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https://mathoverflow.net/questions/94569 | 2 |
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> **Possible Duplicate:**
>
> [Stably isomorphic groups](https://mathoverflow.net/questions/33589/stably-isomorphic-groups)
>
>
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If $G$ and $H$ are two groups (finitely presented, if you wish) with the property that $G\times\mathbb{Z}$ is isomorphic to $H\times\mathbb{Z}$, does that imply that $G$ is isom... | https://mathoverflow.net/users/10273 | Stable isomorphisms of groups | See [Stably isomorphic groups](https://mathoverflow.net/questions/33589/stably-isomorphic-groups) (Hirshon's example is finitely presented).
| 1 | https://mathoverflow.net/users/21684 | 94575 | 55,455 |
https://mathoverflow.net/questions/94490 | 5 | It is well-known that for each planar graph, the number of paths from each source to each sink will give rise to a totally non-negative matrix.
However, did anyone ever come up with a planar network whose paths give a matrix whose entries are Eulerian numbers?
| https://mathoverflow.net/users/23043 | Is there a planar network whose path give a TNN matrix whose entries are Eulerian numbers? | There is a Bratelli diagram which satisfies this. Consider $\mathbb N^2$ as a graph where the edges are given by k parallel directed edges from $(n,k)$ to $(n+1,k)$, and $n-k+1$ parallel directed edges from $(n,k)$ to $(n+1,k+1)$. Here the number of paths from $(1,1)$ (the unique source) to any vertex $(n,k)$ is the [E... | 2 | https://mathoverflow.net/users/2384 | 94581 | 55,458 |
https://mathoverflow.net/questions/94579 | 14 | It is well known that the Stone–Čech compactification $\beta \mathbb N^+$ of the positive natural numbers has the structure of a compact left semitopological semigroup and hence, by Ellis's lemma, has idempotents. The usual proof of Ellis's lemma uses Zorn's lemma. Idempotent ultrafilters are clearly non-principal. It ... | https://mathoverflow.net/users/15934 | Is choice needed to establish the existence of idempotent ultrafilters? | Yes, it's still weaker. To build a model of ZF in which choice fails but $\beta\mathbb N^+$ has idempotents, start with a model of ZFC (which will, of course, have idempotent ultrafilters in $\beta\mathbb N^+$). Add a lot of Cohen-generic subsets of some regular cardinal $\kappa$ well above the cardinal of the continuu... | 18 | https://mathoverflow.net/users/6794 | 94583 | 55,459 |
https://mathoverflow.net/questions/94421 | 6 | If one has a finite dimension simple Lie algebra, one can easily calculate that taking the centralizer of a torus (or toral subalgebra), that is, summing the weight spaces that lie in some proper subspace of the dual Cartan, always gives a finite dimensional reductive Lie algebra; actually almost semi-simple, except th... | https://mathoverflow.net/users/66 | Is the centralizer of a torus in a Kac-Moody algebra always a Borcherds algebra? | The answer to your first question is "yes", and it follows from Theorem 1 in Borcherds's paper *Central extensions of generalized Kac-Moody algebras*, which is available online as number 11 on [his papers page](http://math.berkeley.edu/~reb/papers/index.html). The inner product and involution can be chosen as restricti... | 7 | https://mathoverflow.net/users/121 | 94595 | 55,464 |
https://mathoverflow.net/questions/94539 | 2 | Hello everyone,
though I am quite sure my question is not on a research level, I am a bit helpless as of where to ask it. I posted it on [math.stackexchange here](https://math.stackexchange.com/questions/126559/noether-normalization-in-mathbbcx-1-x-n), but there has been no answer helping me with my actual problem, s... | https://mathoverflow.net/users/22998 | Noether Normalization in $\mathbb{C}[[x_1,...,x_n]]$ | Before proving the theorem I would, using the lemma, prove an intermediate result from which the theorem follows. Actually that intermediate result looks more like Noether normalization to me than the final theorem, but anyway.
>
> **Proposition:** If $A$ is a finite $C[[x\_1,...,x\_n]]$ algebra with structure map... | 4 | https://mathoverflow.net/users/10194 | 94600 | 55,466 |
https://mathoverflow.net/questions/94618 | 2 | Let $\mathcal{C}$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F\_n(\mathcal{C})$ the category with cofibrations consisting of sequences of $n$ cofibrations $A\_0 \rightarrowtail A\_1 \rightarrowtail \dotsc \rightarrowtail A\_n$ in $C$. A cofibration $A \to B... | https://mathoverflow.net/users/2841 | Cube of cofibrations II | For vector spaces 1 does not imply 2. Make a cube by taking a vector space, three subspaces, their pairwise intersections, and the triple intersection. If the big one is two-dimensional and the other three are three distinct lines you have a counterexample.
| 8 | https://mathoverflow.net/users/6666 | 94621 | 55,475 |
https://mathoverflow.net/questions/93765 | 9 | *For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes (Bolzmann-Term depending exponentially on a "Temperature Parameter"). The limit of high Temperature is no-correlation wh... | https://mathoverflow.net/users/22709 | Correlation-Function for Random Graph Ising Model | There is a method to study these physical systems called [**belief propagation (BP)**](http://en.wikipedia.org/wiki/Belief_propagation), which yields exact methods for interaction graphs that are trees, and empirically works pretty fine when you have physical systems with *loopy* interaction graphs that **locally-look-... | 4 | https://mathoverflow.net/users/12793 | 94629 | 55,479 |
https://mathoverflow.net/questions/94620 | 8 | Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G\_m$ and a non-singular scheme $X$ of finite type over $k.$
Let us define the following:
**Condition 1:** $X$ can be covered by $G\_m$-invariant quasi-affine open subschemes.
In the paper "Some theorems on actions of algebraic groups" (Th... | https://mathoverflow.net/users/nan | Bialynicki-Birula decomposition of a non-singular quasi-projective scheme. | It is a theorem of Sumihiro (Equivariant completion, Corollary 2) that a normal variety over an algebraically closed field with an action of a torus is covered by invariant affine open subsets.
Here, the normality hypothesis is necessary : the conclusion does not hold for the action of $\mathbb{G}\_m$ on $\mathbb{P}^... | 14 | https://mathoverflow.net/users/2868 | 94633 | 55,482 |
https://mathoverflow.net/questions/94635 | 0 | Hello,
Thank you for taking a look at this post. I am trying to prove the statement as given in the title. I have managed to disprove it but I am yet not sure if it is correct. I would appreciate it if somebody could comment if the above statement is true or false. Thank you for your time.
| https://mathoverflow.net/users/9870 | For two graphs, $H$ and $G$ let $\bar{H}$ and $\bar{G}$ be their respective compliments. Then if $H$ is a subgraph of $G$, then $\bar{H}$ is a subgraph of $\bar{G}$. | This statement is not true. For example, let $G=K\_4$ and $H=C\_4$, where $G$ and $H$ are complete graph and cycle graph with four vertices, respectively.
It is easy to check that, $H$ is a subgraph of $G$, but $\overline{H}$ is not a subgraph of $\overline{G}$.
It is interesting that, you think about this question:
... | 3 | https://mathoverflow.net/users/19929 | 94636 | 55,484 |
https://mathoverflow.net/questions/94615 | 6 | Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U\_n(\mathbb{C})$. The category of these representations is an exact category, and so we can form the Grothendieck group $K\_0(G)$ of this category. If $H$ is a fin... | https://mathoverflow.net/users/18256 | Induction theorems for finite-dimensional complex representations of infinite groups | I believe that all examples must come from finite quotients:
1. A collection of subgroups is necessarily good if the element $1\in K\_0(G)$ is in the subgroup generated by the images of the induction maps $K\_0(H)\to K\_0(G)$. That's because these images are ideals, using the formula $V\otimes Ind(W)=Ind(Res(V)\otime... | 4 | https://mathoverflow.net/users/6666 | 94637 | 55,485 |
https://mathoverflow.net/questions/94625 | 7 | Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to [this MO post](https://mathoverflow.net/questions/94525/equal-digit-sums).
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal representation of $n$.
>
> Is it true that for any integer ... | https://mathoverflow.net/users/9924 | The digit sum: $s(na)=s(nb)$ | OK, by Seva's request I'm getting somewhat more serious :) Fix $a$, $b$. Take large $M$ to be chosen shortly. Take a $3MN$ digit number $n$ (something from $0$ to $10^{3MN}-1$, written with head zeroes if necessary) and split its decimal representation into pieces of length $3M$ where $M$ is large but fixed. Let us cal... | 10 | https://mathoverflow.net/users/1131 | 94638 | 55,486 |
https://mathoverflow.net/questions/94606 | 12 | Recall that there is an equivalence of categories (Dold-Kan) $$N:\mathrm{s}\mathbf{Ab}\simeq \operatorname{Ch}\_{\geq 0}(\mathbf{Ab}):\Gamma$$ between simplicial abelian groups and (connective) chain complexes, where $N$ sends a simplicial abelian group to its associated normalized chain complex.
Using this equivale... | https://mathoverflow.net/users/1353 | Explicit description of the "simplicial tensor product" of chain complexes | The bad news is that in degree $n$, this tensor product has $3^n$ terms.
The functor $\Gamma$ can be roughly described as follows. If we write $[n]$ for the ordered set $0 < 1 < \cdots < n$, then
$$
\Gamma(C)\_n = \bigoplus\_{k} \bigoplus\_{\phi\colon [n] \twoheadrightarrow [k]} C\_k.
$$
The face maps have two charac... | 10 | https://mathoverflow.net/users/360 | 94640 | 55,488 |
https://mathoverflow.net/questions/94628 | 2 | Let $A$ be an $n\times n$ matrix which depends smoothly on a variable $x\in \mathbb{R}^n$ and such that there are constants $C\_\alpha > 0 $ so that $\| \partial ^\alpha A \| \le C\_\alpha $ for all multi-indices $\alpha \in \mathbb{N}\_0^n$ (i.e. $A$ together with all its derivatives are bounded in matrix norm). Call ... | https://mathoverflow.net/users/19433 | Invertible matrix perturbation | Define $f(x)$ to be the least real eigenvalues of $A(x)$, then $f$ is defined in a neighborhood $B(x\_0,\epsilon)$ of $x\_0$ and $f(x)>1/2$ for any $x\in B(x\_0,\epsilon)$. Suppose $M>\sup\_x||A(x)||$, then choose $\chi(x)$ to be a nonnegative function such that $\chi$ is smooth, $\chi(x)=0$ for any $x\in B(x\_0,\epsil... | 3 | https://mathoverflow.net/users/15289 | 94651 | 55,496 |
https://mathoverflow.net/questions/94632 | 0 | when consider the fractional laplacian $(-\triangle)^\alpha$,is there an essential difference between $0<\alpha<1$ and $\alpha>1$ ? As far as I'm concern,the higer order laplacian ($\alpha>1$ ) ,unlike the lower case, has little connection with stochastic process.(lack of positivy.) Since most paper i have met is the c... | https://mathoverflow.net/users/23078 | Higher order fractional laplacian | The fractional Laplacian can be represented as a hypersingular integral operator, an integral operator with a singularity higher than the space dimension. Its convergence is provided by a regularization whose form depends on the order. From its form the difference in positivity properties is immediately clear. See S. G... | 1 | https://mathoverflow.net/users/12205 | 94655 | 55,499 |
https://mathoverflow.net/questions/94585 | 7 | The title says most. Let $P\_p$ be the cone of positive-definite $p \times p$ matrices.
One can define the Laplace transform of (the distribution of) a random matrix with values in $P\_p$ by (for instance, Muirhead: "Aspects of Multivariate Analysis") the integral
\begin{equation}
\phi(\Theta) = \int\_{P\_p} \exp... | https://mathoverflow.net/users/6494 | Laplace transform on the cone of positive-definite matrices | Laplace and Fourier transforms on the cone of positive definite matrices are somewhat well-studied. The framework is, as already mentioned by Alexander, that of "analysis on symmetric spaces." Here are some references where this subject is developed in greater detail:
1. *Harmonic analysis on symmetric spaces and app... | 3 | https://mathoverflow.net/users/8430 | 94658 | 55,500 |
https://mathoverflow.net/questions/94519 | 1 | I want to solve the follong QCQP problem:
$$
\mbox{Minimize}\quad\beta^TA\beta+\mu\Omega(\beta)$$
$$
\mbox{s.t.}\quad\beta^TB\beta=1 \quad\mbox{and}\quad\beta\ge0
$$
where $A$ and $B$ are both positive definite, and $\Omega(\cdot)$ is a sparse(thus non-smooth, but still convex) norm, like $\ell\_1$ norm or $\ell\_1... | https://mathoverflow.net/users/14910 | Solving a QCQP problem with sparse regularization | There are several methods that one could apply to this problem. I don't have time to write out a full solution, but here's a quick idea. Replace $\Omega$ by $\hat\Omega = \Omega + \delta\_+$, where $\delta\_+$ is the indicator function for the nonnnegative orthant. Now, if you can reduce your problem to a proximal spli... | 3 | https://mathoverflow.net/users/8430 | 94660 | 55,501 |
https://mathoverflow.net/questions/94661 | 5 | This question is motivated by my discussion (via comments) with @fedja regarding [this earlier question.](https://mathoverflow.net/questions/94026/functions-with-same-area) In any case the question is whether there is any concise characterization of finite dimensional subspaces of $L^1.$ I found some papers by our own ... | https://mathoverflow.net/users/11142 | Finite dimensional subspaces of $L^1.$ | I think that $K$ is such a convex body if and only if its dual $K^\*$ is a zonotope, i.e., a Minkowski sum of finitely many line segments, or a limit of zonotopes. I think that every limit $Z$ of zonotopes is characterized by a finite-mass measure $\mu$ on projective space $\mathbb{R}P^{d-1}$. Then the points in $Z$ ar... | 3 | https://mathoverflow.net/users/1450 | 94665 | 55,504 |
https://mathoverflow.net/questions/94675 | 2 | This is the question I had meant to ask when I asked [this question.](https://mathoverflow.net/questions/94661/finite-dimensional-subspaces-of-l1): Is there a concise characterization of finite dimensional subspaces of $L^\infty?$ (that's what the discussion with @fedja was really about...)
| https://mathoverflow.net/users/11142 | Finite imensional subspaces of $L^\infty.$ | That's much easier and more standard than finite-dimensional subspaces of $L^1$. The answer is all norms in finite dimensions, or in the unit ball picture, all centrally symmetric convex bodies. Every polytope is a slice of an $n$-cube, so clearly you get all of those. But then the Banach-Alaoglu theorem in this case l... | 4 | https://mathoverflow.net/users/1450 | 94677 | 55,513 |
https://mathoverflow.net/questions/94537 | 42 | Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$
My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known?
Of course if $X$ is compact $\mathcal O(X)=\mathbb C$ and that dimension is $0$.
There are also quite a lot of no... | https://mathoverflow.net/users/450 | What is the Krull dimension of the ring of holomorphic functions on a complex manifold? | It follows from the proof in Sasane's paper that Krull dimension of a (connected) complex manifold $M$ is infinite iff $M$ admits a nonconstant holomorphic function $F: M\to {\mathbb C}$. Namely, using Sard's theorem find a sequence of points $a\_k \in F(M)$ which are regular values of $F$ and so that $(a\_k)$ converge... | 39 | https://mathoverflow.net/users/21684 | 94684 | 55,516 |
https://mathoverflow.net/questions/94440 | 2 | Define a transformation $T\_s$ of integer sequence $\{ a\_n \}$ by
$$
b\_n=T\_s(a\_n)={n \choose s} \sum\_{i=s}^{n-1} \frac{a\_i}{{i \choose s}},
$$
for a fixed $s \in \mathbb{N}.$
For instance, if we aplly the transformation $T\_2$ to the sequence $a\_n=1$ then we get the sequence $b\_n=n(n-2).$
Maple code for the... | https://mathoverflow.net/users/23022 | Find generating function | This can be done step by step.
First note that $\binom{n}{s}/\binom{i}{s}$ can be written as $n(n-1)\cdots(n-s+1)/i(i-1)\cdots(i-s+1)$
Since we have the generating function (with assuming $a\_i=0$ for $i< s$)
$$a(x)x^{-s}=\sum\_{i=s}^{\infty} a\_ix^{i-s}$$
We obtain the following by integrating $s$ times.
Let $A\_0(x)=... | 3 | https://mathoverflow.net/users/21090 | 94685 | 55,517 |
https://mathoverflow.net/questions/94495 | 1 | Briefly, I'm asking why commuting the metric and the second fundamental form should make a difference.
Normally I wouldn't think much of this, but I came across the issue in the paper "Hypersurfaces of constant curvature in hyperbolic space. II." by Guan and Spruck. They in turn cite another paper, "Nonlinear second... | https://mathoverflow.net/users/13508 | Simple question regarding shape operator/minimal surface equation | Posting this so I can close the problem. Question has been answered above.
| 0 | https://mathoverflow.net/users/13508 | 94686 | 55,518 |
https://mathoverflow.net/questions/94682 | 13 | I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian curvature of a surface is an intrinsic quantity.
For instance, I am fascinated by whether Gauss had imagined that it wa... | https://mathoverflow.net/users/23085 | History surrounding Gauss Theorema Egregium and differential geometry | It seems to me that Volume 2 of Mike Spivak course on differential geometry gives an answer to your question. See Section A of chapter 3 : How to read Gauss?
| 11 | https://mathoverflow.net/users/23086 | 94687 | 55,519 |
https://mathoverflow.net/questions/94688 | 3 | Hi,
i know that the following statement is used extensively, but i cannot find a proof anywhere:
For $\Omega$ a Lipschitz domain with boundary $\Gamma$, the space $H^{1/2}(\Gamma)$ is dense in $L\_2(\Gamma)$.
Here, the space $H^{1/2}(\Gamma)$ is defined as the trace space, i.e. as $\gamma\_0(H^1(\Omega))$, where ... | https://mathoverflow.net/users/23087 | Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$ | Lipschitz continuous functions on $\Gamma$ are dense in $L^2(\Gamma)$ and are contained in $H^{1/2}(\Gamma)$.
| 2 | https://mathoverflow.net/users/7294 | 94698 | 55,523 |
https://mathoverflow.net/questions/94699 | 1 | Dear mathematicians,
in my current research project I came accross this very bothersome sum over a rather simple hypergeometric function, or formulated differently: a sum over squared binomial coefficients:
$\sum\_{b=0}^\infty Y^b\ \_2F\_1(-b,-b;1;X)=\sum\_{b=0}^\infty \sum\_{k=0}^\infty \binom{b}{k}^2 Y^bX^k$
I ... | https://mathoverflow.net/users/23090 | Sum over Hypergeometric function 2F1 (generating function) | The wiki page on [Legendre polynomials](http://en.wikipedia.org/wiki/Legendre_polynomials) has all the identities that I will use below.
Start with
$$P\_n(t)=\sum\_{k=0}^n (-1)^k \binom{n}{k}^2 \left(\frac{1+t}{2}\right)^{n-k}\left(\frac{1-t}{2}\right)^k$$
which can be rewritten as
$$(t-1)^nP\_n\left(\frac{t+1}{t-1}\... | 6 | https://mathoverflow.net/users/2384 | 94703 | 55,526 |
https://mathoverflow.net/questions/94577 | 1 | Let $k$ be a field and let $G$ be an algebraic group over $k$.
I encountered the following notion in an article:
"Let $\psi: \text{Gal}(\overline{k}/k) \to \text{Int}(\overline{G})$ be a cocycle. The **twisted group** $\_\psi G$..."
How is this "twisted group" defined?
| https://mathoverflow.net/users/1107 | Definition of "twisted group" | You can find the definitions in Serre's book "Galois cohomology". The relevant sections are I.5.3 - I.5.7. The idea is that if you have a group $G$ acting on a group $A$ and a cocycle $\psi\in H^1(G,A)$. The twisted group $\_\psi A$ has the same group structure as $A$ but it has a twisted action by $G$, which is given ... | 2 | https://mathoverflow.net/users/2384 | 94705 | 55,527 |
https://mathoverflow.net/questions/85119 | 16 | Hello, everyone.
I want to ask some questions about rigid geometry.
1.what is the motivation of rigid geometry?
2.what is the applications of rigid geometry for solving arithmetic problems, especially for studying the fundamental groups of algebraic curves? what the beautiful theorems which were first proved by r... | https://mathoverflow.net/users/5274 | why we need rigid geometry? | I am really not an expert in the field, so I apologize in advance for omissions or mistakes - I would indeed be glad to get corrections. But let me try, anyhow...
You are asking for a motivation for rigid geometry and here, I guess, Kevin is right when saying that the first historical motivation was may be Tate's th... | 21 | https://mathoverflow.net/users/18238 | 94706 | 55,528 |
https://mathoverflow.net/questions/94341 | 1 | Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and $f'(x)>0$ for all $x \in [-1,0]$.
Notice that given a fixed number $c>0$ (and you may suppose that $c \neq 1$ if necessary), ther... | https://mathoverflow.net/users/21269 | A Fractional Linear Transformation Class Property | You're asking for a one-parameter family $\mathcal{C}$ of maps on a closed interval $I$ such that
1. the endpoints of $I$ are fixed by every map in $\mathcal{C}$;
2. the maps in $\mathcal{C}$ are strictly increasing;
3. $\mathcal{C}$ is closed under composition;
4. every positive real number appears exactly once as ... | 1 | https://mathoverflow.net/users/5701 | 94711 | 55,531 |
https://mathoverflow.net/questions/94715 | 6 | Dear All,
I'm reading a paper (*Residuality of Dynamical Morphisms* by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the extreme points of a compact convex set in a locally convex topological vector space form a $G\_\delta$ subset of the ... | https://mathoverflow.net/users/11054 | Extreme points of a compact convex set are a $G_\delta$? | For a non-metrizable compact convex subset of a locally convex space, extreme points need not even form a Borel set. This has been shown by Bishop-de Leeuw, *The representation of linear functionals by measures on sets of extreme points*, Ann.Inst. Fourier (Grenoble) (1959) .
A very good reference for these topics is P... | 10 | https://mathoverflow.net/users/6101 | 94718 | 55,535 |
https://mathoverflow.net/questions/94719 | 2 | The vertices of G are the k-subsets of some set of size n. Two k-subsets are connected by an edge of G if their symmetric difference is of size 2. The integers n and k are chosen so that the graph is not trivial.
| https://mathoverflow.net/users/22748 | Is this graph well known? | That's the Johnson graph: <http://en.wikipedia.org/wiki/Johnson_graph>
| 5 | https://mathoverflow.net/users/12674 | 94720 | 55,536 |
https://mathoverflow.net/questions/94681 | 3 | Let $X$ be a smooth variety with an action of $\mathbb{C}^\*.$ One has the so-called *Bialynicki-Birula decomposition* of $X$ given by *stable manifolds*: $$X=\bigcup\_N X^+(N),$$ where $N$ varies in the set of connected components of the fixed point locus $X^{\mathbb{C}^\*}.$ Rougly speaking, $X^+(N)$ is related to th... | https://mathoverflow.net/users/nan | Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition | If $X$ is projective, the Bialynicki-Birula decomposition is *filterable*; this means that there is a filtration $Z\_1 \subseteq Z\_2 \subseteq \cdots \subseteq Z\_r = X$ by closed subsets, such each difference $Z\_i \smallsetminus Z\_{i-1}$ is a piece of the Bialynicki-Birula decomposition (Białynicki-Birula, Some pro... | 3 | https://mathoverflow.net/users/4790 | 94722 | 55,537 |
https://mathoverflow.net/questions/94689 | 11 | It is well known that in a given $\delta$-hyperbolic group there are only finitely many conjugacy classes of finite subgroups. This is clearly false for relatively hyperbolic groups since we have no control over the parabolic subgroups. Fix a relatively hyperbolic group $\Gamma$ with parabolic subgroups $P\_i$. Are the... | https://mathoverflow.net/users/2225 | Finite subgroups of relatively hyperbolic groups | Here's an idea for a proof in hyperbolic groups which generalizes to relatively hyperbolic groups. Actually, this will prove something slightly weaker ~~but which easily strengthens to what you want~~: namely, that there are only finitely many conjugacy classes of finite subgroups $F < \Gamma$ such that $F$ is containe... | 5 | https://mathoverflow.net/users/20787 | 94731 | 55,540 |
https://mathoverflow.net/questions/94282 | 6 | Related wikipage: <http://en.wikipedia.org/wiki/Gr%C3%B6tzsch_graph>
Is the crossing number of the Grötzsch graph known? I have heard it conjectured to be 5 (certainly it is no greater), but came up empty-handed in my search of the literature.
| https://mathoverflow.net/users/22971 | Crossing number of the Grötzsch graph | The crossing number of the Grötzsch graph is 5.
Crossing numbers are believed to be difficult to compute in general. (The corresponding decision problem is NP-hard.) However, for small graphs and small crossing numbers, it is possible to find an optimal planar drawing. For example, see Markus Chimani's thesis ["Compu... | 10 | https://mathoverflow.net/users/23092 | 94732 | 55,541 |
https://mathoverflow.net/questions/94700 | 2 | I'm currently reading some stuff on Hopf algebras, specifically *Hopf Algebras: An Introduction* by Sorin Dascalescu, Constantin Nastasescu and Serban Raianu. One proof involves showing that for a Hopf algebra $H$ with antipode $S$ that
$$\Delta(S(h)) = \sum S(h^{(2)})\otimes S(h^{(1)})$$
In part of the proof a map $G... | https://mathoverflow.net/users/22360 | Convolution inverse | As you rightly observed, the poof has much to do with the formalism of the Sweedler notation. I'll try to explain the usuage. Let's start with the identity
$$\Delta\_2 := (\Delta \otimes id) \circ \Delta =(id \otimes \Delta) \circ \Delta$$
Depending on the position of the outer $\Delta$ one has respective structural i... | 4 | https://mathoverflow.net/users/10194 | 94749 | 55,550 |
https://mathoverflow.net/questions/94757 | 4 | A topological space $X$ is weakly Lindelof if every open cover has a countable subfamily $U$ such that $\bigcup \{ V: V\in U\}$ is dense in X.
Question: Are closed subsets of weakly Lindelof spaces necessarily weakly Lindelof?
| https://mathoverflow.net/users/22260 | closed subset of weakly lindelof | The [Niemytzki plane](https://en.wikipedia.org/wiki/Moore_plane) is weakly Lindelöf (the open upper half plane is a dense Lindelöf subspace); the $x$-axis is an uncountable closed and discrete subspace.
| 5 | https://mathoverflow.net/users/5903 | 94760 | 55,556 |
https://mathoverflow.net/questions/94726 | 3 | I have to study/evaluate many determinants of the form
$$
f\_M(J)=\det(J-M),
$$
where $M$ is fixed, and $J$ is a diagonal matrix (with
0/1 on the diagonal, if it helps.) In my problem
$M$ is fixed, and $J$ varies.
Any suggestions?
| https://mathoverflow.net/users/8135 | determinant of diagonal - fixed | Let $J$ be $\mathrm{diag}(a\_1,...,a\_n)$. Let $M\_{i\_1,...,i\_k}$ be the principal submatrix of $M$ with rows and colums number $\{i\_1,...,i\_k\}$ deleted. Then $\det(J+M)=\det(M+\mathrm{diag}(0,a\_2,...,a\_n))+a\_1\det(J\_1+M\_1)$ (expand along the first row). Thus $$\det(J+M)=\sum a\_{i\_1}\cdot\ldots\cdot a\_{i\_... | 3 | https://mathoverflow.net/users/nan | 94762 | 55,558 |
https://mathoverflow.net/questions/94756 | 7 | I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\rightarrow\mathbb{R}$ a smooth function, its symplectic gradient is the unique field $X\_H$ over $M$ satisfying
$$\textrm{d}... | https://mathoverflow.net/users/22339 | is the geodesic flow on Hyperbolic Plane completely integrable? | Yes, the geodesic flow on the hyperbolic plane and, in fact, on any Hadamard manifold (${\mathbb R}^n$ provided with a complete Riemannian metric with non-positive curvature) is integrable.
You can easily construct integrals of motion for the geodesic flow in hyperbolic space as follows:
**1.** Consider the Cayley... | 10 | https://mathoverflow.net/users/21123 | 94764 | 55,560 |
https://mathoverflow.net/questions/93445 | 6 | As a more nontrivial example for my Dissertation thesis, I'd require some example of the following type (of course I'll "cite" ;-) ), so thanx in advance:
Andruskiewitsch/Grana have by a new construction given very interesting new liftings of finite dimensional Nichols algebras e.g. over $S\_4$ ("Examples of liftings... | https://mathoverflow.net/users/22709 | Liftings of Nichols algebras over racks via Doi twist | At the moment it is not easy to see that this will also hold for the non-abelian case, or even for the abelian cases yet to be computed, although there are no counter examples. It holds in most known examples as shown in
A. G. I. and Mombelli, M. Representations of the category of modules over pointed Hopf algebras ... | 6 | https://mathoverflow.net/users/23116 | 94798 | 55,575 |
https://mathoverflow.net/questions/94543 | 13 | EDIT: this question of mine has received little attention, perhaps in part because it was stated in a too general and complicated way. So let me give it a second chance:
>
> Fix an integer $r \geq 0$. Let $E\_r(x)$ be the number of square free integers less than $x$ having exactly $r$ prime factors $p$ congruent to... | https://mathoverflow.net/users/9317 | Density of a set of integers | Here's an attempt to say something about your nice question. It surely follows from Igor Rivin's nice sketch and the Math Overflow question he linked to. Fix $r$. The quantity $E\_r(x)$ in the edited question satisfies
$$
E\_r(x)\gg \sum\_{\substack{m \leq x \\ \omega(m)=r\\ \mu(m)^2=1\\p|m \Rightarrow p \equiv 2 \bm... | 5 | https://mathoverflow.net/users/22202 | 94802 | 55,576 |
https://mathoverflow.net/questions/94807 | 2 | Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative axis. Is $u(z)$ subharmonic on $D\setminus\lbrace z\in D: f(z)=0\rbrace?$ When applying the maximum principle to $u(z)$ mus... | https://mathoverflow.net/users/12337 | Are the real components of s-roots subharmonic? | Assuming you are using the principal branch of the $s$'th root, $u(z)$ is indeed a subharmonic function. You do need to "check the branch cut", but it works out ok.
Consider a point $p$ where $f(p)$ is on the negative real axis (so $p$ is on a branch cut of $f(z)^{1/s}$), and let $v(z)$ be the version of $\Re f(z)^{1... | 2 | https://mathoverflow.net/users/13650 | 94810 | 55,580 |
https://mathoverflow.net/questions/94785 | 2 | Assume $C$ is a locally small category with equalizers in the sense that given any two arrows $f$ and $g$ with a common source $a$ and target $b$, then there is an object $e$ and an arrow
$i\colon e\to a$ such that $fi = gi$ that satisfies the usual universal property. Seems that
this generalizes fairly easily (by ind... | https://mathoverflow.net/users/7779 | How to generalize equilizers in a category to hom-sets? | Look for "equalizer of a family of arrows" in an arbitrary book on category theory. It is a special case of a limit. A typical example for a category which has finite limits, but not arbitrary limits (and therefore not all infinite equalizers), is the category of schemes.
| 5 | https://mathoverflow.net/users/2841 | 94818 | 55,583 |
https://mathoverflow.net/questions/94816 | 3 | Does anyone know if the Johnson graphs are Hamiltonian? I would appreciate any references pertaining to problem.
| https://mathoverflow.net/users/22748 | Are the Johnson graphs Hamiltonian? | Theorem 3.20 of <http://www.science.uva.nl/onderwijs/thesis/centraal/files/f1182505387.pdf>
seems to imply that the answer is yes.
| 4 | https://mathoverflow.net/users/14302 | 94822 | 55,586 |
https://mathoverflow.net/questions/94813 | 2 | Is there some name in ergodic theory or integrable systems theory for a function whose average value on every orbit is zero? (Of course when I say "every orbit" in the context of ergodic theory I mean "modulo a set of measure zero".)
The space of $L^2$ functions with ergodic average zero is the orthocomplement of the... | https://mathoverflow.net/users/3621 | functions whose average along orbits is zero or a constant | Second part: if your system is nontrivial, and function $f$ is "smooth", than condition
of the form
$$
\lim\_{n\to\infty} \frac 1n\sum\_{i=0}^{n-1} f(T^i x) =c
$$
for *all* $\ x$, should imply that
$$
f(x)= g(x)-g(Tx)+c
$$
for some function $g$. In this case, one says that $f$ is cohomologous to a constant.
| 1 | https://mathoverflow.net/users/8135 | 94826 | 55,588 |
https://mathoverflow.net/questions/94811 | 3 | Hello Everyone,
I have a question about smooth embeddings of spheres into larger spheres (by sphere I mean the usual, non-exotic kind, in case that makes a difference).
Let $h : S^n \rightarrow S^{n+k}$ be a smooth embedding (with $n,k > 0$). It's always possible to extend $h$ (or any continuous map from $S^n$ to ... | https://mathoverflow.net/users/23121 | Extending Smooth Embeddings of S^n into S^{n+k} | Your condition can be stated equivalently as the *knot longitude*, as an embedded $S^n$ in the *knot complement* $C\_h = S^{n+k} \setminus h(S^n)$ is null homotopic.
Generally the answer is no when $k=2$:
* When $n=1$ this follows from the *loop theorem* which you can find in the book Mark Grant cites.
* For all $... | 3 | https://mathoverflow.net/users/1465 | 94829 | 55,590 |
https://mathoverflow.net/questions/94838 | 4 | Let $S$ be some (fixed) subset of $\mathbb{Z} [X\_1, \dots , X\_n]$ which contains only homogeneous polynomials, and if $F$ is a field, let $X(F)$ be the set of $ x \in P^{n-1}(F)$ such that $f (x) = 0$ for any $f \in S$.
Now consider the assertion $(E\_p) : X(F)$ is empty for any field $F$ of characteristic $p$.
Fro... | https://mathoverflow.net/users/21724 | Emptyness of a projective variety | No. For example, $3x^3+4y^3+5z^3=0$ has nontrivial solutions mod $p$ for every prime $p$, but it has no nontrivial solutions in $\mathbb{Q}$. Indeed, it has solutions in every $p$-adic field $\mathbb{Q}\_p$ and also solutions in $\mathbb{R}$. This is a famous example of Selmer. One says that the "Hasse principle" fails... | 13 | https://mathoverflow.net/users/11926 | 94839 | 55,595 |
https://mathoverflow.net/questions/94855 | 2 | Hi there. Suppose ${\bf C}\_1$, ${\bf C}\_2$ and $\bf D$ are categories and $F\_i$ is a functor ${\bf C}\_i \to \bf D$. Consider the subcategory of the comma category $( F\_1 \downarrow F\_2)$ whose objects are all triples $(c\_1, c\_2, e)$ where $c\_i \in \mathrm{obj}(\mathbf C\_i)$, $F\_1(c\_1) = F\_2(c\_2)$ and $e$ ... | https://mathoverflow.net/users/16537 | Looking for the name of a particular subcategory of a comma category | The (strict) pullback of $F\_0$ and $F\_1$.
| 8 | https://mathoverflow.net/users/49 | 94856 | 55,603 |
https://mathoverflow.net/questions/94857 | 1 | Hi,
Suppose $G$ is a reductive group over an algebraiclly closed field $k$
(suppose $k$ of char zero if you want at first). Let $X$ be its flag variety.
Question: What is the moduli problem that $X$ represents?
EDIT (to clarify): What is the functor of points of $X$?
Thanks!
| https://mathoverflow.net/users/36285 | moduli problem for flag varieties? | Let $X$ be a space and $H$ be a group such that $X\rightarrow X/H$ is a principal bundle. Then $Hom(Y,X/H)$ is in bijection with $H$ torsors over $Y$ equipped with an equivariant map from their total space to $X$. So maps from $Y$ into the flag variety $G/P$ are in bijection with $P$ torsors on $Y$ equipped with a $P$-... | 5 | https://mathoverflow.net/users/2837 | 94859 | 55,604 |
https://mathoverflow.net/questions/94862 | 0 | Let $V$ be the category of finite dimensional vector spaces and $M$ the category of
smooth finite dimensional Hausdorff manifolds.
Now suppose any finite dimensional vector space is equipped with a smooth structure in such a way that any $n$-dimensional vector space is diffeomorph to $\mathbb{R}^n$
seen as a smooth m... | https://mathoverflow.net/users/21965 | inclusions of linear colimits into smooth manifolds | The canonical map $i(\mathbb{R}^n) \coprod\_M i(\mathbb{R}^m) \to i(\mathbb{R}^n \coprod\_V \mathbb{R}^m)$, where the coproduct index indicates the ambient category, corresponds to the smooth map $\mathbb{R}^n \sqcup \mathbb{R}^m \to \mathbb{R}^{n+m}$. It is neither surjective nor injective (the two zero vectors are ma... | 1 | https://mathoverflow.net/users/2841 | 94872 | 55,612 |
https://mathoverflow.net/questions/94737 | 3 | Given any infinite non-elementary hyperbolic group $G$, a theorem of Gromov asserts that there is a subgroup of $G$ isomorphic to a non-abelian free group on two generators.
Is there a similar result for a quotient of $G$? That is, is there a normal subgroup $N$ of $G$ such that $G/N$ is isomorphic to a non-abelian f... | https://mathoverflow.net/users/18583 | Free groups as quotients of hyperbolic groups | One does not need Kazhdan property (T). Take $\mathrm{PSL}\_2(\mathbb{Z})$. The group is hyperbolic (it has a free subgroup of finite index), and is generated by an element of order 2 and an element of order 3 (it is the free product of two finite cyclic groups). Hence the generators die in every torsion-free homomorph... | 5 | https://mathoverflow.net/users/nan | 94874 | 55,613 |
https://mathoverflow.net/questions/94881 | 5 | Suppose that $p$ is unramified in a number field $K$ of degree $n$. Apparently, Stickelberger proved that $\big( \frac{Disc(K)}{p}\big) = (-1)^{n - g}$, where $g$ is the number of prime ideal factors over $(p)$ in $K$.
Is there a convenient reference for this fact?
Thank you! -Frank
| https://mathoverflow.net/users/1050 | A theorem of Stickelberger on the number of prime ideals in a decomposition | First reduce the question to the local case--This can be done since after base change to $\mathbb{Q}\_{p}$, $K \otimes\_{\mathbb{Q}} \mathbb{Q}\_{p}=K\_1 \times...\times K\_{g}$, the trace form is the orthogonal sum of the local trace forms so the discriminant is the product of the local discriminants, and then use tha... | 6 | https://mathoverflow.net/users/2089 | 94895 | 55,626 |
https://mathoverflow.net/questions/94776 | 1 | Le $X\_1$, $X\_2$ and $Z$ be smooth quasi-projective connected varieties defined over $\mathbf{C}$. Let $p\_1:X\_1\rightarrow Z$ and $p\_2:X\_2\rightarrow Z$ be finite etale maps. Assume that $f:X\_1\rightarrow X\_2$ is an **analytic** isomorphism such that $p\_2\circ f=p\_1$.
Q1: Does it follow that $f$ is regular?
... | https://mathoverflow.net/users/11765 | Analytic isomorphisms above two etale maps | This (from SGA 1) is the proof that the functor is fully faithful: We may suppose
that $X$ is connected. To give an $X$-morphism $Y\rightarrow Y^{\prime}$ is to
give a section to $Y\times\_{X}Y^{\prime}\rightarrow Y$, which is the same as
to give a connected component $\Gamma$ of $Y\times\_{X}Y^{\prime}$ such that
the ... | 1 | https://mathoverflow.net/users/22295 | 94898 | 55,629 |
https://mathoverflow.net/questions/89453 | 13 | In "[Existence theorems...](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=van%2520den%2520Bergh&s5=Existence&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq)" Van den Bergh proposes the ... | https://mathoverflow.net/users/17353 | Injective dimension of graded-injective modules | **Part II:** Answering the question
In the following all gradings are over $\mathbb Z$.
>
> **Theorem:** Let $R$ be a graded left-Noetherian ring and $N$ a graded $R$-module of finite graded injective dimension. Then $\operatorname{injdim}(N)\le \operatorname{gr-injdim}(N) + 1$.
>
>
>
**Remark:** In [1] Ek... | 4 | https://mathoverflow.net/users/10194 | 94911 | 55,636 |
https://mathoverflow.net/questions/94474 | 8 | On a 3-manifold $Y$, oriented 2-plane fields $\xi$ are oriented rank-2 subbundles of $TY$. Denote the set of such *(up to homotopy)* by $\Theta=\pi\_0\lbrace\xi\rbrace$. **What is an explicit canonical map $\Theta\to\mathbb{Z}\_2$ ?**
In particular, I want to see the canonical mod-2 grading on Seiberg-Witten-Floer Ho... | https://mathoverflow.net/users/12310 | $\pi_0${plane fields}$\to\mathbb{Z}_2$ | In light of the paper referenced in Tim Perutz' comment, here is the desired map:
An oriented 2-plane field $\xi$ is equivalent to a pair $(\mathfrak{s},\phi)$ on $Y$, where $\mathfrak{s}$ is a spin-c structure and $\phi$ is unit-length spinor. By Proposition 28.1.2 (of Kronheimer-Mrowka's textbook), there exists an ... | 3 | https://mathoverflow.net/users/12310 | 94912 | 55,637 |
https://mathoverflow.net/questions/94919 | 8 | Let $L\_{E}$ be the language of discretely ordered rings together with an extra predicate symbol $E$. The system $A$ consists of the axioms of $I\Delta\_{0}$ (basic arithmetic plus induction for bounded formulas) together with the statements:
(1) $E(0,1)$
(2) $\forall{x}\exists{y}E(x,y)$
(3) $\forall{x>0,y,z}E(x+... | https://mathoverflow.net/users/15814 | Is there exponentiation in "sufficiently large" models of $I\Delta_{0}$? | Do you intend system $A$ to include only $I\Delta\_0$, or actually $I\Delta\_0(E)$? That is, do you allow induction for formulas involving $E$?
If you do allow it, then then you can prove easily
$$\forall x,y\le w\,(E(x,y)\to x=y=0\lor\exists z\le y\,(z=2^x))$$
by induction on $w$, hence the theory proves exponentiat... | 15 | https://mathoverflow.net/users/12705 | 94928 | 55,644 |
https://mathoverflow.net/questions/89105 | 5 | I have a graph $G$ whose minimum vertex degree is $\delta=7$.
I am seeking an upper bound on the [domination number](http://en.wikipedia.org/wiki/Dominating_set) $\gamma(G)$
in terms of the number of vertices $n$ of $G$.
I found a paper by
Edwin Clark, Boris Shekhtman, Stephen Suen, and David Fisher,
"Upper bounds for... | https://mathoverflow.net/users/6094 | Bound on graph domination number when min degree is 7 | I believe the complete history is as follows. For an arbitrary graph with $n$ vertices and minimum degree $k$, the result has been shown to be at most
$$
\frac{n[1+\ln(k+1)]}{k+1}
$$
by Arnautov1 (1974) and Payan2 (1975), but the articles were written in Russian and French, respectively. For $k=7$, this gives a bound o... | 8 | https://mathoverflow.net/users/23148 | 94942 | 55,655 |
https://mathoverflow.net/questions/94892 | 4 | Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables of the form (homothety):
$$ h(x) = mx, m>0$$
In other words, I would like the following property to hold for any af... | https://mathoverflow.net/users/21269 | What classes of functions are closed under all rescalings? | If $G$ contains the linear group consisting of all functions of the form $h\_m(x) = mx$ then obviously it satisfies your conditions.
Conversely, by the definition of the derivative,
(\*) $lim\_{m \to \infty} (h\_m^{-1} f h\_m(x)) = f'(0) x$
So the closure $\bar G$ of your group (in some appropriate topology) cont... | 2 | https://mathoverflow.net/users/20787 | 94957 | 55,663 |
https://mathoverflow.net/questions/93226 | 0 | let be the differential equation
$ -ixDf(x)-if(x)/2= E\_{n}f(x) $
with the boundary conditions $ f(x)=f(p^{k}x) $ for 'p' prime and $k=...,-2,-1,0,1,2,...$
is this possible to solve this eigenvalue problem ?? thanks
| https://mathoverflow.net/users/21933 | first-order linear differential equation with boundary conditions | As I understand the question, you first fix $p$, and then you search the solutions of $$−ixDf(x)−if(x)/2=E\_nf(x)$$
which satisfy $f(px)=f(x)$ (by reccurence, your $p^k$ conditions is automatically satisfied).
So we solve the differential equation and we find $cx^{iE\_n-1/2}$ and your condition gives $p^{iE\_n-1/2}=1$.... | 1 | https://mathoverflow.net/users/23077 | 94959 | 55,665 |
https://mathoverflow.net/questions/94948 | 7 | Well, the question is probably a rather basic one but I haven't been able to find the answer in literature or come up with it myself so here we go.
Do there exist irreducible representations of the Lie algebra $\mathfrak g$ which aren't a highest weight module with respect to some Borel subalgebra in the case when
... | https://mathoverflow.net/users/19864 | Irreps which aren't highest-weight modules. | Here's a more specific reference for the rank 1 simple Lie algebra; the "classification" by Block shows clearly how sparse the highest weight representations are among all irreducible representations [here](http://www.ams.org/journals/bull/1979-01-01/S0273-0979-1979-14573-7/home.html). Block was basically responding to... | 10 | https://mathoverflow.net/users/4231 | 94965 | 55,668 |
https://mathoverflow.net/questions/94964 | 2 | Is there an n-1 form on $R^n$ which calculates the volume of n-manifolds? Similarly, is there such a 1 form on $S^2$, and $RP^2$? I thought this has something to do with the orientation, is that right?
| https://mathoverflow.net/users/23153 | Questions on calculating volume using n-1 forms | The question is a bit vague so I will try to make the best of it. Suppose that $(M, g)$ is a connected, noncompact oriented, $n$-dimensional Riemann manifold and $dV\_g\in\Omega^n(M)$ is the associated volume form. Suppose that $D\subset M$ is an open, precompact subset of $M$ with smooth boundary $\partial D$. Since $... | 7 | https://mathoverflow.net/users/20302 | 94970 | 55,672 |
https://mathoverflow.net/questions/94605 | 4 | Since the [original question](https://mathoverflow.net/questions/94555/coloring-a-unit-cube) has been answered by Fedor Petrov and Robert Israel, here are more difficult questions (the terminology is preserved).
* Let $m>n$. What is the smallest number $c(m,n)$ of colors needed to color the unit cube $I^m$ so that... | https://mathoverflow.net/users/nan | Coloring a unit cube 2 | The question turned out to be easy: $c(m,n)=2$ as soon as $m\ge 2n$. Indeed, take a vertex $v$ of the cube $I^m$ and let $v'$ be the opposite vertex (i.e. $vv'$ is a long diagonal). Let $B\_n(v)$ be the ball of radius $n$ (in the $l\_1$-metric) around $v$. Let us color the set $B\_n(v)\setminus \{v\}\cup\{v'\}$ in red ... | 2 | https://mathoverflow.net/users/nan | 94971 | 55,673 |
https://mathoverflow.net/questions/94652 | 25 | I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and consider a ball $B \subset M$ centered at a distinguished point $p \in M$ whose radius is no larger than the injectivity r... | https://mathoverflow.net/users/1557 | Relationship between Green's function and geodesic distance? | First observe that on a compact Riemann manifold $(M, g)$ the operator $1+t \Delta$, $t>0$. $\Delta = d^\*d: C^\infty(M)\to C^\infty(M)$ has a unique fundamental solution. Jacques Hadamard has constructed very explicit asymptotic expansions for this fundamental solution which lead to convergent series in the case of re... | 12 | https://mathoverflow.net/users/20302 | 94973 | 55,674 |
https://mathoverflow.net/questions/94972 | 2 | In an article I'm reading, the author is stating :
>
> $O$ is isomorphic to the complement of a zero section of a line bundle over $X$. We have a long exact sequence of (étale) cohomology associated to the fibration :
> $$ \cdots \to H^{i-2}\_c(X) \to H^{i}\_c(X) \to H^{i+1}\_c(O) \to H^{i-1}\_c(X) \to \cdots $$
>... | https://mathoverflow.net/users/15404 | Long exact sequence of cohomology and fibration | Though Mark Grant's comment links to the right answer (what you are looking for is called the Gysin sequence), the Wikipedia page doesn't state it in the form you want. In general, for an open subset $U \subset X$ of an algebraic variety $X$, there is a long exact sequence
$$\cdots \to H^i\_c(U) \to H^i\_c(X) \to H^i\_... | 6 | https://mathoverflow.net/users/1310 | 94975 | 55,675 |
https://mathoverflow.net/questions/94551 | 24 | The set of primes $\mathbb{P}$ has many interesting properties in additive number theory and some of the most famous open problems about $\mathbb{P}$ are the well-known Goldbach's strong and weak conjectures. The weak conjecture was proven by Vinogradov for all sufficiently large integers. Moreover, Chen was able to pr... | https://mathoverflow.net/users/23008 | Are sets with similar asymptotic behavior as the primes necessarily finite additive bases? | Let $A\_n = \{a : a \equiv 1 \mod 2^n \mbox{ and } 2^{2^{n-1}} \leq a < 2^{2^{n}} - 2^{2^{n-1}}\}$, and let $\displaystyle A = \bigcup\_{n=1}^\infty A\_n$.
Then, $A(x) >> \frac{x}{\log x}$, the gap sizes are $<< \sqrt{x}$, and $A$ contains infinitely many non-multiples of $m$ for every $m>1$.
However, $2^{2^n}$ can... | 17 | https://mathoverflow.net/users/8410 | 94977 | 55,677 |
https://mathoverflow.net/questions/94974 | 4 | I am studying slice knots, so for example they say the cone on a trefoil knot can be embedded
in D^4 but it is not locally flat at the vertex of the cone. What I do not understand that I think every surface which is embedded in D^4 is already flat. What prevents the cone to be flat at the vertex? after all in D^4 we h... | https://mathoverflow.net/users/23155 | slice knots, what does the locally flat condition say? | Locally flat is equivalent to flat in this instance, which means the embedding $D^2\subset D^4$ extends to an embedding $D^2\times D^2 \subset D^4$. This is a topological assumption which replaces the theorem (inverse function theorem) that smooth embeddings are always locally flat. As mentioned above, the cone on a tr... | 5 | https://mathoverflow.net/users/3874 | 94986 | 55,682 |
https://mathoverflow.net/questions/95011 | 2 | Suppose I have two multisets, $A=\{a\_1,a\_2,\ldots,a\_n\}$ and $B=\{b\_1,b\_2,\ldots,b\_n\}$. We can construct a (random) bipartite multigraph with the vertex bipartition $\text{set}(A) \cup \text{set}(B)$ by the following process:
* start with the null graph on vertex multiset $A \cup B$,
* pick a (random) permutat... | https://mathoverflow.net/users/2264 | In how many ways can we generate a given bipartite multigraph via the bipartite configuration model? | Hi,
I'm going to try to answer your first question, which is: Given a bipartite multigraph on $set(A)\cup set(B)$, in how many ways could it have resulted from the described procedure?
I think that your question is related to Barvinok's work on contingency tables. In particular, take a look at theorem 1.2 here:
<h... | 4 | https://mathoverflow.net/users/17599 | 95016 | 55,695 |
https://mathoverflow.net/questions/95025 | 7 | Hello everyone
I would like to learn basic theory of the Chevalley Groups. There are several references for this subject, like "Introduction to Lie algebras and representation theory" by Humphreys, and Steinberg notes on the Chevalley groups.
Do you know any other references beside these two books?
Thank you
| https://mathoverflow.net/users/8419 | A reference for the Chevalley Groups | It doesn't contain any representation theory, but I think that Carter's book "Simple Groups of Lie Type" is an excellent place to start. Steinberg's notes have a lot more stuff in them, but Carter's book is an easier read.
| 10 | https://mathoverflow.net/users/317 | 95034 | 55,702 |
https://mathoverflow.net/questions/94947 | 1 | According to the basic rules of symbolic caculus,$[a(x,D),x\_{j}]=-ia^{j}[x,D]$.So we have $[(1-\triangle)^{\frac{1}{2}},x\_i]=\partial\_i(1-\triangle)^{-\frac{1}{2}}$ which is $L^2$ bounded.
It's also true that the $[(1-\triangle)^{\frac{1}{2}},\langle x \rangle]$ is $L^2$ bounded.Can we write the explicit expression ... | https://mathoverflow.net/users/23078 | Does these commutator estimates bound in $L^{2}$ | Here is a technique that shows the commutator is bounded for $\alpha <1$. First note, as I observed in my comment, it is a bounded operator for $\alpha \le 0$. Now note that if $0<\eta<1$ and $A$ is a strictly positive operator then we have the identity
$$ A^\eta = c\_\eta \int\_0^\infty t^\eta \left (\frac{1}{t}- \fra... | 2 | https://mathoverflow.net/users/6781 | 95035 | 55,703 |
https://mathoverflow.net/questions/95031 | 3 | I'm trying to prove the following problem in the *Deformation theory* book by Hartshorne.
>
> Any normalized vector bundle $\mathcal E$ of rank 2 degree 1 on an elliptic curve $\mathcal C$ can be written as a non-split extension
>
>
> $0 \to \mathcal{O\_C} \to \mathcal{E} \to \mathcal{O\_C(p)} \to 0$
>
>
> by a... | https://mathoverflow.net/users/21902 | Vector bundle of rank 2 and degree 1 on an elliptic curve | I guess you mean that any vector bundle with these properties and *which is not a sum of line bundles* can be written in that way.
The following standard proof can be found in Friedman's book "Algebraic surfaces and holomorphic vector bundles", page 35.
Since $\textrm{rank}(\mathcal{E})=2$ and $\det{\mathcal{E}}=1$... | 6 | https://mathoverflow.net/users/7460 | 95037 | 55,704 |
https://mathoverflow.net/questions/95045 | 4 | Cartan's theorem B states that every coherent analytic sheaf on a Stein manifold is acyclic. Hartshorne claims that the converse to this theorem also holds, but doesn't supply a reference. Does anyone where to find a proof of this fact?
Thank you kindly.
| https://mathoverflow.net/users/1106 | Reference for the converse of Cartan's Theorem B | Actually a more general result is true:
If $X$ is a complex *space* (maybe not smooth) and if $H^1(X,\mathcal I)$ is zero for all coherent sheaves of ideals $\mathcal I\subset \mathcal O\_X$, then $X$ is a Stein space.
This is Proposition 52.6 In L.Kaup-B.Kaup's [*Holomorphic Functions of Several Variables*](http... | 9 | https://mathoverflow.net/users/450 | 95051 | 55,711 |
https://mathoverflow.net/questions/95021 | 11 | Quasi-categories (or $\infty$-categories, as they are often called) are a very convenient setting for doing abstract homotopy theory. One of their amazing features is the following: Given a diagram of quasi-categories, we can form its homotopy limit, yielding a quasi-category again. For example, the inverse (homotopy) ... | https://mathoverflow.net/users/2039 | Homotopy limits of quasi-categories | I will address your second question: "one has to prove that the classification diagram functor is sent under this Quillen equivalence to something weakly equivalent to the coherent nerve".
The answer is yes, this is true. First note that the simplicial category $M^\circ$ is equivalent as a simplicial category to $L^H... | 6 | https://mathoverflow.net/users/184 | 95054 | 55,713 |
https://mathoverflow.net/questions/95072 | 1 | For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta\_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta\_R(W) \subseteq W \otimes H$, and note that this implies that $\Delta\_R$ restricts to a coaction $V/W \to V/W \otimes H$. If we denote,
$$
V^H := \lbrace v \in V ~ | ~ \Delta\_R(... | https://mathoverflow.net/users/11206 | Coinvariant Subalgebras of Hopf Comodules and Quotients | Let $\newcommand\Com{\mathsf{Com}^H}\newcommand\Vect{\mathsf{Vect}}\Com$ be the category of right $H$-comodules. This has sufficiently many injectives, so we can compute the right derived functors of the left exact functor $F=\hom\_{\Com}(k,\mathord-):\Com\to\Vect$, where $K$ denotes the trivial, $1$-dimensiona comodul... | 4 | https://mathoverflow.net/users/1409 | 95074 | 55,719 |
https://mathoverflow.net/questions/95068 | 0 | Hello,
I want to solve a linear program using the simplex method, and I know that all my inequalities will pass through the origin (therefore, either my initial solution of (0, ... , 0) is optimal, or the program is unbounded; I'm only really interested in distinguishing between these two cases).
So here's how I th... | https://mathoverflow.net/users/21816 | Is the Simplex Method still polynomial when all inequalities are through the origin? | As far as I can recall, the Simplex Algorithm is *not* running in polynomial time, although when randomly perturbing the input, it runs on average in polynomial time (this is also studied under the name "Smoothed Analysis of Algorithms", see for example the article by Spielman and Teng, Journal of the ACM, Vol. 51, No.... | 2 | https://mathoverflow.net/users/18032 | 95076 | 55,720 |
https://mathoverflow.net/questions/94708 | 4 | Define a new product measure on cantor space as follows:u({0})=a,u({1})=1-a,where a$\in$(0,1/2].
Does any ultrafiter U hasn't measure one?
When a=1/2,I know U hasn't measue one.I guess neither when a$\in$(0,1/2),
but I don't know how to prove.
| https://mathoverflow.net/users/22161 | Does ultrafilter have measure one? | This question is a bit more subtle than I had originally thought (in the comments), but anyway here's an argument that seems to work. I will assume for notational convenience that the ultrafilter is on the first infinite ordinal $\omega$.
Fix $k \in \omega$ with $1/k < a$. The main claim is that any conull subset of ... | 4 | https://mathoverflow.net/users/14913 | 95077 | 55,721 |
https://mathoverflow.net/questions/94115 | 6 | It is known that an abelian category is equivalent to a module category iff it has a finite projective generator and contains arbitary direct sum of that generator by Theorem 1 of Chapter 4 Section 11 of the book "Categories and Functors" by Bodo Pareigis.
We know that arbitary comodule category over a coalgebra is ... | https://mathoverflow.net/users/21505 | when a comodule category is equivalent to a module category? | First, an easy sufficient condition. Let $C$ be a coalgebra over a commutative ring $R$. If $C$ itself is finitely generated and projective as an $R$-module, then $Comod\_C$ is equivalent to the category of modules $Mod\_{C^\ast}$ over the dual algebra $C^\ast = Mod\_R(C, R)$.
First, for $C$ a finite projective modu... | 12 | https://mathoverflow.net/users/2926 | 95080 | 55,723 |
https://mathoverflow.net/questions/95084 | 3 | Let $G$ and $H$ be two non-bipartite graphs. We know that, if $\exists$ homomorphism $\phi : G \rightarrow H$, then $\omega(G) \le \omega(H)$ where $\omega$ is clique number.
$(1)$ Does the converse hold in general?
$(2)$ Under what conditions does the converse hold?
| https://mathoverflow.net/users/10035 | graph homomorphism | You have your inequality backwards, I believe.
If $\omega(G) \le \omega(H)$, it does not follow in general that there is a homomorphism from $G\to H$. There are many triangle-free graphs with chromatic number greater than four, none of these will admit a homomorphism to $K\_4$.
| 8 | https://mathoverflow.net/users/1266 | 95087 | 55,728 |
https://mathoverflow.net/questions/95056 | 2 | I just want to make sure I understand certain things well. I believe my questions are quite simple. Are the following statements true?
1/ One cannot prove Con(ZFC) in ZFC. However ZFC might not be consistent (and in this case its inconsistency would be 'provable' in a some sense).
2/ ZFC + existence of an inaccessi... | https://mathoverflow.net/users/14490 | Consistency and inaccessible cardinals | Your statements are nearly correct, but none of them is fully
exactly right, in that there are in each case missing consistency
hypotheses. So let me explain how each of them can be improved in
some small way.
For the first part of statement (1), the correct thing to say is
that *if ZFC is consistent*, then ZFC does ... | 9 | https://mathoverflow.net/users/1946 | 95089 | 55,729 |
https://mathoverflow.net/questions/95092 | 2 | Is there a linear characterization of being the inverse of a Stieltjes matrix? In other words, if $A$ is a $n \times n$ matrix over the reals, is there a set of linear equations in the entries of $A$ such that $A$ is a the inverse of a Stieltjes matrix if and only if these linear conditions are satisfied?
| https://mathoverflow.net/users/10858 | Linear characterization of inverse of Stieltjes matrix | As far as I know, an exact characterization of the form you want is unknown. A necessary condition, for an inverse M-matrix (weaker than inverse Stieltjes) is the so-called "path product condition" - see <http://www.math.temple.edu/~abed/JS07.pdf>.
Another necessary condition is that the principal minors be all posit... | 1 | https://mathoverflow.net/users/22051 | 95098 | 55,731 |
https://mathoverflow.net/questions/95097 | 3 | I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a way to obtain an explicit expression for the Cholesky factorization of my matrix in this special case. Thanks!
| https://mathoverflow.net/users/22051 | Explicit formula for Cholesky factorization in a special case | The matrix $\alpha J$ is a rank one matrix, so there are simple update/downdate formulas for computing the Choleksy factorization of $Q+sI-\alpha J$ if you start with the factorization of $Q+sI$.
I'm not aware of any update formulas that get you from the Cholesky factorization of $Q$ to a Cholesky factorization of $... | 7 | https://mathoverflow.net/users/9022 | 95100 | 55,732 |
https://mathoverflow.net/questions/95090 | 11 | Given an infinite cardinal $\kappa,$ is there some nice way to construct $2^\kappa$ non-isomorphic groups of that cardinality? In the answer to [this stackexchange question](https://math.stackexchange.com/questions/119642/how-many-non-isomorphic-abelian-groups-of-order-kappa-are-there-for-kappa), there is a fairly high... | https://mathoverflow.net/users/11142 | counting non-isomorphic groups of a given cardinality | If you just want a direct construction which avoids nontrivial set theory such as stationary sets etc., how about this?
Step One: For each subset $S \subseteq \kappa$, let $M(S)$ be the structure $\langle \kappa; < , S \rangle$, where $S$ is regarded as a unary relation. Obviously, if $S \neq T$, then $M(S)$ and $M(T... | 10 | https://mathoverflow.net/users/4706 | 95104 | 55,734 |
https://mathoverflow.net/questions/95036 | 7 | I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are considered to be $G$-, $H$- or $G/H$-module homomorphisms. In particular, in the exact sequence of low degrees
$0\to H^1(G... | https://mathoverflow.net/users/11084 | Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective | You asked for examples when $\text{res}^G\_H\colon H^i(G;A)\to H^i(H;A)$ is not surjective. Here are some examples for $i=1$.
* Let $G=\text{SL}\_2\mathbb{Z}$ and $H=\mathbb{Z}$ generated by $\begin{bmatrix}1&1\\\\0&1\end{bmatrix}$. Then $H^1(G;\mathbb{Z})=0$ but $H^1(H;\mathbb{Z})=\mathbb{Z}$ so $\text{res}$ is not ... | 8 | https://mathoverflow.net/users/250 | 95107 | 55,735 |
https://mathoverflow.net/questions/95096 | 9 | Let $X\_1,\dots,X\_n$ denote i.i.d. standard Gaussian random variables. My question is: Do there exist absolute constants $C,c>0$ such that for every $\epsilon>0$ and positive real numbers $a\_1,\dots,a\_n>0$, we have the following bound
$\quad \Pr(\sum\_{i=1}^n a\_i X\_i^2 \le \epsilon \sum\_{i=1}^n a\_i)\le C\epsil... | https://mathoverflow.net/users/825 | Anti-concentration of Gaussian quadratic form | We can show that
$$
\mathbb{P}\left(\sum\_ia\_iX\_i^2\le\epsilon\sum\_ia\_i\right)\le\sqrt{e\epsilon}
$$
so that the inequality holds with $c=1/2$ and $C=\sqrt{e}$.
For $\epsilon\ge1$ the right hand side is greater than 1, so the inequality is trivial. I'll prove the case with $\epsilon < 1$ now.
Without loss of gene... | 13 | https://mathoverflow.net/users/1004 | 95108 | 55,736 |
https://mathoverflow.net/questions/95103 | 4 | I was thinking about this question a couple of days ago, and got reminded again by a recent question on graph homomorphisms.
Given a graph $G$, we call two vertices $u,v$ indistinguishable if the map which interchanges $u$ and $v$ and is the identity on the rest of the vertices is an isomorphism. Let $\mathbb{Graph}$... | https://mathoverflow.net/users/2384 | Indistinguishable objects in the category of graphs | It's an old result of Lovasz that if $|\mathrm{Hom}(G,X)|=|\mathrm{Hom}(H,X)|$ for all graphs $X$, then $G$ and $H$ are isomorphic. If I understand your quiver correctly, the answer to your question is yes, for the trivial reason that the quiver does not contain a pair of equivalent vertices.
| 7 | https://mathoverflow.net/users/1266 | 95110 | 55,738 |
https://mathoverflow.net/questions/95075 | 2 | Let $A$ be a $C^\*$-algebra. If the second dual of $A$, which is the enveloping von Neumann algebra of $A$, is atomic, can we deduce that $A$ is an ideal in its second dual ?
| https://mathoverflow.net/users/21936 | Atomic enveloping von Neumann algebra | No. Take $A$ to be $c$, the space of convergent sequences. Its second dual is $l^\infty$, which is atomic, but it is not an ideal of $l^\infty$.
| 5 | https://mathoverflow.net/users/23141 | 95112 | 55,740 |
https://mathoverflow.net/questions/94830 | 3 | Suppose that $X$, $Y$ and $Z$ are topological spaces, with $A\subset X$, a map $f:A\rightarrow Y$, and a homotopy equivalence $\phi:Y\rightarrow Z$. It seems fair to think that the adjunction spaces $Y\cup\_{f}X$ and $Z\cup\_{\phi\circ f}X$ will be homotopy equivalent provided that $(X,A)$ has the homotopy extension pr... | https://mathoverflow.net/users/23204 | Homotopy equivalence of certain kinds of adjunction spaces | Thanks to Tom Goodwillie and Ronnie Brown for their answers. Also thanks to Theo for pointing out that I posted the problem on math.stackexchange.com/q/135173/5363.
After some thought it became clear that the pair $(X,A)$ needs to have the homotopy extension property (HEP). A simple counterexample is the following:
... | 2 | https://mathoverflow.net/users/23204 | 95132 | 55,745 |
https://mathoverflow.net/questions/95128 | 5 | The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y\_0(M,N)$. We define $Y\_0(M,N)$ as the moduli space parametrizing an elliptic curve $E$, together with two cyclic subgroups $G$ and $H$, of order respectively $M$ and $N$, of the grou... | https://mathoverflow.net/users/23194 | Modular curve parametrizing two cyclic subgroups of an elliptic curve | $Y\_0(M,N)$ can be reinterpreted as the moduli space of diagrams $E\_1 \to E \leftarrow E\_2$ of elliptic curves, where the arrows are cyclic isogenies of degree $M$ and $N$. From this viewpoint, it is naturally a fiber product of $Y\_0(N)$ with $Y\_0(M)$, as you suggested. It is in general a disjoint union of modular ... | 6 | https://mathoverflow.net/users/121 | 95148 | 55,752 |
https://mathoverflow.net/questions/95073 | 2 | Dear experts,
let $T$ be finite tetrahedral complex in flat 3-dimensional euclidean space. Additionally, let $T$ be 'homogeneous' in a sense that each simplex in $T$ is a face of some tetrahedron from $T$. Are there any known *sufficient* conditions (say in terms of Betti numbers of $T$ and possibly Betti numbers of ... | https://mathoverflow.net/users/23156 | Sufficient conditions for a 3D tetrahedral complex to be homeomorphic to a 3D ball | $\newcommand{\RR}{\mathbb{R}}$The other answers are completely general, but there is simpler way if we use the (given) hypothesis that all the action is taking place in $\RR^3$. So, suppose that $T$ is a finite triangulation contained in $\RR^3$. Let $|T|$ be the *underlying space* for $T$. A necessary and sufficient c... | 6 | https://mathoverflow.net/users/1650 | 95150 | 55,754 |
https://mathoverflow.net/questions/95121 | 2 | So we know many useful theorems that help characterize torsion points on elliptic curves over $\mathbb{Q}$ such as the Nagell–Lutz theorem which provides a useful way to find torsion points on $E/\mathbb{Q}$ and Mazur's theorem which characterizes the torsion subgroup of $E(\mathbb{Q})$.
However, does there exist any... | https://mathoverflow.net/users/22095 | Elliptic Curves and Torsion Points | Probably the set of $E/\mathbb{Q}$ with $E(\mathbb{Q})=\{O\}$ has density 1/2 (in an suitable way of ordering curves). A recent (and very deep) result of Manjul Bhargava and Arul Shankar proves that this set has positive density, so there are provably a lot of curves with the property that you request. There's a nice o... | 8 | https://mathoverflow.net/users/11926 | 95152 | 55,755 |
https://mathoverflow.net/questions/95135 | 5 | Let $S$ be an uncountable set. Consider the subspace $\ell\_\infty(\kappa, S)$ of $\ell\_\infty(S)$ formed by all functions with support of cardinality at most $\kappa$ (here $\kappa<|S|$). Certainly, $\ell\_\infty\subset \ell\_\infty(\kappa, S)$, whence the density character of $\ell\_\infty(\kappa, S)$ is at least $2... | https://mathoverflow.net/users/20746 | Density character of $\ell_\infty(\kappa, S)$ | I understand you to mean that $\ell\_\infty(S)$ is the collection of
bounded functions from $S\to\mathbb{R}$, under the sup norm, and
$\ell\_\infty(\kappa,S)$ consists of those functions which take value
$0$ outside a set of size at most $\kappa$, a fixed infinite
cardinal less than $|S|$. The density character is the ... | 6 | https://mathoverflow.net/users/1946 | 95155 | 55,756 |
https://mathoverflow.net/questions/95033 | 3 | I am currently writing a thesis and got to thinking about the bigger picture of mathematics in the following sense. Both manifolds and groups have highly developed theories in their own rights. When combined, in the appropriate way, we arrive at the theory of Lie groups. This theory is more than just the sum of its par... | https://mathoverflow.net/users/23173 | Examples in the vein of smooth manifold + group = Lie group | You could look up the interaction of groupoids and smooth structures, for example in
Pradines, J. In Ehresmann's footsteps: from group geometries to groupoid geometries. (English summary) Geometry and topology of manifolds, 87 - 157, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007. (arXiv:0711.1608)
There... | 4 | https://mathoverflow.net/users/19949 | 95162 | 55,758 |
https://mathoverflow.net/questions/95160 | 6 | A solid ring is a ring $R$ such that the multiplication
$R\otimes\_{\mathbb{Z}} R \to R$ is an isomorphism.
These were classified by Bousfield and Kan; they are
1. subrings of $\mathbb{Q}$,
2. $\mathbb{Z}/n$,
3. products $R\times \mathbb{Z}/n$ with
$R\subseteq \mathbb{Q}$ and
every divisor of $n$ invertible in $R$... | https://mathoverflow.net/users/3634 | Solid rings and Tor | Let $R^t$ be the torsion submodule and consider the exact sequence
$$0\rightarrow R^t\rightarrow R \rightarrow R/R^t\rightarrow 0$$
Bousfield and Kan show that the ring on the right is a localization of ${\mathbb Z}$, hence flat over ${\mathbb Z}$, so its $Tor$ with $R$ vanishes. Thus if we $Tor$ the above with $R$... | 12 | https://mathoverflow.net/users/10503 | 95166 | 55,760 |
https://mathoverflow.net/questions/95154 | 12 | Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth structures. However $M' = M - pt$ is smoothable by a theorem of Quinn. The question is: does someone know an explicit handleb... | https://mathoverflow.net/users/23193 | Handlebody decomposition of an open 4-manifold | There are not that many explicit handlebody pictures of exotic open 4-manifolds, because they get awfully complex in short order. The ones that I know of are in work of Žarko Bižaca from the mid-90's. I think you probably can work out this particular case by hand. You don't really want to use Quinn's theorem for this, ... | 17 | https://mathoverflow.net/users/3460 | 95167 | 55,761 |
https://mathoverflow.net/questions/95177 | 2 | Fix numbers $m, n, k\in {\Bbb Z}\_+$ and $r\in {\Bbb R}\_+$.
What non-trivial estimates exist for the probability that a random $m\times n$ matrix, with integer entries and with all its rows of Euclidean norm less than or equal to $r$, will have rank $k$?
I'm particularly interested in results asymptotic in the va... | https://mathoverflow.net/users/10909 | Distributions of ranks of random integer matrices | This is addressed in
MR1169034 (94e:11073)
Katznelson, Yonatan R.(1-MSRI)
Integral matrices of fixed rank.
Proc. Amer. Math. Soc. 120 (1994), no. 3, 667–675.
| 5 | https://mathoverflow.net/users/11142 | 95178 | 55,765 |
https://mathoverflow.net/questions/95165 | 16 | In an ordinary category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to Set$ preserves $\kappa$-filtered colimits. We say $C$ is locally presentable if it is cocomplete and "generated" by $\kappa$-compact objects for some $\kappa$.
In an $(\infty,1)$-category $C... | https://mathoverflow.net/users/49 | compact objects in model categories and $(\infty,1)$-categories | If $\mathcal{C}$ is a combinatorial model category, then for all sufficiently large regular cardinals $\kappa$, an object of the underlying $\infty$-category is $\kappa$-compact if and only if it can be represented by a $\kappa$-compact object of $\mathcal{C}$. The meaning of "sufficiently large" might depend on $\math... | 19 | https://mathoverflow.net/users/7721 | 95179 | 55,766 |
https://mathoverflow.net/questions/72959 | 2 | I'm not quite sure about how to define a good measure of the quality of a communication channel with fading and interference. Let us assume the simplest case, where a node in a network receives the following quantity:
$$ y = s + w $$
where $s$ is the amplitude (positive real) of the signal and $w$ the noise (gaus... | https://mathoverflow.net/users/13822 | metric for signal to noise ratio in communication systems | The general case is not known. If you find the correct information theoretic metric for the interference channel, you can find the capacity of interference channels (an open problem since the 1960s).
In your case, it looks like a MAC channel. Decode the stronger signal first and then decode the weaker one. Assuming y... | 1 | https://mathoverflow.net/users/10035 | 95181 | 55,768 |
https://mathoverflow.net/questions/95158 | 1 | When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in $R^{n}\backslash{0}$ .So it's easy to get when $-n< a <0$, $\widehat |x|^a=c\_{a,n}|x|^{-n-a}$.
my question is what's the result when... | https://mathoverflow.net/users/23078 | The fourier transform of homogeneous distribution and related topics | In the first place your presentation must be clarified. An homogeneous distribution $u$ of degree $a$ on $\mathbb R^n$ is characterized by
$$\forall \lambda >0,\quad
u(\lambda x)=\lambda^au(x),\qquad
\text{i.e}\quad\langle u(x),\phi(x/\lambda\rangle\lambda^{-n}=\lambda^{a}
\langle u(x),\phi(x\rangle,
$$
which is also ... | 2 | https://mathoverflow.net/users/21907 | 95189 | 55,770 |
https://mathoverflow.net/questions/95188 | 1 | Consider the weak topology in a von Neumann algebra (weak in the sense of Banach spaces).
Does this topology coincide with the rest of the "weak" topologies when restricted to the unitary group?
| https://mathoverflow.net/users/22789 | Weak topology restricted to the unitary group of a von Neumann algebra | No. Consider the algebra $\ell^\infty$. Let $w$ be an ultrafilter and define $\mu\in(\ell^\infty)^\*$ by $\mu((x\_n)) = \lim\_{n\rightarrow w} x\_n$. Now consider the sequence $x^{(n)}$ in the unitary group of $\ell^\infty$, defined by
$$x^{(n)}\_m = \begin{cases} 1 &: m\leq n, \\ -1 &:m>n. \end{cases}$$
So $\mu(x^{(n)... | 6 | https://mathoverflow.net/users/406 | 95192 | 55,772 |
https://mathoverflow.net/questions/95197 | 6 | Hi,
I am troubled with the following question: Does there exist a finite order automorphism of a free group, $f\in Aut(F)$, such that it fixes no non trivial conjugacy class and no non trivial centralizer, i.e. $f(g)$ is not conjugate to $g$ and $f(g)\neq g^{-1}$ for any $g\in F$ ? Can we find such, so the same holds ... | https://mathoverflow.net/users/23210 | Periodic automorphisms of free groups | No such automorphism exists. Every finite order automorphism of a finite rank free group has a nontrivial fixed conjugacy class. To see why, you can represent the free group automorphism as a simplicial automorphism $f : G \to G$ of some finite connected graph having no vertices of valence 1. Take any vertex $p \in G$ ... | 20 | https://mathoverflow.net/users/20787 | 95198 | 55,774 |
https://mathoverflow.net/questions/95195 | 4 | It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt a-\sqrt bx} + \frac {dx}{\sqrt a+\sqrt bx})$ and integrate both parts.
I am interested in the same equation with $x$ taking its va... | https://mathoverflow.net/users/22620 | ODE in symmetric definite positive matrices | Consider a solution of the linear system
$$ \dot x=by$$ $$\dot y =ax$$
Then $u:=yx^{-1}$ solves your Riccati equation
$$\dot u= a- ubu\, .$$
**rmk.** Note that the solution of the linear system is an exponential, defined for all $t\in\mathbb{R}$; while the solution of the Riccati equation is bounded to the interval... | 7 | https://mathoverflow.net/users/6101 | 95201 | 55,776 |
https://mathoverflow.net/questions/95207 | 13 | I am reading [an introduction to growth of groups](http://arxiv.org/abs/math/0607384). The notions of polynomial and superpolynomial growth are introduced, as are exponential and subexponential growth.
I can prove that the growth of a group is always either exponential or subexponential (it is exercise 1.6). However,... | https://mathoverflow.net/users/19956 | Does every group grow either polynomially or superpolynomially? | Yes. This is a result of Grigorchuk, see his Mittag-Leffler notes.[link text](http://www.mittag-leffler.se/preprints/files/IML-1112s-06.pdf) (Milnor's problem on the growth of groups and its consequences, available on line for free; see page 28).
| 14 | https://mathoverflow.net/users/11142 | 95208 | 55,779 |
https://mathoverflow.net/questions/95176 | 5 | Does the octic,
$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$
for any constant *n* have Galois group of order 1344? Its discriminant *D* is a perfect square,
$D = (1728n^4-341901n^3-11560361n^2+3383044089n+28121497213)^2$
Surely (1) is not an isolated result. How easy is it to find another family wit... | https://mathoverflow.net/users/12905 | Octic family with Galois group of order 1344? | Maple gives Galois group $H$ of your polynomial $f=f\_n$ as the subgroup of $S\_8$ generated by these permutations: $$(1 2)(5 6), (1 2 3)(4 6 5), (1 2 6 3 4 5 7), (1 8)(2 3)(4 5)(6 7), (2 8)(1 3)(4 6)(5 7), (4 8)(1 5)(2 6)(3 7).$$ That can be easily proved by using the standard technique of Galois theory assuming that ... | 5 | https://mathoverflow.net/users/nan | 95216 | 55,785 |
https://mathoverflow.net/questions/95182 | 4 | Hi, mathoverflow, I am currently working on finite element exterior calculus, and right now my concern is the construction of some certain kind of "conforming" basis functions for a finite element space. Guess most of you guys are working on pure math, hence before I asked my questions I would like to briefly introduce... | https://mathoverflow.net/users/13092 | Construction of finite element differential forms based on deRham sequences | As robot suggested, a good place to start would be the work of Arnold, Falk, and Winther -- particularly their 2006 paper in *Acta Numerica* and 2010 paper in *Bulletin of the AMS*. (Both available at <http://ima.umn.edu/~arnold/publications.html>.)
One key insight of their work is that, to get a stable discretizatio... | 3 | https://mathoverflow.net/users/673 | 95219 | 55,787 |
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