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https://mathoverflow.net/questions/41492 | 9 | It seems to me that the results in [this important paper of Kahn-Markovich](http://arxiv.org/abs/0910.5501) imply the following fact. Let $M^3$ be any closed hyperbolic 3-manifold. For every $\epsilon > 0$ there is a natural number $R(\epsilon)$ such that for every $R>R(\epsilon)>0$ there is a closed geodesic whose len... | https://mathoverflow.net/users/6205 | Is the spectrum of closed geodesics in a closed hyperbolic 3-manifold asymptotically homogeneously dense? | I think this should just follow from the exponential mixing of the geodesic flow (due to Pollicott).
Exponential mixing says that there is a constant $q$ such that if you have two smooth functions $f$ and $h$ on the unit tangent bundle, and $g\_t$ is the geodesic flow, then there is a constant $C$ depending on $f$ an... | 14 | https://mathoverflow.net/users/1335 | 41521 | 26,504 |
https://mathoverflow.net/questions/41525 | 16 | When talking about the Eilenberg-Maclane space $K(G,n)$, we usually restrict our attention to the situation where $G$ is abelian. In that case, we get $\Omega K(G,n)=K(G,n-1)$, so we can call $K(G,n)$ a *delooping* of $K(G,n-1)$.
Since $\pi\_n$ is always abelian for $n>1$, it only makes sense to talk about $K(G,1)=BG... | https://mathoverflow.net/users/303 | How should I think about delooping? | One possible answer: Stasheff proved that a (connected) space $X$ is (homotopy equivalent to) a loopspace if and only if $X$ is an algebra over the $A\_\infty$ operad (or rather I should say *an* $A\_\infty$ operad).
See for instance [this article](http://www.ams.org/notices/200406/what-is.pdf).
| 10 | https://mathoverflow.net/users/83 | 41526 | 26,505 |
https://mathoverflow.net/questions/41516 | 0 | Let $X$ a smooth manifold. Is the pullback morphism
$\Omega^\bullet(X)\to\Omega^\bullet(X\times \mathbb{R}^n)$ an acyclic cofibration of differential graded commutative algebras? I guess so, and even that this should be the basic example to have in mind, but being no expert, I don't trust myself too much.
| https://mathoverflow.net/users/8320 | Cofibrations of differential graded commutative algebras | It depends completely on what you mean by cofibrations. The choice is
not quite simple to make as the homotopy category of real commutative dga's is
anti-equivalent to "real homotopy" which would suggest that the cofibrations
should correspond very roughly to fibrations of spaces (judging from your
example this looks l... | 1 | https://mathoverflow.net/users/4008 | 41528 | 26,507 |
https://mathoverflow.net/questions/41522 | 1 | Let $L=L\_1 \cup ... \cup L\_n$ be the union of $n$ distinct lines through the origin in $\mathbb{R}^{3}$. I'd like a convincing argument that $\mathbb{R}^{3} \setminus L$ is homotopy equivalent to a wedge of $n$ circles (if that is true). In fact, I especially care about the case $n=2$.
I know this sounds like a hom... | https://mathoverflow.net/users/6254 | Complement of lines and wedges of spheres | First, deformation retract $\mathbb{R}^3$ minus $L$ to $S^2$ minus $2n$ points (you can do this since you've removed the origin). Stereographically project from one of the punctures, and you've got $\mathbb{R}^2$ minus $2n-1$ points. Choose a point away from the punctures and draw disjoint based loops around each of th... | 6 | https://mathoverflow.net/users/353 | 41529 | 26,508 |
https://mathoverflow.net/questions/41470 | 5 | I'm looking for such pathological examples of uniform spaces which are not metrizable, but whose underlying topology is metrizable. Willard in his General Topology text constructs such a uniformity using ordinals. I am asking for examples which do not rely on ordinals.
EDIT: Below is an example by Daniel Tausk using ... | https://mathoverflow.net/users/6249 | Nonmetrizable uniformities with metrizable topologies | The family of **all** neighbourhoods of the diagonal of $\mathbb{R}$ (with its normal topology) is a uniformity without a countable basis but it generates the normal topology.
The normal topology of $\mathbb{R}$ is the topology generated by the usual distance.
To see that the open sets in the plane that contain the dia... | 5 | https://mathoverflow.net/users/5903 | 41531 | 26,509 |
https://mathoverflow.net/questions/41483 | 2 | I've been reading about reverse mathematics (mostly on wikipedia), and I had been thinking that I understood how to prove the equivalences to WKL0 and ACA0 mentioned in the its article. However, I now realize that my idea of how WKL0 can prove that every continuous real valued function on [0,1] is bounded. My idea woul... | https://mathoverflow.net/users/nan | Proving boundedness of continuous images of [0,1] in WKL0 | Here is a sketch of how the proof might go.
If $f:[0,1]\rightarrow\mathbb R$ is continuous but not bounded then the sets $S\_n=[0,1]\setminus f^{-1}[(-n,n)]$ are closed with $S\_n\ne\emptyset$ for all $n\in\mathbb N$. According to the definition of continuous function the sets $f^{-1}[(-n,n)]$ are represented as $\Si... | 2 | https://mathoverflow.net/users/4600 | 41533 | 26,511 |
https://mathoverflow.net/questions/7492 | 18 | Inspired by [this thread](https://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also-manifolds), which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space underlying a complex variety is a topolo... | https://mathoverflow.net/users/1231 | Algebraic varieties which are topological manifolds | The answer from Dmitri motivates this partial answer from the topological side of the question.
It is [a theorem of Mark Goresky](http://www.math.ias.edu/~goresky/pdf/triangulations.pdf) and others that every stratified space, and in particular every complex variety $V$, has a smooth triangulation. Moreover, I would ... | 14 | https://mathoverflow.net/users/1450 | 41537 | 26,512 |
https://mathoverflow.net/questions/41535 | 17 | Suppose that I have $n$ homogeneous polynomials $f\_1, \dots, f\_n \in \mathbb{C}[x\_1, \dots, x\_m]$ and that $n < m$. Is there a well known method or algorithm to determine if these polynomials are algebraically independent?
As far as I know the Jacobian criterion works only for the case where $n=m$.
| https://mathoverflow.net/users/9899 | How to show a set of polynomials is algebraically independent? | The polynomials are algebraically independent if and only if
$$
df\_1 \wedge df\_2 \wedge \cdots \wedge df\_n
$$
is not identically zero. In other words, you have only to check that
one of the maximal minors of the matrix $\left( \frac{\partial f\_i}{\partial x\_j} \right)$ is nonzero.
| 21 | https://mathoverflow.net/users/605 | 41539 | 26,514 |
https://mathoverflow.net/questions/40403 | 2 | I'm having a hard time finding some references on series solutions for "nonlinear" ODE's, the most I could find was a small excerpt on Wikipedia.
<https://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations>
Most books just say something along the lines of ... and the method is applicable to nonli... | https://mathoverflow.net/users/9404 | Power series solutions for nonlinear ordinary differential equations - references | Nonlinear differential equations is hard to find good references on-partly due to the difficulty of the subject and partly due to the highly specialized nature of most of the research problems connected with them. But a lot of these problems are really problems of numerical approximation-so I think you'll have greater ... | 1 | https://mathoverflow.net/users/3546 | 41549 | 26,520 |
https://mathoverflow.net/questions/41502 | 10 | Let $(A,\alpha, G)$ be a $C^\*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes\_\alpha \Gamma$ and $A\rtimes\_\alpha G$.
Question 1: Is the canonical map $A\rtimes\_\alpha \Gamma \to A\rtimes\_\alpha G$ injective?
Question 2... | https://mathoverflow.net/users/9401 | Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection | The universal case is also injective. What one needs to show is that any covariant representation of $(A,\Gamma)$ extends to a covariant representation of $(A,G)$. This can be shown by using induced representation.
| 7 | https://mathoverflow.net/users/7591 | 41556 | 26,523 |
https://mathoverflow.net/questions/41559 | 4 | Let $X$ be a smooth projective variety and let $D$ be an effective divisor on $X$. Is there a natural way to describe the tangent space to $|D|$ (or $|D|^\vee$, of course) at a divisor $D'$? Preferably as some sort of cohomology group, again ideally on $X$. I would prefer to avoid using the fact that the linear system ... | https://mathoverflow.net/users/622 | Tangent Space to a Linear System | The linear system $|D|$ is the projectivization of $H^0(X,\mathcal O(D)).$ The divisor $D'$ corresponds to a line $\ell\_{D'} \subset H^0(X,\mathcal O(D)).$ (This line consists of all the sections whose zero loci are equal to $D'$.) The space $Hom(\ell\_{D'},H^0(X,\mathcal O(D))/\ell\_{D'})$
is a vector space of the sa... | 9 | https://mathoverflow.net/users/2874 | 41560 | 26,525 |
https://mathoverflow.net/questions/39808 | 7 | The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of [Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-171/issue-2/Amenable-correspondences-and-approximation-properties-for-von-Neumann-algebras/pjm/1102368918.full)) and t... | https://mathoverflow.net/users/6269 | Are the compact and Haagerup approximation properties equivalent? | *(I removed my first answer as it contained an egregious mistake, pointed out by Yemon; here's a second attempt)*
I think that $T\_\phi$ may fail to be compact.
Fix a sequence of projections $\{p\_k\}$ in $M$, pairwise orthogonal, with $\tau(p\_k)=2^{-k}$ ($\tau$ the trace in $M$) and sum 1. Now define
$$
\phi:M\t... | 4 | https://mathoverflow.net/users/3698 | 41562 | 26,526 |
https://mathoverflow.net/questions/41561 | 10 | Liouville's theorem states that all bounded holomorphic functions on $\mathbb{C}^n$ are constant.
I'm wondering which connected complex manifolds have this property ?
Connected compact complex manifolds have it since all holomorphic functions there are constant.
There are no simply connected proper open subsets o... | https://mathoverflow.net/users/9152 | Complex manifolds where bounded holomorphic functions are constant | We have that there are no non-constant bounded functions on $\mathbb C^\*=\mathbb
C\setminus\{0\}$. The easiest way to see that is to notice that such a function
has a removable singularity at the origin and hence comes from a bounded
function on $\mathbb C$ (which incidentally has a removable singularity at
$\infty$ a... | 14 | https://mathoverflow.net/users/4008 | 41565 | 26,527 |
https://mathoverflow.net/questions/41546 | 2 | Hi
Consider Poisson equation with Neumann boundary condition but the right hand side of boundary condition is in term of the unknown function $u$.
How we can solve it?
$\Delta u(x) = f(x)\quad in~ \Omega$
$\frac{\partial u(x)}{\partial n }=g(u(x))\quad on~\partial \Omega$
where n is outward normal vector.
Fo... | https://mathoverflow.net/users/9900 | Poisson equation with special Neumann BC | Set $G$ a primitive of $g$. Then the solution is a critical point of the functional $E:H^1(\Omega)\rightarrow{\mathbb R}$ defined by
$$E[u]:=\int\_\Omega \left(\frac12|\nabla u|^2+fu\right)dx-\int\_{\partial\Omega}G(u)ds.$$
If $G(\pm\infty)=-\infty$, you may look for a minimum of $E$ over $H^1(\Omega)$. When $g$ is a d... | 1 | https://mathoverflow.net/users/8799 | 41570 | 26,531 |
https://mathoverflow.net/questions/41575 | 1 | Given a smooth projective surface $S$ over an algebraically closed field, a sheaf rings or algebras $R$ on $S$ and a simple left $R$-module $M$, i.e. $Hom\_R(M,M)=k$.Then we have $Hom\_R(M,M(-i))=H^{0}(S,\mathcal{H}om\_R(M,M)\otimes O(-i))=0$ for $i>0$.
Now given some $a\in k, a\neq 0$. Then $Hom\_R(M,M(-i))=0$ impl... | https://mathoverflow.net/users/3233 | Why is multiplication with a scalar no global morphism? | The element $a \in k$ has degree zero so it gives a global morphism $M \to M$. These are the only global morphisms because of your simplicity assumption.
If you take instead an element of degree $i > 0$, multiplication with it gives a global morphism $M(-i) \to M$.
Think about $S=\mathbb{P}^2$ and $M=\mathcal{O}$ i... | 3 | https://mathoverflow.net/users/7460 | 41578 | 26,534 |
https://mathoverflow.net/questions/41576 | 4 | Let $k$ be an algebraic closure of a finite field of characteristic $p$. Fix an integer $l\neq p$. For a separated $k$-scheme $X$ of finite type, we define the (compactly supported) Euler characteristic of $X$ to be $$e(X) =\sum\_i (-1)^i \dim\_\mathbf{Q\_l} H^i\_c(X,\mathbf{Q}\_l).$$ Here $H^i\_c(-,\mathbf{Q}\_l)$ den... | https://mathoverflow.net/users/4333 | Behaviour of euler characteristics in characteristic p for finite etale covers | It's not true in general. Over a field of characteristic $p>0$, the map $f:\mathbf{A}^1\to\mathbf{A}^1$ defined by $f(z)=z^p+z$ is etale because its derivative is $1$. The degree of $f$ is $p$, and the Euler characteristic of $\mathbf{A}^1$ is $1$, but $1\neq 1\times p$.
| 5 | https://mathoverflow.net/users/1114 | 41579 | 26,535 |
https://mathoverflow.net/questions/41557 | 11 | Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and $\frac{l(\alpha x)}{l(x)}$ (for all rational $x$) are bounded with some constants depending on $\alpha$ only.
Is this resul... | https://mathoverflow.net/users/5712 | Lengths of continued fractions for the numbers with fixed ratio | It definitely is not new for the length, and I am nearly sure that is not for height either.
See, for example,
Labhalla, Salah; Lombardi, Henri
Transformation homographique appliqu´ee `a un d´eveloppement en fraction continue fini ou
infini. (French) [Fractional linear transformations applied to finite and infinit... | 7 | https://mathoverflow.net/users/4312 | 41589 | 26,543 |
https://mathoverflow.net/questions/41598 | 5 | This is related to my [recent question](https://mathoverflow.net/questions/41597/transfinite-induction-a-theorem-of-pedersen-and-chains-of-subalgebras-of-bh) and would provide a natural positive answer to Question 2. I am sure this must be known to experts.
>
> **Question:** Is there a monotone injection $(\omega\_... | https://mathoverflow.net/users/8176 | Monotone injection of an ordinal into $[0,1]$ | No, because you could use it to construct an injective map $\omega\_1\to\mathbb{Q}$, mapping $\alpha<\omega\_1$ to some rational number between $\alpha$ and $\alpha+1$.
| 18 | https://mathoverflow.net/users/9342 | 41601 | 26,550 |
https://mathoverflow.net/questions/41572 | 1 | Let $X$ be a scheme, $S$ a $K3$ surface and $F$ a flat family of coherent sheaves on $S$ parametrized by $X$. Let us assume that for every $x\in X$ $F\_x$ is locally free, has fixed Chern classes and satisfies $h^1(F\_x)=h^2(F\_x)=0$. Does there exist a scheme $A$ together with a morphism $p:A\to X$ such that $p^{-1}(x... | https://mathoverflow.net/users/33841 | Families of sheaves and automorphisms | There are standard ways of constructing this kind of objects, but I can't immediately think of a reference, so here it goes:
Let $p:Y\to X$ be a projective map (in your case, $Y=S\times X$), let $F$ and $G$ be coherent sheaves on $Y$ with $G$ flat over $X$. Then there is a scheme $H$ of finite type over $X$ whose fib... | 2 | https://mathoverflow.net/users/2653 | 41602 | 26,551 |
https://mathoverflow.net/questions/41606 | 0 | Hello!
*I recently started (it's purely self-education) reading a **"Mathematical programming and optimizations"** book, did a vast part of the exercises related to the theoretical part and at one moment I got the following question about convex sets:*
I'm almost sure this statement is correct, but unfortunately, c... | https://mathoverflow.net/users/6256 | Convex sets and projections | I presume what you want to prove is the following. Let $S$ be a
nonempty closed subset of $\mathbb{R}^n$. Then if there is a point $y\in\mathbb{R}^n$
and there are at least two points $p$ and $q$ in $S$ with Euclidean distance $d$ from $y$
(where $d$ is the distance of $y$ from $S$), then $S$ is not convex. To see
thi... | 4 | https://mathoverflow.net/users/4213 | 41608 | 26,554 |
https://mathoverflow.net/questions/41604 | 8 | Lets say I have a [geometric distribution](http://en.wikipedia.org/wiki/Geometric_distribution) (of the number X of Bernoulli trials needed to get a success) with parameter `p` (success probability of a trial).
Assume I randomly sample `n` elements from this distribution.
My problem is: what is the expected *maximu... | https://mathoverflow.net/users/7083 | What is the expected maximum out of a sample (size N) from a geometric distribution? | Here's an answer: $\sum\_{i=1}^n \binom{n}{i} (-1)^{i+1} \frac{1}{1-q^i}$, where $q = 1-p$. Maybe someone else can help you simplify that further.
The argument: Let $Y = \max\{X\_1, X\_2, \ldots, X\_n\}$, where the $X\_i$ are $n$ sampled values from the geometric distribution.
Then $E[Y] = \sum\_{k=0}^{\infty} P(Y... | 13 | https://mathoverflow.net/users/9716 | 41614 | 26,556 |
https://mathoverflow.net/questions/41597 | 11 | This post is closely related to [this one](https://mathoverflow.net/questions/24440/must-we-close-weakly-to-apply-the-spectral-theorem). (In fact I copied some of its content.)
Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed to be non-degenerate.
T... | https://mathoverflow.net/users/8176 | Transfinite induction, a theorem of Pedersen, and chains of subalgebras of $B(H)$ | The answer to Question 2 is yes; such a chain can be found using only diagonal operators with respect to some fixed basis $(e\_n)\_{n\in{\mathbb N}}$ for $H$. As a starting point, you can find an $\omega\_1$-sequence of subsets $S\_\alpha\subset{\mathbb N}$ such that if $\alpha<\beta$, $S\_\beta\setminus S\_\alpha$ is ... | 7 | https://mathoverflow.net/users/75 | 41617 | 26,558 |
https://mathoverflow.net/questions/41569 | 2 | A neutral Tannakian category over a field $k$ is a rigid $k$-linear abelian tensor category
$\mathcal{C}$ whose unit $1$ satisfies $\mathrm{End}(1) \simeq k$, and is
moreover equipped with an exact faithful tensor functor $\omega : \mathcal{C} \rightarrow
\mathrm{Vect}\_k$ into the category of finite dimensional $k$-v... | https://mathoverflow.net/users/8141 | The condition End(1) = k in Tannakian Categories | If you say the other axioms correctly, then the condition on $\operatorname{End}(1)$ is redundant. Indeed, the word "tensor functor" implies that $\omega: 1 \mapsto k$, and the word "faithful" implies that $\operatorname{End}(1) \hookrightarrow \operatorname{End}(\omega(1))$. What you should include that you don't on y... | 6 | https://mathoverflow.net/users/78 | 41623 | 26,562 |
https://mathoverflow.net/questions/41612 | 5 | Hi, I have a general question about the relationship between exponential sums and differential equations. In particular, I have been trying to read Katz' work on the subject (his book and his lecture notes) but I am having trouble understanding the big idea and getting confused with all of the algebraic geometry backgr... | https://mathoverflow.net/users/9769 | Exponential sums and differential equations | In simplistic terms, exponential sums arise in characteristic p geometry because curves have Artin-Schreier coverings. The appearance of (complete) exponential sums in that context is a reflection in cohomology of that kind of Galois covering, and can be understood pretty much in Weil's terms and technology. (Appendix ... | 2 | https://mathoverflow.net/users/6153 | 41627 | 26,565 |
https://mathoverflow.net/questions/41624 | 7 | Recall that $\operatorname{Sch}$ can be identified with the subcategory of (Zariski-)locally representable (by an affine) étale sheaves on $\operatorname{CRing^{op}}$. In this case, $\operatorname{Spec}(-):\operatorname{CRing^{op}}\to \operatorname{Sch}$ is simply the co-Yoneda embedding $R\mapsto \operatorname{Hom}(R,... | https://mathoverflow.net/users/1353 | A sheaf-theoretic version of the proj construction? | Perhaps I've not understanding your question, but it sounds like you're asking "What is the functor of points of a Proj?"
The answer, of course, is the functor that sends a scheme $X$ to the set of line bundles $L$ on that scheme equipped with a graded map of $R$ to $\Gamma(X;\oplus\_{n\geq 0} L^n)$ whose image gener... | 7 | https://mathoverflow.net/users/66 | 41630 | 26,567 |
https://mathoverflow.net/questions/41616 | 16 | The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\pi\_{1}(BG)$? A reference on this would be great. My initial guess: $\pi\_{1}(BG)$ is the quotient group $\pi\_{0}(G)$ f... | https://mathoverflow.net/users/5801 | What is π_1(BG) for an arbitrary topological group $G$? | If $G$ is homeomorphic to a Cantor set (e.g. $G=\mathbb Z\_p$), then $BG$ contains a copy of the Hawaiian earrings in it. To see this, take a sequence of points of $G$ that converges to the identity element: you'll get a corresponding sequence of 1-cells in $BG$ that converge to the the degenerate 1-cell. The fundament... | 32 | https://mathoverflow.net/users/5690 | 41632 | 26,569 |
https://mathoverflow.net/questions/41638 | 6 | Hello, I was looking for an answer to the following question:
Consider an algebraically closed field $K$ and a map $K \rightarrow K^n$ given by $a \mapsto (p\_1 (a) , \ldots p\_n (a))$ for $p\_i \in K[t]$. Is the image of this map an algebraic set?
Certainly this is false for $K^2 \rightarrow K^n$, for example $(x,... | https://mathoverflow.net/users/18 | Regular Morphism From Affine Line | Dear Damien, let's show that your morphism $f: \mathbb A^1\_K \to \mathbb A^n\_K $ is proper, hence closed, hence certainly has closed image.
For that it is enough to prove that each $f\_i:\mathbb A^1\_K \to \mathbb A^1\_K$ is proper. But this follows from the stronger property that $f\_i$ is finite or dually that t... | 5 | https://mathoverflow.net/users/450 | 41667 | 26,588 |
https://mathoverflow.net/questions/41660 | 5 | Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector
$x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by
$\|\cdot\|$ both the Euclidean norms in $R^m$ and $R^n$ and the induced
operator norm.) The answer may well be negative -- any examples?
---
In case the answer... | https://mathoverflow.net/users/9924 | Is the operator norm always attained on a $\{0,1\}$-vector? | The answer is no. First, to understand the question, WLOG $f$ is symmetric and positive definite; a general $f$ has a polar decomposition $f = os$ and the orthogonal factor $o$ has no effect on any of the norms in question. Then, WLOG $f$ is a rank 1 projection. The second and subsequent eigenvalues of $f$ do not incre... | 9 | https://mathoverflow.net/users/1450 | 41669 | 26,589 |
https://mathoverflow.net/questions/41666 | 15 | Given odd positive integer $n$ and a monic polynomial $f(x)=(x-x\_1)\dots (x-x\_n)$ with $n$ distinct real roots. Is it always true that $\sum f'(x\_i) > 0$? I may prove it for $n=3$ and $n=5$ and it looks plausible.
| https://mathoverflow.net/users/4312 | sum of derivatives in roots of a polynomial of odd degree | If I'm not mistaken this is basically the same question as [this](http://www.artofproblemsolving.com/Forum/viewtopic.php?p=366673&sid=0f77743cc628257f609bb649b7ecfc84#p366673) question from the international mathematical olympiad in 1971. The statement is only true for 3 and 5 variables showing that there is no obvious... | 19 | https://mathoverflow.net/users/2384 | 41674 | 26,590 |
https://mathoverflow.net/questions/41619 | 13 | A nice property of $\mathbb P^n$ is:
>
> Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect.
>
>
>
(for example, any 2 curves in $\mathbb P^2$ intersect)
There are other smooth varieties $X$ when Properties 1 holds. For example, a sufficient cond... | https://mathoverflow.net/users/2083 | Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences | This is a very interesting question and I guess that a general answer is unknown, already in the case $i=1$. Let me just make the following
**Remark.** There exists no upper bound on $\textrm{rank } NS(X)$ which is independent on the dimension.
In fact, let us consider a complex Abelian variety $X$ of dimension $g$... | 7 | https://mathoverflow.net/users/7460 | 41678 | 26,592 |
https://mathoverflow.net/questions/41675 | 11 | Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of operators on Hilbert spaces so please go easy on me :), yet i tried reading parts of the classical Simon and Reeds "Func... | https://mathoverflow.net/users/9419 | How "generalized eigenvalues" combine into producing the spectral measure? | Wouldn't Theorem 4.2 in [here](http://mathserver.neu.edu/~king_chris/GenEf.pdf) answer your question?
**Theorem 4.2.** *(Generalized Eigenfunctions, by Mustafa Kesir)* Let $\mathcal H$ be a Hilbert Space. Given a self-adjoint operator $A$ in $\mathcal H$ and a Hilbert–Schmidt rigging of $\mathcal H$, there exists a ... | 4 | https://mathoverflow.net/users/3698 | 41681 | 26,593 |
https://mathoverflow.net/questions/41679 | 1 | 3 players are playing a game where they get to pick independently without knowing the other players picks one of 2 prizes (A,B) and the payout is (a,b) for the two prizes, divided by how many people picked the specific prize.
For example, if the prizes are (3,1) and 2 people picks A and 1 person picks B, the 2 people... | https://mathoverflow.net/users/9926 | Simple(?) game theory | This looks like a homework exercise on mixed strategy Nash equilibria.
My ASCII art skills are a little rusty, but let's write up a "nice" litte 2x2x2 cube:
```
+--------------+-----------------+
C plays 3 / / / |
+--------------+----------------+ |
C pla... | 2 | https://mathoverflow.net/users/8842 | 41687 | 26,597 |
https://mathoverflow.net/questions/41649 | 4 | I've been searching google and scholar google, but i only have come upon orderings and Hermitian forms on \*-fields.
Has real algebraic geometry been carried over to \*-rings? \*-rings are rings with an involution. For example are there Positivstellensätze and characterizations of sums of squares by total positivity ... | https://mathoverflow.net/users/nan | Doing Real Algebraic Geometry on *-Rings | Konrad Schmüdgen has set up a programme to develop analogues of the basic results in Real Algebraic Geometry in a setup of $\star$-algebras. See [arxiv.org/abs/0709.3170](http://arxiv.org/abs/0709.3170) and for example [arxiv.org/abs/0903.2708](http://arxiv.org/abs/0903.2708). There is also interesting recent work by J... | 1 | https://mathoverflow.net/users/8176 | 41690 | 26,599 |
https://mathoverflow.net/questions/41676 | 7 | I'm reading stalling's article "the augmented ideal in group ring" in Ann. Math. Studies 84(R. H. Fox memorial volume)
In his final remark, he says that Milnor's link invariant could be interpreted by using Spectral sequence.(see Milnor, Isotopy of links, Algebraic geometry and topology, Princeton press)
Are there ... | https://mathoverflow.net/users/7776 | Milnor's isotopy invariant using spectral sequence? | Have you read the Wikipedia page on Massey products?
<http://en.wikipedia.org/wiki/Massey_product>
It mentions Massey products are differentials in the Atiyah-Hirzebruch spectral sequence for a K-theory with local coefficients. The Atiyah-Hirzebruch spectral sequence is to a (co)homology theory what cellular (co)h... | 4 | https://mathoverflow.net/users/1465 | 41692 | 26,601 |
https://mathoverflow.net/questions/41543 | 10 | I cannot find any answer to that apparently simple problem :
On a square lattice, a path is given by a sequence of relative moves in {"move forward", "turn right" and "turn left"}.
Is there a rule that characterizes if a path is self-avoiding (or not) ?
Edit : Let me precise the kind of rule I am looking for:
Let'... | https://mathoverflow.net/users/8779 | How to characterize a Self-avoiding path. | I'm only now starting to understand the question. Let me formulate it slightly differently, but I think it's easy to translate between this and your formulation. I'll imagine a little machine with three possible operations: take a step (which is always in the direction of an arrow on the machine's head, so would be you... | 12 | https://mathoverflow.net/users/1459 | 41694 | 26,602 |
https://mathoverflow.net/questions/41693 | 8 | I expect this to be a small understanding problem and not a real interesting question.
In [Gabriel's thesis](http://ncatlab.org/nlab/show/Des+Cat%25C3%25A9gories+Ab%25C3%25A9liennes) you find a proof of the theorem that every small abelian category $C$ admits a faithful exact functor to the category of abelian groups... | https://mathoverflow.net/users/2841 | projective generator in the category of left-exact functors | The object $U$ is a projective generator in the category of additive functors $C\to Ab$. It is also a generator of the category of left exact functors $C\to Ab$. It is not projective in the latter category, though.
Generally, left exact functors are similar to sheaves (on the category opprosite to $C$; epimorphisms i... | 5 | https://mathoverflow.net/users/2106 | 41699 | 26,606 |
https://mathoverflow.net/questions/41682 | 2 | We know the fact that $K\_0(-)$ and $K\_1(-)$ are continuous under inductive sequence of $C^\*$-algebras (in fact inductive system), i.e. $K\_0(\varinjlim A\_n)=\varinjlim K\_0(A\_n)$ similar for $K\_1(-)$. In fact it is also true that $M\_k(\lim\_{\rightarrow} A\_n)=\varinjlim M\_k(A\_n)$ for $k\in \mathbb N$.
Q1: D... | https://mathoverflow.net/users/9401 | Continuity of functors under inductive sequence of $C^*$-algebras. | The answer to Q1 is a standard fact, which can be found in the standard text books like Bruce Blackadar's book on $C^{\star}$-algebras. The answer to Q2 is "yes" and, again, looking in a text book helps.
For Q3, I think that it is worth noting that the Algebraic K-theory of $C^{\star}$-algebras is very interesting bu... | 3 | https://mathoverflow.net/users/8176 | 41700 | 26,607 |
https://mathoverflow.net/questions/40800 | 25 | Barry Mazur and I have come across the question below, motivated by (but independent
of) issues regarding the Leopoldt conjecture.
Suppose that $\mathbf{C}$ is the complex numbers.
Let $H$ be a finite set, and
let $S$ be a subset of $H$. The vector space $X = \mathbf{C}^{H}$
has a canonical basis consisting of the ... | https://mathoverflow.net/users/nan | How does one compute invariants of certain Grassmannians inside the regular representation? | This is not a complete solution, but it is a way to compute things in sort-of a nice way. The module $U$ has a complementary module $U^\perp$, and the kernel of a generic module map $F:X \to U^\perp$ is isomorphic to $U$. Then you can look at the restriction $F:X\_S \to U^\perp$ and try to compute its generic rank. Thi... | 7 | https://mathoverflow.net/users/1450 | 41707 | 26,611 |
https://mathoverflow.net/questions/41658 | 3 | The $k$th Chebyshev polynomial is denoted by $T\_k$ where
$T\_k(x) = \cos(k\cos^{-1}(x))$
I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the Russian name for Chebyshev) but does anyone know the first use of this notation or verify this is the reason?
| https://mathoverflow.net/users/2011 | Where does the Chebyshev polynomial notation come from? | Great Soviet mathematician [N.I. Akhiezer](http://en.wikipedia.org/wiki/Naum_Akhiezer) mentions in his survey article *"Чебышевское направление в теории функций"* (*"Function Theory According to Chebyshev"*) that the notation
$$T\_n(x)=\frac{1}{2^{n-1}} \cos{(n\arccos x)}$$
was first introduced by S. Bernstein.
I th... | 12 | https://mathoverflow.net/users/5371 | 41717 | 26,615 |
https://mathoverflow.net/questions/41722 | 7 | Is every balanced pre-abelian category abelian? That is, given an additive category $\mathcal{A}$ in which cokernels and kernels exists, such that every morphism, which is a mono- and an epimorphism, is an isomorphism; does it follow that $\mathcal{A}$ is abelian? Note that it would suffice to prove that the canonical ... | https://mathoverflow.net/users/2841 | Is every balanced pre-abelian category abelian? | Let $B$ be the abelian category of 3-term sequences of vector spaces and linear maps $V^{(1)}\to V^{(2)}\to V^{(3)}$ (the composition can be nonzero). There are 6 indecomposable objects in this category; denote them by $E\_1$, $E\_2$, $E\_3$, $E\_{12}$, $E\_{23}$, and $E\_{123}$. Here $dim E^{(i)}\_J=1$ for $i\in J$ an... | 12 | https://mathoverflow.net/users/2106 | 41726 | 26,620 |
https://mathoverflow.net/questions/41736 | -4 | Is there a way to calculate the number of non-repeating digits that precede the periodic repeating portion of a decimal expansion? For example:
1/6 = 0.1666.... (there is 1 non repeating digit) \*\*(Correction)
1/12 = 0.08333... (there are 2 non repeating digits)
7/12 = 0.58333....(there are 2 non repeating digits)
1... | https://mathoverflow.net/users/9934 | How do you calculate/prove the length of n, the number of non-repeating digits preceeding a periodic sequence of a fractional repeating decimal | When one writes an irreducible fraction $m/n$ as a periodic digit number all one does is to write
$m/n=\frac{a}{999...9000.00}$
So the number of digits before the period is the maximum of the power of $2$ and $5$ in $n$,
i.e. wirting $n=2^\alpha 5^\beta k$ with $k$ relatively prime to $10$, the number of digits be... | 6 | https://mathoverflow.net/users/9498 | 41743 | 26,630 |
https://mathoverflow.net/questions/41744 | 3 | A group $G$ is nilpotent if and only if there is a $c\gt 0$ such that the $(c+1)$st term of the lower central series is trivial. A group $G$ is solvable if and only if there is a $c\gt 0$ such that the $c$th term of the derived series is trivial.
Is there some similar criterion for supersolvability, or at least one w... | https://mathoverflow.net/users/3959 | Is there a commutator-theoretic criterion for supersolvability of a group? | Nilpotent groups of a given class and solvable groups of a given class form varieties, each of these varieties is defined by commutator identities. Supersolvable groups of any class do not form a variety of groups. Moreover it is not a union of varieties because there exists a supersolvable group which generates a vari... | 11 | https://mathoverflow.net/users/nan | 41746 | 26,631 |
https://mathoverflow.net/questions/41745 | 5 | Let $U$ be the set of all non-null $n \times 1$ vectors $\mathbf{\mathrm{u}}$, where $u\_i \in \lbrace-1, 0, 1\rbrace$. Let $\mathbf{\mathrm{x}}$ be an $n \times 1$ vector in $\mathbf{R}^n$. Let $\mathbf{\mathrm{u\_x}}$ be the element of $U$ that is closest in angle to $\mathbf{\mathrm{x}}$. Then for any $\mathbf{\math... | https://mathoverflow.net/users/9939 | Maximum distance to nearest-lattice-point on (hyper-)sphere with unit lat-lon lattice. | Given your initial answer for $\mathbf R^2,$ to move to $\mathbf R^3$ consider the vectors $(x,y,z)$ such that $x,y,z \geq 0$ that are equiangular between the plane vectors $(1,0,0)$ and $(1,1,0).$ These make up the plane
$$ y = (\sqrt 2 - 1) x $$ with arbitrary $z \geq 0.$ In order to get the same angle with
$(1,0,1)... | 3 | https://mathoverflow.net/users/3324 | 41748 | 26,633 |
https://mathoverflow.net/questions/41609 | 22 | Any number less than 1 can be expressed in base g as $\sum \_{k=1}^\infty {\frac {D\_k}{g^k}}$, where $D\_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this number, we could equivilantly express it as $\sum \_{k=1}^\infty {\frac {C\_k}{g^{Z(k)}}}$, where $Z(k)$ is the posit... | https://mathoverflow.net/users/9712 | Have all numbers with "sufficiently many zeros" been proven transcendental? | I don't know of a paper proving the result, but I can prove it for you now. In fact, the methods in the paper you link generalize to an arbitrary base $g\gt2$. The authors of the paper don't seem to think that it generalizes quite so easily, as in the Open Problems section they state that "For bases $b\gt2$ there is th... | 22 | https://mathoverflow.net/users/1004 | 41760 | 26,638 |
https://mathoverflow.net/questions/41757 | 0 |
>
> **Possible Duplicate:**
>
> [Eigenvalues of sum of anti-commuting matrices](https://mathoverflow.net/questions/41755/eigenvalues-of-sum-of-anti-commuting-matrices)
>
>
>
I know two anti-commuting (nxn)-matrices A and B, n -even. I know also that +-a and +-b are real eigenvalues among all eigenvalues of A... | https://mathoverflow.net/users/9941 | Eigenvalues of sum of two anti-commuting matrices | Suppose for simplicity's sak that $A$ and $B$ are diagonalizable over $\mathbb{R}$
and are non-singular.
Let $V\_a$ be the $a$-eigenspace of $A$. Then by anti-commutativity,
we find $BV\_a\subseteq V\_{-a}$ etc. As $A$ and $B^2$ commute then there is
an eigenvector $v\in V\_a$ with $B^2v=b^2 v$ for some $v$. If we le... | 1 | https://mathoverflow.net/users/4213 | 41761 | 26,639 |
https://mathoverflow.net/questions/41711 | 1 | One really stupid, trivial question: A Quasi coherent sheaf $F$ on an affine group scheme(Spec R) is simply an R-module. What happens in case R is a Hopf algebra? Will the Q.coherent sheaf $F$ be an algebra in this case?
| https://mathoverflow.net/users/9492 | Quasi coherent sheaf | No. Consider the case of the trivial group scheme over a field $k$ (so $R=k$). In this situation, a quasi-coherent sheaf is just a $k$-vector space. As Lennart Meier said in a comment, you need additional structure to get an algebra, e.g., a multiplication map.
**Added:** If you just want a "pointwise" multiplication... | 1 | https://mathoverflow.net/users/121 | 41762 | 26,640 |
https://mathoverflow.net/questions/41689 | 11 | In Bar-Natan's "Knots at Lunch" seminar at the University of Toronto, we are currently discussing [a talk by Alekseev at Montpellier](http://katlas.math.toronto.edu/drorbn/dbnvp/Alekseev-1006-1.php) about Rouvière's expansion of the [Duflo isomorphism](http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf) to the... | https://mathoverflow.net/users/2051 | Why symmetric spaces? | The algebra of invariant differential operators on a symmetric space is commutative, and this is certainly not true for an arbitrary homogeneous space. While it is not true that the commutativity of the algebra $D^{G}$ of $G$ invariant differential operators on a homogeneous space $X = G/H$ implies that $X$ is symmetri... | 9 | https://mathoverflow.net/users/9471 | 41769 | 26,645 |
https://mathoverflow.net/questions/41776 | 0 | Let $C$ be a complete category, and let $P$ be presheaf on $C$. Let $X\to P$ and $Z\to P$ be objects of $\mathcal{Y}\downarrow P$, where $\mathcal{Y}$ is the Yoneda embedding. Is the pullback $X\times\_P Y$ in $Psh(C)$ representable? This is obvious when $P$ is representable, but I am not sure if it's true otherwise.
... | https://mathoverflow.net/users/1353 | Pulling back representables over a presheaf | No. Take $C$ to be the terminal category, so that the category of presheaves is just $Set$. There is just one representable: the terminal set $1$. Let $P$ be a 2-element set, with elements $a: 1 \to P$ and $b: 1 \to P$. Then the pullback of these two morphisms is the empty set, which is not representable.
| 3 | https://mathoverflow.net/users/2926 | 41779 | 26,651 |
https://mathoverflow.net/questions/41758 | 6 | A *pinching* over $M\_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m\_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is symmetric if the rule is the same for $(i,j)$ and $(j,i)$ whenever $j\ne i$.
R. Bhatia has shown that the pinching $M\map... | https://mathoverflow.net/users/8799 | Pinching and positive definite matrices | R. Bhatia proved (in [Amer. Math. Monthly 107](http://www.ams.org/mathscinet-getitem?mr=1786234)) that the operator $D\_k$ taking $M$ to its $k$-th diagonal ($M\_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive ... | 3 | https://mathoverflow.net/users/6451 | 41782 | 26,653 |
https://mathoverflow.net/questions/41587 | 4 | The following is classical theorem of Ore and Ryser, generalising famous Hall marriage theorem.
Assume that $n$ guys and $m$ girls live in a town, some guys like some girls. Three statements are equivalent:
(i) each guy may get one wife and one mistress (they do differ, but he should like both) so that all wifes ar... | https://mathoverflow.net/users/4312 | union of matroid intersection | The concept of "strongly base orderable" matroids seems to fit the bill, see for example
<http://lemon.cs.elte.hu/egres/open/Base_orderable_matroid>
In particular, partition matroids are strongly base orderable and a "union of intersections" theorem was proven by Davies and McDiarmid.
| 2 | https://mathoverflow.net/users/4020 | 41786 | 26,656 |
https://mathoverflow.net/questions/41784 | 28 | Consider the equation $x^2=x\_0$ in the symmetric group $S\_n$, where $x\_0\in S\_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $x\_0$) is attained when $x\_0$ is the identity permutation? How far may it be generalized?
| https://mathoverflow.net/users/4312 | Roots of permutations | $\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}$
The maximum of the function counting square roots is attained at $x\_0=1$ and this statement generalises quite well.
Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. For the definition, see the edit below. One has $s(... | 47 | https://mathoverflow.net/users/35416 | 41788 | 26,658 |
https://mathoverflow.net/questions/31848 | 5 | Despite the title, this is probably actually a question in linear algebra or algebraic geometry. Let me write the question(s) first, before I explain the background.
**Problems**
Let $h^{\mu\nu}\_{ij}$ represent a map from $\mathbb{R}^4\otimes\mathbb{R}^4$ to $\mathbb{R}^2\otimes\mathbb{R}^2$ (here $\mu\nu$ are in... | https://mathoverflow.net/users/3948 | Characteristic surface for systems of PDE | There is a natural example where an irreducible component of the symbol, a hyperbolic polynomial, is of degree higher than $2$. It occurs in compressible magnetohydrodynamics (MHD). It is a coupling of the Euler system of compressible, inviscid gas dynamics, with Maxwell's equations, when the magnetic part dominates th... | 2 | https://mathoverflow.net/users/8799 | 41796 | 26,662 |
https://mathoverflow.net/questions/41795 | 12 | Every matrix $A\in SL\_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on $\pi\_1(S\_1\times S\_1)$ is given by $A$ (w.r.t the induced basis).
So I am wondering, whether a similar construction a... | https://mathoverflow.net/users/3969 | Self homeomorphisms of $S^2\times S^2$ | The matrix $\text{SL}(2,\mathbb{Z})$ acts on $H^n(S^n \times S^n)$; one interpretation of your question is whether this action lifts to $\text{Diff}(S^n \times S^n)$. There is a simple reason that it doesn't when $n$ is even: The intersection form on $H^n(S^n \times S^n)$ is symmetric rather than anti-symmetric, and an... | 12 | https://mathoverflow.net/users/1450 | 41807 | 26,670 |
https://mathoverflow.net/questions/41742 | 3 | Let $\pi\in S\_n$. I recently needed to understand the permutations $\rho$ such that $\rho\not\leq\pi$ in Bruhat order. Since there are $O(n!)$ of those I really wanted a description of the $O(n^2)$ minimal such.
I have a satisfying (to me) answer now, and so I am asking whether this question is addressed in the lite... | https://mathoverflow.net/users/391 | Reference for: the Bruhat-minimal permutations not less than a fixed permutation pi? | Vic Reiner, Alex Yong, and I spell this out in Sections 4.1 and 4.2 of our paper on the cohomology rings of Schubert varieties: <http://arxiv.org/abs/0809.2981>
This is not really original to us: for type A we refer back to Lascoux and Schutzenberger's paper Trellis et bases des groupes de Coxeter, Elect J. Combin. 3... | 3 | https://mathoverflow.net/users/3077 | 41814 | 26,673 |
https://mathoverflow.net/questions/41802 | 4 | Let $\sigma\_k(n)$ denote the sum of the k-th powers of the divisors of n. For any real value of k, we can find a sequence of numbers $s\_k$ that has increasing values of n at which $\sigma\_k(n)$ attains a new maximum. For k=0, that sequence is called the highly composite numbers, which is [A002182](http://oeis.org/cl... | https://mathoverflow.net/users/2360 | A hierarchy of k-highly composite numbers | It appears you got an answer while I was typing my comment. But let me expand the picture a bit. Ramanujan initiated a technique for finding certain subsequences of yours that possess a parametrization, where the highly composite numbers do not. First, see
two articles,
<http://en.wikipedia.org/wiki/Superior_highly_... | 3 | https://mathoverflow.net/users/3324 | 41818 | 26,676 |
https://mathoverflow.net/questions/41820 | 1 | Let $X$ be a scheme. An open atlas for $X$ is a jointly epimorphic family of Zariski-open immersions $\{X\_i\to X\}$ where each $X\_i$ is an affine scheme.
A morphism $X\to S$ of schemes is called representable by an affine if for any map $Y\to S$ where $Y$ is affine, the pullback $X\times\_S Y$ is itself affine.
... | https://mathoverflow.net/users/1353 | Does every scheme admit an open atlas consisting only of morphisms representable by an affine? | A morphism $X \to Y$ is representable by an affine iff it is an affine morphism (preimages of open affines are open affine), since the latter is stable under base change. Now an open immersion $U \to X$ is affine iff $U \cap V$ is open affine in $X$ for every open affine $V$ in $X$. Thus $X$ has an atlas consisting of ... | 5 | https://mathoverflow.net/users/2841 | 41822 | 26,679 |
https://mathoverflow.net/questions/34213 | 6 | In all instances of convex optimization I know of, the dimension of the vector space is defined beforehand. Is there any work on convex optimization over a vector space of *varying* dimension?
For example,
$$\begin{array}{ll} \text{minimize} & k\\ \text{subject to} & x\_1 + \cdots + x\_k = 10\\ & 1 \leq x\_i \leq 2... | https://mathoverflow.net/users/3609 | Convex optimization over vector space of varying dimension | This reminds me of the compressed sensing literature. Suppose that you know some upper bound for k, let that be K. Then, you can try to solve
$\min||{\bf x}||\_0$
subject to
${\bf 1}^T\_K{\bf x}=10$ and ${\bf x}\_i\in[1,2], \; \forall i\in\{1,\ldots,K\}$. The 0-norm counts the number of nonzero elements in ${\bf x}$.
... | 2 | https://mathoverflow.net/users/2763 | 41824 | 26,681 |
https://mathoverflow.net/questions/41833 | 7 | If $X$ is an algebraic stack fibered in sets (and therefore essentially a sheaf), is it an algebraic space? It seems conceivable that at least when $X$ is Deligne-Mumford, it is actually an algebraic space. However, when $X$ is an Artin stack, it is only required to have an atlas of smooth maps of affines, so it seems ... | https://mathoverflow.net/users/1353 | Is every (Artin/DM) algebraic stack fibered in sets an algebraic space? | Yes. The criterion for an Artin stack to be Deligne-Mumford is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If the stack is fibered in sets, the diagonal is a monomorphism, and a monomorphism is certainly unramified.
| 12 | https://mathoverflow.net/users/4790 | 41834 | 26,686 |
https://mathoverflow.net/questions/41830 | 3 | Let $\varphi : A \to B$ be an isogeny between 2 abelian varieties of dimension $g$. Are there known conditions for the $\ker\varphi$ so that this induces an isomorphism between $A$ and $B$? For example, if $\ker\varphi \cong (\mathbb{Z}/n\mathbb{Z})^{2g}$, then $A \cong B$, because $\varphi$ factors through the multipl... | https://mathoverflow.net/users/5197 | isogenies between abelian varieties that induce isomorphisms? | Kevin's comment is right on the money, but here it is in more detail: I will give a general criterion for an isogeny $\varphi: A \rightarrow B$ of abelian varieties to induce an isomorphism upon passage to the kernel.
Let me work over an unnamed algebraically closed field. Suppose that $A = B$ and $\eta \in \operato... | 3 | https://mathoverflow.net/users/1149 | 41841 | 26,687 |
https://mathoverflow.net/questions/41836 | 28 | Nakayama's lemma is as follows:
>
> Let $A$ be a ring, and $\frak{a}$ an ideal such that $\frak{a}$ is contained in every maximal ideal. Let $M$ be a finitely generated $A$-module. Then if $\frak{a}$$M=M$, we have that $M = 0$.
>
>
>
Most proofs of this result that I've seen in books use some non-trivial linea... | https://mathoverflow.net/users/9960 | Elementary proof of Nakayama's lemma? | There are various forms of the Nakayama lemma. Here is a rather general one; note that it does *not* involve maximal ideals and is a constructive theorem (Atiyah-MacDonald, Commutative Algebra, Prop. 2.4 ff).
>
> Let $M$ be a finitely generated $A$-module, $\mathfrak{a} \subseteq A$ be an ideal and $\phi \in End\_A... | 34 | https://mathoverflow.net/users/2841 | 41842 | 26,688 |
https://mathoverflow.net/questions/41839 | 8 | Say that I have the set $[n] = \{1,2,...,n\}$ and a collection $\mathcal{C} = \{ S\_1, S\_2, ..., S\_k \}$ of subsets of $[n]$. Say that $\mathcal{C}$ is *valid* if it is closed under the superset operation; i.e., if $(S \in \mathcal{C} \wedge S \subseteq S' \subseteq [n]) \implies S' \in \mathcal{C}$. How many valid c... | https://mathoverflow.net/users/9961 | How many collections of subsets of {1,2,...,n} are closed under the superset operation? | Such a set is uniquely determined by its minimal members which form an *antichain*. The first few values and some links are [A000372](http://www.research.att.com/~njas/sequences/A000372) in the OEIS
| 17 | https://mathoverflow.net/users/8008 | 41845 | 26,689 |
https://mathoverflow.net/questions/41844 | 5 | Please consider a central, ordinary 2-sphere $S\_1$, of some radius $r\_1$, and a second ordinary sphere, $S\_2$, of radius $r\_2$, where $r\_2 \leq r\_1$.
My question concerns optimal values for the number of spheres of type $S\_2$ that can be packed in three-dimensional space so that they are non-overlapping and t... | https://mathoverflow.net/users/6588 | Optimal packing of spheres tangent to a central sphere | I believe what you are seeking is sometimes known as a
*spherical packing* or a *spherical code*. Here is
[the MathWorld article](http://mathworld.wolfram.com/SphericalCode.html) on the topic.
Here is [Neil Sloane's webpage](http://neilsloane.com/packings/index.html) on the topic, including "putative
optimal packings... | 5 | https://mathoverflow.net/users/6094 | 41847 | 26,690 |
https://mathoverflow.net/questions/41840 | 15 | Let $(X,d)$ be a separable metric space. Can $X$ always be covered by a sequence of balls $B(x\_i,r\_i) (i=1,2,\dots)$ s.t. radii $r\_i$ tend to 0?
| https://mathoverflow.net/users/4312 | covering a separable metric space by small balls | The answer is no for the Banach space $c\_0$. Suppose $B(x\_i,r\_i)$ is a sequence of balls with $r\_i\to 0$ and WLOG $x\_i$ is supported in $[1,N\_i]$ with
$N\_1<N\_2<...$. Consider a point $x$ in $c\_0$ whose $N\_i+1$ coordinate is $2 r\_i$.
I think the answer is no for any separable Banach space: IIRC, for any s... | 14 | https://mathoverflow.net/users/2554 | 41852 | 26,693 |
https://mathoverflow.net/questions/41862 | 5 | Finite groups are solvable if they have no nontrivial perfect subgroup. But I am sure that for infinite groups, the two notions diverge. Is there standard terminology for groups with no perfect subgroups?
| https://mathoverflow.net/users/3634 | Groups with no perfect subgroups -- terminology? | In the infinite case, there is a close notion of "locally indicable group", i.e. a group where every finitely generated subgroup maps onto $\mathbb Z$ (see, for example, [this paper](https://www.jstor.org/stable/2699673)). Locally indicable groups are left (right) orderable, hence important. Note that in that notion, n... | 11 | https://mathoverflow.net/users/nan | 41863 | 26,699 |
https://mathoverflow.net/questions/41867 | 2 | Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth sections of $V$. In particular, $\Gamma(V)$ is a $\mathcal C^\infty(X)$-module. I am interested in $\mathcal C^\infty(X)$-submo... | https://mathoverflow.net/users/78 | Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated? | The module of all sections of $V$ that vanish to infinite order at a given point of the manifold will not be finitely generated (unless the bundle has rank zero or the manifold has dimension zero).
| 11 | https://mathoverflow.net/users/6666 | 41869 | 26,703 |
https://mathoverflow.net/questions/41868 | 5 | There is at least one result saying that the Mandelbrot set is undecidable, and there might be more, but I think it (or they all) use real computation rather than Turing machines. This makes some sense, as $\mathbb{C}$ is connected, so the only decidable subsets of it are $\{\}$ and $\mathbb{C}$ itself. However, I've b... | https://mathoverflow.net/users/nan | Is the distance function from a point to the Mandelbrot set computable? | Following [Giusto and Simpson, *Located sets and reverse mathematics*, J. Symbolic Logic Volume 65, Issue 3 (2000), 1451-1480], the Mandelbrot set $M$ is *located* if the distance function $f:\mathbb C\rightarrow\mathbb R$, $f(x)=d(x,M)$, exists in the model under consideration, which I assume you take to be the model ... | 5 | https://mathoverflow.net/users/4600 | 41871 | 26,704 |
https://mathoverflow.net/questions/41874 | 2 | A) Given a non-constant polynomial $q\in\mathbb{Z}[\alpha\_1,\alpha\_2,\ldots,\alpha\_n],$ if we pick random $\omega\_i\in\mathbb{F}$ (a finite field) uniformly and independently across $1\leq i\leq n,$ then, we know that $q(\omega\_1,\omega\_2,\ldots,\omega\_n)\neq 0$ with high probability (i.e. the probability goes t... | https://mathoverflow.net/users/7576 | Polynomials over Z evaluated with finite field arguments | This should work out for you, with a bit more geometry and perhaps a little trickery. You are asking for points over finite fields avoiding a given hypersurface, in affine space. It is conceptually a little easier to think of points in a projective space avoiding both a hypersurface and the hyperplane at infinity, beca... | 3 | https://mathoverflow.net/users/6153 | 41879 | 26,707 |
https://mathoverflow.net/questions/41883 | 9 | Let $G$ be an abelian group, $A$ a trivial $G$-module. We know that $\text{Ext}(G,A)$ classifies abelian extensions of $G$ by $A$, whereas $H^2(G,A)$ classifies central extensions of $G$ by $A$. So we have a canonical inclusion $\text{Ext}(G,A)\hookrightarrow H^2(G,A)$. Is there some naturally arising exact sequence/sp... | https://mathoverflow.net/users/5513 | Injection of Ext into H^2 | You get a description from the universal coefficient theorem which gives a (split) exact sequence
$$
0\to \mathrm{Ext}(H\_1(G),A) \to H^2(G,A) \to \mathrm{Hom}(H\_2(G),A) \to 0
$$
and the fact that $H\_1(G)=G$. We have that $H\_2(G)=\Lambda^2G$ and the map $H^2(G,A) \to \mathrm{Hom}(H\_2(G),A)$ associates to an extensi... | 14 | https://mathoverflow.net/users/4008 | 41892 | 26,715 |
https://mathoverflow.net/questions/41902 | 4 | Suppose $\dot{x}=f(x)$ is a dynamical system, with $x$ in $R^n$, and $f:R^n \to R^n$ sufficiently smooth (for example, Lipschitz-continuous).
Assume that $x\_e$ is an *unstable* equilibrium point of the system. Even if $x\_e$ is unstable, the union of all trajectories having $x\_e$ as a limit point could be larger t... | https://mathoverflow.net/users/8460 | Measure of the stable set in a dynamical system | Take for example the equation
$\dot x = \lambda x$
$\dot y = y^2$
For $\lambda < 0$. The origin is an unstable equilibrium point, however, its stable manifold is the whole lower semiplane (including the $y=0$ axis).
If there is one eigenvalue of real part bigger than $0$ for the derivative in the equilibrium... | 4 | https://mathoverflow.net/users/5753 | 41903 | 26,721 |
https://mathoverflow.net/questions/41900 | 3 | Let M be a 2n-dimensional closed symplectic manifold.
Then is there a constant c such that , for any real 2-dimensional embedded J-holomorphic
disk u, the symplectic volume of u is bounded by c ?
If not, is there any result about a condition which makes the statement above to be true?
I really thank you for your a... | https://mathoverflow.net/users/11705 | symplectic volume of embedded J-holomorphic disk | Unless I'm misunderstanding what you're asking, the answer is surely no...consider for instance $\mathbb{C}P^n$ with its standard symplectic and complex structures. This admits embedded $J$-holomorphic curves of arbitrarily large area (take a high-degree curve in a plane $\mathbb{C}P^2\subset \mathbb{C}P^n$), and restr... | 4 | https://mathoverflow.net/users/424 | 41907 | 26,723 |
https://mathoverflow.net/questions/41912 | 5 | In responding to
[Fast computation of multiplicative inverse modulo q](https://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q/41631#41631)
I mentioned an algorithm for computing the inverse of $a \mod p$ different from the extended Euclidean algorithm, hoping that someone coul... | https://mathoverflow.net/users/5373 | Computation of inverses modulo p followup | Here is a tidy way to solve $ax+by=gcd(a,b)$. Start with the matrix
$$\left(\begin{array}{ccc}
1&0&a\\
0&1&b\\
\end{array}\right)
$$
Suppose $a\ge b$. Then replace row 1 by row 1 minus $t$ times row 2, where $t=\lfloor a/b\rfloor$. Repeat this operation until the last entry in one of the two rows is zero. If the oth... | 3 | https://mathoverflow.net/users/5229 | 41925 | 26,730 |
https://mathoverflow.net/questions/41915 | 1 | Hello, i have NP hard problem. Let imagine I have found some polynomial algorithm that find ONLY one of many existing solutions of that problem, but at least one solution (if present in the probem). Is that algorithm considered as solution of NP=P question (if that algorithm transformed to mathematical proof)?
Thanks... | https://mathoverflow.net/users/9992 | NP-Hard solution question | For concreteness, let's pick an NP-hard problem to talk about. Given a graph $G$, the 3-colouring problem asks: "can the vertices of $G$ be painted by three colours such that for any edge $uv$, $u$ and $v$ get different colours?" This is a decision problem --- its possible answers are "yes" or "no" --- but a "yes" answ... | 9 | https://mathoverflow.net/users/785 | 41930 | 26,735 |
https://mathoverflow.net/questions/41932 | 10 | Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s [<http://math.mit.edu/~rstan/pubs/pubfiles/100.pdf>; p.5 of the PDF file]. I would especially like to have an example in which at least one of t... | https://mathoverflow.net/users/8604 | Graphs with the same chromatic symmetric function | I don't think there are any other published examples. I think your best bet is to look at the literature on "chromatically equivalent graphs" (graphs with the same chromatic polynomial) and do your own computations to find examples. (I assume that you wrote some code to compute the chromatic symmetric function when you... | 4 | https://mathoverflow.net/users/3106 | 41938 | 26,738 |
https://mathoverflow.net/questions/41895 | 5 | Let $G$ be a reductive algebraic group over $\mathbb R$ and $K$ a maximal compact subgroup. Then we refer to the conjugacy class in $G$ of some $k \in K$ as an elliptic conjugacy class.
**Question:** Can one characterizes those conjugacy classes in $G$ which contain an elliptic conjugacy class in their closure?
(F... | https://mathoverflow.net/users/9927 | Conjugacy classes with elliptic limit points | For g in G, write g=gsgu as its Jordan decomposition into semisimple and unipotent parts. I claim that the closure of the conjugacy class of g contains an elliptic element if and only if gs is elliptic.
Let us first suppose that gs is not elliptic. Choose an embedding of G into GLn(ℂ). Then by our assumption, gs has ... | 2 | https://mathoverflow.net/users/425 | 41941 | 26,739 |
https://mathoverflow.net/questions/41940 | 7 | Let $\rho\_p : \mbox{Gal}(\overline{\mathbb{Q}}\_p / {\mathbb{Q}\_p}) \to \mbox{GL}\_n(\mathbb{Q}\_p)$ be a de Rham $p$-adic representation. Can one find a representation $\rho : \mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mbox{GL}\_n(\mathbb{Q}\_p)$ such that $\rho$ is geometric (in the sense of Fontaine-Mazur) ... | https://mathoverflow.net/users/10001 | local to global Galois representation | No. There are uncountably many unramified representations from the local Galois group to
$\mathbb{Q}^{\times}\_p$, since Frobenius can be sent to anything in $\mathbb{Z}^{\times}\_p$.
However, there are only countably many global representations of this form, since, by class field theory, they all factor through the G... | 7 | https://mathoverflow.net/users/nan | 41942 | 26,740 |
https://mathoverflow.net/questions/41922 | 18 | Can someone give an explicit example of a group with two generators $a$, $b$, such that $a^2 = b^3 = 1$ and $a b$ has infinite order, but which is not isomorphic to the free product of $\mathbb{Z}\_2$ and $\mathbb{Z}\_3$?
| https://mathoverflow.net/users/2926 | Group which "resembles" the free product of a cyclic group of order two and a cyclic group of order three, but isn't. | It is straightforward to calculate that the commutator subgroup $G' = D$ of $G = \langle a,b \mid a^2, b^3 \rangle$ is a free group on the generators $x=bab^{-1}a$, $y=b^{-1}aba$, where $|G:D|=6$.
Now $(ab)^6$ is equal to the commutator $x^{-1}yxy^{-1}$, which lies in $D'$ but not in $D''$, so if we add any nontrivia... | 18 | https://mathoverflow.net/users/35840 | 41944 | 26,741 |
https://mathoverflow.net/questions/41934 | 5 | I am trying to understand spin structures and am looking at the specific case of complex projective space (viewed as the quotient $SU(N)/U(N-1)$) and more generally the Grassmannians (viewed as the quotient $SU(N)/(U(N-k) \times U(k))$. My questions are as follows:
(1) For what values of $N$ does complex projective $... | https://mathoverflow.net/users/1648 | Spin structures on the Grassmannians | I think there is a slight mistake in the formulation of the question. $\mathbb{CP}^n$ is the homogeneous space $U(n+1)/(U(n) \times U(1))=SU(n+1)/G$ with $G= SU(n+1) \cap (U(n) \times U(1))$. The right formulation of question (2) is: is the spin structure on $\mathbb{CP}^n$ (for odd $n$, there is unique spin structure ... | 9 | https://mathoverflow.net/users/9928 | 41949 | 26,745 |
https://mathoverflow.net/questions/41827 | 3 | I'm reading a paper and here the authors say that a connected 4-manifold with zero rational top homology has a homotopy type of 3-dimensional CW-structure. I can't figure out how it can be done.
| https://mathoverflow.net/users/6569 | a CW-complex homotopic to a manifold | For this to be true, you need to assume that your $4$-manifold $M$ is not a compact nonorientable manifold. Otherwise, you would have $H\_4(M;\mathbb{Q}) = 0$ but $H\_4(M;\mathbb{Z}/2) \neq 0$, so there is no hope that your manifold is homotopy equivalent to a $3$-dimensional CW-complex.
Assuming this, your condition... | 4 | https://mathoverflow.net/users/317 | 41964 | 26,750 |
https://mathoverflow.net/questions/41973 | 3 | In the paper "On the consistency strength of projective uniformization" Woodin proves a lemma "Assume $M$ is a model of ZFC that is $\Sigma^{1}\_{3}$-absolute. Then $M\vDash\forall x\in\mathbb{R}\,[x^{\sharp} \mathrm{\ exists}]$." He then goes on to say after the proof "It in fact now follows by a theorem of Martin-Sol... | https://mathoverflow.net/users/7909 | Question about Woodin's paper "On the consistency strength of projective uniformization" | Rupert, the Martin-Solovay paper shows the absoluteness result follows form the existence of measurable cardinals. The measurables are used in the construction of certain trees (now called Martin-Solovay trees), where some ordinals are chosen by means of the measure to serve as witnesses (of ill-foundedness of some bra... | 7 | https://mathoverflow.net/users/6085 | 41974 | 26,752 |
https://mathoverflow.net/questions/41939 | 21 | A box contains n balls coloured 1 to n. Each time you pick two balls from the bin - the first ball and the second ball, both uniformly at random and you paint the second ball with the colour of the first. Then, you put both balls back into the box. What is the expected number of times this needs to be done so that all ... | https://mathoverflow.net/users/7576 | A balls-and-colours problem | It can probably be done by looking at the sum of squares of sizes of color clusters and then constructing an appropriate martingale. But here's a somewhat elegant solution: reverse the time!
Let's formulate the question like that. Let $F$ be the set of functions from $\{1,\ldots,n\}$ to $\{1,\ldots,n\}$ that are almo... | 29 | https://mathoverflow.net/users/1061 | 41985 | 26,759 |
https://mathoverflow.net/questions/41978 | 32 | I was wondering if anyone could offer some intuition for why [Alexander duality](http://en.wikipedia.org/wiki/Alexander_duality) holds. Of course, the proof is easy enough to check, and it is also easy to work out many examples by hand. However, I don't have any feeling for why it is true.
To give you an example of w... | https://mathoverflow.net/users/10019 | Intuition behind Alexander duality | Let $M$ be a closed orientable $n$-manifold containing the compact set $X$.
Given an $n-q-1$-cocyle on $X$ (I am choosing this degree just to match with the notation
of the Wikipedia article to which you linked), we extend it to some small open neighbourhood $U$ of $X$.
By Lefschetz--Poincare duality on the open manif... | 24 | https://mathoverflow.net/users/2874 | 41986 | 26,760 |
https://mathoverflow.net/questions/41970 | 7 | Let $X$ be an algebraic variety. Consider an exact sequence
$$0\to A\to B\to C\to 0$$
of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ (or $A$)
$$0\to S^kA\to S^{k-1}A\otimes B\to S^{k-2}A\otimes \wedge^2B\to \cdots\to \wedge^kB\to \wedge^k C\to 0.$$
Qu... | https://mathoverflow.net/users/2348 | How to resolve a wedge product of vector bundles | The resolution in question is the $k$th graded piece of the symmetric algebra of the complex $A \rightarrow B$, where $A$ is considered to lie in even degrees (say degree $2$) and $B$ in odd degrees (say degree $1$). (Also, a kind of Koszul complex.)
The way I think of this: forgetting the differential $A \rightarro... | 10 | https://mathoverflow.net/users/4659 | 41990 | 26,764 |
https://mathoverflow.net/questions/41979 | 33 | $\newcommand{\Spec}{\mathrm{Spec}\,}$Let $X=\Spec A$ be a variety over $k$, then we have the definition of the tangent bundle $\hom\_k(\Spec k[\varepsilon]/(\varepsilon^2),X)$ (note that this has the structure of a variety). On the other hand, we have the definition of a tangent sheaf $\hom\_{\mathscr{O}\_X}(\Omega\_{X... | https://mathoverflow.net/users/9035 | Relationship between Tangent bundle and Tangent sheaf | You can always apply the "vector bundle" construction to $\Omega:=\Omega\_{X/k}$ (locally free or not). What you get is a scheme $T=\mathrm{Spec\ Sym}(\Omega)\to X$ which deserves to be called "tangent bundle" (albeit not locally trivial); in particular its $k$-points are what you want and, more generally, if, say, $Z=... | 31 | https://mathoverflow.net/users/7666 | 41992 | 26,766 |
https://mathoverflow.net/questions/41984 | 29 | The "textbook" example of a smooth bijection between smooth manifolds that is not a diffeomorphism is the map $\mathbb{R} \rightarrow \mathbb{R}$ sending $x \mapsto x^3$. However, in this example, the source and target manifolds *are* diffeomorphic -- just not by the given map. Is there an example of a smooth bijection... | https://mathoverflow.net/users/379 | Smooth bijection between non-diffeomorphic smooth manifolds? | Every smooth manifold has a smooth triangulation, which yields a pseudofunctor from the category of smooth manifolds to the category of PL manifolds. (There is no actual functor; that would be crazy.) If two smooth manifolds are PL isomorphic, then the answer is yes. You can start with the PL isomorphism, and then buil... | 22 | https://mathoverflow.net/users/1450 | 41993 | 26,767 |
https://mathoverflow.net/questions/42006 | 12 | Some simple questions, for which I know no precise reference (and would be deeply grateful for a nice one!):
1. Is it true that the category of (pure) polarized Hodge structures is abelian semi-simple, whereas the whole category of pure Hodge structures is not?
2. Should one only consider those morphisms of polarized... | https://mathoverflow.net/users/2191 | On polarized (pure) Hodge structures | Fortunately, these questions are easy to answer. First of all, it helps to distinguish
between polarizable Hodge structures and polarized structures. For polarizable, we merely
require that a polarization exists, but it is not fixed. Let Hodge structure
mean pure rational Hodge structure below.
Then
* The category o... | 21 | https://mathoverflow.net/users/4144 | 42008 | 26,775 |
https://mathoverflow.net/questions/41972 | 13 | Let $\theta \not\in \mathbb{Q}$. We know that $(n\theta)\_{n \geq 1}$ is equidistributed modulo 1.
Let $\epsilon\_n = \mathrm{sign}\bigl(\sin(n\pi \theta)\bigr)$ and $S\_N= \sum\_{n=1}^N \epsilon\_n$.
I'm looking for a "good" asymptotic bound for $|S\_N|$ (not $|S\_N|\leq N$ obviously).
It looks like for any $x>... | https://mathoverflow.net/users/3958 | Consequence of equidistribution or not? | No, you cannot put any better bound than SN = *o*(N). There is a general technique, using the [Baire category theorem](http://en.wikipedia.org/w/index.php?title=Baire_category_theorem&oldid=379733059) of proving the existence of counterexamples to problems like this (which I discovered while trying to find a counterexa... | 9 | https://mathoverflow.net/users/1004 | 42011 | 26,777 |
https://mathoverflow.net/questions/42024 | 0 | hello,
it is often said that a **conformal mapping** preserves the Laplace equetion in 2D.
However, if this is true for the **cartesian coordinates (x,y)**, where the laplacian is:
$$
\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=0
$$
is it true for the **cylindrical coordinates (r,z)** wh... | https://mathoverflow.net/users/9039 | Invariance of the cylindrical Laplace equation under conformal transform | **No**. Take $\phi=r^2-2z^2$ and $f(z+ir)=(z+ir)^2$, thus $u=r^2-z^2$, $v=2rz$. Then $\phi(u,v)=r^4+z^4-10r^2z^2$ is not cylindrical-harmonic.
| 1 | https://mathoverflow.net/users/8799 | 42027 | 26,784 |
https://mathoverflow.net/questions/41998 | 12 | There are various notions of exact categories ([nlab](http://ncatlab.org/nlab/show/exact+category)). In a lecture I've seen the following definition of an exact category, which is basically (exact) = (abelian) − (additive):
A category $C$ is called **exact** if a) it contains a zero object, b) every morphism has a ke... | https://mathoverflow.net/users/2841 | Exact categories which are not additive | These categories are called Puppe-exact or p-exact categories. See paragraph 1.1, of *Jordan-Hölder, modularity and distributivity in non-commutative algebra*, by Francis Borceux and Marco Grandis (JPAA **208** (2007), 665-689 doi:[10.1016/j.jpaa.2006.03.004](https://doi.org/10.1016/j.jpaa.2006.03.004)), for non-abelia... | 12 | https://mathoverflow.net/users/10033 | 42034 | 26,788 |
https://mathoverflow.net/questions/41950 | 2 | I am wondering about approximation and idealization. Specifically, I am wondering if anyone has seen some work on the following. In the semantics of programming languages we find Domains as a place to talk about iteration and approximation. We can define a Scott Topology on the Domain and now our Domain-maps are contin... | https://mathoverflow.net/users/10007 | approximating categories with continuous functors | As it happens, I just saw a paper about this very subject today -- Martin Hyland's ["Some Reaons for Generalizing Domain Theory"](http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2010/srfg10.pdf), which is concerned with precisely the generalization you suggest, in order to clarify the semantics of concurrency.... | 3 | https://mathoverflow.net/users/1610 | 42035 | 26,789 |
https://mathoverflow.net/questions/42010 | 12 | In the next days I have to give a talk in which I need to explain some of the usual singularities of pairs that one meets when dealing with the minimal model program: KLT, DLT and LC pairs.
In particular I would like to give an intuition (also to non specialists) of the reason why an LC pair is much more difficult to t... | https://mathoverflow.net/users/6430 | Singularities of pairs | For references, probably you already know things like Kollar-Mori and Kollar's "Singularities of pairs", for LC-centers and subadjunction, Kollar has some notes on that as well. There's some sections on this in the recent book by Hacon-Kovacs too that I've looked at recently.
With regards to your specific questions I... | 9 | https://mathoverflow.net/users/3521 | 42042 | 26,793 |
https://mathoverflow.net/questions/42016 | 23 | I am looking for algorithms on how to find rational points on an [elliptic curve](http://en.wikipedia.org/wiki/Elliptic_curve) $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, how to find solutions in which the numerators and denominators are bounded,... | https://mathoverflow.net/users/1176 | Algorithms for finding rational points on an elliptic curve? | A good reference to get started from the algorithmic point of view is Chapter 3 of Cremona's [*Algorithms for Modular Elliptic Curves*](http://www.warwick.ac.uk/~masgaj/book/fulltext/index.html). It contains a good deal of pseudocode which explains how Cremona's C++ package mwrank computes rational points on elliptic c... | 10 | https://mathoverflow.net/users/4872 | 42044 | 26,794 |
https://mathoverflow.net/questions/42038 | 3 | Is there example of hypersurface $X \subset \mathbb{P}^n$ satisfying
1. $X$ is of degree 2. (I mean, the Poincare dual PD(X) is 2u, where u is a generator of
$H^2(\mathbb{P}^n, \mathbb{Z})$.
2. Some odd betti number is non-zero.
Thank you for any comment.
| https://mathoverflow.net/users/11705 | Hypersurface of complex projective space | As noticed by Sasha in his comment, the answer is **no**.
The following proof also shows that this result cannot be generalized for higher values of the degree.
For any smooth complex hypersurface of degree $d$, say $X\_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have $$H^k(X\... | 13 | https://mathoverflow.net/users/7460 | 42048 | 26,796 |
https://mathoverflow.net/questions/42040 | 6 | In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine transform (or [discrete] fourier transform) we had again orthogonal bases.
To describe any vector $x \in \mathbb{C}^N$, w... | https://mathoverflow.net/users/5287 | Why do we want to have orthogonal bases in decompositions? | Your first point, non-uniqueness, is definitely false. One of the basic facts in linear algebra is precisely that for any fixed set of basis vectors (we don't even have to work on a vector space endowed with an inner product, so orthogonality doesn't come in at all), a given vector has a unique decomposition.
For if... | 11 | https://mathoverflow.net/users/3948 | 42053 | 26,798 |
https://mathoverflow.net/questions/42037 | 8 | Suppose I have $k$ $n$-dimensional polytopes $P\_1,\ldots,P\_k$, each explicitly specified as the intersection of a collection of hyperplanes. If there was a point $p \in \mathbb{R}^n$ that lay in the intersection of all of these polytopes ($p \in P\_1 \cap \ldots \cap P\_k$), I could efficiently find it by solving a l... | https://mathoverflow.net/users/3027 | Finding a point that lies in a majority of polytopes | This problem is clearly in NP (guess which polytopes) and becomes NP-complete if we replace $2/3$ with $1/2$ and make it a decision problem, dropping the promise that such a $p$ exists. In particular we can reduce integer programming feasibility to this problem.
Say we wish to determine whether $Ax = b$ has a solutio... | 4 | https://mathoverflow.net/users/5963 | 42064 | 26,804 |
https://mathoverflow.net/questions/42065 | 3 | I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?
Fix $1 \leq k \leq n$. Define $X\_{n,k}$ to be the following poset. The elements of $X\_{n,k}$ are ordered sequences $\omega = (x\_1,\ldots,x\_m)$, where the $x\... | https://mathoverflow.net/users/317 | Posets of finite sequences are highly connected | If I understand your question correctly, an answer should appear in the paper "On lexicographically shellable posets" of Anders Bj\"orner and Michelle Wachs, in Transactions of the AMS 277, pp. 323-341.
| 3 | https://mathoverflow.net/users/36466 | 42074 | 26,808 |
https://mathoverflow.net/questions/42056 | 1 | Is there a simple rule to check whether a pretzel link P(n\_1,...,n\_k) is a trivial link?
I am interested in the 2-component case but every information would be helpful.
| https://mathoverflow.net/users/5001 | Trivial pretzel links | A pretzel link has as many components as the pretzel link with coefficients reduced $(\mod 2)$. So it will have two components if and only if there are precisely two even coefficients.
A conceptual classification is given by considering the 2-fold branched cover, or the $\pi$-orbifold, obtained by taking the orbifol... | 7 | https://mathoverflow.net/users/1345 | 42075 | 26,809 |
https://mathoverflow.net/questions/42022 | 15 | Let $A$ be a C\*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^\*$-algebra?
In the case where the answer is false:
Is it true in supposing, in addition, that the range $P(A)$ is an operator system, $A$ unital an... | https://mathoverflow.net/users/5210 | Range of completely positive projection | Yes. $P(A)$ is abstractly a C$^\ast$-algebra $B$ under the Choi-Effros product: $a\circ b := P(ab)$; and $P(A)$ is completely order isomorphic to $B$. See (the proof of) Theorem 3.1 in [M.-D. Choi and E. G. Effros; Injectivity and operator spaces. J. Functional Analysis 24 (1977), 156-–209]. Moreover, there is a surjec... | 21 | https://mathoverflow.net/users/7591 | 42090 | 26,814 |
https://mathoverflow.net/questions/42072 | 2 | Everybody knows that a square matrix $A$ has the same eigenvalues as
$A^T$. And it is clear that if $A^T=BAB^{-1}$ then $B$ maps eigenvectors
of $A$ to those of $A^T$. But I have not found any discussion of the
benefits of knowing $B$. Perhaps it is unusual to know $B$ exactly,
without having analyzed $A$ completely. B... | https://mathoverflow.net/users/6998 | Does knowing a conjugation of A to A^T determine eigenvalues of A? | Generally, over the algebraically closed field, say over $\mathbb{C}$, $A^T$ and $A$ are always conjugate, because they have the same Jordan form. So such $B$ always exists. But the knowledge of the concrete $B$ satisfying to this relation may clarify what the eigenspaces for $A$ are, because ${B^T}B$ always commutes w... | 2 | https://mathoverflow.net/users/10045 | 42092 | 26,816 |
https://mathoverflow.net/questions/42085 | 0 | How to estimate the probability of co-occurrence of the positive integers $c\_i$ and $d\_i$, $1 \leq i \leq t$ drawn from the uniform range $1$ to $2^k-1$, such that $\Sigma^t\_{i=1} c^2\_i = \Sigma^t\_{i=1} c\_i\times d\_i$ and $\forall i, c\_i\neq d\_i$ hold?
$\forall i, c\_i$ and $d\_i$'s are drawn from the same d... | https://mathoverflow.net/users/10041 | Estimating the probability of co-occurrence of a set of positive integers | An approach for the case $t=2$. There are $N^4$ ways of choosing $a,b,c,d$, all between 0 and $N-1$. You want to know how many of them result in $a^2+b^2=ac+bd$. Let's take the case where $\gcd(a,b)=1$ and $a\lt b$. Then you must have $c=a+br$ and $d=b-ar$ for some $r$. This $r$ can't exceed $b/a$, nor can it exceed $(... | 0 | https://mathoverflow.net/users/3684 | 42108 | 26,822 |
https://mathoverflow.net/questions/42113 | 3 | Let $r,s,n$ be positive integers with $r < s < n$. Let $U = \{1,\ldots,n\}$.
>
> Let $S$ contain $s$-element subsets of
> $U$ (of our choosing). What is that smallest we can make $S$ such that every
> $r$-element subset of $U$ is a subset
> of some element in $S$?
>
>
>
I'm curious what the best known upper... | https://mathoverflow.net/users/8574 | A small collection of large subsets covering all small subsets. | What you are looking for is an $(r,s,n)$-covering design. A good starting point might be the [La Jolla Covering Repository](http://www.ccrwest.org/cover.html) or in fact, any book on design theory. In general, the smallest possible size for your covering design is given by $\binom{n}{r}/\binom{s}{r}$ in which case ever... | 9 | https://mathoverflow.net/users/9044 | 42115 | 26,825 |
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