wikipedia.org/wiki/Peano_axioms#First-order_theory_of_arithmetic>](http://en.wikipedia.org/wiki/Peano_axioms#First-order_theory_of_arithmetic)).
MA has arb... | https://mathoverflow.net/users/26766 | Even XOR Odd Infinities? | The answer is no. It is enough to find a model of MA which is an integral domain of characteristic $0$ (whence O1 is true and E1 false) such that $2$ is not invertible (whence E2 is true and O2 false).
One example of such a model is the ring of $2$-adic integers $\mathbb Z\_2$. This is clearly a domain, and $2$ is no... | 19 | https://mathoverflow.net/users/12705 | 119660 | 67,416 |
https://mathoverflow.net/questions/119658 | 12 | Let $G$ be a finite group and let $\mathbb{Q}G=M\_{n\_1}(D\_1)\times\cdots\times M\_{n\_k}(D\_k)$ be the decomposition of $\mathbb{Q}G$ as a product of rings of matrices over divisions rings. Let $Z\_i$ be the center of $D\_i$. Then each $Z\_i$ is an algebraic extension of $\mathbb{Q}$. The question is: how much is it ... | https://mathoverflow.net/users/15235 | Wedderburn's theorem for $\mathbb{Q}G$ | The short answer is yes, the centres $Z\_i$ are contained in the cyclotomic extension $E\_n= {\mathbb Q}(e^{(2\pi i)/n})$ where $n$ is the order of the group. As pranavk says, we need only prove that the simple factors of the centre of ${\mathbb Q}[G]$ are contained on $E\_n$. But the centre is spanned by the averages ... | 12 | https://mathoverflow.net/users/23291 | 119670 | 67,418 |
https://mathoverflow.net/questions/119667 | 7 | I recently read [1], in which Blass exhibits a correspondence between:
* Permutation models of ZFA in which the axiom of choice (AC) fails but the Boolean prime ideal theorem (BPIT) holds; and
* Nontrivial extremely amenable topological groups with small open subgroups.
This correspondence is demonstrated in two th... | https://mathoverflow.net/users/43894 | Models of ZFA corresponding exactly with a particular class of groups | The answer to the first question is negative, because the permutation model generated by $G$ and $\mathcal F$ doesn't really "see" the group $G$. Consider the following "shrinking" operation. Replace $G$ by a subgroup $H$ that belongs to $\mathcal F$ and replace $\mathcal F$ by its restriction to $H$ (i.e., $\{K\in\mat... | 8 | https://mathoverflow.net/users/6794 | 119673 | 67,421 |
https://mathoverflow.net/questions/119586 | 3 | On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a Poisson structure on $M$?
| https://mathoverflow.net/users/12617 | What foliations are symplectic foliations? | Even though each leaf can carry a symplectic structure still there may be no global Poisson structure. Some steps in understanding the problem (and as far as I know the only) were contained in a work by Bertelson <http://arxiv.org/abs/math/0010191>, also Commun. Contemp. Math. 3 441-456 (2001). This paper gives some in... | 2 | https://mathoverflow.net/users/6032 | 119677 | 67,422 |
https://mathoverflow.net/questions/119676 | 2 | If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to
$e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial isometry, will we get partial isometry at the Fock space level?
My main aim is to study $E\_0$ semigroup (on type I factor)... | https://mathoverflow.net/users/30703 | Second quantization of partial isometry | For any contraction $X$ on $H$, the operator $\Gamma(X)$ (in Alain's notation S(X)) is a contraction. It is essentially algebraic to check that a pair of contractions $X,Y$ on $H$ will satisfy $\Gamma(X)\Gamma(Y)=\Gamma(XY)$ and $\Gamma(X)^\\*=\Gamma(X^\\*)$. So yes, you do get a partial isometry.
See Parthasarathy's... | 4 | https://mathoverflow.net/users/10779 | 119682 | 67,423 |
https://mathoverflow.net/questions/119665 | 5 | I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff space. By central I mean of course constant on the conjugacy classes, i.e.
$f(gxg^{-1}) = f(x)$ for all $x,g\in G$.
... | https://mathoverflow.net/users/30364 | "geometric" description of the algebra of central functions on a Lie group | If $G$ is compact, then as explained in comments $G/ad$ is $T/W$. If
further $G$ is simply-connected, hence a product of simple factors,
then $T/W$ is a corresponding product of simplices.
The point is that $T/W = ({\mathfrak t}/\Lambda)/W
= {\mathfrak t}/(\Lambda \rtimes W)$, where $\Lambda$ is the kernel of the
ex... | 7 | https://mathoverflow.net/users/391 | 119687 | 67,425 |
https://mathoverflow.net/questions/119686 | 3 | In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) in the paper) states that the $L^2(\Omega)$-norm of $f$ can be estimated by the $L^2(\partial \Omega)$-norm of its trace ... | https://mathoverflow.net/users/26039 | on an inequality of Brezis-Lieb | No, choose $\Omega=\{z \in \mathbb{C} : |z| \leq 1 \} ,\ f(z)=Re\
z^{n}$ and let $n\rightarrow \infty$ .
| 5 | https://mathoverflow.net/users/17261 | 119688 | 67,426 |
https://mathoverflow.net/questions/119678 | 1 | Consider a set of $n$ real-valued number pairs: $(x\_1,y\_1), (x\_2,y\_2), \dots, (x\_n,y\_n)$. I want to find a permutation $p$ of the indices which minimizes the sum of consecutive absolute differences:
$$\sum\_{j=1}^{n-1} |x\_{p(j+1)} - x\_{p(j)}| + \sum\_{j=1}^{n-1} |y\_{p(j+1)} - y\_{p(j)}|.$$
I suspect this ... | https://mathoverflow.net/users/8719 | sorting two paired lists of real numbers to minimize consecutive absolute differences | This is known in complexity circles as rectilinear TSP. Technically this is a path version TSP rather than the more common tour. In any of these specific settings, this is known to be NP-complete since the 70s (see [Papadimitriou's paper](http://www.sciencedirect.com/science/article/pii/0304397577900123)).
However, u... | 6 | https://mathoverflow.net/users/23373 | 119697 | 67,430 |
https://mathoverflow.net/questions/119696 | 4 | Usually, it is quite easy (if cumbersome) to translate a formula like "if $\varepsilon$ is infinitesimal, and if $f$ is differentiable at $a$, $f'(a)$ is the shadow of $\frac{f(a+\varepsilon)-f(a)}{\varepsilon}$". But I cannot find a standard translation of the conjecture :"There exists a galaxy of non-standard integer... | https://mathoverflow.net/users/17164 | Translation of a non-standard analysis formulation | How about this ...
Let's say a finite set $V \subseteq \mathbb N$ is a **Feldmann set** iff there are infinitely many $n$ such that for all $u \in V$, $n+u$ is prime.
For example, $\{0,2\}$ is a Feldmann set iff there are infinitely many twin primes.
Your equivalent statment: there are arbitrarily large Feldmann... | 1 | https://mathoverflow.net/users/454 | 119700 | 67,432 |
https://mathoverflow.net/questions/119689 | 13 | Let us call a module $M$ over a commutative ring $R$ *symtrivial* if the symmetry $M \otimes M \to M \otimes M, a \otimes b \mapsto b \otimes a$ equals the identity (the same notion applies to arbitrary symmetric monoidal categories). Equivalently, every bilinear map on $M \times M$ is symmetric. Obviously $R$, and mor... | https://mathoverflow.net/users/2841 | Classification of symtrivial modules over a PID | Let $R$ be a Dedekind domain with field of fractions $K$. A module $M$ is symtrivial if and only if its maximal torsion-free quotient $F$ is a submodule of $K$ and its torsion submodule $T$ is a sum over each prime $p$ such that $F$ is $p$-divisible of a $p$-divisible $p$-torsion module and a $p$-power cyclic module $R... | 11 | https://mathoverflow.net/users/18060 | 119705 | 67,435 |
https://mathoverflow.net/questions/119699 | 8 | To define a Grothendieck topology of a category, we usually require that the category is small.
**Question 1: Why do we need to require the category to be small?**
I thought that the problem was that we wanted a sieve to be a set. (In the category of manifolds, the maximal sieve of an arbitrary manifold $X$ is not... | https://mathoverflow.net/users/5206 | Grothendieck topology for a non-small category | 1. We need the category $C$ to be small so as to define the functor category $Cat(C^{op},Set)$, i.e. the category of presheaves, so that it is locally small.
2. The trick works with the category of manifolds because the category of smooth Hausdorff second countable manifolds is equivalent to a small category (use the e... | 10 | https://mathoverflow.net/users/4177 | 119708 | 67,437 |
https://mathoverflow.net/questions/119706 | 2 | Let A be a C\* algebra of operators on a Hilbert space H. Can it happen that for some x in H the set Ax is dense in H but it is not the whole H?
| https://mathoverflow.net/users/30879 | Cyclic vectors for C* algebras | The answer is yes, as per Nik Weaver's hint: E.g. $C[0,1]$ acting by multiplication on $L\_2[0,1]$.
| 3 | https://mathoverflow.net/users/30879 | 119720 | 67,442 |
https://mathoverflow.net/questions/119722 | 3 | For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered by reverse inclusion. The empty intersection, $V$ itself, is the unique minimal element of $L(\mathcal{A})$.
It is kn... | https://mathoverflow.net/users/25028 | Is there a characterization of hyperplane arrangement intersection posets? | Chapters 4 and 8 of *Oriented Matroids
By Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, Gunter M. Ziegler*
reviews the big face lattice of oriented matroids, and when that is realizable as a hyperplane arrangement. Chapter 4 is self contained and I think you can skip to chapter 8 from there
(unfor... | 3 | https://mathoverflow.net/users/26674 | 119724 | 67,445 |
https://mathoverflow.net/questions/119750 | 1 | Hi,
I have a question concerning integration theory I can't figure out, maybe someone can help:
Fix $\varepsilon>0$ and consider $\delta \colon [0,1] \to (0,\infty)$ measurable. Is it then true that
$$\inf\_{N\subset [0,1], \lambda(N)>\varepsilon} \int\_N \delta(t) dt > 0$$
where $\lambda(N)$ is the Lebesgue measu... | https://mathoverflow.net/users/30893 | An infimum of integrals of a positive function. | If it wasn't the case, we would be able to find measurable sets $N\_k$ such that $\lambda(N\_k)>\varepsilon$ and the integral on $N\_k$ of $f$ is smaller than $k^{-1}$. So the sequence $\{f\chi\_{N\_k}\}$ converges in $L^1$ to $0$ (provided $f$ is integral, which can be assumed), and we extract a subsequence $\{f\chi\_... | 1 | https://mathoverflow.net/users/17118 | 119753 | 67,456 |
https://mathoverflow.net/questions/119499 | 16 | Hi,
I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference.
First, recall Voronin's remarkable theorem on the Universality of the Riemann zeta-function :
Let $K$ be a compact subset with connected complement lying in th... | https://mathoverflow.net/users/1162 | On the Universality of the Riemann zeta-function | Here is a cheaper alternative depending on what you mean by a modification. Consider $L(z)$ any Dirichlet L function different from $\zeta$.
>
> **Joint universality theorem**:
> Let $K$ be a compact set in the right half of the critical stripe $1/2< \Re s<1$ with connected complement. For any two functions $f\_1... | 2 | https://mathoverflow.net/users/10400 | 119759 | 67,461 |
https://mathoverflow.net/questions/119756 | 1 | I hope this question is not completely trivial:
Suppose $V$ is an irreducible projective variety and $U\subset V$ is a Zariski open subset isomorphic to an affine variety. Is it true that $V\setminus U$ is a Cartier divisor in $V$? If not, what conditions should we impose on $V$? (I guess if $V$ is smooth, then every... | https://mathoverflow.net/users/13441 | Complement to an open affine subvariety in an irreducible projective one | aglearner, In his article on abelian varieties Bryden Cais proves the result you mention. You can find the statement/proof on the top of page 4. (math.arizona.edu/~cais/Papers/Expos/AbVar.pdf)
Specifically, he proves that if $X$ is separated, normal, noetherian and $U \subset X$ is a nonempty affine open subset the c... | 1 | https://mathoverflow.net/users/25854 | 119764 | 67,464 |
https://mathoverflow.net/questions/119648 | 3 | I know that lots of effort is being put into quantization of geometry(NCG).This effort of course comes with the idea of operator algebra being a powerful machinery. Has any effort been given in the other direction.I mean to make some analogue of geometrical structures in Operator Algebras. This question may be a bit ab... | https://mathoverflow.net/users/30081 | Geometry and quantization | The theory of [Banach manifolds](http://en.wikipedia.org/wiki/Banach_manifold) has many important applications to the theory of operator algebras, and more generally to the theory of Banach spaces. For example, one can use it to very effectively apply the theory of Jordan structures to operator algebraic questions. The... | 2 | https://mathoverflow.net/users/3072 | 119767 | 67,466 |
https://mathoverflow.net/questions/119773 | 7 | I am thinking about the following questions about the cutlocus of a point in a Riemannian manifold or of a hypersurface in the Euclidean space:
1) If all the points of the (nonvoid) cutlocus of a point $p$ in a Riemannian manifold are also conjugate points, then the manifold is rotationally symmetric around the poin... | https://mathoverflow.net/users/24152 | Cutlocus and conjugate points | It is known that for a simply-connected compact
Riemannian symmetric space, the cut locus of a point
coincides with its first conjugate locus, see e.g. the book by Cheeger and Ebin,
"Comparison theorems in Riemannian geometry", Theorem 5.13. So the answer to (1) is no.
Examples of compact symmetric spaces include s... | 5 | https://mathoverflow.net/users/15155 | 119774 | 67,468 |
https://mathoverflow.net/questions/119786 | 2 | Hello,
Goldbach's conjecture states that every even integer greater than $3$ is the sum of two primes. I'm interested in a weaker assertion: has it been proven that every positive integer $n$ such that $6\vert n$ is the sum of two primes?
Thanks in advance.
| https://mathoverflow.net/users/13625 | Is every positive multiple of 6 the sum of two primes? | No. This would imply that every odd number at least $7$ is the sum of $3$ primes, since you can subtract $3$, $5$, or $7$ according to its residue mod 3. But that is not known. The strongest results known are that every sufficiently large odd number is the sum of $3$ primes, and that every odd number at least $11$ is t... | 11 | https://mathoverflow.net/users/18060 | 119787 | 67,477 |
https://mathoverflow.net/questions/119744 | 2 | Given is a constant degree rooted tree of depth $D$. It is also known that the total number of nodes in the tree is at most $D^2$. There is a probabilistic process with discrete time steps on the nodes that work as follows:
1. A node becomes **eligible** to participate in the process when all of its children have **s... | https://mathoverflow.net/users/5873 | Completion time of a process on a tree | It seems the $\log D$ is unnecessary. We can model the process as follows:
Let $v\_1, \dots, v\_j$ be the initial leaves of the tree, and let $p\_1, \dots, p\_j$ be the paths eminating from the leaves to the root vertex. At each step we perform the following:
-Delete the initial vertex of each $p\_i$ independently ... | 1 | https://mathoverflow.net/users/405 | 119790 | 67,479 |
https://mathoverflow.net/questions/119732 | 13 | In [Lee's Introduction to smooth manifolds](https://link.springer.com/book/10.1007/978-1-4419-9982-5) he states that given smooth manifolds $X,Y$ and a *surjective submersion*, $f:X \rightarrow Y$, then $f$ is a *smoothly final map*, that is for any further smooth manifold $Z$, and any map $g:Y\rightarrow Z$, we have $... | https://mathoverflow.net/users/22002 | What is the characteristic property of surjective submersions? | Here's what I had in mind:
**Theorem:** *Suppose $M$ and $N$ are smooth manifolds and $\pi:M\to N$ is a surjective smooth submersion. Then the given topology and smooth structure on $N$ are the only ones that satisfy the characteristic property.*
(That's what Problem 4-7 asks you to prove.)
| 12 | https://mathoverflow.net/users/6751 | 119793 | 67,481 |
https://mathoverflow.net/questions/119796 | 6 | Reading [Lovasz's lecture notes](http://arxiv.org/abs/cs/0205031) on evasive graph properties, I encountered the following extension of Brouwer's fixed point theorem:
**Any continues map from a contractible [finite] simplicial complex to itself has a fixed point.**
Lovasz refers this to Lefschetz. Indeed, it seems ... | https://mathoverflow.net/users/27663 | Any map of a contractible complex to itself has a fixed point | You have to assume that your complex is finite. Then the Lefschetz Fixed Point Theorem definitely says that $f$ must have a fixed point if the (homologically defined) Lefschetz number of $f$ is not zero. If the complex is contractible, then the Lefschetz number of $f$ must be $1$.
This fact about contractible complex... | 12 | https://mathoverflow.net/users/6666 | 119800 | 67,483 |
https://mathoverflow.net/questions/119596 | 0 | We have the following complex integral :
$$\frac{1}{2\pi i}\int\_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$
Where $x\in\mathbb{R}:x>1$. i tried closing the contour to the left, and computing the residues at the essential singularities of the integrand $\left( s=-\frac{1}{n}\... | https://mathoverflow.net/users/20782 | How to evaluate this complex integral !? | Mohammad, I do believe your integral is divergent. Taking for granted the computations that led you to your last comment, I can assure you that the integral
$\int\_{1}^{+\infty}\sin\left(\frac{\ln{x}}{s}-\frac{\pi}{2}\coth{(\pi{s})}\right)\frac{ds}{s}$
does not converge. Indeed for big values of $s$ the integrand i... | 2 | https://mathoverflow.net/users/24309 | 119801 | 67,484 |
https://mathoverflow.net/questions/119797 | 3 | Suppose you take a $G(n,p)$ random graph for a fixed probability $p$ and find a spanning tree using [Kruskal's algorithm](http://en.wikipedia.org/wiki/Kruskal%2527s_algorithm). If you now repeat this process indefinitely, will every tree on $n$ vertices appear with the same frequency?
In other word: are some trees m... | https://mathoverflow.net/users/22051 | How random are random spanning trees? | This is an interesting question and more subtle than it first appears. One question is the behavior of minimal cost spanning tree for the complete graph. For the question at hand it is worth first considering the behavior for $G(n,m)$ the random graph with $n$ vertices and $m$ edges. Then the model $G(n,p)$ gives a wei... | 5 | https://mathoverflow.net/users/8008 | 119804 | 67,486 |
https://mathoverflow.net/questions/119711 | 2 | This item got one answer after some hours on stackexchange, and I have a feeling I should solicit whatever variety of opinions may be out there:
Draw a line from a point on a sphere, which let us call the north pole, through another point on the sphere, to a plane parallel to the plane tangent to the sphere at the no... | https://mathoverflow.net/users/6316 | proper use of the word "stereographic" | My standard reference for elementary geometry is the book M. Berger, Geometry.
In section 18.1.4 he defines
``stereographic projection'' in any dimension. Of course it was originally introduced
for 2-dimensional sphere, and the name comes from this original use. But nowadays this
term is used in any dimension, and I d... | 1 | https://mathoverflow.net/users/25510 | 119805 | 67,487 |
https://mathoverflow.net/questions/119821 | 3 | Hi all,
I was reading a proof that the localization of categories preserves abelianess. Although the author didn't mentioned explicitly, it seems to me that the proof is reduced to the statement that
>
> An additive category is a balanced category(i.e., every monic epimorphism is an isomorphism).
>
>
>
I kn... | https://mathoverflow.net/users/30862 | Is an additive category a balanced category? | An additive category need not be balanced. Consider the full subcategory of abelian groups consisting of all torsion-free groups. Then for any $n\neq0$, the map $n:\mathbb{Z}\to\mathbb{Z}$ is monic and epic, but it is not an isomorphism unless $n=\pm1$. The only nontrivial part of this is that it is epic, and this is s... | 11 | https://mathoverflow.net/users/75 | 119822 | 67,492 |
https://mathoverflow.net/questions/119803 | 9 | Let $G$ be a nonabelian group, with classifying space $BG$.
**Motivation:** We can compute its homology, $H\_\ast(BG)=H\_\ast(G)$. It would be nice to see some equivariant computations, like $H\_\ast^G(BG)$ where $G$ acts by conjugation, but I need a particular model of $BG$ to work with. In any case, if $G$ acted f... | https://mathoverflow.net/users/12310 | Relation between groups and classifying spaces | What you should take as a model is the homotopy quotient $EG \times\_G BG$. From the homotopy sequence of the fibration $EG \times\_G BG \to BG$ (projection on first factor), you get that
$EG \times\_G BG$ is aspherical and a short exact sequence
$$
1 \to \pi\_1 (BG) \to \pi\_1 (EG \times\_G BG) \to \pi\_1 (BG) \to... | 13 | https://mathoverflow.net/users/9928 | 119823 | 67,493 |
https://mathoverflow.net/questions/119808 | 1 | I am looking for reference for following lemma.
Consider a set of hyperplanes $H$ in a N-dimensional Euclidean space. Let $S$ be the set of intersections of all elements of $H$.
Take a point $w$. Repeat as follows: project $w$ on a hyperplane in $H$. If you repeat this, you will get closer to $S$. If you use all hype... | https://mathoverflow.net/users/30911 | Intersecting hyperplanes. | Is this what you want?
Halperin, Israel
The product of projection operators.
Acta Sci. Math. (Szeged) 23 1962 96–99.
MathSciNet review: It is proved that if E1,E2,⋯,En are projections in a Hilbert space H and if T=E1E2⋯En, then Tm converges strongly as m→∞ to the projection E whose range is ⋂ni=1(EiH). For n=2 this... | 2 | https://mathoverflow.net/users/20186 | 119834 | 67,496 |
https://mathoverflow.net/questions/119827 | 14 | My question is the following :
>
> Given an algebraic curvature operator $R\in S^2\_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the curvature operator of symmetric space ?
>
>
>
I would be almost equally happy if someone can point to a way to know i... | https://mathoverflow.net/users/8887 | Algebraic characterization of the curvature operator of symmetric spaces | I suppose that $S^2\_B(\Lambda^2(\mathbb{R}^n))$ means the subspace of $S^2(\Lambda^2(\mathbb{R}^n))$ that satisfies the Bianchi identity, i.e., the kernel of the natural map $S^2(\Lambda^2(\mathbb{R}^n))\longrightarrow \Lambda^4(\mathbb{R}^n)$.
Certainly, you'd need that, if ${\frak{g}}\subset \Lambda^2(\mathbb{R}^n... | 22 | https://mathoverflow.net/users/13972 | 119835 | 67,497 |
https://mathoverflow.net/questions/119836 | 0 | The cycle has this property. For instance, the distance matrix for a 6-cycle is:
$A=\begin{bmatrix}
0 & 1 & 2 & 3 & 2 & 1 \\\\
1 & 0 & 1 & 2 & 3 & 2 \\\\
2 & 1 & 0 & 1 & 2 & 3 \\\\
3 & 2 & 1 & 0 & 1 & 2 \\\\
2 & 3 & 2 & 1 & 0 & 1 \\\\
1 & 2 & 3 & 2 & 1 & 0 \\\\
\end{bmatrix}$
The question is: is the cycle th... | https://mathoverflow.net/users/22051 | Graphs with circulant distance matrices | Any circulant graph has this property. A circulant graph is a graph that has an automorphism that's a cyclic permutation of its vertices. If you apply permute the rows and columns of the distance matrix with the same permutation, you must get the same matrix, so the distance matrix is a circulant matrix.
Graphs other... | 8 | https://mathoverflow.net/users/5340 | 119841 | 67,500 |
https://mathoverflow.net/questions/119850 | 0 | Given a complex vector bundle $V\rightarrow M$, we can form a
fibre bundle $\mathbb{P} V\rightarrow M$, where the fiber over
each point is the corresponding projective space. In particular
consider the space
$ \mathbb{P}(T \mathbb{P}^2)$, the projectivized tangent space
of $\mathbb{P}^2$. Can this complex manifold ... | https://mathoverflow.net/users/4463 | Can one embedd the projectivezed tangent space of CP^2 in a projective space? | Yes to both questions. In particular, $\mathbb P(T\mathbb P^2)$ is isomorphic to a hyperplane section of the Segre embedding $\mathbb P^2\times\mathbb P^2\hookrightarrow \mathbb P^8$. Indeed, $\mathbb P(T\mathbb P^2)$ is isomorphic to the variety of complete flags in the vector space $\mathbb C^3$. Suppose now that in ... | 7 | https://mathoverflow.net/users/29992 | 119853 | 67,504 |
https://mathoverflow.net/questions/119861 | 10 | I was looking in [Ravi Vakil's notes on Intersection Theory](http://math.stanford.edu/~vakil/245/), Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$.
On p. 2, he writes the following which I had no idea about:
>
> **Alarming fact:** This ring is apparently ... | https://mathoverflow.net/users/1310 | Commutativity of the Chow ring in positive characteristic | De Jong's theorem proves that the ring is commutative when tensored with $\mathbb Q$. For the torsion part, I am pretty sure that the question is still open.
[Edit:] Mikhail is absolutely right, with Gabber's theorem (<http://www.math.u-psud.fr/~illusie/refined_uniformization3.pdf>) one can show that the bivariant Ch... | 8 | https://mathoverflow.net/users/4790 | 119864 | 67,509 |
https://mathoverflow.net/questions/119792 | 3 | Let $C$ be a compact convex subset of Euclidean space. Recall that $x\in C$ is an *exposed point* of $C$ if there is a plane $P$ such that $P\cap C = \{x\}$. It is obvious that exposed points are extreme. To see that they are not equivalent, draw two parallel lines and put half circles on each end.
Straszewicz's theo... | https://mathoverflow.net/users/5678 | A quantitative version of Straszewicz's theorem? | $\let\eps\epsilon$
$\def\conv{\mathop{\rm conv}}$
A set is called *$R$-strictly convex* if it is an intersection of balls of radius $R$. Denote by ${\mathop{\rm conv}}\_R C$ the $R$-strict convex hull of $C$, that is --- the intersection of all balls with radius $R$ that contain $C$ (we assume that the diameter of $C$... | 1 | https://mathoverflow.net/users/17581 | 119865 | 67,510 |
https://mathoverflow.net/questions/116452 | 0 | I toss $n$ biased coins and I want to count the number of times you get a H followed by a T or a T followed by a H. I call these switches. So for example if I get HHTTHTHHHT then I have $5$ switches in total. If the coin gives H with prob $p$ and $T$ with prob $1-p$ then how can you find an approximation to the probabi... | https://mathoverflow.net/users/29968 | Bounds for number of coin toss switches | A "head" switch is a tail followed by a head, and a "tail'' switch is a head followed by a tail.
Joint Laplace transform for the number of "head'' switches and "tail'' switches are given
in the short note:
On the Number of Switches in Unbiased Coin-tossing,
<http://www.math.udel.edu/~wli/papers/13-switches-sub.pdf>
... | 4 | https://mathoverflow.net/users/30926 | 119868 | 67,511 |
https://mathoverflow.net/questions/119845 | 3 | Let $T(X)$ be the Teichmuller space of a closed Riemann surface $X$ of genus $g \geq 2,$ and $MF$ the space of equivalence classes of measured foliations. Then we have a length function
$l: T(X) \times MF \to \mathbb R,$ $l\_\sigma (\mathcal F).$ The question is, is there a basic and convenient way to see the continui... | https://mathoverflow.net/users/30776 | continuity of length function $l: T(X) \times MF \to \mathbb R$ | I suppose that to "see" these continuities, you are not asking about a proof, but about some intuition.
For the first question, I like the intuition one gets from Bonahon's paper MR0931208, under which $MF$ and $T(X)$ are both embedded in a common space on which an obviously continuous intersection number is defined... | 3 | https://mathoverflow.net/users/20787 | 119873 | 67,513 |
https://mathoverflow.net/questions/119872 | 4 | Can anyone give me some good references to read geometric K-homology. I know bit of Kasparov's KK theory and analytic K-homology.
| https://mathoverflow.net/users/30703 | References for geometric K-homology | If I remenber correctly one can use, Martin Jakob's geometric approach to generalized homology theories:
Jakob, Martin An alternative approach to homology. Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), 87–97, Contemp. Math., 265, Amer. Math. Soc., Providence, RI,... | 5 | https://mathoverflow.net/users/27816 | 119874 | 67,514 |
https://mathoverflow.net/questions/119867 | 9 | A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See [wikipedia](http://en.wikipedia.org/wiki/Nontransitive_dice))
For some sets, the deviation from transitivity is small in the sense that A beats B beats C be... | https://mathoverflow.net/users/30927 | What is the most extreme set 4 or 5 nontransitive n-sided dice? | If you fix only the number of dice, but let the number of faces be arbitrary (or if you simply find a way to make arbitrary probabitities for different faces), then the answer for $n$ faces is $\displaystyle 1-\frac1{4\cos^2(\pi/(n+2))}$. This is proved (that's quite immodest, I know...) in my paper "intransitive roule... | 16 | https://mathoverflow.net/users/17581 | 119885 | 67,519 |
https://mathoverflow.net/questions/119887 | 16 | The [Erdős–Gallai theorem](http://en.wikipedia.org/wiki/Erd%25C5%2591s%25E2%2580%2593Gallai_theorem) characterizes which degree sequences are graphical (i.e. realizable by a simple graph).
There has been some work on which degree sequences are planar graphical (i.e. realizable by a simple planar graph). See, for exam... | https://mathoverflow.net/users/11598 | Which degree sequences are planar graphical? | It is always difficult to say what is "currently known," but at least around
2008, the paper
>
> "A Characterization of the degree sequences of 2-trees."
> Prosenjit Bose, Vida Dujmovi, Danny Krizanc, Stefan Langerman, Pat Morin, David R. Wood, Stefanie Wuhrer.
> *Journal of Graph Theory*
> Volume 58, Issue 3, p... | 10 | https://mathoverflow.net/users/6094 | 119904 | 67,527 |
https://mathoverflow.net/questions/119879 | 15 | Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, such bound must be at least 18, as the example
$$ G = \left<
\left(
\begin{array}{rr}
5 & 4 \\\
-1 & -1
\end{array}
\ri... | https://mathoverflow.net/users/28104 | Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$ | In Olʹshanskiĭ, A. Yu.; Sapir, M. V. [On $F\_k$-like groups](http://mi.mathnet.ru/al/v48/i2/p245). (Russian) Algebra Logika 48 (2009), no. 2, 245--257, 284, 286--287;
[translation](https://doi.org/10.1007/s10469-009-9044-2) in Algebra Logic 48 (2009), no. 2, 140–146,
we proved (Theorem 2) that every non-virtually cyc... | 17 | https://mathoverflow.net/users/nan | 119909 | 67,529 |
https://mathoverflow.net/questions/119913 | 9 | Say I have a function $f$. Then, if I understand correctly, $f$ can be regarded as a morphism in a suitably chosen category which has the domain of $f$ and the range of $f$ as objects.
So, the other direction. Say I have a morphism $f$ in some category from one object $A$ to another object $B$. Can I not regard $f$ a... | https://mathoverflow.net/users/27287 | What is the difference between a function and a morphism? | You're losing a lot of information. Let's say we have two morphisms $f,g\in \mathrm{Hom}(A,B)$. Using your perspective, we see that both $f$ and $g$ agree at all points (as the only point is $B$). Thus by eta-reduction we should have $f=g$, but this is not necessarily the case. So thinking of morphisms as functions fro... | 4 | https://mathoverflow.net/users/13832 | 119915 | 67,532 |
https://mathoverflow.net/questions/119901 | 0 | This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^\*$ be its Grothendieck group. That is, $T^\* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+e$ for some $e \in T$.
Let $i : T \to T^\*$ be the natural inclusion map, where $i(t) = [(t,0)]$, and let $K$ denot... | https://mathoverflow.net/users/238 | Kernel elements for the Grothendieck group map of a commutative monoid | I don't know a name for $K$ either, but here is a suggestion.
The correct definition of $\sim$ is $(a,b) \sim (c,d) \Leftrightarrow \exists e \in T ~ (a+d+e=b+c+e)$. In particular $(a,b) \sim (0,0) \Leftrightarrow \exists e \in T : a+e=b+e$. I would suggest to call such elements $a,b$ *stably equal*. Remark that this... | 3 | https://mathoverflow.net/users/2841 | 119928 | 67,539 |
https://mathoverflow.net/questions/119942 | 3 | Is there a common presentation of the semigroup of functions from a given (finite) set to itself?
| https://mathoverflow.net/users/16971 | Presentation of the monoid of self-maps of a finite set | There are several well known presentations. One has to first choose your favorite presentation of the symmetric group. Then one usually adds in a rank n-1 idempotent but some people add all of them or other elements. Then one figures out the extra relations. See [page 161](https://books.google.com/books?id=LC3jxfGEcpYC... | 5 | https://mathoverflow.net/users/15934 | 119943 | 67,545 |
https://mathoverflow.net/questions/119940 | 19 | What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?
In addition, I see that one of the requirements for a covariant derivative (in the context of connections) is to commute with contraction. Why is that a natural requirement (like it would b... | https://mathoverflow.net/users/8446 | Tensor contraction and Covariant Derivative | I like the question. Below is a somewhat sketchy version of how I see this.
I think the importance of tensors and contraction of tensors originates from trying to do basic differential geometry or vector calculus from a co-ordinate-free point of view. The most basic objects are curves and velocity vectors of curves. ... | 12 | https://mathoverflow.net/users/613 | 119959 | 67,554 |
https://mathoverflow.net/questions/22885 | 13 |
>
> Are there any non-contractible connected topological rings?
>
>
>
Of course, such a thing cannot be a (topological) algebra over the reals.
(I have a vague memory of having a glance at an erticle by Lurie in which some (for me) rather esoteric theory of higher categorical structures gave rise to topologica... | https://mathoverflow.net/users/4721 | Noncontractible connected topological rings ? | Here is a method for manufacturing such topological rings.
The main technical ingredient is a product-preserving functor
$$\Theta: \mathrm{Set}^{\Delta^{op}} \to \mathrm{CGHaus}$$
from the category of simplicial sets to the category of compactly generated Hausdorff spaces that is **not**, however, the usual ge... | 22 | https://mathoverflow.net/users/2926 | 119962 | 67,556 |
https://mathoverflow.net/questions/119956 | 6 | I am confused by a statement in the very classical paper of A. A. Albert "On the construction of Riemman matrices II", Ann. Math. 1935, Thm 16. If I understand what he saying, the theorem says that given a quaternion algebra $D$ over a totally real field $F$ of degree $t$ over $\mathbb{Q}$, which is not split at any of... | https://mathoverflow.net/users/2290 | Abelian varieties with given endomorphism algebra | Felipe, Albert is right: $r>1$. In particular, there are no complex abelian surfaces, whose endomorphism algebra is a definite quaternion algebra over the rationals. The same is true in any characteristic: see a survey article of Frans Oort entitled "Endomorphism algebras of abelian varieties" (Alg. Geom. and Commut. A... | 8 | https://mathoverflow.net/users/9658 | 119966 | 67,558 |
https://mathoverflow.net/questions/119965 | 2 | Given a weighted graph $G = (V,E)$, the in-degree of a node is defined as
$
k\_{in}(i) = \sum\_{j:j \rightarrow i} A\_{ji}
$
where $A\_{ji}$ is the weight of edge from node $j$ to $i$. My question is if there are $m$ nodes pointing to $i$, what's the terminology for $m$? Is it the number of in-links?
| https://mathoverflow.net/users/30950 | The terminology for a node's number of in-links in weighted directed graph | The quantity you are asking about is usually called the "fan-in" of node $i$. The dual quantity which counts the number of nodes to which $i$ points is called the "fan-out". The terms are standard at least in the circuit design and computer science literature, but I can't seem to find them in graph theory texts at the ... | 0 | https://mathoverflow.net/users/18263 | 119970 | 67,561 |
https://mathoverflow.net/questions/119963 | 7 | Let $K$ be a quadratic extension of $\mathbb Q$ and let $p \neq 2$ be a prime that is **inert** in $K$. Let $X$ be the Shimura variety associated to the unitary group $\operatorname{U}(2,1)$ over $K$ (after a choice of a suitable integral PEL datum). We have an integral model $\mathcal X$ of $X$ defined over $\mathcal ... | https://mathoverflow.net/users/7845 | p-rank stratification in unitary Shimura variety | Yes, both these cases appear. This follows from the Remark, page 92, of my [PhD thesis](http://people.brandeis.edu/~jbellaic/preprint/these.pdf).
This is in the unpublished chapter III, devoted to the study of the Shimura variety for U(2,1)
(a.k.a Picard modular surface) over $W(k)$, $k$ a field of char $p$, with level... | 4 | https://mathoverflow.net/users/9317 | 119974 | 67,563 |
https://mathoverflow.net/questions/119989 | 2 | For two pmf $p=\lbrace p\_i\rbrace$ and $q=\lbrace q\_i\rbrace$ on the same finite alphabet, we know that relateive entropy
$D(p\|q)=\sum p\_i\log\frac{p\_i}{q\_i}$
and 1-norm
$\|p-q\|\_1=\sum |p\_i-q\_i|$
are both measures of their distance. But it is unfortunate that relative entropy is not a norm. My question is: ev... | https://mathoverflow.net/users/5072 | equivalence of 1-norm and relative entropy? | No, that is not true. Let $p^{(n)}\to q$ in $L^1$ such that $p^{(n)}$ lies in the (relative) interior of the probability simplex whereas $q$ is on the boundary (of the simplex), i.e., $q\_i=0$ for some $i$. Then $D(P^{(n)}\|q)=\infty$ for every $n$.
But the other direction is true because of the Pinsker's inequality ... | 3 | https://mathoverflow.net/users/7699 | 119993 | 67,572 |
https://mathoverflow.net/questions/119981 | 9 | Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence
$1 \to \pi\_1^{et}(C \otimes\_\mathbb Q \bar{\mathbb Q}) \to \pi\_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$
gives a homomorphism from $\operatorname{Gal}(\bar{\... | https://mathoverflow.net/users/18060 | Are all anabelian Galois actions faithful? | The answer is "yes" I think, even if you replace $\mathbb Q$ with a number field.
In the affine case this is a result of Matsumoto, as pointed out by Felipe Voloch, see
Matsumoto, Makoto
Galois representations on profinite braid groups on curves.
J. Reine Angew. Math. 474 (1996), 169–219.
In the proper case thi... | 8 | https://mathoverflow.net/users/11682 | 120001 | 67,577 |
https://mathoverflow.net/questions/119895 | 0 | In some situations we have access to a representation like this:
$ f(x,y) = \sum\_i u\_i(x) v\_i(y) $
What is this called? (I know when you jam this into PDE get to call it 'separation of variables' but I'm sure it's got a different name in pure math).
| https://mathoverflow.net/users/8916 | Multivariate expansion in terms of single variate products: what is the name for this? | [This paper](http://www.jstor.org/discover/10.2307/2688535?uid=3737536&uid=2129&uid=2&uid=70&uid=4&sid=21101713553267) calls it "separable of rank $n$" if $n$ is the number of terms in the sum. If the sum is an infinite one, the condition is very weak (for example it includes all functions with a convergent Taylor expa... | 2 | https://mathoverflow.net/users/9025 | 120003 | 67,578 |
https://mathoverflow.net/questions/119985 | 1 | I need to know if a certain Banach space I stumbled upon is reflexive or not. I need to know what are the state of the art techniques to determine if a Banach space is reflexive or not. For example, how does one prove that the space of bounded continuous functions is not reflexive. Same question for Linfinity. A descri... | https://mathoverflow.net/users/30954 | Showing a Banach space is reflexive | A good reference for functional analysis in general is the book by John B. Conway. This book states that if $X$ is a compact space(or more generally a completely regular space), then $C(X)$ is reflexive if and only if $X$ is finite(p. 90). See [This link](https://math.stackexchange.com/questions/250325/space-of-bounded... | 3 | https://mathoverflow.net/users/22277 | 120004 | 67,579 |
https://mathoverflow.net/questions/120002 | 3 | How many lines in $\mathbf{P}^5$ passing through a fixed point $p$ meet in at least two points a fixed smooth surface $S$ given by the intersection of three quadrics?
Or equivalently, calling $T$ the surface obtained projecting from $p$ the surface $S$ onto a $\mathbf{P}^4$, how many nodal singularities are there on ... | https://mathoverflow.net/users/27125 | Counting nodal singularities on a surface | The answer is 4. The argument is the following. Let $L$ be such a line. Each of three quadrics intersects $L$ in two given points. Consider the pencil of quadrics which also contain $p$. Then these intersect $L$ in at least 3 points, hence contain $L$. Vice versa, if $L$ i a line passing through $p$ and contained in th... | 2 | https://mathoverflow.net/users/4428 | 120006 | 67,581 |
https://mathoverflow.net/questions/119980 | 29 | A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, all of the explanations that I have ever seen are either too short or too long. My impression is that the way that a cent... | https://mathoverflow.net/users/1450 | Concise model of modern fiat money and its non-conservation | I think an answer that discusses the actual institutional details of how the Fed controls the money supply would be off-topic here. Also, the Fed works slighlty differently from the ECB in that regard and there is more than one method of influencing the money supply (take a look at the wikipedia page on [money creation... | 21 | https://mathoverflow.net/users/35357 | 120007 | 67,582 |
https://mathoverflow.net/questions/119938 | 2 | Just wonder if there is any known results on the transitive metacyclic groups of $AGL(4,3)$?
Sorry, I should elaborate a bit here. I am working on a graph $\Gamma$ which admits a metacyclic group transitive on its vertices. By some other restricted conditions, I have reduced that $Aut\Gamma$ is contained in $AGL(4,3... | https://mathoverflow.net/users/30610 | Metacyclic groups in $AGL(4,3)$ | There is no such group. Let $G$ be a transitive metacyclic subgroup of $\text{AGL}(4,3)$. Let $C$ be a cyclic normal subgroup of $G$ with $G/C$ cyclic.
As $G$ permutes transitively the orbits of $C$, the kernels of the action of $C$ on its orbits all have the same size, thus they are equal because $C$ is cyclic. But ... | 2 | https://mathoverflow.net/users/18739 | 120018 | 67,588 |
https://mathoverflow.net/questions/120016 | 3 | Let $X\_{K}$ be a curve over a complete DVR $R$, $R/m:=k$ an algebraically closed field. We suppose the minimal field extension $L$ of $K$ such that $X\_{L}$ has stable model $X\_{R\_{L}}$, and the special fiber is $X\_{k}$. We obtain an action of $Gal(L/K)$ on $X\_{k}$.
My question is:
Does $Gal(L/K) \longrightarr... | https://mathoverflow.net/users/5274 | Galois action on special fiber of a stable model | This appears in the literature, see e.g. Theorem 2.2 in Lehr and Matignon, Wild monodromy and automorphisms of curves, Duke Math. 2006. It is attributed there to Deligne and Mumford.
| 4 | https://mathoverflow.net/users/17988 | 120037 | 67,597 |
https://mathoverflow.net/questions/120034 | 10 | Is there any necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds $M$?
| https://mathoverflow.net/users/nan | necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds | Yes, it is well-known that a $6$-manifold has an $\mathrm{SU}(3)$-structure if and only if it is orientable and spinnable (i.e., it has a spin structure).
The necessity of these two conditions is clear, since $\mathrm{SU}(3)$ is both connected and simply-connected.
For the sufficiency, first note that if $M^6$ is ... | 21 | https://mathoverflow.net/users/13972 | 120039 | 67,599 |
https://mathoverflow.net/questions/120036 | 1 | In <http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf> the Arnold conjecture from Symplectic Geometry is shown for the case of montone symplectic manifolds $(M, \omega)$ (i.e. we have $\int\_{S^2} v^\*\omega= \tau \int\_{S^2} v^\*c\_1$ for every smooth map $v: S^2 \rightarrow M$ for the Chern class $c\_1$ and some $... | https://mathoverflow.net/users/29973 | Condition in proof of the Arnold conjecture for monotone manifolds | The normalization
$$\int\_{S^2} v^\ast \omega \in \Bbb Z \text{ for all smooth } v: S^2 \longrightarrow M$$
is achieved by setting $\omega \mapsto \frac{1}{\tau} \omega$ and looking at the monotonicity condition
$$\int\_{S^2} v^\ast c\_1(M) = \tau \int\_{S^2} v^\ast \omega \text{ for all smooth } v: S^2 \longrightarrow... | 1 | https://mathoverflow.net/users/21375 | 120040 | 67,600 |
https://mathoverflow.net/questions/120042 | 2 | Let $G$ be a finite supersolvable group with trivial Frattini subgroup.
Is it true that all Sylow subgroups of $G$ are elementary abelian?
**EDIT**: Many Thanks to Derek for his answer. Let me say some words on the motivation.
I am asked of:
Is it true that in a finite supersolvable group with trivial Frattini subgro... | https://mathoverflow.net/users/19075 | Finite supersolvable groups with trivial Frattini subgroup | What about a Frobenius group of order 20, i.e. $\langle x,y \mid x^5=y^4=1,y^{-1}xy=x^2 \rangle$?
| 3 | https://mathoverflow.net/users/35840 | 120047 | 67,602 |
https://mathoverflow.net/questions/120055 | 6 | Let $G$ be a connected commutative algebraic group over $\mathbb{F}\_q$. If $\text{Fr}\_q : G \to G$ denotes the $q$-Frobenius morphism, we define the *Lang isogeny* $L\_q$ to be the endomorphism of $G$ given by $g \mapsto \text{Fr}\_q(g)g^{-1}$.
I have two questions about this important map.
1. It is not too hard ... | https://mathoverflow.net/users/3544 | The Lang isogeny | 1. Every etale morphism is finite over some nonempty open set. (For instance, locally somewhere write it as a standard etale morphism $(A[t]/f(t))\_{g(t)}$, then consider the open set where the norm of $g$, that is, the resultant of $f$ and $g$, is nonzero.) If a group homomorphism $H \to G$ is finite over some nonempt... | 6 | https://mathoverflow.net/users/18060 | 120057 | 67,607 |
https://mathoverflow.net/questions/120025 | 3 | The segre cubic primal $X\subset P^4$ is the GIT quotient of 6 points on $P^1$. Let $M\_{0,6}$ the DM compactification of the moduli of 6-pointed rational curves. The Segre primal $X$ is a cubic 3-fold with ten double points and there exists a natural map $M\_{0,6}\to X$ that contracts 10 boundary divisors (each iso to... | https://mathoverflow.net/users/4096 | blow up of segre primal and $\mathcal{M}_{0,6}$ | From what you've written you'll get a map $M\_{0,6} \to \tilde{X}$ via the universal property of blowups (see Proposition II.7.14 of Hartshorne's Algebraic Geometry). In fact this map is an isomorphism.
See the paragraph after Theorem 4.2 of
Harvey-Lloyd Philipps, Symmetry and moduli spaces for Riemann surfaces for ... | 2 | https://mathoverflow.net/users/321 | 120059 | 67,609 |
https://mathoverflow.net/questions/119641 | 14 | In first course differential geometry you learn, that Ricci-curvature is something like a mean-value of the curvature endomorphism, because it's a trace, and the scalar curvature is again a mean-value of the Ricci curvature, again because it's a trace.
I'm now interested, what examples of manifolds can be given, with b... | https://mathoverflow.net/users/21779 | Geometric picture of scalar curvature | Have you taken a look at [wikipedia page for Scalar curvature](http://en.wikipedia.org/wiki/Scalar_curvature)? [BTW, always a great resource!] There you can find the standard geometric interpretation of Scalar curvature, as measuring the volume distortion on balls of small radius, compared to Euclidean balls of such ra... | 16 | https://mathoverflow.net/users/15743 | 120065 | 67,612 |
https://mathoverflow.net/questions/120062 | 3 | I would like to understand if the following statement is actually proven in Shafarevich's book "Basic algebraic geometry" (or just learn its proof in the spirit of Shafarevich's book).
**Statement**. Let $K$ be an algebraically closed field and let $X, Y$ be irreducible projective subvarieties in $\mathbb P\_K^n$. Su... | https://mathoverflow.net/users/13441 | Intersection of two projective submanifolds in $P^n$ treatment in Shafarevich book | I don't know how Shafarevich proves it or intended to prove it, but here is a way to do it.
**Claim:** Let $X,Y\subseteq \mathbb A^n$ irreducible of dimension $a,b$. Then every (non-empty!) irreducible component of $X\cap Y$ has dimension at least $a+b-n$.
**Sketch of Proof:** Step 1: using the Krull principle idea... | 14 | https://mathoverflow.net/users/10076 | 120073 | 67,619 |
https://mathoverflow.net/questions/120090 | 2 | Do you think that $\mathbb Z \subset \mathbb R$? On one hand this inclusion is quite handy. We like to write things like:
$$
\sqrt{n} \quad \text{for $n\in \mathbb Z$}
$$
which requires the number $n$ to be a real number (where $\sqrt\cdot$ is defined). On the other hand it is difficult to obtain such an inclusion when... | https://mathoverflow.net/users/15120 | Are integers real? | We do in fact have many different sets isomophic to $\mathbb{Z}$. The correct definition of $\mathbb{Z}$ is a set with some operations on it that satisfy some axioms. This is easier to see with $\mathbb{N}$, which is a set with a 0, a "plus one" operations, that satisfies mathematical induction. There is only one such ... | 7 | https://mathoverflow.net/users/3711 | 120093 | 67,629 |
https://mathoverflow.net/questions/120075 | 6 | Hello!
I'm a student learning the basics of working with the unbounded derived category $D(\mathcal{A})$. I arrived at the natural question, "is every K-projective complex formed out of projective objects?" (having learned that this isn't sufficient to guarantee K-projectivity), and a literature hunt led to the intro... | https://mathoverflow.net/users/30971 | DG-projective vs. K-projective complexes | K-projectivity of a complex is a property of its homotopy equivalence class, i.e., any complex homotopy equivalent to a K-projective complex is K-projective. In particular, any contractible complex is K-projective. Taking the cone of the identity endomorphism of any complex that is not formed out of projective objects ... | 5 | https://mathoverflow.net/users/2106 | 120095 | 67,631 |
https://mathoverflow.net/questions/120096 | 6 | Let $K$ be a nonarchimedean local field with ring of integeres $R\_K$, maximal ideal $m\_K$ and finite residue field $\bf k$. Let $\pi$ be an admissible irreducible complex representation of ${\rm GL}\_2(K)$ with central character $\epsilon$. A fundamental result of Casselman says that there is a largest ideal $J\subse... | https://mathoverflow.net/users/3602 | Explicit Casselman theory: reference needed | For the parabolically induced representation, I suggest to look at Casselman "Restriction of $GL(2, F)$ to to $GL(2,o)$"-paper.
For the Steinberg representations and the super cuspidal representation, I suggest to look at Bushnell-Henniart "Local Langlands conjectures for GL(2)". For the Steinberg, your ideal will be... | 4 | https://mathoverflow.net/users/10400 | 120102 | 67,635 |
https://mathoverflow.net/questions/119919 | 6 | Consider second-order Peano Arithmetic Z2, i.e. the two-sorted first-order theory with induction and comprehension. Remove the assumption about the totality of the successor relationship (the Successor Axiom), i.e. the assumption that every number is successored by a number. Call this theory FPA. FPA has as models the ... | https://mathoverflow.net/users/20716 | Provability in Second-Order Arithmetic without the Successor Axiom | As I already mentioned in another thread for a slightly different theory, it is possible to give a complete description of models of FPA (I mean all models, giving a complete semantics for the many-sorted first-order theory, not just proper second-order models, which abo lists in the question and which I will hencefort... | 9 | https://mathoverflow.net/users/12705 | 120106 | 67,638 |
https://mathoverflow.net/questions/120118 | 11 | I was studying the construction of the modular lambda function and I started thinking about the following question. Suppose that $\Omega\subset \mathbb{C}$ is an open connected set and $f:\Omega\to \mathbb{C}$ is continuous on $\Omega$ and holomorphic except on a set of measure zero (with respect to 2-dimensional Lebes... | https://mathoverflow.net/users/30983 | When does continuity imply holomorphy? | The following is too long for a comment. I will suppose here that your set of measure zero is compact.
In this case, Your question is closely related to so-called *continuous analytic capacity*. Let $K$ be a compact set in the plane, and let $\Omega$ be the complement of $K$ with respect to $\mathbb{C}\_\infty$. The ... | 13 | https://mathoverflow.net/users/1162 | 120122 | 67,644 |
https://mathoverflow.net/questions/120124 | 0 | Hi, I have the following expected value to compute
$E[ \int\_{o}^{T} f(t) dt \int\_{o}^{T} H(s) dW(s)]$,
where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the Brownian motion W.
I think that this expected value could be equal to zero, but I really don't know how to give t... | https://mathoverflow.net/users/19133 | Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral | You can write $h(s)=\int\_0^Tf(t)dtH(s)$ then your expectation could be written as $E[ \int\_{o}^{T} h(s) dW(s)]$. This integral is $0$ if you can prove $(\int\_{o}^{t} h(s) dW(s)) $ to be a martingale, for example $E(\int\_{o}^{T} h^2(s) ds)<+\infty$.
As you noticed in your comment, this procedure is correct if we ... | 0 | https://mathoverflow.net/users/30889 | 120125 | 67,646 |
https://mathoverflow.net/questions/120067 | 61 | The *theta function* is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum\_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.
>
> **Theta reciprocity**: $\theta(\tau)=\frac{1}{\sqrt{\tau}}\theta\le... | https://mathoverflow.net/users/25791 | What do theta functions have to do with quadratic reciprocity? | Going in the direction of more generality:
With $\theta(\tau)=\sum\_n\exp(\pi i n^2 \tau)$, theta reciprocity describes how the function behaves under the linear fractional transformation $[\begin{smallmatrix} 0&1 \\ -1&0\end{smallmatrix}]$. From this one can show it's an automorphic form (of half integral weight, on... | 23 | https://mathoverflow.net/users/6756 | 120129 | 67,647 |
https://mathoverflow.net/questions/120128 | 18 | How do I in general realize a universal C\*-algebra generated by some generators and relation as concrete C\*-algebras? For example, I know that universal C\*-algebra generated by a single unitary is $C(\mathbb{T})$ by functional calculas. I am looking at the following examples to work on:
1. universal C\*-algebra ge... | https://mathoverflow.net/users/13739 | Realizing universal $C^*$-algebras as concrete $C^*$-algebras | **Edit**: as pointed out in the comments, the following answers the question for *unital* C\*-algebras presented in terms of generators and relations. When I say C\*-algebra, I really mean unital C\*-algebra.
It may depend on what exactly you mean by "concrete", but I highly doubt that there is a general solution to ... | 21 | https://mathoverflow.net/users/27013 | 120130 | 67,648 |
https://mathoverflow.net/questions/120121 | 1 | Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems for abstract spaces with measures? (Or at least for Borel measures?)
Later edit:
After reading the first two comments,... | https://mathoverflow.net/users/54780 | Mean value theorems for the Haar integral? | Say $\mu$ is a Borel probability measure on a connected set $A$ in a topological space. Let $f : A \to \mathbb R$ be continuous. Then the mean value
$\int\_A f\;d\mu$ is equal to $f(a)$ for some $a \in A$. Proof: the mean value is between the sup of all values and the inf of all values, so (by connectedness) it is a va... | 8 | https://mathoverflow.net/users/454 | 120132 | 67,649 |
https://mathoverflow.net/questions/120116 | 1 | Hello there, I have a problem and would like to know do anyone have an elementary proof of it and it goes like this:
Show that $ a^n + 1\neq m^2$ when $a=4,7,10$ for every $n$ and $m$ ($n$ and $m$ are natural numbers).
EDIT: I succeded in finding an elementary proof that $ a^n + 1\neq m^2$ when $a$ is of the form $... | https://mathoverflow.net/users/30730 | Numbers of a certain form not expressible as squares | I recommend this [survey](http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf) about the solution of Catalan's conjecture. We learn from here that the special case you are considering, namely $x^p-y^q=1$ for $p=2$, was solved by Chao Ko in 1964. In 1976 Chein published a simpler ... | 3 | https://mathoverflow.net/users/11919 | 120134 | 67,651 |
https://mathoverflow.net/questions/120131 | 1 | Let $X$ be a smooth complex projective variety, $\dim(X)>3$ and $D$ be a very ample possibly reducible divisor on $X$. Is it true that $\textrm{Pic}(X)\cong \textrm{Pic}(D)$? If not, what would be a reasonable condition on $D$ (for example, would it suffice that $D$ is irreducible)?
(if you have a precise reference,... | https://mathoverflow.net/users/13441 | Picard group of a very ample divisor in a smooth variety of dimension >3 | The answer is **yes** under very mild assumptions. In fact there is the following result.
>
> **Proposition.** Let $L$ be a $k$-ample line bundle on a normal, irreducible, projective variety $X$ with at most Cohen-Macauley singularities. Assume that the dimension of the locus of non-rational singularities of $X$ is... | 4 | https://mathoverflow.net/users/7460 | 120135 | 67,652 |
https://mathoverflow.net/questions/120139 | 4 | Let X be a K3 surface and $Y=X/\mathbb{Z}\_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi\_2(X)=\pi\_2(Y)$, while we know $H\_2(X)$ and $H\_2(Y)$ are very different.
What are $\pi\_2(X)$ and $\pi\_2(Y)$?
| https://mathoverflow.net/users/5259 | Homotopy groups of K3 | Hurewicz theorem says that for a simply connected space $X$, $\pi\_2(X)\cong H\_2(X,\mathbb Z)$. So $\pi\_2(K3)\cong H\_2(K3,\mathbb Z)\cong \mathbb Z^{22}$. Here is a link:
<http://en.wikipedia.org/wiki/Hurewicz_theorem>
| 8 | https://mathoverflow.net/users/943 | 120141 | 67,656 |
https://mathoverflow.net/questions/120127 | 6 | Hello everyone.
Let $V$ and $W$ be finite dimensional vector spaces over some field $K$. Consider $\rho:Sym(V)\to End(W)$ a homomorphism of algebras with unit (i.e., a representation of $Sym(V)$ on $W$). Fixing a basis $\{v\_1,\ldots,v\_m\}$ of $V$ there is associated an isomorphism $\Phi:K[x\_1,\ldots,x\_m]\to Sym(V... | https://mathoverflow.net/users/10328 | Representation theory for the exterior algebra | I'm not sure if this answers your question, but if $V$ is finite-dimensional and you are just asking for representations of the algebra structure, then the only irreducible representation of $\Lambda(V)$ is the trivial one-dimensional representation, in which $V$ acts as $0$.
Indeed, let $W$ be an irreducible represe... | 7 | https://mathoverflow.net/users/703 | 120149 | 67,661 |
https://mathoverflow.net/questions/120126 | 2 | In several branches of applied mathematics the problem arises to describe the intersection of two cones in three space.
I have searched and found a few references that discuss the problem for cones with parallel axes. I am interested in the general case.
Assume that one cone has vertex at the origin with a certain ... | https://mathoverflow.net/users/18755 | Intersection of Cones in Three Space | I am not sure I understand the question, see the remarks of Qfwfq. Here is something related which might nevertheless be interesting for some branches of applied mathematics: an algorithm for **parameterizing** the intersection of two *arbitrary* quadrics, given in implicit form with integer coefficients. There are sev... | 4 | https://mathoverflow.net/users/30800 | 120154 | 67,664 |
https://mathoverflow.net/questions/120160 | 10 | The survey paper of Prof. Dan Boneh entitled
"Twenty years of attacks on the RSA cryptosystem" mentioned that (Page 5)
one can attack CRT-RSA in square root of decryption exponent. However no
argument is given. I know this comes from man-in-the-middle attack.
However I cannot understand the idea clearly.
Is there an... | https://mathoverflow.net/users/29295 | Attack on CRT-RSA | It's a good question, since it looks like Boneh's paper doesn't give a reference. It's not actually a man-in-the-middle attack, at least not the attack I've seen. Instead, it's reminiscent of baby-step giant-step but with an extra twist. Here's how it works.
Suppose we are given $N$ and $e$, where $N=pq$ with $p$ and... | 13 | https://mathoverflow.net/users/4720 | 120166 | 67,671 |
https://mathoverflow.net/questions/120172 | 12 | Suppose $f(x\_1,x\_2,\dots)=\frac{P}{Q}$, where $P,Q$ are polynomials in several variables with integer coefficients that have the same degree. Let's denote by $S(f)$ the set of integers $n$ for which $f(x\_1,x\_2,\dots)=n$ is solvable in integers.
Which sets $S\subset \mathbb Z$ can be written as $S(f)$ for some $f$... | https://mathoverflow.net/users/2384 | Sets of integers represented by degree zero rational functions | A set $T \subseteq \mathbb{Z}$ can be written as $S(f)$ if and only if $T$ is effectively enumerable.
Proof: As in zeb's comment, the restriction to degree zero doesn't matter, and we consider rational functions of arbitrary degree.
Suppose $T$ is effectively enumerable. By the MRDP theorem choose a polynomial $f(\... | 8 | https://mathoverflow.net/users/5229 | 120179 | 67,675 |
https://mathoverflow.net/questions/120162 | 3 | (Crossposted from math.stackexchange.)
Studying all 3-regular graphs that have only faces with 5 edges or more (simplified), I empirically found (computer program) that many hypothetically possible graphs, that by Euler's identity may exist ($F5 = 12 + F7 + 2F8 + 3F9 + ...$), do not actually exist. Using a VF2 algori... | https://mathoverflow.net/users/14114 | Question about 3-regular graphs with a restriction (also fullerene and four color theorem) | Use Brinkmann & McKay's program "plantri"...
You will discover that there are
3 on 16 faces (as you said), 4 on 17 faces, 12 on 18 faces, 23 on 19 faces, 73 on 20 faces
and then going to Sloane's online encylopaedia you discover
<https://oeis.org/A081621>
So in short, the answer to your question is "yes, the ... | 7 | https://mathoverflow.net/users/1492 | 120181 | 67,676 |
https://mathoverflow.net/questions/120168 | 3 | Is there a "standard" construction to get a Vitali set in $\mathbb R$ of full outer Lebesgue measure?
| https://mathoverflow.net/users/24152 | Vitali sets of full outer measure | See e.g. <https://groups.google.com/forum/?fromgroups=#!topic/sci.math/ofkao7iugNg>
| 2 | https://mathoverflow.net/users/13650 | 120182 | 67,677 |
https://mathoverflow.net/questions/9951 | 35 | Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection?
Edit ('13): Perhaps it is more interesting to ask whether limits in category theory can be seen as special limits of ultrafilters or nets.
| https://mathoverflow.net/users/2841 | Limits in category theory and analysis | I have asked [this question on math.stackexchange](https://math.stackexchange.com/questions/60590/category-theoretic-limit-related-to-topological-limit) last year, and got satisfying answer.
(So this construction did not come from me.)
Let $(X,\mathcal O)$ be a topological space, $\mathcal F(X)$ the partialy ordered ... | 31 | https://mathoverflow.net/users/15292 | 120183 | 67,678 |
https://mathoverflow.net/questions/119619 | 16 | Previously, I asked a [question on mathoverflow](https://mathoverflow.net/questions/100498/isotopy-inverse-embeddings-vs-diffeomorphisms) comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question about the relation between the... | https://mathoverflow.net/users/21095 | homotopy type of embeddings versus diffeomorphisms | **Personal comment:** It seems the discussion in this question *finally* led me to understand how to modify [Agol's argument](https://mathoverflow.net/questions/100498/isotopy-inverse-embeddings-vs-diffeomorphisms/100660#100660) to answer the present question. In fact, my motivation when asking that question answered b... | 15 | https://mathoverflow.net/users/21095 | 120186 | 67,679 |
https://mathoverflow.net/questions/120185 | 1 | A 6-dimensional ( fundamental) representation of $SU(6)$ becomes (3,2) representation in $SU(3)\times SU(2)$. We can decompose $6\times 6$ of $SU(6)$ into 21-dimensional symmetric and 15-dimensional anti-symmetric representations. What can be the symmetric and anti-symmetric parts of that representation for $SU(3)\time... | https://mathoverflow.net/users/30996 | What are the symmetric and anti-symmetric representations of $6\times6$ of $SU(6)$ in $SU(3)\times SU(2)$? | First we have $3\times 3 = 6+3$ and $2\times 2 = 3+1$. Then combining this we have
$(3,2)\times (3,2) = [ (3,1)+(6,3) ] + [ (3,3)+(6,1) ]$
| 2 | https://mathoverflow.net/users/3992 | 120188 | 67,680 |
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